[1] E. Ising. Beitrag zur Theorie des Ferromagnetismus. Z. Physik, 31, 253-258(1925). https://doi.org/10.1007/BF02980577

[2] H. Nishimori, G. Ortiz. Elements of Phase Transitions and Critical Phenomena(2010).

[3] J. K. Glasbrenner et al. Effect of magnetic frustration on nematicity and superconductivity in iron chalcogenides. Nat. Phys., 11, 953-958(2015). https://doi.org/10.1038/nphys3434

[4] D. P. Landau, K. Binder. Phase diagrams and critical behavior of Ising square lattices with nearest-, next-nearest-, and third-nearest-neighbor couplings. Phys. Rev. B, 31, 5946-5953(1985). https://doi.org/10.1103/PhysRevB.31.5946

[5] D. Pierangeli et al. Observation of replica symmetry breaking in disordered nonlinear wave propagation. Nat. Commun., 8, 1501(2017). https://doi.org/10.1038/s41467-017-01612-2

[6] N. Ghofraniha et al. Experimental evidence of replica symmetry breaking in random lasers. Nat. Commun., 6, 6058(2015). https://doi.org/10.1038/ncomms7058

[7] Y. G. N. Liu et al. Gain-induced topological response via tailored long-range interactions. Nat. Phys., 17, 704-709(2021). https://doi.org/10.1038/s41567-021-01185-4

[8] F. D. M. Haldane. Model for a quantum Hall effect without Landau levels: condensed-matter realization of the ‘parity anomaly’. Phys. Rev. Lett., 61, 2015-2018(1988). https://doi.org/10.1103/PhysRevLett.61.2015

[9] G. Jotzu et al. Experimental realization of the topological Haldane model with ultracold fermions. Nature, 515, 237-240(2014). https://doi.org/10.1038/nature13915

[10] N. Mohseni, P. L. McMahon, T. Byrnes. Ising machines as hardware solvers of combinatorial optimization problems. Nat. Rev. Phys., 4, 363-379(2022). https://doi.org/10.1038/s42254-022-00440-8

[11] E. Small, R. Pugatch, Y. Silberberg. Berezinskii-Kosterlitz-Thouless crossover in a photonic lattice. Phys. Rev. A, 83, 013806(2011). https://doi.org/10.1103/PhysRevA.83.013806

[12] G. Situ, J. W. Fleischer. Dynamics of the Berezinskii–Kosterlitz–Thouless transition in a photon fluid. Nat. Photonics, 14, 517-522(2020). https://doi.org/10.1038/s41566-020-0636-7

[13] T. Inagaki et al. A coherent Ising machine for 2000-node optimization problems. Science, 354, 603-606(2016). https://doi.org/10.1126/science.aah4243

[14] Z. Wang et al. Coherent Ising machine based on degenerate optical parametric oscillators. Phys. Rev. A, 88, 063853(2013). https://doi.org/10.1103/PhysRevA.88.063853

[15] M. Nixon et al. Observing geometric frustration with thousands of coupled lasers. Phys. Rev. Lett., 110, 184102(2013). https://doi.org/10.1103/PhysRevLett.110.184102

[16] M. Parto et al. Realizing spin Hamiltonians in nanoscale active photonic lattices. Nat. Mater., 19, 725-731(2020). https://doi.org/10.1038/s41563-020-0635-6

[17] W. Ettoumi, J. Kasparian, J. P. Wolf. Laser filamentation as a new phase transition universality class. Phys. Rev. Lett., 114, 063903(2015). https://doi.org/10.1103/PhysRevLett.114.063903

[18] N. G. Berloff et al. Realizing the classical XY Hamiltonian in polariton simulators. Nat. Mater., 16, 1120-1126(2017). https://doi.org/10.1038/nmat4971

[19] R. Tao et al. Halide perovskites enable polaritonic XY spin Hamiltonian at room temperature. Nat. Mater., 21, 761-766(2022). https://doi.org/10.1038/s41563-022-01276-4

[20] M. C. Rechtsman et al. Photonic Floquet topological insulators. Nature, 496, 196-200(2013). https://doi.org/10.1038/nature12066

[21] Y. Shen et al. Deep learning with coherent nanophotonic circuits. Nat. Photonics, 11, 441-446(2017). https://doi.org/10.1038/nphoton.2017.93

[22] N. Carlon Zambon et al. Optically controlling the emission chirality of microlasers. Nat. Photonics, 13, 283-288(2019). https://doi.org/10.1038/s41566-019-0380-z

[23] M. Suzuki. Critical exponents for long-range interactions. II: Universality and scaling relations. Prog. Theor. Phys., 49, 1106-1120(1973). https://doi.org/10.1143/PTP.49.1106

[24] D. Dantchev, J. Rudnick. Subleading long-range interactions and violations of finite size scaling. Eur. Phys. J. B, 21, 251-268(2001). https://doi.org/10.1007/s100510170201

[25] J. W. Goodman. Introduction to Fourier Optics(2005).

[26] A. Silva et al. Performing mathematical operations with metamaterials. Science, 343, 160-163(2014). https://doi.org/10.1126/science.1242818

[27] T. Zhu et al. Topological optical differentiator. Nat. Commun., 12, 680(2021). https://doi.org/10.1038/s41467-021-20972-4

[28] J. Liu et al. Intrinsic optical spatial differentiation enabled quantum dark-field microscopy. Phys. Rev. Lett., 128, 193601(2022). https://doi.org/10.1103/PhysRevLett.128.193601

[29] X. Lin et al. All-optical machine learning using diffractive deep neural networks. Science, 361, 1004-1008(2018). https://doi.org/10.1126/science.aat8084

[30] X. Meng et al. Compact optical convolution processing unit based on multimode interference. Nat. Commun., 14, 3000(2023). https://doi.org/10.1038/s41467-023-38786-x

[31] D. Pierangeli, G. Marcucci, C. Conti. Large-scale photonic Ising machine by spatial light modulation. Phys. Rev. Lett., 122, 213902(2019). https://doi.org/10.1103/PhysRevLett.122.213902

[32] D. Pierangeli et al. Scalable spin-glass optical simulator. Phys. Rev. Appl., 15, 034087(2021). https://doi.org/10.1103/PhysRevApplied.15.034087

[33] Y. Fang, J. Huang, Z. Ruan. Experimental observation of phase transitions in spatial photonic ising machine. Phys. Rev. Lett., 127, 043902(2021). https://doi.org/10.1103/PhysRevLett.127.043902

[34] J. Huang, Y. Fang, Z. Ruan. Antiferromagnetic spatial photonic Ising machine through optoelectronic correlation computing. Commun. Phys., 4, 242(2021). https://doi.org/10.1038/s42005-021-00741-x

[35] W. Fan et al. Programmable photonic simulator for spin glass models(2023).

[36] L. Luo et al. Wavelength-division multiplexing optical Ising simulator enabling fully programmable spin couplings and external magnetic fields. Sci. Adv., 9, eadg6238(2023). https://doi.org/10.1126/sciadv.adg6238

[37] H. Yamashita et al. Low-rank combinatorial optimization and statistical learning by spatial photonic Ising machine. Phys. Rev. Lett., 131, 063801(2023). https://doi.org/10.1103/PhysRevLett.131.063801

[38] J. Ouyang et al. On-demand photonic Ising machine with simplified Hamiltonian calculation by phase encoding and intensity detection. Commun. Phys., 7, 168(2024). https://doi.org/10.1038/s42005-024-01658-x

[39] M. Leonetti et al. Optical computation of a spin glass dynamics with tunable complexity. Proc. Natl. Acad. Sci. U.S.A., 118, e2015207118(2021). https://doi.org/10.1073/pnas.2015207118

[40] J. R. Norris. Markov Chains(1998).

[41] N. Metropolis et al. Equation of state calculations by fast computing machines. J. Chem. Phys., 21, 1087-1092(1953). https://doi.org/10.1063/1.1699114

[42] J. Colbois et al. Partial lifting of degeneracy in the J1 - J2 - J3 Ising antiferromagnet on the Kagome lattice. Phys. Rev. B, 106, 174403(2022). https://doi.org/10.1103/PhysRevB.106.174403

[43] J. Oitmaa, R. R. P. Singh. Phase diagram of the J1-J2-J3 Heisenberg model on the honeycomb lattice: a series expansion study. Phys. Rev. B, 84, 094424(2011). https://doi.org/10.1103/PhysRevB.84.094424

[44] J. M. Kosterlitz. The critical properties of the two-dimensional xy model. J. Phys. C: Solid State Phys., 7, 1046-1060(1974). https://doi.org/10.1088/0022-3719/7/6/005

[45] J. Tobochnik. Properties of the q-state clock model for q = 4, 5, and 6. Phys. Rev. B, 26, 6201-6207(1982). https://doi.org/10.1103/PhysRevB.26.6201

[46] Z. Li et al. Critical properties of the two-dimensional q-state clock model. Phys. Rev. E, 101, 060105(2020). https://doi.org/10.1103/PhysRevE.101.060105

[47] A. Jelić, L. F. Cugliandolo. Quench dynamics of the 2D XY model. J. Stat. Mech., 2011, P02032(2011). https://doi.org/10.1088/1742-5468/2011/02/P02032

[48] S. Kumar, H. Zhang, Y. P. Huang. Large-scale Ising emulation with four body interaction and all-to-all connections. Commun. Phys., 3, 108(2020). https://doi.org/10.1038/s42005-020-0376-5

[49] S. Kumar et al. Observation of distinct phase transitions in a nonlinear optical Ising machine. Commun. Phys., 6, 31(2023). https://doi.org/10.1038/s42005-023-01148-6

[50] D. Treves, S. Alexander. Observation of antisymmetric exchange interaction in yttrium orthoferrite. J. Appl. Phys., 33, 1133-1134(1962). https://doi.org/10.1063/1.1728631

[51] T. Moriya. Anisotropic superexchange interaction and weak ferromagnetism. Phys. Rev., 120, 91-98(1960). https://doi.org/10.1103/PhysRev.120.91

[52] E. Maguid et al. Topologically controlled intracavity laser modes based on Pancharatnam-Berry phase. ACS Photonics, 5, 1817-1821(2018). https://doi.org/10.1021/acsphotonics.7b01525

[53] D. Naidoo et al. Controlled generation of higher-order Poincaré sphere beams from a laser. Nat. Photonics, 10, 327-332(2016). https://doi.org/10.1038/nphoton.2016.37

[54] M. Piccardo et al. Vortex laser arrays with topological charge control and self-healing of defects. Nat. Photonics, 16, 359-365(2022). https://doi.org/10.1038/s41566-022-00986-0

[55] B. Wang et al. Visible-frequency dielectric metasurfaces for multi-wavelength achromatic and highly-dispersive holograms. Nano Lett., 16, 5235-5240(2016). https://doi.org/10.1021/acs.nanolett.6b02326

[56] Z. Bomzon et al. Space-variant Pancharatnam–Berry phase optical elements with computer-generated subwavelength gratings. Opt. Lett., 27, 1141-1143(2002). https://doi.org/10.1364/OL.27.001141

[57] E. Maguid et al. Photonic spin-controlled multifunctional shared-aperture antenna array. Science, 352, 1202-1206(2016). https://doi.org/10.1126/science.aaf3417

[58] M. Khorasaninejad, F. Capasso. Metalenses: versatile multifunctional photonic components. Science, 358, eaam8100(2017). https://doi.org/10.1126/science.aam8100

[59] C. Roques-Carmes et al. Heuristic recurrent algorithms for photonic Ising machines. Nat. Commun., 11, 249(2020). https://doi.org/10.1038/s41467-019-14096-z

[60] K. P. Kalinin et al. Analog iterative machine (AIM): using light to solve quadratic optimization problems with mixed variables.