Many-body localization and 3D Anderson transition

Yevgeny Bar Lev, David R. Reichman

Hong-Ze Xu, Shunyao Zhang, Zeyu Rao +3 more

The disordered many-body systems can undergo a transition from the extended ensemble to a localized ensemble, known as many-body localization (MBL), which has been intensively explored in recent years. Nevertheless, the relation between Anderson localization (AL) and MBL is still elusive. Here we show that the MBL can be regarded as an infinite-dimensional AL with the correlated disorder in a virtual lattice. We demonstrate this idea using the disordered XXZ model, in which the excitation of $d$ spins over the fully polarized phase can be regarded as a single-particle model in a $d$ dimensional virtual lattice. With the increasing of $d$, the system will quickly approach the MBL phase, in which the infinite-range correlated disorder ensures the saturation of the critical disorder strength in the thermodynamic limit. From the transition from AL to MBL, the entanglement entropy and the critical exponent from energy level statics are shown to depend weakly on the dimension, indicating that belonging to the same universal class. This work clarifies the fundamental concept of MBL and presents a new picture for understanding the MBL phase in terms of AL.

Léa Ferreira dos Santos, Jonathan Torres-Herrera

Ignacio García-Mata, John Martin, O. Giraud +3 more

The Anderson transition in random graphs has raised great interest, partly out of the hope that its analogy with the many-body localization (MBL) transition might lead to a better understanding of this hotly debated phenomenon. Unlike the latter, many results for random graphs are now well established, in particular, the existence and precise value of a critical disorder separating a localized from an ergodic delocalized phase. However, the renormalization group flow and the nature of the transition are not well understood. In turn, recent works on the MBL transition have made the remarkable prediction that the flow is of Kosterlitz-Thouless type. In this paper, we show that the Anderson transition on graphs displays the same type of flow. Our work attests to the importance of rare branches along which wave functions have a much larger localization length ${\ensuremath{\xi}}_{\ensuremath{\parallel}}$ than the one in the transverse direction ${\ensuremath{\xi}}_{\ensuremath{\perp}}$. Importantly, these two lengths have different critical behaviors: ${\ensuremath{\xi}}_{\ensuremath{\parallel}}$ diverges with a critical exponent ${\ensuremath{\nu}}_{\ensuremath{\parallel}}=1$, while ${\ensuremath{\xi}}_{\ensuremath{\perp}}$ reaches a finite universal value ${\ensuremath{\xi}}_{\ensuremath{\perp}}^{c}$ at the transition point ${W}_{c}$. Indeed, ${\ensuremath{\xi}}_{\ensuremath{\perp}}^{\ensuremath{-}1}\ensuremath{\approx}{{\ensuremath{\xi}}_{\ensuremath{\perp}}^{c}}^{\ensuremath{-}1}+{\ensuremath{\xi}}^{\ensuremath{-}1}$, with $\ensuremath{\xi}\ensuremath{\sim}{(W\ensuremath{-}{W}_{c})}^{\ensuremath{-}{\ensuremath{\nu}}_{\ensuremath{\perp}}}$ associated with a new critical exponent ${\ensuremath{\nu}}_{\ensuremath{\perp}}=\frac{1}{2}$, where $exp(\ensuremath{\xi})$ controls finite-size effects. The delocalized phase inherits the strongly nonergodic properties of the critical regime at short scales, but is ergodic at large scales, with a unique critical exponent $\ensuremath{\nu}=\frac{1}{2}$. This shows a very strong analogy with the MBL transition: the behavior of ${\ensuremath{\xi}}_{\ensuremath{\perp}}$ is identical to that recently predicted for the typical localization length of MBL in a phenomenological renormalization group flow. We demonstrate these important properties for a small-world complex network model and show the universality of our results by considering different network parameters and different key observables of Anderson localization.

Ehud Altman

Jan Šuntajs, J. Bonča, Tomaž Prosen +1 more

Characterizing states of matter through the lens of their ergodic properties is a fascinating new direction of research. In the quantum realm, the many-body localization (MBL) was proposed to be the paradigmatic ergodicity breaking phenomenon, which extends the concept of Anderson localization to interacting systems. At the same time, random matrix theory has established a powerful framework for characterizing the onset of quantum chaos and ergodicity (or the absence thereof) in quantum many-body systems. Here we numerically study the spectral statistics of disordered interacting spin chains, which represent prototype models expected to exhibit MBL. We study the ergodicity indicator g=log_{10}(t_{H}/t_{Th}), which is defined through the ratio of two characteristic many-body time scales, the Thouless time t_{Th} and the Heisenberg time t_{H}, and hence resembles the logarithm of the dimensionless conductance introduced in the context of Anderson localization. We argue that the ergodicity breaking transition in interacting spin chains occurs when both time scales are of the same order, t_{Th}≈t_{H}, and g becomes a system-size independent constant. Hence, the ergodicity breaking transition in many-body systems carries certain analogies with the Anderson localization transition. Intriguingly, using a Berezinskii-Kosterlitz-Thouless correlation length we observe a scaling solution of g across the transition, which allows for detection of the crossing point in finite systems. We discuss the observation that scaled results in finite systems by increasing the system size exhibit a flow towards the quantum chaotic regime.