S S symmetry Article The Generalized OTOC from Supersymmetric Quantum Mechanics—Study of Random Fluctuations from Eigenstate Representation of Correlation Functions Kaushik Y. Bhagat 1, Baibhab Bose 2, Sayantan Choudhury 3,4,5,*,† , Satyaki Chowdhury 4,5, Rathindra N. Das 6, Saptarshhi G. Dastider 7, Nitin Gupta 8 , Archana Maji 6, Gabriel D. Pasquino 9 and Swaraj Paul 10 1 Centre for High Energy Physics, Indian Institute of Science, Bengaluru, Karnataka 560012, India; [email protected] 2 Department of Physics & Astrophysics, University of Delhi, Delhi 11007, India; [email protected] 3 Quantum Gravity and Unified Theory and Theoretical Cosmology Group, Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am Mühlenberg 1, 14476 Potsdam-Golm, Germany 4 National Institute of Science Education and Research, Bhubaneswar, Odisha 752050, India; [email protected] 5 Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400085, India 6 Department of Physics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India; [email protected] (R.N.D.); [email protected] (A.M.) 7 Department of Chemistry, Sree Chaitanya College, Prafullanagar, Habra 743268, India; [email protected] 8 Department of Physical Sciences, Indian Institute of Science Education & Research Mohali, Punjab 140306, India; [email protected] or [email protected] 9 Department of Physics and Astronomy, University of Waterloo, 200 University Ave W, Waterloo, ON N2L 3G1, Canada; [email protected] Citation: Bhagat, K.Y.; Bose, B.; 10 Discipline of Mathematics, Indian Institute of Technology Indore, Indore 453552, India; [email protected] Choudhury, S.; Chowdhury, S.; Das, * Correspondence: [email protected] or [email protected] R.N.; Dastider, S.G.; Gupta, N.; † NOTE: This project is the part of the non-profit virtual international research consortium “Quantum Aspects Maji, A.; Pasquino, G.D.; Paul, S. of Space-Time & Matter” (QASTM). We would like to dedicate this work for the people those who are helping The Generalized OTOC from us to fight against COVID-19 pandemic across the globe. Supersymmetric Quantum Mechanics—Study of Random Abstract: The concept of the out-of-time-ordered correlation (OTOC) function is treated as a very Fluctuations from Eigenstate strong theoretical probe of quantum randomness, using which one can study both chaotic and Representation of Correlation non-chaotic phenomena in the context of quantum statistical mechanics. In this paper, we define a Functions. Symmetry 2021, 13, 44. general class of OTOC, which can perfectly capture quantum randomness phenomena in a better way. https://doi.org/10.3390/ Further, we demonstrate an equivalent formalism of computation using a general time-independent sym13010044 Hamiltonian having well-defined eigenstate representation for integrable Supersymmetric quantum Received: 6 December 2020 systems. We found that one needs to consider two new correlators apart from the usual one to have Accepted: 18 December 2020 a complete quantum description. To visualize the impact of the given formalism, we consider the Published: 30 December 2020 two well-known models, viz. Harmonic Oscillator and one-dimensional potential well within the framework of Supersymmetry. For the Harmonic Oscillator case, we obtain similar periodic time Publisher’s Note: MDPI stays neu- dependence but dissimilar parameter dependences compared to the results obtained from both micro- tral with regard to jurisdictional clai- canonical and canonical ensembles in quantum mechanics without Supersymmetry. On the other ms in published maps and institutio- hand, for the One-Dimensional Potential Well problem, we found significantly different time scales nal affiliations. and the other parameter dependence compared to the results obtained from non-Supersymmetric quantum mechanics. Finally, to establish the consistency of the prescribed formalism in the classical limit, we demonstrate the phase space averaged version of the classical version of OTOCs from a Copyright: © 2020 by the authors. Li- model-independent Hamiltonian, along with the previously mentioned well-cited models. censee MDPI, Basel, Switzerland. This article is an open access article Keywords: OTOC; supersymmetry; out-of-equilibrium quantum statistical mechanics distributed under the terms and con- ditions of the Creative Commons At- tribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Symmetry 2021, 13, 44. https://doi.org/10.3390/sym13010044 https://www.mdpi.com/journal/symmetry Symmetry 2021, 13, 44 2 of 103 Contents 1 Introduction 3 2 Lexicography 12 3 A Short Review of Supersymmetric Quantum Mechanics 12 4 General Remarks on Time Disorder Averaging and Thermal OTOCs 14 5 Eigenstate Representation of thermal OTOCs in Supersymmetric Quantum Mechanics 20 5.1 Partition Function from Supersymmetric Quantum Mechanics......................... 23 (1) 5.2 Representation of 2-Point OTOC: Y (t1, t2) ................................... 24 (2) 5.3 Representation of 2-Point OTOC: Y (t1, t2) ................................... 25 (3) 5.4 Representation of 2-Point OTOC: Y (t1, t2) ................................... 25 (1) 5.5 Representation of 4-Point OTOC: C (t1, t2) ................................... 26 (1) 5.5.1 Un-Normalized: C (t1, t2) ......................................... 26 (1) 5.5.2 Normalized: Ce (t1, t2) ........................................... 27 (2) 5.6 Representation of 4-Point OTOC: C (t1, t2) ................................... 28 (2) 5.6.1 Un-Normalized: C (t1, t2) ......................................... 28 (2) 5.6.2 Normalized: Ce (t1, t2) ........................................... 29 (3) 5.7 Representation of 4-Point OTOC: C (t1, t2) ................................... 30 (3) 5.7.1 Un-Normalized: C (t1, t2) ......................................... 30 (3) 5.7.2 Normalized: Ce (t1, t2) ........................................... 31 5.8 Summary of Results................................................. 32 6 Model I: Supersymmetric Quantum Mechanical Harmonic Oscillator 33 6.1 Eigenspectrum of the Super-Partner Hamiltonian................................ 33 6.2 Partition Function................................................... 34 6.3 Computation of 2-Point OTOCs........................................... 35 (1) 6.3.1 Computation of Y (t1, t2) ......................................... 35 (2) 6.3.2 Computation of Y (t1, t2) ......................................... 37 (3) 6.3.3 Computation of Y (t1, t2) .......................................... 38 6.4 Computation of Un-Normalized 4-Point OTOCs................................. 40 (1) 6.4.1 Computation of C (t1, t2) ......................................... 40 (2) 6.4.2 Computation of C (t1, t2) ......................................... 42 (3) 6.4.3 Computation of C (t1, t2) ......................................... 44 6.5 Computation of Normalized 4-Point OTOCs................................... 46 (1) 6.5.1 Computation of Ce (t1, t2) ......................................... 46 (2) 6.5.2 Computation of Ce (t1, t2) ......................................... 48 (3) 6.5.3 Computation of Ce (t1, t2) ......................................... 49 6.6 Summary of Results................................................. 50 7 Model II: Supersymmetric One-Dimensional Potential Well 52 8 General Remarks on the Classical Limiting Interpretation of OTOCs 54 9 Classical Limit of OTOC for Supersymmetric One-Dimensional Harmonic Oscillator 55 10 Classical Limit of OTOC for Supersymmetric 1D Box 57 11 Numerical Results 60 11.1 Supersymmetric 1D Infinite Potential Well..................................... 61 11.2 Supersymmetric 1D Harmonic Oscillator..................................... 72 12 Conclusions 83 Symmetry 2021, 13, 44 3 of 103 Appendix A Derivation of the Normalization Factors for the Supersymmetric HO 86 Appendix B Poisson Bracket Relation for the Supersymmetric Partner Potential Associated with the 1D Infinite Well Potential 88 Appendix C Derivation of the Eigenstate Representation of the Correlators 91 (1) Appendix C.1 Representation of 2-point Correlator: Y (t1, t2) ............................ 91 (2) Appendix C.2 Representation of 2-point Correlator: Y (t1, t2) ............................ 92 (3) Appendix C.3 Representation of 2-point Correlator: Y (t1, t2) ............................ 93 (1) Appendix C.4 Representation of 4-Point Correlator: C (t1, t2) ............................ 93 (1) Appendix C.4.1 Un-normalized: C (t1, t2) .................................... 93 (2) Appendix C.4.2 Un-normalized: C (t1, t2) .................................... 95 (3) Appendix C.5 Representation of 4-point Correlator: C (t1, t2) ............................ 98 Appendix C.6 Eigenstate Representation of the Normalization Factors for the 4-point Correlators........ 100 References 100 1. Introduction The concept of out-of-time-ordered-correlators (OTOC) was first introduced by the author duo Larkin and Ovchinnikov to describe the semi-classical correlation in the context of superconductivity [1], which was mostly used in various condensed matter systems to study various out-of-equilibrium phenomena in the quantum regime [2]. However, recently, it has attracted the attention of theoretical physicists from other branches in very different contexts, finding applications in the finite-temperature extension of quantum field theories, bulk gravitational theories, quantum black holes, and many more sensational