Title: Duality of deconfined quantum critical point in two dimensional Dirac semimetals,
Abstract: In this paper we discuss the N$\acute{e}$el and Kekul$\acute{e}$ valence bond
solids quantum criticality in graphene Dirac semimetal. Considering the quartic
four-fermion interaction $g(\bar{\psi}_i\Gamma_{ij}\psi_j)^2$ that contains
spin,valley, and sublattice degrees of freedom in the continuum field theory,
we find the microscopic symmetry is spontaneously broken when the coupling $g$
is greater than a critical value $g_c$. The symmetry breaking gaps out the
fermion and leads to semimetal-insulator transition. All possible quartic
fermion-bilinear interactions give rise to the uniform critical coupling, which
exhibits the multicritical point for various orders and the Landau-forbidden
quantum critical point. We also investigate the typical critical point between
N$\acute{e}$el and Kekul$\acute{e}$ valence bond solid transition when the
symmetry is broken. The quantum criticality is captured by the
Wess-Zumino-Witten term and there exist a mutual-duality for
N$\acute{e}$el-Kekul$\acute{e}$ VBS order. We show the emergent spinon in the
N$\acute{e}$el-Kekul$\acute{e}$ VBS transition , from which we conclude the
phase transition is a deconfined quantum critical point. Additionally, the
connection between the index theorem and zero energy mode bounded by the
topological defect in the Kekul$\acute{e}$ VBS phase is studied to reveal the
N$\acute{e}$el-Kekul$\acute{e}$ VBS duality.