The random field Ising model in a shifted bimodal probability distribution

Hadjiagapiou, Ioannis A.; Velonakis, Ioannis N.

doi:10.1016/j.physa.2018.04.018

Cited by 4 publications

(1 citation statement)

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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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