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arXiv:1910.12136 · International Journal of Modern Physics B

Self-consistent effective Hamiltonian theory for fermionic many-body systems

Xindong Wang, Hai‐Ping Cheng

Published Dec 19, 2020• 3 citations
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Abstract

Using a separable many-body variational wavefunction, we formulate a self-consistent effective Hamiltonian theory for fermionic many-body system. The theory is applied to the two-dimensional (2D) Hubbard model as an example to demonstrate its capability and computational effectiveness. Most remarkably for the Hubbard model in 2D, a highly unconventional quadruple-fermion non-Cooper pair order parameter is discovered.

DOI
10.1142/s0217979221500193

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We propose a general variational fermionic many-body wavefunction that generates an effective Hamiltonian in a quadratic form, which can then be exactly solved. The theory can be constructed within the density functional theory framework, and a self-consistent scheme is proposed for solving the exact density functional theory. We apply the theory to structurally-disordered systems, symmetric and asymmetric Hubbard dimers, and the corresponding lattice models. The single fermion excitation spectra show a persistent gap due to the fermionic-entanglement-induced pairing condensate. For disordered systems, the density of states at the edge of the gap diverges in the thermodynamic limit, suggesting a topologically ordered phase. A sharp resonance is predicted as the gap is not dependent on the temperature of the system. For the symmetric Hubbard model, the gap for both half-filling and doped case suggests that the quantum phase transition between the antiferromagnetic and superconducting phases is continuous.
We propose a general variational fermionic many-body wavefunction that generates an effective Hamiltonian in quadratic form which can then be exactly solved. The theory can be constructed within density functional theory framework and a self-consistent scheme is proposed for solving the exact density functional theory. We apply the theory to structurally disordered system and an symmetric and asymmetric Hubbard dimer and corresponding lattice models and the the single fermion excitation spectra show a persistent gap due to the fermionic entanglement induced pairing condensate. For disordered system, density of state at the edge of the gap diverges in the thermodynamic limit, suggesting a topologically ordered phase and a sharp resonance is predicted as the gap is not dependent on the temperature of the system. For the symmetric Hubbard model, the gap for both half filling and doped case suggests quantum phase transition between the AFM and SC is a continuous phase transition.
An effective Hamiltonian with auxiliary bosons and auxiliary fermions is proposed to treat the Hubbard Hamiltonian with strong correlation. Auxiliary particles themselves can not describe any real excitations. In particular, the Bose condensation of auxiliary bosons never occurs, and auxiliary fermions have no Fermi surfaces. Real excitations are described by two kinds of pair-excitations of auxiliary particles, such as fermionic and bosonic pair-excitations. Superconductivity in the effective Hamiltonian is nothing but the condensation of Cooper pairs between fermionic pair-excitations, which are formed by the superexchange interaction mediated by auxiliary fermions. The proposed model of superconductivity is consistent with anisotropic-singlet superconductivity previously examined by the present author (Jpn. J. Appl. Phys. 26 (1987) L652.), which is based on the Fermi liquid theory of the Hubbard Hamiltonian.
We develop a theory of superconductivity (or superfluidity) based on condensed fermion quartets focusing on the dilute spin-12\frac{1}{2}21​ systems at zero temperature. In the spirit of the Bardeen--Cooper--Schrieffer ansatz, a variational wavefunction is constructed such that, within the so-called ``dilute quartet approximation", it is the ground state of an effective quartic Hamiltonian. For a given two-body interaction in favor of quartetting, the gap parameter is suitably defined and the gap equation is also derived. As to the excited states, an intuitive physical picture based on a sixteen-dimensional ``occupation space" is depicted and the associated eigen-energies are obtained. This theory is applied to compute the superfluid fraction, which is found to be the same as in conventional superconductors, despite the interacting nature of the quartet problem.
Recently Wang and Cheng proposed a self-consistent effective Hamiltonian theory (SCEHT) for many-body fermionic systems (Wang & Cheng, 2019). This paper attempts to provide a mathematical foundation to the formulation of the SCEHT that enables further study of excited states of the system in a more systematic and theoretical manner. Gauge fields are introduced and correct total energy functional in relations to the coupling gauge field is given. We also provides a Monte-Carlo numerical scheme for the search of the ground state that goes beyond the SCEHT.