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F. Y. Wu

Publications and source records attributed to F. Y. Wu.

At least 37 records · Page 2Linked to original sources
arXiv2005

Directed percolation in two dimensions: An exact solution

We consider a directed percolation process on an ${\cal M}$ x ${\cal N}$ rectangular lattice whose vertical edges are directed upward with an occupation probability y and horizontal edges directed toward the right with occupation probabilities x and 1 in alternate rows. We deduce a closed-form expression for the percolation probability P(x,y), the probability that one or more directed paths connect the lower-left and upper-right corner sites of the lattice. It is shown that P(x,y) is critical in the aspect ratio $a = {\cal M}/{\cal N}$ at a value $a_c =[1-y^2-x(1-y)^2]/2y^2$ where P(x,y) is discontinuous, and the critical exponent of the correlation length for $a < a_c$ is $ν=2$.

cond-mat.stat-mech↗
arXiv2005

Critical line of an n-component cubic model

We consider a special case of the n-component cubic model on the square lattice, for which an expansion exists in Ising-like graphs. We construct a transfer matrix and perform a finite-size-scaling analysis to determine the critical points for several values of n. Furthermore we determine several universal quantities, including three critical exponents. For n<2, these results agree well with the theoretical predictions for the critical O(n) branch. This model is also a special case of the ($N_α,N_β$) model of Domany and Riedel. It appears that the self-dual plane of the latter model contains the exactly known critical points of the n=1 and 2 cubic models. For this reason we have checked whether this is also the case for 1<n<2. However, this possibility is excluded by our numerical results.

cond-mat.stat-mech↗
arXiv2005

The random cluster model and new summation and integration identities

We explicitly evaluate the free energy of the random cluster model at its critical point for 0 < q < 4 using an exact result due to Baxter, Temperley and Ashley. It is found that the resulting expression assumes a form which depends on whether $π/2\cos^{-1}[\sqrt(q)/2]$ is a rational number, and if it is a rational number whether the denominator is an odd integer. Our consideration leads to new summation identities and, for q = 2, a closed-form evaluation of the integral [1/(4π^2)] \int_0^{2π}dx \int_0^{2π}dy ln[A + B + C - A cos x - B cos y - C cos(x + y)] = -\ln(2S) + (2/π)[Ti_2(AS) + Ti_2(BS) + Ti_2(CS)], where A, B, C >=0 and S = 1/\sqrt{AB+BC+CA}.

cond-mat.stat-mech↗
arXiv2004

Soluble kagome Ising model in a magnetic field

An Ising model on the kagome lattice with super-exchange interactions is solved exactly under the presence of a nonzero external magnetic field. The model generalizes the super-exchange model introduced by Fisher in 1960 and is analyzed in light of a free-fermion model. We deduce the critical condition and present detailed analyses of its thermodynamic and magnetic properties. The system is found to exhibit a second-order transition with logarithmic singularities at criticality.

cond-mat.stat-mech↗
arXiv2004

Theory of resistor networks: The two-point resistance

The resistance between arbitrary two nodes in a resistor network is obtained in terms of the eigenvalues and eigenfunctions of the Laplacian matrix associated with the network. Explicit formulas for two-point resistances are deduced for regular lattices in one, two, and three dimensions under various boundary conditions including that of a Moebius strip and a Klein bottle. The emphasis is on lattices of finite sizes. We also deduce summation and product identities which can be used to analyze large-size expansions of two-and-higher dimensional lattices.

math-ph↗
arXiv2003

On the entropy of spanning trees on a large triangular lattice

The double integral representing the entropy S_{tri} of spanning trees on a large triangular lattice is evaluated using two different methods, one algebraic and one graphical. Both methods lead to the same result S_{tri} = [1/(2 Pi)]^2 \int_0^{2 Pi} dθ\int_0^{2 Pi} dϕln [6-2 cos(θ) - 2 cos(ϕ) -2 cos(θ+ϕ)] = [3(\sqrt 3)/Pi](1 - 5^{-2} + 7^{-2} - 11^{-2} + 13^{-2} - ...)

cond-mat.stat-mech↗
arXiv2003

Dimers on two-dimensional lattices

We consider close-packed dimers, or perfect matchings, on two-dimensional regular lattices. We review known results and derive new expressions for the free energy, entropy, and the molecular freedom of dimers for a number of lattices including the simple-quartic (4^4), honeycomb (6^3), triangular (3^6), kagome (3.6.3.6), 3-12 (3.12^2) and its dual [3.12^2], and 4-8 (4.8^2) and its dual Union Jack [4.8^2] Archimedean tilings. The occurrence and nature of phase transitions are also analyzed and discussed.

cond-mat.stat-mech↗
arXiv2003

Generalized Fibonacci numbers and dimer statistics

We establish new product identities involving the $q$-analogue of the Fibonacci numbers. We show that the identities lead to alternate expressions of generating functions for close-packed dimers on non-orientable surfaces.

cond-mat.stat-mech↗
arXiv2003

The odd eight-vertex model

We consider a vertex model on the simple-quartic lattice defined by line graphs on the lattice for which there is always an odd number of lines incident at a vertex. This is the odd 8-vertex model which has eight possible vertex configurations. We establish that the odd 8-vertex model is equivalent to a staggered 8-vertex model. Using this equivalence we deduce the solution of the odd 8-vertex model when the weights satisfy a free-fermion condition. It is found that the free-fermion model exhibits no phase transitions in the regime of positive vertex weights. We also establish the complete equivalence of the free-fermion odd 8-vertex model with the free-fermion 8-vertex model solved by Fan and Wu. Our analysis leads to several Ising model representations of the free-fermion model with pure 2-spin interactions.

cond-mat.stat-mech↗
arXiv2002

Dimers on a simple-quartic net with a vacancy

A seminal milestone in lattice statistics is the exact solution of the enumeration of dimers on a simple-quartic net obtained by Fisher,Kasteleyn, and Temperley (FKT) in 1961. An outstanding related and yet unsolved problem is the enumeration of dimers on a net with vacant sites. Here we consider this vacant-site problem with a single vacancy occurring at certain specific sites on the boundary of a simple-quartic net. First, using a bijection between dimer and spanning tree configurations due to Temperley, Kenyon, Propp, and Wilson, we establish that the dimer generating function is independent of the location of the vacancy, and deduce a closed-form expression for the generating function. We next carry out finite-size analyses of this solution as well as that of the FKT solution. Our analyses lead to a logarithmic correction term in the large-size expansion for the vacancy problem with free boundary conditions. A concrete example exhibiting this difference is given. We also find the central charge c=-2 in the language of conformal field theory for the vacancy problem, as versus the value c=1 when there is no vacancy.

cond-mat.stat-mech↗
arXiv2002

Duality relation for frustrated spin models

We consider discrete spin models on arbitrary planar graphs and lattices with frustrated interactions. We first analyze the Ising model with frustrated plaquettes. We use an algebraic approach to derive the result that an Ising model with some of its plaquettes frustrated has a dual which is an Ising model with an external field $iπ/2$ applied to the dual sites centered at frustrated plaquettes. In the case that all plaquettes are frustrated, this leads to the known result that the dual model has a uniform field $iπ/2$ whose partition function can be evaluated in the thermodynamic limit for regular lattices. The analysis is extended to a Potts spin glass with analogous results obtained.

cond-mat.stat-mech↗
arXiv2002

The one-dimensional Hubbard model: A reminiscence

In 1968 we published the solution of the ground state energy and wave function of the one-dimensional Hubbard model, and we also showed that there is no Mott transition in this model. Details of the analysis have never been published, however. As the Hubbard model has become increasingly important in condensed matter physics, relating to topics such as the theory of high-T_c superconductivity, it is appropriate to revisit the one-dimensional model and to recall here some details of the solution.

cond-mat.str-el↗
arXiv2002

Close-packed dimers on nonorientable surfaces

The problem of enumerating dimers on an M x N net embedded on non-orientable surfaces is considered. We solve both the Moebius strip and Klein bottle problems for all M and N with the aid of imaginary dimer weights. The use of imaginary weights simplifies the analysis, and as a result we obtain new compact solutions in the form of double products. The compact expressions also permit us to establish a general reciprocity theorem.

cond-mat.stat-mech↗
arXiv2001

Zeroes of the Jones polynomial

We study the distribution of zeroes of the Jones polynomial $V_K(t)$ for a knot $K$. We have computed numerically the roots of the Jones polynomial for all prime knots with $N\leq 10$ crossings, and found the zeroes scattered about the unit circle $|t|=1$ with the average distance to the circle approaching a nonzero value as $N$ increases. For torus knots of the type $(m,n)$ we show that all zeroes lie on the unit circle with a uniform density in the limit of either $m$ or $n\to \infty$, a fact confirmed by our numerical findings. We have also elucidated the relation connecting the Jones polynomial with the Potts model, and used this relation to derive the Jones polynomial for a repeating chain knot with $3n$ crossings for general $n$. It is found that zeroes of its Jones polynomial lie on three closed curves centered about the points $1, i$ and $-i$. In addition, there are two isolated zeroes located one each near the points $t_\pm = e^{\pm 2πi/3}$ at a distance of the order of $3^{-(n+2)/2}$. Closed-form expressions are deduced for the closed curves in the limit of $n\to \infty$.

cond-mat.stat-mech↗
arXiv2001

New correlation duality relations for the planar Potts model

We introduce a new method to generate duality relations for correlation functions of the Potts model on planar graphs. The method extends previously known results, by allowing the consideration of the correlation function for arbitrarily placed vertices on the graph. We show that generally it is linear combinations of correlation functions, not the individual correlations, that are related by dualities. The method is illustrated in several non-trivial cases, and the relation to earlier results is explained. A graph-theoretical formulation of our results in terms of rooted dichromatic, or Tutte, polynomials is also given.

cond-mat↗
arXiv2000

Spanning Trees on Graphs and Lattices in d Dimensions

The problem of enumerating spanning trees on graphs and lattices is considered. We obtain bounds on the number of spanning trees $N_{ST}$ and establish inequalities relating the numbers of spanning trees of different graphs or lattices. A general formulation is presented for the enumeration of spanning trees on lattices in $d\geq 2$ dimensions, and is applied to the hypercubic, body-centered cubic, face-centered cubic, and specific planar lattices including the kagomé, diced, 4-8-8 (bathroom-tile), Union Jack, and 3-12-12 lattices. This leads to closed-form expressions for $N_{ST}$ for these lattices of finite sizes. We prove a theorem concerning the classes of graphs and lattices ${\cal L}$ with the property that $N_{ST} \sim \exp(nz_{\cal L})$ as the number of vertices $n \to \infty$, where $z_{\cal L}$ is a finite nonzero constant. This includes the bulk limit of lattices in any spatial dimension, and also sections of lattices whose lengths in some dimensions go to infinity while others are finite. We evaluate $z_{\cal L}$ exactly for the lattices we considered, and discuss the dependence of $z_{\cal L}$ on d and the lattice coordination number. We also establish a relation connecting $z_{\cal L}$ to the free energy of the critical Ising model for planar lattices ${\cal L}$.

cond-mat.stat-mech↗
arXiv2000

Spanning Trees on Hypercubic Lattices and Non-orientable Surfaces

We consider the problem of enumerating spanning trees on lattices. Closed-form expressions are obtained for the spanning tree generating function for a hypercubic lattice of size N_1 x N_2 x...x N_d in d dimensions under free, periodic, and a combination of free and periodic boundary conditions. Results are also obtained for a simple quartic net embedded on two non-orientable surfaces, a Moebius strip and the Klein bottle. Our results are based on the use of a formula expressing the spanning tree generating function in terms of the eigenvalues of an associated tree matrix. An elementary derivation of this formula is given.

cond-mat.stat-mech↗