Topological Phase Transition and Geometrical Frustration in Fourier Photonic Simulator

Abstract

XY models with continuous spin orientation play a pivotal role in understanding topological phase transitions and emergent frustration phenomena, such as superconducting and superfluid phase transitions. However, the complex energy landscapes arising from frustrated lattice geometries and competing spin interactions make these models computationally intractable. To address this challenge, we design a programmable photonic spin simulator capable of emulating XY models with tunable lattice geometries and spin couplings, allowing systematic exploration of their statistical behavior. We experimentally observe the Berezinskii-Kosterlitz-Thouless (BKT) transition in a square-lattice XY model with nearest-neighbor interactions, accurately determining its critical temperature. Expanding to frustrated systems, we implement the approach in triangular and honeycomb lattices, uncovering sophisticated phase transitions and frustration effects, which are consistent with theoretical predictions. This versatile platform opens avenues for probing unexplored XY model phenomena across diverse geometries and interaction schemes, with potential applications in solving complex optimization and machine learning problems.

Figures (4)

• Figure 1: The principle of FPS. The XY spins $\boldsymbol{S}_j$ with angle $\theta_j$ are encoded to the wavefront phase $\varphi_j$ of a laser by an SLM. The modulated light passes through a lens, which performs an optical Fourier transform, and is finally detected by a camera on the back focal plane of the lens. A pointwise multiplication (Mult.) or Hadamard product is implemented between the light intensity $I$ and the Fourier mask $I_M$, which is equivalent to a convolution (Conv.) operation in the real space. Finally, the Hamiltonian of the XY model is obtained from the summation over all values of the products. Here, the Fourier mask has the effect of a filter for spin interactions, which can select the Hamiltonian of arbitrary range spin interactions. The MCMC algorithm is introduced to update the spin configurations by using the optically obtained Hamiltonian, thereby sampling spin configurations that obey the Boltzmann distribution. These samples can be employed to obtain important quantities of the XY models, such as the correlation function, helicity modulus and specific heat.
• Figure 2: Observing the BKT transition of ferromagnetic XY model on the square lattice. (a) The lattice structure of a square lattice, the spin arrangement on the SLM, and the Fourier mask for NN interactions with periodic boundary conditions, respectively. $a$ is the lattice constant. The blue arrows represent spin coupling. (b) The helicity modulus ($Y$) as functions of temperature across different lattice sizes ($L\times L$=8$\times$8, 10$\times$10, 16$\times$16, 20$\times$20). The solid gray line represents $\tilde{Y} = \frac{2}{\pi}T$. The BKT transition is identified by the intersection of $Y(T)$ and $\tilde{Y}(T)$. The dashed gray line indicates $T\approx0.8922$, which is the $T_\text{BKT}$ of infinite lattice. Insert provides a zoomed-in view near the BKT transition temperature $T_\text{BKT}$. (c) The correlation function $G(r)$ with respect to the distance $r$ between spins at $T = 1.2> T_\text{BKT}$ (top) and $T = 0.8<T_\text{BKT}$ (bottom). Above $T_\text{BKT}$, $G(r)$ fits the exponential decay model better with $R^2=0.9926$, while the power-law decay mode fits better below $T_\text{BKT}$ with $R^2=0.9780$.
• Figure 3: Simulating the antiferromagnetic XY model on a triangular lattice with $N = 12\times12$ spins. (a) The triangular lattice (upper panel), the interacting neighbors on the SLM (lower left), and the Fourier mask for NN interactions with periodic boundary conditions (lower right). (b) The specific heat ($C_V$) and helicity modulus ($Y$) as functions of temperature. $C_V(T)$ peaks at $T_c=0.48J$ indicating a second-order phase transition, while $Y(T)$ intersects $\tilde{Y}(T)=\frac{2}{\pi}T$ at $T_\text{BKT}\approx0.43J$, denoting a BKT transition with $T_\text{BKT}<T_c$.
• Figure 4: Searching the ground state of antiferromagnetic $J_1$-$J_2$ XY model on the honeycomb lattice with $N = 96$ spins. (a) The lattice structure of the honeycomb lattice (upper left), the spin arrangement on the SLM (upper right), and the Fourier masks for NN interactions ($J_1$, lower left) and NNN interactions ($J_2$, lower right) with periodic boundary conditions. In building a honeycomb lattice from a square lattice, we set the light intensity in the dotted box areas in the SLM to be zero. (b) The ground states in reciprocal space with different $J_2/J_1$ ratios. The black dashed lines mark the boundary of the 1BZ, while the white solid lines indicate the theoretically predicted positions of the peaks di_ciolo_spiral_2014.