Signatures of Quantum Phase Transitions in Driven Dissipative Spin Chains

Abstract

Open driven quantum systems have defined a powerful paradigm of nonequilibrium phases and phase transitions; however, quantum phase transitions are generically not expected in this setting due to the decohering effect of dissipation. In this Letter, we consider a quantum Ising model subject to bulk dissipation (at rate $Γ$) and show that, although the correlation length remains finite (hence no phase transition), it develops a pronounced peak close to the ground-state quantum critical point. While standard techniques fail in this regime, we develop a versatile analytical approach that becomes exact with vanishing dissipation ($Γ\to 0$ but finite $Γt$). On a technical level, our approach builds on previous work where the state of the system is described by a slowly evolving generalized Gibbs ensemble that accounts for the integrability of the Hamiltonian while treating dissipation perturbatively. Finally, we demonstrate a kind of universality in that integrability-breaking perturbations of the Hamiltonian lead to the same behavior. To this end, we first show that the steady state of a chaotic Ising Hamiltonian under local Markovian dissipation that preserves the Ising symmetry, and in the limit $Γ\to 0$, is identical to that of quench dynamics in the absence of dissipation. This intriguing connection then allows us to draw on recent findings about quantum phase transition signatures in quench dynamics.

Figures (3)

• Figure 1: Schematic presentation of the correlation length in the steady state of a driven-dissipative Ising chain as a function of a tuning parameter $h$ (e.g., an external field), compared to that of the ground state. While being very distinct, the steady state still features a peak close to the ground-state quantum critical point $h_c$. An analytical approach is lacking in the regime highlighted by the question mark.
• Figure 2: Correlation length $\xi$ calculated in the steady state of the driven-dissipative Ising model. MPS, spin-wave theory, and the free-fermion jump calculations are done at $\Gamma=0.15$. The solution that we developed, plotted in solid line, is asymptotically exact in the limit $\Gamma \to 0$. The vertical dashed line marks the QCP. In the inset, we present extensive MPS calculations for different $h$ and $\Gamma$ and for $L=40$ spins ($L=50$ does not result in a significant change). The dashed line indicates the position of the maximum correlation length for a given value of $\Gamma$. The correlation length decays as $\Gamma$ increases, but the peak remains close to the critical point.
• Figure 3: Correlation length $\xi$ computed in the steady state of the driven-dissipative NNN Ising model ($\Gamma=0.15$) obtained from MPS numerical results. The corresponding QCPs for $0\leq J_2 \leq 0.5$ are at $1\leq h_c \lessapprox 1.78$