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Physical review. B, Condensed matter

First- and second-order transitions in the quantum hard-rod system by Monte Carlo simulation

F. Mallezie

Published Mar 1, 1990• 5 citations
Publisher

Abstract

A one-dimensional quantum Ising model with near-neighbor exclusion (``hard rods'') is mapped onto an anisotropic two-dimensional classical model using the Trotter formula. The critical properties of this classical equivalent model are studied by Monte Carlo techniques and finite-size scaling. A first-order transition is detected if the length of the rods is larger than three, while for dimers a second-order Ising-like transition is found. The type of transition in the borderline case for trimers is not completely settled: It may be second order or very weakly first order, with our data slightly in favor of a first-order transition.

DOI
10.1103/physrevb.41.4475

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A one-dimensional quantum Ising model with near-neighbor exclusion (``hard rods'') is mapped onto an anisotropic two-dimensional classical model using the Trotter formula. The critical properties of this classical equivalent model are studied by Monte Carlo techniques and finite-size scaling. A first-order transition is detected if the length of the rods is larger than three, while for dimers a second-order Ising-like transition is found. The type of transition in the borderline case for trimers is not completely settled: It may be second order or very weakly first order, with our data slightly in favor of a first-order transition.
The critical properties of one-dimensional quantum Ising models with near-neighbor exclusion (hard rods) are studied by finite-size scaling and conformal invariance. For dimers (rods with length m=2) the system exhibits an Ising-type critical point, while for m\ensuremath{\ge}3 the system undergoes a first-order phase transition. The operator content of the critical Hamiltonian of hard dimers has been determined for different boundary conditions and identification with the sectors of the Ising model has been done. Using our results, we propose a phase diagram for the quantum Ising model with multispin interaction in the presence of transverse and longitudinal fields.
The generalized Trotter formula is used to map the two-dimensional spin-l/2 XY model onto a three-dimensional lsing model with complicated many-spin interactions. This relationship sused to construct an efficient Monte Carlo algorithm. Simulation data for the specific heat and vortex correlation functions give strong evidence for the existence of a phase transition. Although it is well known that there cannot exist long-range order in the two-dimensional (2D) spin-1/2 XY model at non-zero temperature [1], high-temper-ature series expansions [2] suggest he existence of critical behavior. Little is known about the quantum model mainly because most techniques which have been succesfully applied to the classical model (also referred to as planar rotator model) are very difficult to extend to the quantum case. In particular, quantum renormalization group analYsis of the 2D spin-1/2 XY model have given either contradictive or unreliable re-suits [3-8]. An alternative method to calculate the thermody-namic properties of this modelis to exploit the formal analogy between d-dimensional quantum spin systems and (d + 1)-dimensional Ising-spin models [9]. Using ~ ' This paper was first submitted to Physical Review Letters
We propose a new efficient scheme for the quantum Monte Carlo study of quantum critical phenomena in quantum spin systems. Rieger and Young's Trotter-number-dependent finite-size scaling in quantum spin systems and Ito et al 's evaluation of the transition point with the non-equilibrium relaxation in classical spin systems are combined and generalized. That is, only one Trotter number and one inverse temperature proportional to system sizes are taken for each system size, and the transition point of the transformed classical spin model is estimated as the point at which the order parameter shows power-law decay. The present scheme is confirmed by the determination of the critical phenomenon of the one-dimensional asymmetric XY model in the ground state. The estimates of the transition point and the critical exponents , and are in good agreement with the exact solutions. The dynamical critical exponent is also estimated as , which is consistent with that of the two-dimensional Ising model.
The scaling of the transition temperature into an ordered phase close to a quantum critical point as well as the order parameter fluctuations inside the quantum critical region provide valuable information about universal properties of the underlying quantum critical point. Here, we employ quantum Monte Carlo simulations to examine these relations in detail for two-dimensional quantum systems that exhibit a finite-temperature Ising-transition line in the vicinity of a quantum critical point that belongs to the universality class of either (i) the three-dimensional Ising model for the case of the quantum Ising model in a transverse magnetic field on the square lattice or (ii) the chiral Ising transition for the case of a half-filled system of spinless fermions on the honeycomb lattice with nearest-neighbor repulsion. While the first case allows large-scale simulations to assess the scaling predictions to a high precision in terms of the known values for the critical exponents at the quantum critical point, for the later case, we extract values of the critical exponents $\ensuremath{\nu}$ and $\ensuremath{\eta}$, related to the order parameter fluctuations, which we discuss in relation to other recent estimates from ground-state quantum Monte Carlo calculations as well as analytical approaches.
We formulate a cluster Monte Carlo method for the anisotropic limit of Ising models on (d+1)-dimensional lattices, which in effect, are equivalent with d-dimensional quantum transverse Ising models. Using this technique, we investigate the transverse Ising models on the square, triangular, Kagome, honeycomb, and simple-cubic lattices. The Monte Carlo data are analyzed by finite-size scaling. In each case we find, as expected, that the critical behavior fits well in the (d+1)-dimensional Ising universality class. For the transverse Ising model on the square lattice, we determine the Binder cumulant of the classical counterpart for a range of aspect ratios between the system sizes in the third or "classical" direction and that in the other two directions. Matching this universal function with the case of the isotropic Ising model yields the length ratio relating the isotropic Ising model with the anisotropic limit. The efficiency of the present algorithm is reflected by the precision of its results, which improves significantly on earlier analyses.
We use Monte Carlo, transfer-matrix and finite-size scaling methods to investigate two-dimensional O (n) models with n > 2, in particular the case n = 3 which includes the classical Heisenberg model. Depending on the type of interaction and the lattice structure, two different types of phase transitions are present. One type resembles the hard-hexagon transition and occurs in the loop representation of the honeycomb O (n) model. The other type is a first-order transition which occurs for spin-spin interactions that are strongly nonlinear in the neighbor-spin products. When the nonlinearity is decreased, the first-order line ends in a critical point. The existence of the first-order line is in agreement with mean-field theory as well as with high- and low-temperature approximations.