Dynamics of dimer and z spin component fluctuations in spin-1/2 XY chain

One-dimensional quantum spin- 1/2 XY models admit the rigorous analysis not only of their static properties (i.e. the thermodynamic quantities and the equal-time spin correlation functions) but also of their dynamic properties (i.e. the different-time spin correlation functions, the dynamic...

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Datum:	2005
Hauptverfasser:	Derzhko, O., Krokhmalskii, T., Hlushak, P.
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Sprache:	English
Veröffentlicht:	Інститут фізики конденсованих систем НАН України 2005
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spelling	irk-123456789-1210602017-06-14T03:04:59Z Dynamics of dimer and z spin component fluctuations in spin-1/2 XY chain Derzhko, O. Krokhmalskii, T. Hlushak, P. One-dimensional quantum spin- 1/2 XY models admit the rigorous analysis not only of their static properties (i.e. the thermodynamic quantities and the equal-time spin correlation functions) but also of their dynamic properties (i.e. the different-time spin correlation functions, the dynamic susceptibilities, the dynamic structure factors). This becomes possible after exploiting the Jordan-Wigner transformation which reduces the spin model to a model of spinless noninteracting fermions. A number of dynamic quantities (e.g. related to transverse spin operator or dimer operator fluctuations) are entirely determined by two-fermion excitations and can be examined in much detail. Одновимірні квантові спін- 1/2 XY моделі допускають строгий аналіз не лише їх статичних властивостей (тобто термодинамічних величин і однаковочасових спінових кореляційних функцій), але також і їх динамічних властивостей (тобто різночасових спінових кореляційних функцій, динамічних сприйнятливостей, динамічних структурних факторів). Це стає можливим після використання перетворення Йордана-Вігнера, яке зводить спінову модель до моделі безспінових невзаємодіючих ферміонів. Ряд динамічних величин (наприклад, пов’язаних з флуктуаціями оператора поперечної спінової компоненти чи димерного оператора) цілком визначаються двоферміонними збудженнями і можуть бути детально вивчені. 2005 Article Dynamics of dimer and z spin component fluctuations in spin-1/2 XY chain / O. Derzhko, T. Krokhmalskii, P. Hlushak // Condensed Matter Physics. — 2005. — Т. 8, № 4(44). — С. 859-867. — Бібліогр.: 12 назв. — англ. 1607-324X PACS: 75.10.-b DOI:10.5488/CMP.8.4.859 http://dspace.nbuv.gov.ua/handle/123456789/121060 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
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language	English
description	One-dimensional quantum spin- 1/2 XY models admit the rigorous analysis not only of their static properties (i.e. the thermodynamic quantities and the equal-time spin correlation functions) but also of their dynamic properties (i.e. the different-time spin correlation functions, the dynamic susceptibilities, the dynamic structure factors). This becomes possible after exploiting the Jordan-Wigner transformation which reduces the spin model to a model of spinless noninteracting fermions. A number of dynamic quantities (e.g. related to transverse spin operator or dimer operator fluctuations) are entirely determined by two-fermion excitations and can be examined in much detail.
format	Article
author	Derzhko, O. Krokhmalskii, T. Hlushak, P.
spellingShingle	Derzhko, O. Krokhmalskii, T. Hlushak, P. Dynamics of dimer and z spin component fluctuations in spin-1/2 XY chain Condensed Matter Physics
author_facet	Derzhko, O. Krokhmalskii, T. Hlushak, P.
author_sort	Derzhko, O.
title	Dynamics of dimer and z spin component fluctuations in spin-1/2 XY chain
title_short	Dynamics of dimer and z spin component fluctuations in spin-1/2 XY chain
title_full	Dynamics of dimer and z spin component fluctuations in spin-1/2 XY chain
title_fullStr	Dynamics of dimer and z spin component fluctuations in spin-1/2 XY chain
title_full_unstemmed	Dynamics of dimer and z spin component fluctuations in spin-1/2 XY chain
title_sort	dynamics of dimer and z spin component fluctuations in spin-1/2 xy chain
publisher	Інститут фізики конденсованих систем НАН України
publishDate	2005
url	http://dspace.nbuv.gov.ua/handle/123456789/121060
citation_txt	Dynamics of dimer and z spin component fluctuations in spin-1/2 XY chain / O. Derzhko, T. Krokhmalskii, P. Hlushak // Condensed Matter Physics. — 2005. — Т. 8, № 4(44). — С. 859-867. — Бібліогр.: 12 назв. — англ.
series	Condensed Matter Physics
work_keys_str_mv	AT derzhkoo dynamicsofdimerandzspincomponentfluctuationsinspin12xychain AT krokhmalskiit dynamicsofdimerandzspincomponentfluctuationsinspin12xychain AT hlushakp dynamicsofdimerandzspincomponentfluctuationsinspin12xychain
first_indexed	2025-07-08T19:07:27Z
last_indexed	2025-07-08T19:07:27Z
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fulltext	Condensed Matter Physics, 2005, Vol. 8, No. 4(44), pp. 859–867 Dynamics of dimer and z spin component fluctuations in spin- 1/2 XY chain O.Derzhko, T.Krokhmalskii, P.Hlushak Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine, 1 Svientsitskii Str., 79011 Lviv, Ukraine Received August 15, 2005 One-dimensional quantum spin- 1/2 XY models admit the rigorous anal- ysis not only of their static properties (i.e. the thermodynamic quantities and the equal-time spin correlation functions) but also of their dynamic properties (i.e. the different-time spin correlation functions, the dynamic susceptibilities, the dynamic structure factors). This becomes possible after exploiting the Jordan-Wigner transformation which reduces the spin mod- el to a model of spinless noninteracting fermions. A number of dynamic quantities (e.g. related to transverse spin operator or dimer operator fluctu- ations) are entirely determined by two-fermion excitations and can be ex- amined in much detail. We consider the spin- 1/2 XY chain in a transverse (‖ z) magnetic field with the Hamiltonian H = ∑ n J ( sx nsx n+1 + sy nsy n+1 ) + ∑ n Ωsz n and calculate the dynamic structure factors SAB(κ, ω) = ∑ n exp (−iκn) ∫ ∞ −∞ dt exp (iωt) 〈Aj(t)Bj+n(0)〉 for the local spin operators {Am, Bm} = { sz m , Dm } where Dm = sx m sx m+1 + sy m sy m+1 is the dimer operator. The results for the dynamic transverse structure factor Szz(κ, ω) and for the dynamic dimer structure factor SDD(κ, ω) are known, whereas the analysis of the dynamic struc- ture factor SzD(κ, ω) = (SDz(κ, ω)) ? has not been reported so far. We compare different two-fermion dynamic quantities contrasting their generic and specific features. Key words: quantum spin chains, dynamic structure factors PACS: 75.10.-b The subject of analysis of the dynamic properties of low-dimensional quantum spin systems has attracted considerable interest for the recent years. On the one c© O.Derzhko, T.Krokhmalskii, P.Hlushak 859 O.Derzhko, T.Krokhmalskii, P.Hlushak hand, quite often the relevant quantities can be examined rigorously, especially if the space dimension is equal to one. This is important even if the models in question are simplified since conventional approximations usually fail after being applied to low-dimensional quantum spin systems. On the other hand, material science provi- des a number of magnetic materials which can be modelled using the spin-1/2 XXZ Heisenberg chains. Therefore, to interpret the experimental data obtained in neu- tron scattering experiments or resonance experiments for such compounds one needs a corresponding theory of the dynamic properties. A particular case of the spin- 1/2 XXZ Heisenberg chain, the XY chain, owing to the Jordan-Wigner fermion- ization trick can be investigated analytically, thus shedding light on the physical effects that may be observed in less tractable cases. In what follows, we consider the one-dimensional spin-1/2 XY model in a trans- verse (‖ z) external field [1] defined by the Hamiltonian H = N ∑ n=1 J ( sx ns x n+1 + sy ns y n+1 ) + N ∑ n=1 Ωsz n . (1) Here J is the exchange interaction constant (we will set further J = −1), Ω is the external magnetic field, sα = 1/2 ·σα, σα with α = x, y, z are the Pauli matrices, and N → ∞ is the number of sites. We imposed periodic boundary conditions in (1). After exploiting the Jordan-Wigner transformation the considered spin model can be presented in terms of noninteracting spinless fermions with the Hamiltonian H = ∑ κ Λκ ( c†κcκ − 1 2 ) , Λκ = Ω + J cos κ. (2) Here periodic boundary conditions are implied (the so-called boundary term is not important for the dynamic quantities examined below) and κ is the quasi-momentum which takes N values in the region from −π to π. Important information about the behavior of the system under small perturba- tions follows from the dynamic susceptibilities [2] χAB(κ, ω) = N ∑ n=1 exp (−iκn) ∫ ∞ 0 dt exp (i (ω + iε) t) 1 i 〈[Aj(t), Bj+n(0)]〉 , ε → +0. (3) Here Am, Bm are the local operators attached to the site m (for example, the spin operators sx m, sy m, sz m, the dimer operator Dm = sx msx m+1 + sy msy m+1 or the trimer operator Tm = sx msx m+2+sy msy m+2 [3–8]). Another quantities which reflect the dynamic properties of the system are the dynamic structure factors SAB(κ, ω) = N ∑ n=1 exp (−iκn) ∫ ∞ −∞ dt exp (iωt) 〈Aj(t)Bj+n(0)〉 . (4) SAB(κ, ω) is connected to the imaginary part of χAB(κ, ω) through the fluctuation- dissipation theorem. The imaginary and real parts of χAB(κ, ω) are connected by the 860 Dynamics of dimer and z spin component fluctuations Kramers-Kronig transformation. In what follows we focus on the dynamic structure factors (4). Introducing the fermionic representation (2) and exploiting the Wick-Bloch-de Dominicis theorem we easily calculate the two-spin correlation functions entering equations (3), (4) 〈sz n(t)sz n+l(0)〉 − 〈sz〉2 = 1 N2 ∑ κ1,κ2 exp (−i (κ1 − κ2) l) exp (i (Λκ1 − Λκ2 ) t) × nκ1 (1 − nκ2 ) , (5) 〈sz n(t)Dn+l(0)〉 − 〈sz〉〈D〉 = 1 N2 ∑ κ1,κ2 exp(−iκ1) + exp(iκ2) 2 exp (−i (κ1 − κ2) l) × exp (i (Λκ1 − Λκ2 ) t) nκ1 (1 − nκ2 ) , (6) 〈Dn(t)Dn+l(0)〉 − 〈D〉2 = 1 N2 ∑ κ1,κ2 cos2 κ1 + κ2 2 exp (−i (κ1 − κ2) l) × exp (i (Λκ1 − Λκ2 ) t) nκ1 (1 − nκ2 ) , (7) where nκ = (1 + exp (βΛκ)) −1 is the Fermi function and 〈sz〉 = 1 N N ∑ n=1 〈sz n〉 = − 1 2N ∑ κ tanh βΛκ 2 , 〈D〉 = 1 N N ∑ n=1 〈Dn〉 = − 1 2N ∑ κ cos κ tanh βΛκ 2 . The associated dynamic structure factors are obtained from equations (5)–(7) by Fourier transform. The resulting expressions for N → ∞ can be brought into the form Szz(κ, ω) − 2πNδκ,0δ(ω)〈sz〉2 = ∫ π −π dκ1nκ1 (1 − nκ1+κ) δ (ω + Λκ1 −Λκ1+κ) = ∑ κ? nκ? (1 − nκ+κ?) 2\|J sin κ 2 cos ( κ 2 + κ? ) \| , (8) SzD(κ, ω)−2πNδκ,0δ(ω)〈sz〉〈D〉 = exp ( i κ 2 ) ∑ κ? cos ( κ 2 + κ? ) nκ? (1 − nκ+κ?) 2\|J sin κ 2 cos ( κ 2 + κ? ) \| , (9) SDz(κ, ω) = (SzD(κ, ω))? , (10) SDD(κ, ω) − 2πNδκ,0δ(ω)〈D〉2 = ∑ κ? cos2 ( κ 2 + κ? ) nκ? (1 − nκ+κ?) 2\|J sin κ 2 cos ( κ 2 + κ? ) \| , (11) where −π 6 κ? 6 π are the solutions of the equation ω = −2J sin κ 2 sin (κ 2 + κ? ) . (12) 861 O.Derzhko, T.Krokhmalskii, P.Hlushak We can combine formulas (8)–(11) rewriting them in the form SAB(κ, ω) − 2πNδκ,0δ(ω)〈A〉〈B〉 = ∫ π −π dκ1dκ2C (2) AB(κ1, κ2)nκ1 (1 − nκ2 ) δ (ω + Λκ1 − Λκ2 ) δκ+κ1−κ2,0 = ∑ κ? C (2) AB(κ, κ?)nκ? (1 − nκ+κ?) 2\|J sin κ 2 cos ( κ 2 + κ? ) \| (13) with C(2) zz (κ, κ?) = 1, (14) C (2) zD(κ, κ?) = exp ( i κ 2 ) cos (κ 2 + κ? ) , (15) C (2) Dz (κ, κ?) = ( C (2) zD(κ, κ?) )? , (16) C (2) DD(κ, κ?) = cos2 (κ 2 + κ? ) . (17) The dynamic transverse structure factor Szz(κ, ω) was obtained by Th. Niemeijer (see [9]), the dynamic dimer structure factor SDD(κ, ω) was examined in [3–5]. Figure 1. The l. h. s. of equation (9) multiplied by exp (−i[κ/2]) for the spin chain (1) with J = −1 and Ω = 0.1 (a), Ω = 0.3 (b), Ω = 0.6 (c), Ω = 0.9 (d) at zero temperature β → ∞. 862 Dynamics of dimer and z spin component fluctuations Figure 2. The same as in figure 1 at β = 10. Note the different vertical scales in figure 1 and figure 2. Figure 3. The same as in figure 1 at β = 1. Note the different vertical scales in figure 1 and figure 3. To the best of our knowledge, the dynamic structure factor SzD(κ, ω) = (SDz(κ, ω))? has not been discussed so far. 863 O.Derzhko, T.Krokhmalskii, P.Hlushak In figures 1–3 we plot the dynamic structure factor SzD(κ, ω) at different tem- peratures. These plots demonstrate not only the specific features of the dynamic structure factor SzD(κ, ω) but also the generic features of all two-fermion dynamic structure factors (13). Figure 4. Two-fermion excitation continuum which governs the ground-state dy- namic structure factors (13). We set \|J \| = 1 and Ω = 0.1 (a) Ω = 0.3 (b), Ω = 0.6 (c), Ω = 0.9 (d). The lower boundary (18) (bold lines), the middle boundary (19) (dashed lines), the upper boundary (20) (thin lines), the line of potential singularities (21) (dotted lines). From equation (13) it is clearly seen that the dynamic structure factors (8), (9), (10), (11) are governed entirely by the two-fermion (particle-hole) excitation continuum the properties of which were examined in [9,10]. Hereinafter we briefly account for these results. At zero temperature β → ∞ the two-fermion excitation continuum exists only if \|Ω\| < \|J \| and has the following lower, middle and upper boundaries in the plane wave-vector κ – frequency ω ωl \|J \| = 2 sin \|κ\| 2 ∣ ∣ ∣ ∣ sin ( \|κ\| 2 − α )∣ ∣ ∣ ∣ , (18) ωm \|J \| = 2 sin \|κ\| 2 sin ( \|κ\| 2 + α ) , (19) 864 Dynamics of dimer and z spin component fluctuations ωu \|J \| = { 2 sin \|κ\| 2 sin ( \|κ\| 2 + α ) , if 0 6 \|κ\| 6 π − 2α, 2 sin \|κ\| 2 , if π − 2α 6 \|κ\| 6 π, (20) respectively; here we have introduced the parameter α = arccos(Ω/\|J \|) which varies from π when Ω = −\|J \| to 0 when Ω = \|J \|. The region ωl 6 ω 6 ωm corresponds to the lower two-fermion excitation continuum, whereas the region ωm 6 ω 6 ωu corre- sponds to the upper two-fermion excitation continuum. The ω-profiles at fixed κ of the two-fermion dynamic structure factors (13) may exhibit Van Hove’s singularities (as the density of one-particle states in one dimension) when ω → ωs − 0, ωs \|J \| = 2 sin \|κ\| 2 . (21) Figure 4 shows expressions (18)–(21) plotted for \|J \| = 1 and various Ω. As temper- ature increases the lower boundary of the two-fermion excitation continuum smears out, i.e. ωl = 0, and the upper boundary becomes ωu = 2\|J \| sin(\|κ\|/2). The specific features of different dynamic structure factors are connected with the explicit form of the function C (2) AB(κ, κ?) (14)–(17). Thus, the dynamic dimer structure factor vanishes (and hence does not diverge) along ωs (21) since C (2) DD(κ, κ?) (17) cancels the zero in the denominator in equation (13). This is contrary to the case of the dynamic transverse structure factor with C (2) zz (κ, κ?) (14). Moreover, the zero- temperature dynamic structure factor SzD(κ, ω) is zero in the upper two-fermion excitation continuum, i.e. for ωm 6 ω 6 ωu. For this region of the κ–ω plane the two roots κ? 1 and κ? 2 of equation (12) yield in equation (13) two contributions of the same value but with opposite signs. At nonzero temperatures the values of these contributions become different and SzD(κ, ω) deviates from zero in the upper two- fermion excitation continuum. In the lower two-fermion excitation continuum one of the roots does not contribute at zero temperature due to the Fermi functions in equation (13). However, when temperature tends to infinity β → 0 the restrictions owing to the Fermi functions in equation (13) disappear and two roots κ? 1 and κ? 2 come into play for all 0 6 ω 6 ωu. They again have the same value but opposite signs and as a result, SzD(κ, ω) disappears within the whole two-fermion excitation continuum in the κ–ω plane. This behavior is illustrated in figures 1–3. Thus, all the considered dynamic quantities exhibit the same generic properties controlled by the δ-functions and the Fermi functions in equation (13) (lower, mid- dle and upper boundaries, soft modes, Van Hove’s singularities) and some specific properties controlled by the function C (2) AB(κ, κ?) (zero values in the accessible region of the κ–ω plane, disappearance of Van Hove’s singularities). Our results may be important from the theoretical point of view helping to understand the dynamic properties of quantum spin chains. Thus, equation (13) gives a hint to the form of multi-fermion excitation continua contributions to dynamic quantities (see [7,8]). On the other hand, we note that spin-1/2 XY chains are realized in some quasi-one- dimensional magnetic insulators [11,12], and hence our findings may have a relation to the experimental data obtained in dynamic experiments. 865 O.Derzhko, T.Krokhmalskii, P.Hlushak This study was supported by the STCU under the project #1673. The authors thank Prof. Joachim Stolze and Prof. Gerhard Müller for collaboration on closely related subjects [7,8]. The paper was partially presented at the International Con- ference on Strongly Correlated Electron Systems (Vienna, July 26 - 30, 2005). O. D. and T. K. thank the Organizing Committee of the SCES’05 for the support to attend the conference. References 1. Lieb E., Schultz T., Mattis D., Ann. Phys. (N.Y.), 1961, 16, 407. 2. Zubarev D.N. Njeravnovjesnaja Statistichjeskaja Tjermodinamika. Nauka, Moskva, 1971 (in Russian). 3. Suzuura H., Yasuhara H., Furusaki A., Nagaosa N., Tokura Y., Phys. Rev. Lett., 1996, 76, 2579. 4. Yongmin Yu, Müller G., Viswanath V.S., Phys. Rev. B, 1996, 54, 9242. 5. Lorenzana J., Eder R., Phys. Rev. B, 1997, 55, R3358. 6. Werner R., Phys. Rev. B, 2001, 63, 174416. 7. Derzhko O., Krokhmalskii T., Stolze J., Müller G., Phys. Rev. B, 2005, 71, 104432. 8. Derzhko O., Krokhmalskii T., Stolze J., Müller G., to appear in Physica B, 2005. 9. Müller G., Thomas H., Beck H., Bonner J.C., Phys. Rev. B, 1981, 24, 1429. 10. Taylor J.H., Müller G., Physica A, 1985, 130, 1. 11. Kenzelmann M., Coldea R., Tennant D.A., Visser D., Hofmann M., Smeibidl P., Tyl- czynski Z., Phys. Rev. B, 2002, 65, 144432. 12. Rainford B.D., private communication (2005). 866 Dynamics of dimer and z spin component fluctuations Динаміка флуктуацій димерного оператора і оператора z спінової компоненти у спін-1/2 XY ланцюжку О.Держко, Т.Крохмальський, П.Глушак Інститут фізики конденсованих систем НАН України, 79011 Львів, вул. Свєнціцького, 1 Отримано 15 серпня 2005 р. Одновимірні квантові спін- 1/2 XY моделі допускають строгий аналіз не лише їх статичних властивостей (тобто термодинамічних величин і однаковочасових спінових кореляційних функцій), але також і їх динамічних властивостей (тобто різночасових спінових кореляційних функцій, динамічних сприйнятливостей, динамічних структурних факторів). Це стає можливим після використання перетворення Йордана-Вігнера, яке зводить спінову модель до моделі безспінових невзаємодіючих ферміонів. Ряд динамічних величин (наприклад, пов’язаних з флуктуаціями оператора по- перечної спінової компоненти чи димерного оператора) цілком визначаються двоферміонними збудженнями і можуть бути деталь- но вивчені. Ми розглядаємо спін- 1/2 XY ланцюжок у поперечному (‖ z) магнітному полі з гамільтоніаном H = ∑ n J ( sx nsx n+1 + sy nsy n+1 ) + ∑ n Ωsz n і обчислюємо динамічні структурні фактори SAB(κ, ω) = ∑ n exp (−iκn) ∫ ∞ −∞ dt exp (iωt) 〈Aj(t)Bj+n(0)〉 для локальних спінових операторів {Am, Bm} = { sz m , Dm } , де Dm = sx m sx m+1 + sy m sy m+1 є димерним оператором. Результати для динамічного поперечного структурного фактора Szz(κ, ω) і для динамічного димерного структурного фактора SDD(κ, ω) є відомі, тоді ж як динамічний структурний фактор SzD(κ, ω) = (SDz(κ, ω)) ? досі не був проаналізований. Ми порівнюємо різні двоферміонні динамічні величини, співставляючи їхні загальні і специфічні влас- тивості. Ключові слова: квантові спінові ланцюжки, динамічні структурні фактори PACS: 75.10.-b 867 868

Dynamics of dimer and z spin component fluctuations in spin-1/2 XY chain

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