Emergent Potts order in the kagom\'e $J_1-J_3$ Heisenberg model

Motivated by the physical properties of Vesignieite BaCu$_3$V$_2$O$_8$(OH)$_2$, we study the $J_1-J_3$ Heisenberg model on the kagom\'e lattice, that is proposed to describe this compound for $J_1<0$ and $J_3\gg|J_1|$. The nature of the classical ground state and the possible phase transitions are investigated through analytical calculations and parallel tempering Monte Carlo simulations. For $J_1<0$ and $J_3>\frac{1+\sqrt{5}}4|J_1|$, the ground states are not all related by an Hamiltonian symmetry. Order appears at low temperature via the order by disorder mechanism, favoring colinear configurations and leading to an emergent $q=4$ Potts parameter. This gives rise to a finite temperature phase transition. Effect of quantum fluctuations are studied through linear spin wave approximation and high temperature expansions of the $S=1/2$ model. For $J_3$ between $\frac14|J_1|$ and $\frac{1+\sqrt{5}}4|J_1|$, the ground state goes through a succession of semi-spiral states, possibly giving rise to multiple phase transitions at low temperatures.