拓扑绝缘体及其研究进展

拓扑绝缘体是当前凝聚态物理领域中的一个热点问题.这类材料的典型特征是体内元激发存在能隙,但在边界上具有受拓扑保护的无能隙边缘激发.从广义上讲,拓扑绝缘体可以分两大类:一类是破坏时间反演的量子霍尔体系,另一类是新近发现的时间反演不变的拓扑绝缘体.这些新材料的奇特物理性质和潜在的应用前景,使其倍受人们关注.文章对这种新奇物态的物理性质和研究进展做了简要的介绍.     

Boundary conformal field theory and tunneling of edge quasiparticles in non-Abelian topological states

We explain how (perturbed) boundary conformal field theory allows us to understand the tunneling of edge quasiparticles in non-Abelian topological states. The coupling between a bulk non-Abelian quasiparticle and the edge is due to resonant tunneling to a zero mode on the quasiparticle, which causes the zero mode to hybridize with the edge. This can be reformulated as the flow from one conformally invariant boundary condition to another in an associated critical statistical mechanical model. Tunneling from one edge to another at a point contact can split the system in two, either partially or completely. This can be reformulated in the critical statistical mechanical model as the flow from one type of defect line to another. We illustrate these two phenomena in detail in the context of the ν=5/2 quantum Hall state and the critical Ising model. We briefly discuss the case of Fibonacci anyons and conclude by explaining the general formulation and its physical interpretation.     

凝聚体中的拓扑量子态

探寻拓扑上非平庸的凝聚体物质状态,特别是其电子结构和输运性质,是当前凝聚体物理 学领域非常重要的前沿研究方向。本文讨论的大多数主题都与电子波函数的拓扑性质有关。全文 除简短的引言外,包括拓扑量子现象、各种拓扑相、拓扑性准粒子的异常输运性质、拓扑性集 体激发和耦合激发,以及继续发展的拓扑量子态研究五个章节。这些章节着重反映拓扑量子态研 究的各个侧面,汇总起来方可以凸显凝聚体中拓扑量子态的全貌。     

Holographic correspondence in topological superconductors

We analytically derive a compatible family of effective field theories that uniquely describe topological superconductors in 3D, their 2D boundary and their 1D defect lines. We start by deriving the topological field theory of a 3D topological superconductor in class DIII, which is consistent with its symmetries. Then we identify the effective theory of a 2D topological superconductor in class D living on the gapped boundary of the 3D system. By employing the holographic correspondence we derive the effective chiral conformal field theory that describes the gapless modes living on the defect lines or effective boundary of the class D topological superconductor. We demonstrate that the chiral central charge is given in terms of the 3D winding number of the bulk which by its turn is equal to the Chern number of its gapped boundary.     

磁性拓扑绝缘体中的量子化反常霍尔效应

文章从平常霍尔效应出发,介绍了反常霍尔效应及其内秉物理机制,并在此基础上介绍了其量子化版本——量子化反常霍尔效应.然后从拓扑有序态的角度,重点讨论了量子化反常霍尔效应与量子霍尔效应、量子自旋霍尔效应、拓扑绝缘体等之间的区别与内在联系.最后介绍了通过在拓扑绝缘体（Bi2Se3, Bi2Te3 和 Sb2Te3）薄膜中掺杂过渡金属元素（Cr 或 Fe）实现量子化反常霍尔效应的方法.     

二维拓扑绝缘体退相干效应研究

拓扑绝缘体是当前凝聚态物理研究的热点.退相干效应对该体系的影响的研究不仅有重要的理论意义,而且也是实现未来量子器件的不可或缺的前期工作.文章作者从理论上研究了退相干对二维拓扑绝缘体特别是量子自旋霍尔效应的影响.研究结果表明,作为量子自旋霍尔效应的标志的量子化纵向电阻平台对不破坏自旋记忆的退相干效应（普通退相干）不敏感,但却对破坏自旋记忆的退相干效应（自旋退相干）非常敏感.因此,该量子化平台只能在尺寸小于自旋退相干长度的介观样品中存在,从而解释了量子自旋霍尔效应实验中所观测到的结果(见Science ,20     

Optical tuning of Berry phase effect in topological insulators

We theoretically discuss the influence of driving laser field on the topological nature, one of the manifestation of the electron Berry phase effect, in two-dimensional electronic systems. Adiabatic change of the laser amplitude with circular polarization alters the “order parameter”, termed the Chern number, in topological insulator with broken time-reversal symmetry, resulting in photo-induced phase transition. The finding is an optical analog of the integer quantum Hall effect, that is triggered by the laser field instead of magnetic field. This parallelism suggests the similarity of effects to electron dynamics between circularly polarized light and magnetic field.     

二维拓扑绝缘体退相干效应研究

拓扑绝缘体是当前凝聚态物理研究的热点.退相干效应对该体系的影响的研究不仅有重要的理论意义,而且也是实现未来量子器件的不可或缺的前期工作.文章作者从理论上研究了退相干对二维拓扑绝缘体特别是量子自旋霍尔效应的影响.研究结果表明,作为量子自旋霍尔效应的标志的量子化纵向电阻平台对不破坏自旋记忆的退相干效应(普通退相干)不敏感,但却对破坏自旋记忆的退相干效应(自旋退相干)非常敏感.因此,该量子化平台只能在尺寸小于自旋退相干长度的介观样品中存在,从而解释了量子自旋霍尔效应实验中所观测到的结果(见Science,2007,318:766).同时,文章作者还定义了一个新的物理量,即自旋霍尔电阻,并发现该自旋霍尔电阻也有量子化平台.特别是该量子化平台对两种类型的退相干都不敏感.这说明在宏观样品中也能观测到自旋霍尔电阻的量子化平台,因此更能全面地反映量子自旋霍尔效应的拓扑特性.     

Effective field theory and integrability in two-dimensional Mott transition

We study the Mott transition in a two-dimensional lattice spinless fermion model with nearest neighbors density–density interactions. By means of a two-dimensional Jordan–Wigner transformation, the model is mapped onto the lattice XXZ spin model, which is shown to possess a quantum group symmetry as a consequence of a recently found solution of the Zamolodchikov tetrahedron equation. A projection (from three to two space–time dimensions) property of the solution is used to identify the symmetry of the model at the Mott critical point as , with deformation parameter q = −1. Based on this result, the low-energy effective field theory for the model is obtained and shown to be a lattice double Chern–Simons theory with coupling constant k = 1 (with the standard normalization). By further employing the effective filed theory methods, we show that the Mott transition that arises is of topological nature, with vortices in an antiferromagnetic array and matter currents characterized by a d-density wave order parameter. We also analyze the behavior of the system upon weak coupling, and conclude that it undergoes a quantum gas–liquid transition which belongs to the Ising universality class.     

Topological mass generation four-dimensional gauge theory

The Lagrangian of non-Abelian tensor gauge fields describes the interaction of the Yang–Mills and massless tensor bosons of increasing helicities. We have found a metric-independent gauge invariant density which is a four-dimensional analog of the Chern–Simons density. The Lagrangian augmented by this Chern–Simons-like invariant describes massive Yang–Mills boson, providing a gauge-invariant mass gap for a four-dimensional gauge field theory. We present invariant densities which can provide masses to the high-rank tensor bosons.     

Chern–Simons theory in the temporal gauge and knot invariants through the universal quantum R-matrix

In temporal gauge A₀=0$A_0=0$ the 3d Chern–Simons theory acquires quadratic action and an ultralocal propagator. This directly implies a 2d R-matrix representation for the correlators of Wilson lines (knot invariants), where only the crossing points of the contours projection on the xy plane contribute. Though the theory is quadratic, P-exponents remain non-trivial operators and R  -factors are easier to guess then derive. We show that the topological invariants arise if additional flag structure R³⊃R²⊃R¹${\mathbb{R}}^3\supset {\mathbb{R}}^2\supset {\mathbb{R}}^1$ (xy plane and a y line in it) is introduced, R is the universal quantum R  -matrix and turning points contribute the “enhancement” factors qρ$q^{\rho }$.     

Topological field patterns in the Yang-Mills theory

We present a formalism where the topological configurations of pure Yang-Mills theory are characterised using gauge fields alone. Here, we obtain an expression for the charges of these topologicalSO(3) gauge field configurations in terms of the Abelian vector potentials. In this formalism we analyse the ’t Hooft-Polyakov monopole solution.     

Skein theory and topological quantum registers: Braiding matrices and topological entanglement entropy of non-Abelian quantum Hall states

We study topological properties of quasi-particle states in the non-Abelian quantum Hall states. We apply a skein-theoretic method to the Read-Rezayi state whose effective theory is the SU(2)_K Chern-Simons theory. As a generalization of the Pfaffian (K = 2) and the Fibonacci (K = 3) anyon states, we compute the braiding matrices of quasi-particle states with arbitrary spins. Furthermore we propose a method to compute the entanglement entropy skein-theoretically. We find that the entanglement entropy has a nontrivial contribution called the topological entanglement entropy which depends on the quantum dimension of non-Abelian quasi-particle intertwining two subsystems.     

A scheme for a topological insulator field effect transistor

We propose a scheme for a topological insulator field effect transistor. The idea is based on the gate voltage control of the Dirac fermions in a ferromagnetic topological insulator channel with perpendicular magnetization connecting to two metallic topological insulator leads. Our theoretical analysis shows that the proposed device displays a switching effect with high on/off current ratio and a negative differential conductance with a good peak to valley ratio.     

Basic and equivariant cohomology in balanced topological field theory

We present a detailed algebraic study of the N=2 cohomological set-up describing the balanced topological field theory of Dijkgraaf and Moore. We emphasize the role of N=2 topological supersymmetry and internal symmetry by a systematic use of superfield techniques and of an covariant formalism. We provide a definition of N=2 basic and equivariant cohomology, generalizing Dijkgraaf’s and Moore’s, and of N=2 connection. For a general manifold with a group action, we show that: (i) the N=2 basic cohomology is isomorphic to the tensor product of the ordinary N=1 basic cohomology and a universal group theoretic factor; (ii) the affine spaces of N=2 and N=1 connections are isomorphic.     

Topological transitions in multi-band superconductors

The search for Majorana fermions has been concentrated in topological insulators or superconductors. In general, the existence of these modes requires the presence of spin–orbit interactions and of an external magnetic field. The former implies in having systems with broken inversion symmetry, while the latter breaks time reversal invariance. In a recent paper, we have shown that a two-band metal with an attractive inter-band interaction has non-trivial superconducting properties, if the k$k$-dependent hybridization is anti-symmetric in the wave-vector. This is the case, if the crystalline potential mixes states with different parities as for orbitals with angular momentum l$l$ and l+1$l+1$. In this paper we take into account the effect of an external magnetic field, not considered in the previous investigation, in a two-band metal and show how it modifies the topological properties of its superconducting state. We also discuss the conditions for the appearance of Majorana fermions in this system.     

纳米尺度下超导体和拓扑绝缘体的电输运特性

超导体和拓扑绝缘体研究是当前凝聚态物理领域中的重大课题.文章重点介绍了作者所在实验室在纳米超导和拓扑绝缘体电输运领域的实验进展,其中包括金属和铁磁纳米线中的超导近邻效应、半金属纳米线中的新奇超导特性、拓扑绝缘体薄膜中的量子输运以及超导态-拓扑量子态的相互作用等,并对该领域的进一步发展进行了展望.     

二维光/声学拓扑态

文章主要介绍光/声学拓扑态的基本概念和研究背景,评述了这一领域中二维体系研究的进展。首先简要回顾了霍尔效应及其与贝利曲率和对称性的关系,接着综合分析了不同对称性条件下的光和声拓扑态,包括光/声量子霍尔效应、Floquet拓扑绝缘体、量子自旋霍尔效应的相关工作,最后讨论了该领域可能的发展方向和前景。