The Generalized OTOC from Supersymmetric Quantum Mechanics—Study of Random Fluctuations from Eigenstate Representation of Correlation Functions

Abstract

The concept of the out-of-time-ordered correlation (OTOC) function is treated as a very strong theoretical probe of quantum randomness, using which one can study both chaotic and non-chaotic phenomena in the context of quantum statistical mechanics. In this paper, we define a general class of OTOC, which can perfectly capture quantum randomness phenomena in a better way. Further, we demonstrate an equivalent formalism of computation using a general time-independent Hamiltonian having well-defined eigenstate representation for integrable Supersymmetric quantum systems. We found that one needs to consider two new correlators apart from the usual one to have a complete quantum description. To visualize the impact of the given formalism, we consider the 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 dependence but dissimilar parameter dependences compared to the results obtained from both microcanonical and canonical ensembles in quantum mechanics without Supersymmetry. On the other hand, for the One-Dimensional PotentialWell problem, we found significantly different time scales 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 model-independent Hamiltonian, along with the previously mentioned well-cited models.

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
	5.2       Representation of 2-Point OTOC: $Y^{\left(1\right)}(t_1,t_2)$................................................................................................................................	24
	5.3       Representation of 2-Point OTOC: $Y^{\left(2\right)}(t_1,t_2)$................................................................................................................................	25
	5.4       Representation of 2-Point OTOC: $Y^{\left(3\right)}(t_1,t_2)$................................................................................................................................	25
	5.5       Representation of 4-Point OTOC: $C^{\left(1\right)}(t_1,t_2)$................................................................................................................................	26
	5.5.1      Un-Normalized: $C^{\left(1\right)}(t_1,t_2)$....................................................................................................................................	26
	5.5.2      Normalized: ${\tilde{C}}^{\left(1\right)}(t_1,t_2)$.....................................................................................................................................	27
	5.6       Representation of 4-Point OTOC: $C^{\left(2\right)}(t_1,t_2)$................................................................................................................................	28
	5.6.1       Un-Normalized: $C^{\left(2\right)}(t_1,t_2)$...................................................................................................................................	28
	5.6.2       Normalized: ${\tilde{C}}^{\left(2\right)}(t_1,t_2)$....................................................................................................................................	29
	5.7      Representation of 4-Point OTOC: $C^{\left(3\right)}(t_1,t_2)$.................................................................................................................................	30
	5.7.1      Un-Normalized: $C^{\left(3\right)}(t_1,t_2)$....................................................................................................................................	30
	5.7.2       Normalized: ${\tilde{C}}^{\left(3\right)}(t_1,t_2)$....................................................................................................................................	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
	6.3.1       Computation of $Y^{\left(1\right)}(t_1,t_2)$...................................................................................................................................	35
	6.3.2      Computation of $Y^{\left(2\right)}(t_1,t_2)$....................................................................................................................................	37
	6.3.3      Computation of $Y^{\left(3\right)}(t_1,t_2)$....................................................................................................................................	38
	6.4      Computation of Un-Normalized 4-Point OTOCs..................................................................................................................................	40
	6.4.1       Computation of $C^{\left(1\right)}(t_1,t_2)$...................................................................................................................................	40
	6.4.2      Computation of $C^{\left(2\right)}(t_1,t_2)$....................................................................................................................................	42
	6.4.3      Computation of $C^{\left(3\right)}(t_1,t_2)$....................................................................................................................................	44
	6.5      Computation of Normalized 4-Point OTOCs.....................................................................................................................................	46
	6.5.1      Computation of ${\tilde{C}}^{\left(1\right)}(t_1,t_2)$..................................................................................................................................	46
	6.5.2      Computation of ${\tilde{C}}^{\left(2\right)}(t_1,t_2)$..................................................................................................................................	48
	6.5.3      Computation of ${\tilde{C}}^{\left(3\right)}(t_1,t_2)$..................................................................................................................................	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
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
	Appendix C.1 Representation of 2-point Correlator: $Y^{\left(1\right)}(t_1,t_2)$..................................................................................................................	91
	Appendix C.2 Representation of 2-point Correlator: $Y^{\left(2\right)}(t_1,t_2)$..................................................................................................................	92
	Appendix C.3 Representation of 2-point Correlator: $Y^{\left(3\right)}(t_1,t_2)$..................................................................................................................	93
	Appendix C.4 Representation of 4-Point Correlator: $C^{\left(1\right)}(t_1,t_2)$..................................................................................................................	93
	Appendix C.4.1 Un-normalized: $C^{\left(1\right)}(t_1,t_2)$.......................................................................................................................................	93
	Appendix C.4.2 Un-normalized: $C^{\left(2\right)}(t_1,t_2)$.......................................................................................................................................	95
	Appendix C.5 Representation of 4-point Correlator: $C^{\left(3\right)}(t_1,t_2)$..................................................................................................................	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 topics [3,4,5,6,7,8]. It is considered to be one of the strongest theoretical probes for quantifying quantum chaos in terms of quantum Lyapunov exponent [9], as well as quantum theories of stochasticity and randomness, among the theoretical physics community. Besides playing a key role in investigating the holographic duality [10,11,12,13] between a strongly correlated quantum system and a gravitational dual system, it also characterizes the chaotic behavior and information scrambling [14,15,16,17,18,19,20] in the context of many-body quantum systems [21,22,23]. The detailed study of OTOCs reveals an intimate relationship between three entirely different physical concepts, namely holographic duality, quantum chaos, and information scrambling. The key idea of OTOCs can be best understood as the growth of the non-commutativity of quantum mechanical operators Specifically, this non-commutative structure of the quantum operators describes the unequal time commutation relations (UETCRs) within the framework of quantum mechanics. However, the mathematical structure, as well as the physical consequences of these correlators in the quantum regime, is completely different from the concept of formulating advanced and the retarded correlators. In the later part of this paper, we will explicitly demonstrate such differences or the correlators within the framework of micro-canonical and canonical quantum statistical systems, which are defined at different time scales, and, hence, can be described using the Poisson Brackets for its classical counterpart. Not only that but also the quantum mechanical thermal ensemble average, or, equivalently, the quantum mechanical trace operation, can be described by using the phase space average in the classical limit. It is considered as the quantum mechanical analogue of the classical sensitiveness to the initial conditions in the time dynamics of a quantum system. The exponential growth of these correlators indicates the presence of chaos in the quantum system, which has led to discussions of the “butterfly effect” in black holes [24,25,26] with a saturation bound on the quantum Lyapunov exponent and for various spin models [27,28].

There has been a growing interest to understand the behavior of OTOC, even for systems where quantum chaos is expected to be absent, the most relevant example being the study of OTOC in the quantum Ising spin chain model, where power law growth of OTOCs is observed, as opposed to the exponential growth in non-integrable models in support of non-chaoticity [29,30,31]. Another interesting revelation came from a recent study of OTOCs for a quantum system with discrete energy levels, weakly coupled to a non-adiabatic dissipative thermal environment. This type of system is commonly known as open quantum system (OQS), where the OTOC was found to saturate exponentially in contrast to the exponential growth for a quantum chaotic system [32]. OTOCs have been of prime theoretical interest for diagnosing also the rate of growth of chaos with respect to the different time scales involved in the quantum system through the operators and, hence, for studying the scrambling of quantum information in black holes and strongly correlated quantum mechanical systems. It serves as a strong theoretical probe for investigating various bulk gravitational dual theories in the framework of AdS/CFT correspondence. Among others, the existence of shock waves inside black holes [33,34,35] and the maximum saturation bound of the quantum version of the Lyapunov exponent are the most famous examples where out of time ordered correlation functions have proven to be useful in the AdS/CFT correspondence. This maximum saturation bound is famously known as the Maldacena-Shenker-Stanford (MSS) bound [36]. The SYK model [37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87] is a well known model which saturates this well known bound and shows the signature of maximal chaos, both in its 1D CFT and gravitational dual 2D black hole counterpart. In Reference [88], this bound on quantum Lyapunov exponent was further generalized for many body systems using the well known Eigenstate Thermalization Hypothesis (ETH). Very recently, in Reference [89], the author used the tools and techniques of computing OTOC in the context of Cosmology by following the underlying slogan Cosmology meets Condensed Matter Physics to study the quantum mechanical correlation functions from random primordial fluctuations appearing in the context of cosmological perturbation theory of background spatially flat FLRW metric. Specifically, these fluctuations are appearing in the context of stochastic particle production during inflation, during the epoch of reheating, and also for the primordial phenomena, which is governed by the quantum generalization of Brownian motion, i.e., for the cosmological epochs in the time line of the universe, where the physics of out-of-equilibrium phenomena play a significant role.

In Reference [90], the earlier discussed fact regarding the exponential growth of the OTOCs in the associated time scales to describe the chaotic fluctuations for non-integrable systems has been established for various well known quantum mechanical systems. A study of the same for integrable models suggest non-chaotic quantum mechanical fluctuations in the quantum regime. In the present context, the phrase quantum randomness describes a physical phenomena which describes chaotic or non-chaotic, i.e., in principle, any random behavior of a system with respect to the associated time scales of the system. For the physical systems, such quantum randomness can be described by the following 2-fold formalisms:

• Formalism I:
  The first approach is based on the construction and the mathematical from of the solution of the Fokker Planck equation, using which various stochastic phenomena in the quantum out-of-equilibrium regime can be studied. One of the famous examples is the stochastic cosmological particle production phenomena, which can be directly mapped to a problem of solving Schr$\ddot{\mathrm{o}}$dinger equation with an impurity potential within the framework of quantum mechanics, which is actually describing propagation of electrons inside an electrical wire in presence of an impurity or defect. Within the framework of quantum statistical mechanics, instead of solving the Schr$\ddot{\mathrm{o}}$dinger problem directly, or maybe solving the dynamical equation for the quantum mechanical fluctuations during the stochastic particle production, one can think about a cumulative probability distribution function of this stochastic process, $\mathcal{P}(n,\tau )$, which depends on two crucial quantities, which are: the number density of the produced particle and the associated time scale of the stochastic process. Using detailed physical arguments and computation, one can explicitly show that this probability distribution function, $\mathcal{P}(n,\tau )$, satisfies the Fokker Planck equation, which is given by:
  $$\begin{array}{ccc}& & \underline{\mathbf{At}\mathbf{infinite}\mathbf{temperature}(\beta \to \mathbf{0}):}\hfill \end{array}$$

  $$\begin{array}{ccc}& & \frac{1}{\mu }\frac{\partial \mathcal{P}(n,\tau )}{\partial \tau }=\underset{\mathbf{Diffusion}}{\underbrace{n(n+1)\frac{{\partial }^2\mathcal{P}(n,\tau )}{\partial n^2}}}+\underset{\mathbf{Drift}}{\underbrace{(1+2n)\frac{\partial \mathcal{P}(n,\tau )}{\partial n}}},\hfill \\ & & \underline{\mathbf{At}\mathbf{finite}\mathbf{temperature}(\beta \ne \mathbf{0}):}\hfill \\ & & \frac{1}{\mu }\frac{\partial \mathcal{P}(n,\tau )}{\partial \tau }=\underset{\mathbf{Diffusion}}{\underbrace{n(n+1)\frac{{\partial }^2\mathcal{P}(n,\tau )}{\partial n^2}}}+\underset{\mathbf{Drift}}{\underbrace{(1+2n)\frac{\partial \mathcal{P}(n,\tau )}{\partial n}}}\hfill \end{array}$$

  (1)

  $$\begin{array}{ccc}& & +\underset{\mathbf{Finite}\mathbf{temperature}\mathbf{contribution}}{\underbrace{\beta \left\{n(n+1)\left[\frac{\partial V\left(n\right)}{\partial n}\frac{\partial \mathcal{P}(n,\tau )}{\partial n}+\frac{{\partial }^{2}V\left(n\right)}{\partial n^2}\mathcal{P}(n,\tau )\right]+(2n+1)\frac{\partial V\left(n\right)}{\partial n}\mathcal{P}(n,\tau )\right\}}},\hfill \end{array}$$

  (2)

  where $\mu$ represents the mean stochastic particle production rate and $V\left(n\right)$ associated potential, which is only significant at finite temperature. By solving this set of equations in the presence of appropriate initial conditions, one can get to know about the related profile of the stochastic process semi-classically in the present context. And not only that, but one can also treat these versions of the Fokker Planck equation as the semi-classical statistical moment generating equations because, by replacing the profile function $\mathcal{P}(n,\tau )$ with the appropriate moment generating function $\mathcal{F}\left(n\right)$, one can compute all the moments. To serve this purpose, one needs to use the following fundamental equation:

  $$\begin{array}{c}\hfill \mathbf{Statistical}\mathbf{average}\mathbf{of}\mathbf{moment}\mathbf{generator}:\langle \mathcal{F}(n)\rangle :=\int dn\mathcal{F}(n)\mathcal{P}(n,\tau ),\end{array}$$

  (3)

  which physically represents the expectation value or the statistical average value of the number density-dependent moment generating any arbitrary function $\mathcal{F}\left(n\right)$. Further substituting the appropriate form of this function in the moment dependent Fokker Planck equations, one can explicitly compute the expression for all the physically relevant statistical moments, i.e., $\langle n\rangle$, $\langle n^2\rangle$, ⋯ explicitly without explicitly knowing about the particular mathematical structure of the profile of the related stochastic dynamical process at infinite or finite temperature for a given structure of number density potential function. These moments are extremely important in the present context of study as all of them semi-classically compute the expressions for all the equal time quantum correlation functions required to study the out-of-equilibrium aspects, such as stochastic effects, disorder, random fluctuations [91], etc., both at infinite and finite temperatures. See References [2,92,93,94,95,96], where all authors have studied the physical impact of this formalism to describe out-of-equilibrium aspects in various different contexts.
  Now, let us speak about some applicability issues related to this particular formalism. Since this formalism only allows us to know about the effect of the semi-classical correlations at equal time that might be not very interesting when we are actually thinking of doing the computation of the correlations and its rate of change at different time scales associated with the quantum mechanical system of study. For example, if we are interested in computing any general N-point semi-classical correlation function as defined by the following expression:

  $$\begin{array}{c}\hfill \langle \prod _{i=1}^{N}n\left({\tau }_i\right)\rangle =\langle n\left({\tau }_1\right)n\left({\tau }_2\right)\cdots n\left({\tau }_N\right)\rangle \forall {\tau }_i(i=1,2,\cdots ,N)\mathbf{contain}\mathbf{disorder},\end{array}$$

  (4)

  then this particular formalism will not work, as using this formalism one cannot capture the effect of disorder effect in the associated time scale of the system. On the other hand, by following the usual tools and techniques, one can only compute the aforementioned N-point correlators either in time ordered sense, where ${\tau }_1>{\tau }_2>\cdots >{\tau }_N$ or in the anti-time ordered sense, where ${\tau }_1<{\tau }_2<\cdots <{\tau }_N$. So, from the technical ground defining this N-point correlator, including the effect of disorder in the time scale at any arbitrary temperature is also a very important topic of research, and here comes the crucial role of the next formalism where we are allowed to define and explicitly compute such quantum effects at the level of correlation functions. Another important aspect we want to point out here is that the present formalism does not bother with any Lagrangian or the Hamiltonian formulation of the associated quantum mechanical system under consideration. So, if we are really interested to know about the effect of time disordering in the expressions for the quantum mechanical correlation functions, which are defined in terms of the fundamental operators appearing in the quantum version of the Lagrangian or the Hamiltonian of the system under consideration, and also want to explicitly know about time variation with respect to different time scales associated with the system, which are actually the source of time disordering, then, instead of the present formalism, it is obviously technically correct and easier to think about the implementation of the second formalism, which gives us the better understanding of time scale disordering. In the next point, and in the rest of the paper, we will follow the second formalism to compute the quantum correlation functions from the fundamental operators from the quantum mechanical systems under study, which can explicitly capture the effect of disordering in the time scale. Not only us, but also the present trend in the research, suggests to make use of the next formalism to get better understanding of time disordering phenomena in quantum mechanical system.
• Formalism II:
  The second approach is based on finding quantum correlation functions, including the time disordering effect, and, throughout the paper, we have followed this formalism to study effects of out-of-equilibrium physics in physical systems [89]. The present computational methodology helps us to know more about the underlying unexplored physical facts regarding the quantum mechanical aspects of various stochastic random process where time ordering or anti-time ordering is not at all important, and, instead of that, disorder in the time scale can be captured in the quantum correlations at very early time scale. This method not only helps us to know about the early time behavior of quantum correlations in the out-of-equilibrium regime of the quantum statistical mechanics but also gives crucial information regarding the late time equilibrium behavior of the quantum correlations of a specific quantum system. However, for all the systems in nature, the above interpretation of the quantum mechanical aspects of the randomness phenomena are not same. Based on all these types of quantum systems, one can categorize the random time disordering phenomena as: A.) a chaotic system which shows exponential growth in the quantum correlators, and B.) a non-chaotic system which shows periodic or aperiodic or irregular random fluctuations in the quantum correlators. The best possible theoretical measure of all such time disorder averaging phenomena for various statistical ensembles, micro-canonical and canonical ensembles, is described by out-of-time-ordered correlation (OTOC) function within the framework of quantum statistical mechanics. Let us define a set of operators, ${\mathcal{O}}_i\left(t_j\right)\forall i,j=1,2$ with $i\ne j$ or $i=j$ possibilities. The time disorder thermal average over statistical ensemble is described by the following expression: -4.6cm0cm

  $$\begin{array}{ccc}\hfill C_N^{\left(ij\right)}(t_1,t_2):=-{\langle {\left[{\mathcal{O}}_i\left(t_1\right),{\mathcal{O}}_j\left(t_2\right)\right]}^N\rangle }_{\beta }& =& -\mathrm{Tr}\left[{\rho }_{\beta }{\left[{\mathcal{O}}_i\left(t_1\right),{\mathcal{O}}_j\left(t_2\right)\right]}^N\right]\forall i,j=1,2,\hfill \end{array}$$

  (5)

  where the thermal density matrix ${\rho }_{\beta }$ is defined as:

  $$\begin{array}{c}\hfill {\rho }_{\beta }:=\frac{1}{Z}exp\left(-\beta H\right)\mathrm{with}Z=\mathrm{Tr}\left[exp(-\beta H)\right].\end{array}$$

  (6)
  In the present context, $C_N^{\left(ij\right)}(t_1,t_2)$ represents three possible types of OTOC, out of which only $i\ne j$ possibility, which will describe only one OTOC, has been explored in earlier works in this area. The other two OTOCs appearing from the $i=j$ possibility will be explicitly studied in this paper. The prime objective is to incorporate two more type of OTOCs, along with the well known other OTOC, to study all of the possible signatures of time disordering average from a quantum mechanical system. Our expectation is all these three types of OTOCs are able to describe the more general structure of stochastic randomness or any simple type of random process in the quantum regime. This idea was revived by Kitaev, then followed by Maldacena, Shenker, and Stanford (MSS) and many more in studying the quantum mechanical signature of chaos, which is actually the $i\ne j$ case in the above definition; but, the mathematical structure of the other two OTOCs represented by the $i=j$ case suggests that any non-chaotic behavior, such as periodic or aperiodic time-dependent behavior, any time-dependent growth in the correlators which is different from any type of exponential growth, and any type of decaying behavior in the correlators, can be explained in a better way compared to just studying the time-dependent behavior from the well known OTOCs which are commonly used in the literature. So, in short, to give a complete picture of any kind of time disordering phenomena, it is better to study all these possible three types of OTOCs to finally comment on the properties of any physical systems in quantum mechanical regime. A few other important things we want to point out here for better understanding the structure of all these OTOCs capturing the underlying physics of disordering averaging phenomena: First of all, here, we have to mention that, in using the definition that we have provided in this paper, we are able to compute three sets of $2N$ point OTOCs. Though, in the further computation, we have restricted our study in the paper by considering $N=1$ and $N=2$ cases, to study the general time disorder averaging process, one may study any even multipoint (i.e., $2N$ point) correlation functions from the present definition. Now, here, the $N=1$ case is basically representing a non-zero UTCR and can be treated as the building block of the full computation, as this particular case is mimicking the computation of the Green’s function in presence of time disordering. More technically, one can interpret that this contribution is made up of two disconnected time disorder averaged thermal correlator. These disconnected contributions are extremely significant if we wait for a large time scale; in literature, we usually identify this time scale as a dissipation time scale on which one can explicitly factorize any higher $2N$ point correlators in terms of the non-vanishing disconnected contributions. For this specific reason, one can treat the $N=1$ case result as the building block of any higher $2N$ point thermal correlators. However, for most of the quantum systems, the $N=1$ case shows random but decaying behavior with respect to the associated time scales which are explicitly appearing in the quantum operators of the theory. For this reason, study of any $N>1$ plays a significant role to give a better understanding of the time disordering phenomena. For this purpose, next, we studied the $N=2$ case, which represents the 4-point thermal correlator in the present context of study, and one of the most significant quantity in the present day research of this area, which can capture better information regarding the time disorder averaging compared to the $N=1$ case. In a future version of this work, we have a plan to extend the present computation to study the physical implications of $N>2$ quantum correlators to better understand the time disorder averaging phenomena. Now, we will comment on the technical side of the present formalism, using which one can explicitly compute these OTOCs in the present context. First of all, we talk about the time-independent Hamiltonians of a quantum system, which have their own eigenstates with a specific energy eigenvalue spectrum. In this case, construction of the OTOCs describing the time disorder thermal averaging over a canonical ensemble is described by two crucial components, the Boltzmann factor on which the general eigenstate dependent spectrum appears and also the temperature independent micro-canonical part of the OTOCs. At the end, we need to take the sum over all possible eigenstates, which will finally give a simplified closed expressions for OTOCs in the present context. Due to the appearance of the eigenstates from the time-independent Hamiltonian, this particular procedure will reduce the job extremely to study the time-dependent behavior of all the previously mentioned OTOCs that we have defined earlier in this paper. In the rest of the paper, we followed this prescription, which is only valid for time-independent Hamiltonian which have their own well-defined eigenstates. For more details, see the rest of the computations and related discussion that we studied in this paper. Most importantly, using this formalism, we can compute all of these OTOCs in a very simple model-independent way. The other technique is more complicated than the previously discussed one. In this case, one starts with a time-dependent Hamiltonian of the theory and uses the well-known Schwinger Keldysh formalism, which is a general path integral framework at finite temperature, for the study of the time evolution of a quantum mechanical system, which is in the out-of-equilibrium state. At the early time scale, once a small perturbation or a response is provided to a quantum system, then it is described by a out-of-equilibrium process within the framework of quantum statistical mechanics, and the present formalism provides us the sufficient tools and techniques, using which one can compute the expressions for the OTOCs. Not only that, the late time behavior of such an OTOC is described by a saturation behavior for chaotic systems, from which one can compute the various characteristic features of large time equilibrium behavior from these OTOCs.
• Formalism III:
  The third approach is based on the circuit quantum complexity [97,98,99,100,101,102,103], which is relatively a very new concept and physically defined as the minimum number of unitary operators, commonly known as quantum gates, that are specifically required to construct the desired target quantum state from a suitable reference quantum mechanical state. In a more generalized physical prescription, quantum mechanical complexity can serve as one of several strong diagnostics for probing the time disorder averaging phenomena of a quantum mechanical system or quantum randomness. The underlying physical concept of circuit complexity can essentially provide essential information regarding various aspects of quantum mechanical randomness, such as the concept of scrambling time, Lyapunov exponent, (The associated time scale when the quantum circuit complexity starts to grow is usually identified to be the scrambling time scale and, in the representative plot with respect to the time scale, particularly, the magnitude of the slope of the linear portion of the curve is physically interpreted as the Lyapunov exponent for the specific systems where the general quantum randomness or the time disorder averaging phenomena is described by quantum chaos.), etc., which are particularly the key features of the study of quantum mechanical chaos. One can further compare between the physical outcomes of the two strongest measures of quantum randomness, which are appearing from out-of-time-order correlators (OTOCs) and the quantum mechanical circuit complexity, and comment further that, for a specific quantum system which one is capturing, there is more information regarding the description of quantum mechanical randomness.

In this paper, we generalize the study of OTOCs for investigating the phenomena of quantum randomness in various Supersymmetric integrable quantum mechanical models. The main motivation behind introducing the concept of Supersymmetry lies in the well-established fact that, for any Supersymmetric quantum mechanical model, the original Hamiltonian is always associated with a partner Hamiltonian, which, in general, is widely different from its original counterpart in terms of eigenstates. Though not always, it is possible that the quantum mechanical model under consideration attains vastly different properties in the context of quantum randomness due to the introduction of Supersymmetry within the framework of quantum mechanics. It is our expectation that this generalization of the study of all the classes of OTOCs would provide an understanding about the significant role that Supersymmetry plays in modifying the randomness behavior of the quantum mechanical models under investigation. Most importantly, from the present study, it will be clear that the additional inclusion of symmetries in the form of Supersymmetry will all affect—and, if so, then how much it will affect—the time disorder averaging phenomena for a canonical and micro-canonical statistical ensemble studied within the framework of out-of-equilibrium quantum statistical mechanics. In the eigenstate representation of the OTOCs, we actually studied three types of OTOC in this paper, out of which one of them is commonly studied in the literature, mostly used to explain the phenomena of quantum mechanical randomness (not only chaotic behavior, but also a general feature including any types of non-chaotic behavior), and the other two OTOCs that we have included in this paper might not be completely independent physical information of each other, but it is extremely important to capture the complete quantum mechanical effect in the quantum mechanical correlation functions to give a more general physical interpretation and a complete and detailed description of time disorder averaging phenomena for various quantum statistical ensembles within the framework of Supersymmetric quantum mechanics. Now, if we cannot able to find out the eigenstate representation of a given time-independent Hamiltonian within the framework of Supersymmetry, then the prescribed methodology for computing the general class of OTOCs in terms of the simplest eigenstate representation will not work. This can happen for the physical systems which are actually described by the time-dependent Hamiltonians. In that case, to compute all of these previously mentioned general classes of OTOCs, one needs to use the quantum mechanical path integral generalization of the present framework, which is commonly described by Schwinger Keldysh formalism, at finite temperature within the framework of Supersymmetric quantum mechanics. The good news is physical outcomes of such a generalization also have not been studied yet, but we have a future plan to look into at this in detail and are also hopeful that we will get a non-trivial and better understanding of various Supersymmetric quantum mechanical models out of these computations.

The mnemonic diagram shows the organization of the entire paper is appearing in Figure 1. Also the out-of-time-ordered correlation (OTOC) team is featured in Figure 2.

Organization of the paper is as follows

• In Section 3, we provide a brief review of the concept of Supersymmetric Quantum Mechanics (QM).
• In Section 4, we explain how the phenomenon of Quantum Randomness can be diagnosed through the out of time ordered correlators.
• In Section 5, we provide a model-independent eigenstate representation of the 2- and the 4-point correlators of all the three kinds defined equally well for any QM model with well defined eigenstates.
• In Section 6, we explicitly calculate the correlators for the Supersymmetric Harmonic Oscillator.
• In Section 7, we provide the numerical calculations of the correlators for the Supersymmetric 1D infinite potnetial well.
• In Section 8, we discuss the semiclassical analogue results for the two Supersymmetric QM models.
• Finally, we conclude with the most important observations from our analysis of the considered Supersymmetric QM models.

2. Lexicography

The lexicographic notations used in this paper is appearing in the Table 1, which will be very helpful to follow various types of notations and its physical interpretations frequently used in this paper.

3. A Short Review of Supersymmetric Quantum Mechanics

The theory of Supersymmetric Quantum Mechanics [104,105,106,107,108] relates quantum eigenfunctions and the corresponding eigenvalues between two partner Hamiltonians through an intertwining relationship using the so-called charge operators. It generally uses the technique of factorizing the Hamiltonian in terms of the intertwining operators, hence determining the superpotential using the well known Riccati equation. The idea of factorizing generally allows one to express the superpotential in terms of the ground state wave-function of the original Hamiltonian of the quantum mechanical model under consideration. The process of factorization is mathematically described by the following equation:

$$\begin{array}{c}\hfill H_1=A^{†}A,\end{array}$$

(7)

where A and $A^{†}$ are the intertwining operators which are defined from the superpotential as:

$$\begin{array}{c}\hfill A=\frac{1}{\sqrt{2}}\frac{d}{dx}+W\left(x\right);A^{†}=-\frac{1}{\sqrt{2}}\frac{d}{dx}+W\left(x\right).\end{array}$$

(8)

Within the framework of Supersymmetry, the ground state energy is usually taken to be zero, which is well justified because it is only the relative energy difference that matters. For a zero energy ground state, the Schr$\ddot{\mathrm{o}}$dinger equation can be written as:

$$\begin{array}{c}\hfill H_1|{\mathsf{\Psi }}_0\rangle =0.\end{array}$$

(9)

Substituting the expressions for A and $A^{†}$ in the above equation it is not very hard to derive the Riccati equation, which further gives a way of writing the potential in terms of the superpotential given by the following expression:

$$\begin{array}{c}\hfill V_1\left(x\right)=W^2\left(x\right)-\frac{1}{\sqrt{2}}{W}^{\prime }\left(x\right),\end{array}$$

(10)

where ${}^{\prime }$ corresponds to $d/dx$ in the above equation. It is often an overrated fact that one needs to know the form of the potential guiding the Hamiltonian to have an idea about the wave functions of the quantum mechanical system, and the fact that the knowledge of the ground state wave function allows one to exactly know the potential associated with the system is overlooked. However, in Supersymmetric quantum mechanics, one generally utilizes this unappreciated fact to construct the potential from the known ground state wave function with zero modes:

$$\begin{array}{c}\hfill W\left(x\right)=-\frac{1}{\sqrt{2m}}\frac{{\psi }_0^{\prime }\left(x\right)}{{\psi }_0\left(x\right)}.\end{array}$$

(11)

The knowledge of superpotential allows one to determine the Supersymmetric partner potential via the following equation:

$$\begin{array}{c}\hfill V_2\left(x\right)=W^2\left(x\right)+\frac{1}{\sqrt{2m}}{W}^{\prime }\left(x\right).\end{array}$$

(12)

The partner Hamiltonian is constructed by reversing the order of the intertwining operators used in the factorization of the original Hamiltonian. The energy eigenvalues and the eigenstates of the original and the partner Hamiltonian are not independent of each other, and that is where the beauty of Supersymmetry lies. Knowing the original Hamiltonian and its ground state, one can easily determine the energy spectrum of the partner Hamiltonian. The eigenvalues of $H_1$ (original Hamiltonian) and $H_2$ (partner Hamiltonian) are related via the following equation:

$$\begin{array}{c}\hfill E_n^{\left(2\right)}=E_{n+1}^{\left(1\right)};E_0^{\left(1\right)}=0\forall n=0,1,2,\cdots \end{array}$$

(13)

It is easy to show that the knowledge of the eigenfunctions of $H_1$ can be used to derive the eigenfunctions of $H_2$ using the A operator and the eigenfunctions of $H_1$ from that of $H_2$ using the $A^{†}$ operator. The role of the operators A and $A^{†}$, apart from the conversion of an eigenfunction of the original Hamiltonian to that of its partner Hamiltonian with the same energy, also destroys or creates one node in the eigenfunction. This justifies the absence of the zero energy or the ground energy state of the partner Hamiltonian. One can put this argument simply by stating that the operator A converts an energy state of the original Hamiltonian into a lower energy state of the partner Hamiltonian keeping the energy value of the state constant. A ${}^{†}$, on the other hand, does the opposite conversion, i.e., takes an energy state of the partner Hamiltonian and converts it into an higher energy state of the original Hamiltonian, keeping the energy eigenvalue fixed.

A Supersymmetric quantum mechanical model is generally described by a Hamiltonian having the following form:

$$\begin{array}{c}\hfill H_{\mathrm{SUSY}}=\left(\begin{array}{ccc}{H}_1& & 0\\ 0& & H_2\end{array}\right).\end{array}$$

(14)

In general, for such a Hamiltonian, a quantum state is represented by:

$$\begin{array}{c}\hfill {|{\mathsf{\Psi }}_n\rangle }^{\mathrm{T}}=\left(|{\psi }^{\left(1\right)}\rangle |{\psi }^{\left(2\right)}\rangle \right),\end{array}$$

(15)

where, $|{\psi }^{\left(1\right)}\rangle$ and $|{\psi }^{\left(2\right)}\rangle$ are the wave functions of the original and the partner Hamiltonian, respectively.

The prime objective of this paper is to provide an eigenstate representation of the desired OTOCs that we have already defined in the introduction, using which we study the various well known quantum mechanical models in the context of Supersymmetry to study the general aspects of time disorder averaging phenomena. With this aim, one would generally look for an eigenstate of the Hamiltonian under inspection. Remembering the relation between the energy eigenvalues and eigenstates of the original and the partner Hamiltonian, it can be easily verified that the wave function given by Equation (15) is not an eigenstate of the Hamiltonian given in Equation (14). We, therefore, take the wave function of the Hamiltonian to be of the form

$$\begin{array}{c}\hfill {|{\mathsf{\Psi }}_n\rangle }^{\mathrm{T}}=\frac{1}{\sqrt{2}}\left(|{\psi }^{\left(1\right)}\rangle |{\psi }^{\left(2\right)}\rangle \right),\end{array}$$

(16)

which indeed represents a normalized eigenfunction of the Hamiltonian of the Supersymmetric quantum mechanical systems considered in this work.

$$\begin{array}{cc}\hfill H_{\mathrm{SUSY}}|{\mathsf{\Psi }}_n\rangle & =\frac{1}{\sqrt{2}}\left(\begin{array}{ccc}{H}_1& & 0\\ 0& & H_2\end{array}\right)\left(\begin{array}{c}|{\psi }_n^{\left(1\right)}\rangle \\ |{\psi }_{n-1}^{\left(2\right)}\rangle \end{array}\right)=\frac{1}{\sqrt{2}}\left(\begin{array}{c}{E}_n^{\left(1\right)}|{\psi }_n^{\left(1\right)}\rangle \\ E_{n-1}^{\left(2\right)}|{\psi }_{n-1}^{\left(2\right)}\rangle \end{array}\right)=\frac{E_n^{\left(1\right)}}{\sqrt{2}}\left(\begin{array}{c}|{\psi }_n^{\left(1\right)}\rangle \\ |{\psi }_{n-1}^{\left(2\right)}\rangle \end{array}\right),\hfill \end{array}$$

where we have used the relation $E_{n-1}^{\left(2\right)}=E_n^{\left(1\right)}$. If we use Equation (15), then $|{\mathsf{\Psi }}_n\rangle$ does not remain an eigenstate of $H_{\mathrm{SUSY}}$, as we show below.

$$\begin{array}{cc}\hfill H_{\mathrm{SUSY}}|{\mathsf{\Psi }}_n\rangle & =\left(\begin{array}{ccc}{H}_1& & 0\\ 0& & H_2\end{array}\right)\left(\begin{array}{c}|{\psi }_n^{\left(1\right)}\rangle \\ |{\psi }_n^{\left(2\right)}\rangle \end{array}\right)=\left(\begin{array}{c}{E}_n^{\left(1\right)}|{\psi }_n^{\left(1\right)}\rangle \\ E_n^{\left(2\right)}|{\psi }_n^{\left(2\right)}\rangle \end{array}\right).\hfill \end{array}$$

Since the energy associated with the nth energy eigenstate of the original Hamiltonian, $H_1$ is not equal to that of its associated partner Hamiltonian $H_2$, i.e., $E_n^{\left(1\right)}\ne E_n^{\left(2\right)}$, $|{\mathsf{\Psi }}_n\rangle$ fails to be an eigenstate of $H_{\mathrm{SUSY}}$.

4. General Remarks on Time Disorder Averaging and Thermal OTOCs

Quantum randomness, using which we have the prime objective to technically demonstrate the time disorder averaging phenomena, is actually a very broad topic of research in theoretical physics, and there are many ways and possibilities, using which one can explicitly quantify this phenomena in the quantum regime. Quantum correlators of different orders are one of them. When a quantum state evolves to reach equilibrium at the late time scales, in that case, the overall amplitude of the correlators also evolves with the evolutionary time scales, which are usually described in terms of the fundamental quantum operators, and the time evolution of these correlators can show the presence of time disorder averaging in the form of chaotic or non-chaotic, periodic, or aperiodic random behavior in the quantum mechanical system under study. Thermal average over the canonical statistical ensemble of a quantum operator is a very powerful technique, using which one can explicitly study the time-dependent exponential growth (chaotic) or some other time-dependent non-chaotic random behavior of an operator for a quantum system that is in out of equilibrium after giving an external response. For a very long time, it was not very clear how one can actually quantify these quantum correlation functions within the framework of out-of-equilibrium quantum statistical physics. Following the previous set of works, the present work helps us to quantify, as well as to physically understand, the impact of them in the present context. In this paper, we are actually interested in three specific kinds of OTOCs, which are described by six sets of correlators, given by:

• 2-point OTOCs: $\langle \left[x\left(t_1\right),x\left(t_2\right)\right]\rangle$, $\langle \left[p\left(t_1\right),p\left(t_2\right)\right]\rangle$, $\langle \left[x\left(t_1\right),p\left(t_2\right)\right]\rangle$,
• 4-point OTOCs: $\langle {\left[x\left(t_1\right),x\left(t_2\right)\right]}^2\rangle$, $\langle {\left[p\left(t_1\right),p\left(t_2\right)\right]}^2\rangle$, $\langle {\left[x\left(t_1\right),p\left(t_2\right)\right]}^2\rangle$,

where x is the quantum position operator, p is the associated canonically conjugate momentum, and, most importantly, both the quantum operators are defined at different time scales, which is one of the prime requirements in studying the effect of time disorder averaging phenomena through the above set of OTOCs. In addition, it is important to note that the symbol $\langle \cdots \rangle$ actually represents the thermal average of a time-dependent quantum operator over a canonical ensemble within the framework of quantum mechanics, which is technically defined as:

$$\begin{array}{c}\hfill \langle \mathcal{O}\left(t\right)\rangle :=\frac{1}{Z\left(t\right)}\mathrm{Tr}\left[exp(-\beta H)\mathcal{O}\left(t\right)\right]=\mathrm{Tr}\left[{\rho }_{\beta }\mathcal{O}\left(t\right)\right],\end{array}$$

(17)

where the partition function Z and thermal density matrix operator ${\rho }_{\beta }$ in terms of a quantum system Hamiltonian, H, are already defined in the introduction of this paper. Since we are dealing with quantum mechanical operators, using which we are trying to understand the impact of random features in the quantum regime, it is quite expected to start with the fact that thermal 1-point function of the position operator x and momentum operator p defined at a specific time scale are zero, which can be technically demonstrated as:

$$\begin{array}{c}\hfill \langle x\left(t\right)\rangle =0,\langle p\left(t\right)\rangle =0,\end{array}$$

(18)

where t is the associated time scale on which both of the quantum operators are evolving. For this specific reason, the explicit study and the computation of these 1-point functions are not at all important in the present context of discussion. On the other hand, due to the time translational symmetry in these thermal correlators, which is actually described by the well known Kubo Martin Schwinger condition, all the odd point OTOCs appearing in the present context will be trivially zero and, for that reason, not the object of interest in the present context of study. This can be further technically demonstrated as:

$$\begin{array}{ccc}& & \underline{\mathbf{Odd}\mathbf{x}\mathbf{correlator}:}\hfill \\ & & \langle x\left(t_1\right)x\left(t_2\right)x\left(t_3\right)\cdots \mathrm{odd}\mathrm{number}\mathrm{of}\mathrm{terms}\rangle =0,\hfill \end{array}$$

(19)

$$\begin{array}{ccc}& & \underline{\mathbf{Odd}\mathbf{p}\mathbf{correlator}:}\hfill \\ & & \langle p\left(t_1\right)p\left(t_2\right)p\left(t_3\right)\cdots \mathrm{odd}\mathrm{number}\mathrm{of}\mathrm{terms}\rangle =0,\hfill \end{array}$$

(20)

$$\begin{array}{ccc}& & \underline{\mathbf{One}\mathbf{p}\&\mathbf{even}\mathbf{x}\mathbf{correlator}:}\hfill \\ & & \langle p\left(t_1\right)x\left(t_2\right)x\left(t_3\right)\cdots \mathrm{odd}\mathrm{number}\mathrm{of}\mathrm{terms}\rangle =0,\hfill \end{array}$$

(21)

$$\begin{array}{ccc}& & \underline{\mathbf{One}\mathbf{x}\&\mathbf{even}\mathbf{p}\mathbf{correlator}:}\hfill \\ & & \langle x\left(t_1\right)p\left(t_2\right)p\left(t_3\right)\cdots \mathrm{odd}\mathrm{number}\mathrm{of}\mathrm{terms}\rangle =0.\hfill \end{array}$$

(22)

This implies that we are only left with all even order OTOCs, out of which, in this paper, we are explicitly computing the physical outcomes from the six sets of time-dependent correlators, described by the previously mentioned 2-point and the 4-point of all possible OTOCs.

Among these correlators, two are made of different operators, which will show the perturbation of one operator measured at one time scale to the other measured at a different time, and vice versa. The other four operators are the new sets of 2- and 4-point correlators that are defined in terms of the same quantum operators, which are basically capturing the quantum effect of the self-correlation of one operator on itself having any arbitrary time-dependent profile in general to describe the phenomena of time disordering within the framework of quantum mechanics. In specific cases, it may happen that these newly defined operators show exponential growth or some other kind of growth, which is periodic or aperiodic, in the corresponding associated time scales of the quantum mechanical operator on which we are interested in the present context. In a more general context, one can see, by studying different kinds of physical systems available in the literature, where the 2-point self-correlation will decay exponentially with an associated time scale of $t_d$, widely known as “dissipation time scale”. This particular time scale is playing the role of transition scale in the present context after reaching that the 4-point correlators can be factorized into the product of two 2-point correlation functions representing disconnected diagrams within the framework of quantum field theory. In addition, it is important to note that, here, at the “dissipation time scale”, all other terms are exponentially suppressed by the factor $e^{-t/t_d}$, which will completely disappear from the factorized version of the 4-point correlators, on which we are interested in this paper, in the large time limit given by $t\gg t_d$ with $t\to \infty$. In usual prescription of the “dissipation time scale”, $t_d$ is identified with the inverse temperature $\beta$, i.e., $t_d\sim \beta =1/T$, where we use Boltzmann constant $k_{\mathrm{B}}=1$, and T physically represents the equilibrium temperature of the quantum statistical ensemble on which we are interested in. So, one can translate the large time limit in terms of the associated equilibrium temperature as $tT\gg 1$, which is not obviously true for a zero temperature case but can be justifiable in any (small or large) temperature of the system under consideration. For example, we now look into a specific 4-point thermal correlation function at the vicinity of the previously mentioned dissipation time scale, around which one can factorize it in the following specific form:

$$\begin{array}{c}\hfill \underset{\mathbf{Unequal}\mathbf{time}\mathbf{4}-\mathbf{point}\mathbf{correlator}}{\underbrace{\langle x\left(t_1\right)x\left(t_1\right)p\left(t_2\right)p\left(t_2\right)\rangle }}\approx \underset{\mathbf{Equal}\mathbf{time}\mathbf{2}-\mathbf{point}\mathbf{correlator}}{\underbrace{\langle x\left(t_1\right)x\left(t_1\right)\rangle }}\underset{\mathbf{Equal}\mathbf{time}\mathbf{2}-\mathbf{point}\mathbf{correlator}}{\underbrace{\langle p\left(t_2\right)p\left(t_2\right)\rangle }}\\ \hfill +\underset{\mathbf{Subdominant}\mathbf{decaying}\mathbf{contribution}}{\underbrace{\mathcal{O}\left(e^{-t/t_d}\right)}},\end{array}$$

(23)

where, in this aforementioned factorization, $\mathcal{O}\left(e^{-t/t_d}\right)$ are the higher order correction terms, which are actually sub-dominant at the vicinity of the dissipation time scale. One more thing we can observe from the aforementioned factorization is that the individual contributions of the 2-point contributions do not mix up the time scales, and, for this reason, they can be written in terms of the product of two equal time 2-point correlators. In addition, for this particular example, when we are thinking of doing the computation with two different operators, after doing the factorization, one can easily observe that the two different operators do not mix with each other at the level of 2-point correlators. So, we are mainly interested in the terms that cannot be written in time ordered form, and those terms provide more insight to the randomness present in a system. Now, we would like to go one step further and normalize the OTOC, which is basically related to this factorization process of the 4-point correlators in terms disconnected equal time 2-point contributions. The process of normalization actually helps us to reduce the unwanted fluctuations from the computed OTOCs, which further allows us to give a clearer picture of the time-dependent behavior of the desired OTOCs in which we are interested in this paper. We do not normalize the 2-point correlation functions as they are the main building blocks of our computation of OTOCs; hence, we only normalize the 4-point OTOC as given by the following expression. We can write $-\langle {\left[x\left(t_1\right),p\left(t_2\right)\right]}^2\rangle$ in the following simplified mathematical form:

$$\begin{array}{c}\hfill -\langle {\left[x\left(t_1\right),p\left(t_2\right)\right]}^2\rangle =2\left\{\left[\langle x\left(t_1\right)x\left(t_1\right)p\left(t_2\right)p\left(t_2\right)\rangle \right]-\mathrm{Re}\left[\langle x\left(t_1\right)p\left(t_2\right)x\left(t_1\right)p\left(t_2\right)\rangle \right]\right\}.\end{array}$$

(24)

After imposiing the previously mentioned constrained obtained at the vicinity of the dissipation time scale, one can further simplify the expression for the mentioned correlator as given by the following expression:

$$\begin{array}{c}\hfill -\langle {\left[x\left(t_1\right),p\left(t_2\right)\right]}^2\rangle =2\left\{\langle x\left(t_1\right)x\left(t_1\right)\rangle \langle p\left(t_2\right)p\left(t_2\right)\rangle -\mathrm{Re}\left[\langle x\left(t_1\right)p\left(t_2\right)x\left(t_1\right)p\left(t_2\right)\rangle \right]\right\}+\mathcal{O}\left(e^{-t/t_d}\right).\end{array}$$

(25)

Now, we normalize this aforementioned quantity using two different equal time 2-point correlators, which we have obtained from the factorization in terms of the disconnected pieces. Consequently, we get the following simplified form of the normalized OTOC:

$$\begin{array}{c}\hfill {\mathcal{C}}^{\left(1\right)}(t_1,t_2)=\frac{-\langle {\left[x\left(t_1\right),p\left(t_2\right)\right]}^2\rangle }{\langle x\left(t_1\right)x\left(t_1\right)\rangle \langle p\left(t_2\right)p\left(t_2\right)\rangle }=2\left[1-\frac{\mathrm{Re}\left[\langle x\left(t_1\right)p\left(t_2\right)x\left(t_1\right)p\left(t_2\right)\rangle \right]}{\langle x\left(t_1\right)x\left(t_1\right)\rangle \langle p\left(t_2\right)p\left(t_2\right)\rangle }\right].\end{array}$$

(26)

The first expression in Equation (26) is a universal contribution, which will always appear for the quantum systems where the previously mentioned factorization process works at the vicinity of the dissipation time scale. On the other hand, the second term of Equation (26) is basically representing a normalized version of the previously mentioned 4-point function, which can only appear after dissipation time. The other two non-trivial OTOCs we are also interested in this paper are of the form $-\langle {\left[x\left(t_1\right),x\left(t_2\right)\right]}^2\rangle$ and $-\langle {\left[p\left(t_1\right),p\left(t_2\right)\right]}^2\rangle$, and, by following the same logical argument at the vicinity of the dissipation time scale, we can normalize them, as well, and write them in the following simplified mathematical forms:

$$\begin{array}{c}\hfill C^{\left(2\right)}(t_1,t_2)=\frac{-\langle {\left[x\left(t_1\right),x\left(t_2\right)\right]}^2\rangle }{\langle x\left(t_1\right)x\left(t_1\right)\rangle \langle x\left(t_2\right)x\left(t_2\right)\rangle }=2\left[1-\frac{\mathrm{Re}\left[\langle x\left(t_1\right)x\left(t_2\right)x\left(t_1\right)x\left(t_2\right)\rangle \right]}{\langle x\left(t_1\right)x\left(t_1\right)\rangle \langle x\left(t_2\right)x\left(t_2\right)\rangle }\right],\end{array}$$

(27)

and

$$\begin{array}{c}\hfill C^{\left(3\right)}(t_1,t_2)=\frac{-\langle {\left[p\left(t_1\right),p\left(t_2\right)\right]}^2\rangle }{\langle p\left(t_1\right)p\left(t_1\right)\rangle \langle p\left(t_2\right)p\left(t_2\right)\rangle }=2\left[1-\frac{\mathrm{Re}\left[\langle p\left(t_1\right)p\left(t_2\right)p\left(t_1\right)p\left(t_2\right)\rangle \right]}{\langle p\left(t_1\right)p\left(t_1\right)\rangle \langle p\left(t_2\right)p\left(t_2\right)\rangle }\right].\end{array}$$

(28)

Now, since these OTOCs acts as a theoretical probe to know about the generic chaotic or non-chaotic time-dependent behavior of quantum system, it has to satisfy the following constraints in terms of the 4-point correlations, which survived in the vicinity of the previously introduced dissipation time scale, as given by:

$$\begin{array}{ccc}& & \frac{\mathrm{Re}\left[\langle x\left(t_1\right)p\left(t_2\right)x\left(t_1\right)p\left(t_2\right)\rangle \right]}{\langle x\left(t_1\right)x\left(t_1\right)\rangle \langle p\left(t_2\right)p\left(t_2\right)\rangle }=\left\{\begin{array}{cc}{\displaystyle 1-{\mathcal{A}}_{12}^{\left(1\right)}{e}^{{\lambda }_1\left(\frac{t_1+t_2}{2}\right)}\mathrm{with}{\lambda }_1\le \frac{2\pi }{{\beta }_1},}\hfill & \hfill \mathbf{Chaotic}\\ \\ {\displaystyle 1-{\mathcal{B}}_{12}^{\left(1\right)}{f}_1(t_1,t_2).}\hfill & \hfill \mathbf{Non-Chaotic}\end{array}\right.\hfill \end{array}$$

(29)

$$\begin{array}{ccc}& & \frac{\mathrm{Re}\left[\langle x\left(t_1\right)p\left(t_2\right)x\left(t_1\right)p\left(t_2\right)\rangle \right]}{\langle x\left(t_1\right)x\left(t_1\right)\rangle \langle p\left(t_2\right)p\left(t_2\right)\rangle }=\left\{\begin{array}{cc}{\displaystyle 1-{\mathcal{A}}_{12}^{\left(2\right)}{e}^{{\lambda }_2\left(\frac{t_1+t_2}{2}\right)}\mathrm{with}{\lambda }_2\le \frac{2\pi }{{\beta }_2},}\hfill & \hfill \mathbf{Chaotic}\\ \\ {\displaystyle 1-{\mathcal{B}}_{12}^{\left(2\right)}{f}_2(t_1,t_2).}\hfill & \hfill \mathbf{Non-Chaotic}\end{array}\right.\hfill \end{array}$$

(30)

$$\begin{array}{ccc}& & \frac{\mathrm{Re}\left[\langle p\left(t_1\right)p\left(t_2\right)p\left(t_1\right)p\left(t_2\right)\rangle \right]}{\langle p\left(t_1\right)p\left(t_1\right)\rangle \langle p\left(t_2\right)p\left(t_2\right)\rangle }=\left\{\begin{array}{cc}{\displaystyle 1-{\mathcal{A}}_{12}^{\left(3\right)}{e}^{{\lambda }_3\left(\frac{t_1+t_2}{2}\right)}\mathrm{with}{\lambda }_3\le \frac{2\pi }{{\beta }_3},}\hfill & \hfill \mathbf{Chaotic}\\ \\ {\displaystyle 1-{\mathcal{B}}_{12}^{\left(3\right)}{f}_3(t_1,t_2).}\hfill & \hfill \mathbf{Non-Chaotic}\end{array}\right.\hfill \end{array}$$

(31)

The aforementioned expression captures all the possibilities which one can observe in different quantum mechanical systems available in nature. Here, ${\mathcal{A}}_{12}^{\left(1\right)}$, ${\mathcal{A}}_{12}^{\left(2\right)}$, and ${\mathcal{A}}_{12}^{\left(3\right)}$ are the quantum mechanical model-dependent pre-factors which show exponential growth (chaotic behavior) in the 4-point correlator with respect to the time scales associated with the system under study. On the other hand, ${\mathcal{B}}_{12}^{\left(1\right)}$, ${\mathcal{B}}_{12}^{\left(2\right)}$, and ${\mathcal{B}}_{12}^{\left(3\right)}$ are the quantum mechanical model-dependent pre-factors which show any type of time-dependent fluctuations (non-chaotic behavior). In addition, for the general prescription, the quantum Lyapunov exponents, ${\lambda }_1$, ${\lambda }_2$, and ${\lambda }_3$, are not the same for which the MSS bound on quantum chaos from these three cases are also not the same. Consequently, the equilibrium saturation temperatures at the late time scales from these three OTOCs also differ from each other, i.e., ${\beta }_1\ne {\beta }_2\ne {\beta }_3$. In addition, it is important to point out that the mathematical structure of the time-dependent functions $f_1(t_1,t_2)$, $f_2(t_1,t_2)$, and $f_3(t_1,t_2)$ are also different for general quantum mechanical set ups. When we are considering the quantum mechanical models described by time-dependent Hamiltonians, in that case, these expectations and all sorts of predictions work very well. However, when we are thinking about particularly quantum mechanical models which are described by the time-independent Hamiltonian and the eigenstate representation of OTOCs, in that situation, one might have further simplifications. There might be a possibility to have an underlying connection between the two functions $f_2(t_1,t_2)$ and $f_3(t_1,t_2)$ in the eigenstate representation of the OTOCs in the non-chaotic case, and, for this reason, they might not be capturing completely independent information of the time disorder averaging. On the other hand, in the chaotic case, if three of the OTOCs independently show exponential growth in the time scale, the quantum Lyapunov exponents and the related equilibrium saturation temperatures are not at all same, even in the eigenstate representation. But, if the first OTOC is showing the chaotic behavior, and other two are not, in that case, the previous connection between the two functions $f_2(t_1,t_2)$ and $f_3(t_1,t_2)$ holds good in the eigenstate representation.

In Figure 3 and Figure 4, we present the diagrammatic representations of all types of OTOCs in which we are interested in this paper. Particularly, in Figure 3, we explicitly depict the possible 2-point OTOCs. Here, we have three possibilities, which are given by $-\langle [x\left(t_1\right),p\left(t_2\right)]\rangle$, $-\langle [x\left(t_1\right),x\left(t_2\right)]\rangle$, and $-\langle [p\left(t_1\right),p\left(t_2\right)]\rangle$. Since each OTOC is made up of a commutator bracket in the quantum mechanical description, for each case, we have two different contributions having overall opposite signatures. Further, to draw the representative diagrams, we need to consider the flow of time scale from $t_1$ to $t_2$ or $t_2$ to $t_1$. In all the representative diagrams, the two vertical solid thick lines correspond to the specific time slice having time $t=t_1$ and $t=t_2$, respectively. It is understandable, from the mathematical structure of the mentioned 2-point OTOCs, that, since the correlator involves only two time scales, that is why drawing two vertical parallel lines are physically justifiable in the present context. Because of the previously mentioned flow of time scale from $t_1$ to $t_2$ or $t_2$ to $t_1$ for each 2-point OTOCs, we have two possible diagrams. So, as a whole, for the 2-point OTOCs, one can draw six possibilities. In addition, by studying each of the diagrammatic representations, we can also observe that each of the contributions of the 2-point functions are represented by separate lines with representative arrows which completely depend on the structure of the individual 2-point correlators. To differentiate between these two contributions, we have used red dotted line and blue solid line in the representative diagrams. Further, in Figure 4, we have explicitly shown the possible 4-point OTOCs in the present context of discussion. Here, we can draw three possible representative diagrams, which are coming from the three possible OTOCs, as given by, $-\langle {[x\left(t_1\right),p\left(t_2\right)]}^2\rangle$, $-\langle {[x\left(t_1\right),x\left(t_2\right)]}^2\rangle$, and $-\langle {[p\left(t_1\right),p\left(t_2\right)]}^2\rangle$. Here, as we can see that each of the OTOCs is made up of commutator bracket squared contributions in the quantum description, for each case, we have four contributions if we expand them. Out of these four 4-point thermal correlators, two of them have a positive signature, and other two have an overall negative signature in the front of each contribution. Just like the 2-point OTOCs, here, we also need to consider the flow of time scale from $t_1$ to $t_2$ or from $t_2$ to $t_1$. It is understandable, from the mathematical structure of the mentioned 4-point OTOCs, that, because the correlator involves only two time scales instead of four different time scales, is why drawing two vertical parallel lines to represent the time slice at $t=t_1$ and $t=t_2$ are physically justifiable in the present context. Now, because of the time scale flow, each of the 4-point OTOCs have two contributions in the diagrammatic representation. In addition, for a given time flow, we have four possible diagrams, which we show in a single diagram, for the sake of simplicity. So, as a whole, to consider both the possibilities of the time flow, we have cumulatively 24 diagrams from the 4-point OTOCs. Like the previous case, here, to differentiate between each of the individual terms for a given OTOC with a specified time flow, we have also used a red dotted line, blue dotted line, and red thick line, respectively, in the representative diagrammatic representations. Last but not least, here, it is important to point that these set of diagrams are the simplest version of the well known Feynman diagrams as appearing in the context of quantum field theory literature. However, within the present framework, we do not have exactly similar Feynman diagrams, but, to understand the structure of the previously mentioned OTOCs, the present version of the diagrammatic representations play a very crucial role. All of the arrows appearing in all the diagrams represent an underlying time disordering upon which we have to take the final average in the present context.

5. Eigenstate Representation of thermal OTOCs in Supersymmetric Quantum Mechanics

In this section, our prime objective is to study various thermal correlators, OTOCs, and see how they can be expressed in a model-independent manner in the framework of quantum mechanics. We also demarcate clearly the effect of Supersymmetry and how it modifies the functional form of the correlators at the end.

To perform this explicit computation, we will follow the prescription outlined in Reference [90] to compute all the thermal OTOCs we have mentioned in the previous section of this paper (Reference [90], particularly, is extremely important for our computation because, here, the authors first have performed the computation of the first OTOC, $-\langle {[x\left(t_1\right),p\left(0\right)]}^2\rangle$ (though, in our computation, we have generalized this to $-\langle {[x\left(t_1\right),p\left(t_2\right)]}^2\rangle$) in the eigenstate formalism for a time-independent quantum mechanical model-independent Hamiltonian.). According to this prescription, for any time-independent Hamiltonian, we define the expectation value of quantum mechanical operators as thermal expectation value in the present context, which one can easily apply for a canonical quantum statistical ensemble. Let us say a quantum mechanical time-dependent operator $A\left(t\right)$ is defined on a specific Hilbert Space $\mathcal{H}$ with an associated Hamiltonian H (which is time-independent, obviously) and the eigenvalues (eigen energy spectrum) of H corresponding to a infinite tower of eigenkets $|{\mathsf{\Psi }}_n\rangle \forall n=0,1,\cdots ,\infty$, where the corresponding energy levels are characterized by the index n. This energy eigenspectrum is represented by $E_n\forall n=0,1,\cdots ,\infty$. Then, for a canonical quantum statistical ensemble, the thermal expectation value of the quantum mechanical time-dependent operator $A\left(t\right)$ at inverse temperature $\beta$, considering the Boltzmann constant $k_B=1$, is defined as:

$$\begin{array}{cc}\hfill {\langle A\left(t\right)\rangle }_{\beta }& =\frac{1}{Z}\mathrm{Tr}\left[e^{-\beta H}A\left(t\right)\right],\hfill \end{array}$$

where the thermal partition function is represented by Z such that $Z=\mathrm{Tr}\left[e^{-\beta H}\right]$. Here, our job is to represent this mathematical trace operation in terms of the sum over all possible eigenstates starting from the ground state ($n=0$). Once we are able to express this operation clearly, then the rest of the story is very typical, and, following this, one can easily compute all of the mentioned OTOCs in the eigenstate representation. It pays to use the thermal representation for expectation values of quantum mechanical operators because there are many physical models and physically relevant toy models which have well-studied structure of time-independent Hamiltonians; hence, we can utilize this property to give an eigenstate representation to the correlators as presented elaborately in this work. Consequently, the analysis presented in this paper is physically justifiable, applicable, and believable, as such, to quantum mechanical models with a well-defined Hamiltonian. In the eigenstate representation, the thermal expectation value of a quantum mechanical time-dependent operator $A\left(t\right)$ can be written as:

$$\begin{array}{c}\hfill {\langle A\left(t\right)\rangle }_{\beta }=\frac{1}{Z}\mathrm{Tr}\left[e^{-\beta H}A\left(t\right)\underset{\mathbf{Identity}\mathbf{operator}}{\underbrace{\sum _n|{\mathsf{\Psi }}_n\rangle \langle {\mathsf{\Psi }}_n|}}\right]=\frac{{\displaystyle \sum _n^{}{e}^{-\beta E_n}\langle {\mathsf{\Psi }}_n\left|A\left(t\right)\right|{\mathsf{\Psi }}_n\rangle }}{{\displaystyle \sum _n^{}{e}^{-\beta E_n}}}.\end{array}$$

Here, we work in the Heisenberg Picture, where the operators evolve with time, as given by ${\mathcal{O}}_H\left(t\right)=e^{iHt}{\mathcal{O}}_S\left(0\right)e^{-iHt}$, where ${\mathcal{O}}_H\left(t\right)$ represents operator ${\mathcal{O}}_H$ at time scale t in Heisenberg picture and ${\mathcal{O}}_S\left(0\right)$ represents its Schr$\ddot{\mathrm{o}}$dinger picture representation at all time scales since operators do not evolve with time in Schr$\ddot{\mathrm{o}}$dinger picture. The latter is denoted as ${\mathcal{O}}_S$ to simplify the notation and make it easier to read.

What do we learn here?:

$$\begin{array}{cc}\hfill & \mathsf{Thermal}\mathsf{Expectation}\mathsf{Value}:{\langle A\left(t\right)\rangle }_{\beta }=\frac{1}{Z}\mathrm{Tr}\left[e^{-\beta H}A\left(t\right)\right]\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill & \mathsf{Heisenberg}\mathsf{Repsentation}\mathsf{of}\mathsf{Operators}:{\mathcal{O}}_H\left(t\right)=e^{iHt}{\mathcal{O}}_S{e}^{-iHt}\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill & \mathsf{Eigenstate}\mathsf{Rep}.\mathsf{of}\mathsf{Thermal}\mathsf{Exp}.:{\langle A\left(t\right)\rangle }_{\beta }=\frac{{\displaystyle \sum _n^{}{e}^{-\beta E_n}\langle {\mathsf{\Psi }}_n\left|A\left(t\right)\right|{\mathsf{\Psi }}_n\rangle }}{{\displaystyle \sum _n^{}{e}^{-\beta E_n}}}\hfill \end{array}$$

(34)

To demonstrate the computation further, let us consider a general time-independent Hamiltonian of the following form:

$$\begin{array}{c}\hfill H(q_i,p_i)=\sum _{i=1}^N{p}_i^2+V\left(q_i\right),\end{array}$$

(35)

where we have used the fact that the mass of all N number of particles are the same and given by $m_i=m=1/2$ to make the further computation simpler. Now, here, it is very easy to prove the following relation (see Reference [90]), which relates the quantum mechanical momentum operator matrix elements with that of the position and the energy operator matrix elements by the following expression:

$$\begin{array}{c}\hfill p_{km}=\frac{i}{2}{E}_{km}{x}_{km}.\end{array}$$

(36)

Furthermore, the role Supersymmetry plays in modification of non-Supersymmetric quantum mechanical observables can be seen vis-a-vis the matrix elements of observables. Let us consider a simple matrix element for the position operator x. In the following chart, we show exactly how the matrix elements in non-Supersymmetric quantum mechanical theories and Supersymmetric quantum mechanical theories are connected:

$$\begin{array}{cc}\hfill x_{mk}& =\langle {\mathsf{\Psi }}_m\left|x\right|{\mathsf{\Psi }}_k\rangle =\underset{\mathsf{neither}\mathbf{m}/\mathbf{k}\mathsf{is}\mathsf{ground}\mathsf{state}}{\underbrace{\frac{1}{2}\left\{\underset{\begin{array}{c}\mathsf{bosonic}\mathsf{part}\\ \mathsf{non}-\mathsf{SUSY}\&\mathsf{SUSY}\end{array}}{{\displaystyle \underbrace{\langle {\psi }_m^B\left|x\right|{\psi }_k^B\rangle }}}+\underset{\begin{array}{c}\mathsf{fermionic}\mathsf{part}\\ \mathsf{SUSY}-\mathsf{only}\end{array}}{{\displaystyle \underbrace{\langle {\psi }_{m-1}^F\left|x\right|{\psi }_{k-1}^F\rangle }}}\right\}}}\begin{array}{c}\\ & \underset{\mathsf{SUSY}}{\overset{\mathsf{switch}\mathsf{off}}{\to }}& \langle {\psi }_m^B\left|x\right|{\psi }_k^B\rangle ,\\ \\ & \underset{\begin{array}{c}\mathsf{if}\mathsf{one}\mathsf{of}\mathbf{m}/\mathbf{k}\mathsf{is}\\ \mathsf{ground}\mathsf{state}\end{array}}{\overset{\mathsf{SUSY}}{\to }}& \frac{1}{\sqrt{2}}\langle {\psi }_m^B\left|x\right|{\psi }_k^B\rangle ,\\ \\ & \underset{\begin{array}{c}\mathsf{both}\mathbf{m}\&\mathbf{k}\mathsf{are}\\ \mathsf{ground}\mathsf{state}\end{array}}{\overset{\mathsf{SUSY}}{\to }}& \langle {\psi }_m^B\left|x\right|{\psi }_k^B\rangle .\end{array}\hfill \end{array}$$

The modifications due to Supersymmetry can clearly be traced back to the fact that we could formulate a wave function of total Hamiltonian as given in Equation (16) from the two partner Hamiltonians $H_1$ and $H_2$, as appearing from the bosonic and the fermion sectors in Supersymmetry. In fact, it is the factorization property of $H_1$ and $H_2$ which generalizes to a more powerful setting in terms of Supersymmetric generalized description of the theory under consideration. One might consider the power of the Supersymmetric description in two ways:

• First, one can merely consider it as a tool to solve non-trivial potentials by means of solutions of their partner ones, provided that the partner ones are easily solvable.
• Secondly, one can consider it as a more unifying description of a more beautiful theory based on the principles of symmetries of nature. It is this second philosophy which has been considered in this work.

We will consider the following six correlators in this work as stated below:

Correlation Functions

$$\begin{array}{cc}\hfill & \mathsf{2}-\mathsf{point}\mathsf{Correlator}:Y^{\left(1\right)}(t_1,t_2)=-{\langle [x\left(t_1\right),p\left(t_2\right)]\rangle }_{\beta }\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill & \mathsf{2}-\mathsf{point}\mathsf{Correlator}:Y^{\left(2\right)}(t_1,t_2)=-{\langle [x\left(t_1\right),x\left(t_2\right)]\rangle }_{\beta }\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill & \mathsf{2}-\mathsf{point}\mathsf{Correlator}:Y^{\left(3\right)}(t_1,t_2)=-{\langle [p\left(t_1\right),p\left(t_2\right)]\rangle }_{\beta }\hfill \\ \hfill \end{array}$$

(39)

$$\begin{array}{cc}\hfill & \mathsf{4}-\mathsf{point}\mathsf{Correlator}:C^{\left(1\right)}(t_1,t_2)=-{\langle {[x\left(t_1\right),p\left(t_2\right)]}^2\rangle }_{\beta }\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill & \mathsf{4}-\mathsf{point}\mathsf{Correlator}:C^{\left(2\right)}(t_1,t_2)=-{\langle {[x\left(t_1\right),x\left(t_2\right)]}^2\rangle }_{\beta }\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill & \mathsf{4}-\mathsf{point}\mathsf{Correlator}:C^{\left(3\right)}(t_1,t_2)=-{\langle {[p\left(t_1\right),p\left(t_2\right)]}^2\rangle }_{\beta }\hfill \\ \hfill & \hfill \end{array}$$

(42)

We will also consider normalized 4-point Correlation Functions, and the discussion pertaining to those is provided in the previous section:

Normalized 4-point Correlation Functions

$$\begin{array}{cc}\hfill & \hfill \\ \hfill & \mathsf{Normalized}\mathsf{4}-\mathsf{point}\mathsf{Correlator}:{\tilde{C}}^{\left(1\right)}(t_1,t_2)\hfill & \hfill =\frac{C^{\left(1\right)}(t_1,t_2)}{{\langle x\left(t_1\right)x\left(t_1\right)\rangle }_{\beta }{\langle p\left(t_2\right)p\left(t_2\right)\rangle }_{\beta }}\\ \hfill & \hfill & \hfill =\frac{-{\langle {[x\left(t_1\right),p\left(t_2\right)]}^2\rangle }_{\beta }}{{\langle x\left(t_1\right)x\left(t_1\right)\rangle }_{\beta }{\langle p\left(t_2\right)p\left(t_2\right)\rangle }_{\beta }}\end{array}$$

(43)

$$\begin{array}{c}\hfill \\ \hfill & \mathsf{Normalized}\mathsf{4}-\mathsf{point}\mathsf{Correlator}:{\tilde{C}}^{\left(2\right)}(t_1,t_2)\hfill & \hfill =\frac{C^{\left(2\right)}(t_1,t_2)}{{\langle x\left(t_1\right)x\left(t_1\right)\rangle }_{\beta }{\langle x\left(t_2\right)x\left(t_2\right)\rangle }_{\beta }}\\ \hfill & \hfill & \hfill =\frac{-{\langle {[x\left(t_1\right),x\left(t_2\right)]}^2\rangle }_{\beta }}{{\langle x\left(t_1\right)x\left(t_1\right)\rangle }_{\beta }{\langle x\left(t_2\right)x\left(t_2\right)\rangle }_{\beta }}\end{array}$$

(44)

$$\begin{array}{c}\hfill \\ \hfill & \mathsf{Normalized}\mathsf{4}-\mathsf{point}\mathsf{Correlator}:{\tilde{C}}^{\left(3\right)}(t_1,t_2)\hfill & \hfill =\frac{C^{\left(3\right)}(t_1,t_2)}{{\langle p\left(t_1\right)p\left(t_1\right)\rangle }_{\beta }{\langle p\left(t_2\right)p\left(t_2\right)\rangle }_{\beta }}\\ \hfill & \hfill & \hfill =\frac{-{\langle {[p\left(t_1\right),p\left(t_2\right)]}^2\rangle }_{\beta }}{{\langle p\left(t_1\right)p\left(t_1\right)\rangle }_{\beta }{\langle p\left(t_2\right)p\left(t_2\right)\rangle }_{\beta }}\end{array}$$

(45)

5.1. Partition Function from Supersymmetric Quantum Mechanics

In the context of Supersymmetric quantum mechanics, the partition function, Z, can be can be expressed in terms of the eigenstate by the following expression:

$$\begin{array}{cc}\hfill Z& =\mathrm{Tr}\left[e^{-\beta H_{\mathrm{SUSY}}}\right]=\sum _m^{}\langle {\mathsf{\Psi }}_m\left|e^{-\beta H_{\mathrm{SUSY}}}\right|{\mathsf{\Psi }}_m\rangle .\hfill \end{array}$$

Here, the underlying relation between Supersymmetric total system and the component (or partner) systems, as discussed in Section 3, is established through the following relations:

$$\begin{array}{cc}& H_{\mathrm{SUSY}}=\left(\begin{array}{cc}{H}_1& 0\\ 0& H_2\end{array}\right)\Rightarrow e^{-\beta H_{\mathrm{SUSY}}}=\left(\begin{array}{cc}{e}^{-\beta H_1}& 0\\ 0& e^{-\beta H_2}\end{array}\right),\hfill \\ & {|{\mathsf{\Psi }}_m\rangle }^{\mathrm{T}}=\frac{1}{\sqrt{2}}\left(|{\psi }_m^{\left(1\right)}\rangle |{\psi }_{m-1}^{\left(2\right)}\rangle \right);{|{\mathsf{\Psi }}_0\rangle }^{\mathrm{T}}=\left(|{\psi }_m^{\left(1\right)}\rangle 0\right).\hfill \end{array}$$

So, the Supersymmetric quantum partition function in the eigenstate representation can be explicitly written as:

$$\begin{array}{cc}\hfill Z& =\langle {\psi }_m^{\left(1\right)}\left|e^{-\beta H_1}\right|{\psi }_m^{\left(1\right)}\rangle {|}_{m=0}+\frac{1}{2}\sum _{m>0}^{}\left\{\langle {\psi }_m^{\left(1\right)}\left|e^{-\beta H_1}\right|{\psi }_m^{\left(1\right)}\rangle +\langle {\psi }_{m-1}^{\left(2\right)}\left|e^{-\beta H_2}\right|{\psi }_{m-1}^{\left(2\right)}\rangle \right\},\hfill \\ \hfill & =\underset{\begin{array}{c}\mathbf{Ground}\mathbf{State}\\ \mathbf{Contribution}\end{array}}{\underbrace{e^{-\beta E_m^{\left(1\right)}}\langle {\psi }_m^{\left(1\right)}|{\psi }_m^{\left(1\right)}\rangle {|}_{m=0}}}+\frac{1}{2}\sum _{m>0}^{}\left\{e^{-\beta E_m^{\left(1\right)}}\langle {\psi }_m^{\left(1\right)}|{\psi }_m^{\left(1\right)}\rangle +e^{-\beta E_{m-1}^{\left(2\right)}}\langle {\psi }_{m-1}^{\left(2\right)}|{\psi }_{m-1}^{\left(2\right)}\rangle \right\}.\hfill \end{array}$$

As shown in Section 3, in Supersymmetric quantum mechanical theories, we have the following constraint:

$$\begin{array}{c}\hfill E_m=E_m^{\left(1\right)}=E_{m-1}^{\left(2\right)},\end{array}$$

(46)

with the requirement that the ground state is eigenstate of $H_1$ only, and all other states are doubly degenerate. It is also important to note that the ground state always has zero energy eigenvalue. It is to be noted that $E_m$ refers to the energy eigenvalue of the total Hamiltonian $H_{\mathrm{SUSY}}$. Using this set of requirements, the aforementioned expression for the quantum partition function within the framework of Supersymmetry can be further simplified as:

$$\begin{array}{cc}\hfill Z& =1+\frac{1}{2}\sum _{m>0}^{}{\left\{e^{-\beta E_m}\left(\langle {\psi }_m^{\left(1\right)}|{\psi }_m^{\left(1\right)}\rangle +\langle {\psi }_{m-1}^{\left(2\right)}|{\psi }_{m-1}^{\left(2\right)}\rangle \right)\right\}}^{}=1+\sum _{m>0}^{}{e}^{-\beta E_m},\hfill \end{array}$$

(47)

which is different compared to usual quantum mechanics results without Supersymmetry, as in the present context ground state energy eigenvalue $E_0=0$, which is not, in general, zero for the other case. So, this implies that, if we separately write down the contribution from the ground state and from all other excited states, instead of writing sum over all eigenstates together, then one can clearly visualize the difference between the results obtained for quantum partition function in the eigenstate representation in both the cases. In the next section, we will provide the summary of all the obtained general model-independent results in which we will implement the aforementioned fact to explicitly show that our obtained results for all types of OTOCs in the framework of Supersymmetric quantum mechanics are different compared to the results obtained from usual quantum mechanics without Supersymmetry.

5.2. Representation of 2-Point OTOC: $Y^{\left(1\right)}(t_1,t_2)$

The first 2-point OTOC is given by the thermal average of the operator $-[x\left(t_1\right),p\left(t_2\right)]$, which is described in the eigenstate representation as:

$$\begin{array}{cc}\hfill Y^{\left(1\right)}(t_1,t_2):& =-{\langle [x\left(t_1\right),p\left(t_2\right)]\rangle }_{\beta }=-\frac{{\displaystyle \sum _m^{}{e}^{-\beta E_m}\langle {\mathsf{\Psi }}_m\left|[x\left(t_1\right),p\left(t_2\right)]\right|{\mathsf{\Psi }}_m\rangle }}{{\displaystyle \left(1+\sum _{m>0}^{}{e}^{-\beta E_m}\right)}}..\hfill \end{array}$$

Using Equation (33) for Heisenberg representation for $x\left(t_1\right)$ and $p\left(t_2\right)$ and inserting the identities between the operators, this 2-point correlator can be written in terms of the micro-canonical correlator, which shows the temperature-independent behavior of the system:

$$\begin{array}{cc}\hfill Y^{\left(1\right)}(t_1,t_2)& =\frac{{\displaystyle \sum _m^{}{e}^{-\beta E_m}{y}_m^{\left(1\right)}(t_1,t_2)}}{{\displaystyle \left(1+\sum _{m>0}^{}{e}^{-\beta E_m}\right)}},\hfill \end{array}$$

(48)

where the micro-canonical 2-point OTOC is defined as:

$$\begin{array}{c}\hfill y_m^{\left(1\right)}(t_1,t_2)=-\langle {\mathsf{\Psi }}_m\left|[x\left(t_1\right),p\left(t_2\right)]\right|{\mathsf{\Psi }}_m\rangle .\end{array}$$

(49)

Expanding the commutator in the definition of the micro-canonical correlator, and using the above relation with appropriate insertion of identities between the operators, it can be shown that the eigenstate representation of the micro-canonical correlator $y_m^{\left(1\right)}(t_1,t_2)$ takes the following form.

$$\begin{array}{c}\hfill y_m^{\left(1\right)}(t_1,t_2)=-i\sum _k^{}{E}_{km}{x}_{mk}{x}_{km}cos{E}_{km}\left(t_1-t_2\right),\end{array}$$

(50)

where we define $E_{mk/m,k}=E_m-E_k$ and $x_{mk/m,k}=\langle {\mathsf{\Psi }}_m\left|x\right|{\mathsf{\Psi }}_k\rangle$.

Substituting the aforementioned expression for the micro-canonical OTOC in the definition of the canonical correlator, $Y^{\left(1\right)}(t_1,t_2)$, we get the following expression:

$$\begin{array}{c}\hfill Y^{\left(1\right)}(t_1,t_2)=-i\frac{{\displaystyle \sum _{m,k}^{}{e}^{-\beta E_m}{E}_{km}{x}_{mk}{x}_{km}cos{E}_{km}(t_1-t_2)}}{{\displaystyle \left(1+\sum _{m>0}^{}{e}^{-\beta E_m}\right)}}.\end{array}$$

(51)

5.3. Representation of 2-Point OTOC: $Y^{\left(2\right)}(t_1,t_2)$

The second 2-point OTOC is given by the thermal average of the operator $-[x\left(t_1\right),x\left(t_2\right)]$, which is described in the eigenstate representation as:

$$\begin{array}{cc}\hfill Y^{\left(2\right)}(t_1,t_2)& =-{\langle [x\left(t_1\right),x\left(t_2\right)]\rangle }_{\beta }=-\frac{{\displaystyle \sum _m^{}{e}^{-\beta E_m}\langle {\mathsf{\Psi }}_m\left|[x\left(t_1\right),x\left(t_2\right)]\right|{\mathsf{\Psi }}_m\rangle }}{{\displaystyle \left(1+\sum _{m>0}^{}{e}^{-\beta E_m}\right)}}.\hfill \end{array}$$

Using Equation (33) for Heisenberg representation for $x\left(t_1\right)$ and $x\left(t_2\right)$ the 2-point correlator in terms of the temperature independent micro-canonical correlator can be written as:

$$\begin{array}{cc}\hfill Y^{\left(2\right)}(t_1,t_2)& =\frac{{\displaystyle \sum _m^{}{e}^{-\beta E_m}{y}_m^{\left(2\right)}(t_1,t_2)}}{{\displaystyle \left(1+\sum _{m>0}^{}{e}^{-\beta E_m}\right)}},\hfill \end{array}$$

(52)

where the micro-canonical 2-point OTOC is defined as:

$$\begin{array}{c}\hfill y_m^{\left(2\right)}(t_1,t_2)=-\langle {\mathsf{\Psi }}_m\left|[x\left(t_1\right),x\left(t_2\right)]\right|{\mathsf{\Psi }}_m\rangle .\end{array}$$

(53)

Expanding the commutator in the definition of the micro-canonical correlator and inserting identity in between the operators, it can be shown that the eigenstate representation for micro-canonical correlator $y_m^{\left(2\right)}(t_1,t_2)$ takes the following form:

$$\begin{array}{c}\hfill y_m^{\left(2\right)}(t_1,t_2)=-2i\sum _k^{}{x}_{mk}{x}_{km}\sin E_{mk}(t_1-t_2).\end{array}$$

(54)

where we define, $E_{mk/m,k}=E_m-E_k$ and $x_{mk/m,k}=\langle {\mathsf{\Psi }}_m\left|x\right|{\mathsf{\Psi }}_k\rangle$.

Substituting Equation (54) in Equation (52), the eigenstate representation for the canonical correlator $Y^{\left(2\right)}(t_1,t_2)$ takes the form:

$$\begin{array}{c}\hfill Y^{\left(2\right)}(t_1,t_2)=-2i\frac{{\displaystyle \sum _{m,k}^{}{e}^{-\beta E_m}{x}_{mk}{x}_{km}\sin E_{mk}(t_1-t_2)}}{{\displaystyle \left(1+\sum _{m>0}^{}{e}^{-\beta E_m}\right)}}.\end{array}$$

(55)

5.4. Representation of 2-Point OTOC: $Y^{\left(3\right)}(t_1,t_2)$

The third two-point OTOC is given by the thermal average of the operator $-[p\left(t_1\right),p\left(t_2\right)]$, which is described in the eigenstate representation as:

$$\begin{array}{cc}\hfill Y^{\left(3\right)}(t_1,t_2)& =-{\langle [p\left(t_1\right),p\left(t_2\right)]\rangle }_{\beta }=-\frac{{\displaystyle \sum _m^{}{e}^{-\beta E_m}\langle {\mathsf{\Psi }}_m\left|[p\left(t_1\right),p\left(t_2\right)]\right|{\mathsf{\Psi }}_m\rangle }}{{\displaystyle \left(1+\sum _{m>0}^{}{e}^{-\beta E_m}\right)}}.\hfill \end{array}$$

Proceeding as previously mentioned, we obtain the following expression for the OTOC, which can be expressed in terms of the temperature independent micro-canonical correlator as:

$$\begin{array}{cc}\hfill Y^{\left(3\right)}(t_1,t_2)& =\frac{{\displaystyle \sum _m^{}{e}^{-\beta E_m}{y}_m^{\left(3\right)}(t_1,t_2)}}{{\displaystyle \left(1+\sum _{m>0}^{}{e}^{-\beta E_m}\right)}},\hfill \end{array}$$

(56)

where the micro-canonical 2-point OTOC is defined as:

$$\begin{array}{c}\hfill y_m^{\left(3\right)}(t_1,t_2)=-\langle {\mathsf{\Psi }}_m\left|[p\left(t_1\right),p\left(t_2\right)]\right|{\mathsf{\Psi }}_m\rangle .\end{array}$$

(57)

Expanding the commutator in the definition of the micro-canonical correlator and inserting identity in between the operators, it can be shown that the eigenstate representation for micro-canonical correlator $y_m^{\left(3\right)}(t_1,t_2)$ is given by the following expression:

$$\begin{array}{c}\hfill y_m^{\left(3\right)}(t_1,t_2)=-\frac{i}{2}\sum _k^{}{E}_{mk}{x}_{mk}{E}_{km}{x}_{km}\sin E_{km}\left(t_1-t_2\right).\end{array}$$

(58)

Substituting the expression for the micro-canonical correlator obtained from Equation (58) in Equation (56), the eigenstate representation for the canonical correlator $Y^{\left(3\right)}(t_1,t_2)$ can be written as:

$$\begin{array}{c}\hfill Y^{\left(3\right)}(t_1,t_2)=-\frac{i}{2}\frac{{\displaystyle \sum _{m,k}^{}{e}^{-\beta E_m}{E}_{mk}{x}_{mk}{E}_{km}{x}_{km}sin{E}_{km}\left(t_1-t_2\right)}}{{\displaystyle \left(1+\sum _{m>0}^{}{e}^{-\beta E_m}\right)}}.\end{array}$$

(59)

5.5. Representation of 4-Point OTOC: $C^{\left(1\right)}(t_1,t_2)$

In this section, we provide the eigenstate representation of the 4-point OTOCs that are mainly used for studying the quantum mechanical analogue of phenomena of time disorder averaging, related randomness in quantum regime. Similar to the 2-point correlators, we define three kinds of 4-point correlators. Generally, in the literature, people study the 4-point correlator $C^{\left(1\right)}(t_1,t_2)$ (as per our notation), which is the thermal expectation value of the square of the commutator made up of the dynamical operators of different kinds at different time scales. The 4-point correlator, $C^{\left(1\right)}(t_1,t_2)$, is proposed as a quantifier of quantum chaos, but chaos is a specific kind of randomness. Hence, it is important to consider the correlators constructed from similar operators at different times to have a complete understanding of the underlying phenomenon of randomness. To understand this time disordering phenomena explicitly, we have also introduced two more new correlators, $C^{\left(2\right)}(t_1,t_2)$ and $C^{\left(3\right)}(t_1,t_2)$, described in detail in the latter half of this paper.

5.5.1. Un-Normalized: $C^{\left(1\right)}(t_1,t_2)$

The first 4-point OTOC is given by the thermal average of the operator $-{[x\left(t_1\right),p\left(t_2\right)]}^2$, which is described in the eigenstate representation as:

$$\begin{array}{cc}\hfill C^{\left(1\right)}(t_1,t_2)& =-{\langle {[x\left(t_1\right),p\left(t_2\right)]}^2\rangle }_{\beta }=\frac{{\displaystyle \sum _m^{}{e}^{-\beta E_m}{c}_m^{\left(1\right)}(t_1,t_2)}}{{\displaystyle \left(1+\sum _{m>0}^{}{e}^{-\beta E_m}\right)}},\hfill \end{array}$$

(60)

where $c_m^{\left(1\right)}(t_1,t_2)$ is the micro-canonical correlator and is responsible for the temperature independent behavior of the correlator. The temperature dependence in the canonical correlator is actually appearing due to the exponential thermal Botzmann factor in the eigenstate representation of the OTOC. Once we take the sum over all possible eigenstates (finite or infinite in number), we get a cumulative dependence on time, temperature, and energy eigenstates. In the present context, the micro-canonical 4-point correlator for Equation (60) is given by the following simplified expression:

$$\begin{array}{c}\hfill c_m^{\left(1\right)}(t_1,t_2)=-\langle {\mathsf{\Psi }}_m\left|{[x\left(t_1\right),p\left(t_2\right)]}^2\right|{\mathsf{\Psi }}_m\rangle =\sum _k^{}{b}_{mk}^{\left(1\right)}(t_1,t_2)b_{mk}^{\left(1\right)*}(t_1,t_2),\end{array}$$

(61)

where we define a time-dependent matrix element, $b_{mk}^{\left(1\right)}(t_1,t_2)$, which is given by:

$$\begin{array}{cc}\hfill b_{mk}^{\left(1\right)}(t_1,t_2)& =-i\langle {\mathsf{\Psi }}_m\left|[x\left(t_1\right),p\left(t_2\right)]\right|{\mathsf{\Psi }}_k\rangle .\hfill \end{array}$$

Using the Heisenberg picture evolution equation for an operator and inserting identity between the operators after expanding the commutator, it is not hard to verify that $b_{mk}^{\left(1\right)}$ can be written as:

$$\begin{array}{c}\hfill b_{mk}^{\left(1\right)*}(t_1,t_2)=\frac{1}{2}\sum _r^{}\left\{E_{rk}{e}^{i{t}_1{E}_{rm}}{e}^{i{t}_2{E}_{kr}}-E_{mr}{e}^{i{t}_2{E}_{rm}}{e}^{i{t}_1{E}_{kr}}\right\}{x}_{rm}{x}_{kr}.\end{array}$$

Substituting the above expression of $b_{mk}^{\left(1\right)*}(t_1,t_2)$ in Equation (61), it can be shown that, after simplification, the eigenstate representation of the micro-canonical correlator for Equation (60) is given by the following expression:

$$\begin{array}{cc}\hfill c_m^{\left(1\right)}(t_1,t_2)& =\frac{1}{4}\sum _{k,l,r}^{}{x}_{ml}{x}_{lk}{x}_{rm}{x}_{kr}\times [E_{rk}{E}_{lk}{e}^{i{E}_{rl}\left(t_1-t_2\right)}+E_{mr}{E}_{ml}{e}^{-i{E}_{rl}\left(t_1-t_2\right)}\hfill \\ \hfill & -E_{rk}{E}_{ml}{e}^{i\left(E_{rm}+E_{lk}\right)\left(t_1-t_2\right)}-E_{mr}{E}_{lk}{e}^{-i\left(E_{rm}+E_{lk}\right)\left(t_1-t_2\right)}].\hfill \end{array}$$

(62)

So, the eigenstate representation of canonical correlator from Equation (60) using Equation (62) is given as:

$$\begin{array}{cc}\hfill C^{\left(1\right)}(t_1,t_2)& =\frac{1}{{\displaystyle 4\left(1+\sum _{m>0}^{}{e}^{-\beta E_m}\right)}}\sum _{m,k,l,r}^{}{e}^{-\beta E_m}{x}_{ml}{x}_{lk}{x}_{rm}{x}_{kr}\hfill \\ \hfill & \times [E_{rk}{E}_{lk}{e}^{i{E}_{rl}\left(t_1-t_2\right)}+E_{mr}{E}_{ml}{e}^{-i{E}_{rl}\left(t_1-t_2\right)}\hfill \\ \hfill & -E_{rk}{E}_{ml}{e}^{i\left(E_{rm}+E_{lk}\right)\left(t_1-t_2\right)}-E_{mr}{E}_{lk}{e}^{-i\left(E_{rm}+E_{lk}\right)\left(t_1-t_2\right)}].\hfill \end{array}$$

(63)

5.5.2. Normalized: ${\tilde{C}}^{\left(1\right)}(t_1,t_2)$

The normalized first 4-point OTOC is from the un-normalized one expressed in Equation (43) can be expressed as:

$$\begin{array}{c}\hfill {\tilde{C}}^{\left(1\right)}(t_1,t_2)=\frac{C^{\left(1\right)}(t_1,t_2)}{{\langle x\left(t_1\right)x\left(t_1\right)\rangle }_{\beta }{\langle p\left(t_2\right)p\left(t_2\right)\rangle }_{\beta }}.\end{array}$$

We have already calculated the numerator $C^{\left(1\right)}(t_1,t_2)$ as given in Equation (63). So, here our only job is compute the two sets of equal time thermal correlators in the eigenstate representation. Since we know the formalism very well that we have developed in this paper, computing these disconnected pieces of equal time correlators, which are extremely significant in the large time dissipation limit, is not at all complicated. Now, we are going to show how one can compute these contributions.

To serve this purpose we need to compute the following expressions in the eigenstate representation:

$$\begin{array}{ccc}& & \langle j|x\left(t_1\right)x\left(t_1\right)|j\rangle =\sum _l{x}_{jl}{x}_{lj},\hfill \end{array}$$

(64)

$$\begin{array}{ccc}& & \langle j|p\left(t_2\right)p\left(t_2\right)|j\rangle =\sum _l{p}_{jl}{p}_{lj}=-\frac{1}{4}\sum _l{E}_{jl}{E}_{lj}{x}_{jl}{x}_{lj}.\hfill \end{array}$$

(65)

Consequently, the desired canonical equal time thermal correlators can be computed as:

$$\begin{array}{ccc}& & {\langle x\left(t_1\right)x\left(t_1\right)\rangle }_{\beta }=\frac{{\displaystyle \sum _j{e}^{-\beta E_j}\langle j|x\left(t_1\right)x\left(t_1\right)|j\rangle }}{{\displaystyle \left(1+\sum _{m>0}^{}{e}^{-\beta E_m}\right)}}=\frac{{\displaystyle \sum _{j,l}{e}^{-\beta E_j}{x}_{jl}{x}_{lj}}}{{\displaystyle \left(1+\sum _{m>0}^{}{e}^{-\beta E_m}\right)}},\hfill \end{array}$$

(66)

$$\begin{array}{ccc}& & {\langle p\left(t_2\right)p\left(t_2\right)\rangle }_{\beta }=\frac{{\displaystyle \sum _j{e}^{-\beta E_j}\langle j|p\left(t_2\right)p\left(t_2\right)|j\rangle }}{{\displaystyle \left(1+\sum _{m>0}^{}{e}^{-\beta E_m}\right)}}=\frac{{\displaystyle \sum _{j,l}{e}^{-\beta E_j}{p}_{jl}{p}_{lj}}}{{\displaystyle \left(1+\sum _{m>0}^{}{e}^{-\beta E_m}\right)}}=-\frac{1}{4}\frac{{\displaystyle \sum _{j,l}{e}^{-\beta E_j}{E}_{jl}{E}_{lj}{x}_{jl}{x}_{lj}}}{{\displaystyle \left(1+\sum _{m>0}^{}{e}^{-\beta E_m}\right)}}.\hfill \end{array}$$

(67)

Finally, the normalized first 4-point OTOC can be expressed by the following simplified expression in the eigenstate representation, as given by:

$$\begin{array}{cc}\hfill {\tilde{C}}^{\left(1\right)}(t_1,t_2)& =\frac{C^{\left(1\right)}(t_1,t_2)}{{\langle x\left(t_1\right)x\left(t_1\right)\rangle }_{\beta }{\langle p\left(t_2\right)p\left(t_2\right)\rangle }_{\beta }}\hfill \\ \hfill & =-\frac{{\displaystyle \left(1+\sum _{j>0}^{}{e}^{-\beta E_j}\right)}}{{\displaystyle \sum _{j,m,l}^{}{e}^{-\beta (E_j+E_m)}{E}_{jl}{E}_{lj}{x}_{jl}{x}_{lj}}}\hfill \\ \hfill & \times \sum _{m,k,l,r}^{}{e}^{-\beta E_m}{x}_{ml}{x}_{lk}{x}_{rm}{x}_{kr}\hfill \\ \hfill & \times [E_{rk}{E}_{lk}{e}^{i{E}_{rl}\left(t_1-t_2\right)}+E_{mr}{E}_{ml}{e}^{-i{E}_{rl}\left(t_1-t_2\right)}\hfill \\ \hfill & -E_{rk}{E}_{ml}{e}^{i\left(E_{rm}+E_{lk}\right)\left(t_1-t_2\right)}-E_{mr}{E}_{lk}{e}^{-i\left(E_{rm}+E_{lk}\right)\left(t_1-t_2\right)}].\hfill \end{array}$$

(68)

5.6. Representation of 4-Point OTOC: $C^{\left(2\right)}(t_1,t_2)$

5.6.1. Un-Normalized: $C^{\left(2\right)}(t_1,t_2)$

The second 4-point OTOC is given by the thermal average of the operator $-{[x\left(t_1\right),x\left(t_2\right)]}^2$, which is described in the eigenstate representation as:

$$\begin{array}{cc}\hfill C^{\left(2\right)}(t_1,t_2)& =-{\langle {[x\left(t_1\right),x\left(t_2\right)]}^2\rangle }_{\beta }=\frac{{\displaystyle \sum _m^{}{e}^{-\beta E_m}{c}_m^{\left(2\right)}(t_1,t_2)}}{{\displaystyle \left(1+\sum _{m>0}^{}{e}^{-\beta E_m}\right)}},\hfill \end{array}$$

(69)

where $c_m^{\left(2\right)}(t_1,t_2)$ is the micro-canonical correlator and is responsible for the temperature independent behavior of the correlator. The temperature dependence in the canonical correlator is actually appearing due to the exponential thermal Botzmann factor in the eigenstate representation of the OTOC. Once we take the sum over all possible eigenstates (finite or infinite in number), we get a cumulative dependence on time, temperature, and energy eigenstates. In the present context, the micro-canonical 4-point correlator for Equation (69) is given by the following simplified expression:

$$\begin{array}{c}\hfill c_m^{\left(2\right)}(t_1,t_2)=-\langle {\mathsf{\Psi }}_m\left|{[x\left(t_1\right),x\left(t_2\right)]}^2\right|{\mathsf{\Psi }}_m\rangle =-\sum _k^{}{b}_{mk}^{\left(2\right)}(t_1,t_2)b_{km}^{\left(2\right)}(t_1,t_2),\end{array}$$

(70)

where we define a time-dependent matrix element, $b_{mk}^{\left(2\right)}(t_1,t_2)$, which is given by:

$$\begin{array}{cc}\hfill b_{mk}^{\left(2\right)}(t_1,t_2)& =\langle {\mathsf{\Psi }}_m\left|[x\left(t_1\right),x\left(t_2\right)]\right|{\mathsf{\Psi }}_k\rangle .\hfill \end{array}$$

Using the Heisenberg picture for the evolution of operators and inserting identity between the operators, it can be shown that $b_{mk}^{\left(2\right)}(t_1,t_2)$ can be written as:

$$\begin{array}{c}\hfill b_{mk}^{\left(2\right)}(t_1,t_2)=\sum _l^{}\left\{e^{i{t}_1{E}_{ml}}{e}^{i{t}_2{E}_{lk}}-e^{i{t}_2{E}_{ml}}{e}^{i{t}_1{E}_{lk}}\right\}{x}_{ml}{x}_{lk}.\end{array}$$

(71)

Substituting Equation (71) in Equation (70) and simplifying the eigenstate representation for the temperature independent micro-canonical correlator can be expressed by the following simplified expression:

$$\begin{array}{cc}\hfill c_m^{\left(2\right)}(t_1,t_2)& =4\sum _{k,l,r}^{}{x}_{ml}{x}_{lk}{x}_{kr}{x}_{rm}\hfill \\ \hfill & \times sin\left[\left(E_r-\frac{E_m+E_k}{2}\right)\left(t_1-t_2\right)\right]sin\left[\left(E_l-\frac{E_m+E_k}{2}\right)\left(t_1-t_2\right)\right].\hfill \end{array}$$

(72)

The canonical or the temperature-dependent correlator can be calculated by substituting the expression of Equation (69) in Equation (72), and, after applying some algebraic manipulation, it can be shown that the canonical correlator can be expressed by the following two expressions:

$$\begin{array}{cc}\hfill C^{\left(2\right)}(t_1,t_2)& =\frac{4}{{\displaystyle \left(1+\sum _{j>0}^{}{e}^{-\beta E_j}\right)}}\sum _{m,k,l,r}^{}{e}^{-\beta E_m}{x}_{ml}{x}_{lk}{x}_{kr}{x}_{rm}\hfill \\ \hfill & \times sin\left[\left(E_r-\frac{E_m+E_k}{2}\right)\left(t_1-t_2\right)\right]sin\left[\left(E_l-\frac{E_m+E_k}{2}\right)\left(t_1-t_2\right)\right].\hfill \end{array}$$

(73)

5.6.2. Normalized: ${\tilde{C}}^{\left(2\right)}(t_1,t_2)$

The normalized first 4-point OTOC is from the un-normalized one expressed in Equation (44) can be expressed as:

$$\begin{array}{c}\hfill {\tilde{C}}^{\left(2\right)}(t_1,t_2)=\frac{C^{\left(2\right)}(t_1,t_2)}{{\langle x\left(t_1\right)x\left(t_1\right)\rangle }_{\beta }{\langle x\left(t_2\right)x\left(t_2\right)\rangle }_{\beta }}.\end{array}$$

We have already calculated the numerator $C^{\left(2\right)}(t_1,t_2)$ as given in Equation (73). So, here our only job is compute the two sets of equal time thermal correlators in the eigenstate representation. Since we know the formalism very well that we have developed in this paper, computing these disconnected pieces of equal time correlators, which are extremely significant in the large time dissipation limit, is not at all complicated. Now, we are going to show how one can compute these contributions.

To serve this purpose, we need to compute the following expressions in the eigenstate representation:

$$\begin{array}{ccc}& & \langle j|x\left(t_1\right)x\left(t_1\right)|j\rangle =\sum _l{x}_{jl}{x}_{lj}=\langle j|x\left(t_2\right)x\left(t_2\right)|j\rangle .\hfill \end{array}$$

(74)

Consequently, the desired canonical equal time thermal correlators can be computed as:

$$\begin{array}{ccc}& & {\langle x\left(t_1\right)x\left(t_1\right)\rangle }_{\beta }=\frac{{\displaystyle \sum _j{e}^{-\beta E_j}\langle j|x\left(t_1\right)x\left(t_1\right)|j\rangle }}{{\displaystyle \left(1+\sum _{m>0}^{}{e}^{-\beta E_m}\right)}}=\frac{{\displaystyle \sum _{j,l}{e}^{-\beta E_j}{x}_{jl}{x}_{lj}}}{{\displaystyle \left(1+\sum _{m>0}^{}{e}^{-\beta E_m}\right)}}={\langle x\left(t_2\right)x\left(t_2\right)\rangle }_{\beta }.\hfill \end{array}$$

(75)

Finally, the normalized first 4-point OTOC can be expressed by the following simplified expression in the eigenstate representation, as given by:

$$\begin{array}{cc}\hfill {\tilde{C}}^{\left(2\right)}(t_1,t_2)& =\frac{C^{\left(2\right)}(t_1,t_2)}{{\langle x\left(t_1\right)x\left(t_1\right)\rangle }_{\beta }{\langle x\left(t_2\right)x\left(t_2\right)\rangle }_{\beta }}\hfill \\ \hfill & =\frac{{\displaystyle 4\left(1+\sum _{j>0}^{}{e}^{-\beta E_j}\right)}}{{\displaystyle \sum _{j,m,l,k}^{}{e}^{-\beta (E_j+E_m)}{x}_{jl}{x}_{lj}{x}_{mk}{x}_{km}}}\sum _{m,k,l,r}^{}{e}^{-\beta E_m}{x}_{ml}{x}_{lk}{x}_{kr}{x}_{rm}\hfill \\ \hfill & \times sin\left[\left(E_r-\frac{E_m+E_k}{2}\right)\left(t_1-t_2\right)\right]sin\left[\left(E_l-\frac{E_m+E_k}{2}\right)\left(t_1-t_2\right)\right].\hfill \end{array}$$

(76)

5.7. Representation of 4-Point OTOC: $C^{\left(3\right)}(t_1,t_2)$

5.7.1. Un-Normalized: $C^{\left(3\right)}(t_1,t_2)$

The third 4-point OTOC is given by the thermal average of the operator $-{[p\left(t_1\right),p\left(t_2\right)]}^2$, which is described in the eigenstate representation as:

$$\begin{array}{cc}\hfill C^{\left(3\right)}(t_1,t_2)& =-{\langle {[p\left(t_1\right),p\left(t_2\right)]}^2\rangle }_{\beta }=\frac{{\displaystyle \sum _m^{}{e}^{-\beta E_m}{c}_m^{\left(3\right)}(t_1,t_2)}}{{\displaystyle \left(1+\sum _{m>0}^{}{e}^{-\beta E_m}\right)}},\hfill \end{array}$$

(77)

where $c_m^{\left(3\right)}(t_1,t_2)$ is the micro-canonical correlator and is responsible for the temperature independent behavior of the correlator. The temperature dependence in the canonical correlator is actually appearing due to the exponential thermal Botzmann factor in the eigenstate representation of the OTOC. Once we take the sum over all possible eigenstates (finite or infinite in number), we get a cumulative dependence on time, temperature, and energy eigenstates. In the present context, the micro-canonical 4-point correlator for Equation (77) is given by the following simplified expression:

$$\begin{array}{c}\hfill c_m^{\left(3\right)}(t_1,t_2)=-\langle {\mathsf{\Psi }}_m\left|{[p\left(t_1\right),p\left(t_2\right)]}^2\right|{\mathsf{\Psi }}_m\rangle =-\sum _k^{}{b}_{mk}^{\left(3\right)}(t_1,t_2)b_{km}^{\left(3\right)}(t_1,t_2),\end{array}$$

(78)

where we define a time-dependent matrix element, $b_{mk}^{\left(2\right)}(t_1,t_2)$, which is given by:

$$\begin{array}{cc}\hfill b_{mk}^{\left(3\right)}(t_1,t_2)& =\langle {\mathsf{\Psi }}_m\left|[p\left(t_1\right),p\left(t_2\right)]\right|{\mathsf{\Psi }}_k\rangle .\hfill \end{array}$$

(79)

Using the Heisenberg picture for the evolution of operators and inserting identity between the operators, it can be shown that $b_{mk}^{\left(3\right)}(t_1,t_2)$ can be simplified into the following form:

$$\begin{array}{c}\hfill b_{mk}^{\left(3\right)}(t_1,t_2)=\sum _l^{}\left\{e^{i{t}_1{E}_{ml}}{e}^{i{t}_2{E}_{lk}}-e^{i{t}_2{E}_{ml}}{e}^{i{t}_1{E}_{lk}}\right\}{p}_{ml}{p}_{lk}.\end{array}$$

Substituting the expression of $b_{mk}^{\left(3\right)}(t_1,t_2)$, i.e., Equation (79) in Equation (78), and simplifying, the eigenstate representation of the micro-canonical correlator can be expressed as:

$$\begin{array}{cc}\hfill c_m^{\left(3\right)}(t_1,t_2)& =\frac{1}{4}\sum _{k,l,r}^{}{E}_{ml}{E}_{lk}{E}_{kr}{E}_{rm}{x}_{ml}{x}_{lk}{x}_{kr}{x}_{rm}\hfill \\ \hfill & \times sin\left[\left(E_r-\frac{E_m+E_k}{2}\right)\left(t_1-t_2\right)\right]sin\left[\left(E_l-\frac{E_m+E_k}{2}\right)\left(t_1-t_2\right)\right].\hfill \end{array}$$

(80)

The eigenstate representation of the canonical correlator from Equation (77) using Equation (80), after simplification, can be expressed by the following two equations:

$$\begin{array}{cc}\hfill C^{\left(3\right)}(t_1,t_2)& =\frac{1}{{\displaystyle 4\left(1+\sum _{m>0}^{}{e}^{-\beta E_m}\right)}}\sum _{m,k,l,r}^{}{e}^{-\beta E_m}{E}_{ml}{E}_{lk}{E}_{kr}{E}_{rm}{x}_{ml}{x}_{lk}{x}_{kr}{x}_{rm}\hfill \\ \hfill & \times sin\left[\left(E_r-\frac{E_m+E_k}{2}\right)\left(t_1-t_2\right)\right]sin\left[\left(E_l-\frac{E_m+E_k}{2}\right)\left(t_1-t_2\right)\right].\hfill \end{array}$$

(81)

5.7.2. Normalized: ${\tilde{C}}^{\left(3\right)}(t_1,t_2)$

The normalized first 4-point OTOC is from the un-normalized one expressed in Equation (45) can be expressed as:

$$\begin{array}{c}\hfill {\tilde{C}}^{\left(3\right)}(t_1,t_2)=\frac{C^{\left(3\right)}(t_1,t_2)}{{\langle p\left(t_1\right)p\left(t_1\right)\rangle }_{\beta }{\langle p\left(t_2\right)p\left(t_2\right)\rangle }_{\beta }}.\end{array}$$

We have already calculated the numerator $C^{\left(3\right)}(t_1,t_2)$ as given in Equation (81). So, here our only job is compute the two sets of equal time thermal correlators in the eigenstate representation. Since we know the formalism very well that we have developed in this paper, computing these disconnected pieces of equal time correlators, which are extremely significant in the large time dissipation limit, is not at all complicated. Now, we are going to show how one can compute these contributions.

To serve this purpose, we need to compute the following expressions in the eigenstate representation:

$$\begin{array}{ccc}& & \langle j|p\left(t_1\right)p\left(t_1\right)|j\rangle =\sum _l{p}_{jl}{p}_{lj}=-\frac{1}{4}\sum _l{E}_{jl}{E}_{lj}{x}_{jl}{x}_{lj}=\langle j|p\left(t_2\right)p\left(t_2\right)|j\rangle .\hfill \end{array}$$

(82)

Consequently, the desired canonical equal time thermal correlators can be computed as:

$$\begin{array}{ccc}& & {\langle p\left(t_1\right)p\left(t_1\right)\rangle }_{\beta }=\frac{{\displaystyle \sum _{j,l}{e}^{-\beta E_j}{p}_{jl}{p}_{lj}}}{{\displaystyle \left(1+\sum _{j>0}^{}{e}^{-\beta E_j}\right)}}=-\frac{1}{4}\frac{{\displaystyle \sum _{j,l}{e}^{-\beta E_j}{E}_{jl}{E}_{lj}{x}_{jl}{x}_{lj}}}{{\displaystyle \left(1+\sum _{j>0}^{}{e}^{-\beta E_j}\right)}}={\langle p\left(t_2\right)p\left(t_2\right)\rangle }_{\beta }.\hfill \end{array}$$

(83)

Finally, the normalized first 4-point OTOC can be expressed by the following simplified expression in the eigenstate representation, as given by:

$$\begin{array}{cc}\hfill {\tilde{C}}^{\left(3\right)}(t_1,t_2)& =\frac{C^{\left(3\right)}(t_1,t_2)}{{\langle p\left(t_1\right)p\left(t_1\right)\rangle }_{\beta }{\langle p\left(t_2\right)p\left(t_2\right)\rangle }_{\beta }}\hfill \\ \hfill & =\frac{{\displaystyle 4\left(1+\sum _{j>0}^{}{e}^{-\beta E_j}\right)}}{{\displaystyle \sum _{j,m,k,l}{e}^{-\beta (E_j+E_m)}{E}_{jl}{E}_{lj}{E}_{mk}{E}_{km}{x}_{jl}{x}_{lj}{x}_{mk}{x}_{km}}}\hfill \\ \hfill & \times \sum _{m,k,l,r}^{}{e}^{-\beta E_m}{E}_{ml}{E}_{lk}{E}_{kr}{E}_{rm}{x}_{ml}{x}_{lk}{x}_{kr}{x}_{rm}\hfill \\ \hfill & \times sin\left[\left(E_r-\frac{E_m+E_k}{2}\right)\left(t_1-t_2\right)\right]sin\left[\left(E_l-\frac{E_m+E_k}{2}\right)\left(t_1-t_2\right)\right].\hfill \\ \hfill \end{array}$$

(84)

5.8. Summary of Results

Related equations are following:

Eigenstate Representation for Micro-Canonical Correlators]

$$\begin{array}{cc}\hfill y_m^{\left(1\right)}(t_1,t_2)& =-i\sum _k^{}{E}_{km}{x}_{mk}{x}_{km}cos\left(E_{km}\left(t_1-t_2\right)\right),\hfill \end{array}$$

(85)

$$\begin{array}{cc}\hfill y_m^{\left(2\right)}(t_1,t_2)& =-2i\sum _k^{}{x}_{mk}{x}_{km}\sin \left(E_{mk}(t_1-t_2)\right),\hfill \end{array}$$

(86)

$$\begin{array}{cc}\hfill y_m^{\left(3\right)}(t_1,t_2)& =-\frac{i}{2}\sum _k^{}{E}_{mk}{x}_{mk}{E}_{km}{x}_{km}\sin \left(E_{km}\left(t_1-t_2\right)\right),\hfill \end{array}$$

(87)

$$\begin{array}{cc}\hfill c_m^{\left(1\right)}(t_1,t_2)& =\frac{1}{4}\sum _{k,l,r}^{}{x}_{ml}{x}_{lk}{x}_{rm}{x}_{kr}\hfill \\ \hfill & \times [E_{rk}{E}_{lk}{e}^{i{E}_{rl}\left(t_1-t_2\right)}+E_{mr}{E}_{ml}{e}^{-i{E}_{rl}\left(t_1-t_2\right)}\hfill \\ \hfill & -E_{rk}{E}_{ml}{e}^{i\left(E_{rm}+E_{lk}\right)\left(t_1-t_2\right)}-E_{mr}{E}_{lk}{e}^{-i\left(E_{rm}+E_{lk}\right)\left(t_1-t_2\right)}],\hfill \end{array}$$

(88)

$$\begin{array}{cc}\hfill c_m^{\left(2\right)}(t_1,t_2)& =4\sum _{k,l,r}^{}{x}_{ml}{x}_{lk}{x}_{kr}{x}_{rm}\hfill \\ \hfill & \times sin\left[\left(E_r-\frac{E_m+E_k}{2}\right)\left(t_1-t_2\right)\right]sin\left[\left(E_l-\frac{E_m+E_k}{2}\right)\left(t_1-t_2\right)\right],\hfill \end{array}$$

(89)

$$\begin{array}{cc}\hfill c_m^{\left(3\right)}(t_1,t_2)& =\frac{1}{4}\sum _{k,l,r}^{}{E}_{ml}{E}_{lk}{E}_{kr}{E}_{rm}{x}_{ml}{x}_{lk}{x}_{kr}{x}_{rm}\hfill \\ \hfill & \times sin\left[\left(E_r-\frac{E_m+E_k}{2}\right)\left(t_1-t_2\right)\right]sin\left[\left(E_l-\frac{E_m+E_k}{2}\right)\left(t_1-t_2\right)\right].\hfill \end{array}$$

(90)

Eigenstate Representation for Canonical Correlators without normalization]

Subjected to (51)

$$\begin{array}{cc}\hfill Y^{\left(2\right)}(t_1,t_2)& =-2i\frac{{\displaystyle \sum _{m,k}^{}{e}^{-\beta E_m}{x}_{mk}{x}_{km}sin\left(E_{mk}\left(t_1-t_2\right)\right)}}{{\displaystyle \left(1+\sum _{m>0}{e}^{-\beta E_m}\right)}},\hfill \end{array}$$

(91)

$$\begin{array}{cc}\hfill Y^{\left(3\right)}(t_1,t_2)& =-\frac{i}{2}\frac{{\displaystyle \sum _{m,k}^{}{e}^{-\beta E_m}{E}_{mk}{x}_{mk}{E}_{km}{x}_{km}sin\left(E_{km}\left(t_1-t_2\right)\right)}}{{\displaystyle \left(1+\sum _{m>0}{e}^{-\beta E_m}\right)}},\hfill \end{array}$$

(92)

$$\begin{array}{cc}\hfill C^{\left(1\right)}(t_1,t_2)& =\frac{1}{{\displaystyle 4\left(1+\sum _{m>0}{e}^{-\beta E_m}\right)}}\sum _{m,k,l,r}^{}{e}^{-\beta E_m}{x}_{ml}{x}_{lk}{x}_{rm}{x}_{kr}\hfill \\ \hfill & \times [E_{rk}{E}_{lk}{e}^{i{E}_{rl}\left(t_1-t_2\right)}+E_{mr}{E}_{ml}{e}^{-i{E}_{rl}\left(t_1-t_2\right)}\hfill \\ \hfill & -E_{rk}{E}_{ml}{e}^{i\left(E_{rm}+E_{lk}\right)\left(t_1-t_2\right)}-E_{mr}{E}_{lk}{e}^{-i\left(E_{rm}+E_{lk}\right)\left(t_1-t_2\right)}],\hfill \end{array}$$

(93)

$$\begin{array}{cc}\hfill C^{\left(2\right)}(t_1,t_2)& =\frac{4}{{\displaystyle \left(1+\sum _{m>0}{e}^{-\beta E_m}\right)}}\sum _{m,k,l,r}^{}{e}^{-\beta E_m}{x}_{ml}{x}_{lk}{x}_{kr}{x}_{rm}\hfill \\ \hfill & \times sin\left[\left(E_r-\frac{E_m+E_k}{2}\right)\left(t_1-t_2\right)\right]sin\left[\left(E_l-\frac{E_m+E_k}{2}\right)\left(t_1-t_2\right)\right],\hfill \end{array}$$

(94)

$$\begin{array}{cc}\hfill C^{\left(3\right)}(t_1,t_2)& =\frac{1}{{\displaystyle 4\left(1+\sum _{m>0}{e}^{-\beta E_m}\right)}}\sum _{m,k,l,r}^{}{e}^{-\beta E_m}{E}_{ml}{E}_{lk}{E}_{kr}{E}_{rm}{x}_{ml}{x}_{lk}{x}_{kr}{x}_{rm}\hfill \\ \hfill & \times sin\left[\left(E_r-\frac{E_m+E_k}{2}\right)\left(t_1-t_2\right)\right]sin\left[\left(E_l-\frac{E_m+E_k}{2}\right)\left(t_1-t_2\right)\right].\hfill \end{array}$$

(95)

Eigenstate Representation for Canonical Correlators with normalization referred to (75), (83), (68), (76) and (84).

6. Model I: Supersymmetric Quantum Mechanical Harmonic Oscillator

6.1. Eigenspectrum of the Super-Partner Hamiltonian

We consider a system described by Hamiltonian $H_1$ with the harmonic oscillator potential as given by:

$$\begin{array}{c}\hfill V_1\left(x\right)=\frac{1}{2}{\omega }^2{x}^2,\end{array}$$

(96)

where $\omega$ is the natural frequency of the oscillator, and we have assumed that the mass is $m=1$, for the sake of algebraic simplification. Now, we represent ${\mathsf{\Psi }}_n^{\left(1\right)}\left(x\right)$ and $E_n^{\left(1\right)}$ as the eigenstates and eigenvalues corresponding to Hamiltonian $H_1$, which can be obtained by solving the Schr$\ddot{\mathrm{o}}$dinger equation as:

$$\begin{array}{cc}\hfill {\mathsf{\Psi }}_n^{\left(1\right)}\left(x\right)& ={\left(\frac{\omega }{\pi }\right)}^{1/4}\frac{1}{\sqrt{2^{n}n!}}{e}^{-\frac{\omega }{2}{x}^2}{H}_n\left(\sqrt{\omega }x\right),\hfill \end{array}$$

(97)

$$\begin{array}{cc}\hfill E_n^{\left(1\right)}& =\left(n+\frac{1}{2}\right)\omega ,\hfill \end{array}$$

(98)

where $H_m\left(y\right)$ appearing in the eigenfunctions are the well-known Hermite polynomials of order m. As discussed in Section 3, the superpotential can be calculated once the ground state wavefunction is known. Hence, using the ground state wavefunction obtained by substituting $n=0$ in Equation (97), the superpotential can be calculated using Equation (11), and, for the case of Supersymmetric one-dimensional harmonic oscillator, it is obtained to be the following:

$$\begin{array}{c}\hfill W\left(x\right)=\frac{x}{\sqrt{2}}.\end{array}$$

(99)

Thus, we can construct the partner Hamiltonian, which is of the following form:

$$\begin{array}{c}\hfill H_2=-\frac{1}{2}\frac{d^2}{d{x}^2}+V_2\left(x\right),\end{array}$$

(100)

where $V_2$ is the partner potential and can be computed using Equation (12). For the Supersymmetric one-dimensional harmonic oscillator, we get the following expression for the partner potential:

$$\begin{array}{c}\hfill V_2\left(x\right)=\left(1+\frac{{\omega }^2{x}^2}{2}\right).\end{array}$$

(101)

More precisely, in the context of Supersymmetric harmonic oscillator, the original Hamiltonian is usually identified as the Bosonic Hamiltonian, and its associated super-partner is identified as the Fermionic Hamiltonian, which are constructed by subtracting off the ground state energy as:

$$\begin{array}{cc}\hfill & \mathbf{Bosonic}\mathbf{Hamiltonian}:H_{\mathrm{B}}=H_1-E_0^{\left(1\right)},\hfill \end{array}$$

(102)

$$\begin{array}{cc}\hfill & \mathbf{Fermionic}\mathbf{Hamiltonian}:H_{\mathrm{F}}=H_2-E_0^{\left(1\right)}.\hfill \end{array}$$

(103)

Now, since this is just a constant shift in the energy, the corresponding eigenstates of the partner Hamiltonians remain the same and can be represented as:

$$\begin{array}{cc}\hfill & \mathbf{Bosonic}\mathbf{eigenstate}:{\mathsf{\Psi }}_n^{\left(1\right)}\left(x\right)\equiv {\mathsf{\Psi }}_n^{\left(B\right)}\left(x\right),\hfill \end{array}$$

(104)

$$\begin{array}{cc}\hfill & \mathbf{Fermionic}\mathbf{eigenstate}:{\mathsf{\Psi }}_n^{\left(2\right)}\left(x\right)\equiv {\mathsf{\Psi }}_n^{\left(F\right)}\left(x\right).\hfill \end{array}$$

(105)

However, in this case, the energy eigenvalues get shifted by an amount $E_0^{\left(1\right)}$ and can be described as:

$$\begin{array}{cc}\hfill & \mathbf{Bosonic}\mathbf{eigenspectrum}:E_n^{\left(B\right)}=n\omega ,\hfill \end{array}$$

(106)

$$\begin{array}{cc}\hfill & \mathbf{Fermionic}\mathbf{eigenspectrum}:E_n^{\left(F\right)}=(n+1)\omega ,\hfill \end{array}$$

(107)

where we have used the property of Supersymmetric quantum mechanical theories:

$$\begin{array}{c}\hfill E_n^{\left(F\right)}=E_{n+1}^{\left(B\right)}\forall n=0,1,2,\dots ..\end{array}$$

(108)

We can compute the energy eigenstates of the partner fermionic system by using the Schr$\ddot{\mathrm{o}}$dinger equation for the Hamiltonian $H_{\mathrm{F}}$, which yields the same wave functions as for $H_{\mathrm{B}}$, as given in Equations (97) and (98).

6.2. Partition Function

The partition function in the eigenstate representation is defined by the following expression:

$$\begin{array}{c}\hfill Z=\mathrm{Tr}\left[e^{-\beta H}\right]=\sum _n\langle {\mathsf{\Psi }}_n\left|e^{-\beta H}\right|{\mathsf{\Psi }}_n\rangle .\end{array}$$

(109)

In the framework of Supersymmetric quantum mechanics, as already discussed, the ground state is only bosonic. The structure of the normalized eigenfunctions are also different for the ground state and the higher energy state. Hence, the partition function can be written in terms of the ground state contribution and in terms of the contribution coming from the excited state as:

$$\begin{array}{c}\hfill Z=\underset{\mathbf{Ground}\mathbf{State}\mathbf{Contribution}}{\underbrace{\langle {\mathsf{\Psi }}_0|{\mathsf{\Psi }}_0\rangle }}+\underset{\mathbf{Excited}\mathbf{State}\mathbf{Contribution}}{\underbrace{\frac{1}{2}\sum _{n>0}[\langle {\mathsf{\Psi }}_n^{\mathrm{B}}\left|e^{-\beta H_{\mathrm{B}}}\right|{\mathsf{\Psi }}_n^{\mathrm{B}}\rangle +\langle {\mathsf{\Psi }}_{n-1}^{\mathrm{F}}\left|e^{-\beta H_{\mathrm{F}}}\right|{\mathsf{\Psi }}_{n-1}^{\mathrm{F}}\rangle ]}}.\end{array}$$

(110)

Further using the fact that the bosonic and the fermionic Hamiltonians nth energy eigenfunction are exactly identical, the partition function for the Supersymmetric one-dimensional harmonic oscillator can be computed as:

$$\begin{array}{c}\hfill Z=1+\frac{1}{2}\sum _{n>0}^{}\left[(e^{-\beta \omega n}+e^{-\beta \omega n})\right]=\frac{1}{1-e^{-\beta \omega }}=\frac{1}{2}{e}^{\frac{\beta \omega }{2}}\mathrm{cosech}\left(\frac{\beta \omega }{2}\right).\end{array}$$

(111)

6.3. Computation of 2-Point OTOCs

6.3.1. Computation of $Y^{\left(1\right)}(t_1,t_2)$

We compute the 2-point correlators for the Supersymmetric Harmonic oscillator. We begin with the correlator constructed from two different dynamical operators, which is defined as:

$$\begin{array}{cc}\hfill Y^{\left(1\right)}(t_1,t_2)& =-{\langle [x\left(t_1\right),p\left(t_2\right)]\rangle }_{\beta }\hfill \\ \hfill & =-\frac{1}{Z}\sum _n^{}{e}^{-\beta E_n}\langle {\mathsf{\Psi }}_n\left|[x\left(t_1\right),p\left(t_2\right)]\right|{\mathsf{\Psi }}_n\rangle \hfill \\ \hfill & =-\frac{1}{Z}[\underset{\mathbf{Ground}\mathbf{State}\mathbf{Contribution}}{\underbrace{\langle {\mathsf{\Psi }}_0\left|[x\left(t_1\right),p\left(t_2\right)]\right|{\mathsf{\Psi }}_0\rangle }}+\underset{\mathrm{Excited}\mathrm{States}\mathrm{Contribution}}{\underbrace{\sum _{n>0}{e}^{-\beta E_n}\langle {\mathsf{\Psi }}_n\left|[x\left(t_1\right),p\left(t_2\right)]\right|{\mathsf{\Psi }}_n\rangle }}]\hfill \\ \hfill & =2e^{-\frac{\beta \omega }{2}}sinh\left(\frac{\beta \omega }{2}\right)\left(y_0^{\left(1\right)}(t_1,t_2)+\sum _{n>0}{e}^{-\beta \omega n}{y}_n^{\left(1\right)}(t_1,t_2)\right),\hfill \end{array}$$

(112)

where $y_0^{\left(1\right)}(t_1,t_2)$ and $y_n^{\left(1\right)}(t_1,t_2)$ represent the micro-canonical correlators for the ground state and the excited states, and the overall temperature-dependent normalization function is appearing from the expression for the inverse of the thermal partition function for canonical quantum statistical ensemble. The purpose of taking the ground state and excited states separately is due to the fact that, in Supersymmetric quantum mechanics, the ground state is always bosonic, having zero energy eigenvalue, whereas, for the excited states, there are contributions from the bosonic, as well as the fermionic, states. It is easy to see that the ground state contribution can be calculated by expanding the commutator and inserting the completeness relation, ${\displaystyle \sum _k|{\mathsf{\Psi }}_k\rangle \langle {\mathsf{\Psi }}_k|=1}$. However, here, it is important to note that there is no contribution appearing from the $k=0$ term, as for the ground state we have $\langle {\mathsf{\Psi }}_0\left|x\left(t_1\right)\right|{\mathsf{\Psi }}_0\rangle =0$. Hence, from the ground state, we finally get the following time-dependent non-trivial contribution, which is given by:

$$\begin{array}{c}\hfill \langle {\mathsf{\Psi }}_0\left|[x\left(t_1\right),p\left(t_2\right)]\right|{\mathsf{\Psi }}_0\rangle =\frac{i}{2}cos\omega (t_1-t_2).\end{array}$$

(113)

Now, to further determine the contribution from the excited states, we need to expand the commutator and insert the completeness relation between the desired time-dependent quantum operators, which gives the following expression:

$$\begin{array}{cc}\hfill \langle {\mathsf{\Psi }}_n\left|[x\left(t_1\right),p\left(t_2\right)]\right|{\mathsf{\Psi }}_n\rangle & =\frac{i}{2}cos\omega (t_1-t_2){\delta }_{n1}\hfill \\ \hfill & +\frac{i}{2}cos\omega (t_1-t_2)\left(1+\sqrt{n(n+1)}-\sqrt{n(n-1)}\right).\hfill \end{array}$$

(114)

Therefore, the micro-canonical correlators for the ground state and the excited states are given by the following temperature independent simplified expressions:

$$\begin{array}{cc}\hfill y_0^{\left(1\right)}& =-\frac{i}{2}cos\omega (t_1-t_2),\hfill \end{array}$$

(115)

$$\begin{array}{cc}\hfill y_n^{\left(1\right)}& =-\frac{i}{2}cos\omega (t_1-t_2){\delta }_{n1}-\frac{i}{2}cos\omega (t_1-t_2)\left(1+\sqrt{n(n+1)}-\sqrt{n(n-1)}\right)\hfill \\ \hfill & =-\frac{i}{2}cos\omega (t_1-t_2)\left[1+{\delta }_{n1}+\sqrt{n(n+1)}-\sqrt{n(n-1)}\right].\hfill \end{array}$$

(116)

Further, using Equations (115) and (116), the canonical 2-point thermal correlator can be obtained by the following expression:

$$\begin{array}{cc}\hfill Y^{\left(1\right)}(t_1,t_2)& =-i{e}^{-\frac{\beta \omega }{2}}sinh\left(\frac{\beta \omega }{2}\right)cos\omega (t_1-t_2)\hfill \\ \hfill & \times \left\{1+e^{-\beta \omega }+\sum _{n>0}^{}{e}^{-\beta \omega n}\left(1+\sqrt{n(n+1)}-\sqrt{n(n-1)}\right)\right\}\hfill \\ \hfill & =-\frac{i}{2}tanh\left(\frac{\beta \omega }{2}\right)cos\omega (t_1-t_2)\hfill \\ \hfill & {\displaystyle \times \left\{1+2cosh\left(\frac{\beta \omega }{2}\right)\sum _{n>0}^{}{e}^{-\beta \omega \left(n+\frac{1}{2}\right)}\left(1+\sqrt{n(n+1)}-\sqrt{n(n-1)}\right)\right\}.}\hfill \end{array}$$

(117)

It can be seen that the obtained OTOC is just a periodic function, indicating the absence of chaos. However, it is quite interesting that the obtained result has an explicit temperature dependence, which mainly comes from the contribution of the ground state. Though the 2-point OTOC is not of much significance, it stills give us a pretty good idea that, at the level 2-point correlation function, the context of Supersymmetric one-dimensional harmonic oscillator is non-chaotic in nature.

Micro-Canonical 2-point Correlator for Ground State:

$$\begin{array}{cc}\hfill y_0^{\left(1\right)}(t_1,t_2)& =-\frac{i}{2}cos\omega (t_1-t_2).\hfill \end{array}$$

(118)

Micro-Canonical 2-point Correlator for Excited States:

$$\begin{array}{c}\hfill y_n^{\left(1\right)}(t_1,t_{}\end{array}$$