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arXiv:1409.8344 · Journal of Physics Conference Series

The generalized t-V model in one dimension

Marcin Szyniszewski, Evgeni Burovski

Published Mar 18, 2015• 4 citations
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Abstract

We develop a systematic strong coupling approach for studying an extended t-V model with interactions of a finite range. Our technique is not based on the Bethe ansatz and is applicable to both integrable and non-integrable models. We illustrate our technique by presenting analytic results for the ground state energy (up to order 7 in t/V ), the current density and density-density correlations for integrable and non-integrable models with commensurate filling factors. We further present preliminary numerical results for incommensurate non- integrable models.

DOI
10.1088/1742-6596/592/1/012057

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We develop a systematic strong coupling approach for studying an extended t-V model with interactions of a finite range. Our technique is not based on the Bethe ansatz and is applicable to both integrable and non-integrable models. We illustrate our technique by presenting analytic results for the ground state energy (up to order 7 in t/V ), the current density and density-density correlations for integrable and non-integrable models with commensurate filling factors. We further present preliminary numerical results for incommensurate non- integrable models.
We use a strong coupling expansion [1] to solve the one-dimensional extended t‑V model of fermions [2,3]. The model is solved for a range of densities, including both commensurate – where a charge density wave is present – and incommensurate densities. The first set consists not only of a trivial case of half filling. The method allows us to trace the transition from a Luttinger liquid phase to a Mott insulating phase. This simple yet powerful method is not based on Bethe ansatz and it works for both integrable and non-integrable systems.References:[1] C.J. Hamer, Phys. Lett. B, 82, 75-78 (1979).[2] G. Gómez-Santos, Phys. Rev. Lett., 70, 3780 (1993).[3] R.G. Dias, Phys. Rev. B, 62, 7791 (2000).
We employ a strong coupling expansion - similar to the one used in the lattice field theory studies [1] - to solve the one-dimensional extended t-V model of fermions on a lattice [2]. This model is solved for a range of filling factors, including both commensurate - where a charge density wave is present - and incommensurate densities. The first set consists not only of a trivial case of half filling. The method allows us to trace the transition from a Luttinger liquid phase to a Mott insulating phase and calculate the critical parameter K. This simple yet powerful method is not based on Bethe ansatz and it works for both integrable and non-integrable systems. Furthermore, we investigate how tailoring the interaction can introduce other ordered phases of the system [3].References:[1] C.J. Hamer, Phys. Lett. B 82, 75-78 (1979); D.P. Crewther, C.J. Hamer, Nucl. Phys. B 170, 353-368 (1980).[2] G. Gómez-Santos, Phys. Rev. Lett. 70, 3780 (1993); R.G. Dias, Phys. Rev. B 62, 7791 (2000).[3] P. Schmitteckert, R. Werner, Phys. Rev. B 69, 195115 (2004); T. Mishra et al., Phys. Rev. B 84, 115135 (2011).
A new analytical method is proposed for solving the Hubbard model in d-dimensions. The method permits, in principle, the study of the Hubbard model at any temperature and any band filling. The accuracy of the method is tested against known Bethe ansatz results of Lieb and Wu (1968), and against known weak- and strong-coupling expansion results. The one-dimensional path integral solution obtained for the ground-state (half-filled band case) is in excellent qualitative agreement with Bethe ansatz results for all couplings and exactly reproduces weak- and strong-coupling expansion results.
An exactly solvable strongly correlated electron model with two independent parameters is constructed in the frame of the quantum inverse scattering method, which can be seen as a generalization of the Bariev model. Through the Bethe ansatz method, a set of Bethe ansatz equations is derived. In the thermodynamic limit, to study the ground state of the model, we obtain the integral equations for the density of Bethe roots. Numerical validation is done to confirm the accuracy of our analytic results.
A density-functional theory (DFT) of lattice fermion models is presented, which uses the single-particle density matrix ${\ensuremath{\gamma}}_{\mathrm{ij}}$ as basic variable. A simple, explicit approximation to the interaction-energy functional $W[\ensuremath{\gamma}]$ of the Hubbard model is derived from exact dimer results, scaling properties of $W[\ensuremath{\gamma}]$ and known limits. Systematic tests on the one-dimensional chain show a remarkable agreement with the Bethe-ansatz exact solution for all interaction regimes and band fillings. New results are obtained for the ground-state energy and charge-excitation gap in two dimensions. A successful description of strong electron correlations within DFT is achieved.