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arXiv:cond-mat/0404157 · Journal of Physics Condensed Matter

New results for thet–Jmodel in ladders: changes in the spin liquid state with applied magnetic field; implications for the cuprates

A. Fabricio Albuquerque, G B Martins

Published Apr 2, 2005• 6 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/0953-8984/17/15/013

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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.
Exact Diagonalization (ED) calculations are presented for the t-J model in the presence of a uniform magnetic field. Results for 4 x 4 clusters and 2 x L ladders (L=8,10,12) with 1 and 2 holes indicate that the diamagnetic response to a perpendicular magnetic field induces a Resonating Valence Bond (RVB) state in the spin background. In the 4 x 4 square cluster, this RVB-like state competes with the zero-field AF quasi-long-range-order(QLRO): the RVB-like state is favored by higher doping and smaller values of J/t. In 2-leg ladders, the zero-field spin-liquid state is reinforced by the magnetic field: a considerable increase of rung AF correlations is observed for J/t < 0.6, for 1 and 2 holes. Pair-breaking is clearly observed in the ladders: this seems to be associated in part with changes promoted by the field in the spin correlations around the zero-fied pair.
We study the ground-state spin correlations in the gapless incommensurate regime of a S=1/2S=1/2S=1/2 XXZ\mathrm{XXZ}XXZ chain and a two-leg antiferromagnetic ladder under a magnetic field, in which the gapless excitations form a Tomonaga-Luttinger (TL) liquid. We calculate numerically the two-spin correlation functions and the local magnetization in the two models using the density-matrix renormalization-group method. By fitting the numerical results for an open XXZ\mathrm{XXZ}XXZ chain of 100 spins to correlation functions of a Gaussian model, we determine the TL-liquid parameter K and the amplitudes of the correlation functions. The value of K estimated from the fits is in excellent agreement with the exact value obtained from the Bethe ansatz. We apply the same method to the open ladder consisting of 200 spins and determine the dependence of K on the magnetization M. The $K\ensuremath{-}M$ relation changes drastically depending on the ratio of the coupling constants in the leg and rung directions. We also discuss implications of these results to experiments on the nuclear spin relaxation rate 1/T1{1/T}_{1}1/T1​ and dynamical spin structure factors.
We study the magnetic orbital effect of a doped two-leg ladder in the presence of a magnetic field component perpendicular to the ladder plane. Combining both low-energy approach (bosonization) and numerical simulations (density-matrix renormalization group) on the strong coupling limit ($t\text{\ensuremath{-}}J$ model), a rich phase diagram is established as a function of hole doping and magnetic flux. Above a critical flux, the spin gap is destroyed and a Luttinger liquid phase is stabilized. Above a second critical flux, a reentrance of the spin gap at high magnetic flux is found. Interestingly, the phase transitions are associated with a change of sign of the orbital susceptibility. Focusing on the small magnetic field regime, the spin-gapped superconducting phase is robust, but immediately acquires algebraic transverse (i.e., along rungs) current correlations which are commensurate with the 4kF4{k}_{F}4kF​ density correlations. In addition, we have computed the zero-field orbital susceptibility for a large range of doping and interaction ratio J∕tJ∕tJ∕t: we found strong anomalies at low J∕tJ∕tJ∕t only in the vicinity of the commensurate fillings corresponding to $\ensuremath{\delta}=1∕4$ and 1∕21∕21∕2. Furthermore, the behavior of the orbital susceptibility reveals that the nature of these insulating phases is different: while for $\ensuremath{\delta}=1∕4$ a 4kF4{k}_{F}4kF​ charge density wave is confirmed, the $\ensuremath{\delta}=1∕2$ phase is shown to be a bond order wave.
Using numerical results from density matrix renormalization group (DMRG) calculations for the t-J model, on systems as large as 10\ifmmode\times\else\texttimes\fi{}7, we examine the structure of the one and two hole ground states in ladder systems and in two dimensional clusters. A simple theoretical framework is used to explain why holes bind in pairs in two-dimensional antiferromagnets. For the case J/t=0.5, which we have studied, the hole pairs reside predominantly on a 2\ifmmode\times\else\texttimes\fi{}2 core plaquette with the probability that the holes are on diagonal sites greater than nearest-neighbor sites. There is a strong singlet bond connecting the spins on the two remaining sites of the plaquette. We find that a general characteristic of dynamic holes in an antiferromagnet is the presence of frustrating antiferromagnetic bonds connecting next-nearest-neighbor sites across the holes. Pairs of holes bind in order to share the frustrating bonds. At low doping, in addition to hole pairs, there are two additional low-energy structures which spontaneously form on certain finite systems. The first is an undoped L\ifmmode\times\else\texttimes\fi{}2 spin-liquid region, or ladder. The second is a hole moving along a one dimensional chain of sites. At higher doping we expect that hole pairing is always favored.
A proper theoretical description of doped antiferromagnets is until now an unresolved issue. In this work we study chain and ladder systems, and in particularly we focus on the physically most interesting two leg ladder. Our starting point is the Spin Fermion model in which an antiferromagnetic background of localized spins interacts with mobile holes via a rotation invariant Kondo-like term. The interesting region of the phase diagram is that close to the Mott-insulator transition where the doping delta is zero. At zero doping in fact the system is an insulator and the spins organize themselves in a spin liquid, a rotational invariant state characterized by a finite correlation length and an energy gap (a spin gap) above the ground state. Such a state is also the ground state of a field theory, the non linear sigma model (NLsigmaM) which is recognized as the low energy effective theory for antiferromagnetic spin ladder. The first question we answer is how such a spin state evolves as one moves off from the Mott phase by increasing the doping. Integrating out the fermions in our model we obtain an effective theory for the spins which we are able to evaluate in the continuum limit. The effective theory is again a NLsigmaM with coupling constants which depend on the concentration of dopant holes. In contrast to existing mean field calculation our theory predicts a lowering of the spin gap with doping and a consequent increase in the correlation length. Indeed a lowering of the spin gap due to doping is also observed in numerical simulation and on NMR experiments on $Sr14-xCaxCu24O41 with which we obtain very good agreement. Secondly we concentrated on the behavior of the fermions. The general paradigm of interacting fermions in one dimension is that of the Luttinger Liquid characterized by bosonic excitations and spin charge separation. By generalizing our approach we are able to access the one particle fermion propagator as a quantity averaged over the NLsigmaM which controls the spin background. We see that in the limit of zero doping the quasiparticle weight Z is non-zero in a neighborhood of the Fermi energy. This in turn implies that the Luttinger liquid parameter Krho goes to one as the doping delta goes to zero as was first argued by Schulz. Our stronger result allows us to assert that in the very low doping regime the fermions constitute a Fermi liquid.
Abstract Doped quantum magnets, especially spin liquids with fractionalized excitations, have long attracted attention as a promising path towards unconventional superconductivity. We investigate the hole-doped Kitaev-Heisenberg ( t – J – K ) model on a two-leg ladder geometry using the density-matrix renormalization group (DMRG). We consider the behavior of the antiferromagnetic Kitaev spin-liquid phase as a function of hopping strength t and doping level. This reveals intriguing pairing tendencies only for tK≲0.65\frac{t}{K}\lesssim 0.65Kt​≲0.65 t K ≲ 0.65 , consistent with prior results on three-leg ladders, and firmly supports the emerging picture that the physics of doped Kitaev spin liquids strongly depends on the kinetic energy of the doped holes. Analysis of one- and two-hole doping uncovers close links between spatial profiles of the plaquette operator and the charge density. We also construct a doping-dependent phase diagram for antiferromagnetic Heisenberg interactions and intermediate hopping t = 1. Upon doping, the rung-singlet region develops dominant superconducting correlations. Spin-density wave-like behavior is found in the Kitaev limits, and in the stripy phase.