Temperature- and Size-dependence of Line shape of ESR spectra of XXZ antiferromagnetic chain
Hiroki Ikeuchi, Hans De Raedt, Sylvain Bertaina, Seiji Miyashita
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] A ug Temperature- and Size-dependence of Line shape of ESR spectra of XXZ antiferromagnetic chain
Hiroki Ikeuchi , Hans De Raedt , Sylvain Bertaina , and Seiji Miyashita ∗ Department of Physics, Graduate School of Science, The University of Tokyo, 7-3-1 Bunkyo-Ku, Tokyo, 113-0033, Japan Department of Applied Physics, Zernike Institute for Advanced Materials,University of Groningen, Nijenborgh 4, NL-9747AG Groningen, The Netherlands Aix-Marseille Universit´e, CNRS, IM2NP UMR7334, F-13397 Marseille Cedex 20, France (Dated: August 29, 2018)The ESR (Electron Spin Resonance) spectrum of the XXZ spin chain with finite length shows a double-peakstructure at high temperatures around the EPR (Electron Paramegnetic Resonance) resonance frequency. Thisfact has been pointed out by direct numerical methods (S. El Shawish, O. C´epas and S. Miyashita: Phys. Rev.B , 224421 (2010); H. Ikeuchi, H. De Raedt, S. Bertaina, and S. Miyashita: Phys. Rev. B , 214431(2015)). On the other hand, at low temperatures. the spectrum has a single peak with a finite shift from thefrequency of EPR as predicted by the analysis of field theoretical works (M. Oshikawa and I. Affleck: Phys.Rev. Lett. , 5136 (1999)). We study how the spectrum changes with the temperature, and also we studythe size-dependence of the line shape including the even-odd effect. In order to understand those dependences,we introduce a decomposition of the spectrum into contributions from transitions specified by magnetization,and we characterize the structure of the spectrum by individual contributions. Applying the moment methodintroduced by M. Brockman et al., to each component, we analyze the size-dependence of the structure ofthe spectrum, which supports the numerical observation that separation of the double-peak structure vanishesinversely with the size. PACS numbers: 05.30.-d,75.10.jm,76.30.-vKeywords: ESR, XXZ model, line shape
I. INTRODUCTION
The ESR (Electron Spin Resonance) is one of the major tools to obtain information about the spin ordering. To understand thespectrum, parameter dependence of a concrete ESR spectrum for a specified system has to be clarified, including the temperature-dependence. To study these aspects theoretically, explicit forms of interactions of magnetic structure of the system such as spatialconfiguration of magnetic ions in the lattice must be taken into account. For example, modeling the ESR spectra of intrinsicdefects in spin chains is an important problem for which data for finite but rather long chains are necessary [1].The one dimensional S = XXZ model is one of the most well-investigated systems in the field of ESR study. As for theresonance shift and the linewidth of the spectrum which include basic information of the system, a lot of theoretical research hasbeen conducted [2, 3]. Oshikawa and Affleck [4] developed an approach based on (1+1)-dimensional field theory, where theyused the bosonization method and successfully derived the shift and the linewidth of the resonance peak at low temperatures inthe thermodynamic limit [4]. This method has been also successfully used to investigate effects of the edge state [5].There are other attempts utilizing the integrability of the XXZ model. Maeda et al. [6] derived the formula for the resonanceshift which is exact up to the first order in anisotropy. By applying the Bethe ansatz technique to the formula, they obtained ananalytic expression of the resonance shift over all the temperature region. Brockmann et al. [7] also obtained consistent resultsfocusing on the moments of the spectral shape. In this way, a lot of information has been found in respect of the resonance shiftand the linewidth.However, when we need more explicit form of the spectrum which may have a complicated structure, e.g., satellite peaks,long tails, etc., we have to evaluate the Kubo formula for the system Hamiltonian of the interest directly. Such attempts havebeen also developed [8–10].It has been pointed out that the line shape of the XXZ spin chain with even number of spins has a double-peak structure athigh temperatures for the lattice with finite length L ≤
16 [8], and the structure has been confirmed up to L =
26 [11]. But, itsdetailed dependence on temperature and size has not been known yet. Such information for finite sizes is important to studydiluted systems which are ensemble of short chains [1], where temperature-dependence of the spectrum is also important.In the present paper, first, we investigate how the high-temperature spectrum with the double-peak structure at the EPR(Electron Paramegnetic Resonance) position, i.e., ¯ h w EPR = g H ( g is the gyromagnetic ratio), changes to the low temperaturespectrum with single peak at a shifted position. We present spectra for the intermediate-temperature region. We find a drasticchange with the temperature. We also study the case of odd-number of spins, in which the high-temperature spectrum has a ∗ Corresponding author. Email address: [email protected] central peak at the EPR position with protuberances beside it. In this case we also found the spectrum changes to the low-temperature shape which does not depend on the parity of the number of spin. The transition is well understood by the energydiagram as a function of the static magnetic field.Next, we study origin of the structure of high-temperature spectrum, i.e., the double peak and small peaks beside it (protuber-rances). We find that each peak can be attributed to a resonance between states with specified magnetizations ( M and M ± M and M ± / N . This observation strongly suggests that the structure would shrink to the center inthe thermodynamic limit. Indeed the numerical study up to N =
28 supports the dependence. But on the other hand, we also findthat its variance converges to a certain finite value as N becomes large. Thus mathematically, there is some possibility of specialshape of the spectrum in which the double peak may remain.The outline of this paper is as follows. In Sec. II, we introduce the model and method. In Sec III, temperature-dependenceof spectra is given. In Sec IV, we analyze structure of spectrum decomposing it into the contributions from transitions betweenspecified sets of magnetizations. In Sec V, we study size-dependence of the structure of spectra by applying the moment methodto individual contributions from the types of transitions. In particular, the estimation of the double peak’s separation is aninteresting problem, which is discussed in Sec V B. A summary and discussion of related problems are given in Sec. VI. II. SYSTEM AND METHOD
We study a one-dimensional S = XXZ model under a static magnetic field along the z -axis. We apply to the system anoscillating magnetic field parallel to the x -axis. The total Hamiltonian of the system is given by H tot = J N − (cid:229) i = SSS i · SSS i + + D N − (cid:229) i = S zi S zi + − H N (cid:229) i = S zi + l cos w t N (cid:229) i = S xi , (1)where D represents the strength of the anisotropy. In this paper, we set J = D = − .
08K (i.e., the XY-like anisotropy) and H =
5K (i.e., a sufficiently strong filed), and we impose the open boundary conditions. In the present study we do not includethe dipole-dipole interaction, and thus the direction of the chain does not affect the results although the dipole-dipole interactioncould cause an interesting angle dependence of the spectrum on the angle between the lattice direction and the fields like theNagata-Tazuke dependence[3]. Thus we do not specify it in the present paper.Here we remind important relations for the the ESR spectrum. According to the Kubo formula [12, 13], the ESR spectrum,i.e., the absorption rate I x ( w ) of the oscillating filed can be obtained with the dynamical susceptibility c ( w ) = c ′ ( w ) + i c ′′ ( w ) as follows: I x ( w ) = wl c ′′ ( w ) , (2) c ′′ ( w ) = − e − bw Z ¥ − ¥ h M x ( ) M x ( t ) i eq e − i w t d t , (3)where M x ( t ) = e i H t M x e − i H t = e i H t (cid:229) i S xi e − i H t , H = J (cid:229) SSS i · SSS i + + D (cid:229) S zi S zi + − H (cid:229) S zi , and h· · · i eq denotes the thermal averagewith respect to H at a temperature b − . By using the set of the eigenvalues and eigenvectors { E n , | n i} Dn = of the Hamiltonian H ( D is the dimension of the Hilbert space of the Hamiltonian), c ′′ ( w ) is readily given by c ′′ ( w ) = (cid:229) m , n D m , n d ( w − w m , n ) , (4)where D m , n ≡ p ( e − b E n − e − b E m ) |h m | M x | n i| / Z , w m , n ≡ E m − E n , Z = D (cid:229) n = e − b E n . (5)Thus, for small systems we obtain c ′′ ( w ) by direct numerical estimation of the fomula (4) as long as we can obtain all theeigenvalues and eigenvectors of the system [10, 11]. However, the method is inevitably limited to small systems. Then time-domain methods have been introduced, where the spectrum is obtained by Fourier transform of the autocorrelation function ofmagnetization (AC method). There it is known that finite observation time causes artificial modification of the spectrum (say,the Gibbs oscillation). Then in our previous study [11], we proposed a new method (WK method) to make use of the Wiener-Khinchin relation with spectral density of magnetization fluctuation, in which the Gibbs oscillation is suppressed. However, inthe same time, we found that that Gibbs oscillation is suppressed in a large system and the AC method works efficiently. Thus,in the present work, we obtained the spectrum with the AC method. In the AC method, we can study the double size of the caseof the diagonalization theoretically. However, not the memory but the CPU time prevents us from treating large systems. In thepresent paper, we calculated up to N =
28 in Sec V. The methods are explained in references [9, 11] in detail.
III. TEMPERATURE-DEPENDENCE OF SPECTRAL SHAPES
In our previous studies [8, 11] we studied spectrum at high temperatures, where a double-peak structure was found aroundthe EPR position, On the other hand, the field theoretical study [4] gives a single peak with a shift from the EPR position at lowtemperatures. Thus, it is interesting to study how the spectrum changes with the temperature. In this section, we investigate thetemperature-dependence of the ESR spectrum.Let us first discuss the case in which the number of spins is even. Spectrum for N =
20 is depicted in Fig. 1(Left) at a hightemperature ( b − = w = b − = w ≃ . N spins is D = N . For the large systems, D is too large to draw the diagramin a figure, and thus we draw the diagram for N = =
64 states. In the caseof no anisotropy D =
0, the eigenstates belong to some multiplets of spin S ( S is a positive integer) which has 2 ( S + ) -folddegenerate states ( M = − S , − S + , · · · , S ) at H =
0. The degenerate states develop with the field according to the Zeemanenergy: E ( H , M ) = E − HM , M = − S , − S + , · · · , S , where E is the energy of the multiplet under no magnetic field.Note that there exists only one state that has the magnetization M = S , which is 3 in the preset case, and the energy is locatedat the lowest in the present system as shown in Fig. 2(Left).When the anisotropy is introduced D =
0, these 2 ( S + ) -fold degenerate levels are split into a single state with M = S pairs of states which have opposite magnetization, i.e., { M , − M } = { , − } , { , − } and { , − } as shown by the red lines.The energy gap D E ( M ) between the doublets with M = , − M = H = D E ( ) .Similarly, D E ( ) and D E ( ) are defined. Note that D E ( ) = M = d E ( = ) , the energy of the state for { M , − M } , M = , , · · · S is given by E ( H , M ) = E + d E + D E ( | M | ) − HM , M = − S , − S + , · · · , S : red lines in Fig. 2 (Right) . (6)In the left panel, detailed structure due to D E ( M ) is hardly seen, and thus the magnified structure is given in Fig. 2 (Right).At H =
5, the ground state is S = M =
3. Thus, the spectrum at low temperatures is mainly given by the resonancebetween this state and the excited state with S = M =
2. The resonant frequency is changed by D E ( M = ) from the EPRvalue g H . This resonance is the peak in Fig. 1(Right) with the shift D w = D E ( M = ) / ¯ h . On the other hand, at high temperaturesall the states are occupied with nearly the same probability, and the corresponding spectrum is that in Fig. 1(Left). Attribution ofeach resonance with D E ( M ) will be investigated in the next section. At intermediate temperatures, the population distributes withthe Boltzmann weight, which gives the change of spectrum of Fig. 1. As the temperature decreases, the double-peak structurebreaks down and the spectrum gradually gets shifted to the right side(Center).Theoretically temperature-dependence of the line shape has not been studied in detail and so far this drastic change has notbeen recognized yet. But, this change is robust and it is expected to be observed in corresponding materials. c ( w ) w / J x temp=100K c ( w ) w / J x temp=100Ktemp=50Ktemp=30Ktemp=10Ktemp=7Ktemp=5Ktemp=3K c ( w ) w / J x temp=0.1Ktemp=1K FIG. 1. (Color online) (Left) The ESR spectrum for N =
20 at b − = N =
20 at b − = , , , , ,
5K and 3K. (Right) The spectra for N =
20 at b − =
1K and 0 .
1K in a large scale of c ( w ) . Next we study the case of odd number of spins. In Fig. 3 the spectra for N =
21 are shown. The spectrum has a single sharppeak at the EPR position with protuberances besides it at the high temperature b − = N = -20-15-10-5 0 5 10 15 20 0 1 2 3 4 5 6 M =3 M =2 M =1 M =0 M =-1 M =-2 M =-3 Single peakat T=0K ( E ne r g y l e v e l ) / J x H / J x other levels7-fold degenerate levels split by anisotropy D E(1) D E(2) D E(3) H + D E(3) H + D E(2) H + D E(1) H - D E(1) H - D E(2) H - D E(3) H / J x other levels split by anisotropy7-fold degenerate levels split by anisotropy FIG. 2. (Color online) (Left) 64 energy levels for a 6-spin system. We accentuate the 7-fold degenerate levels by the red lines. (Right) Theenlarged view of the Left figure. c ( w ) w / J x temp=100K c ( w ) w / J x temp=100Ktemp=50Ktemp=30Ktemp=10Ktemp=7Ktemp=5Ktemp=3K c ( w ) w / J x temp=0.1Ktemp=1K FIG. 3. (Color online) (Left) The spectrum for N =
21 at b − = N =
21 at b − = , , , , , N =
21 at b − =
1K and 0 . As the corresponding energy diagram we give that for N = { M , − M } = (cid:8) , − (cid:9) , (cid:8) , − (cid:9) and (cid:8) , − (cid:9) . The fact that the state with M = M = ± / -15-10-5 0 5 10 15 20 0 1 2 3 4 5 6 M =5/2 M =3/2 M =1/2 M =-1/2 M =-3/2 M =-5/2 Single peakat T=0K ( E ne r g y l e v e l ) / J x H / J x other levels6-fold degenerate levels split by anisotropy D E(3/2) D E(5/2) H + D E(5/2) H + D E(3/2) HH - D E(3/2) H - D E(5/2) H / J x other levels split by anisotropy6-fold degenerate levels slit by anisotropy FIG. 4. (Color online) (Left) 32 energy levels for a 5-spin system. We accentuate the 6-fold degenerate levels by the red lines. (Right) Theenlarged view of the Left figure.
IV. DECOMPOSITION OF THE SPECTRUM INTO CONTRIBUTIONS FROM TRANSITIONS SPECIFIED BYMAGNETIZATION
Without D , the spectrum has a single peak at the EPR position, and thus the structure of high temperature spectrum shouldbe attributed to the energy structure lifted from the degeneracy. In what follows, we will reveal the origins of the characteristicshapes observed in the previous section by focusing on the energy diagrams of the systems given in Figs. 2 and 4In general, the resonance peaks in ESR spectrum are given by transition between states with M and M ′ = M ±
1. The mostdominant contribution comes from the transitions between levels within the same multiplet in the case of D =
0. The breakdownof SU(2) symmetry due to D allows contributions from transitions between different multiplets, but the contributions from themare found very small. Thus, we will ignore those contributions in the analysis of data, although the contributions are included inspectra obtained by numerical method. The contributions from the transitions ( M → M +
1) correspond to emission and it givesspectrum at negative w . We do not study those contributions.Now we decompose the spectrum into contributions from transitions ( M → M − ) of various values of M . Since the multipletis separated into the pairs of { M , − M } and M =
0, the contributions of ( M → M − ) and ( − M + → − M ) has the same matrixelements. Thus we classify the contribution according to M . Here we adopt a system with N =
12 for which there are manystates to form a continuous-like lineshape although we can still calculate the eigenstates by the exact diagonalization method.In Fig. 5(Left), we show the contributions from M =
1, i.e., ( M = → M − = ) (solid line) and ( − M + = → − M = − ) (dotted line). We find the contribution of ( M = → M − = ) gives the right peak of the double peak, and that of ( − M + = → − M = − ) the left one. The separation of the peaks is given by the energy split D E ( M ) . Here we concern themultiplet with maximum spin S = N / =
6, but the transitions of the pair with M = S >
0. Thusnumber of transitions between M = ± M = D E ( M = ) depend on the multiplet which the states belong to, and thus the resonance frequency distributes as we see inFigs. 5(Left).In the same way, the transitions of the pair with M = M > ( − b E ( H , m )) for m= M and − M + w EPR , and consequently the transi-tions between levels with large m = M > M = S is mostly populated under a large filed, and the transition from itto M = S − c ( w ) w / J x (1 to 0)(0 to -1) c ( w ) w / J x (2 to 1)(-1 to -2) c ( w ) w / J x (3 to 2)(-2 to -3)(4 to 3)(-3 to -4)(5 to 4)(-4 to -5)(6 to 5)(-5 to -6) FIG. 5. (Color online) Spectra for N =
12 classified by the magnetizations of transitions at b − = ( → ) (solid) and ( → − ) (dotted). The Center figure is for ( → ) and ( − → − ) . In the Right figure, spectra for larger M ’s are given: Blue: ( → ) , ( − → − ) , Magenta: ( → ) , ( − → − ) , Cyan: ( → ) , ( − → − ) , Black: ( → ) , ( − → − ) . c ( w ) w / J x (1 to 0)(0 to -1) c ( w ) w / J x (2 to 1)(-1 to -2) c ( w ) w / J x (3 to 2)(-2 to -3)(4 to 3)(-3 to -4)(5 to 4)(-4 to -5)(6 to 5)(-5 to -6) FIG. 6. (Color online) Spectra for N =
12 classified by the magnetizations of transitions at b − = ( → ) (solid) and ( → − ) (dotted). The Center figure is for ( → ) and ( − → − ) . In the Right figure, spectra for larger M ’sare given: Blue: ( → ) , ( − → − ) , Magenta: ( → ) , ( − → − ) , Cyan: ( → ) , ( − → − ) , Black: ( → ) , ( − → − ) . c ( w ) w / J x (5 to 4)(6 to 5) FIG. 7. (Color online) Spectra for N =
12 classified by the magnetizations of transitions at b − = ( → ) mainly contributes to the spectrum, and thepeaks from transition between smaller M ’s can hardly be observed in this temperature range. For odd number of spins for which we depicted the energy structure in Fig 4, the multiplets consist of half-odd spins S of M = − S , − S + , · · · , − / , + / , · · · , S . The transitions between M = / − / V. SIZE-DEPENDENCE OF SPECTRAL SHAPESA. Numerical observation
Now, we examine the size-dependence of characteristic shapes of spectra found in the high-temperature region, such as thedouble-peak structure. We depicted spectra of various sizes N in Fig. 8, where we see rather systematic size-dependence. Thesystems with even numbers of spins show that the separation between double peaks D w decreases as the system size becomeslarge. We may anticipate that in the thermodynamic limit, the double peak may get stuck together to become a single centralpeak. In fact, this problem has already posed in the reference [8], but its answer has not been concluded yet. Besides, theseparation between two small protuberances also seems to decrease with the system size increased, and shrink to the center inthe thermodynamic limit.
Central peak DoublepeakSmallprotuberance Dw / J x c ( w ) w / J x N=20N=21N=22N=23N=24N=25N=26N=27N=28
FIG. 8. (Color online) Spectra of N = , , , , , , ,
27 and 28 depicted on the unified scale. Note that the central peaks of oddsystems are cut off in this figure.
The size-dependence of the separation
D w between the double peak is given in Fig. 9 (Left). The error bars in the figuredenote the mesh size of
D w given by the observation time T as 2 p / T . Here we find that in even systems the separation roughlydecreases with the size as 1 / N at least up to N =
22, and that the separations of N = ,
24 and 26 are almost the same. Butat N =
28 it again decreases. Thus, although we can have a rough picture about the finite size effect, we cannot conclude howthe separation behaves in the large-size limit from the figure. In Fig. 9 (Center) we plot the size-dependence of separationsof two small protuberances of both even and odd cases. This shows that the separation of protuberances decreases roughlyproportionally to 1 / N . In Fig. 9 (Right), we find the heights of the peaks of protuberances increase with the system size. Dw / J x N separation of double peaksmesh sizedouble of Eq. (23) with M =1 S epa r a t i on o f t w o p r o t ube r an c e s N N:evenmesh sizedouble of Eq. (23) with M =2N:oddmesh sizedouble of Eq. (23) with M =3/2 H e i gh t o f t w o p r o t ube r an c e s N N:evenN:odd
FIG. 9. (Color online) (Left) Size-dependence of the separation of the double peak. There seems to be a tendency for separation of thedouble peak to decrease as N becomes large. But it might be saturated at some point. The green line denotes theoretical values calculated byEq. (23). (Center) Size-dependences of the separation of two small protuberances. The red points denote odd-spin systems and the blue pointseven-spin systems. We can find that the small protuberances come close to the center w = H as the system size N increases. The error barsdenote the mesh size of D w determined by the observation time T . The green lines denote theoretical values calculated by Eq. (23). (Right)Size-dependences of the height of two small protuberances. The height is related to the the intensity of the absorption. The protuberancesbecome larger with N increasing. B. Estimation of the separation
Now, we examine the size-dependence of characteristic shapes of spectra by making use of the moment method. The methodof moments originated in van Vleck’s paper [15] and has been used as a basic tool to investigate the spectral shapes [12, 16]. Ina recent study [7], the moments of the whole spectrum for the XXZ chain were discussed in detail. The width and position ofa peak in spectrum have been studied by the the moment method. In the moment method, we can grasp characteristics of ESRspectra by focusing on n -th order moment m n of the spectrum.Since we are considering properties at high temperatures, we use S xx ( w ) instead of the susceptibility c ( w ) : S xx ( w ) = Z ¥ − ¥ h M x M x ( t ) i eq e − i w t d t . (7)The moment m n is defined as[7]: m n = Z ¥ − ¥ d ww n Z ¥ − ¥ d t h M x M x ( t ) i eq e − i w t (8) = p h M x ( ad H ) n M x i eq , (9)where ad H · ≡ [ H , · ] . We could estimate these quantities by exact solutions or numerical methods. In particular, at the infinitetemperature, we obtain the moments from combinatorics as we see later.The intensity (i.e., the area of the spectrum), the mean position, and the linewidth are defined asintensity: m , (10)mean position: m m , (11)linewidth: s m m − (cid:18) m m (cid:19) , (12)respectively. There is still room for argument about the definition of linewidth given in (12). Indeed, (12) may not exist for thespectrum with exact Lorentzian shape. But, as will be seen later, (12) always exists for the system we are considering and it isknown that (12) could be regarded as a good approximation to the width of the Lorentzian distribution in some situations [12, 14].In this sense, we adopt (12) as the definition of the linewidth and other problems will be discussed in the last part of this section.So far the method has been used to study the whole shape of the spectrum. For our purpose, however, we need to improve thismethod because moments of the total spectrum are not very helpful in considering the detailed shape of spectrum. In other words,even if we know moments with small n of the whole spectrum, we could not conclude the size-dependence of the double-peak’ssplit.However, as we saw in the previous section, the spectrum can be decomposed to contributions from transitions specified bythe magnetization M . Thus now we extend the method and investigate properties of each contribution to obtain information forthe structure of the spectrum, e.g., the double peak. We will focus on the moments of partial spectrum made by only transitions ( M → M − ) , instead of the whole spectrum. For example, if we know the mean position of the partial spectrum from thetransitions ( → − ) (given by the solid line in Fig. 5(Left)), we can see how the double-peak structure behaves depending onthe size N .From now on, we set the external field H = Z ¥ − ¥ h M x M x ( t ) i ¥ e − i w t = N (cid:229) m , n |h m | M x | n i| pd ( w − ( E m − E n )) , (13)where h· · · i ¥ = Tr [ · · · ] / N . The partial spectrum made by transitions ( M → M − ) is obtained as follows: Z ¥ − ¥ h P ( M z = M ) M x P ( M z = M − ) M x ( t ) P ( M z = M ) i ¥ e − i w t d t = N (cid:229) m ( M zm = M − ) (cid:229) n ( M zn = M ) |h m | M x | n i| pd ( w − ( E m − E n )) , (14)where P ( M z = M ) is a projection operator which projects states onto the subspace where M z = M . The spectrum from the specifiedtransition is given by S Mxx ≡ Z ¥ − ¥ h P ( M z = M ) M x P ( M z = M − ) M x ( t ) P ( M z = M ) i ¥ e − i w t d t , (15)(16)and its n -th order moment m Mn = Z ¥ − ¥ w n S Mxx ( w ) d w (17) = N p (cid:229) sss ( M z sss = M ) h sss | M + ad n H ( M − ) | sss i , (18)where M ± ≡ M x ± i M y . As for the basis set {| sss i} , we may use the up/down-spin representation, such as | ↑↓↓ · · · ↑↓↑i .Because, in the infinite-temperature limit, the Boltzmann factor does not appear, and thus Eq. (18) can be calculated bycounting the number of state (combinatorics). The zeroth, first, and second order moments are explicitly written as m M = N p N (cid:229) sss ( M z sss = M ) h sss | S + S − | sss i , (19) m M = − N p ( N − ) D (cid:229) sss ( M z sss = M ) h sss | S + S − S z | sss i , (20) m M = N p D ( N − ) (cid:229) sss ( M z sss = M ) h sss | S + S − | sss i + ( N − ) (cid:229) sss ( M z sss = M ) h sss | S + S − S z S z | sss i . (21)From these quantities, we obtain the intensity (the area) of the spectrum, the mean position, and the linewidth of partial spectrafrom the transitions ( M → M − ) in the following: m M = N p N (cid:18) N − N − M (cid:19) | M |≪ N ∼ r N p , (22) m M m M = ( − M ) D N , (23) s m M m M − (cid:18) m M m M (cid:19) = | D |√ s(cid:18) − N (cid:19) (cid:18) − ( − M ) N ( N − ) (cid:19) | M |≪ N ∼ | D |√ + O (cid:18) N (cid:19) . (24)Note that these results are valid for any D and J .Let us analyze the numerical results again from the viewpoint of Eqs. (22) ∼ (24). The heights plotted in Fig. 9 (Right)correspond to the intensity of the spectrum and they relate to the quantity given by Eq. (22). Eq. (22) indicates that the areaof the partial spectrum increases with N larger, which supports the size-dependence of the peak’s height. Eq. (23) is consistentwith the observation of 1 / N in the numerical results shown in Fig. 9 (Left) and (Center). We added the green lines representedas 2 ( − M ) D / N in Fig. 9. There are some differences between the numerical results and the green theoretical lines becauseEq. (23) gives just the mean position, not the position of the maximum of the spectrum. Nevertheless, the mean of the peak dueto the transitions between M = − N → ¥ .As for the central peak in odd-spin systems, its mean position turns out to be always zero by substituting M = / / N . This means the position of each peak coming from thetransition ( M → M − S xx ( w ) of N =
11 is decomposed into different values of M as in the previous section.The linewidth of the spectrum due to the transition between M = ± / (cid:214) D S xx ( w ) w / J x (1/2 to -1/2)(3/2 to 1/2),(-1/2 to -3/2)(5/2 to 3/2),(-3/2 to -5/2)(7/2 to 5/2),(-5/2 to -7/2) (cid:214) D S xx ( w ) w / J x (1/2 to -1/2) FIG. 10. (Color online) (Left), (Right) Spectrum for N = b − = ¥ . Note that the scales are different between (Left) and (Right).According to Eq. (24), we consider 2 × D / √ VI. SUMMARY AND DISCUSSION
In the present paper, we studied the temperature- and size-dependence of the ESR spectrum for the XXZ chain. A drasticchange from high temperature spectrum with a structure around the EPR position to the low temperature spectrum with a singlepeak at a shifted position from the EPR position. Temperature-dependence of the line shape has not been studied in detail andthe drastic change has not been recognized yet in experiment. Although the structure may disappear in the thermodynamic limitas we discussed in the present paper, the structure definitely exists up to considerable length of lattice (say N <
30) and we hopethe structure and the temperature-dependence will be observed in experiments.Subsequently, we also investigated the size-dependence of the separation of the double peaks. The double-peak structureobtained by the AC method shows the tendency that the separation shrinks to zero as N → ¥ . The size-dependence was analyzedby making use of an extended moment method. Since the XXZ model has a conserved quantity, i.e., the z -component ofmagnetization M , we can decompose the whole spectrum into the partial spectra specified by types of transitions. We applied themoment method to each contribution of the spectrum from a transition with specified magnetizations. As the results, we foundthat the mean positions of all spectrum approach to the center as D / N and become zero as N → ¥ , which strongly indicates thedouble peak collapses at the center. The linewidth remains finite as D / √ D / J = − .
08) and the transitions between different multiplets can be ignorable, the decomposition of the spectrum into contributionspecified by magnetization was valid. But the analytical results derived with the moment method in the last section is valid forany J and D , and besides, according to our numerical simulation, the decomposition of the spectrum specified by magnetizationsomehow goes well for the wide range of XY-like anisotropy (0 < D / J < D / J > ACKNOWLEDGEMENTS
The present work was supported by Grants-in-Aid for Scientific Research C (25400391) from MEXT of Japan, and theElements Strategy Initiative Center for Magnetic Materials under the outsourcing project of MEXT. The numerical calculationswere supported by the supercomputer center of ISSP of Tokyo University. We also acknowledge the JSPS Core-to-Core Program:Non-equilibrium dynamics of soft matter and information.
Appendix A: Spectral shift with an external field H In Sec. V B, we investigated the spectral shapes by calculating S xx ( w ) under no magnetic field, instead of the susceptibility c ′′ ( w ) under a finite static field H . This is valid for a sufficient strong field H ≫ J ( > ) and a sufficient high temperature b ∼ H . Let the simultaneous eigenvectors of H + H ′ and H z be (cid:8) | E n , M n i (cid:9) Dn = : ( H + H ′ ) | E n , M n i = E n | E n , M n i H z | E n , M n i = − HM n | E n , M n i , n = · · · D . (A1) S xx ( w ) is given by S xx ( w , H ) = p Z (cid:229) m , n |h E n , M n | M x | E m , M m i| d { w − [( E m − E n ) − H ( M m − M n )] } . (A2)Let us divide S xx ( w , H ) into two parts in the following: S xx ( w , H ) = S > xx ( w , H ) + S < xx ( w , H ) , (A3)where S > xx ( w , H ) ≡ p Z (cid:229) m , n |h E n , M n | M + | E m , M m i| d { w − [( E m − E n ) − H ( M m − M n )] } , (A4) S < xx ( w , H ) ≡ p Z (cid:229) m , n |h E n , M n | M − | E m , M m i| d { w − [( E m − E n ) − H ( M m − M n )] } , (A5)and we find S > xx ( − w , H ) = S < xx ( w , H ) . (A6)Noting that we are interested in the absorption, not the emission, we may focus on the region w >
0. The peaks in this regionneed to satisfy the relation E m − E n > H ( M m − M n ) , then it follows that M m = M n −
1, because H ≫ J >
0. Therefore, only S > xx ( w , H ) contributes to the spectrum S xx ( w , H ) in the region w > S xx ( w , H ) = S > xx ( w , H ) , w > , (A7) S xx ( w , H ) = S < xx ( w , H ) , w < . (A8)According to Eqs. (A7) and (A4), we have S xx ( w , H ) = S > xx ( w − H , ) . On the other hand, it is shown that S > xx ( w − H , ) = S > xx ( H − w , ) = S < xx ( w − H , ) , (A9)because of the symmetry of the shape of S > xx as seen in Sec. IV, and Eq. (A6). Then, by using Eq. (A3), we have S > xx ( w − H , ) = S xx ( w − H , ) . Therefore, it follows that S xx ( w , H ) = S xx ( w − H , ) , which is what we wanted to show.The difference between S xx ( w ) and c ′′ ( w ) is just the presence of the factor ( − e − bw ) / ∼ bw / c ′′ ( w ) vanishes at infinitetemperature because of this factor, but we are interested in the spectral shape, not in the exact value of the peak. So we canignore this factor and consider only S xx ( w ) . Strictly speaking, the w -dependence of the factor bw / H is very large. [1] S. Bertaina, C.-E. Dutoit, J. Van Tol, M. Dressel, B. Barbara, and A. Stepanov, Phys. Rev. B , 060404 (2014).[2] J. Kanamori and M. Tachiki, J. Phys. Soc. Jpn , 1384 (1962).[3] K. Nagata and Y. Tazuke, , 337 (1972).[4] M. Oshikawa and I. Affleck, Phys. Rev. Lett. , 5136 (1999); M. Oshikawa and I. Affleck, Phys. Rev. B , 134410 (2002).[5] S. C. Furuya and M. Oshikawa, Phys. Rev. Lett. 109 (2012) 247603.[6] Y. Maeda, K. Sakai, and M. Oshikawa, Phys. Rev. Lett. , 134438 (2012).[8] S. El Shawish, O. C´epas and S. Miyashita, Phys. Rev. B , 224421 (2010).[9] M. Machida, T. Iitaka, and S. Miyashita, J. Phys. Soc. , 107 (2005); M. Machida, T. Iitaka, and S. Miyashita, Phys. Rev. B , 224412(2012).[10] S. Miyashita, T. Yoshino and A. Ogasahara, J. Phys. Soc. Jpn. , 655-661 (1999).[11] H.Ikeuchi, H. De Raedt, S. Bertaina, and S. Miyashita, Phys. Rev. B , 214431 (2015).[12] R. Kubo and K. Tomita, J. Phys. Soc. Jpn. , 888 (1954).[13] R. Kubo, J. Phys. Soc. Jpn. , 570 (1957).[14] J. Choukroun, J.-L. Richard, and A. Stepanov, Phys. Rev. Lett. , 127207 (2001).[15] J. H. van Vleck, Phys Rev. , 1168 (1948).[16] P. W. Anderson and P. R. Weiss, Rev. Mod. Phys.25