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Equations for free fermion correlators out of equilibrium

This article explores the mathematical framework of shock wave physics in the context of free fermion systems. It discusses the KdV equation, the neglect of dispersion, and the application of these concepts to Fermi gases. The paper also introduces integrable equations for more complex correlation functions and highlights their relevance in physical problems such as Fermi edge singularity and counting statistics. Overall, the study aims to derive integrable nonlinear equations for free fermionic correlators and find suitable solutions for different scenarios.

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Equations for free fermion correlators out of equilibrium

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  1. Equations for free fermion correlators out of equilibrium E. B. P. Wiegmann A. Abanov TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAA

  2. Shock wave physics • E. g., The KdV Equation: • Neglecting Dispersion: • How and why should it apply to Fermi systems?

  3. Overturning in Fermi gas • Dynamics E p • Wigner function p vF pF x • Density

  4. Dispersive effects p v pF • A typical Wigner function: x Nonlinearity Dispersion + Hopf

  5. Generating functions • Density too simple • Need more complicated correlation functions: • Appear in physical problems: Fermi Edge singularity, counting statistics r p x x

  6. Math of Integrable shocks Christie • Three fermi points may serve as moduli

  7. Fermi points as moduli Christie • Three fermi points may serve as moduli

  8. Recap • Free Fermions display wave overturning • Integrability may allow to place Fermionic shock waves in general mathematical context • Must obtain integrable equations for more complicated objects than density • Korepin, Izergin, Slavnov, Its, Göhnmann obtained integrable Eqs in equilibrium

  9. Quantum Hopf Equation • Correct in the limit where excitations only scratch the surface • Hopf in components: • Proof: Two Fermion Four Fermionc

  10. The equation • Define: • We prove mKP: • Hirota Derivative: • Can be written in the form: • Semiclassically

  11. Outline of the proof Dynamics: Refermionization:

  12. Conclusion • Integrable nonlinear equation is derived for free fermionic correlators • Must find appropriate solutions relevant to different physical problems • Consistent with the notion that shocks appear which have simple Fermi point moduli

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