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Dyson equation.md

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The Dyson equation and the concept of a fixed point are closely connected, particularly in iterative or self-consistent solutions of quantum systems or many-body problems. The Dyson equation can often be seen as a fixed-point equation because it is an implicit equation that defines a propagator or Green's function in terms of itself through the self-energy.

Dyson Equation: Fixed-Point Perspective

The general Dyson equation is:

[ G = G_0 + G_0 \Sigma G ]

where:

  • ( G ): Full propagator (to be determined).
  • ( G_0 ): Free propagator (known).
  • ( \Sigma ): Self-energy, which may depend on ( G ).

This equation is recursive because ( G ) appears on both sides. Solving this equation is equivalent to finding a fixed point for ( G ), where applying the operation on the right-hand side leaves ( G ) unchanged.

Dyson Equation as a Fixed-Point Problem

  1. Rewrite as a Functional Form: Define an operator ( \mathcal{F} ) such that: [ \mathcal{F}(G) = G_0 + G_0 \Sigma[G] G ] Here, ( \Sigma[G] ) denotes the self-energy, which can itself be a functional of ( G ).

  2. Fixed-Point Definition: ( G ) is a fixed point if: [ \mathcal{F}(G) = G ] This means that substituting ( G ) into the right-hand side yields ( G ) itself.

  3. Iterative Solution: Fixed-point equations like the Dyson equation are often solved iteratively:

    • Start with an initial guess ( G^{(0)} = G_0 ).
    • Update using: [ G^{(n+1)} = G_0 + G_0 \Sigma[G^{(n)}] G^{(n)} ]
    • Repeat until ( G^{(n+1)} \approx G^{(n)} ), indicating convergence to the fixed point.

  4. Convergence Criteria:

    • The convergence depends on properties of ( \Sigma ) and ( G_0 ).
    • If ( \Sigma ) is well-behaved (e.g., Lipschitz continuous), the iteration can converge to the fixed point.

Applications of the Fixed-Point Perspective

  1. Self-Consistent Field Theory:

    • In condensed matter physics, self-energy ( \Sigma ) often depends on ( G ), making the Dyson equation inherently self-consistent.
    • Solving for ( G ) involves finding a fixed point of the self-consistent equations.

  2. Renormalization:

    • Dyson's equation encapsulates renormalization, where ( \Sigma[G] ) represents corrections to bare propagators.
    • The fixed point reflects the renormalized propagator after accounting for all interaction effects.

  3. Quantum Monte Carlo:

    • Fixed-point iterations are used in computational methods, such as solving for Green's functions or spectral properties in quantum Monte Carlo simulations.

  4. Non-Perturbative Methods:

    • The fixed-point formulation is critical in non-perturbative approaches, such as dynamical mean-field theory (DMFT), where the Green's function is updated iteratively until convergence.

Fixed Points and Physics Insights

  1. Stability of Solutions:

    • The fixed point ( G ) of the Dyson equation represents a stable solution to the physical problem. If the system has multiple fixed points, each corresponds to a different phase or state of the system.

  2. Quasiparticles and Poles:

    • The fixed point ( G ) often reveals the properties of quasiparticles. For example, poles of ( G ) in the complex plane correspond to particle-like excitations.

  3. Iterative Refinement:

    • The fixed-point interpretation provides a natural framework for iterative refinement of solutions in systems with complex interactions.

Conclusion

The Dyson equation is naturally a fixed-point equation because the full propagator ( G ) depends recursively on itself. Viewing it as a fixed-point problem not only aids in numerical and analytical solutions but also highlights deep connections between self-consistency, renormalization, and iterative methods in quantum and many-body physics.