Biorthogonal quantum criticality in non-Hermitian many-body systems

Gaoyong Sun, Jia-Chen Tang, Su-Peng Kou

PDF(1093 KB)
PDF(1093 KB)
Front. Phys. ›› 2022, Vol. 17 ›› Issue (3) : 33502. DOI: 10.1007/s11467-021-1126-1
RESEARCH ARTICLE

Biorthogonal quantum criticality in non-Hermitian many-body systems

Author information +
History +

Abstract

We develop the perturbation theory of the fidelity susceptibility in biorthogonal bases for arbitrary interacting non-Hermitian many-body systems with real eigenvalues. The quantum criticality in the non-Hermitian transverse field Ising chain is investigated by the second derivative of the ground-state energy and the ground-state fidelity susceptibility. We show that the system undergoes a second-order phase transition with the Ising universal class by numerically computing the critical points and the critical exponents from the finite-size scaling theory. Interestingly, our results indicate that the biorthogonal quantum phase transitions are described by the biorthogonal fidelity susceptibility instead of the conventional fidelity susceptibility.

Graphical abstract

Keywords

biorthogonal quantum criticality / non-Hermitian systems / fidelity susceptibility

Cite this article

Download citation ▾
Gaoyong Sun, Jia-Chen Tang, Su-Peng Kou. Biorthogonal quantum criticality in non-Hermitian many-body systems. Front. Phys., 2022, 17(3): 33502 https://doi.org/10.1007/s11467-021-1126-1

References

[1]
S. Sachdev , Quantum Phase Transitions, Cambridge University Press, 1999
[2]
M. Levin and X. G. Wen , Detecting topological order in a ground state wave function, Phys. Rev. Lett. 96 (11), 110405 (2006)
CrossRef ADS Google scholar
[3]
M. E. Fisher and M. N. Barber , Scaling theory for finitesize effects in the critical region, Phys. Rev. Lett. 28, 1516 (1972)
CrossRef ADS Google scholar
[4]
M. E. Fisher , The renormalization group in the theory of critical behavior, Rev. Mod. Phys. 46 (4), 597 (1974)
CrossRef ADS Google scholar
[5]
E. J. Bergholtz , J. C. Budich , and F. K. Kunst , Exceptional topology of non-Hermitian systems, Rev. Mod. Phys. 93 (1), 015005 (2021)
CrossRef ADS Google scholar
[6]
Y. Ashida , Z. Gong , and M. Ueda , Non-Hermitian physics, Adv. Phys. 69 (3), 249 (2020)
CrossRef ADS Google scholar
[7]
T. E. Lee , Anomalous edge state in a non-Hermitian lattice, Phys. Rev. Lett. 116 (13), 133903 (2016)
CrossRef ADS Google scholar
[8]
S. Yao , and Z. Wang , Edge states and topological invariants of non-Hermitian systems, Phys. Rev. Lett. 121 (8), 086803 (2018)
CrossRef ADS Google scholar
[9]
F. K. Kunst , E. Edvardsson , J. C. Budich , and E. J. Bergholtz , Biorthogonal bulk-boundary correspondence in non-Hermitian systems, Phys. Rev. Lett. 121 (2), 026808 (2018)
CrossRef ADS Google scholar
[10]
Y. Xiong , Why does bulk boundary correspondence fail in some non-Hermitian topological models, J. Phys. Commun. 2 (3), 035043 (2018)
CrossRef ADS Google scholar
[11]
Z. Gong , Y. Ashida , K. Kawabata , K. Takasan , S. Higashikawa , and M. Ueda , Topological phases of nonHermitian systems, Phys. Rev. X 8 (3), 031079 (2018)
CrossRef ADS Google scholar
[12]
V. M. M. Alvarez , J. E. B. Vargas , and L. E. F. F. Torres , Non-Hermitian robust edge states in one dimension: Anomalous localization and eigenspace condensation at exceptional points, Phys. Rev. B 97, 121401(R) (2018)
CrossRef ADS Google scholar
[13]
K. Yokomizo and S. Murakami , Non-Bloch band theory of non-Hermitian systems, Phys. Rev. Lett. 123 (6), 066404 (2019)
CrossRef ADS Google scholar
[14]
N. Okuma , K. Kawabata , K. Shiozaki , and M. Sato , Topological origin of non-Hermitian skin effects, Phys. Rev. Lett. 124 (8), 086801 (2020)
CrossRef ADS Google scholar
[15]
K. Zhang , Z. Yang , and C. Fang , Correspondence between winding numbers and skin modes in non-Hermitian systems, Phys. Rev. Lett. 125 (12), 126402 (2020)
CrossRef ADS Google scholar
[16]
Z. Yang , K. Zhang , C. Fang , and J. Hu , Non-Hermitian bulk-boundary correspondence and auxiliary generalized Brillouin zone theory, Phys. Rev. Lett. 125 (22), 226402 (2020)
CrossRef ADS Google scholar
[17]
X.-R. Wang , C.-X. Guo , and S.-P. Kou , Defective edge states and number-anomalous bulk-boundary correspondence in non-Hermitian topological systems, Phys. Rev. B 101, 121116(R) (2020)
CrossRef ADS Google scholar
[18]
H. Jiang , R. Lü , and S. Chen , Topological invariants , zero mode edge states and finite size effect for a generalized non-reciprocal Su–Schrieffer–Heeger model, Eur. Phys. J. B 93 (7), 125 (2020)
CrossRef ADS Google scholar
[19]
S. Weidemann , M. Kremer , T. Helbig , T. Hofmann , A. Stegmaier , M. Greiter , R. Thomale , and A. Szameit , Topological funneling of light, Science 368 (6488), 311 (2020)
CrossRef ADS Google scholar
[20]
L. Xiao , T. Deng , K. Wang , G. Zhu , Z. Wang , W. Yi , and P. Xue , Observation of non-Hermitian bulk-boundary correspondence in quantum dynamics, Nat. Phys. 16, 761 (2020)
CrossRef ADS Google scholar
[21]
D. S. Borgnia , A. J. Kruchkov , and R. J. Slager , NonHermitian boundary modes and topology, Phys. Rev. Lett. 124 (5), 056802 (2020)
CrossRef ADS Google scholar
[22]
W. Heiss , The physics of exceptional points, J. Phys. A Math. Theor. 45 (44), 444016 (2012)
CrossRef ADS Google scholar
[23]
V. Kozii and L. Fu , Non-Hermitian topological theory of finite-lifetime quasiparticles: Prediction of bulk Fermi arc due to exceptional point, arXiv: 1708.05841 (2017)
[24]
H. Hodaei , A. U. Hassan , S. Wittek , H. Garcia-Gracia , R. El-Ganainy , D. N. Christodoulides , and M. Khajavikhan , Enhanced sensitivity at higher-order exceptional points, Nature 548 (7666), 187 (2017)
CrossRef ADS Google scholar
[25]
H. Zhou , C. Peng , Y. Yoon , C. W. Hsu , K. A. Nelson , L. Fu , J. D. Joannopoulos , M. Soljacic , and B. Zhen , Observation of bulk Fermi arc and polarization half charge from paired exceptional points, Science 359 (6379), 1009 (2018)
CrossRef ADS Google scholar
[26]
M. A. Miri and A. Alu , Exceptional points in optics and photonics, Science 363 (6422), eaar7709 (2019)
CrossRef ADS Google scholar
[27]
J. H. Park , A. Ndao , W. Cai , L. Y. Hsu , A. Kodigala , T. Lepetit , Y. H. Lo , and B. Kanté , Observation of plasmonic exceptional points, arXiv: 1904.01073 (2019)
[28]
Z. Yang and J. Hu , Non-Hermitian Hopf-link exceptional line semimetals, Phys. Rev. B 99, 081102(R) (2019)
CrossRef ADS Google scholar
[29]
S. Özdemir , S. Rotter , F. Nori , and L. Yang , Parity–time symmetry and exceptional points in photonics, Nat. Mater. 18 (8), 783 (2019)
CrossRef ADS Google scholar
[30]
B. Dóra , M. Heyl , and R. Moessner , The Kibble–Zurek mechanism at exceptional points, Nat. Commun. 10 (1), 2254 (2019)
CrossRef ADS Google scholar
[31]
Y. R. Zhang , Z. Z. Zhang , J. Q. Yuan , M. Kang , and J. Chen , High-order exceptional points in non-Hermitian Moiré lattices, Front. Phys. 14 (5), 53603 (2019)
CrossRef ADS Google scholar
[32]
L. Jin , H. C. Wu , B. B. Wei , and Z. Song , Hybrid exceptional point created from type-Ⅲ Dirac point, Phys. Rev. B 101 (4), 045130 (2020)
CrossRef ADS Google scholar
[33]
L. Xiao , T. Deng , K. Wang , Z. Wang , W. Yi , and P. Xue , Observation of non-Bloch parity-time symmetry and exceptional points, Phys. Rev. Lett. 126 (23), 230402 (2021)
CrossRef ADS Google scholar
[34]
N. Matsumoto , K. Kawabata , Y. Ashida , S. Furukawa , and M. Ueda , Continuous phase transition without gap closing in non-Hermitian quantum many-body systems, Phys. Rev. Lett. 125 (26), 260601 (2020)
CrossRef ADS Google scholar
[35]
M. L. Yang , H. Wang , C. X. Guo , X. R. Wang , G. Sun , and S. P. Kou , Anomalous spontaneous symmetry breaking in non-Hermitian systems with biorthogonal Z2-symmetry, arXiv: 2006.10278 (2020)
[36]
L. Jin and Z. Song , Scaling behavior and phase diagram of a PT-symmetric non-Hermitian Bose–Hubbard system, Ann. Phys. 330, 142 (2013)
CrossRef ADS Google scholar
[37]
Y. Ashida , S. Furukawa , and M. Ueda , Parity–timesymmetric quantum critical phenomena, Nat. Commun. 8 (1), 15791 (2017)
CrossRef ADS Google scholar
[38]
L. Herviou , N. Regnault , and J. H. Bardarson , Entanglement spectrum and symmetries in non-Hermitian fermionic non-interacting models, SciPost Physics 7 (5), 069 (2019)
CrossRef ADS Google scholar
[39]
P. Y. Chang , J. S. You , X. Wen , and S. Ryu , Entanglement spectrum and entropy in topological non-Hermitian systems and nonunitary conformal field theory, Phys. Rev. Res. 2 (3), 033069 (2020)
CrossRef ADS Google scholar
[40]
S. Mu , C. H. Lee , L. Li , and J. Gong , Emergent Fermi surface in a many-body non-Hermitian fermionic chain, Phys. Rev. B 102, 081115(R) (2020)
CrossRef ADS Google scholar
[41]
E. Lee , H. Lee , and B.-J. Yang , Many-body approach to non-Hermitian physics in fermionic systems, Phys. Rev. B 101, 121109(R) (2020)
CrossRef ADS Google scholar
[42]
L. Pan , X. Chen , Y. Chen , and H. Zhai , Non-Hermitian linear response theory, Nat. Phys. 16 (7), 767 (2020)
CrossRef ADS Google scholar
[43]
L. Pan , X. Wang , X. Cui , and S. Chen , Interactioninduced dynamical PT-symmetry breaking in dissipative Fermi–Hubbard models, Phys. Rev. A 102 (2), 023306 (2020)
CrossRef ADS Google scholar
[44]
Z. Xu and S. Chen , Topological Bose–Mott insulators in one-dimensional non-Hermitian superlattices, Phys. Rev. B 102 (3), 035153 (2020)
CrossRef ADS Google scholar
[45]
D. W. Zhang , Y. L. Chen , G. Q. Zhang , L. J. Lang , Z. Li , and S. L. Zhu , Skin superfluid, topological Mott insulators, and asymmetric dynamics in an interacting non-Hermitian Aubry–André–Harper model, Phys. Rev. B 101 (23), 235150 (2020)
CrossRef ADS Google scholar
[46]
C. H. Lee , Many-body topological and skin states without open boundaries, arXiv: 2006.01182 (2020)
[47]
H. Shackleton and M. S. Scheurer , Protection of paritytime symmetry in topological many-body systems: NonHermitian toric code and fracton models, Phys. Rev. Res. 2 (3), 033022 (2020)
CrossRef ADS Google scholar
[48]
T. Liu , J. J. He , T. Yoshida , Z. L. Xiang , and F. Nori , Non-Hermitian topological Mott insulators in one-dimensional fermionic superlattices, Phys. Rev. B 102 (23), 235151 (2020)
CrossRef ADS Google scholar
[49]
K. Yang , S. C. Morampudi , and E. J. Bergholtz , Exceptional spin liquids from couplings to the environment, Phys. Rev. Lett. 126 (7), 077201 (2021)
CrossRef ADS Google scholar
[50]
R. Hanai , A. Edelman , Y. Ohashi , and P. B. Littlewood , Non-Hermitian phase transition from a polariton Bose– Einstein condensate to a photon laser, Phys. Rev. Lett. 122 (18), 185301 (2019)
CrossRef ADS Google scholar
[51]
R. Hamazaki , K. Kawabata , and M. Ueda , NonHermitian many-body localization, Phys. Rev. Lett. 123 (9), 090603 (2019)
CrossRef ADS Google scholar
[52]
W. Xi , Z. H. Zhang , Z. C. Gu , and W. Q. Chen , Classification of topological phases in one dimensional interacting non-Hermitian systems and emergent unitarity, Sci. Bull. (Beijing) 66 (17), 1731 (2021)
CrossRef ADS Google scholar
[53]
K. Yamamoto , M. Nakagawa , K. Adachi , K. Takasan , M. Ueda , and N. Kawakami , Theory of non-Hermitian fermionic superfluidity with a complex-valued interaction, Phys. Rev. Lett. 123 (12), 123601 (2019)
CrossRef ADS Google scholar
[54]
R. Hanai and P. B. Littlewood , Critical fluctuations at a many-body exceptional point, Phys. Rev. Res. 2 (3), 033018 (2020)
CrossRef ADS Google scholar
[55]
R. Arouca , C. H. Lee , and C. M. Smith , Unconventional scaling at non-Hermitian critical points, Phys. Rev. B 102 (24), 245145 (2020)
CrossRef ADS Google scholar
[56]
P. Zanardi and N. Paunkovic , Ground state overlap and quantum phase transitions, Phys. Rev. E 74 (3), 031123 (2006)
CrossRef ADS Google scholar
[57]
L. Campos Venuti and P. Zanardi , Quantum critical scaling of the geometric tensors, Phys. Rev. Lett. 99 (9), 095701 (2007)
CrossRef ADS Google scholar
[58]
W. L. You , Y. W. Li , and S. J. Gu , Fidelity, dynamic structure factor, and susceptibility in critical phenomena, Phys. Rev. E 76 (2), 022101 (2007)
CrossRef ADS Google scholar
[59]
A. F. Albuquerque , F. Alet , C. Sire , and S. Capponi , Quantum critical scaling of fidelity susceptibility, Phys. Rev. B 81 (6), 064418 (2010)
CrossRef ADS Google scholar
[60]
S. J. Gu , Fidelity approach to quantum phase transitions, Int. J. Mod. Phys. B 24 (23), 4371 (2010)
CrossRef ADS Google scholar
[61]
G. Sun , Fidelity susceptibility study of quantum longrange antiferromagnetic Ising chain, Phys. Rev. A 96 (4), 043621 (2017)
CrossRef ADS Google scholar
[62]
Z. Zhu , G. Sun , W. L. You , and D. N. Shi , Fidelity and criticality of a quantum Ising chain with long-range interactions, Phys. Rev. A 98 (2), 023607 (2018)
CrossRef ADS Google scholar
[63]
B. B. Wei and X. C. Lv , Fidelity susceptibility in the quantum Rabi model, Phys. Rev. A 97 (1), 013845 (2018)
CrossRef ADS Google scholar
[64]
B. B. Wei , Fidelity susceptibility in one-dimensional disordered lattice models, Phys. Rev. A 99 (4), 042117 (2019)
CrossRef ADS Google scholar
[65]
S. Chen , L. Wang , Y. Hao , and Y. Wang , Intrinsic relation between ground-state fidelity and the characterization of a quantum phase transition, Phys. Rev. A 77 (3), 032111 (2008)
CrossRef ADS Google scholar
[66]
S. J. Gu , H. M. Kwok , W. Q. Ning , and H. Q. Lin , Fidelity susceptibility, scaling, and universality in quantum critical phenomena, Phys. Rev. B 77 (24), 245109 (2008)
CrossRef ADS Google scholar
[67]
S. Yang , S. J. Gu , C. P. Sun , and H. Q. Lin , Fidelity susceptibility and long-range correlation in the Kitaev honeycomb model, Phys. Rev. A 78 (1), 012304 (2008)
CrossRef ADS Google scholar
[68]
H. M. Kwok , W. Q. Ning , S. J. Gu , and H. Q. Lin , Quantum criticality of the Lipkin–Meshkov–Glick model in terms of fidelity susceptibility, Phys. Rev. E 78 (3), 032103 (2008)
CrossRef ADS Google scholar
[69]
L. Gong and P. Tong , Fidelity, fidelity susceptibility, and von Neumann entropy to characterize the phase diagram of an extended Harper model, Phys. Rev. B 78 (11), 115114 (2008)
CrossRef ADS Google scholar
[70]
W. C. Yu , H. M. Kwok , J. Cao , and S. J. Gu , Fidelity susceptibility in the two-dimensional transverse-field Ising and XXZ models, Phys. Rev. E 80 (2), 021108 (2009)
CrossRef ADS Google scholar
[71]
D. Schwandt , F. Alet , and S. Capponi , Quantum Monte Carlo simulations of fidelity at magnetic quantum phase transitions, Phys. Rev. Lett. 103 (17), 170501 (2009)
CrossRef ADS Google scholar
[72]
Q. Luo , J. Zhao , and X. Wang , Fidelity susceptibility of the anisotropic XY model: The exact solution, Phys. Rev. E 98 (2), 022106 (2018)
CrossRef ADS Google scholar
[73]
M. M. Rams and B. Damski , Quantum fidelity in the thermodynamic limit, Phys. Rev. Lett. 106 (5), 055701 (2011)
CrossRef ADS Google scholar
[74]
S. H. Li , Q. Q. Shi , Y. H. Su , J. H. Liu , Y. W. Dai , and H. Q. Zhou , Tensor network states and ground-state fidelity for quantum spin ladders, Phys. Rev. B 86 (6), 064401 (2012)
CrossRef ADS Google scholar
[75]
V. Mukherjee , A. Dutta , and D. Sen , Quantum fidelity for one-dimensional Dirac fermions and two-dimensional Kitaev model in the thermodynamic limit, Phys. Rev. B 85 (2), 024301 (2012)
CrossRef ADS Google scholar
[76]
B. Damski , Fidelity susceptibility of the quantum Ising model in a transverse field: The exact solution, Phys. Rev. E 87 (5), 052131 (2013)
CrossRef ADS Google scholar
[77]
J. Carrasquilla , S. R. Manmana , and M. Rigol , Scaling of the gap , Scaling of the gap, fidelity susceptibility, and Bloch oscillations across the superfluid-to-Mott-insulator transition in the one-dimensional Bose–Hubbard model, Phys. Rev. A 87 (4), 043606 (2013)
CrossRef ADS Google scholar
[78]
M. Łącki , B. Damski , and J. Zakrzewski , Numerical studies of ground-state fidelity of the Bose–Hubbard model, Phys. Rev. A 89 (3), 033625 (2014)
CrossRef ADS Google scholar
[79]
G. Sun and T. Vekua , Topological quasi-one-dimensional state of interacting spinless electrons, Phys. Rev. B 93 (20), 205137 (2016)
CrossRef ADS Google scholar
[80]
M.-F. Yang , Ground-state fidelity in one-dimensional gapless models, Phys. Rev. B 76, 180403(R) (2007)
CrossRef ADS Google scholar
[81]
J. O. Fjærestad , Ground state fidelity of Luttinger liquids: A wavefunctional approach, J. Stat. Mech. 2008 (07), P07011 (2008)
CrossRef ADS Google scholar
[82]
A. Langari and A. Rezakhani , Quantum renormalization group for ground-state fidelity, New J. Phys. 14 (5), 053014 (2012)
CrossRef ADS Google scholar
[83]
G. Sun , A. K. Kolezhuk , and T. Vekua , Fidelity at Berezinskii–Kosterlitz–Thouless quantum phase transitions, Phys. Rev. B 91 (1), 014418 (2015)
CrossRef ADS Google scholar
[84]
L. Cincio , M. M. Rams , J. Dziarmaga , and W. H. Zurek , Universal shift of the fidelity susceptibility peak away from the critical point of the Berezinskii–Kosterlitz–Thouless quantum phase transition, Phys. Rev. B 100, 081108(R) (2019)
CrossRef ADS Google scholar
[85]
G. Sun , B. B. Wei , and S. P. Kou , Fidelity as a probe for a deconfined quantum critical point, Phys. Rev. B 100 (6), 064427 (2019)
CrossRef ADS Google scholar
[86]
H. Jiang , C. Yang , and S. Chen , Topological invariants and phase diagrams for one-dimensional two-band nonHermitian systems without chiral symmetry, Phys. Rev. A 98 (5), 052116 (2018)
CrossRef ADS Google scholar
[87]
C. Wang , M. L. Yang , C. X. Guo , X. M. Zhao , and S. P. Kou , Effective non-Hermitian physics for degenerate ground states of a non-Hermitian Ising model with RT symmetry, EPL (Europhysics Letters) 128 (4), 41001 (2020)
CrossRef ADS Google scholar
[88]
C. X. Guo , X. R. Wang , and S. P. Kou , Non-Hermitian avalanche effect: Non-perturbative effect induced by local non-Hermitian perturbation on a Z2 topological order, EPL (Europhysics Letters) 131 (2), 27002 (2020)
CrossRef ADS Google scholar
[89]
Y. Nishiyama , Imaginary-field-driven phase transition for the 2D Ising antiferromagnet: A fidelity-susceptibility approach, Physica A 555, 124731 (2020)
CrossRef ADS Google scholar
[90]
Y. Nishiyama , Fidelity-susceptibility analysis of the honeycomb-lattice Ising antiferromagnet under the imaginary magnetic field, Eur. Phys. J. B 93 (9), 174 (2020)
CrossRef ADS Google scholar
[91]
Y. C. Tzeng , C. Y. Ju , G. Y. Chen , and W. M. Huang , Hunting for the non-Hermitian exceptional points with fidelity susceptibility, Phys. Rev. Res. 3 (1), 013015 (2021)
CrossRef ADS Google scholar
[92]
D. D. Solnyshkov , C. Leblanc , L. Bessonart , A. Nalitov , J. Ren , Q. Liao , F. Li , and G. Malpuech , Quantum metric and wave packets at exceptional points in non-Hermitian systems, Phys. Rev. B 103 (12), 125302 (2021)
CrossRef ADS Google scholar
[93]
D. C. Brody , Biorthogonal quantum mechanics, J. Phys. A Math. Theor. 47 (3), 035305 (2014)
CrossRef ADS Google scholar
[94]
M. M. Sternheim and J. F. Walker , Non-Hermitian Hamiltonians, decaying states, and perturbation theory, Phys. Rev. C 6 (1), 114 (1972)
CrossRef ADS Google scholar
[95]
A. Mostafazadeh , Pseudo-hermiticity versus PT symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian, J. Math. Phys. 43 (1), 205 (2002)
CrossRef ADS Google scholar
[96]
A. Mostafazadeh , Pseudo-hermiticity versus PT-symmetry (Ⅱ): A complete characterization of non-Hermitian Hamiltonians with a real spectrum, J. Math. Phys. 43 (5), 2814 (2002)
CrossRef ADS Google scholar
[97]
A. Mostafazadeh , Pseudo-hermiticity versus PT-symmetry (Ⅲ): Equivalence of pseudo-hermiticity and the presence of antilinear symmetries, J. Math. Phys. 43 (8), 3944 (2002)
CrossRef ADS Google scholar
[98]
Y. Y. Fu , Y. Fei , D. X. Dong , and Y. W. Liu , Photonic spin Hall effect in PT-symmetric metamaterials, Front. Phys. 14 (6), 62601 (2019)
CrossRef ADS Google scholar
[99]
Y. Zhao , Equivariant PT-symmetric real Chern insulators, Front. Phys. 15 (1), 13603 (2020)
CrossRef ADS Google scholar
[100]
Y. C. Chen , M. Gong , P. Xue , H. D. Yuan , and C. J. Zhang , Quantum deleting and cloning in a pseudounitary system, Front. Phys. 16 (5), 53601 (2021)
CrossRef ADS Google scholar
[101]
A. Uhlmann , The “transition probability” in the state space of a ∗-algebra, Rep. Math. Phys. 9 (2), 273 (1976)
CrossRef ADS Google scholar
[102]
M. Hauru and G. Vidal , Uhlmann fidelities from tensor networks, Phys. Rev. A 98 (4), 042316 (2018)
CrossRef ADS Google scholar
[103]
G. Gehlen , Critical and off-critical conformal analysis of the Ising quantum chain in an imaginary field, J. Phys. Math. Gen. 24 (22), 5371 (1991)
CrossRef ADS Google scholar
[104]
D. Bianchini , O. Castro-Alvaredo , B. Doyon , E. Levi , and F. Ravanini , Entanglement entropy of non-unitary conformal field theory, J. Phys. A Math. Theor. 48 (4), 04FT01 (2015)
CrossRef ADS Google scholar
[105]
K. L. Zhang and Z. Song , Ising chain with topological degeneracy induced by dissipation, Phys. Rev. B 101 (24), 245152 (2020)
CrossRef ADS Google scholar
[106]
J. Um , S. I. Lee , and B. J. Kim , Quantum phase transition and finite-size scaling of the one-dimensional Ising model, J. Korean Phys. Soc. 50, 285 (2007)
[107]
W. L. You and W. L. Lu , Scaling of reduced fidelity susceptibility in the one-dimensional transverse-field XY model, Phys. Lett. A 373 (16), 1444 (2009)
CrossRef ADS Google scholar
[108]
N. Hatano and H. Obuse , Delocalization of a nonHermitian quantum walk on random media in one dimension, Ann. Phys. 168615 (2021)
CrossRef ADS Google scholar
[109]
T. Liu , S. Cheng , H. Guo , and G. Xianlong , Fate of Majorana zero modes, exact location of critical states, and unconventional real-complex transition in non-Hermitian quasiperiodic lattices, Phys. Rev. B 103 (10), 104203 (2021)
CrossRef ADS Google scholar
[110]
Q. Lin , T. Li , L. Xiao , K. Wang , W. Yi , and P. Xue , Observation of non-Hermitian topological Anderson insulator in quantum dynamics, arXiv: 2108.01097 (2021)

RIGHTS & PERMISSIONS

2022 Higher Education Press
AI Summary AI Mindmap
PDF(1093 KB)

Accesses

Citations

Detail

Sections
Recommended

/