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 $$\hat{n}$$ = 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.
- Publication Date:
- Research Org.:
- Oak Ridge National Laboratory (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. doi: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 = {Tue Nov 17 00:00:00 EST 2020},
month = {Tue Nov 17 00:00:00 EST 2020}
}
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