Modal expansions and the diagonalization of quantum field Hamiltonians

Juan Sebastián Ardenghi, Mario Castagnino, Olímpia Lombardi

Dean Lee, Nathan Salwen, Daniel Lee

Mark Windoloski

The method of modal field theory is a new development in the field of nonperturbative quantum field theory. This approach reduces a quantum field theory to a finite-dimensional quantum mechanical system by expanding field configurations in terms of free-wave modes. In this dissertation we apply this method to three-dimensional &phis;4 theory using two kinds of modal field approaches: a spherical partial wave expansion and a periodic-box mode expansion. The resulting modal-field quantum-mechanical systems are analyzed with the use of the diffusion Monte Carlo method and by calculating the spectrum and eigenstates of the Hamiltonian directly. In the latter approach we employ the recently introduced quasi-sparse eigenvector method which is designed to diagonalize infinite-dimensional yet very sparse matrices. We study the phase structure of three-dimensional &phis;4 theory, computing the critical coupling and the critical exponents ν and β. We also investigate the spectrum of low-lying energy eigenstates and find evidence of a nonperturbative state in the broken-symmetry phase of the theory.

Fang Gao, Yaoxiong Wang, Qing Zhong Gao +2 more

The information of quantum pathways can be extracted in the framework of the Hamiltonian- encoding and Observable-decoding method. For dosed quantum systems, only off-diagonal elements of the Hamiltonian in the Hilbert space is required to be encoded to obtain the desired transitions. For open quantum systems, environment-related terms will appear in the diagonal elements of the Hamiltonian in the Liouville space. Thus diagonal encodings have to be performed to differentiate different pathways, which will lead to self-to-self transitions and inconsistency of pathway amplitudes with Dyson expansion. In this work, a well-designed transformation is proposed to avoid the counterintuitive transitions and the inconsistency, with or without control fields. Numerical simulations show that the method are consistent with Dyson expansion.

Chai Yan

M. H. S. Amin, Anatoly Yu. Smirnov, Neil G. Dickson +1 more

An approximate diagonalization method is proposed that combines exact diagonalization and perturbation expansion to calculate low-energy eigenvalues and eigenfunctions of a Hamiltonian. The method involves deriving an effective Hamiltonian for each eigenvalue to be calculated, using perturbation expansion, and extracting the eigenvalue from the diagonalization of the effective Hamiltonian. The size of the effective Hamiltonian can be significantly smaller than that of the original Hamiltonian, hence the diagonalization can be done much faster. We compare our approximate diagonalization results with those obtained using exact diagonalization and quantum Monte Carlo calculation for random problem instances with up to 128 qubits.