Pure Ising Model Research Articles

Heterogeneous nucleation in the random field Ising model.

We investigate the nucleation dynamics of the three-dimensional random field Ising model under an external field. We use umbrella sampling to compute the free-energy cost of a critical nucleus and use forward flux sampling for the direct estimation of nucleation rates. For moderate to strong disorder, our results indicate that the size of the nucleating cluster is not a good reaction coordinate, contrary to the pure Ising model. We rectify this problem by introducing a coordinate that also accounts for the location of the nucleus. Using the free energy barrier to predict the nucleation rate, we find reasonable agreement, although deviations become stronger as disorder increases. We attribute this effect to cluster shape fluctuations. We also discuss finite-size effects on the nucleation rate.

Classical model emerges in quantum entanglement: Quantum Monte Carlo study for an Ising-Heisenberg bilayer

By developing a cluster sampling of stochastic series expansion quantum Monte Carlo method, we investigate a spin-$1/2$ model on a bilayer square lattice with intra-layer ferromagnetic (FM) Ising coupling and inter-layer antiferromagnetic Heisenberg interaction. The continuous quantum phase transition which occurs at $g_c=3.045(2)$ between the FM Ising phase and the dimerized phase is studied via large scale simulations. From the analyzes of critical exponents we show that this phase transition belongs to the (2+1)-dimensional Ising universality class. Besides, the quantum entanglement is strong between the two layers, especially in dimerized phase. The effective Hamiltonian of single layer seems like a transverse field Ising model. However, we found the quantum entanglement Hamiltonian is a pure classical Ising model without any quantum fluctuations. Furthermore, we give a more general explanation about how a classical entanglement Hamiltonian emerges.

A new correlation inequality for Ising models with external fields

We study ferromagnetic Ising models on finite graphs with an inhomogeneous external field, where a subset of vertices is designated as the boundary. We show that the influence of boundary conditions on any given spin is maximised when the external field is identically 0. One corollary is that spin–spin correlations are maximised when the external field vanishes and the boundary condition is free, which proves a conjecture of Shlosman. In particular, the random field Ising model on $${\mathbb {Z}}^d$$ , $$d\geqslant 3$$ , exhibits exponential decay of correlations in the entire high temperature regime of the pure Ising model. Another corollary is that the pure Ising model in $$d\geqslant 3$$ satisfies the conjectured strong spatial mixing property in the entire high temperature regime.

The two-dimensional continuum random field Ising model

In this paper, we construct the two-dimensional continuum random field Ising model via scaling limits of a random field perturbation of the critical two-dimensional Ising model with diminishing disorder strength. Furthermore, we show that almost surely with respect to the continuum random field given by a white noise, the law of the magnetisation field is singular with respect to that of the two-dimensional continuum pure Ising model constructed by Camia, Garban and Newman (Ann. Probab. 43 (2015) 528–571).

Equilibrium behavior in a nonequilibrium system: Ising-doped voter model on complete graph.

While the Ising model belongs to the realm of equilibrium statistical mechanics, the voter model is an example of a nonequilibrium system. We examine an opinion formation model, which is a mixture of Ising and voter agents with concentrations p and 1-p, respectively. Although in our model for p<1 a detailed balance is violated, on a complete graph the average magnetization in the stationary state for any p>0 is shown to satisfy the same equationas for the pure Ising model (p=1). Numerical simulations confirm such a behavior. Variance of magnetization and susceptibility in our model increase for decreasing p and diverge at the temperature at which magnetization vanishes. Simulations on a random graph also show that a small concentration of Ising agents is sufficient to induce a ferromagnetic ordering.

Comparison of cluster algorithms for the bond-diluted Ising model.

Monte Carlo cluster algorithms are popular for their efficiency in studying the Ising model near its critical temperature. We might expect that this efficiency extends to the bond-diluted Ising model. We show, however, that this is not always the case by comparing how the correlation times τ_{w} and τ_{sw} of the Wolff and Swendsen-Wang cluster algorithms scale as a function of the system size L when applied to the two-dimensional bond-diluted Ising model. We demonstrate that the Wolff algorithm suffers from a much longer correlation time than in the pure Ising model, caused by isolated (groups of) spins which are infrequently visited by the algorithm. With a simple argument we prove that these cause the correlation time τ_{w} to be bounded from below by L^{z_{w}} with a dynamical exponent z_{w}=γ/ν≈1.75 for a bond concentration p<1. Furthermore, we numerically show that this lower bound is actually taken for several values of p in the range 0.5<p<1. Moreover, we show that the Swendsen-Wang algorithm does not suffer from the same problem. Consequently, it has a much shorter correlation time, shorter than in the pure Ising model even. Numerically at p=0.6, we find that its dynamical exponent is z_{sw}=0.09(4).

Spin dynamics of an Ising chain with bond impurity in a tilt magnetic field

Spin dynamics of quantum Ising system has been the subject of condensed matter physics in the past few decades. In this work, the bond-impurity effects on dynamics of the one-dimensional quantum Ising model with both transverse (Bzi) and longitudinal magnetic field (LMF, Bxi) was studied in the high-temperature limit by the recurrence relations method. Three cases, i.e., the LMF effect on the dynamics of the pure Ising model without impurity (Bxi≠0), the weak- and strong-bond impurity effect on the dynamics of the transverse Ising model without LMF (the impurity strength Jj≠Ji, Bxi=0), and their combining effect (Jj≠Ji, Bxi≠0), were investigated respectively. It is found that the dynamical behaviors of the system depend on both the impurity and the LMF. In general, the central-peak behavior (collective-mode behavior) will be enhanced (depressed) when increasing Jj or Bxi properly. However, when Jj reaches a critical value Jjc , the spin motion will be frozen, i.e., the bond impurity will act as a switch, which may turn off or turn on the spin motion. In contrast, the LMF can active the motion of the frozen spin. Therefore, the dynamical behavior of the Ising system depends on the combining effects of the impurity and the LMF.

Two-dimensional super-roughening in the three-dimensional Ising model.

We present a random-interface representation of the three-dimensional (3D) Ising model based on thermal fluctuations of a uniquely defined geometric spin cluster in the 3D model and its 2D cross section. Extensive simulations have been carried out to measure the global interfacial width as a function of temperature for different lattice sizes which is shown to signal the criticality of the model at T_{c} by forming a size-independent cusp in 3D, along with an emergent super-roughening at its 2D cross section. We find that the super-rough state is accompanied by an intrinsic anomalous scaling behavior in the local properties characterized by a set of geometric exponents which are the same as those for a pure 2D Ising model.

Effect of Fluctuation in the Coupling Strength on Critical Dynamics of 1D Transverse Field Quantum Ising Model

The effect of uniformly distributed fluctuation in the coupling strength of a 100 spins transverse field quantum Ising chain is studied using the Density Matrix Renormalization Group on the Matrix Product States formalism. Uniform noise with mean zero and amplitude η is introduced to uniform nearest neighbour coupling of value unity. Disordered averages of thermodynamic quantities are calculated from 100 disordered realizations for each transverse field reading. We show that the system exhibits distinct behaviour of ordered and disordered state separated at η~1.0. For η ≲ 1.0, thermodynamic behaviour of the system gradually deviates from pure quantum Ising model. For η > 1.0 system exhibits highly fluctuating thermodynamic behaviour. Edward-Anderson order parameters are calculated and shown to be much less fluctuating and suitable as an order parameter for η > 1.0. Qualitative behaviour of the system is not affected by finite-size effect and replacing the fluctuation with normally distributed noise. Finally, for noise level between η = 1.0 and 1.5, the system exhibits a faster phase transition and enhances transverse magnetization before the critical point. For certain ranges of noise amplitude before the critical point, the quantum fluctuations are amplified. This suggests a potential improvement of quantum annealing within that regime.

Hysteresis in the Ising model with Glauber dynamics.

We use Glauber dynamics to study time and temperature dependence of hysteresis in the pure (without quenched disorder) Ising model on cubic, square, honeycomb lattices as well as random graphs. Results are discussed in the context of more extensive studies of hysteresis in the random field Ising model.

Restoration of dimensional reduction in the random-field Ising model at five dimensions.

The random-field Ising model is one of the few disordered systems where the perturbative renormalization group can be carried out to all orders of perturbation theory. This analysis predicts dimensional reduction, i.e., that the critical properties of the random-field Ising model in D dimensions are identical to those of the pure Ising ferromagnet in D-2 dimensions. It is well known that dimensional reduction is not true in three dimensions, thus invalidating the perturbative renormalization group prediction. Here, we report high-precision numerical simulations of the 5D random-field Ising model at zero temperature. We illustrate universality by comparing different probability distributions for the random fields. We compute all the relevant critical exponents (including the critical slowing down exponent for the ground-state finding algorithm), as well as several other renormalization-group invariants. The estimated values of the critical exponents of the 5D random-field Ising model are statistically compatible to those of the pure 3D Ising ferromagnet. These results support the restoration of dimensional reduction at D=5. We thus conclude that the failure of the perturbative renormalization group is a low-dimensional phenomenon. We close our contribution by comparing universal quantities for the random-field problem at dimensions 3≤D<6 to their values in the pure Ising model at D-2 dimensions, and we provide a clear verification of the Rushbrooke equality at all studied dimensions.

Critical behavior of models with infinite disorder at a star junction of chains

We study two models having an infinite-disorder critical point—the zero temperature random transverse-field Ising model and the random contact process—on a star-like network composed of M semi-infinite chains connected to a common central site. By the strong disorder renormalization group method, the scaling dimension xM of the local order parameter at the junction is calculated. It is found to decrease rapidly with the number M of arms, but remains positive for any finite M. This means that, in contrast with the pure transverse-field Ising model, where the transition becomes of first order for M > 2, it remains continuous in the disordered models, although, for not too small M, it is hardly distinguishable from a discontinuous one owing to a close-to-zero xM. The scaling behavior of the order parameter in the Griffiths–McCoy phase is also analyzed.

Dynamical Quantum Phase Transitions in Random Spin Chains

Quantum systems can exhibit a great deal of universality at low temperature due to the structure of ground states and the critical points separating distinct states. On the other hand, quantum time evolution of the same systems involves all energies and it is therefore thought to be much harder, if at all possible, to have sharp transitions in the dynamics. In this paper we show that phase transitions characterized by universal singularities do occur in the time evolution of random spin chains. The sharpness of the transitions and integrity of the phases owes to many-body localization, which prevents thermalization in these systems. Using a renormalization group approach, we solve the time evolution of random Ising spin chains with generic interactions starting from initial states of arbitrary energy. As a function of the Hamiltonian parameters, the system is tuned through a dynamical transition, similar to the ground state critical point, at which the local spin correlations establish true long range temporal order. In the state with dominant transverse field, a spin that starts in an up state loses its orientation with time, while in the "ordered" state it never does. As in ground state quantum phase transitions, the dynamical transition has unique signatures in the entanglement properties of the system. When the system is initialized in a product state the entanglement entropy grows as $log(t)$ in the two "phases", while at the critical point it grows as $log^\a(t)$, with $\a$ a universal number. This universal entanglement growth requires generic ("integrability breaking") interactions to be added to the pure transverse field Ising model.

Quantum disorder and local modes of the fully-frustrated transverse field Ising model on a diamond chain

We investigate the transverse field Ising model on a diamond chain using series expansions about the high-field limit and exact diagonalizations. For the unfrustrated case we accurately determine the quantum critical point and its expected 2d Ising universality separating the polarized and the Z2 symmetry broken phase. In contrast, we find strong evidences for a disorder by disorder scenario for the fully-frustrated transverse field Ising model, i.e. except for the pure Ising model, having an extensive number of ground states, the system is always in a quantum disordered polarized phase. The low-energy excitations in this polarized phase are understood in terms of exact local modes of the model. Furthermore, an effective low-energy description for an infinitesimal transverse field allows to pinpoint the quantum disordered nature of the ground state via mapping to an effective transverse field Ising chain and to determine the induced gap to the elementary effective domain wall excitation very accurately.

QUALITATIVE ASPECTS OF THE PHASE DIAGRAM OF J1–J2 MODEL ON THE CUBIC LATTICE

The qualitative aspects of the phase diagram of the Ising model on the cubic lattice, with ferromagnetic (F) nearest-neighbor interactions (J1) and antiferromagnetic (AF) next-nearest-neighbor couplings (J2) are analyzed in the plane temperature versus α, where α = J2/|J1| is the frustration parameter. We used the original Wang–Landau sampling (WLS) and the standard Metropolis algorithm to confront past results of this model obtained by the effective-field theory (EFT) for the cubic lattice. Our numerical results suggest that the predictions of the EFT are in general qualitatively correct, but the low-temperature re-entrant behavior, observed in the frontier separating the F and the collinear order, is an artifact of the EFT approach and should disappear when we consider Monte Carlo (MC) simulations of the model. In addition, our results indicate that the continuous phase transition between the F and the paramagnetic (P) phases, that occurs for 0.0 ≤α&lt; 0.25, belongs to the universality class of the three-dimensional pure Ising Model.

Dynamical barriers of pure and random ferromagnetic Ising models on fractal lattices

We consider the stochastic dynamics of the pure and random ferromagnetic Ising model on a hierarchical diamond lattice of branching ratio K with fractal dimension df = (ln(2K))/ln2. We adapt the real-space renormalization procedure introduced in our previous work (Monthus and Garel, 2013 J. Stat. Mech. P02037) to study the equilibrium time teq(L) as a function of the system size L near zero temperature. For the pure Ising model, we obtain the behavior teq(L) ∼ Lαeβ2JLds, where ds = df − 1 is the interface dimension, and we compute the prefactor exponent α. For the random ferromagnetic Ising model, we derive the renormalization rules for dynamical barriers Beq(L) ≡ (lnteq/β) near zero temperature. For the fractal dimension df = 2, we obtain that the dynamical barrier scales as Beq(L) = cL + L1/2u, where u is a Gaussian random variable of non-zero mean. While the non-random term scaling as L corresponds to the energy cost of the creation of a system-size domain-wall, the fluctuation part scaling as L1/2 characterizes the barriers for the motion of the system-size domain-wall after its creation. This scaling corresponds to the dynamical exponent ψ = 1/2, in agreement with the conjecture ψ = ds/2 proposed by Monthus and Garel (2008 J. Phys. A: Math. Theor. 41 115002). In particular, it is clearly different from the droplet exponent θ ≃ 0.299 involved in the statics of the random ferromagnet on the same lattice.

Dynamics of Ising models near zero temperature: real-space renormalization approach

We consider the stochastic dynamics of Ising ferromagnets (either pure or random) near zero temperature. The master equation satisfying detailed balance can be mapped onto a quantum Hamiltonian which has an exact zero-energy ground state representing the thermal equilibrium. The largest relaxation time teq governing the convergence towards this Boltzmann equilibrium in finite-size systems is determined by the lowest non-vanishing eigenvalue E1 = 1/teq of the quantum Hamiltonian H. We introduce and study a real-space renormalization procedure for the quantum Hamiltonian associated to the single-spin-flip dynamics of Ising ferromagnets near zero temperature. We solve explicitly the renormalization flow for two cases. (i) For the one-dimensional random ferromagnetic chain with free boundary conditions, the largest relaxation time teq can be expressed in terms of the set of random couplings for various choices of the dynamical transition rates. The validity of these RG results in d = 1 is checked by comparison with another approach. (ii) For the pure Ising model on a Cayley tree of branching ratio K, we compute the exponential growth of teq(N) with the number N of generations.

Intrinsic phonon effects on analog quantum simulators with ultracold trapped ions

Linear Paul traps have been used recently to simulate the transverse field Ising model with long-range spin-spin couplings. We study the intrinsic effects of phonon creation (from the initial phonon ground state) on the spin-state probability and spin entanglement for such quantum spin simulators. While it has often been assumed that phonon effects are benign because they play no role in the pure Ising model, they can play a significant role when a transverse field is added to the model. We use a many-body factorization of the quantum time-evolution operator of the system, adiabatic perturbation theory and exact numerical integration of the Schr\"odinger equation in a truncated spin-phonon Hilbert space followed by a tracing out of the phonon degrees of freedom to study this problem. We find that moderate phonon creation often makes the probabilities of different spin states behave differently from the static spin Hamiltonian. In circumstances in which phonon creation is minor, the spin dynamics state probabilities converge to the static spin Hamiltonian prediction at the cost of reducing the spin entanglement. We show how phonon creation can severely impede the observation of kink transitions in frustrated spin systems when the number of ions increases. Many of our results also have implications for quantum simulation in a Penning trap.

The random transverse field Ising model in d = 2: analysis via boundary strong disorder renormalization

To avoid the complicated topology of surviving clusters induced by standard strong disorder RG in dimension d > 1, we introduce a modified procedure called ‘boundary strong disorder RG’ where the order of decimations is chosen a priori. We apply this modified procedure numerically to the random transverse field Ising model in dimension d = 2. We find that the location of the critical point, the activated exponent ψ ≃ 0.5 of the infinite-disorder scaling, and the finite-size correlation exponent νFS ≃ 1.3 are compatible with the values obtained previously using standard strong disorder RG. Our conclusion is thus that strong disorder RG is very robust with respect to changes in the order of decimations. In addition, we analyze the RG flows within the two phases in more detail, to show explicitly the presence of various correlation length exponents: we measure the typical correlation exponent νtyp ≃ 0.64 for the disordered phase (this value is very close to the correlation exponent of the pure two-dimensional quantum Ising model), and the typical exponent νh ≃ 1 for the ordered phase. These values satisfy the relations between critical exponents imposed by the expected finite-size scaling properties at infinite-disorder critical points. We also measure, within the disordered phase, the fluctuation exponent ω ≃ 0.35 which is compatible with the directed polymer exponent in (1 + 1) dimensions.

Nonequilibrium phase transitions and stationary-state solutions of a three-dimensional random-field Ising model under a time-dependent periodic external field

Nonequilibrium behavior and dynamic phase-transition properties of a kinetic Ising model under the influence of periodically oscillating random fields have been analyzed within the framework of effective-field theory based on a decoupling approximation. A dynamic equation of motion has been solved for a simple-cubic lattice (q=6) by utilizing a Glauber-type stochastic process. Amplitude of the sinusoidally oscillating magnetic field is randomly distributed on the lattice sites according to bimodal and trimodal distribution functions. For a bimodal type of amplitude distribution, it is found in the high-frequency regime that the dynamic phase diagrams of the system in the temperature versus field amplitude plane resemble the corresponding phase diagrams of the pure kinetic Ising model. Our numerical results indicate that for a bimodal distribution, both in the low- and high-frequency regimes, the dynamic phase diagrams always exhibit a coexistence region in which the stationary state (ferro or para) of the system is completely dependent on the initial conditions, whereas for a trimodal distribution, the coexistence region disappears depending on the values of the system parameters.