Fermion quantum criticality far from equilibrium

TL;DR

The paper demonstrates a novel non-equilibrium quantum critical point driven by fermionic degrees of freedom in a Lindblad-driven topological insulator. By deriving a Lindblad-Keldysh field theory and uncovering the fermionic dark state symmetry (FDS), it shows that purity is protected and only a single tuning parameter is needed to reach criticality, even though the system is far from equilibrium. A mesoscopic fermion-boson theory reveals a cubic Yukawa coupling to diffusive density modes, with RG analysis near the upper critical dimension $d_c=4$ giving a Wilson-Fisher fixed point and critical exponents $\nu=1/2-\epsilon/8$, $z=2+\epsilon/2$, and $\eta_Z=\epsilon/2$, while the pure-fermion limit remains Gaussian. The work connects this new universality class to directed percolation analogies, highlighting a symmetry-based design principle for robust, universal quantum phenomena in driven open fermionic systems with potential experimental realizations.

Abstract

Driving a quantum system out of equilibrium while preserving its subtle quantum mechanical correlations on large scales presents a major challenge, both fundamentally and for technological applications. At its core, this challenge is pinpointed by the question of how quantum effects can persist at asymptotic scales, analogous to quantum critical points in equilibrium. In this work, we construct such a scenario using fermions as building blocks. These fermions undergo an absorbing-to-absorbing state transition between two topologically distinct and quantum-correlated dark states. Starting from a microscopic, interacting Lindbladian, we derive an effective Lindblad-Keldysh field theory in which critical fermions couple to a bosonic bath with hydrodynamic fluctuations associated with particle number conservation. A key feature of this field theory is an emergent symmetry that protects the purity of the fermions' state even in the presence of the thermal bath. We quantitatively characterise the critical point using a leading-order expansion around the upper critical dimension, thereby establishing the first non-equilibrium universality class of fermions. The symmetry protection mechanism, which exhibits parallels to the problem of directed percolation, suggests a pathway toward a broader class of robust, universal quantum phenomena in fermionic systems.

Figures (2)

• Figure 1: Quantum absorbing-to-absorbing state transition. The Lindblad generator has a unique topological dark state for any value of the tuning parameter $\theta\neq 0$, see Eq. \ref{['eq:X']}. The phase transition is between fermionic dark states, which are topologically distinct by the value of a winding number $n$. The fermionic spectral gap, given by the smallest dissipation rate in the problem, closes at the transition, giving rise to divergent length and time scales characteristic of a critical point. The fermions remain in a pure state across the phase transition, emphasizing the analogy to quantum critical phenomena.
• Figure 7: (a,b) Plot for the beta functions for the complex Yukawa coupling $g$: (a) for $d>4$ and (b) for $d<4$ for $K_{\text{eff}} = K^*_{\text{eff}}$. In $d>4$ the Gaussian fixed point (black dot) is the only stable fixed point. In $d<4$, the Gaussian fixed point is unstable and there is an interacting (WF) fixed point (blue dots) with a single relevant direction (see main text). (c) Schematic of the phase diagram. The purely fermionic dark state model realizes a Gaussian quantum critical point (black dot in (b)), since the bosonic slow mode coupling to the fermions has vanishing noise level. Coupling additionally to $M\to\infty$ bosons at temperature $T>0$ activates the flow to the WF fixed point (blue in (b)), thus realizing a far-from-equilibrium interacting quantum critical point. The WF fixed point features both coherent and dissipative couplings, illustrated in the inset. At 1-loop order, the Yukawa coupling $g$ and the wave function renormalization $Z$ flow to a purely real fixed point. In contrast, the spectral mass and the kinetic coefficients become purely dissipative at the fixed point.