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Physical review. B, Condensed matter

Momentum distribution neark=3 kF in the Tomonaga-Luttinger model

Karlo Penc, Jenő Sólyom

Published Dec 15, 1991• 16 citations
Publisher

Abstract

In contrast to normal Fermi liquids, where the momentum distribution n(k) has a discontinuous jump at k=${\mathit{k}}_{\mathit{F}}$, in Luttinger liquids a power-law singularity appears in n(k). It was argued that this singularity shows up not only at ${\mathit{k}}_{\mathit{F}}$ but at 3${\mathit{k}}_{\mathit{F}}$,5${\mathit{k}}_{\mathit{F}}$, . . . as well, and the anomaly exponent has been detemined for the one-dimensional Hubbard model using numerical methods or assuming conformal invariance. We present an exact calculation of the anomaly exponent for the Tomonaga-Luttinger model and compare it with the results for the Hubbard model.

DOI
10.1103/physrevb.44.12690

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Interacting electron systems in one dimension with spin-split Fermi points are studied theoretically with the aid of the Tomonaga-Luttinger bosonization. The spin splitting of the systems considered is classified into three categories: (i) the momentum-dependent spin splitting without the time-reversal symmetry, (ii) the momentum-dependent spin splitting with the time-reversal symmetry, and (iii) the momentum-independent spin splitting. Calculating analytically the current-current correlation functions in the three cases, the critical exponents of the optical Fermi-edge singularity (FES) are obtained by using the forward-scattering Tomonaga-Luttinger model. The phase diagrams for the sign of the FES exponent are drawn to clarify the relation between the shape of optical edge spectra and the electron-electron or the electron-hole interactions. Response to circularly polarized incident light is also revealed. In the case of the momentum-dependent spin splitting, the power-law divergence of the edge spectra is possible even for repulsive electron-hole interactions. This is in striking contrast to the FES in spin-degenerate (no spin splitting) electron systems. The FES in a spin-split chiral Luttinger liquid is also discussed. \textcopyright{} 1996 The American Physical Society.

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The Luttinger liquid

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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.

Cambridge University Press eBooks•Aug 27, 2015•0 citations
In contrast to normal Fermi liquids, where the momentum distribution n(k) has a discontinuous jump at k=kF{\mathit{k}}_{\mathit{F}}kF​, in Luttinger liquids a power-law singularity appears in n(k). It was argued that this singularity shows up not only at kF{\mathit{k}}_{\mathit{F}}kF​ but at 3kF{\mathit{k}}_{\mathit{F}}kF​,5kF{\mathit{k}}_{\mathit{F}}kF​, . . . as well, and the anomaly exponent has been detemined for the one-dimensional Hubbard model using numerical methods or assuming conformal invariance. We present an exact calculation of the anomaly exponent for the Tomonaga-Luttinger model and compare it with the results for the Hubbard model.
Contrary to the usual Fermi liquids, where the exponent of the momentum distribution at k=kF is fixed to an integer, for the Tomonaga-Luttinger (TL) liquids the anomalous exponent of the momentum distribution changes continuously, resulting in a power-law singularity of the momentum distribution function. It has been said that this power-law singularity appears at kF and 3kF for the Hubbard model as well as at kF for the t-J model. Using the conformal field theory (CFT) technique, we present an exact calculation of the anomalous exponent of the t-J model at kF, 3kF, 5kF and 7kF and compare it with the results for the Hubbard model.
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 ddd 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,d=1,d=1, in higher dimensions the singularity of the momentum distribution nk{n}_{\mathbf{k}}nk​ for wave vectors k close to the Fermi wave vector kF{k}_{F}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>1d>1d>1 the leading singular corrections cancel. Thus, even though singular density-density interactions in d>1d>1d>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 perform a rigorous computation of the spin and charge density correlations of the 1d repulsive Hubbard model at weak coupling, focusing on the properties of the Fourier transform at momentum 0 and ±2 p F , if p F is the Fermi momentum. We prove that the interaction changes the singularity at ±2 p F (the discontinuity in the derivative becomes a power law singularity) while the singularity at 0 is essentially unchanged. Our results show that the logarithmic divergences at zero momentum recently found in [7], which would be in contrast with Luttinger liquid behaviour, are indeed spurious.
Interacting electron systems in one dimension with spin-split Fermi points are studied theoretically with the aid of the Tomonaga-Luttinger bosonization. The spin splitting of the systems considered is classified into three categories: (i) the momentum-dependent spin splitting without the time-reversal symmetry, (ii) the momentum-dependent spin splitting with the time-reversal symmetry, and (iii) the momentum-independent spin splitting. Calculating analytically the current-current correlation functions in the three cases, the critical exponents of the optical Fermi-edge singularity (FES) are obtained by using the forward-scattering Tomonaga-Luttinger model. The phase diagrams for the sign of the FES exponent are drawn to clarify the relation between the shape of optical edge spectra and the electron-electron or the electron-hole interactions. Response to circularly polarized incident light is also revealed. In the case of the momentum-dependent spin splitting, the power-law divergence of the edge spectra is possible even for repulsive electron-hole interactions. This is in striking contrast to the FES in spin-degenerate (no spin splitting) electron systems. The FES in a spin-split chiral Luttinger liquid is also discussed. \textcopyright{} 1996 The American Physical Society.
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.