Dissipation-driven topological phase transitions in open quantum systems independent of system Hamiltonian

TL;DR

Open quantum systems require a reformulation of topology beyond closed-Hamiltonian settings. Using the modular Hamiltonian formalism for 1D Gaussian Lindblad dynamics, the paper proves that the steady-state $Z_2$ invariant in class D depends only on dissipation, not on the Hamiltonian, with $\nu=\mathrm{sgn}[y(0)]\,\mathrm{sgn}[y(\pi)]$. During a dissipative quench, dynamical transitions occur at analytically predictable times $t_{p,k_s}$ that hinge solely on dissipation parameters, enabling multiple transitions even when initial and final invariants coincide. The entanglement-spectrum analysis reveals bulk-edge signatures in non-equilibrium density matrices: SPES gap closures under periodic boundaries and protected zero modes under open boundaries. Collectively, these results offer a practical framework to detect and control dissipation-induced topology in quantum simulators via measurable entanglement observables.

Abstract

We investigate dissipation-driven topological phase transitions in one-dimensional quantum open systems governed by the Lindblad equation with linear dissipation operators, which ensure the density matrix retains its Gaussian form throughout the dynamics. By employing the modular Hamiltonian framework, we rigorously demonstrate that the $\rm{Z}_2$ topological invariant characterizing steady states in one-dimensional class D systems is exclusively dependent on the dissipation operators, rather than the system Hamiltonian. Through a sudden quench protocol where the system evolves from the steady state of one Lindbladian to another, we reveal that topological transitions can occur at analytically predictable critical times, even when the initial and final steady states share identical topological indices. These transitions are shown, both analytically and numerically, to depend solely on dissipation parameters. Entanglement spectrum analysis demonstrates bulk-edge correspondence in non-equilibrium density matrices via coexisting single-particle gap closures (periodic boundaries) and topologically protected zero modes (open boundaries), directly underpinning the detection of dissipation-induced topology in quantum simulators.

Figures (4)

• Figure 1: Topological phase diagram of Lindblad steady state. Different colors denote different $(M_0,M_\pi)$. Red area represents $(M_0,M_\pi)=(-1,-1)$, blue area represents $(M_0,M_\pi)=(-1,1)$, green area represents $(M_0,M_\pi)=(1,-1)$ and white area represents $(M_0,M_\pi)=(1,1)$. Also, we have taken $v_2/v_1=-2$, and the dimensionless parameter $\gamma=0.1$. Note that the system Hamiltonian parameters have no influence on the topological properties of the steady state, although the validity of this phase diagram requires that the Hamiltonian is chosen to ensure a non-vanishing dissipative gap.
• Figure 2: (a)(b)(c)Topological phase transition characterized by the modular Hamiltonian's topological invariant $\nu(t)$ following a dissipative quench from Lindbladian steady state of $(M_0,M_\pi)=(1,-1)$ topological phase to those of $(M_0,M_\pi)=(1,1)$ phase. During the entire quench process, $v_2$ and $v_1$ remain constant, the ratio $v_2/v_1$ is maintained at $-2$, and the dimensionless parameter $\gamma$ takes the value $0.1$. Initial density matrix is the steady state of the Lindbladian with parameters $u_{1,i}/v_1=3$, $u_{2,i}/v_1=3$ and the post-quench Lindbladian parameters are taken as $u_{1,f}/v_1=2.5, u_{2,f}/v_1=-1$ for (a), $u_{1,f}/v_1=3.5, u_{2,f}/v_1=-1$ for (b), $u_{1,f}/v_1=4.5, u_{2,f}/v_1=-1$ for (c). (d)Topological phase transition time, prediected by $t_{p,0}$, varies with the post-quench parameter $u_{1,f}$.
• Figure 3: (a)(b)(c)Topological phase transition characterized by the modular Hamiltonian's topological invariant $\nu(t)$ following a dissipative quench from Lindbladian steady state of $(M_0,M_\pi)=(-1,-1)$ topological phase to those in $(M_0,M_\pi)=(1,1)$ phase. During the entire quench process, the ratio $v_2/v_1$ is maintained at $-2$, and the dimensionless parameter $\gamma$ takes the value $0.1$. Initial density matrix is the steady state of the Lindbladian with parameters $u_{1,i}/v_1=1$, $u_{2,i}/v_1=-1$ and the post-quench Lindbladian parameters are taken as $u_{1,f}/v_1=2.5, u_{2,f}/v_1=-1$ for (a), $u_{1,f}/v_1=3.5, u_{2,f}/v_1=-1$ for (b), $u_{1,f}/v_1=4.5, u_{2,f}/v_1=-1$ for (c). (d)Critical time of topological phase transition prediected by $t_{p,0}$ and $t_{p,\pi}$ vary with the post-quench parameter $u_{1,f}$ .
• Figure 4: Time-evolution of single particle entanglement spectrum. (a)(b) shows the single particle entanglement spectrum under periodical boundary conditions with gap closing dynamics and (c)(d) shows the single particle entanglement spectrum under open boundary conditions (OBC) with the topologically protected zero-mode structure. In (a)(c), the parameters is taken the same as Fig.\ref{['fig:fig2']}(a) and the time-evolving density matrix undergoes one topological phase transitions with its topological invariant changing from $-1$ to $+1$. In (b)(d), the parameters is taken the same as Fig.\ref{['fig:fig3']}(a) and the time-evolving density matrix undergoes two topological phase transitions with its topological invariant changing from $1$ to $-1$, and sequentially from $-1$ to $1$. Also, we have taken the system Hamiltonian of Eq.\ref{['eqlind']} as $\hat{H}=J\hat{c}_m^\dagger\hat{c}_{m+1}+J\hat{c}_{m+1}^\dagger\hat{c}_{m}$ with $J/v_1=1$ and the system size $N=500$ in this figure.