Hierarchy of correlations: Application to Green's functions and interacting topological phases

We study the many-body physics of different quantum systems using a hierarchy of correlations, which corresponds to a generalization of the $1/\mathcal{Z}$ hierarchy. The decoupling scheme obtained from this hierarchy is adapted to calculate double-time Green's functions and due to its nonperturbative nature, we describe quantum phase transition and topological features characteristic of strongly correlated phases. As concrete examples we consider spinless fermions in a dimerized chain and in a honeycomb lattice. We present analytical results which are valid for any dimension and can be generalized to different types of interactions (e.g., long-range interactions), which allows us to shed light on the effect of quantum correlations in a very systematic way. Furthermore, we show that this approach provides an efficient framework for the calculation of topological invariants in interacting systems.

Figure 1

Representation of a particle in the central site of a square lattice. Due to nonlocal terms of the Hamiltonian (either nearest-neighbor hopping $t_1$ or interaction $V_1$), the particle becomes correlated with the neighboring sites. The importance of these correlations is characterized by ${\rho }_{\mathbf{x},\mathbf{y}}^C$, being $\mathbf{x},\mathbf{y}$ nearest neighbors. If the state is strongly correlated, ${\rho }_{\mathbf{x},\mathbf{y}}^C$ has reached a maximum value, and the entanglement with an extra site requires that ${\rho }_{\mathbf{x},\mathbf{y}}^C$ must decrease as $1/\mathcal{Z}$.

Figure 2

Phase diagram in absence of density-density correlations for the real-space hierarchy. We find a HCP and a CDW phase as a function of $t_1/V_1$. We plot the case of a honeycomb lattice (dashed green) and a dimerized chain in the topological ($\lambda =t_1^{\prime }/t_1=1.25$, red) and trivial $(\lambda =0.75$, blue) phase, for $V_2=0$. The addition of density-density correlations in the self-consistency equation shifts the critical point to $t_1/V_1\to \infty$ (Fig. 3).

Figure 3

Full phase diagram for the real-space hierarchy including correlations $\left(V_2=0\right)$. At $T=0$ the CDW phase boundary extends to $t_1/V_1\to \infty$, meaning that the HCP is only present in absence of interactions $V_1=0$, as previously found in absence of correlations. Hybridization only affects the finite temperature region of the phase diagram, lowering the critical temperature from $T_C=\mathcal{Z}{V}_1$.

Figure 4

Correlations in $\mathbf{k}$ space. An initial state with momentum $\mathbf{k}$ interacts with all other particles in the system via the exchange of $\mathbf{q}$. If we assume for simplicity that some ${\mathbf{q}}_0$ dominates the interaction process, we show for a section of the BZ that different states can couple with the same $V_{{\mathbf{q}}_0}=V_{{\mathbf{q}}_0+n\mathbf{Q}}$, as $V_{\mathbf{q}}$ is a periodic function of $\mathbf{q}$. Then for a large system, we find that $n\propto N_p$, and the scaling of the interaction term $V_{{\mathbf{q}}_0}$ must be proportional to $N_p^{-1}$.

Figure 5

Phase diagram for the dimerized chain with nearest-neighbor hopping and interaction $\left(V_2=0\right)$. We have chosen $\lambda =1.25$ and 0.75 (red and blue color, respectively). The continuous lines correspond to the uncorrelated case, while the dashed lines correspond to the one including two-point correlations. It is interesting to see how correlations play an opposite role in the stabilization of the different phases. While the HCP is stabilized for the case $\lambda <1$ (blue), the CDW phase is stabilized for $\lambda >1$ (red). The $T=0$ critical points shift to different values due to correlations, however the critical temperature remains unchanged. It is interesting to see that the correlated phase diagram for the case $\lambda <1$ case shows a bump near $T\simeq 0.4$, where the CDW is stable for larger ratio $t_1/V_1$ than at the critical point.

Figure 6

Phase diagram for the honeycomb lattice $\left(V_2=0\right)$. The continuous line corresponds to the phase boundary between the CDW and the semimetallic phase, when correlations are neglected. The dashed line corresponds to the case including correlations. The shift in the critical point towards a smaller value of $t_1/V_1$ means that the semimetallic phase is stabilized by interactions.

Figure 7

Winding number ${\nu }_1$ of a dimerized chain as a function of $t_1/V_1$ for different ratios of the hopping $\lambda \in \left\{0.75,1.25\right\}$ (blue and red, respectively). For large $t_1/V_1$, we recover the noninteracting winding number, and as interactions increase, the winding number changes. The topological phase (red) changes its winding number to ${\nu }_1=2$ when $t_1/V_1\simeq 0.1$, and the trivial phase (blue) changes to ${\nu }_1=-1$ at $t_1/V_1\simeq 2.2$, and to ${\nu }_1=0$ at $t_1/V_1\simeq 1.15$. The vertical dashed lines indicate the transition to the CDW phase from Fig. 5. This calculation is performed assuming $T=0$.

Figure 8

Evolution of zeros of ${\widehat{G}}_k\left(\omega \right)$ as a function of $t_1/V_1$. The blue(red) line corresponds to the ratio $t_1/t_1^{\prime }=0.75\left(1.25\right)$. Note that all changes in the topological invariant correspond to zeros of the Green's function.

Figure 9

Zoom of Fig. 8 to small $t_1/V_1$. It shows that even the fast change in the winding number from ${\nu }_1=-1$ to ${\nu }_1=1$ is captured by the zeros of the Green's function.