Energy extremes and spin configurations for the one-dimensional antiferromagnetic Ising model with arbitrary-range interaction Jack Douthett Department of Arts and Science, Albuquerque Community College and Technical–Vocational Institute, Albuquerque, New Mexico 87106

Richard Krantz Department of Physics, Metropolitan State College of Denver, Denver, Colorado 80217-3362

~Received 19 October 1995; accepted for publication 18 March 1996! The one-dimensional antiferromagnetic spin- 21 Ising model is investigated using the formalism of Maximally/Minimally Even sets. The salient features of Maximally/ Minimally Even set theory are introduced. Energy and spin content vectors are defined to facilitate the use of interval spectra used in Maximally/Minimally Even set theory. It is shown that Maximally Even sets of up- and down-spins minimize the configurational energy per spin and that Minimally Even sets maximize configurational energy per spin. An exponentially decreasing antiferromagnetic pairwise interaction of arbitrary range is used as an example interaction. The asymptotic (N→`) configurational energy per spin and the energy per spin calculated for seven-near neighbors are compared. © 1996 American Institute of Physics. @S0022-2488~96!00107-7#

I. INTRODUCTION

The simplest description of the pairwise interaction of spins on a lattice is given by the Ising model which yields the following for the configurational energy of a lattice of spin- 21 ‘‘particles:’’ N21

H524

(

i, j50 iÞ j

J ~ u i2 j u ! s iz s jz ,

where N is the total number of lattice sites, and the sum is taken over all pairs of lattice sites. The function J ( u i2 j u ) is the pairwise interaction energy, the absolute value of which decreases with distance. The s z are the z-components of the spins which may take on the values of 6 21. ~For a discussion of lattice gas models, see the excellent reference by Simon.1! In a one-dimensional system, it is convenient to invoke periodic boundary conditions which requires that s 0z 5s Nz . Therefore, a one-dimensional spin- 21 lattice can be thought of as a cycle of lattice sites, some of which are occupied by up-spins ~1 21! and others by down-spins ~2 21!. A simplification may be made by defining s p 52s pz . Then, the configurational energy may be written as N21

H52

(

i, j50 iÞ j

J ~ u i2 j u ! s i s j ,

where the s take on the values of 61. By considering antiferromagnetic pairwise interactions only ~i.e., J ( u i2 j u ),0!, a further simplification results, and the configurational energy is given by

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J. Math. Phys. 37 (7), July 1996

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J. Douthett and R. Krantz: Energy extremes and spin configurations

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FIG. 1. Clockwise distances between pairs of lattice sites.

N21

H anti5

(

i, j50 iÞ j

J ~ u i2 j u ! s i s j ,

where J( u i2 j u )5uJ ( u i2 j u )u. For a given down-spin density ~ratio of the number of down-spin sites to the total number of sites!, the average configurational energy ~energy per site! depends on the distribution of the upand down-spin sites. Since there is a large number of sites, for a given down-spin density, there are many possible values for the average configurational energy. We will focus on the average energy extremes of this cyclic one-dimensional system by exploiting the formalism of Maximally Even and Minimally Even sets as described by Clough and Douthett2 and Block and Douthett.3 Curiously, many of the tools employed in this paper have been developed in the music theory literature. Content vectors are routinely used in the analysis of twentieth-century music as well as in compositional design, and Lewin4 was the first to explore the properties of these vectors in microtonal systems ~systems that have other than 12 divisions to the octave!. Block and Douthett3 developed weighting vectors to be used in conjunction with content vectors to find pitch-class sets ~collections of tones! whose members have particular pairwise intervallic relationships. In doing so, they developed a measure that compares the evenness between any two pitch-class sets of the same size. Clough and Myerson5,6 developed interval spectra to explore combinatorial properties of diatonic scales ~e.g., the white keys on the piano! and to extend these properties into microtonal systems. Clough and Douthett2 expanded on this work and developed the theory of Maximally Even Sets. These sets, along with their iterations, model scale and chord structures in both the usual musical system ~12 divisions to the octave! and microtonal systems. II. ENERGY, CONTENT, AND SPIN CONTENT VECTORS

The method discussed in this section and the next in which the dot product of an energy vector and a content vector is used to determine the configurational energy of a particular lattice configuration is based on related work done by Block and Douthett.3 To apply their method to the problem at hand, let N be the number of sites in the lattice, N 2 be the number of these sites occupied by down-spins, and N 1 be the number occupied by up-spins. Assuming periodic boundary conditions, for each pair of sites, there are two clockwise distances ~with respect to the number of sites that separate them! associated with this pair ~Fig. 1!. If the consecutive sites in the lattice are represented by the consecutive integers 0 through N21, the two distances associated with site 0 and site k are k and N2k. Now assume J is a strictly convex function on the interval ~0,`!. @Later the discussion will be restricted to pairwise interactions for which ( `k51 J(k) converges.# Then the absolute value of the energy contributed by a pair of sites at a distance k is J N ~ k ! 5J ~ k ! 1J ~ N2k ! .

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J. Douthett and R. Krantz: Energy extremes and spin configurations

Thus, if E 2 is the sum of the J N -functions over all down-spin pairs, E 1 is the sum of the J N -functions over all the up-spin pairs, and E 0 is the sum over all the pairs with opposite orientations, the total energy of the lattice is E5E 1 1E 2 2E 0 . It is convenient to construct vectors that allow computation of this energy via dot products. The first vector, the energy vector, is defined as follows: W5„J N ~ 1 ! ,J N ~ 2 ! ,...,J N ~ b N/2c ! …. To construct the content vectors, consider the following example: Suppose N57 and N 2 53. ~Hence, N 1 54.! Consider the distribution in Fig. 2~b! on the complete labeled graph K 7 . Then the set of down-spins is S 2 5 $ 0,2,3% and the set of up-spins is S 1 5 $ 1,4,5,6% . Now consider the complete labeled subgraphs K 2 @Fig. 2~a!# and K 1 @Fig. 2~c!#. The edges of these graphs are weighted as follows: Suppose an edge e has incident vertices whose labels are a and b. Then the weight of this edge is w ~ e ! 5min$ u a2b u ,72 u a2b u % . The content vectors of these graphs are three-tuples that have as their kth entries the number of edges that have weight k. Then the content vectors for K 2 , K 7 , and K 1 are V 2 5(1,1,1), V 7 5(7,7,7), and V 1 5(2,2,2), respectively. It is possible to compute V 1 without its graph. Lewin4 has shown V 1 2V 2 5 ~ n,n,n ! , where n5N 1 2N 2 5N22N 2 . In this case, n5722•351. Thus, V 1 5 ~ 1,1,1! 1V 2 5 ~ 2,2,2! . This characterization will be useful when these results are generalized. Now, consider the complement of K 2 øK 1 , K 2,1 5 ~ K 2 øK 1 ! c whose graph is illustrated in Fig. 2~d!. This graph is a complete bipartite, and the edges in this graph are precisely those that are incident to one up-spin site and one down-spin site. By definition, the edge sets of K 2 , K 1 , and K 2,1 partition the edge set of K 7 . The content vector V 0 of K 2,1 is computed as follows: V 7 5V 2 1V 1 1V 0 ,

V 0 5V 7 2V 2 2V 1 5 ~ 4,4,4! .

Note that this corresponds to the calculation by inspection of K 2,1 in Fig. 2~d!. Finally the spin content vector is defined as V5V 1 1V 2 2V 0 5 ~ 1,1,1! . The above example may be extended to the general case. The lattice may be modelled with the complete labeled graph K N whose labels range from 0 through N21. Then the edge weight of an edge e whose incident vertices have labels a and b is w ~ e ! 5min$ u a2b u ,N2 u a2b u % .

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J. Douthett and R. Krantz: Energy extremes and spin configurations

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FIG. 2. Graphs and content vectors.

The notation (•••n•••) indicates a b N/2c -tuple in which every entry is n. For convenience, assume N is odd. Then the content vector for K N is V N 5(•••N•••). @Note that if N is even, the last entry of this vector must be divided by 2. It is left to the reader to verify that the assumption that N is even will produce the same results as N→`.# Now let S 2 and S 1 be the sets of labels of down-spins and up-spins, respectively. Then the complete labeled subgraphs K 2 and K 1 on these sets and their corresponding content vectors V 2 and V 1 are constructed in the same way as those in the above example. Because of symmetry, it can be assumed without loss of generality that N 2 ,N/2. Furthermore, since N 2 ,N 1 , it can be shown that V 1 2V 2 5(•••N22N 2 •••).4 Thus, V 1 5 ~ •••N22N 2 ••• ! 1V 2 .

~1!

Define the graph K 2,1 5(K 2 øK 1 ) c to be the complete bipartite graph with partitions S 2 and S 1 , and let its content vector be V 0 . Then, since V N 5(•••N•••), and the edge sets of K 2 , K 1 , and K 2,1 partition the edge set of K N , ~1! implies V 2 1V 1 1V 0 5 ~ •••N••• ! ,

V 0 52 ~ •••N 2 ••• ! 22V 2 .

~2!

Then, by ~1! and ~2!, V5V 1 1V 2 2V 0 54V 2 1 ~ •••N24N 2 ••• ! .

J. Math. Phys., Vol. 37, No. 7, July 1996

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J. Douthett and R. Krantz: Energy extremes and spin configurations

FIG. 3. ME and me distributions.

This means that the spin content vector of any configuration can be completely determined by the set of down-spin labels. The total energy can be found by taking the dot product of the spin content vector and the energy vector: E5E 1 1E 2 2E 0 5V 1 •W1V 2 •W2V 0 •W5 ~ V 1 1V 2 2V 0 ! •W5V•W. In the previous example, V 2 5(1,1,1). Thus, V54 ~ 1,1,1! 1 ~ •••724 ~ 3 ! ••• ! 52 ~ 1,1,1! . So the total energy of the lattice is E5V•W52 ~ 1,1,1! •„J 7 ~ 1 ! ,J 7 ~ 2 ! ,J 7 ~ 3 ! …52J 7 ~ 1 ! 2J 7 ~ 2 ! 2J 7 ~ 3 ! . III. MAXIMALLY AND MINIMALLY EVEN DISTRIBUTIONS

Consider the configurations in Fig. 3. The total energy for the configuration in Fig. 3~a! is E525J 7 (1)13J 7 (2)2J 7 (3) and in Fig. 3~b!, E53J 7 (1)2J 7 (2)25J 7 (3). For a given convex interaction J, the configuration in Fig. 3~a! will yield the least energy and in Fig. 3~b!, the greatest. If J is strictly convex, the only other configurations yielding these energy extremes are rotations of the above. We now turn our attention to the unique properties of the classes of these configurations. The interval spectrum of near-neighbor down-spins (up-spins) is defined to be the set of distinct clockwise distances between each pair of consecutive down-spins ~up-spins!. This set is denoted by ^1&2~^1&1!. Thus, for the sets in Fig. 2, ^1&25$1,2,4% and ^1&15$1,2,3%; in Fig. 3~a!, ^1&25$2,3% and ^1&15$1,2%; and in Fig. 3~b!, ^1&25$1,5% and ^1&15$1,4%. This concept may be extended to sets of clockwise distances between pairs of next-neighbor down-spins ~up-spins! @pairs that have precisely one down-spin ~up-spin! site between them#. Then for the sets in Fig. 2, ^2&25$3,5,6% and ^2&15$2,3,4,5%; in Fig. 3~a!, ^2&25$4,5% and ^2&15$3,4%; and in Fig. 3~b!, ^2&25$2,6% and ^2&15$2,5%. In general, the spectrum of k, written ^ k & 6 , is the set of distinct clockwise distances between pairs of down-spins ~up-spins! that have precisely k21 down-spins ~up-spins! between them. A pair of down-spins ~up-spins! is associated with k if the pair has the above property. These spectra are defined for 1