V. Terras

Algebraic Bethe ansatz approach to the asymptotic behavior of correlation functions

We describe a method to derive, from first principles, the long-distance asymptotic behavior of correlation functions of integrable models in the framework of the algebraic Bethe ansatz. We apply this approach to the longitudinal spin–spin correlation function of the XXZ Heisenberg spin- 1/2 chain (with magnetic field) in the disordered regime as well as to the density–density correlation function of the interacting one-dimensional Bose gas. At leading order, the results confirm the Luttinger liquid and conformal field theory predictions.

Correlation functions of the open XXZ chain: I

We derive compact multiple integral formulae for several physical spin correlation functions in the semi-infinite XXZ chain with a longitudinal boundary magnetic field. Our formulae follow from several effective resummations of the multiple integral representation for the elementary blocks obtained in our previous paper (I). In the free fermion point we compute the local magnetization as well as the density of energy profiles. These quantities, in addition to their bulk behavior, exhibit Friedel-type oscillations induced by the boundary; their amplitudes depend on the boundary magnetic field and decay algebraically in terms of the distance to the boundary.

Master equation for spin-spin correlation functions of the XXZ chain

Abstract We derive a new representation for spin–spin correlation functions of the finite X X Z spin- 1 2 Heisenberg chain in terms of a single multiple integral, that we call the master equation . Evaluation of this master equation gives rise on the one hand to the previously obtained multiple integral formulas for the spin–spin correlation functions and on the other hand to their expansion in terms of the form factors of the local spin operators. Hence, it provides a direct analytic link between these two representations of the correlation functions and a complete re-summation of the corresponding series. The master equation method also allows one to obtain multiple integral representations for dynamical correlation functions.

Dynamical correlation functions of the XXZ spin-1/2 chain

Abstract We derive a master equation for the dynamical spin–spin correlation functions of the X X Z spin- 1 2 Heisenberg finite chain in an external magnetic field. In the thermodynamic limit, we obtain their multiple integral representation.

On the thermodynamic limit of form factors in the massless XXZ Heisenberg chain

We consider the problem of computing form factors of the massless XXZ Heisenberg spin-1/2 chain in a magnetic field in the (thermodynamic) limit where the size M of the chain becomes large. For that purpose, we take the particular example of the matrix element of the operator σz between the ground state and an excited state with one particle and one hole located at the opposite ends of the Fermi interval (umklapp-type term). We exhibit its power-law decrease in terms of the size of the chain M and compute the corresponding exponent and amplitude. As a consequence, we show that this form factor is directly related to the amplitude of the leading oscillating term in the long-distance asymptotic expansion of the correlation function ⟨σ1zσm+1z⟩.

On the spin–spin correlation functions of the XXZ spin- infinite chain

We obtain a new multiple integral representation for the spin–spin correlation functions of the XXZ spin- infinite chain. We show that this representation is closely related with the partition function of the six-vertex model with domain wall boundary conditions.

Correlation functions of the openXXZ chain: II

We derive compact multiple integral formulas for several physical spin correlation functions in the semi-infinite XXZ chain with a longitudinal boundary magnetic field. Our formulas follow from several effective re-summations of the multiple integral representation for the elementary blocks obtained in our previous article (I). In the free fermion point we compute the local magnetization as well as the density of energy profiles. These quantities, in addition to their bulk behavior, exhibit Friedel type oscillations induced by the boundary; their amplitudes depend on the boundary magnetic field and decay algebraically in terms of the distance to the boundary.

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

Exact results for the σz two-point function of the XXZ chain at Δ = 1/2

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