Rigorous Analysis of Renormalization Group Pathologies in the 4-State Clock Model

TL;DR

This paper provides an exact renormalization-group analysis of the 4-state clock model, clarifying the nature of phase transitions and flow behavior, and identifying conditions leading to pathological cases where Hamiltonians cannot be defined.

Contribution

It offers a rigorous analysis of RG flow in the 4-state clock model, confirming the continuous flow near phase transitions and explaining the origin of discontinuities in truncated transformations.

Findings

Flow is continuous and single-valued near phase transitions.

Discontinuities occur when a renormalized Hamiltonian cannot be defined.

Truncated transformations can produce discontinuous, multivalued flows.

Abstract

We perform an exact renormalization-group analysis of one-dimensional 4-state clock models with complex interactions. Our aim is to provide a simple explicit illustration of the behavior of the renormalization-group flow in a system exhibiting a rich phase diagram. In particular we study the flow in the vicinity of phase transitions with a first-order character, a matter that has been controversial for years. We observe that the flow is continuous and single-valued, even on the phase transition surface, provided that the renormalized Hamiltonian exist. The characteristics of such a flow are in agreement with the Nienhuis-Nauenberg standard scenario, and in disagreement with the ``discontinuity scenario'' proposed by some authors and recently disproved by van Enter, Fern\'andez and Sokal for a large class of models (with real interactions). However, there are some points in the space of interactions for which a renormalized Hamiltonian cannot be defined. This pathological behavior is similar, and in some sense complementary, to the one pointed out by Griffiths, Pearce and Israel for Ising models. We explicitly see that if the transformation is truncated so as to preserve a Hamiltonian description, the resulting flow becomes discontinuous and multivalued at some of these points. This suggests a possible explanation for the numerical results that motivated the ``discontinuity scenario''.