Transfer operator analysis of the parallel dynamics of disordered Ising chains

# Transfer operator analysis of the parallel dynamics of disordered Ising chains

Anthony C.C. Coolen and Koujin Takeda
Department of Mathematics, King’s College London, The Strand, London WC2R 2LS, UK; London Institute for Mathematical Sciences, 22 South Audley St, Mayfair, London W1K 2NY, UK; Department of Computational Intelligence and Systems Science, Tokyo Institute of Technology, 4259 Nagatsuda, Midori-ku, Yokohama 226-8502, Japan
Corresponding author. Email: ton.coolen@kcl.ac.uk
###### Abstract

We study the synchronous stochastic dynamics of the random field and random bond Ising chain. For this model the generating functional analysis methods of De Dominicis leads to a formalism with transfer operators, similar to transfer matrices in equilibrium studies, but with dynamical paths of spins and (conjugate) fields as arguments, as opposed to replicated spins. In the thermodynamic limit the macroscopic dynamics is captured by the dominant eigenspace of the transfer operator, leading to a relative simple and transparent set of equations that are easy to solve numerically. Our results are supported excellently by numerical simulations.

isordered Ising chains; parallel dynamics; transfer operator

d
\doi\issn

1478-6443 \issnp1478-6435 \jvol00 \jnum00 \jyear2011

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## 1 Introduction

In spite of the absence of equilibrium phase transitions at finite temperature in one-dimensional Ising chains, the dynamics of such systems (solved formally several decades ago [1, 2]) continue to be of interest in the context of ageing phenomena, see e.g. [3]. Disordered versions of such chains, with random bonds and/or random fields, generally require new techniques for solution, unless the disorder can be transformed away as for binary bonds. One method for solving disordered chains in equilibrium is based on iteration of partition functions for growing chains, constrained by the state of the last spin [4, 5, 6, 7, 8]. More recently such models were also solved by diagonalisation of replicated transfer matrices [9, 10]. The situation with the dynamics of disordered Ising chains is less satisfactory. Except for special cases, e.g. [11], our analytical methods are still under development, although it is clear from numerical simulations (e.g. [12, 13]) and from the equilibrium solution that the dynamical phenomenology of disordered Ising chains is rich. A renormalisation group approach was advocated in [14]. In this paper we use the generating functional method of [15] to handle the disorder, and show that this leads to a transfer operator formalism very similar to that found in equilibrium studies; we use parallel dynamics to keep computations simpler, but results for Glauber dynamics will be similar. While developing our study, another study was published [16], also with parallel dynamics but based on the cavity method, which appears to represent an alternative but mathematically equivalent perspective on some of our equations.

## 2 Model definitions

We consider Ising spins on a periodic one-dimensional chain. Their dynamics are given by a synchronous stochastic alignment to local fields of the form , with the convention for all , and with . Upon defining as the probability to find the system at time in state , this Markovian process can be written as

 pt+1(\boldmathσ)=∑\boldmathσ′Wt[\boldmathσ;\boldmathσ′]pt(\boldmathσ′),        Wt[% \boldmathσ;\boldmathσ′]=∏ieβσihi(\boldmathσ′;t)2cosh[βhi(\boldmathσ′;t)]. (1)

The parameter measures the noise in the dynamics, which is fully random for and fully deterministic for . The represent external fields of the form , with random frozen parts and weak time dependent perturbations that serve to define response functions. The bonds and the frozen fields are regarded as quenched disorder, drawn for each site independently from a distribution . We write averages over the process (1) as and averages over the disorder as . Upon removing the time dependent parts of the external fields, so that , the process (1) obeys detailed balance, and the equilibrium state will be of the Peretto [17] form

 p(\boldmathσ) = Z−1e−β~Hβ(\boldmathσ), (2) ~Hβ(\boldmathσ) = −∑iθiσi−1β∑ilog2cosh[βhi(\boldmathσ)]. (3)

The correlation and response functions and will be related by the FDT (fluctuation-dissipation theorem) [18]

 Gij(τ>0)=−β[Cij(τ+1)−Cij(τ−1)],      Gij(τ≤0)=0. (4)

## 3 Generating functional analysis

In order to analyse the macroscopic dynamics of the chain we concentrate on the calculation of the disorder averaged generating functional proposed in [15]:

 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯Z[\boldmathψ] = ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯⟨exp[−i∑i∑t

We isolate the local fields at times in the usual manner via delta functions, using the short-hand , which gives

 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯Z[\boldmathψ] = ∫{dhd^h}∑% \boldmathσ(0)…∑\boldmathσ(tm)p(% \boldmathσ(0))eNF[{\boldmathσ},{^h}] (6) ×∏i∏t

with the disorder dependent exponent

 F[{\boldmathσ},{^h}] = 1Nlog∏i∫dJdθ ~P(J,θ)e−i∑t{θ^hi(t)+J[^hi(t)σi−1(t)+^hi−1(t)σi(t)]}. (7)

To benefit from the linear nature of the chain we write (6) in terms of the single-site objects , , and , with analogous definitions for and . We also introduce the time shift matrix , with entries , and the vector , so that and . For factorised and homogeneous initial conditions we then obtain

 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯Z[\boldmathψ] = ∫∏i[dhid^hi]∑\boldmathσ1…\boldmathσNe−i∑i[^hi⋅~\boldmathθi+\boldmathψi⋅\boldmathσi] (8) ×∏i⟨\boldmathσi,hi,^hi|M|\boldmathσi−1,hi−1,^hi−1]⟩,

with a non-symmetric transfer operator , defined via:

 ⟨\boldmathσ,h,^h|M|\boldmathσ′,h′,^h′⟩ = p(σ(0))ei^h⋅h+β[\boldmathσ⋅Sh+σ(tm)h(tm−1)]∏t

Expression (8) is for dominated by the largest eigenvalue of (9), provided its spectrum is discrete at . In an equilibrium replica analysis [9, 10] the relevant kernel would have replicated spins as arguments; here the arguments are spin ‘paths’, field ‘paths’ and conjugate field ‘paths’ through time. The fields and were only introduced for generating perturbations, so we may expand in powers of these fields. To do this efficiently we define

 Tr[K]=∫dhd^h∑\boldmathσ⟨\boldmathσ,h,^h|K|\boldmathσ,h,^h⟩, (10) ⟨\boldmathσ,h,^h|S(t)|\boldmathσ′,h′,^h′⟩=σ(t)δ(h−h′)δ(^h−^h′)δ\boldmathσ,\boldmathσ′, (11) ⟨\boldmathσ,h,^h|^H(t)|\boldmathσ′,h′,^h′⟩=^h(t)δ(h−h′)δ(^h−^h′)δ% \boldmathσ,\boldmathσ′. (12)

Since for any and for , according to (5) and (8) respectively, we can be sure that and that all terms of order or order in our expansion (which can be written in terms of derivatives with respect to of ) must be zero. Thus we may write our expansion in the form

 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯Z[\boldmathψ] = 1−i∑itψi(t)Tr[S(t)MN]Tr[MN]−12∑itt′ψi(t)ψi(t′)Tr[S(t)S(t′)MN]Tr[MN] (13) −∑i

We may now use the usual relations [15] to express the quantities of interest in the spin chain in terms of derivatives of (5), e.g.
and , giving

 mi(t)=¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯⟨σi(t)⟩ = Tr[S(t)MN]Tr[MN], (14) Cij(t,t′)=¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯⟨σi(t)σj(t′)⟩ = Tr[MN+i−jS(t)Mj−iS(t′)]Tr[MN]    (i≤j), (15) Gij(t,t′)=lim~\boldmathθ→\boldmath0∂¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯⟨σi(t)⟩∂~θj(t′) = −iTr[MN−|i−j|S(t)M|i−j|^H′(t′)]Tr[MN]. (16)

Left- and right eigenvectors with different eigenvalues of (9) are always orthogonal, so we can write (9) in the form , in which the are eigenspace projection operators111If the spectrum of has continuous parts, the eigenvalue sum becomes an integral., with if . The operator exchanges dynamical information between sites at distance . Hence for all , and since for any , must have an eigenvalue . Provided the largest eigenvalue is isolated in the spectrum, it follows that , and the above expressions give

 limN→∞mi(t) = Tr[S(t)U(1)]/Tr[U(1)], (17) limN→∞Cij(t,t′) = Tr[U(1)S(t)Mj−iS(t′)]/Tr[U(1)]    (i≤j), (18) limN→∞Gij(t,t′) = −iTr[U(1)S(t)M|i−j|^H′(t′)]/Tr[U(1)]. (19)

The above quantities are disorder averages of quantities which, by carrying site indices, will not generally be self-averaging. Hence they will not describe the dynamics of an individual realisation of the chain, but averages over many such realisations. In contrast, the following quantities are expected to be self-averaging:

 m(t) =limN→∞1N∑imi(t) =Tr[S(t)U(1)]/Tr[U(1)], (20) C(t,t′) =limN→∞1N∑iCii(t,t′) =Tr[U(1)S(t)S(t′)]/Tr[U(1)], (21) G(t,t′) =limN→∞1N∑iGii(t,t′) =−iTr[U(1)S(t)^H′(t′)]/Tr[U(1)]. (22)

## 4 Spectral properties of the transfer operator

From now on we consider only chains with independently distributed bonds and fields, i.e. . This is the natural and technically easier scenario. To study the spectral properties of it will be helpful to write this operator as

 ⟨\boldmathσ,h,^h|M|\boldmathσ′,h′,^h′⟩ = (2π)−tmP[\boldmathσ|h]ei^h⋅h∫dJdθ ~P(J,θ)e−iθu⋅^h−iJ[^h⋅\boldmathσ′+^h′⋅\boldmathσ], (23)

with the probability of a spin exposed to field path to follow path :

 P[\boldmathσ|h] = p(σ(0))∏t

### 4.1 Reduction of left- and right-eigenvectors

On the right-eigenvectors of (23) we carry out the following transformation:

 uR(\boldmathσ,h,^h) = ∫dx∏t(2π)wR(% \boldmathσ,h,x)ei^h⋅xP[\boldmathσ|h]. (25)

Insertion into the eigenvalue equation reveals that , and after some trivial manipulations we obtain the simplified eigenvalue problem

 λwR(\boldmathσ,h) = ∑\boldmathσ′∫dh′wR(\boldmathσ′,h′)∫dJdθ ~P(J,θ)δ[h−θu−J\boldmathσ′]P[% \boldmathσ′|h′+J\boldmathσ]. (26)

Writing out the left-eigenvector equation immediately reveals that . We now carry out a simple Fourier transformation:

 uL(\boldmathσ,^h) = ∫dx∏t(2π)wL(% \boldmathσ,x)e−i^h⋅x. (27)

Insertion into the left-eigenvalue problem then gives

 λwL(\boldmathσ,h) = ∑\boldmathσ′∫dh′wL(\boldmathσ′,h′)∫dJdθ ~P(J,θ)δ[h−J\boldmathσ′]P[\boldmathσ′|h′+θu+J\boldmathσ]. (28)

The represent distributions of field path contributions, conditioned on spin paths . Given that they are connected via

 wR(\boldmathσ,h) = ∫dθ ~P(θ) wL(% \boldmathσ,h−θu). (29)

To see this we simply define the function and use (28) to establish that it obeys

 λw(\boldmathσ,h) = ∑\boldmathσ′∫dh′wL(\boldmathσ′,h′)∫dJdθdθ′~P(J,θ)~P(θ′)δ[h−θ′u−J% \boldmathσ′]P[\boldmathσ′|h′+θu+J\boldmathσ] (30) = ∑\boldmathσ′∫dh′ w(\boldmathσ′,h′)∫dJdθ ~P(J,θ)δ[h−θu−J\boldmathσ′]P[\boldmathσ′|h′+J\boldmathσ].

Hence obeys (26) and therefore (29) holds. We are now left with only one eigenvalue problem, and upon combining our results we may summarize

 uL(\boldmathσ,h,^h) = ∫dx e−i^h⋅x∏t(2π)ϕ(\boldmathσ,x), (31) uR(\boldmathσ,h,^h) = P[\boldmathσ|h]∫dx ei^h⋅(h−x)∏t(2π)∫dθ ~P(θ)ϕ(\boldmathσ,x−θu), (32)

with to be solved from

 λϕ(\boldmathσ,h) = ∑\boldmathσ′∫dh′ϕ(\boldmathσ′,h′)∫dJdθ ~P(J,θ)δ[h−J\boldmathσ′]P[\boldmathσ′|h′+θu+J\boldmathσ]. (33)

For one easily calculates that and that the only possible eigenvalue of (23) is .

### 4.2 Physical meaning of the λ=1 eigenfunctions

The fields experienced at site can be writen as , where . Apart from the periodicity constraint, all information communicated to site from spins at sites is channeled via . The conditional likelihood to observe at site , given we know that , thus obeys

 Pi(hR|\boldmathσ) = ∑\boldmathσ′∫dh′ Pi−1(h′|\boldmathσ′)P[\boldmathσ′|h′+θi−1u+Ji\boldmathσ]δ(hR−Ji% \boldmathσ′). (34)

The spin path at site is prescribed in , so no longer depends on or . Hence if we average (34) over the disorder we obtain

 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯Pi(hR|\boldmathσ) = ∑\boldmathσ′∫dh′ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯Pi−1(h′|\boldmathσ′) ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯P[\boldmathσ′|h′+θi−1u+Ji\boldmathσ]δ(hR−Ji\boldmathσ′). (35)

Disorder averaging removes any site dependence of , hence , where the latter is now a true conditional probability distribution, although not corresponding to any specific site, giving the final result

 ϕ(h|\boldmathσ) = limN→∞N−1∑i¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯Pi(h|% \boldmathσ)=limN→∞N−1∑i¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯⟨δ(h−Ji\boldmathσi−1)⟩|% \boldmathσi=\boldmathσ, (36) ϕ(h|\boldmathσ) = ∑\boldmathσ′∫dh′ϕ(h′|\boldmathσ′)∫dJdθ ~P(J,θ)δ(h−J\boldmathσ′)P[\boldmathσ′|h′+θu+J\boldmathσ]. (37)

Equation (37) is identical to (33) for . Expression (36) obeys causality, i.e. is independent of . It is reasonable to assume that for there is only one solution of (37) that obeys causality, and that non-causal solutions will be ruled out by time boundary conditions. Thus we may for identify . Similar arguments underly the cavity approach in [16], from which (33) can be recovered upon substituting the characteristics of the 1D chain.

## 5 Calculation of observables

To calculate the observables (20,21,22) we need the projection operator . If we make the reasonable assumption that for the eigenspace is not degenerate, we may use (31,32), , and to write

 ⟨\boldmathσ,h,^h|U(1)|\boldmathσ′,h′,^h′⟩ = γ−1uR(\boldmathσ,h,^h)uL(\boldmathσ′,h′,^h′) (38) =1γP[\boldmathσ|h]∫dxdx′ϕ(x|\boldmathσ)ϕ(x′|\boldmathσ′)∫dθ ~P(θ)ei^h⋅(h−x−θu)−i^h′⋅x′(2π)2tm,

with

 γ = ∑\boldmathσ∫dhd^h uR(\boldmathσ,h,^h)uL(\boldmathσ,h,^h) (39) = (2π)−tm∑\boldmathσ∫dθ ~P(θ)∫dxdx′ P[\boldmathσ|θu+x+x′]ϕ(x|\boldmathσ)ϕ(x′|\boldmathσ).

Since is independent of and is independent of , we can sum in (39) over and integrate over (in that order), resulting in the same expression for but with the replacement . Further iteration of this process leads to . Hence

 ⟨\boldmathσ,h,^h|U(1)|\boldmathσ′,h′,^h′⟩ = P[\boldmathσ|h]∫dxdx′ϕ(x|\boldmathσ)ϕ(x′|\boldmathσ′) (40) ×(2π)−tm∫dθ ~P(θ)ei^h⋅(h−x−θu)−i^h′⋅x′.

We can now write the dynamical observables (20,21,22) (using integration by parts in the response function, where we take ) in the physically transparent form

 m(t) = ∑\boldmathσσ(t)∫dxdx′ ϕ(x|\boldmathσ)ϕ(x′|\boldmathσ)∫dθ ~P(θ)P[\boldmathσ|θu+x+x′], (41) C(t,t′) = ∑\boldmathσσ(t)σ(t′)∫dxdx′ ϕ(x|% \boldmathσ)ϕ(x′|\boldmathσ)∫dθ ~P(θ)P[\boldmathσ|θu+x+x′], (42) G(t,t′) = ∑\boldmathσσ(t)∫dxdx′ ϕ(x|\boldmathσ)ϕ(x′|\boldmathσ)∫dθ ~P(θ)∫dh δ(h−θu−x−x′)∂P[\boldmathσ% |h]∂h(t′) (43) = β{C(t,t′+1)−∑\boldmathσ% σ(t)∫dxdx′ ϕ(x|