Frustrated Ising Systems Research Articles

Itinerant fermions on dilute frustrated Ising lattices

We consider itinerant spinless fermions as moving defects in a dilute two-dimensional frustrated Ising system where they occupy site vacancies. Fermions interact via local spin fluctuations and we analyze coupled self-consistent mean-field equations of the Green functions after expressing the spin and fermion operators in terms of Grassmann variables. The specific heat and effective mass are analyzed with the solutions satisfying the symmetry imposed by the coupling layout. At low temperature, we find that these solutions induce stripes along the lines of couplings with the same sign, and that a low fermion density yields a small effective mass.

On the degeneracy of spin ice graphs, and its estimate via the Bethe permanent

The concept of spin ice can be extended to a general graph. We study the degeneracy of spin ice graph on arbitrary interaction structures via graph theory. We map spin ice graphs to the Ising model on a graph and clarify whether the inverse mapping is possible via a modified Krausz construction. From the gauge freedom of frustrated Ising systems, we derive exact, general results about frustration and degeneracy. We demonstrate for the first time that every spin ice graph, with the exception of the one-dimensional Ising model, is degenerate. We then study how degeneracy scales in size, using the mapping between Eulerian trails and spin ice manifolds, and a permanental identity for the number of Eulerian orientations. We show that the Bethe permanent technique provides both an estimate and a lower bound to the frustration of spin ices on arbitrary graphs of even degree. While such a technique can also be used to obtain an upper bound, we find that in all finite degree examples we studied, another upper bound based on Schrijver inequality is tighter.

Changing the universality class of the three-dimensional Edwards-Anderson spin-glass model by selective bond dilution

The three-dimensional Edwards-Anderson spin-glass model present strong spatial heterogeneities well characterized by the so-called backbone, a magnetic structure that arises as a consequence of the properties of the ground state and the low-excitation levels of such a frustrated Ising system. Using extensive Monte Carlo simulations and finite size scaling, we study how these heterogeneities affect the phase transition of the model. Although, we do not detect any significant difference between the critical behavior displayed by the whole system and that observed inside and outside the backbone, surprisingly, a selective bond dilution of the complement of this magnetic structure induces a change of the universality class, whereas no change is noted when the backbone is fully diluted. This finding suggests that the region surrounding the backbone plays a more relevant role in determining the physical properties of the Edwards-Anderson spin-glass model than previously thought. Furthermore, we show that when a selective bond dilution changes the universality class of the phase transition, the ground state of the model does not undergo any change. The opposite case is also valid, i. e., a dilution that does not change the critical behavior significantly affects the fundamental level.

Ground states, residual entropies, and specific heat capacity properties of frustrated Ising system on pyrochlore lattice in effective field theory cluster approximations

The thermodynamics of the antiferromagnetic spin-1∕2 Ising model in the presence of the external magnetic field on the geometrically frustrated pyrochlore lattice is investigated using various effective field theory cluster approximations up to the cluster size consisting of 7 sites in the form of two site-connected elementary tetrahedrons. The system of all ground states of the model is determined and the dependence of the magnetization, of the specific heat capacity, as well as of the entropy of the model on the used cluster approximation is investigated. It is shown that the thermodynamic properties of the ground states depend on the used cluster approximation and tend to those obtained in the framework of other theoretical approaches with the increasing of the cluster size. The analysis shows that, although it seems that the EFT cluster approximation represents quite effective approach for the investigation of properties of the model for some specific intervals and discrete values of the external magnetic field, in general, one must be very careful when interpreting obtained results. The dependence of the critical temperature, of the spontaneous magnetization, as well as of the specific heat capacity of the ferromagnetic model on the used cluster approximation is also discussed.

Small-network approximations for geometrically frustrated Ising systems

The study of frustrated spin systems often requires time-consuming numerical simulations. As the simplest approach, the classical Ising model is often used to investigate the thermodynamic behavior of such systems. Exploiting the small correlation lengths in frustrated Ising systems, we develop a method for obtaining first approximations to the energetic properties of frustrated two-dimensional Ising systems using small networks of less than 30 spins. These small networks allow much faster numerical simulations, and more importantly, analytical evaluations of their properties are numerically tractable. We choose Ising systems on the triangular lattice, the kagome lattice, and the triangular kagome lattice as prototype systems and find small systems that can serve as good approximations to these prototype systems. Through comparisons between the properties of extended models and small systems, we develop a set of criteria for constructing small networks to approximate general infinite two-dimensional frustrated Ising systems. This method of using small networks provides a different and efficient way to obtain a first approximation to the properties of frustrated spin systems.

Dynamics of magnetization of frustrated ising systems

The dynamics of magnetization in triangle lattices of the Ising chains has been investigated in terms of the Glauber theory. The results of three-dimensional numerical simulation of the magnetic structure and magnetization curves of Ca3Co2O6 are presented. The structures of the low-temperature and high-temperature phases differ significantly: the false frustrated low-temperature phase is transformed into a partially disordered antiferromagnetic honeycomb structure. Two additional magnetization steps at low temperatures are formed at the expense of the domain structure and the ferrimagnetic phase. All fundamental hypotheses used in terms of the two-dimensional model have found their confirmation during the three-dimensional simulation.

String Picture for a Model of Frustrated Quantum Magnets and Dimers

We study the effect of quantum dynamics on geometrically frustrated magnets for a transverse field Ising model at finite temperatures. We develop a microscopic derivation of the Landau-Ginzburg-Wilson action for this model and show that it can be interpreted as the free energy of a 3D elastic lattice of noncrossing strings. As a first application, we quantitatively predict the phase diagram and correlations, confirming excellently a key prediction of recent simulations about the existence of unusual phase transitions and an ordered phase. We discuss the implications of our string picture for the understanding of the effect of quenched disorder in such quantum frustrated systems.

Fully frustrated Ising system on a 3D simple cubic lattice: revisited

Using extensive Monte Carlo simulations, we clarify the critical behaviour of the 3 dimensional simple cubic Ising Fully Frustrated system. We find two transition temperatures and two long range ordered phases. Within the present numerical accuracy, the transition at higher temperature is found to be second order and we have extracted the standard critical exponent using finite size scaling method. On the other hand, the transition at lower temperature is found to be first order. It is argued that entropy plays a major role on determining the low temperature state.

Frustrated Ising systems on Husimi trees

We consider two frustrated Ising model systems. The first is the full frustrated antiferromagnetic Ising model on the triangle lattice. We approximate the system by a Husimi tree. By a “sequential” build up of the tree we get a qualitatively correct phase diagram which quantitatively is close to other approximation methods. Most closed form approximations of this system such as mean field theory give qualitatively incorrect phase diagrams. As a further test of the Husimi tree approach we look at a frustrated Ising model on a checkerboard type lattice. This system has been solved exactly by Azaria et al., Phys. Rev. Lett. 59 (1987) 1629, when h=0. Again the Husimi tree approach gives qualitatively correct results approximating a rather complex phase diagram with e.g. reentrant phases. And in addition this approach allows one to determine the phase diagram for h≠0. Finally, this method should be easily extended to a number of other frustrated lattice spin systems such as the fully frustrated system on the simple cubic lattice.

Studies of the Migdal-Kadanoff renormalization group for Ising systems

We study the critical temperatures obtained from the Migdal-Kadanoff (MK) renormalization transformation scheme for various ferromagnetic and frustrated Ising systems. We point out various problems in previous arguments and methods for obtaining the ordering temperatures for spin glasses using the MK approach, mostly due to the inadequate consideration of the size b of the renormalization cell used. We trace the apparent MK critical temperatures T ∗ as a function of b for the different systems. We find empirically that the optimal value of b = b ∗ , which gives T ∗(b) equal to the ordering temperature, is independent of the details of the set of interactions for a given family of systems at fixed dimension. This allows us to estimate the ordering temperatures for various types of interactions. Our results indicate that frustated system always correspond to higher values of b ∗ than ferromagnetic ones of the same dimension. Finally, we argue that larger values of b ∗ correspond to a larger amount of information that must be incorporated at the level of the basic unit to correctly describe more complex ground states.

Local mean field study of two-sublattice frustrated Ising systems

One- and two-sublattice frustrated Ising systems are studied by the local mean field method. Our results for linear and nonlinear susceptibilities suggest that the reentrancy phenomenon is absent in these systems. A decrease of the irreversibility temperature with the field in the magnetically ordered phases of the two-sublattice system is found to be weaker than in the one-sublattice case. The field-cooled susceptibility of a spin glass is influenced by the two-sublattice structure, but not as much as the mean field theory predicts. The intersublattice frustration is shown to decrease the compensation point in frustrated ferrimagnets whereas the intrasublattice frustration increases it.

A gauge theoretical formulation of a frustrated Ising system

The gauge theoretical formulation of a frustrated Ising model is studied here. In this formulation the Chern–Simons term has played the key role. The inherent anisotropic feature of the system is associated with the order parameter which violates the commutativity of translation.

A STUDY ON THE THERMODYNAMICAL PROPERTIES OF ONE-DIMENSIONAL FIBONACCI QUASIPERIODIC FRUSTRATED ISING MODEL

The calculated formula for partition function of one-Dimensional Fibonacci quasiperiodic frustrated Ising system with free and periodic boundary conditions are given in this paper. Its thermodynamical properties at low temperatures are studied. It is found that when the temperature tends to zero , the thermodynamical quantities as the functions of B/J show discontinuities at B/J=2/(2m+l) (m interger). The ground state configuration of system in various parameter regions are described and the results of calculation are discussed.

Symmetry-breaking fields in frustrated Ising systems on square and cubic lattices: Inclusive and exclusive multiphase lines

A uniform magnetic field is applied to three frustrated Ising systems on square and cubic lattices, breaking the gauge invariance of combined local spin-bond flips. Ordered phases with degeneracy up to 24, accommodating disorder and partial disorder on up to 16 sublattices, are unambiguously identified using the Monte Carlo mean-field method and an approximate transfer-matrix method. The complete phase diagrams for all three systems are obtained in the field-temperature plane and show fluctuation-induced order in two dimensions (2D) and multicritical phenomena and singular lines of degeneracy in three dimensions. These singular lines, which occur in two versions as inclusive and exclusive multiphase lines, are distinct from first-order lines of coexistence and constitute a type of phase behavior that is typical for systems possessing a high degree of degeneracy.

Electrostatic analogy for surfactant assemblies

We develop a concept of frustrating charges to create a theory of self-assembly. In particular, we note that the constraints of stoichiometry frustrate ordinary phase equilibria and lead to self-assembly of systems such as oil-water-surfactant mixtures. Further we note that at long wavelengths, the constraints of stoichiometry are isomorphic to the constraints of charge neutrality in a specific electrostatic analogy. We expand upon this analogy, first noted by Stillinger, and show that it can be used to derive useful analytical estimates. In addition we use the analogy to create a new model for frustrated systems, and we present Monte Carlo results for this charge frustrated Ising system that exhibits varied behaviors of self-assembly. The Monte Carlo calculations are made possible through the development of an algorithm which permits cluster moves. 9 refs., 5 figs.

Lee-Yang Complex-Field Systems and Frustrated Ising Models

It is shown that the Lee-Yang Ising system with an imaginary magnetic field β H = h = i π/2 on the square lattice is equivalent to Villain's fully-frustrated Ising systems. Thus, the present paper unifies the two streams of investigation on Lee-Yang complex-field systems and frustrated Ising systems. This observation suggests a new possibility to study frustrated systems using the corresponding equivalent Lee-Yang complex-field systems. An algebraic approach to dual transformations is given using the quantum σ-τ transformation.

Monte Carlo mean-field theory and frustrated systems in two and three dimensions.

A new method, combining mean-field and Monte Carlo approaches, is applied to frustrated Ising systems in d=2 and 3, in zero and nonzero uniform fields. The method brings to mean-field theory the hard-spin condition and uses much less sampling then the Monte Carlo simulation. The phase diagram of the d=2 triangular antiferromagnet is easily obtained with remarkable global quantitative accuracy. The phase diagram of the d=3 stacked triangular antiferromgnet shows three ordered phases, in a new multicritical topology of lines of XY, Ising, and three-state Potts transitions, accessible to experiments with layered magnets.

Critical behavior of a frustrated Ising system.

Analyse par loi echelle de taille finie d'une simulation du comportement critique d'un systeme frustre tri-dimensionnel non-aleatoire. Ce comportement de type tricritique ne peut etre predit par la simple theorie de Landau-Giuzburg-Wilson. A cause de la frustration, l'energie libre de plusieurs phases est proche de celle observee pour la phase ordonnee et une theorie complete doit prendre en compte les interactions entre les fluctuations des parametres d'ordre de toutes ces phases

On the influence of competing interactions on physical properties of spin systems: numerical exact results for finite Ising and Heisenberg systems

The ground-state properties and low-lying excitations of spin systems with competing interactions are studied for various finite two-dimensional S=1/2 Heisenberg systems of 8 spins and corresponding Ising systems of up to 60 spins. The response of the spin systems on the conflict in the interactions is studied using numerical exact results. In the Ising model the competition of the interactions influences the ground state and the excitations essentially. Contrary to conventional Ising systems in disordered strongly frustrated Ising systems the low-energy density of states is smooth, the gap between the ground state and the first excitations vanishes. The low-lying excitations are characterised by a large portion of multiple spin excitations. Thermodynamic consequences are indicated. In the Heisenberg case the influence of the conflict in the interactions is less obvious, the character of the ground state and of the first excitation has similarities with that of conventional antiferromagnets, the spins have an enlarged number of possibilities to evade the conflict in the interactions.

Correlation functions of a 2D fully frustrated Ising system on a square lattice

Correlation functions of a 2D fully frustrated Ising system on a square lattice