arXiv:2109.14149 · Progress of Theoretical and Experimental Physics

Metal–insulator transition in the (2+1)-dimensional Hubbard model with the tensor renormalization group

Shinichiro Akiyama, Y. Kuramashi, Takumi Yamashita

Published Jan 19, 2022• 13 citations

Abstract

Abstract We investigate the doping-driven metal–insulator transition of the (2+1)-dimensional Hubbard model in the path-integral formalism with the tensor renormalization group method. We calculate the electron density 〈n〉 as a function of the chemical potential μ, choosing three values of the Coulomb potential with U = 80, 8, and 2 as representative cases of the strong, intermediate, and weak couplings. We determine the critical chemical potential at each U, where the Hubbard model undergoes the metal–insulator transition from the half-filling plateau with 〈n〉 = 1 to the metallic state with 〈n〉 > 1. Our results indicate that the model exhibits the metal–insulator transition over a vast region of the finite coupling U.

DOI: 10.1093/ptep/ptac014

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arXiv:2109.14149 · Progress of Theoretical and Experimental Physics

Metal–insulator transition in the (2+1)-dimensional Hubbard model with the tensor renormalization group

Shinichiro Akiyama, Y. Kuramashi, Takumi Yamashita

Published Jan 19, 2022• 13 citations

arXiv HTML PDF

Abstract

DOI: 10.1093/ptep/ptac014

Related papers

Tensor renormalization group approach to ( 1+1 )-dimensional Hubbard model

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We investigate the metal-insulator transition of the ( $1+1$ )-dimensional Hubbard model in the path-integral formalism with the tensor renormalization group method. The critical chemical potential ${\ensuremath{\mu}}_{\mathrm{c}}$ and the critical exponent $\ensuremath{\nu}$ are determined from the $\ensuremath{\mu}$ dependence of the electron density in the thermodynamic limit. Our results for ${\ensuremath{\mu}}_{\mathrm{c}}$ and $\ensuremath{\nu}$ show consistency with an exact solution based on the Bethe ansatz. Our encouraging results indicate the applicability of the tensor renormalization group method to the analysis of higher-dimensional Hubbard models.We investigate the metal-insulator transition of the ($1+1$)-dimensional Hubbard model in the path-integral formalism with the tensor renormalization group method. The critical chemical potential ${\ensuremath{\mu}}_{\mathrm{c}}$ and the critical exponent $\ensuremath{\nu}$ are determined from the $\ensuremath{\mu}$ dependence of the electron density in the thermodynamic limit. Our results for ${\ensuremath{\mu}}_{\mathrm{c}}$ and $\ensuremath{\nu}$ show consistency with an exact solution based on the Bethe ansatz. Our encouraging results indicate the applicability of the tensor renormalization group method to the analysis of higher-dimensional Hubbard models.

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We investigate the metal-insulator transition of the (

1+1

)-dimensional Hubbard model in the path-integral formalism with the tensor renormalization group method. The critical chemical potential ${\ensuremath{\mu}}_{\mathrm{c}}$ and the critical exponent $\ensuremath{\nu}$ are determined from the $\ensuremath{\mu}$ dependence of the electron density in the thermodynamic limit. Our results for ${\ensuremath{\mu}}_{\mathrm{c}}$ and $\ensuremath{\nu}$ show consistency with an exact solution based on the Bethe ansatz. Our encouraging results indicate the applicability of the tensor renormalization group method to the analysis of higher-dimensional Hubbard models.

We show numerically that the nature of the doping-induced metal-insulator transition in the two-dimensional Hubbard model with hopping matrix element

t

and Coulomb repulsion

U

is radically altered by the inclusion of a term

W

that depends upon a square of a single-particle nearest-neighbor hopping. This result is reached by computing the localization length ${\ensuremath{\xi}}_{l}$ in the insulating state. At

W/t=0.05

and

U/t=4

, we find results consistent with ${\ensuremath{\xi}}_{l}\ensuremath{\sim}|\ensuremath{\mu}\ensuremath{-}{\ensuremath{\mu}}_{c}{|}^{\ensuremath{-}1/2}$ where ${\ensuremath{\mu}}_{c}$ is the critical chemical potential. In contrast, ${\ensuremath{\xi}}_{l}\ensuremath{\sim}|\ensuremath{\mu}\ensuremath{-}{\ensuremath{\mu}}_{c}{|}^{\ensuremath{-}1/4}$ for the Hubbard model at

U/t=4

. At half-filling, we calculate the density of states $N(\ensuremath{\omega})$. The large value of $N(\ensuremath{\omega})$ in the vicinity of $\ensuremath{\omega}={\ensuremath{\mu}}_{c}$ present at

W=0

is suppressed with growing values of

W

. At finite doping, the

d

-wave pair-field correlations are enhanced with growing values of

W

. The numerical results imply that at finite values of

W

doping the antiferromagnetic Mott insulator leads to a ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ superconductor.

{t}_{y}

from an insulator to a metal. Therefore, if

{t}_{y}

is fixed in the metallic phase, increasing the interaction

U

leads to a metal-to-insulator transition at a finite critical

U

. This is in contrast to the weak-coupling Hartree-Fock theory which predicts a nesting-induced antiferromagnetic insulator for any

U>0

U_{\rm c} \sim 7.4t

, where t is the transfer integral. A detailed analysis around the transition point shows that the Mott transition is of first order.

U

reaches a critical value

U_c

. Away from half filling, the insulating phase is always gapless and is found to be controlled by the Anderson fixed point at half filling. This result is similar to that obtained in the corresponding one dimensional system and suggests strongly the importance of the electron-electron correlation in this gapless insulating phase.