Magnon-lattice propagation in a Morse chain: the role played by the spin-lattice interaction and the initial condition

: Our research focuses on studying magnon dynamics in a Morse lattice. We used a Heisenberg Hamiltonian to represent the spins while a Morse formalism governed the lattice deformations. The strength of the spin-spin interaction depended on the distance between neighboring spins, which followed an exponential pattern. We explored various initial conditions for the lattice and spin wave function and observed how they affected the magnon-lattice propagation. Additionally, we analyzed the impact of the parameter that controlled the difference in time scales between spin and lattice deformation propagation.


MODEL
Our model is a quantum one-dimensional Heisenberg model with  spin 1/2 on a nonlinear Morse chain.The spin-spin interaction is strongly dependent on the distance between nearest-neighbor spins.The complete quantum Hamiltonian is given by Evangelou & Katsanos (1992), de Moura et al. (2002): (1) The interaction between spins  and  + 1 is given by  ,+1 =  −( +1 −  ) .  is the displacement of spin  from it equilibrium position.We emphasize again that we are dealing with a one-dimensional geometry.Therefore, without lattice vibrations, all spins are in equally spaced positions (in equilibrium, the distance between nearest-neighbor spins is the lattice spacing, an adimensional parameter   = 1).However, our model will consider that the spins can move around their equilibrium position.The spatial movement of the spins produces variations in the value of the spin-spin interaction.Our model will consider that these variations follow this exponential dependence shown earlier.The parameter  characterizes this exponential dependence within this formalism, thus controlling the effective spin-lattice interaction.The lattice dynamics here will be governed by a Morse potential represented by the classical Hamiltonian Hennig et al. (2007), Ikeda et al. (2007), de Lima & de Cavalho (2012), Carrillo et al. (2013): An Acad Bras Cienc (2023) 95(Suppl.2) e20230408 2 | 15 MARCONI SILVA SANTOS JUNIOR et al.

MAGNON-LATTICE PROPAGATION IN A MORSE CHAIN
Here,   represents the particle moment at site .We emphasize that we are using the dimensionless representation considered in ref.Hennig et al. (2007).The time is scaled as  → , with  representing the frequency of oscillations around the minimum of the Morse potential.The energy scale is measured in units of the depth of the Morse potential Hennig et al. (2007).The magnon dynamics is represented by the time-dependent Schrödinger equation for (ℏ = 1) defined as Sales et al. (2018): (3) To clarify, we want to point out that the previous equations were written considering a ferromagnetic ground state denoted as |0⟩ and a set of kets represented by |⟩ =  +  |0⟩.Therefore, the   () value corresponds to the wave function amplitude associated with the spin deviation at position .By utilizing the Hamilton formalism, we have derived the equations governing the dynamics of the lattice: It's important to note that we changed the time scale in the previous equation by rescaling  to , with  representing the frequency of oscillations around the minimum of the Morse potential Hennig et al. (2007).This step is necessary to account for the difference in timescale between electron dynamics (which is faster) and lattice vibrations (which is slower) Hennig et al. (2007), Davydov (1991), Scott (1992).
To put it simply, this framework involves a factor  = /(ℏ) that multiplies the spin equation Hennig  et al. (2007), Davydov (1991), Scott (1992).In our work, we will use  = 0.1, which is in alignment with previous research Hennig et al. (2007), Davydov (1991), Scott (1992), Korotin et al. (2015), Satija et al. (1980), Hutchings et al. (1979), Kadota et al. (1967).The value of  will be adjustable, but in previous works, it was typically chosen to be around 10 Hennig et al. ( 2007), Ranciaro-Neto & de Moura (2016), Sales et al. ( 2018) due to potential differences in time scales between quantum and classical propagation.However, we will explore the effects of varying  around this value.Our initial conditions will be   ( = 0) =  −(−/2) 2 /(4 2  ) ,   ( = 0) = 0, and   ( = 0) =  −(−/2) 2 /(4 2  ) , with  as a normalization constant.We will use a Taylor procedure de Moura (2011) to solve the set of equations 3, and a standard second-order Verlet's like procedure Allen & Tildesley (1987), da Silva et al. (2019) to solve the lattice dynamics.Our analysis will focus on magnon propagation and lattice deformation dynamics along the chain, which can be observed using the quantity   defined as Sales et al. (2018): (5)  The lattice properties can be analyzed using the mean position of the lattice deformation defined as: We want to emphasize that   and   represent the mean position of the spin-wave excitation and the lattice deformation, respectively.These measurements generally are in units of lattice spacing (  = 1).Using these quantities, we can obtain the magnon and the lattice deformation velocities   and   using fittings of the curves   ×  and   × .We stress that here we will use a methodology similar to that was used in the previous literature Hennig et al. (2007), Sales et al. (2018).We will follow the propagation of the magnon and the lattice deformation to describe the existence (or not) of magnon-lattice coupled movement.Generally, stable dynamics with   ≈   and   ≈   indicate the presence of magnon-lattice pair formation.The nonlinear Morse chain considered here contains a solitonic mode propagation along the chain.We can see this solitonic mode by calculating the lattice deformation   ; this quantity represents a generalized probability that deformation around site  occurs.This is obtained by normalizing   = (1 −  [−  + −1 ] ) 2 , that is   =   / ∑  (  ).We will plot    ×  ×  where  =  − /2 (i.e.,  = 0 represents the center of the chain).In fig. 1 we plot our results for  = 0, 1, 2, 3,  0 = 1,   = 0.5,   = 0.5 and  = 10.We can see that independent of the value of , the lattice deformation exhibits a stable solitonic mode propagating along the chain.Therefore, the main focus of our work is investigating the existence of a possible magnon-soliton pair formation and its dependence on all tunable parameters.

RESULTS AND DISCUSSION
Our findings on the velocities   and   in relation to  are presented below.We obtained   and   through the linear fitting of the curves   ×  and   × .Our calculations of   and   suggest that both quantities exhibit long-term linear behavior, consistent with the solitonic dynamics found in references Sales et al. (2018).We performed these calculations using a time limit of   ≈ 10 4 .The linear fitting was conducted using the last 20% of the complete time interval, roughly within the time interval [8000,10000].We used a Taylor expansion up to the tenth order to solve the quantum equations and a second-order Verlet-like method to solve the classical equations.We performed our numerical procedure using a time step of Δ = 0.001.It is important to emphasize that this method is faster than the Runge-Kutta formalism de Moura & Domínguez-Adame ( 2008) for this type of problem.The initial condition was given by : Here,  is a normalization constant,  0 is a tunable parameter, and the   and   are larger than zero.We varied the parameter  within the interval [1,15].We considering initially  0 = 1,   = 0.5,   = 0.5 and several values of .We show our results in figs.2(a-d).We emphasize that the curves indicate the velocities   (black solid line) and   (red dotted line) versus  for several values of .To construct these curves, we calculate the dynamics of the spin and the lattice for long times for several values of  and .We calculate the   and   curves versus  using a linear fitting.We can see that   is roughly independent of .On another side, spin propagation strongly depends on the spin-lattice interaction parameter .Let us clarify this important matter in simpler terms.The lattice's deformation is governed by eq. 4. We can see that when  is small, the nonlinear Morse terms, i.e., the first two terms, dominate over the terms that depend directly on  and the wave functions.Therefore, the soliton velocity remains roughly constant; however, when  increases, the final terms become more significant and have a greater impact on the soliton propagation, causing a slight increase in velocity.On a different note, the behavior of spin dynamics is dictated by equation 3, which shows a significant dependence on the magnitude of  in both the diagonal (first term) and the off-diagonal (last two terms).As such, it was indeed expected that the value of  would influence the magnon's velocity.By analyzing all curves for several of  we have considered, there is a matching of the magnon's and lattice's velocity (  ≈   ) for a specific value of .This result suggests that for this particular value of , the magnon and the lattice deformation travel at the same velocity.We stress that it is the first indication that magnon and lattice may move in a kind of "correlated propagation" (like a magnon-lattice pair formation).We can also see that as the parameter  has increased, this value of  in which the velocities are the same become smaller.To comprehend this phenomenon, we need to emphasize that when  increases, the off-diagonal terms in the Schrödinger equation become more effective.This results in a stronger coupling with the lattice deformation even for smaller values of .
We also calculate the long-time mean distance between the magnon and the lattice deformation.The distance is defined as  = |  ( → ∞) −   ( → ∞)|.We emphasize that  represents also an measurement of the possible existence of the magnon-soliton pair state.In general, bound states exhibit a smaller value of intrinsic internal distances.Dias et al. (2007) used this kind of measure to detect the existence of electron-electron bound states in the low-dimensional two-electron Hubbard model.We emphasize that we will plot (see figs.3(a-d)) /  versus  where   represents the maximum of the distance between the magnon and the lattice position.We can observe that for the same value of  in which that   ≈   , we can see that /  ≈ 0, i.e., the magnon and the lattice position are close, thus suggesting the existence of magnon-lattice pair formation.We can see that the critical value of  in which /  ≈ 0 is in good agreement with the critical value found using the velocity curves versus  (see fig. 2).Therefore all measures of /  ,   , and   are topological quantities that characterize the propagation of the magnon and the lattice deformation.Our calculations numerically demonstrate that for some specific values of  =   , the distance D is small, and the magnon and the lattice deformation travel at the same velocity.This result strongly indicates a magnon-soliton pair formation for these special situations.
In fig.4, we collect the critical value of  versus .We stress that for  =   the system exhibits a magnon-lattice pair formation, i.e., the magnetic excitation moves along with the lattice vibration and at the same velocity.We emphasize again that the decreasing of   with  is a direct consequence of the role played by  at the off-diagonal terms at eq. 3.As the  is increased, the effective off-diagonal term also increases.Increasing the effective spin-spin interaction makes coupling between the spin and the lattice deformations easier.In figures 5 and 6 we consider again  0 = 1 and change the values of   and   respectively to 0.5 and 1; we kept the same range of values of .We can observe that the results are qualitatively the same obtained in figs. 2 and 3 i.e.: as the value of  is increased, the magnon-lattice pair formation is obtained for a specific value of  =   .We also obtained that as  is increased   decreases (see fig. 7).In figures 8 and 9, we show our results considering   = 1 and   = 1, and we kept the same range of values of  and  0 .The results obtained are similar to those packet for the lattice.The velocity intensity and width of these initial Gaussian pulses are adjustable parameters in our model.We also vary the time scales between the magnon and lattice dynamics.We provide a detailed numerical analysis of how magnon-soliton pairs propagate and their dependence on these parameters.Our findings reveal that magnon-soliton propagations are attainable for specific values of the magnon-lattice interaction (called   in our model) and that this critical value is highly reliant on the width of the initial Gaussian pulses.Our numerical calculations indicate that increasing the velocity of the initial Gaussian pulse decreases the critical value of spin-lattice interaction (  ).Furthermore, as the time difference between the magnon and lattice dynamics increases, the intensity of magnon-lattice coupling needed to promote pair formations decreases.Overall, our study underscores the importance of the initial conditions and the specifics of the magnon/lattice dynamics in the existence of magnon-soliton pairs in nonlinear chains.We demonstrate that a time difference of  ≥ 10 yields a more reliable existence of magnon-lattice coupling, consistent with previous research.Our work is intended to inspire further research in this area.