Fermionic quantum criticality and the fractal nodal surface

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Fermionic quantum criticality and the fractal
nodal surface

• Jan Zaanen Frank Krüger

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Plan of talk

• Introduction quantum criticality

• Minus signs and the nodal surface

• Fractal nodal surface and backflow

• Boosting the cooper instability ?

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Quantum criticality

• Scale invariance at the QCP

• quantum critical region characterized by thermal
  fluctuations of the quantum critical state

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QPT in strongly correlated electron systems
Heavy Fermion compounds
High-Tc compounds
Grosche et al., Physica B (1996)
Custers et al., Nature (2003)
Takagi et al., PRL (1992)
Mathur et al., Nature (1998)
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Discontinuous jump of Fermi surface
small FS
large FS
Paschen et al., Nature (2004)
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Fermionic sign problem
Partition function
Density matrix
Imaginary time path-integral formulation
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A bit sharper
Regardless the pretense of your theoretical
friends
Minus signs are mortal !!!
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The nodal hypersurface
Antisymmetry of the wave function
Nodal hypersurface
Pauli surface
Free Fermions
N49, d2
Average distance to the nodes
Free fermions
First zero
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Restricted path integrals
Formally we can solve the sign problem!!
Ceperley, J. Stat. Phys. (1991)
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Temperature dependence of nodes
The nodal hypersurface at finite temperature
Free Fermions
high T
low T
T0
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Reading the worldline picture
Persistence length
Average node to node spacing
Collision time
Associated energy scale
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Key to quantum criticality
At the QCP scale invariance, no EF
Nodal surface has to become fractal !!!
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Turning on the backflow
Nodal surface has to become fractal !!!
Try backflow wave functions
Collective (hydrodynamic) regime
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Fractal nodal surface
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Hydrodynamic backflow
Velocity field
Ideal incompressible (1) fluid with zero
vorticity (2)
Introduce velocity potential (potential flow)
Boundary condition
Cylinder with radius r0,
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Including hydrodynamic backflow in wave functions

• Explanation for mass enhancement in roton
  minimum of 4He

Feynman Cohen, Phy. Rev. (1956)
Simple toy model
Foreign atom (same mass, same forces as 4He
atoms, no subject to Bose statistics) moves
through liquid with momentum
Naive ansatz wave function
Moving particle pushes away 4He atoms,
variational ansatz wave function
Solving resulting differential equation for g

• Backflow wavefunctions in Fermi systems

• Widely used for node fixing in QMC
• Significant improvement of variational GS
  energies

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Extracting the fractal dimension

• The box dimension (capacity dimension)

• The correlation integral

For fractals
Inequality very tight, relative error below 1
Grassberger Procaccia, PRL (1983)
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Fractal dimension of the nodal surface
Calculate the correlation integral
on random d2 dimensional cuts
Backflow turns nodal surface into a fractal !!!
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Just Ansatz or physics?
Mott transition, continuous
Mott insulator
metal
Compressibility 0
Finite compressibility
Neutral QP
Gabi Kotliar
U/W
Quasiparticles turn charge neutral
e
Backflow turns hydrodynamical at the quantum
critical point!
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Boosting the Cooper instability ?

• Fractal nodes hostile to single worldlines

? strong enhancement of Cooper pairing
gap equation
conventional BCS
fractal nodes
? possible explanation for high Tc ???
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Conclusions
Fermi-Dirac statistics is completely encoded in
boson physics and nodal surface constraints.
Hypothesis phenomenology of fermionic matter can
be classified on basis of nodal surface geometry
and bosonic quantum dynamics.
-gt A fractal nodal surface is a necessary
condition for a fermionic quantum critical state.
-gt Fermionic backflow wavefunctions have a
fractal nodal surface Mottness.
Work in progress reading the physics from bosons
and nodal geometry (Fermi-liquids,
superconductivity, criticality ) .