Two-parameter Family-Vicsek scaling in a dissipative XXZ spin chain

1. Problem Statement

The paper investigates the universal features of far-from-equilibrium dynamics in open quantum many-body systems. Specifically, it addresses how Family–Vicsek (FV) scaling, a concept originally developed for classical surface growth, applies to the fluctuations of transferred magnetization in a quantum spin chain subject to both coherent dynamics and bulk dissipation (gain and loss).

• Context: In isolated quantum systems, FV scaling describes how the roughness $W(\ell, t)$ of a fluctuating observable (like segment magnetization) grows in time and saturates due to finite-size effects, governed by exponents $\alpha$ (saturation), $\beta$ (growth), and $z$ (dynamical exponent, $z=\alpha/\beta$).
• Challenge: It is unclear how this scaling behaves in open quantum systems where Markovian dissipation (Lindblad dynamics) competes with coherent unitary evolution. Does the standard one-parameter scaling ($t/\ell^z$) survive, or does dissipation introduce new scaling regimes?
• Model: The authors study a dissipative XXZ spin chain with homogeneous local gain and loss, starting from a non-equilibrium steady state (NESS) with finite magnetization.

2. Methodology

The authors employ a combination of analytical derivations (for the non-interacting limit) and advanced numerical simulations (for interacting cases).

• Model Definition:

  • Hamiltonian: XXZ chain with anisotropy $\Delta$.
  • Dissipation: Modeled via a Lindblad master equation with local jump operators for spin loss ($\sigma^-$) and gain ($\sigma^+$) at rates $\gamma_l$ and $\gamma_p$.
  • Initial State: The system is prepared in the exact NESS of the driven-dissipative system, which is a diagonal product state in the $\sigma^z$ basis with a tunable spin fugacity $\zeta = \gamma_p/\gamma_l$.

• Quantifying Fluctuations:

  • Instead of measuring a single-time observable, they define the transferred segment magnetization $\Delta \Sigma_\ell(t) = \Sigma_\ell(t) - \Sigma_\ell(0)$.
  • They utilize a two-time Quantum Generating Function (QGF), $G_\ell(\lambda, t) = \text{Tr}[e^{i\lambda\Sigma_\ell} e^{\mathcal{L}t}(e^{-i\lambda\Sigma_\ell}\rho_{SS})]$, to compute the second cumulant $\kappa_2$.
  • The roughness is defined as $W(\ell, t) = \sqrt{\kappa_2}$.

• Numerical Techniques:

  • Vectorized TEBD: The density matrix is represented as a Matrix Product State (MPS) in Liouville space. The time evolution is performed using the Time-Evolving Block Decimation (TEBD) algorithm.
  • Counting Field Protocol: To extract cumulants, they evolve a "twisted" initial state $\rho_\lambda(0) = e^{-i\lambda\Sigma_\ell}\rho_{SS}$ and compute the overlap with $e^{i\lambda\Sigma_\ell}$. They use complex phases of $\lambda$ to cancel truncation errors.

3. Key Contributions and Results

A. Non-Interacting Limit ($\Delta = 0$): Analytical Solution

In the XX limit, the model maps to free spinless fermions, allowing for an exact analytical solution.

• Closed-Form Expression: The authors derive an exact formula for the second cumulant:
  $$\kappa_2(\ell, t) = 2\bar{n}(1-\bar{n}) \left[ \ell - e^{-\Gamma t} \sum_{r=-(\ell-1)}^{\ell-1} (\ell-|r|) J_r^2(Jt) \right]$$
  where $\Gamma = \gamma_l + \gamma_p$ is the total dissipation rate, $\bar{n}$ is the filling, and $J_r$ are Bessel functions.
• Two-Parameter Scaling: The presence of dissipation introduces a second dimensionless parameter $y = \Gamma t$ alongside the standard ballistic variable $x = Jt/\ell$. The roughness follows a two-parameter FV scaling:
  $$W(\ell, t) \sim \ell^\alpha f\left( \frac{t}{\ell^z}, \Gamma t \right)$$
• Regimes:
  1. Unitary-like Growth: For short times ($t \ll \min(t^*, t_\Gamma)$), the system exhibits ballistic growth ($W \sim t^{1/2}$) with $z=1$, identical to the closed system.
  2. Dissipation-Dominated Saturation: If the dissipation rate is high ($t_\Gamma \ll t^*$), the roughness saturates at $t \sim t_\Gamma = \Gamma^{-1}$, well before the finite-size time $t^* \sim \ell/J$. In this regime, the standard one-parameter collapse fails, and the dynamics are governed primarily by $\Gamma t$.

B. Interacting Case ($\Delta \neq 0$): Numerical Results

Using TEBD, the authors explore the impact of interactions and integrability breaking.

• Unitary Evolution (Closed System):
  • Finite Magnetization ($\zeta \neq 1$): The system exhibits robust ballistic scaling ($z \approx 1$) across a wide range of anisotropies $\Delta$. Interactions renormalize amplitudes but do not change the universality class, consistent with magnon-like quasiparticle propagation.
  • Infinite Temperature ($\zeta = 1$): The dynamical exponent $z$ depends on $\Delta$: ballistic ($z=1$) for $|\Delta|<1$, KPZ-type superdiffusion ($z \approx 1.5$) at $\Delta=1$, and diffusive ($z \approx 2$) for $\Delta > 1$.
• Lindbladian Evolution (Open System):
  • Breakdown of One-Parameter Scaling: For interacting chains under dissipation, the coherent one-parameter FV collapse (based on $t/\ell^z$) breaks down.
  • Dissipation Control: The roughness saturates at $t \sim t_\Gamma$ regardless of the interaction strength or integrability. The dynamics are dominated by the dissipative relaxation time, leading to a collapse governed by the variable $\Gamma t$.

C. Integrability Breaking

The authors add next-nearest-neighbor (NNN) couplings ($J_2$) to break integrability while preserving $U(1)$ symmetry.

• Unitary Case: At finite magnetization, the ballistic scaling ($z=1$) persists. The NNN term merely renormalizes the effective propagation velocity, shifting the crossover time $t^*$ but preserving the scaling form.
• Dissipative Case: Integrability breaking does not restore the coherent one-parameter scaling. The system remains in the dissipation-dominated regime where $t_\Gamma$ controls the saturation, confirming that bulk gain/loss is the dominant mechanism for memory loss in the QGF.

4. Significance and Conclusions

• Universal Scaling in Open Systems: The paper establishes that FV scaling is a robust framework for open quantum systems but requires a two-parameter generalization when dissipation is significant.
• Competition of Time Scales: The central physical insight is the competition between the coherent finite-size time ($t^* \sim \ell/v$) and the dissipative relaxation time ($t_\Gamma \sim \Gamma^{-1}$).
  • If $t_\Gamma \gg t^*$, the system behaves like a closed system (ballistic transport).
  • If $t_\Gamma \ll t^*$, the system enters a "dissipation-dominated" regime where fluctuations are erased before they can spread ballistically, leading to a collapse governed solely by $\Gamma t$.
• Robustness: The findings are stable against integrability breaking. While interactions and integrability breaking dictate the transport class in closed systems, open systems with bulk gain/loss are generically controlled by the dissipation rate, suppressing the transport universality classes (ballistic, KPZ, diffusive) in favor of a dissipation-controlled saturation.

This work provides a comprehensive theoretical and numerical framework for understanding how environmental noise modifies the universal scaling laws of non-equilibrium quantum dynamics.