Origin of kinks in the energy dispersion of strongly correlated matter

We investigate the origin of ubiquitous low-energy kinks found in angle-resolved photoemission experiments in a variety of correlated matter. Such kinks are unexpected from weakly interacting electrons and hence identifying their origin should lead to fundamental insights in strongly correlated matter. We devise a protocol for extracting the kink momentum and energy from the experimental data which relies solely on the two asymptotic tangents of each dispersion curve, away from the feature itself. It is thereby insensitive to the different shapes of the kinks as seen in experiments. The body of available data are then analyzed using this method. We proceed to discuss two alternate theoretical explanations of the origin of the kinks. Some theoretical proposals invoke local bosonic excitations (Einstein phonons or other modes with spin or charge character), located exactly at the energy of observed kinks, leading to a momentum-independent self-energy of the electrons. A recent alternate is the theory of extremely correlated Fermi liquids (ECFL). This theory predicts kinks in the dispersion arising from a momentum-dependent self-energy of correlated electrons. We present the essential results from both classes of theories, and identify experimental features that can help distinguish between the two mechanisms. The ECFL theory is found to be consistent with currently available data on kinks in the nodal direction of cuprate superconductors, but conclusive tests require higher-resolution energy distribution curve data.

Figure 1

A schematic MDC and EDC spectrum displaying typical features of experiments discussed below. Here, $\widehat{k}=(\vec{k}-{\vec{k}}_F)·\vec{\nabla }{\varepsilon }_{k_F}/\left|\vec{\nabla }{\varepsilon }_{k_F}\right|$ is the momentum component normal to the Fermi surface, and we label EDC variables with a star. [The sketch uses parameters $V_L=2\mathrm{eV}\text{Å},V_H=6$ $\text{eV}\text{Å},r=1.5,{\widehat{k}}_{\mathrm{kink}}=-0.03{\text{Å}}^{-1},{\mathrm{\Delta }}_0=0.03\text{eV}$, and ${\mathrm{\Gamma }}_0=0.01\text{eV}$ in Eqs. (3) and (4).] The tangents in the far zones identify the asymptotic velocities $V_L<V_H$ and $V_L^{*}<V_H^{*}$ that characterize the MDC and EDC spectra. The intersection of the extrapolated MDC tangents fixes the kink momentum ${\widehat{k}}_{\mathrm{kink}}$ and the ideal energy $E_{\mathrm{kink}}^{\mathrm{ideal}}$. The dispersion is rounded with raising $T$, as in the lower (red) curve. We define the MDC kink energy $E_{\mathrm{kink}}^{\text{MDC}}$ as $E\left({\widehat{k}}_{\mathrm{kink}}\right)$, i.e., the binding energy measured at the kink momentum, and similarly the EDC kink energy. In all cases, $V_L=V_L^{*}$. A testable consequence of the ECFL theory is that $V_H^{*}$ is fixed in terms of the two MDC velocities by a strikingly simple relation: $V_H^{*}=\frac{3V_H-V_L}{V_H+V_L}\times V_L$ [see Eq. (10)]. This easily testable prediction is tried against experimental data in Fig. 2 where both EDC and MDC data are available. In contrast, the electron-boson theory predicts a jump in the EDC dispersion at the kink energy, followed by $V_H^{*}=V_H$. Note that the difference between the EDC (MDC) kink energy, $E_{\mathrm{kink}}^{\text{EDC}}=E_{\mathrm{kink}}^{\mathrm{ideal}}-{\mathrm{\Gamma }}_0$ and $E_{\mathrm{kink}}^{\text{MDC}}=E_{\mathrm{kink}}^{\mathrm{ideal}}-{\mathrm{\Gamma }}_0\sqrt{\frac{r}{2-r}\phantom{{}^l}}$, and the ideal kink energy is equal (proportional) to ${\mathrm{\Gamma }}_0$.

Figure 2

ARPES kinks data for OPT Bi2212 from Ref. [4] compared to theoretical ECFL curves (solid lines) using parameters listed in Table 2. (a) The predicted EDC spectrum (blue) from Eq. (3) versus the experimental EDC data (magenta symbols) at $T=115$ K. For reference we also show the MDC data (red dashed curve) and the corresponding ECFL fit (green solid curve). (b) Experimental MDC spectra at 40 K (below $T_c$ in green dashed line) and 115 K (above $T_c$ in red dashed line) yield common asymptotes shown in black lines from the far zone. These determine the parameters displayed in Table 2. (c) At low energy $\pm 60$ meV, the EDCs spectral function (blue solid line) from Eq. (9) is contrasted with the corresponding ARPES data from [4]. (d) At $\omega =0$ we compare the MDCs spectral function (blue solid line) from Eq. (9) with the corresponding ARPES data from Ref. [4]. The range of validity for the theoretical expansion is $\pm {\widehat{k}}_{\mathrm{kink}}\left(0.037{\text{Å}}^{-1}\right)$, the data points in the range are shown in black circle symbols, while the light gray circle symbols are outside this range. The peak position of the theoretical curve has been shifted to left by $0.007{\text{Å}}^{-1}$, a bit less than the instrumental resolution. A similar shift is made in Fig. 3. For analogous reasons, the EDC peak in $A(k,\omega )$ at ${\vec{k}}_F$ is shifted to the left, i.e., $E^{*}\left(0\right)\le 0$. A small shift to the right is made in Fig. 3, in order to compensate for this effect. These shift effects are within the resolution with present setups, but should be interesting to look for in future generation experiments since they give useful insights into the energy momentum dependence of the spectral function.

Figure 3

ARPES kinks data for LSCO data [1] compared to theoretical ECFL curves (solid lines) using parameters listed in Table 3. The doping level $x$ varies between (normal metal) $0.3\le$ $x$ $\le 0.03$ (insulator) in (a)–(j). Each panel shows MDC nodal dispersion data (symbols), whose uncertainties are $\pm 10$ meV. The blue dashed line is the theoretical prediction for EDC dispersion by Eq. (3). (k) We compare the spectral line shape for EDCs at $k_F$ from Eq. (9) (blue solid line) in the range $\pm E_{\mathrm{kink}}^{\mathrm{ideal}}\sim$  65 meV with the corresponding ARPES data (black circles) [12]. (l) At $\omega =0$ we compare the MDCs spectral function (blue solid line) from Eq. (9) with the corresponding ARPES data from Ref. [12]. The range of validity for the theoretical expansion is $\pm {\widehat{k}}_{\mathrm{kink}}$ (0.$037{\text{Å}}^{-1}$ ), the data points in the range are shown in black circle symbols, while the light gray circle symbols are outside this range. The peak position of the theoretical curve MDC has been shifted to left by 0.006 ${\text{Å}}^{-1}$.

Figure 4

ARPES kink analysis for laser ARPES data of Bi2201 at various different doping levels in Ref. [21]. (a)–(f) We predict EDC dispersions (blue dashed lines) using Eq. (3) for various different doping levels of Bi2201 laser ARPES data. (g), (h) First in (g), we present ECFL MDC fit (green solid line) for low-temperature (15 K) laser ARPES dispersion data of OTP Bi2201 from in Fig. 4(a) in Ref. [21] (black circles) and predict low-temperature EDC dispersion (green dashed line). Next, in (g) and (h), we predict high-temperature EDC (blue dashed lines) dispersions (g) 200 K and (h) 100 K for laser ARPES data of OPT Bi2201 [Fig. 4 in Ref. 21], and show the MDC dispersion fits for two temperatures also, blue solid line for 200-K data (red squares) in (g) and brown sold line for 100-K data (yellow circles) in (h). We estimate ${\mathrm{\Gamma }}_0$ from measuring the difference between the ideal kink energy and the MDC kink energy. In order to compare with experiments, the $x$-axis representation in (g) and (h) is given by the physical $k$ (rather than the momentum difference $\widehat{k})$. In (g), the MDC dispersion fit (blue solid line) of 200 K vanishes at $k=0.404\pm 0.002{\text{Å}}^{-1}$, very close to the measured $k=0.405\pm 0.002$ Å${}^{-1}$ of the MDC dispersion data at 200 K. Similarly, in (h) the MDC dispersion fit (brown solid line) at 100 K vanishes at $k=0.398\pm 0.002$ Å${}^{-1}$, close to the measured $k=0.4\pm 0.002$ Å${}^{-1}$ of the MDC dispersion data at 100 K. Note that the true Fermi momentum as estimated from the low-$T$ (15-K) data is $k=0.394\pm 0.002$ Å${}^{-1}$, so that the deviations are bigger than the momentum resolution $\mathrm{\Delta }k\sim 0.004$ Å${}^{-1}$. (i) We plot the temperature dependence of ${\mathrm{\Gamma }}_0$ in Fig. 4(a) in Ref. [21]. Here, the temperature dependence data of ${\mathrm{\Gamma }}_0$ are fitted with Eq. (6), and $\eta$ is determined 5.$3\pm 2$ meV and ${\mathrm{\Omega }}_{\mathrm{\Phi }}=410\pm 100$ meV.