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Journal of the Korean Physical Society

Application of the Functional Schrödinger Picture Theory to the Holstein-Primakoff Boson Theory for Antiferromagnetic Spin Polarons in Two-Dimensional Systems of Strongly Correlated Electrons

Kwangyl Park, Sul-Ah Ahn, Sung‐Ho Suck Salk

Published Jan 15, 2009• 1 citation
Publisher

Abstract

Applying the functional Schrodinger picture to the Holstein-Primakoff boson theory, we obtain a dispersion energy relation for an antiferromagnetic spin polaron or magnon in two-dimensional systems of strongly correlated electrons. We show that our magnon ground-state wave function is equivalent to that of a retraceable path approximation by Reiter in the long-wavelength limit. Further, we demonstrate that the derived dispersion relation of a quasiparticle leads to the self-consistent Born approximation of the Green's function approach only in the large Heisenberg coupling J limit, showing that this new approach is general enough to contain more physical information than the Green's function approach in dealing with the antiferroinagnetic spin polaron of interest.

DOI
10.3938/jkps.54.157

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Applying the functional Schrodinger picture to the Holstein-Primakoff boson theory, we obtain a dispersion energy relation for an antiferromagnetic spin polaron or magnon in two-dimensional systems of strongly correlated electrons. We show that our magnon ground-state wave function is equivalent to that of a retraceable path approximation by Reiter in the long-wavelength limit. Further, we demonstrate that the derived dispersion relation of a quasiparticle leads to the self-consistent Born approximation of the Green's function approach only in the large Heisenberg coupling J limit, showing that this new approach is general enough to contain more physical information than the Green's function approach in dealing with the antiferroinagnetic spin polaron of interest.
We present functional Schrödinger representations of Holstein-Primakoff boson and slave-boson theories for the Heisenberg Hamiltonian and the t−Jt-Jt−J Hamiltonian respectively. Based on these representations we obtain the dispersion relations of magnons for two dimensional antiferromagnets. By applying the functional Schrödinger representation of the Holstein-Primakoff boson theory to the Heisenberg Hamiltonian, the exchange energy is correctly predicted and the self-energy of quasi-hole is obtained. From the use of the functional Schrödinger representation of the t−Jt-Jt−J Hamiltonian it is shown that at half-filling the dispersion relation obtained from the slave-boson theory leads to that obtained from the Holstein-Primakoff boson approach.
The motion of a single hole in a two-dimensional Heisenberg antiferromagnet (hAF) is studied in a representation where holes are described as spinless fermions (holons) and spins as normal bosons. Assuming long-range AF order the spin dynamics is treated in linear spin-wave theory. The formulation highlights the close relation with the conventional polaron problem. The holon Green's function is calculated self-consistently within the Born approximation using finite-cluster geometries for the numerical solution. As a remarkable result we find close agreement with the spectral function A(k,\ensuremath{\omega}) of a hole calculated by exact diagonalization methods. A(k,\ensuremath{\omega}) is characterized by a narrow quasiparticle (QP) peak at the low-energy side of the spectrum, which is well separated from the incoherent part for large enough J values. A complete characterization of our solution is given, including the spectral weight, the dispersion relation, and effective masses of the QP state. A finite-size-scaling study gives a nonvanishing spectral weight of the QP in the thermodynamic limit for values J/t typical for copper oxide superconductors. Our calculations indicate that the self-consistent Born approximation is a valuable scheme for characterizing the dynamics of a hole in the t-J model, even in the strong-coupling regime.
The Holstein-Primakoff boson representation of quantum spins and associated large-SSS expansion have been the standard framework for describing the spin wave excitations in magnetically order phases of quantum spin systems. However, we will show that the omission of projection operators and normal-ordering in this representation can produce incorrect magnon hamiltonians for finite SSS. We will present the exact normal-ordered forms of the finite-SSS projection operators and projected Holstein-Primakoff boson representations of spin and quadrupole operators, which can produce exact two-magnon interaction terms under ferromagnetic or fully polarized states. We will also discuss the difficulties of applying this projected representation to antiferromagnetic spin wave theory.
In this thesis, we investigated the spectral properties of polaronic quasiparticles resulting from the coupling of a charge carrier to the bosonic excitations of ordered environments. Holstein polarons, describing an electron locally coupled to dispersionless phonons, and spin polarons in hole-doped antiferromagnets are considered. The Green's function of the Holstein polaron in a lattice with various kinds of impurities is calculated using the momentum average approximation. The main finding is that the scenario where the mere effect of the coupling to phonons is to enhance the electron's effective mass is incomplete as the phonons are found to also renormalize the impurity potential the polaron interacts with. This formalism is applied first to the case of a single impurity and the range of parameters where the polaron's ground state is localized is identified. The lifetime of the polaron due to scattering from weak but extended disorder is then studied and it is shown that the renormalization of the disorder potential leads to deviation of the strong coupling results from Fermi's golden rule's predictions. Motivated by the hole-doped cuprate superconductors, the motion of a hole in a two-dimensional Ising antiferromagnet and its binding to an attractive impurity is studied next, based on a variational scheme that allows for configurations with a certain maximum number of spin flips. The role of the magnetic sublattices in determining the symmetry of the resulting bound states is discussed. Next, a more realistic model describing a hole in a CuO₂ layer which retains the O explicitly, is considered. By neglecting the fluctuations of the Cu spins and using a variational principle similar to that of the previous chapter, a semi-analytic solution for the Green's function of the hole in an infinite 2D lattice is constructed. The resulting quasiparticle dispersion shows the proper ground state and other features observed in experiments. The lack of importance of the background spin fluctuations is justified based on the hole's ability to move on the O sublattice without disturbing the Cu spins. Finally, the model is generalized to gauge the importance of other relevant O orbitals.
We solved numerically the integral equation for the selfenergy which describes the motion of a single hole in a two-dimensional quantum antiferromagnet (AF) within the selfconsistent Born approximation. This formulation stresses the similarity of the AF-spin polaron with the standard polaron problem. We confine our calculation to finite cluster geometries and compare with results from previous exact diagonalization studies. The spectral function is characterized by a narrow quasiparticle (qp) peak at the low energy side of the spectra, which appears to be well separated from the incoherent band part for large enough J values. For small J we find a reduced width of ~7t for the incoherent band. The bottom of the coherent qp band always occurs at (±π/2, ±π/2). Its bandwidth initially increases with J until J≃t and then decreases as 2t2/J. A complete characterization of our solution is given, including the dispersion relation and effective masses of this quasiparticle. The comparison with exact diagonalization studies for a 4×4 cluster is remarkably good. From our results we see that a lattice of 16×16 sites describes adequately the thermodynamic limit. We conclude that the simple Born approximation is a valuable scheme to characterize the dynamics of one hole in the t-J model. both in the perturbative and the strong coupling regimes.