Edge states and conformal boundary conditions in super spin chains and super sigma models

TL;DR

This paper determines the exact boundary spectrum of super sigma models on projective superspaces using a spin chain approach, revealing irrational exponents and extending understanding of boundary conditions in non-unitary conformal field theories.

Contribution

It provides the first exact spectrum analysis for all boundary conditions in super sigma models with superspace target spaces, generalizing previous specific cases.

Findings

Exact boundary spectrum for super sigma models derived

Exponents are generally irrational numbers

Method connects spin chains, Brauer algebra, and conformal field theory

Abstract

The sigma models on projective superspaces CP^{N+M-1|N} with topological angle theta=pi mod 2pi flow to non-unitary, logarithmic conformal field theories in the low-energy limit. In this paper, we determine the exact spectrum of these theories for all open boundary conditions preserving the full global symmetry of the model, generalizing recent work on the particular case M=0 [C. Candu et al, JHEP02(2010)015]. In the sigma model setting, these boundary conditions are associated with complex line bundles, and are labelled by an integer, related with the exact value of theta. Our approach relies on a spin chain regularization, where the boundary conditions now correspond to the introduction of additional edge states. The exact values of the exponents then follow from a lengthy algebraic analysis, a reformulation of the spin chain in terms of crossing and non-crossing loops (represented as a certain subalgebra of the Brauer algebra), and earlier results on the so-called one- and two-boundary Temperley Lieb algebras (also known as blob algebras). A remarkable result is that the exponents, in general, turn out to be irrational. The case M=1 has direct applications to the spin quantum Hall effect, which will be discussed in a sequel.