We show that a trapped-ion chain interacting with an optical spin-dependent force shows strong frustration effects due to the interplay between long range interactions and the dressing by optical phases. We consider a strong spin-phonon coupling regime and predict a quantum phase diagram with different competing magnetic and structural orders. In a highly frustrated region, the system shows enhanced quantum fluctuations and entanglement, which are characteristic of spin-liquid phases. We propose and describe within a mean-field approach a quantum annealing process to induce a quasi-adiabatic evolution towards the ground state.

Topological insulating phases are primarily associated with condensed-matter systems, which typically feature short-range interactions. Nevertheless, many realizations of quantum matter can exhibit long-range interactions, and it is still largely unknown the effect that these latter may exert upon the topological phases. In this Letter, we investigate the Su-Schrieffer-Heeger topological insulator in the presence of long-range interactions. We show that this model can be readily realized in quantum simulators with trapped ions by means of a periodic driving. Our results indicate that the localization of the associated edge states is enhanced by the long-range interactions, and that the localized components survive within the ground state of the model. These effects could be easily confirmed in current state-of-the-art experimental implementations.Introduction.-Topological phases are one of the most exotic forms of quantum matter. Among their many intriguing traits, we find that they are robust against local decoherence processes, or feature fractional particle excitations with prospective applications in quantum information processing [1, 2]. Some of the simplest systems showcasing non-trivial topological order are the topological insulators [3][4][5][6], gapped phases of non-interacting fermions which present gapless edge states. Despite of several experimental realizations [7, 8], the preparation and measurement of topological insulators is typically difficult in the solid state. Analog quantum simulators [5, 9-12, 14, 15], on the other hand, offer the possibility of exploring and exploiting the topological insulating phases, because of their inherent high degree of controllability. Furthermore, interactions in a quantum simulator can be tuned at will, opening up the possibility of investigating new regimes of the underlying models.Topological edge states usually occur in the insulating phase as long as an associated bulk invariant attains a nontrivial value, and the generic symmetries of the underlying Hamiltonian are preserved [16]. This property -known as the bulk-edge correspondence-is a generic feature of topological insulators. However, if interactions are taken into account, the presence of edge states is no longer guaranteed. For instance, it has been shown that one of the edge states present in the Mott insulating phase of the Bose-Hubbard model on a 1D superlattice is not stable against tunneling [17]. In this work, we extend these considerations to the case of interactions which are explicitly long ranged. Since topological phases are characteristically robust against local perturbations, but longrange interactions may not qualify as such, there is an ongoing effort to elucidate their effect upon the topological states [18][19][20]. This question is not of exclusive theoretical interest, since many experimental systems implementing topological phases of matter feature long-range interactions. In particular, we will show that trapped-ion quantum simulators can realize a long-range interacting version...

Abstract. We present a theoretical description of a system of many spins strongly coupled to a bosonic chain. We rely on the use of a spinwave theory describing the Gaussian fluctuations around the mean-field solution, and focus on spin-boson chains arising as a generalization of the Dicke Hamiltonian. Our model is motivated by experimental setups such as trapped ions, or atoms/qubits coupled to cavity arrays. This situation corresponds to the cooperative (E⊗β) Jahn-Teller distortion studied in solid-state physics. However, the ability to tune the parameters of the model in quantum optical setups opens up a variety of novel intriguing situations. The main focus of this paper is to review the spin-wave theoretical description of this problem as well as to test the validity of mean-field theory. Our main result is that deviations from mean-field effects are determined by the interplay between magnetic order and mesoscopic cooperativity effects, being the latter strongly size-dependent.

Rabi lattice models with discrete gauge symmetry: Phase diagram and implementation in trapped-ion quantum simulators Article (Published Version) http://sro.sussex.ac.uk Nevado, Pedro and Porras, Diego (2015) Rabi lattice models with discrete gauge symmetry: Phase diagram and implementation in trapped-ion quantum simulators. Physical Review A, 92. ISSN 1050-2947This version is available from Sussex Research Online: http://sro.sussex.ac.uk/56957/ This document is made available in accordance with publisher policies and may differ from the published version or from the version of record. If you wish to cite this item you are advised to consult the publisher's version. Please see the URL above for details on accessing the published version. Copyright and reuse:Sussex Research Online is a digital repository of the research output of the University.Copyright and all moral rights to the version of the paper presented here belong to the individual author(s) and/or other copyright owners. To the extent reasonable and practicable, the material made available in SRO has been checked for eligibility before being made available.Copies of full text items generally can be reproduced, displayed or performed and given to third parties in any format or medium for personal research or study, educational, or not-for-profit purposes without prior permission or charge, provided that the authors, title and full bibliographic details are credited, a hyperlink and/or URL is given for the original metadata page and the content is not changed in any way.PHYSICAL REVIEW A 92, 013624 (2015) Rabi lattice models with discrete gauge symmetry: Phase diagram and implementation in trapped-ion quantum simulators We study a spin-boson chain that exhibits a local Z 2 symmetry. We investigate the quantum phase diagram of the model by means of perturbation theory, mean-field theory, and the density matrix renormalization group method. Our calculations show the existence of a first-order phase transition in the region where the boson quantum dynamics is slow compared to the spin-spin interactions. Our model can be implemented with trapped-ion quantum simulators, leading to a realization of minimal models showing local gauge invariance and first-order phase transitions.