Interaction-driven phases in the half-filled honeycomb lattice: An infinite density matrix renormalization group study

The emergence of the Haldane Chern insulator state due to strong short-range repulsive interactions in the half-filled fermionic spinless honeycomb lattice model has been proposed and challenged with different methods and yet it still remains controversial. In this work we revisit the problem using the infinite density matrix renormalization group method and report numerical evidence supporting (i) the absence of the Chern insulator state, (ii) two previously unnoticed charge ordered phases, and (iii) the existence and stability of all the nontopological competing orders that were found previously within mean field. In addition, we discuss the nature of the corresponding phase transitions based on our numerical data. Our work establishes the phase diagram of the half-filled honeycomb lattice model, tilting the balance towards the absence of a Chern insulator phase for this model.

Figure 1

The top left illustration shows a schematic representation of the interacting tight-binding model considered in this work: fermions hopping in a half-filled honeycomb lattice with nearest-neighbor hopping $t$ (solid lines) and interacting via nearest- and next-to-nearest-neighbor interactions, $V_1$ and $V_2$, respectively (curved lines), as defined by (1). The right illustration shows the infinite density matrix renormalization group unit cell used in this work with $3\times 6$ unit cells yielding a cylinder with a circumference of $L_y=12$ sites depicted in bottom left illustration.

Figure 2

Different orders allowed in the half-filled spinless honeycomb lattice considered in this work: (a) semimetal, (b) charge density wave I (CDW I), (c) sublattice charge-modulated (CMs), (d) Kekulé, (e) CDW II, (f) CDW III, and (g) Haldane Chern insulator phases. Phases (a)–(d) and (g) were found within mean-field theory by Refs. [8, 9, 11]. The survival of the Haldane Chern insulator phase (g) was challenged within exact diagonalization with periodic boundary conditions by Refs. [17, 18] but found in Ref. [19] for open boundary conditions. The generic charge imbalance between the sublattices in phase (c) was not found within exact diagonalization [18]. The two sublattices are depicted in blue and orange: $0<\mathrm{\Delta }<\varrho <1/2$ with $\varrho +\mathrm{\Delta }<1/2$ describe the deviations from half filling per site; i.e., $\langle n\rangle =1/2\pm \alpha$ with $\alpha \in \{\varrho ,\varrho \pm \mathrm{\Delta }\}$.

Figure 3

(Left) Phase diagram obtained with iDMRG calculations on an infinite cylinder of circumference $L_y=12,\chi =1600$. The phase boundaries, especially at second-order transitions (see Sec. 4), are to be taken with some error, which can be estimated from Figs. 4 to 8. We draw sharp lines here for better clarity. (Right) Representative charge and bond strength patterns for the four phases with largest unit cell discussed in the main text. The area of the blue circles is proportional to the particle number expectation value on the respective site given by Eq. (2). The thickness of the ellipsoids on the bonds is proportional to the amplitude $t_{ij}$ between nearest neighbors defined by Eq. (3). The unit cells for each phase are depicted by the red polygons. These correspond to (a) CMs phase with $V_1/t=0.8,V_2/t=3.2$; (b) Kekulé phase with $V_1/t=5.6,V_2/t=1.6$; (c) CDW II phase with $V_1/t=5.6,V_2/t=3.2$; (d) CDW III phase with $V_1/t=9.2,V_2/t=2.5$. For (c) dashed lines indicate defect lines of rotated hexagons that have zero energetic cost in the classical limit (see main text).

Figure 4

Correlation length at $V_2=0$, labeled cut A in Fig. 3, probing the transition between the semimetal and CDW I phases. The smooth behavior of $\xi$ indicates a second-order transition. The dashed gray line at $V_1/t=1.356$ shows where the transition has been detected in quantum Monte Carlo simulations [15, 16]. (Inset) Scaling of the entanglement entropy $S$ with the bond dimension $\chi$ for two values of $V_1$ in the semimetal and CDW I phase. Unlike the CDW I phase, the semimetal phase presents the typical critical entanglement scaling $S\propto ln\chi$.

Figure 5

Entanglement entropy at $V_2/t=3.6$, labeled cut B in Fig. 3. The finite discontinuity of $S$ at $V_2/t\approx 2.2$ clearly signals a first-order transition between the CMs and CDW II phases.

Figure 6

Entanglement entropy at $V_1/t=0.4$, labeled cut C in Fig. 3, probing the phase transition between the semimetal and CMs phases. Going towards the phase boundary from high $V_2$, the entanglement entropy smoothly approaches the value in the semimetal, indicating a second-order transition.

Figure 7

Correlation length at $V_1/t=6$, labeled cut D in Fig. 3. In this plot, several phase transitions are made visible. With increasing $V_2$, the CDW I phase turns into the Kekulé phase, which itself goes into the semimetal phase. The transition from semimetal to CDW II can be identified by the peak in the correlation length. (Inset) The vicinity of the Kekulé-semimetal transition showing the discontinuity of the entanglement entropy as a function of $V_2$.

Figure 8

Entanglement entropy at $V_1/t=9.2$, labeled cut E in Fig. 3. The smooth behavior of $S$ at the phase boundary between CDW I and Kekulé phase indicates a second-order transition in accordance with the data of the correlation length in Fig. 7. At $V_2/t\approx 2.8$, the entanglement entropy is discontinuous, which clearly indicates a first-order transition between the CDW III and CDW II phases.