Luttinger liquid physics from the infinite-system density matrix renormalization group

We study one-dimensional spinless fermions at zero and finite temperature $T$ using the density-matrix renormalization group. We consider nearest- as well as next-nearest-neighbor interactions; the latter render the system inaccessible by a Bethe ansatz treatment. Using an infinite-system algorithm we demonstrate the emergence of Luttinger liquid physics at low energies for a variety of static correlation functions as well as for thermodynamic properties. The characteristic power-law suppression of the momentum distribution $n\left(k\right)$ function at $T=0$ can be directly observed over several orders of magnitude. At finite temperature, we show that $n\left(k\right)$ obeys a scaling relation. The Luttinger liquid parameter and the renormalized Fermi velocity can be extracted from the density response function, the specific heat, or the susceptibility without the need to carry out any finite-size analysis. We illustrate that the energy scale below which Luttinger liquid power laws manifest vanishes as the half-filled system is driven into a gapped phase by large interactions.

Figure 1

Zero-temperature single-particle Green's function $G\left(x\right)$ and its Fourier transform, the fermionic momentum distribution function $n\left(k\right)$, for a half-filled [quarter-filled in (b, iii)] lattice of spinless fermions exhibiting a nearest-neighbor interaction $\Delta$. Solid lines show DMRG data; dashed lines show the asymptotic power laws $G\left(x\right)\sim 1/x^{1+\alpha \left(K\right)}$ and $|n\left(k\right)-0.5|\sim |k-k_F{|}^{\alpha \left(K\right)}$ expected for a Luttinger liquid. The LL parameter $K$ is taken from the Bethe ansatz.[16, 17] We use an infinite-system DMRG algorithm where the “block Hilbert space dimension”$\chi$ is the only numerical control parameter.[46] The LL power laws can be observed over several orders of magnitude. As the system is driven toward a gapped phase for $\Delta \to 1$, the momentum scale on which they manifest vanishes.

Figure 2

The same as in Fig. 1 but in the presence of next-nearest-neighbor interactions ${\Delta }_2$ which render the system nonintegrable. Dashed lines show the power law $\sim 1/x^{1+\alpha \left(K\right)}$ with the LL parameter $K$ determined from the density response function.

Figure 3

Single-particle Green's function and momentum distribution at finite temperature $T$. The LL power laws are cut off at long distances (small momenta) and replaced by an exponential (linear) decay. The infinite-system DMRG algorithm is essentially controlled by the discarded weight $\epsilon ={10}^{-11}$ during the imaginary time evolution of the density matrix from $T=\infty$ down to the physical temperature. In (b), we compare our data (thin solid line) with the exact result (thin dashed line) for $\Delta =0$ and $T=1/100$ to demonstrate its accuracy. Left Inset to (b): Derivative ${dn\left(k\right)/dk|}_{k_F}$ as a function of temperature. The solid line shows the expected TL power law $\sim T^{\alpha -1}$. Right Inset to (b): Rescaled variables $y=T^{1-\alpha }{dn\left(k\right)/d\left(\pi k\right)|}_{k_F}$, $x=v_F(k-k_F)/\left(\pi T\right)$. Curves at different $T$ collapse and are consistent with a scaling relation obtained for the TL model.

Figure 4

Zero-temperature spin-correlation function $S\left(j\right)$ defined in Eq. (15) of a spin-$1/2$ Heisenberg chain with $z$-anisotropy $\Delta$. Solid lines shows DMRG data; dashed lines show a field theory result of Ref. 50. The latter yields both the exponents and coefficients of an asymptotic expansion at large distances. Inset: $S_2\left(j\right)=[\delta S\left(j\right)-\delta S(J+1)]/2$ with $\delta S\left(j\right)=S\left(j\right)-S_{\mathrm{ft}}^1\left(j\right)$ and $S_{\mathrm{ft}}^1\left(j\right)=A/j^{\eta }$ being the leading term within field theory. The difference $S_2\left(j\right)$ is governed by the leading alternating term.

Figure 5

DMRG calculation of the zero-temperature instantaneous density response function $C_{NN}\left(x\right)$. Its Fourier transform $C_{NN}\left(k\right)$ exhibits a power-law singularity $C_{NN}(k\approx 2k_F)\sim {(2k_F-k)}^{2K-1}$ for momenta near $2k_F$. For small $k$, $C_{NN}\left(k\right)$ is linear with a slope determined by the Luttinger liquid parameter $K$. The latter can thus be determined accurately through linear fits [see the comparison with Bethe ansatz results in (b)].

Figure 6

Thermodynamic quantities from infinite-system DMRG. The main panel shows the susceptibility defined in Eq. (21); the inset displays: Specific heat $c_V\left(T\right)$. The low-temperature behavior is determined by the LL parameter $K$ and renormalized Fermi velocity $v_F$. At ${\Delta }_2=0$, the dashed lines show the asymptotic behavior predicted by the Bethe ansatz.[54] Our approach provides a way to extract $K$ and $v_F$ for nonintegrable models.

Figure 7

Density-density correlation function $C_{NN,L}\left(k\right)=L^{-1}{\sum }_{x=-L}^L{e}^{-ikx}{C}_{NN}\left(x\right)$. $C_{NN,L}(k_F=\pi /2)$ vanishes for $Ł\to \infty$ in the bond-ordered phase but remains finite in the $k_F$ charge-density wave phase. Moderate values of $\chi =100$ predict that the transition occurs at ${\Delta }_2=1.50\pm 0.01$ in agreement with ${\Delta }_2=1.5$ obtained in Ref. 23 (the transition from the LL to the bond-ordered phase takes place at ${\Delta }_2=1.25$).

Figure 8

Single-particle Green's function and momentum distribution for large interactions that drive the half-filled system into gapped phases (a $2k_F$ charge-density wave phase for $\Delta >1$, ${\Delta }_2=0$ and a bond-ordered phase at $\Delta =0.8$, ${\Delta }_2=1.5$; see Ref. 23 as well as Sec. 5 for details). $G\left(x\right)$ decays exponentially, and $n\left(k\right)$ is linear for close to $k_F$. $\chi$ is chosen such that the results are converged on the scale of this plot.