Staircase of crystal phases of hard-core bosons on the kagome lattice

We study the quantum phase diagram of a system of hard-core bosons on the kagome lattice with nearest-neighbor repulsive interactions, for arbitrary densities, by means of the hierarchical mean-field theory and exact diagonalization techniques. This system is isomorphic to the spin $S=1/2$ XXZ model in presence of an external magnetic field, a paradigmatic example of frustrated quantum magnetism. In the nonfrustrated regime, we find two crystal phases at densities 1/3 and 2/3 that melt into a superfluid phase when increasing the hopping amplitude, in semiquantitative agreement with quantum Monte Carlo computations. In the frustrated regime and away from half-filling, we find a series of plateaux with densities commensurate with powers of 1/3. The broader density plateaux (at densities 1/3 and 2/3) are remnants of the classical degeneracy in the Ising limit. For densities near half-filling, this staircase of crystal phases melts into a superfluid, which displays finite chiral currents when computed with clusters having an odd number of sites. Both the staircase of crystal phases and the superfluid phase prevail in the noninteracting limit, suggesting that the lowest dispersionless single-particle band may be at the root of this phenomenon.

Figure 1

Schematics of a kagome lattice (black lines) and the underlying honeycomb lattice defined by the centers of the triangular plaquettes (gray). Blue arrows mark the two basis vectors of the kagome lattice, ${\mathbf{u}}_1$ and ${\mathbf{u}}_2$. Capital letters label the three-site basis $A,B,C$. Red dashed circles indicate the pattern of the localized resonant hole valence bond crystals (VBCs), as explained in the text.

Figure 2

Parameter regimes of the hard-core boson model on the kagome lattice (4). Regions are classified by the total and third component of the spin of a triangle $(S_p,S_p^z)$ that locally minimize the equivalent XXZ Hamiltonian written in the honeycomb lattice defined by the centers of the triangles comprising the kagome (2). Solid black lines correspond to parameters where the Hamiltonian has an exact solution. Dotted gray lines correspond to $h=0$ (half-filling), Ising, and Heisenberg limits, as explained in the text.

Figure 3

Quantum phase diagram of the hard-core boson model (4) as obtained within HMFT using 27-site clusters (referred in the text as 27-HMFT). Phases are labeled in capital letters as: fully occupied (FO), superfluid (SF), chiral superfluid (CSF) and valence bond crystal of density $\rho$ (${\mathrm{VBC}}_{\rho }$). The prefix $\pi$ is used to distinguish those VBCs with real positive amplitudes in their wave functions, from those containing some negative ones. Nonlabeled phases correspond to crystal phases with densities commensurable with the 27-site, but not with the nine-site cluster. Dotted gray lines correspond to constant density lines within the SF phase.

Figure 4

(a) and (c) Schematic pictures showing the (a) nine-site and (c) 27-site cluster tilings used in the HMFT approach of this work. Both tilings contain the fully packed resonant-hole hexagon pattern of the exact ${\mathrm{VBC}}_{8/9}$ state, represented by dashed red lines. Numbers label the lattice sites within the cluster. Circles represent intra-cluster sites, while squares represent those sites where the auxiliary mean fields are evaluated. (b) First Brillouin zone of the kagome lattice. Empty circles mark the momenta $\left(\mathbf{k}\right)$ consistent with the nine-site and 27-site tilings. In particular, the 9-HMFT contains $\mathrm{\Gamma }, K$, and $K^{\prime }$ points, while the 27-HMFT contains in addition the $J$ point and its rotations. The $M$ point, identified with a solid circle, is contained within the 36-site cluster used for ED.

Figure 5

Clusters used for the ED computations in this work: (a) 27, (b) 36c, (c) 36, and (d) 45.

Figure 6

Quantum phase diagram of model (4) computed with a nine-site (gray) and 27-site (black) CB Gutzwiller ansatz and ED. Also shown are QMC results from Ref. [16] (red), and DMRG and iPEPS plateaux widths at the Heisenberg line ($t/V=+0.5$) from Ref. [7] (blue) and Ref. [8] (orange), respectively. The normalized chemical potential $\left({\mu }^{\prime }\right)$ and hopping amplitude $\left(t^{\prime }\right)$ axis are defined to fit all physical limits. Phases are labeled as superfluid (SF), chiral superfluid (CSF) and valence bond crystal of density $\rho$ (${\mathrm{VBC}}_{\rho }$). The prefix $\pi$ distinguishes frustrated from nonfrustrated regimes. The constant $\rho =2/3$ line within the SF phase (dashed) is computed with 9-HMFT. Due to particle-hole symmetry, the diagram is symmetric around the half-filling line ${\mu }^{\prime }=0$. The trivial fully occupied and empty phases, correspond to ${\mu }^{\prime }>1$ and ${\mu }^{\prime }<-1$, respectively.

Figure 7

Condensate density computed with a nine-site CB Gutzwiller along the $\rho =2/3$ density line, and across the SF-${\mathrm{VBC}}_{2/3}$ transition in the nonfrustrated regime of the phase diagram (Fig. 6). (Inset) Second-order derivative of the ground-state energy along the same line.

Figure 8

Total density and condensate density (inset) in the XY limit ($V=0$) for the (a) nonfrustrated regime $t<0$ and (b) frustrated regime $t>0$ computed with 9-HMFT and 27-HMFT. Total density for the (c) nonfrustrated and (d) frustrated regimes in the same XY limit obtained with ED. In the frustrated regime, the widths of the ED plateaux do not have a monotonic dependence with increasing cluster size, contrary to the nonfrustrated regime.

Figure 9

Energy (in units of $V$) and second-order derivative (inset) across the $\pi {\mathrm{VBC}}_{2/3}$ to ${\mathrm{VBC}}_{2/3}$ transition for $\mu =3V$.

Figure 10

Absolute value of the expectation of the ${\mathcal{J}}_{ij}$ operator defined in Eq. (18) in the CSF phase as computed with (a) 9-HMFT, and (b) 27-HMFT, in the AFM XY limit at $\mu /t=0.02,V=0$. Thickness of the blue lines is proportional to $|\langle {\mathcal{J}}_{ij}\rangle |$, and its maximum value is $|\langle {\mathcal{J}}_{ij}\rangle |=0.1$. The chirality of the loop currents is represented by a circled point (vortex) or a crossed circle (antivortex).

Figure 11

Tower of states representing excitation energies (in units of $2t$) as a function of density $\rho$, obtained by ED on a $N=36$ cluster. States are labelled with respect to their quantum numbers associated to translation and point group symmetry (see Ref. [6] for more details): (a) $t=0.1V$ (near Ising regime) and (b) $V=0$ (AFM XY).

Figure 12

Fidelity susceptibility per site ${\chi }_F/N$ (24) as a function of $V/t$ from the XY regime to the SU(2) point, computed by 27-ED (filled) and 36-ED (empty) for various densities. For $\rho =5/9\text{and}7/9$ filled and empty symbols are superimposed.