Quantum robustness and phase transitions of the 3D Toric Code in a field

We study the robustness of 3D intrinsic topogical order under external perturbations by investigating the paradigmatic microscopic model, the 3D toric code in an external magnetic field. Exact dualities as well as variational calculations reveal a ground-state phase diagram with first and second-order quantum phase transitions. The variational approach can be applied without further approximations only for certain field directions. In the general field case, an approximative scheme based on an expansion of the variational energy in orders of the variational parameters is developed. For the breakdown of the 3D intrinsic topological order, it is found that the (im-)mobility of the quasiparticle excitations is crucial in contrast to their fractional statistics

Emergent Fermions and Anyons in the Kitaev Model

We study the gapped phase of the Kitaev model on the honeycomb lattice using perturbative continuous unitary transformations. The effective low-energy Hamiltonian is found to be an extended toric code with interacting anyons. High-energy excitations are emerging free fermions which are composed of hardcore bosons with an attached string of spin operators. The excitation spectrum is mapped onto that of a single particle hopping on a square lattice in a magnetic field. We also illustrate how to compute correlation functions in this framework. The present approach yields analytical perturbative results in the thermodynamical limit without using the Majorana or the Jordan-Wigner fermionization initially proposed to solve this problem.Comment: 4 pages, 5 figures, published versio

Spectral Properties of Quasi One-dimensional Quantum Antiferromagnets . Perturbative Continuous Unitary Transformations

In this work a perturbative realization of particle conserving continuous unitary transformations is applied to study the energies and the spectral properties of quasi one-dimensional quantum antiferromagnets. The systems considered are defined on a lattice and they allow for a perturbative decomposition. The unperturbed part is chosen to be a fully dimerized state. The groundstate is a product-state of singlets and the excitation spectrum is equidistant. The related energy quantum is called a triplon becaues it has total spin one. The continous unitary transformation leads to an effective triplon-conserving Hamiltonian and effective, experimentally relevant observables. The effective operators are obtained in a high-order series expansion in the perturbation parameters. All calculations are performed on finite clusters in real space and yield exact results in the thermodynamic limit due to the linked cluster theorem. The results are exact in the given order. In order to improve the representation of the results extrapolation techniques are used. A detailed description of extrapolation tools like standard Pade and dlogPade extrapolation, optimized perturbation theory and the use of internal parameters is given. The dimerized and frustrated spin-chain is analysed first. At zero frustration, a detailed investigation of the spectral weights shows that even in the limit of vanishing dimerization, the one-dimensional Heisenberg model, almost the total spectral weight is situated in the two-triplon sector. So, besides spinons, triplons may be used as elementary excitations for the one-dimensional Heisenberg model. The case of strong frustration is not yet settled. An extensive review of one- and two-triplon spectral densities at large and intermediate value of the dimerization for various values of the frustration is presented. The findings are compared with field theoretical results. In addition, the Raman response and the infrared absorption are investigated. Second, the antiferromagnetic two-leg Heisenberg ladder plus additional four-spin interaction is investigated. The transformation starts from the limit of isolated rung dimers. The excitations are rung-triplons. The relative energies of one-triplon states, the two-triplon bound states and the multi-triplon continua are given for various couplings. Optical observables are discussed in detail. The extent of the rung-singlet phase is calculated in the whole parameter space. It is shown that the experimental realizations of two-leg ladder systems are always situated in the rung-singlet phase. In the experimentally relevant regime, most of the spectral weight is captured by the one- and the two-triplon sector, but also three- and four-triplon contributions become sizable. The current understanding of the spectroscopic signatures of magnetic excitations in cuprate ladders measured with inelastic neutron scattering, Raman spectroscopy and infrared absorption is presented. The results obtained are compared with experimental findings. The first experimental evidence of a triplon-triplon bound state in a ladder system is found

Perturbative study of the Kitaev model with spontaneous time-reversal symmetry breaking

We analyze the Kitaev model on the triangle-honeycomb lattice whose ground state has recently been shown to be a chiral spin liquid. We consider two perturbative expansions: the isolated-dimer limit containing Abelian anyons and the isolated-triangle limit. In the former case, we derive the low-energy effective theory and discuss the role played by multi-plaquette interactions. In this phase, we also compute the spin-spin correlation functions for any vortex configuration. In the isolated-triangle limit, we show that the effective theory is, at lowest nontrivial order, the Kitaev honeycomb model at the isotropic point. We also compute the next-order correction which opens a gap and yields non-Abelian anyons.Comment: 7 pages, 4 figures, published versio

Engineering Photon-mediated Long-Range Spin Interactions in Mott Insulators

We investigate the potential to induce long-range spin interactions in a Mott insulator via the quantum electromagnetic field of a cavity. The coupling between light and spins is inherently non-linear, and occurs via multi-photon processes like Raman scattering and two-photon absorption/emission with electronically excited intermediate states. Based on this, two pathways are elucidated: (i) In the absence of external driving, long-range interactions are mediated by the exchange of at least two virtual cavity photons. We show that these vacuum-mediated interactions can surpass local Heisenberg interactions in mesoscopic setups such as sufficiently small split-ring resonators. (ii) In a laser-driven cavity, interactions can be tailored through a hybrid scheme involving both external laser photons and cavity photons. This offers a versatile pathway for Floquet engineering of long-range interactions in macroscopic systems. In general, the derivation of these interactions requires careful consideration: Notably, we demonstrate that a simple phenomenological approach, based on a spin-photon Hamiltonian that captures Raman and two-photon processes with effective matrix elements, can be used only if the cavity is resonantly driven. Outside of these narrow resonant regimes as well as for the undriven case, a fourth-order series expansion within the underlying electronic model is necessary, which we perform to obtain long-range four-spin interactions in the half-filled Hubbard model.Comment: 25 pages, 9 figure

Bound states in two-dimensional spin systems near the Ising limit: A quantum finite-lattice study

We analyze the properties of low-energy bound states in the transverse-field Ising model and in the XXZ model on the square lattice. To this end, we develop an optimized implementation of perturbative continuous unitary transformations. The Ising model is studied in the small-field limit which is found to be a special case of the toric code model in a magnetic field. To analyze the XXZ model, we perform a perturbative expansion about the Ising limit in order to discuss the fate of the elementary magnon excitations when approaching the Heisenberg point.Comment: 21 pages, 18 figures, published versio

Perturbative approach to an exactly solved problem: the Kitaev honeycomb model

We analyze the gapped phase of the Kitaev honeycomb model perturbatively in the isolated-dimer limit. Our analysis is based on the continuous unitary transformations method which allows one to compute the spectrum as well as matrix elements of operators between eigenstates, at high order. The starting point of our study consists in an exact mapping of the original honeycomb spin system onto a square-lattice model involving an effective spin and a hardcore boson. We then derive the low-energy effective Hamiltonian up to order 10 which is found to describe an interacting-anyon system, contrary to the order 4 result which predicts a free theory. These results give the ground-state energy in any vortex sector and thus also the vortex gap, which is relevant for experiments. Furthermore, we show that the elementary excitations are emerging free fermions composed of a hardcore boson with an attached spin- and phase- operator string. We also focus on observables and compute, in particular, the spin-spin correlation functions. We show that they admit a multi-plaquette expansion that we derive up to order 6. Finally, we study the creation and manipulation of anyons with local operators, show that they also create fermions, and discuss the relevance of our findings for experiments in optical lattices.Comment: 28 pages, 25 figure

Robustness of a perturbed topological phase

We investigate the stability of the topological phase of the toric code model in the presence of a uniform magnetic field by means of variational and high-order series expansion approaches. We find that when this perturbation is strong enough, the system undergoes a topological phase transition whose first- or second-order nature depends on the field orientation. When this transition is of second order, it is in the Ising universality class except for a special line on which the critical exponent driving the closure of the gap varies continuously, unveiling a new topological universality class. © 2011 American Physical Society