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Resummation of the Holstein-Primakoff expansion and differential equation approach to operator square roots
Physical Review Research Pub Date : 2020-11-17 , DOI: 10.1103/physrevresearch.2.043243
Michael Vogl , Pontus Laurell , Hao Zhang , Satoshi Okamoto , Gregory A. Fiete

Operator square roots are ubiquitous in theoretical physics. They appear, for example, in the Holstein-Primakoff representation of spin operators and in the Klein-Gordon equation. Often the use of a perturbative expansion is the only recourse when dealing with them. In this paper, we show that under certain conditions, differential equations can be derived which can be used to find perturbatively inaccessible approximations to operator square roots. Specifically, for the number operator n̂=aa we show that the square root n̂ near n̂=0 can be approximated by a polynomial in n̂. This result is unexpected because a Taylor expansion fails. A polynomial expression in n̂ is possible because n̂ is an operator, and its constituents a and a have a non trivial commutator [a,a]=1 and do not behave as scalars. We apply our approach to the zero-mass Klein-Gordon Hamiltonian in a constant magnetic field and, as a main application, the Holstein-Primakoff representation of spin operators, where we are able to find new expressions that are polynomial in bosonic operators. We prove that these new expressions exactly reproduce spin operators. Our expressions are manifestly Hermitian, which offers an advantage over other methods, such as the Dyson-Maleev representation.

中文翻译:

算子平方根的Holstein-Primakoff展开和微分方程方法的恢复

算子平方根在理论物理学中无处不在。例如,它们出现在自旋算子的Holstein-Primakoff表示形式和Klein-Gordon方程中。通常,在与它们打交道时,仅使用扰动扩展即可。在本文中,我们表明在某些条件下,可以导出微分方程,该微分方程可用于找到算子平方根的摄动不可访问的近似。具体来说,对于数字运算符ñ̂=一种一种 我们证明平方根 ñ̂ñ̂=0 可以由以下多项式近似 ñ̂。由于泰勒展开失败,所以此结果是意外的。的多项式表达式ñ̂ 之所以可能是因为 ñ̂ 是一个运算符,及其组成部分 一种一种 有一个非平凡的换向器 [一种一种]=1个并且不表现为标量。我们将方法应用于恒定磁场下的零质量Klein-Gordon哈密顿量,并且作为主要应用是自旋算子的Holstein-Primakoff表示,在其中我们能够找到在玻色子算子中多项式的新表达式。我们证明这些新表达式恰好再现了自旋运算符。我们的表达式显然是Hermitian,这比其他方法(例如Dyson-Maleev表示法)更具优势。
更新日期:2020-11-17
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