Accurate localization of Kosterlitz-Thouless-type quantum phase transitions for one-dimensional spinless fermions

We investigate the charge-density wave (CDW) transition for one-dimensional spinless fermions at half band filling with nearest-neighbor electron transfer amplitude $t$ and interaction $V$. The model is equivalent to the anisotropic XXZ Heisenberg model for which the Bethe Ansatz provides an exact solution. For $V>V_{\mathrm{c}}=2t$, the CDW order parameter and the single-particle gap are finite but exponentially small, as is characteristic for a Kosterlitz-Thouless transition. It is notoriously difficult to locate such infinite-order phase transitions in the phase diagram using approximate analytical and numerical approaches. Second-order Hartree-Fock theory is qualitatively applicable for all interaction strengths, and predicts the CDW transition to occur at $V_{\mathrm{c},2}^{\left(2\right)}\approx 1.5t$. Second-order Hartree Fock theory is almost variational because the density of quasiparticle excitations is small. We apply the density-matrix renormalization group (DMRG) for periodic boundary conditions for system sizes up to 514 sites, which permits a reliable extrapolation of all physical quantities to the thermodynamic limit, apart from the critical region. We investigate the ground-state energy, the gap, the order parameter, the momentum distribution, the quasiparticle density, and the density-density correlation function to locate $V_{\mathrm{c}}$ from the DMRG data. Tracing the breakdown of the Luttinger liquid and the peak in the quasiparticle density at the band edge permits us to reproduce $V_{\mathrm{c}}$ with an accuracy of one percent.

Figure 1

Ground-state energy density for spinless fermions at half band filling as a function of the nearest-neighbor interaction $V$ from Bethe Ansatz (BA), DMRG for $L=512$ sites, and (second-order) Hartree-Fock (HF, HF 2nd). Dashed and dotted lines correspond to the small-$V$ and large-$V$ expansions in Eq. (19).

Figure 2

Single-particle density matrix $B_0\left(V\right)$ between nearest neighbors as a function of the nearest-neighbor interaction $V$ from Bethe Ansatz (BA), including the small and large coupling asymptotes, from DMRG for $L=512$ sites, and from second-order Hartree-Fock (HF 2nd). Dashed and dotted lines correspond to the small-$V$ and large-$V$ expansions in Eq. (22).

Figure 3

Second-order coefficient $c\left(V\right)$ in Eq. (144) from the extrapolation of DMRG data ($L/2$ odd, closed shell) in comparison with the Bethe-Ansatz result (146). The horizontal line indicates the critical value $c\left(V_{\mathrm{c}}\right)=-{\pi }^2/6$.

Figure 4

Single-particle gap ${\mathrm{\Delta }}_1(L,V)$ as a function of $1/L$ for system sizes $L=32,64,128,256,512$ from DMRG for $V=1.4,1.8,2.0,2.3,3$. Lines are second-order polynomial fits, see Eq. (148).

Figure 5

Single-particle gap ${\mathrm{\Delta }}_1\left(V\right)$ in the thermodynamic limit from the polynomial extrapolation of the gap data from DMRG (crosses), in comparison with the exact Bethe Ansatz result (line).

Figure 6

Slope $s_1\left(V\right)/\left(2\pi \right)$ in $1/L$ of half the two-particle gap ${\mathrm{\Delta }}_2\left(V\right)/2$, see Eqs. (151) and (152) from the extrapolation of the DMRG data (dots), in comparison with the Bethe Ansatz result (line) from Eq. (153). The horizontal line indicates the critical value $s_1\left(V_{\mathrm{c}}\right)={\left(2\pi \right)}^2$.

Figure 7

Exact single-particle gap ${\mathrm{\Delta }}_1\left(V\right)$ in the thermodynamic limit in the range $1.8\le V\le 2.2$, in comparison with the result of three extrapolations of the DMRG data for open boundary conditions for $L=64,128,256,512$: polynomial (blue), Eq. (148), logarithmic (red), Eq. (155), and Mishra et al. (green), Eq. (156).

Figure 8

Charge-density wave order parameter $n_a(L,V)$ as a function of the nearest-neighbor interaction $V$ from Bethe Ansatz (BA) and DMRG for $L=128,256,512$ sites. Inset: Comparison of (second-order) Hartree-Fock and DMRG data for $L=512$ sites.

Figure 9

Correlation energy at half-band filling as a function of the nearest-neighbor interaction $V$ from Bethe Ansatz (full line) and from DMRG for $L=512$ sites (crosses).

Figure 10

Momentum distribution $n_k$ for $V=1.4,1.8,2.3,4$ (a) from DMRG for $L=512$ lattice sites and (b) from second-order Hartree-Fock approximation.

Figure 11

Generalized Luttinger liquid exponent $\alpha \left(V\right)$ for spinless fermions at half-band filling as a function of the nearest-neighbor interaction $V$ extrapolated from DMRG and Bethe Ansatz (full line for $0\le V\le V_{\mathrm{c}}=2$). Inset: parameter $b\left(V\right)$.

Figure 12

Quasiparticle distribution function $n_{k,\beta }$ for $V=1.4,1.8,2.3,4$ (a) from DMRG for $L=512$ lattice sites and (b) from second-order Hartree-Fock approximation. Note the factor ten difference in the values on the ordinate. Inset: DMRG and second-order Hartree-Fock for $V=0.8$.

Figure 13

Quasiparticle density $n_{\beta }\left(V\right)$ as a function of $V$ from DMRG for $L=128,256,512$, and from second-order Hartree-Fock theory.

Figure 14

Values of the maxima in the quasiparticle density $n_{\beta }^{\max ,\mathrm{k}}(L,V)$ as a function of $V$ from DMRG for $L=32,64,128,256,512$. The continuous lines are a sixth-order polynomial fit for the region around the maximum.

Figure 15

Extrapolation of the maxima position in the quasiparticle density as a function of $1/L$ from DMRG for $L=64,128,256,512$. The continuous line is a fit to a second-order polynomial in $\sqrt{1/L}$.

Figure 16

Structure factor ${\tilde{C}}^{\mathrm{NN}}(q,V)$ from DMRG for $L=512$ sites for $V=1.4$ (black), $V=2.3$ (green), and $V=4$ (blue). Left inset: Structure factor ${\tilde{C}}_0^{\mathrm{NN}}\left(q\right)$ from DMRG for $L=32$ sites and the analytic result (92) for $V=0$ (red line). Right inset: Structure factor ${\tilde{C}}^{\mathrm{NN}}(q,V)$ from DMRG for $L=512$ sites for $V=6$ and the analytic result (96) for strong coupling (red line).

Figure 17

Luttinger parameter $K\left(V\right)$ as a function of the interaction from the Fourier-transformed density-density correlation function at small $q$ from DMRG, Eq. (54), for $L=256$ sites (blue crosses) and $L=512$ sites (red crosses), in comparison with the exact result (47) from Bethe Ansatz for $0\le V\le V_{\mathrm{c}}=2$ (black line).