Coherence, Transport, and Chaos in 1D Bose-Hubbard Model: Disorder vs. Stark Potential

TL;DR

This study analyzes a finite 1D Bose-Hubbard model under thermal fluctuations, a Stark potential, and quenched disorder using exact diagonalization to map coherence and chaos in the MI–SF landscape. A comprehensive set of observables, including the condensate fraction $f_c$, superfluid fraction $f_s$, visibility $\mathcal{V}$, $\ell_1$-norm of coherence $\mathcal{C}$, number fluctuations $\mathcal{F}$, momentum distribution $n_k$, and spectral metric $\langle r'\rangle$, are computed for ground and thermal states across perturbations. Key findings show that in the clean system the MI–SF crossover occurs near $\tau/U \approx 0.17$, with a non-ergodic spectral structure; a Stark potential induces Wannier-Stark localization that delays superfluidity and preserves local coherence; and disorder drives Anderson localization, suppressing global coherence while enabling thermally enhanced local coherence and a complex, size-dependent spectral statistics landscape. The results reveal distinct localization mechanisms for tilt versus disorder and demonstrate the utility of the $\ell_1$-norm of coherence as a sensitive probe of hidden coherence, with implications for quantum simulation of strongly correlated phases under realistic perturbations.

Abstract

Quantum coherence and phase transitions are studied in a finite one-dimensional Bose--Hubbard model using exact diagonalization under thermal fluctuations, a Stark potential, and disorder. The condensate fraction, superfluid fraction, visibility, number fluctuations, and the $\ell_1$-norm of coherence are computed to characterize the Mott insulator--superfluid transition. Although finite-size effects prevent a sharp transition, ground-state properties reveal signatures of quantum criticality. Thermal fluctuations can enhance coherence via tunneling, a Stark potential promotes localization, and disorder suppresses global superfluidity while preserving local coherence. These results highlight how disorder, tilt, and temperature reshape coherence and offer insights for quantum simulation and strongly correlated phases. For systems up to six sites with unit filling, a spectral analysis is also performed through the metric mean gap ratio (MGR). However, limited statistics due to the small system size and computational constraints prevent a complete characterization of quantum chaos, yielding only approximate signatures.

Figures (16)

• Figure 1: (a)-(c) Phase diagrams in the disordered (dirty) BHM are scaled with the system's linear site-dependent random energy offset strength $\delta/U$. (d)-(i) Single-site occupancy is shown in both the absence and presence of this offset. In the limit of negligible tunneling between lattice sites, distinct SF and MI phases emerge. As the offset becomes significant, bosonic tunneling and hopping disorder give rise to localization effects. These effects become prominent when the interaction strength satisfies $g/U>1$.
• Figure 2: Clean BHM ($g/U = 0$, $\delta/U = 0$) observables at unit filling with $\mu/U = 0.5$ versus $\tau/U$ for the both GS $|\psi_0\rangle$ (black circles) and TS $\varrho_T$ with $T/U = 0.2$ (blue squares), assuming $L=N=6$. (a) Ground state energy $\mathcal{E}_0$ with energy spectrum (the inset) showing gap closure near the transition point $\tau/U \approx 0.17$, which is acceptable for the given system size kuhner1998phasessatoshi2012characterizationcarrasquilla2013scalingkiely2022superfluidity under open boundary conditions. (b) Condensate fraction $f_c$; (c) SF fraction $f_s$, (d) visibility $\mathcal{V}$, (e) $\ell_1$-norm of coherence $\mathcal{C}$, and (f) number fluctuations $\mathcal{F}$.
• Figure 3: Many-body spectral properties of the clean BHM with $L=N=6$ and $\mu/U = 0.5$ at unit filling as a function of normalized hopping $\tau/U$. (top) Eigenenergies $\mathcal{E}_n(\tau/U)$: GS energy $\mathcal{E}_0$ decreases and the excited-state band broadens, with a gap closure at $\tau/U \approx 0.17$, indicating the MI–SF transition. (bottom)$\langle r'\rangle \approx 0.15$ remains constant, indicating persistent spectral correlations.
• Figure 4: System-size dependence of the $\langle r' \rangle$ versus $\tau/U$ for the clean BHM ($g/U = 0.0$, $\mu/U = 0.5$, $\delta/U = 0.0$) at unit filling. Different system sizes ($L = N = 2$ through $L = N = 6$) are shown with their respective colored curves. The dashed and dotted horizontal lines represent the theoretical values for GOE (0.53) and Poisson (0.38) statistics, respectively. While the smallest system ($L = N = 2$) shows increasing $\langle r' \rangle$ with $\tau/U$ that approaches the GOE limit, larger systems maintain consistently low $\langle r' \rangle$ values (0.05-0.25) across all hopping strengths, demonstrating persistent non-ergodic spectral correlations that remain stable through the SF-MI transition at $\tau/U \approx 0.17$. For the minimal two-site case ($L=N=2$), the apparent GOE-like value of the mean gap ratio $\langle r' \rangle$ is a finite-size artifact rather than genuine chaos. With only three many-body states, the model is integrable, but poor spectral statistics, boundary effects, and kinetic-energy dominance mimic level repulsion. For larger systems ($L \geq 3$), these artifacts disappear and $\langle r' \rangle$ settles at sub-Poissonian values, consistent with nonergodic behavior.
• Figure 5: The BHM with strong SP ($g/U = 1.0$, $\delta/U = 0.0$). GS ($|\psi_0\rangle$, black circles) and TS ($\varrho_T$, blue squares) observables versus $\tau/U$ for $L = N = 6$, $\mu/U = 0.5$, and temperature $T/U = 0.2$. (a) GS energy $\mathcal{E}_0$ with spectrum inset showing Wannier-Stark localization, (b) $f_c$, (c) $f_s$, (d) $\mathcal{V}$, (e) $\mathcal{C}$, and (f) $\mathcal{F}$.