Interaction-driven polarization shift in the $t\text{−}V\text{−}{V}^{\prime }$ lattice fermion model at half filling: Emergent Haldane phase

We study the $t\text{−}V\text{−}{V}^{\prime }$ model in one dimension at half filling. It is known that for large enough $V$ fixed, as $V^{\prime }$ is varied, the system goes from a charge-density wave into a Luttinger liquid, then a bond-order phase, and then a second charge-density wave phase. We find that the Luttinger liquid state is further split into two, separating parts with distinct values of the many-body polarization Berry phase. Inside this phase, the variance of the polarization is infinite in the thermodynamic limit, meaning that even if the polarization differs, it would not be measurable. However, in the gapped phases on each side of the Luttinger liquid, the polarization takes a different measurable value, implying topological distinction. The key difference is that the large-$V^{\prime }$ phases are link-inversion symmetric, while the small-$V^{\prime }$ one is site-inversion symmetric. We show that the large-$V^{\prime }$ phase can be related to an $S=1$ spin chain, and exhibits many features of the Haldane phase. The lowest lying states of the entanglement spectrum display different degeneracies in the two cases, and we also find string order in the large-$V^{\prime }$ phase. We also study the system under open boundary conditions, and suggest that the number of defects is related to the topology.

Figure 1

Phase diagram of the $t\text{−}V\text{−}{V}^{\prime }$ model at half filling. The phase lines separating the charge-density wave (CDW), Luttinger liquid (LL), bond-order (BO), and second charge-density wave (CDW-2) phases were determined by Mishra et al. [18]. The thick dashed line inside the LL phase indicates the main finding of this paper, where the polarization undergoes a discrete change. Along this line the polarization distribution is flat. The maximum of the polarization undergoes a discrete shift along this line (Fig. 3). The asterisks at $V/t=6$ indicate the cases for which the polarization is shown in part (a) of Fig. 3.

Figure 2

Heat map of the polarization distribution $P\left(x\right)$ as a function of $V^{\prime }/t$, $V/t=6.0$. Exact diagonalization calculations with periodic boundary conditions.

Figure 3

(a) polarization distributions for systems indicated by asterisks in Fig. 1: $V/t=6;V^{\prime }/t=2.0,2.4,2.59,2.8,3.6$. The points are in the phases CDW, LL (below polarization switch, $V^{\prime }/t<2.59$), LL (where polarization switch occurs), LL (above polarization switch, $V^{\prime }/t>2.59$), BO, respectively. The position of the polarization maximum undergoes a discrete change at $V^{\prime }/t=2.59$. Exact diagonalization calculations with periodic boundary conditions. (b) Size scaling exponent of the variance of the polarization. Arrows indicate the four cases shown in Fig. 4.

Figure 4

Polarization distributions as a function of the rescaled coordinate $x/L$ for systems with $V/t=6$ and different values of $V^{\prime }/t$. For $V^{\prime }/t=1.0$ and $V^{\prime }/t=6.0$ (CDW-1 and CDW-2 phases, respectively) the polarization distributions exhibit sharp peaks, which sharpen with system size. For the cases $V^{\prime }/t=2.4$ and $V^{\prime }/t=2.8$, on either side of the polarization switch, there are no sharp peaks, and the distributions exhibit negligible size dependence. The scaling exponent of the variation of the polarization is indicated by arrows in part (b) of Fig. 3. Exact diagonalization calculations with periodic boundary conditions.

Figure 5

(a) and (b) Low lying states of the entanglement spectrum for a system with $V/t=6$ as a function of $V^{\prime }/t$. In the small CDW-1 phase the degeneracy of the lowest lying state is 2, while in the CDW-2 phase it is 4. These numbers are indicated on the graphs. The upper of these two graphs is a system with $L=24$ with an entanglement cut into a subsystem ($L=12$) and environment ($L=12$). The lower one of the upper two graphs is a system with $L=24$ with an entanglement cut into a subsystem ($L=11$) and environment ($L=13$). Exact diagonalization calculations with periodic boundary conditions. (c) Comparison of energies for a tensor network variational calculation and exact diagonalization. The variational wave function is drawn in the tensor network notation of Ref. [29]. Both calculations use periodic boundary conditions.

Figure 6

(a) Probability of being in a real-space configuration not consistent with $S=1$ hidden antiferromagnetic order. (b) String order correlation function for different $V^{\prime }/t$. The system size is $L=24$. Nearest neighboring sites were paired. $V/t=6$. Exact diagonalization calculations with periodic boundary conditions.

Figure 7

Real space density for four cases: (a) $V/t=6,V^{\prime }/t=0$, (b) $V/t=6,V^{\prime }/t=8$, (c) $V/t=500,V^{\prime }/t=0$, and (d) $V/t=0,V^{\prime }/t=500$. Open boundary conditions. The system size is $L=24$.

Figure 8

Real space density calculated via cluster-mean-field theory for two cases, (a) $V/t=500,V^{\prime }/t=0$ and (b) $V/t=0,V^{\prime }/t=500$, under open boundary conditions. The system size is $L=40$.

Figure 9

(a) Proportion of configurations in the ground state wave-function with number of defects in the ordered state with a given parity. In the CDW-1 cases ($V/t=6,V^{\prime }/t=0$, $V/t=20,V^{\prime }/t=0$, both CDW-1) the weight of configurations with an odd number of defects is shown, whereas in the CDW-2 cases ($V/t=6,V^{\prime }/t=8$, $V/t=0,V^{\prime }/t=20$, both CDW-2) the weight of configurations with an even number of defects is shown. (b) average number of defects (appropriate to CDW-1 or CDW-2) and the variance (error bars) in the ground state. The legend applies to both upper and lower panels. Both calculations are exact diagonalization with open boundary conditions.