SPIN-WAVE THEORY FOR S= ANTIFERROMAGNETIC ISOTROPIC CHAIN D. V. Spiin V.I. Venadsii Tauida National Univesity, Yaltinsaya st. 4, Simfeopol, 957, Cimea, Uaine E-mail: spiin@cimea.edu, spiin@tnu.cimea.ua In the pape we descibe the modification of spin-wave theoy fo one-dimensional isotopic antifeomagnet. This theoy enables to obtain the enegy of magnetic excitations of shot wave length and coelation function in ageement with numeical studies. The Heisenbeg isotopic antifeomagnetic (AF) one-dimensional (d) S= model has been well studied. Haldane conjectued [] that excitations in such a system possess a gap while at any tempeatue thee is no long-ange magnetic ode. This statement was confimed by numeical investigations: exact diagonalization [], Monte-Calo methods [3-4], eal-space enomalization goup [5], Lanczos technique [6]. Geat pogess was achieved with use of nonlinea sigma model [7]. Modified spin-wave theoy (MSWT) fo d S= AF chain was fomulated by Taahashi, Hisch and Tang [8]. Late Rezende [9] extended this theoy taing into account Oguchi coection. Howeve, thei esults diffe substantially fom numeical estimations. The diffeences ae: i) The spectum obtained in the famewos of MSWT has a gap.j fo =,, the wave vecto, while the lowe gap obtained by Nightingale and Blöte [] is = 4J (this value is confimed late =. by numeous studies); ii) MSWT spectum is symmetic with espect to point while numeical methods give asymmetical spectum with gap = =. = = Poviding a simple pictue of a phenomena MSWT cannot pedict impotant popeties of AF S= chain. We popose modification of MSWT. We believe that in such a fomulation it descibes the spectum of excitations of d AF isotopic chain nea the point =.
Conside isotopic AF chain with Hamiltonian: H = S nsn, () n whee summation is taen ove neaest neighbos; spin of a magnetic ion is unitay: S=. Let us setch the MSWT shotly. Using Holstein-Pimaoff tansfomation fo antifeomagnet one can obtain Hamiltonian: ( H SW = an an an an anan ). () n We do not tae into account the Oguchi coection. In the pape [9] it was used Dyson-Maleev epesentation, howeve, since Holstein Pimaoff tansfomation gives simila esults, it does not matte which epesentation is applied. To satisfy the Memin-Wagne theoem [], one should add such condition: N n S z n = a a d =, (3) which is simply the estiction that in aveage the magnetic moment of one sublattice is equal to zeo. Due to the condition (3) the chemical potential aises in Hamiltonian (). It s value detemines a gap in excitation enegy. with One can find the spectum of magnons: µ ( µ cos )( cos) ε = µ, (4) the chemical potential. Eq. (4) is in ageement with that one obtained by Rezende [9] (without Oguchi coection). Solving Eq. (3) numeically we find the gap value: = = =. 7. The spectum (4) is plotted at Fig., a, (solid line); open cicles coespond to esults obtained in Ref. [4]. We can also find coelation functions: a a d, odd S S =. (5) a a d, even
To obtain physical behavio of coelation function we tae into account the foupaticle tem emeging fom z S S z, so that z z S S a a a a a a, (6) due to Eq. (3). Function (5) is plotted in Fig., b (solid squaes). The coelations decay vey slowly in compaison with the function found in numeical investigations []: S S const exp. (7) 6. Second line at Fig., b (solid cicles) depicts the fit of Eq.(5) by function (7). One can conclude that MSWT undeestimates the ole of quantum fluctuations of the system unde study. Because, in fact, MSWT consists in desciption of these fluctuations as waves that peseve condition (3), the slowe decay of coelation function (5) with distance means that MSWT accounts only the waves of sufficiently lage length. Let us conside the spin wave of shotest wavelength λ =. This wave may be conceived as altenating (,, ) spin pojections on the z-axis fo same sublattice. To intoduce such a wave by hands one may use condition: x y z x y z S S, S = ( S, S, S ), S = (, S, S ), (8) S which should eep in an aveage. Eq. (8) means that spins of one sublattice should stongly coelate antifeomagnetically, if the wavelength is small. Using Holstein- Pimaoff epesentation we have: H ( ) a a ( a a a a ) η( µ n n n n n n an an anan ), (9) SW = n whee we intoduced second Lagange multiplie η to satisfy Eq. (8). The magnon enegy should be found fom Eq. (9) using conditions (3) and (8); Eq. (8) assumes the fom: a a d =. ()
The simila esults may be obtained if one uses instead of estiction (8) the following: x y z S S, = ( S, S, S ) 3 x y z S, (, S S ) 3 = S 3 3, 3 S. () The esult fo excitation spectum is pesented at Fig., a. The cosses depict a numeical esult by Taahashi (see Ref. []), the solid line is ou spectum. Fo lying quite nea to the ageement is vey good. We obtain a gap 37 J, Q =. and asymmetical spectum. Suely, we cannot descibe the enegy of excitations fo abitay, because the condition () wos only nea =. The coelation function can be easily found. We calculate coelations fom Eq.(5). Fo odd we tae the fist line, whee the chemical potential η is found fom condition (8), while fo even the second line in Eq. (5) is chosen and potential is calculated fom Eq.(). At Fig., b this function is shown (solid tiangles). Now it can be fitted by Eq. (7) vey well (the fit by (7) almost coincides with obtained coelation function). Fo compaison we epoduce the esult of MSWT fom Fig., a (solid squaes). One can conclude that without conditions lie (8), () MSWT fails to get esults in quantitative ageement with numeical ones. Howeve, using the spin-wave language it is possible to descibe quantitatively popeties of d AF chain. Fo example, fo lage wavelengths one may suppose that spins of one sublattice coelate slightly feomagnetically at lage distance. This enables to estimate the gap value =. ACKNOWLEGEMENTS Autho thans Ministy of Uaine. I also acnowledge the financial suppot of Ministy of Education and Science of Uaine (gant 35/3).
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Figue captions. Fig.. a excitation spectum obtained with MSWT (solid line) and numeical methods (open cicles; Ref. 4). b coelation function calculated with MSWT (solid cicles); solid squaes exponential fit (see Eq. (7)). Fig.. a excitation spectum obtained with pesented vesion of MSWT (solid line); cosses esults of Taahashi (Ref. []). b coelation functions. Solid cicles the same as at Fig., b; solid tiangles ou esults, which can be fit with Eq. (7) vey well.
4 4 3 enegy en( x) Fig.. a.5.5.5 3 x h := C <SS> - - Fig.. b -3 5 5 distance
4 3 enegy en( x) Fig.. a.5.5.5 3 x <SS> - - -3 5 5 distance Fig.. b