New Easy-Plane $\mathbb{C}{\mathbb{P}}^{N-1}$ Fixed Points

We study fixed points of the easy-plane $\mathbb{C}{\mathbb{P}}^{N-1}$ field theory by combining quantum Monte Carlo simulations of lattice models of easy-plane $\mathrm{SU}(N)$ superfluids with field theoretic renormalization group calculations, by using ideas of deconfined criticality. From our simulations, we present evidence that at small $N$ our lattice model has a first-order phase transition which progressively weakens as $N$ increases, eventually becoming continuous for large values of $N$. Renormalization group calculations in $4-\epsilon$ dimensions provide an explanation of these results as arising due to the existence of an $N_{\mathrm{ep}}$ that separates the fate of the flows with easy-plane anisotropy. When $N<N_{\mathrm{ep}}$, the renormalization group flows to a discontinuity fixed point, and hence a first-order transition arises. On the other hand, for $N>N_{\mathrm{ep}}$, the flows are to a new easy-plane $\mathbb{C}{\mathbb{P}}^{N-1}$ fixed point that describes the quantum criticality in the lattice model at large $N$. Our lattice model at its critical point, thus, gives efficient numerical access to a new strongly coupled gauge-matter field theory.

Figure 1

First-order transitions for moderate values of $N$. The upper panels shows Monte Carlo histories (arbitrary units) of the estimator for $m_{\perp }^2$ for $N=6$ and 10. The bottom panel shows histograms of $m_{\perp }^2$ taken at $L=50$ for $J_2/J_1\equiv g=0.250$, 0.876, 1.58 for $N=6$, 8, 10, respectively, clearly showing double-peaked behavior. The double-peaked behavior persists in the $T=0$ and thermodynamic limit [16].

Figure 2

Scaling of the spatial winding number square $⟨W^2⟩$. (a) Crossing for $N=6$. (b) Crossing for $N=21$. (c) Value at the $L$ and $L/2$ crossing of $⟨W^2⟩$ for a range of $N$ normalized to the crossing value at $L=20$ for each $N$. For the smaller $N$, a clear linear divergence is seen as expected for a first-order transition (ergodicity issues limit the system sizes here). For larger $N$, a slow growth is observed very similar to what has been studied in detail for the SU(2) case and interpreted as evidence for a continuous transition with two length scales [25], like we have here. The data were taken at $\beta =6L$, which is in the $T=0$ regime [16].

Figure 3

RG flows of the ep-$\mathbb{C}{\mathbb{P}}^{N-1}$ model for (a) $N_s<N<N_{\mathrm{ep}}$ and (b) $N>N_{\mathrm{ep}}$ at leading order in $4-\epsilon$ dimensions obtained by numerical integration of Eq. (3). Fixed points are shown as bold dots; we have labeled only those significant to our discussion. The flows in the $v=0$ plane have been obtained previously [22] and include the “$s$” fixed point that describes DCP in $\mathrm{SU}(N)$ models (red dot). While the flows have many FPs [16], a DCP of the ep-$\mathrm{SU}(N)$ spin model must have all three eigendirections in the $e^2\text{−}u\text{−}v$ irrelevant. For $N<N_{\mathrm{ep}}$ there are no such FPs; there is hence a runaway flow to a first-order transition. For $N>N_{\mathrm{ep}}$ two FPs emerge: “$m$” is multicritical and “ep” is the new ep-DCP that describes the SF-VBS transition (yellow dot). The Gaussian fixed point at the origin has been labeled “$g$” for clarity. See [16] for further details.

Figure 4

Correlation ratios close to the phase transition for $N=21$. (a) The SF order parameter ratio $R_{m_{\perp }^2}$ shows good evidence for a continuous transition with a nicely convergent crossing point of $g=6.505(5)$. (b) $R_{\mathrm{VBS}}$ shows a crossing point that converges to the same value of the critical coupling. We note, however, that the crossing converges much more slowly (see the text). The inset shows the convergence of the crossings points of $L$ and $L/2$ of SF and VBS ratios. Note their convergence to a common critical coupling indicating a direct transition. The data shown in Figs. 4 and 5 were taken at $\beta =6L$, which is in the $T=0$ regime [16].

Figure 5

Data collapse for the SF order parameter at $N=21$. (a) Finite size data collapsed to $m_{\perp }^2=L^{-(1+{\eta }_{\mathrm{SF}})}\mathbb{M}[(g-g_c)L^{1/\nu }]$ with parameters $g_c=6.511$, $\nu =0.556$ ($1/\nu =1.795$), and $\eta =0.652$. (b) Collapse of ratio $R_{m_{\perp }^2}=\mathbb{R}[(g-g_c)L^{1/\nu }]$ with parameters $g_c=6.518$ and $\nu =0.582$ ($1/\nu =1.719$). The side panels show the convergence of estimates for various quantities from the collapse of $L$ and $L/2$ data: (c) the critical coupling $g_c=6.505(1)$ from pairwise collapses of $m_{\perp }^2$ and $R_{m_{\perp }^2}$, as well as crossings of $⟨W^2⟩$ (see Fig. 2) for data. Panel (d) shows $1/\nu (L)$, which we estimate to converge to $1/\nu =2.3(2)$. Likewise, we estimate $\eta (L)$ to converge to $\eta =0.72(3)$, which is shown in panel (e).