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Weighted Time Averages and Weak Convergence to Equilibrium in Quantum Integrable Systems

Xinyu Liu

Abstract

This paper establishes a natural quantum counterpart of weak equilibration for statistical ensembles in integrable systems. For quantum systems with pure point spectrum, single-time expectation values under unitary evolution are typically quasiperiodic, and hence generally do not admit a pointwise limit as $t\to\infty$. To overcome this difficulty, we introduce a weighted time-averaging procedure and prove that the resulting averaged dynamics converge to the diagonal (dephased) equilibrium state. We further illustrate and validate the theoretical result through a three-spin quantum integrable model.

Weighted Time Averages and Weak Convergence to Equilibrium in Quantum Integrable Systems

Abstract

This paper establishes a natural quantum counterpart of weak equilibration for statistical ensembles in integrable systems. For quantum systems with pure point spectrum, single-time expectation values under unitary evolution are typically quasiperiodic, and hence generally do not admit a pointwise limit as . To overcome this difficulty, we introduce a weighted time-averaging procedure and prove that the resulting averaged dynamics converge to the diagonal (dephased) equilibrium state. We further illustrate and validate the theoretical result through a three-spin quantum integrable model.
Paper Structure (11 sections, 6 theorems, 131 equations, 4 figures)

This paper contains 11 sections, 6 theorems, 131 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quantum systems with pure point spectrum
  4. Weight Functions and Weak Convergence of Weighted Averaged States
  5. Main Results
  6. A General Weighted Dephasing Theorem
  7. Quantitative Acceleration for Quasiperiodic Signals with Finitely Many Frequencies
  8. Explicit Integrable Model and Numerical Verification
  9. An Explicit Quantum Integrable Model
  10. Numerical Verification
  11. Conclusion

Key Result

Theorem 1

Let $H$ be a self-adjoint operator with pure point spectrum, let $\rho_0$ be an initial density operator, and let $A\in \mathcal{B}(\mathcal{H})$ be a bounded observable. Let $\{e_m\}_{m\ge 1}$ be an orthonormal eigenbasis of $H$, satisfying Assume that Then, for any $p,q>0$,

Figures (4)

  • Figure 1: Compactly supported weight functions $w_{p,q}$ corresponding to the four parameter pairs $(p,q)=(0.5,0.5),(1,1),(2,2),(4,4)$. As $\min\{p,q\}$ increases, the weight becomes increasingly concentrated in the middle portion of the time window and suppresses the endpoint data more strongly.
  • Figure 2: Numerical behavior of the running averages. The upper panel shows the unweighted average and the four weighted averages over the intermediate window $40\le N\le 400$; the lower panel displays the signed deviation from the equilibrium value $0$ on a symlog vertical axis. The purpose of this figure is to highlight the process of approaching equilibrium, rather than the final asymptotic error.
  • Figure 3: Semilogarithmic decay plots of the errors $E_N^{\mathrm{unw}}$ and $E_N^{(p,q)}$ over the asymptotic window $650\le N\le 1200$. The weighted averages clearly outperform the unweighted average; for the present model, larger values of $\min\{p,q\}$ correspond to faster error decay.
  • Figure 4: Linearized test of the stretched-exponential convergence law. For each pair $(p,q)$, we plot $\log_{10}E_N^{(p,q)}$ against $N^{\zeta(p,q)}$, where $\zeta(p,q)=(1+1/\min\{p,q\})^{-1}$. The data points exhibit an approximately linear trend overall, thereby supporting the theoretical prediction $E_N^{(p,q)}\approx \exp(-cN^{\zeta(p,q)})$.

Theorems & Definitions (13)

  • Theorem 1
  • proof
  • Corollary 2
  • proof
  • Lemma 3
  • proof
  • Theorem 4
  • proof
  • Remark 5
  • Proposition 6
  • ...and 3 more