Coupled-cluster calculations of nucleonic matter

Coupled-cluster calculations of nucleonic matter

G. Hagen Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996, USA    T. Papenbrock Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996, USA Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA    A. Ekström Department of Physics and Center of Mathematics for Applications, University of Oslo, N-0316 Oslo, Norway National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, MI 48824, USA    K. A. Wendt Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996, USA Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA    G. Baardsen Department of Physics and Center of Mathematics for Applications, University of Oslo, N-0316 Oslo, Norway    S. Gandolfi Theoretical Division, Los Alamos National Laboratory Los Alamos, NM 87545    M. Hjorth-Jensen Department of Physics and Center of Mathematics for Applications, University of Oslo, N-0316 Oslo, Norway National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, MI 48824, USA Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA    C. J. Horowitz Indiana University, Bloomington, IN 47405, USA
Abstract

Background: The equation of state (EoS) of nucleonic matter is central for the understanding of bulk nuclear properties, the physics of neutron star crusts, and the energy release in supernova explosions. Because nuclear matter exhibits a finely tuned saturation point, its EoS also constrains nuclear interactions.
Purpose: This work presents coupled-cluster calculations of infinite nucleonic matter using modern interactions from chiral effective field theory (EFT). It assesses the role of correlations beyond particle-particle and hole-hole ladders, and the role of three-nucleon-forces (3NFs) in nuclear matter calculations with chiral interactions.
Methods: This work employs the optimized nucleon-nucleon () potential NNLO at next-to-next-to leading-order, and presents coupled-cluster computations of the EoS for symmetric nuclear matter and neutron matter. The coupled-cluster method employs up to selected triples clusters and the single-particle space consists of a momentum-space lattice. We compare our results with benchmark calculations and control finite-size effects and shell oscillations via twist-averaged boundary conditions.
Results: We provide several benchmarks to validate the formalism and show that our results exhibit a good convergence toward the thermodynamic limit. Our calculations agree well with recent coupled-cluster results based on a partial wave expansion and particle-particle and hole-hole ladders. For neutron matter at low densities, and for simple potential models, our calculations agree with results from quantum Monte Carlo computations. While neutron matter with interactions from chiral EFT is perturbative, symmetric nuclear matter requires nonperturbative approaches. Correlations beyond the standard particle-particle ladder approximation yield non-negligible contributions. The saturation point of symmetric nuclear matter is sensitive to the employed 3NFs and the employed regularization scheme. 3NFs with nonlocal cutoffs exhibit a considerably improved convergence than their local cousins. We are unable to find values for the parameters of the short-range part of the local 3NF that simultaneously yield acceptable values for the saturation point in symmetric nuclear matter and the binding energies of light nuclei.
Conclusions: Coupled-cluster calculations with nuclear interactions from chiral EFT yield nonperturbative results for the EoS of nucleonic matter. Finite-size effects and effects of truncations can be controlled. For the optimization of chiral forces, it might be useful to include the saturation point of symmetric nuclear matter.

pacs:
21.65.Mn, 21.65.Cd, 21.30.-x, 21.65.-f, 03.75.Ss, 26.60.-c, 26.60.Kp

I Introduction

Bulk nucleonic matter is interesting for several reasons. The EoS of neutron matter, for instance, determines properties of supernova explosions Burrows (2013), and of neutron stars Weber (1999); Heiselberg and Hjorth-Jensen (2000); Lattimer and Prakash (2007); Sammarruca (2010); Lattimer (2012); Hebeler et al. (2013), and it links the latter to neutron radii in atomic nuclei Alex Brown (2000); Horowitz and Piekarewicz (2001); Gandolfi et al. (2012) and symmetry energy Tsang et al. (2012); Steiner and Gandolfi (2012). Likewise, the compressibility of nuclear matter is probed in giant dipole excitations Shlomo and Youngblood (1993), and the symmetry energy of nuclear matter is related to the difference between proton and neutron radii in atomic nuclei Abrahamyan et al. (2012); Reinhard et al. (2013); Erler et al. (2013). The saturation point of nuclear matter determines bulk properties of atomic nuclei, and is therefore an important constraint for nuclear energy-density functionals and mass models (see, e.g., Refs. Kortelainen et al. (2010); Lunney et al. (2003)).

The determination and our understanding of the EoS for nuclear matter is intimately linked with our capability to solve the nuclear many-body problem. Here, correlations beyond the mean field play an important role. Theoretical studies of nuclear matter and the pertinent EoS span back to the very early days of nuclear many-body physics. Early computations are nicely described in the 1967 review by Day Day (1967). These early calculations were performed using Brueckner-Bethe-Goldstone theory Brueckner et al. (1954); Brueckner (1955), see Refs.Heiselberg and Hjorth-Jensen (2000); Baldo and Burgio (2012); Baldo et al. (2012) for recent reviews and developments. In these calculations, mainly particle-particle correlations were summed to infinite order. Other correlations were often included in a perturbative way. Coupled-cluster calculations of nuclear matter were performed already during the late 1970s and early 1980s Kümmel et al. (1978); Day and Zabolitzky (1981). In recent years, there has been a considerable algorithmic development of first-principle methods for solving the nuclear many-body problem. A systematic inclusion of other correlations in a non-perturbative way are nowadays accounted for in Monte Carlo methods Carlson et al. (2003); Gandolfi et al. (2009); Gezerlis and Carlson (2010); Lovato et al. (2012); Gezerlis et al. (2013), self-consistent Green’s function approaches Dickhoff and Barbieri (2004); Somà and Bożek (2008); Rios and Somà (2012); Baldo et al. (2012); Carbone et al. (2013) and nuclear density functional theory Lunney et al. (2003); Erler et al. (2013).

Similar progress has been made in the derivation of nuclear forces based on chiral EFT Machleidt and Entem (2011); Epelbaum et al. (2009a). Nuclear Hamiltonians from chiral EFT are now used routinely in nucleonic matter calculations, with the 3NFs van Kolck (1994); Epelbaum et al. (2002) being front and center of many studies Somà and Bożek (2008); Epelbaum et al. (2009b); Hebeler and Schwenk (2010); Hebeler et al. (2011); Holt et al. (2012); Hammer et al. (2013); Krüger et al. (2013); Carbone et al. (2013); Coraggio et al. (2013). We note finally that there are also approaches to nucleonic matter based on lattice quantum chromodynamics Inoue et al. (2013).

In this work we study the EoS of nucleonic matter, using modern interactions and 3NFs from chiral EFT, and an implementation of the coupled-cluster method Dean and Hjorth-Jensen (2004); Kowalski et al. (2004) that has become a standard in quantum chemistry Bartlett and Musiał (2007); Shavitt and Bartlett (2009). We employ a Cartesian momentum space basis with periodic boundary conditions, similar to the recent coupled-cluster based calculations of the electron gas Shepherd et al. (2012); Roggero et al. (2013). Our calculations are based on coupled cluster with doubles (CCD) approximation Harris et al. (1992); Freeman (1977); Bishop and Lührmann (1978). This is the lowest-order truncation for closed-shell systems in a momentum-space basis, and we will also explore the role of selected triples clusters. We employ a recent parameterization Ekström et al. (2013) of the force from chiral EFT at next-to-next-to-leading order, with inclusion of the 3NF that enters at the same chiral order.

This paper is organized as follows. In the next Section we present the coupled-cluster formalism for infinite matter that includes 3NFs and perturbative triples corrections. The calculations are performed in Cartesian coordinates with a discrete momentum basis and twisted periodic boundary conditions Gros (1992, 1996); Lin et al. (2001). This avoids the tedious partial-wave expansion of the nuclear forces, and it eases considerably the numerical evaluation of 3NFs. Averaging over twisted periodic boundary conditions minimizes finite-size effects and provides us with a good convergence towards the thermodynamic limit. Section II also presents computational results for finite-size effects and a few benchmark calculations. Our results for symmetric nuclear matter and pure neutron matter are presented in Sect. III. Concluding remarks are given in Sect. IV.

Ii Method

In this Section we present the coupled-cluster formalism for infinite matter. We discuss the inclusion and treatment of forces and 3NFs from chiral effective field theory (EFT), correlations up to three-particle-three-hole excitations and finite-size effects. Several benchmark calculations give us confidence in the validity of our approach.

ii.1 Interaction and model space

Our Hamiltonian is

Here, denotes the kinetic energy, and and denote the translationally invariant interaction and 3NF. The interaction and 3NF are from chiral effective field theory Epelbaum et al. (2009a); Machleidt and Entem (2011) at next-to-next-to-leading order (NNLO). We employ the parameterization NNLO for the interaction Ekström et al. (2013), and the local 3NF Navrátil (2007). This 3NF has a local regulator, i.e. the cutoff is in the momentum transfer, and thereby differs from implementations of the 3NF Epelbaum et al. (2002) that employ the cutoff in the relative Jacobi momenta. We note that the numerical implementation of the 3NF in the discrete momentum basis is much simpler than in the harmonic oscillator basis commonly used for finite nuclei, because essentially no transformation of matrix elements is necessary. Nevertheless, the sheer number of matrix elements (and associated function calls) of the 3NF is huge, and this is computationally still a limiting factor.

For the model space, we choose a cubic lattice in momentum space with momentum points. The spin (spin-isospin) degeneracy of each momentum point is () for pure neutron matter (nuclear matter). Thus, filling of the lattice yields shell closures for “Fermi spheres” with fermions, and . We note that one could also use non-cubic lattices. Any periodic lattice permits one to implement momentum conservation exactly. For fixed particle number and density (or Fermi momentum ), one computes the volume of the cubic box , and the box length that determines the lattice spacing . We note that the computed results exhibit a dependence on the shell closure . However, Subsection II.4 shows that shell effects and finite-size effects can be mitigated and controlled, and that the dependence on the parameter becomes very small.

The second parameter of our lattice is . We note that is the momentum cutoff of our single-particle basis. One has to increase until the computed results (e.g. the energy per nucleon) is practically independent of this parameter. For the results reported below we find that is sufficient.

ii.2 Coupled-cluster theory for infinite systems

In this Section we present the coupled-cluster equations for nucleonic matter. Our calculations of nucleonic matter are based on the recently optimized chiral nucleon-nucleon interaction at NNLO Ekström et al. (2013) with the 3NF at the same chiral order. The low-energy constants (LECs) of the 3NF were determined by fitting the constants and to reproduce the experimental half-life and binding energy of the triton. The optimized 3NF LECs are and . With these values the He binding energy is  MeV Navratil et al. (). We employ single-particle states

with momentum , spin projection and isospin projection . Discrete values of the momentum variable result from periodic boundary conditions in a cubic box with length , that is

In this basis, the nuclear Hamiltonian with nucleon-nucleon and three-nucleon interactions is

The kinetic energy is diagonal in the discrete momentum basis . The operators and create and annihilate a nucleon in state , respectively.

The discrete momentum basis allows us to respect translational invariance of the potential and the 3NFs. Momentum is conserved, meaning that the two- and three-body matrix elements of the Hamiltonian (II.2) vanish unless

and

Note also that the chiral nucleon-nucleon and three-nucleon interactions conserve the total isospin projection, but not the total spin projection.

In single-reference coupled-cluster theory the correlated wave-function is written in the form

Here is a product state and serves as the reference. The cluster operator is a linear combination of -particle--hole (-) excitation operators, i.e. . In the discretized momentum basis the reference state is the closed shell Fermi vacuum, and is obtained by filling the states with the lowest kinetic energy. We limit ourselves to spin saturated reference state, meaning that each momentum orbital of the reference state is doubly occupied. In this case the nuclear interaction does not induce - excitations of the reference state, and we have . Thus, the cluster operator becomes

Here and in what follows, indices () label occupied (unoccupied) states. Truncating at the 2-2 excitation level () gives the coupled-cluster doubles (CCD) approximation. The CCD energy and amplitude equations can be written in compact form

(2)
(3)

Here

(4)

is the vacuum expectation value (which in the case of no 1-1 corresponds to the Hartree-Fock energy), is a 2-2 excitation of the reference state, and is the similarity transformation of the normal-ordered Hamiltonian

Here is the normal ordered string of operators with respect to the reference state. The normal-ordered one-body operator is given in terms of the Fock matrix elements

(6)

The normal-ordered two-body operator has matrix elements

(7)

Finally, the normal-ordered three-body operator has matrix elements

(8)

In most of this work, we will neglect all elements of when solving the CCD equations. In this normal-ordered two-body approximation, the 3NF enters in the vacuum expectation value (4), the Fock matrix (6), and the normal-ordered two-body operator (7), but the three-body operator that changes the orbitals of all three nucleons is neglected.

We note that coupled-cluster theory with full inclusion of 3NFs was worked out in the singles and doubles approximation (CCSD)  Hagen et al. (2007), and very recently with triples corrections included Binder et al. (2013).

For an efficient numerical implementation one writes the CCD equations (3) in a factorized (quasi-linear) form,

(9)

Here, is an antisymmetrization operator, and we employed the intermediates

(10)
(11)
(12)
(13)
(14)

In Eqs. (9, 10, 11, 12, 13) and (14) the numerically expensive sums that involve products of two-body operators can all be implemented efficiently as matrix-matrix multiplications. The momentum conservation reduces the computational cost of the CCD equations to , where () is the number of occupied (unoccupied) momentum states. This is a considerable reduction in computational cycles as compared to the normal cost of the CCD equations which is Bartlett and Musiał (2007), and similar to the reduction of computational cost achieved in the angular momentum coupled scheme Hagen et al. (2008, 2010).

The coupled-cluster equations (9) are solved numerically by iteration and yield the matrix elements of . The CCD energy (2) is given in algebraic form by

Below, we will also employ an approximation (denoted as CCD) that only uses the particle-particle and hole-hole ladders in the CCD equations, i.e.

(15)

The CCD approximation was used in Ref. Baardsen et al. (2013) within coupled-cluster theory, and a similar approximation was also employed in other computations of nucleonic matter, see, e.g., Refs. Hebeler and Schwenk (2010); Hebeler et al. (2011).

Let us also discuss the inclusion of three-body clusters. When going beyond the CCD approximation and considering triples excitations, one might question whether the residual three-body part can safely be neglected. After all, three-body forces directly induce excitations of three-body clusters. Below we will include the residual part when considering contributions from triples excitations to the correlation energy, and study the accuracy of the normal-ordered two-body approximation in the presence of triples excitations in neutron and symmetric nuclear matter. Very recently, Binder et al. employed chiral interactions softened via the similarity renormalization group transformation Bogner et al. (2007); Roth et al. (2012), studied the effect of triples corrections in the presence of 3NFs in nuclei such as O and Ca, and found it to be small Binder et al. (2013).

The full inclusion of triples in the presence of three-body forces is demanding and computationally expensive. Some effects of triples can be included in the CCD(T) approximation Raghavachari (1985) that we extend to 3NFs. In CCD(T) the triples excitation amplitude is approximated as

(16)

Here

(17)

The CCD(T) correction to the energy is

(18)

Employing the triples amplitude (16) with the inclusion of yields the energy correction . We also consider the following approximations. Neglecting the residual three-body part yields the normal-ordered two-body approximation to the CCD(T) energy correction, denoted as . Omitting the term in Eq. (16) gives the energy correction . Note that the numerically expensive term in Eq. (16) consist of three distinct diagrams in which one sums over , and intermediate states, respectively. Below we will investigate the contributions of these three diagrams to the CCD(T) energy correction in neutron and symmetric nuclear matter.

ii.3 Ladder approximation in a partial-wave basis

In Ref. Baardsen et al. (2013), the ladder approximation of the coupled-cluster equations for nuclear matter is presented in an alternative formulation. Historically, the equations for nuclear matter, for example in the hole-line approximation Day (1978), have often been expressed explicitly in a partial-wave basis Day (1981); Haftel and Tabakin (1970); Suzuki et al. (2000). Similarly, in the method presented in Ref. Baardsen et al. (2013), the ladder approximation is formulated in a partial-wave basis, assuming that the thermodynamic limit is reached and therefore using integrals over relative and center-of-mass momenta. In the partial-wave expanded equations, the Pauli exclusion operators are treated exactly, using a technique introduced for the Brueckner-Hartree-Fock approximation by Suzuki et al. Suzuki et al. (2000). Apart from the truncation in partial waves, the only approximation in this method is in the single-particle potentials, where an angular-average approximation was used for the laboratory momentum argument Brueckner and Gammel (1958); Baardsen et al. (2013).

ii.4 Finite size effects

We would like to quantify the error due to finite size effects and the accuracy of our coupled-cluster calculations of neutron and nuclear matter. Using periodic boundary conditions (PBC) one could increase the number of particles in the box until convergence to the thermodynamic limit is reached. However, due to variations of the shell effects at different closed shell configurations, there is no guarantee that increasing the number of particles will lead to a systematic and smooth convergence to the thermodynamic limit. Furthermore, the computational cost of many-body methods such as the AFDMC and coupled-cluster methods increases rapidly with increasing particle number, and one would therefore like to employ a method that controls finite size effects already for modest particle numbers. This can be achieved with averaging over phases of Bloch waves that correspond to different boundary conditions Gros (1992, 1996); Lin et al. (2001).

Consider a free particle in a box of size subject to twisted boundary condition, that is, the wave function with momentum fulfills the condition for so-called Bloch waves, namely, . By averaging over the twist angle , shell effects can be eliminated for free Fermi systems Gros (1992), and they are much suppressed for interacting systems Gros (1996); Lin et al. (2001). In this way, one obtains a much more systematic and smooth convergence towards the thermodynamic limit. The twisted boundary conditions are defined by

with the twist angle for systems with time-reversal invariance Lin et al. (2001). This amounts to letting the particles pick up a complex phase when they wrap around the boundary of the cubic box. By integrating or averaging over a finite number of twists in each direction we obtain the twist-averaged boundary conditions (TABC). In our implementation of TABC we integrate over the twist angles using a finite number of Gauss-Legendre quadrature points in . Note that () corresponds to (anti-)periodic boundary conditions.

In order to quantify the finite size effects using PBC and TABC we compute the kinetic and potential energy contribution to the Hartree-Fock energy for several closed shell configurations ranging from tens to several hundreds of nucleons, and compare with the thermodynamic limit for these quantities. In Fig. 1 we show the relative error of the kinetic energy in pure neutron matter for the Fermi momentum computed using standard PBC and TABC. We used 10 Gauss-Legendre points for the twist angle of the direction in the integration interval . Clearly, we obtain a much faster and smoother convergence to the thermodynamic limit using TABC. Generally we get about an order of magnitude reduction in the relative error when using TABC as compared to PBC. Finite size effects are particularly small for PBC and neutrons. This was also seen in AFDMC calculations Gandolfi et al. (2009).

Figure 1: (Color online) Relative finite-size corrections for the kinetic energy in pure neutron matter at the Fermi momentum vs. the neutron number . TABC10 are twist-averaged boundary conditions with 10 Gauss-Legendre points in each spatial direction.

Figure 2 shows the relative error of the potential energy to the Hartree-Fock energy in pure neutron matter for the Fermi momentum computed with TABC. We compute the potential energy from NNLO and from the Minnesota potential. We see that the finite size effects in the potential energy are comparable to the finite size effects in the kinetic energy shown in Fig. 1. We note that finite size effects vanish as the power law in the neutron number .

Figure 2: (Color online) Relative finite-size corrections for the Hartree-Fock energy of the NNLO (full line) and Minnesota (dashed line) potentials in pure neutron matter at the Fermi momentum vs. the neutron number . TABC10 are twist-averaged boundary conditions with 10 Gauss-Legendre points in each spatial direction. The dashed-dotter line shows a power law fit to the NNLO results.

Finally, we would also like to assess the finite-size effects in symmetric nuclear matter. In Fig. 3 we show the relative error of the potential energy to the Hartree-Fock energy in symmetric nuclear matter for the Fermi momentum computed using PBC and TABC. We consider the Hartree-Fock potential energy contribution from the nucleon-nucleon interaction NNLO and the 3NF at order NNLO separately. In particular it is seen that the relative error in the potential energy contribution from the 3NF is about an order of magnitude smaller than the relative error coming from the nucleon-nucleon interaction alone using both PBC and TABC. In the case of symmetric nuclear matter there is no systematic convergence trend using PBC, and for 132 nucleons the relative error for PBC is around , while using TABC the error is reduced to . It is interesting to note that finite size effects for NNLO with TABC decrease as with increasing nucleon number . This exponent is similar to the exponent found in neutron matter (see Fig. 2).

Figure 3: (Color online) Relative finite-size corrections for the Hartree-Fock energy of the potential NNLO and the 3NF potential in symmetric nuclear matter at the Fermi momentum vs. the nucleon number . PBC: periodic boundary conditions. TABC10 and TABC3 are twist-averaged boundary conditions with 10 and 3 Gauss-Legendre points in each spatial direction, respectively.

Coupled-cluster calculations of nucleonic matter using TABC are very expensive. Using 10 twist angles in each direction requires coupled-cluster calculations, although symmetry considerations can reduce this number considerably. In Ref. Lin et al. (2001) it was shown that one can find a specific choice of twist angles (known as special points), in which the Hartree-Fock energy exactly corresponds to the Hartree-Fock energy in the thermodynamic limit. In the following we compute these special points for neutron and nuclear matter using both interactions and 3NFs, and compare with calculations using PBC and TABC.

ii.5 Benchmarks

It is interesting to compare the results for various boundary conditions with the infinite matter results by Baardsen et al. Baardsen et al. (2013). Figure 4 shows the CCD results for neutron matter computed with the nucleon-nucleon potential NNLO. In a finite system, the neutron number is very close to the infinite matter results for both periodic and twist-averaged boundary conditions.

Figure 4: (Color online) Energy per particle of neutron matter for NNLO computed in the CCD approximation with periodic boundary conditions (circles), twist-averaged boundary conditions (squares), and for infinite matter (diamonds). The latter results are from Ref. Ekström et al. (2013). The calculations used neutrons and .

For symmetric nuclear matter, the CCD results are more sensitive to the choice of the boundary conditions, with results shown in Fig. 5. At higher Fermi momenta (), the energy per nucleon for periodic boundary conditions differs by MeV from the result obtained with twist-averaged boundary conditions. A calculation with a special point in the twist is very close to the twist-averaged results. However, for Fermi momenta , the difference between the PBC and TABC is less than  keV per nucleon.

Figure 5: (Color online) Energy per particle of symmetric nuclear matter for NNLO computed in the CCD approximation with periodic boundary conditions (diamonds), twist-averaged boundary conditions (squares), and with a special point and twisted boundary conditions (circles). The calculations used nucleons and .

Figure 6 compares nuclear matter results calculated in the ladder approximation with the CC of Ref. Baardsen et al. (2013). The latter were obtained by taking the thermodynamic limit in the relative and center-of mass frame and by summing over partial waves. The summation over intermediate particle-particle and hole-hole configurations is performed with an exact Pauli operator, while the single-particle energies are computed using an angle-averaging procedure, see Ref. Baardsen et al. (2013) for further details. For these results, the angle-average approximation, together with a truncation in the number of partial waves included, represent the sources of possible errors in the thermodynamic limit. It is therefore very satisfactory that the results from different methods are close to each other.

Figure 6: (Color online) Energy per particle of symmetric nuclear matter for NNLO computed in the CCD approximation in the thermodynamic limit using partial-wave expansion (triangles) (partly adapted from Ref. Baardsen et al. (2013)), with twist averaged boundary conditions (squares), and with a special point and twisted boundary conditions (circles). The calculations used nucleons and .

Let us also consider a simple potential model and benchmark the results of our coupled-cluster calculations against virtually exact results from the auxiliary field diffusion Monte Carlo (AFDMC) method Schmidt and Fantoni (1999). The Minnesota potential Thompson et al. (1977) is a semi-realistic nucleon-nucleon interaction that can be solved accurately with AFDMC. It depends only on the relative momenta and spin, but lacks spin-orbit or tensor contributions. The matrix elements of this potential are real numbers. For the benchmark we employ periodic boundary conditions, neutrons, and .

Figure 7 compares the energy per neutron of our lattice CCD results (circles), and our CCD in the thermodynamic limit, see Ref. Baardsen et al. (2013), to the AFDMC benchmark. Overall, the agreement is good between all methods. As expected, the CCD results are more accurate than the CCD approximation.

Figure 7: (Color online) Energy per particle of neutron matter, computed with the Minnesota potential. Diamonds: AFDMC, circles: CCD, squares: CCD limited to and ladders.

Finally, we turn to 3NFs. The inclusion of 3NFs – even in the normal-ordered approximation – is still numerically expensive due to the large number of required matrix elements. We also study different approximations for 3NFs, and compare the results for symmetric nuclear matter when 3NFs only enter in the normal-ordered approximation as 0-body, 1-body, or up to 2-body forces. Figure 8 clearly shows that normal-ordered 2-body forces are relevant.

Figure 8: (Color online) Energy per particle of symmetric nuclear matter computed in the CCD approximation with NNLO + 3NF (c-0.389, c-0.39). The 3NF is included in the 0-body (black dashed-dotted line), 1-body (red dashed line), and in the two-body (blue solid line) normal-ordered approximations. The calculations used nucleons and .

Iii Results for chiral interactions

In this Section, we present our results for coupled-cluster computations of neutron matter and symmetric nuclear matter. As shown in the previous Section, the finite size effects (and the differences between PBC and TABC) are small for neutrons and nucleons when calculating neutron matter and symmetric nuclear matter, respectively. For this reason, many of the expensive calculations involving 3NFs are only performed with PBC at these specific particle numbers.

iii.1 Neutron matter

Figure 9 shows the energy per neutron as a function of density based on interactions alone and compares various many-body methods. The employed interaction NNLO is perturbative in neutron matter, with second-order many-body perturbation theory (MBPT2), CCD and CCD giving similar results that differ by less than 1 MeV per neutron at nuclear saturation density.

Figure 9: (Color online) Energy per particle in neutron matter with NNLO (NN only). The black dashed line is Hartree-Fock (HF), the green dashed-dotted line is second-order many-body perturbation theory (MBPT2), the blue dashed line is coupled-cluster doubles ladder approximation (CCD), and the red solid line is coupled-cluster doubles (CCD). The calculations used neutrons, , and TABC3.

Figure 10 shows the effect of 3NFs in CCD calculations of the EoS for neutron matter. We consider several approximations involving 3NFs, and it is seen that they yield very similar results. We note that three-nucleon forces act repulsively. The results for neutron matter reported here are consistent with the recent calculations of Krueger et al. Krüger et al. (2013), and our results for the EoS fall within their NNLO uncertainty band. The CCD calculation that includes the normal-ordered 3NFs is shown as diamonds. Triples corrections that are limited to the inclusion of up to two-body terms from the normal-ordered 3NF are shown as circles, while triples corrections that include also the residual 3NF are shown as squares. For neutron matter, the effects of triples are small and account for about 0.3 MeV per neutron at high densities, and the residual 3NFs contribute little to the triples corrections.

Figure 10: (Color online) Energy per particle in neutron matter with NNLO (NN only) and with inclusion of 3NF computed in the CCD and CCD(T) approximations. The 3NF LECs are given by and . The calculations used neutrons, , and PBC and TABC.

iii.2 Nuclear matter

In this Subsection we perform coupled-cluster calculations of symmetric nuclear matter using chiral and 3NF interactions at NNLO. Figure 11 shows the energy per nucleon in symmetric nuclear matter for a wide range of densities computed in MBPT2, the CCD, and in the CCD approximation with the potential NNLO. In these calculations we used nucleons, , and TABC based on angles. We observe that the saturation point is at a too large density, and we get a considerable overbinding. These results for NNLO are in good agreement with the recent self-consistent Green’s function (SCGF) calculations of nuclear matter Carbone et al. (2013), and the CCD calculations of Ref. Baardsen et al. (2013). The difference between MBPT2 and CCD is considerable, indicating that nuclear matter for the NNLO chiral interaction is not perturbative. The difference between the CCD approximation and the full CCD calculations is around 1 MeV per nucleon around saturation density. We can conclude that – in contrast to neutron matter – for nuclear matter and the NNLO interaction (which is rather soft), non-linear terms in the amplitude and particle-hole excitations yield non-negligible contributions. We note also that the coupled-cluster calculations are difficult to converge for Fermi momenta smaller than about . This is presumably due to the clustering of nuclear matter at low densities Horowitz and Schwenk (2006).

Figure 11: (Color online) Energy per particle in symmetric nuclear matter with NNLO (NN only) computed in the MBPT2 (triangles with green dashed-dotted line), CCD (circles with blue dashed line), and CCD (squares with solid red line) approximations. The calculations used 132 nucleons, , and PBC. Diamonds are results from self-consistent Green’s function (SCGF) at the finite temperature  MeV, taken from Ref. Carbone et al. (2013).

Let us turn to 3NFs. Figure 12 shows the energy per nucleon in symmetric nuclear matter for a wide range of densities computed with MBPT2, CCD, and the CCD(T) approximation. The CCD calculations included the 3NF in the normal-ordered two-body approximation. The CCD(T) calculations were performed with 3NFs in the normal-ordered two-body approximation (CCD(T+3NF)), and going beyond the normal-ordered two-body approximation by including the leading-order residual 3NF contribution to the perturbative estimate for the amplitude (CCD(T+3NF)). In these calculations we used nucleons with PBC and . For the densities we consider here, the difference between PBC and TABC is small.

Figure 12: (Color online) Energy per particle in symmetric nuclear matter with NNLO and 3NF computed in the MBPT2 (squares with green dashed-dotted line), CCD (circles with blue dashed line), and CCD(T) with 3NF in the normal-ordered two-body approximation (diamonds with solid red line), and including the residual 3NF in leading order (triangles with dotted black line). The 3NF LECs are given by and . The calculations used 132 nucleons, , and PBC.

In contrast to calculations of neutron matter, the contribution from the perturbative triples corrections is sizable in nuclear matter, and about 1 MeV per nucleon in the range of densities shown when including the 3NF in the normal-ordered two-body approximation. Furthermore, we find that the contribution of the residual 3NF to the CCD(T) energy is significant around saturation density, indicating that the normal-ordered two-body approximation for the 3NF might not be sufficient in symmetric nuclear matter. We checked that the contribution of the residual 3NF to the CCD amplitude equations is negligible, and therefore it might be sufficient to include the full 3NF in the perturbative triples amplitude. In order to check the accuracy of the perturbative triples approximation (CCD(T)) in nuclear matter we also performed non-perturbative, iterative CCDT-1 (see Refs. Lee and Bartlett (1984); Lee et al. (1984)) calculations for and at two different densities and . We found that the difference between CCD(T) and CCDT-1 in this range of densities is at most 0.1 MeV per nucleon. Therefore, we conclude that the CCD(T) approximation is accurate for the potential NNLO and chiral 3NFs in symmetric nuclear matter.

iii.3 Scheme dependence of three-nucleon forces

In this Subsection, we try to further illuminate the role of 3NFs in nucleonic matter. We study different regularization schemes, and compute the energy per particle in pure neutron matter and symmetric nuclear matter. The 3NF employed in the previous Subsections exhibits a cutoff of  MeV. This cutoff is in the momentum transfer, and therefore local in position space Navrátil (2007). This choice of regulator for the 3NF is different from the regularization scheme that is used in the nucleon-nucleon sector, and from other regularizations of the 3NF that exhibit cutoffs on Jacobi momenta Epelbaum et al. (2002). We note that regulators that cut off initial and final Jacobi momenta lead to non-local interactions. Here, the cutoff function is

with and . This regulator reduces to the regulator used in the sector for . In the potential NNLO we use , while for the local regulator of the 3NF defined in Ref. Navrátil (2007) we use in the exponential. In what follows, we compare the NNLO interaction with a 3NF that also uses a local regulator but a lower cutoff of  MeV, and with a 3NF that employs a nonlocal regulator and a cutoff  MeV in relative Jacobi momenta.

Figure 13 shows the energy per particle in pure neutron matter computed in the CCD(T) approximation. Here we included 3NFs in the normal ordered two-body approximation, and in the CCD(T) approximation. For the latter, we went beyond the normal ordered two-body approximation and included the residual three-body term that enters at first order in the triples equation for . In neutron matter the contribution from the residual 3NF to the energy per particle is small. This indicates that the normal-ordered two-body approximation works very well. In the EoS calculation with the local regulator and the lower cutoff  MeV we adjusted the LECs of the three-body contact term to and kept unchanged. Then, the binding energies of the triton and the nuclei He are close to the experimental values. For the non-local regulator with cutoff and power in the exponential, the LECs and reproduce the triton and He binding energies. In pure neutron matter the contributions from the 3NF contact terms with the LECs and vanish for a non-local regulator, and the contribution to the EoS depends only on the pion-nucleon couplings and of the long-range two-pion exchange term of the 3NF Hebeler and Schwenk (2010). However, for a local regulator the 3NF contact terms do not vanish in neutron matter Lovato et al. (2012). The results for the EoS for pure neutron matter show a regulator dependence at densities beyond . The band obtained from the different 3NF regulators are within the corresponding band for neutron matter obtained in Ref. Krüger et al. (2013).

Figure 13: (Color online) Energy per particle in pure neutron matter with NNLO and 3NF computed in the CCD(T) approximation including 3NFs in the normal ordered two-body approximation and including the residual 3NF in the CCD(T) approximation. For the 3NF we used a local regulator with cutoffs and  MeV. The 3NF LECs are given by and for the  MeV local regulator, while for the  MeV local regulator we used and with adjusted to the He binding energy. For the non-local regulator with  MeV cutoff we used and adjusted to the triton and He binding energies. The calculations used 66 neutrons, , and PBC

Figure 14 shows the corresponding plot for the energy per particle in symmetric nuclear matter. Here the results for the local regulator with a cutoff  MeV exhibit a considerable enhancement of the contribution from the residual 3NF to the energy per particle at densities above the saturation densities. The sizeable triples contribution of the residual 3NF questions the usually observed hierarchy of the coupled-cluster approximation. The results from the lower cutoff  MeV are much more satisfactory in the sense that the contribution from the residual three-body part to the binding energy per particle is considerably smaller, and at the order of 0.5 MeV or less for the densities considered. Likewise, the results obtained with the non-local regulator at the cutoff  MeV are also satisfactory in the sense that the contribution from the residual 3NF is at most  MeV to the energy per particle at densities beyond the saturation point. One might speculate whether this problematic feature of the local regulator with a cutoff  MeV is related to the large cutoff dependence found in finite nuclei using this regulator Roth et al. (2012). Naively one would expect that regulator dependencies are higher-order corrections in an EFT. The large scheme dependencies observed in Fig. 14 might therefore suggest that the cutoff  MeV is too close to the EFT breakdown scale.

Figure 14: (Color online) Same caption as in Fig. 13, except that the energy per particle is for symmetric nuclear matter. The calculations used 132 nucleons, , and PBC

For local and non-local regulators we considerably underbind nuclear matter. The saturation density for the local regulators is too high, while for the non-local regulator the saturation density is closer to the empirical value. We tried to adjust the LECs and such that an acceptable result could be obtained simultaneously for the saturation point in symmetric nuclear matter and the triton binding energy. For the non-local regulator the result is shown in Fig. 15. The blue band shows the region where the triton binding energy is reproduced within 5%. The red band shows the region where the saturation Fermi momentum is within 5% of its empirical value, and the green band shows the region where the energy per nucleon is within 5% of the empirical value. The nuclear matter calculations were obtained from MBPT2 calculations using 28 nucleons, and we accounted for about 1 MeV per nucleon in missing correlations energy, and about 0.5 MeV per nucleon due to finite size effects. It thus seems that a simultaneous reproduction of saturation in light nuclei and infinite matter is not possible without adjusting other LECs. As an example we considered the point and . This yields the saturation point and  MeV, while the triton binding energy is  MeV.

Figure 15: (Color online) The blue band shows the region where the triton binding energy is reproduced within 5% of the experimental value. The red band shows the region where the saturation fermi momentum in symmetric nuclear matter is reproduced within 5% of its empirical value, and the green band shows the region where the energy per nucleon is within 5% of the empirical value.

We would like to understand better the role that different regulators and cutoffs play for the chiral 3NF. Unfortunately, it is difficult to visualize 3NFs in momentum space Hebeler (2012); Wendt (2013). We therefore compute the MBPT2 contribution of the residual 3NF and cut off the involved momentum integrations at a single-particle momentum . Figure 16 shows the fractional contribution of the MBPT2 energy correction of the residual 3NF as a function of at the Fermi momentum  fm. The chiral cutoff of  MeV is also shown as a dashed line for comparison. We see that for the local cutoff  MeV most contributions to the MBPT2 result are from high single-particle momenta that are well above the nominal chiral cutoff. The situation is improved for the local regulator with lower cutoff  MeV and even more so for the nonlocal regulator with cutoff  MeV. For a discussion of different cutoff schemes and convergence issues in calculations of the homogeneous electron gas see Ref. Shepherd et al. (2012).

Figure 16: (Color online) Cutoff dependent fraction of the residual 3NF contribution to the MBPT2 energy per particle in symmetric nuclear matter for different regulators. Results are shown for the local regulator with MeV (diamonds), the local regulator with MeV (squares), and for the non-local regulator with MeV (circles). Calculations used 132 nucleons, , and PBC.

Let us finally note that issues with 3NFs also arose in other calculations. Lovato et al. Lovato et al. (2012) pointed out that the equivalence of different chiral 3NF contact terms Epelbaum et al. (2002) is spoiled by local regulators. Roth et al. Roth et al. (2011) used SRG evolution to soften the chiral interaction of Ref. Entem and Machleidt (2003) combined with the local 3NF of Ref. Navrátil (2007), and found that the results in medium-mass nuclei depend considerably on the SRG evolution scale. This dependence is reduced for a cutoff  MeV in the local 3NF Roth et al. (2012). Clearly, more studies of chiral 3NFs are necessary to fully understand regularization scheme dependences.

Iv Summary

We have performed coupled-cluster calculations of nucleonic matter with interactions from chiral EFT at NNLO. The single-particle states consist of a discrete lattice in momentum space, and the implementation of twist-averaged boundary conditions mitigates shell oscillations and finite-size effects. Our benchmark calculations agree well with other well-established methods. We find that neutron matter is perturbative, while symmetric nuclear matter is not perturbative, with significant contributions beyond perturbation theory and particle ladders.

For the employed potential NNLO and 3NFs, the neutron matter results fall within the error estimates of previous calculations for chiral interactions, with 3NFs acting repulsively. For nuclear matter, the empirical saturation could not be reproduced, and the results are very sensitive to the employed regulator (local vs. nonlocal) and cutoff. At larger chiral cutoffs, the nonlocal regulator is preferred over the local one because it corresponds closer to the cutoff generated by the finite single-particle basis. It seems that the variation of the 3NF contact terms alone is insufficient to achieve both an acceptable saturation point of nuclear matter and an acceptable binding of light nuclei.

Acknowledgements.
We thank S. K. Bogner, E. Epelbaum, R. J. Furnstahl, A. Mukherjee, and F. Pederiva for discussions. This work was supported by the Office of Nuclear Physics, U.S. Department of Energy (Oak Ridge National Laboratory), under DE-FG02-96ER40963 (University of Tennessee), DE-FG02-87ER40365 (Indiana University), DE-SC0008499 and DE-SC0008808 (NUCLEI SciDAC collaboration), the Field Work Proposal ERKBP57 at Oak Ridge National Laboratory, the LDRD program at Los Alamos National Laboratory, and the Research Council of Norway under contract ISP-Fysikk/216699. Computer time was provided by the Innovative and Novel Computational Impact on Theory and Experiment (INCITE) program. This research used resources of the Oak Ridge Leadership Computing Facility located in the Oak Ridge National Laboratory, which is supported by the Office of Science of the Department of Energy under Contract No. DE-AC05-00OR22725, and used computational resources of the National Center for Computational Sciences, the National Institute for Computational Sciences, and the Notur project in Norway. Computing time has also been provided by Los Alamos Open Supercomputing. This research also used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.

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