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arXiv:cond-mat/9503085

Strongly Interacting Spinless Fermions in D=1 and 2 Dimensions: A Perturbative-Variational Approach

Miguel Angel Martin-Delgado, Germán Sierra

Published Mar 15, 1995• 1 citation
arXivHTMLPDF

Abstract

We propose a perturbative-variational approach to interacting fermion systems on 1D and 2D lattices at half-filling. We address relevant issues such as the existence of Long Range Order, quantum phase transitions and the evaluation of ground state energy. In 1D our method is capable of unveiling the existence of a critical point in the coupling constant at $(t/U)_c=0.7483$ as in fact occurs in the exact solution at a value of 0.5. In our approach this phase transition is described as an example of Bifurcation Phenomena in the variational computation of the ground state energy. In 2D the van Hove singularity plays an essential role in changing the asymptotic behaviour of the system for large values of $t/U$. In particular, the staggered magnetization for large $t/U$ does not display the Hartree-Fock law $(t/U) e^{-2 π\sqrt{t/U}}$ but instead we find the law $(t/U) e^{- \frac{π^2}{3} t/U}$. Moreover, the system does not exhibit bifurcation phenomena and thus we do not find a critical point separating a CDW state from a fermion "liquid" state.

DOI
10.48550/arxiv.cond-mat/9503085

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We propose a perturbative-variational approach to interacting fermion systems on 1D and 2D lattices at half-filling. We address relevant issues such as the existence of Long Range Order, quantum phase transitions and the evaluation of ground state energy. In 1D our method is capable of unveiling the existence of a critical point in the coupling constant at (t/U)c=0.7483(t/U)_c=0.7483(t/U)c​=0.7483 as in fact occurs in the exact solution at a value of 0.5. In our approach this phase transition is described as an example of Bifurcation Phenomena in the variational computation of the ground state energy. In 2D the van Hove singularity plays an essential role in changing the asymptotic behaviour of the system for large values of t/Ut/Ut/U. In particular, the staggered magnetization for large t/Ut/Ut/U does not display the Hartree-Fock law (t/U)e−2πt/U(t/U) e^{-2 π\sqrt{t/U}}(t/U)e−2πt/U​ but instead we find the law (t/U)e−π23t/U(t/U) e^{- \frac{π^2}{3} t/U}(t/U)e−3π2​t/U. Moreover, the system does not exhibit bifurcation phenomena and thus we do not find a critical point separating a CDW state from a fermion "liquid" state.
We propose an interpolating ansatz between the strong coupling and weak coupling regimes of a system of spinless interacting fermions in 1D and 2D lattices at half-filling. We address relevant issues such as the existence of Long Range Order, quantum phase transitions and the evaluation of ground state energy. In 1D our method is capable of unveiling the existence of a critical point in the coupling constant at (t/U) c =0.7483 as in fact occurs in the exact solution at a value of 0.5. In our approach this phase transition is described as an example of Bifurcation Phenomena in the variational computation of the ground state energy. In 2D the van Hove singularity plays an essential role in changing the asymptotic behaviour of the system for large values of t/U. In particular, the staggered magnetization for large t/U does not display the Hartree–Fock law [Formula: see text] but instead we find the law [Formula: see text]. Moreover, the system does not exhibit bifurcation phenomena and thus we do not find a critical point separating a CDW state from a fermion "liquid" state.
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Abstract The study of ground-state properties of the Fermi-Hubbard model is a long-lasting task in the research of strongly correlated systems, and is crucial for the understanding of notable quantum phenomena including superconductivity and magnetism. Owing to the exponentially growing complexity of the system, a quantitative analysis usually demands high computational cost and is restricted to small samples, especially in two or higher dimensions. Here, we introduce a variational method in the frame of fermionic Gaussian states, and obtain the ground states of one- and two-dimensional attractive Hubbard models via imaginary-time evolution. We calculate the total energy and benchmark the results in a wide range of interaction strength and filling factor with those obtained via exact two-body results, the density matrix renormalization group based on matrix product states (MPS), and projector Quantum Monte Carlo (QMC) method. For both 1D and 2D cases, the Gaussian variational method presents accurate results for total energy with a maximum systematic error ∼ 4 % in the intermediate interaction region. The accuracy of these results has negligible dependence on the system size. We further calculate the double occupancy and find excellent agreement with MPS and QMC, as well as the experimental results of cold quantum gases in optical lattices. The results suggest that the Gaussian pairing state is a good approximation to the ground states of attractive Hubbard model, in particular in the strong and weak coupling limits. Moreover, we generalize the method to the attractive Hubbard model with a finite spin-polarization, which can be mapped to the repulsive interaction case via particle-hole transformation, and obtain accurate results of ground state energy and double occupancy. Our work demonstrates the ability of the Gaussian variational method to extract ground state properties of strongly correlated many-body systems with negligible computational cost, especially of large size and in higher dimensions.
We derive a low energy effective model for a two dimensional (2D) system of spinless fermions on a square lattice with repulsive short-range interactions and away from half filling. This model is a natural 2D analogue of the Luttinger model which describes interacting fermions and can be treated exactly by bosonization. It also includes fermion degrees of freedom which cannot be bosonized but, as we argue, can be treated adequately by mean field theory. Our results generalize to 2D a standard approach to 1D lattice fermion systems making them amenable to non-perturbative computation methods by a suitable continuum limit.
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An analytic expression for the dynamical conductivity sigma ( omega ) of spinless fermions with local disorder and nearest-neighbour repulsion is derived on lattices with infinite coordination number Z. The model is exactly solvable in the whole parameter range assuming two possible phases: a homogeneous phase and a checkerboard charge-density wave (CDW). Away from half filling the system displays anomalous behaviour: weak particle-density fluctuations favour spontaneous symmetry breaking. First, the authors investigate the effects of this anomaly on the conductivity in the AC and DC regimes. Second, they focus on the Mott transition occurring at zero temperature at half filling. The critical exponents for the conductivity are computed for the dependence on the interaction U, the disorder gamma , the filling n, the temperature T and the frequency omega .