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Dirac monopole and spin Hall conductance for anisotropic superconductivities
Physica C 388–389 (2003) 78–79 www.elsevier.com/locate/physc
Dirac monopole and spin Hall conductance for anisotropic superconductivities Yasuhiro Hatsugai a
a,b,*
, Shinsei Ryu
a
Department of Applied Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, Japan b PRESTO, JST, Saitama 332-0012, Japan
Abstract 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 y 2 superconductivitiy with non-zero chemical potential. Non-trivial topological orders on the triangular lattice are also presented and demonstrated by the topological objects. Ó 2003 Elsevier Science B.V. All rights reserved. Keywords: Spin Hall conductance; Topological invariant; Dirac monopole; The Chern number
Topological quantum phase transition is conceptually interesting since it is a zero temperature phase transition without symmetry breaking [1]. It is associated with a change of topological characterization for the quantum mechanical ground state. One of the widely established examples is a quantum Hall plateau transition between different Hall states. Symmetries are the same but only the Hall conductances are different among them. The Hall conductance has an intrinsic topological character and the plateau transition is 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 a selection rule is expected and the possible stability of the phase which puts special importance on the topological transitions discriminates from other quantum phase transitions.
*
Corresponding author. Address: Department of Applied Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, Japan. Tel.: +81-3-3812-2111/6809; fax: +81-3-5689-8253. E-mail address: [email protected] (Y. Hatsugai).
Recently, there have been trials to extend the concept of this topological phase transition to unconventional singlet superconductivity [6,7]. The superconducting system is mapped into the Hall system and analogy between them is used to characterize the superconducting ground state. Here the ‘‘spin’’ Hall conductance plays a main role which is given by the Chern numbers. Recently proposed superconductivity with time-reversal symmetry breaking is a non-trivial example with the topological order [4–7]. The topological phase transition in superconductivity raises important theoretical questions how the topological character restricts transition types. Starting from the generalized lattice Bogoliuvov–de Gennes (B–dG) Hamiltonian for superconductivity, we discuss the topological quantum phase transition and demonstrate the topological character [8]. In this paper how the gap opening and the topological objects are related is focused. Extension to the triangular lattice which is the simplest system with frustration is also discussed. We start from the following lattice B–dG HamiltoP nian for superconducting quasi-particles H ¼ ij ðtij cyir P y y y cjr þ Dij ci" cj# þ Dij cj# ci" Þ l ir cir cir . We assume the system is translational invariant as tij ¼ tij , Dij ¼ Dij and further require tij ¼ tji ððkÞ ¼ ðkÞÞ. Then
0921-4534/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0921-4534(02)02653-9
Y. Hatsugai, S. Ryu / Physica C 388–389 (2003) 78–79
P H ¼ k cy ðkÞhðkÞc with hðkÞ ¼ RðkÞ r where r ¼ ðrx ; ry ; rz Þ are the Pauli matrices and R ¼ RðkÞ ¼ ðRx ; Ry ; Rz Þ ¼ ðRe DðkÞ; Im DðkÞ; where cy ðkÞ ¼ P ðkÞÞ, y ik rj ðc c# ðkÞÞ, ðkÞ ¼ j e tj l and DðkÞ ¼ P" ðkÞ; ik rj Dj . The ‘‘spin’’ Hall conductance of the superje conducting state on a lattice is given by the generalized Thouless–Kohmoto–Nightingale–den R Nijs (TKNN) 2 Rformula as r ¼ ðe =hÞC ¼ ð1=2piÞ T 2 dS k rotk Ak ¼ dk ^ dk ðho kjo ki hoy kjox kiÞ where Ak ¼ hkj$k ki, x y x y T2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hðkÞjki ¼ EðkÞjki, EðkÞ ¼ ðkÞ2 þ jDðkÞj2 [2,3,7]. This expression for the RChern number is rewritten in R space as C ¼ ð1=2piÞ RðT 2 Þ dS R rotR AR where AR ¼ hRj$R Ri [8]. In a particular gauge, jRi ¼ t ð sinðh=2Þei/ cosðh=2ÞÞ and AR ¼ iðsin h=2Rð1 cos hÞÞ^ e/ where ðR; h; /Þ is a polar coordinate of R. The integral region RðT 2 Þ is a closed surface in three dimensional parameter space mapped from the Brillouin zone T 2 . The corresponding vector potential defines a magnetic monopole at the origin as div rot AR ¼ 2pi dR ðRÞ [9]. Therefore, by the GaussÕ theorem, we have another expression for the Chern number as Z C¼ dV dR ðRÞ ¼ N ðRðT 2 Þ; OÞ ¼ Ncovering
79
Fig. 1. The energy dispersion and the mapped surface RðT 2 Þ with Dirac monopole (t ¼ Dx2 y 2 ¼ 1, l ¼ t).
2.5 0
Rz -2.5 -5 -7.5 -5 -2.5
Rx
0 2.5 5 -5
-2.5
0
2.5
5
Ry
RðT 2 Þ
where Ncovering ¼ NðRðT 2 Þ; OÞ is a degree of covering by the closed surface RðT 2 Þ around the origin O [8]. We use the above topological expression to discuss the gap closing (on-critical) condition for the order parameters. We assume a form of the order parameter used in Ref. [8]. It tells ðkÞ ¼ 2tðcos kx þ cos ky Þ l; DðkÞ ¼ D0 þ 2Dx2 y 2 ðcos kx cos ky Þ þ 2iDxy ðcosðkx þ ky Þ cosðkx ky ÞÞ. In the singlet case, the monopole, is doubly covered since RðkÞ ¼ RðkÞ which implies the selection rule DC ¼ 2 [7]. One of the interesting cases is given by the Dxy ¼ 0. In this situation, the energy gap collapses. Then due to the gapless Dirac dispersion, one cannot determine the Chern number without ambiguity (when jlj is small). This situation is clear by the present demonstration since the surface RðT 2 Þ is collapsed into a diamond shaped two dimensional region (see Fig. 1). When the Dirac monopole O is on this region, the Chern number is ill-defined since the dispersion is gapless. This condition is clearly given by jlj 6 2t. Otherwise, there is an energy gap and the Chern number is zero. Another interesting situation can be on the triangular lattice. Starting from the t–J model on the triangular lattice which is a typical correlated system with frustration, we get the B–dG equation on the triangular lattice. We take mean field ansatz, as the hopping are non-zero for ti;iþ^x ¼ tiþ^x;iþ^y ¼ ti;iþ^y ¼ t and a possible order parameters for the superconductivity on the triangular lattice as Di;iþ^x ¼ ei2p=3 Diþ^x;iþ^y ¼ ei4p=3 Di;iþ^y ¼ D. Then ðkÞ ¼ 2tðcoskx þcosky þcosðkx ky ÞÞ and DðkÞ ¼ 2Dðcos kx þ ei2p=3 cos ky þ ei4p=3 cosðkx ky ÞÞ. It clearly
Fig. 2. Mapped Brillouin zone RðT 2 Þ for the triangular lattice. The monopole at the origin O is also shown. (a) is the total surface RðT 2 Þ, kx 2 ð0; 2p, ky 2 ð0; 2p and (b)–(f) are parts of the surface to show how it covers the origin. D0 ¼ 0, Dx ¼ Dy ¼ Dxy ¼ t, l ¼ 0.
shows that the covering degree of mapping and the intersection number is 2 (the Chern number is 2) (see Fig. 2). It suggests there may exist a non-trivial topological order on the possible superconductivity on the triangular lattice. (Detailed discussions will be given elsewhere.) References [1] X.G. Wen, Phys. Rev. B 40 (1989) 7387. [2] D.J. Thouless, M. Kohmoto, P. Nightingale, M. den Nijs, Phys. Rev. Lett. 49 (1982) 405; M. Kohmoto, Ann. Phys. (N.Y.) 160 (1985) 355. [3] Y. Hatsugai, J. Phys. C: Condens. Matter 9 (1997) 2507; Phys. Rev. B 48 (1993) 11851; Phys. Rev. Lett. 71 (1993) 3697. [4] R.B. Laughlin, Phys. Rev. Lett. 80 (1998) 5188. [5] G.E. Volovik, JETP Lett. 66 (1997) 493. [6] T. Senthil, J.B. Marston, M.P.A. Fisher, Phys. Rev. B 60 (1999) 4245. [7] Y. Morita, Y. Hatsugai, Phys. Rev. B 62 (2000) 99; Phys. Rev. Lett. 86 (2000) 51. [8] Y. Hatsugai, S. Ryu, Phys. Rev. B 65 (2002) 212510. [9] T.T. Wu, C.N. Yang, Phys. Rev. D 12 (1975) 3845; M.V. Berry, Proc. R. Soc. 45 (1984) A392.
Dirac monopole and spin Hall conductance for anisotropic superconductivities Yasuhiro Hatsugai a
a,b,*
, Shinsei Ryu
a
Department of Applied Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, Japan b PRESTO, JST, Saitama 332-0012, Japan
Abstract 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 y 2 superconductivitiy with non-zero chemical potential. Non-trivial topological orders on the triangular lattice are also presented and demonstrated by the topological objects. Ó 2003 Elsevier Science B.V. All rights reserved. Keywords: Spin Hall conductance; Topological invariant; Dirac monopole; The Chern number
Topological quantum phase transition is conceptually interesting since it is a zero temperature phase transition without symmetry breaking [1]. It is associated with a change of topological characterization for the quantum mechanical ground state. One of the widely established examples is a quantum Hall plateau transition between different Hall states. Symmetries are the same but only the Hall conductances are different among them. The Hall conductance has an intrinsic topological character and the plateau transition is 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 a selection rule is expected and the possible stability of the phase which puts special importance on the topological transitions discriminates from other quantum phase transitions.
*
Corresponding author. Address: Department of Applied Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, Japan. Tel.: +81-3-3812-2111/6809; fax: +81-3-5689-8253. E-mail address: [email protected] (Y. Hatsugai).
Recently, there have been trials to extend the concept of this topological phase transition to unconventional singlet superconductivity [6,7]. The superconducting system is mapped into the Hall system and analogy between them is used to characterize the superconducting ground state. Here the ‘‘spin’’ Hall conductance plays a main role which is given by the Chern numbers. Recently proposed superconductivity with time-reversal symmetry breaking is a non-trivial example with the topological order [4–7]. The topological phase transition in superconductivity raises important theoretical questions how the topological character restricts transition types. Starting from the generalized lattice Bogoliuvov–de Gennes (B–dG) Hamiltonian for superconductivity, we discuss the topological quantum phase transition and demonstrate the topological character [8]. In this paper how the gap opening and the topological objects are related is focused. Extension to the triangular lattice which is the simplest system with frustration is also discussed. We start from the following lattice B–dG HamiltoP nian for superconducting quasi-particles H ¼ ij ðtij cyir P y y y cjr þ Dij ci" cj# þ Dij cj# ci" Þ l ir cir cir . We assume the system is translational invariant as tij ¼ tij , Dij ¼ Dij and further require tij ¼ tji ððkÞ ¼ ðkÞÞ. Then
0921-4534/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0921-4534(02)02653-9
Y. Hatsugai, S. Ryu / Physica C 388–389 (2003) 78–79
P H ¼ k cy ðkÞhðkÞc with hðkÞ ¼ RðkÞ r where r ¼ ðrx ; ry ; rz Þ are the Pauli matrices and R ¼ RðkÞ ¼ ðRx ; Ry ; Rz Þ ¼ ðRe DðkÞ; Im DðkÞ; where cy ðkÞ ¼ P ðkÞÞ, y ik rj ðc c# ðkÞÞ, ðkÞ ¼ j e tj l and DðkÞ ¼ P" ðkÞ; ik rj Dj . The ‘‘spin’’ Hall conductance of the superje conducting state on a lattice is given by the generalized Thouless–Kohmoto–Nightingale–den R Nijs (TKNN) 2 Rformula as r ¼ ðe =hÞC ¼ ð1=2piÞ T 2 dS k rotk Ak ¼ dk ^ dk ðho kjo ki hoy kjox kiÞ where Ak ¼ hkj$k ki, x y x y T2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hðkÞjki ¼ EðkÞjki, EðkÞ ¼ ðkÞ2 þ jDðkÞj2 [2,3,7]. This expression for the RChern number is rewritten in R space as C ¼ ð1=2piÞ RðT 2 Þ dS R rotR AR where AR ¼ hRj$R Ri [8]. In a particular gauge, jRi ¼ t ð sinðh=2Þei/ cosðh=2ÞÞ and AR ¼ iðsin h=2Rð1 cos hÞÞ^ e/ where ðR; h; /Þ is a polar coordinate of R. The integral region RðT 2 Þ is a closed surface in three dimensional parameter space mapped from the Brillouin zone T 2 . The corresponding vector potential defines a magnetic monopole at the origin as div rot AR ¼ 2pi dR ðRÞ [9]. Therefore, by the GaussÕ theorem, we have another expression for the Chern number as Z C¼ dV dR ðRÞ ¼ N ðRðT 2 Þ; OÞ ¼ Ncovering
79
Fig. 1. The energy dispersion and the mapped surface RðT 2 Þ with Dirac monopole (t ¼ Dx2 y 2 ¼ 1, l ¼ t).
2.5 0
Rz -2.5 -5 -7.5 -5 -2.5
Rx
0 2.5 5 -5
-2.5
0
2.5
5
Ry
RðT 2 Þ
where Ncovering ¼ NðRðT 2 Þ; OÞ is a degree of covering by the closed surface RðT 2 Þ around the origin O [8]. We use the above topological expression to discuss the gap closing (on-critical) condition for the order parameters. We assume a form of the order parameter used in Ref. [8]. It tells ðkÞ ¼ 2tðcos kx þ cos ky Þ l; DðkÞ ¼ D0 þ 2Dx2 y 2 ðcos kx cos ky Þ þ 2iDxy ðcosðkx þ ky Þ cosðkx ky ÞÞ. In the singlet case, the monopole, is doubly covered since RðkÞ ¼ RðkÞ which implies the selection rule DC ¼ 2 [7]. One of the interesting cases is given by the Dxy ¼ 0. In this situation, the energy gap collapses. Then due to the gapless Dirac dispersion, one cannot determine the Chern number without ambiguity (when jlj is small). This situation is clear by the present demonstration since the surface RðT 2 Þ is collapsed into a diamond shaped two dimensional region (see Fig. 1). When the Dirac monopole O is on this region, the Chern number is ill-defined since the dispersion is gapless. This condition is clearly given by jlj 6 2t. Otherwise, there is an energy gap and the Chern number is zero. Another interesting situation can be on the triangular lattice. Starting from the t–J model on the triangular lattice which is a typical correlated system with frustration, we get the B–dG equation on the triangular lattice. We take mean field ansatz, as the hopping are non-zero for ti;iþ^x ¼ tiþ^x;iþ^y ¼ ti;iþ^y ¼ t and a possible order parameters for the superconductivity on the triangular lattice as Di;iþ^x ¼ ei2p=3 Diþ^x;iþ^y ¼ ei4p=3 Di;iþ^y ¼ D. Then ðkÞ ¼ 2tðcoskx þcosky þcosðkx ky ÞÞ and DðkÞ ¼ 2Dðcos kx þ ei2p=3 cos ky þ ei4p=3 cosðkx ky ÞÞ. It clearly
Fig. 2. Mapped Brillouin zone RðT 2 Þ for the triangular lattice. The monopole at the origin O is also shown. (a) is the total surface RðT 2 Þ, kx 2 ð0; 2p, ky 2 ð0; 2p and (b)–(f) are parts of the surface to show how it covers the origin. D0 ¼ 0, Dx ¼ Dy ¼ Dxy ¼ t, l ¼ 0.
shows that the covering degree of mapping and the intersection number is 2 (the Chern number is 2) (see Fig. 2). It suggests there may exist a non-trivial topological order on the possible superconductivity on the triangular lattice. (Detailed discussions will be given elsewhere.) References [1] X.G. Wen, Phys. Rev. B 40 (1989) 7387. [2] D.J. Thouless, M. Kohmoto, P. Nightingale, M. den Nijs, Phys. Rev. Lett. 49 (1982) 405; M. Kohmoto, Ann. Phys. (N.Y.) 160 (1985) 355. [3] Y. Hatsugai, J. Phys. C: Condens. Matter 9 (1997) 2507; Phys. Rev. B 48 (1993) 11851; Phys. Rev. Lett. 71 (1993) 3697. [4] R.B. Laughlin, Phys. Rev. Lett. 80 (1998) 5188. [5] G.E. Volovik, JETP Lett. 66 (1997) 493. [6] T. Senthil, J.B. Marston, M.P.A. Fisher, Phys. Rev. B 60 (1999) 4245. [7] Y. Morita, Y. Hatsugai, Phys. Rev. B 62 (2000) 99; Phys. Rev. Lett. 86 (2000) 51. [8] Y. Hatsugai, S. Ryu, Phys. Rev. B 65 (2002) 212510. [9] T.T. Wu, C.N. Yang, Phys. Rev. D 12 (1975) 3845; M.V. Berry, Proc. R. Soc. 45 (1984) A392.