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Domain wall roughening in dipolar films in the presence of disorder

We derive a low-energy Hamiltonian for the elastic energy of a N\'eel domain wall in a thin film with in-plane magnetization, where we consider the contribution of the long-range dipolar interaction beyond the quadratic approximation. We show that such a Hamiltonian is analogous to the Hamiltonian of a one-dimensional polaron in an external random potential. We use a replica variational method to compute the roughening exponent of the domain wall for the case of two-dimensional dipolar interactions.


I. INTRODUCTION
The roughening properties of elastic manifolds in the presence of quenched disorder is a well studied problem. The subject of study is usually the asymptotic (long-distance) regime in which one wishes to compute the roughening exponent ζ of the D-dimensional elastic manifold (described by a displacement field φ(x)) defined by where x is the position vector in the D-dimensional configuration space and where the brackets represent an average over thermal fluctuations and the overline an average over the realizations of the quenched disorder. Obviously, the study of the crossover between short distances, where the thermal fluctuations dominate and the long-distance behaviour, which is determined by the disorder, is also of interest. Among the methods applied to the study of this problem are the use of Imry-Ma type of arguments [1][2][3] , the mapping of onedimensional interfaces in two dimensions to the noisy Burgers equation 4 , which yields an exact result for the roughening exponent 5,6 , variational approaches involving replica averaging and replica symmetry breaking [7][8][9] and functional renormalization group calculations 10,11 .
The role played by long-range interactions does not seem to be so well understood, although it follows from the one-loop functional RG results of Emig  In particular, if one applies this result to the case of a line domain in a thin magnetic film (D = 1), the effective expansion parameter is ǫ = 3 in both cases, which means that in this case and to one-loop order, the critical exponents obtained are the same in the absence or presence of dipolar interactions. Although one expects the RG calculations to provide a qualitative understanding at low dimensions compared to the upper critical dimension (D = 3 in this case), it is still questionable to what extend one can perform such analytical continuation using just the one-loop results. Furthermore, the results of Emig and Nattermann were based on a series expansion of the dipolar energy 13,14 which meant that, in the absence of disorder, the Hamiltonian describing the low energy degrees of freedom of the domain wall was quadratic in the domain-wall displacement field, i.e. the random potential is the only source of non-linearity in the problem. One still needs to justify that such approximation is sufficient 15 .
Having in mind such difficulties, we wish to discuss the properties of a one-dimensional domain wall in a two-dimensional ferromagnetic film, in the presence of dipolar interactions and a short-range correlated random field, where the dipolar interaction is treated beyond the quadratic approximation. We wish to check whether it is still possible to map such a problem to a (modified) Burgers equation, as was done by Huse et al. 5 for the case of random-bond disorder and by Zhang 6 for the case of random-field disorder, in the absence of long-range interactions. Such a mapping was achieved because one can map the line domain to the world line of a quantum particle in 1+1 dimensions. The free energy of the line domain can then be shown to obey a Schrödinger equation in imaginary time in the presence of a random potential. A Cole-Hopf transformation can then be used to convert this equation into a noisy Burgers equation with additive noise, for which the stationary distribution is known in one-dimension 16 . This allows one to compute the roughening exponent. In our case, the presence of the dipolar interaction does not allow one to write the energy of the domain wall as an action for a single quantum particle, but instead we obtain an action for a quantum particle which interacts with a annealed 'vector potential', whose propagator is determined by the dipolar propagator. In 1+1 dimensions, a gauge transformation allows us to substitute the vector potential by a scalar one, which can then be integrated out, leaving us with an action similar to the action of a polaron as considered by Feynman 17,18 , i.e. the dipolar interaction introduces a 'self-retarded' interaction of the particle with itself.
Despite the fact that such action cannot be integrated exactly, we can still apply a variational method in the spirit of Feynman 7-9 , in which one introduces replicas of the system in order to average the free energy of the domain wall over the realizations of the random field. Using the hierarchical replica symmetry breaking ansatz of Parisi 19 , we were able to derive a set of non-linear equations whose self-consistent solution yields the long-distance physics of the problem within the realm of this approximation. With this method, we are able to treat two different situations. The one of a thin ferromagnetic film grown on a non-magnetic substrate, in which the dipolar interaction has the usual three dimensional form, and the one of a thin ferromagnetic film grown between two type I superconductors, i.e. the case of a superconductor/ferromagnet/superconductor (SC/FM/SC) heterostructure. We have shown elsewhere 20 that in this case the dipolar interaction has a two-dimensional form at small wave-vectors, coming from the renormalization of the magnetic energy due to the Meissner effect in the superconductors. Although we were unable to obtain an analytic solution of the variational non-linear equations, we were able to justify that in the case of two-dimensional dipolar interactions (i.e. for the SC/FM/SC heterostructure), the quadratic approximation is indeed adequate. This case is not only easier to treat analytically, it is also more interesting than the case of three dimensional dipolar interactions, for which one expects the dipolar interaction to be irrelevant (see above). Therefore, we restrict our discussion to the case of two-dimensional dipolar interactions in this paper. Furthermore, we can also show that in this case, if one works within the quadratic approximation, the elastic energy of the domain-wall has the same analytical form as the energy of a domain wall in a ferroelastic material as studied by Kolomeiskii et al. 12 and also of the energy of a liquid-gas interface in a disordered solid, as studied by Hazareesing and Mézard 9 . In both cases, one obtains, in two dimensions, a value of 1/3 for the roughening exponent. This is indeed the value that we obtain for the roughening exponent in our problem. It should be nevertheless noticed that these three problems, though mathematically similar have a different physical origin and motivation.
The structure of this paper is as follows: in section II, we derive a low-energy Hamiltonian describing the Néel domain-wall elastic degrees of freedom. This Hamiltonian can be interpreted as the action in imaginary time of a quantum particle in 1+1 dimensions.
In section III, we use the replica trick to average over the random field realizations, which generates a many particle non-quadratic action. In section IV, we introduce a generalization of the variational ansatz of Mézard and Parisi which allows us to consider the effect of the short-range correlated disorder and of the non-local dipolar term in the many particle action. One obtains a set of non-linear equations which have to be solved self-consistently.
In section V, we perform the analysis of these equations in the particular case of the hierarchical replica symmetry breaking solution of Parisi and we discuss the validity of the quadratic approximation for the dipolar term in these equations, which permits to determine the roughening exponent of the domain wall. Finally, in section VI, we present our conclusions. The thermal and random field fluctuations give rise to deviations of the domain-wall from its straight configuration, i.e. at a given position y, the center of the domain is displaced from x = 0 to x(y) (we neglect overhangs). In this case, equation (1) reduces to

II. ENERGY OF A SINGLE DISTORTED DOMAIN WALL IN
The distance | y − y ′ | is considered to be very large with respect to the lattice spacing and to the length over which the random field is correlated (see below). In the absence of dipolar interactions, one can, as stated above, through a mapping to a Schrödinger equation 6 , show that ζ = 1.
In order to determine the elastic energy of the domain-wall, we consider the underlying system of ferromagnetic spins to be in a square lattice with lattice constant a and overall dimensions L x ×L y . The system has anti-periodic boundary conditions along the x direction (in order that there is a single domain-wall in the system) and periodic boundary conditions along the y direction and is at finite temperature T . Such system is described by a (classical) XY model Hamiltonian, with dipolar interactions and in a random field, For the case of two-dimensional dipolar interactions (i.e. for the SC/FM/SC heterostructure), the constants µ and G are given in the MKS system by µ = µ 0 µ B g L and G = µ 0 where µ 0 is the magnetic permitivity of the vacuum, µ B the Bohr magneton and g L the Landé factor of the spin system. The constants λ L and µ r are, respectively, the London penetration depth and relative magnetic permitivity of the superconductor. In this case, n = 2. In the case of three dimensional dipolar interactions, which we will not consider, G = µ 0 4π (µ B g L ) 2 and n is equal to 3. The quantities J ij and D have the dimensions of energy 21 . The field h i is a random field with Gaussian correlations, i.e. h α i h β j = ∆ ij δ αβ .
In order to stabilize the domain-wall, we have added to the Hamiltonian above an easy-axis anisotropy term, along the y axis, represented by the last term in (3).
For a straight Néel domain-wall, the spin configuration has the form where i is the site label and where φ(x) goes from −π/2 when x → −L x /2 to π/2 when x → L x /2, with S being the value of the spin. The function φ(x) changes its value from −π/2 to π/2 in a region of width w around x = 0. Since w is typically of the order of tens of nanometers 22 and we are only interested in the long-distance properties of the domain-wall, we can write dφ(x) dx = π δ(x). If the center of the domain-wall is displaced from x i = 0 to x i = x(y i ), the spin configuration becomes We wish to compute the partition function Z = T r(e −δH/k B T ), where δH = δH exc + δH rf + δH dip is the energy difference between a configuration described by the displacement x(y) and the straight domain configuration. We define the discrete Fourier transforms where q belongs to the first Brillouin zone. We will approximate the sums over i by integrals, according to the formulas, valid for a square lattice. It can be shown that the (positive) contribution to the exchange energy coming from the domain-wall deformation is given to second order in the displacement field, by 13 where N = L x L y /a 2 is the total number of sites and where we have expanded J(q) ≈ J 0 − Ja 2 q 2 . In real space, equation (11) reads which has the same form as the kinetic part of the action of a particle of mass σ in one dimension, with σ = π 2 JS 2 /a being the line-tension of the domain wall. The contribution coming from the random field can be shown to be equal to 23 whereμ = µ 0 µ B g L πS/a andh y = h y /a 24 . This formula can be made plausible if we consider instead a constant field along y. It can also be easily seen from (11) that the easy-axis anisotropy term does not contribute to the deformation energy of the domain-wall.
Since we wish to compute the partition function, we have to consider the factor where and F (q) = 1 q . Thus, by considering the different dipolar kernels F (q), one can treat the two different cases.
Using a Hubbard-Stratonovich transformation, we write e −H dip /k B T as where N is a normalization factor. We can now show that from which it follows that dy A(x(y), y) dx dy (17) and one can see that there is no contribution to the dipolar energy when x(y) = const. The partition function of the domain wall's degrees of freedom involves the functional integration of the Boltzmann factor e −δH/k B T , over the space of functions x(y).
We shall now make a change in our notation. We will write The subscript m stands for the fact that the wave-vectors ω m are discrete, due to the periodic boundary conditions along τ (y). If we use this notation in the formulae above, we have, collecting all the factors, the following expression for the partition function where the action S is given by This is the action of a single quantum particle in 1+1 dimensions, with 'mass' σ and 'charge' e, at 'inverse temperature' L y and with a 'Planck constant' equal to k B T . Such particle is in a

τ ) and interacts with a 'vector potential'
A(q x , ω m ) 25 characterized by a propagator equal to F (q x , ω m ). This result belongs to the well known class of mappings of 2 dimensional problems of classical statistical mechanics to 1 + 1 dimensional quantum problems. Now, since one has periodic boundary conditions along the τ (y) axis, the following identity holds which corresponds to a gauge transformation of the action 26 . The use of this identity eliminates the dependence on the 'velocity'ẋ(τ ) in all terms of equation (20) except the first one. The action S now reads We can now integrate out the field A(q x , ω m ), obtaining an action entirely in terms of x(τ ) alone. We obtain This action can be used to compute the roughening exponent of the domain wall. Equation (23) also shows that, apart from the interaction with the τ dependent random potential V (x, τ ), the action S is analogous to the action of a one-dimensional polaron, as written by Feynman 17,18 .
However, in order to compute the roughening exponent, one has to determine the following quantity which involves determining the average of the distribution of quenched disorder over a quotient. This type of averaging places the same problem as when one needs to average over the free energy of the system, i.e. one needs to determine the average log Z, instead of simply averaging over the partition function Z. This can be seen by including a source field coupled to x(τ ) in Z. The two-point correlation function appearing in (24) can be computed from log Z by differentiating it with respect to the sources and taking these sources to be zero.
III. THE AVERAGE OVER DISORDER. REPLICA TRICK.
We need to perform the average over the random fieldh y (q x , ω m ), which obeys a Gaussian distribution with a second moment equal to In order to perform the averaging over disorder in the free energy, one introduces n distinct copies (replicas) of the system 27 and makes use of the identity One performs the disorder average over Z n for integer n and makes the analytical continuation to n → 0. One obtains where the 'replicated' action S R is given by and where the last term represents the inter-replica interaction, due to the presence of the random potential. We have included in S R the effect of an applied fieldh y ext (x) = (σΩ 2 0 /μ) x.
This term gives rise to the second term in the action, which is a harmonic potential. The presence of this term is necessary to guarantee that S R is bounded from below, given that the last term of (28) comes with a negative sign. Ω 0 can be set to 0 at the end of the calculations.

EQUATIONS.
The action (28) cannot be integrated exactly. Therefore, we need to develop an approximation scheme. We shall follow Mézard and Parisi 7 and Goldschmidt 8 . We choose a trial quadratic action of the form where we have included a source field η a (τ ), which is useful in the calculation of correlation functions. The Mézard-Parisi ansatz has been generalized by including a matrix function Q ab (τ −τ ′ ), with a non-trivial dependence on τ −τ ′ , necessary to take into account the effect of the non-local dipolar interaction. In the Mézard and Parisi case, since they considered only short-range correlated disorder. Goldschmidt has considered several different forms for the kernel Q ab (τ − τ ′ ), all of which non-local in τ − τ ′ , and which are able to account for the problem of a quantum particle in a disordered static potential and for the related problem of a polymer in a random potential with long-range correlated disorder (see below). Since Q ab (τ ) = Q ab (τ + L y ), we can represent Q ab (τ ) in terms of Matsubara modes Q ab (τ ) = 1/L y m Q m ab e iωmτ . The values of Ω 2 a and Q m ab are to be chosen appropriately. We also demand S 0 to be translationally invariant when Ω a → 0. This gives rise to the constraint It can be shown that the normalized generating function for the correlation functions of such a quadratic action, is given by 28 is given in terms of its Matsubara modes, by whereÎ is the unit matrix in replica space andΩ 2 is the diagonal matrix with elements equal to Ω 2 a . The free energy as computed from the trial action S 0 [0], with the sources η a (τ ) equal to zero, can be shown to be equal to where µ indicates the basis in which the matrix σΩ 2 −Q m is diagonal. One now uses the Feynman inequality for F = −k B T log Z n , which states that, where the average in the rhs of (34) is over the trial quadratic action S 0 [0]. Note that F = −k B T log Z = lim n→0 F/n. One obtains for F var the expression where the following identities, valid for a quadratic action at zero sources, x (Gaa(0)+G bb (0)−2G ab (τ )) were used. Following Hazareesing and Mézard 9 , we choose the random field to have anisotropic correlations, with Q 0 bb given by (30) and where γ aa (τ ) = G aa (0) − G aa (τ ) 31 . We have approximated the sum over q x by an integral, a procedure which becomes exact in the thermodynamic limit, and Erfc(x) is the complementary error function. These equations constitute a closed nonlinear system which has to be solved self-consistently in the limit n → 0. We consider the solution of these equations in this limit in the next section.

V. ANALYSIS OF THE VARIATIONAL EQUATIONS FOR A 2D DIPOLAR
INTERACTION. HIERARCHICAL RSB.
One needs to take the limit n → 0 in the self-consistent equations derived above. This implies dealing with the matrix structure of the equations in this limit. One cannot do this in general. However, there is a particular parametrization of these matrices, due to Parisi 19 , which is sufficiently general for our purposes. This is the so called hierarchical RSB. The diagonal elements of the matrix Q ab (with dimensions n × n where n is arbitrarily large) are taken to be all equal toq and the off-diagonal elements are taken to be equal to q 0 . Q ab is then partioned in block-diagonal submatrices, the elements of Q ab outside these blocks keeping the value q 0 and the elements inside the blocks taking a new value q 1 (except the diagonal elements which keep their valueq). The procedure is repeated in an equal form for every diagonal block submatrix, the off-diagonal elements of these submatrices keeping the value q 1 and the elements inside the diagonal sub-blocks taking the value q 2 . This procedure is repeated 'ad infinitum'. The multiplication rules obeyed by these hierarchical matrices are analytically continued to n → 0. The off-diagonal elements of Q ab are then parametrized by a function q(u), u ∈ [0, 1] and the diagonal elements are parametrized by the numberq.
The constraint (30) takes the formq and one can show that the matrices G m ab are also parametrized by a function g m (u) and bỹ g m which parametrizes the diagonal elements. In the thermodynamic limit L x = L y → ∞, ω m → ω,g m →g(ω) and g m (u) → g(ω, u) and one gets, using equations (36-39), the following relations, whereg(τ ) (g(τ, u)) is the Fourier transform ofg(ω) (g(ω, u)), and where γ(τ ) =g(τ = 0) −g(τ ). One sees from (41) that when n → 0,Ω 2 = Ω 2 0 . The functionsg(ω) and g(ω, u) are in turn given bỹ . (48) The integrand in the first term is regular at small τ (due to the behaviour of γ(τ ) at small τ ) and its contribution is unimportant if ω ξ ≪ 1, which is the set of wave-vectors which determines the roughening exponent ζ. In the second term, we can use, if B ≪ ξ, the asymptotic expansion for Erfc(x). We obtain, omiting the regular contribution of the first term, where the dots stand for terms of order γ n (τ )/τ 2n+2 , with n ≥ 2. Now, since γ(τ ) ∼ τ 2ζ with ζ < 1, then the second term in (49) is much smaller than the first one and the omitted terms are of higher order. We can therefore keep only the first term and we obtain f (ω) ≈ − e 2 2 | ω |, . Now, the second equation can, with the aid of (50), be written in the following form with ν ≡ e 4 16 σ and η ≡ e 2 (k B T ) 3 6μ 4 ∆ 2 σ . This equation is also valid when r(u) < 0 without the need to perform an analytic continuation, which is an advantage with regard to (50). Now, the physical dimensions of ν and η are, according to these definitions, given by [ν] = Jm −3 and [η] = J −1 m 3 (in MKS units). Furthermore, since S 0 has the dimensions of an energy and both x(τ ) and τ have the dimensions of a length, it is also easy to see that the dimensions of q(u) are the same as those of ν. If we define z(u) = u 0 dv q(v), one has q(u) = dz du and [q](u) = u dz du −z(u). Substituting these equations in (53), one obtains a differential algebraic equation for z(u). This equation reads Since z(u) and ν have the same dimensions, it is useful to write z(u) in terms of a dimension- being a dimensionless constant 35 . In terms of l(x), the condition (51) becomes l(0) = 1/2 5/2 , since the x 2 power corresponds to a u 3/2 behaviour at small u. Substituting z(u) in terms of l(x) in (54) we obtain the following differential algebraic equation for l(x) which does not contain any dimensional parameters. The advantage of this equation with regard to (53) or (54) is its scaling form, which means that once we have obtained the scaling function l(x), one simply has to substitute x by its expression in terms of u to obtain z(u) for all ν and α. Furthermore, since we have to solve (54) for u ∈ [0, 1], this means that we need to solve (55) for x ∈ [0, 4 3 α 1/4 ]. When α is large 36 , this interval becomes a small interval around 0 and we can linearize (55), by writing l(x) = 1 2 5/2 + m(x). Substituting this equation above and keeping only terms linear in m(x) and m ′ (x), we obtain the differential with solution where B is an arbitrary constant. Since the solution q ′ (u) = 0 can be valid in a subinterval of [0, 1], the general solution of (42) is where z(u) = ν x 2 l(x) with l(x) being given by the full solution of (55) in the general case or simply by the linear approximation l(x) = 1 From these calculations, we have obtained some important results. Firstly, we were able to represent the dipolar interaction in the domain-wall Hamiltonian as the interaction of the quantum particle with a annealed gauge field. Also, in 1+1 dimensions, one can integrate out such a field, leaving us with polaron like quantum Hamiltonian. Secondly, we have shown that a generalized replica symmetry breaking ansatz allows for the treatment of this problem. Note that this generalization leads to relatively simple equations in our case, since the non-diagonal part of the self-energy in replica space is still wave-vector independent which is due to the short-range correlated nature of the random field. The treatment of long-range correlated disorder is, on the other hand, a much more complicated problem 38 . Finally, we were able to justify that taking the quadratic approximation for two-dimensional dipolar interactions is sufficient to obtain the correct value of the roughening exponent within this approximation. 29 This form of the correlator allows for a smooth introduction of a cut-off in the momentum integrals over q x and leads naturally to the existence of a Larkin length. Note that a cut-off is always provided by the existence of a finite lattice spacing a, but its introduction via the noise correlations makes the equations analitically more tractable. In the limit in which one wants to treat isotropic correlations, one should take B = a. 30 Note that in our case, we have chosen the disorder correlator ∆(q, ω) = ∆(q), which implies that the disorder is locally correlated along the τ (y) axis. Goldschmidt has also considered the case ∆(q, ω) = ∆(q) δ ω,0 , which implies that the disorder correlator is independent of τ (y). For the corresponding problem of a quantum particle in a disordered potential, this just means that such potential is independent of time. 31 In the case of the problem of a quantum particle in a static random potential, Goldschmidt has considered an ansatz in which the only component of Q ab (τ − τ ′ ) which has a non-  36 α large means either that the dipolar interaction is strong or that the temperature is high or that either the line tension σ or the noise strength ∆ are small.
37 W. C. Rheinboldt, Computers Math. Applic. 32, 15 (1996); ibidem 33, 31 (1997). 38 As stated above, one very important example of long-range correlated disorder is the one considered by Goldschmidt which has a direct application to the problem of a quantum particle in a static random potential and to the related problem of a polymer in a medium with long-range correlated disorder. Since Goldschmidt has considered a rather general variational ansatz where the diagonal part of the self-energy is dependent on ω, as we have done above, such variational ansatz could in principle be used to treat the problem of a quantum polaron in a random static potential. We thank M. Mézard for calling our attention to this point.
where e 2 2d is the value of the coupling constant for two-dimensional dipolar interactions (the case treated above) and e 2 3d is the value for three dimensional dipolar interactions.