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.     

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.     

Semi-infinite cohomology in conformal field theory and 2D gravity

We discuss various techniques for computing the semi-infinite cohomology of highest weight modules which arise in the BRST quantization of two dimensional field theories. In particular, we concentrate on two such theories - the G/H coset models and 2D gravity coupled to c ≤ 1 conformal matter.     

A curiosity concerning the role of coherent states in quantum field theory

In the usual Fock quantisation of fields in Minkowski space-time, one has the result that the expectation value of the quantum Hamiltonian in any coherent state equals the energy of the classical field at which the state is peaked. It is shown that this property can be used tocharacterise the usual Fock representation. It is also pointed out that the entire analysis goes through for a substantially more general class of systems including, in particular, Bose fields in arbitrary stationary space-times.     

Tailoring edge and interface states in topological metastructures exhibiting the acoustic valley Hall effect

In this study, we investigate the acoustic topological insulator or topological metastructure, where an acoustic wave can exist only in an edge or interface state instead of propagating in bulk. Breaking the structural symmetry enables the opening of the Dirac cone in the band structure and the generation of a new band gap, wherein a topological edge or interface state emerges.Further, we systematically analyze two types of topological states that stem from the acoustic valley Hall effect mechanism;one type is confined to the boundary, whereas the other type can be observed at the interface between two topologically different structures. Results denote that the selection of different boundaries along with appropriately designed interfaces provides the acoustic waves in the band gap range with abilities of one-way propagation, dual-channel propagation, immunity from backscattering at sharp corners, and/or transition between propagation at interfaces and boundaries. Furthermore, we show that the acoustic wave propagation paths can be tailored in diverse and arbitrary ways by combing the two aforementioned types of topological states.     

Towards a field theory of the plateau transitions in the integer quantum Hall effect

We suggest a procedure for calculating correlation functions of the local densities of states (DOS) at the plateau transitions in the integer quantum Hall effect (IQHE). We argue that their correlation functions are appropriately described in terms of the SL( )/SU(2) WZNW model (at the usual Ka –Moody point and with the level 6≤k≤8 ). In this model we have identified the operators corresponding to the local DOS, and derived the partial differential equation determining their correlation functions. The OPEs for powers of the local DOS obtained from this equation are in agreement with available results.     

Controllable simulation of topological phases and edge states with quantum walk

We simulate various topological phenomena in condense matter, such as formation of different topological phases, boundary and edge states, through two types of quantum walk with step-dependent coins. Particularly, we show that one-dimensional quantum walk with step-dependent coin simulates all types of topological phases in BDI family, as well as all types of boundary and edge states. In addition, we show that step-dependent coins provide the number of steps as a controlling factor over the simulations. In fact, with tuning number of steps, we can determine the occurrences of boundary, edge states and topological phases, their types and where they should be located. These two features make quantum walks versatile and highly controllable simulators of topological phases, boundary, edge states, and topological phase transitions. We also report on emergences of cell-like structures for simulated topological phenomena. Each cell contains all types of boundary (edge) states and topological phases of BDI family.     

Spin polarization of two-dimensional electron system in different Laughlin states and around filling factor

The spin configuration of the ground state of a two-dimensional electron system is investigated for different FQHE states from an analysis of circular polarization of time-resolved luminescence. The method clearly distinguishes between fully spin polarized, partially spin polarized and spin unpolarized FQHE ground states. We demonstrate that FQHE states which are spin unpolarized or partially polarized at low magnetic fields become fully spin polarized at high fields. Temperature dependence of the spin polarization reveals a nonmonotonic behavior at . At and the electron system is found to be fully spin polarized. This result does not indicate the existence of any skyrmionic excitations in high magnetic field limit. However, at the observed spin depolarization of electron system at and becomes broader for lower magnetic fields, so that full spin polarization remains only in a small vicinity of . Such a behavior could be considered as a precursor of skirmionic depolarization, which would dominate for smaller ratios between Zeeman and Coulomb energies.We demonstrate that the spin polarization of 2D-electron system at and can be strongly affected by hyperfine interaction between electrons and optically spin-oriented nuclears. This result is due to the fact that hyperfine interaction can both enhance and suppress effective Zeeman splitting in fixed external magnetic field.     

Enhanced photonic spin Hall effect with subwavelength topological edge states

Photonic structures offer unique opportunities for controlling light‐matter interaction, including the photonic spin Hall effect associated with the transverse spin‐dependent displacement of a light beam that propagates in specially designed optical media. However, due to small spin‐orbit coupling, the photonic spin Hall effect is usually weak at the nanoscale. Here we suggest theoretically and demonstrate experimentally, in both optics and microwave experiments, the photonic spin Hall effect enhanced by topologically protected edge states in subwavelength arrays of resonant dielectric particles. Based on direct near‐field measurements, we observe the selective excitation of the topological edge states controlled by the handedness of the incident light. Additionally, we reveal the main requirements to the symmetry of photonic structures to achieve the topology‐enhanced spin Hall effect, and also analyse the robustness of the photonic edge states against the long‐range coupling.

     

Noninvariant zeta-function regularization in quantum Liouville theory

We consider two possible zeta-function regularization schemes of quantum Liouville theory. One refers to the Laplace–Beltrami operator covariant under conformal transformations, the other to the naive noninvariant operator. The first produces an invariant regularization which however does not give rise to a theory invariant under the full conformal group. The other is equivalent to the regularization proposed by A.B. Zamolodchikov and Al.B. Zamolodchikov and gives rise to a theory invariant under the full conformal group.     

New observables in topological instantonic field theories

Instantonic theories are quantum field theories where all correlators are determined by integrals over the finite-dimensional space (space of generalized instantons). We consider novel geometrical observables in instantonic topological quantum mechanics that are strikingly different from standard evaluation observables. These observables allow jumps of special type for the trajectory (at the point of insertion of such observables). They do not (anti)commute with evaluation observables and raise the dimension of the space of allowed configurations, while the evaluation observables lower this dimension. We study these observables in geometric and operator formalisms. Simple examples are explicitly computed; they depend on the linking of points.     

Bulk and edge properties of the Chern-Simons Ginzburg-Landau theory for the fractional quantum Hall effect

The Chern-Simons Ginzburg-Landau theory for the fractional quantum Hall effect is studied in the presence of a confining potential We review the bulk properties of the model and discuss how the plateau formation emerges without any impurity potential. The effect is related to changes, by accumulation of charge, at the edge when the chemical potential is changed. Fluctuations about the ground state are examined and an expression is found for the velocity of the massless edge mode in terms of the confining potential. The effect of including spin is examined for the case when the system is fully polarized in the bulk. In general a spin texture may appear at the edge, and we examine this effect in the case of a small spin-down component. The low-frequency edge modes are examined and a third-order equation is found for velocities which indicates the presence of three different modes. The discussions are illustrated by numerical studies of the ground states, both for the one- and two-component cases.     

Arithmetic of Calabi–Yau varieties and rational conformal field theory

It is proposed that certain techniques from arithmetic algebraic geometry provide a framework which is useful to formulate a direct and intrinsic link between the geometry of Calabi–Yau manifolds and the underlying conformal field theory. Specifically, it is pointed out how the algebraic number field determined by the fusion rules of the conformal field theory can be derived from the number theoretic structure of the cohomological Hasse–Weil L-function determined by Artin’s congruent zeta function of the algebraic variety. In this context, a natural number theoretic characterization arises for the quantum dimensions in this geometrically determined algebraic number field.     

Inter-layer edge tunneling and transport properties in separately contacted double-layer quantum-Hall systems

A theory of transport in the quantum-Hall regime is developed for separately contacted double-layer electron systems. Inter-layer tunneling provides a channel for equilibration of the distribution functions in the two layers at the edge states. Resistances and transresistances for various configurations of the electrodes are calculated as functions of the inter-layer tunneling amplitude. Induced current in one of the layer by a current in the other is also calculated. It is shown that reflection at the leads causes change in the results for some electrode configurations. The results obtained in this work is consistent with recent experiments.     

Reprint of : Nanomagnet coupled to quantum spin Hall edge: An adiabatic quantum motor

The precessing magnetization of a magnetic islands coupled to a quantum spin Hall edge pumps charge along the edge. Conversely, a bias voltage applied to the edge makes the magnetization precess. We point out that this device realizes an adiabatic quantum motor and discuss the efficiency of its operation based on a scattering matrix approach akin to Landauer–Büttiker theory. Scattering theory provides a microscopic derivation of the Landau–Lifshitz–Gilbert equation for the magnetization dynamics of the device, including spin-transfer torque, Gilbert damping, and Langevin torque. We find that the device can be viewed as a Thouless motor, attaining unit efficiency when the chemical potential of the edge states falls into the magnetization-induced gap. For more general parameters, we characterize the device by means of a figure of merit analogous to the ZT value in thermoelectrics.     

原子霍尔效应中复合粒子和场描述

研究了原子霍尔效应中复合粒子描述方法，并进一步给出Chern－Simon－Gross－Pitaevskii(CSGP)有效场描述。研究结果表明从平均场和复合粒子的角度来看原子霍尔效应和电子霍尔效应是一致的。     

Some aspects of conformal field theories on the plane and higher genus Riemann surfaces

We review some aspects of conformal field theories on the plane as well as on higher genus Riemann surfaces.