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Fermionic quantum criticality and the fractal nodal surface
Jan Zaanen & Frank Krüger
2
Plan of talk
Introduction quantum criticality
Minus signs and the nodal surface
Fractal nodal surface and backflow
Boosting the cooper instability ?
3
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
La1.85Sr0.15CuO4
CePd2Si2
Generic observations:
Non-FL behavior in the quantum critical region
Instability towards SC in the vicinity of the QCP
Takagi et al., PRL (1992) Custers et al., Nature (2003)Grosche et al., Physica B (1996)
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
Boltzmannons or Bosons:
integrand non-negative
probability of equivalent classical system: (crosslinked) ringpolymers
Fermions:
negative Boltzmann weights
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
N=49, d=2
Antisymmetry of the wave function Nodal hypersurface
Pauli surface Free Fermions
Average distance to the nodes
Free fermions
First zero
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Restricted path integrals
Formally we can solve the sign problem!!
Self-consistency problem:
Path restrictions depend on !
Ceperley, J. Stat. Phys. (1991)
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Temperature dependence of nodes
The nodal hypersurface at finite temperature
Free Fermions
high T low TT=0
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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
Mandelbrot set
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
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:
Feynman & Cohen, Phy. Rev. (1956)
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 non-pathological 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?
Gabi KotliarU/W
Mott transition, continuous
Mott insulator
Compressibility = 0
metal
Finite compressibility
Quasiparticles turn charge neutral
Backflow turns hydrodynamical at the quantum critical point!
e
Neutral QP
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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 … ) .