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arXiv:cond-mat/0007196 · Physical review. B, Condensed matter

Indications of spin-charge separation at short distance and stripe formation in the extendedt-Jmodel on ladders and planes

George Balster Martins, J. C. Xavier, C. J. Gazza, Matthias Vojta C, Elbio R. Dagotto

Published Dec 12, 2000• 28 citations
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Abstract

The recently discussed tendency of holes to generate nontrivial spin environments in the extended two-dimensional $t\ensuremath{-}J$ model [G. Martins, R. Eder, and E. Dagotto, Phys. Rev. B 60, R3716 (1999)] is here investigated using computational techniques applied to ladders with several number of legs. This tendency is studied also with the help of analytic spin-polaron approaches directly in two dimensions. Our main result is that the presence of robust antiferromagnetic correlations between spins located at both sides of a hole either along the x or y axis, observed before numerically on square clusters, is also found using ladders, as well as applying techniques based on a string-basis expansion. This so-called ``across-the-hole'' nontrivial structure exists even in the two-leg spin-gapped ladder system, and leads to an effective reduction in dimensionality and spin-charge separation at short-distances, with a concomitant drastic reduction in the quasiparticle weight Z. In general, it appears that holes tend to induce one-dimensional-like spin arrangements to improve their mobility. Using ladders it is also shown that the very small $J/t\ensuremath{\sim}0.1$ regime of the standard $t\ensuremath{-}J$ model may be more realistic than anticipated in previous investigations, since such regime shares several properties with those found in the extended model at realistic couplings. Another goal of the present paper is to provide additional information on the recently discussed tendencies to stripe formation and spin incommensurability reported for the extended $t\ensuremath{-}J$ model. These tendencies are illustrated with several examples.

DOI
10.1103/physrevb.63.014414

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The recently discussed tendency of holes to generate nontrivial spin environments in the extended two-dimensional t\ensuremath−Jt\ensuremath{-}Jt\ensuremath−J model [G. Martins, R. Eder, and E. Dagotto, Phys. Rev. B 60, R3716 (1999)] is here investigated using computational techniques applied to ladders with several number of legs. This tendency is studied also with the help of analytic spin-polaron approaches directly in two dimensions. Our main result is that the presence of robust antiferromagnetic correlations between spins located at both sides of a hole either along the x or y axis, observed before numerically on square clusters, is also found using ladders, as well as applying techniques based on a string-basis expansion. This so-called ``across-the-hole'' nontrivial structure exists even in the two-leg spin-gapped ladder system, and leads to an effective reduction in dimensionality and spin-charge separation at short-distances, with a concomitant drastic reduction in the quasiparticle weight Z. In general, it appears that holes tend to induce one-dimensional-like spin arrangements to improve their mobility. Using ladders it is also shown that the very small J/t\ensuremath∼0.1J/t\ensuremath{\sim}0.1J/t\ensuremath∼0.1 regime of the standard t\ensuremath−Jt\ensuremath{-}Jt\ensuremath−J model may be more realistic than anticipated in previous investigations, since such regime shares several properties with those found in the extended model at realistic couplings. Another goal of the present paper is to provide additional information on the recently discussed tendencies to stripe formation and spin incommensurability reported for the extended t\ensuremath−Jt\ensuremath{-}Jt\ensuremath−J model. These tendencies are illustrated with several examples.
The two-dimensional extended $t\ensuremath{-}J$ model is studied computationally in a broad region of parameter space, motivated by recent photoemission experiments for the undoped cuprate Ca2CuO2Cl2{\mathrm{Ca}}_{2}{\mathrm{CuO}}_{2}{\mathrm{Cl}}_{2}Ca2​CuO2​Cl2​ [F. Ronning et al., Science 282, 2067 (1998)]. The one-hole ground state is shown to develop robust antiferromagnetic (AF) correlations between spins separated by the mobile hole (i.e., across the hole). This effect tends to decouple charge from spin, and the quasiparticle weight becomes negligible, particularly at momenta $(0,\ensuremath{\pi})\ensuremath{-}(\ensuremath{\pi},0).$ Studies with more holes show precursors of metallic stripe formation, with holes sharing their individual spin arrangements, and AF correlations generated across the stripe.
The problem of two holes in the presence of strong antiferromagnetic fluctuations is revisited using computational techniques. Two-dimensional clusters and two-leg ladders are studied with the Lanczos and truncated Lanczos algorithms. Lattices with up to $2\ifmmode\times\else\texttimes\fi{}16$ and $\sqrt{32}\ifmmode\times\else\texttimes\fi{}\sqrt{32}$ sites are studied. The motivation of the paper is the recently discussed spatial distribution of holes in ladders where the maximum probability for the hole-hole distance is obtained at d=2d=\sqrt{2}d=2​ in units of the lattice spacing, a counterintuitive result considering that the overall symmetry of the two-hole bound state is ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$. Here this effect is shown to appear in small ladder clusters that can be addressed exactly, and also in planes. The probability distribution of hole distances ddd was found to be broad with several distances contributing appreciably to the wave function. The existence of holes in the same sublattice is argued to be a consequence of non-negligible retardation effects in the $t\ensuremath{-}J$ model. Effective models with instantaneous interactions nevertheless capture the essence of the hole pairing process in the presence of short-range antiferromagnetic fluctuations (especially regarding the symmetry properties of the condensate), similarly as the (nonretarded) BCS model contains the basic features of the more complicated electron-phonon problem in low-temperature superconductors. The existence of strong spin singlets in the region where the two-hole bound state is located is here confirmed, and a simple explanation for its origin in the case of planes is proposed using the N\'eel state as a background, complementing previous explanations based on a spin-liquid undoped state. It is predicted that these strong singlets should appear regardless of the long distance properties of the spin system under consideration, as long as the bound state is ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$. In particular, it is shown that they are present in an Ising spin background. The time retardation in the family of $t\ensuremath{-}J$ models leads naturally to low-energy hole states with nonzero momentum and spin one, providing a possible explanation for apparent SO(5)-symmetric features observed recently in this context. Finally, the influence of a short-range Coulombic repulsion is analyzed. Rough estimations suggest that at a distance of one lattice spacing this repulsion is larger than the exchange JJJ. The hole distribution in the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ bound state is reanalyzed in the presence of such repulsion. Very short hole-hole distances lose their relevance in the presence of a realistic hole-hole interaction.
Exact diagonalization calculations are presented for the t – J model in the presence of a uniform magnetic field. Results for 2 × L ladders ( L = 8,10,12) and 4 × 4 square clusters with one and two holes indicate that the diamagnetic response to a perpendicular magnetic field tends to induce a spin liquid state in the spin background. The zero-field spin liquid state of a two-leg ladder is reinforced by the magnetic field: a considerable increase of rung antiferromagnetic correlations is observed for J / t up to 0.6, for one and two holes. Pair breaking is also clearly observed in the ladders and seems to be associated in part with changes promoted by the field in the spin correlations around the zero-field pair. In the 4 × 4 cluster, the numerical results seem to indicate that the field-induced spin liquid state competes with the zero-field antiferromagnetic short range order, the spin liquid state being favoured by higher doping and smaller values of J / t . It is interesting to note that the field effect can also be observed in a 2 × 2 plaquette with one and two holes. This opens up the possibility of gaining a qualitative understanding of the effect.
We study the motion of holes in a mixed-dimensional setup of an antiferromagnetic ladder, featuring nearest-neighbor hopping ttt along the ladders and Ising-type spin interactions along, ${J}_{\ensuremath{\parallel}}$, and across, ${J}_{\ensuremath{\perp}}$, the ladder. We determine exact solutions for the low-energy one- and two-hole eigenstates. The presence of the trans-leg spin coupling, ${J}_{\ensuremath{\perp}}$, leads to a linear confining potential between the holes. As a result, holes on separate legs feature a superlinear binding energy scaling as ${({J}_{\ensuremath{\perp}}/t)}^{2/3}$ in the strongly correlated regime of ${J}_{\ensuremath{\perp}},{J}_{\ensuremath{\parallel}}\ensuremath{\le}t$. This behavior is linked to an emergent length scale $\ensuremath{\lambda}\ensuremath{\propto}{(t/{J}_{\ensuremath{\perp}})}^{1/3}$, stemming from the linear confining potential, and which describes how the size of the two-hole molecular state diverges for ${J}_{\ensuremath{\perp}},{J}_{\ensuremath{\parallel}}\ensuremath{\ll}t$. On the contrary, holes on the same leg unbind at sufficiently low spin couplings. This is a consequence of the altered short-range boundary condition for holes on the same leg, yielding an effective Pauli repulsion between them, limiting their kinetic energy and making binding unfavorable. Finally, we determine the exact nonequilibrium quench dynamics following the sudden immersion of initially localized nearest-neighbor holes. The dynamics is characterized by a crossover from an initial ballistic quantum walk to an aperiodic oscillatory motion around a finite average distance between the holes due to the confining potential between them. In the strongly correlated regime of low spin couplings, ${J}_{\ensuremath{\perp}},{J}_{\ensuremath{\parallel}}\ensuremath{\le}t$, we find this asymptotic distance to diverge as $t/{J}_{\ensuremath{\perp}}$, showing a much stronger scaling than the eigenstates. The predicted results should be amenable to state-of-the-art quantum simulation experiments using currently implemented experimental techniques.
We consider a possible mechanism of the charge stripes in the high- T c cuprates, based on some additional interactions in the two-dimensional t – J model; the next-nearest-neighbour and/or four-spin (ring) exchange interactions. The many-hole correlation functions obtained by numerical exact diagonalization of a finite-cluster t – J model including the correction terms indicate the realization of the mechanism and give some preliminary phase diagrams. A realistic combination of these additional terms is also studied.
We study a model of the stripe state in strongly correlated systems consisting of an array of antiferromagnetic spin ladders, each with n(leg) legs, coupled to each other through the spin-exchange interaction to charged stripes in between each pair of ladders. The charged stripes are assumed to be Luttinger liquids in a spin-gap regime. An effective interaction for a pair of neighboring ladders is calculated by integrating out the gapped spin degrees of freedom in the charged stripes. The low energy effective theory of each ladder is a nonlinear sigma model with additional cross couplings of neighboring ladders, which favor either in-phase or antiphase short-range spin orderings depending on the physical parameters of the charged stripe.