Gauge-gravity duality comes to the laboratory: Evidence of momentum-dependent scaling exponents in the nodal electron self-energy of cuprate strange metals

We show that the momentum-dependent scaling exponents of the holographic fermion self-energy of the conformal-to-${\mathrm{AdS}}_2$ Gubser-Rocha model can describe new findings from angle-resolved photoemission spectroscopy experiments on a single-layer (Pb,Bi)${}_2{\mathrm{Sr}}_{2-x}{\mathrm{La}}_x{\mathrm{CuO}}_{6+\delta }$ copper oxide. In particular, it was recently observed in high-precision measurements on constant energy cuts along the nodal direction that the spectral function departs from the Lorentzian line shape that is expected from the power-law-liquid model of a nodal self-energy, with an imaginary part featureless in momentum as ${\mathrm{\Sigma }}_{\text{PLL}}^{\prime \prime }\left(\omega \right)\propto {\left({\omega }^2\right)}^{\alpha }$. By direct comparison with experimental results, we provide evidence that this departure from either a Fermi liquid or the power-law liquid, resulting in an asymmetry of the spectral function as a function of momentum around the central peak, is captured at low temperature and all dopings by a semiholographic model that predicts a momentum-dependent scaling exponent in the electron self-energy as $\mathrm{\Sigma }(\omega ,k)\propto \omega {(-{\omega }^2)}^{\alpha (1-(k-k_F)/k_F)-1/2}$, with $\hslash k_F$ the Fermi momentum.

Figure 1

Comparison of the smoothed experimental data (blue dots) for an overdoped sample with $\alpha =0.65$ [see Eq. (2)] at $T=8\text{K}$, with a Lorentzian fit based on the PLL (green line), and a fit based on the holographic prediction in Eq. (2) (red line). While there is little difference up to energies $E-E_F=-0.1\text{eV}$, (left panel), it is evident that the holographic model accurately captures the peak asymmetry as we move further away from the Fermi surface (right panel). The details of the fit are presented in Sec. 5.

Figure 2

Peak in the fermionic spectral function at the Fermi energy, normalized to peak height, and a zoomed-in version (top right), showing the presence of a Fermi surface. Here we used $m=-0.49, q=0.27$ at $T/\mu =0.0045$.

Figure 3

Spectral function for a fermionic operator computed from holography. It is symmetric in momentum and it shows a linear dispersion expected for a massless fermion, with the cone shifted down by the chemical potential. Here we used $m=-0.49$ and $q=0.27$.

Figure 4

Numerical results for the dependence on the charge $q$ of the quantities determining the Green's function near the Fermi surface. We fixed the mass $m=-0.49$ and we used a value $k_{B}T/\mu =4.5\times {10}^{-3}$.

Figure 5

Imaginary part of the numerical self-energy (circles) for different values of fixed momentum and temperatures in the range of interest, together with a fit to the analytical low-energy result (lines). We can clearly see that the low-energy behavior of the imaginary part of the self-energy is indeed well described by the ${\text{AdS}}_2\times {\mathbb{R}}^2$ solution.

Figure 6

Example of the momentum dependence of the coefficient $\mathcal{C}$ resulting from a fit of the numerical solution at fixed temperature $T/\mu =0.005$, with a function of the form of Eq. (50) with an energy-independent coefficient $\mathcal{C}(\omega ,k,T)\simeq \mathcal{C}(0,k,T)$. It shows that the latter coefficient is linear in $k/\mu$ as expected from the argument in the main text, while temperature corrections are quadratic and thus negligible at sufficiently low temperatures where $\mathcal{C}(\omega ,k,T)\simeq \mathcal{C}(0,k,0)$.

Figure 7

With (semi)holography we obtain the spectral function for the holes (right panel). In an emergent particle-hole symmetry near the Fermi surface, we relate the ARPES MDC along the nodal line for the electrons in a cuprate at $E<E_F$, with momentum origin at the (0,0) point with the holographic distribution at the corresponding $\hslash \omega /\mu =(E_F-E)/\mu >0$ (dashed red line). The origin of the momentum axis is at $2\hslash c{k}_F/\mu$.

Figure 8

Structure of the Fermi surface. The origin of the momentum axis is set at the (0,0) point, with measurements performed along the nodal line. The holographic model describes the response of the hole system from the $(\pi ,\pi )$ point, within an emergent particle-hole symmetry where the $(\pi ,\pi )$ point is assumed at $2k_F$ along the nodal direction.

Figure 9

Plot of the dispersion relation from ARPES measurements along the nodal direction at optimal doping and low-temperature (blue line), and comparison with a linear dispersion $E-E_F=\hslash v_F(k-k_F)$, with $v_F=2.9\text{eV}\text{Å}$ (red line). We can see a kink in the linear dispersion around $E-E_F=-0.07\text{eV}$.

Figure 10

Comparison of the (smoothed) experimental MDC (blue dots) for a sample at optimal doping $\alpha =0.51$ at $T=8\text{K}$, with a Lorentzian fit based on the PLL (green line), and a fit based on the semiholographic prediction for the self-energy in Eq. (62) (red line). Near the Fermi surface both models provide a good fit, but as we move away from the Fermi surface the holographic model provides a better fit, being able to capture the asymmetry in the peak. Deep below the Fermi surface, for $E-E_F\lesssim -0.25$, we start to see deviations even for the semiholographic model. This is, however, where we expect the low-energy approximation to break down, as is also indicated by the dispersion becoming nonlinear in Fig. 11.

Figure 11

(Left) Fit of $G_0\left(\omega \right)$ in Eq. (59) to the electron-phonon model in Eq. (60) for a PLL fit function (green line), giving $G_{\text{ph}}\simeq 0.020$ and $\mathrm{\Omega }\simeq 0.034$, and the semiholographic one (red line), that gives $G_{\text{ph}}\simeq 0.031$ and $\mathrm{\Omega }\simeq 0.030$ (it can be seen that there is a deviation at very small energy, probably related to some leftover disorder). (Right) Comparison between the experimentally observed dispersion relation (blue dots) with the dispersion as expected from the electron-phonon model given the parameters $G_{\text{ph}}$ and $\mathrm{\Omega }$ obtained from the fit. We see that the semiholographic model provides a better fit to the dispersions with $\hslash v_F\simeq 2.9\text{eV}\text{Å}$, while the PLL fit does not properly capture the phonon kink and gives $\hslash v_F\simeq 2.7\text{eV}\text{Å}$.

Figure 12

Comparison of the PLL fit (green line) and semiholographic fit (red line) to MDC data (blue dots) for an overdoped sample with $\alpha =0.61$ at $T=8$ K. Near the Fermi surface, both models provide a good fit to the data, while further away from it the semiholographic model accounts for the asymmetry in the peak.

Figure 13

(Left) Fit to $G_0\left(\omega \right)$ with the electron-phonon model and comparison with the dispersion relation (right), for the overdoped sample with $\alpha =0.65$. We see here as well that the semiholographic fit function provides a description consistent with the electron-phonon model.

Figure 14

Details of the PLL fit (green line) and semiholographic fit (red line) to MDC data blue dots for an overdoped sample with $\alpha =0.82$ at $T=8$ K, both near and deep below the Fermi surface.

Figure 15

(Left) Fit to $G_0\left(\omega \right)$ with the electron-phonon model and comparison with the dispersion relation (right), for the overdoped sample with $\alpha =0.81$.

Figure 16

Plot of the ratio ${({\mathrm{\Sigma }}^{\prime \prime }(\omega =0,k_{B}T=\epsilon ,k_F)/{\mathrm{\Sigma }}^{\prime \prime }(\omega =\epsilon ,k_{B}T=0,k_F))}^{1/2\alpha }$ from the analytical semiholographic model (blue line), and from the full numerical solution (red line), where there are corrections dependent on the mass $m$ in the bulk Dirac equation. In the PLL this ratio is the fit parameter $\beta$, with experimentally determined values between 3 and 4.

Figure 17

Comparison between the Fermi-Dirac distribution (red solid line) and the imaginary part of the approximation in Eq. (66) (black dashed line) with $\mathrm{\Omega }=0$.

Figure 18

Plot of the real part of the approximation to the electron-phonon contribution in Eq. (66), with $\mathrm{\Omega }=0$. We can see a divergence at the phonon frequency $\hslash {\omega }_{\text{ph}}=0.07\text{eV}$ as we approach zero temperature.

Figure 19

Comparison of the PLL fit (green line) and semiholographic fit (red line) to MDC data (blue dots) for an optimally doped sample $\alpha =0.51$ at high temperature $T=205$ K. Here, also added for comparison, is the “holographic inspired” model of Ref. [38] (blue line). We see that the semiholographic model still provides a better fit than the PLL to the asymmetric peak far from the Fermi surface, however, this implies a $G_0\left(\omega \right)$ that cannot be simply described by the electron-phonon model as shown in Fig. 20 below.

Figure 20

(Left) Results for $G_0\left(\omega \right)$ in Eq. (59) from the fit of the MDC data to a PLL fit function (green dots), the “semiholographic inspired model” of Ref. [38], (blue dots), and the semiholographic model (red dots) and comparison with the prediction from the high-temperature generalization of the electron-phonon model (solid lines) as in Eq. (66), with the parameter $G_{\text{ph}}$ and $\mathrm{\Omega }$ as obtained from the fit at low-temperature. While we see that in all three cases away from the Fermi surface there are contributions to $G_0\left(\omega \right)$ that are not explained by the simple model of electron-phonon interactions, the semiholographic model also shows deviations near the Fermi surface, as expected, due to the different prediction of the temperature behavior of the electron self-energy. The deviations from the electron-phonon model are also evident in the discrepancy between the expected and measured dispersion (right).