The band structure of double excited states for a linear chain

Physica B 291 (2000) 29}33

The band structure of double excited states for a linear chain R. Olchawa Institute of Physics, Opole University, ul. Oleska 48, 45-052 Opole, Poland Received 20 April 1999; received in revised form 25 October 1999; accepted 26 October 1999

Abstract The energy band structure in the case of double excited states of "nite spin systems (s") has been investigated.  A geometrical construction based on the Bethe Ansatz method for determining eigenstates has been proposed. The formula for energy spectrum in the center and at the border of Brillouin zone has been obtained. Classi"cation of energy bands has been elaborated on and approximated dispersion law for bounded states given. Some problems with application of the Bethe Ansatz in the case of "nite system has been pointed out.  2000 Elsevier Science B.V. All rights reserved. Keywords: Energy band structure; Spin systems; Double excited states

1. Introduction The model under consideration is the "nite onedimensional Heisenberg model for magnets de"ned by the Hamiltonian , (1) H"J [(S>S\ #S\S> )#eSXSX ] G G> G G>  G G> G with periodic boundary condition S "S ,>  [1}3,6]. Although the history of this model is quite long and lots of results were obtained using the Bethe Ansatz method, in particular, concerning properties of the double excited states [1,4,5], there still remain questions and doubts about the physical signi"cance of the results [7]. Exact analytical solution of eigenproblem for "nite Heisenberg

E-mail address: [email protected] (R. Olchawa).

Hamiltonian (1) is possible only in the case of one-magnon excitations (one-spin deviation) and this solution is the base for the spin-wave theory. The double excited states (two-spin deviation) are very interesting since they constitute the simplest non-trivial case where the interaction of magnons could be investigated. Because the Hamiltonian under consideration is invariant under translation group, it should be possible to describe energy spectrum in terms of band structure, in a similar way as in the case of electron states in crystal potential. To obtain such a picture it is necessary to investigate a form of the energy spectrum for N going to in"nity. As a starting point to approach the problem the Bethe Ansatz is used. Then, introducing some geometrical interpretations of the underlying equation makes it possible to obtain several general properties of the energy spectrum and energy bands. The presented method allows one to obtain exact energy spectrum for any N, but only the results for even N are discussed in detail.

0921-4526/00/$ - see front matter  2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 1 3 8 2 - 4

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R. Olchawa / Physica B 291 (2000) 29}33

where a new quantum number q can still be complex. Conditions (5) for the new quantum numbers take the form

2. The Bethe Ansatz in the case of two-spin deviations Application of Bethe's and HultheH n's method to the case of two-spin deviations leads to the assumption that all eigenvectors can be expressed in the following form [1,3]: "k , k 2" [e I P>I Q>(#e I P>I Q\(]"r, s2,   PQ (2) where r, s"1, 2,2, N are positions of spin deviations, is a phase parameter, and r(s is assumed for uniqueness. Quantum numbers k , k can be   complex, so they cannot be treated as quasimomentum of magnons, in general. Substitution of form (2) into the Heisenberg Hamiltonian (1) one obtains the following expression for energy levels: E"J(N!2#cos k #cos k ),    and the formula for the parameter as

(3)

Nq!2 "2pd,

(8)

where i and d are integers (either odd or even at the same time). It is easy to notice that for i"0,$1,$2,2,$(N/2!1), N/2, k is a vector lying in the "rst Brillouin zone (even N is assumed). Using linear transformation (6), relation (8) and Eq. (4), one can obtain an equation for the quantum number q as






Nq p sin q/2 ! d " , (1/e)cos k/2!cos q/2 22 2

cot

The periodic boundary conditions for period equal to N lead to additional conditions for k , k , and

  Nk ! "2pj ,   Nk # "2pj , (5)   where j , j are integers. In order to obtain the   explicit expression for allowed values of quantum numbers k , k and phase it is necessary to solve   Eq. (4) with conditions (5), which is not a trivial problem.

3. The energy spectrum in the center of the Brillouin zone for isotropic interaction Hamiltonian (1) is invariant under the action of the translation group, which induces a real quantum number k"k #k , the total quasimomen  tum of the magnon system. For this reason it is convenient to introduce a linear transformation (6)

(9)

whereas energy spectrum (3) expressed by the new quantum numbers can be rewritten in the form




(4)

q"k !k ,  

(7)

E"J

sin(k !k )/2   cot " . 2 (1/e)cos(k #k )/2!cos(k !k )/2    

k"k #k ,  

2pi k" , N



1 k q N!2#2 cos cos . 2 2 4

(10)

Eq. (9) can be split into two parts corresponding to even and odd quantum numbers d and i as





cos

N q 1 k Nq !1 " cos cos , d, i-even, 2 22 2 2 e

(11)

sin

N q 1 k Nq !1 " cos sin , d, i-odd. 2 22 2 2 e

(12)

As one can see, the parameter d disappeared from these equations, which means that the original parameters j , j are not unique and they, in fact,   can be replaced by one parameter i"j #j . In   the center of the Brillouin zone (k"0) and for the anisotropy parameter e"1 Eq. (11) simpli"es to




cos



N q Nq !1 "cos 2 2 22

(13)

and could be solved exactly. For further discussion it is very useful to consider a geometrical interpretation of this equation. Solutions for q/2 could be obtained geometrically as shown in Fig. 1. The desirable values of q/2 are here equal to the length of horizontal segments connecting appropriate points on the plot of the cosine function. All nonequivalent solutions could be obtained, if only

R. Olchawa / Physica B 291 (2000) 29}33

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Fig. 1. Geometrical solution of the equation cos((N/2)!1)q/2"cos(N/2)q/2. Values of q/2 are equal to the length of the horizontal segments.

Fig. 2. Geometrical solution of the equation cos((N/2)!1)q/2"p cos(N/2)q/2, where p"(1/e)cos k/2. Values of q/2 are given by projections of the inclined segments on the horizontal axis and a/b"p.

segments with length q/2(p are considered. This, in turn, means that these segments must join points in the same half-period of cosine function. It follows from the geometrical constructions that values of q/2 are given by the expression

more inclined when k goes towards the Brillouin zone border (k"p), where p"0. Values of q/2, which correspond to the border of the Brillouin zone, can be given explicitly as

q 2pa N " , a"0, 1,2, !1. 2 2 N!1

(14)

q (2a!1)p " , 2 N!2

N a"1, 2,2, !1. 2

(17)

(16)

For N going to in"nity the operation of inclining lines becomes a continuous operation and for e"1 the values of q/2 change continuously from values corresponding to the center of the Brillouin zone given by Eq. (14) to the values corresponding to the border given by Eq. (17). Relation (10) allows us to interpret the parameter a as the energy band index. It is very important to note that such a construction is not possible for a"0 and kO0. This special case is considered in the next section. For e'1 one starts with inclined lines in the center of the Brillouin zone and ends with q/2 given by Eq. (17) at the border. This means that for e going to in"nity (the Ising model) the spectrum for q/2 over the Brillouin zone becomes more and more compressed towards values corresponding to the border of the Brillouin zone. Energy (10) at the border of the Brillouin zone is strongly degenerated and does not depend on either band index a or anisotropy parameter e

This idea is presented in Fig. 2. One can conclude from this picture that in the case of isotropy (e"1) the originally horizontal lines corresponding to the center of the Brillouin zone Fig. 1 become more and

E "J(N!2) (18) Ip?$  and corresponds to energy in the Ising model for the case when spin deviations are not the nearest neighbors [1].

The newly introduced quantum number a labels solutions which originate from di!erent half-periods of cosine function. According to formula (10) the energy spectrum of the isotropic magnet in the center of the Brillouin zone takes the form






1 2pa E "J N!2#2 cos . ? N!1 4

(15)

4. The energy bands The geometrical construction changes slightly if kO0 or eO1, as it follows from Eq. (11). Solutions for q/2 could be obtained in a similar way as in the case considered in the preceding section, but the horizontal lines should be replaced by inclined ones. The incline angle depends on the product 1 k p" cos . e 2

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R. Olchawa / Physica B 291 (2000) 29}33

In the case of odd quantum numbers i, Eq. (12) leads to a similar geometrical interpretation and similar constructions, which are based, however, on the sine function. Although solutions which correspond to exactly horizontal lines, given by the formula N q 2pb#p " , b"0, 1,2, !2 N!1 2 2

(19)

are not allowed here, they constitute the limit case for k going to the center of the Brillouin zone if the isotropic model is considered. Like the parameter a in Eq. (14), parameter b could be interpreted as a label of an energy band. For b"1 presented here a geometrical construction is possible only if the following condition is ful"lled: k N!2 cos ' . 2 N

(20)

For anisotropy parameter eP0 (which corresponds to the isotropic "nite XY model) the limit values of q/2 do not depend on the exact value of the parameter i, but are di!erent for even and odd ones q (2a!1)p " , 2 N

N a"1, 2,2, !1, i-even, 2

q (2b!2)p " , N 2

b"1, 2,2,

N !1, 2

i-odd.

5. The bounded states Real solutions outside the center of the Brillouin zone could be obtained only for bands with aO0 and bO1. For kO0 and a"0 the geometrical construction that ful"lls Eq. (11) is possible only if instead of the cosine function we use the hyperbolic one, which obviously corresponds to an imaginary value of q/2. Fig. 3 shows an example of such a construction. From this construction and Eq. (11) one can conclude that if k goes to the border of Brillouin zone q/2 goes to in"nity, so there is no "nite solution for band a"0 at the border of Brillouin zone! The energy value in the singular point could be determined as the limit value for kPp and takes the form E "J(N!1), Ip? 

(21)

Fig. 3. Geometrical solution of the equation cosh((N/2!1)q/2"p cosh(N/2)q/2, where p"(1/e)cos k/2. Values of q/2 are given by projections of the inclined segments on the horizontal axis, a/b"p. For kPp, which corresponds to pP0, solutions for q/2 go to in"nity.

which is the energy for the Ising model for the case when spin deviations are the nearest neighbors [1]. Similarly, solutions for band b"1 when condition (20) is not ful"lled could be obtained using the hyperbolic sine function. Because values of q/2 are imaginary then Bethe's quantum numbers k , k   are complex and cannot be interpreted as quasimomenta of quasiparticles as in the case of real solutions. So, only the total quasimomentum of the system can be used in description of these states. For this reason, the states are interpreted as bounded states. Both hyperbolic cosine and sine functions for arguments which are big enough could be approximated by the exponent function, so Eqs. (11) and (12) can be written as




exp



N q 1 k Nq !1 " cos exp . 2 2 e 2 22

The solution of this equation leads to the approximated dispersion law for bounded states N!2#2 cos k/2 E

. 4

(22)

6. An example In order to show what the band structure looks like, the energy spectrum for small system (N"24) is presented in Fig. 4. The results are obviously the same as those obtained from the diagonalization of Hamiltonian (1). Each energy band was obtained

R. Olchawa / Physica B 291 (2000) 29}33

Fig. 4. The energy spectrum of the double excited states for ferromagnet consisting of 24 nodes. States for even k-vectors are connected by dashed lines. The solid line presents approximated values for bounded states given by formula (22). The encircled state corresponds to in"nite solution for q/2 (see Fig. (3)).

using the presented method. Energy values in the center of the Brillouin zone are given by formula (15). As one could expect, bands for even and odd quantum numbers i are shifted mutually. The lowest-energy band contains the bounded states. The state at the border of Brillouin zone enclosed by circle does not have "nite solution in Bethe's form. For comparison, the approximated values for bounded states given by formula (22) are put on the plot too.

7. Conclusions The above-presented approach to the Heisenberg model of magnet in the case of the double excited states, originates from the Bethe Ansatz method. Applying simple linear transformation it was possible to introduce some geometrical construction for determination of acceptable values of quantum numbers k , k and phase in expres  sions for state vectors and energy levels (2,3). Especially, the exact expression for energy spectrum in the center of the Brillouin zone (15) was obtained. The geometrical construction turned out to be very useful for investigation of the limit behavior of the energy spectrum and allowed arranging it into bands. In fact, the obtained bands are doubly rare"ed [8}10] and contain only either even or odd

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vectors from "rst Brillouin zone (i.e. vectors for even or odd i in expression (7)). The obtained bands could be divided into two categories corresponding to real and complex Bethe's quantum numbers k , k and having di!erent energy limit at   the border of the Brillouin zone (21). The states with complex k , k correspond to bounded states   and their energy spectrum can be approximated by expression (22). The application of the Bethe Ansatz method presented here shows some weaknesses of the method in the case of "nite systems. As it turns out, quantum numbers j , j in Eq. (5) do not label states   uniquely and they are not su$cient; in order to obtain the full spectrum it is necessary to introduce additional quantum numbers a and b(14,19). Moreover, among imaginary solutions which correspond to the bounded states, there is one state that cannot be expressed in the "nite form of Bethe's substitution (2). This state lies on the border of the Brillouin zone and leads to in"nite values for k , k .   Acknowledgements I am very grateful to Professor W.J. Caspers and Professor T. Lulek for inspiring discussion and useful suggestions on this topic. References [1] W.J. Caspers, Spin Systems, World Scienti"c, Singapore, 1989. [2] H.A. Bethe, Z. Phys. 71 (1931) 205. [3] L. Hulthen, Arkiv. Nat. Astron. Fys. 26A (1938) 1. [4] R. Orbach, Phys Rev. 112 (1958) 309. [5] C.J. Thompson, in: C. Domb, M.S. Green (Eds.), Phase Transitions and Critical Phenomena I, Academic Press, London, 1972. [6] W.J. Caspers, G.I. Titlen, in: T. Lulek, W. Florek, B. Lulek (Eds.), Symmetry and Structural Properties of Condensed Matter, World Scienti"c, Singapore, 1997. [7] L.D. Faddeev, L.A. Takhtadzyan, Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta 109 (1981) 134. [8] T. Lulek, J. Phys. 45 (1984) 29. [9] W. Florek, T. Lulek, J. Phys. 20 (1987) 1921. [10] B. Lulek, J. Phys.: Condens. Matter 4 (1992) 8737.