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Nonperturbative flow equations from running expectation values.
F G Scholtz1, B H Bartlett, H B Geyer
1Institute of Theoretical Physics, University of Stellenbosch, Stellenbosch 7600, South Africa.
Physical Review Letters
|October 4, 2003
Summary
Wegner's flow equations for the Lipkin model are solved self-consistently, yielding a nonlinear differential equation for the order parameter. This approach accurately predicts results across phase transitions by expanding in fluctuations.
Area of Science:
- Physics
- Quantum Mechanics
- Statistical Mechanics
Background:
- Wegner's flow equations are a powerful tool for studying quantum systems.
- The Lipkin model provides a simplified yet insightful framework for exploring many-body phenomena.
- Understanding phase transitions in quantum models is crucial for theoretical advancements.
Purpose of the Study:
- To solve Wegner's flow equations self-consistently within the Lipkin model.
- To derive a nonlinear differential equation determining the order parameter.
- To analyze the behavior of the system across phase transitions.
Main Methods:
- Self-consistent solution of Wegner's flow equations.
- Expansion in fluctuations rather than the coupling constant.
- Analysis in the thermodynamic limit.
Main Results:
- A nonlinear differential equation for the order parameter as a function of the dimensionless coupling constant was obtained.
- The derived equation accurately determines the order parameter across the phase transition.
- The expansion in fluctuations ensures convergence to exact results in both phases.
Conclusions:
- The self-consistent solution of Wegner's flow equations provides a robust method for analyzing the Lipkin model.
- This approach offers a unified description of the system's behavior, including across phase transitions.
- The method's convergence properties highlight its validity in the thermodynamic limit.