Ab initio excited states from the in-medium similarity renormalization group

We present two new methods for performing ab initio calculations of excited states for closed-shell systems within the in-medium similarity renormalization group (IMSRG) framework. Both are based on combining the IMSRG with simple many-body methods commonly used to target excited states, such as the Tamm-Dancoff approximation (TDA) and equations-of-motion (EOM) techniques. In the first approach, a two-step sequential IMSRG transformation is used to drive the Hamiltonian to a form where a simple TDA calculation (i.e., diagonalization in the space of $1\mathrm{p}1\mathrm{h}$ excitations) becomes exact for a subset of eigenvalues. In the second approach, EOM techniques are applied to the IMSRG ground-state-decoupled Hamiltonian to access excited states. We perform proof-of-principle calculations for parabolic quantum dots in two dimensions and the closed-shell nuclei ${}^{16}\mathrm{O}$ and ${}^{22}\mathrm{O}$. We find that the TDA-IMSRG approach gives better accuracy than the EOM-IMSRG when calculations converge, but it is otherwise lacking the versatility and numerical stability of the latter. Our calculated spectra are in reasonable agreement with analogous EOM-coupled-cluster calculations. This work paves the way for more interesting applications of the EOM-IMSRG approach to calculations of consistently evolved observables such as electromagnetic strength functions and nuclear matrix elements, and extensions to nuclei within one or two nucleons of a closed shell by generalizing the EOM ladder operator to include particle-number nonconserving terms.

Figure 1

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

Figure 2

Schematic representation of the sequentially decoupled Hamiltonians in the many-body Hilbert space spanned by particle-hole excitations of the reference state. The left-hand panel corresponds to the use of Eq. (21), where the entire 1p1h sector is decoupled, and the right-hand panel corresponds to Eq. (27), where just the valence 1v1h excitations are decoupled. The latter corresponds to the small block in the upper left corner of the full 1p1h block.

Figure 3

Orbital scheme for a six-electron two-dimensional quantum dot in a model space consisting of four major shells. Each orbital has a distinct orbital angular momentum projection $m_l$ and spin projection $m_s$.

Figure 4

Ground-state and low-lying $(M_L,M_S)=(1,0)$ excited states of a six-electron quantum dot in an $\omega =1.0$ trap. The FCI calculations are performed using the flowing TDA-IMSRG(2) Hamiltonian in an $N=3$ model space.

Figure 5

$(M_L,M_S)=(1,0)$ excited states of a six-electron quantum dot in an $\omega =1.0$ trap with an $N=3$ model space. Three FCI calculations are shown: two using the flowing TDA-IMSRG(2) and vTDA-IMSRG(2) Hamiltonians corresponding to each choice of the secondary decoupling $U_x\left(s\right)$ (where $x=2$ or 3), and one using the bare Hamiltonian.

Figure 6

FCI and TDA calculations of the first $(M_L,M_S)=(1,0)$ excited state using the flowing vTDA-IMSRG(2) Hamiltonian for a six-electron quantum dot with $N=3$ and $\omega =1.0$. For reference, the flowing zero-body part of the Hamiltonian $E_{\text{ref}}\left(s\right)$ and the FCI results for the bare Hamiltonian are also shown.

Figure 7

Selected excitation spectra of six-electron quantum dots for $\omega =1.0$ (a) and $\omega =0.5$ (b) performed in an $N=3$ single-particle basis. The quantum numbers of the various states are color coded as ($M_L,M_S$) = (0,0) (red), (1,0) (black), (2,1) (blue), and (3,0) (green). The calculated spectra are displayed for five different many-body approaches, where the subscript indicates which Hamiltonian the respective method is applied to. For example, ${\mathrm{FCI}}_0$ and ${\mathrm{FCI}}_1$ denote FCI calculations on the bare and ground-state-decoupled Hamiltonians, respectively, ${\mathrm{TDA}}_{31}$ denotes a TDA calculation on the vTDA-decoupled Hamiltonian, etc. The lengths of the plotted energy levels indicate the $1\mathrm{p}1\mathrm{h}$ content of the state as defined in Eqs. (51, 52, 53, 54).

Figure 8

Absolute difference between quantum dot excitation energies calculated via IMSRG methods and those calculated with FCI on the bare Hamiltonian. Each point corresponds to an EOM or TDA energy level in Fig. 7.

Figure 9

Same as Fig. 8, except energies are plotted with respect to EOM-IMSRG(2,2) partial norms.

Figure 10

Lowest ${}^{16}\mathrm{O}$ excitation energies plotted for various quantum numbers, calculated with EOM-IMSRG(2,2) and vTDA-IMSRG(2) starting from the ${\mathrm{N}}^3\mathrm{LO}$ (500 MeV) NN interaction of Entem and Machleidt (EM) [44], softened by free-space SRG evolution to $\lambda =2.0{\mathrm{fm}}^{-1}$. The single-particle basis is given by $\hslash \omega =24.0$ MeV and $e_{\text{max}}=8$. Above each plotted energy level from the EOM-IMSRG(2,2) calculation is the $1\mathrm{p}1\mathrm{h}$ partial norm of Eq. (53).

Figure 11

Low-lying states of ${}^{22}\mathrm{O}$ at $\hslash \omega =28.0$ MeV and $e_{\text{max}}=11$ for several values of the Lawson parameter $\beta$, using the ${\mathrm{N}}^3\mathrm{LO}$ (500 MeV) NN interaction of EM [44], softened by free-space SRG evolution to $\lambda =3.0{\mathrm{fm}}^{-1}$. The c.m. frequency $\hslash \tilde{\omega }=17.28$ MeV.

Figure 12

Selected excitation spectra of ${}^{22}\mathrm{O}$ at $\hslash \omega =20.0$ MeV and $e_{\text{max}}=11$ using the ${\mathrm{N}}^3\mathrm{LO}$ (500 MeV) NN interaction of EM [44], softened by free-space SRG evolution to $\lambda =2.0{\mathrm{fm}}^{-1}$. (a) The excitation energies calculated with the intrinsic Hamiltonian, and (b) the result of adding a Lawson center-of-mass term $H=H_{\text{int}}+\beta H_{\text{c.m.}}\left(\tilde{\omega }\right)$, with $\beta =5.0$. Different colors indicate different $J^{\mathrm{\Pi }}$.

Figure 13

Selected excitation spectra of ${}^{22}\mathrm{O}$ at $\hslash \omega =28.0$ MeV and $e_{\text{max}}=11$ using the ${\mathrm{N}}^3\mathrm{LO}$ (500 MeV) NN interaction of EM [44], softened by free-space SRG evolution to $\lambda =3.0{\mathrm{fm}}^{-1}$. (a) The excitation energies calculated with the intrinsic Hamiltonian, and (b) the result of adding a Lawson center-of-mass term $H=H_{\text{int}}+\beta H_{\text{c.m.}}$, with $\beta =5.0$. Different colors indicate different $J^{\mathrm{\Pi }}$.