Recent work suggests that Heisenberg spin glasses may belong to the same universality class than structural glasses. Indeed, finding a lattice equivalent for supercooled liquids would probably allow easier numerical and analytical studies, that may help to answer long-standing questions on the glass transition. Supercooled liquids have many peculiar behaviors that should be found in the paramagnetic phase of Heisenberg spin glasses if the analogy between the two systems holds. It is with this motivation that we undertake a study of the paramagnetic phase of Heisenberg spin glasses. We shall emphasize the role of the energy landscape, with a detailed study of the properties of the inherent structures (by analogy with supercooled liquids, we name inherent structure the local minimum of the energy function which is closest to the current spin configuration). Finding inherent structures will require the development of a new search algorithm. We shall investigate as well the existence of a dynamic transition in the paramagnetic phase. As a matter of fact, both the existence of a complex energy landscape, as well as the existence of a dynamic transition, are distinguished features of the Physics of supercooled liquids.
Spin-glass (SG) and chiral-glass (CG) orderings in three dimensional (3D) Heisenberg spin glass with and without magnetic anisotropy are studied by using large-scale off-equilibrium Monte Carlo simulations. A characteristic time of relaxation, which diverges at a transition temperature in the thermodynamic limit, is obtained as a function of the temperature and the system size. Based on the finite-size scaling analysis for the relaxation time, it is found that in the isotropic Heisenberg spin glass, the CG phase transition occurs at a finite temperature, while the SG transition occurs at a low…
We numerically analyse the landscape governing the evolution of the vibrational dynamics of hard disk glasses as the density increases towards jamming. We find that the dynamics becomes slow, spatially correlated, and starts to display aging dynamics across an avoided Gardner transition, with a phenomenology that resembles three dimensional observations. We carefully analyse the behaviour of single glass samples, and find that the emergence of aging dynamics is controlled by the apparition of a complex organisation of the landscape that splits into a remarkable hierarchy of minima as jamming i…
We investigate the spectral and localization properties of unmagnetized Heisenberg-Mattis spin glasses, in space dimensionalities $d=2$ and 3, at T=0. We use numerical transfer-matrix methods combined with finite-size scaling to calculate Lyapunov exponents, and eigenvalue-counting theorems, coupled with Gaussian elimination algorithms, to evaluate densities of states. In $d=2$ we find that all states are localized, with the localization length diverging as $ω^{-1}$, as energy $ω\to 0$. Logarithmic corrections to density of states behave in accordance with theoretical predictions. In $d=3$ the…
An appropriate model for the random energy landscape in organic glasses is a spatially correlated Gaussian field. We calculated the distribution of the average value of a Gaussian random field in a finite domain. The results of the calculation demonstrate a strong dependence of the width of the distribution on the spatial correlations of the field. Comparison with the simulation results for the distribution of the size of the cluster indicates that the distribution of an average field could serve as a useful tool for the estimation of the asymptotic behavior of the distribution of the size of …
We present the results of a large-scale numerical study of the equilibrium three-dimensional Edwards-Anderson Ising spin glass with Gaussian disorder. Using parallel tempering (replica exchange) Monte Carlo we measure various static, as well as dynamical quantities, such as the autocorrelation times and round-trip times for the parallel tempering Monte Carlo method. The correlation between static and dynamic observables for 5000 disorder realizations and up to 1000 spins down to temperatures at 20% of the critical temperature is examined. Our results show that autocorrelation times are directl…
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.
For a model glass-former we demonstrate via computer simulations how macroscopic dynamic quantities can be inferred from a PEL analysis. The essential step is to consider whole superstructures of many PEL minima, called metabasins, rather than single minima. We show that two types of metabasins exist: some allowing for quasi-free motion on the PEL (liquid-like), the others acting as traps (solid-like). The activated, multi-step escapes from the latter metabasins are found to dictate the slowing down of dynamics upon cooling over a much broader temperature range than is currently assumed.
We have analyzed a fermionic infinite-ranged quantum Heisenberg spin glass with BCS coupling in real space in the presence of a magnetic field. It has been possible to locate the transition line between the normal paramagnetic phase (NP) and the phase where there is a long range order corresponding to pair formation in sites (PAIR). The nature of the transition line is also investigated. This transition ends at Tf, the transition temperature between NP and spin glass phase (SG) where the static approximation and replica symmetry ansatz are reliable.
We propose damage spreading (DS) as a tool to investigate the topological features related to the ruggedness of the free energy landscape. We argue that DS measures the positiveness of the largest Lyapunov exponent associated to the basins of attraction visited by the system during its dynamical evolution. We discuss recent results obtained in the framework of mode-coupling theory and comment how to extend them to the study of realistic glasses. Preliminary results are presented for purely repulsive soft-sphere glasses.
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.
An appropriate model for the random energy landscape in organic glasses is a spatially correlated Gaussian field, generated by randomly located and oriented dipoles and quadrupoles. Correlation properties of energetic disorder directly dictates the mobility dependence on the applied electric field. Electrostatic disorder is significantly modified in the vicinity of the electrode that affects injection properties. Correlated Gaussian field forms clusters. We suggest a simple method to estimate an asymptotics of the cluster distribution on size for deep clusters where a value of the field on eac…
Non-equilibrium dynamics of three dimensional model spin glasses - the Ising system Fe$_{0.50}$Mn$_{0.50}$TiO$_3$ and the Heisenberg like system Ag(11 at% Mn) - has been investigated by measurements of the isothermal time decay of the low frequency ac-susceptibility after a quench from the paramagnetic to the spin glass phase. It is found that the relaxation data measured at different temperatures can be scaled according to predictions from the droplet scaling model, provided that critical fluctuations are accounted for in the analyzes.
Brief review is given on recent numerical research of the ordering of two typical models of spin glasses (SGs), the three-dimensional (3D) Ising SG and the 3D Heisenberg SG models. Particular attention is paid to the questions of whether there is a thermodynamic transition in zero field, what are the associated critical properties, what is the nature of the ordered state, particularly of a possible replica-symmetry breaking, and whether there is a thermodynamic transition in applied fields. The properties of the two models are contrasted, and possible relation to experiments is discussed.
We characterize the phase diagram of anisotropic Heisenberg spin glasses, finding both the spin and the chiral glass transition. We remark the presence of strong finite-size effects on the chiral sector. We find a unique phase transition for the chiral and spin glass sector, in the Universality class of Ising spin glasses. We focus on keeping finite-size effects under control, and we stress that they are important to understand experiments. Thanks to large GPU clusters we have been able to thermalize cubic lattices with up to 64x64x64 spins, over a vast range of temperatures.
We have studied the topology of the energy landscape of a spin-glass model and the effect of frustration on it by looking at the connectivity and disconnectivity graphs of the inherent structure. The connectivity network shows the adjacency of energy minima whereas the disconnectivity network tells us about the heights of the energy barriers. Both graphs are constructed by the exact enumeration of a two-dimensional square lattice of a frustrated spin glass with nearest-neighbor interactions up to the size of 27 spins. The enumeration of the energy-landscape minima as well as the analytical mea…
We examine three-dimensional $\pm J$ Heisenberg models with and without random anisotropies in a magnetic field. We calculate both the stiffness exponent $θ_s$ at absolute zero temperature and spin-glass correlation lengths for the longitudinal and transverse spin components at finite temperatures. We suggest that, contrary to a chirality scenario predicted by Kawamura and his co-workers, a Gabay-Toulouse phase transition occurs when the anisotropies are absent, although no phase transition occurs when they are present.
For five different Heisenberg spin glass systems, torque experiments were performed in applied magnetic fields up to $4 T$. The Dzyaloshinski-Moriya random anisotropy strengths, the in-field torque onset temperatures, and the torque relaxation were measured. Critical exponents were estimated independently using a standard protocol. The data are strong evidence for a true spin glass ordered state which survives under high applied magnetic fields; they can be interpreted consistently in terms of a chiral ordering model with replica symmetry breaking as proposed by Kawamura and coworkers.
We analyse the relationship between dynamics and configuration space structure of Ising spin glass systems. The exact knowledge of the structure of the low--energy landscape is used to study the relaxation of the system by random walk in the configuration space. The influence of the size of the valleys, clusters and energy barriers and the connectivity between them on the spin correlation function is shown.
Large numbers of ground states of two-dimensional Ising spin glasses with periodic boundary conditions in both directions are calculated for sizes up to 40^2. A combination of a genetic algorithm and Cluster-Exact Approximation is used. For each quenched realization of the bonds up to 40 independent ground states are obtained. For the infinite system a ground-state energy of e=-1.4015(3) is extrapolated. The ground-state landscape is investigated using a finite-size scaling analysis of the distribution of overlaps. The mean-field picture assuming a complex landscape describes the situation bet…
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…
Glasses are amorphous solids whose constituent particles are caged by their neighbors and thus cannot flow. This sluggishness is often ascribed to the free energy landscape containing multiple minima (basins) separated by high barriers. Here we show, using theory and numerical simulation, that the landscape is much rougher than is classically assumed. Deep in the glass, it undergoes a "roughness transition" to fractal basins. This brings about isostaticity at jamming and marginality of glassy states near jamming. Critical exponents for the basin width, the weak force distribution, and the spat…
Spin glasses are frustrated magnetic systems due to a random distribution of ferro- and antiferromagnetic interactions. An experimental three dimensional (3d) spin glass exhibits a second order phase transition to a low temperature spin glass phase regardless of the spin dimensionality. In addition, the low temperature phase of Ising and Heisenberg spin glasses exhibits similar non-equilibrium dynamics and an infinitely slow approach towards a thermodynamic equilibrium state. There are however significant differences in the detailed character of the dynamics as to memory and rejuvenation pheno…
A major challenge in the modeling of ionically conducting glasses is to understand how the large variety of possible chemical compositions and specific structural properties influence ionic transport quantities. Here we revisit and extend a theoretical approach for alkali borophosphate glasses, where changes of conductivity activation energies with the borate to phosphate mixing ratio are related to modifications of the ionic site energy landscape. The landscape modifications are caused by varying amounts of different units forming the glassy network, which lead to spatial redistributions of t…
connected spin-glass models with a discontinuous transition. In the thermodynamic limit the equilibrium properties in the high temperature phase are described by the schematic Mode Coupling Theory of super-cooled liquids. We show that {\it finite-size} fully connected spin-glass models do exhibit properties typical of Lennard-Jones systems when both are near the critical glass transition, where thermodynamics is ruled by energy minima distribution. Our study opens the way to consider activated processes in real glasses through finite-size corrections (i.e. calculations beyond the saddle point …
The equation of state for a liquid in equilibrium, written in the potential energy landscape formalism, is generalized to describe out-of-equilibrium conditions. The hypothesis that during aging the system explores basins associated to equilibrium configurations is the key ingredient in the derivation. Theoretical predictions are successfully compared with data from molecular dynamics simulations of different aging processes, such as temperature and pressure jumps.
We study the distribution of overlaps of glassy minima, taking proper care of residual symmetries of the system. Ensembles of locally stable, low lying glassy states are efficiently generated by rapid cooling from the liquid phase which has been equilibrated at a temperature $T_{run}$. Varying $T_{run}$, we observe a transition from a regime where a broad range of states are sampled to a regime where the system is almost always trapped in a metastable glassy state. We do not observe any structure in the distribution of overlaps of glassy minima, but find only very weak correlations, comparable…
The isothermal remanent magnetization is used to investigate dynamical magnetic properties of spatially three dimensional spin glasses with different spin dimensionality (Ising, XY, Heisenberg). The isothermal remanent magnetization is recorded vs. temperature after intermittent application of a weak magnetic field at a constant temperature $T_h$. We observe that in the case of the Heisenberg spin glasses, the equilibrated spin structure and the direction of the excess moment are recovered at $T_h$. The isothermal remanent magnetization thus reflects the directional character of the Dzyaloshin…
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 …
We discuss the slow, nonequilibrium, dynamics of spin glasses in their glassy phase. We briefly review the present theoretical understanding of the spectacular phenomena observed in experiments and describe new numerical results obtained in the first large-scale simulation of the nonequilibrium dynamics of the three dimensional Heisenberg spin glass.
We numerically study the non-equilibrium dynamics of the three dimensional Heisenberg Edwards-Anderson spin glass following a sudden quench to its low temperature phase. The subsequent aging behavior of the system is analyzed in detail, and the scaling behavior of various space-time correlation functions is investigated for both spin and chiral degrees of freedom. We carefully compare our results with those obtained from simulations of the more studied Ising version of the model, and with experiments on real spin glasses in which the spins have vectorial character. Finally, the present dynamic…
The population annealing algorithm introduced by Hukushima and Iba is described. Population annealing combines simulated annealing and Boltzmann weighted differential reproduction within a population of replicas to sample equilibrium states. Population annealing gives direct access to the free energy. It is shown that unbiased measurements of observables can be obtained by weighted averages over many runs with weight factors related to the free energy estimate from the run. Population annealing is well suited to parallelization and may be a useful alternative to parallel tempering for systems …
Disconnectivity graphs are used to visualize the minima and the lowest energy barriers between the minima of complex systems. They give an easy and intuitive understanding of the underlying energy landscape and, as such, are excellent tools for understanding the complexity involved in finding low-lying or global minima of such systems. We have developed a classification scheme that categorizes highly-degenerate minima of spin glasses based on similarity and accessibility of the individual states. This classification allows us to condense the information pertained in different dales of the ener…
Results of Monte Carlo simulations of XY and Heisenberg spin glass models in three dimensions are presented. A finite size scaling analysis of the correlation length of the spins and chiralities of both models shows that there is a single, finite-temperature transition at which both spins and chiralities order.
Fossil amber offers the unique opportunity of investigating an amorphous material which has been exploring its energy landscape for more than 110 Myears of natural aging. By applying different x-ray scattering methods to amber before and after annealing the sample to erase its thermal history, we identify a link between the potential energy landscape and the structural and vibrational properties of glasses. We find that hyperaging induces a depletion of the vibrational density of states in the THz region, also ruling the sound dispersion and attenuation properties of the corresponding acoustic…
We have compared aging phenomena in the Fe_{0.5}Mn_{0.5}TiO_3 Ising spin glass and in the CdCr_{1.7}In_{0.3}S_4 Heisenberg-like spin glass by means of low-frequency ac susceptibility measurements. At constant temperature, aging obeys the same `$ωt$ scaling' in both samples as in other systems. Investigating the effect of temperature variations, we find that the Ising sample exhibits rejuvenation and memory effects which are qualitatively similar to those found in other spin glasses, indicating that the existence of these phenomena does not depend on the dimensionality of the spins. However, sy…
Spin glasses are notoriously difficult to study both analytically and numerically due to the presence of frustration and metastability. Their highly non-convex landscapes require collective updates to explore efficiently. Currently, most state-of-the-art algorithms rely on stochastic spin clusters to perform non-local updates, but such "cluster algorithms" lack general efficiency. Here, we introduce a non-equilibrium approach for simulating spin glasses based on classical dynamics with memory. By simulating various classes of 3d spin glasses (Edwards-Anderson, partially-frustrated, and fully-f…
A method is presented, which allows to sample directly low-temperature configurations of glassy systems, like spin glasses. The basic idea is to generate ground states and low lying excited configurations using a heuristic algorithm. Then, with the help of microcanonical Monte Carlo simulations, more configurations are found, clusters of configurations are determined and entropies evaluated. Finally equilibrium configuration are randomly sampled with proper Gibbs-Boltzmann weights. The method is applied to three-dimensional Ising spin glasses with +- J interactions and temperatures T<=0.5. The…
We study and discuss rejuvenation and memory (numerical) experiments in Ising and Heisenberg three and four dimensional spin glasses. We introduce a quantitative procedure to analyze the results of temperature cycling experiments. We also run, compare and discuss ``twin'' couples of experiments. We find that in our systems aging is always cumulative in nature, and rejuvenation and memory effects are also cumulative: they are very different from the ones observed in experiments on spin glass materials.
We study mean field systems whose free energy landscape is dominated by marginally stable states. We review and develop various techniques to describe such states, elucidating their physical meaning and the interrelation between them. In particular, we give a physical interpretation of the two-group replica symmetry breaking scheme and confirm it by establishing the relation to the cavity method and to the counting of solutions of the Thouless-Anderson-Palmer equations. We show how these methods all incorporate the presence of a soft mode in the free energy landscape and interpret the occurrin…
In this paper we expose the results of our recent work on the dynamical TAP approach to mean field glassy systems. Our aim is to clarify the connection between free energy landscape and out of equilibrium dynamics in solvable models. Frequently qualitative explanations of glassy behaviour are based on the ``Free Energy Landscape Paradigm''. If its relevance for equilibrium properties is clear, the relationship between free energy landscape and out of equilibrium dynamics is not well understood yet. In this paper we clarify this relationship for the class of spin glass models which reproduce ph…
Despite the extreme simplicity in their definition, spin glasses disclose a wide variety of non-trivial behaviors that are not yet fully understood. In this thesis we try to shed light on some of them, focusing on one hand on the search of phase transitions under perturbations of the Hamiltonian, and on the other hand on the zero-temperature properties and responses to external stimuli. After introducing spin glasses through a historiographical review, and useful concepts on phase transitions and numerics, the results of two massive Monte Carlo campaigns on three-dimensional systems are shown.…
Glassy behavior is one of the main open problems in condensed matter physics. In this thesis, we approach the problem by studying spin-glasses and colloids, using several complementary strategies. From the point of view of model building, we propose a new model, the hypercube model, which is mean-field but defined in a metric space. We show that, also in Mean Field, the spin-glass non-equilibrium dynamics is a domain-growth process very similar to its three-dimensional counterpart. We also consider spin-glasses in three spatial dimensions, focusing on temperature chaos. Thanks to the low-tempe…
We argue that the chaotic temperature effect predicted in Ising spin glasses should be stronger when one considers continuous (XY, Heisenberg) kind of spins, due to bigger entropic fluctuations. We then discuss the behavior of 3d spin glasses using the Migdal-Kadanoff Renormalization Group for Ising and XY spins, where we show explicitly that the chaotic length scale, the length beyond which equilibrium configurations are completly reshuffled when one changes temperature, could be far smaller for XY than for Ising spins. These results could thus explain why experiments are seeing a stronger re…
Rare Trajectories in a Prototypical Mean-field Disordered Model: Insights into Landscape and Instantons
Duke University（杜克大学）；Institute for Cross-disciplinary Physics and Complex Systems IFISC (CSIC-UIB)（交叉物理与复杂系统研究所（CSIC-UIB））；Bocconi University（博科尼大学）；Institute for Data Science and Analytics, Bocconi University（博科尼大学数据科学与分析研究所）；Institute of Complex Systems (ISC) - CNR, Rome unit（意大利国家研究委员会复杂系统研究所罗马分部）；Sapienza Università di Roma（罗马大学）
We test for the presence or absence of the de Almeida-Thouless line using one-dimensional power-law diluted Heisenberg spin glass model, in which the rms strength of the interactions decays with distance, r as 1/r^{sigma}. It is argued that varying the power sigma is analogous to varying the space dimension d in a short-range model. For sigma=0.6, which is in the mean field regime regime, we find clear evidence for an AT line. For sigma = 0.85, which is in the non-mean-field regime and corresponds to a space dimension of close to 3, we find no AT line, though we cannot rule one out for very sm…
There is a growing belief that the mode coupling theory is the proper microscopic theory for the dynamics of the undercooled liquid above a critical temperature T_c. In addition, there is some evidence that the system leaves the saddlepoints of the energy landscape to settle in the valleys at this critical temperature. Finally, there is a microscopic theory for the entropy at the calorimetric glass transition T_g by Mezard and Parisi, which allows to calculate the Kauzmann temperature from the atomic pair potentials. The dynamics of the frozen glass phase is at present limited to phenomenologi…
Using Monte Carlo simulations, we compute the spin stiffness of a site-random 3d Heisenberg model with competing ferromagnetic and antiferromagnetic interactions. Our results for the pure limit yield values of the the critical temperature and the critical exponent $ν$ in excellent agreement with previous high precision studies. In the disordered case, a mixed "chiral" phase is found which may be in the same universality class as 3d Heisenberg spin glasses.
We study the problem of glassy relaxations in the presence of an external field in the highly controlled context of a spin-glass simulation. We consider a small spin glass in three dimensions (specifically, a lattice of size L=8, small enough to be equilibrated through a Parallel Tempering simulations at low temperatures, deep in the spin glass phase). After equilibrating the sample, an external field is switched on, and the subsequent dynamics is studied. The field turns out to reduce the relaxation time, but huge statistical fluctuations are found when different samples are compared. After t…
We study the linear excitations around typical energy minima of a mean-field disordered model with continuous degrees of freedom undergoing a Random First Order Transition (RFOT). Contrary to naive expectations, the spectra of linear excitations are ungapped and we find the presence of a pseudogap corresponding to localized excitations with arbitrary low excitation energy. Moving to deeper minima in the landscape, the excitations appear increasingly localized while their abundance decreases. Beside typical minima, there also exist rare ultra-stable minima, with an energy gap and no localised e…
We carefully investigate the two fundamental assumptions in the Stillinger-Weber analysis of the inherent structures (IS's) in the energy landscape and come to conclude that they cannot be validated. This explains some of the conflicting results between their conclusions and some recent rigorous and exact results. Our analysis shows that basin free energies, and not IS's, are useful for understanding glasses.
Ordering properties of XY-like spin-glass magnets with an easy-plane magnetic anisotropy are studied based on a symmetry consideration and the results of recent numerical simulations on the pure Heisenberg and XY spin-glass models. The effects of an easy-plane-type uniaxial anisotropy, a random magnetic anisotropy and an applied magnetic field are investigated. In the XY regime in zero field, the "spin-chirality decoupling" persists even under the random magnetic anisotropy, escaping the "spin-chirality recoupling" phenomenon which inevitably occurs in the Heisenberg regime. Contrast between t…
Population annealing is an efficient sequential Monte Carlo algorithm for simulating equilibrium states of systems with rough free energy landscapes. The theory of population annealing is presented, and systematic and statistical errors are discussed. The behavior of the algorithm is studied in the context of large-scale simulations of the three-dimensional Ising spin glass and the performance of the algorithm is compared to parallel tempering. It is found that the two algorithms are similar in efficiency though with different strengths and weaknesses.
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…
We consider damage spreading transitions in the framework of mode-coupling theory. This theory describes relaxation processes in glasses in the mean-field approximation which are known to be characterized by the presence of an exponentially large number of meta-stable states. For systems evolving under identical but arbitrarily correlated noises we demonstrate that there exists a critical temperature $T_0$ which separates two different dynamical regimes depending on whether damage spreads or not in the asymptotic long-time limit. This transition exists for generic noise correlations such that …
Calorimetric experiments on network glasses provide information on the ergodicity (landscape) temperature of supercooled liquids and can be compared with a recent theory developed by Hall and Wolynes [PRL90, 085505 (2003)]
In the present paper we have analysed a fermionic infinite-ranged quantum Heisenberg spin glass (s=1/2) with a BCS coupling in real space in the presence of an applied magnetic field. This model has been obtained by tracing out the conducting fermions in a superconducting alloy. The magnetic field is applied in the resulting effective model. The problem is formulated in the path integral formalism where the spin variables are defined as bilinear combinations of the Grassmann fields. The static approximation is used to treat both the pairing and the spin terms together with the replica symmetry…
