the corresponding single-particle Hamiltonian $$H_\mathrm{SP}= -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}+V_{\mathrm{ext}}(x),
\label{eq:SP}$$ for which exact values can easily be obtained numerically using finite-difference diagonalization. The bosonic many-body wavefunction can then be obtained from the fermionic one by symmetrising with an anti-symmetric unit function of the form $A(x_1,...,x_N)= \Pi_{1\leq i <  j \leq N} \mathrm{sgn}(x_i-x_j)$. This results in $$\Psi_B(x_1,\dots,x_N)=A(x_1,\dots,x_N)\Psi_F(x_1,\dots,x_N).$$

We chose the external potential to be a periodic lattice of the form $V_\mathrm{ext}(x_n)=V_0 \cos^2(k_l x_n)$, where $V_0$ is the lattice depth and $k_l$ the wavevector. We consider this trapping potential to possess periodic boundary conditions such that it realizes a 1D ring trap of length $L$ with regular intensity modulations and $\Psi(...x_n+L...)=\Psi(...x_n...)$ for any $n$. This allows us to write the lattice wavevector in terms of the number of wells in the lattice potential $M$ as $k_l =  \pi M/L$ and provides an energy scale for the system in terms of the recoil energy $E_r=\frac{\hbar^2 \pi^2 M^2}{2 m L^2}$. For consistency we will use systems with $M=50$ lattice sites for all calculations shown in this manuscript.

It is important to note that the requirement to have periodic boundary conditions for the bosonic TG gas means that the fermionic many-body wavefunction must have periodic boundary conditions 