arXiv subjects

Xiao-Gang Wen

Publications and source records attributed to Xiao-Gang Wen.

At least 127 records · Page 7Linked to original sources

arXiv2013

Classifying gauge anomalies through SPT orders and classifying gravitational anomalies through topological orders

In this paper, we systematically study gauge anomalies in bosonic and fermionic weak-coupling gauge theories with gauge group G (which can be continuous or discrete). We show a very close relation between gauge anomalies and symmetry-protected trivial (SPT) orders [also known as symmetry-protected topological (SPT) orders] in one-higher dimensions. Using such an idea, we argue that, in d space-time dimensions, the gauge anomalies are described by the elements in Free[H^{d+1}(G,R/Z)]\oplus H_π^{d+1}(BG,R/Z). The well known Adler-Bell-Jackiw anomalies are classified by the free part of the group cohomology class H^{d+1}(G,R/Z) of the gauge group G (denoted as Free[H^{d+1}(G,\R/\Z)]). We refer other kinds of gauge anomalies beyond Adler-Bell-Jackiw anomalies as nonABJ gauge anomalies, which include Witten SU(2) global gauge anomaly. We introduce a notion of π-cohomology group, H_π^{d+1}(BG,R/Z), for the classifying space BG, which is an Abelian group and include Tor[H^{d+1}(G,R/Z)] and topological cohomology group H^{d+1}(BG,R/Z) as subgroups. We argue that H_π^{d+1}(BG,R/Z) classifies the bosonic nonABJ gauge anomalies, and partially classifies fermionic nonABJ anomalies. Using the same approach that shows gauge anomalies to be connected to SPT phases, we can also show that gravitational anomalies are connected to topological orders (ie patterns of long-range entanglement) in one-higher dimension.

hep-th↗

arXiv2013

Superconductivity with intrinsic topological order induced by pure Coulomb interaction and time-reversal symmetry breaking

Recently, in certain flat band lattice systems at commensurate fillings, fractional quantum Hall states have been found -- which have anyonic excitations. We study such systems away from commensuration, i.e. the ground state of an anyon gas in such a system. The presence of the underlying lattice allows access to an entirely new regime where the anyon kinetic energy can be larger than their interaction energy. Within the flux-attachment approach, using mean-field then adding fluctuations, we find several possible superfluid states. Two have intrinsic topological order, i.e. fractionalized quasiparticles with a fusion structure of (Z_2)^4 and (Z_8)^2 respectively, and a third has no fractionalized excitations similar to a BCS-type state. This represents a mechanism for superconductivity driven purely by strong repulsion and complex hopping of electrons.

cond-mat.str-el↗

arXiv2013

Projective construction of two-dimensional symmetry-protected topological phases with U(1), SO(3), or SU(2) symmetries

We propose a general approach to construct symmetry protected topological (SPT) states i.e the short-range entangled states with symmetry) in 2D spin/boson systems on lattice. In our approach, we fractionalize spins/bosons into different fermions, which occupy nontrivial Chern bands. After the Gutzwiller projection of the free fermion state obtained by filling the Chern bands, we can obtain SPT states on lattice. In particular, we constructed a U(1) SPT state of a spin-1 model, a SO(3) SPT state of a boson system with spin-1 bosons and spinless bosons, and a SU(2) SPT state of a spin-1/2 boson system. By applying the "spin gauge field" which directly couples to the spin density and spin current of $S^z$ components, we also calculate the quantum spin Hall conductance in each SPT state. The projective ground states can be further studied numerically in the future by variational Monte Carlo etc.

cond-mat.str-el↗

arXiv2013

Quantized topological terms in weak-coupling gauge theories with symmetry and their connection to symmetry enriched topological phases

We study the quantized topological terms in a weak-coupling gauge theory with gauge group $G_g$ and a global symmetry $G_s$ in $d$ space-time dimensions. We show that the quantized topological terms are classified by a pair $(G,ν_d)$, where $G$ is an extension of $G_s$ by $G_g$ and $ν_d$ an element in group cohomology $\cH^d(G,\R/\Z)$. When $d=3$ and/or when $G_g$ is finite, the weak-coupling gauge theories with quantized topological terms describe gapped symmetry enriched topological (SET) phases (i.e. gapped long-range entangled phases with symmetry). Thus, those SET phases are classified by $\cH^d(G,\R/\Z)$, where $G/G_g=G_s$. We also apply our theory to a simple case $G_s=G_g=Z_2$, which leads to 12 different SET phases in 2+1D, where quasiparticles have different patterns of fractional $G_s=Z_2$ quantum numbers and fractional statistics. If the weak-coupling gauge theories are gapless, then the different quantized topological terms may describe different gapless phases of the gauge theories with a symmetry $G_s$, which may lead to different fractionalizations of $G_s$ quantum numbers and different fractional statistics (if in 2+1D).

cond-mat.str-el↗

arXiv2013

Symmetry protected Spin Quantum Hall phases in 2-Dimensions

Symmetry protected topological (SPT) states are short-range entangled states with symmetry. Nontrivial SPT states have symmetry protected gapless edge excitations. In 2-dimension (2D), there are infinite number of nontrivial SPT phases with SU(2) or SO(3) symmetry. These phases can be described by SU(2)/SO(3) nonlinear-sigma models with a quantized topological θ-term. At open boundary, the θ-term becomes the Wess-Zumino-Witten term and consequently the boundary excitations are decoupled gapless left movers and right movers. Only the left movers (if θ>0) carry the SU(2)/SO(3) quantum numbers. As a result, the SU(2) SPT phases have a half-integer quantized spin Hall conductance and the SO(3) SPT phases an even-integer quantized spin Hall conductance. Both the SU(2)/SO(3) SPT phases are symmetric under their U(1) subgroup and can be viewed as U(1) SPT phases with even-integer quantized Hall conductance.

cond-mat.str-el↗

arXiv2013

Constructing gapless spin liquid state for the spin-1/2 J1-J2 Heisenberg model on a square lattice

We construct a class of projected entangled pair states (PEPS) which is exactly the resonating valence bond (RVB) wavefunctions endowed with both short range and long range valence bonds. With an energetically preferred RVB pattern, the wavefunction is simplified to live in a one parameter variational space. We tune this variational parameter to minimize the energy for the frustrated spin 1/2 J1-J2 antiferromagnetic Heisenberg model on the square lattice. Taking a cylindrical geometry, we are able to construct four topological sectors with even or odd number of fluxes penetrating the cylinder and even or odd number of spinons on the boundary. The energy splitting in different topological sectors is exponentially small with the cylinder perimeter. We find a power law decay of the dimer correlation function on a torus, and a lnL correction to the entanglement entropy, indicating a gapless spin liquid phase at the optimum parameter.

cond-mat.str-el↗

arXiv2013

General method for finding ground state manifold of classical Heisenberg model

We investigate classical Heisenberg models with the translation symmetries of infinite crystals. We prove a spiral theorem, which states that under certain conditions there must exist spiral ground states, and propose a natural classification of all manageable models based on some "spectral properties," which are directly related to their ground state manifolds. We demonstrate how the ground state manifold can be calculated analytically for all spectra with finite number of minima and some with extensive minima, and algorithmically for the others. We also extend the method to particular anisotropic interactions.

cond-mat.stat-mech↗

arXiv2013

Symmetry protected topological orders in interacting bosonic systems

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 conservation and time reversal symmetry. Just as group theory allows us to construct 230 crystal structures in 3D, we find that group cohomology theory allows us to construct different interacting bosonic SPT phases in any dimensions and for any symmetry groups. In particular, we are going to show how topological terms in the path integral description of the system can be constructed from nontrivial group cohomology classes, giving rise to exactly soluble Hamiltonians, explicit ground state wave functions and symmetry protected gapless edge excitations.

cond-mat.str-el↗

arXiv2012

Modular transformation and bosonic/fermionic topological orders in Abelian fractional quantum Hall states

The non-Abelian geometric phases of the robust degenerate ground states were proposed as physically measurable defining properties of topological order in 1990. In this paper we discuss in detail such a quantitative characterization of topological order, using generic Abelian fractional quantum Hall states as examples. We find a way to determine even the pure U(1) phase of the non-Abelian geometric phases. We show that the non-Abelian geometric phases not only contain information about the quasi-particle statistics, they also contain information about the chiral central charge Delta c mod 24 of the edge states, as well as the Hall viscosity. Thus, the non-Abelian geometric phases (both the Abelian part and the non-Abelian part) provide a quite complete way to characterize 2D topological order.

cond-mat.str-el↗

arXiv2012

Symmetry protected topological orders and the group cohomology of their symmetry group

Symmetry protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry G. 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 phase which is protected by SO(3) spin rotation symmetry. The topological insulator is another exam- ple of SPT phase which is protected by U(1) and time reversal symmetries. It has been shown that free fermion SPT phases can be systematically described by the K-theory. In this paper, we show that interacting bosonic SPT phases can be systematically described by group cohomology theory: distinct d-dimensional bosonic SPT phases with on-site symmetry G (which may contain anti-unitary time reversal symmetry) can be labeled by the elements in H^{1+d}[G, U_T(1)] - the Borel (1 + d)-group-cohomology classes of G over the G-module U_T(1). The boundary excitations of the non-trivial SPT phases are gapless or degenerate. 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_Ψ, H^{1+d}[G_Ψ, U_T(1)], where G_H is the symmetry group of the Hamiltonian and G_Ψ the symmetry group of the ground states.

cond-mat.str-el↗

arXiv2012

Symmetry Protected Topological Phases in Spin Ladders with Two-body Interactions

Spin-1/2 two-legged ladders respecting inter-leg exchange symmetry and D2 spin rotation symmetry have new symmetry protected topological (SPT) phases which are different from the Haldane phase. Three of the new SPT phases are tx,ty,tz, which all have symmetry protected two-fold degenerate edge states on each end of the open boundaries. However, the edge states in different phases have different response to magnetic field. For example, the edge states in the tz phase will be split by the magnetic field along the z-direction, but not by the fields in the x- and y-directions. We give the Hamiltonian that realizes each SPT phase and demonstrate a proof-of-principle quantum simulation scheme for Hamiltonians of the t0 and tz phases based on coupled-QED-cavity ladder.

cond-mat.str-el↗

arXiv2012

Quantized topological terms in weakly coupled gauge theories and their connection to symmetry protected topological phases

We consider a weakly coupled gauge theory where charged particles all have large gaps (ie no Higgs condensation to break the gauge "symmetry") and the field strength fluctuates only weakly. We ask what kind of topological terms can be added to the Lagrangian of such a weakly coupled gauge theory. In this paper, we systematically construct quantized topological terms which are generalization of the Chern-Simons terms and $F\wedge F$ terms, in space-time dimensions $d$ and for any gauge groups (continuous or discrete), using each element of the topological cohomology classes $H^{d+1}(BG,\Z)$ on the classifying space $BG$ of the gauge group $G$. In 3$d$ or for finite gauge groups above 3$d$, the weakly coupled gauge theories are gapped. So our results on topological terms can be viewed as a systematic construction of gapped topologically ordered phases of weakly coupled gauge theories. In other cases, the weakly coupled gauge theories are gapless. So our results can be viewed as an attempt to systematically construct different gapless phases of weakly coupled gauge theories. Amazingly, the bosonic symmetry protected topological (SPT) phases with a finite on-site symmetry group $G$ are also classified by the same $H^{d+1}(BG,\Z)$. (SPT phases are gapped quantum phases with a symmetry and trivial topological order.) In this paper, we show an explicit duality relation between topological gauge theories with the quantized topological terms and the bosonic SPT phases, for any finite group $G$ and in any dimensions; a result first obtained by Levin and Gu. We also study the relation between topological lattice gauge theory and the string-net states with non-trivial topological order and no symmetry.

cond-mat.str-el↗

arXiv2012

Topological order: from long-range entangled quantum matter to an unification of light and electrons

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 so rich that it is amazing that they can be understood systematically by the symmetry breaking theory of Landau. In this paper, we will review the progress in last 20 -- 30 years, during which we discovered that there are even more interesting phases that are beyond Landau symmetry breaking theory. 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. As a new type of order, topological order requires a new mathematical frame work, such as fusion category and group cohomology, to describe it. More recently, we find that, at microscopical level, topological order is due to long-range quantum entanglements, just like fermion superfluid is due to fermion-pair condensation. Long-range quantum entanglements lead to many amazing emergent phenomena, such as fractional quantum numbers, fractional/non-Abelian statistics, and perfect conducting boundary channels. Long-range quantum entanglements can even provide a unified origin of light and electrons (or more generally, gauge interactions and Fermi statistics): light waves (gauge fields) are fluctuations of long-range entanglements, and electrons (fermions) are defects of long-range entanglements.

cond-mat.str-el↗

arXiv2012

Synthetic Topological Degeneracy by Anyon Condensation

Topological degeneracy is the degeneracy of the ground states in a many-body system in the large-system-size limit. Topological degeneracy cannot be lifted by any local perturbation of the Hamiltonian. The topological degeneracies on closed manifolds have been used to discover/define topological order in many-body systems, which contain excitations with fractional statistics. In this paper, we study a new type of topological degeneracy induced by condensing anyons along a line in 2D topological ordered states. Such topological degeneracy can be viewed as carried by each end of the line-defect, which is a generalization of Majorana zero-modes. The topological degeneracy can be used as a quantum memory. The ends of line-defects carry projective non-Abelian statistics, and braiding them allow us to perform fault tolerant quantum computations.

cond-mat.str-el↗

arXiv2012

Majorana Zero-modes and Topological Phases of Multi-flavored Jackiw-Rebbi model

Motivated by the recent Kitaev's K-theory analysis of topological insulators and superconductors, we adopt the same framework to study the topological phase structure of Jackiw-Rebbi model in 3+1 dimensions. According to the K-theory analysis based on the properties of the charge conjugation and time reversal symmetries, we classify the topological phases of the model. In particular, we find that there exist $\mathbf{Z}$ Majorana zero-modes hosted by the hedgehogs/t'Hooft-Polyakov monopoles, if the model has a $T^2=1$ time reversal symmetry. Guided by the K-theory results, we then explicitly show that a single Majorana zero mode solution exists for the SU(2) doublet fermions in some co-dimensional one planes of the mass parameter space. It turns out we can see the existence of none or a single zero mode when the fermion doublet is only two. We then take a step further to consider four-fermion case and find there can be zero, one or two normalizable zero mode in some particular choices of mass matrices. Our results also indicate that a single normalizable Majorana zero mode can be compatible with the cancellation of SU(2) Witten anomaly.

hep-th↗

arXiv2012

Chiral symmetry on the edge of 2D symmetry protected topological phases

Symmetry protected topological (SPT) states are short-range entangled states with symmetry, which have symmetry protected gapless edge states around a gapped bulk. Recently, we proposed a systematic construction of SPT phases in interacting bosonic systems, however it is not very clear what is the form of the low energy excitations on the gapless edge. In this paper, we answer this question for two dimensional bosonic SPT phases with Z_N and U(1) symmetry. We find that while the low energy modes of the gapless edges are non-chiral, symmetry acts on them in a chiral way, i.e. acts on the right movers and the left movers differently. This special realization of symmetry protects the gaplessness of the otherwise unstable edge states by prohibiting a direct scattering between the left and right movers. Moreover, understanding of the low energy effective theory leads to experimental predictions about the SPT phases. In particular, we find that all the 2D U(1) SPT phases have even integer quantized Hall conductance.

cond-mat.str-el↗

arXiv2012

Complete classification of 1D gapped quantum phases in interacting spin systems

Quantum phases with different orders exist with or without breaking the symmetry of the system. Recently, a classification of gapped quantum phases which do not break time reversal, parity or on-site unitary symmetry has been given for 1D spin systems in [X. Chen, Z.-C. Gu, and X.-G. Wen, Phys. Rev. B \textbf{83}, 035107 (2011); arXiv:1008.3745]. It was found that, such symmetry protected topological (SPT) phases are labeled by the projective representations of the symmetry group which can be viewed as a symmetry fractionalization. In this paper, we extend the classification of 1D gapped phases by considering SPT phases with combined time reversal, parity, and/or on-site unitary symmetries and also the possibility of symmetry breaking. We clarify how symmetry fractionalizes with combined symmetries and also how symmetry fractionalization coexists with symmetry breaking. In this way, we obtain a complete classification of gapped quantum phases in 1D spin systems. We find that in general, symmetry fractionalization, symmetry breaking and long range entanglement(present in 2 or higher dimensions) represent three main mechanisms to generate a very rich set of gapped quantum phases. As an application of our classification, we study the possible SPT phases in 1D fermionic systems, which can be mapped to spin systems by Jordan-Wigner transformation.

cond-mat.str-el↗

arXiv2012

Gutzwiller Projected wavefunctions in the fermonic theory of S=1 spin chains

We study in this paper a series of Gutzwiller Projected wavefunctions for S=1 spin chains obtained from a fermionic mean-field theory for general S>1/2 spin systems [Phys. Rev. B 81, 224417] applied to the bilinear-biquadratic (J-K) model. The free-fermion mean field states before the projection are 1D paring states. By comparing the energies and correlation functions of the projected pairing states with those obtained from known results, we show that the optimized Gutzwiller projected wavefunctions are very good trial ground state wavefunctions for the antiferromagnetic bilinear-biquadratic model in the regime K 0). We find that different topological phases of the free-fermion paring states correspond to different spin phases: the weak pairing (topologically non-trivial) state gives rise to the Haldane phase, whereas the strong pairing (topologically trivial) state gives rise to the dimer phase. In particular the mapping between the Haldane phase and Gutwziller wavefunction is exact at the AKLT point K=1/3. The transition point between the two phases determined by the optimized Gutzwiller Projected wavefunction is in good agreement with the known result. The effect of Z2 gauge fluctuations above the mean field theory is analyzed.

cond-mat.str-el↗