Exact renormalization-group analysis of first order phase transitions in clock models

TL;DR

This paper provides an exact analysis of the renormalization group flow in one-dimensional clock models with complex interactions, revealing continuous flow behavior across phase transitions and identifying unique pathologies due to complex couplings.

Contribution

It offers the first exact, continuous RG flow analysis for first order transitions in clock models with complex interactions, challenging previous discontinuity scenarios.

Findings

RG flow is single-valued and continuous at transition points

Pathologies arise from complex interactions, not discontinuities

Flow patterns inform quantum Hall effect conductivity behavior

Abstract

We analyze the exact behavior of the renormalization group flow in one-dimensional clock-models which undergo first order phase transitions by the presence of complex interactions. The flow, defined by decimation, is shown to be single-valued and continuous throughout its domain of definition, which contains the transition points. This fact is in disagreement with a recently proposed scenario for first order phase transitions claiming the existence of discontinuities of the renormalization group. The results are in partial agreement with the standard scenario. However in the vicinity of some fixed points of the critical surface the renormalized measure does not correspond to a renormalized Hamiltonian for some choices of renormalization blocks. These pathologies although similar to Griffiths-Pearce pathologies have a different physical origin: the complex character of the interactions. We elucidate the dynamical reason for such a pathological behavior: entire regions of coupling constants blow up under the renormalization group transformation. The flows provide non-perturbative patterns for the renormalization group behavior of electric conductivities in the quantum Hall effect.