Nuclear Binding Near a Quantum Phase Transition

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. In this work we present numerical evidence from ab initio lattice simulations showing that nature is near a quantum phase transition, a zero-temperature transition driven by quantum fluctuations. Using lattice effective field theory, we perform Monte Carlo simulations for systems with up to twenty nucleons. For even and equal numbers of protons and neutrons, we discover a first-order transition at zero temperature from a Bose-condensed gas of alpha particles (${}^4\mathrm{He}$ nuclei) to a nuclear liquid. Whether one has an alpha-particle gas or nuclear liquid is determined by the strength of the alpha-alpha interactions, and we show that the alpha-alpha interactions depend on the strength and locality of the nucleon-nucleon interactions. This insight should be useful in improving calculations of nuclear structure and important astrophysical reactions involving alpha capture on nuclei. Our findings also provide a tool to probe the structure of alpha cluster states such as the Hoyle state responsible for the production of carbon in red giant stars and point to a connection between nuclear states and the universal physics of bosons at large scattering length.

Figure 1

Alpha-alpha $S$-wave scattering. We plot $S$-wave phase shifts ${\delta }_0$ for alpha-alpha scattering for interactions $A$ and $B$ versus laboratory energy. We show LO results for interaction $A$ (green triangles), $\mathrm{LO}+\text{Coulomb for}$ $A$ (orange diamonds), LO results for $B$ (blue circles), and $\mathrm{LO}+\text{Coulomb results for}$ $B$ (red squares). The phase shift analysis of experimental data are shown with black asterisks [30]. The theoretical error bars indicate one standard deviation uncertainty due to Monte Carlo errors and the extrapolation to infinite number of time steps.

Figure 2

Tight-binding potential. We plot the tight-binding potential versus radial distance for the LO interactions for $A$, $B$, and $C$, where $C$ is the interaction used in several previous lattice calculations [3, 31]. Interaction $A$ is shown with blue squares and solid line, $B$ is drawn with red crosses and a dashed line, and $C$ is presented with orange circles and a short-dashed line.

Figure 3

Zero-temperature phase diagram. We show the zero-temperature phase diagram as a function of the parameter $\lambda$ in the interaction $V_{\lambda }=(1-\lambda )V_A+\lambda V_B$ without Coulomb included. The blue filled circles indicate neutrons, the red filled circles indicate protons, and the small arrows attached to the circles indicate spin direction. We show a first-order quantum phase transition from a Bose gas to nuclear liquid at the point where the scattering length $a_{\alpha \alpha }$ crosses zero. We have also plotted the alphalike nuclear ground state energies $E_A$ for $A$ nucleons up to $A=20$ relative to the corresponding multialpha threshold $E_{\alpha }A/4$. The last alphalike nucleus to be bound is ${}^8\mathrm{Be}$ at the unitarity point where $|a_{\alpha \alpha }|=\infty$.