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

Abstract

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>i</i>, <i>J</i> is the nearest-neighbor exchange coupling. <i>B<sub>iz</sub></i> and <i>B<sub>ix</sub></i> 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 <i>p</i>, <i>q</i> and <i>r</i> is [0,1], and the coefficients satisfy the constraint condition <i>p</i>+<i>q</i>+<i>r</i>=1. For the case of trimodal random <i>B<sub>iz</sub></i> (consider <i>B<sub>iz</sub></i>≡0 for simplicity), the exchange couplings are supposed to be <i>J</i>≡0 to fix the energy scale. The reference values are set <i>B<sub>p</sub></i>=0.5<<i>J</i> and <i>B<sub>q</sub></i>=1.5><i>J</i>. The coefficient <i>r</i> can be considered as the proportion of non-magnetic impurities. When <i>r</i>=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 <i>q</i>increases, which consistent with the previous work. As <i>r</i> increases, the crossover between different dynamical behaviors changes obviously (e.g., the crossover from central-peak to double-peak when <i>r</i>=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., <i>r</i>=0.4, 0.6 or 0.8. However, the multi-peak behavior disappear when <i>r</i>→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 <i>qB<sub>q</sub></i>=<i>pB<sub>p</sub></i>, the central-peak behavior can be maintained. What makes sense is that the conclusion is universal. For the case of trimodal random <i>B<sub>ix</sub></i>, the coefficient <i>r</i> is no longer represent the proportion of non-magnetic impurities when <i>B<sub>ix</sub></i> and <i>B<sub>iz</sub></i> (<i>B<sub>iz</sub></i>≡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 <i>B<sub>p</sub></i>=0.5<<i>B<sub>iz</sub></i> and <i>B<sub>q</sub></i>=0.5<<i>B<sub>iz</sub></i>. When <i>r</i>is small (<i>r</i>=0, 0.2 or 0.4), the system undergoes a crossover from the collective-mode behavior to the double-peak behavior as <i>q</i> increases. However, the low-frequency responses gradually disappear, while the high-frequency responses are maintained as <i>r</i> increases. Take the case of <i>r</i>=0.8 for example, the system only behaves a collective-mode behavior. The results indicate that increasing <i>r</i> is no longer conducive to the low-frequency response, which is contrary to the case of trimodal random <i>B<sub>ix</sub></i>. The <i>r</i> branch only regulates the intensity of the trimodal random <i>B<sub>ix</sub></i>. Our results indicate that using trimodal random magnetic field to manipulate the spin dynamics of the Ising system may be a new try.

Prediction of tensile strength of basalt continuous fiber from chemical composition using machine learning models

“…Literature research shows that the chemical composition of basalt fiber, calculated by oxide content, has a significant impact on tensile strength 17,22,23 . Furthermore, acidity modulus (Ma), viscosity modulus (Mv), and the degree of depolymerization of the melt, expressed by NBO/T, which is calculated by the ratio of the molar content of non‐bridging oxygen (NBO) to the tetrahedrally coordinated cation (T) are closely associated with the basalt melt and mechanical properties of BCF 42–46 . Therefore, oxides and their derived parameters of Ma, Mv, and NBO/T are chosen as the feature variables to predict the target variable of tensile strength.…”