The half-infinite XXZ chain in Onsager's approach

Abstract

The half-infinite XXZ open spin chain with general integrable boundary conditions is considered within the recently developed `Onsager's approach'. Inspired by the finite size case, for any type of integrable boundary conditions it is shown that the transfer matrix is simply expressed in terms of the elements of a new type of current algebra recently introduced. In the massive regime $-1<q<0$, level one infinite dimensional representation ($q-$vertex operators) of the new current algebra are constructed in order to diagonalize the transfer matrix. For diagonal boundary conditions, known results of Jimbo {\it et al.} are recovered. For upper (or lower) non-diagonal boundary conditions, a solution is proposed. Vacuum and excited states are formulated within the representation theory of the current algebra using $q-$bosons, opening the way for the calculation of integral representations of correlation functions for a non-diagonal boundary. Finally, for $q$ generic the long standing question of the hidden non-Abelian symmetry of the Hamiltonian is solved: it is either associated with the $q-$Onsager algebra (generic non-diagonal case) or the augmented $q-$Onsager algebra (generic diagonal case).

28 pages; Presentation improved; misprints corrected; to appear in Nucl. Phys. B

References in corpus (38)

• Correlation Functions of the XXZ model for $\Delta <-1$
• QKZ equation with |q|=1 and correlation functions of the XXZ model in the gapless regime
• Diagonalization of the XXZ Hamiltonian by Vertex Operators
• Quantum symmetric Kac-Moody pairs
• Bethe Ansatz solution of the open XXZ chain with nondiagonal boundary terms
• Exact spectrum of the XXZ open spin chain from the q-Onsager algebra representation theory
• XXZ chain with a boundary
• Antiperiodic spin-1/2 XXZ quantum chains by separation of variables: Complete spectrum and form factors
• Functional relations from the Yang-Baxter algebra: Eigenvalues of the XXZ model with non-diagonal twisted and open boundary conditions
• Correlation functions of the open XXZ chain I
• Difference Equations in Spin Chains with a Boundary
• Non-diagonal open spin-1/2 XXZ quantum chains by separation of variables: Complete spectrum and matrix elements of some quasi-local operators
• An integrable structure related with tridiagonal algebras
• Deformed Dolan-Grady relations in quantum integrable models
• Curiosities at c=-2
• A new (in)finite dimensional algebra for quantum integrable models
• Coideal subalgebras in quantum affine algebras
• Magic in the spectra of the XXZ quantum chain with boundaries at Delta=0 and Delta=-1/2
• Bethe ansatz for the XXX-S chain with non-diagonal open boundaries
• Quantum affine symmetry in vertex models
• Correlation functions of the open XXZ chain II
• A new current algebra and the reflection equation
• Eigenvectors of open XXZ and ASEP models for a class of non-diagonal boundary conditions
• One-boundary Temperley-Lieb algebras in the XXZ and loop models
• Algebraic Bethe ansatz for open XXX model with triangular boundary matrices
• On the second reference state and complete eigenstates of the open XXZ chain
• Sine-Gordon quantum field theory on the half-line with quantum boundary degrees of freedom
• Exact solutions and elementary excitations in the XXZ spin chain with unparallel boundary fields
• Bosonization of ZF Algebras: Direction Toward Deformed Virasoro Algebra
• A family of tridiagonal pairs and related symmetric functions
• Nested Bethe ansatz for `all' open spin chains with diagonal boundary conditions
• Corner Transfer Matrices and Quantum Affine Algebras
• Boundary non-local charges from the open spin chain
• Nested Bethe ansatz for Y(gl(n)) open spin chains with diagonal boundary conditions
• Generalized q-Onsager Algebras and Dynamical K-matrices
• A remark on ground state of boundary Izergin-Korepin model
• Reflection algebras for sl(2) and gl(1|1)
• Central extension of the reflection equations and an analog of Miki's formula

Cited by in corpus (67)

• The Physicist's Companion to Current Fluctuations: One-Dimensional Bulk-Driven Lattice Gases
• Heisenberg XXX Model with General Boundaries: Eigenvectors from Algebraic Bethe Ansatz
• Modified algebraic Bethe ansatz for XXZ chain on the segment - I - triangular cases
• Modified algebraic Bethe ansatz for XXZ chain on the segment - II - general cases
• The open XXX spin chain in the SoV framework: scalar product of separate states
• Bethe states of the XXZ spin-1/2 chain with arbitrary boundary fields
• Transfer matrix spectrum for cyclic representations of the 6-vertex reflection algebra I
• Bethe Ansatz and Q-operator for the open ASEP
• Exact solution of the Izergin-Korepin model with general non-diagonal boundary terms
• Hall algebras and quantum symmetric pairs II: reflection functors
• Hall algebras and quantum symmetric pairs of Kac-Moody type
• Correlation functions of the half-infinite XXZ spin chain with a triangular boundary
• The open XXZ spin chain in the SoV framework: scalar product of separate states
• The $q$-Onsager Algebra and the Universal Askey-Wilson Algebra
• Exact surface energy and helical spinons in the XXZ spin chain with arbitrary non-diagonal boundary fields
• Non-Abelian symmetries of the half-infinite XXZ spin chain
• Asymptotic representations of augmented q-Onsager algebra and boundary K-operators related to Baxter Q-operators
• Algebraic Bethe ansatz for the six vertex model with upper triangular $K$-matrices
• Algebraic Bethe Ansatz for the XXZ Gaudin Models with Generic Boundary
• Transfer matrix spectrum for cyclic representations of the 6-vertex reflection algebra II
• Algebraic Bethe ansatz for the totally asymmetric simple exclusion process with boundaries
• The alternating central extension of the $q$-Onsager algebra
• Reflection algebra and functional equations
• Counting solutions of the Bethe equations of the quantum group invariant open XXZ chain at roots of unity
• A note on $q$-oscillator realizations of $U_q(gl(M|N))$ for Baxter $Q$-operators
• Conjectures on Hidden Onsager Algebra Symmetries in Interacting Quantum Lattice Models
• Closed hierarchy of Heisenberg equations in integrable models with Onsager algebra
• A Drinfeld type presentation of affine $ı$quantum groups II: split BCFG type
• Boundary quantum Knizhnik-Zamolodchikov equations and fusion
• FRT presentation of the Onsager algebras
• On Separation of Variables for Reflection Algebras
• Onsager algebra and algebraic generalization of Jordan-Wigner transformation
• Off-diagonal Bethe Ansatz on the $so(5)$ spin chain
• An action of the free product ${\mathbb{Z}}_2\star {\mathbb{Z}}_2\star {\mathbb{Z}}_2$ on the $q$-Onsager algebra and its current algebra
• Off-shell scalar products for the $XXZ$ spin chain with open boundaries
• A Q-operator for open spin chains I: Baxter's TQ relation
• Form factors of the half-infinite XXZ spin chain with a triangular boundary
• Universal Baxter TQ-relations for open boundary quantum integrable systems
• The alternating central extension for the positive part of $U_q({\widehat{\mathfrak{sl}}}_2)$
• A conjecture concerning the $q$-Onsager algebra
• Exact solution of an su(n) spin torus
• Exact Large Deviations of the Current in the Asymmetric Simple Exclusion Process with Open Boundaries
• Drinfeld J Presentation of Twisted Yangians
• The $q-$Onsager algebra and multivariable $q-$special functions
• A compact presentation for the alternating central extension of the positive part of $U_q({\widehat{\mathfrak{sl}}}_2)$
• On diagonal solutions of the reflection equation
• Exact solutions of the $C_n$ quantum spin chain
• Relative braid group symmetries on $ı$quantum groups of Kac-Moody type
• Exact solution of the $sp(4)$ integrable spin chain with generic boundaries
• Thermodynamic limit and boundary energy of the spin-1 Heisenberg chain with non-diagonal boundary fields
• Off-diagonal Bethe Ansatz for the $D_3^{(1)}$ model
• Differential approach to on-shell scalar products in six-vertex models
• A representation basis for the quantum integrable spin chain associated with the su(3) algebra
• A bispectral q-hypergeometric basis for a class of quantum integrable models
• T-W relation and free energy of the Heisenberg chain at a finite temperature
• Exact solution of the quantum integrable model associated with the twisted $D_3^{(2)}$ algebra
• An Infinite-Dimensional ${\square }_q$-Module Obtained from the $q$-Shuffle Algebra for Affine ${\mathfrak{sl}}_2$
• Surface energy and elementary excitations of the XYZ spin chain with integrable open boundary fields
• Higher Rank Askey-Wilson Algebras as Skein Algebras
• Inhomogeneous discrete-time exclusion processes
• Exact solution of the trigonometric SU(3) spin chain with generic off-diagonal boundary reflections
• Universal TT- and TQ-relations via centrally extended q-Onsager algebra
• Boundary transfer matrices arising from quantum symmetric pairs
• Bethe States of the integrable spin-s chain with generic open boundaries
• Spectrum of the transfer matrices of the spin chains associated with the $A_3^{(2)}$ Lie algebra
• Using a $q$-shuffle algebra to describe the basic module $V({\Lambda }_0)$ for the quantized enveloping algebra $U_q({\widehat{\mathfrak{sl}}}_2)$
• Graded Off-diagonal Bethe ansatz solution of the $SU(2|2)$ spin chain model with generic integrable boundaries