Seminars from until

I give an overview of the insights we and other people have had into the band structure of magnons and discuss in some detail three main topics from our work: (i) the robustness of topological edge states in the presence of magnon interactions (ii) visualization of spin-momentum locking in magnon systems (iii) the non-Hermitian topology of spontaneous magnon decay.

The localization tensor is a measure of distinguishability between insulators and metals. This tensor is related to the quantum metric tensor associated with the occupied bands in momentum space. In two dimensions and in the thermodynamic limit, it defines a flat Riemannian metric over the twist-angle space, topologically a torus, which endows this space with a complex structure, described by a complex parameter τ . It is shown that the latter is a physical observable related to the anisotropy of the system. The quantity τ and the Riemannian volume of the twist-angle space provide an invariant way to parametrize the flat quantum metric obtained in the thermodynamic limit. Moreover, if by changing the couplings of the theory, the system undergoes quantum phase transitions in which the gap closes, the complex structure τ is still well defined, although the metric diverges (metallic state), and it is fixed by the form of the Hamiltonian near the gap closing points. The Riemannian volume is responsible for the divergence of the metric at the phase transition.

[1] Bruno Mera. Localization anisotropy and complex geometry in two-dimensional insulators. Phys. Rev. B, 101:115128, Mar 2020.

This group of astronomical order is slowly yielding its secrets. It is the symmetry group of a rational conformal field theory. In this introductory talk, I will discuss the functions that constitute monstrous moonshine and explain the importance of the monster group and its connections with better established parts of mathematics.

In this talk we present the Law of Large numbers for three quantities (local density, current and mass) for the Symmetric Simple Exclusion Process (SSEP) on the lattice $\{1, . . . , N − 1\}$ with “nonlinear” boundary dynamics. Informally, we let a particle jump only to its neighbor site if such site is empty. Then, we let the system be in contact with two reservoirs, which inject/remove particles from a window of size $K$ from the boundaries, at rates depending on the site of injection/removal. We let a particle enter to the first free site, and leave from the first occupied site. This boundary dynamics impose strong correlations between particles, which leads to the sudy of most physical quantities of the system being a challenge. Multiplying the boundary rates by $N^{-\theta}$, one observes macroscopically phase transitions on those quantities in the following way. Under a $N^2$ time-scale, macroscopically the local density behaves as a weak solution to the heat equation. For $\theta \in [0, 1)$ we have Dirichlet B.C., nonlinear Robin for $\theta = 1$, and Neumann for $\theta \gt 1$. For the current, we microscopically derive Fick’s Law, which depends on the B.C. for the density, while for the mass, we see that instead the time scale $N^{1+ \theta}$ is the most natural one, and obtain an ODE. We present only results for $\theta \geq 1$. We then show that starting from the stationary measure, we obtain steady state solutions of the aforementioned equations.

Bibliography:

[1] Gonçalves, P., Erignoux, C., Nahum, G.: Hydrodynamics for SSEP with non-reversible slow boundary dynamics: Part I, the critical regime and beyond, arXiv:1912.09841 (2019).

Optimal stopping problems constitute a subset of stochastic control problems in which one is interest in finding the best time to take a given action. This framework has relevant contributions extending across different fields, namely finance, game theory and statistics. Recently the literature on machine learning has grown at a very large pace, specially in what concerns the usage of its techniques in other fields beyond computer science, in the hope that those might shed some light in long persisting problems such as, for instance, the well known curse of dimensionality. In light with this trend the literature on both stochastic control and optimal stopping has presented several contributions by either incorporating reinforcement learning techniques (Machine Learning Control) or by making use of neural networks to estimate the optimal stopping time of a given problem.

Bibliography:

[1] G. Peskir and A. Shiryaev, Optimal stopping and free-boundary problems, 2006, Springer.

[2] W. H. Fleming and H. M. Soner, Controlled Markov processes and viscosity solutions vol. 25, 2006, Springer Science & Business Media.

[3] S. Becker, P. Cheridito and A. Jentzen, Deep optimal stopping, Journal of Machine Learning Research, vol. 20, 2019.

[4] S. Becker, P. Cheridito, A. Jentzen and T. Welti, Solving high-dimensional optimal stopping problems using deep learning, arXiv preprint arXiv:1908.01602.

[5] H.J. Kappen, An introduction to stochastic control theory, path integrals and reinforcement learning, AIP conference proceedings, vol. 887, nr. 1, pp. 149–181, 2007.

[6] E. Theodorou, J. Buchli and S. Schaal, A generalized path integral control approach to reinforcement learning, Journal of Machine Learning Research, vol. 11, Nov, pp. 3137–3181, 2010.

We consider the computation of out-of-time-ordered correlators (OTOCs) in the fishnet theories, with a mass term added. These fields theories are not unitary. We compute the growth exponent, in the planar limit, at any value of the coupling and show that the model exhibits chaos.