Figure 1

The approximate ground state phase diagram of the long-range lattice gas model on a square lattice, based on variational methods as discussed in Sec. 5. On the vertical axis the type $p$ of the long-range interaction $V\left(r\right)=\frac{1}{r^p}$ is given, together with a logarithmic decaying interaction. The horizontal axis represents the particle density. From low to high densities we identify the following phases: The area without name depicts the dilute generalized Wigner crystal, followed by the $1/9$ Wigner crystal, the $1/6$ glassy phase described by Ref. [12], the $1/4$ Wigner crystal, the “checkerboard-in-a-checkerboard” $1/4^{\prime }$ phase, stripe phases (with a plateau for the $1/3$ stripe phase and “C” denotes the channeled stripes as described by Ref. [11]), and finally the checkerboard phase. Phases below $1/4$ filling are discussed in Sec. 3, above $1/4$ filling are discussed in Sec. 4. Below the phase diagram typical particle configurations in six phases are shown enlarged.Reuse & Permissions

Figure 2

Mean field finite temperature phase diagram of the lattice gas model (1) on a square lattice with $V\sim 1/r$ interactions; see Sec. 6. Temperature is in units of the nearest-neighbor interaction. The phases are the same as in the zero-temperature phase diagram of Fig. 1. At low densities we find various Wigner crystalline phases (see Sec. 3) with densities of the form $1/pq$ with $p,q$ integers. Close to half-filling we find checkerboard order which has a smooth crossover to the “checkerboard-in-a-checkerboard” $1/4^{\prime }$ phase. Around $\overline{n}=1/3$ and $3/7$ there are stripe ordered phases (see Sec. 4). The transitions towards the $1/4^{\prime }$ and checkerboard phase are second order, the other transitions are first order.Reuse & Permissions

Figure 3

At density $\rho =1/9$ a triangular crystal of particles is formed, which is not equilateral as would be the case for a perfect Wigner crystal. It is thus a prime example of a generalized Wigner crystal. The unit cell and unit vectors of the Wigner crystal are shown.Reuse & Permissions

Figure 4

A generalized Wigner crystal can be classified according to peaks in the Fourier transformed particle density. For densities close to each specific crystal density (7) the associated crystal structure will be maintained. We put forward the hypothesis that this leads to a devil's staircase of generalized Wigner crystals at low densities. The figure shows the crystal structure versus density obtained by mean field theory (see Sec. 6) at $\beta \to \infty$, displaying a staircase (thick solid lines). The thermodynamic potential (16) relative to the disordered state is shown (decreasing red dashed line), the thin lines underneath indicate the energy of specific ordered states.Reuse & Permissions

Figure 5

Some examples of domain walls between checkerboard phases. (a) A particle on the “wrong” sublattice surrounded by “holes,” which forms the smallest possible domain wall loop. (b) A horizontal domain wall. (c) Instability of a horizontal domain wall, by moving one particle to the other domain. (d) Stable, diagonal domain walls can exist for all long-range interactions.Reuse & Permissions

Figure 6

Some examples of stripe phases. (a) The $\rho =1/3$ state. (b) The $\rho =2/5$ state. (c) The $\rho =3/7$ state, but now doped.Reuse & Permissions

Figure 7

Fourier transformed density at fillings $1/9$, $0.25556$ and $1/3$; on a $90\times 90$ lattice with $1/r$ interactions. The different ordering wave vectors are clearly visible. These peaks are used to identify the phases that lead to the phase diagram of Fig. 1.Reuse & Permissions

Figure 8

Ground state energy of the long-range lattice gas model as a function of density, for various types of interaction. All energies are rescaled such that $E(\rho =1/2)=1$. Starting with a logarithmic interaction in solid red, we computed the energy for interactions of the form $1/r^p$. Notice the scaling behavior in the limit $p\to \infty$ and notice the kink around $\rho =1/3$ signaling stripe order.Reuse & Permissions

Figure 9

Detail of the charge configurations of the lowest energy states at a density of $n=29/77\approx 0.377$ and $L=154$. For logarithmic interactions the perfect stripe order is $9\times {10}^{-5}\%$ lower in energy than the channeled stripe order, in contrast to Ref. [11] which finds channeled stripes here. For $p=1$ interactions the channeled stripes are $0.0014\%$ lower in energy than the perfect stripes.Reuse & Permissions

Figure 10

Thermodynamic potential relative to the disordered state $\Delta \Phi$, in arbitrary units, plotted around various phase transitions. (a) At $n=0.27$ there is a second order phase transition into the $1/4^{\prime }$ state, clearly visible by the Mexican hat potential. The second order phase transition is common for transitions in the Ising universality class. (b) At $n=1/9$ there is a first order transition toward the Wigner crystal, as is customary for solidification transitions. (c) At $n=1/3$ there are two transitions: a second order transition into the $1/4^{\prime }$ state followed by a first order transition towards the stripe phase. The $1/4^{\prime }$ and the stripe phases are locally stable however for a longer range of temperatures. This implies the possibility of supercooling the $1/4^{\prime }$ phase.Reuse & Permissions