On the break in the single-particle energy dispersions and the ‘universal’ nodal Fermi velocity in the high-temperature copper oxide superconductors

Abstract

Recent data from angle-resolved photoemission experiments published by Zhou et al. [Nature, 423, 398 (2003)] concerning a number of hole-doped copper-oxide-based high-temperature superconductors reveal that in the nodal directions of the underlying square Brillouin zones (i.e. the directions along which the d-wave superconducting gap is vanishing) the Fermi velocities for some finite range of k inside the Fermi sea and away from the nodal Fermi wave vector k _F are to within an experimental uncertainty of approximately 20% the same both in all the compounds investigated and over a wide range of doping concentrations and that, in line with earlier experimental observations, at some characteristic wave vector k _☆ away from k _F ( typically amounts to approximately 5% of ) the Fermi velocities undergo a sudden change, with this change (roughly speaking, an increase for ) being the greatest (smallest) in the case of underdoped (overdoped) compounds. We demonstrate that these observations establish four essential facts: firstly, the ground-state momentum distribution function must be discontinuous at k = k _☆; with and denoting the measured velocities close to and , we obtain

in which is the ‘Fermi’ velocity corresponding to the case in which (for two-body interaction potentials of shorter range than the Coulomb potential, whereby ); secondly, the single-particle spectral function must at k = k _☆ possess a coherent contribution corresponding to a well-defined quasiparticle excitation at an energy of approximately 70 meV below the Fermi energy; thirdly, the amount of discontinuity in at the nodal Fermi points must be small and ideally vanishing; fourthly, the long range of the two-body Coulomb potential is of vital importance for realization of a certain aspect of the observed behaviour in the Fermi velocities (specifi- cally) in the underdoped regime. The condition conforms with the observation through an earlier angle-resolved photoemission experiment by Valla et al. [Science, 285, 2110 (1999)], on the optimally doped compound Bi₂Sr₂CaCu₂O_{8 + δ}, which shows that the imaginary part of the self-energy along the nodal directions of the Brillouin zone and in the vicinity of the nodal Fermi points satisfies the scaling behaviour characteristic of marginal Fermi-liquid metallic states, for which is indeed vanishing. We present arguments advocating the viewpoint that the observed ‘kink’ in the measured energy dispersions cannot be a direct consequence of electron-phonon interaction, although a finite may possibly arise from this interaction. In other words, even though possibly vital, the role played by phonons in bringing about the latter ‘kink’ must be indirect. Our approach further provides a consistent interpretation of the observed sudden decrease in the width of the so-called ‘momentum distribution curve’ on ‖k| increasing above .

Acknowledgements

With pleasure I thank Professor Zhi-xun Shen for kindly communicating to me the experimental observations by Zhou et al. (2003) prior to publication, Professor Philip Stamp for bringing the papers by Eskes et al. (1991, 1994) and Eskes and Oleś (1994) to my attention and Professor Fei Zhou (Vancouver) for discussion. With appreciation I acknowledge hospitality and support by Spinoza Institute.

Notes

† On the basis of experimental observations, it has even been advocated (Randeria et al. 1995, Campuzano et al. 1996) (see, however, the cautionary remarks in reference 8 of (Campuzano et al. 1996)) that the Fermi surface be defined as the locus of points in the k space at which is maximal. This definition has been utilized by Putikka et al. (1998) where the pertaining to the GS of the t–J Hamiltonian has been calculated from a high-temperature series expansion (see also Farid (2003)).

† See, for instance, equations (6) and (39) of the paper by Farid (2004).

‡ See, for example, equation (40) in the paper by Farid (2004).

§ For reasons that we have presented earlier (Farid 1999a, 2003), one can identify with (i.e. can be considered as a proper subset of FS_σ); on the other hand, strict distinction has to be made between and k _F; σ. The same applies to k _☆ that we frequently encounter in the text. See §§ 3.1 and 3.2.

† For completeness, for a given there also exists a set , a generic point of which we denote by (see the previous footnote), for which equation (3) is satisfied at , with infinitesimally greater than ϵ_F. See §§ 3.1 and 3.2.

‡ See equation (227) in the paper by Farid (2002a) and equation (A 5) in the paper by Farid (2004). By the assumption of the stability of the underlying GSs, is non-negative.

§ It is evident that, by defining as the expectation value of, for instance, , no need for the introduction of an energy cut-off would arise; this is, however, at the expense of forsaking the simplicity of the expression for corresponding to the standard variance of ϵ.

† We have in various papers elaborated on this subject matter, for which we refer the reader to the paper by Farid (2002a) (§ 3.4) and the references therein. Here and in the following when stating that, for an arbitrary k, in general, equation (3) does not possess a solution, we do not thereby consider the possible solutions of this equation on the non-physical Riemann sheets whose determination requires the knowledge of the analytically continued into these Riemann sheets (for example Farid (1999b)).

† In general, one may have where, for the same reasons that (equation (10)), and are interdependent. As considerations involving only unnecessarily complicate the analysis, in this paper we explicitly deal with cases where .

‡ Since by the arguments presented in the text following equaiton (12) we have , it follows that, in the event that and would differ only by an infinitesimal amount, k _☆ would either be a point of or infinitesimally close to it.

† With reference to our remark in the first footnote in § 3.2, we note that, for cases where has the more general form as presented in that footnote, the counterpart of the result in equation (16) would in principle differ from that which stands in equation (16). However, the considerations in § 3.6 demonstrate that the corresponding to can at most be equal to unity. In other words, equation (16) is more general than the simple form for in equation (9) would suggest. For completeness, in equation (16), , where is the unit vector normal to at , pointing from the inside to outside the Fermi sea.

† With reference to our remark in the first footnote in § 3.2, we point out that equation (27) is specific to cases in which .

† Farid (2003) obtained, for isotropic Fermi-liquid metallic states of fermions of bare mass m _e interacting through a short-range potential, , where is the renormalized (or effective) mass. For the sole purpose of illustration, making the assumption that and employing the data concerning and , corresponding to the Coulomb-interacting homogeneous electron-gas system, as presented in respectively tables 5.6 and 5.7 of the book by Mahan (1990), from for (r _s; λ) we obtain (0;1) (exact), (1;1.12), (2;1.16), (3;1.19) and (4;1.22). Here r _s is the dimensionless Wigner–Seitz density parameter, with r _s → 0 corresponding to the uncorrelated limit.

† With reference to our remark in the first footnote in § 3.2, this implies that, for .

† To this end, one has to use equation (18) in the paper by Farid (2004), which holds for all k, and the fact that and , the Hartree-Fock self-energy, are continuous for all k.

† Let denote the unit vector normal to at pointing to the exterior of the Fermi sea FS_σ. Since, for k in a close neighbourhood of k _F; σ, one has for and for , one observes that, in cases where is divergent for , some useful information can be immediately deduced from equation (54). For instance, for cases in which the interaction potential is as long-ranged as the Coulomb potential and is discontinuous at k = k _F; σ, one can explicitly show that as so that, from equation (54), one directly deduces that, firstly, , for and, secondly, the balance between these diverging contributions must be such that the resultant function on the LHS of equation (54) approaches +∞ as k → k _F; σ, irrespective of whether or .

† In this connection, we point out that according to Belyakov (1961) (see also Sartor and Mahaux (1980)) logarithmically diverges as k → k _F; σ for pertaining to the isotropic GS of fermions interacting through a short-range potential. For some relevant details see the paper by Farid (1999a, § 6).

† This aspect is related to the restriction 0 < γ ≤ 1 for the parameter γ introduced by Farid (2003); γ > 1 is excluded for cases where a _σ > 0.

† It is important to realize that, in contrast with the case where a possible discontinuity in at corresponds to (see equation (16)), there is no a priori restriction on the sign of (viewed as the value of and not as spectral weight) for k _☆ outside . This is appreciated by realizing the fact that all excitations corresponding to the wave vector k _☆ outside can, by the very definition of , only correspond to many-body states whose energies are greater than the energy of the GS of the system under consideration, whereby a possible negative cannot signify an instability of the latter GS. This aspect is already reflected in the fact that, for k _☆ located at a finite distance from , the choice of what we denote by and is in principle arbitrary. That in our present considerations , is related to the fact that we have chosen to be the closer of the two vectors and to the nodal Fermi point and that, in order for , or , to attain the required value ϵ_F at k = k _F; σ, it must (monotonically) increase for k transposed from k _☆ to k _F; σ.

† We point out that the significance of as the subscript of lies in the finite difference between (which lies strictly below μ) and (which lies strictly above μ) for k _☆ located at some finite distance from (see § 4.3) and not in the possibility that is a discontinuous function of k at . In view of the latter, we should emphasize that far from dismissing a discontinuity in , at , as a priori infeasible, our statement here only reflects the confines of our considerations in this paper. We hope to return to this subject matter in a future publication.

† That is, it is linear, but its extension does not pass through the ‘origin’ (1,1) in figure 2. As we shall discuss in some length in § 5 below, interestingly the experimental energy dispersions in figure 1 (a) of the paper by Zhou et al. (2003), when linearly extrapolated from the region (i.e. at ‘high’ binding energies) to k = k _F; σ, all turn out to meet the energy axis at 75 meV above the Fermi energy ϵ_F.

‡ With and (for the specification of the quantities encountered here, see the caption of figure 3), we have employed the following : for ; for ; for , where E ₁(z) is the exponential–integral function and q ₀ > 0 is a constant parameter of dimensions of reciprocal metre.

† With reference to our remarks in the first footnote in § 3.2, here we are implicitly assuming that .

‡ Below (as well as elsewhere in this paper) the in denotes left/right gradients of , defined as the limit of for k approaching k _☆ from the left/right; here left/right is defined by . For the definition of see text preceding equation (72) above.

† Note that here , stands for an entire symbol, that is does not on its own denote an independent operation.

† See the second footnote in § 4.1. Further, for a simple model, such as the electron-gas model, this aspect can be demonstrated by explicit calculation.

† It is interesting to note that the as reported by Paramekanti et al. (2001, 2003) (along the nodal direction of the 1BZ of a square lattice and corresponding to a projected variational wave function for U/t = 12 and t′ = t/4) unequivocally violates the result . Following the analysis by Farid (2003) and in full conformity with the conclusion arrived at herein, we observe that the underlying metallic GS is not a Fermi liquid.

† For the uniform GSs of the single-band Hubbard Hamiltonian, which are dealt with by Randeria et al. (2003), , where is the number of fermions with spin index (the index complementary to σ) per site and U is the on-site interaction energy. The above-mentioned nearly-constant function must therefore be close to the latter limiting value.

† Concerning the role envisaged for the electron–phonon interaction in the context of high-temperature superconductivity in the cuprate compounds see Alexandrov and Mott (1994) and for the interplay between electron–electron and electron–phonon interactions see Kulić (2000).

‡ For completeness we mention that we fail to comprehend the methodology with the aid of which Manske et al. (2003) deduced from their computational results concerning the single-particle spectral function (denoted by Manske et al. (2003) by N(k, ω)) a kink in the single-particle energy dispersion. In our judgement, connecting the maxima of a series of N(k, ω) corresponding to various k (considered as parameter), as Manske et al. (2003) did (see figures 4–6 of the paper by Manske et al. (2003), noting that in these ω is measured in mili-electron volts and k in units of the inverse lattice constant), cannot provide unequivocal information with regard to the dispersion of the underlying single-particle excitation energies.

§ See, for example, Ashcroft and Mermin (1981, figure 26.1); see also Mahan (1990, figure 6.16) and note that the electron–phonon self-energy is vanishing at , or using the notation adopted by Mahan (1990), at u = 0.

† Schafroth (1951) has, however, shown that, as the theory of Fröhlich is based on a finite-order perturbation theory concerning the electron–phonon interaction (explicitly, a second-order theory), it fails to describe the Meissner effect. For an exposition of the work by Schafroth see Rickayzen (1965, appendix A 1.8).

† Anderson (1969) presented the contents of the Fröhlich model in a modern setting, indicating the diagrammatic representations of the underlying exchange processes. Subsequently, with reference to a Migdal (1958) theorem, Anderson (1969, p. 1346) stated: ‘But in fact Migdal resolved the problem even more completely by showing that Σ is a much more sensitive function of ω than of k–-i.e., it is local in space, retarded in time, thus depends on ω much more sharply–-so that the large correction to comes from ∂ Σ/∂ ω: … and Z = 1/[1 − (∂ Σ/∂ ω)] is always positive: the Fröhlich–Bardeen instability simply does not occur. The complete treatment shows that phonon instability and the first singularity of Z actually occur precisely at the same coupling strength Z’. Two comments are in order. Firstly, the electronic state on which the Migdal theorem at issue is based is that of the non-interacting free-fermion model (this is readily verified through inspecting the paper by Migdal (1958) in conjunction with the paper by Fröhlich (1950)); in other words, it is assumed that the scale of the electronic excitation energies is determined by a single electronic mass parameter. For specifically strongly correlated electron systems there is no a priori reason to believe that the GS of this model would be at all an appropriate starting point for dealing with the problem of the electron–phonon interaction (this is exactly the same problem which in fact hinders a rigorous formulation of a theory concerning the superconducting states of the cuprate compounds whose low-lying single-particle excitations in the normal state are not quasiparticle like (see, however, Chakravarty et al. (1993)), so that the above-mentioned Fröhlich–Bardeen instability cannot on its own rule out the existence of the Fröhlich state. Secondly, it is generally, even though perhaps not universally, accepted that non-adiabatic processes are significant in the copper-oxide-based high-temperature superconductors, leading to the ‘failure’ of the Migdal theorem referred to above (Pietronero et al. 1995). For completeness, we have carried out calculations (Farid 1999c, unpublished) on a uniform two-dimensional model in which the GS is the Fröhlich state. From the first-order self-energy in terms of the dynamically screened interaction function we have deduced quasiparticle lifetimes which are generically by one order of magnitude shorter than the lifetimes of quasiparticles in the conventional two-dimensional model at similar densities. We have further studied (Farid 1999c, unpublished) a variety of the properties of this model in the superconducting state within the Bardeen–Cooper–Schrieffer (BCS) framework, despite the fact that the aforementioned short quasiparticle lifetimes undermine the use of the BCS formalism. For instance, in the weak-coupling limit we obtain that is relatively close to 4 (to be compared with the conventional BCS value of 3.52). We point out that a recent (Boronat et al. 2003) non-self-consistent first-order calculation (in which the effective interaction has been deduced from diffusion Monte Carlo results concerning the static charge and spin structure factors and random-phase approximation Ansätze concerning the dynamic density–density and spin–spin susceptibilities) concerning two-dimensional liquid³He indicates that on increasing the aerial density of ³He atoms the effective mass diverges; a subsequent increase in the density suggests what Boronat et al. (2003) refer to as being indicative of a transition to ‘anomalous occupation numbers’ and which we would refer to as the Fröhlich ‘shell’ state. Interestingly, according to Boronat et al. (2003): ‘Both spin and density fluctuations have profound effects’.

† Here is the Fermi surface corresponding to , in which ε_σ has been chosen such that the interior of contains the same number of k points as contained in the interior of , namely N _σ, the total number of fermions in the GS with spin index σ.

† The RHS of equation (A 2) is simply defined in appendix E of the paper by Farid (2002a) (see in particular equations (E 5), (E 12) and (E 13) therein) of which the second contribution is equal to defined in appendix B of the paper by Farid (2002a, equation (B 29)).

† For time-reversal-symmetric GSs, for which holds, one readily verifies that the RHS of equation (A 9) is invariant under the exchange of r and r′ and consequently so is also the LHS.

† The fact that the single-particle density matrix pertaining to a non-interacting GS is idempotent is associated with the very specific condition where coincides with the characteristic function of the non-interacting Fermi sea (equal to unity inside and equal to zero outside the mentioned Fermi sea). The expressions that are presented in equations (A 20) and (A 21) below are therefore of wider applicability than solely to the Γ⁽²⁾ pertaining to non-interacting GSs.