Subtlety of Studying the Critical Theory of a Second Order Phase Transition

TL;DR

This study investigates the quantum phase transition in the extended Bose-Hubbard model on a honeycomb lattice, highlighting the subtlety in accurately determining the critical point and the influence of the dynamical critical exponent z on critical exponent nu.

Contribution

It demonstrates the importance of the inverse temperature parameter beta in Monte Carlo simulations and shows how assumptions about z affect the determination of critical exponents and phase transition points.

Findings

Transition belongs to superfluid-insulator universality class.

Critical point location varies with simulation parameters.

Assuming z=2 yields consistent critical exponents.

Abstract

We study the quantum phase transition from a super solid phase to a solid phase of rho = 1/2 for the extended Bose-Hubbard model on the honeycomb lattice using first principles Monte Carlo calculations. The motivation of our study is to quantitatively understand the impact of theoretical input, in particular the dynamical critical exponent z, in calculating the critical exponent nu. Hence we have carried out four sets of simulations with beta = 2N^{1/2}, beta = 8N^{1/2}, beta = N/2, and beta = N/4, respectively. Here beta is the inverse temperature and N is the numbers of lattice sites used in the simulations. By applying data collapse to the observable superfluid density rho_{s2} in the second spatial direction, we confirm that the transition is indeed governed by the superfluid-insulator universality class. However we find it is subtle to determine the precise location of the critical point. For example, while the critical chemical potential (mu/V)_c occurs at (mu/V)_c = 2.3239(3) for the data obtained using beta = 2N^{1/2}, the (mu/V)_c determined from the data simulated with beta = N/2 is found to be (mu/V)_c = 2.3186(2). Further, while a good data collapse for rho_{s2}N can be obtained with the data determined using beta = N/4 in the simulations, a reasonable quality of data collapse for the same observable calculated from another set of simulations with beta = 8N^{1/2} can hardly be reached. Surprisingly, assuming z for this phase transition is determined to be 2 first in a Monte Carlo calculation, then a high quality data collapse for rho_{s2}N can be achieved for (mu/V)_c ~ 2.3184 and nu ~ 0.7 using the data obtained with beta = 8N^{1/2}. Our results imply that one might need to reconsider the established phase diagrams of some models if the accurate location of the critical point is crucial in obtaining a conclusion.