A model of a physical system that can be solved exactly and the dynamics of which are characterized by regular rather than chaotic motion is called an integrable model. Integrable models can occur in Newtonian mechanics and quantum mechanics, with the harmonic oscillator being an example of such a model in both cases. Integrability has a dimensional component: there are many systems that are integrable in one or two space dimensions but not in three dimensions. Several integrable models have been realized in the laboratory, prompting their use as a test bed for quantum many-body physics, statistical mechanics, and nonequilibrium phenomena.
Integrability in both classical and quantum many-body systems is associated with the existence of conserved quantities. At the quantum level, the latter are related to operators and how they behave in relation to the system Hamiltonian. For simplicity, an operator is just a mathematical symbol that indicates that a mathematical operation should be carried out, in this case, on the Hamiltonian, a function that expresses the energy of a system in terms of its momentum (instead of forces) and positional coordinates. Conserved quantities correspond to operators that commute with the system Hamiltonian and govern the nonequilibrium dynamics and thermalization of a system in isolation.
Even though many of the proposed approaches are closely interrelated, a definite notion of quantum integrability remains under debate. Still, the integrability of a system may be proven by finding the set of conserved quantities.
An important class of quantum systems is characterized by a ground state of Jastrow form, in which the wave function is simply the pairwise product of a pair function. This facilitates the computation of correlation functions in these systems. The family of parent Hamiltonians with Jastrow wave functions can be determined by solving an inverse problem: by acting with the kinetic energy operator in the ground-state wave function, one can recast the resulting terms in the form of a many-body Schrödinger equation, thus identifying the parent Hamiltonian.
How much the ground-state correlations can determine the complete integrability of the system? And, what are the required conditions for this to be the case? Now, a team of researchers shows 1 that the complete family of one-dimensional many-body quantum models with ground state of Jastrow form is integrable.
To this end, the scientists establish the equivalence between this family and models described by the exchange operator formalism, in which the Hamiltonian of the quantum system admits a decoupled form in terms of generalized momenta, which readily allows for the identification of integrals of motion . The team is able to identify explicitly these integrals.
Their construction holds in the presence of an external potential, which allows them to show the integrability of a long-range model, describing bosons confined in a harmonic trap and subject to both contact and Coulomb interactions in one spatial dimension.
These findings advance the study of many-body physics by uncovering the implications of ground-state correlations on integrability. They should lead to manifold applications in the study of quantum solitons, quantum quenches, and the thermalization of isolated integrable systems (governed by integrals of motion), and strongly correlated regimes, among others. The results also bear implications on numerical methods for strongly correlated systems.
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Author: César Tomé López is a science writer and the editor of Mapping Ignorance
Disclaimer: Parts of this article may have been copied verbatim or almost verbatim from the referenced research papers