Stability and Clustering for Lattice Many-Body Quantum Hamiltonians with Multiparticle Potentials

We analyze a quantum system of N identical spinless particles of mass m, in the lattice \(\mathbb {Z}^d\), given by a Hamiltonian \(H_N=T_N+V_N\), with kinetic energy \(T_N\ge 0\) and potential \(V_N=V_{N,2}+V_{N,3}\) composed of attractive pair and repulsive 3-body contact-potentials. This Hamiltonian is motivated by the desire to understand the stability of quantum field theories, with massive single particles and bound states in the energy-momentum spectrum, in terms of an approximate Hamiltonian for their N-particle sector. We determine the role of the potentials \(V_{N,2}\) and \(V_{N,3}\) on the physical stability of the system, such as to avoid a collapse of the N particles. Mathematically speaking, stability is associated with an N-linear lower bound for the infimum of the \(H_N\) spectrum, \(\underline{\sigma }(H_N)\ge -cN\), for \(c>0\) independent of N. For \(V_{N,3}=0\), \(H_N\) is unstable, and the system collapses. If \(V_{N,3}\not =0\), \(H_N\) is stable and, for strong enough repulsion, we obtain \(\underline{\sigma }(H_N)\ge -c' N\), where \(c'N\) is the energy of (N/2) isolated bound pairs. This result is physically expected. A much less trivial result is that, as N varies, we show \([\,\underline{\sigma }(V_N)/N\,]\) has qualitatively the same behavior as the well-known curve for minus the nuclear binding energy per nucleon. Moreover, it turns out that there exists a saturation value \(N_s\) of N at and above which the system presents a clustering: the N particles distributed in two fragments and, besides lattice translations of particle positions, there is an energy degeneracy of all two fragments with particle numbers \(N_r\) and \(N_s-N_r\), with \(N_r=1,\ldots ,N_s-1\).