Level crossings induced by a longitudinal coupling in the transverse field Ising chain

We study the effect of antiferromagnetic longitudinal coupling on the one-dimensional transverse field Ising model with nearest-neighbor couplings. In the topological phase where, in the thermodynamic limit, the ground state is twofold degenerate, we show that, for a finite system of N sites, the longitudinal coupling induces N level crossings between the two lowest-lying states as a function of the field. We also provide strong arguments suggesting that these N level crossings all appear simultaneously as soon as the longitudinal coupling is switched on. This conclusion is based on perturbation theory and a mapping of the problem onto the open Kitaev chain, for which we write down the complete solution in terms of Majorana fermions.

Level crossings and chiral transitions in transverse-field Ising models of adatom and Rydberg chains

Ivo Aguiar Maceira

This thesis is motivated by recent experiments on systems described by extensions of the one-dimensional transverse-field Ising (TFI) model where (1) finite-size properties of Ising-ordered phases -- specifically, ground state level crossings -- were observed and (2) continuous phase transitions related to the p-state chiral clock model were probed, with interesting but only partially conclusive results regarding the nature of the phase transitions and the possible existence of a chiral universality class.For (1), the relation of the level crossings to topologically protected edge modes of Majorana fermion models is discussed, and the implication is made explicit that the auto-correlation time of the edge spins may be infinite even for non-integrable albeit finite systems, and independently of temperature. The level crossings are then reinterpreted in the context of degenerate perturbation theory as being a consequence of destructive quantum interference between tunneling processes involving different numbers of spin-flip operations. It is shown that this phenomenon is independent of the lattice geometry, as long as this one is not geometrically frustrated, and is ubiquitous to TFI-like models, being also found in single spin-S systems, the latter having already been observed in the field of magnetic molecules. The effect of disorder on the crossings is studied in lowest order.For (2), we perform density matrix renormalization group simulations on open chains to investigate the experimentally observed quantum phase transitions and we conclude that isolated conformal critical points exist along the p=3 and p=4 critical boundaries: we accurately locate such points and characterize their universality classes by determining critical exponents numerically, where we find that the p=3 agrees with a 3-state Potts universality class and the p=4 point agrees with an Ashkin-Teller universality class with Îœ â 0.80 (Î» â 0.5). Our results are in favor of the existence of chiral transition lines surrounding the conformal points, beyond which a gapless intermediate phase is expected.

EPFL2022

Level crossings and chiral transitions in transverse-field Ising models of adatom and Rydberg chains

Ivo Aguiar Maceira

EPFL2022