Monte Carlo simulations for two-dimensional quantum systems

The Trotter-Suzuki transformation has been used to obtain the classical representation ford-dimensional lattice systems with boson and fermion degrees of freedom. A Monte Carlo algorithm for the equivalent (d+1)-dimensional classical system is presented. Numerical results are shown for the Heisenberg-spin-glass, the XY model and the spinless fermion lattice gas in two dimensions.     

(3 1)-Dimensional Quantum Mechanics from Monte Carlo Hamiltonian: Harmonic Oscillator

In Lagrangian formulation, it is extremely difficult to compute the excited spectrum and wavefunctions ora quantum theory via Monte Carlo methods. Recently, we developed a Monte Carlo Hamiltonian method for investigating this hard problem and tested the algorithm in quantum-mechanical systems in 1 1 and 2t1 dimensions. In this paper we apply it to the study of thelow-energy quantum physics of the (3 1)-dimensional harmonic oscillator.``     

Quantum statistical monte carlo methods and applications to spin systems

A short review is given concerning the quantum statistical Monte Carlo method based on the equivalence theorem⁽¹⁾ thatd-dimensional quantum systems are mapped onto (d+1)-dimensional classical systems. The convergence property of this approximate tansformation is discussed in detail. Some applications of this geneal appoach to quantum spin systems are reviewed. A new Monte Carlo method, “thermo field Monte Carlo method,” is presented, which is an extension of the projection Monte Carlo method at zero temperature to that at finite temperatures. Invited talk presented at “Frontiers of Quantum Monte Carlo,” Los Alamos National Laboratory, September 3–6, 1985.     

The largest Liapunov exponent for random matrices and directed polymers in a random environment

We study the largest Liapunov exponent for products of random matrices. The two classes of matrices considered are discrete,d-dimensional Laplacians, with random entries, and symplectic matrices that arise in the study ofd-dimensional lattices of coupled, nonlinear oscillators. We derive bounds on this exponent for all dimensions,d, and we show that ifd3, and the randomness is not too strong, one can obtain an explicit formula for the largest exponent in the thermodynamic limit. Our method is based on an equivalence between this problem and the problem of directed polymers in a random environment.     

The absence of phase transition for the classical XY-model on Sierpiński pyramid with fractal dimension D=2

For the spin models with continuous symmetry on regular lattices and finite range of interactions, the lower critical dimension is d?=?2. In two dimensions the classical XY-model displays Berezinskii–Kosterlitz–Thouless (BKT) transition associated with unbinding of topological defects (vortices and antivortices). We perform a Monte Carlo study of the classical XY-model on Sierpiński pyramids (SPs) whose fractal dimension is D = log?4/log?2?=?2 and the average coordination number per site is ≈ 7. The specific heat does not depend on the system size which indicates the absence of a long-range order. From the dependence of the helicity modulus on the cluster size and on boundary conditions, we draw a conclusion that in the thermodynamic limit there is no BKT transition at any finite temperature. This conclusion is also supported by our results for linear magnetic susceptibility. The lack of finite temperature phase transition is presumably caused by the finite order of ramification of SP.     

Monte Carlo generation of self-avoiding walks with fixed endpoints and fixed length

We propose a new class of dynamic Monte Carlo algorithms for generating self-avoiding walks uniformly from the ensemble with fixed endpoints and fixed length in any dimension, and prove that these algorithms are ergodic in all cases. We also prove the ergodicity of a variant of the pivot algorithm.     

Series and Monte Carlo study of high-dimensional Ising models

Ising models in high dimensions are used to compare high-temperature series expansions with Monte Carlo simulations. Simulations of the magnetization on four-, six-, and seven-dimensional hypercubic lattices give consistent values of the critical temperature from both equilibrium and nonequilibrium data ford=6 and 7. We tabulate 15 terms of series expansions for the susceptibility for generald and giveJ/k _B T _c =0.092295 (3) and 0.077706 (2) ford=6 and 7. In contrast to five dimensions, where earlier series found nonanalytic scaling corrections, for d=6 and 7 the leading scaling correction may be analytic inT-T _c . In most cases these expansions gave more accurate results than these simulations.     

Analysis of the Migdal transformation for models with planar spins on a d-dimensional lattice

We show analytically that in the Migdal approximation and at low temperatures planar spins on a d-dimensional lattice (d ⩾ 2) exhibit the same phase transitions as d-dimensional Z(n)-models. For a three-dimensional lattice part of the flow diagram of the Migdal mapping is found numerically, including large families of relevant fixed points which separate the low-temperature fixed points from the trivial high-temperature one. The critical exponents are obtained and for some two-parameter potentials the phase diagram is constructed and explained. After rescaling, quantitative agreement was obtained for one of the Migdal phase diagrams with the Monte Carlo phase diagram.     

Phase Transition and Critical Behavior for XY Model on 2-Dimensional Random Triangle Lattice

The Monte Carlo renormalization group method is applied to discussing the nature of phase transition of XY model on 2-dimensional random triangle lattices. A line of fixed point and un-universal phase transition are found. The results are in agreement with Kosterlitz-Thouless theory. The susceptibility ehows a clear size-dependent behaviorin low temperature region. This means that it should be divergent in this region.     

Nontrivial critical behaviour in a lattice model of random surfaces

A model of “planar random surfaces without spikes” on hypercubical lattices was introduced some years ago as a discretization of quantum string theory. We review some general properties of this model and present results from a Monte Carlo study of its critical behaviour in d = 4, 8 and 10 dimensions. In d = 4 dimensions we find a Hausdorff dimension d_H ≈ 4 and an anomalous dimensions η ≈ 1. These critical exponents imply a deviation from mean field theory in contrast to other lattice random surface models. Furthermore, we find evidence for mean field behaviour in 8 and 10 dimensions, indicating an upper critical dimension d_c^u ⩽ 8.     

Cluster algorithms for anisotropic quantum spin models

We present cluster Monte Carlo algorithms for theXYZ quantum spin models. In the special case ofS=1/2, the new algorithm can be viewed as a cluster algorithm for the 8-vertex model. As an example, we study theS=1/2XY model in two dimensions with a representation in which the quantization axis lies in the easy plane. We find that the numerical autocorrelation time for the cluster algorithm remains of the order of unity and does not show any significant dependence on the temperature, the system size, or the Trotter number. On the other hand, the autocorrelation time for the conventional algorithm strongly depends on these parameters and can be very large. The use of improved estimators for thermodynamic averages further enhances the efficiency of the new algorithms.     

Entropy of random coverings and 4D quantum gravity

We discuss the counting of minimal geodesic ball coverings of n-dimensional (n ≥ 3) riemannian manifolds of bounded geometry, fixed Euler characteristic, and Reidemeister torsion in a given representation of the fundamental group. This counting bears relevance to the analysis of the continuum limit of discrete models of quantum gravity. We establish the conditions under which the number of coverings grows exponentially with the volume, thus allowing for the search of a continuum limit of the corresponding discretized models. The resulting entropy estimates depend on representations of the fundamental group of the manifold through the corresponding Reidemeister torsion. We discuss the sum over inequivalent representations both in the two-dimensional and in the four-dimensional case. Explicit entropy functions as well as significant bounds on the associated critical exponents are obtained in both cases.     

Exact entropy of dimer coverings for a class of lattices in three or more dimensions

We construct a class of lattices in three and higher dimensions for which the number of dimer coverings can be determined exactly using elementary arguments. These lattices are a generalization of the two-dimensional kagome lattice, and the method also works for graphs without translational symmetry. The partition function for dimer coverings on these lattices can be determined also for a class of assignments of different activities to different edges.     

Percolation thresholds and percolation conductivities of octagonal and dodecagonal quasicrystalline lattices

The octagonal and dodecagonal quaislattices were generated by means of the grid method. Monte Carlo simulation and cluster counting procedure were used for numerical determination of the site and bond percolation thresholds. Two types of connectivity called ferromagnetic and chemical were studied. The estimated site percolation thresholds are 0.5435… and 0.585… for octagonal lattice and 0.617… and 0.628… for dodecagonal lattice respectively. The obtained spanning fraction curves (for site percolation) seem to approach the 50% value at the percolation threshold. The site percolation conductivity for these lattices was studied by means of a transfer-matrix approach. The critical behavior was found to be the same as for the periodic lattices.     

Study of transport properties on percolating clusters growth upon very hot dimer deposition

Transport properties are investigated by means of Monte Carlo simulations on percolating clusters obtained depositing very hot dimers in two dimensions. The properties are discussed within the framework of the random walks arguments. The diffusion exponent d_w and the spectral dimension d_s exhibit finite size effects and their asymptotic values in the thermodynamic limit differ from both the corresponding to standard percolation and Euclidean lattices.     

High-temperature series analysis of the p-state Potts glass model on d-dimensional hypercubic lattices

We analyze recently extended high-temperature series expansions for the “Edwards-Anderson” spin-glass susceptibility of the p-state Potts glass model on d-dimensional hypercubic lattices for the case of a symmetric bimodal distribution of ferro- and antiferromagnetic nearest-neighbor couplings . In these star-graph expansions up to order 22 in the inverse temperature , the number of Potts states p and the dimension d are kept as free parameters which can take any value. By applying several series analysis techniques to the new series expansions, this enabled us to determine the critical coupling K_c and the critical exponent of the spin-glass susceptibility in a large region of the two-dimensional (p,d)-parameter space. We discuss the thus obtained information with emphasis on the lower and upper critical dimensions of the model and present a careful comparison with previous estimates for special values of p and d. Received: 25 May 1998 / Revised and Accepted: 11 August 1998     

An analytical study of the Curie temperature in the D-vector model

The successive analytical expressions up to fifth order for the Curie temperature, T_c(D,d), of the D-vector model on d-dimensional hypercubic lattices are evaluated by the Ginzburg-Landau theory based on cumulant expansion. The results by substituting d = 3 and D = 1, 2, 3 into the expressions approach order by order those of the Ising, XY, and Heisenberg models by Monte Carlo simulation (MC). The differences between ours and the other models are about 5.52, 6.86, and 7.94 percent, respectively.     

Spin-glass transition of the three-dimensional Heisenberg spin glass

It is shown, by means of Monte Carlo simulation and finite size scaling analysis, that the Heisenberg spin glass undergoes a finite-temperature phase transition in three dimensions. There is a single critical temperature, at which both a spin glass and a chiral glass ordering develop. The Monte Carlo algorithm, adapted from lattice gauge theory simulations, makes it possible to thermalize lattices of size L = 32, larger than in any previous spin-glass simulation in three dimensions. High accuracy is reached thanks to the use of the Marenostrum supercomputer. The large range of system sizes studied allows us to consider scaling corrections.     

A mean spherical model with coulomb interactions. II. Correlations at a free surface

Studies of the mean spherical model with Coulomb interactions are continued, by considering a system on ad-dimensional lattice which is periodic ind–1 dimensions and has a free surface in the remaining dimension. It is shown explicitly that correlations along the free surface decay asy ^–d ind dimensions and show that the surface properties of this model are those expected for a charged system in its plasma phase.