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arXiv:1611.05563 · Journal of Physics A Mathematical and Theoretical

Central charge from adiabatic transport of cusp singularities in the quantum Hall effect

Tankut Can

Published Mar 29, 2017• 8 citations
arXivHTMLPDF

Abstract

Abstract We study quantum Hall (QH) states on a punctured Riemann sphere. We compute the Berry curvature under adiabatic motion in the moduli space in the large N limit. The Berry curvature is shown to be finite in the large N limit and controlled by the conformal dimension of the cusp singularity, a local property of the mean density. Utilizing exact sum rules obtained from a Ward identity, we show that for the Laughlin wave function, the dimension of a cusp singularity is given by the central charge, a robust geometric response coefficient in the QHE. Thus, adiabatic transport of curvature singularities can be used to determine the central charge of QH states. We also consider the effects of threaded fluxes and spin-deformed wave functions. Finally, we give a closed expression for all moments of the mean density in the integer QH state on a punctured disk.

DOI
10.1088/1751-8121/aa640e

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Abstract We study quantum Hall (QH) states on a punctured Riemann sphere. We compute the Berry curvature under adiabatic motion in the moduli space in the large N limit. The Berry curvature is shown to be finite in the large N limit and controlled by the conformal dimension of the cusp singularity, a local property of the mean density. Utilizing exact sum rules obtained from a Ward identity, we show that for the Laughlin wave function, the dimension of a cusp singularity is given by the central charge, a robust geometric response coefficient in the QHE. Thus, adiabatic transport of curvature singularities can be used to determine the central charge of QH states. We also consider the effects of threaded fluxes and spin-deformed wave functions. Finally, we give a closed expression for all moments of the mean density in the integer QH state on a punctured disk.
Abstract In Can et al (2016 Phys. Rev. Lett. 117 ), quantum Hall states on singular surfaces were shown to possess an emergent conformal symmetry. In this paper, we develop this idea further and flesh out details on the emergent conformal symmetry in holomorphic adiabatic states, which we define in the paper. We highlight the connection between the universal features of geometric transport of quantum Hall states and holomorphic dimension of primary fields in conformal field theory. In parallel we compute the universal finite-size corrections to the free energy of a critical system on a hyperbolic sphere with conical and cusp singularities, thus extending the result of Cardy and Peschel for critical systems on a flat cone (Cardy and Peschel 1988 Nucl. Phys . B 300 377–92), and the known results for critical systems on polyhedra and flat branched Riemann surfaces.
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.
In this paper, we apply techniques of geometric quantization to study the response of the integer and fractional quantum Hall effects to toroidal geometry deformation. The main method is that of using complex time Hamiltonian evolution to induce the geometry change and then the associated generalized coherent state transforms (gCST) to find the evolution of the Laughlin states. We consider two kinds of deformations. The first are flat toroidal deformations. Although Laughlin states for all flat toroidal geometries have been thoroughly studied before, we believe that our approach via the gCST is novel. It also serves as a testing ground to study the non-flat Kähler deformations. The Hamiltonians used in the flat deformations are quadratic in the generators of translations and therefore non periodic. The second kind of deformations involve nonflat Kähler toroidal deformations, generated by global, thus bi-periodic, Hamiltonians on the torus. The corresponding imaginary time flows are (elliptic curve modulus) τττ-preserving Mabuchi geodesics in the space of Kähler metrics on the torus, hitting a curvature singularity in finite imaginary time. By restricting to S1S^1S1-invariant deformations we find explicit analytic expressions for the evolution of the toroidal geometry and of the Laughlin states all the way to the singularity.
We consider area-preserving deformations of the plane, acting on electronic wave functions through “quantomorphisms” that change both the underlying metric and the confining potential. We show that adiabatic sequences of such transformations produce Berry phases that can be written in closed form in terms of the many-body current and density, even in the presence of interactions. For a large class of deformations that generalize squeezing and shearing, the leading piece of the phase is a super-extensive Aharonov-Bohm term ( \propto N^2 ∝ N 2 for N N electrons) in the thermodynamic limit. Its gauge-invariant subleading partner only measures the current, whose dominant contribution to the phase stems from a jump at the edge in the limit of strong magnetic fields. This results in a finite Berry curvature per unit area, reminiscent of the Hall viscosity. We show that the latter is in fact included in our formalism, bypassing its standard derivation on a torus and suggesting realistic experimental setups for its observation in quantum simulators.
Representative wave functions, which encode the topological properties of the spin polarized fractional quantum Hall states in the lowest Landau level, can be expressed in terms of correlation functions in conformal field theories. Until now, the constructions have been restricted to flat geometries, but in this paper we generalize to the simplest curved geometry, namely that of a sphere. Except for being of interest for numerical studies, that usually are performed on a sphere, the response of the fractional quantum Hall (FQH) liquids to curvature can be used to detect a topological quantity, the shift S\mathcal{S}S which is the average orbital spin of the constituent electrons. We give explicit expressions for representative wave functions on the sphere, for the full Abelian FQH hierarchy, and calculate the corresponding shifts. These microscopic results, based on wave functions, agree with the predictions from the effective Chern-Simons field theory. The methods we develop can also be applied to the planar case. It gives simpler expressions for states with both quasiparticle and quasihole condensates, and allows us to give closed form expressions for a general state in the hierarchy, rather than finding the wave function on a case by case basis.
Using methods based on conformal field theory, we construct model wave functions on a torus with arbitrary flat metric for all chiral states in the abelian quantum Hall hierarchy. These functions have no variational parameters, and they transform under the modular group in the same way as the multicomponent generalizations of the Laughlin wave functions. Assuming the absence of Berry phases upon adiabatic variations of the modular parameter τττ, we calculate the quantum Hall viscosity and find it to be in agreement with the formula, given by Read, which relates the viscosity to the average orbital spin of the electrons. For the filling factor ν=2/5ν=2/5ν=2/5 Jain state, which is at the second level in the hierarchy, we compare our model wave function with the numerically obtained ground state of the Coulomb interaction Hamiltonian in the lowest Landau level, and find very good agreement in a large region of the complex τττ-plane. For the same example, we also numerically compute the Hall viscosity and find good agreement with the analytical result for both the model wave function and the numerically obtained Coulomb wave function. We argue that this supports the notion of a generalized plasma analogy that would ensure that wave functions obtained using the conformal field theory methods do not acquire Berry phases upon adiabatic evolution.