arXiv subjects

Xiao-Gang Wen

Publications and source records attributed to Xiao-Gang Wen.

At least 73 records · Page 4Linked to original sources

arXiv2018

A Solution to the 1+1D Gauged Chiral Fermion Problem

We show that the 3450 U(1) chiral fermion theory can appear as the low energy effective field theory of a 1+1D local lattice model, with an on-site U(1) symmetry and finite-range interactions. The on-site U(1) symmetry means that the U(1) symmetry can be gauged (gaugeable for both background probe and dynamical fields), which leads to a non-perturbative definition of chiral gauge theory --- a chiral fermion theory coupled to U(1) gauge theory. Our construction can be generalized to regularize any U(1)-anomaly-free 1+1D gauged chiral fermion theory with a zero chiral central charge (thus no gravitational anomaly) by a lattice, thanks to the recently proven "Poincaré dual" equivalence between the U(1) 't Hooft anomaly free condition and the U(1) symmetric interaction gapping rule, via a bosonization-fermionization technique.

hep-lat↗

arXiv2018

Fermion decoration construction of symmetry protected trivial orders for fermion systems with any symmetries $G_f$ and in any dimensions

We use higher dimensional bosonization and fermion decoration to construct exactly soluble interacting fermion models to realize fermionic symmetry protected trivial (SPT) orders (which are also known as symmetry protected topological orders) in any dimensions and for generic fermion symmetries $G_f$, which can be a non-trivial $Z_2^f$ extension (where $Z_2^f$ is the fermion-number-parity symmetry). This generalizes the previous results from group superconhomology of Gu and Wen (arXiv:1201.2648), where $G_f$ is assumed to be a trivial $Z_2^f$ extension. We find that the SPT phases from fermion decoration construction can be described in a compact way using higher groups.

cond-mat.str-el↗

arXiv2018

Topological non-linear $σ$-model, higher gauge theory, and a realization of all 3+1D topological orders for boson systems

A discrete non-linear $σ$-model is obtained by triangulate both the space-time $M^{d+1}$ and the target space $K$. If the path integral is given by the sum of all the complex homomorphisms $ϕ: M^{d+1} \to K$, with an partition function that is independent of space-time triangulation, then the corresponding non-linear $σ$-model will be called topological non-linear $σ$-model which is exactly soluble. Those exactly soluble models suggest that phase transitions induced by fluctuations with no topological defects (i.e. fluctuations described by homomorphisms $ϕ$) usually produce a topologically ordered state and are topological phase transitions, while phase transitions induced by fluctuations with all the topological defects give rise to trivial product states and are not topological phase transitions. If $K$ is a space with only non-trivial first homotopy group $G$ which is finite, those topological non-linear $σ$-models can realize all 3+1D bosonic topological orders without emergent fermions, which are described by Dijkgraaf-Witten theory with gauge group $π_1(K)=G$. Here, we show that the 3+1D bosonic topological orders with emergent fermions can be realized by topological non-linear $σ$-models with $π_1(K)=$ finite groups, $π_2(K)=Z_2$, and $π_{n>2}(K)=0$. A subset of those topological non-linear $σ$-models corresponds to 2-gauge theories, which realize and classify bosonic topological orders with emergent fermions that have no emergent Majorana zero modes at triple string intersections. The classification of 3+1D bosonic topological orders may correspond to a classification of unitary fully dualizable fully extended topological quantum field theories in 4-dimensions.

cond-mat.str-el↗

arXiv2018

A classification of 3+1D bosonic topological orders (II): the case when some point-like excitations are fermions

In this paper, we classify EF topological orders for 3+1D bosonic systems where some emergent pointlike excitations are fermions. (1) We argue that all 3+1D bosonic topological orders have gappable boundary. (2) All the pointlike excitations in EF topological orders are described by the representations of $G_f=Z_2^f\leftthreetimes_{e_2} G_b$ -- a $Z_2^f$ central extension of a finite group $G_b$ characterized by $e_2\in H^2(G_b,Z_2)$. (3) We find that the EF topological orders are classified by 2+1D anomalous topological orders $\mathcal{A}_b^3$ on their unique canonical boundary. Here $\mathcal{A}_b^3$ is a unitary fusion 2-category with simple objects labeled by $\hat G_b=Z_2^m\leftthreetimes G_b$. $\mathcal{A}_b^3$ also has one invertible fermionic 1-morphism for each object as well as quantum-dimension-$\sqrt 2$ 1-morphisms that connect two objects $g$ and $gm$, where $g\in \hat G_b$ and $m$ is the generator of $Z_2^m$. (4) When $\hat G_b$ is the trivial $Z_2^m$ extension, the EF topological orders are called EF1 topological orders, which is classified by simple data $(G_b,e_2,n_3,ν_4)$. (5) When $\hat G_b$ is a non-trivial $Z_2^m$ extension, the EF topological orders are called EF2 topological orders, where some intersections of three stringlike excitations must carry Majorana zero modes. (6) Every EF2 topological order with $G_f=Z_2^f\leftthreetimes G_b$ can be associated with a EF1 topological order with $G_f=Z_2^f\leftthreetimes \hat G_b$. (7) We find that all EF topological orders correspond to gauged 3+1D fermionic symmetry protected topological (SPT) orders with a finite unitary symmetry group. (8) We further propose that the general classification of 3+1D topological orders with finite unitary symmetries for bosonic and fermionic systems can be obtained by gauging or partially gauging the finite symmetry group of 3+1D SPT phases of bosonic and fermionic systems.

cond-mat.str-el↗

arXiv2018

A Single Right-Moving Free Fermion Mode on an Ultra-Local $1+1$d Spacetime Lattice

Defining a Chiral Fermion Theory on a lattice has presented an ongoing challenge both in Condensed Matter physics and in Lattice Gauge Theory. In this paper, we demonstrate that a chiral free-fermion theory can live on an ultra-local spacetime lattice if we allow the Lagrangian to be non-hermitian. Rather than a violation of unitarity, the non-hermitian structure of our Lagrangian arises because time is discrete, and we show that our model is obeys an elementary unitarity condition: namely, that the norm of the two-point functions conserves probability. Beyond unitarity, our model displays several surprising properties: it is formulated directly in Minkowskian time; it has exactly Lorentz invariant dynamics for all frequencies and momenta (in the large volume limit); and it is free from all gauge anomalies, despite the prediction from field theory that it should suffer one. We show that our model is a discrete time description of a single chiral edge mode of several recently proposed $2+1$d Floquet models. That the chiral edge can be treated without the rest of the $2+1$d system, even when coupled to a gauge field, implies that the Floquet models are radically different from Integer Quantum Hall models, which also support chiral edge modes. Furthermore, the Floquet results imply that our model can be physically realized, which presents an opportunity for gauge theories to be simulated in a condensed matter or cold atom context. Our results present a solution to the `Chiral-fermion problem:' a chiral field theory can indeed be defined on an ultra-local spacetime lattice, and we address how our model avoids several no-go arguments.

hep-lat↗

arXiv2018

Quantum Information Meets Quantum Matter -- From Quantum Entanglement to Topological Phase in Many-Body Systems

This is the draft version of a textbook, which aims to introduce the quantum information science viewpoints on condensed matter physics to graduate students in physics (or interested researchers). We keep the writing in a self-consistent way, requiring minimum background in quantum information science. Basic knowledge in undergraduate quantum physics and condensed matter physics is assumed. We start slowly from the basic ideas in quantum information theory, but wish to eventually bring the readers to the frontiers of research in condensed matter physics, including topological phases of matter, tensor networks, and symmetry-protected topological phases.

cond-mat.str-el↗

arXiv2018

Volume and Topological Invariants of Quantum Many-body Systems

A gapped many-body system is described by path integral on a space-time lattice $C^{d+1}$, which gives rise to a partition function $Z(C^{d+1})$ if $\partial C^{d+1} =\emptyset$, and gives rise to a vector $|Ψ\rangle$ on the boundary of space-time if $\partial C^{d+1} \neq\emptyset$. We show that $V = \text{log} \sqrt{\langleΨ|Ψ\rangle}$ satisfies the inclusion-exclusion property $\frac{V(A\cup B)+V(A\cap B)}{V(A)+V(B)}=1$ and behaves like a volume of the space-time lattice $C^{d+1}$ in large lattice limit (i.e. thermodynamics limit). This leads to a proposal that the vector $|Ψ\rangle$ is the quantum-volume of the space-time lattice $C^{d+1}$. The inclusion-exclusion property does not apply to quantum-volume since it is a vector. But quantum-volume satisfies a quantum additive property. The violation of the inclusion-exclusion property by $V = \text{log} \sqrt{\langleΨ|Ψ\rangle}$ in the subleading term of thermodynamics limit gives rise to topological invariants that characterize the topological order in the system. This is a systematic way to construct and compute topological invariants from a generic path integral. For example, we show how to use non-universal partition functions $Z(C^{2+1})$ on several related space-time lattices $C^{2+1}$ to extract $(M_f)_{11}$ and $\text{Tr}(M_f)$, where $M_f$ is a representation of the modular group $SL(2,\mathbb{Z})$ -- a topological invariant that almost fully characterizes the 2+1D topological orders.

cond-mat.str-el↗

arXiv2018

A classification of 3+1D bosonic topological orders (I): the case when point-like excitations are all bosons

Topological orders are new phases of matter beyond Landau symmetry breaking. They correspond to patterns of long-range entanglement. In recent years, it was shown that in 1+1D bosonic systems there is no nontrivial topological order, while in 2+1D bosonic systems the topological orders are classified by a pair: a modular tensor category and a chiral central charge. In this paper, we propose a partial classification of topological orders for 3+1D bosonic systems: If all the point-like excitations are bosons, then such topological orders are classified by unitary pointed fusion 2-categories, which are one-to-one labeled by a finite group $G$ and its group 4-cocycle $ω_4 \in \mathcal H^4[G;U(1)]$ up to group automorphisms. Furthermore, all such 3+1D topological orders can be realized by Dijkgraaf-Witten gauge theories.

cond-mat.str-el↗

arXiv2017

Pattern of Zeros

In this paper, we provide a set of definitions upon which one can prove in a rigorous way most of the main results achieved in the pattern of zeros classification of fractional quantum Hall states.

math-ph↗

arXiv2017

A mechanism of $\frac{e^2}{2h}$ conductance plateau without 1D chiral Majorana fermions

We address the question about the origin of the $\frac12 \frac{e^2}{h}$ conductance plateau observed in a recent experiment on an integer quantum Hall (IQH) film covered by a superconducting (SC) film. Since 1-dimensional (1D) chiral Majorana fermions on the edge of the above device can give rise to the half quantized plateau, such a plateau was regarded as a smoking-gun evidence for the chiral Majorana fermions. However, in this paper we give another mechanism for the $\frac12 \frac{e^2}{h}$ conductance plateau. We find the $\frac12 \frac{e^2}{h}$ conductance plateau to be a general feature of a good electric contact between the IQH film and SC film, and cannot distinguish the existence or the non-existence of 1D chiral Majorana fermions. We also find that the contact conductance between SC and an IQH edge channel has a non-Ohmic form $σ_\text{SC-Hall} \propto V^2$ in $k_BT \ll eV$ limit, if the SC and IQH bulks are fully gapped.

cond-mat.supr-con↗

arXiv2017

Four revolutions in physics and the second quantum revolution -- a unification of force and matter by quantum information

Newton's mechanical revolution unifies the motion of planets in the sky and falling of apple on earth. Maxwell's electromagnetic revolution unifies electricity, magnetism, and light. Einstein's relativity revolution unifies space with time, and gravity with space-time distortion. The quantum revolution unifies particle with waves, and energy with frequency. Each of those revolution changes our world view. In this article, we will describe a revolution that is happening now: the second quantum revolution which unifies matter/space with information. In other words, the new world view suggests that elementary particles (the bosonic force particles and fermionic matter particles) all originated from quantum information (qubits): they are collective excitations of an entangled qubit ocean that corresponds to our space. The beautiful geometric Yang-Mills gauge theory and the strange Fermi statistics of matter particles now have a common algebraic quantum informational origin.

physics.gen-ph↗

arXiv2017

Hierarchy construction and non-Abelian families of generic topological orders

We generalize the hierarchy construction to generic 2+1D topological orders (which can be non-Abelian) by condensing Abelian anyons in one topological order to construct a new one. We show that such construction is reversible and leads to a new equivalence relation between topological orders. We refer to the corresponding equivalent class (the orbit of the hierarchy construction) as "the non-Abelian family". Each non-Abelian family has one or a few root topological orders with the smallest number of anyon types. All the Abelian topological orders belong to the trivial non-Abelian family whose root is the trivial topological order. We show that Abelian anyons in root topological orders must be bosons or fermions with trivial mutual statistics between them. The classification of topological orders is then greatly simplified, by focusing on the roots of each family: those roots are given by non-Abelian modular extensions of representation categories of Abelian groups.

cond-mat.str-el↗

arXiv2017

Boundary-bulk relation in topological orders

In this paper, we study the relation between an anomaly-free $n+$1D topological order, which are often called $n+$1D topological order in physics literature, and its $n$D gapped boundary phases. We argue that the $n+$1D bulk anomaly-free topological order for a given $n$D gapped boundary phase is unique. This uniqueness defines the notion of the "bulk" for a given gapped boundary phase. In this paper, we show that the $n+$1D "bulk" phase is given by the "center" of the $n$D boundary phase. In other words, the geometric notion of the "bulk" corresponds precisely to the algebraic notion of the "center". We achieve this by first introducing the notion of a morphism between two (potentially anomalous) topological orders of the same dimension, then proving that the notion of the "bulk" satisfies the same universal property as that of the "center" of an algebra in mathematics, i.e. "bulk = center". The entire argument does not require us to know the precise mathematical description of a (potentially anomalous) topological order. This result leads to concrete physical predictions.

cond-mat.str-el↗

arXiv2017

Exactly soluble local bosonic cocycle models, statistical transmutation, and simplest time-reversal symmetric topological orders in 3+1D

We propose a generic construction of exactly soluble \emph{local bosonic models} that realize various topological orders with gappable boundaries. In particular, we construct an exactly soluble bosonic model that realizes a 3+1D $Z_2$ gauge theory with emergent fermionic Kramer doublet. We show that the emergence of such a fermion will cause the nucleation of certain topological excitations in space-time without pin$^+$ structure. The exactly soluble model also leads to a statistical transmutation in 3+1D. In addition, we construct exactly soluble bosonic models that realize 2 types of time-reversal symmetry enriched $Z_2$-topological orders in 2+1D, and 20 types of simplest time-reversal symmetry enriched topological (SET) orders which have only one non-trivial point-like and string-like topological excitations. Many physical properties of those topological states are calculated using the exactly soluble models. We find that some time-reversal SET orders have point-like excitations that carry Kramer doublet -- a fractionalized time-reversal symmetry. We also find that some $Z_2$ SET orders have string-like excitations that carry anomalous (non-on-site) $Z_2$ symmetry, which can be viewed as a fractionalization of $Z_2$ symmetry on strings. Our construction is based on cochains and cocycles in algebraic topology, which is very versatile. In principle, it can also realize emergent topological field theory beyond the twisted gauge theory.

cond-mat.str-el↗

arXiv2017

Gapped spin liquid with $\mathbb{Z}_2$-topological order for kagome Heisenberg model

We apply symmetric tensor network state (TNS) to study the nearest neighbor spin-1/2 antiferromagnetic Heisenberg model on Kagome lattice. Our method keeps track of the global and gauge symmetries in TNS update procedure and in tensor renormalization group (TRG) calculation. We also introduce a very sensitive probe for the gap of the ground state -- the modular matrices, which can also determine the topological order if the ground state is gapped. We find that the ground state of Heisenberg model on Kagome lattice is a gapped spin liquid with the $\mathbb{Z}_2$-topological order (or toric code type), which has a long correlation length $ξ\sim 10$ unit cell length. We justify that the TRG method can handle very large systems with over thousands of spins. Such a long $ξ$ explains the gapless behaviors observed in simulations on smaller systems with less than 300 spins or shorter than 10 unit cell length. We also discuss experimental implications of the topological excitations encoded in our symmetric tensors.

cond-mat.str-el↗

arXiv2017

Zoo of quantum-topological phases of matter

What are topological phases of matter? First, they are phases of matter at zero temperature. Second, they have a non-zero energy gap for the excitations above the ground state. Third, they are disordered liquids that seem have no feature. But those disordered liquids actually can have rich patterns of many-body entanglement representing new kinds of order. This paper will give a simple introduction and a brief survey of topological phases of matter. We will first discuss topological phases that have topological order (ie with long range entanglement). Then we will cover topological phases that have no topological order (ie with only short-range entanglement).

cond-mat.str-el↗

arXiv2017

A Novel Non-Perturbative Lattice Regularization of an Anomaly-Free $1 + 1d$ Chiral $SU(2)$ Gauge Theory

We present a numerical treatment of a novel non-perturbative lattice regularization of a $1+1d$ $SU(2)$ Chiral Gauge Theory. Our approach follows recent proposals that exploit the newly discovered connection between anomalies and topological (or entangled) states to show how to create a lattice regularization of any anomaly-free chiral gauge theory. In comparison to other methods, our regularization enjoys on-site fermions and gauge action, as well as a physically transparent fermion Hilbert space. We follow the `mirror fermion' approach, in which we first create a lattice regularization of both the chiral theory and its mirror conjugate and then introduce interactions that gap out only the mirror theory. The connection between topological states and anomalies shows that such interactions exist if the chiral theory is free of all quantum anomalies. Instead of numerically intractable fermion-fermion interactions, we couple the mirror theory to a Higgs field driven into a symmetry-preserving, disordered, gapped phase.

hep-lat↗

arXiv2017

Loop optimization for tensor network renormalization

We introduce a tensor renormalization group scheme for coarse-graining a two-dimensional tensor network that can be successfully applied to both classical and quantum systems on and off criticality. The key innovation in our scheme is to deform a 2D tensor network into small loops and then optimize the tensors on each loop. In this way, we remove short-range entanglement at each iteration step and significantly improve the accuracy and stability of the renormalization flow. We demonstrate our algorithm in the classical Ising model and a frustrated 2D quantum model.

cond-mat.str-el↗