Thermal destruction of chiral order in a two-dimensional model of coupled trihedra

We introduce a minimal model describing the physics of classical two-dimensional (2D) frustrated Heisenberg systems, where spins order in a nonplanar way at $T=0$. This model, consisting of coupled trihedra (or Ising-$\mathbb{R}{P}^3$ model), encompasses Ising (chiral) degrees of freedom, spin-wave excitations, and ${\mathbb{Z}}_2$ vortices. Extensive Monte Carlo simulations show that the $T=0$ chiral order disappears at finite temperature in a continuous phase transition in the 2D Ising universality class, despite misleading intermediate-size effects observed at the transition. The analysis of configurations reveals that short-range spin fluctuations and ${\mathbb{Z}}_2$ vortices proliferate near the chiral domain walls, explaining the strong renormalization of the transition temperature. Chiral domain walls can themselves carry an unlocalized ${\mathbb{Z}}_2$ topological charge, and vortices are then preferentially paired with charged walls. Further, we conjecture that the anomalous size effects suggest the proximity of the present model to a tricritical point. A body of results is presented, which all support this claim: (i) first-order transitions obtained by Monte Carlo simulations on several related models, (ii) approximate mapping between the Ising-$\mathbb{R}{P}^3$ model and a dilute Ising model (exhibiting a tricritical point), and, finally, (iii) mean-field results obtained for Ising-multispin Hamiltonians, derived from the high-temperature expansion for the vector spins of the Ising-$\mathbb{R}{P}^3$ model.

Figure 1

(Color online) $SO\left(3\right)$ can be represented by a ball of radius $\pi$. A rotation of angle $\theta ∊\left[0,\pi \right]$ around the $\vec{n}$ axis is represented in this ball by the extremity of the vector $\theta \vec{n}$. Two opposite points of the surface (joined by a dashed line) represent the same rotation. A loop with an odd number of crossings, such as $ABFEA$ or $CDEFC$, cannot be continuously shrunk to a point. This is not the case when the number of crossings is even: (a) $ABCDEA$ crosses the surface both at $\left(AB\right)$ and $\left(CD\right)$. [(b) and (c)] It can be continuously deformed to collapse points $B$ on $C$ and $A$ on $D$. (d) $AD$ and $BC$ become contractible closed loops, which collapse on identical points.

Figure 2

(Color online) Uniform frozen chirality model (see Sec. 2B). Top: Vortex density $n_{\Omega }$ versus temperature $T$ from $L=16$ to $L=64$, from bottom to top. Bottom: Thermalized configuration at $T\simeq 1.1$ on a system of size $L=72$ (only a $45\times 45$ square snippet is shown). Square plaquettes carrying a ${\mathbb{Z}}_2$ vortex core are denoted by a (red) bullet. The width of each bond $\left(i,j\right)$ is proportional to $B_{ij}=1-{\left({\vec{v}}_i\cdot {\vec{v}}_j\right)}^2$ (by slices of 1/8). Bonds with $B_{ij}\le 1/8$ are not drawn.

Figure 3

(Color online) Maximum value of the specific heat as a function of system size $L$. The scaling at large $L$ is correctly reproduced by the affine logarithmic fit $A+B\text{ln}\left(L\right)$ (straight line). Inset: Specific heat $C_v$ versus temperature $T$ for $L=64$ and 80 from right to left.

Figure 4

(Color online) Probability distribution $P\left(E,T_c\right)$ of the energy per site at $T_c\left(L\right)$ for increasing linear $L$ from $L=16$ to $L=88$, from bottom to top. $T_c\left(L\right)$ is the temperature at which $C_v$ is maximum. A double peak is visible for $L\lesssim 60$ and disappears for $L>72$.

Figure 5

(Color online) Finite-size scaling at or near the transition temperature. (a) Rescaled chirality [Eq. (10)] versus rescaled temperature for $40\le L$. (b) Log-log plot of the maximum of the Ising susceptibility vs $L$. (c) Scaling of the temperature associated with the maximum of the specific heat vs $1/L$ (log-linear plot). (d) Evolution of the Binder cumulant $B\left(L,T\right)$ with temperature from $L=16$ to $L=88$, from bottom to top.

Figure 6

(Color online) Thermalized configuration at $T\simeq T_c$ of a system of size $L=72$ (only a $45\times 45$ square snippet is shown). The red bullets indicate elementary square plaquettes that: (i) host a ${\mathbb{Z}}_2$ vortex core and (ii) have all four ${\sigma }_i$ equal. Sites with up (down) chiralities are represented by green (white) squares. (a) The width of bonds $\left(i,j\right)$ is proportional to $B_{ij}$ (see text). (b) Bonds are drawn only if $B_{ij}>1/2$.

Figure 7

(Color online) (a) Excess density of vortices vs distance to the domain wall at $T=1.3$ (see text). (b) Vortex density in the bulk of chiral domains vs temperature $T$ from $L=16$ to $L=88$, from bottom to top. (c) Maximum of $\left(\frac{d{n}_{\Omega }}{dT}\right)$ versus specific heat maximum ${\text{max}}_T\left(C_v\right)$ for $16\le L\le 88$.

Figure 8

(Color online) Scalings as $L^2$ (a) of the maximum of the specific heat and (b) of the chiral susceptibility with increasing system size $L$ for $a=1.75$ in Eq. (12).

Figure 9

(Color online) Energy distributions at the transition for model (14).

Figure 10

(Color online) Density of states $g^{ϵ}\left(E\right)$ for a single bond in model (8). The two colors correspond to $ϵ={\sigma }_i{\sigma }_j=1$ (green) and $ϵ=-1$ (red). The lowest energy ($E=-3$ per bond) is attained when ${\left({\vec{v}}_i\cdot {\vec{v}}_j\right)}^2=1$ and ${\sigma }_i{\sigma }_j=1$.

Figure 11

(Color online) Schematic mean-field phase diagram of discrete model (18) on the $\left(T,D\right)$ plane. The two thin lines are trajectories of model (14), with $p=1,2$.

Figure 12

(Color online) Mean-field free energy for model (14) at the transition temperature for $p=1$ (Ising-$\mathbb{R}{P}^3$ model; solid line) and $p=2$ (dashed line). In both cases there are two stable mean-field solutions and the transition is first order. However the energy barrier is much smaller for $p=1$.