We determine the phase transition of the Levy spin glass. A regularized model where the coupling constants smaller than some cutoff $ε$ are neglected can be studied by the cavity method for diluted spin glasses. We show how to handle the $ε\to 0$ limit and determine the de Almeida-Thouless transition temperature in presence of an external field. Contrary to previous findings, in zero external field we do not find any stable replica-symmetric spin glass phase: the spin glass phase is always a replica-symmetry-broken phase.
Using large-scale Monte Carlo simulations that combine parallel tempering with specialized cluster updates, we show that Ising spin glasses with Levy-distributed interactions share the same universality class as Ising spin glasses with Gaussian or bimodal-distributed interactions. Corrections to scaling are large for Levy spin glasses. In order to overcome these and show that the critical exponents agree with the Gaussian case, we perform an extended scaling of the two-point finite size correlation length and the spin glass susceptibility. Furthermore, we compute the critical temperature and c…
We study the Lévy spin-glass model with the replica and the cavity method. In this model each spin interacts through a finite number of strong bonds and an infinite number of weak bonds. This hybrid behaviour of Lévy spin glasses becomes transparent in our solution: the local field contains a part propagating along a backbone of strong bonds and a Gaussian noise term due to weak bonds. Our method allows to determine the complete replica symmetric phase diagram, the replica symmetry breaking line and the entropy. The results are compared with simulations and previous calculations using a Gaussi…
Infinite-range spin-glass models with Levy-distributed interactions show a freezing transition similar to disordered spin systems on finite connectivity random graphs. It is shown that despite diverging moments of the local field distribution this transition can be analyzed within the replica approach by working at imaginary temperature and using a variant of the replica method developed for diluted systems and optimization problems. The replica-symmetric self-consistent equation for the distribution of local fields illustrates how the long tail in the distribution of coupling strengths gives …
Infinite-range spin-glass models with Levy-distributed interactions show a spin-glass transition with similarities to both the Sherrington-Kirkpatrick model and to disordered spin systems on finite connectivity random graphs. Despite the diverging moments of the coupling distribution the transition can be analyzed within the replica approach by working at imaginary temperature. Within the replica-symmetric approximation a self-consistent equation for the distribution of local fields is derived and from the instability of the paramagnetic solution to this equation the glass-transition temperatu…
We investigate the Lévy glass, a mean-field spin glass model with power-law distributed couplings characterized by a divergent second moment. By combining extensively many small couplings with a spare random backbone of strong bonds the model is intermediate between the Sherrington-Kirkpatrick and the Viana-Bray model. A truncated version where couplings smaller than some threshold $\eps$ are neglected can be studied within the cavity method developed for spin glasses on locally tree-like random graphs. By performing the limit $\eps\to 0$ in a well-defined way we calculate the thermodynamic fu…
We revisited, by means of numerical simulations, the one dimensional bond diluted Levy Ising spin glasses outside the limit of validity of mean field theories. In these models the probability that two spins at distance $r$ interact (via a disordered interactions, $J_{ij}=\pm 1$) decays as $r^{-ρ}$. We have estimated, using finite size scaling techniques, the infinite volume correlation length and spin glass susceptibility for $ρ=5/3$ and $ρ=9/5$. We have obtained strong evidence for divergences of the previous observables at a non zero critical temperature. We discuss the behavior of the criti…
A spin-glass transition occurs both in and out of the limit of validity of mean-field theory on a diluted one dimensional chain of Ising spins where exchange bonds occur with a probability decaying as the inverse power of the distance. Varying the power in this long-range model corresponds, in a one-to-one relationship, to change the dimension in spin-glass short-range models. Using different finite size scaling methods evidence for a spin-glass transition is found also for systems whose equivalent dimension is below the upper critical dimension at zero magnetic field. The application of a new…
Ground states of Ising spin glasses on fully connected graphs are studied for a broadly distributed bond family. In particular, bonds $J$ distributed according to a Levy distribution P(J)\propto 1/|J|^{1+α}, |J|>1, are investigated for a range of powers α. The results are compared with those for the Sherrington-Kirkpatrick (SK) model, where bonds are Gaussian distributed. In particular, we determine the variation of the ground state energy densities with α, their finite-size corrections, measure their fluctuations, and analyze the local field distribution. We find that the energies themselves …
Within the replica approach to mean-field spin-glasses the transition from ergodic high-temperature behaviour to the glassy low-temperature phase is marked by the instability of the replica-symmetric saddle-point. For general spin-glass models with non-Gaussian field distributions the corresponding Hessian is a $2^n\times 2^n$ matrix with the number $n$ of replicas tending to zero eventually. We block-diagonalize this Hessian matrix using representation theory of the permutation group and identify the blocks related to the spin-glass susceptibility. Performing the limit $n\to 0$ within these b…
Spin-glass and chiral-glass orderings in three-dimensional Heisenberg spin glasses are studied with and without randaom magnetic anisotropy by dynamical Monte Carlo simulations. In isotropic case, clear evidence of a finite-temperature chiral-glass transition is presented. While the spin autocorrelation exhibits only an interrupted aging, the chirality autocorrelation persists to exhibit a pronounced aging effect reminisecnt of the one observed in the mean-field model. In anisotropic case, asymptotic mixing of the spin and the chirality is observed in the off-equilibrium dynamics.
The three-dimensional $\pm J$ Heisenberg spin-glass model is investigated by the non-equilibrium relaxation method from the paramagnetic state. Finite-size effects in the non-equilibrium relaxation are analyzed, and the relaxation functions of the spin-glass susceptibility and the chiral-glass susceptibility in the infinite-size system are obtained. The finite-time scaling analysis gives the spin-glass transition at $T_{\rm sg}/J=0.21_{-0.02}^{+0.01}$ and the chiral-glass transition at $T_{\rm cg}/J=0.22_{-0.03}^{+0.01}$. The results suggest that both transitions occur simultaneously. The crit…
Campos et al. [Phys. Rev. Lett. 97 (2006) 217204] claim that in the 3d Heisenberg Spin Glass, chiral and spin sector ordering temperatures are identical. We point out that in their analysis of their numerical data key assumptions are made which are unjustified.
For the one-dimensional long-ranged Ising spin-glass with random couplings decaying with the distance $r$ as $J(r) \sim r^{-σ}$ and distributed with the Lévy symmetric stable distribution of index $1 <μ\leq 2$ (including the usual Gaussian case $μ=2$), we consider the region $σ>1/μ$ where the energy is extensive. We study two real space renormalization procedures at zero temperature, namely a simple box decimation that leads to explicit calculations, and a strong disorder decimation that can be studied numerically on large sizes. The droplet exponent governing the scaling of the renormalized c…
Extensive equilibrium Monte Carlo simulations are performed for a three-dimensional Heisenberg spin glass with the nearest-neighbor Gaussian coupling to investigate its spin-glass and chiral-glass orderings. The occurrence of a finite-temperature chiral-glass transition without the conventional spin-glass order is established. Critical exponents characterizing the transition are different from those of the standard Ising spin glass. The calculated overlap distribution suggests the appearance of a peculiar type of replica-symmetry breaking in the chiral-glass ordered state.
We present in this paper exact analytical expressions for the thermodynamical properties and Green functions of a certain family of fermionic Ising spin-glass models with Hubbard interaction, by noticing that their Hamiltonian is a function of the number operator only. The thermodynamical properties are mapped to the classical Ghatak-Sherrington spin-glass model while the the Density of States (DoS) is related to its joint spin-field distribution. We discuss the presence of the pseudogap in the DoS with the help of this mapping.
Detailed account is given of the chirality scenario of experimental spin-glass transitions. In this scenario, the spin glass order of weakly anisotropic Heisenberg-like spin-glass magnets including canonical spin glasses are essentially chirality driven. Recent numerical and experimental results are discussed in conjunction with this scenario.
Spin and chirality orderings of a three-dimensional XY spin glass are studied by extensive Monte Carlo simulations. By calculating an appropriately defined spin-overlap distribution function, we show that the finite-temperature chiral-glass transition does not accompany the standard spin-glass order, giving support to the spin-chirality decoupling picture. Critical behavior of the chiral-glass transition turns out to be different from that of the Ising spin glass. The chiral-glass ordered state exhibits a one-step-like peculiar replica-symmetry breaking.
We present the full phase diagram of the spherical $2+p$ spin glass model with $p\geq 4$. The main outcome is the presence of a new phase with both properties of Full Replica Symmetry Breaking (FRSB) phases of discrete models, e.g, the Sherrington-Kirkpatrick model, and those of One Replica Symmetry Breaking (1RSB). The phase, which separates a 1RSB phase from FRSB phase, is described by an order parameter function $q(x)$ with a continuous part (FRSB) for $x<m$ and a discontinuous jump (1RSB) at $x=m$. This phase has a finite complexity which leads to different dynamic and static properties.
Recent studies on the spin and the chirality orderings of the three-dimensional Heisenberg spin glass and related systems are reviewed with particular emphasis on the possible spin-chirality decoupling phenomena. Chirality scenario of real spin-glass transition and its experimental consequence on the ordering of Heisenberg-like spin glasses are discussed.
It is shown, by means of Monte Carlo simulation and Finite Size Scaling analysis, that the Heisenberg spin glass undergoes a finite-temperature phase transition in three dimensions. There is a single critical temperature, at which both a spin glass and a chiral glass orderings develop. The Monte Carlo algorithm, adapted from lattice gauge theory simulations, makes possible to thermalize lattices of size L=32, larger than in any previous spin glass simulation in three dimensions. High accuracy is reached thanks to the use of the Marenostrum supercomputer. The large range of system sizes studied…
We study analytically the free induction decay and the spin echo decay originating from the localized carriers moving between the sites which host random magnetic fields. Due to disorder in the site positions and energies, the on-site residence times, τ, are widely spread according to the Levy distribution. The power-law tail \propto τ^{-1-α} in the distribution of waiting times does not affect the conventional spectral narrowing for α>2, but leads to a dramatic acceleration of the free induction decay in the domain 2>α>1. The next abrupt acceleration of the decay takes place as the tail param…
Spin-glass and chiral-glass orderings in three-dimensional Heisenberg spin glasses are studied both by equilibrium and off-equilibrium Monte Carlo simulations. Fully isotropic model is found to exhibit a finite-temperature chiral-glass transition without the conventional spin-glass order. Although chirality is an Ising-like quantity from symmetry, universality class of the chiral-glass transition appears to be different from that of the standard Ising spin glass. In the off-equilibrium simulation, while the spin autocorrelation exhibits only an interrupted aging, the chirality autocorrelation …
The ordering of the three-dimensional Heisenberg spin glass with the weak random anisotropy in magnetic fields is studied by extensive equilibrium Monte Carlo simulations. Both the spin and the chirality are monitored. We find strong numerical evidence that a replica symmetry breaking transition occurs in the chiral sector, which accompanies the simultaneous spin-glass order. Despite the similarity in the global symmetry, the ordering behavior of the weakly anisotropic Heisenberg spin glass differs significantly from that of the strongly anisotropic Ising spin glass. The obtained phase diagram…
An approach is proposed to the Hopfield model where the mean-field treatment is made for a given set of stored patterns (sample) and then the statistical average over samples is taken. This corresponds to the approach made by Thouless, Anderson and Palmer (TAP) to the infinite-range model of spin glasses. Taking into account the fact that in the Hopfield model there exist correlations between different elements of the interaction matrix, we obtain its TAP free energy explicitly, which consists of a series of terms exhibiting the cluster effect. Nature of the spin-glass transition in the model …
This review presents various aspects of a mean-field spin glass model known as the p-spin spherical spin glass model, which has raised a lot of interest in the study of spin glasses, and also for its possible links with a mean-field theory of structural glasses. This preprint contains no new results and is therefore not intended to be published, but its aim is to present a collection of results and formulas concerning this very rich model. It is in fact the english translation of one of the chapters of my PhD thesis (``Quelques aspects de la dynamique hors d'equilibre des verres de spin'', ``S…
We report on an extensive study of the influence of spin anisotropy on spin glass aging dynamics. New temperature cycle experiments allow us to compare quantitatively the memory effect in four Heisenberg spin glasses with various degrees of random anisotropy and one Ising spin glass. The sharpness of the memory effect appears to decrease continuously with the spin anisotropy. Besides, the spin glass coherence length is determined by magnetic field change experiments for the first time in the Ising sample. For three representative samples, from Heisenberg to Ising spin glasses, we can consisten…
Spin-glass and chiral-glass orderings of a three-dimensional isotropic Heisenberg spin glass are studied both by equilibrium and off-equilibrium Monte Carlo simulations with emphasis on their dynamical aspects. The model is found to exhibit a finite-temperature chiral-glass transition without the conventional spin-glass order. Although chirality is an Ising-like quantity from symmetry, universality class of the chiral-glass transition appears to be different from that of the standard Ising spin glass. In the off-equilibrium simulation, while the spin autocorrelation exhibits only an interrupte…
It is shown by means of a 1/m expansion about the large-m limit of the m-vector spin glass that the lower critical dimension of the de Almeida-Thouless line in spin glasses is equal to the lower critical dimension of the large-m limit of the m-vector spin glass. Numerical studies suggest that this is close to six.
Using ground state computations, we study the transition from a spin glass to a ferromagnet in 3-d spin glasses when changing the mean value of the spin-spin interaction. We find good evidence for replica symmetry breaking up till the critical value where ferromagnetic ordering sets in, and no ferromagnetic spin glass phase. This phase diagram is in conflict with the droplet/scaling and mean field theories of spin glasses. We also find that the exponents of the second order ferromagnetic transition do not depend on the microscopic Hamiltonian, suggesting universality of this transition.
The large N infinite range spin glass is considered, in particular the number of spin components k needed to form the ground state and the sample-to-sample fluctuations in the Lagrange multiplier field on each site. The physical significance of k for the correlation functions is discussed. The difference between the large N and spherical spin glass is emphasized; a slight difference between the average Lagrange multiplier of the large N and spherical spin glasses is derived, leading to a slight increase in the energy of the ground state compared to the naive expectation. Further, there is a ch…
Ordering of the Heisenberg spin glass in four dimensions (4D) with the nearest-neighbor Gaussian coupling is investigated by equilibrium Monte Carlo simulations, with particular attention to its spin and chiral orderings. It is found that the spin and the chirality order simultaneously with a common correlation-length exponent $ν_{CG}=ν_{SG}\simeq 1.0$, i.e., the absence of the spin-chirality decoupling in 4D. Yet, the spin-glass ordered state exhibits a nontrivial phase-space structure associated with a continuous one-step-like replica-symmetry breaking, different in nature from that of the I…
In this paper we investigate the superspin glass behavior of a concentrated assembly of interacting maghemite nanoparticles and compare it to that of canonical atomic spin glass systems. ac versus temperature and frequency measurements show evidence of a superspin glass transition taking place at low temperature. In order to fully characterize the superspin glass phase, the aging behavior of both the thermo-remanent magnetization (TRM) and ac susceptibility has been investigated. It is shown that the scaling laws obeyed by superspin glasses and atomic spin glasses are essentially the same, aft…
Fluctuation-dissipation (FD) relation of the three-dimensional Heisenberg spin glass with weak random anisotropy is studied by off-equilibrium Monte Carlo simulation. Numerically determined FD ratio exhibits a ``one-step-ike''behavior, the effective temperature of the spin-glass state being about twice the spin-glass transition temperature, $T_{\rm eff}\simeq 2T_g$, irrespective of the bath temperature. The results are discussed in conjunction with the recent experiment by Herisson and Ocio, and with the chirality scenario of spin-glass transition.
We study the four-dimensional Ising spin glass with Gaussian and bond-diluted bimodal distributed interactions via large-scale Monte Carlo simulations and show via an extensive finite-size scaling analysis that four-dimensional Ising spin glasses obey universality.
The ground-state phase diagram of an Ising spin-glass model on a random graph with an arbitrary fraction $w$ of ferromagnetic interactions is analysed in the presence of an external field. Using the replica method, and performing an analysis of stability of the replica-symmetric solution, it is shown that $w=1/2$, correponding to an unbiased spin glass, is a singular point in the phase diagram, separating a region with a spin-glass phase ($w<1/2$) from a region with spin-glass, ferromagnetic, mixed, and paramagnetic phases ($w>1/2$).
Spin and chirality orderings of the three-dimensional Heisenberg spin glass are studied under magnetic fields in light of the recently developed spin-chirality decoupling-recoupling scenario. It is found by Monte Carlo simulations that the chiral-glass transition and the chiral-glass ordered state, which are essentially of the same character as their zero-field counterparts, occur under magnetic fields. Implication to experimental phase diagram is discussed.
The spin and the chirality orderings of the three-dimensional Heisenberg spin glass with the weak random anisotropy are studied under applied magnetic fields by equilibrium Monte Carlo simulations. A replica symmetry breaking transition occurs in the chiral sector accompanied by the simultaneous spin-glass order. The ordering behavior differs significantly from that of the Ising SG, despite the similarity in the global symmetry. Our observation is consistent with the spin-chirality decoupling-recoupling scenario of a spin-glass transition.
The zero-point entropy of a spin glass is a difficult property to experimentally determine and interpret. Spin glass theory provides various predictions, including unphysical ones, for the value of the zero-point entropy, however experimental results have been lacking. We have investigated the magnetic properties and zero-point entropy of two spinel Cu2+ based spin glasses, CuGa2O4 and CuAl2O4. Dc- and ac-susceptibility and specific heat measurements show many characteristic spin glass features for both materials. The spin glass freezing temperature is determined to be Tf = 2.89 +/- 0.05 K for…
The nature of the ordering of the three-dimensional isotropic Heisenberg spin glass with nearest-neighbor random Gaussian coupling is studied by extensive Monte Carlo simulations. Several independent physical quantities are measured both for the spin and for the chirality, including the correlation-length ratio, the Binder ratio, the glass order parameter, the overlap distribution function and the non-self-averageness parameter. By controlling the effect of the correction-to-scaling, we have obtained a numerical evidence for the occurrence of successive chiral-glass and spin-glass transitions …
Glass states of superfluid A-like phase of 3He in aerogel induced by random orientations of aerogel strands are investigated theoretically and experimentally. In anisotropic aerogel with stretching deformation two glass phases are observed. Both phases represent the anisotropic glass of the orbital ferromagnetic vector l -- the orbital glass (OG). The phases differ by the spin structure: the spin nematic vector d can be either in the ordered spin nematic (SN) state or in the disordered spin-glass (SG) state. The first phase (OG-SN) is formed under conventional cooling from normal 3He. The seco…
Numerical simulations on Ising Spin Glasses show that spin glass transitions do not obey the usual universality rules which hold at canonical second order transitions. On the other hand the dynamics at the approach to the transition appear to take up a universal form for all spin glasses. The implications for the fundamental physics of transitions in complex systems are addressed.
By means of parallel tempering Monte Carlo simulations we find strong evidence for a finite-temperature spin-glass transition in a system of diluted classical Heisenberg dipoles randomly placed on the sites of a simple cubic lattice. We perform a finite-size scaling analysis of the spin-glass susceptibility, $χ_{\rm SG}$, and spin-glass correlation length, $ξ_{L}$. For the available system sizes that can be successfully equilibrated, the crossing of $ξ_{L}/L$ versus temperature is strongly affected by corrections to scaling and possibly by a short-length scale ferromagnetic spin blocking. Simi…
We probe the energy landscape of the 3D Edwards-Anderson spin glass in a magnetic field to test for a spin glass ordering. We find that the spin glass susceptibility is anomalously large on the lattice sizes we can reach. Our data suggest that a transition from the spin glass to the paramagnetic phase takes place at B_c=0.65, though the possibility B_c=0 cannot be excluded. We also discuss the question of the nature of the putative frozen phase.
We examine the phase diagram of the $p$-spin mean field glass model in the spin one case, that is when $S=0,+1,-1$. For large $p$ the model is solved exactly. The analysis reveals that the phase diagram is in some way similar to that of Ising spins. However, as we show, the quadrupolar regular ordering as well as qudrupolar glass order are present now. The temperature of the Gardner 1RSB -- FRSB transition is obtained explicitly for large $p$. The case of higher spins is discussed briefly.
We consider identical copies of spin glasses in finite dimension coupled at the boundaries. This allows to identify the spin glass analogous of twisted boundary conditions in ferromagnetic system and leads to the definition of an interface free-energy that is positively defined and that should scale with a positive power of the system size in the spin glass phase. In this note we study the behavior of the interface at the spin glass critical temperature $T_c$ within mean field theory. We show that the leading scaling of the interface free-energy does not depend on replica symmetry breaking, an…
We study a p-spin spin-glass model to understand if the finite-temperature glass transition found in the mean-field regime of p-spin models, and used to model the behavior of structural glasses, persists in the non-mean-field regime. By using a 3-spin spin-glass model with long-range power-law diluted interactions we are able to continuously tune the (effective) space dimension via the exponent of the interactions. Monte Carlo simulations of the spin-glass susceptibility and the two-point finite-size correlation length show that deep in the non-mean-field regime the finite-temperature transiti…
Spin-glass (SG) states of the 3-dimensional Ising Edwards-Anderson model under a static magnetic field $h$ are examined by means of the standard Monte Carlo simulation on the field-shift aging protocol at temperature $T$. For each process with $(T; \tw, h)$, $\tw$ being the waiting time before the field is switched on, we extract the dynamical crossover time, $\tcr(T; \tw, h)$. We have found a nice scaling relation between the two characteristic length scales which are properly determined from $\tcr$ and $\tw$ and then are normalized by the static field crossover length introduced in the SG dr…
The ordering of the one-dimensional Heisenberg spin glass interacting via the long-range power-law interaction is studied by Monte Carlo simulations. Particular attention is paid to the possible occurrence of the ``spin-chirality decoupling'' for appropriate values of the power-law exponent σ. Our result suggests that, for intermediate values of $σ$, the chiral-glass order occurs at finite temperatures while the standard spin-glass order occurs only at zero temperature.
Dilute magnetic nanoparticle systems exhibit slow dynamics [1] due to a broad distribution of relaxation times that can be traced to a correspondingly broad distribution of particle sizes [1]. However, at higher concentrations interparticle interactions lead to a slow dynamics that is qualitatively indistinguishable from that dislayed by atomic spin glasses. A theory is derived below that accounts quantitatively for the spin-glass behaviour. The theory predicts that if the interactions become too strong the spin glass behaviour disappears. This conclusion is in agreement with preliminary exper…
This is a short review about recent methods and results, mostly for mean field spin glasses, based on interpolation and comparison schemes. In particular, the Parisi spontaneous replica symmetry breaking phenomenon is described in the frame of extended variational principles, Derrida-Ruelle probability cascades, and overlap locking.
We study the issue of the spin-chirality decoupling/coupling in the ordering of the Heisenberg spin glass by performing large-scale Monte Carlo simulations on a one-dimensional Heisenberg spin-glass model with a long-range power-law interaction up to large system sizes. We find that the spin-chirality decoupling occurs for an intermediate range of the power-law exponent. Implications to the corresponding $d$-dimensional short-range model is discussed.
Computing the ground state of Ising spin-glass models with p-spin interactions is, in general, an NP-hard problem. In this work we show that unlike in the case of the standard Ising spin glass with two-spin interactions, computing ground states with p=3 is an NP-hard problem even in two space dimensions. Furthermore, we present generic exact and heuristic algorithms for finding ground states of p-spin models with high confidence for systems of up to several thousand spins.
An overview is given of experimental data on Heisenberg Spin Glass materials so as to make detailed comparisons with numerical results on model Heisenberg spin glasses, with particular reference to the chiral driven ordering transition scenario due to Kawamura and collaborators. On weak anisotropy systems, experiments show critical exponents which are very similar to those estimated numerically for the model Heisenberg chiral ordering transition but which are quite different from those at Ising spin glass transitions. Again on weak anisotropy Heisenberg spin glasses, experimental torque data s…
We test for the existence of a spin-glass phase transition, the de Almeida-Thouless line, in an externally-applied (random) magnetic field by performing Monte Carlo simulations on a power-law diluted one-dimensional Ising spin glass for very large system sizes. We find that an Almeida-Thouless line only occurs in the mean field regime, which corresponds, for a short-range spin glass, to dimension d larger than 6.
We review recent findings on spin glass models. Both the equilibrium properties and the dynamic properties are covered. We focus on progress in theoretical, in particular numerical, studies, while its relationship to real magnetic materials is also mentioned.
We report on the ensemble inequivalence in a many-body spin-glass model with integer spin. The spin-glass phase transition is of first order for certain values of the crystal field strength and is dependent whether it was derived in the microcanonical or the canonical ensemble. In the limit of infinitely many-body interactions, the model is the integer-spin equivalent of the random-energy model, and is solved exactly. We also derive the integer-spin equivalent of the de Almeida-Thouless line.
We consider the thermodynamics of an infinite-range Ising p-spin glass model with an additional r-spin ferromagnetic interaction. For r=2 there is a continuous transition to a ferromagnetic phase, while for r>2 the transition is first order. We find both glassy and non-glassy ferromagnetic phases, with replica symmetry breaking of both the one step and full varieties. We obtain new results for the case where r=p>2, demonstrating the existence of a non-glassy ferromagnetic phase, of significance to error-correcting codes.
Guided by old results on simple mode-coupling models displaying glass-glass transitions, we demonstrate, through a crude analysis of the solution with one step of replica symmetry breaking (1RSB) derived by Crisanti and Leuzzi for the spherical $s+p$ mean-field spin glass [Phys. Rev. B 73, 014412 (2006)], that the phase behavior of these systems is not yet fully understood when $s$ and $p$ are well separated. First, there seems to be a possibility of glass-glass transition scenarios in these systems. Second, we find clear indications that the 1RSB solution cannot be correct in the full glassy …
A spin glass is a diluted magnetic material in which the magnetic moments are randomly interacting, with a huge number of metastable states which prevent reaching equilibrium. Spin-glass models are conceptually simple, but require very sophisticated treatments. These models have become a paradigm for the understanding of glassy materials and also for the solution of complex optimization problems. After cooling from the paramagnetic phase, the spin glass remains out of equilibrium, and slowly evolves. This aging phenomenon corresponds to the growth of a mysterious "spin-glass order", whose corr…
