Combining the in-medium similarity renormalization group with the density matrix renormalization group: Shell structure and information entropy

We propose a novel many-body framework combining the density matrix renormalization group (DMRG) with the valence-space (VS) formulation of the in-medium similarity renormalization group. This hybrid scheme admits for favorable computational scaling in large-space calculations compared to direct diagonalization. The capacity of the VS-DMRG approach is highlighted in ab initio calculations of neutron-rich nickel isotopes based on chiral two- and three-nucleon interactions, and allows us to perform converged ab initio computations of ground and excited state energies. We also study orbital entanglement in the VS-DMRG, and investigate nuclear correlation effects in oxygen, neon, and magnesium isotopes. The explored entanglement measures reveal nuclear shell closures as well as pairing correlations.

In other fields of many-body research such as condensed matter physics or quantum chemistry, the density matrix renormalization group (DMRG) is well established as a powerful tool to treat strongly correlated quantum systems [40][41][42][43]. Previous studies in nuclear structure have focused on phenomenological shell-model applications [44][45][46][47] and open quantum systems using a Gamow basis [48,49]. However, the development of the DMRG to medium-mass ab initio calculations has not been explored. This is the goal of this work.
In this Letter, we apply the DMRG approach in ab initio nuclear structure calculations of medium-mass nuclei for the first time. We use the VS-IMSRG to decouple a valence-space Hamiltonian, which is then used as input to large-scale DMRG calculations. The favorable scaling of the DMRG provides an efficient framework for accessing computationally challenging open-shell nuclei in a systematically controllable way. Moreover, entanglement properties of many-body system are accessible from orbital entropies and derived quantities, thus proving a novel perspective to the emergence of structure from nuclear forces.

Valence-space DMRG approach
The central idea of this work is the combination of the DMRG with the valence-space formulation of the IMSRG. This gives rise to a hybrid many-body framework, which we refer to as valence-space density matrix renormalization group (VS-DMRG). Starting from an initial Hamiltonian with two-nucleon (NN) and three-nucleon (3N) interactions, the VS-IMSRG generates a valence-space-decoupled Hamiltonian that is restricted to an active space of limited size [21,23]. While the use of a valence-space Hamiltonian is similar to the phenomenological shell model, with the VS-IMSRG this is derived from chiral EFT interactions without adjustments. During the IMSRG-evolution many-body operators of higher particle rank are truncated at the normal-ordered two-body level, defining the IM-SRG(2) truncation. The valence-space-decoupled Hamiltonian H VS used as input for the DMRG calculation is represented in second-quantized form as where ε p are the single-particle energies and V pqrs the (anti-symmetrized) two-body matrix elements. The collective label p = (n p , l p , j p , m p , t p ) gathers all quantum numbers of a single nucleon: radial quantum number n, orbital angular momentum l, total angular momentum j and its projection m, and isospin projection t distinguishing protons and neutrons. The initial VS-IMSRG decoupling is performed in a single-particle space of 15 major harmonic-oscillator shells, i.e., e max ≡ (2n + l) max = 14, and the 3N interaction matrix elements are restricted to e 1 + e 2 + e 3 ⩽ E 3max = 16. For all our calculations, we employ the 1.8/2.0 NN+3N Hamiltonian from Ref. [50], which is based on chiral EFT interactions. The three-nucleon interactions are taken into account by keeping only two-body contributions after normal ordering [51][52][53].
In the DMRG calculation we use the occupation-number representation of an orbital, yielding a local Hilbert space with dimension d = 2. Therefore, each orbital is represented by two distinct occupation states σ, i.e. σ ∈ {0, 1}. The full Hilbert space of N orbitals is then built from a tensor product of the local spaces, i.e., H N ≡ ⊗ N i=1 H i . The DMRG approach provides a variational procedure for the minimization of the ground-state energy (or the lowest energy for a given total angular momentum and parity) using a matrix product state (MPS) parametrization of the many-body state (see, e.g., Ref. [41]), that eventually converges to the full configuration interaction (FCI) limit for a given Hilbert space. To this end, the nuclear orbitals are mapped onto a one-dimensional chain. This protocol is based on the two-orbital mutual information (see next section) of the orbitals to minimize long-range correlations, i.e., to find a quasi-optimal ordering of the orbitals along the one dimensional DMRG topology [54,55].
The corresponding wave function of N orbitals is an N dimensional tensor, the CI coefficient corresponding to a determinant σ = (σ 1 , σ 2 , . . . , σ i , σ i+1 , . . . , σ N ) is expressed as a product of matrices A σi i associated to each orbital i as |Ψ⟩ = σ C σ |σ⟩, where The dimension of the matrices in the MPS representation scales exponentially with the number of orbitals, such that truncations are required to keep the the dimensions numerically tractable. In the DMRG algorithm the matrices A σi i are iteratively optimized. In an iteration step of the two-site DMRG variant the tensor space is split according ) denote the left (right) blocks that are formed from precontracted A matrices to the left and right of the sites p and p + 1, respectively.
For a given site p, the MPS matrix is updated through a diagonalization of the neighboring block Hamiltonian and the maximal matrix dimension (M ), also known as bond dimension, is kept below a threshold value by keeping only those matrix components which correspond to highest Schmidt weights obtained via singular value decomposition. Therefore, the state's components are obtained through a series of unitary transformations ("sweeps") going through the orbital space forward from left to right, and then backward, until convergence is reached. The method's intrinsic truncation error is thus set by M = dim H (left) = dim H (right) corresponding to the dimension of the left/right blocks. Eventually, the size of the bond dimension to reach an acceptable convergence is in direct correspondence with the amount of quantum entanglement in the many-body state [42]. The DMRG convergence is substantially improved following the configuration-interaction dynamically extended activespace procedure, similar to the calculations performed in Ref. [47].

Entanglement and correlation measures
For the study of correlation effects in nuclear manybody systems, we explore a set of entanglement measures [56,57]. The total entropy [58] I tot ≡ p s p is obtained from the single-orbital entropy s p ≡ −Tr ρ p ln ρ p , where ρ p is the one-orbital-reduced density matrix of the orbital p obtained by tracing out all other orbitals except for p [59]. The single-orbital entropy is directly linked to the natural occupation numbers in the manybody state [60]. Therefore, systems with strong static correlations give rise to increased values for s p and, consequently, I tot . In the case of weakly correlated systems, occupation numbers are either n p ≈ 0 or 1, reflecting the existence of a dominant reference determinant, as obtained in a mean-field calculation, for example. As a consequence, nuclei with shell closures will be accompanied by a local minimum in the total entropy. To more cleanly disentangle correlations for protons and neutrons, we define the proton (neutron) total entropy I (p) where only single-orbital entropies of a given particle species are summed over. Correlations among pairs of orbitals can be further studied from the entanglement entropy s pq ≡ −Tr ρ pq ln ρ pq using the two-orbital reduced density matrix ρ pq . Combing single-and two-orbital entropies leads to the mutual information, I p̸ =q ≡ s p + s q − s pq [54]. Since matrix elements of ρ pq are expressed in terms of two-orbital correlation functions, also known as generalized correlation functions [61], s pq can be viewed as a weighted average of the corresponding correlations. Sub-   traction of s p and s q when I pq is calculated is analogous to the usual subtraction of the unconnected parts of the two-orbital correlation functions. Entanglement studies in nuclear theory have been performed in shell-model applications [47,62] and in no-core calculations of light systems [63]. We emphasize that the entanglement measures are of non-observable character, as they depend on the nuclear Hamiltonian and the many-body basis (see, e.g., Refs. [64,65]). Thus, we focus on their qualitative behavior.

Neutron-rich nickel isotopes from VS-DMRG
To show the power of the VS-DMRG, we apply this new approach to the description of neutron-rich nickel isotopes that are attracting significant experimental attention, e.g., with the recent discovery of the doubly magic nature of 78 Ni [67]. In fact, ab initio calculations approaching 78 Ni require additional truncations of the configuration interaction (CI) or shell model space when exploring a 0ℏω valence space on top of a 60 Ca core 1 . In this work, the CI 1 Reference [67] quotes the 2 + energy for 78 Ni to be E ⋆ 2 + = 3.34 MeV for including up to Tmax = 7 particle-hole excitations. In our studies we confirmed that this was a misprint and calculations were performed up to Tmax = 6. calculations haven been performed using the KSHELL [68] and BIGSTICK [69] codes, while the DMRG calculations together with quantum-information-based analysis tools used the DMRG-Budapest program package [70].
In Fig. 1 we compare large-scale CI and VS-DMRG calculations for 78 Ni based on the same VS-IMSRG interaction as in Ref. [67]. For 78 Ni, the FCI dimension is 2.3·10 11 , while our largest CI calculations involved 1.9 · 10 9 configurations employing a truncation at T max = 7 particle-hole (ph) excitations. In contrast, the dimension of the DMRG space increases only gradually, and is well tractable even for the largest considered bond dimension M = 10240, with corresponding configuration space of ≈ 10 7 , two orders of magnitude below the largest accessible CI dimension. The DMRG dimension is essentially the dimension of the space spanned by the two block spaces and the two orbitals, ∼ M 2 d 2 , further constrained by selection rules for parity, isospin and angular-momentum projection. Figure 1 clearly shows that the VS-DMRG results for the ground and first 2 + excited states reveal a more robust convergence pattern compared to the CI calculation. While the ground-state energy converges systematically in the CI case, there is still a sizeable linear trend present for the first excited 2 + state, making the extrapolation of the excitation energy challenging. This may potentially hint at relevant 8p8h excitations missing in the T max = 7 truncation. In contrast, the VS-DMRG results converge systematically beyond M = 1024. Fitting a quadratic polynomial f extr. (1/M ) = a/M 2 + b/M + c enables a robust extrapolation of the energies [42]. Other sweepbased and truncation error based extrapolation procedures have been successfully applied in condensed-matter and quantum chemistry applications [42,43,71]. Extrapolation uncertainties are obtained by taking into account only the 3, 4, 5 data points corresponding to the largest bond dimensions, yielding a VS-DMRG estimate of E ⋆ 2 + = 3.007 ± 0.017 MeV. At much lower space dimensions, the VS-DMRG approach thus yields much lower uncertainties compared to CI (E ⋆ 2 + = 3.141±0.205 MeV). For a given size of the many-body space the MPS wavefunction includes correlations much more efficiently compared to CI.
Next we study the emergence of shell structure from the perspective of the information entropy from our VS-DMRG calculations. Figure 2 displays neutron, proton and total entropies and 2 + excitation energies for 70−80 Ni. The total entropy shows a pronounced kink for 78 Ni consistent with its doubly magic nature. The proton contribution to the total entropy is small from 70 Ni to 78 Ni and then exhibits a strong increase to 80 Ni. We attribute this sudden increase of proton correlations to the onset of nuclear deformation effects. This is also consistent with the rapid transition from spherical to deformed ground states beyond 78 Ni predicted in Ref. [67]. As expected from the VS-IMSRG results in Ref. [67], the VS-DMRG reproduces nicely the high 2 + excitation energy in 78 Ni, with an improved result of E ⋆ 2 + = 3.01 MeV compared to the published VS-IMSRG excitation energy E ⋆ 2 + ≲ 3.34 MeV. The difference to the experimental value of E ⋆ 2 + = 2.6 MeV is therefore significantly decreased for this 1.8/2.0 NN+3N Hamiltonian, and the difference is attributed to truncated three-body operators in the VS-IMSRG [67,72]. Finally, we note that the convergence with increasing bond dimension M is significantly slower in 78 Ni, which is consistent with the importance of higher n-particle-n-hole (npnh) correlations in the ground and excited states (see also Fig. 1).

Shell structure in sd-shell nuclei
Following 78 Ni, we explore shell structure in the sd shell based on the total entropies obtained from VS-DMRG calculations using the VS-IMSRG decoupled Hamiltonian from the same 1.8/2.0 NN+3N interactions. Figure 3 shows the total neutron and proton entropies for the oxygen isotopes and the N = 16 isotones 26 Ne (Z = 10) and 28 Mg (Z = 12). Since an sd-shell valence space is employed, the proton entropy for the oxygen isotopes is identically zero in all cases. For the even-mass oxygen isotopes one observes a pronounced kink in the single-orbital entropy at N = 16, indicating the strong shell closure for 24 O. A complementary analysis of the CI coefficients reveals that the ground state is dominated by the reference state (≈ 92%) with admixtures from 2p2h-excitations (≈ 7%), thus confirming the weakly correlated nature of the many-body state. A less pronounced kink is observed in 22 O where the d 5/2 shell is closed. For odd-mass nuclei the entropy is lower compared to their neighbors with an additional neutron due to the presence of an unpaired nucleon. Note that the entropy of odd-mass nuclei depends on the particular value of the magnetic quantum number M J in the ground-state multiplet [73]. Here we consistently show the entropy values for M J = 1/2, but differ- ln (2) ln I ij Figure 4: Logarithm of mutual information, ln I ij , for 24 O, 26 Ne, and 28 Mg obtained from VS-DMRG calculations in the sd-shell valence space.
ences for different M J are small, ∆I tot ≈ 0.1, and thus do not affect our general conclusions. Finally, we note that the neutron entropy for 27,28 O vanishes due to the single Slater-determinant ground state in the sd shell. The correlations of 26 Ne and 28 Mg both reveal an enhancement of the neutron total entropy induced by the presence of valence protons (Fig. 3, right panel). Both nuclei admit for more collective many-body states with enhanced mixing from 3p3h excitations (10%, 17% in 26 Ne, 28 Mg, respectively) and 4p4h excitations (12%, 15%). Deformation effects present in neon and magnesium isotopes cannot be captured within a sd-shell valence space but require the inclusion for several major shells [35,74]. However, this poses challenges in the VS-IMSRG decoupling which is beyond the scope of the present paper and left for future studies [75].
A refined understanding of the individual correlation effects is obtained from the mutual information (MI). Figure 4 shows the MI of the sd-shell orbitals for 24 O, 26 Ne, and 28 Mg. In the case of even-mass nuclei with J π = 0 + ground states, the MI for the different m j orbital substates are degenerate. The large diagonal entries (black regions) in the proton-proton and neutron-neutron subblock reflect pairing correlations between time-reversed singleparticle states [47]. In 24 O, the homogeneous strength in the neutron-neutron blocks d 5/2 -d 3/2 and s 1/2 -d 3/2 , as well as the uniform MI background in the d 3/2 -d 3/2 blocks can be understood in terms of nucleon pair fluctuations in generalized seniority-like states [62]. The proton-proton block of the MI in 26 Ne can be similarly understood, and is very similar to the neutron-neutron-block in 18 O (not shown). The emerging structures in the proton-neutron blocks in 26 Ne and 28 Mg share common features, e.g., the formation of neutron-proton pairs built from m j = ±5/2 states. Moreover, both nuclei admit for enhanced couplings between neutron d 3/2 and proton d 5/2 states. Similar pairing correlations were observed in recent no-core studies of 4,6 He [63].

Conclusion and outlook
In this Letter we performed the first ab initio DRMG calculations of medium-mass nuclei based on chiral NN+3N interactions. Combining the DMRG with the VS-IMSRG leads to a powerful hybrid many-body approach, the VS-DMRG, that efficiently accounts for static and dynamic correlation effects. The use of an MPS parametrization of the many-body state is computationally superior to conventional CI expansions, and enables convergence in large-scale valence-space applications. As shown for 78 Ni and in the sd shell, the VS-DMRG through its entropybased entanglement measures also provides new insights to shell structure and correlations in nuclei. Moreover, the VS-DMRG is ideally suited for exploring systems that are not captured starting from a single-reference state, such as deformed nuclei. However, this requires the use of multi-shell decoupling in the VS-IMSRG which is still an open area [75]. While the present focus was on the calculation of energies, the VS-DMRG framework can be naturally extended to other observables such as radii or electroweak transitions. For future developments, the use of a symmetry-restricted, i.e., J-scheme, formulation of the VS-DMRG (see, e.g., Refs. [45,46]) will be helpful to cope with the increasing number of orbitals in largescale applications. Furthermore, the study of multi-partite entanglement [76,77] can provide insights to many-body correlations in nuclei.