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arXiv:hep-th/9508099 · International Journal of Modern Physics A

ON ISING CORRELATION FUNCTIONS WITH BOUNDARY MAGNETIC FIELD

Robert Konik, André LeClair, Giuseppe Mussardo

Published Jun 20, 1996• 40 citations
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Abstract

Exact expressions of the boundary state and the form factors of the Ising model are used to derive differential equations for the one-point functions of the energy and magnetization operators of the model in the presence of a boundary magnetic field. We also obtain explicit formulas for the massless limit of the one-point and two-point functions of the energy operator.

DOI
10.1142/s0217751x96001346

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: We extend the form-factors approach to the quantum Ising model at finite temperature. The two point function of the energy is obtained in closed form, while the two point function of the spin is written as a Fredholm determinant. Using the approach of [1], we obtain, starting directly from the continuum formulation, a set of six differential equations satisfied by this two point function. Four of these equations involve only spacetime derivatives, of which three are equivalent to the equations obtained earlier in [2],[3]. In addition, we obtain two new equations involving a temperature derivative. Some of these results are generalized to the Ising model on the half line with a magnetic field at the origin. 04/96 1. Introduction The form-factors approach has been a surprisingly efficient tool for computing correlation functions in 1 + 1 quantum integrable theories at zero temperature (T ) (see eg. [4],[5],[6],[7],[8]). Recall that a quantum integrable theory in 1+1 is conveniently ...

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Exact expressions of the boundary state and the form factors of the Ising model are used to derive differential equations for the one-point functions of the energy and magnetization operators of the model in the presence of a boundary magnetic field. We also obtain explicit formulas for the massless limit of the one-point and two-point functions of the energy operator.
We extend the form-factors approach to the quantum Ising model at finite temperature. The two-point function of the energy is obtained in closed form, while the two-point function of the spin is written as a Fredholm determinant. Using the approach of Korepin et al., we obtain, starting directly from the continuum formulation, a set of six differential equations satisfied by this two-point function. Four of these equations involve only space-time derivatives, of which three are equivalent to the equations obtained earlier. In addition, we obtain two new equations involving a temperature derivative. Some of these results are generalized to the Ising model on the half line with a magnetic field at the origin.
We compute the exact partition function, the universal ground state degeneracy and boundary state of the 2-D Ising model with boundary magnetic field at off-critical temperatures. The model has a domain that exhibits states localized near the boundaries. We study this domain of boundary bound state and derive exact expressions for the ``g function'' and boundary state for all temperatures and boundary magnetic fields. In the massless limit we recover the boundary renormalization group flow between the conformally invariant free and fixed boundary conditions.
We calculate the two-point correlation function and magnetic susceptibility in the anisotropic 2D Ising model on a lattice with one infinite and the other finite dimension, along which periodic boundary conditions are imposed. Using exact expressions for a part of lattice form factors, we propose the formulas for arbitrary spin matrix elements, thus providing a possibility to compute all multipoint correlation functions in the anisotropic Ising model on cylindrical and toroidal lattices. The scaling limit of the corresponding expressions is also analyzed.
: We extend the form-factors approach to the quantum Ising model at finite temperature. The two point function of the energy is obtained in closed form, while the two point function of the spin is written as a Fredholm determinant. Using the approach of [1], we obtain, starting directly from the continuum formulation, a set of six differential equations satisfied by this two point function. Four of these equations involve only spacetime derivatives, of which three are equivalent to the equations obtained earlier in [2],[3]. In addition, we obtain two new equations involving a temperature derivative. Some of these results are generalized to the Ising model on the half line with a magnetic field at the origin. 04/96 1. Introduction The form-factors approach has been a surprisingly efficient tool for computing correlation functions in 1 + 1 quantum integrable theories at zero temperature (T ) (see eg. [4],[5],[6],[7],[8]). Recall that a quantum integrable theory in 1+1 is conveniently ...