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arXiv:cond-mat/9803145 · Physical Review Letters

Doping of a Spin-1 Chain: An Integrable Model

Holger Frahm, Markus P. Pfannmüller, Alexei M. Tsvelik

Published Sep 7, 1998• 29 citations
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Abstract

An exactly soluble model describing a spin $S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$ antiferromagnetic chain doped with mobile $S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2$ carriers is constructed. In its continuum limit the undoped state is described by three gapless Majorana fermions composing the $\mathrm{SU}(2)$ triplet. Doping adds a scalar charge field and a singlet Majorana fermion with different velocity. We argue that this mode survives when the Haldane gap is added.

DOI
10.1103/physrevlett.81.2116

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An exactly soluble model describing a spin S=1S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1S=1 antiferromagnetic chain doped with mobile S=1/2S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2S=1/2 carriers is constructed. In its continuum limit the undoped state is described by three gapless Majorana fermions composing the SU(2)\mathrm{SU}(2)SU(2) triplet. Doping adds a scalar charge field and a singlet Majorana fermion with different velocity. We argue that this mode survives when the Haldane gap is added.
Different models for doping of two-orbital chains with mobile S=1/2S=1/2S=1/2 fermions and strong, ferromagnetic (FM) Hund's rule couplings stabilizing the S=1 spins are investigated by density matrix renormalization group (DMRG) methods. The competition between antiferromagnetic (AF) and FM order leads to a rich phase diagram with a narrow FM region for weak AF couplings and strongly enhanced triplet pairing correlations. Without a level difference between the orbitals, the spin gap persists upon doping, whereas gapless spin excitations are generated by interactions among itinerant polarons in the presence of a level difference. In the charge sector we find dominant singlet pairing correlations without a level difference, whereas upon the inclusion of a Coulomb repulsion between the orbitals or with a level difference, charge density wave (CDW) correlations decay slowest. The string correlation functions remain finite upon doping for all models.
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.
We present a numerical study of single-chain models of doped spin-1 compounds. We use low-energy effective one-dimensional models for both the cases of paramagnetic and spin-1/2 doping. In the case of paramagnetic doping, the effective model is equivalent to the bond disordered spin-1/2 chain model recently analysed by means of the real space renormalization group by Hyman and Yang. By means of exact diagonalizations in the XX limit, we confirm the stability of the Haldane phase for weak disorder. Above a critical amount of disorder, the effective model flows to the so-called random singlet fixed point. In the case of spin-1/2 doping, we argue that the Haldane phase should be destabilized even for weak disorder. This picture is not in contradiction with existing experimental data. We also discuss the possible occurrence of (unobserved) antiferromagnetically ordered phases.
The low-temperature magnetization process of antiferromagnetic spin- SSS chains doped with mobile spin- $(S\ensuremath{-}1/2)$ carriers is studied in an exactly solvable model. For sufficiently high magnetic fields the system is in a metallic phase with a finite gap for magnetic excitations. In this phase, which exists for a large range of carrier concentrations xxx, the zero-temperature magnetization is determined by xxx alone. This leads to plateaus in the magnetization curve at a tunable fraction of the saturation magnetization. The critical behavior at the edges of these plateaus is studied in detail.
We study itinerant hole doping of the spin-111 Haldane symmetry-protected topological (SPT) phase in two ttt--JJJ chains coupled by ferromagnetic (FM) Hund exchange. Density-matrix renormalization group calculations show that string order, a finite spin gap, and the Haldane entanglement structure coexist over a broad doping range. Combined with entanglement-entropy scaling consistent with a single gapless charge mode, these results support a metallic doped Haldane SPT with a C1S0C1S0C1S0 low-energy description. At low doping, short-range charge correlations reduce the density of spin-1/21/21/2 defect rungs, whereas at larger doping their positions become nearly uncorrelated while their spins are collectively incorporated into the correlated spin sector. The doped Haldane SPT terminates in a fully spin-polarized FM phase, which forms a reentrant pocket at high doping in the global phase diagram. Dilute-limit analyses trace the two FM boundaries to spin--charge factorization at weak Hund coupling and to rung-triplet binding with reduced pair mobility at strong Hund coupling. Our results show how mobile carriers reorganize the Haldane spin structure while preserving its characteristic signatures.
Using the valence-bond-solid (VBS) approach and the Schwinger-boson mean-field approximation, we study the dependence of the Haldane gap of a spin-1 linear-chain Heisenberg antiferromagnet on impurity doping with different spins. The impurity spins affect the singlet pairing order parameter \ensuremath{\Delta} and the constraint factor \ensuremath{\lambda}. As a result, the Haldane gap is reduced by a factor \ensuremath{\sim}ni2/3{\mathit{n}}_{\mathit{i}}^{2/3}ni2/3​, with ni{\mathit{n}}_{\mathit{i}}ni​ as the impurity concentration, and eventually collapses at ni{\mathit{n}}_{\mathit{i}}ni​\ensuremath{\sim}1/\ensuremath{\xi} with \ensuremath{\xi} as the VBS correlation length. This theoretical prediction can be verified by neutron-scattering experiments.