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arXiv:1812.06990 · Physical review. B./Physical review. B

Many-body electric multipole operators in extended systems

William A. Wheeler, Lucas K. Wagner, Taylor L. Hughes

Published Dec 20, 2019• 143 citations
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Abstract

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 $n\mathrm{th}$-order multipole operators whose expectation values can be used to calculate the $n\mathrm{th}$ 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.

DOI
10.1103/physrevb.100.245135

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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−1stn\ensuremath{-}1\mathrm{st}n\ensuremath−1st multipole currents, and encode the adiabatic evolution of the system in the presence of an n\ensuremath−1stn\ensuremath{-}1\mathrm{st}n\ensuremath−1st 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 modern theory of polarization establishes the bulk-boundary correspondence for the bulk polarization. In this paper, we attempt to extend it to a sum rule of the bulk quadrupole moment by employing a many-body operator introduced in Kang et al. [B. Kang, K. Shiozaki, and G. Y. Cho, Phys. Rev. B 100, 245134 (2019)] and Wheeler et al. [W. A. Wheeler, L. K. Wagner, and T. L. Hughes, Phys. Rev. B 100, 245135 (2019)]. The sum rule that we propose consists of the alternating sum of four observables, which are the phase factors of the many-body operator in different boundary conditions. We demonstrate its validity through extensive numerical computations for various noninteracting tight-binding models. We also observe that individual terms in the sum rule correspond to the bulk quadrupole moment, the edge-localized polarizations, and the corner charge in the thermodynamic limit on some models.
The Berry phase provides a modern formulation of electric polarization in crystals. We extend this concept to higher electric multipole moments and determine the necessary conditions and minimal models for which the quadrupole and octupole moments are topologically quantized electromagnetic observables. Such systems exhibit gapped boundaries that are themselves lower-dimensional topological phases. Furthermore, they host topologically protected corner states carrying fractional charge, exhibiting fractionalization at the boundary of the boundary. To characterize these insulating phases of matter, we introduce a paradigm in which "nested" Wilson loops give rise to topological invariants that have been overlooked. We propose three realistic experimental implementations of this topological behavior that can be immediately tested. Our work opens a venue for the expansion of the classification of topological phases of matter.
Insulating systems are characterized by their insensitivity to twisted boundary conditions as quantified by the charge stiffness and charge localization length. The latter quantity was shown to be related to the expectation value of the many-body position operator and serves as a universal criterion to distinguish between metals and insulators. In this work we extend these concepts to a new class of quantum systems having conserved charge and dipole moments. We refine the concept of a charge insulator by introducing notions of multipolar insulators, e.g., a charge insulator could be a dipole insulator or dipole metal. We develop a universal criterion to distinguish between these phases by extending the concept of charge stiffness and localization to analogous versions for multipole moments but with our focus on dipoles. We are able to relate the dipole localization scale to the expectation value of a recently introduced many-body quadrupole operator. This refined structure allows for the identification of phase transitions where charge remains localized but, e.g., dipoles delocalize. We illustrate the proposed criterion using several exactly solvable models that exemplify these concepts and discuss a possible realization in cold-atom systems.
We propose many-body order parameters for bulk multipoles in crystalline systems, whenever the bulk multipole moments can be defined in terms of the nontrivial topology of the nested Wilson loop. The many-body order parameters are designed to measure multipolar charge distribution in a crystalline unit cell, and they match the localized corner charge originating from the multipoles when the symmetries quantizing the multipoles are protected. Our many-body order parameters provide a complementary view to the nested Wilson loop approaches and even go beyond, as our many-body order parameters are readily applicable to interacting quantum many-body systems. Furthermore, even when the symmetries quantizing multipoles are lost so that the nested Wilson loop spectrum does not exactly reproduce the physical multipole moments, our many-body order parameters faithfully measure the physical multipole moments, which is confirmed numerically during Thouless pumping processes. We provide analytic arguments and numerical demonstration of the order parameters for various higher-order insulators having bulk multipole moments. Finally, we discuss the applicability of our many-body order parameters in the cases where the Wannier gap is absent so that the bulk multiple moments cannot be defined in terms of the nested Wilson loop.
We propose dipole and quadrupole indices for interacting insulators with point group symmetries. The proposed indices are defined in terms of many-body quantum multipole operators combined with the generator of the point group symmetry. Unlike the original multipole operators, these combined operators commute with the Hamiltonian under symmetry and therefore their eigenvalues are quantized. This enables a clear identification of nontrivial multipolar states. We calculate the multipole indices in representative models and show their effectiveness as order parameters. Furthermore, we demonstrate a bulk-boundary correspondence: a nonzero index implies the existence of edge/corner states under the point group symmetry.