Slow dynamics of a mobile impurity interacting with an Anderson insulator

Piotr Sierant, Titas Chanda, Maciej Lewenstein, and Jakub Zakrzewski

Phys. Rev. B 107, 144201 – Published 6 April 2023

Abstract

We investigate dynamics of a single mobile impurity immersed in a bath of Anderson localized particles and focus on the regime of relatively strong disorder and interactions. In that regime, the dynamics of the system is particularly slow, suggesting, at short times, an occurrence of many-body localization. Considering longer timescales, we show that the latter is a transient effect and that, eventually, the impurity spreads subdiffusively and induces a gradual delocalization of the Anderson insulator. The phenomenology of the system in the considered regime of slow dynamics includes a subdiffusive growth of mean square displacement of the impurity, power-law decay of density correlation functions of the Anderson insulator, and a power-law growth of entanglement entropy in the system. We observe a similar regime of slow dynamics also when the disorder in the system is replaced by a sufficiently strong quasiperiodic potential.

Received 28 December 2022
Accepted 29 March 2023

DOI:https://doi.org/10.1103/PhysRevB.107.144201

Physics Subject Headings (PhySH)

Anderson localization Cold atoms & matter waves Cold gases in optical lattices Entanglement in quantum gases Localization Quantum quench Quantum transport Random walks Spin dynamics XY model

Statistical Physics & ThermodynamicsAtomic, Molecular & OpticalCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Piotr Sierant^1,*, Titas Chanda ^2,†, Maciej Lewenstein ^1,3,‡, and Jakub Zakrzewski ^4,5,§

¹ICFO-Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, Av. Carl Friedrich Gauss 3, 08860 Castelldefels (Barcelona), Spain
²The Abdus Salam International Centre for Theoretical Physics (ICTP), Strada Costiera 11, 34151 Trieste, Italy
³ICREA, Passeig Lluis Companys 23, 08010 Barcelona, Spain
⁴Instytut Fizyki Teoretycznej, Uniwersytet Jagielloński, Łojasiewicza 11, PL-30-348 Kraków, Poland
⁵Mark Kac Complex Systems Research Center, Uniwersytet Jagielloński, Kraków, Poland

^*piotr.sierant@icfo.eu
^†tchanda@ictp.it
^‡maciej.lewenstein@icfo.eu
^§jakub.zakrzewski@uj.edu.pl

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 107, Iss. 14 — 1 April 2023

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Slow dynamics of the $c$ -boson. The top row show the results for the $c$ -boson starting in the center site of the chain; (a) average density $〈 {\hat{n}}_{i, c} 〉$ of the $c$ -boson for interaction strength $U = 12$ , disorder strength $W = 6.5$ , times $t = 20 and 500$ , and system sizes $L = 24 and 60$ , where $x$ is the number of the site with respect to the initial position of the $c$ -boson, $x \equiv i - i_{0}$ ; (b) the MSD of the $c$ -boson for system size $L = 24$ as function of time $t$ for various interaction amplitudes $U$ at fixed disorder strength $W = 6.5$ , the black dashed lines show the power-law fits $MSD (t) \sim t^{α}$ with $α < 1$ ; (c) MSD for different system sizes $L$ at fixed $U = 12, W = 6.5$ . The dashed line shows a logarithmic fit $MSD (t) \sim log (t)$ , whereas the inset shows data on a log-log scale demonstrating that a power law fit $MSD (t) \sim t^{α}$ works well for $L = 60$ result. (d)–(f) the same as (a)–(c) but for the $c$ -boson initially occupying the leftmost site of the chain. The shaded regions denote statistical uncertainties in the data associated with the finite number of disorder realizations. Data for $L = 60$ were obtained with TDVP with bond dimension $χ = 800$ .
Reuse & Permissions
Figure 2
Slow dynamics of $d$ -bosons, initially, the $c$ -boson is placed on the leftmost site of the chain. (a): the density correlation function $C_{d} (t)$ for system size $L = 24$ as function of time $t$ for various interaction amplitudes $U$ at fixed disorder strength $W = 6.5$ for $l_{1} = 1$ and $l_{2} = 9$ , the red dashed lines show the power-law fits $C_{d} (t) \sim t^{- β}$ with $β > 0$ . The inset compares the correlation function for $U = 3$ and $W = 6.5$ in a left subsystem of the chain ( $l_{1} = 1, l_{2} = 9$ , blue line) and in the central part ( $l_{1} = 9, l_{2} = 18$ , orange line). (b) $C_{d} (t)$ for different system sizes $L$ at fixed $U = 12, W = 6.5$ , and $l_{1} = 1, l_{2} = 9$ . The dashed line shows power-law fits. For clarity, the results for $L = 27$ and $L = 60$ are shifted down respectively by 0.04 and 0.07. Data for $L = 60$ was obtained with TDVP with bond dimension $χ = 800$ .
Reuse & Permissions
Figure 3
Entanglement entropy $S (t)$ in the system. (a) $S (t)$ for cut at bonds $i_{b} = 7, 9, 10, 12, 15$ , solid lines denote data for $L = 27$ , the dashed lines denote data for $L = 21$ , the red dashed lines show power-law fits $S (t) \sim t^{γ}$ . The interaction strength $U = 6$ , disorder strength $W = 6.5$ , the initial state is $| ψ_{left} (0) 〉$ . (b) $S (t)$ for cuts at various $i_{b}$ , colored solid lines denote data for $L = 27$ , the dashed lines denote data for for $L = 21$ , data for $L = 60$ are denoted by black solid lines. The red dashed lines show power-law fits $S (t) \sim t^{γ}$ , the interaction strength is $U = 12$ , disorder strength $W = 6.5$ , the initial state is $| ψ_{left} (0) 〉$ . The inset shows the time $t_{i}$ such that $S (t_{i}) = s_{0}$ versus the position of the cut $i_{b}$ , the threshold value $s_{0}$ is taken as 0.6, the dashed line shows an exponential fit $t_{i} \sim e^{0.42 i_{d}}$ . (c) $S (t)$ for cuts at various $i_{b}$ , colored solid lines denote data for $L = 60$ and $χ = 800$ whereas black dots show data for $L = 60$ and $χ = 256$ , the initial state is $| ψ_{cent} (0) 〉$ . The red dashed lines show power-law fits $S (t) \sim t^{γ}$ , note that a log-log scale is used here.
Reuse & Permissions
Figure 4
Slow dynamics in presence of quasiperiodic potential with amplitude $W_{Q P} = 6$ for initial state $| ψ_{left} (0) 〉$ and interaction strength $U = 12$ . (a) The $MSD (t)$ as function of time for various system sizes (for TDVP data, at $L = 60$ , the bond dimension $χ$ is indicated). While for $L \leq 24$ , we observe a saturation of MSD, the data for $L = 60$ are well approximated by $MSD (t) \sim ln (t)$ . (b) the density correlation function $C_{d} (t)$ calculated around the middle site of the chain (for $i \in [L / 2 - 3, L / 2 + 3]$ ) for various system sizes, fitted with a power-law decay $C_{d} (t) \sim t^{- β}$ . (c): The entanglement entropy $S (t)$ for various cuts $i_{b}$ and system size $L = 60$ . Data obtained with TDVP with bond-dimension $χ = 384$ are denoted by solid lines, whereas data obtained with $χ = 256$ are denoted by dots. The dashed lines show power-law fits $S (t) \sim t^{γ}$ with $γ \approx 0.11, 0.15, 0.20, 0.22, 0.25, 0.29, and 0.27$ respectively for $i_{b} = 30, 32, 34, 35, 37, 40, and 43$ .
Reuse & Permissions
Figure 5
Convergence of TDVP vs numerically exact results from Chebyshev time propagation scheme. System size $L = 30$ , interaction strength $U = 12$ , disorder amplitude $W = 6.5$ , initially, the c-boson ocucpies the left-most site of the chain. (Top) MSD as function of $t$ obtained with Chebyshev method (statistical error, calculated by resampling over disorder realizations is denoted by shaded blue region) is compared against TDVP results. The inset shows the difference between TDVP result ( ${MSD}_{χ}$ ) and Chebyshev data ( ${MSD}_{C H}$ ); Middle: density of the clean boson $n_{c} (x)$ as function of the site number $x$ for various times $t = 10, 70, and 970$ , Chebyshev results denoted by solid lines, TDVP results by dashed dotted lines; Bottom: the same as top panel, but for the density correlation function $C_{d} (t)$ of $d$ -bosons.
Reuse & Permissions
Figure 6
Slow dynamics of $d$ -bosons, initially, the $c$ -boson is placed in the middle of the chain. (a) Density correlation function $C_{d} (t) for$ system size $L = 24$ as function of time $t$ for various interaction amplitudes $U$ at fixed disorder strength $W = 6.5$ , the red dashed lines show the power-law fits $C_{d} (t) \sim t^{- β}$ with $β > 0$ ; (b) $C_{d} (t) for$ different system sizes $L$ at fixed $U = 12, W = 6.5$ . The dashed line shows power-law fits. For clarity, the results for $L = 60$ are shifted down by 0.07.
Reuse & Permissions

Physical Review B

covering condensed matter and materials physics