Bloch Bundles, Marzari-Vanderbilt Functional and Maximally Localized Wannier Functions

Panati, Gianluca; Pisante, Adriano

doi:10.1007/s00220-013-1741-y

Cited by 62 publications

(108 citation statements)

References 45 publications

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“…The following Proposition is a straightforward generalization of a result in [PP,Prop. 2.1], where the case s = 0 is proved.…”

Section: If We Denote Bymentioning

confidence: 78%

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Symmetry and Localization in Periodic Crystals: Triviality of Bloch Bundles with a Fermionic Time-Reversal Symmetry

Monaco

Panati

2014

Acta Appl Math

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Abstract. We describe some applications of group-and bundle-theoretic methods in solid state physics, showing how symmetries lead to a proof of the localization of electrons in gapped crystalline solids, as e. g. insulators and semiconductors. We shortly review the Bloch-Floquet decomposition of periodic operators, and the related concepts of Bloch frames and composite Wannier functions. We show that the latter are almost-exponentially localized if and only if there exists a smooth periodic Bloch frame, and that the obstruction to the latter condition is the triviality of a Hermitian vector bundle, called the Bloch bundle. The rôle of additional Z 2 -symmetries, as time-reversal and space-reflection symmetry, is discussed, showing how time-reversal symmetry implies the triviality of the Bloch bundle, both in the bosonic and in the fermionic case. Moreover, the same Z 2 -symmetry allows to define a finer notion of isomorphism and, consequently, to define new topological invariants, which agree with the indices introduced by Fu, Kane and Mele in the context of topological insulators.

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“…The following Proposition is a straightforward generalization of a result in [PP,Prop. 2.1], where the case s = 0 is proved.…”

Section: If We Denote Bymentioning

confidence: 78%

“…where C is any contour in the complex plane winding once around the set σ * (k) and enclosing no other point in σ(H(k)), allows one to prove [PP,Prop. 2.1] the following Proposition 1.…”

mentioning

confidence: 99%

Symmetry and Localization in Periodic Crystals: Triviality of Bloch Bundles with a Fermionic Time-Reversal Symmetry

Monaco

Panati

2014

Acta Appl Math

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“…Indeed, one can use the well-known Luckhaus interpolation Lemma as in [29], Proposition 4.4, still for a sequence of functionals converging to the Dirichlet integral for maps into a manifold, showing that minimality persist in the limit and the convergence is actually strong in W 1,2 loc . From the monotonicity formula for the Ginzburg-Landau energy,Q ∞ is a degree-zero homogeneous harmonic map, hence it is smooth away from the origin by partial regularity theory [34].…”

Section: Proof Of Theorem 1 (I) It Follows From Propositions 31 and mentioning

confidence: 99%

Uniaxial versus biaxial character of nematic equilibria in three dimensions

2017

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We study global minimizers of the Landau-de Gennes (LdG) energy functional for nematic liquid crystals, on arbitrary three-dimensional simply connected geometries with topologically non-trivial and physically relevant Dirichlet boundary conditions. Our results are specific to an asymptotic limit coined in terms of a dimensionless temperature and material-dependent parameter, t and some constraints on the material parameters, and we work in the t → ∞ limit that captures features of the widely used Lyuksyutov constraint (Kralj and Virga in J Phys A 34:829-838, 2001). We prove (i) that (re-scaled) global LdG minimizers converge uniformly to a (minimizing) limiting harmonic map, away from the singular set of the limiting map; (ii) we have points of maximal biaxiality and uniaxiality near each singular point of the limiting map; (iii) estimates for the size of "strongly biaxial" regions in terms of the parameter t. We further show that global LdG minimizers in the restricted class of uniaxial Q-tensors cannot be stable critical points of the LdG energy in this limit.

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“…defines a smooth family of projectors over C = U × (−µ 0 , µ 0 ) (see [PaPi,Prop. 2.1] for a detailed proof), such that P µ=0 * (k) = P * (k).…”

Section: Proof Of Theorem 43mentioning

confidence: 99%

Topological Invariants of Eigenvalue Intersections and Decrease of Wannier Functions in Graphene

Monaco

Panati

2014

J Stat Phys

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Abstract. We investigate the asymptotic decrease of the Wannier functions for the valence and conduction band of graphene, both in the monolayer and the multilayer case. Since the decrease of the Wannier functions is characterised by the structure of the Bloch eigenspaces around the Dirac points, we introduce a geometric invariant of the family of eigenspaces, baptised eigenspace vorticity. We compare it with the pseudospin winding number. For every value n ∈ Z of the eigenspace vorticity, we exhibit a canonical model for the local topology of the eigenspaces. With the help of these canonical models, we show that the single band Wannier function w satisfies |w(x)| ≤ const · |x| −2 as |x| → ∞, both in monolayer and bilayer graphene.

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Bloch Bundles, Marzari-Vanderbilt Functional and Maximally Localized Wannier Functions

Cited by 62 publications

References 45 publications

Symmetry and Localization in Periodic Crystals: Triviality of Bloch Bundles with a Fermionic Time-Reversal Symmetry

Symmetry and Localization in Periodic Crystals: Triviality of Bloch Bundles with a Fermionic Time-Reversal Symmetry

Uniaxial versus biaxial character of nematic equilibria in three dimensions

Topological Invariants of Eigenvalue Intersections and Decrease of Wannier Functions in Graphene

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