Ising spins on thin graphs

Abstract The Ising model on “thin” graphs (standard Feynman diagrams) displays several interesting properties. For ferromagnetic couplings there is a mean-field phase transition at the corresponding Bethe-lattice transition point. For antiferromagnetic couplings the replica trick gives some evidence for a spin-glass phase. In this paper we investigate both the ferromagnetic and antiferromagnetic models with the aid of simulations. We confirm the Bethe-lattice values of the critical points for the ferromagnetic model on θ 3 and θ 4 graphs and examine the putative spin-glass phase in the antiferromagnetic model by looking at the overlap between replicas in a quenched ensemble of graphs. We also compare the Ising results with those for higher-state Potts models and Ising models on “fat” graphs, such as those used in 2D gravity simulations.

[1]  Finite N analysis of matrix models for an n-Ising spin on a random surface , 1993, hep-th/9309005.

[2]  P. Reinhard.  Zero-point energies in the two-center shell model (II) , 1978 .

[3]  D. Thouless,et al.  Spin-glass on a Bethe lattice. , 1986, Physical review letters.

[4]  Kanter,et al.  Mean-field theory of the Potts glass. , 1985, Physical review letters.

[5]  C. F. Baillie,et al.  Quenching 2D quantum gravity , 1994 .

[6]  T. P. Eggarter.  Cayley trees, the Ising problem, and the thermodynamic limit , 1974 .

[7]  B. Derrida.  Random-energy model: An exactly solvable model of disordered systems , 1981 .

[8]  A. Bray,et al.  Phase diagrams for dilute spin glasses , 1985 .

[9]  The Potts Spin-Glass on the Bethe Lattice: A Solution with Replica Symmetry Breaking , 1988 .

[10]  D. West.  Introduction to Graph Theory , 1995 .

[11]  G. Thorleifsson,et al.  Matter fields with c > 1 coupled to 2d gravity , 1993 .

[12]  L. D. Sèze.  Antiferromagnetic dilute bond Ising model exhibiting a spin-glass phase transition , 1977 .

[13]  R. Baxter.  Exactly solved models in statistical mechanics , 1982 .

[14]  Monte Carlo simulations of the Ising spin glass on lattices with finite connectivity , 1989 .

[15]  Wang,et al.  Replica Monte Carlo simulation of spin glasses. , 1986, Physical review letters.

[16]  Vladimir Kazakov,et al.  The ising model on a random planar lattice: The structure of the phase transition and the exact critical exponents , 1987 .

[17]  J. Zittartz,et al.  New Type of Phase Transition , 1974 .

[18]  C. Domb,et al.  On the theory of cooperative phenomena in crystals , 1960 .

[19]  K. Binder,et al.  Spin glasses: Experimental facts, theoretical concepts, and open questions , 1986 .

[20]  John Baylis.  Introduction to graph theory (3rd edition), by Robin J. Wilson. Pp 166. £5·95. 1985. ISBN 0-582-44685-6 (Longman) , 1986 .

[21]  F. David.  CONFORMAL FIELD THEORIES COUPLED TO 2-D GRAVITY IN THE CONFORMAL GAUGE , 1988 .

[22]  G. Parisi,et al.  Four-dimensional Ising spin glass: scaling within the spin-glass phase , 1993 .

[23]  Berg,et al.  New approach to spin-glass simulations. , 1992, Physical review letters.

[24]  Quenched random graphs , 1994, hep-th/9405068.

[25]  S. Kirkpatrick,et al.  Solvable Model of a Spin-Glass , 1975 .

[26]  A. Fisher,et al.  The Theory of critical phenomena , 1992 .

[27]  G. Parisi,et al.  Simulated tempering: a new Monte Carlo scheme , 1992, hep-lat/9205018.

[28]  Vladimir Kazakov,et al.  Ising model on a dynamical planar random lattice: Exact solution , 1986 .

[29]  D. Sherrington,et al.  Graph bipartitioning and statistical mechanics , 1987 .

[30]  Zero Potts models coupled to two-dimensional quantum gravity , 1992 .

[31]  M. L. Mehta,et al.  A method of integration over matrix variables , 1981 .

[32]  D. Thouless,et al.  A mean field spin glass with short-range interactions , 1986 .

[33]  J. Distler,et al.  Conformal Field Theory and 2D Quantum Gravity , 1989 .

[34]  Alexander M. Polyakov,et al.  Fractal Structure of 2D Quantum Gravity , 1988 .

[35]  Elliott H. Lieb,et al.  Residual Entropy of Square Ice , 1967 .

[36]  P. Mottishaw,et al.  Replica Symmetry Breaking and the Spin-Glass on a Bethe Lattice , 1987 .

[37]  M. Mézard,et al.  Spin Glass Theory and Beyond , 1987 .

[38]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[39]  Y. Goldschmidt,et al.  Replica symmetry breaking in finite connectivity systems: a large connectivity expansion at finite and zero temperature , 1989 .

[40]  David A. Huse,et al.  Absence of many states in realistic spin glasses , 1987 .

[41]  Vortex condensation in a model of random φ4-graphs , 1992, hep-th/9207055.

[42]  Pik-Yin Lai,et al.  Replica symmetry breaking of the ising spin-glass with finite connectivity , 1990 .

[43]  G. Parisi,et al.  Replica symmetry breaking in four-dimensional spin glasses , 1993 .

[44]  Giorgio Parisi,et al.  Order parameter for spin-glasses , 1983 .

[45]  J. Jurkiewicz,et al.  Ising spins on a dynamically triangulated random surface , 1988 .

[46]  S. Edwards,et al.  Theory of spin glasses , 1975 .

[47]  P. Lai,et al.  Application of statistical mechanics to combinatorial optimization problems: The chromatic number problem andq-partitioning of a graph , 1987 .