disentangling orbital magnetic susceptibility with wannier

Journal of the Physical Society of Japan FULL PAPERS

Disentangling Orbital Magnetic Susceptibility with Wannier Functions

Toshikaze Kariyado1, Hiroyasu Matsuura2, and Masao Ogata2,3

1 International Center for Materials Nanoarchitectonics, National Institute for Materials Science,Tsukuba, Japan

2 Department of Physics, Graduate School of Science, University of Tokyo, Tokyo, Japan3 Trans-scale Quantum Science Institute, University of Tokyo, Tokyo, Japan

Orbital magnetic susceptibility involves rich physics such as interband effects despite of its conceptualsimplicity. In order to appreciate the rich physics related to the orbital magnetic susceptibility, it is essentialto derive a formula to decompose the susceptibility into the contributions from each band. Here, we proposea scheme to perform this decomposition using the modified Wannier functions. The derived formula nicelydecomposes the susceptibility into intraband and interband contributions, and from the other aspect, intoitinerant and local contributions. The validity of the formula is tested in a couple of simple models. Interest-ingly, it is revealed that the quality of the decomposition depends on the degree of localization of the usedWannier functions. The formula here complements another formula using Bloch functions, or the formuladerived in the semiclassical theory, which deepens our understanding of the orbital magnetic susceptibilityand may serve as a foundation of a better computational method. The relationship to the Berry curvaturein the present scheme is also clarified.

1. Introduction

The orbital magnetic susceptibility is one of the fun-damental responses from electrons as charged particles.In solids, it is sensitive to band structures1) and affectedby the structure of Bloch wave functions in momentumspace.2,3) Within the semiclassical interpretation, theband structure is reflected in normal velocity, while thewave function structure is reflected in anomalous veloc-ity as Berry curvature. Therefore, experimental measure-ments of the susceptibility is useful in extracting informa-tion of electronic band structure. For instance, it can beused to detect singular band structures in solids such asDirac electrons.4–8) In theory, despite its conceptual sim-plicity, i.e., the magnetic susceptibility is essentially justthe lowest-order magnetic moment in external magneticfield, the formula for the orbital magnetic susceptibilityis very involved as shown in the early studies.9–16) It wasonly when Fukuyama combined the Luttinger-Kohn rep-resentation17) and the Green’s function technique, thatwe had a gauge invariant compact formula for the orbitalmagnetic susceptibility.18)

The Fukuyama formula involves Green’s function, ve-locity operator, and trace over all the states. When weapply this formula to a certain model and calculate theorbital magnetic susceptibility numerically, truncation ofthe trace at some finite number of states is unavoidable.Naively, this truncation by collecting finite number ofenergy bands sounds reasonable. However, it is not asstraightforward as one might think, because it is generi-cally not possible to diagonalize the Green’s function andthe velocity operator simultaneously. In order to havea good formula, good in the sense that it is useful innumerics and easy to comprehend, we have to handlethe offdiagonal matrix elements with great care. Indeed,if we start from the Fukuyama formula in a band ba-sis, we have to make full use of sum rules to isolate the

celebrated Landau-Peierls susceptibility from the correc-tion terms,19) where the correction terms are mostly fromthe local contributions such as the atomic diamagnetismor the van Vleck paramagnetism. Physically, the impor-tance of the offdiagonal matrix elements is from the factthat the orbital magnetic susceptibility is obtained bythe second order perturbation that involves virtual hop-pings to other bands. There is a big difference betweencompletely neglecting the other bands from the begin-ning and taking account of the other bands via the sumrules.

On the other hand, it is currently standard to derivelocalized Wannier functions near the Fermi energy in thefirst principle’s calculation. Therefore, it is necessary toobtain a formula for the orbital magnetic susceptibilityexpressed in terms of the Wannier functions. In this case,however, we have to be careful to use the truncation.

In this paper, we introduce a scheme to perform trun-cation of the orbital magnetic susceptibility, or decom-position of the susceptibility into contribution from eachband, using localized Wannier functions, motivated fromthe observation that the correction to the Landau-Peierlssusceptibility is from local terms.20,21) In order to makethe argument as transparent as possible, we work ona multi-orbital tight-binding model instead of a contin-uum model with periodic potential. Since the numberof bands is finite in tight-binding models, we can haveexact susceptibility free from truncation errors as a ref-erence. Then, we demonstrate decomposition of the sus-ceptibility for multiband models into contribution fromthe subset of the bands (see Fig. 1). We first derive aformula for the susceptibility that includes six terms χ1-χ6 using modified Wannier functions and the Green’sfunction technique. Importantly, there is an intuitive un-derstanding of this decomposition into six terms: χ1-χ4

are intraband contributions and χ5 and χ6 are interband

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J. Phys. Soc. Jpn. FULL PAPERS

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rgy rigorous gauge invariant formula

extract contribution froma certain band group usingWannier functions wiα

Fig. 1. Schematic picture of our scheme. Starting from an N

band tight-binding model, we decompose the susceptibility into the

contributions from each band group consisting of M (< N) bands,and the inter-bandgroup contributions. In this example, N = 4,

and the band group we focus on has two bands (M = 2).

contributions. From the other point of view, χ1 is classi-fied as an itinerant contribution, while χ2, χ3, χ5, and χ6

are local contributions, and χ4 is the cross term betweenlocal and itinerant terms. Then, the formula is appliedfor a couple of simple models. The analysis for the sim-ple models confirms the validity of the formula itself andreveals an interesting feature that the quality of decom-position depends on the degree of localization of the usedWannier functions. We will show that, in some cases, thecalculated orbital magnetic susceptibility has a sizablequantitative error if we apply an inappropriate trunca-tion, or use not well-localized Wannier functions. Finally,we clarify the relationship to the Berry curvature in thepresent scheme of decomposition.

This paper is organized as follows. First, we derive aformula for the orbital magnetic susceptibility in multior-bital tight-binding models using the Wannier functions,and explain the physical meaning of the terms in the for-mula. Next, we apply our formula on three models, thehoneycomb lattice model with sublattice potential, thehoneycomb lattice model with Kekule type distortion,and the decorated square lattice model, and evaluate thevalidity and usefulness of our formula. Then, the paperis closed by giving discussions and summary.

2. Derivation of the formula

2.1 Model and basic notions

We start from the tight-binding Hamiltonian

H =∑ij

H(0)ij c†i cj =

∑αβ

H0,αβd†αdβ , (1)

where c†i (ci) is a creation (annihilation) operator forthe ith orbital, and d†α (dα) is a creation (annihilation)operator for the Wannier state indexed by α. The latticepoints and the spin degrees of freedom are included in iand α. Using the Wannier functions wiα = 〈i|wα〉, d†α iswritten as

d†α =∑i

c†iwiα, (2)

and the matrices H(0) (in the original basis) and H0 (inthe Wannier basis) are related by a unitary transforma-tion as

H0 = W †H(0)W, (W )iα = wiα. (3)

The Wannier functions are constructed so as to have H0

block diagonal in the band group index. Generically, theindex α is a combination of the lattice point index andthe band group index, and possibly the orbital (or, sub-lattice) index when the corresponding band group hasmultiple bands (see Fig. 1).

In this paper, we assume that the magnetic field isadded through the Peierls substitution in the originalbasis as

H =∑ij

H(0)ij e

i q~Arirj c†i cj , (4)

where q is the charge, ri is the position of the ith orbital,and Arirj is defined as

Arirj =

∫Crj→ri

dr ·A(r). (5)

Crj→ri is a path from rj to ri, which is for simplicity,

set to the straight line between ri and rj leading to22)

Arirj =

∫ 1

0

ds(ri − rj) ·A(s(ri − rj) + rj). (6)

Note that the remaining effects of the magnetic fieldother than the Peierls phase are discussed in Ref. 21.

Assuming that the magnetic field is in the z-direction,the orbital magnetic susceptibility χ for free electrons canbe obtained by expanding the thermodynamic potentialwith respect to Bz, resulting in the formula23,24)

χ = kBT∑n

∂2F (iωn + µ,Bz)

(∂Bz)2, (7)

where T is temperature, ωn = (2n+ 1)πkBT (n: integer)is Matsubara frequency, and F (z,Bz) satisfies

∂F (z,Bz)

∂z= TrG(z,Bz), (8)

with G(z,Bz) being the Green function under the mag-netic field associated with Hamiltonian Eq. (4).

2.2 Susceptibility in the original basis

Working with the original basis, it is straightforwardto obtain G(z,Bz) in terms of g = (z1 − H(0))−1.23–25)

Then, the gauge invariant formula for χ is written as

χ =q2

~2

kBT

4

∑n

Tr((γxgγy + γxy)gγxgγy g

)+ (x↔ y),

(9)where γν is the current operator devided by q/~

γν = i[rν , H(0)], (10)

with

(rν)ij = riνδij , (11)

and

γνλ = −[rν , [rλ, H(0)]], (12)

This is exactly the formula for a tight-binding model withthe Peierls phase derived in the previous studies.23–25)

We will use this formula as a reference to evaluate theusefulness of our decomposition by the formula derivedin the following.

2


2.3 Susceptibility in the Wannier basis

Next, we would like to express the Green’s functionunder the magnetic field G(z,Bz) in the Wannier bases,i.e., in terms of g = (z1−H0)−1 where H0 is the WannierHamiltonian that is block diagonal in the band groupindex. For this purpose, we find that it is convenient towrite the Green function using the Wannier functions as

Gjk =∑αβ

wjαGαβw∗kβ , (13)

where wiα is a Wannier function amended by the Peierlsphase22) defined by

wiα = eiq~Arirαwiα, (14)

with the Wannier center rα defined as

rα = 〈wα|r|wα〉 =∑i

riw∗iαwiα. (15)

Expanding the Green function in terms of wiα instead ofwiα helps us in keeping the gauge invariance. However,in turn, the set {wiα} is no longer orthonormalized, andwe have to take care of the overlaps between wiα.

Then, some arithmetic gives us (see Appendix for de-tails)

χ =q2

~2kBT

∑n

(ΞI + ΞII + ΞIII + ΞIV + ΞV) (16)

with

ΞI =1

4Tr((γxgγy + γxy)gγxgγyg

)+ (x↔ y), (17)

ΞII = −Tr(Mz −

i

2([γx, ηy]− [γy, ηx])

)g, (18)

ΞIII = Tr({Sz1 , Lz}+

1

4[Sz1 , [H0, S

z1 ]])g, (19)

ΞIV = −TrLzgLzg, (20)

ΞV = −iTrLzg(γxgγy − γygγx)g. (21)

Here

γν = i[rν , H0], γνλ = −[rν , [rλ, H0]], (22)

with

(rν)αβ = rανδαβ , (23)

and rαν is the ν (= x, y) component of rα. Note that inthe Wannier basis, g, γν , and γνλ are all block diagonalin band groups, which makes it easy to separate intra-and inter-band contributions to χ, as we will do in thefollowing. We have also used

Lz = Hz1 −

1

2{Sz1 , H0}, Mz = Hz

2 −1

2{Sz2 , H0}, (24)

(Sz1 )αβ = i∑i

φzrαrirβw∗iαwiβ , (25)

(Sz2 )αβ = −∑i

(φzrαrirβ )2w∗iαwiβ , (26)

(Hz1 )αβ = i

∑ij

φzrαrirj rβw∗iαH

(0)ij wjβ , (27)

(Hz2 )αβ = −

∑ij

(φzrαrirj rβ )2w∗iαH(0)ij wjβ , (28)

with

φzrαrirβ =1

2ez · [(rα − ri)× (rβ − ri)], (29)

φzrαrirj rβ =1

2ez · [(rα − rj)× (rβ − ri)], (30)

and

ην = i[rν , Sz1 ]. (31)

Note that Sz1,2 takes account for the finite overlaps of theamended Wannier functions.

Because H(0) and H0 are related to each other bya unitary transformation and the formula involves thetrace, one may think that the formula should look thesame in any basis. Actually, the contribution ΞI sharesthe same form with Eq. (9). However, we have to notethat

rν 6= W †rνW (32)

holds since rν only picks up the diagonal elements bydefinition, while (W †rνW )αβ =

∑i riνw

∗iαwiβ is gener-

ically finite for α 6= β. This is the reason why the ad-ditional terms appear in Eq. (16). This means that ΞII,ΞIII, ΞIV, and ΞV are the correction terms that reflectrν 6= W †rνW . We will come back to this point later.

2.4 Intra- and interband contributions

Now, we decompose χ into various kinds of contribu-tions, such as intra- and inter-band contributions. ForΞI, since g, γν , and γνλ are diagonal in band groups, wecan simply extract the contribution from the band groupi by introducing

Ξ[i]1 =

1

4Tr[i]

((γxgγy + γxy)gγxgγyg

)+ (x↔ y), (33)

where Tr[i] denotes the partial trace over the band groupi. ΞII and ΞV also have only intraband contributions be-cause of the trace structure, although Mz, Lz, and ην canhave matrix elements between different band groups.

On the other hand, ΞIII and ΞIV have both intra- andinterband contributions, since they are in the second or-der with respect to Lz or Sz1 (or their cross terms). Notethat it is sufficient to think of pairs of band groups due tothe trace structure. Namely, starting from a band groupi, and going to another band group j by Lz or Sz1 , thenit has to be back on the band group i by the second Lzor Sz1 to complete the trace. Therefore, for the intrabandcontribution of ΞII + ΞIII, we introduce

Ξ[i]2 = −Tr[i]

(Mz −

i

2([γx, ηy]− [γy, ηx])

)g

+ Tr[i]({Sz[i]1 , L[i]

z }+1

4[Sz[i]1 , [H0, S

z[i]1 ]]

)g, (34)

where X [i] means a matrix constructed from X by leav-ing only the matrix elements within the band group i.Similarly, for the intra band contributions of ΞIV andΞV, we introduce

Ξ[i]3 = −Tr[i]L[i]

z gL[i]z g, (35)

3


Table I. Decomposition of the susceptibility. “Intra” and “inter”in the table denote “intra-bandgroup” and “inter-bandgroup”, re-

spectively. “Cross term” means the cross term between itinerant

motion and local moment. The column “in-gap” shows whethereach contribution can be finite when the chemical potential is in

the energy gap between the band groups at zero temperature. Note

that all the terms can be finite in the energy gap which may existinside a band group. (See Fig. 1.)

band type origin in-gap

χ1 intra itinerant electron hoppingχ2 intra local atomic diamagnetism Xχ3 intra local orbital Zeeman & van Vleck

χ4 intra both cross termχ5 inter local atomic diamagnetism Xχ6 inter local van Vleck X

and

Ξ[i]4 = −iTr[i]Lzg(γxgγy − γygγx)g. (36)

Finally, for the interband contributions of ΞIII and ΞIV,we introduce

Ξ[i:j]5 = Tr

({Sz[i:j]1 , L[i:j]

z }+1

4[Sz[i:j]1 , [H0, S

z[i:j]1 ]]

)g,

(37)and

Ξ[i:j]6 = −TrL[i:j]

z gL[i:j]z g, (38)

where X [i:j] denotes a matrix constructed from X byleaving only the matrix elements connecting the bandgroups i and j.

Now, we decompose χ as

χ = χ1 + χ2 + χ3 + χ4 + χ5 + χ6, (39)

where

χa = kBT∑n

∑i

Ξ[i]a (40)

for a ={1,2,3,4}, and

χa = kBT∑n

∑i<j

Ξ[i:j]a (41)

for a ={5,6}. For later use, we also define the suscepti-bility contributed from a set of band groups X as

χ{X} = kBT∑n

(∑i∈X

4∑a=1

Ξ[i]a +

∑i,j∈X,i<j

6∑a=5

Ξ[i:j]a

). (42)

2.5 Physical interpretation of each contribution

Here, we discuss the physical meaning of the decompo-sition. As we have already seen in the derivation, χ1-χ4

are the intraband contributions, while χ5 and χ6 are theinterband contributions.

In particular, χ1 has the same form as Eq. (9), andtherefore, is interpreted as a contirbution from the Peierlssubstitution in the Wannier basis. Since the Peierls phaseaffects the hopping, χ1 is, more or less, a response fromelectrons hopping around, and thus, we regard χ1 as itin-erant contribution.

On the other hand, χ2, χ3, χ5, and χ6 are finite in thepresence of Hz

1,2 and Sz1,2. By definition, Hz1,2 and Sz1,2 are

short ranged if the Wannier functions are well localized.Therefore, we categorize these terms as local contribu-tions. When we look at each term more closely, we cansee that χ2 and χ5 include a single g in the formula, andgive contributions proportional to f(Eα), where f is theFermi distribution function and Eα is eigenenergy in theband group. Namely, these terms depend on the fillingof the band group, and can be interpreted as atomic dia-magnetism generalized to the Wannier orbitals. In par-ticular, χ5 renormalizes atomic diamagnetism via inter-band effects. Here, we use the term “diamagnetism” tomatch it to the conventional terminology in the case ofreal atoms, but in the case of the Wannier functions intight-binding models, the sign of this term is not nec-essarily negative, and there can be “atomic paramag-netism”. Especially for a tight-binding model where theenergy range of the bands is bounded, the orbital mag-netic susceptibility induced by the Peierls phase has tovanish at zero temperature when the chemical potentialis higher than the upper limit of the band energy. Thisis to satisfy the susceptibility sum rule.26) Therefore, theatomic diamagnetism associated with some band has tobe compensated by the atomic “paramagnetism” associ-ated with the other band.

In contrast to χ2 and χ5, χ3 and χ6 include two g’sin the formula, and give contributions proportional to(f(Eα) − f(Eβ))/(Eα − Eβ). These terms can be inter-preted as the contributions from the fluctuation of the lo-cal moment Lz, which account orbital Zeeman type sus-ceptibility and van Vleck type susceptibility. Note thateach band group can have multiple bands in our formal-ism, and thus, χ3, categorized into intraband component,can also contain van Vleck type responses between thebands inside the band group.

Finally, interpreting Lz as a local moment, χ4 can beseen as a cross coupling between the local moment anditinerant motion of electrons by hopping, which is similarto the cross coupling between spins and itinerant motionthat arises in models with spin Zeeman term.27)

These interpretations are summarized in Table I. It isworth noting that among the intraband terms χ1,2,3,4,only χ2 can have finite value in the zero temperaturelimit when the chemical potential is outside of the energyrange of the corresponding band, since χ2 depends onthe orbital filling itself rather than some fluctuations.Therefore, χ at the band gap should be from χ2,5,6 atzero temperature.

3. Application to Tight-Binding Models

Now, we move on to some demonstrations of the de-rived formula applied to some tight-binding models. TheWannier functions play a central role in this study. In thefollowing, the Wannier functions are derived by project-

ing candidate wave functions w(c)il to the Hilbert space

spanned by a specific band group. In w(c)il , i denotes

the ith site just as i in wiα above, while l is for in-dexing the candidates. This procedure is often used inderiving an initial guess for a Wannier function in thewell known method to obtain maximally localized Wan-nier functions.28) Here, we adapt a method to derive

4


−3

−2

−1

0

1

2

3

M Γ K M

Ener

gy[|t|]

Wannier 1

Wannier 2

Candidate 1

Candidate 2

1i

−1−i

tε

−ε

1

2

Fig. 2. Band structure, candidate wave functions, and Wannier

functions for the honeycomb lattice model with a sublattice de-

pendent potential. (Inset in the middle shows H(0)ij in the original

basis.) Finite site potential gaps out the Dirac cone. Candidate 1

and 2 on the left are initial guess to obtain the Wannier functions.

Wanner 1 and 2 on the right represent the Wannier functions forthe lower and the upper bands, respectively.

an initial guess for the tight-binding models. First, wecompute Bloch wave functions ψi,nk for the target bandgroup with the momenta k on a regular grid in the Bril-louin zone, where n specifies a band in the band group.Then, we derive a matrix Ak whose matrix elements are

(Ak)nl =∑i ψ∗i,nkw

(c)il , and perform its singular value

decomposition as Ak = UkΛkV†k . Using these unitary

matrices Uk and Vk, the initial guess Wannier functionscan be fixed as

wi,αR =∑k

∑n

e−ik·Rψi,nk(UkV†k )nα, (43)

with appropriate normalization, where the lattice pointdependence of the Wannier function is explicitly indi-cated by replacing α → αR with α for the degrees offreedom other than the lattice points. The unitarity ofUkV

†k combined with the orthogonality of the Bloch wave

functions ensures the orthogonality of the obtained Wan-nier functions. In the following, we only work with simplemodels, and this “initial guess” already shows fairly goodlocalization as we will see shortly.

3.1 Honeycomb lattice with sublattice potential

The first model we tackle is a honeycomb lattice modelwith a sublattice dependent potential.29) As illustrated

in the inset in the middle of Fig. 2, the matrix H(0)ij in

the original basis consists of the nearest neighbor pairsof sites t, and A (B) sublattice potential +ε (−ε) fori = j. The finite ε induces a gap at the Dirac conesfor the pristine honeycomb lattice model, enabling usto decompose the bands into the lower and the upperbands. We use t = −1 and ε = 0.2, resulting in the gapsize of 0.4|t|. Because the sublattice potential accountsfor the gap, the candidate wave function is chosen to becompletely localized on a single A (or, B) sublattice. Forinstance, in order to have a Wannier function for the

lower band, we set w(c)il = 1 only for the B sublattice

(whose site potential is −ε with ε > 0) in the unit cell

at the origin (R = 0) and w(c)il = 0 for the rest of the

−0.20

−0.15

−0.10

−0.05

0.00

0.05

0.10

0.15

0.20

−3 −2 −1 0 1 2 3

(a)

Susc

eptib

ility

[q2 /~

2 ]

Chemical potential µ [|t|]

χtotχ1χ2+3χ5+6

χ{1}

−3 −2 −1 0 1 2 3

(b)


χlocχ2χ3χ5χ6

Fig. 3. (a) Susceptibility at T = 0.01|t| as a function of chem-

ical potential for the honeycomb lattice model with a sublattice

dependent potential. χ2+3 and χ5+6 are abbreviations of χ2 + χ3

and χ5 + χ6, respectively. Gray shadow of (a) shows the exact χ

obtained by Eq. (9). (b) Chemical potential dependence of χ2, χ3,

χ5, and χ6. Their total is shown as χloc ≡ χ2+3 + χ5+6.

sites. We can see from Fig. 2 that the obtained Wannierfunctions are fairly well localized.

Figure 3 summarizes the calculated susceptibility χat T = 0.01|t| as a function of the chemical potentialµ. As a reference, exact χ obtained with Eq. (9) in theoriginal basis is shown as a gray shadow in Fig. 3(a). It isconfirmed that the total χ obtained with Eqs. (39)-(41)matches with the exact result within the line width inthe scale of Fig. 3(a). (Not shown to avoid making thefigure busy.)

A notable feature in Fig. 3(a) is that χ{1}, which isthe contribution from the band group 1 (lower band),nicely approximates the exact result for µ < 0. The samefeature is also noticed by the smallness of the interbandcontribution χ5 + χ6 over the whole range of µ. Thismeans that the decomposition into the contribution fromeach band is successful in this model with our choice ofthe Wannier functions.

Figures 3(a) and 3(b) also tell us that the diamagneticresponce at the gap (µ ∼ 0) is from χ2, intraband con-tribution corresponding to the atomic diamagnetism ofthe Wannier function. Among the terms in Eq. (34) rel-evant to χ2, the terms involving ([γx, ηy] − [γy, ηx]) and

[Sz[i]1 , [H0, S

z[i]1 ]] are zero in this specific model, and χ2 is

contributed by the terms involving Mz and {Sz[i]1 , L[i]z }.

As we have noted, the atomic diamagnetism associatedwith the lower band is compensated by the atomic para-magnetism associated with the upper band, and this isthe reason that χ2 has positive slope when µ is in theenergy range of the upper band. The same kind of com-pensation is found in the following examples as well.

3.2 Honeycomb lattice with Kekule type modulation

The next model is a honeycomb lattice model withKekule type hopping texture.30–32) Specifically, we in-troduce two types of hopping, t0 and t1, as illustrated inFig. 4(a). With this texture, the six site cluster connectedby t0 bonds forms a unit cell, resulting in the Brillouinzone folding and gapping out the Dirac cones in the pris-tine honeycomb lattice. In this study, we set t0 = t − δ,t1 = t + 2δ, t = −1, and δ = 1/15, which gives a gap

5


t0

t1

(a)

(b) (d) (f)

CandidatesWannier Wannier

1i

−1−i −3

−2

−1

0

1

2

3

M Γ K M

Ener

gy[|t|]

(c)

−3

−2

−1

0

1

2

3

M Γ K M

Ener

gy[|t|]

(e)

2

11

2

3

4

Fig. 4. Band structure, candidate wave functions, and Wannier functions for the honeycomb lattice model with Kekule type distortion.

(a) Schematic picture of the model. (b,f) The obtained Wannier functions for the case (b) with the two band groups and (f) with the four

band groups, respectively. (c,e) Band structure with band groups marked with the numbers. (d) Candidates are initial guess to obtainthe Wannier functions.

−0.08

−0.04

0.00

0.04

0.08

0.12

−3 −2 −1 0 1 2 3

(a)

Susc

eptib

ility

[q2 /~

2 ]


χtotχ1χ2+3χ5+6

χ{1}

−3 −2 −1 0 1 2 3

(b)


χlocχ2χ3χ5χ6

Fig. 5. Susceptibility at T = 0.01|t| as a function of chemical

potential for the honeycomb lattice model with Kekule type dis-

tortion, where the six bands are decomposed into two band groups.

of 0.4|t| at zero energy. The state with δ = 1/15 is adia-batically connected to the state with t1 → 0 (t0 6= 0). Inthis limit, the system consists of six site clusters decou-pled with each other. Then, each cluster can be regardedas a one-dimensional periodic chain with six sites, whoseeigenstates can be expressed as plane waves along thechain ψcluster

n ∼ eiknx with kn = 2πan6 , where a denotes

the unit length and n = {0,±1,±2, 3}.We use ψcluster

n with n = 0,±1 as the candidate wavefunctions to build Wannier functions for the three bandsat E < 0. [See Figs. 4(b)-4(d)] The obtained Wannierfunctions look fairly well localized [Fig. 4(b)]. Then, theseWannier functions are used to decompose the suscepti-bility in the contributions from the band group 1 (bandsat E < 0) and the band group 2 (bands at E > 0), whichare shown in Fig. 5. As in the previous example, the ex-act χ obtained with the original basis is shown as grayshade in Fig. 5(a). Similarly to the previous example,χ{1} approximates the exact result very well for µ < 0[Fig. 5(a)], and the diamagnetism in the gap (µ ∼ 0) ismostly contributed from χ2, signaling that the decompo-sition is effective and useful in this model with our choiceof the Wannier functions.

Closely looking at the band structure, we notice thatthere is a small gap between the lowest band and thesecond lowest band [Fig. 4(e)]. Therefore, in principles,

−0.08

−0.04

0.00

0.04

0.08

0.12

−3 −2 −1 0 1 2 3

Susc

eptib

ility

[q2 /~

2 ]


χtotχ1χ2+3χ5+6

χ{1,2}

−3 −2 −1 0 1 2 3Chemical potential µ [|t|]

χlocχ2χ3

(a) (b) χ5χ6

Fig. 6. Susceptibility at T = 0.01|t| as a function of chemical

potential for the honeycomb lattice model with Kekule type dis-

tortion, where the six bands are decomposed into four band groups.

it is possible to decompose the bands into four bandgroups, the lowest, the second lowest (having two bands),the second highest (having two bands), and the highestband groups, instead of the two band groups (E < 0 andE > 0). In the decomposition into the two groups, thecondition for the Wannier function is that three Wannierfunctions should cover the space spanned by the threebands. On the other hand, in the decomposition into thefour groups, the condition is that one Wannier functionshould strictly generate the space spanned by the lowestband, and two Wannier functions should cover the spacespanned by the second lowest band group. Specifically,we use ψcluster

0 for the lowest band, and ψcluster±1 for the

second lowest band group. That is, there is stronger re-striction in the four-group decomposition than the two-group decomposition. Because of this stronger restric-tion, the Wannier functions for the four-group decompo-sition [Fig. 4(f)] are less localized compared with thosefor the two-group decomposition [Fig. 4(b)].

The susceptibility obtained with the less localizedWannier functions is summarized in Fig. 6. As a sanitycheck, it is confirmed that we still have good agreementwith the exact result if we collect all the terms, whichstrongly supports the validity of our formulation itself.However, when it comes to the quality of the approxima-tion, the quality of the Wannier functions does matter.

6


If we focus on χ{1,2} in Fig. 6(a), which corresponds tothe contribution from the band at E < 0 (including theinterband contribution between the band group 1 and2), we can see some deviations from the exact result forE < 0. Inside the gap (µ ∼ 0), the deviation is from bothχ5 and χ6 [Fig. 6(b)]. Another notable feature is found atµ ∼ −2, where there is a small gap between the lowestand the second lowest band. We find cancellation be-tween the diamagnetic contribution from the intrabandterm χ2 and the paramagnetic contribution from the in-terband term χ6 [Figs. 6(a) and 6(b)]. Because of thesmall gap, the van Vleck type response is enhanced, butwe know from the exact result by the original basis thatthere is no paramagnetic peak, and this discrepancy isresolved by the intraband atomic diamagnetism of theWannier functions.

In short, this example tells us that the Wannier func-tions have to be carefully chosen to make the decomposi-tion into each band effective. Both of χ{1} in Fig. 5 andχ{1,2} in Fig. 6 are for the contribution from the bands inE < 0. [See Figs. 4(c) and 4(e) for the numbering of thebands.] However, looking at the susceptibility at µ = 0,χ{1} in Fig. 5 is much closer to the exact result shownby the gray shadow than χ{1,2} in Fig. 6, and this dif-ference is originated from the difference in the Wannierfunctions [Figs. 4(b) and 5(f)].

3.3 Decorated square lattice model

Before going to the discussion part, we work on a lat-tice other than the honeycomb type. The model we han-dle here is a decorated square lattice model,33,34) whichis a square network of diamond-shape four-site clustersas illustrated in the inset of Fig. 7. We have three param-eters, t0, t1, and d. t0 and t1 are the hopping parameterswithin the four-site cluster and between the clusters, re-spectively, while d determines the size of the diamond-shape cluster, i.e., the distance between the center andthe vertex of the diamond. We set t0 = −1, t1 = −0.5,and d = 1/5 (in the unit of the lattice constant). Withthis choice, the bands are decomposed into three groups.It is noticed that three bands are dominated by s-, px,y-, dx2−y2-like orbitals on the four site cluster, and thus,we use candidate wave functions completely localized ona single cluster with s-, px,y-, dx2−y2 -like symmetry toobtain the Wannier functions (Fig. 7).

The calculated susceptibility is summarized in Fig. 8.As the previous examples, the exact result is shown asgray shade in Fig. 8(a). We notice that χ{1}, which is thecontribution from the band group 1 (the lowest band),gives a good approximation for the lower half of the low-est band (µ < −2), and χ{1,2}, which represents the con-tributions from the band groups 1 and 2 (including theinterband terms between the band groups 1 and 2), nicelyagrees with the exact result for µ < 0. However, χ{1} failsto reproduce the exact result in the gap between the bandgroups 1 and 2, and the deviation is from χ6 [Fig. 8(b)].Another interesting feature found in Fig. 8(b) is that χ4,which is the cross term between the itinerant motion andthe local moment, compensates the sharp peaks in χ3 torestore the exact result, telling us that the cross termcannot be simply neglected.

−3

−2

−1

0

1

2

3

Γ X M Γ

Ener

gy[|t

0|]

1i

−1−i

t0 t1

d

Candidates Wannier

1

2

3

Fig. 7. Band structure, candidate wave functions, and Wannier

functions for the decorated square lattice model. Candidates on

the left are initial guess to obtain the Wannier functions, while theobtained Wannier functions are shown on the right. The inset in

the middle shows the schematic description of the model.

−0.008

−0.004

0.000

0.004

0.008

−3 −2 −1 0 1 2 3

Susc

eptib

ility

[q2 /~

2 ]

Chemical potential µ [|t0|]

χtotχ{1}χ{2}χ{1,2}

−3 −2 −1 0 1 2 3Chemical potential µ [|t0|]

χlocχ2χ3

(a) (b)

×0.5

χ4χ5χ6

Fig. 8. Susceptibility at T = 0.01|t0| as a function of chemical

potential for the decorated square lattice model.

In Fig. 8(a), we focus on the region −1.5 < µ < −0.5where the chemical potential is located inside the gapbetween the lowest and the second-lowest bands. In thisregion, χ{1} has a negative finite value while χ{2} is zero.However, χ{1} shows a significant deviation from the cor-rect susceptibility given by the gray shadow. This meansthat χ{1}+χ{2} is not a good approximation for the sus-ceptibility in this region of µ. On the other hand, χ{1,2}

is a good approximation in the same region. As we cansee from Eq. (42), χ{1,2} differs from χ{1} + χ{2} by

kBT∑n

(Ξ

[1:2]5 + Ξ

[1:2]6

), (44)

namely by the interband contributions. This is a typi-cal example in which the decomposition method has acrucial effect on the calculated susceptibility.

4. Discussion and Summary

4.1 Wannierization dependence

Along the course of this study, we learned that thequality of the Wannier functions has no effect on thetotal susceptibility, while it strongly affects the qualityof the decomposition. Naively, one may find that thisis a condradiction, since generically two sets of Wan-nier functions are related by a unitary transformationand our formula involves matrix traces. The origin of

7


the Wannier function dependence of the decompositioncan be traced back to the definitions of Sz1,2 and Hz

1,2,i.e., Eqs. (25)-(28). Since φzrαrirβ and φzrαrirj rβ dependson the Wannier centers rα,β , which are computed fromthe given Wannier functions, Eqs. (25)-(28) are acutallynot simple unitary transformation from the original ba-sis. Then, Sz1,2 and Hz

1,2 for two different sets of Wannierfunctions may not be related by a unitary transforma-tion, even if the two sets of Wannier functions span thesame band groups.

The Wannierization dependence gives us a chance tooptimize Wannier functions for susceptibility decompo-sition. Generically speaking, the decomposition becomesparticularly powerful if we can suppress the interbandcontribution. Looking at the definitions, we notice thatone of the interband contribution χ5 vanishes if Sz1 = 0.Note that the matrix element (Sz1 )αβ for pairs of Wan-nier functions sharing the Wannier center is zero sinceφzrαrirβ = 0 when rα = rβ . Then, if the given Wannierfunctions are very well localized such that the overlapbetween the pairs of Wannier functions not sharing theWannier center is small, we expect that χ5 is small due tothe small Sz1 . In the case where χ5 is negligible, we haveχ = χ2+χ6 at zero temperature when µ is in some energygap. Using the hermitian nature of Lz, it is straightfor-ward to show that the remaining interband contributionχ6 is always positive. Then, minimizing the interbandcontribution χ6 means minimizing the negative, or dia-magnetic contribution to χ2 since the sum is fixed. Verynaively, we can make the diamagnetic contribution in χ2

small by using well localized Wannier functions, since χ2

corresponds to the atomic diamagnetism of Wannier or-bitals. From all above, we speculate that using compactWannier functions results in better decomposition withsmaller interband contributions. However, we have to domuch more extensive surveys to be conclusive.

4.2 Berry connections

It has been briefly noted that Eq. (32) is a key toobtain our formula. In order to elaborate this argument,we introduce qµ as

W †rµW = rµ + qµ. (45)

By definition, we have (qµ)αβ =∑i riµw

∗iαwiβ (α 6= β),

i.e., qµ is for the offdiagonal matrix elements of the po-sition operator in the Wannier basis. When the systemhas lattice translation symmetry, it is convenient to in-clude lattice point positions in the index to specify theWannier functions as wiα → wi,αR, where R and αdenote the lattice point and the remaining degrees offreedom in the unit cell, respectively. Then, we can con-struct a Bloch wave function corresponding to wi,αR as

ψi,αk =∑

R eik·Rwi,αR. Then, it is known28,35) that we

have

(qµ)αRβR′ =i

N

∑k

eik·(R−R′)∑i

u∗i,αk∂ui,βk

∂kµ, (46)

where ui,αk is the periodic part of the Bloch wave func-tion extracted as ui,αk = e−ik·riψi,αk. Equation (45) also

gives

W †γµW = γµ + i[qµ, H0]. (47)

Comparing Eq. (9) and Eqs. (39)-(41), we know thatχ2-χ6 contains at least one qµ, namely, the quantity qµthat accounts for the difference between W †rµW and rµindeed generates χ2-χ6. Apparently from Eq. (46), qµ isrelated to the Berry connection, and it is interesting thatthe terms χ2-χ6 are generated because of that. Note thatthe second term at the right hand side of Eq. (47) corre-sponds to p``′,µ in Ref. 19, which plays an essential rolein applying the sum rule. Also, it is worth noting thatthe Berry curvature plays an important role in the semi-classical theory of the orbital magnetic susceptibility2)

through the anomalous velocity. These observations sug-gest that our formula complements the previous formu-lae relying on the Berry connection. Equipped with ourformalism, we can use the matrices Sz1,2 and Hz

1,2 to cal-culate the susceptibility, instead of the Bloch wave func-tions and their connections. Which formula, the formulain this paper or the former ones, does perform better maydepend on situations. However, the current formula givesintuitive understanding of the origin of the susceptibilityas summarized in Table I.

4.3 Summary and Outlook

To summarize, we have proposed a new formula to de-compose the orbital magnetic susceptibility in the con-tributions from each band, and we have demonstratedthe decomposition in some simple models. Here, we tooka tight-binding model as a starting point. Since the sizeof the Hilbert space of the tight-binding model is finite,we can always get a rigorous result without any concernsabout truncation error, making the evaluation of the de-composition easy and clear. We believe that almost thesame procedure can be applicable to continuum models.Namely, once the explicit formula for Sz1,2 and Hz

1,2 areobtained, the rest of the derivation will not be changed.Nowadays, it becomes more and more common to useWannier functions obtained in the first-principles calcu-lation for analyzing properties of real materials. In thissituation, it is an interesting and important future workto demonstrate the decomposition in continuum models,having a generic formula working in real materials as anultimate goal.

The work was partially supported by JSPS KAK-ENHI Grants No. JP17K14358 (T.K.), No. JP20K03844(T.K.), and No. JP18H01162. Part of the computa-tions in this work has been done using the facilities ofthe Supercomputer Center, the Institute for Solid StatePhysics, the University of Tokyo.

Appendix: Derivation of the formula

In the original bases, the Green function Gij(z,B) isdefined through the equation∑

j

Lij(z)ei q~ArirjGjk(z,B) = δik (A·1)

with

Lij(z) = zδij −H(0)ij . (A·2)

8


Our goal is to expand the Green function Gij(z,B) upto the second order in B. In the following, we abbreviateGjk(z,B) as Gjk and Lij(z) as Lij .

Now, we introduce an ansatz,

Gjk =∑αβ

wjαGαβw∗kβ , (A·3)

with

wiα = eiq~Arirαwiα. (A·4)

Here, wiα is a Wannier orbital indexed by α, and rα is aWannier center, i.e.,

rα = 〈wα|r|wα〉 =∑i

riw∗iαwiα. (A·5)

Using the ansatz, we have

TrG =∑i

∑αβ

wiαGαβw∗iβ = TrGS = TrG (A·6)

with

Sαβ =∑i

w∗iαwiβ (A·7)

and G = GS. Then, Eq. (A·1) becomes∑j

∑βγ

Lijei q~Arirj wjβGβγw

∗kγ = δik, (A·8)

leading to∑ijkβγ

w∗iαLijei q~Arirj wjβGβγw

∗kγwkδ =

∑i

w∗iαwiδ.

(A·9)This can be rewritten as∑

ijβ

w∗iαLijei q~ Φrαrirj rβwjβe

i q~ Φrαrβ rγ Gβγ = Sαγ ,

(A·10)where G and S are introduced as

Gβγ = eiq~Arβ rγ Gβγ , Sαγ = ei

q~Arαrγ Sαγ , (A·11)

and

Φrαrirj rβ = Arαri +Arirj +Arj rβ +Arβ rα , (A·12)

Φrαrβ rγ = Arαrβ +Arβ rγ +Arγ rα . (A·13)

Note that TrG = TrG because of Arαrα = 0. Very im-portantly, both of Φrαrirj rβ and Φrαrβ rγ are obtainedby line integrating A(r) on a closed path, and therefore,gauge invariant and linear in Bµ.36)

Equation. (A·10) can further be rewritten as∑β

(zSαβ −Hαβ)Gβγeiq~ Φrαrβ rγ = Sαγ , (A·14)

with

Hαβ =∑ij

eiq~ Φrαrirj rβw∗iαH

(0)ij wjβ . (A·15)

Note that Hαβ is related to H0 as we show shortly. Now,our goal is to derive G according to Eq. (A·14), and ex-tract the terms second order in Bµ. Up to the second

order in Φrαrirβ , we have

S ∼ S0 +q

~S1 +

q2

~2S2 (A·16)

with

(S0)αβ = δαβ , (A·17)

(S1)αβ = i∑i

Φrαrirβw∗iαwiβ , (A·18)

(S2)αβ = −1

2

∑i

Φ2rαrirβ

w∗iαwiβ , (A·19)

and similarly, up to the second order in Φrαrirj rβ , wehave

H ∼ H0 +q

~H1 +

q2

~2H2 (A·20)

with

(H0)αβ =∑ij

w∗iαH(0)ij wjβ (A·21)

(H1)αβ = i∑ij

Φrαrirj rβw∗iαH

(0)ij wjβ (A·22)

(H2)αβ = −1

2

∑ij

Φ2rαrirj rβ

w∗iαH(0)ij wjβ . (A·23)

For the later use we define Li as

Li = zSi −Hi, (A·24)

and we write L0 = L for notational simplicity.The heart of this formulation is to expand terms with

structure of ∑β

XαβYβγei q~ Φrαrβ rγ (A·25)

in a series of Φrαrβ rγ , which can also be written as

Φrαrβ rγ =B

2· [(rα − rβ)× (rγ − rβ)]

= −βνλ(rαν − rβν)(rβλ − rγλ)

(A·26)

with βνλ = Bµεµνλ/2 and rαν being ν component of rα.The first order contribution involves a term like∑

β

XαβYβγΦrαrβ rγ

= −iβνλ(rαν − rβν)(rβλ − rγλ)XαβYβγ

=(−iβνλ[rν , X][rλ, Y ]

)αγ

(A·27)

with

(rν)αβ = rανδαβ . (A·28)

Similarly, the second order contribution involves∑β

XαβYβγΦ2rαrβ rγ

=(βνλβν′λ′ [rν , [rν′ , X]][rλ, [rλ′ , Y ]]

)αγ. (A·29)

9


Now, approximating G as

G ∼ g +q

~g1 +

q2

~2g2 (A·30)

where gi is in the ith order in Bµ and g ≡ g0. In thezeroth order in Bµ, Eq. (A·14) leads to

Lg = 1. (A·31)

In the first order in Bµ, Eq. (A·14) gives

L1g + Lg1 − iβνλ[rν ,L][rλ, g] = S1, (A·32)

leading to

g1 = gLg + iβνλgγνgλ, (A·33)

where γν , gλ, and L are introduced as

γν = i[rν , H0], gλ = i[rλ, g], (A·34)

and

L = S1L − L1 = H1 − S1H0. (A·35)

In the second order in Bµ, Eq. (A·14) reduces to

L2g + Lg2 −1

2βνλβν′λ′ [rν , [rν′ ,L]][rλ, [rλ′ , g]]

+ L1g1 − iβνλ[rν ,L1][rλ, g]

− iβνλ[rν ,L][rλ, g1] = S2, (A·36)

giving us

g2 = gMg − 1

2βνλβν′λ′gγνν′gλλ

′− gL1g1

+ βνλg[rν ,L1]gλ − βνλgγν [rλ, g1], (A·37)

where γνν′ , gλλ′, and M are introduced as

γνν′ = −[rν , [rν′ , H0]], gλλ′

= −[rλ, [rλ′ , g]], (A·38)

and

M = S2L − L2 = H2 − S2H0. (A·39)

By subsituting g1 in Eq. (A·33) into Eq. (A·37), we ob-tain g2. For convenience, we decompose g2 as

g2 = g(1)2 + g

(2)2 + g

(3)2 + g

(4)2 (A·40)

with

g(1)2 = gMg, (A·41)

g(2)2 = −gL1gLg, (A·42)

g(3)2 = −iβνλ

(ig[rν ,L1]gλ − igγν [rλ, gLg] + gL1gγνg

λ),

(A·43)

g(4)2 = βνλβν′λ′

(igγν [rλ, gγνg

λ]− 1

2gγνν′gλλ

′). (A·44)

Let us rewrite these first to fourth terms g(1)2 -g

(4)2 , re-

spectively named Term 1 to 4, in convenient forms inthe following.Term 1 Introducing a symmetrized operator M in-

stead of M as

M = H2 −1

2{S2, H0} = M +

1

2[S2,L], (A·45)

we have

Trg(1)2 = TrgMg − 1

2Trg[S2,L]g = − ∂

∂zTrMg (A·46)

where the last identity follows from Trg[S2,L]g =Tr[g, S2] = 0 and

∂g

∂z= −g2. (A·47)

Term 2 Introducing a symmetrized operator L in-stead of L as

L = H1 −1

2{S1, H0}

=1

2{S1,L} − L1 = L− 1

2[S1,L],

(A·48)

we have

g(2)2 =g

(L− 1

2{S1,L}

)g(L+

1

2[S1,L]

)g

=gLgLg − 1

2({S1, g}Lg − gL[g, S1])

− 1

4{S1, g}L[g, S1].

(A·49)

For the first term of the right hand side, we obtain

TrgLgLg =1

2Tr(LgLg2 + Lg2Lg)

= −1

2

∂

∂zTrLgLg.

(A·50)

For the second term, we evaluate

Tr({S1, g}Lg − gL[g, S1])

= Tr(g{S1, L}g + S1gLg − gLgS1)

= − ∂

∂zTr{S1, L}g,

(A·51)

and for the third term, we evaluate

Tr({S1, g}L[g, S1])

= Tr(S1gS1 + gS1S1 − S21g − gS1LS1g)

= −1

2

∂

∂zTr[S1, [H0, S1]]g,

(A·52)

where the last line follows from

S1LS1 =1

2{S2

1 ,L} −1

2[S1, [S1,L]]. (A·53)

Combining the above equations, Trg(2)2 becomes

Trg(2)2 = −1

2

∂

∂zTr(LgLg − {S1, L}g −

1

4[S1, [H0, S1]]g

).

(A·54)Term 3 Using TrA[B,C] = Tr[A,B]C and βνλγνλ =

βνλgνλ = 0, we obtain

Trg(3)2 = iβνλTr(gνL1g

λ − gλγνgLg − gL1gγνgλ)

= iβνλTr(gγλgLgγνg + gγνgγλgLg + gLgγνgγλg)

− iβνλ2

Tr(gλ{S1,L}gν + gλγν [g, S1] + {S1, g}γνgλ).

(A·55)

10


For the first term of the right hand side, we evaluate

Tr(gγλgLgγνg + gγνgγλgLg + gLgγνgγλg)

= Tr(Lgγνg2γλg + Lg2γνgγλg + Lgγνgγλg

2)

= − ∂

∂zTrLgγνgγλg.

(A·56)

For the second term, we evaluate

βνλTr(gλ{S1,L}gν + gλγν [g, S1] + {S1, g}γνgλ)

= βνλTr(gλS1γνg + gγλS1gν − gλγνS1g + gS1γνg

λ)

= βνλTr(gλ[S1, γν ]g + g[S1, γν ]gλ)

= iβνλ∂

∂zTr[rλ, [S1, γν ]]g,

(A·57)

where the second line is from

βνλTr(gλγνgS1 + S1gγνγλ)

= βνλTr(S1gγλgγνg + S1gγνgγλg) = 0, (A·58)

and the last line is from

gλXg + gXgλ = i[rλ, gXg]− ig[rλ, X]g. (A·59)

Combining the above equations, Trg(3)2 becomes

Trg(3)2 = −i ∂

∂zTr(βνλLgγνgγλg −

βνλ2

Tr[γν , i[rλ, S1]]g).

(A·60)

Term 4 For the case B = t(0, 0, Bz), we can simplify

Trg(4)2 following the procedure in Ref. 24, as

Trg(4)2 =

B2z

8

∂

∂zTr(

(γxgγy + γxy)gγxgγyg + (x↔ y)).

(A·61)Collecting the all terms, we eventually obtain for B =

t(0, 0, Bz)

∂2

∂B2z

Trg2

= − ∂

∂zTr(Mz −

i

2([γx, ηy]− [γy, ηx])

)g

+∂

∂zTr({S1, Lz}+

1

4[Sz1 , [H0, S

z1 ]])g

− ∂

∂zTrLzgLzg

− i ∂∂z

TrLzg(γxgγy − γygγx)g

+1

4

∂

∂zTr(

(γxgγy + γxy)gγxgγyg + (x↔ y))

(A·62)

where Sz1 , Sz2 , Lz, and Mz are introduced through

S1 = Sz1Bz, S2 =1

2Sz2B

2z , (A·63)

L = LzBz, M =1

2MzB

2z , (A·64)

and ηλ is defined as

ηλ = i[rλ, Sz1 ]. (A·65)

For the calculation of χ, we need Tr G as shown inEq. (8). Now

Tr G = Tr G = Tr G = Tr

(g +

q

~g1 +

q2

~2g2

). (A·66)

Therefore, Eq. (A·62) leads to Eq. (16).Finally, let us check the susceptibility formula in the

original basis. Comparing Eqs. (A·1) and (A·14), we no-tice that the formula in the original basis can be obtainedby

H0 → H(0), rαν → riν , H1,2 = S1,2 = 0. (A·67)

This means that only the last term of Eq. (A·62) remains,and thus we reproduce the result in Eq. (9).

1) R. Peierls, Zeitschrift fur Physik 80, 763 (1933).

2) Y. Gao, S. A. Yang, and Q. Niu, Phys. Rev. B 91, 214405(2015).

3) F. Piechon, A. Raoux, J.-N. Fuchs, and G. Montambaux,

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disentangling orbital magnetic susceptibility with wannier

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