Download Fermionic quantum criticality and the fractal nodal surface

Fermionic quantum criticality and the
fractal nodal surface
Jan Zaanen & Frank Krüger
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
CePd2Si2
Generic observations:
 Non-FL behavior in the quantum critical region
 Instability towards SC in the vicinity of the QCP
La1.85Sr0.15CuO4
Grosche et al., Physica B (1996)
Mathur et al., Nature (1998)
Custers et al., Nature (2003)
Takagi et al., PRL (1992)
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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
Boltzmannons or Bosons:
Fermions:
 integrand non-negative
 negative Boltzmann weights
 probability of equivalent classical
system: (crosslinked) ringpolymers
 non probablistic!!!
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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
Pauli surface
Free Fermions
N=49, d=2
Nodal hypersurface
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)
Self-consistency problem:
Path restrictions depend on
!
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Temperature dependence of nodes
The nodal hypersurface at finite temperature
Free Fermions
T=0
low T
high T
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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 !!!
Mandelbrot set
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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
Simple toy model:
Feynman & Cohen, Phy. Rev. (1956)
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)
Equality in every nonpathological case !!!
 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 d=2 dimensional cuts
Backflow turns nodal
surface into a fractal !!!
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Just Ansatz or physics?
Mott transition, continuous
metal
Mott insulator
Finite compressibility
Compressibility = 0
U/W
Neutral QP
Gabi Kotliar
Quasiparticles turn charge neutral
Backflow turns hydrodynamical at
the quantum critical point!
e
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Boosting the Cooper instability ?
 Can we understand the „normal“ state (NFL),
e.g.
Relation between
and fractal dimension
?
 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.
-> A fractal nodal surface is a necessary condition for a
fermionic quantum critical state.
-> Fermionic backflow wavefunctions have a fractal
nodal surface: Mottness.
Work in progress: reading the physics from bosons and nodal
geometry (Fermi-liquids, superconductivity, criticality … ) .
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