Stability of flat-band Bose-Einstein condensation from the geometry of compact localized states

Abstract

We consider Bose-Einstein condensation in flat-band models from a real-space perspective. Using a basis of compact localized states, we reformulate the minimization of the mean-field energy as a Euclidian geometry problem. Within Bogoliubov theory, we show that flat-band models where the solutions to this problem are frameworks consisting of triangles with nonzero area are promising for condensation, whereas for instance square frameworks indicate condensation in a single mode is impossible. When restricting the analysis to Bloch states, this approach can be related to a necessary condition for a non-vanishing quantum distance. This work provides a new perspective on how condensation in flat bands is destabilized, and offers principles for the construction of models where flat-band Bose-Einstein condensation is possible.

Figures (11)

• Figure 1: CLSs (hexagons/squares of different colors) and NLSs (thick brown/gray lines) in (a) the kagome lattice and (b) the checkerboard lattice, in a system of $3\times 3$ unit cells with periodic boundary conditions. Full lines indicate a hopping amplitude $t=1$, while dashed lines indicate $t=-1$. Considering the overlaps of CLSs at sites within the region bounded by a thin black line, and requiring uniform $|\varphi_{i\alpha}|$, we obtain a set of constraints $|\omega_i-\omega_j| = 1/\sqrt{N}$ for neighboring $\omega_i$ and $\omega_j$, fixing the distance between these points in the complex plane. (c-d) Examples of possible configurations of coefficients $\omega_i$ in the complex plane in the (c) kagome and (d) checkerboard lattices. Each edge of the frameworks has length $1/\sqrt{N}$, and corresponds to $\varphi_{i\alpha}$ at a given site. The resulting $\ket{\varphi_0}$ are shown in (e-f).
• Figure 2: (a) CLSs used as a starting point for the construction of a model with stable flat-band condensation. Each CLS is centered on a site where there are no overlaps, which constrains $|\omega_i|=1/\sqrt{N}$ to achieve uniform densities. The amplitude of the CLS at other sites is $\pm 1/a$, and the overlaps with adjacent CLSs enforce $|\omega_i-\omega_j|=a/\sqrt{N}$. Together, these constraints then require that $0$, $\omega_i$ and $\omega_j$ form an isosceles triangle. (b) Possible configuration for the coefficients $\omega_i$, under the constraints related to the sites in the gray square in (a). The thin black edges have a length of $1/\sqrt{N}$, and correspond to $\varphi_{i\alpha}$ at the central site of a CLS. The thick black lines have a length of $a/\sqrt{N}$, and give $a\varphi_{i\alpha}$ for the other sites. The $\varphi_{i\alpha}$ corresponding to this configuration are indicated by arrows in (a). (c) Tight-binding model for the Tasaki lattice considered here. (d) Condensate fraction $n_0/n_{\rm tot}$ assuming condensation in the uniform-density state with the lowest ZPE. We set $n_{\rm tot}=1$.
• Figure S1: Band structures of the (a) kagome lattice, (b) checkerboard lattice and (c) modified Tasaki lattice. In the Tasaki lattice, we show band structures for $a=1$ (black), $a=1.5$ (gray) and $a=2$ (light gray). In all panels, the flat band of interest is plotted in red.
• Figure S2: (a) Possible choice of basis of CLSs and NLSs in a kagome lattice with periodic boundary conditions and $3\times 3$ unit cells. (b) Wavefunction corresponding to the $\boldsymbol{k}=(0,0)$ Bloch function. Each arrow represents the complex number $\varphi_{i\alpha}$ and has length $1/\sqrt{N}$. (c) Left: Coefficients $\omega_i$ corresponding to the wavefunction in (b) when NLSs are not included. Some points, such as $\omega_1$, $\omega_2$, $\omega_3$ and even $0$ would need to be at several distinct positions in the complex plane to reproduce the desired phase pattern: the wavefunction in (b) can not be constructed from CLSs alone. Right: Configuration of coefficients when NLSs are also included. In this case, coefficients are well-defined, and some vertices of the framework correspond to linear combination of the various coefficients instead of the coefficients themselves. Note that because of periodic boundary conditions, the numbered black edges need to match the direction of the corresponding numbered gray edges. (d) Example of another candidate wavefunction, which is not a Bloch function, and can only be constructed by allowing for nonzero $\widetilde{\omega}_1$ and $\widetilde{\omega}_2$.
• Figure S3: Numbering of sites and CLSs/NLSs used in the (a) kagome and (b) checkerboard lattice when building the matrix $T$ (see Eq. \ref{['eq.cls_to_sites']}).