Quantum Distillation of Hilbert Spaces, Semi-classics and Anomaly Matching

TL;DR

This paper explores how symmetry-twisted boundary conditions induce quantum distillation, a mechanism that cancels excited states without affecting ground states, aiding the understanding of nonperturbative QFTs and anomaly matching.

Contribution

It introduces quantum distillation as a new concept linking Hilbert space structure, semi-classical analysis, and anomaly matching in quantum field theories.

Findings

Quantum distillation causes cancellations among excited states.

It explains adiabatic continuity under $S^1$ compactification.

The relation between anomaly persistence and quantum distillation is established.

Abstract

A symmetry-twisted boundary condition of the path integral provides a suitable framework for the semi-classical analysis of nonperturbative quantum field theories (QFTs), and we reinterpret it from the viewpoint of the Hilbert space. An appropriate twist with the unbroken symmetry can potentially produce huge cancellations among excited states in the state-sum, without affecting the ground states; we call this effect "quantum distillation". Quantum distillation can provide the underlying mechanism for adiabatic continuity, by preventing a phase transition under $S^1$ compactification. We revisit this point via the 't Hooft anomaly matching condition when it constrains the vacuum structure of the theory on $\mathbb{R}^d$ and upon compactification. We show that there is a precise relation between the persistence of the anomaly upon compactification, the Hilbert space quantum distillation, and the semi-classical analysis of the corresponding symmetry-twisted path integrals. We motivate quantum distillation in quantum mechanical examples, and then study its non-trivial action in QFT, with the example of the 2D Grassmannian sigma model $\mathrm{Gr}(N,M)$. We also discuss the connection of quantum distillation with large-$N$ volume independence and flavor-momentum transmutation.