Superfluid–insulator transition in weakly interacting disordered Bose gases: a kernel polynomial approach

An iterative scheme based on the kernel polynomial method is devised for the efficient computation of the one-body density matrix of weakly interacting Bose gases within Bogoliubov theory. This scheme is used to analyze the coherence properties of disordered bosons in one and two dimensions. In the one-dimensional geometry, we examine the quantum phase transition between superfluid and Bose glass at weak interactions, and we recover the scaling of the phase boundary that was characterized using a direct spectral approach by Fontanesi et al (2010 Phys. Rev. A 81 053603). The kernel polynomial scheme is also used to study the disorder-induced condensate depletion in the two-dimensional geometry. Our approach paves the way for an analysis of coherence properties of Bose gases across the superfluid-insulator transition in two and three dimensions.

Quantum Phase Transition in a Low-Dimensional Weakly Interacting Bose Gas

Luca Fontanesi

This thesis is devoted to the study of the effect of disorder on low-dimensional weakly interacting Bose gases. In particular, the disorder triggers a quantum phase transition in one dimension at zero temperature that is investigated here through the study of the long-range behaviour of the one-body density matrix. An algebraic spatial decay of the coherence marks the quasicondensate, whereas, in the case of strong disorder, an exponential decay is recovered and it characterizes the insulating Bose-glass phase. This analysis is performed using an extended Bogoliubov theory to treat low dimensional Bose gases within a density-phase approach. A systematic numerical study allowed to draw the phase diagram of 1D weakly interacting bosons. The phase boundary obeys two different power laws between interaction and disorder strength depending on the regime of the gas where the transition occurs. These relations can be explained by means of scaling arguments valid in the white noise limit and in the Thomas-Fermi regime of the Bose gas. The phase transition to a quasicondensed phase comes along with the onset of superfluidity: the inspection of the superfluid fraction of the gas is consistent with these predictions for the boundary. The finite temperature case and the scenario in two dimensions are briefly discussed. The quantum phase transition is caused by low-energy phase fluctuations that destroy the quasi-long-range order characterizing the uniform system. Within the approach presented here, the phase fluctuations are identified as the low-lying Bogoliubov modes. Their properties have been investigated in detail to understand which changes trigger the phase transition and we found that the transition to the insulating phase is accompanied by a diverging density of states and a localization length, measured through the inverse participation ratio, that diverges as a power-law with power – 1 for vanishing energy. The fragmentation of the gas is also studied: this notion is very often associated with the onset of the insulating phase. The characterization of the density fragmentation is performed by analyzing the probability distribution of the density. A density profile is defined as fragmented when the probability distribution at vanishing density is finite or divergent and this happens for a gas in the Bose-glass phase. On the contrary, the superfluid phase is characterized by a zero limiting probability of having vanishing densities. This definition is derived analytically, and confirmed by a numerical study. This fragmentation criterion is particularly suited for detecting the phase transition in experiments: when a harmonic trap is included, the transition to the insulating phase can be extracted from the statistics of the local density distribution.

EPFL2011

Quantum phase transitions in low-dimensional weakly interacting disordered Bose gases

Joseph Saliba

In this thesis, we study the quantum phase transition triggered by an external random po- tential in ultra-cold low-dimensional weakly-interacting Bose gases at zero temperature. In one-dimensional systems, the quantum phases are characterized by the decay at long-range of the one-body density matrix. This decay exhibits an algebraic behaviour in the superfluid quasi-condensed phase while in the insulating Bose glass phase this it becomes exponen- tial, signaling the complete loss of coherence in the system. In two-dimensional systems, a characterization based on the long-range behaviour of the one-body density matrix appears to be highly demanding from a computational point of view. We therefore characterized the superfluid-insulator transition in two-dimensional systems by the low-energy behaviour of the cumulative density of states of the Bogoliubov excitations. While in the superfluid condensed phase this quantity grows as the square of the excitation energy in agreement with the existence of a finite velocity of sound, in the insulating Bose glass phase this power law is less than quadratic. This study is performed in the framework of an extended Bogoliubov approach properly adapted to treat low-dimensional Bose gases. Using a systematic numerical study, we draw the interaction-disorder phase diagrams of the superfluid-insulator transition in 1D and 2D. The phase boundary follows two different power- laws depending on the length scales characterizing the spatial correlations of the disorder and the strength of interactions. The power-law exponents were found to be in agreement with the ones predicted by scaling arguments, both in the white noise and the Thomas-Fermi regimes. In the two-dimensional system, the classical percolation threshold in the Thomas-Fermi limit was found to overestimate the critical disorder below which the onset of superfluidity should be observed for a given interaction strength.

EPFL2016

Superfluid to Bose-Glass Transition in a 1D Weakly Interacting Bose Gas

Luca Fontanesi, Vincenzo Savona

We study the one-dimensional Bose gas in spatially correlated disorder at zero temperature, using an extended density-phase Bogoliubov method. We analyze, in particular, the decay of the one-body density matrix and the behavior of the Bogoliubov excitations across the phase boundary. We observe that the transition to the Bose-glass phase is marked by a power-law divergence of the density of states at low energy. A measure of the localization length displays a power-law energy dependence in both regions, with the exponent equal to -1 at the boundary. We draw the phase diagram of the superfluid-insulator transition in the limit of small interaction strength.