Critical phenomena and Kibble–Zurek scaling in the long-range quantum Ising chain

Abstract

We investigate an extension of the quantum Ising (QI) 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 (MPSs) 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 QI model with MPSs. 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 percent. 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.

Keywords

• long-range quantum Ising
• entangled quantum dynamics
• Kibble–Zurek scaling
• matrix product states
• trapped ion arrays
• ultracold molecules and Rydberg gases