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arXiv:cond-mat/9501079 · Physical review. B, Condensed matter

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

Joachim Stolze, Angela Nöppert, Gerhard H. Muller

Published Aug 1, 1995• 52 citations
arXivHTMLPDF

Abstract

Dynamic spin correlation functions 〈${\mathit{S}}_{\mathit{i}}^{\mathit{x}}$(t)${\mathit{S}}_{\mathit{j}}^{\mathit{x}}$〉 for the one-dimensional S=1/2 XX model H=-${\mathit{Jtsum}}_{\mathit{i}}${${\mathit{S}}_{\mathit{i}}^{\mathit{x}}$${\mathit{S}}_{\mathit{i}+1}^{\mathit{x}}$+${\mathit{S}}_{\mathit{i}}^{\mathit{y}}$${\mathit{S}}_{\mathit{i}}$ $_{+1}^{\mathit{y}}\mathrm{}}$ are calculated exactly for finite open chains with up to N=10 000 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=\ensuremath{\infty}, exponential at 0T\ensuremath{\infty}, and power law (\ensuremath{\sim}${\mathit{t}}^{\mathrm{\ensuremath{-}}1/2}$) at T=0, in agreement with exact results. In the boundary regime, a power-law decay is obtained at all temperatures; the characteristic exponent is universal at T=0 (\ensuremath{\sim}${\mathit{t}}^{\mathrm{\ensuremath{-}}1}$) and at 0T\ensuremath{\infty}(\ensuremath{\sim}${\mathit{t}}^{\mathrm{\ensuremath{-}}3/2}$), but is site dependent at T=\ensuremath{\infty}. In the high-temperature regime (T/J\ensuremath{\gg}1) and in the low-temperature regime (T/J\ensuremath{\ll}1), crossovers between different decay laws can be observed in 〈${\mathit{S}}_{\mathit{i}}^{\mathit{x}}$(t)${\mathit{S}}_{\mathit{j}}^{\mathit{x}}$〉. Additional crossovers are found between bulk-tpe and boundary-type decay for i=j near the boundary, and between spacelike and timelike behavior for i\ensuremath{\ne}j.

DOI
10.1103/physrevb.52.4319

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The decay of (disorder-averaged) static spin correlation functions at T=0 for the one-dimensional spin-1/2 XXZ antiferromagnet with uniform longitudinal coupling JΔ and random transverse coupling Jλi is investigated by numerical calculations for ensembles of finite chains. At Δ=0 (XX model) the calculation is based on the Jordan-Wigner mapping to free lattice fermions for chains with up to N=100 sites. At Δ≠0 Lanczos diagonalizations are carried out for chains with up to N=22 sites. The longitudinal correlation function 〈Sz0Szr〉 is found to exhibit a power-law decay with an exponent that varies with Δ and, for nonzero Δ, also with the width of the λi-distribution. The results for the transverse correlation function 〈Sx0Sxr〉 show a crossover from power-law decay to exponential decay as the exchange disorder is turned on.
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We use the functional renormalization approach for quantum spin systems developed by Krieg and Kopietz [Phys. Rev. B 99, 060403(R) (2019)] to calculate the spin-spin correlation function $G(\mathbit{k},\ensuremath{\omega})$ of quantum Heisenberg magnets at infinite temperature. For small wave vectors $\mathbit{k}$ and frequencies $\ensuremath{\omega}$ we find that $G(\mathbit{k},\ensuremath{\omega})$ assumes in dimensions d>2d>2d>2 the diffusive form predicted by hydrodynamics. In three dimensions our result for the spin-diffusion coefficient D\mathcal{D}D is somewhat smaller than previous theoretical predictions based on the extrapolation of the short-time expansion, but is still about 3030%30 larger than the measured high-temperature value of D\mathcal{D}D in the Heisenberg ferromagnet ${\mathrm{Rb}}_{2}{\mathrm{CuBr}}_{4}\ifmmode\cdot\else\textperiodcentered\fi{}2{\mathrm{H}}_{2}\mathrm{O}$. In reduced dimensions $d\ensuremath{\le}2$ we find superdiffusion characterized by a frequency-dependent complex spin-diffusion coefficient $\mathcal{D}(\ensuremath{\omega})$ which diverges logarithmically in d=2d=2d=2, and as a power-law $\mathcal{D}(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{\ensuremath{-}1/3}$ in d=1d=1d=1. Our result in one dimension implies scaling with dynamical exponent z=3/2z=3/2z=3/2, in agreement with recent calculations for integrable spin chains. Our approach is not restricted to the hydrodynamic regime and allows us to calculate the dynamic structure factor $S(\mathbit{k},\ensuremath{\omega})$ for all wave vectors. We show how the short-wavelength behavior of $S(\mathbit{k},\ensuremath{\omega})$ at high temperatures reflects the relative sign and strength of competing exchange interactions.
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