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arXiv:1912.07556 · Physical Review A

Decay rates and decoherence of an interstitial two-level spin impurity in a ferromagnetic lattice

Yamen Hamdouni

Published Jan 13, 2021• 1 citation
arXivHTMLPDF

Abstract

The decay rate of an interstitial two-level spin impurity, located in the center of a unit cell of an anisotropic ferromagnetic lattice subjected to an external magnetic field, is derived. The impurity is coupled to nearest-neighbor spins through a Heisenberg $XY$ interaction. By mapping the lattice spin operators using the Holstein-Primakoff transformation, we establish the similarity with the Fano-Anderson model at low temperatures, and we calculate the retarded Green's function in one and two dimensions analytically for arbitrary coupling strength. It is shown that the reduced density matrix of the impurity satisfies an exact master equation in Lindblad form, from which the decay rate and the Lamb shift are deduced. The evolution in time of the latter together with the excited-state occupation probability is investigated and its dependence on the applied magnetic field is discussed. It is found that there exists a critical resonance-like value of the magnetic field around which the behavior of the decay rate and the density matrix changes drastically. The Markovian decay law, as given by the Fermi golden rule, does not hold in the weak-coupling regime unless the magnetic field is weak, typically less than the critical value. The weak-coupling regime is further treated perturbatively up to second order, and the obtained results are compared with the exact solution. We also discuss the Zeno regime of the dynamics, where it is shown that, at short times, the effective decay rate is twice as small as the exact decay rate, and that when the impurity energy lies outside the lattice continuum, the measurement speeds up the decay of the survival probability.

DOI
10.1103/physreva.103.012209

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The decay rate of an interstitial two-level spin impurity, located in the center of a unit cell of an anisotropic ferromagnetic lattice subjected to an external magnetic field, is derived. The impurity is coupled to nearest-neighbor spins through a Heisenberg XYXYXY interaction. By mapping the lattice spin operators using the Holstein-Primakoff transformation, we establish the similarity with the Fano-Anderson model at low temperatures, and we calculate the retarded Green's function in one and two dimensions analytically for arbitrary coupling strength. It is shown that the reduced density matrix of the impurity satisfies an exact master equation in Lindblad form, from which the decay rate and the Lamb shift are deduced. The evolution in time of the latter together with the excited-state occupation probability is investigated and its dependence on the applied magnetic field is discussed. It is found that there exists a critical resonance-like value of the magnetic field around which the behavior of the decay rate and the density matrix changes drastically. The Markovian decay law, as given by the Fermi golden rule, does not hold in the weak-coupling regime unless the magnetic field is weak, typically less than the critical value. The weak-coupling regime is further treated perturbatively up to second order, and the obtained results are compared with the exact solution. We also discuss the Zeno regime of the dynamics, where it is shown that, at short times, the effective decay rate is twice as small as the exact decay rate, and that when the impurity energy lies outside the lattice continuum, the measurement speeds up the decay of the survival probability.
We consider a finite two dimensional XY spin model. The model consists of a set of seven localized spin- 12 particles in a two dimensional triangular lattice coupled through nearest neighbor exchange interaction in presence of an external magnetic field. We study the effect of a single impurity spin coupled to its nearest neighbor through an exchange interaction J′ on the pairwise entanglement between the different spins in the lattice. We found that when the spin is located at a border site the entanglement between the impurity and its nearest neighbors increases monotonically with J′ reaching a saturation value that depends on the magnetic field strength h. On the other hand the entanglement with the next nearest spins shows a critical behavior where the entanglement increases and then decreases rapidly until it vanishes. The critical coupling value at which the entanglement vanishes increases as h increases. Studying the entanglement between two spins excluding the impurity show that while the entanglement between two spins coupled to the impurity reach a saturation value as J′ increases, it vanishes for spins not coupled to the impurity. Furthermore studying the effect of a central impurity show that the entanglement between nearest neighbor and next nearest neighbor spins reaches an asymptotic value as J′ increases. Interestingly it was demonstrated that the impurity can be used as a switch to control the entanglement between different spins in the lattice turning it on and off.
The mean field approximation is used to investigate the general features of the dynamics of a two-level atom in a ferromagnetic lattice close to the Curie temperature. Various analytical and numerical results are obtained. We first linearize the lattice Hamiltonian, and we derive the self-consistency equation for the order parameter of the phase transition for arbitrary direction of the magnetic field. The reduced dynamics is deduced by tracing out the degrees of freedom of the lattice, which results in the reduction of the dynamics to that of an atom in an effective spin bath whose size is equal to the size of a unit cell of the lattice. It is found that the dephasing and the excited state occupation probability may be enhanced by applying the magnetic field along some specific directions. The dependence on the change of the temperature and the magnitude of spin is also investigated. It turns out that the increase of thermal fluctuations may reduce the occupation probability of the excited state. The entanglement of two such atoms that occupy non-adjacent cells is studied and its variation in time is found to be not much sensitive to the direction of the magnetic field. Entanglement sudden death and revival is shown to occur close to the critical temperature.
Using a nonequilibrium renormalization group method we study the real-time evolution of spin and current in the anisotropic Kondo model (both antiferromagnetic and ferromagnetic) at a finite magnetic field h(0) and bias voltage V. We derive analytic expressions for all times in the weak-coupling regime max{V, h(0),1/t} >> T(c) (T(c) is the strong-coupling scale). We find that all observables decay both with the spin relaxation and decoherence rates Gamma(1/2). Various V-dependent logarithmic, oscillatory, and power-law contributions are predicted. The low-energy cutoff of logarithmic terms is generically identified by the difference of transport decay rates. For small times t << max{V, h(0)}(-1), we obtain universal dynamics for spin and current.
We investigate analytically the magnetization of Heisenberg ferromagnetic lattices in one and two dimensions, and we derive approximate expressions that are valid at high and low temperatures. In the case of the spin-12\frac{1}{2}21​ Heisenberg XX chain in a transverse field, we show that, when the applied magnetic field hhh exceeds its critical value hc=Jh_c=Jhc​=J, where JJJ is the exchange coupling constant, the magnetization per site deviates at low temperatures from its saturation value, 12\frac{1}{2}21​, following a power series with terms involving the power laws T1/2T^{1/2}T1/2, T3/2T^{3/2}T3/2, T5/2T^{5/2}T5/2 ..etc. When h<hch<h_ch<hc​, the zero temperature magnetization per spin turns out to be equal to 1πarcsin⁡(hhc)\frac{1}{\pi}\arcsin(\frac{h}{h_c})π1​arcsin(hc​h​). In this case, the temperature dependence of the magnetization is given by a series expansion with power laws of the form TTT, T3T^3T3, T5T^{5}T5,...etc. In both cases, the coefficients of the expansions are temperature-dependent and are explicitly derived. Using he properties of the Eulerian polynomials, we show that, because of the fast convergence of the derived series for the Fermi-Dirac and the Bose-Einstein distributions, it is possible (in particular in strong magnetic fields) to express the magnetization of the Heisenberg model in a simple analytical form. Furthermore, the analytical results are compared with the exact numerical ones.
The characteristics of the nuclear magnetic resonance (NMR), the contributions to the spin-lattice relaxation rate, and Knight shift of the resonance position are calculated exactly for a model spin-1/21/21/2 XYXYXY chain with the magnetic impurity. It is shown that the deviations of the impurity's coupling to the chain and the effective impurity's ggg factor from the values in the host spin chain yield special features in the temperature and magnetic field behavior for these NMR characteristics. The role of impurity-induced local levels in the NMR low-energy spin dynamics of spin chains is determined.
In the framework of the exactly solvable model, using nonperturbative methods, the temperature and magnetic field behavior of the NMR relaxation rate (and low-energy dynamical structure factor) for the antiferromagnetic Heisenberg spin-1/2 chain with magnetic impurities is calculated. For a single magnetic impurity, totally screened at low temperatures by spinons of the chain, the critical exponents, which determine the low-temperature and magnetic field behavior, get renormalized. For the finite concentration of impurities with the random distribution of the impurity-chain coupling the weak disorder in the distribution does not produce essential changes to the dynamical structure factor. Contrary, the strong disorder yields drastically different behavior of the low-energy dynamical characteristics, comparing to the homogeneous chain. Our results are generic, and they qualitatively agree in limiting cases with the results of perturbative calculations for disordered spin chains and with the data of experiments on quasi-one-dimensional spin compounds.