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The hidden symmetry-breaking picture of symmetry-protected topological order

Dominic V. Else, Stephen D. Bartlett, Andrew C. Doherty

TL;DR

This work extends the Kennedy-Tasaki duality to a broad class of one-dimensional SPT phases protected by a finite abelian on-site symmetry, via a generalized KT transformation $\mathcal{D}_{\omega}$ associated with maximally non-commutative cohomology classes $[\omega]$. It establishes a precise mapping between SPT string order and conventional symmetry-breaking order, and provides explicit formulas for how correlation functions and string operators transform under $\mathcal{D}_{\omega}$. The paper applies the construction to generalized AKLT/MPS states and to continuous symmetry cases such as $SO(2k{+}1)$ and $SU(k)$, showing that SPT order can be viewed as hidden breaking of finite Abelian subgroups (e.g., $Z_2^{\times 2k}$), with a topological disentangler interpretation. The results illuminate the structure of SPT phases, offer a unified framework for identifying phases via string-order signatures, and suggest avenues for higher-dimensional dualities and numerical methods that exploit topological entanglement removal.

Abstract

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 non-local 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 characterise the SPT phases, therefore establishing the perspective in these systems that SPT phases are characterised by hidden symmetry-breaking. Our generalization is also applied to systems with continuous symmetries, including SO(2k+1) and SU(k).

The hidden symmetry-breaking picture of symmetry-protected topological order

TL;DR

This work extends the Kennedy-Tasaki duality to a broad class of one-dimensional SPT phases protected by a finite abelian on-site symmetry, via a generalized KT transformation associated with maximally non-commutative cohomology classes . It establishes a precise mapping between SPT string order and conventional symmetry-breaking order, and provides explicit formulas for how correlation functions and string operators transform under . The paper applies the construction to generalized AKLT/MPS states and to continuous symmetry cases such as and , showing that SPT order can be viewed as hidden breaking of finite Abelian subgroups (e.g., ), with a topological disentangler interpretation. The results illuminate the structure of SPT phases, offer a unified framework for identifying phases via string-order signatures, and suggest avenues for higher-dimensional dualities and numerical methods that exploit topological entanglement removal.

Abstract

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 non-local 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 characterise the SPT phases, therefore establishing the perspective in these systems that SPT phases are characterised by hidden symmetry-breaking. Our generalization is also applied to systems with continuous symmetries, including SO(2k+1) and SU(k).

Paper Structure

This paper contains 19 sections, 1 theorem, 46 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Kennedy-Tasaki transformation
  3. Classification of SPT phases by cohomology classes
  4. The generalized Kennedy-Tasaki transformation
  5. Action of the generalized KT transformation on a generalized AKLT state
  6. String order
  7. Symmetry-breaking phases and two-particle correlations
  8. SPT phases and string correlation functions
  9. Mapping of correlation functions under the generalized Kennedy-Tasaki transformation
  10. Patterns of string order as a "signature" for quantum phases
  11. The Kennedy-Tasaki transformation for continuous symmetries
  12. The $\mathrm{SO}(2k+1)$ Haldane phase
  13. SPT phases for $\mathrm{SU}(k)$
  14. Other continuous symmetry groups
  15. The topological disentangler
  16. ...and 4 more sections

Key Result

Lemma 1

Let $G$ be a finite abelian group, and let $H$ be a subgroup. Then any linear character acting on $H$ can be extended to a linear character on $G$. That is, for any $\xi \in H^{*}$, there exists $\chi \in G^{*}$ such that $\chi_H = \xi$.

Figures (1)

  • Figure 1: (a) The "dimer state" renormalization fixed point for the SPT phase corresponding to a maximally commutative cohomology class $[\omega]$; (b) The result of applying $\mathcal{D}_\omega^{\dagger}$, 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 $\{ | \chi\rangle \}$ 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.

Theorems & Definitions (2)

  • Lemma 1
  • proof