Abstract We discuss the derivation of an effective Hamiltonian for open quantum many-particle systems. The aim is to define an operator that can be used for (molecular) simulations where, through the exchange of energy and matter with the surrounding environment (reservoir), the number of particles, n, becomes a variable of the problem. The Hamiltonian is formally derived from the Von Neumann equation; specifically, we derive an n-hierarchy of equations for the density matrix, ρ ^ n , for near equilibrium situations. Such a hierarchy, in case of stationary equilibrium, delivers the standard grand canonical density matrix as it would be expected. We report that a similar Hamiltonian was conjectured, from empirical considerations, in the field of superconductivity. Thus our result also provide a formal basis for this long-standing hypothesis. Finally, an application is discussed for Path Integral simulations of molecular systems.Abstract We discuss the derivation of an effective Hamiltonian for open quantum many-particle systems. The aim is to define an operator that can be used for (molecular) simulations where, through the exchange of energy and matter with the surrounding environment (reservoir), the number of particles, n, becomes a variable of the problem. The Hamiltonian is formally derived from the Von Neumann equation; specifically, we derive an n-hierarchy of equations for the density matrix, ρ ^ n , for near equilibrium situations. Such a hierarchy, in case of stationary equilibrium, delivers the standard grand canonical density matrix as it would be expected. We report that a similar Hamiltonian was conjectured, from empirical considerations, in the field of superconductivity. Thus our result also provide a formal basis for this long-standing hypothesis. Finally, an application is discussed for Path Integral simulations of molecular systems.

Journal of Physics A Mathematical and TheoreticalMay 24, 20249 citations

Dependence of the one-body density matrix on the superconducting gap parameter Δ(T)

William L. Clinton, Louis J. Massa, Sipra Pal

We show how the one-electron thermally averaged density matrix \ensuremath{\rho}(1',1,T)==〈${\mathrm{\ensuremath{\Psi}}}^{\mathrm{*}}$(1')\ensuremath{\Psi}(1)〉==N〈Fd(2,...,N)${\mathrm{\ensuremath{\Psi}}}^{\mathrm{*}}$ (1',...,n)\ensuremath{\Psi}(1,...,N)〉 varies with the superconducting gap parameter \ensuremath{\Delta}(T). It is suggested how these results can be applied to experiments that probe the density matrix. In particular it is suggested that the x-ray structure factor F(G) should be an ideal probe of the change in \ensuremath{\rho}(r) as a material goes superconducting. We also discuss how our results can be applied to recent analysis of atomic pair distribution functions (PDF) coming from measurements of diffuse scattering of pulsed neutrons on the thallium high- ${\mathit{T}}_{\mathit{c}}$ materials.We show how the one-electron thermally averaged density matrix \ensuremath{\rho}(1',1,T)==〈${\mathrm{\ensuremath{\Psi}}}^{\mathrm{*}}$(1')\ensuremath{\Psi}(1)〉==N〈Fd(2,...,N)${\mathrm{\ensuremath{\Psi}}}^{\mathrm{*}}$ (1',...,n)\ensuremath{\Psi}(1,...,N)〉 varies with the superconducting gap parameter \ensuremath{\Delta}(T). It is suggested how these results can be applied to experiments that probe the density matrix. In particular it is suggested that the x-ray structure factor F(G) should be an ideal probe of the change in \ensuremath{\rho}(r) as a material goes superconducting. We also discuss how our results can be applied to recent analysis of atomic pair distribution functions (PDF) coming from measurements of diffuse scattering of pulsed neutrons on the thallium high-${\mathit{T}}_{\mathit{c}}$ materials.

Physical review. B, Condensed matterOct 1, 19921 citation

Variational Machine Learning Model for Electronic Structure Optimization via the Density Matrix

Luqi Dong, Shuxiang Yang, Su-Huai Wei +1 more

We present a novel approach that combines machine learning with direct variational energy optimization via the density matrix to solve the Kohn-Sham equation in density functional theory. Instead of relying on the conventional self-consistent field method, our approach directly optimizes the ground state by predicting the density matrix using a neural network, thereby bypassing Hamiltonian matrix diagonalization. Our model employs equivariant neural networks to generate a physically constrained density matrix, enabling stable and efficient energy minimization. This method integrates the construction of training sets directly into the model training process and achieves high accuracy in predicting ground-state properties across various molecular and extended systems, thereby establishing a powerful machine learning paradigm for electronic structure optimization and paving the way for large-scale quantum simulations.We present a novel approach that combines machine learning with direct variational energy optimization via the density matrix to solve the Kohn-Sham equation in density functional theory. Instead of relying on the conventional self-consistent field method, our approach directly optimizes the ground state by predicting the density matrix using a neural network, thereby bypassing Hamiltonian matrix diagonalization. Our model employs equivariant neural networks to generate a physically constrained density matrix, enabling stable and efficient energy minimization. This method integrates the construction of training sets directly into the model training process and achieves high accuracy in predicting ground-state properties across various molecular and extended systems, thereby establishing a powerful machine learning paradigm for electronic structure optimization and paving the way for large-scale quantum simulations.

Physical Review LettersNov 25, 20251 citation

Superconductivity & Superfluidity

WORLD SCIENTIFIC eBooksMay 1, 20010 citations

{\mathit{T}}_{\mathit{c}}

materials.