Dissipative free fermions in disguise

Abstract

Recently, a class of spin chains known as ``free fermions in disguise'' (FFD) has been discovered, which possess hidden free-fermion spectra even though they are not solvable via the standard Jordan-Wigner transformation. In this work, we extend this FFD framework to open quantum systems governed by the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation. We establish a general class of exactly solvable open quantum systems within the FFD framework: if the Liouvillian frustration graph is claw-free and has a simplicial clique, the Liouvillian possesses a hidden free-fermion spectrum. In particular, the (even-hole, claw)-free condition automatically guarantees this, enabling exact computation of the Liouvillian gap and an infinite-temperature autocorrelation function. Our results provide the first realization of the FFD mechanism in open quantum systems.

Figures (4)

• Figure 1: Forbidden induced subgraphs for the ECF condition; a subgraph is induced if it contains all edges of the original graph between its vertices. (a) The claw $K_{1,3}$: a center vertex connected to three mutually non-adjacent leaves. (b) Even holes $C_4, C_6, \dots$: induced cycles of even length.
• Figure 2: Examples of ECF frustration graphs. (a) Path graph corresponding to the Ising/XY chain, in which vertices $j$ and $k$ are adjacent when $|j-k|=1$. (b) Zigzag ladder corresponding to the Fendley model \ref{['eq:ffd-rep']} for $M=8$, in which vertices $j$ and $k$ are adjacent when $|j-k| \leq 2$. The shaded vertices provide examples of independent sets, with $|S|=2$ in (a) and $|S|=3$ in (b).
• Figure 3: Simplicial vs. non-simplicial cliques. The claw-free graph $G$ consists of four vertices $A,B,C,D$, and $G_\chi$ is obtained by adding the vertex $\chi$ (green) connected to the clique $K_s = \{A,B\}$ (orange). (a) $G_\chi$ is also claw-free, so $K_s$ is simplicial. (b) Without the edge $C\text{-} D$, $\Gamma[A]\setminus K_s = \{C,D\}$ is not a clique, so $K_s$ is not simplicial; indeed, adding $\chi$ creates a claw centered at $A$ whose leaves are drawn in red.
• Figure 4: Liouvillian frustration graph $\widetilde{G}_M$ for the boundary-driven Fendley model. The left (blue) and right (green) subgraphs are the two copies $G^{(1)}$ and $G^{(2)}$ of the zigzag ladder in Fig. \ref{['fig:ecf-examples']}(b). The central orange vertex is $d=\chi\otimes\chi=\sigma_M^{z}\otimes \sigma_M^{z}$, shown here for the single-vertex boundary clique $K_s=\{M\}$. The extra blue vertex represents the edge operator $\widetilde{\chi}$, and the dotted contour marks the simplicial clique $\widetilde{K}_s^{(1)}=\{d\}\cup K_s^{(1)}$.