Mott collapse and statistical quantum criticality Jan Zaanen J. Zaanen and B.J. Overbosch, arXiv: 0911.4070 (Phil.Trans.Roy.Soc. A, in press) 1
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Who to blame … 3
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
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
Fermionic quantum phase transitions in the heavy fermion metals JZ, Science 319, 1205 (2008) QP effective mass m * = 1 E F E F → 0 ⇒ m * → ∞ Paschen et al., Nature (2004) ‘bad players’
Phase diagram high Tc superconductors Mystery quantum critical ‘Stripy stuff’, spontaneous currents, phase fluctuations .. metal The return of normalcy ( ) vac . + c − k ↓ Ψ BCS = Π k u k + v k c k ↑ + JZ, Science 315 , 1372 (2007) 7
The quantum in the kitchen: Landau’s miracle Kinetic energy Electrons are waves Fermi Pauli exclusion principle: every energy state occupied by one electron Unreasonable: electrons strongly interact !! Fermi momenta k=1/wavelength Landau’s Fermi-liquid: the highly collective low energy quantum excitations are like electrons that do not interact. Fermi surface of copper 8
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 ( ) vac . + c − k ↓ Ψ BCS = Π k u k + v k c k ↑ + 9
Fermions and Hertz-Millis- Moriya-Lonzarich Bosonic (magnetic, etc.) order parameter drives the phase transition Electrons: fermion gas = heat bath damping bosonic critical fluctuations Supercon ductivity Bosonic critical fluctuations ‘back react’ as pairing glue on the electrons Fermi gas 10
Fermion sign problem Imaginary time path-integral formulation Boltzmannons or Bosons: Fermions: integrand non-negative negative Boltzmann weights probability of equivalent classical non probablistic: NP-hard system: (crosslinked) ringpolymers problem (Troyer, Wiese)!!!
Mottness and quantum statistics ‘Weng’ statistics ‘Statistical criticality’ Fermi-Dirac statistics ‘Resonating Valence Bond’ Incompatible statistics Implies Fermi-liquid and orderly physics: stripes merge in scale invariant BCS superconductivity competing with d-wave ‘fractal statistics’ superconductivity Secret of high Tc?? 12
Cuprates start as doped Mott- insulators Anderson + H = t c j σ + U ∑ ∑ c i σ n i ↑ n i ↓ ij i Mott insulator Doped Mott insulator r r ) ) H t − J = t + c j σ + J • ∑ ∑ c S S i σ i j ij ij 13
Mottness and Hilbert space dimensionality 14
Mott-maps and highway ramps Phil Anderson 15
traffic 16
Probing rush hour in the electron world Seamus Davis et al.: Science, march 9 2007 (JZ, Perspective) 17
Mott-maps and highway ramps Phil Anderson 18
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
Mott collapse: Hubbard model Jarrell DCA results for Hubbard model at intermediate couplings (U = 2W): Pseudogaps Superconducting Tc 20
Catherine’s ‘selective Mott transition’ Pepin “RKKY” = f-electrons are “Kondo”= f-electrons are Mott-localized effectively unprojected. 21
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
Cuprates start as doped Mott- insulators Anderson + H = t c j σ + U ∑ ∑ c i σ n i ↑ n i ↓ ij i Mott insulator Doped Mott insulator r r ) ) H t − J = t + c j σ + J • ∑ ∑ c S S i σ i j ij ij 23
Quantum statistics and path integrals Feynman Bose condensation: Partition sum dominated by infinitely long cycles Cycle decomposition 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 ! 24
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 τ c = + 1 magnetic field, can be gauged away on a bipartite lattice (“Marshall signs”) 25
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: [ ] h c The (fermionic) number N h of holon exchanges [ ] ↓ c The number of spinon- N h holon ‘collisions’ arXiv: 0802.0273 26
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
Field theory: “Mutual Chern-Simons” ( ) b j σ ⎛ ⎞ h s − i φ ij 0 + b i σ e + h . c . H t = − t + h j e i σ A ji ⎜ ⎟ ∑ iA ij Coherent state description: h i ⎝ ⎠ ij σ - holes: hard core bosons + s = J h b i σ b j − σ s s ˆ Δ = ˆ ˆ − i σ A ij ∑ ∑ − Δ Δ e - spins: Schwinger bosons H ij J ij ij 2 ij σ π Statistics: mutual flux attachments! ± π h s A A 28
The subtly different d-wave superconductor ( ) b j σ ⎛ ⎞ s = + h ˆ h b i σ b j − σ J Δ s − i φ ij − i σ A ij + b i σ e + h . c . ∑ 0 H t = − t + h j e i σ A ji s s ⎜ ⎟ = ˆ ˆ ∑ iA ij e ∑ h i − Δ Δ H ij ⎝ ⎠ J ij ij 2 σ ij ij σ + ≠ 0 Charge e condensate: h i ) Δ s ≠ 0 Spins Arovas-Auerbach massive RVB: ij => d-wave superconductor supporting massless Bogoliubov excitations! h Topologically subtly different from BCS superconductor: vortex 2 e π carries S=1/2 quantum number. π 29
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
t-J numerics: high T expansions Putikka Singh 12-th order in 1/T, down to T = 0.2 J … (PRL 81, 2966 (1998) ∂ n k - Contrast in momentum ∂ T distribution ( ) tiny ∂ k n k , ∂ T n k compared to equivalent free fermion problem - Fermi-arcs develop with pairing correlations: no big Fermi surface, on its way to the d-wave ground state! ∂ n k ∂ T 31
t-J numerics: high T predictions Take J << t, low hole density: ( ) ≈ r λ free = a 2 zt k B T , λ free T F Free case: 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
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
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
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
Fermionic sign problem Imaginary time path-integral formulation Boltzmannons or Bosons: Fermions: integrand non-negative negative Boltzmann weights probability of equivalent classical non probablistic!!! system: (crosslinked) ringpolymers
The nodal hypersurface Antisymmetry of the wave function Pauli hypersurface Free Fermions d=2 Nodal hypersurface Test particle
Constrained path integrals Formally we can solve the sign problem!! Ceperley, J. Stat. Phys. (1991) Self-consistency problem: Path restrictions depend on !
Reading the worldline picture Average node to node spacing Persistence length Collision time Associated energy scale
Key to fermionic quantum criticality At the QCP scale invariance, no E F Nodal surface has to become fractal !!!
Hydrodynamic backflow Classical fluid: Feynman-Cohen: mass enhancement in 4 He incompressible flow 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
Frank’s fractal nodes … Feynman‘s fermionic backflow wavefunction: Frank Krüger 42
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