The frustrated Ising model on a two-dimensional lattice with open boundary conditions is revisited. A hidden Z2 gauge symmetry relates models with different frustrations which, however, share the same partition function. By means of a duality transformation, it is shown that the partition function only depends on the distribution of gauge invariant vortices on the lattice. We finally show that the exact ground state energy can be calculated in polynomial time using Edmonds' algorithm.
A class of models with self-generated disorder and controlled frustration is studied. Between the trivial case, where frustration is not present at all, and the limit case, where frustration is present over every length scale, a region with local frustration is found where glassy dynamics appears. We suggest that in this region, the mean field model might undergo a p-spin like transition, and increasing the range of frustration, a crossover from a 1-step replica symmetry breaking to a continuous one might be observed.
We study the geometrical frustration scenario of glass formation for simple hard sphere models. We find that the dual picture in terms of defects brings little insight and no theoretical simplification for the understanding of the slowing down of relaxation, because of the strong frustration characterizing these systems. The possibility of a growing static length is furthermore found to be physically irrelevant in the regime that is accessible to computer simulations.
We investigate glassy dynamical properties of one component three-dimensional system of particles interacting via pair repulsive potential by the molecular dynamic simulation in the wide region of densities. The glass state is superfragile and it has high glassforming ability. The glass transition temperature Tg has pronounced minimum at densities where the frustration is maximal.
In spin-glass systems, frustration can be adjusted continuously and considerably, without changing the antiferromagnetic bond probability p, by using locally correlated quenched randomness, as we demonstrate here on hypercubic lattices and hierarchical lattices. Such overfrustrated and underfrustrated Ising systems on hierarchical lattices in d=3 and 2 are studied. With the removal of just 51 % of frustration, a spin-glass phase occurs in d=2. With the addition of just 33 % frustration, the spin-glass phase disappears in d=3. Sequences of 18 different phase diagrams for different levels of fru…
In a finite-connectivity spin-glass at the zero-temperature limit, long-range correlations exist among the unfrozen vertices (whose spin values being non-fixed). Such long-range frustrations are partially removed through the first-step replica-symmetry-broken (1RSB) cavity theory, but residual long-range frustrations may still persist in this mean-field solution. By way of population dynamics, here we perform a perturbation-percolation analysis to calculate the magnitude of long-range frustrations in the 1RSB solution of a given spin-glass system. We study two well-studied model systems, the m…
We conjecture the existence of a relationship between frustration and the transition point at zero temperature of Ising spin glasses. The relation reveals that, in several Ising spin glass models, the concentration of ferromagnetic bonds is close to the critical concentration at zero temperature when the output of a function about frustration is equal to unity. The function is the derivative of the average number of frustrated plaquettes with respect to the average number of antiferromagnetic bonds. This relation is conjectured in Ising spin glasses with binary couplings on two-dimensional lat…
The design of multi-functional BMGs is limited by the lack of a quantitative understanding of the variables that control the glass-forming ability (GFA) of alloys. Both geometric frustration (e.g. differences in atomic radii) and energetic frustration (e.g. differences in the cohesive energies of the atomic species) contribute to the GFA. We perform molecular dynamics simulations of binary Lennard-Jones mixtures with only energetic frustration. We show that there is little correlation between the heat of mixing and critical cooling rate $R_c$, below which the system crystallizes, except that $…
We introduce a hard-core lattice-gas model on generalized Bethe lattices and investigate analytically and numerically its compaction behavior. If compactified slowly, the system undergoes a first-order crystallization transition. If compactified much faster, the system stays in a meta-stable liquid state and undergoes a glass transition under further compaction. We show that this behavior is induced by geometrical frustration which appears due to the existence of short loops in the generalized Bethe lattices. We also compare our results to numerical simulations of a three-dimensional analog of…
Shortened abstract: A mean field theory of long range frustration is constructed for spin glass systems with quenched randomness of vertex--vertex connections and of spin--spin coupling strengths. This theory is applied to a spin glass model of the random $K$-satisfiability problem (K=2 or K=3). The zero--temperature phase diagram of the $\pm J$ Viana--Bray model is also determined, which is identical to that of the random 2-SAT problem. The predicted phase transition between a non-frustrated and a long--rangely frustrated spin glass phase might also be observable in real materials at a finite…
In a spin glass system on a random graph, some vertices have their spins changing among different configurations of a ground--state domain. Long range frustrations may exist among these unfrozen vertices in the sense that certain combinations of spin values for these vertices may never appear in any configuration of this domain. We present a mean field theory to tackle such long range frustrations and apply it to the NP-hard minimum vertex cover (hard-core gas condensation) problem. Our analytical results on the ground-state energy density and on the fraction of frozen vertices are in good agr…
We investigate a possible relation between frustration and phase-transition points in spin glasses. The relation is represented as a condition of the number of frustrated plaquettes in the lattice at phase-transition points at zero temperature and was reported to provide very close points to the phase-transition points for several lattices. Although there has been no proof of the relation, the good correspondence in several lattices suggests the validity of the relation and some important role of frustration in the phase transitions. To examine the relation further, we present a natural extens…
What does the equilibrium atomic, molecular or spin configuration of a glass phase look like? Is there only one unique equilibrium configuration or are there infinitely many configurations of equal energy? The processes and mechanisms governing the path towards equilibrium, i.e. the dynamics of glassy systems, provide insights to these questions. Here we discuss the intrinsic dynamics of different glassy magnets: of spin-glasses, frustrated ferromagnets, superspin-glasses and other nanostructured systems with competing ferro- and antiferromagnetic interactions and randomness in their spatial d…
This theory combines a thermodynamic approach with a dynamic one in order to describe glass transition. Glass transition is regarded as an inaccessible second order phase transition, which is interrupted because of premature critical slowing down, caused by the system's frustration. The frustration-induced vortices are present in the structure besides thermoactivated vortices, and prevent the development of the order parameter fluctuations, that leads to the critical slowing down the system kinetics at some temperature above the phase transition point.
In this work it is studied the Hopfield fermionic spin glass model which allows interpolating from trivial randomness to a highly frustrated regime. Therefore, it is possible to investigate whether or not frustration is an essential ingredient which would allow this magnetic disordered model to present naturally inverse freezing by comparing the two limits, trivial randomness and highly frustrated regime and how different levels of frustration could affect such unconventional phase transition. The problem is expressed in the path integral formalism where the spin operators are represented by b…
We develop a novel method based in the sparse random graph to account the interplay between geometric frustration and disorder in cluster magnetism. Our theory allows to introduce the cluster network connectivity as a controllable parameter. Two types of inner cluster geometry are considered: triangular and tetrahedral. The theory was developed for a general, non-uniform intra-cluster interactions, but in the present paper the results presented correspond to uniform, anti-ferromagnetic (AF) intra-clusters interactions $J_{0}/J$. The clusters are represented by nodes on a finite connectivity ra…
Stage-2 Cu$_{0.93}$Co$_{0.07}$Cl$_{2}$ graphite intercalation compound magnetically behaves like a reentrant ferromagnet near the multicritical point ($c_{MCP} \approx 0.96$). It undergoes two magnetic phase transitions at $T_{RSG}$ ($= 6.64 \pm 0.05$ K) and $T_{c}$ ($= 8.62 \pm 0.05$ K). The static and dynamic nature of the ferromagnetic and reentrant spin glass phase has been studied using DC and AC magnetic susceptibility. Characteristic memory phenomena of the DC susceptibility are observed at $T_{RSG}$ and $T_{c}$. The nonlinear AC susceptibility $χ_{3}^{\prime}$ has a positive local maxi…
The magnetisation relaxations of three different types of geometrically frustrated magnetic systems have been studied with the same experimental procedures as previously used in spin glasses. The materials investigated are Y$_2$Mo$_2$O$_7$ (pyrochlore system), SrCr$_{8.6}$Ga$_{3.4}$O$_{19}$ (piled pairs of Kagomé layers) and (H$_3$O)Fe$_3$(SO$_4$)$_2$(OH)$_6$ (jarosite compound). Despite a very small amount of disorder, all the samples exhibit many characteristic features of spin glass dynamics below a freezing temperature $T_g$, much smaller than their Curie-Weiss temperature $θ$. The ageing …
Exact ground states are calculated for the Sherrington-Kirkpatrick (SK) spin-glass containing up to N=90 spins. A ground-state energy per spin $e^{\infty}_0 = - 0.7637 \pm 0.0004$ is found from the $N$ dependence of the misfit parameter, which is a measure of frustration of the system. The results are compared with those of two related models, which can be introduced by replacing all interactions of the SK model by ferromagnetic or antiferromagnetic ones of the same strengths. A parameter $x$ is introduced, which describes the fraction of antiferromagnetic interactions in these types of models…
In random networks decorated with Ising spins, an increase of the density of frustrations reduces the transition temperature of the spin-glass ordering. This result is in contradiction to the Bethe theory. Here we investigate if this effect depends on the small-world property of the network. The results on the specific heat and the spin susceptibility indicate that the effect appears also in spatial networks.
We study topology of frustration in $d$-dimensional Ising spin glasses with $d\ge 2$ with nearest-neighbor interactions. We prove the following: For any given spin configuration, the domain walls on the unfrustration network are all transverse to a frustrated loop on the unfrustration network, where a domain wall is defined to be a connected element of the collection of all the $(d-1)$-cells which are dual to the bonds having an unfavorable energy, and the unfrustration network is the collection of all the unfrustrated plaquettes. These domain walls are topologically nontrivial because they ar…
We construct a model for qualitative description of the orientational glass transition in C_60 on the spin-glass theory basis. The physical origin of the frustration and the disorder is discussed.
A Bethe-Peierls treatment to dilution in frustrated magnets and spin liquids is given. A spin glass phase is present at low temperatures and close to the percolation point as soon as frustration takes a finite value in the dilute magnet model; the spin glass phase is reentrant inside the ferromagnetic phase. An extension of the model is given, in which the spin glass / ferromagnet phase boundary is shown not to reenter inside the ferromagnetic phase asymptotically close to the tricritical point whereas it has a turning point at lower temperatures. We conjecture similar phase diagrams to exist …
Quenched disorder effects on frustrated systems are explored by considering random fluctuations on the antiferromagnetic (AF) interactions between spins on the checkerboard lattice. The replica framework is adopted within a cluster mean-field approach, resulting in an effective single-cluster model. This effective model is treated within a one-step replica symmetry breaking (RSB) approach with exact evaluations for all intracluster interactions. Competing interactions are introduced by tuning the ratio $J_2/J_1$ (where $J_1$ and $J_2$ are first-neighbour and second-neighbor interactions, respe…
Spin glass (SG) is a typical magnetic system with frozen random spin orientation at low temperatures. The system exhibits rich physical properties, such as infinite number of ground states, memory effect and aging phenomena. There are two main ingredients considered to be pivotal for the existence of SG behavior, namely, frustration and randomness. For the canonical SG system, frustration is led by the presence of competing interaction between ferromagnetic (FM) and antiferromagnetic (AF) couplings. Previously, Bartolozzi {\it et al.} [ Phys. Rev. B{\bf 73}, 224419 (2006)], reported the SG pro…
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…
The origin of the spin-glass state in (Fe,Sn)4N alloys is studied on the basis of a Heisenberg Hamiltonian with parameters derived from first principles within the magnetic force theorem applied in the framework of the disordered local moments method and local spin-density approximation. We show that in the alloy concentration range where the spin-glass state is stable only one Fe sublattice is intrinsically magnetic and the interatomic exchange magnetic interactions are essentially short ranged due to effects of chemical and magnetic disorder. The magnetic Fe atoms with well-localized spin mo…
Motivated by the concept of geometrical frustration, we introduce a class of statistical mechanics lattice models for the glass transition. Monte Carlo simulations in three dimensions show that they display a dynamical glass transition which is very similar to that observed in other off-lattice systems and which does not depend on a specific dynamical rule. Whereas their analytic solution within the Bethe approximation shows that they do have a discontinuous glass transition compatible with the numerical observations.
We present a numerical study on a disordered artificial spin-ice system which interpolates between the long-range ordered square ice and the fully degenerate shakti ice. Starting from the square-ice geometry, disorder is implemented by adding vertical/horizontal magnetic islands to the center of some randomly chosen square plaquettes of the array, at different densities. When no island is added we have ordered square ice. When all square plaquettes have been modified we obtain shakti ice, which is disordered yet in a topological phase corresponding to the Rys F-model. In between, geometrical f…
In the Edwards-Anderson model of spin glasses with a bimodal distribution of bonds, the degeneracy of the ground state allows one to define a structure called backbone, which can be characterized by the rigid lattice (RL), consisting of the bonds that retain their frustration (or lack of it) in all ground states. In this work we have performed a detailed numerical study of the properties of the RL, both in two-dimensional (2D) and three-dimensional (3D) lattices. Whereas in 3D we find strong evidence for percolation in the thermodynamic limit, in 2D our results indicate that the most probable …
We show that a new glassy phase can emerge in presence of strong magnetic frustration and quantum fluctuations. It is a Valence Bond Glass. We study its properties solving the Hubbard-Heisenberg model on a Bethe lattice within the large $N$ limit introduced by Affleck and Marston. We work out the phase diagram that contains Fermi liquid, dimer and valence bond glass phases. This new glassy phase has no electronic or spin gap (although a pseudo-gap is observed), it is characterized by long-range critical valence bond correlations and is not related to any magnetic ordering. As a consequence it …
The gauge principle as applied in field theories for particle physics recognizes redundant degrees of freedom. Some time ago this author introduced a frustration based locally preferred structure (LPS) model of the glass transition. Here we show that gauge freedom plays an essential role in the self organized emergence of two temperature dependent quantities , which connect the time dependence of the bond angle fluctuations as a given LPS approaches the glass temperature. We show that as a choice of gauge these quantities cooperatively converge as , which is equivalent to the stretched exponen…
We propose that glass-forming liquids are intrinsically under the influences of both fluctuating interactions and random fields well-known in the field of spin systems. This is due to the frustration between the isotropic and anisotropic parts of effective intermolecular interactions. Our model indicates the existence of two key temperatures relevant to glass transition, the density ordering point T^*_m and the Vogel-Fulcher temperature T_0. Between T^*_m and T_0, a system has features similar to the `Griffiths phase', while below $T_0$ it has those peculiar to the `spin-glass phase'. This pic…
Hour-glass-shaped magnetic excitation spectra have been detected in a variety of doped transition-metal oxides with stripe-like charge order. Compared to the predictions of spin-wave theory, these spectra display a different intensity distribution and anomalous broadening. Here we show, based on a comprehensive modelling of these phenomena for La5/3Sr1/3CoO4, how quenched disorder in the charge sector causes frustration, and consequently cluster-glass behavior at low temperatures, in the spin sector. This spin-glass physics, which is insensitive to the detailed nature of the charge disorder, b…
Stochastic-Beta-Relaxation (SBR) provides a characterisation of the glass crossover in discontinuous Spin-Glasses and Supercoooled liquid. Notably it can be derived through a rigorous computation from a dynamical Landau theory. In this paper I will discuss the precise meaning of this connection in a language that does not require familiarity with statistical field theory. I will discuss finite-size corrections in mean-field Spin-Glass models and loop corrections in finite-dimensional models that are both described by the dynamical Landau theory considered. Then I will argue that the same Landa…
The glassy behavior observed in the pyrochlore magnet Y2Mo2O7, where the magnetic Mo^{4+} ions interact predominantly via isotropic nearest neighbor antiferromagnetic exchange, possibly with additional weak disorder, is a distinct class of spin glass systems where frustration is mostly geometrical. A model proposed to describe such a spin glass behavior is the Heisenberg model on a pyrochlore lattice with random but strictly antiferromagnetic exchange disorder. In this paper, we provide compelling numerical evidence from extensive Monte Carlo simulations which show that the model exhibits a fi…
We investigate a phenomenological model for the spin glass phase of La_{2-x}Sr_xCuO_4, in which it is assumed that holes doped into the CuO_2 planes localize near their Sr dopant, where they cause a dipolar frustration of the antiferromagnetic environment. In absence of long-range antiferromagnetic order, the spin system can reduce frustration, and also its free energy, by forming a state with an ordered orientation of the dipole moments, which leads to the appearance of spiral spin correlations. To investigate this model, a non-linear sigma model is used in which disorder is introduced via a …
Randomness and frustration are considered to be the key ingredients for the existence of spin glass (SG) phase. In a canonical system, these ingredients are realized by the random mixture of ferromagnetic (FM) and antiferromagnetic (AF) couplings. The study by Bartolozzi {\it et al.} [Phys. Rev. B{\bf 73}, 224419 (2006)] who observed the presence of SG phase on the AF Ising model on scale free network (SFN) is stimulating. It is a new type of SG system where randomness and frustration are not caused by the presence of FM and AF couplings. To further elaborate this type of system, here we study…
We study the low temperature dynamics of a two dimensional short-range spin system with uniform ferromagnetic interactions, which displays glassiness at low temperatures despite the absence of disorder or frustration. The model has a dual description in terms of free defects subject to dynamical constraints, and is an explicit realization of the ``hierarchically constrained dynamics'' scenario for glassy systems. We give a number of exact results for the statics of the model, and study in detail the dynamical behaviour of one-time and two-time quantities. We also consider the role played by th…
We study the far-from-equilibrium dynamical regimes of a many-body spin boson model with disordered couplings relevant for cavity QED and trapped ions experiments, using the discrete truncated Wigner approximation (DTWA). We focus on the dynamics of spin observables upon varying the disorder strength and the frequency of the photons, finding that the latter can considerably alter the structure of the system's dynamical responses. When the photons evolve at a similar rate as the spins, they can induce qualitatively distinct frustrated dynamics characterized by either logarithmic or algebraicall…
We propose a method to study the magnetic properties of a disordered Ising kagome lattice. The model considers small spin clusters with infinite-range disordered couplings and short-range ferromagnetic (FE) or antiferromagnetic interactions. The correlated cluster mean-field theory is used to obtain an effective single-cluster problem. A finite disorder intensity in FE kagome lattice introduces a cluster spin-glass (CSG) phase. Nevertheless, an infinitesimal disorder stabilizes the CSG behavior in the geometrically frustrated kagome system. Entropy, magnetic susceptibility and spin-spin correl…
We solve the q-state Potts model with anti-ferromagnetic interactions on large random lattices of finite coordination. Due to the frustration induced by the large loops and to the local tree-like structure of the lattice this model behaves as a mean field spin glass. We use the cavity method to compute the temperature-coordination phase diagram and to determine the location of the dynamic and static glass transitions, and of the Gardner instability. We show that for q>=4 the model possesses a phenomenology similar to the one observed in structural glasses. We also illustrate the links between …
Hysteresis loops are obtained in the Ising spin-glass phase in d=3, using frustration-conserving hard-spin mean-field theory. The system is driven by a time-dependent random magnetic field H_Q that is conjugate to the spin-glass order Q, yielding a field-driven first-order phase transition through the spin-glass phase. The hysteresis loop area A of the Q-H_Q curve scales with respect to the sweep rate h of magnetic field as A-A_0 = h^b. In the spin-glass and random-bond ferromagnetic phases, the sweep-rate scaling exponent b changes with temperature T, but appears not to change with antiferrom…
We show that introducing long-range Coulomb interactions immediately lifts the massive ground state degeneracy induced by geometric frustration for electrons on quarter-filled triangular lattices in the classical limit. Important consequences include the stabilization of a stripe-ordered crystalline (global) ground state, but also the emergence of very many low-lying metastable states with amorphous "stripe-glass" spatial structures. Melting of the stripe order thus leads to a frustrated Coulomb liquid at intermediate temperatures, showing remarkably slow (viscous) dynamics, with very long rel…
