On the Phase Structure of Many-Flavor QED${}_3$

TL;DR

Using the functional renormalization group, the paper analyzes QED${}_3$ with $N_f$ four-component Dirac flavors to map the phase structure as a function of flavor number. A Fierz-complete set of pointlike four-fermion interactions is tracked together with gauge and fermion sector renormalizations, revealing a conformal-critical flavor number ${N}_{\mathrm{f,cr}}^{\mathrm{qc}}$ where the IR gauge fixed point persists, and showing the possibility of an intermediate vector-channel–dominated phase between chirality-breaking at small $N_f$ and quasi-conformal behavior at large $N_f$. The results indicate ${N}_{\mathrm{f,cr}}^{\mathrm{qc}}$ may lie in a broad, regulator- and momentum-resolution–dependent window roughly ${N}_{\mathrm{f,cr}}^{\mathrm{qc}} \approx 4.1$ to $10.0$, with ${N}_{\mathrm{f,cr}}^{\chi} \approx 4$ as an estimate for the chiral transition; crucially, Fierz completeness dramatically affects these estimates and can reveal or suppress an intermediate phase characterized by vector-channel fluctuations and possible Lorentz symmetry breaking. The work emphasizes the sensitivity of nonperturbative predictions to truncations and regulator choices and motivates cross-method verification, including dynamical bosonization and lattice studies, to robustly determine the phase diagram and its relevance to condensed-matter realizations such as cuprates.

Abstract

We analyze the many-flavor phase diagram of quantum electrodynamics (QED) in 2+1 (Euclidean) space-time dimensions. We compute the critical flavor number above which the theory is in the quasi-conformal massless phase. For this, we study the renormalization group fixed-point structure in the space of gauge interactions and pointlike fermionic self-interactions, the latter of which are induced dynamically by fermion-photon interactions. We find that a reliable estimate of the critical flavor number crucially relies on a careful treatment of the Fierz ambiguity in the fermionic sector. Using a Fierz-complete basis, our results indicate that the phase transition towards a chirally-broken phase occurring at small flavor numbers could be separated from the quasi-conformal phase at larger flavor numbers, allowing for an intermediate phase which is dominated by fluctuations in a vector channel. If these interactions approach criticality, the intermediate phase could be characterized by a Lorentz-breaking vector condensate.

Figures (8)

• Figure 1: 1PI diagram contributing to the vacuum polarization tensor $\Pi_{\mu\nu}$: the double lines represent (full) scale-dependent regularized fermion propagators. The flow of the photon wave-function renormalization is driven by the scale-derivative of this diagram with respect to the regulator.
• Figure 2: (color online) $\beta_{e^2}$ function for three different values of $N_{\text{f}}$ as obtained from the linear regulator evaluated for $\zeta=1$ and $\eta_{\psi}=0$ for simplicity.
• Figure 3: (color online) RG trajectories in the plane spanned by the four-fermion couplings $\tilde{g}$ and $g$ for $N_{\text{f}}=4$ and $e^2=0$ using the linear regulator. The fixed points are depicted by the red dots, where $\mathcal{O}$ is the IR stable Gaußian fixed point, $\mathcal{A}$ and $\mathcal{B}$ are fixed points with one IR attractive and one IR repulsive direction, and $\mathcal{B}$ is an unstable fixed point with two IR repulsive directions. The thin arrows indicate the RG flow towards the IR regime. The dashed line ($g=\tilde{g}$) corresponds to the chiral channel (where $g_V=0$ and $g_\phi$ is nonzero), potentially associated with chiral symmetry breaking, see also Eq. \ref{['eq:coupFierz']}. This channel is typically chosen in Fierz-incomplete studies. The blue/bold arrows attached to the four fixed points indicate the shift of the fixed points induced by an increase of the gauge coupling $e^2 > 0$.
• Figure 4: (color online) The critical value $e^2_{\rm cr}$ for the gauge coupling and the value $e^2_{\ast}$ of the IR fixed point as a function of $N_{\text{f}}$ as obtained from the linear regulator for $\zeta=1$ and $\eta_{\psi}=0$ for simplicity. The intersection point of both lines determines the conformal-critical flavor number $N_{\text{f,cr}}^{\text{qc}}{}$, see Eq. \ref{['eq:critcond']}. Note that the depicted $N_{\text{f}}$ dependence of $e^2_{\ast}$ has been computed with the aid of Eq. \ref{['eq:eFlowsimple']}. However, the associated IR fixed-point $e^2_{\ast}$ is only approached for $N_{\text{f}}\geq N_{\text{f,cr}}^{\text{qc}}{}$. For $N_{\text{f}}<N_{\text{f,cr}}^{\text{qc}}{}$, this fixed point is destabilized due to spontaneous symmetry breaking.
• Figure 5: 1PI diagram contributing to the RG flow of the four-fermion couplings: the double lines represent (full) scale-dependent regularized fermion and photon propagators.