Date: Mon, 6 Dec 2004 08:44:41 -0500 (EST) From: Jeffrey S Pratt Subject: Comment on A086520 I don't think that the published abstract of the Phys. Rev. Lett. paper is very helpful in understanding the origin of sequence A086520, so I'll try to explain that briefly. The paper assesses the quantum mechanical correlations present in magnets at zero and finite temperature, using a measure of quantum correlation called concurrence. Microscopically, magnetism arises from the interactions between particles with spin, that is, the Hilbert space of quantum states associated with each particle carries an irreducible representation of SU(2). For spin-1/2 particles, this representation is two-dimensional; each spin is therefore either 'up' or 'down', or in some complex-linear combination of these basis states. Further, in this case the interactions between the spins are equivalent to permutation operators. If the interactions are all ferromagnetic - that is, tending to align the spins - or zero, then I show that the lowest energy states of the magnet are precisely the completely symmetric states (CSS), i.e. those states that are invariant under any permutation. All such states, except for the trivial states with all spins up or all spins down, contain nonvanishing quantum correlations, which I evaluate analytically. But when all the CSS mix together, the quantum correlations vanish. Why is that? It turns out that the CSS cancel each other's quantum correlations pairwise, except for a special group of states, which I call the core states. The correlations in the core do not vanish even when all the core states mix together. Counting the number of core states in a magnet with N spins is equivalent to counting the number of integers between (N-sqrt(N))/2 and (N+sqrt(N))/2, excluding the end points; for example, when N=16, there are three numbers between 6 and 10 and hence three core states. Starting at N=2, the sequence is 1,2,1,2,3,2,3,2,3,4,3,4,3,4,3,4,5,4,5,4,5,4,5,4... with expanding blocks of alternating integers.