arXiv subjects

Xiao-Gang Wen

Publications and source records attributed to Xiao-Gang Wen.

At least 109 records · Page 6Linked to original sources

arXiv2014

$U(1)\times U(1)$ Symmetry Protected Topological Order in Gutzwiller Wave Functions

Gutzwiller projection is a way to construct many-body wave functions that could carry topological order or symmetry protected topological (SPT) order. However, an important issue is to determine whether or not a given Gutzwiller-projected wave functions (GWF) carries a non-trivial SPT order, and which SPT order is carried by the wavefunction. In this paper, we numerically study the SPT order in a spin $S = 1$ GWF on the Kagome lattice. Using the standard Monte Carlo method, we directly confirm that the GWF has (1) gapped bulk with short-range correlations, (2) a trivial topological order via nondegenerate ground state, and zero topological entanglement entropy, (3) a non-trivial $U(1)\times U(1)$ SPT order via the Hall conductances of the protecting $U(1)\times U(1)$ symmetry, and (4) symmetry protected gapless boundary. This represents numerical evidence of continuous symmetry protected topological order in two-dimensional bosonic lattice systems.

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arXiv2014

Non-Abelian String and Particle Braiding in Topological Order: Modular SL(3,Z) Representation and 3+1D Twisted Gauge Theory

String and particle braiding statistics are examined in a class of topological orders described by discrete gauge theories with a gauge group $G$ and a 4-cocycle twist $ω_4$ of $G$'s cohomology group $\mathcal{H}^4(G,\mathbb{R}/\mathbb{Z})$ in 3 dimensional space and 1 dimensional time (3+1D). We establish the topological spin and the spin-statistics relation for the closed strings, and their multi-string braiding statistics. The 3+1D twisted gauge theory can be characterized by a representation of a modular transformation group SL$(3,\mathbb{Z})$. We express the SL$(3,\mathbb{Z})$ generators $\mathsf{S}^{xyz}$ and $\mathsf{T}^{xy}$ in terms of the gauge group $G$ and the 4-cocycle $ω_4$. As we compactify one of the spatial directions $z$ into a compact circle with a gauge flux $b$ inserted, we can use the generators $\mathsf{S}^{xy}$ and $\mathsf{T}^{xy}$ of an SL$(2,\mathbb{Z})$ subgroup to study the dimensional reduction of the 3D topological order $\mathcal{C}^{3\text{D}}$ to a direct sum of degenerate states of 2D topological orders $\mathcal{C}_b^{2\text{D}}$ in different flux $b$ sectors: $\mathcal{C}^{3\text{D}} = \oplus_b \mathcal{C}_b^{2\text{D}}$. The 2D topological orders $\mathcal{C}_b^{2\text{D}}$ are described by 2D gauge theories of the group $G$ twisted by the 3-cocycles $ω_{3(b)}$, dimensionally reduced from the 4-cocycle $ω_4$. We show that the SL$(2,\mathbb{Z})$ generators, $\mathsf{S}^{xy}$ and $\mathsf{T}^{xy}$, fully encode a particular type of three-string braiding statistics with a pattern that is the connected sum of two Hopf links. With certain 4-cocycle twists, we discover that, by threading a third string through two-string unlink into three-string Hopf-link configuration, Abelian two-string braiding statistics is promoted to non-Abelian three-string braiding statistics.

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arXiv2014

Bosonic Anomalies, Induced Fractional Quantum Numbers and Degenerate Zero Modes: the anomalous edge physics of Symmetry-Protected Topological States

The boundary of symmetry-protected topological states (SPTs) can harbor new quantum anomaly phenomena. In this work, we characterize the bosonic anomalies introduced by the 1+1D non-onsite-symmetric gapless edge modes of 2+1D bulk bosonic SPTs with a generic finite Abelian group symmetry (isomorphic to $G=\prod_i Z_{N_i}=Z_{N_1} \times Z_{N_2} \times Z_{N_3} \times ...$). We demonstrate that some classes of SPTs (termed "Type II") trap fractional quantum numbers (such as fractional $Z_N$ charges) at the 0D kink of the symmetry-breaking domain walls; while some classes of SPTs (termed "Type III") have degenerate zero energy modes (carrying the projective representation protected by the unbroken part of the symmetry), either near the 0D kink of a symmetry-breaking domain wall, or on a symmetry-preserving 1D system dimensionally reduced from a thin 2D tube with a monodromy defect 1D line embedded. More generally, the energy spectrum and conformal dimensions of gapless edge modes under an external gauge flux insertion (or twisted by a branch cut, i.e., a monodromy defect line) through the 1D ring can distinguish many SPT classes. We provide a manifest correspondence from the physical phenomena, the induced fractional quantum number and the zero energy mode degeneracy, to the mathematical concept of cocycles that appears in the group cohomology classification of SPTs, thus achieving a concrete physical materialization of the cocycles. The aforementioned edge properties are formulated in terms of a long wavelength continuum field theory involving scalar chiral bosons, as well as in terms of Matrix Product Operators and discrete quantum lattice models. Our lattice approach yields a regularization with anomalous non-onsite symmetry for the field theory description. We also formulate some bosonic anomalies in terms of the Goldstone-Wilczek formula.

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arXiv2014

Topological quasiparticles and the holographic bulk-edge relation in 2+1D string-net models

String-net models allow us to systematically construct and classify 2+1D topologically ordered states which can have gapped boundaries. We can use a simple ideal string-net wavefunction, which is described by a set of F-matrices [or more precisely, a unitary fusion category (UFC)], to study all the universal properties of such a topological order. In this paper, we describe a finite computational method -- Q-algebra approach, that allows us to compute the non-Abelian statistics of the topological excitations [or more precisely, the unitary modular tensor category (UMTC)], from the string-net wavefunction (or the UFC). We discuss several examples, including the topological phases described by twisted gauge theory (i.e., twisted quantum double $D^α(G)$). Our result can also be viewed from an angle of holographic bulk-boundary relation. The 1+1D anomalous topological orders, that can appear as edges of 2+1D topological states, are classified by UFCs which describe the fusion of quasiparticles in 1+1D. The 1+1D anomalous edge topological order uniquely determines the 2+1D bulk topological order (which are classified by UMTC). Our method allows us to compute this bulk topological order (i.e., the UMTC) from the anomalous edge topological order (i.e., the UFC).

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arXiv2014

Gapped Domain Walls, Gapped Boundaries and Topological Degeneracy

Gapped domain walls, as topological line defects between 2+1D topologically ordered states, are examined. We provide simple criteria to determine the existence of gapped domain walls, which apply to both Abelian and non-Abelian topological orders. Our criteria also determine which 2+1D topological orders must have gapless edge modes, namely which 1+1D global gravitational anomalies ensure gaplessness. Furthermore, we introduce a new mathematical object, the tunneling matrix $\mathcal W$, whose entries are the fusion-space dimensions $\mathcal W_{ia}$, to label different types of gapped domain walls. By studying many examples, we find evidence that the tunneling matrices are powerful quantities to classify different types of gapped domain walls. Since a gapped boundary is a gapped domain wall between a bulk topological order and the vacuum, regarded as the trivial topological order, our theory of gapped domain walls inclusively contains the theory of gapped boundaries. In addition, we derive a topological ground state degeneracy formula, applied to arbitrary orientable spatial 2-manifolds with gapped domain walls, including closed 2-manifolds and open 2-manifolds with gapped boundaries.

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arXiv2014

Modular Matrices as Topological Order Parameter by Gauge Symmetry Preserved Tensor Renormalization Approach

Topological order has been proposed to go beyond Landau symmetry breaking theory for more than twenty years. But it is still a challenging problem to generally detect it in a generic many-body state. In this paper, we will introduce a systematic numerical method based on tensor network to calculate modular matrices in 2D systems, which can fully identify topological order with gapped edge. Moreover, it is shown numerically that modular matrices, including S and T matrices, are robust characterization to describe phase transitions between topologically ordered states and trivial states, which can work as topological order parameters. This method only requires local information of one ground state in the form of a tensor network, and directly provides the universal data (S and T matrices), without any non-universal contributions. Furthermore it is generalizable to higher dimensions. Unlike calculating topological entanglement entropy by extrapolating, which numerical complexity is exponentially high, this method extracts a much more complete set of topological data (modular matrices) with much lower numerical cost.

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arXiv2014

Modular matrices from universal wave function overlaps in Gutzwiller-projected parton wave functions

We implement the universal wave function overlap (UWFO) method to extract modular $S$ and $T$ matrices for topological orders in Gutzwiller-projected parton wave functions (GPWFs). The modular $S$ and $T$ matrices generate a projective representation of $SL(2,\mathbb{Z})$ on the degenerate-ground-state Hilbert space on a torus and may fully characterize the 2+1D topological orders, i.e. the quasi-particle statistics and chiral central charge (up to $E_8$ bosonic quantum Hall states). We used the variational Monte Carlo method to computed the $S$ and $T$ matrices of the chiral spin liquid (CSL) constructed by the GPWF on the square lattice, and confirm that the CSL carries the same topological order as the $ν=\frac{1}{2}$ bosonic Laughlin state. We find that the non-universal exponents in UWFO can be small and direct numerical computation is able to be applied on relatively large systems. We also discuss the UWFO method for GPWFs on other Bravais lattices in two and three dimensions by using the Monte Carlo method. UWFO may be a powerful method to calculate the topological order in GPWFs.

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arXiv2014

Symmetry-protected topological orders for interacting fermions -- Fermionic topological nonlinear $σ$ models and a special group supercohomology theory

Symmetry-protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry $G$, which can all be smoothly connected to the trivial product states if we break the symmetry. It has been shown that a large class of interacting bosonic SPT phases can be systematically described by group cohomology theory. In this paper, we introduce a (special) group supercohomology theory which is a generalization of the standard group cohomology theory. We show that a large class of short-range interacting fermionic SPT phases can be described by the group supercohomology theory. Using the data of super cocycles, we can obtain the ideal ground state wave function for the corresponding fermionic SPT phase. We can also obtain the bulk Hamiltonian that realizes the SPT phase, as well as the anomalous (ie, non-on-site) symmetry for the boundary effective Hamiltonian. The anomalous symmetry on the boundary implies that the symmetric} boundary must be gapless for 1+1D boundary, and must be gapless or topologically ordered beyond 1+1D. As an application of this general result, we construct a new SPT phase in 3D, for interacting fermionic superconductors with coplanar spin order (which have $T^2=1$ time-reversal $Z_2^T$ and fermion-number parity $Z_2^f$ symmetries described by a full symmetry group $Z_2^T\times Z_2^f$). Such a fermionic SPT state can neither be realized by free fermions nor by interacting bosons (formed by fermion-pairs), and thus are not included in the K-theory classification for free fermions or group cohomology description for interacting bosons. We also construct three interacting fermionic SPT phases in 2D with a full symmetry group $Z_2\times Z_2^f$. Those 2D fermionic SPT phases all have central-charge $c=1$ gapless edge excitations, if the symmetry is not broken.

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arXiv2014

Lattice Model for Fermionic Toric Code

The Z_2 topological order in Z_2 spin liquid and in exactly soluble Kitaev toric code model is the simplest topological order for 2+1D bosonic systems. More general 2+1D bosonic topologically ordered states can be constructed via exact soluble string-net models. However, the most important topologically ordered phases of matter are arguably the fermionic fractional quantum Hall states. Topological phases of matter for fermion systems are strictly richer than their bosonic counterparts because locality has different meanings for the two kinds of systems. In this paper, we describe a simple fermionic version of the toric code model to illustrate many salient features of fermionic exactly soluble models and fermionic topologically ordered states.

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arXiv2014

Design local spin models for Gutzwiller-projected parton wave functions

We introduce a method to design a local spin Hamiltonian to realize a Gutzwiller-projected parton wave functions (GPWF) as its ground state. For example, the Dirac spin liquid (DSL) state is quite close to the true ground state of the spin-1/2 Heisenberg model on kagome lattice. We examine what kind of perturbations we should add in order to drive the DSL to more stable chiral spin liquid (CSL), valence bond solid (VBS) or Gutzwiller-projected spin Hall (GSH) states. We compute the two-body reduced-density-matrices (2-RDM) of GPWFs for those target states, and compare them to the 2-RDM of the DSL. This allows us to design local spin models with only two-body interactions that may realize those interesting target states. Our results agree very well with recent numerical calculations for CSL on kagome lattice. We also study spin-1 systems on kagome lattice, and design local spin models that may realize CSL, VBS and symmetry-protected topological (SPT) states. Our work establishes a directional guide for further numerical simulations.

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arXiv2014

Universal symmetry-protected topological invariants for symmetry-protected topological states

Symmetry-protected topological (SPT) states are short-range entangled states with a symmetry G. They belong to a new class of quantum states of matter which are classified by the group cohomology $H^{d+1}(G,\mathbb{R}/\mathbb{Z})$ in d-dimensional space. In this paper, we propose a class of symmetry- protected topological invariants that may allow us to fully characterize SPT states with a symmetry group G (ie allow us to measure the cocycles in $H^{d+1}(G,\mathbb{R}/\mathbb{Z})$ that characterize the SPT states). We give an explicit and detailed construction of symmetry-protected topological invariants for 2+1D SPT states. Such a construction can be directly generalized to other dimensions.

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arXiv2014

Braided fusion categories, gravitational anomalies, and the mathematical framework for topological orders in any dimensions

Gravitational anomalies can be realized on the boundary of topologically ordered states in one higher dimension and are described by topological orders in one higher dimension. In this paper, we try to develop a general theory for both topological order and gravitational anomaly in any dimensions. (1) We introduce the notion of BF category to describe the braiding and fusion properties of topological excitations that can be point-like, string-like, etc. A subset of BF categories -- closed BF categories -- classify topological orders in any dimensions, while generic BF categories classify (potentially) anomalous topological orders that can appear at a boundary of a gapped quantum liquid in one higher dimension. (2) We introduce topological path integral based on tensor network to realize those topological orders. (3) Bosonic topological orders have an important topological invariant: the vector bundles of the degenerate ground states over the moduli spaces of closed spaces with different metrics. They may fully characterize topological orders. (4) We conjecture that a topological order has a gappable boundary iff the above mentioned vector bundles are flat. (5) We find a holographic phenomenon that every topological order with a gappable boundary can be uniquely determined by the knowledge of the boundary. As a consequence, BF categories in different dimensions form a (monoid) cochain complex, that reveals the structure and relation of topological orders and gravitational anomalies in different dimensions. We also studied the simplest kind of bosonic topological orders that have no non-trivial topological excitations. We find that this kind of topological orders form a $\mathbb{Z}$ class in 2+1D (with gapless edge), a $\mathbb{Z}_2$ class in 4+1D (with gappable boundary), and a $\mathbb{Z}\oplus \mathbb{Z}$ class in 6+1D (with gapless boundary).

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arXiv2014

Universal Topological Data for Gapped Quantum Liquids in Three Dimensions and Fusion Algebra for Non-Abelian String Excitations

Recently we conjectured that a certain set of universal topological quantities characterize topological order in any dimension. Those quantities can be extracted from the universal overlap of the ground state wave functions. For systems with gapped boundaries, these quantities are representations of the mapping class group $MCG(\mathcal M)$ of the space manifold $\mathcal M$ on which the systems lives. We will here consider simple examples in three dimensions and give physical interpretation of these quantities, related to fusion algebra and statistics of particle and string excitations. In particular, we will consider dimensional reduction from 3+1D to 2+1D, and show how the induced 2+1D topological data contains information on the fusion and the braiding of non-Abelian string excitations in 3D. These universal quantities generalize the well-known modular $S$ and $T$ matrices to any dimension.

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arXiv2014

Universal Wave Function Overlap and Universal Topological Data from Generic Gapped Ground States

We propose a way -- universal wave function overlap -- to extract universal topological data from generic ground states of gapped systems in any dimensions. Those extracted topological data should fully characterize the topological orders with gapped or gapless boundary. For non-chiral topological orders in 2+1D, this universal topological data consist of two matrices, $S$ and $T$, which generate a projective representation of $SL(2,\mathbb Z)$ on the degenerate ground state Hilbert space on a torus. For topological orders with gapped boundary in higher dimensions, this data constitutes a projective representation of the mapping class group $MCG(M^d)$ of closed spatial manifold $M^d$. For a set of simple models and perturbations in two dimensions, we show that these quantities are protected to all orders in perturbation theory.

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arXiv2014

Modular transformations and topological orders in two dimensions

The string-net approach by Levin and Wen and the local unitary transformation approach by Chen, Gu and Wen provided ways to systematically label non-chiral topological orders in 2D. In those approaches, different topologically ordered many-body wave functions were characterized by different fixed-point tensors. Though extremely powerful, the resulting fixed-point tensors were mathematical abstractions and thus lacked a physical interpretation. As a result it was hard to judge if two different fixed-point tensors described the same quantum phase or not. We want to improve that approach by giving a more physical description of the topological orders. We find that the non-Abelian Berry's phases, $T$- and $S$-matrices, of the topological protected degenerate ground states on a torus give rise to a more physical description of topological orders. Using the Verlinde conjecture, we can even choose the canonical basis for the $T$- and $S$-matrices. It is conjectured that the $T$ and $S$-matrices form a complete and one-to-one characterization of non-chiral topological orders and can replace the fixed-point tensor description to give us a more physical label for topological orders. As a result, all the topological properties can be obtained from the $T$- and $S$-matrices, such as number of quasiparticle types (from the dimension of $T$ or $S$), the quasiparticle statistics (from the diagonal elements of $T$), the quantum dimensions of quasiparticles (from the first row of $S$), \etc.

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arXiv2013

Symmetry-protected topological invariants of symmetry-protected topological phases of interacting bosons and fermions

Recently, it was realized that quantum states of matter can be classified as long-range entangled (LRE) states (i.e. with non-trivial topological order) and short-range entangled (SRE) states (\ie with trivial topological order). We can use group cohomology class ${\cal H}^d(SG,R/Z)$ to systematically describe the SRE states with a symmetry $SG$ [referred as symmetry-protected trivial (SPT) or symmetry-protected topological (SPT) states] in $d$-dimensional space-time. In this paper, we study the physical properties of those SPT states, such as the fractionalization of the quantum numbers of the global symmetry on some designed point defects, and the appearance of fractionalized SPT states on some designed defect lines/membranes. Those physical properties are SPT invariants of the SPT states which allow us to experimentally or numerically detect those SPT states, i.e. to measure the elements in ${\cal H}^d(G, R/Z)$ that label different SPT states. For example, 2+1D bosonic SPT states with $Z_n$ symmetry are classified by a $Z_n$ integer $m \in {\cal H}^3(Z_n, R/Z)=Z_n$. We find that $n$ identical monodromy defects, in a $Z_n$ SPT state labeled by $m$, carry a total $Z_n$-charge $2m$ (which is not a multiple of $n$ in general).

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arXiv2013

Constructing symmetric topological phases of bosons in three dimensions via fermionic projective construction and dyon condensation

Recently, there is a considerable study on gapped symmetric phases of bosons that do not break any symmetry. Even without symmetry breaking, the bosons can still be in many exotic new states of matter, such as symmetry-protected trivial (SPT) phases which are short-range entangled and symmetry-enriched topological (SET) phases which are long-range entangled. It is well-known that non-interacting fermionic topological insulators are SPT states protected by time-reversal symmetry and U(1) fermion number conservation symmetry. In this paper, we construct three-dimensional exotic phases of bosons with time-reversal symmetry and boson number conservation U(1) symmetry by means of fermionic projective construction. We first construct an algebraic bosonic insulator which is a symmetric bosonic state with an emergent U(1) gapless gauge field. We then obtain many gapped bosonic states that do not break the time-reversal symmetry and boson number conservation via proper dyon condensations. We identify the constructed states by calculating the allowed electric and magnetic charges of their excitations, as well as the statistics and the symmetric transformation properties of those excitations. This allows us to show that our constructed states can be trivial SPT states (i.e. trivial Mott insulators of bosons with symmetry), non-trivial SPT states (i.e. bosonic topological insulators) and SET states (i.e. fractional bosonic topological insulators). In non-trivial SPT states, the elementary monopole (carrying zero electric charge but unit magnetic charge) and elementary dyon (carrying both unit electric charge and unit magnetic charge) are fermionic and bosonic, respectively. In SET states, intrinsic excitations may carry fractional charge.

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arXiv2013

Time-reversal symmetry breaking superconducting ground state in the doped Mott insulator on the honeycomb lattice

The emergence of superconductivity in doped Mott insulators has been debated for decades. In this paper, we report the theoretical discovery of a time-reversal symmetry breaking superconducting ground state in the doped Mott insulator (described by the well known t-J model) on honeycomb lattice, based on a recently developed variational method: the Grassmann tensor product state approach. As a benchmark, we use exact diagonalization and density-matrix renormalization methods to check our results on small clusters. We find systematic consistency for the ground-state energy as well as other physical quantities, such as the staggered magnetization. At low doping, the superconductivity coexists with antiferromagnetic ordering.

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