[1] |
De Cipra, B. A., An introduction to the Ising model, Am. Math. Monthly, 94, 937 (1987) · doi:10.1080/00029890.1987.12000742 |
[2] |
Binder, K., Finite size scaling analysis of ising model block distribution functions Z, Phys. B Condens. Matter, 43, 119 (1981) · doi:10.1007/BF01293604 |
[3] |
Newell, G. F.; Montroll, E. W., On the theory of the Ising model of ferromagnetism, Rev. Mod. Phys., 25, 353 (1953) · Zbl 0053.18601 · doi:10.1103/RevModPhys.25.353 |
[4] |
Pfeuty, P., The one-dimensional Ising model with a transverse field, Ann. Phys., 57, 79 (1970) · doi:10.1016/0003-4916(70)90270-8 |
[5] |
Elliott, R. J.; Wood, C., The Ising model with a transverse field. I. High temperature expansion, J. Phys. C: Solid St. Phys., 4, 2359 (1971) · doi:10.1088/0022-3719/4/15/023 |
[6] |
Pfeuty, P.; Elliott, R. J., The Ising model with a transverse field. II. Ground state properties, J. Phys. C: Solid St. Phys., 4, 2370 (1971) · doi:10.1088/0022-3719/4/15/024 |
[7] |
Stinchcombe, R. B., Ising model in a transverse field. I. Basic theory, J. Phys. C: Solid St. Phys., 6, 2459 (1973) · doi:10.1088/0022-3719/6/15/009 |
[8] |
Elliott, R. J.; Pfeuty, P.; Wood, C., Ising model with a transverse field, Phys. Rev. Lett., 25, 443 (1970) · doi:10.1103/PhysRevLett.25.443 |
[9] |
Sachdev, S., Quantum Phase Transitions (1999), Cambridge, UK: Cambridge University Press, Cambridge, UK |
[10] |
Carr, L. D., Understanding Quantum Phase Transitions (2010), Boca Raton: CRC Press, Boca Raton |
[11] |
Suzuki, S.; Inoue, J.; Chakrabarti, B. K., Quantum Ising Phases and Transitions in Transverse Ising Models (2012), Heidelberg: Springer, Heidelberg |
[12] |
Heyl, M.; Polkovnikov, A.; Kehrein, S., Dynamical quantum phase transitions in the transverse-field ising model, Phys. Rev. Lett., 110 (2013) · doi:10.1103/PhysRevLett.110.135704 |
[13] |
Bender, C. M., Making sense of non-Hermitian Hamiltonians, Rep. Prog. Phys., 70, 947 (2007) · doi:10.1088/0034-4885/70/6/R03 |
[14] |
Ashida, Y.; Gong, Z.; Ueda, M., Non-hermitian physics, Adv. Phys., 69, 249 (2020) · doi:10.1080/00018732.2021.1876991 |
[15] |
Brody, D. C., Biorthogonal quantum mechanics, J. Phys. A: Math. Theor., 47 (2014) · Zbl 1283.81070 · doi:10.1088/1751-8113/47/3/035305 |
[16] |
Heiss, W. D., Exceptional points of non-Hermitian operators, J. Phys. A: Math. Theor., 37, 2455-2467 (2004) · Zbl 1044.81039 · doi:10.1088/0305-4470/37/6/034 |
[17] |
Heiss, W. D., The physics of exceptional points, J. Phys. A: Math. Theor., 45 (2012) · Zbl 1263.81163 · doi:10.1088/1751-8113/45/44/444016 |
[18] |
Bender, C. M., Real spectra in non-Hermitian Hamiltonians having PT-symmetry, Phys. Rev. Lett., 80, 5243-5246 (1998) · Zbl 0947.81018 · doi:10.1103/PhysRevLett.80.5243 |
[19] |
Bender, C. M.; Brody, D. C.; Jones, H. F., Complex extension of quantum mechanics, Phys. Rev. Lett., 89 (2002) · Zbl 1267.81234 · doi:10.1103/PhysRevLett.89.270401 |
[20] |
Ahmed, Z., Pseudo-hermiticity of hamiltonians under gauge-like transformation: real spectrum of non-hermitian hamiltonians, Phys. Lett. A, 294, 287-291 (2002) · Zbl 0990.81029 · doi:10.1016/S0375-9601(02)00124-X |
[21] |
Mostafazadeh, A.; Mostafazadeh, A.; Mostafazadeh, A., Pseudo-Hermiticity versus PT symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian. Pseudo-Hermiticity versus PT-symmetry. II. A complete characterization of non-Hermitian Hamiltonians with a real spectrum. Pseudo-Hermiticity versus PT-symmetry III: Equivalence of pseudo-Hermiticity and the presence of antilinear symmetries, J. Math. Phys.. J. Math. Phys.. J. Math. Phys., 43, 3944 (2002) · Zbl 1061.81075 · doi:10.1063/1.1489072 |
[22] |
Rüer, C. E.; Makris, K. G.; El-Ganainy, R.; Christodoulides, D. N.; Segev, M.; Kip, D., Observation of parity-time symmetry in optics, Nat. Phys., 6, 192 (2010) · doi:10.1038/nphys1515 |
[23] |
Hang, C.; Huang, G.; Konotop, V. V., PT-symmetry with a system of three-Level atoms, Phys. Rev. Lett., 110 (2013) · doi:10.1103/PhysRevLett.110.083604 |
[24] |
Peng, P.; Cao1, W.; Shen, C.; Qu, W.; Wen, J.; Jiang, L.; Xiao, Y., Anti-parity-time symmetry with flying atoms, Nat. Phys., 12, 1139-1145 (2016) · doi:10.1038/nphys3842 |
[25] |
Zhang, Z.; Zhang, Z.; Zhang, Y.; Sheng, J.; Yang, L.; Miri, M-A; Christodoulides, D. N.; He, B.; Zhang, Y.; Xiao, M., Observation of Parity-Time symmetry in optically induced atomic lattices, Phys. Rev. Lett., 117 (2016) · doi:10.1103/PhysRevLett.117.123601 |
[26] |
Bender, N.; Factor, S.; Bodyfelt, J. D.; Ramezani, H.; Christodoulides, D. N.; Ellis, F. M.; Kottos, T., Observation of asymmetric transport in structures with active nonlinearities, Phys. Rev. Lett., 110 (2013) · doi:10.1103/PhysRevLett.110.234101 |
[27] |
Assawaworrarit, S.; Yu, X.; Fan, S., Robust wireless power transfer using a nonlinear parity-time-symmetric circuit, Nature, 546, 387 (2017) · doi:10.1038/nature22404 |
[28] |
Choi, Y.; Hahn, C.; Yoon, J. W.; Song, S. H., Observation of an anti-PT-symmetric exceptional point and energy-difference conserving dynamics in electrical circuit resonators, Nat. Commun., 9, 2182 (2018) · doi:10.1038/s41467-018-04690-y |
[29] |
Bittner, S.; Dietz, B.; G¨ther, U.; Harney, H. L.; Miski-Oglu, M.; Richter, A.; Schfer, F., PT-symmetry and spontaneous symmetry breaking in a microwave billiard, Phys. Rev. Lett., 108 (2012) · doi:10.1103/PhysRevLett.108.024101 |
[30] |
Zhu, X.; Ramezani, H.; Shi, C.; Zhu, J.; Zhang, X., PT-symmetric acoustics, Phys. Rev. X, 4 (2014) · doi:10.1103/PhysRevX.4.031042 |
[31] |
Popa, B-I; Cummer, S. A., Non-reciprocal and highly nonlinear active acoustic metamaterials, Nat. Commun., 5, 3398 (2014) · doi:10.1038/ncomms4398 |
[32] |
Fleury, R.; Sounas, D.; Alù, A., An invisible acoustic sensor based on parity-time symmetry, Nat. Commun., 6, 5905 (2015) · doi:10.1038/ncomms6905 |
[33] |
Wu, Y.; Liu, W.; Geng, J.; Song, X.; Ye, X.; Duan, C-K; Rong, X.; Du, J., Observation of parity-time symmetry breaking in a single-spin system, Science, 364, 878 (2019) · Zbl 1431.82002 · doi:10.1126/science.aaw8205 |
[34] |
Kunst, F. K.; Edvardsson, E.; Budich, J. C.; Bergholtz, E. J., Biorthogonal bulk-boundary correspondence in non-Hermitian systems, Phys. Rev. Lett., 121 (2018) · doi:10.1103/PhysRevLett.121.026808 |
[35] |
Yokomizo, K.; Murakami, S., Non-Bloch band theory of non-Hermitian systems, Phys. Rev. Lett., 123 (2019) · doi:10.1103/PhysRevLett.123.066404 |
[36] |
Xiao, L.; Deng, T.; Wang, K.; Zhu, G.; Wang, Z.; Yi, W.; Xue, P., Non-Hermitian bulk¨Cboundary correspondence in quantum dynamics, Nat. Phys., 16, 761-766 (2020) · doi:10.1038/s41567-020-0836-6 |
[37] |
Yang, Z.; Zhang, K.; Fang, C.; Hu, J., Non-Hermitian bulk-boundary correspondence and auxiliary generalized Brillouin zone theory, Phys. Rev. Lett., 125 (2020) · doi:10.1103/PhysRevLett.125.226402 |
[38] |
Xiong, Y., Why does bulk boundary correspondence fail in some non-hermitian topological models, J. Phys. Commun., 2 (2018) · doi:10.1088/2399-6528/aab64a |
[39] |
Yao, S.; Song, F.; Wang, Z., Non-hermitian chern bands, Phys. Rev. Lett., 121 (2018) · doi:10.1103/PhysRevLett.121.136802 |
[40] |
Wang, X-R; Guo, C-X; Kou, S-P, Defective edge states and number-anomalous bulk-boundary correspondence in non-Hermitian topological systems, Phys. Rev. B, 101 (2020) · doi:10.1103/PhysRevB.101.121116 |
[41] |
Yao, S.; Wang, Z., Edge states and topological invariants of non-hermitian systems, Phys. Rev. Lett., 121 (2018) · doi:10.1103/PhysRevLett.121.086803 |
[42] |
Song, F.; Yao, S.; Wang, Z., Non-Hermitian skin effect and chiral damping in open quantum systems, Phys. Rev. Lett., 123 (2019) · doi:10.1103/PhysRevLett.123.170401 |
[43] |
Okuma, N.; Kawabata, K.; Shiozaki, K.; Sato, M., Topological origin of non-Hermitian skin effects, Phys. Rev. Lett., 124 (2020) · doi:10.1103/PhysRevLett.124.086801 |
[44] |
Du, Q.; Cao, K.; Kou, S-P, Physics of PT-symmetric quantum systems at finite temperatures, Phys. Rev. A, 106 (2022) · doi:10.1103/PhysRevA.106.032206 |
[45] |
Deguchi, T.; Ghosh, P. K., The exactly solvable quasi-Hermitian transverse Ising model, J. Phys. A, 42 (2009) · Zbl 1183.82012 · doi:10.1088/1751-8113/42/47/475208 |
[46] |
Kitaev, A. Y., Unpaired Majorana fermions in quantum wires, Phys. Usp., 44, 131 (2001) · doi:10.1070/1063-7869/44/10S/S29 |
[47] |
Li, C.; Zhang, X. Z.; Zhang, G.; Song, Z., Topological phases in a Kitaev chain with imbalanced pairing, Phys. Rev. B, 97 (2018) · doi:10.1103/PhysRevB.97.115436 |
[48] |
Zhao, X-M; Guo, C-X; Yang, M-L; Wang, H.; Liu, W-M; Kou, S-P, Anomalous non-Abelian statistics for non-Hermitian generalization of Majorana zero modes, Phys. Rev. B, 104 (2021) · doi:10.1103/PhysRevB.104.214502 |
[49] |
Bunder, J. E.; McKenzie, R. H., Effect of disorder on quantum phase transitions in anisotropic XY spin chains in a transverse field, Phys. Rev. B, 60, 344 (1999) · doi:10.1103/PhysRevB.60.344 |
[50] |
Campbell, S.; Richens, J.; Gullo, N. L.; Busch, T., Criticality, factorization, and long-range correlations in the anisotropic XY model, Phys. Rev. A, 88 (2013) · doi:10.1103/PhysRevA.88.062305 |
[51] |
Luo, Q.; Zhao, J.; Wang, X., Fidelity susceptibility of the anisotropic XY model: The exact solution, Phys. Rev. A, 98 (2018) · doi:10.1103/PhysRevE.98.022106 |
[52] |
Lieb, E.; Schultz, T.; Mattis, D., Two soluble models of an antiferromagnetic chain, Ann. Phys., 16, 407 (1961) · Zbl 0129.46401 · doi:10.1016/0003-4916(61)90115-4 |