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Determinant Representations of Correlation Functions for the Supersymmetric t-J Model

Shao-you Zhao A, Wen‐Li Yang, Yao-Zhong Zhang

Published Jan 1, 2005• 17 citations
OpenAlex

Abstract

Working in the F-basis provided by the factorizing F-matrix, the scalar products of Bethe states for the supersymmetric t-J model are represented by determinants. By means of these results, we obtain determinant representations of correlation functions The computation of correlation functions is one of the major challenging problems in the theory of quantum integrable models [1, 2]. There are currently two approaches for computing the correlation functions of a quantum integrable model. One is the vertex operator method (see e.g. [3-8]), and another one is based on the detailed analysis of the structure of

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Working in the F-basis provided by the factorizing F-matrix, the scalar products of Bethe states for the supersymmetric t-J model are represented by determinants. By means of these results, we obtain determinant representations of correlation functions The computation of correlation functions is one of the major challenging problems in the theory of quantum integrable models [1, 2]. There are currently two approaches for computing the correlation functions of a quantum integrable model. One is the vertex operator method (see e.g. [3-8]), and another one is based on the detailed analysis of the structure of
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We review the recent progress on the construction of the determinant representations of the correlation functions for the integrable supersymmetric fermion models. The factorizing F-matrices (or the so-called F-basis) play an important role in the construction. In the F-basis, the creation (and the annihilation) operators and the Bethe states of the integrable models are given in completely symmetric forms. This leads to the determinant representations of the scalar products of the Bethe states for the models. Based on the scalar products, the determinant representations of the correlation functions may be obtained. As an example, in this review, we give the determinant representations of the two-point correlation function for the Uq(gl(2|1)) (i.e. q-deformed) supersymmetric t-J model. The determinant representations are useful for analyzing physical properties of the integrable models in the thermodynamical limit.
Determinant Representation of Correlation Functions for the Uq(gl(1∣1))U_q(gl(1|1))Uq​(gl(1∣1)) Free Fermion Model
With the help of the factorizing FFF-matrix, the scalar products of the Uq(gl(1∣1))U_q(gl(1|1))Uq​(gl(1∣1)) free fermion model are represented by determinants. By means of these results, we obtain the determinant representations of correlation functions of the model.
We consider several exactly solvable models of strongly correlated electrons in one dimension, such as the Heisenberg XXX model, the supersymmetric t-J model and the Hubbard model. These models can be solved by using the method of graded algebraic Bethe ansatz. We use it to design graded tensor networks which can be contracted approximately to obtain a Matrix Product State. This overcomes a major shortcoming of current density matrix renormalization group (DMRG) methods which work well on the ground states, but have difficulty working with the excited states of such models. In addition, observables such as correlation functions are important as they are experimentally measurable, but have been analytically described in the double scaling limit only. Moreover, these analytical results are mostly expressed in the form of determinants, which are numerically inefficient to compute. With the tensor network description of the spin models, we can efficiently compute any expectation value of the eigenstates on finite length lattices for direct comparison with laboratory results. As a proof of principle, we calculate correlation functions of ground states and excited states of such models on finite lattices of lengths in an intermediate regime which are of experimental interest. | 108 pages
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