Thermal precondensation in gauge-fermion theories

TL;DR

This work investigates thermal precondensation in gauge-fermion theories with massless fermions in the chiral limit, discovering a finite-range condensate that exists only between $T_{ m crit}$ and $T_{ m pre}$. Using a first-principles functional renormalization group framework and bosonising the scalar channel, the authors show that a competition between fermionic dynamical chiral symmetry breaking and symmetry-restoring bosonic fluctuations yields a momentum-dependent condensate and possible domain structures. They demonstrate how the precondensation regime broadens as the number of flavours $N_f$ increases, linking the effect to enhanced infrared bosonic fluctuations and dimensional reduction, and discuss its relevance for many-flavour theories and physics beyond the Standard Model. The results suggest that precondensation could imprint observable signatures in near-critical thermal dynamics and domain formation, offering connections to moats, inhomogeneous condensates, and potential probes in beyond-Standard-Model contexts.

Abstract

Precondensation is a peculiar phenomenon in phase transitions, characterised by the occurrence of a condensate only over a finite range of length scales. It is closely connected to the emergence of domains, pseudo-gapped phases and spatial inhomogeneities in equilibrium. In this work, we show its occurrence in gauge-fermion theories in the chiral limit, close to the thermal chiral phase transition. We further show that the precondensation regime becomes increasingly pronounced and extends over a wider temperature range as the number of fermion flavours is increased. We analyse the underlying dynamics which is shared by a broad class of fermionic systems, ranging from condensed matter to high-energy physics. Specifically, we discuss the potential relevance of this phenomenon for physics beyond the Standard Model.

Figures (5)

• Figure 1: Chiral order parameter $\sigma_0(r)$ of a QCD-like gauge-fermion theory with $N_c=3$ and $N_f=3$ chiral flavours as a function of the spatial separation $r$ (identified with the inverse fRG cutoff $k^{-1}$) and temperature, normalised to its value at zero temperature in the macroscopic limit ($\sigma_0(\infty)_{\,T\to0}$). The dotted lines correspond to the condensate at high temperatures, where the system remains in the symmetric phase at all length scales. Solid lines show the condensate in the precondensation regime, $T_{\rm crit}$ (vertical dashed line) $<T<T_{\rm pre}$ (vertical dotted line), where it is nonvanishing only over a finite range of length scales. The dashed lines show temperatures in the broken phase, where a macroscopic condensate is obtained at large distances, $r\to\infty$. The blue line indicates the UV length scale ($r_{\rm UV}$), below which the condensate effectively vanishes. The red line marks the domain size ($\xi$) associated with precondensation, see \ref{['fig:cartoondomains']}.
• Figure 2: Cartoon of the thermal evolution of a system with precondensation at a fixed length scale $r\sim (\pi T_{\rm crit})^{-1}$. Top: symmetric phase at high temperatures where only fluctuations are present. Middle: precondensation phase at $T_{\rm crit} < T < T_{\rm pre}$, where a local condensate exists on scales of order $\xi$, while $\sigma_0 = 0$ due to cancellations among domains at large distances. Bottom: broken phase below $T_{\rm crit}$, where the domains align and a homogeneous macroscopic condensate forms.
• Figure 3: Diagrammatic flow of the radial $\sigma$-mode two-point function, which encodes the curvature of the effective potential at small fields and thus the emergence of a condensate. Blue, double-arrowed lines denote full propagators of the $\sigma$ and $\boldsymbol{\pi}$ modes (dark blue), while single-arrowed lines represent fermion propagators. Gray dots indicate full vertices, and crossed circles denote insertions of the scale derivative of the regulator function, as usual in the fRG.
• Figure 4: Domain size as a function of the temperature relative to the critical one. As purple, blue and green dots the results for $N_f=2$, $3$ and $4$ are respectively shown which can also be found in \ref{['fig:OrderParameter', 'fig:precondensationNf2andNf4']}. In the inlay plot we show the reduced precondensation temperature as a function of $N_f^{-1}$.
• Figure 5: Chiral order parameter $\sigma_0(r)$ of two QCD-like gauge-fermion theories at $N_c=3$, with $N_f=2$ (left) and $N_f=4$ (right) chiral flavours as a function of the spatial separation $r=k^{-1}$ and temperature, normalised to its value at zero temperature in the macroscopic limit ($\sigma_0(\infty)_{\,T\to0}$). Labels are the same as in \ref{['fig:OrderParameter']}.