Pairing in spinless fermions and spin chains with next-nearest neighbor interactions

We investigate the phase diagrams of a one-dimensional lattice model of fermions and of a spin chain with interactions extending up to next-nearest-neighbor range. In particular, we investigate the appearance of regions with dominant pairing physics in the presence of nearest-neighbor and next-nearest-neighbor interactions. Our analysis is based on analytical calculations in the classical limit, bosonization techniques and large-scale density-matrix renormalization group numerical simulations. The phase diagram, which is investigated in all relevant filling regimes, displays a remarkably rich collection of phases, including Luttinger liquids, phase separation, charge-density waves, bond-order phases, and exotic cluster Luttinger liquids with paired particles. In relation with recent studies, we show several emergent transition lines with a central charge $c=3/2$ between the Luttinger liquid and the cluster Luttinger liquid phases. These results could be experimentally investigated using highly tunable quantum simulators.

Figure 1

Phase diagram for $n=\frac{1}{5}$. Color map for the background displays the entanglement entropy on $L=70$ chain with PBC. Black lines are classical transition lines obtained neglecting quantum fluctuation. Additional numerical simulations are presented for the points lying on the red lines. Blue lines are a guide to the eye for the main phase boundaries.

Figure 2

DMRG results for $n=\frac{1}{5}$ along the $U=U_2=-U_1$ line of Fig. 1. (a) Ground-state energy density curvature Eq. (6) for various sizes. The apparent nonanalytic behavior suggests a critical point located in the range $3.1\le U\le 3.2$. (b) Extrapolated central charge from Eq. (8). The numerics are compatible with a critical point with $c=3/2$ surrounded by the two $c=1$ LL and CLL phases.

Figure 3

Single-particle gap ${\mathrm{\Delta }}_1\left(\infty \right)$ obtained by extrapolating finite-size gaps Eq. (5) with $L=20,40,60,80$ on the same line as in Fig. 2.

Figure 4

(a) Density fluctuations Fourier transform Eq. (9) and density structure factor Eq. (10). (b) on the same line as in Fig. 2 for $L=40$. $U=1$ for the LL phase and $U=5$ for the CLL phase.

Figure 5

DMRG results for $n=\frac{1}{5}$ along the $U=U_2=U_1$ line. (a) Finite-size scaling of the single-particle gaps Eq. (5). (b) Finite-size scaling of the BO parameter Eq. (13) using a power law $a_0+a_1{L}^{-a_2}$. Inset: extrapolated central charge from systems with $L=60,80,100,120,140$.

Figure 6

Phase diagram for $n=\frac{1}{3}$. Color map for the background displays the entanglement entropy on a $L=42$ chain with PBC. Black lines are classical transition lines obtained neglecting quantum fluctuation. Additional numerical simulations are presented for the points lying on the red lines. Blue lines are a guide to the eye for the main phase boundaries.

Figure 7

DMRG results for $n=\frac{1}{3}$ along the $U=U_2=U_1$ line. (a) Extrapolated central charge from $L=61,82,100,121,142$. (b) Extrapolated single-particle gap from sizes $L=73,97,121$ showing its opening around the critical point.

Figure 8

DMRG results for $\frac{1}{3}$ along the $U=U_2=U_1$ line. (a) Decay of the single-particle and pair correlators $G\left(r\right)$ and $P\left(r\right)$ as a function of separation for $L=121$ and $U=5$, deep in the CDW phase. (b) Density profile Fourier spectrum $n\left(k\right)=\delta n\left(k\right)+n{\delta }_{k,0}$ for $U=3$ (LL phase) and $U=5$ (${\mathrm{CDW}}_3$ phase).

Figure 9

Phase diagram for $n=\frac{2}{5}$. Colormap for the background display the entanglement entropy on a system with $L=30$ and PBC. Black lines are classical transition lines obtained neglecting quantum fluctuation. Additional numerical simulations are presented for the points lying on the red lines. Blue lines are a guide to the eye for the main phase boundaries.

Figure 10

DMRG results for $n=\frac{2}{5}$ along the $U=U_2=-U_1$ line for $L=40$ and PBC. (a) Ground-state energy density curvature Eq. (6). The shape of the profile is likely to be a nonanalytic function of the control parameter $U$, locating the critical point within $-2.5\le U_1\le -2.4$. (b) Extrapolated central charge from Eq. (8). Data once again support a $c=\frac{3}{2}$ critical point, where an additional Ising degree of freedom emerges on top of the bosonic $c=1$ contribution, suggesting the universality class of the 2D Ising model.

Figure 11

DMRG results for $n=\frac{2}{5}$ along the $U=U_2=-U_1$. Extrapolations of the single-particle gap ${\mathrm{\Delta }}_1(N,L)$ across the transition between the two liquid phases. Finite-size results are computed according from Eq. (5) for $L=20,40,60$ and fitted to a linear model in $L^{-1}$. The orange line is a linear fit of the opening of the gap, consistent with the 2D Ising universality class. Inset: Finite-size scaling of the pair gap ${\mathrm{\Delta }}_2(N,L)$ from Eq. (5), both in the LL phase ($U=1.5$) and in the CLL phase ($U=3.5$). They both scale to zero.

Figure 12

DMRG results for $n=\frac{2}{5}$ along the $U=U_2=-U_1$ line for $L=40$ and PBC. (a) Density fluctuations Fourier spectrum Eq. (9) and (b) density structure factor Eq. (10) both in the LL ($U=1.5$) and CLL phase ($U=3.5$). The momentum peak shift at value $k=2\pi \frac{1}{5}$ is incompatible with a standard LL theory and supports that the physics of the CLL phase in the attractive regime is ruled by pair degrees of freedom.

Figure 13

DMRG results for $n=\frac{2}{5}$ along the $U=U_2=U_1$ line for $L=40$ and PBC. (a) Density fluctuations Fourier spectrum Eq. (9) and (b) density structure factor Eq. (10) both in the LL ($U=1$) and the CLL ($U=7$) phases. The momentum peak shift to the value $k=2\pi \frac{3}{10}$ is incompatible with a standard LL and supports that the physics of the CLL phase in the repulsive regime is ruled by the composite cluster degrees of freedom stemming from the classical limit analysis.

Figure 14

DMRG results for $n=\frac{2}{5}$ along the $U=U_2=U_1/4$ line. (a) Finite-size scaling of the single-particle gap ${\mathrm{\Delta }}_1(N,L)$ computed according to Eq. (5). A linear fit points toward zero extrapolated values (three orders of magnitude smaller than the corresponding finite-size values). Such evidence of the gapless nature of single-particle excitations suggests the survival of the LL phase for a wide range of interactions. (b) Finite-size scaling of the BO parameter Eq. (13). Power law fits of the form $a_0+a_1{L}^{-a_2}$ are used and results in extrapolated values at least one order of magnitude smaller than the finite-size values. Such behavior further supports the LL picture. Inset: Extrapolated central charge. The values deviate roughly at most $1.0\%$ from the $c=1$ value predicted for the LL phase.

Figure 15

DMRG results for $n=\frac{2}{5}$ along the $U_1=0$ line for $U_2=20, L=60$ and PBC. (a) Density profile. (b) Entanglement entropy profile $S_L\left(j\right)$.

Figure 16

Phase diagram for $n=\frac{1}{2}$. Color map for the background display the entanglement entropy on a $L=28$ chain with PBC. Black lines are classical transition lines obtained neglecting quantum fluctuation. Additional numerical simulations are presented for the points lying on the red lines. Blue lines are a guide to the eye for the main phase boundaries.

Figure 17

DMRG results for $n=\frac{1}{2}$ along the $U_1=0$ line. (a) Finite-size scaling of the BO parameter (13). A power law fit of the form $a_0+a_1{L}^{-a_2}$ is used. The results show the onset of the BO regime, given that for $U_2=2.2$ the BO parameter extrapolates to a finite nonzero value of an order of magnitude larger than the extrapolated values for smaller $U_2$. (b) Finite-size scaling of the CDW order parameter Eq. (14). The result demonstrates that the CDW order parameter scales to zero for $U_2<2.6$ but acquires a finite value for $U_2>2.6$. Thus, there exists an intervening BO phase.

Figure 18

DMRG results for $n=\frac{1}{2}$ for attractive $U_1$. (a) Luttinger parameter $K$ obtained from fitting the pair correlation functions to a power law. It displays large values close to the PS phase, as it is known on the XXZ line showing that pair correlations are favored. [values inside the phase-separated region are meaningless and shown to visually identify the transition line]. (b) Entanglement entropy (7) of half the system displaying the emergence of a characteristic peak separating the LL phase from phase separation and the appearance of the classical line $U_2=-U_1/2$ as the threshold above which phase separation turns into the ($••\circ \circ$) ${\mathrm{CDW}}_2$ configuration in the infinite coupling limit. (c) Pair kinetic energy providing evidence for the enhancement of pairing fluctuations close to the transition line between the LL phase and phase separation, as expected from the divergent behavior of the Luttinger parameter in the corresponding phase diagram region.