Out of time ordered effective dynamics of a quartic oscillator
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Out of time ordered effective dynamics of a quartic oscillator
Bidisha Chakrabarty † , Soumyadeep Chaudhuri International Centre for Theoretical Sciences (ICTS-TIFR),Tata Institute of Fundamental Research,Shivakote, Hesaraghatta,Bangalore 560089, INDIA † [email protected], * [email protected] Abstract
We study the dynamics of a quantum Brownian particle weakly coupled to a ther-mal bath. Working in the Schwinger-Keldysh formalism, we develop an effectiveaction of the particle up to quartic terms. We demonstrate that this quartic ef-fective theory is dual to a stochastic dynamics governed by a non-linear Langevinequation. The Schwinger-Keldysh effective theory, or the equivalent non-linearLangevin dynamics, is insufficient to determine the out of time order correlators(OTOCs) of the particle. To overcome this limitation, we construct an extendedeffective action in a generalised Schwinger-Keldysh framework. We determine theadditional quartic couplings in this OTO effective action and show their depen-dence on the bath’s 4-point OTOCs. We analyse the constraints imposed on theOTO effective theory by microscopic reversibility and thermality of the bath. Weshow that these constraints lead to a generalised fluctuation-dissipation relationbetween the non-Gaussianity in the distribution of the thermal noise experiencedby the particle and the thermal jitter in its damping coefficient. The quartic effec-tive theory developed in this work provides extension of several results previouslyobtained for the cubic OTO dynamics of a Brownian particle.
Contents qXY model 63 Schwinger-Keldysh effective theory of the oscillator 8 ciPost Physics Submission
A.1 2-point cumulants 39A.2 4-point cumulants 40
B Argument for the validity of the Markovian limit 41C Relations between the OTO effective couplings and the bath’s OTOCs 42D Quartic couplings in the high temperature limit 44References 48
The effective theory framework is a very useful tool in understanding the dynamics of openquantum systems. Such an open system typically interacts with a large environment whichhas a complicated dynamics. This complexity of the environment often makes a microscopicanalysis practically impossible for studying the observables of the system. In such a scenario,one can try to develop an effective theory for only the infrared (low-frequency) degrees offreedom of the system.The construction of such an effective theory usually relies on imposinga) few general conditions based on the unitarity of the microscopic theory, andb) some special conditions based on the symmetries in the particular theory.After identifying these conditions, one can write down the most general effective theory thatsatisfies them. This theory can be then used to determine the observables of the system interms of the effective couplings.A paradigmatic example of an open system where such an effective dynamics can beworked out is a Brownian particle interacting with a thermal bath. The first step towardsconstructing the effective theory is to consider the evolution of the density matrix of the(particle+bath) combined system. One can then trace out the bath’s degrees of freedom toget the evolution of the reduced density matrix of the particle.An efficient way to obtain the evolution of the reduced density matrix is via a pathintegral formalism developed by Feynman-Vernon [1], Schwinger [2] and Keldysh [3], which isnow commonly known as the ‘Schwinger-Keldysh formalism’ [4–6]. In this formalism, tracingout the bath’s degrees of freedom amounts to integrating over them to obtain a correction to2 ciPost Physics Submission the particle’s action. This correction, called the ‘influence phase’ [1], encapsulates the entirecontribution of the interaction with the bath to the effective dynamics of the particle. As wewill discuss later in section 3.2, the form of the influence phase is completely determined interms of correlators of the bath operator that couples to the particle.Generally, such thermal correlators of the bath are hard to determine. Nevertheless, therelationship between the bath’s correlators and the particle’s influence phase is interesting forthe following reasons:1. Such a relation provides a way to extract information about the bath’s correlators bystudying the effective dynamics of the particle.2. The thermality of the bath leads to the Kubo-Martin-Schwinger (KMS) relations [7–10]between its correlators. These relations, in turn, constrain the effective dynamics of theparticle.3. There may be some symmetries in the bath’s dynamics which would lead to furtherrelations between its correlators. Such relations would impose additional constraints onthe particle’s effective theory.4. It may be possible to construct simple toy models where one has analytic handle on themicroscopic dynamics of the bath. In such cases, one can calculate the bath’s correlatorsand then determine the effective couplings of the particle from them. Insights obtainedfrom studying such simple toy models can be used to get a general idea about thedynamics of the particle in more complicated situations.A classic example of such a simple toy model was provided by Caldeira and Leggett in [11]where they considered a bath comprising of a set of harmonic oscillators. These oscillatorscouple to the particle through an interaction which is bilinear in the positions of the particleand the bath oscillators. In this setup, Caldeira-Leggett integrated out the bath oscillatorsto obtain a temporally non-local influence phase which is quadratic in the particle’s position.The simplicity of this quadratic effective theory makes it possible to employ both analyticand numerical techniques to study the evolution of the particle [11–21] . Such studies haveshown that, even in this simple scenario, the Brownian particle demonstrates a rich varietyof phenomena such as decoherence, dissipation, thermalisation, etc.The Caldeira-Leggett model is also suitable for exploring a regime where the particle’sdynamics is approximately local in time. Such a local dynamics can be obtained when theeffect of the interaction with the particle dissipates in the bath much faster than the time-scale in which the particle evolves. As discussed in [22], this can be ensured by taking thetemperature of the bath to be very high and choosing an appropriate distribution for thecouplings between the particle and the bath oscillators. This regime in which the particle hasan approximately local dynamics is often called the ‘Markovian limit’.In this limit, the quadratic effective theory of the Brownian particle is dual to a classicalstochastic dynamics governed by a linear Langevin equation with a Gaussian noise [6]. Underthis Langevin dynamics, the particle experiences a damping as well as a randomly fluctuatingforce. Both these forces originate from the interaction with the bath, and hence their strengthsare related to each other. As we will review in section 4.4, this fluctuation-dissipation relation[23–26] follows from the KMS relations between the 2-point correlators of the bath.Many of the features of the Caldeira-Leggett model that we discussed above are shared bymore complicated systems where the quadratic effective theory holds to a good approximation3 ciPost Physics Submission [22, 27–29]. Nevertheless, it is interesting to explore the corrections to this effective dynamicsby including the cubic and higher degree terms in the analysis. A simple way to obtain suchcorrections is to extend the Caldeira-Leggett model by introducing a non-linear combinationof the bath oscillators’ positions in the operator that couples to the particle [29, 30].One such extension was discussed in [30] where the oscillators in the bath were dividedinto two sets (labelled by X and Y). Then apart from the bilinear interactions (which arepresent in the Caldeira-Leggett model), small cubic interactions were introduced between theparticle (q) and pairs of bath oscillators. Each such pair consisted of one oscillator drawnfrom the set X and the other from the set Y. Due to these cubic interactions, this model wascalled the ‘qXY model’.Introduction of these cubic interactions leads to cubic and higher degree terms in theeffective action of the particle. In [30], this effective theory was studied up to the cubic termsin a Markovian limit. It was shown to be dual to a non-linear Langevin dynamics which hasperturbative corrections over the linear Langevin dynamics in the Caldeira-Leggett model.The additional parameters in this non-linear dynamics receive contributions from the 3-pointcorrelators of the bath. These 3-point correlators lead to an anharmonicity in the particle’sdynamics and a non-Gaussianity in the distribution of the thermal noise experienced by theparticle. In addition, they also give rise to thermal jitters in both the frequency and thedamping of the particle. The thermal jitter in the damping coefficient and the non-Gaussianityin the noise are connected by a generalised fluctuation-dissipation relation [30].A valuable insight gained from the cubic effective theory in [30] is that this generalisedfluctuation-dissipation relation arises from a combined effect of the thermality of the bath andthe microscopic time-reversal invariance in its dynamics. These features of the bath manifestin the form of certain relations between its correlators . As discussed in [9] and [30], theserelations between thermal correlators lead to the inclusion of Out-of-Time-Order Correlators(OTOCs) [31–33] of the bath in the analysis. Such thermal OTOCs have aroused significantinterest in the recent years as they provide insight into the chaotic behaviour of quantumsystems [34–40], and serve as useful diagnostic measures in the study of thermalisation andmany-body localisation [41–44].A convenient way to study the effects of the bath’s OTOCs on the particle’s dynamicsis to extend the effective theory framework to include computation of the particle’s OTOCs.Such an OTO effective theory was developed upto cubic terms in [45]. There, it was seenthat this extension requires introduction of some new couplings which are determined by the3-point OTOCs of the bath. These new couplings in the cubic OTO effective theory getrelated to couplings in the Schwinger-Keldysh effective theory (or the equivalent non-linearLangevin dynamics) due to microscopic reversibility in the bath [30]. At the same time, theKMS relations between bath’s correlators connect the non-Gaussianity in the noise to one ofthe OTO couplings [30]. Combining these relations, one obtains the generalised fluctuation-dissipation relation mentioned above.In this paper, we extend the analysis of the OTO effective theory of the Brownian particleby including the quartic terms in the effective action. There are three reasons for developingthis extension. We enumerate them below. These thermal jitters arise from the same non-Gaussian noise distribution. For example, the thermality of the bath leads to the KMS relations that we mentioned earlier. Such relations between cubic effective couplings can be interpreted as generalisations of the Onsager-Casimir reciprocal relations [46–48] in quadratic effective theories for systems with multiple degrees of freedom. ciPost Physics Submission
1. Such an extension lays the ground for a convenient framework to compute 4-point OTOcorrelators in open quantum systems. This may lead to further insight into the roleplayed by these OTOCs in physical phenomena.2. The quartic couplings in this extended OTO effective theory receives contributions fromthe 4-point thermal OTOCs of the bath. Such thermal OTOCs have been the focus ofmost recent studies on chaos in quantum systems [34–40]. Including the contributionsof these 4-point OTOCs in the effective theory of the particle opens the possibility ofprobing the chaotic behaviour of the bath via the quartic effective couplings [45] (seethe discussion in section 5).3. The relations between the quartic effective couplings and the 4-point OTOCs of the bathallows one to extend the analysis of the constraints imposed on the effective dynamicsof the particle due to the bath’s thermality and microscopic reversibility.To present the construction of the effective theory with a concrete example, we choose towork with the qXY model that we discussed earlier. We simplify this model a bit by switchingoff the Caldeira-Legget-like bilinear interactions. This leads to a symmetry in the particle’sdynamics under q → − q, (1)which results in the vanishing of all the odd degree terms in the effective action.For this simplified qXY model, we first develop the Schwinger-Keldysh effective theory ofthe particle up to quartic terms. Working in a Markovian limit, we determine the dependenceof the quartic couplings on the 4-point Schwinger-Keldysh correlators of the bath. Then weshow that this quartic effective theory is dual to a non-linear Langevin dynamics with a non-Gaussian noise distribution. This stochastic dynamics has a structure quite similar to the onediscussed in [30] for the cubic effective theory.We then extend the analysis to the OTO dynamics of the particle by determining its effec-tive action on a contour with two time-folds . The form of this effective action is constrainedby the microscopic unitarity in the dynamics of the (particle+bath) combined system [45].We figure out all the additional OTO couplings consistent with these constraints, and thendetermine their dependence on the 4-point OTO correlators of the bath. As in case of the 3-point correlators [30] , these 4-point functions of the bath satisfy certain relations imposed bymicroscopic time-reversal invariance and thermality. We show that these relations betweenthe bath’s correlators lead to some constraints on the particle’s effective couplings. Theseconstraints can be interpreted as OTO generalisations [30] of the Onsager reciprocal rela-tions [46, 47] and the fluctuation-dissipation relation [25, 49]. Combining these constraints,one can obtain a generalised fluctuation-dissipation relation between two quartic couplingsin the Schwinger-Keldysh effective theory (or equivalently, the dual non-linear Langevin dy-namics). Just as in case of the cubic effective theory [30], we find that this generalisedfluctuation-dissipation relation connects the thermal jitter in the damping coefficient of theparticle and the non-Gaussianity in the noise distribution. Organisation of the paper
The structure of this paper is as follows: Such a path integral formalism on a contour with multiple time-folds is a straightforward generalisation[32, 33, 45] of the Schwinger-Keldysh formalism. ciPost Physics Submission In section 2, we briefly describe the qXY model which serves as a concrete example in ouranalysis.In section 3, we develop the Schwinger-Keldysh effective action for the particle in a Marko-vian regime. We determine the relations between the effective couplings and the Schwinger-Keldysh correlators of the bath. We also demonstrate a duality between the Schwinger-Keldysh effective theory and a stochastic dynamics governed by a non-linear Langevin equa-tion.In section 4, we extend the effective theory framework to include OTOCs. We deter-mine all the additional quartic OTO couplings appearing in this extension. Exploring theconstraints imposed on the OTO effective theory by the thermality of the bath and its mi-croscopic reversibility, we derive the generalised Onsager relations as well as a generalisedfluctuation-dissipation relation between the quartic effective couplings. By computing theeffective couplings for the qXY model, we verify that all these relations are indeed satisfied.In section 5, we conclude with some discussion on future directions.In appendix A, we show how the cumulants of the bath’s correlators decay when theinterval between any two insertions is increased. In appendix B, we provide an argumentfor the validity of Markov approximation in a certain parameter regime. In appendix C, weexpress the quartic OTO couplings in terms of the 4-point OTO cumulants of the bath. Inappendix D, we provide the forms of the high-temperature limit of all the quartic couplingsin terms of the thermal spectral functions of the bath. qX Y model
In this section, we describe the qXY model which will serve as a concrete example for devel-oping the particle’s effective dynamics in the rest of the paper.In this model, a Brownian particle interacts with a thermal bath (at the temperature β ) comprising of two sets of harmonic oscillators. We represent the positions of the oscillators inthese two sets by X ( i ) and Y ( j ) , and the position of the Brownian particle by q . The particleand the bath couple via cubic interactions involving two of the bath oscillators, one takenfrom each set. The Lagrangian of this model is given by L [ q, X, Y ] = m p q − µ q ) + X i m x,i X ( i )2 − µ x,i X ( i )2 )+ X j m y,j Y ( j )2 − µ y,j Y ( j )2 ) + λ X i,j g xy,ij X ( i ) Y ( j ) q. (2)The bath operator that couples to the particle is λO ≡ λ X i,j g xy,ij X ( i ) Y ( j ) . (3)Notice that all odd point correlators of this operator vanish in the thermal state. We will seelater that this leads to vanishing of all odd degree terms in the effective action of the particle.Among the remaining terms, we would like to restrict our attention to only the quadratic andthe quartic ones in this paper. We will see that the couplings corresponding to these terms Here we are working in units where the Boltzmann constant k B = 1. ciPost Physics Submission receive contributions from the connected parts of the 2-point and 4-point correlators of O atleading order in λ . So, we need to compute these correlators to obtain the leading order formsof the quadratic and the quartic couplings of the particle.While computing the correlators, we assume that there is a large number of oscillators inthe bath, and the frequencies of these oscillators are densely distributed. In such a situation,one can go to the continuum limit of this distribution, and replace the sum over the frequenciesby integrals in the following way: X i,j g xy,ij m x,i m y,j → Z ∞ dµ x π Z ∞ dµ y π DD g xy ( µ x , µ y ) m x m y EE , (4) X i ,j X i ,j g xy,i j g xy,i j g xy,i j g xy,i j m x,i m y,j m x,i m y,j → Z ∞ dµ x π Z ∞ dµ y π Z ∞ dµ ′ x π Z ∞ dµ ′ y π DD g xy ( µ x , µ y ) g xy ( µ x , µ ′ y ) g xy ( µ ′ x , µ y ) g xy ( µ ′ x , µ ′ y ) m x m y m ′ x m ′ y EE , (5)where DD g xy ( µ x , µ y ) m x m y EE ≡ X i,j g xy,ij m x,i m y,j h πδ ( µ x − µ x,i )2 πδ ( µ y − µ y,j ) i , (6) DD g xy ( µ x , µ y ) g xy ( µ x , µ ′ y ) g xy ( µ ′ x , µ y ) g xy ( µ ′ x , µ ′ y ) m x m y m ′ x m ′ y EE ≡ X i ,j X i ,j g xy,i j g xy,i j g xy,i j g xy,i j m x,i m y,j m x,i m y,j h πδ ( µ x − µ x,i )2 πδ ( µ y − µ y,j ) ih πδ ( µ ′ x − µ x,i )2 πδ ( µ ′ y − µ y,j ) i . (7)We choose the functions DD g xy ( µ x ,µ y ) m x m y EE and DD g xy ( µ x ,µ y ) g xy ( µ x ,µ ′ y ) g xy ( µ ′ x ,µ y ) g xy ( µ ′ x ,µ ′ y ) m x m y m ′ x m ′ y EE in amanner which would give us an approximately local effective dynamics of the particle. As wewill see, such a local dynamics can be obtained if the time-scales involved in the evolutionof the particle are much larger than the time-scale in which cumulants of the operator O ( t )decay. Keeping this in mind, we choose the distribution of the couplings to satisfy λ DD g xy ( µ x , µ y ) m x m y EE = Γ µ x Ω µ x + Ω µ y Ω µ y + Ω , (8) λ DD g xy ( µ x , µ y ) g xy ( µ x , µ ′ y ) g xy ( µ ′ x , µ y ) g xy ( µ ′ x , µ ′ y ) m x m y m ′ x m ′ y EE = Γ (cid:16) µ x Ω µ x + Ω (cid:17)(cid:16) µ y Ω µ y + Ω (cid:17)(cid:16) µ ′ x Ω µ ′ x + Ω (cid:17)(cid:16) µ ′ y Ω µ ′ y + Ω (cid:17) , (9)where Ω is a UV regulator.For this distribution of the couplings, we study the high temperature limit of the 2-pointand 4-point cumulants of O ( t ) in appendix A. There, we show that when the time intervals7 ciPost Physics Submission between the insertions are increased, these cumulants decay exponentially at rates which areof the order of Ω. If the natural frequency ( µ ) of the particle and the frequency scalesassociated with the parameters Γ and Γ are taken to be much smaller than Ω, then the bathcorrelators decay much faster than the rate at which the particle evolves. This ensures thatthe effect (via the bath) of some earlier state of the particle on its later dynamics is heavilysuppressed. Consequently, we get an approximately local effective dynamics of the particle.We discuss this effective dynamics in the next section. In this section, we develop the Schwinger-Keldysh effective theory for the dynamics of theparticle. We also demonstrate a duality between this quantum effective theory and a classicalstochastic theory governed by a non-linear Langevin equation.
Let us begin the discussion on the particle’s effective dynamics by specifying the initial stateof the particle and reviewing a path integral formalism which gives its evolution.Consider the situation where the particle is initially unentangled with the bath. Sup-pose the cubic interaction given in (2) is switched on at a time t . Then the state of the(particle+bath) combined system at t is given by ρ = e − βH B Z B ⊗ ρ p , (10)where H B and Z B are the Hamiltonian and the partition function of the bath respectively,and ρ p is the density matrix of the particle at the time t .After the particle starts interacting with the bath, an effective description of its state canbe given in terms of its reduced density matrix which is obtained by tracing out the bath’sdegrees of freedom in the density matrix of the combined system. The evolution of the reduceddensity matrix is given by the quantum master equation [22] of the particle. An equivalentdescription of this evolution can be developed in terms of an effective action of the particle in the Schwinger-Keldysh formalism [1–6].In this formalism, one can first determine the density matrix of the combined system atsome later time t f from a path integral on the contour shown in figure 1. This contour hastwo legs which we label as 1 and 2. For each of these legs, we need to take a copy of theFigure 1: Contour for evolution of the density matrix time t t f degrees of freedom of both the particle and the bath: { q , X , Y } and { q , X , Y } . The Here, we work in units where ~ = 1. We refer the reader to [50] for a detailed discussion on how the quantum master equation is related to theSchwinger-Keldysh effective action. Here, X and Y denote collectively all the oscillators in the two sets. ciPost Physics Submission evolution of the density matrix of the combined system is then obtained from a path integralwith a Lagrangian of the following form L SK = L [ q , X , Y ] − L [ − q , X , Y ] . (11)Let us denote all the degrees of freedom of the combined system collectively by Q . Then, thetwo copies of these degrees of freedom can be expressed as { q , X , Y } → Q , {− q , X , Y } → Q . (12)Now, given the information that the initial density matrix at time t is ρ ( Q , Q ), thedensity matrix at the later time t f is given by the following path integral: ρ f ( Q f , Q f ) = Z dQ Z dQ ρ ( Q , Q ) Q ( t f )= Q f ,Q ( t f )= Q f Z Q ( t )= Q ,Q ( t )= Q [ DQ ][ DQ ] e i R tft dt L SK [ Q ( t ) ,Q ( t )] . (13)To get the reduced density matrix ( ρ pf ) of the particle at the time t f , we need to traceout the bath’s degrees of freedom at t f . This can be achieved in the above path integral bysetting X f = X f = X f and Y f = Y f = Y f , (14)and then integrating over X f and Y f .This path integral representation of the reduced density matrix still involves integrals overthe bath’s degrees of freedom. One can, however, express this integral in terms of only theparticle’s degrees of the freedom by integrating out the bath’s coordinates. As mentionedin the introduction, such an integral over the bath’s coordinates leads to a correction to theparticle’s action. This additional piece in the action which encapsulates the influence of thebath on the particle’s dynamics is called the ‘influence phase’ of the particle [1]. Next, wediscuss the form of this influence phase. In the previous subsection we argued that the expression for the reduced density matrix ρ pf can be obtained from the path integral in (13) by identifying the bath’s degrees of freedomon the two legs at the time t f . Therefore, in this expression, the bath’s coordinates have tobe integrated over a contour of the following form:Figure 2: Schwinger-Keldysh contour t t f This is the usual Schwinger-Keldysh contour where the two legs are connected by a future-turning point. Path integrals over such a contour give correlators where the insertions are Here we find it convenient to put an extra minus sign in q over the standard convention followed in textson the Schwinger-Keldysh formalism [4, 5]. This is consistent with the convention followed in [30, 45]. ciPost Physics Submission contour-ordered according to the arrow indicated in figure 2. Therefore, such contour-orderedcorrelators of the bath would contribute to the effective action of the particle when the bath’scoordinates are integrated out.The contributions of the bath correlators are imprinted in the influence phase ( W SK )which appears in the path integral for ρ pf as follows ρ pf ( q f , − q f ) = Z dq Z dq ρ p ( q , − q ) q ( t f )= q f ,q ( t f )= q f Z q ( t )= q ,q ( t )= q [ Dq ][ Dq ] e i h m p R tft dt n ( ˙ q − µ q ) − ( ˙ q − µ q ) o + W SK i . (15)The influence phase in the above expression can be expanded as a perturbation series in λ : W SK = ∞ X n =1 λ n W ( n ) SK , (16)where W ( n ) SK = i n − X i , ··· ,i n =1 Z t f t dt · · · Z t n − t dt n hT C O i ( t ) · · · O i n ( t n ) i c q i ( t ) · · · q i n ( t n ) . (17)Here hT C O i ( t ) · · · O i n ( t n ) i c is the cumulant (connected part) of a contour-ordered correlatorof O ( t ) where the insertion at time t j is on the i th j leg.We remind the reader that, for the qXY model, such a thermal correlator of O ( t ) is zerowhen the number of insertions is odd. This in turn means that all the odd degree terms inthe perturbative expansion of the influence phase vanish. Among the remaining terms, werestrict our attention to those whose coefficients are up to O( λ ). This leaves us with onlythe quadratic and quartic terms whose expressions are given below: W (2) SK = i X i ,i =1 Z t f t dt Z t t dt hT C O i ( t ) O i ( t ) i c q i ( t ) q i ( t ) , (18) W (4) SK = − i X i , ··· ,i =1 Z t f t dt Z t t dt Z t t dt Z t t dt hT C O i ( t ) O i ( t ) O i ( t ) O i ( t ) i c q i ( t ) q i ( t ) q i ( t ) q i ( t ) . (19)Plugging these expressions for the quadratic and quartic terms in the influence phase intothe path integral given in (15), one can determine the evolution of the reduced density matrixof the particle. The effective action that appears in this path integral can also be used tocompute the correlators of the particle. From the form of the influence phase in (17), wecan see that this effective action is non-local in time. Next, we discuss a certain limit in whichone may get an approximately local form for this effective action. To compute the particle’s correlators from path integrals with this effective action, one needs to put q = − q at some time in the future of all the insertions. ciPost Physics Submission In appendix A, we have shown that, in the high temperature limit ( β Ω ≪ O ( t ) appearing in (18) and (19) decay exponentially when the separationbetween any two insertions is increased. We have also shown that the decay rates of thesecumulants are of the order of the cut-off frequency Ω. Then the values of the coefficientfunctions multiplying the q’s at different times in (18) and (19) become negligible when theinterval between any two instants is O(Ω − ).We choose to work in a regime where this time-scale (Ω − ) is much smaller than all thetime-scales involved in the evolution of the particle. To ensure this, we take the parametersin the qXY model to satisfy the following conditions: β Ω ≪ , µ ≪ Ω , Γ ≪ β ( β Ω) , Γ ≪ (Γ ) . (20)In this regime, the bath’s cumulants die out too fast to transmit any significant effect ofthe history of the particle on its dynamics at a later instant. Consequently, one can getan approximately local form for the influence phase by Taylor-expanding the q’s at differentinstants around t : W (2) SK ≈ i X i ,i =1 Z t f t dt hn Z t t dt hT C O i ( t ) O i ( t ) i c o q i ( t ) q i ( t − ǫ )+ n Z t t dt hT C O i ( t ) O i ( t ) i c t o q i ( t ) ˙ q i ( t − ǫ )+ n Z t t dt hT C O i ( t ) O i ( t ) i c t o q i ( t )¨ q i ( t − ǫ ) i , (21) W (4) SK ≈ − i X i , ··· ,i =1 Z t f t dt hn Z t t dt Z t t dt Z t t dt hT C O i ( t ) O i ( t ) O i ( t ) O i ( t ) i c o q i ( t ) q i ( t − ǫ ) q i ( t − ǫ ) q i ( t − ǫ )+ n Z t t dt Z t t dt Z t t dt hT C O i ( t ) O i ( t ) O i ( t ) O i ( t ) i c t o q i ( t ) ˙ q i ( t − ǫ ) q i ( t − ǫ ) q i ( t − ǫ )+ n Z t t dt Z t t dt Z t t dt hT C O i ( t ) O i ( t ) O i ( t ) O i ( t ) i c t o q i ( t ) q i ( t − ǫ ) ˙ q i ( t − ǫ ) q i ( t − ǫ )+ n Z t t dt Z t t dt Z t t dt hT C O i ( t ) O i ( t ) O i ( t ) O i ( t ) i c t o q i ( t ) q i ( t − ǫ ) q i ( t − ǫ ) ˙ q i ( t − ǫ ) i . (22)Here, we have kept up to second order derivative terms in the quadratic piece to take intoaccount the correction to the kinetic term in the particle’s action. For the quartic piece, we In appendix B, we provide an argument for the validity of Markov approximation in this regime. ciPost Physics Submission have kept only terms with at most a single derivative. To preserve the information aboutthe original ordering of the different time instants, we have kept a small point-split regulator ǫ > Using this approximate local form of the influence phase, one can determine the particle’scorrelators. The same correlators can be obtained from a 1-particle irreducible effective action which is local in time. Next, we discuss the form of this 1-PI effective action. In the previous subsection we discussed an approximately local form of the influence phasewhich is obtained by perturbatively expanding the effects of the particle-bath interaction onthe particle’s dynamics. By putting appropriate initial conditions on path integrals with thisinfluence phase one can calculate the correlators of the particle. However, as we saw, theinfluence phase has a slight non-locality due to the presence of the point-split regulator. Itis cumbersome to keep track of this ordering of the time instants for the different q’s in theeffective action. To avoid this, we extend the framework introduced in [45] and develop alocal 1-PI effective action for the particle which can be employed to compute its correlators. The form of this 1-PI effective action is constrained by some general principles which wediscuss below. • Collapse rule:
The effective action should vanish under the following identification: q = − q = ˜ q. (23)As discussed in [5, 45, 51], this condition is based on the fact that the particle is a partof a closed system governed by a unitary dynamics. At the level of correlators, it makessure that the value of a contour-ordered correlator of the particle just picks up a signwhen one slides the future-most insertion from one leg to the other without changing itstemporal position. The sign is introduced because we are putting an extra minus signin the definition of q over the convention followed in [5, 51]. • Reality conditions:
The effective action should become its own negative under complex conjugation of theeffective couplings along with the following exchange: q ↔ − q . (24)This condition is based on the Hermiticity of the operator q(t) which implies thatthe correlators of q should remain unchanged under a reversal of the ordering of theinsertions followed by a complex conjugation. As discussed in [45, 51], the above realityconditions ensure that such relations between the particle’s correlators are satisfied. It is important to keep this regulator as otherwise one can get wrong answers while computing contributionsof loop diagrams where two of the q’s on the same vertex contract with each other. Only the tree level diagrams obtained from the 1-particle irreducible effective action contribute to thecorrelators of the particle. The issue of contraction of q’s on the same vertex does not arise in the 1-PI effective theory frameworkbecause the correlators receive contributions only from tree-level diagrams. Hence, there is no need for apoint-split regulator in the terms of the 1-PI effective action. ciPost Physics Submission • Symmetry under q → − q , q → − q : The Lagrangian of the qXY mode given in (2) is symmetric under the following twotransformations:. q → − q, X → − X, Y → Y,q → − q, X → X, Y → − Y. (25)Moreover, the bath is in a thermal state which is also invariant under the above trans-formations. Now, if the initial state of the particle obeys the same symmetry, then allcorrelators of the particle with odd number of insertions would vanish. This vanish-ing of the odd point functions of the particle for a class of initial conditions can beensured by demanding that the 1-PI effective action is symmetric under the followingtransformation: q → − q , q → − q . (26)The most general 1-PI effective Lagrangian that is consistent with these conditions can bewritten as a sum of terms with even number of q’s as follows: L SK,1-PI = L (2)SK,1-PI + L (4)SK,1-PI + · · · . (27)In this paper, we will focus only on the quadratic and the quartic terms in this effective action.We give the forms of these terms below: Quadratic terms: L (2)SK,1-PI = 12 ( ˙ q − ˙ q ) − i Z I ( ˙ q + ˙ q ) − µ ( q − q )+ i h f i ( q + q ) − γ ( q + q )( ˙ q − ˙ q ) . (28)Here we consider all quadratic terms up to two derivatives acting on the q’s. We haveincluded the double derivative terms to take into account the renormalisation of the kineticterm in the action. Such a renormalisation introduces a correction to the effective mass of theparticle on top of the bare mass m p . After taking into account this correction, we choose towork in units where the renormalised mass of the particle is unity. Quartic terms: L (4)SK,1-PI = iζ (4) N
4! ( q + q ) + ζ (2) µ q + q ) ( q − q ) − iζ q − q ) − λ
48 ( q − q )( q − q ) + ζ (2) γ q + q ) ( ˙ q − ˙ q ) − λ γ
48 ( ˙ q + ˙ q )( q − q ) + iζ γ q + q ) ( q − q )( ˙ q − ˙ q ) . (29)Among the quartic terms, we keep those with at most a single derivative acting on the q’s. This is consistent with the order at which we truncated the Taylor series expansion of the We are using the convention introduced in [30] for the quadratic couplings. The terms without any derivative were identified for a scalar field theory in [51]. ciPost Physics Submission terms in the influence phase. The reality conditions imply that all the couplings introducedin (28) and (29) are real.Now, using this form of the 1-PI effective action, one can calculate the particle’s correlatorsand match them with the same correlators obtained from the approximately local form of theinfluence phase. The computation of the correlators from both the influence phase and the 1-PI effective action would require specifying the corresponding initial conditions of the particleat some time after the local dynamics has set in. We assume that these initial conditionsare the same in the two approaches up to perturbative corrections in λ . This allows us tocompare the leading order forms of the connected parts of 2-point and 4-point correlators ofq obtained from the two approaches. Such a comparison would yield relations between theeffective couplings and the cumulants of the bath’s correlators up to leading order in λ . Weenumerate these relations in (32), (33) and table 1 below.While expressing these relations, we adopt the following notational conventions:1. We denote the time interval between two instants t i and t j by t ij ≡ t i − t j . (30)2. For connected parts of correlators with nested commutators/ anti-commutators , we putall the insertions within square brackets enclosed by angular brackets. The inner-mostoperator in the nested structure is positioned left-most in the expression within thebrackets. The operators that one encounters as one moves outwards through the nestedstructure are placed progressively rightwards in the expression. The position of eachanti-commutator is indicated by a + sign. For example, h [12] i ≡ h [ O ( t ) , O ( t )] i c , h [12 + ] i ≡ h{ O ( t ) , O ( t ) }i c , h [1234] i ≡ h [[[ O ( t ) , O ( t )] , O ( t )] , O ( t )] i c , h [12 + i ≡ h [[ { O ( t ) , O ( t ) } , O ( t )] , O ( t )] i c , h [12 + + i ≡ h [ {{ O ( t ) , O ( t ) } , O ( t ) } , O ( t )] i c , h [12 + + + ] i ≡ h{{{ O ( t ) , O ( t ) } , O ( t ) } , O ( t ) }i c , etc. (31) Quadratic couplings at leading order in λ : The dependence of all the quadratic cou-plings on the bath’s correlators were obtained in [45]. We provide these expressions below: Z I = λ t − t →∞ h Z t t dt h [12 + ] i t i + O( λ ) , h f i = λ lim t − t →∞ h Z t t dt h [12 + ] i i + O( λ ) , ∆ µ ≡ µ − µ = − i λ lim t − t →∞ h Z t t dt h [12] i i + O( λ ) ,γ = i λ lim t − t →∞ h Z t t dt h [12] i t i + O( λ ) . (32) Quartic couplings at leading order in λ : The leading order form of any quartic coupling g can be expressed as follows: g = λ lim t − t →∞ Z t t dt Z t t dt Z t t dt I [ g ] + O( λ ) . (33)14 ciPost Physics Submission We provide the forms of the integrand I [ g ] for the quartic couplings in table 1.Table 1: Relations between the SK 1-PI effective couplings and the correlators of O ( t ) g I [ g ] ζ (4) N − h [12 + + + ] i λ i h [1234] i λ γ i ( t + t + t ) h [1234] i ζ (2) µ − i (cid:16) h [123 + + ] i + h [12 + + ] i + h [12 + + i (cid:17) ζ (2) γ i (cid:16) (3 t − t − t ) h [123 + + ] i + ( − t + 3 t − t ) h [12 + + ] i + ( − t − t + 3 t ) h [12 + + i (cid:17) ζ (cid:16) h [12 + i + h [123 + i + h [1234 + ] i (cid:17) ζ γ (cid:16) ( − t + t + t ) h [12 + i + ( t − t + t ) h [123 + i + ( t + t − t ) h [1234 + ] i (cid:17) From the expressions of the couplings given in table 1, one can easily check that thesecouplings are real. For this, first note that the operator O given in (3) is Hermitian. Thisimplies that its correlators should satisfy relations of the following form: (cid:16) h O ( t ) O ( t ) O ( t ) O ( t ) i (cid:17) ∗ = h O ( t ) O ( t ) O ( t ) O ( t ) i . (34)Such relations between the bath’s correlators lead to the following conditions on the corre-sponding cumulants: h [1234] i ∗ = −h [1234] i , h [12 + + + ] i ∗ = h [12 + + + ] i , h [12 + i ∗ = h [12 + i , h [123 + i ∗ = h [123 + i , h [1234 + ] i ∗ = h [1234 + ] i , h [12 + + i ∗ = −h [12 + + i , h [12 + + ] i ∗ = −h [12 + + ] i , h [123 + + ] i ∗ = −h [123 + + ] i . (35)From these conditions, the reality of the couplings (see table 1) is manifest.In section 4.5, we will evaluate these couplings (along with the additional OTO couplings)for the qXY model at the high temperature limit of the bath. For now, we would like thereader to just note that the leading order terms in the quadratic and the quartic couplingsare O( λ ) and O( λ ) respectively. In the following subsection, we will show that these leadingorder forms of the couplings enter as parameters in a dual stochastic dynamics. In this subsection, we show that the quartic Schwinger-Keldysh effective theory of the par-ticle is dual to a classical stochastic theory governed by a non-linear Langevin equation. Asmentioned in the introduction, this stochastic dynamics has a structure similar to the oneobtained for the cubic effective theory in [30]. Our analysis will be based on the techniquesdeveloped by Martin-Siggia-Rose [52], De Dominicis-Peliti [53] and Janssen [54] for obtainingsuch dualities between quantum mechanical path integrals and stochastic path integrals. Wewill first propose the form of the dual non-linear Langevin dynamics and then demonstrateits equivalence to the quartic effective theory discussed in the previous subsection.15 ciPost Physics SubmissionThe dual non-linear Langevin dynamics:
Consider a non-linear Langevin equation of the following form: E [ q, N ] ≡ ¨ q + (cid:16) γ + ζ (2) γ N (cid:17) ˙ q + (cid:16) µ + ζ (2) µ N (cid:17) q + N (cid:16) ζ − ζ γ ddt (cid:17) q (cid:16) λ − λ γ ddt (cid:17) q − h f iN = 0 , (36)where N is a noise drawn from a non-Gaussian probability distribution given below P [ N ] ∝ exp h − Z dt (cid:16) h f i N + Z I N + ζ (4) N N (cid:17)i . (37)The non-linearities in this dynamics as well as the non-Gaussianity in the noise are fixedby the following parameters: ζ (4) N , ζ (2) γ , ζ (2) µ , ζ , ζ γ , λ , λ γ . From equation (33) and table 1,we can see that all these parameters are O( λ ). If we ignore these O( λ ) contributions, thenthe dynamics satisfies a linear Langevin equation of the following form:¨ q + γ ˙ q + µ q = h f iN , (38)where the noise is drawn from the Gaussian probability distribution given below P [ N ] ∝ exp h − Z dt (cid:16) h f i N + Z I N (cid:17)i . (39) • Parameters in the linear Langevin dynamics
The parameters appearing in the linear Langevin dynamics given in (38) and (39) canbe interpreted in the following manner:1. h f i is the strength of an additive noise in the dynamics.2. Z I introduces nonzero correlations between the noise at two different times.3. µ is the renormalised frequency.4. γ is the coefficient of damping. • Additional parameters in the non-linear dynamics
If we include the contribution of the O( λ ) parameters in the dynamics, then theseadditional parameters can be interpreted as follows:1. ζ (2) µ is a jitter in the renormalised frequency due to the thermal noise.2. ζ (2) γ is a jitter in the damping coefficient due to the thermal noise.3. ζ (4) N is the strength of non-Gaussianity in the noise distribution.4. λ and λ γ are the strengths of anharmonic terms in the equation of motion.5. ζ and ζ γ are the strengths of anharmonic terms which couple to the noise.Now, let us demonstrate the duality between this non-linear Langevin dynamics and thequartic effective theory that we introduced earlier.16 ciPost Physics SubmissionArgument for the duality: Consider the following stochastic path integral for the non-linear Langevin dynamicsgiven in (36) and (37): Z = Z [ D N ][ Dq ] e R dt (cid:16) ζ (2) µ N q h f i δt + ζ (2) γ N ˙ q h f i δt + ζ (4) N N h f i δt (cid:17) δ ( E [ q, N ]) P [ N ] . (40)Here, δt is a UV-regulator for 2-point functions of N , and h R dt (cid:16) ζ (2) µ N q h f i δt + ζ (2) γ N ˙ q h f i δt + ζ (4) N N h f i δt (cid:17)i isa counter-term introduced to cancel the regulator-dependent contributions arising from loopintegrals of N .Notice that q is a dummy variable in the above path integral. We choose to relabel thisvariable as q a . In addition, we introduce an auxiliary variable q d which replaces the deltafunction for the equation of motion by an integral as shown below Z = Z [ D N ][ Dq a ][ Dq d ] e R dt (cid:16) ζ (2) µ N qa h f i δt + ζ (2) γ N ˙ qa h f i δt + ζ (4) N N h f i δt (cid:17) e − i R dt E [ q a , N ] q d P [ N ]= Z [ D N ][ Dq a ][ Dq d ] exp h i Z dt n − i ζ (2) µ N q a h f i δt − i ζ (2) γ N ˙ q a h f i δt − i ζ (4) N N h f i δt − q d ¨ q a − q d (cid:16) γ + ζ (2) γ N (cid:17) ˙ q a − q d (cid:16) µ + ζ (2) µ N (cid:17) q a − N q d (cid:16) ζ − ζ γ ddt (cid:17) q a − q d (cid:16) λ − λ γ ddt (cid:17) q a h f iN q d + i h f i N + i Z I N + i ζ (4) N N oi . (41)Now, let us introduce the following shift in the noise variable appearing in the above pathintegral: N → N + iq d . (42)Integrating out this shifted noise variable leads to a residual path integral over q a and q d . Inthe action of this residual path integral, we retain all the quadratic terms up to O( λ ), andall the quartic terms up to O( λ ). Up to this approximation, the residual path integral has We ignore the Jacobian det h δ E [ q ( t ) , N ( t )] δq ( t ′ ) i in the path integral as it does not contribute to the quadraticand quartic terms that we eventually get in (47) upto leading orders in λ . While integrating out the noise, we consider the terms associated with Z I and all the O( λ ) parameters inthe action to be small corrections over the term associated with h f i . Then the 2-point function of the noisereduces to hN ( t ) N ( t ) i = 1 h f i δ ( t − t ) + (perturbative corrections) . (43)We ignore the perturbative corrections and regulate the delta function by the UV regulator δt that we intro-duced earlier. Then the equal-time 2-point correlator of N reduces to hN i = 1 h f i δt , (44)The contribution of hN i to the action of the residual path integral exactly cancels the contribution from thecounter-term. ciPost Physics Submission the following form: Z = C Z [ Dq a ][ Dq d ] exp h i Z dt n − q d ¨ q a − γq d ˙ q a − µ q d q a − i Z I q d + i h f i q d + i ζ (4) N q d + ζ (2) µ q d q a + ζ (2) γ q d ˙ q a − iζ q d q a iζ γ q d q a ˙ q a − λ q d q a
3! + λ γ q d q a ˙ q a oi , (45)where C is a constant given by C = Z [ D N ] e R dt ζ (4) N N h f i δt e − R dt (cid:16) h f i N + ZI ˙ N + ζ (4) N N (cid:17) . (46)Integrating by parts the first term and the last term in the action of the above path integral,we get Z = C Z [ Dq a ][ Dq d ] exp h i Z dt n ˙ q d ˙ q a − i Z I q d − γq d ˙ q a + i h f i q d − µ q d q a + i ζ (4) N q d + ζ (2) µ q d q a + ζ (2) γ q d ˙ q a − iζ q d q a iζ γ q d q a ˙ q a − λ q d q a − λ γ ˙ q d q a oi . (47)Now, notice that the action in the above expression is exactly the Schwinger-Keldysh effectiveaction given in (28) and (29) under the following identification: q a ≡ q − q , q d ≡ q + q . (48)This basis { q a , q d } for the Schwinger-Keldysh degrees of freedom is commonly known as theKeldysh basis [3]. The Schwinger-Keldysh effective Lagrangian of the particle in this basis isgiven by L SK,1-PI = ˙ q d ˙ q a − i Z I ˙ q d − µ q d q a + i h f i q d − γq d ˙ q a + iζ (4) N q d + ζ (2) µ q d q a − iζ q d q a − λ q d q a + ζ (2) γ q d ˙ q a − λ γ
3! ˙ q d q a + iζ γ q d q a ˙ q a . (49)Therefore, one can express the stochastic path integral in terms of the Schwinger-Keldysheffective action as shown below Z = C Z [ Dq a ][ Dq d ] e i R dtL SK,1-PI . (50)This concludes our argument for the duality between the quartic effective theory and thenon-linear Langevin dynamics. We refer the reader to [6] for a more detailed discussion onsuch dualities between stochastic and Schwinger-Keldysh path integrals.18 ciPost Physics SubmissionA brief comment on the sign of ζ (4) N : In section 4.5, we will see that, for the qXY model, the value of ζ (4) N is negative. This mayraise concern about the validity of the probability distribution given in (37) since it divergeswhen |N | → ∞ . However, we would like to remind the reader that we have done a perturbativeanalysis here and ignored all possible corrections to the probability distribution beyond thequartic order. Such a perturbative analysis is insufficient to determine the behaviour of theprobabilty density at large values of N . The Schwinger-Keldysh effective theory or the dual non-linear Langevin dynamics developedin this section suffers from the following limitations:1. It allows one to compute only correlators of the particle which can be obtained frompath integrals on the Schwinger-Keldysh contour shown in figure 2.2. The effective couplings receive contributions only from similar Schwinger-Keldysh cor-relators of the bath.However, it is possible to consider more general correlators of the particle which cannot beobtained from path integrals on the Schwinger-Keldysh contour. For example, consider thecorrelator h q ( t ) q (0) q ( t ) q (0) i ≡ Tr[ ρ q ( t ) q (0) q ( t ) q (0)] , (51)where t f > t > > t . Starting from the initial density matrix in the above expression,one has to go forward and backward in time twice to include all the insertions. To get apath integral representation for such correlators, one needs to consider a contour with twotime folds [32, 33] as shown in figure 3. The positions of the insertions that would give thecorrelator in (51) are indicated in this diagram by the crosses. These insertions are contour-ordered according to the arrow indicated in the diagram.Figure 3: Contour-ordered correlator for h q ( t ) q (0) q ( t ) q (0) i t ×××× q ( t ) q ( t ) q (0) q (0) t t f Correlators which can only be obtained by putting insertions on such a contour withmultiple time-folds are known as out-of-time-order correlators (OTOCs). The constructionof an effective theory for computing such OTOCs requires incorporating the effects of similarOTOCs of the bath [45]. As we mentioned in the introduction, such thermal OTOCs in abath have emerged as very important diagnostic measures of chaos, thermalisation, many-body localisation, etc. [34–36, 38, 41–44]. Moreover, it is essential to include these OTOCsof the bath in the study of the analytic properties of its thermal correlators [9, 30]. Theseanalytic properties of the bath’s correlators impose nontrivial constraints [30] on the effectivecouplings of the Brownian particle. 19 ciPost Physics Submission
In the next section, we will first extend the effective theory to include the particle’s OTOCs.Then we will derive the constraints imposed on the quartic couplings in this OTO effectivedynamics by the analytic properties of the bath’s correlators.
In this section, we will extend the effective theory of the particle to one thata) allows computation of the particle’s OTOCs, andb) takes into account the contributions of the bath’s OTOCs.In particular, we will see that the effective couplings in this extended framework receivecontributions from the OTOCs of the operator O that couples to the particle. We will showthat some of these OTO correlators of the bath are related to Schwinger-Keldysh correlatorsdue to the following two reasons:1. microscopic reversibility in the bath’s dynamics,2. thermality of the bath.These relations between the bath’s correlators, in turn, impose certain constraints on thequartic effective couplings. Following the methods illustrated in [30], we will derive theseconstraints, and show that they lead to generalisations of the Onsager reciprocal relations[46, 47] and the fluctuation-dissipation relation [25, 30, 49]. Finally, we will provide the valuesof the effective couplings for the qXY model introduced in section 2 and check that theysatisfy all the constraints. At the end of the last section, we saw that a path integral representation for the OTOCsrequires us to introduce a contour with two time-folds as shown in figure 3. Then followingthe strategy developed for obtaining the Schwinger-Keldysh effective dynamics, here we willhave to consider a copy of the microscopic degrees of freedom for each of the four legs in thiscontour: { q , X , Y } , { q , X , Y } , { q , X , Y } and { q , X , Y } .The Lagrangian of the combined system in this generalised Schwinger-Keldysh path inte-gral is given by L GSK = L [ q , X , Y ] − L [ − q , X , Y ] + L [ q , X , Y ] − L [ − q , X , Y ] . (52)While computing OTO correlators of the particle from path integrals with the above La-grangian, we can first integrate out the bath’s degrees of freedom. This would give us ageneralisation of the influence phase which encodes the effect of the OTOCs of the bath onthe particle’s effective dynamics. This generalised influence phase [45] is given by W GSK = ∞ X n =1 λ n W ( n ) GSK (53) Here we again put extra minus signs while defining the q’s on the even legs. This is consistent with theconvention followed in [45] and [30]. ciPost Physics Submission where W ( n ) GSK = i n − X i , ··· ,i n =1 Z t f t dt Z t t dt · · · Z t n − t dt n hT C O i ( t ) · · · O i n ( t n ) i c q i ( t ) · · · q i n ( t n ) . (54)As in case of the influence phase, one can expand the q’s in (54) at the times t , · · · , t n about t to get an approximately local form for the generalised influence phase. The quadraticand the quartic terms in this approximately local form are given in appendix C.The approximately local generalised influence phase can be used to compute the OTOcorrelators of the particle. But just like the influence phase, it suffers from the problem ofhaving point-split regulators to preserve the original time-ordering in the non-local form. Toavoid this complication, we will follow the method introduced in [45] to construct a local1-PI effective action of the particle on the 2-fold contour. As we will see next, this OTO1-PI effective action is a straightforward extension of the Schwinger-Keldysh effective actiondiscussed in the previous section. The 1-PI effective action for OTOCs has to satisfy some constraints which are based on similarprinciples as those mentioned in section 3.4. These constraints were first discussed in [45] andlater used in [30] to study the quadratic and cubic terms in the OTO effective theory of theqXY model. We summarise these constraints below: • Collapse rules:
The 1-PI effective action should become independent of ˜ q under any of the followingidentifications:1. q = − q = ˜ q ,2. q = − q = ˜ q ,3. q = − q = ˜ q .Moreover, under any of these collapses, the OTO effective action should reduce to theSchwinger-Keldysh effective action introduced earlier. • Reality conditions:
The effective action should become its own negative under complex conjugation of theeffective couplings along with the following exchanges: q ↔ − q , q ↔ − q . (55)In addition, the 1-PI effective action on the 2-fold contour should be invariant under q → − q , q → − q , q → − q , q → − q (56)to respect the symmetries given in (25). 21 ciPost Physics Submission The most general 1-PI effective action which is consistent with these conditions can beexpanded as L = L (2)1-PI + L (4)1-PI + · · · , (57)where L (2)1-PI and L (4)1-PI are the quadratic and the quartic terms in the action respectively. Quadratic terms:
The quadratic terms can be obtained by extending their Schwinger-Keldysh(SK) counterparts given in (28) [45]. We provide the form of these terms below: L (2)1-PI = 12 ( ˙ q − ˙ q + ˙ q − ˙ q ) − µ q − q + q − q ) − γ h ( q + q )( ˙ q − ˙ q − ˙ q − ˙ q ) + ( q + q )( ˙ q + ˙ q + ˙ q − ˙ q ) i + i h f i q + q + q + q ) − iZ I q + ˙ q + ˙ q + ˙ q ) (58)One can easily check that these quadratic terms reduce to the corresponding terms in the SKeffective action under any of the collapses mentioned above. Quartic terms:
The quartic terms can be split into two parts [45]:1. terms which reduce to their SK counterparts given in (29) under any of the collapses,2. terms which go to zero under these collapses.Accordingly, the quartic effective action can be written as L (4)1-PI = L (4)1-PI,SK + L (4)1-PI,OTO (59)where L (4)1-PI,SK and L (4)1-PI,OTO are the two sets of terms mentioned above. Extension of quartic terms in Schwinger-Keldysh effective theory:
The exten-sion of the quartic terms in the SK effective theory can be further divided into different setsaccording to the number of time derivatives in them. Here, we keep terms with up to a singlederivative acting on the q’s. Then L (4)1-PI,SK can be decomposed as L (4)1-PI,SK = L (4 , + L (4 , + · · · , (60)where L (4 , and L (4 , are terms without any derivative and terms with a single derivativerespectively. We give the forms of these terms below: L (4 , = − λ h ( q + q )( q − q − q − q ) + ( q + q )( q + q + q − q ) i + ζ (2) µ q + q + q + q ) ( q − q + q − q ) − iζ q − q + q − q ) + iζ (4) N
24 ( q + q + q + q ) , (61)22 ciPost Physics Submission L (4 , = − λ γ h ( ˙ q + ˙ q )( q − q − q − q ) + ( ˙ q + ˙ q )( q + q + q − q ) i + ζ (2) γ q + q + q + q ) h ( q + q )( ˙ q − ˙ q − ˙ q − ˙ q ) + ( q + q )( ˙ q + ˙ q + ˙ q − ˙ q ) i + iζ γ q + q + q + q ) h ( q + q )( q − q − q − q )( ˙ q − ˙ q − ˙ q − ˙ q )+ ( q + q )( q + q + q − q )( ˙ q + ˙ q + ˙ q − ˙ q ) i . (62) Additional OTO quartic terms:
The additional OTO quartic terms which vanishunder any of the collapses can also be split into terms with different number of derivativesacting on the q’s. As in case of the extension of the SK effective action terms, L (4)1-PI,OTO canbe decomposed as L (4)1-PI,OTO = L (4 , + L (4 , + · · · , (63)where L (4 , and L (4 , are terms without any derivative and terms with a singlederivative respectively. The forms of these terms are given below: L (4 , = ( q + q )( q + q )( q + q ) h A ( q + q ) + A ( q + q ) + A ( q − q ) + A ( q + q ) i , (64) L (4 , =( q + q )( q + q )( ˙ q + ˙ q ) h B ( q − q ) + B ( q + q ) + B ( q + q ) i + ( q + q )( ˙ q + ˙ q )( q + q ) h B ( q − q ) + B ( q + q ) + B ( q + q ) i + ( ˙ q + ˙ q )( q + q )( q + q ) h B ( q − q ) + B ( q + q ) + B ( q + q ) i . (65)The reality conditions impose the following constraints on the quartic OTO couplings: A = − A ∗ , A = A ∗ , A = − A ∗ ,B = B ∗ , B = − B ∗ , B = − B ∗ , B = B ∗ , B = − B ∗ . (66)These additional OTO terms will not contribute to any correlator on the 2-fold contour thatcan also be obtained from the Schwinger-Keldysh contour. This is ensured by the collapserules mentioned above. However, they are essential for computing OTOCs of the particle. Thecouplings appearing in these OTO terms, in turn, receive contributions from the OTOCs ofthe bath. As we will discuss next, some of these OTOCs of the bath are related to Schwinger-Keldysh correlators due to the microscopic reversibility in the dynamics of the bath. We willshow that such relations impose constraints on the quartic couplings in the effective actionwhich can be intepreted as generalisations of the Onsager reciprocal relations [46, 47].23 ciPost Physics Submission In the qXY model, the bath’s dynamics (excluding the perturbation by the particle) has asymmetry under time-reversal: T X ( i ) ( t ) T † = X ( i ) ( − t ) , T Y ( j ) ( t ) T † = Y ( j ) ( − t ) . (67)Here, T is an anti-linear and anti-unitary operator which commutes with the bath’s Hamil-tonian i.e. [ T , H B ] = 0 . (68)Moreover, the bath operator O ≡ P i,j g xy,ij X ( i ) Y ( j ) that couples to the particle has an evenparity under time-reversal which implies T O ( t ) T † = O ( − t ) . (69)Such a microscopic reversibility in the bath’s dynamics introduces certain relations be-tween the couplings in the effective theory of the particle. These relations were first discov-ered by Onsager [46, 47] and later extended by Casimir [48] in the context of the quadraticeffective theory of a system with multiple degrees of freedom. Let us denote these degrees offreedom by { q A } , and assume that q A couples to the bath operator O A with parity η A undertime-reversal. As a concrete example, one can take these degrees of freedom to be the coordi-nates of a Brownian particle moving in multiple dimensions. In such a setup, the frequency,the damping and the noise coefficients in the particle’s effective dynamics will be replaced bymatrices µ AB , γ AB and h f i AB . The first index (A) in these coefficients indicates the degree offreedom in whose equation of motion they appear. The second index (B) indicates the degreeof freedom with which these coefficients get multiplied in the equation of motion of q A . Forthese coefficients, Onsager and Casimir derived reciprocal relations of the following form µ AB = η A η B µ BA , γ AB = η A η B γ BA , h f i AB = η A η B h f i BA . (70)For a system with a single degree of freedom, as in case of the Brownian particle moving in 1dimension, the Onsager-Casimir relations are trivially satisfied by the quadratic couplings. So,at the level of the quadratic effective theory, there is no constraint imposed by the microscopicreversibility in the bath.However, an extension of such relations was found in [30] for the cubic OTO effectivetheory of the Brownian particle. It was observed that, when there is microscopic reversibilityin the bath, all the cubic OTO couplings are related to the cubic Schwinger-Keldysh couplings.These relations between the effective couplings arise from certain relations between the 3-pointcorrelators of the bath which are rooted in its microscopic reversibility. Here, we will firstshow that similar relations exist between the 4-point correlators of the bath. Then we willdiscuss the constraints imposed on the quartic effective couplings of the particle due to theserelations between the 4-point functions of the bath. Relations between the bath’s correlators due to microscopic reversibility:
The microscopic reversibility in the bath leads to the following kind of relations betweenthe 4-point correlators of O : h O ( t ) O ( t ) O ( t ) O ( t ) i = (cid:16) h T O ( t ) T † T O ( t ) T † T O ( t ) T † T O ( t ) T † i (cid:17) ∗ . (71) We refer the reader to [55] for a discussion on these properties of the time-reversal operator. ciPost Physics Submission Using the transformation of O ( t ) under time-reversal (see equation (69)), we get h O ( t ) O ( t ) O ( t ) O ( t ) i = (cid:16) h O ( − t ) O ( − t ) O ( − t ) O ( − t ) i (cid:17) ∗ . (72)Now, due to the Hermiticity of the operator O , the above relation reduces to h O ( t ) O ( t ) O ( t ) O ( t ) i = h O ( − t ) O ( − t ) O ( − t ) O ( − t ) i . (73)Note that such relations can connect two OTO correlators as in the following example: h O ( t ) O (0) O ( t ) O (0) i = h O (0) O ( − t ) O (0) O ( − t ) i . (74)Moreover, they can also connect an OTO correlator to a Schwinger-Keldysh correlator. Forexample, consider the relation h O ( t ) O (0) O (0) O ( t ) i = h O ( − t ) O (0) O (0) O ( − t ) i , (75)where t >
0. The correlator on the left hand side of the above equation is an OTOC which canbe obtained by putting insertions on a 2-fold contour as shown in figure 4. The correlator onFigure 4: Contour-ordered correlator for h O ( t ) O (0) O (0) O ( t ) i where t > t ×××× O ( t ) O ( t ) O (0) O (0) t t f the right-hand side of (75), however, can be obtained from a path integral on the Schwinger-Keldysh contour as shown in figure 5.Figure 5: Contour-ordered correlator for h O ( − t ) O (0) O (0) O ( − t ) i where t > − t ×× ×× O ( − t ) O ( − t ) O (0) O (0) t t f Such relations between the bath’s 4-point correlators can introduce two kinds of constraintson the quartic effective couplings of the particle:1. They can lead to a relation between two different OTO couplings.2. They can also relate an OTO coupling to a Schwinger-Keldysh coupling.We will now derive these constraints on the effective couplings of the particle.25 ciPost Physics SubmissionConstraints on the quartic effective couplings:
To analyse the constraints on the effective couplings, we find it convenient to re-expressthe quartic OTO couplings in terms of some new real parameters as shown below: A = − A ∗ = 112 h ( − λ + e κ ) − i ( ̺ − e ̺ ) i ,A = A ∗ = κ ,A = − A ∗ = i ̺ + e ̺ ) . (76) B = B ∗ = 14 h (2 κ II γ + 4 κ III γ + e κ II γ ) + i ( ̺ II γ + e ̺ I γ − e ̺ II γ ) i ,B = − B ∗ = 116 h ( − λ γ + 24 ζ (2) γ + 12 κ I γ − e κ I γ − e κ II γ )+ i (4 ζ γ − ̺ I γ + 2 ̺ II γ − e ̺ I γ + 2 e ̺ II γ ) i ,B = − B ∗ = 14 h ( − κ II γ + e κ II γ ) + i̺ I γ i ,B = B ∗ = 12 (cid:16) κ II γ + e κ II γ (cid:17) ,B = − B ∗ = 116 h ( λ γ + 8 ζ (2) γ + 4 κ I γ + 4 e κ I γ − e κ II γ )+ i ( − ζ γ + 2 ̺ I γ + 6 ̺ II γ − e ̺ I γ + 6 e ̺ II γ ) i . (77)These new parameters are chosen so that they have definite parities under time-reversal (seethe discussion below). The presence or absence of tilde over any coupling indicates whether ithas odd or even parity respectively. The symbols κ and ̺ are used to represent the couplingswhich get multiplied to real and imaginary terms in the effective action respectively. Thesubscript γ is introduced to distinguish the couplings corresponding to the single derivativeterms in the action.One can determine the dependence of these OTO couplings on the cumulants of theoperator O by comparing the particle’s 4-point OTOCs obtained from the generalised influencephase with those computed using the OTO 1-PI effective action. We provide the expressionsof these couplings in terms of the bath’s OTO cumulants in appendix C.For the purpose of analysing the constraints on the effective couplings, we find it convenientto convert their expressions (see (33),(129) , and tables 1 and 6) into integrals over a frequencydomain where the integrands are determined by some spectral functions of the bath. Thesespectral functions are the Fourier transforms of the 4-point cumulants of O ( t ) as defined bythe relations given in (79), (80) and (81). In these relations, we define the measure for theintegral over the frequencies as Z d ω (2 π ) ≡ Z ∞−∞ dω π Z ∞−∞ dω π Z ∞−∞ dω π Z ∞−∞ dω π . (78) • Spectral functions for Wightman correlators: Z d ω (2 π ) ρ h i e − i ( ω t + ω t + ω t + ω t ) ≡ λ h O ( t ) O ( t ) O ( t ) O ( t ) i c , Z d ω (2 π ) ρ h i e − i ( ω t + ω t + ω t + ω t ) ≡ λ h O ( t ) O ( t ) O ( t ) O ( t ) i c , etc. (79)26 ciPost Physics Submission • Spectral functions for single-nested (anti-)commutators: Z d ω (2 π ) ρ [1234] e − i ( ω t + ω t + ω t + ω t ) ≡ λ h [[[ O ( t ) , O ( t )] , O ( t )] , O ( t )] i c , Z d ω (2 π ) ρ [12 + + + ] e − i ( ω t + ω t + ω t + ω t ) ≡ λ h{{{ O ( t ) , O ( t ) } , O ( t ) } , O ( t ) }i c , Z d ω (2 π ) ρ [1234 + ] e − i ( ω t + ω t + ω t + ω t ) ≡ λ h{ [[ O ( t ) , O ( t )] , O ( t )] , O ( t ) }i c , Z d ω (2 π ) ρ [123 + e − i ( ω t + ω t + ω t + ω t ) ≡ λ h [ { [ O ( t ) , O ( t )] , O ( t ) } , O ( t )] i c , Z d ω (2 π ) ρ [12 + e − i ( ω t + ω t + ω t + ω t ) ≡ λ h [[ { O ( t ) , O ( t ) } , O ( t )] , O ( t )] i c , etc.(80) • Spectral functions for double (anti-)commutators: Z d ω (2 π ) ρ [12][34] e − i ( ω t + ω t + ω t + ω t ) ≡ λ h [ O ( t ) , O ( t )][ O ( t ) , O ( t )] i c Z d ω (2 π ) ρ [12 + ][34 + ] e − i ( ω t + ω t + ω t + ω t ) ≡ λ h{ O ( t ) , O ( t ) }{ O ( t ) , O ( t ) }i c , Z d ω (2 π ) ρ [12 + ][34] e − i ( ω t + ω t + ω t + ω t ) ≡ λ h{ O ( t ) , O ( t ) } [ O ( t ) , O ( t )] i c Z d ω (2 π ) ρ [12][34 + ] e − i ( ω t + ω t + ω t + ω t ) ≡ λ h [ O ( t ) , O ( t )] { O ( t ) , O ( t ) }i c , etc.(81)Any quartic coupling g can be expressed as an integral of these spectral functions in thefollowing manner: g = Z C e I [ g ] + O( λ ) . (82)The domain of integration C is given by Z C ≡ Z ∞− iǫ −∞− iǫ dω π Z ∞− iǫ −∞− iǫ dω π Z ∞ + iǫ −∞ + iǫ dω π Z ∞ + iǫ −∞ + iǫ dω π (83)where ǫ and ǫ are infinitesimally small positive numbers. We provide the dependence of theintegrand e I [ g ] on the spectral functions for all the quartic couplings in tables 2 and 3. The domain C in frequency space is determined by the time domain over which the integrals in (33) and(129) are defined. ciPost Physics Submission Table 2: Couplings with even parity under time-reversal g e I [ g ] λ − e κ ω ω ( ω + ω ) (cid:16) − ρ [1234] + ρ [4321] (cid:17) ζ (2) µ − λ ω ω ( ω + ω ) (cid:16) ρ h i − ρ h i (cid:17) ζ (4) N + 3 e ̺ − i ω ω ( ω + ω ) (cid:16) ρ [12 + + + ] + ρ [43 + + + ] (cid:17) ζ − ζ (4) N iω ω ( ω + ω ) (cid:16) ρ h i + ρ h i (cid:17) κ − ω ω ( ω + ω ) h(cid:16) ρ [1234] − ρ [4321] (cid:17) − (cid:16) ρ [1423] − ρ [4132] + ρ [2314] − ρ [3241] (cid:17)i ̺ i ω ω ( ω + ω ) (cid:16) − ρ [12 + + + ] − ρ [4 + + + + ] + ρ [14 + + + ] + ρ [41 + + + ] + ρ [2314 + ] + ρ [3241 + ] (cid:17) λ γ + 2 e κ I γ iω ω ( ω + ω ) h(cid:16) ω ( ω + ω ) − ω (3 ω + 5 ω ) (cid:17) ρ [1234] − (cid:16) ω ( ω + ω ) − ω (3 ω + 5 ω ) (cid:17) ρ [4321] i ζ γ − e ̺ I γ ω ω ( ω + ω ) h(cid:16) ( ω + ω )( ω − ω ) − ω ω (cid:17)(cid:16) ρ [12 +
34] + ρ [43 + (cid:17) +( ω + ω )( ω − ω ) (cid:16) ρ [123 +
4] + ρ [432 + (cid:17) +( ω + ω )( ω + ω ) (cid:16) ρ [1234 + ] − ρ [4321 + ] (cid:17)i κ I γ i ω ω ( ω + ω ) h(cid:16) ( ω + ω )( ω − ω ) + 2 ω ω (cid:17)(cid:16) ρ [12 + ][34 + ] − ρ [43 + ][21 + ] (cid:17) +( ω + ω )( ω − ω ) (cid:16) ρ [13 + ][24 + ] − ρ [42 + ][31 + ] (cid:17) − ( ω + ω )( ω + ω ) (cid:16) ρ [14 + ][23 + ] − ρ [32 + ][41 + ] (cid:17)i κ II γ i ω ω ( ω + ω ) h(cid:16) ( ω + ω )( ω − ω ) + 2 ω ω (cid:17)(cid:16) ρ [12][34] − ρ [43][21] (cid:17) +( ω + ω )( ω − ω ) (cid:16) ρ [13][24] − ρ [42][31] (cid:17) − ( ω + ω )( ω + ω ) (cid:16) ρ [14][23] − ρ [32][41] (cid:17)i κ III γ i ω ω ( ω + ω ) h(cid:16) ( ω + ω )( ω − ω ) + 2 ω ω (cid:17)(cid:16) ρ [4321] − ρ [1234] + ρ [2314] − ρ [3241] + ρ [2413] − ρ [3142] (cid:17) +( ω + ω )( ω − ω ) (cid:16) ρ [4231] − ρ [1324] + ρ [2341] − ρ [3214] + ρ [3412] − ρ [2143] (cid:17)i ̺ I γ ω ω ( ω + ω ) h − (cid:16) ( ω + ω )( ω − ω ) + 2 ω ω (cid:17)(cid:16) ρ [13][24] + ρ [42][31] + ρ [14][23] + ρ [32][41] (cid:17) − ( ω + ω )( ω − ω ) (cid:16) ρ [12][34] + ρ [43][21] (cid:17)i ̺ II γ ω ω ( ω + ω ) h(cid:16) ( ω + ω )( ω − ω ) + 2 ω ω (cid:17)(cid:16) ρ [12 + ][34] + ρ [43 + ][21] + ρ [12][34 + ] + ρ [43][21 + ] (cid:17) +( ω + ω )( ω − ω ) (cid:16) ρ [13 + ][24] + ρ [42 + ][31] + ρ [13][24 + ] + ρ [42][31 + ] (cid:17) − ( ω + ω )( ω + ω ) (cid:16) ρ [14 + ][23] + ρ [32 + ][41] + ρ [14][23 + ] + ρ [32][41 + ] (cid:17)i ciPost Physics Submission Table 3: Couplings with odd parity under time-reversal g e I [ g ] e κ − ω ω ( ω + ω ) (cid:16) ρ [1234] + ρ [4321] (cid:17)e ̺ − i ω ω ( ω + ω ) h(cid:16) ρ [12 +
34] + ρ [123 +
4] + ρ [1234 + ] (cid:17) − (cid:16) ρ [43 +
21] + ρ [432 +
1] + ρ [4321 + ] (cid:17)i λ γ − ζ (2) γ − iω ω ( ω + ω ) h(cid:16) ( ω + ω )( ω − ω ) + 2 ω ω (cid:17)(cid:16) ρ [12][34 + ] + ρ [43][21 + ] − ρ [12 + ][34] − ρ [43 + ][21] (cid:17) − κ I γ − κ II γ ) +( ω + ω )( ω − ω ) (cid:16) ρ [13][24 + ] + ρ [42][31 + ] − ρ [13 + ][24] − ρ [42 + ][31] (cid:17) − ( ω + ω )( ω + ω ) (cid:16) ρ [14][23 + ] + ρ [32][41 + ] − ρ [14 + ][23] − ρ [32 + ][41] (cid:17)ie κ I γ − i ω ω ( ω + ω ) h(cid:16) ω ( ω + ω ) − ω (3 ω + 5 ω ) (cid:17) ρ [1234]+ (cid:16) ω ( ω + ω ) − ω (3 ω + 5 ω ) (cid:17) ρ [4321] ie κ II γ i ω ω ( ω + ω ) h ( ω − ω ) (cid:16) ρ [23][14] − ρ [14][23] (cid:17) + ( ω + ω ) (cid:16) ρ [12][34] − ρ [43][21] + ρ [13][24] − ρ [42][31] (cid:17)ie ̺ I γ − ω ω ( ω + ω ) h(cid:16) ( ω − ω )( ω + ω ) + 2 ω ω (cid:17)(cid:16) ρ [12 + − ρ [43 + (cid:17) +( ω − ω )( ω + ω ) (cid:16) ρ [123 + − ρ [432 + (cid:17) − ( ω + ω )( ω + ω ) (cid:16) ρ [1234 + ] + ρ [4321 + ] (cid:17)ie ̺ II γ ω ω ( ω + ω ) h − (cid:16) ( ω − ω )( ω + ω ) + ω ω (cid:17)(cid:16) ρ [14][23 + ] + ρ [32 + ][41] (cid:17) + ω ω (cid:16) ρ [14 + ][23] + ρ [32][41 + ] (cid:17) + ω ( ω + 2 ω ) (cid:16) ρ [13 + ][24] + ρ [42][31 + ] (cid:17) − ω ( ω + 2 ω ) (cid:16) ρ [42 + ][31] + ρ [13][24 + ] (cid:17) + ω ( ω + ω ) (cid:16) ρ [12 + ][34] + ρ [43][21 + ] (cid:17) − ω ( ω + ω ) (cid:16) ρ [43 + ][21] + ρ [12][34 + ] (cid:17)i Now, since all the couplings in tables 2 and 3 are real , one can obtain alternativeexpressions for them by taking complex conjugates of the corresponding integrals. Such acomplex conjugation results in the following transformations in the integral:1. All explicit factors of i go to ( − i ).2. In all factors involving the frequencies explicitly, ω i goes to ω ∗ i for i ∈ { , , , } .3. The spectral functions go to their complex conjugates.Now, using the relation (72) which is based on the microscopic reversibility in the bath,one can obtain the following relations between the spectral functions:( ρ h i ) ∗ = ρ h ∗ ∗ ∗ ∗ i , ( ρ [1234]) ∗ = ρ [1 ∗ ∗ ∗ ∗ ] , ( ρ [12][34]) ∗ = ρ [1 ∗ ∗ ][3 ∗ ∗ ] , etc. (84)On the right hand sides of these equations, i ∗ stands for ω ∗ i .Hence, the net effect of the complex conjugation of the couplings is the replacement ofall ω i ’s by ω ∗ i ’s and all explicit factors of i by ( − i ) in the corresponding frequency integrals.The complex conjugation of the frequencies takes the domain of integration to C ∗ , the integralover which is defined as Z C ∗ ≡ Z ∞ + iǫ −∞ + iǫ dω ∗ π Z ∞ + iǫ −∞ + iǫ dω ∗ π Z ∞− iǫ −∞− iǫ dω ∗ π Z ∞− iǫ −∞− iǫ dω ∗ π . (85)Now, notice that the domain C ∗ gets mapped exactly back to C under the following relabelingof the frequencies: ω ∗ → ω , ω ∗ → ω , ω ∗ → ω , ω ∗ → ω . (86) To see the reality of the OTO couplings, one can substitute relations like (34) in the expressions of thesecouplings given in table 6. ciPost Physics Submission Therefore, to summarise, the alternative expressions for the couplings can be obtained byintegrating over the same domain C , but performing the following transformations on theintegrands in the original expressions:1. i → − i for all explicit factors of i ,2. ω ↔ ω , ω ↔ ω .We define the action of time-reversal on the couplings to be the above transformations onthe integrands in their expressions. As we saw, the equality between the couplings and theirtime-reversed counterparts crucially relies on the relations given in (84) which are based onthe microscopic reversibility in the bath’s dynamics.Now, one can compare the expressions of the couplings given in tables 2 and 3 withthe expressions of their time-reversed counterparts. While making these comparisons, it isimportant to bear in mind that the spectral functions include a delta function correspondingto energy conservation in the unperturbed dynamics of the bath. This allows one to imposeconditions such as ( ω + ω ) = − ( ω + ω ) (87)within the expressions of the integrands. Taking this into account one can easily see, that theexpressions for the couplings in table 2 remain unchanged under time-reversal. On the otherhand, the expressions for the couplings in table 3 pick up a minus sign under time-reversal.Thus, all the couplings mentioned in table 2 have even parity under time-reversal, whereasthe couplings in table 3 have odd parity under the same. All the couplings with odd paritymust go to zero due to the microscopic reversibility in the bath’s dynamics. This imposes thefollowing constraints on the effective couplings: e κ = e ̺ = e κ I γ = e κ II γ = e ̺ I γ = e ̺ II γ = 0 , (88) λ γ − ζ (2) γ − κ I γ − κ II γ ) = 0 . (89)Since these constraints are based on the microscopic reversibility in the bath, one can thinkof them as generalisations of the Onsager reciprocal relations [46, 47] . These generalisedOnsager relations are extensions of similar relations obtained between the cubic effectivecouplings in [30].As we mentioned earlier, apart from the microscopic reversibility in the bath’s dynamics,there is another source of relations between the effective couplings of the particle, viz. thethermality of the bath. In the next subsection, we will discuss how the KMS relations [7–9]between the thermal correlators of the bath lead to constraints on the effective couplings.We will see that these constraints, when combined with the relation given in (89), give riseto a generalised fluctuation-dissipation relation between the thermal jitter in the particle’sdamping and the non-Gaussianity in the distribution of the noise. In section 3.5, we saw that if we ignore the quartic couplings in the Schwinger-Keldysh effectivetheory, the corresponding stochastic dual reduces to a linear Langevin dynamics (see (38) and(39)). Under this dynamics, the particle experiences a damping as well as a random force30 ciPost Physics Submission drawn from a Gaussian distribution. These two forces are related to each other as both ofthem arise from the interaction with the bath. More precisely, in the high temperature limitof the bath, the relation between these forces is given by h f i = 2 β γ, (90)where h f i is the strength of the thermal noise experienced by the particle and γ is its dampingcoefficient. This relation is commonly known as the ‘fluctuation-dissipation relation’.The ideas leading to the discovery of this relation emerged at the beginning of the 20 th century with the works of Einstein, Smoluchowski and Sutherland. A similar relation be-tween the thermal fluctuation and the resistance in an electric conductor was experimentallyobserved by Johnson [23] , and then theoretically derived by Nyquist [24]. Later, a proof ofsuch relations for more general systems was provided by Callen and Welton [25], which wasfurther extended by Stratonovich [26, 56].As discussed in [7, 49], the fluctuation-dissipation relation is a consequence of certain re-lations between the 2-point thermal correlators of the bath, which are now commonly knownas the Kubo-Martin-Schwinger (KMS) relations [7, 8]. These relations were studied for higherpoint thermal correlators in [9]. There, it was observed that such KMS relations can con-nect the thermal OTOCs of a bath to its Schwinger-Keldysh correlators. This indicated theneed for including the effects of the bath’s OTOCs while exploring the possibility of findinggeneralisations of the fluctuation-dissipation relation.To include the effects of the OTOCs of the bath, an OTO effective theory of the particlewas developed up to cubic terms in [45]. The cubic couplings in this effective theory receivecontributions from the 3-point correlators of the bath. In [30], it was shown that the KMSrelations between these 3-point thermal correlators lead to a relation between two couplingsin the cubic OTO effective dynamics of the particle. When combined with the constraintsimposed by microscopic reversibility in the bath, this relation leads to a generalisation of thefluctuation-dissipation relation. From the perspective of the dual stochastic theory, this gen-eralised fluctuation-dissipation relation connects the thermal jitter in the particle’s dampingand the non-Gaussianity in the noise.We will see that a similar generalised fluctuation-dissipation relation holds between thequartic effective couplings as well. Before discussing this relation, we will first review theKMS relations between thermal correlators of the bath and then show how they lead to thefluctuation-dissipation relation given in equation (90). Kubo-Martin-Schwinger relations:
The KMS relations [7–9] connect all thermal correlators of the bath which can be obtainedfrom each other by cyclic permutations of insertions. For example, consider the following n-point correlator h O ( t ) O ( t ) · · · O ( t n ) i = Tr h e − βH B Z B O ( t ) O ( t ) · · · O ( t n ) i . (91)In the above expression, if we bring the insertion O ( t n ) from the right-most position to theleft-most position across the thermal density matrix, the argument of the insertion picks upan extra term ( − iβ ) i.e. h O ( t ) O ( t ) · · · O ( t n ) i = h O ( t n − iβ ) O ( t ) · · · O ( t n − ) i . (92)31 ciPost Physics Submission In frequency space, these relations lead to the following kind of relations between the spectralfunctions: ρ h · · · n i = e − βω n ρ h n · · · ( n − i . (93)Here ρ h · · · n i and ρ h n · · · ( n − i are defined in terms of the n -point cumulants of O inthe time domain as follows: Z ∞−∞ dω π · · · Z ∞−∞ dω n π ρ h · · · n i e − i ( ω t + ··· + ω n t n ) ≡ λ n h O ( t ) O ( t ) · · · O ( t n ) i c , Z ∞−∞ dω π · · · Z ∞−∞ dω n π ρ h n · · · ( n − i e − i ( ω t + ··· + ω n t n ) ≡ λ n h O ( t n ) O ( t ) · · · O ( t n − ) i c . (94)In general, there are n ! such spectral functions corresponding to all the n -point Wightmancorrelators of O . However, KMS relations like (93) reduce the number of independent n-pointspectral functions to ( n − Fluctuation-dissipation relation between quadratic couplings:
The couplings in the quadratic terms of the effective action receive contributions fromthe 2-point cumulants of the operator O as shown in (32). These quadratic couplings up toleading order in λ can be re-expressed in terms of two spectral functions ρ [12] and ρ [12 + ]which are defined as Z ∞−∞ dω π Z ∞−∞ dω π ρ [12] e − i ( ω t + ω t ) ≡ λ h [ O ( t ) , O ( t )] i c , Z ∞−∞ dω π Z ∞−∞ dω π ρ [12 + ] e − i ( ω t + ω t ) ≡ λ h{ O ( t ) , O ( t ) }i c . (95)We provide the expressions for these leading order forms of the quadratic couplings below: Z I = − Z C ρ [12 + ] iω , ∆ µ = − Z C ρ [12] ω , h f i = Z C ρ [12 + ] iω , γ = Z C ρ [12] iω . (96)Here, the integrals are performed over the following domain Z C ≡ Z ∞− iǫ −∞− iǫ dω π Z ∞ + iǫ −∞ + iǫ dω π , (97)where ǫ is a small positive number.Now, the KMS relations connect the two spectral functions as follows ρ [12 + ] = coth (cid:16) βω (cid:17) ρ [12] . (98)Then, in the high temperature limit of the bath i.e. the small β limit, the above relationreduces to ρ [12 + ] = (cid:16) βω (cid:17) ρ [12] , (99)32 ciPost Physics Submission where we take the leading order (in β ) forms of the two spectral functions. Plugging thisrelation into the expressions of the couplings given in (96), we get Z I = − β Z C ρ [12] iω , ∆ µ = − Z C ρ [12] ω , h f i = 2 β Z C ρ [12] iω , γ = Z C ρ [12] iω . (100)From these expressions, one can clearly see that at this high temperature limit, h f i = 2 β γ, (101)which is the fluctuation-dissipation relation that we mentioned earlier.Let us now discuss how one can obtain a generalisation of this fluctuation-dissipationrelation for the quartic couplings. Generalised fluctuation-dissipation relation between quartic couplings:
As pointed out in [9], the KMS relations between the 4-point functions of the bath canconnect OTOCs to Schwinger-Keldysh correlators. For example, consider the correlator h O ( t ) O ( t ) O (0) O ( t ) i ≡ Tr h e − βH B Z B O ( t ) O ( t ) O (0) O ( t ) i , (102)where t > t > t >
0. Now, notice that this correlator satisfies the following KMS relation: h O ( t ) O ( t ) O (0) O ( t ) = h O ( t − iβ ) O ( t ) O ( t ) O (0) i . (103)This KMS relation connects the correlator given in (102) to the following correlator by analyticcontinuation: h O ( t ) O ( t ) O ( t ) O (0) i ≡ Tr h e − βH B Z B O ( t ) O ( t ) O ( t ) O (0) i . (104)The correlator given in (102) is an OTOC which can be obtained by putting insertions onthe 2-fold contour as shown in figure 6. On the other hand, the correlator given in (104) is aFigure 6: Contour-ordered correlator for h O ( t ) O ( t ) O (0) O ( t ) i where t > t > t > t t t ×××× O ( t ) O ( t ) O ( t ) O (0) t t f Schwinger-Keldysh correlator as demonstrated in figure 7.We will see that KMS relations like (103) which connect OTO correlators of the bathto Schwinger-Keldysh correlators result in a relation between a quartic OTO coupling and aSchwinger-Keldysh coupling of the particle. To derive this relation, we need to go back to theexpressions of the particle’s effective couplings in terms of the bath’s spectral functions given33 ciPost Physics Submission
Figure 7: Contour-ordered correlator for h O ( t ) O ( t ) O ( t ) O (0) i where t > t > t > t t t ×××× O ( t ) O ( t ) O ( t ) O (0) t t f in tables 2 and 3. As discussed earlier, the KMS relations reduce the number of independent4-point spectral functions to (4 − : ρ [1234] , ρ [4321] , ρ [2314] , ρ [3241] , ρ [2143] , ρ [3142] . (105)We provide the expressions of the quartic effective couplings in terms of these 6 spectralfunctions at the high temperature limit of the bath in appendix D. Among these expressions,we will focus on the forms of two particular couplings here: κ I γ and ζ (4) N . From the expressionsof these couplings given in tables 7 and 8, we can see that the corresponding integrands satisfythe following relation: e I [ κ I γ ] = β e I [ ζ (4) N ]= 6 iβ ω ω ω ω ( ω + ω )( ω + ω )( ω + ω ) h ω ( ω + ω )( ω − ω ω ) ρ [1234] + ω ( ω + ω )( ω + ω )( ω + ω ) ρ [4321] − ω ω ( ω + ω )( ω + ω ) ρ [2143] + ω ω ( ω − ω ) ρ [2314]+ ω ω ( ω + ω )( ω + ω ) ρ [3142] − ω ω ( ω + ω )( ω + ω ) ρ [3241] i , (106)where we take only the leading order (in β ) forms for the spectral functions. From this relationbetween the integrands, we conclude that at leading order in λ and β , these two couplingssatisfy the following relation: κ I γ = β ζ (4) N . (107)This is an example of a generalisation of the fluctuation-dissipation relation which connectsan OTO coupling to a Schwinger-Keldysh coupling. From this analysis, one can see that thisrelation is purely a consequence of KMS relations between the bath’s correlators and holdseven when the bath’s dynamics lacks microscopic reversibility.In the presence of microscopic reversibility in the bath, there is an additional relationwhich involves the coupling κ I γ . This is one of the generalised Onsager relations given in (89)which we quote here once more for convenience: λ γ − ζ (2) γ − κ I γ − κ II γ ) = 0 . (108) See appendix D for a discussion on why these 6 spectral functions form a basis for all 4-point thermalcorrelators of O . ciPost Physics Submission Now, referring to the expressions of the leading order forms of the couplings λ γ and κ II γ givenin tables 7 and 8, we see that the corresponding integrands are as follows: e I [ λ γ ] = 2 iω ω ( ω + ω ) n ω − ω )( ω + ω ) + 2 ω ω o ρ [1234] , e I [ κ II γ ] = i ω ω ( ω + ω ) h ( ω − ω )( ω + ω ) (cid:16) ρ [1234] + ρ [2314] + ρ [3142] (cid:17) + ( ω + ω )( ω + ω ) (cid:16) ρ [2314] + ρ [3241] (cid:17) + (cid:16) ( ω − ω )( ω + ω ) + 2 ω ω (cid:17)(cid:16) ρ [1234] + ρ [2143] (cid:17)i . (109)Notice that these integrands are suppressed by a factor of β compared to the integrand for κ I γ given in (106). Therefore, in the high temperature limit, we can ignore the couplings λ γ and κ II γ in (89) to get κ I γ = − ζ (2) γ . (110)Combining the equations (110) and (107), we get the following relation in the high temperaturelimit: ζ (4) N = − β ζ (2) γ . (111)This is a generalised fluctuation-dissipation relation which connects the non-Gaussianity inthe noise experienced by the particle to the thermal jitter in its damping coefficient. It is acombined effect of microscopic reversibility in the bath and its thermality. In the followingsubsection, we will compute the effective couplings in the qXY model and verify the validityof this relation. In this subsection, we will enumerate the values of the effective couplings of the Brownianparticle in the qXY model.The forms of the quadratic couplings at leading order in λ and β are given in (100) interms of the 2-point function ρ [12]. Similar forms for the quartic couplings are given in tables7 and 8 in terms of the 4-point spectral functions enumerated in (105).We provide the leading order (in β ) forms of the 2-point spectral function ρ [12] and the4-point spectral function ρ [1234] for the qXY model in (112) and (113) respectively. ρ [12] = 2 πδ ( ω + ω ) Γ β ω Ω ω + 4Ω . (112)35 ciPost Physics Submission ρ [1234] =2 πδ ( ω + ω + ω + ω ) (cid:16) i Γ Ω β (cid:17)"( (cid:16) Ω − iω (cid:17)(cid:16) Ω − i ( ω + ω ) (cid:17) ω (cid:16) ω + ω (cid:17)(cid:16) − iω (cid:17)(cid:16)
2Ω + iω (cid:17)(cid:16) − i ( ω + ω ) (cid:17) − (cid:16) Ω + iω (cid:17)(cid:16) Ω + i ( ω + ω ) (cid:17) ω (cid:16) ω + ω (cid:17)(cid:16)
2Ω + iω (cid:17)(cid:16) − iω (cid:17)(cid:16)
2Ω + i ( ω + ω ) (cid:17) + ω Ω (cid:16) Ω − iω (cid:17) ω ω (cid:16) ω + ω (cid:17)(cid:16) − iω (cid:17)(cid:16)
2Ω + iω (cid:17)(cid:16) − i ( ω + ω ) (cid:17) − ω Ω (cid:16) Ω + iω (cid:17) ω ω (cid:16) ω + ω (cid:17)(cid:16)
2Ω + iω (cid:17)(cid:16) − iω (cid:17)(cid:16)
2Ω + i ( ω + ω ) (cid:17) ) − ( ( ω ↔ ω ) ) . (113)The other five 4-point spectral functions can be obtained from (113) by appropriate permu-tations of the frequencies.Substituting these spectral functions in the integrals for the couplings, one can calculatethe values of these couplings in the high temperature limit. We provide the values of thequadratic couplings in (114), the Schwinger-Keldysh quartic couplings in (115), and the OTOquartic couplings in (116) and (117). Quadratic couplings: Z I = Γ β Ω , ∆ µ = − Γ Ω β , h f i = Γ Ω β , γ = Γ Ω2 β . (114) Quartic couplings: A) Schwinger-Keldysh couplings: ζ (4) N = − Ω β , λ = − Ω β , ζ = − Ω β , ζ (2) µ = − Ω β ,λ γ = − Ω β , ζ γ = − Ω β , ζ (2) γ = 5Γ Ω4 β . (115)B) OTO couplings: e κ = e ̺ = e κ I γ = e κ II γ = e ̺ I γ = e ̺ II γ = 0 . (116)36 ciPost Physics Submission κ = − Ω β , ̺ = 7Γ Ω β , κ I γ = − Ω4 β , κ II γ = 5Γ Ω β ,κ III γ = − Ω β , ̺ I γ = − Ω β , ̺ II γ = 5Γ Ω β . (117)From the values of these couplings, one can easily verify the validity of the fluctuation dissipa-tion relation (101), the generalised fluctuation dissipation relation (111), and the generalisedOnsager relations (88) and (110) in the qXY model. In this paper, we have developed the quartic effective dynamics of a Brownian particle weaklyinteracting with a thermal bath. To illustrate the features of this effective dynamics, wehave introduced a simple toy model (the qXY model described in section 2) where the bathcomprises of two sets of harmonic oscillators coupled to the particle through cubic interactions.For this model, we have identified a Markovian regime, where the particle’s effectivedynamics is approximately local in time. Working in this regime, we have constructed a quarticeffective action of the particle in the Schwinger-Keldysh (SK) formalism. Using the techniquesdeveloped in [52–54], we have demonstrated a duality between this quantum effective theoryand a classical stochastic dynamics governed by a non-linear Langevin equation.The SK effective theory and the dual non-linear Langevin dynamics suffer from the limita-tion that they provide no information about the 4-point out of time order correlators (OTOCs)of the particle. To transcend this limitation, we have extended the SK effective action to anout of time ordered effective action defined on a generalised Schwinger-Keldysh contour (seefigure 3). In this extended framework, we have determined the additional quartic couplingswhich encode the effects of the bath’s 4-point OTOCs on the particle’s dynamics. We haveworked out the dependence of these OTO couplings (as well as the SK effective couplings) onthe correlators of the bath up to leading order in the particle-bath interaction.The relations between the particle’s effective couplings and the bath’s correlators providea way to analyse the constraints imposed on the effective dynamics due to thermality andmicroscopic reversibility of the bath. These constraints manifest in the form of certain re-lations between the quartic couplings which can be interpreted as OTO generalisations ofthe well-known Onsager reciprocal relations and fluctuation-dissipation relation (FDR). Bycombining these relations, we have obtained a generalised FDR which connects two of theSchwinger-Keldysh effective couplings. In the dual stochastic dynamics, these two couplingscorrespond to a thermal jitter in the damping coefficient and a non-Gaussianity in the noisedistribution. The generalised FDR between these two quartic couplings is an extension of asimilar relation obtained for the cubic effective dynamics in [30].The generalised FDRs and Onsager relations in both the cubic and the quartic effec-tive theories of the particle suggest that such relations probably hold for even higher degreeterms in the effective action when the bath’s microscopic dynamics is reversible. It would beinteresting to identify the general form of these relations.Although the construction of the quartic effective theory in this paper is demonstratedwith the qXY model, the analysis mostly relies on the validity of the Markov approximation37 ciPost Physics Submission for the particle’s dynamics. Hence, it may be employed to study the effective theory of theparticle when it interacts with more complicated baths. For instance, the bath may even be astrongly coupled system in which case a microscopic analysis of the particle’s dynamics isvery difficult. In such a scenario, the quartic effective theory of the particle would allow oneto determine the particle’s 4-point correlators (including its OTOCs) in terms of the effectivecouplings.For the qXY model studied in this paper, the bath’s 4-point cumulants decay exponentiallywhen the time interval between any two insertions is increased. This allowed us to work in aMarkovian regime by tuning the parameters in the model such that the particle’s evolution ismuch slower than the decay of the bath’s cumulants. However, as pointed out in [45], such anexponential damping of the bath’s cumulants is not strictly necessary in all time regimes forobtaining a nearly local dynamics of the particle. In fact, the Markov approximation for theparticle’s dynamics may be valid even for a chaotic bath [57–60] as long as the bath’s OTOcumulants saturate to sufficiently small values much faster than the particle’s evolution [45].This opens up the possibility of probing the Lyapunov exponents [35] in such chaotic bathsby measuring the OTO effective couplings of the particle [45] (See the relations between theparticle’s OTO effective couplings and the bath’s OTOCs given in (129) and table 6).The applicability of our effective theory framework to the scenario where the bath ischaotic and strongly coupled implies that it may be possible to construct a holographic dualdescription [61–69] of the particle’s non-linear dynamics. The OTO extension of this non-linear dynamics may be useful in determining a holographic prescription for computing theparticle’s OTOCs [67, 70].The non-linear Langevin equation that we discussed in section 3.5 has a structural sim-ilarity with the equations of motion of damped anharmonic oscillators like the Van der Poloscillator [71] and the Duffing oscillator [72] . Such oscillators, under periodic driving, areknown to exhibit chaos in appropriate parameter regimes [72–74]. It would be interestingto see whether one can find a similar regime in the non-linear Langevin dynamics where theparticle undergoes a chaotic motion.It will be useful to formulate a Wilsonian counterpart of the out of time ordered 1-PIeffective action developed in this paper. Such a Wilsonian effective theory can be extended toopen quantum field theories [51,75–82] which show up in the study of quantum cosmology andheavy ion physics. It will be interesting to determine the RG flow [51] of the OTO couplingsin this Wilsonian framework to estimate their relative importance at different energy scales. Acknowledgements
We would like to thank Bijay Kumar Agarwalla, Ahana Chakraborty, Deepak Dhar, ArghyaDas, Abhijit Gadde, G J Sreejith, Sachin Jain, Chandan Jana, Dileep Jatkar, Anupam Kundu,R. Loganayagam, Rajdeep Sensarma and Spenta Wadia for useful discussions. We are gratefulfor the support from International Centre for Theoretical Sciences (ICTS-TIFR), Bangalore.BC would like to thank the Tata Institute of Fundamental Research (TIFR), Mumbai, the Notice that we have assumed a weak coupling only between the particle and the bath. The couplingsbetween the internal degrees of freedom of the bath may be strong. The OTO cumulants in such chaotic baths show an exponentially fast fall-off initially (in the Lyapunovregime) before saturating to some constant values. The major difference is the presence of a thermal noise in the Langevin dynamics. ciPost Physics Submission Indian Institute of Science Education and Research (IISER), Pune and the Harish-ChandraResearch Institute (HRI), Prayagraj for hospitality towards the final stages of this work. BCacknowledges Infosys Program for providing travel support to various conferences. SC wouldlike to thank the Kavli Institute of Theoretical Physics (KITP), UC Santa Barbara and TheAbdus Salam International Centre for Theoretical Physics (ICTP), Trieste for hospitalityduring the course of this work. We acknowledge our debt to the people of India for theirsteady and generous support to research in the basic sciences.
Funding information:
This research was supported in part by the National Science Foun-dation under Grant No. NSF PHY-1748958.
A Cumulants of the bath operator that couples to the particle
In this appendix, we provide the forms of the 2-point and 4-point cumulants of the bathoperator λO ( t ) ≡ λ P i,j g xy,ij X ( i ) ( t ) Y ( j ) ( t ) that couples to the particle. These cumulants arecalculated in the high temperature limit where β Ω ≪ . (118)We will see that, in this limit, the cumulants decay exponentially when the separations betweeninsertions are increased. To show the form of this decay we follow the notational conventionsgiven below: • The interval between two time instants t i and t j is expressed as t ij ≡ t i − t j . (119) • The cumulant of any Wightman correlator h O ( t i ) O ( t i ) · · · O ( t i n ) i is expressed as h i i · · · i n i ≡ h O ( t i ) O ( t i ) · · · O ( t i n ) i c . (120)Keeping these notational conventions in mind, let us now discuss the decaying behaviour ofthe cumulants. A.1 2-point cumulants
From the form of the operator O ( t ) given in (3), we can see that the 2 point cumulants receivecontributions from the Feynman diagram shown in figure 8. Here the blue and red linesFigure 8: Feynman diagram contributing to the 2-point cumulants t t represent X ( i ) - X ( i ) and Y ( j ) - Y ( j ) propagators respectively. To get the 2-point cumulants, one39 ciPost Physics Submission has to sum over all the oscillator frequencies in the bath. In the continuum limit, these sumsreduce to integrals with the appropriate distribution of the couplings given in (8). Performingthese integrals over the oscillator frequencies, we find that the cumulants decay exponentiallywhen the time interval between the insertions is increased. In table 4, we provide the formsof the slowest decaying modes in these cumulants. The decay rates of these modes are of theorder of Ω. All the other modes which we have not mentioned in table 4 decay at much fasterrates which are of the order of β − .Table 4: Decay of 2-point cumulants for t > t Cumulant Slowest decaying mode λ h i − Γ Ω ( e iβ Ω − e − t λ h i − Γ Ω ( − e − iβ Ω ) e − t A.2 4-point cumulants
The Feynman diagrams that contribute to the 4-point cumulants of O ( t ) are given in figure9. Again, integrating over all the oscillator frequencies with the corresponding couplingsFigure 9: Feynman diagrams contributing to the 4-point cumulants t t t t t t t t t t t t t t t t t t t t t t t t following the distribution given in (9), one can obtain the forms of these cumulants. As incase of the 2-point cumulants, these 4-point cumulants also decay exponentially when theseparation between any two insertions is increased. We provide the forms of the slowestdecaying modes of these cumulants in table 5. By looking at these forms, one can see thatthe decay rates of these modes are of the order of Ω as well.40 ciPost Physics Submission Table 5: Decay of 4-point cumulants for t > t > t > t Cumulant Slowest decaying mode λ h i Ω ( e iβ Ω − e − t Ω (cid:16) e − t Ω + 2 (cid:17) λ h i Ω ( e iβ Ω − e − t Ω (cid:16) e − t Ω + 2 e iβ Ω (cid:17) λ h i Ω ( e iβ Ω − e − t Ω (cid:16) e − t Ω+ iβ Ω + e iβ Ω + 1 (cid:17) λ h i Ω ( e iβ Ω − e − t Ω+2 iβ Ω (cid:16) e − t Ω + 2 e − iβ Ω (cid:17) λ h i Ω ( e iβ Ω − e − t Ω+ iβ Ω (cid:16) e − t Ω + e iβ Ω (cid:17) λ h i Ω ( e iβ Ω − e − t Ω+2 iβ Ω (cid:16) e − t Ω (cid:17) λ h i Ω ( e iβ Ω − e − t Ω (cid:16) e − t Ω + 2 e iβ Ω (cid:17) λ h i Ω ( e iβ Ω − e − t Ω (cid:16) e − t Ω + 2 e iβ Ω (cid:17) λ h i Ω ( e iβ Ω − e − t Ω+ iβ Ω (cid:16) e − t Ω + e iβ Ω (cid:17) λ h i Ω ( e iβ Ω − e − t Ω+2 iβ Ω (cid:16) e − t Ω + 2 (cid:17) λ h i Ω ( e iβ Ω − e − t Ω+ iβ Ω (cid:16) e − t Ω + e iβ Ω + e iβ Ω (cid:17) λ h i Ω ( e iβ Ω − e − t Ω+2 iβ Ω (cid:16) e − t Ω + 2 e iβ Ω (cid:17) λ h i Ω ( e iβ Ω − e − t Ω+2 iβ Ω (cid:16) e − t Ω + 2 e − iβ Ω (cid:17) λ h i Ω ( e iβ Ω − e − t Ω+ iβ Ω (cid:16) e iβ Ω + e − t Ω+2 iβ Ω (cid:17) λ h i Ω ( e iβ Ω − e − t Ω+2 iβ Ω (cid:16) e − t Ω (cid:17) λ h i Ω ( e iβ Ω − e − t Ω+2 iβ Ω (cid:16) e iβ Ω + e − t Ω+ iβ Ω (cid:17) λ h i Ω ( e iβ Ω − e − t Ω+2 iβ Ω (cid:16) e − t Ω+2 iβ Ω (cid:17) λ h i Ω ( e iβ Ω − e − t Ω+3 iβ Ω (cid:16) e − t Ω+ iβ Ω (cid:17) λ h i Ω ( e iβ Ω − e − t Ω+2 iβ Ω (cid:16) e − t Ω (cid:17) λ h i Ω ( e iβ Ω − e − t Ω+2 iβ Ω (cid:16) e iβ Ω + e − t Ω+ iβ Ω (cid:17) λ h i Ω ( e iβ Ω − e − t Ω+2 iβ Ω (cid:16) e iβ Ω + e − t Ω (cid:17) λ h i Ω ( e iβ Ω − e − t Ω+3 iβ Ω (cid:16) e iβ Ω + e − t Ω (cid:17) λ h i Ω ( e iβ Ω − e − t Ω+3 iβ Ω (cid:16) e − t Ω+ iβ Ω (cid:17) λ h i Ω ( e iβ Ω − e − t Ω+4 iβ Ω (cid:16) e − t Ω (cid:17) B Argument for the validity of the Markovian limit
In this appendix, we provide an argument for the validity of the Markov approximation forthe parameter regime given in (20). This argument will mainly involve dimensional analysis.In the units where ~ and the renormalised mass of the particle are unity, the dimension ofeach effective coupling can be expressed as some power of time. We provide these dimensionsbelow. Dimensions of quadratic couplings: [ Z I ] = T , [ γ ] = T − , [ µ ] = [ h f i ] = T − . (121) Dimensions of SK quartic couplings: [ ζ (4) N ] = [ λ ] = [ ζ ] = [ ζ (2) µ ] = T − , [ λ γ ] = [ ζ γ ] = [ ζ (2) γ ] = T − . (122)41 ciPost Physics SubmissionDimensions of OTO quartic couplings: [ κ ] = [ e κ ] = [ ̺ ] = [ e ̺ ] = T − , [ κ I γ ] = [ κ II γ ] = [ κ III γ ] = [ e κ I γ ] = [ e κ II γ ] = [ ̺ I γ ] = [ ̺ II γ ] = [ e ̺ I γ ] = [ e ̺ II γ ] = T − . (123)Now, from the value of each of these couplings given in section 4.5, one can obtain a time-scale. For the Markov approximation to be valid, we need all these time-scales to be muchlarger than Ω − . This fixes the following Markovian regime for the parameters: µ ≪ Ω , Γ ≪ β ( β Ω) , Γ ≪ β ( β Ω) . (124)Since, the values of the parameters in section 4.5 are computed in the higher temperaturelimit, we need to impose the following condition as well: β Ω ≪ . (125)In addition to the above hierarchy, we need the following condition for the validity of theperturbative analysis : Γ ≪ (cid:16) Γ (cid:17) . (126)Note that imposing this condition along with the high temperature limit in (125) and theinequality for Γ in (124) automatically ensures that Γ falls in domain given in (124).Combining all these conditions, we get the Markovian regime mentioned in (20). C Relations between the OTO effective couplings and thebath’s OTOCs
In section 4.1, we obtained the form of the non-local generalised influence phase of the parti-cle by integrating out the bath’s degrees of freedom on the 2-fold contour. When the bath’scumulants decay sufficiently fast compared to the time-scales involved in the particle’s evo-lution, one can get an approximately local form for this generalised influence phase. Thisapproximately local form is similar to its Schwinger-Keldysh counterpart (see (21) and (22)).The quadratic and quartic terms in this approximately local generalised influence phase areas follows: W (2) GSK ≈ i X i ,i =1 Z t f t dt hn Z t t dt hT C O i ( t ) O i ( t ) i c o q i ( t ) q i ( t − ǫ )+ n Z t t dt hT C O i ( t ) O i ( t ) i c t o q i ( t ) ˙ q i ( t − ǫ )+ n Z t t dt hT C O i ( t ) O i ( t ) i c t o q i ( t )¨ q i ( t − ǫ ) i , (127) In this perturbative analysis, we assume that the contributions of the quartic terms to the particle’sdynamics are small compared to those of the quadratic terms. ciPost Physics Submission W (4) GSK ≈ − i X i , ··· ,i =1 Z t f t dt hn Z t t dt Z t t dt Z t t dt hT C O i ( t ) O i ( t ) O i ( t ) O i ( t ) i c o q i ( t ) q i ( t − ǫ ) q i ( t − ǫ ) q i ( t − ǫ )+ n Z t t dt Z t t dt Z t t dt hT C O i ( t ) O i ( t ) O i ( t ) O i ( t ) i c t o q i ( t ) ˙ q i ( t − ǫ ) q i ( t − ǫ ) q i ( t − ǫ )+ n Z t t dt Z t t dt Z t t dt hT C O i ( t ) O i ( t ) O i ( t ) O i ( t ) i c t o q i ( t ) q i ( t − ǫ ) ˙ q i ( t − ǫ ) q i ( t − ǫ )+ n Z t t dt Z t t dt Z t t dt hT C O i ( t ) O i ( t ) O i ( t ) O i ( t ) i c t o q i ( t ) q i ( t − ǫ ) q i ( t − ǫ ) ˙ q i ( t − ǫ ) i . (128)One can compare the leading order forms of the 4-point OTO cumulants of the particleobtained from this approximately local generalised influence phase, and the similar forms ob-tained from the out of time ordered 1-PI effective action given in section 4.2. This comparisonyields the leading order form of any quartic OTO coupling g in terms of an integral of the4-point OTO cumulants of O ( t ) as follows g = λ lim t − t →∞ Z t t dt Z t t dt Z t t dt I [ g ] + O( λ ) , (129)where the dependence of I [ g ] on cumulants of the bath are given in table 6 for all the OTOcouplings. In expressing these cumulants , we follow the conventions introduced in section3.4. In addition, we represent the cumulants corresponding to double (anti-)commutators of O ( t ) as follows: h [12][34] i ≡ h [ O ( t ) , O ( t )][ O ( t ) , O ( t )] i c , h [12 + ][34] i ≡ h{ O ( t ) , O ( t ) } [ O ( t ) , O ( t )] i c , h [12][34 + ] i ≡ h [ O ( t ) , O ( t )] { O ( t ) , O ( t ) }i c , h [12 + ][34 + ] i ≡ h{ O ( t ) , O ( t ) }{ O ( t ) , O ( t ) }i c , etc. (130)43 ciPost Physics Submission Table 6: Relations between the OTO couplings and the 4-point OTO cumulants of O ( t ) g I [ g ] κ i h (cid:16) h [1234] i − h [4321] i (cid:17) − (cid:16) h [2314] i − h [3241] i + h [1423] i − h [4132] i (cid:17)ie κ i ( h [1234] i + h [4321] i ) ̺ h − h [12 + + + ] i − h [43 + + + ] i + h [14 + + + ] i + h [41 + + + ] i + h [2314 + ] i + h [3241 + ] i ie ̺ h(cid:16) h [43 + i + h [432 + i + h [4321 + ] i (cid:17) − (cid:16) h [12 + i + h [123 + i + h [1234 + ] i (cid:17)i κ I γ i h ( − t + t + t ) (cid:16) h [34 + ][12 + ] i − h [12 + ][34 + ] i (cid:17) +( t − t + t ) (cid:16) h [24 + ][13 + ] i − h [13 + ][24 + ] i (cid:17) +( t + t − t ) (cid:16) h [23 + ][14 + ] i − h [14 + ][23 + ] i (cid:17)i κ II γ i h ( − t + t + t ) (cid:16) h [34][12] i − h [12][34] i (cid:17) +( t − t + t ) (cid:16) h [24][13] i − h [13][24] i (cid:17) +( t + t − t ) (cid:16) h [23][14] i − h [14][23] i (cid:17)i κ III γ − i h ( − t + t + t ) (cid:16) h [4321] i + h [2314] i + h [2413] i (cid:17) +( t − t + t ) (cid:16) h [4231] i + h [2341] i + h [3412] i (cid:17)ie κ I γ − i h ( t + t + t ) h [1234] i + ( − t − t + 3 t ) h [4321] i ie κ II γ i h ( t − t + t ) (cid:16) h [14][23] − h [32][41] i (cid:17) +( t + t − t ) (cid:16) h [12][34] i − h [43][21] i + h [13][24] i − h [42][31] i (cid:17)i ̺ I γ h ( t − t + t ) (cid:16) h [12][34] + h [43][21] i (cid:17) +( − t + t + t ) (cid:16) h [13][24] i + h [42][31] i + h [14][23] i − h [32][41] i (cid:17)i ̺ II γ ( − t + t + t ) (cid:16) h [12 + ][34] + h [43 + ][21] i + h [12][34 + ] + h [43][21 + ] ii (cid:17) +( t − t + t ) (cid:16) h [13 + ][24] i + h [42 + ][31] i + h [13][24 + ] i − h [42][31 + ] i (cid:17) +( t + t − t ) (cid:16) h [14 + ][23] i + h [32 + ][41] i + h [14][23 + ] i − h [32][41 + ] i (cid:17)e ̺ I γ h ( − t + t + t ) (cid:16) h [12 + i − h [43 + i (cid:17) +( t − t + t ) (cid:16) h [123 + i − h [432 + i (cid:17) +( t + t − t ) (cid:16) h [1234 + ] i + h [4321 + ] i (cid:17)ie ̺ II γ h t (cid:16) h [12][34 + ] + h [43 + ][21] i (cid:17) + t (cid:16) h [43][21 + ] i + h [12 + ][34] (cid:17) + t (cid:16) h [13][24 + ] + h [42 + ][31] i (cid:17) + t (cid:16) h [42][31 + ] i + h [13 + ][24] (cid:17) + t (cid:16) h [14][23 + ] + h [32 + ][41] i (cid:17) + t (cid:16) h [32][41 + ] i + h [14 + ][23] (cid:17)i D Quartic couplings in the high temperature limit
In this appendix, we provide the expressions of all the quartic effective couplings at the hightemperature limit in terms of the 6 spectral functions given in (105). These expressions areobtained from the forms given in tables 2 and 3 by imposing the KMS relations between thespectral functions. Such KMS relations between the bath’s correlators were studied in [10] toexpress the Fourier transforms of all contour-ordered correlators in terms of a different basisof spectral functions which is given below: ρ [1234] , ρ [4321] , ρ [2314] , ρ [12][34] , ρ [13][24] , ρ [14][23] . (131)Notice that the spectral functions corresponding to the nested commutators in this basisare identical to three of the spectral functions given in (105). The remaining three spectralfunctions corresponding to double commutators can be expressed in terms of the spectral44 ciPost Physics Submission functions in our basis as follows: ρ [12][34] = (cid:16) f ( ω ) (cid:17)(cid:16) f ( ω ) (cid:17)(cid:16) f ( ω ) + f ( ω ) (cid:17) (cid:16) ρ [1234] + ρ [2143] (cid:17) ,ρ [13][24] = (cid:16) f ( ω ) (cid:17)(cid:16) f ( ω ) (cid:17)(cid:16) f ( ω ) + f ( ω ) (cid:17) (cid:16) ρ [1234] + ρ [2314] + ρ [3142] (cid:17) ,ρ [14][23] = (cid:16) f ( ω ) (cid:17)(cid:16) f ( ω ) (cid:17)(cid:16) f ( ω ) + f ( ω ) (cid:17) (cid:16) ρ [2314] + ρ [3241] (cid:17) , (132)where f ( ω ) is the Bose-Einstein distribution function given by f ( ω ) ≡ e βω − . (133)This demonstrates that the spectral functions given in (105) indeed form a basis. We chooseto work with this basis rather than the one given in (131) because, in the high temperaturelimit, the β -expansions of all the spectral functions in this basis begin at the same power of β . This simplifies comparisons between the leading order forms of the different couplings.Now, let us provide the expressions of the quartic couplings in terms of this basis. In thehigh temperature limit, any quartic coupling can be expressed as g = Z C e I [ g ] + O( λ ) , (134)where the integrand e I [ g ] for the Schwinger-Keldysh couplings and the OTO couplings aregiven in tables 7 and 8 respectively. In these integrands, the spectral functions are truncatedat their leading order in β -expansion. 45 ciPost Physics Submission Table 7: SK couplings upto leading order in β g e I [ g ] λ − ω ω ( ω + ω ) ρ [1234] ζ (4) N iβ ω ω ω ω ( ω + ω )( ω + ω )( ω + ω ) h ω ( ω + ω )( ω − ω ω ) ρ [1234]+ ω ( ω + ω )( ω + ω )( ω + ω ) ρ [4321] − ω ω ( ω + ω )( ω + ω ) ρ [2143] + ω ω ( ω − ω ) ρ [2314] − ω ω ( ω + ω )( ω + ω ) ρ [3241] + ω ω ( ω + ω )( ω + ω ) ρ [3142] i ζ iβω ω ω ω ( ω + ω )( ω + ω )( ω + ω ) h − ω ω ω ( ω + ω ) (cid:16) ( ω + ω )( ω + ω ) + ω ( ω − ω ) (cid:17) ρ [1234]+ ω ω ω ( ω + ω )( ω + ω ) ρ [2143] − ω ω ω ω ( ω − ω ) ρ [2314] + ω ω ω ( ω + ω )( ω + ω ) ρ [3241] − ω ω ω ( ω + ω )( ω + ω ) ρ [3142] i ζ (2) µ β ω ω ω ω ( ω + ω )( ω + ω )( ω + ω ) h ω ( ω + ω ) (cid:16) ω ( ω − ω )( ω + ω ) + ω ( ω + ω ) (cid:17) ρ [1234]+ ω ( ω + ω )( ω + ω )( ω + ω ) ρ [4321]+ ω ω ( ω + ω )( ω + ω ) ρ [2143] + ω ω ( ω + ω )( ω − ω ) ρ [2314] − ω ω ( ω + ω )( ω + ω ) ρ [3241] + ω ω ( ω + ω )( ω + ω ) ρ [3142] i λ γ iω ω ( ω + ω ) h ω − ω )( ω + ω ) + 2 ω ω i ρ [1234] ζ γ βω ω ω ω ( ω + ω )( ω + ω )( ω + ω ) h − ω ω ω ( ω + ω ) (cid:16) ω ω ( ω + ω ) + 5 ω ω − ω (cid:17) ρ [1234] − ω ω ω ( ω + ω )( ω − ω ) ρ [2143]+ ω ω ω ω ( ω − ω ) (cid:16) ( ω − ω )( ω + ω ) + 2 ω ω (cid:17) ρ [2314] − ω ω ω ( ω + ω ) (cid:16) ( ω − ω )( ω + ω ) + 2 ω ω (cid:17) ρ [3241]+ ω ω ω ( ω + ω ) (cid:16) ( ω − ω )( ω + ω ) + 2 ω ω (cid:17) ρ [3142] i ζ (2) γ i β ω ω ω ω ( ω + ω )( ω + ω )( ω + ω ) h − ω ( ω + ω ) (cid:16) ω ω ( ω + 3 ω ) + ω ω ( ω + 11 ω ω + 10 ω ) − ω (2 ω + 4 ω ω + 3 ω ω + ω ) + ω ω (7 ω + 12 ω ω + 8 ω ω + 5 ω ) (cid:17) ρ [1234] − ω ( ω + ω )( ω + ω )( ω + ω ) (cid:16) − ω ( ω + ω ) + ω (3 ω + 5 ω ) (cid:17) ρ [4321]+ ω ω ( ω + ω )( ω + ω ) (cid:16) ω ω ( ω + ω ) + ω ( ω + 3 ω ) + ω ( ω + 8 ω ω + 9 ω ) (cid:17) ρ [2143]+ ω ω ( ω − ω )( ω + ω ) (cid:16) ω ( ω − ω ) + ω ( ω + ω ) (cid:17) ρ [2314] − ω ω ( ω + ω )( ω + ω ) (cid:16) ω ( ω − ω ) + ω ( ω + ω ) (cid:17) ρ [3241]+ ω ω ( ω + ω )( ω + ω ) (cid:16) ω ( ω − ω ) + ω ( ω + ω ) (cid:17) ρ [3142] i ciPost Physics Submission Table 8: OTO couplings upto leading order in β g e I [ g ] κ − ω ω ( ω + ω ) h ρ [1234] + ρ [2143] + ρ [3142] + ρ [3241] ie κ − ω ω ( ω + ω ) h ρ [1234] + ρ [4321] i ̺ iβω ω ω ω ( ω + ω ) h − ω (cid:16) ω + ( ω + ω )( ω + ω ) (cid:17) ρ [1234] − ω (cid:16) ω + ( ω + ω )( ω + ω ) (cid:17) ρ [4321]+ ω ω ( ω − ω ) (cid:16) ρ [2143] − ρ [3142] (cid:17) − ω ω ( ω − ω ) ρ [2314] + ω ω ( ω − ω ) ρ [3241] ie ̺ − iβω ω ω ω ( ω + ω ) h − ω ( ω + ω ω + ω ω ) ρ [1234] − ω ( ω + ω ω − ω ω ) ρ [4321] − ω ω ( ω + ω ) (cid:16) ρ [2143] + ρ [3142] (cid:17) − ω ω ( ω − ω ) ρ [2314] + ω ω ( ω + ω ) ρ [3241] i κ I γ iβ ω ω ω ω ( ω + ω )( ω + ω )( ω + ω ) h ω ( ω + ω )( ω − ω ω ) ρ [1234]+ ω ( ω + ω )( ω + ω )( ω + ω ) ρ [4321] − ω ω ( ω + ω )( ω + ω ) ρ [2143] + ω ω ( ω − ω ) ρ [2314]+ ω ω ( ω + ω )( ω + ω ) ρ [3142] − ω ω ( ω + ω )( ω + ω ) ρ [3241] i κ II γ i ω ω ( ω + ω ) h ( ω − ω )( ω + ω ) (cid:16) ρ [1234] + ρ [2314] + ρ [3142] (cid:17) +( ω + ω )( ω + ω ) (cid:16) ρ [2314] + ρ [3241] (cid:17) + (cid:16) ( ω − ω )( ω + ω ) + 2 ω ω (cid:17)(cid:16) ρ [1234] + ρ [2143] (cid:17)i κ III γ − i ω ω ( ω + ω ) h ( ω − ω )( ω + ω ) (cid:16) ρ [1234] + ρ [2143] (cid:17) + (cid:16) ( ω − ω )( ω + ω ) + 2 ω ω (cid:17)(cid:16) ρ [1234] + ρ [3241] + ρ [3142] (cid:17)ie κ I γ i ω ω ( ω + ω ) h(cid:16) − ω ( ω + ω ) + ω (3 ω + 5 ω ) (cid:17) ρ [1234]+ (cid:16) − ω ( ω + ω ) + ω (3 ω + 5 ω ) (cid:17) ρ [4321] ie κ II γ i ω ω ( ω + ω ) h ( ω − ω ) (cid:16) ρ [2314] + ρ [3241] (cid:17) + ( ω + ω ) (cid:16) ρ [1234] + ρ [2314] + ρ [2143] + ρ [3142] (cid:17)i ̺ I γ βω ω ( ω + ω )( ω + ω )( ω + ω ) h ( ω + ω )( ω − ω ω + ω ) ρ [1234] + ( ω + ω )( ω − ω ) ρ [2143] − ( ω − ω ) (cid:16) ω ( ω + ω ) − ω ( ω + 3 ω ) (cid:17) ρ [2314] − ( ω + ω ) (cid:16) ω ( ω + ω ) − ω ( ω + 3 ω ) (cid:17) ρ [3241]+( ω + ω ) (cid:16) ω ( ω + ω ) − ω ( ω + 3 ω ) (cid:17) ρ [3142] i ̺ II γ βω ω ω ω ( ω + ω ) h ω (cid:16) − ω ω ( ω + ω ) + ω ( ω + 3 ω ω + ω ) + ω ( ω + 2 ω ω + ω ω + ω ) (cid:17) ρ [1234]+ ω ( ω + ω ) (cid:16) ω + ( ω + ω ω − ω ω )( ω + ω ) (cid:17) ρ [4321] − ω ω (cid:16) ω ω − ω ( ω + ω ) + ω ω ( ω + 2 ω ) (cid:17) ρ [2143] − ω ω ( ω + ω ) ρ [2314] + ω ω ( ω + ω )( − ω + ω ω ) ρ [3241]+ ω ω ( ω + ω ) (cid:16) ω − ω ω + ω ( ω + ω ) (cid:17) ρ [3142] ie ̺ I γ βω ω ω ω ( ω + ω ) h ω (cid:16) ω ( ω + 2 ω ω + 3 ω ) + ω ( ω + 2 ω ω − ω ) + ω ( ω + 3 ω ω + 4 ω ω + 2 ω ) (cid:17) ρ [1234]+ ω (cid:16) ω ω ( ω + ω ) + ω ( ω + 3 ω ) + ω ( ω + 3 ω ω + 2 ω ) + ω ( ω + 3 ω ω + ω ω − ω ) (cid:17) ρ [4321]+ ω ω ( ω + ω )( ω − ω ) ρ [2143] − ω ω ( ω − ω ) (cid:16) ω ( ω + ω ) − ω ( ω + 3 ω ) (cid:17) ρ [2314]+ ω ω (cid:16) ω ω ( ω + ω ) − ω ( ω + 3 ω ) + ω ( − ω − ω ω + ω ) (cid:17) ρ [3241]+ ω ω ( ω + ω ) (cid:16) − ω ( ω + ω ) + ω ( ω + 3 ω ) (cid:17) ρ [3142] ie ̺ II γ βω ω ω ω ( ω + ω ) h(cid:16) ω + 4 ω ω + ω ω − ω + ω ( ω + 4 ω ) + ω ( ω + 3 ω ω + 3 ω ) (cid:17) ρ [1234]+ (cid:16) ω + 5 ω ω + 2 ω ω − ω + 2 ω ( ω + 2 ω ) + ω (2 ω + 5 ω ω + 3 ω ) (cid:17) ρ [4321]+( ω − ω − ω )( ω + ω )( ω + 2 ω ) ρ [2143] + ω ( − ω − ω ω + ω ) ρ [2314]+ ω (cid:16) − ω − ω ω + ω − ω ω − ω ω (cid:17) ρ [3241]+( ω + ω ) (cid:16) ω − ω + 2 ω ω + 4 ω ω (cid:17) ρ [3142] i ciPost Physics Submission References [1] R. 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