Difficulties in operator-based formulation of the bulk quadrupole moment

Abstract

Electric multipole moments are the most fundamental properties of insulating materials. However, the general formulation of bulk multipoles has been a long standing problem. The solution for the electric dipole moment was provided decades ago by King-Smith, Vanderbilt, and Resta. Recently, there have been attempts at generalizing Resta's formula to higher-order multipoles. We point out several issues in the recent proposals.

Thouless Pumps and Bulk-Boundary Correspondence in Higher-Order Symmetry-Protected Topological Phases

“…Our example shows that the tuple of Chern numbers C i can describe Thouless pumps building up both a dipole and a quadrupole moment. This sheds light on previously reported difficulties with defining a pure quadrupole operator in systems without dipole conservation [11]. Our case study also demonstrates that three Chern numbers need to be known to distinguish diagonal from non-diagonal pumps.…”

“…With the recent discovery of higher-order topological insulators (HOTIs) [7,8], efforts were made to generalize these concepts to describe electrical multi-pole moments [7][8][9][10][11][12][13][14] and higher-order Thouless pumps [8,9,[14][15][16]. A n-dimensional bulk with topology of order m can exhibit (n − m)-dimensional corner or hinge states, when open boundary conditions (OBC) are applied.…”

“…Higher-order topological invariants have been proposed for both band insulators in a single-particle picture (HOTIs) and interacting quantum many-body systems (HOSPTs) protected by crystalline symmetries [8,9,13,14,16,[26][27][28][29][30][31][32]]. Yet, there is an ongoing debate on which of the proposed quantities constitute true bulk invariants and how exactly the multipole polarization can be calculated in extended systems with periodic boundary conditions [11,13]. For instance, recent works [9,13] proposed to extend the work of Resta, that connects the polarization to the Zak (Berry) phase [4], by defining a many-body quadrupole operator.…”

“…For instance, recent works [9,13] proposed to extend the work of Resta, that connects the polarization to the Zak (Berry) phase [4], by defining a many-body quadrupole operator. However, these approaches have sparked controversy [11].…”

Rigorous Index Theory for One-Dimensional Interacting Topological Insulators

“…Topological insulators [3] for non-interacting fermions are completely classified according to the "periodic table" [4,5], and are characterized by the indices that are written as an integral over the Brillouin zone when the model has translation invariance (see, e.g., [3,6]), or, in more general, by the indices for projections defined by methods of noncommutative geometry (see, e.g., [7,8]). Although similar classification and characterization of interacting topological insulators have been investigated intensively (see, e.g, [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]), mathematically rigorous results are still limited [24][25][26][27][28][29][30]. See [31][32][33][34][35][36][37] for closely related rigorous index theorems for bosonic (or quantum spin) systems.…”