Physical review. B, Condensed matter

Exact property of the autocorrelation function of a conserved spin component

Edward B. Brown

Published Jul 1, 1995• 4 citations

Abstract

An exact property of the autocorrelation function of a conserved spin component is established. Standard approximate forms of the spin autocorrelation function (power law, exponential, Gaussian) do not have this property.

DOI: 10.1103/physrevb.52.285

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Physical review. B, Condensed matter

Exact property of the autocorrelation function of a conserved spin component

Edward B. Brown

Published Jul 1, 1995• 4 citations

Publisher

Abstract

DOI: 10.1103/physrevb.52.285

Related papers

Autocorrelation Exponent of Conserved Spin Systems in the Scaling Regime Following a Critical Quench

Clément Sire

We study the autocorrelation function of a conserved spin system following a quench at the critical temperature. Defining the correlation length $L(t)\ensuremath{\sim}{t}^{1/z}$, we find that for times ${t}^{\ensuremath{'}}$ and $t$ satisfying $L({t}^{\ensuremath{'}})\ensuremath{\ll}L(t)\ensuremath{\ll}L({t}^{\ensuremath{'}}{)}^{\ensuremath{\phi}}$ well inside the scaling regime, the spin autocorrelation function behaves like $\ensuremath{\langle}s(t)s({t}^{\ensuremath{'}})\ensuremath{\rangle}\ensuremath{\sim}L({t}^{\ensuremath{'}}{)}^{\ensuremath{-}(d\ensuremath{-}2+\ensuremath{\eta})}[L({t}^{\ensuremath{'}})/L(t){]}^{{\ensuremath{\lambda}}_{c}^{\ensuremath{'}}}$. For the $O(n)$ model in the $n\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, we show that ${\ensuremath{\lambda}}_{c}^{\ensuremath{'}}=d+2$ and $\ensuremath{\phi}=z/2$. We give a heuristic argument suggesting that this result is, in fact, valid for any dimension $d$ and spin vector dimension $n$ . We present numerical simulations for the conserved Ising model in $d=1$ and $d=2$ , which are fully consistent with the present theory.We study the autocorrelation function of a conserved spin system following a quench at the critical temperature. Defining the correlation length $L(t)\ensuremath{\sim}{t}^{1/z}$, we find that for times ${t}^{\ensuremath{'}}$ and $t$ satisfying $L({t}^{\ensuremath{'}})\ensuremath{\ll}L(t)\ensuremath{\ll}L({t}^{\ensuremath{'}}{)}^{\ensuremath{\phi}}$ well inside the scaling regime, the spin autocorrelation function behaves like $\ensuremath{\langle}s(t)s({t}^{\ensuremath{'}})\ensuremath{\rangle}\ensuremath{\sim}L({t}^{\ensuremath{'}}{)}^{\ensuremath{-}(d\ensuremath{-}2+\ensuremath{\eta})}[L({t}^{\ensuremath{'}})/L(t){]}^{{\ensuremath{\lambda}}_{c}^{\ensuremath{'}}}$. For the $O(n)$ model in the $n\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, we show that ${\ensuremath{\lambda}}_{c}^{\ensuremath{'}}=d+2$ and $\ensuremath{\phi}=z/2$. We give a heuristic argument suggesting that this result is, in fact, valid for any dimension $d$ and spin vector dimension $n$. We present numerical simulations for the conserved Ising model in $d=1$ and $d=2$, which are fully consistent with the present theory.

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Markus Böhm, V. S. Viswanath, Joachim Stolze +1 more

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Physical review. B, Condensed matterJun 1, 199419 citations

The critical correlation function of the Ising model in a magnetic field and on the co-existence curve

J Hubbard

The critical spin-spin correlation function has been calculated up to and including terms of order epsilon for the Ising model in a magnetic field and on the co-existence curve for T<T c .

Journal of Physics C Solid State PhysicsJun 7, 19742 citations

t

satisfying $L({t}^{\ensuremath{'}})\ensuremath{\ll}L(t)\ensuremath{\ll}L({t}^{\ensuremath{'}}{)}^{\ensuremath{\phi}}$ well inside the scaling regime, the spin autocorrelation function behaves like $\ensuremath{\langle}s(t)s({t}^{\ensuremath{'}})\ensuremath{\rangle}\ensuremath{\sim}L({t}^{\ensuremath{'}}{)}^{\ensuremath{-}(d\ensuremath{-}2+\ensuremath{\eta})}[L({t}^{\ensuremath{'}})/L(t){]}^{{\ensuremath{\lambda}}_{c}^{\ensuremath{'}}}$. For the

O(n)

model in the $n\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, we show that ${\ensuremath{\lambda}}_{c}^{\ensuremath{'}}=d+2$ and $\ensuremath{\phi}=z/2$. We give a heuristic argument suggesting that this result is, in fact, valid for any dimension

d

and spin vector dimension

n

. We present numerical simulations for the conserved Ising model in

d=1

and

d=2

, which are fully consistent with the present theory.

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.

{\mathit{J}}_{\mathit{x}}

{\mathit{J}}_{\mathit{y}}

\ensuremath{\equiv}J,

{\mathit{J}}_{\mathit{z}}

) are investigated on the basis of (i) rigorous bounds in the time domain and (ii) a continued-fraction analysis in the frequency domain. We have found strong numerical evidence for spin diffusion in quantum spin models. For

{\mathit{J}}_{\mathit{z}}

/J increasing from zero, the results of the short-time expansion indicate a change of the bulk-spin xx-autocorrelation function from Gaussian decay to exponential decay. The continued-fraction analysis of the same dynamic quantity signals a change from exponential decay to power-law decay as

{\mathit{J}}_{\mathit{z}}

/J approaches unity and back to a more rapid decay upon further increase of that parameter. By contrast, the change in symmetry at

{\mathit{J}}_{\mathit{z}}

/J=1 has virtually no impact on the bulk-spin zz-autocorrelation function (as expected). Similar contrasting properties are observable in the boundary-spin autocorrelation functions.