We calculate the effect of a quadratic term in the energy dispersion on the low-energy behavior of the Green's function of the spinless Tomonaga–Luttinger model (TLM). Assuming that for small wave-vectors q= k-k F the fermionic excitation energy relative to the Fermi energy is v F q+q2/(2m), we explicitly calculate the single-particle Green's function for finite but small values of λ=q c /(2k F ). Here k F is the Fermi wave-vector, q c is the maximal momentum transfered by the interaction, and v F =k F /m is the Fermi velocity. Assuming equal forward scattering couplings g2=g4, we find that the dominant effect of the quadratic term in the energy dispersion is a renormalization of the anomalous dimension. In particular, at weak coupling the anomalous dimension is [Formula: see text], where γ is the anomalous dimension of the TLM. We also show how to treat the change of the chemical potential due to the interactions within the functional bosonization approach in arbitrary dimensions.
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
nk for wave vectors k close to the Fermi wave vector
kF 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.
We calculate the dynamic structure factor $S(\ensuremath{\omega},q)$ of spinless fermions in one dimension with quadratic energy dispersion
k2/2m and long-range density-density interaction whose Fourier transform
fq is dominated by small momentum transfers $q\ensuremath{\lesssim}{q}_{0}⪡{k}_{F}$. Here
q0 is a momentum-transfer cutoff and
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. 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
q 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 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
v 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 for $q\ensuremath{\rightarrow}0$.
We reinvestigate the momentum-resolved single-particle spectral function of the Tomonaga-Luttinger model. In particular, we focus on the role of the momentum dependence of the two-particle interaction
V(q). Usually,
V(q) is assumed to be a constant and integrals are regularized in the ultraviolet ``by hand'' employing an ad hoc procedure. As the momentum dependence of the interaction is irrelevant in the renormalization group sense, this does not affect the universal low-energy properties of the model, e.g., exponents of power laws, if all energy scales are sent to zero. If, however, the momentum
k is fixed away from the Fermi momentum
kF, with $|k\ensuremath{-}{k}_{\mathrm{F}}|$ setting a nonvanishing energy scale, the details of
V(q) start to matter. We provide strong evidence that any curvature of the two-particle interaction at small transferred momentum
q destroys power-law scaling of the momentum-resolved spectral function as a function of energy. Even for $|k\ensuremath{-}{k}_{\mathrm{F}}|$ much smaller than the momentum-space range of the interaction the spectral line shape depends on the details of
V(q). The significance of our results for universality in the Luttinger liquid sense, for experiments on quasi-one-dimensional metals, and for recent results on the spectral function of one-dimensional correlated systems taking effects of the curvature of the single-particle dispersion into account (``nonlinear LL phenomenology'') is discussed.
We examine the spin-charge separation effects which are manifest in the Tomonaga-Luttinger model (TLM) in the framework of the field theoretical renormalization group (RG) theory. Initially, it is shown how far one can go with the perturbative RG method. Following that, we take into full account the Ward identities [Phys. Rev. 78, 172 (1950)]. This allows us to calculate the spin and charge velocities exactly. Next, we construct the Callan-Symanzik equation (CSE) [Phys. Rev. D 5, 3202 (1972); Commun. Math. Phys. 23, 49 (1971)] for the renormalized fermionic one-particle Green's function. Finally, making use of those perturbative results for the TLM and taking the two characteristic velocities as basic ingredients, we solve this CSE in qualitative good agreement with the exact result.
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.