Conformal Data for the O(3) Wilson-Fisher CFT from Fuzzy Sphere Realization of Quantum Rotor Model

Abstract

We present a model for strongly interacting fermions with internal O(3) symmetry on the fuzzy-sphere that (i) preserves the rotational symmetry of the fuzzy sphere and (ii) undergoes a quantum phase transition in the (2+1)-dimensional O(3) Wilson-Fisher universality class. Using exact diagonalization (ED) and density matrix renormalization group (DMRG), we locate the quantum critical point via conformal perturbation theory and obtain scaling dimensions from finite-size spectra. We identify 24 primary operators and determine some of their operator product expansion coefficients through first-order conformal perturbation theory. The results are benchmarked against conformal bootstrap and large quantum-number expansions and reveal a weakly irrelevant operator that plays a role in dimerized antiferromagnets. Our work establishes the fuzzy sphere as a general framework for quantitatively accessing conformal data in non-Abelian conformal field theories (CFTs).

Figures (4)

• Figure 1: Panel (a) shows the dependence on $h$ of the relevant coupling $g_{\varepsilon}(h)$ for a given $R\propto\sqrt{N}$. The critical point $h_{\mathrm{c}}$ is the root to $g_{\varepsilon}(h)=0$ after the thermodynamic limit $N\uparrow\infty$ has been taken. Panel (b) shows the dependence of the root to $g_{\varepsilon}(h)=0$ as a function of $1/\sqrt{N}\propto 1/R$. Fitting this dependence with an appropriate power law and using the first irrelevant anomalous dimension $0.7688$HENRIKSSON20231, we extract the critical value $h_{\mathrm{c}}=14.992\pm0.018$ and the critical exponent $1/\nu=1.3908\pm0.0852$, which is in close agreement with the known value $1.406$one_by_nu_hasenbuch.
• Figure 2: Dependence on $R^{-\omega}$ of the dimensionless number $\Delta_{\mathrm{o}}(R)$ defined by the scaling Ansatz (\ref{['eq:deltaE_o close to criticality']}) with $h$ chosen to be the root of $g_{\varepsilon}(h)=0$ for each value of $R$. Here, the value $\omega\equiv\Delta_{\varepsilon'}-3\approx0.7668$ is deduced from the CB estimate for the scaling dimension $\Delta_{\varepsilon'}$ of the leading irrelevant CFT perturbation (see Table \ref{['tab:bs_data_table']}). The limiting value $\Delta_{\mathrm{o}}$ of $\Delta_{\mathrm{o}}(R)$ as $R\uparrow\infty$ is interpreted as the scaling dimension of the CFT operator labeled by $\mathrm{o}$. The choice of the eigenstates labeled by $\mathrm{o}$ is organized by $S^{\pm}$ sectors across panels; and by angular momenta $L$ within each panel. Dashed lines indicate conformal bootstrap predictions with the color code green for primary operators and red for descendants. Solid lines represent exact CFT values as applicable to the stress-energy tensor and the conserved Noether current (using the same color code). Operator names, such as $\varepsilon$ and $\varepsilon_{\mu}$, are indicated in red boxes. Only selected sectors are shown here; the complete spectra are available in the Supplementary Material SuppMat. Panel (a) displays the $S=0^{+}$ sector, panel (b) the $S=1^{+}$ sector, panel (c) the $S=1^{-}$ sector, and panel (d) the $S=2^{+}$ sector. Note that, except for the $\sigma$ and $\partial_\mu \sigma$ levels used to determine $c$ and $h_{\mathrm{c}}$, all the plotted values of $\Delta_{\mathrm{o}}(R)$ as $R\uparrow\infty$ are parameter‑free outputs.
• Figure 3: Dependence on $R^{-\omega}$ with $\omega\equiv\Delta_{\varepsilon'}-3\approx0.7668$ of the coefficient $f_{\mathrm{o}\varepsilon\mathrm{o}}(R)= f_{\mathrm{o}\varepsilon\mathrm{o}} + \mathcal{O}(R^{-\omega})$ defined by Eq. (\ref{['eq:def fovarepsilono(R)']}) with $\mathrm{o}$ chosen in the $S=0^{\pm}$ symmetry sectors. Solid lines denote CB estimates. Dashed red lines include CPT-derived descendant factors applied to the bootstrap data Icosahedron. The states (CFT operators) selected by $\mathrm{o}$ include $\varepsilon$, $\varepsilon_{\mu}$, $T_{\mu\nu}$.
• Figure 4: (a) For each quantum number $S=1,2,\cdots$, two scaling dimensions are reported. The lowest one corresponds to the CFT state in the symmetry sector with the spin quantum number $S^{\kappa(S)}=1^{-},2^{+},\cdots$ with $\kappa(S)=(-1)^{S}$ for the internal symmetry group $\mathrm{O}(3)=\mathbb{Z}_{2}\times\mathrm{SO}(3)$ and the angular momenta $L=0$. The larger one corresponds to the CFT state with the quantum numbers $S^{\kappa(S+1)}=1^{+},2^{-},\cdots$ and $L=1$. The scaling dimensions obtained from ED and DMRG of $\widehat{H}_{\mathrm{fzs}}(R)$ and reported as $+(\times)$ for $\kappa(S)=+(-)$. For fuzzy sphere results, number of electrons is 26 (DMRG) for $S=1,2,3,4$ and 12 (ED) for $S=5,6,7$. The scaling dimensions obtained from fitting the large-$S$ expansions (\ref{['eq:appendix large S expansion lowest scalar primaries']}) and (\ref{['eq:appendix large charge expansion odd operator']}) with bootstrap data are reported as $\bigcirc(\square)$ for $\kappa(S)=+(-)$. (b) Scaling dimensions as a function of $R^{-\omega}$ with $\omega\approx0.7668$ for phonon primaries and their descendants obtained from ED of $\widehat{H}_{\mathrm{fzs}}(R)$ in the symmetry sector $S=2^{+}$. The green lines report the scaling dimensions obtained from fitting the large-$S$ expansion (\ref{['eq:appendix large charge expansion phonon modes']}) with bootstrap data.