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arXiv:1309.7387 · Physical Review B

Detecting two-dimensional symmetry-protected topological order in a ground-state wave function

Michael P. Zaletel

Published Dec 5, 2014• 54 citations
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Abstract

Symmetry-protected topological states cannot be deformed to a trivial state so long as the symmetry is preserved, yet there is no local order parameter that can distinguish them from a trivial state. We demonstrate how to detect whether a two-dimensional ground state has symmetry-protected topological order; the measurements play a similar role as the topological entanglement entropy does for detecting anyons. For any finite Abelian onsite symmetry, the measurement completely determines the third cohomology class that characterizes the order. The proposed measurement is validated numerically using the infinite density matrix renormalization group for a model with ${\mathbb{Z}}_{2}$ symmetry-protected order.

DOI
10.1103/physrevb.90.235113

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Symmetry-protected topological states cannot be deformed to a trivial state so long as the symmetry is preserved, yet there is no local order parameter that can distinguish them from a trivial state. We demonstrate how to detect whether a two-dimensional ground state has symmetry-protected topological order; the measurements play a similar role as the topological entanglement entropy does for detecting anyons. For any finite Abelian onsite symmetry, the measurement completely determines the third cohomology class that characterizes the order. The proposed measurement is validated numerically using the infinite density matrix renormalization group for a model with Z2{\mathbb{Z}}_{2}Z2​ symmetry-protected order.
A method of using partial symmetries to distinguish two dimensional symmetry protected topological (SPT) phases of on-site, unitary symmetries is proposed. This novel order parameter takes a wavefunction, such as a ground state of a lattice model, and detects its SPT invariants as expectation values of finitely supported operators, without the need for flux insertion. The construction exploits the rotational symmetry of the lattice to extract on-site SPT invariants, building upon prior work on probing crystalline SPT phases with partial rotations. The method is demonstrated by computing the order parameter analytically on group cohomology models and numerically on a family of states interpolating between the CZX state and a trivial state. Its robustness is suggested by interpreting partial symmetries as generating the topological partition functions of lens spaces.
Symmetry-protected topological (SPT) phases with gapless edge excitations have been shown to exist in strongly interacting bosonic/fermionic systems and it is highly desirable to identify practical systems realizing such phases through numerical simulation. A central question to be addressed is how to determine the SPT order in the system given the simulation result, while no local order parameter can be measured to distinguish the phases from a trivial one. In the tensor network approach to simulate strongly interacting systems, the quantum state renormalization algorithm has been shown to be effective in identifying intrinsic topological orders. Here, we show that a modified algorithm can identify SPT orders by extracting the symmetry-protected fixed point entanglement pattern in the ground-state wave function, which is essential for the existence of SPT order. The key to this approach is to add proper symmetry protection to the renormalization process. We demonstrate the effectiveness of this algorithm with examples of nontrivial SPT phases with internal symmetry in 1D and internal and translation symmetry in 2D.
Edge theories of symmetry-protected topological phases are well known to possess global symmetry anomalies. In this Letter we focus on two-dimensional bosonic phases protected by an on-site symmetry and analyze the corresponding edge anomalies in more detail. Physical interpretations of the anomaly in terms of an obstruction to orbifolding and constructing symmetry-preserving boundaries are connected to the cohomology classification of symmetry-protected phases in two dimensions. Using the tensor network and matrix product state formalism we numerically illustrate our arguments and discuss computational detection schemes to identify symmetry-protected order in a ground state wave function.
Although symmetry-protected topological phases (SPTs) can be distinguished by their entanglement properties, it has been unclear how to extract this information directly from expectation values beyond the 1D case. Here, we close this gap and propose a pair of nonlocal order parameters that can detect and distinguish all bosonic SPTs in 2D protected by internal discrete, Abelian unitary symmetries. The desired topological invariants are extracted by these quantities by effectively simulating the SPT path integral on topologically non-trivial spacetime manifolds. Our order parameters are defined in terms of expectation values of partial symmetry and permutation operations acting on fixed numbers of replicas of the system in finite spatial regions. These expectation values correspond to symmetry-twisted versions of multipartite entanglement quantities known as multi-entropies. We show explicitly that our two order parameters detect symmetry-protected four-party and six-party entanglement, respectively, and we constrain possible "spurious" contributions. We analytically test our proposal in fixed-point lattice models. Our results suggest multipartite entanglement to be a defining feature of SPTs; indeed, we expect our methods to generalize to fermionic and higher-dimensional systems.
Symmetry-protected topological (SPT) phases exhibit nontrivial short-ranged entanglement protected by symmetry and cannot be adiabatically connected to trivial product states while preserving the symmetry. In contrast, intrinsic topological phases do not need ordinary symmetry to stabilize them and their ground states exhibit long-range entanglement. It is known that for a given symmetry group GGG, the 2D SPT phase protected by GGG is dual to the 2D topological phase exemplified by the twisted quantum double model ${D}^{\ensuremath{\omega}}(G)$ via gauging the global symmetry GGG. Recently it was realized that such a general gauging map can be implemented by some local unitaries and local measurements when GGG is a finite, solvable group. Here, we review the general approach to gauging a $G\text{\ensuremath{-}}\mathrm{SPT}$ starting from a fixed-point ground-state wave function and applying a NNN-step gauging procedure. We provide an in-depth analysis of the intermediate states emerging during the NNN-step gauging and provide tools to measure and identify the emerging symmetry-enriched topological order (SET) of these states. We construct the generic lattice parent Hamiltonians for these intermediate states and show that they form an entangled superposition of a twisted quantum double (TQD) with an SPT-ordered state. Notably, we show that they can be connected to the TQD through a finite-depth, local quantum circuit which does not respect the global symmetry of the SET order. We introduce the so-called symmetry branch line operators and show that they can be used to extract the symmetry fractionalization classes (SFC) and symmetry defectification classes (SDC) of the SET phases with the input data GGG and $[\ensuremath{\omega}]\ensuremath{\in}{H}^{3}(G,U(1))$ of the pregauged SPT-ordered state. We illustrate the procedure of preparing and characterizing the emerging SET-ordered states for some Abelian and non-Abelian examples such as dihedral groups Dn{D}_{n}Dn​ and the quaternion group Q8{Q}_{8}Q8​.
Topological insulators in free fermion systems have been well characterized and classified. However, it is not clear in strongly interacting boson or fermion systems what symmetry-protected topological orders exist. In this paper, we present a model in a two-dimensional (2D) interacting spin system with nontrivial onsite Z2{Z}_{2}Z2​ symmetry-protected topological order. The order is nontrivial because we can prove that the one-dimensional (1D) system on the boundary must be gapless if the symmetry is not broken, which generalizes the gaplessness of Wess-Zumino-Witten model for Lie symmetry groups to any discrete symmetry groups. The construction of this model is related to a nontrivial 3-cocycle of the Z2{Z}_{2}Z2​ group and can be generalized to any symmetry group. It potentially leads to a complete classification of symmetry-protected topological orders in interacting boson and fermion systems of any dimension. Specifically, this exactly solvable model has a unique gapped ground state on any closed manifold and gapless excitations on the boundary if Z2{Z}_{2}Z2​ symmetry is not broken. We prove the latter by developing the tool of a matrix product unitary operator to study the nonlocal symmetry transformation on the boundary and reveal the nontrivial 3-cocycle structure of this transformation. Similar ideas are used to construct a 2D fermionic model with onsite Z2{Z}_{2}Z2​ symmetry-protected topological order.