Fermionic phases and their transitions induced by competing finite-range interactions

We identify ground states of one-dimensional fermionic systems subject to competing repulsive interactions of finite range, and provide phenomenological and fundamental signatures of these phases and their transitions. Commensurable particle densities admit multiple competing charge-ordered insulating states with various periodicities and internal structure. Our reference point are systems with interaction range $p=2$, where phase transitions between these charge-ordered configurations are known to be mediated by liquid and bond-ordered phases. For increased interaction range $p=4$, we find that the phase transitions can also appear to be abrupt, as well as being mediated by re-emergent ordered phases that cross over into liquid behavior. These considerations are underpinned by a classification of the competing charge-ordered states in the atomic limit for varying interaction range at the principal commensurable particle densities. We also consider the effects of disorder, leading to fragmentization of the ordered phases and localization of the liquid phases.

Figure 1

Phase diagrams of charge-ordered ground states in the atomic limit of half-filled systems ($Q=1/2$) with interaction range $p=2$ (a) and $p=4$ (b). In both cases we exploit the freedom to choose $U_p$ as the energy scale. The colors indicate the phases listed in Table 2.

Figure 2

Phenomenological signatures of phases and transitions in the kinetic energy density $T$ (a) and bond-order parameter $O_{\mathrm{BO}}$ (b) at finite kinetic energy parameter $t=1$, for interaction range $p=2$ and half-filling ($Q=1/2$). In both panels the next-nearest-neighbor interaction $U_2$ is varied while the nearest-neighbor interaction is set to $U_1=10$. The solid lines indicate the phase transitions in this setting, while the dotted line in (a) indicates the transition between the two charge-ordered phases in the atomic limit. The kinetic energy density only captures a single phase transition at $U_2\approx 7.15$, which coincides with the transition from the liquid phase into the charge-ordered phase $(••\circ \circ )$. The bond-order parameter vanishes in the thermodynamic limit of the phase $(•\circ )$ and the liquid phase, but scales differently with increasing bond dimension $\chi$, thereby providing signatures of all three phase transitions.

Figure 3

kinetic energy density $T$ (a) and bond-order parameter $O_{\mathrm{BO}}$ (b) in analogy to Fig. 2, but for increased interaction range $p=4$ and interaction parameters fixed to $U_1=U_3=4$ and $U_4=1$. In the atomic limit, the transitions between the charge-ordered phases occur at $U_2=3,6$ (dotted lines). Discontinuities in the derivative of the kinetic energy density indicate three clear phase transitions at $U_2\approx \{4.01,5.00,6.57\}$ (solid lines). The bond-order parameter only captures a single transition into the phase $(••\circ \circ )$, which is the only encountered phase where $O_{\mathrm{BO}}$ is finite. Note that the transition between the phases $(•\circ )$ and $(••\circ \circ )$ appears to be abrupt. This is verified in the subsequent figures, which also determine the indicated nature of the mediating region between the phases $(•••\circ \circ \circ )$ and $(••\circ \circ )$. There, we observe a crossover from a re-emergent charge-ordered state $(•\circ )$ to liquid behavior, as indicated by the dashed line.

Figure 4

(a) Range dependence of the correlation function $N_m$ for $p=2$, $Q=1/2$, $t=1$, with the interaction potentials $U_1=10$ and $U_2=8$ set to values where the system is in the charge-ordered state $(••\circ \circ )$. The function displays a clear oscillatory behavior with period 4, reflecting that the charge-ordered character of the phase persists at finite kinetic energy. (b) Extrapolation of the correlation function $N_{4k}$ with increasing $k$, with $U_1$ and $t$ as above but for various values of $U_2$. With increasing range $4k$ this correlation function converges to a well-defined value $N_4^{\infty }$, in accordance to the extrapolated correlators stipulated in Eq. (7).

Figure 5

Extrapolated correlators $N_m^{\infty }$ for (a) $p=2$ and (b) $p=4$, with system parameters specified as in Figs. 2 and 3. The phase transitions indicated by the solid lines coincide with points where the derivative of $N_m^{\infty }$ is discontinuous. The oscillatory behavior of the correlators with the range $m$ agrees with the stipulated charge orders. For $p=4$, the transition between the phases $(•\circ )$ and $(•••\circ \circ \circ )$ remains abrupt. The behavior in the mediating transition region between the phases $(•••\circ \circ \circ )$ and $(••\circ \circ )$ is examined more closely in Fig. 6.

Figure 6

Closeup of the extrapolated correlators $N_m^{\infty }$ from Fig. 5 for the mediating transition region in the system with $p=4$. The correlations gradually approach the value $1/4$, compatible with a gradual crossover from charge order $(•\circ )$ to liquid behavior, where furthermore the results converge only slowly.

Figure 7

Bipartite entanglement entropy for the systems with $p=2$ (a) and $p=4$ (b), with parameters as specified in Figs. 2 and 3. For $p=4$, no critical behavior is detected around the transition between the phases $(•\circ )$ and $(•••\circ \circ \circ )$. However, in the transition region between the phases $(•••\circ \circ \circ )$ and $(••\circ \circ )$ the gradual suppression of charge correlations (Fig. 6) coincides with an onset of finite-size scaling with increasing bond dimension, as expected from a crossover into a liquid phase.

Figure 8

Proposed phase diagram for the model from Eq. (1) with interaction range $p=4$, with interaction parameters $U_1=U_3=4,U_4=1$ while $U_2$ and the kinetic energy parameter $t$ are varied. The phase transitions are determined from the extrapolated correlators and the entanglement entropy, in analogy to Figs. 5 and 7, which correspond to the case $t=1$. The dashed line indicates the detected onset of finite-size scaling with the bond dimension in the crossover from the phase $(•\circ )$ to the liquid phase.

Figure 9

Effect of a small kinetic energy parameter $t=0.1$ on the phase $(•\circ \circ •\circ ••\circ )$ in the half-filled system with $p=4$, as captured by the kinetic energy density $T$ (a), the bond-order parameter $O_{\mathrm{BO}}$ (b), the bipartite entanglement entropy $S$ (c), and the extrapolated correlators $N_m^{\infty }$ (d). In contrast to the case described in Fig. 3, we now set $U_1=4$ and $U_3=U_4=1$, while varying $U_2$ as before. Already at the chosen small kinetic energy parameter, the phase $(•\circ \circ •\circ ••\circ )$ is driven out by a bond-ordered phase, which intervenes in the transition to the other charge-ordered states covered in this parameter range.

Figure 10

Effect of disorder on the kinetic energy density for $p=2$ (a) and $p=4$ (b). The disorder-free case corresponds to Figs. 2 and 3, and is included for guidance. For each finite value of the disorder, the data represent a density plot accumulated over 100 disorder realizations. The size of the disordered unit cell is $L=20$ ($p=2$) and $L=24$ ($p=4$).

Figure 11

Disorder-averaged density plots of the extrapolated correlators $N_m^{\infty }$ for the disordered systems specified in Fig. 10, evaluated at disorder strength $W=1$. These correlators remain nontrivial because of the finite size of the disordered unit cell.

Figure 12

Disorder-averaged density plots of the extrapolated correlators $N_m^{\infty }$ as in Fig. 11, but for disorder strength $W=3$.

Figure 13

Disorder-averaged density plots of the extrapolated correlators $N_m^{\infty }$ as in Figs. 11 and 12, but for disorder strength $W=8$.

Figure 14

Disorder-averaged density plots of the entanglement entropy $S$ for the disordered systems specified in Fig. 10.