Editing Wannier function

{{Short description|Physical function}}
[[Image:N2 Wannier.png|thumb|upright=0.85|Wannier functions of triple- and single-bonded nitrogen dimers in palladium nitride.]]
The '''Wannier functions''' are a complete set of [[orthogonal functions]] used in [[solid-state physics]]. They were introduced by [[Gregory Wannier]] in 1937.<ref name=Wannier1937>{{cite journal | doi = 10.1103/PhysRev.52.191 | volume=52 | issue=3 | title=The Structure of Electronic Excitation Levels in Insulating Crystals | year=1937 | journal=Physical Review | pages=191–197 | author=Wannier Gregory H| bibcode=1937PhRv...52..191W }}</ref><ref name=Wannier1962>{{cite journal | last=Wannier | first=Gregory H. | title=Dynamics of Band Electrons in Electric and Magnetic Fields | journal=Reviews of Modern Physics | publisher=American Physical Society (APS) | volume=34 | issue=4 | date=1 September 1962 | issn=0034-6861 | doi=10.1103/revmodphys.34.645 | pages=645–655 | bibcode=1962RvMP...34..645W}}</ref> Wannier functions are the [[localized molecular orbitals]] of crystalline systems.

The Wannier functions for different lattice sites in a [[crystal]] are orthogonal, allowing a convenient basis for the expansion of [[electron]] states in certain regimes. Wannier functions  have found widespread use, for example, in the analysis of binding forces acting on electrons.

==Definition==

[[Image:WanF-BaTiO3.png|upright=1.2|thumb|Example of a localized Wannier function of titanium in barium titanate (BaTiO3)]]

Although, like [[localized molecular orbitals]], Wannier functions can be chosen in many different ways,<ref>[https://cfm.ehu.es/ivo/publications/marzari-psik03.pdf Marzari ''et al.'': An Introduction to Maximally-Localized Wannier Functions]</ref> the original,<ref name=Wannier1937/> simplest, and most common definition in solid-state physics is as follows. Choose a single [[Electronic band structure|band]] in a perfect crystal, and denote its [[Bloch state]]s by
:<math>\psi_{\mathbf{k}}(\mathbf{r}) = e^{i\mathbf{k}\cdot\mathbf{r}}u_\mathbf{k}(\mathbf{r})</math>
where ''u''<sub>'''k'''</sub>('''r''') has the same periodicity as the crystal. Then the Wannier functions are defined by
:<math>\phi_{\mathbf{R}}(\mathbf{r}) = \frac{1}{\sqrt{N}} \sum_{\mathbf{k}} e^{-i\mathbf{k}\cdot\mathbf{R}} \psi_{\mathbf{k}}(\mathbf{r})</math>,
where
* '''R''' is any lattice vector (i.e., there is one Wannier function for each [[Bravais lattice|Bravais lattice vector]]);
* ''N'' is the number of [[primitive cell]]s in the crystal;
* The sum on '''k''' includes all the values of '''k''' in the [[Brillouin zone]] (or any other [[primitive cell]] of the [[reciprocal lattice]]) that are consistent with [[periodic boundary conditions]] on the crystal. This includes ''N'' different values of '''k''', spread out uniformly through the Brillouin zone. Since ''N'' is usually very large, the sum can be written as an integral according to the replacement rule:
:<math>\sum_{\mathbf{k}} \longrightarrow \frac{\sqrt{N}}{\Omega} \int_\text{BZ} d^3\mathbf{k}</math>
where "BZ" denotes the [[Brillouin zone]], which has volume Ω.

=== Properties ===

On the basis of this definition, the following properties can be proven to hold:<ref name=Bohm>{{cite book |title=The Geometric Phase in Quantum Systems |author=A Bohm, A Mostafazadeh, H Koizumi, Q Niu and J Zqanziger |isbn=978-3-540-00031-0 |publisher=Springer |year=2003 |pages=§12.5, p. 292 ff|doi=10.1007/978-3-662-10333-3 |url=https://cds.cern.ch/record/737299 }}</ref>

* For any lattice vector ''' R' ''',
:<math>\phi_{\mathbf{R}}(\mathbf{r}) = \phi_{\mathbf{R}+\mathbf{R}'}(\mathbf{r}+\mathbf{R}')</math>
In other words, a Wannier function only depends on the quantity ('''r''' − '''R'''). As a result, these functions are often written in the alternative notation
:<math>\phi(\mathbf{r}-\mathbf{R}) := \phi_{\mathbf{R}}(\mathbf{r})</math>

* The Bloch functions can be written in terms of Wannier functions as follows:
:<math>\psi_{\mathbf{k}}(\mathbf{r}) = \frac{1}{\sqrt{N}} \sum_{\mathbf{R}} e^{i\mathbf{k}\cdot\mathbf{R}} \phi_{\mathbf{R}}(\mathbf{r})</math>,
where the sum is over each lattice vector '''R''' in the crystal.

* The set of wavefunctions <math>\phi_{\mathbf{R}}</math> is an [[orthonormal basis]] for the band in question.
:<math>\begin{align}
\int_\text{crystal}  \phi_{\mathbf{R}}(\mathbf{r})^* \phi_{\mathbf{R'}}(\mathbf{r}) d^3\mathbf{r} & = \frac{1}{N} \sum_{\mathbf{k,k'}}\int_\text{crystal} e^{i\mathbf{k}\cdot\mathbf{R}} \psi_{\mathbf{k}}(\mathbf{r})^*  e^{-i\mathbf{k'}\cdot\mathbf{R'}} \psi_{\mathbf{k'}}(\mathbf{r}) d^3\mathbf{r} \\
& =  \frac{1}{N} \sum_{\mathbf{k,k'}} e^{i\mathbf{k}\cdot\mathbf{R}} e^{-i\mathbf{k'}\cdot\mathbf{R'}} \delta_{\mathbf{k,k'}} \\
& = \frac{1}{N} \sum_{\mathbf{k}} e^{i\mathbf{k}\cdot\mathbf{(R-R')}} \\
& =\delta_{\mathbf{R,R'}}
\end{align} </math>

Wannier functions have been extended to nearly periodic potentials as well.<ref name=Kohn0>[http://www.physast.uga.edu/~mgeller/4.pdf MP Geller and W Kohn] ''Theory of generalized Wannier functions for nearly periodic potentials'' Physical Review B 48, 1993</ref>

===Localization===

The Bloch states ''ψ''<sub>'''k'''</sub>('''r''') are defined as the eigenfunctions of a particular Hamiltonian, and are therefore defined only up to an overall phase. By applying a phase transformation ''e''<sup>''iθ''('''k''')</sup> to the functions ''ψ''<sub>'''k'''</sub>('''r'''), for any (real) function ''θ''('''k'''), one arrives at an equally valid choice. While the change has no consequences for the properties of the Bloch states, the corresponding Wannier functions are significantly changed by this transformation.

One therefore uses the freedom to choose the phases of the Bloch states in order to give the most convenient set of Wannier functions. In practice, this is usually the maximally-localized set, in which the Wannier function {{math|''&varphi;''<sub>'''R'''</sub>}} is localized around the point '''R''' and rapidly goes to zero away from '''R'''. For the one-dimensional case, it has been proved by Kohn<ref name=Kohn1>{{cite journal|doi=10.1103/PhysRev.115.809 | volume=115 | issue=4 | title=Analytic Properties of Bloch Waves and Wannier Functions | year=1959| journal=Physical Review  | pages=809–821 | author=W. Kohn| bibcode=1959PhRv..115..809K}}</ref> that there is always a unique choice that gives these properties (subject to certain symmetries). This consequently applies to any [[Separable partial differential equation|separable potential]] in higher dimensions; the general conditions are not established, and are the subject of ongoing research.<ref name=Arxiv-Localization/>

A [[Localized molecular orbitals#Pipek-Mezey|Pipek-Mezey]] style localization scheme has also been recently proposed for obtaining Wannier functions.<ref name=Jonsson2016>{{cite journal|doi=10.1021/acs.jctc.6b00809 | pmid=28099002 | volume=13 | issue=2 | title=Theory and Applications of Generalized Pipek–Mezey Wannier Functions | year=2017 | journal=Journal of Chemical Theory and Computation | pages=460–474 | author=Jónsson Elvar Ö., Lehtola Susi, Puska Martti, Jónsson Hannes| arxiv=1608.06396 | s2cid=206612913 }}</ref> Contrary to the maximally localized Wannier functions (which are an application of the [[Localized molecular orbitals#Foster-Boys|Foster-Boys]] scheme to crystalline systems), the Pipek-Mezey Wannier functions do not mix σ and π orbitals.

====Rigorous results====

The existence of [[Exponential function|exponential]]ly localized Wannier functions in insulators was proved mathematically in 2006.<ref name=Arxiv-Localization>{{cite journal | last1=Brouder | first1=Christian | last2=Panati | first2=Gianluca | last3=Calandra | first3=Matteo | last4=Mourougane | first4=Christophe | last5=Marzari | first5=Nicola | title=Exponential Localization of Wannier Functions in Insulators | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=98 | issue=4 | date=25 January 2007 | issn=0031-9007 | doi=10.1103/physrevlett.98.046402 | page=046402| pmid=17358792 |arxiv=cond-mat/0606726| bibcode=2007PhRvL..98d6402B | s2cid=32812449 }}</ref>

==Modern theory of polarization==
Wannier functions have recently found application in describing the [[Polarization density|polarization]] in crystals, for example, [[Ferroelectricity|ferroelectrics]]. The modern theory of polarization is pioneered by Raffaele Resta and David Vanderbilt. See for example, Berghold,<ref name=Berghold>{{cite journal | last1=Berghold | first1=Gerd | last2=Mundy | first2=Christopher J. | last3=Romero | first3=Aldo H. | last4=Hutter | first4=Jürg | last5=Parrinello | first5=Michele | title=General and efficient algorithms for obtaining maximally localized Wannier functions | journal=Physical Review B | publisher=American Physical Society (APS) | volume=61 | issue=15 | date=15 April 2000 | issn=0163-1829 | doi=10.1103/physrevb.61.10040 | pages=10040–10048| bibcode=2000PhRvB..6110040B }}</ref>  and Nakhmanson,<ref name=Nakhmanson>{{cite journal | last1=Nakhmanson | first1=S. M. | last2=Calzolari | first2=A. | last3=Meunier | first3=V. | last4=Bernholc | first4=J. | last5=Buongiorno Nardelli | first5=M. | title=Spontaneous polarization and piezoelectricity in boron nitride nanotubes | journal=Physical Review B | volume=67 | issue=23 | date=10 June 2003 | issn=0163-1829 | doi=10.1103/physrevb.67.235406 | page=235406|arxiv=cond-mat/0305329v1| bibcode=2003PhRvB..67w5406N | s2cid=119345964 }}</ref> and a power-point introduction by Vanderbilt.<ref name=Vanderbilt>[http://www.physics.rutgers.edu/~dhv/talks/rahman.pdf  D Vanderbilt] ''Berry phases and Curvatures in Electronic Structure Theory''.</ref> The polarization per unit cell in a solid can be defined as the dipole moment of the Wannier charge density:
:<math>\mathbf{p_c} = -e \sum_n \int\ d^3 r \,\, \mathbf{r} |W_n(\mathbf{r})|^2 \ , </math>
where the summation is over the occupied bands, and ''W<sub>n</sub>'' is the Wannier function localized in the cell for band ''n''. The ''change'' in polarization during a continuous physical process is the time derivative of the polarization and also can be formulated in terms of the [[Berry phase]] of the occupied Bloch states.<ref name=Bohm/><ref name=Resta>{{cite book |author=C. Pisani |title=Quantum-mechanical Ab-initio Calculation of the Properties of Crystalline Materials |isbn=978-3-540-61645-0 |year=1994 |publisher=Springer |edition=Proceedings of the IV School of Computational Chemistry of the Italian Chemical Society |page=282 |url=https://books.google.com/books?id=5ak5TwSLreAC&dq=%22Berry+connection%22&pg=PA282}}</ref>

==Wannier interpolation==
Wannier functions are often used to interpolate bandstructures calculated ''ab initio'' on a coarse grid of '''k'''-points to any arbitrary '''k'''-point. This is particularly useful for evaluation of Brillouin-zone integrals on dense grids and searching of Weyl points, and also taking derivatives in the '''k'''-space. This approach is similar in spirit to the [[Tight binding#Connection to Wannier functions|tight binding]] approximation, but in contrast allows for an exact description of bands in a certain energy range. Wannier interpolation schemes have been derived for spectral properties,<ref name="Yates Wang Vanderbilt Souza p. ">{{cite journal | last1=Yates | first1=Jonathan R. | last2=Wang | first2=Xinjie | last3=Vanderbilt | first3=David | last4=Souza | first4=Ivo | title=Spectral and Fermi surface properties from Wannier interpolation | journal=Physical Review B | publisher=American Physical Society (APS) | volume=75 | issue=19 | date=2007-05-21 | page=195121 | issn=1098-0121 | doi=10.1103/physrevb.75.195121 | arxiv=cond-mat/0702554| bibcode=2007PhRvB..75s5121Y | s2cid=31224663 }}</ref>
[[Hall effect#Anomalous Hall effect|anomalous Hall conductivity]],<ref name="Wang Yates Souza Vanderbilt p. ">{{cite journal | last1=Wang | first1=Xinjie | last2=Yates | first2=Jonathan R. | last3=Souza | first3=Ivo | last4=Vanderbilt | first4=David | title=Ab initiocalculation of the anomalous Hall conductivity by Wannier interpolation | journal=Physical Review B | volume=74 | issue=19 | date=2006-11-21 | page=195118 |arxiv=cond-mat/0608257| issn=1098-0121 | doi=10.1103/physrevb.74.195118 | bibcode=2006PhRvB..74s5118W | s2cid=30427871 }}</ref>
[[orbital magnetization]],<ref name="Lopez Vanderbilt Thonhauser Souza p. ">{{cite journal | last1=Lopez | first1=M. G. | last2=Vanderbilt | first2=David | last3=Thonhauser | first3=T. | last4=Souza | first4=Ivo | title=Wannier-based calculation of the orbital magnetization in crystals | journal=Physical Review B | volume=85 | issue=1 | date=2012-01-31 | page=014435 | issn=1098-0121 | doi=10.1103/physrevb.85.014435 | arxiv=1112.1938 | bibcode=2012PhRvB..85a4435L | s2cid=44056938 }}</ref>
thermoelectric and electronic transport properties,<ref name="Computer Physics Communications 2014 pp. 422–429">{{cite journal | title=BoltzWann: A code for the evaluation of thermoelectric and electronic transport properties with a maximally-localized Wannier functions basis | journal=Computer Physics Communications | volume=185 | issue=1 | date=2014-01-01 | issn=0010-4655 | doi=10.1016/j.cpc.2013.09.015 | pages=422–429 |arxiv=1305.1587 | url=https://www.sciencedirect.com/science/article/pii/S0010465513003160 | access-date=2020-07-13| last1=Pizzi | first1=Giovanni | last2=Volja | first2=Dmitri | last3=Kozinsky | first3=Boris | last4=Fornari | first4=Marco | last5=Marzari | first5=Nicola | bibcode=2014CoPhC.185..422P | s2cid=6140858 }}</ref>
[[Magneto-optic effect|gyrotropic effects]],<ref name="Tsirkin Puente Souza p. ">{{cite journal | last1=Tsirkin | first1=Stepan S. | last2=Puente | first2=Pablo Aguado | last3=Souza | first3=Ivo | title=Gyrotropic effects in trigonal tellurium studied from first principles | journal=Physical Review B | volume=97 | issue=3 | date=2018-01-29 | page=035158 | issn=2469-9950 | doi=10.1103/physrevb.97.035158 | arxiv=1710.03204| bibcode=2018PhRvB..97c5158T | s2cid=55517213 }}</ref>
[[Anomalous photovoltaic effect|shift current]],<ref name="Ibañez-Azpiroz Tsirkin Souza p. ">{{cite journal | last1=Ibañez-Azpiroz | first1=Julen | last2=Tsirkin | first2=Stepan S. | last3=Souza | first3=Ivo | title=Ab initio calculation of the shift photocurrent by Wannier interpolation | journal=Physical Review B | volume=97 | issue=24 | date=2018-06-26 | page=245143 | issn=2469-9950 | doi=10.1103/physrevb.97.245143 |arxiv=1804.04030| bibcode=2018PhRvB..97x5143I | s2cid=67751414 }}</ref>
[[Spin Hall effect|spin Hall conductivity]]
<ref name="Qiao Zhou Yuan Zhao p. ">{{cite journal | last1=Qiao | first1=Junfeng | last2=Zhou | first2=Jiaqi | last3=Yuan | first3=Zhe | last4=Zhao | first4=Weisheng | title=Calculation of intrinsic spin Hall conductivity by Wannier interpolation | journal=Physical Review B | volume=98 | issue=21 | date=2018-12-03 | page=214402 |arxiv=1810.07637| issn=2469-9950 | doi=10.1103/physrevb.98.214402 | bibcode=2018PhRvB..98u4402Q | s2cid=119223848 }}</ref>
<ref name="Ryoo Park Souza p. ">{{cite journal | last1=Ryoo | first1=Ji Hoon | last2=Park | first2=Cheol-Hwan | last3=Souza | first3=Ivo | title=Computation of intrinsic spin Hall conductivities from first principles using maximally localized Wannier functions | journal=Physical Review B | volume=99 | issue=23 | date=2019-06-07 | page=235113 | arxiv=1906.07139| issn=2469-9950 | doi=10.1103/physrevb.99.235113 | bibcode=2019PhRvB..99w5113R | s2cid=189928182 }}</ref> 
and other effects.

==See also==

* [[Orbital magnetization]]

==References==
{{reflist}}

==Further reading==
* {{cite book |title=Physics of Ferroelectrics: a Modern Perspective |author1=Karin M Rabe |author1-link= Karin M. Rabe |author2=Jean-Marc Triscone |author3=Charles H Ahn |page=2 |url=https://books.google.com/books?id=CWTzxRCDJdMC&dq=%22Berry+connection%22&pg=PA43 |publisher=Springer |year=2007 |isbn=978-3-540-34590-9}}

==External links==
*{{cite journal | doi = 10.1103/PhysRev.52.191 | volume=52 | issue=3 | title=The Structure of Electronic Excitation Levels in Insulating Crystals | year=1937 | journal=Physical Review | pages=191–197 | author=Wannier Gregory H| bibcode=1937PhRv...52..191W }}
*[http://wannier.org Wannier90 computer code that calculates maximally localized Wannier functions]
*[http://www.wannier-transport.org/ Wannier Transport code that calculates maximally localized Wannier functions fit for Quantum Transport applications]
*[https://www.wanniertools.org/ WannierTools: An open-source software package for novel topological materials]
*[http://wannier-berri.org/ WannierBerri - a python code for Wannier interpolation and tight-binding calculations]

==See also==
*[[Bloch's theorem]]
*[[Hannay angle]]
*[[Geometric phase]]

{{DEFAULTSORT:Wannier Function}}
[[Category:Condensed matter physics]]