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arXiv:1308.4220 · Europhysics Letters (EPL)

Stability of three-fermion clusters with finite range of attraction

P. E. Kornilovitch

Published Jul 1, 2013• 12 citations
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Abstract

Three quantum particles with on-site repulsion and nearest-neighbour attraction on a one-dimensional lattice are considered. The three-body Schrödinger equation is reduced to a set of single-variable integral equations. Energies of three-particle bound complexes (trions) are found from self-consistency of the approximating matrix equation. In the case of spin- fermions, the ground-state trion energy, the excited-state energies, the trion spectra and stability regions are obtained for total spins S = 1/2 and S = 3/2. In the S = 1/2 sector, a narrow but finite parameter region is identified where the ground state consists of a stable fermion pair and an unbound fermion. Also presented is the reference case of spin-0 bosons.

DOI
10.1209/0295-5075/103/27005

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Three quantum particles with on-site repulsion and nearest-neighbour attraction on a one-dimensional lattice are considered. The three-body Schrödinger equation is reduced to a set of single-variable integral equations. Energies of three-particle bound complexes (trions) are found from self-consistency of the approximating matrix equation. In the case of spin- fermions, the ground-state trion energy, the excited-state energies, the trion spectra and stability regions are obtained for total spins S = 1/2 and S = 3/2. In the S = 1/2 sector, a narrow but finite parameter region is identified where the ground state consists of a stable fermion pair and an unbound fermion. Also presented is the reference case of spin-0 bosons.
A three-particle spin-12 fermion problem with on-site repulsion and nearest-neighbor attraction is solved on the two-dimensional square lattice by discretizing a Schrödinger equation in momentum space. Energies of bound complexes (trions) and their binding conditions are obtained. For total spin S=1/2, a wide region of trion instability toward decaying into a stable singlet pair plus a free fermion is identified. The instability is attributed to the formation of a wave function node upon addition of the third fermion. In the S=3/2 sector, trions are found to form in the absence of bound pairs indicating Borromean binding. In the strong coupling limit the system transitions from an S=1/2 ground state to a ferromagnetic S=3/2 ground state in agreement with the Nagaoka theorem for a four-site plaquette.
A three-fermion problem in a three-dimensional lattice with anisotropic hopping is solved by discretizing the Schrödinger equation in momentum space. Interparticle interaction comprises on-site Hubbard repulsion and in-plane nearest-neighbor attraction. By comparing the energy of three-fermion bound clusters (trions) with the energy of one pair plus one free particle, a trion formation threshold is accurately determined, and the region of pair stability is mapped out. It is found that the “close-packed” density of fermion pairs, which is associated with a maximum pair condensation temperature in this model, is the highest in a strongly anisotropic case. It is also argued that pair superconductivity with the highest critical temperature is always close to trion formation, which makes the system prone to phase separation and local charge ordering.
On the basis of a microscopic model of self-consistent field, the thermodynamics of the many-particle Fermi system at finite temperatures with account of three-body interactions is built and the quasiparticle equations of motion are obtained.It is shown that the delta-like three-body interaction gives no contribution into the selfconsistent field, and the description of three-body forces requires their nonlocality to be taken into account.The spatially uniform system is considered in detail, and on the basis of the developed microscopic approach general formulas are derived for the fermion's effective mass and the system's equation of state with account of contribution from three-body forces.The effective mass and pressure are numerically calculated for the potential of "semi-transparent sphere" type at zero temperature.Expansions of the effective mass and pressure in powers of density are obtained.It is shown that, with account of only pair forces, the interaction of repulsive character reduces the quasiparticle effective mass relative to the mass of a free particle, and the attractive interaction raises the effective mass.The question of thermodynamic stability of the Fermi system is considered and the three-body repulsive interaction is shown to extend the region of stability of the system with the interparticle pair attraction.The quasiparticle energy spectrum is calculated with account of three-body forces.
We consider a system of three particles consisting of two identical fermions and one other particle on a one-dimensional lattice. The fermions interact via a nearest-neighbor potential of strength μ1∈R\mu_1\in\mathbb{R}μ1​∈R , while the interaction between a fermion and one other particle is via an on-site potential with strength μ2∈R{\mu_2\in\mathbb{R}}μ2​∈R . We establish existence of bound states of the associated three-body lattice Schrödinger operator for all values of the total quasimomentum K∈T1K\in\mathbb{T}^1K∈T1 . Furthermore, we show that both the bound state fμ1μ2(K; ⋅ , ⋅ )f_{\mu_1\mu_2}(K;\,{\cdot}\,{,}\,{\cdot}\,)fμ1​μ2​​(K;⋅,⋅) and its corresponding eigenvalue Eμ1μ2(K)E_{\mu_1\mu_2}(K)Eμ1​μ2​​(K) depend holomorphically on the quasimomentum.
A one-dimensional system of a finite number of bosons with a δ-like interaction potential is treated. A three-body boson repulsion is taken into account, along with the pairwise attraction. The bound state of N bosons is found in the self-consistent-field approximation. In such a system there is no collapse of the Bose gas, and the ground state in the thermodynamic limit is the bound state of N particles with a practically constant density. Such a state can be considered as a condensate droplet in the coordinate space.
We start with a variational approach and derive a set of coupled integral equations for the bound states of NNN identical spin-$\ensuremath{\uparrow}$ fermions and a single spin-$\ensuremath{\downarrow}$ fermion in a generic multiband Hubbard Hamiltonian with an attractive on-site interaction. As an illustration, we apply our integral equations to the one-dimensional sawtooth lattice up to $N\ensuremath{\le}3$, i.e., to the (3+1)(3+1)(3+1)-body problem, and we reveal not only the presence of tetramer states in this two-band model but also their quasiflat dispersion when formed in a flat band. Furthermore, for $N={4,5,\ensuremath{\cdots},10}$, our density-matrix renormalization-group simulations and exact diagonalization suggest the presence of larger and larger multimers with lower and lower binding energies, conceivably without an upper bound on NNN. These peculiar (N+1)(N+1)(N+1)-body clusters are in sharp contrast with the exact results on the single-band linear-chain model where none of the $N\ensuremath{\ge}2$ multimers appear. Hence their presence must be taken into account for a proper description of the many-body phenomena in flat-band systems, e.g., they may suppress superconductivity especially when there exists a large spin imbalance.