André-Marie Tremblay CENTRE DE RECHERCHE SUR LES PROPRIÉTÉS ÉLECTRONIQUES DE MATÉRIAUX AVANCÉS Sponsors:
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.
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
2. The standard approach : - what it is (FLEX, self-consistent T-matrix...) Φ [G] = + 1 2 +... Σ [G] = δφ [G] / δg = + +... Γ [G] = δσ [G] / δg = + 4
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
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) -5.00 0.00 5.00-5.00 0.00 5.00 ω/t Calc. + QMC: Moukouri et al. P.R. B 61, 7887 (2000). U = + 4 T = 0.2 n=1 8 X 8 Flex -5.00 0.00 5.00 6
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
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
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 4 2 1/2 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).
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
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 = 8 0.0 0.4 0 2 0.0 (0,0) (π,0) q (π,π) (0,0) 0 q QMC + cal.: Vilk et al. P.R. B 49, 13267 (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
Proofs... 2.0 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 0.0 0.50 0.75 1.00 T Calc.: Vilk, et al. J. Phys. I France, 7, 1309 (1997). QMC: Bulut, Scalapino, White, P.R. B 50, 9623 (1994). U = + 4
T X T mf 16 12 U=4 n=1 Monte Carlo 4x4 8 6x6 8x8 10x10 S sp (π,π) 12x12 ξ ~exp ( CT ( )/ T) 4 ξ = 5 0 0.0 0.2 0.4 0.6 0.8 1.0 T Calc.: Vilk et al. P.R. B 49, 13267 (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, 14236 (1996)
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) -5.00 0.00 5.00-5.00 0.00 5.00 ω/t Calc. + QMC: Moukouri et al. P.R. B 61, 7887 (2000). U = + 4 Flex -5.00 0.00 5.00 14
Proofs... 1.0 Monte Carlo Many-Body Σ (s) FLEX a) b) c) U = + 4 L=4 L=6 L=8 0.5 L=10 0.0 0.0-1 0 1-1 0 1-1 0 1 15 G(τ) A(ω) -0.5 0 β/2 β Calc. + QMC: Moukouri et al. P.R. B 61, 7887 (2000).
Proofs... Other approach DCA C.Hushcrof, M. Jarrell et al. P.R.L 86, 139 (2001).
Moving to the attractive case... Calc. : Kyung, Allen,A.M.S.T. P.R.B 64, 075116/1-15 (2001) QMC : Moreo, Scalapino, White, P.R. B. 45, 7544 (1992) Proofs... U = - 4 17
Proofs... U = - 4 Calc. : Kyung, Allen,A.M.S.T. P.R.B 64, 075116/1-15 (2001) QMC : Trivedi and Randeria, P.R. L. 75, 312 (1995) 2 nd order perturbation theory S.C. T-matrix 18
Proofs... 0.4 T = 0.25, n = 0.8, U = -4 QMC Many-Body N( ω ) A(k, ω ) 1.2 (a) 0.8 (b) 0.0-0.2 G(k, τ ) 0.0-0.2 G(k, τ ) (a) (b) 0.2 0.4-0.4 T = 0.25, n = 0.8, U = -4-0.4 QMC Many Body 0.0 0.3-4.0 0.0 4.0 ω T = 0.2, n=0.8, U = -4 QMC Many-Body 0.0 0.6-4.0 0.0 4.0 ω -0.6 0.0 τ -0.6 0.0 (c) (d) τ 0.2 (c) 0.4 (d) -0.2-0.2 0.1 0.2-0.4 T = 0.2, n=0.8, U = -4 QMC -0.4 Many-Body 0.0 0.0-0.6-0.6-4.0 0.0 4.0-4.0 0.0 4.0 Kyung, Allen,A.M.S.T. P.R.B 64, 075116/1-15 (2001) 19 Σ k U = - 4
3. An approach for both U > 0 and U < 0 - How it works 20
General results (Dyson equation) : 1, 1 G 1,2 U 1 1 1 2. 1, 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.
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.
A better approximation for single-particle properties (Ruckenstein) 1 3 1 2 = - 2 + 3 1 3 2 3 4 5 2 1 Σ 2 = 1 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); 1 1 5 2 2 4 N.B.: No Migdal theorem 23
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 2 2 0 k G 1 k e ik n U n n. N k
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
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
4. Results: - Mechanism for pseudogap U = - 4 - (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, 075116/1-15 (2001) 29 27
4. Results: - Mechanism for pseudogap U = - 4 - 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, 075116/1-15 (2001) 28
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=1/5 0.00 0.50 1.00 1.50 2.00 ω Allen, et al. P.R. L 83, 4128 (1999) 29
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 1 2 2 0 2 d q 2 2 0 1 ikn k q 1 ikn k. q v k Analytic continuation R cl k F, ik n UT dd q 1 2 2 0 2 d q 2 2 1 i. q v F
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
4. Results: - Spectral weight rearrangement U = - 4 - 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, 075116/1-15 (2001) 32
4. Results: - Crossover diagram U = - 4 Kyung, Allen,A.M.S.T. P.R.B 64, 075116/1-15 (2001) 33
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, 8190-8192 (1986). Béal Monod, Bourbonnais, Emery P.R. B. 34, 7716 (1986). Miyake, Schmitt Rink, and Varma P.R. B 34, 6554-6556 (1986) Kohn, Luttinger, P.R.L. 15, 524 (1965). Bickers, Scalapino, White P.R.L 62, 961 (1989).
χdx 2 -y 2 d x 2 -y 2-wave susceptibility for 6x6 lattice U = 0 U = 4 U = 6 U = 8 U = 10 4 3 2 1 1.0 0.8 0.6 0.4 0.2 0.0 0.75 0.70 0.65 0.60 0.55 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 δ β Landry, A.-M.S.T. unpublished
0.7 U = 0, β = 4 L = 6 0.6 0.5 0.4 0.3 U = 4 β = 4 β = 3 β = 2 β = 1 χd 0.2 0.1 0.0 0.2 0.4 0.6 0.8 1.0 0.0 Doping QMC: symbols. Solid lines analytical. Kyung, Landry, A.-M.S.T.
d G = - - + - - 2 3 - - - 1 4 G GG 1 G 1 G 0 1 2 3 ~ - - - - - - d G + 1 4 s = - - - 2 3 + - - - 1 4
0.45 0.40 0.35 U = 4, β = 4 0.30 0.25 0.20 0.15 0.10 0.05 Full susceptibility, L=32 Susceptibility without vertex, L=32 Full susceptibility, L=6 0.0 0.2 0.4 0.6 0.8 1.0 0.00 Doping x χd
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) 0.16 0.14 0.12 0.10 0.08 0.06 0.04 χsp (T) Self-consistent T-matrix Crossing 0.02 QMC 0.00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 Temperature
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.
T=0 order parameters From paramagnetic self-energy: B. Kyung, P.R.B. 64, 104512 (2001). Opening of pseudogap (RC regime) From dynamical susceptibility, ξ sc = 4
Pseudogap and total gap B. Kyung, P.R.B. 64, 104512 (2001). Kyung and A.-M.S.T. unpublished t/j = 3, J = 125 mev
Properties of the condensate
Some speculations Weak - coupling T AF fluctuations T pseudogap S.C. δ quasi d = 2 S.C.? d = 3 AF δ
Strong-coupling T AF pseudogap δ Strong-coupling quasi d = 2 T AF S.C.? δ Strong-coupling d = 3
Michel Barrette Elix David Sénéchal A.-M.T. Mehdi Bozzo-Rey Alain Veilleux
Steve Allen François Lemay David Poulin Liang Chen Yury Vilk Bumsoo Kyung Samuel Moukouri Hugo Touchette
Claude Bourbonnais R. Côté D. Sénéchal Sébastien Roy Alexandre Blais Jean-Séastien Landry A-M.T. Bumsoo Kyung