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Locating quantum critical points through singularities of simple observables

Mateusz Łącki, Bogdan Damski, Jakub Zakrzewski

Published May 13, 2016• 0 citations
OpenAlex

Abstract

Quantum critical points are traditionally associated with non-analyticity of the ground state energy reflecting fundamental differences between phases of a system undergoing a quantum phase transition. We discuss how this non-analyticity is passed onto ground state expectation values of different terms of the Hamiltonian. We illustrate our results in the two-dimensional Bose-Hubbard model that can be realized in cold atomic systems. Using quantum Monte Carlo simulations, we show that one can easily extract the position of the critical point in this model from the on-site atom number fluctuations and nearest-neighbor tunnelling. Our results provide a simple experimentally-relevant way of locating critical points in numerous systems undergoing a quantum phase transition.

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Quantum critical points are traditionally associated with non-analyticity of the ground state energy reflecting fundamental differences between phases of a system undergoing a quantum phase transition. We discuss how this non-analyticity is passed onto ground state expectation values of different terms of the Hamiltonian. We illustrate our results in the two-dimensional Bose-Hubbard model that can be realized in cold atomic systems. Using quantum Monte Carlo simulations, we show that one can easily extract the position of the critical point in this model from the on-site atom number fluctuations and nearest-neighbor tunnelling. Our results provide a simple experimentally-relevant way of locating critical points in numerous systems undergoing a quantum phase transition.
We show that the critical point of the two-dimensional Bose-Hubbard model can be easily found through studies of either on-site atom number fluctuations or the nearest-neighbor two-point correlation function (the expectation value of the tunnelling operator). Our strategy to locate the critical point is based on the observation that the derivatives of these observables with respect to the parameter that drives the superfluid-Mott insulator transition are singular at the critical point in the thermodynamic limit. Performing the quantum Monte Carlo simulations of the two-dimensional Bose-Hubbard model, we show that this technique leads to the accurate determination of the position of its critical point. Our results can be easily extended to the three-dimensional Bose-Hubbard model and different Hubbard-like models. They provide a simple experimentally-relevant way of locating critical points in various cold atomic lattice systems.
We discuss how positions of critical points of the three-dimensional Bose-Hubbard model can be accurately obtained from variance of the on-site atom number operator, which can be experimentally measured. The idea that we explore is that the derivative of the variance, with respect to the parameter driving the transition, has a pronounced maximum close to critical points. We show that Quantum Monte Carlo studies of this maximum lead to precise determination of critical points for the superfluid-Mott insulator transition in systems with mean number of atoms per lattice site equal to one, two, and three. We also extract from such data the correlation-length critical exponent through the finite-size scaling analysis and discuss how the derivative of the variance can be reliably computed from numerical data for the variance. The same conclusions apply to the derivative of the nearest-neighbor correlation function, which can be obtained from routinely measured time-of-flight images.
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.
Exactly solvable models provide a unique method, via qualitative changes in the distribution of the ground-state roots of the Bethe ansatz equations, to identify quantum phase transitions. Here we expand on this approach, in a quantitative manner, for two models of Bose–Einstein condensates. The first model deals with the interconversion of bosonic atoms and molecules. The second is the two-site Bose–Hubbard model, widely used to describe tunneling phenomena in Bose–Einstein condensates. For these systems we calculate the ground-state root density. This facilitates the determination of analytic forms for the ground-state energy, and associated correlation functions through the Hellmann–Feynman theorem. These calculations provide a clear identification of the quantum phase transition in each model. For the first model we obtain an expression for the molecular fraction expectation value. For the two-site Bose–Hubbard model we find that there is a simple characterization of condensate fragmentation.
We study the one-dimensional Bose-Hubbard model describing the superfluid-Mott insulator quantum phase transition of cold atoms in optical lattices. We show that derivatives of the variance of the on-site atom number occupation, computed with respect to the parameter driving the transition, have extrema that are located off the critical point even in the thermodynamic limit. We discuss whether such extrema provide solid evidence of the quantum Berezinskii-Kosterlitz-Thouless transition taking place in this system. The calculations are done for systems with the mean number of atoms per lattice site equal to either one or two. They also characterize the nearest-neighbor correlation function, which is typically discussed in the context of time-of-flight images of cold atoms.
We study the one-dimensional Bose-Hubbard model describing the superfluid-Mott insulator quantum phase transition of cold atoms in optical lattices. We show that derivatives of the variance of the on-site atom number occupation, computed with respect to the parameter driving the transition, have extrema that are located off the critical point even in the thermodynamic limit. We discuss whether such extrema provide solid evidence of the quantum Berezinskii-Kosterlitz-Thouless transition taking place in this system. The calculations are done for systems with the mean number of atoms per lattice site equal to either one or two. They also characterize the nearest-neighbor correlation function, which is typically discussed in the context of time-of-flight images of cold atoms.