Spin-spin correlations are calculated in frustrated hierarchical Ising models that exhibit chaotic renormalization-group behavior. The spin-spin correlations, as a function of distance, behave chaotically. The far correlations, but not the near correlations, are sensitive to small changes in temperature or frustration, with temperature changes having a larger effect. On the other hand, the calculated free energy, internal energy, and entropy are smooth functions of temperature. The recursion-matrix calculation of thermodynamic densities in a chaotic band is demonstrated. The leading Lyapunov exponents are calculated as a function of frustration.
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 a first approximation 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 calculation of the properties from the partition…
We perform large scale simulations of the frustrated Ising lattice gas, a three-dimensional lattice model of a structural glass, using the parallel tempering technique. We evaluate the spin and density overlap distributions, and the corresponding non-linear susceptibilities, as a function of the chemical potential. We then evaluate the relaxation functions of the spin and density self-overlap, and study the behavior of the relaxation times. The results suggest that the spin variables undergo a transition very similar to the one of the Ising spin glass, while the density variables do not show a…
We study a one-dimensional chain of corner-sharing triangles with antiferromagnetic Ising interactions along its bonds. Classically, this system is highly frustrated with an extensive entropy at T = 0 and exponentially decaying spin correlations. We show that the introduction of a quantum dynmamics via a transverse magnetic field removes the entropy and opens a gap, but leaves the ground state disordered at all values of the transverse field, thereby providing an analog of the "disorder by disorder" scenario first proposed by Anderson and Fazekas in their search for resonating valence bond sta…
In this paper we show quantum fluctuation effect of fully frustrated Ising spin systems. Quantum annealing has been expected to be an efficient method to find ground state of optimization problems. However it is not clear when to use the quantum annealing. In order to clarify when the quantum annealing works well, we have to study microscopic properties of quantum annealing. In fully frustrated Ising spin systems, there are macroscopically degenerated ground states. When we apply quantum annealing to fully frustrated systems, we cannot obtain each ground state with the same probability. This n…
We study ordering mechanisms which are induced by the quantum fluctuation in fully frustrated Ising spin systems. Since there are many degenerated states in frustrated systems, "order by thermal disorder" often takes place due to a kind of entropy effect. To consider "order by quantum disorder" in fully frustrated Ising spin systems, we apply transverse field as quantum fluctuation. There exists a ferromagnetic correlation in each sublattice. The sublattice correlation at zero temperature is enlarged due to transverse field. The quantum fluctuation enhances the solid order at zero temperatures…
The geometric frustration of the spin-1/2 Ising-Heisenberg model on the triangulated Kagome (triangles-in-triangles) lattice is investigated within the framework of an exact analytical method based on the generalized star-triangle mapping transformation. Ground-state and finite-temperature phase diagrams are obtained along with other exact results for the partition function, Helmholtz free energy, internal energy, entropy, and specific heat, by establishing a precise mapping relationship to the corresponding spin-1/2 Ising model on the Kagome lattice. It is shown that the residual entropy of t…
We study a classical fully-frustrated honeycomb lattice Ising model using Markov chain Monte Carlo methods and exact calculations . The Hamiltonian realizes a degenerate ground state manifold of equal-energy states, where each hexagonal plaquette of the lattice has one and only one frustrated bond, with an extensive residual entropy that grows as the number of spins N. Traditional single-spin flip Monte Carlo methods fail to sample all possible spin configurations in this ground state efficiently, due to their separation by large energy barriers. We develop a non-local "chain-flip" algorithm t…
Quantum annealing method has been widely attracted attention in statistical physics and information science since it is expected to be a powerful method to obtain the best solution of optimization problem as well as simulated annealing. The quantum annealing method was incubated in quantum statistical physics. This is an alternative method of the simulated annealing which is well-adopted for many optimization problems. In the simulated annealing, we obtain a solution of optimization problem by decreasing temperature (thermal fluctuation) gradually. In the quantum annealing, in contrast, we dec…
We study by means of Monte Carlo simulations the off equilibrium properties of a model glass, the Frustrated Ising Lattice Gas (FILG) in three dimensions. We have computed typical two times quantities, like density-density autocorrelations and the autocorrelation of internal degrees of freedom. We find an aging scenario particularly interesting in the case of the density autocorrelations in real space which is very reminiscent of spin glass phenomenology. While this model captures the essential features of structural glass dynamics, its analogy with spin glasses may bring the possibility of it…
We consider discrete spin models on arbitrary planar graphs and lattices with frustrated interactions. We first analyze the Ising model with frustrated plaquettes. We use an algebraic approach to derive the result that an Ising model with some of its plaquettes frustrated has a dual which is an Ising model with an external field $iπ/2$ applied to the dual sites centered at frustrated plaquettes. In the case that all plaquettes are frustrated, this leads to the known result that the dual model has a uniform field $iπ/2$ whose partition function can be evaluated in the thermodynamic limit for re…
A disordered spin glass model where both static and dynamical properties depend on macroscopic magnetizations is presented. These magnetizations interact via random couplings and, therefore, the typical quenched realization of the system exhibit a macroscopic frustration. The model is solved by using a revisited replica approach, and the broken symmetry solution turns out to coincide with the symmetric solution. Some dynamical aspects of the model are also discussed, showing how it could be a useful tool for describing some properties of real systems as, for example, natural ecosystems or huma…
In this paper we consider a 2d random Ising system on a square lattice with nearest neighbour interactions. The disorder is short range correlated and asymmetry between the vertical and the horizontal direction is admitted. More precisely, the vertical bonds are supposed to be non random while the horizontal bonds alternate: one row of all non random horizontal bonds is followed by one row where they are independent dichotomic random variables. We solve the model using an approximate approach that replace the quenched average with an annealed average under the constraint that the number of fru…
The dynamical properties of a three dimensional model glass, the frustrated Ising lattice gas (FILG) are studied by Monte Carlo simulations. We present results of compression experiments, where the chemical potential is either slowly or abruptly changed, as well as simulations at constant density. One time quantities like density and two time ones like correlations, responses and mean square displacements are measured, and the departure from equilibrium clearly characterized. The aging scenario, particularly in the case of density autocorrelations is reminiscent of spin glass phenomenology wit…
We study how the degree of ordering depends on the strength of the thermal and quantum fluctuations in frustrated systems by investigating the correlation function of the order parameter. Concretely, we compare the equilibrium spin correlation function in a frustrated lattice which exhibits a non-monotonic temperature dependence (reentrant type dependence) with that in the ground state as a function of the transverse field that causes the quantum fluctuation. We find the correlation function in the ground state also shows a non-monotonic dependence on the strength of the transverse field. We a…
We investigate the behavior of the frustrated $J_1$-$J_2$ Ising model on a square lattice under the influence of random dilution and spatial anisotropies. Spinless impurities generate a random-field type disorder for the spin-density wave (stripe) order parameter. These random fields destroy the long-range stripe order in the case of spatially isotropic interactions. Combining symmetry arguments, percolation theory and large-scale Monte Carlo simulations, we demonstrate that arbitrarily weak spatial interaction anisotropies restore the stripe phase. More specifically, the transition temperatur…
Recently I proposed a simple dynamical network model for discrete space-time which self-organizes as a graph with Hausdorff dimension d_H=4. The model has a geometric quantum phase transition with disorder parameter (d_H-d_s) where d_s is the spectral dimension of the dynamical graph. Self-organization in this network model is based on a competition between a ferromagnetic Ising model for vertices and an antiferromagnetic Ising model for edges. In this paper I solve a toy version of this model defined on a bipartite graph in the mean field approximation. I show that the geometric phase transit…
No. To illustrate that tensor renormalization group methods are poorly suited for frustrated magnetic systems, we study the thermodynamic properties of the two-dimensional Edwards-Anderson Ising spin-glass model on a square lattice. We show that the limited precision of standard 64-bit data types and not a small cut-off parameter is the main reason for unphysical negative partition function values in spin glasses.
Under certain experimental conditions, the deposition of C60 molecules onto an atomically flat copper surface gives rise to the formation of corrugated islands. This corrugation, which reflects a molecular displacement perpendicular to the surface plane, presents an astonishing pattern: it is well described by a frustrated Ising spin Hamiltonian whose thermodynamics is compatible with a spin liquid about to transit towards an ordered zigzag state. Here we study the statistical properties of such a molecular corrugation using tools generally employed in frustrated magnetism. More specifically, …
