Quantum criticality for few-body systems: Path-integral approach

We present the path-integral approach to treat quantum phase transitions and critical phenomena for few-body quantum systems. Allowing the space and time variables to have discrete values, we turn the quantum problem into an effective classical lattice problem. Imposing the constraint that any change in space time must preserve the scaling invariance of Brownian paths, we show that the mapped classical lattice system has a known scaling behavior when the particle is free, which breaks down when the strength of the interaction potential reaches a certain value. In principle, any quantity with known scaling behavior may be used to determine the transition point. We illustrate the method by numerically evaluating the correlation length and the radial mean distance for a system composed of a single particle in the presence of an attractive Pöschl-Teller potential in one and three dimensions. The method is general and has potential applicability for large systems.