arXiv:cond-mat/9408009 · Physical Review Letters

Anderson-Mott Transition as a Random-Field Problem

T. R. Kirkpatrick, D. Belitz

Published Feb 13, 1995• 26 citations

Abstract

The Anderson-Mott transition of disordered interacting electrons is shown to share many physical and technical features with classical random-field systems. A renormalization group study of an order parameter field theory for the Anderson-Mott transition shows that random-field terms appear at one-loop order. They lead to an upper critical dimension ${d}_{c}^{+}=6$ for this model. For $d>6$ the critical behavior is mean-field-like. For $d<6$ an $\ensuremath{\epsilon}$ expansion yields exponents that coincide with those for the random-field Ising model. Implications of these results are discussed.

DOI: 10.1103/physrevlett.74.1178

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arXiv:cond-mat/9408009 · Physical Review Letters

Anderson-Mott Transition as a Random-Field Problem

T. R. Kirkpatrick, D. Belitz

Published Feb 13, 1995• 26 citations

arXiv HTML PDF

Abstract

{d}_{c}^{+}=6

for this model. For

d>6

the critical behavior is mean-field-like. For

d<6

\ensuremath{\epsilon}

expansion yields exponents that coincide with those for the random-field Ising model. Implications of these results are discussed.

DOI: 10.1103/physrevlett.74.1178

Related papers

Critical behavior of the random-field Ising model at and beyond the upper critical dimension

Björn Ahrens, Alexander K. Hartmann

The disorder-driven phase transition of the random-field Ising model is observed using exact ground-state computer simulations for hypercubic lattices in $d=5,6,7$ dimensions. Finite-size scaling analyses are used to calculate the critical point and the critical exponents of the specific heat, magnetization, susceptibility, and of the correlation length. For dimensions $d=6,7$ which are larger or equal to the assumed upper critical dimension, ${d}_{u}=6$ , mean-field behavior is found, i.e., $\ensuremath{\alpha}=0, \ensuremath{\beta}=1/2, \ensuremath{\gamma}=1, \ensuremath{\nu}=1/2$. For the analysis of the numerical data, it appears to be necessary to include recently proposed corrections to scaling at and beyond the upper critical dimension.The disorder-driven phase transition of the random-field Ising model is observed using exact ground-state computer simulations for hypercubic lattices in $d=5,6,7$ dimensions. Finite-size scaling analyses are used to calculate the critical point and the critical exponents of the specific heat, magnetization, susceptibility, and of the correlation length. For dimensions $d=6,7$ which are larger or equal to the assumed upper critical dimension, ${d}_{u}=6$, mean-field behavior is found, i.e., $\ensuremath{\alpha}=0, \ensuremath{\beta}=1/2, \ensuremath{\gamma}=1, \ensuremath{\nu}=1/2$. For the analysis of the numerical data, it appears to be necessary to include recently proposed corrections to scaling at and beyond the upper critical dimension.

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The Anderson-Mott transition

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We discuss the dependence of the critical properties of the Anderson model on the dimension d in the language of β -function and renormalization group recently introduced in Vanoni et al. [C. Vanoni et al. , Proc. Natl. Acad. Sci. U.S.A. 121 , e2401955121 (2024)] in the context of Anderson transition on random regular graphs. We show how in the delocalized region, including the transition point, the one-parameter scaling part of the β -function for the fractal dimension D 1 evolves smoothly from its d = 2 form, in which β 2 ≤ 0 , to its β ∞ ≥ 0 form, which is represented by the random regular graph (RRG) result. We show how the ϵ = d − 2 expansion and the 1 / d expansion around the RRG result can be reconciled and how the initial part of a renormalization group trajectory governed by the irrelevant exponent y depends on dimensionality. We also show how the irrelevant exponent emerges out of the high-gradient terms of expansion in the nonlinear sigma model and put forward a conjecture about a lower bound for the fractal dimension. The framework introduced here may serve as a basis for investigations of disordered many-body systems and of more general nonequilibrium quantum systems.We discuss the dependence of the critical properties of the Anderson model on the dimension d in the language of β -function and renormalization group recently introduced in Vanoni et al. [C. Vanoni et al. , Proc. Natl. Acad. Sci. U.S.A. 121 , e2401955121 (2024)] in the context of Anderson transition on random regular graphs. We show how in the delocalized region, including the transition point, the one-parameter scaling part of the β -function for the fractal dimension D 1 evolves smoothly from its d = 2 form, in which β 2 ≤ 0 , to its β ∞ ≥ 0 form, which is represented by the random regular graph (RRG) result. We show how the ϵ = d − 2 expansion and the 1 / d expansion around the RRG result can be reconciled and how the initial part of a renormalization group trajectory governed by the irrelevant exponent y depends on dimensionality. We also show how the irrelevant exponent emerges out of the high-gradient terms of expansion in the nonlinear sigma model and put forward a conjecture about a lower bound for the fractal dimension. The framework introduced here may serve as a basis for investigations of disordered many-body systems and of more general nonequilibrium quantum systems.

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Competition between the Mott transition and Anderson localization in one-dimensional disordered interacting electron systems

Satoshi Fujimoto, Norio Kawakami

The competition between the Mott transition and the Anderson localization in one-dimensional electron systems is studied with the bosonization and renormalization-group methods. The $\ensuremath{\beta}$ function is calculated up to second order in the strength of diagonal disorder by using a replica trick. It is found that sufficiently strong forward scattering by random impurities destroys the Mott-Hubbard gap, and backward scattering gives rise to Anderson localization for the resulting gapless state. On the other hand, if the umklapp interaction is strong enough, the Mott insulating state still overwhelms the Anderson localization.The competition between the Mott transition and the Anderson localization in one-dimensional electron systems is studied with the bosonization and renormalization-group methods. The $\ensuremath{\beta}$ function is calculated up to second order in the strength of diagonal disorder by using a replica trick. It is found that sufficiently strong forward scattering by random impurities destroys the Mott-Hubbard gap, and backward scattering gives rise to Anderson localization for the resulting gapless state. On the other hand, if the umklapp interaction is strong enough, the Mott insulating state still overwhelms the Anderson localization.

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We present a theory for disordered interacting electrons that can describe both the Mott and Anderson transitions in the respective limits of zero disorder and zero interaction. We use it to investigate the $T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$ Mott-Anderson transition at a fixed electron density, as the disorder strength is increased. Surprisingly, we find two critical values of disorder ${W}_{\mathrm{nfl}}$ and ${W}_{c}$ . For $W>{W}_{\mathrm{nfl}}$ , the system enters a ``Griffiths'' phase, displaying metallic non-Fermi liquid behavior. At even stronger disorder, ${W\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}W}_{c}>{W}_{\mathrm{nfl}}$ the system undergoes a metal-insulator transition, characterized by the linear vanishing of both the typical density of states and the typical quasiparticle weight.We present a theory for disordered interacting electrons that can describe both the Mott and Anderson transitions in the respective limits of zero disorder and zero interaction. We use it to investigate the $T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$ Mott-Anderson transition at a fixed electron density, as the disorder strength is increased. Surprisingly, we find two critical values of disorder ${W}_{\mathrm{nfl}}$ and ${W}_{c}$. For $W>{W}_{\mathrm{nfl}}$, the system enters a ``Griffiths'' phase, displaying metallic non-Fermi liquid behavior. At even stronger disorder, ${W\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}W}_{c}>{W}_{\mathrm{nfl}}$ the system undergoes a metal-insulator transition, characterized by the linear vanishing of both the typical density of states and the typical quasiparticle weight.

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J.C. Lee, Richard L. Gibbs

For random systems with quenched disorders distributed randomly in ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{R}}$ dimensions and perfectly correlated in the remaining ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{C}}$ dimensions, the renormalization-group theory has been performed in the past in two different ways. In the single-expansion method, one treats ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{C}}$ as finite and expands the perturbation terms only for \ensuremath{\epsilon}= ${\mathit{d}}_{\mathit{c}}$ -d, where ${\mathit{d}}_{\mathit{c}}$ is the upper critical dimension. In the double-expansion method, one treats ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{C}}$ as infinitesimal and performs an additional expansion for ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{C}}$. The former predicts smeared phase transitions, while the latter predicts sharp second-order phase transitions. We wish to find out which of the two predictions holds for a three-dimensional Ising model for which ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{C}}$=2 and ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{R}}$=1. We argue, using the replica method, that the spin correlation is isotropic for large distances as long as the fluctuation among the random bonds (in ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{R}}$ dimensions) is sufficiently smaller than the thermal energy ${\mathit{k}}_{\mathit{B}}$ T. With parameters chosen to satisfy this condition in the model, a Monte Carlo simulation has been performed. The results favor a sharp second-order phase transition.For random systems with quenched disorders distributed randomly in ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{R}}$ dimensions and perfectly correlated in the remaining ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{C}}$ dimensions, the renormalization-group theory has been performed in the past in two different ways. In the single-expansion method, one treats ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{C}}$ as finite and expands the perturbation terms only for \ensuremath{\epsilon}=${\mathit{d}}_{\mathit{c}}$-d, where ${\mathit{d}}_{\mathit{c}}$ is the upper critical dimension. In the double-expansion method, one treats ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{C}}$ as infinitesimal and performs an additional expansion for ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{C}}$. The former predicts smeared phase transitions, while the latter predicts sharp second-order phase transitions. We wish to find out which of the two predictions holds for a three-dimensional Ising model for which ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{C}}$=2 and ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{R}}$=1. We argue, using the replica method, that the spin correlation is isotropic for large distances as long as the fluctuation among the random bonds (in ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{R}}$ dimensions) is sufficiently smaller than the thermal energy ${\mathit{k}}_{\mathit{B}}$T. With parameters chosen to satisfy this condition in the model, a Monte Carlo simulation has been performed. The results favor a sharp second-order phase transition.

Physical review. B, Condensed matterFeb 1, 199217 citations

{d}_{c}^{+}=6

for this model. For

d>6

the critical behavior is mean-field-like. For

d<6

\ensuremath{\epsilon}

expansion yields exponents that coincide with those for the random-field Ising model. Implications of these results are discussed.

The disorder-driven phase transition of the random-field Ising model is observed using exact ground-state computer simulations for hypercubic lattices in

d=5,6,7

dimensions. Finite-size scaling analyses are used to calculate the critical point and the critical exponents of the specific heat, magnetization, susceptibility, and of the correlation length. For dimensions

d=6,7

which are larger or equal to the assumed upper critical dimension,

{d}_{u}=6

, mean-field behavior is found, i.e., $\ensuremath{\alpha}=0, \ensuremath{\beta}=1/2, \ensuremath{\gamma}=1, \ensuremath{\nu}=1/2$. For the analysis of the numerical data, it appears to be necessary to include recently proposed corrections to scaling at and beyond the upper critical dimension.

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

Mott-Anderson transition at a fixed electron density, as the disorder strength is increased. Surprisingly, we find two critical values of disorder

{W}_{\mathrm{nfl}}

and

{W}_{c}

. For

W>{W}_{\mathrm{nfl}}

, the system enters a ``Griffiths'' phase, displaying metallic non-Fermi liquid behavior. At even stronger disorder,

{W\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}W}_{c}>{W}_{\mathrm{nfl}}

the system undergoes a metal-insulator transition, characterized by the linear vanishing of both the typical density of states and the typical quasiparticle weight.

{\mathit{d}}_{\mathit{c}}

-d, where

{\mathit{d}}_{\mathit{c}}

is the upper critical dimension. In the double-expansion method, one treats ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{C}}$ as infinitesimal and performs an additional expansion for ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{C}}$. The former predicts smeared phase transitions, while the latter predicts sharp second-order phase transitions. We wish to find out which of the two predictions holds for a three-dimensional Ising model for which ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{C}}$=2 and ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{R}}$=1. We argue, using the replica method, that the spin correlation is isotropic for large distances as long as the fluctuation among the random bonds (in ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{R}}$ dimensions) is sufficiently smaller than the thermal energy

{\mathit{k}}_{\mathit{B}}

T. With parameters chosen to satisfy this condition in the model, a Monte Carlo simulation has been performed. The results favor a sharp second-order phase transition.