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arXiv:1506.06539 · Journal of Physics A Mathematical and Theoretical

Exact formulas for the form factors of local operators in the Lieb–Liniger model

Lorenzo Piroli, Pasquale Calabrese

Published Oct 20, 2015• 44 citations
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Abstract

We present exact formulas for the form factors of local operators in the repulsive Lieb–Liniger model at finite size. These are essential ingredients for both numerical and analytical calculations. From the theory of algebraic Bethe ansatz, it is known that the form factors of local operators satisfy a particular type of recursive relations. We show that in some cases these relations can be used directly to derive practical expressions in terms of the determinant of a matrix whose dimension scales linearly with the system size. Our main results are determinant formulas for the form factors of the operators and Ψ R (0), for arbitrary integer R , where Ψ, are the usual field operators. From these expressions, we also derive the infinite size limit of the form factors of these local operators in the attractive regime.

DOI
10.1088/1751-8113/48/45/454002

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We present exact formulas for the form factors of local operators in the repulsive Lieb–Liniger model at finite size. These are essential ingredients for both numerical and analytical calculations. From the theory of algebraic Bethe ansatz, it is known that the form factors of local operators satisfy a particular type of recursive relations. We show that in some cases these relations can be used directly to derive practical expressions in terms of the determinant of a matrix whose dimension scales linearly with the system size. Our main results are determinant formulas for the form factors of the operators and Ψ R (0), for arbitrary integer R , where Ψ, are the usual field operators. From these expressions, we also derive the infinite size limit of the form factors of these local operators in the attractive regime.
We derive an exact formula for the field form factor in the anyonic Lieb-Liniger model, valid for arbitrary values of the interaction ccc, anyonic parameter κ\kappaκ, and number of particles NNN. Analogously to the bosonic case, the form factor is expressed in terms of the determinant of a N×NN\times NN×N matrix, whose elements are rational functions of the Bethe quasimomenta but explicitly depend on κ\kappaκ. The formula is efficient to evaluate, and provide an essential ingredient for several numerical and analytical calculations. Its derivation consists of three steps. First, we show that the anyonic form factor is equal to the bosonic one between two special off-shell Bethe states, in the standard Lieb-Liniger model. Second, we characterize its analytic properties and provide a set of conditions that uniquely specify it. Finally, we show that our determinant formula satisfies these conditions.
Abstract We derive an exact formula for the field form factor in the anyonic Lieb–Liniger model, valid for arbitrary values of the interaction c, anyonic parameter κ, and number of particles N. Analogously to the bosonic case, the form factor is expressed in terms of the determinant of an N × N matrix, whose elements are rational functions of the Bethe quasimomenta but explicitly depend on κ. The formula is efficient to evaluate, and provide an essential ingredient for several numerical and analytical calculations. Its derivation consists of three steps. First, we show that the anyonic form factor is equal to the bosonic one between two special off-shell Bethe states, in the standard Lieb–Liniger model. Second, we characterize its analytic properties and provide a set of conditions that uniquely specify it. Finally, we show that our determinant formula satisfies these conditions.
We investigate form factors of local operators in the multi-component Quantum Non-linear Schrödinger model, a prototype theory solvable by the so-called nested Bethe Ansatz. We determine the analytic properties of the infinite volume form factors using the coordinate Bethe Ansatz solution and we establish a connection with the finite volume matrix elements. In the two-component models we derive a set of recursion relations for the "magnonic form factors", which are the matrix elements on the nested Bethe Ansatz states. In certain simple cases (involving states with only one spin-impurity) we obtain explicit solutions for the recursion relations.
Abstract We apply the nested algebraic Bethe ansatz to the models with g l ( 2 | 1 ) and g l ( 1 | 2 ) supersymmetry. We show that form factors of local operators in these models can be expressed in terms of the universal form factors. Our derivation is based on the use of the RTT -algebra only. It does not refer to any specific representation of this algebra. We obtain thus determinant representations for form factors of local operators in the cases where an explicit solution of the quantum inverse scattering problem is not known.