Fermionic quantum criticality and the fractal nodal surface Jan Zaanen & Frank Krüger.

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

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 … ) .