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C. De Dominicis

Publications and source records attributed to C. De Dominicis.

36 records · Page 2Linked to original sources
arXiv2009

Low temperature spin glass fluctuations: expanding around a spherical approximation

The spin glass behavior near zero temperature is a complicated matter. To get an easier access to the spin glass order parameter $Q(x)$ and, at the same time, keep track of $Q_{ab}$, its matrix aspect, and hence of the Hessian controlling stability, we investigate an expansion of the replicated free energy functional around its ``spherical'' approximation. This expansion is obtained by introducing a constraint-field and a (double) Legendre Transform expressed in terms of spin correlators and constraint-field correlators. The spherical approximation has the spin fluctuations treated with a global constraint and the expansion of the Legendre Transformed functional brings them closer and closer to the Ising local constraint. In this paper we examine the first contribution of the systematic corrections to the spherical starting point.

cond-mat.dis-nn↗
arXiv2005

W-T identities and a candidate "droplet" Lagrangean for the Ising Spin Glass

In search for a microscopic description of ``droplet-like'' properties for the Ising spin glass (single component order parameter, zero modes i.e. correlation functions vanishing at infinity) we reconsider the two-packet model of Bray and Moore, which is effectively Replica-Symmetric and enjoys zero modes but only up to one-loop. We show how an appropriate change in the limits of the basic parameters of the model (packet size and replica number) allows for a derivation of Ward-Takahashi (WT) identities, thus ensuring the existence of zero modes to all orders and opening the way for a Lagrangean formulation of a ``droplet-like'' field theory for the Ising spin glass.

cond-mat.dis-nn↗
arXiv2002

Replica field theory and renormalization group for the Ising spin glass in an external magnetic field

We use the generic replica symmetric cubic field-theory to study the transition of short range Ising spin glasses in a magnetic field around the upper critical dimension, d=6. A novel fixed-point is found, in addition to the well-known zero magnetic field fixed-point, from the application of the renormalization group. In the spin glass limit, n going to 0, this fixed-point governs the critical behaviour of a class of systems characterised by a single cubic interaction parameter. For this universality class, the spin glass susceptibility diverges at criticality, whereas the longitudinal mode remains massive. The third mode, the so-called anomalous one, however, behaves unusually, having a jump at criticality. The physical consequences of this unusual behaviour are discussed, and a comparison with the conventional de Almeida-Thouless scenario presented.

cond-mat.dis-nn↗
arXiv2002

Spin glass transition in a magnetic field: a renormalization group study

We study the transition of short range Ising spin glasses in a magnetic field, within a general replica symmetric field theory, which contains three masses and eight cubic couplings, that is defined in terms of the fields representing the replicon, anomalous and longitudinal modes. We discuss the symmetry of the theory in the limit of replica number n to 0, and consider the regular case where the longitudinal and anomalous masses remain degenerate. The spin glass transitions in zero and non-zero field are analyzed in a common framework. The mean field treatment shows the usual results, that is a transition in zero field, where all the modes become critical, and a transition in non-zero field, at the de Almeida-Thouless (AT) line, with only the replicon mode critical. Renormalization group methods are used to study the critical behavior, to order epsilon = 6-d. In the general theory we find a stable fixed-point associated to the spin glass transition in zero field. This fixed-point becomes unstable in the presence of a small magnetic field, and we calculate crossover exponents, which we relate to zero-field critical exponents. In a finite magnetic field, we find no physical stable fixed-point to describe the AT transition, in agreement with previous results of other authors.

cond-mat.dis-nn↗
arXiv2002

Generic replica symmetric field-theory for short range Ising spin glasses

Symmetry considerations and a direct, Hubbard-Stratonovich type, derivation are used to construct a replica field-theory relevant to the study of the spin glass transition of short range models in a magnetic field. A mean-field treatment reveals that two different types of transitions exist, whenever the replica number n is kept larger than zero. The Sherrington-Kirkpatrick critical point in zero magnetic field between the paramagnet and replica magnet (a replica symmetric phase with a nonzero spin glass order parameter) separates from the de Almeida-Thouless line, along which replica symmetry breaking occurs. We argue that for studying the de Almeida-Thouless transition around the upper critical dimension d=6, it is necessary to use the generic cubic model with all the three bare masses and eight cubic couplings. The critical role n may play is also emphasized. To make perturbative calculations feasible, a new representation of the cubic interaction is introduced. To illustrate the method, we compute the masses in one-loop order. Some technical details and a list of vertex rules are presented to help future renormalisation-group calculations.

cond-mat.dis-nn↗
arXiv2002

Twist Free Energy

One may impose to a system with spontaneous broken symmetry, boundary conditions which correspond to different pure states at two ends of a sample. For a discrete Ising-like broken symmetry, boundary conditions with opposite spins in two parallel limiting planes, generate an interface and a cost in free energy per unit area of the interface. For continuum symmetries the order parameter interpolates smoothly between the end planes carrying two different directions of the order parameter. The cost in free energy is then proportional to $L^{d-2}$ for a system of characteristic size L. The power of $L$ is related to the lower critical dimension, and the coefficient of this additional free energy vanishes at the critical temperature. In this note it is shown within a loop expansion that one does find the expected behavior of this twist free energy. This is a preamble to the study of situations where the broken continuum symmetry is believed to be more complex, as in Parisi's ansatz for the Edwards-Anderson spin glass.

cond-mat↗
arXiv2002

Twist Free Energy in a Spin Glass

The field theory of a short range spin glass with Gaussian random interactions, is considered near the upper critical dimension six. In the glassy phase, replica symmetry breaking is accompanied with massless Goldstone modes, generated by the breaking of reparametrization invariance of a Parisi type solution. Twisted boundary conditions are thus imposed at two opposite ends of the system in order to study the size dependence of the twist free energy. A loop-expansion is performed to first order around a twisted background. It is found, as expected but it is non trivial, that the theory does renormalize around such backgrounds, as well as for the bulk. However two main differences appear, in comparison with simple ferromagnetic transitions : (i) the loop expansion yields a (negative) anomaly in the size dependence of the free energy, thereby lifting the lower critical dimension to a value greater than two given by $d_c = 2-η(d_c)$ (ii) the free energy is lowered by twisting the boundary conditions. This sign may reflect a spontaneous spatial non-uniformity of the order parameter.

cond-mat↗
arXiv2000

Reparametrization invariance: a gauge-like symmetry of ultrametrically organised states

The reparametrization transformation between ultrametrically organised states of replicated disordered systems is explicitly defined. The invariance of the longitudinal free energy under this transformation, i.e. reparametrization invariance, is shown to be a direct consequence of the higher level symmetry of replica equivalence. The double limit of infinite step replica symmetry breaking and n=0 is needed to derive this continuous gauge-like symmetry from the discrete permutation invariance of the n replicas. Goldstone's theorem and Ward identities can be deduced from the disappearence of the second (and higher order) variation of the longitudinal free energy. We recall also how these and other exact statements follow from permutation symmetry after introducing the concept of "infinitesimal" permutations.

cond-mat.dis-nn↗
arXiv1998

On Ward-Takahashi identities for the Parisi spin glass

The introduction of ``small permutations'' allows us to derive Ward-Takahashi identities for the spin-glass, in the Parisi limit of an infinite number of steps of replica symmetry breaking. The first identities express the emergence of a band of Goldstone modes. The next identities relate components of (the Replica Fourier Transformed) 3-point function to overlap derivatives of the 2-point function (inverse propagator). A jump in this last function is exhibited, when its two overlaps are crossing each other, in the special simpler case where one of the cross-overlaps is maximal.

cond-mat.dis-nn↗
arXiv1998

Scaling and infrared divergences in the replica field theory of the Ising spin glass

Replica field theory for the Ising spin glass in zero magnetic field is studied around the upper critical dimension d=6. A scaling theory of the spin glass phase, based on Parisi's ultrametrically organised order parameter, is proposed. We argue that this infinite step replica symmetry broken (RSB) phase is nonperturbative in the sense that amplitudes of scaling forms cannot be expanded in term of the coupling constant w^2. Infrared divergent integrals inevitably appear when we try to compute amplitudes perturbatively, nevertheless the ε-expansion of critical exponents seems to be well-behaved. The origin of these problems can be traced back to the unusual behaviour of the free propagator having two mass scales, the smaller one being proportional to the perturbation parameter w^2 and providing a natural infrared cutoff. Keeping the free propagator unexpanded makes it possible to avoid producing infrared divergent integrals. The role of Ward-identities and the problem of the lower critical dimension are also discussed.

cond-mat.dis-nn↗
arXiv1998

New phenomena in the random field Ising model

We reconsider Ising spins in a Gaussian random field within the replica formalism. The corresponding continuum model involves several coupling constants beyond the single one which was considered in the standard $ϕ^4$ theory approach. These terms involve more than one replica, and therefore in a mean field theory they do not contribute to the zero-replica limit. However the fluctuations involving those extra terms are singular on the Curie line below eight dimensions, and by the time one reaches the dimension six, it is necessary to keep them in the renormalization group analysis. As a result it is found that there is no stable fixed point of order $(6-d)$. Whether this means that there is no expansion in powers of $(6-d)$, or that the transition is driven to first order by these fluctuations, is difficult to decide at this level, but it explains the failure of the $(d, d-2)$ correspondence. In this work we have not considered the possibility of a glassy phase in the would-be paramagnetic region, but in a subsequent work we shall present evidence for replica symmetry breaking in dimension lower than six.

cond-mat↗
arXiv1998

Dynamics versus replicas in the random field Ising model

In a previous article we have shown, within the replica formalism, that the conventional picture of the random field Ising model breaks down, by the effect of singularities in the interactions between fields involving several replicas, below dimension eight. In the zero-replica limit several coupling constants have thus to be considered, instead of just one. As a result we had found that there is no stable fixed point in the vicinity of dimension six. It is natural to reconsider the problem in a dynamical framework, which does not require replicas, although the equilibrium properties should be recovered in the large time limit. Singularities in the zero-replica limit are a priori not visible in a dynamical picture. In this note we show that in fact new interactions are also generated in the stochastic approach. Similarly these interactions are found to be singular below dimension eight. These critical singularities require the introduction of a time origin $t_0$ at which initial data are given. The dynamical properties are thus dependent upon the waiting time. It is shown here that one can indeed find a complete correspondence between the equilibrium singularities in the $n=0$ limit, and the singularities in the dynamics when the initial time $t_0$ goes to minus infinity, with $n$ replaced by $-\frac{1}{t_0}$. There is thus complete coherence between the two approaches.

cond-mat↗
arXiv1997

On the Replica Fourier Transform

The Replica Fourier Transform introduced previously is related to the standard definition of Fourier transforms over a group. Its use is illustrated by block-diagonalizing the eigenvalue equation of a four-replica Parisi matrix.

cond-mat.dis-nn↗
arXiv1997

Replica Fourier Transforms on Ultrametric Trees, and Block-Diagonalizing Multi-Replica Matrices

The analysis of objects living on ultrametric trees, in particular the block-diagonalization of 4-replica matrices $M^{αβ; γδ}$, is shown to be dramatically simplified through the introduction of properly chosen operations on those objects. These are the Replica Fourier Transforms on ultrametric trees. Those transformations are defined and used in the present work.

cond-mat.dis-nn↗
arXiv1996

Stability of the Mezard-Parisi solution for random manifolds

The eigenvalues of the Hessian associated with random manifolds are constructed for the general case of $R$ steps of replica symmetry breaking. For the Parisi limit $R\to\infty$ (continuum replica symmetry breaking) which is relevant for the manifold dimension $D<2$, they are shown to be non negative.

cond-mat↗
arXiv1995

Random field Ising model: dimensional reduction or spin-glass phase?

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 free-energy {\sl 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 {\sl on the Curie line,\/} one is able to use a virial-like relationship generated by scaling the {\sl 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 {\sl distinct\/} replicas, it follows that, despite the repulsive coupling of the RFIM the effective pair coupling is {\sl attractive\/} (at least for small values of the parameter $ g\bar Δ, $ $ g $ the coupling and $ \bar Δ$ the effective random field fluctuation). As a result, \lq\lq bound states\rq\rq\ associated with replica pairs (negative eigenvalues) provide the instability signature. {\sl 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.

cond-mat↗