arXiv:2105.00372 · Physical review. D/Physical review. D.

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

Shinichiro Akiyama, Y. Kuramashi

Published Jul 7, 2021• 16 citations

Abstract

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.

DOI: 10.1103/physrevd.104.014504

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arXiv:2105.00372 · Physical review. D/Physical review. D.

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

Shinichiro Akiyama, Y. Kuramashi

Published Jul 7, 2021• 16 citations

arXiv HTML PDF

Abstract

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.

DOI: 10.1103/physrevd.104.014504

Related papers

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

Shinichiro Akiyama, Y. Kuramashi, Takumi Yamashita

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.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.

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Xiao Luo, Y. Kuramashi

We study (1+1)-dimensional O(3) nonlinear sigma model using the tensor renormalization group method with the infinite limit of the bond dimension $D_{\rm cut}\rightarrow \infty$ . At the vanishing chemical potential $μ=0$ , we investigate the von Neumann and Rényi types of entanglement entropies. The central charge is determined to be $c=1.97(9)$ by using the asymptotic scaling properties of the entropies. We also examine the consistency between two entropies. In the finite density region with $μ\ne 0$ , where this model suffers from the sign problem in the standard Monte Carlo approach, we investigate the properties of the quantum phase transition. We determine the transition point $μ_{\rm c}$ and the critical exponent of the correlation length $ν$ from the $μ$ dependence of the number density in the thermodynamic limit. The dynamical critical exponent $z$ is also extracted from the scaling behavior of the temporal correlation length as a function of $μ$ . This is the first successful calculation of the dynamical critical exponent with the TRG method.We study (1+1)-dimensional O(3) nonlinear sigma model using the tensor renormalization group method with the infinite limit of the bond dimension $D_{\rm cut}\rightarrow \infty$. At the vanishing chemical potential $μ=0$, we investigate the von Neumann and Rényi types of entanglement entropies. The central charge is determined to be $c=1.97(9)$ by using the asymptotic scaling properties of the entropies. We also examine the consistency between two entropies. In the finite density region with $μ\ne 0$, where this model suffers from the sign problem in the standard Monte Carlo approach, we investigate the properties of the quantum phase transition. We determine the transition point $μ_{\rm c}$ and the critical exponent of the correlation length $ν$ from the $μ$ dependence of the number density in the thermodynamic limit. The dynamical critical exponent $z$ is also extracted from the scaling behavior of the temporal correlation length as a function of $μ$. This is the first successful calculation of the dynamical critical exponent with the TRG method.

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We study the quantum phase transition of the (1+1)-dimensional O(3) nonlinear sigma model at finite density using the tensor renormalization group method. This model suffers from the sign problem, which has prevented us from investigating the properties of the phase transition. We investigate the properties of the phase transition by changing the chemical potential mu at a fixed coupling of beta. We determine the transition point mu c and the critical exponent nu from the mu dependence of the number density in the thermodynamic limit. The dynamical critical exponent z is also extracted from the scaling behavior of the temporal correlation length as a function of mu.We study the quantum phase transition of the (1+1)-dimensional O(3) nonlinear sigma model at finite density using the tensor renormalization group method. This model suffers from the sign problem, which has prevented us from investigating the properties of the phase transition. We investigate the properties of the phase transition by changing the chemical potential mu at a fixed coupling of beta. We determine the transition point mu c and the critical exponent nu from the mu dependence of the number density in the thermodynamic limit. The dynamical critical exponent z is also extracted from the scaling behavior of the temporal correlation length as a function of mu.

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Quantum phase transition of (1+1)-dimensional O(3) nonlinear sigma model at finite density with tensor renormalization group

Xiao Luo, Y. Kuramashi

A bstract We study the quantum phase transition of the (1+1)-dimensional O(3) nonlinear sigma model at finite density using the tensor renormalization group method. This model suffers from the sign problem, which has prevented us from investigating the properties of the phase transition. We investigate the properties of the phase transition by changing the chemical potential μ at a fixed coupling of β . We determine the transition point μ c and the critical exponent ν from the μ dependence of the number density in the thermodynamic limit. The dynamical critical exponent z is also extracted from the scaling behavior of the temporal correlation length as a function of μ .A bstract We study the quantum phase transition of the (1+1)-dimensional O(3) nonlinear sigma model at finite density using the tensor renormalization group method. This model suffers from the sign problem, which has prevented us from investigating the properties of the phase transition. We investigate the properties of the phase transition by changing the chemical potential μ at a fixed coupling of β . We determine the transition point μ c and the critical exponent ν from the μ dependence of the number density in the thermodynamic limit. The dynamical critical exponent z is also extracted from the scaling behavior of the temporal correlation length as a function of μ .

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Quantum Field Theories with Tensor Renormalization Group

Shinichiro Akiyama, Y. Kuramashi, Yusuke Yoshimura

We report recent progress on the application of the tensor renormalization group (TRG) to quantum field theories pursued by the Tsukuba group. We explain how to treat the scalar, fermion, and gauge theories with the TRG method presenting the results for the phase transitions in the (3+1)-dimensional ((3+1) $d$ ) complex $\phi^4$ theory at finite density, (1+1) $d$ pure U(1) lattice gauge theory with a $\theta$ term, (3+1) $d$ Nambu--Jona-Lasinio model at finite density and (1+1) $d$ and (2+1) $d$ Hubbard models at an arbitrary chemical potential. It is demonstrated that the TRG method is free from the sign problem in practical calculations and applicable to the four-dimensional models.We report recent progress on the application of the tensor renormalization group (TRG) to quantum field theories pursued by the Tsukuba group. We explain how to treat the scalar, fermion, and gauge theories with the TRG method presenting the results for the phase transitions in the (3+1)-dimensional ((3+1)$d$) complex $\phi^4$ theory at finite density, (1+1)$d$ pure U(1) lattice gauge theory with a $\theta$ term, (3+1)$d$ Nambu--Jona-Lasinio model at finite density and (1+1)$d$ and (2+1)$d$ Hubbard models at an arbitrary chemical potential. It is demonstrated that the TRG method is free from the sign problem in practical calculations and applicable to the four-dimensional models.

arXiv:2111.04240International Symposium on Lattice Field Theory (LATTICE)May 16, 20224 citations

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}

We study (1+1)-dimensional O(3) nonlinear sigma model using the tensor renormalization group method with the infinite limit of the bond dimension

D_{\rm cut}\rightarrow \infty

. At the vanishing chemical potential

\mu=0

, we investigate the von Neumann and R\'enyi types of entanglement entropies. The central charge is determined to be

c=1.97(9)

by using the asymptotic scaling properties of the entropies. We also examine the consistency between two entropies. In the finite density region with

\mu\ne 0

, where this model suffers from the sign problem in the standard Monte Carlo approach, we investigate the properties of the quantum phase transition. We determine the transition point

\mu_{\rm c}

and the critical exponent of the correlation length

\nu

from the

\mu

dependence of the number density in the thermodynamic limit. The dynamical critical exponent

z

is also extracted from the scaling behavior of the temporal correlation length as a function of

\mu

. This is the first successful calculation of the dynamical critical exponent with the TRG method.

We study (1+1)-dimensional O(3) nonlinear sigma model using the tensor renormalization group method with the infinite limit of the bond dimension

D_{\rm cut}\rightarrow \infty

. At the vanishing chemical potential

μ=0

, we investigate the von Neumann and Rényi types of entanglement entropies. The central charge is determined to be

c=1.97(9)

by using the asymptotic scaling properties of the entropies. We also examine the consistency between two entropies. In the finite density region with

μ\ne 0

, where this model suffers from the sign problem in the standard Monte Carlo approach, we investigate the properties of the quantum phase transition. We determine the transition point

μ_{\rm c}

and the critical exponent of the correlation length

ν

from the

μ

dependence of the number density in the thermodynamic limit. The dynamical critical exponent

z

is also extracted from the scaling behavior of the temporal correlation length as a function of

μ

. This is the first successful calculation of the dynamical critical exponent with the TRG method.

d

) complex

\phi^4

theory at finite density, (1+1)

d

pure U(1) lattice gauge theory with a

\theta

term, (3+1)

d

Nambu--Jona-Lasinio model at finite density and (1+1)

d

and (2+1)

d

Hubbard models at an arbitrary chemical potential. It is demonstrated that the TRG method is free from the sign problem in practical calculations and applicable to the four-dimensional models.