and we obtain Substituting Eq. (C6) into Eqs. (C3) and solving the lat- ter we obtain the eigenvalue ˜ E Eq. (C7) demonstrates that the value of ˜ E near the point λ = 0 strongly depends on the ratio γ/δ . In the limit | λ | ≫ δ ≫ γ we obtain We see from Eqs. (C7, C8) that a finite value of δ that de- termines the splitting of the times makes the eigenvalues ˜ E complex. The imaginary part, being infinitesimally small is very important for small λ generating a finite imaginary part of the integral over u in Eq. (2.54). Work- ing in the limit of γ ≪ δ one would inevitably encounter the sign problem. In contrast, putting δ = 0 and work- ing with small but finite γ gives strictly positive Z [ φ ] and one avoids the sign problem. 31 ˜ λ ∗ 2 ≃ ˜ λ$_{1}$ = ˜ λ (C6) ˜ E = $_{±}$√ √ γ 2 + | λ | 2 + 2 δiIm ( ˜ λλ ∗ ) x$_{1}$x$_{2}$ (C7) ˜ E = ±   | λ | + iδ Im ( ˜ λλ ∗ ) | λ |   1 ( x$_{1}$x$_{2}$ ) 1 / 2 (C8) 33 G. Parisi, N. Sourlas, Phys. Rev. Lett. 43 , 744 (1979) 32 L.D. Faddeev, Theor. Math. Phys. 1 , 3, (1969). 31 J. Zinn-Justin, chap. 16, Quantum Field Theory and Crit- ical Phenomena, Clarendon Press, Oxford (1989). 30 L.V. Keldysh, Zh. Eksp. Teor. Fiz. 47 , 1515 (1964) (Sov. Phys. JETP, 20 , 1018 (1965)) 29 K.B. Efetov, C. Pepin, and H. Meier, Phys. Rev. Lett. 103 ,186403 (2009) 28 C. Becchi, A. Rouet, and A. Stora, Ann. Physics, 98 , 287 (1976); I.V. Tyutin, unpublished , Lebedev preprint, FIAN, 39 (1975) 27 S. Cook, in Proceedings of the Third Annual ACM Sym- posium on Theory of Computing, p. 151, ACM, New York (1971). 26 M. Troyer, U-J. Wiese, Phys. Rev. Lett. 94 , 170201 (2005). 25 R.R. dos Santos, Braz. J. Phys. 33 , 36 (2003). 24 W. von der Linden, Phys. Rep. 220 , 53 (1992). 23 J.E. Hirsch, Phys. Rev. B 28 , 4059 (1983); ibid 31 , 4403 (1985). 22 R. Blankenbecler, D.J. Scalapino, and R.L. Sugar, Phys. Rev. D 24 , 2278 (1981); S.R. White, D. J. Scalapino, R. L. Sugar, N. E. Bickers, R.T. Scalettar, Phys. Rev. B 39 , 839 (1989). 21 A.V. Chubukov, D.L. Maslov, Phys. Rev. B 76 , 165111 (2007) 20 A.V. Chubukov, D. L. Maslov, and R. Saha, Phys. Rev. B 77 , 085109 (2008) 19 I.E. Dzyaloshinskii and A.I. Larkin, Zh. Eksp. Teor. Fiz. 61 , 791 (1971) (Sov. Phys. JETP 34 , 422 (1972)) 18 S. Lukyanov, Nucl. Phys. B 522, 533 (1998) 17 I. Affleck, D. Gepner, H.J. Schulz, and T. Ziman, J.Phys. A, 22 , 511 (1989) 1 A.A. Abrikosov, L.P. Gorkov, and I.E. Dzyaloshinskii, Methods of Quantum Field Theory in Statistical Physics, Prentice Hall, New York (1963) 2 P. A. Lee, N. Nagaosa, and X-G Wen, Rev. Mod. Phys. 78 , 17 (2006) 3 M. Norman and C. Pepin, Rep. Prog. Phys. 66, 1547 (2003) 4 A.O. Gogolin, A.A. Nersesyan, and A.M. Tsvelik, Bosonization and Strongly Correlated Systems, Cambridge University Press, Cambridge (1998) 5 T. Giamarchi, Quantum Physics in One Dimension, Ox- ford University Press, Oxford (2003) 6 A. Luther, Phys. Rev. B 19 , 320 (1979) 7 F.D.M. Haldane, Helv. Phys. Acta. 65 , 152 (1992); Pro- ceedings of International School of Physics “Enrico Fermi”, Course CXXI: “Perspectives in Many-Particle Physics”, p.5, eds. R.A. Broglia and J.R. Schrieffer, North-Holland, Amsterdam (1994) 8 A. Houghton and J.B. Marston, Phys. Rev. B 48 , 7790 (1993); A. Houghton, H.-J. Kwon, and J.B. Marston, Phys. Rev. B 50 , 1351 (1994) 9 A.H. Castro Neto and E. Fradkin, Phys. Rev. Lett. 72 , 1393 (1994); Phys. Rev. B 49 , 10877 (1994) 10 P. Kopietz and K. Sch¨ onhammer, Z. Phys. B 100 , 259 (1996) 11 P. Kopietz, Bosonization of Interacting Fermions in Arbi- trary Dimensions, Springer Verlag, Berlin (1997) 12 D.V. Khveshchenko, R. Hlubina, and T.M. Rice, Phys. Rev. B 48 , 10766 (1993) 13 D.V. Khveshchenko, Phys. Rev. B 49 , 16893 (1994); ibid. 52 , 4833 (1995) 14 C. Castellani, C. Di Castro, and W. Metzner, Phys. Rev. Lett. 72 , 316 (1994); Adv. Phys. 47 , 317 (1998). 15 I.L. Aleiner, K.B. Efetov, Phys. Rev. B 74 , 075102 (2006) 16 G. Schwiete, K.B. Efetov, Phys. Rev. B 74 , 165108 (2006)