Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1904.06928 · Physical review. B./Physical review. B

Slow dynamics and strong finite-size effects in many-body localization with random and quasiperiodic potentials

Felix Weiner, Ferdinand Evers, Soumya Bera

Published Sep 24, 2019• 76 citations
arXivHTMLPDF

Abstract

We investigate charge relaxation in disordered and quasiperiodic quantum wires of spinless fermions $(t\ensuremath{-}V$ model) at different inhomogeneity strength $W$ in the localized and nearly localized regime. Our observable is the time-dependent density correlation function, $\mathrm{\ensuremath{\Phi}}(x,t)$, at infinite temperature. We find that disordered and quasiperiodic models behave qualitatively similar: Although even at longest observation times the width $\mathrm{\ensuremath{\Delta}}x(t)$ of $\mathrm{\ensuremath{\Phi}}(x,t)$ does not exceed significantly the noninteracting localization length, ${\ensuremath{\xi}}_{0}$, strong finite-size effects are encountered. Our findings appear difficult to reconcile with the rare-region physics (Griffiths effects) that often is invoked as an explanation for the slow dynamics observed by us and earlier computational studies. Motivated by our numerical data we discuss a scenario in which the MBL-phase splits into two subphases: in ${\mathrm{MBL}}_{\text{A}}\phantom{\rule{4pt}{0ex}}\mathrm{\ensuremath{\Delta}}x(t)$ diverges slower than any power, while it converges towards a finite value in ${\mathrm{MBL}}_{\text{B}}$. Within the scenario the transition between ${\mathrm{MBL}}_{\text{A}}$ and the ergodic phase is characterized by a length scale, $\ensuremath{\xi}$, that exhibits an essential singularity $ln\ensuremath{\xi}\ensuremath{\sim}1/|W\ensuremath{-}{W}_{{\text{c}}_{1}}|$. Relations to earlier numerics and proposals of two-phase scenarios will be discussed.

DOI
10.1103/physrevb.100.104204

Related papers

Slow dynamics and strong finite-size effects in many-body localization with random and quasiperiodic potentials

Felix Weiner, Ferdinand Evers, Soumya Bera

We investigate charge relaxation in disordered and quasiperiodic quantum wires of spinless fermions (t -V model) at different inhomogeneity strength W in the localized and nearly localized regime. Our observable is the time-dependent density correlation function, Phi(x, t), at infinite temperature. We find that disordered and quasiperiodic models behave qualitatively similar: Although even at longest observation times the width Delta x(t) of Phi(x, t) does not exceed significantly the noninteracting localization length, xi(0), strong finite-size effects are encountered. Our findings appear difficult to reconcile with the rare-region physics (Griffiths effects) that often is invoked as an explanation for the slow dynamics observed by us and earlier computational studies. Motivated by our numerical data we discuss a scenario in which the MBL-phase splits into two subphases: in MBLA Delta x(t) diverges slower than any power, while it converges towards a finite value in MBLB. Within the scenario the transition between MBLA and the ergodic phase is characterized by a length scale, xi, that exhibits an essential singularity In xi similar to 1/vertical bar W - W-c1 vertical bar. Relations to earlier numerics and proposals of two-phase scenarios will be discussed.

•Jan 1, 2019•63 citations

Density Propagator for Many-Body Localization: Finite-Size Effects, Transient Subdiffusion, and Exponential Decay

Soumya Bera, Giuseppe De Tomasi, Felix Weiner +1 more

We investigate charge relaxation in quantum wires of spinless disordered fermions ($t\text{\ensuremath{-}}V$ model). Our observable is the time-dependent density propagator ${\mathrm{\ensuremath{\Pi}}}_{ϵ}(x,t)$, calculated in windows of different energy density $ϵ$ of the many-body Hamiltonian and at different disorder strengths $W$, not exceeding the critical value ${W}_{c}$. The width ${\mathrm{\ensuremath{\Delta}}x}_{ϵ}(t)$ of ${\mathrm{\ensuremath{\Pi}}}_{ϵ}(x,t)$ exhibits a behavior $d\mathrm{ln}{\mathrm{\ensuremath{\Delta}}x}_{ϵ}(t)/d\mathrm{ln}t={\ensuremath{\beta}}_{ϵ}(t)$, where the exponent function ${\ensuremath{\beta}}_{ϵ}(t)\ensuremath{\lesssim}1/2$ is seen to depend strongly on $L$ at all investigated parameter combinations. (i) We confirm the existence of a region in phase space that exhibits subdiffusive dynamics in the sense that ${\ensuremath{\beta}}_{ϵ}(t)<1/2$ in a large window of times. However, subdiffusion might possibly be transient, only, finally giving way to a conventional diffusive behavior with ${\ensuremath{\beta}}_{ϵ}=1/2$. (ii) We cannot confirm the existence of many-body mobility edges even in regions of the phase diagram that have been reported to be deep in the delocalized phase. (iii) (Transient) subdiffusion $0<{\ensuremath{\beta}}_{ϵ}(t)\ensuremath{\lesssim}1/2$ coexists with an enhanced probability for returning to the origin ${\mathrm{\ensuremath{\Pi}}}_{ϵ}(0,t)$, decaying much slower than $1/{\mathrm{\ensuremath{\Delta}}x}_{ϵ}(t)$. Correspondingly, the spatial decay of ${\mathrm{\ensuremath{\Pi}}}_{ϵ}(x,t)$ is far from Gaussian, being exponential or even slower. On a phenomenological level, our findings are broadly consistent with the effects of strong disorder and (fractal) Griffiths regions.

arXiv:1610.03085•Physical Review Letters•May 8, 2017•121 citations

Can we observe the many-body localization?

Piotr Sierant, Jakub Zakrzewski

We study time dynamics of 1D disordered Heisenberg spin-1/2 chain focusing on a regime of large system sizes and a long time evolution. This regime is relevant for observation of many-body localization (MBL), a phenomenon that is expected to freeze the dynamics of the system and prevent it from reaching thermal equilibrium. Performing extensive numerical simulations of the imbalance, a quantity often employed in the experimental studies of MBL, we show that the regime of a slow power-law decay of imbalance persists to disorder strengths exceeding by at least a factor of 2 the current estimates of the critical disorder strength for MBL. Even though we investigate time evolution up to few thousands tunneling times, we observe no signs of the saturation of imbalance that would suggest freezing of system dynamics and provide a smoking gun evidence of MBL. We demonstrate that the situation is qualitatively different when the disorder is replaced by a quasiperiodic potential. In this case, we observe an emergence of a pattern of oscillations of the imbalance that is stable with respect to changes in the system size. This suggests that the dynamics of quasiperiodic systems remain fully local at the longest time scales we reach provided that the quasiperiodic potential is sufficiently strong. Our study identifies challenges in an unequivocal experimental observation of the phenomenon of MBL.

•Sep 28, 2021•5 citations

Finite-size scaling analysis of the many-body localization transition in quasiperiodic spin chains

Adith Sai Aramthottil, Titas Chanda, Piotr Sierant +1 more

We analyze the finite-size scaling of the average gap ratio and the entanglement entropy across the many-body localization (MBL) transition in one dimensional Heisenberg spin chain with quasiperiodic (QP) potential. By using the recently introduced cost-function approach, we compare different scenarios for the transition using exact diagonalization of systems up to 22 lattice sites. Our findings suggest that the MBL transition in the QP Heisenberg chain belongs to the class of the Berezinskii-Kosterlitz-Thouless transition, the same as in the case of uniformly disordered systems as advocated in recent studies. Moreover, we observe that the critical disorder strength shows a clear sublinear drift with the system size as compared to the linear drift seen in random disordered models, suggesting that the finite-size effects in the MBL transition for the QP systems are less severe than that in the random disordered scenario. Moreover, deep in the ergodic regime, we find an unexpected double-peak structure of distribution of on-site magnetizations that can be traced back to the strong correlations present in the QP potential.

arXiv:2109.08408•Physical review. B./Physical review. B•Dec 2, 2021•48 citations

Observation of Slow Dynamics near the Many-Body Localization Transition in One-Dimensional Quasiperiodic Systems

Henrik P. Lüschen, Pranjal Bordia, Sebastian Scherg +4 more

In the presence of sufficiently strong disorder or quasiperiodic fields, an interacting many-body system can fail to thermalize and become many-body localized. The associated transition is of particular interest, since it occurs not only in the ground state but over an extended range of energy densities. So far, theoretical studies of the transition have focused mainly on the case of true-random disorder. In this work, we experimentally and numerically investigate the regime close to the many-body localization transition in quasiperiodic systems. We find slow relaxation of the density imbalance close to the transition, strikingly similar to the behavior near the transition in true-random systems. This dynamics is found to continuously slow down upon approaching the transition and allows for an estimate of the transition point. We discuss possible microscopic origins of these slow dynamics.

arXiv:1612.07173•Physical Review Letters•Dec 27, 2017•281 citations

Density propagator for many-body localization: finite size effects, transient subdiffusion, (stretched-)exponentials

Soumya Bera, Giuseppe De Tomasi, Felix Weiner +1 more

We investigate charge relaxation in the spin-less disordered fermionic Hubbard chain ($t{-}V$-model). Our observable is the time-dependent density propagator, $\Pi_{\varepsilon}(x,t)$, calculated in windows of different energy density, $\varepsilon$, of the many-body Hamiltonian and at different disorder strengths, $W$, not exceeding the critical value $W_\text{c}$. The width $\Delta x_\varepsilon(t)$ of $\Pi_\varepsilon(x,t)$ exhibits a behavior $d\ln \Delta x_\varepsilon(t) / d\ln t {=} \beta_\varepsilon(t)$, where the exponent function $\beta_\varepsilon(t){\lesssim}1/2$ is seen to depend strongly on $L$ at all investigated parameter combinations. (i) We cannot confirm the existence of a region in phase space that exhibits (genuine) subdiffusive dynamics in the sense that $\beta_\varepsilon{<}1/2$ is numerically fixed in the limit of large $L$. Instead, subdiffusion might possibly be transient, only, finally giving way to conventional diffusive behavior with $\beta_\varepsilon{=}1/2$. (ii) Similarly, we cannot confirm the existence of many-body mobility edges deep in the delocalized phase. (iii) (Transient) subdiffusion $0<\beta_\varepsilon(t)\lesssim 1/2$, coexists with an enhanced probability for returning to the origin, $\Pi_\varepsilon(0,t)$, decaying much slower than $1/\Delta x_\varepsilon (t)$. Correspondingly, the spatial decay of $\Pi_\varepsilon(x,t)$ is far from Gaussian being exponential or even slower. On a phenomenological level, our findings are broadly consistent with effects of strong disorder and (fractal) Griffiths regions.

•Oct 10, 2016•4 citations
We investigate charge relaxation in disordered and quasiperiodic quantum wires of spinless fermions (t\ensuremath−V(t\ensuremath{-}V(t\ensuremath−V model) at different inhomogeneity strength WWW in the localized and nearly localized regime. Our observable is the time-dependent density correlation function, \ensuremathΦ(x,t)\mathrm{\ensuremath{\Phi}}(x,t)\ensuremathΦ(x,t), at infinite temperature. We find that disordered and quasiperiodic models behave qualitatively similar: Although even at longest observation times the width \ensuremathΔx(t)\mathrm{\ensuremath{\Delta}}x(t)\ensuremathΔx(t) of \ensuremathΦ(x,t)\mathrm{\ensuremath{\Phi}}(x,t)\ensuremathΦ(x,t) does not exceed significantly the noninteracting localization length, \ensuremathξ0{\ensuremath{\xi}}_{0}\ensuremathξ0​, strong finite-size effects are encountered. Our findings appear difficult to reconcile with the rare-region physics (Griffiths effects) that often is invoked as an explanation for the slow dynamics observed by us and earlier computational studies. Motivated by our numerical data we discuss a scenario in which the MBL-phase splits into two subphases: in MBLA\ensuremathΔx(t){\mathrm{MBL}}_{\text{A}}\phantom{\rule{4pt}{0ex}}\mathrm{\ensuremath{\Delta}}x(t)MBLA​\ensuremathΔx(t) diverges slower than any power, while it converges towards a finite value in MBLB{\mathrm{MBL}}_{\text{B}}MBLB​. Within the scenario the transition between MBLA{\mathrm{MBL}}_{\text{A}}MBLA​ and the ergodic phase is characterized by a length scale, \ensuremathξ\ensuremath{\xi}\ensuremathξ, that exhibits an essential singularity ln\ensuremathξ\ensuremath∼1/∣W\ensuremath−Wc1∣ln\ensuremath{\xi}\ensuremath{\sim}1/|W\ensuremath{-}{W}_{{\text{c}}_{1}}|ln\ensuremathξ\ensuremath∼1/∣W\ensuremath−Wc1​​∣. Relations to earlier numerics and proposals of two-phase scenarios will be discussed.
We investigate charge relaxation in disordered and quasiperiodic quantum wires of spinless fermions (t -V model) at different inhomogeneity strength W in the localized and nearly localized regime. Our observable is the time-dependent density correlation function, Phi(x, t), at infinite temperature. We find that disordered and quasiperiodic models behave qualitatively similar: Although even at longest observation times the width Delta x(t) of Phi(x, t) does not exceed significantly the noninteracting localization length, xi(0), strong finite-size effects are encountered. Our findings appear difficult to reconcile with the rare-region physics (Griffiths effects) that often is invoked as an explanation for the slow dynamics observed by us and earlier computational studies. Motivated by our numerical data we discuss a scenario in which the MBL-phase splits into two subphases: in MBLA Delta x(t) diverges slower than any power, while it converges towards a finite value in MBLB. Within the scenario the transition between MBLA and the ergodic phase is characterized by a length scale, xi, that exhibits an essential singularity In xi similar to 1/vertical bar W - W-c1 vertical bar. Relations to earlier numerics and proposals of two-phase scenarios will be discussed.
We investigate charge relaxation in quantum wires of spinless disordered fermions ($t\text{\ensuremath{-}}V$ model). Our observable is the time-dependent density propagator ${\mathrm{\ensuremath{\Pi}}}_{ϵ}(x,t)$, calculated in windows of different energy density ϵϵϵ of the many-body Hamiltonian and at different disorder strengths WWW, not exceeding the critical value Wc{W}_{c}Wc​. The width ${\mathrm{\ensuremath{\Delta}}x}_{ϵ}(t)$ of ${\mathrm{\ensuremath{\Pi}}}_{ϵ}(x,t)$ exhibits a behavior $d\mathrm{ln}{\mathrm{\ensuremath{\Delta}}x}_{ϵ}(t)/d\mathrm{ln}t={\ensuremath{\beta}}_{ϵ}(t)$, where the exponent function ${\ensuremath{\beta}}_{ϵ}(t)\ensuremath{\lesssim}1/2$ is seen to depend strongly on LLL at all investigated parameter combinations. (i) We confirm the existence of a region in phase space that exhibits subdiffusive dynamics in the sense that ${\ensuremath{\beta}}_{ϵ}(t)<1/2$ in a large window of times. However, subdiffusion might possibly be transient, only, finally giving way to a conventional diffusive behavior with ${\ensuremath{\beta}}_{ϵ}=1/2$. (ii) We cannot confirm the existence of many-body mobility edges even in regions of the phase diagram that have been reported to be deep in the delocalized phase. (iii) (Transient) subdiffusion $0<{\ensuremath{\beta}}_{ϵ}(t)\ensuremath{\lesssim}1/2$ coexists with an enhanced probability for returning to the origin ${\mathrm{\ensuremath{\Pi}}}_{ϵ}(0,t)$, decaying much slower than $1/{\mathrm{\ensuremath{\Delta}}x}_{ϵ}(t)$. Correspondingly, the spatial decay of ${\mathrm{\ensuremath{\Pi}}}_{ϵ}(x,t)$ is far from Gaussian, being exponential or even slower. On a phenomenological level, our findings are broadly consistent with the effects of strong disorder and (fractal) Griffiths regions.
We study time dynamics of 1D disordered Heisenberg spin-1/2 chain focusing on a regime of large system sizes and a long time evolution. This regime is relevant for observation of many-body localization (MBL), a phenomenon that is expected to freeze the dynamics of the system and prevent it from reaching thermal equilibrium. Performing extensive numerical simulations of the imbalance, a quantity often employed in the experimental studies of MBL, we show that the regime of a slow power-law decay of imbalance persists to disorder strengths exceeding by at least a factor of 2 the current estimates of the critical disorder strength for MBL. Even though we investigate time evolution up to few thousands tunneling times, we observe no signs of the saturation of imbalance that would suggest freezing of system dynamics and provide a smoking gun evidence of MBL. We demonstrate that the situation is qualitatively different when the disorder is replaced by a quasiperiodic potential. In this case, we observe an emergence of a pattern of oscillations of the imbalance that is stable with respect to changes in the system size. This suggests that the dynamics of quasiperiodic systems remain fully local at the longest time scales we reach provided that the quasiperiodic potential is sufficiently strong. Our study identifies challenges in an unequivocal experimental observation of the phenomenon of MBL.
We analyze the finite-size scaling of the average gap ratio and the entanglement entropy across the many-body localization (MBL) transition in one dimensional Heisenberg spin chain with quasiperiodic (QP) potential. By using the recently introduced cost-function approach, we compare different scenarios for the transition using exact diagonalization of systems up to 22 lattice sites. Our findings suggest that the MBL transition in the QP Heisenberg chain belongs to the class of the Berezinskii-Kosterlitz-Thouless transition, the same as in the case of uniformly disordered systems as advocated in recent studies. Moreover, we observe that the critical disorder strength shows a clear sublinear drift with the system size as compared to the linear drift seen in random disordered models, suggesting that the finite-size effects in the MBL transition for the QP systems are less severe than that in the random disordered scenario. Moreover, deep in the ergodic regime, we find an unexpected double-peak structure of distribution of on-site magnetizations that can be traced back to the strong correlations present in the QP potential.
In the presence of sufficiently strong disorder or quasiperiodic fields, an interacting many-body system can fail to thermalize and become many-body localized. The associated transition is of particular interest, since it occurs not only in the ground state but over an extended range of energy densities. So far, theoretical studies of the transition have focused mainly on the case of true-random disorder. In this work, we experimentally and numerically investigate the regime close to the many-body localization transition in quasiperiodic systems. We find slow relaxation of the density imbalance close to the transition, strikingly similar to the behavior near the transition in true-random systems. This dynamics is found to continuously slow down upon approaching the transition and allows for an estimate of the transition point. We discuss possible microscopic origins of these slow dynamics.
We investigate charge relaxation in the spin-less disordered fermionic Hubbard chain (t−Vt{-}Vt−V-model). Our observable is the time-dependent density propagator, Πε(x,t)\Pi_{\varepsilon}(x,t)Πε​(x,t), calculated in windows of different energy density, ε\varepsilonε, of the many-body Hamiltonian and at different disorder strengths, WWW, not exceeding the critical value WcW_\text{c}Wc​. The width Δxε(t)\Delta x_\varepsilon(t)Δxε​(t) of Πε(x,t)\Pi_\varepsilon(x,t)Πε​(x,t) exhibits a behavior dln⁡Δxε(t)/dln⁡t=βε(t)d\ln \Delta x_\varepsilon(t) / d\ln t {=} \beta_\varepsilon(t)dlnΔxε​(t)/dlnt=βε​(t), where the exponent function βε(t)≲1/2\beta_\varepsilon(t){\lesssim}1/2βε​(t)≲1/2 is seen to depend strongly on LLL at all investigated parameter combinations. (i) We cannot confirm the existence of a region in phase space that exhibits (genuine) subdiffusive dynamics in the sense that βε<1/2\beta_\varepsilon{<}1/2βε​<1/2 is numerically fixed in the limit of large LLL. Instead, subdiffusion might possibly be transient, only, finally giving way to conventional diffusive behavior with βε=1/2\beta_\varepsilon{=}1/2βε​=1/2. (ii) Similarly, we cannot confirm the existence of many-body mobility edges deep in the delocalized phase. (iii) (Transient) subdiffusion 0<βε(t)≲1/20<\beta_\varepsilon(t)\lesssim 1/20<βε​(t)≲1/2, coexists with an enhanced probability for returning to the origin, Πε(0,t)\Pi_\varepsilon(0,t)Πε​(0,t), decaying much slower than 1/Δxε(t)1/\Delta x_\varepsilon (t)1/Δxε​(t). Correspondingly, the spatial decay of Πε(x,t)\Pi_\varepsilon(x,t)Πε​(x,t) is far from Gaussian being exponential or even slower. On a phenomenological level, our findings are broadly consistent with effects of strong disorder and (fractal) Griffiths regions.