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Exact bounds for dynamical critical exponents of transverse-field Ising chains with a correlated disorder

Tatsuhiko Shirai, Shu Tanaka

TL;DR

This work addresses how correlated disorder in a transverse-field Ising chain affects the dynamical critical exponent $z$ at quantum criticality. By correlating site-dependent transverse fields to neighboring couplings and analyzing weak and strong disorder regimes, the authors map the problem to a free-fermion formulation and derive exact bounds on $z$, showing $z=1$ for weak disorder and $z$ bounded by $Dig( frac{1}{2}+|s- frac{1}{2}|ig)+ frac{1}{2}$ and $D+1$ in the strong-disorder regime, with $s$ tuning the field distribution. They complement analytic bounds with numerical exact diagonalization, revealing that $z$ depends on the transverse-field tuning process and tends toward the lower bound in several regimes, though finite-size effects are significant near symmetric tuning. The results demonstrate that appropriate correlation in disorder can convert infinite dynamical scaling into a finite, tunable $z$, offering potential benefits for adiabatic quantum computation and suggesting directions for higher-dimensional generalizations.

Abstract

This study investigates the dynamical critical exponent of disordered Ising chains under transverse fields to examine the effect of a correlated disorder on quantum phase transitions. The correlated disorder, where the on-site transverse field depends on the nearest-neighbor coupling strengths connecting the site, gives a qualitatively different result from the uncorrelated disorder. In the uncorrelated disorder cases where the transverse field is either homogeneous over sites or random independently of the nearest-neighbor coupling strengths, the dynamical critical exponent is infinite. In contrast, in the presence of the correlated disorder, we analytically show that the dynamical critical exponent is finite. We also show that the dynamical critical exponent depends on the tuning process of the transverse field strengths.

Exact bounds for dynamical critical exponents of transverse-field Ising chains with a correlated disorder

TL;DR

This work addresses how correlated disorder in a transverse-field Ising chain affects the dynamical critical exponent at quantum criticality. By correlating site-dependent transverse fields to neighboring couplings and analyzing weak and strong disorder regimes, the authors map the problem to a free-fermion formulation and derive exact bounds on , showing for weak disorder and bounded by and in the strong-disorder regime, with tuning the field distribution. They complement analytic bounds with numerical exact diagonalization, revealing that depends on the transverse-field tuning process and tends toward the lower bound in several regimes, though finite-size effects are significant near symmetric tuning. The results demonstrate that appropriate correlation in disorder can convert infinite dynamical scaling into a finite, tunable , offering potential benefits for adiabatic quantum computation and suggesting directions for higher-dimensional generalizations.

Abstract

This study investigates the dynamical critical exponent of disordered Ising chains under transverse fields to examine the effect of a correlated disorder on quantum phase transitions. The correlated disorder, where the on-site transverse field depends on the nearest-neighbor coupling strengths connecting the site, gives a qualitatively different result from the uncorrelated disorder. In the uncorrelated disorder cases where the transverse field is either homogeneous over sites or random independently of the nearest-neighbor coupling strengths, the dynamical critical exponent is infinite. In contrast, in the presence of the correlated disorder, we analytically show that the dynamical critical exponent is finite. We also show that the dynamical critical exponent depends on the tuning process of the transverse field strengths.

Paper Structure

This paper contains 8 sections, 83 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Model
  3. Dynamical critical exponent
  4. Analytic results
  5. Numerical results
  6. Conclusion and outlook
  7. A fermionic view of the eigenvalue problem for the energy gap and the correlation function
  8. Average and typical correlation lengths

Figures (2)

  • Figure 1: (Color online) (a) System-size dependences of the energy gap $[\Delta]_{\mathrm{av}}$ at $D=1$ for various values of $s$ [$s=0$ (open squares), $s=0.1$ (open circles), $s=0.2$ (open triangles), $s=0.3$ (filled squares), $s=0.4$ (filled circles), $s=0.5$ (filled triangles)]. Data for $s=0.6, 0.7, 0.8, 0.9,$ and $1$ are omitted since the data for $s$ are overlapped with those for $1-s$. The bold lines and the dotted lines are obtained by fitting the data for $N\ge 10^3$ and the data for $10^2 \leq N \leq 10^3$, respectively. Each data point is obtained by averaging $1000$ realizations of disorder. (b) $s$-Dependence of the dynamical critical exponent $z$. $z$ is estimated by the regression analysis in Fig. (a) and the error bars denote the standard deviation for the slopes of the regression lines. Squares and Circles denote $z$ obtained by using the data for $N\ge 10^3$ and the data for $10^2 \leq N \leq 10^3$, respectively. Dotted lines give the lower bound of $z$ [see eq. (\ref{['zbound_strong']})].
  • Figure 2: (Color online) Average and typical correlation lengths $\xi$ and $\bar{\xi}$ as a function of $\Gamma-\Gamma_{\rm c}$ in the strong-disorder case. The parameters $N=8000$, $D=1$, and $s=0.5$ are used. The error bars denote the standard deviation for fitting the correlation functions, which are obtained by averaging $20$ realizations of disorder. Both correlation lengths are well described by the bold line with slope $-1$ in the log-log plot (i.e., $\nu=\bar{\nu}=1$).