Conformal Truncation of Chern-Simons Theory at Large $N_f$

TL;DR

The paper demonstrates that lightcone conformal truncation can be effectively applied to a 3d abelian Chern-Simons theory coupled to fermions in the large-$N_f$ limit, by truncating to primary fermion bilinears up to a cutoff and analytically diagonalizing the resulting Hamiltonian. A gauge-boson mass counterterm $m_a$ is introduced to cancel regulator-induced UV effects and preserve Lorentz and gauge invariance, enabling exact (in truncation) computation of current spectral functions that reproduce the known Lagrangian results as $Δ_{ m max} o ext{infty}$. An intermediate ${ m O}(N)$ scalar warm-up illustrates an optimal, separable basis for the interaction that yields analytic solutions, while the CS analysis at infinite $N_f$ confirms an exactly solvable two-state mixing problem for current correlators, with convergence rates quantified by integrated spectral densities. The work also discusses regulator-induced Lorentz symmetry breaking, convergence properties, and future extensions to finite $N_f$, non-Abelian CS theories, and finite-temperature/density settings, offering a path toward nonperturbative Hamiltonian studies of gauge-matter QFTs.

Abstract

We set up and analyze the lightcone Hamiltonian for an abelian Chern-Simons field coupled to $N_f$ fermions in the limit of large $N_f$ using conformal truncation, i.e. with a truncated space of states corresponding to primary operators with dimension below a maximum cutoff $Δ_{\rm max}$. In both the Chern-Simons theory, and in the $O(N)$ model at infinite $N$, we compute the current spectral functions analytically as a function of $Δ_{\rm max}$ and reproduce previous results in the limit that the truncation $Δ_{\rm max}$ is taken to $\infty$. Along the way, we determine how to preserve gauge invariance and how to choose an optimal discrete basis for the momenta of states in the truncation space.

Figures (4)

• Figure 1: Comparison with exact analytic result (red, dashed) and truncation result (black, solid) for various values of the coupling $\lambda$ and truncation level $\ell_{\rm max}$. The residuals (defined as $\frac{\rm truncation}{\rm exact}-1$) are shown in the insets.
• Figure 2: Left: Difference $\delta I_{--}$ between exact analytic result and truncation result at $\lambda=0$ and $q=3m_f$ as a function of truncation level $\ell_{\rm max}$. The black solid line is 0.1 $\ell_{\rm max}^{-3/2}$ for comparison. Right: Analogous plot, but for the integrated value of $\langle j_- | \tilde{D} | j_-\rangle$.
• Figure 3: Left: Dependence of $\log\left( -\log \Delta \tilde{\rho}_{\rm free}\right)$ versus $\log \ell_{\rm max}$, where $\Delta \tilde{\rho} \equiv \frac{\tilde{\rho}-\tilde{\rho}^{\rm num}}{\tilde{\rho}}$ and $\tilde{\rho}^{\rm num}$ is evaluated at finite $\ell_{\rm max}$, at $x = 0.2 + 0.01 i$. The linear best fit is shown above the plot. There is a trend $\Delta \tilde{\rho} \sim e^{- a L^b}$ with $b \sim 0.6$. Right: Dependence of $\log \Delta \tilde{\rho}$ versus $\ell_{\rm max}$ for $\lambda = 24$ and $x = 0.16 e^{0.2 i}$. Exponential best fit is shown above the plot.
• Figure 4: Symbolic Bethe-Salpeter equation for the large $N_f$ CS theory