Thermal properties of vortices on curved surfaces

We use Monte Carlo simulations to study the finite temperature behavior of vortices in the XY model for tangent vector order on curved backgrounds. Contrary to naive expectations, we show that the underlying geometry does not affect the proliferation of vortices with temperature respect to what is observed on a flat surface. Long-range order in these systems is analyzed by using two-point correlation functions. As expected, in the case of slightly curved substrates these correlations behave similarly to the plane. However, for high curvatures, the presence of geometry-induced unbounded vortices at low temperatures produces the rapid decay of correlations and an apparent lack of long-range order. Our results shed light on the finite-temperature physics of soft-matter systems and anisotropic magnets deposited on curved substrates.

Figure 1

Low temperature configuration ($k_{B}T=0.1J$) of the modified XY model obtained by Monte Carlo simulations on a Gaussian bump surface. In this relaxed configuration, a positive vortex (red circle) locates on the top of the bump, and a negative vortex (yellow circle) down the surface. The inset shows a detail of the curved grid used in the simulations and the tangent vector field on this grid.

Figure 2

Typical vector field configurations obtained by Monte Carlo simulations on a substrate of $\alpha =3$, for low $k_{B}T=1.0J$ (a) and high $k_{B}T=1.75J$ (b) temperatures. The left panels show the vector field and the vortices, and the right panels show the corresponding short-range order maps and the color code used.

Figure 3

(a) Local vortex density as a function to the distance $r$ to the center of the surface, for a substrate of $\alpha =3$ and different temperatures (error bars are of the order of symbol size and omitted for clarity). (b) Local vortex density as a function of $r$ at a low temperature, for a substrate of higher curvature with $\alpha =5$. In this case, there is a geometrically unbounded dipole at low temperatures, such that the density of positive and negative vortices is inhomogeneous. These local density curves show that the positive and negative vortices locate at $r\sim 0$ and $r\sim 2r_0$ (see arrow), as expected. As temperature increases, more dipoles are excited, and the density of positive and negative vortices become homogeneous as in panel (a).

Figure 4

Mean local order correlator $S$ and vortex density $\rho$ (obtained by averaging over the whole substrate) as a function of temperature $T$ and for surfaces of different aspect ratios $\alpha$. Note that all the curves superimpose showing that the systems disorder in an equal form, irrespective of the underlying curvature.

Figure 5

Behavior of long-range correlation functions at a low temperature $k_{B}T=0.13J$, for a system of low curvature $\alpha =0.5$. (a) Relaxed spin configuration at this temperature. Note that spins align along a particular direction and there are no vortices in the pattern. (b) Two-point correlation function $C\left(s\right)$ as a function of geodesic distance, calculated along different meridians (black lines), and the average (red line). Here the correlation $C\left(r\right)$ accurately captures the long-range order of the system as observed in panel (a), where $C\left(s\right)\sim 0.9$ even at long distances. The inset is an scheme showing the different meridians and the characteristic circle $r=r_0$ of the Gaussian surface.

Figure 6

Behavior of long-range correlations at a low temperature $k_{B}T=0.13J$, for a highly curved system with $\alpha =7.0$. (a) Two-point correlation function $C\left(s\right)$ as a function of geodesic distance $s$, calculated along different meridians (black lines) of the Gaussian bump. Color lines corresponds to correlations calculated along specific meridians, which are indicated with a dashed line in panels (b) and (c). The inset is an scheme illustrating the different meridian paths used to get the correlations and the location of the vortices. Notice the huge dispersion of correlations calculated along different meridians. (b, c) Views of the configuration of spins at this temperature. The dashed pink line of panel (b) indicates a meridian where the spin orientation slightly changes and two-point correlations depicts long-range order [the correlation calculated along this path is shown in panel (a) with the pink line]. In panel (c) we show a meridian in dashed blue where the orientations of spins changes due to the negative vortex (shown as the yellow sphere). Along this path the two-point correlation decays abruptly [shown in panel (a) as a blue line]. Thus, in non-Euclidean systems the two-point correlation function may not correctly capture long-range ordered configurations.

Figure 7

Energy landscapes for a dipole and pair of dipoles on a Gaussian bump of $\alpha =7$ (left panels), and the energy level curves (right panels), where we use the same color code as in the left panel to identify the intensities. (a) Energy landscape for a first dipole, where the $r_{+}$ and $r_{-}$ coordinates corresponds to the position of positive and negative vortex, respectively, in opposite directions to the Gaussian bump (see schematic inset). The energy minima corresponds to $r_{+}\sim 0$ and $r_{-}\sim 2r_0$, such that the dipole unbinds due to the geometric force given by the geometric potential. (b) Energy landscape for a first dipole, but now the vortex and antivortex are in the same meridian (see schematic inset). The energy minima still corresponds to $r_{+}\sim 0$ and $r_{-}\sim 2r_0$. Note also that the dipole configurations having $r_{+}\sim r_{-}$ have always roughly the same energy, irrespective of the location of the dipole in the surface. (c) Energy landscape for a second dipole, in the same substrate, but when the first dipole is already unbounded (see inset). Now the energy minima corresponds to the second dipole bound, because the geometry has been already largely screened through the unbinding of the first dipole.