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Title: Resummation of the Holstein-Primakoff expansion and differential equation approach to operator square roots

Abstract

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 $$\hat{n}$$ = aa we show that the square root √$$\hat{n}$$ near $$\hat{n}$$ = 0 can be approximated by a polynomial in $$\hat{n}$$. This result is unexpected because a Taylor expansion fails. A polynomial expression in $$\hat{n}$$ is possible because $$\hat{n}$$ 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.

Authors:
ORCiD logo; ORCiD logo; ; ORCiD logo; ORCiD logo
Publication Date:
Research Org.:
Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); USDOE Office of Science (SC), Basic Energy Sciences (BES). Materials Sciences & Engineering Division
OSTI Identifier:
1716543
Alternate Identifier(s):
OSTI ID: 1720239
Grant/Contract Number:  
AC05-00OR22725; FG02-13ER41967; DMR-1720595; DMR-1949701
Resource Type:
Published Article
Journal Name:
Physical Review Research
Additional Journal Information:
Journal Name: Physical Review Research Journal Volume: 2 Journal Issue: 4; Journal ID: ISSN 2643-1564
Publisher:
American Physical Society (APS)
Country of Publication:
United States
Language:
English
Subject:
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY

Citation Formats

Vogl, Michael, Laurell, Pontus, Zhang, Hao, Okamoto, Satoshi, and Fiete, Gregory A. Resummation of the Holstein-Primakoff expansion and differential equation approach to operator square roots. United States: N. p., 2020. Web. https://doi.org/10.1103/PhysRevResearch.2.043243.
Vogl, Michael, Laurell, Pontus, Zhang, Hao, Okamoto, Satoshi, & Fiete, Gregory A. Resummation of the Holstein-Primakoff expansion and differential equation approach to operator square roots. United States. https://doi.org/10.1103/PhysRevResearch.2.043243
Vogl, Michael, Laurell, Pontus, Zhang, Hao, Okamoto, Satoshi, and Fiete, Gregory A. Tue . "Resummation of the Holstein-Primakoff expansion and differential equation approach to operator square roots". United States. https://doi.org/10.1103/PhysRevResearch.2.043243.
@article{osti_1716543,
title = {Resummation of the Holstein-Primakoff expansion and differential equation approach to operator square roots},
author = {Vogl, Michael and Laurell, Pontus and Zhang, Hao and Okamoto, Satoshi and Fiete, Gregory A.},
abstractNote = {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 $\hat{n}$ = a†a we show that the square root √$\hat{n}$ near $\hat{n}$ = 0 can be approximated by a polynomial in $\hat{n}$. This result is unexpected because a Taylor expansion fails. A polynomial expression in $\hat{n}$ is possible because $\hat{n}$ 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.},
doi = {10.1103/PhysRevResearch.2.043243},
journal = {Physical Review Research},
number = 4,
volume = 2,
place = {United States},
year = {2020},
month = {11}
}

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