Abstract
We study low-temperature properties of boson density matrices when macroscopic occupation of quantum energy levels can occur. An important novelty of our approach is the definition of order indices that characterize the behaviour of largest eigenvalues of density matrices in the thermodynamic limit. Continuous variation of the order indices gives rise to a whole spectrum of possible order types: long-range order, mid-range order and short-range order. A relation between order indices is found. As an illustration of the general scheme a new model exemplifying mid-range order is constructed.
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