Journal of the Physical Society of Japan

Variational Monte-Carlo Studies of Superconductivity in Strongly Correlated Electron Systems

Hisatoshi Yokoyama, Hiroyuki Shiba

Published Jul 15, 1988• 207 citations

Abstract

A variational Monte-Carlo method is formulated to study superconductivity in strongly correlated electrons. An application is made to the strongly correlated Hubbard model on two-dimensional square lattice, by employing the simplest mean-field-type trial wave function, i.e. the Gutzwiller-projected BCS wave function. Among the various choices examined, the d-wave-type superconductivity turns out to be the most favorable. The nature of the superconducting order parameter and spin correlations is also examined.

DOI: 10.1143/jpsj.57.2482

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Journal of the Physical Society of Japan

Variational Monte-Carlo Studies of Superconductivity in Strongly Correlated Electron Systems

Hisatoshi Yokoyama, Hiroyuki Shiba

Published Jul 15, 1988• 207 citations

Publisher

Abstract

DOI: 10.1143/jpsj.57.2482

Related papers

Antiferromagnetism, Superconductivity and Phase Diagram in the Two-Dimensional Hubbard Model —Off-Diagonal Wave Function Monte Carlo Studies of Hubbard Model III—

Takashi Yanagisawa

We investigate the ground-state phase diagram of the two-dimensional Hubbard model based on the optimization variational Monte Carlo method. We use a wave function that is an off-diagonal type given as $ψ=\exp(-λK)P_Gψ_0$ , where $ψ_0$ is a one-particle state, $P_G$ is the Gutzwiller operator, $K$ is the kinetic operator, and $λ$ is a variational parameter. The many-body effect plays an important role as an origin of spin correlation and superconductivity in correlated electron systems. We examine the competition between the antiferromagnetic state and superconducting state by varying the Coulomb repulsion $U$ , the band parameter $t'$ and the electron density $n_e$ . We show a phase diagram that includes superconducting and antiferromagnetic phases and that $t'=0$ is most favorable for superconductivity.We investigate the ground-state phase diagram of the two-dimensional Hubbard model based on the optimization variational Monte Carlo method. We use a wave function that is an off-diagonal type given as $ψ=\exp(-λK)P_Gψ_0$, where $ψ_0$ is a one-particle state, $P_G$ is the Gutzwiller operator, $K$ is the kinetic operator, and $λ$ is a variational parameter. The many-body effect plays an important role as an origin of spin correlation and superconductivity in correlated electron systems. We examine the competition between the antiferromagnetic state and superconducting state by varying the Coulomb repulsion $U$, the band parameter $t'$ and the electron density $n_e$. We show a phase diagram that includes superconducting and antiferromagnetic phases and that $t'=0$ is most favorable for superconductivity.

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We investigate the ground-state properties of the two-dimensional Hubbard model, based on the off-diagonal wave function variational Monte Carlo method. We use an optimized wave function that is improved from an initial one-body wave function by multiplying by multiple correlation operators that are given by the form of exp( $-S$ )-type where $S$ is a suitable operator. We examine the inhomogeneous ground state at near 1/8 doping in the strongly correlated region where the on-site Coulomb interaction is larger than the bandwidth. The $d$ -wave superconductivity with a spatially oscillating gap function can coexist with the charge ordering in the ground state without magnetic ordering, and superconducting condensation energy increases by this coexistence. We also show the $d$ -wave pair correlation function as a function of lattice sites. The correlation function indicates that the long-range superconducting order indeed exists, and also that the strong electron correlation suppresses the pair correlation function.We investigate the ground-state properties of the two-dimensional Hubbard model, based on the off-diagonal wave function variational Monte Carlo method. We use an optimized wave function that is improved from an initial one-body wave function by multiplying by multiple correlation operators that are given by the form of exp($-S$)-type where $S$ is a suitable operator. We examine the inhomogeneous ground state at near 1/8 doping in the strongly correlated region where the on-site Coulomb interaction is larger than the bandwidth. The $d$-wave superconductivity with a spatially oscillating gap function can coexist with the charge ordering in the ground state without magnetic ordering, and superconducting condensation energy increases by this coexistence. We also show the $d$-wave pair correlation function as a function of lattice sites. The correlation function indicates that the long-range superconducting order indeed exists, and also that the strong electron correlation suppresses the pair correlation function.

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Magnetism and superconductivity of strongly correlated electrons on the triangular lattice

Cédric Weber, Andreas Martin Läuchli, Frédéric Mila +1 more

We investigate the phase diagram of the $t\text{\ensuremath{-}}J$ model on a triangular lattice using a variational Monte Carlo approach. We use an extended set of Gutzwiller projected fermionic trial wave functions allowing for simultaneous magnetic and superconducting order parameters. We obtain energies at zero doping for the spin- $1∕2$ Heisenberg model in very good agreement with the best estimates. Upon electron doping (with a hopping integral $t<0$ ) this phase is surprisingly stable variationally up to $n\ensuremath{\approx}1.4$, while the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}+i{d}_{xy}$ order parameter is rather weak and disappears at $n\ensuremath{\approx}1.1$. For hole doping, however, the coplanar magnetic state is almost immediately destroyed and ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}+i{d}_{xy}$ superconductivity survives down to $n\ensuremath{\approx}0.8$. For lower $n$ , between $0.2$ and $0.8$ , we find saturated ferromagnetism. Moreover, there is evidence for a narrow spin density wave phase around $n\ensuremath{\approx}0.8$. Commensurate flux phases were also considered, but these turned out not to be competitive at finite doping.We investigate the phase diagram of the $t\text{\ensuremath{-}}J$ model on a triangular lattice using a variational Monte Carlo approach. We use an extended set of Gutzwiller projected fermionic trial wave functions allowing for simultaneous magnetic and superconducting order parameters. We obtain energies at zero doping for the spin-$1∕2$ Heisenberg model in very good agreement with the best estimates. Upon electron doping (with a hopping integral $t<0$) this phase is surprisingly stable variationally up to $n\ensuremath{\approx}1.4$, while the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}+i{d}_{xy}$ order parameter is rather weak and disappears at $n\ensuremath{\approx}1.1$. For hole doping, however, the coplanar magnetic state is almost immediately destroyed and ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}+i{d}_{xy}$ superconductivity survives down to $n\ensuremath{\approx}0.8$. For lower $n$, between $0.2$ and $0.8$, we find saturated ferromagnetism. Moreover, there is evidence for a narrow spin density wave phase around $n\ensuremath{\approx}0.8$. Commensurate flux phases were also considered, but these turned out not to be competitive at finite doping.

arXiv:cond-mat/0509520Physical Review BJan 20, 200653 citations

ψ=\exp(-λK)P_Gψ_0

, where

ψ_0

is a one-particle state,

P_G

is the Gutzwiller operator,

K

is the kinetic operator, and

λ

is a variational parameter. The many-body effect plays an important role as an origin of spin correlation and superconductivity in correlated electron systems. We examine the competition between the antiferromagnetic state and superconducting state by varying the Coulomb repulsion

U

, the band parameter

t'

and the electron density

n_e

. We show a phase diagram that includes superconducting and antiferromagnetic phases and that

t'=0

is most favorable for superconductivity.

-S

)-type where

S

is a suitable operator. We examine the inhomogeneous ground state at near 1/8 doping in the strongly correlated region where the on-site Coulomb interaction is larger than the bandwidth. The

d

-wave superconductivity with a spatially oscillating gap function can coexist with the charge ordering in the ground state without magnetic ordering, and superconducting condensation energy increases by this coexistence. We also show the

d

-wave pair correlation function as a function of lattice sites. The correlation function indicates that the long-range superconducting order indeed exists, and also that the strong electron correlation suppresses the pair correlation function.

1∕2

Heisenberg model in very good agreement with the best estimates. Upon electron doping (with a hopping integral

t<0

) this phase is surprisingly stable variationally up to $n\ensuremath{\approx}1.4$, while the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}+i{d}_{xy}$ order parameter is rather weak and disappears at $n\ensuremath{\approx}1.1$. For hole doping, however, the coplanar magnetic state is almost immediately destroyed and ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}+i{d}_{xy}$ superconductivity survives down to $n\ensuremath{\approx}0.8$. For lower

n

, between

0.2

and

0.8

, we find saturated ferromagnetism. Moreover, there is evidence for a narrow spin density wave phase around $n\ensuremath{\approx}0.8$. Commensurate flux phases were also considered, but these turned out not to be competitive at finite doping.