The DOI matches its Crossref record; 184 of the 184 references it cites were found in a registry or an index.
Making sense of non-Hermitian Hamiltonians
Open accessAbstract
The Hamiltonian H specifies the energy levels and time evolution of a quantum theory. A standard axiom of quantum mechanics requires that H be Hermitian because Hermiticity guarantees that the energy spectrum is real and that time evolution is unitary (probability-preserving). This paper describes an alternative formulation of quantum mechanics in which the mathematical axiom of Hermiticity (transpose +complex conjugate) is replaced by the physically transparent condition of space–time reflection ( ) symmetry. If H has an unbroken symmetry, then the spectrum is real. Examples of -symmetric non-Hermitian quantum-mechanical Hamiltonians are and . Amazingly, the energy levels of these Hamiltonians are all real and positive! Does a -symmetric Hamiltonian H specify a physical quantum theory in which the norms of states are positive and time evolution is unitary? The answer is that if H has an unbroken symmetry, then it has another symmetry represented by a linear operator . In terms of , one can construct a time-independent inner product with a positive-definite norm. Thus, -symmetric Hamiltonians describe a new class of complex quantum theories having positive probabilities and unitary time evolution. The Lee model provides an excellent example of a -symmetric Hamiltonian. The renormalized Lee-model Hamiltonian has a negative-norm 'ghost' state because renormalization causes the Hamiltonian to become non-Hermitian. For the past 50 years there have been many attempts to find a physical interpretation for the ghost, but all such attempts failed. The correct interpretation of the ghost is simply that the non-Hermitian Lee-model Hamiltonian is -symmetric. The operator for the Lee model is calculated exactly and in closed form and the ghost is shown to be a physical state having a positive norm. The ideas of symmetry are illustrated by using many quantum-mechanical and quantum-field-theoretic models.
Overview
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Integrity signals
Registry
Cites
Statistics
Verification
Claims the paper makes
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
The cited source doesn’t support the claim as written.
In the citing paper
“In the so-called PT -symmetric formulation [39, 40, 41, 43], for example, an operator qualifies as an observable if it is simultaneously invariant with respect to both parity ( P ) and time-reversal ( T ) operations.”
From Fundamentals of quantum mechanics in Liouville space (2020) · cites this paper as [43]
Checked by a reasoning model — may be wrong.
Experimental — accuracy not yet measuredThe cited source says what the claim attributes to it.
In the citing paper
“In non-Hermitian quantum mechanics [39, 40, 41, 43], the formal definition of an operator representing a physical observable posits on entirely different criteria and are more relaxed with respect to the Hermiticity requirement in Hermitian quantum mechanics.”
From Fundamentals of quantum mechanics in Liouville space (2020) · cites this paper as [43]
In the citing paper
“This inner product, however, results in indefinite norms for the non-Hermitian, but PT -symmetric case [238].This observation will become crucial below, since this means that for the geometric approach the angle (67) will have to be re-defined.”
From Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control (2017) · cites this paper as [238]
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
The self-check needs the paper’s full text.
Citations
Cites
(1998) · doi:10.1103/physrevlett.80.5243 · Cited by 6,280
Self-citationAn integrable shallow water equation with peaked solitons
Camassa, R., & Holm, D. D. (1993) · Physical Review Letters · doi:10.1103/physrevlett.71.1661 · Cited by 3,732
Complex Extension of Quantum Mechanics
Bender, C. M., Brody, D. C., & Jones, H. F. (2002) · Physical Review Letters · doi:10.1103/physrevlett.89.270401 · Cited by 1,803
Self-citationMostafazadeh, A. (2002) · Journal of Mathematical Physics · doi:10.1063/1.1418246 · Cited by 1,645
Localization Transitions in Non-Hermitian Quantum Mechanics
Hatano, N., & Nelson, D. R. (1996) · Physical Review Letters · doi:10.1103/physrevlett.77.570 · Cited by 1,541
𝓟𝓣-symmetric quantum mechanics
Bender, C. M., Boettcher, S., & Meisinger, P. N. (1999) · Journal of Mathematical Physics · doi:10.1063/1.532860 · Cited by 1,379
Self-citationBender, C. M., & Wu, T. T. (1969) · Physical Review · doi:10.1103/physrev.184.1231 · Cited by 1,117
Self-citationDivergence of Perturbation Theory in Quantum Electrodynamics
Dyson, F. J. (1952) · Physical Review · doi:10.1103/physrev.85.631 · Cited by 1,017
Quantum Gravity In De Sitter SpacePreprint
Witten, E. (2001) · arXiv (Cornell University) · doi:10.48550/arxiv.hep-th/0106109 · Cited by 886
Mostafazadeh, A. (2002) · Journal of Mathematical Physics · doi:10.1063/1.1461427 · Cited by 849
Research works printed but not linked: 15
- Barton G, 1963, Introduction to Advanced Field Theory
- Streater R F, 1964, PCT, Spin and Statistics, and All That
- Fring A, 2007, J. Phys. A: Math. Theor.
- Bender C M, 1978, Advanced Mathematical Methods for Scientists and Engineers
- Bender C M, 2007, J. Math. Phys.
- Federov Y N, 2007, Physica
- Bleuler K, 1950, Helv. Phys. Acta., 23, 567
- Faria C F de M, 2006, Laser Phys.
- Ushveridze A G, 1993, Quasi-Exactly Solvable Models in Quantum Mechanics
- Turbiner A V, 1988, Sov. Phys.-JETP, 67, 230
- Curtright T, 2006, J. Math. Phys.
- Henkel M, 2003, Classical and Quantum Nonlinear Integrable Systems: Theory and Applications
- Källén G, 1955, Mat.-Fys. Medd., 30
- Schweber S S, 1961, An Introduction to Relativistic Quantum Field Theory
- Symanzik K, 1973, Nuovo Cim., 6, 77
Cited by
Sorted by citations
Items 301–400 of 1,001
Non-Hermitian chiral degeneracy of gated graphene metasurfaces
Baek, S., Park, S. H., Oh, D., Lee, K., Lee, S., Lim, H., Ha, T., Park, H. S., Zhang, S., Yang, L., et al. (2023) · Light Science & Applications · doi:10.1038/s41377-023-01121-6 · Cited by 73
Symmetry of Open Quantum Systems: Classification of Dissipative Quantum Chaos
Kawabata, K., Kulkarni, A., Li, J., Numasawa, T., & Ryu, S. (2023) · PRX Quantum · doi:10.1103/prxquantum.4.030328 · Cited by 90
Jarzynski Equality in PT -Symmetric Quantum Mechanics
Deffner, S., & Saxena, A. (2015) · Physical Review Letters · doi:10.1103/physrevlett.114.150601 · Cited by 73
Classical simulation of relativistic quantum mechanics in periodic optical structures
Longhi, S. (2011) · Applied Physics B · doi:10.1007/s00340-011-4628-7 · Cited by 72
Two-dimensional accessible solitons in PT-symmetric potentials
Zhong, W., Belić, M. R., & Huang, T. (2012) · Nonlinear Dynamics · doi:10.1007/s11071-012-0594-9 · Cited by 72
Three-dimensional topological solitons in PT-symmetric optical lattices
Kartashov, Y. V., Hang, C., Huang, G., & Torner, L. (2016) · Optica · doi:10.1364/optica.3.001048 · Cited by 72
Quantum Concepts in the Social, Ecological and Biological Sciences
Bagarello, F. (2019) · Cambridge University Press eBooks · doi:10.1017/9781108684781 · Cited by 71
Unambiguous scattering matrix for non-Hermitian systems
Novitsky, A. V., Lyakhov, D. A., Michels, D. L., Pavlov, A. A., Shalin, A. S., & Novitsky, D. V. (2020) · Physical Review A · doi:10.1103/physreva.101.043834 · Cited by 72
Ju, R., Xu, G., Xu, L., Qi, M., Wang, D., Cao, P., Xi, R., Shou, Y., Chen, H., Qiu, C., et al. (2023) · Advanced Materials · doi:10.1002/adma.202209123 · Cited by 72
Macroscopic magnetic structures with balanced gain and loss
Lee, J. M., Kottos, T., & Shapiro, B. (2015) · Physical Review B · doi:10.1103/physrevb.91.094416 · Cited by 72
Ghatak, A., & Das, T. (2018) · Physical review. B./Physical review. B · doi:10.1103/physrevb.97.014512 · Cited by 72
Non-Hermitian extension of gauge theories and implications for neutrino physics
Alexandre, J., Bender, C. M., & Millington, P. (2015) · Journal of High Energy Physics · doi:10.1007/jhep11(2015)111 · Cited by 71
Self-citationBerry curvature and quantum metric in N -band systems: An eigenprojector approach
Graf, A., & Piéchon, F. (2021) · Physical review. B./Physical review. B · doi:10.1103/physrevb.104.085114 · Cited by 89
Non-Hermitian Swanson model with a time-dependent metric
Fring, A., & Moussa, M. H. Y. (2016) · Physical Review A · doi:10.1103/physreva.94.042128 · Cited by 71
Proposal for enhanced photon blockade in parity-time-symmetric coupled microcavities
Li, J., Yu, R., & Wu, Y. (2015) · Physical Review A · doi:10.1103/physreva.92.053837 · Cited by 69
Parity-time symmetry in optical microcavity systems
Wen, J., Jiang, X., Jiang, L., & Xiao, M. (2018) · Journal of Physics B Atomic Molecular and Optical Physics · doi:10.1088/1361-6455/aae42f · Cited by 69
Light transport in PT -invariant photonic structures with hidden symmetries
Teimourpour, M. H., El‐Ganainy, R., Eisfeld, A., Szameit, A., & Christodoulides, D. N. (2014) · Physical Review A · doi:10.1103/physreva.90.053817 · Cited by 69
Analytic results for aPT-symmetric optical structure
Jones, H. F. (2012) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8113/45/13/135306 · Cited by 69
Enhancement of Second-Order Non-Hermitian Skin Effect by Magnetic Fields
Li, C.-A., Trauzettel, B., Neupert, T., & Zhang, S.-B. (2023) · Physical Review Letters · doi:10.1103/physrevlett.131.116601 · Cited by 71
Stable localized modes in asymmetric waveguides with gain and loss
Tsoy, E. N., Allayarov, I., & Abdullaev, F. K. (2014) · Optics Letters · doi:10.1364/ol.39.004215 · Cited by 69
On the quantisation of complex higher derivative theories and avoiding the Ostrogradsky ghost
Raidal, M., & Veermäe, H. (2017) · Nuclear Physics B · doi:10.1016/j.nuclphysb.2017.01.024 · Cited by 68
Synthetic exceptional points and unidirectional zero reflection in non-Hermitian acoustic systems
Shen, C., Li, J., Peng, X., & Cummer, S. A. (2018) · Physical Review Materials · doi:10.1103/physrevmaterials.2.125203 · Cited by 67
Xu, L., Wang, J., Dai, G., Yang, S., Yang, F., Wang, G., & Huang, J. (2020) · International Journal of Heat and Mass Transfer · doi:10.1016/j.ijheatmasstransfer.2020.120659 · Cited by 68
Non-Hermitian topological magnonics
Yu, T., Zou, J., Zeng, B., Rao, J., & Xia, K. (2024) · Physics Reports · doi:10.1016/j.physrep.2024.01.006 · Cited by 75
A note on the invariant periodic potential
Midya, B., Roy, B., & Roychoudhury, R. (2010) · Physics Letters A · doi:10.1016/j.physleta.2010.04.046 · Cited by 68
Quantum master equation with balanced gain and loss
Dast, D., Haag, D., Cartarius, H., & Wunner, G. (2014) · Physical Review A · doi:10.1103/physreva.90.052120 · Cited by 67
Geometric phase in PT -symmetric quantum mechanics
Gong, J., & Wang, Q. (2010) · Physical Review A · doi:10.1103/physreva.82.012103 · Cited by 68
Exact time dependence of solutions to the time-dependent Schrödinger equation
Lohe, M. A. (2008) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8113/42/3/035307 · Cited by 71
PT symmetry breaking in multilayers with resonant loss and gain locks light propagation direction
Novitsky, D. V., Karabchevsky, A., Lavrinenko, A. V., Shalin, A. S., & Novitsky, A. V. (2018) · Physical review. B./Physical review. B · doi:10.1103/physrevb.98.125102 · Cited by 67
PT spectroscopy of the Rabi problem
Joglekar, Y. N., Marathe, R., Durganandini, P., & Pathak, R. K. (2014) · Physical Review A · doi:10.1103/physreva.90.040101 · Cited by 67
Scattering theory with localized non-Hermiticities
Znojil, M. (2008) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.78.025026 · Cited by 67
Families of stationary modes in complex potentials
Konotop, V. V., & Zezyulin, D. A. (2014) · Optics Letters · doi:10.1364/ol.39.005535 · Cited by 67
4 Families of stationary modes in complex potentialsPreprint
Konotop, V. V., & Zezyulin, D. A. (2014) · Cited by 67
Nonlocal homogenization of PT -symmetric multilayered structures
Novitsky, D. V., Shalin, A. S., & Novitsky, A. V. (2019) · Physical Review A · doi:10.1103/physreva.99.043812 · Cited by 66
Arkhipov, I. I., Miranowicz, A., Stefano, O. D., Stassi, R., Savasta, S., Nori, F., & Özdemir, Ş. K. (2019) · Physical Review A · doi:10.1103/physreva.99.053806 · Cited by 66
Photon blockade in a double-cavity optomechanical system with nonreciprocal coupling
Wang, D., Bai, C., Liu, S., Zhang, S., & Wang, H. (2020) · New Journal of Physics · doi:10.1088/1367-2630/abaa8a · Cited by 66
Knot topology of exceptional point and non-Hermitian no-go theorem
Hu, H., Sun, S.-K., & Chen, S. (2022) · Physical Review Research · doi:10.1103/physrevresearch.4.l022064 · Cited by 67
Parity-Time Symmetry and Exceptional Points [Electromagnetic Perspectives]
Krasnok, A., Nefedkin, N. E., & Alù, A. (2021) · IEEE Antennas and Propagation Magazine · doi:10.1109/map.2021.3115766 · Cited by 66
Multiple quantum Mpemba effect: Exceptional points and oscillations
Chatterjee, A. K., Takada, S., & Hayakawa, H. (2024) · Physical Review A · doi:10.1103/physreva.110.022213 · Cited by 66
Stabilizing non-Hermitian systems by periodic driving
Gong, J., & Wang, Q. (2015) · Physical Review A · doi:10.1103/physreva.91.042135 · Cited by 66
Non-Hermiticity-induced flat band
Ramezani, H. (2017) · Physical Review A · doi:10.1103/physreva.96.011802 · Cited by 66
How to Compute Spectra with Error Control
Colbrook, M. J., Roman, B., & Hansen, A. C. (2019) · Physical Review Letters · doi:10.1103/physrevlett.122.250201 · Cited by 65
Deconstructing effective non-Hermitian dynamics in quadratic bosonic Hamiltonians
Flynn, V. P., Cobanera, E., & Viola, L. (2020) · New Journal of Physics · doi:10.1088/1367-2630/ab9e87 · Cited by 72
Simulating Exceptional Non-Hermitian Metals with Single-Photon Interferometry
Wang, K., Xiao, L., Budich, J. C., Yi, W., & Xue, P. (2021) · Physical Review Letters · doi:10.1103/physrevlett.127.026404 · Cited by 68
Signatures of Liouvillian Exceptional Points in a Quantum Thermal Machine
Khandelwal, S., Brunner, N., & Haack, G. (2021) · PRX Quantum · doi:10.1103/prxquantum.2.040346 · Cited by 79
Parity-Time Symmetry in Acoustics: Theory, Devices, and Potential Applications
Fleury, R., Sounas, D. L., & Alù, A. (2016) · IEEE Journal of Selected Topics in Quantum Electronics · doi:10.1109/jstqe.2016.2549512 · Cited by 65
Emergence of PT-symmetry breaking in open quantum systems
Huber, J., Kirton, P., Rotter, S., & Rabl, P. (2020) · SciPost Physics · doi:10.21468/scipostphys.9.4.052 · Cited by 64
Eigenvalue structure of a Bose–Einstein condensate in a $\mathcal {PT}$-symmetric double well
Dast, D., Haag, D., Cartarius, H., Main, J., & Wunner, G. (2013) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8113/46/37/375301 · Cited by 64
PT symmetry via electromagnetically induced transparency
Li, H., Dou, J.-P., & Huang, G. (2013) · Optics Express · doi:10.1364/oe.21.032053 · Cited by 64
Exciting flavored bound states
Rojas, E., El-Bennich, B., & de Melo, J. P. B. C. (2014) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.90.074025 · Cited by 64
Bender, C. M., & Mannheim, P. D. (2008) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8113/41/30/304018 · Cited by 64
Self-citationQuantum Interference, Graphs, Walks, and Polynomials
Tsuji, Y., Estrada, E., Movassagh, R., & Hoffmann, R. (2018) · Chemical Reviews · doi:10.1021/acs.chemrev.7b00733 · Cited by 64
Asymmetric scattering of flexural waves in a parity-time symmetric metamaterial beam
Wu, Q., Chen, Y., & Huang, G. (2019) · The Journal of the Acoustical Society of America · doi:10.1121/1.5116561 · Cited by 64
High-order exceptional points in supersymmetric arrays
Zhang, S. M., Zhang, X. Z., Jin, L., & Song, Z. (2020) · Physical Review A · doi:10.1103/physreva.101.033820 · Cited by 63
Nearby states in non-Hermitian quantum systems I: Two states
Eleuch, H., & Rotter, I. (2015) · The European Physical Journal D · doi:10.1140/epjd/e2015-60389-7 · Cited by 63
Parity‐Time‐Symmetric Optical Lattice with Alternating Gain and Loss Atomic Configurations
Zhang, Z., Yang, L., Feng, J., Sheng, J., Zhang, Y., Zhang, Y., & Xiao, M. (2018) · Laser & Photonics Review · doi:10.1002/lpor.201800155 · Cited by 63
Asymmetric diffraction based on a passive parity-time grating
Zhu, X.-Y., Xu, Y.-L., Zou, Y., Sun, X., He, C., Lu, M., Liu, X.-P., & Chen, Y. (2016) · Applied Physics Letters · doi:10.1063/1.4962639 · Cited by 62
Barashenkov, I. V., & Gianfreda, M. (2014) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8113/47/28/282001 · Cited by 62
Non-Hermitian photonic lattices: tutorial
Wang, Q., & Chong, Y. D. (2023) · Journal of the Optical Society of America B · doi:10.1364/josab.481963 · Cited by 64
Non-Hermitian topological phases: principles and prospects
Banerjee, A., Sarkar, R., Dey, S., & Narayan, A. (2023) · Journal of Physics Condensed Matter · doi:10.1088/1361-648x/acd1cb · Cited by 70
Spectral singularities in PT -symmetric periodic finite-gap systems
Correa, F., & Plyushchay, M. S. (2012) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.86.085028 · Cited by 62
Goos-Hänchen shift in complex crystals
Longhi, S., Valle, G. D., & Staliūnas, K. (2011) · Physical Review A · doi:10.1103/physreva.84.042119 · Cited by 62
Non-Hermitian Kitaev chain with complex on-site potentials
Zeng, Q.-B., Zhu, B., Chen, S., You, L., & Lü, R. (2016) · Physical Review A · doi:10.1103/physreva.94.022119 · Cited by 62
Asymmetric light diffraction of an atomic grating with PT symmetry
Liu, Y.-M., Gao, F., Fan, C.-H., & Wu, J. (2017) · Optics Letters · doi:10.1364/ol.42.004283 · Cited by 62
Geometric Origin of Non-Bloch P T Symmetry Breaking
Hu, Y.-M., Wang, H., Wang, Z., & Song, F. (2024) · Physical Review Letters · doi:10.1103/physrevlett.132.050402 · Cited by 61
Anomalous Transient Amplification of Waves in Non-normal Photonic Media
Makris, K. G., Ge, L., & Türeci, H. E. (2014) · Physical Review X · doi:10.1103/physrevx.4.041044 · Cited by 63
PT -symmetric quantum field theory in D dimensions
Bender, C. M., Hassanpour, N., Klevansky, S. P., & Sarkar, S. (2018) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.98.125003 · Cited by 57
Self-citationBound states, scattering states, and resonant states in PT -symmetric open quantum systems
Garmon, S. S., Gianfreda, M., & Hatano, N. (2015) · Physical Review A · doi:10.1103/physreva.92.022125 · Cited by 59
Speeding Up Entanglement Generation by Proximity to Higher-Order Exceptional Points
Li, Z.-Z., Chen, W., Abbasi, M., Murch, K. W., & Whaley, K. B. (2023) · Physical Review Letters · doi:10.1103/physrevlett.131.100202 · Cited by 60
Hang, C., Li, W., & Huang, G. (2019) · Physical Review A · doi:10.1103/physreva.100.043807 · Cited by 60
Clustering of exceptional points and dynamical phase transitions
Eleuch, H., & Rotter, I. (2016) · Physical Review A · doi:10.1103/physreva.93.042116 · Cited by 59
Time correlation functions for non-Hermitian quantum systems
Sergi, A., & Zloshchastiev, K. G. (2015) · Physical Review A · doi:10.1103/physreva.91.062108 · Cited by 59
Quantum noise effects with Kerr-nonlinearity enhancement in coupled gain-loss waveguides
He, B., Yan, S., Wang, J., & Xiao, M. (2015) · Physical Review A · doi:10.1103/physreva.91.053832 · Cited by 58
Li, J., Yu, R., Ding, C., & Wu, Y. (2016) · Physical Review A · doi:10.1103/physreva.93.023814 · Cited by 59
Yang, Y., Jia, H., Bi, Y., Zhao, H., & Yang, J. (2019) · Physical Review Applied · doi:10.1103/physrevapplied.12.034040 · Cited by 59
Non-Hermitian topological metamaterials with odd elasticity
Zhou, D., & Zhang, J. (2020) · Physical Review Research · doi:10.1103/physrevresearch.2.023173 · Cited by 60
Symmetry protected topological phases characterized by isolated exceptional points
Lin, S., Jin, L., & Song, Z. (2019) · Physical review. B./Physical review. B · doi:10.1103/physrevb.99.165148 · Cited by 58
Scale-free localization and PT symmetry breaking from local non-Hermiticity
Li, B., Wang, H.-R., Song, F., & Wang, Z. (2023) · Physical review. B./Physical review. B · doi:10.1103/physrevb.108.l161409 · Cited by 58
Complex Correspondence Principle
Bender, C. M., Hook, D. W., Meisinger, P. N., & Wang, Q. (2010) · Physical Review Letters · doi:10.1103/physrevlett.104.061601 · Cited by 58
Self-citationRelativistic non-Hermitian quantum mechanics
Jones-Smith, K., & Mathur, H. (2014) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.89.125014 · Cited by 57
Finite-size effects in non-Hermitian topological systems
Chen, R., Chen, C., Zhou, B., & Xu, D.-H. (2019) · Physical review. B./Physical review. B · doi:10.1103/physrevb.99.155431 · Cited by 58
Bell-state generation for spin qubits via dissipative coupling
Zou, J., Zhang, S., & Tserkovnyak, Y. (2022) · Physical review. B./Physical review. B · doi:10.1103/physrevb.106.l180406 · Cited by 61
Experimental Demonstration of Controllable PT and Anti- PT Coupling in a Non-Hermitian Metamaterial
Li, C., Yang, R., Huang, X., Fu, Q., Fan, Y., & Zhang, F. (2024) · Physical Review Letters · doi:10.1103/physrevlett.132.156601 · Cited by 56
Transverse periodic PT symmetry for modal demultiplexing in optical waveguides
Benisty, H., Lupu, A., & Degiron, A. (2015) · Physical Review A · doi:10.1103/physreva.91.053825 · Cited by 57
Semi-rational solutions for the (2+1) -dimensional nonlocal Fokas system
Cao, Y., Rao, J., MIHALACHE, D. B., & He, J. (2018) · Applied Mathematics Letters · doi:10.1016/j.aml.2017.12.026 · Cited by 57
Non-Hermitian generalizations of extended Su–Schrieffer–Heeger models
He, Y., & Chien, C.-C. (2020) · Journal of Physics Condensed Matter · doi:10.1088/1361-648x/abc974 · Cited by 57
Fundamental limits for reciprocal and nonreciprocal non-Hermitian quantum sensing
Bao, L., Qi, B., Dong, D., & Nori, F. (2021) · Physical Review A · doi:10.1103/physreva.103.042418 · Cited by 59
Topological phase transition independent of system non-Hermiticity
Zhang, K. L., Wu, H., Jin, L., & Song, Z. (2019) · Physical review. B./Physical review. B · doi:10.1103/physrevb.100.045141 · Cited by 55
Dimensional transmutation in gravity and cosmology
Salvio, A. (2021) · International Journal of Modern Physics A · doi:10.1142/s0217751x21300064 · Cited by 61
Complex PT -symmetric nonlinear Schrödinger equation and Burgers equation
Yan, Z. (2013) · Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences · doi:10.1098/rsta.2012.0059 · Cited by 56
Competition between band topology and non-Hermiticity
Cheng, J., Zhang, X., Lu, M., & Chen, Y. (2022) · Physical review. B./Physical review. B · doi:10.1103/physrevb.105.094103 · Cited by 55
Evading the sign problem in the mean-field approximation through Lefschetz-thimble path integral
Tanizaki, Y., Nishimura, H., & Kashiwa, K. (2015) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.91.101701 · Cited by 55
PT symmetry as a necessary and sufficient condition for unitary time evolution
Mannheim, P. D. (2013) · Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences · doi:10.1098/rsta.2012.0060 · Cited by 55
Chen, Y., Yan, Z., MIHALACHE, D. B., & Malomed, B. A. (2017) · Scientific Reports · doi:10.1038/s41598-017-01401-3 · Cited by 54
Quantum effects in classical systems having complex energy
Bender, C. M., Brody, D., & Hook, D. (2008) · UCL Discovery (University College London) · Cited by 55
Self-citationCornelius, J., Xu, Z., Saxena, A., Chenu, A., & del Campo, A. (2022) · Physical Review Letters · doi:10.1103/physrevlett.128.190402 · Cited by 56
Time-dependent Hamiltonians with 100% evolution speed efficiency
Uzdin, R., Günther, U., Rahav, S., & Moiseyev, N. (2012) · Journal of Physics A: Mathematical and Theoretical · doi:10.1088/1751-8113/45/41/415304 · Cited by 54
Variational quantum algorithm for nonequilibrium steady states
Yoshioka, N., Nakagawa, Y. O., Mitarai, K., & Fujii, K. (2020) · Physical Review Research · doi:10.1103/physrevresearch.2.043289 · Cited by 54
Quantum simulation of parity–time symmetry breaking with a superconducting quantum processor
Dogra, S., Melnikov, A. A., & Paraoanu, G. S. (2021) · Communications Physics · doi:10.1038/s42005-021-00534-2 · Cited by 54
Unidirectional invisibility in a two-layer non-PT-symmetric slab
Shen, Y., Deng, X. H., & Chen, L. (2014) · Optics Express · doi:10.1364/oe.22.019440 · Cited by 54
Metadata
- Authors
- Bender, C. M.
- Affiliations
- Los Alamos National Laboratory (US)
- Corresponding author
- Bender, Carl M
- ORCID
- Bender, Carl M
- Venue
- Reports on Progress in Physics
- Publisher
- IOP Publishing
- Published
- 2007-05-30
- Volume
- 70
- Issue
- 6
- Pages
- 947-1018
- ISSN
- 0034-4885 (ISSN-L)
- 1361-6633
- OpenAlex
- W1997323544
- Cited by
- 2,984
- References
- 162
- Citations by year
- 2026: 167 · 2025: 204 · 2024: 237 · 2023: 233 · 2022: 260 · 2021: 199 · 2020: 211 · 2019: 198 · 2018: 183 · 2017: 172 · 2016: 170 · 2015: 137 · 2014: 128 · 2013: 111 · 2012: 113
- FWCI
- 38.76
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Reports on Progress in Physics (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- U.S. Department of Energy
- Los Alamos National Laboratory
- Topics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Non-Hermitian Physics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
- Physical Sciences › Mathematics › Mathematical Physics › Spectral Theory in Mathematical Physics
- Keywords
- non-Hermitian Hamiltonian, PT symmetry, real energy spectrum, unitary evolution, Lee model, ghost states, renormalization
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
