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.
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.
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.
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 nth-order multipole operators whose expectation values can be used to calculate the 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.
Many-body higher-order topological invariant for Cn-symmetric insulators
Higher-order topological insulators in two spatial dimensions display fractional corner charges. While fractional charges in one dimension are known to be captured by a many-body bulk invariant, computed by the Resta formula, a many-body bulk invariant for higher-order topology and the corresponding fractional corner charges remains elusive despite several attempts. Inspired by recent work by Tada and Oshikawa, we propose a well-defined many-body bulk invariant for Cn symmetric higher-order topological insulators, which is valid for both non-interacting and interacting systems. Instead of relating them to the bulk quadrupole moment as was previously done, we show that in the presence of Cn rotational symmetry, this bulk invariant can be directly identified with quantized fractional corner charges. In particular, we prove that the corner charge is quantized as e/n with Cn symmetry, leading to a Zn classification for higher-order topological insulators in two dimensions.
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 bulk-boundary correspondence relates quantized edge states to bulk topological invariants in topological phases of matter. In one-dimensional symmetry-protected topological systems, quantized topological Thouless pumps directly reveal this principle and provide a sound mathematical foundation. Symmetry-protected higher-order topological phases of matter (HOSPTs) also feature a bulk-boundary correspondence, but its connection to quantized charge transport remains elusive. Here, we show that quantized Thouless pumps connecting C_{4}-symmetric HOSPTs can be described by a tuple of four Chern numbers that measure quantized bulk charge transport in a direction-dependent fashion. Moreover, this tuple of Chern numbers allows to predict the sign and value of fractional corner charges in the HOSPTs. We show that the topologically nontrivial phase can be characterized by both quadrupole and dipole configurations, shedding new light on current debates about the multipole nature of the HOSPT bulk. By employing corner-periodic boundary conditions, we generalize Restas's theory to HOSPTs. Our approach provides a simple framework for understanding topological invariants of general HOSPTs and paves the way for an in-depth description of future dynamical experiments.
Higher-order topological insulators in two spatial dimensions display fractional corner charges. While fractional charges in one dimension are known to be captured by a many-body bulk invariant, computed by the Resta formula, a many-body bulk invariant for higher-order topology and the corresponding fractional corner charges remains elusive despite several attempts. Inspired by recent work by Tada and Oshikawa [arXiv:2302.00800], we propose a well-defined, many-body bulk invariant for Cn-symmetric higher-order topological insulators that is valid for both noninteracting and interacting systems. Instead of relating them to the bulk quadrupole moment as was previously done, we show that in the presence of Cn rotational symmetry, this bulk invariant can be directly identified with quantized fractional corner charges. In particular, we prove that the corner charge is quantized as e/n with Cn symmetry, leading to a Zn classification for higher-order topological insulators in two dimensions.