Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Spectral form factors and late time quantum chaos

Junyu Liu

Abstract

This is a collection of notes that are about spectral form factors of standard ensembles in the random matrix theory, written for the practical usage of current study of late time quantum chaos. More precisely, we consider Gaussian Unitary Ensemble (GUE), Gaussian Orthogonal Ensemble (GOE), Gaussian Symplectic Ensemble (GSE), Wishart-Laguerre Unitary Ensemble (LUE), Wishart-Laguerre Orthogonal Ensemble (LOE), and Wishart-Laguerre Symplectic Ensemble (LSE). These results and their physics applications cover a three-fold classification of late time quantum chaos in terms of spectral form factors.

Spectral form factors and late time quantum chaos

Abstract

This is a collection of notes that are about spectral form factors of standard ensembles in the random matrix theory, written for the practical usage of current study of late time quantum chaos. More precisely, we consider Gaussian Unitary Ensemble (GUE), Gaussian Orthogonal Ensemble (GOE), Gaussian Symplectic Ensemble (GSE), Wishart-Laguerre Unitary Ensemble (LUE), Wishart-Laguerre Orthogonal Ensemble (LOE), and Wishart-Laguerre Symplectic Ensemble (LSE). These results and their physics applications cover a three-fold classification of late time quantum chaos in terms of spectral form factors.

Paper Structure

This paper contains 36 sections, 7 theorems, 167 equations, 10 figures.

Table of Contents

  1. Overview
  2. GUE spectral form factor
  3. Random matrix theory overview
  4. Two point form factor
  5. The disconnected piece
  6. The connected piece: box approximation
  7. The connected piece: an improvement
  8. Higher point form factor: theorem
  9. Four point form factor
  10. The first term
  11. The second term
  12. The final result
  13. Finite temperature result
  14. GOE/GSE spectral form factor
  15. Random matrix theory review
  16. ...and 21 more sections

Key Result

Theorem 2.1

We have the following formula to compute the convolution of the sine kernel: where $s$ is the sine kernel and the principle valued Fourier transform of the sine kernel is given by

Figures (10)

  • Figure 1: GOE, GUE, GSE two point form factors $\mathcal{R}_2(t)$ with box cutoff and infinite temperature. We choose $L=100$. Up: full form factor; Down: connected form factor.
  • Figure 2: GOE, GUE, GSE four point form factors $\mathcal{R}_4(t)$ with box cutoff and infinite temperature. We choose $L=1000$.
  • Figure 3: LOE, LUE, LSE two point form factors $\mathcal{R}_2(t)$ with box cutoff and infinite temperature. We choose $L=100$. Up: full form factor; Down: connected form factor.
  • Figure 4: LOE, LUE, LSE four point form factors $\mathcal{R}_4(t)$ with box cutoff and infinite temperature. We choose $L=1000$.
  • Figure 5: A direct comparison between Gaussian ensembles and Wishart-Laguerre ensembles in terms of two point form factor $\mathcal{R}_2(t)$ with box cutoff and infinite temperature. We choose $L=100$. Up GOE/LUE; Middle: GUE/LUE; Down: GSE/LSE.
  • ...and 5 more figures

Theorems & Definitions (8)

  • Theorem 2.1: Convolution formula for infinite $L$, in eq.5.2.23, book2
  • proof
  • Theorem 2.2: Convolution formula for finite large $L$
  • Theorem 3.1: Convolution formula for GOE
  • Theorem 3.2: Convolution formula for GSE
  • Theorem 4.1: Convolution formula for LUE
  • Theorem 4.2: Convolution formula for LOE
  • Theorem 4.3: Convolution formula for LSE