Star junctions and watermelons of pure or random quantum Ising chains : finite-size properties of the energy gap at criticality

We consider $M \geq 2$ pure or random quantum Ising chains of $N$ spins when they are coupled via a single star junction at their origins or when they are coupled via two star junctions at the their two ends leading to the watermelon geometry. The energy gap is studied via a sequential self-dual real-space renormalization procedure that can be explicitly solved in terms of Kesten variables containing the initial couplings and and the initial transverse fields. In the pure case at criticality, the gap is found to decay as a power-law $\Delta_M \propto N^{-z(M)} $ with the dynamical exponent $z(M)=\frac{M}{2}$ for the single star junction (the case $M=2$ corresponds to $z=1$ for a single chain with free boundary conditions) and $z(M)=M-1$ for the watermelon (the case $M=2$ corresponds to $z=1$ for a single chain with periodic boundary conditions). In the random case at criticality, the gap follows the Infinite Disorder Fixed Point scaling $\ln \Delta_M = -N^{\psi} g$ with the same activated exponent $\psi=\frac{1}{2}$ as the single chain corresponding to $M=2$, and where $g$ is an $O(1)$ random positive variable, whose distribution depends upon the number $M$ of chains and upon the geometry (star or watermelon).