249 research outputs found

### The 8-vertex model with quasi-periodic boundary conditions

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, $\sigma^x$-twisted, $\sigma^y$-twisted or
$\sigma^z$-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, in all the twisted cases, the vertex-IRF transformation
is bijective. This allows us to completely characterize, from our previous
results on the antiperiodic dynamical 6-vertex model, the twisted 8-vertex
transfer matrix spectrum (proving that it is simple) and eigenstates. We also
consider the periodic case for $\mathsf{N}$ odd. In this case we can define two
independent vertex-IRF transformations, both not bijective, and by using them
we show that the 8-vertex transfer matrix spectrum is doubly degenerate, and
that it can, as well as the corresponding eigenstates, also be completely
characterized in terms of the spectrum and eigenstates of the dynamical
6-vertex antiperiodic transfer matrix. In all these cases we can adapt to the
8-vertex case the reformulations of the dynamical 6-vertex transfer matrix
spectrum and eigenstates that had been obtained by $T$-$Q$ functional
equations, where the $Q$-functions are elliptic polynomials with
twist-dependent quasi-periods. Such reformulations enables one to characterize
the 8-vertex transfer matrix spectrum by the solutions of some Bethe-type
equations, and to rewrite the corresponding eigenstates as the multiple action
of some operators on a pseudo-vacuum state, in a similar way as in the
algebraic Bethe ansatz framework.Comment: 35 page

### Antiperiodic XXZ chains with arbitrary spins: Complete eigenstate construction by functional equations in separation of variables

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-type
equations. More precisely, we consider here two such reformulations. The first
one is given in terms of Q-solutions, in the form of trigonometric polynomials
of a given degree $N_s$, of a one-parameter family of T-Q functional equations
with an extra inhomogeneous term. The second one is given in terms of
Q-solutions, again in the form of trigonometric polynomials of degree $N_s$ but
with double period, of Baxter's usual (i.e. without extra term) T-Q functional
equation. In both cases, we prove the precise equivalence of the discrete SOV
characterization of the transfer matrix spectrum with the characterization
following from the consideration of the particular class of Q-solutions of the
functional T-Q equation: to each transfer matrix eigenvalue corresponds exactly
one such Q-solution and vice versa, and this Q-solution can be used to
construct the corresponding eigenstate.Comment: 38 page

### Multi-point local height probabilities of the CSOS model within the algebraic Bethe Ansatz framework

We study the local height probabilities of the exactly solvable cyclic
solid-on-solid model within the algebraic Bethe Ansatz framework. We more
specifically consider multi-point local height probabilities at adjacent sites
on the lattice. We derive multiple integral representations for these
quantities at the thermodynamic limit, starting from finite-size expressions
for the corresponding multi-point matrix elements in the Bethe basis as sums of
determinants of elliptic functions.Comment: 39 page

### Quantum Dynamical coBoundary Equation for finite dimensional simple Lie algebras

For a finite dimensional simple Lie algebra g, the standard universal
solution R(x) in $U_q(g)^{\otimes 2}$ of the Quantum Dynamical Yang--Baxter
Equation can be built from the standard R--matrix and from the solution F(x) in
$U_q(g)^{\otimes 2}$ of the Quantum Dynamical coCycle Equation as
$R(x)=F^{-1}_{21}(x) R F_{12}(x).$ It has been conjectured that, in the case
where g=sl(n+1) n greater than 1 only, there could exist an element M(x) in
$U_q(sl(n+1))$ such that $F(x)=\Delta(M(x)){J}
M_2(x)^{-1}(M_1(xq^{h_2}))^{-1},$ in which $J\in U_q(sl(n+1))^{\otimes 2}$ is
the universal cocycle associated to the Cremmer--Gervais's solution. The aim of
this article is to prove this conjecture and to study the properties of the
solutions of the Quantum Dynamical coBoundary Equation. In particular, by
introducing new basic algebraic objects which are the building blocks of the
Gauss decomposition of M(x), we construct M(x) in $U_q(sl(n+1))$ as an explicit
infinite product which converges in every finite dimensional representation. We
emphasize the relations between these basic objects and some Non Standard Loop
algebras and exhibit relations with the dynamical quantum Weyl group.Comment: 46 page

### Antiperiodic dynamical 6-vertex model by separation of variables II: Functional equations and form factors

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 discrete
system and show, in the case of a model with an even number of sites, that the
complete spectrum and eigenstates of the antiperiodic transfer matrix can
equivalently be described in terms of a particular class of its $Q$-solutions,
hence leading to a complete system of Bethe equations. Finally, we compute the
form factors of local operators for which we obtain determinant representations
in finite volume.Comment: 52 page

### The open XXX spin chain in the SoV framework: scalar product of separate states

We consider the XXX open spin-1/2 chain with the most general non-diagonal
boundary terms, that we solve by means of the quantum separation of variables
(SoV) approach. We compute the scalar products of separate states, a class of
states which notably contains all the eigenstates of the model. As usual for
models solved by SoV, these scalar products can be expressed as some
determinants with a non-trivial dependance in terms of the inhomogeneity
parameters that have to be introduced for the method to be applicable. We show
that these determinants can be transformed into alternative ones in which the
homogeneous limit can easily be taken. These new representations can be
considered as generalizations of the well-known determinant representation for
the scalar products of the Bethe states of the periodic chain. In the
particular case where a constraint is applied on the boundary parameters, such
that the transfer matrix spectrum and eigenstates can be characterized in terms
of polynomial solutions of a usual T-Q equation, the scalar product that we
compute here corresponds to the scalar product between two off-shell Bethe-type
states. If in addition one of the states is an eigenstate, the determinant
representation can be simplified, hence leading in this boundary case to direct
analogues of algebraic Bethe ansatz determinant representations of the scalar
products for the periodic chain.Comment: 39 page

### On determinant representations of scalar products and form factors in the SoV approach: the XXX case

In the present article we study the form factors of quantum integrable
lattice models solvable by the separation of variables (SoV) method. It was
recently shown that these models admit universal determinant representations
for the scalar products of the so-called separate states (a class which
includes in particular all the eigenstates of the transfer matrix). These
results permit to obtain simple expressions for the matrix elements of local
operators (form factors). However, these representations have been obtained up
to now only for the completely inhomogeneous versions of the lattice models
considered. In this article we give a simple algebraic procedure to rewrite the
scalar products (and hence the form factors) for the SoV related models as
Izergin or Slavnov type determinants. This new form leads to simple expressions
for the form factors in the homogeneous and thermodynamic limits. To make the
presentation of our method clear, we have chosen to explain it first for the
simple case of the $XXX$ Heisenberg chain with anti-periodic boundary
conditions. We would nevertheless like to stress that the approach presented in
this article applies as well to a wide range of models solved in the SoV
framework.Comment: 46 page

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