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arXiv:0903.0305 · Nuclear Physics B

Quantum field theory solution for a short-range interacting SO ( 3 ) quantum spin-glass

C. M. S. da Conceição, Eduardo C. Marino

Published Feb 26, 2009• 5 citations
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Abstract

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 antiferromagnetic coupling, J¯>0 and with a width ΔJ. 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 J¯ and having a quartic spin interaction proportional to the disorder (ΔJ). After deriving the CP1 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 CP1 fields zi. We, thereby obtain the phase diagram of the system in terms of (T,J¯,ΔJ). 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.

DOI
10.1016/j.nuclphysb.2009.02.023

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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 antiferromagnetic coupling, J¯>0 and with a width ΔJ. 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 J¯ and having a quartic spin interaction proportional to the disorder (ΔJ). After deriving the CP1 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 CP1 fields zi. We, thereby obtain the phase diagram of the system in terms of (T,J¯,ΔJ). 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 antiferromagnetic (AF) coupling J[over ]>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 Néel 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.
In this work we study the effects of quantum fluctuations in a disordered magnetic system. We consider a cluster-spin Ising model with antiferromagnetic short range interactions (J0) and disordered long range interactions (J) between clusters, where the system is under the influence of a transverse magnetic field Γ. The analytic treatment of the model is done by the replica method and Trotter formalism, in which a cluster mean field, replica symmetric and static approximations are used. This allows us to obtain an effective model of a single cluster that is exactly diagonalized. Thermodynamic quantities like order parameter and free energy are obtained then, the phase diagrams are analyzed. They are shown in temperature versus J0=J phase diagrams for some intensities of Γ. With Γ = 0, when we decrease the temperature (for lower intensities of J0=J) a continuous phase transition between the paramagnetic and cluster spin glass state (CSG) is found. When we increase the ratio J0=J, the system presents the antiferromagnetic phase (AF). The CSG state and the AF phase are separated by first order transitions. In the intermediate region of disorder, there exist a small interval in the AF-PM transition that is first order. When Γ is increased, the transitions PM-AF and PM-VSC decrease toward critical points in the lower and higher disorder, respectively. It is worth to note, in the intermediate range of disorder, that the transition CSG-AF disappears, letting a PM phase between them, even for zero temperature. Besides that, the first-order region, in the PM-AF transition, turns into a second order phase transition, changing its criticality in the low temperature regime. Our results show that quantum fluctuations affects the phase transitions of the model being capable to suppress and/or change his criticality. The present work suggests a model and a mechanism of the change in the criticality away from the quantum critical point, helping the investigation of this interesting and recent topic.
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. The 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 nontriviality 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 $T\ensuremath{\ne}0$, whereas the AF phase is washed out to T=0T=0T=0 by the quantum fluctuations. Our results suggest a more symmetric way of characterizing a SG phase: one for which both the order and disorder parameters vanish, namely, $⟨\ensuremath{\sigma}⟩=0$ and $⟨\ensuremath{\mu}⟩=0$, where $\ensuremath{\sigma}$ is the spin and $\ensuremath{\mu}$ is the topological excitation operators.
In the present work, a fermionic infinite-range Ising spin glass (SG) model with a transverse field Γ is studied. The model is analyzed in a one-lattice structure (quantum SG) and in a two-sublattice structure (SG/antiferromagnetism (AF) competition). In these two cases, the exchange interactions are random variables that follow Gaussian probability distributions (SK interactions). For the two-sublattice model, spins of distinct sublattices can couple with an SK-type interaction with an AF average. For a general case of this model, the disordered interactions can also occur with spins of same sublattices. The problem is formulated in a Grassmann path integral formalism within the static ansatz. The replica method is applied to treat the disorder, in which the replica symmetry (RS) and one-step replica symmetry breaking (1P-RSB) approximations are used. The one-lattice model results provide evidences that the 1S-RSB can be used to improve the physical description of the ordered quantum SG phase treated by the replica method. In the two-sublattice model, a parallel magnetic field H breaks the symmetry of the sublattices. It destroys the antiferromagnetic (AF) order, but it can favor the nonergodic mixed phase (SG+AF) characterizing an asymmetric RSB region. In this region, intra-sublattice disordered interactions V increase the difference between the RSB solutions of each sublattice. The freezing temperature (Tf ) shows a higher increase with H when V enhances. A discontinuous phase transition from the replica symmetry (RS) solution to the RSB solution can appear with the presence of an intra-sublattice ferromagnetic average coupling. The Γ field introduces a quantum spin flip mechanism that suppresses the magnetic orders leading them to quantum critical points. Results suggest that the quantum effects are not able to restore the RS solution. Therefore, the presence of both fields, Γ and H, produces effects that are in competition on Tf for instance. However, in the asymmetric RSB region, Γ can produce a stable RS solution at any finite temperature for a particular sublattice while the other sublattice still presents RSB solution for the special case in which only the intra-sublattice spins couple with disordered interactions.
We use quantum Monte Carlo simulations to study a disordered S=1/2S=1/2S=1/2 Heisenberg quantum spin model with three different nearest-neighbor interactions, ${J}_{1}\ensuremath{\le}{J}_{2}\ensuremath{\le}{J}_{3}$, on the square lattice. We consider the regime in which J1{J}_{1}J1​ represents weak bonds, and J2{J}_{2}J2​ and J3{J}_{3}J3​ correspond to two kinds of stronger bonds (dimers) which are randomly distributed on columns forming coupled two-leg ladders. When increasing the average intradimer coupling (J2+J3)/2({J}_{2}+{J}_{3})/2(J2​+J3​)/2, the system undergoes a N\'eel to quantum glass transition of the ground state and later a second transition into a quantum paramagnet. The quantum glass phase is of the gapless Mott glass type (i.e., in boson language it is incompressible at temperature T=0T=0T=0), and we find that the temperature dependence of the uniform magnetic susceptibility follows the stretched exponential form $\ensuremath{\chi}\ensuremath{\sim}exp(\ensuremath{-}b/{T}^{\ensuremath{\alpha}})$, with $0<\ensuremath{\alpha}<1$. At the N\'eel-glass transition, we observe the standard O(3) critical exponents, which implies that the Harris criterion for the relevance of the disorder is violated in this system.