The large $N$ vector model with angular velocity

Technical Summary: The Large N Vector Model with Angular Velocity

Problem Statement
This work investigates the thermodynamics of the critical $O(N)$ vector model in $(2+1)$ dimensions at large $N$ on the geometry $S^1_\beta \times S^2_r$, subject to an angular velocity $\hat{\mu}$ along the $z$-axis. The study focuses on the system without the singlet constraint. A primary motivation is to test recent conjectures regarding the behavior of Conformal Field Theories (CFTs) at finite temperature and angular velocity. Specifically, it is conjectured that the thermal free energy of CFTs develops a pole as the angular velocity approaches the speed of light (in units of the sphere radius), i.e., at $\hat{\mu}^2 r^2 = 1$. While this pole's existence and residue have been argued via thermal effective field theory and modular invariance in specific dimensions, explicit verification in interacting, non-trivial thermal CFTs at large $N$ remains a critical test.

Methodology
The authors employ the large $N$ expansion to analyze the critical $O(N)$ model. The methodology proceeds through the following steps:

1. Hubbard-Stratonovich Transformation: The quartic interaction of the $O(N)$ model is decoupled using an auxiliary field $\zeta$. In the strict large $N$ limit, the path integral over the auxiliary field reduces to a saddle-point approximation determined by its zero mode, $\zeta_0$. This introduces a thermal mass $\tilde{m}$ for the scalar fields.
2. Gap Equation Formulation: The thermal mass $\tilde{m}$ is determined by a gap equation derived from minimizing the free energy (or the effective action) with respect to $\tilde{m}$. The free energy is expressed as a sum of vacuum and thermal contributions, regularized using standard techniques.
3. Analytic Expansions: The authors derive analytic expansions for the free energy and the thermal mass in two distinct regimes:
   • Small Angular Velocity ($\hat{\mu}r \ll 1$): An expansion around $\hat{\mu}r = 0$ (equivalently $x^2 = 1 + \hat{\mu}^2 r^2 \approx 1$).
   • Critical Angular Velocity ($\hat{\mu}^2 r^2 \to 1$): An expansion around the pole location, defined by $x \to 0$ where $x^2 = 1 - \hat{\mu}^2 r^2$. This regime requires a non-perturbative treatment of the gap equation due to the non-analytic behavior of the thermal mass.
4. Numerical Analysis: The gap equation is solved numerically to interpolate between the two analytic limits, verifying the smoothness of the transition.
5. pp-Wave Geometry Check: To independently verify the residue of the pole at $\hat{\mu}^2 r^2 = 1$, the authors compute the partition function of the $O(N)$ model on a pp-wave geometry. This geometry is related to the $S^1 \times S^2$ limit via a specific scaling, allowing the extraction of the pole residue without solving the full gap equation on the sphere.

Key Results

• Free Theory Baseline: For the free scalar theory, the leading high-temperature free energy is dominated by a simple pole at $\hat{\mu}^2 r^2 = 1$ with a residue proportional to $\zeta(3)$.
• Critical $O(N)$ Model at $\hat{\mu}r \to 0$: At the non-trivial IR fixed point (critical point), the free energy expansion differs from the free theory. The leading term is $8/5 \zeta(3)$, and the coefficients of the expansion in $\hat{\mu}^2 r^2$ involve logarithmic terms dependent on the golden ratio $\rho_g = (1+\sqrt{5})/2$. The ratio of the critical free energy to the free theory free energy in this limit is $4/5$.
• Critical $O(N)$ Model at $\hat{\mu}^2 r^2 \to 1$:
  • Thermal Mass: The thermal mass $\tilde{m}$ exhibits non-analytic behavior, scaling as $\tilde{m}\beta \sim \log^2(1/x)$ where $x = \sqrt{1-\hat{\mu}^2 r^2}$.
  • Free Energy Pole: The free energy develops a simple pole at $\hat{\mu}^2 r^2 = 1$. Crucially, the residue of this pole coincides exactly with that of the free theory ($2\zeta(3)$).
  • Sub-leading Terms: Unlike the pole residue, the sub-leading terms in the expansion around $\hat{\mu}^2 r^2 = 1$ are non-analytic (involving $\log^6(1/x)$ and other logarithmic powers) and distinguish the critical fixed point from the free theory.
  • Interpolation: Numerical solutions confirm a smooth interpolation of the free energy ratio between the limits $\hat{\mu}r = 0$ (ratio $4/5$) and $\hat{\mu}^2 r^2 = 1$ (ratio $1$).
• pp-Wave Consistency: The calculation on the pp-wave geometry reveals that the only critical point for the large $N$ model in this background is the free theory ($m=0$). Consequently, the residue of the pole derived from the pp-wave limit must match the free theory result. This provides an independent consistency check confirming the residue found in the direct $S^1 \times S^2$ calculation.

Significance and Claims
The paper claims to provide further evidence for the conjecture that CFT partition functions develop poles as the angular velocity approaches unity. Specifically, it demonstrates that:

1. The critical $O(N)$ model, a non-trivial interacting thermal CFT, exhibits this pole behavior.
2. The residue of the pole at the critical fixed point is identical to that of the free theory, supporting the proposal that this residue can be extracted via the pp-wave geometry limit.
3. The non-analytic sub-leading terms in the free energy expansion near the pole serve as a distinguishing feature between the critical and free fixed points, originating from the non-analytic nature of the thermal mass solution to the gap equation.

The authors note that while the pole residue matches the free theory, the presence of non-analytic logarithmic terms in the sub-leading expansion suggests a need for generalizing thermal effective field theory techniques to incorporate such features. The work serves as a benchmark for thermal bootstrap programs and effective field theory methods in curved geometries with rotation.