Tuning the Shape of the Condensate in Spontaneous Symmetry Breaking

We investigate the conditions which determine the shape of a particle condensate in situations when it emerges as a result of spontaneous breaking of translational symmetry. We consider a model with particles hopping between sites of a one-dimensional grid and interacting if they are at the same site or at neighboring sites. We predict the envelope of the condensate and the scaling of its width with the system size for various interaction potentials and show how to tune the shape from a delta peak to a rectangular or paraboliclike form.

Figure 1

Comparison between the envelope of the condensate from Eq. (17) and rescaled MC data for the weights (6) with $J=U=1$ for $N=1000$, $\rho =M/N=1$ and 3 (circles, squares) and $N=4000$, $\rho =1$, 3 (diamonds, triangles).

Figure 2

Phase diagram for $K(x)\sim e^{-x^{\beta }}$ and $p(m)\sim e^{-m^{\gamma }}$. The exponent ${\alpha }_{\mathrm{rect}}=(\beta -\gamma )/(\beta -\gamma +1)$ for rectangular and ${\alpha }_{\mathrm{smooth}}=(\beta -\gamma )/(2\beta -\gamma )$ for smooth condensates. The dotted lines show $\alpha =0.05,0.1,\dots ,0.45$ (from left to right).