Magic Gap Ratio for Optimally Robust Fermionic Condensation and Its Implications for $\mathrm{High}\text{−}{T}_c$ Superconductivity

Bardeen-Schrieffer-Cooper (BCS) and Bose-Einstein condensation (BEC) occur at opposite limits of a continuum of pairing interaction strength between fermions. A crossover between these limits is readily observed in a cold atomic Fermi gas. Whether it occurs in other systems such as the high temperature superconducting cuprates has remained an open question. We uncover here unambiguous evidence for a BCS-BEC crossover in the cuprates by identifying a universal magic gap ratio $2\mathrm{\Delta }/k_B{T}_c\approx 6.5$ (where $\mathrm{\Delta }$ is the pairing gap and $T_c$ is the transition temperature) at which paired fermion condensates become optimally robust. At this gap ratio, corresponding to the unitary point in a cold atomic Fermi gas, the measured condensate fraction $N_0$ and the height of the jump $\delta \gamma (T_c)$ in the coefficient $\gamma$ of the fermionic specific heat at $T_c$ are strongly peaked. In the cuprates, $\delta \gamma (T_c)$ is peaked at this gap ratio when $\mathrm{\Delta }$ corresponds to the antinodal spectroscopic gap, thus reinforcing its interpretation as the pairing gap. We find the peak in $\delta \gamma (T_c)$ also to coincide with a normal state maximum in $\gamma$, which is indicative of a pairing fluctuation pseudogap above $T_c$.

Figure 1

(a) Schematic $\gamma (T)$ with (solid lines) and without (dotted lines) a phase transition. Because of the transition, the normal state maximum is visible only when $T_{\gamma }>T_c$ (or $1/k_{F}a>0$ or $p<p^{*}$). Also shown is a schematic of resonant pairing, occurring when the bound state energy coincides with the Fermi level [29], producing a sharp peak in $\delta \gamma (T_c)$. (b) Unitary regime of a Fermi gas [8, 14]. Upper panel: $T_c$ from Ref. [14] and $T_{\gamma }=2\mathrm{\Delta }/6.5k_B$ and $T_{\chi }=2\mathrm{\Delta }/3k_B$ (using $\mathrm{\Delta }$ at the lowest $T$ from Ref. [14]). Lower panel: $\delta \gamma (T_c)$ (lower and upper bound estimates extracted [29] from $S(T)$ in Fig. 5 of Ref. [14]), $\delta S$ [from Fig. 6 of Ref. [14]; this closely follows $\delta \gamma (T_c)$, providing a guide to the eye], and $N_0$ (brown [13] and yellow [12] circles). (c) Cuprates. Upper panel: $T_c(p)$ [58, 84, 85, 86, 87, 88, 89, 90, 94] and $T_{\gamma }$ and $T_{\chi }$ from Fig. 2. Lower panel: $\delta \gamma (T_c)$; spline fits connect points. In (b) and (c), dotted lines indicate $p=p^{*}$ (for each cuprate family [90]) and $1/k_{F}a=0$, at which $T_{\gamma }=T_c$, coinciding approximately with peaks in $\delta \gamma (T_c)$.

Figure 2

(a) $\delta \gamma (T_c)$, $N_0$, and $\delta S$ [rescaled to unity from Figs. 1 and 1] versus $2\mathrm{\Delta }/k_B{T}_c$. (b) $1/k_{F}a$ versus $2\mathrm{\Delta }/k_B{T}_c$ [14]. (c) Spectroscopic and thermal antinodal gap measurements [73, 102, 103]. (d) Maxima in $\gamma$ (grey symbols) and $\chi$ (white symbols) from the raw data [29]. $T_{\gamma }$ (green line) is a polynomial fit to the grey symbols [104], from which we obtain $\mathrm{\Delta }=6.5k_B{T}_{\gamma }/2$ [i.e., the purple line in (d)] and $T_{\chi }=2\mathrm{\Delta }/3k_B$ (red line). Symbol shapes identify the cuprate family in (a), (c), and (d). The down triangle in (c) refers to HBCO [90].

Figure 3

(a) Left-hand axes: $\gamma$ for $T>T_c$ [29, 37, 64, 84, 87, 110] (colored lines; including 2% Zn substitution for the highest 2 YBCO dopings [110]) versus $2\mathrm{\Delta }/k_{B}T$; $p$ and $1/k_{F}a$ values are indicated throughout. Right-hand axes: $\delta \gamma (T_c)$ at $T_c$ versus $2\mathrm{\Delta }/k_B{T}_c$ for the cuprates [58, 84, 85, 86, 87, 88] (center-dot circles; shifted by $\pm 10\mathrm{mJ}{\mathrm{mol}}^{-1}{\mathrm{K}}^{-2}$ for YBCO and Ca-YBCO) and a Fermi gas from Fig. 1 (center-dot squares). (b) ${\chi }_m$ [87, 92, 112, 113], $K$ [91, 111, 114, 115, 116] and $\chi$ [108] versus $2\mathrm{\Delta }/k_{B}T$ [117]. Included are $\gamma$ and $\chi$ for model $d$- (black) and $s$-wave (grey) gaps of magnitude $\mathrm{\Delta }$ [29]. Some YBCO $K$ curves are shifted vertically for clarity and spline fits connect coarsely spaced points. Dotted lines indicate $2\mathrm{\Delta }/k_B{T}_c=6.5$ and $2\mathrm{\Delta }/k_{B}T=3$.