Higher symmetries can emerge at low energies in a topologically ordered state with no symmetry, when some topological excitations have very high energy scales while other topological excitations have low energies. The low energy properties of topological orders in this limit, with the emergent higher symmetries, may be described by higher symmetry protected topological order. This motivates us, as a simplest example, to study a lattice model of $\mathbb{Z}_n$-1-symmetry protected topological (1-SPT) states in 3+1D for even $n$. We write down an exactly solvable lattice model and study its boundary transformation. On the boundary, we show the existence of anyons with non-trivial self-statistics. For the $n=2$ case, where the bulk classification is given by an integer $m$ mod $4$, we show that the boundary can be gapped with double semion topological order for $m=1$ and toric code for $m=2$. The bulk ground state wavefunction amplitude is given in terms of the linking numbers of loops in the dual lattice. Our construction can be generalized to arbitrary 1-SPT protected by finite unitary symmetry.
Topological edge states in two-dimensional $\mathbb{Z}_4$ Potts paramagnet protected by the $\mathbb{Z}_4^{\times 3}$ symmetry
A.Alikhanyan National Science Laboratory (Yerevan Physics Institute)（阿·阿里哈尼扬国家科学实验室（埃里温物理研究所））；Yerevan State University（埃里温国立大学）；Department of Physics, University of Massachusetts, Amherst（马萨诸塞大学阿默斯特分校物理系）
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…
We construct exactly solved commuting projector Hamiltonian lattice models for all known 2+1d fermionic symmetry protected topological phases (SPTs) with on-site unitary symmetry group $G_f = G \times \mathbb{Z}_2^f$, where $G$ is finite and $\mathbb{Z}_2^f$ is the fermion parity symmetry. In particular, our models transcend the class of group supercohomology models, which realize some, but not all, fermionic SPTs in 2+1d. A natural ingredient in our construction is a discrete form of the spin structure of the 2d spatial surface $M$ on which our model is defined, namely a `Kasteleyn' orientati…
We construct an exactly solvable commuting projector model for a $4+1$ dimensional ${\mathbb Z}_2$ symmetry-protected topological phase (SPT) which is outside the cohomology classification of SPTs. The model is described by a decorated domain wall construction, with "three-fermion" Walker-Wang phases on the domain walls. We describe the anomalous nature of the phase in several ways. One interesting feature is that, in contrast to in-cohomology phases, the effective ${\mathbb Z}_2$ symmetry on a $3+1$ dimensional boundary cannot be described by a quantum circuit and instead is a nontrivial quan…
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…
Majorana zero modes are a promising platform for topologically protected quantum information processing. Their non-Abelian nature, which is key for performing quantum gates, is most prominently exhibited through braiding. While originally formulated for two-dimensional (2d) systems, it has been shown that braiding can also be realized using one-dimensional (1d) wires by forming an essentially two-dimensional network. Here, we show that in driven systems far from equilibrium, one can do away with the second spatial dimension altogether by instead using quasienergy as the second dimension. To re…
It is demonstrated by explicit construction that three-dimensional bosonic crystalline symmetry protected topological (cSPT) phases are classified by $H_ϕ^{5}(G;\mathbb{Z})\oplus H_ϕ^{1}(G;\mathbb{Z})$ for all 230 space groups $G$, where $H^n_ϕ(G;\mathbb{Z})$ denotes the $n$th twisted group cohomology of $G$ with $\mathbb{Z}$ coefficients, and $ϕ$ indicates that $g\in G$ acts non-trivially on coefficients by sending them to their inverses if $g$ reverses spacetime orientation and acts trivially otherwise. The previously known summand $H_ϕ^{5}(G;\mathbb{Z})$ corresponds only to crystalline phas…
Recently, it has been found that there exist symmetry-protected topological phases of fermions, which have no realizations in non-interacting fermionic systems or bosonic models. We study the edge states of such an intrinsically interacting fermionic SPT phase in two spatial dimensions, protected by $\mathbb{Z}_4\times\mathbb{Z}_2^T$ symmetry. We model the edge Hilbert space by replacing the internal $\mathbb{Z}_4$ symmetry with a spatial translation symmetry, and design an exactly solvable Hamiltonian for the edge model. We show that at low-energy the edge can be described by a two-component …
We propose the $\mathbb{Z}_Q$ Berry phase as a topological invariant for higher-order symmetry-protected topological (HOSPT) phases for two- and three-dimensional systems. It is topologically stable for electron-electron interactions assuming the gap remains open. As a concrete example, we show that the Berry phase is quantized in $\mathbb{Z}_4$ and characterizes the HOSPT phase of the extended Benalcazar-Bernevig-Hughes (BBH) model, which contains the next-nearest neighbor hopping and the intersite Coulomb interactions. In addition, we introduce the $\mathbb{Z}_4$ Berry phase for the spin-mod…
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…
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…
Symmetry is fundamental to topological phases. In the presence of a gauge field, spatial symmetries will be projectively represented, which may alter their algebraic structure and generate novel topological phases. We show that the $\mathbb{Z}_2$ projectively represented translational symmetry operators adopt a distinct commutation relation, and become momentum dependent analogous to twofold nonsymmorphic symmetries. Combined with other internal or external symmetries, they give rise to many exotic band topology, such as the degeneracy over the whole boundary of the Brillouin zone, the single …
We report that a spin-1/2 tetrahedral Heisenberg chain realizes a gapless symmetry-protected topological (gSPT) phase characterized by the coexistence of the Tomonaga-Luttinger-liquid criticality due to chirality degrees of freedom and the symmetry-protected edge state due to spin degrees of freedom. This gSPT phase has an interesting feature that no symmetry forbids the trivial spin gap opening but a discrete symmetry, $\mathbb Z_3\times\mathbb Z_2^T$, forbids the unique gapped ground state. In the first part of the paper, we numerically show the coexistence of a critical entanglement entropy…
We derive a necessary and sufficient criterion for when a two dimensional gapped many-body system with Abelian anyons and a unitary $\mathbb{Z}_2$ symmetry has a protected gapless edge mode. Our criterion is phrased in terms of edge theories --- or more specifically, chiral boson edge theories with $\mathbb{Z}_2$ symmetry --- and it applies to any bosonic or fermionic system whose boundary can be described by such an edge theory. At an operational level, our criterion takes as input a chiral boson edge theory with $\mathbb{Z}_2$ symmetry, and then produces as output a prediction as to whether …
Using the decorated domain wall procedure, we construct Finite Depth Local Unitaries (FDLUs) that realize Fermionic Symmetry-Protected Topological (SPT) phases. This results in explicit 'full' commuting projector Hamiltonians, where 'full' implies the fact that the ground state, as well as all excited states of these Hamiltonians, realizes the nontrivial SPT phase. We begin by constructing explicit examples of 1+1D phases protected by symmetry groups $G=\mathbb Z_2^T \times \mathbb Z_2^F$ , which also has a free fermion realization in class BDI, and $G=\mathbb Z_4 \times \mathbb Z_4^F$, which …
It is demonstrated that fermionic/bosonic symmetry-protected topological (SPT) phases across different dimensions and symmetry classes can be organized using geometric constructions that increase dimensions and symmetry-forgetting maps that change symmetry groups. Specifically, it is shown that the interacting classifications of SPT phases with and without glide symmetry fit into a short exact sequence, so that the classification with glide is constrained to be a direct sum of cyclic groups of order 2 or 4. Applied to fermionic SPT phases in the Wigner-Dyson class AII, this implies that the co…
We propose a diagnostic tool for detecting non-trivial symmetry protected topological (SPT) phases protected by a symmetry group $G$ in 2+1 dimensions. Our method is based on directly studying the 1+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 in the in-existence of boundary states that preserves both the conformal symmetry and the global symmetry $G$. We discuss the relation between edgeability, the ability to find a consiste…
Classification and construction of symmetry protected topological (SPT) phases in interacting boson and fermion systems have become a fascinating theoretical direction in recent years. It has been shown that the (generalized) group cohomology theory or cobordism theory can give rise to a complete classification of SPT phases in interacting boson/spin systems. Nevertheless, the construction and classification of SPT phases in interacting fermion systems are much more complicated, especially in 3D. In this work, we revisit this problem based on the equivalent class of fermionic symmetric local u…
We show how angular momentum conservation can stabilise a symmetry-protected quasi-topological phase of matter supporting Majorana quasi-particles as edge modes in one-dimensional cold atom gases. We investigate a number-conserving four-species Hubbard model in the presence of spin-orbit coupling. The latter reduces the global spin symmetry to an angular momentum parity symmetry, which provides an extremely robust protection mechanism that does not rely on any coupling to additional reservoirs. The emergence of Majorana edge modes is elucidated using field theory techniques, and corroborated b…
Quantum anomalies, breakdown of classical symmetries by quantum effects, provide a sharp definition of symmetry protected topological phases. In particular, they can diagnose interaction effects on the non-interacting classification of fermionic symmetry protected topological phases. In this paper, we identify quantum anomalies in two kinds of (3+1)-dimensional fermionic symmetry protected topological phases: (i) topological insulators protected by CP (charge conjugation $\times$ reflection) and electromagnetic $\mathrm{U}(1)$ symmetries, and (ii) topological superconductors protected by refle…
From symmetry-protected topological order to Landau order
针对离散对称群Z_N x Z_N的特殊情况，建立了一维自旋系统的对称保护拓扑相与对称破缺相之间的映射.它是实现在一个非局部酉变换，保持局部的哈密顿量。我们推导出涉及的各个阶段的映射的图像，包括对称性破缺和拓扑保护的混合物。我们的分析也适用于任意连续对称的酉，正交和辛型自旋系统的拓扑相位。这是通过在所有这些群中识别合适的子群Z_N x Z_N，以及各个投射表示类之间的双射来实现的。
Whether self correcting quantum memories can exist at non-zero temperature in a physically reasonable setting remains a great open problem. It has recently been argued [1] that symmetry protected topological (SPT) systems in three space dimensions subject to a strong constraint -- that the quantum dynamics respect a 1-form symmetry -- realize such a quantum memory. We illustrate how this works in Walker-Wang codes, which provide a specific realization of these desiderata. In this setting we show that it is sufficient for the 1-form symmetry to be enforced on a sub-volume of the system which is…
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…
We classify and construct models for two-dimensional (2D) interacting fermionic symmetry-protected topological (FSPT) phases with general finite Abelian unitary symmetry $G_f$. To obtain the classification, we couple the FSPT system to a dynamical discrete gauge field with gauge group $G_f$ and study braiding statistics in the resulting gauge theory. Under reasonable assumptions, the braiding statistics data allows us to infer a potentially complete classification of 2D FSPT phases with Abelian symmetry. The FSPT models that we construct are simple stacks of the following two kinds of existing…
The classification and lattice model construction of symmetry protected topological (SPT) phases in interacting fermion systems are very interesting but challenging. In this paper, we give a systematic fixed point wave function construction of fermionic SPT (FSPT) states for generic fermionic symmetry group $G_f=\mathbb{Z}_2^f \times_{ω_2} G_b$ which is a central extension of bosonic symmetry group $G_b$ (may contain time reversal symmetry) by the fermion parity symmetry group $\mathbb{Z}_2^f = \{1,P_f\}$. Our construction is based on the concept of equivalence class of finite depth fermionic …
We present a series of models of three-dimensional rotation-symmetric fragile topological insulators in class AI (time-reversal symmetric and spin-orbit-free systems), which have gapless surface states protected by time-reversal ($T$) and $n$-fold rotation ($C_n$) symmetries ($n=2,4,6$). Our models are generalizations of Fu's model of a spinless topological crystalline insulator, in which orbital degrees of freedom play the role of pseudo-spins. We consider minimal surface Hamiltonian with $C_n$ symmetry in class AI and discuss possible symmetry-protected gapless surface states, i.e., a quadra…
Abelian Chern-Simons theory, characterized by the so-called $K$ matrix, has been quite successful in characterizing and classifying Abelian fractional quantum hall effect (FQHE) as well as symmetry protected topological (SPT) phases, especially for bosonic SPT phases. However, there are still some puzzles in dealing with fermionic SPT(fSPT) phases. In this paper, we utilize the Abelian Chern-Simons theory to study the fSPT phases protected by arbitrary Abelian total symmetry $G_f$. Comparing to the bosonic SPT phases, fSPT phases with Abelian total symmetry $G_f$ has three new features: (1) it…
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…
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.
We consider compact $U^κ(1)$ gauge theory in 3+1D with the $2π$-quantized topological term ${\sum_{I, J =1}^κ\frac{K_{IJ}}{4π}\int_{M^4}F^I\wedge F^J}$. At energies below the gauge charges' gaps but above the monopoles' gaps, this field theory has an emergent ${\mathbb{Z}_{k_1}^{(1)}\times\mathbb{Z}_{k_2}^{(1)}\times\cdots}$ 1-symmetry, where $k_i$ are the diagonal elements of the Smith normal form of $K$ and $\mathbb{Z}_{0}^{(1)}$ is regarded as $U(1)^{(1)}$. In the $U^κ(1)$ confined phase, the boundary's IR properties are described by Chern-Simons field theory and has a ${\mathbb{Z}_{k_1}^{(…
Periodic driving has the longstanding reputation for generating exotic phases of matter with no static counterparts. This work explores the interplay among periodic driving, interaction effects, and $\mathbb{Z}_2$ symmetry that leads to the emergence of Floquet symmetry protected second-order topological phases in a simple but insightful two-dimensional spin-1/2 lattice. Through a combination of analytical and numerical treatments, we verify the formation of 0 and $π$ modes, i.e., corner localized $\mathbb{Z}_2$ symmetry broken operators that respectively commute and anticommute with the one-p…
Recently, it has been shown that two-dimensional bosonic symmetry-protected topological(SPT) phases with on-site unitary symmetry $G$ can be completely classified by the group cohomology class $H^3(G, \mathrm{U}(1))$. Later, group super-cohomology class was proposed as a partial classification for SPT phases of interacting fermions. In this work, we revisit this problem based on the mathematical framework of $G$-extension of unitary braided tensor category(UBTC) theory. We first reproduce the partial classifications given by group super-cohomology, then we show that with an additional $H^1(G, …
We propose the definitions of many-body topological invariants to detect symmetry-protected topological phases protected by point group symmetry, using partial point group transformations on a given short-range entangled quantum ground state. Partial point group transformations $g_D$ are defined by point group transformations restricted to a spatial subregion $D$, which is closed under the point group transformations and sufficiently larger than the bulk correlation length $ξ$. By analytical and numerical calculations,we find that the ground state expectation value of the partial point group t…
Duality viewpoint of noninvertible symmetry protected topological phases
Southern Denmark University（南丹麦大学）；Niels Bohr Institute, University of Copenhagen（哥本哈根大学尼尔斯玻尔研究所）；University of Tokyo（东京大学）；University of Ghent（根特大学）；The Pennsylvania State University（宾夕法尼亚州立大学）
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）
We study Abelian braiding statistics of loop excitations in three-dimensional (3D) gauge theories with fermionic particles and the closely related problem of classifying 3D fermionic symmetry-protected topological (FSPT) phases with unitary symmetries. It is known that the two problems are related by turning FSPT phases into gauge theories through gauging the global symmetry of the former. We show that there exist certain types of Abelian loop braiding statistics that are allowed only in the the presence of fermionic particles, which correspond to 3D "intrinsic" FSPT phases, i.e., those that d…
