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4. Symmetries and representations¶
Chapter 2 showed that a symmetry group produces a von Neumann algebra, the span of the symmetry operators, and its commutant, the symmetric observables (Proposition 2.7), and decomposed one example by hand. This chapter supplies the representation theory behind it, turns it into an algorithm that the library runs on any list of matrices, and applies it to the two symmetries of spin systems: \(U(1)\) (a particle number or a magnetisation) and \(SU(2)\) (the rotations of spins).
You will learn complete reducibility and Schur's lemma, how the decomposition into irreducibles determines the commutant and the generated algebra, how characters count multiplicities, why a random element of the commutant finds the decomposition numerically, what charge sectors are, the irreducible spin representations and how \(L\) spins \(\frac12\) decompose, why \(SU(2)\)-invariant Hamiltonians have much smaller blocks than \(U(1)\)-invariant ones, how complex conjugation sorts the sectors of an algebra generated by real matrices into real, complex and quaternionic type, and which tolerances make the numerical decomposition of noisy generators correct and how its output certifies itself. The numerical checks of this chapter are in Tutorial 4.
Throughout, \(S'\) is the commutant of \(S\subseteq M_n(\mathbb C)\) (Chapter 2). If \(S^*=S\), \(S''\) is the von Neumann algebra generated by \(S\): a \(*\)-algebra containing \(S\), and every \(*\)-algebra \(\mathcal B\supseteq S\) satisfies \(\mathcal B=\mathcal B''\supseteq S''\) (Theorem 2.2). \(\mathcal A'_h=\{K\in\mathcal A':K=K^*\}\) is a real vector space. \(\mathbb C^k\otimes\mathbb C^m\) carries the Kronecker order of Chapter 7 (\(e_a\otimes f_j\) is basis vector \(am+j\), \(0\)-based), so \(1_k\otimes x\) is block diagonal with \(k\) copies of \(x\). \(|G|\) is the order of a finite group, and \(\hbar=1\).
4.1 Representations and Schur's lemma¶
Definition. Let \(X\) be a set with an involution \(x\mapsto x^*\) (\(x^{**}=x\)). A representation of \(X\) on \(\mathbb C^k\) is a map \(\rho:X\to M_k(\mathbb C)\) with \(\rho(x^*)=\rho(x)^*\). A subspace \(K\) is invariant if \(\rho(x)K\subseteq K\) for all \(x\); \(\rho\) is irreducible if \(k\ge1\) and \(0,\mathbb C^k\) are the only invariant subspaces. An intertwiner from \(\rho\) to a representation \(\rho'\) on \(\mathbb C^{k'}\) is a \(k'\times k\) matrix \(T\) with \(T\rho(x)=\rho'(x)T\) for all \(x\); they form a vector space \(\mathrm{Hom}(\rho,\rho')\), and \(\mathrm{Hom}(\rho,\rho)=\rho(X)'\). \(\rho\) and \(\rho'\) are equivalent, \(\rho\cong\rho'\), if there is an invertible intertwiner.
Three cases occur: a group \(G\) with \(g^*=g^{-1}\), where we require \(\rho\) to be a homomorphism \(G\to U(k)\) (a unitary representation; then \(\rho(g^{-1})=\rho(g)^{-1}=\rho(g)^*\)); a \(*\)-algebra with its adjoint, where \(\rho\) is a unital \(*\)-homomorphism; and labels of generators with \(x^*=x\), e.g. \(\{x,y,z\}\), represented by tuples of Hermitian matrices such as spin matrices (Section 4.5). If \(K\) is invariant and the columns of \(Q\in M_{k\times d}\) are an orthonormal basis of \(K\), then \(\rho(x)Q=QQ^*\rho(x)Q\), so \(\rho_K(x)=Q^*\rho(x)Q\) is again a representation (a homomorphism if \(\rho\) is one), the restriction of \(\rho\) to \(K\).
Lemma 4.1 (complete reducibility). Let \(\rho\) be a representation of \(X\) on \(\mathbb C^n\). (a) If \(K\) is invariant, so is \(K^\perp\). (b) \(\mathbb C^n\) is an orthogonal direct sum of invariant subspaces on each of which \(\rho\) is irreducible.
Proof. (a) For \(y\in K^\perp\) and \(z\in K\), \(\langle z,\rho(x)y\rangle=\langle\rho(x^*)z,y\rangle=0\), since \(\rho(x^*)z\in K\). (b) Induction on \(n\). A nonzero invariant subspace \(K_1\) of minimal dimension is irreducible; \(K_1^\perp\) is invariant by (a), and the invariant subspaces of the restriction to \(K_1^\perp\) are invariant subspaces of \(\rho\), so the induction hypothesis applies to it. \(\square\)
For a finite group unitarity costs nothing: if the \(\rho(g)\) are merely invertible, \(P=\sum_g\rho(g)^*\rho(g)\) satisfies \(P\ge\rho(e)^*\rho(e)=1\), so \(P>0\), and \(\rho(h)^*P\rho(h)=\sum_g\rho(gh)^*\rho(gh)=P\) (substitute \(g\to gh^{-1}\)); hence \((P^{1/2}\rho(h)P^{-1/2})^*(P^{1/2}\rho(h)P^{-1/2})=P^{-1/2}\rho(h)^*P\rho(h)P^{-1/2}=1\), and \(h\mapsto P^{1/2}\rho(h)P^{-1/2}\) is a unitary representation equivalent to \(\rho\) (Weyl's trick).
Lemma 4.2 (Schur's lemma). Let \(\rho,\rho'\) be irreducible representations of \(X\) on \(\mathbb C^k,\mathbb C^{k'}\) and \(T\in\mathrm{Hom}(\rho,\rho')\). (a) \(T=0\) or \(T\) is invertible. (b) \(\mathrm{Hom}(\rho,\rho)=\mathbb C1\). (c) \(T^*\in\mathrm{Hom}(\rho',\rho)\), and if \(T\neq0\), then \(T=\lambda U\) with \(\lambda>0\) and \(U\) unitary. Hence \(\dim\mathrm{Hom}(\rho,\rho')\) is \(1\) if \(\rho\cong\rho'\) and \(0\) otherwise, and equivalent irreducible representations are unitarily equivalent.
Proof. (a) and (b) were proved for groups after Proposition 2.7, and the proof only uses that \(\ker T\) is invariant under \(\rho\) (\(T\rho(x)v=\rho'(x)Tv=0\) if \(Tv=0\)) and the range of \(T\) under \(\rho'\): so \(\ker T\in\{0,\mathbb C^k\}\) and \(\mathrm{ran}\,T\in\{0,\mathbb C^{k'}\}\); and for (b), \(T-\lambda1\) with \(\lambda\) an eigenvalue of \(T\) is a non-invertible intertwiner, hence \(0\). (c) The adjoint of \(T\rho(x^*)=\rho'(x^*)T\) is \(\rho(x)T^*=T^*\rho'(x)\). Then \(T^*T\in\mathrm{Hom}(\rho,\rho)\), so \(T^*T=\lambda^21\) with \(\lambda>0\) if \(T\neq0\) (\(T^*T\ge0\) is nonzero), and \(U=T/\lambda\) is square by (a) with \(U^*U=1\). Finally, if \(T_1,T_2\in\mathrm{Hom}(\rho,\rho')\) are nonzero, then \(T_1^{-1}T_2\in\mathrm{Hom}(\rho,\rho)=\mathbb C1\). \(\square\)
If all \(\rho(x)\) commute with each other (an abelian group), an irreducible \(\rho\) is one-dimensional: every \(\rho(x)\) lies in \(\mathrm{Hom}(\rho,\rho)=\mathbb C1\), so every subspace is invariant.
4.2 The isotypic decomposition and the commutant¶
Theorem 4.3 (isotypic decomposition). Let \(\rho\) be a representation of \(X\) on \(\mathbb C^n\).
There are irreducible, pairwise inequivalent representations \(\rho_1,\dots,\rho_C\) of \(X\) on
\(\mathbb C^{k_c}\), integers \(m_c\ge1\) with \(\sum_ck_cm_c=n\), and a unitary
\(V:\bigoplus_c\mathbb C^{k_c}\otimes\mathbb C^{m_c}\to\mathbb C^n\) with
\(V^*\rho(x)V=\bigoplus_c\rho_c(x)\otimes1_{m_c}\) for all \(x\). For every such decomposition
math
V^*\rho(X)'V=\bigoplus_{c=1}^C1_{k_c}\otimes M_{m_c}(\mathbb C),\qquad
V^*\rho(X)''V=\bigoplus_{c=1}^CM_{k_c}(\mathbb C)\otimes1_{m_c},
and \(m_c=\dim\mathrm{Hom}(\rho_c,\rho)\); an irreducible representation \(\sigma\) of \(X\) has
\(\mathrm{Hom}(\sigma,\rho)\neq0\) iff \(\sigma\cong\rho_c\) for exactly one \(c\).
Proof. Existence. Lemma 4.1 gives pairwise orthogonal invariant subspaces \(W_1,\dots,W_M\) with sum \(\mathbb C^n\) and irreducible restrictions \(\sigma_i(x)=Q_i^*\rho(x)Q_i\). Group the \(W_i\) into classes of equivalent restrictions and choose in each class \(c\) a reference \(\rho_c=\sigma_{i_0}\). For each other \(i\) in the class, Lemma 4.2(c) gives a unitary \(U_i\) with \(\sigma_i(x)=U_i\rho_c(x)U_i^*\); replacing \(Q_i\) by \(Q_iU_i\) makes \(Q_i^*\rho(x)Q_i=\rho_c(x)\). Number the members of class \(c\) by \(j=0,\dots,m_c-1\), write \(Q_{c,j}\) for their bases, and let \(V(e_a\otimes f_j)=Q_{c,j}e_a\) on the \(c\)-th summand. As the \(W_i\) are invariant and orthogonal, \(Q_{c,j}^*\rho(x)Q_{d,l}=\delta_{cd}\delta_{jl}\rho_c(x)\), which is the claimed form.
Commutant. Now let \(V\) be any decomposition as in the statement, and let \(Q_{c,j}\) be the \(n\times k_c\) isometry \(e_a\mapsto V(e_a\otimes f_j)\) (in the \(c\)-th summand); for the \(V\) just constructed these are the \(Q_{c,j}\) above. The identity \(V^*\rho(x)V=\bigoplus_c\rho_c(x)\otimes1_{m_c}\) says \(\rho(x)Q_{d,l}=Q_{d,l}\rho_d(x)\), and its adjoint applied to \(x^*\) gives \(Q_{c,j}^*\rho(x)=\rho_c(x)Q_{c,j}^*\). Since \(V\) is unitary, \(Y\in M_n\) commutes with all \(\rho(x)\) iff every block \(Y_{cj,dl}=Q_{c,j}^*YQ_{d,l}\) satisfies \(Y_{cj,dl}\rho_d(x)=\rho_c(x)Y_{cj,dl}\), i.e. lies in \(\mathrm{Hom}(\rho_d,\rho_c)\), which is \(0\) for \(c\neq d\) and \(\mathbb C1\) for \(c=d\) (Lemma 4.2). So \(Y\in\rho(X)'\) iff \(Y_{cj,dl}=\delta_{cd}(y_c)_{jl}1_{k_c}\), i.e. \(V^*YV=\bigoplus_c1_{k_c}\otimes y_c\).
Double commutant. If \(Z\) commutes with \(\rho(X)'\), it commutes with the projections onto the summands (take \(y_c=1\) and \(y_d=0\) for \(d\neq c\)), so \(V^*ZV=\bigoplus_cZ_c\), and \(Z_c\) commutes with \(1_{k_c}\otimes M_{m_c}\); by Lemma 2.5, Step 5 (with the two tensor factors exchanged), \(Z_c\in M_{k_c}\otimes1_{m_c}\). Conversely these operators commute with \(\rho(X)'\).
Multiplicities. For \(T\in\mathrm{Hom}(\sigma,\rho)\) with \(\sigma\) irreducible, the same argument shows \(Q_{c,j}^*T\in\mathrm{Hom}(\sigma,\rho_c)\), which is \(\mathbb Ct_{c}\) with \(t_c\) invertible if \(\sigma\cong\rho_c\) and \(0\) otherwise; and \(T=\sum_{c,j}Q_{c,j}Q_{c,j}^*T\). As the \(\rho_c\) are pairwise inequivalent, \(\sigma\) is equivalent to at most one of them, say \(\rho_c\) (otherwise \(T=0\)), and then \(T=\sum_ja_jQ_{c,j}t_c\); conversely each \(Q_{c,j}t_c\) lies in \(\mathrm{Hom}(\sigma,\rho)\), and these \(m_c\) maps have orthogonal ranges, so \(\dim\mathrm{Hom}(\sigma,\rho)=m_c\). \(\square\)
In the library: a decomposition as in Theorem 4.3 is computed by decompose.from_group and decompose.from_generators — tutorial, Section 4.3.
For a group, \(\rho(G)''\) is the span of \(\rho(G)\) (Section 2.4), and Theorem 4.3 is Proposition 2.7 read in the other direction: there the structure theorem produced the irreducible representations, here they produce the structure; in both, \(\dim\rho(X)'=\sum_cm_c^2\) and \(\dim\rho(X)''=\sum_ck_c^2\). The range of \(P_c=\sum_jQ_{c,j}Q_{c,j}^*\), a minimal central projection of \(\rho(X)''\), is the isotypic component of \(\rho_c\). By Theorem 4.3 every Hermitian \(H\in\rho(X)'\) is \(V(\bigoplus_c1_{k_c}\otimes h_c)V^*\) with Hermitian \(h_c\in M_{m_c}\), and \(1_{k_c}\otimes h_c\) is block diagonal with \(k_c\) copies of \(h_c\): the spectrum of \(H\) is the union of the spectra of the \(m_c\times m_c\) blocks \(h_c\), each eigenvalue of \(h_c\) repeated \(k_c\) times.
Physical meaning (physical interpretation, not a theorem of this course). If \(\rho(X)\) is a symmetry and \(H\in\rho(X)'\) the Hamiltonian, the \(k_c\)-fold degenerate levels just described are the multiplets of the symmetry, and the blocks one has to diagonalise have the sizes \(m_c\) of the multiplicities, not the sizes of the multiplets.
For a finite group the multiplicities can be computed without decomposing anything.
Theorem 4.4 (Schur orthogonality). Let \(\rho,\rho'\) be irreducible unitary representations of a
finite group \(G\) on \(\mathbb C^k,\mathbb C^{k'}\). Then
math
\frac1{|G|}\sum_{g\in G}\rho(g)_{ij}\,\overline{\rho'(g)_{i'j'}}=
\begin{cases}\delta_{ii'}\delta_{jj'}/k&\text{if }\rho=\rho',\\0&\text{if }\rho\not\cong\rho'.\end{cases}
Proof. For a \(k\times k'\) matrix \(Y\) let \(T=\frac1{|G|}\sum_g\rho(g)Y\rho'(g)^*\). For \(h\in G\), substituting \(g\to hg\) and using \(\rho'(hg)^*=\rho'(g)^*\rho'(h)^{-1}\) gives \(\rho(h)T=T\rho'(h)\), so \(T\in\mathrm{Hom}(\rho',\rho)\). If \(\rho\not\cong\rho'\), \(T=0\); if \(\rho=\rho'\), \(T=\lambda1\) with \(k\lambda=\mathrm{Tr}\,T=\mathrm{Tr}\,Y\) (Lemma 4.2). For \(Y=|j\rangle\langle j'|\), the entry \(T_{ii'}\) is the left side, and \(\mathrm{Tr}\,Y=\delta_{jj'}\). \(\square\)
The character of a representation is \(\chi_\rho(g)=\mathrm{Tr}\,\rho(g)\); equivalent representations have equal characters, since \(\mathrm{Tr}(T\rho(g)T^{-1})=\mathrm{Tr}\,\rho(g)\). Write \(\langle\varphi,\psi\rangle_G=\frac1{|G|}\sum_g\overline{\varphi(g)}\psi(g)\).
Corollary 4.5 (characters). Let \(G\) be finite.
(a) For irreducible unitary \(\rho,\rho'\): \(\langle\chi_\rho,\chi_{\rho'}\rangle_G\) is \(1\) if \(\rho\cong\rho'\) and \(0\) otherwise.
(b) For a unitary representation \(\pi\) decomposed as in Theorem 4.3 and an irreducible \(\rho\), the
multiplicity of \(\rho\) in \(\pi\) is
math
m_\rho=\frac1{|G|}\sum_{g\in G}\overline{\chi_\rho(g)}\,\chi_\pi(g)
(\(m_\rho=m_c\) if \(\rho\cong\rho_c\), and \(m_\rho=0\) if \(\rho\) is equivalent to no \(\rho_c\)).
(c) \(\dim\pi(G)'=\sum_cm_c^2=\frac1{|G|}\sum_{g\in G}|\chi_\pi(g)|^2\). In particular \(\pi\) is irreducible iff
this number is \(1\).
Proof. (a) Replace \(\rho'\) by an equivalent representation (same character). Sum Theorem 4.4 over \(i=j\), \(i'=j'\) and conjugate: the result is \(\sum_{i,i'}\delta_{ii'}/k=1\) if \(\rho=\rho'\), and \(0\) if \(\rho\not\cong\rho'\). (b) The trace is unitarily invariant and \(\mathrm{Tr}(a\otimes1_m)=m\mathrm{Tr}\,a\), so \(\chi_\pi=\sum_cm_c\chi_{\rho_c}\); apply (a). (c) By the same expansion and (a), \(\langle\chi_\pi,\chi_\pi\rangle_G=\sum_cm_c^2\), which is \(\dim\pi(G)'\) by Theorem 4.3. \(\square\)
Computation: Corollary 4.5 for \(S_3\) permuting three qubits — tutorial, Section 4.2; Theorem 4.4 for the standard representation of \(S_3\) — tutorial, Section 4.3.
A second proof of (c) is how the library samples commutants. The twirl \(P(Y)=\frac1{|G|}\sum_g\pi(g)Y\pi(g)^*\) is idempotent with range \(\pi(G)'\) (Chapter 2, Exercise 3) and self-adjoint for \(\langle Y,Z\rangle_{HS}=\mathrm{Tr}(Y^*Z)\) (substitute \(g\to g^{-1}\)), hence the orthogonal projection onto \(\pi(G)'\), and \(\dim\pi(G)'\) is its trace. The map \(Y\mapsto AYB\) has trace \(\mathrm{Tr}\,A\mathrm{Tr}\,B\) (its diagonal entries are \(A_{ii}B_{jj}\)), which gives \(\frac1{|G|}\sum_g|\chi_\pi(g)|^2\).
In the library: the twirl is decompose.twirl — tutorial, Section 4.3; \(\dim\pi(G)'\) is decompose.commutant_dimension — tutorial, Section 4.2.
We redo the example of Section 2.4 with characters. A permutation matrix has as trace the number of basis vectors it fixes, and \(\pi(\sigma)\) fixes \(|b_1b_2b_3\rangle\) iff \(b\) is constant on the cycles of \(\sigma\) (Lemma B.1(c)), so \(\chi_\pi(\sigma)=2^{\#\text{cycles}(\sigma)}\). Three unitary representations of \(S_3\) are at hand. Let \(P_\sigma e_i=e_{\sigma(i)}\) be the permutation representation on \(\mathbb C^3\) (a homomorphism into the permutation matrices). The trivial representation is \(\sigma\mapsto1\). The sign is \(\mathrm{sgn}\,\sigma=\det P_\sigma\), a homomorphism into \(\{\pm1\}\subseteq U(1)\) because the determinant is multiplicative and the determinant of a permutation matrix is \(\pm1\). The line \(\mathbb C(1,1,1)\) is invariant, hence so is \(E=(1,1,1)^\perp\) (Lemma 4.1(a)); the standard representation is the restriction of \(P\) to \(E\). In an orthonormal basis adapted to \(\mathbb C(1,1,1)\oplus E\), \(P_\sigma=1\oplus\mathrm{std}(\sigma)\), so \(\chi_{\rm std}(\sigma)=\mathrm{Tr}\,P_\sigma-1\), the number of fixed points minus one. The six elements of \(S_3\) are the identity, three transpositions and two \(3\)-cycles, with \(3,2,1\) cycles and \(3,1,0\) fixed points; a transposition swaps two columns of \(1_3\), so its sign is \(-1\), and a \(3\)-cycle is a product of two transpositions (\((1\,2\,3)=(1\,3)(1\,2)\)), so its sign is \(+1\). This gives the table
| identity (\(1\)) | transpositions (\(3\)) | \(3\)-cycles (\(2\)) | |
|---|---|---|---|
| \(\chi_{\rm triv}\) | \(1\) | \(1\) | \(1\) |
| \(\mathrm{sgn}\) | \(1\) | \(-1\) | \(1\) |
| \(\chi_{\rm std}\) | \(2\) | \(0\) | \(-1\) |
| \(\chi_\pi\) | \(8\) | \(4\) | \(2\) |
(so \(\mathrm{sgn}\,\sigma=(-1)^{3-\#\text{cycles}(\sigma)}\); all characters are real). With \(\langle\varphi,\psi\rangle_G\)
summed class by class:
\(\langle\chi_{\rm triv},\chi_{\rm triv}\rangle_G=\frac{1+3+2}6=1\), \(\langle\mathrm{sgn},\mathrm{sgn}\,\rangle_G=\frac{1+3+2}6=1\) and
\(\langle\chi_{\rm std},\chi_{\rm std}\rangle_G=\frac{4+0+2}6=1\), so the three representations are irreducible by
Corollary 4.5(c), and they are pairwise inequivalent because their characters differ. Corollary 4.5(b) gives the
multiplicities in \(\pi\):
math
m_{\rm triv}=\tfrac{8+3\cdot4+2\cdot2}6=4,\qquad m_{\rm sgn}=\tfrac{8-3\cdot4+2\cdot2}6=0,\qquad
m_{\rm std}=\tfrac{2\cdot8+0-2\cdot2}6=2 .
So the trivial and the standard representation are equivalent to constituents \(\rho_c\) of \(\pi\) (Theorem 4.3) with
\(m_c=4\) and \(m_c=2\); they account for \(4\cdot1+2\cdot2=8=\sum_ck_cm_c\) dimensions, so there is no other constituent:
\(k=(1,2)\), \(m=(4,2)\), and \(\dim\pi(S_3)'=4^2+2^2=\frac16(64+3\cdot16+2\cdot4)=20\) (Corollary 4.5(c)).
This is the decomposition of Section 2.4, now obtained from a table of six numbers per representation.
Computation: this table, recomputed from \(\pi\) — tutorial, Section 4.2.
4.3 From generators to the algebra: a randomised algorithm¶
Theorem 4.3 describes the decomposition but does not compute it: its proof needs the irreducible invariant subspaces. Murota, Kanno, Kojima and Kojima (Japan J. Indust. Appl. Math. 27 (2010) 125–160; see also Maehara and Murota, ibid. 263–293) observed that the eigenspaces of a random element of the commutant find them. Let \(\mathcal A\) be the von Neumann algebra generated by given matrices \(S_1,\dots,S_J\) and their adjoints, with a decomposition \(V^*\mathcal AV=\bigoplus_cM_{k_c}\otimes1_{m_c}\), \(V^*\mathcal A'V=\bigoplus_c1_{k_c}\otimes M_{m_c}\) (Theorem 2.6); only the analysis uses \(V\).
Lemma 4.6. Let \(K\in\mathcal A'_h\) with \(V^*KV=\bigoplus_c1_{k_c}\otimes\kappa_c\). Every eigenspace of \(K\) is \(\mathcal A\)-invariant. If the \(N=\sum_cm_c\) eigenvalues of all the \(\kappa_c\) together are pairwise distinct, the eigenspaces of \(K\) are the \(N\) subspaces \(V(\mathbb C^{k_c}\otimes e)\) (in the \(c\)-th summand, \(e\) an eigenvector of \(\kappa_c\)), and \(\mathcal A\) acts on each of them irreducibly, as \(M_{k_c}\).
Proof. If \(Kv=\lambda v\) and \(a\in\mathcal A\), then \(Kav=aKv=\lambda av\). Write \(\kappa_c=\sum_j\mu_{cj}|e_{cj}\rangle\langle e_{cj}|\) with orthonormal \(e_{cj}\) (Theorem 1.3). Then \(V^*KV=\sum_{c,j}\mu_{cj}R_{cj}\), where \(R_{cj}\) is the projection onto \(\mathbb C^{k_c}\otimes e_{cj}\); the \(R_{cj}\) are orthogonal with sum \(1\) and the \(\mu_{cj}\) distinct, so these are the spectral projections. On \(\mathbb C^{k_c}\otimes e_{cj}\) the element \(V^*aV=\bigoplus_da_d\otimes1\) acts as \(a_c\), and the \(a_c\) fill \(M_{k_c}\), which has no invariant subspaces but \(0\) and \(\mathbb C^{k_c}\). \(\square\)
Proposition 4.7 (genericity). Let \(K\) and \(Y\) be independent random elements of \(\mathcal A'_h\) and of \(\mathcal A'\) whose laws have densities with respect to Lebesgue measure on these real vector spaces (Lemma A.3). With probability one: (a) the hypothesis of Lemma 4.6 holds; (b) \(\Pi_iY\Pi_j\neq0\) for any two distinct spectral projections \(\Pi_i,\Pi_j\) of \(K\) whose ranges lie in the same summand.
Proof. (a) The Hermitian matrix \(\kappa=\bigoplus_c\kappa_c\in M_N\) depends real-linearly on \(K\). The Gram determinant \(\delta(K)=\det[\mathrm{Tr}\,\kappa^{i+j}]_{i,j=0}^{N-1}\) of \(1,\kappa,\dots,\kappa^{N-1}\) for \(\langle\cdot,\cdot\rangle_{HS}\) is a real polynomial in the coordinates of \(K\), nonzero iff these powers are linearly independent. As \(p(\kappa)=U\mathrm{diag}(p(\lambda_l))U^*\) for \(\kappa=U\mathrm{diag}(\lambda)U^*\), a nonzero polynomial of degree \(<N\) annihilates \(\kappa\) iff it vanishes on \(\mathrm{spec}\,\kappa\), which is possible iff \(\kappa\) has fewer than \(N\) distinct eigenvalues (take \(\prod_\mu(t-\mu)\) over them). So \(\delta(K)\neq0\) is the hypothesis of Lemma 4.6, and \(\delta\not\equiv0\): it is nonzero at \(V(\bigoplus_c1\otimes\mathrm{diag}(\text{distinct reals}))V^*\in\mathcal A'_h\). Hence \(\mathbb P(\delta(K)=0)=0\) by Corollary A.7(b).
(b) The set \(O=\{\delta\neq0\}\subseteq\mathcal A'_h\) is open (\(\delta\) is continuous). For \(K\in O\), \(K\) has exactly \(N\) distinct eigenvalues \(\mu_1(K)<\dots<\mu_N(K)\) (those of \(\kappa\), of multiplicities \(k_c\)), so its spectral projections \(\Pi_1(K),\dots,\Pi_N(K)\), numbered by increasing eigenvalue, are continuous on \(O\) (Proposition A.13). Fix \(l\neq l'\) and let \(\mathcal N_{ll'}\) be the set of \((K,Y)\in O\times\mathcal A'\) with \(P_c\Pi_l(K)=\Pi_l(K)\) and \(P_c\Pi_{l'}(K)=\Pi_{l'}(K)\) for some \(c\) (\(P_c\) the central projections) and \(\Pi_l(K)Y\Pi_{l'}(K)=0\); it is defined by continuous equations on the Borel set \(O\times\mathcal A'\), hence Borel. Fix \(K\in O\) for which the two ranges lie in the summand \(c\). By Lemma 4.6, \(V^*\Pi_lV\) and \(V^*\Pi_{l'}V\) project onto \(\mathbb C^{k_c}\otimes e\) and \(\mathbb C^{k_c}\otimes e'\) with orthonormal eigenvectors \(e,e'\) of \(\kappa_c\), and for \(V^*YV=\bigoplus_d1\otimes y_d\), \(\Pi_lY\Pi_{l'}=0\) iff \(\langle e,y_ce'\rangle=0\). The map \(Y\mapsto\langle e,y_ce'\rangle\) is real-linear \(\mathcal A'\to\mathbb C\) and nonzero (its value at \(V(0\oplus\dots\oplus1\otimes|e\rangle\langle e'|\oplus\dots\oplus0)V^*\) is \(1\)), so its kernel is a proper subspace, a null set (Corollary A.7(c)). So every section \(\{Y:(K,Y)\in\mathcal N_{ll'}\}\) is null (it is empty if the ranges lie in different summands), and as \(Y\) is independent of \(K\), \(\mathbb P((K,Y)\in\mathcal N_{ll'})=0\) by Corollary A.7(d). The event of (b) fails only on \(\{\delta(K)=0\}\cup\bigcup_{l\neq l'}\{(K,Y)\in\mathcal N_{ll'}\}\), a finite union of events of probability \(0\). \(\square\)
Lemma 4.8 (compressions; aligning the copies). (a) Let \(W\) be an \(\mathcal A\)-invariant subspace with orthonormal basis \(Q\in M_{n\times d}\) and \(\sigma(a)=Q^*aQ\) the restriction. Then \(\sigma(\mathcal A)\) is a \(*\)-algebra, \(\{Q^*YQ:Y\in\mathcal A'\}=\sigma(\mathcal A)'\) and \(\{Q^*KQ:K\in\mathcal A'_h\}=\sigma(\mathcal A)'_h\). \(\mathcal A\) acts irreducibly on \(W\) iff \(Q^*YQ\in\mathbb C1\) for every \(Y\in\mathcal A'\). If it does not, and \(X\) is a random element of \(\mathcal A'\) or of \(\mathcal A'_h\) with a density, then \(Q^*XQ\notin\mathbb C1\) with probability one.
(b) Let \(W_i,W_j\) be invariant subspaces on which \(\mathcal A\) acts irreducibly, with orthonormal bases \(Q_i,Q_j\) and restrictions \(\sigma_i(a)=Q_i^*aQ_i\), and let \(Y\in\mathcal A'\). Then \(T=Q_i^*YQ_j\in\mathrm{Hom}(\sigma_j,\sigma_i)\). If \(T\neq0\), then \(T=\lambda U\) with \(\lambda>0\), \(U\) unitary, and \(Q_j'=Q_jU^*\) satisfies \(Q_j'^*aQ_j'=\sigma_i(a)\) for all \(a\in\mathcal A\). If \(W_i,W_j\) lie in different summands (\(W_i\subseteq P_c\mathbb C^n\), \(W_j\subseteq P_d\mathbb C^n\), \(c\neq d\)), then \(T=0\).
Proof. (a) As in the proof of Theorem 4.3 (\(W\) is invariant under \(a\) and \(a^*\)), \(aQ=Q\sigma(a)\) and \(Q^*a=\sigma(a)Q^*\) for \(a\in\mathcal A\). Hence \(\sigma(ab)=Q^*aQ\sigma(b)=\sigma(a)\sigma(b)\), \(\sigma(a^*)=\sigma(a)^*\) and \(\sigma(1)=Q^*Q=1_d\), so \(\sigma\) is a unital \(*\)-homomorphism and \(\sigma(\mathcal A)\) is a \(*\)-algebra; and for \(Y\in\mathcal A'\), \(Q^*YQ\sigma(a)=Q^*YaQ=Q^*aYQ=\sigma(a)Q^*YQ\). Conversely, for \(T\in\sigma(\mathcal A)'\) let \(Y=QTQ^*\); then \(aY=Q\sigma(a)TQ^*=QT\sigma(a)Q^*=Ya\), so \(Y\in\mathcal A'\), with \(Q^*YQ=T\), and \(Y\) is Hermitian if \(T\) is. If \(\sigma\) is irreducible, \(\sigma(\mathcal A)'=\mathrm{Hom}(\sigma,\sigma)=\mathbb C1\) (Lemma 4.2(b)). If \(E\subseteq\mathbb C^d\) is a \(\sigma\)-invariant subspace other than \(0,\mathbb C^d\), so is \(E^\perp\) (Lemma 4.1(a)), and the orthogonal projection onto \(E\) is a non-scalar element of \(\sigma(\mathcal A)'_h\). In that case \(\mathbb C1\cap\sigma(\mathcal A)'\) and \(\mathbb R1\cap\sigma(\mathcal A)'_h\) are proper real subspaces, their preimages under the surjective real-linear maps \(X\mapsto Q^*XQ\) are proper subspaces of \(\mathcal A'\) and \(\mathcal A'_h\), and these are null sets (Corollary A.7(c)).
(b) As in (a), \(aQ_j=Q_j\sigma_j(a)\) and \(Q_i^*a=\sigma_i(a)Q_i^*\), so \(T\sigma_j(a)=Q_i^*YaQ_j=Q_i^*aYQ_j=\sigma_i(a)T\). Lemma 4.2 gives \(T=\lambda U\) and \(U\sigma_j(a)U^*=\sigma_i(a)\), which is \(Q_j'^*aQ_j'\). Finally \(P_c\in\mathcal A\cap\mathcal A'\), \(Q_i=P_cQ_i\) and \(P_cQ_j=0\), so \(T=Q_i^*P_cYQ_j=Q_i^*YP_cQ_j=0\). \(\square\)
Computation: Lemma 4.6 and Lemma 4.8(b) for a random element of \(\pi(S_3)'\) — tutorial, Section 4.3.
The algorithm (decompose.wedderburn, with the front ends from_group and from_generators) is now:
- Sample the commutant. For all elements of a finite group, \(Y=P(Z)\) and \(K=P(Z'+Z'^*)\) with the twirl \(P\) and independent Ginibre matrices \(Z,Z'\). \(P\) is a linear projection of \(M_n\) onto \(\mathcal A'\) with \(P(Z)^*=P(Z^*)\), so \(Z\mapsto P(Z)\) and \(Z'\mapsto P(Z'+Z'^*)\) are real-linear surjections onto \(\mathcal A'\) and onto \(\mathcal A'_h\) (for \(H\in\mathcal A'_h\), \(P(\frac H2+\frac H2)=H\)); a Ginibre matrix is a non-degenerate centred Gaussian vector of \(M_n\) (Proposition A.16), so \(Y\) and \(K\) are non-degenerate centred Gaussian vectors of \(\mathcal A'\) and \(\mathcal A'_h\) and have densities (Theorem A.15). For generators, \(\mathcal A'=\{X:[S_j,X]=[S_j^*,X]=0\}\) is the null space of a linear map of \(M_n\) (the library uses the Hermitian parts \(\frac12(S_j+S_j^*)\), \(\frac1{2i}(S_j-S_j^*)\), which span the same complex space as \(S_j,S_j^*\) and so have the same commutant); with a basis \((B_r)_{r\le D}\) of it, \(Y=\sum_rc_rB_r\) and \(K=Y'+Y'^*\) with \(Y'\) an independent copy of \(Y\) and independent standard complex Gaussian \(c_r\): images of a non-degenerate Gaussian vector of \(\mathbb C^D\) under the real-linear bijection \(c\mapsto\sum_rc_rB_r\) onto \(\mathcal A'\) and the real-linear surjection \(c\mapsto X+X^*\) (\(X=\sum_rc_rB_r\)) onto \(\mathcal A'_h\), hence again with densities. Every random element is a fresh draw, independent of the earlier ones; the library rescales them to unit norm, which changes no eigenspace and no vanishing of a block.
- Split. The eigenspaces of \(K\) are irreducible (Lemma 4.6, Proposition 4.7(a)); their dimensions are the \(k_c\).
- Group and align. The blocks \(Q_i^*YQ_j\) between eigenspaces of equal dimension are nonzero exactly within a summand (Proposition 4.7(b), Lemma 4.8(b)); the classes of the relation "linked by a nonzero block" give the sectors, their sizes the \(m_c\), and the polar factors align the copies to one reference copy per class (Lemma 4.8(b); a copy aligned to an aligned copy is aligned to the reference).
- Assemble \(V\) in Kronecker order, so that \(V^*aV=\bigoplus_ca_c\otimes1_{m_c}\).
So, in exact arithmetic, with the eigenspaces of \(K\) separated exactly, the algorithm returns with probability one a decomposition as in Theorem 4.3: by Lemma 4.6 the summand \(c\) contains \(m_c\) eigenspaces of dimension \(k_c\), and after alignment the construction is the existence part of the proof of Theorem 4.3. In floating point the library treats eigenvalues closer than a tolerance as equal, so a computed "eigenspace" \(W\) may be a sum of several eigenspaces of \(K\); it is still \(\mathcal A\)-invariant (Lemma 4.6), and the library compresses a fresh random \(X\in\mathcal A'_h\) to it. By Lemma 4.8(a), \(Q^*XQ\) is a multiple of \(1\) whenever \(\mathcal A\) acts irreducibly on \(W\), and otherwise it is not one with probability one; then its eigenspaces, being invariant under \(\sigma(\mathcal A)\) (Lemma 4.6 applied to \(\sigma(\mathcal A)\)), split \(W\) into smaller invariant subspaces, and the library repeats until every compression is scalar. Whatever happens in floating point, the result is certified at the end:
Proposition 4.9 (certificate). Let \(\mathcal A\) be generated by \(S_1,\dots,S_J\) and their adjoints, and \(\mathcal B=V(\bigoplus_cM_{k_c}\otimes1_{m_c})V^*\) with \(V\) unitary. If all \(S_j\in\mathcal B\) and \(\dim\mathcal A'=\sum_cm_c^2\), then \(\mathcal A=\mathcal B\).
Proof. \(\mathcal B\) is a \(*\)-algebra containing all \(S_j,S_j^*\), so \(\mathcal A\subseteq\mathcal B\) and \(\mathcal B'\subseteq\mathcal A'\). Theorem 4.3, applied to the representation of \(\mathcal B\) on \(\mathbb C^n\) (the block maps \(b\mapsto b_c\) are irreducible, and inequivalent, as the \(c\)-th central projection is mapped to \(1\) by one and to \(0\) by the others), gives \(\dim\mathcal B'=\sum_cm_c^2=\dim\mathcal A'\). So \(\mathcal B'=\mathcal A'\) and \(\mathcal B=\mathcal B''=\mathcal A''=\mathcal A\). \(\square\)
In the library: the algorithm, decompose.wedderburn (front ends from_group, from_generators), the certificate of Proposition 4.9, Decomposition.verify(), and their cost — tutorial, Section 4.3.
Section 4.9 makes the floating-point part quantitative: which tolerances give the right commutant for noisy generators (Theorem 4.25), how small the eigenvalue gaps of the random elements are (Proposition 4.27), how close the computed decomposition is to an exact one (Theorem 4.28), and what the output certifies by itself (Proposition 4.26).
4.4 \(U(1)\) and charge sectors¶
Proposition 4.10. Let \(Q\) be a Hermitian matrix on \(\mathbb C^n\) with integer eigenvalues \(q\) (the charges), spectral projections \(P_q\) and \(d_q=\mathrm{rank}\,P_q\), and \(U_\theta=e^{i\theta Q}\); we write \(U(1)'=\{U_\theta:\theta\in\mathbb R\}'\) and \(U(1)''\) for its commutant.
- \(\theta\mapsto U_\theta\) is a unitary representation of \(U(1)=\mathbb R/2\pi\mathbb Z\), and every continuous unitary representation of \(U(1)\) on \(\mathbb C^n\) has this form.
- For \(X\in M_n\) the following are equivalent: (i) \(XU_\theta=U_\theta X\) for all \(\theta\); (ii) \([X,Q]=0\); (iii) \(X=\sum_qP_qXP_q\). Hence \(U(1)'\cong\bigoplus_qM_{d_q}(\mathbb C)\) (the block-diagonal operators) and \(U(1)''=\mathrm{span}\{P_q\}\): all \(k_c=1\), and the multiplicities are the \(d_q\).
- (Conserved charge.) For a Hermitian \(H\), \(e^{itH}Qe^{-itH}=Q\) for all \(t\) iff \([H,Q]=0\), i.e. iff \(H\in U(1)'\). Then every sector weight \(\mathrm{Tr}(P_q\rho(t))\) of \(\rho(t)=e^{-itH}\rho e^{itH}\) is constant.
Proof. (1) \(U_\theta=\sum_qe^{iq\theta}P_q\) is unitary, \(U_\theta U_{\theta'}=U_{\theta+\theta'}\) (Theorem 1.5(c)) and \(U_{2\pi}=1\). Conversely, for a continuous representation, \((U_\theta)_{\theta\ge0}\) is a continuous semigroup of matrices, so \(U_\theta=e^{\theta B}\) for \(\theta\ge0\) by Lemma 8.5 (proved in Chapter 8; the proof uses only finite dimension), and for \(\theta<0\), \(U_\theta=U_{-\theta}^{-1}=e^{\theta B}\). Differentiating \(U_\theta^*U_\theta=1\) at \(0\) (Lemma 8.1) gives \(B^*+B=0\), so \(B=iQ\) with \(Q\) Hermitian. The exponential series \(e^{i\theta Q}\) equals the functional calculus (Proposition 8.2(1) with \(H=-Q\)), so \(U_{2\pi}=\sum_qe^{2\pi iq}P_q=1\) forces \(e^{2\pi iq}=1\), \(q\in\mathbb Z\), for every eigenvalue \(q\) of \(Q\). (2) (i)\(\Rightarrow\)(ii): differentiate at \(\theta=0\). (ii)\(\Rightarrow\)(iii): \(X\) commutes with \(P_q=f(Q)\) (Theorem 1.5(d)), so \(P_qXP_{q'}=XP_qP_{q'}=0\) for \(q\neq q'\). (iii)\(\Rightarrow\)(i): \(XU_\theta=\sum_qe^{iq\theta}P_qXP_q=U_\theta X\). The block-diagonal operators form \(\bigoplus_qM_{d_q}\otimes1_1\) (in an orthonormal basis adapted to the ranges of the \(P_q\)), whose commutant is \(\bigoplus_q1_{d_q}\otimes M_1=\mathrm{span}\{P_q\}\): an operator commuting with them commutes with every \(P_q\), so it is block diagonal, and each block commutes with \(M_{d_q}\), hence is a scalar (Lemma 2.5, Step 5, with \(m=1\)). (3) \(\frac d{dt}e^{itH}Qe^{-itH}=ie^{itH}[H,Q]e^{-itH}\) (Proposition 8.2) vanishes for all \(t\) iff \([H,Q]=0\). Then \(e^{itH}\) commutes with \(P_q\) (Theorem 1.5(d), twice), and \(\mathrm{Tr}(P_q\rho(t))=\mathrm{Tr}(e^{itH}P_qe^{-itH}\rho)=\mathrm{Tr}(P_q\rho)\). \(\square\)
The irreducible representations of \(U(1)\) are thus the characters \(\theta\mapsto e^{iq\theta}\), \(q\in\mathbb Z\)
(one-dimensional, as for every abelian group), and the multiplicity of the charge \(q\) is \(d_q\). Charges add
under tensor products: if \(u_k\) and \(w_l\) are orthonormal eigenbases of \(Q_1\) and \(Q_2\) with eigenvalues \(\alpha_k\) and
\(\beta_l\), the orthonormal basis \(u_k\otimes w_l\) consists of eigenvectors of \(Q_1\otimes1+1\otimes Q_2\) (eigenvalue
\(\alpha_k+\beta_l\)) and of \(e^{i\theta Q_1}\otimes e^{i\theta Q_2}\) (eigenvalue \(e^{i\theta\alpha_k}e^{i\theta\beta_l}\)), so by the definition
of the functional calculus
math
e^{i\theta(Q_1\otimes1+1\otimes Q_2)}=e^{i\theta Q_1}\otimes e^{i\theta Q_2},
and it acts on the range of \(P_{q_1}\otimes P_{q_2}\) with charge \(q_1+q_2\). This is the fusion rule \((q_1,q_2)\mapsto q_1+q_2\) of the fused
tensor products of Chapter 7.
For \(n\) qubits the charge is the number operator \(\hat N=\sum_jn_j\), \(n_j=|1\rangle\langle1|\) on qubit
\(j\): \(\hat N|b\rangle=|b|\,|b\rangle\) with the Hamming weight \(|b|\). Its sectors have dimension
\(\binom nw\), so the operators that conserve the Hamming weight form
math
U(1)'\cong\bigoplus_{w=0}^nM_{\binom nw}(\mathbb C),
which is the fused product of \(n\) copies of \(M_1\oplus M_1\) with charges \(0,1\)
(bit \(1\) = spin up, \(\hat N=S^z+\frac n2\)). Every XXZ Hamiltonian
\(\sum_{\langle ij\rangle}J(s^x_is^x_j+s^y_is^y_j+\Delta s^z_is^z_j)\) conserves \(\hat N\): \(s^z_is^z_j\) is diagonal in the basis
\(|b\rangle\), and the hopping term \(s^x_is^x_j+s^y_is^y_j=\frac12(s^+_is^-_j+s^-_is^+_j)\) (expand
\(s^\pm=s^x\pm is^y\); the cross terms cancel) maps \(|b\rangle\) to \(0\) or to the configuration in which an up spin has
moved from one of the sites \(i,j\) to the other, so it maps every eigenspace of \(\hat N\) into itself.
Computation: Proposition 4.10 for four spins: the charge sectors found from \(\hat N\) alone, and an XXZ ring diagonalised sector by sector — tutorial, Section 4.4. In the library: \(U(1)'\) for \(n\) qubits is SpinChain(n).algebra, and the XXZ Hamiltonian is SpinChain.xxz — tutorial, Section 4.4.
Physical meaning (physical interpretation, not a theorem of this course). \(U(1)\) symmetry is the conservation of a particle number, a charge or a magnetisation. Proposition 4.10(3) is Noether's theorem in this setting: the charge is conserved iff the Hamiltonian is symmetric, and then the dynamics never leaves a sector (Theorem 8.3). If in addition nothing can measure coherences between sectors, the sectors are superselection sectors (Chapter 3).
4.5 \(SU(2)\): spin¶
Definition. A spin representation on \(\mathbb C^d\) is a triple of Hermitian matrices
\(S_x,S_y,S_z\) with
math
[S_x,S_y]=iS_z,\qquad[S_y,S_z]=iS_x,\qquad[S_z,S_x]=iS_y,
i.e. a representation of the labels \(\{x,y,z\}\) (Section 4.1) with these relations. The basic example is
\(s_a=\frac12\sigma_a\) on \(\mathbb C^2\) (Pauli matrices). If \((S_a)\) and \((T_a)\) are spin representations, so is
\(S_a\otimes1+1\otimes T_a\) (the cross terms commute); hence the total spin \(S_a=\sum_is_a^{(i)}\) of \(L\)
spins \(\frac12\) (decompose.total_spin(L)) is one. The rotations \(e^{-i\theta\,\mathbf n\cdot\mathbf S}\) (\(\theta\in\mathbb R\),
\(\mathbf n\in\mathbb R^3\) a unit vector, functional calculus) commute with \(X\) for all \(\theta,\mathbf n\) iff \(X\) commutes
with the three \(S_a\) (differentiate at \(\theta=0\), Proposition 8.2(1); conversely Theorem 1.5(d)), so the commutant
of "the \(SU(2)\) action" is \(\{S_x,S_y,S_z\}'\), and by Theorem 2.2 the rotations and the \(S_a\) generate the same
algebra (both sets are closed under adjoints). For \(L\) spins \(\frac12\) the rotations are the \(V^{\otimes L}\),
\(V\in SU(2)\). Indeed, \(\mathbf n\cdot\mathbf S=\sum_ih^{(i)}\) with \(h=\mathbf n\cdot\mathbf s\), and as for charges in Section 4.4
(induction on \(L\)), \(e^{-i\theta\,\mathbf n\cdot\mathbf S}=(e^{-i\theta h})^{\otimes L}\), with \(e^{-i\theta h}\in SU(2)\) (its eigenvalues
are \(e^{\mp i\theta/2}\), as \(h\) is traceless with \(h^2=\frac14\)). Conversely, a \(V\in SU(2)\) is normal with eigenvalues
of modulus \(1\) and product \(1\), so \(V=W\mathrm{diag}(e^{i\varphi},e^{-i\varphi})W^*\) with \(W\) unitary, and \(V=e^{iA}\) for the
traceless Hermitian \(A=W\mathrm{diag}(\varphi,-\varphi)W^*\). A traceless Hermitian \(2\times2\) matrix is a real combination
\(A=2\,\mathbf c\cdot\mathbf s\) of the Pauli matrices, so \(V=e^{-i\theta\,\mathbf n\cdot\mathbf s}\) with \(\theta=2|\mathbf c|\),
\(\mathbf n=-\mathbf c/|\mathbf c|\) (and \(\theta=0\) if \(\mathbf c=0\)). Theorem B.6(a),(c) and Corollary B.7 (with \(d=2\)) show
directly that the \(V^{\otimes L}\), \(V\in SU(2)\), span the algebra generated by the \(S_a\).
Put \(S_\pm=S_x\pm iS_y\), so \(S_-=S_+^*\), and \(\mathbf S^2=S_x^2+S_y^2+S_z^2\). The relations give
math
[S_z,S_\pm]=\pm S_\pm,\qquad[S_+,S_-]=2S_z,\qquad\mathbf S^2=S_-S_++S_z^2+S_z,
and \(\mathbf S^2\) commutes with every \(S_a\) (e.g.
\([\mathbf S^2,S_z]=S_x[S_x,S_z]+[S_x,S_z]S_x+S_y[S_y,S_z]+[S_y,S_z]S_y=-i\{S_x,S_y\}+i\{S_y,S_x\}=0\)).
Theorem 4.11 (irreducible spin representations). For every \(s\in\{0,\frac12,1,\frac32,\dots\}\) the
matrices of decompose.spin_matrices(s) — on \(\mathbb C^{2s+1}\) with basis \(|m\rangle\), \(m=s,s-1,\dots,-s\),
math
S_z|m\rangle=m|m\rangle,\qquad S_+|m\rangle=\sqrt{s(s+1)-m(m+1)}\;|m+1\rangle,\qquad S_-=S_+^*,
form an irreducible spin representation (spin \(s\)) with \(\mathbf S^2=s(s+1)1\). Every irreducible spin
representation is unitarily equivalent to exactly one of them.
Proof. Spin \(s\) is a spin representation. \([S_z,S_+]=S_+\) is clear. \(S_+S_-|m\rangle=(s(s+1)-(m-1)m)|m\rangle\) and \(S_-S_+|m\rangle=(s(s+1)-m(m+1))|m\rangle\), so \([S_+,S_-]=2S_z\); these two relations and their adjoints are equivalent to the three defining ones. \(\mathbf S^2|m\rangle=(s(s+1)-m(m+1)+m^2+m)|m\rangle=s(s+1)|m\rangle\). Irreducible: an invariant \(W\neq0\) is invariant under \(S_z\), whose eigenvalues are simple, so it contains some \(|m\rangle\); the coefficients of \(S_\pm\) vanish only at \(m=\pm s\), so applying \(S_\pm\) gives all \(|m'\rangle\).
Classification. Let \((S_a)\) be irreducible on \(\mathbb C^d\), \(\lambda\) the largest eigenvalue of \(S_z\) and
\(v_0\) a unit eigenvector. \(S_zS_+v_0=S_+(S_z+1)v_0=(\lambda+1)S_+v_0\), so \(S_+v_0=0\) by maximality. Let
\(v_j=S_-^jv_0\); then \(S_zv_j=(\lambda-j)v_j\), and by induction, using \(S_+S_-=S_-S_++2S_z\),
math
S_+v_j=j(2\lambda-j+1)\,v_{j-1}
(\(S_+v_j=S_-S_+v_{j-1}+2S_zv_{j-1}=[(j-1)(2\lambda-j+2)+2(\lambda-j+1)]v_{j-1}\)). Nonzero \(v_j\) are eigenvectors of
\(S_z\) for distinct eigenvalues, so there is a last nonzero one, \(v_r\). The span of \(v_0,\dots,v_r\) is invariant
under \(S_z,S_\pm\), hence equals \(\mathbb C^d\), and \(d=r+1\). From \(0=S_+v_{r+1}=(r+1)(2\lambda-r)v_r\),
\(\lambda=r/2=:s\). Finally \(\|v_j\|^2=\langle v_{j-1},S_+v_j\rangle=j(2s-j+1)\|v_{j-1}\|^2\), so the unit vectors
\(|s-j\rangle=v_j/\|v_j\|\) satisfy \(S_+|m\rangle=\sqrt{(s-m)(s+m+1)}\,|m+1\rangle\) with \(m=s-j\), which is the formula
above. Different \(s\) have different dimensions. \(\square\)
In the library: spin \(s\) is decompose.spin_matrices(s), checked against Theorem 4.11 in tutorial, Section 4.6; the total spin of \(L\) spins \(\frac12\) is decompose.total_spin(L) — tutorial, Section 4.5.
Corollary 4.12. Every spin representation on \(\mathbb C^n\) is unitarily equivalent to
\(\bigoplus_s(\text{spin }s)\otimes1_{m_s}\), and with \(d_M=\dim\ker(S_z-M)\),
math
m_s=d_s-d_{s+1}.
Proof. Theorem 4.3 and Theorem 4.11. Each copy of spin \(s'\) contains exactly one eigenvector of \(S_z\) for the eigenvalue \(M\) if \(|M|\le s'\) and \(s'-M\in\mathbb Z\), and none otherwise; so \(d_M=\sum_{s'\ge|M|,\,s'-M\in\mathbb Z}m_{s'}\), and the difference of two consecutive values is \(m_s\). \(\square\)
Proposition 4.13 (\(L\) spins \(\frac12\)). The total spin on \((\mathbb C^2)^{\otimes L}\) contains spin \(S\) iff
\(S\in\{\frac L2,\frac L2-1,\dots\}\), \(S\ge0\), with multiplicity
math
m_S=\binom L{\frac L2-S}-\binom L{\frac L2-S-1}\qquad\Big(\tbinom L{-1}=0\Big),
so the algebra generated by \(S_x,S_y,S_z\) is \(\bigoplus_SM_{2S+1}\otimes1_{m_S}\) and its commutant
\(\bigoplus_S1_{2S+1}\otimes M_{m_S}\).
Proof. \(S_z=\sum_is_z^{(i)}\) is diagonal in the product basis, with eigenvalue \(u-\frac L2\) on a product of \(u\) spins up and \(L-u\) spins down, so \(d_M=\binom L{L/2+M}=\binom L{L/2-M}\) if \(\frac L2+M\in\{0,\dots,L\}\) and \(0\) otherwise. Apply Corollary 4.12 and Theorem 4.3. The values are positive because \(\binom L{j+1}/\binom Lj=\frac{L-j}{j+1}>1\) for \(j<\frac{L-1}2\). \(\square\)
For \(L=2\): \(\frac12\otimes\frac12=1\oplus0\) (Clebsch–Gordan): the triplet \(S_-^j|{\uparrow\uparrow}\rangle\), \(j=0,1,2\),
symmetric under the swap \(F\) of the two spins, and the singlet
\(\frac1{\sqrt2}(|{\uparrow\downarrow}\rangle-|{\downarrow\uparrow}\rangle)\), antisymmetric and annihilated by \(S_+\) and \(S_z\).
For \(L=4\): \(m_2=1\), \(m_1=4-1=3\), \(m_0=6-4=2\), so
math
\{S_x,S_y,S_z\}''\cong M_5\oplus M_3\oplus M_1\ \text{ with multiplicities }1,3,2,\qquad
\{S_x,S_y,S_z\}'\cong M_1\oplus M_3\oplus M_2,
of dimensions \(35\) and \(14\) — the numbers found in Chapter 10.
Computation: the decomposition for \(L=4\) from the three generators alone, with Proposition 4.13 and \(\mathbf S^2=S(S+1)\) on spin \(S\) — tutorial, Section 4.5.
Which operators commute with all rotations? The site permutations do: conjugation by \(\pi(\sigma)\) maps \(s^{(i)}_a\) to \(s^{(\sigma(i))}_a\) (Lemma B.1(b)), so it fixes \(\sum_is^{(i)}_a\). Conversely, nothing else does:
Theorem 4.14 (Schur–Weyl duality). Let \(\pi\) be the representation of \(S_L\) permuting the factors of \((\mathbb C^2)^{\otimes L}\). (a) \(\mathrm{span}\,\pi(S_L)\subseteq\{S_x,S_y,S_z\}'\). (b) Equality holds; equivalently (Theorem 2.2), the algebra generated by the total spin is \(\pi(S_L)'\).
Proof. Part (a) was just proved. Part (b) is Corollary B.7, the case \(d=2\) of Theorem B.6, which proves the duality for \((\mathbb C^d)^{\otimes L}\) and the diagonal action of \(U(d)\) for all \(d\) and \(L\). \(\square\)
(For further reading: W. Fulton and J. Harris, Representation Theory, Springer 1991, Lecture 6; R. Goodman and N. Wallach, Symmetry, Representations, and Invariants, Springer 2009, Section 9.1.)
Computation: Theorem 4.14 for \(L\le5\), the dimensions \(2,5,14,42\) of \(\{S_a\}'\) (Corollary B.10), and \(\pi(S_4)''\), whose pairs \((k_c,m_c)\) are those of the total spin exchanged (Corollary B.9) — tutorial, Section 4.5.
Physical meaning (physical interpretation, not a theorem of this course). A spin-\(S\) multiplet consists of \(2S+1\) states of equal energy for every rotation-invariant Hamiltonian, distinguished by \(S^z=M\). Two spins \(\frac12\) form a triplet and a singlet; four spins form one quintet, three triplets and two singlets. The multiplicity spaces of the two singlets and of the three triplets are the decoherence-free qubit and the noiseless qutrit of Chapter 10.
4.6 Spin systems¶
The Heisenberg model on \(L\) spins \(\frac12\) is \(H=\sum_{i<j}J_{ij}\,\mathbf s^{(i)}\cdot\mathbf s^{(j)}\) with real couplings \(J_{ij}\).
Proposition 4.15. (a) (Dirac's identity) \(\mathbf s^{(i)}\cdot\mathbf s^{(j)}=\frac12F_{ij}-\frac14\), where \(F_{ij}\) swaps the spins \(i\) and \(j\). Hence every Heisenberg Hamiltonian lies in \(\mathrm{span}\,\pi(S_L)\subseteq\{S_a\}'\), and \(V^*HV=\bigoplus_S1_{2S+1}\otimes H_S\) with blocks \(H_S\in M_{m_S}\) in the decomposition of Proposition 4.13: each eigenvalue of \(H_S\) is a \((2S+1)\)-fold degenerate level of \(H\). (b) The space of highest-weight vectors \(\ker S_+\cap\ker(S_z-S)\) has dimension \(m_S\), is invariant under \(H\), and \(H\) acts on it as \(H_S\) (up to a unitary change of basis).
Proof. (a) On two spins, \(\mathbf s^{(1)}\cdot\mathbf s^{(2)}=\frac12(\mathbf S^2-\mathbf s^{(1)2}-\mathbf s^{(2)2})=\frac12\mathbf S^2-\frac34\) (Theorem 4.11 with \(s=\frac12\)). The triplet vectors \(S_-^j|{\uparrow\uparrow}\rangle\) are \(|{\uparrow\uparrow}\rangle\), \(|{\downarrow\uparrow}\rangle+|{\uparrow\downarrow}\rangle\), \(2|{\downarrow\downarrow}\rangle\), and with the singlet they form an orthogonal basis of \(\mathbb C^2\otimes\mathbb C^2\). On the triplet, \(\mathbf S^2=2\) and \(F=1\): \(\mathbf S^2=S_-S_++S_z^2+S_z\) is \(0+1+1=2\) on \(|{\uparrow\uparrow}\rangle\), and \(\mathbf S^2\) and \(F\) commute with \(S_-\), while \(F\) fixes \(|{\uparrow\uparrow}\rangle\). On the singlet, \(\mathbf S^2=0\) (it is annihilated by \(S_+\) and \(S_z\)) and \(F=-1\). Both sides agree on these four vectors. For sites \(i,j\) of \(L\) tensor with the identity. The rest is Theorem 4.14(a) and Theorem 4.3. (b) We prove more: for any spin representation decomposed as in Corollary 4.12 and any \(\mu\in\frac12\mathbb Z\), \(\ker S_+\cap\ker(S_z-\mu)\) is (the image under the unitary of Corollary 4.12 of) \(|\mu,\mu\rangle\otimes\mathbb C^{m_\mu}\) if \(\mu\ge0\), and \(0\) if \(\mu<0\). Indeed, in that basis a vector with \(S_z=\mu\) is \(\sum_{S'}|S',\mu\rangle\otimes w_{S'}\) over the \(S'\ge|\mu|\) with \(S'-\mu\in\mathbb Z\), and by Theorem 4.11 \(S_+\) maps it to \(\sum_{S'}\sqrt{(S'-\mu)(S'+\mu+1)}\,|S',\mu+1\rangle\otimes w_{S'}\), where \(S'+\mu+1\ge1\); so it vanishes iff \(w_{S'}=0\) for all \(S'\neq\mu\). For the Heisenberg \(H\), \(1\otimes H_S\) acts on \(|S,S\rangle\otimes\mathbb C^{m_S}\) as \(H_S\). \(\square\)
Computation: Proposition 4.15 on a ring of four spins and on a ring of eight spins — tutorial, Section 4.6.
Part (b) is how the module su2 (spin chains, Chapter 9) builds \(H_S\) without numerical decomposition: it
constructs an orthonormal basis of the highest-weight vectors, adding one site at a time, with the
Clebsch–Gordan coefficients of \(j\otimes\frac12\). We derive them by lowering from the highest weight. By Theorem 4.11
and \(S_-=S_+^*\), spin \(j\) satisfies
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S_\pm|j,m\rangle=\sqrt{(j\mp m)(j\pm m+1)}\;|j,m\pm1\rangle ,
and we put \(|j,m\rangle=0\) for \(|m|>j\) (consistent with the formula, whose coefficient vanishes at \(m=\pm j\)).
Theorem 4.16 (Clebsch–Gordan coefficients of \(j\otimes\frac12\)). Let \(j\in\{\frac12,1,\frac32,\dots\}\), let
\(S_a=S^{(j)}_a\otimes1+1\otimes s_a\) on \(\mathbb C^{2j+1}\otimes\mathbb C^2\) (spin \(j\) of Theorem 4.11, with basis \(|j,m\rangle\),
coupled to a spin \(\frac12\) with \(|{\uparrow}\rangle=|\frac12,\frac12\rangle\), \(|{\downarrow}\rangle=|\frac12,-\frac12\rangle\)), and put
\(\alpha_M=\sqrt{(j+M+\frac12)/(2j+1)}\), \(\beta_M=\sqrt{(j-M+\frac12)/(2j+1)}\). For \(J=j\pm\frac12\) and \(M=J,J-1,\dots,-J\) let
math
\begin{aligned}
|j+\tfrac12,M\rangle\!\rangle&=\alpha_M\,|j,M-\tfrac12\rangle\otimes|{\uparrow}\rangle+\beta_M\,|j,M+\tfrac12\rangle\otimes|{\downarrow}\rangle,\\
|j-\tfrac12,M\rangle\!\rangle&=-\beta_M\,|j,M-\tfrac12\rangle\otimes|{\uparrow}\rangle+\alpha_M\,|j,M+\tfrac12\rangle\otimes|{\downarrow}\rangle .
\end{aligned}
(a) These \(2(2j+1)\) vectors are an orthonormal basis of \(\mathbb C^{2j+1}\otimes\mathbb C^2\).
(b) \(|j+\frac12,j+\frac12\rangle\!\rangle=|j,j\rangle\otimes|{\uparrow}\rangle\), and \(|j-\frac12,j-\frac12\rangle\!\rangle\) is, up to a factor of modulus one, the
only unit vector \(x\) with \(S_zx=(j-\frac12)x\) and \(S_+x=0\); the factor is fixed by a positive coefficient of
\(|j,j\rangle\otimes|{\downarrow}\rangle\) (the Condon–Shortley convention).
(c) \(S_z|J,M\rangle\!\rangle=M|J,M\rangle\!\rangle\) and \(S_-|J,M\rangle\!\rangle=\sqrt{(J+M)(J-M+1)}\,|J,M-1\rangle\!\rangle\): the states \(|J,M\rangle\!\rangle\)
are obtained from \(|J,J\rangle\!\rangle\) by lowering and normalising.
(d) Hence \(S_a|J,M\rangle\!\rangle=\sum_{M'}(S^{(J)}_a)_{M'M}|J,M'\rangle\!\rangle\) for \(a=x,y,z\): the unitary \(|J,M\rangle\mapsto|J,M\rangle\!\rangle\) maps
spin \((j+\frac12)\oplus\) spin \((j-\frac12)\) onto \(S_a\), so \(j\otimes\frac12\cong(j+\frac12)\oplus(j-\frac12)\), and the
Clebsch–Gordan coefficients \(\langle j,m;\frac12,\sigma|J,M\rangle\) (the coordinates of \(|J,M\rangle\!\rangle\) in the product basis) are
\(\pm\alpha_M,\beta_M\) as displayed.
Proof. (a) If a basis vector in the display is \(0\), its coefficient is \(0\) too: \(|j,M-\frac12\rangle=0\) only for
\(M=-j-\frac12\), where \(\alpha_M=0\), and \(|j,M+\frac12\rangle=0\) only for \(M=j+\frac12\), where \(\beta_M=0\). So each vector has norm
\(\alpha_M^2+\beta_M^2=1\). Vectors with different \(M\) involve different product basis vectors, and for equal \(M\) the
inner product is \(-\alpha_M\beta_M+\beta_M\alpha_M=0\). There are \((2j+2)+2j=2(2j+1)\) of them.
(b) The first statement is \(\alpha_{j+1/2}=1\), \(\beta_{j+1/2}=0\). The eigenspace \(S_z=j-\frac12\) is spanned by
\(|j,j-1\rangle\otimes|{\uparrow}\rangle\) and \(|j,j\rangle\otimes|{\downarrow}\rangle\), and with \(s_+|{\uparrow}\rangle=0\), \(s_+|{\downarrow}\rangle=|{\uparrow}\rangle\),
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S_+\big(p\,|j,j-1\rangle\otimes|{\uparrow}\rangle+q\,|j,j\rangle\otimes|{\downarrow}\rangle\big)=(p\sqrt{2j}+q)\,|j,j\rangle\otimes|{\uparrow}\rangle ,
which vanishes iff \(q=-\sqrt{2j}\,p\); with \(|p|^2+|q|^2=1\) and \(q>0\) (hence \(p<0\)) this is \(p=-\beta_{j-1/2}=-1/\sqrt{2j+1}\),
\(q=\alpha_{j-1/2}=\sqrt{2j/(2j+1)}\). (c) \(S_z\): both terms have \(S_z=(M\mp\frac12)\pm\frac12=M\). For \(S_-\), let
\(x_M=p\,|j,M-\frac12\rangle\otimes|{\uparrow}\rangle+q\,|j,M+\frac12\rangle\otimes|{\downarrow}\rangle\); since \(s_-|{\uparrow}\rangle=|{\downarrow}\rangle\) and \(s_-|{\downarrow}\rangle=0\),
math
S_-x_M=p\sqrt{(j+M-\tfrac12)(j-M+\tfrac32)}\;|j,M-\tfrac32\rangle\otimes|{\uparrow}\rangle
+\Big(p+q\sqrt{(j+M+\tfrac12)(j-M+\tfrac12)}\Big)|j,M-\tfrac12\rangle\otimes|{\downarrow}\rangle .
We use \(\beta_M\sqrt{(j+M+\frac12)(j-M+\frac12)}=(j-M+\frac12)\alpha_M\) and \(\alpha_M\sqrt{(j+M+\frac12)(j-M+\frac12)}=(j+M+\frac12)\beta_M\).
For \(J=j+\frac12\), \((p,q)=(\alpha_M,\beta_M)\) and \(c=\sqrt{(J+M)(J-M+1)}=\sqrt{(j+M+\frac12)(j-M+\frac32)}\): the first coefficient
is \(\sqrt{(j+M+\frac12)(j+M-\frac12)(j-M+\frac32)/(2j+1)}=c\,\alpha_{M-1}\), the second \(\alpha_M(j-M+\frac32)=c\,\beta_{M-1}\).
For \(J=j-\frac12\), \((p,q)=(-\beta_M,\alpha_M)\) and \(c=\sqrt{(j+M-\frac12)(j-M+\frac12)}\): the first coefficient is
\(-\sqrt{(j-M+\frac12)(j+M-\frac12)(j-M+\frac32)/(2j+1)}=-c\,\beta_{M-1}\), the second \(\beta_M(j+M-\frac12)=c\,\alpha_{M-1}\).
In both cases \(S_-|J,M\rangle\!\rangle=c\,|J,M-1\rangle\!\rangle\) for \(M>-J\). For \(M=-J\) both coefficients vanish (\(p=\alpha_M=0\)
if \(J=j+\frac12\); \(j+M-\frac12=0\) if \(J=j-\frac12\)), so \(S_-|J,-J\rangle\!\rangle=0\). (d) By (a) and (c), in the orthonormal basis
\(|J,M\rangle\!\rangle\) the matrices of \(S_z\) and \(S_-\) are those of spin \((j+\frac12)\oplus\) spin \((j-\frac12)\) (Theorem 4.11
and the formula above); so is the matrix of \(S_+=S_-^*\), hence of \(S_x=\frac12(S_++S_-)\) and \(S_y=\frac1{2i}(S_+-S_-)\). \(\square\)
Computation: Theorem 4.16 for \(j\le2\), against dense matrices and against su2.clebsch_gordan — tutorial, Section 4.6.
Exercise 2 extends the decomposition (but not the coefficients) to \(j_1\otimes j_2\). For the library only the two highest-weight vectors of Theorem 4.16 matter, and they work in any spin representation, irreducible or not:
Corollary 4.17 (highest-weight vectors one site at a time). Let \((T_a)\) be a spin representation on \(E\), let
\(\mathcal H_\mu(E)=\ker T_+\cap\ker(T_z-\mu)\), with an orthonormal basis \(B_\mu\) (\(B_\mu=\emptyset\) if \(\mathcal H_\mu(E)=0\)), and
let \(S_a=T_a\otimes1+1\otimes s_a\) on \(E\otimes\mathbb C^2\). For every \(S\in\{0,\frac12,1,\dots\}\) the vectors
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v\otimes|{\uparrow}\rangle\quad(v\in B_{S-1/2}),\qquad
\sqrt{\tfrac{2S+1}{2S+2}}\;v\otimes|{\downarrow}\rangle-\tfrac1{\sqrt{(2S+1)(2S+2)}}\;T_-v\otimes|{\uparrow}\rangle\quad(v\in B_{S+1/2})
form an orthonormal basis of \(\ker S_+\cap\ker(S_z-S)\) (the first family is empty for \(S=0\)). Starting from one spin
(\(E=\mathbb C^2\), \(B_{1/2}=\{|{\uparrow}\rangle\}\), all other \(B_\mu\) empty) and adding \(L-1\) spins, this gives orthonormal bases
of the spaces of highest-weight vectors of \(L\) spins \(\frac12\), of dimensions \(m_S\) (Proposition 4.15(b)).
Proof. Write \(a=\sqrt{(2S+1)/(2S+2)}\), \(b=-1/\sqrt{(2S+1)(2S+2)}\), \(u(v)=a\,v\otimes|{\downarrow}\rangle+b\,T_-v\otimes|{\uparrow}\rangle\). For \(v\in\mathcal H_{S+1/2}(E)\), \(T_+T_-v=(T_-T_++2T_z)v=(2S+1)v\), and \(a+(2S+1)b=0\). Highest weight: \(S_z\) is \(S\) on both families (\(T_z=S\mp\frac12\) on \(v\) and \(S-\frac12\) on \(T_-v\)); \(S_+(v\otimes|{\uparrow}\rangle)=T_+v\otimes|{\uparrow}\rangle=0\), and \(S_+u(v)=a\,T_+v\otimes|{\downarrow}\rangle+a\,v\otimes|{\uparrow}\rangle+b\,T_+T_-v\otimes|{\uparrow}\rangle=(a+(2S+1)b)\,v\otimes|{\uparrow}\rangle=0\). Orthonormal: \(\langle v\otimes|{\uparrow}\rangle,u(v')\rangle=b\langle v,T_-v'\rangle=b\langle T_+v,v'\rangle=0\) for \(v\in B_{S-1/2}\), and \(\langle u(v),u(v')\rangle=a^2\langle v,v'\rangle+b^2\langle v,T_+T_-v'\rangle=(a^2+(2S+1)b^2)\langle v,v'\rangle=\langle v,v'\rangle\). Spanning: let \(S_zw=Sw\) and \(S_+w=0\), and write \(w=x\otimes|{\uparrow}\rangle+y\otimes|{\downarrow}\rangle\). Then \(T_zx=(S-\frac12)x\), \(T_zy=(S+\frac12)y\) and \(0=S_+w=(T_+x+y)\otimes|{\uparrow}\rangle+T_+y\otimes|{\downarrow}\rangle\), so \(T_+y=0\), i.e. \(y\in\mathcal H_{S+1/2}(E)\), and \(T_+x=-y\). Then \(x'=x+\frac1{2S+1}T_-y\) satisfies \(T_+x'=-y+y=0\), so \(x'\in\mathcal H_{S-1/2}(E)\), which is \(0\) for \(S=0\) (proof of Proposition 4.15(b)). As \(b/a=-1/(2S+1)\), \(w=x'\otimes|{\uparrow}\rangle+a^{-1}u(y)\), with \(u\) extended linearly, lies in the span of the two families. For \(L\) spins apply this with \(E=(\mathbb C^2)^{\otimes(n-1)}\) and its total spin \(T_a\), \(n=2,\dots,L\); on one spin, \(\mathcal H_{1/2}=\mathbb C|{\uparrow}\rangle\) and all other \(\mathcal H_\mu=0\) (the only spin is \(\frac12\)). \(\square\)
The second family consists of the vectors \(|S,S\rangle\!\rangle\) of Theorem 4.16 with \(j=S+\frac12\), written inside the copy of spin \(j\) spanned by the \(T_-^kv\) (proof of Theorem 4.11, \(|j,j-1\rangle=T_-v/\sqrt{2j}\)), and the first family is \(|j+\frac12,j+\frac12\rangle\!\rangle\) with \(j=S-\frac12\). Labelling each basis vector by its sequence of intermediate spins \(S_1=\frac12,S_2,\dots,S_L=S\), \(|S_n-S_{n-1}|=\frac12\), gives the Yamanouchi basis.
In the library: su2.SU2Chain builds the Yamanouchi basis with these coefficients, inside the \(U(1)\) sector \(\hat N=\frac L2+S\) of SpinChain; Corollary 4.17 is checked for eight spins, with the \(U(1)\) and \(SU(2)\) blocks of the Heisenberg ring and their cost — tutorial, Section 4.6.
Physical meaning (physical interpretation, not a theorem of this course). The Heisenberg model describes the exchange interaction of localised electron spins in magnetic insulators; \(J_{ij}>0\) favours antiparallel neighbours. Its spectrum consists of multiplets, and a magnetic field \(h\,S^z\) splits each multiplet into \(2S+1\) equidistant levels (Exercise 4). Theorem 9.9 (bipartite couplings; for even antiferromagnetic chains and rings a singlet ground state) goes back to E. Lieb and D. Mattis (J. Math. Phys. 3 (1962) 749).
4.7 Real algebras: real, complex and quaternionic type¶
Symmetries are often given by real matrices: the rotations of \(\mathbb R^3\), permutations, Hamiltonians that are real in some basis. The complex algebra they generate is decomposed by Theorem 2.6, but the real matrices in it form a real algebra, whose simple pieces are of three kinds, \(M_k(\mathbb R)\), \(M_k(\mathbb C)\) and \(M_k(\mathbb H)\). We derive them from the complex decomposition and one more datum: what complex conjugation does to the sectors.
For \(a\in M_n(\mathbb C)\) let \(\bar a\) be the entrywise complex conjugate (likewise for vectors, and \(\bar S=\{\bar s:s\in S\}\) for sets). Conjugation is antilinear and bijective, \(\overline{ab}=\bar a\bar b\), \(\overline{a^*}=\bar a^*\) and \(\bar{\bar a}=a\); \(\bar a=a\) iff \(a\) is real, and then \(a^*=a^T\). For a set \(\mathcal B\) of matrices (or of vectors) let \(\mathcal B_{\mathbb R}=\mathcal B\cap M_n(\mathbb R)\) (or \(\mathcal B\cap\mathbb R^n\)). The inner product is \(\langle u,v\rangle=\sum_i\bar u_iv_i\), so \(\langle\bar u,\bar v\rangle=\overline{\langle u,v\rangle}=\langle v,u\rangle\).
Lemma 4.18 (real parts). (a) Let \(\mathcal B\) be a complex subspace of \(M_n(\mathbb C)\) or of \(\mathbb C^n\) with \(\bar{\mathcal B}=\mathcal B\). Then \(\mathcal B=\mathcal B_{\mathbb R}\oplus i\mathcal B_{\mathbb R}\) and \(\dim_{\mathbb R}\mathcal B_{\mathbb R}=\dim_{\mathbb C}\mathcal B\). (b) Let \(\mathcal A\) be a \(*\)-algebra on \(\mathbb C^n\) with \(\bar{\mathcal A}=\mathcal A\). Then \(\mathcal A_{\mathbb R}\) is a real algebra with unit, closed under transposition; \(\overline{\mathcal A'}=\mathcal A'\); and \((\mathcal A')_{\mathbb R}\) is the set of real \(X\) with \(Xa=aX\) for all \(a\in\mathcal A_{\mathbb R}\). (c) If \(S\subseteq M_n(\mathbb C)\) satisfies \(S^*=S\) and \(\bar S=S\) — for instance \(S=\{S_1,\dots,S_J,S_1^T,\dots,S_J^T\}\) with real \(S_j\), or a group of real orthogonal matrices — then \(\overline{S''}=S''\).
Proof. (a) For \(b\in\mathcal B\), \(\mathrm{Re}\,b=\frac12(b+\bar b)\) and \(\mathrm{Im}\,b=\frac1{2i}(b-\bar b)\) are real elements of \(\mathcal B\) with \(b=\mathrm{Re}\,b+i\mathrm{Im}\,b\), and an element of \(\mathcal B_{\mathbb R}\cap i\mathcal B_{\mathbb R}\) is real and purely imaginary, hence \(0\). A basis \(b_1,\dots,b_r\) of the real space \(\mathcal B_{\mathbb R}\) therefore spans \(\mathcal B\) over \(\mathbb C\), and it is linearly independent over \(\mathbb C\): if \(\sum_j(\alpha_j+i\beta_j)b_j=0\) with real \(\alpha_j,\beta_j\), the real and imaginary parts give \(\sum_j\alpha_jb_j=\sum_j\beta_jb_j=0\). (b) \(\mathcal A_{\mathbb R}\) contains \(1\) and is closed under products, and under \(a\mapsto a^T=a^*\). \(Y\) commutes with all \(t\in T\) iff \(\bar Y\) commutes with all \(\bar t\), so \(\overline{T'}=\bar T'\) for every set \(T\), and \(\overline{\mathcal A'}=\bar{\mathcal A}'=\mathcal A'\). A real \(X\) that commutes with \(\mathcal A_{\mathbb R}\) commutes with \(\mathcal A_{\mathbb R}+i\mathcal A_{\mathbb R}=\mathcal A\) (by (a)). (c) Apply \(\overline{T'}=\bar T'\) twice: \(\overline{S''}=\bar S''=S''\). \(\square\)
Standing notation. For the rest of this section \(\mathcal A\) is a \(*\)-algebra on \(\mathbb C^n\) with
\(\bar{\mathcal A}=\mathcal A\), and \(V\) is a decomposition as in Theorem 2.6. \(V_c\) is the \(n\times k_cm_c\) block of
columns of \(V\) that belongs to the \(c\)-th summand, so that \(V_c^*V_d=\delta_{cd}1\) and \(V_cV_c^*=P_c\), the minimal
central projections, and
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aV_c=V_c\big(\pi_c(a)\otimes1_{m_c}\big)\quad(a\in\mathcal A),\qquad XV_c=V_c\big(1_{k_c}\otimes x_c\big)\quad(X\in\mathcal A'),
with \(\pi_c(a)\in M_{k_c}\) and \(x_c\in M_{m_c}\). Each \(\pi_c\) is a unital \(*\)-homomorphism of \(\mathcal A\) onto
\(M_{k_c}\): for \(x\in M_{k_c}\) the element \(V_c(x\otimes1)V_c^*\) of \(\mathcal A\) has \(\pi_c=x\) and \(\pi_e=0\) for \(e\neq c\).
Finally \(\Omega_2=\begin{pmatrix}0&-1\\1&0\end{pmatrix}\) and \(\Omega_{2r}=1_r\otimes\Omega_2\) (block diagonal with \(r\)
copies of \(\Omega_2\)).
A map \(\theta:E\to E\) on a subspace \(E\subseteq\mathbb C^k\) is antiunitary if \(\theta(\alpha u+\beta v)=\bar\alpha\theta u+\bar\beta\theta v\) and \(\langle\theta u,\theta v\rangle=\langle v,u\rangle\). For a unitary \(J\in U(k)\), \(\theta_Jv=J\bar v\) is antiunitary on \(\mathbb C^k\) (\(\langle J\bar u,J\bar v\rangle=\langle\bar u,\bar v\rangle=\langle v,u\rangle\)), with \(\theta_J^2v=J\bar Jv\).
Lemma 4.19 (antiunitary maps). Let \(\theta\) be antiunitary on \(E\neq0\) with \(\theta^2=\varepsilon\,\mathrm{id}\), \(\varepsilon\in\{1,-1\}\). (a) If \(\varepsilon=1\), \(E\) has an orthonormal basis of vectors \(w\) with \(\theta w=w\). (b) If \(\varepsilon=-1\), then \(\langle u,\theta u\rangle=0\) for all \(u\in E\); for every unit vector \(u\), \(\mathrm{span}\{u,\theta u\}\) is a \(\theta\)-invariant plane; and \(E\) has an orthonormal basis \(w_1,\theta w_1,\dots,w_r,\theta w_r\). In particular \(\dim E=2r\) is even. (c) Let \(J\in U(k)\) with \(J\bar J=\varepsilon1\). If \(\varepsilon=1\), there is a unitary \(W\) with \(J\bar W=W\), i.e. \(J=WW^T\). If \(\varepsilon=-1\), \(k\) is even and there is a unitary \(W\) with \(J\bar W=W\Omega_k\), i.e. \(J=W\Omega_kW^T\).
Proof. (a) Let \(F=\{v\in E:\theta v=v\}\), a real subspace. Every \(v\in E\) is \(v_1+iv_2\) with \(v_1=\frac12(v+\theta v)\) and \(v_2=\frac1{2i}(v-\theta v)\), and both lie in \(F\) (\(\theta v_2=\frac1{-2i}(\theta v-v)=v_2\)). For \(u,v\in F\), \(\langle u,v\rangle=\langle\theta u,\theta v\rangle=\langle v,u\rangle\) is real, so \(\langle\cdot,\cdot\rangle\) is a real inner product on \(F\); let \(w_1,\dots,w_s\) be an orthonormal basis of \(F\) (Gram–Schmidt). The \(w_j\) are orthonormal in \(E\), hence linearly independent over \(\mathbb C\), and they span \(E\) over \(\mathbb C\) since \(E=F+iF\). (b) Applying \(\langle\theta x,\theta y\rangle=\langle y,x\rangle\) to \(x=\theta u\), \(y=u\): \(\langle u,\theta u\rangle=\langle\theta^2u,\theta u\rangle=-\langle u,\theta u\rangle\). So for a unit vector \(u\), \(\theta u\) is a unit vector orthogonal to \(u\), and \(E_1=\mathrm{span}\{u,\theta u\}\) is invariant: \(\theta(\alpha u+\beta\theta u)=\bar\alpha\theta u-\bar\beta u\). The orthogonal complement of \(E_1\) in \(E\) is invariant too: for \(x\perp E_1\) and \(y\in E_1\), \(y=-\theta(\theta y)\) and \(\langle\theta x,y\rangle=-\langle\theta x,\theta(\theta y)\rangle=-\langle\theta y,x\rangle=0\), as \(\theta y\in E_1\). Induction on \(\dim E\) gives the basis. (c) \(\theta_J\) is antiunitary on \(\mathbb C^k\) with \(\theta_J^2=J\bar J=\varepsilon\). Let \(W\) be the unitary whose columns are the basis of (a), resp. the basis \(w_1,\theta_Jw_1,w_2,\theta_Jw_2,\dots\) of (b). The columns of \(J\bar W\) are the images under \(\theta_J\) of the columns of \(W\). In case (a) this says \(J\bar W=W\). In case (b) the images are \(\theta_Jw_j\) and \(\theta_J^2w_j=-w_j\), which are the columns \(2j-1\) and \(2j\) of \(W\Omega_k\) (as \(\Omega_2e_1=e_2\), \(\Omega_2e_2=-e_1\)). Since \(\bar W^{-1}=\overline{W^*}=W^T\), \(J=WW^T\), resp. \(J=W\Omega_kW^T\). \(\square\)
Theorem 4.20 (conjugation of sectors). (a) There are a permutation \(\sigma\) of the sectors with
\(\sigma\circ\sigma=\mathrm{id}\) and unitaries \(J_c\in U(k_c)\), \(M_c\in U(m_c)\) such that \(\bar P_c=P_{\sigma(c)}\),
\(k_{\sigma(c)}=k_c\), \(m_{\sigma(c)}=m_c\) and, for every \(c\) and all \(a\in\mathcal A\),
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\overline{V_c}=V_{\sigma(c)}\,(J_c\otimes M_c),\qquad\pi_{\sigma(c)}(\bar a)=J_c\,\overline{\pi_c(a)}\,J_c^*.
\(J_c\otimes M_c\) is unique, and \(J_c,M_c\) are unique up to \((J_c,M_c)\mapsto(\lambda J_c,\bar\lambda M_c)\) with \(|\lambda|=1\).
(b) If \(\sigma(c)=c\), there is \(\varepsilon_c\in\{1,-1\}\) with \(J_c\bar J_c=\varepsilon_c1_{k_c}\) and
\(M_c\bar M_c=\varepsilon_c1_{m_c}\). If \(\varepsilon_c=-1\), then \(k_c\) and \(m_c\) are even.
(c) \(\sigma\) and the \(\varepsilon_c\) do not depend on the choice of \(V\).
(d) For \(y\in M_{m_c}\), \(\overline{V_c(1\otimes y)V_c^*}=V_{\sigma(c)}\big(1\otimes M_c\bar yM_c^*\big)V_{\sigma(c)}^*\). Hence, if
\(X\in\mathcal A'\) has the blocks \(x_c\), then \(\bar X\in\mathcal A'\) has the blocks \((\bar X)_{\sigma(c)}=M_c\bar x_cM_c^*\): on the
commutant, \(M_c\) plays the role that \(J_c\) plays on \(\mathcal A\).
Proof. (a) Conjugation maps \(\mathcal A\) onto \(\mathcal A\) and preserves sums, products and adjoints, so it maps the
centre \(Z(\mathcal A)\) onto itself, and the set of nonzero projections of \(Z(\mathcal A)\) bijectively onto itself,
preserving the order \(p\le q\) (\(pq=p\)). Hence it permutes the minimal ones, which are the \(P_c\) (Lemma 2.3):
\(\bar P_c=P_{\sigma(c)}\), and \(\sigma\circ\sigma=\mathrm{id}\) because \(\bar{\bar P}_c=P_c\). Let \(d=\sigma(c)\). The range of
\(\overline{V_c}\) is the conjugate of the range of \(P_c\), which is the range of \(\bar P_c=P_d\); so
\(\overline{V_c}=P_d\overline{V_c}=V_dT\) with \(T=V_d^*\overline{V_c}\), and
\(T^*T=\overline{V_c}^*P_d\overline{V_c}=\overline{V_c^*V_c}=1\). As \(k_cm_c=\mathrm{rank}\,P_c=\mathrm{rank}\,\bar P_c=k_dm_d\),
the isometry \(T\) is square, hence unitary. Conjugating \(aV_c=V_c(\pi_c(a)\otimes1)\) and using
\(\bar aV_d=V_d(\pi_d(\bar a)\otimes1)\) gives \(V_d(\pi_d(\bar a)\otimes1)T=V_dT(\overline{\pi_c(a)}\otimes1)\); multiplying by
\(V_d^*\),
math
\big(\pi_d(\bar a)\otimes1_{m_d}\big)\,T=T\,\big(\overline{\pi_c(a)}\otimes1_{m_c}\big)\qquad(a\in\mathcal A).
Write \(T=\sum_{j,l}T_{jl}\otimes|j\rangle\langle l|\) with \(k_d\times k_c\) blocks \(T_{jl}\). The display says
\(\pi_d(\bar a)T_{jl}=T_{jl}\overline{\pi_c(a)}\): every \(T_{jl}\) is an intertwiner from \(\rho(a)=\overline{\pi_c(a)}\) to
\(\rho'(a)=\pi_d(\bar a)\). These are representations of \(\mathcal A\) with its adjoint in the sense of Section 4.1
(\(\rho(a^*)=\rho(a)^*\), \(\rho'(a^*)=\rho'(a)^*\)), and they are irreducible, as their images are all of \(M_{k_c}\) and of
\(M_{k_d}\). Since \(T\neq0\), some \(T_{jl}\neq0\), so \(k_c=k_d\) and \(\mathrm{Hom}(\rho,\rho')=\mathbb CJ_c\) for a unitary \(J_c\)
(Lemma 4.2). Hence \(T_{jl}=\mu_{jl}J_c\), i.e. \(T=J_c\otimes M_c\) with \(M_c=(\mu_{jl})\); \(1=T^*T=1\otimes M_c^*M_c\) and
\(m_c=k_cm_c/k_c=m_d\), so \(M_c\) is unitary, and the display becomes the formula for \(\pi_d(\bar a)\). \(T\) is determined by
\(V\); if also \(T=J'\otimes M'\) with \(J'\) unitary, the blocks \(T_{jl}=M'_{jl}J'\) show \(J'\in\mathrm{Hom}(\rho,\rho')\), so
\(J'=\lambda J_c\) with \(|\lambda|=1\), and then \(M'=\bar\lambda M_c\).
(b) Let \(\sigma(c)=c\) and \(J=J_c\), \(M=M_c\). Conjugating \(\overline{V_c}=V_c(J\otimes M)\) gives
\(V_c=\overline{V_c}(\bar J\otimes\bar M)=V_c(J\bar J\otimes M\bar M)\), so \(J\bar J\otimes M\bar M=1\). Applying the formula of
(a) twice, with \(\overline{J^*}=J^T\),
math
\pi_c(a)=\pi_c(\bar{\bar a})=J\,\overline{J\,\overline{\pi_c(a)}\,J^*}\,J^*=(J\bar J)\,\pi_c(a)\,(J\bar J)^*,
so the unitary \(J\bar J\) commutes with \(\pi_c(\mathcal A)=M_{k_c}\) and is a scalar \(\lambda1\) (Lemma 2.5, Step 5, with
\(m=1\)), \(|\lambda|=1\). Then \(\bar JJ=\overline{J\bar J}=\bar\lambda1\) and \(\lambda J=(J\bar J)J=J(\bar JJ)=\bar\lambda J\), so
\(\lambda=\bar\lambda\in\{1,-1\}\); call it \(\varepsilon_c\). It does not change under \(J\mapsto\lambda J\), and
\(\varepsilon_c1\otimes M\bar M=1\) gives \(M\bar M=\varepsilon_c1\). If \(\varepsilon_c=-1\), then
\((-1)^{k_c}=\det(J\bar J)=\det J\cdot\overline{\det J}=|\det J|^2>0\), so \(k_c\) is even, and likewise \(m_c\).
(c) The \(P_c\) do not depend on \(V\) (Theorem 2.6), so neither does \(\sigma\). Let \(V'\) be another decomposition, numbered so that \(V'_cV_c'^*=P_c\) (Theorem 2.6), with homomorphisms \(\pi'_c\) and data \(J'_c,M'_c\), and let \(\sigma(c)=c\). Then \(V'_c=P_cV'_c=V_cY\) with \(Y=V_c^*V'_c\), a square isometry, hence unitary, and \((\pi_c(a)\otimes1)Y=Y(\pi'_c(a)\otimes1)\) for \(a\in\mathcal A\). The argument of (a), with the irreducible representations \(\pi'_c,\pi_c\) in place of \(\rho,\rho'\), gives \(Y=u\otimes w\) with unitaries \(u,w\). Hence \(\overline{V'_c}=\overline{V_c}(\bar u\otimes\bar w)=V_c(J_c\bar u\otimes M_c\bar w)=V'_c(u^*J_c\bar u\otimes w^*M_c\bar w)\), so \(J'_c=\lambda u^*J_c\bar u\) with \(|\lambda|=1\) by the uniqueness in (a), and, as \(\bar uu^T=\overline{uu^*}=1\), \(J'_c\bar J'_c=u^*J_c\bar uu^T\bar J_cu=u^*J_c\bar J_cu=\varepsilon_c1\).
(d) By (a), \(\overline{V_c(1\otimes y)V_c^*}=V_{\sigma(c)}(J_c\otimes M_c)(1\otimes\bar y)(J_c\otimes M_c)^*V_{\sigma(c)}^*\), and \((J_c\otimes M_c)(1\otimes\bar y)(J_c\otimes M_c)^*=J_cJ_c^*\otimes M_c\bar yM_c^*=1\otimes M_c\bar yM_c^*\). Sum over \(c\) for \(X=\sum_cV_c(1\otimes x_c)V_c^*\). \(\square\)
A sector \(c\) is of real type if \(\sigma(c)=c\) and \(\varepsilon_c=1\), of quaternionic type if \(\sigma(c)=c\) and \(\varepsilon_c=-1\), and of complex type if \(\sigma(c)\neq c\); by Theorem 4.20(c) this is a property of \(\mathcal A\) and \(P_c\). A sector with \(\sigma(c)=c\) and odd \(k_c\) or odd \(m_c\) is of real type, by Theorem 4.20(b).
In the library: \(\sigma\), \(J_c\), \(M_c\), \(\varepsilon_c\) and the types of the sectors are computed by Decomposition.real_structure() — tutorial, Section 4.7.
Theorem 4.21 (the real algebra). Let \(\Gamma\) contain every sector \(c\) with \(\sigma(c)=c\) and one sector of every
pair \(\{c,\sigma(c)\}\) with \(\sigma(c)\neq c\). For \(\sigma(c)=c\) let \(R_c=\{x\in M_{k_c}(\mathbb C):xJ_c=J_c\bar x\}\). Then
\(a\mapsto(\pi_c(a))_{c\in\Gamma}\) is an isomorphism of real algebras, which maps the transpose to the adjoint,
math
\mathcal A_{\mathbb R}\;\cong\bigoplus_{c\in\Gamma,\ \sigma(c)=c}R_c\;\oplus\bigoplus_{c\in\Gamma,\ \sigma(c)\neq c}M_{k_c}(\mathbb C),
the last summands regarded as real algebras (complex type). Moreover:
(a) (real type) If \(\varepsilon_c=1\), write \(J_c=W_JW_J^T\) and \(M_c=W_MW_M^T\) with \(J_c\bar W_J=W_J\), \(M_c\bar W_M=W_M\)
(Lemma 4.19(c)). Then \(x\mapsto W_J^*xW_J\) maps \(R_c\) isomorphically onto \(M_{k_c}(\mathbb R)\). The columns of
\(O_c=V_c(W_J\otimes W_M)\) are real and orthonormal, they span the range of \(P_c\) over \(\mathbb C\), and
\(aO_c=O_c(\alpha_c(a)\otimes1_{m_c})\) with the real matrix \(\alpha_c(a)=W_J^*\pi_c(a)W_J\) for \(a\in\mathcal A_{\mathbb R}\).
(b) (quaternionic type) If \(\varepsilon_c=-1\), write \(J_c=W_J\Omega_{k_c}W_J^T\) with \(J_c\bar W_J=W_J\Omega_{k_c}\)
(Lemma 4.19(c)). Then \(x\mapsto\mathbb M^{-1}(W_J^*xW_J)\) is an isomorphism \(R_c\cong M_{k_c/2}(\mathbb H)\) of real algebras
mapping the adjoint to the quaternionic adjoint \((q^*)_{ij}=\bar q_{ji}\); here \(\mathbb M:M_r(\mathbb H)\to M_{2r}(\mathbb C)\)
replaces every entry \(q\) by the \(2\times2\) block \(M(q)\) of Lemma E.15(2), and its image is
\(\{y\in M_{2r}(\mathbb C):y\Omega_{2r}=\Omega_{2r}\bar y\}\).
In particular \(\dim_{\mathbb R}\mathcal A_{\mathbb R}=\sum_ck_c^2=\dim_{\mathbb C}\mathcal A\).
Proof. For real \(a\), \(\bar a=a\), and Theorem 4.20(a) reads \(\pi_{\sigma(c)}(a)=J_c\overline{\pi_c(a)}J_c^*\). For \(\sigma(c)=c\) this says \(\pi_c(a)\in R_c\). So \(\Phi(a)=(\pi_c(a))_{c\in\Gamma}\) maps \(\mathcal A_{\mathbb R}\) into the right side; it is a homomorphism of real algebras, and \(\Phi(a^T)=\Phi(a^*)=\Phi(a)^*\). If \(\Phi(a)=0\), the same formula gives \(\pi_{\sigma(c)}(a)=0\) for \(c\in\Gamma\), so all \(\pi_e(a)=0\) and \(a=0\). Onto: let \(c\in\Gamma\), \(x\in R_c\) if \(\sigma(c)=c\) and \(x\in M_{k_c}\) otherwise, and \(a=V_c(x\otimes1)V_c^*\in\mathcal A\). By Theorem 4.20(a) for the sector \(\sigma(e)\), \(\pi_e(\bar a)=J_{\sigma(e)}\overline{\pi_{\sigma(e)}(a)}J_{\sigma(e)}^*\), which is \(0\) unless \(\sigma(e)=c\). If \(\sigma(c)=c\), \(\pi_c(\bar a)=J_c\bar xJ_c^*=x\) (definition of \(R_c\)), so the real matrix \(\mathrm{Re}\,a=\frac12(a+\bar a)\in\mathcal A_{\mathbb R}\) has \(\Phi(\mathrm{Re}\,a)=x\) in the slot \(c\) and \(0\) elsewhere. If \(\sigma(c)\neq c\), \(\pi_c(\bar a)=0\), and \(2\mathrm{Re}\,a\) has \(\pi_c=x\) and \(\pi_e=0\) for \(e\notin\{c,\sigma(c)\}\); as \(\sigma(c)\notin\Gamma\), \(\Phi(2\mathrm{Re}\,a)\) is \(x\) in the slot \(c\) and \(0\) elsewhere. Sums of these elements give the whole right side.
(a) Multiply \(xW_JW_J^T=W_JW_J^T\bar x\) by \(W_J^*\) on the left and by \(\bar W_J\) on the right; as \(W_J^T\bar W_J=\overline{W_J^*W_J}=1\), it becomes \(W_J^*xW_J=W_J^T\bar x\bar W_J=\overline{W_J^*xW_J}\), and the steps can be reversed. So \(R_c=W_JM_{k_c}(\mathbb R)W_J^*\), and \(x\mapsto W_J^*xW_J\) is an isomorphism of real algebras compatible with the adjoint. Next \(\overline{O_c}=\overline{V_c}(\bar W_J\otimes\bar W_M)=V_c(J_c\bar W_J\otimes M_c\bar W_M)=V_c(W_J\otimes W_M)=O_c\), \(O_c^*O_c=1\), and \(aO_c=V_c(\pi_c(a)W_J\otimes W_M)=O_c(W_J^*\pi_c(a)W_J\otimes1)\), with \(W_J^*\pi_c(a)W_J\) real for \(a\in\mathcal A_{\mathbb R}\).
(b) The same computation with \(J_c=W_J\Omega W_J^T\) (\(\Omega=\Omega_{k_c}\)) shows \(xJ_c=J_c\bar x\) iff \(y=W_J^*xW_J\) satisfies \(y\Omega=\Omega\bar y\). Write \(y\) in \(2\times2\) blocks \(y_{ij}\); as \(\Omega\) is block diagonal, \(y\Omega=\Omega\bar y\) iff \(y_{ij}\Omega_2=\Omega_2\bar y_{ij}\) for all \(i,j\), and for \(y_{ij}=\begin{pmatrix}\alpha&\beta\\\gamma&\delta\end{pmatrix}\) this reads \(\begin{pmatrix}\beta&-\alpha\\\delta&-\gamma\end{pmatrix}=\begin{pmatrix}-\bar\gamma&-\bar\delta\\\bar\alpha&\bar\beta\end{pmatrix}\), i.e. \(\delta=\bar\alpha\) and \(\beta=-\bar\gamma\): \(y_{ij}=\begin{pmatrix}\alpha&-\bar\gamma\\\gamma&\bar\alpha\end{pmatrix}=M(\alpha,\bar\gamma)\) with \(M(a,b)=\begin{pmatrix}a&-b\\\bar b&\bar a\end{pmatrix}\). So the image of \(\mathbb M\) is as stated. By Lemma E.15(2), \(M\) is an injective real-linear homomorphism \(\mathbb H\to M_2(\mathbb C)\) with \(M(\bar q)=M(q)^*\); since a matrix product uses only sums and products of entries, \(\mathbb M\) is an injective homomorphism of real algebras with \(\mathbb M(q^*)=\mathbb M(q)^*\) (as in the proof of Theorem E.24(1)). Hence \(x\mapsto\mathbb M^{-1}(W_J^*xW_J)\) is the claimed isomorphism. Dimensions: \(\dim_{\mathbb R}R_c=k_c^2\) in both cases (\(\dim_{\mathbb R}M_r(\mathbb H)=4r^2\) with \(2r=k_c\)), and a pair contributes \(\dim_{\mathbb R}M_{k_c}(\mathbb C)=2k_c^2=k_c^2+k_{\sigma(c)}^2\). \(\square\)
The same holds for the commutant, with the factors exchanged: \((\mathcal A')_{\mathbb R}\) is the direct sum of \(M_{m_c}(\mathbb R)\), \(M_{m_c/2}(\mathbb H)\) and \(M_{m_c}(\mathbb C)\) over the same sectors, with the same type. Indeed, \(\mathcal A'\) is a \(*\)-algebra with \(\overline{\mathcal A'}=\mathcal A'\) (Lemma 4.18(b)), and the proof above used only three facts, which hold for \(\mathcal A'\) with \(X\mapsto x_c\) in place of \(\pi_c\) and \(M_c\) in place of \(J_c\): \(X\mapsto x_c\) is a unital \(*\)-homomorphism onto \(M_{m_c}\) (\(V_c(1\otimes y)V_c^*\in\mathcal A'\) has the block \(y\) in sector \(c\) and \(0\) elsewhere), \((\bar X)_{\sigma(c)}=M_c\bar x_cM_c^*\) (Theorem 4.20(d)), and \(M_c\bar M_c=\varepsilon_c1\) for \(\sigma(c)=c\) (Theorem 4.20(b)).
In the library: a real orthogonal basis adapted to Theorem 4.21, Decomposition.real_basis() — tutorial, Section 4.7.
Three examples. (i) The generators \((L_a)_{bc}=-\epsilon_{abc}\) of the rotations of \(\mathbb R^3\), acting on \(\mathbb R^3\otimes\mathbb R^2\) as \(L_a\otimes1_2\): there is one sector, \(k=3\), \(m=2\) (\(iL_x,iL_y,iL_z\) is a spin representation on \(\mathbb C^3\) in which \(iL_z\) has the simple eigenvalues \(1,0,-1\), hence spin \(1\) by Corollary 4.12); it is of real type because \(k\) is odd, and \(\mathcal A_{\mathbb R}\cong M_3(\mathbb R)\). (ii) The rotation generator \(\Omega_2\) of \(\mathbb R^2\), acting on \(\mathbb R^2\otimes\mathbb R^2\) as \(\Omega_2\otimes1_2\): since \(\Omega_2^2=-1\) and \(\Omega_2^*=-\Omega_2\), the algebra is \(\mathrm{span}\{1,\Omega_2\otimes1\}\); the eigenvectors \((1,\mp i)/\sqrt2\) of \(\Omega_2\) for \(\pm i\), which are conjugate to each other, give two sectors with \(k=1\), \(m=2\), exchanged by conjugation: a pair of complex type, and \(\mathcal A_{\mathbb R}=\{\alpha+\beta\,\Omega_2\otimes1\}\cong\mathbb C\). (iii) Left multiplication \(L_q\) by a quaternion \(q=(a,b)\in\mathbb H\) (Section E.6) on \(\mathbb H=\mathbb R^4\), with coordinates \((\mathrm{Re}\,a,\mathrm{Im}\,a,\mathrm{Re}\,b,\mathrm{Im}\,b)\). \(q\mapsto L_q\) is an injective homomorphism of real algebras (\(\mathbb H\) is associative, Lemma E.15(2)), and \(L_{\bar q}=L_q^T\), because \(\langle qx,y\rangle=\langle x,\bar qy\rangle\) (Lemma E.14(2)) for the norm \(N(a,b)=|a|^2+|b|^2\), which is the Euclidean norm of \(\mathbb R^4\). So the real algebra generated by \(L_I,L_J,L_K\) (\(I=(i,0)\), \(J=(0,1)\), \(K=(0,i)\)) and their transposes is \(\{L_q:q\in\mathbb H\}\cong\mathbb H\); the complex algebra \(\mathcal A\) is the complex span of the products of the generators, \(\{L_q\}+i\{L_q\}\), so \(\mathcal A_{\mathbb R}=\{L_q\}\cong\mathbb H\) (Lemma 4.18(a)). This already fixes the decomposition: \(\sum_ck_c^2=\dim_{\mathbb C}\mathcal A=\dim_{\mathbb R}\mathcal A_{\mathbb R}=4\) (Theorem 4.21), and not all \(k_c=1\), since then \(\mathcal A\), hence \(\mathcal A_{\mathbb R}\cong\mathbb H\), would be commutative; so there is one sector, with \(k=2\) and \(\sigma(c)=c\), and it is not of real type, as Theorem 4.21(a) would then give \(\mathcal A_{\mathbb R}\cong M_2(\mathbb R)\), which has zero divisors, while \(\mathbb H\) has none (Lemma E.15(3)): it is of quaternionic type, with \(m=4/2=2\) on \(\mathbb R^4\). (The unit quaternions act here as \(SU(2)\): \(M(q)\) is unitary with determinant \(|a|^2+|b|^2=1\), and every \(U=\begin{pmatrix}\alpha&\beta\\\gamma&\delta\end{pmatrix}\in SU(2)\) is of this form, as \(U^*=U^{-1}=\begin{pmatrix}\delta&-\beta\\-\gamma&\alpha\end{pmatrix}\) gives \(\delta=\bar\alpha\), \(\beta=-\bar\gamma\), so \(U=M(\alpha,\bar\gamma)\).) On \(\mathbb H^2=\mathbb R^8\), with \(q\) acting as \(1_2\otimes L_q\), the same argument gives one sector with \(k=2\), \(m=4\), of quaternionic type, \(\mathcal A_{\mathbb R}\cong M_1(\mathbb H)=\mathbb H\).
Computation: the three examples, with Decomposition.real_structure() and Decomposition.real_basis() — tutorial, Section 4.7.
In the representation theory of a group by real matrices, the following is the real form of Schur's lemma; it shows that the three types can be read off the real representation itself.
Theorem 4.22 (real Schur lemma). Let \(U\subseteq\mathbb R^n\) be a nonzero subspace with \(aU\subseteq U\) for all \(a\in\mathcal A_{\mathbb R}\) that is irreducible: \(0\) and \(U\) are its only subspaces with this property. Let \(D_U\) be the set of real-linear maps \(T:U\to U\) with \(Tau=aTu\) for all \(a\in\mathcal A_{\mathbb R}\), \(u\in U\). (a) \(D_U\) is an associative real algebra with unit and without zero divisors, hence isomorphic to \(\mathbb R\), \(\mathbb C\) or \(\mathbb H\) (Theorem E.19). (b) For exactly one orbit \(\mathcal O=\{c,\sigma(c)\}\) of \(\sigma\), \(U\) lies in the range of \(P_{\mathcal O}=\sum_{e\in\mathcal O}P_e\), and \(\dim_{\mathbb R}D_U=1,2,4\) and \(\dim_{\mathbb R}U=k_c,2k_c,2k_c\) according as \(c\) is of real, complex or quaternionic type. So \(D_U\cong\mathbb R,\mathbb C,\mathbb H\), respectively. (c) For every orbit \(\mathcal O\) there is such a \(U\) in the range of \(P_{\mathcal O}\). \(\mathbb R^n\) itself is irreducible iff \(\sigma\) has a single orbit \(\{c,\sigma(c)\}\) and \(m_c=1\) (real and complex type), resp. \(m_c=2\) (quaternionic type).
Proof. (a) For \(T\in D_U\), \(\ker T\) and \(TU\) are invariant subspaces, so \(T=0\) or \(T\) is bijective, and then \(T^{-1}\in D_U\). \(D_U\) is closed under composition, which is associative, contains \(\mathrm{id}_U\neq0\), and \(ST=0\) with \(S\neq0\) forces \(T=S^{-1}ST=0\).
(b) Invariant subspaces and real projections. If \(U'\subseteq\mathbb R^n\) is invariant, so is its orthogonal complement in \(\mathbb R^n\) (for \(y\perp U'\), \(u\in U'\) and \(a\in\mathcal A_{\mathbb R}\), \(u\cdot ay=a^Tu\cdot y=0\), as \(a^T\in\mathcal A_{\mathbb R}\)), so the orthogonal projection \(e\) onto \(U'\), a real symmetric idempotent, commutes with \(\mathcal A_{\mathbb R}\); by Lemma 4.18(b), \(e\in(\mathcal A')_{\mathbb R}\). Conversely \(e\mathbb R^n\) is invariant for every projection \(e\in(\mathcal A')_{\mathbb R}\). So \(U'\mapsto e\) is an inclusion-preserving bijection from the invariant subspaces onto the set \(\mathcal P\) of projections in \((\mathcal A')_{\mathbb R}\), and \(U\) is irreducible iff its projection \(e\) is a minimal nonzero element of \(\mathcal P\). By Lemma 4.18(a) applied to the range \(e\mathbb C^n\) (whose real vectors are \(e\mathbb R^n\)), \(\dim_{\mathbb R}U=\mathrm{rank}\,e\).
\(D_U\cong e(\mathcal A')_{\mathbb R}e\). For \(T\in D_U\) let \(\hat T\) be the real \(n\times n\) matrix \(x\mapsto T(ex)\). For \(a\in\mathcal A_{\mathbb R}\), \(\hat Tax=T(eax)=T(aex)=aT(ex)=a\hat Tx\), so \(\hat T\in(\mathcal A')_{\mathbb R}\) (Lemma 4.18(b)) and \(\hat T=e\hat Te\). Conversely, for \(X\in(\mathcal A')_{\mathbb R}\), \(eXe\) restricted to \(U\) lies in \(D_U\). These maps are inverse to each other and multiplicative. Moreover \(e\mathcal A'e\) is closed under conjugation, and its real elements are the \(e(\mathcal A')_{\mathbb R}e\) (if \(eXe\) is real, \(eXe=e\mathrm{Re}(X)e\) with \(\mathrm{Re}\,X\in(\mathcal A')_{\mathbb R}\)), so \(\dim_{\mathbb R}D_U=\dim_{\mathbb C}e\mathcal A'e\) by Lemma 4.18(a).
Projections in the decomposition. A projection \(e\in\mathcal A'\) is \(e=\sum_cV_c(1\otimes f_c)V_c^*\) with projections \(f_c\in M_{m_c}\) (Theorem 2.6). Then \(\mathrm{rank}\,e=\sum_ck_c\mathrm{rank}\,f_c\), \(e\mathcal A'e=\sum_cV_c(1\otimes f_cM_{m_c}f_c)V_c^*\) has complex dimension \(\sum_c(\mathrm{rank}\,f_c)^2\), and by Theorem 4.20(d), \(\bar e=e\) iff \(f_{\sigma(c)}=M_c\bar f_cM_c^*\) for all \(c\).
Minimal elements. Let \(e\) be minimal in \(\mathcal P\). For each orbit \(\mathcal O\) of \(\sigma\), \(P_{\mathcal O}\) is a real central projection (\(\bar P_{\mathcal O}=P_{\mathcal O}\) by Theorem 4.20(a)) in \(Z(\mathcal A)\subseteq\mathcal A'\), which commutes with \(e\), so \(eP_{\mathcal O}\in\mathcal P\) and \(eP_{\mathcal O}\le e\); as \(e=\sum_{\mathcal O}eP_{\mathcal O}\), exactly one orbit \(\mathcal O\) has \(eP_{\mathcal O}=e\), i.e. \(U=e\mathbb R^n\) lies in the range of \(P_{\mathcal O}\), and \(f_c=0\) for \(c\notin\mathcal O\). If \(\mathcal O=\{c,d\}\) with \(d\neq c\), then \(f_d=M_c\bar f_cM_c^*\) has the rank of \(f_c\); for a rank-one projection \(g\le f_c\), \(g'=Y+\bar Y\) with \(Y=V_c(1\otimes g)V_c^*\) is a sum of two orthogonal projections of \(\mathcal A'\) (in the sectors \(c\) and \(d\)), real, and \(g'\le e\) (as \(Y\le e\) and \(\bar Y\le\bar e=e\)); so \(g'=e\), \(\mathrm{rank}\,f_c=\mathrm{rank}\,f_d=1\), \(\dim_{\mathbb C}e\mathcal A'e=2\) and \(\mathrm{rank}\,e=2k_c\). If \(\mathcal O=\{c\}\), let \(f=f_c\), \(M=M_c\) and \(\theta v=M\bar v\), antiunitary on \(\mathbb C^{m_c}\) with \(\theta^2=M\bar M=\varepsilon_c\) (Theorem 4.20(b)). From \(fM=M\bar f\), \(\theta(fv)=M\bar f\bar v=fM\bar v=f\theta v\), so the range \(F\) of \(f\) is \(\theta\)-invariant. If \(F_1\subseteq F\) is \(\theta\)-invariant with projection \(g\), then \(M\bar gM^*\) is the projection onto \(M\bar F_1=\theta F_1=F_1\) (\(\theta\) is injective), so \(g=M\bar gM^*\) and \(V_c(1\otimes g)V_c^*\in\mathcal P\) lies below \(e\). By minimality, \(F\) has no \(\theta\)-invariant subspaces but \(0\) and \(F\). If \(\varepsilon_c=1\), a fixed unit vector of \(\theta\) in \(F\) (Lemma 4.19(a)) spans one, so \(\mathrm{rank}\,f=1\), \(\dim_{\mathbb C}e\mathcal A'e=1\) and \(\mathrm{rank}\,e=k_c\). If \(\varepsilon_c=-1\), the plane \(\mathrm{span}\{u,\theta u\}\) of a unit vector \(u\in F\) is one (Lemma 4.19(b)), so \(\mathrm{rank}\,f=2\), \(\dim_{\mathbb C}e\mathcal A'e=4\) and \(\mathrm{rank}\,e=2k_c\). Since \(\mathbb R\), \(\mathbb C\), \(\mathbb H\) have dimensions \(1,2,4\), (a) gives \(D_U\cong\mathbb R,\mathbb C,\mathbb H\), respectively.
(c) \(P_{\mathcal O}\in\mathcal P\) is nonzero, and an element of \(\mathcal P\setminus\{0\}\) below it of minimal rank is minimal. \(\mathbb R^n\) is irreducible iff \(e=1\) is minimal; its blocks are \(f_c=1_{m_c}\), so by the proof of (b) this requires a single orbit and \(m_c=1,1,2\) in the three cases. Conversely, under these conditions a nonzero \(e'\in\mathcal P\) has \(f'_c\neq0\) for the sectors of the orbit (\(f'_{\sigma(c)}=M_c\bar f'_cM_c^*\)), so \(f'_c=1\) if \(m_c=1\); in the quaternionic case the range of \(f'_c\) is \(\theta\)-invariant, hence of even dimension (Lemma 4.19(b)), so \(f'_c=1_2\). Thus \(e'=1\). \(\square\)
For a group \(G\) acting by real orthogonal matrices, \(\mathcal A=\mathrm{span}\,\pi(G)\) is closed under conjugation (Lemma 4.18(c)), \(\mathcal A_{\mathbb R}\) is the real span of \(\pi(G)\) (Lemma 4.18(a): \(\mathcal A_{\mathbb R}\) consists of the real parts of the elements of \(\mathcal A\)), and \(D_U\) is the commutant of an irreducible real representation of \(G\): it is \(\mathbb R\), \(\mathbb C\) or \(\mathbb H\). By the proof of (b), the complexification \(U+iU=e\mathbb C^n=V_c(\mathbb C^{k_c}\otimes F)\) (plus its conjugate for a pair) carries \(\pi_c\), \(\pi_c\oplus\pi_{\sigma(c)}\) or \(\pi_c\oplus\pi_c\) in the three cases: it is irreducible, the sum of two inequivalent irreducible representations, or the sum of two equivalent ones.
Computation: Theorem 4.22 on the irreducible real modules \(\mathbb R^3\), \(\mathbb R^2\), \(\mathbb R^4\) of the three examples — tutorial, Section 4.7.
Proposition 4.23 (real elements of the commutant). Let \(X\in\mathcal A'\) be real, with blocks \(x_c\). Then \(x_{\sigma(c)}=M_c\bar x_cM_c^*\) for every \(c\). If moreover \(X=X^T\): (a) for \(c\) of real type, \(y_c=W_M^*x_cW_M\) (\(W_M\) as in Theorem 4.21(a)) is a real symmetric matrix, and \(XO_c=O_c(1_{k_c}\otimes y_c)\) with the real orthonormal columns \(O_c\) of Theorem 4.21(a); (b) for \(c\) of quaternionic type, every eigenvalue of \(x_c\) has even multiplicity, so every eigenvalue of \(X\) on the range of \(P_c\) has a multiplicity divisible by \(2k_c\) (Kramers degeneracy); (c) for \(c\) of complex type, \(x_c\) and \(x_{\sigma(c)}\) have the same eigenvalues with the same multiplicities. Without \(X=X^T\), \(y_c\) in (a) is still real.
Proof. \(X=\bar X\), so Theorem 4.20(d) gives \(x_{\sigma(c)}=M_c\bar x_cM_c^*\). If \(X=X^T\), then \(X^*=X\) and every \(x_c\) is Hermitian. (a) With \(M_c=W_MW_M^T\), \(M_c^*=\bar W_MW_M^*\) and \(W_M^T\bar W_M=1\): \(\bar y_c=W_M^T\bar x_c\bar W_M=W_M^TM_c^*x_cM_c\bar W_M=W_M^*x_cW_M=y_c\), and \(y_c\) is Hermitian, hence real symmetric. Further \(XO_c=XV_c(W_J\otimes W_M)=V_c(W_J\otimes x_cW_M)=O_c(1\otimes y_c)\). (b) Let \(\theta v=M_c\bar v\), antiunitary with \(\theta^2=-1\) (Theorem 4.20(b)). From \(x_cM_c=M_c\bar x_c\), \(x_c\theta v=M_c\bar x_c\bar v=\theta(x_cv)\), so for an eigenvalue \(\lambda\in\mathbb R\) of \(x_c\) the eigenspace \(E_\lambda\) satisfies \(\theta E_\lambda\subseteq E_\lambda\) (\(x_c\theta v=\theta(\lambda v)=\lambda\theta v\)), and \(\dim E_\lambda\) is even by Lemma 4.19(b). On the range of \(P_c\), \(X\) acts as \(V_c(1_{k_c}\otimes x_c)V_c^*\), whose eigenvalues are those of \(x_c\), each repeated \(k_c\) times. (c) \(\bar x_c\) has the eigenvalues \(\bar\lambda=\lambda\) of \(x_c\), with the conjugate eigenvectors, and \(x_{\sigma(c)}\) is unitarily equivalent to \(\bar x_c\). \(\square\)
In the quaternionic case \(y_c=W_M^*x_cW_M\), with \(M_c=W_M\Omega_{m_c}W_M^T\) as in Lemma 4.19(c), satisfies \(y_c\Omega=\Omega\bar y_c\) (the computation of Theorem 4.21(b) with \(M_c\) in place of \(J_c\)), so \(y_c=\mathbb M(h)\) for a quaternionic matrix \(h\); if \(X=X^T\), then \(h^*=h\), since \(\mathbb M(h^*)=y_c^*=y_c\) and \(\mathbb M\) is injective.
Computation: Proposition 4.23 for a random real symmetric element of the commutant in each of the three examples — tutorial, Section 4.7.
Physical meaning (physical interpretation, not a theorem of this course). If a Hamiltonian commutes with a symmetry and is real in a basis in which the symmetry is real — the effect of a time-reversal symmetry, which acts there as complex conjugation — Proposition 4.23 says what its blocks look like: real symmetric in sectors of real type, complex Hermitian with a partner of the same spectrum in pairs of complex type, and quaternion Hermitian with doubly degenerate levels in sectors of quaternionic type, as for the time reversal of an odd number of spins \(\frac12\), whose square is \(-1\) (Kramers' theorem). Dyson's threefold way (F. J. Dyson, J. Math. Phys. 3 (1962) 1199) models the blocks of a generic Hamiltonian of each kind by random matrices of the corresponding class — the GOE and the GUE of Section 5.3, and the Gaussian symplectic ensemble (GSE), which this course does not define — and predicts the level statistics of each block accordingly.
4.8 What to compare with¶
In the library: comparison with other software — tutorial, Section 4.8.
4.9 Stability of the numerical decomposition¶
Section 4.3 analysed the algorithm in exact arithmetic. In practice the generators are known only approximately — measured, or rounded — and a symmetry is decided by a tolerance: the commutant is spanned by the singular vectors of the commutator map whose singular values lie below a threshold \(\theta\). This section says which thresholds are correct, how far the computed objects are from the exact ones, and what can be certified from the output alone.
Throughout, \(H_1,\dots,H_J\) are Hermitian and \(\mathcal A=\{H_j\}''\) (the library first replaces the generators by
their Hermitian parts, which generate the same algebra, Section 4.3). The commutator map is
math
L:M_n(\mathbb C)\to M_n(\mathbb C)^J,\qquad LX=([H_1,X],\dots,[H_J,X]),
with the norm \(\|X\|_2\) on \(M_n\) and \(\|(X_j)_j\|_2=(\sum_j\|X_j\|_2^2)^{1/2}\) on \(M_n^J\); \(\ker L=\mathcal A'\) has
dimension \(d=\sum_cm_c^2\). For a linear map \(T\) between finite-dimensional complex inner product spaces whose domain
has dimension \(p\), \(\sigma_1(T)\le\dots\le\sigma_p(T)\) denote the square roots of the eigenvalues of \(T^*T\ge0\), in
increasing order (for a square matrix these are its singular values, Section 1.4, in reverse order). As
\(|Tx|^2=\langle x,T^*Tx\rangle\), \(\ker T=\ker T^*T\), and \(\sigma_p(T)=\|T\|\). The gap of the commutator map is
\(\gamma=\sigma_{d+1}(L)\), its smallest nonzero singular value (\(\gamma=\infty\) if \(d=n^2\)). We use
\(\|AX\|_2\le\|A\|\|X\|_2\) (apply \(|Ax|\le\|A\||x|\) to the columns of \(X\)) and \(\|XA\|_2=\|A^*X^*\|_2\le\|X\|_2\|A\|\),
hence \(\|[E,X]\|_2\le2\|E\|\|X\|_2\).
Lemma 4.24 (perturbation). (a) Let \(T,\tilde T\) be linear maps between the same finite-dimensional complex inner product spaces and \(\Delta=\|\tilde T-T\|\). Then \(|\sigma_i(\tilde T)-\sigma_i(T)|\le\Delta\) for all \(i\). If moreover \(\sigma_d(T)=0<\gamma=\sigma_{d+1}(T)\) and \(\Delta<\gamma/2\), let \(\Pi\) be the orthogonal projection onto \(\ker T\) and \(\tilde\Pi\) the spectral projection of \(\tilde T^*\tilde T\) for its eigenvalues \(\le\Delta^2\); then \(\tilde\Pi\) has rank \(d\) and \(\|\tilde\Pi-\Pi\|\le2\Delta/\gamma\).
(b) Let \(A,\tilde A\in\mathrm{Herm}_n\) with \(\|\tilde A-A\|\le\Delta\), and let \(\mu\) be an eigenvalue of \(A\) of multiplicity \(r\), with spectral projection \(P\), at distance \(g>2\Delta\) from the other eigenvalues of \(A\). Then \(\tilde A\) has exactly \(r\) eigenvalues in \([\mu-\Delta,\mu+\Delta]\), all others at distance \(\ge g-\Delta\) from \(\mu\), and the spectral projection \(\tilde P\) of \(\tilde A\) for these \(r\) eigenvalues satisfies \(\|\tilde P-P\|\le2\Delta/g\).
(c) Let \(P,\tilde P\) be orthogonal projections of rank \(k\) with \(\|\tilde P-P\|\le\varepsilon\le\frac12\), and let the columns of \(\tilde Q\in M_{n\times k}\) be an orthonormal basis of \(\mathrm{ran}\,\tilde P\). Then \(S=\tilde Q^*P\tilde Q\) is positive definite, the columns of \(Q=P\tilde QS^{-1/2}\) are an orthonormal basis of \(\mathrm{ran}\,P\), and \(\|Q-\tilde Q\|\le2\varepsilon\).
(d) For invertible \(M\in M_k\), \(\mathrm{pol}(M)=M(M^*M)^{-1/2}\) is unitary (the unitary of Theorem 1.8). If \(\|M-\lambda U\|\le\zeta<\lambda\) with \(\lambda>0\) and \(U\) unitary, then \(M\) is invertible and \(\|\mathrm{pol}(M)-U\|\le2\zeta/\lambda\).
Proof. (a) By Theorem A.8 for \(T^*T\), and as \(t\mapsto t^{1/2}\) is increasing, \(\sigma_i(T)=\min_{\dim U=i}\max_{x\in U,|x|=1}|Tx|\). For unit \(x\), \(|\tilde Tx|\le|Tx|+\Delta\); taking the maximum over \(U\) and then the minimum gives \(\sigma_i(\tilde T)\le\sigma_i(T)+\Delta\), and exchanging \(T,\tilde T\) gives the reverse. Hence \(\sigma_i(\tilde T)\le\Delta\) for \(i\le d\) and \(\sigma_i(\tilde T)\ge\gamma-\Delta>\Delta\) for \(i>d\), so \(\tilde\Pi\) has rank \(d\). For unit \(x\in\mathrm{ran}\,\tilde\Pi\), expanding in eigenvectors of \(\tilde T^*\tilde T\) gives \(|\tilde Tx|^2=\langle x,\tilde T^*\tilde Tx\rangle\le\Delta^2\), so \(|Tx|\le2\Delta\); and \(Tx=T(1-\Pi)x\) with \((1-\Pi)x\perp\ker T^*T\), where the eigenvalues of \(T^*T\) are \(\ge\gamma^2\), so \(|Tx|\ge\gamma|(1-\Pi)x|\). Hence \(\|(1-\Pi)\tilde\Pi\|\le2\Delta/\gamma\). For unit \(x\in\ker T\), \(|\tilde Tx|\le\Delta\) and \(|\tilde Tx|^2=\langle\tilde\Pi x,\tilde T^*\tilde T\tilde\Pi x\rangle+\langle(1-\tilde\Pi)x,\tilde T^*\tilde T(1-\tilde\Pi)x\rangle\ge(\gamma-\Delta)^2|(1-\tilde\Pi)x|^2\) (\(\tilde\Pi\) commutes with \(\tilde T^*\tilde T\), whose eigenvalues on \(\mathrm{ran}(1-\tilde\Pi)\) are \(\ge(\gamma-\Delta)^2\)), so \(\|(1-\tilde\Pi)\Pi\|\le\Delta/(\gamma-\Delta)\le2\Delta/\gamma\). Finally \((\Pi-\tilde\Pi)x=\Pi(1-\tilde\Pi)(1-\tilde\Pi)x-(1-\Pi)\tilde\Pi\,\tilde\Pi x\) is a sum of two orthogonal vectors, and \(\|\Pi(1-\tilde\Pi)\|=\|(1-\tilde\Pi)\Pi\|\) (adjoints, Lemma 1.1), so \(|(\Pi-\tilde\Pi)x|^2\le(2\Delta/\gamma)^2(|(1-\tilde\Pi)x|^2+|\tilde\Pi x|^2)=(2\Delta/\gamma)^2|x|^2\).
(b) Number the eigenvalues increasingly. By Theorem A.10, the \(r\) eigenvalues \(\lambda_i(\tilde A)\) with \(\lambda_i(A)=\mu\) lie in \([\mu-\Delta,\mu+\Delta]\), and the others are at distance \(\ge g-\Delta>\Delta\) from \(\mu\). So the \(r\) smallest eigenvalues of \((\tilde A-\mu)^2\), which are \(\le\Delta^2\), are those of these \(r\) eigenvalues, and (a) with \(T=A-\mu\), \(\tilde T=\tilde A-\mu\) (so \(\ker T=\mathrm{ran}\,P\) and \(\sigma_{r+1}(T)=g\)) gives the bound.
(c) \((1-P)\tilde Q=(1-P)\tilde P\tilde Q=(1-P)(\tilde P-P)\tilde Q\) has norm \(\le\varepsilon\), so \(S=1-\tilde Q^*(1-P)\tilde Q\) has eigenvalues in \([1-\varepsilon^2,1]\) (Theorem A.10) and \(S^{-1/2}\) exists (Theorem 1.5). \(Q^*Q=S^{-1/2}\tilde Q^*P\tilde QS^{-1/2}=1\), and \(\mathrm{ran}\,Q\subseteq\mathrm{ran}\,P\) has dimension \(k\). From \(Q-\tilde Q=\tilde Q(S^{-1/2}-1)-(1-P)\tilde QS^{-1/2}\), \(\|Q-\tilde Q\|\le(1-\varepsilon^2)^{-1/2}-1+\varepsilon(1-\varepsilon^2)^{-1/2}=f(\varepsilon)\) with \(f(\varepsilon)=((1+\varepsilon)/(1-\varepsilon))^{1/2}-1\). Its derivative \((1-\varepsilon)^{-1}(1-\varepsilon^2)^{-1/2}\) is increasing, so \(f\) is convex with \(f(0)=0\), and \(f(\varepsilon)\le2\varepsilon f(\frac12)=2\varepsilon(\sqrt3-1)<2\varepsilon\).
(d) \(M^*M>0\), and \(\mathrm{pol}(M)^*\mathrm{pol}(M)=(M^*M)^{-1/2}M^*M(M^*M)^{-1/2}=1\); \(\mathrm{pol}(M)|M|=M\), so it is the unitary of Theorem 1.8. Write \(M=\lambda U(1+G)\) with \(\|G\|=\|M-\lambda U\|/\lambda\le\zeta/\lambda<1\). By (a) (with \(T=1\), \(\tilde T=1+G\)), the eigenvalues of \(R=((1+G)^*(1+G))^{1/2}\) lie in \([1-\|G\|,1+\|G\|]\), so \(R\), \(1+G\) and \(M\) are invertible; \(M^*M=\lambda^2R^2\), so \(\mathrm{pol}(M)=UW\) with the unitary \(W=(1+G)R^{-1}\), and \(\|W-1\|\le\|W-WR\|+\|WR-1\|=\|1-R\|+\|G\|\le2\|G\|\). \(\square\)
Theorem 4.25 (the commutant at a tolerance). Let \(\tilde H_1,\dots,\tilde H_J\) be Hermitian, \(\eta=(\sum_j\|\tilde H_j-H_j\|^2)^{1/2}\), \(\tilde L\) their commutator map, and for \(\theta\ge0\) let \(\mathcal C_\theta\) be the span of the eigenvectors of \(\tilde L^*\tilde L\) with eigenvalues \(\le\theta^2\) — the commutant at tolerance \(\theta\) (\(\mathcal C_0=\{\tilde H_j\}'\)).
(a) \(|\sigma_i(\tilde L)-\sigma_i(L)|\le2\eta\) for all \(i\); in particular \(\sigma_i(\tilde L)\le2\eta\) for \(i\le d\) and \(\sigma_{d+1}(\tilde L)\ge\gamma-2\eta\).
(b) If \(2\eta<\theta<\gamma-2\eta\), then \(\dim\mathcal C_\theta=d\), \(\mathcal C_\theta\) is closed under \(X\mapsto X^*\), and the orthogonal projections \(\tilde\Pi\) onto \(\mathcal C_\theta\) and \(\Pi\) onto \(\mathcal A'\) satisfy \(\|\tilde\Pi-\Pi\|\le4\eta/\gamma\).
(c) If \(\theta\ge\gamma+2\eta\), then \(\dim\mathcal C_\theta>d\). If \(\theta<2\eta\), \(\dim\mathcal C_\theta<d\) can happen: for \(H_1=\mathrm{diag}(1,1,0)\) and \(\tilde H_1=H_1+\eta(E_{12}+E_{21})\), \(d=5\), \(\gamma=1\), the singular values of \(\tilde L\) are \(0,0,0,2\eta,2\eta,1-\eta,1-\eta,1+\eta,1+\eta\), and \(\dim\mathcal C_\theta=3\) for \(\theta<2\eta\).
So the tolerance must exceed twice the noise level and stay below the gap minus twice the noise level; such a \(\theta\) exists iff \(\eta<\gamma/4\), and then the computed commutant has the right dimension and is within \(4\eta/\gamma\) of the true one. The bound \(2\eta\) is attained in the example of (c).
Proof. (a) \(\|(\tilde L-L)X\|_2^2=\sum_j\|[\tilde H_j-H_j,X]\|_2^2\le4\eta^2\|X\|_2^2\), and \(\sigma_i(L)=0\) for \(i\le d\); apply Lemma 4.24(a). (b) By (a), \(\sigma_i(\tilde L)\le2\eta<\theta\) for \(i\le d\) and \(\sigma_i(\tilde L)\ge\gamma-2\eta>\theta\) for \(i>d\), so \(\mathcal C_\theta\) is the range of the projection \(\tilde\Pi\) of Lemma 4.24(a) with \(\Delta=2\eta<\gamma/2\). For Hermitian \(H\), \(\langle Y,[H,X]\rangle_{HS}=\mathrm{Tr}(Y^*HX-Y^*XH)=\langle[H,Y],X\rangle_{HS}\), so \(\tilde L^*\tilde LX=\sum_j[\tilde H_j,[\tilde H_j,X]]\), and taking adjoints, \((\tilde L^*\tilde LX)^*=\sum_j[[X^*,\tilde H_j],\tilde H_j]=\tilde L^*\tilde L(X^*)\). So \(X\mapsto X^*\) maps each eigenspace of \(\tilde L^*\tilde L\) (its eigenvalues are real) into itself. (c) \(\sigma_{d+1}(\tilde L)\le\gamma+2\eta\le\theta\) by (a). In the example write \(X=\begin{pmatrix}X_{11}&x_{12}\\x_{21}&x_{22}\end{pmatrix}\) for \(\mathbb C^3=\mathbb C^2\oplus\mathbb C\) and \(\tilde H_1=(1+\eta\sigma)\oplus0\), \(\sigma=\begin{pmatrix}0&1\\1&0\end{pmatrix}\). Then \([\tilde H_1,X]\) has the blocks \(\eta[\sigma,X_{11}]\), \((1+\eta\sigma)x_{12}\), \(-x_{21}(1+\eta\sigma)\) and \(0\), an orthogonal decomposition. \(X_{11}\mapsto[\sigma,X_{11}]\) vanishes on \(1,\sigma\) and maps the orthonormal pair \(\mathrm{diag}(1,-1)/\sqrt2\), \(\begin{pmatrix}0&-i\\i&0\end{pmatrix}/\sqrt2\) to \(\mp2i\) times each other; \(1+\eta\sigma\) has the eigenvalues \(1\pm\eta\). This gives the singular values, \(\eta=0\) gives \(d=5\) and \(\gamma=1\), and for \(\theta<2\eta\) only the three zeros remain. \(\square\)
In the library \(\theta\) is tol times \(\max(1,\sigma_{\max})\) (module docstring); the null space is computed after a reduction of
the unknowns that is exact for exact generators (Section 4.3), and the singular values it reports are those of the reduced map.
Theorem 4.25 needs \(\eta\) and \(\gamma\), which are unknown in practice. The output can nevertheless be certified:
Proposition 4.26 (a-posteriori certificate). Let \(\tilde H_1,\dots,\tilde H_J\) be Hermitian, \(\tilde V\) unitary and \(\sum_ck_cm_c=n\); let \(\tilde{\mathcal B}=\tilde V(\bigoplus_cM_{k_c}\otimes1_{m_c})\tilde V^*\), \(E\) the orthogonal projection of \(M_n\) onto \(\tilde{\mathcal B}\) for \(\langle\cdot,\cdot\rangle_{HS}\), \(H_j=E(\tilde H_j)\) with \(\tilde V^*H_j\tilde V=\bigoplus_ca_{j,c}\otimes1_{m_c}\), \(r=(\sum_j\|\tilde H_j-H_j\|_2^2)^{1/2}\) (the backward error), \(d=\sum_cm_c^2\), and \(\gamma_{\mathcal B}=\sigma_{d+1}\) of the commutator map of \(H_1,\dots,H_J\).
(a) \(\gamma_{\mathcal B}\) is the minimum of \(\sigma_2(\Phi_{cc})\) over the \(c\) with \(k_c\ge2\) and of \(\sigma_1(\Phi_{ce})\) over \(c\neq e\) (\(\infty\) if there are none), where \(\Phi_{ce}:M_{k_c\times k_e}\to M_{k_c\times k_e}^J\), \(Y\mapsto(a_{j,c}Y-Ya_{j,e})_j\).
(b) If \(\gamma_{\mathcal B}>0\), the \(H_j\) generate \(\tilde{\mathcal B}\): \(\tilde V\) and \((k_c,m_c)\) are an exact decomposition of the algebra generated by Hermitian matrices at distance \(r\) from the input.
(c) If \(2r<\gamma_{\mathcal B}\), then for every \(\theta\in(2r,\gamma_{\mathcal B}-2r)\) the commutant at tolerance \(\theta\) of the input has dimension \(d\) and is within \(4r/\gamma_{\mathcal B}\) of \(\tilde{\mathcal B}'\); and Hermitian \(H^0_j\) with \((\sum_j\|H^0_j-\tilde H_j\|^2)^{1/2}<\gamma_{\mathcal B}/2-r\) generate an algebra whose commutant has dimension \(\le d\).
Proof. \(\tilde{\mathcal B}\) is closed under \(X\mapsto X^*\), a conjugate-linear map preserving \(\mathrm{Re}\langle\cdot,\cdot\rangle_{HS}\), so it maps \(\tilde{\mathcal B}^\perp\) to itself and commutes with \(E\); hence the \(H_j\) and the \(a_{j,c}\) are Hermitian, and \(\|\tilde H_j-H_j\|\le\|\tilde H_j-H_j\|_2\). (a) Write \(\tilde V^*X\tilde V\) in blocks \(X_{ce}\in M_{k_cm_c\times k_em_e}=M_{k_c\times k_e}\otimes M_{m_c\times m_e}\) (Kronecker order); \(X\mapsto\tilde V^*X\tilde V\) preserves \(\|\cdot\|_2\), these blocks are orthogonal, and the \((c,e)\) block of \(\tilde V^*[H_j,X]\tilde V\) is \((a_{j,c}\otimes1)X_{ce}-X_{ce}(a_{j,e}\otimes1)\), which is \(\Phi_{ce}(Y)_j\otimes Z\) for \(X_{ce}=Y\otimes Z\). So the commutator map acts as \(\Phi_{ce}\otimes\mathrm{id}\) on the \((c,e)\) block, its \(T^*T\) is \(\bigoplus_{c,e}\Phi_{ce}^*\Phi_{ce}\otimes1\), and its singular values are those of the \(\Phi_{ce}\), each repeated \(m_cm_e\) times. Since \(\Phi_{cc}(1)=0\), at least \(\sum_cm_c^2=d\) of them vanish, and exactly \(d\) iff \(\sigma_2(\Phi_{cc})>0\) for all \(c\) and \(\sigma_1(\Phi_{ce})>0\) for \(c\neq e\); then \(\sigma_{d+1}\) is the smallest of these numbers, and otherwise both sides are \(0\). (b) The \(H_j\in\tilde{\mathcal B}\) commute with \(\tilde{\mathcal B}'\), which has dimension \(d\) (Theorem 4.3, as in the proof of Proposition 4.9), so \(\tilde{\mathcal B}'\subseteq\{H_j\}'\); if \(\gamma_{\mathcal B}>0\), at most \(d\) singular values vanish, so \(\{H_j\}'=\tilde{\mathcal B}'\) and \(\{H_j\}''=\tilde{\mathcal B}''=\tilde{\mathcal B}\) (Theorem 2.2). (c) Theorem 4.25 with the exact generators \(H_j\) (gap \(\gamma_{\mathcal B}\), commutant \(\tilde{\mathcal B}'\) by (b)) and the perturbed ones \(\tilde H_j\), at distance \(\eta\le r\). For the \(H^0_j\), \((\sum_j\|H^0_j-H_j\|^2)^{1/2}<\gamma_{\mathcal B}/2\), and Theorem 4.25(a) gives \(\sigma_{d+1}>0\) for their commutator map. \(\square\)
So the output certifies itself: a small backward error \(r\) and a gap \(\gamma_{\mathcal B}\) well above \(2r\) mean that the computed structure is exact for slightly modified generators, that every threshold in \((2r,\gamma_{\mathcal B}-2r)\) gives the same commutant dimension, and that no model with more symmetry lies within \(\gamma_{\mathcal B}/2-r\) of the input.
In the library: the certificate is Decomposition.certificate(): \(r\), \(\gamma_{\mathcal B}\) from the blocks as in (a), the window
\((2r,\gamma_{\mathcal B}-2r)\) and the singular values on both sides of the threshold actually used, with a warning when that threshold is
not in the window — tutorial, Section 4.9.
The remaining steps — splitting by a random \(K\) and grouping by a random \(Y\) — need quantitative genericity. The library's samples are rescaled Gaussian vectors (Section 4.3, step 1): with an orthonormal basis \((B_r)\) of the commutant and independent standard complex Gaussian \(c_r\), \(Y=\sum_rc_rB_r\) is a standard complex Gaussian vector of \(\mathcal A'\) (its law does not depend on the basis, Proposition A.16(a)), and \(K=Y'+Y'^*=\sqrt2\,G\) for an independent copy \(Y'\), where \(G\) is the standard Gaussian vector of the real space \(\mathcal A'_h\) with \(\langle X,Y\rangle=\mathrm{Tr}(XY)\) (\(\sqrt2\,Y'\) is the standard Gaussian vector of the real space \(\mathcal A'\) with \(\mathrm{Re}\langle\cdot,\cdot\rangle_{HS}\), and \(X\mapsto\frac12(X+X^*)\) is the orthogonal projection onto \(\mathcal A'_h\); Propositions A.16(a), A.14(b)). The same holds for the twirl sampler, since \(Z+Z^*=\sqrt2\,H\) with \(H\) a GUE matrix, the standard Gaussian vector of \(\mathrm{Herm}_n\), whose orthogonal projection onto \(\mathcal A'_h\) is the twirl, and the twirl of a Ginibre matrix is a standard complex Gaussian vector of \(\mathcal A'\).
Proposition 4.27 (random elements of the commutant). Let \(G\) and \(Z\) be independent standard Gaussian vectors of \(\mathcal A'_h\) and standard complex Gaussian vectors of \(\mathcal A'\).
(a) \(V^*GV=\bigoplus_c1_{k_c}\otimes k_c^{-1/2}H_c\) and \(V^*ZV=\bigoplus_c1_{k_c}\otimes k_c^{-1/2}Z_c\) with independent GUE matrices \(H_c\) and Ginibre matrices \(Z_c\) of size \(m_c\); \(\mathbb E\|G\|_2^2=\mathbb E\|Z\|_2^2=d\).
(b) Let \(\kappa_c=k_c^{-1/2}H_c\) and \(\delta_\times=\min|\lambda_i(\kappa_c)-\lambda_j(\kappa_e)|\) over \(c\neq e\) and all \(i,j\). For \(x>0\),
math
\mathbb P(\delta_\times<x)\le\sqrt{\tfrac2\pi}\;x\sum_{c<e}m_cm_e\Big(\frac1{k_cm_c}+\frac1{k_em_e}\Big)^{-1/2}.
(c) For \(m_c\ge2\) let \(\delta_c\) be the smallest spacing of \(\kappa_c\). Then \(\mathbb P(\delta_c<x)\le C_{m_c}k_c^{3/2}x^3\) for \(k_c^{1/2}x\le1\), with the constant \(C_{m_c}\) of Proposition C.9 (\(\beta=2\)); for \(m_c=2\), \(C_2=1/(6\sqrt\pi)\) for all \(x\).
(d) Let \(X\) be a standard Gaussian vector of a real inner product space of dimension \(D\), \(W\) a random variable independent of \(X\), and \(f\ge0\) Borel with \(f(w,tx)=tf(w,x)\) for \(t>0\). Then \(\mathbb P(f(W,X)<\varepsilon|X|)\le2\,\mathbb P(f(W,X)<\varepsilon(2D)^{1/2})\). For \(X=G\), \(D=d\); for \(X=Z\) the same holds with \((2d)^{1/2}\).
(e) If \(G\) satisfies the hypothesis of Lemma 4.6 and \(\Pi,\Pi'\) are spectral projections of \(G\) for two distinct eigenvalues of the same \(\kappa_c\), then conditionally on \(G\), \(\|\Pi Z\Pi'\|_2^2\) has the density \(e^{-t}\) on \((0,\infty)\): \(\mathbb P(\|\Pi Z\Pi'\|_2<x)=1-e^{-x^2}\le x^2\).
Consequently, with \(\delta(G)=\min(\delta_\times,\min_c\delta_c)\) (the smallest distance between distinct eigenvalues of \(G\) when
these are \(N=\sum_cm_c\), and \(0\) otherwise) and \(\beta(G,Z)=\min\|\Pi_l Z\Pi_{l'}\|_2\) over pairs of distinct
spectral projections in the same summand,
math
\mathbb P\big(\delta(G)<\varepsilon\|G\|_2\big)\le\frac{4\sqrt d}{\sqrt\pi}\,\varepsilon\sum_{c<e}\frac{m_cm_e}{(\frac1{k_cm_c}+\frac1{k_em_e})^{1/2}}
+2\sum_{c:m_c\ge2}C_{m_c}(2dk_c)^{3/2}\varepsilon^3=:p_\delta(\varepsilon),\qquad
\mathbb P\big(\beta(G,Z)<\beta'\|Z\|_2\big)\le4d\beta'^2\sum_cm_c(m_c-1)=:p_\beta(\beta'),
the first for \(\varepsilon(2dk_c)^{1/2}\le1\). The order \(\varepsilon\) cannot be improved: levels of different sectors do not repel
(Theorem 5.15), while those of one sector do (order \(\varepsilon^3\)).
Proof. (a) The matrices \(k_c^{-1/2}V(1\otimes b)V^*\), with \(b\) in an orthonormal basis of \(\mathrm{Herm}_{m_c}\) for \(\mathrm{Tr}(xy)\) and \(c=1,\dots,C\), form an orthonormal basis of \(\mathcal A'_h\) (\(\mathrm{Tr}((1\otimes b)(1\otimes b'))=k_c\mathrm{Tr}(bb')\)), so by Proposition A.14(b) \(\kappa_c=k_c^{-1/2}\sum_bg_bb\) with independent \(N(0,1)\) coefficients \(g_b\). For the basis \(e_{ii}\), \((e_{ij}+e_{ji})/\sqrt2\), \(i(e_{ij}-e_{ji})/\sqrt2\), \(\sum_bg_bb\) has independent entries \(h_{ii}\sim N(0,1)\) and \(\mathrm{Re}\,h_{ij},\mathrm{Im}\,h_{ij}\sim N(0,\frac12)\): it is a GUE matrix (Proposition A.17). Likewise the orthonormal basis \(k_c^{-1/2}V(1\otimes e_{jj'})V^*\) of \(\mathcal A'\) gives \(Z_c\) with independent standard complex Gaussian entries. Finally \(\mathbb E\|G\|_2^2=\sum\mathbb Eg_b^2=d\), and likewise for \(Z\).
(b) Fix \(c\neq e\), put \(s=(\frac1{k_cm_c}+\frac1{k_em_e})^{1/2}\) and let \(u\in\mathcal A'_h\) have the blocks \(\alpha1\) in summand \(c\), \(-\beta1\) in summand \(e\) and \(0\) elsewhere, \(\alpha=1/(k_cm_cs)\), \(\beta=1/(k_em_es)\). Then \(\|u\|_2^2=\alpha^2k_cm_c+\beta^2k_em_e=1\) and \(\alpha+\beta=s\). By Proposition A.14(b), \(G=G_0+tu\) with \(t=\langle u,G\rangle\sim N(0,1)\) independent of \(G_0\). The blocks are \(\kappa_c=\kappa^0_c+t\alpha\) and \(\kappa_e=\kappa^0_e-t\beta\), so \(\lambda_i(\kappa_c)-\lambda_j(\kappa_e)=a+ts\) with \(a=\lambda_i(\kappa^0_c)-\lambda_j(\kappa^0_e)\) a function of \(G_0\) (Corollary A.11). For fixed \(a\), \(\{t:|a+ts|<x\}\) is an interval of length \(2x/s\), of probability \(\le2x/(s\sqrt{2\pi})\); by independence (Fubini) this bounds \(\mathbb P(|\lambda_i(\kappa_c)-\lambda_j(\kappa_e)|<x)\). Sum over \(c<e\), \(i\le m_c\), \(j\le m_e\).
(c) \(\delta_c<x\) iff the smallest spacing of \(H_c\) is \(<k_c^{1/2}x\); Proposition C.9. For \(m_c=2\) the spacing has the density \(s^2e^{-s^2/4}/(2\sqrt\pi)\le s^2/(2\sqrt\pi)\) (proof of Proposition 5.13), whose integral over \((0,y)\) is \(\le y^3/(6\sqrt\pi)\).
(d) In orthonormal coordinates \(X\) is a standard Gaussian vector of \(\mathbb R^D\); by Theorem A.20(a), \(|X|\) is independent of \(X/|X|\), hence of \((W,X/|X|)\). By Markov's inequality \(\mathbb P(|X|^2\ge2D)\le\mathbb E|X|^2/2D=\frac12\). As \(f(W,X)=|X|f(W,X/|X|)\), \(\mathbb P(f(W,X)<\varepsilon(2D)^{1/2})\ge\mathbb P(f(W,X/|X|)<\varepsilon)\,\mathbb P(|X|^2<2D)\ge\frac12\mathbb P(f(W,X)<\varepsilon|X|)\). For \(Z\), \(\sqrt2Z\) is a standard Gaussian vector of dimension \(2d\), and \(\varepsilon(4d)^{1/2}\) for \(\sqrt2Z\) is \(\varepsilon(2d)^{1/2}\) for \(Z\).
(e) By (a) and Lemma 4.6, \(V^*\Pi V\) and \(V^*\Pi'V\) project onto \(\mathbb C^{k_c}\otimes e\) and \(\mathbb C^{k_c}\otimes e'\) for orthonormal eigenvectors \(e,e'\) of \(H_c\), which depend on \(G\) only. Then \(\|\Pi Z\Pi'\|_2^2=k_c|\langle e,k_c^{-1/2}Z_ce'\rangle|^2=|\langle e,Z_ce'\rangle|^2\), and for fixed \(e,e'\), \(\langle e,Z_ce'\rangle\) is the coordinate of \(Z_c\) along the unit vector \(|e\rangle\langle e'|\), a standard complex Gaussian (Proposition A.16(a),(b)), whose squared modulus has the density \(e^{-t}\) (Theorem A.20(b), \(N=1\)). As \(Z\) is independent of \(G\), Fubini's theorem gives the conditional statement. Consequences. (d) with \(W\) trivial, then (b), (c) and a union bound, with \(x=\varepsilon(2d)^{1/2}\); and (d) with \(W=G\), then (e) and a union bound over the \(\sum_cm_c(m_c-1)\) ordered pairs, with \(x^2=2d\beta'^2\). The events are Borel by Corollary A.11 and Proposition A.13. \(\square\)
For the algorithm the commutant samples are drawn from the computed commutant, not from \(\mathcal A'\); we compare them with their projections onto \(\mathcal A'\).
Theorem 4.28 (stability of the decomposition). Let \(\tilde{\mathcal A}'\subseteq M_n\) be a subspace of dimension \(d\), closed under
\(X\mapsto X^*\), with orthogonal projection \(\tilde\Pi\) and \(\|\tilde\Pi-\Pi\|\le\varepsilon_0<1\) (by Theorem 4.25(b), \(\varepsilon_0=4\eta/\gamma\)
for the commutant at a tolerance in the window). Let \(\tilde K\in\tilde{\mathcal A}'_h\), \(\tilde Y\in\tilde{\mathcal A}'\), \(K=\Pi(\tilde K)\in\mathcal A'_h\)
and \(Y=\Pi(\tilde Y)\in\mathcal A'\). Suppose \(K\) satisfies the hypothesis of Lemma 4.6, with eigenvalues \(\mu_1<\dots<\mu_N\), spectral
projections \(P_l\) of rank \(k_{c(l)}\) in the summand \(c(l)\), and \(\delta=\min_l(\mu_{l+1}-\mu_l)\); let
\(\beta=\min\{\|P_lYP_{l'}\|_2:l\neq l',\,c(l)=c(l')\}\) (\(\infty\) if there is no such pair), \(h=\max_cm_c-1\), and
math
\rho=\varepsilon_0\|\tilde K\|_2,\qquad\varepsilon_1=\frac{2\rho}\delta,\qquad\zeta=(\varepsilon_0+2\varepsilon_1)\|\tilde Y\|_2,\qquad
\xi=\frac{k_{\max}^{1/2}(\varepsilon_0+4\varepsilon_1)\|\tilde Y\|_2}\beta .
(a) If \(4\rho<\delta\), then for every \(\tau\in[2\rho,\delta-2\rho)\) the increasing eigenvalues of \(\tilde K\), cut where two consecutive ones differ by more than \(\tau\), form \(N\) groups; the \(l\)-th has \(k_{c(l)}\) elements, and its spectral projection \(\tilde P_l\) satisfies \(\|\tilde P_l-P_l\|\le\varepsilon_1\).
(b) If moreover \(2\zeta<\beta\), then for every \(\theta'\in[\zeta,\beta-\zeta)\) two groups \(l\neq l'\) of equal size satisfy \(\|\tilde P_l\tilde Y\tilde P_{l'}\|_2>\theta'\) iff \(c(l)=c(l')\). So the linked classes of step 3 are the summands, and the algorithm returns the exact \((k_c,m_c)\).
(c) If moreover \(3^h\xi\le1\), let \(\tilde V\) be assembled as in steps 3–4 from orthonormal bases of the \(\mathrm{ran}\,\tilde P_l\), the copies of each summand aligned with polar factors along any tree of depth \(\le h\). There is a unitary \(\hat V\) with \(\hat V^*a\hat V=\bigoplus_c\rho_c(a)\otimes1_{m_c}\) for all \(a\in\mathcal A\) (a decomposition as in Theorem 4.3) and \(\|\tilde V-\hat V\|\le N^{1/2}(2\varepsilon_1+3^h\xi)\).
(d) Let \(\tilde K\) be a positive multiple of a standard Gaussian vector of \(\tilde{\mathcal A}'_h\) and \(\tilde Y\) an independent positive multiple of a standard complex Gaussian vector of \(\tilde{\mathcal A}'\), as in the library. (The library normalises them to unit norm; the hypotheses and conclusions of (a)–(c) do not change under positive multiples of \(\tilde K\) and of \(\tilde Y\), with \(\tau\) and \(\theta'\) scaled accordingly.) For every Borel set \(\mathcal E\) of pairs, \(\mathbb P((K,Y)\in\mathcal E)\le(1-\varepsilon_0^2)^{-3d/2}\,\mathbb P((cG,c'Z)\in\mathcal E)\) for the same multiples \(c,c'\) of \(G,Z\) as in Proposition 4.27. Consequently, if \(\varepsilon,\beta'\in(0,1]\) and \(\varepsilon_0\le\varepsilon\beta'/(14\cdot3^hk_{\max}^{1/2})\), then outside an event of probability \(\le(1-\varepsilon_0^2)^{-3d/2}(p_\delta(\varepsilon)+p_\beta(\beta'))\) the hypotheses of (a)–(c) hold, and \(\|\tilde V-\hat V\|\le17\cdot3^h(Nk_{\max})^{1/2}\varepsilon_0/(\varepsilon\beta')\). If \(\varepsilon_0^2\le1/(3d)\), the factor is \(\le2\).
Proof. \(\Pi\) commutes with \(X\mapsto X^*\) (as \(E\) in Proposition 4.26), so \(K\) is Hermitian. For \(X\in\mathrm{ran}\,\tilde\Pi\), \(X-\Pi X=(1-\Pi)\tilde\Pi X=(\tilde\Pi-\Pi)\tilde\Pi X\), so \(\|X-\Pi X\|_2\le\varepsilon_0\|X\|_2\) and \(\|\Pi X\|_2\le\|X\|_2\). (a) Hence \(\|\tilde K-K\|\le\|\tilde K-K\|_2\le\rho\). Lemma 4.24(b) for each \(\mu_l\) (with \(g\ge\delta>4\rho\), \(\Delta=\rho\)) gives exactly \(k_{c(l)}\) eigenvalues of \(\tilde K\) in \([\mu_l-\rho,\mu_l+\rho]\); these intervals are disjoint, consecutive eigenvalues in one of them differ by \(\le2\rho\le\tau\), and eigenvalues in different ones by \(\ge\delta-2\rho>\tau\); and \(\|\tilde P_l-P_l\|\le2\rho/\delta\). (b) By \(\|AXB\|_2\le\|A\|\|X\|_2\|B\|\), \(|\|\tilde P_l\tilde Y\tilde P_{l'}\|_2-\|P_lYP_{l'}\|_2|\le\|\tilde Y-Y\|_2+\|(\tilde P_l-P_l)Y\|_2+\|Y(\tilde P_{l'}-P_{l'})\|_2\le\zeta\), while \(P_lYP_{l'}=0\) if \(c(l)\neq c(l')\) (Lemma 4.8(b)) and \(\|P_lYP_{l'}\|_2\ge\beta\) otherwise. All groups of one summand are therefore linked to each other and to no other group, and summand \(c\) gives \(m_c\) groups of size \(k_c\).
(c) Let \(\tilde Q_l\) be the basis of \(\mathrm{ran}\,\tilde P_l\) and \(Q_l\) the basis of \(\mathrm{ran}\,P_l\) given by Lemma 4.24(c) (\(\varepsilon_1<\frac12\)), so \(\|\tilde Q_l-Q_l\|\le2\varepsilon_1\). By Lemma 4.8(b), \(T_{ll'}=Q_l^*YQ_{l'}=\lambda_{ll'}U_{ll'}\) for \(l,l'\) in one summand \(c\), with \(U_{ll'}\) unitary and \(\lambda_{ll'}=k_c^{-1/2}\|P_lYP_{l'}\|_2\ge\beta k_{\max}^{-1/2}\), and \(\tilde T_{ll'}=\tilde Q_l^*\tilde Y\tilde Q_{l'}\) satisfies \(\|\tilde T_{ll'}-T_{ll'}\|\le\varepsilon_0\|\tilde Y\|_2+4\varepsilon_1\|Y\|_2\le\xi\lambda_{ll'}\). The algorithm puts \(\tilde W_{l_0}=1\) at the root and \(\tilde W_{l'}=\mathrm{pol}(\tilde W_l^*\tilde T_{ll'})^*\) along an edge from \(l\) to \(l'\); define \(W_l\) in the same way from the \(T_{ll'}\), so \(W_{l'}=U_{ll'}^*W_l\). By Lemma 4.8(b) and induction along the tree, the bases \(Q_lW_l\) of the copies of summand \(c\) carry the same matrices \((Q_lW_l)^*a(Q_lW_l)\), so the \(Q_lW_l\), arranged as the \(\tilde Q_l\tilde W_l\) are in \(\tilde V\), form a decomposition \(\hat V\) as in the existence part of the proof of Theorem 4.3. Let \(e_l=\|\tilde W_l-W_l\|\). Then \(\|\tilde W_l^*\tilde T_{ll'}-W_l^*T_{ll'}\|\le e_l\|\tilde T_{ll'}\|+\|\tilde T_{ll'}-T_{ll'}\|\le\lambda_{ll'}(e_l(1+\xi)+\xi)\), and if this is \(<\lambda_{ll'}\), Lemma 4.24(d) (with \(W_l^*T_{ll'}=\lambda_{ll'}W_l^*U_{ll'}\)) gives \(e_{l'}\le2e_l(1+\xi)+2\xi\le3e_l+2\xi\). By induction on the depth \(t\), \(e_l\le\xi(3^t-1)\): the step is \(3\xi(3^t-1)+2\xi=\xi(3^{t+1}-1)\), and it applies since for \(t\le h-1\), \(e_l(1+\xi)+\xi\le\xi3^t(1+\xi)\le\frac13\cdot\frac43<1\) (as \(\xi\le3^{-h}\le\frac13\)). So \(\|\tilde Q_l\tilde W_l-Q_lW_l\|\le2\varepsilon_1+3^h\xi\); the matrix \(\tilde V-\hat V\) consists of these \(N\) blocks of columns, and \(|\sum_lA_lx_l|\le\sum_l\|A_l\||x_l|\le(\sum_l\|A_l\|^2)^{1/2}|x|\) gives the bound.
(d) In orthonormal coordinates of \(\tilde{\mathcal A}'_h\) and \(\mathcal A'_h\) (both of real dimension \(d\): a subspace closed under adjoints is \(\mathcal S_h\oplus i\mathcal S_h\)), \(\Pi\) restricts to a linear map \(B:\tilde{\mathcal A}'_h\to\mathcal A'_h\) with \((1-\varepsilon_0^2)|x|^2\le|Bx|^2\le|x|^2\). So \(B\) is bijective, \(|B^{-1}y|\ge|y|\), and \(|\det B|=\det(B^TB)^{1/2}\ge(1-\varepsilon_0^2)^{d/2}\) (Theorem A.8). By Lemma A.3(c), \(K/c=B(\tilde K/c)\) has the density \(\varphi_d(B^{-1}y)/|\det B|\le(1-\varepsilon_0^2)^{-d/2}\varphi_d(y)\), where \(\varphi_d\) is the density of \(G\). The same argument on the real space \(\tilde{\mathcal A}'\) of dimension \(2d\) gives the factor \((1-\varepsilon_0^2)^{-d}\) for \(Y\), and \(K,Y\) are independent. On the complement of the two events of Proposition 4.27 (for \(K\) and \(Y\); they are invariant under the multiples), and with \((1-\varepsilon_0^2)^{1/2}\ge\frac{\sqrt3}2\): \(\delta\ge\varepsilon\|K\|_2\ge\frac{\sqrt3}2\varepsilon\|\tilde K\|_2\), so \(\varepsilon_1\le\frac4{\sqrt3}\frac{\varepsilon_0}\varepsilon<\frac12\); \(\beta\ge\beta'\|Y\|_2\ge\frac{\sqrt3}2\beta'\|\tilde Y\|_2\), and \(2\zeta\le2(1+\frac8{\sqrt3\varepsilon})\varepsilon_0\|\tilde Y\|_2\le11.24\,\frac{\varepsilon_0}\varepsilon\|\tilde Y\|_2\le\frac{11.24}{14}\beta'\|\tilde Y\|_2<\beta\); and \(3^h\xi\le3^hk_{\max}^{1/2}\frac{2}{\sqrt3\beta'}(1+\frac{16}{\sqrt3\varepsilon})\varepsilon_0\le11.83\cdot3^hk_{\max}^{1/2}\frac{\varepsilon_0}{\varepsilon\beta'}\le\frac{11.83}{14}<1\). The bound on \(\|\tilde V-\hat V\|\) follows from \(2\varepsilon_1\le4.62\,\varepsilon_0/\varepsilon\), the bound on \(3^h\xi\) and \(4.62+11.83<17\). Finally \((1-\frac1{3d})^{-3d/2}\le(\frac23)^{-3/2}<2\), as \((1-\frac1m)^{-m}\) decreases in \(m\ge2\). \(\square\)
Together: if the noise is small compared with the gap, \(\eta\le\gamma\varepsilon\beta'/(56\cdot3^hk_{\max}^{1/2})\), and the thresholds lie in the windows of Theorems 4.25 and 4.28, the algorithm recovers \((k_c,m_c)\) and a \(V\) within \(O(\eta/\gamma)\) of an exact decomposition, except with probability \(O(\varepsilon+\beta'^2)\). The constants are worst-case bounds; typical errors are much smaller. The library cannot know \(\eta\) and \(\gamma\), so it estimates the relative error of its commutant samples by \(\nu=\max(\sigma_d(\tilde L),\epsilon_{\rm mach}\max(1,\sigma_{\max}))/\sigma_{d+1}(\tilde L)\) (of the order of \(\varepsilon_0\) by Lemma 4.24(a), as \(\sigma_d(\tilde L)\le\Delta\)) and puts its clustering floor and its linking threshold at \(\min(10^3\nu,\nu^{1/2})\) (the latter at least \(\epsilon_{\rm mach}^{1/2}\)), between the noise and the unit scale of the structure; Proposition 4.26 then checks the outcome.
Computation: Theorem 4.25 on noisy generators (the gap of the singular values against the tolerance, and the example of (c)), Proposition 4.26 for the certificate, and Proposition 4.27 by Monte Carlo — tutorial, Section 4.9.
Exercises¶
- Show that for the representation of \(S_L\) on \((\mathbb C^d)^{\otimes L}\), \(\chi_\pi(\sigma)=d^{\#\text{cycles}(\sigma)}\) and \(\dim\pi(S_L)'=\binom{d^2+L-1}L\). Check \(20\) (Section 2.4) and \(35\) (\(=\dim\{S_a\}''\) for four spins).
- (Clebsch–Gordan.) Show \(j_1\otimes j_2\cong\bigoplus_jj\), \(j=|j_1-j_2|,|j_1-j_2|+1,\dots,j_1+j_2\), for the spin representation \(S^{(j_1)}_a\otimes1+1\otimes S^{(j_2)}_a\) — the non-abelian fusion rule of \(SU(2)\) that Chapter 7 mentions.
- Let \(\theta/2\pi\) be irrational. Show that \(\{U_\theta\}'=U(1)'\) for a single \(U_\theta=e^{i\theta Q}\), but that \(\{e^{i\pi\hat N}\}'\cong M_{2^{n-1}}\oplus M_{2^{n-1}}\) for \(n\) qubits.
- (Zeeman splitting.) Let \(H\) be a Heisenberg Hamiltonian on \(L\ge1\) spins and \(H_h=H+hS_z\). Show that \(H_h\) commutes with \(S_z\) and \(\mathbf S^2\) but, for \(h\neq0\), not with \(S_x\), and that its eigenvalues are \(E+hM\), \(M=-S,\dots,S\), for the eigenvalues \(E\) of \(H_S\).
- (The quaternion group.) Let \(Q_8=\{\pm1,\pm I,\pm J,\pm K\}\subseteq\mathbb H\) with \(I=(i,0)\), \(J=(0,1)\), \(K=(0,i)\) (Section E.6), and let \(\pi\) be its regular representation on \(\mathbb R^8\), \(\pi(g)e_h=e_{gh}\). Show that \(\mathcal A=\mathrm{span}\,\pi(Q_8)\) has four sectors of real type with \(k=m=1\) and one of quaternionic type with \(k=m=2\), so that \(\mathcal A_{\mathbb R}\cong\mathbb R^4\oplus\mathbb H\).
Solutions
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By Lemma B.1(c), \(\pi(\sigma)\) is a permutation matrix fixing \(|b_1\dots b_L\rangle\) iff \(b\) is constant on the cycles of \(\sigma\), and there are \(d^{\#\text{cycles}(\sigma)}\) such \(b\). By Corollary 4.5(c), \(\dim\pi(S_L)'=\frac1{L!}\sum_\sigma(d^2)^{\#\text{cycles}(\sigma)}=\frac{x(x+1)\cdots(x+L-1)}{L!}\) with \(x=d^2\) (the identity \(\sum_\sigma x^{\#\text{cycles}(\sigma)}=x(x+1)\cdots(x+L-1)\) is proved in the solution of Chapter 5, Exercise 2), which is \(\binom{d^2+L-1}L\) (Proposition B.3 proves the same dimension by counting multisets): \(\binom63=20\) and \(\binom74=35\). That \(35\) is also \(\dim\{S_a\}''\) is Theorem 4.14(b) for \(L=4\): \(\{S_a\}''=\pi(S_4)'\).
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\(S_a=S^{(j_1)}_a\otimes1+1\otimes S^{(j_2)}_a\) is a spin representation (Section 4.5), and the unitary \(x\otimes y\mapsto y\otimes x\) intertwines it with the one for \((j_2,j_1)\); so let \(j_1\ge j_2\). \(S_z\) has the eigenvalue \(m_1+m_2\) on \(|m_1\rangle\otimes|m_2\rangle\), so its eigenvalues lie in \(j_1+j_2+\mathbb Z\), and for such \(M\), \(d_M\) is the number of \(m_2\in\{-j_2,\dots,j_2\}\) with \(|M-m_2|\le j_1\) (then \(m_1=M-m_2\in j_1+\mathbb Z\) automatically); \(d_M=0\) for \(M\notin j_1+j_2+\mathbb Z\). For \(M\ge0\) the condition reads \(M-j_1\le m_2\le M+j_1\), and \(M+j_1\ge j_2\), so \(m_2\) runs from \(\max(-j_2,M-j_1)\) to \(j_2\): \(d_M=2j_2+1\) for \(0\le M\le j_1-j_2\), \(d_M=j_1+j_2-M+1\) for \(j_1-j_2\le M\le j_1+j_2\), and \(d_M=0\) for \(M>j_1+j_2\). By Corollary 4.12 the multiplicity of spin \(j\ge0\) is \(m_j=d_j-d_{j+1}\). It is \(0\) if \(j\notin j_1+j_2+\mathbb Z\) or \(j>j_1+j_2\); \(m_{j_1+j_2}=1-0=1\); for \(j_1-j_2\le j<j_1+j_2\) (with \(j\in j_1+j_2+\mathbb Z\), so \(j+1\le j_1+j_2\)), \(m_j=(j_1+j_2-j+1)-(j_1+j_2-j)=1\); and for \(0\le j<j_1-j_2\) (so \(j+1\le j_1-j_2\)), \(m_j=(2j_2+1)-(2j_2+1)=0\). Hence \(j_1\otimes j_2\cong\bigoplus_{j=j_1-j_2}^{j_1+j_2}j\), each spin once. For \(j_2=\frac12\) this is Theorem 4.16(d), which in addition gives the coefficients.
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\(U_\theta=\sum_qe^{iq\theta}P_q\), and \(e^{iq\theta}=e^{iq'\theta}\) iff \((q-q')\theta\in2\pi\mathbb Z\), which for irrational \(\theta/2\pi\) forces \(q=q'\). So the eigenspaces of \(U_\theta\) are the ranges of the \(P_q\), and an operator commutes with the normal matrix \(U_\theta\) iff it preserves its eigenspaces (Theorem 1.5(d) and Lagrange interpolation as in Section 2.2), i.e. iff \(X=\sum_qP_qXP_q\); this is \(U(1)'\) by Proposition 4.10. The argument only uses that the \(e^{iq\theta}\) for the charges \(q\) that occur are distinct. For \(\theta=\pi\) the eigenvalue \((-1)^w\) only sees the parity of the Hamming weight: two eigenspaces of dimension \(2^{n-1}\) (for \(n\ge1\), the configurations of even and of odd weight; flipping the first bit is a bijection between them), and the commutant is \(M_{2^{n-1}}\oplus M_{2^{n-1}}\).
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\(S_z\) and \(\mathbf S^2\) commute with \(H\) (Proposition 4.15(a)) and with \(S_z\), so with \(H_h\); and \([H_h,S_x]=h[S_z,S_x]=ihS_y\neq0\). In the decomposition of Proposition 4.13, \(V^*S_zV=\bigoplus_SS^{(S)}_z\otimes1_{m_S}\) and \(V^*HV=\bigoplus_S1_{2S+1}\otimes H_S\) (Proposition 4.15), so on the spin-\(S\) summand \(V^*H_hV=hS^{(S)}_z\otimes1+1\otimes H_S\), a sum of commuting operators diagonalised by the products of eigenvectors, with eigenvalues \(hM+E\).
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By the proof of Theorem E.19, \(I^2=J^2=K^2=-1\) and \(IJ=-JI=K\), \(JK=-KJ=I\), \(KI=-IK=J\), so \(Q_8\) is a group, and \(h\mapsto gh\) is a bijection of \(Q_8\): the \(\pi(g)\) are permutation matrices, a unitary representation by real matrices, and \(\mathcal A=\pi(Q_8)''\) is closed under conjugation (Lemma 4.18(c)). \(\chi_\pi(g)\) is the number of \(h\) with \(gh=h\), i.e. \(8\) for \(g=1\) and \(0\) otherwise, so by Corollary 4.5(b) an irreducible unitary \(\rho\) has multiplicity \(m_\rho=\frac18\overline{\chi_\rho(1)}\cdot8=\dim\rho\). One-dimensional: modulo signs, \(I,J,K\) multiply like the non-identity elements of \(\mathbb Z_2\times\mathbb Z_2\), so for \(s,t\in\{\pm1\}\), \(\chi(\pm1)=1\), \(\chi(\pm I)=s\), \(\chi(\pm J)=t\), \(\chi(\pm K)=st\) is a homomorphism; these are four distinct characters, hence four inequivalent irreducible representations, each with \(m=1\). Two-dimensional: \(\rho(g)=M(g)\) (Lemma E.15(2)) is a unitary representation (\(M(g)^*M(g)=M(\bar gg)=1\)) with \(\chi_\rho(\pm1)=\pm2\) and \(\chi_\rho=0\) elsewhere (\(\mathrm{Tr}\,M(a,b)=2\mathrm{Re}\,a\)), so \(\langle\chi_\rho,\chi_\rho\rangle_{Q_8}=\frac{4+4}8=1\) and \(\rho\) is irreducible (Corollary 4.5(c)), with \(m=2\). For a decomposition \(V\) as in Theorem 2.6, the \(\pi_c\circ\pi\) are irreducible and pairwise inequivalent (Proposition 2.7), so \(V\) is a decomposition as in Theorem 4.3, and each of the five representations above, having \(m_\rho\ge1\), is equivalent to exactly one \(\pi_c\circ\pi\), different ones to different \(c\) (Corollary 4.5(b)). As \(4\cdot1\cdot1+2\cdot2=8=\sum_ck_cm_c\), there are no other sectors. By Theorem 4.20(c) we may use a decomposition with \(\pi_c(\pi(g))=\rho_c(g)\) for these representations (Lemma 4.2(c) gives unitaries \(u\) with \(\rho_c=u\,\pi_c(\pi(\cdot))\,u^*\); replace \(V_c\) by \(V_c(u^*\otimes1)\)). Since \(\pi(g)\) is real, Theorem 4.20(a) gives \(\pi_{\sigma(c)}(\pi(g))=J_c\overline{\rho_c(g)}J_c^*\). For a character \(\chi\) this is \(\chi(g)\), so \(\pi_{\sigma(c)}\circ\pi\cong\pi_c\circ\pi\), \(\sigma(c)=c\) (Proposition 2.7: different sectors carry inequivalent representations), and \(J_c\bar J_c=|J_c|^2=1\): real type. For \(\rho=M\), \(\sigma(c)=c\) (the only sector with \(k=2\)), and \(\overline{M(g)}=\Omega_2^*M(g)\Omega_2\) (by the proof of Theorem 4.21(b), \(M(q)\Omega_2=\Omega_2\overline{M(q)}\)), so \(M(g)=(J_c\Omega_2^*)M(g)(J_c\Omega_2^*)^*\). The matrices \(M(1)=1\), \(M(I)=\mathrm{diag}(i,-i)\), \(M(J)=\Omega_2\), \(M(K)=\begin{pmatrix}0&-i\\-i&0\end{pmatrix}\) are linearly independent, so they span \(M_2(\mathbb C)\); hence \(J_c\Omega_2^*=\lambda1\), \(J_c=\lambda\Omega_2\) and \(J_c\bar J_c=|\lambda|^2\Omega_2^2=-1\): quaternionic type. By Theorem 4.21, \(\mathcal A_{\mathbb R}\cong\mathbb R^4\oplus M_1(\mathbb H)\), of dimension \(8=|Q_8|\).
Computation: Exercises 1–5 numerically — tutorial, Exercises.
Next: Chapter 5 draws random matrices in an algebra: Haar unitaries of \(\prod_cU(k_c)\), the Weingarten calculus — whose order-two case rests on the commutant of \(\{V\otimes V\}\), the smallest instance of Schur–Weyl duality — and level statistics sector by sector.