Explicit Gauge Fixing for Degenerate Multiplets: A Generic Setup for Topological Orders

Full text:
PDF (101 kB)

+ Affiliations

¹Department of Applied Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656

Received May 13, 2004; Revised June 17, 2004

We supply basic tools for the study of the topological order of a multiplet which is an eigenspace of a finite-dimensional normal operator with continuous parameters. We allow intrinsic degeneracies within the multiplet where a well-known standard procedure does not work. As an important example, we give novel expressions for a spin Hall conductance for unitary superconductors with equal spin pairing. Generic topological orders will be treated in this unified manner particularly with nontrivial topological degeneracies.

https://doi.org/10.1143/JPSJ.73.2604

KEYWORDS: Chern numbers, degeneracy, topological orders

References

1 Geometric Phases in Physics, ed. A.Shapere and F.Wilczek (World Scientific, Singapore, 1989). Google Scholar
2 X. G.Wen:Phys. Rev. B 40 (1989) 7387. Crossref, Google Scholar
3 K. V.Klitzing, G.Dorda and M.Pepper:Phys. Rev. Lett. 45 (1980) 494. Crossref, Google Scholar
4 T.Ando:J. Phys. Soc. Jpn. 37 (1974) 622. Link, Google Scholar
5 H.Aoki and T.Ando:Solid State Commun. 38 (1981) 1079. Crossref, Google Scholar
6 R. B.Laughlin:Phys. Rev. B 23 (1981) 5632. Crossref, Google Scholar
7 D. J.Thouless, M.Kohmoto, P.Nightingale and M.den Nijs:Phys. Rev. Lett. 49 (1982) 405. Crossref, Google Scholar
8 M.Kohmoto:Ann. Phys. (N.Y.) 160 (1985) 355. Crossref, Google Scholar
9 R. B.Laughlin:Phys. Rev. Lett. 50 (1983) 1395. Crossref, Google Scholar
10 Q.Niu, D. J.Thouless and Y.Wu:Phys. Rev. B 31 (1985) 3372. Crossref, Google Scholar
11 S.Girvin, A. H.MacDonald and P. M.Platzman:Phys. Rev. Lett. 54 (1985) 581. Crossref, Google Scholar
12 P.Strêda:J. Phys. C 15 (1982) L717. Crossref, Google Scholar
13 F. D. M.Haldane:Phys. Rev. Lett. 50 (1983) 1153. Crossref, Google Scholar
14 W. P.Su, J. R.Schrieffer and A. J.Heeger:Phys. Rev. Lett. 42 (1979) 1698. Crossref, Google Scholar
15 T.Senthil, J. B.Marston and M.Fisher:Phys. Rev. B 60 (1999) 4245. Crossref, Google Scholar
16 Y.Morita and Y.Hatsugai:Phys. Rev. B 62 (2000) 99. Crossref, Google Scholar
17 Y.Hatsugai and S.Ryu:Phys. Rev. B 65 (2002) 212510. Crossref, Google Scholar
18 G. E.Volovik:JETP Lett. 66 (1997) 522. Crossref, Google Scholar
19 G. E.Volovik:Exotic Properties of Superfluid³He (World Scientific, Singapore, 1992). Crossref, Google Scholar
20 N.Read and D.Green:Phys. Rev. B 61 (2000) 10267. Crossref, Google Scholar
21 Y.Taguchi, Y.Oohara, H.Yoshizawa, N.Nagaosa and Y.Tokura:Science 291 (2001) 2573. Crossref, Google Scholar
22 D. J.Thouless:Phys. Rev. B 27 (1983) 6083. Crossref, Google Scholar
23 S.Murakami, N.Nagaosa and S.-C.Zhang:Science 301 (2003) 1348. Crossref, Google Scholar
24 R. D.King-Smith and D.Vanderblit:Phys. Rev. B 47 (1993) 1651. Crossref, Google Scholar
25 S.Ryu and Y.Hatsugai:Phys. Rev. Lett. 89 (2002) 077002. Crossref, Google Scholar
26 X. G.Wen, F.Wilczek and A.Zee:Phys. Rev. B 39 (1989) 11413. Crossref, Google Scholar
27 I.Affleck and J. B.Marston:Phys. Rev. B 37 (1988) 3774. Crossref, Google Scholar
28 J.Jain:Phys. Rev. Lett. 63 (1989) 199. Crossref, Google Scholar
29 B. I.Halperin:Phys. Rev. B 25 (1982) 2185. Crossref, Google Scholar
30 Y.Hatsugai:Phys. Rev. B 48 (1993) 11851. Crossref, Google Scholar
31 T.Kennedy:J. Phys.: Condens. Matter 2 (1990) 5737. Crossref, Google Scholar
32 P.Dirac:Lectures on Quantum Mechanics (Dover, New York, 1964). Google Scholar
33 M. V.Berry: Proc. R. Soc. London, Ser. A 392 (1984) 45. Crossref, Google Scholar
34 F.Wilczek and A.Zee:Phys. Rev. Lett. 52 (1984) 2111. Crossref, Google Scholar
35 T.Eguchi, P. B.Gilkey and A. J.Hanson:Phys. Rep. 66 (1980) 213. Crossref, Google Scholar
36 B.Zumino:Current Algebra and Anomaly (World Scientific, Singapore, 1991) p. 362. Google Scholar
37 Y.Hatsugai:Phys. Rev. Lett. 71 (1993) 3697. Crossref, Google Scholar
38 Y.Hatsugai:J. Phys.: Condens. Matter 9 (1997) 2507. Crossref, Google Scholar
39 M.Sigrist and K.Ueda:Rev. Mod. Phys. 63 (1991) 239. Crossref, Google Scholar
40 F. D. M.Haldane and E. H.Rezayi:Phys. Rev. B 31 (1985) 2529. Crossref, Google Scholar
41 We require dim V=d ≥2 for the following discussions. However, one can freely supplement additional parameters. Then the case dimV =1 is also within the analysis. Google Scholar
42 Y.Hatsugai, S.Ryu and M.Kohmoto:Phys. Rev. B 70 (2004) 054502. Crossref, Google Scholar
43 B. I.Halperin:Proc. 18th Conf. Low Temperature Physics, 1987, Kyoto Google Scholar ,Jpn. J. Appl. Phys. 26 (1987) Suppl. 26-3, p. 1913. Google Scholar
44 M.Kohmoto, B. I.Halperin and Y. S.Wu:Phys. Rev. B 45 (1992) 13488. Crossref, Google Scholar
45 The surface S can be a genus g Riemann surface Σg. The simplest and the most common case is the torus T2=Σ1 which appears as a two-dimensional Brillouin zone or a couple of fluxes in the multiflux AB effect. Google Scholar
46 We have to note that integral theorems cannot be applied here since log det ω and \( \mathop{\mathrm{Arg}} \det \tilde{\mathbf{\Phi}}^{\dagger} \mathbf{P}\mathbf{\Phi}\) are not global single valued functions at the boundaries ∂S0Φ. We use d' to indicate it. Google Scholar
47 T.Kato:Prog. Theor. Phys. 4 (1949) 514. Crossref, Google Scholar
48 J. E.Avron, R.Seiler and B.Simon:Phys. Rev. Lett. 51 (1983) 51. Crossref, Google Scholar
49 J. E.Avron and R.Seiler:Phys. Rev. Lett. 54 (1985) 259. Crossref, Google Scholar

« Previous Article | Next Article »

957 total downloads

Explicit Gauge Fixing for Degenerate Multiplets: A Generic Setup for Topological Orders

References

Article Tools