The random field Ising model in an asymmetric and anisotropic linearly field-dependent bimodal probability distribution

Hadjiagapiou, Ioannis A.; Velonakis, Ioannis N.

doi:10.1016/j.physa.2021.126112

Cited by 2 publications

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References 52 publications

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“…相变 [1] ，诱导 Ising-like 反铁磁体 BaCo 2 V 2 O 8 发生铁磁-反铁磁相变 [2] ，可以快速抑制 Ising-like 反铁磁体 SrCo 2 V 2 O 8 的 Né el 温度等 [3] 。在 Ising-like 光学晶格中，通过改变纵向磁场(纵场)可以引起顺磁-反铁磁相变 [4] 。而横场和纵场共同存在时的混合磁场(又称倾斜磁场)对自旋系统性质的调控更为明显，研究发现混合磁场对系统的基态相图 [5,6] 、热力学性质 [7] 、临界行为 [8,9] 、量子相变 [10] 和动力学行为 [11] 等现象的影响明显不同于外加磁场只有横场或纵场时的结果。截至目前，对于纯自旋系统已有大量的研究，且取得了丰硕的成果。但对随机自旋系统的研究仍然处于发展阶段，这是由于实际材料中的无序效应非常复杂，有时根本没有规律可循。不同于纯自旋系统，在随机自旋系统中，系统的自旋-自旋耦合作用或外加磁场不再为常数，而是被取为满足某种概率分布的随机数。从数学角度来看，随机分布的选取是任意的，常被采用的几种典型随机分布是离散型的双模分布 [12][13][14][15] 、连续型的高斯分布 [16,17] 和更广义的双高斯分布 [18,19] 等。但从物理角度来看，随机分布的选取还要考虑实际材料的性质。比如晶格中的非磁性杂质(非磁性自旋、空位或缺陷)不会受到外磁场的影响，且非磁性杂质的存在会降低系统的随机性等 [20] 。因此可以将纯数学的随机分布进行扩展，间接引入某些实际因素，例如改变版的双模分布 [21] 、各项异性场依赖的双模分布 [22] 、对称或非对称的三模随机分布等 [23][24][25] 。由于三模分布可以蜕化为双模分布，且在特殊参数下(p=1/3)是高斯分布的一个很好的近似 [26,27] ，因此备受人们关注。一些稀释反铁磁体(如 Fe x Zn 1-x F 2 )，氰化物晶体(如 X(CN) x Y 1-x )， X =K,Na,Rb，Y=Br,Cl,I)等材料在外磁场下的性质，可以采用三模分布进行描述 [20,23] 。目前人们对于 Ising 模型在三模型随机外场下…”

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Effects of trimodal random magnetic field on spin dynamics of quantum Ising chain

Yuan¹

2023

Acta Phys. Sin.

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It is of fundamental importance to know the dynamics of quantum spin systems immersed in external magnetic fields. In this work, the dynamical properties of one-dimensional quantum Ising model with trimodal random transverse and longitudinal magnetic fields are investigated by the recursion method. The spin correlation function \[C\left( t \right) = \overline {\left\langle {\sigma _j^x\left( t \right)\sigma _j^x\left( 0 \right)} \right\rangle } \] and the corresponding spectral density \[\emptyset \left( \omega \right) = \int_{ -\infty }^{ + \infty } {dt{e^{iwt}}} C\left( t \right)\] are calculated. The model Hamiltonian can be written as \[H = -\frac{1}{2}J\sum\limits_i^N {\sigma _i^x\sigma _{i + 1}^x} -\frac{1}{2}\sum\limits_i^N {{B_{iz}}\sigma _i^z} -\frac{1}{2}\sum\limits_i^N {{B_{ix}}\sigma _i^x} \]. Where <img border=0 > are Pauli matrices at site i, J is the nearest-neighbor exchange coupling. Biz and Bix denote the transverse and longitudinal magnetic field, respectively. They satisfy the following trimodal distribution, \[\rho = \left( {{B_{iz}}} \right) = p\delta \left( {{B_{iz}} -{B_p}} \right) + q\delta \left( {{B_{iz}} -{B_p}} \right) + r\delta \left( {{B_{iz}}} \right)\] and \[\rho = \left( {{B_{i{\rm{x}}}}} \right) = p\delta \left( {{B_{ix}} -{B_p}} \right) + q\delta \left( {{B_{ix}} -{B_p}} \right) + r\delta \left( {{B_{ix}}} \right)\].The value interval of the coefficients p, q and r is [0,1], and the coefficients satisfy the constraint condition p+q+r=1. For the case of trimodal random Biz (consider Biz≡0 for simplicity), the exchange couplings are supposed to be J≡0 to fix the energy scale. The reference values are set Bp=0.5<J and Bq=1.5>J. The coefficient r can be considered as the proportion of non-magnetic impurities. When r=0, the trimodal distribution degrades into the bimodal distribution. The dynamics of the system exhibits a crossover from the central-peak behavior to the collective-mode behavior as qincreases, which consistent with the previous work. As r increases, the crossover between different dynamical behaviors changes obviously (e.g., the crossover from central-peak to double-peak when r=0.2), and the presence of non-magnetic impurities favors low-frequency response. Due to the competition between the non-magnetic impurities and transverse magnetic field, the system tends to exhibit multi-peak behavior in most cases, e.g., r=0.4, 0.6 or 0.8. However, the multi-peak behavior disappear when r→1. That is because the system's response to the transverse field is limited when the proportion of non-magnetic impurities are large enough. Interestingly, when the parameters satisfy qBq=pBp, the central-peak behavior can be maintained. What makes sense is that the conclusion is universal. For the case of trimodal random Bix, the coefficient r is no longer represent the proportion of non-magnetic impurities when Bix and Biz (Biz≡1) coexist here. In the case of weak exchange coupling, the effects of longitudinal magnetic field on spin dynamics are obvious, so J≡0.5 is set here. The reference values are set Bp=0.5<Biz and Bq=0.5<Biz. When ris small (r=0, 0.2 or 0.4), the system undergoes a crossover from the collective-mode behavior to the double-peak behavior as q increases. However, the low-frequency responses gradually disappear, while the high-frequency responses are maintained as r increases. Take the case of r=0.8 for example, the system only behaves a collective-mode behavior. The results indicate that increasing r is no longer conducive to the low-frequency response, which is contrary to the case of trimodal random Bix. The r branch only regulates the intensity of the trimodal random Bix. Our results indicate that using trimodal random magnetic field to manipulate the spin dynamics of the Ising system may be a new try.

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Effects of trimodal random magnetic field on spin dynamics of quantum Ising chain

Yuan¹

2023

Acta Phys. Sin.

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Effects of Link-impurity on spin dynamics of the one-dimensional quantum Ising model

Yuan Xiao-Juan

2025

Acta Phys. Sin.

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It is of considerable theoretical interest to study the effects of impurity on spin dynamics of quantum spin systems. In this paper, the dynamical properties of the one-dimensional quantum Ising model with symmetric and asymmetric link-impurity are investigated by the recursion method, respectively. The autocorrelation function $C(t)=\overline{\left\langle\sigma_j^x(t) \sigma_j^x(0)\right\rangle}$ and the associated spectral density $\Phi(\omega)=\int_{-\infty}^{+\infty} d t e^{i \omega t} C(t)$ are calculated. The Hamiltonian of the Ising model with link-impurity can be written as $H=-\frac{1}{2}\left(J_{j-1} \sigma_{j-1}^x \sigma_j^x+J_j \sigma_j^x \sigma_{j+1}^x\right)-\frac{1}{2} J \sum_{i \neq j, j-1}^N \sigma_i^x \sigma_{i+1}^x-\frac{1}{2} B \sum_i^N \sigma_i^z$. Where J is the nearest-neighbor exchange coupling of the main spin chain, B denotes the external transverse magnetic field, $\sigma_i^\alpha(\alpha=x, y, z)$ are Pauli matrices at site i.The constant 1/2 is introduced for the convenience of theoretical deduction, and N is the number of spins. The so-called link-impurity Jj(Jj-1) is randomly introduced, which denotes the exchange coupling between the jth spin and the (j+1)th spin (the (j-1)th spin).The symmetric and asymmetric link-impurity correspond to the case of Jj-1=Jj and Jj-1≠Jj, respectively. The periodic boundary conditions are assumed in the theoretical calculation. After introducing the link-impurity, the original competition between B and J in the pure Ising model was broken. The dynamics of the system depends on synergistic effect of multiple factors, such as the mean spin coupling $\bar{J}$ between j and the link-impurity, the asymmetry degree between Jj-1 and Jj,and the strength of the external magnetic field. In calculation, the exchange couplings of the main spin chain are set $J \equiv 1$ to fix the energy scale. We first consider the effects of symmetric link-impurity, the reference values can be set Jj-1=Jj＜J(e.g., 0.4, 0.6 or 0.8) or Jj-1=Jj＞J(e.g., 1.2, 1.6, 2.0),which are called weak or strong impurity coupling. When the magnetic field B≥J(e.g.B=1, 1.5 or 2.0),it is found that the dynamics of the system exhibits a crossover from a collective-mode behavior to a central-peak behavior as the impurity strength Jj-1=Jj increase. Interestingly, for B＜J(e.g., B=4 or 0.7),there are two crossovers that is a collective-mode-like behavior to a double-peak behavior, then to a central-peak one as Jj-1=Jj increase. For the case of asymmetric link-impurity, the impurity configuration is more complex. Using the cooperation between Jj-1 and Jj,more freedom of regulation can be provided and the dynamical properties are more abundant. For the case of B≤J(e.g., B=0.5, 1.0),the system tends to exhibit a collective-mode behavior when the mean spin coupling $\bar{J}$,are weak, and a central-peak behavior when $\bar{J}$ are strong. However, when the asymmetry between Jj-1 and Jj is obvious, the system tends to exhibit a double- or multi-peak behavior. For the case of B＞J(e.g., B=1.5, 2.0),when $\bar{J}$ are strong, it tends to exhibit a central-peak behavior. However, when the asymmetry between Jj-1 and Jj is evident, the bispectral feature (two spectral peak appear at $\omega_1 \neq 0$ and $\omega_2 \neq 0$) dominates the dynamics. Under the regulating effect of link-impurities, the crossover between different dynamic behaviors can be easily realized, and it is easier to stimulate new dynamic modes, such as the double-peak behavior, the collective-mode-like behavior or bispectral feature one. The results indicate that using link-impurity to manipulate the dynamics of quantum spin systems may be a new try.

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The random field Ising model in an asymmetric and anisotropic linearly field-dependent bimodal probability distribution

Cited by 2 publications

References 52 publications

Effects of trimodal random magnetic field on spin dynamics of quantum Ising chain

Effects of trimodal random magnetic field on spin dynamics of quantum Ising chain

Effects of Link-impurity on spin dynamics of the one-dimensional quantum Ising model

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