Phase transitions and reflection positivity. II. Lattice systems with short-range and Coulomb interactions

Abstract

We discuss applications of the abstract scheme of part I of this work, in particular of infrared bounds and chessboard estimates, to proving the existence of phase transitions in lattice systems. Included are antiferromagnets in an external field, hard-core exclusion models, classical and quantum Coulomb lattice gases, and six-vertex models.

On the flux phase conjecture at half-filling: An improved proof

“…Strictly speaking however, even in one dimension, it is different. A slightly different transformations was employed previously by several authors, e.g., in [9]. Note however that in [9] the paragraph about fermions contains a mistake.…”

“…A slightly different transformations was employed previously by several authors, e.g., in [9]. Note however that in [9] the paragraph about fermions contains a mistake. With the transformation employed there the hopping terms on the right acquire the opposite sign of the hopping terms on the left, and thus the Hamiltonian is not in reflection positive form.…”

Phase transitions in quantum spin systems with isotropic and nonisotropic interactions

“…This is discussed further in Ref. 50. We will restrict ourselves to nearest neighbor interactions on a simple cubic lattice, but no special commutation properties are required.…”

A Model with Simultaneous First and Second Order Phase Transitions

“…Finally we prove the Peierls condition: first we sum over all the specifications of the contours with a fixed support using the estimates (14), (15), (17), (19), and (21); next we use the Koenigsberg's lemma to get an upper bound on the entropy for the contours. This concludes our proof of part I of the Theorem.…”