Static and dynamic properties of Single-Chain Magnets with sharp and broad domain walls
Orlando V. Billoni, Vivien Pianet, Danilo Pescia, Alessandro Vindigni
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J u l Single-Chain Magnets from sharp to broad domain walls
Orlando V. Billoni , ∗ Vivien Pianet , , Danilo Pescia , and Alessandro Vindigni † Facultad de Matemática, Astronomía y Física (IFEG-CONICET),Universidad Nacional de Córdoba, Ciudad Universitaria, 5000 Córdoba, Argentina CNRS, UPR 8641, Centre de Recherche Paul Pascal (CRPP), Equipe "Matériaux Moléculaires Magnétiques",115 avenue du Dr. Albert Schweitzer, Pessac, F-33600, France Université de Bordeaux, UPR 8641, Pessac, F-33600, France and Laboratorium für Festkörperphysik, Eidgenössische Technische Hochschule Zürich, CH-8093 Zürich, Switzerland (Dated: July 29, 2018)We discuss time-quantified Monte-Carlo simulations on classical spin chains with uniaxialanisotropy in relation to static calculations. Depending on the thickness of domain walls, controlledby the relative strength of the exchange and magnetic anisotropy energy, we found two distinctregimes in which both the static and dynamic behavior are different. For broad domain walls, theinterplay between localized excitations and spin waves turns out to be crucial at finite temperature.As a consequence, a different protocol should be followed in the experimental characterization ofslow-relaxing spin chains with broad domain walls with respect to the usual Ising limit.
I. INTRODUCTION
The interest in the physics of domain walls (DWs)in 1d magnetic systems has been renewed by the capa-bility of controlling their motion by means of an elec-tric current . The technological relevance of this topicmainly derives from the possibility of employing DWs innovel magneto-storage and spintronic devices . From amore fundamental point of view, the synthesis of the firstslow-relaxing spin chains gave the chance to reconsiderthermally-induced DW diffusion in 1d classical spin mod-els. The systems displaying such a behavior have beennamed Single-Chain Magnets (SCMs), by analogy withSingle-Molecule Magnets (SMMs) : both classes of mate-rials show a magnetic hysteresis at finite temperature dueto slow dynamics rather than cooperative 3d ordering.Thanks to the remarkable property of being bistable at amolecular level, SMMs and SCMs have been proposed aspossible magnetic storage units. However, the advantageof having identical units whose arrangement might – inprinciple – be tailored with chemical methods is coun-terbalanced by the relatively poor thermal stability. Infact, the relaxation time becomes macroscopic only attemperatures of the order of one Kelvin or lower. Thequest to improve thermal stability of molecular magnetshas called for a better understanding of their physicalproperties. The investigation and development of SCMshas become an independent and very active field of re-search during the last decade . At odds with SMMs,which can be considered 0d magnetic systems, SCMs de-velop short-range order over a distance comparable tothe correlation length, ξ . The latter is expected to di-verge when the temperature approaches zero. However,defects and lattice dislocations – whose role is particu-larly dramatic in 1d – typically hinder the divergence of ξ below a certain temperature. The physics of SCMsis thus different depending on whether the correlationlength exceeds or not the average size of connected spincenters. Finite-size effects will be neglected in our theo-retical investigation. This means that our results should apply to the temperature region for which the correla-tion length is smaller than the average distance betweentwo defects in a real spin chain. Under these hypothe-ses, some DWs will be always present in the system atfinite temperature. A simple random-walk argument re-lates the relaxation time, τ , to the correlation length :within a time τ a domain wall performs a random walkover a distance ξ . The characterization of SCMs basicallyconsists in measuring both these quantities ( ξ and τ ) asfunction of temperature. In ideal cases, the observed be-havior is then reproduced by fitting the parameters of anappropriate spin model to the experimental data.Following the experimental procedure, we studiedthe temperature dependence of the correlation lengthand the relaxation time in a representative model forclassical spin chains with uniaxial anisotropy. ξ hasbeen computed with the transfer-matrix technique .The relaxation of the magnetization and DW diffu-sion have been studied by using time-quantified MonteCarlo (TQMC) . The latter is a recently devel-oped algorithm which, in classical spin systems, simu-lates a dynamics equivalent to the stochastic Landau-Lifshitz-Gilbert equation. TQMC has been previouslyemployed to model classical spin chains in the high damp-ing limit , but the most recent developments allowusing it for low-damping calculations .Most of SCMs show a large single-site magneticanisotropy. The reference model is thus represented bythe 1d Ising model or – more specifically – its kinetic ver-sion proposed by Glauber . In order to better capturethe physics of experimental systems, the Glauber modelhave been extended to take into account finite-size ef-fects , ferrimagnetism , the effect of a strong exter-nal magnetic field and the reciprocal non-collinearity oflocal anisotropy axes . Here we report about a lin-ear ferromagnetic spin chain and we address the questionof how physics changes when the single-site anisotropy isprogressively reduced. We found that the Glauber sce-nario needs to be revisited for SCMs which do not possessa large single-site anisotropy so that DWs acquire a finitethickness (more than one lattice unit). Our theoreticalpredictions are in agreement with the few available exper-imental results on molecular spin chains with low single-site anisotropy . Note that in metallic nonowires (Co,Ni, Fe, Permalloy) DWs always extend over several lat-tice units. Therefore, for materials traditionally usedin magneto-storage manufactory the Glauber’s pictureof magnetization reversal and relaxation is not expectedto hold exactly, nor the correlation length is expectedto show an Ising-like behavior over a wide temperaturerange.In section II, we present the model and define thephysical questions that we want to address. In sectionIII, we study the temperature dependence of the corre-lation length by means of the transfer-matrix techniqueand Polyakov renormalization. In section IV, we intro-duce the time-quantified Monte Carlo method and studythe temperature dependence of the relaxation time in thebroad-wall regime. In section V, we use TQMC to studytemperature-induced diffusion of both sharp and broadDWs. In section VI, we provide some phenomenologicalarguments which support our findings and discuss howthey compare with experiments. In the conclusions, wesummarize our main results and highlight their relevancefor further work which could possibly include an electriccurrent driving DW motion. Broad DW Sharp DW
FIG. 1. Domain-wall energy (in J units) as a function ofthe ratio D/J , adapted from Ref. 33: discrete-lattice cal-culation ε dw (solid line), continuum-limit solution √ DJ (dashed line). Crosses and circles correspond to the ratiosused in transfer-matrix and time-quantified Monte-Carlo cal-culations respectively. The vertical line indicates the value( D/J = 2 / ) at which the transition from broad- to sharp-wall regime occurs. Insets: sketch of the spatial dependenceof S zi as a function of i in the broad-wall (up left) and in thesharp-wall (down right) regime. II. THE SYSTEM
As a reference model for SCMs, we consider the fol-lowing classical Heisenberg Hamiltonian: H = − N X i =1 h J ~S i · ~S i +1 + D ( S zi ) + ~H · ~S i i (1)where D represents the anisotropy energy, J the exchangecoupling and ~H an external applied field. Each spin vari-able ~S i is a three-component unit vector associated withthe i –th node of the lattice. In this paper either periodicor open boundary conditions will be considered depend-ing on the calculation. D and J are assumed positive sothat Hamiltonian (1) describes a spin chain with uniaxialanisotropy pointing in the z direction. This model andvariations of it have been investigated extensively fromthe theoretical point of view . Moreover, Hamilto-nian (1) has been employed to reproduce the experimen-tal behavior of some SCMs . Many physical proper-ties are related to the energy increase due to the creationof a domain wall (DW) in one of the two ground-state,uniform configurations with S zi = ± ( ∀ i ). In Fig. 1 weplot this energy obtained from a discrete-lattice calcula-tion ε dw (solid line) for different values of the ratio D/J .For
D/J ≤ / , the minimal DW energy is realized by aspin profile in which several spins are not aligned alongthe easy axis, z (see the sketch in the inset up on theleft). In this case, DWs spread over more than one latticespacing: broad DWs. For
D/J > / , the minimum DWenergy is realized if the transition between S zi = +1 to S zi = − occurs within one lattice spacing; in this way allthe spins are aligned along the easy axis: sharp DWs (seethe sketch in the inset down on the right). The transi-tion between broad- and sharp-wall regime is highlightedby a singularity in the log-linear plot, which evidencesa different functional dependence of ε dw on the param-eters D and J for anisotropy-to-exchange ratios smalleror larger than 2/3. For D/J ≪ , the analytic expression √ DJ can be obtained by minimizing the DW energyin the continuum-limit approximation (see Eq. (11)).This function is plotted in Fig. 1 as a dashed line. Ap-proaching the transition ratio, D/J = 2 / , from belowthe continuum-limit prediction starts to deviate from thediscrete-lattice calculation, ε dw (solid line). This is rea-sonable since the continuum approximation is more ac-curate for smaller anisotropy-to-exchange ratios, namelythe broader DWs are. For D/J > / , the DW energytakes the constant value ε dw = 2 J . The transition regionwhere none of the two analytic expressions holds exactlyis narrow, meaning that the function ε w = ( √ DJ for D/J ≤ / J for D/J > / (2)describes the DW energy accurately for most values of D and J . In Fig. 1, symbols represent the DW energy ε dw for the values of D/J at which static (circles) anddynamic (crosses) calculations have been performed.The main question we want to address is the following:are SCMs ruled by different laws depending on whetherDWs are sharp or broad? Our results show that physicsis significantly different in these two limits. This factwill eventually affect the experimental characterizationof SCMs which can be modeled by Hamiltonian (1) (orvariations of it ).Two key quantities characterizing a specific SCM are thecorrelation length, ξ , and the relaxation time, τ . For ourstudy, the most relevant correlations are those relatingto spin projections along the easy axis. Therefore, thecorrelation length shall be defined from pair-spin corre-lations along z as: h S zi + r S zi i = h ( S zi ) i e − rξ , (3)(where h . . . i stands for thermal average). The static sus-ceptibility along z scales with the correlation length asfollows: χ z ( T ) ∼ ξT . (4)The relaxation time can be obtained from the dynamicsusceptibility, χ ( ω, T ) = χ ( T )1 − iωτ , (5)where ω is the frequency of the oscillating applied fieldand χ ( T ) is the static susceptibility. Both the real andthe imaginary part of χ ( ω, T ) display a maximum for ωτ = 1 . In the following we will use an alternative defini-tion of τ in terms of the relaxation of the magnetization.Eqs. (4) and (5) relate ξ and τ to measurable quanti-ties, the static and dynamic susceptibility. Based on arandom-walk argument , the relation between the cor-relation length and the relaxation time ξ ≃ D s τ (6)is usually assumed for such temperatures that ξ < N (bulk regime). D s is the DW diffusion coefficient but itcan also be interpreted as the attempt frequency for asingle-spin flip. The temperature dependence of D s willbe discussed in more details in Sect. V. Within the ki-netic Ising model, it is possible to deduce Eq. (6) analyt-ically . The basic experimental characterization of aSCM essentially reduces to determining the temperaturedependence of the three quantities involved in Eq. (6): τ , ξ and D s . The scenario is well-established in the sharp-wall regime, in which one expects that all these quantitiesobey a thermally activated mechanism: ξ ∼ e ∆ ξT τ ∼ e ∆ τT D s ∼ e − ∆ AT . (7) k B = 1 will be assumed throughout the manuscript.Eq. (6) relates ξ and τ – two experimentally accessiblequantities – with each other so that the following relationbetween energy scales holds: ∆ τ = 2∆ ξ + ∆ A . (8) The above relation has been confirmed by several exper-imental works on sharp-wall SCMs using the reason-able assumption ∆ A ≃ D proposed in Ref. 25 and 39.In Sect. V we will see that our numerical results confirmthe validity of Eq. (8) for sharp DWs. On the contrary,the scenario turns out to be significantly different in thebroad-wall regime. In fact, for D/J < / , we found that: • ∆ ξ is not constant but is reduced by more than 30%of the value that it takes at low T with increasingtemperature. • D s does not follow a thermally activated mecha-nism as suggested by Eq. (7).These two important findings indeed affect the experi-mental characterization of a SCM for which D/J < / . III. STATIC PROPERTIESA. Transfer matrix calculations
The static properties of a classical spin chain canbe computed efficiently with the transfer-matrix (TM)method . Here we use this technique to compute thethermodynamic properties of an infinite chain ( N → ∞ )but finite systems could also be considered . The essen-tial ideas of the TM approach are recalled in Appendix A.In particular, we used this method to compute the cor-relation length. The dependence on r can be eliminatedfrom Eq. (3) by summing over all the lattice separations X r ≥ h S zi + r S zi i = h ( S zi ) i X r ≥ e − rξ = h ( S zi ) i − e − /ξ . (9)In practice, we evaluated the summation numerically andinverted the previous formula as follows ξ = − (cid:20) ln (cid:18) − P r ≥ h S zi + r S zi ih ( S zi ) i (cid:19)(cid:21) − . (10)Fig. 2 shows the logarithm of the correlation length vs. the energy of the DW divided by the temperature as ob-tained by TM calculations. Different curves correspondto different values of D/J . Since we set J = 1 , the caseswith D = 1 − fall in the sharp-wall regime, while thecases with D = 0 . − . correspond to the broad-wallregime (see Fig. 1). We will discuss the behavior of ξ inthese two regimes separately. Sharp-wall regime – In this regime, the DW energyamounts to ε w = 2 J . This value has been used to plotthe logarithm of the correlation length as a function of ε w /T for D = 1 − in Fig. 2. At low temperatures, allthe solid curves have the same slope, equal to one, re-vealing that ln( ξ ) ∼ ε w /T . A reference line with slopeone is plotted with short dashes. For D = 1 , some devia-tions from this straight line occur at high temperatures.Later on, we will show that in the broad-wall regime the FIG. 2. Color online. Logarithm of the correlation lengthobtained by TM calculations as function of the inverse tem-perature in units of the DW energy ε w (defined in Eq. (2)) for J = 1 and different values of D : Dotted lines, broad domainwalls, D = sharp domain walls, D = interplay between spin waves and DWs leads to an effec-tive decrease of ∆ ξ with increasing T . It is reasonableto think that a similar phenomenon may take place inthe sharp-wall regime as well, when the transition ratio D/J = 2 / is approached from above. The horizontaldotted lines indicate the values ξ = 10 , , . Dueto lattice dislocations or impurities (in molecular com-pounds) or intrinsic problems in the deposition procedure(in mono-atomic nanowires ), the average length of spinchains is typically of − magnetic centers in realSCMs. The actual length of a spin chain sets an upperbound to the low-temperature divergence of ξ . Such anupper bound can be reduced by introducing additionalnon-magnetic impurities . Most of the SCMs reportedin the literature fall in the sharp-wall regime . Forthem, in the temperature range where the correlation-length divergence is relevant ( ξ ≫ ), ∆ ξ can be assumedequal to J , independently of T . Broad-wall regime – For
D/J < / , the DW energy isdescribed well by the function ε w = 2 √ DJ (see Eq. (2)).As already pointed out when discussing Fig. 1, this ex-pression obtained in the continuum limit deviates fromthe discrete-lattice calculation in the vicinity of the tran-sition region from broad to sharp DWs. In order tomake the comparison with experiments easier, in themain frame of Fig. 2 the correlation length is plotted vs. ε w /T , with ε w = 2 √ DJ for all values of D fallingin the broad-wall regime (instead of using ε dw , i.e. , valueobtained from the discrete-lattice calculation). At lowtemperature the slope of all the dotted lines is about 0.9,which suggests an effective ∆ ξ = 1 . √ DJ . This tenper cent of reduction with respect to the DW energy canbe accounted for using the low-temperature expansion of the correlation given in Ref. 36: ξ ∼ ( T /ε w ) exp ( ε w /T ) .More strikingly, all the dotted curves bend when thetemperature increases till their slope becomes roughly0.6 at high temperature. A tentative fit of the temper-ature dependence of ξ using the corresponding formulain Eq. (7) would give a value of the activation energy . √ DJ < ∆ ξ < . √ DJ depending on the tempera-ture range in which the fitting has been performed. Thisis nothing but the standard procedure followed in the ex-perimental characterization of SCMs . Normally, theextracted energy is then related to the barrier observedin the relaxation time by means of Eq. (8). It is worthremarking that the temperature ranges where ∆ ξ and ∆ τ are extracted usually do not overlap. In fact, to measure ξ the characteristic time scale of the experiment has to bemuch longer than the relaxation time; while for measur-ing τ the characteristic time scale of the experiment hasto be comparable to the relaxation time itself, or shorter.Thus, the effective variation of ∆ ξ with T – which mayreach 30% of its value – has to be considered when onetries to relate this quantity to ∆ τ by means of Eq. (8). Inparticular, we remark again that ∆ ξ can only be accessedat relatively high temperature because finite-size effectsor three-dimensional inter-chain interactions prevent thecorrelation length from diverging indefinitely .As an example of how this first theoretical result maybe used to better characterize real SCMs, we refer to thesystem reported in Ref. 31. That spin chain is betterdescribed by the Seiden model with anisotropy ratherthan Hamiltonian (1). Apart from a multiplicative factortwo in front of the exchange-energy term, the Hamilto-nian of the Seiden model and the one we study here takethe same form in the continuum limit (see Eq. (11)).Yet, the experimental system shows ∆ expξ ≃ K and J = 108 . K (adapting the fitted value to our model).Thus1. assuming that the fit of ∆ expξ has been performedin the region where ∆ ξ = 0 . ε w , one would obtain ε w ≃ K and, accordingly, D = 11 . K;2. assuming that in the fitted region ∆ ξ = 0 . ε w , onewould get D = 5 . K.The first estimate of D falls in the suggested range D = 8 . − . K (estimated by EPR measurementson equivalent isolated magnetic units ) while the sec-ond estimate is clearly wrong. The experimental cor-relation length displays a clear exponential divergencein the range 25 K < T <
50 K, which corresponds to < ε w /T < (if ε w = 100 K is assumed). Indeed, thesevalues of the reduced variable ε w /T relate to the high-temperature region where ∆ ξ ≃ . ε w for D/J ≃ . (seeFig. 2).In the following we will give a justification of the im-portant variation of ∆ ξ with T observed in the broad-wallregime in terms of Polyakov renormalization . B. Polyakov renormalization
We focus now in the broad-wall regime in which ∆ ξ decreases with increasing temperature. A deeper insightin such a phenomenon can be achieved by consideringthe effect of spin-wave renormalization on the couplingconstants D and J . To this aim, we work with the con-tinuum version of the Hamiltonian in Eq. (1): H = − JN + Z (cid:20) J | ∂ x ~S | − D ( S z ( x )) (cid:21) dx , (11)where a unitary lattice spacing has been assumed. Fol-lowing Polyakov , we represent ~S ( x ) as a superposi-tion of a field fluctuating over short spatial scales, ~φ ( x ) ,and a field varying smoothly and over large spatial scales, ~n ( x ) . More explicitly, we write ~S ( x ) = ~n ( x ) p − φ ( x ) + ~φ ( x ) . (12)Requiring | ~S ( x ) | = 1 and | ~n ( x ) | = 1 , one necessarily has ~n ( x ) · ~φ ( x ) = 0 ; thus ~φ ( x ) can be expressed on a basisorthonormal to ~n ( x ) : ~φ ( x ) = X a φ a ~e a with a = 1 , (13)with | ~e a ( x ) | = 1 . The terms appearing in Hamilto-nian (11) are affected by the averaging procedure overthe ~φ ( x ) field as follows: ( h| ∂ x ~S | i = (cid:2) − h φ a i (cid:3) ( ∂ x ~n ) + P a h ( ∂ x φ a ) ih ( S z ) i = (cid:2) − h φ a i (cid:3) ( n z ) + h φ a i . (14)The latter equations represent a well-known result ,however we will recall their derivation in Appendix B forconvenience.In the isotropic Heisenberg chain ( D = 0) and in the easy-plane case ( D < excited spin waves suffice to destroythe long-range order present in the ground state at finitetemperatures (because of the existence of a Goldstonemode). When D > , the spectrum of spin-wave excita-tions acquires a gap so that the long-range order is ratherdestroyed by DW proliferation at finite temperatures. Tofix the ideas, one can think the field ~n ( x ) to be associ-ated with localized excitations (DWs) while ~φ ( x ) beingassociated with spin waves. Within a distance separat-ing two successive DWs one can assume ( n z ) ≃ and | ∂ x ~n | ≃ . Therefore, up to quadratic terms, the ~φ -fieldHamiltonian reads: H φ = N Λ X a Z Λ / − Λ / (cid:20) J ∂ x φ a ) + Dφ a (cid:21) dx , (15) Λ being the average distance between two successive DWs(see Appendix B for the derivation of Eq. (15)). Follow-ing the typical procedure of the renormalization group (RG), we express the Hamiltonian (15) in the Fourierspace H φ = N Λ X a π Z (cid:18) J q + D (cid:19) | ˜ φ a ( q ) | dq , (16)and apply equipartition h| ˜ φ a ( q ) | i = TJq + 2 D . (17)The integration of the fast -fluctuating excitations in therange πL ′ ≤ q ≤ πL , with L ′ = L + dL , yields h φ a i = T π Z π/Lπ/L ′ dqJq + 2 D = T D dLL + (cid:0) πwL (cid:1) , (18)where w = J/ D . The prefactors of ( ∂ x ~n ) and ( n z ) inEqs. (14) together with Eq. (18) define the RG equations: ( J ( L + dL ) = J ( L ) (cid:2) − h φ a i (cid:3) D ( L + dL ) = D ( L ) (cid:2) − h φ a i (cid:3) . (19)As only short-range order is present in a 1d system, it isreasonable to integrate Eqs. (19) only up to the averagedistance separating two successive DWs, Λ . The inverseof Λ can be identified with the average density of DWs.A low-temperature estimate of the DW density is givenin Ref. 36:
1Λ = 1 w ε w T exp (cid:16) − ε w T (cid:17) (20)where ε w = 2 √ JD again. Note that in Ref. 36 the DWdensity is given in terms of bare constants J and D . Ourgoal is to see how Λ has to be corrected at finite temper-ature to account for spin-wave renormalization. This hasbeen done with the following procedure:1. fix the temperature T
2. compute numerically the renormalized constants atan infinitesimal dL starting from the bare values of J and D
3. compute Λ( dL ) from Eq. (20) using the renormal-ized constants J ( dL ) and D ( dL )
4. iterate the procedure till L + dL > Λ( L )
5. at this point we defined ˜Λ( T ) = Λ( L ) and anal-ogously the renormalized constants: ˜ D ( T ) , ˜ J ( T ) , ˜ ε w ( T ) and ˜ w ( T ) .Apart from a multiplicative constant, the final value of ˜Λ( T ) and the correlation length, ξ , are supposed to de-pend on temperature likewise.Before proceeding with the analysis of the RG equations,we specify the convention on the notation we use. In thissection, we introduced a dependence on L in the quanti-ties D ( L ) , J ( L ) , ε w ( L ) and w ( L ) in order to renormalize D ( L ) / D ( ) L FIG. 3. Color online. The ratio between renormalized con-stant D ( L ) and the bare constant D (0) is plotted as a func-tion of some selected intermediate values of L , for differentvalues of D (0) = 0 . (black squares) ,0.1 (red circles), 0.2(blue crosses) and different temperatures ε w (0) /T = L = ˜Λ( T ) . them. This dependence is somewhat technical. The rel-evant values of those renormalized quantities are ˜ D ( T ) , ˜ J ( T ) , ˜ ε w ( T ) and ˜ w ( T ) , namely the values assumed whenthe renormalization stops. These last values only dependon temperature, not on L anymore. In the other sec-tions, when not specified differently, we always refer tothe bare constants; these equal the values of the corre-sponding renormalized quantities at T = 0 and those ofthe “technical” D ( L ) , J ( L ) , ε w ( L ) and w ( L ) for L = 0 . C. Symmetry properties ofthe renormalization flux
By noting that wD = ε w one can rewrite the averageof the ~φ -field components as: h φ a i = Tε w wL
11 + (cid:0) πwL (cid:1) dLL . (21)From the above expression, one can immediately see that h φ a i depends on L/w (0) and ε w /T only. This symmetryproperty is directly transferred to the RG equations. Thiscan be made more transparent by rewriting Eqs. (19) as dJJ = − Tε w wL ( πwL ) dLLdDD = − Tε w wL ( πwL ) dLL . (22)Note that the renormalized constants will, however, de-pend on the values they take at zero temperature ˜ J (0) and ˜ D (0) , i.e. the bare constants. Without loss of gen-erality, we can set J = 1 (as a unit for the energies and D ( L ) / D ( ) L/w
FIG. 4. Color online. Same data as in Fig. 3 but plotted vs.
L/w (0) . temperature) and focus on the renormalization of D withincreasing L . In order to compare the renormalization ofthe anisotropy energy with L for different values of thebare D (0) , we plot the ratio D ( L ) /D (0) (Fig. 3). Thedifferent temperatures have been chosen so that the ra-tios ε w (0) /T are the same for different initial anisotropyvalues. Fig. 4 shows how all the curves corresponding tothe same ε w (0) /T ratio but different D (0) collapse ontoa single one when plotted as a function of L/w (0) (in-stead of L ). Such a data collapsing evidences that dueto the symmetry of Eqs. (22) the renormalization flux, i.e. the relative change of renormalized constants withrespect to the bare ones, eventually depends only on the initial values ε w (0) /T and L/w (0) . In other words, whatmatters are • the temperature in units of the DW energy, ε w (0) • and length scales in units of the DW width, w (0) .Besides that, from Eq. (20) it is clear that Λ /w only de-pends on ε w /T . Therefore, applying the same argumentas for the renormalization flux displayed in Fig. 4, weexpect ˜Λ( T ) /w (0) to be a function of ε w (0) /T only. Thecorrelation length is expected to fulfill the same scalingproperty as ˜Λ : ξ/w (0) has to be a universal function of ε w (0) /T . In the following we will see that this propertyholds true provided that ε w (0) is replaced by the DWenergy obtained from the discrete-lattice calculation ε dw (we remind that Fig. 1 highlights some deviations of ε w from ε dw in the vicinity of D/J = 2 / ). D. Transfer Matrix versus
Polyakov Renormalization
As stated before, the temperature dependence of thecorrelation length is expected to follow the behavior of ˜Λ( T ) . In Fig. 5, we compare ξ ( T ) computed by means of ε dw /T l n [ ξ / w ( ) ] D =0 D =0.05 ε w /T ln FIG. 5. Color online. Logarithm of the correlation length vs. ε w (0) /T : TM calculation for D = 0 . (solid red line); ˜Λ( T ) multiplied by a constant to match the TM results at low T (circles); analytic result for D = 0 given in Ref. 12 (dashedline). The straight lines are guide to the eye to highlight thechange of slope due to spin-wave renormalization. The dot-ted line corresponds to the renormalized DW width ln[ ˜ w ( T )] .Inset: Logarithm of ξ/w (0) vs. ε dw (0) /T for D = the TM technique with ˜Λ( T ) as a function of ε w (0) /T .We chose the specific value D = 0 . because for largervalues of D the correction to the DW energy due to thediscreteness of the lattice is not negligible and the identi-fication ε dw = 2 √ DJ is not totally justified (see Fig. 1).The values of ˜Λ( T ) (crosses) have been shifted by a con-stant factor to match the TM results (line-symbols) atlow temperature. Indeed, the temperature behavior of ˜Λ( T ) closely follows that of the correlation length till rel-atively high temperatures. The change in the slope oc-curring at intermediate temperatures is well reproducedby the RG calculation. This means that spin-wave renor-malization is the main physical reason for the reductionof ∆ ξ with increasing temperature observed in the broad-wall regime (dotted lines in Fig 2). In the RG language,such a change in the slope can also be interpreted as acrossover towards the temperature at which ˜ D ( T ) van-ishes. This phenomenon is the analogous of the reori-entation transition in magnetic films . The progressivevanishing of the uniaxial anisotropy with increasing tem-perature reflects in the fact that for the highest temper-atures reported in Fig. 5 the TM calculation recovers theanalytic solution for the isotropic Heisenberg chain, with D (0) = 0 (solid line) . In such a region, the behaviorof ˜Λ( T ) deviates from the one of the correlation length.However, at such short distances the RG treatment losesits meaning since the length scale at which the renor-malization stops, ˜Λ( T ) , becomes of the order of the DWwidth ˜ w ( T ) (dotted blue line) or even smaller.According to the RG analysis discussed in subsec-tion III C one expects that the ratio ξ/w (0) be given by a universal scaling function of ε w (0) /T . However, thewhole RG approach is based on the continuum approxi-mation, Eq. (11), which is expected to hold for D ≪ J , i.e. when DWs are significantly broad. But in fact, the dis-crepancies between ε w (0) = 2 √ JD and the correspond-ing values obtained from the discrete-lattice calculation, ε dw , are significant for most of the D and J used to pro-duce the curves in Fig. 2. For D/J < / but not D ≪ J ,we can tentatively extend the validity of the scaling prop-erty of ξ beyond the continuum limit by replacing ε w withthe DW energy obtained from the discrete-lattice calcu-lation. Therefore we propose that ξ ( T ) w (0) = f (cid:16) ε dw T (cid:17) (23)with f universal scaling function. In the inset of Fig. 5we check the validity of Eq. (23) by plotting ξ ( T ) /w (0) as a function of the ratio ε dw /T for J = 1 and differentvalues of D = 0 . − . . The correlation length data arethe same as the ones plotted in Fig. 2. The data collaps-ing predicted by Eq. (23) is indeed observed in the insetof Fig. 5. Remarkably, the log-linear plot of the scalingfunction f ( ε dw /T ) vs. ε dw /T does not show a constantslope as a consequence of the intrinsic temperature de-pendence of ∆ ξ characterizing the broad-wall regime. IV. DYNAMIC PROPERTIES: TQMC
In order to study the dynamics of the model definedby Eq. (1), we used a time-quantified Monte-Carlo al-gorithm (TQMC) fixing N = 100 and implement-ing both periodic and open boundary conditions. In theTQMC scheme, Monte-Carlo steps (MCS) are mappedinto real-time units through the following relation , ∆ t [ τ K ] = D T R ∆ t [ MCS ] . (24)This relation gives the variation of real time in units ofthe damping time τ K where R is the size of the coneused for updating single-spin configurations . Notethat the factor that relates MCS with the real time is di-vided by the temperature T , therefore the time spannedin a given simulation increases when the temperatureis decreased, provided the factor DR / (40 T ) is smallenough. We set R = 0 . in all simulations. Thus,for typical values D/J = 0 . and T /J = 0 . one has DR / (40 T ) ∼ . × − . On the other hand, the damp-ing time is given by τ K = 1 + a a γµ H K , (25)where µ H K is the anisotropy field, a the adimensionaldamping constant and γ the gyromagnetic ratio . Sincethe anisotropy field associated with Hamiltonian (1) canbe expressed as µ H K = 2 D/gµ B and γ = gµ B / ¯ h ,then γµ H K = 2 D/ ¯ h . Henceforth, the quantity ¯ h/ D TABLE I. Parameters and physical quantities.
D/J ε w /J w T /J range ξ simulation0.1 0.894 2.23 0.116 - 0.26 86 - 5 Relaxation0.2 1.264 1.58 0.153 - 0.3 89 - 5 Relaxation0.3 1.549 1.29 0.181 - 0.33 83 - 5 Relaxation0.1 0.894 2.23 0.01 - 0.04 86 - 5 Diffusion0.2 1.264 1.58 0.014 - 0.056 89 - 5 Diffusion1 · - 165 Diffusion2 w and the correlation length ξ are given inlattice units. The latter has been rounded to integer values. will be assumed as time unit. In real SCMs D is ofthe order K. We used an intermediate value for thedamping constant, a = 0 . , therefore τ K is in therange of picoseconds ( ¯ h = 7 . K ps), with these param-eters. Moreover, the maximum number of MCS thatwe spanned in our simulations is of the order of [MCS], which would correspond to a total time ∼ . µ sfor D/J = T /J . For TQMC simulations we set the fol-lowing values for the ratio between the anisotropy andthe exchange strength
D/J = 0 . , . , . , and 2. Theseratios are comparable to values of real SCMs and cor-respond to different regimes of the DW profile (seeFig. 1). We studied magnetic relaxation for the selectedvalues D/J = 0 . , . and . , all falling in the broad-wall regime. The temperatures were ranged between thevalues indicated in Table I. In relaxation simulations,such values of T were chosen in order that the corre-lation length was smaller than the system size, even iffor the lowest temperatures ξ and N = 100 are of thesame order of magnitude. The applied field was cho-sen so that H = 0 . D in every simulated relaxation.In the next section we will present results concerning adetailed study of the trajectory displayed by DWs be-cause of thermal fluctuations. This investigation wasalso performed by means of TQMC simulations for dif-ferent temperatures and anisotropy values. For the dif-fusion studies in the broad-wall regime we set D/J = 0 . and D/J = 0 . ; temperatures were chosen in the ranges T /J = 0 . − . and T /J = 0 . − . , respectively(see Table I). For the studies in the sharp-wall limit weused D/J = 1 and while temperatures were chosen inthe ranges T /J = 0 . − . and T /J = 0 . − . respec-tively. Table I summarizes the parameters and physicalquantities related to the different types of simulations. A. Relaxation curves
For studying how the magnetization relaxes towardsequilibrium we used the following protocol. In each sim-ulation, the system was cooled down in zero applied field
Time [h/2D] M z T/J M z FIG. 6. Relaxation curves at different temperatures for
D/J = 0 . . Inset: FC curve obtained by TM calculations(solid line); open circles are the values of M obtained byfitting the relaxation curves with Eq. (26) at different tem-peratures. at a cooling rate r = 5 · − ; this means that thetemperature was decreased according to the expression T = T − r t , with T /J = 1 being the initial temperatureand t the time measured in MCS. Once the temperatureat which we wanted to simulate relaxation was reached,we let the system equilibrate in zero field. Further on,we switched the field on and let the magnetization evolvetowards its equilibrium value. The equilibration time inzero field was chosen to be equal to the relaxation time(in H = 0 ). Fig. 6 shows the relaxation of the magne-tization under a homogeneous applied field of strength H/J = 0 . along the z direction for different tempera-tures. The anisotropy value is D/J = 0 . .All these curves can be fitted well with a stretched expo-nential, like in Ref. 25: M z ( t ) = M h − e − ( tτ ) α i . (26)For all the relaxation processes we found that . ≤ α ≤ . In the inset of Fig. 6 we report field-cooling curve (FC)as a function of T /J obtained by TM calculations for aninfinite system (full continuous line). The applied field isthe same as in relaxation simulations,
H/J = 0 . . Thepoints represent the values of M obtained from fittingthe computed relaxation curves with Eq. (26). The verygood agreement confirms that the magnetization indeedrelaxed to its equilibrium value in all the simulated re-laxation processes. Fig. 7 shows the relaxation times, τ ,obtained by fitting different relaxation curves, analogousto the ones shown in Fig. 6, for three different values ofthe anisotropy D/J = τ on the tempera-ture is indeed evidenced for any value of D and for bothopen and periodic boundary conditions (o.b.c. and p.b.c.respectively). The values of τ obtained with o.b.c lie on ε w /T10 τ [ h / D ] D=0.3 (o.b.c.)D=0.2 (o.b.c.)D=0.1 (o.b.c.)D=0.1 (p.b.c.)
FIG. 7. Color online. Arrhenius plot of the relaxation timesobtained at different temperatures and for three values of theanisotropy,
D/J = 0 . (full circles), 0.2 (squares) and 0.3(triangles), with open boundary conditions. The dashed lineis guide to the eyes and has slope . . Empty circles: periodicboundary conditions for D/J = 0 . . the same line whose slope is roughly 1.1. The same calcu-lation was repeated with p.b.c. for D/J = 0 . only. Thecorresponding points (empty circles in Fig. 7) can be as-sumed to lie on the same line as for o.b.c. up to the value ε w /T ∼ . . The last point at lower temperature devi-ates significantly from the dashed line with slope 1.1. Atthis temperature, the correlation length is comparable tothe system size, which explains the discrepancy betweenthe calculation with p.b.c and o.b.c. (see Table I). Morespecifically, it is ξ = N = 100 for ε w /T = 7 . . On theother hand, for ε w /T < . the relaxation time computedwith open and periodic boundary conditions depends onthe temperature likewise. This fact suggests that for suchtemperatures the relaxation is not affected by the detailsof boundary conditions, namely it is a bulk process.We cannot provide any trivial explanation for the uni-versal slope observed for ε w /T < . : ∆ τ = 1 . ε w . Weremark that in the sharp-wall limit the formula given inEq. (8), ∆ τ = 2∆ ξ + ∆ A , has been confirmed by a num-ber of experiments on SCMs . Due to the fact thatfor sharp DWs ∆ ξ = ε w (with ε w = 2 J ), the same formulaadapted to the broad-wall limit would predict a slope ofabout D/ε w . The value obtained by fitting the datain Fig. 7 is therefore about 50% smaller than what wouldpredict Eq. (8). We will come back to this importantpoint when comparing our results with the few availableexperimental data on SCMs in the broad-wall limit. V. DOMAIN-WALL DIFFUSION
In this section we discuss the kind of trajectory dis-played by DWs at finite temperature. Particular atten-tion will be given to the temperature dependence of thediffusion coefficient both in the broad- and sharp-wall limit. For the latter case we will provide a numerical con-firmation of the phenomenological law D s ∼ e − D/T (seeEq. (7)) which – to the best of our knowledge – was stillmissing in SCM literature. The Arrhenius-like depen-dence of the diffusion coefficient highlights that each ele-mentary move of a DW occurs – in the average – througha thermally activated mechanism, in the sharp-wall limit.We will show that this is not the case in the broad-wallregime.
A. Analysis of domain-wall trajectories
As already mentioned, the assumption that DWs per-form a random walk induced by thermal fluctuations isthe basic ingredient to relate the correlation length to re-laxation time. Such assumption can be verified directlyin TQMC by analyzing the microscopic configurationsexplored during a simulation. Details about this anal-ysis can be found in Appendix C. The outcome is theaverage trajectory followed by each DW at a given tem-perature. We found that, the diffusion relation h x i ∝ t isnot obeyed at short times (see Eq. (C1)). However, DWtrajectories could be fitted well with a more general ex-pression which describes a random walk with correlatedsteps : h x i = σ tτ c + e − tτc − ! . (27) τ c is the characteristic crossover time from the ballisticregime at short times to the diffusive regime at longer times. When t ≪ τ c we are in the regime of correlatedsteps. Expanding Eq. (27) accordingly, for t/τ c ≪ , oneobtains the ballistic relation between the displacementand time: h x i = σ τ c t . (28)When t ≫ τ c , the diffusion equation is recovered: h x i = σ τ c t = 2 D s t, (29)with D s being the diffusion coefficient ( D s = σ /τ c ).By fitting the mean-square displacement of DWs withEq. (27) both the diffusion coefficient and the crossovertime, τ c , can be obtained. In Table II we report the valuesof such parameters for different temperatures, D/J = 0 . and H = 0 . Note that both D s and τ c decrease whenthe damping constant increases. For low applied fields,like those used in relaxation simulations, D s and τ c areindependent of the field itself.In conclusion, over some time window – generallylarger the lower the temperature is – each DW performsa ballistic motion before the genuine diffusion processstarts.0 TABLE II. Diffusion parameters.
T /J a τ c σ D s B. Temperature dependence of the diffusioncoefficient
In order to obtain quantitative results about thetemperature dependence of diffusion coefficient D s (seeEq. (29)) we introduced a DW at the center of the spinchain with anti-periodic boundary conditions (the spinsat each end were kept anti-parallel to each other alongthe z axis: S z = 1 and S zN = − ). In each numericalexperiment, we thermalized the system at a given tem-perature and then followed the DW trajectories. Theregimes with D/J > / and D/J < / will be analyzedseparately. Sharp-wall regime – In Fig. 8 we plot in a log-linearscale the diffusion coefficient as a function of
D/T foranisotropy
D/J = 1 and 2. In this scale the points sim-ulated can be fitted well by a straight line indicating anArrhenius behavior, D s ∼ e − ∆ A /T . This fact confirmsthe validity of the expression given in Eq. (7) for the dif-fusion coefficient in the sharp-wall regime: the slope is ∆ A = 0 . D and ∆ A = 0 . D for D/J = 1 and 2, respec-tively. The slight reduction of the energy barrier withrespect to the prediction of Eq. (7) ( ∆ A = D ) may be dueto the high-temperature points in Fig. 8. Even for theattempt frequency of a single nanoparticle with uniaxialanisotropy , an Arrhenius behavior is expected only inthe limit D/T ≪ . Another possibility is that ∆ A be-comes smaller than D as the crossover ratio, D/J = 2 / ,is approached. In this sense one would expect the renor-malizing effect of spin waves to be more important for D/J = 1 than for
D/J = 2 (see Sec. III). Therefore,within the numerical accuracy, our TQMC simulationsconfirm the phenomenological law proposed in Ref. 25and 39 with ∆ A = D . Broad-wall regime – The tempera-ture dependence of the diffusion coefficient changes whenthe anisotropy-to-exchange ratio is reduced. In Fig. 9 weplot the diffusion coefficient vs. temperature for thetwo ratios
D/J = 0 . and . (broad-wall regime). Thetemperature is now expressed in units of the DW energy ε w , given in Eq. (2). In this case the diffusion coeffi-cients increase linearly with temperature. This behavioris at odds with the Arrhenius dependence predicted byEq. (7) for sharp DWs: it rather reminds the behaviorof the diffusion coefficient of a massive particle in a vis- D s D = 1D = 2
FIG. 8. Color online. Temperature dependence of the diffu-sion coefficient for D = T/E dw D s / w MC D=0.1MC D=0.2
T/E w D s MC D=0.1MC D=0.2
FIG. 9. Color online. Temperature dependence of the diffu-sion coefficient for two values of the anisotropy D = 0 . and0.2 (broad-wall regime). The inset shows the scaled curves. cous medium. The slope of the diffusion coefficient asa function of T /ε w is larger the smaller the anisotropyis. Following analogous considerations to those that al-lowed us to derive the scaling relation in Eq. (23), wecan attempt to propose a scaling ansatz for the diffusioncoefficient. Note that the units of D s are square unitlengths divided by a unit time. The time unit assumedthroughout the paper is ( γ µ H k ) − = ¯ h/ D . This unitcorresponds, in general, to different physical time scalesfor different values of D . Nevertheless, this specific de-pendence on D has been already eliminated from D s bymeasuring the time in units ¯ h/ D . Then we need to ex-press the lengths in unit of w and the energies in units ε dw , which yields D s ( T ) w = g (cid:16) ε dw T (cid:17) . (30)From Fig. 9 and from the analogy with a particle in vis-cous medium, we conclude that the scaling function g isjust a line passing through the origin in the T - D s plane1so that D s ( T ) w = A Tε dw , (31) A being some constant with units D/ ¯ h . In the insetof Fig. 9 we plot D s /w vs. T /ε dw using the same dataas in the main frame. Indeed, the scaling prediction ofEq. (31) is well-obeyed, giving A = 0 .
17 [2 D/ ¯ h ] . VI. PHENOMENOLOGICAL ARGUMENTS
The analysis performed for both static and dynamicproperties allows stating that the relevant energy scalesin our problem are the DW energy ε w and the width and position of the spin-wave spectrum. Such energies,and their relationship with the thermal energy, determinethe physics of SCMs described by the model Hamilto-nian (1). The full spin-wave spectrum can be obtainedby linearizing the Landau-Lifshitz equation correspond-ing to Hamiltonian (1). The energy of a spin wave withfrequency ω and wave vector q is ¯ hω ( q ) = | S z | [2 J (1 − cos( q )) + 2 D ] . (32)At low enough temperature, in a spatial region delimitedby two DWs one essentially has | S z | = 1 . In the left panelof Fig. 10 the spectrum of fluctuations, ¯ hω ( q ) , is plotted vs. q for D = 2 . J (the sharp-wall limit). The dashedhorizontal line indicates the corresponding DW energy ε w = 2 J . Clearly the energies of the two family of exci-tations – spin waves and DWs – are well separated fromeach other. Moreover, the genuine 1d character of a spinchain is evident when the energy of thermal fluctuationsis lower than the DW energy: T < ε w . At these tem-peratures some short-range correlations develop, i.e. , ξ exceeds some lattice units. The dotted horizontal linehighlights a realistic reference for such a thermal en-ergy. On the right panel, the same plot is displayed for D = 0 . J corresponding to broad DWs. In this case thedashed horizontal line, representing the DW energy ε w , -3 -2 -1 0 1 2 301234 q E w S.W. -3 -2 -1 0 1 2 301234 q-3 -2 -1 0 1 2 301234 q E w S.W. T E w E w S.W. -3 -2 -1 0 1 2 301234 q E w B.W. -3 -2 -1 0 1 2 301234 q-3 -2 -1 0 1 2 301234 q E w B.W. E w E w B.W.
FIG. 10. Color online. The solid lines represent the energyof spin waves ¯ hω ( q ) (in J units) as a function of the wavevector q . Horizontal lines correspond to the energy of oneDW in the sharp- and in the broad-wall regime: for the leftpanel D/J = 2 while for right panel
D/J = 0 . . The horizontaldashed line represents an indicative temperature smaller than ε w in both cases (see the text). passes through the spectrum of spin waves. As a con-sequence, in the broad-wall regime one expects the in-terplay between DWs and spin waves to affect cruciallythe finite-temperature properties of the system. On theother hand, as in Fig. 10 (left) the spin-wave spectrumlies well above ε w and T , spin waves are expected to playno essential role in sharp-wall limit (when T < ε w < D ).The fact that ∆ ξ computed in Sect. III is independent of T for sharp DWs while it effectively depends on tempera-ture for broad DWs, confirms the heuristic argument evi-denced by Fig. 10. In particular, we have shown throughPolyakov renormalization that the interplay between spinwaves and DWs at finite temperature gives a quantita-tive explanation for the dependence of ∆ ξ on T in thebroad-wall limit. Parenthetically, we note that in thetime domain it is easy to see that averaging over spinwaves (the scalar fields φ a in the language of Polyakovrenormalization) corresponds to an integration over fast fluctuations. In fact, according to Eq. (32) the typicaltime periods of spin waves, π/ω ( q ) , are of the order ofour reference time unit ¯ h/ D or even smaller. The typ-ical time scale for the creation or annihilation of DWsis, instead, of the order of the relaxation time τ , i.e. ,several orders of magnitude larger than ¯ h/ D . Thus, inthe experimental situation relevant for SCMs it is clearly τ ≫ /ω ( q ) , meaning that ~n ( x ) represents a slow varyingfield and ~φ ( x ) a fast varying field.For what concerns the relaxation time we cannot pro-vide an effective argument as Polyakov renormalizationto justify our numerical findings. However, it seems rea-sonable that the very same interplay between DWs andspin waves affects the temperature dependence of the re-laxation time in a similar way as it affects the correlationlength. A natural consequence of this is that Eq. (8) doesnot necessarily hold true in the broad-wall limit. One hasto be very cautious even in trying to generalize the rela-tion ∆ τ = 2∆ ξ + ∆ A (Eq. (8)) to the case of broad DWs.In this regard, we recall that for D/J < / ∆ ξ depends on the temperature range in which itis measured2. the time window in which the DW motion is ballis-tic, and not diffusive, becomes larger while loweringthe temperature3. in the temperature range that we investigated withTQMC it is ∆ A = 0 , meaning that each single DWmove is not thermally activated.For ε w < T < ε w and ξ < N , the numerical data re-ported in Fig. 7 suggest that Eq. (8) has to be modifiedinto ∆ τ = 1 . ε w (33)for broad DWs. In order to check this prediction againstexperiment we refer to two Mn(III)-based spin chainswith D/J < / (broad DW) reported in Ref. 31 and 32.Both chains are better described by the Seiden model S = 2 ) alternate with an organic radical (TCNE orTCNQ) whose magnetic contribution is essentially thesame as a free electron: spin s = 1 / and Landé factor g s = 2 . In the broad-wall limit, the Seiden model withanisotropy can be mapped into the Heisenberg model de-scribed by Hamiltonian (1) with an halved exchange cou-pling . For the Mn(III)-TCNE spin chain , we haveestimated in Sect. III ε w ≃ K. Adapting to our con-vention the values of D and J given in Ref. 32 we ob-tain ε w ≃ K for the Mn(III)-TCNQ spin chain. ThusEq. (33) would predict the following activation barriersfor relaxation • for the Mn(III)-TCNE spin chain ∆ τ ≃
110 K, tocompare with the experimental value ∆ expτ =
117 K • for the Mn(III)-TCNQ spin chain ∆ τ ≃
95 K, tocompare with the experimental value ∆ expτ =
94 K.The prediction of Eq. (33) agrees well with the measuredenergies in both cases. We note, however, that in exper-iments the validity of the empirical formula ∆ τ = 1 . ε w seems to extend down to a temperature region in which ξ > N . In fact, the lowest temperature for which the ∆ expτ = 94 K for the Mn(III)-TCNQ spin chain isroughly T = 4 . K. The corresponding correlation length,extrapolated from Fig. 2, should be of the order of Mn(III)-TCNQ units. The same estimate gives a corre-lation length of the order of units for the Mn(III)-TCNE spin chain . As already stated in Sect. III, de-fects and dislocations typically limit the length of spinchains to − units (see the dotted horizontal linesin Fig. 2). Therefore, at the lowest temperatures at whichEq. (33) seems to apply, ξ should exceed the average dis-tance between two defects in both molecular spin chains.A conclusive analysis would require a more accurate fit-ting of the model parameters to the experimental datafor each sample. At this stage, we have no qualitativeexplanation nor a numerical confirmation for the validityof Eq. (33) in the regime ξ > N . Simulating a relaxationexperiment at lower temperatures, in the region where ξ ≫ N , is computationally very expensive due to theArrhenius dependence of the relaxation time. This issue,indeed, deserves further theoretical investigation but thisis beyond the scope of the present work. VII. CONCLUSIONS
We studied a model paradigmatic for classical spinchain or magnetic nanowires with uniaxial anisotropyand identified two distinct regimes for static and dynamicproperties. Such differences in the finite-temperature be-havior are closely related to the thickness of DWs at zerotemperature. We distinguished, accordingly, between thesharp- and broad-DW regimes. In the sharp-wall regime(
D/J > / ) the correlation length obtained by TM cal-culations shows an activated behavior as function of the inverse of the temperature. The corresponding activa-tion energy is equal to the DW energy ∆ ξ = ε w ( ε w = 2 J in this regime). In fact, for large anisotropy-to-exchangeratios, DWs extend only over one lattice spacing so thatthe anisotropy energy does not affect two-spin correla-tions. At variance, when DWs develop over several lat-tice units ( D/J < / ), the correlation length still showsan exponential divergence with the inverse of the tem-perature, but with a temperature-dependent ∆ ξ . At lowtemperatures the activation energy is larger ∆ ξ = 0 . ε w (here ε w = 2 √ DJ ), whereas at higher temperaturesit is ∆ ξ = 0 . ε w . The first result agrees with a low-temperature expansion available in the literature . Be-sides that, we provided a physical argument – based onPolyakov renormalization – to justify the 30% of reduc-tion of ∆ ξ observed at high temperature. This allowedus to conclude that the appearance of the lower activa-tion energy at higher temperatures is due to the interplaybetween DWs and spin waves. This interplay is not signif-icant in the sharp-wall regime where static properties arepractically determined by the energy cost to create a DWat zero temperature. The reason why physics is remark-ably different in the sharp- and broad-wall regime lies onthe relative difference among the energy scales involvedin the problem and the thermal energy corresponding totemperatures at which short-range correlations develop(see Sect. VI).In SCMs, relaxation is usually assumed to be driven byDWs diffusion . Based on this assumption, the acti-vation energy for the correlation length, ∆ ξ , and that ofthe relaxation time, ∆ τ , are then related with each other.We tested, with TQMC simulations, that DWs indeedperform a random walk for time intervals much longerthan the typical precessional time of a single spin and de-termined the temperature dependence of the diffusion co-efficient. In the sharp-wall regime, we found that the dif-fusion coefficient D s follows an Arrhenius behavior withan activation energy close to the anisotropy value, D , asassumed in most of the experimental works . Inthe broad-wall limit, the diffusion coefficient does not fol-low an activated mechanism but it rather grows linearlywith the temperature, reminding the behavior of a par-ticle in a viscous medium. The results of this analysisconfirm the robustness of the random-walk argument re-lating the correlation length to the relaxation time and suggest a dynamic critical exponent z = 2 . Neverthe-less, in the broad-wall regime, the relation between the ∆ ξ and ∆ τ is not trivial due to spin-wave renormaliza-tion. As a consequence, the joint theoretical and exper-imental characterization of SCMs falling in this regimecannot be based on the simple Glauber model or gen-eralizations of it .The symmetry of renormalization-group equations sug-gests the existence of scaling laws specific to the broad-wall regime: the natural unit for length scales is the DWwidth w = p J/ D while for energies it is ε w = 2 √ DJ (the DW energy at zero temperature). Thus, one ex-pects that the physical observables which depend only3on these quantities be universal functions of properlyrescaled variables (e.g. ε w /T ). Static TM calculationsand dynamic TQMC simulations confirm the validity ofthis conjecture for the correlation length, the diffusion co-efficient of DW motion and the relaxation time (for thelatter, scaling is obeyed apart from a residual tempera-ture dependence in the time unit intrinsic of the TQMCmethod).In view of possible magneto-storage applications, in-creasing the thermal stability of the SCMs would be de-sirable. To this aim, we note that designing novel com-pounds with a larger J as possible would not be a goodstrategy for two reasons: for D/J < / i) the DW energyis the sole quantity which controls thermal stability andit scales as ε w ∼ √ J , instead of ∼ J like for sharp DWs;ii) spin-waves renormalization progressively lowers the ef-fective energy barrier for relaxation as the temperatureis increased (by renormalizing ε w ).Temperature is often neglected in models employedto study the current-induced DW motion in magneticnanowires or it is taken into account in the phe-nomenological parameters of the Landau-Lifshitz-Gilbertequation . The basic hypothesis is that the considerednanowire behaves as a 3d magnet below its critical tem-perature . However, recent experiments suggestthat Joule heating may induce the formation of domainsin the nanowires, which highlights the restorations of agenuine 1d magnetic character. In this situation, the DWtrajectory may result from a delicate combination of ther-mal diffusion (stochastic) and the deterministic motioninduced by the electric current. Our study of the tem-perature dependence of the diffusion coefficient can beconsidered a preliminary contribution to this problem,of technological relevance , that indeed deserves fur-ther investigation. Typically in metallic nanowires (Co,Ni, Fe, Permalloy) D ≃ − K ( ∼ . − meV) and J ≃ − K ( ∼ − meV), meaning that theygenerally fall in the broad-wall regime where the tem-perature dependence of any observable is expected to beaffected by the non-trivial interplay between DWs andspin waves. ACKNOWLEDGMENTS
A. V. would like to thank Claude Coulon and RodolpheClérac for drawing this problem under his attention andfor fruitful discussions. Hitoshi Miyasaka is also acknowl-edged for sharing with us unpublished experimental re-sults on broad-wall SCMs. We acknowledge the financialsupport of ETH Zurich and the Swiss National ScienceFoundation.
Appendix A: The transfer-matrix approach
Given a general classical spin-chain Hamiltonian withnearest-neighbor interactions H = − N X i =1 V ( ~S i , ~S i +1 ) (A1)the partition function Z is given by Z = R d Ω R d Ω . . . R e βV ( ~S , ~S ) e βV ( ~S ~S ) . . . e βV ( ~S N , ~S ) d Ω N (A2)where the integrals, d Ω i , are performed over any possibledirection of the unit vectors ~S i and β = 1 /T . Definingthe transfer kernel K as K ( ~S i , ~S i +1 ) = e βV ( ~S i , ~S i +1 ) (A3)and taking periodic boundary conditions ( N +1 = 1) , thepartition function Z takes the form of the trace of N -thpower of K ( ~S i , ~S i +1 ) : Z = R d Ω R d Ω . . . R K ( ~S , ~S ) K ( ~S , ~S ) . . . K ( ~S N , ~S ) d Ω N = Tr (cid:8) K N (cid:9) . (A4)The computation of such a trace, as well as other physicalobservables, is simplified if one first solves the followingintegral eigenvalue problem: Z K ( ~S i , ~S i +1 ) ψ n ( ~S i +1 ) d Ω i +1 = λ m ψ m ( ~S i ) (A5)The eigenfunctions fulfill the properties X m ψ m ( ~S i ) ψ m ( ~S j ) = δ ( ~S i − ~S j ) completeness(A6)and Z ψ n ( ~S ) ψ m ( ~S ) d Ω = δ n,m orthonormality(A7)where δ ( ~S i − ~S j ) is the Dirac δ -function and δ n,m is the Kronecker symbol. For non-symmetric ker-nels, K ( ~S i , ~S i +1 ) = K ( ~S i +1 , ~S i ) , properties similar toEqs. (A6) and (A7) hold for the left and right eigen-functions, but this is not our case. Using Eq. (A6) thekernel can be rewritten as K ( ~S i , ~S i +1 ) = X m λ m ψ m ( ~S i ) ψ m ( ~S i +1 ) . (A8)Combining Eqs. (A4), (A8) and (A7) we get Z = X m λ Nm . (A9)The eigenvalues λ m are all real and positive, as the ker-nel operator (A3) is a positive defined function of ~S i and ~S i +1 . For symmetric kernels, K ( ~S i , ~S i +1 ) = K ( ~S i +1 , ~S i ) ,4the reality of the eigenvalues descends from the analo-gous of the spectral theorem for real symmetric matri-ces. Moreover, it is possible to show that the spectrumof (A5) is upper bounded so that the eigenvalues λ m canbe ordered from the largest to the smallest one: λ > λ > λ > . . . In the thermodynamic limit the asymptotic behavior ofthe partition function (A9) is dominated by the largesteigenvalue λ , yielding Z ∼ N →∞ λ N . (A10)Once the largest eigenvalue λ is known, the free energyand its derivative can be computed from the relation F = − T ln Z .
1. Pair-spin correlations
Here we recall how pair-spin correlations can beevaluated by means of the eigenvalues and the eigenfunc-tions defined in Eq. (A5). Consider the µ component ofthe spin located at site i and the ν component of the spinlocated at the site ( i + r ) , with µ, ν = x, y, z . The averagewe have to evaluate is (cid:10) S µi S νi + r (cid:11) = Z R d Ω R d Ω . . . R K ( ~S , ~S ) . . . K ( ~S i − , ~S i ) S µi K ( ~S i , ~S i +1 ) . . . (A11) K ( ~S i + r − , ~S i + r ) S νi + r K ( ~S i + r , ~S i + r +1 ) . . . K ( ~S N , ~S ) d Ω N . Following the procedure of the previous section we obtain (cid:10) S µi S νi + r (cid:11) = Z P m ,m ...,m N λ m λ m (A12) . . . λ m N δ m ,m δ m ,m . . . δ m i − ,m i − R ψ m i − ( ~S i ) S µi ψ m i ( ~S i ) d Ω i δ m i ,m i +1 . . . δ m i + r − ,m i + r − R ψ m i + r − ( ~S i + r ) S νi + r ψ m i + r ( ~S i + r ) d Ω i + r δ m i + r ,m i + r +1 . . . δ m N − ,m N δ m N ,m . Considering all the repeated indices in the Kroneckersymbols, we have (cid:10) S µi S νi + r (cid:11) = Z P m i ,m i + r λ N − rm i + r R ψ m i + r ( ~S i ) S µi ψ m i ( ~S i ) d Ω i λ rm i R ψ m i ( ~S i + r ) S νi + r ψ m i + r ( ~S i + r ) d Ω i + r . (A13)If we substitute to Z its asymptotic expansion,Eq. (A10), we need to evaluate the products (cid:16) λ m i + r λ (cid:17) N × (cid:16) λ m i λ m i + r (cid:17) r (A14)for N → ∞ . It is straightforward to conclude that onlythe terms for which m i + r = 0 will not vanish. Thus,pair-spin correlations are given by (cid:10) S µi S νi + r (cid:11) = P m i (cid:16) λ mi λ (cid:17) r R ψ ( ~S i ) S µi ψ m i ( ~S i ) d Ω i R ψ m i ( ~S i + r ) S νi + r ψ ( ~S i + r ) d Ω i + r . (A15) In the previous formula, i is a dummy index but theresult obviously depends on the separation between thetwo considered spins, r . Note that not only the λ ψ but all the eigenvalues and eigenfunctions enter Eq. (A15).Eq. (A15) can be further simplified when the µ - µ corre-lation function is considered (cid:10) S µi S µi + r (cid:11) = P m i (cid:16) λ mi λ (cid:17) r (cid:12)(cid:12)(cid:12) R ψ ( ~S i ) S µi ψ m i ( ~S i ) d Ω i (cid:12)(cid:12)(cid:12) . (A16)
2. Discretization of the unitary sphere
The eigenvalue problem defined in Eq. (A5) can bemapped into a linear algebra problem by sampling theunitary sphere with a finite number of points. Given ageneric function of two angles θ and φ , say f ( θ, φ ) , theintegral over the solid angle ( d Ω = dφ sin θdθ ) can beapproximated as: Z f ( φ, θ ) d Ω ≃ P X h =1 w h f ( u h ) , (A17)where u h represent the special points that sample the uni-tary sphere, w h are the relative weights and P is numberof points themselves . After discretizing the kernel K in this way, the eigenvalue problem (A5) transforms intothe following linear algebra problem P X h =1 w h K ( u l , u h ) ψ m ( u h ) = λ m ψ m ( u l ) (A18)which is usually symmetrized as ( K l,h = √ w l w h K ( u l , u h )Ψ nh = √ w h ψ n ( u h ) to yield P X h =1 K l,h Ψ nh = λ n Ψ nl . (A19)The number of special points, P , defines the size of thematrix that has to be diagonalized. For the calculationsreported in this work we have chosen P = 72 , , (Mc Laren formula 14-th degree and Gauss sphericalproduct formulae of 15-th and 16-th degree ). The com-parison between the results obtained for different sam-pling allows estimating the accuracy of the calculationat a given temperature. We used the routine DSPEVof the LAPACK package to solve the eigenvalue problemdefined in Eq. (A19). Appendix B: Polyakov renormalization
We rewrite here for convenience the continuum Hamil-tonian given in Eq. (11): H = − JN + Z (cid:20) J | ∂ x ~S | − D ( S z ( x )) (cid:21) dx (B1)5and represent ~S ( x ) as a superposition of tow fields: ~S ( x ) = ~n ( x ) p − φ ( x ) + ~φ ( x ) . (B2)As mentioned in the main text, requiring that | ~n ( x ) | = 1 one necessarily has ~n ( x ) · ~φ ( x ) = 0 so that ~φ ( x ) can beexpressed on a basis orthonormal to ~n ( x ) : ~φ ( x ) = X a φ a ~e a with a = 1 , (B3)with | ~e a ( x ) | = 1 . Moreover, the fact that | ~n ( x ) | = 1 implies that ∂ x ~n ( x ) is orthogonal to ~n ( x ) itself, thus ∂ x ~n ( x ) = X a c a ~e a . (B4)The gradient term in Hamiltonian (B1) reads | ∂ x ~S | = | p − φ ∂ x ~n + ∂ x p − φ ~n + X a ∂ x φ a ~e a + X a φ a ∂ x ~e a | . (B5)Exploiting the orthogonality of the basis ( ~n, ~e a ) , thederivatives ∂ x ~e a can be written as ∂ x ~e a = − c a ~n + f ab ~e b (B6)where f ab is an antisymmetric two-by-two tensor whosecomponents can be made negligible with a proper choiceof the reference frame ( ~e , ~e ) . Finally, one gets | ∂ x ~S | = (cid:0) − φ (cid:1) ( ∂ x ~n ) + X a ( ∂ x φ a ) + X ab c a c b φ a φ b + 2 p − φ X a ∂ x φ a c a − ∂ x p − φ X a c a φ a + (cid:16) ∂ x p − φ (cid:17) . (B7)The two terms at the second line of Eq. (B7) vanish afterspatial integration; while the last term (third line) onlycontains powers of φ a higher than the square after thederivation (thus we neglect it). P a ( ∂ x φ a ) representsthe “kinetic” term of the field ~φ ( x ) . Our goal is to re-tain all the other quadratic terms in the field ~φ ( x ) anduse them to perform thermal averages. After thermalaveraging, h . . . i , we will have h φ a φ b i = δ ab h φ a i (B8)which allows rewriting h X ab c a c b φ a φ b i = h φ a i X a c a = h φ a i ( ∂ x ~n ) . (B9)Besides this, one has h φ i = h φ + φ i = 2 h φ a i . (B10) Thus, the Polyakov’s result is readily recovered h| ∂ x ~S | i = (cid:2) − h φ a i (cid:3) ( ∂ x ~n ) + X a h ( ∂ x φ a ) i . (B11)Let us consider now the anisotropy term in Hamilto-nian (B1): ( S z ) = n z p − φ + X a φ a e za ! = ( n z ) (cid:0) − φ (cid:1) + X ab φ a φ b e za e zb + O ( φ a ) , (B12)where O ( φ a ) stands for terms linear in φ a that vanishafter spatial integration; while after thermal averagingone has h X ab φ a φ b e za e zb i = h φ a i X a ( e za ) = h φ a i h − ( n z ) i ; (B13)In this case, we obtain h ( S z ) i = (cid:2) − h φ a i (cid:3) ( n z ) + h φ a i . (B14)Now we want to derive the quadratic Hamiltonian forthe ~φ field written in Eq. (15) and used to perform ther-mal averages. We first rewrite the Hamiltonian (B1) interms of the fields ~φ ( x ) and ~n ( x ) . In doing this, we willkeep the approximations we made previously: we will ne-glect the terms of order higher than the second in φ a aswell as what is supposed to vanish after the spatial inte-gration. A delicate point of the latter approximation isthe assumption φ a φ b = δ ab φ + φ (B15)based on symmetry reasons. The simplified continuumHamiltonian reads H = − JN + J Z (cid:20) − φ + φ (cid:21) | ∂ x ~n | dx + X a J Z ( ∂ x φ a ) dx − D Z (cid:20) ( n z ) (cid:18) − φ + φ (cid:19) + φ + φ (cid:21) dx = − JN + J Z | ∂ x ~n | dx − D Z ( n z ) dx + X a J Z ( ∂ x φ a ) dx + X a Z (cid:20) D n z ) − D − J | ∂ x ~n | (cid:21) φ a dx . (B16)Within a distance separating two successive DWs we as-sume ( ( n z ) ≃ | ∂ x ~n | ≃ , (B17)6which allows expressing the ~φ -field Hamiltonian as H φ = N Λ X a Z Λ / − Λ / (cid:20) J ∂ x φ a ) + Dφ a (cid:21) dx , (B18) Λ being the average distance between two successiveDWs. Appendix C: Domain-wall motion and random walk
In order to check how the choice of boundary condi-tions could affect DW diffusion, we performed a prelimi-nary analysis of a standard one-dimensional random walkconstrained to a finite segment. Such a geometrical con-straint is equivalent to the one experienced by a DWwhich diffuses in the Heisenberg spin chain with o.b.c..In a random walk process the relation between the meansquare displacement h x i and the time t elapsed duringthe walk is: h x i = σ τ c t, (C1)where σ is the mean square displacement and τ c is themean-time for each step. We considered a discrete ran-dom walk in a linear system of N sites using differentboundary and initial conditions. Since we set the latticeunit a = 1 and take τ c = 1 then σ = 1 . In Fig. 11, weplot the square root of the mean-square displacement ina system of the same size of our spin chain, N = 100 . Weused both reflecting and absorbing boundary conditions.In addition, we chose for each case two initial conditions:in one we started the random walk from the middle ofthe chain and in the other one at a random position. Inthe four studied cases we observed that any significantdeviation of the relation given by Eq. (C1) started at h x i > . Below this value boundary conditions are notexpected to affect the analysis of DW trajectories either.For what concerns the spin chain, we began by study-ing the diffusion of DWs which were naturally present inthe system at the chosen temperature (if ξ < N ). Thesetup was the same as the one used to obtain the curvesof Fig. 6. We chose three representative temperatures, T /J = 0 . , T /J = 0 . , and T /J = 0 . using two valuesfor the damping constant: a = 0 . and a = 4 . Fig. 12shows the mean-square DW displacement when the sys-tem was equilibrated at T /J = 0 . , for D/J = 0 . . Thedata reported in the two figures on the left-hand-side cor-respond to zero applied field, whereas for the two figureson the right-hand-side the field was H/J = 0 . . Instead,figures in the first (Figs. 12 (a) and (b)) and in the secondraw (Figs. 12 (c) and (d)) correspond to the two differentvalues of the damping constant. For all the curves, thediffusion relation h x i ∝ t is not obeyed at short times(see Eq. (C1)). Thus we fitted those trajectories with amore general expression which describes a random walk ( < x ( t ) > ) . absorb N/2reflecting N/2reflecting randomabsorv random FIG. 11. Color online. Square root of the mean square devi-ation of a random walker obtained using different boundaryand initial conditions.
Time[h/2D] ( < x ( t ) > ) . FIG. 12. Average dispersion of the DW trajectories at
T /J =0 . for two damping constant and applied fields In Figs. (a)and (c) the applied field is H = 0 and the damping constant a = 0 . and a = 4 , respectively. In Figs. (b) and (d) theapplied field H = 0 . and the damping constant a = 0 . and a = 4 , respectively. with correlated steps : h x i = σ tτ c + e − tτc − ! . (C2)the meaning of the parameters Eq. (C2) is explained inthe main text in Sect. V.Fig. 13 displays some individual trajectories that wereused to produce the curves in Fig. 12. Some correla-tion emerges for small displacements, of the order of theDW width, which confirms the occurrence of a ballisticregime for short times. At very short time scales thereis an uncertainty in the DW position of the order of alattice parameter. In fact, fast fluctuations of the DWstructure produce an offset in the initial position of thediffusing DW, intrinsic to the method used to detect such7 (a) (b)(c) (d) Time[h/2D] x FIG. 13. Color online. Domain-wall trajectories at
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