Berry Phases in Electronic Structure Theory

Vanderbilt, David

doi:10.1017/9781316662205

Cited by 539 publications

(303 citation statements)

References 216 publications

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“…In this exact relation, the ambiguity modulo 2π in θ is consistent with a freedom to prepare insulating surfaces with values of σ surf AHC differing by the quantum e 2 /h, e.g., by changing the Chern number of some surface bands, or of adding or deleting a surface layer with a nonzero Chern number. 22,[35][36][37][38] If all surfaces adopt the same branch choice -i.e., the same value of σ surf AHC -then the sample as a whole exhibits a true magnetoelectric response of −α CS , where the quantized part of the response has been absorbed into the branch choice for α CS . This phenomenon is a higher-dimensional analog of the modern theory of electric polarization, 39 where the 2π ambiguity of the Berry phase reflects the inability to define the bulk polarization, or to predict the bound charge density of an insulating surface, except modulo a quantum.…”

Section: A Axion Coupling In the Bloch Representationmentioning

confidence: 99%

Axion coupling in the hybrid Wannier representation

2020

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Many magnetic point-group symmetries induce a topological classification on crystalline insulators, dividing them into those that have a nonzero quantized Chern-Simons magnetoelectric coupling ("axion-odd" or "topological"), and those that do not ("axion-even" or "trivial"). For time-reversal or inversion symmetries, the resulting topological state is usually denoted as a "strong topological insulator" or an "axion insulator" respectively, but many other symmetries can also protect this "axion Z2" index. Topological states are often insightfully characterized by choosing one crystallographic direction of interest, and inspecting the hybrid Wannier (or equivalently, the non-Abelian Wilson-loop) band structure, considered as a function of the two-dimensional Brillouin zone in the orthogonal directions. Here, we systematically classify the axion-quantizing symmetries, and explore the implications of such symmetries on the Wannier band structure. Conversely, we clarify the conditions under which the axion Z2 index can be deduced from the Wannier band structure. In particular, we identify cases in which a counting of Dirac touchings between Wannier bands, or a calculation of the Chern number of certain Wannier bands, or both, allows for a direct determination of the axion Z2 index. We also discuss when such symmetries impose a "flow" on the Wannier bands, such that they are always glued to higher and lower bands by degeneracies somewhere in the projected Brillouin zone, and the related question of when the corresponding surfaces can remain gapped, thus exhibiting a half-quantized surface anomalous Hall conductivity. Our formal arguments are confirmed and illustrated in the context of tight-binding models for several paradigmatic axion-odd symmetries including time reversal, inversion, simple mirror, and glide mirror symmetries.

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Section: A Axion Coupling In the Bloch Representationmentioning

confidence: 99%

Axion coupling in the hybrid Wannier representation

2020

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“…An important step towards a generic discus-sion of Q B has been undertaken within the so-called modern theory of polarization (MTP) [33][34][35][36][37][38][39][40], see Ref. [41] for a recent textbook review over the field. In summary, the MTP is based on two fundamental ingredients put forward in Refs.…”

Section: Introductionmentioning

confidence: 99%

“…This is fundamentally related to the fact that Q B is defined via a macroscopic average on scales much larger than Za, analog to the definition of the macroscopic charge density in classical electrodynamics (see, e.g., Chapter 4.5.1 in Ref. [41]). As a result, for large Z or, equivalently, in the large wave-length limit of the potential, one expects that Q B (ϕ) will be almost a linear function with a universal slope Zρ 2π = ν 2π on average.…”

Section: Introductionmentioning

confidence: 99%

Surface charge theorem and topological constraints for edge states: Analytical study of one-dimensional nearest-neighbor tight-binding models

et al. 2020

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arXiv:1911.06886v2 [cond-mat.mes-hall] 31 Jan 2020Besides the explicit solution for all eigenstates of the considered class of tight-binding models and the rigorous proof of the central results (1) and (4), we elaborate on a number of further issues in this work. Besides the standard case where the wavelength λ of the modulations is identical to the length Za of the unit cell, we will also discuss the case λ = Za/p with a rational number Z/p. This is a standard choice within the discussion of the IQHE and is also covered by our analysis for the boundary charge. We find that the topological constraint (1) and the universal form of the phase-dependence of the boundary charge is not changed but, in accordance with the IQHE [54][55][56], we find that the Chern number is given by the Diophantine equation pC ν = ν − s ν Z.Another result concerns a useful universal relationship between the boundary charges Q R,(α) Band Q

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“…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 .…”

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

et al. 2020

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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

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Berry Phases in Electronic Structure Theory

Cited by 539 publications

References 216 publications

Axion coupling in the hybrid Wannier representation

Axion coupling in the hybrid Wannier representation

Surface charge theorem and topological constraints for edge states: Analytical study of one-dimensional nearest-neighbor tight-binding models

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

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