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arXiv:2310.10757 · Physical review. B./Physical review. B

Fractal quasicondensation in one dimension

Flavio Elias Riche, Miguel Gonçalves, Bruno Amorim, Eduardo V. Castro, Pedro Ribeiro

Published Dec 26, 2024• 0 citations
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Abstract

We unveil a mechanism for quasicondensation of hard-core bosons in the presence of quasiperiodicity-induced multifractal single-particle states. The observed phase, here dubbed a fractal quasicondensate, is characterized by natural orbitals with multifractal properties and by an occupancy of the lowest natural orbital, ${\ensuremath{\lambda}}_{0}\ensuremath{\simeq}{L}^{\ensuremath{\gamma}}$, which grows with system size but with a nonuniversal scaling exponent, $\ensuremath{\gamma}<1/2$. In contrast to fractal quasicondensates obtained when the chemical potential lies in a region of multifractal single-particle states, placing the chemical potential in regions of localized or delocalized states yields, respectively, no condensation or the usual 1D quasicondensation with $\ensuremath{\gamma}=1/2$. Our findings are established by studying one-dimensional hard-core bosons subjected to various quasiperiodic potentials, including the well-known Aubry-Andr\'e model, employing a mapping to noninteracting fermions that allows for numerically exact results. We discuss how to test our findings in state-of-the-art ultracold atom experiments.

DOI
10.1103/physrevb.110.224523

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We unveil a mechanism for quasicondensation of hard-core bosons in the presence of quasiperiodicity-induced multifractal single-particle states. The observed phase, here dubbed a fractal quasicondensate, is characterized by natural orbitals with multifractal properties and by an occupancy of the lowest natural orbital, \ensuremathλ0\ensuremath≃L\ensuremathγ{\ensuremath{\lambda}}_{0}\ensuremath{\simeq}{L}^{\ensuremath{\gamma}}\ensuremathλ0​\ensuremath≃L\ensuremathγ, which grows with system size but with a nonuniversal scaling exponent, \ensuremathγ<1/2\ensuremath{\gamma}<1/2\ensuremathγ<1/2. In contrast to fractal quasicondensates obtained when the chemical potential lies in a region of multifractal single-particle states, placing the chemical potential in regions of localized or delocalized states yields, respectively, no condensation or the usual 1D quasicondensation with \ensuremathγ=1/2\ensuremath{\gamma}=1/2\ensuremathγ=1/2. Our findings are established by studying one-dimensional hard-core bosons subjected to various quasiperiodic potentials, including the well-known Aubry-Andr'e model, employing a mapping to noninteracting fermions that allows for numerically exact results. We discuss how to test our findings in state-of-the-art ultracold atom experiments.
We study localization and many-body localization transition in one-dimensional systems in the presence of deterministic quasiperiodic potential. We use single-particle excitations obtained through single-particle Green's function in real space to characterize the localization to delocalization transition. A single parameter scaling analysis of the ratio of the typical to average value of the local density of states (LDOS) of single-particle excitations shows that the critical exponent with which the correlation length $\ensuremath{\xi}$ diverges at the transition point $\ensuremath{\xi}\ensuremath{\sim}|h\ensuremath{-}{h}_{c}{|}^{\ensuremath{-}\ensuremath{\nu}}$, coming from the localized side, satisfies the inequality $\ensuremath{\nu}\ensuremath{\ge}1$ for the noninteracting Aubry-Andre (AA) model. For the interacting system with AA potential, we study single-particle excitations produced in highly excited many-body eigenstates across the MBL transition and found that the critical exponent obtained from finite-size scaling of the ratio of the typical to average value of the LDOS satisfies $\ensuremath{\nu}\ensuremath{\ge}1$ here as well. This analysis of the local density of states shows that the localization and MBL transition in systems with quasiperiodic potential belong to a different universality class than the localization and MBL transition in systems with random disorder where $\ensuremath{\nu}\ensuremath{\ge}2$. In complete contrast to this, finite-size scaling of the level spacing ratio is known to support the same universality class for MBL transitions in systems with quasiperiodic as well as random disorder potentials. For the interacting systems with quasiperiodic potentials, though finite-size scaling of the level spacing ratio shows a transition at hclsr{h}_{c}^{\mathrm{lsr}}hclsr​ which is close to the transition point obtained from LDOS within numerical precision, the critical exponent obtained from finite-size scaling of level spacing ratio is $\ensuremath{\nu}\ensuremath{\sim}0.54$ in close similarity to the MBL systems with random disorder.
We study the dynamics and the resulting state after relaxation in a quasidisordered integrable lattice system after a sudden quench. Specifically, we consider hard-core bosons in an isolated one-dimensional geometry in the presence of a quasiperiodic potential whose strength is abruptly changed to take the system out of equilibrium. In the delocalized regime, we find that the relaxation dynamics of one-body observables, such as the density, the momentum distribution function, and the occupation of the natural orbitals, follow, to a good approximation, power laws. In that regime, we also show that the observables after relaxation can be described by the generalized Gibbs ensemble, while such a description fails for the momentum distribution and the natural orbital occupations in the presence of localization. At the critical point, the relaxation dynamics is found to be slower than in the delocalized phase.
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One-dimensional quasiperiodic systems with power-law hopping, 1/r^{a}, differ from both the standard Aubry-André (AA) model and from power-law systems with uncorrelated disorder. Whereas in the AA model all single-particle states undergo a transition from ergodic to localized at a critical quasidisorder strength, short-range power-law hops with a>1 can result in mobility edges. We find that there is no localization for long-range hops with a≤1, in contrast to the case of uncorrelated disorder. Systems with long-range hops rather present ergodic-to-multifractal edges and a phase transition from ergodic to multifractal (extended but nonergodic) states. Both mobility and ergodic-to-multifractal edges may be clearly revealed in experiments on expansion dynamics.