# Nonequilibrium quantum criticality in bilayer itinerant ferromagnets

## TL;DR

This paper develops a theoretical framework for understanding nonequilibrium quantum criticality in a bilayer system of itinerant electron magnets, revealing multi-scale critical behavior and the dominance of a z=3 dynamical exponent due to interlayer coupling.

## Contribution

It introduces a microscopic model and effective action for coupled bilayer magnets under nonequilibrium conditions, highlighting the emergence of a low energy fixed point with a specific dynamical critical exponent.

## Key findings

- Multi-scale quantum criticality with different dynamical exponents in uncoupled layers.
- Interlayer coupling leads to a dominant z=3 dynamical critical exponent.
- Nonequilibrium drive does not simply mimic thermal temperature effects.

## Abstract

We present a theory of nonequilibrium quantum criticality in a coupled bilayer system of itinerant electron magnets. The model studied consists of the first layer subjected to an inplane current and open to an external substrate. The second layer is closed and subject to no direct external drive, but couples to the first layer via short-ranged spin exchange interaction. No particle exchange is assumed between the layers. Starting from a microscopic fermionic model, we derive an effective action in terms of two coupled bosonic fields which are related to the magnetization fluctuations of the two layers. When there is no interlayer coupling, the two bosonic modes possess different dynamical critical exponents z with z=2 (z=3) for the first (second) layer. This results in multi-scale quantum criticality in the coupled system. It is shown that the linear coupling between the two fields leads to a low energy fixed point characterized by the larger dynamical critical exponent z=3. The perturbative renormalization group is used to compute the correlation length in the quantum disordered and quantum critical regimes. We also derive the stochastic dynamics obeyed by the critical fluctuations in the quantum critical regime. Comparing the nonequilibrium situation to the thermal equilibrium scenario, where the whole system is at a temperature T, we find that the nonequilibrium drive does not always play the role of temperature.