Stability of a two-sublattice spin-glass model

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Stability of a two-sublattice spin-glass model

Carlos S. O. Yokoi^I; Francisco A. da Costa^II

^IInstituto de Física, Universidade de São Paulo, Caixa Postal 66318, 05315-970 São Paulo, SP, Brazil

^IIDepartamento de Física Teórica e Experimental, Universidade Federal do Rio Grande do Norte, Caixa Postal 1641, 59072-970 Natal, RN, Brazil

ABSTRACT

We study the stability of the replica-symmetric solution of a two-sublattice infinite-range spin-glass model, which can describe the transition from an antiferromagnetic to a spin-glass state. The eigenvalues associated with replica-symmetric perturbations are in general complex. The natural generalization of the usual stability condition is to require the real part of these eigenvalues to be positive. The necessary and sufficient conditions for all the roots of the secular equation to have positive real parts is given by the Hurwitz criterion. The generalized stability condition allows a consistent analysis of the phase diagram within the replica-symmetric approximation.

1 Introduction

The infinite-range Sherrington-Kirkpatrick (SK) model [1] for a spin glass has attracted considerable attention over the past decades [2,3,4]. These investigations have revealed highly non-trivial properties such as the instability of the replica-symmetric (RS) solution [5] and the replica-symmetry-breaking scheme to produce a stable solution [6-9]. Most studies have concentrated on situations where the exchange distributions are either symmetric or with an additional ferromagnetic interaction. More recently a two-sublattice version of the SK model was introduced [10-13] to allow for antiferromagnetic interactions between different sublattices. Such extension is quite natural in view of the existence of many experimental systems such as Fe_xMg_1-xCl₂[14-16] and Fe_xMn_1-xTiO₃[17,18], which exhibit a transition from an Ising antiferromagnetic into an Ising spin glass state for a certain range of x values. In contrast to the standard SK model, in the two-sublattice SK model with antiferromagnetic intersublattice interactions, the ordered (antiferromagnetic) phase extends to finite fields and the de Almeida-Thouless instability line [5] has distinct branches in the paramagnetic and antiferromagnetic phases, which do not meet at a first-order transition [10-13]. Experimental determination of the field-temperature phase diagram in Fe_xMn_1-xTiO₃, as well as the de Almeida-Thouless instability line [19], are in qualitative agreement with mean-field results [13].

In the previous studies of this model the stability of the RS solution against transversal fluctuations, i.e., outside the RS space, has already been investigated [10-13], and the stability against longitudinal fluctuations, i.e., inside the RS space, was also briefly considered [12]. The stability of the RS solution against transversal fluctuations is important to establish whether replica symmetry breaking is necessary. The stability against longitudinal fluctuations, however, is also necessary to ensure the validity of RS solution. For certain parameter values of the two-sublattice SK model there may be up to three RS solutions, all of them stable against transversal fluctuations. In such a situation the analysis of the stability against longitudinal fluctuations is important for a consistent study of the phase diagram by eliminating unstable solutions.

In this work we remedy the lack of such investigation by a detailed numerical and analytical study of the eigenvalues associated with longitudinal fluctuations. Surprisingly, these eigenvalues are in general complex. It is natural to assume that stability condition should require the real part of these eigenvalues to be positive. The necessary and sufficient condition for all the roots of the secular equation to have positive real parts is given by the Hurwitz criterion. We show that this generalized stability condition allows a consistent study of the phase diagram within the RS approximation.

2 The model

We consider a system of 2N Ising spins S_i = ±1 located at the sites of two identical sublattices A and B. The interactions are described by the Hamiltonian

where the first sum is over all distinct pairs of spins belonging to different sublattices, the second and third ones refer to all distinct pairs of spins belonging to the same sublattices, and the last sum is over all spins in the two sublattices. J_ij is the exchange interaction between spins in different sublattices, is the exchange interaction between spins in the same sublattice, and H is the applied magnetic field. The exchange interactions are independent, quenched, Gaussian random variables with mean values

and variances

The mean intrasublattice interactions will always assumed to be ferromagnetic ( > 0), whereas the mean intersublattice interactions may be ferromagnetic (J₀ > 0) or antiferromagnetic (J₀ < 0).

The standard approach to compute the quenched average is to introduce n non-interacting replicas a = 1,2,...,n of the system, calculate the annealed averages and then take the limit n ® 0 [2,4]. In this replica method the free energy per spin f is given by

where b = 1/k_BT and Zⁿ is the partition function of n replicas of the system. Performing the average of Zⁿ over the random couplings we find

where (ab) denotes distinct pairs of replicas and we have introduced the sublattice magnetization and sublattice overlap function of the replicas,

The trace over the spin variables in (5) can be performed by taking into account the constraints (6) by means of the identities

and

We then obtain

where

with

_A and _B denoting the ''effective sublattice Hamiltonians'',

In the limit of large N the integrations over the l variables in (9) can be performed by the saddle-point method. The saddle point is given by

These equations determine l variables in terms of m and q variables. The remaining integrations over the m and q variables in (9) can be performed by the Laplace method in the limit of large N. The stationary-point equations are given by

where is the sublattice complementary to X, i.e., if X = A then = B, and vice versa. Substituting these results in the expression of f given by Eq. (10) we find

where we have discarded terms that vanish in the limit of large N. Analogously, the effective sublattice Hamiltonians (11) become

To evaluate the general expressions obtained thus far it is necessary to impose some structure on m and q variables. The simplest assumption corresponds to the RS solution[2,4] obtained by assuming order parameters independent of replica indices,

Proceeding in the usual way [2,4], one finds that the saddle-point equations (12) and stationary-point equations (13) give the equations of state

where

and the brackets without subscript á...ñ denote Gaussian averages,

Analogously, the free energy per spin (4) becomes

3 The stability of replica-symmetric solution

The validity of the RS solution (17) rests on the applicability of Laplace method used to perform the integrations over m and q variables for large N. The integral converges only if the stationary point (13) is a minimum of f, i.e., only if the eigenvalues of the Hessian matrix formed by the second derivatives of the function f given by equation (10) with respect to the m and q variables are all positive. We can equivalently consider f as a function of l variables, related to m and q variables by means of Eq. (12). We will follow the latter approach because it leads to simpler calculations. The Hessian is a [n(n+1)/2] ×[n(n+1)/2] matrix whose elements are 2 × 2 matrices given by

where

At the stationary point of the RS solution (17) there are seven different types of 2 × 2 elements of the Hessian matrix. We denote these elements by [5]

where the indices a, b, g and d are all distinct and the tilde denotes the transpose of the matrix. We do not quote the lengthy expressions for these elements because only their linear combinations are needed in the calculation of the eigenvalues.

The eigenvalues of the Hessian matrix can now be determined by finding the eigenvectors that divide the space into orthogonal subspaces closed to the permutation operation. The procedure are analogous to the case of the SK model [5] except that now the elements of the Hessian matrix are 2 × 2 matrices (23). These eigenvectors are [20]: n(n - 3) transversal or replicon eigenvectors depending on two replica indices, 4 (n - 1) anomalous eigenvalues depending on a single replica index, and 4 longitudinal eigenvectors independent of replica indices.

The eigenvalues associated with the transversal eigenvectors are found to be the eigenvalues of the 2 × 2 matrix

with elements

where

The necessary and sufficient condition for all the eigenvalues to be positive are

which are equivalent to the conditions

in agreement with previous studies [10,13]. A RS solution satisfying these conditions will be called transversally (T) stable, and T unstable otherwise.

The eigenvalues associated with anomalous and longitudinal eigenvectors are the same in the limit n ® 0. They are found to be the eigenvalues of the 4 × 4 matrix

where

The characteristic equation has the form

where the coefficients a_n are n-th order traces of the matrix L. A numerical study of equation (43) shows that the eigenvalues are complex for some values of the parameters of the model, in contrast with one-sublattice SK model in which the anomalous and longitudinal eigenvalues never become complex [5]. Even though the Hessian matrix (21) for n > 1 is real and symmetric, in the limit n ® 0 there is no guarantee that the eigenvalues will be real. In fact, complex longitudinal and anomalous eigenvalues also arise in the spin 1 one-sublattice infinite-range spin-glass model with crystal-field anisotropy [21,22]. In general, therefore, the stability condition should require the real part of the eigenvalues to be positive. According to the Hurwitz criterion [23], the necessary and sufficient condition for all the roots of equation (43) to have positive real parts are

These condition are equivalent to the following four conditions:

A RS solution satisfying these conditions will be called longitudinally (L) stable, and L unstable otherwise.

4 Results of the stability analysis

In this section we present the results of the stability analysis of the RS solution for different values of the parameters of the model. Since the Hamiltonian (1) is invariant under the simultaneous transformations

it is sufficient to consider fields H > 0. For H = 0 only one of the two solutions related by the global inversion symmetry has to be considered.

4.1 Zero applied field

4.1.1 Ferromagnetic intersublattice interaction

In zero applied field (H = 0) and ferromagnetic intersublattice interactions (J₀ > 0) the solutions of the set of equations (17) are of the form

Three types of solutions are possible:

• Paramagnetic (P) solution: m = 0, q = 0.

• Spin Glass (SG) solution: q > 0, m = 0.

• Ferromagnetic (F) solution: q > 0, m > 0.

Fig. 1 shows the lines delimiting the regions where different types of solutions can be found in the plane of temperature versus J₀ + .

The P solution is always possible. However it is L stable only above the line (b) and the left portion of line (a), and T stable above line (a). The L instability of P solution occurs due to the violation of the condition (49), which is given in the case of P solution by

For (J₀ + )/< 1/2, the second factor of in (52) becomes negative below line (a). Thus the left portion of line (a) is determined by

On the other hand, for (J₀ + )/ > 1/2 the fourth factor in (52) becomes negative below line (b). Thus the equation for line (b) is

The T instability of the P solution is due to the violation of the condition (31), which is given in the case of P solution by

The second factor in (55) becomes negative below line (a) for all values of J₀ + . Thus line (a) is given by equation (53) for all J₀ + . The T and L instabilities of the P solution occur simultaneously on the line (a) for (J₀ + )/< 1/2.

The SG solution is possible only below line (a). It is T unstable throughout this region and L stable to the left of line (c). The L instability of the SG solution occurs due to the violation of the condition (49), which is given in the case of SG solution by

For (J₀ + )/ > 1/2 the last factor in (56) becomes negative to the left of line (c). Thus the equation determining line (c) is

The F solution is possible only between lines (b) and (c). It is L stable throughout this region but T stable only above line (d). The T instability of the F solution occurs due to the violation of the condition (31), which is given in the case of F solution by

For (J₀ + )/ > 1/2 the second factor in (58) becomes negative below line (d). Thus line (d) is described by equation

Rejecting solutions that are L unstable, we conclude that the P phase is located above lines (b) and left portion of line (a), the SG phase between the left portion of line (a) and line (c), and finally the F phase between lines (b) and (c). The SG phase, and the F phase between lines (c) and (d), are T unstable, indicating the need for a replica-symmetry-breaking solution in this region. The transition line (c) will change to a vertical line if such a solution is considered [24]. We mention that, as should be expected, in the case J₀ = 0 and J = 0, or = 0 and J' = 0, these results reduce to those of one-sublattice SK model [2,4].

4.1.2 Antiferromagnetic intersublattice interaction

The Hamiltonian (1) in zero applied field (H = 0) is invariant under simultaneous transformations

In fact, we can check explicitly that all the expressions for the RS solution, including those of stability conditions, are invariant under simultaneous transformations

Thus the case of antiferromagnetic intersublattice interaction J₀ < 0 is completely equivalent to the case of ferromagnetic intersublattice interaction -J₀ > 0 by replacing m_B by -m_B. This means that the F solution is replaced by the antiferromagnetic (AF) solution

The results displayed in Fig. 1 remains valid, with J₀ replaced by -J₀ and F solution by AF solution.

4.2 Non-zero applied field

4.2.1 Ferromagnetic intersublattice interaction

In non-zero applied field (H > 0) and ferromagnetic intersublattice interactions (J₀ > 0), only the paramagnetic (P) solution is possible for the set of equations (17), which are of the form

This solution is always L stable, but becomes T unstable for low temperatures due to the violation of the condition (31), which in this case is also given by Eq. (58). The instability line is given by Eq. (59), illustrated in Fig. 2 for the case J'/J = 1, /J₀ = 1/2 and (J₀ + )/ = 2 . As should be expected, in the case J₀ = 0 and J = 0, or = 0 and J' = 0, these results reduce to the de Almeida-Thouless line of one-sublattice SK model[5].

4.2.2 Antiferromagnetic intersublattice interaction

In non-zero applied field (H > 0) and antiferromagnetic intersublattice interactions (J₀ < 0), two types of solutions to the set of equations (17) are possible:

• Paramagnetic (P) solution: m_A = m_B = m > 0,q_A = q_B = q > 0.

• Antiferromagnetic (AF) solution: m_A ¹ m_B, q_A ¹ q_B.

For (-J₀ + )/< 1/2 only P solution is possible. This solution is always L stable but becomes T unstable at low temperatures due to the to the violation of the condition (31), which in this case it is also given by (58). The instability line is given by Eq. (59).

For (-J₀ + )/ > 1/2, AF solution also becomes possible. Fig. 3 shows the lines delimiting the regions of existence and stability of each type of solution for -/J₀ = 1/2. The P solution becomes L unstable below line (a) due to the violation of the condition (49). In the case of P solution this condition is given by

The first factor in (64) becomes negative inside line (a). Therefore the equation determining line (a) is

which is in agreement with previous study [12]. The P solution is T unstable below line (b). This instability occurs due to the violation of condition (31), given in this case by Eq. (58), caused by the second factor. Therefore the line (b) is determined by equation (59). The AF solution is possible only inside line (a). It is L stable throughout this region and T unstable below line (c). This instability is due to the violation of condition (31). Therefore line (c) obeys the equation

Rejecting solutions that are L unstable, we conclude that P phase exists outside and AF phase inside line (a). The P solution becomes T unstable below line (b) and AF solution below line (c), which meet smoothly on the line (a). In the region below lines (b) and (c) it is necessary to consider replica symmetry breaking solution, which will presumably change line (a) in this region.

For sufficiently large values of the ratio -/J₀ the model can exhibit first-order transition from AF phase to the P phase [12]. As an example, we consider the case -/J₀ = 5 shown in Fig. 4. The P solution is L stable inside line (a) given by Eq. (65), and T stable above line (b) given by Eq. (66). There is one AF solution inside line (a) and two distinct AF solutions between lines (a) and (d), as illustrated in Figs. 5(a) and 5(b) for k_BT/ = 1. One of the AF solutions, corresponding to dotted lines in Fig. 5, is L unstable due to the violation of the condition (49), as shown in Fig. 5(c). The transition between AF phase and P phase is first order, determined by equating the free energies of L stable AF and P phases, as shown in Fig. 5(d). The first-order transition line is shown as dotted line in Fig. 4, which ends at the tricritical point TCP. The L stable AF solution becomes T unstable below line (c) due to the violation of condition (31), and it is determined by Eq. (66). We conclude that in Fig. 4 the P phase exists outside and AF phase inside lines (a) and (e), which meet smoothly at the tricritical point TCP. The P phase becomes T unstable below line (b), and AF phase below line (c). Notice that the lines (b) and (c) are discontinuous across first-order transition [12]. It is likely that the first-order transition line will change in this part of phase diagram once the replica-symmetry-breaking solutions are considered for P and AF phases.

5 Conclusions

In this paper we have investigated the stability of the RS symmetric solution of the two-sublattice generalization of the SK infinite-range spin-glass model. We have derived stability conditions for transversal fluctuations in agreement with previous investigations, and we have extended previous study of the stability against longitudinal or anomalous fluctuations. The eigenvalues associated with such perturbations are in general complex. We generalized the usual stability condition by requiring the real part of these eigenvalues to be positive. The necessary and sufficient stability conditions were found using the Hurwitz criterion for all the roots of the secular equation to have positive real parts. These conditions allowed us to select one RS solution among those that are transversally stable. We believe that the generalized stability condition should also be useful in other spin glass models where eigenvalues associated with longitudinal and anomalous perturbations become complex.

Acknowledgments

The authors acknowledge partial financial support from the Brazilian Government Agencies PROCAD/CAPES and CNPq.

Received on 6 October, 2003

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Publication Dates

• Publication in this collection
  25 Nov 2005
• Date of issue
  Dec 2003

History

• Received
  06 Oct 2003
• Accepted
  06 Oct 2003