Lattice Size Dependence Research Articles

The sign problem in quantum Monte Carlo simulations of the Hubbard model in a weak interaction region

Determinant quantum Monte Carlo (QMC) simulations of fermion models normally face the sign problem, which has a strong dependence of the interaction strength, electron filling, temperature, lattice geometry and lattice size. It is useful to set up the data set of such dependence, to make it clear which parameter region is accessible or not. To supplement the previous study, here we report the behavior of the average sign in the weak interaction region, as functions of lattice geometry, lattice size, electron filling and temperature. Furthermore, we find that in the weak interaction region, for several interesting lattice geometries (except for the kagome lattice) the average sign gets improved when the lattice sizes increase, which is in contrast to the general understanding of the lattice size dependence of the average sign. This finding points out that when the sign problem is present in the weak interaction region, the simulation of a larger lattice size can be better under control.

The spectral gap and the dynamical critical exponent of an exact solvable probabilistic cellular automaton

We obtained the exact solution of a probabilistic cellular automaton related to the diagonal-to-diagonal transfer matrix of the six-vertex model on a square lattice. The model describes the flow of ants (or particles), traveling on a one-dimensional lattice whose sites are small craters containing sleeping or awake ants (two kinds of particles). We found the Bethe ansatz equations and the spectral gap for the time-evolution operator of the cellular automaton. From the spectral gap we show that in the asymmetric case it belongs to the Kardar–Parisi–Zhang (KPZ) universality class, exhibiting a dynamical critical exponent value z=32. This result is also obtained from a direct Monte Carlo simulation, by evaluating the lattice-size dependence of the decay time to the stationary state.

Monte Carlo Simulation of First Order Phase Transitions

Monte Carlo simulation with Metropolis algorithm has been performed on a cubic Ising lattice with four-spin couplings. Temperature dependence of the spin average and average energy has been obtained. We found that the phase transition is of first order when the strength of four-spin coupling is stronger a critical value, and this critical value is of lattice size dependence. Binder cumulants still can be used to determine the second order phase transition temperature. For first order phase transition, the spin average disappears abruptly at the temperature of the average energy being a third of the ground state energy.

Dependence of film surface roughness and slope on surface migration and lattice size in thin film deposition processes

This work focuses on the study of the dependence of film surface roughness and slope on lattice size in thin film deposition processes. Two different models of thin film deposition processes, in both 1D and 2D, are considered: random deposition with surface relaxation model and deposition/migration model. Surface roughness and surface slope are defined as the root-mean-squares of the surface height profile and of the surface slope profile, respectively. Both surface roughness and slope evolve to steady-state values at large times but are subject to different dynamics and scaling properties. A linear and a logarithmic dependence of surface roughness square on lattice size are observed in the 1D and 2D lattice models, respectively, in both the random deposition with surface relaxation model and the deposition/migration model with zero activation energy contribution from each neighboring particle. Furthermore, a stronger lattice-size dependence is found in the deposition/migration model when the migration activation energy contribution from each neighboring particle becomes significant. On the other hand, a weak lattice-size dependence is found for the surface mean slope in all growth models considered, especially at large lattice sizes. Finally, the dynamics of surface roughness and surface slope is studied with respect to different characteristic length scales.

Dynamics and Lattice-Size Dependence of Surface Mean Slope in Thin-Film Deposition

This work focuses on the study of the dynamic behavior and lattice-size dependence of the surface root-mean-square slope of thin-film deposition processes that involve thermal balance between film ...

Lattice-size Dependence and Dynamics of Surface Mean Slope in a Thin Film Deposition Process

This work focuses on the study of the dynamic behavior and lattice size dependence of the surface root-mean-square slope in a porous thin film deposition process taking place on a triangular lattice. The simulation results indicate that the expected mean slope square reaches quickly a steady-state value and exhibits a very weak dependence with respect to lattice size variation. The simulation findings are corroborated by an analysis of appropriate finite-difference discretizations of surface height profiles computed by an Edwards-Wilkinson-type partial differential equation that can be used to describe the dynamics of surface height profile in the thin film deposition process under consideration.

Studies of bosons in optical lattices in a harmonic potential

We present a theoretical study of Bose condensation and specific heat of non-interacting bosons in finite lattices in harmonic potentials in one, two, and three dimensions. We numerically diagonalize the Hamiltonian to obtain the energy levels of the systems. Using the energy levels thus obtained, we investigate the temperature dependence, dimensionality effects, lattice size dependence, and evolution to the bulk limit of the condensate fraction and the specific heat. Some preliminary results on the specific heat of fermions in optical lattices are also presented. The results obtained are contextualized within the current experimental and theoretical scenario.

Superconducting condensation energy computed for the 2D Hubbard model with Bi2212-type band

The 2D Hubbard model is the simplest model that possibly explains the occurrence of high-Tc superconductivity (SC) in cuprates. To demonstrate this possibility with a condition of moderate on-site Coulomb energy U, we have computed the SC condensation energy Econd by means of the variational Monte Carlo method employing a slightly modified Gutzwiller trial wave function which also takes account of the Bi2212-type band, i.e., t′∼−0.34 and t″∼0.23 (t, t′ and t″ are defined in the main text; energy unit is t). The competing SDW Econd strongly depended on the employed lattice size L×L but approached the bulk-limit value when L⩾16. With proper assumption on the value of t″, the t′ region where the SDW Econd are larger than the SC Econd shrank to −0.16⩽t′⩽−0.08. In −0.40<t′<−0.16 the SC Econd was found predominant. Concerning the lattice-size dependence the SC Econd in the neighborhood of the Bi2212-type set was found to tend to increase with increasing L when L⩾16, taking about 1/3 or 1 times the experimental SC Econd of YBCO. This was computed using periodic boundary conditions (bc’s) for the two directions. The average of two results for periodic and antiperiodic bc’s imposed to both x- and y-directions for t′=−0.31 and t″=0.21 proved to make the L-dependence so mild that the extrapolation to the finite bulk limit looks plausible. Together with the previous positive result with t′∼−0.05 and t″=0, this result clearly supports the applicability of the 2D Hubbard model to the SC in the whole group of cuprates. Incidentally, more elaborate Gutzwiller–Jastrow-type trial functions were also examined but found to bring in only minor differences.

Fluctuations and correlations in two-dimensional Bak$ndash$Sneppen model

The self-organized criticality in the two-dimensional Bak–Sneppen model is studied from the fluctuations of the mean fitness and the jump of the minimum sites. An ensemble having the same number of updates from the initial state is used in the investigation. From the Gaussian distribution of the mean fitness in the ensemble, the lattice size dependence of the ensemble-averaged fitness is found. From the dependence, a solution to the gap equation is found and the critical gap and an exponent are calculated. A spatial–temporal correlation function is investigated from the jump of the minimum site.

Detailed structure of configuration space and its importance on ergodic separation of ± J Ising lattices

A complete and exact characterization of the configuration space of 2-D ± J Ising lattices is performed. A new algorithm is introduced here representing advantages for reaching all states for small samples and doing a non-biased sampling of ground states for larger samples. We report efficient procedures to find all ground states grouped in local ensembles of ground states (LEGs) and also a convenient way of storing and comparing states. Properties of such LEGs differ from some approximate descriptions reported in the literature. The onset of lattice size dependence of properties is discussed. Four different ways of performing ergodic separation are used to calculate order parameters. The most significant way of doing ergodic separation requires previous classification of states in LEGs.

Do large abelian monopole loops survive the continuum limit?

An analysis of the monopole loop length distribution is performed in Wilson-action SU(2) lattice gauge theory. A pure power law in the inverse length is found, at least for loops of length, l, less than the linear lattice size N. This power shows a definite β dependence, passing 5 around β = 2.9, and appears to have very little finite lattice size dependence. It is shown that when this power exceeds 5, no loops any finite fraction of the lattice size will survive the infinite lattice limit. This is true for any reasonable size distribution for loops larger than N. The apparent lack of finite size dependence in this quantity would seem to indicate that abelian monopole loops large enough to cause confinement do not survive the continuum limit. Indeed they are absent for all β > 2.9.

CHARGED FERMION STATES IN THE QUENCHED U(1) CHIRAL WILSON–YUKAWA MODEL

The property of charged fermion states is investigated in the quenched U(1) chiral Wilson–Yukawa model. Fitting the charged fermion propagator with a single hyperbolic cosine does not yield reliable results. On the other hand the behavior of the propagator including large lattice size dependence is well described by the large Wilson–Yukawa coupling expansion, providing strong evidence that no charged fermion state exists as an asymptotic particle in this model.

Hybrid Monte-Carlo Spin-Dynamics Simulation of Short-Range J Heisenberg Models with and without Anisotropy

An efficient method of simulating Heisenberg models is developed combining the Monte-Carlo· method and spin-dynamics inherent to the models. The method is applied to three-dimensional short-range ±J models with the concentration of ferromagnetic bonds P, and with and without random anisotropy to reveal effects of the anisotropy. To study their equilibrium properties, the spin-glass susceptibility XSg and the absolute magnetization are calculated together with the spin-glass order parameter Q(K) for a finite Monte-Carlo step K. When the anisotropy is absent, Q(K) vanishes as K increases even at very low temperatures, suggesting the absence of the spin-glass phase. This is confirmed by studying lattice size dependence of XSg. The phase diagram is also given in which the critical concentration is estimated as Pc~0.79. When the anisotropy is present, the things change drastically. Q(K) at low temperatures reduces very little as K increases and XSg'S for different sizes of the lattice are scaled well using a scaling function with a finite transition temperature. It is concluded, hence, that a spin-glass phase transition occurs at a finite temperature. In addition, for P > Pc, a reentrant mixed phase of the ferromagnetism and the transverse spin-glass is found. Transverse random fields induced by the ferromagnetic long-range order smear the reentrant phase transition. The phase diagram thus obtained qualitatively explains experimental ones. We conclude, hence, that the anisotropy plays an essential role for the occurrence of not only the spin-glass phase but also the reentrant mixed phase.

Spectral properties and lattice-size dependences in cluster algorithms.

Simulation results of Ising systems for several update rules, observables, and dimensions are analyzed. The lattice-size dependence is discussed for the autocorrelation times and for the weights of eigenvalues, giving fit results in the case of power laws. Implications of spectral properties are pointed out and the behavior of a particular observable not governed by detailed balance is explained.

Fermion spectrum in the quenched U(1) chiral Wilson-Yukawa model

Nature of charged fermion state is investigated in the quenched U(1) chiral Wilson-Yukawa model. Fitting the charged fermion propagator with a single hyperbolic cosine does not yield reliable result. On the other hand the behaviour of the propagator including large lattice size dependence is well described by the large Wilson-Yukawa coupling expansion, providing strong evidence that no charged state exists as an asymptotic particle in this model.

Effects of flipping rules in cluster algorithms

Studying rules for flipping cluster spins, one with the fastest decay of autocorrelations is found. It turns out that connectivity of clusters is an additional crucial parameter. Another result is that there are only a few eigenvalues of the transition matrix which play a role. One eigenvalue dominates if the flipped cluster distribution is extended. The lattice size dependence of the weights of the eigenvalues is observed for the first time.

On the helicity modulus, the critical temperature and Monte Carlo simulations for the two-dimensional XY-model

A lattice size dependence of the helicity modulus for the two-dimensional XY-model is inferred by aid of the Coulomb gas-vortex analogy and an explicit form of this dependence is obtained from a set of self-consistent equations for the Coulomb gas. High precision Monte Carlo simulations in conjunction with this inferred lattice size dependence is used to determine the critical temperature. Some other recent results in the literature are discussed in the light of the present investigation.

Numerical study of the phase structure of compact QED with dynamical fermions

We report numerical results of a simulation of compact QED with dynamical fermions which show a peak in the correlation length. The lattice size dependence of our results suggests strongly a crossover phenomenon against a second order phase transition.

Finite-size-scaling estimation of critical exponents in the two-dimensional axial next-nearest-neighbor Ising model.

The magnetic critical exponent of the two-dimensional axial next-nearest-neighbor Ising model is estimated by means of the finite-size scaling of the order parameter. For a finite system without ordering field, this magnitude is computed from the two leading eigenvectors of the transfer matrix. The ferromagnetic-paramagnetic transition temperatures estimated with this method exhibit a small lattice-size dependence. Moreover, they are in good agreement with those obtained using phenomenological renormalization, and also with analytical approximations. The correlation-length critical exponent is calculated by means of phenomenological renormalization. The results found with these numerical techniques indicate that the ferromagnetic-paramagnetic transition of the model seems to be continuous, in contradiction with a previous result obtained using the persistence-length criterion, which indicated that the transition is always first order. We found that the phase transition is first order only when the ratio of the next-nearest-neighbor and nearest-neighbor interactions equals one-half.

Finite-size effects for lattice glueball masses.

We have measured the lattice size dependence of the lightest glueball mass in SU(2) and SU(3) gauge theory at \ensuremath{\beta}=2.2 and 2.3 [SU(2)] and \ensuremath{\beta}=6.0 [SU(3)]. We observe an increase in the glueball mass when the lattice size shrinks below a critical value. The magnitude of this length scale, which we interpret as the physical size of the glueball, is roughly the same for SU(2) and SU(3) when expressed in units of the string tension.