Generalizing Deconfined Criticality to 3D $N$-Flavor $\mathrm{SU}(2)$ Quantum Chromodynamics on the Fuzzy Sphere

TL;DR

The paper investigates a 3D conformal window for SU$(2)$ QCD with $N$ fermion flavors by mapping to an $ ext{Sp}(N)$-symmetric NLSM with a level-1 WZW term realized on a fuzzy sphere. Using sign-problem-free auxiliary-field QMC and LLL projection, it accesses large $N$ (up to $N=16$) and identifies a continuous transition between a symmetry-broken phase and a conformal QCD phase for $N eq2$, with emergent conformal symmetry evidenced by conformal correlators and the state-operator spectrum. The extracted scaling dimensions satisfy $oldsymbol extDelta_oldsymbol extphi=1.10(1)$ for $ ext{Sp}(4)$ and $1.75(2)$ for $ ext{Sp}(10)$, in agreement with large-$N$ expectations, and the operator multiplet structure follows $oldsymbol extDelta_{A,l}=oldsymbol extDelta_oldsymbol extphi+l$ and $oldsymbol extDelta_{T,l}=1+l$. The results place the conformal window boundary between $2<N_c<4$, offering quantitative insight into deconfined criticality beyond $N=2$ and highlighting the fuzzy-sphere approach as a powerful, sign-problem-free tool for probing 3D CFTs and gauge dynamics.

Abstract

The infra-red behaviour of gauge theories coupled to matter remains an open problem in quantum field theory. For a given gauge group, such theories are expected to flow to an interacting conformal fixed point over a range of fermion or scalar flavours, known as the `conformal window.' Their nature is important for understanding critical phases and phase transitions beyond the Landau paradigm like the deconfined quantum critical point (DQCP), yet remains challenging for conventional non-perturbative approaches. In this work, we study a family of fuzzy-sphere models corresponding to non-linear sigma models with $\mathrm{Sp}(N)$ global symmetry extended to the strongly-coupled region. These theories are expected have an infra-red fixed point described by $\mathrm{SU}(2)$ quantum chromodynamics (QCD) in three space-time dimensions with $N$ flavours of fermions. They can be viewed as a generalisation of the $\mathrm{SO}(5)$ DQCP, corresponding to $N=2$. We investigate them using quantum Monte Carlo for $N$ up to $16$. We find evidence that for $N\geq4$ the phase diagram contains a critical phase that appears to be absent for $N=2$. Within this phase, we measure the two-point correlation function and the excitation spectrum, which exhibit emergent conformal symmetry. We also extract the scaling dimension $Δ_φ$ of a leading operator and find consistency with large-$N$ expectations.

Figures (10)

• Figure 1: The putative RG-flow diagram of the NLSM extended to the strongly-coupled region. The colour and the intervals between arrows mark the rate of the RG flow. The red and black bars denote the conformal and non-conformal fixed points.
• Figure 2: The equal-time correlation functions in the $\mathrm{Sp}(10)$ model at (a) $V=0.05$ in the spontaneous-symmetry-broken phase and (b) $V=3.0$ in the critical QCD phase. The linear distance is taken as $r=2R\sin(\gamma_{12}/2)$. The values of $C^a(r)$ are scaled so that $C^a(0)=1$ for each of the system sizes. The red dashed line in (b) marks the a power law fit to data for $r>2.5$.
• Figure 3: Main: RG-invariant observables $\mathcal{C}_A(\pi) / \mathcal{C}_A(\pi/2)$ as a function of the coupling $V$ for $\mathrm{Sp}(10)$, $\mathrm{Sp}(4)$, and $\mathrm{Sp}(2)$ models. Insets of (b) and (c): The $V$-value at which the RG-invariant observable $\mathcal{C}_A(\pi) / \mathcal{C}_A(\gamma'_{12}=m\pi/12)$ has a crossing as a function of $1/\sqrt{N_{\mathrm{orb}}}$ for pairs $(N_{\mathrm{orb}}-2,N_{\mathrm{orb}})$.
• Figure 4: The conformal correlator $\mathcal{C}_A(\gamma_{12})$ and the extracted scaling dimension $\Delta_\phi$ for (a--c) the $\mathrm{Sp}(10)$ model and (d--f) the $\mathrm{Sp}(4)$ model. (a,d) The real-space correlator \ref{['eq:conf_corr']}$\mathcal{C}_A(\gamma_{12})$ of the density operator $n_A$ as a function of the angular distance $\gamma_{12}$, measured at (a) $V=10$ and (d) $V=12$. The dashed lines mark the best-fit conformal 2-pt function. Insets of (a,d). The scaling dimension extracted through Eq. \ref{['eq:dim_corr']} as a function of $\gamma_{12}$. The dashed line marks the best-fit conformal 2-pt function. (b,e) The scaling dimension $\Delta_\phi$ as a function of $V$ extracted through $\gamma_{12}$-dependence \ref{['fssscale']}. (c,f) The scaling dimension $\Delta_\phi$ as a function of $V$ extracted through $R$-dependence \ref{['eq:dim_corr']}. The dashed grid-line marks and the grey shade marks the finite-size scaling result of $\Delta_\phi$ and its error-bar (see Appendix \ref{['app:fit_dim']}). The red grid-line marks the unitarity bound $\Delta=1/2$.
• Figure 5: The scaling dimension $\Delta_\phi$ as a function of $N$ calculated from the conformal correlator. The dashed line is the large-$N$ expansion result Xu2008LargeN.