Density-dependent tunneling in the extended Bose–Hubbard model

For ease of comparison and for later reference, figure 1(a) reproduces the phase diagram of U = 20 that has been thoroughly investigated in [8]. It is well known that for $\rho < \frac {1}{2}$, there exist only two distinct regions, the SF phase and a PS region. For sufficiently low values of V , the system stays SF across the entire density range until unit filling, where it becomes an MI state. At half filling, a CDW I phase appears at a critical value of V , which in the present case lies around V =2.5. A system in a checkerboard phase (CDW I) can be doped by holes or particles. When it is doped with holes, these create domain walls and cause the system to phase separate, preventing the appearance of an SS phase. In the case of hardcore bosons, this behavior would be mirrored for $\rho > \frac {1}{2}$, due to particle–hole symmetry. In the case of softcore bosons, such particle–hole symmetry can break down. At sufficiently low V , a region of PS appears and the system does present a hardcore-like behavior, but as the NN repulsion is increased, this PS region disappears. Since now the particles can occupy either an empty or occupied site, it is no longer necessary for the domain walls to form and the system can move into an SS state. Moreover, at a certain value of V , upon increasing ρ the SS phase is followed by a region of PS, instead of going into an SF phase and then becoming an MI. At unit filling, this PS region then changes to the CDW II phase, which is characterized by a checkerboard pattern consisting of an alternation of doubly filled sites and empty ones.

Figure 2 shows, for a fixed V =3, the observables described in section 2.1 that we used to determine the various phases. The boson density as a function of the chemical potential displays clear plateaus, corresponding to gapped insulating phases (figure 2(a)). As mentioned above, jumps in figure 2(a) correspond to PS regions in figure 1. The structure factor and the SF stiffness are shown in figure 2(b). For low chemical potential (density), the system is in an SF state with non-zero SF stiffness and vanishing structure factor. At ρ ≈ 0.43, the system phase separates and there are no values for these observables. At half filling, when the system moves to CDW I phase, the structure factor becomes finite. There is a small region, roughly around 0.5 < ρ < 0.51, where the system is in an SS phase—here both the SF stiffness and the structure factor are non-zero. This phase is followed by a second region of PS that extends up to ρ ≈ 0.61. At higher densities, an SF phase is observed up to unit filling, where an MI state follows, as revealed by vanishing SF density and structure factor.

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We now consider U = 5, a case of weaker repulsion that has not been studied earlier. For the moment, we still retain T = 0. The phase diagram figure 1(b) seems a bit simpler than for U = 20. Importantly, the particle-doped side now has to deal with much 'softer' bosons allowing for multiple occupancy on any given site (in the numerical calculations we allow for up to four bosons per site, which is sufficient for lattice fillings below unity). The hole-doped side is much less affected since the on-site repulsion has a lesser influence on lower densities. For weak NN repulsion V , the system stays SF across the entire range of densities from empty to unit filling and then goes into the MI state. At V ≈2.3 up to 3.1, the system goes directly from an SF phase into an SS phase, which ends at a CDW II phase at unit filling. At larger V , a PS region appears. The biggest difference between U = 20 and 5 cases appears for higher values of V , where the PS region at the particle-doped side disappears and the SS phase occupies the entire region between the CDW I at half filling and the CDW II at unit filling.

The different transitions are revealed by slices through the phase diagram at fixed V (exemplified for a few values in figure 3). At V =3.0, the density is strictly increasing across the entire range of μ (figure 3(a)). Notice, however, a change of the slope around μ/U = 4, corresponding to ρ = 0.6. As seen in figure 3(d), the structure factor starts to rise in a similar parameter range, namely around ρ = 0.65. At the same time, the SF stiffness only has a peak at ρ = 0.65, but remains finite for all values of μ considered. Therefore, the increase of the structure factor is a clear sign of a second-order transition from an SF to an SS. Also, since the structure factor does not drop back to zero at ρ = 1, the unit-filling phase will be a CDW II and not an MI.

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The next slice is taken at V =4.5, where the state changes from SF to PS to SS without ever settling into the CDW I phase at half filling. In the density graph, figure 3(b), we can see a small jump that bypasses $\rho =\frac {1}{2}$. This explains why the CDW I phase does not appear at this value of V . The SF at small ρ is identified by a non-zero SF fraction and vanishing structure factor (figure 3(e)). This phase is followed by the PS region from ρ ≈ 0.435 to 0.51. At higher densities, an SS state appears as characterized by non-zero structure factor and SF stiffness. Finally, the system settles into the CDW II state at unit filling.

The last slice at V =6.0 is similar to the previous one at V =4.5 with one major difference, the appearance of the CDW I phase at half filling. As before, we can see a jump (this time slightly larger) in the density, (figure 3(c)), but now it is followed by a plateau that signifies the CDW I phase. In figure 3(f), we see again the three distinct phases, SF up to ρ ≈ 0.35, then a region of PS up to ρ = 0.5 and from half filling to unit filling there is the SS phase, once again ending in the CDW II state.

The first case we study is U = 20 when the two tunneling amplitudes t and T have the same sign. Comparison of figure 4 with figure 2 shows that in the presence of density-dependent tunneling, the PS region at low V values has disappeared and there is no PS region between the SS and CDW II phases. One can explain this behavior by the increase in the total hopping due to the additional tunneling term T. Thus, the on-site repulsion U behaves as if it were effectively rescaled to a smaller value. Similar arguments explain the shift of the point where the $\rho = \frac {1}{2}$ plateau first appears and, therefore, the CDW I phase moves from V ≈2.5 (with T = 0, figure 2) to V ≈3.5 (figure 4). As a consequence, the phase diagram at U = 20, with t and T of the same sign, looks very similar to the one at U = 5 with vanishing T.

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The behavior in the U = 20 phase diagram becomes more interesting when the two tunneling terms compete due to opposite signs, T < 0. The phase diagram is presented in figure 4(b). The CDW I phase now starts at a lower value of |T| than in the previously discussed case. Similarly, the region of PS at the lower values of |T| (and thus V ) now becomes much larger. This shows that the system has a hardcore behavior for a larger range of parameters. Additionally, the SS region diminishes and finally disappears as V gets larger. These findings can be explained through the competition between t and T, which decreases the effective, overall tunneling strength. This decrease can alternatively be seen as an effective relative increase of the interaction parameters U and V . As a result, the hardcore behavior of the system becomes more pronounced and the PS regions become more important.

The observed phases may again be analyzed in detail via the cuts at fixed T (and therefore V ) presented in figure 5. The first slice we present is for T = −0.3 (V =3.0). As seen in figure 5(a), the plateau at half filling—a CDW I, as indicated by the finite structure factor, figure 5(d)—is surrounded by discontinuities in the density, thus implying regions of PS. These are surrounded by SF phases, with an MI appearing at unit filling.

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The next slice cuts through the phase diagram at T = −0.52 (V =5.2) and this time shows also a region of the SS phase for densities just above half filling (figure 5(e)). This SS may also be observed in the density plot, (figure 5(b)); above half filling, there is a small interval of steady increase before a discontinuity occurs around ρ = 0.65. After this PS region, there is a small region where the system becomes SF before once again phase separating. At unit filling, the system finally transitions into a CDW II phase. Below half filling, another jump in the density indicates yet another PS.

The final cut is taken at T = −0.6 (V =6.0). Again, at low densities the system starts in an SF phase and then jumps through a region of PS to reach the CDW I phase at half filling. For higher densities, the system first enters an SS phase and around ρ = 0.72 a transition to PS occurs. This time, the system ends in the CDW II phase when unit filling is reached.

In the previous section, we saw that the additional density-dependent tunneling term T can increase or decrease the effective importance of the interactions U and V , depending on whether it competes with or supports the single-particle tunneling t. In this section, we study this effect for weaker on-site interaction U = 5. The corresponding phase diagrams are presented in figure 6.

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The positive T diagram reveals that the CDW I phase, present for T = 0, disappears completely (figure 6(a)). This means that at no point does there exist a plateau in the density graphs at $\rho =\frac {1}{2}$. Instead, a discontinuity bypasses half filling altogether. The rest of the behavior is rather similar to the system without the density-dependent term. There are still only three phases below unit filling, i.e. the SF phase at low densities and low T (and therefore at low V ), the PS region near half filling for larger T and finally the SS phase for still higher T and larger densities. As can be expected, when the SF phase persists through the entire range of densities, the system ends in an MI state at unit filling. Instead, when the system at fixed T passes through the SS state, the final phase at unit filling is, as before, the CDW II phase.

Consider now the phase diagram of a system with U = 5 when the tunneling terms have opposite signs (figure 6(b)). Here, contrary to the case of positive T, the CDW I exists at half filling. This indicates that the relative importance of the effective total tunneling is suppressed for T < 0. Moreover, now a second region of PS appears above half filling. As a result, for T ≲ − 0.8 there is no stable phase with a density between the CDW I and the CDW II.

These observations about the phase diagram are supported by an analysis of cuts at a few chosen values of T (and thus V ); see figure 7. At T = −0.3 (V =3.0), one observes a smooth density increase all the way until unit filling, where a plateau appears (figure 7(a)). The structure factor starts increasing near half filling, indicating the transition from the SF to the SS phase (figure 7(d)); at unit filling, the system lands in the CDW II phase.

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A cut at the slightly higher absolute value T = −0.4 (V =4.0) reveals a plateau at half filling (CDW I) and a second one at unit filling (CDW II). Comparing the density plot (figure 7(b)) with those of SF stiffness and structure factor (figure 7(e)), we see that upon increasing the chemical potential, the SF phase appears at low densities, followed by the PS which transitions into the CDW I at half filling. For higher densities, there is a region of SS, where both the structure factor and the SF stiffness are non-zero. Finally, there is the jump caused by the PS region directly to the CDW II phase at unit filling.

Let us finally consider stronger density-dependent tunneling T and inter-site repulsion V , namely T = −0.8 (V =8.0). Below half filling, the density gradually increases up to the value of ρ ≈ 0.27 and then jumps to the CDW I phase (figure 7(c)). After this phase, the density behaves step-like, jumping directly into the CDW II phase at ρ = 1. This behavior is seen clearly in the data presented in figure 7(f), where the SF phase for low densities is followed by two distinct regions of PS. These regions are only interrupted by the CDW I phase at half filling and the CDW II phase at full filling.

As these results show, for the lower on-site interaction U = 5, the density-dependent term T does not change much the overall behavior of the phase diagram if it has the same sign as the single-particle tunneling t. Instead, if the two tunneling terms have opposite signs, a large part of the SS phase disappears into a phase-separated region, due to the increased relative importance of the interaction terms.