Path Integral Monte Carlo on a Sphere

Abstract

We solve numerically exactly a simple toy model to quantum general relativity or more properly to path integral on a curved space. We consider the thermal equilibrium of a quantum many body problem on the sphere, the surface of constant positive curvature. We use path integral Monte Carlo to measure the kinetic energy, the internal energy and the static structure of a bosons, fermions and anyons fluid at low temperatures on the sphere. For bosons we also measure the superfluid fraction and compare its behavior at the critical temperature with the universal jump predicted by Nelson and Kosterlitz in flat space in the thermodynamic limit at the superfluid phase transition. For fermions and anyons it is necessary to use the restricted path integral recipe in order to overcome the sign problem. Even if this recipe is exact for the non interacting fluid it reduces to just an approximation for an interacting system. And we make the example of the electron gas at low temperature. Snapshots of the many body path configuration during the evolution of the computer experiment show that the ``speed'' of the single particle path near the poles slows down as a consequence of the ``hairy ball theorem'' of Poincaré. The influence of curvature on the thermodynamic and structural properties of the many body fluid is also studied.

Figures (7)

• Figure 1: Snapshot of the macroscopic path during the simulation for $N=10$ non interacting distinguishable particles with $M=50, a=5, \beta=10$. Case A in Table \ref{['tab:tq']}. The simulation started with all bodies distributed uniformly randomly on the equator. The different paths have different colors. In the left panel the top view and in the right panel the front view. In the simulation we measured $e_K=\langle{\cal K}\rangle=1.089(4)$. Reducing $\beta$ each path shrinks and tends to form a ring enclosing less amount of area.
• Figure 2: The radial distribution function for the non interacting distinguishable particles gas with $N=10$ on a sphere of radius $a=5$ at an inverse temperature $\beta=10$. We use $M=50$ (case A in Table \ref{['tab:tq']}) with only the displace move of appendix \ref{['app:dmove']} and with both the displace and the bridge move as described in Appendix \ref{['app:bmove']}. The dashed line is for $g(r)=1-1/N=0.9$.
• Figure 3: The supefluid fraction (\ref{['eq:sfarea']}) for the condensate of non interacting bosons with $N=10,a=5,M=50$. In this case $\tau\leq 0.4$ and $T_c=T_D\approx 0.0318$. The universal jump in $f_s$ which would be expected at the superfluid phase transition is $(4m^2T_c/\sigma)/2\pi\approx 0.636$.
• Figure 4: Snapshot of the macroscopic path during the simulation for $N=10$ non interacting bosons with $M=50, a=5, \beta=100/3$. Case B in Table \ref{['tab:tq']}. The simulation started with all bodies distributed uniformly randomly on the equator. Paths corresponding to different permutations cycles have different colors. In the left panel the top view and in the right panel the front view. In the simulation we measured $e_K=\langle{\cal K}\rangle=0.224(2)$. Reducing $\beta$ each path shrinks and tends to form a ring enclosing less amount of area.
• Figure 5: The radial distribution function for the non interacting bosons and fermions gas with $N=10$ on a sphere of radius $a=5$ at an inverse temperature $\beta=100/3$ with $M=50$. Case B and C in Table \ref{['tab:tq']} respectively. The dashed line is for $g(r)=1-1/N=0.9$. The bump at $r=0$ for bosons is a manifestation of their tendency to like themselves. The exchange hole at $r=0$ for fermions is a manifestation of their tendency to dislike themselves due to the Pauli exclusion principle and it requires $g(0)=0$.