Hidden symmetry-breaking picture of symmetry-protected topological order

We generalize the hidden symmetry-breaking picture of symmetry-protected topological (SPT) order developed by Kennedy and Tasaki in the context of the Haldane phase. Our generalization applies to a wide class of SPT phases in one-dimensional spin chains, protected by an on-site representation of a finite Abelian group. This generalization takes the form of a nonlocal unitary map that relates local symmetry-respecting Hamiltonians in an SPT phase to local Hamiltonians in a symmetry-broken phase. Using this unitary, we establish a relation between the two-point correlation functions that characterize fully symmetry-broken phases with the string-order correlation functions that characterize the SPT phases, therefore establishing the perspective in these systems that SPT phases are characterized by hidden symmetry breaking. Our generalization is also applied to systems with continuous symmetries, including $\mathrm{SO}(2k+1)$ and $\mathrm{SU}\left(k\right)$.

Figure 1

(a) The “dimer state” renormalization fixed point for the SPT phase corresponding to a maximally commutative cohomology class $\left[\omega \right]$. (b) The result of applying ${\mathcal{D}}_{\omega }^{†}$, for a particular choice of boundary conditions. Each shaded area represents one coarse-grained site. The black dots transform under irreducible projective representations with factor systems $\omega$ and ${\omega }^{-1}$ under the symmetry, and the diamonds do not transform at all under the symmetry. Note two adjacent black dots transform linearly under the symmetry; therefore, we can introduce the simultaneous eigenbasis $\left\{\right|\chi \rangle \}$ of the symmetry (with the states labelled by linear characters $\chi$; from Schur's lemma, it follows that they must be maximally entangled). For $\chi =1,$ this gives the state $|1\rangle$ appearing in (a). In (b), we have defined $|*\rangle ={\sum }_{\chi }|\chi \rangle$. The state $|\lambda \rangle$ is not universal and depends on the specific point in the phase.