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Quantum Hall States response to toroidal geometry deformation

Bruno Mera, José M. Mourão, João P. Nunes, Carolina Paiva

Abstract

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) $τ$-preserving Mabuchi geodesics in the space of Kähler metrics on the torus, hitting a curvature singularity in finite imaginary time. By restricting to $S^1$-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.

Quantum Hall States response to toroidal geometry deformation

Abstract

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) -preserving Mabuchi geodesics in the space of Kähler metrics on the torus, hitting a curvature singularity in finite imaginary time. By restricting to -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.

Paper Structure

This paper contains 18 sections, 3 theorems, 134 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Complex symplectomorphisms and Kähler structure deformations
  4. Geometric Quantization
  5. Generalized Coherent State Transform
  6. Theta Functions
  7. The Lowest Landau Level for flat geometry and non-periodic quadratic Hamiltonian
  8. One-Particle States
  9. Evolution of one-particle states under the gCST
  10. Integer Quantum Hall Effect
  11. Evolution of the fully filled LLL states under the gCST
  12. Fractional Quantum Hall Effect
  13. Evolution of Laughlin states under the gCST and the Tao-Thouless state
  14. Non-Flat Geometry and Periodic Hamiltonian
  15. One-Particle States and evolution under the gCST
  16. ...and 3 more sections

Key Result

Proposition 1

Let P be a polarization on a symplectic manifold $(N,\omega)$ with prequantization $(L,h(.,.),\nabla)$. Let $\delta_{P}$ be a square root of $K_{P}$. The tensor product partial connection $\nabla: \Gamma(\overline{P})\times \Gamma(L\otimes\delta_{P})\rightarrow \Gamma(L\otimes\delta_{P})$ can be wri and it does not depend on the choice of $s,\nu$ and thus it is defined globally.

Figures (11)

  • Figure 1: Density Profile for single particle states- $N_{\phi}=3$- $\tau = 5i$
  • Figure 2: Varying the number of particles while keeping $\tau = i$.
  • Figure 3: Varying $\tau_{s}$ while keeping $N=2$.
  • Figure 4: Varying the number of particles of the system while keeping $\tau = i$.
  • Figure 5: Varying $\tau_{s}$ while keeping $N_{e}=2$.
  • ...and 6 more figures

Theorems & Definitions (9)

  • Remark 1
  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 1: (see BrianHall
  • Definition 4
  • Proposition 2
  • Proposition 3
  • proof