Investigating magnetic materials at realistic conditions with first-principles methods is a challenging task due to the interplay of vibrational and magnetic degrees of freedom. The most difficult contribution to include in simulations is represented by the longitudinal magnetic degrees of freedom [longitudinal spin fluctuation (LSF)] due to their inherent many-body nature; nonetheless, schemes that enable to take into account this effect on a semiclassical level have been proposed and employed in the investigation of magnetic systems. However, assessment of the effect of vibrations on LSF is lacking in the literature. For this reason, in this work we develop a supercell approach within the framework of constrained density functional theory to calculate self-consistently the size of local-environment-dependent magnetic moments in the paramagnetic, high-temperature state in the presence of lattice vibrations and for liquid Fe in different conditions. First, we consider the case of bcc Fe at the Curie temperature and ambient pressure. Then, we perform a similar analysis on bcc Fe at Earth's inner-core conditions, and we find that LSFs stabilize nonzero moments which affect atomic forces and the electronic density of states of the system. Finally, we employ the present scheme on liquid Fe at the melting point at ambient pressure and at Earth's outer-core conditions ($p\approx 200$ GPa, $T\approx 6000$ K). In both cases, we obtain local magnetic moments of sizes comparable to the solid-state counterparts.

中文翻译：

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使用超级电池方法计算的高温和高压下bcc和液态Fe的纵向自旋波动

由于振动和磁自由度的相互作用，使用第一原理方法在现实条件下研究磁性材料是一项艰巨的任务。由于其固有的多体性质，最难包括在仿真中的是纵向磁自由度[纵向自旋涨落（LSF）]。然而，已经提出了能够在半经典水平上考虑到这种影响的方案，并在磁性系统的研究中采用了这些方案。但是，文献中缺乏对振动对LSF的影响的评估。因此，在这项工作中，我们在约束密度泛函理论的框架内开发了一种超级单元方法，可以自洽计算顺磁中与局部环境有关的磁矩的大小，在晶格振动的存在下以及在不同条件下的液态铁的高温状态。首先，我们考虑在居里温度和环境压力下bcc Fe的情况。然后，我们在地球内核条件下对bcc Fe进行了类似的分析，我们发现LSF稳定了影响原子力和系统状态电子密度的非零矩。最后，我们在环境压力和地球外核条件下在熔点上使用液态铁的本方案（并且我们发现LSF使非零阶矩稳定下来，这些矩会影响原子力和系统状态的电子密度。最后，我们在环境压力和地球外核条件下在熔点上的液态铁上采用本方案（并且我们发现LSF使非零阶矩稳定下来，这些矩会影响原子力和系统状态的电子密度。最后，我们在环境压力和地球外核条件下在熔点上的液态铁上采用本方案（$p\approx 200$ GPa， $Ť\approx 6000$ K）。在这两种情况下，我们得到的局部磁矩的大小都可以与固态对应的磁矩相媲美。