Phase transitions and mass generation in 2+1 dimensions

G. W. Semenoff; P. Suranyi; L. C. R. Wijewardhana

doi:10.1103/PhysRevD.50.1060

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Phase transitions and mass generation in 2+1 dimensions

G. W. Semenoff

P. Suranyi and L. C. R. Wijewardhana

Department of Physics, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z1

Department of Physics, University of Cincinnati, Cincinnati, Ohio 45221

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Phys. Rev. D 50, 1060 – Published 15 July, 1994

DOI: https://doi.org/10.1103/PhysRevD.50.1060

Abstract

The possibility that the ε expansion can predict the order of phase transitions in three-dimensional field theories is examined. For a Hermitian matrix-value and order parameter, the ε expansion predicts fluctuation-induced first order phase transitions. We analyze two (2+1)-dimensional quantum field theories which exhibit spontaneous symmetry breaking and have matrix order parameters. Using the large N expansion, we show that these models exhibit second order transitions and discuss the implications for the chiral-symmetry-breaking transition in (2+1)-dimensional QCD for a critical number of quark flavors.

References (26)

Y. Nambu, in New Theories in Physics, Proceedings of the XI International Symposium on Elementary Particle Physics, Kazimierz, Poland, 1988, edited by Z. Adjuk, S. Pokorski, and A. Trautmann (World Scientific, Singapore, 1989); T. Appelquist, T. Takeuchi, M. Einhorn and L.C.R. Wijewardhana, Phys. Lett. B 220, 223 (1989); V.A. Miransky, M. Tanabashi and K. Yamawaki, ibid. 221, 1043 (1989); Mod. Phys. Lett. A 4, 1043 (1989) R.S. Chivukula, A.G. Cohen and K. Lane, Nucl. Phys. B343, 554 (1990); R.S. Chivukula, M. Golden and E.H. Simmons, Phys. Rev. Lett. 70, 1587 (1993); W.A. Bardeen, C.T. Hill and D U. Jungnickel, Phys. Rev. D 49, 1437 (1994).
K. Wilson, Phys. Rev. D 14, 2911 (1974).
T. Appelquist and D. Nash, Phys. Rev. Lett. 64, 721 (1990).
This is somewhat analogous to four dimensional QCD where the dimensional transmutation which accompanies asymptotic freedom changes the coupling constant into the mass scale. Both theories are weakly coupled at high energies.
T. Appelquist, D. Nash and L.C.R. Wijewardhana, Phys. Rev. Lett. 60, 2575 (1988); D. Nash, ibid. 62, 3024 (1989).
G. Ferretti and S.G. Rajeev, Phys. Rev. Lett. 69, 2033 (1992); G. Ferretti, S.G. Rajeev and Z. Yang, Int. J. Mod. Phys. A 7, 7989 (1992); ibid. 7, 8001 (1992).
M. C. Diamantini, G.W. Semenoff and P. Sodano, Phys. Rev. Lett. 70, 3438 (1993).
E. Brezin, S. Hikami and J. Zinn Justin, Nucl. Phys. B165, 128 (1980).
G. Semenoff, Mod. Phys. Lett. A 17, 2811 (1992); E. Langmann and G. Semenoff, Phys. Lett. B 297, 175 (1992); M.C. Diamantini, P. Sodano, E. Langmann, and G. Semenoff, in Proceedings of Field Theory and Collective Phenomena, Perugia, Italy, 1992 (unpublished); Nucl. Phys. B406, 595 (1993).
This is, strictly speaking, only true when $N_{f}$ is a multiple of four. The reduction of the flavor symmetry group from SU( $N_{f}$ ) to SU( $N_{f}$ /2) occurs by explicit symmetry breaking by the lattice regularization. In that case, the continuous part of what we refer to as chiral symmetry is actually broken explicitly. However, a discrete chiral symmetry remains and is sufficient to render fermions massless. It is therefore reasonable to ask whether fermion mass is generated in the lattice models. This mass generation is found to occur as Néel order of the effective antiferromagnet.
S. Dagotto, J. Kogut and A. Kocic, Phys. Rev. Lett. 62, 1083 (1989); Nucl. Phys. B334, 229 (1990).
R. Pisarski and F. Wilczek, Phys. Rev. D 29, 338 (1984); ibid. 29, 1222 (1984); ibid. 29, 2423 (1984).
F. Wilczek, ``Remarks on the phase transition in QCD, '' Institute for Advanced Study Report No. IASSNS HEP 92/23, 1992 (unpublished); Int. J. Mod. Phys. A 7, 3911 (1992).
S. Coleman and E. Weinberg, Phys. Rev. D 7, 1888 (1973).
R. Pisarski, Phys. Rev. D 44, 1866 (1991).
C. Vafa and E. Witten, Nucl. Phys. B234, 173 (1984); A.P. Polychronakos, Phys. Rev. Lett. 60, 1920 (1988).
H. Yamagishi, Phys. Rev. D 23, 1880 (1981).
T. Appelquist, J. Terning, and L.C.R. Wijewardhana (unpublished).
B. Rosenstein, B. Warr and S. Park, Phys. Lett. B 218, 465 (1989); ibid. 219, 469 (1989); Phys. Rev. D 39, 3088 (1989).
G.W. Semenoff and L.C.R. Wijewardhana, Phys. Rev. D 45, 1342 (1992).
B.I. Halperin, T. Lubensky and S.K. Ma, Phys. Rev. Lett. 32, 292 (1974).
C. Dasgupta and B. I. Halperin, Phys. Rev. Lett. 47, 1556 (1981).
J. March Russell, Phys. Lett. B 296, 364 (1992).
The method for computing the integral is outlined in [7].
One could alternatively view this expansion as a large $N_{c}$ expansion where $g^{2}$ is also of order 1/ $N_{c}$ .
W. Chen, G. Semenoff and Yong Shi Wu, Phys. Rev. D 44, 1625 (1991).

[c1] Y. Nambu, in New Theories in Physics, Proceedings of the XI International Symposium on Elementary Particle Physics, Kazimierz, Poland, 1988, edited by Z. Adjuk, S. Pokorski, and A. Trautmann (World Scientific, Singapore, 1989); T. Appelquist, T. Takeuchi, M. Einhorn and L.C.R. Wijewardhana, Phys. Lett. B 220, 223 (1989); V.A. Miransky, M. Tanabashi and K. Yamawaki, ibid. 221, 1043 (1989); Mod. Phys. Lett. A 4, 1043 (1989) R.S. Chivukula, A.G. Cohen and K. Lane, Nucl. Phys. B343, 554 (1990); R.S. Chivukula, M. Golden and E.H. Simmons, Phys. Rev. Lett. 70, 1587 (1993); W.A. Bardeen, C.T. Hill and D U. Jungnickel, Phys. Rev. D 49, 1437 (1994).

[c2] K. Wilson, Phys. Rev. D 14, 2911 (1974).

[c3] T. Appelquist and D. Nash, Phys. Rev. Lett. 64, 721 (1990).

[c4] This is somewhat analogous to four dimensional QCD where the dimensional transmutation which accompanies asymptotic freedom changes the coupling constant into the mass scale. Both theories are weakly coupled at high energies.

[c5] T. Appelquist, D. Nash and L.C.R. Wijewardhana, Phys. Rev. Lett. 60, 2575 (1988); D. Nash, ibid. 62, 3024 (1989).

[c6] G. Ferretti and S.G. Rajeev, Phys. Rev. Lett. 69, 2033 (1992); G. Ferretti, S.G. Rajeev and Z. Yang, Int. J. Mod. Phys. A 7, 7989 (1992); ibid. 7, 8001 (1992).

[c7] M. C. Diamantini, G.W. Semenoff and P. Sodano, Phys. Rev. Lett. 70, 3438 (1993).

[c8] E. Brezin, S. Hikami and J. Zinn Justin, Nucl. Phys. B165, 128 (1980).

[c9] G. Semenoff, Mod. Phys. Lett. A 17, 2811 (1992); E. Langmann and G. Semenoff, Phys. Lett. B 297, 175 (1992); M.C. Diamantini, P. Sodano, E. Langmann, and G. Semenoff, in Proceedings of Field Theory and Collective Phenomena, Perugia, Italy, 1992 (unpublished); Nucl. Phys. B406, 595 (1993).

[c10] This is, strictly speaking, only true when $N_{f}$ is a multiple of four. The reduction of the flavor symmetry group from SU( $N_{f}$ ) to SU( $N_{f}$ /2) occurs by explicit symmetry breaking by the lattice regularization. In that case, the continuous part of what we refer to as chiral symmetry is actually broken explicitly. However, a discrete chiral symmetry remains and is sufficient to render fermions massless. It is therefore reasonable to ask whether fermion mass is generated in the lattice models. This mass generation is found to occur as Néel order of the effective antiferromagnet.

[c11] S. Dagotto, J. Kogut and A. Kocic, Phys. Rev. Lett. 62, 1083 (1989); Nucl. Phys. B334, 229 (1990).

[c12] R. Pisarski and F. Wilczek, Phys. Rev. D 29, 338 (1984); ibid. 29, 1222 (1984); ibid. 29, 2423 (1984).

[c13] F. Wilczek, ``Remarks on the phase transition in QCD, '' Institute for Advanced Study Report No. IASSNS HEP 92/23, 1992 (unpublished); Int. J. Mod. Phys. A 7, 3911 (1992).

[c14] S. Coleman and E. Weinberg, Phys. Rev. D 7, 1888 (1973).

[c15] R. Pisarski, Phys. Rev. D 44, 1866 (1991).

[c16] C. Vafa and E. Witten, Nucl. Phys. B234, 173 (1984); A.P. Polychronakos, Phys. Rev. Lett. 60, 1920 (1988).

[c17] H. Yamagishi, Phys. Rev. D 23, 1880 (1981).

[c18] T. Appelquist, J. Terning, and L.C.R. Wijewardhana (unpublished).

[c19] B. Rosenstein, B. Warr and S. Park, Phys. Lett. B 218, 465 (1989); ibid. 219, 469 (1989); Phys. Rev. D 39, 3088 (1989).

[c20] G.W. Semenoff and L.C.R. Wijewardhana, Phys. Rev. D 45, 1342 (1992).

[c21] B.I. Halperin, T. Lubensky and S.K. Ma, Phys. Rev. Lett. 32, 292 (1974).

[c22] C. Dasgupta and B. I. Halperin, Phys. Rev. Lett. 47, 1556 (1981).

[c23] J. March Russell, Phys. Lett. B 296, 364 (1992).

[c24] The method for computing the integral is outlined in [7].

[c25] One could alternatively view this expansion as a large $N_{c}$ expansion where $g^{2}$ is also of order 1/ $N_{c}$ .

[c26] W. Chen, G. Semenoff and Yong Shi Wu, Phys. Rev. D 44, 1625 (1991).

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Phase transitions and mass generation in 2+1 dimensions

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