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arXiv:2212.12935 · SciPost Physics

Adiabatic deformations of quantum Hall droplets

Blagoje Oblak, Benoit Estienne

Published Oct 13, 2023• 3 citations
arXivHTMLPDF

Abstract

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.

DOI
10.21468/scipostphys.15.4.159

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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.
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.
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.
Geometric phases in quantum mechanics play an extraordinary role in broadening our understanding of fundamental significance of geometry in nature. One of the best known examples is the Berry phase [M.V. Berry (1984), Proc. Royal. Soc. London A, 392:45], which naturally emerges in quantum adiabatic evolution. So far the applicability and measurements of the Berry phase were mostly limited to systems of weakly interacting quasi-particles, where interference experiments are feasible. Here we show how one can go beyond this limitation and observe the Berry curvature, and hence the Berry phase, in generic systems as a nonadiabatic response of physical observables to the rate of change of an external parameter. These results can be interpreted as a dynamical quantum Hall effect in a parameter space. The conventional quantum Hall effect is a particular example of the general relation if one views the electric field as a rate of change of the vector potential. We illustrate our findings by analyzing the response of interacting spin chains to a rotating magnetic field. We observe the quantization of this response, which we term the rotational quantum Hall effect.
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.
The Berry curvature and its descendant, the Berry phase, play an important role in quantum mechanics. They can be used to understand the Aharonov-Bohm effect, define topological Chern numbers, and generally to investigate the geometric properties of a quantum ground state manifold. While Berry curvature has been well studied in the regimes of few-body physics and noninteracting particles, its use in the regime of strong interactions is hindered by the lack of numerical methods to solve for it. In this paper I fill this gap by implementing a quantum Monte Carlo method to solve for the Berry curvature, based on interpreting Berry curvature as a leading correction to imaginary time ramps. I demonstrate my algorithm using the transverse-field Ising model in one and two dimensions, the latter of which is nonintegrable. Despite the fact that the Berry curvature gives information about the phase of the wave function, I show that the algorithm has no sign or phase problem for standard sign-problem-free Hamiltonians. My algorithm scales similarly to conventional methods as a function of system size and energy gap, and therefore should prove a valuable tool in investigating the quantum geometry of many-body systems.