[Paper Review] Vacuum stability, fixed points, and phases of QED$_3$ at large $N_f$

Lorenzo Di Pietro, Edoardo Lauria+1 more|arXiv (Cornell University)|Jan 11, 2023

Black Holes and Theoretical Physics4 citations

TL;DR

This paper investigates vacuum stability, renormalization group (RG) fixed points, and phase structure in three-dimensional quantum electrodynamics (QED₃) with a Chern-Simons term and large $N_f$ flavors. Using a large-$N_f$ expansion to all orders in $k/N_f$ and couplings, it computes effective potentials and $eta$-functions for classically marginal couplings in critical and tricritical bosonic QED₃, Gross-Neveu, and fermionic QED₃. The key result is that only specific $k/N_f$ values allow stable vacua in tricritical and Gross-Neveu theories, while critical and fermionic QED₃ remain stable across all $k/N_f$, with phase diagrams fully mapped under relevant deformations.

ABSTRACT

We consider three-dimensional Quantum Electrodynamics in the presence of a Chern-Simons term at level $k$ and $N_f$ flavors, in the limit of large $N_f$ and $k$ with $k/N_f$ fixed. We consider either bosonic or fermionic matter fields, with and without quartic terms at criticality: the resulting theories are critical and tricritical bosonic QED$_3$, Gross-Neveu and fermionic QED$_3$. For all such theories we compute the effective potentials and the $β$ functions of classically marginal couplings, at the leading order in the large $N_f$ limit and to all orders in $k/N_f$ and in the couplings. We determine the RG fixed points and discuss the quantum stability of the corresponding vacua. While critical bosonic and fermionic QED$_3$ are always stable CFTs, we find that tricritical bosonic and Gross-Neveu QED$_3$ exist as stable CFTs only for specific values of $k/N_f$. Finally, we discuss the phase diagrams of these theories as a function of their relevant deformations.

Motivation & Objective

• To determine the quantum stability of vacua in large-$N_f$ QED₃ with a Chern-Simons term at level $k$.
• To identify RG fixed points for classically marginal couplings in critical, tricritical, Gross-Neveu, and fermionic QED₃ theories.
• To map the phase diagrams of these theories under relevant deformations such as mass and quartic couplings.
• To establish the interplay between RG fixed point existence and vacuum stability, particularly in tricritical and Gross-Neveu models.

Proposed method

• Computes the effective potential $\mathcal{V}_{\text{eff}}$ at leading order in $1/N_f$ and exactly in $k/N_f$ and coupling constants for bosonic and fermionic QED₃ models.
• Derives the $\beta$-functions of classically marginal couplings to all orders in $k/N_f$ and couplings, using large-$N_f$ resummation techniques.
• Imposes vacuum stability by requiring the effective potential to be bounded from below, identifying critical bounds on couplings.
• Analyzes gap equations to determine existence and nature (stable/metastable) of vacua across phase diagrams.
• Maps phase diagrams using a dictionary between tricritical and Gross-Neveu models, relating $\Sigma$, $h$, $\lambda$, and $m^2$ to physical observables.
• Uses the $\Lambda = e^2 N_f$ and $\kappa = k/N_f$ expansion to take the large-$N_f$ limit while retaining non-perturbative $\kappa$ dependence.

Experimental results

Research questions

• RQ1 For which values of $k/N_f$ are tricritical bosonic QED₃ and Gross-Neveu QED₃ quantum mechanically stable CFTs?
• RQ2 How do the $\beta$-functions of marginal couplings in large-$N_f$ QED₃ depend on $k/N_f$ and the coupling constants?
• RQ3 What is the structure of the phase diagram of tricritical QED₃ under relevant deformations by mass and quartic coupling?
• RQ4 Why does the sextic coupling in critical QED₃ remain irrelevant even when tuned in tricritical QED₃?
• RQ5 How do the low-energy descriptions differ between phases with positive and negative $\Sigma$ in Gross-Neveu QED₃?

Key findings

• Tricritical bosonic QED₃ is quantum mechanically stable only if the sextic coupling $h$ satisfies $h < 16\pi^2/3$, which corresponds to the value of $h$ in critical QED₃.
• The effective potential for tricritical QED₃ is bounded from below only when $h \leq 16\pi^2/3$, confirming vacuum stability under this bound.
• For $h$ outside this bound, the gap equations have no solution in certain regions of the phase diagram, and the potential becomes unbounded from below, indicating metastability or instability.
• Gross-Neveu QED₃ has a stable vacuum only when $|y| < 1/(6\pi)$, and this stability condition is satisfied only for $\kappa \gtrsim 0.273$, where $y$ is the cubic coupling.
• The phase diagram of tricritical QED₃ features a second-order transition line at $m^2 = 0$, $\lambda > 0$, and a multicritical point at the origin, with distinct Higgsed and unHiggsed phases separated by critical lines.
• Fermionic QED₃ and Gross-Neveu QED₃ share the same phase diagram structure under a precise mapping of parameters, with $\Sigma > 0$ and $\Sigma < 0$ phases corresponding to $U(1)_{k \pm N_f/2}$ Chern-Simons theories with unbroken $SU(N_f)$ symmetry.