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arXiv:2206.06085 · Physical review. B./Physical review. B

Finite-temperature study of correlations in a bilayer band insulator

Yogeshwar Prasad

Published Nov 15, 2022• 5 citations
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Abstract

We perform the finite-temperature determinant quantum Monte Carlo simulation for the attractive Hubbard model on the half-filled bilayer square lattice. Recent progress on optical lattice experiments lead us to investigate various single-particle properties such as momentum distribution and double occupancies which should be easily measured in cold-atom experiments. The pair-pair and the density-density correlations have been studied in detail and, through finite-size scaling, we show that there is no competing charge density wave order in the bilayer band-insulator model and that the superfluid phase is the stable phase for the interaction range $|U|/t=5--10$. We show the existence of two energy scales in the system as we increase the attractive interaction, one governing the phase coherence and the other one corresponding to the molecule formation. In the end, we map out the full $T\ensuremath{-}U$ phase diagram and compare the ${T}_{c}$ obtained through the mean-field analysis. We observe that the maximum ${T}_{c}/t(=0.27)$ occurs for $|U|/t=6$, which is roughly twice the reported ${T}_{c}$ of the single-layer attractive Hubbard model.

DOI
10.1103/physrevb.106.184506

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We perform the finite-temperature determinant quantum Monte Carlo simulation for the attractive Hubbard model on the half-filled bilayer square lattice. Recent progress on optical lattice experiments lead us to investigate various single-particle properties such as momentum distribution and double occupancies which should be easily measured in cold-atom experiments. The pair-pair and the density-density correlations have been studied in detail and, through finite-size scaling, we show that there is no competing charge density wave order in the bilayer band-insulator model and that the superfluid phase is the stable phase for the interaction range ∣U∣/t=5−−10|U|/t=5--10∣U∣/t=5−−10. We show the existence of two energy scales in the system as we increase the attractive interaction, one governing the phase coherence and the other one corresponding to the molecule formation. In the end, we map out the full T\ensuremath−UT\ensuremath{-}UT\ensuremath−U phase diagram and compare the Tc{T}_{c}Tc​ obtained through the mean-field analysis. We observe that the maximum Tc/t(=0.27){T}_{c}/t(=0.27)Tc​/t(=0.27) occurs for ∣U∣/t=6|U|/t=6∣U∣/t=6, which is roughly twice the reported Tc{T}_{c}Tc​ of the single-layer attractive Hubbard model.
In previous studies of the half-filled bilayer attractive Hubbard model [Prasad et al., Phys. Rev. A 89, 043605 (2014); Prasad, Phys. Rev. B 106, 184506 (2022)], it has been shown that the clean system has a band-insulator (BI) to superfluid (SF) quantum phase transition. In this paper, we append the effects of random on-site disorder on the kinetic energy, double occupancy, and the pair-pair correlations in the bilayer model. Using the determinant quantum Monte Carlo simulation, we observe that the on-site random disorder plays a significant role in the localization of on-site pairs, and hence in the reduction of the effective hopping. This results in an increase of the double occupancy, which is an effect that is similar to the attractive interaction. We find no change in the critical value of the interaction at which the model undergoes a transition from the BI to SF regime, even though the pair-pair correlations get suppressed for finite on-site disorder strengths Vd/t=0.1{V}_{d}/t=0.1Vd​/t=0.1--0.8. We also confirm that the weak-disorder suppresses the SF phase largely in the strong-coupling limit. Hence the region of the SF phase reduces in the presence of random on-site disorder. Finally, through finite-size scaling, we have estimated the critical disorder strength ${V}_{d}^{c}/t\ensuremath{\sim}1.44$ at ∣U∣/t=5|U|/t=5∣U∣/t=5.
Employing a combination of functional renormalization group calculations and projective determinantal quantum Monte Carlo simulations, we examine the Hubbard model on the square lattice bilayer at half filling. From this combined analysis, we obtain a comprehensive account of the ground-state phase diagram with respect to the extent of the system's metallic and (antiferromagnetically ordered) Mott-insulating as well as band-insulating regions. By means of an unbiased functional renormalization group approach, we exhibit the antiferromagnetic Mott-insulating state as the relevant instability of the free metallic state, induced by any weak finite on-site repulsion. Upon performing a careful analysis of the quantum Monte Carlo data, we resolve the difficulty of identifying this antiferromagnetic ground state for finite interlayer hopping in the weak-coupling regime, where nonmonotonic finite-size corrections are shown to relate to the two-sheeted Fermi surface structure of the metallic phase. On the other hand, quantum Monte Carlo simulations are well suited to identify the transition between the Mott-insulating phase and the band insulator in the intermediate-to-strong coupling regime. Here, we compare our numerical findings to indications for the transition region obtained from the functional renormalization group procedure.
We study the attractive fermionic Hubbard model on a honeycomb lattice using determinantal quantum Monte Carlo simulations. By increasing the interaction strength UUU (relative to the hopping parameter ttt) at half filling and zero temperature, the system undergoes a quantum phase transition at 5.0<Uc/t<5.15.0<{U}_{c}/t<5.15.0<Uc​/t<5.1 from a semimetal to a phase displaying simultaneously superfluid behavior and density order. Doping away from half filling, and increasing the interaction strength at finite but low temperature TTT, the system always appears to be a superfluid exhibiting a crossover between a BCS and a molecular regime. These different regimes are analyzed by studying the spectral function. The formation of pairs and the emergence of phase coherence throughout the sample are studied as UUU is increased and TTT is lowered.
The attractive Hubbard model has become a model readily realizable with ultracold atoms on optical lattices. However, the superconducting (superfluid) critical temperatures Tc{T}_{c}Tc​'s are still somewhat smaller than the lowest temperatures achieved in experiments. Here, we consider two possible routes, generically called layering, to increase Tc{T}_{c}Tc​: a bilayer and a simple cubic lattice, both with tunable hopping tz{t}_{z}tz​ between attractive Hubbard planes. We have performed minus-sign--free determinant quantum Monte Carlo simulations to calculate response functions such as pairing correlation functions, uniform spin susceptibility, and double occupancy, through which we map out some physical properties. We have found that by a judicious choice of fillings and intensity of on-site attraction, a bilayer can exhibit Tc{T}_{c}Tc​'s between 1.5 and 1.7 times those of the single layer; for the simple-cubic lattice the enhancement can be 30% larger than the maximum for the single layer. We also check the accuracy of both a BCS-like estimate for Tc{T}_{c}Tc​ in the attractive Hubbard model as well as of an upper bound for Tc{T}_{c}Tc​ based on the superfluid density.
From the condensed matter physics perspective, the most natural single orbital tight-binding Hamiltonians, and hence the most widely studied, contain two fermionic species, corresponding to spin up and spin down electrons. In cold atom systems, however, SU(N) symmetry, in which N>2N > 2N>2 fermionic species reside within a single band, also occurs. In order to understand such experiments, the SU(N) Hubbard model has been increasingly studied. Here we present determinant Quantum Monte Carlo simulations of the SU(N) {\it Holstein} Hamiltonian, in which NNN fermionic species couple to a single local phonon mode. We show that at half filling it has an insulating charge density wave phase (CDW) at low temperatures, in which empty sites alternate with sites with NNN particles. We determine the N=3N=3N=3 CDW phase diagram in the temperature, TTT, versus electron-phonon coupling, ααα, plane at fixed phonon frequency ω0ω_0ω0​ and half-filling ρ=1.5ρ=1.5ρ=1.5. The critical temperature TcT_cTc​ for N=3N=3N=3 can be as high as twice the maximum attainable for N=2N=2N=2. We also obtain the NNN dependence of TcT_cTc​ for a representative, fixed, ω0ω_0ω0​ and ααα.
We present a numerically exact study of the Hubbard model with spin-dependent anisotropic hopping on the square lattice using auxiliary-field quantum Monte Carlo method. At half filling, the system undergoes Ising phase transitions on cooling, leading to the formation of Ising-type antiferromagnetic order for repulsive interactions and charge-density wave order for attractive interactions at finite temperatures. By elegantly implementing the sign-problem-free condition and Hubbard-Stratonovich transformations, we achieve significant improvements in precision control of the numerical calculations and obtain highly accurate results of the transition temperatures from weak to strong interactions across representative anisotropies. We further characterize the system by examining the temperature dependence of various thermodynamic properties, including the energy, double occupancy, specific heat, and charge susceptibility. Specifically, we provide unbiased numerical results of the entropy map on temperature-interaction plane, the critical entropy, and the spin, singlon and doublon correlations, all of which are directly measurable in optical lattice experiments. Away from half filling, we explore the behavior of the sign problem and investigate the possible emergence of stripe spin-density wave order in the system with repulsive interaction.