Abstract
We study lattice models of two-dimensional membranes of interest in statistical physics. The energy functional of a membrane is expressed as a sum of terms proportional to the surface area of the membrane, an extrinsic curvature and an intrinsic curvature quantity, respectively, but we neglect excluded volume effects. We introduce a renormalization transformation for these models which preserves the form of the energy functional, up to nonlocal terms. Our renormalization group construction is used to analyze the phase diagram and the different critical regimes of our models. We find evidence for a crumpling transition, separating a regime where surfaces are “crystalline” from one where the surfaces collapse to branched polymers, and we find a third genuine random-surface regime.
Similar content being viewed by others
References
D. R. Nelson, T. Piran, and S. Weinberg, eds.Statistical Mechanics of Membranes and Surfaces (World Scientific, Singapore, to be published).
A. M. Polyakov,Nucl. Phys. B 268:406 (1986); D. C. Brydges, B. Durhuus, J. Fröhlich, and C. Giffen,Nucl. Phys. B 275[FS17]:459 (1986), and references therein; F. David and E. Guitter,Europhys. Lett. 3:1169 (1987); P. Olesen and S.-K. Yang,Nucl. Phys. B 283:73 (1987); E. Braten, R. D. Pisarksi, and S.-M. Tse,Phys. Rev. Lett. 58:93 (1987); A. M. Polyakov, Gauge fields and strings, (Harwood, Chur, London 1987).
J. Ambjørn, B. Durhuus, and T. Jónsson,Phys. Rev. Lett. 58:2619 (1987).
J. Ambjørn, B. Durhuus, J. Fröhlich, and T. Jónsson,Nucl. Phys. B 290[FS20]:480 (1987).
B. Durhuus, J. Fröhlich, and T. Jónsson,Nucl. Phys. B 257[FS14]:779 (1985).
B. Durhuus, J. Fröhlich, and T. Jónsson,Nucl. Phys. B 240[FS12]:453 (1984).
B. Durhuus, J. Fröhlich, and T. Jónsson,Phys. Lett. 137B:93 (1984).
B. Durhuus and T. Jónsson,Phys. Lett. 108B:385 (1986).
T. Jónsson,Commun. Math. Phys. 106:679 (1986).
W. Helfrich,J. Phys. (Paris)46:1263 (1985); A. M. Polyakov,Nucl. Phys. B 268:406 (1986); L. Peliti and S. Leibler,Phys. Rev. Lett. 54:1690 (1985); D. Förster,Phys. Lett. 114A:115 (1986); H. Kleinert,Phys. Lett. 174B:335 (1986); F. David,Europhys. Lett. 2:577 (1986); R. D. Pisarski,Phys. Rev. Lett. 58:1300 (1987); F. Alonso and D. Espriu,Nucl. Phys. B 283:393 (1987).
J. Fröhlich and T. Spencer,Commun. Math. Phys. 81:527 (1981).
M. Wortis, R. Pandit, and M. Schick, inMelting, Localization and Chaos, R. K. Kalia and P. Vashishta, eds. (Elsevier, Houston, Texas, 1982); K. Binder, inPhase Transitions and Critical Phenomena, Vol. 8, C. Domb and J. L. Lebowitz, eds. (Academic Press, 1983); M. E. Fisher,J. Stat. Phys. 34:667 (1984); P.-G. De Gennes,Rev. Mod. Phys. 57:827 (1985); D. B. Abraham, inPhase Transitions and Critical Phenomena, Vol. 10, C. Domb and J. L. Lebowitz, eds.
J. Bricmont, A. El Mellouki, and J. Fröhlich,J. Stat. Phys. 42:743 (1987); J. Fröhlich and C- E. Pfister,Europhys. Lett. 3:845 (1987);Commun. Math. Phys. 109:493,112:51 (1987).
H. Kawai and Y. Okamoto,Phys. Lett. 130B:415 (1983).
J. Fröhlich, inApplications of Field Theory to Statistical Mechanics, L. Garido, ed. (Springer-Verlag, Berlin, 1985).
B. Durhuus, J. Fröhlich, and T. Jónsson,Nucl. Phys. B 225:185 (1983).
B. Baumann and B. Berg,Phys. Lett. 164B:131 (1985); B. Baumann,Nucl. Phys. B 285[FS19]:391 (1987).
J. Fröhlich, inProceedings of the 1984 Les Houches Summer School, K. Osterwalder and R. Stora, eds. (Plenum Press, New York, 1986).
D. Ruelle,Statistical Mechanics, Rigorous Results (Benjamin, New York, 1969).
E. Seiler,Gauge Theories as a Problem of Constructive Quantum Field Theory and Statistical Mechanics (Springer-Verlag, Berlin, 1982).
C. Cammarota,Commun. Math. Phys. 85:517 (1982); R. Kotecky and D. Preiss,Commun Math. Phys. 103:491 (1986).
M. Aizenman, J. T. Chayes, L. Chayes, J. Fröhlich, and L. Russo,Commun. Math. Phys. 92:19 (1983).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Ambjørn, J., Durhuus, B., Fröhlich, J. et al. A renormalization group analysis of lattice models of two-dimensional membranes. J Stat Phys 55, 29–85 (1989). https://doi.org/10.1007/BF01042591
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01042591