Abstract
This is a general and exact study of multiple Hamiltonian walks (HAW) filling the two-dimensional (2D) Manhattan lattice. We generalize the original exact solution for a single HAW by Kasteleyn to a system ofmultiple closed walks, aimed at modeling a polymer melt. In 2D, two basic nonequivalent topological situations are distinguished. (1) the Hamiltonian loops are allrooted andcontractible to a point:adjacent one to another, and, on a torus,homotopic to zero. (2) the loops can encircle one another and, on a torus, canwind around it. Forcase 1, the grand canonical partition function and multiple correlation functions are calculated exactly as those of multiple rooted spanningtrees or of a massive 2Dfree field, critical at zero mass (zero fugacity). The conformally invariant continuum limit on a Manhattantorus is studied in detail. The melt entropy is calculated exactly. We also consider the relevant effect of free boundary conditions. The number of single HAWs on Manhattan lattices with other perimeter shapes (rectangular, Kagomé, triangular, and arbitrary) is studied and related to the spectral theory of the Dirichlet Laplacian. This allows the calculation of exact shape-dependent configuration exponents y. An exact surface critical exponent is obtained. Forcase 2, nested and winding Hamiltonian circuits are allowed. An exact equivalence to thecritical Q-state Potts model exists, whereQ 1/2 is the walk fugacity. The Hamiltonian system is then always critical (forQ<-4). The exact critical exponents, in infinite numbers, are universal and identical to those of theO(n=Q 1/2) model in its low-temperature phase, i.e. are those of dense polymers. The exact critical partition functions on the torus are given from conformai invariance theory. These models 1 and 2 yield the two first exactly solved models of polymer melts.
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References
W. Hamilton, Tour around the world: Twenty towns on the earth are figured by the 20 vertices of a regular dodecahedron (a polyhedron with 12 pentagonal faces and 20 vertices).
C. Berge,Graphes et hypergraphes (Dunod, Paris, 1970).
L. Euler,Commentationes Arithmeticae Collectae (St. Peterburg, 1766), p. 337.
P. W. Kasteleyn,Physica 29:1329 (1963).
F. Harary, ed.,Graph Theory and Theoretical Physics (Academic Press, London, 1967).
T. van Aardenne-Ehrenfest and N. G. de Bruijn,Simon Stevin 28:203 (1951).
W. T. Tutte,Proc. Camb. Phil. Soc. 44:463 (1948).
G. Kirchhoff,Ann. Phys. Chem. 72:497 (1847).
M. N. Barber,Physica 48:237 (1970).
A. Malakis,Physica 84A:256 (1976).
W. J. C. Orr,Trans. Faraday Soc. 43:12 (1947).
C. Domb,Polymer 15:259 (1974).
J. F. Nagle,Proc. R. Soc. Lond. A 337:569 (1974).
M. Gordon, P. Kapadia, and A. Malakis,J. Phys. A 5:751 (1976).
P. D. Gujrati and M. Goldstein,J. Chem. Phys. 74:2596 (1981).
T. G. Schmaltz, G. E. Hite, and D. J. Klein,J. Phys. A 17:445 (1984).
H. Orland, C. Itzykson, and C. De Dominicis,J. Phys. Lett. (Paris) 46:L353 (1985).
J. F. Nagle, P. D. Gujrati, and M. Goldstein,J. Phys. Chem. 88:4599 (1984); J. F. Nagle,J. Stat. Phys. 38:531 (1985).
M. G. Bawendi and K. F. Freed,J. Chem. Phys. 84:7036 (1986).
B. Duplantier,J. Phys. A 19:L1009 (1986).
H. Saleur,Phys. Rev. B 35:3657 (1987).
B. Duplantier and H. Saleur,Nucl. Phys. B 290[FS20]:291 (1987).
M. den Nijs,Phys. Rev. B 27:1674 (1983), and references therein.
B. Nienhuis,Phys. Rev. Lett. 49:1062 (1982);J. Stat. Phys. 34:781 (1984); inPhase Transitions and Critical Phenomena, Vol. 11, C. Domb and J. L. Lebowitz, eds. (Academic Press, London, 1987).
B. Duplantier,J. Stat. Phys. 49:411 (1987).
F. Y. Wu,Rev. Mod. Phys. 54:239 (1982).
J. L. Cardy, inPhase Transitions and Critical Phenomena, Vol. 11, C. Domb and J. L. Lebowitz, eds. (Academic Press, London, 1987).
J. L. Cardy,Nucl. Phys. B 275[FS17]:580 (1986).
A. Weil,Elliptic Functions According to Eisenstein and Kronecker (Springer-Verlag, 1976).
C. Itzykson and J. B. Zuber,Nucl. Phys. B 275:561 (1986).
N. L. Balazs, C. Schmit, and A. Voros,J. Stat. Phys. 46:1067 (1987), and references therein.
A. E. Ferdinand and M. E. Fisher,Phys. Rev. B 185:832 (1969).
H. Saleur and C. Itzykson,J. Stat. Phys. 48:449 (1987).
C. Itzykson, Marseille Lecture Notes, SPhT/86 (1986).
R. G. Petschek and P. Pfeuty,Phys. Rev. Lett. 58:1096 (1987).
H. Blöte, M. P. Nightingale, and J. L. Cardy,Phys. Rev. Lett. 56:742 (1986); I. Affleck,Phys. Rev. Lett. 56:746 (1986).
M. Gaudin,J. Phys. (Paris) 48:1633 (1987).
M. Kac,Am. Math. Monthly 73:1 (1966); H. P. McKean and I. M. Singer,J. Diff. Geom. 1:43 (1967); H. P. Baltes and E. R. Hilf,Spectra of Finite Systems (Bibliographisches Institut, Mannheim, 1976), Chapter VI.
B. Duplantier and H. Saleur,Phys. Rev. Lett. 57:3179 (1986).
R. J. Baxter, S. B. Kelland, and F. Y. Wu,J. Phys. A 9:397 (1976).
R. J. Baxter,J. Phys. C 6:L445 (1973).
H. Saleur and B. Duplantier,Phys. Rev. Lett. 58:2325 (1987).
H. Saleur,J. Phys. A 19:L807 (1986).
C. M. Fortuin and P. W. Kasteleyn,Physica 57:536 (1972).
P. Di Francesco, H. Saleur, and J. B. Zuber,Nucl. Phys. B 285[FS19]:454 (1987);J. Stat. Phys. 49:57 (1987).
V. L. S. Dotsenko and V. A. Fateev,Nucl. Phys. B 240:312 (1984).
B. Duplantier,Phys. Rev. Lett. 57:941 (1986).
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Duplanticr, B., David, F. Exact partition functions and correlation functions of multiple Hamiltonian walks on the Manhattan lattice. J Stat Phys 51, 327–434 (1988). https://doi.org/10.1007/BF01028464
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DOI: https://doi.org/10.1007/BF01028464