Abstract
Certain features in Frenkel-Kontorova and other models of phases with a one-dimensional modulation can be analyzed by assuming parallel interfaces separating sets of lattice planes belonging to two different phases, and treating the free energy σ to create interfaces, as well as the interaction of two, three, or more interfaces, as phenomenological parameters. A strategy employed by Fisher and Szpilka for interacting defects can be extended to the case of interfaces, allowing a systematic study of the phase diagram by ignoring all interface interactions, and then successively taking into account pair, triple, and higher-order terms. The possible phase diagrams which can occur near the point where σ=0 include: various sorts of endpoints analogous to critical endpoints, an accumulation point of first-order transitions and triple points, and a self-similar structure which we call an upsilon point, which turns out to be an accumulation point of an infinite number of segments of first-order transition lines, each of which terminates in two upsilon points.
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References
R. Blinc and A. P. Levanyuk, eds.,Incommensurate Phases in Dielectrics. 1.Fundamentals.
Materials (North-Holland, Amsterdam, 1986).
P. Bak,Rep. Prog. Phys. 45:587 (1982).
V. L. Pokrovsky and A. L. Talapov,Theory of Incommensurate Crystals (Harwood, New York, 1984).
H. Z. Cummins, Experiental studies of structurally incommensurate crystal phases,Phys. Rep. 185:211 (1990).
W. Selke,Phys. Rep. 170:213 (1988).
J. Yeomans, inSolid State Physics, Vol. 14, H. Ehrenreich and D. Turnbull, eds. (Academic Press, New York, 1988), p. 151.
D. G. Sannikov, inIncommensurate Phases in Dielectrics, Vol. 1, R. Blinc and A. P. Levanyuk, eds. (North-Holland, Amsterdam, 1986), p. 43.
T. Janssen, inIncommensurate Phases in Dielectrics, Vol. 1, R. Blinc and A. P. Levanyuk, eds. (North-Holland, Amsterdam, 1986), p. 67.
S. Aubry,J. Phys. (Paris)44:147 (1983).
S. Aubry, inStructures et Instabilités, C. Godrèche, ed. (Editions de Physique, Les Ulis, France, 1985), p. 73.
S. Aubry and P. Y. Le Daeron,Physica D 8:381 (1983).
R. B. Griffiths, inFundamental Problems in Statistical Mechanics VII, H. Van Beijeren, ed. (North-Holland, Amsterdam, to appear).
W. Chou and R. B. Griffiths,Phys. Rev. B 34:6219 (1986).
K. Sasaki and L. M. Floria,J. Phys. Cond. Mat. 1:2179 (1989).
K. E. Bassler, Ph.D. Dissertation, Carnegie-Mellon University, Pittsburgh, Pennsylvania (1990).
S. Aubry, F. Axel, and F. Vallet,J. Phys. C 18:753 (1985).
M. E. Fisher and A. M. Szpilka,Phys. Rev. B 36:644 (1987).
K. Sasaki, to be published.
L. H. Tang and R. B. Griffiths,J. Stat. Phys. 53:853 (1988).
C. S. O. Yokoi, L. H. Tang, and W. Chou,Phys. Rev. B 37:2173 (1988).
W. Chou,J. Stat. Phys. 50:207 (1988).
M. Marchand, K. Hood, and A. Caillé,Phys. Rev. Lett. 58:1660 (1987);Phys. Rev. B 37:1898 (1988).
M. Marchand and A. Caillé,Phys. Rev. B 38:4845 (1988).
F. Vallet, R. Schilling, and S. Aubry,J. Phys. C 21:67 (1988).
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Bassler, K.E., Sasaki, K. & Griffiths, R.B. Interface interactions in modulated phases, and upsilon points. J Stat Phys 62, 45–88 (1991). https://doi.org/10.1007/BF01020859
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DOI: https://doi.org/10.1007/BF01020859