Fixed-phase correlation-function quantum Monte Carlo calculations for ground and excited states of helium in neutron-star magnetic fields

We apply the correlation-function quantum Monte Carlo (CFQMC) method to the calculation of the energies of ground and excited states for helium in neutron-star magnetic fields. The method has been successfully applied by Jones, Ortiz, and Ceperley to the calculation of helium in white dwarf magnetic fields [Phys. Rev. E 55, 6202 (1997)]. We extend the accessible range of magnetic field strengths by introducing a fixed-phase variant of the CFQMC method. We find that with growing magnetic field strength the variances increase significantly and put a limit to the applicability of the method for atoms in strong magnetic fields. The behavior of the variances is traced back to the logarithmic divergence of the energy of the bosonic ground state with increasing magnetic field strength. We use basis sets, which account for the growing dominance of the cylindrical symmetry as the magnetic field is increased and incorporate them into the CFQMC algorithm. These basis sets are taken from Hartree-Fock calculations, performed using a $B$-Spline and Landau expansion beyond the adiabatic approximation.

Figure 1

Fixed-phase and released-phase (inset) correlation-function quantum Monte Carlo calculation for the second excited state of the (${\pi }_z=+,L_z=-1$) symmetry subspace of neutral helium in a magnetic field of strength $\beta =8$. In this range the variance is too high for the released-phase method to gain results. See the text for an interpretation of the released-phase result. The fixed-phase calculation has the same parameters and the same sequence of random numbers as the released-phase (inset). The statistical noise is significantly lower, one notes a fast relaxation to the true energy, which thus can be obtained successfully. The black line is the eigenvalue corresponding to the state, estimated at projection times each $0.005$ hartree${}^{-1}$. The green area marks the estimated $\pm 1\sigma$ area. The horizontal blue line is the variational quantum Monte Carlo energy of the two-dimensional Hartree-Fock-Roothaan trial function the correlation-function quantum Monte Carlo method starts from within its $\pm 1\sigma$ area of confidence. The calculation was carried out with 12 million correlation-function quantum Monte Carlo steps per walker with $\Delta \tau ={10}^{-4}$ hartree${}^{-1}$, an equilibration of $800000$ VQMC steps per walker, an ensemble of 2560 walkers and the $c$ parameters for the guiding function being $\{0.2,1.5,2.25,3.375\}$. Estimates of the eigenvalues have been generated every $0.005$ hartree${}^{-1}$, the line serves as a guide to the eye.

Figure 2

Convergence properties of the fixed-phase correlation-function quantum Monte Carlo method (top) compared with the released-phase method (bottom). The figure shows a calculation for the ground state of the $({\pi }_z=-,L_z=0)$ symmetry subspace for neutral helium at $\beta =21.27$. Intermediate results taken at 2 million steps, 6 million steps, and 12 million steps are shown. The horizontal line is the variational quantum Monte Carlo result of the two-dimensional Hartree-Fock-Roothaan trial function used.

Figure 3

Energies of the bosonic (absolute) and the fermionic ground state over increasing magnetic field strength. The bosonic ground state has the (field-free) electronic configuration $1s^2$, the fermionic $1s2p_{-1}$, which is the ground state of helium for the displayed range of magnetic field strength. One easily sees the increasing difference proportional to $\sqrt{\beta }$, which poses a problem on the CFQMC method, as this energy difference increases variance. For excited fermionic states, energy difference is even larger. Results for individual magnetic field strengths were calculated using DQMC, the lines serve as guides to the eye.