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Atoms

Phase Diagram of the Attractive Kane-Mele-Hubbard Model at Half Filling

Zlatko G. Koinov

Published Sep 14, 2020• 1 citation
Publisher

Abstract

Motivated by recent developments in the experimental study of ultracold atoms in graphene-like honeycomb optical lattices, we investigate superconductivity of the attractive Kane-Mele-Habbard (KMH) model with the next-nearest-neighbor (NNN) hoping at half filling. The mean-field approximation is used to study the phase diagram which interpolates the trivial and the non-trivial topological states. It is shown that: (a) when the NNN hoping is taken into account, one has to introduce two mean-field gap equations for the two sublattices, instead of a single gap when the NNN hopping is neglected, and (b) in the non-trivial topological region the phase diagram with the NNN hopping is significantly different compared to the phase diagram calculated previously, but without the NNN term. We also discuss the superconducting instability of the attractive KMH model that is driven by condensation of Cooperons.

DOI
10.3390/atoms8030058

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Motivated by recent developments in the experimental study of ultracold atoms in graphene-like honeycomb optical lattices, we investigate superconductivity of the attractive Kane-Mele-Habbard (KMH) model with the next-nearest-neighbor (NNN) hoping at half filling. The mean-field approximation is used to study the phase diagram which interpolates the trivial and the non-trivial topological states. It is shown that: (a) when the NNN hoping is taken into account, one has to introduce two mean-field gap equations for the two sublattices, instead of a single gap when the NNN hopping is neglected, and (b) in the non-trivial topological region the phase diagram with the NNN hopping is significantly different compared to the phase diagram calculated previously, but without the NNN term. We also discuss the superconducting instability of the attractive KMH model that is driven by condensation of Cooperons.
Recent experiment on MoTe2_22​-WTe2_22​ twisted bi-layer demonstrated physics of Kane-Mele (KM) as well as Haldane models. Although topological properties of KM model has been studied extensively, effects of interaction are still less explored beyond half filling. In this work we study the effect of Hubbard interaction in KM model at small hole doping around half filling. We use density matrix renormalization group method to characterize different phase of cylinders with width of Ly=3L_y=3Ly​=3 unit cells. We identify {a possible} superconducting (SC) phase and transition at different strength of spin-orbit coupling (SOC) and show the strong dependence of critical interaction for SC transition. We also calculate the single-particle Green's function, spin-spin and density-density correlations and compare with SC correlation to establish the dominance of SC correlations. Our result highlights striking behaviour of SC transition and its dependence on SOC which could be useful in understanding different ways to achieve unconventional SC.
The dimerized Kane-Mele model with/without strong interaction is studied using analytical methods. The boundary of the topological phase transition of the model without strong interaction is obtained. Our results show that the occurrence of the transition only depends on dimerized parameter [Formula: see text]. From the one-particle spectrum, we obtain the completed phase diagram including the quantum spin Hall state and the topologically trivial insulator. Then, using different mean field methods, we investigate the Mott transition and the magnetic transition of the strongly correlated dimerized Kane-Mele model. In the region between the two transitions, the topological Mott insulator with characteristics of Mott insulators and topological phases may be the most interesting phase. In this work, the effects of hopping anisotropy and Hubbard interaction U on the boundaries of the two transitions are observed in detail. The completed phase diagram of the dimerized Kane-Mele-Hubbard model is also obtained in this work. Quantum fluctuations have extremely important influences on a quantum system. However, investigations are under the framework of mean field treatment in this work and the effects of fluctuations in this model will be discussed in the future.
We investigate the ground-state phase diagram of the half-filled one-dimensional Hubbard model with next-nearest-neighbor hopping using the density-matrix renormalization-group technique as well as an unrestricted Hartree-Fock approximation. We find commensurate and incommensurate disordered magnetic insulating phases and a spin-gapped metallic phase in addition to the one-dimensional Heisenberg phase. At large on-site Coulomb repulsion U, we make contact with the phase diagram of the frustrated Heisenberg chain, which has spin-gapped phases for sufficiently large frustration. For weak U, sufficiently large next-nearest-neighbor hopping t2{t}_{2}t2​ leads to a band structure with four Fermi points rather than two, producing a spin-gapped metallic phase. As U is increased in this regime, the system undergoes a Mott-Hubbard transition to a frustrated antiferromagnetic insulator.
Superconducting phase diagram of the extended Hubbard model supplemented with interaction and hopping terms exceeding the nearest-neighbour distance in range is analysed systematically at different band filling and temperature values in a mean-field approximation. The obtained results clearly emphasize the importance of next-nearest-neighbour terms in developing the main superconducting phases of different symmetries at a given point of the phase diagram; in particular, critical temperatures TT , zero temperature gap amplitude values Δ0, Δ0/T c ratios, doping and temperature dependences are all strongly influenced by next-nearest-neighbour contributions.
We prove that the Kane-Mele-Hubbard model with purely imaginary next-nearest-neighbor hoppings has a particle-hole symmetry at half filling. Such a symmetry has interesting consequences including the absence of charge and spin currents along open edges, and the absence of the sign problem in the determinant quantum Monte Carlo simulations. Consequentially, the interplay between band topology and strong correlations can be studied at high numeric precisions. The process that the topological band insulator evolves into the antiferromagnetic Mott insulator as interaction strength increases is studied by calculating both the bulk and edge electronic properties.
We investigate the phase diagram of spinless fermions with nearest- and next-nearest-neighbor density-density interactions on the honeycomb lattice at half-filling. Using exact diagonalization techniques of the full Hamiltonian and constrained subspaces, combined with a careful choice of finite-size clusters, we determine the different charge orderings that occur for large interactions. In this regime, we find a two-sublattice N\'eel-like state, a charge modulated state with a tripling of the unit cell, a zigzag phase, and a charge ordered state with a 12-site unit cell we call N\'eel domain wall crystal, as well as a region of phase separation for attractive interactions. A sizable region of the phase diagram is classically degenerate, but it remains unclear whether an order-by-disorder mechanism will lift the degeneracy. For intermediate repulsion, we find evidence for a Kekul\'e or plaquette bond-order wave phase. We also investigate the possibility of a spontaneous Chern insulator phase (dubbed topological Mott insulator), as previously put forward by several mean-field studies. Although we are unable to detect convincing evidence for this phase based on energy spectra and order parameters, we find an enhancement of current-current correlations with the expected spatial structure compared to the noninteracting situation. While for the studied $t\ensuremath{-}{V}_{1}\ensuremath{-}{V}_{2}$ model, the phase transition to the putative topological Mott insulator is preempted by the phase transitions to the various ordered states, our findings might hint at the possibility for a topological Mott insulator in an enlarged Hamiltonian parameter space, where the competing phases are suppressed.