Static kinks in chains of interacting atoms

We theoretically analyse the equation of topological solitons in a chain of particles interacting via a repulsive power-law potential and confined by a periodic lattice. Starting from the discrete model, we perform a gradient expansion which delivers the kink equation in the continuum limit. The power-law interaction modifies the sine-Gordon equation, giving rise to a rescaling of the coefficient in from of the second derivative (the kink width) and to an additional integral term. We argue that the integral term does not affect the local properties of the kink, but it governs the behaviour at the asymptotics. The kink behaviour at the center is dominated by a sine-Gordon equation, where the kink width tends to increase with the power law exponent. When the interaction is the Coulomb repulsion, in particular, the kink width depends logarithmically on the chain size. We define an appropriate thermodynamic limit and compare our results with existing studies performed for infinite chains. Our formalism allows one to systematically take into account the finite-size effects and also slowly varying external potentials, such as for instance the curvature in an ion trap.


I. INTRODUCTION
The Frenkel-Kontorova model reproduces in one dimension the essential features of stick-slip motion between two surfaces [1][2][3]. The ground state is expected to describe the structure of a one-dimensional crystal monolayer growing on top of a substrate crystal. In one dimension the elastic crystal is modelled by a periodic chain of classical particles with uniform equilibrium distance a, which interact with a sinusoidal potential with periodicity b [4,5]. Frustration emerges from the competition between the two characteristic lengths: Depending on the mismatch between a and b and on the strength of the substrate potential, a continuous transition occurs between a structure with the substrate's period (commensurate), to an incommensurate structure [6]. The transition is characterized by proliferation of kinks, namely, of local distributions of excess particles (or holes) in the substrate potential. When the interactions of the elastic crystal are nearest-neighbour, in the long-wavelength limit the dynamics of a single kink is governed by the integrable sine-Gordon equation [6][7][8].
The study of kinks and of nanofriction in these systems requires one to analyse the effects of the tails of the interactions on the sine-Gordon equation. Specifically, in one dimension the energy is non additive in Coulomb systems [32]. Yet, long-range effects are marginal: the exponent of the Coulomb interaction formally constitutes a critical value, such that for slower power-law decays the dynamical equations are characterized by fractional derivatives, while for faster decays the derivatives are integer [33,34]. The effect of long-range interactions on the commensurate-incommensurate transition have been discussed [35,37], the kink solutions in a periodic potentials have been analysed numerically for the long-range Kac-Baker interactions [36] and for dipolar and Coulomb potentials [38]. Analytic studies of the kink solutions have been performed in the thermodynamic limit [36][37][38].
The aim of the present work is to review the analytical derivation of the kink equation for power-law interacting potentials by means of a gradient expansion, which is implemented following the lines of the study of Ref. [39,40]. This derivation allows us to determine the local properties of the kink as a function of the interaction range for integer exponents when this decay as the Coulomb or faster, its asymptotic behaviour, as well as to systematically take into account the finite-size effects, thus setting the basis of a study where these effects can be included in a perturbative fashion. This manuscript is organised as follows. In Sec. II we introduce the Lagrangian of an atomic chain in a periodic potential, where the atoms interact via a power-law potential. In Sec. III we consider the long-wavelength limit and derive the equation for the static kink. We discuss separately the case of Coulomb interactions. We analyse then the thermodynamic limit and compare our results with the ones of Ref. [38]. The conclusions are drawn in Sec. IV.

II. AN ATOMIC CHAIN IN A PERIODIC SUBSTRATE POTENTIAL
A chain of N interacting atoms with mass m is confined in a finite volume and is parallel to the x axis. Their atomic positions and canonically-conjugated momenta are x j and p j , with j = −N/2, N/2+1 . . . , N/2−1 and x j < x j+1 . Their Lagrangian L reads where V pot is the potential energy and is thus the sum of a periodic substrate potential and of the harmonic interaction between pairs of particles, V pot = V opt + V n . The periodic substrate potential V opt is a sinusoidal lattice with periodicity b and amplitude V 0 : The atomic interaction V n couple atoms at distance ra with strength scaling with 1/r n+2 : with n an integer number, n ≥ 1, and K n is the spring constant between nearest neighbour. The interaction term vanishes when the atoms are at the equilibrium positions x (0) j = ja. We note that interactions of this form are obtained by expanding the interaction potential till second order in the displacements about the equilibrium positions of the interaction forces, and discarding anharmonicities. Let the interaction between two particles at distance x be given by where W n is a constant which depends on n. Then, the spring constant takes the form The textbook limit, where the particles interact via nearest-neighbour interactions, is recovered by letting n → ∞. In this case W n shall be appropriately rescaled in order to warrant that the spring constant K ∞ remains finite, with K ∞ = lim n→∞ K n . In the following we discuss the cases of n > 1 (with dipolar and van der Waals interactions corresponding to the particular values n = 3 and n = 6 respectively), as well as the Coulomb interaction n = 1. For Coulomb interaction between particles of charge q, the spring constant is K = 2q 2 /(4πε 0 a 3 ), with q the charge of the particles and ε 0 the vacuum's permittivity [39].
Here and in what follows we assume motion along one axis and open boundary conditions. This can be realised by means of anisotropic traps and sufficiently cold atoms, such that one can assume that the motion in the transverse direction is frozen out. Typically the trap curvature gives rise to inhomogeneity in the equilibrium particle distribution, which leads to a position-dependent spring constant K n . Below we assume that the atoms are uniformly distributed. We note, however, that this formalism can systematically include the trap inhomogeneity, as shown in Ref. [39], as long as the trap curvature can be treated in the continuum limit [41].

A. Equilibrium configuration in the discrete chain
The equilibrium configurations of the discrete chain are solutions of the equation of motion which satisfy mẍ j = 0 for all j. For V 0 = 0, namely, in the absence of the substrate potential, the chain is uniform and the ground state is the uniform chain with equilibrium positions whereū j are static displacements that solve the set of equations: A static kink describes particles displacements which are localized in a region of the chain, such that the chain is uniform at the edges. The solution interpolates between the two boundary values u j → 0 for j → −N/2 and u j → b for j → +N/2 and describes a topological soliton. The antikink is the topological soliton of opposite charge and interpolates between u j → 0 for j → −N/2 and It is convenient to introduce the phase variable θ j of particle j, which is a dimensionless scalar and is defined as or, alternatively:x The phase function θ j gives the shift of the ion j from the commensurate configuration, such that for every 2π change in the phase the respective ion position is shifted by one period b of the periodic potential. The equation of the phase function is then given by where and represents the mismatch between the former equilibrium positions of the crystal and the nodes of the periodic potential. The equation depends now on the power-law exponent, on the mismatches δ j , and more specifically, on the ratio a/b, and on the dimensionless ratio which scales the weight of the interactions with the respect to the localizing potential. We now conveniently rewrite the mismatch by introducing the ratio = a/b , with x the nearest integer to x. The configuration is commensurate when the ratio a/b = , then the groundstate equilibrium positions are x which vanishes when both length scales are commensurate with each other. In general, δ is a periodic function of the ratio a/b, in fact δ(a/b + p) = δ(a/b), p ∈ Z, and its value ranges in the interval −π < δ < π. We use this expression and rewrite Eq. (10) in terms of the phase function: In the rest of this manuscript we will consider structures whose ground state is commensurate, and thus take δ = 0 in Eq. (14). We analyse the equation of motion of single topological solitons in the long-wavelength limit when the periodic substrate potential is a small perturbation to the elastic crystal, m K 1.

III. EQUATION OF THE STATIC KINK IN THE LONG-WAVELENGTH LIMIT
In this section we derive the continuum limit of Eq. (10) extending the procedure of Ref. [39] to kink's equation. In our treatment we keep n finite and consider the commensurate case, a = b, when the mismatch δ vanishes. For completeness, we first shortly review the derivation of the sine-Gordon equation in the case of nearest-neighbour interactions and briefly discuss the properties of its exact solution.

A. Nearest-neighbour interactions
The sine-Gordon equation of the Frenkel-Kontorova model in our model is found by considering the limit n → ∞ in Eq. (14), thus keeping only the nearest neighbour terms (r = 1) and taking K ∞ = lim n→∞ K n to be constant. The long-wavelength limit is determined by first taking the Fourier transform of the phase function, θ j →θ(k) = j e ikxj θ j / √ N and expanding the equation of motion for small k [33]. In k space the interaction term takes the form −2K ∞ (1−cos(ka))θ(k) ∼ −k 2 a 2 K ∞θ (k). Going back to position space, this term is cast in terms of second-order derivative of the phase function. Equation (14) takes then the form of a sine-Gordon equation [8]: One can rewrite the equation as where d is the width of the kink [38], Equation (16) admits the solution [8] θ with σ = ±1 the topological charge and x 0 the position of the kink. Specifically, for σ = 1, the solution is a single kink with θ(x) → 2π for x → ∞, and θ(x) → 0 for x → −∞. The antikink corresponds to σ = −1 and is thus the mirror reflection of the kink at x 0 . We note that the continuum limit is consistent when the width of the kink is much larger than the interparticle distance, d a, and the potential is a small perturbation to the elastic crystal.
The constant m K scales the mass m SG of the sine-Gordon kink. We recall that the kink's mass m kink is defined as m kink = m j (∂ū j /∂x 0 ) 2 . In the continuum limit and using Eq. (18), one obtains [38] Anharmonicities of the potential give rise to higherorder derivatives, which give rise to an asymmetry between kink and antikink [38,42]. In this treatment we discard these terms, restricting the expansion of the interaction potential to the harmonic terms. In this limit the kink and antikink in the continuous limit just differ because of the topological charge σ. We further note that, by performing the continuum limit, we discarded higher order terms in the gradient expansion. These terms account for discreteness effects and give rise to an effective kink narrowing [38].

B. Power-law interactions
We now perform a gradient expansion for power-law interactions. This is done by means of a manipulation of Eq. (14) which is equivalent to a Taylor expansion for low momenta in Fourier space, and which consists is singling out the terms contributing to the second derivative. The equation of motion for the static kink then becomes an integro-differential equation. In the following we will assume m K 1, unless otherwise specified. In order to study the long-wavelength limit we use the prescription 2i/N → ξ and θ j → θ(ξ). For N 1, then ξ can be treated as a continuous variable defined in the interval [−1/2, +1/2]. With this prescription Eq. (14) takes the form: where I n is dimensionless: (21) and the discrete nature of the chain at atomic distances enters through the (high-frequency) cutof a = 1/N .

In Eq. (20) we have also introduced the characteristic length
which would correspond to the kink's width when simply truncating the sum in Eq. (14) till the nearestneighbours. We note that in this formalism the chain length is a low-frequency cutoff, and in particular the limit N 1 is equivalent to small k in Fourier space. We will first keep N constant, thus consider a finite chain, and take the thermodynamic limit only after performing the gradient expansion.
We now integrate Eq. (21) by parts applying the procedure as in Ref. [39] and rewrite it as the sum of three terms [39]: where the first term on the RHS contains the contributions of the chain's edges: (1/2) n+1 , (24) and θ (ξ) ≡ ∂θ(ξ)/∂ξ. The second integral describes the contribution of the nearest-neighbour terms: By making a Taylor expansion about ξ, Iā can be cast into the form : with θ (ξ) ≡ ∂ 2 θ(ξ)/∂ξ 2 and θ j (ξ) ≡ ∂ j θ(ξ)/∂ξ j . Finally, the term I n reads: and still contains second derivatives of the phase function. We note that it can be also rewritten in the form: so that the equation for the static kink is the integrodifferential equation: − d 2 n n + 2 n(n + 1) Here, the contribution due to I edge has been discarded, and we have used Nā = 1 and x = N aξ. Moreover, we have discarded higher order derivatives θ (2j) (a), which account for discreteness effects. If we would take now the thermodynamic limit, and thus let N → ∞, then this expression would coincide with the one reported in Ref. [38], apart for the definition of the scaling coefficients and for higher order local derivatives. We note, however, that the integral term in Eq. (29) still contains terms which can be of the same order as k 2 a 2 . In order to single them out, we perform a further step of the partial integration. We identify two qualitatively different cases, the case n > 1 and the Coulomb case n = 1, which we discuss individually below.

Power-law interactions with n > 1
Performing partial integration of I n for n > 1 we obtain and I n = 1 n(n 2 − 1) We now collect the edge contributions and verify that they scale like 1/N n−1 , thus we neglect them under the reasonable assumption that the kink derivatives vanish at the edges. Using that Nā = 1 and going back to dimensional coordinates (x = N aξ), we obtain the integrodifferential equation for a static kink in a chain of atoms interacting via power-law interactions: = d 2 n n − 1 a n−1 n(n + 1) where L = N a is the chain's length, and now the thirdorder derivative is taken with respect to the dimensional coordinates.

Coulomb interactions
Fore n = 1 partial integration of Eq. (27) leads to the expression where Neglecting the contributions from the edges we obtain the equation for a static kink in a sufficiently long chain of single-component charges:

C. Discussion
The integro-differential equations (32) and (35) are characterised by a left-hand side (LHS), which is a SG equation, and a the right-hand side (RHS), which is an integral term depending on the kink. We first observe that a priori the integral term on the RHS cannot be discarded. This term, in particular, is responsible for the behaviour at the edges of the chain, far away from the kink's core: Simple considerations show that the tail of the kink decays algebraically with 1/x n+1 . This behaviour is in contrast to the exponential decay of the sine-Gordon kink at the asymptotics in Eq. (18). It is recovered by inspecting the behaviour of the integral term at distances much larger than the kink core, where one can replace the kink's first derivative with a Dirac-delta function [38], or can be derived from the general properties of the interactions [37]. In this limit, in particular, the second derivative on the LHS can be neglected. Thus, the integral term is majorly responsible for the non-local properties of the kink and gives rise to a power-law interactions between distant kinks [37,38].
At the kink's core the integral term is negligible for n > 1 as long as the ratio a/d is sufficiently small. Analytical estimates and numerical calculations indicate that the RHS of Eq. (32) scales approximately like a/d for n = 2 and has a sharper decay for larger n. Therefore, for n > 1 the integral term scales like discreteness effects at the kink's core, and in the continuum limit the kink's core is determined to good approximation by a sine-Gordon equation with a rescaled kink's width It is also interesting to analyse the scaling of d n with n: For n finite it increases as n decreases: In fact, the repulsive interactions become locally increasingly strong. The nearest-neighbour case can be recovered by appropriately rescaling the elastic constant K n , such as K n ∼ nK ∞ . In this limit the term on the right-hand side of Eq. (32) tends to zero and one recovers the sine-Gordon equation. The behaviour at the kink's core for n = 1, corresponding to repulsive Coulomb interactions, shall be discussed apart. For this case we have considered a SG kink and numerically verified that this slowly converges to the solution of Eq. (35) as a/d becomes smaller. For finite but large chain the integral term can be approximated by the function −α sin θ, plus a correction which is negligible at the kink's core. The kink's equation at the core is then given by a SG equation, with the kink's width where α = α(a/d) and 1 > α > 0. This coefficient monotonously decreases with a/d for the values we checked. These predictions are in excellent agreement with the numerical results with a discrete chain of ions. Figure 1 displays the kink's solution for N = 100, 300, 500. The solid blue line corresponds to a Sine-Gordon kink whose width is given by Eq. (37), the convergence of the behaviour at the kink's center with the SG kink is visible.On the basis of these numerical analysis we conclude that the kink at the core is described by a SG kink whose width is proportional to √ log N and thus weakly depends on the chain's length.
This dependence of the kink's width on the ions number is a consequence of the weak non-additivity of Coulomb systems in one dimension. Extensivity can be formally re-established for instance, by rescaling the spring constant as which is an implementation of Kac's scaling for onedimensional Coulomb interactions [32,40]. For a static kink, this is equivalent to increasing the depth of the potential as V 0 → V 0 log N . This rescaling leads to the definition:d Thus, in the thermodynamic limit the core of the static kink is described by a SG kink with width (39), while at the tails the kink decays as 1/x 2 .

IV. CONCLUSIONS
We have shown a procedure that permits to determine the equation for a static kink in a chain of atoms interacting with a repulsive, power-law potential, and to infer the properties of the solution. Starting from the discrete equation we have taken a continuum limit and cast the equation into the sum of a local term, which has the form of a SG equation, and a integral term. We have argued that in the continuum limit the SG equation determines the properties at the kink's core, while the integral gives the behaviour at the tails. The correction of the integral term to the behaviour at the kink's core, in particular, are of the same order of the discreteness effects. These effects modify the kink's width and form. Moreover, they significantly modify the dynamical properties [42,43].
The formalism discussed here can be extended by including a non-homogeneous density distribution, as it is the case in the presence of a harmonic trap. In this case the gradient expansion shall be performed including the density in the continuum limit and will generally give rise to first-order derivatives of the phase function [39]. We thus expect that trap inhomogeneities may give rise to asymmetry of the kink-antikink solutions even in the limit in which the anharmonicities of the interaction potential can be neglected.