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arXiv:2205.02377 · Physical review. B./Physical review. B

Bilayer Hubbard model: Analysis based on the fermionic sign problem

Yingping Mou, Rubem Mondaini, Richard T. Scalettar

Published Sep 12, 2022• 13 citations
arXivHTMLPDF

Abstract

The bilayer Hubbard model describes the antiferromagnet to spin singlet transition and, potentially, aspects of the physics of unconventional superconductors. Despite these important applications, significant aspects of its phase diagram in the interplane hopping ${t}_{\ensuremath{\perp}}$ versus on-site interaction $U$ parameter space, at half filling, are largely in disagreement. Here we provide an analysis making use of the average sign of weights over the course of the importance sampling in quantum Monte Carlo simulations to resolve several central open questions. Specifically, this metric of the weights clarifies the finite-sized metallic regimes at small $U$. Furthermore, at strong interactions, it points to the existence of a crossover from a correlated to uncorrelated band insulator not yet explored in a variety of existing, unbiased numerical methods. Our paper demonstrates the versatility of using properties of the weights in quantum Monte Carlo simulations to reveal important physical characteristics of the models under study.

DOI
10.1103/physrevb.106.125116

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The bilayer Hubbard model describes the antiferromagnet to spin singlet transition and, potentially, aspects of the physics of unconventional superconductors. Despite these important applications, significant aspects of its phase diagram in the interplane hopping t\ensuremath⊥{t}_{\ensuremath{\perp}}t\ensuremath⊥​ versus on-site interaction UUU parameter space, at half filling, are largely in disagreement. Here we provide an analysis making use of the average sign of weights over the course of the importance sampling in quantum Monte Carlo simulations to resolve several central open questions. Specifically, this metric of the weights clarifies the finite-sized metallic regimes at small UUU. Furthermore, at strong interactions, it points to the existence of a crossover from a correlated to uncorrelated band insulator not yet explored in a variety of existing, unbiased numerical methods. Our paper demonstrates the versatility of using properties of the weights in quantum Monte Carlo simulations to reveal important physical characteristics of the models under study.
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 Hubbard model on a bilayer with repulsive on-site interactions U in which fermions undergo both intraplane (t) and interplane (tz{\mathit{t}}_{\mathit{z}}tz​) hopping. This situation is what one would expect in high-temperature superconductors such as YBa2{\mathrm{YBa}}_{2}YBa2​Cu3{\mathrm{Cu}}_{3}Cu3​${\mathrm{O}}_{7\mathrm{\ensuremath{-}}\mathit{y}}$, with two adjacent CuO2{\mathrm{CuO}}_{2}CuO2​ planes. Magnetic and pairing properties of the system are investigated through quantum Monte Carlo simulations for both half- and quarter-filled bands. We find that in all cases interplanar pairing with dx2{\mathit{d}}_{\mathit{x}}^{2}dx2​-z2{\mathit{z}}^{2}z2 symmetry is dominant over planar pairing with dx2{\mathit{d}}_{\mathit{x}}^{2}dx2​-y2{\mathit{y}}^{2}y2 symmetry, and that for a large enough tz{\mathit{t}}_{\mathit{z}}tz​ pair formation is possible through antiferromagnetic correlations. However, another mechanism is needed to make these pairs condense into a superconducting state at lower temperatures. We identify the temperature for pair formation with the spin-gap crossover temperature.
We investigate the phase diagram of the square lattice bilayer Hubbard model at half-filling with the variational Monte Carlo method for both the magnetic and the paramagnetic case as a function of the interlayer hopping and on-site Coulomb repulsion U . With this study we resolve some discrepancies in previous calculations based on the dynamical mean-field theory, and we are able to determine the nature of the phase transitions between metal, Mott insulator and band insulator. In the magnetic case we find only two phases: an antiferromagnetic Mott insulator at small for any value of U and a band insulator at large . At large U values we approach the Heisenberg limit. The paramagnetic phase diagram shows at small a metal to Mott insulator transition at moderate U values and a Mott to band insulator transition at larger U values. We also observe a re-entrant Mott insulator to metal transition and metal to band insulator transition for increasing in the range of . Finally, we discuss the phase diagrams obtained in relation to findings from previous studies based on different many-body approaches.
Abstract. We investigate the phase diagram of the square lattice bilayer Hubbard model at half filling with the variational Monte Carlo method for both the magnetic and the paramagnetic case as a function of the interlayer hopping t ⊥ and on-site Coulomb repulsion U. With this study we resolve some discrepancies in previous calculations based on the dynamical mean field theory, and we are able to determine the nature of the phase transitions between metal, Mott insulator and band insulator. In the magnetic case we find only two phases: An antiferromagnetic Mott insulator at small t ⊥ for any value of U and a band insulator at large t⊥. At large U values we approach the Heisenberg limit. The paramagnetic phase diagram shows at small t ⊥ a metal to Mott insulator transition at moderate U values and a Mott to band insulator transition at larger U values. We also observe a re-entrant Mott insulator to metal transition and metal to band insulator transition for increasing t ⊥ in the range of 5.5t < U < 7.5t. Finally, we discuss the obtained phase diagrams in relation to previous studies based on different many-body approaches. PACS numbers: 71.10.Fd, 71.27.+a, 71.30.+h 1.
The Hubbard model at temperatures above the Néel transition, despite being a paramagnet, can exhibit rich physics due to the interplay of Fermi surface, on-site interaction U and thermal fluctuations. Nevertheless, the understanding of the crossover physics remains only at a qualitative level, because of the intrinsically smooth behavior. Employing an improved variant of the numerically exact auxiliary-field quantum Monte Carlo algorithm equipped with numerical analytic continuation, we obtain a broad variety of thermodynamic and dynamical properties of the three-dimensional Hubbard model at half filling, quantitatively determine the crossover boundaries, and observe that the metal-insulator crossover regime, in which antiferromagnetic spin correlations appear strongest, exists over an extended regime in between the Fermi liquid for small U and the Mott insulator for large U. In particular, the location of the most rapid suppression of double occupancy as U increases, is found to fully reside in the metallic Fermi liquid regime, in contrast to the conventional intuition that it is a representative feature for entering the Mott insulator. Beside providing a reliable methodology for numerical study of crossover physics, our work can also serve as a timely and important guideline for the most recent optical lattice experiments.
We compute transport and thermodynamic properties of a two-band spin-fermion model describing itinerant fermions in two dimensions interacting via Z2{Z}_{2}Z2​ antiferromagnetic quantum critical fluctuations by means of a sign-problem-free quantum Monte Carlo approach. We show that the phase diagram of this model indeed contains a ddd-wave superconducting phase at low enough temperatures. However, a crucial question that arises is whether a non-Fermi-liquid metallic regime exists above Tc{T}_{c}Tc​, exhibiting hallmark strange-metal transport phenomenology. Interestingly, we find that this version of the model describes a non-Fermi-liquid metallic regime that displays an approximately TTT-linear resistivity above Tc{T}_{c}Tc​ for a strong fermion-boson interaction. Using the Nernst-Einstein relation, our QMC results also show that this strange metal phase exhibits a crossover from being characterized by a charge compressibility given approximately by ${\ensuremath{\chi}}_{c}\ensuremath{\sim}1/T$ at high temperatures to being described by a charge diffusivity consistent with the scaling ${D}_{c}\ensuremath{\sim}1/T$ at low temperatures. Therefore, our paper adds support to the view that the Z2{Z}_{2}Z2​ antiferromagnetic spin-fermion model at strong coupling can be considered a minimal model that describes both unconventional superconductivity and strange metallicity, which are fundamentally interconnected in many important strongly correlated quantum materials.