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Register a new userCompactly-supported Wannier functions and algebraic $K$-theory
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In a tight-binding lattice model with $n$ orbitals (single-particle states) per site, Wannier functions are $n$-component vector functions of position that fall off rapidly away from some location, and such that a set of them in some sense span all states in a given energy band or set of bands; compactly-supported Wannier functions are such functions that vanish outside a bounded region. They arise not only in band theory, but also in connection with tensor-network states for non-interacting fermion systems, and for flat-band Hamiltonians with strictly short-range hopping matrix elements. In earlier work, it was proved that for general complex band structures (vector bundles) or general complex Hamiltonians---that is, class A in the ten-fold classification of Hamiltonians and band structures---a set of compactly-supported Wannier functions can span the vector bundle only if the bundle is topologically trivial, in any dimension $d$ of space, even when use of an overcomplete set of such functions is permitted. This implied that, for a free-fermion tensor network state with a non-trivial bundle in class A, any strictly short-range parent Hamiltonian must be gapless. Here, this result is extended to all ten symmetry classes of band structures without additional crystallographic symmetries, with the result that in general the non-trivial bundles that can arise from compactly-supported Wannier-type functions are those that may possess, in each of $d$ directions, the non-trivial winding that can occur in the same symmetry class in one dimension, but nothing else. The results are obtained from a very natural usage of algebraic $K$-theory, based on a ring of polynomials in $e^{pm ik_x}$, $e^{pm ik_y}$, . . . , which occur as entries in the Fourier-transformed Wannier functions.
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Orbital magnetic susceptibility involves rich physics such as interband effects despite of its conceptual simplicity. In order to appreciate the rich physics related to the orbital magnetic susceptibility, it is essential to derive a formula to decompose the susceptibility into the contributions from each band. Here, we propose a scheme to perform this decomposition using the modified Wannier functions. The derived formula nicely decomposes the susceptibility into intraband and interband contributions, and from the other aspect, into itinerant and local contributions. The validity of the formula is tested in a couple of simple models. Interestingly, it is revealed that the quality of the decomposition depends on the degree of localization of the used Wannier functions. The formula here complements another formula using Bloch functions, or the formula derived in the semiclassical theory, which deepens our understanding of the orbital magnetic susceptibility and may serve as a foundation of a better computational method. The relationship to the Berry curvature in the present scheme is also clarified.
We present an algorithm for the adaptive tetrahedral integration over the Brillouin zone of crystalline materials, and apply it to compute the optical conductivity, dc conductivity, and thermopower. For these quantities, whose contributions are often localized in small portions of the Brillouin zone, adaptive integration is especially relevant. Our implementation, the woptic package, is tied into the wien2wannier framework and allows including a many-body self energy, e.g. from dynamical mean-field theory (DMFT). Wannier functions and dipole matrix elements are computed with the DFT package Wien2k and Wannier90. For illustration, we show DFT results for fcc-Al and DMFT results for the correlated metal SrVO$_3$.
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