Showing papers on "Ising model published in 2001"

Quantum Phase Transitions

Abstract: Part I. Introduction: 1. Basic concepts 2. The mapping to classical statistical mechanics: single site models 3. Overview Part II. Quantum Ising and Rotor Models: 4. The Ising chain in a transverse field 5. Quantum rotor models: large N limit 6. The d = 1, 0 (N greater than or equal to 3) rotor models 7. The d = 2 (N greater than or equal to 3) rotor models 8. Physics close to and above the upper-critical dimension 9. Transport in d = 2 Part III. Other Models: 10. Boston Hubbard model 11. Dilute Fermi and Bose gases 12. Phase transitions of Fermi liquids 13. Heisenberg spins: ferromagnets and antiferromagnets 14. Spin chains: bosonization 15. Magnetic ordering transitions of disordered systems 16. Quantum spin glasses.

Determining the density of states for classical statistical models: a random walk algorithm to produce a flat histogram.

Abstract: We describe an efficient Monte Carlo algorithm using a random walk in energy space to obtain a very accurate estimate of the density of states for classical statistical models. The density of states is modified at each step when the energy level is visited to produce a flat histogram. By carefully controlling the modification factor, we allow the density of states to converge to the true value very quickly, even for large systems. From the density of states at the end of the random walk, we can estimate thermodynamic quantities such as internal energy and specific heat capacity by calculating canonical averages at any temperature. Using this method, we not only can avoid repeating simulations at multiple temperatures, but we can also estimate the free energy and entropy, quantities that are not directly accessible by conventional Monte Carlo simulations. This algorithm is especially useful for complex systems with a rough landscape since all possible energy levels are visited with the same probability. As with the multicanonical Monte Carlo technique, our method overcomes the tunneling barrier between coexisting phases at first-order phase transitions. In this paper, we apply our algorithm to both first- and second-order phase transitions to demonstrate its efficiency and accuracy. We obtained direct simulational estimates for the density of states for two-dimensional ten-state Potts models on lattices up to 200 x 200 and Ising models on lattices up to 256 x 256. Our simulational results are compared to both exact solutions and existing numerical data obtained using other methods. Applying this approach to a three-dimensional +/-J spin-glass model, we estimate the internal energy and entropy at zero temperature; and, using a two-dimensional random walk in energy and order-parameter space, we obtain the (rough) canonical distribution and energy landscape in order-parameter space. Preliminary data suggest that the glass transition temperature is about 1.2 and that better estimates can be obtained with more extensive application of the method. This simulational method is not restricted to energy space and can be used to calculate the density of states for any parameter by a random walk in the corresponding space.

Discrete Riemann Surfaces and the Ising Model

Abstract: We define a new theory of discrete Riemann surfaces and present its basic results. The key idea is to consider not only a cellular decomposition of a surface, but the union with its dual. Discrete holomorphy is defined by a straightforward discretisation of the Cauchy–Riemann equation. A lot of classical results in Riemann theory have a discrete counterpart, Hodge star, harmonicity, Hodge theorem, Weyl's lemma, Cauchy integral formula, existence of holomorphic forms with prescribed holonomies. Giving a geometrical meaning to the construction on a Riemann surface, we define a notion of criticality on which we prove a continuous limit theorem. We investigate its connection with criticality in the Ising model. We set up a Dirac equation on a discrete universal spin structure and we prove that the existence of a Dirac spinor is equivalent to criticality.

Ising models of quantum frustration

Abstract: We report on a systematic study of two-dimensional, periodic, frustrated Ising models with quantum dynamics introduced via a transverse magnetic field. The systems studied are the triangular and kagome{prime} lattice antiferromagnets, fully frustrated models on the square and hexagonal (honeycomb) lattices, a planar analog of the pyrochlore antiferromagnet, a pentagonal lattice antiferromagnet, as well as two quasi-one-dimensional lattices that have considerable pedagogical value. All of these exhibit a macroscopic degeneracy at T=0 in the absence of the transverse field, which enters as a singular perturbation. We analyze these systems with a combination of a variational method at weak fields, a perturbative Landau-Ginzburg-Wilson approach from large fields, as well as quantum Monte Carlo simulations utilizing a cluster algorithm. Our results include instances of quantum order arising from classical criticality (triangular lattice) or classical disorder (pentagonal and probably hexagonal) as well as notable instances of quantum disorder arising from classical disorder (kagome{prime}). We also discuss the effect of finite temperature, as well as the interplay between longitudinal and transverse fields{emdash}in the kagome{prime} problem the latter gives rise to a nontrivial phase diagram with bond-ordered and bond-critical phases in addition to the disordered phase. We also note connections to quantum-dimer models and therebymore » to the physics of Heisenberg antiferromagnets in short-ranged resonating valence-bond phases that have been invoked in discussions of high-temperature superconductivity.« less

Thermal concurrence mixing in a one-dimensional Ising model

Abstract: We investigate the entanglement arising naturally in a one-dimensional Ising chain in a magnetic field in an arbitrary direction We find that for different temperatures, different orientations of the magnetic field give maximum entanglement In the high-temperature limit, this optimal orientation corresponds to the magnetic field being perpendicular to the Ising orientation $(z$ direction) In the low-temperature limit, we find that varying the angle of the magnetic field very slightly from the z direction leads to a rapid rise in entanglement We also find that the orientation of the magnetic field for maximum entanglement varies with the field amplitude Furthermore, we have derived a simple rule for the mixing of concurrences (a measure of entanglement) due to the mixing of pure states satisfying certain conditions

Long-range order at low temperatures in dipolar spin ice.

Abstract: It has recently been suggested that long-range magnetic dipolar interactions are responsible for spin ice behavior in the Ising pyrochlore magnets Dy2Ti2O7 and Ho2Ti2O7. We report here numerical results on the low temperature properties of the dipolar spin ice model, obtained via a new loop algorithm which greatly improves the dynamics at low temperature. We recover the previously reported missing entropy in this model, and find a first order transition to a long-range ordered phase with zero total magnetization at very low temperature. We discuss the relevance of these results to Dy2Ti2O7 and Ho2Ti2O7.

Ising Field Theory in a Magnetic Field: Analytic Properties of the Free Energy

Abstract: We study the analytic properties of the scaling function associated with the 2D Ising model free energy in the critical domain T→T c , H→0 The analysis is based on numerical data obtained through the Truncated Free Fermion Space Approach We determine the discontinuities across the Yang–Lee and Langer branch cuts We confirm the standard analyticity assumptions and propose “extended analyticity;” roughly speaking, the latter states that the Yang–Lee branching point is the nearest singularity under Langer's branch cut We support the extended analyticity by evaluating numerically the associated “extended dispersion relation”

Freezing in Ising ferromagnets.

Abstract: We investigate the final state of zero-temperature Ising ferromagnets that are endowed with single-spin-flip Glauber dynamics. Surprisingly, the ground state is generally not reached for zero initial magnetization. In two dimensions, the system reaches either a frozen stripe state with probability $\ensuremath{\approx}1/3$ or the ground state with probability $\ensuremath{\approx}2/3.$ In greater than two dimensions, the probability of reaching the ground state or a frozen state rapidly vanishes as the system size increases; instead the system wanders forever in an isoenergy set of metastable states. An external magnetic field changes the situation drastically---in two dimensions the favorable ground state is always reached, while in three dimensions the field must exceed a threshold value to reach the ground state. For small but nonzero temperature, relaxation to the final state proceeds first by formation of very long-lived metastable states, as in the zero-temperature case, before equilibrium is reached.

A cluster Monte Carlo algorithm for 2-dimensional spin glasses

Abstract: A new Monte Carlo algorithm for 2-dimensional spin glasses is presented. The use of clusters makes possible global updates and leads to a gain in speed of several orders of magnitude. As an example, we study the 2-dimensional ±J Edwards-Anderson model. The new algorithm allows us to equilibrate systems of size 1002 down to temperature T = 0.1. Our main result is that the correlation length diverges as an exponential ( ξ∼e2βJ) and not as a power law as T↦Tc = 0.

Fate of zero-temperature Ising ferromagnets.

Abstract: We investigate the relaxation of homogeneous Ising ferromagnets on finite lattices with zero-temperature spin-flip dynamics. On the square lattice, a frozen two-stripe state is apparently reached approximately 3/10 of the time, while the ground state is reached otherwise. The asymptotic relaxation is characterized by two distinct time scales with the longer stemming from the influence of a long-lived diagonal stripe defect. In greater than two dimensions, the probability to reach the ground state rapidly vanishes as the size increases and the system typically ends up wandering forever within an iso-energy set of stochastically ``blinking'' metastable states.

Phase transitions in the quantum Ising and rotor models with a long-range interaction

Abstract: We investigate the zero-temperature and finite-temperature phase transitions of quantum Ising and quantum rotor models. We here assume a long-range (falling off as ${1/r}^{d+\ensuremath{\sigma}},$ where r is the distance between two spins/rotors in units of lattice spacing) ferromagnetic interaction among the spins or rotors. We find that the long-range behavior of the interaction drastically modifies the universal critical behavior of the system. The corresponding upper critical dimension and the hyperscaling relation and exponents associated with the quantum transition are modified and, as expected, they attain values of short-range system when $\ensuremath{\sigma}=2.$ The dynamical exponent varies continuously as the parameter \ensuremath{\sigma} and is unity for $\ensuremath{\sigma}=2.$ The one-dimensional long-range quantum Ising system shows a phase transition at $T=0$ for all values of \ensuremath{\sigma}. The most interesting observation is that the phase diagram for $\ensuremath{\sigma}=d=1$ shows a line of Kosterlitz-Thouless transition at finite temperature even though the $T=0$ transition is a simple order-disorder transition. These finite temperature transitions are studied near the phase boundary using renormalisation group equations and a region with diverging susceptibility is located. We have also studied one-dimensional quantum rotor model which exhibits a rich and interesting transition behavior depending upon the parameter \ensuremath{\sigma}. We explore the phase diagram extending the short-range quantum nonlinear \ensuremath{\sigma} model renormalisation group equations to the present case.

A ferromagnet with a glass transition

Abstract: We introduce a finite-connectivity ferromagnetic model with a three-spin interaction which has a crystalline (ferromagnetic) phase as well as a glass phase. The model is not frustrated, it has a ferromagnetic equilibrium phase at low temperature which is not reached dynamically in a quench from the high-temperature phase. Instead it shows a glass transition which can be studied in detail by a one-step replica-symmetry-broken calculation. This spin model exhibits the main properties of the structural glass transition at a solvable mean-field level.-1

Mean-field solution of the mixed spin-1 and spin- Ising system with different single-ion anisotropies

Abstract: The mixed spin-1 and spin- 3 2 Ising ferrimagnetic system with different anisotropies is studied within the mean-field theory based on Bogoliubov inequality for the Gibbs free energy. Global phase diagrams are obtained in the temperature-anisotropy plane. In particular we find first-order transition lines separating different low-temperature ordered phases (characterized by different values of the sublattice magnetizations) each one terminating at an isolated critical point. The existence and dependence of a compensation temperature on single-ion anisotropies is also investigated.

Introduction to statistical physics

Abstract: (1) Introduction to statistical methods.- (2) Statistical description of a physical system.- (3) Overview of classical thermodynamics.- (4) Microcanonical ensemble.- (5) Canonical ensemble.- (6) The classical gas in the canonical formalism.- (7) The grand canonical and pressure ensembles.- (8) The ideal quantum gas.- (9) The ideal Fermi gas.- Free bosons: Bose-Einstein condensations Gas of photons.- Phonons and magnons.- Phase transitions and critical phenomena: classical theories.- The Ising model.- Scaling theories and the renormalization group.- Non-equilibrium phenomena. I. Kinetic methods.- Non-equilibrium phenomena. II. Stochastic methods.

Mutation-selection models solved exactly with methods of statistical mechanics.

Abstract: We reconsider deterministic models of mutation and selection acting on populations of sequences, or, equivalently, multilocus systems with complete linkage. Exact analytical results concerning such systems are few, and we present recent and new ones obtained with the help of methods from quantum statistical mechanics. We consider a continuous-time model for an infinite population of haploids (or diploids without dominance), with N sites each, two states per site, symmetric mutation and arbitrary fitness function. We show that this model is exactly equivalent to a so-called Ising quantum chain. In this picture, fitness corresponds to the interaction energy of spins, and mutation to a temperature-like parameter. The highly elaborate methods of statistical mechanics allow one to find exact solutions for non-trivial examples. These include quadratic fitness functions, as well as 'Onsager's landscape'. The latter is a fitness function which captures some essential features of molecular evolution, such as neutrality, compensatory mutations and flat ridges. We investigate the mean number of mutations, the mutation load, and the variance in fitness under mutation-selection balance. This also yields some insight into the 'error threshold' phenomenon, which occurs in some, but not all, examples.

The Low-Temperature Expansion of the Wulff Crystal in the 3D Ising Model

Abstract: We compute the expansion of the surface tension of the 3D random cluster model for q≥ 1 in the limit where p goes to 1. We also compute the asymptotic shape of a plane partition of n as n goes to ∞. This same shape determines the Wulff crystal to order o(ɛ) in the 3D Ising model (and more generally in the 3D random cluster model for q≥ 1) at temperature ɛ.

Aging, phase ordering and conformal invariance

Abstract: In a variety of systems which exhibit aging, the two-time response function scales as R(t,s) approximately s(-1-a)f(t/s). We argue that dynamical scaling can be extended towards conformal invariance, thus obtaining the explicit form of the scaling function f. This quantitative prediction is confirmed in several spin systems, both for T

Modeling substorm dynamics of the magnetosphere: from self-organization and self-organized criticality to nonequilibrium phase transitions.

Abstract: Earth's magnetosphere during substorms exhibits a number of characteristic features such as the signatures of low effective dimension, hysteresis, and power-law spectra of fluctuations on different scales. The largest substorm phenomena are in reasonable agreement with low-dimensional magnetospheric models and in particular those of inverse bifurcation. However, deviations from the low-dimensional picture are also quite considerable, making the nonequilibrium phase transition more appropriate as a dynamical analog of the substorm activity. On the other hand, the multiscale magnetospheric dynamics cannot be limited to the features of self-organized criticality (SOC), which is based on a class of mathematical analogs of sandpiles. Like real sandpiles, during substorms the magnetosphere demonstrates features, that are distinct from SOC and are closer to those of conventional phase transitions. While the multiscale substorm activity resembles second-order phase transitions, the largest substorm avalanches are shown to reveal the features of first-order nonequilibrium transitions including hysteresis phenomena and a global structure of the type of a temperature-pressure-density diagram. Moreover, this diagram allows one to find a critical exponent, that reflects the multiscale aspect of the substorm activity, different from the power-law frequency and scale spectra of autonomous systems, although quite consistent with second-order phase transitions. In contrast to SOC exponents, this exponent relates input and output parameters of the magnetosphere. Using an analogy to the dynamical Ising model in the mean-field approximation, we show the connection between the data-derived exponent of nonequilibrium transitions in the magnetosphere and the standard critical exponent $\ensuremath{\beta}$ of equilibrium second-order phase transitions.

Ising ferromagnetism and domain morphology in the fractional quantum Hall regime.

Abstract: The density driven quantum phase transition between the unpolarized and fully spin polarized nu = 2/3 fractional quantum Hall state is accompanied by hysteresis in accord with 2D Ising ferromagnetism and domain formation. The temporal behavior is reminiscent of the Barkhausen and time-logarithmic magnetic after-effects ubiquitous in familiar ferromagnets. It too suggests domain morphology and, in conjunction with NMR, intricate domain dynamics, which is partly mediated by the contact hyperfine interaction with nuclear spins of the host semiconductor.

Multiplicity of Phase Transitions and Mean-Field Criticality on Highly Non-Amenable Graphs

Abstract: We consider independent percolation, Ising and Potts models, and the contact process, on infinite, locally finite, connected graphs.

Metastability in Glauber dynamics in the low-temperature limit: Beyond exponential asymptotics

Abstract: We consider Glauber dynamics of classical spin systems of Ising type in the limit when the temperature tends to zero in finite volume. We show that information on the structure of the most profound minima and the connecting saddle points of the Hamiltonian can be translated into sharp estimates on the distribution of the times of metastable transitions between such minima as well as the low lying spectrum of the generator. In contrast with earlier results on such problems, where only the asymptotics of the exponential rates is obtained, we compute the precise pre-factors up to multiplicative errors that tend to 1 as $T\downarrow 0$. As an example we treat the nearest neighbor Ising model on the 2 and 3 dimensional square lattice. Our results improve considerably earlier estimates obtained by Neves-Schonmann and Ben Arous-Cerf. Our results employ the methods introduced by Bovier, Eckhoff, Gayrard, and Klein.

Resistance spikes and domain wall loops in Ising quantum Hall ferromagnets.

Abstract: We explain the recent observation of resistance spikes and hysteretic transport properties in Ising quantum Hall ferromagnets in terms of the unique physics of their domain walls. Self-consistent RPA/Hartree-Fock theory is applied to microscopically determine properties of the ground state and domain wall excitations. In these systems domain wall loops support one-dimensional electron systems with an effective mass comparable to the bare electron mass and may carry charge. Our theory is able to account quantitatively for the experimental Ising critical temperature and to explain qualitative characteristics of the resistive hysteresis loops.

Glauber slow dynamics of the magnetization in a molecular Ising chain

Abstract: The slow dynamics (10^-6 s - 10^4 s) of the magnetization in the paramagnetic phase, predicted by Glauber for 1d Ising ferromagnets, has been observed with ac susceptibility and SQUID magnetometry measurements in a molecular chain comprising alternating Co{2+} spins and organic radical spins strongly antiferromagnetically coupled. An Arrhenius behavior with activation energy Delta=152 K has been observed for ten decades of relaxation time and found to be consistent with the Glauber model. We have extended this model to take into account the ferrimagnetic nature of the chain as well as its helicoidal structure.

Mean-field renormalization group study of the long-range Ising model

Abstract: In this paper, we study the one-dimensional long-range Ising model with a spin-spin interaction that decays as 1/r 1+σ using the mean-field renormalization group. The critical coupling K c and critical exponents y t and y h are computed using cluster sizes of up to sixteen spins. We then apply the alternating-alpha VBS (Vanden Broeck-Schwartz) transformation to accelerate the convergence of results for different cluster sizes.

Chapter 4 Geometrical frustration

Abstract: Publisher Summary This chapter discusses percolation theory as well as spin-glass phenomena, the former of which is used to understand subjects as diverse as fluid flow in porous media as well as electronic transport in granular metals. The central role of magnetic models for developing ideas in condensed matter derives from the unique attributes of magnetic materials: (1) the variability of the size and dimensionality of the fundamental degree of freedom, the effective atomic spin. The dimensionality, n , can be 1 for Ising, and 2 (x-y) or 3 (Heisenberg) for continuous spin degrees of freedom. Low spin values lead to enhanced quantum effects, (2) the variability of spin-spin interactions. These include dipole-dipole coupling, direct exchange, indirect exchange, super exchange, itinerant exchange, and anisotropic exchange. The range of these interactions can extend beyond nearest neighbor distances, and can be anisotropic, leading to low effective spatial dimensionality, and (3) the ability to couple directly to individual spins with magnetic field.

Criticality in one dimension with inverse square-law potentials.

Abstract: We demonstrate that the scaled order parameter for ferromagnetic Ising and three-state Potts chains with inverse square interactions exhibits a universal critical jump, in analogy with the superfluid density in helium films. Renormalization-group arguments are combined with numerical simulations of systems containing up to 10{sup 6} lattice sites to accurately determine the critical properties of these models. In strong contrast with earlier work, compelling quantitative evidence for the Kosterlitz-Thouless{endash}like character of the phase transition is provided.

Damage spreading, coarsening dynamics and distribution of political votes in sznajd model on square lattice

Abstract: In the Sznajd model of sociophysics on the square lattice, neighbors having the same opinion convince their neighbors of this opinion. We study scaling of the cluster growth. The spreading-of-damage technique is applied for the spread of opinions. We study the time evolution of the damage and compare it with the magnetization evolution. We also compare this model with the Ising model at low temperatures. It was recently shown that the distribution of votes in Brazilian elections follows a power law behavior with exponent ≃ -1.0. A model for elections based on the Sznajd model is proposed. The exponent obtained for the distribution of votes during the transient agrees with that obtained for elections.