We have applied the recently developed theory of topological Uhlmann numbers to a representative model of a topological insulator in two dimensions, the Qi-Wu-Zhang model. We have found a stable symmetry-protected topological (SPT) phase under external thermal fluctuations in two-dimensions. A complete phase diagram for this model is computed as a function of temperature and coupling constants in the original Hamiltonian. It shows the appearance of large stable phases of matter with topological properties compatible with thermal fluctuations or external noise and the existence of critical lines separating abruptly trivial phases from topological phases. These novel critical temperatures represent thermal topological phase transitions. The initial part of the paper comprises a self-contained explanation of the Uhlmann geometric phase needed to understand the topological properties that it may acquire when applied to topological insulators and superconductors.
Edge theories of symmetry-protected topological phases are well-known to possess global symmetry anomalies. In this work we focus on two-dimensional bosonic phases protected by an on-site symmetry and analyse 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 compu…
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 2D interacting spin system with nontrivial on-site $Z_2$ symmetry protected topological order. The order is nontrivial because we can prove that the 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 …
Symmetry protected topological phases (SPTs) have universal degeneracies in the entanglement spectrum in one dimension (1D). Here, we formulate this phenomenon in the framework of symmetry-resolved entanglement (SRE) using cohomology theory. We develop a general approach to compute entanglement measures of SPTs in any dimension and specifically SRE via a discrete path integral on multi-sheet Riemann surfaces with generalized defects. The resulting path integral is expressed in terms of group cocycles describing the topological actions of SPTs. Their cohomology classification allows to identify…
One-dimensional valence bond solid (VBS) states represent the simplest symmetry protected topological phases. We show that their ground state entanglement spectrum contains both topological and non-topological structures. For the $SO(3)$ symmetric VBS states with odd-integer spins, the two-fold topological degeneracy is associated with an underlying $Z_{2}\times Z_{2}$ symmetry that protects the corresponding topological phase. In general, for the $SO(2S+1)$ symmetric VBS states with integer spins $S$, the corresponding protecting symmetry is identified as the $(Z_{2}\times Z_{2})^{S}$ symmetr…
Measurement-based quantum computation is a model for quantum information processing utilizing local measurements on suitably entangled resource states for the implementation of quantum gates. A complete characterization for universal resource states is still missing. It has been shown that symmetry-protected topological order in one dimension can be exploited for the protection of certain quantum gates in measurement-based quantum computation. In this paper we show that the two-dimensional plaquette states on arbitrary lattices exhibit nontrivial symmetry-protected topological order in terms o…
Gutzwiller projection is a way to construct many-body wave functions that could carry topological order or symmetry protected topological (SPT) order. However, an important issue is to determine whether or not a given Gutzwiller-projected wave functions (GWF) carries a non-trivial SPT order, and which SPT order is carried by the wavefunction. In this paper, we numerically study the SPT order in a spin $S = 1$ GWF on the Kagome lattice. Using the standard Monte Carlo method, we directly confirm that the GWF has (1) gapped bulk with short-range correlations, (2) a trivial topological order via n…
We study symmetry-protected topological (SPT) phases of matter in 2D protected by symmetries acting on fractal subsystems of a certain type. Despite the total symmetry group of such systems being subextensively large, we show that only a small number of phases are actually realizable by local Hamiltonians. Which phases are possible depends crucially on the spatial structure of the symmetries, and we show that in many cases no non-trivial SPT phases are possible at all. In cases where non-trivial SPT phases do exist, we give an exhaustive enumeration of them in terms of their locality.
We present systematic constructions of tensor-network wavefunctions for bosonic symmetry protected topological (SPT) phases respecting both onsite and spatial symmetries. From the classification point of view, our results show that in spatial dimensions $d=1,2,3$, the cohomological bosonic SPT phases protected by a general symmetry group $SG$ involving onsite and spatial symmetries are classified by the cohomology group $H^{d+1}(SG,U(1))$, in which both the time-reversal symmetry and mirror reflection symmetries should be treated as anti-unitary operations. In addition, for every SPT phase pro…
We discuss the symmetry-protected topological (SPT) orders for bosonic systems from an information-theoretic viewpoint. We show that with a proper choice of the onsite basis, the degenerate ground-state space of SPT orders (on a manifold with boundary) is a quantum error-correcting code with macroscopic classical distance, hence is stable against any local bit-flip errors. We show that this error-correcting property of the SPT orders has a natural connection to that of the symmetry-breaking orders, whose degenerate ground-state space is a classical error-correcting code with a macroscopic dist…
Symmetry-protected topological (SPT) phases exhibit nontrivial order if symmetry is respected but are adiabatically connected to the trivial product phase if symmetry is not respected. However, unlike the symmetry-breaking phase, there is no local order parameter for SPT phases. Here we employ a tensor-network method to compute the topological invariants characterized by the simulated modular $S$ and $T$ matrices to study transitions in a few families of two-dimensional (2D) wavefunctions which are $\mathbb{Z}_N$ ($N=2\, \&3$) symmetric. We find that in addition to the topologically ordered ph…
Measurement-based quantum computation describes a scheme where entanglement of resource states is utilized to simulate arbitrary quantum gates via local measurements. Recent works suggest that symmetry-protected topologically non-trivial, short-ranged entanged states are promising candidates for such a resource. Miller and Miyake [NPJ Quantum Information 2, 16036 (2016)] recently constructed a particular $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry-protected topological state on the Union-Jack lattice and established its quantum computational universality. However, they sugg…
Identifying topological properties is a major challenge because, by definition, topological states do not have a local order parameter. While a generic solution to this challenge is not available yet, a broad class of topological states, namely symmetry-protected topological (SPT) states, can be identified by distinctive degeneracies in their entanglement spectrum. Here, we propose and realize two complementary protocols to probe these degeneracies based on, respectively, symmetry-resolved entanglement entropies and measurement-based computational algorithms. The two protocols link quantum inf…
We systematically investigate the robustness of symmetry protected topological (SPT) order in open quantum systems by studying the evolution of string order parameters and other probes under noisy channels. We find that one-dimensional SPT order is robust against noisy couplings to the environment that satisfy a strong symmetry condition, while it is destabilized by noise that satisfies only a weak symmetry condition, which generalizes the notion of symmetry for closed systems. We also discuss "transmutation" of SPT phases into other SPT phases of equal or lesser complexity, under noisy channe…
Symmetry fractionalization, generating a large amount of symmetry-protected topological phases, provides scenarios for continuous phase transitions different from spontaneous symmetry breaking. However, it is hard to detect these symmetry-protected topological phase transitions experimentally. Motivated by the recent development of highly tunable ultracold polar molecules, we show that the setup in a zigzag optical lattice of this system provides a perfect platform to realize symmetry-protected topological phase transitions. By using infinite time-evolving block decimation, we obtain the phase…
Symmetry-protected topological (SPT) states are short-range entangled states with a symmetry G. They belong to a new class of quantum states of matter which are classified by the group cohomology $H^{d+1}(G,\mathbb{R}/\mathbb{Z})$ in d-dimensional space. In this paper, we propose a class of symmetry- protected topological invariants that may allow us to fully characterize SPT states with a symmetry group G (ie allow us to measure the cocycles in $H^{d+1}(G,\mathbb{R}/\mathbb{Z})$ that characterize the SPT states). We give an explicit and detailed construction of symmetry-protected topological …
We argue that symmetry-broken phases proximate in phase space to symmetry-protected topological phases can exhibit dynamical signatures of topological physics. This dynamical, symmetry-protected "topological" regime is characterized by anomalously long edge coherence times due to the topological decoration of quasiparticle excitations, even if the underlying zero-temperature ground state is in a non-topological, symmetry-broken state. The dramatic enhancement of coherence can even persist at infinite temperature due to prethermalization. We find exponentially long edge coherence times that are…
We introduce the concepts of a symmetry-protected sign problem and symmetry-protected magic to study the complexity of symmetry-protected topological (SPT) phases of matter. In particular, we say a state has a symmetry-protected sign problem or symmetry-protected magic, if finite-depth quantum circuits composed of symmetric gates are unable to transform the state into a non-negative real wave function or stabilizer state, respectively. We prove that states belonging to certain SPT phases have these properties, as a result of their anomalous symmetry action at a boundary. For example, we find t…
Higher-order topological phases protected by noninvertible and subsystem symmetries
C. N. Yang Institute for Theoretical Physics, State University of New York at Stony Brook（石溪州立大学杨振宁理论物理研究所）；Department of Physics and Astronomy, State University of New York at Stony Brook（石溪州立大学物理与天文系）；Center for Quantum Phenomena, Department of Physics, New York University（纽约大学量子现象中心）
Applying the results of Ref.[1], we carry out the non-abelian bosonization for a class of free fermion symmetry-protected topological states (SPTs). The resulting boson theories are non-linear sigma models with topological $θ$ term, thus unifying the field theory description of the (free) fermionic and bosonic SPTs. Tuning $θ$ from $0$ to $2π$ triggers the topological trivial to non-trivial phase transitions. In addition, applying the same idea to the symmetry-protected critical spin liquids, we obtain the NL$\s$ models for spin SPTs living in one spatial dimension higher.
In this paper we systematically study a simple class of translation-symmetry protected topological orders in quantum spin systems using slave-particle approach. The spin systems on square lattice are translation invariant, but may break any other symmetries. We consider topologically ordered ground states that do not spontaneously break any symmetry. Those states can be described by Z2A or Z2B projective symmetry group. We find that the Z2A translation symmetric topological orders can still be divided into 16 sub-classes corresponding to 16 new translation-symmetry protected topological orders…
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 3rd cohomology class that characterizes the order. The proposed measurement is validated numerically usi…
We study the topological properties of an antiferromagnetic phase with reflection symmetry in three-dimensional heavy-fermion systems. We here propose a reflection-symmetric topological state in the three-dimensional antiferromagnetic phase and demonstrate how the paramagnetic phase changes into the antiferromagnetic topological phase of f -electron materials such as SmB6.
Symmetry-protected topological (SPT) phases are short-range entangled phases of matter with a non-local order parameter which are preserved under a local symmetry group. Here, by using unsupervised learning algorithm, namely the diffusion maps, we demonstrate that can differentiate between symmetry broken phases and topologically ordered phases, and between non-trivial topological phases in different classes. In particular, we show that the phase transitions associated with these phases can be detected in different bosonic and fermionic models in one dimension. This includes the interacting SS…
Symmetry protected topological phase is one type of nontrivial quantum disordered many-body state of matter. In this work we study one class of symmetry protected topological phases in two dimensional space, with both PSU(N) and time reversal symmetry. These states can be described by a principal chiral model with a topological Theta-term. As long as the time-reversal symmetry and PSU(N) symmetry are both preserved, the 1+1 dimensional boundary of this system must be either gapless or degenerate. We will also construct a wave function of a spin-1 system on the honeycomb lattice, which is a can…
Strong interactions can give rise to new fermionic symmetry protected topological phases which have no analogs in free fermion systems. As an example, we have systematically studied a spinless fermion model with $U(1)$ charge conservation and time reversal symmetry on a three-leg ladder using density-matrix renormalization group. In the non-interacting limit, there are no topological phases. Turning on interactions, we found two gapped phases. One is trivial and is adiabatically connected to a band insulator, while another one is a nontrivial symmetry protected topological phase resulting from…
The study of continuous phase transitions triggered by spontaneous symmetry breaking has brought revolutionary ideas to physics. Recently, through the discovery of symmetry protected topological phases, it is realized that continuous quantum phase transition can also occur between states with the same symmetry but different topology. Here we study a specific class of such phase transitions in 1+1 dimensions -- the phase transition between bosonic topological phases protected by $Z_n\times Z_n$. We find in all cases the critical point possesses two gap opening relevant operators: one leads to a…
A 3d symmetry protected topological phase, by definition must have symmetry protected nontrivial boundary states, namely its 2d boundary must be either gapless or degenerate. In this work we demonstrate that once we couple a 3d SPT phase to a lattice dynamical Z2 gauge field, in many cases the Z2 vison loop excitation (line defect) can be viewed as a "1d boundary" of the 3d SPT phase, and this line defect is guaranteed to have gapless or degenerate spectrum, which is also protected by the symmetry of the SPT phase.
We present a fully many-body formulation of topological invariants for various topological phases of fermions protected by antiunitary symmetry, which does not refer to single particle wave functions. For example, we construct the many-body $\mathbb{Z}_2$ topological invariant for time-reversal symmetric topological insulators in two spatial dimensions, which is a many-body counterpart of the Kane-Mele $\mathbb{Z}_2$ invariant written in terms of single-particle Bloch wave functions. We show that an important ingredient for the construction of the many-body topological invariants is a fermioni…
Floquet symmetry protected topological (FSPT) phases are non-equilibrium topological phases enabled by time-periodic driving. FSPT phases of 1d chains of bosons, spins, or qubits host dynamically protected edge states that can store quantum information without decoherence, making them promising for use as quantum memories. While FSPT order cannot be detected by any local measurement, here we construct non-local string order parameters that directly measure general 1d FSPT order. We propose a superconducting-qubit array based realization of the simplest Ising-FSPT, which can be implemented with…
Fusion category symmetries are finite symmetries in 1+1 dimensions described by unitary fusion categories. We classify 1+1d time-reversal invariant bosonic symmetry protected topological (SPT) phases with fusion category symmetry by using topological field theories. We first formulate two-dimensional unoriented topological field theories whose symmetry splits into time-reversal symmetry and fusion category symmetry. We then solve them to show that SPT phases are classified by equivalence classes of quintuples $(Z, M, i, s, ϕ)$ where $(Z, M, i)$ is a fiber functor, $s$ is a sign, and $ϕ$ is the…
In two-dimensional topological phases, quasiparticle excitations can carry fractional symmetry quantum numbers. We generalize this notion of symmetry fractionalization to three-dimensional topological phases, in particular to loop excitations, and propose a partial classification for symmetry-enriched $\mathbb{Z}_2$ toric code phase. We apply the results to the classification of fermionic symmetry-protected topological phases in three dimensions.
We construct a 2D quantum spin model that realizes an Ising paramagnet with gapless edge modes protected by Ising symmetry. This model provides an example of a "symmetry-protected topological phase." We describe a simple physical construction that distinguishes this system from a conventional paramagnet: we couple the system to a Z_2 gauge field and then show that the π-flux excitations have different braiding statistics from that of a usual paramagnet. In addition, we show that these braiding statistics directly imply the existence of protected edge modes. Finally, we analyze a particular mic…
We classify subsystem symmetry-protected topological (SSPT) phases in $3+1$D protected by planar subsystem symmetries, which are dual to abelian fracton topological orders. We distinguish between weak SSPTs, which can be constructed by stacking $2+1$D SPTs, and strong SSPTs, which cannot. We identify signatures of strong phases, and show by explicit construction that such phases exist. A classification of strong phases is presented for an arbitrary finite abelian group. Finally, we show that fracton orders realizable via $p$-string condensation are dual to weak SSPTs, while strong SSPTs do not…
Symmetry-Protected Topological Phases in the Triangular Majorana-Hubbard Ladder
Western Washington University（西华盛顿大学）；Stewart Blusson Quantum Matter Institute, University of British Columbia（不列颠哥伦比亚大学斯图尔特·布拉斯量子物质研究所）；Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）
Symmetry protected topological (SPT) phases are gapped quantum phases which host symmetry-protected gapless edge excitations. On the other hand, the edge states can be gapped by spontaneously breaking symmetry. We show that topological defects on the symmetry-broken edge cannot proliferate due to their fractional statistics. A gapped symmetric boundary, however, can be achieved between an SPT phase and certain fractionalized phases by condensing the bound state of a topological defect and an anyon. We demonstrate this by two examples in two dimensions: an exactly solvable model for the boundar…
In this paper, we investigate topological properties of the ground state of the SU(N) Heisenberg chain, which is argued to be relevant to the Mott-insulating phase of alkaline-earth cold fermions in a one-dimensional optical lattice. By calculating the entanglement spectrum, we show that the ground state is in one of the topological phases protected by SU(N) symmetry. We then discuss an alternative characterization of it with non-local string order parameters. We also consider how the reduction of the protecting symmetry affects the topological phase paying particular attention to the entangle…
We study a two-dimensional fermionic square lattice, which supports the existence of two-dimensional Weyl semimetal, quantum anomalous Hall effect, and $2π$-flux topological semimetal in different parameter ranges. We show that the band degenerate points of the two-dimensional Weyl semimetal and $2π$-flux topological semimetal are protected by two distinct novel hidden symmetries, which both corresponds to antiunitary composite operations. When these hidden symmetries are broken, a gap opens between the conduction and valence bands, turning the system into a insulator. With appropriate paramet…
We study two-dimensional bosonic symmetry protected topological (SPT) phases which are protected by reflection symmetry and local symmetry [$Z_N\rtimes R$, $Z_N\times R$, U(1)$\rtimes R$, or U(1)$\times R$], in the search for two-dimensional bosonic analogs of topological crystalline insulators in integer-$S$ spin systems with reflection and spin-rotation symmetries. To classify them, we employ a Chern-Simons approach and examine the stability of edge states against perturbations that preserve the assumed symmetries. We find that SPT phases protected by $Z_N\rtimes R$ symmetry are classified a…
It has recently been demonstrated that protected supersymmetry emerges on the boundaries of one-dimensional intrinsically fermionic symmetry protected trivial (SPT) phases. Here we investigate the boundary supersymmetry of one-dimensional fermionic phases beyond SPT phases. Using the connection between Majorana edge modes and real supercharges, we compute, in terms of the bulk phase invariants, the number of protected boundary supercharges.
Symmetry protected topological (SPT) states are bulk gapped states with gapless edge excitations protected by certain symmetries. The SPT phases in free fermion systems, like topological insulators, can be classified by the K-theory. However, it is not known what SPT phases exist in general interacting systems. In this paper, we present a systematic way to construct SPT phases in interacting bosonic systems, which allows us to identify many new SPT phases, including three bosonic versions of topological insulators in three dimension and one in two dimension protected by particle number conserv…
We construct a series of bosonic symmetry protected topological (BSPT) states beyond group cohomology classification using "decorated defects" approach. This construction is based on topological defects of ordinary Landau order parameters, decorated with the bosonic short range entangled (BSRE) states in $(4k+3)d$ and $(4k+5)d$ space-time (with $k$ being nonnegative integers), which do not need any symmetry. This approach not only gives these BSPT states an intuitive physical picture, it also allows us to derive the effective field theory for all these BSPT states beyond group cohomology.
Anfuso and Rosch [Phys. Rev. B 75, 144420 (2007)] showed that the "topological" Haldane phase in a fermionic spin-1/2 ladder can be continuously deformed into a "trivial" phase without explicitly breaking symmetries when local charge fluctuations are taken into account. Within the framework of symmetry protected topological phases, we revisit the model and demonstrate how the Haldane phase can be adiabatically connected to a trivial phase due to charge fluctuations. Furthermore, we show that the Haldane phase remains stable as long as the system is symmetric under particular reflection symmetr…
The second law of thermodynamics points to the existence of an `arrow of time', along which entropy only increases. This arises despite the time-reversal symmetry (TRS) of the microscopic laws of nature. Within quantum theory, TRS underpins many interesting phenomena, most notably topological insulators and the Haldane phase of quantum magnets. Here, we demonstrate that such TRS-protected effects are fundamentally unstable against coupling to an environment. Irrespective of the microscopic symmetries, interactions between a quantum system and its surroundings facilitate processes which would b…
We demonstrate the existence of topologically nontrivial phase in a one-dimensional fermionic lattice system subjected to synthetic gauge fields, which is beyond the standard Altland-Zirnbauer classification of topological insulators. The topological phase can be characterized by the presence of degenerate zero-mode edge states or a quantized Berry phase of the occupied Bloch band. By analyzing symmetries of the system, we identify that the topological phase and zero-mode edge states are protected by two hidden symmetries. An extended model with hidden symmetry breaking is also studied in orde…
We study different phases of the one-dimensional bond-alternating spin-$1/2$ Heisenberg model by using the symmetry fractionalization mechanism. We employ the infinite matrix-product state representation of the ground state (through the infinite-size density matrix renormalization group algorithm) to obtain inequivalent projective representations of the (unbroken) symmetry groups of the model, which are used to identify the different phases. We find that the model exhibits trivial as well as symmetry-protected topological phases. The symmetry-protected topological phases are Haldane phases on …
We study bosonic symmetry protected topological (SPT) phases with $C_{2}$ rotational symmetry in four spatial dimensions which is not captured by the group cohomology classification. By using the topological crystal approach, we show that the topological crystalline state of this SPT phase is given by placing an $E_{8}$ state on the two dimensional rotational invariant plane, which provides a simple physical picture of this phase. Based on this understanding, we show that a variant of QED4 with charge-1 and charge-3 Dirac fermions is a field theoretical description of the three dimensional bou…
We describe recent progress in our understanding of the interplay between interactions, symmetry, and topology in states of quantum matter. We focus on a minimal generalization of the celebrated topological band insulators to interacting many particle systems, known as Symmetry Protected Topological (SPT) phases. In common with the topological band insulators these states have a bulk gap and no exotic excitations but have non-trivial surface states that are protected by symmetry. We describe the various possible such phases and their properties in three dimensional systems with realistic symme…
Quantum phase transitions out of a symmetry-protected topological (SPT) phase in (1+1) dimensions into an adjacent, topologically distinct SPT phase protected by the same symmetry or a trivial gapped phase, are typically described by a conformal field theory (CFT). At the same time, the low-lying entanglement spectrum of a gapped phase close to such a quantum critical point is known(Cho et al., arXiv:1603.04016), very generally, to be universal and described by (gapless) boundary conformal field theory. Using this connection we show that symmetry properties of the boundary conditions in bounda…
In an earlier work we developed a holographic theory for the phase transition between bosonic symmetry-protected topological (SPT) states. This paper is a continuation of it. Here we present the holographic theory for fermionic SPT phase transitions. We show that in any dimension $ d $, the critical states of fermionic SPT phase transitions has an emergent $Z_2^T$ symmetry and can be realized on the boundary of a $ d+1 $-dimensional bulk SPT with an extra $Z_2^T$ symmetry.
The study of symmetry-protected topological states in presence of electron correlations has recently aroused great interest as rich and exotic phenomena can emerge. Here, we report a concrete example by employing large-scale unbiased quantum Monte Carlo study of the Kane-Mele model with cluster charge interactions. The ground-state phase diagram for the model at half filling is established. Our simulation identifies the coexistence of a symmetry-protected topological order with a symmetry-breaking Kekul$\acute{e}$ valence bond order and shows that the spontaneous symmetry-breaking is accompani…
Symmetry-protected topological edge modes are one of the most remarkable phenomena in topological physics. Here, we formulate and quantitatively examine the effect of a quantum bath on these topological edge modes. Using the density matrix renormalization group method, we study the ground state of a composite system of spin-1 quantum chain, where the system and the bath degrees of freedom are treated on the same footing. We focus on the dependence of these edge modes on the global/partial symmetries of system-bath coupling and on the features of the quantum bath. It is shown that the time-reve…
Recently, it was realized that quantum states of matter can be classified as long-range entangled (LRE) states (i.e. with non-trivial topological order) and short-range entangled (SRE) states (\ie with trivial topological order). We can use group cohomology class ${\cal H}^d(SG,R/Z)$ to systematically describe the SRE states with a symmetry $SG$ [referred as symmetry-protected trivial (SPT) or symmetry-protected topological (SPT) states] in $d$-dimensional space-time. In this paper, we study the physical properties of those SPT states, such as the fractionalization of the quantum numbers of th…
We define and compute many-body topological invariants of interacting fermionic symmetry-protected topological phases, protected by an orientation-reversing symmetry, such as time-reversal or reflection symmetry. The topological invariants are given by partition functions obtained by a path integral on unoriented spacetime which, as we show, can be computed for a given ground state wave function by considering a non-local operation, "partial" reflection or transpose. As an application of our scheme, we study the $\mathbb{Z}_8$ and $\mathbb{Z}_{16}$ classification of topological superconductors…
It has been shown that the symmetry-protected topological (SPT) phases with finite Abelian symmetries can be described by Chern-Simons field theory. We propose a topological response theory to uniquely identify the SPT orders, which allows us to obtain a systematic scheme to classify bosonic SPT phases with any finite Abelian symmetry group. We point out that even for finite Abelian symmetry, there exist bosonic SPT phases beyond the current Chern-Simons theory framework. We also apply the theory to fermionic SPT phases with $\mathbb{Z}_m$ symmetry and find the classification of SPT phases dep…
A platform for constructing microscopic Hamiltonians describing bosonic symmetry-protected topological (SPT) states is presented. The Hamiltonians we consider are examples of frustration-free Rokhsar-Kivelson models, which are known to be in one-to-one correspondence with classical stochastic systems in the same spatial dimensionality. By exploring this classical-quantum mapping, we are able to construct a large class of microscopic models which, in a closed manifold, have a non-degenerate gapped symmetric ground state describing the universal properties of SPT states. Examples of one and two …
We explore novel topological phases realized in a superlattice system based on the Wilson-Dirac model. Our main focus is on a two-dimensional analogue of weak topological insulator phases. We find such phases as those characterized by gapless edge states that are protected by symmetry but sensitive to the orientation of the edge relative to the superlattice structure. We show that manifest and hidden reflection symmetries protect such weak topological phases, and propose bulk Z2 indices responsible for the topological protection of the edge states.
