André-Marie Tremblay

Transcription

1 André-Marie Tremblay CENTRE DE RECHERCHE SUR LES PROPRIÉTÉS ÉLECTRONIQUES DE MATÉRIAUX AVANCÉS Sponsors:

2 Pairing vs antiferromagnetism in the cuprates 1. Motivation, model 2. The standard approach : - what it is - a qualitatively incorrect result - limitations of the approach 3. A non-perturbative approach (U > 0 and U < 0 ) - Proof that it works - How it works 4. Results: - Mechanism for pseudogap - Spectral weight rearrangement - Pairing in the repulsive model 5. Perspectives.

3 1. Motivation, model -Here, -W < U < W with (W = 8t) -Relevant for high Tc where U > W? - A question of threshold (and continuity) U SDW T δ - Importance of quantitative predictions - Location of QCP - No ferromagnetism - Effective U < 0 model may be not too strongly interacting - Suppose we find new quasiparticles in strong coupling. How do we study residual interactions? - Do we give up calculating Landau parameters? - How to predict when the theory is bad? - Standard method gives qualitatively incorrect results 3

4 2. The standard approach : - what it is (FLEX, self-consistent T-matrix...) Φ [G] = Σ [G] = δφ [G] / δg = Γ [G] = δσ [G] / δg = + 4

5 2. The standard approach : - what it is (FLEX, self-consistent T-matrix...) - Thermodynamically consistent: df/dµ = Tr[G] - Satisfies Luttinger theorem (Volume of Fermi surface at T = 0 preserved) - Satisfies Ward identities (conservation laws): G2(1,1;2,3) appropriately related to G(1,2) 5

6 2. The standard approach : - a qualitatively incorrect result ( π,0) Monte Carlo Many-Body (0,0) (0, π ) π π (, ) 4 2 π π 4 4 (, ) (, ) π 2 π 2 (0, π ) (0,0) ω/t Calc. + QMC: Moukouri et al. P.R. B 61, 7887 (2000). U = + 4 T = 0.2 n=1 8 X 8 Flex

7 T X T = 0 T n c E A(ω) k F I k ω k F A(ω) k F II ω E A(ω) k F III 2 k ω k F 2 99-aps/Explication... 7

8 2. The standard approach : - limitations of the approach - Integration over coupling constant of potential energy does not give back the starting Free energy. - The Pauli principle in its simplest form is not satisfied (It is used in defining the Hubbard model in the first place) - There is an infinite number of conserving approximations (How do we pick up the diagrams?) - Inconsistency: Strongly frequency-dependent self-energy, constant vertex No Migdal theorem, so vertex corrections should be included 8

9 Singular Σ (k F,ikn) U T 4 N X q 1 Uspχ sp (q, 0) ikn ² k+q Σ (k F +q,ikn) Σ (ikn) = 2 ikn Σ (ikn) ReΣ R (ω) = ω 2 ω 2 ω θ ( ω 2 ) ³ ω /2 ImΣ R (ω) = 1 2 θ (2 ω ) ³ 4 2 ω 2 1/2 9 Non Fermi-liquid but not singular at ω = 0 Vilk, et al. J. Phys. I France, 7, 1309 (1997).

10 2. The standard approach : - problem... - We do not know how to properly solve even the «standard model» for heavy fermions (Coleman). - Lee-Rice-Anderson simpler approach seems qualitatively better, why? - GW approach to improve band structure calculations? 10

11 3. An approach for both U > 0 and U < 0 - Proofs that it works Notes: -F.L. parameters -Self also Fermi-liquid 0.4 n = 0.20 n = 0.45 (b) (c) U = 8 2 n = 0.20 n = 0.45 n = 1.0 U = (0,0) (π,0) q (π,π) (0,0) 0 q QMC + cal.: Vilk et al. P.R. B 49, (1994) 11 n S ch (q) n S sp (q) n = 0.26 n = 0.60 n = 0.94 (a) U = 4 n = 0.19 n = 0.33 n = 0.80 (d) U = 4 U > 0

12 Proofs U=4.0 n=0.87 Monte Carlo 8x8 (BWS) 1.5 8*8 (BW) 1.0 χ ( π,π sp,0) 0.5 This work FLEX no AL FLEX T Calc.: Vilk, et al. J. Phys. I France, 7, 1309 (1997). QMC: Bulut, Scalapino, White, P.R. B 50, 9623 (1994). U = + 4

13 T X T mf U=4 n=1 Monte Carlo 4x4 8 6x6 8x8 10x10 S sp (π,π) 12x12 ξ ~exp ( CT ( )/ T) 4 ξ = T Calc.: Vilk et al. P.R. B 49, (1994) QMC: S. R. White, et al. Phys. Rev. 40, 506 (1989). O ( N = ) A.-M. Daré, Y.M. Vilk and A.-M.S.T Phys. Rev. B 53, (1996)

14 Proofs... ( π,0) Monte Carlo Many-Body (0,0) (0, π ) π π (, ) 4 2 π π 4 4 (, ) (, ) π 2 π 2 (0, π ) (0,0) ω/t Calc. + QMC: Moukouri et al. P.R. B 61, 7887 (2000). U = + 4 Flex

15 Proofs Monte Carlo Many-Body Σ (s) FLEX a) b) c) U = + 4 L=4 L=6 L=8 0.5 L= G(τ) A(ω) β/2 β Calc. + QMC: Moukouri et al. P.R. B 61, 7887 (2000).

16 Proofs... Other approach DCA C.Hushcrof, M. Jarrell et al. P.R.L 86, 139 (2001).

17 Moving to the attractive case... Calc. : Kyung, Allen,A.M.S.T. P.R.B 64, /1-15 (2001) QMC : Moreo, Scalapino, White, P.R. B. 45, 7544 (1992) Proofs... U =

18 Proofs... U = - 4 Calc. : Kyung, Allen,A.M.S.T. P.R.B 64, /1-15 (2001) QMC : Trivedi and Randeria, P.R. L. 75, 312 (1995) 2 nd order perturbation theory S.C. T-matrix 18

19 Proofs T = 0.25, n = 0.8, U = -4 QMC Many-Body N( ω ) A(k, ω ) 1.2 (a) 0.8 (b) G(k, τ ) G(k, τ ) (a) (b) T = 0.25, n = 0.8, U = QMC Many Body ω T = 0.2, n=0.8, U = -4 QMC Many-Body ω τ (c) (d) τ 0.2 (c) 0.4 (d) T = 0.2, n=0.8, U = -4 QMC -0.4 Many-Body Kyung, Allen,A.M.S.T. P.R.B 64, /1-15 (2001) 19 Σ k U = - 4

20 3. An approach for both U > 0 and U < 0 - How it works 20

21 General results (Dyson equation) : 1, 1 G 1,2 U , 1 G 1,1 U n 1 n 1 Hartree-Fock approximation H 1, 1 G H H 1,2 UG 1,1 G H 1,2. Step one of our approximation 1 1, 1 G 1 1,2 AG 1 1, 1 G 1 1, 2 A U n n n n.

22 Find n n from (a) fluctuation-dissipation theorem (b)pauli principle Usp 1 1 n n. U n n G 1 G 1 n n 2 n n 2 n n T 0 q N q 1 1 U 2 sp 0 q n 2 n n T 0 q N q 1 1 U 2 ch 0 q n 2 n n n 2.

23 A better approximation for single-particle properties (Ruckenstein) = Σ 2 = Y.M. Vilk and A.-M.S. Tremblay, J. Phys. Chem. Solids 56, 1769 (1995). Y.M. Vilk and A.-M.S. Tremblay, Europhys. Lett. 33, 159 (1996); N.B.: No Migdal theorem 23

24 Improved self-energy (step (2)) 2 k sym Un U 8 T N q 3Usp sp 1 1 q Uch ch q 1 G k q. Consistency check 1 Tr 2 G 1 lim T k G 1 k e ik n U n n. N k

25 Results of the analogous procedure for U < 0 Upp = U h(1 n )n i h1 n ihn i. (5) χ (1) p (q) = χ (1) 0 (q) 1+Uppχ (1) (6) 0 (q) T N X q χ (1) p (q)exp( iqn0 )=h i = hn n i. (7) Σ (1) ' U 2 U pp (1 n) 2 (8) Σ (2) σ (k) =Un σ U T N X q Uppχ (1) p (q)g (1) σ (q k) (9) 25

26 SatisÞes Pauli principle and generalization of f sum rule Z dω Dh ie π Im χ(1) (q, ω) = Dh q (0), q (0) ie =1 n ; q (10) Z dω π ω Im χ(1) (q, ω) = 1 X ³ε N k + ε k+q ³1 2 D E n k (11) k µ 2 µ (1) U (1 n) ; q (12) 2 Internal accuracy check (For both U>0 and U<0). 1 2 Tr h Σ (2) G (1)i T = lim τ 0 N X k Σ (2) σ (k) G (1) σ (k) e ik D nτ = U Check : Tr h Σ (2) G (1)i Tr h Σ (2) G (2)i n n E 26

27 4. Results: - Mechanism for pseudogap U = (analogous to U > 0 ) : Vilk et al. Europhys. Lett. 33, 159 (1996) Pines, Schmalian (98) - Enter the renormalized-classical regime. N.B. d = 2 Kyung, Allen,A.M.S.T. P.R.B 64, /1-15 (2001) 29 27

28 4. Results: - Mechanism for pseudogap U = Pairing correlation length larger than single-particle thermal de Broglie wavelength (vf / T) ξ 1.3 ξ th Kyung, Allen,A.M.S.T. P.R.B 64, /1-15 (2001) 28

29 Mechanism for pseudogap formation in the attractive model: U = - 4 d = 2 is crucial Even part of the pair susceptibility at q = 0, for different temperatures T=1/3 T=1/4 T=1/ ω Allen, et al. P.R. L 83, 4128 (1999) 29

30 Mechanism for pseudogap in RC regime k,ik n Un U 4 N T Upp pp q, q Large correlation length limit cl k, ikn UT dd q d q ikn k q 1 ikn k. q v k Analytic continuation R cl k F, ik n UT dd q d q i. q v F

31 Imaginary part: compare Fermi liquid, limt 0 R kf,0 0 R kf,0 T d d 1 q vf 1 T 2 q 2 vf 3 d d 2 th Why leads to pseudogap A k, 2 R k R 2 R 2

32 4. Results: - Spectral weight rearrangement U = Pseudogap appears first in total density of states - Fills in instead of opening up - Rearrangement over huge frequency scale compared with either T or T. ( T ~ 0.03, T ~ 0.2, ω 1 ) Kyung, Allen,A.M.S.T. P.R.B 64, /1-15 (2001) 32

33 4. Results: - Crossover diagram U = - 4 Kyung, Allen,A.M.S.T. P.R.B 64, /1-15 (2001) 33

34 4. Results: - Pairing in the repulsive model Historical reminder: Spin fluctuation induced d-wave superconductivity U/t = 4 D. J. Scalapino, E. Loh, Jr., and J. E. Hirsch P.R. B 34, (1986). Béal Monod, Bourbonnais, Emery P.R. B. 34, 7716 (1986). Miyake, Schmitt Rink, and Varma P.R. B 34, (1986) Kohn, Luttinger, P.R.L. 15, 524 (1965). Bickers, Scalapino, White P.R.L 62, 961 (1989).

35 χdx 2 -y 2 d x 2 -y 2-wave susceptibility for 6x6 lattice U = 0 U = 4 U = 6 U = 8 U = δ β Landry, A.-M.S.T. unpublished

36 0.7 U = 0, β = 4 L = U = 4 β = 4 β = 3 β = 2 β = 1 χd Doping QMC: symbols. Solid lines analytical. Kyung, Landry, A.-M.S.T.

37 d G = G GG 1 G 1 G ~ d G s =

38 U = 4, β = Full susceptibility, L=32 Susceptibility without vertex, L=32 Full susceptibility, L= Doping x χd

39 Analogous procedure for attractive case can be used to compute particle-hole properties (Spin and charge susceptibilities) U=-4, n=0.2, N=12X12 (Fig.1 in the paper) χsp (T) Self-consistent T-matrix Crossing 0.02 QMC Temperature

40 5. Perspectives H tx c i, i,j, Phenomenological model cj, h. c J S i S j 1 4 n inj i,j i, c i, ci,. Computation of local correlation function from the mean-field solution HMF k, k c k, c k, 4Jmm Jd d d Jy y F 0 Compute dynamical susceptibilities with renormalized vertices obtained from T N pp q e i m0 d 0 2. q pp k 1 4 V effvpp T N q Compute single-particle properties from d k d q k 2 pp q G 0 q k.

41 T=0 order parameters From paramagnetic self-energy: B. Kyung, P.R.B. 64, (2001). Opening of pseudogap (RC regime) From dynamical susceptibility, ξ sc = 4

42 Pseudogap and total gap B. Kyung, P.R.B. 64, (2001). Kyung and A.-M.S.T. unpublished t/j = 3, J = 125 mev

43 Properties of the condensate

44 Some speculations Weak - coupling T AF fluctuations T pseudogap S.C. δ quasi d = 2 S.C.? d = 3 AF δ

45 Strong-coupling T AF pseudogap δ Strong-coupling quasi d = 2 T AF S.C.? δ Strong-coupling d = 3

46 Michel Barrette Elix David Sénéchal A.-M.T. Mehdi Bozzo-Rey Alain Veilleux

47 Steve Allen François Lemay David Poulin Liang Chen Yury Vilk Bumsoo Kyung Samuel Moukouri Hugo Touchette

48 Claude Bourbonnais R. Côté D. Sénéchal Sébastien Roy Alexandre Blais Jean-Séastien Landry A-M.T. Bumsoo Kyung