Boundary conditions and the residual entropy of ice systems

In this work we address the classical statistical mechanical problem of calculating the residual entropy of ice models. The numerical work found in the literature is usually based on extrapolating to infinite-size results obtained for finite-size systems with periodic boundary conditions. In this work we investigate how boundary conditions affect the calculation of the residual entropy for square, cubic, and hexagonal lattices using periodic, antiperiodic, and open boundary conditions. We show that periodic boundary conditions lead to noticeable oscillations in the entropy as a function of lattice size, and we calculate in open finite systems the contribution to the entropy from the open boundary. For our calculations we introduce a variation on multicanonical simulation methods that directly calculate the number of states in the ground state without the need of a Hamiltonian.

Figure 1

Logarithm of the density of states, $log\left(g\right)$, as a function of $\rho$ as obtained from WL simulations on a $30\times 30$ square lattice. Inset: A schematic view of the six possible ice configurations of a vertex.

Figure 2

$w$ as a function of the lattice size $L$ for different square lattices with periodic boundary conditions (PBCs). The blue circles and lines correspond to the exact calculation for small lattices. The exact value for infinite lattices, $w_{\mathrm{Lieb}}$, is indicated by a black line. The inset shows an ice configuration on a square lattice: while PBCs cannot be imposed on the $3\times 3$ lattice, if the system is reduced to $2\times 2$ the configuration satisfies the constraints of PBCs.

Figure 3

The residual entropy density under periodic boundary conditions, $s_{\mathrm{PBC}}$, as a function of the inverse number of sites, $1/N$, for lattices with $L>5$. The black lines correspond to fits to even and odd lattice sizes. The red dashed line is the one determined in Herrero and Ramírez [23] by integrating simulated specific heat.

Figure 4

The entropy density under antiperiodic bounday conditions, $s_{\mathrm{APBC}}$, as a function of the inverse number of sites, $1/N$. The black line corresponds to power law fit $s=s_0\left(\infty \right)+A{(1/N)}^{\alpha }$, with $s_0=0.4313\left(3\right), A=0.43126\left(4\right)$, and $\alpha =0.8157\left(3\right)$

Figure 5

A comparison between the residual entropy density under periodic (red squares), antiperiodic (green pentagons), and open (blue circles) boundary conditions as a function of lattice size for square ice. The inset shows the open boundary contribution to the entropy $\delta$ as a function of lattice size [see Eq. (8) and text for details].

Figure 6

$w$ as a function of the number of vertices, $N$, for three-dimensional lattices: red squares correspond to hexagonal lattices and blue circles to cubic ones. The value calculated by Nagle [15] is shown as a black horizontal line. The inset shows the same data as a function of the inverse number of vertices, $1/N$. The solid (dotted) line is a fit to the cubic (hexagonal) data using Eq. (7) (see text). The value calculated by Nagle for an infinite lattice is shown as a black dot.