Non-Abelian fusion rules from an Abelian system

We demonstrate the emergence of non-Abelian fusion rules for excitations of a two dimensional lattice model built out of Abelian degrees of freedom. It can be considered as an extension of the usual toric code model on a two dimensional lattice augmented with matter fields. It consists of the usual C(Z_p)$\mathbb{C}\left({\mathbb{Z}}_p\right)$ gauge degrees of freedom living on the links together with matter degrees of freedom living on the vertices. The matter part is described by a n$n$ dimensional vector space which we call H_n$H_n$. The Z_p${\mathbb{Z}}_p$ gauge particles act on the vertex particles and thus H_n$H_n$ can be thought of as a C(Z_p)$\mathbb{C}\left({\mathbb{Z}}_p\right)$ module. An exactly solvable model is built with operators acting in this Hilbert space. The vertex excitations for this model are studied and shown to obey non-Abelian fusion rules. We will show this for specific values of n$n$ and p$p$, though we believe this feature holds for all n>p$n>p$. We will see that non-Abelian anyons of the quantum double of C(S₃)$\mathbb{C}\left(S_3\right)$ are obtained as part of the vertex excitations of the model with n=6$n=6$ and p=3$p=3$. Ising anyons are obtained in the model with n=4$n=4$ and p=2$p=2$. The n=3$n=3$ and p=2$p=2$ case is also worked out as this is the simplest model exhibiting non-Abelian fusion rules. Another common feature shared by these models is that the ground states have a higher symmetry than Z_p${\mathbb{Z}}_p$. This makes them possible candidates for realizing quantum computation.     

Quantum spherical model with competing interactions

We analyse the phase diagram of a quantum mean spherical model in terms of the temperature T$T$, a quantum parameter g$g$, and the ratio p=−_J2/_J1$p=-J_2/J_1$, where _J1>0$J_1>0$ refers to ferromagnetic interactions between first-neighbour sites along the d$d$ directions of a hypercubic lattice, and _J2<0$J_2<0$ is associated with competing antiferromagnetic interactions between second neighbours along m≤d$m\le d$ directions. We regain a number of known results for the classical version of this model, including the topology of the critical line in the g=0$g=0$ space, with a Lifshitz point at p=1/4$p=1/4$, for d>2$d>2$, and closed-form expressions for the decay of the pair correlations in one dimension. In the T=0$T=0$ phase diagram, there is a critical border, _gc=_gc(p)$g_c=g_c\left(p\right)$ for d≥2$d\ge 2$, with a singularity at the Lifshitz point if d<(m+4)/2$d<(m+4)/2$. We also establish upper and lower critical dimensions, and analyse the quantum critical behavior in the neighborhood of p=1/4$p=1/4$.     

Probing Planckian physics in de Sitter space with quantum correlations

We study the quantum correlation and quantum communication channel of both free scalar and fermionic fields in de Sitter space, while the Planckian modification presented by the choice of a particular α$\alpha$-vacuum has been considered. We show the occurrence of degradation of quantum entanglement between field modes for an inertial observer in curved space, due to the radiation associated with its cosmological horizon. Comparing with standard Bunch–Davies choice, the possible Planckian physics causes some extra decrement on the quantum correlation, which may provide the means to detect quantum gravitational effects via quantum information methodology in future. Beyond single-mode approximation, we construct proper Unruh modes admitting general α$\alpha$-vacua, and find a convergent feature of both bosonic and fermionic entanglements. In particular, we show that the convergent points of fermionic entanglement negativity are dependent on the choice of α$\alpha$. Moreover, an one-to-one correspondence between convergent points H_c$H_c$ of negativity and zeros of quantum capacity of quantum channels in de Sitter space has been proved.     

Third-harmonic generation in asymmetric coupled quantum wells

The third-harmonic generation (THG) in asymmetric coupled quantum wells (ACQWs) for different values of the well parameter Δ$\Delta$ and width of barrier (_WB)$\left(W_B\right)$ are theoretically studied. The analytical expression of the third-harmonic generation is derived by using the compact density-matrix approach and the iterative method. Finally, the numerical calculations are presented for typical GaAs/Al_xGa_1−xAs asymmetric coupled quantum wells. Results obtained show that the third-harmonic generation in the asymmetric coupled quantum wells can be importantly modified by the parameter Δ$\Delta$ and _WB$W_B$. Moreover, third-harmonic generation also depends on the relaxation rate of the asymmetric coupled quantum wells.     

Cyclic groups and quantum logic gates

We present a formula for an infinite number of universal quantum logic gates, which are 4$4$ by 4$4$ unitary solutions to the Yang–Baxter (Y–B) equation. We obtain this family from a certain representation of the cyclic group of order n$n$. We then show that this discrete   family, parametrized by integers n$n$, is in fact, a small sub-class of a larger continuous   family, parametrized by real numbers θ$\theta$, of universal quantum gates. We discuss the corresponding Yang-Baxterization and related symmetries in the concomitant Hamiltonian.     

Random matrix theory for transition strength densities in finite quantum systems: Results from embedded unitary ensembles

Embedded random matrix ensembles are generic models for describing statistical properties of finite isolated interacting quantum many-particle systems. For the simplest spinless fermion (or boson) systems, with say m$m$ fermions (or bosons) in N$N$ single particle states and interacting via k$k$-body interactions, we have EGUE(k$k$) [embedded GUE of k$k$-body interactions] with GUE embedding and the embedding algebra is U(N)$U\left(N\right)$. A finite quantum system, induced by a transition operator, makes transitions from its states to the states of the same system or to those of another system. Examples are electromagnetic transitions (then the initial and final systems are same), nuclear beta and double beta decay (then the initial and final systems are different), particle addition to or removal from a given system and so on. Towards developing a complete statistical theory for transition strength densities (transition strengths multiplied by the density of states at the initial and final energies), we have derived formulas for the lower order bivariate moments of the strength densities generated by a variety of transition operators. Firstly, for a spinless fermion system, using EGUE(k$k$) representation for a Hamiltonian that is k$k$-body and an independent EGUE(t$t$) representation for a transition operator that is t$t$-body and employing the embedding U(N)$U\left(N\right)$ algebra, finite-N$N$ formulas for moments up to order four are derived, for the first time, for the transition strength densities. Secondly, formulas for the moments up to order four are also derived for systems with two types of spinless fermions and a transition operator similar to beta decay and neutrinoless beta decay operators. In addition, moments formulas are also derived for a transition operator that removes k₀$k_0$ number of particles from a system of m$m$ spinless fermions. In the dilute limit, these formulas are shown to reduce to those for the EGOE version derived using the asymptotic limit theory of Mon and French (1975). Numerical results obtained using the exact formulas for two-body (k=2$k=2$) Hamiltonians (in some examples for k=3$k=3$ and 4$4$) and the asymptotic formulas clearly establish that in general the smoothed (with respect to energy) form of the bivariate transition strength densities take bivariate Gaussian form for isolated finite quantum systems. Extensions of these results to bosonic systems and EGUE ensembles with further symmetries are discussed.     

Sharing of classical and quantum correlations via XY interaction

The sharing of classical and quantum correlations via XY interaction is investigated. The model includes two identical networks consisting of n$n$ nodes, the i$i$th node of one network sharing a correlated state with the j$j$th node of the other network, while all other nodes are initially unconnected. It is shown that classical correlation, quantum discord as well as entanglement can be shared between any two nodes of the network via XY interaction and that quantum information can be transferred effectively between them. It is found that there is no simple dominating relation between the quantum correlation and entanglement in inertial system.     

Quasiprobability distributions in open quantum systems: Spin-qubit systems

We study nonclassical features in a number of spin-qubit systems including single, two and three qubit states, as well as an N$N$ qubit Dicke model and a spin-1 system, of importance in the fields of quantum optics and information. This is done by analyzing the behavior of the well known Wigner, P$P$, and Q$Q$ quasiprobability distributions on them. We also discuss the not so well known F$F$ function and specify its relation to the Wigner function. Here we provide a comprehensive analysis of quasiprobability distributions for spin-qubit systems under general open system effects, including both pure dephasing as well as dissipation. This makes it relevant from the perspective of experimental implementation.     

Entanglement thresholds for displaying the quantum nature of teleportation

A protocol for transferring an unknown single qubit state evidences quantum features when the average fidelity of the outcomes is, in principle, greater than 2/3$2/3$. We propose to use the probabilistic and unambiguous state extraction scheme   as a mechanism to redistribute the fidelity in the outcome of the standard teleportation when the process is performed with an X$X$-state as a noisy quantum channel. We show that the entanglement of the channel is necessary but not sufficient in order for the average fidelity f_X$f_X$ to display quantum features, i.e., we find a threshold C_X$C_X$ for the concurrence of the channel. On the other hand, if the mechanism for redistributing fidelity is successful then we find a filterable outcome with average fidelity f_X,0$f_{X,0}$ that can be greater than f_X$f_X$. In addition, we find the threshold concurrence of the channel C_X,0$C_{X,0}$ in order for the average fidelity f_X,0$f_{X,0}$ to display quantum features and surprisingly, the threshold concurrence C_X,0$C_{X,0}$ can be less than C_X$C_X$. Even more, we find some special cases for which the threshold values become zero.     

Precision of estimation and entropy as witnesses of non-Markovianity in the presence of random classical noises

We investigate in detail the quantum Fisher information (QFI) behavior by examining a single qubit model in the presence of random classical noises in both Markovian and non-Markovian regimes. In particular, we precisely study the effects of noise switching rate ξ$\xi$ and qubit–environment coupling strength ν$\nu$ on the precision of estimation, when the qubit is subjected to random telegraph noise with a Lorentzian spectrum or colored noise with a spectrum of the form 1/f^α$1/f^{\alpha }$. In the Markovian regime, a monotone decay of the QFI with time is found, whereas for non-Markovian noise sudden death and revivals may occur. Despite these oscillations of the QFI in non-Markovian regime, we find that non-Markovian parameter γ=ξ/ν$\gamma =\xi /\nu$ is not the principal parameter controlling the collapse and revival of the QFI. In fact, in both Markovian and non-Markovian regimes, parameters ξ$\xi$ and ν$\nu$independently determine how the QFI varies. We also find that the QFI in the case of colored environments decreases when the number of fluctuators realizing the noise increases, and therefore the parameter estimation becomes more inaccurate. Furthermore, by analyzing the von Neumann entropy of the system density matrix, we illustratively unveil a fundamental relationship between the dynamics of this quantity and non-Markovian behavior in the presence of random classical noises. We also show that this result may lead to a better non-Markovianity interpretation, based on quantum memory effects. Moreover, we demonstrate the connection between the precision of parameter estimation and rising the non-Markovianity in our model where the environment is modeled as classical.     

Superconducting cosmic strings and one dimensional extended supersymmetric algebras

In this article we study in detail the supersymmetric structures that underlie the system of fermionic zero modes around a superconducting cosmic string. Particularly, we extend the analysis existing in the literature on the one dimensional N=2$N=2$ supersymmetry and we find multiple N=2$N=2$, d=1$d=1$ supersymmetries. In addition, compact perturbations of the Witten index of the system are performed and we find to which physical situations these perturbations correspond. More importantly, we demonstrate that there exists a much more rich supersymmetric structure underlying the system of fermions with N_f$N_f$ flavors and these are N$N$-extended supersymmetric structures with non-trivial topological charges, with “N$N$” depending on the fermion flavors.     

Multi-spin-subband structure of dilute magnetic semiconductor quantum wells: Feedback mechanism

Using a fully self-consistent envelope function approach, we focus on wide conduction band NMS (non-magnetic semiconductor)/DMS (dilute magnetic semiconductor)/NMS quantum wells, under weak external parallel magnetic field, where many spin-subbands are usually present. We concentrate on small values of the magnetic field because we want to investigate the influence of the feedback mechanism due to the difference of the concentrations of spin-up and spin-down carriers which could induce spontaneous spin-polarization i.e. in the absence of a magnetic field. We study the spin-subband structure, the spin-subband populations and the spin-polarization as functions of the sheet carrier concentration, _Ns$N_s$, for different values of the magnitude of the exchange interaction, |J|$\left|J\right|$, between the itinerant carriers and the magnetic impurities. Our calculations for 0.01 T show that at 20 K the values of |J|$\left|J\right|$ necessary to make this feedback mechanism sufficiently strong are too high compared to the |J|$\left|J\right|$ values of common Mn-doped systems in the conduction band. However, the feedback mechanism will be sufficiently strong at low enough temperatures below 20 K for realistic values of |J|$\left|J\right|$. Moreover, we explain how increasing the sheet carrier concentration the heterostructure is transformed from an almost square quantum well to a system of two coupled heterojunctions with an intermediate soft barrier.     

Thermodynamics properties of an anisotropic Ising antiferromagnet in both external longitudinal and transverse fields

We have studied the anisotropic two-dimensional nearest-neighbor Ising model with competitive interactions in both uniform longitudinal field H$H$ and transverse magnetic field Ω$\Omega$. Using the effective-field theory (EFT) with correlation in cluster with N=1$N=1$ spin we calculate the thermodynamic properties as a function of temperature with values H$H$ and Ω$\Omega$ fixed. The model consists of ferromagnetic interaction _Jx$J_x$ in the x$x$ direction and antiferromagnetic interaction _Jy$J_y$ in the y$y$ direction, and it is found that for H/_Jy∈[0,2]$H/J_y\in [0,2]$ the system exhibits a second-order phase transition. The thermodynamic properties are obtained for the particular case of λ=_Jx/_Jy=1$\lambda =J_x/J_y=1$ (isotropic square lattice).     

Spin-1 Blume–Capel model with random crystal field effects

The random-crystal field spin-1 Blume–Capel model is investigated by the lowest approximation of the cluster-variation method which is identical to the mean-field approximation. The crystal field is either turned on randomly with probability p$p$ or turned off with q=1−p$q=1-p$ in a bimodal distribution. Then the phase diagrams are constructed on the crystal field (Δ$\Delta$)–temperature (kT/J)$(kT/J)$ planes for given values of p$p$ and on the (kT/J,p$kT/J,p$) planes for given Δ$\Delta$ by studying the thermal variations of the order parameters. In the latter, we only present the second-order phase transition lines, because of the existence of irregular wiggly phase transitions which are not good enough to construct lines. In addition to these phase transitions, the model also yields tricritical points for all values of p$p$ and the reentrant behavior at lower p$p$ values.     

Bertrand curves in the three-dimensional sphere

A curve α$\alpha$ immersed in the three-dimensional sphere S³${\mathbb{S}}^3$ is said to be a Bertrand curve if there exists another curve β$\beta$ and a one-to-one correspondence between α$\alpha$ and β$\beta$ such that both curves have common principal normal geodesics at corresponding points. The curves α$\alpha$ and β$\beta$ are said to be a pair of Bertrand curves in S³${\mathbb{S}}^3$. One of our main results is a sort of theorem for Bertrand curves in S³${\mathbb{S}}^3$ which formally agrees with the classical one: “Bertrand curves in S³${\mathbb{S}}^3$ correspond to curves for which there exist two constants λ≠0$\lambda \ne 0$ and μ$\mu$ such that λκ+μτ=1$\lambda \kappa +\mu \tau =1$”, where κ$\kappa$ and τ$\tau$ stand for the curvature and torsion of the curve; in particular, general helices in the 3-sphere introduced by M. Barros are Bertrand curves. As an easy application of the main theorem, we characterize helices in S³${\mathbb{S}}^3$ as the only twisted curves in S³${\mathbb{S}}^3$ having infinite Bertrand conjugate curves. We also find several relationships between Bertrand curves in S³${\mathbb{S}}^3$ and (1,3)-Bertrand curves in R⁴${\mathbb{R}}^4$.     

Force free projective motions of the sphere

Suppose that the sphere Sⁿ$S^n$ has initially a homogeneous distribution of mass and let G$G$ be the Lie group of orientation preserving projective diffeomorphisms of Sⁿ$S^n$. A projective motion of the sphere, that is, a smooth curve in G$G$, is called force free if it is a critical point of the kinetic energy functional. We find explicit examples of force free projective motions of Sⁿ$S^n$ and, more generally, examples of subgroups H$H$ of G$G$ such that a force free motion initially tangent to H$H$ remains in H$H$ for all time (in contrast with the previously studied case for conformal motions, this property does not hold for H=S_On+1$H=S{O}_{n+1}$). The main tool is a Riemannian metric on G$G$, which turns out to be not complete (in particular not invariant, as happens with non-rigid motions), given by the kinetic energy.