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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 ? 2 Quantum criticality Scale invariance at the QCP quantum critical region characterized by thermal fluctuations of the quantum critical state 3 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) 4 Discontinuous jump of Fermi surface small FS large FS Paschen et al., Nature (2004) 5 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!!! 6 A bit sharper Regardless the pretense of your theoretical friends: -- - - - - - Minus signs are mortal !!! 7 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 8 Restricted path integrals Formally we can solve the sign problem!! Ceperley, J. Stat. Phys. (1991) Self-consistency problem: Path restrictions depend on ! 9 Temperature dependence of nodes The nodal hypersurface at finite temperature Free Fermions T=0 low T high T 10 Reading the worldline picture Persistence length Average node to node spacing Collision time Associated energy scale 11 Key to quantum criticality At the QCP scale invariance, no EF Nodal surface has to become fractal !!! Mandelbrot set 12 Turning on the backflow Nodal surface has to become fractal !!! Try backflow wave functions Collective (hydrodynamic) regime: 13 Fractal nodal surface 14 Hydrodynamic backflow Velocity field Ideal incompressible (1) fluid with zero vorticity (2) Introduce velocity potential (potential flow) Boundary condition Cylinder with radius r0, 15 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 16 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) 17 Fractal dimension of the nodal surface Calculate the correlation integral on random d=2 dimensional cuts Backflow turns nodal surface into a fractal !!! 18 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 19 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 ??? 20 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 … ) . 21