Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 343 / 3

For a unique gapped ground state, one-dimensional bosonic phases protected by an on-site symmetry $G$ are classified by the cohomology classof a projective virtual symmetry of a matrix product state. After blocking finitely many sites, an injective matrix product state tensor satisfiesChanging the virtual gauge or rephasing $V(g)$ changes $\omega$ by a coboundary and leaves $[\omega ]$ fixed.

If symmetry breaking and degenerate ground states are allowed, the additional data are the unbroken subgroup $H\subseteq G$, the way $G$ permutes the broken-symmetry sectors, and a class in $H^2(H,U(1))$ within one sector. Acting with ${⨂}_i{U}_i(g)$ preserves a symmetric ground state up to phase in the unbroken case; in a broken phase it generally permutes the different ground states.

A uniform matrix product state is specified by matrices $A^i\in M_D$ and, on a periodic chain, has amplitudesIt is injective when, after some blocking length $\ell$, the products $A^{i_1}\cdots A^{i_{\ell }}$ span $M_D$.

The fundamental theorem of matrix product states says that two injective tensors generating the same states for all sufficiently large $N$ satisfyfor one invertible matrix $X$; conversely this relation plainly gives the same periodic states up to the overall phase $e^{iN\theta }$.

For the proof, block enough sites that both tensors are injective. Injectivity gives left inverses from physical blocks to arbitrary virtual matrices. Equality of the states then implies that replacing one blocked tensor inside any sufficiently long network defines an invertible linear map on its two virtual boundary indices. Applying the replacement at two adjacent blocks in either order shows that this boundary map preserves multiplication: $\Phi (MN)=\Phi (M)\Phi (N)$. Every automorphism of the full matrix algebra is inner, so $\Phi (M)=XM{X}^{-1}$. Undoing the blocking gives $A^i=e^{i\theta }X{B}^i{X}^{-1}$, with only an $N$th-root phase left by periodic closure. Equality for consecutive sufficiently large lengths makes that phase independent of $N$ and completes the result.

Apply the theorem to a symmetric injective MPS. The physically transformed tensor $A_g^i={\sum }_{j}U(g{)}_{ij}{A}^j$ generates the same state, so the theorem forcesApplying $g$ and then $h$ shows that $V(g)V(h)$ and $V(gh)$ implement the same gauge transformation. Injectivity makes that gauge unique up to a scalar, hence $V(g)V(h)=\omega (g,h)V(gh)$. Associativity gives the two-cocycle equation for $\omega$.

A continuous symmetry-preserving gapped path changes the tensor and $V(g)$ continuously but cannot change the discrete cohomology class $[\omega ]$ without losing injectivity, breaking the symmetry, or closing the gap. Thus the fundamental theorem of matrix product states turns the virtual projective representation into the invariant classifying one-dimensional symmetry-protected topological phases. With broken symmetry, one first records the permuted ground-state sectors and then applies the same argument to their unbroken subgroup.

For the Affleck--Kennedy--Lieb--Tasaki state, take the dihedral subgroup of spin rotations generated by $\pi$ rotations about the $x$ and $z$ axes. On each physical spin-one site these generators commute and give an ordinary representation of ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$. On the virtual spin-one-half space they may be represented, up to phases, byThey anticommute,so the virtual action is the nontrivial projective representation of ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$.

The reduced density operator across a cut commutes with this irreducible projective action. By Schur lemma it is proportional to the identity within each virtual doublet. Consequently every level in the entanglement spectrum of a matrix product state is at least twofold degenerate, the characteristic protected edge-spin degeneracy of the AKLT phase.