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

AIP conference proceedings

Disorder Operator In 2D Insulating States

Ryuichi Shindou, Ken‐Ichiro Imura, Masao Ogata

Published Jan 1, 2006• 0 citations
Publisher

Abstract

In this brief report, we introduce a disorder operator or (dual) vortex field compatible with the periodic boundary condition (p.b.c.) and estimate its expectation value with respect to non‐interacting wavefunctions (w.f.) and gapped mean‐field ordered w.f. on a quite general footing. For metallic cases, its expectation value reduces to zero in the thermodynamic limit for almost arbitrary Fermi surfaces (F.S.). Meanwhile, it remains finite for the band insulating case/gapped mean‐field order case, whose value is characterized by the localization length.

DOI
10.1063/1.2354842

Related papers

Characterization of two-dimensional fermionic insulating states

Ryuichi Shindou, Ken‐Ichiro Imura, Masao Ogata

Inspired by the duality picture between superconductivity (SC) and insulator in two spatial dimension (2D), we conjecture that the order parameter, suitable for characterizing 2D fermionic insulating state, is the disorder operator, usually known in the context of statistical transformation. Namely, the change of the phase of the disorder operator along a closed loop measures the particle density accommodating inside this loop. Thus, identifying this (doped) particle density with the dual counterpart of the magnetic induction in 2D SC, we can naturally introduce the disorder operator as the dual order parameter of 2D insulators. The disorder operator has a branch cut emitting from this ``vortex'' to the single infinitely far point. To test this conjecture against arbitrary two-dimensional lattice models, we have chosen this branch cut to be compatible with the periodic boundary condition and obtain a general form of its expectation value for noninteracting metal and/or insulator wave function, including gapped mean-field order wave function. Based on this expression, we observed analytically that it indeed vanishes for a wide class of band metals in the thermodynamic limit. On the other hand, it takes a finite value in insulating states, which is quantified by the localization length or the real-valued gauge invariant two form dubbed as the quantum metric tensor. When successively applied along a closed loop, our disorder operator plays the role of twisting the boundary condition of a periodic system. We argue this point by highlighting the Aharonov-Bohm phase associated with this nonlocal operator.

arXiv:0706.4258•Physical Review B•Dec 12, 2006•4 citations

Mean Field Theory of the Mott-Anderson Transition

V. Dobrosavljević, Gabriel Kotliar

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.

arXiv:cond-mat/9611100•Physical Review Letters•May 19, 1997•195 citations

A simple position operator for periodic systems

Emília Valença Ferreira de Aragão, Diego Moreno Martinez, Stefano Battaglia +5 more

We present a position operator that is compatible with periodic boundary conditions (PBCs). It is a one-body operator that can be applied in calculations of correlated materials by simply replacing the traditional position vector by the new definition. We show that it satisfies important fundamental as well as practical constraints. To illustrate the usefulness of the PBC position operator, we apply it to the localization tensor, a key quantity that is able to differentiate metallic from insulating states. In particular, we show that the localization tensor given in terms of the PBC position operator yields the correct expression in the thermodynamic limit. Moreover, we show that it correctly distinguishes between finite precursors of metals and insulators.

arXiv:1811.05190•Physical review. B./Physical review. B•May 24, 2019•32 citations

Mean-field hopping solution in the Anderson-Hubbard model

R. Allub

The tight-binding diagonally disordered Anderson model with on-site Coulomb interaction is solved in the mean-field hopping approach. Phase diagrams of the metal-insulator transition are obtained from the free energy for the half-filled band and the binary-alloy distribution. For attractive interaction (U<0), the Coulomb term gives additional localizing effect and for some values of the microscopic parameters the temperature induced two transitions (insulator to metal to insulator). For U>0 two different types of behavior are distinguished. For small disorder (W\ensuremath{\le}U/2), the metallic phase can be induced by disorder, while for large W, the disorder always destroys the phase of extended states.

Physical review. B, Condensed matter•May 1, 1988•3 citations

Two-dimensional disordered Mott metal-insulator transition

Martha Villagrán, Nikolaos Mitsakos, Tsung-Han Lee +3 more

We studied several aspects of the Mott metal-insulator transition in the disordered case. The model on which we based our analysis is the disordered Hubbard model, which is the simplest model capable of capturing the Mott metal-insulator transition. We investigated this model through statistical dynamical mean-field theory (statDMFT). This theory is a natural extension of dynamical mean-field theory (DMFT), which has been used with relative success in the past several years with the purpose of describing the Mott transition in the clean case. As is the case for the latter theory, statDMFT incorporates the electronic correlation effects only in their local manifestations. Disorder, on the other hand, is treated in such a way as to incorporate Anderson localization effects. With this technique, we analyzed the disordered two-dimensional Mott transition, using the quantum Monte Carlo algorithm to solve the associated single-impurity problems. We found spinodal lines at which the metal and insulator ceased to be metastable. We also studied spatial fluctuations of local quantities, such as self-energy and local Green's function, and showed the appearance of metallic regions within the insulator and vice versa. We carried out an analysis of finite-size effects and showed that, in agreement with the theorems of Imry and Ma [Y. Imry and S. K. Ma, Phys. Rev. Lett. 35, 1399 (1975).], the first-order transition is smeared in the thermodynamic limit. We analyzed transport properties by means of a mapping to a random classical resistor network and calculated both the average current and its distribution across the metal-insulator transition.

arXiv:1908.04144•Physical review. B./Physical review. B•Jun 2, 2020•19 citations

Single-site and cellular dynamical mean-field theory of the Anderson-Hubbard model

Ara Go, Gun Sang Jeon

We study the Anderson-Hubbard model to examine the interplay between the disordered potential and the Hubbard interactions. We use the dynamical mean-field theory (DMFT) with an exact diagonalization as an impurity solver to effectively deal with the quantum fluctuations due to the local interactions. In order to characterize the paramagnetic metal-insulator transition in disordered systems, we compute the typical local density of states, as well as the arithmetically-averaged local density of states. Our approach is found to reproduce the characteristic features of the metal-insulator transition in infinite dimensions, such as the existence of the disorder-driven metallic phase for moderate values of disorder strength and interactions, as observed in earlier studies. In two dimensions, the system turns out to exhibit more insulating behavior when we increase the cluster size in the cellular DMFT. The metallic phase, which is induced from weak Mott insulators by the introduction of a disordered potential, is not observed in two dimensions.

Journal of the Korean Physical Society•Mar 1, 2013•0 citations
In this brief report, we introduce a disorder operator or (dual) vortex field compatible with the periodic boundary condition (p.b.c.) and estimate its expectation value with respect to non‐interacting wavefunctions (w.f.) and gapped mean‐field ordered w.f. on a quite general footing. For metallic cases, its expectation value reduces to zero in the thermodynamic limit for almost arbitrary Fermi surfaces (F.S.). Meanwhile, it remains finite for the band insulating case/gapped mean‐field order case, whose value is characterized by the localization length.
Inspired by the duality picture between superconductivity (SC) and insulator in two spatial dimension (2D), we conjecture that the order parameter, suitable for characterizing 2D fermionic insulating state, is the disorder operator, usually known in the context of statistical transformation. Namely, the change of the phase of the disorder operator along a closed loop measures the particle density accommodating inside this loop. Thus, identifying this (doped) particle density with the dual counterpart of the magnetic induction in 2D SC, we can naturally introduce the disorder operator as the dual order parameter of 2D insulators. The disorder operator has a branch cut emitting from this ``vortex'' to the single infinitely far point. To test this conjecture against arbitrary two-dimensional lattice models, we have chosen this branch cut to be compatible with the periodic boundary condition and obtain a general form of its expectation value for noninteracting metal and/or insulator wave function, including gapped mean-field order wave function. Based on this expression, we observed analytically that it indeed vanishes for a wide class of band metals in the thermodynamic limit. On the other hand, it takes a finite value in insulating states, which is quantified by the localization length or the real-valued gauge invariant two form dubbed as the quantum metric tensor. When successively applied along a closed loop, our disorder operator plays the role of twisting the boundary condition of a periodic system. We argue this point by highlighting the Aharonov-Bohm phase associated with this nonlocal operator.
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=0T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0T=0 Mott-Anderson transition at a fixed electron density, as the disorder strength is increased. Surprisingly, we find two critical values of disorder Wnfl{W}_{\mathrm{nfl}}Wnfl​ and Wc{W}_{c}Wc​. For W>WnflW>{W}_{\mathrm{nfl}}W>Wnfl​, the system enters a ``Griffiths'' phase, displaying metallic non-Fermi liquid behavior. At even stronger disorder, W=Wc>Wnfl{W\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}W}_{c}>{W}_{\mathrm{nfl}}W=Wc​>Wnfl​ 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 position operator that is compatible with periodic boundary conditions (PBCs). It is a one-body operator that can be applied in calculations of correlated materials by simply replacing the traditional position vector by the new definition. We show that it satisfies important fundamental as well as practical constraints. To illustrate the usefulness of the PBC position operator, we apply it to the localization tensor, a key quantity that is able to differentiate metallic from insulating states. In particular, we show that the localization tensor given in terms of the PBC position operator yields the correct expression in the thermodynamic limit. Moreover, we show that it correctly distinguishes between finite precursors of metals and insulators.
The tight-binding diagonally disordered Anderson model with on-site Coulomb interaction is solved in the mean-field hopping approach. Phase diagrams of the metal-insulator transition are obtained from the free energy for the half-filled band and the binary-alloy distribution. For attractive interaction (U<0), the Coulomb term gives additional localizing effect and for some values of the microscopic parameters the temperature induced two transitions (insulator to metal to insulator). For U>0 two different types of behavior are distinguished. For small disorder (W\ensuremath{\le}U/2), the metallic phase can be induced by disorder, while for large W, the disorder always destroys the phase of extended states.
We studied several aspects of the Mott metal-insulator transition in the disordered case. The model on which we based our analysis is the disordered Hubbard model, which is the simplest model capable of capturing the Mott metal-insulator transition. We investigated this model through statistical dynamical mean-field theory (statDMFT). This theory is a natural extension of dynamical mean-field theory (DMFT), which has been used with relative success in the past several years with the purpose of describing the Mott transition in the clean case. As is the case for the latter theory, statDMFT incorporates the electronic correlation effects only in their local manifestations. Disorder, on the other hand, is treated in such a way as to incorporate Anderson localization effects. With this technique, we analyzed the disordered two-dimensional Mott transition, using the quantum Monte Carlo algorithm to solve the associated single-impurity problems. We found spinodal lines at which the metal and insulator ceased to be metastable. We also studied spatial fluctuations of local quantities, such as self-energy and local Green's function, and showed the appearance of metallic regions within the insulator and vice versa. We carried out an analysis of finite-size effects and showed that, in agreement with the theorems of Imry and Ma [Y. Imry and S. K. Ma, Phys. Rev. Lett. 35, 1399 (1975).], the first-order transition is smeared in the thermodynamic limit. We analyzed transport properties by means of a mapping to a random classical resistor network and calculated both the average current and its distribution across the metal-insulator transition.
We study the Anderson-Hubbard model to examine the interplay between the disordered potential and the Hubbard interactions. We use the dynamical mean-field theory (DMFT) with an exact diagonalization as an impurity solver to effectively deal with the quantum fluctuations due to the local interactions. In order to characterize the paramagnetic metal-insulator transition in disordered systems, we compute the typical local density of states, as well as the arithmetically-averaged local density of states. Our approach is found to reproduce the characteristic features of the metal-insulator transition in infinite dimensions, such as the existence of the disorder-driven metallic phase for moderate values of disorder strength and interactions, as observed in earlier studies. In two dimensions, the system turns out to exhibit more insulating behavior when we increase the cluster size in the cellular DMFT. The metallic phase, which is induced from weak Mott insulators by the introduction of a disordered potential, is not observed in two dimensions.