Abstract

The topological phases in materials have been studied in recent decades for their unique boundary states and transport properties. Photonic systems with band structures embrace the topological phases closely, where they not only provide platforms to testify the topological band theory, but also shed light on designing novel optical devices. In this review, we present exciting developments, supported by brief descriptions of prominent milestones of topological phases in photonic systems in recent years. These studies may sustain further developments of optical devices and offer novel methods for light manipulations.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2018 (19)

Y. Yang, Y. F. Xu, T. Xu, H. X. Wang, J. H. Jiang, X. Hu, and Z. H. Hang, “Visualization of a Unidirectional Electromagnetic Waveguide Using Topological Photonic Crystals Made of Dielectric Materials,” Phys. Rev. Lett. 120(21), 217401 (2018).
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Y. Yang, H. Jiang, and Z. H. Hang, “Topological Valley Transport in Two-dimensional Honeycomb Photonic Crystals,” Sci. Rep. 8(1), 1588 (2018).
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X. Ni, D. Purtseladze, D. A. Smirnova, A. Slobozhanyuk, A. Alù, and A. B. Khanikaev, “Spin- and valley-polarized one-way Klein tunneling in photonic topological insulators,” Sci. Adv. 4(5), eaap8802 (2018).
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B. Yang, Q. Guo, B. Tremain, R. Liu, L. E. Barr, Q. Yan, W. Gao, H. Liu, Y. Xiang, and J. Chen, “Ideal Weyl points and helicoid surface states in artificial photonic crystal structures,” Science ( 11), eaaq1221 (2018).

Q. Yan, R. Liu, Z. Yan, B. Liu, H. Chen, Z. Wang, and L. Lu, “Experimental discovery of nodal chains,” Nat. Phys. 14(5), 461–464 (2018).
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W. Gao, B. Yang, B. Tremain, H. Liu, Q. Guo, L. Xia, A. P. Hibbins, and S. Zhang, “Experimental observation of photonic nodal line degeneracies in metacrystals,” Nat. Commun. 9(1), 950 (2018).
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H. Shen, B. Zhen, and L. Fu, “Topological band theory for non-Hermitian Hamiltonians,” Phys. Rev. Lett. 120(14), 146402 (2018).
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T. Bar, S. K. Choudhary, M. A. Ashraf, K. S. Sujith, S. Puri, S. Raj, and B. Bansal, “Kinetic Spinodal Instabilities in the Mott Transition in V_{2}O_{3}: Evidence from Hysteresis Scaling and Dissipative Phase Ordering,” Phys. Rev. Lett. 121(4), 045701 (2018).
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H. Zhou, C. Peng, Y. Yoon, C. W. Hsu, K. A. Nelson, L. Fu, J. D. Joannopoulos, M. Soljačić, and B. Zhen, “Observation of bulk Fermi arc and polarization half charge from paired exceptional points,” Science 11, 9859 (2018).

E. Khalaf, “Higher-order topological insulators and superconductors protected by inversion symmetry,” Phys. Rev. B 97(20), 205136 (2018).
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F. Schindler, A. M. Cook, M. G. Vergniory, Z. Wang, S. S. Parkin, B. A. Bernevig, and T. Neupert, “Higher-order topological insulators,” Phys. Rev. B 97(20), 205136 (2018).
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M. Geier, L. Trifunovic, M. Hoskam, and P. W. Brouwer, “Second-order topological insulators and superconductors with an order-two crystalline symmetry,” Phys. Rev. B 97(20), 205135 (2018).
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R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14(1), 11–19 (2018).
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C. W. Peterson, W. A. Benalcazar, T. L. Hughes, and G. Bahl, “A quantized microwave quadrupole insulator with topologically protected corner states,” Nature 555(7696), 346–350 (2018).
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J. Noh, W. A. Benalcazar, S. Huang, M. J. Collins, K. P. Chen, T. L. Hughes, and M. C. Rechtsman, “Topological protection of photonic mid-gap defect modes,” Nat. Photon.,  12(7) 408-415 (2018).

M. Pan, H. Zhao, P. Miao, S. Longhi, and L. Feng, “Photonic zero mode in a non-Hermitian photonic lattice,” Nat. Commun. 9(1), 1308 (2018).
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F. Zhong, J. Li, H. Liu, and S. Zhu, “Controlling surface plasmons through covariant transformation of the spin-dependent geometric phase between curved metamaterials,” Phys. Rev. Lett. 120(24), 243901 (2018).
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2017 (22)

G. Salerno, T. Ozawa, H. M. Price, and I. Carusotto, “Propagating edge states in strained honeycomb lattices,” Phys. Rev. B 95(24), 245418 (2017).
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Y. Xu, S. T. Wang, and L. M. Duan, “Weyl Exceptional Rings in a Three-Dimensional Dissipative Cold Atomic Gas,” Phys. Rev. Lett. 118(4), 045701 (2017).
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W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, “Quantized electric multipole insulators,” Science 357(6346), 61–66 (2017).
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W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, “Electric multipole moments, topological multipole moment pumping, and chiral hinge states in crystalline insulators,” Phys. Rev. B 96(24), 245115 (2017).
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J. Y. Lin, N. C. Hu, Y. J. Chen, C. H. Lee, and X. Zhang, “Line nodes, Dirac points, and Lifshitz transition in two-dimensional nonsymmorphic photonic crystals,” Phys. Rev. B 96(7), 075438 (2017).
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S. Longhi, “Parity-time symmetry meets photonics: A new twist in non-Hermitian optics,” EPL 120(6), 64001 (2017).
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Z. Yan, R. Bi, H. Shen, L. Lu, S. C. Zhang, and Z. Wang, “Nodal-link semimetals,” Phys. Rev. B 96(4), 041103 (2017).
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R. Bi, Z. Yan, L. Lu, and Z. Wang, “Nodal-knot semimetals,” Phys. Rev. B 96(20), 201305 (2017).
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B. Yang, Q. Guo, B. Tremain, L. E. Barr, W. Gao, H. Liu, B. Béri, Y. Xiang, D. Fan, A. P. Hibbins, and S. Zhang, “Direct observation of topological surface-state arcs in photonic metamaterials,” Nat. Commun. 8(1), 97 (2017).
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H. X. Wang, Y. Chen, Z. H. Hang, H. Y. Kee, and J. H. Jiang, “Type-II Dirac photons,” npj Quantum Materials 2(1), 54 (2017).
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J. Noh, S. Huang, D. Leykam, Y. D. Chong, K. P. Chen, and M. C. Rechtsman, “Experimental observation of optical Weyl points and Fermi arc-like surface states,” Nat. Phys. 13(6), 611–617 (2017).
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C. Liu, W. Gao, B. Yang, and S. Zhang, “Disorder-Induced Topological State Transition in Photonic Metamaterials,” Phys. Rev. Lett. 119(18), 183901 (2017).
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M. L. Chang, M. Xiao, W. J. Chen, and C. T. Chan, “Multiple Weyl points and the sign change of their topological charges in woodpile photonic crystals,” Phys. Rev. B 95(12), 125136 (2017).
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F. Gao, H. Xue, Z. Yang, K. Lai, Y. Yu, X. Lin, Y. Chong, G. Shvets, and B. Zhang, “Topologically protected refraction of robust kink states in valley photonic crystals,” Nat. Phys. 14(2), 140–144 (2017).
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S. Mukherjee, A. Spracklen, M. Valiente, E. Andersson, P. Öhberg, N. Goldman, and R. R. Thomson, “Experimental observation of anomalous topological edge modes in a slowly driven photonic lattice,” Nat. Commun. 8, 13918 (2017).
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F. Liu and K. Wakabayashi, “Novel topological phase with a zero Berry curvature,” Phys. Rev. Lett. 118(7), 076803 (2017).
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A. Slobozhanyuk, S. H. Mousavi, X. Ni, D. Smirnova, Y. S. Kivshar, and A. B. Khanikaev, “Three-dimensional all-dielectric photonic topological insulator,” Nat. Photonics 11(2), 130–136 (2017).
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A. B. Khanikaev and G. Shvets, “Two-dimensional topological photonics,” Nat. Photonics 11(12), 763–773 (2017).
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S. Mansha and Y. Chong, “Robust edge states in amorphous gyromagnetic photonic lattices,” Phys. Rev. B 96(12), 121405 (2017).
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M. Xiao and S. Fan, “Photonic Chern insulator through homogenization of an array of particles,” Phys. Rev. B 96(10), 100202 (2017).
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Z. Yang, M. Xiao, F. Gao, L. Lu, Y. Chong, and B. Zhang, “Weyl points in a magnetic tetrahedral photonic crystal,” Opt. Express 25(14), 15772–15777 (2017).
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2016 (26)

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C. He, X. C. Sun, X. P. Liu, M. H. Lu, Y. Chen, L. Feng, and Y.-F. Chen, “Photonic topological insulator with broken time-reversal symmetry,” Proc. Natl. Acad. Sci. U.S.A. 113(18), 4924–4928 (2016).
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K. Shiozaki, M. Sato, and K. Gomi, “Topology of nonsymmorphic crystalline insulators and superconductors,” Phys. Rev. B 93(19), 195413 (2016).
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A. Bansil, H. Lin, and T. Das, “Colloquium: Topological band theory,” Rev. Mod. Phys. 88(2), 021004 (2016).
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C. He, X. Ni, H. Ge, X. C. Sun, Y.-B. Chen, M.-H. Lu, X.-P. Liu, and Y.-F. Chen, “Acoustic topological insulator and robust one-way sound transport,” Nat. Phys. 12(12), 1124–1129 (2016).
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K. Lai, T. Ma, X. Bo, S. Anlage, and G. Shvets, “Experimental realization of a reflections-free compact delay line based on a photonic topological insulator,” Sci. Rep. 6(1), 28453 (2016).
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B. Xiao, K. Lai, Y. Yu, T. Ma, G. Shvets, and S. M. Anlage, “Exciting reflectionless unidirectional edge modes in a reciprocal photonic topological insulator medium,” Phys. Rev. B 94(19), 195427 (2016).
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2015 (24)

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S. Y. Xu, I. Belopolski, N. Alidoust, M. Neupane, G. Bian, C. Zhang, R. Sankar, G. Chang, Z. Yuan, C. C. Lee, S. M. Huang, H. Zheng, J. Ma, D. S. Sanchez, B. Wang, A. Bansil, F. Chou, P. P. Shibayev, H. Lin, S. Jia, and M. Z. Hasan, “Discovery of a Weyl fermion semimetal and topological Fermi arcs,” Science 349(6248), 613–617 (2015).
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T. Ma, A. B. Khanikaev, S. H. Mousavi, and G. Shvets, “Guiding electromagnetic waves around sharp corners: topologically protected photonic transport in metawaveguides,” Phys. Rev. Lett. 114(12), 127401 (2015).
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2014 (7)

S. A. Skirlo, L. Lu, and M. Soljačić, “Multimode one-way waveguides of large Chern numbers,” Phys. Rev. Lett. 113(11), 113904 (2014).
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H. Pan, Z. Li, C.-C. Liu, G. Zhu, Z. Qiao, and Y. Yao, “Valley-polarized quantum anomalous Hall effect in silicene,” Phys. Rev. Lett. 112(10), 106802 (2014).
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Y. Zhao and Z. Wang, “Topological connection between the stability of Fermi surfaces and topological insulators and superconductors,” Phys. Rev. B 89(7), 075111 (2014).
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M. Hafezi, “Measuring topological invariants in photonic systems,” Phys. Rev. Lett. 112(21), 210405 (2014).
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2013 (17)

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T. Morimoto and A. Furusaki, “Topological classification with additional symmetries from Clifford algebras,” Phys. Rev. B 88(12), 125129 (2013).
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2012 (7)

M. Hafezi and P. Rabl, “Optomechanically induced non-reciprocity in microring resonators,” Opt. Express 20(7), 7672–7684 (2012).
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2011 (13)

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2010 (6)

C. He, X. L. Chen, M. H. Lu, X. F. Li, W. W. Wan, X. S. Qian, R. C. Yin, and Y. F. Chen, “Tunable one-way cross-waveguide splitter based on gyromagnetic photonic crystal,” Appl. Phys. Lett. 96(11), 111111 (2010).
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C. He, X. L. Chen, M. H. Lu, X. F. Li, W. W. Wan, X. S. Qian, R. C. Yin, and Y. F. Chen, “Left-handed and right-handed one-way edge modes in a gyromagnetic photonic crystal,” J. Appl. Phys. 107(12), 123117 (2010).
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S. Liu, W. Lu, Z. Lin, and S. Chui, “Magnetically controllable unidirectional electromagnetic waveguiding devices designed with metamaterials,” Appl. Phys. Lett. 97(20), 201113 (2010).
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2009 (2)

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009).
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2008 (3)

S. Raghu and F. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78(3), 033834 (2008).
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Figures (6)

Fig. 1
Fig. 1 The experimental setup and numerical simulation results. (a). The layout of the PC consisted of gyromagnetic rods which are confined by metal walls. (b). The photograph of the PC with the top plate removed. (c). Numerical simulation shows that there is a unidirectional edge state propagating along the interface without backscattering which is robust against an obstacle. (d). The transmission spectra of the PC without confinement and the projected PC band structure. There is a clear suppress of transmission at the frequencies in the PBG. The red line represents the gapless edge state. (e). Transmission spectra of the confined PC where the spectra are non-reciprocal which shows the edge states propagate unidirectionally at the interface. (a)-(e) are reprinted with permission from [31]. Copyright 2009, Springer Nature.
Fig. 2
Fig. 2 QSHE in CROW. (a). Two coupled resonators with different lengths of the upper and lower branches. (b). If we introduce scatterers in resonators and waveguides, we can achieve in-plane magnetic field and spin-flip hopping terms. (c). 2D CROW which can be described by a tight-binding model. (d). Demonstration of forward- and backward-propagating edge states with different pseudo-spin components. The edge states are robust against disorders. (e). Projected band structure of CROW which shows two edge states in the band gaps. (a)-(e) are reprinted with permission from [60]. Copyright 2013, Springer Nature.
Fig. 3
Fig. 3 QSHE in all-dielectric PCs. (a). Schematic plot of PC with the lattice vectors a 1 and a 2 , the lattice constant a 0 , the dielectric constants of rods and background ε d and ε A , the diameter of rods d . (b). Band inversion process induced by reducing the lattice constant from extended phase to shrunk phase. (c). Numerical calculation of the projected band structures shows that there are two topological edge states with opposite group velocities emerging at the band gap. (d). Real space distribution of the E z field at point A and B indicated in (c). Two pseudo-spin components propagate along opposite directions. (e). Experimental setup of PC with extended and shrunk configurations. A square-shaped antenna array is used to selectively excite certain EM pseudo-spin state indicated by circular arrows. (f). Clockwise pseudo-spin state is excited and propagates along the interface of two topologically inequivalent configurations where the edge states are robust against sharp corners. (g). Transmission measured at two different positions indicated in (f) which shows that there is no backscattering of the edge states. (a)-(d) are reprinted with permission from [70]. Copyright 2015, American Physical Society. (e)-(g) are reprinted with permission from [71]. Copyright 2018, American Physical Society.
Fig. 4
Fig. 4 Floquet topological gapped phases in PCs. (a). Image of the PC at the input facet. (b). Schematic of the helical waveguides. (c). Band structure of the PC when rotation radius is not zero where there is a bandgap. (d). Projected band structure of (c). There are two edge states with non-zero group velocity. (e). The group velocity of edge states is relevant to the radius of rotation. (a)-(e) are reprinted with permission from [92]. Copyright 2013, Springer Nature.
Fig. 5
Fig. 5 WPs in PCs. (a). Double-gyroid (DG) structure of PC and the Brillouin zone (BZ). A sphere defect may be introduced in the structure to break T or P symmetry. (b). Band structures along a certain path in BZ of PC with different symmetries. The Weyl points (WPs) appear when P or T symmetry is broken. (c). Schematic of a saddle-shaped metallic inclusion and its BZ with WPs and surface-state arcs. (d). Band structures with ideal WPs. (e). Helicoid surface states plotted using Jacobi elliptic function. The arcs with different colors represent the evolution of equi-frequency arcs which connect WPs with opposite chirality. (a)-(b) are reprinted with permission from [106]. Copyright 2013, Springer Nature. (c)-(e) are reprinted with permission from [118]. Copyright 2018 Springer Science
Fig. 6
Fig. 6 Second-order photonic topological insulator with corner states. (a). Schematic of the PC and its Brillouin zone. (b). The eigenmodes of a square supercell as shown in (c). There are four degenerate states in the middle of the PBG. (c). Corner states which are strongly localized. (a)-(c) are reprinted from [157].

Tables (2)

Tables Icon

Table 1 - Classification of topological insulators and topological superconductors with respect to time-reversal symmetry (T), particle-hole symmetry (C) and chiral symmetry (S). Number 0 (1) for S means no (have) chiral symmetry. Since time-reversal operator and particle-hole operators are anti-unitary operators and square to ± 1 , there are three types: even, odd and absent which are represented by + 1, −1 and 0 respectively. d represents the spatial dimension of the system. The notation of ten classes follows the notation invented by Cartan, Altland and Zirnbauer (CAZ) [28, 29]. The entries 0, , 2 and 2 represents the topological invariants of each Hamiltonians which correspond to 0, an integer, an integer of mod 2 and an even integer respectively. Table. 1 is reproduced with permission from [18]. Copyright 2010, IOP Publishing.

Tables Icon

Table 2 Topological classification of gapless structures. p is the codimension of a gapless structure. There are totally ten distinct topological classes in which the topological charges have 8-fold periodicity with respect to codimensions of the gapless structures. The topological charges are defined as same as those in Table. 1. Table. 2 is reproduced with permission from [95]. Copyright 2013, American Physical Society.

Equations (11)

Equations on this page are rendered with MathJax. Learn more.

C n = 1 2 π B Z d 2 k F n
T H ( k ) T 1 = H ( k ) C H ( k ) C 1 = H ( k ) S H ( k ) S 1 = H ( k )
× [ μ 1 ( r ) × E ] = ϵ ( r ) ω 2 E
A n n ( k ) = i E n k | k | E n k
C n = 1 2 π i B Z d 2 k ( A y n n k x A x n n k y )
F x y ( k ) = A y n n k x A x n n k y
μ = [ μ i κ 0 i κ μ 0 0 0 μ 0 ]
H 0 = κ ( σ , x , y a σ x + 1 , y a σ x , y e i 2 π α y σ + a σ x , y a σ x + 1 , y e i 2 π α y σ + a σ x , y + 1 a σ x , y + a σ x , y a σ x , y + 1 )
i z ψ ( x , y , z ) = 1 2 k 0 2 ψ ( x , y , z ) k 0 Δ n ( x , y , z ) n 0 ψ ( x , y , z )
H D P = ν ( q x σ x + q y σ y )
H W P = ν ( q x σ x + q y σ y + q z σ z )

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