ELEMENTARY PARTICLE PHYSICS

How do neutrons and protons bind to form atomic nuclei? Why do we observe alpha-particle clustering in light and medium-mass nuclei but not in heavy ones? These questions can be tackled in the framework of nuclear lattice effective field theory. These investigations have revealed some intriguing features of nuclei related to much discussed quantum phenomena such as entanglement and quantum phase transitions.

How do protons and neutrons bind to form nuclei? This is the central question of ab initio nuclear structure theory. While the answer may seem as simple as the fact that nuclear forces are attractive, the full story is more complex and interesting. Using the approach of nuclear lattice effective field theory, we have presented numerical evidence showing that nature is near a quantum phase transition, a zero-temperature transition driven by quantum fluctuations [1].

Before discussing this phase transition, let us briefly discuss the underlying computational tool. Nuclear lattice effective field theory combines the modern approach to the nuclear force problem based on a chiral effective field theory with high-performance computing methods. It defines a completely new method to exactly solve the nuclear A-body problem (with A the number of nucleons, that is protons and neutrons, in a nucleus). The first ingredient of this method is a systematic and precise effective field theory description of the forces between two and three nucleons, that has been worked out in the last decades by various groups worldwide. To go beyond atomic number four, one has to devise a method to exactly solve the A-body problem. Such a method is given by nuclear lattice simulations. Space-time is discretized with spatial length L_s and temporal length L_t , and nucleons are placed on the lattice sites. The minimal length on the lattice, the so-called lattice spacing a, entails a maximum momentum, p_max = π/a. Such a lattice representation is ideally suited for parallel computing. Breakthrough calculations have been the ab initio calculation of the Hoyle state in ¹²C [2] and of low-energy alpha-alpha scattering [3].

Let us come back to the quantum phase transition. Initially, we consider two types of two-nucleon interactions with a different degree of non-locality (velocity-dependence). Both interactions give an identical description of neutron-proton scattering and the properties of the lightest nucleus, the deuteron. However, when one computes the groundstate energies of alpha-like nuclei (⁸Be, ¹²C, ¹⁶O, ²⁰Ne), a surprise emerges. The groundstate for the purely non-local interaction V_A in each case turns out to be a weakly-interacting Bose gas of alpha-particles, whereas the binding energies for the more local interaction V_B are within a few percent of the empirical values. In order to discuss the many-body limit, we switch off the Coulomb interaction and define a one-parameter family of interactions, V_λ = (1 − λ)V_A + λV_B, 0 ≤ λ ≤ 1. While the properties of the two, three, and four nucleon systems vary only slightly with λ, the many-body ground state of V_λ undergoes a quantum phase transition from a Bose-condensed gas to a nuclear liquid, as sketched in the phase diagram in Fig. 1. The phase transition occurs when the alpha-alpha s-wave scattering length a_αα crosses zero, and the Bose gas collapses due to the attractive interactions. At slightly larger λ, finite alpha-like nuclei also become bound, starting with the largest nuclei first. The last alpha-like nucleus to be bound is ⁸Be at the so-called unitarity point where |a_αα | = ∞. Superimposed on the phase diagram, we have sketched the alpha-like nuclear ground state energies E_A for A nucleons up to A = 20 relative to the corresponding multi-alpha threshold E_α A/4.

Another intriguing phenomenon in nuclear physics is the observation of the formation of alpha- clusters in light and medium-mass nuclei and the disappearance of it in heavier systems. The alpha-particle is formed as a substructure because of its relatively strong binding and its resistance against external perturbations. While clustering is observed in many approaches such as density functional theory [4] or fermion molecular dynamics [5], a truly quantitative measure of clustering was not available until the work of Ref. [6].

Measuring the distribution of three or four nucleons on the lattices sites allows one to test the amount of clustering. Indeed, in case of non-interacting clusters, this measure called ρ₄ would lead to integer numbers, like e.g. four for oxygen and its neutron-rich even-even isotopes. However, it was found that with increasing atomic number, the value calculated for ρ₄ was increasing steadily when going from systems with one alpha-particle (the He isotopes) to the oxygen chain (four alpha-particles). This increase can be interpreted in terms of quantum entanglement, that is the nucleons feel the presence of the other nucleons and this eventually leads to the disappearance of the individual clusters. Stated differently, these results show that the transition from cluster-like states in light systems to nuclear liquid-like states in heavier systems should not be viewed as a simple suppression of multi-nucleon short-distance correlations, but rather an increasing entanglement of the nucleons involved in the multi-nucleon correlations, as depicted schematically in Fig. 2.