Random Field Ising Model: Dimensional Reduction or Spin-Glass Phase?

Abstract

The stability of the random field Ising model (RFIM) against spin glass (SG) fluctuations, as investigated by Mézard and Young, is naturally expressed via Legendre transforms, stability being then associated with the non-negativeness of eigenvalues of the inverse of a generalized SG susceptibility matrix. It is found that the signal for the occurrence of the SG transition will manifest itself in freeenergy fluctuations only, and not in the free energy itself.Eigenvalues of the inverse SG susceptibility matrix is then approached by the Rayleigh Ritz method which provides an upper bound. Coming from the paramagnetic phase on the Curie line, one is able to use a virial-like relationship generated by scaling the single unit length (D < 6; in higher dimension a new length sets in, the inverse momentum cut off). Instability towards a SG phase being probed on pairs of distinct replicas, it follows that, despite the repulsive coupling of the RFIM the effective pair coupling is attractive (at least for small values of the parameter g∆, g the coupling and∆ the effective random field fluctuation). As a result, "bound states" associated with replica pairs (negative eigenvalues) provide the instability signature. Away from the Curie line, the attraction is damped out till the SG transition line is reached and paramagnetism restored. In D < 6, the SG transition always precedes the ferromagnetic one, thus the domain in dimension where standard dimensional reduction would apply (on the Curie line) shrinks to zero. ABSTRACTThe stability of the random field Ising model (RFIM) against spin glass (SG) fluctuations, as investigated by Mézard and Young, is naturally expressed via Legendre transforms, stability being then associated with the non-negativeness of eigenvalues of the inverse of a generalized SG susceptibility matrix. It is interesting to note that the signal for the occurrence of the SG transition will manifest itself in free-energy fluctuations only, i.e., in the n-replica formulation, in terms of order n 2

Effect of second-rank random anisotropy on critical phenomena of a random-fieldO(N)spin model in the largeNlimit

“…Following Mézard and Young, the instability of the replica-symmetric solution against replica symmetry breaking has been reported in several papers. 11,12 However, the physical meaning of the instability in the SCSA equation is still unclear.…”

Spontaneous versus explicit replica symmetry breaking in the theory of disordered systems

“…In this case, the spin-glass susceptibility diverges when the replicon eigenvalue goes to zero (see Appendix B). The more general setting considered here also encompasses the random field O(N ) model as previously discussed 2,32 . The instability associated with the vanishing of the smallest eigenvalue of the replicon operator is harder to interpret than in mean-field-like cases and does not necessarily implies the divergence of the spin-glass susceptibility (see below).…”

“…involves a 2-point, 2-replica function, as in the Parisi solution of the SK model 2 . As developed in the theory of spin glasses and discussed more recently for the random field O(N ) model 30,31,32,33 , spontaneous RSB is signaled by an instability of the replica-symmetric solution which is characterized by the appearance of a zero eigenvalue in an appropriate stability operator. The natural framework to investigate such phenomena involving 2-point, 2-replica functions is the so-called 2-particle irreducible (2PI) formalism 35,36,37 : by introducing sources that couple not only to the fundamental fields but also to composite bilinear fields and then performing a double Legendre transform, one obtains a generating functional that has for argument both fields (magnetizations) and 2-point correlation functions.…”

“…The natural framework to investigate such phenomena involving 2-point, 2-replica functions is the so-called 2-particle irreducible (2PI) formalism 35,36,37 : by introducing sources that couple not only to the fundamental fields but also to composite bilinear fields and then performing a double Legendre transform, one obtains a generating functional that has for argument both fields (magnetizations) and 2-point correlation functions. When coupled to a replica approach, this formalism allows one to write the above mentioned stability operator as the jacobian of the double Legendre transform 32,38 . (On the other hand, the FRG approach to disordered systems is commonly expressed within the 1-particle irreducible (1PI) formalism in which sources are linearly coupled to the fundamental fields; the 1PI formalism can be recovered from the 2PI one, which makes the latter a suitable starting point for studying the relation between spontaneous and explicit RSB.)…”

How to break the replica symmetry in structural glasses

“…In order to grasp the glassy (or 1-RSB) solution , F {g, g 0 , ∆} has been expanded up to the 4-th order with respect to ∆(k) around the RS-solution (14). As a result the increment of the free energy connected with the non-zero order parameter ∆(k) is determined by the Landau expansion…”