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Journal of the Physical Society of Japan

Antiferromagnetic Phase Transition and Crossover to Fermi Liquid Phase in a Weakly Coupled Half-Filled Chain System

Jun‐ichiro Kishine, Kenji Yonemitsu

Published Aug 15, 1998• 23 citations
Publisher

Abstract

Effects of the intrachain electron-electron umklapp process and the interchain one-particle hopping, t ⊥ , in a weakly coupled half-filled chain system have been studied, based on the perturbative renormalization-group approach. By solving the scaling equations, we found that the intrachain umklapp process strongly suppresses an interchain one-particle process and causes a finite critical value for t ⊥ , t ⊥ c . For t ⊥ < t ⊥ c , as the temperature decreases, the system undergoes a phase transition into an antiferromagnetic long-range-ordered phase at a transition temperature T N which increases with the increasing t ⊥ . On the other hand, for t ⊥ > t ⊥ c , the system undergoes a crossover to the Fermi liquid phase where the interchain coherent propagation of quasi-particles occurs.

DOI
10.1143/jpsj.67.2590

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Effects of the intrachain electron-electron umklapp process and the interchain one-particle hopping, t ⊥ , in a weakly coupled half-filled chain system have been studied, based on the perturbative renormalization-group approach. By solving the scaling equations, we found that the intrachain umklapp process strongly suppresses an interchain one-particle process and causes a finite critical value for t ⊥ , t ⊥ c . For t ⊥ < t ⊥ c , as the temperature decreases, the system undergoes a phase transition into an antiferromagnetic long-range-ordered phase at a transition temperature T N which increases with the increasing t ⊥ . On the other hand, for t ⊥ > t ⊥ c , the system undergoes a crossover to the Fermi liquid phase where the interchain coherent propagation of quasi-particles occurs.
We study two (very) weakly coupled Hubbard chains in the half-filled case, and especially the situation where the intrachain Mott scale m is much larger than the (bare) single-electron interchain hopping ${t}_{\ensuremath{\perp}}.$ First, we find that the divergence of the intrachain umklapp channel at the Mott transition results in the complete vanishing of the single-electron interchain hopping: this is significant of a strong confinement of coherence along the chains. Excitations are usual charge fermionic solitons and spinon-(anti)spinon pairs of the Heisenberg chain. Then, we show rigorously how the tunneling of spinon-(anti)spinon pairs produces an antiferromagnetic interchain exchange of the order of ${J}_{\ensuremath{\perp}}{=t}_{\ensuremath{\perp}}^{2}/m.$ In the ``confined'' phase and in the far infrared, the system behaves as a pure spin ladder. The final result is an insulating ground state with spin-gapped excitations exactly as in the opposite ``delocalized'' limit (i.e., for rather large interchain hoppings) where the two-leg ladder is in the well-known insulating $D\ensuremath{-}\mathrm{Mott}$ phase. Unlike materials with an infinite number of coupled chains (Bechgaard salts), the confinement/deconfinement transition at absolute zero is here a simple crossover: no metallic phase is found in undoped two-leg ladders. This statement might be generalized for N-leg ladders with N=3,4,…N=3,4,\dots{}N=3,4,… (but not too large).
We study two (very) weakly-coupled Hubbard chains in the half-filled case, and especially the situation where the intrachain Mott scale mmm is much larger than the bare single-electron interchain hopping t⊥t_{\perp}t⊥​. First, we find that the divergence of the intrachain Umklapp channel at the Mott transition results in the complete vanishing of the single-electron interchain hopping: This is significant of a strong confinement of coherence along the chains. Excitations are usual charge fermionic solitons and spinon pairs of the Heisenberg chain. Then, we show rigorously how the tunneling of spinon-pairs produces a magnetic interchain exchange J⊥=t⊥2/m>0J_{\perp}={t_{\perp}}^2/m>0J⊥​=t⊥​2/m>0. The result is an insulating state with spin-gapped excitations as in the delocalized limit (i.e. for rather large interchain hoppings), where the two-leg ladder is in the well-known D-Mott phase. Unlike for Bechgaard salts, the confinement/deconfinement transition at absolute zero is here a simple crossover: no metallic phase is found in undoped two-leg ladders. This statement might be generalized for N-leg ladders with N=3,4... (but not too large).
Quasi-one-dimensional quantum antiferromagnets formed by a ddd-dimensional hypercubic lattice of weakly coupled spin- 12\frac{1}{2}21​ antiferromagnetic Heisenberg chains are studied by combining exact results in one-dimension and renormalization group analyses of the interchain correlations. It is shown that ddd-dimensional magnetic long-range order develops at zero temperature for infinitesimal antiferromagnetic or ferromagnetic interchain couplings. In the presence of weak bond alternations, the order-disorder transition occurs at a finite interchain coupling. Relevances to the lightly doped quantum antiferromagnets and multilayer quantum Hall systems are discussed.
We use higher-dimensional bosonization to study the normal state of electrons in weakly coupled metallic chains interacting with long-range Coulomb forces. Particular attention is paid to the crossover between Luttinger and Fermi-liquid behavior as the interchain hopping ${t}_{\ensuremath{\perp}}$ is varied. Although in the physically interesting case of finite but small ${t}_{\ensuremath{\perp}}$ the quasiparticle residue does not vanish, the single-particle Green's function exhibits the signature of Luttinger-liquid behavior (i.e., anomalous scaling and spin-charge separation) in a large intermediate parameter regime. Using realistic parameters, we find that the scaling behavior in this regime is characterized by an anomalous dimension of the order of unity, as suggested by recent experiments on quasi-one-dimensional conductors. Our calculation also gives insights into the approximations inherent in higher-dimensional bosonization; in particular, we show that the replacement of a curved Fermi surface by a finite number MMM of flat patches can give rise to unphysical nesting singularities in the single-particle Green's function, which disappear only in the limit $M\ensuremath{\rightarrow}\ensuremath{\infty}$ or if curvature effects are included. We also compare our approach with other methods. This work extends our recent letter [Phys. Rev. Lett. 74, 2997 (1995)].
We study the problem of the crossover from one- to higher-dimensional metals by considering an array of Luttinger liquids (one-dimensional chains) coupled by a weak interchain hopping ${t}_{\ensuremath{\perp}}.$ We evaluate the exact asymptotic low-energy behavior of the self-energy in the anisotropic infinite-dimension limit. This limit extends the dynamical mean-field concept to the case of a chain embedded in a self-consistent medium. The system flows to a Fermi-liquid fixed point for energies below the dimensional crossover temperature, and the anomalous exponent \ensuremath{\alpha} renormalizes to zero, in the case of equal spin and charge velocities. In particular, the single-particle spectral function shows sharp quasiparticle peaks with nonvanishing weight along the whole Fermi surface, in contrast to the lowest-order result. Our result is obtained by carrying out a resummation of all diagrams of the expansion in ${t}_{\ensuremath{\perp}}$ contributing to the anisotropic $\stackrel{\ensuremath{\rightarrow}}{D}\ensuremath{\infty}$ limit. This is done by solving, in an almost completely analytic way, an asymptotically exact recursive equation for the renormalized vertices, within a skeleton expansion. Our outcome shows that perturbation expansions in ${t}_{\ensuremath{\perp}}$ restricted to lowest orders are unreliable below the crossover temperature. The extension to finite dimensions is discussed. This work extends our recent letter [Phys. Rev. Lett. 83, 128 (1999)], and includes all mathematical details.