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arXiv:cond-mat/9609101 · Physical review. B, Condensed matter

Charge and spin dynamics in the one-dimensional t- Jz and t-J models

Shu Zhang, Michael Karbach, Gerhard H. Muller, Joachim Stolze

Published Mar 1, 1997• 18 citations
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Abstract

The impact of the spin-flip terms on the (static and dynamic) charge and spin correlations in the Luttinger-liquid ground state of the one-dimensional (1D) t-J model is assessed by comparison with the same quantities in the 1D t-${\mathrm{J}}_{\mathrm{z}}$ model, where spin-flip terms are absent. We employ the recursion method combined with a weak-coupling or a strong-coupling continued-fraction analysis. At ${\mathrm{J}}_{\mathrm{z}}$/t=${0}^{+}$ we use the Pfaffian representation of dynamic spin correlations. The changing nature of the dynamically relevant charge and spin excitations on approach of the transition to phase separation is investigated in detail. At the transition point, the t-${\mathrm{J}}_{\mathrm{z}}$ ground state has zero (static) charge correlations and very short-ranged (static) spin correlations, whereas the t-J ground state is critical. The t-${\mathrm{J}}_{\mathrm{z}}$ charge excitations (but not the spin excitations) at the transition have a single-mode nature, whereas charge and spin excitations have a complicated structure in the t-J model. A major transformation of the t-J spin excitations takes place between two distinct regimes within the Luttinger-liquid phase, while the t-${\mathrm{J}}_{\mathrm{z}}$ spin excitations are found to change much more gradually. In the t-${\mathrm{J}}_{\mathrm{z}}$ model, phase separation is accompanied by N\'eel long-range order, caused by the condensation of electron clusters with an already existing alternating up-down spin configuration (topological long-range order). In the t-J model, by contrast, the spin-flip processes in the exchange coupling are responsible for continued strong spin fluctuations (dominated by two-spinon excitations) in the phase-separated state.

DOI
10.1103/physrevb.55.6491

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The impact of the spin-flip terms on the (static and dynamic) charge and spin correlations in the Luttinger-liquid ground state of the one-dimensional (1D) t-J model is assessed by comparison with the same quantities in the 1D t-Jz{\mathrm{J}}_{\mathrm{z}}Jz​ model, where spin-flip terms are absent. We employ the recursion method combined with a weak-coupling or a strong-coupling continued-fraction analysis. At Jz{\mathrm{J}}_{\mathrm{z}}Jz​/t=0+{0}^{+}0+ we use the Pfaffian representation of dynamic spin correlations. The changing nature of the dynamically relevant charge and spin excitations on approach of the transition to phase separation is investigated in detail. At the transition point, the t-Jz{\mathrm{J}}_{\mathrm{z}}Jz​ ground state has zero (static) charge correlations and very short-ranged (static) spin correlations, whereas the t-J ground state is critical. The t-Jz{\mathrm{J}}_{\mathrm{z}}Jz​ charge excitations (but not the spin excitations) at the transition have a single-mode nature, whereas charge and spin excitations have a complicated structure in the t-J model. A major transformation of the t-J spin excitations takes place between two distinct regimes within the Luttinger-liquid phase, while the t-Jz{\mathrm{J}}_{\mathrm{z}}Jz​ spin excitations are found to change much more gradually. In the t-Jz{\mathrm{J}}_{\mathrm{z}}Jz​ model, phase separation is accompanied by N'eel long-range order, caused by the condensation of electron clusters with an already existing alternating up-down spin configuration (topological long-range order). In the t-J model, by contrast, the spin-flip processes in the exchange coupling are responsible for continued strong spin fluctuations (dominated by two-spinon excitations) in the phase-separated state.
The recursion method is used to determine the T=0 spin dynamic structure factor Szz(q,ω) in the Luttinger liquid state and in the phase-separated state of the one-dimensional t–J model. As the exchange coupling increases from zero, the dispersions and line shapes of the dominant spin excitations are observed to undergo a major metamorphosis between the free-fermion limit and the onset of phase separation. The familiar two-spinon spectrum of the Heisenberg antiferromagnetic chain emerges gradually in the strongly phase-separated state.
It is well established that at low energies one-dimensional (1D) fermionic systems are described by the Luttinger-liquid (LL) theory, which predicts phenomena such as spin-charge separation and charge fractionalization into chiral modes. Here we show through the time evolution of an electron injected into a 1D ttt-JJJ model, obtained with time-dependent density matrix renormalization group, that a further fractionalization of both charge and spin takes place beyond the hydrodynamic limit. Its dynamics can be understood at the supersymmetric point (J=2tJ=2tJ=2t) in terms of the excitations of the Bethe-ansatz solution. Furthermore we show that fractionalization with similar characteristics extends to the whole region corresponding to a repulsive LL.
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.
One of the central issues in the study of high-temperature superconducting cuprates which are composed of two-dimensional (2D) CuO 2 planes is whether the 2D systems with strong electron correlation behave as a Fermi liquid or a non-Fermi-liquid-like one-dimensional (1D) system with electron correlation. In this article, we start with the detailed examination of the electronic structure in cuprates and study theoretically the spin and charge dynamics in 1D and 2D cuprates. The theoretical background of spin-charge separation in the 1D model systems including the Hubbard and t - J models is presented. The first direct observation of collective modes of spin and charge excitations in a 1D cuprate, which are called spinons and holons respectively, in angle-resolved photoemission spectroscopy (ARPES) experiments is reviewed in the light of the theoretical results based on the numerically exact-diagonalization method. The charge and spin dynamics in 1D insulating cuprates is also discussed in connection with the spin-charge separation. The arguments are extended to the 2D cuprates, and the unique aspects of the electronic properties of high-temperature superconductors are discussed. Special emphasis is placed on the d-wave-like excitations in insulating 2D cuprates observed in ARPES experiments. We explain how the excitations are caused by the spin-charge separation. The charge stripes observed in the underdoped cuprates are examined in connection with spin-charge separation in real space.
A fermion-spin transformation is used to implement the charge-spin separation, and is developed to study the low-dimensional t-J model. In this approach, the charge and spin degrees of freedom of the physical electron are separated, and the charge degree of freedom is represented by a spinless fermion while the spin degree of freedom is representd by a hard-core boson. The on-site local constraint for single occupancy is satisfied even in the mean-field approximation and the sum rule for the physical electron is obeyed. This approach can be applied to both one and two-dimensional systems. In the one-dimensional case, the spinon as well as the physical electron behave like Luttinger liquids. We have obtained a gapless charge and spin excitation spectrum, a good ground state energy, and a reasonable electron-momentum distribution within the mean-field approximation. The correct exponents of the correlation functions and momentum distribution are also obtained if the squeezing effect and rearrangement of the spin configurations are taken into account. In the two-dimensional case, within the mean-field approximation the magnetized flux state with a gap in the spinon spectrum has the lowest energy at half filling. The antiferromagnetic long-range order is destroyed by hole doping of the order \ensuremath{\sim}10-15 % for t/J=3-5 and a disordered flux state with gapless spinon spectrum becomes stable. The calculated specific heat is roughly consistent with observed results on copper oxide superconductors. The possible phase separation is also discussed at the mean-field level.
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.