Seminars

Shaffique Adam (National University of Singapore): Localization in two-dimensional materials

In this talk I will survey some of our work in localization in two dimensions. I will start the discussion by considering graphene [1] where the role of topology guarantees that the system never localizes, even in the presence of strong (smooth) disorder. The second example is two-dimensional transition metal dichalcogenides [2] where we show how the crossover from weak localization to weak antilocalization allow us to infer that spin relaxation occurs via the Dyakanov-Perel mechanism. Finally, we show numerical evidence of many-body localization in two dimensions [3], and give arguments on how to evade proliferation of thermal avalanches in this situation [4].

[1] S. Adam, P. Brouwer, and S. Das Sarma, “Crossover from quantum to Boltzmann transport in graphene”, Phys. Rev. B. 79 201404R (2009); [2] H. Schmidt et al. “Quantum Transport and Observation of Dyakonov-Perel Spin-Orbit Scattering in Monolayer MoS2”, Phys. Rev. Lett. 116, 046803 (2016); [3] Ho-Kin Tang, Nyayabanta Swain, Darryl Foo, Brian Khor, Gabriel Lemarié, Fakher Assaad, Shaffique Adam, and P. Sengupta, “Evidence of many-body localization in 2D from quantum Monte Carlo simulation”, arXiv:2106.08587 (2021); [4] Darryl Foo, Nyayabanta Swain, Pinaki Sengupta, Gabriel Lemarié, and Shaffique Adam, “A stabilization mechanism for many-body localization in two dimensions”, arXiv:2202.09072 (2022).

Nicolas Dupuis (Sorbonne Université): One-dimensional disordered Bose fluid: from Bose glass to Mott glass

In a one-dimensional Bose fluid, disorder can induce a quantum phase transition between a superfluid phase (Luttinger liquid) and a localized phase (Bose glass). Using bosonization, the replica method and a nonperturbative functional renormalization-group approach, we find that the Bose-glass phase is described by a fully attractive strong-disorder fixed point characterized by a singular disorder correlator whose functional dependence assumes a cuspy form. This reveals the glassy properties (pinning, "shocks" and "avalanches") due to the existence of metastable states, as well as the crucial role of quantum tunneling between different metastable configurations. We also show that long-range interactions can stabilize a Mott glass, i.e. a state intermediate between a Mott insulator and a Bose glass, and characterized by a vanishing compressibility and a gapless optical conductivity.

Olivier Giraud (Université Paris-Saclay): Perturbation theory approach for multifractal dimensions

We consider simple random matrix models with multifractal eigenstates. We will calculate explicitly multifractal dimensions for some simple examples of random matrices, using perturbation theory.  This will allow us to demonstrate relations such as the symmetry relation for multifractal exponents, the connection between information dimension and compressibility, or the logarithmic correlations in multifractal matrix ensembles.

Ivan Khaymovich (Nordita Stokholm): Random-matrix approach to ergodicity breaking and slow dynamics in quantum systems

Ergodicity breaking based on the coherent quantum nature of interacting systems inevitably appear and play a crucial role in technological applications, such as computing, classical or quantum, and machine learning. Great efforts have been applied to ergodicity breaking over the last decades, motivated by the fact that ergodic systems keep no information about their initial state. Currently, the attention is focused on many-body localization (MBL) where all degrees of freedom are localized via disorder in the on-site potential of the system and break ergodicity, being a prototype of a quantum memory. However, the complexity of quantum many-body systems prevents one from the analytical description of it. Thus, it is of particular concern and high demand to model essential attributes of these phenomena on a universal level. Random matrix theory (RMT) provides such a universal and powerful approach to describe the thermalizing many-body quantum systems and sets the basis for a quantum eigenstate thermalization hypothesis, (ETH). However, unlike the thermalizing case, there is no framework of the same generality to ETH to describe them. In this talk, I will focus on an alternative RMT framework [1-3] for description of ergodicity breaking phenomena, basing on the random-matrix theory and statistical mapping of the Hilbert-space structure of many-body systems to RMT. The similarity between non-ergodic delocalized phase, found in a so-called Rosenzweig-Porter model, with the wave-function structure of the MBL phase in the Hilbert space allows one to develop the mapping between these systems. The same is also true for the graph prototypes of MBL [4-5]. This statistical mapping opens a simple and universal way to describe the ergodicity-breaking physics of the many-body models in the Hilbert space and on the disordered  ierarchical graphs, similar in spirit of how RMT describes the ergodic behavior. As a generic example of this approach, we consider the static [1-2] and the dynamical [3] phases in a Rosenzweig-Porter random matrix ensemble with a distribution of off-diagonal matrix elements of the form of the large-deviation ansatz, relevant for the MBL description [4]. We show that our model contains four possible phases [1]. These are not only the localized and the fully ergodic (RMT-like) phases which existence is known long ago. Our model supports also the genuine multifractal phase which is currently vigorously discussed in connection with MBL and which local spectrum is multifractal, and a "bad metal" one related to slow dynamics. We present a general theory of dynamical properties (survival probability) in such a random-matrix model [3] and show that the averaged survival probability may decay with time as a simple exponent, stretch-exponent, and power-law or slower. Correspondingly, we identify the exponential, the stretch-exponential and the frozen-dynamics phases. We consider the mapping of the Anderson localization model on Random Regular Graph [2, 6], the known proxy of MBL, onto the RP model and find exact values of the stretch-exponent kappa in the thermodynamic limit. Our theory allows to describe analytically the finite-size multifractality and to compute the critical length with the exponent 1 associated with it.

[1] I. M. Khaymovich, V. E. Kravtsov, B. L. Altshuler, L. Ioffe, “Fragile ergodic phases in logarithmically-normal Rosenzweig-Porter model”, Phys. Rev. Res. 2, 043346 (2020) [arXiv:2006.04827]; [2] V. E. Kravtsov, I. M. Khaymovich, B. L. Altshuler, L. B. Ioffe, “Localization transition on the Random Regular Graph as an unstable tricritical point in a log-normal Rosenzweig-Porter random matrix ensemble”, [arXiv:2002.02979]; [3] I. M. Khaymovich, V. E. Kravtsov, “Dynamical phases in a "multifractal" Rosenzweig-Porter model”, SciPost Phys. 11(2), 045 (2021) [arXiv:2106.01965]; [4] M. Tarzia, “Many-Body localization transition in Hilbert space”, Phys. Rev. B 102, 014208 (2020);  [5] G. Biroli and M. Tarzia, “Lévy-Rosenzweig-Porter random matrix ensemble”, Phys. Rev. B 103, 104205 (2021) [arXiv: 2012.12841];  [6] P. A. Nosov, I. M. Khaymovich, A. Kudlis, V. E. Kravtsov, “Statistics of Green's functions on a disordered Cayley tree and the validity of forward scattering approximation”, SciPost Phys. 12(2), 048 (2022) [arXiv:2108.10326].

Vladimir Kravtsov (ICTP Trieste): Non-ergodic extended states in the Anderson localization problem and the singular-continuous spectrum

We consider the random matrix models which possess multifractal statistics of eigenvectors and show how the random Cantor set or a mini-band of levels appear in the local spectrum provided that the global spectrum is everywhere dense. We consider two examples: the Power-law banded random matrices and the Rosenzweig-Porter random matrix theory which are shown to be an arch-typical examples of the random Cantor set and the mini-band. We show that the correlation function of the local densities of states at coincident sites and different energies is a proper measure of a singular-continuous spectra. Namely, in the limit of infinite matrix size the correlation function in the non-ergodic extended phase must have a singularity at the energy difference tending to zero. This singularity has a form of a branch-cut for random Cantor set and is a delta-function for a mini-band. Next we suggest a general criterion of a random Cantor set and consider an inner and an outer fractality of local spectra.

Jorge Kurchan (ENS Paris): Designs, Free Probability, and the Eigenstate Thermalization Hypothesis

These three different approaches to quantum ergodicity have several bridges connecting them. I will introduce the three ideas, and describe the connections between them.

Markus Muller (PSI Villigen): Electron glasses – Coulomb frustration, localization and quantum glassiness

I will review salient properties of electron glasses – localized electrons subject to disorder and Coulomb repulsion. Surprisingly, despite the lightness of electrons, those can exhibit extremely slow glassy dynamics, including memory effects and aging. I will discuss the how the electron glass quantum melts into a disordered metal, and what we can say about this transition in synthetic model systems of cold fermionic atoms in multi-mode cavities. An issue of particular interest are the low energy excitations in the classical and the quantum regime and their impact on transport.

Piotr Sierant (ICFO Barcelona): Possible interpretations of numerical results for MBL

The status of many-body localization (MBL) as of a stable non-ergodic phase of matter has been recently debated. In this talk, I will explore different interpretations of numerical results obtained with classical simulations of strongly disordered quantum many-body systems. Starting from the definition of MBL phase (which will be contrasted with an MBL regime), I will emphasize the role of interactions in slow dynamics of disordered many-body systems [1], highlighting the connections with other approaches [2,3]. These findings will be linked to the spectral properties of many-body systems, as reflected by the so called Thouless time [4,5]. Subsequently, I will introduce a simple method of analysis of the ergodic-MBL crossover in exact diagonalization results. This method involves the introduction of two system size dependent disorder strengths: the first one delineates departure from ergodic behavior at given system size L, while the second one is the crossing point that estimates, at given L, the position of the putative ergodic-MBL transition. I will present results of this method for 1D disordered systems: Heisenberg spin chain [6] (and compare these results with earlier interpretations [4,7,8], constrained spin chains [9], and Floquet models [10]. Finally, I will draw comparisons between the observed finite size drifts at the ergodic-MBL crossover in interacting many-body systems and the finite size effects observed in the Anderson localization transition on random regular graphs [11]. Overall, this talk aims to shed light on the ongoing discussions surrounding the MBL phase and its characterization in disordered many-body systems.

[1] PS, J. Zakrzewski, Phys. Rev. B 105, 224203 (2022); [2] M. Kiefer-Emmanouilidis, R. Unanyan, M. Fleischhauer, J. Sirker, Phys. Rev. Lett. 124, 243601 (2020); [3] F. Evers, S. Bera, arXiv:2302.11384; [4] J. Šuntajs, J. Bonča, T. Prosen, L. Vidmar, Phys. Rev. E 102, 062144 (2020); [5] PS, D. Delande, J. Zakrzewski, Phys. Rev. Lett. 124, 186601 (2020); [6] PS, M. Lewenstein, J. Zakrzewski, Phys. Rev. Lett. 125, 156601 (2020); [7] V. Oganesyan, D. Huse, Phys. Rev. B 75, 155111 (2007); [8] D. Luitz, N. Laflorencie, F. Alet, Phys. Rev. B 91, 081103(R) (2015); [9] PS, E. Lazo, M. Dalmonte, A. Scardicchio, J. Zakrzewski, Phys. Rev. Lett. 127, 126603 (2021); [10] PS, M. Lewenstein, A. Scardicchio, J. Zakrzewski, Phys. Rev. B 107, 115132 (2023); [11] PS, M. Lewenstein, A. Scardicchio, arXiv:2205.14614 (accepted in SciPost Physics)