Determining the underlying Fermi surface of strongly correlated superconductors
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Contributed by P. W. Anderson, July 25, 2006
Abstract
The notion of a Fermi surface (FS) is one of the most ingenious concepts developed by solidstate physicists during the past century. It plays a central role in our understanding of interacting electron systems. Extraordinary efforts have been undertaken, by both experiment and theory, to reveal the FS of the hightemperature superconductors, the most prominent class of strongly correlated superconductors. Here, we discuss some of the prevalent methods used to determine the FS and show that they generally lead to erroneous results close to halffilling and at low temperatures, because of the large superconducting gap (pseudogap) below (above) the superconducting transition temperature. Our findings provide a perspective on the interplay between strong correlations and superconductivity and highlight the importance of strong coupling theories for the characterization and determination of the underlying FS in angleresolved photoemission spectroscopy experiments.
During the last decade, angleresolved photoemission spectroscopy (ARPES) has emerged as a powerful tool (1, 2) for studying the electronic structure of the hightemperature superconductor (HTSC) (3). ARPES is a direct method for probing the Fermi surface (FS), the locus in momentum space where the oneelectron excitations are gapless (4). However, because the lowtemperature phase of the HTSC has a superconducting or pseudogap with dwave symmetry, a FS can be defined only along the nodal directions or along the socalled Fermi arcs (1, 2, 5–7). The full “underlying FS” emerges only when the pairing interactions are turned off, either by a Gedanken experiment or by raising the temperature. Its experimental determination presents a great challenge because ARPES is more accurate at lower temperatures. Because the FS plays a key role in our understanding of condensed matter, it is important to know what is exactly measured by ARPES in a superconducting or a pseudogap state. The problem becomes even more acute in HTSC because of the presence of strong correlation effects (8–11). Hence, it is desirable to examine a reference dwave superconducting state with aspects of strong correlation built explicitly in its construction. Motivated by these considerations, we study the FS of a strongly correlated dwave superconductor (8, 11) and discuss our results in the context of ARPES in HTSC.
Fermi vs. Luttinger Surface
We begin by highlighting the differences between a FS and a Luttinger surface. The FS is determined by the poles of the oneelectron Green's function (4). The Luttinger surface is defined as the locus of points in reciprocal space, where the oneparticle Green's function changes sign (12). In the Fermi liquid state of normal metals, the Luttinger surface coincides with the FS. In a MottHubbard insulator the Green's function changes sign because of a characteristic 1/ω divergence of the singleparticle selfenergy (13, 14) at momenta k of the noninteracting FS. In the HTSC the gapped states destroy the FS but only mask the Luttinger surface. Hence, it seems natural to relate the Luttinger surface of the superconducting and the pseudogap states with the concept of an underlying FS and ask whether such a surface can be determined by ARPES.
To answer this question, we recall that the elementary excitations in a superconductor are given by the dispersion relation, where ε _{k} are the momentumdependent orbital energies of electrons in the absence of a superconducting order parameter Δ _{k} , and μ is the chemical potential. The corresponding Luttinger surface is determined by the condition ξ _{k} ≡ 0, which is also the definition of the normalstate FS when Δ _{k} ≡ 0. In the following, we discuss two methods commonly used to determine the underlying FS, namely, the Luttinger surface, of the HTSC by ARPES (1, 2, 15).
FS Determination
In the socalled “maximal intensity method” the intensity of ARPES spectra at zero frequency is used to map out the underlying FS. It can be shown that this quantity is where Γ _{k} is determined both by the experimental resolution and the width of the quasiparticle peak. When the momentum dependence of Γ _{k} is small compared with that of E_{k} (as is usually the case), the maximal intensity is given by the set of momenta ℏk for which E_{k} is minimal.
To examine the accuracy of this method in determining the underlying FS, we calculate this quantity for a strongly correlated dwave superconducting state. All calculations are done with model parameters for HTSC by using the renormalized mean field theory (RMFT) (8, 11), for which the quasiparticle dispersion E_{k} retains the form of Eq. 1 . In Fig. 1, we show our results for the spectral intensity at zero frequency and the locus of the Luttinger surface. The former is deduced from the inverse of E_{k} .
For large hole doping, x = 0.25, the superconducting gap is small and the Luttinger surface is close to the points in momentum space for which the zero frequency intensity is maximal. But for smaller doping, x = 0.05, the gap is substantial and the Luttinger surface deviates qualitatively from the maximal intensity surface because of the momentum dependence of Δ _{k} (see ridges in Fig. 1). We have verified that this behavior persists for a wide range of Δ _{k}  and not just the values estimated from RMFT. Fig. 1 a also reveals that the maximum intensity splits into two ridges (orange inner, red outer). Although not widely discussed in the literature, this splitting may be deduced from experimental data, e.g., the intensity plots in E − k space along symmetric lines (0, 0) → (π, 0) → (π, π) in ref. 16. It follows that when the gap or the pseudogap is large the criterion of maximal spectral intensity alone does not suffice to identify the correct FS, and it is necessary to supplement the analysis of the zerofrequency ARPES intensity (Eq. 2 ) with a dispersion relation such as Eq. 1 . These considerations explain why the (outer) maximal intensity ridges seen in ARPES (at low temperatures in the underdoped regime) may yield an underlying FS whose volume is too large. In particular, this effect is seen in Ca_{2x}Na _{x} CuO_{2}Cl_{2} (17), which also exhibits quite a large pseudogap (18).
Another method used in extracting the Luttinger surface is the “maximal gradient method.” The method is based on the fact that the FS is given by the set of k values for which the momentum distribution function n_{k} shows a jump in discontinuity. When this discontinuity is smeared out, say, by thermal broadening or a small gap, the gradient of n_{k} , ∇n_{k} , is assumed to be maximal at the locus of the underlying FS.
We calculated ∇n_{k}  within RMFT and show our results in Fig. 2. We see that the maximal gradient surface is very sensitive to the presence of even small gaps. For example, the superconducting gap at x = 0.25 is quite small. Nonetheless, the electronlike Luttinger surface (determined by ξ _{k} ≡ 0) is not clearly revealed by the ridges in ∇n_{k} . Similar deviations of ∇n_{k}  from the underlying surface are also obtained from a hightemperature expansion of the tJ model (19) and dynamical cluster approximation in the Hubbard model (20). We conclude that the maximal gradient method alone cannot be used to determine the underlying FS unambiguously from numerical (19, 20) or ARPES data (21, 22).
The notion that the underlying FS of a pseudogapped or a superconducting state is identical to the Luttinger surface is only approximately correct (12, 23). In the Fermi liquid state of normal metals, the FS satisfies the Luttinger sum rule; the volume enclosed by the FS is identical to the total number of conducting electrons. But, in a superconductor, the chemical potential is generally renormalized and is a function of the superconducting order parameter, μ = μ _{SC} (Δ). The number of states n_{Lutt} (Δ) enclosed by the resulting Luttinger surface, ξ _{k} ≡ 0, then deviates from the true particle number n, as the results in Fig. 3 show. However, this effect is small (a few percent) and unlikely to be discerned experimentally. The discrepancy between n_{Lutt} (Δ) and n vanishes when particlehole symmetry is present. Further, it changes sign when the geometry of the Luttinger surface changes from holelike to electronlike, as seen in Fig. 3.
FS Renormalization
Finally, we focus on the influence of the strong electron–electron interactions on the geometry of the Luttinger surface close to half filling. The Cu–O planes of the HTSC are characterized by a nearest neighbor (NN) hopping parameter t ≈ 300 meV and a next NN hopping parameter t′ ≈ −t/4. These parameters are the bare parameters and determine the dispersion relation in the absence of any electron–electron interaction. On the other hand, true hopping processes are influenced by the Coulomb interaction U ≈ 12 t, leading to a renormalization of the effective hopping matrix elements, Close to halffilling we find t̃ ∝ J = 4t ^{2}/U and t̃′ → 0, i.e., the next NN hopping is renormalized to zero. This behavior is illustrated in Fig. 4. The resulting Luttinger surface renormalizes to perfect nesting. A similar behavior has been observed in recent variational studies of organic charge transfer–salt superconductors (24).
At halffilling the Hubbard model reduces to a spin model with NN J = 4t ^{2}/U and a frustrating next NN J′ = 4(t′)^{2}/U. The groundstate wave function obeys the socalled Marshall sign rule in the absence of frustration, J′ = 0, namely, when the underlying FS is perfectly nested by the reciprocal magnetic ordering vector Q = (π, π) (in units of the inverse lattice constant). Hence, any deviation from the Marshall sign rule as a function of the frustrating J′ can be used to determine the degree of effective frustration present in the ground state. We emphasize this is a qualitative statement of the groundstate wave function. A numerical study has found that the Marshall sign rule remains valid even for small but finite J′, namely, the effective frustration renormalizes to zero (25). Such a behavior is in agreement with the results presented in Fig. 4. This renormalization of the underlying FS to perfect nesting close to halffilling is unique to strong coupling theories such as RMFT.
In summary, we showed that the accurate determination of the underlying FS in underdoped HTSC is a difficult task and that analysis of the experimental data alone is often insufficient for an unambiguous determination of the FS. Commonly used methods like the zerofrequency spectral intensity or the gradient of n_{k} can yield significant deviations from the true Luttinger surface as shown in Figs. 1 and 2. Indeed, a clear distinction between electron and holelike underlying FS cannot be made solely from analyses of spectral intensity maps when the gaps are large. Such analyses have to be supplemented by a minimal modeling of the gapped states. Furthermore, the underlying FS in the pseudogapped or superconducting state fulfills Luttinger theorem only approximately, owing to the dependence of the chemical potential on the superconducting gap. We also demonstrated that the strong correlations renormalize the ratio t̃′/t̃ near halffilling, yielding a Luttinger surface that is perfectly nested. This result suggests in a very natural way that the strong coupling mean field superconducting state is unstable to antiferromagnetism at low doping. Our findings resulting from the combined effects of strong correlations and dwave superconductivity allow for a more precise interpretation of experiments that determine the FS of HTSC.
Note in Proof
Note.
After submission of this article, we learned that Sensarma et al. (26) recently obtained similar results.
Footnotes
 ^{§}To whom correspondence should be addressed. Email: pwa{at}princeton.edu

Author contributions: C.G., B.E., V.N.M., and P.W.A. performed research.

The authors declare no conflict of interest.
 Abbreviations:
 FS,
 Fermi surface;
 ARPES,
 angleresolved photoemission spectroscopy;
 HTSC,
 hightemperature superconductor;
 RMFT,
 renormalized mean field theory;
 NN,
 nearest neighbor.
 © 2006 by The National Academy of Sciences of the USA
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