Quantum spin chains with quantum group symmetry

TL;DR

This paper investigates the actions of quantum groups on lattice spin systems, showing that local actions are classical and exploring the nature of semi-local actions, their invariance properties, and implications for ground states.

Contribution

It demonstrates that local quantum group actions on lattice systems are essentially classical and introduces the concept of semi-local actions that preserve localized invariants.

Findings

Local actions of quantum groups are equivalent to classical groups.

Semi-local actions can exist with quantum group invariance.

Ground states invariant under quantum groups have no unitary representation in GNS.

Abstract

We consider actions of quantum groups on lattice spin systems. We show that if an action of a quantum group respects the local structure of a lattice system, it has to be an ordinary group. Even allowing weakly delocalized (quasi-local) tails of the action, we find that there are no actions of a properly quantum group commuting with lattice translations. The non-locality arises from the ordering of factors in the quantum group C*-algebra, and can be made one-sided, thus allowing semi-local actions on a half chain. Under such actions, localized quantum group invariant elements remain localized. Hence the notion of interactions invariant under the quantum group and also under translations, recently studied by many authors, makes sense even though there is no global action of the quantum group. We consider a class of such quantum group invariant interactions with the property that there is a unique translation invariant ground state. Under weak locality assumptions, its GNS representation carries no unitary representation of the quantum group.