One-dimensional Ising ferromagnet frustrated by long-range interactions at finite temperatures

One-dimensional Ising ferromagnet frustrated by long-range interactions at finite temperatures

F. Cinti

*

Dipartimento di Fisica, Università di Firenze, I-50019 Sesto Fiorentino (FI), Italy and CNR-INFM S3 National Research Center, I-41100 Modena, Italy

O. Portmann, D. Pescia, and A. Vindigni†

Laboratorium für Festkörperphysik, Eidgenössische Technische Hochschule Zürich, CH-8093 Zürich, Switzerland 共Received 3 December 2008; revised manuscript received 30 April 2009; published 26 June 2009兲 We consider a one-dimensional lattice of Ising-type variables where the ferromagnetic exchange interaction J between neighboring sites is frustrated by a long-ranged antiferromagnetic interaction of strength g between the sites i and j, decaying as兩i− j兩−␣, with␣⬎1. For ␣ smaller than a certain threshold ␣0, which is larger than 2 and depends on the ratio J/g, the ground state consists of an ordered sequence of segments with equal length and alternating magnetization. The width of the segments depends on both␣ and the ratio J/g. Our Monte Carlo study shows that the on-site magnetization vanishes at finite temperatures and finds no indication of any phase transition. Yet, the modulation present in the ground state is recovered at finite temperatures in the two-point correlation function, which oscillates in space with a characteristic spatial period. The latter depends on ␣ and J/g and decreases smoothly from the ground-state value as the temperature is increased. Such an oscillation of the correlation function is exponentially damped over a characteristic spatial scale, the correlation length, which asymptotically diverges roughly as the inverse of the temperature as T = 0 is approached. This suggests that the long-range interaction causes the Ising chain to fall into a universality class consistent with an underlying continuous symmetry. The e⌬/Ttemperature dependence of the correlation length and the uniform ferromagnetic ground state, characteristic of the g = 0 discrete Ising symmetry, are recovered for␣⬎␣0.

DOI:10.1103/PhysRevB.79.214434 PACS number共s兲: 75.60.Ch, 64.60.De, 75.10.Hk

I. INTRODUCTION

The competition between a short-ranged interaction favor-ing local order and a long-range interaction frustratfavor-ing it on larger spatial scales is often used to explain pattern formation in chemistry, biology, and physics.1,2 _{The role of the}

long-range interaction is to avoid the global phase separation fa-vored by the short-ranged interaction and promote a state of phase separation at mesoscopic or nanoscales. Thus, the long-range interaction is not, in general, a small perturbation3–8 _{but must be considered as precisely as}

pos-sible. From a computational point of view, this means that the frustrating interaction has to be accounted for by involv-ing all the lattice sites in the computation, which in turn limits the actual system size that can be handled in, e.g., Monte Carlo 共MC兲 simulations.9–14 _{Few exact results on}

multiscale and multi-interaction3,4 _{systems are present—to}

our knowledge—in literature. For one-dimensional systems, rigorous proof of absence of a phase transition in the pure long-range antiferromagnetic model has been obtained.15 Be-sides, rigorous results concerning the ground-state phase dia-gram can be found in Ref. 3. Regarding two-dimensional 共2D兲 lattice models with restricted spin orientation and dipole-dipole interaction competing with ferromagnetic nearest-neighbor exchange interaction, Giuliani et al.16

showed that the ground state is periodic striped while a zero-temperature reorientation transition共from in-plane to out-of-plane magnetization兲 occurs at a given relative strength of the short-range and long-range interactions when both are antiferromagnetic. Finally, a generalization of this periodic ground state in some continuum versions has been rigorously proved.17–19

In this paper, we perform MC simulations on a one-dimensional 共1D兲 lattice with sites occupied by Ising-type

classical variables assuming values ␴j=⫾1. The

nearest-neighbor sites interact by a short-ranged ferromagnetic inter-action of strength J which favors the same sign for two ad-jacent variables共in the language of magnetism, the exchange interaction favors parallel alignment of neighboring spins兲. In addition, any two variables located at sites i and j interact by means of a long-range interaction of strength g decaying according to a power law兩i− j兩−␣and favoring, instead, anti-parallel alignment. In the present study, selected values of

␣⬎1 and in the vicinity of 2 are investigated. This range turns out to be representative of the different physical re-gimes. We are aware of the apparently academic nature of共i兲 a one-dimensional model and of共ii兲 this choice of values for

␣. In fact, point charges interact via the Coulomb interaction, which has ␣= 1, while the dipolar interaction between two localized magnetic moments has ␣= 3. On the other side, imposing a monodimensional modulation to two- or three-dimensional 共3D兲 arrangement of charges and spins 共a sym-metry often realized in experiments1,2,20,21兲 produces an ef-fective one-dimensional long-ranged interaction potential with an effective value of ␣which can differ from 1 and 3, respectively. As an example, elementary magnetic moments arranged into stripes and located on a two-dimensional array of sites interact with an effective one-dimensional dipolar long-range interaction which, asymptotically, is proportional to兩i− j兩−2.20_{Accordingly, a systematic study for values of}␣

in this range might reveal properties that can be used to explain physically relevant situations, such as those repre-sented by the two-dimensional system of stripes quoted above or similar models of frustration discussed in connec-tion with electronic phase separaconnec-tion.5_{A 1D model has great}

computational advantages compared to its 2D and 3D coun-terparts, such as the possibility of simulating lattices of larger

linear dimensions, which in turn allows larger modulation lengths than already reported,9–14_{which are, indeed, closer to}

experimental situations. Later in the paper, we will single out the relevance of our results for understanding realistic spin and charged systems. Besides, variations in the 1D Ising model including long-ranged potentials have been widely ap-plied to biological problems,22_{such as protein folding}23 _and

helix-coil transitions.24

This paper is organized as follows. In Sec. II, we intro-duce the model and its known ground-state phase diagram3

and then present our main results on the oscillatory character of the two-point correlation function on the temperature de-pendence of the corresponding modulation period and on the correlation length. These facts point to the persistence of the modulated structure emerging in the ground state even if, strictly speaking, the on-site order is completely lost in the thermodynamic limit.25 _{In Sec. III, we provide some}

argu-ments aiming at explaining, within an analytic approach, the crossover from the g = 0 Ising universality class to a continuous-symmetry behavior for gⱖ0 and␣ⱕ␣₀ as well as the temperature dependence of some physical observables in comparison with MC simulations. In Sec.IV, we provide a summary of the most relevant results and indicate possible directions for further work. Technical aspects of the MC simulations and of the analytical computations are presented in Appendixes A–D.

II. MONTE CARLO RESULTS A. Model and the ground state

The Hamiltonian with Ising variables ␴i=⫾1 on a 1D

lattice reads as H = − J

兺

j=1 N ␴j␴j+1+ g 2_兵i⫽j其

兺

␴i␴j 兩i − j兩␣, 共1兲 where N is the number of spins in the chain and兵i⫽ j其 indi-cates a sum over all the couples in the chain; periodic bound-ary conditions ␴i+N=␴i are assumed. The ground state3 of

this Hamiltonian is uniform for ␣⬎␣0, where ␣0ⱖ2 de-pends on the ratio J/g. For␣ⱕ␣0, the ground state consists of a regular sequence of groups of h adjacent spins with positive 共␴j= +1兲 and negative 共␴j= −1兲 orientations. The

zero-temperature phase diagram in the共␣, g/J兲 plane is sche-matically reported in Fig.1. The inset zooms into the region of the parameter space关共␣, J/g兲 plane in this case兴 in which MC simulations have been performed: J/g=2.5 and ␣ = 1.6, . . . , 3.2. The main thermodynamic observable we ad-dress is the two-point correlation function at temperature T and fixed␣

C_␣共r兲 = 具具␴j+r␴j典j典T 共2兲

and its Fourier transformS_␣共q兲 共commonly named structure factor兲

S␣共q兲 =

兺

r=−⬁

+⬁

具具␴j+r␴j典j典Te−iqr. 共3兲

As the system cannot be assumed to have translational in-variance, an average over the lattice sites j is needed关具¯典j

in Eqs. 共2兲 and 共3兲 and henceforth兴; 具¯典Tdenotes the

ther-mal average. The physical quantities computed with the MC approach actually correspond to the double average 具¯典 =具具¯典j典T.

The lowest-energy spin profiles are known to be square waves Sq共k0j兲 with a modulation period 2h=2␲/k0.3 The total energy can be parametrized with h by inserting the square profile into the Hamiltonian 共1兲. The ground-state equilibrium value of h—let us call it hgs, corresponding to

kgs—is then determined by minimizing the resulting energy 共B3兲 with respect to h. hgs depends on ␣ and J/g; some values are reported in Fig.5. The two-point correlation func-tion for a generic square-wave profile reads as共see Appendix B for details兲 具␴j+r␴j典j= 1 N

兺

_j=1 N Sq关k0共j + r兲兴Sq共k0j兲 = 1 2_m=0

兺

⬁ am2 cos共kmr兲 ⬟ Tr共k0r兲, 共4兲 where k0=␲/h, km=共2m+1兲k0, am= 4关␲共2m+1兲兴−1, and

Tr共k0r兲 is a symmetric triangular wave of period 2h. Accord-ing to Eq. 共4兲 evaluated in h=h_gs, the ground-state structure factor only takes nonzero values in the points located at q =⫾共2m+1兲kgsfor whichS␣关共2m+1兲kgs兴=N·4关␲共2m+1兲兴−2. The structure factor of a uniform state takes a finite value at

q = 0 only: S_␣共0兲=N. This case can be regarded as the limit h→⬁ so that k₀= 0 and all the peaks ofS_␣共km兲 collapse into

the peak at q = 0.

B. Finite temperature

Fig.2 shows the two-point correlation function共2兲 com-puted by MC simulations for ␣= 1.8共domain ground state兲 and ␣= 3.0 共uniform ground state兲 at different temperatures 共see Appendix A for details about the computational meth-ods兲. In spite of the fact that the single-spin average 具具␴j典j典T

J/g g/J Domain ground state Uniform ground state

FIG. 1. 共Color online兲 Ground-state phase diagram in the 共␣,g/J兲 plane. For 1ⱕ␣ⱕ2, the ground state always consists of domains共grey region兲. For␣⬎2, the crossover to a uniform ground state共white region兲 occurs when␣ exceeds the threshold value ␣0 indicated by the solid line. MC calculations have been performed for values of␣ between 1.6 and 3.2 and g/J=0.4 共horizontal line兲. Inset: zoom of the region where MC simulations have been per-formed in the共␣,J/g兲 plane, J/g=2.5 共horizontal line兲.

is zero at any finite temperature, the correlation function re-produces the essential aspects of the ground-state spin con-figurations. For␣= 1.8关Fig.2共a兲兴, C_␣共r兲 displays an oscilla-tory decay as a function of r, indicating that the loss of on-site magnetization proceeds in such a way that the ground-state segment order is maintained. In the regime in which the ground state is uniform共␣= 3兲, instead, the corre-lation function decays smoothly and, in general, monotoni-cally 关Fig.2共b兲兴. A closer look at the highest reported tem-peratures 共T/g=0.8,1.4兲 reveals a small interval at short distances in which C_␣共r兲 becomes negative 关inset in Fig. 2共b兲兴. This might be taken as an indication that, even starting from a uniform ground state, when the temperature is in-creased the system can spontaneously produce a phase with reduced symmetry in which the short-range order occurs with a well-defined modulation. We will come back to this point at the end of Sec.III.

In Fig.3, the structure factor corresponding to␣= 1.8 and

T = 0.02 is plotted. The set of discrete peaks of the ground

state has broadened to Lorentzians centered at q =共2m + 1兲q␣,max. Here, q␣,max means the position of the highest peak of the simulatedS_␣共q兲 at finite T and does not, in gen-eral, coincide with kgs共the temperature dependence of q␣,max will be discussed below兲. The occurrence of multiple peaks in the finite-temperature structure factor not only indicates that the periodic structure of the ground state propagates at finite temperatures but also shows that some memory of the detailed square-wave spin profile is retained. As T is in-creased, peaks with m⬎0 rapidly lose weight and, for T ⲏ0.1, basically only one peak is detectable. This implies a change in the correlation profile from triangular wavelike共all harmonics兲 at low temperatures to cosinelike 共single har-monic兲 at higher temperatures. The same crossover is pre-dicted to occur for the equilibrium mean-field spin profile within a 2D stripe-domain pattern and observed experimen-tally in the striped phase of ultrathin Fe films grown epitaxi-ally on Cu共001兲.20_{Note that the height of the peaks ofS}

␣共q兲 in the ground-state scales such as 共2m+1兲−2_{, while the ratio} between the peaks at m = 1 and m = 0 in Fig. 3 is about an order of magnitude smaller at finite temperatures. In the next section, we will give a simple explanation for this observa-tion. The Lorentzian shape of the peaks and the m depen-dence of their width 共inset兲—which are related to the expo-nential spatial dumping of the correlation shown in Fig. 2共a兲—will also be discussed in Sec.III.

A typical temperature dependence of the structure factor is shown in Fig. 4using a linear scale where only the most prominent peak m = 0 is evident. Two facts are visible:共1兲 the location of the maximum q_␣,maxvaries with temperature and 共2兲 the peak broadens considerably when the temperature is increased. We will discuss these two features more thor-oughly.

1. Temperature and␣ dependence of q_␣,max

具h␣典⬟␲/q␣,maxis plotted as a function of the temperature in Fig. 5 for the set of values ␣= 1.6, 1.8, 2 , 2.2, all in the regime␣⬍␣₀for the chosen J/g. The ground-state value h_gs found by minimizing the total energy共B3兲 with respect to h is also indicated. When ␣ is increased—approaching the transition line to the uniform state—both ground-state and finite-temperature values also increase. A strongly decaying -0.5 0 0.5 1 C α=1.8 (r) T/g = 0.04 T/g = 0.1 T/g = 0.6 T/g = 1.2 0 10 20 30 40 50 -0.02 0 0.02 0.04 0 50 100 150 200 250 300 r 0 0.5 1 Cα=3.0 (r) T/g = 0.3 T/g = 0.4 T/g = 0.8 T/g = 1.4 (a) (b)

FIG. 2. 共a兲 Correlation function for ␣=1.8 共domain ground state兲 at different temperatures for L=1000 and J=2.5. 共b兲 Correla-tion funcCorrela-tion for␣=3.0 共uniform ground state兲 at different tempera-tures for L = 1000 and J = 2.5. Inset: reminiscence of the competing dipolar interaction共see text兲.

0 0.5 1 1.5 2 2.5 3 3.5 q 0.01 1 100 S α=1.8 (q,T/g=0.02) q_m1 0.01 0.1 λα, m 3 0.5

FIG. 3. 共Color online兲 Structure factor obtained by MC simula-tions at T/g=0.02 for␣=1.8 with Lorentzian fitting for the peaks located at q = qm=共2m+1兲q␣,max 共and J=2.5兲. Inset: the HWHM ␭␣,mobtained by Lorentzian fitting of the MC data 共full squares兲 and given by formula 共9兲 共line兲 is plotted versus qm in a log-log scale, which clearly reveals the power-law behavior.

2.4 q T/g 1 0.8 0.6 0.4 0.2 0 0.05 0.1 0.15 0.2 0.25 0.3

FIG. 4. 共Color online兲 Evolution of S_␣共q兲 as a function of tem-perature: the dotted line represents the peak position q_␣,max at dif-ferent temperatures 共parameters: J/g=2.5 and ␣=2.4, giving hgs = 51兲. Inset: schematic view of a square-wave spin profile.

long-range interaction favors longer periods. To be more quantitative, two temperature regions have to be considered. 共i兲 For T/Jⲏ0.3, the period of modulation decreases with temperature, in a similar way to what is found for the stripe width in the mean-field approximation 共MFA兲 of a similar but 2D model and in line with experimental results.20

共ii兲 The temperature range T/Jⱗ0.3 is more difficult to explain. The modulation period saturates at the ground-state value for ␣= 1.6, 1.8 and remains below the ground-state value for ␣= 2.0, 2.2. We interpret the convergence of 具h_␣典

→hgswith T→0 as a positive indication that our MC calcu-lations capture the essential equilibrium properties of the model; although we note that for larger periods, in this tem-perature range, the MC acceptance rate approaches zero 共“blocked condition”兲. A further investigation should be re-quired to decide whether this is due to a technical limitation or rather to the set in of intrinsic slow dynamics by analogy with similar systems.26–30

In Appendix D we will introduce an energy functional for finite T which depends parametrically on the period of modu-lation 2h. Within some approximations, there we show that the minimum of such a functional is found for smaller h as the temperature is increased, thus, reproducing qualitatively the dependence of q_␣,max on T.

2. Temperature and␣ dependence of the correlation length

␰␣(T)’␭␣,max −1 _,␭

␣,maxbeing the half-width at half maximum of

the Lorentzian centered at q_␣,max

In Fig. 6,␰_␣ is plotted versus T/g for␣= 1.6, 1.8, 2 , 2.2 共all falling in the region ␣ⱕ␣0 for J/g=2.5兲 in a log-log scale. Dots correspond to MC data while the solid lines rep-resent fits with the function A_␣/TB␣_{, with fitting parameters}

A_␣ and B_␣. The best fit yields the same exponent B_␣ = 1.10共5兲 for each␣, while A_␣ has a more complicated de-pendence on␣关see squares in the inset of Fig.6共the zig-zag line will be discussed in the next section兲兴. We conclude that the dependence of the correlation length on T is better de-scribed by A_␣/TB␣ _{than the Ising exponential relation e}⌬/T_, which holds for g = 0. A deeper understanding of this differ-ence will be provided in Sec.III. For a comparison with the

uniform regime 共i.e.,␣= 2.6, 2.8, 3 , 3.2兲, let us consider the correlation function for ␣= 3.0 displayed in Fig.2共b兲. Note that the plot is limited to low-enough temperatures in order to avoid the anomalous range of spatial decay where the correlation function becomes negative 关see inset of Fig. 2共b兲兴. Besides, in the temperature range T=0.3, ... ,0.8, the

g = 0 behavior is recovered. In fact, looking at the correlation

length␰_␣for␣= 2.6, 2.8, 3 , 3.2, we find that it is better fitted by ␰_␣⬃exp共⌬_␣/T兲, see the log-linear plot of␰_␣ versus g/T in Fig.7. Remarkably, the energy barrier⌬_␣does depend on

␣ 关see squares in the inset of Fig.7共the dashed line will be discussed at the end of the next section兲兴.

Regarding the possibility to observe long-range order at finite temperature, our MC results seem to exclude such a

0 0.5 1 1.5 2 2.5 3 T / g 4 5 6 7 8 9 10 11 12 13 14 〈hα 〉 h_gs= 7 h_gs= 8 h_gs= 11 h_gs= 17

FIG. 5. 共Color online兲 Plot of 具h_␣典 versus T/g in the domain-ground-state region. The computation parameters are L = 1000, J/g=2.5, and␣=1.6 共circles兲, 1.8 共squares兲, 2 共diamonds兲, and 2.2 共triangles兲. The statistical errors are smaller than the data symbols.

1 T / g 10 〈ξ α 〉 _{1.6 1.8 2 2.2} α 3 4 5 Aα /g 0.2 0.5 5 30

FIG. 6. 共Color online兲 Plot of ␰_␣ versus T/g in the domain-ground-state region. The computation parameters are L = 1000, J/g=2.5, and␣=1.6 共circles兲, 1.8 共squares兲, 2 共diamonds兲, and 2.2 共triangles兲. The continuous lines represent the fit function y = A_␣/xB_{␣共see text兲. The statistical errors are smaller than the data} symbols. Inset: the values of A_␣/g obtained by fitting the results of the MC simulations 共main frame兲 are reported as a function of␣ 共full squares connected by dotted line兲. The theoretical value of the prefactor of 1/T in formula 共11兲, 共4/␲2_兲h4_共⳵2_Egs/⳵h2_{兲, is indicated} by the solid “zig-zag” line共see text兲.

1 1.5 2 2.5 3 3.5 g / T 10 100 1000 ξα 2.6 2.8 3.0 3.2 α 0.5 1 1.5 2 ∆α

FIG. 7. 共Color online兲 Log-linear plot of␰_␣ versus g/T in the uniform-ground-state region. The computation parameters are L = 1000, J/g=2.5, and ␣=2.6 共circles兲, 2.8 共squares兲, 3.0 共dia-monds兲, and 3.2 共triangles兲. The continuous lines are best-fit func-tions. The statistical errors are smaller than the data symbols. Inset: the dependence of the barrier⌬_w= 2J − 2g␨共␣−1兲 derived in Sec.III

on␣ 共dashed line兲 is compared with that of ⌬_␣obtained by fitting ␰␣computed in MC simulations共full squares兲.

hypothesis. In fact, in a correct analysis of the structure fac-tor, beyond the usual intensive term connected with the cor-relations 共Lorenzian-type functions兲, one should also take into account an extensive factor associated with the occur-rence of long-range order.31_{This last component has always}

been considered in the fitting procedure but it has never given a significant contribution to the simulated structure factor共3兲. The occurrence of long-range order at T=0 only is also supported by the low-temperature divergence of ␰_␣ for both␣ⱕ␣₀共Fig.6兲 and␣⬎␣₀共Fig.7兲. The whole scenario confirms some recent theoretical works. In fact, in Ref. 15 the absence of long-range order at any temperature for g ⬎0, J=0, and ␣⬎1 is rigorously proved. Even if in that specific case a short-range ferromagnetic term was not in-cluded, it seems reasonable to extend such a result to the case

J⬎0 and conclude that long-range order should not occur

with the model Eq.共1兲 for␣⬎1.3_{The occurrence of a phase}

transition has been suggested, instead, for 0⬍␣ⱕ1 so that it would be particularly interesting to investigate the finite-temperature properties of the model共1兲 in this regime. How-ever, several other issues are related to the divergence of the energy per spin for pair-spin interaction decaying as 1/r␣ with␣⬍d, d being the dimension of the space in which the spin system is embedded such as energy nonadditivity and

ensemble inequivalence.32–34 For this reason, our canonical MC method would not necessarily provide the unique and correct results in this context, but further analyses involving the comparison of different computational approaches would be, most probably, required.

III. DISCUSSION

In this section, we provide an explanation for some of the MC results on the basis of a simple physical model for the excited states of the Hamiltonian 共1兲. In particular, we will provide a physical picture for the temperature dependence of the correlation length in the two distinct regimes␣ⱕ␣0and ␣⬎␣0.

A. Case␣ⱕ␣₀

We construct excited states of the Hamiltonian 共1兲 by modifying the square-wave profile to

␴j= Sq关k0共j + uj兲兴 =

兺

m=0

⬁

amsin关km共j + uj兲兴 共5兲

with uj being a displacement field. This perturbation

corre-sponds to displacing the position of the wall between adja-cent segments which creates a generally nonperiodic spin configuration. The quantity we need to compute is the incre-ment of energy due to the displaceincre-ment field

⌬Eh⬟ Eh关u兴 − Eh关u = 0兴, 共6兲

with Eh being defined as 具H典j. ⌬Eh is computed

perturba-tively, i.e., in the limit of small u˜q关uj⬟共1/N兲兺q˜uqeiqj兴 共see

Appendix C兲. For qⰆk0 and setting k0= kgs, with kgs being ␲/hgs, one has ⌬Egs= 1 N

兺

q

冋

1 2kgs 2 ⳵ 2_E gs ⳵k₀2 q 2_兩u˜ q兩2

册

. 共7兲

Equation共7兲 describes the spectrum of the excited states 共see also Refs.35and36for a model in 2D兲 and the coefficient of

q2 _{is a stiffness k} gs 2_共⳵2_E

gs/⳵k02兲 against fluctuations from the ground-state spin configuration 共see Appendix C for

⳵2_E gs/⳵k0

2 _{definition兲. Note the gapless quasicontinuum} na-ture of the spectrum of fluctuations, in clear contrast to the gapped spectrum of fluctuations in a pure共g=0兲 Ising model. Equation 共7兲 is the central result of this section as it al-lows computing the structure factorS_␣共q兲 and the correlation length ␰_␣共T兲. The resulting structure factor 共3兲 consists of a series of Lorentzian peaks centered at q =⫾km

S␣共q兲 = 1 2_m=0

兺

⬁

再

am 2

冋

␭␣,m 共q − km兲2+␭_␣,m2 + ␭␣,m 共q + km兲2+␭_␣,m2

册

冎

, 共8兲 with a half-width at half maximum 共HWHM兲 given by

␭␣,m=共2m + 1兲2 T 2⳵ 2_E gs ⳵k₀2 . 共9兲

The reader is referred to Appendix C for the details. The same behavior is observed in the MC results plotted in Fig. 3. Our analysis finds the origin of the multiple peaks of

S␣共q兲 in the quasicontinuum spectrum of gapless excitations 关see Eq. 共7兲兴 appearing in the frustrated model for␣ⱕ␣₀. A remarkable feature is the nontrivial scaling of the maxima with the higher-harmonic index 2m + 1

S␣共q = ⫾ km兲 = 1 2 am 2 ␭␣,m= 16 ␲2 ⳵2_E gs ⳵k₀2 1 T 1 共2m + 1兲4, 共10兲 which accounts for the strong reduction detected for the ratio between the peak heights for m = 1 and m = 0 in the MC re-sults at finite temperatures. Note also the square-power de-pendence of the HWHM␭_␣,mon 2m + 1 in formula共9兲. This theoretical prediction 共solid line in the inset of Fig. 3 with log-log scale兲 is in excellent agreement with the behavior of the S_␣共q兲 simulated for ␣= 1.8 and J/g=2.5 at T/g=0.02 共squares in the inset of Fig. 3兲. From the assumption k0 = kgs 关Eq. 共7兲兴 it follows that within our analytic model, the highest peak ofS_␣共q兲 is expected to occur at q_␣,max= k_gsand the correlation length is defined as ␰_␣=␭_␣,0−1 consistently. Even if we already know that in MC simulations q_␣,maxdoes not remain constant as T is varied 共see Fig.5兲, this assump-tion produces a 1/T dependence of the correlation length

␰␣= 1 ␭␣,0 = 2⳵ 2_E gs ⳵k₀2 1 T, 共11兲

which is in good agreement 关B_␣= 1.10共5兲兴 with the corre-sponding quantity computed again with the MC technique 共see Fig. 6兲. Finally, the analytic model predicts that A_␣ = 2共⳵2_E

gs/⳵k0

2_{兲. Computing this expression numerically} pro-duces the solid curve in the inset of Fig. 6. The steplike behavior of 2共⳵2_E

gs/⳵k0

opti-mal domain width h_gs and the second derivative of the energy—computed in h = hgs—are discontinuous functions of ␣共Ref.3兲 in virtue of the discreteness of the lattice. Both the order of magnitude and the scaling with ␣ agree with MC calculations 共squares in the inset of Fig.6兲: A_␣ decreases as

␣ increases approaching the uniform-ground-state region. In summary, the agreement between numerical and analytical results indicates that the distortion of the ground-state spin profile due to the displacement of domain walls represents the main disordering mechanism when␣ⱕ␣₀.

To the aim of reproducing the temperature dependence of

q_␣,max, the expansion for qⰆk0—performed in Appendix C to get from Eq. 共C12兲 to Eq. 共C13兲—is not expected to be accurate anymore. However, in Appendix D we show that letting h be an adjustable parameter at finite T with an ap-propriate 共temperature-dependent兲 stiffness we are able to reproduce qualitatively the decrease in the modulation period with increasing temperature observed in MC simulations. This, indeed, happens because in the correlation function 共C21兲, higher harmonics are progressively more suppressed as the temperature increases. As a result, the competition between the ferromagnetic exchange and the antiferromag-netic long-range interaction turns out to be biased with re-spect to the zero-temperature case and the period of modu-lation decreases subsequently. This close remodu-lationship between the suppression of higher-harmonic components and the decrease in the characteristic period of modulation has been already highlighted experimentally and by mean-field calculations in an equivalent 2D system, suggesting that it might be a general property of such models.

The pure Ising Hamiltonian is invariant with respect to any operation that changes the variable␴j to −␴j. It has the

discrete symmetry group Z2. In the next subsection, we will discuss this case in connection with␣⬎␣₀. The 1/T depen-dence of the correlation length obtained by introducing a long-range interaction共g⫽0兲 suggests that, in the regime of

␣ⱕ␣0, the frustrated system crosses over to the completely different universality class of one-dimensional chains hosting a planar spin field with SO共2兲 continuous symmetry.37,38

B. Case␣⬎␣₀

In this regime, the ground state is uniform and the Ising universality class is restored at low enough temperatures, as shown by the correlation length diverging exponentially as

e⌬/T共see Fig.7兲. Specific to this case is that ⌬=⌬_␣共see inset Fig.7兲. We try to explain this result by considering that, in the pure Ising model共g=0兲, the barrier ⌬ equals the energy cost to reverse half of the spins starting from a uniform con-figuration. Were the general arguments which associate such an energy with the low-temperature expansion of␰共Ref.39兲 applicable in the presence of long-range interaction, the en-ergy of a single wall would be expected to equal⌬_␣. When half of the spins in the chain is reversed, the exchange energy increases by 2J. To compute the variation due to the long-range interaction, note that this interaction energy is just given by twice the interaction energy between the two parts of the chains lying on opposite sides with respect to the domain wall共as the self-energy in each domain remains the

same before and after the flip of half of the spins兲. This interaction energy is given by

⌬g= − 2g

兺

jⱖ0

兺

iⱖ1 1 兩j + i兩␣= − 2g

兺

rⱖ1 r r␣= − 2g␨共␣− 1兲, 共12兲 where␨共x兲 is the Riemann zeta function, while i and j are the site indices of spins lying on opposite sides of the domain wall. The energy to create a wall becomes explicitly depen-dent on␣and amounts to⌬w= 2J − 2g␨共␣− 1兲. In the inset of

Fig.7, one can appreciate how this estimate actually repro-duces both the order of magnitude and the dependence on␣ of the energy barrier of the exponentially diverging ␰_␣ ob-tained from MC simulations. To be rigorous, one should point out that this approach is not completely justified in this context since, when a long-range interaction is present, the creation of a new domain wall is not statistically independent of the number and the location of the pre-existing domain walls in the chain; such a hypothesis is indeed a basic as-sumption to put the correlation length in relationship with the cost to create a single wall in the system.39_{Letting ␣ go to}

infinity effectively reduces the spin-spin interaction to near-est neighbors only so that our system becomes equivalent to the usual Ising model provided that the exchange interaction is replaced by J − g.

At T = 0, the condition⌬w= 0 defines ␣0. In fact, as soon as ⌬wⱕ0 the uniform configuration has no more the lowest

energy and the system prefers to split into domains. For a given ratio J/g, ␣0 fulfills the condition␨共␣0− 1兲=J/g. Us-ing the integral definition of the Riemann zeta function, the previous condition can be rewritten as

J g =␨共␣0− 1兲 = 1 ⌫共␣0− 1兲

冕

0 ⬁ dxx ␣0−2 ex− 1 = 1 ␣0− 1 1 ⌫共␣0− 1兲

冕

0 ⬁ dx x ␣0−1_ex 共ex_{− 1兲}2 = 1 ⌫共␣0兲

冕

0 ⬁ dx x ␣0−1_e−x 共1 − e−2x_兲2, 共13兲

this implicit equation for␣₀turns out to be exact3_共the

solu-tion being the solid line in Fig.1兲.

For completeness, we recall that at relatively high tem-peratures a well-defined period of modulation seems to emerge in the correlation function also for␣⬎␣₀关see inset of Fig. 2共b兲兴. A naïve, but essentially correct, interpretation of the temperature dependence of q_␣,max in the regime ␣ ⱕ␣0suggests that thermal fluctuations effectively reduce the ratio J/g 共the antiferromagnetic long-range interaction is fa-vored in the competition with the ferromagnetic exchange interaction, which finally leads to decrease the modulation period with respect to the T = 0 case兲. In this sense, one may think that even when the uniform pattern has the minimum energy at T = 0 关e.g., for J/g=2.5 and␣= 3 as in Fig. 2共b兲兴, thermal fluctuations induce an effective decrease in the ratio

J/g=2.5 so that a modulated phase eventually has lower free

only be evident if such a crossover occurs when there is still enough correlation between spins to develop—at least—half period of modulation, i.e., roughly for␰_␣⬎1/q_␣,max. In fact, if the period of the underlying modulated phase is much larger than the correlation length ␰_␣, two-point correlations just display a monotonic decay as a function of the lattice separation. A detailed investigation of this phenomenon would be, indeed, intriguing but it is beyond the purpose of the present work.

IV. CONCLUSIONS

The mean-field approximation reported, e.g., in Ref. 20 provides some straightforward results concerning ferromag-netic Ising system frustrated by a long-range interaction. However, the MFA fails in one important instance. It predicts that the modulated order in the ground state propagates at finite temperatures up to a second-order transition tempera-ture Tc, while the Landau-Peierls instability forbids a finite

on-site具␴j典Tat any finite temperature.25,40On the other side,

MC simulations are much more accurate than the MFA but very difficult to perform under experimentally realistic con-ditions. For instance, the large modulation lengths often ob-served in experiments are practically inaccessible to MC simulations. We concentrated on a model 关Eq. 共1兲兴 that is highly simplified but captures some essential characteristics of some physically relevant two-dimensional frustrated sys-tems. Within this model, we have been able to enlarge the modulation length with respect to full two-dimensional MC simulations.41,42 _{With this model, we have obtained a set of}

results that might help to shed light onto some experimental outcomes. In particular, the modulation length appearing in the ground state is found to remain a characteristic length at finite temperatures, where it appears as the length modulat-ing the oscillatory part of the correlation function. Strikmodulat-ingly, it decays with temperature in a way that is similar to the temperature dependence of the stripe-domain width observed in MFA and experimentally on Fe/Cu共001兲 films.20,21 _In

ad-dition, the spatial profile of the correlation function contains the same kind of higher harmonics appearing in the MFA spin profile, with only one fundamental harmonic remaining at sufficiently high temperatures, as specified within the MFA and found experimentally.20 _{In contrast to the MFA,}

which predicts a second-order phase transition also in 1D, we do not find any trace of a phase transition—and this is a major deviation from full two-dimensional MC simulations41,42 _{or experimental findings. When the spatial}

decay of the long-range interaction is too short ranged, the ground state and the finite-temperature state lose the modu-lated character and become uniform. Correspondingly, the system crosses over from the universality class proper of 1D systems with continuous symmetry37,38 _{to the standard 1D}

Ising-type universality class.43,44

For future work, a more accurate treatment of the dis-placement field ujbeyond the qⰆk0approximation共see Ap-pendix C兲 is certainly to be considered.

ACKNOWLEDGMENTS

We would like to thank S. Cannas, A. Rettori, P. Politi, D.

Stariolo, N. Saratz, and M. G. Pini for fruitful discussions. The financial support by ETH Zurich and the Swiss National Science Foundation is acknowledged.

APPENDIX A: MONTE CARLO METHOD

In this appendix, we discuss the technical details of the MC method we used to study the finite-temperature proper-ties of the Hamiltonian 共1兲. A first important issue for the system under investigation is the treatment of finite-size ef-fects. In the presence of long-range interactions, they need to be handled with particular care both numerically and analytically.45 Some techniques to tackle the problem nu-merically are given, for instance, in Ref.46. We perform our simulations on a system containing L spins and treat the long-range effects by replicating many identical copies of the “simulation box.”47 _{More explicitly, the interaction between}

two spins separated by r lattice sites reads as

G_␣共r兲 = 1 r␣+

兺

n

1

兩r + nL兩␣, 共A1兲

where the index n accounts for the number N/L of replicated boxes. Since we have in mind the thermodynamic limit N

→⬁, for numerical evaluation of G␣共r兲 we let n go to ⫾⬁ in order to account for the copies of the system lying on both the left-hand and right-hand sides of the simulated segment, containing just L spins. The effective coupling共A1兲 can be rewritten, in a way that is more suitable for computational purposes G_␣共r兲 = 1 r␣+_{n=⫾1¯⫾⬁}

兺

1 兩r + nL兩␣= 1 r␣+_n=1

兺

⬁

冋

1 兩r + nL兩␣ + 1 兩r − nL兩␣

册

= 1 r␣+ 1 L␣

冦

_n=1

兺

M

冤

1

冏

n + r L

冏

␣+ 1

冏

n − r L

冏

␣

冥

+

兺

n=M+1 ⬁

冤

1

冏

n + r L

冏

␣+ 1

冏

n − r L

冏

␣

冥

冧

⯝ 1 r␣ + 1 L␣

冦

兺

n=1 M

冤

1

冏

n + r L

冏

␣+ 1

冏

n − r L

冏

␣

冥

+ 2

兺

n=M+1 ⬁ 1 n␣

冧

= 1 r␣+ 2␨共␣兲 L␣ + 1 L␣

兺

_n=1 M

冤

1

冏

n + r L

冏

␣+ 1

冏

n − r L

冏

␣− 2 n␣

冥

, 共A2兲 in the third passage we have neglected r/L with respect to

M; the error of the whole approximation can be estimated

following Ref. 47. This approximation reduces the main computational task to evaluating the finite sum over n which is—however—rapidly convergent. Finally, the working Hamiltonian, restricted to our simulation box, is given by

H = − J

兺

i=1 L ␴i␴i+1+ g 2

兺

_i=1 L

兺

j=1 L ␴i␴jG␣共i − j兲, 共A3兲

which descends directly from Eq. 共1兲 with the replica as-sumption ␴i⫾nL=␴i共n= ⫾1¯ ⫾⬁兲, periodic boundary

con-ditions on the simulation box␴L+1=␴1, and setting r =兩i− j兩. Note that the indices i and j now vary in the range关1,L兴 and are allowed to be equal, G_␣共0兲 being representative of the interaction between different spins in the original Hamil-tonian 共1兲; in this particular case 共r=0兲, there is no interac-tion inside the simulainterac-tion box but the ith spin still interacts with its own copies lying in the different replicas,␴i⫾nL, so

that G_␣共0兲 =

兺

n=⫾1¯⫾⬁ 1 兩nL兩␣= 2␨共␣兲 L␣ . 共A4兲

The MC simulations have been performed using the simu-lated annealing 共SA兲 共Ref. 48兲 paradigm. The SA is exten-sively applied in statistical physics with the intent to study systems where both the ground-state energy and the equilib-rium at low temperatures are inaccessible through the basic Metropolis criterion.46 _{Certainly, spin glasses,}49 _frustrated

magnetic spin structures,50 _{and models with long-range}

interactions45 _{are some typical examples of systems where}

the SA and related methods51_{are largely exploited.}

We also have to remind that in literature some cluster methods were employed in order to reach a correct thermo-dynamic equilibrium for a simple model where ferromag-netic long-range interactions are only present.52 _However,

the strong frustration due to the competition between the antiferromagnetic long-range interactions and the nearest-neighbor ferromagnetic exchange interaction renders the generalization of such cluster MC technique to the present case nontrivial. For this reason, we have followed in this work the main idea of Kirkpatrick et al.48 _{A random initial}

configuration共which should be considered as a paramagnetic state兲 is picked up. Subsequently, the thermodynamic equi-librium at a high enough temperature T0 is established. We remember that the MC steps per spin considered here only comprise Metropolis moves at the analyzed temperature. T₀ is usually chosen in order to have a high MC acceptance ratio per spin. Then the temperature is decreased gradually

T→T−⌬T共⌬T⬎0兲, and a fixed number of MC steps per

spin␶is run, starting with the last configuration sampled at the previous higher temperature. So, the main assumption is to force a constant and sufficiently slow cooling rate defined as r =⌬T/␶. We have taken ⌬T=0.1, ... ,0.001 and ␶ = 1 , . . . , 5⫻105 _{depending on the studied value of ␣. The} procedure is completed when the ground state is approached. We have considered simulation boxes of size L = 100, 200, 500, 1000, and 2000. After discarding the first 1⫻105 MC steps, we have collected between 5⫻105 _{and 1⫻10}6 mea-surements of the thermodynamic observables, repeating the simulation for each temperature at least three times. The es-timation of the statistical errors has been achieved by apply-ing the usual blockapply-ing technique.46

APPENDIX B: CORRELATIONS IN THE GROUND STATE

In this appendix, we compute the two-point correlations for a generic square-wave spin profile representative of the regime in which the ground state consists of domains: ␣ ⱕ␣0. The lowest-energy configurations, at T = 0, are known to be given3 _{by square-wave spin profiles}

␴j= Sq共k0j兲 =

兺

m=0

⬁

amsin共kmj兲, 共B1兲

with k0=␲/h, km=␲共2m+1兲/h, and am= 4/关␲共2m+1兲兴. With

the orthogonality relation兺N_j=1e−i共k−k⬘兲j= N␦k,k⬘, the two-point

correlations averaged over the site variables j can be com-puted 具␴j+r␴j典j= 1 N

兺

_j=1 N Sq关k0共j + r兲兴Sq共k0j兲 = 1 N

兺

_j=1 N

兺

m,m⬘=0 ⬁ am⬘amsin关km共j + r兲兴sin共km⬘j兲 = 1 2_m=0

兺

⬁ am2 cos共kmr兲. 共B2兲

The Fourier coefficients of the series obtained in the final passage of Eq. 共B2兲 happen to be the same as for the sym-metric triangular wave of period 2h so that in the text we use the compact notation具␴j+r␴j典j= Tr共k0r兲.

Equation 共B2兲 allows writing the energy per spin for a general square-wave profile

Eh= − J Tr共k0兲 + g 2m=0

兺

⬁ am 2

兺

r_ⱖ1 cos共kmr兲 r␣ =m=0

兺

⬁ am 2 f_␣共km兲, 共B3兲 which depends parametrically on the half period of modula-tion h. The ground-state energy for a given ratio J/g and ␣ can be obtained by minimizing Eq. 共B3兲 with respect to h numerically, which consequently defines the equilibrium do-main width hgs at T = 0. The exchange term in Eq. 共B3兲 straightforwardly gives −J Tr共k0兲=−J共1−2/h兲, also deduc-ible by counting the number of walls present in the domain configuration with modulation period 2h. The function

f_␣共km兲=−共J/2兲cos共km兲+g/2兺rⱖ1关cos共kmr兲/r␣兴 introduced

above will be used to write the perturbed energy in a more compact form.

APPENDIX C: PERTURBATIVE TREATMENT OF CORRELATIONS AT FINITE TEMPERATURES

In this appendix, we develop a perturbative elastic model which allows us to compute the two-point correlations in the regime ␣ⱕ␣₀ at finite temperatures. Let us consider a dis-placement field, uj, of the whole square-wave profile共B1兲

␴j= Sq关k0共j + uj兲兴 =

兺

m=0

⬁

amsin关km共j + uj兲兴. 共C1兲

To compute how the energy共B3兲 is modified by the presence of this elementary perturbation, we introduce the constants

再

a = km共j + r兲

b = km⬘j

冎

再

␥= kmuj+r

␤= km⬘uj,

冎

共C2兲 where the two greek letters will henceforth be assumed in-finitesimal. Equation共B2兲 then involves terms such as

sin共a +␥兲sin共b +␤兲 = sin a sin b cos␥cos␤ + sin a cos b cos␥sin␤ + cos a sin b sin␥cos␤ + cos a cos b sin␥sin␤. 共C3兲 We will further assume that the average over the lattice in-dices j,具¯典j, can be performed independently for the rigid

pattern variables 共latin letters兲 and for the fluctuating dis-placement field uj共Ref.53兲,

具sin a sin b cos␥cos␤典j=具sin a sin b典j具cos␥cos␤典j.

共C4兲 The average 具¯典j for elementary trigonometric functions

with arguments a and b gives

具sin a sin b典j=具cos a cos b典j=

1

2␦m,m⬘cos共kmr兲 具sin a cos b典j= −具cos a sin b典j=

1

2␦m,m⬘sin共kmr兲, 共C5兲 which can be exploited to average Eq.共C3兲 with respect to j

具sin共a +␥兲sin共b +␤兲典j=

1

2␦m,m⬘cos共kmr兲具cos␥cos␤ + sin␥sin␤典j+

1

2␦m,m⬘sin共kmr兲 ⫻具cos␥sin␤− sin␥cos␤典j

=1

2␦m,m⬘兵cos共kmr兲具Re关e

i共␤−␥兲_兴典 j

+ sin共kmr兲具Im关ei共␤−␥兲兴典j其, 共C6兲

then, recalling that ␤−␥= km共uj− uj+r兲, we get

具␴j+r␴j典j= 1 2_m=0

兺

⬁ 兵am 2_关cos共k

mr兲具Re关eikm共uj−uj+r兲兴典j+ sin共kmr兲

⫻具Im关eik_m共uj−uj+r兲兴典

j兴其. 共C7兲

The introduction of the displacement field brings an incre-ment to the energy of a general square-wave profile 共B3兲 equal to ⌬Eh= − J 1 2_m=0

兺

⬁ 具am 2_关sin共k

m兲sin关km共uj− uj+1兲兴 + cos共km兲共cos关km共uj− uj+1兲兴 − 1兲兴典j+

g 2_m=0

兺

⬁

冓

am 2

兺

rⱖ1

冋

sin共kmr兲 r␣ sin关km共uj− uj+r兲兴 +cos共kmr兲 r␣ 共cos关km共uj− uj+r兲兴 − 1兲

册

冔

_j⯝ − J 1 2_m=0

兺

⬁

冓

am 2

冋

_{− sin共k} m兲km共uj+1− uj兲 − 1 2cos共km兲km 2_共u j+1− uj兲2

册

冔

j +g 2_m=0

兺

⬁

冓

am2

兺

rⱖ1

冋

−sin共kmr兲 r␣ km共uj+r− uj兲 − 1 2 cos共kmr兲 r␣ km 2_共u j+r− uj兲2

册

冔

j , 共C8兲

where we have expanded the energy for small displacement differences u_j+r− uj.

To proceed in our calculation, it is convenient to express the displacement field in terms of its Fourier transform u˜q

uj= 1 N

兺

q u ˜qe iqj with u˜q=

兺

j uje −iqj , 共C9兲

the sum is performed over the Fourier wave numbers qm

=⫾共2␲m兲/N with m苸关−N/2,N/2兴, but we drop the index

m for simplicity. From Eq.共C9兲, it follows that the averaged

square difference is

具共uj+r− uj兲2典j=

2

N

兺

q

兩u˜q兩2关1 − cos共qr兲兴, 共C10兲

while具uj+r− uj典j= 0. The previous results and the elementary

trigonometric relation cos共x兲cos共y兲=共1/2兲关cos共x−y兲+cos共x + y兲兴 allow writing Eq. 共C8兲 as

⌬Eh= 1 N

兺

q J 2_m=0

兺

⬁

再

a_m2k_m2

冋

cos共km兲 − 1 2cos共km− q兲 − 1 2cos共km+ q兲

册

兩u˜q兩 2

冎

− 1 N

兺

q g 2_m=0

兺

⬁

再

am2km2 1 2r

兺

ⱖ1 1 r␣

冋

cos共kmr兲 − 1 2cos关共km− q兲r兴 − 1 2cos关共km+ q兲r兴

册

兩u˜q兩 2

冎

_{. 共C11兲}

Recalling the definition of f_␣共km兲 in Eq. 共B3兲 one can rewrite the perturbed energy 共C11兲 as

⌬Eh= 1 N

兺

q m=0

兺

⬁

再

am2km2

冋

1 2f␣共km− q兲 + 1 2f␣共km+ q兲 − f␣共km兲

册

兩u˜q兩 2

冎

_{. 共C12兲}

As far as the large-distance behavior is concerned—like for the computation of the correlation length—one can expand the energy共C12兲 for qⰆk0 to get

⌬Eh= 1 N

兺

q m=0

兺

⬁

冋

am 2 km 21 2 ⳵2_f ␣ ⳵2_k m q2兩u˜q兩2

册

, 共C13兲

where the derivatives are formally defined by assuming kmto

be a continuum variable k and taking the limit ⳵n_f

␣/⳵km n

= limk→k_m共⳵nf␣/⳵kn兲, which is, of course, more justified the

larger h is. The fact that ⳵km/⳵k0= km/k0 and ⳵2km/⳵2k0= 0 implies ⳵2_E h ⳵k₀2 =m=0

兺

⬁

再

am 2

冋

⳵2km ⳵2_k 0 ⳵f_␣ ⳵km +

冉

⳵km ⳵k0

冊

2_⳵2_f ␣ ⳵2_k m

册

冎

=

兺

m=0 ⬁

冋

am 2

冉

km k₀

冊

2_⳵2_f ␣ ⳵km 2

册

共C14兲

so that the perturbed energy共C13兲 finally reads as ⌬Eh= 1 N

兺

q

冋

1 2k0 2⳵ 2_E h ⳵k₀2q 2_兩u˜ q兩2

册

. 共C15兲

Equation共C15兲 specialized to h=hgsfor the ground-state en-ergy essentially matches the result obtained in Ref.36for a 2D system with an analogous Hamiltonian. Within the range of validity of Eq. 共C15兲 and with restriction to h=h_gs, an analytical formula for the structure factor共3兲 can be derived. The thermal averages 具¯典T of the displacement field uj,

which appears in the perturbed two-point correlation 共C7兲, have to be performed first. Those thermal averages can easily be evaluated since the Hamiltonian 共C15兲 is quadratic for small perturbations of the ground state 共h=hgs兲. The well-known theorem for Gaussian distributed physical quantities25

readily gives 具具␴j+r␴j典j典T= 1 2_m=0

兺

⬁

再

am 2 _cos共k mr兲exp

冋

− 1 2km 2_具具共u j − u_j+r兲2典j典T

册

冎

. 共C16兲

On top of the site average Eq.共C10兲 one has to perform the thermal average

具具共uj+r− uj兲2典j典T=

2

N

兺

q

具u˜q2典T关1 − cos共qr兲兴, 共C17兲

in particular 具u˜q2典T can be computed applying the

equiparti-tion theorem to Eq.共C15兲: 共k_gs2/2兲共⳵2Egs/⳵k0 2_兲具u˜ q 2_典 Tq2= T/2 so that 具具共uj+r− uj兲2典j典T= T k_gs2 ⳵ 2_E gs ⳵k₀2 2 N

兺

q 1 − cos共qr兲 q2 . 共C18兲

By writing the wave numbers explicitly, the sum in the pre-vious formula can be evaluated analytically in the thermody-namic limit共N→⬁兲 2 N

兺

q 1 − cos共qr兲 q2 = 2 N_m=−N

兺

_/2 N_/2 1 − cos

冉

2␲r N m

冊

冉

2␲ Nm

冊

2 = N 2␲2

冦

1 2

冉

2␲r N

冊

2 + 2

兺

m=1 N_/2 1 m2− 2_m=1

兺

N_/2 cos

冉

2␲r N m

冊

m2

冧

⯝ N 2␲2

冦

1 2

冉

2␲r N

冊

2 + 2

兺

m=1 ⬁ 1 m2 − 2

兺

m=1 ⬁ cos

冉

2␲r N m

冊

m2

冧

= N 2␲2

再

1 2

冉

2␲r N

冊

2 + 2

冋

␲ 2 6 − 1 4

冉

2␲r N

冊

2 +␲ 2

冉

2␲r N

冊

− ␲2 6

册

冎

= N 2␲2 2␲2r N = r. 共C19兲

The thermal average 共C17兲 is finally obtained 具具共uj+r− uj兲2典j典T= T k_gs2 ⳵ 2_E gs ⳵k₀2 r. 共C20兲

Combining Eqs. 共C20兲 and 共C16兲, the sought-for quantity reads as 具具␴j+r␴j典j典T= 1 2m=0

兺

⬁ 关am 2 _cos共k mr兲e−␭␣,mr兴, 共C21兲 with ␭␣,m= 1 2km 2 T k_gs2⳵ 2_E gs ⳵k₀2 =共2m + 1兲2 T 2⳵ 2_E gs ⳵k₀2 . 共C22兲

The structure factor 共3兲 is thus expected to have a series of Lorentzian peaks at q =⫾共2m+1兲kgs,␭␣,m being the corre-sponding HWHM. More explicitly

S␣共q兲 =1 2_m=0

兺

⬁

再

am2

冋

␭␣,m 共q − km兲2+␭_␣,m2 + ␭␣,m 共q + km兲2+␭_␣,m2

册

冎

. 共C23兲

APPENDIX D: OPTIMAL PERIOD OF MODULATION AT FINITE TEMPERATURES

In this appendix we provide a qualitative explanation for the dependence of q_␣,maxon the temperature. First, we show that the decrease in the modulation period with increasing temperature is not reproduced just letting k₀be an adjustable parameter at any temperature. In fact, formula 共C21兲 can be used to compute the two-point correlations associated with any square-wave profile provided that the appropriate stiff-ness against deviations from the given period h⫽hgs is ac-cordingly employed: k₀2共⳵2Eh/⳵k0

2_{兲 关see Eq. 共B3兲 for the} defi-nition of Eh兴. In this way, one can account for the effect of

thermal fluctuations on a square-wave profile of an arbitrary half period h and construct the functional

具Hh典 = − NJ具具␴j+1␴j典j典T+ N g 2r

兺

ⱖ1 具具␴j+r␴j典j典T r␣ , 共D1兲 where 具具␴j+r␴j典j典T= 1 2_m=0

兺

⬁ 关am 2 _cos共k mr兲e−␭␣,mr兴 共D2兲

with km=共2m+1兲␲/h 共h⫽hgs are here allowed兲 and ␭␣,m = km

2

T/关2k₀2共⳵2Eh/⳵k0

2_{兲兴. The functional 具H}

h典 关Eq. 共D1兲兴 can

then be minimized with respect to h to obtain an effective equilibrium period of modulation at finite temperatures. This procedure produces the dashed line in Fig. 8: for J/g=2.5 and␣= 2, the optimal half-period of modulation corresponds the ground-state value hgs= 11 for T⬍0.07, while the func-tional共D1兲 has a minimum in h=10 for higher temperatures. All these indicate that the constant decrease in the

modula-tion period observed in the MC simulamodula-tions is not reproduced just by including thermal fluctuations through a displacement field into the different square-wave profiles and by further minimizing the functional Eq.共D1兲 with respect to h. Such a failure might be due to the assumption that the stiffness

k₀2共⳵2_E

h/⳵k02兲 remains the same at any temperature. In order to circumvent this limitation, we propose a heuristic exten-sion of our elastic model. Let us first compute the two-point correlations at an infinitesimal temperature␦T

具具␴j+r␴j典j典␦T= 1 2_m=0

兺

⬁ 关am 2 _cos共k mr兲e−␦␭␣,mr兴, 共D3兲

with km=共2m+1兲␲/h and␦␭␣,m= km2␦T/关2k02共⳵2Eh/⳵k02兲兴. The correlation 共D3兲 can be thought of as resulting from a rigid spin profile

␴j=

兺

m=0

⬁

amsin共qmj兲, 共D4兲

in which the wave numbers qm=共2m+1兲q0 are statistically distributed. In particular, if a Lorentzian distribution

P共qm兲 = ␦␭␣,m ␲ 1 ␦␭_␣,m2 +共qm− km兲2 共D5兲 is assumed, the corresponding averages—performed after the site average 具¯典j—mimic the effect of thermal fluctuations

such that Eq. 共D3兲 can then be rewritten as 具具␴j+r␴j典j典␦T=具具␴j+r␴j典j典q_m =1 2_m=0

兺

⬁

冋

am2

冕

−⬁ +⬁ dqmP共qm兲cos共qmr兲

册

. 共D6兲 The corresponding energy functional reads as

0 0.5 1 1.5 2 T / g 5 6 7 8 9 10 11 12 〈hα 〉

FIG. 8. 共Color online兲 Plot of 具h_␣典 versus T/g in the domain-ground-state region with J/g=2.5 and␣=2: MC simulations 共dia-monds兲, elastic model with constant 共dashed line兲, and temperature-dependent 共solid line兲 stiffness, k₀2共⳵2_Eh/⳵k02_{兲 and} k₀2关⳵2_具H˜

具H˜ h典␦T=具H˜h典q_m=

冕

−⬁ +⬁ dqmP共qm兲

冋

− NJ具␴j+1␴j典j + Ng 2r

兺

ⱖ1 具␴j+r␴j典j r␣

册

=

兺

m=0 ⬁

冋

am2

冕

−⬁ +⬁ dqmP共qm兲f␣共qm兲

册

. 共D7兲

To the aim of computing the correlation function at an infini-tesimally higher temperature, the spin profile 共D4兲 can be further perturbed with a displacement field, which brings an increment to the energy functional共D7兲 equal to

具⌬H˜ h典q_m= 1 N

兺

q m=0

兺

⬁

再

am2

冕

−⬁ +⬁ dqmP共qm兲qm2

冋

1 2f␣共qm− q兲 +1 2f␣共qm+ q兲 − f␣共qm兲

册

兩u˜q兩 2

冎

_{. 共D8兲} By analogy with what done in the previous section, we per-form an expansion for qⰆk₀共since q₀’s follow a Lorentzian distribution with maximum in k0, qⰆq0 as well兲

具⌬H˜ h典q_m= 1 N

兺

q m=0

兺

⬁

再

am 2

冕

−⬁ +⬁ dqmP共qm兲qm 21 2 ⳵2_f ␣ ⳵qm2 q2兩u˜q兩2

冎

. 共D9兲 The fact that qm=共2m+1兲q0 implies ⳵qm/⳵q0= qm/q0, ⳵2_q m/⳵2q0= 0, and, consequently, 具⌬H˜ h典q_m= 1 N

兺

q 1 2

冓

q0 2⳵ 2_H˜ h ⳵q₀2

冔

q_m q2_兩u˜ q兩2. 共D10兲

In the present case, the effective stiffness 具q₀2共⳵2H˜h/⳵q0 2_兲典

q_m

has a more complicated dependence on q₀with respect to Eq. 共C15兲. However, we can simplify its computation signifi-cantly with the approximation

冓

q₀2⳵ 2_H˜ h ⳵q₀2

冔

q_m ⯝ k02

冏

⳵2_具H˜ h典q_m ⳵k₀2

冏

_cos= k0 2

冏

⳵ 2_具H˜ h典␦T ⳵k₀2

冏

_cos, 共D11兲 ⳵2_具H˜ h典␦T

⳵k02 兩cosmeaning that the derivative with respect to k0

in-volves only the fluctuating functions cos共kmr兲. The

correla-tion funccorrela-tion at the new temperature共T=␦T +␦T兲 is given by

具具␴j+r␴j典j典T= 1 2_m=0

兺

⬁

冤

am 2

冕

−⬁ +⬁ dqmP共qm兲cos共qmr兲exp

冢

−qm 2 2 ␦T k₀2

冏

⳵ 2_具H˜ h典␦T ⳵k₀2

冏

_cos r

冣冥

⯝ 1 2_m=0

兺

⬁

冤

a_m2

冕

−⬁ +⬁ dqmP共qm兲cos共qmr兲exp

冢

−km 2 2 ␦T k₀2

冏

⳵ 2_具H˜ h典␦T ⳵k₀2

冏

_cos r

冣冥

, 共D12兲

where in the last passage we have substituted qm

2

inside the exponential with its maximum km2. Such an approximation

allows writing the energy functional at the new temperature again in the form 共D7兲 provided that the HWHM of the Lorentzian distribution P共qm兲 is changed into

␦␭m= km 2 2

冤

␦T k₀2⳵ 2_E h ⳵k₀2 + ␦T k₀2

冏

⳵ 2_具H˜ h典␦T ⳵k₀2

冏

_cos

冥

r. 共D13兲

The whole process can then be iterated to obtain correlations at any temperature 具具␴j+r␴j典j典T= 1 2m=0

兺

⬁ 关am 2 _cos共k mr兲e−␭␣,m共T兲r兴, 共D14兲

and the corresponding energy functional

具H˜ h典T=具H˜h典q_m=

兺

m=0 ⬁

冋

am2

冕

−⬁ +⬁ dqmP共qm兲f␣共qm兲

册

, 共D15兲 with HWHM of P共qm兲 共letting␦T→dT兲 equal to

␭␣,m共T兲 = km2

冕

0 T d␭_␣,m=1 2 km 2 k₀2

冕

0 T dT

冏

⳵2_具H˜ h典T ⳵k₀2

冏

_cos . 共D16兲 共Remember that the derivative with respect to k0 involves only the fluctuating functions cos共kmr兲 and not the dumping

terms兲. By minimizing numerically the functional 共D15兲 with respect to h, we obtain the steplike curve in Fig. 8. In this case, a decrease with increasing temperature is indeed ob-served throughout the investigated range. Such a qualitative agreement with MC results suggests that the change in the