Spin-Peierls instability in the spin-1 transverse XX chain 2 with Dzyaloshinskii-Moriya interaction 0 O. Derzhkoa,b, J. Richterc and O. Zaburannyia 0 0 aInstitute for Condensed Matter Physics, 2 1 Svientsitskii St., L’viv-11, 290011, Ukraine n a bChair of Theoretical Physics, Ivan Franko State University of L’viv, J 12 Drahomanov St., L’viv-5, 290005, Ukraine 3 cInstitut fu¨r Theoretische Physik, Universita¨t Magdeburg, 1 v P.O. Box 4120, D-39016 Magdeburg, Germany 4 1 February 1, 2008 0 1 0 0 Abstract 0 / We calculate exactly the density of magnon states of the regularly alternating t a spin-1 XX chain with Dzyaloshinskii-Moriya interaction. The obtained result per- m 2 mitustoexaminethestabilityofthechainwithrespecttospin-Peierlsdimerization. - We found that depending on the dependences of Dzyaloshinskii-Moriya interaction d n on distortion amplitude it may act either in favour of the dimerization or against o the dimerization. c : v Xi PACS numbers: 75.10.-b r a Keywords: spin-1 XY chain, Dzyaloshinskii-Moriya interaction, spin-Peierls dimeriza- 2 tion Postal addresses: Dr. Oleg Derzhko (corresponding author) Oles’ Zaburannyi Institute for Condensed Matter Physics 1 Svientsitskii Street, L’viv-11, 290011, Ukraine tel/fax: (0322) 76 19 78 email: [email protected] Prof. Johannes Richter Institut fu¨r Theoretische Physik, Universit¨at Magdeburg P.O. Box 4120, D-39016 Magdeburg, Germany tel: (0049) 391 671 8841 fax: (0049) 391 671 1217 email: [email protected] The discovery of the inorganic spin-Peierls compound CuGeO [1, 2] renewed an in- 3 terest in the investigation of spin-Peierls instability of quantum spin chains. Up to date a large number of papers concerning the quantum spin chains which are believed to model appropriately the spin degrees of freedom of the spin-Peierls compounds has appeared. As a rule the considered models are rather complicated and have been examined with exploiting different approximations. On the other hand, some generic features of the spin-Peierls systems can be illustrated in the simplified but exactly solvable models. As anexample ofsuch modelone canrefer tothespin-1 XX chainthat was studied inseveral 2 papers [3, 4, 5]. The purpose of the present paper is to examine the influence of intro- ducing Dzyaloshinskii-Moriya interspin coupling on the spin-Peierls dimerization within the frames of the one-dimensional spin-1 XX model in a transverse field. The presence 2 of Dzyaloshinskii-Moriya interaction for CuGeO was proposed in Refs. [6, 7, 8] in order 3 to explain the EPR and ESR experimental data. On the other hand, the multisublattice spin-1 XX chain with Dzyaloshinskii-Moriya interaction was introduced in [9], however, 2 that paper was not devoted to the study of spin-Peierls instability. In our paper we closely follow the idea of Ref. [3] and compare the total ground state energy of the dimerized and uniform chains. In contrast to previous works [3, 4, 5, 9] we use the continued-fraction representation for one-fermion Green functions [10, 11, 12, 13] that allows one to con- sider in a similar way not only the dimerized lattice but also more complicated lattice distortions. Based on the performed calculations we found that Dzyaloshinskii-Moriya interaction may act both in favour of the dimerization and against the dimerization. The result of its influence depends on the dependence of Dzyaloshinskii-Moriya interaction on the distortion amplitude in comparison with such a dependence of the isotropic exchange interaction. We consider N spins 1 on a circle with the Hamiltonian → ∞ 2 N H = 2 I sxsx +sysy n n n+1 n n+1 nX=1 (cid:16) (cid:17) N +2 D sxsy sysx n n n+1 − n n+1 nX=1 (cid:16) (cid:17) N + Ω sz. (1) n n n=1 X Here I and D are the isotropic exchange coupling and Dzyaloshinskii-Moriya coupling n n between the neighbouring sites n and n+1 and Ω is the transverse field at site n. We n restrictedourselvestotheHamiltonian(1)sinceaftertheJordan-Wignertransformationit reducestotight-bindingspinlessfermions. Weintroducethetemperaturedouble-timeone- fermion Green functions that yield the density of magnon states by the relation ρ(E) = 1 N ImG∓ , G∓ G∓ (E iǫ). In the case of the tight-binding fermions the ∓πN n=1 nn nm ≡ nm ± P required diagonal Green functions can be expressed by means of continued fractions [10, 11, 12, 13] 1 G∓ = , (2) nn E iǫ Ω ∆− ∆+ ± − n − n − n I2 +D2 ∆− = n−1 n−1 , n E iǫ Ω In2−2+Dn2−2 ± − n−1 − E±iǫ−Ωn−2−... I2 +D2 ∆+ = n n . n E iǫ Ω In2+1+Dn2+1 ± − n+1 − E±iǫ−Ωn+2−... For any periodic configuration of the intersite couplings and transverse field the fractions ∆− and ∆+ involved into G∓ (2) become finite and can be evaluated exactly yielding n n nn obviously the exact result for the density of states ρ(E) and hence for the thermodynamic quantities of spin model (1). In what follows we shall use the result for the periodic chain having period 2 that is characterized by a sequence I D Ω I D Ω I D Ω I D Ω ... . For such a case we have 1 1 1 2 2 2 1 1 1 2 2 2 0, if E b , b E b , b E, 4 3 2 1 ≤ ≤ ≤ ≤ ρ(E) =  (3)  21π|2E√−ΩB(1E−)Ω2|, if b4 < E < b3, b2 < E < b1,  (E) = 4 2 2 (E Ω )(E Ω ) 2 2 2 B I1I2 − − 1 − 2 −I1 −I2 h i = (E b )(E b )(E b )(E b ), 4 3 2 1 − − − − − 1 1 b b b b = (Ω +Ω ) b , (Ω +Ω ) b , 4 3 2 1 1 2 1 1 2 2 { ≤ ≤ ≤ } 2 ± 2 ± (cid:26) (cid:27) 1 b = (Ω Ω )2 +4( + )2, 1 1 2 1 2 2 − |I | |I | q 1 b = (Ω Ω )2 +4( )2, 2 1 2 1 2 2 − |I |−|I | q 2 = I2 +D2. In n n Letusexaminetheinstabilityoftheconsideredspinchainwithrespecttodimerization. To do this we assume I = I (1 + δ), I = I (1 δ), D = D (1 + kδ), D = 1 2 1 2 | | | | | | | | − | | | | | | D (1 kδ), where 0 δ 1 is the dimerization parameter. Putting k = 0 one has a | | − ≤ ≤ chaininwhichDzyaloshinskii-Moriya interactiondoesnotdependonthelatticedistortion, whereas for k = 1 the dependence of Dzyaloshinskii-Moriya interaction on the lattice distortion is as that for the isotropic exchange interaction. We consider a case of zero temperature and look for the total energy per site (δ) that consists of the magnetic part E e (δ) and the elastic part αδ2. From (3) one finds that 0 1 ∞ e (δ) = dEρ(E) E 0 −2 | | Z−∞ 1 b2 b2 1 1 ψ = b E ψ, 1 − 2 Ω +Ω , (4) −π 1 b21 !− 2 | 1 2| 2 − π! 0 if b 1 Ω +Ω , 1 ≤ 2| 1 2|  ψ =  arcsinrb21−14b(21Ω−1b+22Ω2)2 if b2 ≤ 12|Ω1 +Ω2| < b1, π if 1 Ω +Ω < b , 1  2 2| 1 2| 2 2 b = (Ω Ω )2 +4 I2(1+δ)2 +D2(1+kδ)2 + I2(1 δ)2 +D2(1 kδ)2 , 1 1 2 2s − − − (cid:20)q q (cid:21) 1 2 b = (Ω Ω )2 +4 I2(1+δ)2 +D2(1+kδ)2 I2(1 δ)2 +D2(1 kδ)2 , 2 1 2 2s − − − − (cid:20)q q (cid:21) and E(ψ,a2) ψdφ 1 a2sin2φ is the elliptic integral of the second kind. We also ≡ 0 − q seek for a nonzerRo solution δ⋆ = 0 of the equation ∂E(δ) = 0. Using (4) one gets 6 ∂δ 1 b2 b2 ∂b E ψ, 1 − 2 1 − π b21 ! ∂δ 1 b2∂b1 b b ∂b2 b2 b2 b2 b2 2 ∂δ − 1 2 ∂δ E ψ, 1 − 2 F ψ, 1 − 2 −π b21 −b22 " b21 !− b21 !# +2αδ = 0, (5) ∂b 1 1,2 = I2(1+δ)2 +D2(1+kδ)2 I2(1 δ)2 +D2(1 kδ)2 ∂δ b ± − − 1,2 (cid:20)q q (cid:21) I2(1+δ)+kD2(1+kδ) I2(1 δ)+kD2(1 kδ) − − , × I2(1+δ)2 +D2(1+kδ)2 ∓ I2(1 δ)2 +D2(1 kδ)2 − − q q  and F(ψ,a2) ψdφ/ 1 a2sin2φ is the elliptic integral of the first kind. ≡ 0 − q R Until the end of the paper we shall consider a case of the uniform transverse field Ω = Ω = Ω . In the interesting for application limit δ 1 valid for hard lattices having 1 2 0 ≪ large values of α one finds b = 2 , b = 2 δ with = √I2 +D2 and = I2+kD2 1 |I| 2 |I|ℵ |I| ℵ I2+D2 and instead of Eqs. (4), (5) one has 2 1 ψ e (δ) = |I|E(ψ,1 2δ2) Ω , (6) 0 0 − π −ℵ −| | 2 − π! 0 if 2 < Ω , 0 |I| | |  ψ =  arcsinr4I42I(12−−ℵΩ220δ2) if 2|I|ℵδ ≤ |Ω0| < 2|I|, π if Ω < 2 δ; πα 2 2 | 0| |I|ℵ = ℵ (F(ψ,1 2δ2) E(ψ,1 2δ2)) (7) 1 2δ2 −ℵ − −ℵ |I| −ℵ Consider the case Ω = 0. After rescaling I, α α, δ δ one finds that Eq. 0 I → ℵ2 → ℵ → (7) is exactly as that considered in Ref. [3] and thus δ⋆ 1 exp 1 πα . Thus for k = 1 ∼ ℵ −ℵ2 |I| (cid:16) (cid:17) when = 1 Dzyaloshinskii-Moriya interaction leads to increasing of the dimerization ℵ parameter δ⋆, whereas for k = 0 when 1 Dzyaloshinskii-Moriya interaction leads to ℵ ≤ decreasing of the dimerization parameter δ⋆. Consider further the case 0 < Ω < 2 . Varying δ in the r.h.s. of Eq. (7) from 0 to 0 | | |I| 1 one calculates a lattice parameter α for which the taken value of δ realizes an extremum of (δ). One immediately observes that for |Ω0| δ the dependence α versus δ remains E 2|I| ≤ ℵ as that in the absence of the field, whereas for 0 δ < |Ω0| the calculated quantity ≤ ℵ 2|I| α starts to decrease. From this one concludes that the field |Ω0| = exp 1 πα makes 2|I| −ℵ2 |I| (cid:16) (cid:17) the dimerization unstable against the uniform phase. The latter relation tells us that the Dzyaloshinskii-Moriya interaction increases the value of that field for k = 1 and decreases it for k = 0. In Figs. 1, 2 we presented the changes of the total energy (δ) (0) (4) vs the E − E dimerization parameter δ with switching on Dzyaloshinskii-Moriya interaction and the nonzero solution δ⋆ of Eq. (5) vs α with switching on Dzyaloshinskii-Moriya interaction, respectively. These results agree with the above ones valid in the limit δ 1. ≪ Let us emphasize that one cannot treat rigorously within the Jordan-Wigner picture complete Dzyaloshinskii-Moriya interaction that for neighbouring sites n and n+1 reads x(sysz szsy )+ y(szsx sxsz )+ z(sxsy sysx ) except the latter term Dn n n+1− n n+1 Dn n n+1− n n+1 Dn n n+1− n n+1 being included in (1). The effects of x, y may be examined numerically performing D D finite-chain calculations. To conclude, we have analysed a stability of the spin-1 transverse XX chain with 2 respect to dimerization in the presence of the Dzyaloshinskii-Moriya interaction calcu- lating for this purpose with the help of continued fractions the ground state energy for an arbitrary value of the dimerization parameter. Depending on the dependence of Dzyalosahinskii-Moriya interaction on the amplitude of lattice distortion it acts either in favour of dimerization or against it. It is generally known [2] that the increasing of the external field 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 distortions δ δ δ δ ... , δ +δ = 0. To 1 2 1 2 1 2 clarify a possibility of more complicated distortions the chains with longer periods should be examined. The present study was partly supported by the DFG (projects 436 UKR 17/20/98 and Ri 615/6-1). O. D. acknowledges the kind hospitality of the Magdeburg University in the spring of 1999 when a part of the paper was done. He is also indebted to Mrs. Olga Syska for continuous financial support. References [1] M. Hase, I. Terasaki, and K. Uchinokura, Phys. Rev. Lett. 70, 3651 (1993). [2] For a review see: J. P. Boucher and L. P. Regnault, J. Phys. I France 6, 1939 (1996). [3] P. Pincus, Solid State Commun. 9, 1971 (1971). [4] J. H. Taylor and G. Mu¨ller, Physica A 130, 1 (1985) (and references therein). [5] K. Okamoto and K. Yasumura, J. Phys. Soc. Jpn. 59, 993 (1990) (and references therein); K. Okamoto, J. Phys. Soc. Jpn. 59, 4286 (1990); K. Okamoto, Solid State Commun. 83, 1039 (1992). [6] I. Yamada, M. Nishi, and J. Akimitsu, J. Phys.: Condens. Matter 8, 2625 (1996). c [7] V. N. Glazkov, A. I. Smirnov, O. A. Petrenko, D. M K. Paul, A. G. Vetkin, and R. M. Eremina, J. Phys.: Condens. Matter 10, 7879 (1998). [8] H. Nojiri, H. Ohta, S. Okubo, O. Fujita, J. Akimitsu, and M. Motokawa, cond- mat/9906074. [9] A. A. Zvyagin, Phys. Lett. A 158, 333 (1991). [10] S. W. Lovesey, J. Phys. C 21, 2805 (1988). [11] J. K. Freericks and L. M. Falicov, Phys. Rev. B 41, 2163 (1990). [12] R. L yz˙wa, Physica A 192, 231 (1993). [13] O. Derzhko, cond-mat/9809018. FIGURE1. Dependence (δ) (0)vs δ forthespin-1 XX chain with Dzyaloshinskii- E −E 2 Moriya interaction. I = 1, Ω = 0, α = 0.8, a: k = 1 ( D = 0, 0.2, 0.4, 0.6, 0.8, 1 0 | | | | | | from top to bottom), b: k = 0 ( D = 0, 0.2, 0.4, 0.6, 0.8, 1 from bottom to top), c: | | D = 0.5 (k = 1, 0.8, 0.6, 0.4, 0.2, 0 from bottom to top). | | FIGURE2. Dependenceδ⋆ vsαforthespin-1 transverseXX chainwithDzyaloshinskii- 2 Moriya interaction. I = 1, α = 0.8, Ω = 0 (a, d, g), Ω = 0.1 (b, e, h), Ω = 0.3 0 0 0 | | (c, f, i), k = 1 (a, b, c) ( D = 0, 0.2, 0.4, 0.6, 0.8, 1 from left to right), k = 0 | | (d, e, f) ( D = 0, 0.2, 0.4, 0.6, 0.8, 1 from right to left), D = 0.5 (g, h, i) | | | | (k = 1, 0.8, 0.6, 0.4, 0.2, 0 from right to left). (δ)− (0) E E 0.00 -0.01 a 0.01 0.00 b 0.01 0.00 c δ 0.0 0.2 Figure 1: Dependence (δ) (0) vs δ for the spin-1 XX chain with Dzyaloshinskii- E − E 2 Moriya interaction. I = 1, Ω = 0, α = 0.8, a: k = 1 ( D = 0, 0.2, 0.4, 0.6, 0.8, 1 0 | | | | | | from top to bottom), b: k = 0 ( D = 0, 0.2, 0.4, 0.6, 0.8, 1 from bottom to top), c: | | D = 0.5 (k = 1, 0.8, 0.6, 0.4, 0.2, 0 from bottom to top). | | δ* 0.7 0.3 a d g 0.7 0.3 b e h 0.7 0.3 c f i 0.5 1.0 1.5 0.5 1.0 1.5 0.5 1.0 1.5 α Figure 2: Dependence δ⋆ vs α for the spin-1 transverse XX chain with Dzyaloshinskii- 2 Moriya interaction. I = 1, α = 0.8, Ω = 0 (a, d, g), Ω = 0.1 (b, e, h), Ω = 0.3 (c, f, 0 0 0 | | i), k = 1 (a, b, c) ( D = 0, 0.2, 0.4, 0.6, 0.8, 1 from left to right), k = 0 (d, e, f) ( D = | | | | 0, 0.2, 0.4, 0.6, 0.8, 1 from right to left), D = 0.5 (g, h, i) (k = 1, 0.8, 0.6, 0.4, 0.2, 0 | | from right to left).