Harmonically confined long-ranged interacting gas in the presence of a hard wall

In this paper, we compute exactly the average density of a harmonically confined Riesz gas $N$ particles for large in presence hard wall. gas, repel each other via pairwise interaction that behaves as $|x_i - x_j|^{-k}$ $k>-2$, with $x_i$ denoting position $i^{\rm th}$ particle. This can be classified into three different regimes $k$. For $k \geq 1$, where interactions are effectively short-ranged, appropriately scaled has finite support over $[-l_k(w),w]$ $w$ is While vanishes at left edge support, it approaches nonzero constant right $w$. $-1<k<1$, weakly long-ranged, find again supported $[-l_k(w),w]$. still diverges algebraically an exponent $(k-1)/2$. $-2<k< -1$, strongly long-ranged leads to rather exotic profile extended bulk part and delta-peak wall, separated by hole between. Exactly $k=-1$ disappears. interesting first-order phase transition when wall decreases through critical value $w=w^*(k)$. $w<w^*(k)$, pure located The amplitude plays role order parameter which jumps $1$ decreased $w^*(k)$. Our analytical results very good agreement our Monte-Carlo simulations.