On the theory of microwave absorption by the spin-1/2 Heisenberg-Ising magnet

# On the theory of microwave absorption by the spin-1/2 Heisenberg-Ising magnet

Michael Brockmann    Frank Göhmann    Michael Karbach    Andreas Klümper Fachbereich C – Physik, Bergische Universität Wuppertal, 42097 Wuppertal, Germany    Alexander Weiße Max-Planck-Institut für Mathematik, P.O. Box 7280, 53072 Bonn, Germany
###### Abstract

We analyze the problem of microwave absorption by the Heisenberg-Ising magnet in terms of shifted moments of the imaginary part of the dynamical susceptibility. When both, the Zeeman field and the wave vector of the incident microwave, are parallel to the anisotropy axis, the first four moments determine the shift of the resonance frequency and the line width in a situation where the frequency is varied for fixed Zeeman field. For the one-dimensional model we can calculate the moments exactly. This provides exact data for the resonance shift and the line width at arbitrary temperatures and magnetic fields. In current ESR experiments the Zeeman field is varied for fixed frequency. We show how in this situation the moments give perturbative results for the resonance shift and for the integrated intensity at small anisotropy as well as an explicit formula connecting the line width with the anisotropy parameter in the high-temperature limit.

The total magnetization in an isotropic system of interacting spins rotates as a whole about the axis of a homogeneous external field (see e.g. Oshikawa and Affleck (1999); *OsAf02). We consider spins-, combining to a total spin , , in a magnetic field of strength in -direction. The Heisenberg equation of motion for with a Zeeman term is solved by , a rotation counterclockwise about the -axis. In ESR experiments this can be probed by circularly polarized microwave radiation propagating along the -direction. Since its wavelength is large compared to typical distances in spin systems, we may assume a magnetic field component of the form , . It couples to the total spin as and produces a sharp resonance at (see (4) below). If the spin system is perturbed by anisotropic interactions, this resonance is broadened, shifted or even split in a way that is characteristic of the microscopic interactions between the spins.

For any spin system with Hamiltonian linear response theory relates the observed absorbed intensity to the (imaginary part of the) dynamical susceptibility Kubo and Tomita (1954)

 χ′′+−(ω,h)=12L∫∞−∞dteiωt⟨[S+(t),S−]⟩T. (1)

Here and stands for the canonical average at temperature calculated by means of the statistical operator . Through this average the dynamical susceptibility depends on and . The absorbed intensity per spin, normalized by the intensity of the incident wave and averaged over a half-period of the microwave field, is

 I(ω,h)=ω2χ′′+−(ω,h). (2)

In current ESR experiments in solids is measured as a function of for fixed . Of particular interest are experiments on quasi one-dimensional compounds (reviewed e.g. in Ajiro (2003); Krug von Nidda et al. (2010)) which provide prototypical realizations of interacting many-body systems with strong quantum fluctuations. Still, the data are not always easy to interpret, because of a lack of reliable theoretical predictions.

Most of the existing theories are based on a priori assumptions about the line shape and typically apply for limited ranges of temperature and magnetic field. Field theoretical approaches Oshikawa and Affleck (1999); *OsAf02 are restricted to small temperatures and small (but not too small) magnetic fields. The more traditional approaches Kubo and Tomita (1954); Nagata and Tazuke (1972) rely on the high temperature approximation. Purely numerical approaches Miyashita et al. (1999); *OgMi03; *ECM10 are unbiased, but the extrapolation of the data to the thermodynamic limit of large chains may be difficult.

A remarkable result for the resonance shift in one-dimensional antiferromagnetic chains, being valid at arbitrary temperatures, was obtained in Maeda et al. (2005). It utilizes the exact nearest-neighbor correlation functions of the isotropic spin- Heisenberg chain. Here we present an alternative framework for the derivation of the resonance shift which, in the limit of small anisotropy, reproduces Maeda et al. (2005). In our approach the anisotropy is treated non-perturbatively, and it allows us to derive an exact formula for the line width ‘in frequency direction’ at fixed magnetic field, as well as a new explicit expression for the ESR-line width in the high temperature regime.

We consider an important example of anisotropic interactions described by the Heisenberg-Ising Hamiltonian

 H=J∑⟨ij⟩(sxisxj+syisyj+(1+δ)sziszj). (3)

Here the sum is over nearest neighbors, and is the anisotropy parameter. If , then (1), (2) imply that the normalized absorbed intensity is

 I(ω,h)=πδ(ω−h)hm(T,h). (4)

It is proportional to the magnetic energy per lattice site. This case includes the familiar paramagnetic resonance (Zeeman effect) for , for which the magnetization is .

For non-zero the function is unknown and hard to calculate. Still, some more elementary spectral characteristics, such as the position of the resonance or the line width, may be expressed in terms of certain static correlation functions that determine the moments of the normalized intensity function.

Let us assume for a while that our chain is large but finite. Then the spectrum is bounded and the integrals

 In=∫∞−∞dωωnI(ω,h) (5)

exist for all non-negative integers . Since is non-negative everywhere and since , we may interpret as a probability distribution and the as its moments. As we shall see, it is convenient to express the in terms of another closely related sequence of integrals

 mn(T,h)=J−n∫∞−∞dω2π(ω−h)nχ′′+−(ω,h) (6)

which, by slight abuse of language, will be called (shifted) moments. Again they exist for every finite chain.

By definition the shift of the resonance for fixed is

 δω=I1I0−h=JJm2+hm1Jm1+hm0. (7)

A measure for the line width is the mean square deviation

 Δω2=I2I0−I21I20=J2Jm3+hm2Jm1+hm0−δω2. (8)

Hence, in order to calculate the resonance shift and the line width, we need to know the first four shifted moments , , , of the dynamic susceptibility .

In the following we shall employ the notation for the adjoint action of an operator . Then , since and , and it follows with (1) and (6) that

 (9)

The latter formula shows that the moments are static correlation functions whose complexity grows with growing . The first few of them can be easily calculated. We shall show the results for the one-dimensional model, for which

 m0=12L⟨[S+,S−]⟩T=1L⟨Sz⟩T, (10)

which is the magnetization per lattice site. The subsequent moments are less intuitive,

 m1 =δ⟨s+1s−2−2sz1sz2⟩T, (11a) m2 =12δ2⟨sz1+4sz1sz2sz3−4sz1s+2s−3⟩T, (11b) m3 =14δ2⟨2s+1s+2s−3s−4+4s+1s−2s+3s−4−2s+1s−2s−3s+4 −8sz1sz2s+3s−4−4sz1s+2sz3s−4+8sz1s+2s−3sz4−4s+1s−2 −s+1s−3+8sz1sz2sz3sz4+2sz1sz3−4sz1sz2 +δ(8sz1s+2s−3sz4+2s+1s−2−8sz1sz2)⟩T. (11c)

They are certain combinations of static short-range correlation functions. This implies, in particular, that they all exist in the thermodynamic limit . It follows that the line shape for fixed cannot be strictly Lorentzian, as is often assumed in the literature.

All static correlation functions of the one-dimensional Heisenberg-Ising model are polynomials in the derivatives of three functions , and Jimbo et al. (2009) which, as is common in integrable models, can be expressed in terms of the solutions of certain well behaved linear and non-linear integral equations Boos and Göhmann (2009). This is the reason why in this case the moments can be calculated exactly by means of the techniques developed in Boos et al. (2007); *BDGKSW08; *TGK10a.

We parameterize the anisotropy as . Then, with the shorthand notations , , for , we obtain

 m0 =−12φ(0),m1=(q−1)2(q2+4q+1)ω′(0,1)16q2−(q3−1)ω(0,0)4q(q+1), m2 =(q−1)2256q4[4q(q+1)(q3−1)(ω(0,2)φ(0)−2ω(1,1)φ(0)−ω(0,0)φ(2)) +(q2−1)2(q2+4q+1)(ω′(1,2)φ(0)+ω′(0,1)φ(2))−16q2(q−1)2φ(0)], m3 =(q−1)498304q8(q4−1)(q6−1) [16q2(q2−1)3(q4−1)(q6−1)(q2+4q+1)(ω(0,2)ω′(0,1)+ω(0,0)ω′(1,2)) +64q2(q2−1)4(2q10−q9+4q8−4q7−12q6−14q5−12q4−4q3+4q2−q+2)ω(0,0)ω(1,1) −16q2(q2−1)2(q4−1)(q6−1)(3q4+14q2+3)ω′(0,3) +8q2(q2−1)(q4−1)2(q6−1)(q+1)2(8ω′(1,2)−ω′(2,3)) +192q4(q2−1)2(q4−1)(q6+18q4+8q3+18q2+1)ω(0,2) +64q2(q2−1)2(q4−1)(q+1)2(2q8−5q7+26q6−49q5+28q4−49q3+26q2−5q+2)ω(1,1) −16q2(q2−1)2(q4−1)(q+1)2(q8−q7+q6+q5+2q4+q3+q2−q+1)(2ω(1,3)−3ω(2,2)) +64q2(q4−1)(q6−1)(3q8+2q6+24q5−130q4+24q3+2q2+3)ω′(0,1) +(q4−1)(q6−1)(q+1)2(q10−2q9+25q8+16q7+118q6+164q5 +118q4+16q3+25q2−2q+1)(ω′(0,3)ω′(1,2)+ω′(0,1)ω′(2,3)) −1536q5(q4−1)(4q8−9q7−2q6−6q5+8q4−6q3−2q2−9q+4)ω(0,0) +4q2(q6−1)(q+1)2(q2+1)(5q8−2q7+32q6+50q5+70q4+50q3+32q2−2q+5) (2ω(1,3)ω′(0,1)−3ω(2,2)ω′(0,1)+ω(0,2)ω′(0,3)−2ω(1,1)ω′(0,3)−3ω(0,2)ω′(1,2)−ω(0,0)ω′(2,3)) −16q2(q+1)2(q16−q15+8q14+9q13+47q12+45q11+96q10+91q9+128q8+91q7+96q6 +45q5+47q4+9q3+8q2−q+1)(3ω2(0,2)−6ω(1,1)ω(0,2)+2ω(0,0)ω(1,3)−3ω(0,0)ω(2,2))]. (12)

These functions represent the moments in the thermodynamic limit. Since they can be calculated to arbitrary precision, we obtain numerically accurate results for the resonance shift and for the line width as functions of temperature or magnetic field over the whole range of the phase diagram. In particular, our approach is not restricted to small anisotropies. Examples for are shown in figures 1 and 2. We find a broadening of the line width as defined by (8) for small temperatures in the critical () as well as in the massive () regime (latter case not shown here).

At first sight this seems to contradict experimental results Krug von Nidda et al. (2010) which claim a narrowing. Still, one has to take into account that usually in the analysis of experimental data rather different definitions of the line width, as e.g. the distance between the turning points right and left to the maximum of the intensity, are used. In particular, if the intensity distribution has long shallow tails the definition (8) will give considerably larger values than the distance between the turning points. We have performed a detailed numerical study of the dynamical susceptibility for chains of 16, 20 and 24 lattice sites (to be published elsewhere) and we see indeed such long tails at low temperature (compare also El Shawish et al. (2010)). In experiments they may be misinterpreted as background stemming from couplings of the spin chain to other degrees of freedom, but in fact they are due to the spin-spin interactions and are part of the true absorption line. For the determination of the resonance shift tails are expected to have less influence. As long as they are not too much asymmetric the shift calculated by means of (7) should agree with the shift of the maximum of the absorbed intensity.

In current ESR experiments the microwave frequency is kept fixed and the Zeeman field is modulated. This means that, as opposed to most of the theoretical treatments, including our considerations above, the absorbed intensity is determined as a function of for fixed , and the resonance shift and line width are measured in ‘-direction’. Away from the isotropic point (), where is symmetric and the absorption line is extremely narrow, this may clearly lead to rather different values. To be closer to present-day ESR experiments one should calculate resonance shift and line width in terms of the moments of the dynamical susceptibility in ‘-direction’.

We define

 Mn(T,ω)=J−n∫∞−∞dh2π(h−ω)nχ′′+−(ω,h). (13)

For these functions we obtain the representation

 Mn(T,ω)=(−1)n∞∑k=0(−J)kk!m(k)k+n(T,ω), (14)

where the superscript denotes the th derivative with respect to the second argument. We see that the are determined by infinitely many of the and their derivatives, i.e., they depend on static correlation functions for arbitrarily large distances. For this reason they cannot be calculated by our exact method above. Yet, in certain cases only finitely many terms of the series are needed for a good approximation.

We first of all express the resonance shift and the mean square deviation from the center of the absorption peak in terms of the ,

 δhJ=M1M0,Δh2J2=M2M0−M21M20. (15)

We have identified two cases, where these formulae simplify and finitely many of the are enough to determine and approximately.

The equation for the resonance shift simplifies for small anisotropy . Since , and, generically, itself is of order (see (11), (14)) we obtain to linear order in

 δhJ=−m1m0. (16)

In Nagata and Tazuke (1972); Maeda et al. (2005) the same equation was obtained by a more intuitive reasoning. It leads to results which compare rather well to experiments Maeda et al. (2005). However, some care is necessary with the interpretation of (16). vanishes at . It follows that with some coefficients , whence must be large compared to for (16) to be applicable. Note that all higher moments are of order . Hence, there is no simplification for small anisotropy, like in (16), for the line width. But the integrated intensity has again a finite approximation to first order in , .

The representation (14) is a series in ascending powers of (with still temperature dependent coefficients). This can be used to evaluate (15) asymptotically for high temperatures. It turns out that that the leading terms in the expansion of and cancel each other: in the high-temperature limit , and

 ΔhJ=|δ|√2 (17)

for arbitrary microwave frequency . This formula provides a simple means to directly measure the anisotropy parameter .

It may be instructive to illustrate our formula with one of the few explicit results, namely with the formula for the intensity in the free fermion case at Brandt and Jacoby (1976). In this case and, in agreement with (17), we obtain the line width , whereas the width in omega direction depends on .

Our work is the first exact result for the resonance shift and the line width in microwave absorption experiments on the Heisenberg-Ising chain. The reduction to moments is not restricted to the integrable case and may be interesting for the two- and three-dimensional models as well. Our approach is unbiased. It makes no a priori assumptions about the shape of the spectral line. As opposed to all other approaches it is valid for all temperatures and magnetic fields and in addition for arbitrary values of . For small close to the isotropic point we recover the result of Maeda et al. (2005) for the line shift. The resonance shift or and the line width or defined in terms of moments show a simple scaling behavior. They depend on the exchange interaction only through the ratios and . In this sense the curves in figures 1 and 2 are universal.

The intensity is a function of and . With our definitions of and we determine the resonance shift and the line width in -direction as functions of , while in standard ESR experiments the resonance shift and line width in -direction are measured as functions of , which should be clearly distinguished. For the resonance shift it follows from (7), (16) that to linear order in . For the line width there is no such simple relation between and . However, for we obtained the simple high-temperature formula (17) which we suggest to be useful to measure the anisotropy directly. We are further convinced that it may be worth trying to measure , which is now known exactly, directly in multi-frequency ESR experiments.

###### Acknowledgements.
The authors would like to thank F. Anders, H. Bomsdorf, H. Boos, B. Lenz, K. Sakai, J. Stolze, A. Zvyagin, and, in particular, Y. Maeda for stimulating discussions.

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