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.