There is life in the old horse yet or what else we can learn studying spin-1/2 XY chains

We review some recent results on statistical mechanics of the one-dimensional spin-1/2 XY systems paying special attention to the dynamic and thermodynamic properties of the models with Dzyaloshinskii-Moriya interaction, correlated disorder, and regularly alternating Hamiltonian parameters.

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Дата:	2002
Автор:	Derzhko, O.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут фізики конденсованих систем НАН України 2002
Назва видання:	Condensed Matter Physics
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/120686
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Цитувати:	There is life in the old horse yet or what else we can learn studying spin-1/2 XY chains / O. Derzhko // Condensed Matter Physics. — 2002. — Т. 5, № 4(32). — С. 729-749. — Бібліогр.: 51 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine

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spelling	irk-123456789-1206862017-06-13T03:03:17Z There is life in the old horse yet or what else we can learn studying spin-1/2 XY chains Derzhko, O. We review some recent results on statistical mechanics of the one-dimensional spin-1/2 XY systems paying special attention to the dynamic and thermodynamic properties of the models with Dzyaloshinskii-Moriya interaction, correlated disorder, and regularly alternating Hamiltonian parameters. Зроблено огляд деяких недавніх результатів з статистичної механіки одновимірних спін-1/2 XY систем. Особлива увага звернута на динамічні і термодинамічні властивості моделей з взаємодією Дзялошинського-Морія, скорельованим безладом і регулярно змінними параметрами гамільтоніана. 2002 Article There is life in the old horse yet or what else we can learn studying spin-1/2 XY chains / O. Derzhko // Condensed Matter Physics. — 2002. — Т. 5, № 4(32). — С. 729-749. — Бібліогр.: 51 назв. — англ. 1607-324X PACS: 75.10.-b DOI:10.5488/CMP.5.4.729 http://dspace.nbuv.gov.ua/handle/123456789/120686 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We review some recent results on statistical mechanics of the one-dimensional spin-1/2 XY systems paying special attention to the dynamic and thermodynamic properties of the models with Dzyaloshinskii-Moriya interaction, correlated disorder, and regularly alternating Hamiltonian parameters.
format	Article
author	Derzhko, O.
spellingShingle	Derzhko, O. There is life in the old horse yet or what else we can learn studying spin-1/2 XY chains Condensed Matter Physics
author_facet	Derzhko, O.
author_sort	Derzhko, O.
title	There is life in the old horse yet or what else we can learn studying spin-1/2 XY chains
title_short	There is life in the old horse yet or what else we can learn studying spin-1/2 XY chains
title_full	There is life in the old horse yet or what else we can learn studying spin-1/2 XY chains
title_fullStr	There is life in the old horse yet or what else we can learn studying spin-1/2 XY chains
title_full_unstemmed	There is life in the old horse yet or what else we can learn studying spin-1/2 XY chains
title_sort	there is life in the old horse yet or what else we can learn studying spin-1/2 xy chains
publisher	Інститут фізики конденсованих систем НАН України
publishDate	2002
url	http://dspace.nbuv.gov.ua/handle/123456789/120686
citation_txt	There is life in the old horse yet or what else we can learn studying spin-1/2 XY chains / O. Derzhko // Condensed Matter Physics. — 2002. — Т. 5, № 4(32). — С. 729-749. — Бібліогр.: 51 назв. — англ.
series	Condensed Matter Physics
work_keys_str_mv	AT derzhkoo thereislifeintheoldhorseyetorwhatelsewecanlearnstudyingspin12xychains
first_indexed	2025-07-08T18:24:13Z
last_indexed	2025-07-08T18:24:13Z
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fulltext	Condensed Matter Physics, 2002, Vol. 5, No. 4(32), pp. 729–749 There is life in the old horse yet or what else we can learn studying spin-1 2 XY chains O.Derzhko 1,2 1 Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine, 1 Svientsitskii Str., 79011 Lviv, Ukraine 2 Chair of Theoretical Physics, Ivan Franko National University of Lviv, 12 Drahomanov Street, 79005 Lviv, Ukraine Received September 2, 2002 We review some recent results on statistical mechanics of the one-dimens- ional spin- 1 2 XY systems paying special attention to the dynamic and ther- modynamic properties of the models with Dzyaloshinskii-Moriya interac- tion, correlated disorder, and regularly alternating Hamiltonian parameters. Key words: spin- 1 2 XY chains, Dzyaloshinskii-Moriya interaction, correlated disorder, magnetization plateaus, spin-Peierls instability PACS: 75.10.-b 1. Introductory remarks One-dimensional spin- 1 2 XY model in a transverse field defined by the Hamilto- nian H = ∑ n Ωszn + ∑ n ( Jxsxns x n+1 + Jysyns y n+1 ) (1) is known as the simplest quantum many-body system for which many statistical me- chanical calculations can be performed exactly, i.e., without making any simplifying approximations. For more than forty years this model has been a standard testing- ground for checking various conjectures or new calculation schemes and approaches in statistical mechanics and condensed matter physics. The aim of the present paper is to elucidate some recent results derived for spin- 1 2 XY chains and to foresee some further problems which are attractive to study. The interest in spin- 1 2 XY chains may be enforced nowadays because of the progress in material sciences (see, for ex- c© O.Derzhko 729 O.Derzhko ample, the recent report on Cs2CoCl4, the compound which is a good realization of the famous spin-1 2 isotropic XY chain [1]). 2. Dynamic properties in fermionic picture Spin-1 2 XY chains contain a hidden symmetry which was discovered by applying the Jordan-Wigner transformation: the system of interacting spins (1) can be de- scribed in terms of noninteracting spinless fermions (E.Lieb, T.Schultz, D.Mattis). As a result many (although by no means all) statistical mechanical calculations can be performed rigorously. As an example of notorious problems in the statis- tical mechanics of spin- 1 2 XY chains we may mention the analysis of the time- dependent correlation functions of x (y) spin components 〈sxn(t)sxn+m〉, sxn(t) = exp(iHt)sxn exp(−iHt), 〈(. . .)〉 = Tr(exp(−βH)(. . .))/Tr(exp(−βH)). Since the rela- tion between the x spin component attached to a certain site and the on-site creation and annihilation operators of fermions is nonlocal and involves the occupation- number operators of fermions at all previous sites, the problem of applying the Wick-Bloch-de Dominicis theorem to a product of a huge number of multipliers arises. The result can be written compactly as the Pfaffian of an antisymmetric matrix (generally speaking of huge sizes) constructed from the elementary contrac- tions and hence a further analytical analysis becomes not simple. The problem has been solved to some extent by elaborating the numerical schemes for computation of Pfaffians [2–6]. The numerically derived1 results for xx (yy, xy, yx) dynamics supplemented by the analytical results for zz dynamics [8] permit to work out the theory of dynamic properties of spin- 1 2 XY chains in the fermionic picture (G.Müller with coworkers) and thus to explain the peculiarities of responses of the spin system to small external perturbations [8,9]. Let us sketch briefly the linear response theory of the spin chain in the fermionic language considering for simplicity the isotropic XY model (Jx = Jy = J). The zz dynamic structure factor for this model which is given by Szz(κ, ω) = ∫ π −π dκ1nκ1 (1 − nκ1−κ) δ (ω + Λκ1 − Λκ1−κ) , (2) Λκ = Ω + J cos κ, nκ = 1 1 + exp(βΛκ) suggests the following interpretation [8,10]. Consider at first the high-temperature limit β → 0 when nκ → 1 2 and hence Szz(κ, ω) (2) becomes independent of Ω. Ap- plying the infinitesimally small external field (directed along z axis) characterized by the wave vector κ and frequency ω we observe that the responsive magnetiza- tion (directed along z axis) is determined by generation of the two fermions with energies Λκ1 and Λκ2 under the restrictions κ = κ1 − κ2 and ω = −Λκ1 + Λκ2 = 1The numerical approach is not restricted to the uniform chains and can be easily applied to the nonuniform chains in which the Hamiltonian parameters vary regularly along the chain with a finite period or are random variables with a given probability distribution [7]. 730 Studying spin- 1 2 XY chains Szz(κ,ω) κ ω a 1 2 1 2 0 1 2 Szz(κ,ω) κ ω b 1 2 1 2 0 1 2 Szz(κ,ω) κ ω c 1 2 1 2 0 1 2 Szz(κ,ω) κ ω d 1 2 1 2 0 1 2 Figure 1. Szz(κ, ω) for the isotropic XY chain in a transverse field (J = 1) at infinite temperature β = 0 (a) and zero temperature β → ∞ (b: Ω = 0, c: Ω = 0.5, d: Ω = 0.9). 2J sin κ 2 sin ( κ1 − κ 2 ) . The “dummy” wave vector κ1 in (2) varies within the region −π 6 κ1 6 π. As a result, such an experimental probe “measures” a continuum of the two-fermion excitations in the κ-ω plane. The upper boundary of the two- fermion continuum is given by ω = 2\|J sin κ 2 \| and Szz(κ, ω) exhibits divergence along this line as it follows from (2). At low temperatures β → ∞, the Fermi factors in (2) come into play and Szz(κ, ω) becomes dependent on Ω. The additional conditions Λκ1 < 0 and Λκ2 > 0 lead to the appearance of the lower boundary (for example, ω = \|J sin κ\| if Ω = 0) at which a finite value of Szz(κ, ω) jumps to zero. The trans- verse field (which plays a role of the chemical potential in the fermionic picture) effects the lower boundary of the two-fermion continuum in the κ-ω plane and the redistribution of the values of Szz(κ, ω) in the κ-ω plane at low temperatures. For \|Ω\| > \|J \| Szz(κ, ω) vanishes everywhere in the κ-ω plane. The above-said can be seen in figure 1 where some typical results illustrating zz dynamics are reported. Contrary to the zz dynamics the xx dynamics is more involved: we do not know the explicit expression for Sxx(κ, ω) similar to (2). Equation (2) arises after compu- tation of the average of the product of four Fermi operators 〈( 1 − 2c+n (t)cn(t) ) ( 1 − 2c+n+mcn+m )〉 731 O.Derzhko that is obviously the two-fermion quantity. The xx dynamic structure factor contains the averages like 〈( 1 − 2c+1 (t)c1(t) ) . . . ( 1 − 2c+n−1(t)cn−1(t) ) ( c+n (t) + cn(t) ) × ( 1 − 2c+1 c1 ) . . . ( 1 − 2c+n+m−1cn+m−1 ) ( c+n+m + cn+m )〉 and thus it is a many-fermion quantity. However, the numerical calculations show that the two-fermion continuum dominates the low-temperature behaviour of the xx dynamic structure factor. Although Sxx(κ, ω) is not restricted to the two-fermion continuum region in the κ-ω plane and has nonzero value above the upper bound- ary of the two-fermion continuum (demonstrating the effects of the many-fermion continua [11]) its value outside the two-fermion continuum is rather small. Sxx(κ, ω) may be described by several washed-out excitation branches following roughly the two-fermion continuum boundaries. Further studies are required to clarify why two- fermion features rule the many-fermion quantity Sxx(κ, ω). For \|Ω\| > \|J \| the zero- temperature xx dynamic structure factor Sxx(κ, ω) shows a single δ-peak along the fermion branch ω = Λκ. In the high-temperature limit the low-temperature struc- tures in κ-ω plane disappear and Sxx(κ, ω) becomes κ-independent in agreement with exact calculations for β = 0 [12]. Alternatively the xx dynamics can be exam- ined using a bosonization treatment [13], however, such an analysis is restricted to low-energy physics and only a small region in the κ-ω plane can be explored by this approach. The described analysis of the dynamic properties may be extended to the ani- sotropic XY interaction (Jx 6= Jy) [8] and the dimerized isotropic XY interaction (Jx = Jy → J(1 − (−1)nδ), 0 6 δ 6 1 is the dimerization parameter) [8,9]. Appar- ently, the exhaustive study of the zz dynamics for the former case [8] should be still supplemented by the corresponding analysis of the xx dynamics whereas the case when both anisotropy and dimerization are present requires a separate study. Re- cently, the effects of periodic inhomogeneity on the dynamic susceptibility χzz(κ, ω) (but not χxx(κ, ω)) have been reported in the paper on dynamics of isotropic XY model on one-dimensional superlattices [14]. The xx dynamic quantities for the case of extremely anisotropic XY interaction (J y = 0), i.e., for the spin- 1 2 transverse Ising chain, are of interest for interpreting the experimental data on the dynamic dielectric permittivity of the quasi-one-dimensional hydrogen-bonded ferroelectric compound CsH2PO4 [15,16]. To end up this section, let us note that the dynamic properties of two-dimensional quantum spin models can be also explained in terms of the two-fermion continuum, however, such a picture may have only an approximate meaning [17]. Another in- teresting question is to contrast the results for dynamic structure factors of spin- 1 2 and spin-1 chains [18]. 3. Dzyaloshinskii-Moriya interaction The Dzyaloshinskii-Moriya interaction is often present in the low-dimensional quantum magnets (see, for example, a recent paper [19]). It is generally known 732 Studying spin- 1 2 XY chains that the Dzyaloshinskii-Moriya interaction ∑ nD · (sn × sn+1) being added to the Hamiltonian (1) does not destroy the rigorous treatment if D = (0, 0, D) [20]2. The main effect of the Dzyaloshinskii-Moriya interaction is the loss of the sym- metry of elementary excitation energies with respect to the change κ→ −κ: Λκ = D sin κ+ √ ( Ω + Jx + Jy 2 cos κ )2 + ( Jx − Jy 2 )2 sin2 κ 6= Λ−κ [20,22]. In the presence of the Dzyaloshinskii-Moriya interaction some remarkable changes in the thermodynamic and dynamic properties of spin chains may occur. Consider, for example, the isotropic XY chain. Since the Dzyaloshinskii-Moriya interaction for such a chain (even inhomogeneous one) can be eliminated by the local rotations in spin space around the z axis sxn cosφn + syn sin φn → sxn, −sxn sinφn + syn cosφn → syn , φn = ϕ1 + . . .+ ϕn−1, tanϕm = Dm Jm (3) resulting in a model with isotropic XY interaction √ J2 n +D2 n the zz dynamics remains as for a chain without Dzyaloshinskii-Moriya interaction, however, with renormalized energy scale J → √ J2 +D2 3. In contrast, the xx dynamic quantities according to (3) involve 〈sxn(t)sxn+m〉J,D = cosφn cosφn+m〈sxn(t)sxn+m〉√J2+D2,0 − cosφn sinφn+m〈sxn(t)syn+m〉√J2+D2,0 − sinφn cosφn+m〈syn(t)sxn+m〉√J2+D2,0 + sinφn sinφn+m〈syn(t)syn+m〉√J2+D2,0 (4) (for the homogeneous chain φn = (n−1)ϕ, tanϕ = D/J). In view of (4) the relation between the zz and xx dynamics discussed in the previous section (Sxx(κ, ω) at low temperatures exhibits the washed-out excitation branches which roughly follow the boundaries of the two-fermion continuum which determines Szz(κ, ω)) may appear to be more intricate [23]. In the next sections 4 and 5 we give further examples of how the Dzyaloshinskii-Moriya interaction manifests itself in the properties of spin- 1 2 XY chains. 4. Correlated off-diagonal and diagonal disorder The Jordan-Wigner transformation maps the spin- 1 2 isotropic XY chain in a transverse field onto the chain of tight-binding spinless fermions with the on-site 2Let us note that in some cases the Dzyaloshinskii-Moriya interaction can be eliminated by the corresponding spin-coordinate transformation [21] (see, for example, equation (3)). 3This is not the case if XY interaction is anisotropic; the zz dynamics of the model with extremely anisotropic XY interaction, i.e., the transverse Ising chain with Dzyaloshinskii-Moriya interaction, was considered in [22]. 733 O.Derzhko energy Ω and hopping I = J/2. If the transverse fields are independent random variables (diagonal disorder) each with the Lorentzian probability distribution p(Ωn) = 1 π Γ (Ωn − Ω0) 2 + Γ2 the resulting fermionic model is the one-dimensional version of the Lloyd model. The density of states, ρ(E) = 1 N N ∑ k=1 δ(E − Λk), (. . .) = . . . ∫ dΩnp(Ωn) . . . (. . .), for the Lloyd model can be found exactly [24]. Going far beyond the idea of H.Ni- shimori we may consider a spin model with the correlated off-diagonal and diagonal Lorentzian disorder which after fermionization reduces to the one-dimensional ver- sion of the extended Lloyd model introduced by W.John and J.Schreiber. Namely, we consider the isotropic XY model with independent random exchange interactions (off-diagonal disorder) given by the Lorentzian probability distribution p(. . . , Jn, . . .) = ∏ n p(Jn) = ∏ n 1 π Γ (Jn − J0) 2 + Γ2 . (5) Moreover, we consider the correlated off-diagonal and diagonal disorder assuming that the on-site transverse fields in the chain are determined by the surrounding exchange interactions according to the relation Ωn − Ω0 = a 2 (Jn−1 + Jn − 2J0) , a is real, \|a\| > 1. (6) Then the density of states ρ(E) yielding the thermodynamic quantities of the in- troduced random spin chain can be calculated exactly [25,26]. To get ρ(E) we must calculate the diagonal Green functions G∓ nn(E), since ρ(E) = ∓ 1 π =G∓ nn(E). The set of equations of motion for G∓ nm(E ± iε), ε → +0 can be averaged using contour integration in complex planes of random (Lorentzian) variables Jn. Using the Gershgorin criterion we find the set of equations for the averaged Green func- tions which has the same structure as before averaging but possesses translational symmetry. As a result we obtain the desired quantities G∓ nm(E) and hence all ther- modynamic quantities. In figures 2a and 2b we present ρ(E) in the most interesting region \|a\| → 1, when ρ(E) becomes not symmetric with respect to the change E−Ω0 → − (E − Ω0). Such asymmetry immediately yields a nonzero (average) transverse magnetization mz = −1 2 ∫ dEρ(E) tanh βE 2 6= 0 734 Studying spin- 1 2 XY chains 0.1 0.3 -2 0 2 ρ(E) a -2 0 2 E-Ω0 b -0.5 0.0 -2 0 2 -mz c -2 0 2 Ω0 d Figure 2. The density of states (a, b) and the transverse magnetization (c, d) of the isotropic XY chain with correlated Lorentzian disorder (J0 = 1, Γ = 1, a = −1 (a, c), a = 1 (b, d)). The dotted curves correspond to the nonrandom case (Γ = 0). at low temperatures β → ∞ at zero (average) transverse field Ω0 = 0 (figures 2c and 2d). Let us consider more closely this somewhat unexpected magnetic property of the introduced random spin chain. For a certain random realization of the chain defined by (5), (6) one may expect the same numbers of sites surrounded by stronger than J0 exchange interactions as the sites surrounded by weaker than J0 exchange interactions. Because of (6) for Ω0 = 0 the transverse fields at the former and at the latter sites have the same value but the opposite signs giving as a result ∑ n Ωn = 0. On the other hand, one may expect that the sites surrounded by strong isotropic XY exchange interactions exhibit small z magnetization whereas the sites surrounded by weak isotropic XY exchange interactions exhibit large z magnetization (in the opposite direction). As a result, the average transverse magnetization has a nonzero value. As \|a\| increases, a difference in the oppositely directed z magnetizations be- comes smaller. Thus, a nonzero mz at Ω0 = 0 appears owing to the imposed relation (6) which expresses the condition of correlated disorder. 735 O.Derzhko Some further insight into the origin of the asymmetry of ρ(E) can be obtained after examining the moments of the density of states M (r) ≡ ∫ dEErρ(E) = 1 N N ∑ n=1 〈{ [. . . [cn, H] , . . . , H] , c+n }〉 (7) (here H is the Hamiltonian of fermions which represent the spin chain). It is just the correlated disorder that yields a nonzero third moment M (3) 6= 04 at Ω0 = 0. For not correlated off-diagonal and diagonal disorders one gets M (3) = 0 at Ω0 = 0. The moments of the density of states can be calculated for any probability distribution of random variables p(Jn) (not necessarily for the Lorentzian probability distribution), for example, for the rectangle probability distribution. These calculations explicitly demonstrate the cause of the asymmetry appearance in ρ(E) [27]. Some other results on the effects of correlated disorder can be found in [10,28,29]. Finally, let us remark that the considered spin model with correlated Lorentzian disorder may be extended by introducing the nonrandom Dzyaloshinskii-Moriya in- teraction D. Another extension is to assume the exchange interaction to be nonran- dom Jn = J whereas the Dzyaloshinskii-Moriya interactions Dn to be independent random Lorentzian variables determining the transverse fields according to (6). Both models are related to each other through a certain sequence of rotations of spin axes around the z axis (being applied to the Hamiltonian with the exchange interactions Dn and the Dzyaloshinskii-Moriya interactions −Jn it gives the Hamiltonian with the exchange interactions Jn and the Dzyaloshinskii-Moriya interactions Dn) and hence it is sufficient to consider only one of them. Considering, for example, the former model one finds that the nonrandom Dzyaloshinskii-Moriya interaction may lead to the recovery of the symmetry with respect to the change E−Ω0 → − (E − Ω0) and hence to a decrease of the nonzero value of mz at Ω0 = 0. Such an effect becomes also apparent after calculating the moments of the density of states M (2) and M (3) (7) for this model. To end up this section, let us note that a different random spin- 1 2 XY chains were rigorously analytically examined by Th.M.Nieuwenhuizen and coauthors [30]. 5. Effects of regularly alternating bonds and fields 5.1. Continued fractions The quantum spin chains with regularly alternating Hamiltonian parameters can model dimerized (l-merized) chains, one-dimensional superlattices, one-dimensional 4Here (. . .) denotes the random averaging either for the correlated off-diagonal and diagonal disorder or for the independent off-diagonal (with the probability distribution p(Jn)) and diagonal (with the probability distribution p(Ωn) = ∫ dJn−1 ∫ dJnp(Jn−1)p(Jn)δ ( Ωn − Ω0 − a 2 (Jn−1 + Jn −2J0))) disorders. 736 Studying spin- 1 2 XY chains decorated chains etc. The case of spin- 1 2 XY chains supplemented by the contin- ued fraction approach (for other approaches see [31]) remains amenable for rigorous analysis of the thermodynamic properties if the exchange interaction is isotropic (Jxn = Jyn) [32], extremely anisotropic (Jyn = 0) [33] or if Ωn = 0 [34]. The thermo- dynamic quantities of the regularly alternating isotropic XY chain in a transverse field can be obtained through the density of states ρ(E) = 1 N ∑N k=1 δ(E − Λk). The thermodynamic quantities of the regularly alternating transverse Ising chain and the regularly alternating anisotropic XY chain without transverse field can be obtained through the density of states R(E2) = 1 N ∑N k=1 δ(E 2 − Λ2 k). Let us recall that after fermionization of the isotropic XY chain in a transverse field (Jxn = Jyn = 2In) one faces the Hamiltonian which is a bilinear fermion form. While making it diagonal one arrives at the set of equations In−1gk,n−1 + (Ωn − Λk) gkn + Ingk,n+1 = 0, (8) here gkn are the coefficients of the linear transformation which make the initial bilin- ear fermion form diagonal and Λk are the resulting elementary excitation energies. Therefore, introducing the Green functions Gnm(E) according to −In−1Gn−1,m(E) + (E − Ωn)Gnm(E) − InGn+1,m(E) = δnm (9) one gets the density of states ρ(E) = ∓ 1 πN N ∑ n=1 =Gnn(E ± iε), ε→ +0. For the transverse Ising chain (Jxn = 2In, J y n = 0) instead of (8) and (9) we have Ωn−1In−1Φk,n−1 + ( Ω2 n + I2 n−1 − Λ2 k ) Φkn + ΩnInΦk,n+1 = 0 (10) and −Ωn−1In−1Gn−1,m(E2) + ( E2 − I2 n−1 − Ω2 n ) Gnm(E2) − ΩnInGn+1,m(E2) = δnm , (11) respectively, whereas for the anisotropicXY chain without field (Jxn = 2Ixn, J y n = 2Iyn, Ωn = 0) instead of (8) and (9) we have Iyn−2I x n−1Φk,n−2 + ( Ixn−1 2 + Iyn 2 − Λ2 k ) Φkn + IynI x n+1Φk,n+2 = 0 (12) and −Iyn−2I x n−1Gn−2,m(E2) + ( E2 − Ixn−1 2 − Iyn 2 ) Gnm(E2) − IynI x n+1Gn+2,m(E2) = δnm , (13) respectively. For the last two models the Green functions Gnm(E2) yield the density of states R(E2) = ∓ 1 πN N ∑ n=1 =Gnn(E2 ± iε), ε→ +0. 737 O.Derzhko For a general case of the anisotropic XY chain in a transverse field a set of equations like (10) or (12) is five diagonal banded (but not three diagonal banded as (8), (10) or (12)) and the next step, i.e., the continued fraction representation for the diagonal Green functions is less evident. According to equation (9), (11) or (13) the diagonal Green functions for all these models can be represented in terms of continued fractions. For example, from (13) it immediately follows that Gnn(E2) = 1 E2 − Ixn−1 2 − Iyn 2 − ∆− n − ∆+ n , ∆− n = Iyn−2 2Ixn−1 2 E2 − Ixn−3 2 − Iyn−2 2 − Iy n−4 2Ix n−3 2 E2−Ix n−5 2−Iy n−4 2−... , ∆+ n = Iyn 2Ixn+1 2 E2 − Ixn+1 2 − Iyn+2 2 − Iy n+2 2 Ix n+3 2 E2−Ix n+3 2−Iy n+4 2−... . (14) If now the Hamiltonian parameters are periodic with any finite period p the contin- ued fractions ∆∓ n in (14) become periodic and can be evaluated by solving square equations. As a result, we get the exact expressions for Gnn(E2) (or Gnn(E) for the isotropic XY chain) and hence for all thermodynamic quantities of the regularly alternating spin chains in question. Let us remark, that the thermodynamic quantities of the anisotropic XY chain in a transverse field of period 2 were obtained in [35] in a different manner on a quite general background. This scheme, however, becomes cumbersome if the pe- riod of nonuniformity increases. Further development of a general approach to the study of thermodynamic quantities and spin correlation functions has been report- ed by L.L.Gonçalves with coworkers in connection with spin- 1 2 XY models on one- dimensional superlattices. 5.2. Magnetization processes The most spectacular manifestation of the effects of regular alternation can be seen in the magnetization processes at low temperatures. Consider at first the isotropic XY chain. Owing to regularly alternating parameters of the Hamiltonian the fermion band splits into several subbands the number of which does not exceed the period of regular inhomogeneity. This circumstance immediately suggests that the zero-temperature dependence mz = −1 2 ∫ dEρ(E) tanh βE 2 on Ω (we have assumed that Ωn = Ω + ∆Ωn) consists of horizontal parts (plateaus) which appear when E = 0 remains in between subbands (given by ρ(E)) with changing of Ω separated by the parts of varying mz which appear when E = 0 remains within subbands with Ω changing. Clearly, the number of plateaus does 738 Studying spin- 1 2 XY chains not exceed the period of nonuniformity. Moreover, their heights −s 6 mz 6 s are in agreement with the condition: p (s−mz) = integer with s = 1 2 [36]. This conjecture was suggested on the model-independent background for a general spin- s chain with axial symmetry in a uniform magnetic field using the Lieb-Schultz- Mattis theorem and the bosonization arguments. Obviously, spin- 1 2 XY chain is an easy case (represented by noninteracting fermions) contrary to the chains with Heisenberg exchange interaction or higher spins s > 1 2 . On the other hand, the magnetization curve for spin- 1 2 XY chain can be obtained explicitly. Moreover, the elaborated scheme also permits to obtain the local (on-site) magnetizations. The on- site magnetizations exhibit plateaus which begin and end up at the same values of Ω as for the total magnetization, however, the plateaus heights (i.e., the values of the on-site magnetizations) are not universal quantities no longer obeying the above- mentioned condition and strongly depend on a concrete set of the Hamiltonian parameters. Moreover, a sequence of sites n1, n2, . . . , np satisfying the inequalities mzn1 6 mzn2 6 . . . 6 mznp depends on the value of the applied field Ω. The step-like behaviour of the zero-temperature dependence mz (mzn) vs. Ω is accompanied by the corresponding zero-temperature behaviour χz = ∂mz/∂Ω (χzn = ∂mzn/∂Ω) vs. Ω. The zero-temperature dependence χz on Ω behaves like −ρ(E = 0) vs. Ω thus reproducing the density of states ρ(E). In contrast to the isotropic XY chain, a regular alternation of the parameters of the transverse Ising chain Hamiltonian does not lead to plateaus in the zero- temperature dependence mz on Ω. The difference arises owing to the anisotropy of XY interaction: if Jxn 6= Jyn one finds that [ ∑ n s z n, H] 6= 0. Thus, even if the spin system remains in the same ground state \|0〉 with varying Ω the magnetization 〈0\| ∑ n s z n\|0〉, nevertheless, varies with varying Ω. As a result the dependence mz vs. Ω does not exhibit horizontal parts. On the other hand, in the vicinity of some fields Ω? (critical fields) the magnetization may behave as mz ∼ (Ω − Ω?) ln \|Ω−Ω?\| and hence the susceptibility as χz ∼ ln \|Ω − Ω?\|. For the uniform chain there are two critical fields Ω? = ±\|I\|. If the bonds become regularly alternating only the values of Ω? change. For example, Ω? = ± √ \|I1I2\| for p = 2, Ω? = ± 3 √ \|I1I2I3\| for p = 3 etc. If the fields become regularly alternating not only the values of the critical fields vary but the number of the critical fields may change. For example, assuming Ω1,2 = Ω ± ∆Ω, ∆Ω > 0 one has either two critical fields Ω? = ± √ ∆Ω2 + \|I1I2\| if ∆Ω < √ \|I1I2\| or four critical fields Ω? = ± √ ∆Ω2 ± \|I1I2\| if ∆Ω > √ \|I1I2\|. The transverse field is a controlling parameter of the second-order quantum phase transition in the spin- 1 2 Ising chain in a transverse field [37]. Thus, a number of the quantum phase transition points governed by Ω in this model may increase due to a regular alternation of the transverse fields presuming the strength of inhomogeneity is sufficiently strong. For example, for p = 2 there may be either two or four quantum phase transition points, for p = 3 there may be either two, or four, or six quantum phase transition points etc. The critical behaviour remains unchanged and is just like for the temperature driven phase transition in the square-lattice Ising model. For example, the zero-temperature dependence χz vs. Ω always exhibits logarithmic singularities the number of which depends on a concrete set of the Hamiltonian 739 O.Derzhko parameters. It seems interesting to trace the changes in magnetization processes as anisotropy of the exchange interaction varies between the isotropic and extremely anisotropic limits. 0.2 0.4 -mz a 0.2 0.4 b 0.2 0.4 0 1 2 Ω c Figure 3. The zero-temperature magnetization curves for the isotropic XY (a), transverse Ising (b) and classical (c) chains of period 2 (Jn = 1, Ω1,2 = Ω ± ∆Ω, ∆Ω = 0 (solid curves), ∆Ω = 0.5 (long-dashed curves), ∆Ω = 1 (short-dashed curves), ∆Ω = 1.5 (dotted curves)). It is instructive to compare the zero- temperature magnetization processes in the quantum and classical XY chains5. A classical chain consists of arrows (vectors) s = (s, θ, φ), s = 1 2 , 0 6 θ 6 π, 0 6 φ < 2π which in- teract with each other and an external (trans- verse) field. The classical isotropic XY chain is described by the Hamiltonian H = ∑ n Ωs cos θn + ∑ n Js2 sin θn sin θn+1 cos (φn − φn+1) (15) (compare with (1)). The Hamiltonian of the classical transverse Ising chain contains cosφn cosφn+1 instead of cos (φn − φn+1) in the interspin interaction terms in (15). Consider fur- ther a chain of period 2 with Ω1,2 = Ω ± ∆Ω, ∆Ω > 0, J1,2 = J . The ground-state energy ansatz for both the isotropic XY and transverse Ising chains reads E0(θ1, θ2) = N 2 s ((Ω + ∆Ω) cos θ1 + (Ω − ∆Ω) cos θ2) −N \|J \|s2 sin θ1 sin θ2 (16) and θ1, θ2 are determined from the equations (Ω + ∆Ω) sin θ1 + 2\|J \|s cos θ1 sin θ2 = 0, (Ω − ∆Ω) sin θ2 + 2\|J \|s sin θ1 cos θ2 = 0 (17) to provide a minimum of E0(θ1, θ2) (16). The on-site magnetizations have the component ar- bitrary directed in the xy plane with \|m⊥n\| = s sin θn (n = 1, 2) for the isotropic XY case or the component directed along the x axis with \|mxn\| = s sin θn for the transverse Ising case and the z component given by mz n = s cos θn in both cases. The components of the total mag- netization per site are then as follows m⊥ = 5Rigorous calculations of the initial static susceptibility tensor for another nonuniform classical chain, i.e., the spin- 1 2 Ising chain, can be found in [38]. 740 Studying spin- 1 2 XY chains 1 2 (m⊥1 +m⊥2) or mx = 1 2 (mx1 +mx2) (if J < 0) and mz = 1 2 (mz1 +mz2). Equa- tions (17) may be solved analytically yielding the ground-state energy and the on- site magnetizations m⊥1 (mx1), m⊥2 (mx2), mz1, mz2. In figure 3 we contrast the zero-temperature magnetization curves for the isotropic XY , transverse Ising and classical chains of period 2. Note, that the classical isotropic XY chain similarly to its quantum counterpart may exhibit plateaus in the dependence mz vs. Ω. Ev- idently, it would be interesting to examine a quantum-to-classical crossover in the magnetization curves. Probably a real-space renormalization-group method aimed on the study of quantum fields on a lattice and applied in [39] to the spin- 1 2 trans- verse Ising chain may be used to analyse the spin-1 chain, that is, an intermediate case between the quantum (s = 1 2 ) and the classical (s→ ∞) cases. Exact results for the magnetization curves of spin- 1 2 XY chains may be employed to test the strong-coupling approach that is a widely used approximate approach in the theory of low-dimensional spin systems consisting of periodically repeating units (spin chains with a periodic modulation of the intersite interactions or spin ladders) [40]. 5.3. Spin-Peierls instability Since the paper of P.Pincus [41] we know that the spin- 1 2 isotropic XY chain is unstable with respect to a lattice dimerization due to the spin-Peierls mechanism6. Really, the ground-state energy per site of the dimerized spin chain (i.e., with Jn = J (1 − (−1)nδ)) is given by e0(δ) = −\|J \| π ∫ ψ 0 dϕ √ 1 − (1 − δ2) sin2 ϕ− \|Ω\| ( 1 2 − ψ π ) , (18) ψ =      0, if \|J \| 6 \|Ω\|, arcsin √ 1−Ω2/J2 1−δ2 , if δ\|J \| 6 \|Ω\| < \|J \|, π 2 , if \|Ω\| < δ\|J \|. At Ω = 0 the ground-state energy per site (18) (e0(δ) ≈ e0(0) + \|J \| 2π δ2 ln δ, δ � 1) decreases rapidly enough in comparison with the increase of the elastic energy per site αδ2 as δ increases to provide a minimum of the total energy per site e0(δ)+αδ2 at a nonzero value of the dimerization parameter δ?. The external field may destroy dimerization: if Ω exceeds a certain value the dimerized phase cannot survive and the uniform phase becomes favourable. The phase diagram shown in figure 4a specifies the region of stability/metastability of the dimerized phase at zero temperature. It is generally known (see, for example, a review on CuGeO3 [43]) that the in- crease of external field leads to a transition from the dimerized phase to the incom- mensurate phase rather than to the uniform phase that contradicts to what is seen in figure 4. Obviously, the incommensurate phase cannot appear within the frames of the adopted ansatz for lattice distortion δ1δ2δ1δ2 . . . , δ1 + δ2 = 0. Therefore, we 6This conclusion was obtained within the adiabatic approximation; the corresponding treatment within the nonadiabatic approximation is more sophisticated [42]. 741 O.Derzhko 0.4 0.8 Ω a A B1 B2 C 0.4 0.8 b A B1 B2 C 0.4 0.8 0.2 0.5 α c A B1 B2 C Figure 4. The phase diagram of the isotropic XY chain (J = 1) in the plane lattice stiffness α – transverse field Ω which indicates the regions of stability/metastability of the dimer- ized and uniform phases at zero tem- perature. A (C) – only dimerized (uniform) phase occurs; B1 (B2) both phases are possible but the dimerized (uniform) is the favourable one. The phase diagram also illustrates the possible effects of the Dzyaloshinskii- Moriya interaction (a: D = 0, b: D = 0.5, k = 1, c: D = 0.5, k = 0). should assume Jn = J (1 + δn) and exam- ine the total energy E0 ({δn})+ ∑ n αδ 2 n for different lattice distortions {δn}. We intro- duce a trial distortion of the form δn = −δ cos ( 2π p n ) , (19) where p is the period of modulation and analyse numerically (N = 1000) the total energy to reveal which spin-Peierls phase is realized in the presence of external field (for details see [44]). Comparing the behaviour of the total energy for p = 2 and p = 1.9, p = 2.1 as Ω increases we conclude that a long-period structure does arise if Ω ex- ceeds a certain value. The dimerized phase transforms into a long-period phase rather than into the uniform phase while the field increases. The important conclusion of fur- ther computations using (19) is that the dimerized phase persists up to a certain characteristic field. Further, in moderate fields the lattice parameterized by (19) may exhibit short-period phases, for example, the trimerized phase with p = 3 [45] (an- other possible lattice distortion which pre- serves the chain length is δ1 = −δ2). How- ever, a behaviour of the trimerized phase is essentially different in comparison with that of the dimerized phase: for any small devia- tion of the field from the value at which the trimerized phase occurs there exists such a long-periodic structure for which the lattice distortion (19) gives smaller total energy than for p = 3. Thus, contrary to the dimer- ized phase, the trimerized phase does not persist with the field varying and it contin- uously transforms into a certain long-period phase with the field varying. In strong fields the uniform lattice can be expected. Clear- ly, such a study is restricted to the adopt- ed ansatz (19) (although any other (but not all) distortion pattern may be assumed) and therefore we can say for sure which lat- 742 Studying spin- 1 2 XY chains tice distortion is not realized rather than to point out which lattice distortion should occur. It is interesting to discuss the effect of the Dzyaloshinskii-Moriya interaction on the spin-Peierls dimerization. For this purpose we should find the ground-state energy of the chain of period 2 with a sequence of parameters ΩJ1D1ΩJ2D2ΩJ1D1ΩJ2D2 . . . . Moreover, J1,2 = J(1 ± δ) and D1,2 = D(1 ± kδ). Putting k = 0 one has a chain in which D does not depend on the lattice distortion, whereas for k = 1 the dependence of D on the lattice distortion is the same as that for the isotropic XY exchange interaction. In the limit of small δ � 1 (valid for lattices with large values of α and corresponding to the experimental situation) the analysis becomes extremely simple. It is convenient to introduce the parameter ℵ = J2 + kD2 J2 +D2 . Consider, for example, the case Ω = 0. After a simple rescaling of variables one arrives at the equations considered in [41] and as a result δ? ∼ 1 ℵ exp ( − 2πα√ J2 +D2ℵ2 ) . We immediately conclude that for k = 1 nonzero D leads to the increase of the dimerization parameter δ?, whereas for k = 0 nonzero D leads to the decrease of δ?. Thus, the Dzyaloshinskii-Moriya interaction may act either in favour of the dimerization or against it. The actual result of its effect depends on the dependence of the Dzyaloshinskii-Moriya interaction on the amplitude of the lattice distortion in comparison with the corresponding dependence of the isotropic XY exchange interaction. Comparing figure 4a and figures 4b, 4c one can see the described effects of the Dzyaloshinskii-Moriya interaction on the spin-Peierls dimerization. Finally, let us make some remarks on the exchange interaction anisotropy effects on the spin-Peierls dimerization. Consider the anisotropic XY chain (without field). In the isotropic limit it exhibits the spin-Peierls dimerized phase, whereas in the extremely anisotropic limit (Ising chain) the ground-state energy does not depend on δ and the spin-Peierls dimerization cannot occur. Therefore, it is attracting to follow how anisotropy of the exchange interaction destroys the dimerized phase and to find, in particular, a critical value of anisotropy above which the dimerized phase cannot appear. The consideration becomes even more complicated in the presence of an external (transverse) field. Note, however, that such a study could be performed many years ago on the basis of the results for the anisotropic XY chain in a transverse field of period 2 reported in [35]. Another issue which deserves to be studied in detail is related to a spin-Peierls instability of the spin- 1 2 transverse Ising chain [33]. By direct inspection one can make sure that although the ground-state energy of the dimerized Ising chain in a constant 743 O.Derzhko transverse field Ω decreases as δ increases, however, not sufficiently rapidly (for any Ω) to get a gain in the total energy. As a result no spin-Peierls dimerization should be anticipated. On the other hand, the transverse Ising chain is unitary equivalent to the anisotropic XY chain without field. By comparison of the diagonal Green functions (see equation (14) in the present paper and equation (8) in [33, J. Phys. A]) it can be found that the Helmholtz free energy of the anisotropic XY chain without field of N sites IxnI y nI x n+1I y n+1 . . . is the sum of the Helmholtz free energies of two transverse Ising chains ΩnInΩn+1In+1 . . . of N/2 sites, namely, . . . Ixn−1I y nI x n+1I y n+2 . . . and . . . Iyn−1I x nI y n+1I x n+2 . . . . Consider at first the following isotropic XY chain Ixn = Iyn = I (1 − (−1)nδ) which exhibits the spin-Peierls dimerization. In view of the mentioned correspondence the introduced model is thermodynamically equivalent to two uniform transverse Ising chains (both with the transverse field I(1 + δ) (or I(1 − δ)) and the Ising exchange interaction I(1− δ) (or I(1+ δ))) and thus we conclude that the critical Ising chain (Ω = I) is unstable with respect to a uniform extending (or shortening) accompa- nied by the corresponding increase (decrease) of the transverse field. Further, it is known that the quadrimerized isotropic XY chain without field may be energetically favourable in comparison with the uniform chain (although yielding a smaller gain in the total energy than the dimerized chain). Consider, therefore, such a chain of period 4 with In = I ′(1 + δ), In+1 = I ′′(1 + δ), In+2 = I ′(1 − δ), In+3 = I ′′(1 − δ) which is unstable with respect to a lattice distortion characterizing by nonzero value of δ for certain values of I ′, I ′′. This model exhibits the same thermodynamics as two identical transverse Ising chains of period 2 ΩnInΩn+1In+1ΩnInΩn+1In+1 . . . with Ωn = I ′′(1−δ), Ωn+1 = I ′′(1+δ) and In = I ′(1+δ), In+1 = I ′(1−δ). These arguments indicate a possibility of the spin-Peierls bond dimerization of the spin- 1 2 transverse Ising chain accompanied by the coherent modulation of the on-site transverse fields. 6. Skipped items and summary Needless to say that many important contributions have appeared out of the scope of this brief survey. Let us discuss in a telegraph-style manner some of them. Numerous works were devoted to the analysis of the properties of spin- 1 2 XY chains with aperiodic or random Hamiltonian parameters [46] (as, for example, an extensive real-space renormalization-group treatment of the random transverse Ising chain by D.S.Fisher). One of the interesting subjects in the theory of magnetic materials is the magnetic relaxation in spin systems and, in particular, the impurity spin relaxation in spin systems. Spin-1 2 XY chains provide a possibility to rigorously study the relaxation phenomena in one-dimensional spin models containing impurities [47,3]. A single impurity may be assumed in the sense that the interaction with its neighbours is 744 Studying spin- 1 2 XY chains different in strength (J ′ 6= J). The impurity spin may be located either at the boundary of the system (J1 = J ′, J2 = J3 = . . . = J) or in the bulk (e.g., JN 2 −1 = JN 2 = J ′ and Jn = J for all other n). After the Jordan-Wigner transformation one faces a chain of tight-binding fermions with the impurity site (in the sense that the hopping amplitude(s) surrounding this site is (are) different in value) the energy spectrum of which is well-known (if J ′ < Jc the elementary excitation energies form a band whereas if J ′ > Jc two states emerge from the band; moreover, Jc = √ 2J (Jc = J) for the boundary (bulk) impurity). Nevertheless, the time dependence of the equilibrium autocorrelation functions 〈sαn(t)sαn〉 is not obvious since they are two-fermion (zz) or, generally speaking, many-fermion (xx) quantities. The time- dependent autocorrelation functions 〈sαn(t)sαn〉 for the impurity spin or for the spins in its vicinity may exhibit new types of asymptotic behaviour depending on the relation between J and J ′ and temperature. The impurity relaxation may become even more complicated in the presence of an external (transverse) field. Spin-1 2 XY chains have been used to study the nonequilibrium properties of quantum systems [48]. Z.Rácz with coworkers suggested to consider a nonequilib- rium system imposing a current on a system and investigating the steady states. Alternatively, we can gain an understanding of the nonequlibrium properties exam- ining the dynamics of an initial state (for example, a kink or droplet configuration of z on-site magnetizations). Recently a study of the transport properties (for example, of the thermal con- ductivity) of low-dimensional spin systems which are significantly determined by magnetic excitations have attracted much interest [49]. A simple case of the isotrop- ic XY chain emerges in such studies as a milestone providing reference results for more sophisticated models. Finally, some exotic applications of spin- 1 2 XY chains have appeared recent- ly. For example, in connection with quantum information processing the numeri- cal/analytical computations of entanglement in spin- 1 2 XY models on one-dimen- sional lattices of small/infinite number of sites were carried out [50]. A study of the correlation function which is called the emptiness formation probability (i.e., the probability of the formation of a ferromagnetic string in the antiferromagnet- ic ground state) yields interesting links between statistical mechanics and number theory. The case of the isotropic XY chain provides a valuable background for cal- culation of such correlation functions [51]. We hope that the present review shows that spin- 1 2 XY chains still contain quite unexplored properties which deserve to be discussed. 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There is life in the old horse yet or what else we can learn studying spin-1/2 XY chains

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