Spin-12 periodic nonuniform XX chains and the spin-Peierls instability

Spin-12 periodic nonuniform XX chains and the spin-Peierls instability
Spin-12 periodic nonuniform XX chains and the spin-Peierls instability

a r X i v :c o n d -m a t /9908385v 1 26 A u g 1999

Vol.97(2000)ACTA PHYSICA POLONICA A No.1(2)

2

PERIODIC NONUNIFORM XX CHAINS AND THE SPIN-PEIERLS INSTABILITY

O.Derzhko a,b ,J.Richter c and O.Zaburannyi b

a

Institute for Condensed Matter Physics,1Svientsitskii St.,L’viv-11,290011,Ukraine

b

Chair of Theoretical Physics,Ivan Franko State University of L’viv,

12Drahomanov St.,L’viv-5,290005,Ukraine c

Institut f¨u r Theoretische Physik,Universit¨a t Magdeburg,

P.O.Box 4120,D-39016Magdeburg,Germany

(Version date:April 27,1999)

Using continued fractions we obtained the exact result for the den-sity of magnon states of the regularly alternating spin-1

2

XX

chain that was studied in several papers [2,3,4](note,however,that in the non-adiabatic limit such a spin chain does not permit exact analysis [5]).The aim of the present study is to examine the in?uence of an additional Dzyaloshinskii-Moriya coupling on the spin-Peierls dimerization.The presence of such a term for CuGeO 3was proposed in [6,7].The multisublattice spin-1

2

on a circle with the Hamiltonian

H =

n

?n s z n

+2

n

I n s x n s x

n +1+s y n s y

n +1

+2

n

D n s x n s y

n +1?s y n s x

n +1 .

(1)

(1)

2

After the Jordan-Wigner transformation one comes to tight-binding spinless fer-mions on a circle with complex hopping integrals.We introduce the temperature double-time one-fermion Green functions that yield the density of magnon states ρ(E)=?1

E±i???n???n??+n,

??n=

I2n?1+D2n?1

E±i???n?2?.

..

,

?+n=

I2n+D2n

E±i???n+2?.

..

.(2)

One immediately notes that for any?nite period of varying?n,I n,D n the contin-ued fractions??n,?+n involved into G?nn(2)become periodic and thus can be cal-culated exactly yielding the exact result for the density of states and hence for the thermodynamic quantities of spin model(1).For example,for the periodic chain having period2?1I1D1?2I2D2?1I1D1?2I2D2...the described scheme gives

ρ(E)= 0,if E≤b4,b3≤E≤b2,b1≤E,

1√

2(?1+?2)±b1,

1 2 2

2 ∞?∞dEρ(E)|E|=?2|I|2?ψ

I2+D2has been introduced, E(ψ,a2)≡ ψ0dφ

3 if2|I|≤|?0|,ψ=arcsin 4I2(1?δ2)if2δ|I|≤|?0|<2|I|,ψ=π?δ=0

may have a nonzero oneδ?=0at moderate and weak?elds(i.e.|?0|<2|I|). This nontrivialδ?comes from the equation that follows from(4)

πα

1?δ2 F(ψ,1?δ2)?E(ψ,1?δ2) (5) where F(ψ,a2)≡ ψ0dφ/

δ2+x2 (x=cosφ)that yieldsδ?~exp ?πα|I|.

One also notes that the obtained result coincides with the one reported in[2]

up to a renormalization of the e?ective interspin coupling|I|→|I|=

for which the taken value ofδ

|I|

realizes an extremum of E(δ).One immediately observes that for|?0|

versusδremains as that in the absence of the?eld,whereas for |I|

starts to decrease.From this one concludes that for 0≤δ<|?0|

|I|

hard lattices the?eld|?0|

|I| makes the dimerization unstable against the uniform phase.The latter relation tells us that the Dzyaloshinskii-Moriya interaction increases the value of that?eld.

It is generally known[1]that the increasing of the external?eld leads to a transition from the dimerized phase to the incommensurate phase rather than to the uniform phase.Evidently,the incommensurate phase cannot appear in the presented treatment within the frames of the adopted ansatz for the lattice distortionsδ1δ2δ1δ2...,δ1+δ2=0.To clarify a possibility of more complicated distortions the chains with longer periods should be examined.

Alternatively,we may also assume di?erent dependences onδfor the isotro-pic coupling and the Dzyaloshinskii-Moriya coupling,for example,|I1|=|I|(1+δ), |D1|=|D|,|I2|=|I|(1?δ),|D2|=|D|.Supposing thatδ?1after simple rescal-ing arguments one?nds that the dimerization parameterδ?~I2|I|I4

=exp ?πI4α decreases as well.

2|I|

To conclude,we have analysed a stability of the spin-1

4

the ratio of the Dzyaloshinskii-Moriya coupling to the isotropic coupling does not depend on the dimerization parameter we have found that the Dzyaloshinskii-Moriya interaction leads to an increasing of the e?ective interspin coupling and thus to some quantitative changes,i.e.to an increasing of the value of the dimer-ization parameter which characterizes the dimerized phase and the value of the ?eld which destroys the dimerized phase.In the other limiting case when the Dzyaloshinskii-Moriya coupling does not depend on the dimerization parameter it has an opposite e?ect leading to a decreasing of the value of the dimerization parameter and the value of the?eld which destroys dimerization.The obtained results are in agreement with some earlier studies of the thermodynamic proper-ties of spin-1

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