Vertex Corrections in the Spin-fluctuation-induced Superconductivity

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Vertex Corrections in the Spin-?uctuation-induced Superconductivity

arXiv:cond-mat/9910154v1 [cond-mat.supr-con] 11 Oct 1999

Takuya Okabe?
Faculty of Engineering, Gunma University, Kiryu, Gunma 376-8515. (Received June 28, 1999)

We evaluate vertex corrections to Tc on the basis of the antiferromagnetic spin-?uctuation model of the high-Tc superconductivity. It is found that the corrections are attractive in the dx2 ?y 2 channel, and they become appreciable as we go through an intermediate-coupling regime ? of Tc ? 100K, the maximum Tc attainable in the one-loop Eliashberg calculation.
KEYWORDS: high-Tc superconductor, antiferromagnetic spin ?uctuation, vertex correction

As a model for the high-Tc superconductivity, the spin ?uctuation mechanism has been one of the most widely discussed.1, 2) The model assumes that quasiparticle is coupled with antiferromagnetic spin ?uctuation, represented by a peculiar low-energy expression for the magnetic susceptibility.1, 3) The phenomenological coupling constant to ?t the transition temperature Tc is used to explain, among others, the anomalous transport properties consistently.1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13) On the other side, from a microscopic point of view, numerical studies based on the ?uctuation exchange (FLEX) approximation14) have been carried out by many authors to estimate Tc as well as to explain the deviations from the normal Fermi liquid behavior.14, 15, 16, 17) Computational feasibility of these strong coupling theories rests on the e?ective use of a fast Fourier transform (FFT) algorithm. To asses the quantitative aspect of the theories, corrections coming from higher-order terms are investigated for the vertex function at some ?xed external momenta, e.g., on the basis of the spin-?uctuation model, and the qualitative features of the e?ect are emerging to some extent.18, 19, 20, 21, 22, 23, 24) However, the total e?ect of the vertex corrections on the physical observables is yet to be estimated numerically. Indeed to do this is generally formidable because of the inapplicability of the FFT to a required additional sum on internal frequency and momentum. In this paper, we manage to evaluate the vertex corrections to Tc , and discuss questions of convergence of the formal perturbation theory with respect to the coupling constant. To put it concretely, the following investigation is based on the model of Monthoux and Pines (MP),4, 6, 7) in which the self-energy Σ(p, iωn ) is determined as a selfconsistent solution of the equations, Σ(p, iωn ) = g and G(p, iωn )?1 = G(0) (p, iωn )?1 ? Σ(p, iωn ) + δ?, (2)
2T

where G(0) (p, iωn ) = and εp = ?2t(cos px + cos py ) ? 4t′ cos px cos py . (4) 1 , iωn ? εp + ?(0) (3)

In eq. (1), χ(q, iνm ) as a function of the Matsubara frequency νm = 2mπT is inferred from the low-energy form of the magnetic susceptibility,3) χQ ωsf χ(q, ω) = , (5) ωq ? iω where ωq ≡ ωsf (1 + ξ 2 (q ? Q)2 ), for qx > 0 and qy > 0. We assume χ(q, iνn ) = ? 2 = π 1 π
0 ω0 ?ω0 ω0

Q = (π, π),

(6)

Imχ(q, ω) dω iνn ? ω (n = 0) (n = 0) (7)

ωImχ(q, ω) dω 2 νn + ω 2

= χ(q, ω = 0).

Here it is noted that a cuto? ω0 has to be arti?cially 2 introduced so as to meet the condition χ(q, iνn ) → 1/νn 4) as |νn | → ∞. For eq. (5), the integral in eq. (7) is analytically evaluated; χ(q, iνn ) = 2χQ ωsf /π 2 2 νn ? ωq |νn | tan?1 ω0 ω0 ? ωq tan?1 |νn | ωq .

(8) To estimate the critical temperature Tc , the linearized gap equation is used, Φ(p, iωn ) = ? T ? V (p, iωn ; p′ , iωn′ )
p′ ,n′

?

χ(q, iνm )G(p ? q, iωn ? iνm ), (1)
q,m

×|G(p′ , iωn′ )|2 Φ(p′ , iωn′ ),

(9)

where Φ(p, iωn ) is the anomalous self-energy. The pairing potential V (p, iωn ; p′ , iωn′ ) reads V (p, iωn ; p′ , iωn′ ) = V (1) (p ? p′ , iωn ? iωn′ ) +Vv(2) (p, iωn ; p′ , iωn′ ) + Vc(2) (p, iωn ; p′ , iωn′ ),

?

E-mail: okabe@phys.eg.gunma-u.ac.jp

(10)

2

Takuya OKABE

where V (1) (p ? p′ , iωn ? iωn′ ) = g 2 χ(p ? p′ , iωn ? iωn′ ). (11) The second and third terms in eq. (10) originate from the vertex corrections that we discuss below. As we are concerned about the d-wave instability, introducing a notation f (p)
p



1 ?

(cos(px ) ? cos(py ))f (p),
p

(12)

we put eq. (9) into ? Φ(iωn ) =
n′

? K(iωn , iωn′ )Φ(iωn′ ),

(13)

where ? Φ(iωn ) = Φ(p, iωn )
p

,

(14) (15)

(2) (2) K(iωn , iωn′ ) = K (1) + Kv + Kc ,

Fig. 1. The coupling constant g 2 to account for Tc = 90K as a function of the cuto? ω0 , calculated on a 16×16 lattice with periodic boundary conditions.

and Ki
(2) (1,2)

= ?T Vi

(1,2)

|G(p′ , iωn′ )|2

p,p′

.

(16)

Here Ki (iωn , iωn′ ) (i = v, c) come from the vertex corrections. As is clear from eq. (13), the condition that the largest eigenvalue of K(iωn , iωn′ ) reaches unity pro? vides a nonzero solution Φ(iωn ), thus de?nes Tc . Below ? we look for a real solution Φ(iωn ), for which the imaginary part of K(iωn , iωn′ ), namely, Im V (p, iωn ; p′ , iωn′ ), is neglected. As for the parameters, we assume t = 0.25eV and t′ = ?0.45t for eq. (4), and we take ξ = 2.3, χQ = 75/eV, ωsf = 14meV, (17)

to describe χ(q, ω) of YBa2 Cu3 O7 at Tc = 90K, according to MP.7) The chemical potential ?(0) is ?xed by , e(εp ??(0) )/T + 1 (18) in which we assume n = 0.75 throughout this paper. The shift δ? in eq. (2) is adjusted in every iteration to assure
p,n p

2T n= ?

G

(0)

(p, iωn )e

+iωn 0

2 = ?

1

Fig. 2. Critical temperature Tc is shown as a function of g 2 for the cuto? ω0 = 0.4eV. The other parameters are given in the text. The results are shown for a 16×16 (circles), 32×32 (squares), 64×64 (diamonds) and 128×128 (triangles) square lattice. The e?ect of Σ(p, iωn ) is not taken into account for the open symbols, while it is included for the closed symbols. A closed triangle for 128×128 is not shown.

δn =

2T ?

G(p, iωn ) ? G(0) (p, iωn ) = 0,
p,n

(19) is not taken into account, but the e?ect of Σ(p, iωn ) is. As we see from the ?gure, we cannot ?x g 2 just from Tc (g 2 ) = 90K without the knowledge of the cuto? ω0 in eq. (7). The strong dependence on ω0 in the low-energy region ω0 ≤ 0.3eV re?ects that the transition temperature Tc , unlike the transport properties controlled by the quasiparticle damping, is not determined solely from the low-energy expression, eq. (5). Indeed, this is one of the general problems to infer the full structure of χ(q, iνn ) entirely from the low-energy ‘observable’ χ(q, ω), and to settle ω0 may be a key point in the discrepancy between Radtke et al.5) and MP.9) As it is not our purpose to discuss this point further, for the time being, we assume eq. (5) up to the cuto? energy ω0 in eq. (7), and arbitrarily set ω0 = 0.4eV? 4.6 × 103K, following MP.7) This value is used throughout in the following. Then we obtain g 2 = 0.57eV2 for Tc = 90K, still in disagreement

as this is easier to handle than the formally divergent sum, n = (2T /?) p,n G(p, iωn ). For practical purposes, the Matsubara sums in eqs. (1), (9) and (19) are restricted within a ?nite range |ωn |, |νm | ≤ ωc . To avoid spurious temperature dependences, the cuto? is ?xed at ωc = 6.2eV ? 3 times the bandwidth, for which we have |νm | ? ωc for m = ±27 at T = 90K.7) The above, except for the cuto? ω0 in eq. (7), are all the necessary ingredients to reproduce the results of MP.7) Nevertheless, we could not derive them precisely, though qualitative features are consistently reproduced. Let us discuss the point brie?y. First we have to note the ω0 dependence of Tc as a function of the coupling constant g 2 . In Fig. 1, g 2 to give Tc = 90K is shown as a function of ω0 . In this result, K (1) in eq. (16) is used for (2) the eigenequation (13), i.e., the vertex correction Ki

Vertex Corrections in the Spin-?uctuation-induced Superconductivity

3

Fig. 3. The diagram (a) for the vertex corrected pairing potential (2) (2) Vv and (b) for Vc . Fig. 4. Tc as a function of g 2 . Triangles; calculated without Σ(p, iωn ). Circles; including the e?ect of Σ(p, iωn ). Diamonds; including the e?ect of Σ(p, iωn ) as well as the vertex corrections (2) (2) Vv and Vc . Two squares at T = 90K and 45K are calculated (2) (2) with Σ(p, iωn ) and Vc but without Vv .

with g 2 = 0.41eV2 of MP.7) This is not due to the size of a square lattice, as we see in Fig. 2, where Tc is shown as a function of g 2 . At T = Tc = 90K, the 16×16 lattice is large enough for us to conclude g 2 = 0.57eV2 in the selfconsistent calculation including the e?ect of Σ(p, iωn ). A close inspection indicates that the disagreement originates in details of χ(q, iνm ). In fact, we ?nd g 2 = 0.34, smaller than g 2 = 0.41 of MP, if we adopt the second line, instead of the third line, of eq. (7) for n = 0 too. This means that Tc depends sensitively on how we prepare χ(q, iνn ) in the low-energy regime. This is complementary to the above remark on the high-energy contribution to Tc . As this quantitative di?erence is not of our primal concern either, deferring this problem, we choose to use our own de?nition, i.e., χ(q, iωn ) for n = 0 is speci?ed separately by eq. (7). The qualitative results presented below are not a?ected by this choice. Now let us discuss how we evaluate the vertex cor(2) rection. The pairing potentials Vv (p, iωn ; p′ , iωn′ ) and (2) ′ Vc (p, iωn ; p , iωn′ ) including the vertex correction are diagrammatically represented by Fig. 3(a) and Fig. 3(b), respectively. These potentials at low frequencies ωn = ωn′ = πT (for n = n′ = 0) are particularly studied by Monthoux.24) To see the e?ect on Tc precisely, however, (2) we have to evaluate the kernel Ki (iωn , iωn′ ), eq. (16), for a full set of the Matsubara frequencies ωn and ωn′ , then the kernel must be diagonalized. In e?ect, this is not practical at present. Thus, as a tractable method, we set up perturbation theory to evaluate the vertex corrections to the eigenvalue κ of the kernel. We shall make e?ective use of the results obtainable by means of the FFT. Let us introduce ? the eigenfunction Φ(1) (iωn ) for the largest eigen? value κ(1) of the kernel K (1) (iωn , iωn′ ); Φ(1) (iωn ) = (1) ? ? (iωn , iωn′ )Φ(1) (iωn′ ) = κ(1) Φ(1) (iωn ). Then the ′ K n eigenvalue κ including the vertex corrections is given by κ = κ(1) + κ(2) + κ(2) , v c where κi
(2) (2) ? ? Φ(1) (iωn )Ki (iωn , iωn′ )Φ(1) (iωn′ ). n,n′

? grounds, the norm of Φ(1) (iωn ) decreases quite rapidly as |ωn | increases. Therefore, the sum over the Matsubara frequencies in eq. (21) is allowed to be restricted in a narrow region around (n, n′ ) ? (0, 0). In e?ect, (2) we evaluate Ki (iωn , iωn′ ) for a 16 × 16 mesh around the Fermi energy. Moreover, in the remainder of the paper, the results are calculated on a 16 × 16 square 2 ? lattice. Measured in terms of the weight Φ(1) (iωn ) , 2 ? we ?nd |ωn |≤15πT Φ(1) (iωn ) = 0.98, 0.91 and 0.78 at T = Tc =90K, 45K and 22K, respectively. Even at low Tc , the error involved is not appreciable, for the coupling constant itself is small there. As Fig. 2 shows, a 16 × 16 mesh in the momentum space is large enough to grasp the qualitative features caused by the vertex corrections. (2) To prepare Vi (p, iωn ; p′ , iωn′ ) is most timeconsuming. Therefore, ?rst we use the bare Green’s function G(0) (p, iωn ) instead of G(p, iωn ) to provide (2) (2) Vi (p, iωn ; p′ , iωn′ ). With Vi (p, iωn ; p′ , iωn′ ) thus calculated beforehand and G(p, iωn ) of the solution of (2) eqs. (1) and (2), we calculate Ki (iωn , iωn′ ) in eq. (16). (2) Then, to evaluate κi , eq. (21), is straightforward, and the critical coupling g 2 at Tc is determined. Results thus obtained are shown in Fig. 4, where the triangles (without Σ(p, iωn )) and circles (with Σ(p, iωn )) denote the results without the vertex corrections. (See Fig. 2). The (2) diamonds include the vertex correction Vv as well as (2) (2) Vc , while only the e?ect of Vc is taken into account for the two squares at T = 90K and 45K. Several points are noted from the ?gure. In the ?rst (2) (2) place, both the e?ects of Vv and Vc are attractive on the whole, or enhances Tc of the d-wave instability. (2) The e?ect of Vc (Fig. 3(b)),19) however, is negligibly small, as noted by Monthoux.24) On the other hand, the (2) e?ect of Vv (Fig. 3(a)) is prominent. In particular, it a?ects the result of MP, denoted by the circles interpolated with the solid line in Fig. 4, that the maximum transition temperature attainable in this model is about
p

(20)

=

(21)

? We assume Φ(1) (iωn ) is normalized.

On physical

4

Takuya OKABE

Fig. 5. At T = 90K, κ(1) and κ(1) + κ(2) are shown as a function (2) of g 2 . Squares, including only the e?ect of Vc , are overlapping with circles to denote κ(1) without the vertex corrections.

100K.4) In fact, Tc as a function of g 2 shows no sign of saturation, and keeps increasing beyond 200K when (2) the vertex correction Vv is taken into account. In this regard, the vertex correction has an e?ect more than a mere scale-up of the e?ective coupling constant g 2 . (2) Next, the e?ect of Σ(p, iωn ) on Vi has to be investigated. To this end, G(p, iωn ) to meet eqs. (1) and (2) (2) is used to evaluate Vi (p, iωn ; p′ , iωn′ ). The maximum eigenvalues calculated for T = 90K are shown in Fig. 5 as a function of g 2 . The e?ect of Σ(p, iωn ) is to weaken the vertex corrections. The e?ect, however, is not appreciable for Tc = 90K, as we see from Fig. 5 in which we see g 2 = 0.36 while we have g 2 = 0.32 in Fig. 4 in the case including the vertex corrections. Comparing these with g 2 = 0.57 without the vertex corrections, (2) we conclude that the correction due to Σ(p, iωn ) in Vi is not important at least at Tc = 90K. In other words, if the coupling constant g should be evaluated to account for Tc , our result is that the vertex correction is not negligibly small at this temperature,22) at variance with previous results.21, 23) The discrepancy may be due to a high-energy contribution included in our calculation, or it is traced back to the above ?nding of a slight (2) renormalization e?ect on Vi . On the other hand, for Tc = 180K, we ?nd that g 2 = 0.73 in Fig. 4 is modi?ed to g 2 = 1.57. The large modi?cation in this case is due to a large coupling constant to realize that high transition temperature. The results in this regime must be taken with care. To the extent that the vertex corrections that we found for the pairing potentials are not negligible, the vertex corrections to eq. (1) should have to be investigated next.23) The latter e?ect on Σ(p, iωn ) will reduce Tc somewhat particularly through the pair propagator |G(p′ , iωn′ )|2 in eq. (16), according to the above note, as a result of enhanced quasiparticle damping. Therefore, we will be ultimately led to a convergent result of Tc (g 2 ), somewhere in between the dashed and solid lines of Fig. 4. The results, however, would then indicate that Tc ? 100K is on the verge of practical applicability of this kind of perturbation theory in g 2 , as inferred from

Fig. 4. Note that, for us in this context, to su?er a small correction is more important than to ?nd out a high Tc . In summary, a result of this paper is presented in Fig. 4, though the result at high temperature is somewhat modi?ed as stated above. Applying perturbation theory to the eigenvalue of the kernel K(iωn , iωn′ ), we estimated the vertex corrections to Tc as a function of the coupling constant g 2 on the basis of the spin?uctuation model of the high-Tc superconductivity. We found that the e?ect of Fig. 3(b) is numerically negligible as far as the dx2 ?y2 pairing instability is concerned, while Fig. 3(a) enhances Tc appreciably. For Tc ? 100K, the e?ect of Σ(p, iωn ) mainly comes in through the pair propagator |G(p′ , iωn′ )|2 , dressing the vertex functions is not so important. In a strong-coupling regime at high temperatures, the vertex corrections become even qualitatively important, particularly in case where Tc in ? the one-loop Eliashberg calculation is substantially suppressed by lifetime e?ects. We would like to thank J. Igarashi, M. Takahashi, T. Nagao, T. Yamamoto and N. Ishimura for valuable discussion. This work was supported by the Japan Society for the Promotion of Science for Young Scientists.
[1] T. Moriya, Y. Takahashi and K. Ueda: J. Phys. Soc. Jpn. 59 (1990) 2905. [2] P. Monthoux, A. V. Balatsky and D. Pines: Phys. Rev. B 46 (1992) 14803. [3] A. J. Millis, H. Monien and D. Pines: Phys. Rev. B 42 (1990) 167. [4] P. Monthoux and D. Pines: Phys. Rev. Lett. 67 (1992) 961. [5] R. J. Radtke, S. Ullah, K. Levin and M. R. Norman: Phys. Rev. B 46 (1992) 11975; R. J. Radtke, K. Levin, H.-B. Sch¨ttler and M. R. Norman: Phys. Rev. B 48 (1993) 15957. u [6] P. Monthoux and D. Pines: Phys. Rev. B 47 (1993) 6069. [7] P. Monthoux and D. Pines: Phys. Rev. B 49 (1994) 4261. [8] R. Hlubina and T. M. Rice: Phys. Rev. B 51 (1995) 9253. [9] H.-B. Sch¨ttler and M. R. Norman: Phys. Rev. B 54 (1996) u 13295. [10] B. P. Stojkovi? and D. Pines: Phys. Rev. Lett. 76 (1996) 811. c [11] Y. Yanase and K. Yamada: J. Phys. Soc. Jpn. 68 (1999) 548. [12] K. Kanki and H. Kontani: J. Phys. Soc. Jpn. 68 (1999)1614. [13] T. P. Devereaux and A. P. Kampf: Phys. Rev. B 59 (1999) 6411. [14] N. E. Bickers, D. J. Scalapino and S. R. White: Phys. Rev. Lett. 62 (1989) 961; N. E. Bickers and S. R. White: Phys. Rev. B 43 (1991) 8044. [15] A. I. Liechtenstein, O. Gunnarsson, O. K. Andersen and R. M. Martin: Phys. Rev. B 54 (1996) 12505; P. Monthoux and D. J. Scalapino: Phys. Rev. Lett. 72 (1994) 1874; T. Dahm and L. Tewordt: Phys. Rev. B 52 (1995) 1297. [16] K. Yonemitsu: J. Phys. Soc. Jpn. 59 (1990) 2183; T. Hotta: J. Phys. Soc. Jpn. 63 (1994) 4126; S. Koikegami, S. Fujimoto and K. Yamada: J. Phys. Soc. Jpn. 66 (1997) 1438; T. Takimoto and T. Moriya: J. Phys. Soc. Jpn. 66 (1997) 2459. [17] H. Kontani, K. Kanki and K. Ueda: Phys. Rev. B 59 (1999) 14723. [18] I. Grosu and M. Crisan: Phys. Rev. B 49 (1994) 1269. [19] J. R. Schrie?er: J. Low Temp. Phys. 99 (1995) 397. [20] A. V. Chubukkov: Phys. Rev. B 52 (1995) R3840. [21] B. L. Altshuler, L. B. Io?e and A. J. Millis: Phys. Rev. B 52 (1995) 5563. [22] M. H. S. Amin and P. C. Stamp: Phys. Rev. Lett. 77 (1996) 3017. [23] A. V. Chubukkov, P. Monthoux and D. K. Morr: Phys. Rev. B 56 (1997) 7789. [24] P. Monthoux: Phys. Rev. B 55 (1997) 15261.

2

Tc=90 K
1.5

g (eV )

2

1

2

0.5

0

0.2

0.4

ω0 (eV)

0.6

0.8

1

200 16 2 32 2 64 2 128
2

150 Tc (K)

100

50

0

0

0.2

0.4 2 2 g (eV )

0.6

0.8

(a)

(b) p p’

-p’

-p

200

150

Tc (K)

100

50

0

0

0.2

0.4
2

g (eV )

2

0.6

0.8

1

1.4
κ (1) (2) κ +κ
(1)

1.2

κ

1

0.8 T=90 K 0.6

0.2

0.3

0.4 2 2 g (eV )

0.5

0.6


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