Exact 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 work 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 $\hat{n}={\hat{a}}^{†}a$ we show that the square-root $\sqrt{\hat{n}}$ 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 offer an advantage over other methods, such as the Dyson-Maleev representation.

References in corpus (78)

• Anyons in an exactly solved model and beyond
• Quantum Spin Liquid States
• Quantum Spin Liquids
• Kitaev-Heisenberg Model on Honeycomb Lattice: Possible Exotic Phases in Iridium Oxides $A_2$IrO${}_3$
• Kitaev quantum spin liquid - concept and materialization
• Relevance of the Heisenberg-Kitaev model for the honeycomb lattice iridates A_2IrO_3
• Neutron tomography of magnetic Majorana fermions in a proximate quantum spin liquid
• Noncommuting mixed states cannot be broadcast
• Fidelity approach to quantum phase transitions
• Linear spin wave theory for single-Q incommensurate magnetic structures
• A Field Guide to Spin Liquids
• Dynamics of a two-dimensional quantum spin liquid: signatures of emergent Majorana fermions and fluxes
• Excitations in the field-induced quantum spin liquid state of alpha-RuCl3
• Interplay between Kitaev interaction and single ion anisotropy in ferromagnetic CrI${}_3$ and CrGeTe${}_3$ monolayers
• Kitaev magnetism in honeycomb RuCl3 with intermediate spin-orbit coupling
• Spontaneous Magnon Decays
• Hunting Majorana Fermions in Kitaev Magnets
• Spin-wave excitations evidencing the Kitaev interaction in single crystalline $\alpha$-RuCl${}_3$
• Mixed-state fidelity and quantum criticality at finite temperature
• Fundamental Spin Interactions Underlying the Magnetic Anisotropy in the Kitaev Ferromagnet CrI${}_3$
• Field-induced quantum criticality in the Kitaev system $\alpha$-RuCl${}_3$
• Generalized Jordan-Wigner Transformations
• Kitaev-Heisenberg-J2-J3 model for the iridates A2IrO3
• Emergence of a Field-Driven U(1) Spin Liquid in the Kitaev Honeycomb Model
• Spin-orbit excitations and electronic structure of the putative Kitaev magnet $\alpha$-RuCl${}_3$
• Microscopic mechanism for higher-spin Kitaev model
• Uhlmann fidelity between two-mode Gaussian states
• Observation of two types of anyons in the Kitaev honeycomb magnet
• Fidelity susceptibility and long-range correlation in the Kitaev honeycomb model
• Dynamical Quantum Phase Transitions in the Kitaev Honeycomb Model
• Relation between Kitaev magnetism and structure in $\alpha$-RuCl${}_3$
• Average fidelity between random quantum states
• Mixed state sensitivity of several quantum information benchmarks
• Finite field regime for a quantum spin liquid in $\alpha$-RuCl${}_3$
• The nature of spin excitations in the one-third magnetization plateau phase of Ba${}_3$CoSb${}_2$O${}_9$
• Time Evolution of Many-Body Localized Systems with the Flow Equation Approach
• Frustrated two dimensional quantum magnets
• Site dilution in Kitaev's honeycomb model
• Alternative fidelity measure for quantum states
• Flow equations for band--matrices
• Anisotropic susceptibilities in the honeycomb Kitaev system $\alpha$-RuCl${}_3$
• Ultralow-temperature thermal conductivity of the Kitaev honeycomb magnet $\alpha$-RuCl${}_3$ across the field-induced phase transition
• Physical solutions of the Kitaev honeycomb model
• Dynamical and thermal magnetic properties of the Kitaev spin liquid candidate $\alpha$-RuCl${}_3$
• A fidelity measure for quantum channels
• Fidelity Between Partial States as Signature of Quantum Phase Transitions
• Perturbative approach to an exactly solved problem: the Kitaev honeycomb model
• SU(2) Slave Fermion Solution of the Kitaev Honeycomb Lattice Model
• Topological magnonics in the two-dimensional van der Waals magnet CrI3
• Quantum Fidelity and Thermal Phase Transitions
• Flow Equations for Electron-Phonon Interactions
• Resonant inelastic X-ray scattering response of the Kitaev honeycomb model
• Bose-Fermi Transformation In Three Dimensional Space
• SU(2) slave-boson formulation of spin nematic states in S=1/2 frustrated ferromagnets
• Kitaev honeycomb and other exotic spin models with polar molecules
• Perturbative study of the Kitaev model with spontaneous time-reversal symmetry breaking
• Operator fidelity susceptibility: an indicator of quantum criticality
• A flow equation approach to periodically driven quantum systems
• Flow Equations and Normal Ordering. A Survey
• Deriving models for the Kitaev spin-liquid candidate material $\alpha$-RuCl${}_3$ from first principles
• Doping a topological quantum spin liquid: slow holes in the Kitaev honeycomb model
• Exploring many-body localization in quantum systems coupled to an environment via Wegner-Wilson flows
• Flow equations for the Anderson Hamiltonian
• Embedding dissipation and decoherence in unitary evolution schemes
• Localization transition in one dimension using Wegner flow equations
• Phonon-mediated dimensional crossover in bilayer CrI3
• Flow equations for QED in the light front dynamics
• Quantum information approach to the quantum phase transition in the Kitaev honeycomb model
• Chemical- and hydrostatic-pressure effects on the Kitaev honeycomb material Na${}_2$IrO${}_3$
• Kitaev honeycomb tensor networks: exact unitary circuits and applications
• Reduced fidelity in Kitaev honeycomb model
• Thermal states of the Kitaev honeycomb model: a Bures metric analysis
• Spin Gap in Chains with Hidden Symmetries
• Rigorous Calculations of Non-Abelian Statistics in the Kitaev Honeycomb Model
• Realization of the Exactly Solvable Kitaev Honeycomb Lattice Model in a Spin Rotation Invariant System
• Analogue of Hamilton-Jacobi theory for the time-evolution operator
• Umbrella-coplanar transition in the triangular XXZ model with arbitrary spin
• Wave dispersion derived from the square-root Klein-Gordon-Poisson system

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