Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 45 min ago
Peer-reviewedCC BY 4.0Quantum

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

  • Matias Ginzburg1
  • Simone Rademacher2
  • Giacomo De Palma3
  1. 1Scuola Internazionale Superiore di Studi Avanzati, Via Bonomea, 265, 34136 Trieste TS, Italy
  2. 2Department of Mathematics, LMU Munich, Theresienstrasse 39, 80333 Munich, Germany
  3. 3University of Bologna, Department of Mathematics, Piazza di Porta San Donato 5, 40126 Bologna BO, Italy
Open at sourcePDFDOICite

Abstract

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 → ∞ 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 → ∞ .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.

Related

  1. Conserved quantities in a bosonic tight-binding chain with non-Hermitian quartic terms
    Tetsuya Iwasaki, Hosho Katsura
    arXivPreprintarXiv licenceFull textMathematical physicsAtomic, molecular & optical physicsarXiv
  2. Structure-Preserving Full Approximation Scheme for Real-Space Hartree-Fock Equations
    Yuejia Zhang
    arXivPreprintarXiv licenceFull textComputational physicsarXiv
  3. Dynamical quantum phase transitions in a pseudo-Hermitian Hamiltonian: The imbalanced-pairing Kitaev model
    R. Jafari, Alireza Akbari, Shukhrat Mardonov +1
    Physical Review ResearchPeer-reviewedCC BY 4.0Atomic, molecular & optical physicsCrossref
  4. When is a bosonic operator a Hamiltonian?
    Felix Fischer, Hermann Schulz-Baldes, Daniel Burgarth +1
    arXivPreprintarXiv licenceFull textMathematical physicsQuantum physics & quantum informationarXiv
  5. Grid-state deformation in a no-jump non-Hermitian bosonic dimer
    B. M. Rodriguez-Lara, H. Ghaemi-Dizicheh, S. Dehdashti +3
    arXivPreprintarXiv licenceFull textQuantum physics & quantum informationarXiv

Cite

BibTeXRIS
BibTeX
@article{ginzburg2026failure,
  title = {{Failure of the mean-field Hartree approximation for a bosonic many-body system with non-Hermitian Hamiltonian}},
  author = {Matias Ginzburg and Simone Rademacher and Giacomo De Palma},
  journal = {Quantum},
  year = {2026},
  month = oct,
  doi = {10.22331/q-2026-10-22-2244},
  url = {https://quantum-journal.org/papers/q-2026-10-22-2244/},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Failure of the mean-field Hartree approximation for a bosonic many-body system with non-Hermitian Hamiltonian
AU  - Matias Ginzburg
AU  - Simone Rademacher
AU  - Giacomo De Palma
PY  - 2026
DA  - 2026/10/22
JO  - Quantum
DO  - 10.22331/q-2026-10-22-2244
UR  - https://quantum-journal.org/papers/q-2026-10-22-2244/
AB  - 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 → ∞ 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 → ∞ .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.
SN  - 2521-327X
ER  -