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arXiv:2107.10787 · Physical review. B./Physical review. B

Universal properties of boundary and interface charges in multichannel one-dimensional models without symmetry constraints

Niclas Müller, Kiryl Piasotski, Dante Marvin Kennes, Herbert Schoeller, Mikhail Pletyukhov

Published Sep 30, 2021• 5 citations
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Abstract

The boundary charge that accumulates at the edge of a one-dimensional single-channel insulator is known to possess the universal property that its change under a lattice shift towards the edge by one site is given by the sum of the average bulk electronic density and a topologically invariant contribution, restricted to the values 0 and $\ensuremath{-}1$ [Pletyukhov et al., Phys. Rev. B 101, 165304 (2020)]. This quantized contribution is associated with particle-hole duality, ensures charge conservation, and fixes the mod(1) ambiguity appearing in the modern theory of polarization. In the present paper we generalize the above-mentioned single-channel results to the multichannel case by employing the technique of boundary Green's functions. We show that the topological invariant associated with the change in boundary charge under a lattice shift in multichannel models can be expressed as a winding number of a certain combination of components of bulk Green's functions as a function of the complex frequency, as it encircles the section of the energy axis that corresponds to the occupied part of the spectrum. We observe that this winding number is restricted to values ranging from $\ensuremath{-}{N}_{c}$ to zero, where ${N}_{c}$ is the number of channels (orbitals) per site. Furthermore, we consider translationally invariant one-dimensional multichannel models with an impurity and introduce topological indices which correspond to the quantized charge that accumulates around said impurity. These invariants are again given in terms of winding numbers of combinations of components of bulk Green's functions. Through this construction we provide a rigorous mathematical proof of the so-called nearsightedness principle formulated by Kohn [Kohn, Phys. Rev. Lett. 76, 3168 (1996)] for noninteracting multichannel lattice models.

DOI
10.1103/physrevb.104.125447

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The boundary charge that accumulates at the edge of a one-dimensional single-channel insulator is known to possess the universal property that its change under a lattice shift towards the edge by one site is given by the sum of the average bulk electronic density and a topologically invariant contribution, restricted to the values 0 and \ensuremath−1\ensuremath{-}1\ensuremath−1 [Pletyukhov et al., Phys. Rev. B 101, 165304 (2020)]. This quantized contribution is associated with particle-hole duality, ensures charge conservation, and fixes the mod(1) ambiguity appearing in the modern theory of polarization. In the present paper we generalize the above-mentioned single-channel results to the multichannel case by employing the technique of boundary Green's functions. We show that the topological invariant associated with the change in boundary charge under a lattice shift in multichannel models can be expressed as a winding number of a certain combination of components of bulk Green's functions as a function of the complex frequency, as it encircles the section of the energy axis that corresponds to the occupied part of the spectrum. We observe that this winding number is restricted to values ranging from \ensuremath−Nc\ensuremath{-}{N}_{c}\ensuremath−Nc​ to zero, where Nc{N}_{c}Nc​ is the number of channels (orbitals) per site. Furthermore, we consider translationally invariant one-dimensional multichannel models with an impurity and introduce topological indices which correspond to the quantized charge that accumulates around said impurity. These invariants are again given in terms of winding numbers of combinations of components of bulk Green's functions. Through this construction we provide a rigorous mathematical proof of the so-called nearsightedness principle formulated by Kohn [Kohn, Phys. Rev. Lett. 76, 3168 (1996)] for noninteracting multichannel lattice models.
Recently, it has been shown how topological phases of matter with crystalline symmetry and U(1)U(1)U(1) charge conservation can be partially characterized by a set of many-body invariants, the discrete shift So{\mathcal{S}}_{\mathrm{o}}So​ and electric polarization ${\stackrel{P\vec}{\mathcal{P}}}_{\mathrm{o}}$, where o\mathrm{o}o labels a high-symmetry point. Crucially, these can be defined even with nonzero Chern number and/or magnetic field. One manifestation of these invariants is through quantized fractional contributions to the charge in the vicinity of a lattice disclination or dislocation. In this paper, we show that these invariants can also be extracted from the length and corner dependence of the total charge (mod 1) on the boundary of the system. We provide a general formula in terms of So{\mathcal{S}}_{\mathrm{o}}So​ and ${\stackrel{P\vec}{\mathcal{P}}}_{\mathrm{o}}$ for the total charge of any subregion of the system which can include full boundaries or bulk lattice defects, unifying boundary, corner, disclination, and dislocation charge responses into a single general theory. These results hold for Chern insulators, despite their gapless chiral edge modes, and for which an unambiguous definition of an intrinsically two-dimensional electric polarization has been unclear until recently. We also discuss how our theory can fully characterize the topological response of quadrupole insulators.
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Topological insulators are noninteracting, gapped fermionic systems which have gapless boundary excitations. They are characterized by topological invariants, which can be written in many different ways, including in terms of Green's functions. Here we show that the existence of the edge states directly follows from the existence of the topological invariant written in terms of the Green's functions, for all ten classes of topological insulators in all spatial dimensions. We also show that the resulting edge states are characterized by their own topological invariant, whose value is equal to the topological invariant of the bulk insulator. This can be used to test whether a given model Hamiltonian can describe an edge of a topological insulator. Finally, we observe that the results discussed here apply equally well to interacting topological insulators, with certain modifications.
We establish a quantitative bulk-boundary correspondence in interacting topological insulators by relating many-body topological invariants to characteristic degeneracy structures in the entanglement spectrum. Focusing on generalized Su-Schrieffer-Heeger chains with higher winding number, we construct a gauge-invariant many-body winding invariant based on Pancharatnam geometric phases that remains well defined in the presence of interactions. We show that this invariant uniquely determines the low-lying entanglement-spectrum degeneracy, which exhibits a universal 4ν4^ν4ν scaling with the winding number ννν, providing a concrete formulation of bulk-boundary correspondence beyond single-particle topology. Using exact diagonalization, we demonstrate the robustness of this correspondence under interactions and symmetry-preserving disorder, and identify inversion symmetry as a minimal protecting symmetry that stabilizes both the quantization of the invariant and the associated entanglement degeneracies. Our results unify geometric-phase invariants and entanglement diagnostics within a many-body framework and provide a route to identifying interacting topological phases beyond band theory.
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 2 ρ and 1 2 ( ρ -1), where ρ 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-filling ρ = 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.
Abstract We outline a theory of symmetry protected topological phases of one-dimensional quantum walks. We assume spectral gaps around the symmetry-distinguished points +1 and −1, in which only discrete eigenvalues are allowed. The phase classification by integer or binary indices extends the classification known for translation invariant systems in terms of their band structure. However, our theory requires no translation invariance whatsoever, and the indices we define in this general setting are invariant under arbitrary symmetric local perturbations, even those that cannot be continuously contracted to the identity. More precisely we define two indices for every walk, characterizing the behavior far to the right and far to the left, respectively. Their sum is a lower bound on the number of eigenstates at +1 and −1. For a translation invariant system the indices add up to zero, so one of them already characterizes the phase. By joining two bulk phases with different indices we get a walk in which the right and left indices no longer cancel, so the theory predicts bound states at +1 or −1. This is a rigorous statement of bulk-edge correspondence. The results also apply to the Hamiltonian case with a single gap at zero.