Quantifying fermionic interactions from the violation of Wick’s theorem

Jiannis K. Pachos1 and Chrysoula Vlachou2,3

1School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United Kingdom
2Instituto de Telecomunicações, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal
3Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.


In contrast to interacting systems, the ground state of free systems has a highly ordered pattern of quantum correlations, as witnessed by Wick's decomposition. Here, we quantify the effect of interactions by measuring the violation they cause on Wick's decomposition. In particular, we express this violation in terms of the low entanglement spectrum of fermionic systems. Moreover, we establish a relation between the Wick's theorem violation and the interaction distance, the smallest distance between the reduced density matrix of the system and that of the optimal free model closest to the interacting one. Our work provides the means to quantify the effect of interactions in physical systems though measurable quantum correlations.

► BibTeX data

► References

[1] K. Byczuk, J. Kuneš, W. Hofstetter, and D. Vollhardt. Quantification of correlations in quantum many-particle systems. Phys. Rev. Lett., 108: 087004, 2012. 10.1103/​PhysRevLett.108.087004.

[2] P. Calabrese and J. Cardy. Entanglement entropy and conformal field theory. Journal of Physics A: Mathematical and Theoretical, 42 (50): 504005, 2009. 10.1088/​1751-8113/​42/​50/​504005.

[3] A. Chakraborty, P. Gorantla, and R. Sensarma. Nonequilibrium field theory for dynamics starting from arbitrary athermal initial conditions. Phys. Rev. B, 99: 054306, 2019. 10.1103/​PhysRevB.99.054306.

[4] C. Chamon, A. Hamma, and E. R. Mucciolo. Emergent irreversibility and entanglement spectrum statistics. Phys. Rev. Lett., 112: 240501, 2014. 10.1103/​PhysRevLett.112.240501.

[5] G. De Chiara and A. Sanpera. Genuine quantum correlations in quantum many-body systems: a review of recent progress. Reports on Progress in Physics, 81 (7): 074002, 2018. 10.1088/​1361-6633/​aabf61.

[6] M. Dalmonte, B. Vermersch, and P. Zoller. Quantum simulation and spectroscopy of entanglement hamiltonians. Nature Physics, 14: 827–831, 2018. 10.1038/​s41567-018-0151-7.

[7] G. De Chiara, L. Lepori, M. Lewenstein, and A. Sanpera. Entanglement spectrum, critical exponents, and order parameters in quantum spin chains. Phys. Rev. Lett., 109: 237208, 2012. 10.1103/​PhysRevLett.109.237208.

[8] M. Endres, M. Cheneau, T. Fukuhara, C. Weitenberg, P. Schauß, C. Gross, L. Mazza, M. C. Bañuls, L. Pollet, I. Bloch, and S. Kuhr. Observation of correlated particle-hole pairs and string order in low-dimensional Mott insulators. Science, 334 (6053): 200–203, 2011. 10.1126/​science.1209284.

[9] J. J. Fernández-Melgarejo and J. Molina-Vilaplana. Entanglement entropy: non-gaussian states and strong coupling. Journal of High Energy Physics, 2021: 106, 2021. 10.1007/​JHEP02(2021)106.

[10] A. Hamma, R. Ionicioiu, and P. Zanardi. Ground state entanglement and geometric entropy in the Kitaev model. Physics Letters A, 337 (1): 22–28, 2005. https:/​/​doi.org/​10.1016/​j.physleta.2005.01.060.

[11] K. Hettiarachchilage, C. Moore, V. G. Rousseau, K.-M. Tam, M. Jarrell, and J. Moreno. Local density of the bose-glass phase. Phys. Rev. B, 98: 184206, 2018. 10.1103/​PhysRevB.98.184206.

[12] A. Y. Kitaev. Anyons in an exactly solved model and beyond. Annals of Physics, 321 (1): 2–111, 2006. https:/​/​doi.org/​10.1016/​j.aop.2005.10.005. January Special Issue.

[13] R. B. Laughlin. Anomalous quantum hall effect: An incompressible quantum fluid with fractionally charged excitations. Phys. Rev. Lett., 50: 1395–1398, 1983. 10.1103/​PhysRevLett.50.1395.

[14] H. Li and F. D. M. Haldane. Entanglement spectrum as a generalization of entanglement entropy: Identification of topological order in non-abelian fractional quantum Hall effect states. Phys. Rev. Lett., 101: 010504, 2008. 10.1103/​PhysRevLett.101.010504.

[15] E. M. Lifshitz, L. D. Landau, and L. P. Pitaevskii. Statistical Physics, Part 2: Theory of the Condensed State. Pergamon Press, 1980.

[16] D. Markham, J. A. Miszczak, Z. Puchała, and K. Życzkowski. Quantum state discrimination: A geometric approach. Phys. Rev. A, 77: 042111, 2008. 10.1103/​PhysRevA.77.042111.

[17] G. Matos, A. Hallam, A. Deger, Z. Papić, and J. K. Pachos. Emergence of gaussianity in the thermodynamic limit of interacting fermions. Phys. Rev. B, 104: L180408, 2021. 10.1103/​PhysRevB.104.L180408.

[18] K. Meichanetzidis, C. J. Turner, A. Farjami, Z. Papić, and J. K. Pachos. Free-fermion descriptions of parafermion chains and string-net models. Phys. Rev. B, 97: 125104, 2018. 10.1103/​PhysRevB.97.125104.

[19] B. Mera, C. Vlachou, N. Paunković, and V. R. Vieira. Uhlmann connection in fermionic systems undergoing phase transitions. Phys. Rev. Lett., 119: 015702, 2017. 10.1103/​PhysRevLett.119.015702.

[20] B. Mera, C. Vlachou, N. Paunković, V. R. Vieira, and O. Viyuela. Dynamical phase transitions at finite temperature from fidelity and interferometric Loschmidt echo induced metrics. Phys. Rev. B, 97: 094110, 2018. 10.1103/​PhysRevB.97.094110.

[21] S. Moitra and R. Sensarma. Entanglement entropy of fermions from Wigner functions: Excited states and open quantum systems. Phys. Rev. B, 102: 184306, 2020. 10.1103/​PhysRevB.102.184306.

[22] R. Nandkishore and D. A. Huse. Many-body localization and thermalization in quantum statistical mechanics. Annual Review of Condensed Matter Physics, 6 (1): 15–38, 2015. 10.1146/​annurev-conmatphys-031214-014726.

[23] J. K. Pachos. Introduction to Topological Quantum Computation. Cambridge University Press, 2012. 10.1017/​CBO9780511792908.

[24] J. K. Pachos and Z. Papić. Quantifying the effect of interactions in quantum many-body systems. SciPost Phys. Lect. Notes, page 4, 2018. 10.21468/​SciPostPhysLectNotes.4.

[25] K. Patrick, V. Caudrelier, Z. Papić, and J. K. Pachos. Interaction distance in the extended XXZ model. Phys. Rev. B, 100: 235128, 2019a. 10.1103/​PhysRevB.100.235128.

[26] K. Patrick, M. Herrera, J. Southall, I. D'Amico, and J. K. Pachos. Efficiency of free auxiliary models in describing interacting fermions: From the Kohn-Sham model to the optimal entanglement model. Phys. Rev. B, 100: 075133, 2019b. 10.1103/​PhysRevB.100.075133.

[27] I. Peschel. Calculation of reduced density matrices from correlation functions. Journal of Physics A: Mathematical and General, 36 (14): L205–L208, 2003. 10.1088/​0305-4470/​36/​14/​101.

[28] I. Peschel and M.-C. Chung. On the relation between entanglement and subsystem hamiltonians. EPL (Europhysics Letters), 96 (5): 50006, 2011. 10.1209/​0295-5075/​96/​50006.

[29] I. Peschel and V. Eisler. Reduced density matrices and entanglement entropy in free lattice models. Journal of Physics A: Mathematical and Theoretical, 42 (50): 504003, 2009. 10.1088/​1751-8113/​42/​50/​504003.

[30] H. Pichler, G. Zhu, A. Seif, P. Zoller, and M. Hafezi. Measurement protocol for the entanglement spectrum of cold atoms. Phys. Rev. X, 6: 041033, 2016. 10.1103/​PhysRevX.6.041033.

[31] N. Read and G. Moore. Fractional quantum Hall effect and nonabelian statistics. Progress of Theoretical Physics Supplement, 107: 157–166, 1992. 10.1143/​PTPS.107.157.

[32] T. Schweigler, V. Kasper, S. Erne, I. Mazets, B. Rauer, F. Cataldini, T. Langen, T. Gasenzer, J. Berges, and J. Schmiedmayer. Experimental characterization of a quantum many-body system via higher-order correlations. Nature, 545: 323–326, 2017. 10.1038/​nature22310.

[33] T. Schweigler, M. Gluza, M. Tajik, S. Sotiriadis, F. Cataldini, S.-C. Ji, F. S. Møller, J. Sabino, B. Rauer, J. Eisert, and J. Schmiedmayer. Decay and recurrence of non-gaussian correlations in a quantum many-body system. Nature Physics, 17: 559–563, 2021. 10.1038/​s41567-020-01139-2.

[34] B. Swingle. Entanglement entropy and the Fermi surface. Phys. Rev. Lett., 105: 050502, 2010. 10.1103/​PhysRevLett.105.050502.

[35] D. C. Tsui, H. L. Stormer, and A. C. Gossard. Two-dimensional magnetotransport in the extreme quantum limit. Phys. Rev. Lett., 48: 1559–1562, 1982. 10.1103/​PhysRevLett.48.1559.

[36] C. J. Turner, K. Meichanetzidis, Z. Papić, and J. K. Pachos. Optimal free descriptions of many-body theories. Nature Communications, 8: 14926, 2017. 10.1038/​ncomms14926.

[37] C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn, and Z. Papić. Quantum scarred eigenstates in a rydberg atom chain: Entanglement, breakdown of thermalization, and stability to perturbations. Phys. Rev. B, 98: 155134, 2018. 10.1103/​PhysRevB.98.155134.

[38] F. Verstraete, M. Popp, and J. I. Cirac. Entanglement versus correlations in spin systems. Phys. Rev. Lett., 92: 027901, 2004. 10.1103/​PhysRevLett.92.027901.

[39] G. Vidal, J. I. Latorre, E. Rico, and A. Y. Kitaev. Entanglement in quantum critical phenomena. Phys. Rev. Lett., 90: 227902, 2003. 10.1103/​PhysRevLett.90.227902.

[40] G. C. Wick. The evaluation of the collision matrix. Phys. Rev., 80: 268–272, 1950. 10.1103/​PhysRev.80.268.

[41] P. Zanardi and N. Paunković. Ground state overlap and quantum phase transitions. Phys. Rev. E, 74: 031123, 2006. 10.1103/​PhysRevE.74.031123.

Cited by

On Crossref's cited-by service no data on citing works was found (last attempt 2022-11-30 01:53:08). On SAO/NASA ADS no data on citing works was found (last attempt 2022-11-30 01:53:09).