Nonperturbative Methods in Low Dimensional Quantum Field Theories

Giuseppe Mussardo

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.

Klaus Fredenhagen

Andrés Schlief, Peter Lunts, Sung-Sik Lee

In this work, the quantum field theory that describes non-Fermi liquids realized at antiferromagnetic quantum critical points in space dimensions between two and three has been solved nonperturbatively. Using the solvable theory as a model system, the dimensional regularization scheme is put to the test. While the perturbative solution proves to be useful in constructing the nonperturbative solution, it reveals one subtlety: the low-energy limit and the integer dimension limits do not commute due to crossover energy scales that vanish at the physical dimensions.

Kara Farnsworth

<!--HTML-->Hamiltonian truncation is a non-perturbative numerical method for calculating observables of a quantum field theory by truncating the Hilbert space to states with energy below a maximum energy cutoff. In this talk I will present an effective field theory approach to Hamiltonian truncation, which provides a systematic way of improving the calculations without increasing the energy cutoff. I will demonstrate this with numerical results for the two dimensional phi^4 theory, and talk about future applications of this method to more complicated theories.

Mrinmoy Basak

We employ the wavelet formalism of quantum field theory to study field theories in the nonperturbative Hamiltonian framework. Specifically, we make use of Daubechies wavelets in momentum space. These basis elements are characterised by a resolution and a translation index that provides for a natural nonperturbative infrared and ultraviolet truncation of the quantum field theory. As an application, we consider the $φ^4$ theory and demonstrate the emergence of the well-known nonperturbative strong-coupling phase transition in the $m^2>0$ sector.