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

arXiv:1811.05190 · Physical review. B./Physical review. B

A simple position operator for periodic systems

Emília Valença Ferreira de Aragão, Diego Moreno Martinez, Stefano Battaglia, Gian Luigi Bendazzoli, Stefano Evangelisti, Thierry Leininger, Nicolas Suaud, J. A. Berger

Published May 24, 2019• 32 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.99.205144

Related papers

Localization tensor revisited: geometric-probabilistic foundations and a structure factor criterion under periodic boundaries

Zhe-Hao Zhang, Xiaoming Cai, Yi-Cong Yu

Abstract We revisit the localization tensor (LT) from geometric and probabilistic perspectives and construct extensions that are naturally compatible with periodic boundary conditions (PBC), without redefining the position operator. In open boundary conditions, we show that the LT can be written exactly as the covariance of a bivariate probability distribution built from density–density correlations. This leads to two conceptually distinct extensions to PBC: (i) a geometric one based on the Riemannian center (Frëchet mean) on the circle, and (ii) a metric-free one based on the mutual information I , which treats the configuration space purely as a probability space. We then relate the LT to the static structure factor by identifying the diagonal part, C pp , as a ‘localization function’ C ( p ), whose small-momentum behavior determines the LT in the thermodynamic limit. This clarifies why the LT is sensitive to transitions out of the extended phase but by itself cannot distinguish Anderson-type localization from dimerization: both share the same low-momentum asymptotics. We show that the finite-momentum behavior of C ( p ), together with an inverse participation ratio (IPR) based upper bound valid in localized phases, provides a sharp criterion that discriminates localization from dimerization. These results are illustrated on the Su–Schrieffer–Heeger and Aubry–Andrë models, with and without interactions, and suggest that structure factor based probes offer robust and experimentally accessible diagnostics of localized and dimerized phases under PBC.

Physica Scripta•Feb 16, 2026•0 citations

Many-body position operator in lattice fermionic systems with periodic boundary conditions

Balázs Hetényi

A total position operator $X$ in the position representation is derived for lattice fermionic systems with periodic boundary conditions. The operator is shown to be Hermitian, the generator of translations in momentum space, and its time derivative is shown to correspond to the total current operator in a periodic system. The operator is such that its moments can be calculated up to any order. To demonstrate its utility finite size scaling is applied to the Brinkman-Rice transition as well as metallic and insulating Gutzwiller wavefunctions.

arXiv:0909.0211•Journal of Physics A Mathematical and Theoretical•Sep 22, 2009•7 citations

Localization Tensor Revisited: Geometric-Probabilistic Foundations and a Structure-Factor Criterion under Periodic Boundaries

Zhe-Hao Zhang, Xiaoming Cai, Yi-Cong Yu

We revisit the localization tensor (LT) from geometric and probabilistic perspectives and construct extensions that are naturally compatible with periodic boundary conditions (PBC), without redefining the position operator. In open boundary conditions, we show that the LT can be written exactly as the covariance of a bivariate probability distribution built from density-density correlations. This leads to two conceptually distinct extensions to PBC: (i) a geometric one based on the Riemannian center (Frechet mean) on the circle, and (ii) a metric-free one based on the mutual information I, which treats the configuration space purely as a probability space. We then relate the LT to the static structure factor by identifying the diagonal part, Cpp, as a "localization function" C(p), whose small-momentum behavior determines the LT in the thermodynamic limit. This clarifies why the LT is sensitive to transitions out of the extended phase but by itself cannot distinguish Anderson-type localization from dimerization: both share the same low-momentum asymptotics. We show that the finite-momentum behavior of C(p), together with an inverse participation ratio (IPR)-based upper bound valid in localized phases, provides a sharp criterion that discriminates localization from dimerization. These results are illustrated on the Su-Schrieffer-Heeger and Aubry-Andre models, with and without interactions, and suggest that structure factor-based probes offer robust and experimentally accessible diagnostics of localized and dimerized phases under PBC.

quant-pharXiv:2602.14779•Feb 17, 2026•0 citations

The localization spread and polarizability of rings and periodic chains

Celestino Angeli, Gian Luigi Bendazzoli, Stefano Evangelisti +1 more

The localization spread gives a criterion to decide between metallic and insulating behavior of a material. It is defined as the second moment cumulant of the many-body position operator, divided by the number of electrons. Different operators are used for systems treated with open or periodic boundary conditions. In particular, in the case of periodic systems, we use the complex position definition, which was already used in similar contexts for the treatment of both classical and quantum situations. In this study, we show that the localization spread evaluated on a finite ring system of radius R with open boundary conditions leads, in the large R limit, to the same formula derived by Resta and co-workers [C. Sgiarovello, M. Peressi, and R. Resta, Phys. Rev. B 64, 115202 (2001)] for 1D systems with periodic Born-von Kármán boundary conditions. A second formula, alternative to Resta's, is also given based on the sum-over-state formalism, allowing for an interesting generalization to polarizability and other similar quantities.

arXiv:2105.08174•The Journal of Chemical Physics•Sep 23, 2021•7 citations

Unique one-body position operator for periodic systems

Stefano Evangelisti, Faten Abu-Shoga, Celestino Angeli +2 more

In this work we prove that the one-body position operator for periodic systems that we have recently proposed [Phys. Rev. B 99, 205144 (2019)] is unique modulo a phase factor and an additive constant. The proof uses several general physical constraints that a periodic one-body position operator should satisfy. We show that these constraints are sufficient to uniquely define a position operator that is compatible with periodic boundary conditions.

arXiv:2111.12538•Physical review. B./Physical review. B•Jun 6, 2022•11 citations

Disorder Operator In 2D Insulating States

Ryuichi Shindou, Ken‐Ichiro Imura, Masao Ogata

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.

AIP conference proceedings•Jan 1, 2006•0 citations
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.
Abstract We revisit the localization tensor (LT) from geometric and probabilistic perspectives and construct extensions that are naturally compatible with periodic boundary conditions (PBC), without redefining the position operator. In open boundary conditions, we show that the LT can be written exactly as the covariance of a bivariate probability distribution built from density–density correlations. This leads to two conceptually distinct extensions to PBC: (i) a geometric one based on the Riemannian center (Frëchet mean) on the circle, and (ii) a metric-free one based on the mutual information I , which treats the configuration space purely as a probability space. We then relate the LT to the static structure factor by identifying the diagonal part, C pp , as a ‘localization function’ C ( p ), whose small-momentum behavior determines the LT in the thermodynamic limit. This clarifies why the LT is sensitive to transitions out of the extended phase but by itself cannot distinguish Anderson-type localization from dimerization: both share the same low-momentum asymptotics. We show that the finite-momentum behavior of C ( p ), together with an inverse participation ratio (IPR) based upper bound valid in localized phases, provides a sharp criterion that discriminates localization from dimerization. These results are illustrated on the Su–Schrieffer–Heeger and Aubry–Andrë models, with and without interactions, and suggest that structure factor based probes offer robust and experimentally accessible diagnostics of localized and dimerized phases under PBC.
A total position operator XXX in the position representation is derived for lattice fermionic systems with periodic boundary conditions. The operator is shown to be Hermitian, the generator of translations in momentum space, and its time derivative is shown to correspond to the total current operator in a periodic system. The operator is such that its moments can be calculated up to any order. To demonstrate its utility finite size scaling is applied to the Brinkman-Rice transition as well as metallic and insulating Gutzwiller wavefunctions.
We revisit the localization tensor (LT) from geometric and probabilistic perspectives and construct extensions that are naturally compatible with periodic boundary conditions (PBC), without redefining the position operator. In open boundary conditions, we show that the LT can be written exactly as the covariance of a bivariate probability distribution built from density-density correlations. This leads to two conceptually distinct extensions to PBC: (i) a geometric one based on the Riemannian center (Frechet mean) on the circle, and (ii) a metric-free one based on the mutual information I, which treats the configuration space purely as a probability space. We then relate the LT to the static structure factor by identifying the diagonal part, Cpp, as a "localization function" C(p), whose small-momentum behavior determines the LT in the thermodynamic limit. This clarifies why the LT is sensitive to transitions out of the extended phase but by itself cannot distinguish Anderson-type localization from dimerization: both share the same low-momentum asymptotics. We show that the finite-momentum behavior of C(p), together with an inverse participation ratio (IPR)-based upper bound valid in localized phases, provides a sharp criterion that discriminates localization from dimerization. These results are illustrated on the Su-Schrieffer-Heeger and Aubry-Andre models, with and without interactions, and suggest that structure factor-based probes offer robust and experimentally accessible diagnostics of localized and dimerized phases under PBC.
The localization spread gives a criterion to decide between metallic and insulating behavior of a material. It is defined as the second moment cumulant of the many-body position operator, divided by the number of electrons. Different operators are used for systems treated with open or periodic boundary conditions. In particular, in the case of periodic systems, we use the complex position definition, which was already used in similar contexts for the treatment of both classical and quantum situations. In this study, we show that the localization spread evaluated on a finite ring system of radius R with open boundary conditions leads, in the large R limit, to the same formula derived by Resta and co-workers [C. Sgiarovello, M. Peressi, and R. Resta, Phys. Rev. B 64, 115202 (2001)] for 1D systems with periodic Born-von Kármán boundary conditions. A second formula, alternative to Resta's, is also given based on the sum-over-state formalism, allowing for an interesting generalization to polarizability and other similar quantities.
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.