Search Talks

Many-body localization is the paradigm for how interacting quantum systems can resist thermalization in the presence of strong disorder.
After a brief recap on the main ideas of many-body localization,
I will present new results in the limit of weakly interacting systems, where our numerical simulations indicate that below a certain disorder threshold, weak interactions necessarily lead to ergodic instabilities.


Reference: Jeanne Colbois, Fabien Alet, Nicolas Laflorencie, Phys. Rev. Lett. 133, 116502 (2024)

We have used Langevin dynamics simulations to study the effects of activity in a two-dimensional athermal glass-forming system of Lennard-Jones particles. We consider the limit of infinite persistence time in which the self-propulsion forces on the particles have the same magnitude but different directions that do not change with time. This system exhibits a liquid state for large values of the self-propulsion force and a force-balanced jammed state if the self-propulsion force is smaller than a threshold value. The liquid state is found to exhibit long-range correlations. A length scale extracted from spatial correlations of the velocity field increases with system size as a power law with exponent close to one. Spatial correlations of the self-propulsion forces also exhibit a similar length scale, indicating that the particles self-organize to form a steady state in which particles with similar directions of self-propulsion forces come close to one another and move together. This state is “critical” in the sense that it exhibits a correlation length that diverges in the limit of infinite system size. The velocity pattern in the steady state exhibit an intriguing asymmetry. The development of correlations in time, starting from an initial state with random velocities and forces, is analogous to that in domain growth and coarsening in spin systems after a quench from the disordered to the ordered state. However, quantitative features of this process appear to be different from those in domain growth in spin systems with the same symmetry.

This work was done in collaboration with Suman Dutta, Atharva Shukla, Pinaki Chaudhuri and Madan Rao.

The study of Langevin dynamics within highly non-convex random landscapes has been crucial for understanding fundamental aspects of glassy dynamics such as aging, violations of fluctuation-dissipation relations, the emergence of effective temperatures, and the role of high dimensionality and entropy in slowing down relaxation. A comprehensive understanding of the activated regime of the dynamics, where the system transitions between metastable states by overcoming energy barriers, remains however elusive. In the talk I will consider a prototypical model of a high-dimensional energy landscape with Gaussian statistics, exhibiting plenty of metastable states. After recalling the main reasons why activated dynamics is theoretically challenging, I will consider effective processes in which the system jumps in the landscape, visiting the closest metastable states at given energy. Understanding this effective dynamics requires to analyze the geometry of the landscape, particularly the joint distribution of triplets of metastable states that lie in the same region of the high-dimensional configuration space. I will report on the results of this geometrical analysis, and comment on implication for activated dynamics.    

We will discuss open questions in particulate flows, and why present methods fall short in answering the basic questions. We'll then talk about ways forward, and I will present some of my recent results.

Multicomponent quantum mixtures in one dimension can be characterized by their symmetry under particle exchange. For a strongly interacting Bose-Bose mixture, we show that the time evolution of the momentum distribution from an initially symmetry-mixed state is quasiconstant for a SU(2) symmetry conserving Hamiltonian, while it displays large oscillations in time for the symmetry-breaking case where inter- and intraspecies interactions are different. Using the property that the momentum distribution operator at strong interactions commutes with the class-sum operator, the latter acting as a symmetry witness, we show that the momentum distribution oscillations correspond to symmetry oscillations, with a mechanism analogous to neutrino flavor oscillations.

We explore the dynamics of the simplest worm algorithm for a class of two-dimensional dimer models and argue that the dynamics of the worm head represents an example of fractional Brownian motion whenever dimer correlations in equilibrium have power-law character. Numerical estimates of the corresponding Hurst exponent and persistence exponent are obtained, and it is further argued that the Hurst exponent is completely determined by equilibrium dimer correlations, while the persistence exponent is additionally influenced by equilibrium correlations between test monomers.

The Weil representation zeta function of a group G is a generating function counting the absolutely irreducible representations of G over all finite fields. It is reminiscent of the Hasse-Weil zeta function of algebraic varieties and converges for the large class of UBERG groups. We give a short introduction, discuss the abscissa of convergence and present some examples. Even for procyclic groups it can be difficult to determine the abscissa of convergence due to close relations to open problems in number theory. We will explain how to calculate the Weil abscissa for random procyclic groups. (based on joint work with Ged Corob Cook and Matteo Vannacci)

We study a variant of the Iwahori-Hecke algebra of a Coxeter group, whose generators T_i satisfy the braid relations but are assumed to be nilpotent (in parallel to Coxeter groups where the T_i are involutions, and 0-Hecke algebras where they are idempotent). Motivated by Coxeter (1957) and Broue-Malle-Rouquier (1998), we classify the finite-dimensional among these "generalized nil-Coxeter algebras". These turn out to be the usual nil-Coxeter algebras, and exactly one other type-A family of algebras, which have a finite "word basis" in the T_i and a unique longest word.

In the remaining time I will present joint work with Sutanay Bhattacharyya, in which we explore the "Temperley-Lieb" variant of the above, wherein all sufficiently long braid words are also killed. Now the finite-dimensional algebras obtained include ones with bases indexed by:
(a) words with a unique reduced expression (any Coxeter type),
(b) fully commutative words (counted by Stembridge),
(c) Catalan numbers (via the XYX algebras of Postnikov), and
(d) the \bar{12} avoiding signed permutations (in type B=C).

Let G be a nice (connected reductive) Lie group. An irreducible representation of G, when restricted to a maximal torus, decomposes into weights with multiplicity. We outline a procedure to compute symmetric polynomials (e.g., power sums) of this multiset of weights in terms of the highest weight. This is joint work with Rohit Joshi.

Similar to standard growth of (finitely generated) groups, one can define conjugacy growth of groups which, informally, counts the number of conjugacy classes in a ball of radius n in a Cayley graph. This was first studied by Riven for free groups, and techniques from geometry, combinatorics and formal language theory have proven to be useful for determining information about the conjugacy growth series for a variety of groups.
This talk will provide a survey on the key tools and results about conjugacy growth. Time permitting, I’ll also discuss joint work with Laura Ciobanu, where we studied conjugacy growth in dihedral Artin groups.

In this talk, I will discuss zeta functions that count the number of twisted conjugacy classes of a fixed group.

Twisted conjugacy is a generalisation of the usual conjugacy, where we introduce a twist by an endomorphism. Specifically, given a group G and an automorphism f, we consider the action gx = gx f(g)^{-1}. The orbits of this action are known as twisted conjugacy classes, or Reidemeister classes.

Recent years have seen intensive investigation into the sizes of these classes. A major goal in this area is to classify groups where all classes are infinite. For groups that do not possess this property, the focus shifts to understanding the possible sizes of the classes, among all automorphisms.

In this talk, we will see that, as typical, these zeta functions admit Euler product decompositions with rational local factors, and we will explore how these zeta functions can be utilised to understand twisted conjugacy classes of certain nilpotent groups.