This is a supplementary material for arxiv:1507.04712,
This is a supplementary material for arxiv:1507.04712. See the published version in Phys. Rev. Lett.
Introduction
The two-component Ginzburg-Landau relevant in the case of an interband dominated pairing, in dimensionless units reads as \begin{align*} {\cal F}&= \frac{\B^2}{8\pi} +\sum_{j=1}^2k_j|\D\psi_j|^2 + \alpha_j|\psi_j|^2+\frac{1}{2}\beta_j|\psi_j|^4 \\ &+ k_{12,a}(\D_a^*\psi_1^*\D_a\psi_2+c.c. ) %\\& + \gamma|\psi_1|^2|\psi_2|^2 + \frac{\delta}{2}(\psi_1^*\psi_2+c.c) \end{align*} with $\D=\Grad-2i\tilde{e}\A$. The components $\psi_{1,2}$ are determined by a superposition of the different gap functions $\Delta_i$. All the coefficients of the model are consistently determined from the microscopic coupling matrix. The general GL functional describes both the $s+is$ and $s+id$ states which differ by the structure of mixed gradient terms. In the case of $s+is$ symmetry, $k_{12,x}=k_{12,y}$, while for $s+id$ symmetry mixed-gradients satisfy $k_{12,x}=-k_{12,y}$. In the later case, these are responsible for the explicit breaking of the crystalline $C_4$ symmetry which is unbroken in the $s+is$ case. To study magnetic and electric responses of both $s+is$ and $s+id$ states, we use a Time-Dependent Ginzburg-Landau (TDGL) approach generalized to a multiband case \cite{Supplementary} \begin{equation} \Gamma\Big(\partial_t-\frac{2ie}{\hbar}\varphi\Big)\psi_k= -\frac{\delta\F}{\delta\psi_k^*} \,,~~~ \frac{c}{4\pi}\Curl\B -\sigma_n \E={\bm j}_s \,, \end{equation} where $\Gamma$ is the damping constant which we assume for simplicity to be the same for all $k$ components, $\varphi$ is the electrostatic potential and $\sigma_n$ the normal state conductivity, ${\bs j}_s = - c\delta\F/\delta\A$ is the superconducting current.

In order to investigate the response to spatial modulations of the components of the order parameter induced by a local heat source, the fields $\psi_{1,2}$ and $\A$ are discretized using a finite-element framework provided by Freefem++ library.

Time-varying hotspot and details for simulations: To model the local heating, the temperature profile is found by solving the (stationary) heat equation for an heat source while the boundaries are kept at $T_0=0.7T_c$. Once the temperature profile is found, the coefficient $\alpha_k$ vary in space and the TDGL equations are evolved for $\Delta t=80$ (in dimensionless). The temperature of the heat source is then modified to $T^\prime_s$, and the TDGL equations are further evolved for the new temperature profile for a period $\Delta t$, thus corresponding to an adiabatic process. The temperature of the source is initially set to $T_0$, sequentially ramped-up to $0.95T_c$ and then ramped-down back to $T_0$. In our simulations, we chose the dimensionless conductivity $\sigma_n=0.1$ and the gauge coupling constant $\tilde{e}=0.113$. The coefficients in GL functional are determined using the microscopic coupling matrix with coefficients $\eta=5$ and $\lambda=4.5$.
1. Magnetic response to a time-varrying hotspot
The displayed quantities are : The leftmost panel shows the time evolution of the source's temperature and the red dot denotes the position in the time series.The two upper panels show the response of an $s+is$ superconducting state respectively with a domain wall and the homogeneous state. The two lower panels show the response of an $s+id$ superconducting state respectively with a domain wall and the homogeneous (anisotropic) state. The color plot shows the magnitude of the out-of-plane induced magnetic field $B_z$ in unit of the second critical field.


Movie 1 - Evolution of the magnetic response that originates in time-varying local heating of the sample. The overall time sequence runs for $t/t_0=2400$. This corresponds to Figure 2 of the main text.
2. Electric response to a time-varrying hotspot
The displayed quantities are : The leftmost panel shows the time evolution of the source's temperature and the red dot denotes the position in the time series. The color plot shows the voltage induced in the normal detector by the charge imbalance due to the non-stationary heating. The two upper panels show the response of an $s+is$ superconducting state respectively with a domain wall and the homogeneous state. The two lower panels show the response of an $s+id$ superconducting state respectively with a domain wall and the homogeneous (anisotropic) state.


Movie 2 - This shows Evolution of the electric response that originates in time-varying local heating of the sample. The overall time sequence runs for $t/t_0=2400$. This corresponds to Figure 3 of the main text.
References
  1. J. Garaud, M. Silaev and E. Babaev
    Thermoelectric Signatures of Time-Reversal Symmetry Breaking States in Multiband Superconductors
    Phys. Rev. Lett. 116, 097002, (2016). Link , arXiv