Quantum topological excitations (skyrmions) are analyzed from the point of view of their duality to spin excitations in the different phases of a disordered two-dimensional, short-range interacting, SO(3) quantum magnetic system of Heisenberg type. T
he phase diagram displays all the phases, which are allowed by the duality relation. We study the large distance behavior of the two-point correlation function of quantum skyrmions in each of these phases and, out of this, extract information about the energy spectrum and non-triviality of these excitations. The skyrmion correlators present a power-law decay in the spin-glass(SG)-phase, indicating that these quantum topological excitations are gapless but nontrivial in this phase. The SG phase is dual to the AF phase, in the sense that topological and spin excitations are respectively gapless in each of them. The Berezinskii-Kosterlitz-Thouless mechanism guarantees the survival of the SG phase at $T eq 0$, whereas the AF phase is washed out to T=0 by the quantum fluctuations. Our results suggest a new, more symmetric way of characterizing a SG-phase: one for which both the order and disorder parameters vanish, namely $<sigma > = 0 $, $<mu > =0 $, where $sigma$ is the spin and $mu$ is the topological excitation operators.
We study the quenched disordered magnetic system, which is obtained from the 2D SO(3) quantum Heisenberg model, on a square lattice, with nearest neighbors interaction, by taking a Gaussian random distribution of couplings centered in an antiferromag
netic coupling, $bar J>0$ and with a width $Delta J$. Using coherent spin states we can integrate over the random variables and map the system onto a field theory, which is a generalization of the SO(3) nonlinear sigma model with different flavors corresponding to the replicas, coupling parameter proportional to $bar J$ and having a quartic spin interaction proportional to the disorder ($Delta J$). After deriving the CP$^1$ version of the system, we perform a calculation of the free energy density in the limit of zero replicas, which fully includes the quantum fluctuations of the CP$^1$ fields $z_i$. We, thereby obtain the phase diagram of the system in terms of ($T, bar J, Delta J$). This presents an ordered antiferromagnetic (AF) phase, a paramagnetic (PM) phase and a spin-glass (SG) phase. A critical curve separating the PM and SG phases ends at a quantum critical point located between the AF and SG phases, at T=0. The Edwards-Anderson order parameter, as well as the magnetic susceptibilities are explicitly obtained in each of the three phases as a function of the three control parameters. The magnetic susceptibilities show a Curie-type behavior at high temperatures and exhibit a clear cusp, characteristic of the SG transition, at the transition line. The thermodynamic stability of the phases is investigated by a careful analysis of the Hessian matrix of the free energy. We show that all principal minors of the Hessian are positive in the limit of zero replicas, implying in particular that the SG phase is stable.
We present a mean-field solution for a quantum, short-range interacting, disordered, SO(3) Heisenberg spin model, in which the Gaussian distribution of couplings is centered in an AF coupling $bar J>0$, and which, for weak disorder, can be treated as
a perturbation of the pure AF Heisenberg system. The phase diagram contains, apart from a Neel phase at T=0, spin-glass and paramagnetic phases whose thermodynamic stability is demonstrated by an analysis of the Hessian matrix of the free-energy. The magnetic susceptibilities exhibit the typical cusp of a spin-glass transition.
