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arXiv:1312.4186 · Physical Review Letters

Ferromagnetism and Borromean Binding in Three-Fermion Clusters

P. E. Kornilovitch

Published Feb 19, 2014• 14 citations
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Abstract

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.

DOI
10.1103/physrevlett.112.077202

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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.
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-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.
The t-t'-J model (strongly interacting limit of a particular Hubbard model) is solved exactly on small clusters of eight sites with periodic boundary conditions for the simple-, body-centered-, and face-centered-cubic lattices and for the two-dimensional square lattice. The symmetry, k vector, and spin of the ground state are studied as functions of crystalline environment, interaction strength, and electron concentration. Phase diagrams are presented for stable solutions, and regions of parameter space that exhibit ferromagnetism and heavy-fermionic behavior are identified.
Numerical calculations [D. Mattis and S. Rudin, Phys. Rev. Lett. 52, 755 (1984)] indicate the absence of bound states of three identical spin-(1/2) fermions moving on a two-dimensional lattice and interacting with attractive two-body zero-range potentials. Here, we give a proof of this result in the strong-coupling limit, and we also prove that, in the weak-coupling limit, the ground state of six fermions consists of three bound pairs. We discuss the relevance of this result to superconductivity.
We investigate the strong-coupling limit of a three-component Fermi mixture in an optical lattice with attractive interactions. In this limit bound states (trions) of the three components are formed. We study the effective Hamiltonian for these composite fermions and show that it is asymptotically equivalent at the leading order to an antiferromagnetic Ising model. By using Monte Carlo simulations, we investigate the spatial arrangement of the trions in this regime and the formation of a trionic density wave, both in a homogeneous lattice and in the presence of an additional harmonic confinement. Depending on the strength of the confinement and on the temperature, we found several scenarios for the trionic distribution, including the coexistence of disordered trions with a density wave and band insulator phases. Our results show that, due to a proximity effect, staggered density modulations are induced in regions of the trap where they would not otherwise be present according to the local density approximation.
A random antiferromagnetic spin-1 chain with both Heisenberg and biquadratic nearest neighbor couplings is found to exhibit a novel ferromagnetic-like instability at low energies, if there is competition between the Heisenberg and biquadratic couplings, even though each individual nearest neighbor coupling is such as to produce a singlet ground state for the isolated pair. This instability leads to the generation of ferromagnetic couplings, and thus to the formation of weakly coupled large moments at low temperatures. A perturbative calculation (up to 4th order) is performed to investigate whether such a situation is physically realizable in spin-1 systems.