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However, both directions of communication can be beneficial. Financial econometrics focuses a lot on theoretical properties of estimators as well as their restrictions and assumptions. As shown in the text above, the power-law cross-correlations literature has missed a lot of it since the very beginning. The econophysi... |
With increasing computational power, the non-existence (or negligence) of the asymptotic properties for the econophysics methods is becoming less of a problem as the properties can be simulated even for very long time series with various dynamic properties. As it turns out, many of these econophysics methods can compet... |
In this text, it has been shown that the power-law cross-correlations setting is inherently problematic and the bivariate Hurst exponent alone does not give any information about the relationship between two analyzed series. Only a comparison of [MATH] with its separate counterparts gives any information. However, unle... |
Acknowledgements Ladislav Kristoufek gratefully acknowledges financial support of the Czech Science Foundation (project 17-12386Y). References |
# Source: arxiv 1806.02082 # Title: Traveling domain walls in chiral ferromagnets # Sections: all # Downloaded: 2026-03-02T08:56:08.653246+00:00 |
Traveling domain walls in chiral ferromagnets Abstract We show that chiral symmetry breaking enables traveling domain wall solution for the conservative Landau-Lifshitz equation of a uniaxial ferromagnet with Dzyaloshinskii-Moriya interaction. In contrast to related domain wall models including stray-field based anisot... |
keywords: Micromagnetics , Dzyaloshinskii-Moriya interaction , chiral symmetry breaking , domain walls , traveling waves Introduction |
Magnetic domain walls (DW) are transition layers separating domains of different magnetizations in magnetic materials. They are fundamental for understanding domain structure. In mathematical idealization, they are a special form of kinks, connecting different asymptotic equilibrium states on the unit sphere. Static do... |
Antisymmetric exchange, also called the Dzyaloshinskii-Moriya interaction (DMI) , is present in a class of ferromagnetic materials whose crystal structure lacks inversion symmetry. The DMI has profound consequences for the equilibrium domain structure |
It breaks the symmetry of the interaction energy and the degeneracy of the solution space. In a DM material, if anisotropy is strong enough, the fully aligned ferromagnetic state is the ground state and a domain wall is an excitation. In the presence of DMI two specific meridians are selected, on which domain walls, st... |
The dynamics of domain walls is particularly interesting because a moving domain wall corresponds to a varying domain structure. Also, a domain wall can play the role of a unit of information that is transmitted via wall propagation. In general, domain wall dynamics is governed by the (conservative) Landau-Lifshitz (LL... |
or the (dissipative) Landau-Lifshitz-Gilbert (LLG) equation , derived from an energy functional. In a magnet with symmetric Heisenberg exchange and uniaxial anisotropy alone, traveling domain wall solutions for the corresponding LL equation are not possible. A traveling domain wall solution of the LL equation, called t... |
, is obtained in a model with an extra anisotropy, stemming from magnetostatic stray-field interaction, that is breaking the symmetry around the original uniaxial anisotropy axis. These solutions are typically discussed in a model with damping and external field, but they are actually exact solutions of the conservativ... |
A common mathematical feature of the models which have been shown to support traveling domain walls (without temporal oscillations) is a breaking of rotational symmetry through stray fields, see e.g. |
and for the extreme case of a traveling Néel wall. We notice that this effect can also be achieved by chiral symmetry breaking. Therefore, we are motivated to explore the possibility of traveling domain walls in the LL equation in the presence of DMI. |
The possibility for freely (unforced) traveling domain walls in chiral magnets is indicated by numerical and analytical studies for the wall mobility, as a response to an applied magnetic field. It has been shown that this is enhanced by the DMI compared to the standard Walker domain wall |
The increased mobility for chiral DW is also manifested in the case of motion due to an applied electrical current The dynamics of the DW has been discussed largely within collective coordinate approaches, called the [MATH] model, and numerical simulations |
We study the effect of chiral symmetry breaking on the laws of dynamics for magnetic domain walls. The dynamics of the magnetization can be linked to the symmetries of the magnetic interactions via associated conservation laws. We show that the symmetry-breaking introduced by the DMI in a film with perpendicular anisot... |
At the stable meridian ( [MATH] ) the wall is static and deviation from this at the symmetry point (center of wall) gives rise to a tilting angle [MATH] Tilting, however, induces a dynamic response. Here, we prove the existence of propagating domain wall solutions in the form of traveling waves in the conservative mode... |
Theorem 1 Let the easy-axis anisotropy parameter [MATH] and the DMI parameter [MATH] be sufficiently small. Then, for sufficiently small tilting angle, there exists a traveling domain wall. |
The existence and stability of traveling domain walls in the presence of damping and an external driving magnetic field can also be proven by means of a perturbative approach |
Details of the domain wall profiles come as a product of the proof of their existence. The propagating DMI walls have a profile that is substantially more complicated than the Walker DWs. For the case of bulk DMI a Bloch wall is static but a Néel wall is propagating with maximum velocity in stark contrast to the Walker... |
The Landau-Lifshitz equation We consider a ferromagnet described by the magnetization vector [MATH] that is a function of space and time but it has a constant magnitude [MATH] , where [MATH] is called the saturation magnetization. Statics and dynamics of the magnetization are governed by the Landau-Lifshitz equation, w... |
[EQUATION] where [MATH] is the normalized magnetization, in component form [MATH] The variable [MATH] is the dimensionless time that is measured in units of [MATH] , where [MATH] is the gyromagnetic ratio and [MATH] the permeability of vacuum. The effective field [MATH] contains the interactions in the material. We wil... |
[EQUATION] where [MATH] is the unit vector for the magnetization in the [MATH] direction. We measure distance in units of exchange length [MATH] where [MATH] is the exchange constant. There are two length scales in the model, [MATH] , where [MATH] is the anisotropy constant, and [MATH] , where [MATH] is the DMI constan... |
[EQUATION] In the energy ( ) we have only kept the Lifshitz invariant [MATH] in the [MATH] direction corresponding to cubic DMI given by [MATH] . Replacing |
[MATH] by [MATH] (interfacial DMI) or a linear combination of both yields a model that is mathematically equivalent modulo a rigid rotation around the [MATH] axis. |
The effective field entering ( ) is obtained by varying the energy, [EQUATION] The uniform (ferromagnetic) states [MATH] are the simplest time-independent (static) solutions of the LL equation ( ). For large anisotropy, such that |
[EQUATION] the ferromagnetic is the ground state of the system, while for [MATH] spiral configuration becomes the ground state The period of the spiral increases for increasing anisotropy and goes to infinity as [MATH] |
In this work we assume a material with [MATH] and we are looking for domain wall solutions as excitations of the ferromagnetic ground state |
A standard Bloch wall [EQUATION] for any combination of the signs, is a solution of Eq. ( ) also in the presence of DMI ( [MATH] ) as the contribution of the DM term on the right-hand-side of Eq. ( ) vanishes identically for these configurations. As the DMI is chiral, the walls with the same signs for [MATH] in Eq. ( )... |
One can easily prove that traveling DWs are not possible in model ( ) when the DMI is not included in the effective field ( ). This is one of the results obtained in where a direct and complete solution for the domain walls of the system for [MATH] is given. For a more intuitive proof let us consider the total magnetiz... |
[EQUATION] in the sense of the Cauchy principle value and calculate its time derivative using Eq. ( [EQUATION] This result is a reflection of the fact that the exchange and anisotropy interactions are invariant with respect to rotations around the third axis of the magnetization, and therefore the total magnetization [... |
In the model with effective field ( ) it is entirely due to the DMI that the symmetry is broken and the associated conservation law is not valid, thus allowing for the possibility of propagating domain walls. If we assume a rigid wall connecting the south pole ( [MATH] ) at [MATH] to the north pole ( [MATH] ) at [MATH]... |
[EQUATION] This gives an upper bound for the speed [EQUATION] More generally, a Lifshitz invariant [MATH] gives rise to an integrand [MATH] in Eq. ( ). In particular, no non-trivial traveling DW solution is possible in the case [MATH] corresponding to a wire along [MATH] with cubic DMI and stray-field induced anisotrop... |
Lastly, considering the conditions of Eq. ( ) and Eq. ( 10 ), we have, for positive [MATH] , the ordering [EQUATION] Derivation of a dynamical system for traveling waves |
Let us assume a rigid domain wall configuration propagating with a constant velocity [MATH] We substitute the traveling wave ansatz, [MATH] , in Eq. ( ) and this reduces to |
[EQUATION] Our aim is to prove the existence of domain wall solutions for Eq. ( 12 ) and to understand in detail their profiles. We begin by writing explicitly Eq. ( 12 |
[EQUATION] where we have the traveling wave [MATH] and the prime denotes differentiation with respect to [MATH] We define the orthonormal system of the three unit vectors |
[EQUATION] where we have defined [MATH] Both sides of Eq. ( 13 ) are orthogonal to the magnetization vector [MATH] , hence, they lie on the tangent plane of the unit sphere at point [MATH] Thus, Eq. ( 13 ) produces at most two independent scalar equations. We obtain these equations by projecting Eq. ( 13 ) on the two u... |
Projection of Eq. ( 13 ) on the vector [MATH] We obtain [EQUATION] The DM term does not contribute as this is proportional to [MATH] For the first term we have |
[EQUATION] For the second term we have [EQUATION] Inserting these results in Eq. ( 15 ), we obtain, [EQUATION] which integrates to |
[EQUATION] where the choice of the integration constant arises from the requirement that the north and south poles [MATH] constitute constant equilibrium solutions. |
We will make use in the following of the spherical parametrization for the magnetization vector [EQUATION] Eq. ( 18 ) reads [EQUATION] |
Furthermore, we may write [EQUATION] As a result of the last relation, the equality [MATH] occurs only at the poles. The solutions constructed in this paper assume the poles as values only in the limit [MATH] Thus, there is no conflict with Eq. ( 14 ) where [MATH] appears in the denominator. |
Projection of Eq. ( 13 ) on [MATH] We obtain [EQUATION] It is advantageous at this point to define the unit tangent vectors [MATH] and [MATH] in the direction of increasing [MATH] and [MATH] respectively, |
[EQUATION] and to let [EQUATION] The “linear velocity” components [MATH] and [MATH] are, respectively, along a parallel with radius equal to [MATH] (counter-clockwise) and a meridian (from north to south). They are given by |
[EQUATION] Note the relation [MATH] The second term on the right of Eq. ( 22 ) is calculated using the expressions of [MATH] [MATH] from Eqs. ( 21 ), as well as Eqs. ( 20 ) and ( 25 ), |
[EQUATION] In order to simplify the first term on the right of ( 22 ) we differentiate both sides of Eq. ( 24 ) and obtain [EQUATION] |
where the variable after the comma indicates the partial derivative with respect to that variable. The term [MATH] equals zero and the term [MATH] is parallel to [MATH] and does not contribute to the triple product [MATH] The remaining vector derivatives are given by [MATH] |
The contributing terms of [MATH] to be inserted in Eq. ( 22 ) are [EQUATION] where we have inserted [MATH] from Eqs. ( 20 ), ( 25 ). In the cross product [MATH] , the term [MATH] is orthogonal to [MATH] , thus, has vanishing projection on [MATH] We are left with the two combinations [MATH] , and we obtain |
[EQUATION] Inserting the results of Eqs. ( 26 ) and ( 29 ) into Eq. ( 22 ), we finally obtain [EQUATION] Let us introduce a new variable [MATH] |
[EQUATION] such that [MATH] corresponds to motion from the south to the north pole. We insert Eqs. ( 31 ) in Eq. ( 30 ) to obtain |
[EQUATION] Two equations connecting [MATH] with [MATH] and [MATH] are derived using Eqs. ( 25 ), ( 31 ) and ( 20 ), [EQUATION] Eqs. ( 32 ) and ( 33 ) have the equilibrium solutions [MATH] that correspond to [MATH] via Eq. ( 20 ). We obtain the autonomous system of ordinary differential equations (ODEs) |
[EQUATION] Existence of traveling domain walls 4.1 The system of equations for traveling chiral domain walls The presence of the DM interaction ( [MATH] ) in Eqs. ( 34 ) allows us to prove the existence of traveling domain wall solutions, that connect the two equilibrium points [MATH] and [MATH] , representing constant... |
We make a change of variable that transforms the sphere to a cylinder, allowing for simpler calculation. We define a new variable [MATH] by [MATH] Making use of the formulae |
[EQUATION] the system of Eqs. ( 34 ) becomes [EQUATION] The gradient of the right side of these equations with respect to the three dependent variables remains globally bounded. Therefore the ODE system exhibits global existence and uniqueness of solutions as well as continuous dependence of the solution on their initi... |
[EQUATION] The circular cylinder (of radius unity) formed in the new coordinates when the points [MATH] and [MATH] are identified, is topologically equivalent to the original sphere [MATH] , punctured at its north and the south poles. The north pole of the sphere corresponds to [MATH] , the south pole to [MATH] The mer... |
In the absence of the DM term ( [MATH] ), the system ( 36 ) yields a standing domain-wall ( [MATH] and, thus, [MATH] ). The solution is [MATH] [MATH] and [MATH] Taking the hyperbolic tangent of both sides in the latter relation and recalling that [MATH] , obtains the standing domain walls respectively for [MATH] and [M... |
[EQUATION] The plus sign corresponds to [MATH] and the minus sign corresponds to [MATH] The domain-wall ( 38 ) survives even in the presence of the DM term for [MATH] , which eliminates the DM term in Eqs. ( 36 ). |
4.2 Symmetry and choices We require the functions [MATH] and [MATH] to be odd and the function [MATH] to be even. This is consistent with the fact that when the sign of [MATH] is reversed the substitution |
[EQUATION] leaves the system of Eqs ( 36 ) invariant. Effectively, this restricts the analysis of the system to the domain [MATH] adopting the initial data |
[EQUATION] The condition that [MATH] at the wall center ( [MATH] ) means that the domain wall goes from the south to the north pole, i.e. [MATH] as [MATH] More general domain walls can be easily obtained as explained in Sec. 5.1 We narrow our search to strictly monotone traveling domain walls. As a domain wall connects... |
(we have tacitly assumed that [MATH] ), monotonicity means that [MATH] is strictly increasing and [MATH] , in the limit [MATH] Adopting the further conditions |
[EQUATION] guarantees that, for the velocity [MATH] , the DM term counteracts the velocity term in Eq. ( 36a ) as [MATH] increases from its zero value. From the same equation it is clear that there must hold [MATH] in order to have a monotone domain wall. This velocity bound has been also derived in Eq. ( 10 ). |
Theorem 2 A strictly monotone domain wall exhibiting symmetries ( 39 ) and taking the values ( 40 ) at its center, satisfies the relations |
[EQUATION] In all limits, the convergence is exponential. The magnetization vector converges to the north pole as [MATH] increases ( [MATH] ) and the domain wall travels to the right. |
Proof. Since [MATH] is strictly increasing, we combine Eqs. ( 36a ) and ( 36b ) to make a change of the independent variable from [MATH] to [MATH] |
[EQUATION] Multiplying both sides of the equation by the integrating factor [MATH] obtains [EQUATION] Integrating from an arbitrary [MATH] to positive infinity, multiplying both sides by [MATH] and doing simple algebra obtains |
[EQUATION] We note that [EQUATION] We now obtain Eq. ( 42a ), where [EQUATION] Inserting the limit of Eq. ( 42a ) in Eqs. ( 36b ) and ( 36c ) we obtain Eqs. ( 42b ) and ( 42c ) respectively. |
If we integrate Eq. ( 44 ) from zero to [MATH] , we obtain [EQUATION] The right side of this equation represents a balance between two terms that tend to infinity as [MATH] increases. The balance is delicate; controlled by the angle [MATH] , it forces the right side to stay between [MATH] The rate of change of [MATH] |
[EQUATION] is obtained by combining Eqs. ( 36b ) and ( 36c ). These formulae are valid as long as [MATH] is monotone. The boundary of monotonicity is reached at [MATH] |
Remark: One may need the integral of [MATH] e.g. for the purpose of producing bounds. The integral is computable by exact formula |
[EQUATION] 4.3 Existence of domain-wall solutions: A topological approach We begin by proving the following technical theorem. Lemma 1 |
Let [MATH] and let the function [MATH] be defined by the formula [EQUATION] where [EQUATION] Let the ordering ( 11 ) of the parameters [MATH] be satisfied. Then, for all [MATH] for which [MATH] , the orbit with initial conditions ( 40 ) either never reaches the boundary [MATH] |
or it reaches the boundary transversely ( [MATH] ). Proof. We prove the theorem with the aid of two claims Claim 1. If [MATH] , then the equalities [MATH] and [MATH] cannot hold simultaneously. |
Proof of claim 1. Inserting [MATH] and [MATH] in the first Eq. ( 36 ), we obtain [EQUATION] which is expressed as a quadratic equation in [MATH] |
[EQUATION] The roots are [EQUATION] Considering the constraints ( 11 ) the following estimate applies, [EQUATION] Taking the logarithm on both sides, produces exactly the inequality [MATH] The exponent [MATH] can be made arbitrarily small by having values of [MATH] sufficiently small. |
With claim 1 now proved, we still need to exclude the simultaneous holding of [MATH] and [MATH] . Claim 2 does more than this. Claim 2. [MATH] when [MATH] |
Proof of Claim 2. When [MATH] , we obtain easily that [MATH] is absolutely greater than the right hand side of Eq. ( 46 ) and hence than the left side, [MATH] The requirement [MATH] of the theorem, implies [MATH] , hence [MATH] |
Theorem 3 [Local existence of traveling domain walls] Let [MATH] satisfy the condition of Lemma Then, there is a neighborhood [MATH] of [MATH] , such that for [MATH] there is a strictly increasing domain wall solution of the system of Eqs. ( 36 ). The domain wall velocity is [MATH] if [MATH] and [MATH] if [MATH] |
Proof. We fix the value [MATH] so that it satisfies the condition of Lemma We assume that the initial value of the angle coordinate [MATH] is less than [MATH] The case [MATH] will be proved by symmetry. We then consider the solution trajectories of the system of Eqs. ( 36 ) with initial values ( 40 ) and with velocity ... |
1. Subset A: contains the values of [MATH] for which the first occurrence of [MATH] is at [MATH] 2. Subset B: contains the values of [MATH] for which the first occurrence of [MATH] is at [MATH] |
3. Subset C: contains the remaining points of the set [MATH] , that is all the velocities [MATH] for which [MATH] for all [MATH] |
The sets A and B are open in [MATH] , due to the continuous dependence of the orbits of system ( 36 ) on the velocity [MATH] and due to the fact that orbits reaching the boundary planes [MATH] in phase space do so transversely, as shown in Lemma In the claims below we show that the velocity [MATH] and, hence, a neighbo... |
The sets A and B are disjoint and open in the closed interval [MATH] Therefore, the complement C of their union is nonempty. As a result, there is a velocity [MATH] with a trajectory for which [MATH] for [MATH] and thus [MATH] is a monotone function that converges to infinity. This solution represents a domain wall. |
Claim 1. The set A is nonempty. We consider the orbit with [MATH] . As a result, the right side of Eq. ( 36a ) remains positive, when [MATH] , and [MATH] converges to positive infinity due to its first term. Thus, the orbit crosses the value [MATH] |
Claim 2. The set B is nonempty. We consider the orbit with [MATH] As the point [MATH] evolves from [MATH] , we will prove that, for suitable values of [MATH] and [MATH] , the variable [MATH] will reach the value [MATH] monotonically; this orbit will then belong to set B, thus proving that it is nonempty. Our proof cons... |
Making the following simplifying changes of notation and introducing the function [MATH] [EQUATION] we rewrite Eqs. ( 43 ) and ( 47 ), respectively, as the system |
[EQUATION] and we impose initial conditions [MATH] and [MATH] As [MATH] the solutions of the linearized system [EQUATION] approximate the solutions of the full system ( 56 ) uniformly in compact sets [MATH] The substitution [MATH] transforms the system ( 57 ) to |
[EQUATION] where dots on top indicate derivatives taken with respect to [MATH] and [EQUATION] We make the following observations: (i) [MATH] is negative above the graph of [MATH] and below the graph of [MATH] ; it is positive between the two graphs, (ii) [MATH] lies between the two graphs, and (iii) [MATH] monotonicall... |
Returning to the original variables, we have [EQUATION] For the linear approximation to be valid, we need [MATH] In this way a large [MATH] is reached, while [MATH] and [MATH] are still small and nearly equal to each other ( [MATH] approaches [MATH] ). In Eq. ( 56a ), the second term on the right becomes negligible, as... |
In order to prove the theorem, all we have to do is to select a [MATH] sufficiently small. More traveling domain walls 5.1 Parity related domain walls |
The domain walls whose existence has been proved in Theorem go from the south to the north pole, i.e. [MATH] as [MATH] , and the velocity is [MATH] i.e. , they travel to the right. For any such domain wall solution we can obtain further traveling domain walls with the following simple transformations. These are easy to... |
1. South to north pole, positive velocity. We have obtained in Theorem a domain wall [MATH] that goes from the south to the north pole and has positive velocity [MATH] i.e. , it is traveling to the right. |
2. South to north pole, negative velocity. Apply [MATH] and [MATH] In terms of the variables used in Eqs. ( 36 ) we obtain the same transformation by [MATH] |
3. North to south pole, positive velocity. Apply [MATH] The same transformation is obtained by [MATH] 4. North to south pole, negative velocity. |
Apply [MATH] and [MATH] The same transformation is obtained by [MATH] For the last two cases the value [MATH] at the wall center is [MATH] , while we have assumed [MATH] in Sec. |
Our results carry over to the case of the so-called interfacial DMI, in which the bulk DMI term in Eq. ( ) is replaced by the term [MATH] Any solution [MATH] of the original model is a solution of the interfacial DMI model if we rotate the magnetization vector by [MATH] |
5.2 Higher velocity domain walls If we obtain a traveling domain wall solution for specific parameter values [MATH] , we can obtain further solutions by a straightforward scaling. Let us assume [MATH] a traveling domain wall solution with velocity [MATH] for specific values of the parameters [MATH] such that [MATH] , s... |
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