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Journal of the Physical Society of Japan

Entanglement Entropy in the Bilayer Hubbard Model on the Bethe Lattice

Hunpyo Lee

Published Nov 5, 2018• 0 citations
Publisher

Abstract

Using dynamical mean field theory (DMFT) in combination with a continuous-time quantum Monte Carlo (DMFT+CTQMC) tool, we studied the von Neumann entanglement entropy (S) in a bilayer Hubbard model on the Bethe lattice. Jump behaviors in S are observed at the critical points among the Fermi liquid, the Mott insulator, and the band insulator at a low temperature, similar to the results in a double occupancy which the DMFT+CTQMC method can easily calculate. As novel results that the double occupancy cannot account for, the behaviors of S in the weak region with metallic states can supply information on the coherence temperature and the Fermi temperature without the need to analyze the transport properties through analytical continuation of the self-energy or differentiation of the energy. In addition, when the strength of the interlayer hopping is tuned in the strong interaction region, S increases in the Mott phase owing to the short-range spatial fluctuations between impurities of bilayers while it decreases in the band phase. These behaviors mean that S might provide a criterion for separation between the Mott insulator and the band insulator as an order parameter in the strong interaction region.

DOI
10.7566/jpsj.87.124702

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Using dynamical mean field theory (DMFT) in combination with a continuous-time quantum Monte Carlo (DMFT+CTQMC) tool, we studied the von Neumann entanglement entropy (S) in a bilayer Hubbard model on the Bethe lattice. Jump behaviors in S are observed at the critical points among the Fermi liquid, the Mott insulator, and the band insulator at a low temperature, similar to the results in a double occupancy which the DMFT+CTQMC method can easily calculate. As novel results that the double occupancy cannot account for, the behaviors of S in the weak region with metallic states can supply information on the coherence temperature and the Fermi temperature without the need to analyze the transport properties through analytical continuation of the self-energy or differentiation of the energy. In addition, when the strength of the interlayer hopping is tuned in the strong interaction region, S increases in the Mott phase owing to the short-range spatial fluctuations between impurities of bilayers while it decreases in the band phase. These behaviors mean that S might provide a criterion for separation between the Mott insulator and the band insulator as an order parameter in the strong interaction region.
We study the critical behavior of the single-site entanglement entropy S at the Mott metal-insulator transition in infinite-dimensional Hubbard model. For this model, the entanglement between a single site and rest of the lattice can be evaluated exactly, using the dynamical mean-field theory (DMFT). Both the numerical solution using exact diagonalization and the analytical one using two-site DMFT gives S - Sc∝ α log2[(1/2 - Dc)/Dc](U - Uc), with Dcthe double occupancy at Ucand α < 0 being different on two sides of the transition.
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.
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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.
Abstract The Mott transition is a paradigmatic phenomenon where Coulomb interactions between electrons drive a metal‐insulator phase transition. It is extensively studied within the Hubbard model, where a quantum critical transition occurs at a finite temperature second‐order critical point. This work investigates the Mott transition in a modified periodic Anderson model that may be viewed as a three‐orbital lattice model including an interacting, localized orbital coupled to a delocalized conduction orbital via a second conduction orbital. Within the dynamical mean field theory, this model possesses a strictly zero temperature quantum critical point separating a Fermi liquid and a Mott insulating phase. By employing a simplified version of the dynamical mean field theory, namely, the two‐site or linearized dynamical mean field theory, an analytical estimate is provided for the critical parameter strengths at which the transition occurs at zero temperature. An analytical estimate of the single‐site von Neumann entanglement entropy is also provided. This measure can be used as a robust identifier for the phase transition. These calculations are extended to their cluster version to incorporate short‐range, spatial correlations and discuss their effects on the transition observed in this model.
We study the Hubbard model at half band-filling on a Bethe lattice with infinite coordination number in the paramagnetic insulating phase at zero temperature. We use the dynamical mean-field theory (DMFT) mapping to a single-impurity Anderson model with a bath whose properties have to be determined self-consistently. For a controlled and systematic implementation of the self-consistency scheme we use the fixed-energy (FE) approach to the DMFT. In FE-DMFT the onset and the width of the Hubbard bands are adjusted self-consistently but the energies of the bath levels are kept fixed relatively to both band edges during the calculation of self-consistent hybridization strengths between impurity and bath sites. Using the dynamical density-matrix renormalization group method (DDMRG) we calculate the density of states with a resolution ranging from 3% of the bare bandwidth W=4tW=4tW=4t at high energies to 0.5% in the vicinity of the gap. The DDMRG resolution and accuracy for the density of states and the gap is superior to those obtained with other numerical methods in previous DMFT investigations. We find that the Mott gap closes at a critical coupling Uc/t=4.45±0.05U_{\rm c}/t=4.45 \pm 0.05Uc​/t=4.45±0.05. At U=4.5tU=4.5tU=4.5t, we observe prominent shoulders near the onset of the Hubbard bands. They are the remainders of the quasi-particle resonance in the metallic phase which appears to split when the gap opens at UcU_{\rm c}Uc​.