Critical point of the transition between $s_\pm$ and $s_{++}$ states of a two-band superconductor with nonmagnetic impurities

Abstract

Behavior of the Grand thermodynamic potential along with its derivatives, entropy and specific heat, is considered within a two-band model of an unconventional $s_\pm$ superconductor with nonmagnetic impurities. The transition $s_\pm \to s_{++}$ is shown to be a smooth crossover at high temperatures, while it becomes the first order phase transition at low temperatures. Thus, on a phase diagram `temperature'-`impurity scattering rate' there appears to be a critical end point. Temperature at which the behavior of the transition is changed is maximal in the Born limit and tends to zero away from the limit, which points out to the possible realization of a quantum phase transition.

Figures (4)

• Figure 1: (Color online) The surface of the minimum energy $\Delta\Omega(\Gamma_a, T)$ (a) in the Born limit, $\sigma = 0$, has the kink within the region of coexistence of two solutions of the Eliashberg equations. For clarity, constant level lines are drawn on the surface, with the kink being denoted with circle markers. The presence of the kink in $\Delta\Omega$ leads to the line of peaks in entropy $\Delta{S(\Gamma_a, T)}$ (b) along with line of divergencies in electronic specific heat $\Delta{C(\Gamma_a, T)}/T$ (c). Free energy, entropy and specific heat are given in units of eV, J/(K$\cdot$mol) and J/(K$^2\cdot$mol), respectively
• Figure 2: (Color online) Surfaces of $\Delta\Omega$ (a), $\Delta S$ (b) and $\Delta C/T$ (c) in the axes ($\Gamma_a$, $T$) near the Born limit at $\sigma = 0.06$. The main difference from the results shown in Fig. \ref{['fig:Deltas']} is the decrease of the critical end point temperature $T_{ \mathrm{CEP} }$. All values are given in the same units as in Fig. \ref{['fig:Deltas']}
• Figure 3: (Color online) Graphs for $\Delta{C}(\Gamma_a)/T$ (a) and $\Delta{C}(T)/T$ (b) plotted for the temperature above the critical end point $T_{ \mathrm{CEP} }$ in the Born limit. The panel (a) shows also plot for $\Delta{C}(\Gamma_a)/T$ at $T = 0.06$$T_{c0}$ revealing how specific heat diverges due to the first order phase transition. In panel (b), marked and unmarked lines correspond to the families of curves located to the left and to the right of the critical end point $\Gamma_a^{ \mathrm{CEP} }$, respectively
• Figure 4: (Color online) Location of the critical end point at the ($\Gamma_a$, $T$) plane for different values of the generalized cross-section $0.0 < \sigma < 0.18$. The dashed line corresponds to the critical end point location extrapolated to the zero temperature, where it may become the quantum critical point at $\Gamma_a^{\mathrm{QCP}}\approx 1.15$$T_{c0}$