行政院國家科學委員會專題研究計畫 成果報告
Potts 與 Ising 模型的解析計算
計畫類別： 個別型計畫 計畫編號： NSC93-2119-M-006-009- 執行期間： 93 年 11 月 01 日至 94 年 10 月 31 日 執行單位： 國立成功大學物理學系（所） 計畫主持人： 張書銓 報告類型： 精簡報告 報告附件： 出席國際會議研究心得報告及發表論文 處理方式： 本計畫可公開查詢中 華 民 國 94 年 11 月 7 日
中文摘要：
在 Fortuin-Kasteleyn 表象下的轉移矩陣法已經被用於計算 q 態 Potts 模型在有限寬、無限 長之晶格長條的配分函數，其中 q 可任意（不需是整數）。我們過去已將這個方法由開放及 圓筒邊界條件擴展到具有週期和 Möbius 邊界條件，正方、三角及蜂巢晶格的平面長條。考 慮具有環形和 Klein bottle 邊界條件所具有的旋轉對稱性，如今我們巳成功地將這個方法進 一步推廣到這些非平面的長條。特別是我們得到係數和的表示式，和轉移矩陣的大小。除 了任意溫度的 q 態 Potts 模型外，我們也得到零度反鐵磁 Potts 模型，也就是色多項式，的 轉移矩陣的結構。另一方面，我們引入一個新的方法來推導 Ising 模型在寬度為 L、長度無 限之矩形長條的邊界關連函數。在 L 趨近無限大的熱力學極限下，邊界關連函數為邊界自 發磁化強度的平方。我們也研究在溫度高於、等於和低於臨界溫度下，這個邊界關連函數 的漸近行為。由於最近鄰關連函數也可在同樣的形式下推導，這便提供了可以同時考慮長 程邊界關連函數和短程關連函數的統一方法。英文摘要：
The transfer matrix method in the Fortuin-Kasteleyn representation has been used to calculate the partition function of the q-state Potts model for arbitrary (not necessarily integer) q on lattice strips of fixed width and arbitrarily great length. We had extended the method for planar strips of the square, triangular and honeycomb lattices with free and cylindrical boundary conditions to cyclic and Möbius boundary conditions. Now we have succeeded in generalizing the method to non-planar strips with toroidal and Klein bottle strips by taking into account the rotational symmetry of these strips. Especially, the expression for the sum of coefficients is obtained, and the size of the transfer matrix is determined. In addition to the full-temperature q-state Potts model, the structure of the transfer matrix for the zero-temperature Potts antiferromagnet, i.e. the chromatic polynomial, is also obtained. In another direction, we introduced a novel method to derive the boundary-boundary correlation function for the rectangular Ising lattice strip with infinite length and finite width L. In the thermodynamic limit such that L approaches to infinity, the boundary-boundary correlation function yields the square of the boundary spontaneous magnetization. We have also studied the asymptotic behavior of the boundary-boundary
correlation function when the temperature is above, at, and below the critical temperature. As the nearest-neighbor correlation function could be derived in the same formalism, this provides a unified way to consider both long-range boundary correlation function and the short-range correlation function at once.
關鍵詞：
Potts 模型, 色多項式, Ising 模型, 臨界現象, 邊界效應, 精確解
報告內容：
My research work has been in statistical mechanics. A major part of my research has been concerned with a very fundamental topic, namely, 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] is a generalization of the Ising model, and has been a subject of central interest. The famous review paper [2] on this model given by Professor F. Y. Wu has been cited for more than a thousand times. This subject also has a deep connection with mathematical graph theory, since the q-state Potts model partition function Z(G,q,v) on a lattice, or more generally, 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 of coloring the vertices of the lattice G with q colors such that no two adjacent vertices have the same color.
Together with my thesis advisor, Professor Robert Shrock, when I was a Ph.D. student in the C. N. Yang Institute for Theoretical Physics and Department of Physics and Astronomy at the State University of New York at Stony Brook, we obtained exact results for the zero-temperature partition functions of the q-state Potts antiferromagnet for several lattice strips of finite width and arbitrarily great length, with a variety of different boundary conditions, including free, cylindrical, cyclic, Möbius, toroidal, and Klein bottle [10-17]. Once the polynomial P(G,q) is calculated, q can be generalized from a positive integer to a complex variable. We calculated the zeros of P(G,q), called chromatic zeros, for the strips of the square, triangular and honeycomb lattices. Since our exact results apply for arbitrary length, we were able to study the infinite-length limit in which a subset of these zeros accumulate to form curves, denotedB, across which the degeneracy per site is nonanalytic. This is analogous to the generalization that Yang and Lee carried out of the magnetic field from real to complex values [18] and their study of the zeros of the Ising model partition function. Another analogy is to the corresponding generalization carried by M. E. Fisher of the temperature from physical to complex values and his study of the continuous
accumulation set of Fisher zeros [19]. We found that although the thermodynamics of these infinite strips is quasi-one-dimensional, a disorder quantity such as the ground state degeneracy per site can, for even moderately large widths, be very close to its value for the infinite 2D lattice.
In a series of papers, we performed exact calculations of the full temperature-dependent partition function of the q-state Potts model, Z(G,q,v) for arbitrary (not necessarily integer) q on lattice strips of fixed width and arbitrarily great length, again with a variety of boundary conditions [20-27]. Our results apply for both the ferromagnetic and antiferromagnetic cases of the Potts model. With these exact results in hand, we went on to determine the singular locusB where the
variables (q,v). We showed how the zeros of the partition function in the q plane for fixed v and in the v plane for fixed q accumulate to form this locus.
The transfer matrix method for lattice strips of the square and triangular lattices with free longitudinal boundary conditions and free or cylindrical transverse boundary conditions in the Fortuin-Kasteleyn representation was explained in detail by Salas and Sokal [28]. By using the bases for the chromatic polynomials in terms of the compatibility matrix [29-31], we succeeded in extending the method to cyclic and Möbius strips for various planar lattices, i.e., square, triangular and honeycomb lattices [32, 33]. We also proved a number of theorems concerning the general structure of the Potts model partition function for these strips with not just arbitrary length, but also arbitrary width L [13, 22, 25, 26]. The partition functions for cyclic strips can be written as , ( , ) ( ) , , , 0 1 ( , , ) ( ) Z cyc n L d L d m Z L d j d j Z G q v c
,where m is the number of repeated subgraph in the longitudinal direction. We found the coefficient of degree d in q is c( d) = U2d( q/2) , where U2d(x) is the Chebyshev
polynomial of the second kind. The transfer matrix has a block structure, and each block has the
size nZ cyc, ( , )L d with Z,L,d,j as its eigenvalues. Although the structure of the Potts model
partition function for cyclic strips has been discussed by Saleur [34, 35] by a different approach, our results extend his to consider the structure of the chromatic polynomial for both cyclic and Möbius strips. In addition, our determination of the structure of the partition function for self-dual cyclic strips of the square lattice filled out the relevant Bratteli diagrams.
One purpose of the proposed project is to continue the research results given above. As we have already extended the transfer matrix method to cyclic and Möbius strips [32, 33], it is desirable to apply the method to toroidal and Klein bottle strips that are not planar and more difficult. I am happy to report here that we have succeeded in this extension, and the results are in press to be published in Physica A. For toroidal and Klein bottle strips, we knew the coefficient of degree d is not unique [14, 23, 36], in contrast to the coefficient for cyclic and Möbius strips. As the slice of the toroidal strips is a circle, the rotational symmetry should be taken into account. We found that the partition functions for toroidal strips can be written as
( ) , , , 0 ( , , ) ( ) L d m j Z L d j d j Z G q v b
,where there is only one coefficient b( )jd for both d=0 and d=1, but there are two coefficients for d=2 and d=3. The reason is due to the possible permutation of the color assignments for the bases. A set of d! eigenvalues should have their coefficients summed to be equal to b( )d which can be determined by the sieve formula [29-31]. For d=0, 1, 2, b( )d are the same as c( )d , and
( ) ( ) ( 1) (1)
( 1)
d d d
b c c c for d larger or equal to two. We have also determined the reduced size of the transfer matrix nZ tor, ( , )L d without the permutation of color assignments. For d equal
to zero and one, , ( , 0) 1 2 1 Z tor L n L L L and , 2 1 ( ,1) 1 Z tor L n L L
that are obtained by the Narayana number, N n k( , ), the number of partitions of size n with k components. For d between
two and L, , , ' 2 ( , ) ( , ') L Z tor Z cyc d d L n L d n L d L d
that is obtained through the comparison with the corresponding size of the transfer matrix for cyclic strips. When the longitudinal boundary condition is changed from toroidal to Klein bottle, the eigenvalues remain the same and we observed the changes of coefficients. Certain coefficients become zero for the Klein bottle strips so that the number of eigenvalues for Klein bottle strips always appears to be less than the number for the corresponding toroidal strips. We also worked out the corresponding results for the zero-temperature Potts antiferromagnet as follows. It is necessary that adjacent vertices are not assigned the same color for the chromatic polynomials. Denote the reduced size of the transfer matrix for the square and triangular lattices as nP tor, ( , )L d . For d=0, nP tor, ( , 0)L is thenumber of non-crossing non-nearest-neighbor partitions of L vertices on a circle, namely, the Riordan number. We found
[ / 2] , 1 1 ( ,1) 1 L P tor k L L k n L k k
that is obtained by the number of diagonal dissections of a convex n-gon into k regions. For d between two and L, nP tor, ( , )L d isagain obtained through the comparison with the corresponding size of the transfer matrix for cyclic strips. For a strip of the honeycomb lattice with toroidal boundary conditions, the width L must be even. In each transverse slice of the strip, the L vertices are connected in a pairwise manner. That is, compared with the square and triangular lattices, only half edges are kept. Denote the reduced size of the transfer matrix as nP tor hc, , ( , )L d without the permutation of the
color assignments. nP tor hc, , ( , 0)L is the same as the corresponding number nP cyc hc, , ( , 0)L for
cyclic strips. We found
1 , , , , ' 1 ( ,1) ( , ') L P tor hc P cyc hc odd d n L n L d
and , , , , ' ( , ) ( , ') L P tor hc P cyc hc d d n L d n L d
for d larger or equal to two. In addition to the above general structure results which apply to arbitrary width and length for the strips with toroidal boundary conditions, we also performed explicit calculations for the square and honeycomb lattices with L=4 and for the triangular lattice with L=3. These reduce to the zero-temperature partition function of the q-state Potts
antiferromagnet that we had in Ref. [12].
I have been also working with Professor Masuo Suzuki, a world authority in statistical mechanics, with a new research direction, namely, the mesoscopic phenomena in infinite systems. It is
interaction on only a part of an infinite system, but keep the size of the affected part infinite. The result is then compared with both macroscopic and microscopic phenomena. It is well known that the Ising model is one of the simplest models in the study of critical phenomena since Onsager's celebrated work on the free energy of the rectangular lattice without magnetic field [37], followed by Yang's derivation of the spontaneous magnetization [38]. One interesting direction is to study boundary effects in an infinite system, and the approach of the boundary result to the bulk result. In Ref. [39], we calculated the long-range correlation functions of the rectangular Ising model between two spins on the same row along the direction with free boundary conditions. Expressing the final result as a low-temperature series expansion with the same spin-spin couplings in both directions and applying D Log Padé approximant to the series [40], we showed that if one spin is on the m-th row from one boundary, and the other spin is on the n-th row from the other boundary, their long-range correlation function is the product of the corresponding m-th and n-th row
spontaneous magnetizations. In terms of low-temperature series expansions, the approach of the correlation function between two m-th row spins to the bulk correlation function could be
understood as follows: the dominant terms of their series expansions are the same and the number of these terms increases monotonically as m increases.
One purpose of the proposed project is to investigate a special mesoscopic phenomenon, the boundary effect of an infinite system where the size of boundaries is infinite. In collaboration with Professor Masuo Suzuki and his student (now postdoc) Hidenori Suzuki, we proposed a novel method to derive the boundary-boundary correlation function. The results have been published in Journal of Mathematical Physics, 46, 1 (2005). Consider the rectangular Ising lattice strip with infinite length and finite width L and ferromagnetic interactions. Take periodic
boundary conditions in the horizontal direction and free boundary conditions in the transverse direction, then impose a topological spin-spin interaction J’between the spins on the free boundaries. The boundary-boundary correlation function CL is essentially the derivative of the free energy with respect to J’, followed by taking the limit J’=0. When the width L approaches to infinity, the boundary-boundary correlation function is equal to square of the boundary
spontaneous magnetization mb given by McCoy and Wu [41] as expected. Furthermore, we determined the asymptotic behavior of the boundary-boundary correlation functions as follows. When the temperature is above the critical temperature T ,C CL is proportional to
/ L e L , where ξis the same as the ordinary correlation length with exponent ν=1. At the critical temperature, we found CL is proportional to 1/L. It becomes more complicated when the temperature is below
C
T and we found that CLmb2 is proportional to eL/ '. As temperature is low, it is interesting to note that CL is nonmonotonic with respect to the width L. On the other hand, L approaching to infinity limit can be taken first before the limit J’=0. We found these two limits are
interchangeable for the two-dimensional rectangular lattice. That is, if L approaching to infinity is taken first for the two-dimensional lattice, then the limit J’=0 gives the same long-range
the regular spin-spin interaction, we rederived the ordinary nearest-neighbor correlation function [42, 43]. That is, J’changes the topology of the system according to its value. If J’is not equal to zero, we have the short-range correlations; while for J’=0, we have the long-range
boundary-boundary correlations for the two-dimension lattice. This procedure was checked first on the Ising chain by imposing a topological spin-spin interaction J’between the edge spins. It is straightforward to calculate the partition function of this Ising chain for non-zero magnetic field and we compared the boundary magnetization where a magnetic field only applies on an edge spin with the uniform magnetization per site where the magnetic field applies to all the spins.
References:
[1] R. B. Potts, Proc. Camb. Phil. Soc. 48 (1952) 106. [2] F. Y. Wu, Rev. Mod. Phys. 54 (1982) 235.
[3] W. T. Tutte, Can. J. Math. 6, 80 (1954). [4] W. T. Tutte, J. Combin. Theory 2, 301 (1967).
[5] W. T. Tutte, ``Chromials'', in Lecture Notes in Math. v. 411 (1974) 243; Graph Theory, vol. 21 of Encyclopedia of Mathematics and Applications (Addison-Wesley, Menlo Park, 1984).
[6] F. Y. Wu, J. Stat. Phys. 52 (1988) 99.
[7] R. C. Read, J. Combin. Theory 4 (1968) 52.
[8] R. C. Read and W. T. Tutte, ``Chromatic Polynomials'', in Selected Topics in Graph Theory, 3, (Academic Press, New York, 1988), p. 15.
[9] N. L. Biggs, Algebraic Graph Theory (Cambridge Univ. Press, Cambridge, 1st ed. 1974, 2nd ed. 1993).
[10] S.-C. Chang and R. Shrock, Phys. Rev. E 62 (2000) 4650. [11] S.-C. Chang and R. Shrock, Physica A 290 (2001) 402. [12] S.-C. Chang and R. Shrock, Physica A 292 (2001) 307. [13] S.-C. Chang and R. Shrock, Physica A 296 (2001) 131. [14] S.-C. Chang and R. Shrock, Ann. Phys. 290 (2001) 124. [15] S.-C. Chang, Physica A 296 (2001) 495.
[16] S.-C. Chang and R. Shrock, Physica A 301 (2001) 301. [17] S.-C. Chang, Physica A 313 (2002) 397.
[18] T. D. Lee and C. N. Yang, Phys. Rev. 87 (1952) 410.
[19] M. E. Fisher, Lectures in Theoretical Physics, vol. 7C (Univ. Colorado Press, Boulder, CO, 1965), p. 1.
[20] S.-C. Chang and R. Shrock, Physica A 286 (2001) 189.
[21] S.-C. Chang and R. Shrock, Int. J. Mod. Phys. B 15 (2001) 443. [22] S.-C. Chang and R. Shrock, Physica A 296 (2001) 183.
[23] S.-C. Chang and R. Shrock, Physica A 296 (2001) 234. [24] S.-C. Chang and R. Shrock, Phys. Rev. E 64 (2001) 066116.
[25] S.-C. Chang, J. Salas, and R. Shrock, J. Stat. Phys. 107 (2002) 1207.
[26] S.-C. Chang, J. Jacobsen, J. Salas, and R. Shrock, J. Stat. Phys. 114 (2004) 763. [27] S.-C. Chang and R. Shrock, Advances in Applied. Math. 32 (2004) 44.
[28] J. Salas and A. Sokal, J. Stat. Phys. 104 (2001) 609. [29] N. L. Biggs, J. Combin. Theory B 82 (2001) 19. [30] N. L. Biggs, Bull. London Math. Soc. 34 (2002) 129.
[31] N. L. Biggs, London School of Economics Centre for Discrete and Applicable Mathematics, Report LSE-CDAM-2000-04.
[32] S.-C. Chang and R. Shrock, Physica A, 346 (2005) 400. [33] S.-C. Chang and R. Shrock, Physica A, 347 (2005) 314. [34] H. Saleur, Nucl. Phys. B 360 (1991) 219.
[35] H. Saleur, Commun. Math. Phys. 132 (1990) 657. [36] N. L. Biggs and R. Shrock, J. Phys. A 32 (1999) L489. [37] L. Onsager, Phys. Rev. 65 (1944) 117.
[38] C. N. Yang, Phys. Rev. 85 (1952) 808.
[39] S.-C. Chang and M. Suzuki, Int. J. Mod. Phys. B 18 (2004) 275.
[40] D. S. Gaunt and A. J. Guttmann, Phase Transitions and Critical Phenomena, Vol. 3, eds. C. Domb and M. S. Green (Academic Press, London, 1974).
[41] B. M. McCoy and T. T. Wu, Phys. Rev. 162 (1967) 436.
[42] E. W. Montroll, R. B. Potts and J. C. Ward, J. Math. Phys. 4, 308 (1963). [43] B. Kaufman, Phys. Rev. 76, 1232 (1949).
計畫成果自評：
In retrospect, the main projects proposed were completed in the due period. We have succeeded in generalizing the transfer matrix method in the Fortuin-Kasteleyn representation to the strips of the square, triangular and honeycomb lattices with toroidal and Klein bottle boundary conditions. The structure of the transfer matrix was obtained for both the full-temperature q-state Potts model and the zero-temperature Potts antiferromagnet, i.e. the chromatic polynomial. Now the method is very general to be applied to both planar and non-planar two-dimensional lattice strips with all the boundary conditions. We also proposed to investigate the special interval 0 ≤q ≤4 when the phase transition for two-dimension lattices is known to be second-order from a high-temperature paramagnetic phase to a low-temperature phase with ferromagnetic long-range order. The result is out of our expectation, and this work is still in progress. In another direction, we introduced a novel method to derive both the nearest-neighbor and boundary-boundary correlation functions for the rectangular Ising lattice strip with infinite length and finite width L. The thermodynamic limit, infinite L, is interchangeable with the topological spin-spin interaction J’=0 to give the long-range boundary-boundary correlation function. We also showed the asymptotic behavior of the boundary-boundary correlation function for large L when the temperature is above, at, and below the critical temperature.
