Color-flavor reflection in the continuum limit of two-dimensional lattice gauge theories with scalar fields

We address the interplay between local and global symmetries in determining the continuum limit of two-dimensional lattice scalar theories characterized by $\text{SO}\left(N_c\right)$ gauge symmetry and non-Abelian $\text{O}\left(N_f\right)$ global invariance. We argue that, when a quartic interaction is present, the continuum limit of these models corresponds in some cases to the gauged nonlinear $\sigma$ model field theory associated with the real Grassmannian manifold $\text{SO}\left(N_f\right)/(\text{SO}\left(N_c\right)\times \text{SO}(N_f-N_c)$), which is characterized by the invariance under the color-flavor reflection $N_c\leftrightarrow N_f-N_c$. Monte Carlo simulations and finite-size scaling analyses, performed for $N_f=7$ and several values of $N_c$, confirm the emergence of the color-flavor reflection symmetry in the scaling limit and support the identification of the continuum limit.

Figure 1

Graphical representation of the different low-temperature continuum limits expected from the study of the minimum-energy configurations when $N_f>N_c$. Minima of type (i) ($w<0$) and type (ii) ($w>0)$ are associated to the $\mathbb{R}{P}^{N_f-1}$ and Grassmannian models, respectively. The case $w=0$ is numerically found to have the same asymptotic behavior of the $w<0$ cases (see Ref. [11]). For $\gamma \to +\infty$ we recover the $\text{O}\left(N_f\right)$ vector model when $w<0$, the $\text{O}\left(N_c{N}_f\right)$ vector model when $w=0$, and the Stiefel model $V_{(N_f,N_c)}$ when $w>0$. If $N_f\le N_c$ no critical behavior is expected for $w>0$

Figure 2

Top: Binder cumulant $U$ versus $R_{\xi }$ for the case $N_c=2, N_f=7$ and quartic coupling $w=10$ ($\gamma =0$). The solid line is a polynomial interpolation of the $L=64$ and $L=128$ data, and it is our estimate of the asymptotic curve $\mathcal{U}\left(R_{\xi }\right)$ for the Grassmannian $\text{SO}\left(7\right)/\left(\text{SO}\right(2)\times \text{SO}(5\left)\right)$ (note that $L=64$ and $L=128$ data are consistent with each other). Bottom: Binder cumulant $U$ versus $R_{\xi }$ for the case $N_c=5, N_f=7$ and quartic coupling $w=10$ ($\gamma =0$). The solid line is the estimate of $\mathcal{U}\left(R_{\xi }\right)$ obtained from $N_c=2$ data.

Figure 3

Top: Binder cumulant $U$ versus $R_{\xi }$ for the case $N_c=3, N_f=7$ and quartic coupling $w=10$ ($\gamma =0$). The solid line is a polynomial interpolation of the $L=64$ and $L=128$ data. The dashed line is our estimate of $\mathcal{U}\left(R_{\xi }\right)$ for the $N_c=2, N_f=7$ model. Bottom: Binder cumulant $U$ versus $R_{\xi }$ for the case $N_c=4, N_f=7$ and quartic coupling $w=10$ ($\gamma =0$). The solid line is our estimate of $\mathcal{U}\left(R_{\xi }\right)$ for the $N_c=3, N_f=7$ model.

Figure 4

Binder cumulant $U$ versus $R_{\xi }$ for the case $N_c=6, N_f=7$ and quartic coupling $w=10$ ($\gamma =0$). For reference we also report our estimate of $\mathcal{U}\left(R_{\xi }\right)$ for spin-2 observables in the vector $\text{O}\left(7\right)$ model, obtained by a polynomial interpolation of data coming from $L=32$ and $L=64$ lattices (consistent with each other within statistical uncertainties). In the inset, scaling corrections are shown to be roughly consistent with an $L^{-1}$ behavior.

Figure 5

Binder cumulant $U$ versus $R_{\xi }$ for the Stiefel models $V_{(7,2)}$ and $V_{(7,5)}$.

Figure 6

Binder cumulant $U$ versus $R_{\xi }$ for the case $N_c=2, N_f=7$, quartic coupling $w=20$, and gauge coupling $\gamma =15$. The dashed line represents our estimate of $\mathcal{U}\left(R_{\xi }\right)$ for the model with $N_c=2, N_f=7, w=10$, and $\gamma =0$, while the dot-dashed line is an estimate of $\mathcal{U}\left(R_{\xi }\right)$ for the Stiefel model $V_{(7,2)}$.