Gaussian, exponential, and power-law decay of time-dependent correlation functions in quantum spin chains

TL;DR

This paper provides exact calculations of time-dependent spin correlation functions in a 1D quantum spin chain, revealing Gaussian, exponential, and power-law decay behaviors depending on temperature and boundary conditions.

Contribution

It offers the first exact numerical analysis of correlation decay laws in the 1D XX model across various temperature regimes and boundary conditions.

Findings

Gaussian decay at infinite temperature

Exponential decay at finite temperatures

Power-law decay at zero temperature

Abstract

Dynamic spin correlation functions $<S_i^x (t)S_j^x>$ for the 1D $S=1/2$ $XX$ model $H = -J\Sigma_i \{S_i^x S_{i+1}^x + S_i^y S_{i+1}^y \}$ are calculated exactly for finite open chains with up to N=10000 spins. Over a certain time range the results are free of finite-size effects and thus represent correlation functions of an infinite chain (bulk regime) or a semi-infinite chain (boundary regime). In the bulk regime, the long-time asymptotic decay as inferred by extrapolation is Gaussian at $T=\infty$, exponential at $0 < T < \infty$, and power-law $(\sim t^{-1/2})$ at T=0, in agreement with exact results. In the boundary regime, a power-law decay obtains at all temperatures; the characteristic exponent is universal at T=0 $(\sim t^{-1})$ and at $0 < T < \infty$ $(\sim t^{-3/2})$, but is site-dependent at $T=\infty$. In the high-temperature regime $(T/J \gg 1)$ and in the low-temperature regime $(T/J \ll 1)$, crossovers between different decay laws can be observed in $<S_i^x (t)S_j^x>$. Additional crossovers are found between bulk-type and boundary-type decay for $i=j$ near the boundary, and between space-like and time-like behavior for $i \neq j$.