Physical Review B

Quantum critical behavior in the insulating region of the two-dimensional metal-insulator transition

D. J. W. Geldart, D. Neilson

Published Nov 13, 2007• 4 citations

Abstract

We show that the quantum critical point associated with the metal-insulator transition phenomenon in two dimensions controls an extended critical region encompassing not only the usual quantum critical sector but also a range of the low-temperature insulator region. The extended range of criticality permits a unified analysis of data from the insulating region and quantum critical sector, allowing us to determine both the dynamical critical exponent $z$ and the correlation length critical exponent $\ensuremath{\nu}$ from published data from a single experiment in the insulator critical region. We show that the critical exponents determined from the insulator sector consistently describe the temperature dependence of the resistance data from the same experiment in the quantum critical sector. This provides evidence for the presence of a quantum critical point in these systems.

DOI: 10.1103/physrevb.76.193304

Related papers

Metal–insulator transition in 2D as a quantum phase transition

D. J. W. Geldart, D. Neilson

Journal of Physics A Mathematical and TheoreticalMay 8, 20090 citations

Quantum scaling for the metal–insulator transition in a two-dimensional electron system

V. Kagalovsky, S. V. Kravchenko, Demitry Nemirovsky

Scientific ReportsMay 31, 20244 citations

Critical Exponents of the Metal-Insulator Transition in the Two-Dimensional Hubbard Model

Nobuo Furukawa, Fakher F. Assaad, Masatoshi Imada

arXiv:cond-mat/9605047Journal of the Physical Society of JapanJul 15, 19963 citations

Quantum criticality around metal-insulator transitions of strongly correlated electron systems

Takahiro Misawa, Masatoshi Imada

arXiv:cond-mat/0612632Physical Review BMar 21, 200754 citations

Universal Scaling, Beta Function, and Metal-Insulator Transitions

D. N. Sheng, Zheng-Yu Weng

arXiv:cond-mat/9902231Physical Review LettersJul 5, 19997 citations

Temperature dependent resistivity in the low-resistance region for diffusive transport in two dimensions

D. J. W. Geldart, D. Neilson

Physical Review BDec 28, 20041 citation

Physical Review B

Quantum critical behavior in the insulating region of the two-dimensional metal-insulator transition

D. J. W. Geldart, D. Neilson

Published Nov 13, 2007• 4 citations

Publisher

Abstract

z

and the correlation length critical exponent

\ensuremath{\nu}

from published data from a single experiment in the insulator critical region. We show that the critical exponents determined from the insulator sector consistently describe the temperature dependence of the resistance data from the same experiment in the quantum critical sector. This provides evidence for the presence of a quantum critical point in these systems.

DOI: 10.1103/physrevb.76.193304

Related papers

Metal–insulator transition in 2D as a quantum phase transition

D. J. W. Geldart, D. Neilson

We discuss the metal–insulator transition phenomenon in two dimensions in terms of a quantum critical point that controls a range of the low temperature insulator region as well as the usual quantum critical sector. We show that this extended range of criticality permits a determination of both the dynamical critical exponent z and the correlation length critical exponent ν from published data from a single experiment in the insulator critical region. The resulting value of the product zν is consistent with the temperature dependence of the resistance in the quantum critical sector. This provides strong quantitative evidence for the presence of a quantum critical point.We discuss the metal–insulator transition phenomenon in two dimensions in terms of a quantum critical point that controls a range of the low temperature insulator region as well as the usual quantum critical sector. We show that this extended range of criticality permits a determination of both the dynamical critical exponent z and the correlation length critical exponent ν from published data from a single experiment in the insulator critical region. The resulting value of the product zν is consistent with the temperature dependence of the resistance in the quantum critical sector. This provides strong quantitative evidence for the presence of a quantum critical point.

Journal of Physics A Mathematical and TheoreticalMay 8, 20090 citations

Quantum scaling for the metal–insulator transition in a two-dimensional electron system

V. Kagalovsky, S. V. Kravchenko, Demitry Nemirovsky

The quantum phase transition observed experimentally in two-dimensional (2D) electron systems has been a subject of theoretical and experimental studies for almost 30 years. We suggest Gaussian approximation to the mean-field theory of the second-order phase transition to explain the experimental data. Our approach explains self-consistently the universal value of the critical exponent 3/2 (found after scaling measured resistivities on both sides of the transition as a function of temperature) as the result of the divergence of the correlation length when the electron density approaches the critical value. We also provide numerical evidence for the stretched exponential temperature dependence of the metallic phase's resistivities in a wide range of temperatures and show that it leads to correct qualitative results. Finally, we interpret the phase diagram on the density-temperature plane exhibiting the quantum critical point, quantum critical trajectory and two crossover lines. Our research presents a theoretical description of the seminal experimental results.The quantum phase transition observed experimentally in two-dimensional (2D) electron systems has been a subject of theoretical and experimental studies for almost 30 years. We suggest Gaussian approximation to the mean-field theory of the second-order phase transition to explain the experimental data. Our approach explains self-consistently the universal value of the critical exponent 3/2 (found after scaling measured resistivities on both sides of the transition as a function of temperature) as the result of the divergence of the correlation length when the electron density approaches the critical value. We also provide numerical evidence for the stretched exponential temperature dependence of the metallic phase's resistivities in a wide range of temperatures and show that it leads to correct qualitative results. Finally, we interpret the phase diagram on the density-temperature plane exhibiting the quantum critical point, quantum critical trajectory and two crossover lines. Our research presents a theoretical description of the seminal experimental results.

Scientific ReportsMay 31, 20244 citations

Critical Exponents of the Metal-Insulator Transition in the Two-Dimensional Hubbard Model

Nobuo Furukawa, Fakher F. Assaad, Masatoshi Imada

We study the filling-controlled metal-insulator transition in the two-dimensional Hubbard model near half-filling with the use of zero temperature quantum Monte Carlo methods. In the metallic phase, the compressibility behaves as $κ\propto |μ- μ_c|^{-0.58\pm0.08}$ where $μ_c$ is the critical chemical potential. In the insulating phase, the localization length follows $ξ_l \propto |μ- μ_c|^{-ν_l}$ with $ν_l = 0.26 \pm 0.05$ . Under the assumption of hyperscaling, the compressibility data leads to a correlation length exponent $ν_κ= 0.21 \pm 0.04$ . Our results show that the exponents $ν_κ$ and $ν_l$ agree within statistical uncertainty. This confirms the assumption of hyperscaling with correlation length exponent $ν= 1/4$ and dynamical exponent $z = 4$ . In contrast the metal-insulator transition in the generic band insulators in all dimensions as well as in the one-dimensional Hubbard model satisfy the hyperscaling assumption with exponents $ν= 1/2$ and $z = 2$ .We study the filling-controlled metal-insulator transition in the two-dimensional Hubbard model near half-filling with the use of zero temperature quantum Monte Carlo methods. In the metallic phase, the compressibility behaves as $κ\propto |μ- μ_c|^{-0.58\pm0.08}$ where $μ_c$ is the critical chemical potential. In the insulating phase, the localization length follows $ξ_l \propto |μ- μ_c|^{-ν_l}$ with $ν_l = 0.26 \pm 0.05$. Under the assumption of hyperscaling, the compressibility data leads to a correlation length exponent $ν_κ= 0.21 \pm 0.04$. Our results show that the exponents $ν_κ$ and $ν_l$ agree within statistical uncertainty. This confirms the assumption of hyperscaling with correlation length exponent $ν= 1/4$ and dynamical exponent $z = 4$. In contrast the metal-insulator transition in the generic band insulators in all dimensions as well as in the one-dimensional Hubbard model satisfy the hyperscaling assumption with exponents $ν= 1/2$ and $z = 2$.

arXiv:cond-mat/9605047Journal of the Physical Society of JapanJul 15, 19963 citations

Quantum criticality around metal-insulator transitions of strongly correlated electron systems

Takahiro Misawa, Masatoshi Imada

Quantum criticality of metal-insulator transitions in correlated electron systems is shown to belong to an unconventional universality class with violation of the Ginzburg-Landau-Wilson (GLW) scheme formulated for symmetry breaking transitions. This unconventionality arises from an emergent character of the quantum critical point, which appears at the marginal point between the Ising-type symmetry breaking at nonzero temperatures and the topological transition of the Fermi surface at zero temperature. We show that Hartree-Fock approximations of an extended Hubbard model on square lattices are capable of such metal-insulator transitions with unusual criticality under a preexisting symmetry breaking. The obtained universality is consistent with the scaling theory formulated for Mott transitions and with a number of numerical results beyond the mean-field level, implying that preexisting symmetry breaking is not necessarily required for the emergence of this unconventional universality. Examinations of fluctuation effects indicate that the obtained critical exponents remain essentially exact beyond the mean-field level. It further clarifies the whole structure of singularities by a unified treatment of the bandwidth-control and filling-control transitions. Detailed analyses of the criticality, containing diverging carrier density fluctuations around the marginal quantum critical point, are presented from microscopic calculations and reveal the nature as quantum critical ``opalescence.'' The mechanism of emerging marginal quantum critical point is ascribed to a positive feedback and interplay between the preexisting gap formation present even in metals and kinetic energy gain (loss) of the metallic carrier. Analyses of crossovers between GLW type at nonzero temperature and topological type at zero temperature show that the critical exponents observed in ${(\mathrm{V},\mathrm{Cr})}_{2}{\mathrm{O}}_{3}$ and $\ensuremath{\kappa}\text{\ensuremath{-}}\mathrm{ET}$-type organic conductors provide us with evidence for the existence of the present marginal quantum criticality.Quantum criticality of metal-insulator transitions in correlated electron systems is shown to belong to an unconventional universality class with violation of the Ginzburg-Landau-Wilson (GLW) scheme formulated for symmetry breaking transitions. This unconventionality arises from an emergent character of the quantum critical point, which appears at the marginal point between the Ising-type symmetry breaking at nonzero temperatures and the topological transition of the Fermi surface at zero temperature. We show that Hartree-Fock approximations of an extended Hubbard model on square lattices are capable of such metal-insulator transitions with unusual criticality under a preexisting symmetry breaking. The obtained universality is consistent with the scaling theory formulated for Mott transitions and with a number of numerical results beyond the mean-field level, implying that preexisting symmetry breaking is not necessarily required for the emergence of this unconventional universality. Examinations of fluctuation effects indicate that the obtained critical exponents remain essentially exact beyond the mean-field level. It further clarifies the whole structure of singularities by a unified treatment of the bandwidth-control and filling-control transitions. Detailed analyses of the criticality, containing diverging carrier density fluctuations around the marginal quantum critical point, are presented from microscopic calculations and reveal the nature as quantum critical ``opalescence.'' The mechanism of emerging marginal quantum critical point is ascribed to a positive feedback and interplay between the preexisting gap formation present even in metals and kinetic energy gain (loss) of the metallic carrier. Analyses of crossovers between GLW type at nonzero temperature and topological type at zero temperature show that the critical exponents observed in ${(\mathrm{V},\mathrm{Cr})}_{2}{\mathrm{O}}_{3}$ and $\ensuremath{\kappa}\text{\ensuremath{-}}\mathrm{ET}$-type organic conductors provide us with evidence for the existence of the present marginal quantum criticality.

arXiv:cond-mat/0612632Physical Review BMar 21, 200754 citations

Universal Scaling, Beta Function, and Metal-Insulator Transitions

D. N. Sheng, Zheng-Yu Weng

We demonstrate a universal scaling form of longitudinal resistance in the quantum critical region of metal-insulator transitions. This is based on numerical results of three-dimensional Anderson transitions (with and without magnetic field), two-dimensional quantum Hall plateau to insulator transition, as well as experimental data of the recently discovered two-dimensional metal-insulator transition. The associated reflection symmetry and a peculiar logarithmic form of the beta function exist over a wide range in which the resistance can change by more than 1 order of magnitude. Interesting implications for the two-dimensional metal-insulator transition are discussed.We demonstrate a universal scaling form of longitudinal resistance in the quantum critical region of metal-insulator transitions. This is based on numerical results of three-dimensional Anderson transitions (with and without magnetic field), two-dimensional quantum Hall plateau to insulator transition, as well as experimental data of the recently discovered two-dimensional metal-insulator transition. The associated reflection symmetry and a peculiar logarithmic form of the beta function exist over a wide range in which the resistance can change by more than 1 order of magnitude. Interesting implications for the two-dimensional metal-insulator transition are discussed.

arXiv:cond-mat/9902231Physical Review LettersJul 5, 19997 citations

Temperature dependent resistivity in the low-resistance region for diffusive transport in two dimensions

D. J. W. Geldart, D. Neilson

The interpretation of the metal-insulator transition phenomena in disordered two-dimensional electron systems in terms of density-dependent scaling variables suggests the existence of a quantum critical point at some critical electron density. However a first principles scaling theory based on renormalization group (RG) methods predicts a strong temperature dependence of the dimensionless resistivity $\mathcal{R}(T)$ , even at small $\mathcal{R}(T)$ , that is not observed. The observed properties are in fact consistent with a weakly disordered Fermi liquid, and there are no indications of strong temperature dependence induced by scaling. While the RG expansion in a power series in $\mathcal{R}(T)$ has only been evaluated to lowest order, this should be sufficient to describe experiments in the region of very small $\mathcal{R}$ . A further apparent anomaly is a return from metal-like to insulating-like behavior for increasing density. We explain these fundamental discrepancies between the first principles theory and experiment. We find that the $\mathcal{R}⪡1$ data in the currently attainable temperature range are in a weak scaling regime described by the logarithmic approximation. We independently determine the density dependent prefactor of the logarithm using data for the spin susceptibility and effective mass. We find good agreement between theory and experiment for $\mathcal{R}(T)$ in the diffusive regime. We point out that there are corrections to the leading logarithm approximation that should be observable at still lower temperatures.The interpretation of the metal-insulator transition phenomena in disordered two-dimensional electron systems in terms of density-dependent scaling variables suggests the existence of a quantum critical point at some critical electron density. However a first principles scaling theory based on renormalization group (RG) methods predicts a strong temperature dependence of the dimensionless resistivity $\mathcal{R}(T)$, even at small $\mathcal{R}(T)$, that is not observed. The observed properties are in fact consistent with a weakly disordered Fermi liquid, and there are no indications of strong temperature dependence induced by scaling. While the RG expansion in a power series in $\mathcal{R}(T)$ has only been evaluated to lowest order, this should be sufficient to describe experiments in the region of very small $\mathcal{R}$. A further apparent anomaly is a return from metal-like to insulating-like behavior for increasing density. We explain these fundamental discrepancies between the first principles theory and experiment. We find that the $\mathcal{R}⪡1$ data in the currently attainable temperature range are in a weak scaling regime described by the logarithmic approximation. We independently determine the density dependent prefactor of the logarithm using data for the spin susceptibility and effective mass. We find good agreement between theory and experiment for $\mathcal{R}(T)$ in the diffusive regime. We point out that there are corrections to the leading logarithm approximation that should be observable at still lower temperatures.

Physical Review BDec 28, 20041 citation

z

and the correlation length critical exponent

\ensuremath{\nu}

κ\propto |μ- μ_c|^{-0.58\pm0.08}

where

μ_c

is the critical chemical potential. In the insulating phase, the localization length follows

ξ_l \propto |μ- μ_c|^{-ν_l}

with

ν_l = 0.26 \pm 0.05

. Under the assumption of hyperscaling, the compressibility data leads to a correlation length exponent

ν_κ= 0.21 \pm 0.04

. Our results show that the exponents

ν_κ

and

ν_l

agree within statistical uncertainty. This confirms the assumption of hyperscaling with correlation length exponent

ν= 1/4

and dynamical exponent

z = 4

. In contrast the metal-insulator transition in the generic band insulators in all dimensions as well as in the one-dimensional Hubbard model satisfy the hyperscaling assumption with exponents

ν= 1/2

and

z = 2

{(\mathrm{V},\mathrm{Cr})}_{2}{\mathrm{O}}_{3}

and $\ensuremath{\kappa}\text{\ensuremath{-}}\mathrm{ET}$-type organic conductors provide us with evidence for the existence of the present marginal quantum criticality.

\mathcal{R}(T)

, even at small

\mathcal{R}(T)

, that is not observed. The observed properties are in fact consistent with a weakly disordered Fermi liquid, and there are no indications of strong temperature dependence induced by scaling. While the RG expansion in a power series in

\mathcal{R}(T)

has only been evaluated to lowest order, this should be sufficient to describe experiments in the region of very small

\mathcal{R}

. A further apparent anomaly is a return from metal-like to insulating-like behavior for increasing density. We explain these fundamental discrepancies between the first principles theory and experiment. We find that the

\mathcal{R}⪡1

data in the currently attainable temperature range are in a weak scaling regime described by the logarithmic approximation. We independently determine the density dependent prefactor of the logarithm using data for the spin susceptibility and effective mass. We find good agreement between theory and experiment for

\mathcal{R}(T)

in the diffusive regime. We point out that there are corrections to the leading logarithm approximation that should be observable at still lower temperatures.