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Critical temperature and density of spin-flips in the anisotropic random field Ising model

We present analytical results for the strongly anisotropic random field Ising model, consisting of weakly interacting spin chains. We combine the mean-field treatment of interchain interactions with an analytical calculation of the average chain free energy (``chain mean-field'' approach). The free energy is found using a mapping on a Brownian motion model. We calculate the order parameter and give expressions for the critical random magnetic field strength below which the ground state exhibits long range order and for the critical temperature as a function of the random magnetic field strength. In the limit of vanishing interchain interactions, we obtain corrections to the zero-temperature estimate by Imry and Ma [Phys. Rev. Lett. 35, 1399 (1975)] of the ground state density of domain walls (spin-flips) in the one-dimensional random field Ising model. One of the problems to which our model has direct relevance is the lattice dimerization in disordered quasi-one-dimensional Peierls materials, such as the conjugated polymer trans-polyacetylene.


I. INTRODUCTION
Owing to the wide range of physical systems that can be described by Ising-like models, it is believed that the random field Ising model (RFIM), i.e., the Ising model with a spatially random magnetic field, captures the essential physics of many disordered systems. Therefore, over the past two decades this model has been extensively studied, both theoretically [1] and experimentally [2]. The most studied realization of the RFIM is the dilute antiferromagnet in a uniform field. This realization was first found by Fishman and Aharony [3] in 1979 and is still under current investigation [4,5]. Other equally interesting as diverse problems to which the RFIM has been found to be applicable are, e.g., the Anderson-Mott transition of disordered interacting electrons [6] and aspects of protein folding [7].
Much of the theoretical interest in the RFIM has originated from the early work by Imry and Ma [8]. In 1975, these authors noticed that in the presence of a random magnetic field the lowest energy state of the Ising model may have no long range order (LRO).
They considered the energetics of creating a "wrong" domain (i.e., a domain in which all spins have reversed signs) of linear size R in the ordered phase. The typical domain size R caused by the disorder fluctuations is determined by the balance between the energy cost of domain walls and the possible energy gain due to the decrease of the interaction energy between the reversed spins and the random field. In the one-dimensional case, this balance reads: where I is the energy of one domain wall and h 2 is the strength of the random magnetic field. This zero temperature estimate implies that in the one-dimensional random field Ising model LRO does not exist, as in an infinite system it is always possible to choose R large enough to make the creation of a "wrong" domain favorable. From Eq.(1), the zero temperature density of spin-flips is estimated to be [8,9]: The lower critical dimension above which LRO in the RFIM is possible is 2 [8,10].
In most previous studies of the RFIM, the interactions between the Ising spins were assumed to be isotropic, i.e., independent of the direction. These studies focused on the universal critical properties of the model, such as the critical exponents and the lower critical dimension, which do not depend on the degree of anisotropy of the interactions.
On the other hand, the anisotropic RFIM has several nonuniversal properties, such as the critical temperature, that are very interesting in relation to disordered quasi-onedimensional systems.
For example, we recently showed [11] that disordered Peierls chains with a doubly degenerate ground state can under certain conditions be described by the one-dimensional RFIM. In this mapping, the Ising variables describe the local lattice dimerization, while the random field corresponds to disorder in the electron hopping amplitudes. The spinflips induced by the random field then correspond to disorder-induced neutral solitons in Peierls chains. The density of spin-flips is related to the magnetic properties of the disordered Peierls systems, since the neutral solitons have spin 1/2 [12]. The conjugated polymer trans-polyacetylene is a well-known example of a Peierls system for which this mapping may be used. Taking into account the weak interchain interactions in quasione-dimensional crystalline trans-polyacetylene, the thermodynamics of solitons in this material may be described by the anisotropic RFIM [13]. In this context, the interesting properties of the anisotropic RFIM are the temperature of the three-dimensional Peierls transition, the order parameter, and the density of spin-flips as a function of the random field strength and (or) temperature. These properties strongly depend on the degree of anisotropy of the interactions.
In the present paper we study the d-dimensional anisotropic RFIM (d > 2), which consists of regularly arranged chains of spins in a random magnetic field (Sec. II). The interaction between spins in neighboring chains is supposed to be much smaller than the interaction within chains and will be treated in the mean-field approximation; the intrachain interactions will be treated exactly. Using the transfer matrix formalism and a mapping on a Brownian motion model, we find an analytical expression for the average free energy of the continuum version of our model (Sec. III). This key result is used in Sec. IV to obtain the order parameter, the critical temperature, and the density of spinflips. In Sec. IV A, we present analytical results for these quantities in the limit where thermal creation of spin-flips dominates their creation by the random magnetic field; in Sec. IV B we consider the opposite limit. We find the critical random field strength below which the ground state exhibits LRO; well below the critical strength, the density of spin-flips is found to be exponentially suppressed.
In the absence of the interchain interactions, the anisotropic model reduces to the one-dimensional RFIM. In this limiting case, our solution is exact. It shows that the estimate for the density of spin-flips Eq.(2) in a single chain is correct to lowest order in the random magnetic field strength and gives higher-order corrections to this expression.
Previous attempts to obtain an exact solution for the one-dimensional case [14] did not yield closed expressions for the free energy and the spin-flip density.
We conclude this paper by giving the physical interpretation of our results in Sec. V and summarizing in Sec. VI.

II. MODEL
We consider the anisotropic RFIM on a d-dimensional lattice with a total number of N d sites. The Ising spins occupying the lattice, interact with their nearest neighbors and with a magnetic field that has a random value at each lattice site. The anisotropy results from the fact that the exchange interaction, I, along one of the lattice directions is assumed to be much stronger than the interaction, I ⊥ , perpendicular to this direction: 0 < I ⊥ ≪ I (for definiteness, the interactions are chosen positive). The lattice is thus The energy of our model is given by: Here, σ iα = ±1 is the classical spin variable describing the two possible spin directions, up and down, at site i of chain α. The summations over i, i ′ and α, α ′ are restricted to nearest neighbor pairs. The random magnetic field h iα is assumed to be uncorrelated for different lattice sites, with vanishing mean and Gaussian correlator of strength ǫ: Here, . . . denotes the average over the random field realizations. Finally, we impose open boundary conditions on the model. This choice does not affect our results, as these will be derived for a large system, N → ∞.
For vanishing magnetic field strength ǫ, the ground state of the spin system Eq.(3) is ferromagnetically ordered. This is not necessarily the case when a finite random field is present: For a number of spins it may then be favorable to flip and align with the local direction of h iα to minimize the total energy.
To calculate the partition function of the model we treat the interchain interactions in the mean-field approximation. Then the energy Eq.(3) becomes a sum of single-chain energies and can be written as where B is the homogeneous mean field,Ĩ ⊥ equals I ⊥ times the number of nearestneighbour chains of one particular chain, and we took into account that the contributions of all N d−1 chains are on average equal. The mean field B satisfies the self-consistency equation: where . . . denotes both the thermal average and the average over random field realizations. The order parameter σ , which is the normalized difference between the average number of sites with spin up (N ↑ ) and spin down (N ↓ ), can be found from Here, F denotes the average over the random field realization of the free energy of a single chain in the mean field B. For a particular random field realization: where the chain energy E[σ, h] is given by the relevant part of Eq. (5), Using Eq.(8), the explicit form of the self-consistency condition (6) becomes: In view of the relative weakness of the interchain interactions (I ⊥ ≪ I), this so-called chain mean-field approximation [15] is expected to be accurate.
The disorder-averaged free energy, F , which we calculate analytically in the next section, can also be used to find other relevant thermodynamic properties. In particular, the average density of spin-flips inside a chain as a function of temperature and random field strength is obtained through: since I is the spin-flip creation energy. In the context of the degenerate ground state conjugated polymers, mentioned in the Introduction, n s directly gives the density of solitons within chains.

III. THE AVERAGE FREE ENERGY
To derive the continuum version of the discrete model Eq.(10), we follow the standard path [16,17] by first rewriting the chain partition function in the transfer matrix formalism. The partition function is thus considered as the expectation value of an ordered product of operatorsT i along the chain axis: Here, the final summation over σ, σ ′ accounts for all possible boundary conditions for the spins on the chain's ends. On the basis spanned by the two vectors | + and | − (corresponding to σ = +1 and σ = −1, respectively), the transfer matrixT i reads: T i depends on the site-index i through the random magnetic field.
A relation between the transfer matrix Eq. (14) In the continuum approximation the deviation ofT from the identity matrix is assumed to be small, so that the last expression can approximately be written aŝ whereĤ, obtained from Eq. (14), has the form: Here,σ 1 andσ 3 are the Pauli matrices and the random magnetic field is now Gaussian in time: and to be fulfilled simultaneously. While the first condition results in an upper limit for the temperature, T ≪ I, the latter implies a lower temperature limit: The continuum limit allows us to write the partition function Eq.(13) as a sum over matrix elements for the time-ordered propagation of a spin-1/2 particle in imaginary time: Here,T is the time-ordering operator and the integration over the dimensionless timevariable t has to be performed from the initial time t i = 0 to the final time t f = N corresponding to the chain length.
It is now convenient to rotate the spinor basis over an angle π/2 around the σ 2 -axis, which yields the transformed Hamiltonian: and To make this expression more compact, we introduce the wave function of the spin, which obeys the time-dependent Schrödinger-like equation, Together with the inital condition Ψ(t = 0) = | + , the partition function Eq.(23) now simply reads: We start by introducing the two time-dependent functions u(t) and v(t), such that the spin wave function Eq.(24) reads: 10 The initial condition, Ψ(t = 0) = | + , requires that u(t = 0) = 1 and v(t = 0) = 0, and from Eq.(25) we obtain two coupled differential equations for u(t) and v(t), and While Eq.(28) can be easily integrated, yielding Eq.(29) can be interpreted as the Langevin equation for the velocity v of a particle undergoing a one-dimensional Brownian motion. The last term in Eq.(29) is the random force acting on the particle, while the first and the second terms describe, respectively, a constant (bias) force and a friction.
In the absence of the random force (no random magnetic field), the velocity v approaches a time-independent value for t → ∞: In the presence of the random force, v(t) becomes a stochastic variable with a timedependent distribution, P (v, t), which may be obtained from the corresponding Fokker-Planck equation [18]. In the limit of long times (N → ∞), the time-independent equilibrium distribution, P eq (v), is reached, obeying: Here, the first and second moments are given by, respectively, and The solution of the Fokker-Planck equation (32) is easily found to be: where we introduced and the normalization coefficient N determined by: The equilibrium distribution Eq.(35) is centered around v ∞ (Eq.(31)), and in the limit Using our result for P eq (v), the average free energy of the RFIM can easily be obtained.
We first note that Eq.(30) enables us to write the partition function Eq.(26) in terms of v(t) alone: Omitting in the limit of long chains all terms that do not grow proportional to N, we can now replace the average of ln Z[h] over the random field realizations by the average over P eq (v). This leads to where K ν (z) denotes the modified Bessel function of order ν and K ′ ν (z) its derivative with respect to z [19].
The analytical solution Eq.(40) of the average free energy is the key result of this paper, from which all further results follow. In Fig. 1, this solution as a function of temperature is compared to results of a numerical simulation of the discrete RFIM at ǫ = 0.01I 2 for three values of the mean field B. Apart from the presence of the mean field, the simulation method is identical to the one described in Ref. [11]. Figure 1 shows that in the temperature region T ≪ I, the analytical solution is in excellent agreement with the simulation, except for very low temperatures, where the condition Eq. (20) is not met. It is observed that the analytical solution for the free energy reaches a maximum at T = T 0 (ǫ) = 0 (indicated by the thin line). The meaning of T 0 (ǫ) will be discussed in Sec. IV B.

IV. ANALYSIS OF RESULTS
Using Eq.(40), it is straightforward to solve numerically the self-consistency equation (11), from which the order parameter σ is found through Eq.(6). Figure 2 gives the thus obtained order parameter as a function of temperature and random-field strength for I ⊥ /I = 0.02. A low-temperature cut-off T 0 (ǫ) was used, as will be explained in detail in Here, two limiting cases should be distinguished. In Sec. IV A, we will consider the case of very weak disorder, where thermally induced spin-flips dominate the ones induced by the random magnetic field. In Sec. IV B, we consider the opposite limit.

A. Dominant thermal spin-flips
For a weak random magnetic field (ǫ → 0) the modified Bessel function in Eq. (40) has to be evaluated at large order and argument (ν, z ≫ 1). Using standard expansions [19], we obtain to first order in ǫ: The first term in the right hand side of Eq.(41) is the continuum approximation for the free energy of the discrete model Eq.(10) at ǫ = 0. Under the conditions specified by Eqs. (19) and (20) this term agrees with the exact expression, which can be obtained by diagonalization of the transfer-matrix Eq. (14) at h i = 0.
At the transition temperature, T c (ǫ), the order parameter vanishes. Therefore, by putting B = 0 in Eq.(43), we obtain an equation for T c (ǫ): In particular, at ǫ = 0 we obtain: This equation can be compared with the exact result for the critical temperature of the two-dimensional Ising model [20,21] : For I ⊥ ≪ I, both Eq.(46) and Eq. (47) give approximately: which shows that the chain mean-field approximation works well for the strongly anisotropic Ising model.
From Eq.(45) we find that at weak disorder the transition temperature decreases linearly with ǫ, where α is: Finally, the average density of spin-flips [see Eq. (12)] for weak disorder is given by . e −2βI 2 I ⊥ β + ǫβ 2 y 2 for T < T c (ǫ) We note, that in the disordered phase (above T c (ǫ)), no correction to the density of spinflips linear in ǫ occurs. This is related to the fact that in the weak disorder limit, the domain size R in Eq.(1) is limited by the distance between thermally induced spin-flips, which is too small to allow for the creation of spin-flips by the random field. In this case the lowest-order correction is ǫ 2 β 4 exp(βI)/8.

B. Dominant disorder-induced spin-flips
We now turn to the limit exp[−βI] ≪ β 2 ǫ, or equivalently: z ≪ 1, where the thermal creation of spin-flips is negligible compared to creation by the random magnetic field. To lowest order in z, we then have [19] in Eq.(40): To find the critical temperature, at which B → 0, it is sufficient to consider only small values of ν ∝ B, where Γ(1 ± ν) ≃ e ∓γν (γ ≃ 0.577 is Euler's constant), so that Eq.(52) further reduces to: Using this result and the definitions Eq.(36) and Eq.(37) of ν and z, respectively, the free energy Eq.(40) finally becomes: Here, we introduced with a characteristic temperature that depends on the strength of the random magnetic field.
Furthermore, we defined x, At T = T 0 , the free energy reaches a maximum (cf. Fig. 1), implying that the entropy vanishes. The general expression for the entropy in our model is: which becomes negative for T < T 0 . This unphysical behavior is also observed for isolated chains (I ⊥ = 0) [11]. Its origin is the continuum approximation, which breaks down at low temperatures. As we discussed in Ref. [11], the problem arises due to the fact that in confirmed that for the one-dimensional RFIM this indeed is a reasonable identification [11].
Using Eq.(54), the self-consistency condition Eq.(11) now takes the form: For σ ≪ 1, the non-trivial solution of Eq.(59) for the order parameter reads: At the critical temperature T c (ǫ), the left hand side of Eq.(60) vanishes, yielding  49)). The dash-dotted curve in Fig. 3 indicates the lower temperature limit T 0 (ǫ), which we argued to correspond to T = 0 in the discrete model.
The critical disorder strength ǫ c below which LRO exists in the ground state is a function of the interchain interaction and can be calculated from Eq.(61) by requiring that T c (ǫ) = T 0 (ǫ). This yields where It is seen that ǫ c is of the order ofĨ ⊥ I. For a random field strength ǫ ≪ ǫ c , the ground state (T = T 0 ) of the anisotropic RFIM is nearly perfectly ordered, σ ≃ 1, as is clear from Fig. 2.
We now turn to the average density of spin-flips within the chains, which using Eqs. (12) and (54), is obtained as: In the disordered phase (x = 0), Eq.(64) reduces to In particular, the latter expression holds for the one-dimensional RFIM, which is disordered at all temperatures [8,9]. Equation (65) was obtained previously in Ref. [11], where we discussed the density of kinks in isolated polymer chains. The temperature region where Eq.(65) is valid is specified by Eqs. (19) and (20). However, as we showed in Ref. [11], the density of spin-flips at zero temperature in the discrete RFIM is very close to the continuum result Eq.(65) at T = T 0 (ǫ):  Fig. 4. At this particular value of the interchain interaction, the critical disorder strength is ǫ c ≃ 0.0071I 2 . It is observed from Fig. 4 that well below this critical value, the spin-flip density is strongly suppressed. In fact, for ǫ ≪ ǫ c , Eq.(64) reduces to which shows exponential suppression for small ǫ. Equation (67) is represented in Fig. 4 by the dashed curve. It should be noted that, strictly speaking, Eq.(67) is only valid for ν = B/(βǫ) ≪ 1 which was assumed in deriving Eq.(53) from Eq.(52). However, straightforward analysis shows that forĨ ⊥ /I = 0.02 and T = T 0 , this condition on ν implies ǫ ≫ O(10 −4 ), which holds for the overwhelming part of Fig. 4. We first discuss the density of thermally induced spin-flips in the absence of a random magnetic field (Eq.(51) with ǫ = 0). As we assumed the temperature to be much smaller than the spin-flip creation energy, I, the density of the thermally-induced spin-flips is small. In the disordered phase (above T c ), one can neglect the correlations between the positions of spin-flips in different chains and the density of the spin-flips (kinks and antikinks) is given by the Boltzmann formula: n s = e −βI . In the ordered phase, the interchain interactions bind kinks and antikinks within one chain into pairs, which is clear from the fact that n s ∝ e −2βI below T c . As follows from Eq.(3) the binding energy V of such a pair is proportional to the distance R between kink and antikink: as over this distance the spin sign is opposite to σ . Thus, below T c (0), the density of the thermally induced kink-antikink pairs is: only has a solution if ǫ > ǫ c ∼Ĩ ⊥ I, in accordance with Eq.(62). For ǫ < ǫ c the distances between disorder-induced spin-flips in isolated chains are very large, so that they are suppressed by the interchain interactions and the system at zero temperature is in the ordered state. If, on the other hand, ǫ > ǫ c the system is disordered at all temperatures.
The dependence of the critical temperature on the disorder strength in the RFIM can be compared with that in another Ising-type model of disordered (spin-) Peierls systems considered in Ref. [22]. There it was assumed that impurities randomly cut the chains into finite segments, some of which, e.g., the ones with an odd number of units, contain cannot be made arbitrarily small, which results in the existence of a critical random field strength.

VI. SUMMARY AND CONCLUSIONS
In this paper, we studied the anisotropic RFIM, consisting of spin chains with interchain exchange interactions that are much weaker than the intrachain interactions.
Treating the interchain interactions in the mean-field approximation, we found an analytical solution Eq.(40) for the free energy. The crux of our approach lies in the fact that in this chain mean-field approximation the free energy of the continuum version of the model can be related to a one-dimensional Brownian motion. We found the stationary solution of the corresponding Fokker-Planck equation, which enabled us to perform the average of the free energy over the random magnetic field realizations. From this, we obtained the order parameter as a function of temperature and random field strength and, more specifically, the critical temperature below which LRO occurs as a function of the random field strength. We also calculated the density of spin-flips in the chains. As we decribed in Sec. V, our results have a clear physical interpretation.
The chain mean-field approximation restricts the validity of our results to d > 2, where thermodynamic quantities suffer much less from fluctuations than in lower dimensional systems. We believe that the critical temperature T c (ǫ) given by Eq.(61) describes the phase transition in three dimensions qualitatively correct and is quantitatively correct for a strongly anisotropic realization of the RFIM. Clearly, our mean-field treatment of interchain interactions is not applicable in two dimensions, where LRO is lacking for all temperatures and (or) random-field strengths [10].
The one-dimensional RFIM had already been solved by us in the context of disorderinduced solitons in a single chain of the polymer trans-polyacetylene [11]. Our present treatment of the anisotropic d-dimensional RFIM allows us to extend that study to include the soliton-antisoliton confinement that is imposed by electron hopping between neighboring polyacetylene chains in the bulk material. In particular, the exponential suppression of the density of spin-flips well below the critical random field strength (see Eq.(67)) may be relevant to explain the absence of experimental signatures of these solitons. In a forthcoming paper [13], we will focus more particularly on this application of the anisotropic RFIM and discuss various experimental observables. We finally note that if B is considered an external (rather than self-consistent) field, our model describes the lattice dimerization of a disordered polymer chain with a non-degenerate ground state [12]. B then is proportional to the energy difference between the two phases with different signs of the dimerization.