Kramers' degeneracy for open systems in thermal equilibrium

Kramers' degeneracy theorem underpins many interesting effects in quantum systems with time-reversal symmetry. We show that the generator of dynamics for Markovian open fermionic systems can exhibit an analogous degeneracy, protected by a combination of time-reversal symmetry and the microreversibility (detailed balance) property of systems at thermal equilibrium—the degeneracy is lifted if either condition is not met. We provide simple examples of this phenomenon and show that the degeneracy is reflected in the single-particle Green's functions. Furthermore, we show that certain experimental signatures of topological edge modes in open many-body systems can be protected by microreversibility in the same way. Our results highlight the importance of detailed balance in characterizing open topological matter.

Figure 1

(a) A fermionic system (three small circles) coupled to an environment (big circle) in thermal equilibrium at temperature ${\beta }^{-1}$ has a Kramers' degeneracy. (b) The same system (three small circles) coupled to two baths (two big circles) at different temperatures ${\beta }_1^{-1}, {\beta }_2^{-1}$ will not be Kramers' degenerate. This leads to a spin splitting of the Green's function (13), which can be experimentally detected using, e.g., spin-resolved tunneling spectroscopy (bottom panels). For both (a) and (b), the system-bath Hamiltonian obeys TRS: $[H_{SB},T]=0$.

Figure 2

(a) Three doubly degenerate energy levels ($N=3$) labeled by ${\epsilon }_{i,\pm }$. TRS-respecting dissipators cause jumps between energy levels at a rate consistent with detailed balance in thermal equilibrium at temperature ${\beta }^{-1}$. (b) Lindblad spectrum for the model described in the main text (for states with 0, 1, or 2 occupied fermions) at a temperature $\beta \mathrm{\Delta }=1$, with $N=3, {\mathrm{\Delta }}_d/\mathrm{\Delta }=0.1$, and $g/\mathrm{\Delta }=1$. The odd-superparity sector (blue and red dots) is twofold degenerate, while the even sector (black dots) is not. (c) Real part of the retarded Green's function for same parameters as in (b). The Kramers' degeneracy ensures that $G_{i,\sigma }=G_{i,-\sigma }$ in the expectation value of the thermal steady state with one occupied fermionic mode: black line and red dashed line correspond to $\text{Re}\left[G_{1,+}\right]$ and $\text{Re}\left[G_{1,-}\right]$, respectively. In general, Green's function pairs other than $G_{i,\pm }$ are nondegenerate: blue line corresponds to $\text{Re}\left[G_{2,+}\right]$. (d) A system that is coupled to two thermal baths $g_1/\mathrm{\Delta }=1,g_2/\mathrm{\Delta }=0.4$ at different temperatures: ${\beta }_1\mathrm{\Delta }=1,{\beta }_2\mathrm{\Delta }=10$ [otherwise same parameters as in (b)]. Microreversibility is broken, hence the Green's functions for $i,+$ and $i,-$ split.

Figure 3

Steady-state, spin-resolved spectral function at the boundary of the Majorana chain (14) coupled to two baths found via exact diagonalization. (a) Zero-bias peak due to the Majorana modes. The width of the peaks is set by the temperature. (b) Splitting between spin-up and spin-down spectral functions due to breaking of microreversibility via a nonequilibrium environment. Parameters: $N=3,u_1/v_1=0.25,u_2/v_1=0.25,v_2/v_1=0.5,\beta v_1=3.3,g/v_1=0.05,g_{\text{neq}}/v_1=2$.