We consider the XXZ spin-1/2 Heisenberg chain with antiperiodic boundary conditions. The inhomogeneous version of this model can be solved by Separation of Variables (SoV), and the eigenstates can be constructed in terms of Q-functions, solution of a Baxter TQ-equation, which have double periodicity compared to the periodic case. We compute in this framework the scalar products of a particular class of separate states which notably includes the eigenstates of the transfer matrix. We also compute the form factors of local spin operators, i.e. their matrix elements between two eigenstates of the transfer matrix. We show that these quantities admit determinant representations with rows and columns labelled by the roots of the Q-functions of the corresponding separate states, as in the periodic case, although the form of the determinant are here slightly different. We also propose alternative types of determinant representations written directly in terms of the transfer matrix eigenvalues.
We study the thermodynamic limit of the antiperiodic XXZ spin chain with the anisotropic parameter $η=\frac{πi}{3}$. We parameterize eigenvalues of the transfer matrix by their zero points instead of Bethe roots. We obtain patterns of the distribution of zero points. Based on them, we calculate the ground state energy and the elementary excitations in the thermodynamic limit. We also obtain the two-body scattering matrix of elementary excitations. Two types of elementary excitations and three types of scattering processes are discussed in detailed.
We investigate the thermodynamic limit of the inhomogeneous T-Q relation of the antiferromagnetic XXZ spin chain with antiperiodic boundary condition. It is shown that the contribution of the inhomogeneous term at the ground state can be neglected when the system-size N tends to infinity, which enables us to reduce the inhomogeneous Bethe ansatz equations (BAEs) to the homogeneous ones. Then the quantum numbers at the ground states are obtained, by which the system with arbitrary size can be studied. We also calculate the twisted boundary energy of the system.
Finding out root patterns of quantum integrable models is an important step to study their physical properties in the thermodynamic limit. Especially for models without $U(1)$ symmetry, their spectra are usually given by inhomogeneous $T-Q$ relations and the Bethe root patterns are still unclear. In this paper with the antiperiodic $XXZ$ spin chain as an example, an analytic method to derive both the Bethe root patterns and the transfer-matrix root patterns in the thermodynamic limit is proposed. Based on them the ground state energy and elementary excitations in the gapped regime are derived.…
In this paper we consider the spin 1/2 highest weight representations for the 6-vertex Yang-Baxter algebra on a finite lattice and analyze the integrable quantum models associated to the antiperiodic transfer matrix. For these models, which in the homogeneous limit reproduces the XXZ spin 1/2 quantum chains with antiperiodic boundary conditions, we obtain in the framework of Sklyanin's quantum separation of variables (SOV) the following results: I) The complete characterization of the transfer matrix spectrum (eigenvalues/eigenstates) and the proof of its simplicity. II) The reconstruction of …
Generic inhomogeneous integrable XXZ chains with arbitrary spins are studied by means of the quantum separation of variables (SOV) method. Within this framework, a complete description of the spectrum (eigenvalues and eigenstates) of the antiperiodic transfer matrix is derived in terms of discrete systems of equations involving the inhomogeneity parameters of the model. We show here that one can reformulate this discrete SOV characterization of the spectrum in terms of functional T-Q equations of Baxter's type, hence proving the completeness of the solutions to the associated systems of Bethe-…
The scalar products, form factors and correlation functions of the XXZ spin chain with twisted (or antiperiodic) boundary condition are obtained based on the inhomogeneous $T-Q$ relation and the Bethe states constructed via the off-diagonal Bethe Ansatz. It is shown that the scalar product of two off-shell Bethe states, the form factors and the two-point correlation functions can be expressed as the summation of certain determinants. The corresponding homogeneous limits are studied. The results are also checked by the numerical calculations.
We pursue our study of the antiperiodic dynamical 6-vertex model using Sklyanin's separation of variables approach, allowing in the model new possible global shifts of the dynamical parameter. We show in particular that the spectrum and eigenstates of the antiperiodic transfer matrix are completely characterized by a system of discrete equations. We prove the existence of different reformulations of this characterization in terms of functional equations of Baxter's type. We notably consider the homogeneous functional $T$-$Q$ equation which is the continuous analog of the aforementioned discret…
We investigate the thermodynamic limit of the one-dimensional ferromagnetic XXZ model with twisted (or antiperiodic ) boundary condition. It is shown that the distribution of the Bethe roots of the inhomogeneous Bethe Ansatz equations (BAEs) for the ground state as well as for the low-lying excited states satisfy the string hypothesis, although the inhomogeneous BAEs are not in the standard product form which has made the study of the corresponding thermodynamic limit nontrivial. We also obtain the twisted boundary energy induced by the non-trivial twisted boundary conditions in the thermodyna…
Based on its off-diagonal Bethe ansatz solution, we study the thermodynamic limit of the spin-$\frac{1}{2}$ XYZ spin chain with the antiperiodic boundary condition. The key point of our method is that there exist some degenerate points of the crossing parameter $η_{m,l}$, at which the associated inhomogeneous $T-Q$ relation becomes a homogeneous one. This makes extrapolating the formulae deriving from the homogeneous one to an arbitrary $η$ with $O(N^{-2})$ corrections for a large $N$ possible. The ground state energy and elementary excitations of the system are obtained. By taking the trigono…
The antiperiodic transfer matrix associated to higher spin representations of the rational 6-vertex Yang-Baxter algebra is analyzed by generalizing the approach introduced recently in [1], for the cyclic representations, in [2], for the spin-1/2 highest weight representations, and in [3], for the spin 1/2 representations of the reflection algebra. Here, we derive the complete characterization of the transfer matrix spectrum and we prove its simplicity in the framework of Sklyanin's quantum separation of variables (SOV). Then, the characterization of local operators by Sklyanin's quantum separa…
We propose a non-standard separation of variables for the classical integrable XXX and XXZ spin chains with degenerate twist matrix. We show that for the case of such twist matrices one can interchange the role of classical separating functions $A(u)$ and $B(u)$ and construct a new full set of separated variables, satisfying simpler equation of separation and simpler Abel equations in comparison with the standard separated variables of Sklyanin. We show that for certain cases of the twist matrices the constructed separated variables can be directly identified with action-angle coordinates.
We construct the generalised Eigenfunctions of the entries of the monodromy matrix of the $N$-site modular XXZ magnet and show, in each case, that these form a complete orthogonal system in $L^2(\mathbb{R}^N)$. In particular, we develop a new and simple technique, allowing one to prove the completeness of such systems. As a corollary of out analysis, we prove the Bystko-Teschner conjecture relative to the structure of the spectrum of the $\boldsymbol{ \texttt{B} }(\la)$-operator for the odd length lattice Sinh-Gordon model.
In this paper we analyze the classical XXZ spin chain with reflecting boundaries. We exhibit a system of log-canonical coordinates on the phase space generalizing Sklyanin's separation of variables for the periodic XXZ chain, and use these coordinates to construct action-angle variables for the system. We also integrate the flows of the reflection Hamiltonians explicitly in terms of Riemann theta functions. Central to our analysis is the algebraic integrability of the model.
We use the quantum separation of variable (SOV) method to construct the eigenstates of the open XXZ chain with the most general boundary terms. The eigenstates in the inhomogeneous case are constructed in terms of solutions of a system of quadratic equations. This SOV representation permits us to compute scalar products and can be used to calculate form factors and correlation functions.
We extend Sklyanin's method of separation of variables to quantum integrable models associated to elliptic curves. After reviewing the differential case, the elliptic Gaudin model studied by Enriquez, Feigin and Rubtsov, we consider the difference case and find a class of transfer matrices whose eigenvalue problem can be solved by separation of variables. These transfer matrices are associated to representations of the elliptic quantum group $E_{τ,η}(sl_2)$ by difference operators. One model of statistical mechanics to which this method applies is the IRF model with antiperiodic boundary condi…
The spin-1/2 highest weight representations of the dynamical 6-vertex and the standard 8-vertex Yang-Baxter algebra on a finite chain are considered in this paper. For the antiperiodic dynamical 6-vertex transfer matrix defined on chains with an odd number of sites, we adapt the Sklyanin's quantum separation of variable (SOV) method and explicitly construct SOV representations from the original space of representations. We provide the complete characterization of eigenvalues and eigenstates proving also the simplicity of its spectrum. Moreover, we characterize the matrix elements of the identi…
We study the inhomogeneous 8-vertex model (or equivalently the XYZ Heisenberg spin-1/2 chain) with all kinds of integrable quasi-periodic boundary conditions: periodic, $σ^x$-twisted, $σ^y$-twisted or $σ^z$-twisted. We show that in all these cases but the periodic one with an even number of sites $\mathsf{N}$, the transfer matrix of the model is related, by the vertex-IRF transformation, to the transfer matrix of the dynamical 6-vertex model with antiperiodic boundary conditions, which we have recently solved by means of Sklyanin's Separation of Variables (SOV) approach. We show moreover that,…
On the gauge invariant boundary condition in open XXZ chains: SoV bases, elementary blocks and overlaps
Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique（里昂大学，巴黎高等师范学院，克劳德·贝尔纳大学，法国国家科学研究中心，物理实验室）；Université Paris-Saclay, CNRS, LPTMS（巴黎萨克雷大学，法国国家科学研究中心，理论与模型物理实验室）
In our previous paper [1] we have obtained, for the XXX spin-1/2 Heisenberg open chain, new determinant representations for the scalar products of separate states in the quantum separation of variables (SoV) framework. In this article we perform a similar study in a more complicated case: the XXZ open spin-1/2 chain with the most general integrable boundary terms. To solve this model by means of SoV we use an algebraic Vertex-IRF gauge transformation reducing one of the boundary K-matrices to a diagonal form. As usual within the SoV approach, the scalar products of separate states are computed…
An orthogonal basis of the Hilbert space for the quantum spin chain associated with the su(3) algebra is introduced. Such kind of basis could be treated as a nested generalization of separation of variables (SoV) basis for high-rank quantum integrable models. It is found that all the monodromy-matrix elements acting on a basis vector take simple forms. With the help of the basis, we construct eigenstates of the su(3) inhomogeneous spin torus (the trigonometric su(3) spin chain with antiperiodic boundary condition) from its spectrum obtained via the off-diagonal Bethe Ansatz (ODBA). Based on sm…
We solve the longstanding problem to define a functional characterization of the spectrum of the transfer matrix associated to the most general spin-1/2 representations of the 6-vertex reflection algebra for general inhomogeneous chains. The corresponding homogeneous limit reproduces the spectrum of the Hamiltonian of the spin-1/2 open XXZ and XXX quantum chains with the most general integrable boundaries. The spectrum is characterized by a second order finite difference functional equation of Baxter type with an inhomogeneous term which vanishes only for some special but yet interesting non-d…
After the separation of the center-of-mass motion, a new privileged class of canonical Darboux bases is proposed for the non-relativistic N-body problem by exploiting a geometrical and group theoretical approach to the definition of {\it body frame} for deformable bodies. This basis is adapted to the rotation group SO(3), whose canonical realization is associated with a symmetry Hamiltonian {\it left action}. The analysis of the SO(3) coadjoint orbits contained in the N-body phase space implies the existence of a {\it spin frame} for the N-body system. Then, the existence of appropriate non-sy…
