Similarity renormalization group evolution of hypernuclear Hamiltonians
Unitary transformations of a Hamiltonian generally induce interaction terms beyond the particle rank present in the untransformed Hamiltonian that have to be captured and included in a many-body calculation. In systems with strangeness such as hypernuclei, the three-body terms induced by the hyperon-nucleon interaction are strong, so their inclusion is crucial.
We present in detail a procedure for computing hyperon-nucleon-nucleon interaction terms that are induced during a similarity renormalization group (SRG) flow. The SRG is carried out in a basis spanned by antisymmetric harmonic-oscillator states with respect to three-body Jacobi coordinates. We discuss basis construction, antisymmetrization, numerical evaluation of the flow equations, and separation of the genuine three-body terms.
We then use the hypernuclear no-core shell model with SRG-evolved Hamiltonians, addressing the sensitivity of hypernuclear states and hyperon separation energies to changes in the nucleonic Hamiltonian by example of , , , and the hyper-helium chain. We also present a survey of the hyper-hydrogen chain, exploring the structure of hypernuclei with extreme neutron-proton asymmetries.
The understanding of strangeness in finite and infinite strongly-interacting systems is key to not only gaining insight into the low-energy limit of the strong interaction itself, but also to understanding the structure of neutron stars Hiyama and Yamada (2009); Chatterjee and Vidaña (2016); Gal et al. (2016); Vidaña (2018). Recently, we developed the hypernuclear no-core shell model (NCSM) Wirth et al. (2014, 2018), the first ab initio method able to calculate hypernuclei beyond the shell with nonlocal interactions, such as those derived from chiral effective field theory Polinder et al. (2006); Haidenbauer et al. (2013). With that, we have established a link between the low-energy effective field theory of QCD and the phenomenology of hypernuclei.
The NCSM and all other basis-expansion approaches to the nuclear many-body problem rely on model spaces spanned by finite basis sets. The convergence of many-body observables with increasing model-space size contributes to the theoretical uncertainties and eventually limits the range of ab initio calculations in terms of mass number. Therefore, the acceleration of convergence via unitary or similarity transformations of the Hamiltonian and other relevant operators is a key ingredient in ab initio nuclear and hypernuclear structure theory. The similarity renormalization group (SRG) Głazek and Wilson (1993); Wegner (1994); *Wegner2000 has proven to be a very versatile and effective tool to achieve this convergence acceleration. The tradeoff that comes with using unitary transformations is that these transformations induce many-body interactions beyond those that are present in the initial Hamiltonian. Contrary to other methods, the SRG allows for the explicit computation of induced many-body terms and of consistently transformed operators in a conceptually straight-forward manner.
In our previous works we have presented the first NCSM calculations for -shell hypernuclei with chiral two- and three-baryon interactions Wirth et al. (2014); Wirth and Roth (2018) and we have demonstrated that hyperon-nucleon-nucleon (YNN) terms induced by the SRG transformation are strong and cannot be neglected when working with transformed interactions Wirth and Roth (2016). The size of SRG-induced YNN interactions is remarkable and highlights a special feature of the hyperon-nucleon (YN) interactions, the conversion of a to a hyperon in an interaction process with a nucleon. We have shown that the elimination of this conversion, i.e. the decoupling of and channels, through an SRG evolution leads to strong repulsive NN interactions Wirth and Roth (2016). As a consequence, models for hyperons in matter that only include the hyperon and omit the conversion have to include strong repulsive NN forces. This has direct impact on the hyperon puzzle in neutron-star physics Wirth and Roth (2016); Chatterjee and Vidaña (2016); Lonardoni et al. (2015).
In Refs. Wirth and Roth (2016, 2018) we have focused on the applications and the physics discussion of the hypernuclear SRG. In the present paper we provide a detailed discussion of the formalism and, particularly, the extension of the SRG to the YNN three-baryon sector with all elements necessary for the practical implementation. In addition we present new calculations for light hypernuclei, including the hydrogen and helium isotopic chains up to the driplines, and investigate the impact of the nucleonic part of the Hamiltonian on hypernuclear spectra.
This paper is organized as follows: Section II considers the necessary steps for computing the YNN terms induced during the SRG evolution of a two-body Hamiltonian. We give a short overview over the core ideas of the hypernuclear no-core shell model in section III. In section IV we present NCSM calculations for a sample set of hypernuclei using state-of-the-art nucleonic Hamiltonians, which provide better saturation properties than the one used before.
Ii Similarity renormalization group
Most models of baryon-baryon interactions display strong repulsion at short distances where the baryons overlap. This repulsive core strongly couples two-baryon states with low and high relative momentum. In order to accommodate these couplings, which is necessary to achieve convergence of the many-body calculation, large many-body model spaces are needed. For all but the lightest systems, the sizes needed are beyond the capabilities of current high-performance computers. To accelerate convergence and reduce the required model-space sizes, methods based on unitary transformations of the Hamiltonian have been devised that suppress the coupling between low- and high-momentum states. A simple and very versatile method is the SRG.
The SRG is a continuous unitary transformation of a Hamiltonian, depending on a flow parameter . It is governed by the flow equation
where denotes the derivative with respect to . The anti-Hermitian generator can be chosen freely to achieve a desired behavior. A very general and convenient ansatz for is a commutator so that the flow stops when the Hamiltonian commutes with the Hermitian operator . In nuclear physics, we commonly use , the intrinsic kinetic energy, which drives the Hamiltonian to band-diagonal form in HO basis. The squared nucleon mass sets the unit of to .
Other observables can be evolved consistently by solving the same differential equation creftype 1 for the observable. Note, however, that the generator depends on the Hamiltonian, so that the observables have to be evolved simultaneously. When considering multiple observables, it is more economical to compute the unitary transformation by solving
while evolving the Hamiltonian and transform the observables afterwards.
ii.2 Evolution in two-body space
The abstract operator differential equation creftype 1 needs to be converted to a flow equation for matrix elements that can be evaluated numerically. We begin by discussing the evolution in two-body space to show the necessary steps without introducing the complexity of a three-body system.
For a general -body system with charge and strangeness111Strangeness is defined as the number of anti-strange minus the number of strange quarks so hypernuclei have . , the starting point of the evolution is a Hamiltonian . The first term is the intrinsic kinetic energy
with single-particle momenta , masses , and center-of-mass kinetic energy . The second term,
where is the total rest mass of the noninteracting system of protons, neutrons, and hyperons, accounts for the higher rest mass of the hyperon. The remaining terms are two-body NN and YN interactions, the three-nucleon (NNN) interaction, and higher interaction terms that are neglected. For the evolution in two-body space, the NNN interaction can also be omitted.
To carry out the evolution, we take matrix elements of the Hamiltonian between harmonic-oscillator (HO) states. These states are defined with respect to the center-of-mass and relative Jacobi coordinates and ,
The and are the masses and positions of the particles, respectively, and the nucleon mass is used as a global scaling factor to make the have units of length. The Jacobi coordinates coincide with the ones commonly used for nucleonic systems if for all particles.
Since the interactions are translationally invariant, we can separate the center-of-mass degrees of freedom and define a basis of antisymmetric relative HO states , with the spins of the particles, coupled to total spin , the radial quantum number and orbital angular momentum of the relative motion, total angular momentum with projection , and with denoting the species, i.e. strangeness, isospin and isospin projection, of the two particles. To make this two-body basis finite, we introduce an energy truncation . In the finite space spanned by the basis states, the SRG flow equation becomes an ordinary matrix differential equation that can be solved numerically. Due to symmetries of the Hamiltonian, only certain subsets of states can produce nonvanishing matrix elements, such that the Hamiltonian decomposes into blocks of equal total angular momentum , charge and strangeness , which can be evolved separately. Also, the matrix elements are independent of the angular momentum projection . The SRG transformation on the two-body level is given in more detail in Ref. Wirth et al. (2018).
ii.3 Evolution in three-body space
The evolution in two-body space is, by construction, unable to capture induced terms beyond the two-body level. To determine the induced three-body terms, the evolution has to be carried out in three-body space. There, the initial Hamiltonian can already contain three-body forces. However, the NNN and YNN forces act on the and sectors of the three-particle Hilbert space, respectively, so only one of them needs to be included in the evolution. Moreover, initial YNN forces do not enter the chiral expansion until N2LO and are, thus, absent from the calculation presented here.
ii.3.1 Coordinate systems, basis sets, and transformations
with the center-of-mass coordinate of particles and (which coincides with creftype 5). One can also define a different set of coordinates
where the first coordinate is defined by the first and third particle. This set is needed for antisymmetrization.
The , , and are connected via orthogonal transformations of the type
parametrized by the nonnegative number , which relate two general pairs of coordinates and . The transformation from to the single-particle coordinates is effected in two steps by first transforming to () and then to with . Transforming between the and is needed to antisymmetrize the three-body states. Since the center-of-mass coordinate is the same in both sets, it suffices to find the orthogonal transformation connecting and , which after some algebra turns out to have .
For HO states, the overlap between states and defined with respect to two coordinate pairs and related by an orthogonal transformation creftype 16 is given by a harmonic-oscillator bracket (HOB) . The HOBs can be computed analytically, e.g., using the expressions given by Kamuntavičius et al. (2001).
In the following we will work in a basis spanned by HO states with respect to the Jacobi coordinates , coupled to total angular momentum and isospin. We choose to work in an isospin-coupled basis and neglect isospin breaking of the induced YNN terms because it is computationally too demanding to take this effect into account.
For the upcoming derivations we define a basis set that is antisymmetric only under exchange of the first two particles, indicated by a subscript “”,
The intrinsic quantum numbers are collected into and the notation denotes the set where the quantum numbers of particles and have been exchanged. The Kronecker delta is short notation for .
Contrary to the two-body sector, antisymmetrizing a three-body state is not trivial except for product states. We achieve antisymmetrization in the Jacobi HO basis through explicit projection onto the antisymmetric subspace. The antisymmetric subspace is the space spanned by the eigenvectors of the antisymmetrizer to eigenvalue . Diagonalizing the antisymmetrizer, represented as a matrix with respect to the partially-antisymmetric Jacobi HO basis creftype 17, gives a basis of the antisymmetric subspace in terms of these basis states.
The antisymmetrizer for a three-body system is
and with respect to the -antisymmetric basis, exploiting that and the eigenvalue relation of , its matrix elements
are trivially related to the matrix elements of of the transposition operator . We separate the spin, isospin and spatial parts of the matrix element and consider them separately (for simplicity, we consider a nonantisymmetric matrix element and apply creftype 17 to get the final result):
The spin and isospin parts have the same structure apart from an additional constraint on the strangeness quantum numbers. The application of the permutation changes the coupling order to a scheme where particles 1 and 3 are coupled first and particle 2 couples to the resulting spin. The matrix element is
and the isospin part gets an additional factor .
The effect on the spatial part is similar: the result of the permutation is a state with the same quantum numbers, but in the coordinate system where the first Jacobi coordinate is defined by particles and . Hence,
with transformation parameter . This relation shows a unique property of the antisymmetrizer in a Jacobi HO basis: the HOB conserves the intrinsic energy quantum number , so the antisymmetrizer is block diagonal not only in and , but also in and antisymmetrization can be carried out separately for each (finite-dimensional) block.
To shorten the following formulae, we introduce
assume that all particles have spin , and omit the center-of-mass degrees of freedom of which the antisymmetrizer is independent. Employing creftype 17 we get
with . The antisymmetrizer is block-diagonal with respect to the quantum numbers and the orderless set of quantum numbers , , defining the species of the participant particles. It is also independent of the projection quantum numbers and .
The eigenvectors of the matrix representation of to eigenvalue define an orthonormal basis
spanning the antisymmetric subspace. The index enumerates the different vectors emerging from the highly-degenerate eigenvalue problem and carries no physical significance. The components of the eigenvectors are the coefficents of fractional parentage (CFPs), where .
ii.3.3 Transformation to scheme
In order to exploit the symmetries of the Hamiltonian and to limit the number of matrix elements that need to be stored in memory during the many-body calculation tractable we store the three-body matrix elements in a -coupled scheme. Hence, we need to express a Slater determinant in terms of antisymmetric Jacobi-HO states as follows:
The sum is over those with fixed and . The coefficient
is the overlap of a -antisymmetric Jacobi-HO basis state and a product state of single-particle HO states coupled to total isospin and angular momentum. Here, refers to the isospin quantum numbers of the set. It can be represented in terms of an overlap between nonantisymmetric states
The coefficients do not depend on the quantum numbers , and . In addition to that, we get energy conservation constraints from the HOBs that force and effectively eliminate the sum over .
For matrix elements of a scalar operator independent of the center-of-mass degrees of freedom we get
and we precompute and store the expression in brackets. The final decoupling to the -scheme is done on-the-fly during the many-body calculation.
ii.3.4 Embedding of two-body matrix elements
The evolution in three-body space requires matrix elements of and with respect to the antisymmetric Jacobi-HO basis. For the two-body parts of these operators, we compute
where denotes a general two-body operator, embedded into three-body space. The operator acts only on the first two particles. The -antisymmetric Jacobi-HO basis has the same quantum numbers and coupling scheme as a -coupled two-body basis. Thus, assuming a scalar-isoscalar operator, the three-body matrix elements are calculated from the two-body ones by
For nonscalar operators, one needs to decouple the second Jacobi coordinate and the dependence on the projection quantum numbers introduces three-body matrix elements coupling different (or ).
ii.3.5 Numerical solution
If we neglect isospin breaking, the Hamiltonian of the three-body system decomposes into blocks with good with intrinsic parity and the SRG evolution acts on each block separately. This makes a straight-forward solution of the flow equation feasible. We solve the matrix differential equation creftype 1 with standard numerical methods, calculating the derivative by computing the double matrix commutator. Since all matrices and intermediates are (anti-) hermitian
with (), the matrix commutator is
and we can compute it by a single matrix multiplication followed by an addition or subtraction of the adjoint of the result.
After the evolution the three-body matrix elements contain a mixture of intrinsic kinetic energy, two- and induced three-body interactions. The induced three-body terms need to be separated because two- and three-body interactions scale differently in many-body calculations, i.e., one cannot simply embed the three-body matrix elements of the Hamiltonian into -body space. We achieve the separation by subtracting the intrinsic kinetic energy and a two-body interaction evolved in two-body space to the same flow parameter.
Here, one has to carefully consider the truncations of the two- and three-body SRG model spaces: to subtract truncation artifacts, one has to, in priciple, truncate two-body matrix elements identically in both model spaces. The three-body space is truncated by , so the maximum relative energy of the first two particles (the first Jacobi coordinate) depends on the energy of the spectator particle. There is hence no single truncation for the two-body evolution to describe this truncation. So, to capture the main effect and to minimize truncation artifacts in the first place, we set and choose sufficiently large SRG model spaces by setting and the basis frequency to sufficiently high values.
A consequence of this discussion is that the optimal basis frequencies for the SRG evolution and for the many-body calculation are in general different. Before converting the matrix elements to single-particle coordinates we, therefore, change the basis frequency from the former to the latter. The method is the same as described in Roth et al. (2014) and the overlap of two states with different basis frequencies and is given by
The functions are HO wave functions with oscillator lengths , and are the reduced masses corresponding to the Jacobi coordinates. The integrals can be transformed so they only depend on the ratio of the basis frequencies. Analytical expressions for the overlaps are given in appendix A.
As a final remark we note that the antisymmetrization and embedding procedures presented here actually comprise a Jacobi-coordinate formulation of the NCSM (J-NCSM) for the three-body system. This formulation is very economical because it makes extensive use of the symmetries of the system. An extension to larger particle numbers is feasible, and has been employed for nuclear Barrett et al. (2013); Liebig et al. (2016) and hypernuclear Wirth et al. (2018) systems, to a point where antisymmetrization becomes too cumbersome.
Iii No-core shell model
The SRG-evolved Hamiltonian can be used in any many-body method, either directly or through additional approximations for the three-body force like the normal-ordered two-body approximation Roth et al. (2012). One of the conceptually simplest many-body methods able to include the full three-body part of the Hamiltonian is the no-core shell model (NCSM). In the NCSM, a matrix representation of the Hamiltonian is computed in a model space spanned by Slater determinants of HO single-particle states. The model space is truncated by limiting the number of HO excitation quanta to a maximum value of . Since the basis states are Slater determinants, antisymmetrization is trivial and computation of many-body matrix elements using Slater-Condon rules is simple. Also, working with single-particle states allows for an exact treatment of isospin-breaking mass differences, the Coulomb interaction and charge-symmetry-breaking parts of the baryon-baryon interactions.
Due to the - conversion present in the YN interaction, the model space for hypernuclei contains determinants with different numbers of protons, neutrons and hyperons. Only the total number of particles, total charge and strangeness are conserved. This increases the size of the model spaces significantly and, in combination with the rapid growth with total particle number and , limits the range of tractable hypernuclei. This limitation is mitigated by the introduction of an importance-truncation scheme Roth (2009), where, starting from a reference state from a small model space, the importance of each basis state for the description of the target state is estimated perturbatively. Only those states whose importance exceeds a given threshold are included in the (importance-truncated) model space. The effect of the neglected states on observables is taken into account by computing them for multiple values of the threshold and subsequently extrapolating to vanishing threshold.
Starting from a small- calculation, one can thus build an iterative scheme, where the eigenstates computed in each step are used as reference states for constructing the model space for the next step. During this procedure, all basis states are reassessed and the model space dynamically adapts to the structure of the targeted states.
iv.1 Interaction dependence of hypernuclear states
Our previous calculations did not explore the effect of the nucleonic Hamiltonian on hypernuclear observables. Instead, we only used a single Hamiltonian that provides a good reproduction of experimental energy levels in the and shells. This Hamiltonian, however, has certain deficiencies. First, it predicts nuclear radii that are too small by approximately . Second, there was an error in the formula for determining the three-nucleon low-energy constant from the triton beta-decay half life Marcucci et al. (2018); Gazit et al. (2019). Recently, new families of interactions became available that use the correct formula and predict larger radii.
Larger nuclear radii imply lower nucleon densities in the interior of the nucleus. Since the YN interaction essentially couples the hyperon to the nucleon density, we expect that the hyperon will experience less attraction, leading to a lower hyperon separation energy.
In the following, we will use three different Hamiltonians. The first is the previously used one, consisting of a two-nucleon interaction at next-to-next-to-next-to-leading order (N3LO) by Entem and Machleidt (2003) and a three-nucleon interaction at N2LO with local regulator Navrátil (2007); Gazit et al. (2009). We will refer to this Hamiltonian as N3LOEM+N2LOL. for the second Hamiltonian, called N3LOEM+N2LONL, we use a nonlocal regulator for the three-nucleon interaction with the corrected value Gazit et al. (2019). Finally, we consider a Hamiltonian that is built from the recent N4LO two-nucleon interaction by Entem, Machleidt, and Nosyk (2017) and also uses a nonlocal three-nucleon interaction. In what follows, we call this Hamiltonian N4LOEMN+N2LONL. All Hamiltonians use a regulator cutoff of in the two- and three-body sector. For hypernuclei, we combine them with a LO hyperon-nucleon interaction Polinder et al. (2006) with a cutoff of . The calculations are carried out with an oscillator frequency of , which is close to the variational minimum. For the N3LOEM+N2LOL Hamiltonian, we use an SRG flow parameter of as in our previous publications. The other Hamiltonians are evolved to wich provides faster convergence and an improved description of the ground-state energy. A detailed discussion of these Hamiltonians and applications to non-strange nuclei is presented elsewhere Hüther et al. (2019).
To investigate the effect of the nucleonic Hamiltonian on the properties of hypernuclei, we consider the hypernuclei , , and , as well as their nonstrange parent nuclei. Here, we compare the N3LOEM+N2LOL and the N4LOEMN+N2LONL Hamiltonians in order to assess the effect of switching to the new generation of chiral Hamiltonians.
Fig. 1 shows the absolute and excitation energies of and . In the absolute energies for the parent nucleus, we see that the N4LOEMN+N2LONL Hamiltonian provides approx. less binding for both the ground state and the first excited state. This feature carries over to the hypernucleus and cancels some of the overbinding generated by the YN interaction, bringing the calculated absolute energies closer to the experimental ones. Excitation energies of the parent and the hypernucleus show very little variation between the two Hamiltonians. Only the excitation energy changes by approx. , which is reflected in the excitation energies of the doublet in the hypernucleus. The excitation energy shows almost no variation at all.
The situation for and , shown in Fig. 2, is very similar to , just that the N4LOEMN+N2LONL Hamiltonian now provides less binding energy. In (cf. Fig. 3), the difference increases to more than . Additionally, the excitation energy of the is almost lower for the N4LOEMN+N2LONL, increasing the difference to the experimental value. The lower excitation energy translates to the excited-state doublet in the hypernucleus, where we also notice slightly different doublet spacings and a different convergence behavior.