Puiseux Series Expansions for the Eigenvalues of Transfer Matrices
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ysical h Ma f P t o h e l m a n a r Schmidt and Karanth, J Phys Math 2014, 5:1 t i u c o s J Journal of Physical Mathematics DOI: 10.4172/2090-0902.1000125 ISSN: 2090-0902 Research Artilce Article OpenOpen Access Access Puiseux Series Expansions for the Eigenvalues of Transfer Matrices and Partition Functions from the Newton Polygon Method for Nanotubes and Ribbons Jeffrey R Schmidt* and Dileep Karanth Department of Physics, University of Wisconsin-Parkside, USA Abstract For certain classes of lattice models of nanosystems the eigenvalues of the row-to-row transfer matrix and the components of the corner transfer matrix truncations are algebraic functions of the fugacity and of Boltzmann weights. Such functions can be expanded in Puiseux series using techniques from algebraic geometry. Each successive term in the expansions in powers a Boltzmann weight is obtained exactly without modifying previous terms. We are able to obtain useful analytical expressions for any thermodynamic function for these systems from the series in circumstances in which no exact solutions can be found. Introduction of interesting cases have not yet been solved exactly (hard squares, the Ising model with external fields, to name two). In such cases series Transfer matrices and partition functions expansions can provide some answers to questions about th physical The partition function Z for vertex and face models of lattice properties. There is a vast body of literature on series expansions for systems in statistical mechanics (such as those with nearest neighbor the row-to row Z and the free energy per particle for interacting lattice interactions) can be computed by application of matrix mathematics models [2], and the subject continues to receive attention [3]. to the problem in at least two ways. For models with nearest neighbor Corner transfer matrices can be used to define a variety of partition interactions only, a row-to-row transfer matrix [1] can be constructed functions, on a triangular lattice three partition functions ZZA , 2 , whose eigenvalues determine Z. For example let the states of a lattice site (Figure 2) and Zw (Supplementary Figure 1 in the appendix), can be i in a row of length m of the lattice be enumerated by a state-variable constructed [1,4,5] from matrices Aa( ), Fab ( , ) which are functions of ti , 1≤≤im. The state-variables of a row in the lattice form a vector the state-variables ab, of central sites (we discuss a few details in the t =(tt12 , , , tm ), and the Hamiltonian for the system decomposes as appendix); 6 Hh= (tt , ')+ k ( t ) (1) ZA =∑ Tr A () a ∑∑ a (tt , ') t 22 in which we sum over all pairs (tt , ') of adjacent rows. We will use t and Z2 =∑ Tr A ()(,) a F a b A ()(,) b F b a ab, s for state-variables appearing in row-to-row transfer matrices, and a,b (the third partition function Zw =∑ wabcFabAbFbcAcFcaAa (,,)(,)()(,)()(,)() and c for state-variables appearing in corner transfer matrices. abc,, is given in the appendix, w(a,b,c) is the weight of the face of the lattice If the lattice has a periodic boundary condition on its N rows, the whose vertices are a,b,c) and from these one can express the partition partition function is writable in terms of eigenvalues xi of the row-to- ZZ2 κ Aw function per lattice site as = 3 as the lattice size becomes infinite. row transfer matrix T, Z2 By taking variations of Boltzmann parameters that leave the partition Z =T T==, trTNN x T = e−β (h (tt , ') ++ kk ( t ) ( t ')2) ∑∑ ∑ tt,' t ','′ t ′′ ∑ i tt,' (2) κ {}tt {'} {'} t′′ i function per site unchanged, two corner transfer matrix equations (on the triangular lattice) can be obtained [4,5]. in which the sums are over all of the possible states of each row. If Z N is very large, the free energy per site will be determined by a single ∑wabcFbcAcFca(,,)(,)()(,)=w AbFbaAa ()(,)()≡η AbFbaAa ()(,)() (4) c Z2 eigenvalue, the largest, which we will label as x1 , xx1 > i , for i >1 2 2 Z2 44 η N FabAbFba(,) () (,)= Aa ()≡ ξκ Aa (), = ∑ x2 b Z A ξ f=−lim kTZN ln = − kTx ln 11 −lim kTN ln 1 + +− =kT ln x (3) NN→∞ →∞ x1 Computation of N-point functions ( n ≥ 2 ) require additional eigenvalues [1]. The characteristic equation 0 = det(TX−⋅ 1) for *Corresponding author: Dr. Jeffrey R Schmidt, Associate Professor–Physics, the row-to-row transfer matrix T= Tz () will be a multi-variable Mathematics and Physics, University of Wisconsin-Parkside, 900 Wood Road, Kenosha, Wisconsin 53141, USA, Tel: 202-537-3645; E-mail: schmidt@uwp.edu polynomial in X and Boltzmann weights z (or fugacity z if Z is the Received July 27, 2013; Accepted February 28, 2014; Published March 10, 2014 grand canonical partition function), so the solutions Xx= i will be algebraic functions of these weights. It is quite a challenge to compute Citation: Schmidt JR, Karanth D (2014) Puiseux Series Expansions for the these eigenvalues exactly in dimensions higher than one. Eigenvalues of Transfer Matrices and Partition Functions from the Newton Polygon Method for Nanotubes and Ribbons. J Phys Math 5: 125 doi: 10.41722090- The most extensively studied two-dimensional lattice models are 0902.1000125 the mN× lattice vertex and face models in the thermodynamic limit Copyright: © 2014 Schmidt JR, et al. This is an open-access article distributed of both mN, →∞, the row-to-row transfer matrix eigenvalues are under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the obtained by the Bethe Ansatz or methods descended from it. A number original author and source are credited. J Phys Math ISSN: 2090-0902 JPM, an open access journal Volume 5 • Issue 1 • 1000125 Citation: Schmidt JR, Karanth D (2014) Puiseux Series Expansions for the Eigenvalues of Transfer Matrices and Partition Functions from the Newton Polygon Method for Nanotubes and Ribbons. J Phys Math 5: 125 doi: 10.41722090-0902.1000125 Page 2 of 13 t1 t’1 t6 t’6 t2 t’2 t5 t’5 t3 t’3 t4 t’4 Figure 1: Adjacent rows of sites in a 6× N Ising model tube. } j { j } { k } } j { j } { k } b a a 6 Z A = T r A (a) Σ Z = T r A 2 (a) F (a, b) A 2 (b) F (b, a) a 2 Σ a,b Figure 2: Two partition functions using corner transfer matrices for a triangular lattice. The square lattice case is a bit more complex, with eight partition such as whiskers, filaments, and tubes, in which one dimension is small functions. Computation of the partition function per site κ from and one is large. the corner transfer matrix equations is even more difficult than the row-to-row matrices because of their nonlinearity. Approximation Many symbolic-numerical methods for finding eigenvalues of methods truncate the infinite-dimensional matrices A(a) and F(a,b) to matrices or roots of polynomials can fail on degenerate roots [7,8] but some finite size in representations with A(a) diagonal, and even the this does not appear to happen for the problems that we investigated most extreme approximation (truncation to one by one matrices, the by the techniques described in this article. We found the largest root Kramers-Wannier approximation) gives surprisingly good results for is always non-degenerate (a consequence of the Frobenius-Perron κ , for a non-interacting lattice gas this approximation is in fact exact. theorem), and we are able to find expansions of degenerate “less significant’’ roots without any difficulties, in fact there were tell-tale We have found that for a number of important models on mN× signs at stages in the calculations that revealed what the degeneracies lattices with m finite and N →∞, the eigenvalues of the row-to-row transfer matrix T, and the corner transfer-matrix “eigenvalues’’ η and were. The way in which these series expansions are constructed is ξ can be expanded in Puiseux series in the Boltzmann weights, by a fundamentally different from traditional developments of power series powerful method from the theory of algebraic geometry, the Newton expansions of functions such as Z and f in that it is not a perturbation polygon. This is a type of series expansion that does not appear to have or cluster expansion or a counting of graphs. The expansions gotten by been considered before in statistical physics (but has been used recently successive iterations of the Newton polygon method are not progressive in cosmological calculations [6]). These ideas allow us to obtain refinements; rather they are extensions, each of which gives one new analytical formulas for thermodynamical quantities for nanosystems term in the series exactly, and does not alter preceding terms. J Phys Math ISSN: 2090-0902 JPM, an open access journal Volume 5 • Issue 1 • 1000125 Citation: Schmidt JR, Karanth D (2014) Puiseux Series Expansions for the Eigenvalues of Transfer Matrices and Partition Functions from the Newton Polygon Method for Nanotubes and Ribbons. J Phys Math 5: 125 doi: 10.41722090-0902.1000125 Page 3 of 13 γγ22 pp γ The Geometry of Algebraic Functions P=( a01 () z+ a () zcz + a 2 () zcz ++ ap () zc z ) + gX (1 ()) z (11) We will first introduce the Puiseux series of an algebraic function, This can be zero if the terms with like powers of z cancel; begin and the procedure for obtaining them using the method of the Newton with the leading term according to our monomial ordering and let polygon.