arXiv subjects

Xiao-Gang Wen

Publications and source records attributed to Xiao-Gang Wen.

At least 163 records · Page 9Linked to original sources

arXiv2010

Grassmann tensor network states and its renormalization for strongly correlated fermionic and bosonic states

The projective construction (the slave-particle approach) has played an very important role in understanding strongly correlated systems, such as the emergence of fermions, anyons, and gauge theory in quantum spin liquids and quantum Hall states. Recently, fermionic Projected Entangled Pair States (fPEPS) have been introduced to effciently represent many-body fermionic states. In this paper, we show that the strongly correlated bosonic/fermionic states obtained both from the projective construction and the fPEPS approach can be represented systematically as Grassmann tensor product states. This construction can also be applied to all other tensor network states approaches. The Grassmann tensor product states allow us to encode many-body bosonic/fermionic states effciently with a polynomial number of parameters. We also generalize the tensor-entanglement renormalization group (TERG) method for complex tensor networks to Grassmann tensor networks. This allows us to approximate the norm and average local operators of Grassmann tensor product states in polynomial time, and hence leads to a variational approach for describing strongly correlated bosonic/fermionic systems in higher dimensions.

cond-mat.str-el↗

arXiv2010

Emergence of helicity +/- 2 modes (gravitons) from qubit models

The general equivalence principle and the associated diffeomorphism gauge symmetry are regarded as a founding principles of nature. But, one may wonder, can diffeomorphism gauge symmetry emerge as a low energy property of certain topological/quantum order in a qbit model that has no diffeomorphism gauge symmetry? In this paper, we showed that, at least, the linearized diffeomorphism gauge symmetry h_{μν}\to h_{μν} +\prt_μf_ν+\prt_νf_μcan indeed emerge from some qbit models (or quantum spin models). Physically, the emergence of the (linearized) diffeomorphism gauge symmetry implies the emergence of gapless helicity +/- 2 excitations (ie the gravitons). In the first qbit model (called the L-type model), we show that helicity +/- 2 gapless excitations appear as the only type of low energy excitations using a reliable semiclassical approach. The dispersion of the gapless helicity +/- 2 is found to be \eps_k \propto k^3. The appearance of the gapless helicity +/- 2 modes suggests that the ground state of the qbit model is a new state of matter. In the second model (called the N-type model) the collective modes are strongly interacting and there is no reliable approach to understand its low energy dynamics. Using a spin-wave/quantum-freeze approach (which is shown to reproduce the correct emergent U(1) gauge theory in a quantum rotor model), we argue that the second model may contain helicity +/- 2 gapless excitations as the only type of low energy excitations with a linear dispersion \om \propto k.

gr-qc↗

arXiv2010

Tensor product representation of topological ordered phase: necessary symmetry conditions

The tensor product representation of quantum states leads to a promising variational approach to study quantum phase and quantum phase transitions, especially topological ordered phases which are impossible to handle with conventional methods due to their long range entanglement. However, an important issue arises when we use tensor product states (TPS) as variational states to find the ground state of a Hamiltonian: can arbitrary variations in the tensors that represent ground state of a Hamiltonian be induced by local perturbations to the Hamiltonian? Starting from a tensor product state which is the exact ground state of a Hamiltonian with $\mathbb{Z}_2$ topological order, we show that, surprisingly, not all variations of the tensors correspond to the variation of the ground state caused by local perturbations of the Hamiltonian. Even in the absence of any symmetry requirement of the perturbed Hamiltonian, one necessary condition for the variations of the tensors to be physical is that they respect certain $\mathbb{Z}_2$ symmetry. We support this claim by calculating explicitly the change in topological entanglement entropy with different variations in the tensors. This finding will provide important guidance to numerical variational study of topological phase and phase transitions. It is also a crucial step in using TPS to study universal properties of a quantum phase and its topological order.

cond-mat.str-el↗

arXiv2010

Topological Properties of Tensor Network States From Their Local Gauge and Local Symmetry Structures

Tensor network states are capable of describing many-body systems with complex quantum entanglement, including systems with non-trivial topological order. In this paper, we study methods to calculate the topological properties of a tensor network state from the tensors that form the state. Motivated by the concepts of gauge group and projective symmetry group in the slave-particle/projective construction, and by the low-dimensional gauge-like symmetries of some exactly solvable Hamiltonians, we study the $d$-dimensional gauge structure and the $d$-dimensional symmetry structure of a tensor network state, where $d\leq d_{space}$ with $d_{space}$ the dimension of space. The $d$-dimensional gauge structure and $d$-dimensional symmetry structure allow us to calculate the string operators and $d$-brane operators of the tensor network state. This in turn allows us to calculate many topological properties of the tensor network state, such as ground state degeneracy and quasiparticle statistics.

cond-mat.str-el↗

arXiv2010

Topological Entanglement Renyi Entropy and Reduced Density Matrix Structure

We generalize the topological entanglement entropy to a family of topological Renyi entropies parametrized by a parameter alpha, in an attempt to find new invariants for distinguishing topologically ordered phases. We show that, surprisingly, all topological Renyi entropies are the same, independent of alpha for all non-chiral topological phases. This independence shows that topologically ordered ground-state wavefunctions have reduced density matrices with a certain simple structure, and no additional universal information can be extracted from the entanglement spectrum.

cond-mat.str-el↗

arXiv2009

String-nets, single and double-stranded quantum loop gases for non-Abelian anyons

String-net condensation can give rise to non-Abelian anyons whereas loop condensation usually gives rise to Abelian anyons. It has been proposed that generalized quantum loop gases with non-orthogonal inner products can produce non-Abelian anyons. We detail an exact mapping between the string-net and the generalized loop models and explain how the non-orthogonal products arise. We also introduce a loop model of double-stranded nets where quantum loops with an orthogonal inner product and local interactions supports non-Abelian Fibonacci anyons. Finally we emphasize the origin of the sign problem in such systems and its consequences on the complexity of their ground state wave functions.

cond-mat.str-el↗

arXiv2009

Non-Abelian Quantum Hall States and their Quasiparticles: from the Pattern of Zeros to Vertex Algebra

In the pattern-of-zeros approach to quantum Hall states, a set of data {n;m;S_a|a=1,...,n; n,m,S_a in N} (called the pattern of zeros) is introduced to characterize a quantum Hall wave function. In this paper we find sufficient conditions on the pattern of zeros so that the data correspond to a valid wave function. Some times, a set of data {n;m;S_a} corresponds to a unique quantum Hall state, while other times, a set of data corresponds to several different quantum Hall states. So in the latter cases, the patterns of zeros alone does not completely characterize the quantum Hall states. In this paper, We find that the following expanded set of data {n;m;S_a;c|a=1,...,n; n,m,S_a in N; c in R} provides a more complete characterization of quantum Hall states. Each expanded set of data completely characterize a unique quantum Hall state, at least for the examples discussed in this paper. The result is obtained by combining the pattern of zeros and Z_n simple-current vertex algebra which describes a large class of Abelian and non-Abelian quantum Hall states Φ_{Z_n}^sc. The more complete characterization in terms of {n;m;S_a;c} allows us to obtain more topological properties of those states, which include the central charge c of edge states, the scaling dimensions and the statistics of quasiparticle excitations.

cond-mat.str-el↗

arXiv2009

Effective Field Theory and Projective Construction for the Z_k Parafermion Fractional Quantum Hall States

The projective construction is a powerful approach to deriving the bulk and edge field theories of non-Abelian fractional quantum Hall (FQH) states and yields an understanding of non-Abelian FQH states in terms of the simpler integer quantum Hall states. Here we show how to apply the projective construction to the Z_k parafermion (Laughlin/Moore-Read/Read-Rezayi) FQH states, which occur at filling fraction ν= k/(kM+2). This allows us to derive the bulk low energy effective field theory for these topological phases, which is found to be a Chern-Simons theory at level 1 with a U(M) \times Sp(2k) gauge field. This approach also helps us understand the non-Abelian quasiholes in terms of holes of the integer quantum Hall states.

cond-mat.str-el↗

arXiv2009

$U(1) \times U(1) \rtimes Z_2$ Chern-Simons Theory and Z_4 Parafermion Fractional Quantum Hall States

We study $U(1) \times U(1) \rtimes Z_2$ Chern-Simons theory with integral coupling constants (k,l) and its relation to certain non-Abelian fractional quantum Hall (FQH) states. For the $U(1) \times U(1) \rtimes Z_2$ Chern-Simons theory, we show how to compute the dimension of its Hilbert space on genus g surfaces and how this yields the quantum dimensions of topologically distinct excitations. We find that Z_2 vortices in the $U(1) \times U(1)\rtimes Z_2$ Chern-Simons theory carry non-Abelian statistics and we show how to compute the dimension of the Hilbert space in the presence of n pairs of Z_2 vortices on a sphere. These results allow us to show that l=3 $U(1)\times U(1) \rtimes Z_2$ Chern-Simons theory is the low energy effective theory for the Z_4 parafermion (Read-Rezayi) fractional quantum Hall states, which occur at filling fraction $ν= 2/(2k - 3)$. The $U(1)\times U(1) \rtimes Z_2$ theory is more useful than an alternative $SU(2)_4\times U(1)/ U(1)$ Chern-Simons theory because the fields are more closely related to physical degrees of freedom of the electron fluid and to an Abelian bilayer phase on the other side of a two-component to single-component quantum phase transition. We discuss the possibility of using this theory to understand further phase transitions in FQH systems, especially the $ν= 2/3$ phase diagram.

cond-mat.str-el↗

arXiv2009

Translation-symmetry protected topological orders on lattice

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. We introduced four $Z_2$ topological indices $ζ_{ǩ}=0,1$ at $\v {k}=(0,0)$, $(0,π)$, $(π, 0)$, $(π,π)$ to characterize those 16 new topological orders. We calculated the topological degeneracies and crystal momenta for those 16 topological phases on even-by-even, even-by-odd, odd-by-even, and odd-by-odd lattices, which allows us to physically measure such topological orders. We predict the appearance of gapless fermionic excitations at the quantum phase transitions between those symmetry protected topological orders. Our result can be generalized to any dimensions. We find 256 translation-symmetry protected Z2A topological orders for a system on 3D lattice.

cond-mat.str-el↗

arXiv2009

Translation invariant topological superconductors on lattice

In this paper we introduce four Z_2 topological indices zeta_k=0,1 at k=(0,0), (0,pi), (pi, 0), (pi, pi) characterizing 16 universal classes of 2D superconducting states that have translation symmetry but may break any other symmetries. The 16 classes of superconducting states are distinguished by their even/odd numbers of fermions on even-by-even, even-by-odd, odd-by-even, and odd-by-odd lattices. As a result, the 16 classes topological superconducting states exist even for interacting systems. For non-interacting systems, we find that zeta_k is the number of electrons on k=(0,0), (0,pi), (pi, 0), or (pi,pi) orbitals (mod 2) in the ground state. For 3D superconducting states with only translation symmetry, there are 256 different types of topological superconductors.

cond-mat.supr-con↗

arXiv2009

Non-Abelian Two-component Fractional Quantum Hall States

A large class of fractional quantum Hall (FQH) states can be classified according to their pattern of zeros, which describes the order of zeros in ground state wave functions as various clusters of electrons are brought together. The pattern-of-zeros approach can be generalized to systematically classify multilayer/spin-unpolarized FQH states, which has led to the construction of a class of non-Abelian multicomponent FQH states. Here we discuss some of the simplest non-Abelian two-component states that we find and the possibility of their experimental realization in bilayer systems at $ν= 2/3$, 4/5, 4/7, 4/9, 1/4, etc.

cond-mat.mes-hall↗

arXiv2009

Classification of Abelian and Non-Abelian Multilayer Fractional Quantum Hall States Through the Pattern of Zeros

A large class of fractional quantum Hall (FQH) states can be classified according to their pattern of zeros, which describes the way ideal ground state wave functions go to zero as various clusters of electrons are brought together. In this paper we generalize this approach to classify multilayer FQH states. Such a classification leads to the construction of a class of non-Abelian multilayer FQH states that are closely related to $\hat{g}_k$ parafermion conformal field theories, where $\hat{g}_k$ is an affine simple Lie algebra. We discuss the possibility of some of the simplest of these non-Abelian states occuring in experiments on bilayer FQH systems at $ν= 2/3$, 4/5, 4/7, etc.

cond-mat.mes-hall↗

arXiv2009

Doped Kagome System as Exotic Superconductor

A Chern--Simons theory for the doped spin-1/2 kagomé system is constructed, from which it is shown that the system is an exotic superconductor that breaks time-reversal symmetry. It is also shown that the system carries minimal vortices of flux $hc/4e$ (as opposed to the usual $hc/2e$ in conventional superconductors) and contains fractional quasiparticles (including fermionic quasiparticles with \emph{semionic} mutual statistics and spin-1/2 quasiparticles with \emph{bosonic} self-statistics) in addition to the usual spin-1/2 fermionic Bougoliubov quasiparticle. Two Chern--Simons theories--one with an auxiliary gauge field kept and one with the auxiliary field and a redundant matter field directly eliminated--are presented and shown to be consistent with each other.

cond-mat.str-el↗

arXiv2009

Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order

We study the renormalization group flow of the Lagrangian for statistical and quantum systems by representing their path integral in terms of a tensor network. Using a tensor-entanglement-filtering renormalization (TEFR) approach that removes local entanglement and coarse grain the lattice, we show that the resulting renormalization flow of the tensors in the tensor network has a nice fixed-point structure. The isolated fixed-point tensors $T_{inv}$ plus the symmetry group $G_{sym}$ of the tensors (i.e. the symmetry group of the Lagrangian) characterize various phases of the system. Such a characterization can describe both the symmetry breaking phases and topological phases, as illustrated by 2D statistical Ising model, 2D statistical loop gas model, and 1+1D quantum spin-1/2 and spin-1 models. In particular, using such a $(G_{sym}, T_{inv}) $ characterization, we show that the Haldane phase for a spin-1 chain is a phase protected by the time-reversal, parity, and translation symmetries. Thus the Haldane phase is a symmetry protected topological phase. The $(G_{sym}, T_{inv})$ characterization is more general than the characterizations based on the boundary spins and string order parameters. The tensor renormalization approach also allows us to study continuous phase transitions between symmetry breaking phases and/or topological phases. The scaling dimensions and the central charges for the critical points that describe those continuous phase transitions can be calculated from the fixed-point tensors at those critical points.

cond-mat.str-el↗

arXiv2008

Topological surface states in three-dimensional magnetic insulators

An electron moving in a magnetically ordered background feels an effective magnetic field that can be both stronger and more rapidly varying than typical externally applied fields. One consequence is that insulating magnetic materials in three dimensions can have topologically nontrivial properties of the effective band structure. For the simplest case of two bands, these "Hopf insulators" are characterized by a topological invariant as in quantum Hall states and Z_2 topological insulators, but instead of a Chern number or parity, the underlying invariant is the Hopf invariant that classifies maps from the 3-sphere to the 2-sphere. This paper gives an efficient algorithm to compute whether a given magnetic band structure has nontrivial Hopf invariant, a double-exchange-like tight-binding model that realizes the nontrivial case, and a numerical study of the surface states of this model.

cond-mat.mes-hall↗

arXiv2008

Spontaneous spin ordering of Dirac spin liquid in a magnetic field

The Dirac spin liquid was proposed to be the ground state of the spin-1/2 Kagome antiferromagnets. In a magnetic field $B$, we show that the state with Fermi pocket is unstable to the Landau level (LL) state. The LL state breaks the spin rotation around the axis of the magnetic field. We find that the LL state has an in-plane 120$^{\circ}$ $q=0$ magnetization $M$ which scales with the external field $M\sim B^α$, where $α$ is an intrinsic calculable universal number of the Dirac spin liquid. We discuss the related experimental implications which can be used to detect the possible Dirac spin liquid phase in Herbertsmithite ZnCu$_3$(OH)$_6$Cl$_2$.

cond-mat.str-el↗

arXiv2008

Spin-triplet p-wave pairing in a 3-orbital model for iron pnictide superconductors

We examine the possibility that the superconductivity in the newly discovered FeAs materials may be caused by the Coulomb interaction between d-electrons of the iron atoms. We find that when the Hund's rule ferromagnetic interaction is strong enough, the leading pairing instability is in spin-triplet p-wave channel in the weak coupling limit. The resulting superconducting gap has nodal lines on the 3D Fermi surfaces. The k dependent hybridization of several orbitals around a Fermi pocket is the key for the appearance of the spin-triplet p-wave pairing.

cond-mat.supr-con↗