$${\mathbb{Z}_{2}}$$ Z 2 Invariants of Topological Insulators as Geometric Obstructions

Fiorenza, Domenico; Monaco, Domenico; Panati, Gianluca

doi:10.1007/s00220-015-2552-0

Cited by 34 publications

(38 citation statements)

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“…The possibility to enforce also a TRS constraint on the frame is investigated. This leads to a topological obstruction in dimension 2, related to Z2 topological phases.We review several proposals for Z2 indices that distinguish these topological phases, including the ones by Fu-Kane [FK], Prodan [Pr2], Graf-Porta [GP] and Fiorenza-Monaco-Panati [FMP2]. We show that all these formulations are equivalent.…”

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confidence: 99%

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Wannier functions and ℤ2 invariants in time-reversal symmetric topological insulators

2017

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Abstract. We provide a constructive proof of exponentially localized Wannier functions and related Bloch frames in 1-and 2-dimensional time-reversal symmetric (TRS) topological insulators. The construction is formulated in terms of periodic TRS families of projectors (corresponding, in applications, to the eigenprojectors on an arbitrary number of relevant energy bands), and is thus model-independent. The possibility to enforce also a TRS constraint on the frame is investigated. This leads to a topological obstruction in dimension 2, related to Z2 topological phases.We review several proposals for Z2 indices that distinguish these topological phases, including the ones by Fu-Kane [FK], Prodan [Pr2], Graf-Porta [GP] and Fiorenza-Monaco-Panati [FMP2]. We show that all these formulations are equivalent. In particular, this allows to prove a geometric formula for the the Z2 invariant of 2-dimensional TRS topological insulators, originally indicated in [FK], which expresses it in terms of the Berry connection and the Berry curvature.

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confidence: 99%

“…We review several proposals for Z2 indices that distinguish these topological phases, including the ones by Fu-Kane [FK], Prodan [Pr2], Graf-Porta [GP] and Fiorenza-Monaco-Panati [FMP2]. We show that all these formulations are equivalent.…”

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confidence: 99%

Wannier functions and ℤ2 invariants in time-reversal symmetric topological insulators

2017

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“…Mathematically, these different types are characterised by θ 2 = 1 (BTRS) or θ 2 = −1 (FTRS). In the FTRS case, but not in the BTRS case, a further topological obstruction appears when trying to find Wannier functions respecting the natural symmetry of the problem [FMP16b]. In d = 2, there are two classes of systems: those for which one can find localised symmetric Wannier functions and those for which this is not possible.…”

Section: Introductionmentioning

confidence: 99%

Numerical construction of Wannier functions through homotopy

Gontier

Levitt

Siraj-Dine

2019

Journal of Mathematical Physics

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We provide a mathematically proven, simple and efficient algorithm to build localised Wannier functions, with the only requirement that the system has vanishing Chern numbers. Our algorithm is able to build localised Wannier for topological insulators such as the Kane-Mele model. It is based on an explicit and constructive proof of homotopies for the unitary group. We provide numerical tests validating the methods for several systems, including the Kane-Mele model. arXiv:1812.06746v1 [math-ph] 17 Dec 2018 locality of Wannier functions, starting from an initial guess. This algorithm is able to yield exponentially localised Wannier functions [PP13], but depends strongly on the choice of the initial guess. Recent advances, based on the use of matrix logarithms [CLPS17], selected columns of the density matrix (SCDM [DLY15, DLY17]) or an extended set of projections [MCCL15], provide ways to automatically construct initial projections, without any specific physical input.However, in the topologically non-trivial FTRS case, such as the Kane-Mele model of topological insulators, substantial difficulties appear. Since no symmetric Bloch frame can exist, algorithms that do not explicitly break this symmetry fail. This means that the initial guess for the method of [MV97] has to break this symmetry manually, which often proves challenging in practice. In the method of [CLPS17], this manifests as a crossing of eigenvalues, making the logarithm ill-defined (see Appendix of [CLPS17]). In the SCDM method, this appears as a loss of rank, unless a system-specific choice of columns is imposed [Lin18].The goal of this paper is to provide an automatic method that constructs localised Wannier functions even in the FTRS case. Our method is based on a standard reduction of the problem of finding Wannier functions to that of finding homotopies in the unitary group U(N ). This problem was solved using matrix logarithms in [CLPS17], and using a multi-step logarithm based on a perturbation argument in [CHN16,CM17]. In this paper, we instead use a iterative method where the columns of the unitaries are contracted one by one. This method, which is natural and robust, implements an idea hinted at, but not detailed, in [FMP16a, p.81]. Unlike the similar method of [CHN16, CM17], it does not exploit the eigenstructure, which proves unstable in practice.We emphasise that methods to construct Wannier functions specifically for the case of Z2 insulators were proposed in [SV11], [SV12], [WST16] and [MCCL16]. These methods however require model-specific information, while our method is completely automatic.The paper is organised as follows. We present in Section 2 the definition of Wannier functions, and the equivalence between localised Wannier functions and smooth Bloch frames. Then, we recall in Section 3 the standard reduction from the problem of finding smooth Bloch frames to that of finding homotopies of unitary matrices. We explain in Section 4 the difficulties of this problem and our solution, which we illustrate numerically in Section 5.

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“…Giving a full account of the geometric nature of this invariant has been a primary objective for mathematical physicists in the last decade, and a plethora of mathematical tools has been used in this endeavour, ranging from K-theory to homotopy theory, from functional analysis to noncommutative geometry, from equivariant cohomology to operator theory. We refer to [10,25,7] for recent accounts on the ever-growing literature on the subject.…”

Section: Introductionmentioning

confidence: 99%

Advances in Quantum Mechanics

Michelangeli¹,

Dell’Antonio²

2017

Springer INdAM Series

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The use of topological invariants to describe geometric phases of quantum matter has become an essential tool in modern solid state physics. The first instance of this paradigmatic trend can be traced to the study of the quantum Hall effect, in which the Chern number underlies the quantization of the transverse Hall conductivity. More recently, in the framework of time-reversal symmetric topological insulators and quantum spin Hall systems, a new topological classification has been proposed by Fu, Kane and Mele, where the label takes value in Z 2 . We illustrate how both the Chern number c ∈ Z and the Fu-Kane-Mele invariant δ ∈ Z 2 of 2-dimensional topological insulators can be characterized as topological obstructions. Indeed, c quantifies the obstruction to the existence of a frame of Bloch states for the crystal which is both continuous and periodic with respect to the crystal momentum. Instead, δ measures the possibility to impose a further time-reversal symmetry constraint on the Bloch frame.

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$${\mathbb{Z}_{2}}$$ Z 2 Invariants of Topological Insulators as Geometric Obstructions

Cited by 34 publications

References 40 publications

Wannier functions and ℤ2 invariants in time-reversal symmetric topological insulators

Wannier functions and ℤ2 invariants in time-reversal symmetric topological insulators

Numerical construction of Wannier functions through homotopy

Advances in Quantum Mechanics

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