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arXiv:cond-mat/9808019 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Hamiltonian dynamics and the phase transition of the XY model

Xavier Leoncini, Alberto Verga, Stefano Ruffo

Published Jun 1, 1998• 50 citations
arXivHTMLPDF

Abstract

A Hamiltonian dynamics is defined for the $\mathrm{XY}$ model by adding a kinetic energy term. Thermodynamical properties (total energy, magnetization, vorticity) derived from microcanonical simulations of this model are found to be in agreement with canonical Monte Carlo results in the explored temperature region. The behavior of the magnetization and energy as functions of the temperature are thoroughly investigated, taking into account finite size effects. By representing the spin field as a superposition of random phased waves, we derive a nonlinear dispersion relation whose solutions allow the computation of thermodynamical quantities, which agree quantitatively with those obtained in numerical experiments, up to temperatures close to the transition. At low temperatures the propagation of phonons is the dominant phenomenon, while above the phase transition the system splits into ordered domains separated by interfaces populated by topological defects. In the high temperature phase, spins rotate, and an analogy with an Ising-like system can be established, leading to a theoretical prediction of the critical temperature ${T}_{\mathrm{KT}}\ensuremath{\approx}0.855.$

DOI
10.1103/physreve.57.6377

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52•Jan 1, 2015•0 citations
A Hamiltonian dynamics is defined for the XY\mathrm{XY}XY model by adding a kinetic energy term. Thermodynamical properties (total energy, magnetization, vorticity) derived from microcanonical simulations of this model are found to be in agreement with canonical Monte Carlo results in the explored temperature region. The behavior of the magnetization and energy as functions of the temperature are thoroughly investigated, taking into account finite size effects. By representing the spin field as a superposition of random phased waves, we derive a nonlinear dispersion relation whose solutions allow the computation of thermodynamical quantities, which agree quantitatively with those obtained in numerical experiments, up to temperatures close to the transition. At low temperatures the propagation of phonons is the dominant phenomenon, while above the phase transition the system splits into ordered domains separated by interfaces populated by topological defects. In the high temperature phase, spins rotate, and an analogy with an Ising-like system can be established, leading to a theoretical prediction of the critical temperature TKT\ensuremath≈0.855.{T}_{\mathrm{KT}}\ensuremath{\approx}0.855.TKT​\ensuremath≈0.855.
A class of infinite-ranged random model Hamiltonians is defined as a limiting case in which the appropriate form of mean-field theory, order parameters and phase diagram to describe spin-glasses may be established. It is believed that these Hamiltonians may be exactly soluble, although a complete solution is not yet available. Thermodynamic properties of the model for Ising and XY\mathrm{XY}XY spins are evaluated using a many-replica procedure. Results of the replica theory reproduce properties at and above the ordering temperature which are also predicted by high-temperature expansions, but are in error at low temperatures. Extensive computer simulations of infinite-ranged Ising spin-glasses are presented. They confirm the general details of the predicted phase diagram. The errors in the replica solution are found to be small, and confined to low temperatures. For this model, the extended mean-field theory of Thouless, Anderson, and Palmer gives physically sensible low-temperature predictions. These are in quantitative agreement with the Monte Carlo statics. The dynamics of the infinite-ranged Ising spin-glass are studied in a linearized mean-field theory. Critical slowing down is predicted and found, with correlations decaying as ${e}^{{\ensuremath{-}[\frac{(T\ensuremath{-}{T}_{c})}{T}]}^{2}t}$ for TTT greater than Tc{T}_{c}Tc​, the spin-glass transition temperature. At and below Tc{T}_{c}Tc​, spin-spin correlations are observed to decay to their long-time limit as ${t}^{\ensuremath{-}\frac{1}{2}}$.
A class of infinite-ranged random model Hamiltonians is defined as a limiting case in which the appropriate form of mean-field theory, order parameters and phase diagram to describe spin-glasses may be established. It is believed that these Hamiltonians may be exactly soluble, although a complete solution is not yet available. Thermodynamic properties of the model for Ising and XY\mathrm{XY}XY spins are evaluated using a "many-replica" procedure. Results of the replica theory reproduce properties at and above the ordering temperature which are also predicted by high-temperature expansions, but are in error at low temperatures. Extensive computer simulations of infinite-ranged Ising spin-glasses are presented. They confirm the general details of the predicted phase diagram. The errors in the replica solution are found to be small, and confined to low temperatures. For this model, the extended mean-field theory of Thouless, Anderson, and Palmer gives physically sensible low-temperature predictions. These are in quantitative agreement with the Monte Carlo statics. The dynamics of the infinite-ranged Ising spin-glass are studied in a linearized mean-field theory. Critical slowing down is predicted and found, with correlations decaying as ${e}^{{\ensuremath{-}[\frac{(T\ensuremath{-}{T}_{c})}{T}]}^{2}t}$ for TTT greater than Tc{T}_{c}Tc​, the spin-glass transition temperature. At and below Tc{T}_{c}Tc​, spin-spin correlations are observed to decay to their long-time limit as ${t}^{\ensuremath{-}\frac{1}{2}}$.
A simulation of two-dimensional XY spin model has been done using Monte Carlo methods written in C. In this simulation, the behaviour of spins on a certain temperature had been visually learned, and a quantitative result of data containing the values of Hamiltonian energy, specific heat, magnetization, susceptibility, and amount of vortices in a system with temperature below and above the critical temperature Tc has also been obtained. Moreover, different critical temperature value obtained for both lattice size tested; Tc = 1.1 for system with lattice size 16×16 and Tc = 1.0 for system with lattice size 256×256, proving that critical temperature of a finite-size lattice would be shifted. The obtained Tc values for both finite-size systems are close to the ideal critical temperature for Kosterlitz-Thouless transition, TKT = 0.892.
The phase of matter is the state of a system in a region on a macroscopic scale. At this macroscopic scale, in equilibrium, physical properties of the system are homogeneous. Statistical physics tries to explain the homogeneity and the associated properties from microscopic interactions and by laws of nature. On a macroscopic scale, the systems to investigate are many-body systems like gases, liquids or solids. They are characterized by thermodynamic functions like the free energy. Order parameters, like for example the magnetization in magnetic systems, play a special role here. Most of the time they are associated with a breaking of symmetry when transitioning from one phase to another. This however is not a universal observation. One example for which this is not the case is the Berezinskii-Kosterlitz-Thouless phase transition (KT-transition). This particular transition can be studied in two-dimensional XY-models, where there is a phase transition without the breaking of continuous symmetries. First theoretical descriptions trace back to the works of Berezinskii [Ber71], as well as Kosterlitz and Thouless [Kos73, Kos74, Kos16]. They initially investigated a system of classical spins fixed on the nodes of a square lattice. Consequently, only the rotational degree of freedom remains. In this work we start with the model of [LB16] who introduces a coupling of spins to velocities of mobile particles. After a numerical analysis of different integration methods for the equations of motion and investigations of this system, we remove the coupling. By that, we mimic an XY- model with additional translational degrees of freedom. For this system we examine the system with respect to the occurrence of the KT-transition by computer simulations. We apply several general tools for the analysis of magnetic systems. Subsequently we look into topological defects in the system, namely vortices and anti-vortices. For temperatures lower than the transition temperature T < T KT , defects are closely bound together, while for temperatures T > T KT they can proliferate in the system and can be treated like a gas of unbound defects. Moreover, we investigate spatial and temporal spin-correlations to estimate the critical exponent in the system, as well as the critical temperature. For a reference we first simulate an XY-model on a trigonal lattice as a comparison for our results in the mobile model. In both systems we do find the KT-transition by the occurrence of unbound topological defects and from the critical exponents we calculate from the spin-correlations functions.
Experimentally derived physical properties of 3d Ising spin glasses are discussed and compared to what is anticipated from theory and simulations of model systems. It is found that the spin glass experiences a zero-field phase transition to a low-temperature spin-glass phase, that there is no phase transition in a magnetic field and that nonequilibrium and chaos effects often dominate the dynamics of the low-temperature glassy phase. These results are in agreement with findings from Monte Carlo simulations and with properties derived from the droplet model.
In this master thesis the criticality of the 3D XY model with cubic anisotropy has been examined by the means of Monte Carlo simulation. The 3D XY model is a special case of the O(n)O(n)O(n) spin-vector model which describe numerous phenomena's. Here, nnn is the number of vector dimensions and for the XY model n=2n=2n=2. The O(n)O(n)O(n)-model is invariant for global rotations. This means that it is arbitrary which direction in the symmetry breaking occurs. In real system one should always expect perturbations from a perfect system, here it is done by adding a cubic anisotropy (order p=4p=4p=4) then the symmetry of the Hamiltonian is lowered and it will be advantageous for the spins to be ordered in directions 90o90^o90o apart. This symmetry is called C4C_{4}C4​, which is a part of the cyclic group, and it restricts the symmetry breaking to one of these four directions. \n\nThe criticality of this model is when the spins suddenly align as the temperature is lowered to a critical temperature. Sufficiently close to the critical temperature quantities like the magnetization, which is the sum of spins, will show a power law growth characterised by critical exponents specified by the class of universality of the transition. The critical exponents can be obtained by finite-size scaling. By applying renormalization theory on the critical region one can show that some of parameters in the Hamiltonian are: relevant and effect the scaling, irrelevant and does not effect the scaling and dangerously irrelevant meaning that they only affect some of the critical exponents. Previous Monte Carlo and renormalization calculus show that anisotropy which is relevant for p<4p<4p<4 becomes dangerously irrelevant for pgeq4p\\geq4pgeq4. The question of interest is what kind of effect the dangerously irrelevant parameter in the Hamiltonian has on the scaling. This is interesting because the effect of dangerously irrelevant parameters is hardly studied and anisotropy arise in many crystal systems because of the lattice geometry. \n\nThe results are obtained by Monte Carlo simulations and the critical slowing down close to the phase transition is solved by the method of parallel tempering. This method was chosen because it is simple to implement in a parallel code.\n\nThe Binder cumulant for different quantities and the peak of the magnetic susceptibility was used support a critical temperature Tc=2.2019textnormalkT_c = 2.2019 \\ \\textnormal kTc​=2.2019textnormalk for no field applied and no anisotropy. The known critical exponents for the no field and no anisotropy was used. The scaling made a good collapse, meaning that the right TcT_cTc​ and the correct critical exponents where used. The scaling did not take into account a magnetic field and therefore a good collapse of curves where not obtained in this case. When the cubic anisotropy was added the critical exponent for the magnetic susceptibility and the cubic order parameter changed, while the other exponents where unchanged. This supports the view that cubic anisotropy is dangerously irrelevant for the 3D XY model. The scaling of the magnetic susceptibility showed a possible new kind of scaling, a scaling which is different on the two sides of the transition. Although a more accurate determination of the critical exponent in the ordered phase require more extensive simulations. It appears that this exponent is larger than the exponent in the disordered phase, where one recovered the isotropic exponent. The difference in the critical exponents has been predicted to increase with ppp and an investigation of the hexagonal anisotropy p=6p=6p=6 has been started.