# Quantum Kibble-Zurek physics in long-range transverse-field Ising models

## TL;DR

This paper investigates quantum phase transitions and Kibble-Zurek scaling in long-range transverse-field Ising models, providing theoretical and simulation insights for experimental observation with trapped ions.

## Contribution

It offers a detailed analysis of quantum critical behavior and non-equilibrium dynamics in long-range Ising models, including a scheme for digital quantum simulation with trapped ions.

## Key findings

- Critical exponents and phase diagram determined via DMRG.
- Universal Kibble-Zurek scaling laws identified in non-equilibrium dynamics.
- Feasible experimental parameters for observing these phenomena in trapped ion systems.

## Abstract

We analyze the quantum phase transitions taking place in a one-dimensional transverse field Ising model with long-range couplings that decay algebraically with distance. We are interested in the Kibble-Zurek universal scaling laws emerging in non-equilibrium dynamics and in the potential for the unambiguous observation of such behavior in a realistic experimental setup based on trapped ions. To this end, we determine the phase diagram of the model and the critical exponents characterizing its quantum phase transitions by means of density-matrix renormalization group calculations and finite-size scaling theory, which allows us to obtain good estimates for different range of ferro- and antiferromagnetic interactions. Beyond critical equilibrium properties, we tackle a non-equilibrium scenario in which quantum Kibble-Zurek scaling laws may be retrieved. Here it is found that the predicted non-equilibrium universal behavior, i.e. the scaling laws as a function of the quench rate and critical exponents, can be observed in systems comprising an experimentally feasible number of spins. Finally, a scheme is introduced to simulate the algebraically decaying couplings accurately by means of a digital quantum simulation with trapped ions. Our results suggest that quantum Kibble-Zurek physics can be explored and observed in state-of-the-art experiments with trapped ions realizing long-range Ising models.