Magnus expansion and in-medium similarity renormalization group

We present an improved variant of the in-medium similarity renormalization group (IM-SRG) based on the Magnus expansion. In the new formulation, one solves flow equations for the anti-Hermitian operator that, upon exponentiation, yields the unitary transformation of the IM-SRG. The resulting flow equations can be solved using a first-order Euler method without any loss of accuracy, resulting in substantial memory savings and modest computational speedups. Since one obtains the unitary transformation directly, the transformation of additional operators beyond the Hamiltonian can be accomplished with little additional cost, in sharp contrast to the standard formulation of the IM-SRG. Ground state calculations of the homogeneous electron gas (HEG) and ${}^{16}\mathrm{O}$ nucleus are used as test beds to illustrate the efficacy of the Magnus expansion.

Figure 1

Schematic representation of the initial and final Hamiltonians, $H\left(0\right)$ and $H\left(\infty \right)$, in the many-body Hilbert space spanned by particle-hole excitations of the reference state.

Figure 2

$|H_{11}\left(s\right)-E_{\mathrm{gs}}|$ versus $s$ for different Euler step sizes calculated via direct integration of the SRG flow equation, Eq. (3), and using the Magnus expansion, Eqs. (28) and (30).

Figure 3

Relative importance of the $k\mathrm{th}$ term in the Magnus derivative as defined by the left-hand side of Eq. (36) evaluated in the NO2B approximation. The top row is for the homogeneous electron gas at Wigner-Seitz radii of (a) $r_s=0.5$ and (b) $r_s=5.0$. The bottom row is for ${}^{16}\mathrm{O}$, starting from the chiral NN potential of Entem and Machleidt [38], softened by a free-space SRG evolution to (c) $\lambda =2.0{\mathrm{fm}}^{-1}$ and (d) $\lambda =3.0{\mathrm{fm}}^{-1}$. The electron gas calculations were done for $N=14$ electrons in a periodic box with $M=114$ single-particle orbitals. The ${}^{16}\mathrm{O}$ calculations were done in an $e_{\max }=8$ model space, with $\hslash \omega =24$ MeV for the underlying harmonic oscillator basis.

Figure 4

Magnitude of the 0-body contributions of the $k\mathrm{th}$ term in Eq. (30) evaluated in the NO2B approximation. The top row is for the electron gas at Wigner-Seitz radii of (a) $r_s=0.5$ and (b) $r_s=5.0$. The bottom row is for ${}^{16}\mathrm{O}$, starting from the chiral NN potential of Entem and Machleidt [38], softened by a free-space SRG evolution to (c) $\lambda =2.0{\mathrm{fm}}^{-1}$ and (d) $\lambda =3.0{\mathrm{fm}}^{-1}$. The electron gas calculations were done for $N=14$ electrons in a periodic box with $M=114$ single-particle orbitals. The ${}^{16}\mathrm{O}$ calculations were done in an $e_{\max }=8$ model space, with $\hslash \omega =24$ MeV for the underlying harmonic oscillator basis.

Figure 5

Flowing IM-SRG(2) and Magnus(2) HEG correlation energy, $E_0\left(s\right)-E_{\mathrm{HF}}$, for Wigner-Seitz radii of (a) $r_s=5.0$ and (b) $r_s=0.5$. The solid black line corresponds to IM-SRG(2) results using an adaptive solver based on the Adams-Bashforth method, while the other lines correspond to Magnus(2) and IM-SRG(2) results using different Euler step sizes. The red circles denote the quasi-exact FCIQMC results of Ref. [35].

Figure 6

Flowing IM-SRG(2) and Magnus(2) ground state energy, $E_0\left(s\right)$, for ${}^{16}\mathrm{O}$ starting from the ${\mathrm{N}}^3\mathrm{LO}$ NN interaction of Entem and Machleidt [38] evolved by the free-space SRG to (a) $\lambda =2.7{\mathrm{fm}}^{-1}$ and $\lambda =2.0{\mathrm{fm}}^{-1}$. The solid black line corresponds to IM-SRG(2) results using an adaptive solver based on the Adams-Bashforth method, while the other lines correspond to Magnus(2) and IM-SRG(2) results using different Euler step sizes. The calculations were done in an $e_{\max }=8$ model space, with $\hslash \omega =24\mathrm{MeV}$ for the underlying harmonic oscillator basis.

Figure 7

Electron gas momentum distributions calculated in the Magnus(2) approximation. The calculations were done for $N=14$ electrons in a periodic box with $M=778$ single-particle orbitals.

Figure 8

Timing for Magnus(2) and IM-SRG(2) HEG calculations as the single-particle basis is increased. The two bottom curves are for $r_s=0.5$ and the top for $r_s=5$. For the Magnus(2) timing, this includes the calculations of the momentum distributions.

Figure 9

Center-of-mass diagnostics for Magnus(2) calculations of ${}^{16}\mathrm{O}$ starting from the ${\mathrm{N}}^3\mathrm{LO}$ NN interaction of Entem and Machleidt [38] evolved by the free-space SRG to $\lambda =2.0{\mathrm{fm}}^{-1}$. See the text for details. The calculations were done in an $e_{\max }=9$ model space.