Equations for free fermion correlators out of equilibrium

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Equations for free fermion correlators out of
equilibrium
E. B. P. Wiegmann A. Abanov
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Shock wave physics

• E. g., The KdV Equation
• Neglecting Dispersion

• How and why should it apply to Fermi systems?

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Overturning in Fermi gas

• Wigner function

p
vF
pF
x

• Density

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Dispersive effects

• A typical Wigner function

x
Dispersion
Nonlinearity

Hopf
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Generating functions

• Density too simple
• Need more complicated correlation functions
• Appear in physical problems Fermi Edge
  singularity, counting statistics

r
p
x
x
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Math of Integrable shocks
Christie

• Three fermi points may serve as moduli

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Fermi points as moduli
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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

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Quantum Hopf Equation

• Correct in the limit where excitations only
  scratch the surface
• Hopf in components
• Proof

Two Fermion
Four Fermionc
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The equation

• Define
• We prove mKP
• Hirota Derivative
• Can be written in the form
• Semiclassically

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Outline of the proof

• Dynamics
• Refermionization

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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