arXiv subjects

Zheng-Cheng Gu

Publications and source records attributed to Zheng-Cheng Gu.

At least 91 records · Page 5Linked to original sources

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

Topological flat band models with arbitrary Chern numbers

We report the theoretical discovery of a systematic scheme to produce topological flat bands (TFBs) with arbitrary Chern numbers. We find that generically a multi-orbital high Chern number TFB model can be constructed by considering multi-layer Chern number C=1 TFB models with enhanced translational symmetry. A series of models are presented as examples, including a two-band model on a triangular lattice with a Chern number C=3 and an $N$-band square lattice model with $C=N$ for an arbitrary integer $N$. In all these models, the flatness ratio for the TFBs is larger than 30 and increases with increasing Chern number. In the presence of appropriate inter-particle interactions, these models are likely to lead to the formation of novel Abelian and Non-Abelian fractional Chern insulators. As a simple example, we test the C=2 model with hardcore bosons at 1/3 filling and an intriguing fractional quantum Hall state is observed.

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

Braiding statistics approach to symmetry-protected topological phases

We construct a 2D quantum spin model that realizes an Ising paramagnet with gapless edge modes protected by Ising symmetry. This model provides an example of a "symmetry-protected topological phase." We describe a simple physical construction that distinguishes this system from a conventional paramagnet: we couple the system to a Z_2 gauge field and then show that the π-flux excitations have different braiding statistics from that of a usual paramagnet. In addition, we show that these braiding statistics directly imply the existence of protected edge modes. Finally, we analyze a particular microscopic model for the edge and derive a field theoretic description of the low energy excitations. We believe that the braiding statistics approach outlined in this paper can be generalized to a large class of symmetry-protected topological phases.

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

Non-Abelian Quantum Hall Effect in Topological Flat Bands

Inspired by recent theoretical discovery of robust fractional topological phases without a magnetic field, we search for the non-Abelian quantum Hall effect (NA-QHE) in lattice models with topological flat bands (TFBs). Through extensive numerical studies on the Haldane model with three-body hard-core bosons loaded into a TFB, we find convincing numerical evidence of a stable $ν=1$ bosonic NA-QHE, with the characteristic three-fold quasi-degeneracy of ground states on a torus, a quantized Chern number, and a robust spectrum gap. Moreover, the spectrum for two-quasihole states also shows a finite energy gap, with the number of states in the lower energy sector satisfying the same counting rule as the Moore-Read Pfaffian state.

cond-mat.str-el↗

arXiv2011

Nearly-flat bands with nontrivial topology

We report the theoretical discovery of a large class of 2D tight-binding models containing nearly-flat bands with nonzero Chern numbers. In contrast with previous studies, where nonlocal hoppings are usually required, the Hamiltonians of our models only require short-range hopping and have the potential to be realized in cold atomic gases. Due to the similarity with 2D continuum Landau levels, these topologically nontrivial nearly-flat bands may lead to the realization of fractional anomalous quantum Hall states and fractional topological insulators in real materials. Among the models we discover, the most interesting and practical one is a square-lattice three-band model which has only nearest-neighbor hopping. To understand better the physics underlying the topological flat band aspects, we also present the studies of a minimal two-band model on the checkerboard lattice.

cond-mat.str-el↗

arXiv2011

Fractional Quantum Hall Effect of Hard-Core Bosons in Topological Flat Bands

Recent proposals of topological flat band (TFB) models have provided a new route to realize the fractional quantum Hall effect (FQHE) without Landau levels. We study hard-core bosons with short-range interactions in two representative TFB models, one of which is the well known Haldane model (but with different parameters). We demonstrate that FQHE states emerge with signatures of even number of quasi-degenerate ground states on a torus and a robust spectrum gap separating these states from higher energy spectrum. We also establish quantum phase diagrams for the filling factor 1/2 and illustrate quantum phase transitions to other competing symmetry-breaking phases.

cond-mat.str-el↗

arXiv2011

Fractional quantum Hall effect in the absence of Landau levels

It has been well-known that topological phenomena with fractional excitations, i.e., the fractional quantum Hall effect (FQHE) \cite{Tsui1982} will emerge when electrons move in Landau levels. In this letter, we report the discovery of the FQHE in the absence of Landau levels in an interacting fermion model. The non-interacting part of our Hamiltonian is the recently proposed topologically nontrivial flat band model on the checkerboard lattice \cite{sun}. In the presence of nearest-neighboring repulsion ($U$), we find that at 1/3 filling, the Fermi-liquid state is unstable towards FQHE. At 1/5 filling, however, a next-nearest-neighboring repulsion is needed for the occurrence of the 1/5 FQHE when $U$ is not too strong. We demonstrate the characteristic features of these novel states and determine the phase diagram correspondingly.

cond-mat.str-el↗

arXiv2011

Possible proximity of the Mott insulating Iridate Na2IrO3 to a topological phase: Phase diagram of the Heisenberg-Kitaev model in a magnetic field

Motivated by the recent experimental observation of a Mott insulating state for the layered Iridate Na2IrO3, we discuss possible ordering states of the effective Iridium moments in the presence of strong spin-orbit coupling and a magnetic field. For a field pointing in the [111] direction - perpendicular to the hexagonal lattice formed by the Iridium moments - we find that a combination of Heisenberg and Kitaev exchange interactions gives rise to a rich phase diagram with both symmetry breaking magnetically ordered phases as well as a topologically ordered phase that is stable over a small range of coupling parameters. Our numerical simulations further indicate two exotic multicritical points at the boundaries between these ordered phases.

cond-mat.str-el↗

arXiv2010

Classification of Gapped Symmetric Phases in 1D Spin Systems

Quantum many-body systems divide into a variety of phases with very different physical properties. The question of what kind of phases exist and how to identify them seems hard especially for strongly interacting systems. Here we make an attempt to answer this question for gapped interacting quantum spin systems whose ground states are short-range correlated. Based on the local unitary equivalence relation between short-range correlated states in the same phase, we classify possible quantum phases for 1D matrix product states, which represent well the class of 1D gapped ground states. We find that in the absence of any symmetry all states are equivalent to trivial product states, which means that there is no topological order in 1D. However, if certain symmetry is required, many phases exist with different symmetry protected topological orders. The symmetric local unitary equivalence relation also allows us to obtain some simple results for quantum phases in higher dimensions when some symmetries are present.

cond-mat.str-el↗

arXiv2010

Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order

Two gapped quantum ground states in the same phase are connected by an adiabatic evolution which gives rise to a local unitary transformation that maps between the states. On the other hand, gapped ground states remain within the same phase under local unitary transformations. Therefore, local unitary transformations define an equivalence relation and the equivalence classes are the universality classes that define the different phases for gapped quantum systems. Since local unitary transformations can remove local entanglement, the above equivalence/universality classes correspond to pattern of long range entanglement, which is the essence of topological order. The local unitary transformation also allows us to define a wave function renormalization scheme, under which a wave function can flow to a simpler one within the same equivalence/universality class. Using such a setup, we find conditions on the possible fixed-point wave functions where the local unitary transformations have \emph{finite} dimensions. The solutions of the conditions allow us to classify this type of topological orders, which generalize the string-net classification of topological orders. We also describe an algorithm of wave function renormalization induced by local unitary transformations. The algorithm allows us to calculate the flow of tensor-product wave functions which are not at the fixed points. This will allow us to calculate topological orders as well as symmetry breaking orders in a generic tensor-product state.

cond-mat.str-el↗

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↗

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

Tensor-product representations for string-net condensed states

We show that general string-net condensed states have a natural representation in terms of tensor product states (TPS) . These TPS's are built from local tensors. They can describe both states with short-range entanglement (such as the symmetry breaking states) and states with long-range entanglement (such as string-net condensed states with topological/quantum order). The tensor product representation provides a kind of 'mean-field' description for topologically ordered states and could be a powerful way to study quantum phase transitions between such states. As an attempt in this direction, we show that the constructed TPS's are fixed-points under a certain wave-function renormalization group transformation for quantum states.

cond-mat.str-el↗

arXiv2008

Tensor-entanglement renormalization group approach to topological phases

The tensor-entanglement renormalization group approach is applied to Hamiltonians that realize a class of topologically ordered states -- string-net condensed states. We analyze phase transitions between phases with and without string-net condensation. These phase transitions change topological order without changing any symmetries. This demonstrates that the tensor-entanglement renormalization group approach can be used to study the phase diagram of a quantum system with topologically ordered phases.

cond-mat.str-el↗