一○三 學年度
休假研究報告書(封面)
休假研究計畫名稱：在碎形和晶格上的統計模型
單位： 物理系
姓名： 張書銓
中華民國 一○四 年 九 月 七 日
It is my pleasure to visit Professor Robert Shrock in the C. N. Yang Institute for Theoretical Physics at State University of New York at Stony Brook during my sabbatical year, so that we can continue our long-term collaboration. Let me first review what we had accomplished, then describe the results that we obtain and submit for publication during this year. The ongoing projects will also be mentioned.
A part of our research has been concerned with the phenomenon of nonzero ground state entropy. Physical examples of this include ice and certain other hydrogen-bonded molecular crystals. A particularly simple model exhibiting ground state entropy (without the complication of frustration) is the q-state Potts antiferromagnet for sufficiently large values of q. The Potts model [1, 2] is a generalization of the Ising model, and has been a subject of central interest. The Potts model partition function Z G q v( , , ) on a graph G, is, up to a prefactor, the same as the Tutte-Whitney polynomial in modern mathematical graph theory [3-5], where v is the temperature variable. Specifically, the zero-temperature partition function of the
q-state Potts antiferromagnet is identical to the chromatic polynomial P G q( , ) [6-9]. This polynomial yields the number of ways to color the vertices of the graph G with q colors such that no adjacent vertices have the same color. In the earlier works with Professor Robert Shrock, we obtained exact results for the zero-temperature partition functions of the q-state Potts antiferromagnet and the full temperature-dependent partition function of the q-state Potts model for the square, triangular, honeycomb lattice strips of finite width and arbitrarily great length, with a variety of different boundary conditions, including free, cylindrical, cyclic, Möbius, torus, and Klein bottle [10-26]. We extended the transfer matrix method in the Fortuin-Kasteleyn representation from free and cylindrical strips to cyclic and Möbius strips [27, 28] and then to toroidal and Klein bottle strips [29] for various lattices. We also investigated the zero distributions of the q-state Potts model partition function in the large q limit for the square, triangular, honeycomb and kagomé lattices with various boundary conditions [30]. We showed that for large q these zeros take on approximately circular patterns in the complex temperature plane except for the strips with free boundary conditions. In addition, we calculated exactly the partition function Z G q v( , , ) of the
q-state Potts model on various lattice strips with q and v restricted to satisfy
conditions corresponding to the ferromagnetic phase transition on the associated two-dimensional lattices [31, 32]. We then made a breakthrough to include an external magnetic field in the q-state Potts model partition function Z G q v w( , , , ) with the field variable w [33, 34]. With no loss of generality, we took the external field to pick out a specific spin value from the q possibilities. A positive field, i.e., w>1, gives a weighting that favors spin configurations in which spins have the specific value; while
a negative field, i.e., 0≤ <w 1, disfavors such configurations. Analogous to the relation between zero-field Potts model partition function and the Tutte polynomial, we defined a rational function that generalizes the Tutte polynomial and is able to distinguish the graphs which have the same Tutte polynomial. For cyclic strip graphs of regular lattices with fixed width and arbitrary length, we have obtained a general structural determination of the partition function. We have generalized the methods for the zero-field case to construct the full transfer matrix for these strips that has a block structure consisting of submatrices. The dimensions of these submatrices, and therefore that of the full transfer matrix, were determined for arbitrary strip width. The multiplicity of their eigenvalues is given by the Chebyshev polynomial of the second kind. The corresponding results for Möbius strips of the square lattice or the honeycomb lattice whose number of width vertices is even were also derived. The determinants of the full transfer matrices and their submatrices for the square, triangular, honeycomb lattices have simple expressions. The transfer submatrix with the highest degree in multiplicity is simply a scalar for each of these lattices. We have obtained the general expressions of the transfer submatrices with the next highest degree in multiplicity for these lattices. We also considered families of self-dual strip graphs of the square lattice with fixed width and arbitrarily great length, having periodic longitudinal boundary conditions, such that all vertices on one side of the strip are joined by edges to a single external vertex. Recently, we calculated the zeros of the q-state Potts model partition function and chromatic polynomials on the two-dimensional Sierpinski gasket and its generalization. The exact expressions of the chromatic polynomial and the nonzero ground-state entropy per site were obtained for the q=4 Potts antiferromagnet. Our results are consistent with the inference that in the n→ ∞ limit a subset of chromatic zeros forms a closed loop on the q plane that crosses the real axis at q=0 and at a maximal point qc= . Equivalently, this 3 implies that q=3 Potts antiferromagnet is critical on the two-dimensional Sierpinski gasket at zero temperature. More generally, we analyzed the zeros of the q-state Potts model partition function in the complex q plane at finite temperature for both antiferromagnetic and ferromagnetic cases, and the zeros in the plane of a temperature-like Boltzmann variable (i.e., Fisher zeros) for various values of q [35]. The consideration for other fractals, e.g. diamond hierarchical graphs, is in progress. It is interesting to gain insight of the critical behavior using the plots of zeros in the q plane for the antiferromagnetic or ferromagnetic cases and in the y plane for various q values.
Because the partition function of the q-state Potts model on two-dimensional lattices cannot be solve exactly in general, one purpose of our early studies of the q-state Potts
model on various lattice strips is to understand how the properties and structure of the
q-state Potts model approach from quasi-one-dimensional strips to two-dimensional
lattices. Especially, we would like to see how the value of ground state entropy or ground state degeneracy per site of the zero-temperature Potts antiferromagnet for the two-dimensional lattices can be approached from that of the quasi-one-dimensional strips. In the previous studies we found that when we considered a lattice strip with free boundary conditions and took the chromatic polynomial to the power of one over the vertex number, the value at a given positive q is larger than that for the two-dimensional lattice. As we considered free strips with wider widths, the ground state degeneracy per site becomes smaller monotonically as a function of width and can be used as an upper bound for that of the two-dimensional lattice. However, this approach to the value for the two-dimensional lattice is rather slow. On the other hand, when we considered a lattice strip with cylindrical boundary conditions and took the chromatic polynomial to the power of one over the vertex number, the value at a certain large q remains larger than that for the two-dimensional lattice when the number of vertices in a transverse slice is even but becomes smaller than that for the two-dimensional lattice when the number of vertices in a transverse slice is odd. The reason to have such fluctuation for the cylindrical strips can be attributed to the fact that the chromatic number is two for the circle graph with even number of vertices while the chromatic number is three for the circle graph with odd number of vertices. As the cylindrical strips have periodic boundary conditions in the transverse direction, the approach of the ground state degeneracy per site to the value for the two-dimensional lattice is faster than that of the free strips, and these ground state degeneracies per site can be used as a better upper bound when the number of vertices in a transverse slice is even and as a lower bound when the number of vertices in a transverse slice is odd. As a two-dimensional lattice can be constructed as starting from a line graph or circle graph depending on the boundary conditions then one applies a transfer matrix to the next line graph or circle graph repeatedly, the lower bound of the ground state degeneracy per site for the square lattice was derived by Biggs [36] using the classical theory of Perron and Frobenius. Although the infinite large transfer matrix is unsolvable, the theory claims that there is a unique eigenvalue with greatest positive value and not less than the mean row sum of the matrix because the matrix has non-negative (coloring) elements. The mean sum of all the elements of the transfer matrix was calculated as the chromatic polynomial of the ladder graph divided by the chromatic polynomial of a line graph or circle graph. As the thermodynamic limit is always taken, it does not matter one uses free or cyclic boundary conditions because the dominant eigenvalue is the same when q is larger than or equal to the critical value. On the other hand, the upper bound of the ground
state degeneracy per site was harder to derive and the inequality of the coloring from one vertex to the next one was used in [36]. Shrock and Tsai adopted the method to consider the upper and lower bounds of the ground state degeneracies per site on the honeycomb and triangular lattices and noticed that the lower bounds are closer to the two-dimensional values than the upper bounds in terms of large-q series [37]. In a subsequent publication, Shrock and Tsai found that the lower bound for the honeycomb lattice could be improved by considering a different transfer direction, and the lower bound for the square lattice with next-nearest-neighbor couplings which is not a planar graph was also obtained and compared with Monte Carlo measurements [38]. Subsequently, the lower bounds and large-q series for the ground state degeneracies per site on all the Archimedean lattices and their duals were derived and a theorem for the lower bound applicable to all the Archimedean lattices was stated [39]. As we have calculated chromatic polynomials for various lattice strips with certain widths, we are able to improve their lower bounds on some lattices with the corresponding transfer matrices applied more than once. When the transfer matrix is symmetric and nonnegative, the theorem given by Merikoski [40] guarantees that the more times the transfer matrix applies the better lower bounds one gets. On the contrary, for the lattices with asymmetric transfer matrices, we are not able to obtain better lower bounds. For the square lattice, a further improvement is obtained using the ladder graph or even wider strips rather than a line graph as the basis of the transfer matrix, and similarly for the other lattices, including the triangular, honeycomb, kagomé, (4.82) lattices, and the square lattice with next-nearest-neighbor couplings. We show that such generalizations do increase the lower bounds for these lattices strips. Notice that the main purpose here is to see how the ground state degeneracies per site of various two-dimensional lattices can be approached by strips with larger and larger width, but not to estimate their numerical values as that can be computed by Monte Carlo measurements. The improvement is better when q is equal to or just above the critical value, and becomes insignificant when q is large as the bound approaches to the exact value rapidly as q increases. Because the dominant eigenvalue for the strip with width equal to four is a solution of a cubic equation for the square lattice and a solution of a quartic equation for the triangular lattice, they can be solved analytically even though the expressions are messy. It follows that the lower bound of the ground state degeneracy per site can be expanded as a series using the strips with width smaller than or equal to four, to be compared against with the known large-q series expansion. For the square lattice, the large-q series expansion had been considered more than four decades ago [41] and carried out to higher orders in [42-44]. That has been further extended to 80 terms recently [45]. For the triangular lattice, the general form of the ground state degeneracy per site had been derived by
Baxter [46, 47] as an infinite product when q is larger than or equal to four. For the honeycomb lattice, the large-q series expansion in [42] was defined per two-cell rather than per site, and is the square of the per site series. We find that using the same transfer matrix for a certain lattice, no matter how many times it applies, the order that it coincides with the series of the two-dimensional lattice remains the same, although the value of the next order becomes closer to the exact series as the transfer matrix is applied more times. For the square lattice, comparison of the series using the transfer matrix of the ladder graph once with that using the transfer matrix of the line graph once shows that the former coincides with the series of the two-dimensional lattice for two more orders. The same statement applies to the triangular and honeycomb lattices, while the known series of other two-dimensional lattices that we study, i.e. kagomé and (4.82) lattices, are not long enough and agree with the series of our lower bounds. I am happy to report here that the above results have been published as Physical Review E 91, 052142 (2015) with 17 pages.
In addition to consider the lower bound on the ground-state entropy of the antiferromagnetic Potts model with q larger than or equal to the critical value, it is also interesting to investigate the corresponding results with q smaller than or equal to zero. In our previous studies of chromatic polynomials and the accumulation of zeros in the complex q plane for a variety of lattice strips with periodic boundary conditions, the resultant singular loci are finite and never intersect the negative real q axis. It follows that the negative real q axis can be analytically continued from the positive large q axis, and the dominant eigenvalue of the transfer matrix remains the same in the negative q axis as that when q is a positive and large value. Although it looks unphysical to have negative q, the absolute value of the zero-temperature antiferromagnetic Potts model partition function, i.e. the chromatic polynomial, with
1
q= − gives the number of acyclic orientations [48], which is of course positive by definition. To be precise, the number of acyclic orientations is equal to the chromatic polynomial P G q( , ) evaluated at q= −1 multiplied by ( 1)− to the power of the number of vertices. Alternatively, we can convert the transfer matrix in terms of the Tutte polynomial variables x and y, then there is no sign ambiguity when x is set to two and y to zero. Furthermore, the chromatic polynomial divided by q in the limit
0
q→ gives the number of acyclic orientations in which there is exactly one source [49]. Notice that this number is independent of the vertex chosen to be the source. Unlike the numbers of spanning trees and dimer coverings on the regular lattices whose entropies per site can be calculated exactly, these numbers of acyclic orientations cannot be solved analytically on two dimensional lattices. The asymptotics, i.e. these numbers of acyclic orientations raised to the power of one over
the vertex number, cannot be approximated by the large-q series either which are only valid for large q. While we used the ground state degeneracy per site of a free strip as an upper bound for that of the two-dimensional lattice when q is large and positive as discussed above, we find that the asymptotics of these numbers of acyclic orientations of free strips become lower bounds for that of the two-dimensional lattice when
1
q= − and q=0. This can be verified using the exact expression for the triangular lattice [46, 47] that should be valid not only when q is larger or equal to four but also when q is less than or equal to zero. The asymptotic of a cylindrical strip also becomes a lower bound for that of the two-dimensional lattice and increases monotonically as the strip width increases no matter the width has even or odd vertices, in contrast to the behavior for the ground state degeneracy per site with positive q mentioned above. It is not hard to show that the value of the asymptotic is greater using the strips with periodic transverse boundary conditions than those with free transverse boundary conditions. We had already used this fact to obtain the lower bounds of the asymptotics of the acyclic orientations for the square and triangular lattices [25], which was quoted in [50] and better than the lower bound for the square lattice derived there. Although the upper and lower bounds of the asymptotics of the acyclic orientations were further improved recently [51], the authors overlooked our paper [25] and their lower bound fell far behind our value. It is expected that the upper bounds of these asymptotics can also be derived using our explicit calculation of the Potts model partition function for various lattice strips. The number of totally cyclic orientations of a two-dimensional lattice is equal to the number of acyclic orientations of the dual lattice, and corresponds to the Potts model partition function with q= −1 and v=1. For the honeycomb lattice, the asymptotic of totally cyclic orientations should be evaluated exactly by duality. The dominant eigenvalue in the region contains q= −1 and v=1 depends on the boundary conditions and we will choose the boundary condition that gives the best upper and lower bounds. There are other quantities whose asymptotic values cannot be derived exactly for the two-dimensional lattices. For example, the number of spanning forests corresponds to the Potts model partition function with v→0 , q→0 satisfying q v/ →1. Alternatively, we can convert the transfer matrix in terms of the Tutte polynomial variables x and y, then set x equal to two and y equal to one [52]. The upper and lower bounds for the asymptotic of spanning forests were obtained in [49] then improved in [50]. Currently the best upper bound was estimated in [53], while the best lower bound was given in [51]. The number of connected spanning subgraphs of a two-dimensional lattice is equal to the number of spanning forests of the dual lattice, and corresponds to the Potts model partition function with v=1 and q=0. Alternatively, we can convert the transfer matrix in terms of the Tutte polynomial
variables x and y, then set x equal to one and y equal to two [52]. As we will need the full-temperature Potts model rather than just chromatic polynomial, the upper and lower bounds for these asymptotics may not be as close to the corresponding two-dimensional values compared to those for the numbers of acyclic orientations. Unlike the case of acyclic orientations where the exact result was known for the two-dimensional triangular lattice, there is no exact value to be checked against for the number of spanning forests or the number of connected spanning subgraphs. The upper and lower bounds of all these quantities on various two-dimensional lattices are under investigation.
Professor Robert Shrock and one of his previous students, Yan Xu, had generalized our early results to consider an external magnetic field H in the q-state Potts model partition function Z G q s v w( , , , , ) affecting not just one spin state but s spin sates [54, 55], where the integer s is larger or equal to one and less or equal to q. They had calculated the q-state Potts model partition function in such generalized external field that favors or disfavors a subset of states on line graphs, star graphs, complete graphs, and the corresponding weighted-set chromatic polynomial, i.e., the zero-temperature antiferromagnet was also studied. While the general properties of this partition function had been derived for cyclic strips, including the coefficients and the number of eigenvalues, only the partition function for the circuit graphs was obtained explicitly. It is difficult to calculate for the strips with large width because the sizes of the transfer matrices depend on the variable s, especially for the sector d =0 where it is a polynomial in s with degree L . While the transfer matrix for each sector and y
certain s value can be derived individually, its size increases rapidly as s increases. I must calculate the transfer matrices for the first few s values in order to derive the general characteristic polynomial for the eigenvalues as a function of s. I succeed in calculating the partition function of this model on ladder strips of the square, triangular and honeycomb lattices with arbitrary length and cyclic boundary conditions. In addition to the coefficient that is a function of q−s and can be expressed in terms of the Chebyshev polynomial of the second kind, each eigenvalue also has a multiplicity which is a function of s. The form of the multiplicity is related to that of the coefficient, such that terms interchange with each other under the symmetric replacements s→ −q s , w→w−1 and multiplication by Ly
w . It is instructive to see how the exact solutions reduce to the zero field cases given in [18, 20, 56] and the results for s=1 in [34]. For the case of antiferromagnetic spin-spin coupling, our calculations provide exactly solved models that exhibit an onset of
frustration and competing interactions in the context of a novel type of tensor-product
s q s
S ⊗S − global symmetry, where S is the permutation group on s objects. Namely, s
when s is larger than zero and less than the chromatic number, the application of a positive magnetic field involves competing interactions and causes frustration. Similarly, when q−s is larger than zero and less than the chromatic number, the application of a negative magnetic field also involves competing interactions. This frustration becomes especially strong as the temperature goes to zero. For the infinite strip, it leads to a nonanalytic change in the properties of the ground state as a function of H/ |J|, where the spin-spin exchange interaction J is negative and s is less than the chromatic number. We consider the circuit graph as the simplest illustration for this phenomenon. As the temperature is close to zero, the spin-spin interaction requires that the spins on adjacent vertices should be different, while the magnetic field term biases them to lie in the same state when s=1. These requirements conflict with each other. We find that when H >2 |J| the ferromagnetic tendency due to the external field dominates over the proper q-coloring tendency due to the spin-spin interaction such that the magnetization per vertex is one, while the magnetization per vertex is a half when H <2 |J |. As H decreases through the value 2 |J|, there is a discontinuous decrease in the magnetization. All these results, including the transfer matrix of sector d =0 for the ladder graph of the square lattice and the transfer matrices of sectors d =0, d =1 for the ladder graph of the triangular lattice, are expected to be published soon.
In another direction, I continue to work on directed percolation, or oriented percolation problem, that can be thought of simply as a percolation process on a directed lattice in which connections are allowed only in a preferred direction [57, 58]. It was first studied by Broadbent and Hammersley in 1957 [59, 60] and it has remained to this day as one of the most outstanding interesting problems in probability and statistical mechanics. It is known that directed percolation is closely related to the Reggeon field theory in high-energy physics [61] and the Markov processes with branching, recombination and absorption that occur in chemistry and biology [62], etc. Very little is known in the way of exact solutions for the directed percolation problem. In 1981, Domany and Kinzel [63] defined a solvable version of compact directed percolation on the square lattice such that each vertical bond is directed upward with occupation probability p∈(0,1) (independently of the other bonds) while each horizontal bond is directed rightward with occupation probability 1. The boundary of the Domany-Kinzel model has the same distribution as the one-dimensional last passage percolation model [64]. A three-dimensional version of
Domany-Kinzel model with occupation probability 1 along two spatial directions was considered in [65]. As the Domany-Kinzel model had been formulated as a random-walk problem [66], a generalization has been done to consider the directed percolation on the square lattice whose vertical edges occupied with probability p v
and horizontal edges in the n-th row occupied with probability 1 if n is even and p h
if n is odd [67], and its asymptotic behavior was analyzed [68]. Particularly for 0
h
p = or 1, that model reduces to the original Domany-Kinzel model. However,
some of the arguments in [68] were not perfect and should be amended. Collaborated with Professor Lung-Chi Chen who has recently moved to Department of Mathematical Sciences in National Chengchi University, we generalized further to consider the directed percolation problem on the triangular lattice [69]. It should be clarified that the model we studied is not a compact directed percolation as holes may exist. The work of the directed percolation problem on the honeycomb lattice, with vertical probability y∈(0,1] and horizontal probabilities 1 and x∈(0,1]
alternatively, is completed when I visit Stony Brook University. A critical value of aspect ratio exists as for the square and triangular lattices, such that the probability of percolation becomes one, zero, or one-half when the aspect ratio above, below, or equal to the critical value. The critical exponent is found to be the same as that for the square and triangular lattices as expected. When x=1, the model reduces to the Domany-Kinzel model on the honeycomb lattice. It is easier to consider the honeycomb lattice as a brick lattice with the long axis of the bricks horizontal, and the corresponding results for the original honeycomb lattice composed of regular hexagons are obtained with proper scaling. Comparing the brick lattice against the square lattice, some edges of the square lattice are missing, therefore it is adequate to consider only the aspect ratio larger than or equal to one. In terms of one-dimensional independent and identically distributed random variables, the unit step is two for the brick lattice, in contrast to one for the square and triangular lattices. Furthermore, we use large deviation argument and the Berry-Esseen theorem to quantify the asymptotic behavior when the aspect ratio approaches to the critical point from below or from above in the thermodynamic limit. These results for the honeycomb lattice have been published as Physica A 436 (2015) 547-557 that includes Stony Brook University as one of my addresses.
We shall further consider directed percolation on a more general lattice such that there are two kinds of vertical edges with different occupation probabilities p , 1 p that 2
are alternative in each row and are located differently for even and odd rows. For simplicity, all the horizontal probabilities are set to one. The reason to consider such lattice is that when p1= p2 it reduces to the Domany-Kinzel model on the square
lattice while it reduces to the Domany-Kinzel model on the honeycomb lattice as bricks when one of the occupation probabilities is set to zero. It is expected that a critical value of aspect ratio exists as for the three homogeneous two-dimensional lattices, such that the probability of percolation becomes one, zero, or one-half when the aspect ratio above, below, or equal to the critical value. The critical exponent should also be the same as what we found previously for the square, triangular and honeycomb lattices. Nevertheless, the critical value of aspect ratio is now a function of p , 1 p , and so is the variance. For a percolation from one row to the next row, the 2
starting point can have either probability p or 1 p upward, and so is the ending 2
point. It is conceivable that the problem should be formulated in term of a two by two matrix. We shall also study the asymptotic behavior when the aspect ratio is close to the critical value, where the calculation would be involved. This directed percolation model on such general lattice remains under investigation.
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