On the Number of Positive Eigenvalues in a Class of Two-Body Quantum Systems with Nonlocal interaction.

We consider the quantum optics of a single photon interacting with a system of two level atoms. This leads to the study of a nonlinear eigenproblem for a system of nonlocal partial differential equations. Two classes of solutions to these equations are studied. Bound states correspond to negative eigenvalues and resonances to eigenvalues with positive real parts. We have found necessary and sufficient conditions for the existence of bound states, along with an upper bound on the number of such states. We have also considered the eigenproblem for atomic models with small high contrast inclusions. In this setting, we have derived asymptotic formulas for the eigenvalues. Our results are illustrated with numerical computations.We consider the quantum optics of a single photon interacting with a system of two level atoms. This leads to the study of a nonlinear eigenproblem for a system of nonlocal partial differential equations. Two classes of solutions to these equations are studied. Bound states correspond to negative eigenvalues and resonances to eigenvalues with positive real parts. We have found necessary and sufficient conditions for the existence of bound states, along with an upper bound on the number of such states. We have also considered the eigenproblem for atomic models with small high contrast inclusions. In this setting, we have derived asymptotic formulas for the eigenvalues. Our results are illustrated with numerical computations.

arXiv:2306.10431Jun 17, 20231 citation

New Class of Solvable and Integrable Many-Body Models

R. W. Richardson

Integrability conditions for systems of bosons or fermions with seniority conserving hamiltonians are derived. The conditions are shown to be invariant under a large class of transformations of the interaction matrix elements. Previously published integrable models are shown to satisfy these conditions and the existence of a new class of integrable models is demonstrated. The number of free parameters in the interaction in these models equals the number of single particle levels plus 3 . Equations for the energy eigenstates and eigenvalues are derived and the eigenvalues of the complete set of two-body integrals of the motion are given for the new class. Some two-body correlations in these eigenstates are derived from the integrals of the motion.Integrability conditions for systems of bosons or fermions with seniority conserving hamiltonians are derived. The conditions are shown to be invariant under a large class of transformations of the interaction matrix elements. Previously published integrable models are shown to satisfy these conditions and the existence of a new class of integrable models is demonstrated. The number of free parameters in the interaction in these models equals the number of single particle levels plus 3 . Equations for the energy eigenstates and eigenvalues are derived and the eigenvalues of the complete set of two-body integrals of the motion are given for the new class. Some two-body correlations in these eigenstates are derived from the integrals of the motion.

arXiv:cond-mat/0203512Mar 25, 20021 citation

Asymptotic completeness of N-body quantum systems with long-range interactions

Daniel Schenker

Jan 1, 19960 citations

Conserved quantities from entanglement Hamiltonian

Biao Lian

We show that the subregion entanglement Hamiltonians of excited eigenstates of a quantum many-body system are approximately linear combinations of subregionally (quasi)local approximate conserved quantities, with relative commutation errors $\mathcal{O}(\frac{\text{subregion}\phantom{\rule{4.pt}{0ex}}\text{boundary}\phantom{\rule{4.pt}{0ex}}\text{area}}{\text{subregion}\phantom{\rule{4.pt}{0ex}}\text{volume}})$ . By diagonalizing an entanglement Hamiltonian superdensity matrix (EHSM) for an ensemble of eigenstates, we can obtain these conserved quantities as the EHSM eigenoperators with nonzero eigenvalues. For free fermions, we find the number of nonzero EHSM eigenvalues is cut off around the order of subregion volume, and some of their EHSM eigenoperators can be rather nonlocal, although subregionally quasilocal. In the interacting XYZ model, we numerically find the nonzero EHSM eigenvalues decay roughly as a power law if the system is integrable, with the exponent $s\ensuremath{\approx}1$ $(s\ensuremath{\approx}1.5--2)$ if the eigenstates are extended (many-body localized). For fully chaotic systems, only two EHSM eigenvalues are significantly nonzero, the eigenoperators of which correspond to the identity and the subregion Hamiltonian.We show that the subregion entanglement Hamiltonians of excited eigenstates of a quantum many-body system are approximately linear combinations of subregionally (quasi)local approximate conserved quantities, with relative commutation errors $\mathcal{O}(\frac{\text{subregion}\phantom{\rule{4.pt}{0ex}}\text{boundary}\phantom{\rule{4.pt}{0ex}}\text{area}}{\text{subregion}\phantom{\rule{4.pt}{0ex}}\text{volume}})$. By diagonalizing an entanglement Hamiltonian superdensity matrix (EHSM) for an ensemble of eigenstates, we can obtain these conserved quantities as the EHSM eigenoperators with nonzero eigenvalues. For free fermions, we find the number of nonzero EHSM eigenvalues is cut off around the order of subregion volume, and some of their EHSM eigenoperators can be rather nonlocal, although subregionally quasilocal. In the interacting XYZ model, we numerically find the nonzero EHSM eigenvalues decay roughly as a power law if the system is integrable, with the exponent $s\ensuremath{\approx}1$ $(s\ensuremath{\approx}1.5--2)$ if the eigenstates are extended (many-body localized). For fully chaotic systems, only two EHSM eigenvalues are significantly nonzero, the eigenoperators of which correspond to the identity and the subregion Hamiltonian.

arXiv:2104.11753Physical review. B./Physical review. BJan 4, 20228 citations

On the negative point spectrum of quantum mechanical three-body systems

Annals of PhysicsSep 1, 19760 citations

\mathcal{O}(\frac{\text{subregion}\phantom{\rule{4.pt}{0ex}}\text{boundary}\phantom{\rule{4.pt}{0ex}}\text{area}}{\text{subregion}\phantom{\rule{4.pt}{0ex}}\text{volume}})

. By diagonalizing an entanglement Hamiltonian superdensity matrix (EHSM) for an ensemble of eigenstates, we can obtain these conserved quantities as the EHSM eigenoperators with nonzero eigenvalues. For free fermions, we find the number of nonzero EHSM eigenvalues is cut off around the order of subregion volume, and some of their EHSM eigenoperators can be rather nonlocal, although subregionally quasilocal. In the interacting XYZ model, we numerically find the nonzero EHSM eigenvalues decay roughly as a power law if the system is integrable, with the exponent $s\ensuremath{\approx}1$ $(s\ensuremath{\approx}1.5--2)$ if the eigenstates are extended (many-body localized). For fully chaotic systems, only two EHSM eigenvalues are significantly nonzero, the eigenoperators of which correspond to the identity and the subregion Hamiltonian.