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

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 $i \neq j$.