Universal Quantization of the Magnetic Susceptibility Jump at a Topological Phase Transition

PHYSICAL REVIEW RESEARCH 3, 013058 (2021)

Universal quantization of the magnetic susceptibility jump at a topological phase transition

Soshun Ozaki * and Masao Ogata Department of Physics, University of Tokyo, Bunkyo, Tokyo 113-0033, Japan

(Received 21 April 2020; accepted 16 November 2020; published 19 January 2021)

We examine the magnetic susceptibility of topological insulators microscopically and ﬁnd that the orbital- Zeeman (OZ) cross term, the cross term between the orbital effect and the spin Zeeman effect, is directly related to the Berry curvature when the z component of spin is conserved. In particular, the OZ cross term reﬂects the spin Chern number, which results in the quantization of the magnetic susceptibility jump at the topological

phase transition. The magnitude of the jump is in units of the universal value 4|e|μB/h. We also apply the obtained formula to an explicit model and demonstrate the quantization. For this model, the physical origin of this quantization is clariﬁed.

DOI: 10.1103/PhysRevResearch.3.013058

I. INTRODUCTION g0 = 2. (Here we neglect the relativistic correction of g0.) Thus, their conclusion means that the jump in χ is not Topological insulators (TIs) [1–13] show anomalous phe- OZ quantized in a universal value. nomena such as electric conduction on sample surfaces. In the present paper, we study χ microscopically based Experimentally, the search for candidate materials for TIs OZ on the Green’s function formalism and show that, in contrast is one of the most important problems. In particular, two- to the results of Nakai and Nomura, the jump in χ is exactly dimensional (2D) TIs are predicted to show unique phenom- OZ quantized in units of the universal value 4|e|μ /h [see Eq. (9) ena, such as the spin Hall effect and robust edge states against B below] even in the presence of SOI as long as the z component nonmagnetic impurities, only a few of which have been found of the spin is conserved. Here μ =|e|h¯/2m is the Bohr mag- [7–11]. So far, the confirmation of topological materials has B neton. When we study a model microscopically [as in Eq. (1)], been achieved by finding the edge state by angle-resolved the modification of the g factor does not occur explicitly, photoemission spectroscopy or from the transport coefficients. and instead the effect of SOI appears in the deformation of Since both methods detect anomalous electronic states at the the Bloch wave functions and the energy dispersion, which edge, it is desirable to develop some bulk-sensitive methods eventually leads to the orbital-dependent g factors. We show that enable us to confirm the topological nature of a material. below that the effect of SOI is exactly canceled out in χ , In this paper, we propose that the quantization of the bulk OZ which leads to the quantization of jump with a universal value. magnetic susceptibility jump can be used as strong evidence In the latter part of this paper, we apply the obtained formula for the topological phase transition in 2D TIs. to a low-energy effective model for a topological insulator Usually, the magnetic susceptibility is discussed in terms to show the validity of the present proposal. For this model, of the orbital effect of the magnetic field [14–30] and spin we clarify the physical origin of the quantization: It turns out Zeeman effect independently. In general, however, there can that the quantization is associated with the chiral edge current, be a cross term between the orbital and Zeeman effects which is characteristic of the topologically nontrivial state. [5,6,31–35], which we call the orbital-Zeeman (OZ) cross term χOZ in the following. Recently, Nakai and Nomura [34] II. GENERAL FORMALISM discussed the jump in χOZ at the topological phase transition using the formula of the orbital magnetization [36–42] and First, we develop microscopically a general formula for the Stredaˇ formula [43]. They calculated the OZ cross term magnetic susceptibility including orbital magnetism, Pauli in the Bernevig-Hughes-Zhang model [4] and concluded that paramagnetism, and the OZ cross term in terms of thermal the width of the jump depends on the g factors of the involved Green’s functions in the presence of SOI. Let H be the general orbitals introduced phenomenologically. In general, spin-orbit Hamiltonian derived from the Dirac equation in the presence interaction (SOI) modifies the g factor from its bare value of a general 3D periodic potential V (r) and a magnetic field, which is given by 1 eh¯ H = [p − eA(r)]2 − σ · B(r) + V (r) *[email protected] 2m 2m h¯2 h¯ Published by the American Physical Society under the terms of the + ∇2V + σ ·∇V × [p − eA(r)], (1) 2 2 2 2 Creative Commons Attribution 4.0 International license. Further 8m c 4m c distribution of this work must maintain attribution to the author(s) where A(r) is a vector potential, e < 0 for electrons, σ = and the published article’s title, journal citation, and DOI. (σx,σy,σz )are2× 2 Pauli matrices, and B(r) =∇×A(r)

2643-1564/2021/3(1)/013058(9) 013058-1 Published by the American Physical Society SOSHUN OZAKI AND MASAO OGATA PHYSICAL REVIEW RESEARCH 3, 013058 (2021) represents a magnetic field. The last term represents the SOI. perpendicular to the 2D x-y plane. Eventually, we obtain the It is to be noted that the second term representing the Zee- formula for 2D systems by substituting the 3D k summation man interaction has the bare value g0 = 2. As performed for a 2D k summation and by including trace over the subband by Fukuyama [20], we implement a perturbative calculation index for the z direction wave function. Hereafter, we use bold of the free energy in terms of the vector potential A(r)via for two-component vectors, e.g., k = (kx, ky ). the Luttinger-Kohn representation [44]. As a result, we ob- Then, to discuss the quantization, we rewrite Eq. (2c) tain the expression for each contribution as follows (the details in terms of the Bloch wave functions in a similar way to of the derivation are shown in the Appendix): Ref. [22]. The periodic part of the Bloch wave function t , = ↑ , , ↓ , 2 uˆlk(r z) (ulk (r z) ulk (r z)) satisfies χ = e kBT γ Gγ Gγ Gγ G, orbit Tr x y x y (2a) , = ε , , = −ikr ikr, 2¯h2 L3 Hkuˆlk(r z) l (k)ûlk(r z) Hk e He (3) nk where l represents both the band index in the 2D Brillouin kBT χ =− Tr MsGMsG, (2b) zone and the subband index in the z direction. In the fol- Pauli L3 z z nk lowing, we abbreviate ulkσ (r, z)asulkσ . When the SOI is / 2 2 σ ∂ − − ∂ − i|e| k T given by (¯h 4m c ) z[( xV )(py eAy ) ( yV )(px eAx )], χ =− B sGγ Gγ G − sGγ Gγ G , OZ Tr M x y M y x the z component of spin is conserved. In this case, up- and h¯ L3 z z nk down-spin electrons are independent, and ulk↑ and ulk↓ are the (2c) eigenstates of σz with eigenvalues εl↑(k) and εl↓(k) (denoted ε ↑ ε ↓ G as l and l in the following). In this case, the matrices and where G is the thermal Green’s function G(k, iεn ), whose (ll ) s −1 M are diagonal and given by [Gσ ]ll = δll (iεn − εlσ + μ) component is the matrix element between the lth and lth z and bands. Each band index includes the pseudospin degrees of s ∗ freedom in the case with SOI. εn is the Matsubara frequency, = −μ σ σ Mz lσ,lσ ulkσ ( B z )ul k drdz γμ represents the current operator in the μ direction divided by / s −μ σ e h¯, and Mz is the matrix for the operator B z. The effect =−μBσδll δσσ , (4) of SOI is included in G and γμ. Tr is the trace over the band indices and the spin degrees of freedom. In Eq. (2), χ and respectively. Note that the normalization is fixed by the equa- orbit tion χPauli represent the orbital and Pauli magnetic susceptibility, respectively. These are the same expressions as were obtained ∗ u u σ drdz = δ δσσ , (5) before [20,45] even in the presence of SOI. On the other hand, lkσ l k ll χ OZ is the OZ cross term, which we focus on in this paper. where the spatial integral is over the entire crystal. On the other hand, the matrix γμ has off-diagonal matrix elements III. UNIVERSAL QUANTIZATION OF χ between the different bands and it becomes [20,46], OZ ∗ ∂Hk In the following, we consider the case where the z com- γ = [ μσ ]ll ulkσ ul kσ drdz ponent of spin is conserved even in the presence of SOI. ∂kμ This situation will be realized more easily in 2D systems ∂εlσ ∗ ∂ulkσ = δ + ε − ε . than in 3D systems. Therefore, we focus on 2D systems in ll ( l σ lσ ) ulkσ drdz (6) ∂kμ ∂kμ the following. As shown in Appendix, it is straightforward to extend Eq. (2) in 2D systems where V (r) gives a con- Substituting these quantities into Eq. (2c) and carrying out the finement potential in the z direction and the magnetic field Matsubara summation, we obtain

| |μ | |μ ∂ ∗ ∂ 2 e B z i e B ulkσ ulkσ χ =− f (ε σ )σ + σ f (ε ) (ε σ − H ) drdz − (x ↔ y) , (7) OZ 2 l lkσ 2 l l k hL¯ hL¯ ∂kx ∂ky lkσ lkσ

z χ where lσ is the Berry curvature in the z direction, OZ is written as 4|e|μ χ 2D =− B , OZ Chs,l (9) ∗ ∗ h ∂u ∂u σ ∂u ∂u σ l:occ z = lkσ lk − lkσ lk , lkσ i drdz (8) ∂kx ∂ky ∂ky ∂kx where the summation l:occ is taken for the occupied bands and Chs,l is the spin Chern number for the lth band deﬁned by 1 2π (ε−μ)/k T −1 Ch , = ( ↑ − ↓). (10) f (ε) = [1 + e B ] , and the completeness condition s l 2 lk lk ∗ 2 L , , = δ − δ − k lσ ul kσ (r z)u σ (r z ) (r r ) (z z ) has been used l k to take the summation over the intermediate state l . Espe- At a topological phase transition, Chs,l changes from one cially for 2D insulator at zero temperature, the second term integer to another. Therefore, Eq. (9) leads to the quantiza- in Eq. (7) vanishes because the Fermi surface is absent. Then tion of the magnetic susceptibility jump at the topological

013058-2 UNIVERSAL QUANTIZATION OF THE MAGNETIC … PHYSICAL REVIEW RESEARCH 3, 013058 (2021) phase transition. The magnitude of the jump is in units of χ0 = 4|e|μB/h, which is universal. Although the effect of SOI is included in ulkσ , it does not affect the coefﬁcient in Eq. (9) since Chs,l is a topological number, which leads to the universal quantization of jump in χOZ.

IV. EXPLICIT CALCULATION OF χOZ IN A TOPOLOGICAL INSULATOR In the rest of this paper, we consider the low-energy effective model of the Kane-Mele model [1,2,47–51], one of the models for 2D TIs, to show that χOZ actually has a jump at the topological phase transition and that other contributions do not conceal the quantized jump. We introduce the Kane-Mele model, FIG. 1. Energy dispersion for σ = 1 (up spin) of the model in =− † + † − † → → → H tciαc jα 0 ciαciα ciαciα Eq. (11) along the path K K for two typical√ choices / = / , / = / i, j α i∈A,α i∈B,α of parameters: (a) solid line, 0 t 1 4 t2 t 3 36√ (topologically trivial), and (b) dashed line, /t = 1/4, t /t = 3/12 + ν † z , 0 2 it2 ijciαsαβc jβ (11) (topologically nontrivial). The energy dispersion for σ =−1(down i, j ,αβ spin) is obtained by exchanging K for K points. Inset: Phase diagram † α of this model. where ciα is the creation operator of an electron with spin at site i, and the summation i, j (i, j ) runs over all the √ nearest- (next-nearest-) neighbor sites of the 2D honeycomb momentum√ space, gaps open at K = (4π/3 3a, 0)√ and K = lattice. The first term represents the usual nearest-neighbor (−4π/3 3a, 0). Their√ magnitudes are 2| 0 + (3 3/2)σt2| hopping with transfer integral t. The second term represents a at K and 2| 0 − (3 3/2)σt2| at K , respectively. + − staggered on-site potential, 0 for A sublattice and 0 for In this model, the ratio of 0 to t2 determines the topolog-√ B sublattice. The last term represents the hopping originating ical order [49–51]: topologically trivial for |t2/ √ 0| < 2/3 3 z from SOI. We take account of only the s component as in and topologically nontrivial for |t2/ 0| > 2/3 3. The phase Ref. [2] and set νij =−ν ji =+1(−1) if the electron makes diagram is shown in the inset of Fig. 1. a left (right) turn to propagate to the next-nearest sites. This Before calculating magnetic susceptibility, let us examine model is known as one for silicene [47–51]. We can control the Berry curvature. Figure 2 shows the distribution of the 0 by changing the electric field applied perpendicular to the Berry curvature in the momentum space for the valence band layer due to its buckled structure. electrons with up spin for the two choices of the parameters Performing the Fourier transform, we obtain the Hamilto- in Fig. 1. It is seen that the Berry curvature is localized near nian in a matrix form, K and K points. After numerical integration, we find that the ∗ Chern numbers are 0 for Fig. 2(a) and 1 for Fig. 2(b), which 0 + σλk −tγ H σ = k , (12) k −tγ − − σλ is consistent with the fact that Figs. 2(a) and 2(b) belong to k 0 k the topologically trivial and nontrivial phases, respectively. where σ = 1 is for up-spin and σ =−1 is for down-spin. The According to Figs. 1 and 2, the low-energy excitations in complex factor γk, the vicinity of K and K points are important when μ 0. √ √ 3 1 3 1 Therefore, we approximate the Hamiltonian by the expansion −ikya i( kx + ky )a i(− kx + ky )a γk = e + e 2 2 + e 2 2 , (13) comes from the Fourier transform of hopping with a being the distance between the nearest-neighbor sites, and √ √ 3 3 3 λ = 2t sin k a cos k a − cos k a (14) k 2 2 x 2 y 2 x comes from the spin-dependent hopping, t2. The energy dispersion of this model is given by ± =± + σλ 2 + 2|γ |2, Eσ k ( 0 k ) t k (15) √ which is shown in Fig. 1(a) with 0/t = 1/4, t2/t = 3/36 (topologically√ trivial; solid line) and in Fig. 1(b) with 0/t = 1/4, t2/t = 3/12 (topologically nontrivial; dashed line). In FIG. 2. Distribution of Berry curvature in the momentum space the following, we use Figs. 1(a) and 1(b) as typical cases. for the valence band electrons with up spin for case of (a) (topolog- Since the space inversion symmetry is broken, the energy ically trivial) and (b) (topologically nontrivial). The Chern numbers dispersions for up and down spins can be different. In the are 0 for (a) and 1 for (b).

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FIG. 4. Schematic pictures for the edge state. (a) Ground state without an external magnetic field. (b) Change caused by the Zeeman interaction. (c) New ground state in the magnetic field. √FIG. 3. Contributions to magnetic susceptibility as a function of / / μ = = / (3 3 2)t2 0 for 0 with 0 t 4. They are normalized by √ χ 0. The values of vF and a are chosen to be the same as those from Eq. (17c). When (3 3/2)t2/ 0 > 1, on the other hand, K of graphene.√ The system is topologically trivial (nontrivial) in the the system is topologically nontrivial and the signs of ↑ and region (3 3/2)t2/ 0 < 1(> 1). K ↑ are the same, which leads to χOZ =−χ0. As a result, χOZ has a universal√ jump at the topological phase transition around K and K points, i.e., k · p perturbation. In this way, at t2/ 0 = 2/3 3. On the other hand, χorbit diverges at the we obtain a low-energy effective model, phase transition due to the gap closing. However, we can see 2 2 √ √ e vF −1 −1 that χ =− (| − 3 3t /2| +| + 3 3t /2| ) hK/K = K/K − σαk2 τ + h¯v k τ + h¯v k τ , (16) orbit 6π 0 2 0 2 σ σ z F x x F y y and the magnitude of divergence is the same on both sides where τx, τy, and τz are the Pauli matrices representing the of the phase transition. Therefore, when we subtract the di- K χ χ degrees of freedom of sublattices A and B, σ = 0 + vergence of orbit, we will be able to detect the jump in OZ. √ √ √ K 2 Note that the effect of SOI represented by t2 appears only in σ (3 3/2)t2, σ =− 0 + σ (3 3/2)t2, α = (9 3/8)a t2, χ = / K/K the magnitude of orbit and does not affect the magnitude of andh ¯vF (3 2)at. Note that the signs of the mass term σ χ at K and K points with different spins are opposite, i.e., OZ. K K K K ↑ =− ↓ and ↓ =− ↑ . This effective Hamiltonian is , → / justified in the limit of t2 0 0 with t2 0 fixed. V. PHYSICAL ORIGIN OF QUANTIZATION Let us discuss the magnetic susceptibility. Note that Ezawa [50] calculated the orbital magnetism but the OZ cross term Let us consider the physical origin of the quantization. was not taken into account. In the model Eq. (16), the ther- χOZ is interpreted as the sum of the correction to the orbital K/K K/K −1 mal Green’s function is defined as Gσ = (iεn − hσ ) . magnetic moment induced by the magnetic field that couples The current operator in the μ direction is given by γμ = to the spin magnetic moment and the correction to the spin h¯vF τμ − 2σαkμτz. Substituting these quantities into Eq. (2), magnetic moment induced by the magnetic field that couples carrying out the Matsubara summation, and performing the to the orbital magnetic moment. Here we estimate the former 2D momentum integration at T = 0, we obtain correction. In the edge state, there are spin-polarized linear dispersions and a spin current flows. Figure 4(a) corresponds 2 2 e v 1 η F to the state with Chs,l =+1. (Note that the number of pairs χorbit =− η θ(| ↑|−|μ|), (17a) 6π | | | , | η=K,K ↑ of dispersion coincides with Chs l .) When a magnetic field is applied through the Zeeman interaction μBσzB, the up- 2 μ η χ = B |μ|θ(|μ|−| |), (17b) spin (down-spin) band moves upward (downward) as shown Pauli 2 2 ↑ π in Fig. 4(b). The width of change is μBB. Then, in the h¯ vF η=K,K lowest energy state [Fig. 4(c)], the number of down-spin 2μ |e| η η νμ ν χ =− B sgn( )θ(| |−|μ|), (17c) (up-spin) electrons increases (decreases) by BB, where OZ h ↑ ↑ is the density of states ν = L/ch and c is the velocity of η=K,K the edge current. This change leads to an electric current of in the limit of t2, 0 → 0 with t2/ 0 fixed. Here χorbit is the −2|e|μBB/h in the right direction, which causes the orbital orbital diamagnetic susceptibility of the 2D Dirac electrons magnetic moment of −2|e|μBB/h per area. The coefficient discussed in the preceding studies [23–29]. χPauli is the Pauli of B is half of the quantization of magnetic susceptibility, paramagnetism proportional to the density of states (∝|μ|). −χ0/2. We can also estimate the contribution from the other To observe the quantization of the jump, we focus on correction (i.e., the spin magnetic moment induced by an en- the case of μ = 0, an insulating case, where χPauli vanishes. ergy shift originating from an orbital magnetic moment made Figure√ 3 shows χorbit and χOZ as a function of t2/ 0. When by a circular electric current), which gives the same value. (3 3/2)t2/ 0 < 1, the system is topologically trivial and the Combining these two contributions, we obtain the OZ cross K K signs of ↑ and ↑ are opposite, which leads to χOZ = 0 term as −χ0, which is consistent with Eq. (9).

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VI. DISCUSSION AND CONCLUSION APPENDIX: DERIVATION OF THE MAGNETIC SUSCEPTIBILITY FORMULA FOR PERIODIC SYSTEMS The expression for the magnetic susceptibility including the spin Zeeman effect was obtained in terms of 1. 3D cases χ = Bloch wave functions [35], which contains a term occ:2 We consider Bloch electrons in a 3D periodic potential − e ε z z. χ χ h¯ Re lk f ( k )Mll l Actually, occ:2 gives half of OZ in V (r), Zeeman interaction, and SOI. For this purpose, we use Eq. (9). The origin of this difference is as follows. In Ref. [35], the following Hamiltonian derived from the Dirac equation in the effect of Zeeman interaction is distributed among several the presence of a magnetic field: terms for total magnetic susceptibility including χocc:2. There- fore, if we collect all the effects of Zeeman interaction in the 1 2 eh¯ formalism of Ref. [35], then we can recover χ . H = [p − eA(r)] − σ · B(r) + V (r) OZ 2m 2m Based on the microscopic theory, we have derived a new h¯2 h¯ simple formula for magnetic susceptibility in a Bloch system + ∇2V + σ ·∇V × [p − eA(r)], (A1) with SOI and Zeeman interaction to show that the OZ cross 8m2c2 4m2c2 term, one of the three contributions to magnetic susceptibility, < is always quantized in units of the universal value 4|e|μ /h where A(r) is a vector potential, e 0 for electrons, and B =∇× for 2D spin-conserving insulators at zero temperature. We B(r) A(r) represents a magnetic field. The g factor for = have also applied the formula to a model for a 2D TI and electrons is assumed to be g 2. demonstrated the quantization. For this model, we have clari- The magnetic susceptibility per volume is defined as fied that this quantization originates from the redistribution of 1 ∂2 the chiral edge state due to the magnetic field. We expect that χ =− lim , (A2) this discussion is also qualitatively applicable to other 2D TIs. L3 B→0 ∂B2 Our results clearly show that the magnetic response reflects the topological nature of a material. Therefore, it should be where is the thermodynamic potential of the system with possible to make a bulk-sensitive confirmation of the topolog- chemical potential μ, i.e., ical phase transition. =−kBT ln Tr exp[−β(H − μN)]. (A3) ACKNOWLEDGMENTS Therefore, it is sufficient to evaluate the second-order deviation of due to a magnetic field, defined as (2). We thank H. Matsuura, H. Maebashi, I. Tateishi, To perform the pertubative calculations with respect to A T. Hirosawa, N. Okuma, and V. Könye for very fruitful and B as was done by Fukuyama [20], we rewrite the Hamil- discussions. This work was supported by Grants-in-Aid for tonian in a second quantized form as Scientific Research from the Japan Society for the Promotion of Science (Grant No. JP18H01162). S.O. was supported by H = H0 + H1 + H2 (A4) the Japan Society for the Promotion of Science through the Program for Leading Graduate Schools (MERIT). with

2 2 † h¯ 2 h¯ 2 h¯ H = ψ (r) − ∇ + V (r) + ∇ V (r) + σαβ ·∇V × p ψβ (r)dr 0 α 2 2 2 2 αβ 2m 8m c 4m c † eh¯ H1 =− j(r) · A(r)dr − ψα (r) σαβ · B(r)ψβ (r)dr αβ 2m 2 e 2 † H2 = A (r)ψα (r)ψβ (r)dr, (A5) α 2m where j(r) is the current operator without a vector potential, Hebborn and Sondheimer [16],

iqr −iqr e † † A(r) =−iAq(e − e ), (A7) j = {ψα (pψα ) − (pψα )ψα} 2m α and let q = 0 in the ﬁnal expressions. eh¯ † To make the pertubative calculation easier, we employ the + ψασαβ ×∇V ψβ . (A6) 4m2c2 Luttinger-Kohn representation [44], αβ ikr ψα (r) = χ α (r)a α,χα (r) = e u α (r), (A8) The subscript of each Hamiltonian represents the order of lk lk lk lk0 lkα A. To avoid unphysical contributions due to the unbounded t character of the coordinate operators, we introduce the vec- where k0 is a ﬁxed wave vector andu ˆlk(r) = (ulk↑(r), ulk↓(r)) tor potential A with Fourier component q, as was done by is the periodic part of the Bloch wave function with the wave

013058-5 SOSHUN OZAKI AND MASAO OGATA PHYSICAL REVIEW RESEARCH 3, 013058 (2021) vector k of the lth band.u ˆlk satisﬁes By using the Luttinger-Kohn representation, the thermodynamic potential becomes H uˆ (r) = ε uˆ (r)(A9) k lk lk lk 1 with dλ λ λ = + k T k, iε G k, iε , 0 B λ Tr[ ( n ) ( n )] h¯2 h¯2 0 H = (k − i∇)2 + V (r) + ∇2V kn k 2m 8m2c2 (A16) 2 h¯ + σ ·∇V × (k − i∇). (A10) where 0 is the thermodynamic potential without a vector 4m2c2 potential. λ and Gλ are the self-energy matrix and ther- In this representation, H0, H1, and H2 are rewritten as mal Green’s function corresponding to the Hamiltonian H0 + λ H + H † ( 1 2 ), respectively. We expand the self-energy as = , H0 Ell αβalkαal kβ λ llαβk = λ (1) + λ2 (2) + O(A3). (A17) ie † † = · γ q − q H1 Aq (k) alk+ q αal k− β alk− q αal k+ β Making use of Dyson’s equation, h¯ 2 2 2 2 llαβk Gλ(k, iε ) = G(0)(k, iε ) + G(0)(k, iε )λ(k, iε )Gλ(k, iε ), s † † n n n n n − × · q + q , q Aq M a + q αalk− β a − q αalk+ β (A18) lk 2 2 lk 2 2 llαβk 2 with G(0) being Green’s function for the nonperturbative e 2 H2 =− A (ρ−2qα − 2ρ0α + ρ2qα ), (A11) Hamiltonian H , we can carry out the integral as 2m q 0 α 1 where − = k T Tr (1)G(0) + (2)G(0) + O(A3). 0 B 2 ∗ kn E αβ = u H u β dr, (A12) ll lk0α k l k0 (A19) ∗ ∂Hk To estimate Eq. (A19), we ﬁnd nine types of contributions [γ (k)]lα,lβ = u α ulk β dr, (A13) lk0 ∂k 0 of the order of A2, whose diagrams are shown in Fig. 5. Figures 5(1a)–5(1c), 5(2a)–5(2d), and 5(3a) and 5(3b) give s ∗ eh¯ [M ] α, β = u σu β dr, (A14) the orbital contribution, the orbital-Zeeman cross term con- l l lk0α l k0 2m tribution, and the Zeeman contribution to the free energy, ρ = † . q alk−qαalkα (A15) respectively. Each contribution is calculated as lkα

2 2 (2) 1 e q q e 2 = kBT AqμAqν Tr γμG k + , iεn γν G k − , iεn + (q ↔−q) + kBT A TrG(k, iεn ), (A20) orbit 2 h¯2 2 2 m q nk nk (2) 1 ie q s q =− k T Tr A μγμG k + , iε q × A · M G k − , iε + (q ↔−q) OZ 2 B h¯ q 2 n q 2 nk 1 s q ie q + k T Tr q × A · M G k + , iε A μγμG k − , iε + (q ↔−q) , (A21) 2 B q 2 n h¯ q 2 n nk 1 q q (2) = k T Tr q × A · MsG k + , iε q × A · MsG k − , iε + (q ↔−q) , (A22) Zeeman 2 B q 2 n q 2 n nk respectively. By expanding the expression for small q with the help of the Ward identity,

∂G(k, iεn ) G(k, iεn )γμG(k, iεn ) = , (A23) ∂kμ we obtain 2 (2) e 2 =− (q × Aq) kBT Tr[γxGγyGγxGγyG] orbit 2¯h2 z nk ie (2) =− (q × A )2k T Tr MsGγ Gγ G − MsGγ Gγ G OZ h¯ q z B z x y z y x nk (2) = × 2 sG sG . Zeeman (q Aq)z kBT Tr Mz Mz (A24) nk

013058-6 UNIVERSAL QUANTIZATION OF THE MAGNETIC … PHYSICAL REVIEW RESEARCH 3, 013058 (2021)

FIG. 5. Feynman diagrams for the second-order deviation of the free energy (2).

× 2 2 2. 2D cases Since 2(q Aq)z corresponds to Bz , the magnetic susceptibility becomes In the derivation in the previous section, we have assumed a 3D periodic potential. However, when the magnetic field is χ = χorbit + χOZ + χPauli, (A25) applied in the z direction, the assumption of periodicity in the z direction is unnecessary, as shown in this section. Therefore, where the same discussion as in the 3D case is applicable to 2D systems, where electrons are confined in the z direction. 2 e kBT We consider the case where the elctronic potential χorbit = Tr[γxGγyGγxGγyG] 2¯h2 L3 V (x, y, z) is periodic in the x-y plane perpendicular to the nk magnetic field. Hereafter, we use bold for two-component ie k T B s s vectors, e.g., k = (kx, ky ). From Bloch’s theorem, the wave χ = Tr M GγxGγyG − M GγyGγxG OZ h¯ L3 z z function is characterized by k, which is written as nk kBT s s ψˆ , , = ikr , , , χ =− Tr M GM G . (A26) k(x y z) e uˆk(x y z) (A27) Pauli L3 z z nk whereu ˆk(x, y, z) is a function which has the same periodicity Since the Luttinger-Kohn representation is associated with as V (x, y, z)inthex-y plane. Since Eq. (A27) is the solution the Bloch representation by a unitary transformation [44], of the Schrödinger equation the expressions in Eq. (A26) can be regarded as written in terms of Green’s functions and current operators for the Bloch Hψˆk(x, y, z) = Ekψˆk(x, y, z), (A28) representation.

013058-7 SOSHUN OZAKI AND MASAO OGATA PHYSICAL REVIEW RESEARCH 3, 013058 (2021) uˆk(x, y, z) satisﬁes the equation ⎡ h¯2 ∂ 2 ∂ 2 ∂2 ⎣ k − i + k − i − + V (x, y, z) 2m x ∂x y ∂y ∂z2 ⎛ ⎞⎤ − ∂/∂ h¯2 h¯2 kx i x + ∇2V + σ ·∇V × ⎝k − i∂/∂y⎠⎦uˆ (x, y, z) = E uˆ (x, y, z). (A29) 2 2 2 2 y k k k 8m c 4m c −i∂/∂z

Since the wave functionu ˆk(x, y, z) is defined on a region [0, L] × [0, L] × (−∞, ∞) and is normalizable on the region, the energy eigenvalues take descrete values. When V (x, y, z) gives a confinement potential in the z direction, the wave functions are characterized by the band index in the kx-ky Brillouin zone and the subband index in the z direction. Therefore, we denote the wave function asu ˆlk(x, y, z), where l represents the above two indices. Usingu ˆlk’s, we can apply the same discussion as in the previous section and derive a similar formula. The Luttinger-Kohn representation is modified: ψα (x, y, z) = χlkα (x, y, z)alkα, lkα χ , , = ikr , , . lkα (x y z) e ulk0α (x y z) (A30)

When the magnetic ﬁeld is parallel to the z direction, we can assume without loss of generarity that both the vector potential Aq and q have only x and y components. Then, H1 is expressed in this 2D scheme as ie † † = · γ q − q H1 Aq (k) alk+ q αal k− β alk− q αal k+ β h¯ 2 2 2 2 llαβk s † † − − q + q , (qxAqy qyAqx ) Mz a + q αalk− β a − q αalk+ β (A31) lk 2 2 lk 2 2 llαβk

γ s while H0 and H2 do not change formally. We note that the spatial integrals in matrix elements Ell αβ, (k), and Mz are 3D integrals. After all, we obtain a very similar formula 2 e kBT χorbit = Tr[γxGγyGγxGγyG] 2¯h2 L2 nk ie k T χ = B Tr MsGγ Gγ G − MsGγ Gγ G OZ h¯ L2 z x y z y x nk k T χ =− B Tr MsGMsG (A32) Pauli L2 z z nk where the value is normalized by L2, different from the 3D case, and Tr is the trace over l and α.

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