Heat capacity double transitions in time-reversal symmetry broken superconductors

Standard superconductors display a ubiquitous discontinuous jump in the electronic specific heat at the critical superconducting transition temperature. In a growing class of unconventional superconductors, however, a second order parameter component may get stabilized and produce a second heat capacity jump at a lower temperature, typically associated with the spontaneous breaking of time-reversal symmetry. The splitting of the two specific heat discontinuities can be controlled by external perturbations such as chemical substitution, hydrostatic pressure, or uniaxial strain. We develop a theoretical quantitative multiband framework to determine the ratio of the heat capacity jumps, given the band structure and the order parameter momentum structure. We discuss the conditions of the gap profile which determine the amplitude of the second jump. We apply our formalism to the case of ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$, and using the gap functions from a microscopic random phase approximation calculation, we show that recently proposed accidentally degenerate order parameters may exhibit a strongly suppressed second heat capacity jump. We discuss the origin of this result and consider also the role of spatial inhomogeneity on the specific heat. Our results provide a possible explanation of why a second heat capacity jump has so far evaded experimental detection in ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$.

Figure 1

Sketch of the order parameter temperature dependence, where $\mathrm{\Delta }(T,\mathit{p})={\mathrm{\Delta }}_0\left(T\right)[f_a\left(\mathit{p}\right)+iX\left(T\right)f_b\left(\mathit{p}\right)]$. When the two components have critical temperatures $T_c≔T_{c1}$ and $T_{c2}$, repulsion leads to the onset of the second component at $T_{\mathrm{TRSB}}<T_{c2}$, for which time-reversal symmetry is broken (blue line). Due to repulsion ${\mathrm{\Delta }}_0\left(T\right)$ (green line) becomes nonanalytic at $T_{\mathrm{TRSB}}$. The nonanalyticity leads to a reduction of the second heat capacity jump.

Figure 2

Heat capacity for $s+i{d}_{xy}$ [panel (a)] and $d_{x^2-y^2}+i{g}_{xy(x^2-y^2)}$ [panel (b)] order parameters per temperature per normal-state value in a one-band model. We consider the square lattice with nearest-neighbor hopping $t=100\mathrm{meV}$ at filling $\langle n\rangle =0.6$, with critical temperatures $T_{c1}=1.5\text{K}, T_{c2}=1.2\text{K}$, and Ginzburg-Landau parameters fixed to ${\alpha }_a^0={\alpha }_b^0=30{\text{meV}}^{-1}$ and ${\beta }_a={\beta }_b=1{\text{meV}}^{-3}$. We take ${\beta }_{ab}=0$ (blue curves), ${\beta }_{ab}=0.2{\beta }_a$ (green curves), and the opposite order of both components with ${\beta }_{ab}=0$ (red curves), using the fundamental harmonic order parameters shown in the insets.

Figure 3

Fermi surface from the three-band tight-binding model for ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ described in Appendix pp3. Left panel: Unstained case, and right panel: uniaxial (100) strain just below the Lifshitz transition on the $\gamma$ band. The bands have been color coded by their Ru $t_{2g}$ orbital content.

Figure 4

Leading even-parity order parameters (labeled $s^{\prime }$ and $d_{xy}$ at $p=0$) belonging to irreducible representations $A_1$ and $B_1$ of the point group $D_2$ at zero and finite (100) uniaxial strain from the RPA calculations of Ref. [35]. The interaction parameters were set to $U=100$ meV, $J/U=0.1$, and $V/U=0.2$.

Figure 5

Next-to-leading even-parity order parameters (labeled $d_{x^2-y^2}$ and $g_{xy(x^2-y^2)}$ at $p=0$) belonging to irreducible representations $A_1$ and $B_1$ of the point group $D_2$ at zero and finite (100) uniaxial strain from the RPA calculations of Ref. [35] and for the same interaction parameters as in Fig. 4.

Figure 6

Split transition temperatures as a function of the dimensionless uniaxial (100) strain parameter $p$, as measured with combined heat capacity (teal diamonds) and $\mu \mathrm{SR}$ (blue circles) from Ref. [51]. Gray lines are quadratic fits to the data points, used to parametrize $T_c\left(p\right)$ and $T_{\mathrm{TRSB}}\left(p\right)$ as used in the calculations. The light blue and red colored bands indicate transition widths of $\sigma =100$ mK; cf. the simulated smearing in Fig. 7.

Figure 7

Heat capacity per temperature and normal-state value for the (left column) $s^{\prime }+i{d}_{xy}$ and (right column) $d_{x^2-y^2}+i{g}_{xy(x^2-y^2)}$ order parameters from Ref. [35]. Panels (a) and (b): The unstrained case. We use (blue lines) the order parameters from Figs. 4 and 5 with ${\mathrm{\Delta }}_0\left(T\right)$ and $X\left(T\right)$ (insets) that minimize the free energy of Eq. (8). The experimental data from Refs. [27, 63] is plotted alongside. The black dashed line shows the result for a single-component $d_{x^2-y^2}$-wave order parameter, ${\mathrm{\Delta }}_{\mu }\left(\mathit{p}\right)={\mathrm{\Delta }}_{\mu }(cos{p}_x-cos{p}_y)$ with ${\mathrm{\Delta }}_{\alpha }/{\mathrm{\Delta }}_{\gamma }=0.55$ and ${\mathrm{\Delta }}_{\beta }/{\mathrm{\Delta }}_{\gamma }=0.45$, for reference. Panels (c) and (d): Calculated heat capacity at finite strain with $X\left(T\right)$ shown in the insets.

Figure 8

Simulated effect of smearing in the (a) $s^{\prime }+i{d}_{xy}$ and (b) $d_{x^2-y^2}+i{g}_{xy(x^2-y^2)}$ order parameter scenario for the same strain values as considered in Figs. 7 and 7. Smearing is here modeled by averaging over realizations where we draw transition temperatures from normal distributions with means $T_c\left(p\right)$ and $T_{\mathrm{TRSB}}\left(p\right)$ from Fig. 6 and standard deviation $\sigma =100\mathrm{mK}$. For each realization we calculate the Ginzburg-Landau coefficients of Eqs. (9, 10, 11) and obtain the heat capacity from Eq. (A6) with the order parameter of Eq. (1). Panel (c): The experimental data from Ref. [27], providing an indirect measurement of the heat capacity.

Figure 9

Cooper pair operator in the (pseudo)spin singlet channel.

Figure 10

The (a) second-order and (b) fourth-order diagrams contributing to the free energy of Eq. (B3). Corresponding algebraic expressions are given by Eqs. (B6) and (B7).

Figure 11

Order parameters from Fig. 4 labeled by the respective irreps of $D_2$, as a function of polar angle [on the $\alpha$ band the polar angle is measured from $(\pi ,\pi )$]. The colored data points are the raw data from Ref. [35], and the black dashed lines show the data passed through a Savitzky-Golay filter to reduce finite-size effects. The filtered order parameters were used in the calculations of Sec. 3b.

Figure 12

Order parameters from Fig. 5 labeled by the respective irreps of $D_2$. See the caption of Fig. 11 for further details.

Figure 13

Comparison of the heat capacity per temperature and normal-state value for the (a) $s^{\prime }+i{d}_{xy}$ and (b) $d_{x^2-y^2}+i{g}_{xy(x^2-y^2)}$ order parameters from Ref. [35] at zero strain using two different temperature dependencies. The blue lines show results using the Ginzburg-Landau solution, while the red lines use simple square root temperature dependencies and a constant $X_0=0.8$. In both cases it is seen that the nonanalyticity of ${\mathrm{\Delta }}_0(T=T_{\mathrm{TRSB}})$ reduces the second jump.

Figure 14

Comparison of the heat capacity per temperature and normal-state value at zero strain using the Ginzburg-Landau solution with form factors obtained from microscopic RPA calculations [35] (blue lines) to a case where the band-resolved gap magnitudes have been redistributed to obtain a better fit (purple lines) with the experimental data of Ref. [63] (gray diamonds). Panel (a): The $s^{\prime }+i{d}_{xy}$ scenario. Here we redistributed ${\mathrm{\Delta }}_{\alpha }/{\mathrm{\Delta }}_{\gamma }=0.7$ and ${\mathrm{\Delta }}_{\beta }/{\mathrm{\Delta }}_{\gamma }=0.6$ as compared to the bare RPA form factors to obtain the purple line. Panel (b): The $d_{x^2-y^2}+i{g}_{xy(x^2-y^2)}$ scenario, where ${\mathrm{\Delta }}_{\alpha }/{\mathrm{\Delta }}_{\gamma }=0.4$ and ${\mathrm{\Delta }}_{\beta }/{\mathrm{\Delta }}_{\gamma }=0.5$ characterizes the purple line.

Figure 15

Supplement to Fig. 7: Heat capacity for the $d_{xy}+i{s}^{\prime }$ order parameter. We used the order parameter ansatz of Eq. (1) with ${\mathrm{\Delta }}_0\left(T\right)$ and $X\left(T\right)$ (inset) resulting from minimization of the free energy of Eq. (8), and with the momentum-dependent form factors of Fig. 4.

Figure 16

Simulated effect of smearing with a transition width of $\sigma =50\mathrm{mK}$ in the (a) $s^{\prime }+i{d}_{xy}$ and (b) $d_{x^2-y^2}+i{g}_{xy(x^2-y^2)}$ order parameter scenario. See the caption of Fig. 8 for further details.