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arXiv:cond-mat/9805005 · Physical review. B, Condensed matter

Correlation functions of higher-dimensional Luttinger liquids

Lorenz Bartosch, Peter Kopietz

Published Feb 15, 1999• 12 citations
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Abstract

Using higher-dimensional bosonization we study correlation functions of fermions with singular forward scattering. Following Bares and Wen [Phys. Rev. B 48, 8636 (1993)] we consider density-density interactions in $d$ dimensions that diverge in Fourier space for small momentum transfers |q| as $|\mathbf{q}{|}^{\ensuremath{-}\ensuremath{\eta}}$ with $\ensuremath{\eta}=2(d\ensuremath{-}1).$ In this case the single-particle Green's function shows Luttinger-liquid behavior. However, in contrast to $d=1,$ in higher dimensions the singularity of the momentum distribution ${n}_{\mathbf{k}}$ for wave vectors k close to the Fermi wave vector ${k}_{F}$ is characterized by a different exponent than the singularity of the density of states \ensuremath{\rho}(\ensuremath{\omega}) for frequencies \ensuremath{\omega} close to zero. We also calculate the irreducible polarization ${\ensuremath{\Pi}}_{\ensuremath{\star}}(\mathbf{q},\ensuremath{\omega})$ for $|\mathbf{q}|\ensuremath{\approx}{2k}_{F}.$ Whereas in the one-dimensional Tomonaga-Luttinger model ${\ensuremath{\Pi}}_{\ensuremath{\star}}(\ifmmode\pm\else\textpm\fi{}{2k}_{F}+q,\ensuremath{\omega})$ exhibits anomalous scaling, we show that in $d>1$ the leading singular corrections cancel. Thus, even though singular density-density interactions in $d>1$ lead to Luttinger liquid behavior of the single-particle Green's function, gauge invariant correlation functions such as the polarization or the conductivity show conventional Fermi-liquid behavior. We discuss consequences for the effect of disorder on higher-dimensional Luttinger liquids.

DOI
10.1103/physrevb.59.5377

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We present a consistent bosonization treatment for interacting fermion systems in dimensions higher than one. In our analysis, all physically relevant excitations corresponding to forward scattering are included and physical results do not depend on the cutoff parameter introduced in the bosonization process. We calculate the fermion momentum distribution function, from which a condition for the appearance of a Luttinger-liquid state is derived. We find that, for nonsingular interactions, the zero-frequency particle-hole pair excitations directly destroy the orthogonality catastrophe so that there is no Luttinger-liquid fixed point. We further consider the unscreened Coulomb interaction and find that the system remains in a Fermi-liquid state. We observe that the class of singular interactions which are expected to generate a Luttinger-liquid state is very tightly constrained by these zero-frequency excitations.
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 compute the spectral function rho (q, omega ) of the one-dimensional Luttinger model. We discuss the distinct influences of charge-spin separation and of the anomalous dimensions of the fermion operators and their evolution with correlation strength. Charge-spin separation shows up in finite spectral weight at frequencies between v sigma q and v rho q, where v sigma and v rho are the velocities of spin and charge fluctuations, while spectral weight above v rho q and below -v rho q is generated by the hybridization of the Fermi surface at +or-k F by interactions. There are non-universal power-law singularities at these special frequencies. We discuss the consistency of recent photoemission experiments on low-dimensional conductors with a Luttinger-liquid picture, which then would suggest very strong long-range interactions. It is pointed out that many-particle correlation functions in principle exhibit similar singularities, but they probe different and complementary aspects of the Fermi-surface interactions.
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For pure fermionic systems, and in the absence of symmetry breaking (e.g. magnetism, superconductivity, …), two types of metallic phases are well established: (i) the “normal” Fermi liquid phase in d = 3 presented in Chapters 12 and 19 via the response and the Green functions techniques; and (ii) the “anomalous” Luttinger liquid phase in d = 1 originally formulated with the bosonization technique by which the fermionic operators are represented in terms of “bosonic” density operators (Luther and Peschel (1974); Haldane (1981)) with the exact solution (Mattis and Lieb (1965)) of the Tomonaga–Luttinger model (Tomonaga (1950); Luttinger (1963)) discussed inmany reviews and books (see for instance the book by Giamarchi (2004)). Here, to stress similarities and differences between the Fermi liquid and the Luttinger liquid, we will follow the alternative route of response and Green functions techniques (Dzyaloshinskii and Larkin (1973), Di Castro and Metzner (1991), Metzner et al. (1998)) and add a few comments on the renormalization group approach (Metzner and Di Castro (1993)). As we have seen in Chapters 12 and 19, the Fermi liquid is described asymptotically by the free low-lying single-particle excitations, i.e., the quasiparticles with a zero-kelvin discontinuous occupation number in momentum space n k (see Eq. (18.54)) at a well defined Fermi surface (k = k F ): where here k F> and k F< indicate the limit k → k F from above and below. This discontinuity still marks the Fermi surface in the interacting system. The finite reduction of the single-particle spectral weight z kF with respect to the Fermi gas (z kF = 1) is given by the finite (i.e. non-critical) “wavefunction renormalization” (see Eq. (18.94)). The presence of the discontinuity at the Fermi surface, together with the Pauli principle, compel the inverse quasiparticle lifetime to be τ −1 ≈ max( T 2 , ϵ 2 ), where _ is the deviation of the energy of the quasiparticle from the Fermi energy. The energy uncertainty ħτ −1 due to the finite lifetime of a quasiparticle near the Fermi surface is small compared to its energy ϵ , and the quasiparticle concept is well defined. All the momentum transferring scattering processes become asymptotically ineffective, as derived phenomenologically in Chapter 12 and confirmed microscopically in Chapter 19.
We calculate the dynamic structure factor $S(\ensuremath{\omega},q)$ of spinless fermions in one dimension with quadratic energy dispersion k2/2m{k}^{2}/2mk2/2m and long-range density-density interaction whose Fourier transform fq{f}_{q}fq​ is dominated by small momentum transfers $q\ensuremath{\lesssim}{q}_{0}⪡{k}_{F}$. Here q0{q}_{0}q0​ is a momentum-transfer cutoff and kF{k}_{F}kF​ is the Fermi momentum. Using functional bosonization and the known properties of symmetrized closed fermion loops, we obtain an expansion of the inverse irreducible polarization to second order in the small parameter q0/kF{q}_{0}/{k}_{F}q0​/kF​. In contrast to perturbation theory based on conventional bosonization, our functional bosonization approach is not plagued by mass-shell singularities. For interactions which can be expanded as ${f}_{q}={f}_{0}+{f}_{0}^{\ensuremath{''}}{q}^{2}/2+O({q}^{4})$ with ${f}_{0}^{\ensuremath{''}}\ensuremath{\ne}0$, we show that the momentum scale ${q}_{c}=1/|m{f}_{0}^{\ensuremath{''}}|$ separates two regimes characterized by a different qqq dependence of the width ${\ensuremath{\gamma}}_{q}$ of the collective zero sound mode and other features of $S(\ensuremath{\omega},q)$. For qc⪡q⪡kF{q}_{c}⪡q⪡{k}_{F}qc​⪡q⪡kF​ we find that the line shape is non-Lorentzian with an overall width ${\ensuremath{\gamma}}_{q}\ensuremath{\propto}{q}^{3}/(m{q}_{c})$ and a threshold singularity ${[(\ensuremath{\omega}\ensuremath{-}{\ensuremath{\omega}}_{q}^{\ensuremath{-}}){\text{ln}}^{2}(\ensuremath{\omega}\ensuremath{-}{\ensuremath{\omega}}_{q}^{\ensuremath{-}})]}^{\ensuremath{-}1}$ at the lower edge $\ensuremath{\omega}\ensuremath{\rightarrow}{\ensuremath{\omega}}_{q}^{\ensuremath{-}}=vq\ensuremath{-}{\ensuremath{\gamma}}_{q}$, where vvv is the velocity of the zero sound mode. Assuming that higher orders in perturbation theory transform the logarithmic singularity into an algebraic one, we find for the corresponding threshold exponent ${\ensuremath{\mu}}_{q}=1\ensuremath{-}2{\ensuremath{\eta}}_{q}$ with ${\ensuremath{\eta}}_{q}\ensuremath{\propto}{q}_{c}^{2}/{q}^{2}$. Although for $q\ensuremath{\lesssim}{q}_{c}$ we have not succeeded to explicitly evaluate our functional bosonization result for $S(\ensuremath{\omega},q)$, we argue that for any one-dimensional model belonging to the Luttinger liquid universality class, the width of the zero sound mode scales as q2/m{q}^{2}/mq2/m for $q\ensuremath{\rightarrow}0$.