Workshop Recent Advances in Disordered Systems (RAD)

April 7-9 2025

Place: Maison de la Recherche, CY Advanced Studies, Campus Neuville

5 mail Gay-Lussac CS 20601 Neuville

95031 Cergy-Pontoise cedex

The topics include (but are not restricted to) long-range random models, correlated random models, random spin systems and many-body localization, and unitary systems and random quantum walks.

Organizers: H. Boumaza (Sorbonne Nord), C. Rojas-Molina (CY Cergy Paris)

Contributed talks by Matteo Gallone (SISSA), Fabrizio Caragiulo (SISSA), Yulin Cao (Warwick)

Registration is free but compulsory, for organizational purposes. Please fill this form.

There is some limited funding for young researchers. For more information contact crojasmo - at- cyu.fr

Program

Titles and abstracts

Houssam Abdul Rahman (UAE University)

Title: Localization and entanglement in disordered oscillator systems

Abstract: Many-body localization (MBL) is the analogue of Anderson localization in large quantum systems where both disorder and interactions are present. Many-body localized systems do not thermalize under their own dynamics and, as a result, can serve as quantum storage devices. In this talk, we discuss various indicators of the MBL phenomenon. We then present recent results on MBL in a class of disordered oscillator systems. Specifically, we examine the non-equilibrium dynamics of a disordered quantum system composed of coupled harmonic oscillators on a high-dimensional lattice. If the system is sufficiently localized, we demonstrate that, starting from a broad class of initial product states, associated with a tiling (decomposition) of the lattice, the dynamical evolution of entanglement follows an area law after disorder averaging. Furthermore, we present recent entanglement bounds for the low-lying eigenstates of the oscillator system.

Fabrizio Caragiulo (SISSA)

Title: Quantum Hall Effect and quasi-periodic deformations

Abstract: (Integer) Quantum Hall effect has many facets. One is the exact and robust quantization of the transverse conductivity of a 2-dimensional incompressible electron gas. Another is a relationship between indices defined in the bulk of the sample and excitations at the edge: the so-called bulk-edge correspondence. We investigate the validity of this picture under weak quasi-periodic deformations of a non-interacting model, when only one edge-mode is present. Such perturbations are physically relevant in a variety of situations as in Moiré-super-lattices. We use a combination of rigorous Renormalization Group techniques and lattice Ward identities. Partial progress toward the case of multiple edge-modes is also reported.

Christopher Cedzich (Duesseldorf)

Title: A new view on quasiperiodic CMV matrices and the cocycle characterization for the unitary almost Mathieu operator.

Abstract: In this talk, we introduce the unitary almost-Mathieu operator (UAMO) and discuss its connections to other model systems in physics and mathematics. We draw parallels to the self-adjoint almost Mathieu

operator and discuss how the UAMO originates in a two-dimensional quantum walk in a uniform magnetic field as well as its connection to one-dimensional split-step quantum walks and CMV matrices. We exhibit a

version of Aubry–André duality for this model, which partitions the parameter space into three regions: a supercritical region and a subcritical region that are dual to one another, and a critical regime that is self-dual. In each parameter region, we characterize the cocycle dynamics of the transfer matrix cocycle generated by the associated generalized eigenvalue equation. As a practical application, we show how to obtain a lower bound on the Lyapunov exponent, providing a first step for a (complete) spectral characterization of the UAMO.

Margherita Disertori (Bonn)

Title: Fractional Anderson model and long-range self-avoiding walks.

Abstract: The fractional Anderson model has been subject to increasing interest in recent years. I will review some properties and recent results connecting the corresponding Green’s function with the two point function of a self-avoiding walk with long-range jumps. These results adapt a strategy proposed by Schenker in 2015 and are joint work with Constanza Rojas-Molina and Roberto Maturana Escobar. 

Benoît Douçot (LPTHE, CNRS and Sorbonne University)

Title: Quantum transport in disordered Chern insulators

Abstract: the famous bulk-edge correspondence is one of the striking properties that has strongly motivated enormous interest for topological insulators in the past decades. However, in a rather surprising way, recent experiments imaging the non-equilibrium current pattern in disordered Chern insulators have shown that most of the Hall current happens to flow inside the bulk of the spectrum, and not on narrow edge channels, contrary to the most common expectation. I will present a recent attempt (in a collaboration with D. Kovrizhin and R. Moessner) to account for these striking observations, which relies on two key ingredients: a smooth confining electrostatic potential and quenched disorder.

Our main result is the existence of wide meandering conduction channels, that carry the quantized Hall current. Preliminary numerical results suggest, possibly for the first time, an intriguing coexistence of extended energy eigenstates with sharply localized states, at all energies.

Matteo Gallone (SISSA)

Title: Quasi-Periodic Ising model and zero-energy properties of quasi-periodic lattice Dirac Operators

Abstract: Quasi-periodic modulations in statistical mechanics models describe a range of physical systems, including quasi-crystals and experimental setups where quasi-periodicity effectively mimics a disordered background. These modulations are often tunable, leading to transitions between delocalized and localized phases. The two-dimensional classical Ising model, a fundamental example of a critical system, provides an ideal framework for studying the effects of quasi-periodic perturbations on exactly solvable models.

A powerful approach to analyzing the Ising model is through the Grassmann representation of its partition function, which allows critical properties to be understood via the decay of the Green function of a (1+1)-dimensional lattice Dirac operator. Using rigorous renormalization group techniques and the fermionic representation of the model, we demonstrate that small-amplitude quasi-periodic disorder is irrelevant for a broader class of two-dimensional disorders than previously studied in the literature.

This work is in collaboration with Vieri Mastropietro.

Yilun Gao (Warwick)

Title: Spectral and entanglement properties of the random-exchange Heisenberg chain

Abstract: Disordered quantum systems have become an important research topic in modern condensed matter physics ever since the discovery of Anderson localization. The investigation of many-body localization in quantum interacting systems has received much recent attention following the increase of computational power and improvement in numerical methods. We focus on a Heisenberg spin chain with SU(2) symmetry where the exchange couplings between neighboring spins are considered disordered. Both exact diagonalization and sparse matrix diagonalization methods are applied when calculating eigenvalues and eigenvectors of the Hamiltonian matrix. By understanding the structure of eigenvalues and eigenvectors in terms of spin symmetry, we investigate the consecutive gap ratio, participation ratio, and entanglement entropy as a function of disorder strengths. We average over many disorder realizations and compare the results for different disorder distributions. We find, for system size up to L=24, a clear distinction between the SU(2)-invariant random exchange model and the more often studied random field model. In particular, the regime of seemingly localized behavior is much less pronounced in the random exchange model than in the field model case.

Alain Joye (Grenoble) - online

Title : Scattering Quantum Walks on Graphs

Abstract : We consider a class of Unitary Quantum Walks on arbitrary graphs, parameterized by a family of scattering matrices.  After explaining that these Scattering Quantum Walks encompass several known Quantum Walks, we further introduce two classes of Scattering Open Quantum Walks on arbitrary graphs based on that construction, whose asymptotic states we discuss. 

Rudolf Roemer (Warwick)

Title: Some recent results on MBL, random coined quantum walks and random finite-range hopping in 1D 

Valentina Ros (CNRS)

Minicourse on Many-Body Localization

Mostafa Sabri (NYU Abu Dhabi)

Title:The curious spectra and dynamics of non-locally finite crystals

Abstract: I will discuss the spectra and dynamics of Z^d-periodic graphs which are not locally finite but carry non-negative, symmetric and summable edge weights. These graphs are shown to exhibit rather intriguing behaviour. We construct a periodic graph whose Laplacian has purely singularly continuous spectrum. Regarding the point spectrum, we construct a graph with a partly flat band whose eigenvectors must have infinite support. Concerning dynamical aspects, under some assumptions we prove that motion remains ballistic along at least one layer. We also construct a graph whose Laplacian has purely absolutely continuous spectrum, exhibits ballistic transport, yet fails to satisfy a dispersive estimate. This provides a negative answer to an open question in this context. The fractional Laplacian is viewed as a special  case, and we prove for it a phase transition in its dynamical behaviour. Many questions still remain open, and we believe that this class of graphs can serve as a playground to better understand exotic spectra and dynamics.

Xiaolin Zeng (Strasbourg)

Title: Spectral Anderson Transition for a Random Operator Associated to Vertex-Reinforced Jump Process in One Dimension with a Decaying Potential

Abstract: We study the spectral transition between localization and delocalization of a Schrödinger operator with a particular dependent random potential in a one-dimensional setting with decaying parameters. This transition governs the transition of long-term behavior of the vertex-reinforced jump process (VRJP); and that of the \(H^{2|2}\) supersymmetric hyperbolic sigma model. We will discuss the phase transitions of these three models in a comparative context. This work is in collaboration with Fumihiko Nakano and Laure Marêché.