Superconductivity and Magnetic Order in ${\mathrm{CeRhIn}}_5$: Spectra of Coexistence

We discuss the fixed-point Hamiltonian and the spectrum of excitations of a quasi-bidimensional electronic system supporting simultaneously antiferromagnetic ordering and superconductivity. The coexistence of these two order parameters in a single phase is possible because the magnetic order is linked to the formation of a metallic spin density wave, and its order parameter is not associated to a spectral gap but to an energy shift of the paramagnetic bands. This peculiarity entails several distinct features in the phase diagram and the spectral properties of the model, which may have been observed in ${\mathrm{CeRhIn}}_5$. Apart from the coexistence, we find an abrupt suppression of the spin density wave when the superconducting and magnetic ordering temperatures are equal. The divergence of the cyclotron mass extracted from de Haas–van Alphen experiments is also analyzed in the same framework.

Figure 1

Schematic experimental pressure-temperature phase diagram of ${\mathrm{CeRhIn}}_5$ at zero-magnetic field adapted from Refs. 6, 9, 21.

Figure 2

Calculated electronic structure of paramagnetic ${\mathrm{CeRhIn}}_5$. (a) Density of states and (b) band structure. The arrow indicates a saddle point singularity at the $X$ point of the Brillouin zone. The inset shows the variation of the saddle point energy along the $z$ $R\mathrm{\text{−}}X\mathrm{\text{−}}R$ direction [it crosses the Fermi level at approximately the ($\pi /a$, 0, $0.6\pi /c$) point].

Figure 3

Phase diagram (temperature versus pressure) obtained using the model Hamiltonian (1) and a two-dimensional band structure with the FL close to a saddle point. The variables are in units of the half bandwidth $W_0$. $T_N$ and $T_{\mathrm{SC}}$ stand for the SDW and SC critical temperatures, respectively. Panels (a), (b), and (c) are the calculated results for the parameters given in the text and $U=2.25W_0$, $U=2.50W_0$, and $U=2.75W_0$ respectively. The inset in panel (c) indicates the metallic DOS in the three (SDW, $\mathrm{SDW}+\mathrm{SC}$, and SC) different regimes. The shaded area indicates occupied levels. Panel (d) represents the experimental results of Ref. 9.

Figure 4

Cyclotron mass ($m_c$) as a function of pressure. Triangles are the experimental values from Ref. 4 showing the $m_c$ divergence close to $P\sim 2.4\mathrm{GPa}$, where the magnetic order disappears. The solid line is a fit to the theoretical model. Close to a Van Hove singularity the cyclotron mass diverges logarithmically as the difference between $E_F$ and the $E_{\mathrm{VHS}}$ vanishes (see lower inset). According to the model this energy difference is proportional to $T_N$ and its pressure dependence can be extracted, for instance, from Ref. 6 (see upper inset).