Zeeman and orbital limiting fields: Separated spin and charge degrees of freedom in cuprate superconductors

Recent in-plane thermal (Nernst) and interlayer (tunnelling) transport experiments in ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+y}$ high temperature superconductors report hugely different limiting magnetic fields. Based on pairing (and the uncertainty principle) combined with the definitions of the Zeeman energy and the magnetic length, we show that in the underdoped regime both fields convert to the same (normal state) pseudogap energy scale $T^{*}$ upon transformation as orbital and spin (Zeeman) critical fields, respectively. We reconcile these seemingly disparate findings invoking separated spin and charge degrees of freedom residing in different regions of a truncated Fermi surface.

Figure 1

(Color online) Doping dependence of the peak field $H_{\mathrm{sc}}$ (half-purple squares) in ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+y}$ in the $T\to 0$ limit. $H_{\mathrm{sc}}(0,p)$ (in Tesla) $\sim 1.4T_c$ (in Kelvin, shown as half-orange squares) shapes the “superconducting dome” defined through the presence of large interplane Josephson currents and hosting the “conventional” superconducting phase with recombined quasiparticles. A similar dome-shape is derived from the magnetization measurements of systematically doped ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$, shown as red dots (from Ref. 16). Inset illustrates the nearly $T$-exponential temperature dependence of $H_{\mathrm{sc}}\left(T\right)$ in ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+y}$ with $p=0.225$, pointing to the $T=0$ value of $H_{\mathrm{sc}}$. See also Ref. 5.

Figure 2

(Color online) Comparison of $H_{\mathrm{pg}}$, $H_{c2}^N$ in ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+y}$, and the transformation from $H_{\mathrm{pg}}$ to $H_{c2}^{*}$ via Eq. (3), where we used (Ref. 30) $v_F\approx 1.8\pm 0.2\mathrm{eV}\AA$ and $\alpha \approx 0.6$; note the single “$H_{c2}$” scale in the low doping regime. As sketched for emphasis in the inset, the lines $H_{c2}^N\left(0\right)$ and $H_{c2}^{*}\left(0\right)$ split apart upon reaching the superconducting dome (shaded sky-blue) $H_{\mathrm{sc}}(0,p)$. $H_{c2}^N$ data (half-green diamonds) is from scaling near $T_c$ (Ref. 7). Light yellow band follows $H_{c2}^N\left(0\right)$ (half-green squares) obtained as described in the text.

Figure 3

(Color online) The breakup of the Fermi surface (FS) into regions describing spin-singlet pairs and charged holes: the spin-pairing opens up gaps at the $(0,\pi )$ points (the FS corners)—the corresponding pseudogap energy $T^{*}$ establishes correlations on the scale ${\xi }^{*}\sim \hslash v_F∕T^{*}$. Upon doping, a truncated FS appears around the $(\pi ,\pi )$ diagonals. When charges pair up, they draw correlations from the spin-singlet background, hence spin-singlet pairing at the FS corners and hole-pairing at the diagonals derive from the same energy scale $T^{*}$. The separation of spins and holes will eventually become “fuzzy” in the overdoped regime and there may be an overlap on the Fermi surface. In this limit the two (in-plane and out-of-plane) experimental transport probes will not detect the differences.