Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical review. B, Condensed matter

Interaction-driven metal-insulator transitions in disordered fermion systems

Claudio Castellani, C. Di Castro, Patrick A. Lee, Michael Ma

Published Jul 15, 1984• 382 citations
Publisher

Abstract

We study the effects of electron-electron interactions in disordered metals in and close to two dimensions (2D). We consider physical situations in which localization effects are suppressed. The field-theoretical renormalization-group (RG) calculation performed recently by Finkelstein is interpreted and rederived in terms of perturbative results. Surprisingly, except for the density of states, the scaling behavior is independent of the interaction range. We further extend the model to several new universality classes. In the presence of a strong magnetic field the metal is unstable in 2D and undergoes a metal-insulator transition in $d=2+\ensuremath{\epsilon}$. The conductivity exponent, defined by $\ensuremath{\sigma}\ensuremath{\sim}{(n\ensuremath{-}{n}_{c})}^{\ensuremath{\mu}}$, is universal with $\ensuremath{\mu}=1+O(\ensuremath{\epsilon})$ but $N({E}_{F})$ depends not only on the range of the interaction but also on its strength for short-ranged interactions. In 2D the conductivity has a universal temperature dependence [$\ensuremath{\delta}\ensuremath{\sigma}(T)={\ensuremath{\sigma}}_{N}(2\ensuremath{-}2\mathrm{ln}2)\mathrm{ln}(T\ensuremath{\tau})$, ${\ensuremath{\sigma}}_{N}=\frac{{e}^{2}}{2{\ensuremath{\pi}}^{2}\ensuremath{\hbar}}$] if the interaction is Coulombic. If magnetic impurities (or strong spin-orbit scattering with a weak magnetic field) are present instead, the noninteracting fixed point is stable for short-ranged interactions ($\ensuremath{\mu}=\frac{1}{2}$). For the Coulomb interaction the interaction is relevant and drives a metal-insulator transition in $d=2+\ensuremath{\epsilon}$ with universal critical properties ($\ensuremath{\mu}=1$). In 2D the conductivity also has a universal temperature dependence [$\ensuremath{\delta}\ensuremath{\sigma}(T)={\ensuremath{\sigma}}_{N}\mathrm{ln}(T\ensuremath{\tau})$]. We also discuss the behavior of the dielectric constant on the insulator side and the frequency (temperature) dependence of the conductivity at criticality. Remarks are made on the relationship of the above to experiments.

DOI
10.1103/physrevb.30.527

Related papers

Non-Fermi-liquid theory for disordered metals near two dimensions

M. A. Baranov, I. S. Burmistrov, Adrianus M. M. Pruisken

We consider the Finkelstein action describing a system of spin-polarized or spinless electrons in $2+2\ensuremath{\epsilon}$ dimensions, in the presence of disorder as well as the Coulomb interactions. We extend the renormalization-group analysis of our previous work and evaluate the metal-insulator transition of the electron gas to second order in an $\ensuremath{\epsilon}$ expansion. We obtain the complete scaling behavior of physical observables like the conductivity and the specific heat with varying frequency, temperature, and/or electron density. We extend the results for the interacting electron gas in $2+2\ensuremath{\epsilon}$ dimensions to include the quantum critical behavior of the plateau transitions in the quantum Hall regime. Although these transitions have a very different microscopic origin and are controlled by a topological term in the action $(\ensuremath{\theta}$ term), the quantum critical behavior is in many ways the same in both cases. We show that the two independent critical exponents of the quantum Hall plateau transitions, previously denoted as $\ensuremath{\nu}$ and p, control not only the scaling behavior of the conductances ${\ensuremath{\sigma}}_{\mathrm{xx}}$ and ${\ensuremath{\sigma}}_{\mathrm{xy}}$ at finite temperatures T, but also the non-Fermi-liquid behavior of the specific heat ${(c}_{v}\ensuremath{\propto}{T}^{p}).$ To extract the numerical values of $\ensuremath{\nu}$ and p it is necessary to extend the experiments on transport to include the specific heat of the electron gas.

arXiv:cond-mat/0106446•Physical review. B, Condensed matter•Aug 9, 2002•32 citations

Metal-insulator transition from combined disorder and interaction effects in Hubbard-like electronic lattice models with random hopping

Matthew S. Foster, Andreas W. W. Ludwig

We uncover a disorder-driven instability in the diffusive Fermi liquid phase of a class of many-fermion systems, indicative of a metal-insulator transition of first-order type, which arises solely from the competition between quenched disorder and interparticle interactions. Our result is expected to be relevant for sufficiently strong disorder in $d=3$ spatial dimensions. Specifically, we study a class of half-filled, Hubbard-like models for spinless fermions with (complex) random hopping and short-ranged interactions on bipartite lattices in $d\ensuremath{\geqslant}2$. In a given realization, the hopping disorder breaks time-reversal invariance but preserves the special ``nesting'' symmetry responsible for the charge density wave instability of the ballistic Fermi liquid. This disorder may arise, e.g., from the application of a random magnetic field to the otherwise clean model. We derive a low-energy effective field theory description for this class of disordered, interacting fermion systems, which takes the form of a Finkel'stein nonlinear sigma model $(\mathrm{FNL}\ensuremath{\sigma}\mathrm{M})$ (A. M. Finkel'stein, Zh. Eksp. Teor. Fiz. 84, 168 (1983) [Sov. Phys. JETP 57, 97 (1983)]). We analyze the $\mathrm{FNL}\ensuremath{\sigma}\mathrm{M}$ using a perturbative, one-loop renormalization group analysis controlled via an $ϵ$ expansion in $d=2+ϵ$ dimensions. We find that in $d=2$ dimensions, the interactions destabilize the conducting phase known to exist in the disordered, noninteracting system. The metal-insulator transition that we identify in $d>2$ dimensions $(ϵ>0)$ occurs for disorder strengths of order $ϵ$, and is therefore perturbatively accessible for $ϵ⪡1$. We emphasize that the disordered system has no localized phase in the absence of interactions, so that a localized phase and the transition into it can only appear due to the presence of the interactions.

arXiv:0710.0400•Physical Review B•Apr 3, 2008•37 citations

Chapter 2 Metal-Insulator Transition in a Strongly Correlated Two-Dimensional Electron System

A. A. Shashkin, S. V. Kravchenko

This chapter focuses on experimental results obtained in low-disordered strongly interacting two-dimensional (2D) electron systems; in particular in (100)-silicon metal-oxide-semiconductor field-effect transistors (MOSFET). It describes the additional confirmation in favor of zero-temperature zero-field metal-insulator transition is provided by magnetic and thermopower measurements. The chapter argues that the metal-insulator transition in silicon samples with very low level of disorder is driven by interactions. In a low-disordered 2D electron system in silicon MOSFETs, the resistivity at a certain "critical" electron density shows virtually no temperature dependence over a wide range of temperatures. In 2000, it was experimentally found that the ratio between the spin and the cyclotron splittings in silicon MOSFETs strongly increases at low electron densities. The strong enhancement of the spin susceptibility has indicated that at low electron densities, the system behaves well beyond the weakly interacting Fermi liquid.

•May 25, 2017•3 citations

Approaching the metal-insulator transition

T. R. Kirkpatrick, D. Belitz

The problem of disordered interacting electrons is considered. We study a model for a generic disordered Fermi liquid without cooper pairs and in the absence of any spin-flip mechanisms. We prove to within logarithmic accuracy that there is a stable renormalization-group fixed point to all orders in a loop expansion. We prove that the conductivity exponent at this fixed point is identically equal to zero, while the spin-diffusion constant scales to zero with an exponent \ensuremath{\gamma}=4+O(d-2). By an explicit two-loop calculation we then show that this fixed point is suppressed by logarithmic terms, but a sizeable scaling region persists. The main conclusion is that the metal-insulator transition is preceded on the metallic side by a near instability in the spin system. There is a scaling region leading from a spin-diffusion phase to a region of very slow spin transport. Throughout this scaling region the charge transport is unaffected. Experiments are discussed in the light of these results, and further experiments to test the theory are proposed.

Physical review. B, Condensed matter•Jun 1, 1990•34 citations

Two-body metal-insulator transitions in the Anderson-Hubbard model

Filippo Stellin, Giuliano Orso

Abstract We review our recent results on Anderson localization in systems of two interacting particles coupled by contact interactions. Based on an exact mapping to an effective single-particle problem, we numerically investigate the occurrence of metal-insulator phase transitions for the pair in two-(2D) and three-dimensional (3D) disordered lattices. In two dimensions, we find that interactions cause an exponential enhancement of the pair localization length with respect to its single-particle counterpart, but do not induce a delocalization transition. In particular we show that previous claims of 2D interaction-induced Anderson transitions are the results of strong finite-size effects. In three dimensions we find that the pair undergoes a metal-insulator transition belonging to the same (orthogonal) universality class of the noninteracting model. We then explore the phase diagram in the space of energy E, disorder W and interaction strength U, which reveals a rich and counterintuitive structure, endowed with multiple metallic and insulating phases. We point out that this phenomenon originates from the molecular and scattering-like nature of the pair states available at given energy and disorder strength.

arXiv:2012.01103•Journal of Physics Conference Series•Jan 1, 2021•0 citations

The Role of Cooperons in the Disordered Electron Problem: Logarithmic Corrections to Scaling Near the Metal-Insulator Transition

T. R. Kirkpatrick, D. Belitz

The effect of Cooperons on metal-insulator transitions (MIT) in disordered interacting electronic systems is studied. It is argued that previous results which concluded that Cooperons are qualitatively unimportant near the MIT might not be correct, and that the problem is much more complicated than had previously been realized. Although we do not completely solve the Cooperon problem, we propose a new approach that is at least internally consistent. Within this approach we find that in all universality classes where Cooperons are present, i.e. in the absence of magnetic impurities and magnetic fields, there are logarithmic corrections to scaling at the MIT. This result is used for a possible resolution of the so-called exponent puzzle for the conductivity near the MIT. A discussion of the relationship between theory and experiment is given. We also make a number of predictions concerning crossover effects which occur when a magnetic field is applied to a system with Cooperons.

arXiv:cond-mat/9310035•Oct 15, 1993•0 citations
We study the effects of electron-electron interactions in disordered metals in and close to two dimensions (2D). We consider physical situations in which localization effects are suppressed. The field-theoretical renormalization-group (RG) calculation performed recently by Finkelstein is interpreted and rederived in terms of perturbative results. Surprisingly, except for the density of states, the scaling behavior is independent of the interaction range. We further extend the model to several new universality classes. In the presence of a strong magnetic field the metal is unstable in 2D and undergoes a metal-insulator transition in d=2+\ensuremathϵd=2+\ensuremath{\epsilon}d=2+\ensuremathϵ. The conductivity exponent, defined by \ensuremathσ\ensuremath∼(n\ensuremath−nc)\ensuremathμ\ensuremath{\sigma}\ensuremath{\sim}{(n\ensuremath{-}{n}_{c})}^{\ensuremath{\mu}}\ensuremathσ\ensuremath∼(n\ensuremath−nc​)\ensuremathμ, is universal with \ensuremathμ=1+O(\ensuremathϵ)\ensuremath{\mu}=1+O(\ensuremath{\epsilon})\ensuremathμ=1+O(\ensuremathϵ) but N(EF)N({E}_{F})N(EF​) depends not only on the range of the interaction but also on its strength for short-ranged interactions. In 2D the conductivity has a universal temperature dependence [\ensuremathδ\ensuremathσ(T)=\ensuremathσN(2\ensuremath−2ln2)ln(T\ensuremathτ)\ensuremath{\delta}\ensuremath{\sigma}(T)={\ensuremath{\sigma}}_{N}(2\ensuremath{-}2\mathrm{ln}2)\mathrm{ln}(T\ensuremath{\tau})\ensuremathδ\ensuremathσ(T)=\ensuremathσN​(2\ensuremath−2ln2)ln(T\ensuremathτ), \ensuremathσN=e22\ensuremathπ2\ensuremathℏ{\ensuremath{\sigma}}_{N}=\frac{{e}^{2}}{2{\ensuremath{\pi}}^{2}\ensuremath{\hbar}}\ensuremathσN​=2\ensuremathπ2\ensuremathℏe2​] if the interaction is Coulombic. If magnetic impurities (or strong spin-orbit scattering with a weak magnetic field) are present instead, the noninteracting fixed point is stable for short-ranged interactions (\ensuremathμ=12\ensuremath{\mu}=\frac{1}{2}\ensuremathμ=21​). For the Coulomb interaction the interaction is relevant and drives a metal-insulator transition in d=2+\ensuremathϵd=2+\ensuremath{\epsilon}d=2+\ensuremathϵ with universal critical properties (\ensuremathμ=1\ensuremath{\mu}=1\ensuremathμ=1). In 2D the conductivity also has a universal temperature dependence [\ensuremathδ\ensuremathσ(T)=\ensuremathσNln(T\ensuremathτ)\ensuremath{\delta}\ensuremath{\sigma}(T)={\ensuremath{\sigma}}_{N}\mathrm{ln}(T\ensuremath{\tau})\ensuremathδ\ensuremathσ(T)=\ensuremathσN​ln(T\ensuremathτ)]. We also discuss the behavior of the dielectric constant on the insulator side and the frequency (temperature) dependence of the conductivity at criticality. Remarks are made on the relationship of the above to experiments.
We consider the Finkelstein action describing a system of spin-polarized or spinless electrons in $2+2\ensuremath{\epsilon}$ dimensions, in the presence of disorder as well as the Coulomb interactions. We extend the renormalization-group analysis of our previous work and evaluate the metal-insulator transition of the electron gas to second order in an $\ensuremath{\epsilon}$ expansion. We obtain the complete scaling behavior of physical observables like the conductivity and the specific heat with varying frequency, temperature, and/or electron density. We extend the results for the interacting electron gas in $2+2\ensuremath{\epsilon}$ dimensions to include the quantum critical behavior of the plateau transitions in the quantum Hall regime. Although these transitions have a very different microscopic origin and are controlled by a topological term in the action $(\ensuremath{\theta}$ term), the quantum critical behavior is in many ways the same in both cases. We show that the two independent critical exponents of the quantum Hall plateau transitions, previously denoted as $\ensuremath{\nu}$ and p, control not only the scaling behavior of the conductances ${\ensuremath{\sigma}}_{\mathrm{xx}}$ and ${\ensuremath{\sigma}}_{\mathrm{xy}}$ at finite temperatures T, but also the non-Fermi-liquid behavior of the specific heat ${(c}_{v}\ensuremath{\propto}{T}^{p}).$ To extract the numerical values of $\ensuremath{\nu}$ and p it is necessary to extend the experiments on transport to include the specific heat of the electron gas.
We uncover a disorder-driven instability in the diffusive Fermi liquid phase of a class of many-fermion systems, indicative of a metal-insulator transition of first-order type, which arises solely from the competition between quenched disorder and interparticle interactions. Our result is expected to be relevant for sufficiently strong disorder in d=3d=3d=3 spatial dimensions. Specifically, we study a class of half-filled, Hubbard-like models for spinless fermions with (complex) random hopping and short-ranged interactions on bipartite lattices in $d\ensuremath{\geqslant}2$. In a given realization, the hopping disorder breaks time-reversal invariance but preserves the special ``nesting'' symmetry responsible for the charge density wave instability of the ballistic Fermi liquid. This disorder may arise, e.g., from the application of a random magnetic field to the otherwise clean model. We derive a low-energy effective field theory description for this class of disordered, interacting fermion systems, which takes the form of a Finkel'stein nonlinear sigma model $(\mathrm{FNL}\ensuremath{\sigma}\mathrm{M})$ (A. M. Finkel'stein, Zh. Eksp. Teor. Fiz. 84, 168 (1983) [Sov. Phys. JETP 57, 97 (1983)]). We analyze the $\mathrm{FNL}\ensuremath{\sigma}\mathrm{M}$ using a perturbative, one-loop renormalization group analysis controlled via an ϵϵϵ expansion in d=2+ϵd=2+ϵd=2+ϵ dimensions. We find that in d=2d=2d=2 dimensions, the interactions destabilize the conducting phase known to exist in the disordered, noninteracting system. The metal-insulator transition that we identify in d>2d>2d>2 dimensions (ϵ>0)(ϵ>0)(ϵ>0) occurs for disorder strengths of order ϵϵϵ, and is therefore perturbatively accessible for ϵ⪡1ϵ⪡1ϵ⪡1. We emphasize that the disordered system has no localized phase in the absence of interactions, so that a localized phase and the transition into it can only appear due to the presence of the interactions.
This chapter focuses on experimental results obtained in low-disordered strongly interacting two-dimensional (2D) electron systems; in particular in (100)-silicon metal-oxide-semiconductor field-effect transistors (MOSFET). It describes the additional confirmation in favor of zero-temperature zero-field metal-insulator transition is provided by magnetic and thermopower measurements. The chapter argues that the metal-insulator transition in silicon samples with very low level of disorder is driven by interactions. In a low-disordered 2D electron system in silicon MOSFETs, the resistivity at a certain "critical" electron density shows virtually no temperature dependence over a wide range of temperatures. In 2000, it was experimentally found that the ratio between the spin and the cyclotron splittings in silicon MOSFETs strongly increases at low electron densities. The strong enhancement of the spin susceptibility has indicated that at low electron densities, the system behaves well beyond the weakly interacting Fermi liquid.
The problem of disordered interacting electrons is considered. We study a model for a generic disordered Fermi liquid without cooper pairs and in the absence of any spin-flip mechanisms. We prove to within logarithmic accuracy that there is a stable renormalization-group fixed point to all orders in a loop expansion. We prove that the conductivity exponent at this fixed point is identically equal to zero, while the spin-diffusion constant scales to zero with an exponent \ensuremath{\gamma}=4+O(d-2). By an explicit two-loop calculation we then show that this fixed point is suppressed by logarithmic terms, but a sizeable scaling region persists. The main conclusion is that the metal-insulator transition is preceded on the metallic side by a near instability in the spin system. There is a scaling region leading from a spin-diffusion phase to a region of very slow spin transport. Throughout this scaling region the charge transport is unaffected. Experiments are discussed in the light of these results, and further experiments to test the theory are proposed.
Abstract We review our recent results on Anderson localization in systems of two interacting particles coupled by contact interactions. Based on an exact mapping to an effective single-particle problem, we numerically investigate the occurrence of metal-insulator phase transitions for the pair in two-(2D) and three-dimensional (3D) disordered lattices. In two dimensions, we find that interactions cause an exponential enhancement of the pair localization length with respect to its single-particle counterpart, but do not induce a delocalization transition. In particular we show that previous claims of 2D interaction-induced Anderson transitions are the results of strong finite-size effects. In three dimensions we find that the pair undergoes a metal-insulator transition belonging to the same (orthogonal) universality class of the noninteracting model. We then explore the phase diagram in the space of energy E, disorder W and interaction strength U, which reveals a rich and counterintuitive structure, endowed with multiple metallic and insulating phases. We point out that this phenomenon originates from the molecular and scattering-like nature of the pair states available at given energy and disorder strength.
The effect of Cooperons on metal-insulator transitions (MIT) in disordered interacting electronic systems is studied. It is argued that previous results which concluded that Cooperons are qualitatively unimportant near the MIT might not be correct, and that the problem is much more complicated than had previously been realized. Although we do not completely solve the Cooperon problem, we propose a new approach that is at least internally consistent. Within this approach we find that in all universality classes where Cooperons are present, i.e. in the absence of magnetic impurities and magnetic fields, there are logarithmic corrections to scaling at the MIT. This result is used for a possible resolution of the so-called exponent puzzle for the conductivity near the MIT. A discussion of the relationship between theory and experiment is given. We also make a number of predictions concerning crossover effects which occur when a magnetic field is applied to a system with Cooperons.