# Information Geometry of Complex Hamiltonians and Exceptional Points

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## Abstract

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## 1. Introduction

## 2. Information Geometry and Statistical Mechanics

## 3. Statistical Geometry in Complex Vector Spaces

## 4. Eigengeometry of Hermitian Hamiltonians

## 5. Information Geometry for Complex Hamiltonians

## 6. Geometry Close to Exceptional Points

## 7. Discussion and Summary

## Acknowledgments

## Conflicts of Interest

## References

- Brody, D.C.; Hook, D.W.; Hughston, L.P. Quantum phase transitions without thermodynamic limits. Proc. R. Soc. Lond. A
**2007**, 463, 2021–2030. [Google Scholar] [CrossRef][Green Version] - Kato, T. Perturbation Theory for Linear Operators, 2nd ed.; Springer: Berlin, Germany, 1976. [Google Scholar]
- Yang, C.N.; Lee, T.D. Statistical theory of equations of state and phase transitions. I. Theory of condensation. Phys. Rev.
**1952**, 87, 404–409. [Google Scholar] [CrossRef] - Lee, T.D.; Yang, C.N. Statistical theory of equations of state and phase transitions. II. Lattice gas and Ising model. Phys. Rev.
**1952**, 87, 410–419. [Google Scholar] [CrossRef] - Blythe, R.A.; Evans, M.R. The Lee-Yang theory of equilibrium and nonequilibrium phase transitions. Braz. J. Phys.
**2003**, 33, 464–475. [Google Scholar] [CrossRef] - Cejnar, P.; Heinze, S.; Macek, M. Coulomb analogy for non-Hermitian degeneracies near quantum phase transitions. Phys. Rev. Lett.
**2007**, 99, 100601. [Google Scholar] [CrossRef] [PubMed] - Heiss, W.D. The physics of exceptional points. J. Phys. A
**2012**, 45, 444016. [Google Scholar] [CrossRef] - Brody, D.C.; Hook, D.W. Information geometry in vapour-liquid equilibrium. J. Phys. A
**2009**, 42, 023001. [Google Scholar] [CrossRef] - Zanardi, P.; Giorda, P.; Cozzini, M. Information-theoretic differential geometry of quantum phase transitions. Phys. Rev. Lett.
**2007**, 99, 100603. [Google Scholar] [CrossRef] [PubMed] - Pancharatnam, S. The propagation of light in absorbing biaxial crystals. II. Experimental. Proc. Indian Acad. Sci. A
**1955**, 42, 235–248. [Google Scholar] - Dembowski, C.; Gräf, H.D.; Harney, H.L.; Heine, H.L.; Heiss, W.D.; Rehfeld, H.; Richter, A. Experimental observation of the topological structure of exceptional points. Phys. Rev. Lett.
**2001**, 86, 787–790. [Google Scholar] [CrossRef] [PubMed] - Dembowski, C.; Dietz, B.; Gräf, H.D.; Harney, H.L.; Heine, H.L.; Heiss, W.D.; Richter, A. Observation of a chiral state in a microwave cavity. Phys. Rev. Lett.
**2003**, 90, 034101. [Google Scholar] [CrossRef] [PubMed] - Lee, S.B.; Yang, J.; Moon, S.; Lee, S.Y.; Shim, J.B.; Kim, S.W.; Lee, J.H.; An, K. Observation of an exceptional point in a chaotic optical microcavity. Phys. Rev. Lett.
**2009**, 103, 134101. [Google Scholar] [CrossRef] [PubMed] - Bender, C.M.; Boettcher, S. Real spectra in non-Hermitian Hamiltonians having PT symmetry. Phys. Rev. Lett.
**1998**, 80, 5243–5246. [Google Scholar] [CrossRef] - Makris, K.G.; El-Ganainy, R.; Christodoulides, D.N.; Musslimani, Z.H. Beam dynamics in PT symmetric optical lattices. Phys. Rev. Lett.
**2008**, 100, 103904. [Google Scholar] [CrossRef] [PubMed] - Klaiman, S.; Günther, U.; Moiseyev, N. Visualization of branch points in PT-symmetric waveguides. Phys. Rev. Lett.
**2008**, 101, 080402. [Google Scholar] [CrossRef] [PubMed] - Mostafazadeh, A. Spectral singularities of complex scattering potentials and infinite reflection and transmission coefficients at real energies. Phys. Rev. Lett.
**2009**, 102, 220402. [Google Scholar] [CrossRef] [PubMed] - Guo, A.; Salamo, G.J.; Duchesne, D.; Morandotti, R.; Volatier-Ravat, M.; Aimez, V.; Siviloglou, G.A.; Christodoulides, D.N. Observation of PT-symmetry breaking in complex optical potentials. Phys. Rev. Lett.
**2009**, 103, 093902. [Google Scholar] [CrossRef] [PubMed] - Rüter, C.E.; Makris, K.G.; El-Ganainy, R.; Christodoulides, D.N.; Kip, D. Observation of parity-time symmetry in optics. Nat. Phys.
**2010**, 6, 192–195. [Google Scholar] [CrossRef] - Ge, L.; Chong, Y.D.; Rotter, S.; Türeci, H.E.; Stone, A.D. Unconventional modes in lasers with spatially varying gain and loss. Phys. Rev. A
**2011**, 84, 023820. [Google Scholar] [CrossRef] - Schindler, J.; Li, A.; Zheng, M.C.; Ellis, F.M.; Kottos, T. Experimental study of active LRC circuits with PT symmetries. Phys. Rev. A
**2011**, 84, 040101. [Google Scholar] [CrossRef] - Liertzer, M.; Ge, L.; Cerjan, A.; Stone, A.D.; Türeci, H.E.; Rotter, S. Pump-induced exceptional points in lasers. Phys. Rev. Lett.
**2012**, 108, 173901. [Google Scholar] [CrossRef] [PubMed] - Ramezani, H.; Schindler, J.; Ellis, F.M.; Günther, U.; Kottos, T. Bypassing the bandwidth theorem with PT symmetry. Phys. Rev. A
**2012**, 85, 062122. [Google Scholar] [CrossRef] - Bittner, S.; Dietz, B.; Günther, U.; Harney, H.L.; Miski-Oglu, M.; Richter, A.; Schäfer, F. PT symmetry and spontaneous symmetry breaking in a microwave billiard. Phys. Rev. Lett.
**2012**, 108, 024101. [Google Scholar] [CrossRef] [PubMed] - Brody, D.C.; Graefe, E.M. Mixed-state evolution in the presence of gain and loss. Phys. Rev. Lett.
**2012**, 109, 230405. [Google Scholar] [CrossRef] [PubMed] - Bender, C.M.; Berntson, B.K.; Parker, D.; Samuel, E. Observation of PT phase transition in a simple mechanical system. Am. J. Phys.
**2013**, 81, 173–179. [Google Scholar] [CrossRef] - Rao, C.R. Information and the accuracy attainable in the estimation of statistical parameters. Bull. Calcutta Math. Soc.
**1945**, 37, 81–91. [Google Scholar] - Fisher, R.A. Theory of statistical estimation. Proc. Camb. Philos. Soc.
**1925**, 22, 700–725. [Google Scholar] [CrossRef] - Brody, D.C.; Hughston, L.P. Geometry of quantum statistical inference. Phys. Rev. Lett.
**1996**, 77, 2851–2854. [Google Scholar] [CrossRef] [PubMed] - Mandelbrot, B. The role of sufficiency and of estimation in thermodynamics. Ann. Math. Stat.
**1962**, 33, 1021–1038. [Google Scholar] [CrossRef] - Brody, D.C.; Hughston, L.P. Geometrisation of statistical mechanics. Proc. R. Soc. Lond. A
**1999**, 455, 1683–1715. [Google Scholar] [CrossRef] - Anandan, J.; Aharonov, Y. Geometry of quantum evolution. Phys. Rev. Lett.
**1990**, 65, 1697–1700. [Google Scholar] [CrossRef] [PubMed] - Holevo, A. Probabilistic and Statistical Aspects of Quantum Theory; Edizioni della Normale: Pisa, Italy, 2011. [Google Scholar]
- Brody, D.C.; Graefe, E.M. Coherent states and rational surfaces. J. Phys. A
**2010**, 43, 255205. [Google Scholar] [CrossRef] - Zhu, S.L. Scaling of geometric phases close to the quantum phase transition in the XY spin chain. Phys. Rev. Lett.
**2006**, 96, 077206. [Google Scholar] [CrossRef] [PubMed] - Hamma, A. Berry phases and quantum phases transitions.
**2006**. arXiv:quant-ph/0602091. [Google Scholar] - Berry, M.V. Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. Lond. A
**1984**, 392, 45–57. [Google Scholar] [CrossRef] - Brody, D.C.; Hughston, L.P. Statistical geometry in quantum mechanics. Proc. R. Soc. Lond. A
**1998**, 454, 2445–2475. [Google Scholar] [CrossRef] - Mahaux, C.; Weidenmüller, H.A. Shell Model Approach to Nuclear Reactions; North Holland Publishing Company: Amsterdam, The Netherlands, 1969. [Google Scholar]
- Sternheim, M.M.; Walker, J.F. Non-Hermitian Hamiltonians, decaying states, and perturbation theory. Phys. Rev. C
**1972**, 6, 114–121. [Google Scholar] [CrossRef] - Dattoli, G.; Torre, A.; Mignani, R. Non-Hermitian evolution of two-level quantum systems. Phys. Rev. A
**1990**, 42, 1467–1475. [Google Scholar] [CrossRef] [PubMed] - Okołowicz, J.; Płoszajczak, M.; Rotter, I. Dynamics of quantum systems embedded in a continuum. Phys. Rep.
**2003**, 374, 271–383. [Google Scholar] [CrossRef] - Moiseyev, N. Non-Hermitian Quantum Mechanics; Cambridge University Press: Cambridge, UK, 2011. [Google Scholar]
- Pell, A.J. Biorthogonal systems of functions. Trans. Am. Math. Soc.
**1911**, 12, 135–164. [Google Scholar] [CrossRef] - Des Cloizeaux, J. Extension d’une formule de Lagrange à des problèmes de valeurs propres (In French). Nucl. Phys.
**1960**, 20, 321–346. [Google Scholar] [CrossRef] - More, R.M. Theory of decaying states. Phys. Rev. A
**1971**, 4, 1782–1790. [Google Scholar] [CrossRef] - Curtright, T.; Mezincescu, L. Biorthogonal quantum systems. J. Math. Phys.
**2007**, 48, 092106. [Google Scholar] [CrossRef] - Fyodorov, Y. V.; Savin, D. V. Statistics of resonance width shifts as a signature of eigenfunction nonorthogonality. Phys. Rev. Lett.
**2012**, 108, 184101. [Google Scholar] [CrossRef] [PubMed] - Cui, X.D.; Zheng, Y. Geometric phases in non-Hermitian quantum mechanics. Phys. Rev. A
**2012**, 86, 064104. [Google Scholar] [CrossRef] - Garrison, J.C.; Wright, E.M. Complex geometrical phases for dissipative systems. Phys. Lett. A
**1988**, 128, 177–181. [Google Scholar] [CrossRef] - Mailybaev, A.A.; Kirillov, O.N.; Seyranian, A.P. Geometric phase around exceptional points. Phys. Rev. A
**2005**, 72, 014104. [Google Scholar] [CrossRef] - Mehri-Dehnavi, H.; Mostafazadeh, A. Geometric phase for non-Hermitian Hamiltonians and its holonomy interpretation. J. Math. Phys.
**2008**, 49, 082105. [Google Scholar] [CrossRef] - Akimov, A.V.; Sinitsyn, N.A. Sensitivity field for nonautonomous molecular rotors. J. Chem. Phys.
**2011**, 135, 224104. [Google Scholar] [CrossRef] [PubMed] - Günther, U.; Rotter, I.; Samsonov, B.F. Projective Hilbert space structures at exceptional points. J. Phys. A
**2007**, 40, 8815–8833. [Google Scholar] [CrossRef] - Graefe, E.M.; Günther, U.; Korsch, H.J.; Niederle, A.E. A non-Hermitian symmetric Bose-Hubbard model: Eigenvalue rings from unfolding higher-order exceptional points. J. Phys. A
**2008**, 41, 255206. [Google Scholar] [CrossRef] - Arnold, V.I. On matrices depending on parameters. Russ. Math. Surv.
**1971**, 26, 29–43. [Google Scholar] [CrossRef] - Seyranian, A.P.; Mailybaev, A.A. Multiparameter Stability Theory with Mechanical Applications; World Scientific: Signapore, Signapore, 2003. [Google Scholar]
- Demange, G.; Graefe, E.M. Signatures of three coalescing eigenfunctions. J. Phys. A
**2012**, 45, 025303. [Google Scholar] [CrossRef] - Gutöhrlein, R.; Main, J.; Cartarius, H.; Wunner, G. Bifurcations and exceptional points in dipolar Bose-Einstein condensates. J. Phys. A
**2013**, 46, 305001. [Google Scholar] [CrossRef] - Ma, Y.; Edelman, A. Nongeneric eigenvalue perturbations of Jordan blocks. Linear Algebra Appl.
**1998**, 273, 45–63. [Google Scholar] [CrossRef] - Brody, D.; Rivier, N. Geometrical aspects of statistical mechanics. Phys. Rev. E
**1995**, 51, 1006–1011. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

Brody, D.C.; Graefe, E.-M.
Information Geometry of Complex Hamiltonians and Exceptional Points. *Entropy* **2013**, *15*, 3361-3378.
https://doi.org/10.3390/e15093361

**AMA Style**

Brody DC, Graefe E-M.
Information Geometry of Complex Hamiltonians and Exceptional Points. *Entropy*. 2013; 15(9):3361-3378.
https://doi.org/10.3390/e15093361

**Chicago/Turabian Style**

Brody, Dorje C., and Eva-Maria Graefe.
2013. "Information Geometry of Complex Hamiltonians and Exceptional Points" *Entropy* 15, no. 9: 3361-3378.
https://doi.org/10.3390/e15093361