arXiv subjects

Shu-Chiuan Chang

Publications and source records attributed to Shu-Chiuan Chang.

At least 55 records · Page 3Linked to original sources

arXiv2004

Exact Results for Average Cluster Numbers in Bond Percolation on Lattice Strips

We present exact calculations of the average number of connected clusters per site, $ $, as a function of bond occupation probability $p$, for the bond percolation problem on infinite-length strips of finite width $L_y$, of the square, triangular, honeycomb, and kagomé lattices $Λ$ with various boundary conditions. These are used to study the approach of $ $, for a given $p$ and $Λ$, to its value on the two-dimensional lattice as the strip width increases. We investigate the singularities of $ $ in the complex $p$ plane and their influence on the radii of convergence of the Taylor series expansions of $ $ about $p=0$ and $p=1$.

cond-mat.stat-mech↗

arXiv2004

Exact Potts Model Partition Functions for Strips of the Triangular Lattice

We present exact calculations of the Potts model partition function Z(G,q,v) for arbitrary q and temperature-like variable v on n-vertex strip graphs G of the triangular lattice for a variety of transverse widths equal to L vertices and for arbitrarily great length equal to m vertices, with free longitudinal boundary conditions and free and periodic transverse boundary conditions. These have the form Z(G,q,v)=\sum_{j=1}^{N_{Z,G,λ}} c_{Z,G,j}(λ_{Z,G,j})^{m-1}. We give general formulas for N_{Z,G,j} and its specialization to v=-1 for arbitrary L. The free energy is calculated exactly for the infinite-length limit of the graphs, and the thermodynamics is discussed. It is shown how the internal energy calculated for the case of cylindrical boundary conditions is connected with critical quantities for the Potts model on the infinite triangular lattice. Considering the full generalization to arbitrary complex q and v, we determine the singular locus {\cal B}, arising as the accumulation set of partition function zeros as m\to\infty, in the q plane for fixed v and in the v plane for fixed q.

cond-mat.stat-mech↗

arXiv2004

Transfer Matrices for the Partition Function of the Potts Model on Cyclic and Mobius Lattice Strips

We present a method for calculating transfer matrices for the $q$-state Potts model partition functions $Z(G,q,v)$, for arbitrary $q$ and temperature variable $v$, on cyclic and Möbius strip graphs $G$ of the square (sq), triangular (tri), and honeycomb (hc) lattices of width $L_y$ vertices and of arbitrarily great length $L_x$ vertices. For the cyclic case we express the partition function as $Z(Λ,L_y \times L_x,q,v)=\sum_{d=0}^{L_y} c^{(d)} Tr[(T_{Z,Λ,L_y,d})^m]$, where $Λ$ denotes lattice type, $c^{(d)}$ are specified polynomials of degree $d$ in $q$, $T_{Z,Λ,L_y,d}$ is the transfer matrix in the degree-$d$ subspace, and $m=L_x$ ($L_x/2$) for $Λ=sq, tri (hc)$, respectively. An analogous formula is given for Möbius strips. We exhibit a method for calculating $T_{Z,Λ,L_y,d}$ for arbitrary $L_y$. Explicit results for arbitrary $L_y$ are given for $T_{Z,Λ,L_y,d}$ with $d=L_y$ and $d=L_y-1$. In particular, we find very simple formulas the determinant $det(T_{Z,Λ,L_y,d})$, and trace $Tr(T_{Z,Λ,L_y})$. Corresponding results are given for the equivalent Tutte polynomials for these lattice strips and illustrative examples are included. We also present formulas for self-dual cyclic strips of the square lattice.

cond-mat.stat-mech↗

arXiv2004

Some Exact Formulas on Long-Range Correlation Functions of the Rectangular Ising Lattice

We study long-range correlation functions of the rectangular Ising lattice with cyclic boundary conditions. Specifically, we consider the situation in which two spins are on the same column, and at least one spin is on or near free boundaries. The low-temperature series expansions of the correlation functions are presented when the spin-spin couplings are the same in both directions. The exact correlation functions can be obtained by D log Pade for the cases with simple algebraic resultant expressions. The present results show that as the two spins are infinitely far from each other, the correlation function is equal to the product of the row magnetizations of the corresponding spins as expected. In terms of low-temperature series expansions, the approach of this m-th row correlation function to the bulk correlation function for increasing m can be understood from the observation that the dominant terms of their series expansions are the same successively in the above two correlation functions. The number of these dominant terms increases monotonically as m increases.

cond-mat.stat-mech↗

arXiv2003

Reliability Polynomials and their Asymptotic Limits for Families of Graphs

We present exact calculations of reliability polynomials $R(G,p)$ for lattice strips $G$ of fixed widths $L_y \le 4$ and arbitrarily great length $L_x$ with various boundary conditions. We introduce the notion of a reliability per vertex, $r(\{G\},p) = \lim_{|V| \to \infty} R(G,p)^{1/|V|}$ where $|V|$ denotes the number of vertices in $G$ and $\{G\}$ denotes the formal limit $\lim_{|V| \to \infty} G$. We calculate this exactly for various families of graphs. We also study the zeros of $R(G,p)$ in the complex $p$ plane and determine exactly the asymptotic accumulation set of these zeros ${\cal B}$, across which $r(\{G\})$ is nonanalytic.

cond-mat.stat-mech↗

arXiv2002

Flow Polynomials and their Asymptotic Limits for Lattice Strip Graphs

We present exact calculations of flow polynomials $F(G,q)$ for lattice strips of various fixed widths $L_y$ and arbitrarily great lengths $L_x$, with several different boundary conditions. Square, honeycomb, and triangular lattice strips are considered. We introduce the notion of flows per face $fl$ in the infinite-length limit. We study the zeros of $F(G,q)$ in the complex $q$ plane and determine exactly the asymptotic accumulation sets of these zeros ${\cal B}$ in the infinite-length limit for the various families of strips. The function $fl$ is nonanalytic on this locus. The loci are found to be noncompact for many strip graphs with periodic (or twisted periodic) longitudinal boundary conditions, and compact for strips with free longitudinal boundary conditions. We also find the interesting feature that, aside from the trivial case $L_y=1$, the maximal point, $q_{cf}$, where ${\cal B}$ crosses the real axis, is universal on cyclic and Möbius strips of the square lattice for all widths for which we have calculated it and is equal to the asymptotic value $q_{cf}=3$ for the infinite square lattice.

cond-mat.stat-mech↗

arXiv2002

Exact Potts Model Partition Function for Strips of the Square Lattice

We present exact calculations of the Potts model partition function $Z(G,q,v)$ for arbitrary $q$ and temperature-like variable $v$ on $n$-vertex square-lattice strip graphs $G$ for a variety of transverse widths $L_t$ and for arbitrarily great length $L_\ell$, with free longitudinal boundary conditions and free and periodic transverse boundary conditions. These have the form $Z(G,q,v)=\sum_{j=1}^{N_{Z,G,λ}} c_{Z,G,j}(λ_{Z,G,j})^{L_\ell}$. We give general formulas for $N_{Z,G,j}$ and its specialization to $v=-1$ for arbitrary $L_t$ for both types of boundary conditions, as well as other general structural results on $Z$. The free energy is calculated exactly for the infinite-length limit of the graphs, and the thermodynamics is discussed. It is shown how the internal energy calculated for the case of cylindrical boundary conditions is connected with critical quantities for the Potts model on the infinite square lattice. Considering the full generalization to arbitrary complex $q$ and $v$, we determine the singular locus ${\cal B}$, arising as the accumulation set of partition function zeros as $L_\ell \to \infty$, in the $q$ plane for fixed $v$ and in the $v$ plane for fixed $q$.

cond-mat.stat-mech↗

arXiv2002

General Structural Results for Potts Model Partition Functions on Lattice Strips

We present a set of general results on structural features of the $q$-state Potts model partition function $Z(G,q,v)$ for arbitrary $q$ and temperature Boltzmann variable $v$ for various lattice strips of arbitrarily great width $L_y$ vertices and length $L_x$ vertices, including (i) cyclic and Möbius strips of the square and triangular lattice, and (ii) self-dual cyclic strips of the square lattice. We also present an exact solution for the chromatic polynomial for the cyclic and Möbius strips of the square lattice with width $L_y=5$ (the greatest width for which an exact solution has been obtained so far for these families). In the $L_x \to \infty$ limit, we calculate the ground-state degeneracy per site, $W(q)$ and determine the boundary ${\cal B}$ across which $W(q)$ is singular in the complex $q$ plane.

cond-mat.stat-mech↗

arXiv2002

Tutte Polynomials and Related Asymptotic Limiting Functions for Recursive Families of Graphs

We prove several theorems concerning Tutte polynomials $T(G,x,y)$ for recursive families of graphs. In addition to its interest in mathematics, the Tutte polynomial is equivalent to an important function in statistical physics, the Potts model partition function of the $q$-state Potts model, $Z(G,q,v)$, where $v$ is a temperature-dependent variable. We determine the structure of the Tutte polynomial for a cyclic clan graph $G[(K_r)_m,L=jn]$ comprised of a chain of $m$ copies of the complete graph $K_r$ such that the linkage $L$ between each successive pair of $K_r$'s is a join $jn$, and $r$ and $m$ are arbitrary. The explicit calculation of the case $r=3$ (for arbitrary $m$) is presented. The continuous accumulation set of the zeros of $Z$ in the limit $m \to \infty$ is considered. Further, we present calculations of two special cases of Tutte polynomials, namely, flow and reliability polynomials, for cyclic clan graphs and discuss the respective continuous accumulation sets of their zeros in the limit $m \to \infty$. Special valuations of Tutte polynomials give enumerations of spanning trees and acyclic orientations. Two theorems are presented that determine the number of spanning trees on $G[(K_r)_m,jn]$ and $G[(K_r)_m,id]$, where $L=id$ means that the identity linkage. We report calculations of the number of acyclic orientations for strips of the square lattice and use these to obtain an improved lower bound on the exponential growth rate of the number of these acyclic orientations.

math-ph↗

arXiv2001

Exact Chromatic Polynomials for Toroidal Chains of Complete Graphs

We present exact calculations of the partition function of the zero-temperature Potts antiferromagnet (equivalently, the chromatic polynomial) for graphs of arbitrarily great length composed of repeated complete subgraphs $K_b$ with $b=5,6$ which have periodic or twisted periodic boundary condition in the longitudinal direction. In the $L_x \to \infty$ limit, the continuous accumulation set of the chromatic zeros ${\cal B}$ is determined. We give some results for arbitrary $b$ including the extrema of the eigenvalues with coefficients of degree $b-1$ and the explicit forms of some classes of eigenvalues. We prove that the maximal point where ${\cal B}$ crosses the real axis, $q_c$, satisfies the inequality $q_c \le b$ for $2 \le b$, the minimum value of $q$ at which ${\cal B}$ crosses the real $q$ axis is $q=0$, and we make a conjecture concerning the structure of the chromatic polynomial for Klein bottle strips.

math-ph↗

arXiv2001

Complex-Temperature Phase Diagrams for the q-State Potts Model on Self-Dual Families of Graphs and the Nature of the $q \to \infty$ Limit

We present exact calculations of the Potts model partition function $Z(G,q,v)$ for arbitrary $q$ and temperature-like variable $v$ on self-dual strip graphs $G$ of the square lattice with fixed width $L_y$ and arbitrarily great length $L_x$ with two types of boundary conditions. Letting $L_x \to \infty$, we compute the resultant free energy and complex-temperature phase diagram, including the locus ${\cal B}$ where the free energy is nonanalytic. Results are analyzed for widths $L_y=1,2,3$. We use these results to study the approach to the large-q limit of ${\cal B}$.

cond-mat.stat-mech↗

arXiv2001

Potts Model Partition Functions for Self-Dual Families of Strip Graphs

We consider the $q$-state Potts model on families of self-dual strip graphs $G_D$ of the square lattice of width $L_y$ and arbitrarily great length $L_x$, with periodic longitudinal boundary conditions. The general partition function $Z$ and the T=0 antiferromagnetic special case $P$ (chromatic polynomial) have the respective forms $\sum_{j=1}^{N_{F,L_y,λ}} c_{F,L_y,j} (λ_{F,L_y,j})^{L_x}$, with $F=Z,P$. For arbitrary $L_y$, we determine (i) the general coefficient $c_{F,L_y,j}$ in terms of Chebyshev polynomials, (ii) the number $n_F(L_y,d)$ of terms with each type of coefficient, and (iii) the total number of terms $N_{F,L_y,λ}$. We point out interesting connections between the $n_Z(L_y,d)$ and Temperley-Lieb algebras, and between the $N_{F,L_y,λ}$ and enumerations of directed lattice animals. Exact calculations of $P$ are presented for $2 \le L_y \le 4$. In the limit of infinite length, we calculate the ground state degeneracy per site (exponent of the ground state entropy), $W(q)$. Generalizing $q$ from ${\mathbb Z}_+$ to ${\mathbb C}$, we determine the continuous locus ${\cal B}$ in the complex $q$ plane where $W(q)$ is singular. We find the interesting result that for all $L_y$ values considered, the maximal point at which ${\cal B}$ crosses the real $q$ axis, denoted $q_c$ is the same, and is equal to the value for the infinite square lattice, $q_c=3$. This is the first family of strip graphs of which we are aware that exhibits this type of universality of $q_c$.

cond-mat.stat-mech↗

arXiv2001

Zeros of Jones Polynomials for Families of Knots and Links

We calculate Jones polynomials $V_L(t)$ for several families of alternating knots and links by computing the Tutte polynomials $T(G,x,y)$ for the associated graphs $G$ and then obtaining $V_L(t)$ as a special case of the Tutte polynomial. For each of these families we determine the zeros of the Jones polynomial, including the accumulation set in the limit of infinitely many crossings. A discussion is also given of the calculation of Jones polynomials for non-alternating links.

math-ph↗

arXiv2001

Upper Bound for the Coefficients of Chromatic polynomials

This paper describes an improvement in the upper bound for the magnitude of a coefficient of a term in the chromatic polynomial of a general graph. If $a_r$ is the coefficient of the $q^r$ term in the chromatic polynomial $P(G,q)$, where $q$ is the number of colors, then we find $a_r \le {e \choose v-r} - {e-g+2 \choose v-r-g+2} + {e-k_g-g+2 \choose v-r-g+2} - \sum _{n=1}^{k_g-\ell_g}\sum_{m=1}^{\ell_g-1} {e-g+1-n-m \choose v-r-g} - δ_{g,3}\sum_{n=1}^{k_g+\ell_{g+1}^*-\ell_g} {e-\ell_g-g+1-n \choose v-r-g}$, where $k_g$ is the number of circuits of length $g$ and $\ell_g$ and $\ell_{g+1}^*$ are certain numbers defined in the text.

math.CO↗

arXiv2000

Exact Potts Model Partition Functions on Wider Arbitrary-Length Strips of the Square Lattice

We present exact calculations of the partition function of the q-state Potts model for general q and temperature on strips of the square lattice of width L_y=3 vertices and arbitrary length L_x with periodic longitudinal boundary conditions, of the following types: (i) (FBC_y,PBC_x)= cyclic, (ii) (FBC_y,TPBC_x)= Möbius, (iii) (PBC_y,PBC_x)= toroidal, and (iv) (PBC_y,TPBC_x)= Klein bottle, where FBC and (T)PBC refer to free and (twisted) periodic boundary conditions. Results for the L_y=2 torus and Klein bottle strips are also included. In the infinite-length limit the thermodynamic properties are discussed and some general results are given for low-temperature behavior on strips of arbitrarily great width. We determine the submanifold in the {\mathbb C}^2 space of q and temperature where the free energy is singular for these strips. Our calculations are also used to compute certain quantities of graph-theoretic interest.

cond-mat.stat-mech↗

arXiv2000

Chromatic Polynomials for Lattice Strips with Cyclic Boundary Conditions

The zero-temperature $q$-state Potts model partition function for a lattice strip of fixed width $L_y$ and arbitrary length $L_x$ has the form $P(G,q)=\sum_{j=1}^{N_{G,λ}}c_{G,j}(λ_{G,j})^{L_x}$, and is equivalent to the chromatic polynomial for this graph. We present exact zero-temperature partition functions for strips of several lattices with $(FBC_y,PBC_x)$, i.e., cyclic, boundary conditions. In particular, the chromatic polynomial of a family of generalized dodecahedra graphs is calculated. The coefficient $c_{G,j}$ of degree $d$ in $q$ is $c^{(d)}=U_{2d}(\frac{\sqrt{q}}{2})$, where $U_n(x)$ is the Chebyshev polynomial of the second kind. We also present the chromatic polynomial for the strip of the square lattice with $(PBC_y,PBC_x)$, i.e., toroidal, boundary conditions and width $L_y=4$ with the property that each set of four vertical vertices forms a tetrahedron. A number of interesting and novel features of the continuous accumulation set of the chromatic zeros, ${\cal B}$ are found.

cond-mat.stat-mech↗

arXiv2000

Exact Potts Model Partition Functions on Strips of the Honeycomb Lattice

We present exact calculations of the partition function of the $q$-state Potts model on (i) open, (ii) cyclic, and (iii) Möbius strips of the honeycomb (brick) lattice of width $L_y=2$ and arbitrarily great length. In the infinite-length limit the thermodynamic properties are discussed. The continuous locus of singularities of the free energy is determined in the $q$ plane for fixed temperature and in the complex temperature plane for fixed $q$ values. We also give exact calculations of the zero-temperature partition function (chromatic polynomial) and $W(q)$, the exponent of the ground-state entropy, for the Potts antiferromagnet for honeycomb strips of type (iv) $L_y=3$, cyclic, (v) $L_y=3$, Möbius, (vi) $L_y=4$, cylindrical, and (vii) $L_y=4$, open. In the infinite-length limit we calculate $W(q)$ and determine the continuous locus of points where it is nonanalytic. We show that our exact calculation of the entropy for the $L_y=4$ strip with cylindrical boundary conditions provides an extremely accurate approximation, to a few parts in $10^5$ for moderate $q$ values, to the entropy for the full 2D honeycomb lattice (where the latter is determined by Monte Carlo measurements since no exact analytic form is known).

cond-mat.stat-mech↗

arXiv2000

Exact Partition Function for the Potts Model with Next-Nearest Neighbor Couplings on Strips of the Square Lattice

We present exact calculations of partition function $Z$ of the $q$-state Potts model with next-nearest-neighbor spin-spin couplings, both for the ferromagnetic and antiferromagnetic case, for arbitrary temperature, on $n$-vertex strip graphs of width $L_y=2$ of the square lattice with free, cyclic, and Möbius longitudinal boundary conditions. The free energy is calculated exactly for the infinite-length limit of these strip graphs and the thermodynamics is discussed. Considering the full generalization to arbitrary complex $q$ and temperature, we determine the singular locus ${\cal B}$ in the corresponding ${\mathbb C}^2$ space, arising as the accumulation set of partition function zeros as $n \to \infty$.

cond-mat.stat-mech↗