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Zeitschrift für Naturforschung A

Nearest Neighbour Particle-Particle Interaction in Fermionic Quasi One-Dimensional Flat Band Lattices

Simon Tilleke, Mirko Daumann, T. Dahm

Published Mar 25, 2020• 26 citations
Publisher

Abstract

Abstract In this paper, we have studied spinless fermions in four specific quasi one-dimensional systems that are known to host flat bands in the noninteracting limit: the triangle lattice, the stub lattice, the diamond lattice, and the diamond lattice with transverse hopping. The influence of the nearest neighbour interaction on the flat bands was investigated. We used exact diagonalization of finite size lattices employing the Lanczos technique and determine the single particle spectral functions of the interacting system. Our results are compared with mean field calculations. In the cases of the triangle lattice and the stub lattice we found that the flat bands become dispersive in the presence of a finite interaction. For the diamond lattice and the diamond lattice with transverse hopping, we demonstrated that the flat bands are robust under the influence of the interaction in certain parameter ranges. Such systems could be realised experimentally with cold atoms in optical lattices.

DOI
10.1515/zna-2019-0371

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Abstract In this paper, we have studied spinless fermions in four specific quasi one-dimensional systems that are known to host flat bands in the noninteracting limit: the triangle lattice, the stub lattice, the diamond lattice, and the diamond lattice with transverse hopping. The influence of the nearest neighbour interaction on the flat bands was investigated. We used exact diagonalization of finite size lattices employing the Lanczos technique and determine the single particle spectral functions of the interacting system. Our results are compared with mean field calculations. In the cases of the triangle lattice and the stub lattice we found that the flat bands become dispersive in the presence of a finite interaction. For the diamond lattice and the diamond lattice with transverse hopping, we demonstrated that the flat bands are robust under the influence of the interaction in certain parameter ranges. Such systems could be realised experimentally with cold atoms in optical lattices.
The difference between boson and fermion dynamics in quasi-one-dimensional lattices is studied by calculating the persistent current in small quantum rings and by exact simulations of the time evolution of the many-particle state in two cases: expansion of a localized cloud and collisions in a Newton's cradle. We consider three different lattices which in the tight-binding model exhibit flat bands. The physical realization is considered to be an optical lattice with bosonic or fermionic atoms. The atoms are assumed to interact with a repulsive short-range interaction. The different statistics of bosons and fermions lead to different dynamics. Spinless fermions are easily trapped in the flat-band states due to the Pauli exclusion principle, which prevents them from interacting, while bosons are able to push each other out from the flat-band states.
According to the exact analytical Bethe-ansatz solution of the one-dimensional spin-1/2 XXZ model, we perform a numerical study of one-dimensional spinless fermions in an optical lattice in the presence of harmonic potential by using a local density approximation. We study the density profile, and obtain the -u phase diagram (here =NfV2/t is the characteristic dimensionless density and u=V/t is the interaction strength scaled in units of the hopping parameter t). With the increases of particle density and nearest-neighbour interaction, the system undergoes five different quantum phases. Through calculating the thermodynamic stiffness S, we find that it can be used as an order parameter. Its singular points can measure the quantum phase transitions in such a confined system.
We systematically study the effect of disorder and interactions on a quasi-one-dimensional diamond chain possessing flat bands.Disorder localizes all the single-particle eigenstates, while at low disorder strengths, we obtain weak flat-band based localization (FBL), at high disorder strengths, we see conventional Anderson localization (AL).The compactly localized (CL) eigenstates of flat bands show a persisting oscillatory recurrence in the study of single-particle wave-packet dynamics.For low disorder, a damped oscillatory recurrence behavior is observed which is absent for high disorder.Noninteracting many-particle fermion states also follow the same trend except showing a delocalizing tendency at intermediate disorder due to the fermionic statistics in the system.As interactions are switched on, for the finite sizes that we are able to study, a nonergodic "mixed phase" is observed at low disorder which is separated from the MBL phase at high disorder by a thermal phase at intermediate disorder.A study of many-body nonequilibrium dynamics reinforces these findings.
We study single-particle properties of a spinless ppp-band correlated fermionic gas in an optical lattice by means of a variational cluster approach (VCA). The single-particle spectral function is almost flat at half filling and develops a strongly dispersive behavior at lower fillings. The competition between different orbital orderings is studied as a function of filling. We observe that an ``antiferromagnetic'' orbital order develops at half filling and is destroyed by doping the system evolving into a disordered orbital state. At low-filling limit, we discuss the possibility of ``ferromagnetic'' orbital order by complementing the VCA result with observations based on a trial wave function. We also study the behavior of the momentum distribution for different values of the on-site interaction. Finally, we introduce an integration contour in the complex plane which allows to efficiently carry out Matsubara frequency sums.
Electrons may form inter-site pairs (dimers) by a number of mechanisms. For example, long-range (Fröhlich) electron-phonon interactions and strong on-site Hubbard U allow formation of small light bipolarons in some lattices. We identify circumstances under which triplet dimers are strictly localised by interference in certain one- and two-dimensional lattices. We assume a U-V Hamiltonian with nearest- and next-nearest-neighbour hopping integrals t and t ', large positive U and attractive nearest- and next-nearest-neighbour interactions V and V '. In the square ladder and some two-dimensional bilayers, if the dimer Hilbert space is restricted to nearest- and next-nearest-neighbour dimers, triplet dimers become strictly localised for certain values of these parameters. For example, in a square ladder with t ' = t and V ' = V , all triplet bands become flat due to exact cancellation of hopping paths. We identify the localised eigenstates for all flat bands in each lattice. We show that many of the flat bands persist for arbitrary t/t ' so long as other restrictions still apply.
We use the density-matrix renormalization group method to investigate the superconductor-insulator transition (SIT) in disordered quasi-one-dimensional systems. Focusing on the case of an interacting spinful Hamiltonian at quarter filling, we contrast the differences arising in the SIT when the parent noninteracting model features flat or dispersive bands. Furthermore, we unveil the critical disorder amplitude that triggers insulating behavior by comparing disorder distributions that preserve or not SU(2) symmetry. While scaling analysis suggests the transition be of a Berezinskii-Kosterlitz-Thouless type for all models (two lattices and two disorder types), only in the flat-band model with Zeeman-like disorder is the critical disorder nonvanishing. In this sense, the flat-band structure does strengthen superconductivity in the presence of attractive interactions. For both flat- and dispersive-band models, (i) in the presence of SU(2)-symmetric random chemical potentials, the disorder-induced transition is from superconductor to insulator of singlet pairs; (ii) for the Zeeman-type disorder, the transition is from superconductor to insulator of unpaired fermions. In all cases, our numerical results suggest no intermediate disorder-driven metallic phase.