Condensed Matter and Statistical Mechanics

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.

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.

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.

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.

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.