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arXiv:1604.03988 · Physical review. B./Physical review. B

Geometric defects in quantum Hall states

Andrey Gromov

Published Aug 10, 2016• 29 citations
arXivHTMLPDF

Abstract

We describe a geometric (or gravitational) analog of the Laughlin quasiholes in fractional quantum Hall states. Analogously to the quasiholes, these defects can be constructed by an insertion of an appropriate vertex operator into the conformal block representation of a trial wave function; however, unlike the quasiholes these defects are extrinsic and do not correspond to true excitations of the quantum fluid. We construct a wave function in the presence of such defects and explain how to assign an electric charge and a spin to each defect and calculate the adiabatic, non-Abelian statistics of the defects. The defects turn out to be equivalent to the genons in that their adiabatic exchange statistics can be described in terms of representations of the mapping class group of an appropriate higher genus Riemann surface. We present a general construction that, in principle, makes it possible to calculate the statistics of ${\mathbb{Z}}_{n}$ genons for any ``parent'' topological phase. We illustrate the construction on the example of the Laughlin state and perform an explicit calculation of the braiding matrices. In addition to non-Abelian statistics, geometric defects possess a universal Abelian overall phase, determined by the gravitational anomaly.

DOI
10.1103/physrevb.94.085116

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We describe a geometric (or gravitational) analog of the Laughlin quasiholes in fractional quantum Hall states. Analogously to the quasiholes, these defects can be constructed by an insertion of an appropriate vertex operator into the conformal block representation of a trial wave function; however, unlike the quasiholes these defects are extrinsic and do not correspond to true excitations of the quantum fluid. We construct a wave function in the presence of such defects and explain how to assign an electric charge and a spin to each defect and calculate the adiabatic, non-Abelian statistics of the defects. The defects turn out to be equivalent to the genons in that their adiabatic exchange statistics can be described in terms of representations of the mapping class group of an appropriate higher genus Riemann surface. We present a general construction that, in principle, makes it possible to calculate the statistics of Zn{\mathbb{Z}}_{n}Zn​ genons for any ``parent'' topological phase. We illustrate the construction on the example of the Laughlin state and perform an explicit calculation of the braiding matrices. In addition to non-Abelian statistics, geometric defects possess a universal Abelian overall phase, determined by the gravitational anomaly.
It has recently been realized that a general class of non-Abelian defects can be created in conventional topological states by introducing extrinsic defects, such as lattice dislocations or superconductor-ferromagnet domain walls in conventional quantum Hall states or topological insulators. In this paper, we begin by placing these defects within the broader conceptual scheme of extrinsic twist defects associated with symmetries of the topological state. We explicitly study several classes of examples, including Z2{Z}_{2}Z2​ and Z3{Z}_{3}Z3​ twist defects, where the topological state with NNN twist defects can be mapped to a topological state without twist defects on a genus $g\ensuremath{\propto}N$ surface. To emphasize this connection we refer to the twist defects as genons. We develop methods to compute the projective non-Abelian braiding statistics of the genons, and we find the braiding is given by adiabatic modular transformations, or Dehn twists, of the topological state on the effective genus ggg surface. We study the relation between this projective braiding statistics and the ordinary non-Abelian braiding statistics obtained when the genons become deconfined, finite-energy excitations. We find that the braiding is generally different, in contrast to the Majorana case, which opens the possibility for fundamentally novel behavior. We find situations where the genons have quantum dimension 2 and can be used for universal topological quantum computing (TQC), while the host topological state is by itself nonuniversal for TQC.
We propose a geometric description of all gapped excitations in fractional quantum Hall phases that reveals several fundamental understandings with experimental consequences. These include a duality between the Hilbert space of multiple gapped ``graviton-like" spin 2 excitations at ν=1/3ν=1/3ν=1/3 Laughlin phase, and that of non-Abelian quasiholes of an irrational Haffnian conformal field theory. This leads us to construct microscopic wave functions for multiple higher-spin gapped neutral modes in the Laughlin phase. Carrying spin s≥2s \ge 2s≥2, they emerge from higher-order geometric deformations of the topological ground state, and live within the Gaffnian conformal Hilbert space asymptotically. We show that the full many-body Hilbert space of an FQH phase can be generated from superpositions of such geometric deformations, supporting a concrete geometric interpretation of all gapped excitations. We analyse the scattering between these higher spin modes and conjecture that they can have a long lifetime, and propose methods for their experimental detection.
The quantum Hall effect has been the prototypical example for many emerging fields of Physics, including the study of topological phases of matter, topological order, and topological insulators. The objects of interest in this thesis are anyons, which are elementary excitations in fractional quantum Hall (FQH) fluids. Anyons are quasiparticles with fractional charge and fractional statistics. The latter prop- erty makes it a subject of immense interest due to application in fault-tolerant quantum computing. Here, we employ a perspective that anyons are the appro- priate degree of freedom to describe the relevant physics of FQH systems. This is because when a topological phase is assumed and the energy gap goes to infinity, anyons (“quasiholes”) are the only possible elementary excitations, since they are not energetically punished by the correspondng model Hamiltonian. With respect to a realistic Hamiltonian, however, this choice of degree of freedom depends on the exact effective interaction in the system. For example, while the Laughlin state is the model state at filling factor n + 1/3 (where n is some integer), when the effective electron-electron interaction is more long-ranged, the quasiholes of a Gaffnian state may be excited and become the better degree of freedom to describe physics in the same filling factor. This perspective results in a surprising observa- tion that a Laughlin quasihole may fractionalize under certain conditions, leading to a quasiparticle of charge e/6 that can potentially be detected in experiments. Another important aspect of anyons is their intrinsic spin and the corresponding spin-statistics relation, which must account for the fact that FQH anyons are not point particles, but objects with finite sizes and intricate internal structures. Our investigation leads to a rigorous derivation of the general spin-statistics relation for anyons in Abelian FQH phases. Furthermore, we show it is possible to fully capture the important features of the statistics by a conformal Hilbert space (CHS) description. Our analysis provides the necessary tools to study such geometric effects, which we use to investigate how disorder in a realistic system may affect different experimental schemes for measuring anyon statistics.
Many trial wave functions for fractional quantum Hall states in a single Landau level are given by functions called conformal blocks, taken from some conformal field theory. Also, wave functions for certain paired states of fermions in two dimensions, such as px+ipy{p}_{x}+i{p}_{y}px​+ipy​ states, reduce to such a form at long distances. Here we investigate the adiabatic transport of such many-particle trial wave functions using methods from two-dimensional field theory. One context for this is to calculate the statistics of widely separated quasiholes, which has been predicted to be non-Abelian in a variety of cases. The Berry phase or matrix (holonomy) resulting from adiabatic transport around a closed loop in parameter space is the same as the effect of analytic continuation around the same loop with the particle coordinates held fixed (monodromy), provided the trial functions are orthonormal and holomorphic in the parameters so that the Berry vector potential (or connection) vanishes. We show that this is the case (up to a simple area term) for paired states (including the Moore-Read quantum Hall state) and present general conditions for it to hold for other trial states (such as the Read-Rezayi series). We argue that trial states based on a nonunitary conformal field theory do not describe a gapped topological phase, at least in many cases. By considering adiabatic variation in the aspect ratio of the torus, we calculate the Hall viscosity, a nondissipative viscosity coefficient analogous to Hall conductivity, for paired states, Laughlin states, and more general quantum Hall states. Hall viscosity is an invariant within a topological phase and is generally proportional to the ``conformal spin density'' in the ground state.
We investigate extrinsic wormholelike twist defects that effectively increase the genus of space in lattice versions of multicomponent fractional quantum Hall systems. Although the original band structure is distorted by these defects, leading to localized midgap states, we find that a new lowest flat band representing a higher genus system can be engineered by tuning local single-particle potentials. Remarkably, once local many-body interactions in this new band are switched on, we identify various Abelian and non-Abelian fractional quantum Hall states, whose ground-state degeneracy increases with the number of defects, i.e, with the genus of space. This sensitivity of topological degeneracy to defects provides a "proof of concept" demonstration that genons, predicted by topological field theory as exotic non-Abelian defects tied to a varying topology of space, do exist in realistic microscopic models. Specifically, our results indicate that genons could be created in the laboratory by combining the physics of artificial gauge fields in cold atom systems with already existing holographic beam shaping methods for creating twist defects.
We propose a method of realizing non-Abelian braiding of fractionalized quasiholes in the Laughlin fractional quantum Hall phase at ν=1/3ν=1/3ν=1/3 with realistic two-body interactions within the lowest Landau level. It is numerically shown that low-lying gapped excitations near ν=1/3ν=1/3ν=1/3 are contained almost entirely within the null space of the three-body Moore-Read model Hamiltonian. They are thus quantum fluids of non-Abelian quasiholes that are in principle physically accessible. In particular, Laughlin ground state can be described as a fluid of ``ψψψ-type" quasiholes formed by binding a magnetic flux with a Majorana fermion (MF), and the Laughlin quasiholes are described by the ``111-type'' quasiholes, which are magnetic fluxes without a MF attached. Within the Laughlin phase, Laughlin quasiholes can be locally fractionalized into non-Abelian quasiholes, when the strong attraction between them is overcome by properly designed one-body electronstatic trapping potentials. Extensive numerics with proper finite-size scaling corroborate this physical picture, and our study points to the possibility of realizing non-Abelian braiding within an Abelian topological phase in experiment without the need for fine-tuning realistic electron-electron interaction.