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arXiv:cond-mat/9505001 · Physical review. B, Condensed matter

Dynamic susceptibility and photoemission in thet-t’-Jmodel

Bruce Normand, Patrick A. Lee

Published Jun 1, 1995• 9 citations
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Abstract

We consider the slave-fermion--Schwinger-boson decomposition of a t-J model for a two-dimensional antiferromagnet in which the next-nearest-neighbor hopping is taken into account by an explicit t' term. This technique is particularly suitable to describe systems with low hole density, as in the parent phases of high-${\mathit{T}}_{\mathit{c}}$ superconducting (HTSC) materials, and we examine the experimental consequences of the model. Using mean-field states which admit finite spiral and off-diagonal order, we compute first the dynamic susceptibility \ensuremath{\chi}(q,\ensuremath{\omega}) from fluctuations in the boson (spin) degrees of freedom, and investigate a selected parameter regime for the model to seek features consistent with inelastic neutron scattering experiments on underdoped HTSC crystals. We find peaks in the scattered intensity at incommensurate wave vectors (\ensuremath{\pi}\ifmmode\pm\else\textpm\fi{}2${\mathit{k}}_{0}$,\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}) and (\ensuremath{\pi},\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}\ifmmode\pm\else\textpm\fi{}2${\mathit{k}}_{0}$), but within our approximation the q-integrated susceptibility as a function of frequency and temperature resembles that from spin-wave theory, which does not contain the scaling behavior over all values of \ensuremath{\omega}/T found experimentally. We show also the form of the nuclear magnetic resonance relaxation time ${\mathit{T}}_{1}^{\mathrm{\ensuremath{-}}1}$(T), which measures the zero-frequency limit of the susceptibility. The imaginary part of the physical electron Green function is a convolution of both fermionic (hole) and bosonic Green functions, and so incorporates the effect of spin dynamics on the charge motion. We calculate the photoemitted intensity, a direct experimental measurement of this quantity, focusing on its behavior near the hole pockets at (\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2,\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2) which characterize this type of model. The peak photocurrent occurs not at the hole Fermi surface, but at points separated from it by ${\mathbf{k}}_{0}$ due to the boson incommensuration.

DOI
10.1103/physrevb.51.15519

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We consider the slave-fermion--Schwinger-boson decomposition of a t-J model for a two-dimensional antiferromagnet in which the next-nearest-neighbor hopping is taken into account by an explicit t' term. This technique is particularly suitable to describe systems with low hole density, as in the parent phases of high-Tc{\mathit{T}}_{\mathit{c}}Tc​ superconducting (HTSC) materials, and we examine the experimental consequences of the model. Using mean-field states which admit finite spiral and off-diagonal order, we compute first the dynamic susceptibility \ensuremath{\chi}(q,\ensuremath{\omega}) from fluctuations in the boson (spin) degrees of freedom, and investigate a selected parameter regime for the model to seek features consistent with inelastic neutron scattering experiments on underdoped HTSC crystals. We find peaks in the scattered intensity at incommensurate wave vectors (\ensuremath{\pi}\ifmmode\pm\else\textpm\fi{}2k0{\mathit{k}}_{0}k0​,\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}) and (\ensuremath{\pi},\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}\ifmmode\pm\else\textpm\fi{}2k0{\mathit{k}}_{0}k0​), but within our approximation the q-integrated susceptibility as a function of frequency and temperature resembles that from spin-wave theory, which does not contain the scaling behavior over all values of \ensuremath{\omega}/T found experimentally. We show also the form of the nuclear magnetic resonance relaxation time T1\ensuremath−1{\mathit{T}}_{1}^{\mathrm{\ensuremath{-}}1}T1\ensuremath−1​(T), which measures the zero-frequency limit of the susceptibility. The imaginary part of the physical electron Green function is a convolution of both fermionic (hole) and bosonic Green functions, and so incorporates the effect of spin dynamics on the charge motion. We calculate the photoemitted intensity, a direct experimental measurement of this quantity, focusing on its behavior near the hole pockets at (\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2,\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2) which characterize this type of model. The peak photocurrent occurs not at the hole Fermi surface, but at points separated from it by k0{\mathbf{k}}_{0}k0​ due to the boson incommensuration.
We have calculated within the fluctuation exchange approximation the full momentum and real-frequency dependence of the quasiparticle self-energy, the gap function, and the dynamic spin susceptibility for the two-dimensional Hubbard model near half filling. The spin-density-wave (SDW) instability is reached only asymptotically in the limit of infinite U or zero temperature. Near the SDW instability we find approximately marginal Fermi-liquid behavior, a pseudogap in the density of states, and a singlet dx2{\mathit{d}}_{\mathit{x}}^{2}dx2​-y2{\mathit{y}}^{2}y2 pairing instability. Below Tc{\mathit{T}}_{\mathit{c}}Tc​, gaps open rapidly in the frequency spectra of the gap function, the quasiparticle scattering rate, and the density of states. The spectral density of the dynamic spin susceptibility agrees qualitatively with the neutron-scattering data on ${\mathrm{La}}_{2\mathrm{\ensuremath{-}}\mathit{x}}$Srx{\mathrm{Sr}}_{\mathit{x}}Srx​CuO4{\mathrm{CuO}}_{4}CuO4​. The quasiparticle scattering rate and spin-lattice relaxation rate ${\mathit{T}}_{1}^{\mathrm{\ensuremath{-}}1}$ decrease rapidly below Tc{\mathit{T}}_{\mathit{c}}Tc​. The photoemission intensities along the antinode and node of the gap are in good agreement with the angle-resolved photoemission spectroscopy data on Bi 2212. Including next-nearest-neighbor hopping we obtain for sufficiently large coupling (U/t\ensuremath{\gtrsim}7) a broad commensurate neutron-scattering peak like the observed one in YBa2{\mathrm{YBa}}_{2}YBa2​Cu3{\mathrm{Cu}}_{3}Cu3​O6+x{\mathrm{O}}_{6+\mathit{x}}O6+x​.
A relaxation-function theory for the dynamic spin susceptibility in the $t\text{\ensuremath{-}}J$ model is presented. By a sum-rule-conserving generalized mean-field approximation (GMFA), the two-spin correlation functions of arbitrary range, the staggered magnetization, the uniform static susceptibility, and the antiferromagnetic correlation length are calculated in a wide region of hole doping and temperatures. A good agreement with available exact diagonalization (ED) data is found. The correlation length is in reasonable agreement with neutron-scattering experiments on ${\text{La}}_{2\ensuremath{-}\ensuremath{\delta}}{\text{Sr}}_{\ensuremath{\delta}}{\text{CuO}}_{4}$. Going beyond the GMFA, the self-energy is calculated in the mode-coupling approximation. The spin dynamics at arbitrary frequencies and wave vectors is studied for various temperatures and hole doping. At low doping a spin-wave-type behavior is found as in the Heisenberg model, while at higher doping a strong damping caused by hole hopping occurs, and a relaxation-type spin dynamics is observed in agreement with the ED results. The local spin susceptibility and its $\ensuremath{\omega}/T$ scaling behavior are calculated in a reasonable agreement with experimental and ED data.
We consider the slave-fermion Schwinger-boson decomposition of an effective model obtained through a systematic low-energy reduction of the three-band Hubbard Hamiltonian. The model includes a three-site term t ′′ similar to that obtained in the large- U limit of the Hubbard model but of opposite sign for realistic or large O-O hopping. For parameters close to the most realistic ones for the cuprates, the mean-field solution exhibits d + s superconductivity (predominantly d x 2 − y 2 ) with a dependence on doping x very similar to the experimentally observed. We also obtained incommensurate peaks at wave vectors ∼ π(1,1 ± 2 x ) in the spin structure factor, which also agree with experiment.
The dynamical spin susceptibility χ( q , ω), of the t-J model in two-dimensional square lattice have been studied based on the mean field approximation to the slave-boson formalism. The static spin susceptibility, χ( q , 0), and Im χ( q , ω) at various energy, ω, have been calculated. We found; 1) χ(0, 0) has only weak temperature dependence 2) T 1 T ∝( T +θ) for T≳0.1JT\gtrsim0.1{J}T≳0.1J with θ linearly dependent on the doping rate, δ 3) T 1 T has a minimum at T =(0.1∼0.2) J 4) Im χ( q , ω) at q = Q ≡(π, π) has a gap like structure as a function of ω if the temperature is low. All these are consistent with experimental results.
Using exact diagonalization techniques, we study one dynamical hole in the two-dimensional t-J and Hubbard models on a square lattice including a next-nearest-neighbor hopping t'. We present the phase diagram in the parameter space (J/t,t'/t), discussing the ground-state properties of the hole. At J=0, a crossing of levels exists at some value of t' separating a ferromagnetic from an antiferromagnetic ground state. For nonzero J, at least four different regions appear where the system behaves like an antiferromagnet or a (not fully saturated) ferromagnet. We study the quasiparticle behavior of the hole, showing that for small values of \ensuremath{\Vert}t'\ensuremath{\Vert} the previously presented string picture is still valid. We also find that, for a realistic set of parameters derived from the Cu-O Hamiltonian, the hole has momentum (\ensuremath{\pi}/2,\ensuremath{\pi}/2), suggesting an enhancement of the p-wave superconducting mode due to the second-neighbor interactions in the spin-bag picture. Results for the t-t'-U model are also discussed with conclusions similar to those of the t-t'-J model. In general we found that t'=0 is not a singular point of these models.
We study a system of doped antiferromagnet in three dimensions at finite temperatures using the ttt-JJJ model, a canonical model of strongly correlated electrons. We employ the slave-fermion representation of electrons, in which an electron is described as a composite of a charged spinless holon and a chargeless spinon. We introduce two kinds of U(1) gauge fields on links as auxiliary fields, one describing resonating valence bonds of antiferromagnetic nearest-neighbor spin pairs and the other for nearest-neighbor hopping amplitudes of holons and spinons in the ferromagnetic channel. To perform a numerical study of the system, we integrate out the fermionic holon field by using the hopping expansion in powers of the hopping amplitude, which is legitimate for the region in and near the insulating phase. The resultant effective model is described in terms of bosonic spinons, two U(1) gauge fields, and a collective field for hole pairs. We study this model by means of Monte Carlo simulations, calculating the specific heat, spin correlation functions, and instanton densities. We obtain a phase diagram in the hole concentration-temperature plane, which is in good agreement with that observed recently for clean and homogeneous underdoped samples.