#### Exact Self-Consistent Effective Hamiltonian Theory

*Xindong Wang, Xiao Chen, Liqin Ke, Hai-Ping Cheng, and Bruce Harmon*

We propose a general variational fermionic many-body wavefunction that generates an effective Hamiltonian in a quadratic form, which can then be exactly solved. The theory can be constructed within the density functional theory framework, and a self-consistent scheme is proposed for solving the exact density functional theory. We apply the theory to structurally-disordered systems, symmetric and asymmetric Hubbard dimers, and the corresponding lattice models. The single fermion excitation spectra show a persistent gap due to the fermionic-entanglement-induced pairing condensate. For disordered systems, the density of states at the edge of the gap diverges in the thermodynamic limit, suggesting a topologically ordered phase. A sharp resonance is predicted as the gap is not dependent on the temperature of the system. For the symmetric Hubbard model, the gap for both half-filling and doped case suggests that the quantum phase transition between the antiferromagnetic and superconducting phases is continuous.

#### Many-Body Fermions and Riemann Hypothesis

*Xindong Wang and Alex Shulman*

We study the algebraic structure of the eigenvalues of a Hamiltonian that corresponds to a many-body fermionic system. As the Hamiltonian is quadratic in fermion creation and/or annihilation operators, the system is exactly integrable and the complete single fermion excitation energy spectrum is constructed using the non-interacting fermions that are eigenstates of the quadratic matrix related to the system Hamiltonian. Connection to the Riemann Hypothesis is discussed.

#### Low Frequency Electrical Resonance in Water

*Xindong Wang and Qiang Fu*

We report the observation of sharp electrical resonance of water with width ~2 neV in the low radio frequency range at room temperature. The neV level of the resonant width under room temperature (~25 meV) is consistent with the theory in Wang et al (2020) that predicts a macroscopic long-range coherent quantum mechanical excited states, Majorana fermions, resulting from quantum entanglement of proton hopping at hydrogen bonds.

#### Mathematical Theory of Locally Coherent Quantum Many-Body Fermionic System

*Xindong Wang*

We provide a mathematical foundation to the formulation of the SCEHT (“Self-Consistent Effective Hamiltonian Theory”) that enables further study of excited states of the system in a more systematic and theoretical manner. Gauge fields are introduced and correct total energy functional in relations to the coupling gauge field is given. We also provides a Monte-Carlo numerical scheme for the search of the ground state that goes beyond the SCEHT.

#### Self-Consistent Effective Hamiltonian Theory for Fermionic Many Body Systems

*Xindong Wang and Hai-Ping Cheng*

Accepted and forthcoming in International Journal of Modern Physics B

Using a separable many-body variational wavefunction, we formulate a self-consistent effective Hamiltonian theory for fermionic many-body system. The theory is applied to the two-dimensional Hubbard model as an example to demonstrate its capability and computational effectiveness. Most remarkably for the Hubbard model in 2-d, a highly unconventional quadruple-fermion non-Cooper-pair order parameter is discovered.