Bose condensation in flat bands

We derive effective Hamiltonians for lattice bosons with strong geometrical frustration of the kinetic energy by projecting the interactions on the flat lowest Bloch band. Specifically, we consider the Bose Hubbard model on the one-dimensional sawtooth lattice and the two-dimensional kagome lattice. Starting from a strictly local interaction the projection gives rise to effective long-range terms stabilizing a supersolid phase at densities above ${\nu }_c=1/9$ of the kagome lattice. In the sawtooth lattice on the other hand we show that the solid order, which exists at the magic filling ${\nu }_c=1/4$, is unstable to further doping. The universal low-energy properties at filling $1/4+\delta \nu$ are described by the well-known commensurate-incommensurate transition. We support the analytic results by detailed numerical calculations using the density-matrix renormalization group and exact diagonalization. Finally, we discuss possible realizations of the models using ultracold atoms as well as frustrated quantum magnets in high magnetic fields. We compute the momentum distribution and the noise correlations, that can be extracted from time of flight experiments or neutron scattering, and point to signatures of the unique supersolid phase of the kagome lattice.

Figure 1

(Color online) (a) Single-particle dispersion on the kagome lattice along high-symmetry lines in the Brillouin zone (gray). (b) The kagome geometry with its lattice vectors ${\mathbf{a}}_{1/2}$ and the basis sites $A$, $B$, and $C$ in each unit cell (gray). (c) Single-particle dispersion on the sawtooth lattice as a function of momentum $k$ for $t^{\prime }=\sqrt{2}t$. (d) The sawtooth geometry with couplings $t$ and $t^{\prime }$ and the basis sites $A$ and $B$ in the unit cell (gray).

Figure 2

(Color online) Strictly local eigenstates of the kinetic energy (local magnon states). (a) Resonating hexagon state on the kagome lattice where the $+$ and $-$ refer to the wave-function amplitude on each site around the hexagon. (b) Localized V-state on the sawtooth chain. (c) Ground state of the many-magnon system on the kagome lattice at density ${\nu }_c=1/9$. (d) Ground state of the many-magnon system on the sawtooth chain at ${\nu }_c=1/4$.

Figure 3

Processes in the spin-1/2 Hamiltonian (10). Arrows denote hopping (or spin-exchange) and dashed lines stand for density $\left({\sigma }^z\right)$ interactions.

Figure 4

(Color online) Fourier transform of the density correlations in the sawtooth chain. (a) Prediction of the long-wavelength theory. The delta-function peak at $q=\pi /a$ (red dashed line), which signals the CDW at $\nu ={\nu }_c$, splits with increased density $\nu ={\nu }_c+\delta \nu$ into two power-law peaks (blue full line) at $q_{\pm }=\pi \left(1\pm 2\delta \nu \right)$. The effective Luttinger parameter, which controls the power law is given in Eq. (14). (b) Density correlations computed using DMRG. The splitting of the peak by one quanta $\delta k=\frac{2\pi }{L}$ per added particle confirms the predicted $q_{\pm }$ of the long-wavelength theory.

Figure 5

(Color online) The effective Luttinger parameter $\tilde{K}$ as a function of density $\delta \nu =\nu -{\nu }_c$ extracted from the off-diagonal correlation function calculated using DMRG. The black line is a guide to the eyes.

Figure 6

(Color online) Illustration of the processes appearing in the effective Hamiltonian (23) for the kagome lattice. A dot represents a density operator, lines denote either a nearest-neighbor interaction $\left(I^z\right)$ or a hopping $\left(I^{xy}\right)$ assisted by the nearby density operator. Note that $I_{1/3}^{xy}$ have negative effective hopping amplitudes (red) while $I_{2/4}^{xy}$ have positive hopping (blue).

Figure 7

(Color online) Three patches of the CDW (different colors/grayscale) with two types of domain walls: straight (left) and zigzag (right). The thick lines denote the violated bonds which lead to the interaction energy cost. The yellow (shaded) ellipses around half of the particles in the zigzag domain wall indicate resonating particles which gain the delocalization $I_2^{xy}$.

Figure 8

(Color online) Order parameters in the kagome system as a function of the density $\nu ={\nu }_c+\delta \nu$. The lines are the result of the mean-field analysis of the effective Hamiltonian on the flat band. The CDW order parameter is shown as a full (red) line while the superfluid order parameter is the dashed (blue) line. Circles (CDW) and triangles (superfluid) mark the values obtained from exact diagonalization of the original kagome system on a small cluster. The (blue) star at $\delta \nu =0.15$ is the fluctuation reduced superfluid order parameter. The mean-field result shows a critical density where the CDW order vanishes.

Figure 9

(Color online) Boundary conditions for the finite kagome lattice in the presence of a magnetic flux ${\Phi }_{\pm }=\pm \pi /2$ through the openings of the torus. The hopping over a blue bond (dashed) is associated with a phase $e^{i\pi /2}$ (the bonds are directed as indicated by the arrow). Particles hopping over green bonds (dashed-dotted) bonds acquire a phase $e^{-i\pi /2}$. The red (dashed-dotted-dotted) bond connecting site 9 and 20 winds around both circumferences and picks up a phase $-1$.

Figure 10

The compressibility as a function of the density $\delta \nu =\nu -{\nu }_c$. The mean-field compressibility is given by the solid line while the dots indicate the results of numerical exact diagonalization. The nonvanishing compressibility of the CDW state at $\nu ={\nu }_c$ $\left(\delta \nu =0\right)$ is due to finite size of the cluster. Compressibility at $\nu >{\nu }_c$ is due to the condensate fraction. The fact that the mean-field compressibility is higher than the exact diagonalization result is consistent with the higher condensate density found in the mean-field theory.

Figure 11

(Color online) Predictions for time of flight experiments, including momentum distribution and noise correlations for the different phases of the kagome system. (a) The momentum distribution $⟨n_{\mathbf{k}}⟩$ in the CDW state at $\nu ={\nu }_c$. The smooth structure stems from the off-diagonal correlations inside hexagons. Destructive interference due to the resonating hexagon wave function leads to vanishing of $⟨n_{\mathbf{k}}⟩$ on the reciprocal-lattice vectors of the original kagome lattice (marked by yellow crosses). The (yellow) hexagon marks the boundaries of the first Brillouin zone. (b) In the supersolid state at $\nu >{\nu }_c$, sharp delta-function peaks appear in $⟨n_{\mathbf{k}}⟩$ in addition to the smooth background contribution of the CDW. These condensate peaks appear at the $K$ and $K^{\prime }$ points and points related to them by the CDW wave vector. However, points in the reciprocal lattice of the original kagome lattice remain exact zeros of the momentum distribution. In panel (c) we show the noise correlations for the CDW state as a function of the relative momentum $\delta \mathbf{k}=\mathbf{k}-{\mathbf{k}}^{\prime }$ for fixed zero average momentum. The delta peaks on the reciprocal vectors of the CDW lattice (peaks away from the yellow crosses that indicate the underlying kagome lattice) indicate the nontrivial ordering in an emergent lattice structure.

Figure 12

(Color online) Wannier functions (probability distribution) as a function of position on the sawtooth lattice with $t^{\prime }/t=\sqrt{2}$. The excited band has an oscillatory slowly decaying tail while the flat-band Wannier functions decay with a localization length $\xi \approx \text{log}\left(2.15\right)a$. The inset shows that locally, the Wannier function of the flat band is almost identical to the strictly localized V states indicated by the filled circles.

Figure 13

(Color online) Wannier function for the flat band in the kagome lattice. (a) Wave-function amplitudes shown on the two-dimensional lattice, where the size of the dots represents the weight and the color (shading) the denotes the sign $+$ or $-$ of the wave function. The definitions of the weights $w_1\dots w_5$ are indicated. (b) Algebraic decay of the Wannier function as $\sim 1/\left|r\right|$ along the high-symmetry direction marked by the dashed line in (a).