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arXiv:cond-mat/9802004 · AIP conference proceedings

The quantum-mechanical position operator and the polarization problem

Raffaele Resta

Published Jan 1, 1998• 4 citations
arXivHTMLPDF

Abstract

The position operator (defined within Schrödinger representation as usual) becomes meaningless when the usual Born-von Kármán periodic boundary conditions are adopted: this fact is at the root of the polarization problem. I show how to define the position expectation value by means of rather peculiar many-body (multiplicative) operator acting on the wavefunction of the extended system. This definition can be regarded as the generalization of a precursor work, apparently unrelated to the polarization problem. For uncorrelated electrons, the present finding coincides with the so-called “single-point Berry phase” formula, which can hardly be regarded as the approximation of a continuum integral, and is computationally very useful for disordered systems. Simulations which are based on this concept are being performed by several groups.

DOI
10.1063/1.56271

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The position operator (defined within Schrödinger representation as usual) becomes meaningless when the usual Born-von Kármán periodic boundary conditions are adopted: this fact is at the root of the polarization problem. I show how to define the position expectation value by means of rather peculiar many-body (multiplicative) operator acting on the wavefunction of the extended system. This definition can be regarded as the generalization of a precursor work, apparently unrelated to the polarization problem. For uncorrelated electrons, the present finding coincides with the so-called “single-point Berry phase” formula, which can hardly be regarded as the approximation of a continuum integral, and is computationally very useful for disordered systems. Simulations which are based on this concept are being performed by several groups.
The quantum mechanical position operators, and their products, are not well-defined in systems obeying periodic boundary conditions. Here we extend the work of Resta [Phys. Rev. Lett. 80, 1800 (1998)], who developed a formalism to calculate the electronic polarization as an expectation value of a many-body operator, to include higher multipole moments, e.g., quadrupole and octupole. We define nthn\mathrm{th}nth-order multipole operators whose expectation values can be used to calculate the nthn\mathrm{th}nth multipole moment when all of the lower moments are vanishing (modulo a quantum). We show that changes in our operators are tied to flows of $n\ensuremath{-}1\mathrm{st}$ multipole currents, and encode the adiabatic evolution of the system in the presence of an $n\ensuremath{-}1\mathrm{st}$ gradient of the electric field. Finally, we test our operators on a set of tight-binding models to show that they correctly determine the phase diagrams of topological quadrupole and octupole models, capture an adiabatic quadrupole pump, and distinguish a bulk quadrupole moment from other mechanisms that generate corner charges.
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.
The expectation value definitions over an extended space from the considered Hilbert space of the system under consideration is given in another paper of the second author in this symposium. There, in that paper, the conceptuality rather than specification is emphasized on. This work uses that conceptuality to investigate the time evolutions of the position related operators' expectation values not in its standard meaning but rather in a new version of the definition over not the original Hilbert space but in the space obtained by extensions via introducing the images of the given initial wave packet under the positive integer powers of the system Hamiltonian. These images may not be residing in the same space of the initial wave packet when certain singularities appear in the structure of the system Hamiltonian. This may break down the existence of the integrals in the definitions of the expectation values. The cure is the use of basis functions in the abovementioned extended space and the sandwiching of the target operator whose expectation value is under questioning by an appropriately chosen operator guaranteeing the existence of the relevant integrals. Work specifically focuses on the hydrogen-like quantum systems whose Hamiltonians contain a polar singularity at the origin.
The expectation value of the twist operator has been widely used as a polarization-based index for gapped and gapless phases in interacting quantum many-body systems. Although numerous studies support this usage in specific settings and rigorous results have established the validity of the criterion in important settings, the precise assumptions required for it to sharply distinguish gapped and gapless phases under more general conditions have not been fully clarified. In this work, we clarify the logical status of polarization-based indices by formulating symmetry-based statements under explicitly stated assumptions. We identify the role of ground-state degeneracy in the statements for gapped systems and clarify the distinct assumptions required to exclude gapless scenarios that could otherwise mimic gapped behavior in the thermodynamic limit. Building on this controlled framework, we construct a meaningful extension beyond one dimension, emphasizing that such an extension is nontrivial and cannot be obtained by a straightforward generalization of the one-dimensional twist operator. Our results delineate the regime in which polarization-based quantities are justified as sharply defined many-body indices.