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home | Authors | Referees | Browse | Announcement | Download | Subscription | Contact us | CPS Journals | Chinese Dear authors, Thank you very much for your contribution to Chinese Physics B. Your paper has been published in Chinese Physics B, 2014, Vol.23, No.8. Attached is the PDF offprint of your published article, which will be convenient and helpful for your communication with peers and coworkers. Readers can download your published article through our website http://www.iop.org/cpb or http://cpb.iphy.ac.cn What follows is a list of related articles published recently in Chinese Physics B. Dynamics of one-dimensional random quantum XY system with Dzyaloshinskii–Moriya interaction Li Yin-Fang, Kong Xiang-Mu Chin. Phys. B , 2013, 22(3): 037502.Full Text: PDF (363KB) Effective-field and Monte Carlo studies of mixed spin-2 and spin-1/2 Ising diamond chain Liu Wei-Jie, Xin Zi-Hua, Chen Si-Lun, Zhang Cong-Yan Chin. Phys. 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B Vol. 23, No. 8 (2014) 087505 The spin dynamics of the random transverse Ising chain with a double-Gaussian disorder∗ Liu Zhong-Qiang(刘中强)†, Jiang Su-Rong(姜素蓉), and Kong Xiang-Mu(孔祥木) College of Physics and Engineering, Qufu Normal University, Qufu 273165, China (Received 3 January 2014; revised manuscript received 26 March 2014; published online 10 June 2014) The dynamical properties of one-dimensional random transverse Ising model (RTIM) with a double-Gaussian disorder is investigated by the recursion method. Based on the first twelve recurrences derived analytically, the spin autocorrelation function (SAF) and associated spectral density at high temperature were obtained numerically. Our results indicate that when the standard deviation sJ (or sB) of the exchange couplings Ji (or the random transverse fields Bi) is small, no long-time tail appears in the SAF. The spin system undergoes a crossover from a central-peak behavior to a collective- mode behavior, which is the dynamical characteristics of RTIM with the bimodal disorder. However, when sJ (or sB) is large enough, the system exhibits similar dynamics behaviors to those of the RTIM with the Gaussian disorder, i.e., the system exhibits an enhanced central-peak behavior for large sJ or a disordered behavior for large sB. In this instance, −2 SAFs exhibit a similar long-time tail, i.e., C(t) ∼ t for large t. Similar properties are obtained when Ji (or Bi) satisfy the double-exponential distribution or the double-uniform distribution. Besides, when both the standard deviations and the mean values of the exchange couplings are small, the effects of the Gaussian random bonds may drive the system undergo two crossovers from a triplet state to a doublet state, and then to a collective-mode state. Keywords: random transverse Ising model, spin autocorrelation function, spectral density, long-time tail PACS: 75.10.Pq, 75.40.Gb, 75.10.Jm, 75.50.Lk DOI: 10.1088/1674-1056/23/8/087505 1. Introduction sults indicate that the disorder effects may drastically slow down the behavior of the disordered averaged spin–spin cor- The time-dependent behavior of quantum spin systems relation function,[25] including the SAFs. For example, as has been the subject of theoretical studies for quite a long time. xx However, most literature deals with the dynamic correlation the couplings vary, the Gaussian behavior of -relaxation of functions of the pure quantum spin systems. It is found that the single impurity in XY chain slows down to the stretched [27] the spin autocorrelation functions (SAFs) of these pure sys- exponential-like relaxation at T = ¥. However, as far as tems are associated with the temperature. The transverse cor- we know, although the short-time behaviors of the SAFs have [15–19,21–23] relation functions show a power-law behavior at T = 0,[1–3] an been reported, the long-time behaviors of them are exponential behavior at 0 < T < ¥,[4,5] and a Gaussian behav- still not understood. An important issue in this work is to ior at T = ¥,[6–9] respectively. As to the longitudinal correla- explore whether it is possible to realize a situation in which tion functions, Niemeijer[10] found that SAF of XY spin chain one may obtain the long-time behavior of SAFs of the ran- is a damped oscillation function which may be expressed by dom transverse Ising chain, where the exchange couplings and the square of the Bessel function at T = ¥, but it has a more the transverse fields are independent random variable satisfy- complex expression for other temperature.[11] For the spin-1/2 ing a double-Gaussian distribution. The answer is positive. van der Waals model, Dekeyser and Lee[12] pointed out that Moreover, the spectral density of SAF, reflecting the dynami- the power-law long-time tails are found in the transverse com- cal properties of spin systems, was also calculated in this work. ponent only at high temperature. Calculational results exhibit more details of the dynamics of Only recently, since the disorder effects besides the ef- RTIM. fects of the temperature has been shown to affect the dynam- The remainder of this paper is organized as follows. The ics behavior of spin systems in a drastic way providing a very RTIM with the double-Gaussian disorder and the recursion rich area of investigation, more and more attention has been method will be introduced in Section 2. Section 3 presents the paid to the random quantum spin systems, especially on the calculational results of the SAFs and the corresponding spec- low-dimensional systems, such as the random transverse Ising tral densities for two types models: a random-bonds model and model (RTIM),[13–20] XY chain,[21–24] XX and XXZ chain,[25] a random-fields model. The related discussion is also given in one-dimensional Blume–Capel model,[26] etc. The known re- this section. Finally, the summary is presented in Section 4. ∗Project supported by the National Natural Science Foundation of China (Grant Nos. 11302118 and 11275112), the Natural Science Foundation of Shan- dong Province of China (Grant Nos. ZR2013AQ015 and ZR2011AM018), and the Postdoctoral Science Foundation of Qufu Normal University (Grant No. BSQD2012053). †Corresponding author. E-mail: [email protected] © 2014 Chinese Physical Society and IOP Publishing Ltd http://iopscience.iop.org/cpb http://cpb.iphy.ac.cn 087505-1 Chin. Phys. B Vol. 23, No. 8 (2014) 087505 2. Model and the recursion method When p = 1 (or p = 0), equation (2) becomes a standard Gaus- 2.1. RTIM with the double-Gaussian disorder sian distribution. Of course, r1(bi) and r2(bi) may be other distributions, such as an exponential distribution or a uniform One-dimensional (1D) RTIM is one of simplest but non- distribution. In this way, r(bi) becomes a double-exponential trivial example of the random quantum spin chains. It is de- distribution or a double-uniform distribution. fined by the following Hamiltonian: 2.2. Recursion method 1 1 H = − J s xs x − B s z; (1) ∑ i i i+1 ∑ i i The basic tool employed in this work is the recursion 2 i 2 i method,[34] which is very powerful in the study on many-body where 1=2 is the constant number introduced for the conve- dynamics.[9,15,16,21–23,35–46] It may help us obtain the SAF and x z nience of the derivation. si , si denote Pauli matrices at site associated spectral density. We will review this method briefly i. The exchange couplings Ji (or the transverse fields Bi) are as follows. independent random variables obeying the distribution r(Ji) x For RTIM, the time evolution of s j (t) is governed by the [or r(Bi)]. This model has a well-known physical interpreta- Liouville equation tion in connection not only with some quasi-one-dimensional ds x (t)=dt = iLs x (t); (6) hydrogen-bonded ferroelectric crystals like Cs(H1−xDx)2PO4, j j [28–31] PbH1−xDxPO4, etc., but also with Ising spin glasses where L is the Liouville operator defined by LA = [H;A] ≡ [32] LiHo0:167Y0:833F4. Besides, this model may capture vital HA−AH. H denote the Hamiltonian of the spin system. Then [20,33] features of the neutron scattering experiment in LiHoF4. the solution of Eq. (6) can be formally given as In this work, we mainly investigate the dynamics of RTIM ¥ with the double-Gaussian disorder, i.e., J or B are indepen- x i i s j (t) = ∑ an (t) fn ; (7) dent variables, which satisfy respectively a double-Gaussian n=0 distribution where f fn g is a complete orthogonal sequence in the Hilbert space S, fan (t)g is a set of time-dependent real functions. r(b ) = pr (b ) + (1 − p)r (b ); (2) i 1 i 2 i They respectively satisfy two recurrence relations,[34] i.e., where fn+1 = iL fn + Dn fn−1; n ≥ 0; (8) 1 h 2 2i r1(bi) = p exp −(bi − b1m) =2sb (3) and 2psb D a (t) = −da (t)=dt + a (t); n ≥ 0; (9) and n+1 n+1 n n−1 1 h i where f−1 ≡ 0, a−1 (t) ≡ 0, the coefficients (recurrences) are ( ) = p −( − )2= 2 r2 bi exp bi b2m 2sb (4) defined as 2psb Dn = ( fn ; fn )=( fn−1; fn−1) (10) are the standard Gaussian distributions, wherein b1m (J1m or B1m) and b2m (J2m or B2m) denote the corresponding mean val- with D0 ≡ 1.