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

Abstract

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 $N_r=1,\ldots ,N_s-1$.