Simulating strongly correlated multiparticle systems in a truncated Hilbert space

Representing a strongly interacting multiparticle wave function in a finite product basis leads to errors. Simple rescaling of the contact interaction can preserve the low-lying energy spectrum and long-wavelength structure of wave functions in one-dimensional systems and thus correct for the basis set truncation error. The analytic form of the rescaling is found for a two-particle system where the rescaling is exact. A detailed comparison between finite Hilbert space calculations and exact results for up to five particles show that rescaling can significantly improve the accuracy of numerical calculations in various external potentials. In addition to ground-state energies, the low-lying excitation spectrum, density profile, and correlation functions are studied. The results give a promising outlook for numerical simulations of trapped ultracold atoms.

Figure 1

Relative deviation of the approximate ground-state energy from the exact Lieb-Liniger (LL) result [18] before rescaling (white area of panel a) and after rescaling (gray area of panels a and b) for five particles in a box of length $L$ with periodic boundary conditions. Rescaling reduces the maximum errors from 23% to 0.6% ($M=13$). The inset shows the relative energy deviation vs the number of modes, $M$, for three particles at $g=1.9099$. Results for the truncated Hamiltonian without rescaling (squares) converge very slowly while rescaling (circles) significantly reduces the errors already for small $M$.

Figure 2

Ground-state energies for two interacting bosons in a periodic box before (lines) and after (symbols) rescaling according to Eqs. (11) and (15) compared with the exact (Lieb-Liniger) energy (solid line). The calculated energies have been obtained via exact diagonalization of the truncated Hamiltonian matrix with different numbers of modes, $M$. Clearly, rescaling of the interaction reduces the errors in the energies significantly.

Figure 3

Density-density correlation function for five particles and $M=13$ single-particle functions in a periodic box. This plot compares the reduced two-particle density ${\rho }_2^{\left(g\right)}$ of Eq. (18) for different interaction strengths from MCTDH calculations without rescaling with the exact TG expression (the area below this curve is shaded). We find a good agreement with the exact solution close to $g_0=5.3958E_{0}L$. This is shown in the inset, where the overlap $F^{\left(g\right)}({\rho }_2^{\left(g\right)},{\rho }_2^{TG})$ of Eq. (19) is plotted as a function of the interaction $g$. There, the maximum occurs close to $g_0$.

Figure 4

Maximal relative deviation of the energy after rescaling from the exact result $E_{\text{LL}}$ over the range of interaction strengths $0<g<\infty$. The particle number and box length $L$ are varied while the density is kept fixed.

Figure 5

Density of five particles in a harmonic potential for different interaction strengths without rescaling and $M=13$. Analogous to Fig. 3, the best agreement with the exact TG result, which is given by the shaded areas, can be found at $g=g_0$, where $g_0=15.516E_{0}L$.

Figure 6

Energies of the ground state and the sixth to eighth excited states with $M=15$ of two bosons in a harmonic well after rescaling compared to the exact results from Eq. (21). The results clearly show the high accuracy of the rescaled results. The largest error occurs for the seventh excited state, where it is decreased from $\sim$2% before rescaling (not shown on this graph) to $\sim$0.3% after rescaling.

Figure 7

Excited states of the Tonks-Girardeau (TG) gas in a double-well trap. Shown are the energies of the ground state and the first four excited states minus the exact TG ground-state energy for five bosons in a double-well potential vs barrier height $h/E_{0,H}$. The graph compares rescaled MCTDH calculations with $M=13$ single-particle functions (solid line with bullets) and exact TG energies (dashed line).