Quantum spin chains with quantum group symmetry

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.

plain TeX, 42 pages, no figures

References in corpus (6)

• Diagonalization of the XXZ Hamiltonian by Vertex Operators
• Analytical Bethe Ansatz for Quantum-Algebra-Invariant Spin Chains
• Braided Matrix Structure of the Sklyanin Algebra and of the Quantum Lorentz Group
• Ground states of the infinite q-deformed Heisenberg ferromagnet
• q-deformations of quantum spin chains with exact valence-bond ground states
• Positive representations of general commutation relations allowing Wick ordering

Cited by in corpus (20)

• Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, and Theorems
• A Density Matrix Algorithm for 3D Classical Models
• Developments in the Tensor Network -- from Statistical Mechanics to Quantum Entanglement
• On topological phases of spin chains
• Stochastic Models on a Ring and Quadratic Algebras. The Three Species Diffusion Problem
• N-species stochastic models with boundaries and quadratic algebras
• Ground states of the infinite q-deformed Heisenberg ferromagnet
• Structure of irreducibly covariant quantum channels for finite groups
• Symmetry protected topological phases beyond groups: The q-deformed Affleck-Kennedy-Lieb-Tasaki model
• The spectral gap of the ferromagnetic XXZ chain
• Entanglement in valence-bond-solid states on symmetric graphs
• Product Wave Function Renormalization Group: construction from the matrix product point of view
• A Classical Background for the Wave Function Prediction in the Infinite System DMRG Method
• Two-Site shift Product Wave Function Renormalization Group Method Applied to Quantum Systems
• Symmetry protected topological phases beyond groups: The q-deformed bilinear-biquadratic spin chain
• Duality and hidden symmetry breaking in the q-deformed Affleck-Kennedy-Lieb-Tasaki model
• Quantum Chains of Hopf Algebras with Order-Disorder Fields and Quantum Double Symmetry
• $Q$-Deformed Rainbows: a Universal Simulator of Free Entanglement Spectra
• A brief note on the G${}_2$ Affleck-Kennedy-Lieb-Tasaki chain
• Quantum symmetry groups of noncommutative tori