@article{marzari-prb97,
   Author = {Marzari, N. and Vanderbilt, D.},
   Title = {Maximally localized generalized Wannier functions for
                  composite energy bands},
   Journal = {Phys. Rev. B},
   Volume = {56},
   Pages = {12847},
   Abstract = {We discuss a method for determining the optimally
                  localized set of generalized Wannier functions
                  associated with a set of Bloch bands in a
                  crystalline solid. By ''generalized Wannier
                  functions'' we mean a set of localized orthonormal
                  orbitals spanning the same space as the specified
                  set of Bloch bands. Although we minimize a
                  functional that represents the total spread
                  Sigma(n)(r(2))(n) - (r)(n)(2) of the Wannier
                  functions in real space, our method proceeds
                  directly from the Bloch functions as represented on
                  a mesh of k points, and carries out the minimization
                  in a space of unitary matrices U-mn((k)) describing
                  the rotation among the Bloch bands at each k
                  point. The method is thus suitable for use in
                  connection with conventional electronic-structure
                  codes. The procedure also returns the total electric
                  polarization as well as the location of each Wannier
                  center. Sample results for Si, GaAs, molecular C2H4,
                  and LiCl will be presented.},
    Year = {1997},
}

@article{souza-prb01,
Author = {Souza, I. and Marzari, N. and Vanderbilt, D.},
Title = {Maximally localized Wannier functions for entangled energy bands},
Journal = {Phys. Rev. B},
Volume = {65},
Pages = {035109},
Abstract = {We present a method for obtaining well-localized
                  Wannier-like functions (WF's) for energy bands that
                  are attached to or mixed with other bands. The
                  present scheme removes the limitation of the usual
                  maximally localized WF's method [N. Marzari and
                  D. Vanderbilt, Phys. Rev. B 56, 12 847 (1997)] that
                  the bands of interest should form an isolated group,
                  separated by gaps from higher and lower bands
                  everywhere in the Brillouin zone. An energy window
                  encompassing N bands of interest is specified by the
                  user, and the algorithm then proceeds to disentangle
                  these from the remaining bands inside the window by
                  filtering out an optimally connected N-dimensional
                  subspace. This is achieved by minimizing a
                  functional that measures the subspace dispersion
                  across the Brillouin zone. The maximally localized
                  WF's for the optimal subspace are then obtained via
                  the algorithm of Marzari and Vanderbilt. The method,
                  which functions as a postprocessing step using the
                  output of conventional electronic-structure codes,
                  is applied to the s and d bands of copper, and to
                  the valence and low-lying conduction bands of
                  silicon. For the low-lying nearly-free-electron
                  bands of copper we find WF's which are centered at
                  the tetrahedral-interstitial sites, suggesting an
                  alternative tight-binding parametrization.},
Year = {2001} 
}

@article{mostofi-cpc08,
Author = {Mostofi, A. A. and Yates, J. R. and Lee, Y.-S. and Souza,
                  I. and Vanderbilt, D. and Marzari, N.},
Title = {wannier90: A tool for obtaining maximally-localised Wannier
                  functions},
Journal = {Comput. Phys. Commun.},
Volume = {178},
Pages = {685},
Abstract = {We present wannier90, a program for calculating
                  maximally-localised Wannier functions (MLWF) from a
                  set of Bloch energy bands that may or may not be
                  attached to or mixed with other bands. The formalism
                  works by minimising the total spread of the MLWF in
                  real space. This is done in the space of unitary
                  matrices that describe rotations of the Bloch bands
                  at each k-point. As a result, wannier90 is
                  independent of the basis set used in the underlying
                  calculation to obtain the Bloch states. Therefore,
                  it may be interfaced straightforwardly to any
                  electronic structure code. The locality of MLWF can
                  be exploited to compute band-structure, density of
                  states and Fermi surfaces at modest computational
                  cost. Furthermore, wannier90 is able to output MLWF
                  for visualisation and other post-processing
                  purposes. Wannier functions are already used in a
                  wide variety of applications. These include analysis
                  of chemical bonding in real space; calculation of
                  dielectric properties via the modem theory of
                  polarisation; and as an accurate and minimal basis
                  set in the construction of model Hamiltonians for
                  large-scale systems, in linear-scaling quantum Monte
                  Carlo calculations, and for efficient computation of
                  material properties, such as the anomalous Hall
                  coefficient. wannier90 is freely available under the
                  GNU General Public License from
                  http://www.wannier.org/. Program summary Program
                  title: wannier90 Catalogue identifier: AEAK_v1_0
                  Program summary URL:
                  http://cpc.cs.qub.ac.uk/summaries/AEAK_v1_0.html
                  Program obtainable from: CPC Program Library,
                  Queen's University, Belfast, N. Ireland Licensing
                  provisions: Standard CPC licence,
                  http://cpc.cs.qub.ac.uk/licence/licence.html No. of
                  lines in distributed program, including test data,
                  etc.: 556 495 No. of bytes in distributed program,
                  including test data, etc.: 5 709 419 Distribution
                  format: tar.gz Programming language: Fortran 90,
                  perl Computer: any architecture with a Fortran 90
                  compiler Operating system: Linux, Windows, Solaris,
                  AIX, Tru64 Unix, OSX RAM: 10 MB Word size: 32 or 64
                  Classification: 7.3 External routines: BLAS
                  (http://www/netlib.oi-g/blas). LAPACK
                  (http://www.netlib.org/lapack). Both available under
                  open-source licenses. Nature of problem: Obtaining
                  maximally-localised Wannier functions from a set of
                  Bloch energy bands that may or may not be
                  entangled. Solution method: In the case of entangled
                  bands, the optimally-connected subspace of interest
                  is determined by minimising a functional which
                  measures the subspace dispersion across the
                  Brillouin zone. The maximally-localised Wannier
                  functions within this subspace are obtained by
                  subsequent minimisation of a functional that
                  represents the total spread of the Wannier functions
                  in real space. For the case of isolated energy bands
                  only the second step of the procedure is
                  required. Unusual features: Simple and user-friendly
                  input system. Wannier functions and interpolated
                  band structure output in a variety of file formats
                  for visualisation. Running time: Test cases take 1
                  minute. References: [1] N. Marzari, D. Vanderbilt,
                  Maximally localized generalized Wannier functions
                  for composite energy bands, Phys. Rev. B 56 (1997)
                  12847. [2] I. Souza, N. Marzari, D. Vanderbilt,
                  Maximally localized Wannier functions for entangled
                  energy bands, Phys. Rev. B 65 (2001) 035109. (C)
                  2007 Elsevier B.V. All rights reserved.},
Year = {2008} 
}

@article{wang-prb06,
Author = {Wang, X. and Yates, J. R. and Souza, I. and Vanderbilt, D.},
Title = {Ab initio calculation of the anomalous Hall conductivity by
                  Wannier interpolation},
Journal = {Phys. Rev. B},
Volume = {74},
Pages = {195118},
Abstract = {The intrinsic anomalous Hall conductivity in ferromagnets
                  depends on subtle spin-orbit-induced effects in the
                  electronic structure, and recent ab initio studies
                  found that it was necessary to sample the Brillouin
                  zone at millions of k-points to converge the
                  calculation. We present an efficient
                  first-principles approach for computing this
                  quantity. We start out by performing a conventional
                  electronic-structure calculation including
                  spin-orbit coupling on a uniform and relatively
                  coarse k-point mesh. From the resulting Bloch
                  states, maximally localized Wannier functions are
                  constructed which reproduce the ab initio states up
                  to the Fermi level. The Hamiltonian and
                  position-operator matrix elements, needed to
                  represent the energy bands and Berry curvatures, are
                  then set up between the Wannier orbitals. This
                  completes the first stage of the calculation,
                  whereby the low-energy ab initio problem is
                  transformed into an effective tight-binding
                  form. The second stage only involves Fourier
                  transforms and unitary transformations of the small
                  matrices setup in the first stage. With these
                  inexpensive operations, the quantities of interest
                  are interpolated onto a dense k-point mesh and used
                  to evaluate the anomalous Hall conductivity as a
                  Brillouin zone integral. The present scheme, which
                  also avoids the cumbersome summation over all
                  unoccupied states in the Kubo formula, is applied to
                  bcc Fe, giving excellent agreement with
                  conventional, less efficient first-principles
                  calculations. Remarkably, we find that about 99\% of
                  the effect can be recovered by keeping a set of
                  terms depending only on the Hamiltonian matrix
                  elements, not on matrix elements of the position
                  operator.},
Year = {2006} 
}


@Article{yao-prl04,
  author = {Y. Yao and L. Kleinman and A. H. MacDonald and J. Sinova and
T. Jungwirth and D.-S. Wang and E. Wang and Q. Niu},
  title = {},
  journal = {Phys. Rev. Lett.},
  volume = {92},
  pages = {037204},
  year = {2004},
}

@article{lopez-prb12,
Author = {M.~G. Lopez and D. Vanderbilt and T. Thonhauser and I. Souza},
Title = {},
Journal = {Phys. Rev. B},
Volume = {85},
Pages = {014435},
Year = {2012} 
}

@article{yates-prb07,
Author = {Yates, J. R. and Wang, X. and Vanderbilt, D. and Souza, I.},
Title = {Spectral and Fermi surface properties from Wannier interpolation},
Journal = {Phys. Rev. B},
Volume = {75},
Pages = {195121},
Abstract = {We present an efficient first-principles approach for
                  calculating Fermi surface averages and spectral
                  properties of solids, and use it to compute the
                  low-field Hall coefficient of several cubic metals
                  and the magnetic circular dichroism of iron. The
                  first step is to perform a conventional
                  first-principles calculation and store the low-lying
                  Bloch functions evaluated on a uniform grid of k
                  points in the Brillouin zone. We then map those
                  states onto a set of maximally localized Wannier
                  functions, and evaluate the matrix elements of the
                  Hamiltonian and the other needed operators between
                  the Wannier orbitals, thus setting up an "exact
                  tight-binding model." In this compact representation
                  the k-space quantities are evaluated inexpensively
                  using a generalized Slater-Koster
                  interpolation. Owing to the strong localization of
                  the Wannier orbitals in real space, the smoothness
                  and accuracy of the k-space interpolation increases
                  rapidly with the number of grid points originally
                  used to construct the Wannier functions. This allows
                  k-space integrals to be performed with ab initio
                  accuracy at low cost. In the Wannier representation,
                  band gradients, effective masses, and other k
                  derivatives needed for transport and optical
                  coefficients can be evaluated analytically,
                  producing numerically stable results even at band
                  crossings and near weak avoided crossings.},
Year = {2007} 
}

@article{marzari-rmp12,
Author = {N. Marzari and A. A. Mostofi and J. R. Yates and I. Souza
                  and D. Vanderbilt},
Title = {},
Journal = {Rev. Mod. Phys.},
Volume = {84},
Pages = {1419},
Year = {2012} 
}

@article{marzari-arxiv98,
Author = {N. Marzari and D. Vanderbilt},
Title = {},
Journal = {arXiv:9802210},
Volume = {},
Pages = {},
Year = {1998}
}

@article{vanderbilt-prb90,
Author = {D. Vanderbilt},
Title = {},
Journal = {Phys. Rev. B},
Volume = {41},
Pages = {7892},
Year = {1990} 
}

@article{yao-prb07,
Author = {Y. Yao and Y. Liang and D. Xiao and Q. Niu and S.-Q. Shen
                  and X. Dai and Z. Fang},
Title = {},
Journal = {Phys. Rev. B},
Volume = {75},
Pages = {020401},
Year = {2007} 
}

@article{lee-prl05,
Author = {Lee, Y.-S. and Nardelli, M. B. and Marzari, N.},
Title = {Band structure and quantum conductance of nanostructures from
                  maximally localized wannier functions: The case of
                  functionalized carbon nanotubes},
Journal = {Phys. Rev. Lett.},
Volume = {95},
Pages = {076804},
Abstract = {We have combined large-scale, Gamma-point
                  electronic-structure calculations with the maximally
                  localized Wannier functions approach to calculate
                  efficiently the band structure and the quantum
                  conductance of complex systems containing thousands
                  of atoms while maintaining full first-principles
                  accuracy. We have applied this approach to study
                  covalent functionalizations in metallic
                  single-walled carbon nanotubes. We find that the
                  band structure around the Fermi energy is much less
                  dependent on the chemical nature of the ligands than
                  on the sp(3) functionalization pattern disrupting
                  the conjugation network. Common aryl
                  functionalizations are more stable when paired with
                  saturating hydrogens; even when paired, they still
                  act as strong scattering centers that degrade the
                  ballistic conductance of the nanotubes already at
                  low degrees of coverage.},
Year = {2005} 
}


@misc{UserGuide,
  author = {A.~A.~Mostofi and G.~Pizzi and I.~Souza and J.~R.~Yates},
  year = {},
  howpublished = " User Guide to {\tt wannier90}, available at
                  \url{http://www.wannier.org/user\_guide.html}"
}


@Article{xiao-rmp10,
  title = {Berry phase effects on electronic properties},
  author = {Xiao, Di  and Chang, Ming-Che  and Niu, Qian },
  journal = {Rev. Mod. Phys.},
  volume = {82},
  pages = {1959--2007},
  numpages = {48},
  year = {2010},
  month = {Jul},
  doi = {10.1103/RevModPhys.82.1959},
  publisher = {American Physical Society}
}

@article{ceresoli-prb06,
Author = {Ceresoli, D. and Thonhauser, T. and Vanderbilt, D. and
                  Resta, R.},
Title = {Orbital magnetization in crystalline solids: Multi-band
                  insulators, Chern insulators, and metals},
Journal = {Phys. Rev. B},
Volume = {74},
Pages = {024408},
Abstract = {We derive a multi-band formulation of the orbital
                  magnetization in a normal periodic insulator (i.e.,
                  one in which the Chern invariant, or in two
                  dimensions (2D) the Chern number,
                  vanishes). Following the approach used recently to
                  develop the single-band formalism [Thonhauser,
                  Ceresoli, Vanderbilt, and Resta,
                  Phys. Rev. Lett. 95, 137205 (2005)], we work in the
                  Wannier representation and find that the
                  magnetization is comprised of two contributions, an
                  obvious one associated with the internal circulation
                  of bulklike Wannier functions in the interior and an
                  unexpected one arising from net currents carried by
                  Wannier functions near the surface. Unlike the
                  single-band case, where each of these contributions
                  is separately gauge invariant, in the multi-band
                  formulation only the sum of both terms is gauge
                  invariant. Our final expression for the orbital
                  magnetization can be rewritten as a bulk property in
                  terms of Bloch functions, making it simple to
                  implement in modern code packages. The
                  reciprocal-space expression is evaluated for 2D
                  model systems and the results are verified by
                  comparing to the magnetization computed for finite
                  samples cut from the bulk. Finally, while our formal
                  proof is limited to normal insulators, we also
                  present a heuristic extension to Chern insulators
                  (having nonzero Chern invariant) and to metals. The
                  validity of this extension is again tested by
                  comparing to the magnetization of finite samples cut
                  from the bulk for 2D model systems. We find
                  excellent agreement, thus providing strong empirical
                  evidence in favor of the validity of the heuristic
                  formula.},
Year = {2006} 
}


@article{nardelli-prb99,
        title = {Electronic transport in extended systems: Application
                  to carbon nanotubes},
        volume = {60},
        journal = {Phys. Rev. B},
        author = {Marco Buongiorno Nardelli},
        year = {1999},
        pages = {7828}
}


@article{gygi-cpc03,
Author = {Gygi, F. and Fattebert, J. L. and Schwegler, E.},
Title = {Computation of Maximally Localized Wannier Functions using a
                  simultaneous diagonalization algorithm},
Journal = {Comput. Phys. Commun.},
Volume = {155},
Pages = {1-6},
Abstract = {We show that a simultaneous diagonalization algorithm used
                  in signal processing applications can be used in the
                  context of electronic structure calculations to
                  efficiently compute Maximally Localized Wannier
                  Functions (MLWFs). Applications to calculations of
                  MLWFs in molecular and solid systems demonstrate the
                  efficiency of the approach. We also present and
                  discuss a parallel version of the algorithm. An
                  extension of the concept of MLWF to generalized
                  minimum spread wavefunctions is proposed. (C) 2003
                  Elsevier B.V. All rights reserved.},
Year = {2003} 
}


@article{posternak-prb02,
Author = {Posternak, M. and Baldereschi, A. and Massidda, S. and
                  Marzari, N.},
Title = {Maximally localized Wannier functions in antiferromagnetic
                  MnO within the FLAPW formalism},
Journal = {Phys. Rev. B},
Volume = {65},
Pages = {184422},
Abstract = {We have calculated the maximally localized Wannier
                  functions of MnO in its antiferromagnetic (AFM)
                  rhombohedral unit cell, which contains two formula
                  units. Electron Bloch functions are obtained with
                  the linearized-augmented-plane-wave method within
                  both the local-spin density (LSD) and the LSD+U
                  schemes. The thirteen uppermost occupied spin-up
                  bands correspond in a pure ionic scheme to the five
                  Mn 3d orbitals at the Mn-1 (spin-up) site and the
                  four O 2s/2p orbitals at each of the O-1 and O-2
                  sites. Maximal localization identifies uniquely four
                  Wannier functions for each O, which are trigonally
                  distorted sp(3)-like orbitals. They display a weak
                  covalent bonding between O 2s/2p states and
                  minority-spin d states of Mn-2, which is absent in a
                  fully ionic picture. This bonding is the fingerprint
                  of the interaction responsible for the AFM ordering,
                  and its strength depends on the one-electron scheme
                  being used. The five Mn Wannier functions are
                  centered on the Mn-1 site, and are atomic orbitals
                  modified by the crystal field. They are not uniquely
                  defined by the criterion of maximal localization and
                  we choose them as the linear combinations that
                  diagonalize the r(2) operator, so that they display
                  the D-3d symmetry of the Mn-1 site.},
Year = {2002} 
}

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  author       = "G. Grosso and G. P. Parravicini",
  title        = "Solid State Physics",
  publisher    = "Academic Press",
  address      = "",
  year         = 2000}

@book{ziman-book72,
  author       = "J. Ziman",
  title        = "Principles of the Theory of Solids",
edition = "2nd ed.",
  publisher    = "Cambridge University Press",
  address      = "",
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@Article{blount-ssp62,
author= {E. I. Blount},
journal = {Solid State Physics},
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pages = {305},
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@Article{pizzi-cpc14,
author= {G. Pizzi and D. Volja and B. Kozinsky and M. Fornari and N. Marzari},
journal = {Comput. Phys. Commun.},
title = {BoltzWann: A code for the evaluation of thermoelectric and electronic transport properties with a maximally-localized Wannier functions basis},
volume = {185},
pages = {422},
year= {2014},
doi= {doi:10.1016/j.cpc.2013.09.015}
}


@InCollection{mahan-itc06,
author= {G. D. Mahan},
booktitle = {Intern. Tables for Crystallography},
volume = {D},
chapter = {1.8},
pages = {7828},
year= {2006},
}

@article{sakuma-prb13,
  title = {Symmetry-adapted Wannier functions in the maximal localization procedure},
  author = {Sakuma, R.},
  journal = {Phys. Rev. B},
  volume = {87},
  pages = {235109},
  year = {2013}
}

@ARTICLE{tsirkin-arxiv17,
   author = {{Tsirkin}, S.~S. and {Aguado Puente}, P. and {Souza}, I.},
    title = "{Gyrotropic effects in trigonal tellurium studied from first principles}",
  journal = {ArXiv e-prints},
archivePrefix = "arXiv",
   eprint = {1710.03204},
 primaryClass = "cond-mat.mtrl-sci",
 keywords = {Condensed Matter - Materials Science},
     year = 2017,
    month = oct,
   adsurl = {http://adsabs.harvard.edu/abs/2017arXiv171003204T},
  adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}

@Article{yoda-sr15,
  author = {T. Yoda and T. Yokoyama and S. Murakami},
  title = {Current-induced Orbital and Spin Magnetizations in Crystals with Helical Structure},
  url={http://dx.doi.org/10.1038/srep12024},
  doi={10.1038/srep12024},	
  journal = {Sci. Rep.},
  volume = {5},
  pages = {12024},
  year = {2015}
}

@article{zhong-prl16,
  title = {Gyrotropic Magnetic Effect and the Magnetic Moment on the Fermi Surface},
  author = {Zhong, S. and Moore, J.~E. and Souza, I.},
  journal = {Phys. Rev. Lett.},
  volume = {116},
  issue = {7},
  pages = {077201},
  numpages = {6},
  year = {2016},
  month = {Feb},
  publisher = {American Physical Society},
  doi = {10.1103/PhysRevLett.116.077201},
  url = {https://link.aps.org/doi/10.1103/PhysRevLett.116.077201}
}

@article{malashevich-prb10,
  title = {Band theory of spatial dispersion in magnetoelectrics},
  author = {Malashevich, A. and Souza, I.},
  journal = {Phys. Rev. B},
  volume = {82},
  pages = {245118},
  year = {2010},
  doi = {10.1103/PhysRevB.82.245118}
}

@article{ivchenko-spss75,
author={Ivchenko, E.L. and Pikus, G.E.},
title={Natural optical activity of semiconductors (tellurium).},
journal={Sov. Phys. Solid State},
year={1975},
volume={16},
number={7},
pages={1261},
url={https://www.scopus.com/inward/record.uri?eid=2-s2.0-0016444392&partnerID=40&md5=d44204b27eb4356a166f389a0f8c8a4e},
source={Scopus}
}

@article{LinLin-ArXiv2017,
author = {{Damle}, A. and {Lin}, L.},
title = "{Disentanglement via entanglement: A unified method for Wannier localization}",
journal = {ArXiv e-prints},
eprint = {1703.06958},
year = 2017,
month = mar,
url = {http://adsabs.harvard.edu/abs/2017arXiv170306958D},
adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}

@article{ibanez-azpiroz_ab_2018,
	title = {Ab initio calculation of the shift photocurrent by Wannier interpolation},
        journal = {ArXiv e-prints},
        archivePrefix = "arXiv",
	author = {Iba\~nez{-}Azpiroz, Julen and Tsirkin, Stepan S. and Souza, Ivo},
	urldate = {2018-05-02},
	year = {2018},
	eprinttype = {arxiv},
	eprint = {1804.04030},
}

@misc{pythtb,
  author = {T. Yusufaly and D. Vanderbilt and S. Coh},
title={{Tight-Binding Formalism in the Context of the PythTB Package}},
howpublished={\url{http://physics.rutgers.edu/pythtb/formalism.html}}
}

@article{Marianetti,
title = {Selectively localized Wannier functions},
author = {Wang, Runzhi and Lazar, Emanuel A. and Park, Hyowon and Millis, Andrew J. and Marianetti, Chris A.},
abstractNote = {},
doi = {10.1103/PhysRevB.90.165125},
journal = {Physical Review B},
number = 16,
volume = 90,
place = {United States},
year = {2014},
month = {10}
}
