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Personne# Ivo Aguiar Maceira

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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.

Ivo Aguiar Maceira, Frédéric Mila

Collective tunneling is a ubiquitous phenomenon in finite-size spin clusters that shows up in systems as diverse as molecular magnets or spin clusters adsorbed at surfaces. The basic problem we explore is to understand how small flipping terms can cooperate to flip a large spin to the opposite direction or a cluster of interacting elementary Ising spins into the time-reversed state. These high-order processes will involve at least two channels, a single spin-flip channel due to a transverse field and a two-spin flip channel due to exchange or other pairwise interactions or due to single-ion anisotropies. In view of the complexity of high-order perturbation theory, nonperturbative approaches based on large-spin path integrals were developed when this problem was first addressed in the context of single spin models. In the present paper, we show that high-order perturbation theory can in fact be formulated and evaluated with the help of simple recurrence relations, leading to a compact theory of tunneling in macroscopic spins, in one-dimensional clusters, as well as in small higher-dimensional clusters. This is demonstrated explicitly in the case of the Ising model with a transverse field and transverse exchange, and in the case of macroscopic spins with uniaxial anisotropy. Our approach provides a transparent theory of level crossings, where the tunneling between time-reversed configurations vanishes as a function of the external field. Those crossings result from the destructive quantum interferences between competing flipping channels. Destructive interferences are expected to be present as soon as the two-spin flip channels have an overall positive amplitude and thus compete with the intrinsically negative second-order processes due to the transverse field. Our theory consistently predicts N crossings in chains of N Ising spins and 2S crossings in single spins of magnitude S and yields explicit analytical formulas for the level crossings of open chains and macroscopic spins. Disorder can be easily implemented in this perturbative formalism. Leading disorder effects can be treated analytically for spin rings. We find that at the smallest transverse field crossing the suppression of tunneling is most robust with respect to disorder and fluctuations in the parameters. We briefly discuss the implications of our findings for the use of realistic spin clusters on surfaces to store information.

Ivo Aguiar Maceira, Natalia Chepiga, Frédéric Mila

Using density matrix renormalization group simulations on open chains, we map out the wave vector in the incommensurate disordered phase of a realistic model of Rydberg chains with 1/r6 interactions, and we locate and characterize the points along the commensurate lines where the transition out of the period 3 and 4 phases is conformal. We confirm that it is three-state Potts for the period-3 phase, and we show that it is Ashkin-Teller with v similar or equal to 0.80 for the period-4 phase. We further show that close to these points, the transition is still continuous, but with a completely different scaling of the wave vector, in agreement with a chiral transition. Finally, we propose to use the conformal points as benchmarks for Kibble-Zurek experiments, defining a roadmap towards a conclusive identification of the chiral universality class.