Dense neutrino gases form in extreme astrophysical sites, and the flavor content of the neutrinos likely has an important impact on the subsequent dynamical evolution of their environment. Through coherent forward scattering among neutrinos, the flavor content of the gas evolves under a time-dependent potential which can be modeled in a quantum many-body formalism as an all-to-all coupled spin-spin interaction. This two-body potential generically introduces entanglement and greatly complicates the study of these systems. In this work we study the evolution of the quantum many-body problem as well as the typically employed mean-field approximation to it for a small number of neutrinos ( $N = 16$ ). We consider randomly chosen one- and two-body couplings in the Hamiltonian, and the resulting evolution of several initial product states. We subsequently compare many-body and mean-field predictions for one-body observables, and we consider one- and two-body entanglement to assess under what conditions the many-body and mean-field predictions are likely to disagree. Except for a special category of prototypical initial conditions, we find that the typically employed mean-field approximation is insufficient to capture the evolution of one-body operators in the systems we consider. We also observe a loss of coherence in one- and two-body trace-reduced subsystems which suggests that the evolution may be well approximated as a classical mixture of separable states.Dense neutrino gases form in extreme astrophysical sites, and the flavor content of the neutrinos likely has an important impact on the subsequent dynamical evolution of their environment. Through coherent forward scattering among neutrinos, the flavor content of the gas evolves under a time-dependent potential which can be modeled in a quantum many-body formalism as an all-to-all coupled spin-spin interaction. This two-body potential generically introduces entanglement and greatly complicates the study of these systems. In this work we study the evolution of the quantum many-body problem as well as the typically employed mean-field approximation to it for a small number of neutrinos ($N = 16$). We consider randomly chosen one- and two-body couplings in the Hamiltonian, and the resulting evolution of several initial product states. We subsequently compare many-body and mean-field predictions for one-body observables, and we consider one- and two-body entanglement to assess under what conditions the many-body and mean-field predictions are likely to disagree. Except for a special category of prototypical initial conditions, we find that the typically employed mean-field approximation is insufficient to capture the evolution of one-body operators in the systems we consider. We also observe a loss of coherence in one- and two-body trace-reduced subsystems which suggests that the evolution may be well approximated as a classical mixture of separable states.

arXiv:2301.07049Jan 17, 20231 citation

Dynamics of Two-Body Hamiltonian Systems

Xiang‐Guo Meng, Ji‐Suo Wang, Bao‐Long Liang

Springer eBooksJan 1, 20230 citations

Time-dependent coupled-cluster approximation to nuclear dynamics. I. Application to a solvable model

Pervez Amirali Hoodbhoy, John W. Negele

A general hierarchy of approximations to the many-body Schr\"odinger equation is presented which reduces to the time-dependent mean-field approximation in lowest order and provides systematic corrections in subsequent orders. The theory is applied to two interacting systems described by the Lipkin model Hamiltonian. Comparison of the results of the lowest two orders of approximation with the exact solution demonstrates the practicality of the method and its potential for generalizing nuclear dynamics beyond the mean field theory.NUCLEAR STRUCTURE Time-dependent approximation to nuclear dynamics. Corrections to time-dependent mean-field theory. Solution of Lipkin model for two interacting systems.A general hierarchy of approximations to the many-body Schr\"odinger equation is presented which reduces to the time-dependent mean-field approximation in lowest order and provides systematic corrections in subsequent orders. The theory is applied to two interacting systems described by the Lipkin model Hamiltonian. Comparison of the results of the lowest two orders of approximation with the exact solution demonstrates the practicality of the method and its potential for generalizing nuclear dynamics beyond the mean field theory.NUCLEAR STRUCTURE Time-dependent approximation to nuclear dynamics. Corrections to time-dependent mean-field theory. Solution of Lipkin model for two interacting systems.

Physical Review CNov 1, 197865 citations

Relativistic two-body interactions: a Hamiltonian formulation

R. Giachetti, E. Sorace

Il Nuovo cimento della Società italiana di fisica. B/Il Nuovo cimento BApr 1, 198012 citations

Failure of the mean-field Hartree approximation for a bosonic many-body system with non-Hermitian Hamiltonian

Matias Ginzburg, Simone Rademacher, Giacomo De Palma

Mean-field Hartree theory is a central tool for reducing many-body dynamics to an effective one-particle evolution. This approximation has been employed when the Hamiltonian governing the many-body dynamics is not Hermitian. Indeed, non-Hermitian Hamiltonians model particle gain/loss or the evolution of open quantum systems between consecutive quantum jumps. Furthermore, the validity of the Hartree approximation for generic non-Hermitian Hamiltonians lies at the basis of a quantum algorithm for nonlinear differential equations. In this work, we show that this approximation can fail. First, we consider the most general bosonic mean-field Hamiltonian with two-body interactions and we prove that, for non-Hermitian Hamiltonian, the Hartree equation does not capture the correct second time derivative of the one-particle marginal state at $t=0$ in the thermodynamic limit. Then, we analytically solve a model of $N$ bosonic qubits with a purely anti-Hermitian Hamiltonian, compute the limit for $N\to\infty$ of the one-particle marginal state and show that it does not agree with the solution of the non-Hermitian Hartree equation. We further show that there exists an initial condition such that the exact one-particle marginal state undergoes a finite-time transition to a mixed state, a phenomenon that is completely absent in the case of Hermitian Hamiltonians. Finally, we simulate a non-Hermitian Hamiltonian with real spectrum and show that, also in this case, there are initial conditions for which the Hartree equation does not reproduce the correct one-particle marginal state in the limit $N\to\infty$ . Our findings challenge the validity of the mean-field Hartree approximation for non-Hermitian Hamiltonians, and call for additional conditions for the validity of the mean-field regime to model the dynamics of particle gain and loss and the open-system dynamics in bosonic many-body systems.Mean-field Hartree theory is a central tool for reducing many-body dynamics to an effective one-particle evolution. This approximation has been employed when the Hamiltonian governing the many-body dynamics is not Hermitian. Indeed, non-Hermitian Hamiltonians model particle gain/loss or the evolution of open quantum systems between consecutive quantum jumps. Furthermore, the validity of the Hartree approximation for generic non-Hermitian Hamiltonians lies at the basis of a quantum algorithm for nonlinear differential equations. In this work, we show that this approximation can fail. First, we consider the most general bosonic mean-field Hamiltonian with two-body interactions and we prove that, for non-Hermitian Hamiltonian, the Hartree equation does not capture the correct second time derivative of the one-particle marginal state at $t=0$ in the thermodynamic limit. Then, we analytically solve a model of $N$ bosonic qubits with a purely anti-Hermitian Hamiltonian, compute the limit for $N\to\infty$ of the one-particle marginal state and show that it does not agree with the solution of the non-Hermitian Hartree equation. We further show that there exists an initial condition such that the exact one-particle marginal state undergoes a finite-time transition to a mixed state, a phenomenon that is completely absent in the case of Hermitian Hamiltonians. Finally, we simulate a non-Hermitian Hamiltonian with real spectrum and show that, also in this case, there are initial conditions for which the Hartree equation does not reproduce the correct one-particle marginal state in the limit $N\to\infty$. Our findings challenge the validity of the mean-field Hartree approximation for non-Hermitian Hamiltonians, and call for additional conditions for the validity of the mean-field regime to model the dynamics of particle gain and loss and the open-system dynamics in bosonic many-body systems.

quant-pharXiv:2601.13038QuantumJan 20, 20260 citations

N = 16

). We consider randomly chosen one- and two-body couplings in the Hamiltonian, and the resulting evolution of several initial product states. We subsequently compare many-body and mean-field predictions for one-body observables, and we consider one- and two-body entanglement to assess under what conditions the many-body and mean-field predictions are likely to disagree. Except for a special category of prototypical initial conditions, we find that the typically employed mean-field approximation is insufficient to capture the evolution of one-body operators in the systems we consider. We also observe a loss of coherence in one- and two-body trace-reduced subsystems which suggests that the evolution may be well approximated as a classical mixture of separable states.

t=0

in the thermodynamic limit. Then, we analytically solve a model of

N

bosonic qubits with a purely anti-Hermitian Hamiltonian, compute the limit for

N\to\infty

of the one-particle marginal state and show that it does not agree with the solution of the non-Hermitian Hartree equation. We further show that there exists an initial condition such that the exact one-particle marginal state undergoes a finite-time transition to a mixed state, a phenomenon that is completely absent in the case of Hermitian Hamiltonians. Finally, we simulate a non-Hermitian Hamiltonian with real spectrum and show that, also in this case, there are initial conditions for which the Hartree equation does not reproduce the correct one-particle marginal state in the limit

N\to\infty

. Our findings challenge the validity of the mean-field Hartree approximation for non-Hermitian Hamiltonians, and call for additional conditions for the validity of the mean-field regime to model the dynamics of particle gain and loss and the open-system dynamics in bosonic many-body systems.