Topological invariants to characterize universality of boundary charge in one-dimensional insulators beyond symmetry constraints

Abstract

In the absence of any symmetry constraints we address universal properties of the boundary charge QB for a wide class of tight-binding models with non-degenerate bands in one dimension.We provide a precise formulation of the bulk-boundary correspondence by splitting QB via a gauge invariant decomposition in a Friedel, polarisation, and edge part. We reveal the topological nature of QB by proving the quantization of a topological index I = ∆QB −ρ, where ∆QB is the change of QB when shifting the lattice by one site towards a boundary andρ is the average charge per site. For a single band we find this index to be given by the winding number of the fundamental phase difference of the Bloch wave function between two adjacent sites. For a given chemical potential we establish a central topological constraint I ∈ {−1, 0} related to charge conservation and particlehole duality. Our results are shown to be stable against disorder and we propose generalizations to multi-channel and interacting systems.Introduction-Motivated by the discovery of the Quantum Hall effect [1,2], the search for materials with topological edge states (TESs) has become a very important field of condensed matter physics and quantum optics [3-9], see Refs. [10][11][12][13][14] for reviews and textbooks. Routinely, topological insulators are classified via their symmetry class and dimension [15][16][17][18][19][20][21][22][23][24]. Topological invariants like Chern and winding numbers are established and can be used to predict TESs at the boundary of two materials with different topological indices. Recently, the classification has been extended to include inversion symmetry within the field of topological crystalline insulators (TCIs) [25][26][27][28][29][30]. Here, the Zak phase [31] is the topological invariant which, via the so-called modern theory of polarization [32][33][34][35][36], can be related to the boundary charge Q B [37][38]. However, since the Zak phase of an individual band is not gauge invariant an unknown integer of topological nature occurs in Q B . Away from symmetry restrictions, finite one-dimensional (1D) tightbinding models with a sinusoidal on-site potential were studied [39,40], where a continuous phase variable ϕ controls the offset of the potential. Surprisingly, in the long wavelength limit, Q B (ϕ) reveals a universal linear slope which was shown to be stable against disorder and to be related to the quantized Hall conductance. The linear behavior can be explained from classical charge conservation which, however, leaves again an unknown integer undetermined. Shifting the lattice adiabatically by one site towards a boundary of a half-infinite system, the boundary charge changes by the constant amount ∆Q B =ρ, whereρ is the average charge per site. This is a generalization of charge pumping [41,42], where the lattice is shifted by a whole unit cell such that the charge νe, given by the number ν of occupied bands, is shifted into

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

“…According to Refs. [88,89] this invariant fulfils the topological constraint I ∈ {−1, 0} due to charge conservation of particles and holes. We note that this property is not changed when a gap closing point appears during the shift of the lattice by one site.…”

“…These relations are exact and do not depend on the presence or absence of short-ranged electronelectron interaction or random disorder, see Appendix B and bosonization studies in Sec. IV C. They are based on the same arguments as charge pumping [84,85] and have been extensively used recently for noncyclic adiabatic processes to analyze the universal average slope of the phase-dependence of the boundary charge [39,[87][88][89]. The unknown integer arises since bound states (at the boundaries) can cross the chemical potential during the adiabatic process leading to discrete integer jumps of the boundary charge.…”

“…In Refs. [88,89], the case of single-channel and nearestneighbor tight-binding models has been studied analytically for any value of Z and for generic modulations of the onsite potentials v m = h m (0) and the hoppings t m = −h m (1) (which can all be chosen real and positive t m > 0, see Appendix A of Ref. [89]).…”

“…Equation 2is a straightforward consequence of charge conservation since on average the charge nρ is moved into the boundary when the translation is described via an adiabatic process (up to an integer charge arising from edge states crossing the chemical potential during the adiabatic process). It has been used in a variety of recent works on single-channel and nearest-neighbor tight-binding models to analyze the universal phase dependence Q B (ϕ) as a function of a phase ϕ describing a continuous shift of the lattice towards the boundary [39,[87][88][89][90]. Equation 3is a fundamental transformation which is based on the simple observation that local inversion of a semi-infinite system with a left boundary turns it to the same semi-infinite system with a right boundary [39].…”

Topological and nontopological features of generalized Su-Schrieffer-Heeger models

“…where the sum is over the two occupied eigenstates |u k of the Bloch Hamiltonian. The Zak phase is not invariant with respect to spatial translations or gauge transformations [33,34], but it can be defined in such a way that its values reflect the number of edge modes [32,49]. This bulk-boundary correspondence remains valid if one adds the spin-orbit term.…”