# Condensed Matter and Statistical Mechanics

## 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.

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## Random walks and graphs in materials, biology, and quantum information science

What does mathematics, materials science, biology, and quantum

information science have in common? It turns out, there are many

connections worth exploring. I this talk, I will focus on graphs and random

walks, starting from the classical mathematical constructs and moving on to

quantum descriptions and applications. We will see how the notions of graph

entropy and KL divergence appear in the context of characterizing

polycrystalline material microstructures and predicting their performance

under mechanical deformation, while also allowing to measure adaptation in

cancer networks and entanglement of quantum states. We will discover

unified conditions under which master equations for classical random walks

exhibit nonlocal and non-diffusive behavior and see how quantum walks allow

to realize the coveted exponential speedup in quantum Hamiltonian

simulations. Recent classical and quantum breakthroughs and open questions

will be discussed.

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## Crystallography of hyperbolic lattices: from children's drawings to Fuchsian groups

yperbolic Lattices are tessellations of the hyperbolic plane

using, for instance, heptagons or octagons. They are relevant for quantum

error correcting codes and experimental simulations of quantum physics in

curved space. Underneath their perplexing beauty lies a hidden and,

perhaps, unexpected periodicity that allows us to identify the unit cell

and Bravais lattice for a given hyperbolic lattice. This paves the way for

applying powerful concepts from solid state physics and, potentially,

finding a generalization of Bloch's theorem to hyperbolic lattices. In my

talk, I will explain some of the mathematics underlying this hyperbolic

crystallography.

For other events in this series see the quanTA events website

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## The topology and geometry of the space of gapped lattice systems

Recently there has been a lot of progress in classifying phases of gapped quantum many-body systems. From the mathematical viewpoint, a phase of a quantum system is a connected component of the “space” of gapped quantum systems, and it is natural to study the topology of this space. I will explain how to probe it using generalizations of the Berry curvature. I will focus on the case of lattice systems where all constructions can be made rigorous. Coarse geometry plays an important role in these constructions.

## The Infinite HaPPY Code

I will construct an infinite-dimensional analog of the HaPPY code as a growing series of stabilizer codes defined respective to their Hilbert spaces. These Hilbert spaces are related by isometries that will be defined during this talk. I will analyze its system in various aspects and discuss its implications in AdS/CFT. Our result hints that the relevance of quantum error correction in quantum gravity may not be limited to the CFT context.

For other events in this series see the quanTA events website.

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## Conformal field theories and quantum phase transitions: an entanglement perspective

Quantum phase transitions occur when a quantum system undergoes a sharp change in its ground state, e.g. between a ferro- and para-magnet. I will present a remarkable set of transitions, called quantum critical, that are described by conformal field theories (CFTs). I will focus on 2 and 3 spatial dimensions, where the conformal symmetry is powerful yet less constraining than in 1 dimension. We will probe these scale-invariant theories via the structure of their quantum entanglement. The methods will include large-N expansions, the AdS/CFT duality from string theory, and large-scale numerical simulations. Finally, we’ll see that certain quantum Hall states, which are topological in nature, possess very similar entanglement properties. This hints at broader principles that relate very different quantum states.

For other events in this series see the quanTA events website.

## Conformal Invariance and Universality in the 2D Ising Model

It is conjectured that many 2D lattice models of physical phenomena (percolation, Ising model of a ferromagnet, self avoiding polymers, ...) become invariant under rotations and even conformal maps in the scaling limit (i.e. when "viewed from far away"). A well-known example is the Random Walk (invariant only under rotations preserving the lattice) which in the scaling limit converges to the conformally invariant Brownian Motion.

Assuming the conformal invariance conjecture, physicists were able to make a number of striking but unrigorous predictions: e.g. dimension of a critical percolation cluster is almost surely 91/48; the number of simple length N trajectories of a Random Walk is about N11/32·mN, with m depending on a lattice, and so on.

We will discuss the recent progress in mathematical understanding of this area, in particular for the Ising model. Much of the progress is based on combining ideas from probability, complex analysis, combinatorics.

## Modelling Aperiodic Solids: Concepts and Properties of Tilings and their Physical Interpretation

Topics: Quasicrystals, Quasiperiodicity, Translation module, Repetitivity, Local Isomorphism, Mutual Local Derivability, Matching Rules, Covering Rules, Maximal Coverings