Non-hermitian random matrix theory: Method of hermitian reduction

Nuclear Physics B

Volumn 504, Issue 3, 1997, Pages 579-608

  Feinberg, Joshua ^a   Zee, A ^a,b  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b INSTITUTE FOR ADVANCED STUDY   (United States)

Author keywords

Deterministic plus random; Non gaussian ensembles; Non hermitian hamiltonians and localization; Non hermitian random matrices; The method of hermitization

Indexed keywords

EID: 0031551792     PISSN: 05503213     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0550-3213(97)00502-6     Document Type: Article

References (48)

1. [1] E.P. Wigner, Can. Math. Congr. Proc. (University of Toronto Press, Toronto, 1957) p. 174, reprinted in C.E. Porter, Statistical Theories of Spectra: Fluctuation (Academic Press, New york, 1965); See also M.L. Mehta, Random Matrices (Academic Press, New York, 1991).
2. [1] E.P. Wigner, Can. Math. Congr. Proc. (University of Toronto Press, Toronto, 1957) p. 174, reprinted in C.E. Porter, Statistical Theories of Spectra: Fluctuation (Academic Press, New york, 1965); See also M.L. Mehta, Random Matrices (Academic Press, New York, 1991).
3. [1] E.P. Wigner, Can. Math. Congr. Proc. (University of Toronto Press, Toronto, 1957) p. 174, reprinted in C.E. Porter, Statistical Theories of Spectra: Fluctuation (Academic Press, New york, 1965); See also M.L. Mehta, Random Matrices (Academic Press, New York, 1991).
4. [2] M.A. Stephanov, Phys. Rev. Lett. 76 (1996) 4472.
5. [3] F. Haake, F. Izrailev, N. Lehmann, D. Saher and H.J. Sommers, Z. Phys. B 88 (1992) 359.
6. [4] E. Gudowska-Nowak, G. Papp and J. Brickmann, preprint cond-mat/9701187.
7. [5] H.J. Sommers, A. Crisanti, H. Sompolinsky and Y. Stein, Phys. Rev. Lett. 60 (1988) 1895.
8. [6] N. Hatano and D.R. Nelson, Phys. Rev. Lett. 77 (1997) 570.
9. [7] K.B. Efetov, preprint cond-mat/9702091.
10. [8] Y.V. Fyodorov, B.A. Khoruzhenko and H.J. Sommers, preprint, cond-mat/9606173; B.A. Khoruzhenko J. Phys. A. 29 (1996) L165.
11. [8] Y.V. Fyodorov, B.A. Khoruzhenko and H.J. Sommers, preprint, cond-mat/9606173; B.A. Khoruzhenko J. Phys. A. 29 (1996) L165.
12. [9] R.A. Janik, M.A. Nowak, G. Papp and I. Zahed, preprints cond-mat/9612240, hep-ph/9606329; R.A. Janik, M.A. Nowak, G. Papp J. Wambach and I. Zahed, preprint hep-ph/9609491.
13. [9] R.A. Janik, M.A. Nowak, G. Papp and I. Zahed, preprints cond-mat/9612240, hep-ph/9606329; R.A. Janik, M.A. Nowak, G. Papp J. Wambach and I. Zahed, preprint hep-ph/9609491.
14. [10] E. Brézin, C. Itzykson, G. Parisi and J.-B. Zuber, Commun. Math. Phys. 59 (1978) 35.
15. [11] J. Ginibre, J. Math. Phys. 6 (1965) 440
16. [12] See for example, E. Brézin and A. Zee, Phys. Rev. E 49 (1994) 2588.
17. [13] E. Brézin and A. Zee, Comp. Rend. Acad. Sci. (Paris) 317 (1993) 735.
18. [14] E. Brézin and J. Zinn-Justin, Phys. Lett. B 288 (1992) 54; S. Higuchi, C. Itoh, S. Nishigaki and N. Sakai, Phys. Lett. B 318 (1993) 63; Nucl. Phys. B 434 (1995) 283; B 441 (1995) 405 (E).
19. [14] E. Brézin and J. Zinn-Justin, Phys. Lett. B 288 (1992) 54; S. Higuchi, C. Itoh, S. Nishigaki and N. Sakai, Phys. Lett. B 318 (1993) 63; Nucl. Phys. B 434 (1995) 283; B 441 (1995) 405 (E).
20. [14] E. Brézin and J. Zinn-Justin, Phys. Lett. B 288 (1992) 54; S. Higuchi, C. Itoh, S. Nishigaki and N. Sakai, Phys. Lett. B 318 (1993) 63; Nucl. Phys. B 434 (1995) 283; B 441 (1995) 405 (E).
21. [14] E. Brézin and J. Zinn-Justin, Phys. Lett. B 288 (1992) 54; S. Higuchi, C. Itoh, S. Nishigaki and N. Sakai, Phys. Lett. B 318 (1993) 63; Nucl. Phys. B 434 (1995) 283; B 441 (1995) 405 (E).
22. [15] J. D'Anna and A. Zee, Phys. Rev. E 53 (1996) 1399.
23. [16] J. Feinberg and A. Zee, J. Stat. Phys., to appear (cond-mat/9609190).
24. [17] L.A. Pastur Theor. Math. Phys. 10 (1972) 67.
25. [18] See the lectures by U. Haagerup and others in Free Probability Theory, Fields Institute Communications Vol. 12, ed. D.V. Voiculescu (American Mathematical Society, Providence RI, 1997).
26. [19] A. Zee, Nucl. Phys. B 474 (1996) 726.
27. [20] E. Brézin, S. Hikami and A. Zee, Phys. Rev. E 51 (1995) 5442.
28. [21] M.A. Nowak, G. Papp and I. Zahed, Phys. Lett. B 389 (1996) 137, 341; Nucl. Phys. B 478 (1996) 605.
29. [21] M.A. Nowak, G. Papp and I. Zahed, Phys. Lett. B 389 (1996) 137, 341; Nucl. Phys. B 478 (1996) 605.
30. [22] E. Brézin, S. Hikami and A. Zee, Nucl. Phys. B 464 (1996) 411.
31. [23] There are many papers on chiral matrices. A partial list: J.J.M. Verbaarschot, Nucl. Phys. B 426 (1994) 559; J.J.M. Verbaarschot and I. Zahed, Phys. Rev. Lett. 70 (1993) 3852; J. Ambjørn, Quantization of geometry, in Les Houches 1990, ed. J. Dalibard et al., see Section 4.3 and references therein; J. Ambjørn, J. Jurkiewicz and Yu.M. Makeenko, Phys. Lett. B 251 (1990) 517; S. Nishigaki, Phys. Lett. B 387 (1996) 139; A. Anderson, R.C. Myers and V. Periwal, Phys. Lett. B 254 (1991) 89; Nucl. Phys. B 360 (1991) 463; R.C. Myers and V. Periwal, Nucl. Phys. B 390 (1991) 716; K. Slevin and T. Nagao, Phys. Rev. Lett. 70 (1993) 635; Phys. Rev. B 50 (1994) 2380; T. Nagao and K. Slevin, J. Math. Phys. 34 (1993) 2075, 2317; T. Nagao and P.J. Forrester, Nucl. Phys. B 435 (FS) (1995) 401; A.V. Andreev, B.D. Simons and N. Taniguchi, Nucl. Phys. B 432 (1994) 485; S. Hikami and A. Zee, Nucl. Phys. B 446 (1995) 337; S. Hikami, M. Shirai and F. Wegner, Nucl. Phys. B 408 (1993) 415; C.B. Hanna, D.P. Arovas, K. Mullen and S.M. Girvin, cond-mat/9412102.
32. [23] There are many papers on chiral matrices. A partial list: J.J.M. Verbaarschot, Nucl. Phys. B 426 (1994) 559; J.J.M. Verbaarschot and I. Zahed, Phys. Rev. Lett. 70 (1993) 3852; J. Ambjørn, Quantization of geometry, in Les Houches 1990, ed. J. Dalibard et al., see Section 4.3 and references therein; J. Ambjørn, J. Jurkiewicz and Yu.M. Makeenko, Phys. Lett. B 251 (1990) 517; S. Nishigaki, Phys. Lett. B 387 (1996) 139; A. Anderson, R.C. Myers and V. Periwal, Phys. Lett. B 254 (1991) 89; Nucl. Phys. B 360 (1991) 463; R.C. Myers and V. Periwal, Nucl. Phys. B 390 (1991) 716; K. Slevin and T. Nagao, Phys. Rev. Lett. 70 (1993) 635; Phys. Rev. B 50 (1994) 2380; T. Nagao and K. Slevin, J. Math. Phys. 34 (1993) 2075, 2317; T. Nagao and P.J. Forrester, Nucl. Phys. B 435 (FS) (1995) 401; A.V. Andreev, B.D. Simons and N. Taniguchi, Nucl. Phys. B 432 (1994) 485; S. Hikami and A. Zee, Nucl. Phys. B 446 (1995) 337; S. Hikami, M. Shirai and F. Wegner, Nucl. Phys. B 408 (1993) 415; C.B. Hanna, D.P. Arovas, K. Mullen and S.M. Girvin, cond-mat/9412102.
33. [23] There are many papers on chiral matrices. A partial list: J.J.M. Verbaarschot, Nucl. Phys. B 426 (1994) 559; J.J.M. Verbaarschot and I. Zahed, Phys. Rev. Lett. 70 (1993) 3852; J. Ambjørn, Quantization of geometry, in Les Houches 1990, ed. J. Dalibard et al., see Section 4.3 and references therein; J. Ambjørn, J. Jurkiewicz and Yu.M. Makeenko, Phys. Lett. B 251 (1990) 517; S. Nishigaki, Phys. Lett. B 387 (1996) 139; A. Anderson, R.C. Myers and V. Periwal, Phys. Lett. B 254 (1991) 89; Nucl. Phys. B 360 (1991) 463; R.C. Myers and V. Periwal, Nucl. Phys. B 390 (1991) 716; K. Slevin and T. Nagao, Phys. Rev. Lett. 70 (1993) 635; Phys. Rev. B 50 (1994) 2380; T. Nagao and K. Slevin, J. Math. Phys. 34 (1993) 2075, 2317; T. Nagao and P.J. Forrester, Nucl. Phys. B 435 (FS) (1995) 401; A.V. Andreev, B.D. Simons and N. Taniguchi, Nucl. Phys. B 432 (1994) 485; S. Hikami and A. Zee, Nucl. Phys. B 446 (1995) 337; S. Hikami, M. Shirai and F. Wegner, Nucl. Phys. B 408 (1993) 415; C.B. Hanna, D.P. Arovas, K. Mullen and S.M. Girvin, cond-mat/9412102.
34. [23] There are many papers on chiral matrices. A partial list: J.J.M. Verbaarschot, Nucl. Phys. B 426 (1994) 559; J.J.M. Verbaarschot and I. Zahed, Phys. Rev. Lett. 70 (1993) 3852; J. Ambjørn, Quantization of geometry, in Les Houches 1990, ed. J. Dalibard et al., see Section 4.3 and references therein; J. Ambjørn, J. Jurkiewicz and Yu.M. Makeenko, Phys. Lett. B 251 (1990) 517; S. Nishigaki, Phys. Lett. B 387 (1996) 139; A. Anderson, R.C. Myers and V. Periwal, Phys. Lett. B 254 (1991) 89; Nucl. Phys. B 360 (1991) 463; R.C. Myers and V. Periwal, Nucl. Phys. B 390 (1991) 716; K. Slevin and T. Nagao, Phys. Rev. Lett. 70 (1993) 635; Phys. Rev. B 50 (1994) 2380; T. Nagao and K. Slevin, J. Math. Phys. 34 (1993) 2075, 2317; T. Nagao and P.J. Forrester, Nucl. Phys. B 435 (FS) (1995) 401; A.V. Andreev, B.D. Simons and N. Taniguchi, Nucl. Phys. B 432 (1994) 485; S. Hikami and A. Zee, Nucl. Phys. B 446 (1995) 337; S. Hikami, M. Shirai and F. Wegner, Nucl. Phys. B 408 (1993) 415; C.B. Hanna, D.P. Arovas, K. Mullen and S.M. Girvin, cond-mat/9412102.
35. [23] There are many papers on chiral matrices. A partial list: J.J.M. Verbaarschot, Nucl. Phys. B 426 (1994) 559; J.J.M. Verbaarschot and I. Zahed, Phys. Rev. Lett. 70 (1993) 3852; J. Ambjørn, Quantization of geometry, in Les Houches 1990, ed. J. Dalibard et al., see Section 4.3 and references therein; J. Ambjørn, J. Jurkiewicz and Yu.M. Makeenko, Phys. Lett. B 251 (1990) 517; S. Nishigaki, Phys. Lett. B 387 (1996) 139; A. Anderson, R.C. Myers and V. Periwal, Phys. Lett. B 254 (1991) 89; Nucl. Phys. B 360 (1991) 463; R.C. Myers and V. Periwal, Nucl. Phys. B 390 (1991) 716; K. Slevin and T. Nagao, Phys. Rev. Lett. 70 (1993) 635; Phys. Rev. B 50 (1994) 2380; T. Nagao and K. Slevin, J. Math. Phys. 34 (1993) 2075, 2317; T. Nagao and P.J. Forrester, Nucl. Phys. B 435 (FS) (1995) 401; A.V. Andreev, B.D. Simons and N. Taniguchi, Nucl. Phys. B 432 (1994) 485; S. Hikami and A. Zee, Nucl. Phys. B 446 (1995) 337; S. Hikami, M. Shirai and F. Wegner, Nucl. Phys. B 408 (1993) 415; C.B. Hanna, D.P. Arovas, K. Mullen and S.M. Girvin, cond-mat/9412102.
36. [23] There are many papers on chiral matrices. A partial list: J.J.M. Verbaarschot, Nucl. Phys. B 426 (1994) 559; J.J.M. Verbaarschot and I. Zahed, Phys. Rev. Lett. 70 (1993) 3852; J. Ambjørn, Quantization of geometry, in Les Houches 1990, ed. J. Dalibard et al., see Section 4.3 and references therein; J. Ambjørn, J. Jurkiewicz and Yu.M. Makeenko, Phys. Lett. B 251 (1990) 517; S. Nishigaki, Phys. Lett. B 387 (1996) 139; A. Anderson, R.C. Myers and V. Periwal, Phys. Lett. B 254 (1991) 89; Nucl. Phys. B 360 (1991) 463; R.C. Myers and V. Periwal, Nucl. Phys. B 390 (1991) 716; K. Slevin and T. Nagao, Phys. Rev. Lett. 70 (1993) 635; Phys. Rev. B 50 (1994) 2380; T. Nagao and K. Slevin, J. Math. Phys. 34 (1993) 2075, 2317; T. Nagao and P.J. Forrester, Nucl. Phys. B 435 (FS) (1995) 401; A.V. Andreev, B.D. Simons and N. Taniguchi, Nucl. Phys. B 432 (1994) 485; S. Hikami and A. Zee, Nucl. Phys. B 446 (1995) 337; S. Hikami, M. Shirai and F. Wegner, Nucl. Phys. B 408 (1993) 415; C.B. Hanna, D.P. Arovas, K. Mullen and S.M. Girvin, cond-mat/9412102.
37. [23] There are many papers on chiral matrices. A partial list: J.J.M. Verbaarschot, Nucl. Phys. B 426 (1994) 559; J.J.M. Verbaarschot and I. Zahed, Phys. Rev. Lett. 70 (1993) 3852; J. Ambjørn, Quantization of geometry, in Les Houches 1990, ed. J. Dalibard et al., see Section 4.3 and references therein; J. Ambjørn, J. Jurkiewicz and Yu.M. Makeenko, Phys. Lett. B 251 (1990) 517; S. Nishigaki, Phys. Lett. B 387 (1996) 139; A. Anderson, R.C. Myers and V. Periwal, Phys. Lett. B 254 (1991) 89; Nucl. Phys. B 360 (1991) 463; R.C. Myers and V. Periwal, Nucl. Phys. B 390 (1991) 716; K. Slevin and T. Nagao, Phys. Rev. Lett. 70 (1993) 635; Phys. Rev. B 50 (1994) 2380; T. Nagao and K. Slevin, J. Math. Phys. 34 (1993) 2075, 2317; T. Nagao and P.J. Forrester, Nucl. Phys. B 435 (FS) (1995) 401; A.V. Andreev, B.D. Simons and N. Taniguchi, Nucl. Phys. B 432 (1994) 485; S. Hikami and A. Zee, Nucl. Phys. B 446 (1995) 337; S. Hikami, M. Shirai and F. Wegner, Nucl. Phys. B 408 (1993) 415; C.B. Hanna, D.P. Arovas, K. Mullen and S.M. Girvin, cond-mat/9412102.
38. [23] There are many papers on chiral matrices. A partial list: J.J.M. Verbaarschot, Nucl. Phys. B 426 (1994) 559; J.J.M. Verbaarschot and I. Zahed, Phys. Rev. Lett. 70 (1993) 3852; J. Ambjørn, Quantization of geometry, in Les Houches 1990, ed. J. Dalibard et al., see Section 4.3 and references therein; J. Ambjørn, J. Jurkiewicz and Yu.M. Makeenko, Phys. Lett. B 251 (1990) 517; S. Nishigaki, Phys. Lett. B 387 (1996) 139; A. Anderson, R.C. Myers and V. Periwal, Phys. Lett. B 254 (1991) 89; Nucl. Phys. B 360 (1991) 463; R.C. Myers and V. Periwal, Nucl. Phys. B 390 (1991) 716; K. Slevin and T. Nagao, Phys. Rev. Lett. 70 (1993) 635; Phys. Rev. B 50 (1994) 2380; T. Nagao and K. Slevin, J. Math. Phys. 34 (1993) 2075, 2317; T. Nagao and P.J. Forrester, Nucl. Phys. B 435 (FS) (1995) 401; A.V. Andreev, B.D. Simons and N. Taniguchi, Nucl. Phys. B 432 (1994) 485; S. Hikami and A. Zee, Nucl. Phys. B 446 (1995) 337; S. Hikami, M. Shirai and F. Wegner, Nucl. Phys. B 408 (1993) 415; C.B. Hanna, D.P. Arovas, K. Mullen and S.M. Girvin, cond-mat/9412102.
39. [23] There are many papers on chiral matrices. A partial list: J.J.M. Verbaarschot, Nucl. Phys. B 426 (1994) 559; J.J.M. Verbaarschot and I. Zahed, Phys. Rev. Lett. 70 (1993) 3852; J. Ambjørn, Quantization of geometry, in Les Houches 1990, ed. J. Dalibard et al., see Section 4.3 and references therein; J. Ambjørn, J. Jurkiewicz and Yu.M. Makeenko, Phys. Lett. B 251 (1990) 517; S. Nishigaki, Phys. Lett. B 387 (1996) 139; A. Anderson, R.C. Myers and V. Periwal, Phys. Lett. B 254 (1991) 89; Nucl. Phys. B 360 (1991) 463; R.C. Myers and V. Periwal, Nucl. Phys. B 390 (1991) 716; K. Slevin and T. Nagao, Phys. Rev. Lett. 70 (1993) 635; Phys. Rev. B 50 (1994) 2380; T. Nagao and K. Slevin, J. Math. Phys. 34 (1993) 2075, 2317; T. Nagao and P.J. Forrester, Nucl. Phys. B 435 (FS) (1995) 401; A.V. Andreev, B.D. Simons and N. Taniguchi, Nucl. Phys. B 432 (1994) 485; S. Hikami and A. Zee, Nucl. Phys. B 446 (1995) 337; S. Hikami, M. Shirai and F. Wegner, Nucl. Phys. B 408 (1993) 415; C.B. Hanna, D.P. Arovas, K. Mullen and S.M. Girvin, cond-mat/9412102.
40. [23] There are many papers on chiral matrices. A partial list: J.J.M. Verbaarschot, Nucl. Phys. B 426 (1994) 559; J.J.M. Verbaarschot and I. Zahed, Phys. Rev. Lett. 70 (1993) 3852; J. Ambjørn, Quantization of geometry, in Les Houches 1990, ed. J. Dalibard et al., see Section 4.3 and references therein; J. Ambjørn, J. Jurkiewicz and Yu.M. Makeenko, Phys. Lett. B 251 (1990) 517; S. Nishigaki, Phys. Lett. B 387 (1996) 139; A. Anderson, R.C. Myers and V. Periwal, Phys. Lett. B 254 (1991) 89; Nucl. Phys. B 360 (1991) 463; R.C. Myers and V. Periwal, Nucl. Phys. B 390 (1991) 716; K. Slevin and T. Nagao, Phys. Rev. Lett. 70 (1993) 635; Phys. Rev. B 50 (1994) 2380; T. Nagao and K. Slevin, J. Math. Phys. 34 (1993) 2075, 2317; T. Nagao and P.J. Forrester, Nucl. Phys. B 435 (FS) (1995) 401; A.V. Andreev, B.D. Simons and N. Taniguchi, Nucl. Phys. B 432 (1994) 485; S. Hikami and A. Zee, Nucl. Phys. B 446 (1995) 337; S. Hikami, M. Shirai and F. Wegner, Nucl. Phys. B 408 (1993) 415; C.B. Hanna, D.P. Arovas, K. Mullen and S.M. Girvin, cond-mat/9412102.
41. [23] There are many papers on chiral matrices. A partial list: J.J.M. Verbaarschot, Nucl. Phys. B 426 (1994) 559; J.J.M. Verbaarschot and I. Zahed, Phys. Rev. Lett. 70 (1993) 3852; J. Ambjørn, Quantization of geometry, in Les Houches 1990, ed. J. Dalibard et al., see Section 4.3 and references therein; J. Ambjørn, J. Jurkiewicz and Yu.M. Makeenko, Phys. Lett. B 251 (1990) 517; S. Nishigaki, Phys. Lett. B 387 (1996) 139; A. Anderson, R.C. Myers and V. Periwal, Phys. Lett. B 254 (1991) 89; Nucl. Phys. B 360 (1991) 463; R.C. Myers and V. Periwal, Nucl. Phys. B 390 (1991) 716; K. Slevin and T. Nagao, Phys. Rev. Lett. 70 (1993) 635; Phys. Rev. B 50 (1994) 2380; T. Nagao and K. Slevin, J. Math. Phys. 34 (1993) 2075, 2317; T. Nagao and P.J. Forrester, Nucl. Phys. B 435 (FS) (1995) 401; A.V. Andreev, B.D. Simons and N. Taniguchi, Nucl. Phys. B 432 (1994) 485; S. Hikami and A. Zee, Nucl. Phys. B 446 (1995) 337; S. Hikami, M. Shirai and F. Wegner, Nucl. Phys. B 408 (1993) 415; C.B. Hanna, D.P. Arovas, K. Mullen and S.M. Girvin, cond-mat/9412102.
42. [23] There are many papers on chiral matrices. A partial list: J.J.M. Verbaarschot, Nucl. Phys. B 426 (1994) 559; J.J.M. Verbaarschot and I. Zahed, Phys. Rev. Lett. 70 (1993) 3852; J. Ambjørn, Quantization of geometry, in Les Houches 1990, ed. J. Dalibard et al., see Section 4.3 and references therein; J. Ambjørn, J. Jurkiewicz and Yu.M. Makeenko, Phys. Lett. B 251 (1990) 517; S. Nishigaki, Phys. Lett. B 387 (1996) 139; A. Anderson, R.C. Myers and V. Periwal, Phys. Lett. B 254 (1991) 89; Nucl. Phys. B 360 (1991) 463; R.C. Myers and V. Periwal, Nucl. Phys. B 390 (1991) 716; K. Slevin and T. Nagao, Phys. Rev. Lett. 70 (1993) 635; Phys. Rev. B 50 (1994) 2380; T. Nagao and K. Slevin, J. Math. Phys. 34 (1993) 2075, 2317; T. Nagao and P.J. Forrester, Nucl. Phys. B 435 (FS) (1995) 401; A.V. Andreev, B.D. Simons and N. Taniguchi, Nucl. Phys. B 432 (1994) 485; S. Hikami and A. Zee, Nucl. Phys. B 446 (1995) 337; S. Hikami, M. Shirai and F. Wegner, Nucl. Phys. B 408 (1993) 415; C.B. Hanna, D.P. Arovas, K. Mullen and S.M. Girvin, cond-mat/9412102.
43. [23] There are many papers on chiral matrices. A partial list: J.J.M. Verbaarschot, Nucl. Phys. B 426 (1994) 559; J.J.M. Verbaarschot and I. Zahed, Phys. Rev. Lett. 70 (1993) 3852; J. Ambjørn, Quantization of geometry, in Les Houches 1990, ed. J. Dalibard et al., see Section 4.3 and references therein; J. Ambjørn, J. Jurkiewicz and Yu.M. Makeenko, Phys. Lett. B 251 (1990) 517; S. Nishigaki, Phys. Lett. B 387 (1996) 139; A. Anderson, R.C. Myers and V. Periwal, Phys. Lett. B 254 (1991) 89; Nucl. Phys. B 360 (1991) 463; R.C. Myers and V. Periwal, Nucl. Phys. B 390 (1991) 716; K. Slevin and T. Nagao, Phys. Rev. Lett. 70 (1993) 635; Phys. Rev. B 50 (1994) 2380; T. Nagao and K. Slevin, J. Math. Phys. 34 (1993) 2075, 2317; T. Nagao and P.J. Forrester, Nucl. Phys. B 435 (FS) (1995) 401; A.V. Andreev, B.D. Simons and N. Taniguchi, Nucl. Phys. B 432 (1994) 485; S. Hikami and A. Zee, Nucl. Phys. B 446 (1995) 337; S. Hikami, M. Shirai and F. Wegner, Nucl. Phys. B 408 (1993) 415; C.B. Hanna, D.P. Arovas, K. Mullen and S.M. Girvin, cond-mat/9412102.
44. [23] There are many papers on chiral matrices. A partial list: J.J.M. Verbaarschot, Nucl. Phys. B 426 (1994) 559; J.J.M. Verbaarschot and I. Zahed, Phys. Rev. Lett. 70 (1993) 3852; J. Ambjørn, Quantization of geometry, in Les Houches 1990, ed. J. Dalibard et al., see Section 4.3 and references therein; J. Ambjørn, J. Jurkiewicz and Yu.M. Makeenko, Phys. Lett. B 251 (1990) 517; S. Nishigaki, Phys. Lett. B 387 (1996) 139; A. Anderson, R.C. Myers and V. Periwal, Phys. Lett. B 254 (1991) 89; Nucl. Phys. B 360 (1991) 463; R.C. Myers and V. Periwal, Nucl. Phys. B 390 (1991) 716; K. Slevin and T. Nagao, Phys. Rev. Lett. 70 (1993) 635; Phys. Rev. B 50 (1994) 2380; T. Nagao and K. Slevin, J. Math. Phys. 34 (1993) 2075, 2317; T. Nagao and P.J. Forrester, Nucl. Phys. B 435 (FS) (1995) 401; A.V. Andreev, B.D. Simons and N. Taniguchi, Nucl. Phys. B 432 (1994) 485; S. Hikami and A. Zee, Nucl. Phys. B 446 (1995) 337; S. Hikami, M. Shirai and F. Wegner, Nucl. Phys. B 408 (1993) 415; C.B. Hanna, D.P. Arovas, K. Mullen and S.M. Girvin, cond-mat/9412102.
45. [23] There are many papers on chiral matrices. A partial list: J.J.M. Verbaarschot, Nucl. Phys. B 426 (1994) 559; J.J.M. Verbaarschot and I. Zahed, Phys. Rev. Lett. 70 (1993) 3852; J. Ambjørn, Quantization of geometry, in Les Houches 1990, ed. J. Dalibard et al., see Section 4.3 and references therein; J. Ambjørn, J. Jurkiewicz and Yu.M. Makeenko, Phys. Lett. B 251 (1990) 517; S. Nishigaki, Phys. Lett. B 387 (1996) 139; A. Anderson, R.C. Myers and V. Periwal, Phys. Lett. B 254 (1991) 89; Nucl. Phys. B 360 (1991) 463; R.C. Myers and V. Periwal, Nucl. Phys. B 390 (1991) 716; K. Slevin and T. Nagao, Phys. Rev. Lett. 70 (1993) 635; Phys. Rev. B 50 (1994) 2380; T. Nagao and K. Slevin, J. Math. Phys. 34 (1993) 2075, 2317; T. Nagao and P.J. Forrester, Nucl. Phys. B 435 (FS) (1995) 401; A.V. Andreev, B.D. Simons and N. Taniguchi, Nucl. Phys. B 432 (1994) 485; S. Hikami and A. Zee, Nucl. Phys. B 446 (1995) 337; S. Hikami, M. Shirai and F. Wegner, Nucl. Phys. B 408 (1993) 415; C.B. Hanna, D.P. Arovas, K. Mullen and S.M. Girvin, cond-mat/9412102.
46. [23] There are many papers on chiral matrices. A partial list: J.J.M. Verbaarschot, Nucl. Phys. B 426 (1994) 559; J.J.M. Verbaarschot and I. Zahed, Phys. Rev. Lett. 70 (1993) 3852; J. Ambjørn, Quantization of geometry, in Les Houches 1990, ed. J. Dalibard et al., see Section 4.3 and references therein; J. Ambjørn, J. Jurkiewicz and Yu.M. Makeenko, Phys. Lett. B 251 (1990) 517; S. Nishigaki, Phys. Lett. B 387 (1996) 139; A. Anderson, R.C. Myers and V. Periwal, Phys. Lett. B 254 (1991) 89; Nucl. Phys. B 360 (1991) 463; R.C. Myers and V. Periwal, Nucl. Phys. B 390 (1991) 716; K. Slevin and T. Nagao, Phys. Rev. Lett. 70 (1993) 635; Phys. Rev. B 50 (1994) 2380; T. Nagao and K. Slevin, J. Math. Phys. 34 (1993) 2075, 2317; T. Nagao and P.J. Forrester, Nucl. Phys. B 435 (FS) (1995) 401; A.V. Andreev, B.D. Simons and N. Taniguchi, Nucl. Phys. B 432 (1994) 485; S. Hikami and A. Zee, Nucl. Phys. B 446 (1995) 337; S. Hikami, M. Shirai and F. Wegner, Nucl. Phys. B 408 (1993) 415; C.B. Hanna, D.P. Arovas, K. Mullen and S.M. Girvin, cond-mat/9412102.
47. [24] P.M. Morse and H. Feshbach, Methods of Theoretical Physics, Vol. 1 (McGraw-Hill, New York. 1953) p. 884.
48. [25] T.P. Eggarter and R. Riedinger, Phys. Rev. B 18 (1978) 569.