PT Symmetry on the Lattice: The Quantum Group Invariant XXZ Spin-Chain

Abstract

We investigate the PT-symmetry of the quantum group invariant XXZ chain. We show that the PT-operator commutes with the quantum group action and also discuss the transformation properties of the Bethe wavefunction. We exploit the fact that the Hamiltonian is an element of the Temperley-Lieb algebra in order to give an explicit and exact construction of an operator that ensures quasi-Hermiticity of the model. This construction relys on earlier ideas related to quantum group reduction. We then employ this result in connection with the quantum analogue of Schur-Weyl duality to introduce a dual pair of C-operators, both of which have closed algebraic expressions. These are novel, exact results connecting the research areas of integrable lattice systems and non-Hermitian Hamiltonians.

32 pages with figures, v2: some minor changes and added references, version published in JPA

References in corpus (8)

• Physical Aspects of Pseudo-Hermitian and $PT$-Symmetric Quantum Mechanics
• PT-symmetric Deformations of the Korteweg-de Vries Equation
• PT-Symmetric Extension of the Korteweg-de Vries Equation
• Non-Hermitian Quantum Mechanics of Non-diagonalizable Hamiltonians: puzzles with self-orthogonal states
• Non-Hermitian Hamiltonians with real eigenvalues coupled to electric fields: from the time-independent to the time dependent quantum mechanical formulation
• PT-symmetry and its spontaneous breakdown explained by anti-linearity
• Completeness of ``Good'' Bethe Ansatz Solutions of a Quantum Group Invariant Heisenberg Model
• On the Two-Point Correlation Function for the $U_q[SU(2)]$ Invariant Spin One-Half Heisenberg Chain at Roots of Unity

Cited by in corpus (56)

• Non-Hermitian Physics
• Parity-time-symmetric topological superconductor
• Solutions of PT symmetric tight-binding chain and its equivalent Hermitian counterpart
• Entanglement Entropy of Non Unitary Conformal Field Theory
• Kondo effect in a parity-time-symmetric non-Hermitian Hamiltonian
• A spin chain model with non-Hermitian interaction: The Ising quantum spin chain in an imaginary field
• The robust $\mP\mT$-symmetric chain and properties of its Hermitian counterpart
• Degrees and signatures of broken PT-symmetry in (non-uniform) lattices
• Gegenbauer-solvable quantum chain model
• Spontaneous PT symmetry breaking and quantum phase transitions in dimerized spin chains
• Non-Hermitian anisotropic XY model with intrinsic rotation-time reversal symmetry
• Entanglement Entropy of Non-Unitary Integrable Quantum Field Theory
• PT-symmetric deformations of integrable models
• Non-Hermitian Hamiltonians of Lie algebraic type
• Origin of maximal symmetry breaking in even PT-symmetric lattices
• Irreversibility of the renormalization group flow in non-unitary quantum field theory
• Self-sustained emission in semi-infinite non-Hermitian systems at the exceptional point
• Hermitian dynamics in a class of pseudo-Hermitian networks
• Exactly Solvable Quasi-hermitian Transverse Ising Model
• PT-symmetry breaking and universal chirality in a PT-symmetric ring
• PT Symmetry of the non-Hermitian XX Spin-Chain: Non-local Bulk Interaction from Complex Boundary Fields
• Nonadiabatic robust excitation transfer assisted by an imaginary gauge field
• Scaling behavior and phase diagram of a PT-symmetric non-Hermitian Bose-Hubbard system
• Non-Hermitian dynamics and $\mathcal{P}\mathcal{T}$-symmetry breaking in interacting mesoscopic Rydberg platforms
• From PT-symmetric quantum mechanics to conformal field theory
• Generation of Bell, W and GHZ states via exceptional points in non-Hermitian quantum spin systems
• Scale-free non-Hermitian skin effect in a boundary-dissipated spin chain
• An exactly solvable quantum-lattice model with a tunable degree of nonlocality
• An introduction to PT-symmetric quantum mechanics -- time-dependent systems
• The Jordan Structure of Two Dimensional Loop Models
• Probing phase transitions in non-Hermitian systems with Multiple Quantum Coherences
• Exact solution of a non-Hermitian $\mathcal{PT}$-symmetric Heisenberg spin chain
• Discrete quantum square well of the first kind
• Counting solutions of the Bethe equations of the quantum group invariant open XXZ chain at roots of unity
• On the reality of spectra of ${\mathbf{U}}_{\mathbf{q}}(\mathbf{s}{\mathbf{l}}_2)$-invariant XXZ Hamiltonians
• Non linear pseudo-bosons versus hidden Hermiticity
• Quantum entanglement and non-Hermiticity in free-fermion systems
• The Floquet Baxterisation
• On the construction of pseudo-hermitian quantum system with a pre-determined metric in the Hilbert space
• Solvable non-Hermitian discrete square well with closed-form physical inner product
• Anomalous bulk behaviour in the free parafermion $Z(N)$ spin chain
• Physics of PT-Symmetric Quantum Systems at Finite Temperature
• Effective non-Hermitian physics for degenerate ground states of a nonHermitian Ising model with $\mathcal{R}\mathcal{T}$ symmetry
• Constructing Exactly Solvable Pseudo-hermitian Many-particle Quantum Systems by Isospectral Deformation
• Metric operators for non-Hermitian quadratic su(2) Hamiltonians
• Fusion rules for the Temperley-Lieb algebra and its dilute generalisation
• Rational Q-systems at Root of Unity I. Closed Chains
• Semi-localization transition driven by a single asymmetrical tunneling
• Turning the Quantum Group Invariant XXZ Spin-Chain Hermitian: A Conjecture on the Invariant Product
• Mixedness timescale in non-Hermitian quantum systems
• A solvable non-unitary fermionic long-range model with extended symmetry
• Exact solution of an integrable anisotropic $J_1-J_2$ spin chain model
• Non-Hermitian Parent Hamiltonian from Generalized Quantum Covariance Matrix
• Quantum group deformation of the Kittel--Shore model
• Determining non-Hermitian parent Hamiltonian from a single eigenstate
• Interferometric Geometric Phases of $\mathcal{P}\mathcal{T}$-symmetric Quantum Mechanics