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arXiv:cond-mat/9901234

The Effective Theory of Hole Doped Spin-1 Chain

Xi Dai, Bin Chen, Zhao-Bin Su

Published Jan 22, 1999• 0 citations
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Abstract

An effective theory for the hole doped spin-1 antiferromagnetic chain is proposed in this paper. The two branches of low energy quasipaticle excitation is obtained by the diagrammic technique. In the large t limit(in which t is the hole hopping term), the lower band is essentially the bound state of one hole and one magnon and the other band is the sigle hole state. We find a critical value of t, $t_c=0.21Δ_H$(in which $Δ_H$ is the Haldane gap).For $t>t_c$, with the decrement of t, the mixing of these two bands become stronger and stronger, and at the same time the effective band mass becomes larger and larger. When $t

DOI
10.48550/arxiv.cond-mat/9901234

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arXiv:cond-mat/9808345•Physical review. B, Condensed matter•Feb 1, 1999•1 citation
An effective theory for the hole doped spin-1 antiferromagnetic chain is proposed in this paper. The two branches of low energy quasipaticle excitation is obtained by the diagrammic technique. In the large t limit(in which t is the hole hopping term), the lower band is essentially the bound state of one hole and one magnon and the other band is the sigle hole state. We find a critical value of t, tc=0.21ΔHt_c=0.21Δ_Htc​=0.21ΔH​(in which ΔHΔ_HΔH​ is the Haldane gap).For t>tct>t_ct>tc​, with the decrement of t, the mixing of these two bands become stronger and stronger, and at the same time the effective band mass becomes larger and larger. When $t
An effective theory for the hole-doped spin-1 antiferromagnetic chain is proposed in this paper. The two branches of low-energy quasiparticle excitation is obtained by the diagrammatic technique. In the large t limit (in which t is the hole hopping term), the lower band is essentially the bound state of one hole and one magnon and the other band is the single-hole state. We find a critical value of t, ${t}_{c}=0.21{\ensuremath{\Delta}}_{H}$ (in which ${\ensuremath{\Delta}}_{H}$ is the Haldane gap). For t>tc,t>{t}_{c},t>tc​, with the decrement of t, the mixing of these two bands become stronger and stronger, and at the same time the effective band mass becomes larger and larger. When t<tct<{t}_{c}t<tc​ the minimum of the lower band moves away from zero to the other point between zero and $\ensuremath{\pi}/2.$The spin structure factor is also calculated in this paper, and we find that for the large t limit the main contribution is from the interband transition, which induces a resonant peak in the Haldane gap. For the small t limit the main contribution is from the intraband transition, which only causes a diffusionlike broad bump in the Haldane gap.
Identifying the correct low-energy effective theory for magnons and holes in an antiferromagnet has remained an open problem for a long time. In analogy to the effective theory for pions and nucleons in QCD, based on a symmetry analysis of Hubbard and $t\text{\ensuremath{-}}J$-type models, we construct a systematic low-energy effective field theory for magnons and holes located inside pockets centered at lattice momenta $(\ifmmode\pm\else\textpm\fi{}\frac{\ensuremath{\pi}}{2a},\ifmmode\pm\else\textpm\fi{}\frac{\ensuremath{\pi}}{2a})$. The effective theory is based on a nonlinear realization of the spontaneously broken spin symmetry and makes model-independent universal predictions for the entire class of lightly doped antiferromagnetic precursors of high-temperature superconductors. The predictions of the effective theory are exact, order by order in a systematic low-energy expansion. We derive the one-magnon exchange potentials between two holes in an otherwise undoped system. Remarkably, in some cases the corresponding two-hole Schr\"odinger equations can even be solved analytically. The resulting bound states have ddd-wave characteristics. The ground state wave function of two holes residing in different hole pockets has a ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-like symmetry, while for two holes in the same pocket the symmetry resembles dxy{d}_{xy}dxy​.
We develop an extended self-consistent mean-field approach to study a spin-1 antiferromagnetic (AF) chain with the Hamiltonian H=tsumi{\mathit{tsum}}_{\mathit{i}}tsumi​ [Six{\mathit{S}}_{\mathit{i}}^{\mathit{x}}Six​Si+1x{\mathit{S}}_{\mathit{i}+1}^{\mathit{x}}Si+1x​+Siy{\mathit{S}}_{\mathit{i}}^{\mathit{y}}Siy​Si+1y{\mathit{S}}_{\mathit{i}+1}^{\mathit{y}}Si+1y​ +\ensuremath{\lambda}Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​Si+1z{\mathit{S}}_{\mathit{i}+1}^{\mathit{z}}Si+1z​+D(Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​)2{)}^{2})2] based on Anderson's spin-wave theory. The ground-state energy changes little compared with the traditional spin-wave theory and the excitation spectrum shows a gap which lends support to Haldane's conjecture. At the Heisenberg point (\ensuremath{\lambda}=1,D=0), the ground-state energy and the gap are -1.3505 and 0.3914, respectively. In the Heisenberg-Ising case (\ensuremath{\lambda}>0,D=0), the gap decreases when \ensuremath{\lambda} goes down from 1 and at a critical ${\ensuremath{\lambda}}_{\mathit{c}}$>0 the gap goes to 0, while for the single-ion anisotropy (\ensuremath{\lambda}=1,D>0) the gap decreases with the increasing of D and at a critical value Dc{\mathit{D}}_{\mathit{c}}Dc​ the gap goes to 0. These results agree well with those of other theoretical methods and experiments. We study the one-magnon excitation spectrum carefully and suggest an explanation for its crossing with the two-magnon excitation spectrum at low momentum. We estimate the crossing occurring at k\ensuremath{\simeq}0.3\ensuremath{\pi}. The results agree well with the very recent theoretical and experimental results. The picture of the Haldane gap is discussed from our extended self-consistent mean-field ground state and many interesting properties of the spin-1 AF chain are discussed.
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.
Based on a symmetry analysis of the microscopic Hubbard and ttt-JJJ models, a systematic low-energy effective field theory is constructed for hole-doped antiferromagnets on the honeycomb lattice. In the antiferromagnetic phase, doped holes are massive due to the spontaneous breakdown of the SU(2)s\text{SU}{(2)}_{s}SU(2)s​ symmetry, just as nucleons in Quantum Chromodynamics (QCD) pick up their mass from spontaneous chiral symmetry breaking. In the broken phase, the effective action contains a single-derivative term, similar to the Shraiman-Siggia term in the square lattice case. Interestingly, an accidental continuous spatial rotation symmetry arises at leading order. As an application of the effective field theory, we consider one-magnon exchange between two holes and the formation of two-hole bound states. As an unambiguous prediction of the effective theory, the wave function for the ground state of two holes bound by magnon exchange exhibits fff-wave symmetry.
A large-nnn approximation to the S=1S=1S=1 antiferromagnetic chain, using the symmetric tensor representation and its conjugate, is developed to order 1/n1/n1/n in order to calculate the magnon wave function and to study the effect of modifying the exchange coupling from J to ${J}^{\ensuremath{'}}$ on a single link. It is shown that a magnon bound state exists below the Haldane gap for arbitrarily small negative ${J}^{\ensuremath{'}}\ensuremath{-}J$ but only above a certain critical value of ${J}^{\ensuremath{'}}\ensuremath{-}J$ for positive values. In the former case the binding energy vanishes as $(J\ensuremath{-}{J}^{\ensuremath{'}}{)}^{2}.$