Long Range Topological Order, the Chiral Condensate, and the Berry Connection in QCD

TL;DR

This paper explores the analogy between topological insulators and the QCD vacuum, highlighting the role of Berry connection, topological order, and Chern-Simons membranes in understanding long-range topological phenomena in quantum chromodynamics.

Contribution

It proposes a novel topological framework for QCD vacuum structure using Berry phases and Chern-Simons membranes, linking condensed matter concepts to high-energy physics.

Findings

QCD vacuum resembles a topological insulator with discrete vacua.

Chern-Simons membranes support delocalized surface modes for Goldstone pions.

Berry phase describes polarization of Chern-Simons membranes in QCD.

Abstract

Topological insulators are substances which are bulk insulators but which carry current via special "topologically protected" edge states. The understanding of long range topological order in these systems is built around the idea of a Berry connection, which is a gauge connection obtained from the phase of the electron wave function transported over momentum space rather than coordinate space. The phase of a closed Wilson loop of the Berry connection around the Brillouin zone defines a topological order parameter which labels discrete flux vacua. The conducting states are surface modes on the domain walls between discrete vacua. Evidence from large-$N_c$ chiral dynamics, holographic QCD, and Monte Carlo observations has pointed to a picture of the QCD vacuum that is very similar to that of a topological insulator, with discrete quasivacua labelled by $\theta$ angles that differ by mod $2\pi$. In this picture, the domain walls are membranes of Chern-Simons charge, and the quark condensate consists of surface modes on these membranes, which are delocalized and thus support the long range propagation of Goldstone pions. The Berry phase in QED2 describes charge polarization of fermion-antifermion pairs, while in 4D QCD it describes the polarization of Chern-Simons membranes.