Microscopic theory of strain-controlled split superconducting and time-reversal symmetry-breaking transitions in $s+id$ superconductor

TL;DR

This work develops a microscopic tight-binding framework to study strain-controlled $s+id$ superconductivity with $U(1)\times\mathbb{Z}_2$ symmetry on a square lattice. By incorporating uniaxial, shear, and isotropic strains (including Poisson effects), the authors show that $T_c^{U(1)}$ and $T_c^{\mathbb{Z}_2}$ can split nontrivially, with cases where a linear kink appears at zero strain under [100] compression when the two critical temperatures coincide, but not for shear, and with nonmonotonic $T_c^{\mathbb{Z}_2}$ in certain parameter regimes. Extending to models with on-site interactions, they find robust BTRS regions ($s+id$) and complex phase structure, including nontrivial phase differences between gap components induced by orthorhombicity. Boundary and finite-size effects significantly broaden the BTRS region and introduce boundary-localized currents, offering practical routes to probe and stabilize BTRS states in mesoscopic samples. Overall, the results highlight substantial deviations from simple Ginzburg-Landau predictions and demonstrate that strain and boundaries can be used to tune and detect $s+id$ BTRS superconductivity.

Abstract

We study conditions of the appearance of $U(1)\times \mathbb{Z}_2$ superconducting states that spontaneously break time-reversal symmetry (BTRS) on a square lattice as a function of applied stress. Calculations show that if critical temperatures coincide at zero stress, they exhibit a linear kink and no kink otherwise for uniaxial and isotropic strain. Linear kink is absent for shear strain. We find that in general, the microscopic calculations show a complex phase diagram, for example, non-monotonic behavior of BTRS transition. Another beyond-Ginzburg-Landau theory result is that $U(1)$ critical temperature can decrease under compressional [100] uniaxial strain for small Poisson ratio materials. In the second part of the paper, we consider the effects of boundaries and finiteness of the sample on the strain-induced splitting of $T_c^{U(1)}$ and $T_c^{\mathbb{Z}_2}$ transitions. A finite sample has BTRS boundary states with persistent superconducting currents over a wide range of band filling. Overall, the BTRS dome occupies a larger band filling--temperature phase space region for a mesoscopic sample with [110] surface compared to an infinite system. Hence, the presence of boundaries helps to stabilize the BTRS phase.

Figures (17)

• Figure 1: Superconducting and BTRS critical temperatures as a function of [100] uniaxial strain (a measure of orthorhombicity) within Ginzburg-Landau formalism. Note the linear kink for $U(1)$ and $\mathbb{Z}_2$ critical temperatures at zero strain when system is tuned that $T_c^{U(1)}(0)=T_c^{\mathbb{Z}_2}(0)$. The kink is absent when $U(1)$ and $\mathbb{Z}_2$ critical temperatures are split at zero stress. Microscopic calculations are presented in Fig. \ref{['fig:microscopic critical temperatures small stress']}.
• Figure 2: (a) The dependence of the superconducting gap components $|\Delta_\alpha|$ on the band filling $n$ for the model with gap $\Delta(\boldsymbol{k}) = \Delta_{d} (\cos k_x - \cos k_y) + \Delta_{s_\text{ext}} (\cos k_x + \cos k_y)$. The blue (orange) line corresponds to the $d$-wave ($s_\text{ext}$-wave) gap irreducible representation. (b) The dependence of the phase difference between gap components $\arg (\Delta_d^* \Delta_{s_\text{ext}})$ on the band filling $n$. Solid and dashed lines correspond to the tetragonal (change of hopping integral $\delta t = 0$) and orthorhombic ([100] uniaxial strain $\delta t = 0.1$, Poisson ratio $\nu=0.3$, $\delta t_y = -\nu \delta t= -0.03$) system, respectively. (c) Phase diagram for an unstrained system. The $s+id$ state has $\pi/2$ phase difference between components. (d) Phase diagram for orthorhombic ($\delta t = 0.1$, $\nu=0.3$) system. Both gap components are always non-zero. The $s_\text{ext}+d$ state has coinciding phases of both components. The $s_\text{ext}+id$ state has phase difference $\in(0;\pi/2)$ between components. Nearest-neighbor interaction strength $V_1=2$, $T=0$.
• Figure 3: Superconducting phase diagrams in band filling $n$, temperature $T$ coordinates for different measures of orthorhombicity $\delta t$ for Poisson ratios (a) $\nu=0$, (b) $\nu=0.3$, (c) $\nu=1$. Dashed and solid lines correspond to the $U(1)$ and $\mathbb{Z}_2$ symmetry-breaking critical temperatures, respectively. Above the dashed line is a normal metal phase, between the dashed and solid line is $U(1)$ symmetry-breaking phase, and below the solid line is the BTRS phase. BTRS dome moves monotonously to a smaller (larger) band filling under stress for $\nu=0$ ($\nu=1$). BTRS dome first moves towards lower and then towards a larger band filling under stress for $\nu=0.3$. Vertical black dashed line corresponds to band filling $n=1.58518$ where $T_c^{U(1)}=T_c^{\mathbb{Z}_2}$ for tetragonal system ($\delta t = 0$). Nearest-neighbor interaction strength $V_1=2$.
• Figure 4: Superconducting and BTRS critical temperatures as a function of change of hopping $\delta t$ (a measure of orthorhombicity) due to external [100] uniaxial compressional stress for different Poisson ratios $\nu$. Dashed and solid lines correspond to the $U(1)$ and $\mathbb{Z}_2$ symmetry-breaking critical temperatures, respectively. Ginzburg-Landau behavior of critical temperatures in Fig. \ref{['fig:GL critical temperatures']} qualitatively describes microscopic calculations only for $\nu=1$. The horizontal black line indicates the critical temperature for the tetragonal system without stress $T_c^{U(1)}=T_c^{\mathbb{Z}_2}=0.02832$, $\delta t =0$, band filling $n=1.58518$. Nearest-neighbor interaction strength $V_1=2$.
• Figure 5: Superconducting and BTRS critical temperatures as a function of tensile ($\delta t <0$) and compressional ($\delta t >0$) [100] uniaxial strain within microscopic model. (a) System is tuned that $T_c^{U(1)}=T_c^{\mathbb{Z}_2}$ at zero external strain, band filling $n=1.58518$. Note the linear kink for $U(1)$ and $\mathbb{Z}_2$ critical temperatures at zero strain. (b) System has $T_c^{U(1)}>T_c^{\mathbb{Z}_2}$ at zero external strain, $n=1.58458$. Note the absence of kink in critical temperatures at zero stress for the case (b). Compressive and tensile stress are not symmetric for $\nu \neq 1$.