Glauber Dynamics for the Quantum Ising Model in a Transverse Field on a Regular Tree

Motivated by a recent use of Glauber dynamics for Monte Carlo simulations of path integral representation of quantum spin models (Krzakala et al. in Phys. Rev. B 78(13):134428, 2008), we analyse a natural Glauber dynamics for the quantum Ising model with a transverse field on a finite graph G. We establish strict monotonicity properties of the equilibrium distribution and we extend (and improve) the censoring inequality of Peres and Winkler to the quantum setting. Then we consider the case when G is a regular b-ary tree and prove the same fast mixing results established in Martinelli et al. (Commun. Math. Phys. 250(2):301–334, 2004) for the classical Ising model. Our main tool is an inductive relation between conditional marginals (known as the “cavity equation”) together with sharp bounds on the operator norm of the derivative at the stable fixed point. It is here that the main difference between the quantum and the classical case appear, as the cavity equation is formulated here in an infinite dimensional vector space, whereas in the classical case marginals belong to a one-dimensional space.     

The Mixing Time Evolution of Glauber Dynamics for the Mean-Field Ising Model

We consider Glauber dynamics for the Ising model on the complete graph on n vertices, known as the Curie-Weiss model. It is well-known that the mixing-time in the high temperature regime (β < 1) has order n log n, whereas the mixing-time in the case β > 1 is exponential in n. Recently, Levin, Luczak and Peres proved that for any fixed β < 1 there is cutoff at time with a window of order n, whereas the mixing-time at the critical temperature β = 1 is Θ(n ^3/2). It is natural to ask how the mixing-time transitions from Θ(n log n) to Θ(n ^3/2) and finally to exp (Θ(n)). That is, how does the mixing-time behave when β = β(n) is allowed to tend to 1 as n → ∞. In this work, we obtain a complete characterization of the mixing-time of the dynamics as a function of the temperature, as it approaches its critical point β _c  = 1. In particular, we find a scaling window of order around the critical temperature. In the high temperature regime, β = 1 − δ for some 0 < δ < 1 so that δ ² n → ∞ with n, the mixing-time has order (n/δ) log(δ ² n), and exhibits cutoff with constant and window size n/δ. In the critical window, β = 1± δ, where δ ² n is O(1), there is no cutoff, and the mixing-time has order n ^3/2. At low temperature, β = 1 + δ for δ > 0 with δ ² n → ∞ and δ = o(1), there is no cutoff, and the mixing time has order . Research of J. Ding and Y. Peres was supported in part by NSF grant DMS-0605166.     

The Mean Field Ising Model trough Interpolating Techniques

Aim of this paper is to illustrate how some recent techniques developed within the framework of spin glasses do work on simpler model, focusing on the method and not on the analyzed system. To fulfill our will the candidate model turns out to be the paradigmatic mean field Ising model. The model is introduced and investigated with the interpolation techniques. We show the existence of the thermodynamic limit, bounds for the free energy density, the explicit expression for the free energy with its suitable expansion via the order parameter, the self-consistency relation, the phase transition, the critical behavior and the self-averaging properties. At the end a formulation of a Parisi-like theory is tried and discussed.     

Mean Field Theory of Ising Model on Sierpinski

We study Ising model on Sierpinski carpets by using mean field theory. We find a phase transition at T_c > 0 which is dependent on the geometrical factors. The critical exponents are calculated and found to be the same as the values for translationally invariant lattices.     

Glauber Dynamics for the Mean-Field Potts Model

We study Glauber dynamics for the mean-field (Curie-Weiss) Potts model with q??3 states and show that it undergoes a critical slowdown at an inverse-temperature ?? _s (q) strictly lower than the critical ?? _c (q) for uniqueness of the thermodynamic limit. The dynamical critical ?? _s (q) is the spinodal point marking the onset of metastability. We prove that when ??<?? _s (q) the mixing time is asymptotically C(??,q)nlogn and the dynamics exhibits the cutoff phenomena, a sharp transition in mixing, with a window of order n. At ??=?? _s (q) the dynamics no longer exhibits cutoff and its mixing obeys a power-law of order n ^4/3. For ??>?? _s (q) the mixing time is exponentially large in n. Furthermore, as ?????? _s with n, the mixing time interpolates smoothly from subcritical to critical behavior, with the latter reached at a scaling window of O(n ^?2/3) around ?? _s . These results form the first complete analysis of mixing around the critical dynamical temperature??including the critical power law??for a model with a first order phase transition.     

Glauber Dynamics of the Random Energy Model

 We investigate the long-time behavior of the Glauber dynamics for the random energy model below the critical temperature. We give very precise estimates on the motion of the process to and between the states of extremal energies. We show that when disregarding time, the consecutive steps of the process on these states are governed by a Markov chain that jumps uniformly on all possible states. The mean times of these jumps are also computed very precisely and are seen to be asymptotically independent of the terminal point. A first indicator of aging is the observation that the mean time of arrival in the set of states that have waiting times of order T is itself of order T. The estimates proven in this paper will furnish crucial input for a follow-up paper where aging is analysed in full detail. Received: 9 October 2001 / Accepted: 17 October 2002 Published online: 28 February 2003 RID="*" ID="*" Work partially supported by the Swiss National Science Foundation under contract 21-65267.01 RID="⋆⋆" ID="⋆⋆" On leave from CPT-CNRS, Luminy, Case 907, 13288 Marseille Cedex 9, France. E-mail:veronique.gayrard@epfl.ch Communicated by M. Aizenman     

Finite-Volume Glauber Dynamics in a Small Magnetic Field

We consider Glauber dynamics on a finite cube in d-dimensional lattice (d2), which is associated with basic Ising model at temperature T=1/1 under a magnetic field h > 0. We prove that if the effective magnetic field is positive, then the relaxation of the Glauber dynamics in the uniform norm is exponentially fast, uniformly over the size of underlying cube. The result covers the case of the free-boundary condition with arbitrarily small positive magnetic field. This paper is a continuation of an attempt initiated earlier by Schonmann and Yoshida to shed more light on the relaxation of the finite-volume Glauber dynamics when the thermodynamic parameter (, h) is so near the phase transition line, (, h); _c < &h = 0, that the Dobrushin–Shlosman mixing condition is no longer available.     

A Simple Mean Field Model for Social Interactions: Dynamics,Fluctuations, Criticality

We study the dynamics of a spin-flip model with a mean field interaction. The system is non reversible, spacially inhomogeneous, and it is designed to model social interactions. We obtain the limiting behavior of the empirical averages in the limit of infinitely many interacting individuals, and show that phase transition occurs. Then, after having obtained the dynamics of normal fluctuations around this limit, we analyze long time fluctuations for critical values of the parameters. We show that random inhomogeneities produce critical fluctuations at a shorter time scale compared to the homogeneous system.     

Slow Dynamics for the Dilute Ising Model in the Phase Coexistence Region

In this paper we consider the Glauber dynamics for a disordered ferromagnetic Ising model, in the region of phase coexistence. It was conjectured several decades ago that the spin autocorrelation decays as a negative power of time [Huse and Fisher, in Phys Rev B 35(13):6841–6846, 1987]. We confirm this behavior by establishing a corresponding lower bound in any dimensions d ≥ 2, together with an upper bound when d = 2. Our approach is deeply connected to the Wulff construction for the dilute Ising model. We consider initial phase profiles with a reduced surface tension on their boundary and prove that, under mild conditions, those profiles are separated from the (equilibrium) pure plus phase by an energy barrier.     

Two-Site Effective Field Approximation for Mixed Spin Ising Model

A simple and powerful method (two-site effective field approximation) for mixed spin Ising model was presented. Our result about transition temperature of mixed Ising spin system is much better than that by making use of other approximate methods.     

Dynamics of the Random Ising Model with Long-Range Interaction

Critical dynamics of the random Ising model with long-range interaction decaying as r-(d σ) where d is the dimensionality) is studied by the theoretic renormalization-group approach. The system is released to an evolution within a model A dynamics. Asymptotic scaling laws are studied in a frame of the expansion in = 2σ - d. In dimensions d ＜ 2σ. the dynamic exponent z is calculated to the second order in at the random fixed point.``     

Efficient Algorithms for the Two-Dimensional Ising Model with a Surface Field

The bond propagation and site propagation algorithms are extended to the two-dimensional (2D) Ising model with a surface field. With these algorithms we can calculate the free energy, internal energy, specific heat, magnetization, correlation functions, surface magnetization, surface susceptibility and surface correlations. The method can handle continuous and discrete bond and surface-field disorder and is especially efficient in the case of bond or site dilution. To test these algorithms, we study the wetting transition of the 2D Ising model, which was solved exactly by Abraham. We can locate the transition point accurately with a relative error of \(10^{-8}\) . We carry out the calculation of the specific heat and surface susceptibility on lattices with sizes up to \(200^2 \times 200\) . The results show that a finite jump develops in the specific heat and surface susceptibility at the transition point as the lattice size increases. For lattice size \(320^2 \times 320\) the parallel correlation length exponent is \(1.86\) , while Abraham’s exact result is \(2.0\) . The perpendicular correlation length exponent for lattice size \(160^2\times 160\) is \(1.05\) , whereas its exact value is \(1.0\) .     

Super-sensitivity in Dynamics of Ising Model with Transverse Field: From Perspective of Franck-Condon Principle

We study the role of Franck-Condon (F-C) principle in the dynamics of a central spin system, which is coupled to an Ising chain in transverse field. The transition process of energy levels caused by the excited central spin is studied to manifest the quantum critical effect through the Franck-Condon principle. The super-sensitivity of this quantum critical system is demonstrated clearly from the properties of Franck-Condon factors. We analytically show how spin numbers, coupling strength and order parameter of the Ising chain sensitively effect on the energy level populations in dynamical evolution near the critical point. This super-sensitivity and criticality are explicitly displayed in absorption spectrum.     

Generalized Ising Model in a Magnetic Field

Journal of Experimental and Theoretical Physics - Results on the generalization of the Ising model to an arbitrary number of translations of a linear chain in an external magnetic field, taking...     

An extended chain Ising model and its Glauber dynamics

It was first proposed that an extended chain Ising (ECI) model contains the Ising chain model, single spin double-well potentials and a pure phonon heat bath of a specific energy exchange with the spins. The extension method is easy to apply to high dimensional cases. Then the single spin-flip probability (rate) of the ECI model is deduced based on the Boltzmann principle and general statistical principles of independent events and the model is simplified to an extended chain Glauber-Ising (ECGI) model. Moreover, the relaxation dynamics of the ECGI model were simulated by the Monte Carlo method and a comparison with the predictions of the special chain Glauber-Ising (SCGI) model was presented. It was found that the results of the two models are consistent with each other when the Ising chain length is large enough and temperature is relative low, which is the most valuable case of the model applications. These show that the ECI model will provide a firm physical base for the widely used single spin-flip rate proposed by Glauber and a possible route to obtain the single spin-flip rate of other form and even the multi-spin-flip rate.     

Generalized Ising Model in the Absence of Magnetic Field

Journal of Experimental and Theoretical Physics - The Ising model on an one-dimensional monoatomic equidistant lattice with different exchange interactions between atomic spins at the sites of...     

On the Ising Model with Strongly Anisotropic External Field

In this paper we analyze the equilibrium phase diagram of the two-dimensional ferromagnetic n.n. Ising model when the external field takes alternating signs on different rows. We show that some of the zero-temperature coexistence lines disappear at every positive sufficiently small temperature, whereas one (and only one) of them persists for sufficiently low temperature.     

Short-Time Dynamics of the Three-Dimensional Ising Model with Competing Interactions

The critical relaxation from the low-temperature ordered state of the three-dimensional Ising model with competing interactions on a simple cubic lattice has been studied for the first time using the short-time dynamics method. Competition between exchange interactions is due to the ferromagnetic interaction between the nearest neighbors and the antiferromagnetic interaction between the next nearest neighbors. Particles containing 262144 spins with periodic boundary conditions have been studied. Calculations have been performed by the standard Metropolis Monte Carlo algorithm. The static critical exponents of the magnetization and correlation radius have been calculated. The dynamic critical exponent of the model under study has been calculated for the first time.