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Acta Physica Sinica

Simulation of two-dimensional many-particle hardcore bosons by using the quantum Monte Carlo method

Ying Xu, Jinbin Li

Published Jan 1, 2012• 3 citations
Publisher

Abstract

In this paper, the stochastic series expansion quantum Monte Carlo method is employed to investigate the thermodynamic properties of hardcore Bose-Hubbard model in two-dimensional space. The two-dimensional hardcore Bose-Hubbard model can be mapped into the two-dimensional antiferromagnetic quasi-Heisenberg model under transform of bosonic operators. There is an additional term which is proportional to the total number of sites compared with real Heisenberg model and it is difficult for simulation. Using a nonlocal operator-loop update, it allows one to simulate thousands of sites. Our simulation results show that, first, energy decreases with the increase of density of particles in a range from 0 to 0.5, and finally approaches to a fixed value. Moreover, with the size of square lattice increasing, energy also increases. Second, when we fix the system size, energy and magnetization increase with temperature, but not with of chemical potential. When we increase the system size, energy increases, while, the magnetization decreases. Third, specific heat is independent of chemical potential, but it dramatically increases with temperature and approaches to a peak, then decreases slowly. According to Landau theory of superfluidity, the tends of curve for energy and specific heat fit the research of He II in the Landau two-fluid model. Fourth, different square lattice linear system sizes have a little influence on tiny differences to the reciprocal of uniform susceptibility. There are small fluctuations in a range from 0 to 0.5(J/kB), where J is the coupling energy, kB is the Boltzmann constant, but the reciprocal of uniform susceptibility increases with temperature increasing in a range from 0.5 to 2(J/kB). The tends of curve are similar to those of Kondo effect.

DOI
10.7498/aps.61.110207

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In this paper, the stochastic series expansion quantum Monte Carlo method is employed to investigate the thermodynamic properties of hardcore Bose-Hubbard model in two-dimensional space. The two-dimensional hardcore Bose-Hubbard model can be mapped into the two-dimensional antiferromagnetic quasi-Heisenberg model under transform of bosonic operators. There is an additional term which is proportional to the total number of sites compared with real Heisenberg model and it is difficult for simulation. Using a nonlocal operator-loop update, it allows one to simulate thousands of sites. Our simulation results show that, first, energy decreases with the increase of density of particles in a range from 0 to 0.5, and finally approaches to a fixed value. Moreover, with the size of square lattice increasing, energy also increases. Second, when we fix the system size, energy and magnetization increase with temperature, but not with of chemical potential. When we increase the system size, energy increases, while, the magnetization decreases. Third, specific heat is independent of chemical potential, but it dramatically increases with temperature and approaches to a peak, then decreases slowly. According to Landau theory of superfluidity, the tends of curve for energy and specific heat fit the research of He II in the Landau two-fluid model. Fourth, different square lattice linear system sizes have a little influence on tiny differences to the reciprocal of uniform susceptibility. There are small fluctuations in a range from 0 to 0.5(J/kB), where J is the coupling energy, kB is the Boltzmann constant, but the reciprocal of uniform susceptibility increases with temperature increasing in a range from 0.5 to 2(J/kB). The tends of curve are similar to those of Kondo effect.
The Antiferromagnetic Heisenberg Model with spin 1/2 on a square lattice was simulated by using the Stochastic Series Expansion(SSE) Quantum Monte Carlo Method.The temperature dependence of the internal energy,specific heat and uniform susceptibility was observed.The results showed that specific heat has a maximum at kT/J=0.6 and uniform susceptibility saturates at kT/J=1 for the isotropy case.And the size of lattice also affects the temperature dependence of these thermodynamic quantities.For the anisotropy case,energy decreases with anisotropic parameter g increasing.Susceptibility changes slowly with temperature for g1 and exponentially decay for g1 in the low temperature region and its behaviors approach consistent for different g in the high temperature region.
The hardcore-Bose-Hubbard model with random chemical potential is investigated using quantum Monte Carlo simulation. We consider two cases of random distribution of the chemical potential: a uniformly random distribution and a correlated distribution. The temperature dependences of the superfluid density, the specific heat, and the correlation functions are calculated. If the distribution of the randomness is correlated, there exists an intermediate state, which can be thought of as a localized condensate state of bosons, between the superfluid state and the normal state.
Spin-1/2 nearest neighbor Heisenberg antiferromagnet and XY model on a square lattice are studied via large scale quantum Monte Carlo simulations using a fast and efficient multispin coding algorithm on the Caltech/JPL MarkIIIfp parallel supercomputer, based on the Suzuki-Trotter transformation. We performed simulations with very good statistics on lattices as large as 128x128 spins, in the temperature range from 0.1 to 2.5 in units of the effective exchange coupling J. We calculated energy, specific heat, magnetic susceptibilities and also spin correlation functions from which we deduce the correlation lengths. For the Heisenberg model, at temperatures higher than J the results are in excellent agreement with high-temperature series expansion. At low temperatures the long wavelength behavior is essentially classical. Our data show that the correlation length and staggered susceptibility are quantitatively well described by the renormalized classical picture at the 2-loop level of approximation. From the divergence of the correlation length, we deduce the value of the quantum renormalized spin stiffness, [rho][subscript s]/J = 0.199(2). We give evidence that the correlation function is of the Ornstein-Zernicke type. By comparing the largest measured correlation lengths with neutron scattering experiments on La2CuO4, we deduce the value of effective exchange coupling J = 1450±30 K. By measuring the imaginary time-dependent correlation functions, we show that the dynamics of the model can be well understood within a Bose liquid-type picture. The spin waves are rather sharp throughout most of the Brillouin zone and the damping is weakly dependent on the wave vector. In the case of the XY model, convincing numerical evidence is obtained on square lattices as large as 96x96 that the spin-1/2 XY model undergoes a Kosterlitz-Thouless (KT) phase transition at kTc/J = 0.350(4). The correlation length and in-plane susceptibility diverge at Tc precisely according to the form predicted by Kosterlitz and Thouless for the classical XY model. The specific heat increases very rapidly on heating near Tc and exhibits a peak around kT/J = 0.45. We also measure the spin stiffness and the correlation function exponent below the transition temperature. Within the statistical accuracy of the measurements, the results are well described by the square root singularity (with a nonuniversal amplitude) below Tc, and they have the universal values in agreement with KT theory at Tc.
We study the effect of disorder on hardcore bosons in two dimensions at the SU(2) symmetric ``Heisenberg point''. We obtain our results with quantum Monte Carlo simulations using the directed loop algorithm. In the absence of disorder, the system has no long-range order at finite temperature due to the enhanced symmetry. However, the introduction of a disordered potential, uniformly distributed from -D to D, induces a finite-temperature superfluid phase. In particular the diagonal correlation length ξdecreases but the superfluid order-parameter correlation function becomes a power-law. A non-monotonic finite-size behavior is noted and explained as arising due to ξ. We provide evidence that at long distances the effects of a weak disordered potential can be mimicked by an effective uniform potential with a root-mean-square value: mu_eff = D/sqrt{3}. For strong disorder, the system becomes a Bose glass insulator.
The Quantum Monte-Carlo simulations of the two-dimensional Hubbard model are presented for the half filling. The method based on the direct-space proposed by Suzuki and al., and Hirsch and al. was used. The states generated by this method are basis states in occupation number representation built with Wannier states localised on each site of the square array. The configurations of fermions can be observed on the real 2D array. An antiferromagnetic factor is defined and calculated for each temperature. The curves of energy, specific heat, conducivity and antiferromagnetic factor are presented for different values of the repulsive coulombian on site interaction U. There is a metal-insulator transition at low temperature for small values of U. This transition corresponds with a paramagnetic-ferromagnetic first order transition. Indeed, for these interaction values, the energy curves show a gap which is a characteristic of a first order transition. An hysteresis phenomenon appears on the conductivity curves. There is a behaviour change for U/t=3.5. For the values U>3.5 there is ferromagnetic-paramagnetic change without observable effect on the energy and the specific heat. The metal-insulator transition does not exist any more, the conductivity stays very small. Isotherms of the physical quantities versus U/t show a transition which seems to be the metal-insulator Mott transition. These results allow to draw a phase diagram with two first order transition lines.
The Quantum Monte Carlo simulation of the two-dimensional Emery model of the CuO2 plane of hight Tc superconductors were performed. The method based on the direct-space proposed by Suzuki and Hirsch was used. Contrary to the method based on the Hubbard-Stratonovich transformation, the states generated by this method are basis states in occupation number representation, i. e. configurations of fermions can be observed on real two-dimensional array.Energy and specific heat were computed for different dopings. Specific heat curves show peaks at low temperature which could be assigned to electronic transitions. Quantity similar to current-current correlation function were computed. The static electric conductivity curves obtained by this way show metal-insulator transitions and two different metallic behaviours. On the direct-space states generated at low temperature and zero doping, the fermions form antiferromagnetic loops while they form antiferromagnetic chains for other dopings. The loops seem to appear when the conductivity becomes zero while yhe conductivity increases with the numbers of chains but superconductivity is not unambiguously evident.