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arXiv:1512.04238 · Physical Review A

Finite-temperature effects on interacting bosonic one-dimensional systems in disordered lattices

Lorenzo Gori, Thomas Barthel, Avinash Kumar, E. Lucioni, Luca Tanzi, M. Inguscio, Giovanni Carlo Modugno, Thierry Giamarchi, Chiara D’Errico, Guillaume Roux

Published Mar 29, 2016• 20 citations
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Abstract

We analyze the finite-temperature effects on the phase diagram describing the insulating properties of interacting one-dimensional bosons in a quasiperiodic lattice. We examine thermal effects by comparing experimental results to exact diagonalization for small-sized systems and to density-matrix renormalization group (DMRG) computations. At weak interactions, we find short thermal correlation lengths, indicating a substantial impact of temperature on the system coherence. Conversely, at strong interactions, the obtained thermal correlation lengths are significantly larger than the localization length, and the quantum nature of the $T=0$ Bose-glass phase is preserved up to a crossover temperature that depends on the disorder strength. Furthermore, in the absence of disorder, we show how quasiexact finite-$T$ DMRG computations, compared to experimental results, can be employed to estimate the temperature, which is not directly accessible in the experiment.

DOI
10.1103/physreva.93.033650

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We analyze the finite-temperature effects on the phase diagram describing the insulating properties of interacting one-dimensional bosons in a quasiperiodic lattice. We examine thermal effects by comparing experimental results to exact diagonalization for small-sized systems and to density-matrix renormalization group (DMRG) computations. At weak interactions, we find short thermal correlation lengths, indicating a substantial impact of temperature on the system coherence. Conversely, at strong interactions, the obtained thermal correlation lengths are significantly larger than the localization length, and the quantum nature of the T=0T=0T=0 Bose-glass phase is preserved up to a crossover temperature that depends on the disorder strength. Furthermore, in the absence of disorder, we show how quasiexact finite-TTT DMRG computations, compared to experimental results, can be employed to estimate the temperature, which is not directly accessible in the experiment.
One of the most important issues in disordered systems is the interplay of the disorder and repulsive interactions. Several recent experimental advances on this topic have been made with ultracold atoms, in particular the observation of Anderson localization and the realization of the disordered Bose–Hubbard model. There are, however, still questions as to how to differentiate the complex insulating phases resulting from this interplay, and how to measure the size of the superfluid fragments that these phases entail. It has been suggested that the correlation function of such a system can give new insights, but so far very little experimental investigation has been performed. Here, we show the first experimental analysis of the correlation function for a weakly interacting, bosonic system in a quasiperiodic lattice. We observe an increase in the correlation length as well as a change in the shape of the correlation function in the delocalization crossover from Anderson glass to coherent, extended state. In between, the experiment indicates the formation of progressively larger coherent fragments, consistent with a fragmented BEC, or Bose glass.
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One of the most important issues in disordered systems is the interplay of the disorder and repulsive interactions. Several recent experimental advances on this topic have been made with ultracold atoms, in particular the observation of Anderson localization and the realization of the disordered Bose–Hubbard model. There are, however, still questions as to how to differentiate the complex insulating phases resulting from this interplay, and how to measure the size of the superfluid fragments that these phases entail. It has been suggested that the correlation function of such a system can give new insights, but so far very little experimental investigation has been performed. Here, we show the first experimental analysis of the correlation function for a weakly interacting, bosonic system in a quasiperiodic lattice. We observe an increase in the correlation length as well as a change in the shape of the correlation function in the delocalization crossover from Anderson glass to coherent, extended state. In between, the experiment indicates the formation of progressively larger coherent fragments, consistent with a fragmented BEC, or Bose glass.
The study of the disordered Bose-Hubbard model is key to understanding the interplay of disorder and interactions. Despite many studies with uniform diagonal disorder, few have inquired into experimental realizations with an additional correlated off-diagonal disorder. The presence of a trap and finite temperature effects in experiments lead to multiple do- mains of the Superfluid, Mott-Insulator/normal and the Bose-Glass phase. Previous studies using approximate theories produced results that are not in accordance with experiments. Stochastic Series Expansion is a finite temperature technique that can solve Bosonic lattice Hamiltonians exactly for large systems. Here, studies are performed for an extensive range of parameters using disorder distributions that are similar to experiments. Insights are first acquired by studying trap-free situations. Constant density calculations show that, although the qualitative features of the phase diagram remain robust between speckle disorder and uniform box disorder, there are quantitative differences. Studies of the Bose-glass phase ex- plicitly show that it is composed of superfluid puddles that are stable to finite temperature effects for large temperature ranges. Finite temperature behavior of a strongly correlated sys- tem reveals that at unit filling, the transition temperature of the superfluid is increased due to the addition of disorder. Inquires are then extended to discern the properties of trapped systems. Extensive calculations show that domain-like structures that develop can be rig- orously demarcated using the single-particle eigenstates extracted from the single-particle density matrix. Observables are calculated for the system at the single-site and global scales, showing that intermediate length scales provide the correct description of the physics of the domains in these systems. These techniques are used to conclusively show the possibility of the re-entrant superfluid that should be accessible to experiments. The temperature de- pendence of the re-entrant domain is explicitly calculated to be within experimental limits provided interactions are not too large. Comparisons with the local density approximation show reasonable agreement at low disorder strengths. At large disorder strengths there can be quantitative errors and can also result in qualitative errors. The phase diagram due to speckle disorder is presented for a range of values that are readily accessible to experiments. It is quantitatively shown that the effects of off-diagonal disorder are minimal. The superfluid remains unaffected despite large disorder in the tunneling term. Full scale ab initio calcu- lations of the largest trapped disordered systems to date are performed in order to identify the superfluid-Bose-glass phase boundary in collaboration with experiments. Results show remarkable agreement, but there are open questions with regards to the possibility of glassy dynamics.
We use quantum Monte Carlo (QMC) simulations to study the combined effects of harmonic confinement and temperature for bosons in a two-dimensional(2D) optical lattice. The scale-invariant, finite temperature state diagram is presented for the Bose-Hubbard model in terms of experimental parameters -- the particle number, confining potential and interaction strength. To distinguish the nature of the spatially separated superfluid, Mott insulator, and normal Bose liquid phases, we examine the local density, compressibility, superfluid density, and Green's function. In the annular superfluid rings, as the width of the ring decreases, the long-range superfluid correlations start to deviate from an equivalent homogeneous 2D system. At zero temperature, the correlation decay is intermediate between one-dimension and two-dimensions, while at a finite temperature, the decay is similar to that in one-dimension at a much lower temperature. The calculations reveal shortcomings of the local density approximation in describing superfluid properties of trapped bosons. We also present the finite-temperature phase diagram for the homogeneous two dimensional Bose-Hubbard model. We compare our state diagram with the results of a recent experiment at NIST on a harmonically trapped 2D lattice [Phys. Rev. Lett. 105, 110401 (2010)], and identify a finite-temperature effect in the experiment.
We study the occupation numbers and number fluctuations of ultracold atoms in deep optical lattices for finite-temperatures within the Bose-Hubbard model. Simple analytical expressions for the mean occupation number and number fluctuations are obtained in the weak-hopping regime using an interpolation between results from different perturbation approaches in the Mott-insulator and superfluid phases. With this approach the magnitude of number fluctuations under a wide range of experimental conditions can be estimated and the properties of the finite-temperature phase diagram can be studied. These analytical results are compared to exact one-dimensional numerical calculations using a finite temperature variant of the density-matrix renormalization group (DMRG) method and found to have a high degree of accuracy. We find very good agreement, also in the crossover ``thermal'' region. We also analyze the influence of finite temperature on the behavior of the system in the vicinity of the zero-temperature phase transition, in one, two, and three dimensions.