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

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## Abstract

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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

**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:$$\begin{array}{ccc}& & \underline{\mathbf{At}\phantom{\rule{3.33333pt}{0ex}}\mathbf{infinite}\phantom{\rule{3.33333pt}{0ex}}\mathbf{temperature}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}(\beta \to \mathbf{0}):}\hfill \end{array}$$$$\begin{array}{ccc}& & \phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\frac{1}{\mu}\frac{\partial \mathcal{P}(n,\tau )}{\partial \tau}=\underset{\mathbf{Diffusion}}{\underbrace{n(n+1)\frac{{\partial}^{2}\mathcal{P}(n,\tau )}{\partial {n}^{2}}}}+\underset{\mathbf{Drift}}{\underbrace{(1+2n)\phantom{\rule{3.33333pt}{0ex}}\frac{\partial \mathcal{P}(n,\tau )}{\partial n}}},\phantom{\rule{3.33333pt}{0ex}}\hfill \\ & & \underline{\mathbf{At}\phantom{\rule{3.33333pt}{0ex}}\mathbf{finite}\phantom{\rule{3.33333pt}{0ex}}\mathbf{temperature}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}(\beta \ne \mathbf{0}):}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\hfill \\ & & \phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\frac{1}{\mu}\frac{\partial \mathcal{P}(n,\tau )}{\partial \tau}=\underset{\mathbf{Diffusion}}{\underbrace{n(n+1)\frac{{\partial}^{2}\mathcal{P}(n,\tau )}{\partial {n}^{2}}}}+\underset{\mathbf{Drift}}{\underbrace{(1+2n)\phantom{\rule{3.33333pt}{0ex}}\frac{\partial \mathcal{P}(n,\tau )}{\partial n}}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\hfill \end{array}$$$$\begin{array}{ccc}& & \phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}+\underset{\mathbf{Finite}\phantom{\rule{3.33333pt}{0ex}}\mathbf{temperature}\phantom{\rule{3.33333pt}{0ex}}\mathbf{contribution}}{\underbrace{\beta \left\{n(n+1)\left[\frac{\partial V\left(n\right)}{\partial n}\phantom{\rule{3.33333pt}{0ex}}\frac{\partial \mathcal{P}(n,\tau )}{\partial n}+\frac{{\partial}^{2}V\left(n\right)}{\partial {n}^{2}}\phantom{\rule{3.33333pt}{0ex}}\mathcal{P}(n,\tau )\right]+(2n+1)\phantom{\rule{3.33333pt}{0ex}}\frac{\partial V\left(n\right)}{\partial n}\phantom{\rule{3.33333pt}{0ex}}\mathcal{P}(n,\tau )\right\}}},\hfill \end{array}$$$$\begin{array}{c}\hfill \mathbf{Statistical}\phantom{\rule{3.33333pt}{0ex}}\mathbf{average}\phantom{\rule{3.33333pt}{0ex}}\mathbf{of}\phantom{\rule{3.33333pt}{0ex}}\mathbf{moment}\phantom{\rule{3.33333pt}{0ex}}\mathbf{generator}:\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\langle \mathcal{F}(n)\rangle :=\int dn\phantom{\rule{3.33333pt}{0ex}}\mathcal{F}(n)\phantom{\rule{3.33333pt}{0ex}}\mathcal{P}(n,\tau ),\end{array}$$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:$$\begin{array}{c}\hfill \langle \prod _{i=1}^{N}n\left({\tau}_{i}\right)\rangle =\langle n\left({\tau}_{1}\right)n\left({\tau}_{2}\right)\cdots n\left({\tau}_{N}\right)\rangle \phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\forall \phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\tau}_{i}\phantom{\rule{3.33333pt}{0ex}}(i=1,2,\cdots ,N)\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathbf{contain}\phantom{\rule{3.33333pt}{0ex}}\mathbf{disorder},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\end{array}$$**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, ${\mathcal{O}}_{i}\left({t}_{j}\right)\phantom{\rule{3.33333pt}{0ex}}\forall \phantom{\rule{3.33333pt}{0ex}}i,j=1,2$ with $i\ne j$ or $i=j$ possibilities. The time disorder thermal average over statistical ensemble is described by the following expression: -4.6cm0cm$$\begin{array}{ccc}\hfill {C}_{N}^{\left(ij\right)}({t}_{1},{t}_{2}):=-{\langle {\left[{\mathcal{O}}_{i}\left({t}_{1}\right),{\mathcal{O}}_{j}\left({t}_{2}\right)\right]}^{N}\rangle}_{\beta}& =& -\mathrm{Tr}\left[{\rho}_{\beta}\phantom{\rule{3.33333pt}{0ex}}{\left[{\mathcal{O}}_{i}\left({t}_{1}\right),{\mathcal{O}}_{j}\left({t}_{2}\right)\right]}^{N}\right]\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\forall \phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}i,j=1,2,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\hfill \end{array}$$$$\begin{array}{c}\hfill {\rho}_{\beta}:=\frac{1}{Z}\phantom{\rule{3.33333pt}{0ex}}exp\left(-\beta H\right)\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{with}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}Z=\mathrm{Tr}\left[exp(-\beta H)\right].\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\end{array}$$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.

**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

## 3. A Short Review of Supersymmetric Quantum Mechanics

## 4. General Remarks on Time Disorder Averaging and Thermal OTOCs

**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 $,

## 5. Eigenstate Representation of thermal OTOCs in Supersymmetric Quantum Mechanics

**What do we learn here?:**

- 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.

**Correlation Functions**

**Normalized 4-point Correlation Functions**

#### 5.1. Partition Function from Supersymmetric Quantum Mechanics

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

#### 5.3. Representation of 2-Point OTOC: ${Y}^{\left(2\right)}({t}_{1},{t}_{2})$

#### 5.4. Representation of 2-Point OTOC: ${Y}^{\left(3\right)}({t}_{1},{t}_{2})$

#### 5.5. Representation of 4-Point OTOC: ${C}^{\left(1\right)}({t}_{1},{t}_{2})$

#### 5.5.1. Un-Normalized: ${C}^{\left(1\right)}({t}_{1},{t}_{2})$

#### 5.5.2. Normalized: ${\tilde{C}}^{\left(1\right)}({t}_{1},{t}_{2})$

#### 5.6. Representation of 4-Point OTOC: ${C}^{\left(2\right)}({t}_{1},{t}_{2})$

#### 5.6.1. Un-Normalized: ${C}^{\left(2\right)}({t}_{1},{t}_{2})$

#### 5.6.2. Normalized: ${\tilde{C}}^{\left(2\right)}({t}_{1},{t}_{2})$

#### 5.7. Representation of 4-Point OTOC: ${C}^{\left(3\right)}({t}_{1},{t}_{2})$

#### 5.7.1. Un-Normalized: ${C}^{\left(3\right)}({t}_{1},{t}_{2})$

#### 5.7.2. Normalized: ${\tilde{C}}^{\left(3\right)}({t}_{1},{t}_{2})$

#### 5.8. Summary of Results

**Eigenstate Representation for Micro-Canonical Correlators]**

**Eigenstate Representation for Canonical Correlators without normalization]**

## 6. Model I: Supersymmetric Quantum Mechanical Harmonic Oscillator

#### 6.1. Eigenspectrum of the Super-Partner Hamiltonian

#### 6.2. Partition Function

#### 6.3. Computation of 2-Point OTOCs

#### 6.3.1. Computation of ${Y}^{\left(1\right)}({t}_{1},{t}_{2})$

**Micro-Canonical 2-point Correlator for Ground State**:

**Micro-Canonical 2-point Correlator for Excited States**: