Fu Z, Guo W, Blöte HWJ. Ising-like transitions in the O(n) loop model on the square lattice.
PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013;
87:052118. [PMID:
23767498 DOI:
10.1103/physreve.87.052118]
[Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/29/2013] [Indexed: 06/02/2023]
Abstract
We explore the phase diagram of the O(n) loop model on the square lattice in the (x,n) plane, where x is the weight of a lattice edge covered by a loop. These results are based on transfer-matrix calculations and finite-size scaling. We express the correlation length associated with the staggered loop density in the transfer-matrix eigenvalues. The finite-size data for this correlation length, combined with the scaling formula, reveal the location of critical lines in the diagram. For n>>2 we find Ising-like phase transitions associated with the onset of a checkerboardlike ordering of the elementary loops, i.e., the smallest possible loops, with the size of an elementary face, which cover precisely one-half of the faces of the square lattice at the maximum loop density. In this respect, the ordered state resembles that of the hard-square lattice gas with nearest-neighbor exclusion, and the finiteness of n represents a softening of its particle-particle potentials. We also determine critical points in the range -2≤n≤2. It is found that the topology of the phase diagram depends on the set of allowed vertices of the loop model. Depending on the choice of this set, the n>2 transition may continue into the dense phase of the n≤2 loop model, or continue as a line of n≤2 O(n) multicritical points.
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