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

4 QPT in strongly correlated electron systems Heavy Fermion compounds High-T c compounds La 1.85 Sr 0.15 CuO 4 CePd 2 Si 2 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)

5 Discontinuous jump of Fermi surface small FSlarge FS Paschen et al., Nature (2004)

6 Fermionic sign problem Partition functionDensity 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!!!

7 A bit sharper Regardless the pretense of your theoretical friends: Minus signs are mortal !!!

8 The nodal hypersurface N=49, d=2 Antisymmetry of the wave functionNodal hypersurface Pauli surface Free Fermions Average distance to the nodes Free fermions First zero

9 Restricted path integrals Formally we can solve the sign problem!! Self-consistency problem: Path restrictions depend on ! Ceperley, J. Stat. Phys. (1991)

10 Temperature dependence of nodes The nodal hypersurface at finite temperature Free Fermions high T low TT=0

11 Reading the worldline picture Persistence lengthAverage node to node spacing Collision time Associated energy scale

12 Key to quantum criticality Mandelbrot set At the QCP scale invariance, no E F Nodal surface has to become fractal !!!

13 Turning on the backflow Nodal surface has to become fractal !!! Try backflow wave functions Collective (hydrodynamic) regime:

14 Fractal nodal surface

15 Hydrodynamic backflow Velocity field Ideal incompressible (1) fluid with zero vorticity (2) Introduce velocity potential (potential flow) Boundary condition Cylinder with radius r 0,

16 Including hydrodynamic backflow in wave functions  Explanation for mass enhancement in roton minimum of 4 He Simple toy model: Foreign atom (same mass, same forces as 4 He atoms, no subject to Bose statistics) moves through liquid with momentum Naive ansatz wave function: Moving particle pushes away 4 He 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

17 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)

18 Fractal dimension of the nodal surface Calculate the correlation integral on random d=2 dimensional cuts Backflow turns nodal surface into a fractal !!!

19 Just Ansatz or physics? Gabi Kotliar U/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

20 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 T c ???

21 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 … ).