Quantum Symmetry from Enhanced Sampling Methods

We address the problem of the minus sign sampling for two-electron systems using the path integral approach. We show that this problem can be reexpressed as one of computing free energy differences and sampling the tails of statistical distributions. Using metadynamics, a realistic problem like that of two electrons confined in a quantum dot can be solved. We believe that this is a strategy that can possibly be extended to more complex systems.

Figure 1

Schematic visualization of two particles in the path integral description. Each particle is represented by a necklace of beads connected by harmonic interactions. To describe indistinguishable particles, one needs to include contributions from both disconnected necklaces (a) and connected necklaces (b). The weight of the connected necklaces carries a positive sign for bosons and a negative sign for fermions.

Figure 2

Nonbiased simulation of the indistinguishable particles in a harmonic well: (a) pair distribution function for bosons (blue) and fermions (red). Our results are denoted by circles, the analytical results are the continuous lines. The squares are the result of a simulation in which exchange is ignored. (b) Logarithm of the $p_{oo}(s)$ distribution. (c) Plot of the integrand in $⟨W_I{⟩}_{oo}=\int p_{oo}(s)W_I(s)ds$ (the height of the peaks is irrelevant and has been rescaled for visualization purposes). The simulation has been conducted in the quantum regime $(\beta \hslash \omega =3)$.

Figure 3

Results of a metadynamics simulation for two particles in a harmonic potential at $\beta \hslash \omega =3$. For a definition of the symbols, see Fig. 2.

Figure 4

Total energy as function of temperature for two electrons in a quantum dot with $\hslash {\omega }_0=5.1\mathrm{meV}$, $R_W=1.34$, and $\eta =1.38$. Dashed lines are the energies obtained from exact diagonalization of the Hamiltonian. When not otherwise indicated, the error bars are smaller than the size of the dots. The computation was done from $10^8$ samples with 15 beads at $T=11.6\mathrm{K}$ and otherwise, such that $\beta /P$ was constant.

Figure 5

Electron density of singlet and triplet states for a two-particle quantum dot with $R_W=1.34$ for different ellipticities $\eta$ of the confining potential (reported in the figure). The third column shows the contribution to the triplet density given by the first excited single-particle state, computed as the difference between the triplet and singlet densities, ${\rho }_T-\frac{1}{2}{\rho }_S$. These results where computed from $10^8$ samples at $T=11.6\mathrm{K}$ with $P=15$ beads.