Physica B 186-188 (1993) 822-827 North-Holland
Spin-Peierls fluctuation and dimerization: a possible mechanism of superconductivity Masatoshi Imada Institute for Solid State Physics, University of Tokyo, Minato-ku, Tokyo 106, Japan
A characteristic and unique feature of electron-phonon coupling in strongly correlated metals is examined. Significant properties due to spin-Peierls fluctuation near the Mott transition recently clarified are reviewed, especially in terms of the pairing mechanism and spin gap formation in the metallic phase. A superconducting state appears when the singlet ground state of a Mott insulator with a spin gap is doped with metallic carriers. The spin-Peierls t-J model is shown to be a relevant model. Exotic features of the spin-Peierls pairing mechanism are discussed. Experimental consequences and possible relevance in high Tc oxides and fullerenes are also discussed.
Discovery of superconductors in highly correlated electron systems such as f-electron systems and high T c cuprates has provoked recent intensive studies on mechanisms of superconductivity from a purely electronic origin. The possibility of a superconducting ground state has been extensively investigated in several correlated lattice fermions. Although mean field approximations often lead to various types of superconductivity, we do not have any conclusive evidence for the occurrence of the superconducting state in these models. For example, in the square-lattice Hubbard model, numerical studies show various indications for the absence of superconductivity [1,2]. A conventional BCS pairing mechanism mediated by ordinary electron-phonon coupling has also been examined as a possibility for f-electrons and high T c cuprates by taking less serious account of the strong correlation effect. However, from the theoretical point of view, extension of the conventional BCS formalism to the case with strong electron-phonon coupling and the consequential very short coherence length is difficult. For example, the retardation effect and the possible lattice instability have never been estimated on a reliable basis. From the experimental point of view, we have no explicit evidence to support the
Correspondence to: M. Imada, Institute for Solid State Physics, University of Tokyo, Roppongi 7-22-1, Minato-ku, Tokyo 106, Japan.
ordinary electron-phonon interaction as a main mechanism of pairing in such strongly correlated systems. In low-dimensional conductors, electron-phonon coupling shows unique features due to strong quantum fluctuation . The Peierls transition is a typical example of the strong effect of electron-phonon coupling in low-dimensional systems. The electron-phonon coupling relevant to the Peierls transition shares essentially the same form as the ordinary electron-phonon coupling which drives the BCS pairing, as discussed below. In quasi-one-dimensional conductors, interplay between electron-phonon coupling and the electronelectron interaction has been the subject of intensive study. For example, the origin of charge density wave ordering has been discussed in terms of a strong electron-electron repulsion as well as in terms of the electron-phonon coupling. In high Tc oxides, the presence of strong electronelectron correlation is now widely accepted as a key feature of these materials. Under these circumstances, we may expect an aspect qualitatively different from the conventional electron-phonon interaction. In this paper, we discuss a new feature of the electronphonon coupling in strongly correlated metals. It is emphasized that spin-Peierls-type fluctuation and distortion can play an important role. A novel type of pairing mechanism due to a combined effect of strong electron-lattice coupling and strong electron-electron correlation proposed recently is reviewed. We also discuss a possible candidate for the phonon mode responsible for spin-Peierls fluctuation in high T, oxides.
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M. Imada / Pairing mechanism by spin-Peierls fluctuation
2. Spin-Peierls fluctuation Ordinary electron-phonon coupling diagrammatically expressed in fig. 1 is also the origin of the Peierls transition in low-dimensional conductors. The PeierlsHubbard model defined by the Hamiltonian, = - ~'~ (t o + A" (x i - xj))(c~cjo + c*j~ci~)
included not only through the transfer term t but also through J. Then a relevant model is given by the following spin-Peierls t - J model : H:-•
+ E (Jo + O t ' ( x i - x j ) )
( S i ' S j - ~1 nin j )
+UZ ni~ni++ Z Vqninj +~'i Pl i
k + ~'~ ~ ( x i - xi) 2 ,
Here ?~*, creates fermions in the restricted Hilbert space where the double occupancy on the same site is strictly excluded. In the high T c cuprates, J is even larger than 103 K. In this case, the electron-phonon coupling should be
It is known that the spin-Peierls coupling proportional to a in (3) may cause spin-Peierls instability and static lattice dimerization in quasi-one-dimensional systems. Generally speaking, we expect that spin-Peierls fluctuation is strong in low-dimensional systems with large spin exchange coupling. In high T c cuprates, no direct measurement to determine a has yet been made. However, an estimation for the dependence of J on C u - O bond length in different compounds  allows us an indirect evaluation of a leading to a - 3 J o / a where a is the C u - C u lattice constant. This turns out to be rather large coupling. Direct evaluation of a is highly desirable. The term proportional to t~ is certainly a higher order term in the diagrammatic sense, as in fig. 2. However, it may become much more important in the strong correlation regime, as we show below. Even in the weak correlation approach, spin-Peierlstype fluctuation exists through the paramagnonphonon coupling. It may be continuously connected with the spin-Peierls coupling in eq. (3). Because the qualitative features are expected to be smoothly connected, we only consider the strong correlation case, as in eq. (3). In the following, we investigate how spin-Peierls fluctuation is related to the appearance of spin gap and superconductivity.
Fig. 1. Ordinary electron-phonon coupling. Wavy line represents phonons,
Fig. 2. Electron-phonon coupling due to spin-Peierls fluctuation.
has been extensively used to represent this instability . Here, interplay among the on-site repulsion U, the Coulomb repulsion Vq between different sites i and j, and electron-phonon coupling it has been the main subject of Peierls transition and physics of low-dimensional conductors. We note that it is expressed through the transfer coupled to the lattice distortion xi of ions with mass M and the conjugate momentum p~. If the electron-electron Coulomb repulsion given by U is strong enough as in high Tc cuprates, the t-J model is a relevant effective Hamiltonian in the absence of the coupling to phonons:
( i j ) ,~r
+ J E (s,.s,-~ 1
M. lmada / Pairing mechanism by spin-Peierls fluctuation
3. Dimerized t-J model We first consider the case where spin-Peierls transition actually takes place. Studies on the case of the static dimerization provide insight and are of great importance for the following reason: when the optical phonon frequency responsible for spin-Peierls fluctuation becomes smaller than a characteristic electronic energy scale, the lattice distortion may be viewed as static for that electronic process. In this limit, a relevant Hamiltonian is primarily expressed by static spin-Peierls distortion. Therefore, even in the absence of spin-Peierls symmetry breaking, electronic properties at relatively small phonon frequency are expected to be smoothly connected with the symmetry broken spin-Peierls state. In the half-filled system, a spin gap obviously opens between the singlet ground state and triplet excitation in the spin-Peierls insulating phase. This is the case with, for example, MEM-(TCNQ)2 . What will happen if the insulating spin-Peierls state is doped with carriers? If one starts from the small transfer limit, it is clear that the spin gap remains up to a certain level of doping. To make the discussion more concrete, a dimerized t-J model has been examined [4,7-9], the Hamiltonian of which is given by
(~;j,~ + ~ , ~ )
H = -t E
+JoE (s,.s~ _ ~1 n~n~) (ij)
+ J1 ~ , ' ( S i . S I (ij)
- 41 nins,./
more realistic, we should take account of the dimerization fluctuation on t, as discussed below. In one dimension, it is known that the spin gap opens for arbitrarily small J] at the half-filling. The spin gap is defined by the energy difference between the lowest triplet state and the singlet ground state in the thermodynamic limit. Away from the half-filling, the spin gap is plotted in fig. 4 as a function of the dimerization parameter. It shows penetration of the spin gap formation into non-half-filled systems. When a spin gap opens without opening of the charge gap, the fixed point of interacting fermions belongs to the so-called L u t h e r - E m e r y universality, where the charge degrees of freedom follow a TomonagaLuttinger liquid. In this case, the singlet pairing and charge density wave (CDW) correlation at distance r show power law decays as Cse(r ) oc r-l/°c
Ccow(r ) oc r-°c ,
respectively. The exponent 0c increases far beyond the unity near the half-filling, indicating the dominance of the pairing correlation over C D W . It is known that the half-filled system is unstable to the spin-Peierls distortion, because the exponent a defined by the electronic energy gain AE c~y", with y being the dimerization, is smaller than 2 (in fact a - - 1.4)  and hence IAEI is larger than the elastic loss of the lattice. The smooth behavior of" the spin gap near the half-filling may mean that the system is
The summation E' represents the sum only over the dimerized stronger bonds as in fig. 3. Here we have neglected the eleetron-phonon coupling in the transfer term for simplicity, because it does not affect the essential point of the present issue. Of course, to be
. . . . . -::;F--~6-.. '0.3
0.6 ', ,
J1 Fig. 3. A n example of a dimerized pattern. Stronger bonds are shown by thicker lines.
Fig. 4. Spin gap in the dimerized t-J model plotted in the filling p versus dimerization J~ plane. The spin gap values are given aside the symbols. The parameters are t = 1.0 and J0 = 0.0.
M. lmada / Pairing mechanism by spin-Peierls fluctuation also unstable to the distortion even away from the half-filling. In two dimensions, the phase boundary of the spin gap region has not yet been clarified. We expect that competition between the antiferromagnetic long-range order and the singlet formation with a spin gap makes the phase diagram rich and complicated. Even in this case, it is clear that the spin gap region exists in some extended area. Irrespective of the dimensionality, the dimerized t - J model is shown to be mapped to the attractive Hubbard model in a strongly dimerized region. If J1 is sufficiently larger than J0 and t, singlet pairs are formed on stronger bonds given by J1 and they hop through pair tunneling to other empty stronger bonds. Such a boson representation of the dimerized t - J model in the strongly dimerized limit is equivalent to the strong coupling limit of the attractive Hubbard model. The existence of the spin gap in the strongly dimerized region is, therefore, rather clear even away from the half-filling. The ground state should be superconducting in two and three dimensions if the system is away from the half-filled and the quarterfilled. (The quarter-filled system has been shown to have a charge gap as in the half-filled case of the attractive Hubbard model .) The region with a spin gap and the superconducting ground state may extend to the weaker dimerization region, as in the spin gap region in one dimension. It competes with the antiferromagnetically ordered state near the half-filling in two and three dimensions. It seems likely that the spin gap region exists outside the antiferromagnetically ordered region away from the half-filling for some intermediate dimerization region. Detailed analysis of this issue remains for further study.
4. Dynamical spin-Peierls coupling A more complicated situation arises when spinPeierls symmetry breaking does not take place but a large fluctuation exists. We may assume that the characteristic phonon frequency of the spin-Peierls fluctuation is relatively small compared to the Fermi energy E F in realistic cases. In the usual BCS formalism, effectively attractive interaction of the electrons occurs for those within a thin shell of the width of Debye frequency toD around the fermi surface. Contrary to the BCS case, from the study in the previous section, it has been clarified that the pairing occurs from the spin-Peierls fluctuation even in the case of static distortion, i.e., top-->0. Therefore, the spinPeierls pairing mechanism is apparently quite different
from the BCS mechanism. Because the pairing is not restricted within the shell, there is no reason to expect the isotope effect proportional to the phonon frequency shift in marked contrast with the ordinary BCS mechanism. For electrons within the shell around the Fermi level, as in the BCS formalism, a weak coupling theory may be constructed in the second-order perturbation of a by eliminating the phonon degrees of freedom. The effective Hamiltonian for the electronic part reads 1 Hs _ ph = -N
2 , 1 a q F ( k , q ) F ( k ,q) h w q - 8e
x \S," S~ x
o _ 1 n~n_k_q)
\ S k,.S_k,_q - 4 nk,n_k, q ) ,
where F ( k , q ) = f ( k x , q x ) q . + f ( k y , q r ) q y and f ( k , q ) "= sin(q/2) cos(k + (q/2)) for the square lattice. Here aq is proportional to a in (3) (i.e., Olq ~ a/qx/-~q for the optical phonon frequency tOq). The electronic excitation energy of the intermediate state is 8e. The term in eq.(7) is illustrated diagrammatically in fig. 5. Because the exchange coupling is rewritten by a singlet pairing operator Oij =- (citcj~ - ciscjt)/X/-2 as
S i "Sj - ~ nin j = O~Oii ,
eq. (7) means attractive interaction of the singlet pair and favors the singlet pairing for 8e < htoq. The weak coupling picture of the pairing appropriate for small a described above looks quite different from the picture for strong a with small tOq, described through the spin-Peierls distortion as in the previous section. A t the moment, we do not have a reliable way of representing these apparently different views in a unified manner. Because we have to treat the phonon degrees of freedom in a highly dynamical way, this
Fig. 5. Effective attraction due to the spin-Peierls fluctuation in the weak coupling picture.
M. lmada / Pairing mechanism by spin-Peierlsfluctuation
remains a challenging problem for further investigation. Nevertheless, we can speculate upon the qualitative aspect of spin-Peierls pairing to some extent as follows. If spin-Peierls dimerization takes place, the pairing should also have a dimerized pattern. It leads to the pairing given by the linear combination of A~ ~-(Ck~C_k~) and An =- (Ck~C_k+e~), where O = ('n/a, "n/ a) or Q = ('n/a, 0) etc., according to the dimerization pattern. Here a is the lattice constant of the original undimerized lattice. Because the Brillouin zone is halved in a dimerized system, - k + Q represents another branch of the mode at the wave number - k . It has been shown that the dimer pairing state has s-wave symmetry in its fundamental sense because the gap is nodeless . However, because of its highly anisotropic pairing structure, it can show similar behavior to the d-wave symmetry near and intermediately below the transition temperature. In fact, it has been shown that the coherence peak in the NMR relaxation rate is absent in typical cases . The dimer pairing state shows variety from a more s-wave-like state to a more d-wave-like state according to the ratio of An to A~. If An is small, the nature of the s-wave symmetry appears at relatively low temperatures and the d-wave-like properties govern from T~ to somewhat lower temperatures. On the contrary, s-wave properties can be seen at a higher temperature if An becomes large. If the static spin-Peierls dimerization does not take place but dynamical fluctuation is strong, the pairing order parameter should have a more smeared and complicated structure such as
g(p) X~k f(k)(akta_k+p$) ,
where g(p) has, for example, peaks at p = (0, 0) and (~r, rr) but has symmetric tails around them. In this case, the relevant phonon mode which we discuss in the next section is expected to show anomalous fluctuation. Although the order parameter is more complicated than in the case of the static dimerization, we expect that the relaxation in the ordered state shows similarities. The pairing symmetry is of the s-wave type in the sense of a nodeless gap. However, it may have d-wave-like character near To, such as in the absence of the coherence peak.
Peierls fluctuation must be dynamical as a possible origin of the mechanism in the high Tc oxides. The spin-Peierls fluctuation is relevantly coupled to optical phonon modes which cause the fluctuation of C u - C u distance in the CuO2 plane. Therefore, the zone boundary mode of E u phonons is a promising candidate for this mechanism. In the Eo phonon modes, roughly speaking, we have two cases. In one case, in-plane oxygens around the copper oscillate in the same direction as the copper as in fig. 6(a). In this case, the fluctuation of t and J in eq. (3) occurs in phase. In the other case, the copper and the oxygens oscillate out of phase as in fig. 6(b). In this case, it is expected that bonds with larger coupling in J correspond to weaker t, because the wave function of the doped holes contributing to the transfer has larger component at the oxygen sites, while the superexchange is dominantly determined by the C u - C u distance. If this mechanism works, we expect that the zone boundary modes of Eo phonons as sketched in figs. 6(a) and (b) may show some anomalous feature. In fact, because the explicit symmetry breaking of the dimer order is unlikely in the absence of static dimerization, we expect that the primary effect is anomalous increase of the phonon damping around the zone boundary of E u phonons. Detailed experimental examination is necessary. It should be noted that the recently discovered superconducting state in fullerenes is also interesting in terms of this spin-Peierls mechanism. In C6o, it is known that the lattice dimerization actually exists . Even under the doping of carriers into C6o such as in K3C60, the dimerization pattern does not disappear. Furthermore, electron-electron correlation on the same "rr orbital of a carbon site may not be small. Therefore, the superconducting mechanism discussed in this paper would also be relevant in doped C6o.
5. Possible experimental relevance in high T~ oxides and fullerenes Because we have no explicit evidence for the structural phase transition of the dimerization, the spin-
Fig. 6. Phonon modes relevantly coupled to the spin-Peierls fluctuation in the CuO 2 plane.
M. lmada / Pairing mechanism by spin-Peierls fluctuation
Acknowledgements This work is financially supported by a Grant-in-Aid for Scientific Research on Priority Areas, Computational Physics as a New Frontier in Condensed Matter Research, from the Ministry of Education, Science and Culture, Japan.
References  M. Imada, J. Phys. Soc. Jpn. 60 (1991) 2740.  N. Furukawa and M. Imada, J. Phys. Soc. Jpn. 61 (1992) 3331.
 For references, see for example, D. Baeriswyl and D.K. Campbell (eds.), Interacting Electrons in Reduced Dimensions (Plenum, New York, 1989).  M. Imada, J. Phys. Soc. Jpn. 60 (1991) 1877.  Y. Tokura, Jpn. J. Appl. Phys. 7 (1992) 14.  S. Huizinga et al., Phys. Rev. B 19 (1979) 4733.  M. Imada, preprint.  M. Imada, Physica C 185-189 (1991) 1421.  M. Imada, J. Phys. Soc. Jpn. 61 (1992) 423.  See for example, K. Okamoto, H. Nishimori and Y. Taguchi, J. Phys. Soc. Jpn. 55 (1986) 1458.  C.S. Yannoni et al., J. Am. Chem. Soc. 113 (1991) 3190.