Onsager symmetries in $U(1)$-invariant clock models

Abstract

We show how the Onsager algebra, used in the original solution of the two-dimensional Ising model, arises as an infinite-dimensional symmetry of certain self-dual models that also have a $U(1)$ symmetry. We describe in detail the example of nearest-neighbour $n$-state clock chains whose ${\mathbb{Z}}_n$ symmetry is enhanced to $U(1)$. As a consequence of the Onsager-algebra symmetry, the spectrum of these models possesses degeneracies with multiplicities $2^N$ for positive integer $N$. We construct the elements of the algebra explicitly from transfer matrices built from non-fundamental representations of the quantum-group algebra $U_q(s{l}_2)$. We analyse the spectra further by using both the coordinate Bethe ansatz and a functional approach, and show that the degeneracies result from special exact $n$-string solutions of the Bethe equations. We also find a family of commuting chiral Hamiltonians that break the degeneracies and allow an integrable interpolation between ferro- and antiferromagnets.

arXiv commentv2: 39 pages

References in corpus (24)

• Quantum Inverse Scattering Method and Correlation Functions
• Parafermionic edge zero modes in Z_n-invariant spin chains
• Non-equilibrium spin transport in integrable spin chains: persistent currents and emergence of magnetic domains
• Quasilocal conservation laws in XXZ spin-1/2 chains: open, periodic and twisted boundary conditions
• Continuity of the phase transition for planar random-cluster and Potts models with $1\le q\le 4$
• Bethe's equation is incomplete for the XXZ model at roots of unity
• Representations of the Virasoro algebra from lattice models
• Quasilocal conservation laws from semicyclic irreducible representations of $U_q({\mathfrak{sl}}_2)$ in $XXZ$ spin-$1/2$ chains
• Conformal data and renormalization group flow in critical quantum spin chains using periodic uniform matrix product states
• Quasi-local conserved charges and spin transport in spin-$1$ integrable chains
• Spin chains with dynamical lattice supersymmetry
• The L(sl_2) symmetry of the Bazhanov-Stroganov model associated with the superintegrable chiral Potts model
• A Q-Operator Identity for the Correlation Functions of the Infinite XXZ Spin-Chain
• On integrable Hamiltonians for higher spin XXZ chain
• The Many Faces of the Chiral Potts Model
• Auxiliary matrices for the six-vertex model at roots of 1 and a geometric interpretation of its symmetries
• Physics beyond quasi-particles: Spectrum and completeness of the 3 state superintegrable chiral Potts model
• Auxiliary matrices for the six-vertex model at roots of unity II
• Loop symmetry of integrable vertex models at roots of unity
• R-matrices of three-state Hamiltonians solvable by Coordinate Bethe Ansatz
• FRT presentation of the Onsager algebras
• Auxiliary matrices on both sides of the equator
• The Transfer Matrix of Superintegrable Chiral Potts Model as the Q-operator of Root-of-unity XXZ Chain with Cyclic Representation of $U_q(s{l}_2)$
• Classification of three-state Hamiltonians solvable by Coordinate Bethe Ansatz

Cited by in corpus (27)

• Onsager's Scars in Disordered Spin Chains
• Systematic construction of scarred many-body dynamics in 1D lattice models
• Large Classes of Quantum Scarred Hamiltonians from Matrix Product States
• From tunnels to towers: quantum scars from Lie Algebras and q-deformed Lie Algebras
• Probing symmetries of quantum many-body systems through gap ratio statistics
• Lack of symmetry restoration after a quantum quench: an entanglement asymmetry study
• The "not-A", RSPT and Potts phases in an $S_3$-invariant chain
• Quantized axial charge of staggered fermions and the chiral anomaly
• Lattice T-duality from non-invertible symmetries in quantum spin chains
• The alternating central extension of the $q$-Onsager algebra
• Conjectures on Hidden Onsager Algebra Symmetries in Interacting Quantum Lattice Models
• Generalised Onsager Algebra in Quantum Lattice Models
• Closed hierarchy of Heisenberg equations in integrable models with Onsager algebra
• Integrable Quantum Circuits from the Star-Triangle Relation
• Self-dual $S_3$-invariant quantum chains
• Disorder enhanced quantum many-body scars in Hilbert crystals
• Pivoting through the chiral-clock family
• Hidden quasi-local charges and Gibbs ensemble in a Lindblad system
• Exact Chiral Symmetries of 3+1D Hamiltonian Lattice Fermions
• ${\mathbb{Z}}_2$ gauging and self-dualities of the $XX$ model and its cousins
• Systematic construction of asymptotic quantum many-body scar states and their relation to supersymmetric quantum mechanics
• Defining relations for quantum symmetric pair coideals of Kac-Moody type
• Charge pumps, pivot Hamiltonians and symmetry-protected topological phases
• Symmetry-Enforced Fermi Surfaces
• On the Q operator and the spectrum of the XXZ model at root of unity
• Equivalence class of Emergent Single Weyl fermion lattice models in 3 dimensions: gapless superconductors and superfluids versus chiral fermions
• Parity anomaly from LSM: exact valley symmetries on the lattice