Critical Phenomena and Kibble-Zurek Scaling in the Long-Range Quantum Ising Chain

TL;DR

This paper explores the effects of long-range interactions on the quantum Ising model, demonstrating how they influence critical points and defect scaling during quantum phase transitions.

Contribution

It introduces numerical and theoretical methods to analyze long-range quantum Ising chains, revealing significant shifts in critical points and altered Kibble-Zurek scaling behavior.

Findings

Finite size effects shift the critical point by 15%

Longer-range interactions slow defect density increase

Critical exponent changes by 25% with interaction range

Abstract

We investigate an extension of the quantum Ising model in one spatial dimension including long-range $1 / r^{\alpha}$ interactions in its statics and dynamics with possible applications from heteronuclear polar molecules in optical lattices to trapped ions described by two-state spin systems. We introduce the statics of the system via both numerical techniques with finite size and infinite size matrix product states and a theoretical approaches using a truncated Jordan-Wigner transformation for the ferromagnetic and antiferromagnetic case and show that finite size effects have a crucial role shifting the quantum critical point of the external field by fifteen percent between thirty-two and around five-hundred spins. We numerically study the Kibble-Zurek hypothesis in the long-range quantum Ising model with Matrix Product States. A linear quench of the external field through the quantum critical point yields a power-law scaling of the defect density as a function of the total quench time. For example, the increase of the defect density is slower for longer-range models and the critical exponent changes by twenty-five per cent. Our study emphasizes the importance of such long-range interactions in statics and dynamics that could point to similar phenomena in a different setup of dynamical systems or for other models.