Structural scale $q-$derivative and the LLG-Equation in a scenario with fractionality

TL;DR

This paper introduces a deformed version of the Landau-Lifshitz-Gilbert equation using structural derivatives to incorporate long-range interactions and fractional effects, resulting in natural damping phenomena without explicit damping terms.

Contribution

It develops a novel deformed LLG equation employing scale q-derivative and Mittag-Leffler functions, capturing fractional and long-range effects in magnetic dynamics.

Findings

Damping appears naturally without explicit Gilbert damping.

The deformed equation accounts for long-range forces and fractional space effects.

The approach generalizes classical LLG to fractional and non-local interactions.

Abstract

In the present contribution, we study the Landau-Lifshitz-Gilbert equation with two versions of structural derivatives recently proposed: the scale $q-$derivative in the non-extensive statistical mechanics and the axiomatic metric derivative, which presents Mittag-Leffler functions as eigenfunctions. The use of structural derivatives aims to take into account long-range forces, possible non-manifest or hidden interactions and the dimensionality of space. Having this purpose in mind, we build up an evolution operator and a deformed version of the LLG equation. Damping in the oscillations naturally show up without an explicit Gilbert damping term.

Equations

$$\frac{dy}{dx}=y^q,$$

$$D_{(q)}^{\lambda }f(\lambda x)\equiv [1+(1-q)\lambda x]\frac{df(x)}{dx}.$$

$$D_{(q)}^{\lambda }f(\lambda x)=\lambda f(\lambda x).$$

$$\frac{d^q}{d{t}^q}=(1+(1-q)\lambda x\frac{d}{dx}$$

$$i\hslash D_{q,t}^{\lambda }\psi =-\frac{{\hslash }^2}{2m}{\nabla }^2\psi -V\psi =H\psi ,$$

$$U_q(t,t_0)=e_q^{(-\frac{i}{\hslash M_q}{\mathcal{H}}_{q}t)}.$$

$$D_{t,q}^{\lambda }\hat{A}(t)=-\frac{i}{\hslash M_q}[\hat{A},\mathcal{H}],$$

$$D_{t,q}^{\lambda }{\hat{S}}_q(t)=-\frac{i}{\hslash M_q}[{\hat{S}}_q,\mathcal{H}],$$

$$\mathcal{H}=-g_q\frac{{\mu }_B}{\hslash M_q}{\hat{S}}_q\circ {\vec{H}}_{eff}.$$

$$\left[{\hat{x}}_i^q,{\hat{p}}_j^q\right]=ı[1+\lambda (1-q^{\prime })x]\hslash M_{q^{\prime }}{\delta }_{ıj}I$$

$$\left[{\hat{L}}_i^q,{\hat{L}}_j^q\right]=ı[1+\lambda (1-q^{\prime })x]\hslash M_q{\hat{L}}_k^q.$$

$$D_{t,q}^{\lambda }{\hat{S}}_q(t)=-\frac{[1+\lambda (1-q^{\prime })x]g_q{\mu }_B}{\hslash M_q}\widehat{S_q}\times {\vec{H}}_{eff}.$$

$$D_{t,q}^{\lambda }\widehat{m_q}(t)=-\left|\gamma \right|\widehat{m_q}\times {\vec{H}}_{eff}.$$

$$m_{x,q}=\rho {\cos }_q({\theta }_0){\cos }_q(\gamma H_{0}t)+\rho {\sin }_q({\theta }_0){\sin }_q(\gamma H_{0}t).$$

$$U_{\alpha }(t,t_0)=E_{\alpha }(-\frac{i}{{\hslash }^{\alpha }}\mathcal{H}{t}^{\alpha }),$$

$$D_t^{\alpha }{A}_{\alpha }^H(t)=-\frac{i}{{\hslash }^{\alpha }}[A_{\alpha }^H,\mathcal{H}],$$

$$D_t^{\alpha }{\hat{S}}_{\alpha }(t)=-\frac{i}{{\hslash }^{\alpha }}[{\hat{S}}_{\alpha },\mathcal{H}],$$

$$\mathcal{H}=-g_{\alpha }\frac{{\mu }_B}{{\hslash }^{\alpha }}{\hat{S}}_{\alpha }\circ {\vec{H}}_{eff}.$$

$${\hat{p}}^{\alpha }=-i{\left(\hslash \right)}^{\alpha }{M}_{x,\alpha }\frac{{\partial }^{\alpha }}{\partial x^{\alpha }}.$$

$$\left[{\hat{x}}_i^{\alpha },{\hat{p}}_j^{\alpha }\right]=ı\Gamma (\alpha +1){\hslash }^{\alpha }{M}_{\alpha }{\delta }_{ıj}I$$

$$\left[{\hat{L}}_i^{\alpha },{\hat{L}}_j^{\alpha }\right]=ı\Gamma (\alpha +1){\hslash }^{\alpha }{M}_{\alpha }{\hat{L}}_k^{\alpha }.$$

$${}_0^J{D}_t^{\alpha }{\hat{S}}_{\alpha }(t)=-\frac{M_{\alpha }\Gamma (\alpha +1)g_{\alpha }{\mu }_B}{{\hslash }^{\alpha }}\widehat{S_{\alpha }}\times {\vec{H}}_{eff}.$$

$${}_0^J{D}_t^{\alpha }{\hat{m}}_{\alpha }(t)=-\left|{\gamma }_{\alpha }\right|{\hat{m}}_{\alpha }\times {\vec{H}}_{eff},$$

$$m_{\alpha x}=A\cos {\theta }_0{E}_{2\alpha }(-{\omega }_0^2{t}^{2\alpha })+A\sin {\theta }_0.x.E_{2\alpha ,1+\alpha }(-{\omega }_0^2{t}^{2\alpha }).$$

Full text

Keywords: Structural Derivatives, Deformed Heisenberg Equation, LLG Equation, Non-extensive Statistics, Axiomatic Deformed Derivative

I Introduction

In recent works, we have developed connections and a variational formalism to treat deformed or metric derivatives, considering the relevant space-time/ phase space as fractal or multifractal Nosso On conection2015 and presented a variational approach to dissipative systems, contemplating also cases of a time-dependent mass Variational Deformed .

The use of deformed-operators was justified based on our proposition that there exists an intimate relationship between dissipation, coarse-grained media and a limit energy scale for the interactions. Concepts and connections like open systems, quasi-particles, energy scale and the change in the geometry of space–time at its topological level, nonconservative systems, noninteger dimensions of space–time connected to a coarse-grained medium, have been discussed. With this perspective, we argued that deformed or, we should say, Metric or Structural Derivatives, similarly to the Fractional Calculus (FC), could allows us to describe and emulate certain dynamics without explicit many-body, dissipation or geometrical terms in the dynamical governing equations. Also, we emphasized that the paradigm we adopt was different from the standard approach in the generalized statistical mechanics context Tsallis1 ; Tsallis BJP- 20 anos ; Tsallis2 , where the modification of entropy definition leads to the modification of the algebra and, consequently, the concept of a derivative Nosso On conection2015 ; Variational Deformed . This was set up by mapping into a continuous fractal space Balankin PRE85-Map-2012 ; Balankin Rapid Comm ; Balankin-Towards a physics on fractals which naturally yields the need of modifications in the derivatives, that we named deformed or, better, metric derivatives Nosso On conection2015 ; Variational Deformed . The modifications of the derivatives, accordingly with the metric, brings to a change in the algebra involved, which, in turn, may lead to a generalized statistical mechanics with some adequate definition of entropy.

The Landau-Lifshitz-Gilbert (LLG) equation sets out as a fundamental approach to describe physics in the field of Applied Magnetism. It exhibits a wide spectrum of effects stemming from its non-linear structure, and its mathematical and physical consequences open up a rich field of study. We pursue the investigation of the LLG equation in a scenario where complexity may play a role. The connection between LLG and fractionality, represented by an $\alpha-$deformation parameter in the deformed differential equations, has not been exploited with due attention. Here, the use of metric derivatives aims to take into account long-range forces, possible non-manifest or hidden interactions and/or the dimensionality of space.

In this contribution, considering intrinsically the presence of complexity and possible dissipative effects, and aiming to tackle these issues, we apply our approach to study the LLG equation with two metric or structural derivatives, the recently proposed scale $-q-$derivative Variational Deformed in the nonextensive statistical mechanics and, as an alternative, the axiomatic metric derivative (AMD) that has the Mittag-Leffler function as eigenfunction and where deformed Leibniz and chain rule hold - similarly to the standard calculus - but in the regime of low-level of fractionality. The deformed operators here are local. We actually focus our attention to understand whether the damping in the LLG equation can be connected to some entropic index, the fractionality or even dimensionality of space; in a further step, we go over into anisotropic Heisenberg spin systems in (1+1) dimensions with the purpose of modeling the weak anisotropy effects by means of some representative parameter, that depends on the dimension of space or the strength of the interactions with the medium. Some considerations about an apparent paradox in the magnetization or angular damping is given.

Our paper is outlined as follows: In Section 2, we briefly present the scale$-q-$derivative in a nonextensive context, building up the $q-$ deformed Heisenberg equation and applying to tackle the problem of the LLG equation; in Section 3, we apply the axiomatic derivative to build up the $\alpha-$deformed Heisenberg equation and to tackle again the problem of LLG equation. We finally present our Conclusions and Outlook in Section 4.

II Applying scale$-q-$derivative in a nonextensive context

Here, in this Section, we provide some brief information to recall the main forms of scale $-q-$ derivative. The readers may see ref. Nosso On conection2015 ; Variational Deformed ; Balankin PRE85-Map-2012 for more details.

Some initial claims here coincide with our work of Refs. Nosso On conection2015 ; Variational Deformed and the approaches here are in fact based on local operators Nosso On conection2015 .

The local differential equation,

[TABLE]

with convenient initial condition, yields the solution given by the q-exponential, $y=e_{q}(x)$ Tsallis1 ; Tsallis BJP- 20 anos ; Tsallis2 .

The key of our work here is the Scale$-q-$derivative (Sq-D) that we have recently defined as

[TABLE]

The eigenvalue equation holds for this derivative operator, as the reader can verify:

[TABLE]

II.1 $q-$ deformed Heisenberg Equation in the Nonextensive

Statistics Context

With the aim to obtain a scale$-q-$ deformed Heisenberg equation, we now consider the $scale-q-$ derivative Variational Deformed

[TABLE]

and the Scale -$q-$ Deformed Schrödinger Equation Variational Deformed ,

[TABLE]

that, as we have shown in Variational Deformed , is related to the nonlinear Schrödinger equation referred to in Refs. NRT-PRL-2011 as NRT-like Schrödinger equation (with $q=q^{\prime}-2$ compared to the $q-$index of the reference) and can be thought as resulting from a $time-scale-q-$deformed-derivative applied to the wave function $\psi$.

Considering in eq.(5), $\psi(\vec{r,t})=U_{q}(t,t_{0})\psi(\vec{r,t}_{0})$, the $q-$ evolution operator naturally emerges if we take into account a $time-scale-q-$ deformed-derivative (do not confuse with formalism of discrete scale time derivative):

[TABLE]

Here, $M_{q}$ is a constant for dimensional regularization reasons. Note that the q-deformed evolution operator is neither Hermitian nor unitary, the possibility of a $q-$unitary as $U_{q}^{\dagger}(t,t_{0})\otimes_{q}U_{q}(t,t_{0})=\mathbf{1}$ could be thought to come over these facts. In this work, we assume the case where the commutativity of $U_{q}$ and $\mathcal{H}$ holds, but the $q-$unitarity is also a possibility.

Now, we follow similar reasonings that can be found in Ref.Zitter and considering the Sq-D.

So, with these considerations, we can now write a nonlinear $Scale-q-$deformed Heisenberg Equation as

[TABLE]

where we supposed that $U_{q}$ and $\mathcal{H}$ commute and $M_{q}$ is some factor only for dimensional equilibrium.

II.2 $q-$deformed LLG Equation

To build up the scale$-q-$ deformed Landau-Lifshitz-Gilbert Equation, we consider eq.(7), with $\hat{A}(t)=\hat{S}_{q}$

[TABLE]

where we supposed that $U_{q}$ and $\mathcal{H}$ commute.

[TABLE]

Here, $\vec{H}_{eff}$ is some effective Hamiltonian whose form that we shall clearly write down in the sequel.

The scale$-q-$deformed momentum operator is here defined as $\widehat{p}_{q^{\prime}}^{\lambda}=-i\hbar M_{q^{\prime}}[1+\lambda(1-q^{\prime})x]\frac{\partial^{q}}{\partial x^{q}}.$

Considering this operator, we obtain a deformed algebra, here in terms of commutation relation between coordinate and momentum

[TABLE]

and, for angular momentum components, as

[TABLE]

The $q^{\prime}$ factor in $\hat{x}_{\imath}^{q^{\prime}},\hat{p}_{j}^{q^{\prime}},\hat{L}_{i}^{q^{\prime}},\hat{L}_{j}^{q^{\prime}},M_{q^{\prime}}$ is only an index and $q$ is not necessarily equal to $q^{\prime}$.

The resulting scale$-q-$deformed LLG equation can now be written as

[TABLE]

Take $\hat{m}_{q}\equiv\gamma_{q}\hat{S}_{q},$ $\gamma_{q^{\prime}}\equiv\frac{[1+\lambda(1-q^{\prime})x]g_{q}\mu_{B}}{\hbar M_{q}}$.

If we consider that the spin algebra is nor affected by any emergent effects, we can take $q^{\prime}=1$.

Considering the eq.(7) with $\hat{A}(t)=\hat{S_{q}}$ and $\hat{m_{q}}=\left|\gamma_{q}\right|\hat{S_{q}}$ and $q^{\prime}=1$; we obtain the $q-$time deformed LLG dynamical equation for magnetization as

[TABLE]

Considering $\vec{H}_{eff}=H_{0}\hat{k},$ we have the solution:

[TABLE]

In the figure, $\theta_{0}=0.$

III Applying Axiomatic Derivative and the $\alpha-$Deformed Heisenberg

Equation

Now, to compare results with two different local operators, we apply the axiomatic metric derivative. Following the steps on Zitter and considering the axiomatic MD Axiomatic Metric , there holds the eigenvalue equation $D_{x}^{\alpha}E_{\alpha}(\lambda x^{\alpha})=\lambda E_{\alpha}(\lambda x^{\alpha}),$ where $E_{\alpha}(\lambda x^{\alpha})$ is the Mittag-Leffler function that is of crucial importance to describe the dynamics of complex systems. It involves a generalization of the exponential function and several trigonometric and hyperbolic functions. The eigenvalue equation above is only valid if we consider $\alpha$ very close to $1.$ This is what we call low-level fractionality Axiomatic Metric . Our proposal is to allow the use o Leibniz rule, even if it would result in an approximation. So, we can build up an** **evolution operator:

[TABLE]

and for the deformed Heisenberg Equation

[TABLE]

where we supposed that $U_{\alpha}$ and $\mathcal{H}$ commute.

To build up the deformed Landau-Lifshitz-Gilbert Equation, we use the eq. (16), and considering and spin operator $\hat{S}_{\alpha}(t)$, in such a way that we can write the a deformed Heisenberg equation as

[TABLE]

whith

[TABLE]

Here, $\vec{H}_{eff}$ is some effective Hamiltonian whose form that we will turn out clear forward.

Now, consider the deformed momentum operator as nosso AHEP-g-factor ; Aspects-nosso ; Zitter

[TABLE]

Taking this operator, we obtain a deformed algebra, here in terms of commutation relation for coordinate and momentum

[TABLE]

and for angular momentum components as

[TABLE]

The resulting the $\alpha-$deformed LLG equation can now be written as

[TABLE]

If we take $\hat{m}_{\alpha}\equiv\gamma_{\alpha}\hat{S}_{\alpha}$, $\gamma_{\alpha}\equiv\frac{M_{\alpha}\Gamma(\alpha+1)g_{\alpha}\mu_{B}}{\hbar^{\alpha}}$, we can re-write the equation as the $\alpha-$deformed LLG

[TABLE]

with $\vec{H}_{eff}=H_{0}\hat{k}$. We have the Solution of eq.(23):

[TABLE]

In the figure below, the reader may notice the behavior of the magnetization, considering $\theta_{0}=0$.

For $\alpha=1,$ the solution reduces to $m_{x}=A\cos(\omega_{0}t+\theta_{0})$, the standard Simple Harmonic Oscillator solution for the precession of magnetization.

The presence of complex interactions and dissipative effects that are not explicitly included into the Hamiltonian can be seen with the use of deformed metric derivatives. Without explicitly adding up the Gilbert damping term, the damping in the oscillations could reproduce the damping described by the Gilbert term or could it disclose some new extra damping effect. Also, depending on the relevant parameter, the $q-$ entropic parameter or for $\alpha$, the increasing oscillations can signally that it is sensible to expect fractionality to interfere on the effects of polarized currents as the Slonczewski term describes. We point out that there are qualitative similarities in both cases, as the damping or the increasing of the oscillations, depending on the relevant control parameters. Despite that, there are also some interesting differences, as the change in phase for axiomatic derivative application case.

Here, we cast some comments about an apparent paradox: If we make, as usually done in the literature for LLG, the scalar product in eq. (13) with, $\hat{m}_{q},$ we obtain an apparent paradox that the modulus of $\hat{m}_{q}$does not change. On the other hand, if instead of $\hat{m}_{\alpha},$we proceed now with a scalar product with $\vec{H}_{eff}$ and we obtain thereby the indications that the angle between $\hat{m}_{\alpha}$and $\vec{H}_{eff}$ does not change. So, how to explain the damping in osculations for $\hat{m}_{q}?$ This question can be explained by the the following arguments. Even the usual LLG equation, with the term of Gilbert, can be rewritten in a form similar to eq. LLG without term of Gilbert. See eq. (2.7) in the Ref. The fascinating world of the Landau=00003D002013Lifshitz=00003D002013Gilbert . The effective $\vec{H}_{eff}$ field now stores information about the interactions that cause damping. In our case, when carrying out the simulations, we have taken $\vec{H}_{eff}$ as a constant effective field. Here, we can argue that the damping term, eq. (2.8) in Ref. The fascinating world of the Landau=00003D002013Lifshitz=00003D002013Gilbert being small, this would cause the effective field $\vec{H}_{eff}=\overrightarrow{H}(t)+\overrightarrow{k}(\overrightarrow{S}\times\overrightarrow{H})$ to be approximately $\overrightarrow{H}(t)$. In this way, the scalar product would make dominate over the term of explicit dissipation. This could, therefore, explain the possible inconsistency.

IV Conclusions and Outlook

In short:

Here, we tackle the problem of LLG equations considering the presence of complexity and dissipation or other interactions that give rise to the term proposed by Gilbert or the one by Slonczewski.

With this aim, we have applied scale -$q-$derivative and the axiomatic metric derivative to build up deformed Heisenberg equations. The evolution operator naturally emerges with the use of each case of the structural derivatives. The deformed LLG equations are solved for a simple case, with both structural or metric derivatives.

Also, in connection with the LLG equation, we can cast some final considerations for future investigations:

Does fractionality simply reproduce the damping described by the Gilbert term or could it disclose some new effect extra damping?

Is it sensible to expect fractionality to interfere on the effects of polarized currents as the Slonczewski term describes?

These two points are relevant in connection with fractionality and the recent high precision measurements in magnetic systems may open up a new venue to strengthen the relationship between the fractional properties of space-time and Condensed Matter systems.

The authors wish to express their gratitude to FAPERJ-Rio de Janeiro and CNPq-Brazil for the partial financial support.