Action at a distance in classical uniaxial ferromagnetic arrays

We examine in detail the theoretical foundations of striking long-range couplings emerging in arrays of fluid cells connected by narrow channels by using a lattice gas (Ising model) description of a system. We present a reexamination of the well known exact determination of the two-point correlation function along the edge of a channel using the transfer matrix technique and a different interpretation is provided. The explicit form of the correlation length is found to grow exponentially with the cross section of the channels at the bulk two-phase coexistence. The aforementioned result is recaptured by a refined version of the Fisher-Privman theory of first order phase transitions in which the Boltzmann factor for a domain wall is decorated with a contribution stemming from the point tension originated at its end points. The Boltzmann factor for a domain wall together with the point tension is then identified exactly thanks to two independent analytical techniques, providing a critical test of the Fisher-Privman theory. We then illustrate how to build up the network model from its elementary constituents, the cells and the channels. Moreover, we are able to extract the strength of the coupling between cells and express them in terms of the length and width and coarse-grained quantities such as surface and point tensions. We then support our theoretical investigation with a series of corroborating results based on Monte Carlo simulations. We illustrate how the long-range ordering occurs and how the latter is signaled by the thermodynamic quantities corresponding to both planar and three-dimensional Ising arrays.

Figure 1

The network of boxes from top view (right) and from side view (left). The typical dimensions considered in the experimental setup (see text) are $M\sim 1\mu \mathrm{m}$, $L\sim 2\mu \mathrm{m}$, $L_0\sim 2\mu \mathrm{m}$, $h\sim 30\mathrm{nm}$.

Figure 2

Ising model on a planar lattice with free boundary conditions and nearest-neighbor interactions $K_j>0,j=1,2$, is shown. The transfer direction is indicated. In the applications considered here, we impose cyclic boundary conditions in the $(1,0)$ direction.

Figure 3

A schematic representation of a typical domain wall configuration on the Ising strip. Typically, the domains are widely separated along $(1,0)$ and thus infrequent.

Figure 4

Side view of an Ising system comprised of two cubic lattice boxes of a side $L_0$ connected by a $L\times M$ strip with $L\gg M$. We assume $L_0\gg M$.

Figure 5

(a) The critical length $L_c$ as a function of the surface tension $\tau$ and the strip width $M$. Given $\tau$ and $M$, a system with a length smaller than $L_c$ is ordered and corresponds to a point in the phase diagram below the surface of the graph. The iso-$L_c$ contour lines are highlighted in red. Note the existence, for a given $M$, of ordered configurations for a pair of values of $\tau$ (reentrant phenomenon). (b) The phase diagram in terms of the scaling variables $\tau L$ and $e^{-\tau M}$. The critical line $\tau L_c{e}^{-\tau M}=2^{-1}ln(1+\sqrt{2})$ separates ordered and disordered configurations, as illustrated in the shadowed regions. Notice that $e^{-\tau M}$ is bounded from above by unity since $\tau$ is non-negative.

Figure 6

Cylindrical lattice with a line of reversed bonds.

Figure 7

(a) Geometry of a 1D array of $\mathcal{N}$ squares of size $L_0$ connected by strips (channels) of length $L$ and width $M$. (b) The equivalent 1D system, that consists of $\mathcal{N}$ coarse-grained spin variables $S_j$ connected by bonds with effective interaction $K_{\mathrm{eff}}$.

Figure 8

(a) The scheme for the computation of the spin-spin correlation function $G\left(x\right)$ along three different lines: line 1 at the middle of the channel (red solid line), line 2 at the side of the channel (cyan dashed-dotted line), and line 3 at the side of the square (magenta dashed line). (b) Spin-spin correlation function $G\left(x\right)$ for the 1D array of $\mathcal{N}=100$ squares of a side length $L_0=100$ as a function of the distance $x$ along the three lines: the line passing through center of the channel (red solid line), through the side of the channel (cyan dashed-dotted line) and through the side of the square (magenta dashed line) for $K=0.6$; the channel length is $L=100$ and the width is $M=10$. (c) The plateau values of $G\left(x\right)$ computed along the middle line of the channel as a function of $x/L$ (symbols) follow the 1D Ising correlation function law given by Eq. (45) (dashed blue line) with $m_0^{*}=0.97$.

Figure 9

Spin-spin correlation function $G\left(x\right)$ along the centers of the squares (line 1) as a function of the distance $x$ for the same system as in Fig. 8 and various couplings, $K=0.5,0.56,0.6,0.64,0.68$.

Figure 10

Thermodynamic quantities for a 1D array of $\mathcal{N}=100$ squares of size $L_0=100$ connected (in a periodic way) by strips of size $100\times M$ as a function of the coupling $K$: energy per spin $e$ (upper panel, left), heat capacity $C$ (upper panel, right), magnetization per spin $m$ (lower panel, left), and magnetic susceptibility $\chi$ (lower panel, right). Results for a single OBC square with $L_0=100$ are plotted by black dashed line for comparison (except for the plot of $\chi$).

Figure 11

(a) Geometry of a 2D array of a linear size $\mathcal{N}$ consisting of ${\mathcal{N}}^2$ squares of size $L_0$ connected by strips (channels) of the length $L$ and the width $M$. The scheme for computation of the spin-spin correlation function $G\left(x\right)=〈\sigma \left(0\right)\sigma \left(x\right)〉$ for three different lines: line 1 at the middle of the channel (red solid line), line 2 at the side of the channel (cyan dashed-dotted line), and line 3 at the side of the square (magenta dashed line). (b) Geometry of the equivalent network model.

Figure 12

Thermodynamic quantities for the system shown in Fig. 11 with $\mathcal{N}=10$ and $L_0=100$ connected (in a periodic way) by strips of size $100\times M$ as a function of the coupling $K$: energy per spin (upper panel, left) $e$, heat capacity $C$ (upper panel, right), magnetization per spin $m$ (lower panel, left), and magnetic susceptibility $\chi$ (lower panel, right). Results for a single OBC square $L_0=100$ are plotted with a black dashed line for comparison.

Figure 13

Left panel: the spin-spin correlation function $G\left(x\right)$ as a function of the distance $x$ for a system with $L_0=100$, $L=100$, $M=10$ for $K=0.56$ along the three different lines: centers of the channels (red solid line), sides of the channels (cyan dashed-dotted line), and along the sides of the squares (magenta dashed line). Right panel: the correlation function along the centers of the channels for various couplings $K=0.46,0.53,0.55,0.56,0.58$.

Figure 14

The effective interaction constant $K_{\mathrm{eff}}$ mediated by the Ising strip (channel) of the size $L\times M$ as a function of $K$ for $L=100$ and $M=4,6,10,20,30,40$. Lines correspond to Eq. (30), symbols correspond to the MC results (48).

Figure 15

Thermodynamic quantities for the 2D array of $\mathcal{N}\times \mathcal{N}$ cubes of side $L_0=20$ connected (in a periodic way) by channels of size $40\times 4\times 4$ as functions of the coupling $K$: heat capacity $C$ (left panel), magnetic susceptibility $\chi$ (right panel).

Figure 16

Thermodynamic quantities for the 2D array of $10\times 10$ cubes of the side $L_0$ connected (in a periodic way) by channels of size $40\times 4\times 4$ as functions of the coupling $K$: heat capacity $C$ (left panel), magnetic susceptibility $\chi$ (right panel).

Figure 17

Thermodynamic quantities for the 2D array of $10\times 10$ cubes of the side $L_0=20$ connected (in a periodic way) by channels of the size $L\times 4\times 4$ as functions of the coupling $K$: heat capacity $C$ (left panel), magnetic susceptibility $\chi$ (right panel).

Figure 18

Left panel: the spin-spin correlation function $G\left(x\right)$ as a function of the distance $x$ along the center of the channel [line 1 of Fig. 11], for the 1D array of $\mathcal{N}=100$ 3D cubes of the linear size $L_0=40$ (periodically) connected by channels of the size $40\times 4\times 4$ for several values of the coupling $K$. Right panel: the corresponding values of $G\left(x\right)$ at the centers of the cubes.

Figure 19

The integration contour $\mathfrak{C}$ in the complex $k$ plane. The vertical lines have $\mathrm{Re}\left(k\right)=\pi \left[1+1/\left(4M\right)\right]$ and $\mathrm{Re}\left(k\right)=-\pi \left[1-1/\left(4M\right)\right]$, so that they pass between zeros of $e^{ikM}=\pm 1$, and $\pi$ is inside the contour, but not $-\pi$. Note that $g_N\left(k\right)$ is even in $k$ and there is $2\pi$ periodicity. Thus, the side line contributions cancel.