Fermi-Surface Curvature and Hall Conductivity in Metals
aa r X i v : . [ c ond - m a t . o t h e r] N ov Fermi-Surface Curvature and Hall Conductivityin Metals
Osamu Narikiyo ∗ Abstract
The Tsuji formula which relates the Fermi-surface curvature and the weak-fieldHall conductivity in metals is discussed in Haldane’s framework.
The Tsuji formula [1] relates the Fermi-surface curvature and the weak-field Hallconductivity in metals. Although it is widely known as a geometrical formula, itsintelligible derivation has not been reported. For example, in the Appendix of aninfluential textbook [2] the author reported the correspondence with Tsuji but it doesnot seem a derivation. I think that not only the derivation but also the meaning of theTsuji formula becomes clear when we employ Haldane’s framework. [3] Thus we willdiscuss the Tsuji formula in Haldane’s framework.While the Tsuji formula [1] was derived under the assumption of the cubic symmetry,Haldane [3] tried to eliminate the assumption. However, we will criticize the trial.This paper is organized as follows. In the section 2 we start from the Boltzmanntransport theory. In the section 3 we focus on the contribution from the Fermi surface.We review and criticize Haldane’s result in the section 4. In the section 5 we analyzethe Tsuji formula in Haldane’s framework. Another geometrical formula by Ong [4] isbriefly mentioned in the section 6. We give short summary in the last section.
The weak-field DC Hall conductivity 𝜎 𝑥𝑦 per spin is given by 𝜎 𝑥𝑦 = 𝑒 Õ 𝒌 (cid:18) − 𝜕 𝑓𝜕𝜀 (cid:19) 𝑙 𝑥 ( 𝑩 × 𝒗 ) · 𝜕𝜕 𝒌 𝑙 𝑦 , (1)using the solution of the linearized Boltzmann equation. [2, 5] Here the magnetic field 𝑩 is a constant vector. The mean free path vector 𝒍 = ( 𝑙 𝑥 , 𝑙 𝑦 , 𝑙 𝑧 ) is given as 𝒍 = 𝜏 𝒗 ∗ Department of Physics, Kyushu University 𝒗 = ( 𝑣 𝑥 , 𝑣 𝑦 , 𝑣 𝑧 ) is the velocity of the quasi-particle and 𝜏 is its renormalizedtransport life-time. 𝜕 𝑓 / 𝜕𝜀 is the derivative of the Fermi distribution function 𝑓 withrespect to the quasi-particle energy 𝜀 . Although we have suppressed the argument of 𝒗 , 𝜏 and 𝜀 , they are the functions of the position 𝒌 = ( 𝑘 𝑥 , 𝑘 𝑦 , 𝑘 𝑧 ) in 𝒌 -space. Thecomponent of the quasi-particle velocity 𝑣 𝑎 ( 𝑎 = 𝑥, 𝑦, 𝑧 ) is given as 𝑣 𝑎 = 𝜕𝜀 / 𝜕 𝑘 𝑎 ≡ 𝜀 𝑎 .It should be noticed that Eq. (1) is not the consequence of the relaxation-time approx-imation but the effects of the collision term is fully taken into account by renormalizingthe life-time. [6] Thus Eq. (1) is in accordance with the result of the Fermi-liquidtheory. [7]The off-diagonal conductivity, Eq. (1), satisfies the Onsager’s reciprocal relation: 𝜎 𝑥𝑦 = − 𝜎 𝑦𝑥 with 𝜎 𝑦𝑥 = 𝑒 Õ 𝒌 (cid:18) − 𝜕 𝑓𝜕𝜀 (cid:19) 𝑙 𝑦 ( 𝑩 × 𝒗 ) · 𝜕𝜕 𝒌 𝑙 𝑥 . (2)Actually the integration by parts [5] for 𝜎 𝑥𝑦 leads to − 𝜎 𝑦𝑥 , since 𝜕𝜕 𝒌 𝜕 𝑓𝜕𝜀 · ( 𝑩 × 𝒗 ) = , (3)and 𝜕𝜕 𝒌 · ( 𝑩 × 𝒗 ) = . (4)Eq. (3) and Eq. (4) are the consequences of 𝒗 = 𝜕𝜀 / 𝜕 𝒌 .In the following we switch the factor ( 𝑩 × 𝒗 ) · 𝜕 / 𝜕 𝒌 to its equivalence ( 𝒗 × 𝜕 / 𝜕 𝒌 ) · 𝑩 .Then Eq. (1) is equivalent to 𝜎 𝑥𝑦 = 𝑒 Õ 𝒌 (cid:18) − 𝜕 𝑓𝜕𝜀 (cid:19) 𝑙 𝑥 (cid:18) 𝒗 × 𝜕𝜕 𝒌 (cid:19) · 𝑩 𝑙 𝑦 . (5)If we measure the Hall current in 𝑥 -direction, 𝐽 𝑥 , under the electric field in 𝑦 -direction, 𝑬 = ( , 𝐸 𝑦 , ) , and the magnetic field in 𝑧 -direction, 𝑩 = ( , , 𝐵 𝑧 ) , 𝐽 𝑥 = 𝜎 𝑥𝑧𝑦 𝐵 𝑧 𝐸 𝑦 with 𝜎 𝑥𝑧𝑦 = 𝑒 Õ 𝒌 (cid:18) − 𝜕 𝑓𝜕𝜀 (cid:19) 𝑙 𝑥 (cid:18) 𝒗 × 𝜕𝜕 𝒌 (cid:19) 𝑧 𝑙 𝑦 . (6)Since 𝜎 𝑥𝑧𝑦 = − 𝜎 𝑦𝑧𝑥 , we use 𝑒 𝛾 𝑥𝑧𝑦 ≡ ( 𝜎 𝑥𝑧𝑦 − 𝜎 𝑦𝑧𝑥 )/ 𝜎 𝑥𝑧𝑦 . The anti-symmetric property, 𝛾 𝑥𝑧𝑦 = − 𝛾 𝑦𝑧𝑥 , is evident in 𝛾 𝑥𝑧𝑦 = Õ 𝒌 (cid:18) − 𝜕 𝑓𝜕𝜀 (cid:19) (cid:20) 𝑙 𝑥 (cid:18) 𝒗 × 𝜕𝜕 𝒌 (cid:19) 𝑧 𝑙 𝑦 − 𝑙 𝑦 (cid:18) 𝒗 × 𝜕𝜕 𝒌 (cid:19) 𝑧 𝑙 𝑥 (cid:21) . (7)Using 𝜎 𝑥𝑧𝑦 = 𝑒 Õ 𝒌 (cid:18) − 𝜕 𝑓𝜕𝜀 (cid:19) 𝜏𝑣 𝑥 (cid:18) 𝑣 𝑥 𝜕𝜕 𝑘 𝑦 − 𝑣 𝑦 𝜕𝜕 𝑘 𝑥 (cid:19) 𝜏𝑣 𝑦 . (8)we obtain 𝛾 𝑥𝑧𝑦 = Õ 𝒌 (cid:18) − 𝜕 𝑓𝜕𝜀 (cid:19) ( 𝑣 𝑥 , 𝑣 𝑦 ) (cid:18) 𝑀 − 𝑦𝑦 − 𝑀 − 𝑦𝑥 − 𝑀 − 𝑥𝑦 𝑀 − 𝑥𝑥 (cid:19) (cid:18) 𝑣 𝑥 𝑣 𝑦 (cid:19) 𝜏 (9)2here 𝑀 − 𝑎𝑏 ≡ 𝜕 𝜀𝜕 𝑘 𝑎 𝜕 𝑘 𝑏 ≡ 𝜀 𝑎𝑏 , (10)is the effective mass tensor. Through the subtraction, 𝜎 𝑥𝑦𝑧 − 𝜎 𝑦𝑥𝑧 , the derivatives of 𝜏 cancel out. This result (9) is equivalent to that of the diagrammatic analysis [7] ofthe linear response of the Fermi liquid. Thus the expression for the Hall conductivitywhich contains the derivatives of 𝜏 , for example (2.53) in Ref. [2], is a bad expression.Eq. (9) is the general result for the Hall conductivity so that you have only toestimate it numerically if you are not interested in its geometrical interpretation. In the case of Fermi degeneracy we can estimate − 𝜕 𝑓 / 𝜕𝜀 by the delta function: ∫ d 𝒌 (cid:18) − 𝜕 𝑓𝜕𝜀 (cid:19) = ∫ d 𝑆 | 𝒗 | , (11)where 𝒗 = 𝜕𝜀 / 𝜕 𝒌 and the integral in the right-hand-side is over the Fermi surface.Thus 𝛾 𝑥𝑧𝑦 is determined by the integration over the Fermi surface: 𝛾 𝑥𝑧𝑦 = ∫ d 𝑆 ( 𝜋 ) ( 𝑣 𝑥 , 𝑣 𝑦 ) (cid:18) 𝑀 − 𝑦𝑦 − 𝑀 − 𝑦𝑥 − 𝑀 − 𝑥𝑦 𝑀 − 𝑥𝑥 (cid:19) (cid:18) 𝑣 𝑥 𝑣 𝑦 (cid:19) 𝜏 | 𝒗 | , (12)and reflects the geometry of the Fermi surface. Actually it is shown in the followingthat 𝛾 𝑥𝑧𝑦 is a sampling of the local curvatures of the Fermi surface.Throughout this paper we only consider the contribution from a single sheet of theFermi surface. In the case of multi-sheets we should sum the contributions from all thesheets. [3] Haldane used the framework 𝜎 𝑥𝑦 ≡ 𝑒 Õ 𝑎 Õ 𝑏 𝜖 𝑥𝑦𝑎 𝛾 𝑎𝑏 𝐵 𝑏 , (13)where 𝜖 𝑎𝑏𝑐 is the antisymmetric rank-3 Levi-Civita symbol. Since Haldane derived 𝛾 𝑎𝑏 from (1), the last equation in p. 2 of Ref. 3, 𝜖 𝑥𝑦𝑧 𝛾 𝑧𝑧 should be equal to our 𝛾 𝑥𝑧𝑦 .However, it is different from ours: 𝛾 𝑧𝑧 = ∫ d 𝑆 ( 𝜋 ) ( 𝑛 𝑥 , 𝑛 𝑦 ) (cid:18) 𝜅 𝑦𝑦 − 𝜅 𝑦𝑥 − 𝜅 𝑥𝑦 𝜅 𝑥𝑥 (cid:19) (cid:18) 𝑛 𝑥 𝑛 𝑦 (cid:19) 𝑙 , (14)where 𝒏 is the unit vector, 𝒏 ≡ 𝒗 /| 𝒗 | ≡ ( 𝑛 𝑥 , 𝑛 𝑦 , 𝑛 𝑧 ) , which is normal to the Fermisurface, 𝜅 𝑎𝑏 ≡ 𝜕𝑛 𝑎 / 𝜕 𝑘 𝑏 and 𝑙 = | 𝒍 | = | 𝒗 | 𝜏 . For example, ours needs only ( 𝜕𝑣 𝑥 / 𝜕 𝑘 𝑦 )/| 𝒗 | but Haldane’s needs 𝜕 ( 𝑣 𝑥 /| 𝒗 |)/ 𝜕 𝑘 𝑦 . As discussed in the Appendices3 𝑎𝑏 is a key ingredient of the geometrical description but its appearance in (14) isincorrect. Since it has no geometrical meaning, the life-time 𝜏 is safely separated fromthe other factors related to the shape of the Fermi surface as (12). On the other hand,in Eq. (14) the mean free path 𝑙 which still contains the geometrical information of theFermi surface is separated. Such a separation is a bad strategy and cannot be justified.Anyway Eq. (14) is not derived from the solution of the Boltzmann equation (1).The information on the geometry of the Fermi surface can be obtained from the3 × © « 𝜅 𝑥𝑥 𝜅 𝑦𝑥 𝜅 𝑧𝑥 𝜅 𝑥𝑦 𝜅 𝑦𝑦 𝜅 𝑧𝑦 𝜅 𝑥𝑧 𝜅 𝑦𝑧 𝜅 𝑧𝑧 ª®¬ . (15)It can be diagonalized as © « 𝜅 𝜅
00 0 0 ª®¬ . (16)The pair of the eigenvalues ( 𝜅 and 𝜅 ) is the basis of the geometrical interpretation: 𝐺 = 𝜅 𝜅 and 2 𝐻 = 𝜅 + 𝜅 where 𝐺 is the Gaussian curvature and 𝐻 is the meancurvature.Especially the trace 2 𝐻 , which is independent of the choice of the local coordinate,is the target in the next section. Our master equation (12) is rewritten as 𝛾 𝑥𝑧𝑦 = ∫ d 𝑆 ( 𝜋 ) ℎ 𝑧 𝜏 , (17)where ℎ 𝑧 is determined by the derivative of the function 𝜀 which determines the shapeof the Fermi surface: ℎ 𝑧 = | 𝒗 | (cid:0) 𝜀 𝑥 𝜀 𝑥 𝜀 𝑦𝑦 + 𝜀 𝑦 𝜀 𝑦 𝜀 𝑥𝑥 − 𝜀 𝑥 𝜀 𝑦 𝜀 𝑦𝑥 − 𝜀 𝑦 𝜀 𝑥 𝜀 𝑥𝑦 (cid:1) , (18)with 𝜀 𝑎 = 𝜕𝜀 / 𝜕 𝑘 𝑎 and 𝜀 𝑎𝑏 = 𝜕 𝜀 / 𝜕 𝑘 𝑎 𝜕 𝑘 𝑏 . This ℎ 𝑧 reflects the Fermi-surfacegeometry but 𝜏 has no geometrical meaning.On the other hand, the trace 2 𝐻 = 𝜅 𝑥𝑥 + 𝜅 𝑦𝑦 + 𝜅 𝑧𝑧 of the 3 × 𝐻 = | 𝒗 | h 𝜀 𝑥 𝜀 𝑥 ( 𝜀 𝑦𝑦 + 𝜀 𝑧𝑧 ) + 𝜀 𝑦 𝜀 𝑦 ( 𝜀 𝑧𝑧 + 𝜀 𝑥𝑥 )+ 𝜀 𝑧 𝜀 𝑧 ( 𝜀 𝑥𝑥 + 𝜀 𝑦𝑦 ) − 𝜀 𝑥 ( 𝜀 𝑦 𝜀 𝑦𝑥 + 𝜀 𝑧 𝜀 𝑧𝑥 )− 𝜀 𝑦 ( 𝜀 𝑥 𝜀 𝑥𝑦 + 𝜀 𝑧 𝜀 𝑧𝑦 ) − 𝜀 𝑧 ( 𝜀 𝑥 𝜀 𝑥𝑧 + 𝜀 𝑦 𝜀 𝑦𝑧 ) i , (19)as shown in the Appendix. The mean curvature 𝐻 is given by this expression (19) forany shape of the Fermi surface. 4y comparing (18) and (19) we see that ℎ 𝑧 is a piece of 𝐻 . By summing threepieces we can construct 𝐻 : 𝐻 = ( ℎ 𝑧 + ℎ 𝑥 + ℎ 𝑦 )/| 𝒗 | where ℎ 𝑥 = | 𝒗 | (cid:0) 𝜀 𝑦 𝜀 𝑦 𝜀 𝑧𝑧 + 𝜀 𝑧 𝜀 𝑧 𝜀 𝑦𝑦 − 𝜀 𝑦 𝜀 𝑧 𝜀 𝑧𝑦 − 𝜀 𝑧 𝜀 𝑦 𝜀 𝑦𝑧 (cid:1) , (20)and ℎ 𝑦 = | 𝒗 | ( 𝜀 𝑧 𝜀 𝑧 𝜀 𝑥𝑥 + 𝜀 𝑥 𝜀 𝑥 𝜀 𝑧𝑧 − 𝜀 𝑧 𝜀 𝑥 𝜀 𝑥𝑧 − 𝜀 𝑥 𝜀 𝑧 𝜀 𝑧𝑥 ) . (21)In experiments ℎ 𝑧 is related to the measurement: 𝐽 𝑥 = 𝑒 𝛾 𝑥𝑧𝑦 𝐵 𝑧 𝐸 𝑦 . In thesame manner ℎ 𝑥 and ℎ 𝑦 are related to the measurements: 𝐽 𝑦 = 𝑒 𝛾 𝑦𝑥𝑧 𝐵 𝑥 𝐸 𝑧 and 𝐽 𝑧 = 𝑒 𝛾 𝑧𝑦𝑥 𝐵 𝑦 𝐸 𝑥 where 𝛾 𝑦𝑥𝑧 = ∫ d 𝑆 ( 𝜋 ) ℎ 𝑥 𝜏 , (22)and 𝛾 𝑧𝑦𝑥 = ∫ d 𝑆 ( 𝜋 ) ℎ 𝑦 𝜏 . (23)By summing three experimental results with different configurations we obtain 𝛾 𝑥𝑧𝑦 + 𝛾 𝑦𝑥𝑧 + 𝛾 𝑧𝑦𝑥 = ∫ d 𝑆 ( 𝜋 ) 𝐻𝑙 , (24)where 𝑙 = | 𝒗 | 𝜏 . This relation holds for any shape of the Fermi surface.In the case of cubic symmetry (24) reduces to the Tsuji formula 𝛾 𝑥𝑧𝑦 = 𝛾 𝑦𝑥𝑧 = 𝛾 𝑧𝑦𝑥 = ∫ d 𝑆 ( 𝜋 ) 𝐻 𝑙 . (25)Now we are at the position from which we can guess why Haldane expected (14).In the case of cubic symmetry the integrand in (25) is the product of the scalar 𝐻 andthe square of the mean free path 𝑙 . Haldane introduced a generalized expression byreplacing the scalar with a tensor keeping 𝑙 in the integrand. However, in the case ofgeneral symmetry, 𝑙 cannot be factored out. We can only factor out 𝜏 . Although Haldane [3] discusses the relation between 2D and 3D formulae, we shalldiscuss the 2D case as a separate issue.For simplicity we put 𝑩 = ( , , 𝐵 ) and set the 2D system in 𝑥𝑦 -plane. The 2Dversion of (1) on the Fermi line is given as 𝜎 𝑥𝑦 = 𝑒 ∫ d 𝑘 𝑡 ( 𝜋 ) 𝑙 𝑥 (cid:20) ( 𝑩 × 𝒏 ) · 𝜕𝜕 𝒌 (cid:21) 𝑧 𝑙 𝑦 , (26)where d 𝑘 𝑡 is the length along the Fermi line. Since 𝑩 × 𝒏 = 𝐵 𝒕 and 𝒕 · ( 𝜕 / 𝜕 𝒌 ) = 𝜕 / 𝜕 𝑘 𝑡 , ( 𝑩 × 𝒏 ) · ( 𝜕 / 𝜕 𝒌 ) = 𝐵𝜕 / 𝜕 𝑘 𝑡 where 𝒕 is the unit tangent vector along the Fermi line, Eq.(26) is written as 𝜎 𝑥𝑦 = 𝑒 𝐵 ( 𝜋 ) ∫ 𝑙 𝑥 d 𝑙 𝑦 , (27)5here d 𝑙 𝑦 = ( 𝜕𝑙 𝑦 / 𝜕 𝑘 𝑡 ) d 𝑘 𝑡 . After the anti-symmetrization we obtain the Ong formulaˆ 𝜎 𝑥𝑦 = 𝑒 𝐵 ( 𝜋 ) ∫ h 𝑙 𝑥 d 𝑙 𝑦 − 𝑙 𝑦 d 𝑙 𝑥 i = 𝑒 𝐵 ( 𝜋 ) ∫ h 𝒍 × d 𝒍 i 𝑧 , (28)where ˆ 𝜎 𝑥𝑦 ≡ ( 𝜎 𝑥𝑦 − 𝜎 𝑦𝑥 )/
2. Moreover, Ong [4] discussed the “Stokes" area in 𝒍 -space. In this paper we have discussed the geometrical formulae for the Hall conductivity.In 2D the Ong formula is expressed by the area in 𝒍 -space. In 3D the Tsuji formulais expressed by the curvature of the Fermi surface in 𝒌 -space. A Curvature in differential forms
In this appendix we review the minimum fundamentals of a smooth surface Σ in 3DEuclidean space. (See, for example, §4.5 of Ref. [8].)Let us choose a point 𝒙 on the surface Σ and consider the vector 𝒏 normal to Σ at 𝒙 . Then we move to another point 𝒙 ′ on Σ and consider the normal vector 𝒏 ′ there.Both 𝒏 and 𝒏 ′ are unit vectors. We assume that the movement is infinitesimally smallso that both d 𝒙 ≡ 𝒙 ′ − 𝒙 and d 𝒏 ≡ 𝒏 ′ − 𝒏 are in the tangent plane at the point 𝒙 . Thenormalization 𝒏 · 𝒏 = 𝒏 · 𝒏 = 𝒙 = 𝜎 𝒆 + 𝜎 𝒆 and d 𝒏 = 𝜔 𝒆 + 𝜔 𝒆 where 𝒆 and 𝒆 are the basis vectors of the tangent plane. Here 𝜎 , 𝜎 , 𝜔 and 𝜔 are 1-forms. The 2-form 𝜎 𝜎 represents the element of area of Σ . The2-form 𝜔 𝜔 represents the element of area of the unit sphere. The Gaussian curvature 𝐾 is introduced as the magnification factor between two areas: 𝜔 𝜔 = 𝐾𝜎 𝜎 .Two sets of 1-forms are related by a symmetric matrix ( 𝑐 = 𝑏 ) as (cid:18) 𝜔 𝜔 (cid:19) = (cid:18) 𝑎 𝑏𝑐 𝑑 (cid:19) (cid:18) 𝜎 𝜎 (cid:19) . (29)The determinant of this matrix is the Gaussian curvature: 𝐾 = 𝑎𝑑 − 𝑏𝑐 . The trace isrelated to the mean curvature 𝐻 : 2 𝐻 = 𝑎 + 𝑑 . If this 2 × (cid:18) 𝜅 𝜅 (cid:19) , (30) 𝐾 = 𝜅 𝜅 and 2 𝐻 = 𝜅 + 𝜅 . Here the eigenvalues, 𝜅 and 𝜅 , are principal curvatures. B Components of × representation In this appendix we calculate the components of the 2 × Σ as 𝒙 = ( 𝑥 , 𝑥 , 𝑢 ) with 𝑢 = 𝑢 ( 𝑥 , 𝑥 ) . Accordinglyd 𝒙 = ( d 𝑥 , d 𝑥 , d 𝑢 ) where d 𝑢 = 𝑝 d 𝑥 + 𝑝 d 𝑥 with 𝑝 𝑖 ≡ 𝜕𝑢 / 𝜕𝑥 𝑖 ( 𝑖 = , 𝒕 = ( , , 𝑝 ) and 𝒕 = ( , , 𝑝 ) the small tangent vector is written as d 𝒙 = 𝒕 d 𝑥 + 𝒕 d 𝑥 . The unit normal vector is given by 𝒏 = 𝒘 /| 𝒘 | with 𝒘 = (− 𝑝 , − 𝑝 , ) .It is apparent that 𝒘 · 𝒕 = 𝒘 · 𝒕 = 𝐴 : d 𝒙 → d 𝒏 and introduce the 2 × ( d 𝒏 · 𝒕 𝑖 ) = Õ 𝑗 = 𝑎 𝑖 𝑗 ( d 𝒙 · 𝒕 𝑗 ) , (31)where the inner product is defined as 𝒙 · 𝒕 = ( 𝑥, 𝑦, 𝑧 ) · ( 𝑡 𝑥 , 𝑡 𝑦 , 𝑡 𝑧 ) t = ( 𝑥, 𝑦, 𝑧 ) · © « 𝑡 𝑥 𝑡 𝑦 𝑡 𝑧 ª®¬ = 𝑥𝑡 𝑥 + 𝑦𝑡 𝑦 + 𝑧𝑡 𝑧 . (32)Since d 𝒏 · 𝒕 𝑖 = −( /| 𝒘 |) Í 𝑗 𝑟 𝑖 𝑗 d 𝑥 𝑗 and d 𝒙 · 𝒕 𝑗 = Í 𝑘 ( 𝛿 𝑗𝑘 + 𝑝 𝑗 𝑝 𝑘 ) d 𝑥 𝑘 with 𝑟 𝑖 𝑗 = 𝜕 𝑢 / 𝜕𝑥 𝑖 𝜕𝑥 𝑗 , 𝑎 𝑖 𝑗 satisfies Õ 𝑗 = 𝑎 𝑖 𝑗 (cid:0) 𝛿 𝑗𝑘 + 𝑝 𝑗 𝑝 𝑘 (cid:1) = − | 𝒘 | 𝑟 𝑖𝑘 . (33)In terms of Monge’s notation ( 𝑝 = 𝜕𝑢 / 𝜕𝑥, 𝑞 = 𝜕𝑢 / 𝜕 𝑦, 𝑟 = 𝜕 𝑢 / 𝜕𝑥 , 𝑠 = 𝜕 𝑢 / 𝜕𝑥𝜕 𝑦, 𝑡 = 𝜕 𝑢 / 𝜕 𝑦 with 𝑥 = 𝑥 and 𝑥 = 𝑦 ) Eq. (33) is written asˆ 𝐴 (cid:18) + 𝑝 𝑝𝑞𝑝𝑞 + 𝑞 (cid:19) = − | 𝒘 | (cid:18) 𝑟 𝑠𝑠 𝑡 (cid:19) . (34)Thus ˆ 𝐴 = | 𝒘 | (cid:18) 𝑝𝑞𝑠 − ( + 𝑞 ) 𝑟 𝑝𝑞𝑟 − ( + 𝑝 ) 𝑠𝑝𝑞𝑡 − ( + 𝑞 ) 𝑠 𝑝𝑞𝑠 − ( + 𝑝 ) 𝑡 (cid:19) . (35)The trace is readily obtained as2 𝐻 = trace (cid:16) ˆ 𝐴 (cid:17) = | 𝒘 | h 𝑝𝑞𝑠 − ( + 𝑝 ) 𝑡 − ( + 𝑞 ) 𝑟 i . (36)After some calculations the determinant is obtained as 𝐺 = det (cid:16) ˆ 𝐴 (cid:17) = | 𝒘 | h 𝑟𝑡 − 𝑠 i . (37)The components 𝑎 𝑖 𝑗 are also obtained by the derivative of the unit normal vector 𝒏 = ( 𝑛 𝑥 , 𝑛 𝑦 , 𝑛 𝑧 ) where 𝒏 = 𝒘 / 𝑤 with 𝒘 = (− 𝑝, − 𝑞, ) and 𝑤 ≡ 𝑝 + 𝑞 +
1. Here weput 𝑝 𝑥 ≡ 𝑟 , 𝑞 𝑦 ≡ 𝑡 and 𝑠 ≡ 𝑝 𝑦 = 𝑞 𝑥 for the convenience of the calculation. If we set theview point at ( , , ∞) , the identification, 𝑎 = 𝜕𝑛 𝑥 / 𝜕𝑥 , 𝑎 = 𝜕𝑛 𝑥 / 𝜕 𝑦 , 𝑎 = 𝜕𝑛 𝑦 / 𝜕𝑥 and 𝑎 = 𝜕𝑛 𝑦 / 𝜕 𝑦 , is naturally understood. The results are 𝑎 = 𝜕𝑛 𝑥 𝜕𝑥 = − 𝑤 𝑝 𝑥 + 𝑝𝑤 ( 𝑝 𝑝 𝑥 + 𝑞𝑞 𝑥 ) = 𝑤 h 𝑝𝑞𝑠 − ( + 𝑞 ) 𝑟 i , (38)7 = 𝜕𝑛 𝑥 𝜕 𝑦 = − 𝑤 𝑝 𝑦 + 𝑝𝑤 ( 𝑝 𝑝 𝑦 + 𝑞𝑞 𝑦 ) = 𝑤 h 𝑝𝑞𝑡 − ( + 𝑞 ) 𝑠 i , (39) 𝑎 = 𝜕𝑛 𝑦 𝜕𝑥 = − 𝑤 𝑞 𝑥 + 𝑞𝑤 ( 𝑝 𝑝 𝑥 + 𝑞𝑞 𝑥 ) = 𝑤 h 𝑝𝑞𝑟 − ( + 𝑝 ) 𝑠 i , (40) 𝑎 = 𝜕𝑛 𝑦 𝜕 𝑦 = − 𝑤 𝑞 𝑦 + 𝑞𝑤 ( 𝑝 𝑝 𝑦 + 𝑞𝑞 𝑦 ) = 𝑤 h 𝑝𝑞𝑠 − ( + 𝑝 ) 𝑡 i . (41)Here we should take care that d 𝒏 = d 𝒙 ˆ 𝐴 . C × representation Here we move from 𝒙 -space to 𝒌 -space. In the Appendix B we have assumed that the 𝑧 -component is given by the function 𝑢 ( 𝑥, 𝑦 ) explicitly. In the following we assumethat the point 𝒌 = ( 𝑘 𝑥 , 𝑘 𝑦 , 𝑘 𝑧 ) on the Fermi surface is given by 𝜀 ( 𝒌 ) = × (cid:18) 𝜅 𝑥𝑥 𝜅 𝑦𝑥 𝜅 𝑥𝑦 𝜅 𝑦𝑦 (cid:19) , (42)which is a part of the 3 × © « 𝜅 𝑥𝑥 𝜅 𝑦𝑥 𝜅 𝑧𝑥 𝜅 𝑥𝑦 𝜅 𝑦𝑦 𝜅 𝑧𝑦 𝜅 𝑥𝑧 𝜅 𝑦𝑧 𝜅 𝑧𝑧 ª®¬ , (43)where 𝜅 𝑎𝑏 ≡ 𝜕𝑛 𝑎 𝜕 𝑘 𝑏 , (44)with 𝑎, 𝑏 = 𝑥, 𝑦, 𝑧 .If the 2 × (cid:18) 𝜅 𝜅 (cid:19) , (45)then the 3 × © « 𝜅 𝜅