Spiral cutoff-flow of quantum quartic oscillator

Theory of the quantum quartic oscillator is developed with close attention to the cutoff one needs to impose on the system in order to approximate the smallest eigenvalues and corresponding eigenstates of its Hamiltonian by diagonalizing matrices of limited size. The matrices are obtained by evaluating matrix elements of the Hamiltonian between the associated harmonic-oscillator eigenstates and by correcting the computed matrices to compensate for their limited dimension, using the Wilsonian renormalization-group procedure. When the oscillator is used to represent the zero-momentum mode of a scalar quantum field, the cutoff limits the number of quanta one includes in the dynamics, in analogy to the Tamm-Dancoff approach to solving Hamiltonian eigenvalue problems in quantum field theory. The cutoff dependence of the corrected matrices is found to be described by a spiral motion of a three-dimensional vector. This behavior is shown to result from a combination of a limit-cycle and a floating fixed-point behaviors, an unexpected phenomenon that warrants further study for the purpose of learning how the limits on the number of quanta can be accounted for in more complex systems. A brief discussion of the research directions concerning polynomial interactions of degree higher than four, spontaneous symmetry breaking and coupling of more than one oscillator, is included.