Exact ground state for a class of matrix Hamiltonian models: quantum phase transition and universality in the thermodynamic limit

TL;DR

This paper derives the exact ground state for a class of matrix Hamiltonian models in the thermodynamic limit, revealing a universal behavior and a quantum phase transition related to the potential levels and degeneracies.

Contribution

It introduces a probabilistic approach to determine the exact ground state for a broad class of matrix Hamiltonian models, including fully connected and random potential systems.

Findings

Universal thermodynamic limit characterized by potential levels and degeneracies.

Identification of a quantum phase transition when the lowest potential level degeneracy tends to zero.

Ground state condensates into the subspace of the lowest potential level in the frozen phase.

Abstract

By using a recently proposed probabilistic approach, we determine the exact ground state of a class of matrix Hamiltonian models characterized by the fact that in the thermodynamic limit the multiplicities of the potential values assumed by the system during its evolution are distributed according to a multinomial probability density. The class includes i) the uniformly fully connected models, namely a collection of states all connected with equal hopping coefficients and in the presence of a potential operator with arbitrary levels and degeneracies, and ii) the random potential systems, in which the hopping operator is generic and arbitrary potential levels are assigned randomly to the states with arbitrary probabilities. For this class of models we find a universal thermodynamic limit characterized only by the levels of the potential, rescaled by the ground-state energy of the system for zero potential, and by the corresponding degeneracies (probabilities). If the degeneracy (probability) of the lowest potential level tends to zero, the ground state of the system undergoes a quantum phase transition between a normal phase and a frozen phase with zero hopping energy. In the frozen phase the ground state condensates into the subspace spanned by the states of the system associated with the lowest potential level.