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arXiv:cond-mat/9408047 · Physical Review Letters

One-Particle and Two-Particle Instability of Coupled Luttinger Liquids

Daniel Boies, C. Bourbonnais, A.–M. S. Tremblay

Published Feb 6, 1995• 116 citations
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Abstract

It is shown that the Luttinger liquid is unstable to arbitrarily small transverse hopping. The crossover temperatures below which either transverse coherent band motion or long-range order start to develop can be finite even when spin and charge velocities differ. Explicit scaling relations for the one-particle and two-particle crossover temperatures are derived in terms of transverse hopping, spin and charge velocities, and anomalous exponents. The special case of infinite-range transverse hopping is treated exactly and yields a Fermi liquid down to $T=0$, unless the anomalous exponent $\ensuremath{\theta}>1$.

DOI
10.1103/physrevlett.74.968

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It is shown that the Luttinger liquid is unstable to arbitrarily small transverse hopping. The crossover temperatures below which either transverse coherent band motion or long-range order start to develop can be finite even when spin and charge velocities differ. Explicit scaling relations for the one-particle and two-particle crossover temperatures are derived in terms of transverse hopping, spin and charge velocities, and anomalous exponents. The special case of infinite-range transverse hopping is treated exactly and yields a Fermi liquid down to T=0T=0T=0, unless the anomalous exponent \ensuremathθ>1\ensuremath{\theta}>1\ensuremathθ>1.
The stability of the Luttinger liquid to small transverse hopping has been studied from several points of view. The renormalization group approach in particular has been criticized because it does not take explicitly into account the difference between spin and charge velocities and because the interaction should be turned-on before the transverse hopping if the stability of the Luttinger liquid is a non-perturbative effect. An approach that answers both of these objections is explained here. It shows that the Luttinger liquid is unstable to arbitrarily small transverse hopping. The crossover temperatures below which either transverse coherent band motion or long-range order start to develop can be finite even when spin and charge velocities differ. Explicit scaling relations for these one-particle and two-particle crossover temperatures are derived in terms of transverse hopping, spin and charge velocities and anomalous exponents. The renormalization group results are recovered as special cases when spin and charge velocities are identical. The results compare well with recent experiments presented at this conference. Magnetic field effects are alluded to.
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.
We demonstrate that for general spin-charge separated Luttinger liquids there exists a critical value of the inter-Luttinger-liquid single particle hopping, ${\mathit{t}}_{\mathrm{\ensuremath{\perp}}}^{\mathit{c}}$, below which there is no coherent single particle hopping between the liquids. In the absence of coherent single particle hopping, two Luttinger liquids coupled by ${\mathit{t}}_{\mathrm{\ensuremath{\perp}}}$ will not exhibit split Fermi surfaces. For many Luttinger liquids, no band dipsersing between the liquids will form, and thus the system will retain a one-dimensional Fermi surface. This will have dramatic implications for the physical properties of such a system.
We consider the low-energy region of an array of Luttinger liquids coupled by a weak interchain hopping. The leading-logarithmic divergences can be resummed to all orders, within a self-consistent perturbative expansion in the hopping, in the large-dimension limit. The anomalous exponent scales to zero below the one-particle crossover temperature. As a consequence, coherent quasiparticles with finite weight appear along the whole Fermi surface. Extending the expansion self-consistently to all orders turns out to be crucial to restore the correct Fermi-liquid behavior.
One-particle interchain hopping in a system of coupled Luttinger liquids is investigated using exact diagonalizations techniques. First, the two-chain problem of spinless fermions is studied in order to see the behavior of the band splitting as a function of the exponent $\ensuremath{\alpha}$ which characterizes the one-dimensional (1D) Luttinger liquid. Moderate intrachain interactions can lead to a strong reduction of this splitting. The onset of the confinement within the individual chains (defined by a vanishing splitting) seems to be governed by $\ensuremath{\alpha}$. We give numerical evidence that interchain coherent hopping can be totally suppressed for $\ensuremath{\alpha}\ensuremath{\sim}0.4$ or even smaller $\ensuremath{\alpha}$ values. The transverse conductivity is shown to exhibit a strong incoherent part. Even when coherent interchain hopping is believed to occur (at small $\ensuremath{\alpha}$ values), it is shown that the coherent Drude weight is always significantly smaller than the incoherent weight. Implications for the optical experiments in quasi-1D organic or high-Tc{T}_{c}Tc​ superconductors are outlined.