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

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.

Figure 1

Experimental setup. Two horizontal optical lattices provide a tight confinement forming an array of 1D vertical potential tubes for the ${}^{39}\mathrm{K}$ atoms with tunable interaction energy $U$. The vertical quasiperiodic potential is formed by superimposing two incommensurate optical lattices: the main lattice (${\lambda }_1=1064\mathrm{nm}$), which is related to the tunneling energy $J$, and the secondary one (${\lambda }_2=859.6\mathrm{nm}$), which is related to the disorder amplitude $\mathrm{\Delta }$. The harmonic trapping confinement makes the 1D systems inhomogeneous.

Figure 2

Measured rms width $\mathrm{\Gamma }$ of the momentum distribution $P\left(k\right)$ in the $U-\mathrm{\Delta }$ diagram. Without interaction ($U=0$), increasing $\mathrm{\Delta }$ induces the transition from the superfluid (SF) to the Anderson insulator (AI). In the absence of disorder ($\mathrm{\Delta }=0$), increasing $U$ leads to the superfluid to Mott-insulator (MI) transition. For increasing $\mathrm{\Delta }$ at large interaction, according to $T=0$ DMRG calculations, MI domains exist only at the right of the dashed line (i.e., $U>2\mathrm{\Delta }$ for large $U$), where they coexist with SF or Bose-glass (BG) domains, respectively, below and above $\mathrm{\Delta }=2J$. The diagram is generated from 94 data points (crosses). Standard deviations of $\mathrm{\Gamma }$ are between 2% and 5%. Data taken from Ref. [15].

Figure 3

$U-\mathrm{\Delta }$ diagram for experimental estimates of the entropy per particle, $S/\left(N{k}_B\right)$. The white crosses show the data points from which the 2D diagram was generated by interpolation. The measurements are performed at the lowest possible experimental temperature as in the measurements of Fig. 2. Data taken from Supplemental Material of Ref. [15].

Figure 4

Inverse correlation length $1/{\xi }_{\text{expt}}$ as a function of the entropy $S$, for $\mathrm{\Delta }=6.6J$ and for two interaction strengths (a) $U=2.3J$ and (b) $U=23.4J$. In the regime of strong interaction, where the Bose-glass and the disordered Mott-insulating phases coexist, $1/{\xi }_{\text{expt}}$ starts increasing only above a certain entropy value. The uncertainties are the standard deviation of typically five measurements. The lines are a guide to the eye.

Figure 5

(a) Finite-$T$ DMRG data for the number of atoms $N\left(\mu \right)$ in a tube, as a function of the tube chemical potential $\mu$. (b) Distribution of the tube particle numbers. Compared to the grand-canonical distribution of particles among tubes, the Thomas-Fermi distribution favors more highly filled tubes in the center of the trap.

Figure 6

Theoretical rms width $\mathrm{\Gamma }$ of the momentum distribution $P\left(k\right)$ at $T=0$, averaged over all tubes. The diagram is built from 94 data points as in the experimental diagram in Fig. 2. For few representative points, $P\left(k\right)$ is also shown at the side of the diagram: the theoretical result for $T=0$ (blue, dotted-dashed) is compared to the experimental finite-$T$ data (black, solid). The labels indicate the different quantum phases: superfluid (SF), Mott insulator (MI), Anderson insulator (AI), and Bose glass (BG). The dashed line delimits the region for the existence of the MI phase. Data taken from Ref. [15].

Figure 7

Density profiles obtained from $T=0$ DMRG calculations for $U=26J$ and $\mathrm{\Delta }=0$ (top) or $\mathrm{\Delta }=6.5J$ (bottom). From left to right: blue, red, and black curves refer to tubes with $N=20,55,96$ atoms, respectively. The shaded areas represent the regions with noninteger filling where the superfluid (top) becomes Bose glass (bottom). Data taken from Supplemental Material of Ref. [15].

Figure 8

$U-\mathrm{\Delta }$ diagram for the rms width $\mathrm{\Gamma }$ of the phenomenological $P\left(k\right)$ (red, dashed). The $T=0$ momentum distribution (blue, dotted-dashed) is thermally broadened in such a way that the phenomenological $P\left(k\right)$ fits the experimental one (black, solid). The thermal correlation length ${\xi }_T$ is the fitting parameter that phenomenologically accounts for thermal effects according to the ansatz (5). The full diagram is generated by interpolation from the same $U-\mathrm{\Delta }$ points as in Fig. 2. Data taken from Ref. [15].

Figure 9

$U-\mathrm{\Delta }$ diagram of the thermal correlation length ${\xi }_T$ resulting from the phenomenological ansatz (5) by fitting it to the experimental momentum distribution $P\left(k\right)$. Thermal effects are significantly more relevant for small $U$. Data taken from Supplemental Material of Ref. [15].

Figure 10

Temperature dependence of the inverse correlation length $1/\xi \left(T\right)$, calculated by exact diagonalization for a small system ($L=12d$) in the superfluid regime ($\mathrm{\Delta }=0, U=2J$), for various site occupancies $n$ (increasing from top to bottom). The dashed line is a fit of the numerical high-$T$ data with Eq. (8). Inset: density dependence of $1/\xi \left(T\right)$. As shown in the main panel, the change of density $n$ can be taken into account by a scaling factor such that, when $\xi \left(T\right)$ is plotted versus $k_{B}T/\left(J{n}^{1/2}\right)$, all curves overlap for $k_{B}T\gtrsim 2J$. Data taken from Supplemental Material of Ref. [15].

Figure 11

Temperature dependence of the inverse correlation length $1/\xi \left(T\right)$, calculated by exact diagonalization and derived from a Lorentzian fit of $P\left(k\right)$ for a weakly interacting system ($U=2.3J$) with $L=13d$ and $n=0.46$, for three values of the disorder strength: $\mathrm{\Delta }=0,6J,16J$.

Figure 12

Temperature dependence of the inverse correlation length, calculated by exact diagonalization for a strongly interacting system with $U=44J$, disorder strength $\mathrm{\Delta }=10J$, and for both the commensurate case of a Mott insulator ($n=1$) and the incommensurate case of a Bose glass ($n=0.46$). The system lengths are $L=9d$ and $13d$, respectively. Arrows indicate the crossover temperatures $T_0$ below which ${\xi }^{-1}\left(T\right)$ is rather constant before starting increasing.

Figure 13

Crossover temperature $T_0$ as a function of the disorder strength $\mathrm{\Delta }$, calculated by exact diagonalization for a strongly interacting system with $U=44J$, for both the commensurate case of a Mott insulator ($n=1$) and the incommensurate case of a Bose glass ($n=0.46$). The system lengths are $L=9d$ and $13d$, respectively. The error bars for $T_0$ are related to the finite number of simulations performed for $1/\xi \left(T\right)$ in proximity of the crossover temperature.

Figure 14

Thermometry in the superfluid regime ($U=3.5J$) from finite-$T$ DMRG calculations for both the hypotheses of a grand-canonical (GC) or Thomas-Fermi (TF) distribution of the particles among the tubes. In the legend, “GC distr. @ $k_{B}T=X$” indicates that the GC distribution for temperature $X$ was used (also in Figs. 16, 17, 18). The TF distribution does not depend on temperature. See Sec. 4b.

Figure 15

Temperature dependence of the rms width $\mathrm{\Gamma }$ of $P\left(k\right)$ in the superfluid regime from finite-$T$ DMRG calculations under both the GC and TF assumptions. In the former case, the effect of varying $N_{\text{tot}}$ is also shown.

Figure 16

Testing the phenomenological approach (5) for $U=3.5J$ and $T=5.3J/k_B$. To this purpose, $T=0$ DMRG data for the momentum distribution are folded with a Lorentzian that corresponds to the effective thermal correlation length ${\xi }_T=d/0.65$. The results are compared with the quasiexact finite-$T$ DMRG data.

Figure 17

Thermometry in the strong-interaction regime ($U=21J$) from finite-$T$ DMRG calculations, for both the hypotheses of a GC or TF distribution of the particles among the tubes.

Figure 18

Testing the phenomenological approach (5) for $U=21J$ and $T=2J/k_B$. To this purpose, $T=0$ DMRG data for the momentum distribution are folded with a Lorentzian that corresponds to thermal correlation lengths ${\xi }_T=d/0.4$ and $d/0.5$, respectively, for the GC and TF cases. The results are compared with the quasiexact finite-$T$ DMRG data.

Figure 19

DMRG density profile in a tube at $T=0$ and at the estimated experimental temperature for $U=21J$. Comparison with the classical model is also given.