Chiral bosonic phases on the Haldane honeycomb lattice

Recent experiments in ultracold atoms and photonic analogs have reported the implementation of artificial gauge fields in lattice systems, facilitating the realization of topological phases. Motivated by such advances, we investigate the Haldane honeycomb lattice tight-binding model, for bosons with local interactions at the average filling of one boson per site. We analyze the ground-state phase diagram and uncover three distinct phases: a uniform superfluid (SF), a chiral superfluid (CSF), and a plaquette Mott insulator with local current loops (PMI). Nearest-neighbor and next-nearest-neighbor currents distinguish CSF from SF, and the phase transition between them is first order. We apply bosonic dynamical mean-field theory and exact diagonalization to obtain the phase diagram, complementing numerics with calculations of excitation spectra in strong and weak coupling perturbation theory. The characteristic density fluctuations, current correlation functions, and excitation spectra are measurable in ultracold atom experiments.

Figure 1

(a) Lattice vectors on the honeycomb lattice and hopping integrals the Haldane model of Eq. (1). (b) Phase diagram of the model (5) at unit filling, containing plaquette Mott insulator (PMI), uniform superfluid (SF) and chiral superfluid (CSF) phases. Solid (dashed) lines represent DMFT (Gutzwiller mean-field) results. (c) Local condensate order parameter in the uniform superfluid. (d) In CSF the condensate order parameters on sublattices $A$ and $B$ are determined up to a relative phase. (e)–(g) The lowest band of Eq. (1) for $t_1=1,\varphi =\pi /2$, and $t_2=0,1/\sqrt{3},1$, from the left to the right. Band minima move from the center $\mathit{\Gamma }$ to the corners ${\mathbf{K}}_A,{\mathbf{K}}_B$ of the first Brillouin zone (depicted as a solid line). At $t_1=\sqrt{3}{t}_2$, there are three degenerate minima.

Figure 2

Aligning the phases of order parameters on the two sublattices: $U=1,t_1=10,t_2=10$, and average filling $n=1$. In (a), the top left link hosts a defect $t_1=4t_2$. In (b), we impose open boundary conditions in $x$ direction. Arrows represent local (plaquette) currents with amplitude $|J_{\mathrm{AB}}|\approx |2t_{1}nsin\frac{2\pi }{3}|=\sqrt{3}n{t}_1$. Weaker currents that should vanish in the bulk, far away from the defect are not plotted. Results are obtained by minimizing the energy functional (12) with respect to ${\psi }_i$ for the 120 sites shown in the plot. In both plots, there are density modulations, for example, in (a) there are more particles sitting on the sites linked by $4t_1$.

Figure 3

Local density fluctuations $\langle {\widehat{n}}_i^2\rangle -{\langle {\widehat{n}}_i\rangle }^2$ vs $t_2$ at unit filling $n=1$. We see a competing effect of $t_1$ and $t_2$, as indicated by arrows: for small values of $t_2,t_1$ enhances fluctuations and drives the transition into SF. The opposite effect is found for strong enough $t_2$ in CSF, where eventually the effect of $t_1$ is washed out.

Figure 4

(Top) Absolute value of the next-nearest-neighbor bond current $|J_{AA}|/U$ deep in MI. The dashed line $t_1=\sqrt{2}{t}_2$ marks the region $J_{AA}=0$ according to Eq. (33). (Bottom) DMFT data for $J_{AA}$ (dots) agree very well with the result Eq. (33) (lines) in this region of the phase diagram.

Figure 5

The $J_{AA}$ current vs $t_2$ for $n=1$ and several values of $t_1$. Deep in SF (upper left) the current is positive and exhibits linear increase with $t_2$. In CSF (bottom right), the current is negative, strongly dependent on $t_2$ and only weakly affected by $t_1$. The absolute value $|J_{AA}|$ is much weaker in MI (intermediate regions).

Figure 6

Comparison of DMFT [(a) and (c)] and ED results [(b) and (d)]. Absolute value of the current on the bond between two next-nearest neighbors $|J_{AA}|/U$ is shown in (a) and (b). In (c) and (d), we plot local density fluctuations $\langle {\widehat{n}}_i^2\rangle -{\langle {\widehat{n}}_i\rangle }^2$. The inset in (a) gives the sign of $J_{AA}$; in the superfluid, the sign changes for $t_1=\sqrt{3}{t}_2$ (solid line), while deep in the Mott domain the boundary is given by $t_1=\sqrt{2}{t}_2$ (dashed line). DMFT data shown in these plots are for the fixed value of chemical potential $\mu /U=0.4$, while ED results are at fixed filling $n=1$. However, the main features discussed in the text are clearly visible.

Figure 7

Low-lying energy levels in the spectra of the $3\times 3$ unit cell lattice at unit filling, for: PMI $(t_1/U=0.04,t_2/U=0.02)$, SF $(t_1/U=0.14,t_2/U=0.02)$, and CSF $(t_1/U=0.04,t_2=0.08)$. The first eight energy levels in each sector of total momentum $\mathbf{Q}$ are plotted. The nine momentum sectors in the Brillouin zone spanned by the vectors ${\mathbf{g}}_1$ and ${\mathbf{g}}_2$ are represented in the top left panel. For example, sectors labeled 0, 7, 5 correspond to $\mathit{\Gamma },{\mathbf{K}}_A$, and ${\mathbf{K}}_B$, respectively.

Figure 8

Bogoliubov dispersion relation $\omega \left(\mathbf{k}\right)/U$ for $nU=1$ and (a) $t_1=1,t_2=0$, (b) $t_1=1,t_2=0.1$, (c) $t_1=1,t_2=1$, (d) $t_1=0,t_2=1$. Black solid lines give the corresponding noninteracting dispersion relation shifted upwards by $2Un-\mu$. The inset in (b) defines the path in the Brillouin zone.

Figure 9

(a) Contribution of quantum fluctuations to the ground-state energy $\delta E(\mathbf{k},\varphi )-\delta E(\mathbf{k},0),t_1=t_2=1,nU=1,\varphi ={\varphi }_A-{\varphi }_B$. For a single value of $\mathbf{k}$, we obtain the thick curve. The qualitative behavior of $\delta E(\mathbf{k},\varphi )$ is similar throughout the Brillouin zone (thin lines). In (b), we illustrate the six configurations of order parameters on the two sublattices favoured by quantum fluctuations. In (c), the corresponding emerging plaquette currents are depicted.

Figure 10

Typical spectral functions deep in the Mott domain, for $t_1=3U/100,t_2=U/100,\mu =U/2$: (a) and (b) quasihole and quasiparticle branches of the spectral function obtained from DMFT for a cylinder geometry exhibiting edge modes for $\delta =U/1000$, (c) density of states $A\left(\omega \right)$ (arbitrary units) in a torus geometry for two values of $\delta$: $U/1000$ (solid black line) and $U/100$ (thick dashed red line). Quasiparticle and quasihole bands are centered at $U-\mu$ and $-\mu$, respectively.