Quantum Monte Carlo methods for nuclear physics

Quantum Monte Carlo methods have proved valuable to study the structure and reactions of light nuclei and nucleonic matter starting from realistic nuclear interactions and currents. These ab initio calculations reproduce many low-lying states, moments, and transitions in light nuclei, and simultaneously predict many properties of light nuclei and neutron matter over a rather wide range of energy and momenta. The nuclear interactions and currents are reviewed along with a description of the continuum quantum Monte Carlo methods used in nuclear physics. These methods are similar to those used in condensed matter and electronic structure but naturally include spin-isospin, tensor, spin-orbit, and three-body interactions. A variety of results are presented, including the low-lying spectra of light nuclei, nuclear form factors, and transition matrix elements. Low-energy scattering techniques, studies of the electroweak response of nuclei relevant in electron and neutrino scattering, and the properties of dense nucleonic matter as found in neutron stars are also described. A coherent picture of nuclear structure and dynamics emerges based upon rather simple but realistic interactions and currents.

Figure 1

Three-nucleon force diagrams for (a) two-pion $P$-wave, (b) two-pion $S$-wave, and (c), (d) three-pion ring terms.

Figure 2

GFMC energies of four $5/2^{-}$ states in ${}^7\mathrm{Li}$ vs imaginary time $\tau$. The solid symbols show the computed energies at each $\tau$, and the open symbols show the results of rediagonalization.

Figure 3

GFMC energies of light nuclear ground and excited states for the AV18 and $\mathrm{AV}18+\mathrm{IL}7$ Hamiltonians compared to experiment. See Table 1 for references.

Figure 4

GFMC excitation energies of light nuclei for the AV18 and $\mathrm{AV}18+\mathrm{IL}7$ Hamiltonians compared to experiment. See Table 1 for references.

Figure 5

GFMC point proton (up triangles) and neutron (down triangles) densities (upper panel) and magnetic spin densities (lower panel) for the chain of lithium isotopes; also shown are proton magnetic orbital density (diamonds), and total magnetic density in IA (circles). From [272].

Figure 6

GFMC $pp$ densities for the chain of lithium isotopes. From [272].

Figure 7

VMC proton momentum distributions in $T=0$ light nuclei. From [278].

Figure 8

VMC $pn$ (diamonds) and $pp$ (circles) back-to-back ($Q=0$) pair momentum distributions for $T=0$ nuclei. From [278].

Figure 9

VMC proton-proton momentum distributions in ${}^4\mathrm{He}$ averaged over the directions of $\mathbf{q}$ and $\mathbf{Q}$ as a function of $q$ for several fixed values of $Q$ from 0 to $1.25{\mathrm{fm}}^{-1}$. From [278].

Figure 10

VMC proton-neutron momentum distributions in ${}^4\mathrm{He}$ averaged over the directions of $\mathbf{q}$ and $\mathbf{Q}$ as a function of $q$ for several fixed values of $Q$ from 0 to $1.25{\mathrm{fm}}^{-1}$. From [278].

Figure 11

VMC and GFMC calculations of the $⟨{}^6\mathrm{He}(0^{+})+p(p_{3/2})|{}^7\mathrm{Li}({\frac{3}{2}}^{-})⟩$ overlap; see text for details. From [31].

Figure 12

The ratios $C(r)$ of the VMC and GFMC $⟨{}^6\mathrm{He}(0^{+})+p(p_{3/2})|{}^7\mathrm{Li}({\frac{3}{2}}^{-})⟩$ overlaps to the asymptotic Whittaker function; see text for details. From [31].

Figure 13

Overlaps for various bound states as computed by (1) VMC sampling (points with error bars), (2) a bound-state integral relation with the VMC as input but imposing experimental separation energies (solid curves) evaluated by [176], and (3), GFMC overlaps (dashed curves) from [31].

Figure 14

Phase shifts for $n-\alpha$ scattering. Solid symbols (with statistical errors smaller than the symbols) are GFMC results, dashed curves are polynomial fits, and solid curves are from an $R$-matrix fit to data. From [177].

Figure 15

GFMC ${}^4\mathrm{He}$ binding energies at LO, NLO, and ${\mathrm{N}}^2\mathrm{LO}$ compared with experiment (dashed line) and with the Argonne ${\mathrm{AV}8}^{\prime }$ energy. Also shown is a first-order perturbation-theory calculation of the ${\mathrm{N}}^2\mathrm{LO}$ binding energy using the NLO wave function. From [151].

Figure 16

Diagrams illustrating one- and two-body electromagnetic current operators at ${(P/{\mathrm{\Lambda }}_{\chi })}^{-2}$ (LO), ${(P/{\mathrm{\Lambda }}_{\chi })}^{-1}$ (NLO), ${(P/{\mathrm{\Lambda }}_{\chi })}^0$ (${\mathrm{N}}^2\mathrm{LO}$), and ${(P/{\mathrm{\Lambda }}_{\chi })}^1$ (${\mathrm{N}}^3\mathrm{LO}$). Nucleons, pions, and photons are denoted by solid, dashed, and wavy lines, respectively. The square in (d) represents the relativistic correction to the LO one-body current, suppressed relative to it by an additional ${(P/{\mathrm{\Lambda }}_{\chi })}^2$ factor; the solid circle in (j) is associated with a $\gamma \pi N$ vertex in $H_{\gamma \pi N}$ involving the low-energy constants (LECs) $d_8^{\prime }$, $d_9^{\prime }$, and $d_{21}^{\prime }$; the solid circle in (k) denotes two-body contact terms of minimal and nonminimal nature, the latter involving the LECs $C_{15}^{\prime }$ and $C_{16}^{\prime }$. Only one among all possible time orderings is shown for the NLO and ${\mathrm{N}}^3\mathrm{LO}$ currents, so that both direct- and crossed-box contributions are retained.

Figure 17

The ${}^6\mathrm{Li}$ longitudinal elastic (upper left panel), inelastic (bottom left), and transverse elastic (upper right), and inelastic (bottom right) calculated with VMC in the impulse approximation (IA), and with the addition of MEC contributions ([277]). The results are compared to the experimental data indicated in the legend. See [277] and references therein.

Figure 18

The longitudinal elastic form factor of ${}^{12}\mathrm{C}$ including one- (empty circles) and one- plus two-body operators (filled circles) calculated with GFMC. The results are compared to the experimental data. From [146].

Figure 19

GFMC propagated energy vs imaginary time for the first two $0^{+}$ states of ${}^{12}\mathrm{C}$.

Figure 20

VMC and GFMC point-proton densities for the first two $0^{+}$ states of ${}^{12}\mathrm{C}$. The experimental band was unfolded from electron scattering data. From [64].

Figure 21

VMC and GFMC $E0$ transition form factors between the first two $0^{+}$ states of ${}^{12}\mathrm{C}$ in the impulse approximation. From [57].

Figure 22

Magnetic moments in nuclear magnetons for $A\le 10$ nuclei. Stars indicate the experimental values ([252], [253]), while dots (diamonds) represent GFMC calculations which include the IA one-body electromagnetic current (full $\chi \mathrm{EFT}$ current up to ${\mathrm{N}}^3\mathrm{LO}$); asterisks denote first excited states. From [192] and [189].

Figure 23

Ratio of calculated to experimental $M1$, $E2$ ([192]), and GT reduced transition probabilities ([195]) in $A\le 9$ nuclei. Symbols are as in Fig. 22.

Figure 24

The longitudinal sum rule of ${}^{12}\mathrm{C}$ obtained with GFMC from the $\mathrm{AV}18+\mathrm{IL}7$ Hamiltonian with one-body only (empty circles, dashed line) and one- and two-body (solid circles, solid line) terms in the charge operator is compared to experimental data without (empty squares), and with (solid squares), the tail contribution. From [146].

Figure 25

Same as in Fig. 26, but for the transverse sum rule. The open symbols do not contain derivative terms while a VMC evaluation of the derivative terms is included for the solid dots. The inset shows $S_T(q)/C_T$ in the small-$q$ region. From [146].

Figure 26

The GFMC sum rules $S_{\alpha \beta }$ in ${}^{12}\mathrm{C}$, corresponding to the $\mathrm{AV}18+\mathrm{IL}7$ Hamiltonian and obtained with one-body only (dashed lines) and one- and two-body (solid lines) terms in the NC. From [147].

Figure 27

The GFMC $S_{xx}/C_{xx}$ sum rules obtained with the NC (curves labeled NC) and either its vector (curves labeled VNC) or axial-vector (curves labeled ANC) parts only. The corresponding one-body (one- and two-body) contributions are indicated by dashed (solid) lines. Note that the normalization factor $C_{xx}$ is not included. From [147].

Figure 28

The longitudinal (upper panel) and transverse (lower panel) electromagnetic Euclidean responses for ${}^{12}\mathrm{C}$ at $q=570\mathrm{MeV}/c$. The bands represent the transform of the experimental data, and the calculations with single-nucleon and two-nucleon currents are shown as open and filled symbols, respectively.

Figure 29

The longitudinal (upper panel) and transverse (lower panel) electromagnetic response of ${}^4\mathrm{He}$ at $q=600\mathrm{MeV}/c$ reconstructed from the Euclidean response compared to experimental data. The experimental results are shown as symbols with error bars, and the bands show the reconstructed responses and errors associated with the maximum entropy reconstruction. From [148].

Figure 30

The neutral current weak response of ${}^{12}\mathrm{C}$ at $q=570\mathrm{MeV}/c$. Calculations with single-nucleon currents are shown as open symbols and with the full currents as filled symbols. The upper panel shows the transverse response and its vector-vector and axial-axial contributions, while the lower panel shows the interference vector-axial-vector response.

Figure 31

The EoS of neutron matter as a function of the density, obtained using the $\mathrm{AV}{8}^{\prime }$ $NN$ interaction alone (lower symbols and line) and combined with the UIX $3N$ force ([90]).

Figure 32

The EoS of neutron matter as a function of the density, calculated by [99] using AFDMC with chiral $NN$ interactions at LO, NLO, and ${\mathrm{N}}^2\mathrm{LO}$ for the two different cutoffs indicated in the figure (three-body forces have not been included at ${\mathrm{N}}^2\mathrm{LO}$). Also shown are the results obtained by [281] using lattice QMC at ${\mathrm{N}}^2\mathrm{LO}$, by including the $3N$ interaction (upper dot-dashed line) and without (lower dot-dashed line), and the results of [219] using the ${\mathrm{N}}^2{\mathrm{LO}}_{\text{opt}}$ without $3N$ (dashed line).

Figure 33

The energy per particle of neutron matter for different values of the nuclear symmetry energy ($E_{\text{sym}}$). For each value of $E_{\text{sym}}$ the corresponding band shows the effect of different spatial and spin structures of the three-neutron interaction. The bottom and top lines show the same result of Fig. 31, where just the two-body alone and with the original UIX three-body forces has been used. The inset shows the linear correlation between $E_{\text{sym}}$ and its density derivative $L$. From [89].

Figure 34

Predicted neutron-star masses as a function of stellar radius ([89]; [90]). Different EoSs are considered: those obtained with the $\mathrm{AV}{8}^{\prime }$ $NN$ (bottom curve) and with the UIX $3N$ interactions (solid line) presented in Sec. 6a. The two shaded bands show the results obtained using different $3N$ forces constrained to have a particular value of the symmetry energy (indicated by numbers near the bands and curves).

Figure 35

Energies divided by $\hslash \omega N^{4/3}$ for neutrons in HO fields with $\hslash \omega =10\mathrm{MeV}$ (top) and 5 MeV (bottom). Filled symbols indicate ab initio calculations; the dashed lines are Thomas-Fermi results. Other results have been obtained using a variety of Skyrme forces indicated in the legend. From [88].

Figure 36

Energies per particle for neutrons in the Woods-Saxon field. The symbols are as in Fig. 35. From [88].

Figure 37

Calculated radii of neutrons confined in HO (upper) and WS (lower) fields compared to original and adjusted Skyrme models (see text). From [88].

Figure 38

Calculated densities of neutrons in HO potentials compared to Skyrme models (see text). From [88].

Figure 39

Energies divided by $\hslash \omega N^{4/3}$ for neutrons in HO fields with $\hslash \omega =10\mathrm{MeV}$ obtained using ${\mathrm{AV}8}^{\prime }$ with and without three-neutron forces with AFDMC, and using JISP16 with the NCFC method. From [162].

Figure 40

Energies of neutron drops predicted using the UNEDF0, UNEDF1, and UNEDF2 Skyrme energy-density functionals, compared to the AFDMC results. From [127].