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arXiv:cond-mat/0008394 · Physical Review Letters

Antiferromagnetic Spin Ladders Effectively Coupled by One-Dimensional Electron Liquids

W. Vincent Liu, Eduardo Fradkin

Published Feb 26, 2001• 9 citations
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Abstract

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.

DOI
10.1103/physrevlett.86.1865

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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.
We study a one-dimensional electron liquid coupled by a weak spin-exchange interaction to an antiferromagnetic spin- SSS ladder with nnn legs. A perturbative renormalization group analysis in the semiclassical limit reveals the opening of a spin gap, driven by the local magnetic correlations on the ladder. The effect, which we argue is present for any gapful ladder or gapless ladder with $\mathrm{nS}\ensuremath{\gg}1$, is enhanced by the repulsive interaction among the conduction electrons but is insensitive to the sign of the spin exchange interaction with the ladder. Possible implications for the striped phases of the cuprates are discussed.
We study a one-dimensional (1D) electron liquid embedded in a 2D antiferromagnetic insulator, and coupled to it via a weak antiferromagnetic spin-exchange interaction. We argue that this model may qualitatively capture the physics of a single charge stripe in the cuprates on length and time scales shorter than those set by its fluctuation dynamics. Using a local mean-field approach we identify the low-energy effective theory that describes the electronic-spin sector of the stripe as that of a sine-Gordon model. We determine its phases via a perturbative renormalization-group analysis. For realistic values of the model parameters we obtain a phase characterized by enhanced spin density and composite charge-density-wave correlations, coexisting with subleading triplet and composite singlet-pairing correlations. This result is shown to be independent of the spatial orientation of the stripe on the square lattice. We argue that slow transverse fluctuations of the stripes tend to suppress the density correlations, thus promoting the pairing instabilities. The largest amplitudes for the composite instabilities appear when the stripe forms an antiphase domain wall in the antiferromagnet. For twisted spin alignments the amplitudes decrease and leave room for a new type of composite pairing correlation, breaking parity but preserving time-reversal symmetry.
We propose a low-energy effective theory of lightly doped two-leg $t\ensuremath{-}J$ ladders with the help of slave fermion technique. The continuum limit of this model consists of two kinds of Dirac fermions, which are coupled to the O(3) nonlinear sigma model in terms of the gauge coupling with opposite sign of ``charges.'' In addition to the gauge interaction, there is another kind of attractive force between these Dirac fermions, which arises from the short-ranged antiferromagnetic order. We show that the latter is essential to determine the low energy properties of lightly doped two-leg $t\ensuremath{-}J$ ladders. The effective Hamiltonian we obtain is a bosonic Gaussian model and the boson field basically describes the particle-density fluctuation. We also find two types of gapped spin excitations. Finally, we discuss the possible instabilities: charge density wave and singlet superconductivity (SC). We find that the SC instability dominates in our approximation. Our results indicate that lightly doped ladders fall into the universality class of Luther-Emery model.
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.
Motivated by the co-existing charge and spin order found in strongly correlated ladder systems, we study an effective pseudospin model on a coupled two-leg ladder. A bosonisation analysis yields a rich phase diagram showing Wigner/Peierls charge order and Neel/dimer spin order. In a broad parameter regime, the orbital antiferromagnetic phase is found to be stable. An intermediate gapless phase of finite width is found to lie in between two charge-ordered gapped phases. Our work is potentially relevant for a unified description of a class of strongly correlated, quarter-filled ladder systems.
Interacting one-dimensional electron systems are generally referred to as "Luttinger liquids", after the effective low-energy theory in which spin and charge behave as separate degrees of freedom with independent energy scales. The "spin-incoherent Luttinger liquid" describes a finite-temperature regime that is realized when the temperature is very small relative to the Fermi energy, but larger than the characteristic spin energy scale, and it is realized for instance in the strongly interacting Hubbard chain (with U → ∞). Similar physics can take place in the ground-state, when a Luttinger Liquid is coupled to a spin bath, which effectively introduces a "spin temperature" through its entanglement with the spin degree of freedom. We show that the spin-incoherent state can be exactly written as a factorized wave-function, with a spin wave-function that can be described within a valence bond formalism. This enables us to calculate exact expressions for the momentum distribution function and the entanglement entropy. This picture holds not only for two antiferromagnetically coupled t-J chains, but also for the t-J-Kondo chain with strongly interacting conduction electrons. In chapter 3 we argue that this theory is quite universal and may describe a family of problems that could be dubbed "spin-incoherent".