Physical Review Letters

Metal-insulator transition in 1D Hubbard model

M. Gulácsi, Kevin S. Bedell

Published Apr 25, 1994• 27 citations

Abstract

For the one-dimensional Hubbard model we extended the bosonization technique, away from half-filling, in such a way that a general formula is obtained for the zero temperature single-particle Green function with validity over the whole doping range. Our method allows us to calculate, for the first time, the one-body Green function in both Tomonaga-Luttinger and Luther-Emery universality classes, to characterize the crossover between these two behaviors and thus, to describe the metal-insulator transition. We have found that the transition can be characterized as a Brinkman-Rice transition with a diverging effective mass.

DOI: 10.1103/physrevlett.72.2765

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Physical Review Letters

Metal-insulator transition in 1D Hubbard model

M. Gulácsi, Kevin S. Bedell

Published Apr 25, 1994• 27 citations

Publisher

Abstract

DOI: 10.1103/physrevlett.72.2765

Related papers

Metal-insulator transition in the one-dimensional SU(N) Hubbard model

Roland Assaraf, P. Azaria, Michel Caffarel +1 more

We investigate the metal-insulator transition of the one-dimensional $\mathrm{SU}(N)$ Hubbard model for repulsive interaction. Using the bosonization approach a Mott transition in the charge sector at half filling ${(k}_{F}=\ensuremath{\pi}{/Na}_{0})$ is conjectured for $N>2.$ Expressions for the charge and spin velocities as well as for the Luttinger-liquid parameters and some correlation functions are given. The theoretical predictions are compared with numerical results obtained with an improved zero-temperature quantum Monte Carlo approach. The method used is a generalized Green's function Monte Carlo scheme in which the stochastic time evolution is partially integrated out. Very accurate results for the gaps, velocities, and Luttinger-liquid parameters as a function of the Coulomb interaction $U$ are given for the cases $N=3$ and $N=4.$ Our results strongly support the existence of a Mott-Hubbard transition at a nonzero value of the Coulomb interaction. We find ${U}_{c}\ensuremath{\sim}2.2$ for $N=3$ and ${U}_{c}\ensuremath{\sim}2.8$ for $N=4.$ We investigate the metal-insulator transition of the one-dimensional $\mathrm{SU}(N)$ Hubbard model for repulsive interaction. Using the bosonization approach a Mott transition in the charge sector at half filling ${(k}_{F}=\ensuremath{\pi}{/Na}_{0})$ is conjectured for $N>2.$ Expressions for the charge and spin velocities as well as for the Luttinger-liquid parameters and some correlation functions are given. The theoretical predictions are compared with numerical results obtained with an improved zero-temperature quantum Monte Carlo approach. The method used is a generalized Green's function Monte Carlo scheme in which the stochastic time evolution is partially integrated out. Very accurate results for the gaps, velocities, and Luttinger-liquid parameters as a function of the Coulomb interaction $U$ are given for the cases $N=3$ and $N=4.$ Our results strongly support the existence of a Mott-Hubbard transition at a nonzero value of the Coulomb interaction. We find ${U}_{c}\ensuremath{\sim}2.2$ for $N=3$ and ${U}_{c}\ensuremath{\sim}2.8$ for $N=4.$

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We use quantum Monte Carlo methods to determine $T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$ Green functions, $G(\stackrel{\ensuremath{\rightarrow}}{r},\ensuremath{\omega})$, on lattices up to $16\ifmmode\times\else\texttimes\fi{}16$ for the 2D Hubbard model at $U/t\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}4$ . For chemical potentials $\ensuremath{\mu}$ within the Hubbard gap $|\ensuremath{\mu}|<{\ensuremath{\mu}}_{c}$ and at long distances $\stackrel{\ensuremath{\rightarrow}}{r}$, $G(\stackrel{\ensuremath{\rightarrow}}{r},\ensuremath{\omega}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\mu})\ensuremath{\sim}{e}^{\ensuremath{-}|\stackrel{\ensuremath{\rightarrow}}{r}|/{\ensuremath{\xi}}_{l}}$ with critical behavior ${\ensuremath{\xi}}_{l}\ensuremath{\sim}|\ensuremath{\mu}\ensuremath{-}{\ensuremath{\mu}}_{c}{|}^{\ensuremath{-}\ensuremath{\nu}}$, $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.26\ifmmode\pm\else\textpm\fi{}0.05$. This result stands in agreement with the assumption of hyperscaling with correlation exponent $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/4$ and dynamical exponent $z\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}4$ . In contrast, the generic band insulator as well as the metal-insulator transition in the 1D Hubbard model are characterized by $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2$ and $z\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2$ .We use quantum Monte Carlo methods to determine $T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$ Green functions, $G(\stackrel{\ensuremath{\rightarrow}}{r},\ensuremath{\omega})$, on lattices up to $16\ifmmode\times\else\texttimes\fi{}16$ for the 2D Hubbard model at $U/t\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}4$. For chemical potentials $\ensuremath{\mu}$ within the Hubbard gap $|\ensuremath{\mu}|<{\ensuremath{\mu}}_{c}$ and at long distances $\stackrel{\ensuremath{\rightarrow}}{r}$, $G(\stackrel{\ensuremath{\rightarrow}}{r},\ensuremath{\omega}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\mu})\ensuremath{\sim}{e}^{\ensuremath{-}|\stackrel{\ensuremath{\rightarrow}}{r}|/{\ensuremath{\xi}}_{l}}$ with critical behavior ${\ensuremath{\xi}}_{l}\ensuremath{\sim}|\ensuremath{\mu}\ensuremath{-}{\ensuremath{\mu}}_{c}{|}^{\ensuremath{-}\ensuremath{\nu}}$, $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.26\ifmmode\pm\else\textpm\fi{}0.05$. This result stands in agreement with the assumption of hyperscaling with correlation exponent $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/4$ and dynamical exponent $z\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}4$. In contrast, the generic band insulator as well as the metal-insulator transition in the 1D Hubbard model are characterized by $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2$ and $z\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2$.

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Filling-Control Metal-Insulator Transition in the Hubbard Model Studied by the Operator Projection Method

Shigeki Onoda, Masatoshi Imada

Thermodynamic and dynamical properties of filling-control metal-insulator transition (MIT) in the Hubbard model are studied by the operator projection method, especially in two dimensions. This is a non-perturbative analytic approach to many-body systems. The present theory incorporates the Mott-Hubbard, Brinkman-Rice and Slater pictures of the MIT into a unified framework, together with reproducing low-energy narrow band arising from spin-charge fluctuations. At half filling, single-particle spectra $A(ω, k)$ show formation of two Hubbard bands and their antiferromagnetic shadows separated by a Mott gap in the plane of energy $ω$ and momentum $k$ with lowering temperatures. These four bands produce splitting to two low-energy narrow bands and two SDW-like bands in the dispersion. Near half filling, the low-energy narrow band persists at low temperatures. This narrow band has a particularly weak dispersion and large weights around $(π, 0)$ and $(0, π)$ momenta. The velocity of these low-energy excitations is shown to vanish towards the MIT, indicating the mass divergence as in the Brinkman-Rice picture, but most prominently around $(π,0)$ and $(0,π)$ with strong momentum dependence. This reflects the suppression of the coherence near the MIT. Main structures in $A(ω, k)$ show remarkable similarities to quantum Monte-Carlo results in two dimensions as well as in the one-dimensional Hubbard model. The charge compressibility appears to diverge with decreasing doping concentration in both one and two dimensions in agreement with the exact and quantum Monte-Carlo results. We also discuss implications of the flat dispersion formed near the Fermi level to the observations in high-$\Tc$ cuprate superconductors.Thermodynamic and dynamical properties of filling-control metal-insulator transition (MIT) in the Hubbard model are studied by the operator projection method, especially in two dimensions. This is a non-perturbative analytic approach to many-body systems. The present theory incorporates the Mott-Hubbard, Brinkman-Rice and Slater pictures of the MIT into a unified framework, together with reproducing low-energy narrow band arising from spin-charge fluctuations. At half filling, single-particle spectra $A(ω, k)$ show formation of two Hubbard bands and their antiferromagnetic shadows separated by a Mott gap in the plane of energy $ω$ and momentum $k$ with lowering temperatures. These four bands produce splitting to two low-energy narrow bands and two SDW-like bands in the dispersion. Near half filling, the low-energy narrow band persists at low temperatures. This narrow band has a particularly weak dispersion and large weights around $(π, 0)$ and $(0, π)$ momenta. The velocity of these low-energy excitations is shown to vanish towards the MIT, indicating the mass divergence as in the Brinkman-Rice picture, but most prominently around $(π,0)$ and $(0,π)$ with strong momentum dependence. This reflects the suppression of the coherence near the MIT. Main structures in $A(ω, k)$ show remarkable similarities to quantum Monte-Carlo results in two dimensions as well as in the one-dimensional Hubbard model. The charge compressibility appears to diverge with decreasing doping concentration in both one and two dimensions in agreement with the exact and quantum Monte-Carlo results. We also discuss implications of the flat dispersion formed near the Fermi level to the observations in high-$\Tc$ cuprate superconductors.

arXiv:cond-mat/0107291Journal of the Physical Society of JapanNov 1, 200133 citations

We investigate the metal-insulator transition of the one-dimensional

\mathrm{SU}(N)

Hubbard model for repulsive interaction. Using the bosonization approach a Mott transition in the charge sector at half filling ${(k}_{F}=\ensuremath{\pi}{/Na}_{0})$ is conjectured for

N>2.

Expressions for the charge and spin velocities as well as for the Luttinger-liquid parameters and some correlation functions are given. The theoretical predictions are compared with numerical results obtained with an improved zero-temperature quantum Monte Carlo approach. The method used is a generalized Green's function Monte Carlo scheme in which the stochastic time evolution is partially integrated out. Very accurate results for the gaps, velocities, and Luttinger-liquid parameters as a function of the Coulomb interaction

U

are given for the cases

N=3

and

N=4.

Our results strongly support the existence of a Mott-Hubbard transition at a nonzero value of the Coulomb interaction. We find ${U}_{c}\ensuremath{\sim}2.2$ for

N=3

and ${U}_{c}\ensuremath{\sim}2.8$ for

N=4.

We use quantum Monte Carlo methods to determine

T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0

Green functions, $G(\stackrel{\ensuremath{\rightarrow}}{r},\ensuremath{\omega})$, on lattices up to $16\ifmmode\times\else\texttimes\fi{}16$ for the 2D Hubbard model at

U/t\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}4

. For chemical potentials $\ensuremath{\mu}$ within the Hubbard gap $|\ensuremath{\mu}|<{\ensuremath{\mu}}_{c}$ and at long distances $\stackrel{\ensuremath{\rightarrow}}{r}$, $G(\stackrel{\ensuremath{\rightarrow}}{r},\ensuremath{\omega}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\mu})\ensuremath{\sim}{e}^{\ensuremath{-}|\stackrel{\ensuremath{\rightarrow}}{r}|/{\ensuremath{\xi}}_{l}}$ with critical behavior ${\ensuremath{\xi}}_{l}\ensuremath{\sim}|\ensuremath{\mu}\ensuremath{-}{\ensuremath{\mu}}_{c}{|}^{\ensuremath{-}\ensuremath{\nu}}$, $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.26\ifmmode\pm\else\textpm\fi{}0.05$. This result stands in agreement with the assumption of hyperscaling with correlation exponent $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/4$ and dynamical exponent

z\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}4

. In contrast, the generic band insulator as well as the metal-insulator transition in the 1D Hubbard model are characterized by $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2$ and

z\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2

U

, temperature

T

, and filling

n

shows that, upon increasing

n

toward the Mott insulator, there is a surface of first-order transition between two metals at nonzero doping. That surface ends at a finite temperature critical line originating at the half-filled Mott critical point. Associated with this transition, there is a maximum in scattering rate as well as thermodynamic signatures. These findings suggest a new scenario for the normal-state phase diagram of the high temperature superconductors. The criticality surmised in these systems can originate not from a T

=

0 quantum critical point, nor from the proximity of a long-range ordered phase, but from a low temperature transition between two types of metals at finite doping. The influence of Mott physics therefore extends well beyond half-filling.

A(ω, k)

show formation of two Hubbard bands and their antiferromagnetic shadows separated by a Mott gap in the plane of energy

ω

and momentum

k

with lowering temperatures. These four bands produce splitting to two low-energy narrow bands and two SDW-like bands in the dispersion. Near half filling, the low-energy narrow band persists at low temperatures. This narrow band has a particularly weak dispersion and large weights around

(π, 0)

and

(0, π)

momenta. The velocity of these low-energy excitations is shown to vanish towards the MIT, indicating the mass divergence as in the Brinkman-Rice picture, but most prominently around

(π,0)

and

(0,π)

with strong momentum dependence. This reflects the suppression of the coherence near the MIT. Main structures in

A(ω, k)

show remarkable similarities to quantum Monte-Carlo results in two dimensions as well as in the one-dimensional Hubbard model. The charge compressibility appears to diverge with decreasing doping concentration in both one and two dimensions in agreement with the exact and quantum Monte-Carlo results. We also discuss implications of the flat dispersion formed near the Fermi level to the observations in high-$\Tc$ cuprate superconductors.