Quantum Monte Carlo studies of the ground states of heavy atoms in neutron-star magnetic fields

The “released-phase” diffusion quantum Monte Carlo method is used to calculate the ground-state energies of atoms with nuclear charges from $Z=2,3,4,\dots ,26$ for magnetic field strengths relevant for neutron stars. The feature of our study is the use of adiabatic approximation wave functions as guiding wave functions to initialize the quantum Monte Carlo procedure. Our calculations are motivated by the discovery of broad features in the thermal spectra of isolated neutron stars, which may be due to heavy atoms. Our results confirm previous results for nuclear charge numbers up to 10, and are the most accurate ones available in the literature to date for $Z>10$.

Figure 1

Nassi-Shneiderman diagram of the released-phase DQMC method. The method is initialized by a variational quantum Monte Carlo simulation for the guiding wave function. The index $b$ counts the number of calculations (“blocks”), in each block $s$ steps are carried out, $j$ is the walker index. In our simulations the total number of blocks was 700, the number of steps 200, and 500 walkers were used. The averaging over the block energies only starts at block $b_s$, where the walkers have reached dynamical equilibrium. The $m=1$ flag takes care of the rare cases where because of very low acceptance probabilities a walker becomes locked in some part of configuration space. Setting $P_B=0.9$ offers the possibility of either deleting or keeping this walker.

Figure 2

(Color online) Ground state energy $E_{\mathrm{VQMC}}$ and the standard deviation $\sigma$ of the local energy in dependence of the Jastrow parameter $b^{\mathrm{JF}}$ for neon $(Z=10)$ and $B={10}^8\mathrm{T}$. $E(b^{\mathrm{JF}}\to \infty )$ (upper horizontal line) corresponds to the case where no Jastrow factor is present, i.e., to the energy value in adiabatic approximation. $E_{\mathrm{RPDQMC}}$ (lower horizontal line) is the final result of the released-phase simulation.

Figure 3

(Color online) Behavior of the block energy $E_B$ (ragged curve) and the averaged block energy $⟨E_B⟩$ (smooth curve) in the DQMC simulation for the ground state energy of neon $(Z=10)$ at $B={10}^7\mathrm{T}$ as a function of the number of blocks. The simulation is divided in three stages, indicated by the black vertical lines: a VQMC, a FPDQMC, and a RPDQMC simulation. The value at the end of the simulation is $E_0=-4.675\mathrm{keV}$. The dashed vertical lines indicate the blocks where dynamical equilibrium of the walkers is reached. In each block, 200 time steps $\mathrm{\Delta }\tau ={10}^{-4}\text{a.u.}$ were performed. [HFFEM (top horizontal line), energy value in adiabatic approximation; 2DHF (bottom horizontal line), energy value of Ivanov and Schmelcher [29].]

Figure 4

(Color online) Same as Fig. 3, but for the ground state energy of iron $(Z=26)$ at $B={10}^8\mathrm{T}$. The value at the end of the simulation is $E_0=-59.366\mathrm{keV}$. Here in each block 200 time steps $\mathrm{\Delta }\tau =2\times {10}^{-5}\text{a.u.}$ were performed. [HFFEM (top horizontal line), energy value in adiabatic approximation; DF (second horizontal line from top), density-functional result of Jones [9].]