Condensates, crystals, and renormalons in the Gross-Neveu model at finite density

TL;DR

The paper analyzes the $O(2N)$ Gross–Neveu model at finite density with a chemical potential $h$ for $a \le N-2$ fermions, developing a unified picture from perturbation theory, large-$N$ semiclassics, and integrability. It identifies two dynamical scales, $Λ_\mathrm{n}$ and $Λ_\mathrm{c}$, that govern neutral and charged sectors and drive nonperturbative corrections to the free energy, connecting condensates to renormalon phenomena. At large $h$ the ground state forms an inhomogeneous crystal with $\langle \sigma(x) \rangle \approx Λ_\mathrm{n} + Λ_\mathrm{c} \sin(2 h x^1)$, where $m_{\text{neutral}} = Λ_\mathrm{n}$ and $m_{\text{charged}} = (1/2) Λ_\mathrm{c}$; a massless mode (phonon) emerges in the crystal and persists as a quasi-long-range order in finite-$N$ systems. Integrability confirms these scales at finite $N$ and reveals a precise renormalon structure via a transseries in the condensation energy $\Delta F(h)$, with pole data $\xi_k$ and $\xi'_k$ governing nonperturbative terms and reproducing known results for special $a$ (e.g., $a=1$). Overall, the work explicitly links perturbative logs, condensate formation, and renormalon phenomena in a controlled 2d model, offering a template for similar analyses in other theories.

Abstract

We study the $O(2N)$ symmetric Gross-Neveu model at finite density in the presence of a $U(1)$ chemical potential $h$ for a generic number $a \leq N-2$ of fermion fields. By combining perturbative quantum field theory, semiclassical large $N$, and Bethe ansatz techniques, we show that at finite $N$ two new dynamically generated scales $Λ_\mathrm{n}$ and $Λ_\mathrm{c}$ appear in the theory, governing the mass gap of neutral and charged fermions, respectively. Above a certain threshold value for $h$, $a$-fermion bound states condense and form an inhomogeneous configuration, which at infinite $N$ is a crystal spontaneously breaking translations. At large $h$, this crystal has mean $Λ_\mathrm{n}$ and spatial oscillations of amplitude $2Λ_\mathrm{c}$. The two scales also control the nonperturbative corrections to the free energy, resolving a puzzle concerning fractional-power renormalons and predicting new ones.

Figures (5)

• Figure 1: One-loop diagrams that contribute to $\Gamma$. Vertices are slightly separated to track spinor contractions.
• Figure 2: The two-band structure of $\omega(p)$ in the Brillouin zone obtained from \ref{['eq:disp-rel']} for $y=2/3$ and $h= 0.85 \, \Lambda$ (dashed line).
• Figure 3: The profile of $\sigma(x)$ for $y=1/2$, for very low density ($h=1.0001 \, h_*$, in red), medium density ($h=0.95 \, \Lambda$, in purple) and high density ($h= 1.3 \, \Lambda$, in blue). The dotted line is the single bound state from Dashen:1975xh. Note how the high-density approximation \ref{['eq:sigma-osc']} extends up to values of $h\gtrsim \Lambda$.
• Figure 4: The sequence $L_k$ in \ref{['eq:seq-Lk']} for $N=5$ and $a=2$, for charged fermions/holes (in blue, with Richardson transform Bender1978 in orange) and neutral fermions (in green, with Richardson transform in red). The dashed and dotted lines correspond to $(2N-2)/a$ and $(N-1)/(N-a-1)$, respectively.
• Figure 5: Plot of $dp/d\omega$ for $N=102$, $a=68$, $h=1.5 \, \Lambda$. The points are the numerical solution of \ref{['eq:TBA-probe']} for charged fermions (in blue), charged holes (in cyan), and neutral fermions (in red). The dashed line is \ref{['eq:disp-rel']}.