Emergent geometry and duality in the carbon nucleus

The carbon atom provides the backbone for the complex organic chemistry composing the building blocks of life. The physics of the carbon nucleus in its predominant isotope, 12C, is similarly full of multifaceted complexity. Here we provide a model-independent density map of the geometry of the nuclear states of 12C using the ab initio framework of nuclear lattice effective field theory. We find that the well-known but enigmatic Hoyle state is composed of a “bent-arm” or obtuse triangular arrangement of alpha clusters. We identify all of the low-lying nuclear states of 12C as having an intrinsic shape composed of three alpha clusters forming either an equilateral triangle or an obtuse triangle. The states with the equilateral triangle formation also have a dual description in terms of particle-hole excitations in the mean-field picture.

We convert the eigenvalues λ i (N t ) to energies using the relation exp(−α t E i (N t )) = λ i (N t ). In the calculation, the two-body interactions among A nucleons are replaced by one-body interactions between single nucleons with auxiliary fields that are sampled by Monte Carlo methods [4].

Pinhole algorithm
We use the pinhole algorithm (PA) to determine a classical distribution for the nucleon positions, spins, and isospins. For this, we compute the amplitude Z kl (n 1 , . . . , n A , N t ) = Φ k |M Nt/2 ρ(n 1 , . . . , with the A-body density operator ρ(n 1 , . . . , n A ) constructed from the normal-ordered product of density operators ρ(n i ) = a † i (n)a i (n). The positions for A nucleons (or "pinholes") n i are sampled stochastically. The finite size of the nucleons is accounted for by a random Gaussian smearing of the nucleon positions.

Nucleon density operators
The definition of the density operatorρ(n) that appears in main text as Eq. (1) is given by [5]: Here s L is the local smearing parameter, and the smeared annihilation operator is where s NL is the non-local smearing parameter.

Euclidean time extrapolation
In NLEFT, energies and operator expectation values are computed using auxiliary-field lattice MC at finite Euclidean projection time. As an example, the low-energy spectrum is obtained from the transient energies E i (N t ) = −log(λ i (N t ))/α t . Some extrapolation is then usually required to obtain the corresponding values at infinite projection time. For instance, we obtain energies at infinite projection time by fitting the lattice data using the ansatz [6,7] E i (t) =  (4) interaction. The data shown corresponds to a three-channel auxiliary field lattice MC calculation using alpha-cluster trial states and A + 1 projection. The data are fitted and extrapolated using Eq. (S5). The widths of the horizontal lines reflect the extrapolation error.
where E i , ∆E i,k , c i,k are fit parameters. The number of exponentials k max should be kept as small as possible [6], in order to avoid fitting statistical fluctuations. For expectation value of operators of the type i|Ô|i , or transition operators i|Ô|j , we use the lowest order extrapolation formulas [7] with only one ∆E i,k (or ∆E j,k ). In Fig. S1, we show the extrapolation using Eq. (S5) for the lowest three 0 + states of 12 C. With k max = 2, the convergence of the auxiliary-field MC data is well described.

SU(4) interaction parameters and systematic errors
In Table S1, we summarize the interaction parameters for the SU(4) interaction, together with the observables used to determine them. Our preferred set of parameters is denoted V1. It is then of interest to assess the systematic error associated with this choice of parameters. Since we are here considering an SU(4) symmetric interaction without a specific EFT power counting scheme, our strategy to estimate the systematic errors is to perform a number of fits with a different choice of the non-local smearing parameter s NL . The remaining three parameters C 2 , C 3 , s L are again, in each case, fitted to the ground state energies of 4 He and 12 C, and to the electromagnetic radius of the 12 C ground state. While this procedure allows for a large number of possibilities, we have chosen two interactions denoted V2 and V3, for closer comparison. The results of our systematic error analysis are given in Table S1. For simplicity, calculated energies and matrix elements are given at finite Euclidean projection time, as the additional computational effort required for a full extrapolation would not substantially alter the results nor the conclusions. Based on earlier experience, we take t = 0.4 MeV −1 for 3 H and 12 C, and t = 0.2 MeV −1 for 4 He. Such projection times should suffice for the purposes of the error analysis. From Table S1, we find that all three interactions give a similarly good description of the fitted observables. While the energy of the Hoyle state and the 4 He radius appear insensitive to the choice of interaction, the triton energy is influenced more. In particular, we observe that smaller values of s NL give a more deeply bound 3 H. We find that such systematic errors for the 12 C spectrum do not exceed 2%, while for the alpha particle they are about 2 . . . 3%. The radii of 4 He and 12 C are even less affected (≤ 1%). For the bound states (0 + 1 and 2 + 1 ) and those near threshold (0 + 2 , 2 + 2 , 3 − 1 ), adding the 3NF has little influence. For the higher-lying states, the 3NF in general contributes more repulsion. Overall, the agreement with experiment is quite good.
In Fig. S2, we show the 12 C form factors obtained at finite Euclidean projection time t = 0.4 MeV −1 for the three interactions V1, V2, and V3, characterized by a different strength of the non-local smearing. As s L is varied, the value of s NL is adjusted so that ground state radius is kept constant. This results in the ground state form factor remaining almost the same for the three different interactions. There is also not a significant change to the overall charge density Supplementary Table S1: Summary of interaction parameters C 2 , C 3 , s L , and s NL , and estimate of systematic uncertainty due to the choice of SU(4) interaction. The column labeled V1 denotes the preferred interaction used in this work, while V2 and V3 are used to quantify the systematic uncertainty. Observables shown are the ground state energies of 3 H, 4 He and 12 C, the energy of the Hoyle state, and the charge radii of 4 He and 12 C calculated by r c = r 2 + (0.84 fm) 2 . The interaction parameters are fitted to E4 He , E12 C and r12 C . As a measure of the resulting systematic error, we give Q (  Results are shown for Euclidean projection times t = 0.4 MeV −1 with three SU(4) interactions V1, V2 and V3 corresponding to s NL = 0.1, 0.2, 0.05, respectively. The comparison with experimental data is also shown [8,9,10,5]. The error bars are one standard deviation uncertainties associated with stochastic errors.
of the Hoyle. However, the transition from the ground state to the Hoyle state is more sensitive to changes in s L . The shapes and positions of the maxima and minima of the transition form factor have some dependence on s L , and this is also reflected in significant changes to the electromagnetic transitions between the Hoyle state and members of the ground state rotational band.
In a previous study of the 12 C spectrum using NLEFT [3], two different lattice spacings were used. The differences between the a = 1.97 fm results and a = 1.64 fm results can be explained by the significantly larger lattice errors at a = 1.97 fm. As shown in Ref. [11] for calculations of 8 Be, the lattice artifacts become significantly larger for a > 1.7 fm. This explains the choice of lattice spacing a = 1.64 fm for this analysis. Work currently in progress using a smaller lattice spacing of a = 1.32 fm provides confirmation that the lattice artifacts at a = 1.64 fm are less than 1 or 2 MeV in binding energy for the spectrum of 12 C. This is reflected in the excellent agreement with the observed binding energies.

N2LO chiral interaction parameters and results
The chiral interaction up to N2LO is defined as where Q is the low-energy scale (external momentum or pion mass), Λ χ 700 MeV is the hard scale, V OPE is the one-pion-exchange potential, and V 3N gives the leading three-nucleon force that appears at N2LO. Note that we have explicitly separated out the SU(4) symmetric pieces of the two-and three-nucleon contact terms that together with the one-pion exchange are treated non-perturbatively as leading order. All other operators appearing beyond LO are then treated in perturbation theory [12]. Note further that the two-pion exchange operators are expanded in powers of (Q/Λ χ ) 2 and absorbed in the corresponding contact interactions [13]. Furthermore, the two-nucleon contact interactions are smeared with a Gaussian regulator to improve the description of the two-and other fewnucleon systems. In the S-waves, the corresponding cut-off is chosen as Λ = 400 MeV. The parameters of SU(4) and contact terms are determined by fitting to the phase shift of Nijmegen partial wave analysis (PWA93) [14], and are listed in Table S2. Galilean invariance is restored using the method in Ref. [15], and the coefficients are also listed in Table S2. The two low-energy constants of the three-nucleon interaction are determined then by fitting to the ground state energies of 4 He and 12 C, the corresponding LECs are also collected in Table S2. Supplementary Table S2: Summary of N2LO chiral interaction parameters, all in lattice units.
The obtained phase shifts are given in Fig. S3, in comparison with Nijmegen partial wave analysis (PWA93) [14]. A good convergence is obtained at (Q/Λ χ ) 2 order and a reasonably good accuracy can be achieved for the S-, P -, D-waves over the relative momentum range 0 < p rel < 200 MeV. Note that due to the treatment of the two-pion exchange, these phase shifts are identical at NLO and N2LO.
Besides the Fig. 3 in the main text, we also show in Figs. S4 and S5 that the SU(4) interaction and the N2LO chiral interactions produce the same nuclear structures. In Fig. S4 we compare SU(4) and N2LO chiral results for the 3 − 1 and 4 + 1 states, both of which have good descriptions in terms of shell model states. In Fig. S5 we show the SU(4) and N2LO chiral results for the radial charge densities for each of these states.  [14]. Panels a, b, c, d, e, f, g, h, i, j, k, l show phase shifts and mixing angles for the 1

Electromagnetic observables
Details on the calculation of electromagnetic observables can be found in the literature [16]. Here, we only give the final results and discuss issues directly relevant to the lattice calculation. The quadrupole moment Q for a given state where the quadrupole moment operator is given by with e the electric charge, ρ c the charge density distribution and dτ is the integration measure for the whole space. The reduced transition probability B(E2) is defined as The electric quadrupole operator for different µ components is given by with Y 2µ (θ, ϕ) the pertinent spherical harmonics. In the above expressions, the coordinate r for each nucleon is defined in the CM frame of the nucleus. For the quadrupole moment and the B(E2) strength, the operator to be evaluated is r 2 (or x 2 , . . . , xy, . . . , depending on the choice of µ), as for the RMS radius. We evaluate expressions such as where we consider the proton density with Z the number of protons and A the number of nucleons. For the quadrupole moment and the B(E2) matrix element have a similar form. For example, when µ = 0 the operator is proportional to which can be treated along the lines of Eq. (S12).

Density projection of lattice states
In the NLEFT simulation for a given state with good angular momentum, the wave function is projected onto a given irrep of the cubic group. For example, the case of J π = 0 + corresponds to the A + 1 irrep [1], which entails an equal superposition of all possible rotations of the wave function. In other words, for 0 + states there is no preference in the angular distribution and the density distribution should be spherical. However, one may inquire as to the intrinsic shape of a 12 C nucleus, without such a superposition of all possible spatial rotations. Here, we adopt the following strategy: With the PA we can obtain the superposition of a large number of coordinates r i (i = 1, . . . , A) for A nucleons in all possible rotations. For the ground state, we already have the (model-independent) information from Fig. 2 (a) that an equilateral triangular configuration is preferred. Hence, for each configuration r i we calculate its principal axis and align it in such a way that the shortest axis coincides with the x-axis. In this way, the three alpha clusters will be located on the y − z plane. This can be achieved by calculating the matrix and solving the eigenvalue problem Rv = λv. (S15) The resulting three eigenvalues correspond to the length along the longest, shortest, and intermediate principal axes, respectively. The eigenvectors can be used to rotate the original distribution so that the longest, shortest, and intermediate principal axes coinciding with the coordinate axes. For example, if we want to align the shortest axis with the x-axis, the intermediate axis with the y-axis, and the longest axis with the z-axis, we let v min , v mid , and v max be the corresponding eigenvectors, and the rotation matrix can be constructed as Then one can rotate all particles i = 1, . . . , A from the old coordinates to the new ones as (x, y, z) new = (x, y, z) old × O. (S17) For nuclear states that we have already identified as having an equilateral shape, we rotate the configuration along the x-axis so that one of the alpha clusters is positioned on the positive z axis. We then symmetrize with respect to 0 • , 120 • , and 240 • rotations. For nuclear states that we have already identified as having an obtuse isosceles shape, we identify the z axis as the direction with the longest RMS deviation of the nucleon positions relative to the CM. We then rotate the density configurations so that the alpha cluster closest to the CM lies on the positive y axis.
In Fig. S6, we show the density distribution for the two inner angles of the triangle in the top panel and the twodimensional projection of the nuclear density in the middle panel for the states 1 − 1 , 4 + 1 , 1 + 2 , 2 + 3 , 4 − 2 obtained using shell model initial states. The color scales are same as in Fig. 2. The corresponding orbitals are shown in the lower panel of Fig. S6. The red circles are for protons and blue circles for neutrons, with solid ones for particles and hollow ones for holes. For those configurations not showing neutrons explicitly, the neutrons are occupied in the lowest possible levels (that is, fully s 1/2 and 1p 3/2 ). They all show a similar equilateral triangle structure as the ground state, but in some cases with a less of a pronounced alpha cluster structure. It is also interesting to see that among those selected states, the alpha cluster structure is less pronounced when the particle excitation is in s − d shell rather than the 1p 1/2 level. Furthermore, the separation between the clusters is larger when the particle excitation reaches the 1d 5/2 level.
For completeness, the density distributions for the other six states calculated in this work are shown in Fig. S7 and are ordered by excitation energies. The color scales are the same as in Fig. 2. The 4 + 2 appears to be a rotational excitation of the Hoyle state, while the 1 + 1 is similar to the Hoyle state but with a smaller angle and more compact structure. The 2 − 1 , 1 + 1 , 4 + 2 , and 4 − 1 are obtained with cluster wave functions, while 2 − 2 is obtained with shell model wave function 2s 1/2 ⊗ 1p 3/2 . The 0 + 4 is obtained using a shell model state with two-particle and two-holes in the 1p 1/2 ⊗ 1p 3/2 orbitals (see the schematic plot in Fig. S6).
We should emphasize that the intrinsic shapes we are showing are not the result of initial state bias. This can be seen clearly in Fig. S8 for the ground state of 12 C. In the upper panel (a)-(d), the initial state is a shell model wave function with harmonic oscillator orbitals. In the lower panel (e)-(h), the initial state is a cluster wave function. We see that the density distributions from very different initial wave functions begin to look similar as the projection time increases. This independence with respect to the choice of initial state was also demonstrated in previous work [17,18,3]. At Euclidean projection time t = 0.2 MeV −1 each of the distributions in the upper and lower panels look similar to the one shown for the converged ground state in Fig. 2 the intrinsic density for the shell model initial wave function is quite different from the converged ground state. In particular, there is a higher probability at the center and less probability to be restricted to the y − z plane, and it is therefore more spherical.

Model-independent probes of cluster geometry
To more clearly assess the angular distribution of the various states, we show in Fig. S9 the probability of θ 3 when θ 1 = θ 2 , which corresponds to the diagonal line along Fig. 2. For the ground state, we find a clear peak around 60 • as discussed above, but there is also a non-vanishing probability for an obtuse triangle (θ 3 > 90 • ). For the Hoyle state, there is little probability for an acute triangle, with a peak probability around 170 • . The probability to form an exactly linear chain appears to be strongly suppressed (θ 3 ∈ [176 • , 180 • ] with 4 • resolution). This supports the notion that the intrinsic shape of Hoyle state is a large-angle obtuse triangle, but not exactly a linear chain. The lowest two 2 + states show the same intrinsic shapes as the corresponding 0 + states, which gives credit to the assertion made e.g. in Ref. [18] that the 2 + 2 state is a rotational excitation of the Hoyle state. We also note that the lowest negative-parity state, the 3 − 1 , shows a distribution similar to the ground state. The angular distribution of the 0 +  note the stunning similarity in this distribution for the 0 + 1 , 2 + 1 and 0 + 2 , 2 + 2 states, reinforcing the notion of the 2 + states being rotational excitations of the corresponding 0 + states. Note also that the 3 − 1 state is somewhat broader than the ground state, but by far less extended than the Hoyle state. The similarity of 0 + 3 and Hoyle state, but with a slightly larger separation distance distribution, again supports the notion of a small amplitude vibrational excitation.

Electromagnetic density distributions
Here, we display the electromagnetic density distribution not shown in the main text. In Fig. S11 the proton radial density distribution of the 0 + 1 , 0 + 2 states is displayed, and the transition of 0 + 1 → 0 + 2 in comparison with the available experimental data [19,5]. The density distribution of the ground state is nicely reproduced. The slight decrease in the center at r = 0 fm reflects the fact that the three alpha clusters are equally far away from the center. For the Hoyle state, the density distribution shown in Fig. S11 (a) has no such decrease. This is also consistent from the shape information, that one alpha cluster is closely located near the center. The decrease of the density distribution as r increases has a visible slow down from r = 2 fm to r = 6 fm, and this corresponds to the separation between the two alpha clusters forming the longest side of the obtuse triangle. The pattern of the transition density 0 + 1 → 0 + 2 is also reproduced, with a minimum near r = 2 fm, which is the approximate distance from the center to the α clusters in the ground state. The maximum occurs near r = 4 fm, which is the approximate distance from the center of the Hoyle state to the two α clusters furthest from the center.