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Physical review. B, Condensed matter

Two-loop scattering amplitudes for thet-Jmodel: Evidence for a spin-exchange collective mode

J. W. Rasul

Published Jan 15, 1995• 1 citation
Publisher

Abstract

We study contributions to the spin-exchange scattering amplitude at two-loop (1/${\mathit{N}}^{2}$) order for the metallic phase of the two-dimensional t-J model, for which a well-defined 1/N expansion is possible. We find for small hole dopings and small exchange couplings (t\ensuremath{\gg}J) a positive spin-exchange Landau parameter, which suggests that the spin-density fluctuations show a well-defined collective mode near the metal-insulator filling. This behavior results from the fact that the chirality fluctuations decouple from and are suppressed relative to the charge fluctuations for large mass enhancements. The result is interpreted as paralleling the collective spin and charge modes characteristic of the Luttinger liquid for low wave vectors.

DOI
10.1103/physrevb.51.2632

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We study contributions to the spin-exchange scattering amplitude at two-loop (1/N2{\mathit{N}}^{2}N2) order for the metallic phase of the two-dimensional t-J model, for which a well-defined 1/N expansion is possible. We find for small hole dopings and small exchange couplings (t\ensuremath{\gg}J) a positive spin-exchange Landau parameter, which suggests that the spin-density fluctuations show a well-defined collective mode near the metal-insulator filling. This behavior results from the fact that the chirality fluctuations decouple from and are suppressed relative to the charge fluctuations for large mass enhancements. The result is interpreted as paralleling the collective spin and charge modes characteristic of the Luttinger liquid for low wave vectors.
We study the spin-exchange scattering amplitude at two-loop (1/N2{\mathit{N}}^{2}N2) order for the infinite-U Hubbard model away from the metal-insulator filling, with finite electron charge, for which a well-defined 1/N expansion is possible, the mean-field level being equivalent to Gutzwiller enhanced quasiparticles. We find a positive spin-exchange Landau parameter which suggests that the spin-density fluctuations show a well-defined collective mode near the metal-insulator filling. This behavior is interpreted as paralleling the spin and charge separation in the Luttinger liquid.
We investigate the charge density correlation function in the slave fermion representation of the two-dimensional t - J model using the self-consistent perturbation approach developed for that model by Li et al (1992 Phys. Rev. B 45 5428), which to lowest order in (t, J) gives a diagrammatic equivalent to the usual mean field theory. At the lowest order the equal-time correlation function (interpreted as the holon density correlation function) shows similar features to the results of recent high-temperature series calculations. However the loop corrections, representing scattering of holes by the antiferromagnetic spin fluctuations, lead to a divergent compressibility at low temperatures for low hole dopings and small J/t, which we interpret as indicating a tendency towards phase separation at a temperature .
A real spin-charge separation scheme is found based on a saddle-point state of the t-J model. In the one-dimensional (1D) case, such a saddle-point reproduces the correct asymptotic correlations at the strong-coupling fixed point of the model. In the two-dimensional (2D) case, the transverse gauge field confining spinon and holon is shown to be gapped at finite doping so that a spin-charge deconfinement is obtained for its first time in 2D. The gap in the gauge fluctuation disappears at half-filling limit, where a long-range antiferromagnetic order is recovered at zero temperature and spinons become confined. The most interesting features of spin dynamics and transport are exhibited at finite doping where exotic residual couplings between spin and charge degrees of freedom lead to systematic anomalies with regard to a Fermi-liquid system. In spin dynamics, a commensurate antiferromagnetic fluctuation with a small, doping-dependent energy scale is found, which is characterized in momentum space by a Gaussian peak at (\ensuremath{\pi}/a,\ensuremath{\pi}/a) with a doping-dependent width (\ensuremath{\propto} \ensuremath{\surd}\ensuremath{\delta} , \ensuremath{\delta} is the doping concentration). This commensurate magnetic fluctuation contributes a non-Korringa behavior for the NMR spin-lattice relaxation rate. There also exists a characteristic temperature scale below thich a pseudogap behavior appears in the spin dynamics. Furthermore, an incommensurate magnetic fluctuation is also obtained at a finite energy regime. In the transport, a strong-range phase intereference leads to an effective holon Lagrangian which can give rise to a series of interesting phenomena including linear-T resistivity and a T2{\mathit{T}}^{2}T2 Hall angle. We discuss the striking similarities of these theoretical features with those found in the high-Tc{\mathit{T}}_{\mathit{c}}Tc​ cuprates and give a consistent picture for the latter. Electronic properties like Fermi surface and superconducting pairing in this framework are also discussed.
The behavior of holes in the one-dimensional (1D) t-J model differs substantially from the one in planar correlated systems. We show that within the 1D model one can recover some essential properties of 2D systems by including longer-range spin exchange. Be means of analytical calculations for the anisotropic exchange and the exact diagonalization of small systems, it is established that this generalization leads to the large mass enhancement for a single hole, to the hole binding at strongly reduced J/t, and to substantial changes in the phase diagram at electron densities n\ensuremath{\lesssim}1. It is shown that the longer spin exchange changes qualitatively the phase-separation line Jc{\mathit{J}}_{\mathit{c}}Jc​(n), which obtains an n dependence analogous to the 2D systems, i.e., opposite to the one in the 1D t-J model. The correlation exponent ${\mathit{K}}_{\mathrm{\ensuremath{\rho}}}$ calculated in the Luttinger-liquid regime JJc{\mathit{J}}_{\mathit{c}}Jc​(n) indicates also that the region of dominant pairing correlations can move even to J/t1.
In the first part of the present Thesis, the ground state (GS) properties of the quasi-one-dimensional AB₂ Hubbard model are investigated taking the effects of charge and spin quantum fluctuations on equal footing. Using a functional integral approach, combined with a perturbative expansion in the strong-coupling regime, we obtain the Lagrangian density associated with the charge (Grassmann fields) and spin [SU(2) gauge fields] degrees of freedom. In the strong-coupling regime, we derive a perturbative low-energy theory suitable to describe the ferrimagnetic phase at half filling and the phases in the hole-doped regime. At half filling, a perturbative spin-wave analysis allows us to find the GS energy, sublattice magnetizations, and total spin per unit cell in the Lieb ferrimagnetic GS of the effective quantum Heisenberg model, in very good agreement with previous results. In the challenging hole doping regime away from half filling, we derive the corresponding t-J Hamiltonian. Under the assumption that charge and spin quantum correlations are decoupled, the evolution of the second-order spin-wave modes in the doped regime unveils the occurrence of spatially modulated spin structures and the emergence of phase separation in the presence of resonatingvalence-bond (RVB) states. We also calculate the doping-dependent GS energy and total spin per unit cell, including both Zeeman and orbital contributions, in which case it is shown that the spiral ferrimagnetic order collapses at a critical hole concentration. Notably, our analytical results in the doped regime are in very good agreement with density matrix renormalization group (DMRG) studies, where our assumption of spin-charge decoupling is numerically supported by the formation of charge-density waves in anti-phase with the modulation of the magnetic structure. In the second part, motivated by analogy with photonic lattices, we examine the edge states of a one-dimensional trimer lattice in the phases with and without inversion symmetry protection. In contrast to the Su-Schrieffer-Heeger model, we show that the edge states in the inversion-symmetry broken phase of the trimer model turn out to be chiral, i.e., instead of appearing in pairs localized at opposite edges they can appear at a single edge. Interestingly, these chiral edge states remain robust to large amounts of disorder. In addition, we use the Zak phase to characterize the emergence of degenerate edge states in the inversion-symmetric phase of the trimer model. Furthermore, we capture the essentials of the whole family of trimers through a mapping onto the commensurate off-diagonal Aubry-André-Harper model, which allows us to establish a direct connection between chiral edge modes in the two models, including the calculation of Chern numbers.We thus suggest that the chiral edge modes of the trimer lattice have a topological origin inherited from this effective mapping. Also, we find a nontrivial connection between the topological phase transition point in the trimer lattice and the one in its associated two-dimensional parent system, in agreement with results in the context of Thouless pumping in photonic lattices.
We study the ``t-J'' model with strongly correlated electrons (infinite-U Hubbard-model kinetic energy plus superexchange J) by exact diagonalization on small two-dimensional clusters of six, eight, and twelve sites on a square lattice with periodic boundary conditions. We find an attractive pairing potential for two holes when J/t\ensuremath{\ge}(J/t)c{)}_{\mathit{c}})c​\ensuremath{\simeq}0.2, with t the hopping matrix element. Ground-state calculations show that extended s-wave and d-wave pairing susceptibilities and equal-time pairing correlations grow weakly with increasing J and system size at fixed filling n. Pairing correlations may be enhanced for (J/t)>(J/t)c{)}_{\mathit{c}})c​\ensuremath{\approxeq}0.2. We compare equal-time pairing correlations to a mean-field theory for J=0 and note potential difficulties with interpreting numerical results for such small systems. The widths of one- and two-hole energy bands in low-spin states are of order J; the one-hole effective mass is larger than the two-hole effective mass in such states. Spin-charge correlation functions for a single hole in the eight- and twelve-site lattices are consistent with the spin-charge decoupling scenario for resonating-valence-bond states.