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  • Letter
  • Open Access

Quantum geometry of bosonic Bogoliubov quasiparticles

Isaac Tesfaye* and André Eckardt†

  • *Contact author: i.tesfaye@tu-berlin.de
  • †Contact author: eckardt@tu-berlin.de
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Phys. Rev. Research 7, L042052 – Published 3 December, 2025

DOI: https://doi.org/10.1103/1sz1-nwzf

Abstract

Bosonic Bogoliubov-de Gennes (BBdG) Hamiltonians describe the excitations of weakly interacting Bose condensates as well as photonic systems under parametric driving. Their topological features have been studied mainly by utilizing a generalized symplectic version of the Berry curvature and related Chern numbers. However, a full characterization of geometrical features in BBdG systems is still lacking. Here, we propose a symplectic quantum geometric tensor (SQGT), whose imaginary part leads to the previously studied symplectic Berry curvature, while the real part gives rise to a symplectic quantum metric, providing a natural distance measure in the space of bosonic Bogoliubov modes. The SQGT is directly related to observable properties of BBdG systems. We show how to measure all components of the SQGT by extracting excitation rates in response to periodic modulations of the system’s parameters. Moreover, we connect the symplectic Berry curvature to a generalized symplectic anomalous velocity term for Bogoliubov-Bloch wave packets. We test our results for a bosonic Bogoliubov-Haldane model.

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References (143)

  1. N. N. Bogolyubov, On the theory of superfluidity, J. Phys. (USSR) 11, 23 (1947).
  2. J. Colpa, Diagonalization of the quadratic boson Hamiltonian, Physica A 93, 327 (1978).
  3. Y. Kawaguchi and M. Ueda, Spinor Bose–Einstein condensates, Phys. Rep. 520, 253 (2012).
  4. R. Ozeri, N. Katz, J. Steinhauer, and N. Davidson, Colloquium: Bulk Bogoliubov excitations in a Bose-Einstein condensate, Rev. Mod. Phys. 77, 187 (2005).
  5. A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, Introduction to quantum noise, measurement, and amplification, Rev. Mod. Phys. 82, 1155 (2010).
  6. V. Peano, M. Houde, F. Marquardt, and A. A. Clerk, Topological quantum fluctuations and traveling wave amplifiers, Phys. Rev. X 6, 041026 (2016).
  7. V. Peano, M. Houde, C. Brendel, F. Marquardt, and A. A. Clerk, Topological phase transitions and chiral inelastic transport induced by the squeezing of light, Nat. Commun. 7, 10779 (2016).
  8. D. Smirnova, D. Leykam, Y. Chong, and Y. Kivshar, Nonlinear topological photonics, Appl. Phys. Rev. 7, 021306 (2020).
  9. T. Shi, H. J. Kimble, and J. I. Cirac, Topological phenomena in classical optical networks, Proc. Natl. Acad. Sci. USA 114, E8967 (2017).
  10. T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, Topological photonics, Rev. Mod. Phys. 91, 015006 (2019).
  11. R. Shindou, R. Matsumoto, S. Murakami, and J.-I. Ohe, Topological chiral magnonic edge mode in a magnonic crystal, Phys. Rev. B 87, 174427 (2013).
  12. R. Shindou, J.-I. Ohe, R. Matsumoto, S. Murakami, and E. Saitoh, Chiral spin-wave edge modes in dipolar magnetic thin films, Phys. Rev. B 87, 174402 (2013).
  13. P. A. McClarty, Topological magnons: A review, Annu. Rev. Condens. Matter Phys. 13, 171 (2022).
  14. L. Zhang, J. Ren, J.-S. Wang, and B. Li, Topological nature of the phonon Hall effect, Phys. Rev. Lett. 105, 225901 (2010).
  15. E. E. Wollman, C. U. Lei, A. J. Weinstein, J. Suh, A. Kronwald, F. Marquardt, A. A. Clerk, and K. C. Schwab, Quantum squeezing of motion in a mechanical resonator, Science 349, 952 (2015).
  16. M.-W. Xiao, Theory of transformation for the diagonalization of quadratic Hamiltonians, arXiv:0908.0787.
  17. M. Z. Hasan and C. L. Kane, Topological insulators, Rev. Mod. Phys. 82, 3045 (2010).
  18. X.-L. Qi and S.-C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys. 83, 1057 (2011).
  19. Y. Hatsugai, Chern number and edge states in the integer quantum Hall effect, Phys. Rev. Lett. 71, 3697 (1993).
  20. Y. Hatsugai, Edge states in the integer quantum Hall effect and the Riemann surface of the Bloch function, Phys. Rev. B 48, 11851 (1993).
  21. R. Barnett, Edge-state instabilities of bosons in a topological band, Phys. Rev. A 88, 063631 (2013).
  22. B. Galilo, D. K. K. Lee, and R. Barnett, Selective population of edge states in a 2D topological band system, Phys. Rev. Lett. 115, 245302 (2015).
  23. G. Engelhardt and T. Brandes, Topological Bogoliubov excitations in inversion-symmetric systems of interacting bosons, Phys. Rev. A 91, 053621 (2015).
  24. S. Furukawa and M. Ueda, Excitation band topology and edge matter waves in Bose–Einstein condensates in optical lattices, New J. Phys. 17, 115014 (2015).
  25. A. Altland and M. R. Zirnbauer, Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures, Phys. Rev. B 55, 1142 (1997).
  26. S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. W. Ludwig, Topological insulators and superconductors: Tenfold way and dimensional hierarchy, New J. Phys. 12, 065010 (2010).
  27. A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig, Classification of topological insulators and superconductors in three spatial dimensions, Phys. Rev. B 78, 195125 (2008).
  28. A. Kitaev, Periodic table for topological insulators and superconductors, AIP Conf. Proc. 1134, 22 (2009).
  29. C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu, Classification of topological quantum matter with symmetries, Rev. Mod. Phys. 88, 035005 (2016).
  30. V. Peano and H. Schulz-Baldes, Topological edge states for disordered bosonic systems, J. Math. Phys. 59, 031901 (2018).
  31. M. Lein and K. Sato, Krein-Schrödinger formalism of bosonic Bogoliubov–de Gennes and certain classical systems and their topological classification, Phys. Rev. B 100, 075414 (2019).
  32. Z. Zhou, L.-L. Wan, and Z.-F. Xu, Topological classification of excitations in quadratic bosonic systems, J. Phys. A: Math. Theor. 53, 425203 (2020).
  33. Z. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Higashikawa, and M. Ueda, Topological phases of non-Hermitian systems, Phys. Rev. X 8, 031079 (2018).
  34. K. Kawabata, K. Shiozaki, M. Ueda, and M. Sato, Symmetry and topology in non-Hermitian physics, Phys. Rev. X 9, 041015 (2019).
  35. H. Zhou and J. Y. Lee, Periodic table for topological bands with non-Hermitian symmetries, Phys. Rev. B 99, 235112 (2019).
  36. S. Lieu, Topological symmetry classes for non-Hermitian models and connections to the bosonic Bogoliubov–de Gennes equation, Phys. Rev. B 98, 115135 (2018).
  37. D. Bernard and A. LeClair, A classification of non-Hermitian random matrices, in Statistical Field Theories, edited by A. Cappelli and G. Mussardo (Springer Netherlands, Dordrecht, 2002), pp. 207–214.
  38. X.-Q. Wang, G.-Q. Luo, J.-Y. Liu, W. V. Liu, A. Hemmerich, and Z.-F. Xu, Evidence for an atomic chiral superfluid with topological excitations, Nature (London) 596, 227 (2021).
  39. Z. Jalali-mola, T. Grass, V. Kasper, M. Lewenstein, and U. Bhattacharya, Topological Bogoliubov quasiparticles from Bose-Einstein condensate in a flat band system, Phys. Rev. Lett. 131, 226601 (2023).
  40. H. Kondo, Y. Akagi, and H. Katsura, Z2 topological invariant for magnon spin Hall systems, Phys. Rev. B 99, 041110 (2019).
  41. T. Ohashi, S. Kobayashi, and Y. Kawaguchi, Generalized Berry phase for a bosonic Bogoliubov system with exceptional points, Phys. Rev. A 101, 013625 (2020).
  42. C.-E. Bardyn, T. Karzig, G. Refael, and T. C. H. Liew, Chiral Bogoliubov excitations in nonlinear bosonic systems, Phys. Rev. B 93, 020502 (2016).
  43. A. McDonald, T. Pereg-Barnea, and A. A. Clerk, Phase-dependent chiral transport and effective non-Hermitian dynamics in a bosonic Kitaev-Majorana chain, Phys. Rev. X 8, 041031 (2018).
  44. N. Okuma, Bosonic Andreev bound state, arXiv:2310.09197.
  45. L.-L. Wan, Z. Zhou, and Z.-F. Xu, Squeezing-induced topological gap opening on bosonic Bogoliubov excitations, Phys. Rev. A 103, 013308 (2021).
  46. G. Chaudhary, M. Levin, and A. A. Clerk, Simple approach to characterizing band topology in bosonic pairing Hamiltonians, Phys. Rev. B 103, 214306 (2021).
  47. Q.-R. Xu, V. P. Flynn, A. Alase, E. Cobanera, L. Viola, and G. Ortiz, Squaring the fermion: The threefold way and the fate of zero modes, Phys. Rev. B 102, 125127 (2020).
  48. G. Massarelli, I. Khait, and A. Paramekanti, Krein-unitary Schrieffer-Wolff transformation and band touchings in bosonic Bogoliubov–de Gennes and other Krein-Hermitian Hamiltonians, Phys. Rev. B 106, 144434 (2022).
  49. M. Di Liberto, A. Hemmerich, and C. Morais Smith, Topological Varma superfluid in optical lattices, Phys. Rev. Lett. 117, 163001 (2016).
  50. V. P. Flynn, E. Cobanera, and L. Viola, Deconstructing effective non-Hermitian dynamics in quadratic bosonic Hamiltonians, New J. Phys. 22, 083004 (2020).
  51. G.-H. Huang, G.-Q. Luo, Z. Wu, and Z.-F. Xu, Interaction-induced topological Bogoliubov excitations in a spin-orbit-coupled Bose-Einstein condensate, Phys. Rev. A 103, 043328 (2021).
  52. G.-H. Huang, Z.-F. Xu, and Z. Wu, Intrinsic anomalous Hall effect in a bosonic chiral superfluid, Phys. Rev. Lett. 129, 185301 (2022).
  53. C. Chen, G.-H. Huang, and Z. Wu, Intrinsic anomalous Hall effect across the magnetic phase transition of a spin-orbit-coupled Bose-Einstein condensate, Phys. Rev. Res. 5, 023070 (2023).
  54. S. Ravets, N. Pernet, N. Mostaan, N. Goldman, and J. Bloch, Thouless pumping in a driven-dissipative Kerr resonator array, Phys. Rev. Lett. 134, 093801 (2025).
  55. A. Julku, G. M. Bruun, and P. Törmä, Quantum geometry and flat band Bose-Einstein condensation, Phys. Rev. Lett. 127, 170404 (2021).
  56. A. Julku, G. M. Bruun, and P. Törmä, Excitations of a Bose-Einstein condensate and the quantum geometry of a flat band, Phys. Rev. B 104, 144507 (2021).
  57. A. Julku, G. Salerno, and P. Törmä, Superfluidity of flat band Bose–Einstein condensates revisited, Low Temp. Phys. 49, 701 (2023).
  58. G. Salerno, T. Ozawa, and P. Törmä, Drude weight and the many-body quantum metric in one-dimensional Bose systems, Phys. Rev. B 108, L140503 (2023).
  59. R. Resta, The insulating state of matter: A geometrical theory, Eur. Phys. J. B 79, 121 (2011).
  60. M. Kolodrubetz, D. Sels, P. Mehta, and A. Polkovnikov, Geometry and non-adiabatic response in quantum and classical systems, Phys. Rep. 697, 1 (2017).
  61. J. P. Provost and G. Vallee, Riemannian structure on manifolds of quantum states, Commun. Math. Phys. 76, 289 (1980).
  62. We note that non-Hermitian generalizations of the quantum geometric tensor have also recently been put forward [116, 117, 118], although not explicitly applied to the case of bosonic BdG systems.
  63. D. T. Tran, A. Dauphin, A. G. Grushin, P. Zoller, and N. Goldman, Probing topology by “heating”: Quantized circular dichroism in ultracold atoms, Sci. Adv. 3, e1701207 (2017).
  64. T. Ozawa, Steady-state Hall response and quantum geometry of driven-dissipative lattices, Phys. Rev. B 97, 041108 (2018).
  65. D. Xiao, M.-C. Chang, and Q. Niu, Berry phase effects on electronic properties, Rev. Mod. Phys. 82, 1959 (2010).
  66. G. Sundaram and Q. Niu, Wave-packet dynamics in slowly perturbed crystals: Gradient corrections and Berry-phase effects, Phys. Rev. B 59, 14915 (1999).
  67. M.-C. Chang and Q. Niu, Berry phase, hyperorbits, and the Hofstadter spectrum, Phys. Rev. Lett. 75, 1348 (1995).
  68. Throughout this Letter, we only consider thermodynamically stable systems, i.e., the coefficient matrix M is positive definite [30], such that a diagonalization of the BdG Hamiltonian (1) satisfying the bosonic commutation relations with strictly positive eigenvalues is always possible [2, 69, 119, 120].
  69. V. P. Flynn, E. Cobanera, and L. Viola, Restoring number conservation in quadratic bosonic Hamiltonians with dualities, Europhys. Lett. 131, 40006 (2020).
  70. See Supplemental Material at http://link.aps.org/supplemental/10.1103/1sz1-nwzf for details regarding the paraunitary Bogoliubov transformation, the symmetries in BBdG systems and Krein space structure, the associated indefinite inner product space, the derivation of alternative representations of the SQGT, the comparison between the SQGT and QGT, the PHS of the SQGT, the symplectic quantum metric as an infinitesimal distance measure, the local conservation law for the symplectic Berry curvature, the derivation of the equation of motion for Bogoliubov modes and its solution using time-dependent perturbation theory, the application of the measurement protocol to BBdG lattice systems, the multiquasiparticle case in Fock space, adiabatic perturbation theory of the equation of motion, the derivation of the symplectic anomalous velocity, the Bogoliubov-Haldane model including its topological classification and symmetries, and observable signatures for the original particles, which includes Refs. [2, 3, 6, 7, 11, 16, 23, 24, 25, 29, 30, 31, 32, 41, 47, 59, 60, 61, 63, 64, 65, 66, 67, 69, 71, 72, 73, 73, 96, 97, 101, 102, 103, 107, 119, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142].
  71. A. Ferraro, S. Olivares, and M. G. A. Paris, Gaussian States in Quantum Information (Bibliopolis, Napoli, 2005).
  72. J.-P. Blaizot and G. Ripka, Quantum Theory of Finite Systems (MIT Press, Cambridge, MA, 1986).
  73. H. Schulz-Baldes and C. Villegas-Blas, Signatures for J-Hermitians and J-unitaries on Krein spaces with real structures, Math. Nachrichten 290, 1840 (2017).
  74. We note that the particle-hole symmetry (PHS) should correctly be called a particle-hole constraint as it does not describe an actual symmetry of BBdG systems [31] (see also Supplemental Material [70]). Nevertheless, we still use the terminology PHS to connect to previous literature [6, 23, 24, 30].
  75.  A BBdG system is called dynamic stable if all eigenvalues of the dynamical matrix D are real-valued, which implies a bounded time evolution of the system. Note that thermodynamic stability, which follows from a positive definite Hamiltonian (1), is a stronger condition that implies dynamical stability [30, 46].
  76. G. Fubini, Sulle Metriche Definite da una Forma Hermitiana: Nota (Office graf. C. Ferrari, 1904).
  77. Y. Aharonov and J. Anandan, Phase change during a cyclic quantum evolution, Phys. Rev. Lett. 58, 1593 (1987).
  78. J. Anandan and Y. Aharonov, Geometry of quantum evolution, Phys. Rev. Lett. 65, 1697 (1990).
  79. P. Zanardi, P. Giorda, and M. Cozzini, Information-theoretic differential geometry of quantum phase transitions, Phys. Rev. Lett. 99, 100603 (2007).
  80. L. Campos Venuti and P. Zanardi, Quantum critical scaling of the geometric tensors, Phys. Rev. Lett. 99, 095701 (2007).
  81. A. T. Rezakhani, D. F. Abasto, D. A. Lidar, and P. Zanardi, Intrinsic geometry of quantum adiabatic evolution and quantum phase transitions, Phys. Rev. A 82, 012321 (2010).
  82. A. Carollo, D. Valenti, and B. Spagnolo, Geometry of quantum phase transitions, Phys. Rep. 838, 1 (2020).
  83. J. Liu, H. Yuan, X.-M. Lu, and X. Wang, Quantum Fisher information matrix and multiparameter estimation, J. Phys. A: Math. Theor. 53, 023001 (2020).
  84. Y.-Q. Ma, S.-J. Gu, S. Chen, H. Fan, and W.-M. Liu, The Euler number of Bloch states manifold and the quantum phases in gapped fermionic systems, Europhys. Lett. 103, 10008 (2013).
  85. P. Hauke, M. Heyl, L. Tagliacozzo, and P. Zoller, Measuring multipartite entanglement through dynamic susceptibilities, Nat. Phys. 12, 778 (2016).
  86. T. Ozawa and N. Goldman, Probing localization and quantum geometry by spectroscopy, Phys. Rev. Res. 1, 032019 (2019).
  87. C.-R. Yi, J. Yu, H. Yuan, R.-H. Jiao, Y.-M. Yang, X. Jiang, J.-Y. Zhang, S. Chen, and J.-W. Pan, Extracting the quantum geometric tensor of an optical Raman lattice by Bloch-state tomography, Phys. Rev. Res. 5, L032016 (2023).
  88. M. Yu, Y. Liu, P. Yang, M. Gong, Q. Cao, S. Zhang, H. Liu, M. Heyl, T. Ozawa, N. Goldman, and J. Cai, Quantum Fisher information measurement and verification of the quantum Cramér–Rao bound in a solid-state qubit, npj Quantum Inf. 8, 56 (2022).
  89. M. Yu, X. Li, Y. Chu, B. Mera, F. N. Ünal, P. Yang, Y. Liu, N. Goldman, and J. Cai, Experimental demonstration of topological bounds in quantum metrology, Natl. Sci. Rev. 11, nwae065 (2024).
  90. X. Tan, D.-W. Zhang, Z. Yang, J. Chu, Y.-Q. Zhu, D. Li, X. Yang, S. Song, Z. Han, Z. Li, Y. Dong, H.-F. Yu, H. Yan, S.-L. Zhu, and Y. Yu, Experimental measurement of the quantum metric tensor and related topological phase transition with a superconducting qubit, Phys. Rev. Lett. 122, 210401 (2019).
  91. M. Yu, P. Yang, M. Gong, Q. Cao, Q. Lu, H. Liu, S. Zhang, M. B. Plenio, F. Jelezko, T. Ozawa, N. Goldman, and J. Cai, Experimental measurement of the quantum geometric tensor using coupled qubits in diamond, Natl. Sci. Rev. 7, 254 (2020).
  92. A. Gianfrate, O. Bleu, L. Dominici, V. Ardizzone, M. De Giorgi, D. Ballarini, G. Lerario, K. W. West, L. N. Pfeiffer, D. D. Solnyshkov, D. Sanvitto, and G. Malpuech, Measurement of the quantum geometric tensor and of the anomalous Hall drift, Nature (London) 578, 381 (2020).
  93. W. Zheng, J. Xu, Z. Ma, Y. Li, Y. Dong, Y. Zhang, X. Wang, G. Sun, P. Wu, J. Zhao, S. Li, D. Lan, X. Tan, and Y. Yu, Measuring quantum geometric tensor of non-Abelian system in superconducting circuits, Chin. Phys. Lett. 39, 100202 (2022).
  94. J. Cuerda, J. M. Taskinen, N. Källman, L. Grabitz, and P. Törmä, Observation of quantum metric and non-Hermitian Berry curvature in a plasmonic lattice, Phys. Rev. Res. 6, L022020 (2024).
  95. L. Asteria, D. T. Tran, T. Ozawa, M. Tarnowski, B. S. Rem, N. Fläschner, K. Sengstock, N. Goldman, and C. Weitenberg, Measuring quantized circular dichroism in ultracold topological matter, Nat. Phys. 15, 449 (2019).
  96. D. M. Stamper-Kurn, A. P. Chikkatur, A. Görlitz, S. Inouye, S. Gupta, D. E. Pritchard, and W. Ketterle, Excitation of phonons in a Bose-Einstein condensate by light scattering, Phys. Rev. Lett. 83, 2876 (1999).
  97. 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 (1988).
  98. G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif, and T. Esslinger, Experimental realization of the topological Haldane model with ultracold fermions, Nature (London) 515, 237 (2014).
  99. M. Tarnowski, F. N. Ünal, N. Fläschner, B. S. Rem, A. Eckardt, K. Sengstock, and C. Weitenberg, Measuring topology from dynamics by obtaining the Chern number from a linking number, Nat. Commun. 10, 1728 (2019).
  100. C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Feshbach resonances in ultracold gases, Rev. Mod. Phys. 82, 1225 (2010).
  101. J. M. Vogels, K. Xu, C. Raman, J. R. Abo-Shaeer, and W. Ketterle, Experimental observation of the Bogoliubov transformation for a Bose-Einstein condensed gas, Phys. Rev. Lett. 88, 060402 (2002).
  102. A. Brunello, F. Dalfovo, L. Pitaevskii, and S. Stringari, How to measure the Bogoliubov quasiparticle amplitudes in a trapped condensate, Phys. Rev. Lett. 85, 4422 (2000).
  103. C. Zhang, A. M. Dudarev, and Q. Niu, Berry phase effects on the dynamics of quasiparticles in a superfluid with a vortex, Phys. Rev. Lett. 97, 040401 (2006).
  104. N. R. Cooper, J. Dalibard, and I. B. Spielman, Topological bands for ultracold atoms, Rev. Mod. Phys. 91, 015005 (2019).
  105. C. Gross and I. Bloch, Quantum simulations with ultracold atoms in optical lattices, Science 357, 995 (2017).
  106. F. Schäfer, T. Fukuhara, S. Sugawa, Y. Takasu, and Y. Takahashi, Tools for quantum simulation with ultracold atoms in optical lattices, Nat. Rev. Phys. 2, 411 (2020).
  107. I. Bloch, J. Dalibard, and W. Zwerger, Many-body physics with ultracold gases, Rev. Mod. Phys. 80, 885 (2008).
  108. C.-E. Bardyn, L. Wawer, A. Altland, M. Fleischhauer, and S. Diehl, Probing the topology of density matrices, Phys. Rev. X 8, 011035 (2018).
  109. L. Wawer and M. Fleischhauer, Chern number and Berry curvature for Gaussian mixed states of fermions, Phys. Rev. B 104, 094104 (2021).
  110. J. C. Budich and S. Diehl, Topology of density matrices, Phys. Rev. B 91, 165140 (2015).
  111. A. Bouhon, Q. Wu, R.-J. Slager, H. Weng, O. V. Yazyev, and T. Bzdušek, Non-Abelian reciprocal braiding of Weyl points and its manifestation in ZrTe, Nat. Phys. 16, 1137 (2020).
  112. F. N. Ünal, A. Bouhon, and R.-J. Slager, Topological Euler class as a dynamical observable in optical lattices, Phys. Rev. Lett. 125, 053601 (2020).
  113. J. Ahn, S. Park, and B.-J. Yang, Failure of Nielsen-Ninomiya theorem and fragile topology in two-dimensional systems with space-time inversion symmetry: Application to twisted bilayer graphene at magic angle, Phys. Rev. X 9, 021013 (2019).
  114. P. Weinberg, M. Bukov, L. D’Alessio, A. Polkovnikov, S. Vajna, and M. Kolodrubetz, Adiabatic perturbation theory and geometry of periodically-driven systems, Phys. Rep. 688, 1 (2017).
  115. P. Weinberg and M. Bukov, QuSpin: A python package for dynamics and exact diagonalisation of quantum many body systems. Part II: Bosons, fermions and higher spins, SciPost Phys. 7, 020 (2019).
  116. D.-J. Zhang, Q.-h. Wang, and J. Gong, Quantum geometric tensor in PT-symmetric quantum mechanics, Phys. Rev. A 99, 042104 (2019).
  117. Y.-Q. Zhu, W. Zheng, S.-L. Zhu, and G. Palumbo, Band topology of pseudo-Hermitian phases through tensor Berry connections and quantum metric, Phys. Rev. B 104, 205103 (2021).
  118. C. Chen Ye, W. L. Vleeshouwers, S. Heatley, V. Gritsev, and C. Morais Smith, Quantum metric of non-Hermitian Su-Schrieffer-Heeger systems, Phys. Rev. Res. 6, 023202 (2024).
  119. R. Simon, S. Chaturvedi, and V. Srinivasan, Congruences and canonical forms for a positive matrix: Application to the Schweinler–Wigner extremum principle, J. Math. Phys. 40, 3632 (1999).
  120. R. Rossignoli and A. M. Kowalski, Complex modes in unstable quadratic bosonic forms, Phys. Rev. A 72, 032101 (2005).
  121. M. Ueda, Fundamentals and New Frontiers of Bose-Einstein Condensation (World Scientific, Singapore, 2010).
  122. C. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases, 2nd ed. (Cambridge University Press, New York, 2008).
  123. J. Dereziński, Bosonic quadratic Hamiltonians, J. Math. Phys. 58, 121101 (2017).
  124. J. Bognár, Indefinite Inner Product Spaces (Springer, Berlin, 1974).
  125. Edited by R. S. MacKay and J. D. Meiss, Hamiltonian Dynamical Systems: A Reprint Selection (CRC Press, Boca Raton, 2020).
  126. B. Diu, F. Laloe, and C. Cohen-Tannoudji, Quantum Mechanics: Volume II: Angular Momentum, Spin, and Approximation Methods, 2nd ed. (Wiley-VCH, Weinheim, Germany, 2019).
  127. R. Karplus and J. M. Luttinger, Hall effect in ferromagnetics, Phys. Rev. 95, 1154 (1954).
  128. G. Rigolin, G. Ortiz, and V. H. Ponce, Beyond the quantum adiabatic approximation: Adiabatic perturbation theory, Phys. Rev. A 78, 052508 (2008).
  129. C. De Grandi and A. Polkovnikov, Adiabatic perturbation theory: From Landau–Zener problem to quenching through a quantum critical point, in Quantum Quenching, Annealing and Computation, edited by A. K. Chandra, A. Das, and B. K. Chakrabarti (Springer, Berlin, 2010), pp. 75–114.
  130. M. V. Berry, Quantal phase factors accompanying adiabatic changes, Proc. R. Soc. Lond. A 392, 45 (1984).
  131. M. Kolodrubetz, V. Gritsev, and A. Polkovnikov, Classifying and measuring geometry of a quantum ground state manifold, Phys. Rev. B 88, 064304 (2013).
  132. L. Bracci, G. Morchio, and F. Strocchi, Wigner’s theorem on symmetries in indefinite metric spaces, Commun. Math. Phys. 41, 289 (1975).
  133. H. M. Price, O. Zilberberg, T. Ozawa, I. Carusotto, and N. Goldman, Measurement of Chern numbers through center-of-mass responses, Phys. Rev. B 93, 245113 (2016).
  134. J. Großmann and H. Schulz-Baldes, Index pairings in presence of symmetries with applications to topological insulators, Commun. Math. Phys. 343, 477 (2016).
  135. F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys. 71, 463 (1999).
  136. Y.-J. Wu, W.-Y. Zhou, and S.-P. Kou, Bogoliubov excitations in the Bose-Hubbard extension of a Weyl semimetal, Phys. Rev. A 95, 023620 (2017).
  137. J. Williamson, On the algebraic problem concerning the normal forms of linear dynamical systems, Am. J. Math. 58, 141 (1936).
  138. F. Nicacio, Williamson theorem in classical, quantum, and statistical physics, Am. J. Phys. 89, 1139 (2021).
  139. P. Hauke, M. Lewenstein, and A. Eckardt, Tomography of band insulators from quench dynamics, Phys. Rev. Lett. 113, 045303 (2014).
  140. V. Gurarie and J. T. Chalker, Bosonic excitations in random media, Phys. Rev. B 68, 134207 (2003).
  141. A. Serafini, Quantum Continuous Variables: A Primer of Theoretical Methods, 2nd ed. (CRC Press, Boca Raton, FL, 2023).
  142. D. Xiao, J. Shi, and Q. Niu, Berry phase correction to electron density of states in solids, Phys. Rev. Lett. 95, 137204 (2005).
  143. I. Tesfaye and A. Eckardt, Quantum geometry of bosonic Bogoliubov quasiparticles (v1.0.0) [Data set], Zenodo (2025), doi: 10.5281/zenodo.17593200.

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Phys. Rev. Research 7, L042052– Published 3 December, 2025
  • Received 19 July 2024
  • Accepted 14 October 2025
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