Many-body electric multipole operators in extended systems

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

Phys. Rev. B 100, 245135 – Published 20 December 2019

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 $n th$ -order multipole operators whose expectation values can be used to calculate the $n th$ 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 $n - 1 st$ multipole currents, and encode the adiabatic evolution of the system in the presence of an $n - 1 st$ 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.

Received 17 December 2018
Revised 28 May 2019

DOI:https://doi.org/10.1103/PhysRevB.100.245135

Physics Subject Headings (PhySH)

Electric polarization Geometric & topological phases Symmetry protected topological states

Tight-binding model

Condensed Matter, Materials & Applied PhysicsGeneral Physics

Authors & Affiliations

William A. Wheeler¹, Lucas K. Wagner^2,*, and Taylor L. Hughes^2,†

¹Department of Materials Science and Engineering, University of Illinois at Urbana-Champaign, Illinois 61801, USA
²Department of Physics and Institute for Condensed Matter Theory, University of Illinois at Urbana-Champaign, Illinois 61801, USA

^*Corresponding author: lkwagner@illinois.edu
^†Corresponding author: hughest@illinois.edu

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Seishiro Ono, Luka Trifunovic, and Haruki Watanabe

Phys. Rev. B 100, 245133 (2019)

Many-body order parameters for multipoles in solids

Byungmin Kang, Ken Shiozaki, and Gil Young Cho

Phys. Rev. B 100, 245134 (2019)

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Vol. 100, Iss. 24 — 15 December 2019

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Images

Figure 1
Tight-binding model having four spinless orbitals per cell with intercell tunneling $λ_{x, y},$ intracell tunneling $γ_{x, y},$ and onsite potential $δ$ that takes opposite signs on the filled and empty circles. There is a flux $ϕ$ between each unit cell and within each unit cell in a gauge where the dotted lines have a relative phase factor of $e^{i Φ} .$
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Figure 2
(a) Phase diagram for model in Eq. (17) with $L_{x} = L_{y} = 40$ as a function of $(γ_{x} / λ, γ_{y} / λ, Φ) .$ The green and purple regions have a values of $Q^{x y}$ that differ by $e / 2,$ as calculated using Eq. (2). Dependence on $Φ$ is only shown along the diagonal lines that maintain $C_{4}$ symmetry. (b) The evolution of the quadrupole moment [Eq. (2)] during a pumping process as compared with the corner charge and edge polarization. All three match exactly, as expected. (c) Phase diagram for Eq. (17) with $Φ = 0$ as a function of $(γ_{y} / λ_{y}, λ_{x} / λ_{y}) .$ Equation (2) does not distinguish two different phases that have corner charge and a trivial insulator with no corner charge. This is as expected as in this regime the model has a vanishing quadrupole moment even in the phases with corner charge/modes [13]. (d) Phase diagram for an octupole model [12] with $L_{x} = L_{y} = L_{z} = 10$ as a function of $(γ_{x} / λ, γ_{y} / λ, γ_{z} / λ) .$ The green and purple regions have values of $O^{x y z}$ that differ by $e / 2$ as determined by Eq. (3).
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Figure 3
Phase diagram for the model in Eq. (18) with $L_{x} = L_{y} = 40$ as a function of $(m, Δ) .$ The green and purple regions have values of $Q^{x y}$ that differ by $e / 2$ , according to Eq. (2). The data along the line $Δ = 0$ is not shown because the values of $Q^{x y}$ are highly fluctuating, and the quadrupole moment is not well-defined since the edges are gapless.
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Figure 4
Phase diagrams in $γ_{x} − γ_{y}$ of $Im ln 〈 Φ_{0} | {\hat{U}}_{x y}^{Q} | Φ_{0} 〉$ for $N_{x} = N_{y} = 10$ (a), 20 (b), 40 (c). Teal lines at $| γ_{i} | = 1$ indicate the phase boundary of the model. Our operator gives a quadrupole moment of zero in the green regions and 0.5 in the purple regions. In larger systems, our operator's quadrupole phase more closely approaches the theoretical phase boundary.
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Figure 5
Log of the magnitude $| 〈 Φ_{0} | {\hat{U}}_{x y}^{Q} | Φ_{0} 〉 |$ at different points in the $γ_{x} − γ_{y}$ phase diagram. The magnitude goes to zero with increasing system size at all points in the phase diagram.
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Figure 6
Comparison of determinant (solid lines) and product states (dashed lines) of Gaussian orbitals as a function of system size, keeping $N_{x} = N_{y}$ (always even) and $σ_{x} = σ_{y}$ . For all these calculations, we fixed $x_{0} = y_{0} = 0.2$ . Top left: Magnitude of the polarization operator goes to zero for the determinant with larger values of $σ_{x}$ , but goes to one for the product state, regardless of $σ_{x}$ . Top right: Phase of the polarization operator is always zero. Bottom left: Magnitude of the quadrupole operator plateaus in the infinite limit to a value depending on $σ_{x}$ for both determinant and product states. For large enough $σ$ , the magnitude from the determinant state tends toward zero in the infinite limit. Bottom right: The phase of the quadrupole operator is consistent for all calculations. For odd values of $N_{x} = N_{y}$ (not shown), the quadrupole phase is shifted by 0.5.
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