# QM3 Quantum Matter meets Maths

## Planned sessions

### Deformed Airy kernel determinants: from KPZ tails to initial data for KdV

Fredholm determinants associated to deformations of the Airy kernel are closely connected to the solution to the Kardar-Parisi-Zhang (KPZ) equation with narrow wedge initial data, and they also appear as largest particle distribution in models of positive-temperature free fermions. I will explain how logarithmic derivatives of the Fredholm determinants can be expressed in terms of a $2\times 2$ Riemann-Hilbert problem.

This Riemann-Hilbert representation can be used to derive precise lower tail asymptotics for the solution of the KPZ equation with narrow wedge initial data, refining recent results by Corwin and Ghosal, and it reveals a remarkable connection with a family of unbounded solutions to the Korteweg-de Vries (KdV) equation and with an integro-differential version of the Painlevé II equation.

### Eigenstate Thermalization, random matrices and (non)local operators in many-body systems

The eigenstate thermalization hypothesis (ETH) is a cornerstone in our understanding of quantum statistical mechanics. The extent to which ETH holds for nonlocal operators (observables) is an open question. I will address this question using an analogy with random matrix theory. The starting point will be the construction of extremely non-local operators, which we call Behemoth operators. The Behemoths turn out to be building blocks for all physical operators. This construction allow us to derive scalings for both local operators and different kinds of nonlocal operators.

### Berry's Phase, $\operatorname{TKN}^2$ Integers and All That: My work on Topology in Condensed Matter Physics 1983-1993

I will give an overview of my work on topological methods in condensed matter physics almost 40 years ago. Include will be Homotopy and $\operatorname{TKN}^2$ integers, holonomy and Berry's phase and quarternions and Berry's phase for Fermions. If time allows, I'll discuss supersymmetry and pairs of projections.

### Hyperbolic band theory

The notions of Bloch wave, crystal momentum, and energy bands are commonly regarded as unique features of crystalline materials with commutative translation symmetries. Motivated by the recent realization of hyperbolic lattices in circuit QED, I will present a hyperbolic generalization of Bloch theory, based on ideas from Riemann surface theory and algebraic geometry. The theory is formulated despite the non-Euclidean nature of the problem and concomitant absence of commutative translation symmetries. The general theory will be illustrated by examples of explicit computations of hyperbolic Bloch wavefunctions and bandstructures.

### Superuniversality of superdiffusion

Emergence of anomalous transport laws in deterministic interacting many-body systems has become a subject of intense study in the past few years. One of the most prominent examples is the unexpected discovery of superdiffusive spin dynamics in the isotropic Heisenberg quantum spin chain with at half filling, which falls into the universality class of the celebrated Kardar-Parisi-Zhang equation. In this talk, we will theoretically justify why the observed superdiffusion of the Noether charges with anomalous dynamical exponent $z=3/2$ is indeed superuniversal, namely it is a feature of all integrable interacting lattice models or quantum field theories which exhibit globally symmetry of simple Lie group $G$, in thermal ensembles that do not break $G$-invariance. The phenomenon can be attributed to thermally dressed giant quasiparticles, whose properties can be traced back to fusion relations amongst characters of quantum groups called Yangians. Giant quasiparticles can be identified with classical solitons, i.e. stable nonlinear solutions to certain integrable PDE representing classical ferromagnet field theories on certain types of coset manifolds. We shall explain why these inherently semi-classical objects are in one-to-one correspondence with the spectrum of Goldstone modes. If time permits, we shall introduce another type of anomalous transport law called undular diffusion that generally occurs amongst the symmetry-broken Noether fields in $G$-invariant dynamical systems at finite charge densities.

### Less is more: effective description of topological spin liquids

It is widely accepted that topological superconductors can only have an effective interpretation in terms of curved geometry rather than gauge fields due to their charge neutrality. This approach is commonly employed in order to investigate their properties, such as the behaviour of their energy currents, though we do not know how accurate it is. I will show that the low-energy properties of the Kitaev honeycomb lattice model, such as the shape of Majorana zero modes or the deformations of the correlation length, are faithfully described in terms of Riemann-Cartan geometry. Moreover, I will present how effective axial gauge fields can couple to Majorana fermions, thus giving a unified picture between vortices and lattice dislocations that support Majorana zero modes.

### Universality of dimers via imaginary geometry

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