Correlationinduced Suppression of Bilayer Splitting in HighT _{ c } Cuprates: A Variational Cluster Approach
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DOI: 10.1007/s109480121537x
 Cite this article as:
 Fulterer, A.M. & Arrigoni, E. J Supercond Nov Magn (2012) 25: 1769. doi:10.1007/s109480121537x
Abstract
We carry out a theoretical study of the bilayer singleband Hubbard model in the undoped and in the superconducting phases by means of the variational cluster approach. In particular, we focus on the splitting between the “bonding” and “antibonding” bands induced by the interlayer hopping, as well as its interplay with strong correlation effects. We find that the splitting is considerably suppressed in both the normal and superconducting phases, in qualitative agreement with experiments on Bi_{2}Sr_{2}CaCu_{2}O_{8+δ }. In addition, in the superconducting phase, the shape of the splitting in k space is modified by correlations.
Keywords
Hightemperature superconductivity Electronic correlations1 Introduction
It is widely accepted that the fundamental physics of HighT _{ c } superconductors (HTSC) takes place in the twodimensional CuO_{2}layers. On the other hand, several classes of HTSC exist with a different number of CuO_{2}layers per unit cell, their transition temperature being strongly related to this number [1]. There have been several explanations for this phenomenon. Among them, one could mention interlayer interactions, charge imbalance, or quantum tunneling of Cooper pairs [2–4].
Experimental measurements, supported by theoretical investigations [5], show that the interlayer coupling and the third dimension more generally have a strong impact on angleresolved photoemission spectroscopy (ARPES) results [6–8]. Depending on photon energy and polarization, different features are accentuated in the measured spectra [9, 10], while the “real” underlying quasiparticle spectrum remains hidden. In the last decade, the BiSrCuO compounds BSCO2212 and BSCO2201 have been studied thoroughly, and several conclusions have been drawn from the results: High resolution ARPES on BSCO2212 with suppressed superstructure reveals the presence of two Fermi surface pieces: one holelike, the other changing from electron to holelike [10]. Heavily overdoped BSCO2212 shows a difference in bilayer band splitting for the normal and superconducting case [11]. In the normal state, this is about 88 meV and gets renormalized to about 20 meV in the superconducting state. In the superconducting state, each one of the two split band develops its own peakdiphump structure (PDH). This is most probably due to the strong renormalization at about 60 meV produced by the interactions with spin fluctuations [11].
Bilayer splitting in the normal state only weakly depends on doping [12]. In optimally doped BSCO2212 (bilayer), the quasiparticle in the (π,0) region should look similar to that of BSCO2201 (monolayer) [9]; the enhanced linewidth in the bilayer material is attributed to correlation effects, more specifically (π,π) scattering due to antiferromagnetic fluctuations. In order to unravel the underlying mechanisms producing these effects, different theoretical methods have been applied. LDA calculation done for YBCO [13] show that the interlayer hopping comes from copper s electrons. Different models were used to describe the system of coupled 2D CuO planes, e.g., the bilayer Hubbard Model [14, 15], coupled twoleg spin ladders [16], tight binding extended Hubbard Model [17, 18], bilayer tJ model [19]. From these calculations, the following conclusions can be drawn. The PDH structure can be explained by a coupling of the electronic excitations to magnetic resonances or spin fluctuations [20, 21]. At low doping, the coupling between the layers should be antiferromagnetic [15], and there might be contributions to superconductivity by interlayer Cooper pairs, being formed by holes belonging to different layers. The reduction of the bilayer splitting with respect to the noninteracting tight binding model is attributed to the formation of spin bags in the layers [19], which increases the quasiparticle weight or/and antiferromagnetic interlayer order.
In this paper, we address these issues by an alternative approach in which correlations are evaluated exactly at a shortrange level of a cluster, and thus is expected to capture the interplay between shortrange antiferromagnetic coupling and quasiparticle excitations. Specifically, we use the Variational Cluster Approach (VCA) [22, 23] to solve the bilayer Hubbard model. VCA is an extension of Cluster Perturbation Theory (CPT) [24, 25]. Due to its variational nature, it allows for a treatment of symmetry breaking phases, in our case antiferromagnetism and/or superconductivity. The method has already been successfully been applied to a wide range of problems [23, 26–29] and is based on the SelfEnergy Functional Theory (SFT) [30, 31].
We will illustrate the effects of bilayer splitting by displaying the spectral functions for the two bands. Finally, we will discuss the reduction of the splitting due to correlation in both the normal as well as in the superconducting state.
2 Model and Method

staggered magnetic fieldwith Q=(π,π).$$H_M=h_M\sum_{i\sigma}(1)^\sigma e^{i\vec{Q} \vec{r}} c_{i\sigma}^+c_{i\sigma},$$(6)

superconducting fieldwhere η is the form factor which determines the symmetry of the superconducting order parameter, in our case dwave.$$H_{SC}=h_{SC}\sum_{i,j}\frac{\eta_{i,j}}{2}(c_{i\uparrow}c_{j\downarrow }+c_{j\uparrow}c_{i\downarrow}),$$(7)

onsite energywhich is needed for thermodynamic consistency [29].$$H_{n}=\epsilon\sum_{i\sigma} n_{i\sigma}$$(8)
The nearestneighbor hopping t=1 sets the energy scale, and we take typical values U=8 and t′=0.3t (see, e.g., [33]). The interlayer hopping is chosen to be \(\tilde{t}\approx0.2\) close to the value estimated for BSCO2212 in [18].
3 Results
Half Filling
Optimal Doping
Moreover, it was found that the shape of the quasiparticle peak in the (π,0) region of the optimally doped monolayer (BSCO2201) and bilayer material (BSCO2212) are similar [9]. This is also very well reproduced in our data, as can be seen in Fig. 3(b).
Overdoping
The values of the splitting for U=8 plotted in Fig. 6 are obtained in the following way: In the normal state, there is just one prominent dispersing peak for each k _{ z } defining a bonding and antibonding band. The k dependent splitting is defined as the distance between the maxima of these peaks for k _{ z }=0,π. For the superconducting state, we determine the splitting for the quasiparticle states below the Fermi level. We have checked that it very close to the splitting of the mirror states above it. When going away from the antinodal point both in the normal state as well as in the superconducting state, each quasiparticle peak first broadens, which introduces an error in the determination of Δ, and then evolves into a two peak structure, which resembles the peakdiphump structure that is observed in ARPES [11]. Measuring the distance between the second pair of peaks gives a second set of data points, which is also displayed in Fig. 6.
4 Conclusion
We have studied the bilayer Hubbard model by means of the variational cluster approach, a method appropriate to capture short range correlation in strongly interacting lattice systems. As expected, the interlayer hopping splits the spectrum into a bonding and an antibonding band. However, the corresponding bilayer splitting is strongly renormalized due to correlations. This is evident in the overdoped case in both the normal and superconducting phases. In qualitative agreement with ARPES measurements, the suppression effect is stronger in the superconducting phase. Surprisingly, for optimal doping, the bilayer splitting vanishes completely, as found in ARPES [9].
For simplicity, we refer to k as the component of the crystal momentum parallel to the CuO_{2} layers (in units of the inverse lattice spacing 1/a). k _{ z } is the perpendicular component and takes the values 0 for the bonding and π for the antibonding band, respectively.
Acknowledgements
We gratefully acknowledge L. Chioncel, H. Allmaier, C. Heil, M. Knapp, M. Nuss, and B. Kollmitzer for fruitful discussions. This work was supported by the Austrian Science Fund (FWF) P18551N16.
Open Access
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