Generalized Pauli constraints (GPCs) impose constraints in the form of inequalities on the natural orbital occupation numbers of the one electron reduced density matrix (1-RDM), defining the set of pure N-representable 1-RDMs, or 1-RDMs that can be derived from an N-electron wave function. Saturation of these constraints is termed “pinning” and implies a significant simplification of the N-electron wave function as the number of Slater determinants required to fully describe the system is reduced. Recent research has shown pinning to occur for the ground states of atoms and molecules with N = 3 and r = 6, where N is the number of electrons and r is the number of spin orbitals. For N = 4 and r = 8, however, pinning occurs not to the GPCs but rather to inequalities defining the pure N-representable two-electron reduced density matrices (2-RDMs). Using these more general inequalities, we derive a wave function ansatz for a system with four electrons in eight spin orbitals. We apply the ansatz to the isoelectronic series of the carbon atom and the dissociation of linear H4 where the correlation energies are recovered to fractions of a kcal/mol. These results provide a foundation for further developments in wave function and RDM theories based on “pinned” solutions, and elucidate a fundamental physical basis for the emergence of non-orthogonal bases in electronic systems of N ≥ 4.Generalized Pauli constraints (GPCs) impose constraints in the form of inequalities on the natural orbital occupation numbers of the one electron reduced density matrix (1-RDM), defining the set of pure N-representable 1-RDMs, or 1-RDMs that can be derived from an N-electron wave function. Saturation of these constraints is termed “pinning” and implies a significant simplification of the N-electron wave function as the number of Slater determinants required to fully describe the system is reduced. Recent research has shown pinning to occur for the ground states of atoms and molecules with N = 3 and r = 6, where N is the number of electrons and r is the number of spin orbitals. For N = 4 and r = 8, however, pinning occurs not to the GPCs but rather to inequalities defining the pure N-representable two-electron reduced density matrices (2-RDMs). Using these more general inequalities, we derive a wave function ansatz for a system with four electrons in eight spin orbitals. We apply the ansatz to the isoelectronic series of the carbon atom and the dissociation of linear H4 where the correlation energies are recovered to fractions of a kcal/mol. These results provide a foundation for further developments in wave function and RDM theories based on “pinned” solutions, and elucidate a fundamental physical basis for the emergence of non-orthogonal bases in electronic systems of N ≥ 4.

The Journal of Chemical PhysicsApr 8, 20198 citations

Properties of Reduced Density Matrices

Theoretical chemistryJan 1, 19760 citations

Universal spectral correlations in the chaotic wave function and the development of quantum chaos

Xiao Chen, Andreas W. W. Ludwig

We investigate the appearance of quantum chaos in a single many-body wave function by analyzing the statistical properties of the eigenvalues of its reduced density matrix ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{A}$ of a spatial subsystem $A$ . We find that (i): the spectrum of the density matrix is described by so-called Wishart random matrix theory, which (ii): exhibits besides level repulsion, spectral rigidity, and universal spectral correlations between eigenvalues separated by distances ranging from one up to many mean level spacings, which we investigate. We use these universal spectral characteristics of the reduced density matrix as a definition of chaos in the wave function. A simple and precise characterization of such universal correlations in a spectrum is a segment of strictly linear growth at sufficiently long times, recently called the ``ramp,'' of the spectral form factor which is the Fourier transform of the correlation function between a pair of eigenvalues. It turns out that Wishart and standard random matrix theory have the same universal ``ramp.'' Specifically, here numerical results for the spectral form factor of the density matrix of generic nonintegrable many-body systems, such as one-dimensional quantum Ising and Floquet spin models, are found to exhibit a universal ``ramp'' identical to that appearing for a ``random pure state'' (``Page state,'' or ``Haar state''). The density matrix of the latter is precisely the Wishart random matrix, the reduced density matrix of a completely random wave function. In addition, we study the development of chaos in the wave function by letting an initial direct product state evolve under the unitary time evolution. We find that the universal spectral correlations as manifested by the ``ramp'' set in as soon as the entanglement entropy begins to grow, and first develop for the eigenvalues at the top of the spectrum of the density matrix ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{A}$, subsequently spreading over the entire spectrum at later times. Finally, we study a prethermalized regime described by a generalized Gibbs ensemble, which develops in a rapidly driven Floquet model at intermediate times. We find that the prethermalized regime exhibits no chaos, as evidenced by the absence of a ``ramp'' in the spectral form factor of the density matrix, while the universal spectral correlations start to develop when the prethermalized regime finally relaxes at late times to the fully thermalized (infinite temperature) chaotic regime.We investigate the appearance of quantum chaos in a single many-body wave function by analyzing the statistical properties of the eigenvalues of its reduced density matrix ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{A}$ of a spatial subsystem $A$. We find that (i): the spectrum of the density matrix is described by so-called Wishart random matrix theory, which (ii): exhibits besides level repulsion, spectral rigidity, and universal spectral correlations between eigenvalues separated by distances ranging from one up to many mean level spacings, which we investigate. We use these universal spectral characteristics of the reduced density matrix as a definition of chaos in the wave function. A simple and precise characterization of such universal correlations in a spectrum is a segment of strictly linear growth at sufficiently long times, recently called the ``ramp,'' of the spectral form factor which is the Fourier transform of the correlation function between a pair of eigenvalues. It turns out that Wishart and standard random matrix theory have the same universal ``ramp.'' Specifically, here numerical results for the spectral form factor of the density matrix of generic nonintegrable many-body systems, such as one-dimensional quantum Ising and Floquet spin models, are found to exhibit a universal ``ramp'' identical to that appearing for a ``random pure state'' (``Page state,'' or ``Haar state''). The density matrix of the latter is precisely the Wishart random matrix, the reduced density matrix of a completely random wave function. In addition, we study the development of chaos in the wave function by letting an initial direct product state evolve under the unitary time evolution. We find that the universal spectral correlations as manifested by the ``ramp'' set in as soon as the entanglement entropy begins to grow, and first develop for the eigenvalues at the top of the spectrum of the density matrix ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{A}$, subsequently spreading over the entire spectrum at later times. Finally, we study a prethermalized regime described by a generalized Gibbs ensemble, which develops in a rapidly driven Floquet model at intermediate times. We find that the prethermalized regime exhibits no chaos, as evidenced by the absence of a ``ramp'' in the spectral form factor of the density matrix, while the universal spectral correlations start to develop when the prethermalized regime finally relaxes at late times to the fully thermalized (infinite temperature) chaotic regime.

arXiv:1710.02686Physical review. B./Physical review. BAug 30, 201858 citations

A

. We find that (i): the spectrum of the density matrix is described by so-called Wishart random matrix theory, which (ii): exhibits besides level repulsion, spectral rigidity, and universal spectral correlations between eigenvalues separated by distances ranging from one up to many mean level spacings, which we investigate. We use these universal spectral characteristics of the reduced density matrix as a definition of chaos in the wave function. A simple and precise characterization of such universal correlations in a spectrum is a segment of strictly linear growth at sufficiently long times, recently called the ``ramp,'' of the spectral form factor which is the Fourier transform of the correlation function between a pair of eigenvalues. It turns out that Wishart and standard random matrix theory have the same universal ``ramp.'' Specifically, here numerical results for the spectral form factor of the density matrix of generic nonintegrable many-body systems, such as one-dimensional quantum Ising and Floquet spin models, are found to exhibit a universal ``ramp'' identical to that appearing for a ``random pure state'' (``Page state,'' or ``Haar state''). The density matrix of the latter is precisely the Wishart random matrix, the reduced density matrix of a completely random wave function. In addition, we study the development of chaos in the wave function by letting an initial direct product state evolve under the unitary time evolution. We find that the universal spectral correlations as manifested by the ``ramp'' set in as soon as the entanglement entropy begins to grow, and first develop for the eigenvalues at the top of the spectrum of the density matrix ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{A}$, subsequently spreading over the entire spectrum at later times. Finally, we study a prethermalized regime described by a generalized Gibbs ensemble, which develops in a rapidly driven Floquet model at intermediate times. We find that the prethermalized regime exhibits no chaos, as evidenced by the absence of a ``ramp'' in the spectral form factor of the density matrix, while the universal spectral correlations start to develop when the prethermalized regime finally relaxes at late times to the fully thermalized (infinite temperature) chaotic regime.