Corrections to scaling in the critical theory of deconfined criticality

Inspired by recent conflicting views on the order of the phase transition from an antiferromagnetic Néel state to a valence bond solid, we use the functional renormalization group to study the underlying quantum critical field theory which couples two complex matter fields to a noncompact gauge field. In our functional renormalization group approach, we only expand in covariant derivatives of the fields and use a truncation in which the full field dependence of all wave-function renormalization functions is kept. While we do find critical exponents which agree well with some quantum Monte Carlo studies and support the scenario of deconfined criticality, we also obtain an irrelevant eigenvalue of small magnitude, leading to strong corrections to scaling and slow convergence in related numerical studies.

Figure 1

Graphical representation of the flow equations of the effective potential $U_{\ell }\left(\rho \right)$ and the wave-function renormalization functions $Z_{\ell }\left(\rho \right)$, ${\tilde{Z}}_{\ell }\left(\rho \right)$, and $Z_{\ell }^A\left(\rho \right)$, as mathematically described by Eqs. (A5) and (A9)–(A11) in Appendix . While the longitudinal and transverse propagators are depicted by solid and dashed lines, respectively, the gauge field propagator is represented by a wiggly line. The crosses in the flow of $U_{\ell }\left(\rho \right)$ stand for regulator insertions. The regulator insertions in the flow equations for the wave-function renormalization functions are generated by the derivative operator ${\tilde{\partial }}_{\ell }$ which only acts on regulator terms.

Figure 2

Derivative $w^{*}\left(\tilde{\rho }\right)=d{u}^{*}\left(\tilde{\rho }\right)/d\tilde{\rho }$ of the rescaled effective potential and field-dependent wave-function renormalization functions $z^{*}\left(\tilde{\rho }\right)$, ${\tilde{z}}^{*}\left(\tilde{\rho }\right)$, and $z^{A*}\left(\tilde{\rho }\right)$ at the charged critical point for $N=2$.

Figure 3

Evolution of the rescaled charge ${\tilde{e}}_{\ell }^2$, the anomalous dimensions of the matter and gauge field ${\eta }_{\ell }$ and ${\eta }_{A,\ell }$, the coupling constant ${\tilde{\rho }}_{\ell }^{*}$ denoting the position of the minimum of the rescaled effective potential, and ${\tilde{\lambda }}_{\ell }$ for $N=1$. Using a bisection method, we fine tune the initial value ${\tilde{\rho }}_0^{*}$ (or equivalently ${\tilde{r}}_0=-{\tilde{\lambda }}_0{\tilde{\rho }}_0^{*}$) to the value ${\tilde{\rho }}_0^{c*}$ leading to criticality. Choosing ${\tilde{\rho }}_0={\tilde{\rho }}_0^{c*}\pm {10}^{-10},{\tilde{\rho }}_0^{c*}\pm {10}^{-12},{\tilde{\rho }}_0^{c*}\pm {10}^{-14}$, we obtain an exponential runaway flow for large $\ell$, as indicated by the light-colored lines.

Figure 4

Evolution of ${\tilde{e}}_{\ell }^2$, ${\eta }_{\ell }$, ${\eta }_{A,\ell }$, ${\tilde{\rho }}_{\ell }^{☆}$, and ${\tilde{\lambda }}_{\ell }$, for initial values chosen as in Fig. 3, but for $N=2$, at and near criticality. In contrast to the case $N=1$, the critical point is approached much more slowly.