Condensed Matter Theory

Abstract:

We derive exact strong zero mode (ESZM) operators for integrable spin- S <math display="inline"> <mi>S</mi> </math> chains with open boundary conditions and a boundary field. Their locality properties are generally weaker than in the previously known cases, but they still imply infinite coherence times in the vicinity of the edges. We explain how such integrable chains possess multiple ground states describing a first-order quantum phase transition, and that the odd number of such states for integer S <math display="inline"> <mi>S</mi> </math> makes the weaker locality properties necessary. We make contact with more traditional approaches by showing how the ESZM for S=\tfrac12 <math display="inline"> <mrow> <mi>S</mi> <mo>=</mo> <mstyle displaystyle="false"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </math> acts on energy eigenstates given by solutions of the Bethe equations.

Abstract:

These notes accompany lectures given in June 2025 at the summer school Fundamental Problems in Statistical Physics XVI. They offer a short introduction to random quantum circuits as simple models for generic many-body quantum systems. They give an outline of the motivation for introducing these models, starting from ideas of random matrix theory. They also provide a sketch of calculations of some of the quantities of most physical interest, based on an average over an ensemble of systems. These quantities give insights into operator spreading, entanglement dynamics and spectral correlations.