Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:nucl-th/0407031 · International Journal of Modern Physics E

TO WHAT EXTENT DOES THE SELF-CONSISTENT MEAN-FIELD EXIST?

LU GUO, FUMIHIKO SAKATA, En-Guang Zhao, J. A. MARUHN

Published Jun 14, 2006• 2 citations
arXivHTMLPDF

Abstract

A non-convergent difficulty near level-repulsive region is discussed within the self-consistent mean-field theory. It is shown by numerical and analytic studies that the mean-field is not realized in the many-fermion system when quantum fluctuations coming from two-body residual interaction and quadrupole deformation are larger than an energy difference between two avoided crossing orbits. An analytic condition indicating a limitation of the mean-field concept is derived for the first time.

DOI
10.1142/s0218301306004739

Related papers

SELF-CONSISTENT MEAN FIELD IN MANY-BODY CORRELATION DYNAMICS

Shun-Jin Wang, 中国高等科学技术中心(世界实验室)理论物理中心,兰州大学

The self-consistent mean field in the many-body correlation dynamics is discussed in detail. It is shown that the self-consistent mean field exists in general in many-body correlation dynamics for both stationary and non-stationary cases. The many-body correlations enter the mean field through the two-body correlation c2 and the corresponding collision term I. The role played by I is twofold i.e., dynamical effects and statistical effects on single particle states. The many-body correlations also manifest their effects on the many-body energy and yield an energy expression different from the conventional expression in the HF-Bruc-kner theory.

Acta Physica Sinica•Jan 1, 1988•0 citations

Multi-orbital two-particle self-consistent approach – Strengths and limitations

Jonas B. Profe, Jiawei Yan, Karim Zantout +2 more

Extending many-body numerical techniques which are powerful in the context of simple model calculations to the realm of realistic material simulations can be a challenging task. Realistic systems often involve multiple active orbitals, which increases the complexity and numerical cost because of the large local Hilbert space and the large number of interaction terms or sign-changing off-diagonal Green’s functions. The two-particle self-consistent approach (TPSC) is one such many-body numerical technique, for which multi-orbital extensions have proven to be involved due to the substantially more complex structure of the local interaction tensor. In this paper we extend earlier multi-orbital generalizations of TPSC by setting up two different variants of a fully self-consistent theory for TPSC in multi-orbital systems. We first investigate the strengths and limitations of the approach analytically and then benchmark both variants against dynamical mean-field theory (DMFT) and D-TRILEX results. We find that the exact behavior of the system can be faithfully reproduced in the weak-coupling regime, while at stronger couplings the performance of the two TPSC variants strongly depends on details of the system.

arXiv:2410.00962•SciPost Physics•Jul 21, 2025•4 citations

Stochastic mean-field dynamics for fermions in the weak-coupling limit

Denis Lacroix A

Assuming that the effect of the residual interaction beyond the mean field is weak and has a short memory time, two approximate treatments of correlation in fermionic systems by means of the Markovian quantum jump are presented. A simplified scenario for the introduction of fluctuations beyond the mean field is presented first. In this theory, part of the quantum correlations between the residual interaction and the one-body density matrix are neglected and jumps occur between many-body densities formed of pairs of states $D=|{\ensuremath{\Phi}}_{a}\ensuremath{\rangle}\ensuremath{\langle}{\ensuremath{\Phi}}_{b}|/\ensuremath{\langle}{\ensuremath{\Phi}}_{b}|{\ensuremath{\Phi}}_{a}\ensuremath{\rangle}$, where $|{\ensuremath{\Phi}}_{a}\ensuremath{\rangle}$ and $|{\ensuremath{\Phi}}_{b}\ensuremath{\rangle}$ are antisymmetrized products of single-particle states. The underlying stochastic mean-field theory is discussed and is applied to the monopole vibration of a spherical $^{40}\mathrm{Ca}$ nucleus under the influence of a statistical ensemble of two-body contact interactions. This framework is however too simplistic to account for both fluctuation and dissipation. In the second part of this work, an alternative quantum jump method is obtained without making the approximation on quantum correlations. By restricting to two-particle--two-hole residual interactions, the evolution of the one-body density matrix of a correlated system is transformed into a Lindblad equation. The associated dissipative dynamics can be simulated by quantum jumps between densities written as $D=|\ensuremath{\Phi}\ensuremath{\rangle}\ensuremath{\langle}\ensuremath{\Phi}|$, where $|\ensuremath{\Phi}\ensuremath{\rangle}$ is a normalized Slater determinant. The associated stochastic Schr\"odinger equation for single-particle wave functions is given.

arXiv:quant-ph/0509038•Physical Review C•Apr 17, 2006•7 citations

Introduction

Yoshio Kuramoto, Yusuke S. Kato

Interactions in many-body systems bring about collective phenomena such as superconductivity and magnetism. In many cases, simple mean-field theory provides a basic understanding of these phenomena. In fermion systems in one dimension, however, neither the mean-field theory nor perturbation theory works if it starts from the non-interacting fermions. This is because the interaction effects in one dimension are much stronger than those in higher dimensions. Intuitively speaking, two particles cannot avoid collision in a single-way track in contrast with two and three dimensions. Thus the interaction effects appear in a drastic way in one dimension. Another aspect in one dimension, which overcompensates the difficulty of perturbation and mean-field theories, is that a complete account of interaction effects is possible under certain conditions. The class of systems satisfying such conditions is referred to as exactly solvable. Soon after the establishment of quantum mechanics, Bethe solved exactly the Heisenberg spin model in one dimension [28]. The basic idea of the solution is now called the Bethe ansatz. Since then, theoretical physics in one dimension has developed into a magnificent edifice, including sophisticated mathematical techniques. In many cases, the eigenfunctions derived by the Bethe ansatz consist of plane waves that are defined stepwise for each spatial configuration of particles. Since the coefficients of plane waves depend on the configuration, the property of the wave function cannot be made explicit without detailed knowledge of these coefficients. We mention some of the recent monographs on the Bethe ansatz and its extensions [54, 118, 179]. A comprehensive account on exactly solvable models has recently been given by Sutherland [178].

Cambridge University Press eBooks•Aug 6, 2009•0 citations

Solitons, vortices and shell structure in ultracold atomic quantum systems

Gunnar Eriksson

This dissertation deals with finite-size effects in a few different quantum many-body phenomena in ultracold atomic systems. The finite-sized systems were simulated numerically using both mean-field methods and methods beyond mean-field, e.g. quadratic configuration interaction and exact diagonalization. The thesis is based on five scientific papers:Paper I is about the finite-size effects in the dynamics of a few-body soliton-like state. The collapses and revivals of the solitary wavefront in the particle density is characterized both analytically and numerically in the limit of weak interactions.Paper II analyzes a scheme to renormalize the contact interaction when used for exact diagonalization in a two-dimensional space. By relating the coupling strength in the diagonalization using the two-particle system, for which regularized solutions exists, converged results are obtained.Paper III deals with the formation of quantized vortices when stirring a harmonically trapped finite-sized system with a quadrupole deformation. I the energy spectrum the avoided crossing is found to be described by either of two types of correlated states. The type of avoided crossing can be controlled by the inclusion of an extra quartic deformation. Hysteresis in the mean-field description is found to be related to the types of avoided crossings exhibited and the full many-body time-evolution is compared to mean-field results. Paper IV investigates vortices in rotating fermionic droplets with dipole-dipole interactions. The vortex structure is found to still be present after the interaction is made anisotropic by having the dipoles tilt. Paper V contains an investigation of the effects on shell structure in fermionic droplets with dipole-dipole interactions. For a anisotropic harmonic oscillator, by tilting the dipoles, the shell structure of an isotropic oscillator can be restored.

•Mar 26, 2020•0 citations

PERTURBATION THEORY ABOUT GENERALIZED SELF-CONSISTENT FIELD SOLUTION

Chia‐Wei Yang, ARMEN N. KOCHARIAN, Y. L. Chiang +1 more

A new efficient converging perturbation technique valid for arbitrary interaction strength U/t and electron concentration n is applied to study strongly correlated electrons within the repulsive Hubbard model. We test the new perturbation expansion theory in terms of interacting quasi-particles about generalized self-consistent field (GSCF) solution in an entire parameter space. The developed perturbation formalism for strong interaction and large perturbation region differs from the traditional theory on the vicinity of non-interacting electrons by incorporating systematically fluctuations of quasi-particles around self-consistent solution. Performed analytical calculations of the ground state properties in the extreme conditions of one dimensionality provide quite reasonable numerical agreement with the Bethe-ansatz results in the intermediate range of U/t and n. The results are compared with the predictions of traditional perturbation theory.

International Journal of Modern Physics B•Jun 10, 2005•1 citation
A non-convergent difficulty near level-repulsive region is discussed within the self-consistent mean-field theory. It is shown by numerical and analytic studies that the mean-field is not realized in the many-fermion system when quantum fluctuations coming from two-body residual interaction and quadrupole deformation are larger than an energy difference between two avoided crossing orbits. An analytic condition indicating a limitation of the mean-field concept is derived for the first time.
The self-consistent mean field in the many-body correlation dynamics is discussed in detail. It is shown that the self-consistent mean field exists in general in many-body correlation dynamics for both stationary and non-stationary cases. The many-body correlations enter the mean field through the two-body correlation c2 and the corresponding collision term I. The role played by I is twofold i.e., dynamical effects and statistical effects on single particle states. The many-body correlations also manifest their effects on the many-body energy and yield an energy expression different from the conventional expression in the HF-Bruc-kner theory.
Extending many-body numerical techniques which are powerful in the context of simple model calculations to the realm of realistic material simulations can be a challenging task. Realistic systems often involve multiple active orbitals, which increases the complexity and numerical cost because of the large local Hilbert space and the large number of interaction terms or sign-changing off-diagonal Green’s functions. The two-particle self-consistent approach (TPSC) is one such many-body numerical technique, for which multi-orbital extensions have proven to be involved due to the substantially more complex structure of the local interaction tensor. In this paper we extend earlier multi-orbital generalizations of TPSC by setting up two different variants of a fully self-consistent theory for TPSC in multi-orbital systems. We first investigate the strengths and limitations of the approach analytically and then benchmark both variants against dynamical mean-field theory (DMFT) and D-TRILEX results. We find that the exact behavior of the system can be faithfully reproduced in the weak-coupling regime, while at stronger couplings the performance of the two TPSC variants strongly depends on details of the system.
Assuming that the effect of the residual interaction beyond the mean field is weak and has a short memory time, two approximate treatments of correlation in fermionic systems by means of the Markovian quantum jump are presented. A simplified scenario for the introduction of fluctuations beyond the mean field is presented first. In this theory, part of the quantum correlations between the residual interaction and the one-body density matrix are neglected and jumps occur between many-body densities formed of pairs of states $D=|{\ensuremath{\Phi}}_{a}\ensuremath{\rangle}\ensuremath{\langle}{\ensuremath{\Phi}}_{b}|/\ensuremath{\langle}{\ensuremath{\Phi}}_{b}|{\ensuremath{\Phi}}_{a}\ensuremath{\rangle}$, where $|{\ensuremath{\Phi}}_{a}\ensuremath{\rangle}$ and $|{\ensuremath{\Phi}}_{b}\ensuremath{\rangle}$ are antisymmetrized products of single-particle states. The underlying stochastic mean-field theory is discussed and is applied to the monopole vibration of a spherical 40Ca^{40}\mathrm{Ca}40Ca nucleus under the influence of a statistical ensemble of two-body contact interactions. This framework is however too simplistic to account for both fluctuation and dissipation. In the second part of this work, an alternative quantum jump method is obtained without making the approximation on quantum correlations. By restricting to two-particle--two-hole residual interactions, the evolution of the one-body density matrix of a correlated system is transformed into a Lindblad equation. The associated dissipative dynamics can be simulated by quantum jumps between densities written as $D=|\ensuremath{\Phi}\ensuremath{\rangle}\ensuremath{\langle}\ensuremath{\Phi}|$, where $|\ensuremath{\Phi}\ensuremath{\rangle}$ is a normalized Slater determinant. The associated stochastic Schr\"odinger equation for single-particle wave functions is given.
Interactions in many-body systems bring about collective phenomena such as superconductivity and magnetism. In many cases, simple mean-field theory provides a basic understanding of these phenomena. In fermion systems in one dimension, however, neither the mean-field theory nor perturbation theory works if it starts from the non-interacting fermions. This is because the interaction effects in one dimension are much stronger than those in higher dimensions. Intuitively speaking, two particles cannot avoid collision in a single-way track in contrast with two and three dimensions. Thus the interaction effects appear in a drastic way in one dimension. Another aspect in one dimension, which overcompensates the difficulty of perturbation and mean-field theories, is that a complete account of interaction effects is possible under certain conditions. The class of systems satisfying such conditions is referred to as exactly solvable. Soon after the establishment of quantum mechanics, Bethe solved exactly the Heisenberg spin model in one dimension [28]. The basic idea of the solution is now called the Bethe ansatz. Since then, theoretical physics in one dimension has developed into a magnificent edifice, including sophisticated mathematical techniques. In many cases, the eigenfunctions derived by the Bethe ansatz consist of plane waves that are defined stepwise for each spatial configuration of particles. Since the coefficients of plane waves depend on the configuration, the property of the wave function cannot be made explicit without detailed knowledge of these coefficients. We mention some of the recent monographs on the Bethe ansatz and its extensions [54, 118, 179]. A comprehensive account on exactly solvable models has recently been given by Sutherland [178].
This dissertation deals with finite-size effects in a few different quantum many-body phenomena in ultracold atomic systems. The finite-sized systems were simulated numerically using both mean-field methods and methods beyond mean-field, e.g. quadratic configuration interaction and exact diagonalization. The thesis is based on five scientific papers:Paper I is about the finite-size effects in the dynamics of a few-body soliton-like state. The collapses and revivals of the solitary wavefront in the particle density is characterized both analytically and numerically in the limit of weak interactions.Paper II analyzes a scheme to renormalize the contact interaction when used for exact diagonalization in a two-dimensional space. By relating the coupling strength in the diagonalization using the two-particle system, for which regularized solutions exists, converged results are obtained.Paper III deals with the formation of quantized vortices when stirring a harmonically trapped finite-sized system with a quadrupole deformation. I the energy spectrum the avoided crossing is found to be described by either of two types of correlated states. The type of avoided crossing can be controlled by the inclusion of an extra quartic deformation. Hysteresis in the mean-field description is found to be related to the types of avoided crossings exhibited and the full many-body time-evolution is compared to mean-field results. Paper IV investigates vortices in rotating fermionic droplets with dipole-dipole interactions. The vortex structure is found to still be present after the interaction is made anisotropic by having the dipoles tilt. Paper V contains an investigation of the effects on shell structure in fermionic droplets with dipole-dipole interactions. For a anisotropic harmonic oscillator, by tilting the dipoles, the shell structure of an isotropic oscillator can be restored.
A new efficient converging perturbation technique valid for arbitrary interaction strength U/t and electron concentration n is applied to study strongly correlated electrons within the repulsive Hubbard model. We test the new perturbation expansion theory in terms of interacting quasi-particles about generalized self-consistent field (GSCF) solution in an entire parameter space. The developed perturbation formalism for strong interaction and large perturbation region differs from the traditional theory on the vicinity of non-interacting electrons by incorporating systematically fluctuations of quasi-particles around self-consistent solution. Performed analytical calculations of the ground state properties in the extreme conditions of one dimensionality provide quite reasonable numerical agreement with the Bethe-ansatz results in the intermediate range of U/t and n. The results are compared with the predictions of traditional perturbation theory.