Theory of superconductivity with non-Hermitian and parity-time reversal symmetric Cooper pairing symmetry

Recently developed parity ($\mathcal{P}$) and time-reversal ($\mathcal{T}$) symmetric non-Hermitian systems govern a rich variety of new and characteristically distinct physical properties, which may or may not have a direct analog in their Hermitian counterparts. We study here a non-Hermitian, $\mathcal{PT}$-symmetric superconducting Hamiltonian that possesses a real quasiparticle spectrum in the $\mathcal{PT}$-unbroken region of the Brillouin zone. Within a single-band mean-field theory, we find that real quasiparticle energies are possible when the superconducting order parameter itself is either Hermitian or anti-Hermitian. Within the corresponding Bardeen-Cooper-Schrieffer (BCS) theory, we find that several properties are characteristically distinct and novel in the non-Hermitian pairing case than its Hermitian counterpart. One of our significant findings is that while a Hermitian superconductor gives a second-order phase transition, the non-Hermitian one produces a robust first-order phase transition. The corresponding thermodynamic properties and the Meissner effect are also modified accordingly. Finally, we discuss how such a $\mathcal{PT}$-symmetric pairing can emerge from an antisymmetric potential, such as the Dzyloshinskii-Moriya interaction, but with an external bath, or complex potential, among others.

Figure 1

The splitting of the BZ into the “paired region” (gray color) and “unpaired region” (white) in the NH-SC case. The shape of the “unpaired region” is determined by the pairing symmetry at hand, while the area is proportional to the gap amplitude (${\mathrm{\Delta }}_0$). As $T\to T_c$, the white region vanishes smoothly. $k_{x,y}$ are defined in units of $1/a$ where $a$ is the lattice constant.

Figure 2

(a) and (b) Self-consistent values of the SC gap ${\mathrm{\Delta }}_0$ for the NH-SC and H-SC cases, respectively, are plotted for different values of the pairing potential $V_0$ and chemical potential $\mu$ (where $t=1$ and $k_B=1$ in all the calculations and figures). The NH-SC gap values are plotted as a dashed line near $T_c$ to emphasize the fact that due to the first-order phase transition here, the gap discontinuously drop to zero without tracing the smooth curve to reach zero. The exact value where the phase transition occurs is not attainable from the gap function in Eq. (20). The temperature where a solid to dashed line transition occurs is chosen here arbitrarily and for illustration purposes only.

Figure 3

The computed free energy for the H-SC case in (a) and the NH-SC case in (b) is plotted in both normal (dashed line) and SC (solid line) states. Here we choose the same parameter $V=1.5$ and $\mu =0.03$ in both cases (in units of $t$). The entropy computed from the free energies for the H-SC and the NH-SC cases is plotted in (c) and (d), respectively. Normal and SC free energies obtain a similar slope in the H-SC case, and thus their first derivative (entropy) does not obtain any jump, and their second derivative becomes discontinuous at $T_c$ (not shown). On the other hand, in the NH-SC case, these two free energies acquire opposite slopes, and thus obtain a discontinuity in the first derivative at $T_c$. The exact location where the jump occurs is arbitrarily chosen, as in Fig. 2. In fact, as mentioned in Fig. 2, the actual phase transition occurs before reaching the predicted $T_c$ from Eq. (20). The entropy for the NH case is plotted after adding a constant entropy of a yet unknown source (see text).

Figure 4

Plot of Ginzburg-Landau free energy as a function of the order parameter $\mathrm{\Delta }$ for the H-SC and NH-SC cases, with the same parameter sets of $a=-1.5, b=1, c=2$, and $d=0.5$ [Eq. (23)]. Note that both the first- and second-order phase transitions can be reproduced for some values of these parameters. For the H-SC case (gray line), it shows two minima at finite $\pm {\mathrm{\Delta }}_0^c$ and they merge to a single minimum at ${\mathrm{\Delta }}_0=0$ as $a$ smoothly increases and crosses zero. For the same parameter set, the NH-SC system (in which all the odd powers of ${\left|\mathrm{\Delta }\right|}^2$ become negative) gives a first-order phase transition (black line). This situation is analogous to making $a=-a$ and $c=-c$, while keeping $b$ and $d$ the same.

Figure 5

Plots of superfluid “kernels” (proportional to currents) are shown with paramagnetic ($K_p$, black dashed), diamagnetic ($K_d$, magenta), and total ($K_d+K_p$, blue) components (along the $xx$ and $yy$ directions), see Eq. (25). The parameter set is kept the same as in Fig. 3. For the NH-SC, all plots are drawn in dashed lines near $T_c$ to emphasize that the location of the discontinuous phase transition is unknown.

Figure 6

The self-consistent values of the SC gap ${\mathrm{\Delta }}_0$ for different PT-symmetric NH-SC cases are, respectively, plotted for the cases (i), (iii), and (iii), with $V_0=2$ for all cases. For all the cases, the temperature dependence of ${\mathrm{\Delta }}_0$ is similar (note that the second case is our considered case in the main text) and thus the other relevant characteristics of $\mathcal{PT}$-symmetric NH-SC according to our results are also similar.