Nearly perfect fluidity in a high-temperature superconductor

Perfect fluids are characterized as having the smallest ratio of shear viscosity to entropy density, $\eta /s$, consistent with quantum uncertainty and causality. So far, nearly perfect fluids have only been observed in the quark-gluon plasma and in unitary atomic Fermi gases, exotic systems that are amongst the hottest and coldest objects in the known universe, respectively. We use angle resolved photoemission spectroscopy to measure the temperature dependence of an electronic analog of $\eta /s$ in an optimally doped cuprate high-temperature superconductor, finding it too is a nearly perfect fluid around, and above, its superconducting transition temperature $T_c$.

Figure 1

Scaled intensity maps for the Bi2212 Fermi surface at 95 K. From left to right: $\omega =-33\mathrm{meV}$, $\omega =0$, and $\omega =+33\mathrm{meV}$. The integration window is $\pm 3\mathrm{meV}$. The experimental details are described in the Methods section of the main text. The red line shows the antiferromagnetic zone boundary and the black line shows the visible portion of the Fermi surface for $x=0.16$ calculated using the YRZ model as described above.

Figure 2

Scaled intensity maps for the Bi2212 Fermi surface at 60 K. From left to right: $\omega =-20\mathrm{meV}$, $\omega =0$, and $\omega =+20\mathrm{meV}$. The integration window is $\pm 3\mathrm{meV}$. The experimental details are described in the Methods section of the main text. The red line shows the antiferromagnetic zone boundary and the black line shows the visible portion of the Fermi surface for $x=0.16$ calculated using the YRZ model as described above.

Figure 3

(a) Experimental DOS for Bi2212. Raw DOS (solid black lines) are generated by integrating data over regions of the BZ demarcated by the Fermi surface maps in (b). Raw DOS are fit above $E_F$ by FD distributions (solid red lines) ensuring a smooth approach to zero intensity at high energies. ${\tilde{g}}_0\left(\omega \right)$ (solid blue lines) and symmetrized DOS (black lines) are also shown. All DOS in the figure are normalized to unity at high $(-\omega )$ and offset as indicated by the horizontal dashed black lines. Vertical dashed black lines indicate $\pm 4k_{B}T$ for each $T$. (b) Fermi surface maps corresponding to regions of the BZ measured in ARPES used to generate the DOS in (a). Red dashed lines show the zone boundary of the underlying antiferromagnetic spin lattice. The 95 K and 60 K maps are shown in Figs. 1 and 2, respectively. The upper-left panel shows a schematic of the underlying tight binding Bi2212 FS (black line) as described in the Appendixes. (c) $T$ dependence of $T_{\eta }/T$ [Eq. (3)]. (Solid blue line) Theoretical $T_{\eta }/T$ for the tight binding model. (Black circles, a diamond, and a square point) $T_{\eta }/T$ derived by applying Eq. (3) to the experimental DOS in (a). Different symbols apply to different samples. Error bars reflect the uncertainty of the chemical potential, which was 0.5 meV. Dotted red and blue lines demarcate $T_{\eta }/T$ equal to $(2/3)$ and $(1/2)$, respectively. (Black dashed line) Phenomenological fit to the data used to scale ARPES scattering rates for Fig. 8.

Figure 4

Panel (a) shows the Gaussian distribution that weights the full DOS when calculating $s\left(T\right)$. Panel (b) shows the factor multiplying the experimental “occupied states” DOS actually measured directly in ARPES. In both panels the vertical dotted lines denote $\pm 4k_{B}T$.

Figure 5

Panel (a) shows the weighting factor for $u\left(T\right)$, $\omega f_T\left(\omega \right)$ including at $T=0$. Panel (b) shows $\omega (f_T\left(\omega \right)-f_0\left(\omega \right))$. Vertical lines in both panels denote $\pm 4k_{B}T$.

Figure 6

(a) (Black) raw DOS, $g_T\left(\omega \right)$, divided by $f(T,\omega )$ after subtracting just the minimum intensity value, (red) DOS fit and replaced at high $\omega$ by a FD function, background subtracted to fit zero, then divided by the FD function, and (blue) raw DOS with an average background subtraction of $6\times {10}^{-5}$ before FD division. (b) DOS before FD division (red) with fit and subtracted, (blue) minimum value subtracted, and (black) minimum value $+6\times {10}^{-5}$ subtracted.

Figure 7

Temperature dependence of $\hslash /{\tau }_p$ derived from nodal ARPES data [30] for Bi2212 ($T_c=91$ K) (red circles). Original data [30], in which inverse mean free paths $\Delta k={\ell }^{-1}$ are measured directly, has been rescaled into scattering rates using the temperature-dependent Fermi velocities $v_F$ (in units of [eV Å]) of Ref. [32] such that $\hslash /{\tau }_{\vec{k}}=v_F\Delta k$. The final scattering rate (red circles) is $\hslash /{\tau }_p=\hslash /{\tau }_{\vec{k}}-\hslash /{\tau }_0$ where for simplicity we take $\hslash /{\tau }_0=\text{min}[\hslash /{\tau }_{\vec{k}}(T\to 0)]=17.7\mathrm{meV}$. $4\pi k_B{T}_{\eta }$ lines are also plotted as a reference; the closer $\hslash /{\tau }_p$ approaches $T_{\eta }\left(T\right)$ from below, the closer $\eta /s$ is to the holographic bound after scaling by Eq. (2). Relevant ideal bounds consistent with Eq. (1) include $T_{\eta }=(1/2)T$ (dotted blue line) and $T_{\eta }=(2/3)T$ (dotted red line) from Eq. (15), yielding $\hslash /{\tau }_p=2\pi k_{B}T$ and $\hslash /{\tau }_p=(8/3)\pi k_{B}T$, respectively, as well as the $T_{\eta }$ including $d$-wave superconductivity in the simple tight binding model (solid blue line) and $T_{\eta }$ from the present experiment on Bi2212 (black circles). The phenomenological fit to the experimental $T_{\eta }$ values is shown as the dashed black line. The navy line is a linear fit to $\hslash /{\tau }_p(T>T_c)$.

Figure 8

$T_{\eta }{\tau }_p$ in units of the KSS bound, $\hslash /\left(4\pi k_B\right)$, versus $T^{\prime }=(T-T_c)/T_c$ evaluated using Eq. (2) (right axis) and $\eta /s$ for the QGP and UFG (left axis). (Filled red circles) ARPES for Bi2212 scaled by the phenomenological fit to $T_{\eta }$ (black dashed line in Fig. 3). $T_{\eta }{\tau }_p$'s at lower $T^{\prime }$ are too small to provide a reliable result. (Blue bar) The range of $\eta /s$ consistent with RHIC data on the QGP at $T_c$ from Au+Au collisions is encompassed by the blue vertical band [39]. (Gray points) Data from the UFG [35]. (Open red circle) The point at $T\gg T_c$ is the extrapolated high $T$ value for $\eta /s\sim \hslash /2A$ of Bi2212. The KSS bound is marked by the solid green line and $T_c$ for Bi2212 by the vertical black dotted line.

Figure 9

$T_{\eta }/T=U/ST$ for the tight binding (orange line) and YRZ model at $x=0.16$ (purple line), respectively. The red and blue dashed lines correspond to $T_{\eta }/T=(2/3)$ and $(1/2)$, respectively. The vertical dashed black line marks $T_c=91\mathrm{K}$.