Rational boundary charge in one-dimensional systems with interaction and disorder

Abstract

We study the boundary charge Q B of generic semi-infinite one-dimensional insulators with translational invariance and show that nonlocal symmetries (i.e., including translations) lead to rational quantizations p/q of Q B. In particular, we find that (up to an unknown integer) the quantization of Q B is given in integer units of 1 2ρ and 1 2 (ρ − 1), whereρ is the average charge per site (which is a rational number for an insulator). This is a direct generalization of the known half-integer quantization of Q B for systems with local inversion or local chiral symmetries to any rational value. Quite remarkably, this rational quantization remains valid even in the presence of short-ranged electron-electron interactions as well as static random disorder (breaking translational invariance). This striking stability can be traced back to the fact that local perturbations in insulators induce only local charge redistributions. We establish this result with complementary methods including density matrix renormalization group calculations, bosonization methods, and exact solutions for particular lattice models. Furthermore, for the special case of half-fillingρ = 1 2 , we present explicit results in single-channel and nearest-neighbor hopping models and identify Weyl semimetal physics at gap closing points. Our general framework also allows us to shed new light on the well-known rational quantization of soliton charges at domain walls.

Interacting Rice-Mele model: Bulk and boundaries

“…Our first important step is thus to compute the boundary charge for the noninteracting RM model and illustrate the above mentioned characteristics resulting from bulk properties. We show that results obtained from an effective lowenergy theory for gaps much smaller than the band width [45] hold in a surprisingly large parameter regime. In addition, we find an interesting 1 4 quantization of the boundary charge in the limit of large gaps.…”

“…As shown for noninteracting and clean systems via the polarization in terms of the Zak-Berry phase [36][37][38][39][40][41][42][43][44] and recently also for disordered and interacting systems [45] the fractional part of the boundary charge shows characteristics which follow directly from bulk properties. Furthermore it is an interesting observable in its own right as it indicates various universal properties, such as the linear phase-dependence against continuous translations of the lattice [46][47][48][49], the possibility to realize rational quantization in the presence of symmetries [45], and a universal low-energy behavior for very small gaps [45]. Moreover, the fractional part of the boundary charge can be related to the bulk polarization which can be defined generically for any many-body system in terms of the phase of the ground-state expectation value of an exponential containing the position operator [50,51].…”

“…They severely limit its applicability in the limit of small gaps as corrections to the leading term become exceedingly large. Thus, as for metallic systems [52,53], in the past low-energy field theories and field-theoretical methods, such as bosonization, were employed [45][46][47][54][55][56][57]. They circumvent logarithmic terms.…”

“…They circumvent logarithmic terms. However, if being interested in the properties of a microscopic lattice model, such as the interacting RM model, their application requires the additional approximate step of mapping the lattice model to a continuum field theory [45][46][47]57]. They are furthermore bound to the low-energy limit.…”

“…We benchmark our functional RG results for the above observables to numerical density matrix renormalization group data (DMRG) [60]. DMRG was earlier applied to models of topological insulators [14,16,45,[61][62][63][64][65][66][67]. However, to reach the low-energy regime for systems with boundaries requires the study of exceedingly large systems which provides a computational challenge to this approach.…”