Bose-Einstein condensation in exotic lattice geometries

TL;DR

This work shows that exotic lattice geometries, including fractals and hyperbolic tilings, fundamentally modify Bose-Einstein condensation and quantum phase transitions. By analyzing non-interacting bosons and interacting Bose-Hubbard systems on these geometries, the authors find that fractals dramatically lower Tc and induce strong condensate fragmentation and fluctuations, whereas hyperbolic lattices exhibit 3D-like condensation despite two-dimensional embedding. The study combines tight-binding spectra, density-of-states analyses, and cluster Gutzwiller plus field-theoretic methods to reveal how spectra and DoS underpin condensation and Mott physics, with phase diagrams that interpolate between 1D and 2D in fractals and show finite Tc in the thermodynamic limit for hyperbolic lattices. These insights suggest lattice geometry can serve as a powerful control knob for quantum phases and motivate experiments in photonic and Rydberg-atom simulators.

Abstract

Modern quantum engineering techniques allow for synthesizing quantum systems in exotic lattice geometries, from self-similar fractal networks to negatively curved hyperbolic graphs. We demonstrate that these structures profoundly reshape Bose-Einstein condensation. Fractal lattices dramatically lower the condensation temperature and enhance condensation fluctuations. In a Sierpiński carpet, quasi-degeneracies in the tight-binding spectrum fragment the condensate. Hyperbolic lattices, on the other hand, exhibit condensation features similar to regular three-dimensional lattices, despite their embedding in only two dimensions: The critical temperature increases as the system grows, and the temperature-dependence of the condensate fraction follows the same power-law as for cubic lattices. We explain these similarities through the similarity of the densities of state at low energies. When strong repulsive interactions are included, the gas enters a Mott insulating state. Using a multi-site Gutzwiller approach as well as a simple strong-coupling expansion, for the Sierpiński triangle we find a smooth interpolation between the characteristic insulating lobes of one-dimensional and two-dimensional systems. Our findings establish lattice geometry as a powerful tuning knob for quantum phase phenomena and pave the way for experimental exploration in photonic waveguide arrays and Rydberg-atom tweezer arrays.

Figures (13)

• Figure 1: Illustrations of lattice geometries used in this work: (a) Sierpiński gasket with dimension $d \approx 1.585$, (b) triangular lattice with $d=2$, (c) Sierpiński carpet with $d \approx 1.893$, (d) square lattice with $d=2$, (e) Sierpiński tetrahedron with $d = 2$, and (f) tetrahedral lattice with $d=3$, as well as hyperbolic lattices with: $\{p, q\} = \{3,7\}$, with $V = 96$ sites (g), $V=4264$ (i) and $\{p, q\} = \{7,3\}$, with $V = 112$ (h), and $V=3481$ (j) sites. Lattices in Fig \ref{['fig:latticeGraphs']}(i,j) have a vertex as their center, while lattices in Fig \ref{['fig:latticeGraphs']}(g,h) are constructed around a central $p$-polygon.
• Figure 2: Condensate fraction in regular and hyperbolic lattice geometries. We plot the condensate fraction as a function of the temperature in units of the critical temperature $T_{\rm c}$ for (a) a three dimensional simple cubic lattice with $V = 20^3$ sites, and (b) a two dimensional square lattice with $V = 90^2$ sites, both with open boundary conditions, as well as a $\{p,q\}=\{3,7\}$ hyperbolic lattice with $V = 11173$ sites (c), and a $\{p,q\}=\{7,3\}$ hyperbolic lattice with $V = 9136$ sites (d). The different colored points correspond to different lattice fillings. The solid black line corresponds to the theoretical curve of Eq. (\ref{['n0fit']}), for the regular geometries. In the hyperbolic lattices, the condensate fraction behaves similarly to the 3D cubic lattice.
• Figure 3: Condensate fraction in fractal lattices vs temperature. Panel (a) shows, on a linear scale, the condensate fraction in a Sierpiński triangle lattice ($V=3282$ sites) in units of the critical temperature. The condensate fraction drops linearly at small temperatures with a heavy tail above $T_{\rm c}$, regardless of the filling. In panels (b-d), the behavior in different fractal lattices is compared to their non-fractal counterparts with a similar number of sites: (b) Sierpiński triangle lattice ($V=3282$ sites), a standard triangular lattice ($V=3321$ sites) and a 1D lattice ($V = 3282$), (c) Sierpiński carpet ($V=4096$ sites) and square lattice ($V=4096$ sites); for the carpet, both the standard condensate and the fragmented condensate (including the four lowest eigenstates) are shown. Panel (d) compares the Sierpiński tetrahedron lattice ($V=2050$ sites) to a regular tetrahedral lattice ($V=2024$ sites). In all plots, unit filling is chosen, and the condensate fraction is plotted vs a logarithmic temperature scale in units of the hopping constant $J$. The vertical lines mark the critical temperatures of the fractal and reference lattices at unit filling.
• Figure 4: Critical temperature $T_{\rm c}$ at a unit filling as a function of the system size, on a double logarithmic scale. The results from a cubic, tetrahedral, triangular, square and 1D lattices are compared to results from fractal geometries: the Sierpiński gasket ($d \approx 1.585$), Sierpiński tetrahedron ($d = 2$) and Sierpiński carpet ($d \approx 1.893$). In the Sierpiński carpet, the condensate fragments between the lowest $4$ eigenstates, which results in rapid decay of $T_{\rm c}$. One can consider BEC through the occupancy of any of these $4$ states, which produces different results (cyan). In general, the $T_{\rm c}$ decreases and decays faster in lattices with smaller dimension, and does not decay in the 3D tetrahedral and cubic lattices.
• Figure 5: Critical temperature as a function of the system size in hyperbolic lattices. The $\{p,q\}=\{3,7\}$ lattice (a) displays a small increase in critical temperatures with the system size (similarly to a cubic lattice with OBC). The $\{p,q\}=\{7,3\}$ lattice (b) displays a small decrease followed by an increase in the critical temperature. Critical temperature not decreasing monotonically with the number of sites suggests a finite value in the thermodynamic limit.