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arXiv:1411.2839 · Physical Review B

Nested algebraic Bethe ansatz for the supersymmetric t−J model and tensor networks

You Quan Chong, V. Murg, V. E. Korepin, Frank Verstraete

Published May 20, 2015• 13 citations
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Abstract

We consider a model of strongly correlated electrons in 1D called the $t\text{\ensuremath{-}}J$ model, which was solved by the graded algebraic Bethe ansatz. We use it to design graded tensor networks which can be contracted approximately to obtain a matrix product state. As a proof of principle, we calculate observables of ground states and excited states of finite lattices up to 18 lattice sites.

DOI
10.1103/physrevb.91.195132

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Algebraic Bethe ansatz and tensor networks

You Quan Chong

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

•Sep 28, 2017•0 citations

Algebraic Bethe ansatz and tensor networks

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The algebraic Bethe ansatz is a prosperous and well-established method for solving one-dimensional quantum models exactly. The solution of the complex eigenvalue problem is thereby reduced to the solution of a set of algebraic equations. Whereas the spectrum is usually obtained directly, the eigenstates are available only in terms of complex mathematical expressions. This makes it very hard, in general, to extract properties from the states, for example, correlation functions. In our work, we apply the tools of tensor-network states to describe the eigenstates approximately as matrix product states. From the matrix product state expression, we then obtain observables like the structure factor, dimer-dimer correlation functions, chiral correlation functions, and one-particle Green function directly.

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Diagonalization of a coupled supersymmetrict-Jmodel and its ground-state properties

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We obtain an integrable two-leg supersymmetric t - J model through the algebraic Bethe ansatz scheme in the BFF grading. In this model, the two t - J chains interact with each other via a coupling constant κ. The model reduces to a two-chain Heisenberg spin model when the bosonic degrees of freedom are turned off and to two decoupled t - J chains when κ = 0. The construction of the diagonalized Hamiltonian yields the Bethe ansatz equations. We also obtain thermodynamic Bethe ansatz equations and calculate the magnetic susceptibility at a weak magnetic field and at zero temperature.

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In this work we investigate the supersymmetric t-J model in one dimension. The spectrum of the hamiltonian is obtained by means of the algebraic nested Bethe-ansatz method. Furthermore, we present a detailed analysis of the algebraic structure of the states. By combining the Bethe ansatz with the spl(2,1) supersymmetry of the t-J hamiltonian a complete set of eigenstates is constructed. The ground state and the elementary excitations of the model are investigated. We also introduce a new integrable vertex model, the anisotropic t-J model, which possesses splq(2, 1) "quantum supergroup" invariance.

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Frank Göhmann, Er Seel, Fachbereich Physik

We solve the gl(1|2) generalized model by means of the algebraic Bethe ansatz. The resulting eigenvalue of the transfer matrix and the Bethe ansatz equations depend on three complex functions, called the parameters of the generalized model. Specifying the parameters appropriately, we obtain the Bethe ansatz equations of the supersymmetric t-J model, the Hubbard model, or of Yang’s model of electrons with delta interaction. This means that the Bethe ansatz equations of these (and many other) models can be obtained from a common algebraic source, namely from the Yang-Baxter algebra generated by the gl(1|2) invariant R-matrix.

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A new algebraic Bethe ansatz forgl(2,1) invariant vertex models

Markus P. Pfannmüller, Holger Frahm

The algebraic Bethe ansatz for the integrable vertex model constructed from the four-dimensional representation of the superalgebra gl(2,1) is calculated using a ferromagnetic reference state. This Bethe ansatz was known only for the three-dimensional representation leading to the supersymmetric t-J model. The necessary modification of the nested algebraic Bethe ansatz scheme and generalizations to related models are discussed.

Journal of Physics A Mathematical and General•Aug 7, 1997•18 citations
We consider a model of strongly correlated electrons in 1D called the t\ensuremath-Jt\text{\ensuremath{-}}Jt\ensuremath-J model, which was solved by the graded algebraic Bethe ansatz. We use it to design graded tensor networks which can be contracted approximately to obtain a matrix product state. As a proof of principle, we calculate observables of ground states and excited states of finite lattices up to 18 lattice sites.
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
The algebraic Bethe ansatz is a prosperous and well-established method for solving one-dimensional quantum models exactly. The solution of the complex eigenvalue problem is thereby reduced to the solution of a set of algebraic equations. Whereas the spectrum is usually obtained directly, the eigenstates are available only in terms of complex mathematical expressions. This makes it very hard, in general, to extract properties from the states, for example, correlation functions. In our work, we apply the tools of tensor-network states to describe the eigenstates approximately as matrix product states. From the matrix product state expression, we then obtain observables like the structure factor, dimer-dimer correlation functions, chiral correlation functions, and one-particle Green function directly.
We obtain an integrable two-leg supersymmetric t - J model through the algebraic Bethe ansatz scheme in the BFF grading. In this model, the two t - J chains interact with each other via a coupling constant κ. The model reduces to a two-chain Heisenberg spin model when the bosonic degrees of freedom are turned off and to two decoupled t - J chains when κ = 0. The construction of the diagonalized Hamiltonian yields the Bethe ansatz equations. We also obtain thermodynamic Bethe ansatz equations and calculate the magnetic susceptibility at a weak magnetic field and at zero temperature.
In this work we investigate the supersymmetric t-J model in one dimension. The spectrum of the hamiltonian is obtained by means of the algebraic nested Bethe-ansatz method. Furthermore, we present a detailed analysis of the algebraic structure of the states. By combining the Bethe ansatz with the spl(2,1) supersymmetry of the t-J hamiltonian a complete set of eigenstates is constructed. The ground state and the elementary excitations of the model are investigated. We also introduce a new integrable vertex model, the anisotropic t-J model, which possesses splq(2, 1) "quantum supergroup" invariance.
We solve the gl(1|2) generalized model by means of the algebraic Bethe ansatz. The resulting eigenvalue of the transfer matrix and the Bethe ansatz equations depend on three complex functions, called the parameters of the generalized model. Specifying the parameters appropriately, we obtain the Bethe ansatz equations of the supersymmetric t-J model, the Hubbard model, or of Yang’s model of electrons with delta interaction. This means that the Bethe ansatz equations of these (and many other) models can be obtained from a common algebraic source, namely from the Yang-Baxter algebra generated by the gl(1|2) invariant R-matrix.