Puiseux Series Expansions for the Eigenvalues of Transfer Matrices

hysical M P a f th o e l m a

n a r Schmidt and Karanth, J Phys Math 2014, 5:1

t i

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: [email protected] 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.

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. Next we explain what the series represent, and why the f= min (( O a () z cjj zγ ))= min (( O a )+ jγ ) method works the way it does. jj (12) In() a Puiseux series be the lowest power of z appearing on the right side, then if  is the coefficient of the leading term of ax ()

If K is a field then we can define the closed non-Archimedean field Oa( ( z ))+γ = f Kz[[ ]]  (13) of Puiseux series over K (for our purposes K=C or R) with ∑ In( a ) c =0  coefficients fKk ∈ as the set of formal expressions of the form [9-11]. ∞ k which we solve for c. We now iterate the process on what is left over, n using the next term in the series solution to kill the next lowest powers fz( )= ∑ fzk (5) PXz( ,) kk= 0 of z. The lowest powers of z will come from those terms of where n and k are a nonzero natural number and an integer corresponding to the very lowest points of the Newton diagram of 0 PXz( ,) respectively (which are part of the datum of f): in other words, Puiseux . series differ from formal Laurent series in that they allow for fractional The Newton diagram of P (X, z) exponents of the indeterminate as long as these fractional exponents The algorithm for carrying out this construction of the series is have bounded denominator (here n), and like Laurent series, Puiseux ξ γ i,1 series allow for negative exponents of the indeterminate as long as these called Newton’s method. The first term cz= xi,1 z of the series negative exponents are bounded (here by k0 ). can be found by purely geometric considerations, the next term will be obtained by applying Newton’s method to the remainder after the ξ This type of series will arise in statistical physics when we i,1 leading term elimination; PX(+ xi,1 z , z )=0, and higher terms computed seek expansions of the roots (eigenvalues) Xx= , ip= 1,2, , of i by further iterations [11,12]. characteristic polynomials of the row-to-row or corner transfer matrix

2 p For small z , candidates for the dominant term are found det(()Tz−⋅ X 1)= PXz ( ,)= a01 () z+ a () zX+ a 2 () zX + ap () zX =0 (6) geometrically. For every term cXir z in PXz( ,) draw a dot (,ir ) in the ∈ r ∈ PXz( ,) which defines Xx= i as an algebraic function of z (the fugacity or a i  ,  plane. These constitute the Newton diagram of , which Boltzmann weight) will be columns of dots at integer intervals horizontally. TheNewton ξ ξξ+ ξξ++ ξ i,1 ii,1 ,2 ii,1 ,2 i ,3 polygon is the convex hull of the dots at the bottom of the columns. xii= xz,1 ++ xz i,2 xzi,3 + (7) x The Newton polygon will give us the numbers 1 and ξ1 in the Puiseux We introduce some terminology used in algebraic geometry: if expansion of the roots above. For example [11] the polynomial ∈ xi Kz[[ ]] with xi,1 ≠ 0 and with the exponents ξξ= ordered 22345 i∑ j ij, P( X , z )= z− 2 z X −+ X zX + zX (14)

ξξ12<<, we call the order of x the function Ox( )=ξ (exponent i ii,1 has Newton diagram consisting of the points {(0,1),(2, 2),(3,0),(4,1),(5,1)} of the lowest power of the indeterminate), and the initial coefficient (Figure 3). To create the convex lower hull, find the set of points at the of xi is In( xii )= x ,1 . These would correspond to the exponents and bottom of each column, which in this case is all of the points. Order coefficient of the leading term with respect to a local ordering in the them by increasing first component to get points {MM01 , ,} , and draw case of polynomials or power series. a vertical line downwards from M 0 (in this case (0,1) ) to serve as an The polynomial (in X ) Eq. 6 defines an algebraic curve in the (Xz ,) plane. The problem of parameterizing such a curve can be summarized 5 in a few steps [10]. First a theorem; in some suitable coordinate system any parameterization of the ith branch is equivalent to one of the form

n nn12+++n3  z= zt ()= t , X = xii = x,1 t x i ,2 t x i ,3 t, 0< n , 0< n1 < n2 < (8) 4

(introducing parameter t ). Fractional power series occur when we eliminate t and put X∈⊂ Kz[1n ] K [[ z ]] . The second step is the development of the expansion with respect to some monomial ordering (which is a 3 local ordering in Kz[]). Let P be a polynomial in X (integer powers of X) z 1 * ∞  with coefficients in K[][] z = Kzn . Write it as n=1  2 p p j  PX( , z ) =∑ ajp ( zX ) = a ( z )∏ ( X− x j ( z )), Px ( i , z ) = 0, i = 1,2, , p j=0 j=1 (9) 1 Then solutions are of the form Eq. 7 (suppressing the subscripti labeling the branches) γγ+ γγ ++ γ γ ′1 ′′ 12 X= cz++ c z c z +,γγ =OX ( ),i > 0 (10) 0 0 1 2 3 4 5 and the series is built term by term. The idea of the proof of this X γ γ statement is as follows; write X= zc (+ Xz1 ( )) with OX(11 )= >0 substitute and expand Eq. 9 Figure 3: Newton diagram of Eq. 14

Page 4 of 13

ξξ indicator as to how the next line is drawn. You must always turn left. ii,1 ,1 X→+ X xzii,1 , P′ = PX (+ xz,1 ,) z (20) Draw a connecting line from M 0 to M i if the vertical line from M 0 and applies the polygon procedure to P′, one obtains the next term downwards hits M i as it is slowly rotated counterclockwise. Then repeat the process but now the indicator line that you rotate lies along in the Puiseux series for this solution, of the form seen in Eq. 16. This

MM0 i . There are systematic algorithms, we state the Jarvis Gift Wrap process is now repeated until the series attains the desired precision, in th − other words for the i solution algorithm [13]: create the vectors (0,0) M 0 , and MMi − 0 , in this case (0,− 1) {(2,1),(3,− 1),(4,0),(5,0)} ξξ++ξξ these are and . Now compute the angle X→+ X xzii,1 ,2 , P′′ = PX ′ ( + xzii,1 ,2 ,) z (21) between the line connecting the previous two points of the polygon ii,2 ,2 ξξ++ ξ with the last point added to the polygon, and each of the remaining ii,1 ,2 i ,3 results in the next correction xzi,3 . lowest points in each column of points, and select as the next point the one that gives the smallest positive value of this angle. This will ensure The calculations are straightforward, but there can be vast numbers that every point of the Newton diagram lies on or above the Newton of terms in these polynomials. We use the REDUCE computer algebra polygon. The algorithm may not be the most efficient, but it is simple system [14] to perform the calculations. Singular [15] also has a Puiseux to program. expansion (Hamburger-Noether expansion) library “hnoether.lib”, but we prefer our own in REDUCE because we have designed it to make Once the Newton polygon is found, the algorithm for construction it easier to follow a specific branch. Our REDUCE program NPplus.  ′ of the roots of Eq. 6 is as follows [11]; each segment EM=[ , , M ] red returns a list of Newton polygon lines {slope , intercept , Q ( x , z )}. We of the Newton polygon of P corresponds to a new homogeneous sub- illustrate its use by expanding the roots of the multivariate polynomial polynomial (a slight departure from the notation of [11], we want to 32 display the z-dependence) Q of P of the form P=4 X−+ Xz z (22)

Oa()h h mmhh Q() P,, E X , z =∑ In() ah z X, M hh∈ E , M = h , , O() a h = (in each iteration we follow the top of first branch =QXz ( ,) mm h  selection) (15) Q( X , z )= X3 − 4 Xz with h running over the exponents of X for points of the Newton First iteration P( X , z )= X32− 4 X z +→ z  ξ 2 γ 1 Q( X , z )= z− 4 Xz diagram on the corresponding segment. The substitution X== cz x1 z 1 ξ 1,1 2 for a unique γ lets one factor out z from Q , transforming Eq. 15 into xz1,1 =2 z Q( X , z )= z2 + 8 Xz Eq. 13, to be solved for c . [10,11], we have only tried to summarize the Second iteration P( X ,)=( z P X+→ 2 z ,) z  32++ basic ideas that distinguish the series from the more familiar series such Q( X , z )= X 8 Xz 6 X z ξξ+ 1 xz1,1 1,2 = − z as Taylor-Maclaurin. The statement in theorem form Basu et al. [11] is; 1,2 8  3 5 let x be a nonzero root of one of these characteristic polynomials Q Q() X,= z z2 + 8 Xz Third iteration P()() X, z = P X−→ z / 8, z  32 −ξ of a segment E with slope 1 , of multiplicity r, then a Puiseux series  32++ ξ Q() X,= z X 8 Xz 6 X z 1 3 xz ξξ++ ξ for a root of P starting with 1 is 1,1 1,2 1,3 3 2 xz1,3 = − z ξ ξξ+ 256 x= xz1++ xz 12 ,P ( x , z )=0 (16)  1 12 3 Q( X , z )=8 Xz+ z3 3 2  We find that this first term alone is a very good approximation to Fourth iteration P( X ,)=( z P X−→ z ,) z  64 256  32 the roots. The two Newton polynomials for the polynomial PXz( ,) of Q( X , z )= X++ 8 Xz 6 X z Eq. 14, [11] from which we deduce that a Puiseux expansion of one of the roots of Q= z−− X3 , Q = X32 ( zX 1) (17) this polynomial is 13 2213 12 xz1 =2 −− z z − z + (23) (the two segments in Figure 1) which have roots 8 256 512 12ππii 14 1 1 2 1 33 33 3 ± ⋅⋅ ξ = − 1 and a simple test for numerical values 0≤≤z 0.5 shows that these few X= x =1⋅⋅⋅ ze , ze , z and Xz=1 , and slopes 3 and 2 , so that the roots of the polynomial have Puiseux series beginning with terms give a precision in the value of a root of five decimal places 7 1 21ππii 41 1 − x32−+4 zx z = 0.00320434570313z2 + (z4 ) (24) xz=,,,3 ez 33 ez 33 ± z2 (18) 11 The first term in the Puiseux series expansions for the eigenvalues The series that are being obtained are not expansions about a point, of the transfer matrix can be computed from a small fraction of the total like a Taylor-Maclaurin or Laurent series, but are instead series that number of terms in the secular equation, all we need is the lower convex are the solutions to an algebraic equation correct to a particular order hull. This means that the first terms can be computed by hand even in z. These series may actually truncate or terminate, we will see this for fairly large transfer matrices. Obtaining the secular (characteristic) occur in our study of mN× Ising models. It should be pointed out that equation computationally is more intensive than computing the first these series solutions are formal, and so are very useful as generating terms for the Puiseux series of its roots. A few words are said about this functions for combinatorial data, but convergence under substitution step in the appendix. of explicit z-values is not guaranteed. As we see here and will see again shortly, convergence for sufficiently large/smallz -values is actually Iteration quite spectacular. Suppose that we start with a transfer matrix secular equation Eq. 9 Application to m×N Lattice Gases with Exclusion with z a Boltzmann weight, and obtain the Newton polygon solutions (the first terms of the series expansions) x for X Now we turn our attention to the application of these ideas to ξξ ξ statistical physics. The procedure described in the previous section x={ xz1,1 , x z 2,1 , , x zp,1 } (19) 1,1 2,1 p,1 is perfectly suited to computation of high and low temperature If one performs the substitution expansions for such systems as planar lattice gas ribbons and tubes; the

Page 5 of 13

mN× former being a lattice gas on an lattice with periodic boundary Z1/ N = x = 1++ 5 zz 103 − 55 z 4 + 325 z5 − 2025 z6 + 13120 z7 − 87545 z8 + conditions in the “long dimension’’ N only, and the latter case with 5,×N tube max 1 1124 6791 periodic boundary conditions on both “long’’ and “short’’ dimensions. κ =Z5N = 1+− zz 223 + 8 z − 40 z 4 + z 5 − z 6 + 55 Hard squares at low fugacity z<<1 For the 6× N ribbon and tube, respectively

Consider the case of hard squares, a square lattice with site 1/ N 2 3 4 5 6 7 8 Z6,×N rib= x max = 1++ 6 zz 4 + 6 z − 32 z + 190 z − 1164 z + 7352 z − 47630 z ; (33) occupancy ti = 0,1 such that no two adjacent sites can be simultaneously 1 11 41 2309 4081 1260845 occupied. This model has not yet been solved exactly, but in the κ =Z16N =1+− zz23 + z − z 4 + z 5 − z 6 + 6 6 72 24 1296 thermodynamic limit ( NN× lattice, N →∞) corner transfer matrices 1/ N 2 3 4 5 6 7 8 provide us with some information [5,16]. On a square lattice κ= ηξ is Z6,×N tube= x max = 1++ 6 zz 3 + 8 z − 45 z + 276 z − 1760 z + 11610 z − 78681 z + 1 the partition function per particle 23 4 58171 6 51997 7 171577 8 κ =Z16N = 1+− zz 2 + 8 z − 40 z + 225 z − z + z − z + 66 3 ∑FabA(,)22 ()(,)= bFbaξ A () a (25) b=0,1 (note that six must be a good approximation to infinity for eighth order; the z8 term agrees with the corner transfer matrix result for ∑wacdbFacAcFcdAdFdb(,,,)(,)()(,)()(,)=η AaFabAb ()(,)() NN× , N →∞ to within 11 parts in 10,000 ). We can go on cd, 1/ N 2 3 4 5 6 7 8 9 Z7,×N tube = 1++ 7 zz 7 + 7 z − 35 z + 224 z − 1463 z + 9807 z − 67270 z + 470442 z + (34) For hard squares the face weight function is 1 23 4 5 660689 7 400744 8 (abcd+++ )/4 κ =Z17 N = 1+− zz 2 + 8 z − 40 z + 225 z − 1362 z + z −7 z + wabcd(,,, )= zεεεε (,)(,)(, ab bc cd )(,), da ε (,)=1 ab− ab (26) 77 1/ N ++2 + 3 − 4 +5 −6 +7 −8 +9 + In the Kramers-Wannier approximation one treats the matrices Z8,×N tube = 1 8 zzz 12 8 26 z 176 z 1180 z 8048 z 55886 z 394488 z 1 458023 as numbers (one by one matrices), the equations Eq. 25 simplify to 18N 23 4 5 6 7 8 2 24κ =Z = 1+− zz 2 + 8 z − 40 z + 225 z − 1362 z + 8670 z − z + A(0) = 1, AA(1) = , F(1,1) = 0 , FF(0,1)⋅ (1,0) = ξ A, FA(0,0) =ξ (1− ) , 8 η= κξ , elimination of η and ξ results in REDUCE is built for comfort, not for speed, nevertheless each 1 iteration takes only a second or two even for the 64× 64 transfer matrix κ 4 34+− (27) =2zA() 1 A of the 6× N case. 11 4243 κ A= z (1−+ A ) zA Hard squares at high fugacity z>>1 Eliminate κ and apply the Newton polygon method to expand A The high fugacity expansions for the lattice gases do in general in a Puiseux series in z 1 result in true Puiseux series, with both positive and negative fractional Az=4 (1−+ z 4 z23 − 21 z + 125 z 4 − 800 z 5 + 5368 z 6 − 37240 z 7 + 264828 z 8 − 1919690 z 9 + ) (28) exponents. For the simplest case, which can be worked out by hand, the characteristic polynomial for the 1× N hard-squares tube is and use the first of Eqs. 27 to get the partition function per particle [16] + ′ 1 ′ (tt11 )/2 2 2 23 4 5 6 7 8 9 T1++tt ,1 ′ = (1− ttz1 1 ) , PXz ( , ) = det( TX−⋅ 1) = XXz −− (35) κ =limZN = 1+− zz 2 + 8 z − 40 z + 225 z − 1362 z + 8670 z − 57254 z + 388830 z +(29) 11 N →∞ where agreement of this result with computer calculations that perform To obtain the high-fugacity expansion we transform zu→1 , and graph counting is discussed. Since the hard squares model has not been multiply the entire polynomial by a sufficient number q of u factors solved analytically, series provide the only information available for the (the order of Pxz(,) as a function of z) in order to make all u-exponents thermodynamic limit case. positive; q −−1 12 For a mN× hard-square ribbon lattice gas with m finite and P(,) x z→ u P (,)=(,)= x u uP X u uX−− uX 1 (36) N →∞ we order the (row to row) transfer matrix rows and columns as and apply the Newton construction, the largest eigenvalue will be m −1 (tt+ ′ )/2 seeded by the lowest power of u, and finally perform uz→ to obtain m ∑ ii i=1 (30) T − −= ( (1−−tt′ )(1 tt++ )(1 − tt′′ )) z 12++tt + + 2m1 t ,122 ++ tt′′ + +m 1 t ′ ∏ jj jj11 jj 1/ N 2 3 4 5 6 7 8 12 mm12 j=1 Z1×N = 1+− zz + 2 z − 5 z + 14 z − 42 z + 132 z − 429 z + z<< 1 (37) 13 5 Xx= −− − and solve for the largest eigenvalue max of polynomial 1/ N 1122 1 1 2 Zz× = ++ z − z + z + z>> 1 PXz( , ) = det( Tz ( )−⋅ X 1) using our simple Newton polygon library. Our 1 N 2 8 128 1024 results for the N →∞ partition function per lattice site for the 4× N We get a better picture of what transformation Eq. 36 does for the ribbon and tube, respectively 3× N hard squares tube, compare the Newton diagrams (Figure 4) in

1/ N 2 3 4 5 6 7 8 each regime Z4,×N rib= x max = 1+−+ 4 zz 8 z − 39 z + 206 z − 1152 z + 6710 z − 40277 z + (31) 1 7 25 899 4557 99441 For z << 1 the characteristic polynomial is κ =Z4N =1+− zz23 + z − z 4 + z 5 − z 6 + 4 4 32 32 128 4 4 3 22 1/ N 2 3 4 5 6 7 8 PXz( , )= X ( X−− X 3 X ( z + z ) Z4,×N tube= x max = 1+− 4 zz 2 + 12 z − 62 z + 352 z − 2122 z + 13344 z − 86566 z + 3 23 1 159 885 2639 −Xz(2 +− 3 z ) z ) (38) κ =Z4N =1+− zz 223 + 8 z − z 4 + z 5 + z 6 + 44 2 and for uz=1 <<1 It is reassuring to see that this is a good match for the NN× case in thermodynamic limit, at least for the lowest order terms. For the 5× N uPXu3 ( , )= X4 ( Xu 43−− Xu 333 Xu ( 2 + u ) ribbon and tube respectively we obtain −Xu(3 +− 2) 1) (39) Z1/ N = x = 1+++ 5 zz2 7 z 3 − 37 z 4 + 208 z5 − 1225 z6 − 7485 z7 ; (32) 5,×N rib max The lowest level construction results in x =1 and a doubly- −−−111 1 9 33 762 111814 degenerate −z for the former case and x=−− 12, uuu , − ,2 for the κ =Z5N = 1+− zz23 + z − z 4 +159 z 5 − z 6 + 5 5 25 125 latter. Since the Puiseux series solutions Eq. 7 have ascending positive

Page 6 of 13

4 4

3 3 u z 2 2

1 1

0 0 0 1 2 3 4 0 1 2 3 4 X X

q −1 Figure 4: Pxz(,)→ uPxu (, ) is a symmetry transforming upper convex hull of one diagram into lower convex hull.

1/ Nm exponents, when z << 1 the solution starting with x =1 will be largest, lattice ZmN× = κ and when uz−1 = >> 1 or u << 1 the series starting with 2u−1 will be − − −−− − − − − largest; z12κ ( z 1 ) = 1++−+ zzz 1 3 45 z 5 − 10 z 6 + 39 z 7 − 95 z 8 (41)

1/ N 2 3 3 32 573 754 843 5 63 2 −−1 2 − 4 Z3,×N tube = +−uu + − u + u − u + (40) κ =zzz+ 3 + 3 ++ 4 3 z + 12 z + 69 z + u 2 8 8 128 128 1024 3 3 3 57 75 843 κ 84=zzzz+ 4 3 + 6 2 + 8 ++ 9 20 z−−1 + 18 z 2 + 92 z − 3 + 36 z − 4 + = 2z+− zz−−12 + − z − 3 + z − 4 − z − 5 + z>> 1 2 8 8 128 128 1024 1/ N 2 3 4 5 6 Hard hexagons Z3,×N tube = 1+− 3 zz 3 + 12 z − 57 z + 300 z + 1686 z + z<< 1 The hard hexagon model can be realized as a lattice gas on a square Z1/ N = zz2 + 2 +− 4 2 z−−1 − 8 z 2 + 22 z − 3 − 190 z −5 + 120 z −6 + 1442 z −7 − 3268 z −8 , z >> 1 4,×N tube lattice with an additional interaction along one diagonal [1,16] 1/ N +−2 + 3 − 4 +5 −6 +7 −8 + m Z4,×N tube = 1 4 zz 2 12 z 62 z 352 z 2122 z 13344 z 86566 z z<< 1 (tt+ ′ )2 m ∑ jj (42) T = ( (1−−−−tt )(1 tt′′ )(1 tt ′ )(1 tt ′ )) zj=1 1/ N 3 4 5 6 7 8 ++ + +m−1 ++′′ + +m − 1 ′ ∏ jj++11 jj jj jj+1 12tt12 2 tmm ,122 tt12 t Z5,×N tube = 1++ 5 zz 10 − 55 z + 325 z − 20225 z + 13120 z − 87545 z + , z << 1 j=1 9 11 15 339 1765 16831 72975 Z1/ N = 2 zz2 + +− z−−1 + z2 − z −3 + z −4 − z −5 + z>> 1 5,×N tube 2 8 16 128 256 1024 2048 This matrix is considerably sparser than the hard squares transfer

1/ N 2 3 4 5 6 7 8 matrix, with many more null vectors. We find that for low fugacity Z6,×N tube = 1++ 6 zz 3 + 8 z − 45 z + 276 z − 1760 z + 11610 z − 78681 z + , z << 1 z << 1, mN× periodic boundary conditions (tubes) result in 1/ N 3 2 −−1 2 −3 −4 −5 −6 Z6,×N tube = zzz+ 3 ++− 3 14 9 z + 45 z − 371 z + 1134 z − 4959 z + 27798 z 1/ N 2 3 4 5 6 7 8 9 Z4,×N tube = 1+− 4 zz 6 + 32 z − 212 z + 1568 z − 12400 z + 102592 z − 876944 z + 7683456 z + (43) −−78 −++121002z 570303 zz >> 1 1 423 3125 24707 13987571 122615141 κ =Z4 = 1+− zz 323 + 16 z − z 4 + z 5 − z 6 +51115 z 7 − z8 + z9 + 4 4 4 32 32 1/ N ++2 + 3 − 4 +5 −6 +7 −8 +9 + Z8,×N tube = 1 8 zzz 12 8 26 z 176 z 1180 z 8048 z 55886 z 39448 z z << 1 1/ N 2 3 4 5 6 7 8 9 Z6,×N tube = 1+− 6 zz 3 + 26 z − 186 z + 1440 z − 11836 z + 101586 z − 900342 z + 8178268 z + 1/ N 4 3 2 −−1 2 −3 −4 Z= zzzz+ 4 + 6 ++− 8 44 12 z + 166 z − 612 z − 1444 z + z >> 1 1 8,×N tube 37907 319129 25008499 κ =Z6 = 1+− zz 323 + 16 z − 106 z 4 + 789 z 5 − z 6 + z 7 −464388 z 9 + z9 + We take a moment to address convergence. These formulas are in 66 6 1/ N 2 3 4 5 6 7 8 9 excellent agreement with high-precision numerical computation of the Z8,×N tube = 1++ 8 zz 4 + 16 z − 134 z + 1104 z − 9384 z + 82312 z − 741276 z + 6821040 z + 1 7317383 33392609 eigenvalues (obtained using Gnu octave). κ =Z8 = 1+− zz 323 + 16 z − 106 z 4 + 789 z 5 − 6318 z 6 + 53198 z 7 − z 8 + z9 + 88 HS 5× N tube HS 6× N tube These agree exactly up to and including the zm−1 term with the

z xmax (octave) xmax (Eq. 40) xmax (octave) xmax (Eq. 40) known low-fugacity series expansions for the NN× model with N →∞, 0.01 1.05000948 1.0500094 6 1.06030757595 1.06030757595 we begin to see finite size deviations beyond that point. We compare 0.025 1.12513751 1.12513 307 1.15198475 1.15198475 with the corner transfer matrix computation in the Kramers-Wannier 0.05 1.25098387 1.250 69862 1.30828428 1.30828 350 approximation [16]: 5 7.37564404e+01 7.375 75383 e+01 2.27193463e+02 2.27 949742 e+02 ∑FabA(,)24 ()(,)= bFbaξ A () a (44) 10 2.46302186e+02 2.463021 95 e+02 1.34326223e+03 1.34326 421 e+03 b 20 8.91333977e+02 8.91333977+02 9.27362202e+03 9.27362202e+03 ∑wabcFacAcFcb(,,) (,) () (,)=η AaFabAb () (,) () 40 3.38135312e+03 3.38135312e+03 6.89337977288e+04 6.89337977288e+04 c abc++ η 2 wabc(,,)= z6 εεε (,)(,)(,), ab bc ca κ = The last of Eqs. 40 are approaching the infinite lattice results ξ for z >> 1 reported in Appendix B of [4] computed by numerical in which ε (a , b )=1− ab , ab,∈ {0,1} . Truncating to 11× matrices we get computations of corner transfer matrix truncations, since for the mn× ξ 4 2 + 22 A(0) = AF (1) (0,1) F (1,0) AF (0) (0,0) (45)

Page 7 of 13

ξ A4(1) = AF 2 (0) (0,1) F (1,0)+ AF22 (1) (1,1) the 6× N tube and truncated corner transfer matricies for the infinite 1 lattice η AF(0) (0,0) A (0) = zAF6 (1) (0,1) F (1,0)+ AF (0)2 (0,0) 1 κ +−34 + − 5 − 6 + 7 + (51) η A(0) F (0,1) A (1) = zA6 (0) F (0,1) F (0,0) 6×N = (1zz 2 7 z 12 z 15 z 194 z ) 1 κ = (1+−zz 234 + 7 z − 12 z 5 − 15 z 6 + 193 z 7 + ) η A(1) F (1,0) A (0) = zA6 (0) F (1,0) F (0,0) TCTM

η AF(1) (1,1) A (1) = 0 Application to m×N Lattice Ising Models from which we deduce Ising mN× ribbons and tubes have transfer matrices with all matrix elements non-zero, and one finds that eigenvalues occur in F(1,1) = 0, FF (0,1) = (1,0) (46) approximate ± -pairs, a spectral feature revealed by the first level Set A(0)=1, A(1)=A, F(0,1)=f, F(0,0)=g, the corner transfer matrix Newton polygon segments, with very large gaps between largest and equations define a one-dimensional affine variety in (,Afg ,,,,)ξκ z second largest eigenvalues of the same sign. space [17] Ising ribbons on mN× lattices and tubes on 2mN× lattices have ξ = Af22+ g 2 (47) transfer matrices with the property −−β ξ Af42= PTz⋅⋅( ) P′ = Tz (1 ), z = e J (52) 1 η A= zg6 in which PP, ′ are permutation (orthogonal) matrices with additional 1 properties η g= z6 Af22+ g [PP ,′′ ]=0, PP⋅ = S , [ Tz ( ), S ]=0 (53) Eliminating f, g and ξ we obtain the elimination ideal generated by 1 1 1 6 3 Consider a 3× N ribbon (Figure 5) of magnetic dipoles si each 6 (1− Az ) 6 66−− − κ (48) zA(1 A )( z A ) = 0, = 2 of which can be in two states si =1± , with magnetic dipole-dipole A interactions between neighboring spins only. Expand the first of these in a Puiseux series by the Newton polygon κ The transfer matrix for this ribbon is 88× since each row of three method, and use the second to obtain : 3 1 spins has 2 states, and the states of two adjacent rows label the rows Az=6 (1−+ z 5 z23 − 34 z + 265 z 4 − 2232 z 5 + 19766 z 6 − 181300 z 7 + 1706737 z 8 + ) (49) and columns of the transfer matrix;

−β J(( ss + s s + ss′′ + s ′ s ′)/2 ++ ss ′ s s ′ + ss ′) 1 23 4 5 6 7 8 12 23 12 23 11 22 33 β ≥ κ +− + − + − + − + Tz()(,ss ,),(,,) s ss′′ s ′ =e , = ,T 0 (54) = 1zz 3 16 z 106 z 789 z 6319 z 53228 z 465258 z 123 123 kT ((ss+ s s + ss′′ + s ′ s ′)/2 ++ ss ′ s s ′ + ss ′) a result in excellent agreement [16] with series computations produced =z12 23 12 23 11 22 33 ,=ze−β J in the 60’s and 70’s. The Newton polygon method is very well-suited to the study of corner transfer matrices if one can establish some of the If J <0 ( 0> 1: (,ss123 ,),(,,) s ss 123 s (,ss123 ,),(,,) s ss 123 s

1/ N −−1 2 −3 −4 −5 −6 −7 Zz4×N =+− 4 20 z + 236 z − 3468 z + 57044 z − 1005124 z + 18552156 z − 354084764 z+ The inversion property Eq. 52 is due to the exponent being negated (50) by flipping alternating spins in adjacent rows according to the pattern

1/N 2 −−1 2 −3 −4 −5 −6 −7 (for example again the 3× N case) Z6×N = zz+ 2 ++ 15 62 z − 92 z − 3392 z − 17462 z + 95678 z + 1919540 z + 7151412 z + 53 1 1 5 −1 65 −−593 Z1/ N = zz22+2 + 4 z + 7 z + 21 z 2 −− 5 z 2 − 325 z−−1 − z32 +919 z −2 + 6221 z2 + Tz()(,ss ,),(,,) s ss′′ s ′ =Tz ( )(,s− s,),(, s −− ss′′ , s′ ) (56) 8×N 22 123 123 1 23 12 3 We also examined a variation (perhaps unstudied) on The matrices P,P’, and S the hard hexagon model with Boltzmann weights for a face w( a , b , c ) = z(abc++ )/6 (1− abc ) , and found excellent agreement between Switch to a “bit-ordered’’ presentation of the states of each row of

s3 s’3

s2 s’2

s1 s’1

Figure 5: A section of the 3×N Ising ribbon.

Page 8 of 13 the lattice, let Let v be a vector of length one, all of whose components are positive. There are numbers {ai ,| = 1,2, ,2N } and {bi ,| = 1,2, ,2N } (Tz() 0if s =1− i i i and Tz()−1 are Hermitian) such that ti= , stii=2− 1 (57) 1if si =1 ′ (70) Pa⋅ v=∑∑ii uv ,= bi w i and label the rows and columns of the transfer matrix accordingly in ii binary, for the 3× N example where {xzii ( ),u } are the ordered eigenvalue/eigenvector pairs of Tz() −1 Tz()′′ ′ =Tz ()++ + ++′′′ + =Tz () , ij , =1,2,,8 and {yzii ( ),w } are the ordered eigenvalue/eigenvector pairs of Tz() (,ss123 ,),(,,) s ss 123 s 1t1 2 t 2 4,12 t 3 t 1 t 2 4 t 3 ij, (58) Define xxii≥≥++11, yy ii (71)

1 if ti =0 By the Frobenius-Perron theorem the largest eigenvalues x and t =  (59) 1 i y are non-degenerate, positive and correspond to eigenvectors with 0 if ti =1 1 non-negative components. then symmetry Eq. 52 can be expressed as | (Tz (−1 ))NN⋅v |=| P ⋅ ( Tz ( )) ⋅⋅ P′ vwu |, | by NN |=| P⋅ ax | −1 ∑∑ii i ii i (72) ii Tz()1++++++tt 24,124 t tt′′ t ′=Tz ( )1++++++ttt 24,1'24' t t′ t (60) 123123 123 12 3 As N becomes very large, the two sides become yy xx Let P and P’ be stochastic (permutation) matrices that are zeroes yN| bwu |()=+2NNy b + ()=| 2 NNx aP⋅+ |()= 22NNx a + () N 1 11 yy11 1 1 1 xx11 (73) everywhere except 1 1 11 and therefore for any z , by dividing these equations for consecutive PP++++++ =1, ′+++ +++′ =1 (61) 1tt123123 24,124 t ttt 1tt12 24,1'24' t3 t 12 t t 3 odd integers N and N + 2 as N →∞

22 Note that [P, P’]=0, and P. P’=S is the matrix with all ones down x1=, y 1 xy 11 = (74) the upper right to lower left diagonal. The significance is that the −1 the phases being set to unity by the Frobenius-Perron theorem. transformation zz→ is a linear transformation of the transfer − matrix Even more can be said about the spectra of Tz() and Tz()1 (ferromagnetic/anti-ferromagnetic Ising models). It is easy to show −1 ⋅⋅′ Tz( )= PTz ( ) P (62) that and from the form of P and P’ one can see that yii=± xi , >1 (75) [PP ,′′ ] = 0, ( PP⋅ )22 = 1 = S (63) Since [Tz (−1 ), S ] = 0 there is a unitary matrix U such that and the spin-negation symmetry Eq. 55 is simply UTz⋅(−−11 ) ⋅ U = ∆ ( z ), U ⋅⋅ SU−1= Σ (76) STz⋅⋅() S = Tz () (64) in which ∆(z )= diag ( y , y , ) and Σ is diagonal with half of its Proof of the inversion property of largest eigenvalue 12 diagonal entries being 1, the other half being −1. Then sinceP is Let Tz() be a symmetric matrix ( 22nn× ) all of whose matrix orthogonal, PTz⋅⋅() P−1 and Tz() are similar q elements are of the form z where q is a finite rational number, andz ′ −1 (77) is a non-negative real variable. PTz⋅⋅() P = Tz ( ) −−11′ The matrix Tz() has the property that there exists two stochastic PTz⋅⋅⋅⋅⋅⋅⋅()( P PP ) = PTzP () S matrices P, P’ (each row/column are unit vectors with only a single PTz⋅⋅() P−−11 = Tz ( ) ⋅ S component non-zero, they are permutation matrices) −− − − U⋅( PTz ⋅ () ⋅ P11 ) ⋅ U = UTz ⋅ ( 1 ) ⋅⋅ SU 1 −1 ′ −−11 −− 11 − 1 PTz⋅⋅() P = Tz ( ) (65) U⋅( PTz ⋅ ( ) ⋅ P ) ⋅ U =( UTz ⋅ ( ) ⋅ U ) ⋅ ( USU ⋅⋅ ) 2 [PP ,′′ ] = 0, PP⋅ = S , S = 1, [ Tz ( ), S ] = 0 (66) −1 (UP⋅ ) ⋅ Tz ()( ⋅ UP ⋅ ) = ∆ () z ⋅Σ −1 however Tz() and Tz() are not similar. Then it is easy to show that The right side of this expression is a diagonal matrix, therefore the −1 NN left side must be as well, call it Λ(z )= diag ( x , x , ) (Tz ( )) = P⋅⋅ ( Tz ( )) P′ (67) ii12 Λ()=zz ∆ () ⋅Σ (78) for any odd integer N . An interesting bit of algebraic geometry is that models whose (Tz (−1 ))=(N PTz⋅⋅ () P′′ )( PTz ⋅⋅ () P ) ( PTz ⋅⋅ () P ′′ )( PTz ⋅⋅ () P ) (68) transfer matrix Tz() has property Eq. 52 have characteristic polynomials Pxz(,)=det( Tz ()− x ) which have Newton diagrams with convex hulls ′′′′′ =PTz⋅⋅⋅⋅⋅⋅ ()( P P ) Tz ()( P P ) ( P ⋅⋅⋅⋅⋅⋅ P ) Tz ()( P P ) Tz () P possessing a reflection symmetry about a horizontal line Figure 6. ⋅⋅⋅⋅ ⋅⋅⋅⋅′ =PTz ()() S Tz ()() S () S Tz ()() S Tz () P Results for the Ising ribbons and tubes =P⋅⋅ ( S )NN−1 ( Tz ( )) ⋅ P′ The ribbons possess the inversion property for all m , but the tubes =P⋅⋅ ( Tz ( ))N P′ only for m even (and recall that ze= −β J ).

1/ N −5 ++3 + 5 + 7 + 9 − 11 −13 −15 −17 + For any vector v we have Z3,×N rib = z 3 zz 7 12 z 18 z 13 z 41 z 222 z 648 z 1295 z z << 1 (79)

Z1/ N = zz5+++ 3−− 1 7 z 3 12 z − 5 + 18 z − 7 + 13 z − 9 − 41 z − 11 − 222 z −13 − 648 z −15 − 1295 z −17  z >> 1 |PP⋅⋅v |=|′ vv |=| | (69) 3,×N rib

Page 9 of 13

Ising 2xN Hard squares 2xN 20 20

15 15 z z 10 10

5 5

0 0 0 1 2 3 4 0 2 4 6 8 x x

Figure 6: A comparison of models with Newton diagram hulls with and without the reflection symmetry.

0.3 4.58099200670 4.5809920 1152 1.52467452875790 1.5246745287 7609 1/ N −−4 2 2 4 10 12 14 16 18 20 Z3,×N tube = z+ 2 z +++−−+ 2 zzz 3 3 3 z 12 z − 18 z + 45 z − 69 z z << 1 e+03 e+03 e+04 e+04 4 1.63934372289 1.639343722 91 6.55410791084417 6.554107910844 64 Z1/ N = z6 ++++++ 1 3 zzzzz−−−−−2 6 4 6 6 9 8 9 10 − 6 z − 12 + 36 z − 14 − 99 z − 16 − 213 z −18 + z >> 1 3,×N tube e+04 e+03 e+04 e+04 10 1.00000205102855 1.00000205103 1.00000005002001 1.00000005002001 × × Ising 3 N ribbon Ising 3 N tube e+07 e+07 e+08 e+08 z x x x x 20 1.28000004025126 1.28000004025 2.56000000050001 2.56000000050001 max (octave) max (Eq. 79) max (octave) max (Eq. 79) e+09 e+09 e+10 e+10 0.05 3.20000015087876 3.20000015087876 1.60802002518750 1.60802002518750 1/ N −−−9 3 1 3 5 7 9 (81) e+06 e+06 e+05 e+05 Z5,×N rib = z+ 2 z + 5 z ++ 5 zz 20 + 52 z + 129 z + 245 z + z << 1 0.1 1.0000030712181 1.0000030712181 1.02020102999997 1.02020102999997 −−− − − 3 e+05 2 e+05 e+04 e+04 1/ N 9 3 1 1 3 5 7 9 Z5,×N rib = z+++ 2 zzz 5 5 + 20 z + 52 z + 129 z + 245 z + z >> 1 0.2 3.125660076011 3.125660076011 6.77044799682371 6.77044799682371 46 e+03 53 e+03 e+02 e+02 Z1/ N = zzzz−−−−8++++++ 2 6 2 4 2 2 7 11 z2 20 zz 4 +−+ 10 6 25 z 8  z << 1 0.25 1.0248722282 0783 1.0248722282 1178 2.900742157512 2.900742157512 5,×N tube e+03 e+03 70 e+02 68 e+02 1/ N 10+ 2 ++−−−2 + 4 + 6 + −8 + −10 + −12 + 0.3 4.12644867 125248 4.12644867 229159 1.47793293548 279 1.47793293548 156 Z5,×N tube = zz 5 1 10 z 20 z 35 z 105 z 110 z 280 z z >> 1 e+02 e+02 e+02 e+02 5× N 5× N 4 1.0248722282 0783 1.0248722282 1178 4.09721254 773772 4.09721254 800625 Ising ribbon Ising tube e+03 e+03 e+03 e+03 z x (octave) x (Eq. 81) x (octave) x (Eq. 81) 10 1.00000307121813 1.00000307121813 1.00000103060609 1.00000103060609 max max max max e+05 e+05 e+06 e+06 0.05 5.12000016100253 5.12000016100253 2.5728320807 2.5728320807 e+11 e+11 e+10 e+10 20 3.20000015087876 3.20000015087876 6.40000010075376 6.40000010075376 e+06 e+06 e+07 e+07 0.1 1.00000205052053 1.00000205052053 1.0202020711 1.0202020711 e+09 e+09 e+08 e+08 1/ N −−7+ 1 ++3 + 5 + 7 + 9 + 11 − 13 −15 + Z4,×N rib = z 2 z 5 zz 10 28 z 54 z 84 z 86 z 56 z 698 z z << 1 (80) 0.2 1.9534011784 2540 1.9534011784 1664 4.2318247279 4.23182472 58 e+06 e+06 e+05 e+05 1/ N 7 −−1 3 − 5 − 7 − 9 − 11 − 13 −15 0.25 2.62293622 193879 2.62293622 089386 7.4279768402 7.427976 7685 Z4,×N rib = z++ 2 zz 5 + 10 z + 28 z + 54 z + 84 z + 86 z − 56 z − 698 z + z >> 1 e+05 e+05 e+04 e+04 Z1/ N = z−8 ++ 5 20 zz4 + 64 8 + 116 z 12 − 140 z 16 + 2556 z20 + z << 1 0.3 5.089820435 87032 5.089820356 06666 1.8262359688 1.826235 67048 4,×N tube e+04 e+04 e+04 e+04 4 2.62293622 193880 2.62293622 089386 1.0486577134 1.048657713 3 1/ N 8 −−4 8 −12 −16 −20 Z4,×N tube = z++ 5 20 zz + 64 + 116 z − 140 z + 2556 z + z >> 1 e+05 e+05 e+06 e+06 10 1.00000205052053 1.00000205052053 1.0000000501 1.0000000501 e+09 e+09 e+10 e+10

Ising 4× N ribbon Ising 4× N tube 20 5.12000016100253 5.12000016100253 1.0240000002 1.0240000002 z e+11 e+11 e+13 e+13 xmax (octave) xmax (Eq. 80) xmax (octave) xmax (Eq. 80) 0.05 1.28000004025 1.28000004025 2.56000000050001 2.56000000050001 The precision of the series expansions become astounding at e+09 e+09 e+10 e+10 m =6; 0.1 1.00000205103 1.00000205103 1.00000005002001 1.00000005002001 1/ N −−12+ 4 ++4 +8 +12 +16 + e+07 e+07 e+08 e+08 Z6,×N tube = zz 6 13 75 z 378 z 1420 z 3933 z z << 1 (82) 0.2 7.81360896959 7.81360896959 3.90630032164314 3.90630032164314 1/ N 12 4 −−4 8 −12 −16 e+04 e+04 e+05 e+05 Z6,×N tube = zz+ 6 ++ 13 75 z + 378 z + 1420 z + 3933 z + z >> 1 0.25 1.63934372289 1.639343722 91 6.55410791084417 6.554107910844 64 e+04 e+04 e+04 e+04

Page 10 of 13

−− Ising 6× N tube Following another branch of the twelve-fold x→+ xu42 →− x2 u segment, namely z xmax (octave) xmax (Eq. 82) 0.05 4.09600000096002e+15 4.09600000096001e+15 Qxu=(− 222 ) (90) 0.1 1.00000006001301e+12 1.00000006001301e+12 0.2 2.44144388120973e+08 2.44144388120973e+08 another step to 0.25 1.67787652988221e+07 1.67785092988221e+07 Q=( xu+ 42 ) (91) 0.3 1.88243079697202e+06 1.88230734018170e+06 4 1.67787652988222e+07 1.67787652988221e+07 we come to another terminating line, giving the double root − − −− 10 1.00000006001301e+12 1.00000006001301e+12 xu=4− 2 u 2 +− 2 u 24 u = z 4 −+ 22 z 2 z 2 − z 4 (92) 20 4.09600000096001e+15 4.09600000096001e+15 Return to second node, and follow the branch x→+ xu−4 →− x2 1mN We report the partition functions per particle κmN××= Z mN to yet another fork

21 −−−−−− 4 6 8 1710 7 12 2 14 2 44 κ3×N =2zzzzzzz+++++++ (83) −− (93) 3 9 33 Q= x 4, Q = ( xu ) 25 −− 6 10 437 −14 41 −18 the second branch of this fork promptly terminates on the quadruple κ4×N =5zzz+++ z + z + 4 32 4 root 2− 6 −− 8 10 − 12 − 14 −16 κ × =zz++++++ 15 z 2 z 4 z 5 z 1015 z + 5 N xu=−−4−+ 2=2 u 44 z −+ z 4 (94) 13 313 κ = zz2++−−− 6 z10 +10 z 14 + z −18 + 6×N 66 The significance of polynomial roots, and especially degenerate Polynomial and degenerate eigenvalues polynomial roots, is that they are rather easy to find by the Newton polygon method since they require few iterations. They appear to We discover some interesting surprises scattered among the follow a clear pattern. Once they are found, they can be factored out of eigenvalues computed by these methods. Some Ising systems possess the characteristic polynomial exactly, reducing its order substantially. × eigenvalues with truncated series expansion, polynomials. For the 3 N This not only simplifies calculation of the remaining, more significant Ising tube we find that there are two pairs of degenerate eigenvalues roots, but makes it possible to tackle even larger systems. whose series truncate after a few terms. The signal for this termination is that the Newton polygon construction gives no new segments. This example contains the first case that we have illustrated in which a root has non-rational coefficients: look at the branch beginning with 6 −−−−−2 4 6 8 10 − 12 − 14 − 16 xxzzzzzz= =++++++ 1 3 6 6 9 9 − 6 z − 36 z − 99 z + (84) 42 max 1 the factor (ux − 8) . The best way to deal with such a case is by treating 6 −−−−−2 4 6 8 10 − 12 − 14 − 16 it symbolically [12] xz2 =−+ 1 3 z − 6 z + 6 z − 9 z + 9 z + 6 z − 36 z + 99 z +

2 −−−−−2 4 6 8 10 − 12 − 14 − 16 let !C^2-8=0; xz3 =+− 2 z − 5 z − 6 z − 9 z − 9 z + 6 z + 36 z + 99 u + and develop the root in a series in the two variables C and u, obtaining 2−− 24 x4,5 =1 z+− zz − in fact two roots (the third largest and third from smallest) since there

2−− 24 are two solutions to 8=0 x6,7 =1 z−− zz + −4 −− 1 2 −1 2 4 −1 6 8 −1 10 12 −1 14 2 −−−−−2 4 6 8 10 − 12 − 14 − 16 x= u++−−−−−−−+ 8 Cu 2 8 Cu 7 u 48 Cu 32 u 144 Cu 56 u 112 Cu  (95) xzzzzzz8 =−− 2 + 5 − 6 + 9 − 9 − 6 z + 36 z − 99 u + −1 4 2 −−2 4 −6 − 8 −10 − 12 −14 For the 4× N Ising tube, zu→ we obtain three branches at the xz3 =+ 8 z +− 2 8 z − 7 z − 6 8 z − 32 z − 18 8 z − 56 z − 14 8 z + first level 4 2 −−2 4 −6 − 8 −10 − 12 −14 xz14 =− 8 z ++ 2 8 z − 7 z + 6 8 z − 32 z + 18 8 z − 56 z + 14 8 z + Q= ( xu8− 1) 2 , x 2 ( xu 4 −− 1) 12 , (x 1)2 (85) the third largest eigenvalue of the transfer matrix. indicating two (large) roots starting with uz−88= , twelve starting −44 with uz= and two starting with 1. umP(x,u−1) Take the twelve-fold branch and substitute x→+ xu−4 , the new Newton polygon exhibits branches (u8x−1)2 x2(u4x−1)12 (x−1)2

Q= ( ux2− 2) 22 ( ux + 2) 242 ( ux − 8), x − 3 u4 , Q = xx ( ++ 4)( x 2)4 (86)

4 2 2 2 2 4 2 4 −4 x−3 u (u x−2) (u x+2) (u x −8) x(x+4)(x+2) Follow the branch x→+ xu, it does not fork, there is only one new (degenerate) Newton polygon segment (x+2 u2)2 (x−2 u2)2 x2−4 (x−u4)4 Qxu=(+ 222 ) (87) 2 which we follow with xxu→−2 , again it does not fork, there is only 4 2 4 2 one new (degenerate) segment (x+u ) (x+u ) 0 Q=( xu+ 42 ) (88) 0 0 which terminates at the next level (no new segments are produced), giving the (double) polynomial root (far left branch set of Figure 7). Figure 7: Illustration of the lineage ( Q -polynomials) of degenerate polynomial eigenvalues for the 4× N Ising tube. xu=−4+ 2 u − 2 −− 2 u 24 u = z 4 +− 22 z 2 z−− 2 − z 4 (89)

Page 11 of 13

The full set of eigenvalue expansions for the 4× N ferromagnetic constructions can be used to construct the corresponding eigenvector Ising tube are as follows: components in Puiseux series for non-degenerate eigenvalues [18].

8 −−4 8 −12 −16 −20 xz1 =++ 5 20 z + 64 z + 116 z − 140 z + 2556 z + (96) The Baxter-Wu model

8 −−−4 8 12 − 16 − 20 The Baxter-Wu model, with Hamiltonian xz2 =+− 3 4 z − 12 z + 12 z + 84 z − 84 z + H= − J∑ sssi,1 i ,2 i ,3, sin , ∈− { 1,1} (100) 4 2 −−2 4 −6 − 8 −10 − 12 −14 − 16 facesi xz3 =+ 8 z +− 2 8 z − 7 z − 6 8 z − 32 z − 18 8 z − 56 z − 14 8 z + 64 z +

4 2−− 24 in which we sum over all faces of the (triangular) lattice, each face x4,5 = z+− 22 z zz − having three vertices (with states sin, , n = 1,2,3) provides us with an example in which second-largest magnitude eigenvalues are complex. xz=4+−− 3 z−− 4 4 z 8 12 z − 12 + 12 z − 16 + 84 z − 20 + 6 For the 4× N tube we find

44− x= z−8++ 4 z 4 13 z 8 + 64 z 12 + 356 z16 + 1980 z20 + 10792 z24 + , ze = β J x7,8,9,10 =2 zz−+ max (101)

4 −−−4 8 12 − 16 − 20 xz11 =−+ 4 3 z + 12 z − 12 z − 84 z + 84 z + Baxter-Wu 4× N tube Octave Series −− z x= z4−+ 22 z 2 zz 24 − 12,13 0.1 1.00000000000400e+08 1.00000000000e+0.8 0.2 3.90625006433545e+05 3.90625006434e+5 xz=4− 8 z 2 ++ 2 8 z−−2 − 7 z 4 + 6 8 z −6 − 32 z − 8 + 18 8 z −10 − 56 z − 12 + 14 8 z −14 + 64 z − 16 + 14 0.3 1.52416123161342e+04 15241.6123161 0.4 1.52599107786097e+03 1525.99107738 x= 1−++ 4 zz−−4 3 8 12 z − 12 − 12 z − 16 − 84 z − 20 + 15 0.5 256.324688942023101 256.324369907 x= 1− 4 zz−−4 +− 8 4 z − 12 + 12 z − 16 + 140 z −20 + 560 z −24 + 16 with eight complex second-largest roots. The secular equation hasa branch ω −6 with ω8 , 2πip /8 . The primitives ( p = 1,3,5,7) The 4× N hard-squares model on a tube (low fugacity) also has xz= =1 ω = e give four roots polynomial (in fact monomial) eigenvalues, as well as a large null-space 1 1 17 3 −6−−6 −− 4 + 2 − 2 +−64 x= 1+− 4 zz 223 + 12 z − 62 z 4 + 352 z 5 − 2122 z 6 + 13344 z 7 − 86566 z 8 + z<< 1 x( p ) = wz (1 w ) z wz wz(1 w ) z 1 (97) 2 2 32 2 337 21915 2345−+−+ 7 − 8 + 9 + +wz6 +−4(1 w6 ) z 8 + wz10 + (102) xz2 = 2 z 4 z 6 z 50 z 268 z 990 z 128 2048 all of which have equal magnitude real and imaginary parts. The roots xz=234−+− 2 z 6 z 22 z 5 + 90 z 6 − 394 z 7 + 1806 z 8 − 8558 z 9 + 3 for which p = 0,4 (ω =1± ) give eigenvalues

−−643 − 2 67 2 x4− 12 =0 xp( )=ω z+ z + ωω z ++4 z 28 359 15261 x=−+ zz 223 − 10 z + 58 z 4 − 346 z 5 + 2122 z 6 − 13394 z 7 + 86834 z 8 + ++15z4 ωω zz6 −−10 8 z10 +(103) 13 16 128 and p = 2,6 (ω = ±i ) result in polynomials xz14,15 = − −−6 226 234 5 6 7 8 xp( )=ω ( z+ z −− z z) (104) x16 =−− zz 2 + 2 z − 6 z + 22 z − 90 z + 394 z − 1806 z + and for high fugacity we also find polynomial (monomial) roots The 4× N Baxter-Wu ribbon is interesting for having its four largest-magnitude eigenvalues all of the same leading order in z, the xz=2+ 2 z +− 4 2 z−−12 − 8 z + 22 z − 3 − 190 z − 5 + 120 z − 6 + 1442 z − 7 − 3268 z − 8 , z >> 1 64 63 62 −6 1 (98) first level has four branches Q=()()()1 xz−−+ xz xz , or xz= , ω −6 with ω 2πip /3 , p = 0,1,2; −−−123 − 4 − 5 − 6 − 7 − 8 xz= = e xz2 =−+ 2 6 z − 22 z + 90 z − 394 z + 1806 z − 8558 z + 41586 z − 206098 z + −+−−12 + − − 3 − − 4 + − 5 − − 6 − − 7 + − 8 + −6 2 4 6 8 10 xz3 = 2 2 z 6 z 22 z 10 z 186 z 178 z 1390 z 3630 z xxzmax =1 =++ 4 10 zzz + 30 + 78 + 174 z + 307 z +

x4→ 12 =0 −6 4 22 4 34 6 26 8 59 10 x2,3,4 =2ω  z+− zz − − z − z + z + (105) −2 −− 45 − 6 − 7 − 8 − 9 33 9 9 9 x13 =−+ 1 2 z − 10 zz + 4 + 58 z − 52 z − 362 z + 544 z + In all of the examples in this subsection, there is agreement with xz= − β J 14,15 purely numerical determination of the eigenvalues for ze= = 0.1,0.2 2−−− 123 − 4 − 5 − 6 − 7 − 8 to ten decimal places or more, for both real and imaginary parts. Our x16 =− zz − 2 +− 2 6 z + 22 z − 90 z + 394 z − 1806 z + 8558 z − 41586 z + 206098 z + purpose here is not to tout the precision (but it is a welcome feature of Correlation functions and other interesting properties can be the results), but rather to point out that our methods do not suffer from computed if one has all of the eigenvalues of the transfer matrix. any of the problems associated with eigenvalue degeneracy or near- Let U be a unitary matrix that diagonalizes Tz(), in other words let degeneracy that plague many purely numerical approaches. −1 U⋅⋅ T() z U = Λ ()= z diag ( x12 , x , ), then ( x1 is the largest eigenvalue) − with σ = U⋅⋅ sU1 Conclusions and Conjectures N−−() jk jk− jk− Tr (σσ⋅Λ()z ⋅ ⋅Λ ) x →  →∞ The Newton polygon algorithm offers some distinct advantages ssjk =,N ∑a  N (99) Tr()Λ () z ≠1 x1 for the study of nanotubes and ribbons. In the first place series expansions for all eigenvalues of the transfer matrix can be found the coefficients a being given by the various components of U. Hensel

Page 12 of 13

{ j } { k }

} j { j } { k } t6 t’6

t5 t’5 t t’ 4 4 c t3 t’3 a b t2 t’2

t1 t’1

Figure 8: Graphical representation of Aa()= Akj, () a (left) and Zw (right).

Ferromagnetic Ising m×N tube Level 1 leading term Tr (T) s1 s2 s3 m (degeneracy) t t’ z6 (2), 2Z2 (3+z4) 4 4 3 y z z2 (6) 4 8 8 2(1+6z +z ) t3 t’3 z (2), 4 4 z (12), x z0 (2) t2 t’2 z10 (2), 2z2(z8+10z4+5) 6 5 z (20), z2 (10) t1 t’1 z12 (2), 2(z12+ 15z8+15z4 +1) z8 (30), 6 z4 (30), a z0 (2)

Figure 9: An explicit case of A(a) for N = 4. The trace is rather easy to calculate in the general case ( mN× Ising tube): m PBC s2 + ss symbolic-numerically: as functions of some quantity such as fugacity ∑∑i ii+1 Tr( T )= ∑∑ z ii=1 (106) or a Boltzmann weight. Both very large and very low fugacity regimes, ±± ss1=1m =1 and the cases of ferromagnetic or anti-ferromagnetic systems, are PBC ss equally approachable. Secondly the technique is not an approximation. = zzm ∑∑ ∏ ii+1 ±± Each iteration gives the next order in the series expansion exactly. ss1=1m =1 i Furthermore if a root truncates to a finite sum of terms, the Newton PBC mJ β = z ∑∑∏(cosh(ββJ )+ ssii+1 sinh( J )), z = e polygon process itself simply halts, no new segments are produced once ±± ss1=1m =1 i the last term in the series is reached. A third point is that degeneracy mm mm 22m m of a root presents none of the problems normally encountered by =z 2 (cosh (ββ J )+ sin ( Jz )) = (++− 1) ( z 1) numerical root extraction methods in this case. The level of degeneracy For hard squares we can make no such conjecture, since the trace of a root has tell-tale signs in the calculation. of the transfer matrix (the sum of Boltzmann weights for all allowed configurations with adjacent “rungs’’ of the ladder in identical states) is The technique is not limited to a single expansion variablez , Inaba simply one for all m. [19] has used an extended Hensel construction based upon the Newton polygon algorithm to factor multivariate polynomials. Hard squares m×N tube m Level 1 leading term (degeneracy) Tr (T) Leading exponents 4 0(9), z0(1), z1(4), z2(2) 1 There appears to be a correlation between the leading terms of 5 0(21), z0(1), z1(5), z2(5) 1 eigenvalues, the number of eigenvalues with this leading term, and the 6 0(46), z0(1), z1(6), z2(9), z3(2) 1 trace of the transfer matrix for the ferromagnetic Ising tubes. 7 z0(1), z1(7), z2(14), z3(7) 1 8 0(209), z0(1), z1(8), z2(20), z3(16), z4(2) 1

Page 13 of 13

s1 s2 s3 s4 s5 s6 s7 s8 s9 s10

t1 t2 t3 t4 t5 t6 t7 t8 t9 t10

Figure 10: Two rows of the lattice, with sites labeled appropriately for constructing the row-to-row transfer matrix.

References 10. Robert JW (1950) Algebraic Curves. Dover, New York. 1. RJ Baxter (1982) Exactly Solved Models in Statistical Mechanics. Dover, New 11. S Basu, R Pollack, Marie-Françoise, C Roy (2006) Algorithms in Real Algebraic York. Geometry (2nd edn), Springer. 2. Domb C, Green MS (1974) Phase Transitions and Critical Phenomena (Volume 3). Series Expansions for Lattice Models. Academic Press, London. 12. Kung HT, Traub JF (1978)All Algebraic Functions Can Be Computed Fast. JACM 25: 245-260. 3. Jaan Oitmaa, Chris Hamer, Weihong Zheng (2006) Series Expansion Methods For Strongly Interacting Lattice Models. Cambridge University Press, 13. Thomas H Cormen, Charles E Leisterson, Ronald L Rivest, Clifford Stein Cambridge. (2009) Introduction to Algorithms (3rd edn). The MIT Press, Cambridge.

4. Baxter RJ (1981) Corner transfer matrices. Physica 106A: 18-27. 14. http://reduce-algebra.sourceforge.net/.

5. Baxter RJ (2007) Corner transfer matrices in statistical mechanics. J Phys A: 15. http://www.singular.uni-kl.de Math Theor40: 12577-12588. 16. Baxter RJ (1999) Planar lattice gases with nearest-neighbor exclusion. Annals 6. Celine Cattoen, Matt Visser (2006) Generalized Puisieux series expansion for of Combinatorics 3: 191-203. cosmological milestones. arXiv:gr-qc/0609073. 17. David Cox, John Littel, Donal O’Shea (1997) Ideals, Varieties and Algorithms. 7. Takua Kitamoto (1994) Approximate eigenvalues, eigenvectors and inverse of Springer, New York, USA. a matrix with polynomial entries. J Indust Appl Math 11: 73-85. 18. Takua Kitamoto (1994) Approximate eigenvalues, eigenvectors and inverse of 8. Takeaki Sasaki, Fujio Kako (1999) Solving multivariate algebraic equation by a matrix with polynomial entries. J Indust Appl Math 11: 73-85. Hensel construction.J Indust Appl Math16: 257-285. 19. Daiju Inaba (2005) Factorization of multivariate polynomials by extended 9. David A Cox, John Little, Donal O’Shea (2005) Using Algebraic Geometry (2nd Hensel construction. ACM SIGSAM Bulletin. 39: 2-14. edn). Springer.

J Phys Math ISSN: 2090-0902 JPM, an open access journal Volume 5 • Issue 1 • 1000125