# QM3 Quantum Matter meets Maths

## Planned sessions

### Lattice Geometry Dependence and Independence: Important Applications of a Simple Law

The ability to create and manipulate optical lattices for cold atoms, with a view towards studying topological matter, has brought renewed focus to the physics of Bloch waves and the role of the lattice in governing their properties. We consider generic tight binding models where particle motion is described in terms of hopping amplitudes between orbitals. The physical attributes of the orbitals, including their locations in space, are independent pieces of information. We identify a notion of geometry-independence: any physical quantity that depends only on the tight-binding parameters (and not on the explicit information about the orbital geometry) is said to be "geometry-independent." Identification of geometry-dependent vs. independent quantities can be used as a novel principle for constraining a variety of results in both non-interacting and interacting systems. We show, e.g., how Hall measurements based on accelerated lattices or tilted potentials, and those based on applying a chemical potential imbalance between reservoirs, give different results due to the fact that one is geometry-dependent, while the other is geometry-independent. Similar considerations apply for thermal Hall responses in electronic, cold atomic, and spin systems.

Ref:

Steven H. Simon and Mark S. Rudner, Phys. Rev. B 102, 165148, 2020.

### Liquid crystal director fields in three-dimensional non-Euclidean geometries

This work investigates nematic liquid crystals in three-dimensional curved space, and determines which director deformation modes are compatible with each possible type of non-Euclidean geometry. Previous work by Sethna et al. [1] showed that double twist is frustrated in flat space $\mathbb{R}^3$, but can fit perfectly in the hypersphere $\mathbb{S}^3$. Here, we extend that work to all four deformation modes (splay, twist, bend, and biaxial splay) and all eight Thurston geometries [2]. Each pure mode of director deformation can fill space perfectly, for at least one type of geometry. This analysis shows the ideal structure of each deformation mode in curved space, which is frustrated by the requirements of flat space.

1. Sethna J. P., Wright D. C. and Mermin N. D., 1983 Phys. Rev. Lett. 51 467–70.
2. J.-F. Sadoc, R. Mosseri and J. Selinger, New Journal of Physics 22 (2020) 093036.

Projecto FCT UIDB/04459/2020.

### Indicators of quantum chaos and the transition from few- to many-body systems

Quantum chaos, especially when caused by particle interactions, is closely related with topics of high experimental and theoretical interest, from the thermalization of isolated systems to the difficulties to reach a localized phase, and the emergence of quantum scars. In this talk, various indicators of quantum chaos will be compared, including level statistics, structure of eigenstates, matrix elements of observables, out-of-time ordered correlators, and the correlation hole (ramp). These indicators are then employed to identify the minimum number of interacting particles required for the onset of strong chaos in quantum systems with short-range and also with long-range interactions.

Refs:

### Mathematics of magic angles for twisted bilayer graphene

Magic angles are a hot topic in condensed matter physics: when two sheets of graphene are twisted by those angles the resulting material is superconducting. Please do not be scared by the physics though: I will present a very simple operator whose spectral properties are thought to determine which angles are magical. It comes from a recent PR Letter by Tarnopolsky–Kruchkov–Vishwanath. The mathematics behind this is an elementary blend of representation theory (of the Heisenberg group in characteristic three), Jacobi theta functions and spectral instability of non-self-adjoint operators (involving Hoermander’s bracket condition in a very simple setting). The results will be illustrated by colourful numerics which suggest some open problems. This is joint work with M. Embree, J. Wittsten, and M. Zworski.

### Quantum many-body dynamics in two dimensions with artificial neural networks

In the last two decades the field of nonequilibrium quantum many-body physics has seen a rapid development driven, in particular, by the remarkable progress in quantum simulators, which today provide access to dynamics in quantum matter with an unprecedented control. However, the efficient numerical simulation of nonequilibrium real-time evolution in isolated quantum matter still remains a key challenge for current computational methods especially beyond one spatial dimension. In this talk I will present a versatile and efficient machine learning inspired approach. I will first introduce the general idea of encoding quantum many-body wave functions into artificial neural networks. I will then identify and resolve key challenges for the simulation of real-time evolution, which previously imposed significant limitations on the accurate description of large systems and long-time dynamics. As a concrete example, I will consider the dynamics of the paradigmatic two-dimensional transverse field Ising model, where we observe collapse and revival oscillations of ferromagnetic order and demonstrate that the reached time scales are comparable to or exceed the capabilities of state-of-the-art tensor network methods.

### 17/05/2021, 17:00 — 18:00 Europe/Lisbon — Online David Tong, University of Cambridge

The goal of QM3 is to discuss recent topics in quantum matter from a mathematical perspective while building bridges between the physics and mathematics communities.

Organizers: Bruno Mera, João Pimentel Nunes, José Mourão, Pedro Ribeiro, Roger Picken, Vitor Rocha Vieira