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Shan-Ho Tsai

Publications and source records attributed to Shan-Ho Tsai.

35 records · Page 2Linked to original sources
arXiv1998

Fourth-Order Cumulants to Characterize the Phase Transitions of a Spin-1 Ising Model

Fourth-order cumulants of physical quantities have been used to characterize the nature of a phase transition. In this paper we report some Monte Carlo simulations to illustrate the behavior of fourth-order cumulants of magnetization and energy across second and first-order transitions in the phase diagram of a well known spin-1 Ising model. Simple ideas from the theory of thermodynamic fluctuations are used to account for the behavior of these cumulants.

cond-mat.stat-mech↗
arXiv1998

Ground State Entropy of Potts Antiferromagnets: Cases with Noncompact W Boundaries Having Multiple Points at 1/q = 0

We present exact calculations of the zero-temperature partition function, $Z(G,q,T=0)$, and ground-state degeneracy (per site), $W({G},q)$, for the $q$-state Potts antiferromagnet on a number of families of graphs ${G}$ for which the boundary ${\cal B}$ of regions of analyticity of $W$ in the complex $q$ plane is noncompact and has the properties that (i) in the $z=1/q$ plane, the point $z=0$ is a multiple point on ${\cal B}$ and (ii) ${\cal B}$ includes support for $Re(q) < 0$. These families are generated by the method of homeomorphic expansion. Our results give further insight into the conditions for the validity of large--$q$ series expansions for the reduced function $W_{red.}=q^{-1}W$.

cond-mat.stat-mech↗
arXiv1998

Chromatic Polynomials for $J(\prod H)I$ Strip Graphs and their Asymptotic Limits

We calculate the chromatic polynomials $P$ for $n$-vertex strip graphs of the form $J(\prod_{\ell=1}^m H)I$, where $J$ and $I$ are various subgraphs on the left and right ends of the strip, whose bulk is comprised of $m$-fold repetitions of a subgraph $H$. The strips have free boundary conditions in the longitudinal direction and free or periodic boundary conditions in the transverse direction. This extends our earlier calculations for strip graphs of the form $(\prod_{\ell=1}^m H)I$. We use a generating function method. From these results we compute the asymptotic limiting function $W=\lim_{n \to \infty}P^{1/n}$; for $q \in {\mathbb Z}_+$ this has physical significance as the ground state degeneracy per site (exponent of the ground state entropy) of the $q$-state Potts antiferromagnet on the given strip. In the complex $q$ plane, $W$ is an analytic function except on a certain continuous locus ${\cal B}$. In contrast to the $(\prod_{\ell=1}^m H)I$ strip graphs, where ${\cal B}$ (i) is independent of $I$, and (ii) consists of arcs and possible line segments that do not enclose any regions in the $q$ plane, we find that for some $J(\prod_{\ell=1}^m H)I$ strip graphs, ${\cal B}$ (i) does depend on $I$ and $J$, and (ii) can enclose regions in the $q$ plane. Our study elucidates the effects of different end subgraphs $I$ and $J$ and of boundary conditions on the infinite-length limit of the strip graphs.

cond-mat.stat-mech↗
arXiv1998

End Graph Effects on Chromatic Polynomials for Strip Graphs of Lattices and their Asymptotic Limits

We report exact calculations of the ground state degeneracy per site (exponent of the ground state entropy) $W(\{G\},q)$ of the $q$-state Potts antiferromagnet on infinitely long strips with specified end graphs for free boundary conditions in the longitudinal direction and free and periodic boundary conditions in the transverse direction. This is equivalent to calculating the chromatic polynomials and their asymptotic limits for these graphs. Making the generalization from $q \in {\mathbb Z}_+$ to $q \in {\mathbb C}$, we determine the full locus ${\cal B}$ on which $W$ is nonanalytic in the complex $q$ plane. We report the first example for this class of strip graphs in which ${\cal B}$ encloses regions even for planar end graphs. The bulk of the specific strip graph that exhibits this property is a part of the $(3^3 \cdot 4^2)$ Archimedean lattice.

cond-mat.stat-mech↗
arXiv1998

Ground State Entropy of Potts Antiferromagnets on Homeomorphic Families of Strip Graphs

We present exact calculations of the zero-temperature partition function, and ground-state degeneracy (per site), $W$, for the $q$-state Potts antiferromagnet on a variety of homeomorphic families of planar strip graphs $G = (Ch)_{k_1,k_2,Σ,k,m}$, where $k_1$, $k_2$, $Σ$, and $k$ describe the homeomorphic structure, and $m$ denotes the length of the strip. Several different ways of taking the total number of vertices to infinity, by sending (i) $m \to \infty$ with $k_1$, $k_2$, and $k$ fixed; (ii) $k_1$ and/or $k_2 \to \infty$ with $m$, and $k$ fixed; and (iii) $k \to \infty$ with $m$ and $p=k_1+k_2$ fixed are studied and the respective loci of points ${\cal B}$ where $W$ is nonanalytic in the complex $q$ plane are determined. The ${\cal B}$'s for limit (i) are comprised of arcs which do not enclose regions in the $q$ plane and, for many values of $p$ and $k$, include support for $Re(q) < 0$. The ${\cal B}$ for limits (ii) and (iii) is the unit circle $|q-1|=1$.

cond-mat.stat-mech↗
arXiv1997

Chromatic Polynomials for Families of Strip Graphs and their Asymptotic Limits

We calculate the chromatic polynomials $P((G_s)_m,q)$ and, from these, the asymptotic limiting functions $W(\{G_s\},q)=\lim_{n \to \infty}P(G_s,q)^{1/n}$ for families of $n$-vertex graphs $(G_s)_m$ comprised of $m$ repeated subgraphs $H$ adjoined to an initial graph $I$. These calculations of $W(\{G_s\},q)$ for infinitely long strips of varying widths yield important insights into properties of $W(Λ,q)$ for two-dimensional lattices $Λ$. In turn, these results connect with statistical mechanics, since $W(Λ,q)$ is the ground state degeneracy of the $q$-state Potts model on the lattice $Λ$. For our calculations, we develop and use a generating function method, which enables us to determine both the chromatic polynomials of finite strip graphs and the resultant $W(\{G_s\},q)$ function in the limit $n \to \infty$. From this, we obtain the exact continuous locus of points ${\cal B}$ where $W(\{G_s\},q)$ is nonanalytic in the complex $q$ plane. This locus is shown to consist of arcs which do not separate the $q$ plane into disconnected regions. Zeros of chromatic polynomials are computed for finite strips and compared with the exact locus of singularities ${\cal B}$. We find that as the width of the infinitely long strips is increased, the arcs comprising ${\cal B}$ elongate and move toward each other, which enables one to understand the origin of closed regions that result for the (infinite) 2D lattice.

cond-mat.stat-mech↗
arXiv1997

Complex-Temperature Partition Function Zeros of the Potts Model on the Honeycomb and Kagomé Lattices

We calculate complex-temperature (CT) zeros of the partition function for the $q$-state Potts model on the honeycomb and kagomé lattices for several values of $q$. These give information on the CT phase diagrams. A comparison of results obtained for different boundary conditions and a discussion of some CT singularities are given. Among other results, our findings show that the Potts antiferromagnet with $q=4$ and $q=5$ on the kagomé lattice has no phase transition at either finite or zero temperature.

cond-mat.stat-mech↗
arXiv1997

A Mapping Relating Complex and Physical Temperatures in the 2D $q$-state Potts Model and Some Applications

We show an exact equivalence of the free energy of the $q$-state Potts antiferromagnet on a lattice $Λ$ for the full temperature interval $0 \le T \le \infty$ and the free energy of the $q$-state Potts model on the dual lattice for a semi-infinite interval of complex temperatures (CT). This implies the existence of two quite different types of CT singularities: the generic kind, which does not obey universality or various scaling relations, and a special kind which does obey such properties and encodes information of direct physical relevance. We apply this observation to characterize CT properties of the Potts model on several lattices, to rule out two existing conjectures, and to determine the critical value of $q$ above which the Potts antiferromagnet on the diced lattice has no phase transition.

cond-mat.stat-mech↗
arXiv1997

Lower Bounds and Series for the Ground State Entropy of the Potts Antiferromagnet on Archimedean Lattices and their Duals

We prove a general rigorous lower bound for $W(Λ,q)=\exp(S_0(Λ,q)/k_B)$, the exponent of the ground state entropy of the $q$-state Potts antiferromagnet, on an arbitrary Archimedean lattice $Λ$. We calculate large-$q$ series expansions for the exact $W_r(Λ,q)=q^{-1}W(Λ,q)$ and compare these with our lower bounds on this function on the various Archimedean lattices. It is shown that the lower bounds coincide with a number of terms in the large-$q$ expansions and hence serve not just as bounds but also as very good approximations to the respective exact functions $W_r(Λ,q)$ for large $q$ on the various lattices $Λ$. Plots of $W_r(Λ,q)$ are given, and the general dependence on lattice coordination number is noted. Lower bounds and series are also presented for the duals of Archimedean lattices. As part of the study, the chromatic number is determined for all Archimedean lattices and their duals. Finally, we report calculations of chromatic zeros for several lattices; these provide further support for our earlier conjecture that a sufficient condition for $W_r(Λ,q)$ to be analytic at $1/q=0$ is that $Λ$ is a regular lattice.

cond-mat.stat-mech↗
arXiv1997

Families of Graphs with W_r({G},q) Functions That Are Nonanalytic at 1/q=0

Denoting $P(G,q)$ as the chromatic polynomial for coloring an $n$-vertex graph $G$ with $q$ colors, and considering the limiting function $W(\{G\},q) = \lim_{n \to \infty}P(G,q)^{1/n}$, a fundamental question in graph theory is the following: is $W_r(\{G\},q) = q^{-1}W(\{G\},q)$ analytic or not at the origin of the $1/q$ plane? (where the complex generalization of $q$ is assumed). This question is also relevant in statistical mechanics because $W(\{G\},q)=\exp(S_0/k_B)$, where $S_0$ is the ground state entropy of the $q$-state Potts antiferromagnet on the lattice graph $\{G\}$, and the analyticity of $W_r(\{G\},q)$ at $1/q=0$ is necessary for the large-$q$ series expansions of $W_r(\{G\},q)$. Although $W_r$ is analytic at $1/q=0$ for many $\{G\}$, there are some $\{G\}$ for which it is not; for these, $W_r$ has no large-$q$ series expansion. It is important to understand the reason for this nonanalyticity. Here we give a general condition that determines whether or not a particular $W_r(\{G\},q)$ is analytic at $1/q=0$ and explains the nonanalyticity where it occurs. We also construct infinite families of graphs with $W_r$ functions that are non-analytic at $1/q=0$ and investigate the properties of these functions. Our results are consistent with the conjecture that a sufficient condition for $W_r(\{G\},q)$ to be analytic at $1/q=0$ is that $\{G\}$ is a regular lattice graph $Λ$. (This is known not to be a necessary condition).

cond-mat.stat-mech↗
arXiv1997

Ground State Entropy of Potts Antiferromagnets: Bounds, Series, and Monte Carlo Measurements

We report several results concerning $W(Λ,q)=\exp(S_0/k_B)$, the exponent of the ground state entropy of the Potts antiferromagnet on a lattice $Λ$. First, we improve our previous rigorous lower bound on $W(hc,q)$ for the honeycomb (hc) lattice and find that it is extremely accurate; it agrees to the first eleven terms with the large-$q$ series for $W(hc,q)$. Second, we investigate the heteropolygonal Archimedean $4 \cdot 8^2$ lattice, derive a rigorous lower bound, on $W(4 \cdot 8^2,q)$, and calculate the large-$q$ series for this function to $O(y^{12})$ where $y=1/(q-1)$. Remarkably, these agree exactly to all thirteen terms calculated. We also report Monte Carlo measurements, and find that these are very close to our lower bound and series. Third, we study the effect of non-nearest-neighbor couplings, focusing on the square lattice with next-nearest-neighbor bonds.

cond-mat.stat-mech↗
arXiv1997

Asymptotic Limits and Zeros of Chromatic Polynomials and Ground State Entropy of Potts Antiferromagnets

We study the asymptotic limiting function $W({G},q) = \lim_{n \to \infty}P(G,q)^{1/n}$, where $P(G,q)$ is the chromatic polynomial for a graph $G$ with $n$ vertices. We first discuss a subtlety in the definition of $W({G},q)$ resulting from the fact that at certain special points $q_s$, the following limits do not commute: $\lim_{n \to \infty} \lim_{q \to q_s} P(G,q)^{1/n} \ne \lim_{q \to q_s} \lim_{n \to \infty} P(G,q)^{1/n}$. We then present exact calculations of $W({G},q)$ and determine the corresponding analytic structure in the complex $q$ plane for a number of families of graphs ${G}$, including circuits, wheels, biwheels, bipyramids, and (cyclic and twisted) ladders. We study the zeros of the corresponding chromatic polynomials and prove a theorem that for certain families of graphs, all but a finite number of the zeros lie exactly on a unit circle, whose position depends on the family. Using the connection of $P(G,q)$ with the zero-temperature Potts antiferromagnet, we derive a theorem concerning the maximal finite real point of non-analyticity in $W({G},q)$, denoted $q_c$ and apply this theorem to deduce that $q_c(sq)=3$ and $q_c(hc) = (3+\sqrt{5})/2$ for the square and honeycomb lattices. Finally, numerical calculations of $W(hc,q)$ and $W(sq,q)$ are presented and compared with series expansions and bounds.

cond-mat.stat-mech↗
arXiv1997

Families of Graphs With Chromatic Zeros Lying on Circles

We define an infinite set of families of graphs, which we call $p$-wheels and denote $(Wh)^{(p)}_n$, that generalize the wheel ($p=1$) and biwheel ($p=2$) graphs. The chromatic polynomial for $(Wh)^{(p)}_n$ is calculated, and remarkably simple properties of the chromatic zeros are found: (i) the real zeros occur at $q=0,1,...p+1$ for $n-p$ even and $q=0,1,...p+2$ for $n-p$ odd; and (ii) the complex zeros all lie, equally spaced, on the unit circle $|q-(p+1)|=1$ in the complex $q$ plane. In the $n \to \infty$ limit, the zeros on this circle merge to form a boundary curve separating two regions where the limiting function $W(\{(Wh)^{(p)}\},q)$ is analytic, viz., the exterior and interior of the above circle. Connections with statistical mechanics are noted.

cond-mat.stat-mech↗
arXiv1997

Complex-Temperature Phase Diagrams of 1D Spin Models with Next-Nearest-Neighbor Couplings

We study the dependence of complex-temperature phase diagrams on details of the Hamiltonian, focusing on the effect of non-nearest-neighbor spin-spin couplings. For this purpose, we consider a simple exactly solvable model, the 1D Ising model with nearest-neighbor (NN) and next-to-nearest-neighbor (NNN) couplings. We work out the exact phase diagrams for various values of $J_{nnn}/J_{nn}$ and compare these with the case of pure nearest-neighbor (NN) couplings. We also give some similar results for the 1D Potts model with NN and NNN couplings.

cond-mat.stat-mech↗