Spectral function of the Tomonaga-Luttinger model revisited: Power laws and universality

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 $V\left(q\right)$. Usually, $V\left(q\right)$ 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 $k_{\mathrm{F}}$, with $|k-k_{\mathrm{F}}|$ setting a nonvanishing energy scale, the details of $V\left(q\right)$ 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 $|k-k_{\mathrm{F}}|$ much smaller than the momentum-space range of the interaction the spectral line shape depends on the details of $V\left(q\right)$. 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.

Figure 1

Spectral function of the spinful $g_4$ model as a function of energy for (a) $(k-k_{\mathrm{F}})/q_{\mathrm{c}}=0.01$, (b) $(k-k_{\mathrm{F}})/q_{\mathrm{c}}=0.1$, and (c) $(k-k_{\mathrm{F}})/q_{\mathrm{c}}=0.3$. Spectra for the different potentials are shown. In addition to the FFT data with broadening $\chi /\left(v_{\mathrm{F}}{q}_{\mathrm{c}}\right)=5\times {10}^{-5}$ we show unbroadened results for the box potential [“box (ana)”] and the broadened results of the ad hoc procedure (iii) [“broadened (iii)”]. The other parameters are $g_{4,c}/\left(\pi v_{\mathrm{F}}\right)=\frac{1}{2}=-g_{4,s}/\left(\pi v_{\mathrm{F}}\right)$ and $n_{\mathrm{c}}=5\times {10}^4$. The curves are partly hidden by others (see the text). The inset of (c) shows the position of the spin peak extracted from the data (stars) in comparison to the collective spin mode dispersion ${\omega }_s(k-k_{\mathrm{F}})$ (lines). The dashed line displays the unrenormalized dispersion $v_{\mathrm{F}}(k-k_{\mathrm{F}})$.

Figure 2

Logarithmic derivative (56) of the spectral function of the spinful $g_4$ model close to the spin peak for (a) $(k-k_{\mathrm{F}})/q_{\mathrm{c}}=0.01$ and (b) $(k-k_{\mathrm{F}})/q_{\mathrm{c}}=0.1$. In addition to the broadened FFT data for the different potentials and the unbroadened results for the box potential [“box (ana)”], we show broadened [“broadened pl”] as well as unbroadened [“pl”] data for the product of single-sided square-root singularities [Eq. (57)]. The parameters are as in Fig. 1.

Figure 3

Total spectral function of the $g_2-g_4$ model as a function of energy for (a) $(k-k_{\mathrm{F}})/q_{\mathrm{c}}=0$, (b) $(k-k_{\mathrm{F}})/q_{\mathrm{c}}=0.01$, and (c) $(k-k_{\mathrm{F}})/q_{\mathrm{c}}=0.1$. Spectra for the different potentials are shown. In addition to the FFT data with broadening $\chi /\left(v_{\mathrm{F}}{q}_{\mathrm{c}}\right)={10}^{-3}$ we show unbroadened results for the box potential [“box (ana)”]. The interaction at $q=0$ is chosen such that $v_c=2v_{\mathrm{F}}$ and ${\gamma }_c\left(0\right)=\frac{1}{8}$. The system size is set by $n_{\mathrm{c}}=2\times {10}^4$.

Figure 4

Logarithmic derivative (56) of the spectral function of the spinless $g_2-g_4$ model close to the inverse photoemission peak for (a) $(k-k_{\mathrm{F}})/q_{\mathrm{c}}=0$, (b) $(k-k_{\mathrm{F}})/q_{\mathrm{c}}=0.01$, and (c) $(k-k_{\mathrm{F}})/q_{\mathrm{c}}=0.1$. The full lines labeled by “box (ana)” show the unbroadened results of the box potential. The dotted lines in (a) result from broadened spectra. The other full lines are obtained from the numerical spectra after a deconvolution (see the text). The dashed lines indicate the expected exponent. The parameters are as in Fig. 3.

Figure 5

Photoemission part of the spectral function of the spinful TLM at $(k-k_{\mathrm{F}})/q_{\mathrm{c}}=-0.1$. A spin-independent interaction with equal interbranch and intrabranch amplitude is assumed. The interaction strength is chosen such that ${\gamma }_c(q=0)$ assumes the values given in the legend. The other parameters are $n_{\mathrm{c}}={10}^3$ (system size) and $\chi /\left(v_{\mathrm{F}}{q}_{\mathrm{c}}\right)=3\times {10}^{-3}$ (broadening). In (a) the spectra for the Gaussian potential are shown, in (b) the ones for the exponential potential. The small “high-energy” oscillations for larger interactions in (a) are a finite-size effect.