Self-Dualities and Renormalization Dependence of the Phase Diagram in 3d $O(N)$ Vector Models

TL;DR

This paper explores the self-duality and renormalization scheme dependence of phase diagrams in 3d $O(N)$ vector models, revealing how fixed points merge or become complex, affecting phase transitions and gap properties.

Contribution

It provides exact analytic expressions for the renormalization dependence of critical couplings and demonstrates the merging and complexification of fixed points in 3d $O(N)$ models.

Findings

Critical couplings depend analytically on the renormalization scheme.

Fixed points merge at a specific scheme parameter and become complex beyond that.

Numerical evidence supports the self-duality and the gapped nature in certain schemes.

Abstract

In the classically unbroken phase, 3d $O(N)$ symmetric $\phi^4$ vector models admit two equivalent descriptions connected by a strong-weak duality closely related to the one found by Chang and Magruder long ago. We determine the exact analytic renormalization dependence of the critical couplings in the weak and strong branches as a function of the renormalization scheme (parametrized by $\kappa$) and for any $N$. It is shown that for $\kappa=\kappa_*$ the two fixed points merge and then, for $\kappa<\kappa_*$, they move into the complex plane in complex conjugate pairs, making the phase transition no longer visible from the classically unbroken phase. Similar considerations apply in 2d for the $N=1$ $\phi^4$ theory, where the role of classically broken and unbroken phases is inverted. We verify all these considerations by computing the perturbative series of the 3d $O(N)$ models for the vacuum energy and for the mass gap up to order eight, and Borel resumming the series. In particular, we provide numerical evidence for the self-duality and verify that in renormalization schemes where the critical couplings are complex the theory is gapped. As a by-product of our analysis, we show how the non-perturbative mass gap at large $N$ in 2d can be seen as the analytic continuation of the perturbative one in the classically unbroken phase.