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Xiao Gang Wen - One of the best experts on this subject based on the ideXlab platform.
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Introduction to Topological Order
Quantum Information Meets Quantum Matter, 2019Co-Authors: Bei Zeng, Xie Chen, Duan-lu Zhou, Xiao Gang WenAbstract:In primary school, we were told that there are four states of matter: solid, liquid, gas, and plasma. In college, we learned that there are much more than four states of matter. For example, there are ferromagnetic states as revealed by the phenomenon of magnetization and superfluid states as defined by the phenomenon of zero-viscosity. The various phases in our colorful world are extremely rich. So it is amazing that they can be understood systematically by the symmetry-breaking theory of Landau. However, in the past 20–30 years, we discovered that there are even more interesting phases that are beyond Landau symmetry-breaking theory. In this chapter, we discuss new “Topological” phenomena, such as Topological degeneracy, that reveal the existence of those new phases—Topologically Ordered phases. Just like zero-viscosity defines the superfluid Order, the new “Topological” phenomena define the Topological Order at macroscopic level.
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gapped quantum liquids and Topological Order stochastic local transformations and emergence of unitarity
Physical Review B, 2015Co-Authors: Bei Zeng, Xiao Gang WenAbstract:In this work we present some new understanding of Topological Order, including three main aspects: (1) It was believed that classifying Topological Orders corresponds to classifying gapped quantum states. We show that such a statement is not precise. We introduce the concept of \emph{gapped quantum liquid} as a special kind of gapped quantum states that can "dissolve" any product states on additional sites. Topologically Ordered states actually correspond to gapped quantum liquids with stable ground-state degeneracy. Symmetry-breaking states for on-site symmetry are also gapped quantum liquids, but with unstable ground-state degeneracy. (2) We point out that the universality classes of generalized local unitary (gLU) transformations (without any symmetry) contain both Topologically Ordered states and symmetry-breaking states. This allows us to use a gLU invariant -- Topological entanglement entropy -- to probe the symmetry-breaking properties hidden in the exact ground state of a finite system, which does not break any symmetry. This method can probe symmetry- breaking Orders even without knowing the symmetry and the associated Order parameters. (3) The universality classes of Topological Orders and symmetry-breaking Orders can be distinguished by \emph{stochastic local (SL) transformations} (i.e.\ \emph{local invertible transformations}): small SL transformations can convert the symmetry-breaking classes to the trivial class of product states with finite probability of success, while the Topological-Order classes are stable against any small SL transformations, demonstrating a phenomenon of emergence of unitarity. This allows us to give a new definition of long-range entanglement based on SL transformations, under which only Topologically Ordered states are long-range entangled.
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Boundary Degeneracy of Topological Order
Physical Review Letters, 2015Co-Authors: Juven Wang, Xiao Gang WenAbstract:We introduce the notion of boundary degeneracy of Topologically Ordered states on a compact orientable spatial manifold with boundaries, and emphasize that it provides richer information than the bulk degeneracy. Beyond the bulk-edge correspondence, we find the ground state degeneracy of fully gapped edge states depends on boundary gapping conditions. We develop a quantitative description of different types of boundary gapping conditions by viewing them as different ways of non-fractionalized particle condensation on the boundary. Via Chern-Simons theory, this allows us to derive the ground state degeneracy formula in terms of boundary gapping conditions, which reveals more than the fusion algebra of fractionalized quasiparticles. We apply our results to Toric code and Levin-Wen string-net models. By measuring the boundary degeneracy on a cylinder, we predict Zk gauge theory and U(1)k×U(1)k non-chiral fractional quantum hall state at even integer k can be experimentally distinguished. Our work refines definitions of symmetry protected Topological Order and intrinsic Topological Order.
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gapped quantum liquids and Topological Order stochastic local transformations and emergence of unitarity
Physical Review Letters, 2015Co-Authors: Bei Zeng, Xiao Gang WenAbstract:In this work, we present some new understanding of Topological Order, including three main aspects. (1) It was believed that classifying Topological Orders corresponds to classifying gapped quantum states. We show that such a statement is not precise. We introduce the concept of gapped quantum liquid as a special kind of gapped quantum states that can “dissolve” any product states on additional sites. Topologically Ordered states actually correspond to gapped quantum liquids with stable ground-state degeneracy. Symmetry-breaking states for on-site symmetry are also gapped quantum liquids, but with unstable ground-state degeneracy. (2) We point out that the universality classes of generalized local unitary (gLU) transformations (without any symmetry) contain both Topologically Ordered states and symmetry-breaking states. This allows us to use a gLU invariant—Topological entanglement entropy—to probe the symmetry-breaking properties hidden in the exact ground state of a finite system, which does not break any symmetry. This method can probe symmetry- breaking Orders even without knowing the symmetry and the associated Order parameters. (3) The universality classes of Topological Orders and symmetry-breaking Orders can be distinguished by stochastic local (SL) transformations (i.e., local invertible transformations): small SL transformations can convert the symmetry-breaking classes to the trivial class of product states with finite probability of success, while the Topological-Order classes are stable against any small SL transformations, demonstrating a phenomenon of emergence of unitarity. This allows us to give a definition of long-range entanglement based on SL transformations, under which only Topologically Ordered states are long-range entangled.
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Models with on-site symmetry protected Topological Order in two spatial dimension
2011Co-Authors: Xie Chen, Zheng-xin Liu, Xiao Gang WenAbstract:Topological Order in one dimensional systems has been well characterized and classified. However in higher dimensions, what kind of Topological Order exist is still an open question, especially in interacting systems. In this paper, we focus on short range entangled systems and present an exactly solvable spin model on two dimensional square lattice with nontrivial on-site symmetry protected Topological Order (SPT Order). We find a close connection between SPT Order and the 3-cocycle of the on-site symmetry group, which we expect to generalize into a more complete understanding of SPT Order in any dimension. Specifically, our model has an on-site Z2 symmetry which is not broken in the ground state. The ground state is short range entangled but is inequivalent to a product state if the on-site Z2 symmetry is kept. The nontrivialness of the system shows up on the boundary where the effective degrees of freedom has a symmetry which does not take an on-site form and cannot have a gapped symmetric ground state. We show this by developing the tool of Matrix Product Unitary Operators and through its relation to the 3-cocycles of the on-site symmetry group. We use similar ideas to generalize this model to a two-dimensional fermionic system with on-site Z2 symmetry protected Topological Order.
Subir Sachdev - One of the best experts on this subject based on the ideXlab platform.
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Topological Order, emergent gauge fields, and Fermi surface reconstruction.
Reports on Progress in Physics, 2018Co-Authors: Subir SachdevAbstract:This review describes how Topological Order associated with the presence of emergent gauge fields can reconstruct Fermi surfaces of metals, even in the absence of translational symmetry breaking. We begin with an introduction to Topological Order using Wegner's quantum [Formula: see text] gauge theory on the square lattice: the Topological state is characterized by the expulsion of defects, carrying [Formula: see text] magnetic flux. The interplay between Topological Order and the breaking of global symmetry is described by the non-zero temperature statistical mechanics of classical XY models in dimension D = 3; such models also describe the zero temperature quantum phases of bosons with short-range interactions on the square lattice at integer filling. The Topological state is again characterized by the expulsion of certain defects, in a state with fluctuating symmetry-breaking Order, along with the presence of emergent gauge fields. The phase diagrams of the [Formula: see text] gauge theory and the XY models are obtained by embedding them in U(1) gauge theories, and by studying their Higgs and confining phases. These ideas are then applied to the single-band Hubbard model on the square lattice. A SU(2) gauge theory describes the fluctuations of spin-density-wave Order, and its phase diagram is presented by analogy to the XY models. We obtain a class of zero temperature metallic states with fluctuating spin-density wave Order, Topological Order associated with defect expulsion, deconfined emergent gauge fields, reconstructed Fermi surfaces (with 'chargon' or electron-like quasiparticles), but no broken symmetry. We conclude with the application of such metallic states to the pseudogap phase of the cuprates, and note the recent comparison with numerical studies of the Hubbard model and photoemission observations of the electron-doped cuprates. In a detour, we also discuss the influence of Berry phases, and how they can lead to deconfined quantum critical points: this applies to bosons on the square lattice at half-integer filling, and to quantum dimer models.
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Topological Order in the pseudogap metal.
Proceedings of the National Academy of Sciences of the United States of America, 2018Co-Authors: Mathias S Scheurer, Shubhayu Chatterjee, Michel Ferrero, Antoine Georges, Subir SachdevAbstract:We compute the electronic Green's function of the Topologically Ordered Higgs phase of a SU(2) gauge theory of fluctuating antiferromagnetism on the square lattice. The results are compared with cluster extensions of dynamical mean field theory, and quantum Monte Carlo calculations, on the pseudogap phase of the strongly interacting hole-doped Hubbard model. Good agreement is found in the momentum, frequency, hopping, and doping dependencies of the spectral function and electronic self-energy. We show that lines of (approximate) zeros of the zero-frequency electronic Green's function are signs of the underlying Topological Order of the gauge theory and describe how these lines of zeros appear in our theory of the Hubbard model. We also derive a modified, nonperturbative version of the Luttinger theorem that holds in the Higgs phase.
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Topological Order and Fermi surface reconstruction
2018Co-Authors: Subir SachdevAbstract:This review describes how Topological Order can reconstruct Fermi surfaces of metals, even in the absence of translational symmetry breaking. We begin with an introduction to Topological Order using Wegner's quantum $\mathbb{Z}_2$ gauge theory on the square lattice: the Topological state is characterized by the expulsion of defects, carrying $\mathbb{Z}_2$ magnetic flux. The interplay between Topological Order and the breaking of global symmetry is described by the non-zero temperature statistical mechanics of classical XY models in dimension $D=3$; such models also describe the zero temperature quantum phases of bosons with short-range interactions on the square lattice at integer filling. The Topological state is again characterized by the expulsion of certain defects, in a state with fluctuating symmetry-breaking Order. The phase diagrams of the $\mathbb{Z}_2$ gauge theory and the XY models are obtained by embedding them in U(1) gauge theories, and by studying their Higgs and confining phases. These ideas are then applied to the single-band Hubbard model on the square lattice. A SU(2) gauge theory describes the fluctuations of spin-density-wave Order, and its phase diagram is presented by analogy to the XY models. We obtain a class of zero temperature metallic states with fluctuating spin-density wave Order, Topological Order associated with defect expulsion, reconstructed Fermi surfaces (with `chargon' or electron-like quasiparticles), but no broken symmetry. We conclude with the application of such metallic states to the pseudogap phase of the cuprates, and note the recent comparison with numerical studies of the Hubbard model. In a detour, we also discuss the influence of Berry phases, and how they can lead to deconfined quantum critical points: this applies to bosons on the square lattice at half-integer filling, and to quantum dimer models.
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intertwining Topological Order and broken symmetry in a theory of fluctuating spin density waves
Physical Review Letters, 2017Co-Authors: Shubhayu Chatterjee, Subir Sachdev, Mathias S ScheurerAbstract:: The pseudogap metal phase of the hole-doped cuprate superconductors has two seemingly unrelated characteristics: a gap in the electronic spectrum in the "antinodal" region of the square lattice Brillouin zone and discrete broken symmetries. We present a SU(2) gauge theory of quantum fluctuations of magnetically Ordered states which appear in a classical theory of square lattice antiferromagnets, in a spin-density wave mean field theory of the square lattice Hubbard model, and in a CP^{1} theory of spinons. This theory leads to metals with an antinodal gap and Topological Order which intertwines with the observed broken symmetries.
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Insulators and Metals With Topological Order and Discrete Symmetry Breaking
Physical Review B, 2017Co-Authors: Shubhayu Chatterjee, Subir SachdevAbstract:Numerous experiments have reported discrete symmetry breaking in the high temperature pseudogap phase of the hole-doped cuprates, including breaking of one or more of lattice rotation, inversion, or time-reversal symmetries. In the absence of translational symmetry breaking or Topological Order, these conventional Order parameters cannot explain the gap in the charged fermion excitation spectrum in the anti-nodal region. Zhao et al. (1601.01688) and Jeong et al. (arXiv:1701.06485) have also reported inversion and time-reversal symmetry breaking in insulating Sr2IrO4 similar to that in the metallic cuprates, but co-existing with Neel Order. We extend an earlier theory of Topological Order in insulators and metals, in which the Topological Order combines naturally with the breaking of these conventional discrete symmetries. We find translationally-invariant states with Topological Order co-existing with both Ising-nematic Order and spontaneous charge currents. The link between the discrete broken symmetries and the Topological-Order-induced pseudogap explains why the broken symmetries do not survive in the confining phases without a pseudogap at large doping. Our theory also connects to the O(3) non-linear sigma model and CP1 descriptions of quantum fluctuations of the Neel Order. In this framework, the optimal doping criticality of the cuprates is primarily associated with the loss of Topological Order.
Mathias S Scheurer - One of the best experts on this subject based on the ideXlab platform.
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identifying Topological Order through unsupervised machine learning
Nature Physics, 2019Co-Authors: Joaquin F Rodrigueznieva, Mathias S ScheurerAbstract:The Landau description of phase transitions relies on the identification of a local Order parameter that indicates the onset of a symmetry-breaking phase. In contrast, Topological phase transitions evade this paradigm and, as a result, are harder to identify. Recently, machine learning techniques have been shown to be capable of characterizing Topological Order in the presence of human supervision. Here, we propose an unsupervised approach based on diffusion maps that learns Topological phase transitions from raw data without the need for manual feature engineering. Using bare spin configurations as input, the approach is shown to be capable of classifying samples of the two-dimensional XY model by winding number and capture the Berezinskii–Kosterlitz–Thouless transition. We also demonstrate the success of the approach on the Ising gauge theory, another paradigmatic model with Topological Order. In addition, a connection between the output of diffusion maps and the eigenstates of a quantum-well Hamiltonian is derived. Topological classification via diffusion maps can therefore enable fully unsupervised studies of exotic phases of matter. Machine learning techniques have latterly gained currency in condensed-matter physics, for example by identifying phase transitions. An unsupervised machine learning algorithm that identifies Topological Order is now demonstrated.
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Identifying Topological Order through unsupervised machine learning
arXiv: Statistical Mechanics, 2019Co-Authors: Joaquin F. Rodriguez-nieva, Mathias S ScheurerAbstract:The Landau description of phase transitions relies on the identification of a local Order parameter that indicates the onset of a symmetry-breaking phase. In contrast, Topological phase transitions evade this paradigm and, as a result, are harder to identify. Recently, machine learning techniques have been shown to be capable of characterizing Topological Order in the presence of human supervision. Here, we propose an unsupervised approach based on diffusion maps that learns Topological phase transitions from raw data without the need of manual feature engineering. Using bare spin configurations as input, the approach is shown to be capable of classifying samples of the two-dimensional XY model by winding number and capture the Berezinskii-Kosterlitz-Thouless transition. We also demonstrate the success of the approach on the Ising gauge theory, another paradigmatic model with Topological Order. In addition, a connection between the output of diffusion maps and the eigenstates of a quantum-well Hamiltonian is derived. Topological classification via diffusion maps can therefore enable fully unsupervised studies of exotic phases of matter.
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Topological Order in the pseudogap metal.
Proceedings of the National Academy of Sciences of the United States of America, 2018Co-Authors: Mathias S Scheurer, Shubhayu Chatterjee, Michel Ferrero, Antoine Georges, Subir SachdevAbstract:We compute the electronic Green's function of the Topologically Ordered Higgs phase of a SU(2) gauge theory of fluctuating antiferromagnetism on the square lattice. The results are compared with cluster extensions of dynamical mean field theory, and quantum Monte Carlo calculations, on the pseudogap phase of the strongly interacting hole-doped Hubbard model. Good agreement is found in the momentum, frequency, hopping, and doping dependencies of the spectral function and electronic self-energy. We show that lines of (approximate) zeros of the zero-frequency electronic Green's function are signs of the underlying Topological Order of the gauge theory and describe how these lines of zeros appear in our theory of the Hubbard model. We also derive a modified, nonperturbative version of the Luttinger theorem that holds in the Higgs phase.
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intertwining Topological Order and broken symmetry in a theory of fluctuating spin density waves
Physical Review Letters, 2017Co-Authors: Shubhayu Chatterjee, Subir Sachdev, Mathias S ScheurerAbstract:: The pseudogap metal phase of the hole-doped cuprate superconductors has two seemingly unrelated characteristics: a gap in the electronic spectrum in the "antinodal" region of the square lattice Brillouin zone and discrete broken symmetries. We present a SU(2) gauge theory of quantum fluctuations of magnetically Ordered states which appear in a classical theory of square lattice antiferromagnets, in a spin-density wave mean field theory of the square lattice Hubbard model, and in a CP^{1} theory of spinons. This theory leads to metals with an antinodal gap and Topological Order which intertwines with the observed broken symmetries.
Bei Zeng - One of the best experts on this subject based on the ideXlab platform.
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Introduction to Topological Order
Quantum Information Meets Quantum Matter, 2019Co-Authors: Bei Zeng, Xie Chen, Duan-lu Zhou, Xiao Gang WenAbstract:In primary school, we were told that there are four states of matter: solid, liquid, gas, and plasma. In college, we learned that there are much more than four states of matter. For example, there are ferromagnetic states as revealed by the phenomenon of magnetization and superfluid states as defined by the phenomenon of zero-viscosity. The various phases in our colorful world are extremely rich. So it is amazing that they can be understood systematically by the symmetry-breaking theory of Landau. However, in the past 20–30 years, we discovered that there are even more interesting phases that are beyond Landau symmetry-breaking theory. In this chapter, we discuss new “Topological” phenomena, such as Topological degeneracy, that reveal the existence of those new phases—Topologically Ordered phases. Just like zero-viscosity defines the superfluid Order, the new “Topological” phenomena define the Topological Order at macroscopic level.
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gapped quantum liquids and Topological Order stochastic local transformations and emergence of unitarity
Physical Review B, 2015Co-Authors: Bei Zeng, Xiao Gang WenAbstract:In this work we present some new understanding of Topological Order, including three main aspects: (1) It was believed that classifying Topological Orders corresponds to classifying gapped quantum states. We show that such a statement is not precise. We introduce the concept of \emph{gapped quantum liquid} as a special kind of gapped quantum states that can "dissolve" any product states on additional sites. Topologically Ordered states actually correspond to gapped quantum liquids with stable ground-state degeneracy. Symmetry-breaking states for on-site symmetry are also gapped quantum liquids, but with unstable ground-state degeneracy. (2) We point out that the universality classes of generalized local unitary (gLU) transformations (without any symmetry) contain both Topologically Ordered states and symmetry-breaking states. This allows us to use a gLU invariant -- Topological entanglement entropy -- to probe the symmetry-breaking properties hidden in the exact ground state of a finite system, which does not break any symmetry. This method can probe symmetry- breaking Orders even without knowing the symmetry and the associated Order parameters. (3) The universality classes of Topological Orders and symmetry-breaking Orders can be distinguished by \emph{stochastic local (SL) transformations} (i.e.\ \emph{local invertible transformations}): small SL transformations can convert the symmetry-breaking classes to the trivial class of product states with finite probability of success, while the Topological-Order classes are stable against any small SL transformations, demonstrating a phenomenon of emergence of unitarity. This allows us to give a new definition of long-range entanglement based on SL transformations, under which only Topologically Ordered states are long-range entangled.
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gapped quantum liquids and Topological Order stochastic local transformations and emergence of unitarity
Physical Review Letters, 2015Co-Authors: Bei Zeng, Xiao Gang WenAbstract:In this work, we present some new understanding of Topological Order, including three main aspects. (1) It was believed that classifying Topological Orders corresponds to classifying gapped quantum states. We show that such a statement is not precise. We introduce the concept of gapped quantum liquid as a special kind of gapped quantum states that can “dissolve” any product states on additional sites. Topologically Ordered states actually correspond to gapped quantum liquids with stable ground-state degeneracy. Symmetry-breaking states for on-site symmetry are also gapped quantum liquids, but with unstable ground-state degeneracy. (2) We point out that the universality classes of generalized local unitary (gLU) transformations (without any symmetry) contain both Topologically Ordered states and symmetry-breaking states. This allows us to use a gLU invariant—Topological entanglement entropy—to probe the symmetry-breaking properties hidden in the exact ground state of a finite system, which does not break any symmetry. This method can probe symmetry- breaking Orders even without knowing the symmetry and the associated Order parameters. (3) The universality classes of Topological Orders and symmetry-breaking Orders can be distinguished by stochastic local (SL) transformations (i.e., local invertible transformations): small SL transformations can convert the symmetry-breaking classes to the trivial class of product states with finite probability of success, while the Topological-Order classes are stable against any small SL transformations, demonstrating a phenomenon of emergence of unitarity. This allows us to give a definition of long-range entanglement based on SL transformations, under which only Topologically Ordered states are long-range entangled.
Ashvin Vishwanath - One of the best experts on this subject based on the ideXlab platform.
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Constraints on Topological Order in mott insulators.
Physical Review Letters, 2015Co-Authors: Michael P. Zaletel, Ashvin VishwanathAbstract:We point out certain symmetry induced constraints on Topological Order in Mott insulators (quantum magnets with an odd number of spin $\frac{1}{2}$ moments per unit cell). We show, for example, that the double-semion Topological Order is incompatible with time reversal and translation symmetry in Mott insulators. This sharpens the Hastings-Oshikawa-Lieb-Schultz-Mattis theorem for 2D quantum magnets, which guarantees that a fully symmetric gapped Mott insulator must be Topologically Ordered, but is silent about which Topological Order is permitted. Our result applies to the kagome lattice quantum antiferromagnet, where recent numerical calculations of the entanglement entropy indicate a ground state compatible with either toric code or double-semion Topological Order. Our result rules out the latter possibility.
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exactly soluble model of a three dimensional symmetry protected Topological phase of bosons with surface Topological Order
Physical Review B, 2014Co-Authors: F J Burnell, Lukasz Fidkowski, X Chen, Ashvin VishwanathAbstract:We construct an exactly soluble Hamiltonian on the D=3 cubic lattice, whose ground state is a Topological phase of bosons protected by time-reversal symmetry, i.e., a symmetry-protected Topological (SPT) phase. In this model, excitations with anyonic statistics are shown to exist at the surface but not in the bulk. The statistics of these surface anyons is explicitly computed and shown to be identical to the three-fermion Z2 model, a variant of Z2 Topological Order which cannot be realized in a purely D=2 system with time-reversal symmetry. Thus the model realizes a novel surface termination for three-dimensional (3D) SPT phases, that of a fully symmetric gapped surface with Topological Order. The 3D phase found here was previously proposed from a field theoretic analysis but is outside the group cohomology classification that appears to capture all SPT phases in lower dimensions. Such phases may potentially be realized in spin-orbit-coupled magnetic insulators, which evade magnetic Ordering. Our construction utilizes the Walker-Wang prescription to create a 3D confined phase with surface anyons, which can be extended to other Topological phases.
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Symmetry Enforced Non-Abelian Topological Order at the Surface of a Topological Insulator
Physical Review B, 2014Co-Authors: Xie Chen, Lukasz Fidkowski, Ashvin VishwanathAbstract:The surfaces of three dimensional Topological insulators (3D TIs) are generally described as Dirac metals, with a single Dirac cone. It was previously believed that a gapped surface implied breaking of either time reversal $\mathcal T$ or U(1) charge conservation symmetry. Here we discuss a novel possibility in the presence of interactions, a surface phase that preserves all symmetries but is nevertheless gapped and insulating. Then the surface must develop Topological Order of a kind that cannot be realized in a 2D system with the same symmetries. We discuss candidate surface states - non-Abelian Quantum Hall states which, when realized in 2D, have $\sigma_{xy}=1/2$ and hence break $\mathcal T$ symmetry. However, by constructing an exactly soluble 3D lattice model, we show they can be realized as $\mathcal T$ symmetric surface states. The corresponding 3D phases are confined, and have $\theta=\pi$ magnetoelectric response. Two candidate states have the same 12 particle Topological Order, the (Read-Moore) Pfaffian state with the neutral sector reversed, which we term T-Pfaffian Topological Order, but differ in their $\mathcal T$ transformation. Although we are unable to connect either of these states directly to the superconducting TI surface, we argue that one of them describes the 3D TI surface, while the other differs from it by a bosonic Topological phase. We also discuss the 24 particle Pfaffian-antisemion Topological Order (which can be connected to the superconducting TI surface) and demonstrate that it can be realized as a $\mathcal T$ symmetric surface state.
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non abelian Topological Order on the surface of a 3d Topological superconductor from an exactly solved model
arXiv: Strongly Correlated Electrons, 2013Co-Authors: Lukasz Fidkowski, X Chen, Ashvin VishwanathAbstract:Three dimensional Topological superconductors (TScs) protected by time reversal (T) symmetry are characterized by gapless Majorana cones on their surface. Free fermion phases with this symmetry (class DIII) are indexed by an integer n, of which n=1 is realized by the B-phase of superfluid Helium-3. Previously it was believed that the surface must be gapless unless time reversal symmetry is broken. Here we argue that a fully symmetric and gapped surface is possible in the presence of strong interactions, if a special type of Topological Order appears on the surface. The Topological Order realizes T symmetry in an anomalous way, one that is impossible to achieve in purely two dimensions. For odd n TScs, the surface Topological Order must be non-Abelian. We propose the simplest non-Abelian Topological Order that contains electron like excitations, SO(3)_6, with four quasiparticles, as a candidate surface state. Remarkably, this theory has a hidden T invariance which however is broken in any 2D realization. By explicitly constructing an exactly soluble Walker-Wang model we show that it can be realized at the surface of a short ranged entangled 3D fermionic phase protected by T symmetry, with bulk electrons trasforming as Kramers pairs, i.e. T^2=-1 under time reversal. We also propose an Abelian theory, the semion-fermion Topological Order, to realize an even n TSc surface, for which an explicit model is derived using a coupled layer construction. We argue that this is related to the n=2 TSc, and use this to build candidate surface Topological Orders for n=4 and n=8 TScs. The latter is equivalent to the three fermion state which is the surface Topological Order of a Z2 bosonic Topological phase protected by T invariance. One particular consequence of this is that an n=16 TSc admits a trivially gapped T-symmetric surface.
