Mott collapse and statistical quantum criticality

Transcription

1 Mott collapse and statistical quantum criticality Jan Zaanen J. Zaanen and B.J. Overbosch, arxiv: (Phil.Trans.Roy.Soc. A, in press) 1

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3 Who to blame 3

4 Plan 1. The idea of Mott collapse 2. Mottness versus Weng statistics 3. Statistics and t-j numerics 4. Fermions, scale invariance and Ceperley s path integral 4

5 Quantum Phase transitions Quantum scale invariance emerges naturally at a zero temperature continuous phase transition driven by quantum fluctuations: JZ, Science 319, 1205 (2008) 5

6 m * = 1 E F Fermionic quantum phase transitions in the heavy fermion metals JZ, Science 319, 1205 (2008) QP effective mass E F 0 m * Paschen et al., Nature (2004) bad players

7 Phase diagram high Tc superconductors Stripy stuff, spontaneous currents, phase fluctuations.. Mystery quantum critical metal The return of normalcy JZ, Science 315, 1372 (2007) Ψ BCS = Π k u k + v k c + c + ( k ) vac. k 7

8 The quantum in the kitchen: Landau s miracle Fermi energy Kinetic energy Fermi momenta k=1/wavelength Fermi surface of copper Electrons are waves Pauli exclusion principle: every state occupied by one electron Unreasonable: electrons strongly interact!! Landau s Fermi-liquid: the highly collective low energy quantum excitations are like electrons that do not interact. 8

9 BCS theory: fermions turning into bosons Fermi-liquid + attractive interaction Bardeen Cooper Schrieffer Quasiparticles pair and Bose condense: Ground state D-wave SC: Dirac spectrum Ψ BCS = Π ( k u k + v k c + c + ) k vac. k 9

10 Fermions and Hertz-Millis- Moriya-Lonzarich Bosonic (magnetic, etc.) order parameter drives the phase transition Supercon ductivity Electrons: fermion gas = heat bath damping bosonic critical fluctuations Fermi gas Bosonic critical fluctuations back react as pairing glue on the electrons 10

11 Fermion sign problem Imaginary time path-integral formulation Boltzmannons or Bosons: integrand non-negative probability of equivalent classical system: (crosslinked) ringpolymers Fermions: negative Boltzmann weights non probablistic: NP-hard problem (Troyer, Wiese)!!!

12 Mottness and quantum statistics Weng statistics Statistical criticality Fermi-Dirac statistics Resonating Valence Bond orderly physics: stripes competing with d-wave superconductivity Incompatible statistics merge in scale invariant fractal statistics Secret of high Tc?? Implies Fermi-liquid and BCS superconductivity 12

13 Cuprates start as doped Mottinsulators Anderson H = t + c iσ c jσ + U n i ij Mott insulator i n i Doped Mott insulator ) H t J = t + c ) r iσ c jσ + J S i r ij ij S j 13

14 Mottness and Hilbert space dimensionality 14

15 Mott-maps and highway ramps Phil Anderson 15

16 traffic 16

17 Probing rush hour in the electron world Seamus Davis et al.: Science, march (JZ, Perspective) 17

18 Mott-maps and highway ramps Phil Anderson 18

19 Mott collapse: Hubbard model Phillips Jarrell DCA results for Hubbard model at intermediate couplings (U = 0.75W): Non-fermi liquid Mott fluid Quantum critical state, very unstable to d-wave superconductivity Fermi-liquid at high dopings 19

20 Mott collapse: Hubbard model Jarrell DCA results for Hubbard model at intermediate couplings (U = 2W): Pseudogaps Superconducting Tc 20

21 Catherine s selective Mott transition Pepin RKKY = f-electrons are Mott-localized Kondo = f-electrons are effectively unprojected. 21

22 Plan 1. The idea of Mott collapse 2. Mottness versus Weng statistics 3. Statistics and t-j numerics 4. Fermions, scale invariance and Ceperley s path integral 22

23 Cuprates start as doped Mottinsulators Anderson H = t + c iσ c jσ + U n i ij Mott insulator i n i Doped Mott insulator ) H t J = t + c ) r iσ c jσ + J S i r ij ij S j 23

24 Quantum statistics and path integrals Feynman Bose condensation: Partition sum dominated by infinitely long cycles Fermions: infinite cycles set in at T F, but cycles with length w and w+1 cancel each other approximately. Free energy pushed to E F! Cycle decomposition 24

25 Mott insulator: the vanishing of Fermi-Dirac statistics Mott-insulator: the electrons become distinguishable, stay at home principle! Spins live in tensor-product space. Spin signs are like hard core bosons in a magnetic field, can be gauged away on a bipartite lattice ( Marshall signs ) τ c = +1 25

26 Doped Mott-insulator: Weng statistics Zheng-Yu Weng t-j model: spin up is background, spin down s ( spinons ) and holes ( holons ) are hard core bosons. Exact Partition sum: The sign of any term is set by: N h h c [ ] N h c [ ] The (fermionic) number of holon exchanges The number of spinonholon collisions arxiv:

27 RVB: the statistical rational Resonating valence bond states: Quantum liquid organizing away the Weng collision signs, lowers the energy! Weng statistics compatible with a d-wave superconducting ground state 27

28 Field theory: Mutual Chern-Simons Coherent state description: - holes: hard core bosons - spins: Schwinger bosons Statistics: mutual flux π ( ) b jσ H t = t h i + h j e ia ij s iφ ij 0 H J attachments! ±π = ij σ + b iσ e iσa h ji + h. c. J 2 ij Δˆ s ij + Δˆ s ij Δ ˆ s ij = e iσa ij h b iσ b j σ σ s A h A 28

29 The subtly different d-wave superconductor ( ) b jσ H t = t h i + h j e ia ij s iφ ij 0 ij σ + b iσ e iσa h ji s + + h. c. H J = Δˆ Δˆ J 2 ij ij s ij Δ ˆ s ij = e iσa ij h b iσ b j σ σ Charge e condensate: Spins Arovas-Auerbach massive RVB: h i + ) s Δ ij 0 0 => d-wave superconductor supporting massless Bogoliubov excitations! Topologically subtly different from BCS superconductor: carries S=1/2 quantum π number. π h 2e vortex 29

30 Plan 1. The idea of Mott collapse 2. Mottness versus Weng statistics 3. Statistics and t-j numerics 4. Fermions, scale invariance and Ceperley s path integral 30

31 t-j numerics: high T expansions Putikka Singh 12-th order in 1/T, down to T = 0.2 J (PRL 81, 2966 (1998) - Contrast in momentum distribution ( k n k, T n k ) tiny compared to equivalent free fermion problem n k T - Fermi-arcs develop with pairing correlations: no big Fermi surface, on its way to the d-wave ground state! n k T 31

32 t-j numerics: high T Take J << t, low hole density: predictions Free case: λ free = a 2zt k B T, λ free( T F ) r s t-j model: hole thermal de Broglie wavelength limited by spin-spin correlation length through collision signs! Free case: below the Fermi temperature the high T expansion is strongly oscillating because of hard Fermion signs. t-j model: positive contributions increasingly outnumber negative ones since signs are organized away! 32

33 t-j numerics: the DMRG stripes White T=0 Crystallized RVB Weng statistics implies much less delocalization pressure compared to Fermi-Dirac: competing localizing instabilities spoil the superconducting fun! 33

34 Plan 1. The idea of Mott collapse 2. Mottness versus Weng statistics 3. Statistics and t-j numerics 4. Fermions, scale invariance and Ceperley s path integral 34

35 Statistical quantum criticality Weng- and Fermi-Dirac statistics microscopically incompatible: Mott collapse should turn into a first order phase separation transition But Mark claims a quantum critical end point!? 35

36 Fermionic sign problem Imaginary time path-integral formulation Boltzmannons or Bosons: integrand non-negative probability of equivalent classical system: (crosslinked) ringpolymers Fermions: negative Boltzmann weights non probablistic!!!

37 The nodal hypersurface Antisymmetry of the wave function Free Fermions Pauli hypersurface d=2 Nodal hypersurface Test particle

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

39 Reading the worldline picture Persistence length Average node to node spacing Collision time Associated energy scale

40 Key to fermionic quantum criticality At the QCP scale invariance, no E F Nodal surface has to become fractal!!!

41 Hydrodynamic backflow Classical fluid: incompressible flow Feynman-Cohen: mass enhancement in 4 He Wave function ansatz for foreign atom moving through He superfluid with velocity small compared to sound velocity: Backflow wavefunctions in Fermi systems Widely used for node fixing in QMC Significant improvement of variational GS energies

42 Frank s fractal nodes Feynman s fermionic backflow wavefunction: Frank Krüger 42

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45 Fermionic quantum phase transitions in the heavy fermion metals JZ, Science 319, 1205 (2008) Paschen et al., Nature (2004)

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

47 MC calculation of n(k) m m 1 a * a c 3 Divergence of effective mass as a a c

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49 Further reading. Overview: J. Zaanen and B.J. Overbosch, arxiv: (Phil.Trans.Roy.Soc. A, in press). Weng statistics: K. Wu, Z.Y. Weng and J. Zaanen, PRB 77, (2008); arxiv: Mott collapse and conformal superconductivity: S.-X. Yang et al, PRL 106, (2011). Fractal nodes: F. Kruger and J. Zaanen, PRB 78, (2008) 49

50 In summary 1. Mott collapse: cuprates, heavy fermions? DCA is convincing! 2. Mottness and Weng statistics 3. Statistics and t-j numerics 4. Ceperley s path integral: quantum statistics can go scale invariant 50

51 Why Tc is high JZ, Nature 430, 512(2004) Fermionic quantum critical state BCS type transition: pairs form at Tc But BCS wisdoms like: 2Δ = hω boson e 1/ λ 2Δ 3.5k B T c Need the Fermi energy! 51

52 Why is Tc high? Because there is superglue binding the electrons in pairs Wrong! The superfluid density is tiny, it is very easy to bend and twist a high Tc superconductor. Its cohesive energy sucks. Tc s are set by the competition between the two sides The theory of the mechanism should explain why the free energy of the metal is seriously BAD. 52

53 Superconductivity born from a fermionic critical state 53

54 The glue-ish essence of BCS Let s believe retarded glue => Gap equation: 1 gχ pp (k B T,Δ,hω B ) = 0 Glue strength SC gap Glue frequency The pair susceptibility of the Fermi liquid is a logarithm because of EF: χ pp ( k B T,L) = ln E hω F B k B T,L k B T c = hω B e 1/ λ, 2Δ 3.5k B T c 54

55 Hitting the critical stuff with glue J.-H. She Gap equation: 1 g χ pp (k B T,Δ,hω B ) = 0 Glue strength SC gap Cooper instability of the fermionic quantum critical state? Glue frequency The pair susceptibility has just to be the most divergent one! Form largely fixed by scaling (non-conserved currents): χ pp ( ω,t) Z ( ) Φ hω / z pp k B T T 2 η pp hω k B T : hω k B T : χ pp χ pp 1 ( ) / z T 2 η pp 1 ( ) / z ( iω) 2 η pp 1 1 iωτ h 55

56 Scaling versus the BCS gap Gap equation: Fermi-liquid: Critical case: Huang s equation : ω c = E F, λ = g E F,α =1 λ = g, α = 2 η pp ω c z 1 g ω c +1 Δ 0 = 2hω B λ + 2ω B λ ( ) 2 η pp ω c 2hω B ( ) / z J.-H. She dω = 0 2Δ 0 ω α Δ 0 = 2hω B e 1/ λ 56 z 2 η pp

57 Huang s equation at work λ Δ 0 = 2hω B λ + 2ω B ( ) 2 η pp ω c ( ) / z J.-H. She Strongly interacting critical state, e.g. 1+1D Ising: η pp =1/4, z =1 z 2 η pp Increasing retardation: more bang for the bucks! Standard BCS 57

58 Huang s equation versus high Tc E.g. 1+1D Ising: η pp =1/4, z =1 J.-H. She Typical phonon-, cut-off energy: ω B ω c = 50 mev 500 mev Typical gap: Fermi-liquid: Critical case: Δ 0 = 40meV λ 1.1 λ 0.13!!! Davis, Balatsky 58

59 In summary 1. High Tc s normal state, heavy fermions: experiment demonstrates the existence of a mysterious scale invariant state formed from fermions. 2. Fermionic quantum criticality is not governed by Wilsonian RNG: the fractal nodal surface, 3. Quantum critical BCS: moderate glue yields a high Tc! Δ 0 = 2hω B λ / λ + 2ω B /ω c ( ( ( ) 2 η pp ) / z ) z 2 η pp 59

60 Quantum criticality or conformal fields 60

61 Fractal Cauliflower (romanesco)

62 Where is the cuprate quantum critical point? High precision resistivities La2-xSrxCuO4. Science 323, 603 (2009) ρ = ρ + α T + α T Hussey ρ T α et al. τ h = h k B T 62

63 Fermion hunting club She Sadri Overbosch Krueger, Urbana Mukhin, Moscow Weng, Beijing Mitas, Raleigh Fisher, Microsoft Ceperley, Urbana Further reading/playing: arxiv: , , ,

64 Senthil s critical Fermi-surface A( K r,ω,t,g) = c 0 ω α ( k // ) / z k // ( ) F c 1ω z( k k // ),ω T,k ν k g g // c ( ) arxiv:

65 BCS nodal structure Fermi liquid: Ψ FL = Π k c + + k c k vac. BCS state: Ψ BCS = Π ( k u k + v k c + c + ) k vac. k Nodal holes => BCS log Conjecture: the thermodynamic measure of the BCS nodal holes is larger for fractal nodal structure, because the latter is more configuration space filling. 65

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

67 Fermi liquid s nodal pocket Average distance to the nodes Free fermions First zero 67

68 Just Ansatz or physics? Mott transition, continuous metal Mott insulator Finite compressibility Compressibility = 0 Quasiparticles turn charge neutral U/W Backflow turns hydrodynamical at the quantum critical point! Neutral QP e Gabi Kotliar

69 Scanning tunneling spectroscopy Seamus Davis, Cornell 69

70 Zaanen-Gunnarsson (1987) Large S on 3-band Hubbard model: Hole density on oxygen 70

71 Ceperley path integral: Fermi gas in momentum space Single particle propagator: single particle momentum conserved N particle density matrix: Sergei Mukhin harmonic potential

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

73 Ceperley path integral in 1+1D Nodal surface (Nd-1)= Pauli surface (Nd-d) Statistical physics of hard core polymers (Pokrovsky, Talapov) Ordering is preserved, statistics changes in N! relabellings x 1 ( τ) < x 2 ( τ) <L < x N ( τ), τ Particles become distinguishable. v Fermi energy: E F = h /τ collisions Entropic repulsions: algebraic order, sound velocity = Fermi velocity. JZ, PRL

74 Fermi gas = cold atom Mott insulator in harmonic trap! ( ) ρ F (K,K";τ) = 2πδ k i k i " k 1 k 2 L k N e k i 2 τ 2hm Mukhin, JZ,..., Iranian J. Phys. (2008)

75 Switching on interactions Single particle spectral function: Configuration space: all Mott configurations of particles in trap Fermi-gas: all configurations isolated = nodal surface J.-H. She ( ) d Fermi-liquid: n 1 F n n F pieces of (d-1) dimensional antinodal surfaces each of which has volume ( k r = 2π r n ) a Fermi-surface protection: antinodal surface shrinks to a point! ( ) n F d 2 n n F 75

76 Extracting the fractal dimension The correlation integral: For fractals: Inequality very tight, relative error below 1% Grassberger & Procaccia, PRL (1983)

77 The Gross list: the 14 Big Questions 1. The origin of the universe? 2. What is dark matter? What is space-time? 14. New states of matter: are there generic non-fermi liquid states of interacting condensed matter? Solution of the Fermion sign problem??

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