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Physical review. B./Physical review. B

One-dimensional lattice models for the boundary of two-dimensional Majorana fermion symmetry-protected topological phases: Kramers-Wannier duality as an exact Z2 symmetry

Robert A. Jones, Max A. Metlitski

Published Dec 20, 2021• 10 citations
Publisher

Abstract

It is well known that symmetry-protected topological (SPT) phases host nontrivial boundaries that cannot be mimicked in a lower-dimensional system with a conventional realization of symmetry. However, for SPT phases of bosons (fermions) within the cohomology (supercohomology) classification the boundary can be recreated without the bulk at the cost of a non-onsite-symmetry action. This raises the following question: Can one also mimic the boundaries of SPT phases which lie outside the (super)cohomology classification? In this paper, we study this question in the context of $(2+1)$-dimensional $[(2+1)\mathrm{D}]$ fermion SPTs. We focus on the root SPT phase for the symmetry group $G={Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}^{f}$. Starting with an exactly solvable model for the bulk of this phase constructed by Tarantino and Fidkowski, we derive an effective one-dimensional (1D) lattice model for the boundary. Crucially, the Hilbert space of this 1D model does not have a local tensor product structure, but rather is obtained by placing a local constraint on a local tensor product Hilbert space. We derive the action of the ${Z}_{2}$ symmetry on this Hilbert space and find a simple three-site Hamiltonian that respects this symmetry. We study this Hamiltonian numerically using exact diagonalization and DMRG and find strong evidence that it realizes an Ising conformal field theory where the ${Z}_{2}$ symmetry acts as the Kramers-Wannier duality; this is the expected stable gapless boundary state of the present SPT. A simple modification of our construction realizes the boundary of the $(2+1)\mathrm{D}$ topological superconductor protected by time-reversal symmetry $\mathcal{T}$ with ${\mathcal{T}}^{2}={(\ensuremath{-}1)}^{\mathcal{F}}$.

DOI
10.1103/physrevb.104.245130

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It is well known that symmetry-protected topological (SPT) phases host nontrivial boundaries that cannot be mimicked in a lower-dimensional system with a conventional realization of symmetry. However, for SPT phases of bosons (fermions) within the cohomology (supercohomology) classification the boundary can be recreated without the bulk at the cost of a non-onsite-symmetry action. This raises the following question: Can one also mimic the boundaries of SPT phases which lie outside the (super)cohomology classification? In this paper, we study this question in the context of (2+1)(2+1)(2+1)-dimensional [(2+1)D][(2+1)\mathrm{D}][(2+1)D] fermion SPTs. We focus on the root SPT phase for the symmetry group G=Z2\ifmmode×\else\texttimes\fiZ2fG={Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}^{f}G=Z2​\ifmmode×\else\texttimes\fiZ2f​. Starting with an exactly solvable model for the bulk of this phase constructed by Tarantino and Fidkowski, we derive an effective one-dimensional (1D) lattice model for the boundary. Crucially, the Hilbert space of this 1D model does not have a local tensor product structure, but rather is obtained by placing a local constraint on a local tensor product Hilbert space. We derive the action of the Z2{Z}_{2}Z2​ symmetry on this Hilbert space and find a simple three-site Hamiltonian that respects this symmetry. We study this Hamiltonian numerically using exact diagonalization and DMRG and find strong evidence that it realizes an Ising conformal field theory where the Z2{Z}_{2}Z2​ symmetry acts as the Kramers-Wannier duality; this is the expected stable gapless boundary state of the present SPT. A simple modification of our construction realizes the boundary of the (2+1)D(2+1)\mathrm{D}(2+1)D topological superconductor protected by time-reversal symmetry T\mathcal{T}T with T2=(\ensuremath−1)F{\mathcal{T}}^{2}={(\ensuremath{-}1)}^{\mathcal{F}}T2=(\ensuremath−1)F.
1d lattice models for the boundary of 2d "Majorana" fermion SPTs: Kramers-Wannier duality as an exact Z2Z_2Z2​ symmetry
It is well known that symmetry protected topological (SPT) phases host non-trivial boundaries that cannot be mimicked in a lower-dimensional system with a conventional realization of symmetry. However, for SPT phases of bosons (fermions) within the cohomology (supercohomology) classification the boundary can be recreated without the bulk at the cost of a non-onsite symmetry action. This raises the question: can one also mimic the boundaries of SPT phases which lie outside the (super)cohomology classification? In this paper, we study this question in the context of 2+1D fermion SPTs. We focus on the root SPT phase for the symmetry group G=Z2×Z2fG =Z_2 \times Z^f_2G=Z2​×Z2f​. Starting with an exactly solvable model for the bulk of this phase constructed by Tarantino and Fidkowski, we derive an effective 1d lattice model for the boundary. Crucially, the Hilbert space of this 1d model does not have a local tensor product structure, but rather is obtained by placing a local constraint on a local tensor product Hilbert space. We derive the action of the Z2Z_2Z2​ symmetry on this Hilbert space and find a simple 3-site Hamiltonian that respects this symmetry. We study this Hamiltonian numerically using exact diagonalization and DMRG and find strong evidence that it realizes an Ising CFT where the Z2Z_2Z2​ symmetry acts as the Kramers-Wannier duality; this is the expected stable gapless boundary state of the present SPT. A simple modification of our construction realizes the boundary of the 2+1D topological superconductor protected by time-reversal symmetry T{\cal T}T with T2=(−1)F{\cal T}^2 = (-1)^{\cal F}T2=(−1)F.
Symmetry-protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry GGG, which can all be smoothly connected to the trivial product states if we break the symmetry. It has been shown that a large class of interacting bosonic SPT phases can be systematically described by group cohomology theory. In this paper, we introduce a (special) group supercohomology theory which is a generalization of the standard group cohomology theory. We show that a large class of short-range interacting fermionic SPT phases can be described by the group supercohomology theory. Using the data of supercocycles, we can obtain the ideal ground state wave function for the corresponding fermionic SPT phase. We can also obtain the bulk Hamiltonian that realizes the SPT phase, as well as the anomalous (i.e., non-onsite) symmetry for the boundary effective Hamiltonian. The anomalous symmetry on the boundary implies that the symmetric boundary must be gapless for (1+1)-dimensional [(1+1)D] boundary, and must be gapless or topologically ordered beyond (1+1)D. As an application of this general result, we construct a new SPT phase in three dimensions, for interacting fermionic superconductors with coplanar spin order (which have T2=1{T}^{2}=1T2=1 time-reversal Z2T{Z}_{2}^{T}Z2T​ and fermion-number-parity Z2f{Z}_{2}^{f}Z2f​ symmetries described by a full symmetry group ${Z}_{2}^{T}\ifmmode\times\else\texttimes\fi{}{Z}_{2}^{f}$). Such a fermionic SPT state can neither be realized by free fermions nor by interacting bosons (formed by fermion pairs), and thus are not included in the KKK-theory classification for free fermions or group cohomology description for interacting bosons. We also construct three interacting fermionic SPT phases in two dimensions (2D) with a full symmetry group ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}^{f}$. Those 2D fermionic SPT phases all have central-charge c=1c=1c=1 gapless edge excitations, if the symmetry is not broken.
Symmetry protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry GGG. They can all be smoothly connected to the same trivial product state if we break the symmetry. The Haldane phase of spin-1 chain is the first example of SPT phases which is protected by SO(3)\mathit{SO}(3)SO(3) spin rotation symmetry. The topological insulator is another example of SPT phases which are protected by U(1)U(1)U(1) and time-reversal symmetries. In this paper, we show that interacting bosonic SPT phases can be systematically described by group cohomology theory: Distinct ddd-dimensional bosonic SPT phases with on-site symmetry GGG (which may contain antiunitary time-reversal symmetry) can be labeled by the elements in H1+d[G,UT(1)]{\mathcal{H}}^{1+d}[G,{U}_{T}(1)]H1+d[G,UT​(1)], the Borel (1+d)(1+d)(1+d)-group-cohomology classes of GGG over the GGG module UT(1){U}_{T}(1)UT​(1). Our theory, which leads to explicit ground-state wave functions and commuting projector Hamiltonians, is based on a new type of topological term that generalizes the topological $\ensuremath{\theta}$ term in continuous nonlinear $\ensuremath{\sigma}$ models to lattice nonlinear $\ensuremath{\sigma}$ models. The boundary excitations of the nontrivial SPT phases are described by lattice nonlinear $\ensuremath{\sigma}$ models with a nonlocal Lagrangian term that generalizes the Wess-Zumino-Witten term for continuous nonlinear $\ensuremath{\sigma}$ models. As a result, the symmetry GGG must be realized as a non-on-site symmetry for the low-energy boundary excitations, and those boundary states must be gapless or degenerate. As an application of our result, we can use ${\mathcal{H}}^{1+d}[U(1)\ensuremath{\rtimes}{Z}_{2}^{T},{U}_{T}(1)]$ to obtain interacting bosonic topological insulators (protected by time reversal Z2T{Z}_{2}^{T}Z2T​ and boson number conservation), which contain one nontrivial phase in one-dimensional (1D) or 2D and three in 3D. We also obtain interacting bosonic topological superconductors (protected by time-reversal symmetry only), in term of H1+d[Z2T,UT(1)]{\mathcal{H}}^{1+d}[{Z}_{2}^{T},{U}_{T}(1)]H1+d[Z2T​,UT​(1)], which contain one nontrivial phase in odd spatial dimensions and none for even dimensions. Our result is much more general than the above two examples, since it is for any symmetry group. For example, we can use ${\mathcal{H}}^{1+d}[U(1)\ifmmode\times\else\texttimes\fi{}{Z}_{2}^{T},{U}_{T}(1)]$ to construct the SPT phases of integer spin systems with time-reversal and U(1)U(1)U(1) spin rotation symmetry, which contain three nontrivial SPT phases in 1D, none in 2D, and seven in 3D. Even more generally, we find that the different bosonic symmetry breaking short-range-entangled phases are labeled by the following three mathematical objects: $({G}_{H},{G}_{\ensuremath{\Psi}},{\mathcal{H}}^{1+d}[{G}_{\ensuremath{\Psi}},{U}_{T}(1)])$, where GH{G}_{H}GH​ is the symmetry group of the Hamiltonian and ${G}_{\ensuremath{\Psi}}$ the symmetry group of the ground states.
We construct an exactly sovable commuting projector Hamiltonian for (2+1)D bosonic topological insulator which is one of symmetry-protected topological (SPT) phases protected by U(1) and time-reversal Z2T\mathbb{Z}_2^TZ2T​ symmetry, where the symmetry group is U(1)⋊Z2T\rtimes\mathbb{Z}_2^T⋊Z2T​. The model construction is based on the decorated domain-wall interpretation of the E∞E_{\infty}E∞​-page of a spectral sequence of a cobordism group that classifies the SPT phases in question. We demonstrate nontriviality of the model by showing an emergence of a Kramers doublet when the system is put on a semi-infinite cylinder (−∞,0]×S1(-\infty,0]\times S^1(−∞,0]×S1 with an inserted πππ-flux. The surface anomaly manifests itself as a non-onsite representation of the U(1)⋊Z2T\rtimes\mathbb{Z}_2^T⋊Z2T​ symmetry. Anomaly matching on a boundary is discussed within a simple boundary theory.
We propose a diagnostic tool for detecting nontrivial symmetry-protected topological (SPT) phases protected by a symmetry group GGG in 2+12+12+1 dimensions. Our method is based on directly studying the 1+11+11+1-dimensional anomalous edge conformal field theory (CFT) of SPT phases. We claim that if the CFT is the edge theory of an SPT phase, then there must be an obstruction to cutting it open. This obstruction manifests as the in-existence of boundary states in the CFT that preserves both the conformal symmetry and the global symmetry GGG. We discuss the relation between edgeability, i.e., the ability to find a consistent boundary state, and gappability, i.e., the ability to gap out a CFT, in the presence of GGG. We study several cases including time-reversal symmetric topological insulators, ZN{\mathbb{Z}}_{N}ZN​ symmetric bosonic SPT phases, and ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetric topological superconductors.
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.