Towards ab-initio simulation of high temperature superconductors

Transcription

1 Towards ab-initio simulation of high temperature superconductors Sandro Sorella SISSA, DEMOCRITOS, Trieste Santa Barbara 1 September 2009

2 Motivations Model systems, nice reduction schemes but how much can we trust them? Do they explain or fit the experiments? Variational approach is always approximate and no exact method exists for fermions in 2D /3D strongly correlated systems. Still waiting for PEPS, MERA, Sign problem solution in quantum Monte Carlo Why do not we try to use the variational scheme for realistic systems with full Coulomb e-e?

3 Outline RVB theory, short review on lattice models Testing RVB theory: assume RVB wave function is a correct and exhaustive description of correlation reasonable agreement with experiments Starting from 2003 still a lot of work to be done Testing RVB on small molecules, comparison with existing quantum chemistry methods. HTc: few preliminary results with huge computational resources

4 Cuprates quasi-2d structure Phase diagram: temperature vs. doping J = 1500 K >> Other δ (doping) = 0.2 << 1 materials

5 Cu S S S J H j Cu n n i / 2 1 Cu - plane 2D.. = = r r The Anderson s wavefunction 0 = δ ] 2 ) ( exp[ ) ( 0 Singlet bond ˆ ˆ ) ( exp,,,,, = = = = < i i n g i n i n i I J J j i j i j i j i c c c c f P BCS ,,,,, The pairing function :...),.(. k k k k k k k k k k k k k BCS f stripes AF h c c c c c H + Δ + Δ = + + Δ + = ε ε ε σ σ σ

6 Anderson s variational wavefunction for spin models P BCS Jˆ = exp fi, j ( c c c c ) i, j, i, j, i< j Singlet bond 0 But J Resonance Valence Bond P BCS = All p Weight: #Sites/2 ( 1) fi k, jk k= 1 ( 1) p is the sign (+/-) of the permutation of i, j1, i2, j

7 Best variational PBCS in Heisenberg model Δ Δ = BCS (cos k cos k ) x y 2 k Nodal point gapless Hot point gap = Δ BCS Energy of BCS excitations: E k = ε 2 k + Δ 2 k

8 The BCS d-wave variational parameter shows the pseudogap feature observed in experiments Δ BCS From Pamarekanti, Randeria, Trivedi, PRB 00

9 From RVB to superconductivity [ N( # particles), θ (phase coherence) ] = i The presence of holes (empty sites) allows charge (super-) current and superconductivity

10 Instead the actual order parameter ~ x (doping) Φ = ( ) c + c + 0 only for x > 0 d wave This is the most important feature of an RVB superconductor

11 Weak coupling theory and RVB are not in conflict t-j(singlet attraction)-v,u(charge-repulsion): V (nearest neigbour) U (on site) Plehanov-Fabrizio-Sorella PRL 06

12 Rational behind High-Tc in Cuprates M. Capone et al., Science (2002). We need antiferromagnetic fluctuations J as an electronic attractive channel for supercond. On the other hand we need a small quasiparticle weight Z (close to a Mott insulator) in order to overscreen the long range Coulomb repulsion: V V Z 2 J J Pnictides, another way to have small Z?

13 Classification of the possible RVB insulators Paired RVB, i.e. Δ BCS 0 Nodal RVB Δ BCS 0 Fermi RVB Δ BCS on a = 0 Fermi gapped gapless gapless surface with a gap in the mean field, and broken symmetry (e.g. AF) RVB spin liquid, gap but no kind of order(dimer,magnetic,nematic. Spin liquid (algebraic) Nodal fermions quasiparticles Fermi gas spin properties: e.g. finite susceptibility

14 The Cooper pairing in an insulator: 2D Hubbard model N=#sites E. Plehanov F. Becca and S.S. PRB 05 F R = N 2 c i c + i+ R c i+ R c i N U/t=4 Fermi sea (Δ BCS ->0) 0.01 F(R) sites R// (1.0)

15 Similar conclusions in Bulut, Scalapino, White, PRB 93 Half filled Hubbard model testing instabilities λ~1 of the quasiparticles interactions AF D-wave singlet

16 For a lattice model ψ J ˆ RVB = exp f ( c c + c c ) i, j 14 i, 4 j, 24 j, 4 i 3, i, j Singlet bond 0

17 ) ( )], ( exp[, < = r i r j j j i i f Det r r v RVB x TurboRVB : the RVB for real electrons (see qeforge web page) ore general than Hartree-Fock, the most important correlation is included for free with a single determinant..., (, centered on the ion orbital,...), (, the is ) ( where : ) ( ) ( ) ( ) (,,,,,,,,,, c b a k j i r r r g f th th th i a j b i a j i b a j i b a j i r r r r ψ ψ ψ λ υ = contracted p ), ( ) ( 2 1, r Z p j j k m l j A e r Y r = = φ θ ψ

18 Technical achievements used for TurboRVB I) Optimization of several ~10000 Variational parameters now possible Using the recent Hessian method : S.S. PRB 05, C. Umrigar PRL 05, C. Umrigar, J. Toulouse, C, Filippi, S.S. and R. Henning PRL 07 S. Sorella, M. Casula and D. Rocca JCP 07 II) DMC: the lowest energy state with the same nodes of the Variational wf with pseudo: Old i) often unstable, ii) non variational energy upper bound New (M.Casula C.Filippi and S.S.) PRL05 LatticeRegularizedDiffusionMonteCarlo Very stable variational upper bounds of the pseudo Hamiltonian energy. Key idea: on a lattice all interactions are nonlocal III) Very general parametrization of the Jastrow factor, in principle complete (so vdw correct)

19 Small systems where electron correlation is very important and TurboRVB was successful Hydrogen and benzene (JCP 08) Benzene dimer (JCP 05) Beryllium dimer (Archive) Iron dimer (CPL 09)

20 Binding energy of aromatic molecules 7 + Molecule Within RVB+LRDMC Estimated from Exp. C (7)eV 6.34(14)eV C 6 H (2)eV 59.24(11)eV

21 The Beryllium dimer The weakest bounded dimer (~0.1eV). Without vdw unbounded (HF) LDA binds (it is a luck? )

22 Beryllium dimer: a challenging molecule

23 V The solution: a constrained energy minimization Realize that the chemical bond ~ev << Atomic energy ~10H To have an accurate description of the chemical bond inter-atomic correlations are more important than intra-atomic correlations We constrain the energy minimization to have a number n* of molecular orbitals n * = RVB resonance in the same active space n HF (A) N /2 A JHF level of correlation Valence bond energy consistent RVB r

24 Start with a simple case: the H dimer 1σ g 1σ u Bonding orbital Antibonding orbital A B A r B r r r e e r r r r + A r B r r r e e r r r r At equilibrium, at large distance (strong correlation) = 0.25 λ 1: λ Heitler-London: [ ] = 2 1 A B A B ) ( )1 ( 1 ) ( )1 ( 1, r r r r f u u g g r r + = r r r r r r σ σ λ σ σ 1) (RHF 2 * = = HF n n

25 Beryllium dimer

26 Fluorine dimer F 2 molecule S=0 F atom S=1/2 JAGPn* size consistent! (bonding-antibonding resonance) CEEIS-FCI: correlation energy extrapolation by intrinsic scaling - full CI L. Bytautas, T. Nagata, M. S. Gordon, K. Ruedenberg, J. Chem. Phys. 127, (2007)

27 Systematic tests on simple molecules In several important molecules: all the first raw dimers and the most difficult ones of the G1 set: CN, LiH, SO and others (calculation in progress) the accuracy in the binding energy and bond length, is from 4 to 10 times better than the previous best DMC results on single Slater determinant (J. Grossman, JCP 2002, Toulouse, Umrigar, 2008) Often the JRVB improves the accuracy of the chemical bond by an order of magnitude (e.g. F2): The resonance valence bond really works!

28 Consistent description of the spectrum of the Iron dimer Configuration (quantum numbers) 8 u Charge +e (Anion) JAGPn* (ev) -0.89(12) EXP (ev) (8) 9 g 0 (Neutral) 0 (Reference JAGPn*) 0 (Reference photoemission) 7 g 0 (Neutral) +0.64(7) (4) 7 0 no accessible from (Neutral) photoemission Δ u -0.44(9)??? Equilibrium distance ( 7 Δ u ) :3.89(2) Exp.: 3.82(2)

29 From chemistry to solid state physics &HTc Problems (NB sign problem is not in the list): 1)Pseudopotential for QMC are often not available e.g. La is a problem. 2) Correlated periodic systems: difficult to reach thermodynamic properties. DFT simple Average over k vectors (linear). QMC system large enough. #el^2 complexity 3) Error bar small enough for large systems e.g. condensation energy is ~10K/Cu Error bar so far reached is 100K/Cu. QMC huge computer time, next 5 years?

30 CaCuO the parent compund of HTc superconductors 2 4x4 unit cell 64 atoms and 544electrons by QMC

31 Calculation details Dolg-Filippi pseudopotentials neon core spd non local components scalar relativistic corrections included Gaussian basis set for orbitals wave function Gaussian basis set for orbitals wave function Cu (8s7p5d2f) O (4s5p2d2f) Ca (6s6p2d1f) Total one particle basis: ~5000 gaussians.

32 4x4 supercell with antiperiodic boundary cond. ) 4, 4 ( π ± π ± ) 4 3, 4 ),( 4, 4 3 ( π π π π ± ± ± ± There are 16 states occupied 4x2(spin) so the 8 ) 4 3, 4 3 ( π π ± ± ), ( π π Q =

33 Characterizing the DFT orbitals Overlap 2 Cu-d x 2 -y Fermi energy 4x4 one band states E-E Fermi (ev)

34 One can classify the 8 degenerate DFT orbitals in terms of point spatial group symmetries 1 s (x 2 +y 2 ) defined on A sublattice 1s (odd x<->y) = B = 1d (xy) = B 1d (x 2 -y 2 ) = A = 2E (x+iy p-wave) linear combination in A or B Thus 4 molecular orbitals are in the A and 4 in B Easy to build AF (4 up in A and 4 down in B)

35 A typical molecular orbital at the Fermi energy: localized 2 2 character on the Copper. d x y

36 We use DFT orbitals and combine them in an AF fashion to gain energy: Lattice model u k e ikr ± v k e i( k + Q) R Real electrons u k k DFT ± v k k + Q DFT We optimize all possible u k v k and include a Jastrow factor fully optimized and local (analogous to Gutzwiller wavefunction). Total of ~200 variational parameters.

37 By considering the LDA molecular orbitals and optimizing the energy with AF order and Jastrow correlation we obtain (4x4 unit cell point): Magnetic moment CaCuO2 ( μb ) LSDA HF LDA+U B3Lyp VMC RVB Exp ± ±0.05 It is clear that the missing ingredient of LSDA is the correct exchange energy.

38 4x4 unit cell 64 atoms and 544electrons by QMC The spin density in CaCuO2 parent compound HTc DFT-LDA no magnetic moment This QMC, is takes a promising into account result, electron probably correlation the initial and step to leads understand to AF order HTc-superconducitvity compatible with exp. by ~0.51±0.05 simulation

39 As in the one band HM can we gain energy in the insulator by allowing pairing? d 2 x J(correlation)+DFT Slater Det y 2 d x y J(correlation)+ 2 2 ~0.4eV/Cu d x y J(correlation)+ 2 2+AntiFerro order

40 Comparison with the Hubbard model Gain /Gain +AF dx 2 -y 2 dx 2 -y U/t

41 The true pairing condensation energy is the BCS gain in energy with respect to the AF state Gain d x 2 -y 2on AF/Gain d x 2 -y 2+AF U/t Not accessible in realistic simulation too small error bars. Probably with correlated sampling.

42 How the d-wave pairing looks like in a realistic calculation? CuO

43 Remark: There is a clear energy gain to have a pairing function with d-wave symmetry. This is clearly not a direct evidence of RVB HTc But at least does not exclude this theory, as if no d-wave energy gain RVB death.

44 Conclusions The RVB wavefunction is a new paradigm for Mott insulators: paired, gapless, Fermi From lattice model to realistic simulations: A new accurate and cheap method for physics/chemistry RVB=JAGP with constrained number of molecular orbitals: Accurate description of dispersive forces by means of a more accurately parametrized Jastrow First consistent description of Iron dimer spectrum (Chem. Phys. Lett.) It looks in all cases studied accurate <0.1eV/Atom and is cheap because requires 1Det + QMC Application to HTc materials in progress, promising

45 Acknowledgments Main developers of the TurboRVB, the QMC code used in this work: Michele Casula (Ecole Polytechnique, Palaiseau, France), Claudio Attaccalite (San Sebastian, Spain) Leonardo Spanu (UC Davis) H benzene Todd Beaudet (UIUC) Jeongnim Kim (UIUC) Richard Martin (UIUC) benzene dimer Dario Rocca (UC Davis) first row dimers Mariapia Marchi (SISSA) Sam Azadi (SISSA) iron dimer Leonardo Guidoni (Universita dell Aquila) Lattice models Federico Becca, Luca Tocchio, Evgeny Plehanov, M. Lugas (SISSA).\ Grants: Cineca INFM, Cineca SISSA, QMCHTC DEISA (2009), hours on sp6.

46 The generalized Langevin dynamics

47 Discretization of the Langevin dynamics

48

49 The 16 H case with PBC, MD with friction RVB liquid phase possible at high pressure 21

50 A benchmark correlated dimer Fe 2 Method HF DFT RVB Exp. R (a.u.)? (1) 3.82(20) 0 ω 0 (cm Type -1 )?? (18) Δ u 9 9 g g It is possible to explain the photoemission spectrum in the anion Leopold JPC (1988) Fe 2 22

51 DFT occupation molecular orbitals 23

52 The right occupation is due to correlation 8 u Explains the exp. Fe 2 Within our RVB wf 7 Δ u is 0.7 ev higher Confirmed also by recent CI, Hubner JPC 02 24

53 Iron dimer (II) 9 Σ g LRDMC Harmonic equilibrium frequency: distance: 284 (24) cm 4.08(5) -1 (Indirect) Experimental experimental value: ~ 300 value: (15) ~ cm 3.8-1

54 Conclusions and Perspectives -d-wave superconductivity in strongly correlated models -exploiting the RVB=BCS+J for molecular calculations M. Casula and S. Sorella JCP 01 M. Casula C. Attaccalite and S.S. JCP 04 The Iron dimer a successful test case relevant for biophysics -Possible stable low-temperature high-pressure liquid phase for hydrogen Final goal: simulation of complex correlated electron systems by Monte Carlo calculation and beyond DFT 26

55 Lattice GFMC Green function: Lattice hamiltonian: + + = + j i j i ij a i i a i n n V h c c c t H,, 2 1.). ( ) ( ) ( ) (,,, x x H G T T x x x x x x Ψ Ψ Λ = δ importance sampling Markov chain x x transition probability ) (,,,, x e G G G p L x x x x x x x x x Λ = = weight )) ( ( 1 x e w w L i i Λ = + Transition probability well defined? NO, for the fermionic sign problem FN approximation (see D.M.Ceperley et al. PRB 51, (1995))

56 From continuous to lattice Kinetic term ˆ) ( ) ( ) ( 2 ˆ a x x T a O a I T T T T a d i i a a a i i + = Ψ Ψ + + = Δ Δ = hopping term t 1/a 2 Potential term For a faster convergence a 0 regularisation Potential energy ) ( ) ( ) ( ) (, x E x x H G x e L T T x x x L = Ψ Ψ = = Ψ ΔΨ Ψ Ψ Δ + = ) ( ) ( ) ( ) ( ) ( ) ( ) ( x x x x x V x V x V T T T T a a ( 2 ) a O V V a +

57 From lattice to pseudo lattice (dense continuum) Separation of core and valence dynamics for heavier atoms and molecules two hopping terms in the kinetic part ΔΨ( x) pδ Ψ( x) + (1 p) Δ Ψ( x) + a b O( 2 a ) p can depend on the distance from the nucleus if a < b, p(0) = 1 and p( ) = 1 Our choice: p( r) = 2 1+ γ r Moreover, if b is not a multiple of a, the random walk can sample all over the space! 0

58 Comparison with the best DMC (Umrigar,Nightingale,Runge 93) C atom all electrons Δτ (a)

59 Non local pseudo possible!!! For heavy atoms Z>20 it is impossible to avoid them (see L.Mitas PRB 49(6), 4411 (1994)) 1) No localization approximation employed 2) Still variational upper bound theorem holds exactly as in the lattice fixed node 3) It works also without Jastrow optimization 4) The fixed node energy depends only on the nodal structure and weakly on the amplitudes

60 The disease of the localization approximation C pseudoatom with 4 electrons ( 2 core)

61 Why is that? Nodal surface + + non local move By neglecting the allowed non local moves the localization approximation infinitely negative attractive potential close to the nodal surface. It works only for very good trial function.

62 Targets AIM look for Monte Carlo algorithms that can deal with atoms beyond the first row (all electrons) find a good trial wave function able to get correlation and to treat molecular bonds The pseudo-lattice approach can improve the efficiency? Possible use of pseudopotentials within fully variational DMC calculations even for heavier atoms?

63 Accuracy in the total energy (~76Ry) of C as compared with the ionization energy 11.26eV HF 38% HF+J 14% AGP+J 6.5% DMC+AGP+J 1% For poor accuracy also the HF is enough

64 1) For given energy accuracy per ion a simple algorithm (N^3) is enough: no (sign) problem 2) For correlation functions we need an accuracy ~1/N (below the gap) unfortunately 3) I do not see any hope for this, so far any improvement (like DMC) reduces the energy accuracy by a factor at most. 4) The realistic hope is the effective Hamiltonian

65 A short review of fixed node approximation 1) It works in configuration space x: electrons and spins given H 2 h = Δ+ V( x) 2m 2) Given any wave function an Hamiltonian is found ψ G( x ) x H H ψ = 0 choice: H = H + δv( x) δv( x) = G G G 3) An effective hamiltonian is studied closer to H: x ψ eff H = H δv( x) with constraint x ψ > 0 G Note: exact for bosons and in the classical limit h 0 G ψ

66 Philosophy of the approach Assume there are physical Hamiltonian that describe a phase and are therefore stable away from critical points: H H +δv The phase remains stable for physical perturbation With lattice fixed node we can simulate H with several ψ δv δv If G is stable than we can say that G may represent a ground state of some stable hamiltonian (not necessarily H) ψ For practical purposes ψ G is taken by minimizing the energy of H

67 Solution of model Hamiltonians?????? Ground state Properties of stable Hamiltonians???????????? Properties of reasonable wave functions

68 Effective hamiltonian approach for strongly correlated lattice models References M. Calandra & S. Sorella S. Sorella S. Sorella PRL '98 & L. Capriotti S. Sorella Phys. Rev. B Phys.Rev.B.'98 Phys.Rev.B The Beyond Lanczos Generalized Outline of the lecture: the algorithm and variational Lanczos by variance approach: Stochastic the extrapolation fixed node Reconfiguration approximation d - wavesuperconductivity for J/t > ~ 0.2 Application to the t - J model: effective t - J model for La2CuO4 Kerkrade 24/2-1/3-2002

69 The Lanczos algorithm in QMC: From lattice model to continuous models? ) 4 1 (.. j i j i j i n n S S J c h P c c P t H + + = + σ σ P Strong correlation O

70 Lanczos with QMC on lattice models (L sites): For p>1 Lanczos steps #operation /MC ~ L p+1 Always polynomial at fixed p. Probably improvement to p! # operations The question is how much computer effort is required for prescribed accuracy at given L.

71 Variational energy for various QMC vs. variance by VMC wavefunction with p=0,1,2 Lanczos iterations Energy per site Variational Fixed node Stochastic Reconf. 4 holes 26 sites J/t=1 Exact <H 2 >-<H> 2 The improvement in energy for both fixed node and present method (best) is irrelevant as far as energy.

72 On a 6x6 (not possible exactly) SR convergence is evident for p=2 VMC p=1 p=2 p= From optimal Δ BCS From Δ BCS --> Manhattan Distance= x + y SR p=3 pairing consistent within 3% (error bars) FN+2LS ============ 20% VMC + 2LS ========= 70%

73 Lanczos method for continuous models? Unfortunately for the first Lanczos step: Ψ T Ψ T (1 + α (1 + H ) H (1 + α H )(1 + α α H H ) ) Ψ Ψ T T and Ψ T H 3 Ψ + α Only a statistical method known with Caffarel & Ceperley or 0 e ΔtH backflow wavefunctions (poor scaling)

74 Projected BCS wave function on triangular lattice P-BCS = P G BCS L /2 = f rc C k k r, k r r, k BCS 0 : projected BCS state ( h.c. ) H = t C C + μ C C r r r r BCS rur r, σ r, σ r r, σ r, σ rr,, σ r, σ L ( ) + Δr Cr Cr r Cr Cr r + h.c. r r l r, r+ l, r, r+ l, r= 1 l : ground state of BCS Hamiltonian 2 2 ( ) ( μ ) r r ( ) f r =Δr / ε r μ + ε r +Δ Δ μ k k k k k k L /2 BCS ( ) Δr Cr Cr r Cr Cr r 0 r r l r, r+ l, r, r+ l, r l

75 Marshall sign rule H = J S S x = i (W.Marshall, Proc.R.Soc.London Ser. A 232,48 (1955)) uur uur J ij i j ij 0 : if i and j on the same sub-lattice i, j Jij 0 : if i and j on the different sub-lattice ( Z $ ) U = exp i m iπ s+ S : bases i B uur i 2 S m = s( s+ 1) m Z S$ i mi = mi mi i i i Ground state of H : ( M ) ( M ) ( ) = ( + ) ( M ) H U HU = H dig + H off Z H = J S$ S$ Z dig ij i j i, j 1 ( + + $ $ $ $ ) H off = Jij Si S j + Si S j 2 i, j Ψ 0 = fx x with fx > 0 ( ) ( ) Ψ = U Ψ = f x N x N x s m i B i x x x k=0 A1 : Marshall sign rule

76 Fixed node approximation (D.F.B.ten Haaf et al., PRB 51, ( 95)) Effective Hamiltonian with no negative sign problem H xx if H xx < 0 and x x eff H xx = γ H xx if H xx > 0 and x x H xx + ( 1+ γ) νsf ( x) if x = x ( ) ( ) H xx ΨG x Hxx ΨG x x ( x) H xx γ : positive constant x H x : spin configurations : matrix elements ΨG x Ψ : variational (guiding) wave function G A standard Green function MC for effective Hamiltonian ( eff ) n n eff ϕ Ψ ( ) Φ ( ) init G 0 x G x x G eff eff xx =Λδ xx H x x eff Φ 0 : ground state eff of H

77 Fixed node approximation (II) eff 1. Φ 0 x same phase as ( ) Ψ ( x G ) eff 2. Φ ( ) 0 x variational state for H better than Ψ ( x ) G Ψ H Ψ E Φ H Φ E eff eff eff G G H H eff Φ = E Φ eff eff eff Φ = E Φ 0 0 0

78 1D limit (J =0) P-BCS Projected BCS wave function: BCS>: ground state of BCS Hamiltonian = P G BCS BCS,, i, j, σ ( h.c. ) i σ j σ H = t C C + L ( + ) Δl C C C C + i, i+ l, i, i+ l, i= 1 l h.c. Δ l 3 l up to 3 rd neighbors, Ground state properties well described (Gros et.al.) Low-lying excited states (spinon): k = PG γ k, BCS

79

80 RVB variational wavefunction for lattice models ψ BCS P N Δ f = k = exp fi, j ( c c + c c ) 0 k i, j, j, i, i, j ε k Singlet bond where Δ is k ε the free k = AGP = A f ( r, r ) i i Pairs BCS the BCS electron + ε 2 + Δ 2 k k gap function dispersion eneral: ε k = Single particle dispersion metal (no pairing) Band Insulator Superconductor f f k k = Θ( ε < ε F ) k ε ~ 0 k ε F

81 P N BCS = AGP = A f ( r, r i i Pairs ) General: ε k = Single particle dispersion metal (no pairing) Band Insulator f k = Θ( ε < ε F ) k Superconductor New phase JAGP = J x AGP J = exp( v( r, r i i< j j )) RVB f k f k ε ~ 0 k ε ε ~ 0 k But insulator F ε F

82 Definition of spin liquid A spin state with Neel no magnetic order (classical trivial) no broken translation symmetry (less trivial): no Dimer state (Read,Sachdev) is a spin liquid

83 Experiments from: Coldea et al (PRL 01) PRB 03 J /J=1/3 J=0.375meV J between planes ~1K

84 κ (ET) 2Cu 2(CN) 3 J Shimizu et al. PRL 03 / J 0.9 J / J 1.8 Spin Liquid? κ J=250K!!!

85 Methods (S. Sorella, PRB 64, 01) Variational quantum Monte Carlo (QMC) method Projected BCS wave function: BCS = i, j e f i, j C + i, C + j, 0 P-BCS = P : GS of BCS Hamiltonian G BCS H BCS = i, j, σ t i, j Notice : C + i, σ C t i, σ i, i + i, j ( Δ i, j C + i, C + i, + h.c.) = μ Chemical potential Resonating valence bond states from PBCS QMC with Fixed node appr., (D. ten Haaf et al. PRB 95) to study the stability of the spin liquid state.

86 2D with J /J=0.33 ξ = ε μ k k Bogoliubov QP spectrum: E = ξ +Δ 2 2 k k k Gapless excitations S=1/2 Fractionalization in 2D! see e.g. X.G. Wen or M. Fisher No particular d-wave or s-wave symmetry due to anisotropy

87 Isotropic triangular lattice with J /J=1.0 18x18 Green function Monte Carlo Spin correlation function: v r ur C () l = S () r S ( r+ lτ ) i Z Z i Spin liquid state unstable toward classical Neel state 8

88 Spin structure factor for J /J=1.0 Order parameter: GFMC? Starting from a spin liquid : m/ m max = 0.351± Starting from ordered state : Linear spin wave GFMC [1] 0.41(2) Exp. ~0.45 (organic) at J /J=1.8 [1] Capriotti, Trumper, SS, PRL 82, 3899 ( 99)

89 Spin structure factor for J /J=0.7 Incommensurate peaks at Q*=(q*,0) No long range order 5

90 Stability against dimerization

91 Summary Possible spin liquid (SL) state in 2D Two different SL? Gapless vs. gaped?

92 Correlation plays a crucial role: 1) No way to have superconductivity in a model with repulsive interaction. HTc not explained, HeIII, spin liquid (organic) 2) No way to obtain insulating behavior with a model with 1el/unit cell (Mott Insulator). This is instead possible with correlated Jastrow We should optimize the RVB wavefunction in presence of its Jastrow. QMC only for correcting the HF is meaningless 12

93 Why RVB wavefunction should work for molecules? A molecule has a gap insulator Why not RVB insulator? Van der Waals forces are included by Jastrow In a complex system the molecular orbitals are often nearly degenerate Resonance Valence Bond approach OK 13

94 Computational complexity now N^4 In QMC for given accuracy (e.g. Kcal/Mol) Cost= N^4, as sampling length=m~n. One has to solve: sx = f where s has linear dimension N^2 N^6??? No!!!One can exploit conj.grad.and that s + = M M where M = ( M ~ N, N 2 ) :

95 DMC on the lowest energy JAGP wf. Old technique non variational (often unstable) with nonlocal pseudopotential New (M.Casula C.Filippi and S.S.) PRL05 LatticeRegularizedDiffusionMonteCarlo ery stable variational upper bounds of the pseudo Hamiltonian energy. ey idea: on a lattice all interactions are nonlocal

96 Linus Pauling: the concept of resonance is old Benzene C6H6 + r r H = JS S a b 6 valence electrons occupy the 2p z orbital then strong correlation Heisenberg model a,b nearest Carbon sites 1 a b = ψ2p ( r) ψ2 ( ) z p r + a b z 2

97 In the old formulation RVB was expensive 1) Use of non orthogonal configurations 2) The number of VB grows exponentially with the number of atoms The molecular orbital approach won but Now (after Htc) we have a better tool

98 On a given electron configuration: 7 { } 1, 2, 3 1, 2, 3 x = r r r r r r The pairing function can be computed: f f f 1, 1 1, 2 1, 3 r r r r r r x RVB = f f f = AGP 2, 1 2, 2 2, 3 r r r r r r f f f 3, 1 3, 2 3, 3 r r r r r r With a single determinant N/2 x N/2, N=# el. even when RVB = many Slater Determinants

99 Mapping to a simple model: the 2-site Hubbard U H, σ = t( R) ca, σ cb, σ + U ( R) c c c c A, A, A, A + A B We use the Singlet-Triplet gap and the optimized pairing function 1σ g ( r)1σ ( r g ) λ 1σ u ( r)1σ u ( r ) Effective parameter U/t Strong correlation 2 4 R (a.u.) Effective parameter (K) J=4t 2 /U R (a.u.)

100 The crucial difference between an Htc superconductor and a RVB insulator is: The long distance Jastrow factor 1/R or log(r) λ i,j Jastrow RVB liq. N=128 bcc solid N=128 bcc solid N= R (a.u.)

101 Phase Diagram of Hydrogen LogT [K] Molecular Liquid Electron Proton Plasma Clustered liquid Electron Proton Liquid Molecular Solid Proton Solid LogP[GPa] Superconductivity? Ashcroft Nature

102 Indications of an anomalous melting line S.A. Bonev,..,G. Galli Nature 2004 Another quantum T=0 liquid phase?

103 Simple test case: solid-metal (bcc) = r s r s H 2 bond length Energy per H at high-pressure (Hartree) 2 Gaussians per protons (Det) 1 Gaussian per proton (Jastrow) Comparison with previous works (a) (b) C. Pierleoni at al. PRA

104 The basic steps for moving atoms Forces can be computed efficiently with VMC we use Caffarell et al. JCP 2000 Optimization of the electronic VMC parameters: 1s Gaussian for Geminal and Jastrow ~200 parameters for 16 H We use Hessian, much progress done in QMC: C.Umrigar & C. Filippi PRL (2005), S.S. PRB (2005), C. Umrigar et al (also SS) PRL, (2007) At each step we move ions with MD and VMC parameters (with hessian), ab initio

105 Energy/N A (H) Temperature (K) New ab-initio Molecular-dynamics with QMC H 1step = QMC opt. ~10000 par.! Δt = 1fs t (fs) 16 H Proton Classical Internal energy decreases at 1300K!!! With RVB wf QMC possible for ~100 atoms

106 Why MD can be so efficient for QMC? The simulation at finite T requires some external noise to the forces e.g. Langevin dynamics &r r r r r R = f + η with < η( t) η( t ' ) >= 2Tδ ( t t ' ) But the noise is given for free within QMC!!! Compared with methods based only on energy we use 3N entries (forces) with the same cost. Expected at least a factor N speed-up improvement

107 Proton-proton pair correlation function RVB molecular liquid g(r) 1.4 N= T=365K P=340Gpa R (a.u.)

108 Snap-shot of the protons at the last iteration

109 The pairing function in the liquid phase Remind :in a λ i, j = k k F e r ik ( R λ i,j 1.2 bcc 0.8- metal i r R 0.4 ) j 0.0 RVB Liquid cos(k F R)/R 2 we expect occupied all r r r r cos( k R R ) / R R F i j i 2 j k k F, i. e.: R i -R j (a.u.)

110 Optimization with the s-wave constraint λ Det. i,j E s-projected -E RVB =0.3Hartree =0.8Hartree RVB Liquid s-wave proj. projection R> R (a.u.) Our energy is below any published one for the solid

111 A new possibility HTc in Hydrogen at 300Gpa? R (a.u.) J at the broad peak of g(r) is about 10000K In Copper Oxide J is 1500K, Tc~100K Tc > Room temperature? At rs=1.31 the solid phase simple hexagonal is competing in energy (Natoli et al. PRL 93) under current investigation. 0 J=4t 2 /U (K)

112 Conclusions The Jastrow+RVB (AGP) gives an accuate description of the chemical bond. It is described by a single determinant and is computationally convenient for QMC. Reproduced several experiments on simple molecules, benzene and its dimer, water, C2. Due to important achievements in the energy optimization. Realistic MD with most of the correlation HTc physics in hydrogen at 300Gpa?

113 The drawback of many VMC parameters. Energy (H) Silicon Diamond lattice ~1200 #Samples/Iter parameters Iterations Sick when #parameters > # QMC Samples

114 The Berillium dimer: a challenging molecule

115 The F2 molecule and the problem of size-consistent results