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Physical review. B, Condensed matter

Intermediate-coupling theory of the spin polaron in thet-Jmodel

H. Barentzen

Published Mar 1, 1996• 15 citations
Publisher

Abstract

The spin polaron in the t-J model, i.e., a hole dressed by a cloud of virtual magnons of the antiferromagnetic spin background, is treated within the framework of intermediate-coupling theory. To this end the original t-J model is first reformulated in terms of spinless fermions and bosons by means of the generalized Dyson-Maleev representation (DMR). The latter may be regarded as the natural extension of the ordinary DMR of pure (undoped) spin systems to the case where holes are present, and is similar to the one originally proposed by Schmitt-Rink, Varma, and Ruckenstein. The reformulated t-J model, which is reminiscent of the Fr\"ohlich Hamiltonian, is then subjected to a series of unitary transformations, analogous to those employed by Lee, Low, and Pines in their treatment of the Fr\"ohlich polaron. Our approach yields an approximate quasiparticle energy E(k) as well as the corresponding eigenvector. To explore the range of validity of the theory presented here, the analytic expressions are then further analyzed for intermediate (J/t=0.4) and strong (J/t=0.08) coupling, where special attention is paid to the quasiparticle bandwidth W. The intermediate-coupling result for E(k) is in excellent agreement with the dispersion curve recently obtained by Dagotto and co-workers by means of a Green function Monte Carlo method. Surprisingly, even in the strong-coupling range the bandshape remains qualitatively correct.The bandwidth W is rather accurate for weak coupling (J/t\ensuremath{\gtrsim}3), as expected, and still reasonable in the intermediate range 0.4\ensuremath{\lesssim}J/t\ensuremath{\le}3, where it deviates from the correct values by some 10--20 %. Our theory fails, however, to describe the proper behavior of W in the strong-coupling regime. This shows that the limitations of our approach manifest themselves in the bandwidths rather than in the shapes of the dispersion curves. Our conclusion is that intermediate-coupling theory is appropriate for J/t\ensuremath{\gtrsim}0.4, whereas a genuine strong-coupling theory is required for all other cases. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.53.5598

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Physical review. B, Condensed matter•Mar 15, 1998•6 citations
The spin polaron in the t-J model, i.e., a hole dressed by a cloud of virtual magnons of the antiferromagnetic spin background, is treated within the framework of intermediate-coupling theory. To this end the original t-J model is first reformulated in terms of spinless fermions and bosons by means of the generalized Dyson-Maleev representation (DMR). The latter may be regarded as the natural extension of the ordinary DMR of pure (undoped) spin systems to the case where holes are present, and is similar to the one originally proposed by Schmitt-Rink, Varma, and Ruckenstein. The reformulated t-J model, which is reminiscent of the Fr"ohlich Hamiltonian, is then subjected to a series of unitary transformations, analogous to those employed by Lee, Low, and Pines in their treatment of the Fr"ohlich polaron. Our approach yields an approximate quasiparticle energy E(k) as well as the corresponding eigenvector. To explore the range of validity of the theory presented here, the analytic expressions are then further analyzed for intermediate (J/t=0.4) and strong (J/t=0.08) coupling, where special attention is paid to the quasiparticle bandwidth W. The intermediate-coupling result for E(k) is in excellent agreement with the dispersion curve recently obtained by Dagotto and co-workers by means of a Green function Monte Carlo method. Surprisingly, even in the strong-coupling range the bandshape remains qualitatively correct.The bandwidth W is rather accurate for weak coupling (J/t\ensuremath{\gtrsim}3), as expected, and still reasonable in the intermediate range 0.4\ensuremath{\lesssim}J/t\ensuremath{\le}3, where it deviates from the correct values by some 10--20 %. Our theory fails, however, to describe the proper behavior of W in the strong-coupling regime. This shows that the limitations of our approach manifest themselves in the bandwidths rather than in the shapes of the dispersion curves. Our conclusion is that intermediate-coupling theory is appropriate for J/t\ensuremath{\gtrsim}0.4, whereas a genuine strong-coupling theory is required for all other cases. \textcopyright{} 1996 The American Physical Society.
The t - J model for the doped two-dimensional Heisenberg quantum antiferromagnet is studied in the generalized Dyson-Maleev representation, applying a new continuous unitary transformation which eliminates the coupling of spin and charge degrees of freedom. The analytical solutions of the resulting flow equations are derived in the weak-coupling regime where t / J is small. This continuous transformation yields a new weak-coupling result for the dispersion of the spin-polaron, if the elimination of both the non-diagonal spin-wave contributions and the terms coupling holes and spin-waves is performed simultaneously . The associated one-particle ground state is lower in energy than the corresponding perturbative result, which is reproduced upon application of subsequent transformations.
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 report discusses the slave-fermion representations of the t-J model and describes another representation, in which fermions and bosons are completely commuting and in which the properties of fermions are directly related to the properties of physical holes. For a study of the system in the new representation at half-filling, interaction of fermions with two magnons is treated in mean-field theory. The obtained effective model, in comparison to that of the usual slave-fermion representation, has an additional bare hole dispersion due to the hole moving by using quantum spin fluctuations present in the undoped antiferromagnetic ground state. The single-hole Green's function at half-filling is then found numerically using the self-consistent Born approximation. For all studied quantities good or excellent agreement with numerical data is observed in the entire parameter range, noticeably better than in the studies with the slave-fermion representation. Using the same effective model, the two-hole problem is also studied by solving numerically the Bethe-Salpeter equation with noncrossing diagrams.
After deriving the quantum mechanical model polaron Hamiltonian, Frohlich, Pelzar and Zineau presented a perturbative solution to the problem. This chapter examines the perturbation theory from Frohlich et al. and presents other important weak and intermediate coupling polaron theories. It discusses several polaronic properties like average number of virtual phonons in the polaron cloud, electron life-time, lattice polarization potential using the first-order perturbative wave function and obtained the polaron energy and the polaron mass using the second-order Rayleigh-Schrodinger perturbation theory (RSPT). The second-order RSPT is known to have a limited range of validity. In the one-phonon Tamm-Dancoff approximation, the exact ground state energy appears in the denominator instead of the unperturbed energy that appears in RSPT. The chapter also discusses the work of Lee and Pines who have used the Tomonaga's approximation made in the formulation of the meson-theoretic problem to the case of polaron.
The report discusses the slave-fermion representations of the t-J model and describes another representation, in which fermions and bosons are completely commuting and in which the properties of fermions are directly related to the properties of physical holes. For a study of the system in the new representation at half-filling, interaction of fermions with two magnons is treated in mean-field theory. The obtained effective model, in comparison to that of the usual slave-fermion representation, has an additional bare hole dispersion due to the hole moving by using quantum spin fluctuations present in the undoped antiferromagnetic ground state. The single-hole Green's function at half-filling is then found numerically using the self-consistent Born approximation. For all studied quantities good or excellent agreement with numerical data is observed in the entire parameter range, noticeably better than in the studies with the slave-fermion representation. Using the same effective model, the two-hole problem is also studied by solving numerically the Bethe-Salpeter equation with noncrossing diagrams.
This report discusses the slave-fermion representations of the $t\ensuremath{-}J$ model and describes another representation in which fermions and bosons are completely commuting. For a study of the system in the new representation at half-filling, the interaction of fermions with two magnons is treated in mean-field theory. The obtained effective model, in comparison to that of the usual slave-fermion representation, has an additional bare hole dispersion due to the hole moving by using quantum spin fluctuations present in the undoped antiferromagnetic ground state. The single-hole Green's function at half-filling is then found numerically using the self-consistent Born approximation. For all studied quantities good or excellent agreement with numerical data is observed in the entire parameter range, noticeably better than in the studies with the slave-fermion representation. Using the same effective model, the two-hole problem is also studied by solving numerically the Bethe-Salpeter equation with noncrossing diagrams.