Concept of the topological order is useful to characterize anisotropic
superconductivities. The spin Hall conductance distinguishes
superconductivities with the same symmetry. The Chern number for the spin Hall
conductance is given by the covering degree of a closed surface around the
Dirac monopole. It gives a clear illustration for the on-critical condition for
the $d_{x^2-y^2}$ supreconductivitiy with non zero chemical potential. Non
trivial topological orders on the triangular lattice are also presented and
demonstrated by the topological objects.Comment: 3 figures. Appeared in proceedings of the LT2

Berry

Hatsugai

Kohmoto

Laughlin

Morita

Senthil

Shinsei Ryu

Thouless

Volovik

Yasuhiro Hatsugai

English

arXiv

Crossref

Dirac monopole and spin Hall conductance for anisotropic superconductivities

Concept of the topological order is useful to characterize anisotropic superconductivities. The spin Hall conductance distinguishes superconductivities with the same symmetry. The Chern number for the spin Hall conductance is given by the covering degree of a closed surface around the Dirac monopole. It gives a clear illustration for the on-critical condition for the dx2−y2 superconductivitiy with non-zero chemical potential. Non-trivial topological orders on the triangular lattice are also presented and demonstrated by the topological objects.othe

HATSUGAI Yasuhiro

Ryu Shinsei

Tsukuba Repository  (Tulips-R)

Dirac monopole and spin Hall conductance foranisotropic superconductivities著者 Hatsugai Yasuhiro, Ryu Shinseijournal orpublication titlePhysica. C, Superconductivity and itsapplicationsvolume 388-389page range 78-79year 2003-05URL http://hdl.handle.net/2241/100829doi: 10.1016/S0921-4534(02)02653-9arXiv:cond-mat/0311258v1  [cond-mat.supr-con]  12 Nov 2003Dirac Monopole and Spin Hall Conductance for AnisotropicSuperconductivitiesYasuhiro Hatsugai a,1, Shinsei Ryu ba Dept. of Applied Physics Univ. of Tokyo 7-3-1 Hongo, Bunkyo-ku, Tokyo, JAPAN andPRESTO, JST, Saitama 332-0012, JAPANbDept. of Applied Physics Univ. of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, JAPANAbstractConcept of the topological order is useful to characterize anisotropic superconductivities. The spin Hall conductancedistinguishes superconductivities with the same symmetry. The Chern number for the spin Hall conductance isgiven by the covering degree of a closed surface around the Dirac monopole. It gives a clear illustration for theon-critical condition for the dx2−y2 supreconductivitiy with non zero chemical potential. Non trivial topologicalorders on the triangular lattice are also presented and demonstrated by the topological objects.Key words: Spin Hall Conductance; Topological Invariant; Dirac Monopole; The Chern numberTopological quantum phase transition is conceptu-ally interesting since it is a zero temperature phasetransition without symmetry breaking[1]. It is asso-ciated with a change of topological characterizationfor the quantum mechanical ground state. One of thewidely established examples is a quantum Hall plateautransition between different Hall states. Symmetriesare the same but only the Hall conductances are dif-ferent among them. The Hall conductance has an in-trinsic topological character and the plateau transitionis specified by the change of such topological objects.Typical realizations of them are the Chern numbers,vortices and edges states[2,3]. Also special form of aselection rule is expected and the possible stability ofthe phase which puts special importance on the topo-logical transitions discriminates from other quantumphase transitions.Recently, there have been trials to extend the con-cept of this topological phase transition to uncon-ventional singlet superconductivity.[6,7]The super-conducting system is mapped into the Hall system1 Corresponding author. E-mail:hatsugai@pothos.t.u-tokyo.ac.jpand analogy between them is used to characterize thesuperconducting ground state. Here the “spin” Hallconductance plays a main role which is given by theChern numbers. Recently proposed superconductivitywith time-reversal symmetry breaking is a non trivialexample with the topological order. [5,4,6,7].The topological phase transition in superconduc-tivity raises important theoretical questions how thetopological character restricts transition types. Start-ing from the generalized lattice Bogoliuvov-de Gennes(B-dG) hamiltonian for superconductivity, we discussthe topological quantum phase transition and demon-strate the topological character[8]. In this paper howthe gap opening and the topological objects are relatedis focused. Extension to the triangular lattice which isthe simplest system with frustration is also discussed.We start from the following lattice B-dG hamil-tonian for superconducting quasi-particles H =∑ij(tijc†iσcjσ +∆ijc†i↑c†j↓ +∆∗ijcj↓ci↑)− µ∑iσ c†iσciσ.We assume the system is translational invari-ant as tij = ti−j , ∆ij = ∆i−j and further re-quire tij = tji ( ǫ(k) = ǫ(−k)). Then H =∑k c†(k)h(k)c with h(k) = R(k) · σ where σ =(σx, σy , σz) are the Pauli matrices and R = R(k) =Preprint submitted to LT23 Proceedings 25 October 2008(Rx, Ry , Rz) = (Re∆(k),−Im∆(k), ǫ(k)), wherec†(k) = (c†↑(k), c↓(k)), ǫ(k) =∑j e−ik·rj tj − µ and∆(k) =∑j e−ik·rj∆j . The “spin” Hall conductanceof the superconducting state on a lattice is given by thegeneralized Thouless-Kohmoto-Nightingale-den Nijs(TKNN) formula as σ = − e2hC = 12pii∫T2dSk ·rotkAk=∫T2dkx ∧ dky(〈∂xk|∂yk〉 − 〈∂yk|∂xk〉) whereAk = 〈k|∇kk〉, h(k)|k〉 = −E(k)|k〉, E(k) =√ǫ(k)2 + |∆(k)|2 [2,3,7].This expression for theChern number is rewritten in R space as C =12pii∫R(T2)dSR · rotRAR where AR = 〈R|∇RR〉. [8]In a particular gauge, |R〉 = t(− sin θ2eiφ cos θ2) andAR = isin θ2R(1−cos θ)eˆφ where (R, θ, φ) is a polar coor-dinate of R. The integral region R(T 2) is a closedsurface in three dimensional parameter space mappedfrom the Brillouin zone T 2. The corresponding vectorpotential defines a magnetic monopole at the originas div rotAR = −2πi δR(R) [9]. Therefore, by theGauss’ theorem, we have another expression for theChern number asC =−∫R(T2)dV δR(R) = −N(R(T2), O) = −Ncoveringwhere Ncovering = N(R(T2), O) is a degree of coveringby the closed surface R(T 2) around the origin O [8].We use the above topological expression to discussthe gap closing ( on-critical ) condition for the orderparameters. We assume a form of the order parameterused in ref[8]. It tells ǫ(k) = −2t(cos kx + cos ky)− µ,∆(k) = ∆0+2∆x2−y2(cos kx−cos ky)+2i∆xy(cos(kx+ky)− cos(kx − ky)). In the singlet case, the monopole,is doubly covered since R(k) = R(−k) which impliesthe selection rule ∆C = ±2[7]. One of the interestingcases is given by the ∆xy = 0. In this situation, theenergy gap collapses. Then due to the gapless Diracdispersion, one can not determine the Chern numberwithout ambiguity ( when |µ| is small). This situationis clear by the present demonstration since the surfaceR(T 2) is collapsed into a diamond shaped two dimen-sional region ( See Fig.1). When the Dirac monopole Ois on this region, the Chern number is ill-defined sincethe dispersion is gapless. This condition is clearly givenby |µ| ≤ 2t. Otherwise, there is an energy gap and theChern number is zero.Another interesting situation can be on the triangu-lar lattice. Starting from the t-J model on the trian-gular lattice which is a typical correlated system withfrustration, we get the B-dG equation on the triangu-lar lattice. We take mean field ansatz, as the hoppingare non zero for ti,i+xˆ = ti+xˆ,i+yˆ = ti,i+yˆ = t anda possible order parameters for the superconductivityon the triangular lattice as ∆i,i+xˆ = ei2pi/3∆i+xˆ,i+yˆ =ei4pi/3∆i,i+yˆ = ∆. Then ǫ(k) = −2t(cos kx + cos ky +cos(kx − ky)) and ∆(k) = 2∆(cos kx + ei2pi/3 cos ky +kxkyE-4-2024Rx-4-2024Ry-2024RzFig. 1. The energy dispersion and the mapped surface R(T 2)with Dirac monopole. (t = ∆x2−y2= 1, µ = −t )-5-2.502.55Rx-5 -2.50 2.55Ry-7.5-5-2.502.5RzFig. 2. Mapped Brillouin zone R(T 2) for the triangular lat-tice. The monopole at the origin O is also shown. (a) is thetotal surface R(T 2), kx ∈ (0, 2pi], ky ∈ (0, 2pi] and (b)-(f)are parts of the surface to show how it covers the origin.∆0 = 0, ∆x = ∆y = ∆xy = t, µ = 0.ei4pi/3 cos(kx − ky)). It clearly shows that the coveringdegree of mapping and the intersection number is 2 (The Chern number is 2). It suggests there may exista non trivial topological order on the possible super-conductivity on the triangular lattice. (Detailed dis-cussions will be given elsewhere.)References[1] X. G. Wen,Phys. Rev. B 40, 7387 (1989).[2] D. J. Thouless, M. Kohmoto, P. Nightingale, M. den Nijs,Phys. Rev. Lett. 49, 405 (1982), M. Kohmoto, Ann. Phys.(N. Y. ) 160, 355 (1985).[3] Y. Hatsugai, J. Phys. C, Condens. Matter, 9, 2507,(1997), Phys. Rev. B48, 11851, (1993), Phys. Rev. Lett.71, 3697 ( 1993).[4] R. B.Laughlin, Phys. Rev. Lett. 80, 5188 (1998)[5] G. E. Volovik,JETP Lett. 66, 493 (1997)[6] T. Senthil, J.B.Marston, M.P.A. Fisher, Phys. Rev. B 60,4245, (1999).[7] Y. Morita, Y. Hatsugai, Phys. Rev. B62, 99 (2000), Phys.Rev. Lett. 86, 51 (2000)[8] Y. Hatsugai, S. Ryu, to appear in Phys. Rev. B (2002).2[9] T. T. Wu, C. N. Yang, Phys. Rev. D12, 3845 (1975), M.V. Berry, Proc. R. Soc. 45, A392, (1984)3

Physica C Superconductivity

Dirac Monopole and Spin Hall Conductance for Anisotropic
  Superconductivities

Dirac Monopole and Spin Hall Conductance for Anisotropic Superconductivities

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