Nodal ARPES data: power-law analysis with k-independent self-energy
In Fig.1 we show the imaginary part of the nodal self-energy of (Pb,Bi)2Sr2−xLaxCuO6+δ over a large range in frequency, doping and temperature. The commonly adopted assumption is made that the bare dispersion is linear, ε(k) = vF(k − kF), with vF being the Fermi velocity and kF the Fermi wave vector. Under the crucial, additional assumption of negligible k dependence for the self-energy - Σ(k, ω) ≃ Σ(ω) - the single-particle spectral function reduces to a symmetric, Lorentzian lineshape as a function of momentum k at each fixed frequency ω, i.e., \(L(k)=\frac{W}{\pi }\frac{\Gamma /2}{{(k-{k}_{*})}^{2}+{(\Gamma /2)}^{2}}\). Here W(ω) is the intensity, k*(ω) the peak position and Γ(ω) its width. For the results presented in the rest of this paper, W(ω) is not a key parameter and will not be discussed further. In practice, the peaks are better fit using a Voigt lineshape, the Gaussian part of which accounts for experimental resolution. In this framework, the imaginary part of the self-energy is then proportional to the width of the Lorentzian component as Σ″(ω) = vFΓ(ω)/214.
a–e Temperature-dependent self-energies from ARPES for five different doping levels, extracted using symmetric Voigt fits to MDCs, plotted using colour-coded solid lines for temperature. The dashed, black lines are results to two-dimensional (in T and ω) fits employing three parameters (α, β, λ), using the power-law-liquid formalism introduced in Ref. 6 and given in Eq. (1). Fitting parameters used are indicated in each data panel, and gathered together with the analogous parameters for Bi-22126 in (h). The red dashed lines indicate the marginal Fermi liquid (α = 0.5 at optimal doping), which is shown above the canonical Presland dome (i) used to determine the doping level of the measured samples18. g Shows a typical measured ARPES dataset, containing both the positive-k and negative-k nodal branches, along the k-space direction indicated in the schematic Fermi surface in (f). Supplementary Table1 in the Supplementary Note7 lists all temperatures measured for each doping level. The error bars in (h) reflect the sensitivity of α, β and λ to details of the fitting procedure, in particular the energy and temperature range over which the fits are performed.
The extraction of the self-energy from these ARPES MDC widths can then be approached in different ways. One is to fit the data using pre-defined scenarios for the (ω, T)-dependence of the self-energy, such as linear- or quadratic-in-(ω, T) behaviour14,15,16. Another is to allow the power-law exponent to take on whichever value best matches the data at that doping level, conducting fits working simultaneously in frequency and temperature space6.
In Fig.1, we show the results of a first benchmarking of our (Pb,Bi)2Sr2−xLaxCuO6+δ ARPES data, adopting the latter approach, in which the self-energy is given by Eq. (1), which contains three dimensionless parameters α, β and λ:
$$\Gamma=\frac{2{\Sigma }^{{''} }(\omega,T)}{{v}_{F}}={G}_{0}(\omega,T)+\lambda \frac{{[{(\hslash \omega )}^{2}+{(\beta {k}_{B}T)}^{2}]}^{\alpha }}{{(\hslash {\omega }_{N})}^{2\alpha -1}}.$$
(1)
Here λ is a coupling constant describing the strength of the interaction, normalised to an energy scale ℏωN = 0.5eV for all dopings and the parameter β sets the balance between the relative influence of temperature and frequency. G0(ω, T) is an extra term, combining a self-energy contribution from electron-phonon coupling, that is most clearly seen at low T around an energy of 70 meV, with impurity scattering, described in Supplemenary Note1.
Note that Eq. (1) -which has been dubbed the power-law liquid, or PLL -describes the marginal Fermi liquid17 with a power of unity (α = 1/2) at optimal doping and the quadratic temperature and frequency behaviour of a Fermi liquid emerges when α = 1.
Figure1a–e show the experimentally extracted self-energies (coloured lines), together with the result of a two-dimensional (ω, T) fit to the data using Eq. (1) (dashed lines). An overview of the parameters extracted from the fits is given in Fig.1h, together with the parameters previously determined for Bi-22126. Compared to the latter, the ARPES data presented here cover a complementary doping range from near optimal doping (p = 0.14) to such overdoping that superconductivity disappears and also includes doping levels on either side of p* 18, showing that neither superconductivity nor the opening of the pseudogap influence the observed trend in the nodal self-energy. Across this doping interval, this analysis shows the exponent to increase smoothly from 1.02 (α = 0.51) - consistent with the marginal Fermi liquid expectation - to 1.68 (α = 0.84) for the highest hole doping measured. This means that even at the edge of the superconducting dome, the quadratic power of the Fermi liquid (α = 1) is not reached.
The remarkable degree of continuity in the α values evident in Fig.1h highlights good agreement between the nodal self-energy behaviour in both single- and bi-layer cuprates.
Asymmetric ARPES MDCs: evidence of k-dependent power-law exponents
Crucial to the analysis just carried out is that the MDC lineshape of each nodal branch is a symmetric Lorentzian over an extended range of frequency, signalling negligible k dependence of the self-energy. In Fig.2a–d we show an examination of the Lorentzian fits in detail.
a–c A trio of MDCs at energies indicated measured at 8K, with symmetric peak fits in red. The residuals grow as the binding energy grows. Here (d) shows the resulting 2Σ″(ω)/vF = Γ(ω) from the symmetric fits. e–g The same three MDCs now fit in purple using our model given by Eq. (3), including the holographically-predicted k dependence, with V set to −1. The residuals are clearly superior to the symmetric fit for energies further from EF. h The imaginary part of the self-energy 2Σ″(k, ω)/vF = ΓH(k, ω) at k*(ω) (red), which includes the k-dependent self-energy (blue) and the free fitting parameter G0 (green).
Figure 2a shows that a Lorentzian spectral function gives a very good fit close to EF, and yields a small residual. However, at higher binding energy, such as at 200 and 300 meV, the foot of the peaks clearly show that the data (black crosses) are not adequately captured using two symmetric peaks (red lines). The experimental bottom line is that the MDC peaks in Fig.2b, c are asymmetric: showing more spectral weight at large momenta ∣k∣ > ∣k*(ω)∣, compared to ∣k∣ < ∣k*(ω)∣, thus leading to residual values of differing signs inside and outside the MDC peak pair.
In theSupplementary Information, we discuss in turn a number of possible ‘trivial’ sources that could lead to non-Lorentzian MDCs. These include non-zero curvature of an underlying ‘bare band’ (Supplementary Note2), averaging over local doping variations or local facetting at the sample surface (Supplementary Note3), inhom*ogeneous detector response (Supplementary Note4), or contributions from background signals (Supplementary Notes5 and 6). None of these are able to adequately reproduce the observed asymmetry of the MDC lineshapes (reduced/increased intensity on the low/high ∣k∣ side of the peak) and that this asymmetry grows with k − kF. Therefore, after this due diligence, we propose a simple yet profound explanation of the MDC asymmetry as being an expression of k dependence of the electronic self-energy itself.
One model-independent approach to describe this would be to linearly expand the measured self-energy around kF, reported in Supplementary Fig.10. This significantly enlarges the set of fitting parameters, by means of an additional frequency and temperature dependent function. In the following we show that theoretical input provided by AdS/CFT enables the data to be fitted across the full frequency range as proficiently as those near EF could be, all without any expansion in the number of free parameters.
Semi-holographic theoretical description
Physically, a semi-holographic model describes an electron that interacts with a CFT (accounting for a quantum critical state deformed by non-zero T and chemical potential) via linear coupling to a fermionic operator \({{{{{{{\mathcal{O}}}}}}}}\) that has a unique scaling dimension. As a result, the self-energy becomes proportional to the correlation function \(\langle {{{{{{{{\mathcal{OO}}}}}}}}}^{{{{\dagger}}} }\rangle (k,\omega,T)\) in the CFT. The latter can be determined from the holographic dictionary7,8, and automatically inherits certain scaling properties from the ‘critical’ CFT. To be applicable to the cuprate strange metals, the model must have a dynamical critical exponent z → ∞ in order to recover the behaviour that is well-described by Eq. (1) when k is close to kF. From this imposition of local quantum criticality, there follows a fundamental condition from the theory side that the scaling exponents have to be momentum dependent19.
Here, for the CFT we use an Einstein-Maxwell-Dilaton model of holography, specifically the Gubser-Rocha model20,21, as it offers an analytical treatment of the gravitational spacetime. In the long-wavelength limit, within the framework of an emergent particle-hole symmetry, our semi-holography model then gives at T = 0:
$$\frac{2{\Sigma }^{{''} }(k,\omega )}{{v}_{F}}=\lambda \frac{{[{(\hslash \omega )}^{2}]}^{\alpha (k)}}{{(\hslash {\omega }_{N})}^{2\alpha (k)-1}}\,,$$
(2a)
$$\alpha (k)=\alpha \left(1-\frac{k-{k}_{F}}{{k}_{F}}\right).$$
(2b)
The semi-holographic self-energy can also be generalised to non-zero temperature19,21, and under our conditions is well approximated by replacing (ℏω)2 with \({(\hslash \omega )}^{2}+{(\beta {k}_{B}T)}^{2}\) in Eq. (2a). This also eases comparison to the PLL in Eq. (1), and highlights the key new insight that as frequency increases, and k ≃ kF − ω/vF departs from kF, a k dependence emerges in the exponent describing the (ω, T)-dependence of the self-energy. For low frequencies, where k ≃ kF, one returns to Eq. (1) of the PLL. The behaviour predicted by holography should leave a clear experimental fingerprint, namely that the ARPES MDCs are asymmetric: the data shown in Fig.2 show this is the case.
Figure3a shows a simulated spectral function, generated using the semi-holographic self-energy shown in Fig.3c, d. The resulting MDC asymmetry at non-zero energy is illustrated in Fig.3b, visible as an increased intensity at the ∣k∣ > ∣k*∣ side of the peak maximum, just like in the experimental data of Fig. 2c. The next step is to test whether this holographic approach can yield superior quantitative fits to the ARPES data.
a Spectral function with a linear bare band (vF = 4 eV Å, kF = 0.45 Å−1), convoluted with the experimental k resolution and (b) MDC at non-zero binding energy generated using the full ω and k-dependent self-energy plotted in (c, d) from the semi-holographic model at T=20 K. Note that (c) includes the imaginary part of a k-independent phonon self-energy.
Fits to the ARPES data inspired by the semi-holographic model
As non-Lorentzian MDCs force a major departure from conventional ARPES data analysis methodology, we now describe how to deal with this in the analysis of real data from (Pb,Bi)2Sr2−xLaxCuO6+δ. The non-zeroT version of the self-energy with the k dependence in Eq. (2b), suggests a modified fit-function LH(k) at the fixed ω and T relevant for each MDC:
$${L}_{H}(k)=\frac{W}{\pi }\frac{\frac{{\Gamma }_{H}}{2}}{{(k-{k}_{*})}^{2}+{\left(\frac{{\Gamma }_{H}}{2}\right)}^{2}},$$
(3a)
$${\Gamma }_{H}(k)={G}_{0}+\lambda \frac{{[{(\hslash \omega )}^{2}+{(\beta {k}_{B}T)}^{2}]}^{\alpha (k)}}{\hslash {\omega }_{N}^{2\alpha (k)-1}},$$
(3b)
$$\alpha (k)=\alpha \left(1+V\left[\frac{k-{k}_{F}}{{k}_{F}}\right]\right).$$
(3c)
ΓH(k) in Eq. (3b) captures both the peak width and its asymmetry via momentum dependence built into the exponent α(k) in Eq. (3c). Moreover, the Gubser-Rocha holographic model discussed above inEq. (2), actually requires that V = −1 for all frequencies. Therefore by fixing α, β, λ and ωN to the PLL values at low energies and V to the value in the Gubser-Rocha model, only G0 in Eq. (3b) remains free to vary in the fitting process for each MDC. Then the resulting fit function in Fig.2e–g has exactly the same number of free parameters as the PLL6.
We reiterate that the k-dependence of the exponent α describing the (ω, T)-dependence of the self-energy in Eqs. (3b, 3c) used to fit to the ARPES datafollows directly from the semi-holographic prediction given in Eqs. (2a, 2b), itself a consequence of the requirement of local quantum criticality19.
Comparing Fig.2b, c (PLL, k-independent) with Fig.2f, g (k-dependent scaling exponents), it is clear that on adopting Eq. (3), the residuals in the panels f and g for energies well below EF drop, becoming as low as they were for the PLL at EF. In particular, the zoom to Fig.2c, g illustrates clearly that the asymmetry of the MDC peaks is now captured almost perfectly. We emphasise that the non-zero value of V, which accounts for the experimentally observed MDC asymmetry, is a non-trivial result and is deeply rooted in the holography. Figure2h shows the total self-energy along the loci of the MDC peak-maxima in red, with the holography-inspired, k-dependent part in blue and the free fitting parameter G0(ω, T) in green. The latter can be seen to automatically take on the combined form of an offset (impurity scattering) plus a step function centred at the phonon energy of 70 meV.
Testing the semi-holographic prediction
Having determined the free parameters in the Gubser-Rocha model (see Methods), the validity of the fits using Eq. (3) can be tested. For the Gubser-Rocha model adopted here, the asymmetry parameter should have a frequency-independent value of V = −1. The experimental data can now be used to test this in a second set of fits, now leaving V as a free MDC-fitting parameter for each ω. Figure3 shows that—with no guiding restraint at all applied to V(ω)—the experimental data yield for all dopings, temperatures (see Supplementary Fig.9) and frequencies a value of V in the vicinity of −1, which is close to the semi-holographic prediction.
The Gubser-Rocha model also yields values for β. In Fig.1h, the experimental β value shows a lightly upward trend vs. hole doping in an interval between β = 3–4 for the doping levels studied. The Gubser-Rocha model itself (see Supplementary Fig.9d) predicts a similar trend vs. doping, but with lower β values running between 2 and 3. A detailed derivation of parameters such as β falls outside the experimental focus of this paper, and will be presented in a separate publication.
There is still some room for improvement, in particular as regards β output by the Gubser Rocha semi-holographic model. The spectral function fits to the ARPES data that are so successful in Fig.2e–g used β values from the PLL parameterisation (not those directly predicted by the Gubser Rocha model), and the Gubser-Rocha inspired V = −1 in Eq. (3) admirably captures the asymmetry of the MDCs and when tested in Fig.4 is found to be a stably representative value. These ARPES data and their parametrization presented here are an invitation to the many existing z = ∞ holographic models besides the analytical Gubser-Rocha model—or to completely different theoretical methods—to take the next step in capturing the modest frequency dependence of V seen in experiment, and the observed β values. We stress however that according to our experimental data, the non-zero value of V which underpins the observed MDC asymmetry must be accounted for by any theoretical approaches aiming to describe the spectral function of the nodal charge carriers in cuprates.
Asymmetry parameter V of the momentum-dependent scaling exponent extracted from the experimental ARPES data over the full doping range measured at 8K, obtained by performing MDC fits to the data using Eq. (3), where the leading scaling exponent α, the coupling constant λ and the quantity \({(\hslash \omega )}^{2}+{(\beta {k}_{B}T)}^{2}\) are fixed. A value V = 0 means there is no asymmetry, and V = −1 is the prediction from the semi-holographic Gubser-Rocha model. Data from analogous fits at high temperature are shown in Supplementary Fig.9.
Before providing a discussion of other contexts in which k-dependent self-energies have been discussed, we re-iterate our main results. In the ARPES data of nodal carriers in the normal state of the single-layer cuprate (Pb,Bi)-2201, asymmetric, non-Lorentzian MDCs are observed for energies away from EF. As Supplementary Notes2–6 show, we cannot explain this due to bare band non-linearity, doping inhom*ogeneity, surface roughness or technical problems with detector or backgrounds. Instead, analysing these in terms of the ω and T dependence of the electronic self-energy, an excellent fit for each doping level is obtained using a single power law with a k-dependent scaling exponent. Given our experimental data, power-law scaling is well supported over a significantly wider energy and temperature range than in ref. 6, and for the whole overdoped portion of the phase diagram. At k = kF, the k-dependent power-law exponent is found to take a nearly marginal Fermi-liquid value of unity (α = 0.51) at optimal doping, growing to 1.68 (α = 0.84) for the most overdoped, non-superconducting system studied. For k = kF, these results connect smoothly to and greatly extend those of Bi-2212 in Ref. 6. Significantly, for values k ≠ k − kF these powers change. Such a k dependence is a key prediction from (semi)holographic models of strange metals as a quantum critical phase, and the excellent qualitative agreement between the experimental data and these family of models provides a benchmark for future work. In particular, any competing theoretical model for strange metals must be able to account for the observed ARPES MDC peak-asymmetry that is well described here by a momentum-dependent power-law self-energy.
In the cuprate context, numerous ARPES studies focus on the k dependence of the self-energy in terms of the position on the FS, both in the normal15,22,23,24 and superconducting states25,26,27. A dependence of the self-energy on k − kF has been linked to the doping dependence of the high-energy velocity of the nodal states in ARPES28, that has been argued to possibly arise from strong electron-phonon coupling29. Moreover, asymmetric MDCs in ARPES data from LSCO have been proposed to be due to a k-dependent self-energy in phenomenological models based on ‘extremely correlated Fermi liquids’30 or have been proposed to be a more general signal of strong correlations31.
In a different condensed-matter realisation, k-dependent power laws have been observed in the zero-bias conduction anomaly in transport spectroscopy of nanowires32 and linked to non-linear Luttinger liquid theory33. One might therefore wonder if there is a link between the 1D physics in the nanowire case, and the carriers in the nodal k-space direction of the quasi-2D (Pb,Bi)-2201 system that seem well described by the 1D CFT encoded semi-holographically in the AdS2 infra-red geometry of the z = ∞ models.
The results presented here herald future comparisons of the experimentally-tested, theoretical spectral function to other experimental probes such as optical and DC conductivities34. Alternatively, many-body condensed-matter theories can be guided by both the experimental self-energies presented here and the results from the AdS/CFT analogue, opening up possible pathways out of phenomenology and into a microscopic understanding of strange metals.
