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Convergence to equilibrium under a random Hamiltonian

Fernando G. S. L. Brandão, Piotr Ćwikliński, Michał Horodecki, Paweł Horodecki, Jarosław Korbicz, Marek Mozrzymas

TL;DR

The paper shows that, for subsystems evolving under a Hamiltonian whose eigenbasis is Haar-random, equilibration times are governed by the mean inverse energy gaps rather than the smallest gap. By reducing Haar averages to traces over four-fold tensor products and employing the representation theory of the permutation group $S_4$, the authors derive an explicit formula for $\langle\|\rho_S(t)-\omega_S\|_2^2\rangle$ and identify conditions on degeneracies under which fast equilibration occurs. When averaging over Gaussian energy spectra with variance $\sigma^2 \sim (\log d)^2$, the distance decays roughly as $e^{-t^2\sigma^2}$, yielding a scale $t\sim 1/\log d$ for equilibration, which contrasts with smallest-gap bounds and highlights the importance of eigenvector structure. The work further shows that Haar randomness can be relaxed to unitary 4-designs, indicating broad applicability to random-like Hamiltonians, though real systems with locality may require additional constraints.

Abstract

We analyze equilibration times of subsystems of a larger system under a random total Hamiltonian, in which the basis of the Hamiltonian is drawn from the Haar measure. We obtain that the time of equilibration is of the order of the inverse of the arithmetic average of the Bohr frequencies. To compute the average over a random basis, we compute the inverse of a matrix of overlaps of operators which permute four systems. We first obtain results on such a matrix for a representation of an arbitrary finite group and then apply it to the particular representation of the permutation group under consideration.

Convergence to equilibrium under a random Hamiltonian

TL;DR

The paper shows that, for subsystems evolving under a Hamiltonian whose eigenbasis is Haar-random, equilibration times are governed by the mean inverse energy gaps rather than the smallest gap. By reducing Haar averages to traces over four-fold tensor products and employing the representation theory of the permutation group , the authors derive an explicit formula for and identify conditions on degeneracies under which fast equilibration occurs. When averaging over Gaussian energy spectra with variance , the distance decays roughly as , yielding a scale for equilibration, which contrasts with smallest-gap bounds and highlights the importance of eigenvector structure. The work further shows that Haar randomness can be relaxed to unitary 4-designs, indicating broad applicability to random-like Hamiltonians, though real systems with locality may require additional constraints.

Abstract

We analyze equilibration times of subsystems of a larger system under a random total Hamiltonian, in which the basis of the Hamiltonian is drawn from the Haar measure. We obtain that the time of equilibration is of the order of the inverse of the arithmetic average of the Bohr frequencies. To compute the average over a random basis, we compute the inverse of a matrix of overlaps of operators which permute four systems. We first obtain results on such a matrix for a representation of an arbitrary finite group and then apply it to the particular representation of the permutation group under consideration.

Paper Structure

This paper contains 10 sections, 21 theorems, 87 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Averaging over a random choice of the eigenbasis
  3. Average over time/energies
  4. Conclusions
  5. Appendix A: Proof of Proposition \ref{['prop:dist-U4']}
  6. Appendix b: Averages
  7. Appendix C: Inverse of the matrix $M$
  8. Properties of $M$ matrix for general representations
  9. Applications
  10. Miscellaneous facts about matrix $M^D$

Key Result

Proposition 1

The following relation holds where with $\sigma=\rho_{SB}(0)$, $\mathbb{F}_{34}=\mathbb{V}_{S_3:S_4}\otimes \mathbb{I}_{B_3:B_4}$, and the label $i=1,2,3,4,$ denoting a copy of the composite system $S_iB_i$.

Figures (1)

  • Figure 1: The composite system consisting of a system and a bath, governed by a random Hamiltonian with the eigenbasis drawn according to the Haar measure.

Theorems & Definitions (27)

  • Proposition 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Proposition 5
  • Proposition 6
  • Proposition 7
  • Definition 8
  • Definition 9
  • Proposition 10
  • ...and 17 more