# Universal dynamical scaling of long-range topological superconductors

## TL;DR

This paper investigates the out-of-equilibrium dynamics of long-range p-wave superconducting wires, revealing how the heat generated during a phase transition scales with the quench rate and identifying regimes with universal behavior.

## Contribution

It introduces a scaling law for heat production during quenches in long-range superconductors and uncovers anomalous dynamical universality in certain parameter regimes.

## Key findings

- Heat scales with quench rate as δ^θ, with θ depending on interaction range.
- Universal scaling can be linked to equilibrium critical exponents in specific regimes.
- Anomalous dynamical universality occurs when hopping dominates over pairing.

## Abstract

We study the out-of-equilibrium dynamics of $p$-wave superconducting quantum wires with long-range interactions, when the chemical potential is linearly ramped across the topological phase transition. We show that the heat produced after the quench scales with the quench rate $\delta$ according to the scaling law $\delta^\theta$, where the exponent $\theta$ depends on the power law exponent of the long-range interactions. We identify the parameter regimes where this scaling can be cast in terms of the universal equilibrium critical exponents and can thus be understood within the Kibble-Zurek framework. When the electron hopping decays more slowly in space than pairing, it dominates the equilibrium scaling. Surprisingly, in this regime the dynamical critical behaviour arises only from paring and, thus, exhibits anomalous dynamical universality unrelated to equilibrium scaling. The discrepancy from the expected Kibble-Zurek scenario can be traced back to the presence of multiple universal terms in the equilibrium scaling functions of long-range interacting systems close to a second order critical point.