 93181 Michael Aizenman, Bruno Nachtergaele
 Geometric Aspects of Quantum Spin States
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Jun 16, 93

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Abstract. A number of interesting features of the ground
states of quantum spin chains are analized with the help of a functional
integral representation of the system's equilibrium states. Methods of general
applicability are introduced in the context of the SU($2S+1$)invariant
quantum spin$S$ chains with the interaction $P^{(0)}$, where $P^{(0)}$
is the projection onto the singlet state of a pair of nearest neighbor spins.
The phenomena discussed here include: the absence of N\'eel order,
the possibility of dimerization,
conditions for the existence of a spectral gap, and a dichotomy
analogous to one found by Affleck and Lieb, stating that the systems
exhibit either slow decay of correlations or translation symmetry breaking.
Our representation elucidates the relation, evidence for which
was found earlier, of the $P^{(0)}$ spin$S$ systems with the Potts and
the FortuinKasteleyn randomcluster models in one more dimension.
The method reveals the geometric aspects of the listed phenomena,
and gives a precise sense to a picture of the ground state in which
the spins are grouped into random clusters of zero total spin. E.g.,
within such structure the dichotomy is implied by a topological argument,
and the alternatives correspond to whether, or not, the clusters are of
finite mean length.
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