Numerical construction of Wannier functions for crystals

Sami Siraj-dineSupervisor: A. Levitt

July 31, 2017

Contents

I Some theoretical background on Wannier functions 3

1 Formalism 31.1 Bravais lattice of a crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Reciprocal lattice and Brillouin Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Schrodinger equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Decomposition of the Hamiltonian on the Bloch waves 52.1 Description of the system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Bloch waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Band Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 Bloch Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.5 Decomposition of the Hamiltonian into the Hkk . . . . . . . . . . . . . . . . . . . 10

3 Wannier functions 103.1 Wannier functions for an isolated band . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Composite Wannier functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4 Topological obstruction to the existence of Wannier functions 124.1 Some elements of bundle theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.2 The Bloch bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.3 The Chern numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5 Numerical construction of Wannier functions and application 165.1 Marzari-Vanderbilt localisation functional . . . . . . . . . . . . . . . . . . . . . . . . 165.2 Application: Wannier interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5.2.1 Fourier interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.2.2 Application to band interpolation . . . . . . . . . . . . . . . . . . . . . . . . . 18

II Construction of continuous Bloch frames 19

6 Logarithm algorithm 196.1 Algorithm in 1D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.2 Algorithm in 2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.3 The Kane-Mele model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

7 Recursive interpolation algorithm 237.1 Recursive interpolation of a single matrix . . . . . . . . . . . . . . . . . . . . . . . . 237.2 Recursive interpolation of a matrix path . . . . . . . . . . . . . . . . . . . . . . . . . 237.3 Numerical results for the Kane-Mele model . . . . . . . . . . . . . . . . . . . . . . . 23

1

IntroductionCrystals are highly ordered and regular arrangements of atoms that can have a variety of interestingphysical properties, such as conducting electricity, being an insulator, or a semi-conductor. To studythese properties, we use the framework of quantum chemistry, in which the system is represented bya wavefunction, a complex function of space and time. Because the nuclei of atoms are far heavierthan the electrons (and thus much slower), the Born-Oppenheimer approximation assumes that thenuclei are fixed, and considers only the motion of the electrons.

In this framework, the behaviour of the electrons is determined by the wavefunction that Schrodingerequation. This means that for every additional particle in the system, we add a few dimensions tothe base space. However, it is possible to greatly reduce the dimensionality by considering onlythe density of particles. Due to the Hohenberg-Kohn theorem, the density suffices to determine thelowest energy state of the system.

In general, the electrons interact with each other, so that their dynamics are coupled, makingdirect computations more difficult. However, the Kohn-Sham equations allow us to find the densityof interacting particles as the density of a fictitious “Kohn-Sham system” of non-interacting particles.Hence, we focus here on the case of non-interacting particles, and the methods extend readily to thecase of coupled particles.

The regularity of the crystal implies some invariances on the Hamiltonian: for example, a periodiccrystal implies that the Hamiltonian is periodic in some sense. This structure allows us to use Bloch-Floquet theory to determine a localised basis of the wavefunctions, called the Wannier functions.

To build these Wannier functions in practice, one can minimise a localisation functional, proposedin [7]. However, this procedure requires a good initial guess to be manually chosen with physicalconsiderations. If the procedure is badly initialised, it may not converge to maximally-localisedWannier functions.

In [2], the authors proposed a practical method to produce a robust initial guess. This methodhas produced stable numerical results on common materials like Silicon, but a theoretical counter-example was found, where the method failed. This counter-example was not encountered in practiceuntil the method was tried on the Kane-Mele topological insulator model, a simple case that exhibitsa similar behaviour to Graphene.

The contribution of this internship is the discovery of a variation of the previous method, that isshown to yield robust results in all cases, and particularly on the Kane-Mele model. To the best ofour knowledge, this variation makes it the first systematic, proven method to yield localised Wannierfunctions.

2

Part I

Some theoretical background on Wannierfunctions

1 Formalism

In this report, we aim to describe the behaviour of electrons in a periodic crystal within the frameworkof the Born-Oppenheimer approximation. A periodic crystal is an ordered arrangement of particles(atoms, ions, molecules) that is assumed to be infinite in all directions.

Figure 1: Crystal structure of Sodium Chloride

1.1 Bravais lattice of a crystal

Periodic crystals are invariant by translations of a period along their principal axes. Hence, one canrepresent their structure as a repetition of a small building block called the unit cell. The geometryof this unit cell is described using the primitive vectors a1, . . . ,ad, where d ≤ 3 is the dimension ofthe crystal. The primitive vectors form an independent set of vectors that span the lattice.

Definition 1. A Bravais lattice R is a discrete set of points obtained by translations of the originalong integer linear combinations of the basis of primitive vectors a1, . . .ad

R = n1a1 + · · ·+ ndad, n1, . . . , nd ∈ Z.

We define the (centred) unit cell A of R as

A =

d∑i=1

αiai, αidi=1 ∈[−1

2,

1

2

].

1.2 Reciprocal lattice and Brillouin Zone

One can associate a dual object to the Bravais lattice R (generated by the basis aidi=1 ) called thereciprocal lattice.

Definition 2. We call R′ reciprocal lattice of R the lattice generated by the primitive vectors bidi=1,where

ai · bj = 2πδij .

It is well defined, as one can find the bj by solving the well-posed matrix equation

[b1 . . .bd] = 2π[a1 . . .ad]−T ,

where [a1 . . .ad] is the matrix of columns ai, invertible because the primitive vectors form a basis.The reciprocal lattice exhibits in fact a Fourier-type duality, as shown below.

Proposition 1. The reciprocal lattice R is the support of the Fourier transforms of R-periodicfunctions.

3

Proof. Let f be a periodic function of R, such that for x ∈ Rn, and R ∈ R, f(x + R) = f(x). ItsFourier series reads

f(x) =

∞∑n1,...,nd=−∞

cn1,...,ndexp

2πi

d∑j=1

njx · aj|aj |

.

Hence, denoting K =∑dj=1 njbj , and using ai · bj = 2πδij , we have

f(x) =∑

K∈R′

cKeiK·x,

and the Fourier transform of f is∑

K∈R′ cKδK, where δK is the Dirac function of support K.

Definition 3. The first Brillouin Zone B is the Voronoı cell of the reciprocal lattice around theorigin, that is, the locus of the points that are closer to the origin than to any other point in thelattice.

Figure 2: First Brillouin zone of face centred cubic structure

Abusing notation, in what follows, we also call Brillouin Zone the primitive cell [0, 1]d. Thedefinition of the primitive cell is a convention, as many choices are possible. However, this particularchoice is convenient in the context of this paper.

1.3 Schrodinger equation

We now consider the behaviour of electrons in the crystal. As postulated in quantum theory, thepossible states of the system are represented by the unit sphere of a Hilbert space H. The wholesystem (containing all the atoms and electrons of the crystal) can be described up to a phase by awave function Ψ ∈ H, of unit norm. In the simple case of a free particle without spin, we can choosethe representation

Ψ : Rn × [0, T ]→ C(x, t) 7→ Ψ(x, t),

where ‖Ψ(·, t)‖L2(Rn) = 1, ∀t, Rn is the configuration space, and T ∈ (0,+∞] is the time horizon.One can think of the configuration space as the space describing the position of the particle. Inthe Copenhagen interpretation, the function x → |Ψ(x)|2 represents the probability of finding theparticle in the configuration x.

We adopt here the Born-Oppenheimer approximation, which assumes, in the case of a crystal,that the nuclei are fixed, so that we can reduce the configuration space to the electrons only. Let Ψbe the wave function of the system of electrons in the crystal. It satisfies the general Schrodingerequation

i∂tΨ = HΨ (1)

where H is the Hamiltonian operator of the system.Observables are Hermitian operators A, which are measured experimentally through the action

〈Ψ| A |Ψ〉. Equation (1) admits stationary solutions of the system, for which all observables areconstant in time. Consequently, the stationary states are such that the wave function Ψ is constantup to a phase.

4

These states satisfy the time-independent Schrodinger equation:

HΨ = εΨ (2)

where H is the Hamiltonian operator, and ε is energy of the state. Note that this equation is aneigenvalue problem for the Hamiltonian.

2 Decomposition of the Hamiltonian on the Bloch waves

2.1 Description of the system

From here on, we consider the following system:

• The electrons are independent from each other,

• Each electron is subjected to a L2unif scalar potential V (·).

The independence assumption gives us the explicit form of the Hamiltonian operator:(HΨ

)(x) =

(−∆

2+ V (x)

)Ψ(x),

described in atomic units. Because of the periodicity of the lattice R, the potential is also periodic:

∀x ∈ Rd, ∀R ∈ R, V (x + R) = V (x).

L2unif(Rd) is defined as the space of functions v that verify

supx∈Rd

∫B(x,1)

v2 <∞,

where B(x, 1) is the ball of radius 1 around x.To properly define the Hamiltonian operator, its explicit form has to be completed by the defini-

tion of the domain of the operator, that is, a dense subset of H. According to Rellich-Kato theory,with these assumptions, the Hamiltonian is a self-adjoint operator on the domain

D(H) = H2(Rd).

Hence, its spectrum is real. However, its spectrum can very well be a continuous subset of R. Tocharacterise the spectrum in a more precise way, we first introduce a set of functions, the Blochwaves, that play a particular role in the study of the Hamiltonian.

2.2 Bloch waves

We have defined the Hamiltonian on the domain H2(Rd), on which it is self-adjoint. In what follows,we derive formally a convenient basis of eigenfunctions, the Bloch waves, without consideration forthe well-posedness of such a problem, or their belonging to the domain of the operator. However,this “physicist’s” approach is useful to give some intuition for the rigorous presentation done insections 2.4 and 2.5.

Let us define the translation operator τR, for R ∈ R, as

τRf(x) = f(x−R), ∀f.

The translation operators satisfy τRτR′ = τR′τR, and τ∗R = τ−R = τ−1R . Then the periodicity of the

potential can be seen as a commutation relation, as

τR (V (·)Ψ(·)) (x) = V (x + R)Ψ(x + R) = V (x)Ψ(x + R) = ((V (·)τR) Ψ) (x).

Of course, the same applies to the Laplacian operator ∆, and so we conclude that the Hamiltoniancommutes with translations along vectors of the lattice,

τR

(HΨ

)= H (τRΨ) .

In analogy with the finite-dimensional cases, where commutation between matrices implies that thematrices are co-diagonalisable, we consider here that the commutation between the Hamiltonian and

5

the each of translation operators (as well as the commutation of translation operators between eachother) indicates that these co-diagonalisable, i.e. that there exists a Hilbert base of orthonormalvectors that diagonalises them both. It follows that for such an eigenstate Ψ of the Hamiltonian, wehave

HΨ = εΨ

τRΨ = λ(R)Ψ.

A natural property of the translation operator is that τRτR′ = τR′τR = τR+R′ , implying that theeigenvalues of the translation operators satisfy

λ(R)λ(R′) = λ(R + R′).

Moreover, we have λ(0) = 1, and |λ(R)| = 1 because a translation along 0 is the identity, andtranslation operators are unitary, hence have eigenvalues of modulus one. Along each primitivevector aj , we then have

λ(aj)n = λ(naj),

which means that we can choose numbers kjdj=1 in [0, 1)d, such that λ(aj) = e2πikj . We recall

that aj · bi = 2πδij , so, for any lattice vector R =∑dj=1 njaj , denoting k =

∑di=1 kibi,

λ(R) =

d∏j=1

λ(aj)nj = exp

d∑j=1

2πikjnj

= eik·R.

It follows that the eigenstate Ψ is such that

Ψ(x + R) = eik·RΨ(x).

Let us now defineu(x) = e−ik·xΨ(x).

This function is R−periodic, as

u(x + R) = e−ik·xe−ik·RΨ(x + R) = e−ik·xe−ik·Reik·RΨ(x) = e−ik·xΨ(x) = u(x).

Definition 4. A Bloch wave is an eigenfunction of the translation operators τRR∈R, that is, afunction of the form

Ψk(x) = eik·xu(x),

where u is a R-periodic function,

Rigorously, Bloch waves are not in the domain of the Hamiltonian operator. For this reason, theyare not actual eigenfunctions. However, they can be recombined to give elements of the domain.This is similar to the Fourier decomposition, in which one “diagonalises” the Laplacian operator onthe Fourier basis, which are then recombined to give elements of the proper domain, which is licitbecause the Fourier Transform is a unitary operator. In the same way, these formal calculations willbe made rigorous by the introduction of the Bloch Transform in section 2.4.

For now, we stay within a formal setting, in which the previous calculations hint at a resultknown as Bloch’s theorem.

Bloch’s “Theorem”. The functions of a Hilbert space H, defined on a domain V, endowed witha lattice R, can be expressed as a combination of Bloch waves

Ψn,k = eik·xun,k(x), un,k is R-periodic ,k ∈ B,

and each Ψn,k is an energy eigenstate, HΨnk = εn,kΨk.This result is useful in practice, as it suggests the Ansatz eigenstates eik·xu(x), where u is

R-periodic. We now consider the action of the Hamiltonian on such eigenstates.

H(eik·xu(x)

)= −∆

2

(eik·xun,k(x)

)+ V (x)

(eik·xu(x)

)= eik·x

(−∆

2− ik · ∇+

|k|2

2

)u(x) + V (x)

(eik·xu(x)

).

6

Hence, it is natural to define the family of operators

Hk =1

2(−i∇ + k)2 + V (x), ∀k ∈ B,

acting on the R-periodic part of Bloch waves.We have therefore reduced the problem of determining eigenstates on the whole space to deter-

mining eigenfunctions of a modified operator on the primitive cell A for each k.This operator is self-adjoint on a well-chosen domain, and has compact resolvent. The mathe-

matical aspects are detailed in section 2.5. This implies that each Hk has a sequence of eigenvaluesdiverging to +∞ that are bounded below. We can order them, and denote them

ε1(k) ≤ ε2(k) ≤ · · ·

These εn(·) are also continuous and R′-periodic functions of k.

2.3 Band Structure

The behaviour of the eigenvalues of Hk, which we previously denoted εn(k), contain some meaningfulphysical information. Thus, it is useful to plot their variation with k. Each k → εn(k) is called aband, and they are usually plotted in 1D, even if we are considering a system of greater dimension.This is done by choosing an appropriate path in the Brillouin zone, and representing the value ofeach band along this path. The structure of the bands determines some physical properties. Animportant feature is their position with respect to the Fermi level.

Definition 5. The Fermi level εF is the thermodynamic work required to add one electron to thesystem.

The Fermi level has a

Proposition 2. The Fermi level εF verifies the following relation to the number of electrons in theunit cell N

N =∑n

∫B1(εn(k) ≤ εF ) dk.

Because the electrons follow Schrodinger’s equation, their energy can only jump from one bandto another, and a band is said to be occupied if an electron has that energy. The significance of theFermi level in the context of solid-state physics is that the ground state energies of the electrons areon the bands below the Fermi level.

Electric properties of the crystal on the macro-scale are given by the response to low energyexcitations (small perturbations). Hence, we can restrict the study to a small window of energyaround the Fermi level.

For example, if the Fermi level is in a wide gap between the lowest group of bands and the upperones, it indicates that the system is an insulator.

Figure 3: Band Structure of Silicon crystal

7

An example of such a band structure is given in Figure 3, for the Silicon crystal. The energybands are represented on a 1 dimensional path in the Brillouin Zone, starting with Γ, the origin. Onthe y-axis, the energy is expressed relatively to the Fermi level (set to 0).

The electronic structure of a Silicon atom is [Ne]3s23p2, which means that it has four valenceelectrons on the 3s and 3p subshells, and [Ne] denotes the electronic structure of Neon, that containsthe 10 core electrons.

The Silicon crystal has a diamond cubic structure, and hence, each unit cell contains two atoms.Hence, there are 8 valence electrons in each unit cell, filling 4 valence bands (as each contains 2electrons of opposite spin).

The lowest 4 bands are the valence bands, the higher bands are the conduction bands (notoccupied in the ground state).

2.4 Bloch Transform

To motivate the definition of the Bloch Transform, we start by a formal calculation showing itsrelationship to the Fourier Transform. We define the Fourier Transform of a function Ψ ∈ L2(Rd)as the function Ψ ∈ L2(Rd) that verifies

Ψ(x) =

∫Rd

Ψ(ξ)e−iξ·x dξ.

It satisfies the reciprocal Fourier identity

Ψ(x) =

∫Rd

Ψ(ξ)eiξ·x dξ.

In our case, the space is endowed with a Bravais latticeR, and the frequency space with the reciprocallattice R′. Hence, a natural decomposition for ξ is

ξ = k + K, k ∈ B,K ∈ R′.

Substituting into the reciprocal Fourier identity,

Ψ(x) =∑

K∈R′

∫k∈B

eik·xΨ(k + K)eiK·x dk.

By exchanging sum and integral, we can group together the dependence in K,

Ψ(x) =

∫k∈B

eik·x∑

K∈R′

Ψ(k + K)eiK·x dk.

Let us now pose

uk(x) =∑

K∈R′

Ψ(k + K)eiK·x.

We also poseΨk(x) = eik·xuk(x).

Note the direct relation with the Bloch waves in section 2.2, which were defined as common eigen-functions of all the translation operators. This property can in fact be verified here as well. First,we remark that

τReiξ·x = ei(k+K)·(x−R) = e−iK·Re−ik·Rei(k+K)·x = e−ik·Reiξ·x.

Hence, all the x 7→ ei(k+K)·x, for any K ∈ R′, are in the same eigenspace of τR. This readily extendsto the periodic part uk, and to the Bloch wave Ψk. Hence, we have in fact grouped together theparts of the integrand that are in the same eigenspace of the operator τR.

We have therefore decomposed Ψ as a continuous collection (indexed by k) of functions on A,or equivalently, as a single function of the two variables (r,k) on (A× B).

To state the following theorem, we remind the reader that the Schwartz space is defined as follows

S(Rd) =

f ∈ C∞(Rd)

∣∣∣∣∣∣ supx

∣∣∣∣∣∣(1 + ‖x‖2n) d∏

j=1

∂ij

∂ijxijj

f(x)

∣∣∣∣∣∣ <∞ ∀n, i1, i2, . . . , id ∈ N

.

8

We also define

P(A× B) =

u ∈ C∞(A× B)

∣∣∣∣ u(r + R,k) = u(r,k) ∀R ∈ R,

u(r,k + K) = e−iK·ru(r,k) ∀K ∈ R′

Theorem 1. (Bloch Transform) For Ψ ∈ S(Rd), the Bloch Transform of Ψ, denoted u, is thefunction of P(A× B) such that

u(r,k) =∑R∈R

e−ik·(r+R)Ψ(r + R), ∀r ∈ A, ∀k ∈ B.

Moreover, it verifies ∀r ∈ A, ∀R ∈ R,

Ψ(r + R) =

∫k∈B

u(r,k)eik·(r+R) dk.

We denote the Bloch Transform operator

U : S(Rd)→ P(A× B)

Ψ 7→ u

and its inverse

U−1 : P(A× B)→ S(Rd)u 7→ Ψ

Proof. The expression of the Bloch transform in terms of the real-space sum derives from the Poissonsummation formula, ∑

R∈R

f(R) =∑

K∈R′

f(K).

Applying this formula for f(R) = e−ik·(r+R)Ψ((r + R)), and computing its Fourier transform

f(K) =(δk ∗ Ψ

)(K)eiK·r = eiK·rΨ(K + k),

it follows that

u(r,k) =∑

K∈R′

Ψ(k + K)eiK·r

=∑R∈R

e−ik·(r+R)Ψ(r + R).

Theorem 2. The operators U and U−1 can be extended to unitary operators, such that

U : L2(Rd) → L2(A× B)

U−1 : L2(A× B)→ L2(Rd).

Moreover, we haveU−1 = U∗, U =

(U−1

)∗,

where U∗ is the adjoint of U .

For a proof of this result, see [3].

9

2.5 Decomposition of the Hamiltonian into the HkkLet us define the family of operators

Hk =1

2(−i∇ + k)2 + V (x),∀k ∈ B,

acting on a domain of L2(A).

Lemma 1. • Hk is self-adjoint on D.

• Hk is essentially self-adjoint on D0.

• If λ is not in the spectrum of Hk, the resolvent (Hk − λI)−1

is compact.

• If Im(λ) 6= 0 or λ < −‖V (x)‖∞, then λ is not in the spectrum of Hk.

For a proof, we refer the reader to [3].These results have proven that Hk has compact resolvent, hence its spectrum is a sequence of

eigenvalues diverging to +∞, and it is bounded below, with

σ (Hk) ⊂[− sup

x|V (x)| ,+∞

).

Let us order the eigenvalues and denote them by

e1(k) ≤ e2(k) ≤ . . . ,

where the dependence in k is denoted functionally, for the eigenvalues have good properties asfunctions of k.

Proposition 3. For all n, en(k) is continuous in k and R′-periodic. Moreover,

en(k) −−−−−→n→+∞

+∞, where the limit is uniform in k.

Theorem 3. If V is a Lpunif(Rd) function that is R-periodic, withp = 2 if d ≤ 3

p > 2 if d = 4

p = d2 if d ≥ 5.

Then H is decomposed by the Bloch Transform into the Hkk, , i.e.(UHU−1

)un,k = Hkun,k

and the spectrum of H = −∆2 + V is the union of the spectra of the Hk for all k ∈ B. That is,

σ(H) = en(k) | k ∈ B, n ∈ N .

Remark 1. A more general framework for this kind of decomposition uses the notion of directintegrals, which is an extension of the decomposition by direct sums. Using this notation, we wouldwrite

UHU−1 =

∫ ⊕BHk dk.

For more information on direct integrals, see [12].

3 Wannier functions

From here on, we denote the Bloch waves

Ψn,k(x) = eik·xun,k(x),

where we add the band index n, the number of the corresponding eigenvalue εn, i.e.

HkΨn,k = εn(k)Ψn,k.

10

Moreover, we normalise the Bloch waves according to the condition∫r∈A|Ψn,k(r)|2 dr = 1.

We remind the reader of the Dirac notation, which denotes by |.〉 an element of the Hilbert space (a”ket”), and by 〈.| an element of the dual space (a ”bra”).

In practice, we are given a set of such Bloch waves by a DFT code, which determines theirR-periodic part so that they are eigenvalues of the modified Hamitonian Hk of the system we arestudying.

3.1 Wannier functions for an isolated band

In this section, we consider the case of an isolated band εn, such that

εm(k′) 6= εn(k), ∀m 6= n, ∀k′ 6= k.

Definition 6. (Wannier Functions) Using the previous notation, for R ∈ R, we let

wn,R(x) =

∫k∈B

e−ik·RΨn,k (x) dk.

Let us remark that this is the inverse Bloch transform of the Bloch wave with a change of phase.In the special case R = 0, we obtain

wn,0 =

∫k∈B

un,k(x) dk.

Moreover, it follows from the Fourier duality between translation and exponential modulation that

wn,R(x) = wn,0(x−R).

For convenience, we will use the Dirac notation |Rn〉 = wn,R. The inverse Fourier transform entailsthat

|Ψn,k〉 =∑R∈R

eik·R |Rn〉 .

As the Bloch transform is unitary, it follows that the |Rn〉 form an orthonormal set. It is thusstraightforward to see the unitary character of the transformation from Bloch waves to Wannierfunctions. The equivalence between the two formulations is made clear by the expression of theprojection operator on the eigenspace associated with band n

Pn =

∫k∈B|Ψn,k〉 〈Ψn,k| dk =

∑R∈R

|Rn〉 〈Rn| .

Let us note that the Wannier functions are not eigenvalues of the decomposed Hamiltonian, andhence do not have a fixed energy, in contrast to the Bloch waves.

Theorem 4. If the Bloch functions k 7→ Ψn,k are complex analytic on a strip Ωβ,

Ωβ =

z ∈ Cd

∣∣∣∣|Im(zi)| < β, ∀1 ≤ i ≤ d,

then the Wannier functions are exponentially localised, i.e. x 7→ eβ‖x‖wn,R(x) are L2(Rd).

This can be interpreted as trading localisation in energy for localisation in space. For a proof ofthis result, see [10].

The Wannier functions are not uniquely defined. Indeed, we can define the alternative Blochwave ∣∣∣Ψn,k

⟩= eiφn,k |Ψn,k〉 ,

where φn,k is a real R′-periodic function, called a gauge. There is no preferential gauge stemmingfrom the physics of the system. This other Bloch wave would result in different Wannier functions,and the choice of gauge would impact the localisation of the Wannier functions. Hence, one wouldlike to choose a gauge that ensures the regularity of the Bloch waves, so as to have well-localisedWannier functions. We come back to this point in later sections.

We have assumed here that the band n is isolated. This means that the eigenvalue en(k) is neverdegenerate, so that the associated eigenspace is always one-dimensional, ensuring the existence ofa regular Bloch wave, and hence guaranteeing the localisation of the associated Wannier function.However, one can also define regular Bloch functions for an isolated set of bands, which is the subjectof the next section.

11

3.2 Composite Wannier functions

We now consider a set of the N lowest bands

σN = en(k), k ∈ BNn=1 ,

and we assume that this set of bands is isolated from the others:

dist(σN , em) > 0, ∀m > N.

There is no loss in generality in assuming that we consider only the lowest bands, as we took themfor convenience of notation. However, taking the lowest bands also carries some physical importance,because it corresponds to the case of an insulator.

Naturally, there is a set of Bloch waves|Ψm,k〉

Nm=1

, each |Ψm,k〉 being associated with a bandem(k). However, these Bloch waves are not regular in general, especially when two bands intersect.

To solve this issue, we extend the definition of the Bloch waves for such a multi-band set, byallowing a mix of eigenstates of different energies, leaving more degrees of gauge freedom. For thesegeneralised Bloch waves, the choice of a phase becomes the definition of a unitary matrix for eachpoint of the Brillouin zone, denoted U(k) ∀k ∈ B, with

∣∣∣Ψn,k

⟩=

N∑m=1

|Ψm,k〉U (k)mn.

Let us remark that these generalised Bloch waves are no longer eigenstates of the Hamiltonian, butthey still span the same vector space, as shown by the form of the spectral projector

Pk =

N∑n=1

|Ψn,k〉 〈Ψn,k| =N∑n=1

∣∣∣Ψn,k

⟩⟨Ψn,k

∣∣∣ .The definition implies that the generalised Bloch waves are orthonormal. In what follows, we call

the basis set|Ψm,k〉

Nm=1

a Bloch frame, and the gauge freedom means that we are just choosingan appropriate change of orthonormal basis to ensure the regularity of the Bloch frame with respectto k.

With the previous notation, we define the Wannier functions associated to the∣∣∣Ψn,k

⟩by

|Rn〉 =

∫k∈B

e−ik·RN∑m=1

U (k)mn |Ψm,k〉 dk.

The choice of the gauge, i.e. the unitary matrix U(k) for each k remains, and is the subject of thefollowing sections.

4 Topological obstruction to the existence of Wannier func-tions

We now discuss the regularity of the Bloch waves in k. Although this question does not seem toodifficult at first, it has only been solved within the framework of topology. It may seem surprisingto need such tools, but framing the problem differently makes the connection more straightforward.Indeed, the condition of (quasi-)periodicity of the Bloch waves endows the Brillouin zone with thetopology of a torus, to which the eigenspace associated with the bands ε1, . . . εN is added, generatingthe structure of a fibre bundle.

A simple example of this kind of structure is the Mobius strip, in Figure 4. It can be decomposedas a simple line (the fibre) relying on a circle. In this case, if we consider the real vector space onthe fibre, and our problem is to find a regular normed vector that spans the fibre. As we consideronly a real vector space, our only choice is the orientation of the tangent vector. However, becauseof the topology of the Mobius strip, after one turn around the circle, the vector is upside-down. Itis thus impossible to have a normed continuous basis of the associated fibre bundle.

In bundle theory, the question of finding a gauge (that is, a change of basis on each fibre) thatguarantees the regularity of the |Ψn,k〉 with respect to k amounts to what is called the triviality ofthe bundle (see [11]).

12

Figure 4: Mobius strip as a fibre bundle

4.1 Some elements of bundle theory

First, we define a few bundle-theoretic concepts.

Definition 7. Let E, B, and F be topological spaces, where B is connected. Let π : E → B be acontinuous surjection.

The structure (E,B, π, F ) is a fibre bundle if it admits a local trivialisation, as defined below.The space B is called the base space of the bundle, E the total space, and F the fibre. The map π iscalled the projection map (or bundle projection).

A bundle admits a local trivialisation if, for every x ∈ E, there is an open neighbourhood U ⊂ Bof π(x) such that there is a homeomorphism φ : π−1(U)→ U × F , with

∀y ∈ π−1(U), projU (φ(y)) = π(y).

where for (u, f) ∈ U × F , projU ((u, f)) = u. It means that locally, the bundle can be identified tothe product space of the base and the fibre.

In the case of the Mobius strip, the base space is the circle, the fibre is the orthogonal line, andit is easy to see that it admits a local trivialisation, as it is possible to flatten any small part ofthe strip. However, we cannot flatten the whole strip: it does not admit a global trivialisation, asdefined below.

Definition 8. A fibre bundle is called trivial if it admits a global trivialisation, that is, a homeo-morphic map

φ : π−1(B)→ B × F,which means that the fibre bundle is equivalent to a global Cartesian product B × F .

An example of a trivial bundle is the cylinder, which can be written as a global Cartesian product.More structure can be added to fibre bundles.

Definition 9. A vector bundle is a fibre bundle for which the fibre can be endowed with a vectorspace structure.

A fibre bundle is smooth if E and B are smooth manifolds, π is a smooth map, and the localtrivialisations are diffeomorphisms.

A smooth complex vector bundle is Hermitian if it admits a smooth field of Hermitian innerproducts.

4.2 The Bloch bundle

Let Pkk∈B be the spectral projector on the span of the Bloch waves. We show that the spectralprojectors satisfy three conditions.

Proposition 4. • Analyticity: the map k 7→ Pk is real-analytic on RN .

• τ -covariance: the map k 7→ Pk satisfies:

Pk+K = τ−1K PkτK, ∀K ∈ R′

for τ the unitary translation operators.

13

• Time-reversal symmetry: the map k 7→ Pk satisfies

P−k = ΘPkΘ−1,

for an antiunitary operator Θ such that Θ2 = ±I.

Proof. • Let γ be a complex contour around the energy bands, as per Figure 5. Then the Riesz

Figure 5: Contour enclosing the lower energy bands of Silicon

formula applied to the projector on the enclosed bands gives

Pk =1

2πi

∮γ

(z −Hk)−1

dz.

As Hk is polynomial in k, (z −Hk)−1 is analytic, and therefore Pk is analytic.

• First, remark that Hk is τK-covariant, in the sense that

τ−1K HkτK = Hk+K, K ∈ R′.

This property is readily transmitted to PK, so that

τ−1K PkτK = Pk+K, K ∈ R′.

• In the case of particles without spin, we have Θ2 = 1, Θ is just the complex conjugation, andthis property means that conjugating a plane wave is like reversing time. This property iscalled bosonic time-reversal symmetry.

If spins are taken into account, one can show that Θ2 = −1, and the property is called fermionictime-reversal symmetry. See [11] for a more precise derivation.

Definition 10. The Bloch bundle associated with the Pkk∈B is the Hermitian vector bundle overthe Brillouin zone B, with fibre Ran(Pk) over each point k.

In section 3.1 and 3.2, we stated results for a frame of Bloch functions that is complex analyticin k. Such a frame gives in fact a trivialisation of the Bloch bundle, since this frame can be used tountwist the fibre, giving a homeomorphic equivalence between the bundle and the Cartesian productB × Ran(P0).

4.3 The Chern numbers

There are characteristic numbers that determine whether a Hermitian bundle is trivial: the Chernnumbers. For simplicity of presentation, we restrict ourselves to the case d = 2, the only applicationin this report.

In that context, the first Chern number (the only relevant one) of the Bloch bundle is expressedas

c1(P ) =1

2πi

∫B

Tr (Pk[∂1Pk, ∂2Pk]) dk1dk2,

14

Figure 6: Schematic representation of the Brillouin Zone

where Pk is the projector of section 4.2.It is possible to interpret the first Chern number as a topological obstruction, by the following

method. To build a gauge that makes the Bloch frame continuous and τ -covariant, we proceedconstructively.

• Start by a continuous Bloch frame k 7→ Ψk inside of the Brillouin Zone. The problem is onlythe periodicity.

• First, we find a choice of gauge that makes the Bloch frame periodic on edge 1 (E1 on thefigure). By periodicity of the Hamiltonian, the Bloch frame at v1 and v2 spans the same vectorspace, but the frames are not equal in general. Two bases of the same complex vector spacebeing related by a unitary transformation, there exists an obstruction matrix Obs ∈ U(N),such that

Ψv1 = Ψv2Obs∗,

where A∗ denotes the conjugate transpose of A. Because the group of unitary matrices U(N) ispath connected, it is always possible to unwind the obstruction. This can be done for examplewith the matrix power

Φk = ΨkObsk1 ,

with the convention k1 = 0 at v1, and k1 = 1 at v2.

• We apply the same method to the second edge E2.

• Define the frame on the other edges E3 and E4 to satisfy the condition of τ -covariance, statedin 4.2. It follows that the modified frame Φk is continuous on the boundary. It remains to seewhether it can be extended to the interior of the Brillouin Zone.

In the simplest case, we take the example of U(1): the parametrisation ei2πtt∈[0,1] of a circleis smooth, and yet, it cannot be extended continuously to the disk.

Imagine that we have such a parametrisation z 7→ e2πiθ(z), with θ(e2πit) = t. Consider all thepaths between z = 1 and z = −1. By the theorem of intermediary values, the set

R1 =z∣∣ Re(e2πiθ(z) = 0

is a curve that separates the disk into two parts. Remark that z = i and z = −i both belong toR1. However, the imaginary part must go from 1 to −1 along R1. Hence, there is a point such thatboth the imaginary and the real parts are 0, which contradicts our hypothesis.

In more topological terms, the parametrisation of the circle is not homotopically trivial, thatis, it cannot be deformed continuously to a single point. Hence, to know whether we can extendthe gauge to the interior of the domain, it suffices to count the number of turns it does along thecontour, the winding number.

Definition 11. Let z(t), t ∈ [0, 1] be a parametrisation of the complex unit circle. The windingnumber, noted deg(z) of this parametrisation reads

deg(z) =1

2πi

∮t∈[0,1]

z′(t)

z(t)dt.

15

Proposition 5. The winding number is an integer.

Proof. Let z(t) = e2πiθ(t) be a parametrisation of the unit circle. Then

e2πiθ(0) = e2πiθ(1), i.e. θ(0) ≡ θ(1)[1].

Moreover,

deg(z) =1

2πi

∮t∈[0,1]

z′(t)

z(t)dt

=1

2πi

∮t∈[0,1]

2πiθ′(t)dt = θ(1)− θ(0).

It follows that deg(z) ∈ N.

This definition can be generalised to unitary gauges of higher dimension. In fact, the topologyof U(N) can be reduced by the following decomposition:

U(N) ' SU(N)× U(1),

where ' denotes an isomorphic relation.In simpler words, one can decompose any unitary matrix uniquely into a special unitary matrix

(of determinant 1), and an element of U(1), its determinant. This is useful because SU(N) is simplyconnected, which means that it is homotopically trivial, i.e. any closed path in SU(N) can bedeformed to a single point. However, U(1) is not simply connected, as closed paths with differentwinding numbers are not homotopically equivalent.

This observation suggests that the topology of the Bloch bundle depends only on the determinantof the gauge, which is confirmed by the following characterisation of the first Chern number.

Proposition 6. If U(·) is a gauge that makes the Bloch frame continuous and (quasi)-periodicon the boundary of the Brillouin Zone, then the Chern number of the associated Bloch bundle (ofprojector P ) is the winding number of its determinant.

c1(P ) = deg (det (U(·))) =1

2πi

∮∂B∂k(det(U(k))) dk

=1

2πi

∮∂B

Tr(U(k)−1

∂kU(k))

dk

Hence, we can compute the Chern number of the Bloch bundle as the topological obstruction tothe extension of a continuous Bloch frame on the boundary to the interior of the Brillouin Zone (see[8]).

5 Numerical construction of Wannier functions and applica-tion

Wannier functions can be built numerically, based on ab-initio computations that provide a basisof Bloch waves on a grid of the Brillouin Zone. Then localising the Wannier functions is equivalentto choosing a regular gauge of these Bloch waves, a post-processing step that requires much lesscomputational power than ab-initio calculations, and yet has very useful properties.

5.1 Marzari-Vanderbilt localisation functional

For Wannier functions |wn,R〉 = |Rn〉, we define the localisation functional

Ω(〈0n|n) =

N∑n=1

〈0n| r2 |0n〉 − 〈0n| r |0n〉2 .

It measures the quadratic spreads of the Wannier functions around their centres. This functional iswell-defined only for appropriately localised Wannier functions, which we assume in this section.

16

One can split the localisation into two parts

Ω = ΩI + Ω,

where

ΩI(〈0n|n) =∑n

(〈0n| r2 |0n〉 −

∑Rm

| 〈Rm| r |0n〉 |2),

andΩ(〈0n|n) =

∑n

∑Rm 6=0n

| 〈Rm| r |0n〉 |2.

ΩI can be shown to be gauge-invariant, by rewriting its components in terms of projection operators.Moreover, both ΩI and Ω are positive definite (see [6] for a more in-depth presentation).

Each part of the localisation functional can be alternatively written in the k-space representation.Using the fact that

〈Rn| r |0m〉 = i|B|

(2π)3

∫eik·R 〈un,k| ∇k |um,k〉 dk,

and

〈Rn| r2 |0m〉 = − |B|(2π)3

∫eik·R 〈un,k| ∇2

k |um,k〉 dk,

one can rewrite Ω(〈0n|n) only in terms of un,k. This form makes it clear that the regularity ofthe Bloch frame is directly linked to the localisation of the Wannier functions. Moreover, a finite-difference scheme yields a straightforward discrete version of the functional. Then, using a gradientdescent in the space of unitary matrices, one can refine the gauge choice to produce a more localisedset of Wannier functions.

In [10], it is shown that the minimisers of the localisation functional are maximally-localisedWannier functions (almost exponentially localised functions), and a smooth Bloch frame k 7→ un,k.The rationale is that the algorithm should converge to these minimisers, provided we have a goodinitial guess that satisfies Ω <∞.

In the original paper [7], the initial guess is obtained by a projection method. Reference orbitalsare given as a manual input, and result from physical considerations. They are then projectedonto the eigenspace of the Hamiltonian, and renormalised. The issue with this method is that it isneither systematic, nor fully reliable: it can produce initial guesses that do not allow the procedureto converge.

5.2 Application: Wannier interpolation

Finding an analytic frame of Bloch functions on the Brillouin Zone (or equivalently, exponentially-localised Wannier functions) is useful in practice, for example to construct accurate band diagramswith sparse sampling. This can be done through Fourier interpolation of the Hamiltonian operator.

5.2.1 Fourier interpolation

Let f be a periodic function of [0, T ]. Without loss of generality, we take T = 2π here.Assume that we are given the values of f on a set of N equispaced points tj = 2πj

n , 0 ≤ j ≤ n−1.To produce a periodic interpolation of this function, it is natural to consider approaching it by aFourier series.

p(x) =∑k

ckeikx.

By changing variables z = eix, the interpolation becomes

q(z) =

K∑k=−K

ckzk.

This series contains both positive and negative indices k, and multiplication by zK gives a polynomialof degree 2K + 1. Hence, the interpolation problem amounts to a polynomial interpolation ofz 7→ zKf(arg(z)) on the complex unit circle.

Because the values of the function are only known on N points, the interpolation polynomial isuniquely determined if its degree is equal to N . However, if N is even, this is not possible, and wewill have a free parameter.

If N is odd, the interpolation can be easily computed with the Discrete Fourier Transform.

17

Definition 12. Discrete Fourier Transform: To a set of N complex numbers x0, . . . xN−1, theDiscrete Fourier Transform associates the complex numbers X0, X1, . . . XN−1

Xk =

N−1∑j=0

xj · e−i2πjkN .

Proposition 7. Let (Y0, Y1, . . . YN−1) be the Discrete Fourier Transform of the data points

Yk =

N−1∑j=0

f(tj) e−i2π jk

N .

Then p(x) =1

N

K∑k=−K

Yk eikx satisfies p(tj) = f(tj), for all 0 ≤ j ≤ N−1.

The proof is quite straightforward, by plugging the value of Yk in the expression of p and ex-changing the summations.

This result can be adapted to the case N even, for example with the Dirichlet kernel.

5.2.2 Application to band interpolation

The naive way to construct a band diagram is to calculate and order the eigenvalues of the Hamilto-nian, and then to interpolate between these values. However, when eigenvalues collide, this methodis not optimal, as the eigenvalues exhibit cusps at the collisions, which makes them locally non-differentiable, thus greatly reduces the precision of interpolation, because of a Gibbs phenomenonon the derivative.

There is however a way to systematically work around this issue (and increase the overall precisionin so doing). Let us define some useful notation to that extent.

The periodic operator Hk is expressed in the basis of the Bloch waves as

Amn(k) = 〈um,k|Hk |un,k〉 .

If k 7→ uk is smooth and periodic, then k 7→ A is smooth and periodic as well, and in particular,it does not have cusps. Thus, a Fourier interpolation of the coefficients of A(k) does not exhibit theGibbs phenomenon, which yields a very accurate band diagram for very few points.

Figure 7 presents a result from [13] that computes such an interpolation for a metal with only5 points in the Brillouin Zone, that shows a remarkable agreement to the referential and computa-tionally intensive ab-initio method.

Figure 7: Wannier Interpolation for metallic body-centred cubic Fe.

18

Part II

Construction of continuous Bloch framesWe have seen previously that localising Wannier functions amounts to regularising the correspondingBloch frame, and that one can apply the Marzari-Vanderbilt procedure to make the frame incre-mentally smoother. However, this procedure requires that the localisation functional for the initialguess be well-defined.

In this section, we present an algorithm that builds a continuous Bloch frame, which can thenbe used as a robust initial guess for the Marzari-Vanderbilt algorithm.

6 Logarithm algorithm

This method was proposed in [2]. It constructs a continuous Bloch frame iteratively, unwinding theobstruction to periodicity on boundaries of increasing dimension. We present here the 1D and 2Dcases, but the method is readily generalised to higher dimensions.

6.1 Algorithm in 1D

In 1D, there is no topological obstruction to building a continuous frame. The following procedureis therefore always valid.

We present the discrete procedure below. The Brillouin Zone is discretised with a step size ofδk. The input Bloch frame is denoted vjδkj . We first build a regularised frame ujδkj , and thencorrect it to be periodic.

1. Choose u0 = v0.

2. Project u0 on Ran(vδk) and renormalise to obtain uδk.

uδk = ‖Pδku0‖−1Pδku0,

where Pk denotes the projector on Ran(vk).

3. Iterate the projections until (j + 1)δk = 1.

u(j+1)δk = ‖P(j+1)δkujδk‖−1P(j+1)δkujδk,

4. u0 can be different from u1, but they span the same space. Hence, we can find a unitarymatrix Obs, such that u1 = u0Obs.

5. We can then unwind the frame byuk = ukObs−k.

Then k 7→ uk in the continuum limit is continuous and periodic.

6.2 Algorithm in 2D

We assume that the Brillouin Zone is discretised on a grid of step size δk1 along the first axis, andδk2 along the second axis. For simplicity, we take here δk1 = δk2 = δk, but the procedure is general.As before, the input is a Bloch frame vk1,k2 .

1. Apply the 1D algorithm to v0,k2 , and put the result in u0,k2 .

2. Propagate uk1,k2 as before, by

u(j+1)δk,k2 = ‖P(j+1)δk,k2ujδk,k2‖−1P(j+1)δkk2ujδk,k2 ,

until (j + 1)δk = 1. Then uk1,k2 is k2-periodic, but not k1-periodic.

3. Define an obstruction matrix such that u1,k2 = u0,k2Obs(k2).

4. Build a continuous interpolation (homotopy) Obs(k1, k2) s.t.

Obs(0, k2) = I, Obs(1, k2) = Obs(k2).

19

5. Unwind the frame byuk1,k2 = Obs(k1, k2)∗uk1,k2 .

To ensure the continuity of the result, we need to devise a systematic construction of a continuousinterpolation map (k1, k2) 7→ Obs(k1, k2).

The first approach to perform such an interpolation is to use the matrix exponential and thelogarithm, as follows

Obs(k1, k2) = Obs(k2)k1 = exp (k1 log Obs(k2)) .

However, the logarithm has an intrinsic discontinuity, called a branch-cut, when an eigenvaluemakes a full turn around the complex unit circle. As an extension to the usual definition of thecomplex logarithm, it is discontinuous if the eigenvalues pass through −1.

0.4 0.2 0.0 0.2 0.4k

4

3

2

1

0

1

2

3

4

Eig

envalu

es

of

L

Figure 8: Discontinuous eigenvalues of the logarithm of an obstruction matrix path

A natural idea to attempt to solve this issue is to use the phase freedom of the exponential, byadding or subtracting 2π to the logarithm so as to ensure continuity.

0.4 0.2 0.0 0.2 0.4k

4

3

2

1

0

1

2

3

4

Eig

envalu

es

of

L

Figure 9: Fixed eigenvalues of the logarithm of an obstruction matrix path

20

This procedure produces a continuous logarithm on the interior of the Brillouin Zone, but doesnot necessarily create a continuous and periodic logarithm. Indeed, eigenvalues of the obstructionmatrix can make turns, which is the case if the Chern number is non-zero. In this case, it isimpossible to have a continuous and periodic logarithm.

Even when the Chern is zero, if the top eigenvalue of the logarithm collides with the lowest one(up to 2π), then one cannot find a continuous and periodic branch of the logarithm.

In both cases, continuity and periodicity of the eigenvalues are irreconcilable.We are not considering cases with non-zero Chern number here, but the second case is problematic

for the logarithm method. If there are eigenvalue collisions in an application, the method fails. Thisis precisely what happens in the Kane-Mele model in the QSH phase.

6.3 The Kane-Mele model

This model, first proposed in [5], is a toy model of a topological insulator, where a 2D crystal withtwo electrons is considered. Hence, its Hamiltonian can be written in C4, and for a certain range ofparameters, it exhibits a Quantum Spin Hall effect.

Hk =

5∑a=1

da(k)Γa +

5∑a<b=1

dab(k)Γab,

where Γab = [Γa,Γb], and Γa are the Dirac matrices (σx ⊗ I,σz ⊗ I,σy ⊗ sx,σy ⊗ sy,σy ⊗ sz), σj

and sj Pauli matrices of sublattice and spin. The Pauli matrices are

σx = sx =

(0 11 0

), σy = sy =

(0 −ii 0

), σz = sz =

(1 00 −1

).

d1 t(1 + 2 cosx cos y) d12 −2t cosx sin yd2 λν d15 λSO(2 sin 2x− 4 sinx cos y)d3 λR(1− cosx cos y) d23 −λR cosx sin y

d4 −√

3λR sinx sin y d24

√3λR sinx cos y

Table 1: Non-zero coefficients of the Hamiltonian in k-space, for x = k1a/2 and y = sqrt3k2a/2.

Here, we fix the parameters a = 1, t = 1, λR = 0, λSO = 1, and only vary λν .

• For λν > 3√

3, the material is in a regular insulating phase, and one can see the gap betweenthe two lower eigenvalues and the two upper ones in Figure 10.

• For λν < 3√

3, the material is in the Quantum Spin Hall (QSH) phase, where it is stillinsulating in the bulk, but where the edge states are conducting. One can see the insulatinggap in Figure 11.

• For λν = 3√

3, the material is in a transitional metallic phase: the gap closes (see Figure 12),which means that the material is conducting.

In both insulating phases, the Chern number of the Kane-Mele model is 0. This is important,because models with non-zero Chern number (as the Haldane model, [4]) break the time-reversalsymmetry (defined in section 4.2), which is very hard to realise in practice. However, the Kane-Melemodel satisfies the time-reversal symmetry in both insulating phases, which seems to correspondbetter to our physical reality, and is thus closer to applications.

The topological nature of the QSH phase is different from the normal insulating phase. To de-scribe the topology of the Bloch bundle in this case, the authors of [5] introduce the Z2 characteristicindex (we refer the reader to the original paper [5]).

21

Figure 10: Band diagram of the Kane-Mele Hamiltonian at λν = 6: insulating phase

Figure 11: Band diagram of the Kane-Mele Hamiltonian at λν = 4: QSH phase

Figure 12: Band diagram of the Kane-Mele Hamiltonian at λν = 3√

3: metallic phase

22

7 Recursive interpolation algorithm

This section describes some new contributions, developed in the context of the internship.Building on the method of [2], we modify the interpolation method, so that it always produces

continuous frames when the Chern number is zero.

7.1 Recursive interpolation of a single matrix

Let, for z 6= −e1,

Φ(z) =

z1 −bz2 . . . −bzmz2

... δij − zizj1+z1

zm

, b =1 + z1

1 + z1.

Φ(z) is the unitary matrix that sends e1 to z, and that preserves Ran(e1, z)⊥. As a rotation matrix,its eigenvalues are never -1.

This map is a continuous injection of Cmr−e1 into U(m), that produces a unitary matrix withfirst column z.

7.2 Recursive interpolation of a matrix path

We start again at step 4 of the 2D algorithm described in section 6.2. At this step, we have obtaineda regular frame on the interior of the Brillouin Zone that is k2-periodic, but not k1-periodic, and wehave constructed a path of obstruction matrices on the edge k1 = 1 so that

u1,k2 = u0,k2Obs(k2).

In order to make the frame regular and periodic on the whole Brillouin Zone, we need to unwind theobstruction, that is, find a continuous interpolation map from the identity path to k2 7→ Obs(k2).

The input of this algorithm is therefore only Obs(k2), with possibly Obs(k2)e1 = −e1 for somek2.

1. Find R unitary such that RObs(k2)e1 6= −e1

2. Set Obs(k2) = RObs(k2), and z(k2) = Obs(k2)e1 its first column.

3. Then U1(k2) = Φ(z(k2)

)never has −1 as eigenvalue. We can take its logarithm, a build a

continuous interpolationU(k1, k2) = (R∗)k1U(k2)k1 .

which satisfies U(0, k2) = I,

U(1, k2)e1 = R∗Φ (RObs(k2)e1) e1 = Obs(k2)e1.

As U1 unitary,

U1(1, k2)∗Obs(k2) =

(1 00 ∗

).

4. Define U2(k2) = Obs(k2)∗U1(1, k2). It can be interpolated easily because it does not turn(Chern number c = 0).

5. Then U(k1, k2) = U1(k1, k2)U2(k1, k2) interpolates continuously between I and Obs(k2).

This interpolation method was presented here in dimension N = 2 for simplicity, but it is readilygeneralised in higher dimensions recursively. Hence, this method produces systematically (in thecontinuum limit) a continuous and periodic frame of the Bloch bundle.

7.3 Numerical results for the Kane-Mele model

An implementation of the previously described algorithm was tested against the log interpolation, onthe Kane-Mele model, with λν = 4 (QSH phase). The Brillouin zone was discretised by a 100× 100grid.

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Figure 13: Local contribution to Ω obtained by log interpolation

Figure 14: Local contribution to Ω obtained by recursive map interpolation

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One can see that the log interpolation method fails at constructing a continuous map, as itexhibits lines of discontinuity. However, the recursive interpolation method works around the dis-continuity, and produces a locally regular and periodic frame.

This frame can be used as an initial guess for the Marzari-Vanderbilt algorithm. Table 2 presentssome results for a 50× 50 grid.

Log method Recursive map interpolationLocal Ω before MV 0.3235 0.1645Maximum of local Ω before MV 2.122 0.5139Ω after MV 218.8 79.90

Table 2: Local and global spreads Ω before and after the Marzari-Vanderbilt optimisation, to com-pare the initial guesses of the log method and the recursive map interpolation.

Not only does the recursive map method produce a smoother initial guess than the log method,but it also allows the Marzari-Vanderbilt algorithm to converge to a much smaller spread. Thismeans that the Marzari-Vanderbilt algorithm was not able to compensate for the discontinuity ofthe initial guess, and did not converge to the real global minimum.

Conclusion

The physical properties of a crystal are mainly determined by its band structure, that is, the eigen-value manifold of its Hamiltonian operator. In order to compute it, it is useful to use a localised basisof Wannier functions. Although it is not always possible to construct such a basis, the theoreticalrequirement for its existence is well-defined: it is a topological obstruction, encoded in the Chernnumber.

However, even if the topology permits it, the numerical construction of Wannier functions is notimmediate. One well-studied algorithm, the Marzari-Vanderbilt procedure, was proposed in [7]. Itbuilds maximally-localised Wannier functions, provided that it starts from an adequate initial guess.

Previous work in [2] proposed a robust method to provide such an initial guess, but the particularcase of topological insulators was not covered, and known to cause issues in the method.

Building on that method, we have presented a systematic method to build continuous and pe-riodic Bloch frames (in the continuum limit), that can be used as a robust initial guess to theMarzari-Vanderbilt localisation algorithm, so as to provide numerically maximally-localised Wan-nier functions without the need for initial input.

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Sources of figures

• Figure 1: By Benjah-bmm27 - Own work, Public Domain, https://commons.wikimedia.

org/w/index.php?curid=702423

• Figure 1.2: By Inductiveload - Own work, Public Domain https://commons.wikimedia.org/

wiki/File:Brillouin_Zone_(1st,_FCC).svg

• Figure 3: Public Domain https://upload.wikimedia.org/wikipedia/commons/thumb/0/

04/Band_structure_Si_schematic.svg/580px-Band_structure_Si_schematic.svg.png

• Figure 4: From [1].

• Figure 5: From [1].

• Figure 6: From Panati & Monaco, http://cond-math.it/panati_napoli1.pdf

• Figure 7: From [13].

• Figures 8 and 9: From [2].

References

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[2] E. Cances, A. Levitt, G. Panati, and G. Stoltz. Robust determination of maximally localizedwannier functions. Phys. Rev. B, 95:075114, Feb 2017.

[3] J. Feldman. The Spectrum of Periodic Schrodinger Operators.

[4] F. D. M. Haldane. Model for a quantum hall effect without landau levels: Condensed-matterrealization of the ”parity anomaly”. Phys. Rev. Lett., 61:2015–2018, Oct 1988.

[5] C. L. Kane and E. J. Mele. Z2 topological order and the quantum spin hall effect. Phys. Rev.Lett., 95:146802, Sep 2005.

[6] N. Marzari, A. A. Mostofi, J. R. Yates, I. Souza, and D. Vanderbilt. Maximally localized Wan-nier functions: Theory and applications. Reviews of Modern Physics, 84:1419–1475, October2012.

[7] Nicola Marzari and David Vanderbilt. Maximally localized generalized wannier functions forcomposite energy bands. Phys. Rev. B, 56:12847–12865, Nov 1997.

[8] D. Monaco. Chern and Fu-Kane-Mele invariants as topological obstructions. ArXiv e-prints,May 2017.

[9] G. Nenciu. Dynamics of band electrons in electric and magnetic fields: rigorous justification ofthe effective hamiltonians. Rev. Mod. Phys., 63:91–127, Jan 1991.

[10] G. Panati and A. Pisante. Bloch Bundles, Marzari-Vanderbilt Functional and Maximally Lo-calized Wannier Functions. Communications in Mathematical Physics, 322:835–875, September2013.

[11] Gianluca Panati. Triviality of Bloch and Bloch-Dirac bundles. Annales Henri Poincare,8(5):995–1011, 8 2007.

[12] Michael Reed and Barry Simon. IV: Analysis of Operators. Elsevier, 1978.

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