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arXiv:2301.07899 · Physical review. B./Physical review. B

Noninvertible duality transformation between symmetry-protected topological and spontaneous symmetry breaking phases

Linhao Li, Masaki Oshikawa, Yunqin Zheng

Published Dec 22, 2023• 89 citations
arXivHTMLPDF

Abstract

In 1992, Kennedy and Tasaki constructed a nonlocal unitary transformation that maps between a ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ spontaneously symmetry breaking phase and the Haldane gap phase, which is a prototypical symmetry-protected topological (SPT) phase in modern framework, on an open spin chain. In this work, we propose a way to define it on a closed chain by sacrificing unitarity. The operator realizing such a nonunitary transformation satisfies noninvertible fusion rule and implements a generalized gauging of the ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ global symmetry. These findings connect the Kennedy-Tasaki transformation to numerous other concepts developed for SPT phases and opens a way to construct SPT phases systematically using the duality mapping.

DOI
10.1103/physrevb.108.214429

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In 1992, Kennedy and Tasaki constructed a nonlocal unitary transformation that maps between a Z2\ifmmode×\else\texttimes\fiZ2{\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}Z2​\ifmmode×\else\texttimes\fiZ2​ spontaneously symmetry breaking phase and the Haldane gap phase, which is a prototypical symmetry-protected topological (SPT) phase in modern framework, on an open spin chain. In this work, we propose a way to define it on a closed chain by sacrificing unitarity. The operator realizing such a nonunitary transformation satisfies noninvertible fusion rule and implements a generalized gauging of the Z2\ifmmode×\else\texttimes\fiZ2{\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}Z2​\ifmmode×\else\texttimes\fiZ2​ global symmetry. These findings connect the Kennedy-Tasaki transformation to numerous other concepts developed for SPT phases and opens a way to construct SPT phases systematically using the duality mapping.
We give an explicit operator representation (via a sequential circuit and projection to symmetry subspaces) of Kramers-Wannier duality transformation in higher-dimensional subsystem symmetric models, generalizing the construction in the 1D transverse-field Ising model. Using the Kramers-Wannier duality operator, we also construct the Kennedy-Tasaki transformation that maps subsystem symmetry-protected topological phases to spontaneous subsystem symmetry-breaking phases, where the symmetry group for the former is either ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ or Z2{\mathbb{Z}}_{2}Z2​. This also generalizes the recently proposed picture of the one-dimensional Kennedy-Tasaki transformation as a composition of manipulations involving gauging and stacking symmetry-protected topological phases to higher dimensions.
The Kennedy-Tasaki (KT) transformation was used to construct the gapped symmetry protected topological (SPT) phase from the symmetry breaking phase with open boundary condition, and was generalized in our proceeding work [Phys. Rev. B 108, 214429 (2023)] on a ring by sacrificing the unitarity, and should be understood as a non-invertible duality transformation. In this work, we further apply the KT transformation to systematically construct gapless symmetry protected topological phases. This construction reproduces the known examples of (intrinsically) gapless SPT where the non-trivial topological features come from the gapped sectors by means of decorated defect constructions. We also construct new (intrinsically) purely gapless SPTs where there are no gapped sectors, hence are beyond the decorated defect construction. This construction elucidates the field theory description of the various gapless SPTs, and can also be applied to analytically study the stability of various gapless SPT models on the lattice under certain symmetric perturbations.
We extend non-invertible duality concepts from one-dimensional systems to two spatial dimensions by constructing a web of non-invertible dualities for lattice models with subsystem symmetries. For the Z2×Z2\mathbb{Z}_2 \times \mathbb{Z}_2Z2​×Z2​ subsystem symmetry on the square lattice, we build two complementary dualities: a map that sends spontaneous subsystem symmetry-broken (SSSB) phases to the trivial phase (the analogue of the Kramers-Wannier (KW) duality in 1+1D), and a generalized subsystem Kennedy-Tasaki (KT) transformation that maps SSSB phases to subsystem symmetry-protected topological (SSPT) phases while leaving the trivial phase invariant. These dualities are boundary-sensitive. On open lattices, both subsystem KW and KT transformations act as unitary, invertible operators. In particular, the KT map not only matches the bulk Hamiltonians of the dual phases but also carries the spontaneous ground-state degeneracy of the SSSB phase onto the protected boundary degeneracy of the SSPT phase. On closed manifolds, however, both maps become intrinsically non-unitary and non-invertible when restricted to the original Hilbert space. We demonstrate this non-invertibility via ground-state degeneracy matching (in two copies of the Xu-Moore/Ising-plaquette model), analysis of symmetry-twist sectors mapping, and the fusion algebra of the duality operator. Enlarging the Hilbert space to include twisted sectors allows the subsystem KW map to be formulated as a projective unitary preserving quantum transition probabilities, consistent with generalized Wigner-theorem-based constructions. We also show that the KT map faithfully transmits the algebraic content of bulk and edge invariants diagnosing strong SSPT order: although strictly local SSPT repair operators map to highly nonlocal objects in the dual SSSB phase, the essential commutation algebra and the bulk-edge correspondence remain intact.
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 SO(2k+1)\mathrm{SO}(2k+1)SO(2k+1) and SU(k)\mathrm{SU}(k)SU(k).
We extend the hidden symmetry breaking picture, first proposed by Kennedy and Tasaki in the context of the Haldane phase, to a wider class of symmetry-protected topological (SPT) phases. We construct a generalization of the Kennedy-Tasaki transformation that transforms SPT phases into symmetry-breaking phases and relates long-range order in the latter to the more subtle “string order” in the former. In doing so we directly connect the form of the Kennedy-Tasaki transformation to the modern formulation of SPT order. We apply our generalized Kennedy-Tasaki transformation to solve the following problem in quantum information theory. We consider the 2-D cluster state, a simple “toy model” of a locally interacting system whose ground state is a universal resource for MBQC. We prove that, in the presence of a perturbation to the interaction Hamiltonian, the perturbed ground state remains a universal resource. We do this by using the generalized Kennedy-Tasaki transformation to prove that, if we employ the techniques of fault-tolerant quantum computation, the ground states of models in an appropriate SPT phases can serve as universal resources for MBQC provided that the symmetry-breaking is sufficiently strong in the symmetry-breaking phase obtained through the generalized Kennedy-Tasaki transformation.
The interplay between topology and quantum criticality has established symmetry enriched criticality as a new paradigm beyond conventional frameworks in condensed matter and statistical physics. However, the conditions under which such criticality emerges remain poorly understood. In this work, we show that a broad class of symmetry enriched critical points separating symmetry protected topological (SPT) and spontaneous symmetry breaking (SSB) phases is governed by Kennedy Tasaki (KT) self duality. At these criticalities, the KT transformation that interchanges the SPT and SSB phases acts as a self-dual noninvertible symmetry, endowing the critical points with nontrivial topological characteristics. We numerically demonstrate this mechanism in Zn×Zn\mathbb{Z}_{n} \times \mathbb{Z}_{n}Zn​×Zn​ cluster qudit chains and further argue that it naturally extends to higher dimensions, providing a duality based criterion for identifying symmetry-enriched criticality across dimensions.