# Fast Scrambling at the Boundary

Ancel Larzul  JEIP, UAR 3573 CNRS, Collège de France, PSL Research University,
11 Place Marcelin Berthelot, 75321 Paris Cedex 05, France  Anirvan M. Sengupta
Department of Physics and Astronomy, Rutgers University, Piscataway, New
Jersey 08854, USA  Center for Computational Quantum Physics, Flatiron
Institute, New York, New York 10010, USA  Center for Computational
Mathematics, Flatiron Institute, New York, New York 10010, USA  Antoine
Georges  Center for Computational Quantum Physics, Flatiron Institute, New
York, New York 10010, USA  Collège de France, PSL University, 11 place
Marcelin Berthelot, 75005 Paris, France  Department of Quantum Matter Physics,
University of Geneva, 24 quai Ernest-Ansermet, 1211 Geneva, Switzerland  CPHT,
CNRS, Ecole Polytechnique, IP Paris, F-91128 Palaiseau, France  Marco Schirò
JEIP, UAR 3573 CNRS, Collège de France, PSL Research University, 11 Place
Marcelin Berthelot, 75321 Paris Cedex 05, France

###### Abstract

Many-body systems which saturate the quantum bound on chaos are attracting
interest across a wide range of fields. Notable examples include the Sachdev-
Ye-Kitaev model and its variations, all characterised by some form or
randomness and all to all couplings. Here we study many-body quantum chaos in
a quantum impurity model showing Non-Fermi-Liquid physics, the overscreened
multichannel $SU(N)$ Kondo model. We compute exactly the low-temperature
behavior of the out-of time order correlator in the limit of large $N$ and
large number of channels $K$, at fixed ratio $\gamma=K/N$. Due to strong
correlations at the impurity site the spin fractionalizes in auxiliary
fermions and bosons. We show that all the degrees of freedom of our theory
acquire a Lyapunov exponent which is linear in temperature as $T\rightarrow
0$, with a prefactor that depends on $\gamma$. Remarkably, for $N=K$ the
impurity spin displays maximal chaos, while bosons and fermions only get up to
half of the maximal Lyapunov exponent. Our results highlights two new
features: a non-disordered model which is maximally chaotic due to strong
correlations at its boundary and a fractionalization of quantum chaos.

##  I Introduction

Unitarity of quantum mechanics implies that information initially encoded in
local degrees of freedom cannot be lost under time evolution but only
scrambled into non-local ones, inaccessible to external probes Hayden and
Preskill (2007); Sekino and Susskind (2008); Shenker and Stanford (2014);
Hosur  _et al._ (2016). The ability of quantum many-body systems to act as
scramblers has emerged in recent years as a new lens to classify phases of
matter. A powerful proxy to quantify quantum scrambling is provided by out-of-
time ordered correlators (OTOC) Larkin and Ovchinnikov (1969); Aleiner  _et
al._ (2016); Swingle (2018); Xu and Swingle (2024) originally introduced in
the quasi-classical theory of superconductivity and more recently extended to
a variety of different settings, from field theories Stanford (2016);
Chowdhury and Swingle (2017); Patel  _et al._ (2017) to random circuits Nahum
_et al._ (2018); Khemani  _et al._ (2018); Chan  _et al._ (2018); Kos  _et
al._ (2018); von Keyserlingk  _et al._ (2018) to models with local bounded
Hilbert space Kukuljan  _et al._ (2017). Whenever the early-time growth of the
OTOC is exponential in time one can define a quantum Lyapunov exponent
$\lambda_{L}$, which generalizes the butterfly effect to the quantum domain
and it is often taken as a measure of many-body quantum chaos. Quantum
mechanics fundamentally influences this exponent by setting an upper bound on
chaos Maldacena  _et al._ (2016); Tsuji  _et al._ (2018); Pappalardi  _et al._
(2022), set essentially by the temperature of the system, $\lambda_{L}\leq
2\pi k_{B}T/\hbar$. Systems which saturate this bound, so called fast
scramblers, such as the celebrated Sachdev-Ye-Kitaev (SYK) model Sachdev and
Ye (1993); Parcollet and Georges (1999); Georges  _et al._ (2001); Kitaev
(2015); Kitaev and Suh (2018), are therefore particularly interesting for a
broad range of communities Maldacena and Stanford (2016); Polchinski and
Rosenhaus (2016); Jensen (2016); Facoetti  _et al._ (2019); Bera  _et al._
(2022).

A natural place to look for fast scramblers and maximal chaos are quantum
phases of matter without well-defined quasiparticles. These violate Landau’s
Fermi Liquid paradigm and display unusual transport properties, which have
attracted continuous interest over the past decades Chowdhury  _et al._
(2022). Recently a fresh new view on these Non-Fermi-Liquids has emerged which
is rooted in their ability to thermalize in the fastest possible way Sachdev
(2015); Hartnoll  _et al._ (2018); Hartnoll and Mackenzie (2022) and to
efficiently (maximally) scramble quantum information Sekino and Susskind
(2008); Shenker and Stanford (2014). Understanding the chaotic properties of
Non-Fermi Liquids has become therefore an urgent problem and recent literature
has been devoted to the study of OTOC for different types of models, including
variations of the SYK model Gu  _et al._ (2017); Banerjee and Altman (2017);
Guo  _et al._ (2019); Davis and Wang (2023) or interacting fermions coupled to
critical bosons Patel and Sachdev (2017); Kim  _et al._ (2021); Tikhanovskaya
_et al._ (2022). A common thread in the search for Non-Fermi Liquids and
maximally chaotic quantum matter is the presence of quenched randomness in the
form of all to all couplings, which are solvable Patel  _et al._ (2023).
Whether randomness is a fundamental requirement is however not understood.
This raises the question of what is the simplest model displaying maximal
chaos. In this work we show that, quite surprisingly, randomness is not needed
for maximal chaos. Indeed we will be able to display a model for maximal chaos
which only describes a quantum spin strongly coupled to bath, i.e. a quantum
impurity.

Quantum impurity models have been long known to display exotic Non-Fermi
Liquid behavior Nozières, Ph. and Blandin, A. (1980); Cox and Zawadowski
(1998), whenever for example a single magnetic impurity possibly with non-
trivial orbital character interacts with multiple channels of conduction
electrons Andrei and Destri (1984); Affleck and Ludwig (1991) or in presence
of magnetic interactions between impurities Jayaprakash  _et al._ (1981);
Jones  _et al._ (1988); Georges and Sengupta (1995, 1997). The two-channels
Kondo model in this respect is among the most studied example of NFL behavior,
also experimentally relevant for mesoscopic physics Matveev (1995); Potok  _et
al._ (2007); Mebrahtu  _et al._ (2012); Iftikhar  _et al._ (2015, 2018). Here
the break-down of Fermi Liquid theory occurs because two symmetrically coupled
conduction electron channels compete to screen the impurity, which is left
partially unscreened, leading to a residual entropy at zero temperature and a
NFL groundstate Emery and Kivelson (1992); Affleck and Ludwig (1993); Sengupta
and Georges (1994); Fabrizio  _et al._ (1995); Han  _et al._ (2022).

A rich NFL physics is known to emerge upon enlarging the symmetry of the Kondo
problem to a $SU(N)$ spin and $SU(K)$ channels (flavours of conduction
electrons) leading to an overscreening of the impurity. In this regime a
controlled theory can be obtained by sending $N,K\rightarrow\infty$ at fixed
ratio $\gamma=K/N$. The universal thermodynamic, spectral and transport
properties of this multichannel Kondo (MCK) model have been computed and
demonstrated emergent conformal invariance and residual entropy Parcollet and
Georges (1997); Parcollet  _et al._ (1998), features known to occur in the SYK
model. However, quantum chaos properties of this class of NFL are not known.
Here we fill this gap, specifically by computing the leading low temperature
behaviour of the OTOC for the MCK and extract the associated Lyapunov
exponent. In the MCK model the impurity spin fractionalizes in auxiliary
fermions and bosons which are coupled to conduction electrons. Using the
ladder diagrammatics Gu and Kitaev (2019) we compute the OTOC of bulk
fermions, auxiliary fermions and bosons. We show that all the degrees of
freedom have Lyapunov exponent $\lambda_{L}$ linear in temperature as
$T\rightarrow 0$, with a prefactor that is tunable with $\gamma$. For
$\gamma=1$, corresponding to a number of channels equal to the size of the
spin, the Lyapunov exponent reaches its maximum value, equal to half of the
bound value. We develop a diagrammatic technique to compute the OTOC of the
physical impurity spin, which is a composite object. Remarkably, we show that
the spin OTOC saturates the bound on chaos. The MCK model emerges therefore as
the simplest model of maximally chaotic quantum matter, comprising only a
strongly coupled quantum spin at the boundary of a non-chaotic system. This
result, in addition to the structure of the large $N,K$ theory and its
similarity with the SYK model hints towards a gravity analogue for the MCK.

The paper is organised as follows. In the next section II we summarise the
main concepts and results of this work, which will be presented in more detail
in the rest of the paper. In Sec. III we introduce the multichannel Kondo
model and review its large $N$ limit solution leading to Non-Fermi Liquid
scaling. In Sec. IV we discuss the chaos properties of the model and compute
in particular the leading low-temperature behaviour of the OTOC for the
fundamental degrees of freedom of the theory. In Sec. V we extend the
diagrammatic calculation of the OTOC to the physical spin and obtain the low
temperature behaviour of its Lyapunov exponent. In Sec. VI we discuss our
result in the broader perspective of integrability and chaos of quantum
impurity models. Finally, Sec. VII is devoted to conclusions. In the
Appendixes we provide technical details on our calculations.

##  II Summary of Main Results

In this section we present a summary of the main results of this work, which
will be discussed more in detail in the rest of the paper. The class of models
that we consider in this work are quantum impurity models where a single
quantum spin is coupled to a bath of non-interacting fermions. We consider a
$SU(N)$ spin coupled to a conduction electron bath featuring $K$ channels of
spinful electrons, leading to the multichannel Kondo Model (MCK). The
Hamiltonian of the model is

$\hat{H}=\sum_{\vec{k},i,\alpha}\epsilon_{\vec{k}}\,\hat{c}^{\dagger}_{\vec{k}i\alpha}\hat{c}_{\vec{k}i\alpha}+\frac{J}{N}\sum_{\vec{k},\vec{k}^{\prime},i,\alpha,\beta}\hat{S}_{\alpha\beta}\,\hat{c}^{\dagger}_{\vec{k}i\alpha}\hat{c}_{\vec{k}^{\prime}i\beta}$
(1)

where $\hat{c}^{\dagger}_{\vec{k}i\alpha}$ creates a conduction electron with
moment $\vec{k}$, channel index $i=1,\cdots,K$ and SU$(N)$ spin index
$\alpha=1,\cdots,N$. We use a fermionic (antisymmetric) representation for the
impurity spin with $N$ auxiliary fermions $\hat{f}_{\alpha}$ with a constraint
such that

$\hat{S}_{\alpha\beta}=\hat{f}_{\alpha}^{\dagger}\hat{f}_{\beta}-\frac{Q}{N}\delta_{\alpha\beta},\qquad\sum_{\alpha=1}^{N}\hat{f}_{\alpha}^{\dagger}\hat{f}_{\alpha}=Q$
(2)

We will consider the limit where both $N$ and $K$ are large keeping the ratio
$\gamma=K/N$ fixed. To ensure that the large-$N$ limit is well-defined we also
need to take a large spin size $Q\equiv q_{0}N$ Parcollet  _et al._ (1998).
When written in terms of Abrikosov fermions the Kondo coupling describes
interaction between impurity fermions and conduction electrons, which can be
described by the exchange of a bosonic excitation, leading to a simple
diagrammatics in the large $N,K$ limit, that we review in Sec. III. We are
interested in the many-body chaos as encoded in the OTOC of our composite
degrees of freedom

$\displaystyle C_{O}(t_{1},t_{2})$
$\displaystyle\sim\theta(t_{1})\theta(t_{2})\sum_{\mu,\nu}\textrm{Tr}\Big{[}\sqrt{\rho}\big{[}\hat{O}_{\mu}(t_{1}),\hat{O}_{\nu}^{\dagger}(0)\big{]}_{\pm}\sqrt{\rho}\big{[}\hat{O}_{\mu}(t_{2}),\hat{O}_{\nu}^{\dagger}(0)\big{]}_{\pm}^{\dagger}\Big{]}$
(3)

where $\hat{O}$ can be auxiliary fermions and bosons, conduction electrons as
we all as the physical spin, $\mu,\nu$ represent the corresponding degrees of
freedom and $[,]_{\pm}$ denotes a commutator or anticommutator whether we
consider bosonic or fermionic operators. We are interested in the regime where
the OTOC grows exponentially in time,

$C_{O}(t_{1},t_{2})\sim\left(\frac{1}{N}\right)e^{\lambda_{L}t}$

with $t=(t_{1}+t_{2})/2$ the central time and where we can identify a quantum
Lyapunov exponent $\lambda_{L}$ that we want to compute in the low temperature
limit.

From the point of view of auxiliary fermions the large $N,K$ theory turns out
to be structurally similar to the complex SYK model Chowdhury  _et al._ (2022)
and we show indeed in Sec. IV that the OTOC of the auxiliary fermions can be
computed exactly at low temperature by resumming ladder diagrams. We find that
the Lyapunov exponent is linear in temperature with a prefactor that depends
on $\gamma=K/N$ and which is maximal at $\gamma=1$. Intriguingly, at this
optimal value the Lyapunov exponent is half of maximal bound. Similarly we
compute the OTOC of the auxiliary bosons and show that the ladder structure
gives rise to the same equations as the fermionic auxiliary degrees of
freedom. As a result we conclude that the Lyapunov exponent of bosons and
fermions is identical and reaches its maximum at $\gamma=1$, where we get
$\lambda_{L}=\pi T$, only half-value away from the bound. Auxiliary fermions
and bosons however are not physical (gauge invariant) degrees of freedom in
our theory. It is therefore interesting to compute the chaos properties of the
physical degrees of freedom. In Section IV we compute the OTOC of conduction
electrons. We show that the OTOC of the latter can be obtained directly from
the OTOC of fermions and bosons and it is also lead to a linear in temperature
Lyapunov exponent, with the same prefactor as auxiliary fermions and bosons.
This is a remarkable result: conduction electrons in our model are essentially
free degrees of freedom in the bulk which interact only with the impurity at
their boundary. Such a strong interaction is however sufficient to generate a
linear in temperature quantum Lyapunov exponent, at least close to the
impurity site. Finally in section IV we compute the OTOC of the spin, which is
composite object in our theory of fermions coupled to bosons. We show that the
OTOC of the impurity spin can be still computed in closed form and that,
strikingly, its Lyapunov exponent saturates the bound on chaos for
$\gamma=K/N=1$. Strong correlations at the boundary of our otherwise non-
interacting system lead therefore to maximal quantum chaos. This appears to be
equally shared by strongly coupled auxiliary degrees of freedom our quantum
spin fractionalises into, a sort of equipartition of many-body quantum chaos.
We present a detailed discussion of this remarkable result in Sec. VI and
present our conclusions in Sec. VII.

###  II.1 Relation to Prior Works

Here we wish to connect our work to previous literature on many-body chaos in
Non-Fermi Liquids and quantum many-body systems. Models of coupled bosons and
fermions have been considered before, in the context of Yukawa models for
random Dirac Fermions Kim  _et al._ (2021); Davis and Wang (2023),
antiferromagnetic quantum critical points Lunts and Patel (2019) or fermions
coupled to gauge fields Patel and Sachdev (2017); Tikhanovskaya  _et al._
(2022). We note that these works have typically focused only on OTOC of single
particle operators. Models of fractionalised degrees of freedom have displayed
similar large N theories Joshi  _et al._ (2020); Christos  _et al._ (2022a,
b), but their quantum chaos content is not known. In this respect our work
goes beyond those studies in that we compute explicitly the OTOC of a
fractionalised state of matter. This approach opens up the possibilities for
non-perturbative treatments of those couplings in lattice models. Recently the
information scrambling of the 2CK model has been investigated within the
Emery-Kivelson solution Dóra  _et al._ (2017), leading to an OTOC that reaches
a constant at long time. Similarly, the OTOC of a Bose-Fermi Kondo model was
evaluated at intermediate temperatures Han and Yu (2021). With respect to
these works here we consider the large N,M limit as well as the low
temperature limit and extract the leading low-temperature behaviour of the
Lyapunov exponent. Similarly, our work differs from the recent concept of
boundary chaos introduced in Ref. Fritzsch and Prosen (2022), where a quantum
circuit with an impurity is studied from the point of view of correlation
functions and entanglement.

##  III Model, Large $N,K$ solution and Non-Fermi Liquid behaviour

We start reviewing the equilibrium properties of the MCK model, Eq. (1), in
the large $N,K$ limit Parcollet  _et al._ (1998). The partition function for
the model, written as a path integral reads

$Z=\int\mathcal{D}[\overline{f},f,\overline{c},c,\overline{B},B,\mu]e^{-S}$

where the action $S=S_{0}+S_{int}$ is

$\displaystyle S_{0}$ $\displaystyle=\int_{0}^{\beta}d\tau
d\tau^{\prime}\sum_{i\alpha}\overline{c}_{i\alpha}(\tau)(G_{c}^{(0)})^{-1}(\tau,\tau^{\prime})c_{i\alpha}(\tau^{\prime})+$
$\displaystyle+\int_{0}^{\beta}d\tau\Big{\\{}\sum_{\alpha}\overline{f}_{\alpha}(\tau)\big{[}\partial_{\tau}+i\mu(\tau)\big{]}f_{\alpha}(\tau)-iq_{0}N\mu(\tau)\Big{\\}}$
(4)
$\displaystyle+\frac{1}{J}\int_{0}^{\beta}d\tau\sum_{i}\overline{B}_{i}(\tau)B_{i}(\tau)$
$\displaystyle S_{int}$
$\displaystyle=-\frac{1}{\sqrt{N}}\int_{0}^{\beta}d\tau\sum_{i\alpha}\Big{(}\overline{B}_{i}(\tau)\overline{c}_{i\alpha}(\tau)f_{\alpha}(\tau)$
(5)
$\displaystyle+B_{i}(\tau)\overline{f}_{\alpha}(\tau)c_{i\alpha}(\tau)\Big{)}$

In this expression we have introduced auxiliary fermions
$f_{\alpha},\overline{f}_{\alpha}$ to represent the spin, as in Eq. (2),
auxiliary bosons $B_{i},\overline{B}_{i}$ to decouple the Kondo interaction in
Eq. (1) and we have denoted with $\mu(\tau)$ the Lagrange multiplier used to
impose the constraint on the number of fermions. Furthermore we have
integrated out the conduction electrons in the bulk so that
$c_{i\alpha}\equiv\sum_{\vec{k}}c_{\vec{k}i\alpha}$ is the fermionic field at
the impurity site and
$G_{c}^{0}=\sum_{\vec{k}}1/(i\omega_{n}-\epsilon_{\vec{k}})$ the non-
interacting on-site Green’s function. In the large $N$ limit, the Green’s
functions

$\displaystyle G_{f}(\tau,\tau^{\prime})$
$\displaystyle=-\frac{1}{N}\sum_{\alpha=1}^{N}\big{\langle}f_{\alpha}(\tau)\overline{f}_{\alpha}(\tau^{\prime})\big{\rangle}$
(6) $\displaystyle G_{B}(\tau,\tau^{\prime})$
$\displaystyle=+\frac{1}{K}\sum_{i=1}^{K}\big{\langle}B_{i}(\tau)\overline{B}_{i}(\tau^{\prime})\big{\rangle}$
(7)

satisfy the Schwinger-Dyson equations Parcollet  _et al._ (1998)

$\displaystyle G_{f}^{-1}(i\omega_{n})$
$\displaystyle=i\omega_{n}+\lambda-\Sigma_{f}(i\omega_{n})$ (8) $\displaystyle
G_{B}^{-1}(i\nu_{n})$ $\displaystyle=\frac{1}{J}-\Sigma_{B}(i\nu_{n})$ (9)

with $\Sigma_{f}$ and $\Sigma_{B}$ the self-energies

$\Sigma_{f}(\tau)=\gamma\,G_{c}^{0}(\tau)\,G_{B}(\tau)\qquad\Sigma_{B}=G_{c}^{0}(\tau)\,G_{f}(\tau)$
(10)

Here $\omega_{n}=(2n+1)\pi/\beta$ and $\nu_{n}=2n\pi/\beta$ are respectively
the fermionic and bosonic frequencies. Equations (8-10) can be obtained by the
saddle point method after introducing the self-energies as Lagrange
multipliers to enforce the definitions of the Green’s functions $G_{f}$ and
$G_{B}$ and integrating out the fields $f$, $B$ and $c$ to get an effective
action written only in terms of the bilocal fields $G$ and $\Sigma$. Here,
$\lambda$ is the saddle-point value of the field $\mu(\tau)$ and is fixed by
the condition $G_{f}(\tau=0^{-})=q_{0}$. Another way of deriving the
Schwinger-Dyson equations (8 \- 10) is to expand the two-point functions
(6),(7) in powers of the interaction vertex (5) and keep only the diagrams of
order $\mathcal{O}(1)$ in the $1/N$ expansion. The corresponding Feynamn
diagrams are shown in Fig. 1. Notice that if one expands the conduction
electron propagator $G_{c}(\tau,\tau^{\prime})=-\langle
c(\tau)\,\overline{c}(\tau^{\prime})\rangle$, one finds that it is equal to
the bare propagator $G_{c}^{0}$ in the large $N$ limit (since in each bubble
correction the loop is of order $\mathcal{O}(1)$ and cannot cancel the $1/N$
factor coming from the two interaction vertices.)

Figure 1:  Feynman diagrams for the Dyson equations (8) and (9) in the large
$N$ limit. Double lines represent dressed propagators and simple lines non-
interacting propagators. Black solid lines correspond to the impurity fermion
$f$, dashed lines to the conduction electron at the impurity site $c$ and red
lines to the auxiliary boson $B$. In these diagrams, each interaction vertex
is of order $1/\sqrt{N}$ and each loop contributes a factor of $N$, so overall
the bubbles are of order $\mathcal{O}(1)$ in $N$ and survive in the large $N$
limit.

At low energies $\omega,T\ll T_{K}$, where $T_{K}$ is the Kondo temperature,
the Schwinger-Dyson equations (8 \- 10) have an emergent conformal and U(1)
gauge symmetries and one can derive an explicit expression for the Green’s
functions Parcollet  _et al._ (1998); Florens (2004)

$G_{f,B}(\tau)=\left\\{\begin{array}[]{ll}-A_{f,B}\frac{e^{\alpha\big{(}\frac{\tau}{\beta}-\frac{1}{2}\big{)}}}{\cosh{\frac{\alpha}{2}}}\bigg{(}\frac{\pi}{\beta\sin\big{(}\frac{\pi\tau}{\beta}\big{)}}\bigg{)}^{2\Delta_{f,B}}&0<\tau<\beta\\\
\xi
A_{f,B}\frac{e^{\alpha\big{(}\frac{\tau}{\beta}+\frac{1}{2}\big{)}}}{\cosh{\frac{\alpha}{2}}}\bigg{(}\frac{\pi}{\beta\sin\big{(}-\frac{\pi\tau}{\beta}\big{)}}\bigg{)}^{2\Delta_{f,B}}&-\beta<\tau<0\end{array}\right.$
(11)

$\xi=1$ (resp. $-1$) for fermions (resp. bosons). To get these expressions,
one needs to use the fact that the local non-interacting Green’s function of
the conduction electron takes the scaling form
$G_{c}^{0}(\tau)=-\rho_{0}\pi/(\beta\sin(\pi\tau/\beta))$ (since the
corresponding density of states is independent of temperature). The scaling
dimensions of the fermions and bosons are given by

$2\Delta_{f}=\frac{1}{1+\gamma}\qquad 2\Delta_{B}=\frac{\gamma}{1+\gamma}$
(12)

$\alpha$ is an asymmetry paramater which is linked to the spin representation
parameter $q_{0}$ by

$\alpha=\ln\frac{\sin\Big{(}\frac{\pi
q_{0}}{1+\gamma}\Big{)}}{\sin\Big{(}\frac{\pi(1-q_{0})}{1+\gamma}\Big{)}}$
(13)

Finally, the product of amplitudes $A_{f}A_{B}$ satisfies the constraint

$2\Delta_{B}=-2\gamma
A_{f}A_{B}\rho_{0}\Gamma\big{(}1-2\Delta_{B}\big{)}\Gamma\big{(}2\Delta_{B}\big{)}\frac{\Big{|}\sin\Big{(}\pi\Delta_{B}-i\frac{\alpha}{2}\Big{)}\Big{|}^{2}}{\cosh^{2}\Big{(}\frac{\alpha}{2}\Big{)}}$
(14)

Details about the origin of these formulas can be found in Parcollet  _et al._
(1998). The saddle point large $N,M$ solution of the MCK model describes
therefore a Non-Fermi Liquid associated with the overscreening of the impurity
spin by the $K$ channels of the bath. In the next section we will compute the
OTOC properties of this model.

##  IV Out of Time Order Correlator and Lyapunov Exponent

In this section we compute the OTOC for the different degrees of freedom
entering our theory, namely auxiliary fermions and bosons as well as
conduction electrons. In particular we will derive Bethe-Salpeter equations
for these objects and extract the leading low-temperature behaviour of the
Lyapunov exponent $\lambda_{L}$, or scrambling rate, describing the short-time
exponential growth of the OTOC.

###  IV.1 f-fermion and B-boson OTOC

Figure 2:  Two-fold Keldysh contour used to define the regularised OTOC that
we consider in this work. The imaginary time evolution is broken down into two
time-intervals, each with an evolution for an interval $-i\beta/2$ and the
real-time evolution can occur on the up ($u$) or down ($d$) double contour.

We start with the regularized retarded OTOC of the auxiliary fermions

$\begin{split}C_{f}(t_{1},t_{2})&=\frac{\theta(t_{1})\theta(t_{2})}{N^{2}}\sum_{\alpha,\beta=1}^{N}\textrm{Tr}\Big{[}\sqrt{\rho}\big{\\{}\hat{f}_{\alpha}(t_{1}),\hat{f}_{\beta}^{\dagger}(0)\big{\\}}\\\
&\times\sqrt{\rho}\big{\\{}\hat{f}_{\alpha}(t_{2}),\hat{f}_{\beta}^{\dagger}(0)\big{\\}}^{\dagger}\Big{]}\,.\end{split}$
(15)

This correlator can be computed using the two-fold Keldysh contour shown on
Fig. 2. This complex time contour consists of two real time folds separated by
an imaginary time segment of length $-i\beta/2$. The position of the fields on
the contour is specified by a fold index $\sigma=u,d$ and a branch index
$s=+,-$ for forward and backward time evolution respectively. To proceed it is
convenient to perform a Keldysh rotation on each fold and define the classical
and quantum components

$\displaystyle f_{\alpha}^{\sigma,cl/q}(t)$
$\displaystyle=\frac{f_{\alpha}^{\sigma,+}(t)\pm
f_{\alpha}^{\sigma,-}(t)}{\sqrt{2}}\,,$ (16)
$\displaystyle\overline{f}_{\alpha}^{\sigma,cl/q}(t)$
$\displaystyle=\frac{\overline{f}_{\alpha}^{\sigma,+}(t)\mp\overline{f}_{\alpha}^{\sigma,-}(t)}{\sqrt{2}}\,.$
(17)

A similar transformation is done for the auxiliary bosons and the conduction
electrons. Within the two-fold Keldysh formalism, the regularized retarded
OTOC can be rewritten as the four-point function (see Appendix A)

$C_{f}(t_{1},t_{2})=\frac{1}{N^{2}}\sum_{\alpha,\beta=1}^{N}\big{\langle}f_{\alpha}^{d,cl}(t_{1})\overline{f}_{\beta}^{d,cl}(0)\overline{f}_{\alpha}^{u,q}(t_{2})f_{\beta}^{u,q}(0)\big{\rangle}\,,$
(18)

where $\langle\cdots\rangle$ denotes the average over the two-fold Keldysh
contour. We now expand this correlator in powers of the interaction vertex and
keep only the diagrams at leading order $1/N$. For convenience, we define
$C_{f}(t_{1},t_{2})\equiv F_{f}(t_{1},t_{2})/N$. The $1/N$ diagrams form and
infinite series of ladder diagrams, which after resummation are represented in
Fig. 3. In these Feynman diagrams, horizontal lines represent
retarded/advanced Green’s functions $G^{R/A}$ and vertical lines are Wightman
Green’s functions $G^{K}_{ud}$ or $G^{K}_{du}$ (see Appendix B). The
resummation of all the ladder diagrams shown in Fig. 3 corresponds to the
Bethe-Salpeter (BS) equation

$\begin{split}F_{f}(t_{1},t_{2})&=F_{f}^{(0)}(t_{1},t_{2})+\frac{\gamma}{4}\int
dt_{3}\,dt_{4}\,K_{f}(t_{1},t_{2},t_{3},t_{4})\\\ &\times
F_{f}(t_{3},t_{4})\end{split}$ (19)

where $F_{f}^{(0)}(t_{1},t_{2})=G_{f}^{R}(t_{1})G_{f}^{A}(-t_{2})$ and the
kernel $K_{f}$ is given by

$\displaystyle K_{f}(t_{1},t_{2},t_{3},t_{4})$ $\displaystyle=\int
dt\,dt^{\prime}K^{(1)}_{f}(t_{1},t_{2},t,t^{\prime})$ $\displaystyle\times
K^{(2)}_{f}(t,t^{\prime},t_{3},t_{4})$ (20) $\displaystyle
K^{(1)}_{f}(t_{1},t_{2},t_{3},t_{4})$
$\displaystyle=G_{f}^{R}(t_{1},t_{3})\,G_{f}^{A}(t_{4},t_{2})\,G_{c,du}^{K}(t_{3},t_{4})$
(21) $\displaystyle K^{(2)}_{f}(t_{1},t_{2},t_{3},t_{4})$
$\displaystyle=G_{B}^{R}(t_{1},t_{3})\,G_{B}^{A}(t_{4},t_{2})\,G_{c,ud}^{K}(t_{4},t_{3})$
(22)

Figure 3:  Resummation of ladder diagrams for the OTOC of the auxiliary
fermions, leading to a Bethe Salpether equation for $F_{f}(t_{1},t_{2})$.
Dashed lines are conduction electrons propagators while full red-lines are
bosonic ones. The resulting expression is given in Eq. (19).

Explicit expressions for the Green’s functions entering the BS equation are
given in Appendix B. With respect to the standard ladder diagrammatics of the
OTOC for the SYK model Kitaev and Suh (2018) we note that here the kernel
$K_{f}$ is itself expressed as a convolution over product of fermionic,
bosonic and conduction electrons Green’s function. This is a direct
consequence of the structure of the Feynman vertex of the theory. Similar
structure appears in other theories with fermions coupled to bosons Lunts and
Patel (2019); Kim  _et al._ (2020, 2021); Tikhanovskaya  _et al._ (2022). We
note that a similar equation was written for the Bose-Fermi Kondo model Han
and Yu (2021) and solved numerically to extract the Lyapunov exponent at high
temperature. Here instead we derive the scrambling rate $\lambda_{L}$ in the
$T\rightarrow 0$ limit, by making use of the conformal expressions of the
Green’s functions given in Appendix B. Since we expect $F_{f}$ to grow
exponentially, we neglect the first term $F_{f}^{(0)}$ in the ladder equation.
The Bethe-Salpeter equation becomes

$\begin{split}F(t_{1}&,t_{2})=Ce^{i\frac{\alpha}{\beta}t_{12}}\int
dt_{3}\,dt_{4}\,dt_{5}\,dt_{6}\tilde{G}_{f}^{R}(t_{13})\tilde{G}_{f}^{R}(t_{24})\\\
&\times\tilde{G}_{c}^{W}(t_{34})\tilde{G}_{B}^{R}(t_{35})\tilde{G}_{B}^{R}(t_{46})\tilde{G}_{c}^{W}(t_{56})e^{-i\frac{\alpha}{\beta}t_{56}}F(t_{5},t_{6})\,,\end{split}$
(23)

where we have used the notation $t_{ij}\equiv t_{i}-t_{j}$ and with
$\tilde{G}_{f,B}^{R}(t)$ and $\tilde{G}_{c}^{W}(t)$ defined by

$\tilde{G}_{f,B}^{R}(t)=\theta(t)\bigg{(}\frac{\pi}{\beta\sinh\big{(}\frac{\pi
t}{\beta}\big{)}}\bigg{)}^{2\Delta_{f,B}},\tilde{G}_{c}^{W}(t)=\frac{\pi}{\beta\cosh\big{(}\frac{\pi
t}{\beta}\big{)}}\,,$ (24)

and with

$C=16\frac{\Delta_{f}\Delta_{B}}{\Gamma(1-2\Delta_{f})^{2}\Gamma(1-2\Delta_{B})^{2}}\,.$
(25)

To get rid of the exponential phase factors we define
$\tilde{F}(t_{1},t_{2})=e^{-i\frac{\alpha}{\beta}t_{12}}F(t_{1},t_{2})$. The
BS equation takes the form of an eigenvalue problem
$\tilde{K}\cdot\tilde{F}=k\,\tilde{F}$ with $k=1$. To find the Lyapunov
exponent, we assume that $\tilde{F}$ has an exponential growth

$\tilde{F}(t_{1},t_{2})=e^{\lambda_{L}\frac{t_{1}+t_{2}}{2}}\tilde{f}(t_{1}-t_{2})\,,$
(26)

and we determine the eigenvalues and eigenvectors of the kernel $\tilde{K}$ in
this subspace. $\lambda_{L}$ should be seen as a free parameter, which is
fixed at the end of the computation by setting $k(\lambda_{L})=1$. Plugging
the exponential ansatz in the BS equation and taking the Fourier transform
with respect to $t_{1}-t_{2}$ we obtain

$\begin{split}k(\lambda_{L})\,\tilde{f}(\omega)&=C\,\tilde{G}_{f}^{R}\Big{(}\omega+i\frac{\lambda_{L}}{2}\Big{)}\tilde{G}_{f}^{R}\Big{(}-\omega+i\frac{\lambda_{L}}{2}\Big{)}\\\
&\times\int\frac{d\omega^{\prime}}{2\pi}\tilde{K}(\omega,\omega^{\prime})\tilde{f}(\omega^{\prime})\end{split}$
(27)

with

$\begin{split}\tilde{K}(\omega,\omega^{\prime})&=\int\frac{d\Omega}{2\pi}\tilde{G}_{B}^{R}\Big{(}\Omega+i\frac{\lambda_{L}}{2}\Big{)}\tilde{G}_{B}^{R}\Big{(}-\Omega+i\frac{\lambda_{L}}{2}\Big{)}\\\
&\times\tilde{G}_{c}^{W}(\omega-\Omega)\tilde{G}_{c}^{W}(\Omega-\omega^{\prime})\end{split}$
(28)

Using the Fourier transforms of (24), we can eventually rewrite the BS
equation like

$\begin{split}k(h)\tilde{f}(u)&=16\frac{\Delta_{f}\Delta_{B}}{(2\pi)^{2}}\,\bigg{|}\frac{\Gamma\big{(}\Delta_{f}+\frac{h}{2}-iu\big{)}}{\Gamma\big{(}1-\Delta_{f}+\frac{h}{2}-iu\big{)}}\bigg{|}^{2}\\\
&\times\int
du^{\prime}\,dv\left[\bigg{|}\frac{\Gamma\big{(}\Delta_{B}+\frac{h}{2}-iv\big{)}}{\Gamma\big{(}1-\Delta_{B}+\frac{h}{2}-iv\big{)}}\bigg{|}^{2}\right.\\\
&\times\left.\Big{|}\Gamma\Big{(}\frac{1}{2}+i(u-v)\Big{)}\Big{|}^{2}\Big{|}\Gamma\Big{(}\frac{1}{2}+i(u^{\prime}-v)\Big{)}\Big{|}^{2}\tilde{f}(u^{\prime})\right]\end{split}$
(29)

Here we have defined $h=\lambda_{L}/(2\pi T)$ and made the change of variable
$u=\omega/(2\pi T)$. Using the identity on gamma functions

$\begin{split}\int_{-\infty}^{+\infty}du^{\prime}\,\Big{|}&\Gamma\Big{(}\frac{1}{2}+i(u-u^{\prime})\Big{)}\Big{|}^{2}\,\Big{|}\Gamma\Big{(}\frac{a}{2}+iu^{\prime}\Big{)}\Big{|}^{2}\\\
&=\frac{2\pi}{a}\,\Big{|}\Gamma\Big{(}\frac{1}{2}+\frac{a}{2}+iu\Big{)}\Big{|}^{2}\end{split}$
(30)

we see that the function $\tilde{f}(u)$

$\tilde{f}(u)=\Big{|}\Gamma\Big{(}\Delta_{f}+\frac{h}{2}+iu\Big{)}\Big{|}^{2}$
(31)

is solution of the eigenproblem with eigenvalue

$k(h)=16\,\frac{\Delta_{f}\Delta_{B}}{(2\Delta_{f}+h)(2\Delta_{B}+h)}$ (32)

Setting $k(h)=1$ and keeping only the positive solution we finally obtain

$\lambda_{L}=\varphi(\gamma)\,2\pi
T,\qquad\varphi(\gamma)=\frac{-1+\sqrt{1+12\,\frac{\gamma}{(1+\gamma)^{2}}}}{2}$
(33)

Figure 4:  Normalised Lyapunov exponent $h=\lambda_{L}/(2\pi T)$ of auxiliary
fermions and bosons in the MCK model, as a function of $\gamma=K/N$, the ratio
between number of channels and spin indexes. We see that the Lyapunov exponent
vanishes for both small and large $\gamma$ and has a maximum for $\gamma=1$.
The maximal value obtained is $h=0.5$, corresponding to $\lambda_{L}=\pi T$.

We find that the scrambling rate is linear in temperature, with a prefactor
which depends on the ratio $\gamma$. The function $\varphi(\gamma)$ is plotted
in Fig. 4. It vanishes for $\gamma=0$ and $\gamma=\infty$ which corresponds to
the scaling dimensions of the fields $(\Delta_{f},\Delta_{B})=(\frac{1}{2},0)$
and $(\Delta_{f},\Delta_{B})=(0,\frac{1}{2})$ respectively. It is maximal when
$\gamma=1$ for which $\Delta_{f}=\Delta_{B}=\frac{1}{4}$ and reach the value
$\varphi(1)=\frac{1}{2}$. Notice that at the maximum $\gamma=1$ both fields
$f$ and $B$ have the same scaling dimension as fermions described by the SYK
Hamiltonian. However, differently from the SYK case, here the Lyapunov
exponent does not saturate the chaos bound but only reaches a value that is
exactly half of the maximal chaos, $\lambda_{L}=\lambda_{L}^{\rm max}/2=\pi
T$. We now turn to the computation of the OTOC of the auxiliary bosons $B$. As
done previously, we define the regularized retarded OTOC

$\begin{split}C_{B}(t_{1},t_{2})&=\frac{\theta(t_{1})\theta(t_{2})}{K^{2}}\sum_{i,j=1}^{K}\textrm{Tr}\Big{[}\sqrt{\rho}\big{[}\hat{B}_{i}(t_{1}),\hat{B}_{j}^{\dagger}(0)\big{]}\\\
&\times\sqrt{\rho}\big{[}\hat{B}_{i}(t_{2}),\hat{B}_{j}^{\dagger}(0)\big{]}^{\dagger}\Big{]}\,.\end{split}$
(34)

The OTOC can be rewritten as a four-point function on the on the two-fold
Keldysh contour introduced above

$C_{B}(t_{1},t_{2})=\frac{-1}{K^{2}}\sum_{i,j=1}^{K}\big{\langle}B_{i}^{d,cl}(t_{1})\overline{B}_{j}^{d,q}(0)\overline{B}_{i}^{u,cl}(t_{2})B_{j}^{u,q}(0)\big{\rangle}$
(35)

Doing a similar diagrammatic expansion as for the auxiliary $f$-fermions, we
find that the $\frac{1}{N}$ part $F_{B}$ of $C_{B}$ satisfies the following
ladder equation in the limit $N\rightarrow\infty$

$\begin{split}F_{B}(t_{1},t_{2})&=F^{(0)}_{B}(t_{1},t_{2})+\frac{\gamma}{4}\int
dt_{3}\,dt_{4}K_{B}(t_{1},t_{2},t_{3},t_{4})\\\ &\times
F_{B}(t_{3},t_{4})\end{split}$ (36)

where

$\displaystyle K_{B}(t_{1},t_{2},t_{3},t_{4})$ $\displaystyle=\int
dt\,dt^{\prime}K_{B}^{(1)}(t_{1},t_{2},t,t^{\prime})$ (37)
$\displaystyle\times K_{B}^{(2)}(t,t^{\prime},t_{3},t_{4})$ (38)
$\displaystyle K_{B}^{(1)}(t_{1},t_{2},t_{3},t_{4})$
$\displaystyle=G_{B}^{R}(t_{1},t_{3})G_{B}^{A}(t_{4},t_{2})G_{c,ud}^{K}(t_{4},t_{3})$
(39) $\displaystyle K_{B}^{(2)}(t_{1},t_{2},t_{3},t_{4})$
$\displaystyle=G_{f}^{R}(t_{1},t_{3})G_{f}^{A}(t_{4},t_{2})G_{c,du}^{K}(t_{3},t_{4})$
(40)

Figure 5:  Resummation of ladder diagrams for the OTOC of the auxiliary
bosons, leading to a Bethe Salpether equation for $F_{B}(t_{1},t_{2})$. Dashed
lines are conduction electrons propagators while full red-lines are bosonic
ones. The resulting expression is given in Eq. (36).

The corresponding Feynman diagrams are shown in Fig. 5. We note that the
kernel $K_{B}$ entering the bosonic OTOC has a similar structure with respect
to the fermionic one, namely two loops made with fermionic (bosonic) Green’s
function and vertical (Wightman) conduction electons one. In the low-
temperature limit, using the conformal expressions for the Green’s function,
we conclude that the two kernels $K_{f}$ and $K_{B}$ are equal upon changing
the scaling dimensions of auxiliary fermions and bosons, i.e. upon sending
$\Delta_{f}\leftrightarrow\Delta_{B}$. Then the computation of the Lyapunov
exponent proceeds in exactly the same way as above. The eigenvector (31) has
the same form with $\Delta_{f}$ replaced by $\Delta_{B}$. Remarkably, the
eigenvalue $k(h)$ is symmetric under exchange of $\Delta_{f}$ and
$\Delta_{B}$. As a result the scrambling rate for the auxiliary bosons $B$ is
the same as for impurity fermions $f$ and given again by Eq. (33) and plotted
in Fig. 4. To summarize in our MCK the physical spin fractionalizes in
strongly coupled auxiliary fermions and bosons. Their scrambling properties at
low temperature turns out to be identical, a Lyapunov exponent $\lambda_{L}$
linear in temperature with a prefactor that depends on $\gamma$. Intriguingly
for $\gamma=K/N=1$ both modes acquire their maximal scrambling rate which is
however only half of the maximal value allowed by the bound on chaos. This
result is therefore different from other models of fermions coupled to bosons
in which the two sectors acquire a Lyapunov exponent parametrically smaller
one from the other, as well as from the maximum allowed value Lunts and Patel
(2019)

We should remark at this stage that these auxiliary degrees of freedom are not
physical, gauge invariant objects but only serve to represent the quantum
spin. In order to fully elucidate the many-body chaos content of our MCK model
it is therefore crucial to evaluate the OTOC of physical degrees of freedom,
such as conduction electrons and the impurity spin as we will do in the next
sections.

Figure 6:  Resummation of ladder diagrams for the OTOC of the conduction
electrons, $F_{c}(t_{1},t_{2})$. Dashed lines are conduction electrons
propagators while full red-lines are bosonic ones. The resulting expression is
given in Eq. (44-45).

###  IV.2 Conduction electrons

We now turn to the computation of the OTOC of the conduction electrons at the
impurity site

$\begin{split}C_{c}(t_{1},t_{2})&=\frac{\theta(t_{1})\theta(t_{2})}{N^{2}K^{2}}\sum_{i,j=1}^{K}\sum_{\alpha,\beta=1}^{N}\textrm{Tr}\Big{[}\sqrt{\rho}\big{\\{}\hat{c}_{i\alpha}(t_{1}),\hat{c}_{j\beta}^{\dagger}(0)\big{\\}}\\\
&\times\sqrt{\rho}\big{\\{}\hat{c}_{i\alpha}(t_{2}),\hat{c}_{j\beta}^{\dagger}(0)\big{\\}}^{\dagger}\Big{]}\end{split}$
(41)

which can be represented as the four-point function on the two-fold Keldysh
contour like

$\begin{split}C_{c}(t_{1},t_{2})&=\frac{1}{N^{2}K^{2}}\sum_{i,j=1}^{K}\sum_{\alpha,\beta=1}^{N}\Big{\langle}c_{i\alpha}^{d,cl}(t_{1})\overline{c}_{j\beta}^{d,cl}(0)\\\
&\times\overline{c}_{i\alpha}^{u,q}(t_{2})c_{j\beta}^{u,q}(0)\Big{\rangle}\end{split}$
(42)

We expand this correlator in powers of the interaction vertex including $1/N$
corrections

$C_{c}(t_{1},t_{2})=\frac{1}{NK}G^{R,0}_{c}(t_{1})G_{c}^{A,0}(-t_{2})+\frac{1}{N}\frac{1}{NK}F_{c}(t_{1},t_{2})+\cdots$
(43)

In the large $N$ limit, $F_{c}(t_{1},t_{2})$ satisfies the ladder equation
shown on Fig. 6.

$F_{c}(t_{1},t_{2})=F_{c}^{(0)}(t_{1},t_{2})+F^{(B)}_{c}(t_{1},t_{2})+F^{(f)}_{c}(t_{1},t_{2})$
(44)

The first two diagrams, which correspond to $F_{c}^{(0)}$, are the $1/N$
bubble corrections to the bare electronic propagator. The exponential growth
will come from the last two diagrams $F_{c}^{(B)}$ and $F_{c}^{(f)}$ and we
will focus on these two. Notice that this ladder equation is not a self-
consistent equation as the previous BS equations for fermions and bosons, i.e.
$F_{c}$ only appears on the left hand side at the leading order. This is
ultimately due to the structure of the diagrammatics in our MCK model, with a
vertex coupling fermions, bosons and conduction electrons. As such we can
evaluate the OTOC of conduction electrons directly since we already know
$F_{f}$ and $F_{B}$ from the previous section.

We detail the derivation for the diagram involving $F_{f}$; it is then
straightforward to adapt the computation to the other one. The fermionic
contribution $F^{(f)}_{c}(t_{1},t_{2})$ can be read off from Fig.6 and it is
given by

$\begin{split}F^{(f)}_{c}(t_{1},t_{2})&=\int\prod_{i=3}^{i=6}dt_{i}G_{c}^{R}(t_{1},t_{3})G_{c}^{A}(t_{4},t_{2})G_{B,ud}^{K}(t_{4},t_{3})\\\
&\times
F_{f}(t_{3},t_{4},t_{5},t_{6})G_{c}^{R}(t_{5})G_{c}^{A}(-t_{6})G_{B,du}^{K}(t_{5},t_{6})\end{split}$
(45)

In this expression, $F_{f}(t_{1},t_{2},t_{3},t_{4})$ is a generalisation of
$F_{f}(t_{1},t_{2})$ where we replace the times $0$ by $t_{3}$ and $t_{4}$ on
the right end of the ladders in (4). To compute this quantity, we replace the
exponential ansatz (26) by
$\tilde{F}(t_{1},t_{2},t_{3},t_{4})=e^{\lambda_{L}\frac{t_{1}+t_{2}-t_{3}-t_{4}}{2}}f_{1}(t_{1}-t_{2})f_{2}(t_{3}-t_{4})$.
After Fourier transform with respect to $t_{1}-t_{2}$ and $t_{3}-t_{4}$,
$f_{2}$ drops out of the ladder equation and $f_{1}$ satisfies (27) and so is
equal to (31). Taking the inverse Fourier transform we obtain

$\displaystyle
F_{f}(t_{1},t_{2},t_{3},t_{4})=e^{i\frac{\alpha}{\beta}(t_{1}-t_{2})}e^{\lambda_{L}\frac{t_{1}+t_{2}-t_{3}-t_{4}}{2}}\times$
$\displaystyle\times\frac{\Gamma\big{(}\Delta_{f}+\frac{h}{2}\big{)}}{(2\pi
T)^{2\Delta_{f}+h-1}}\bigg{(}\frac{\pi}{\beta\cosh\big{(}\frac{\pi
t_{12}}{\beta}\big{)}}\bigg{)}^{2\Delta_{f}+h}f_{2}(t_{3}-t_{4})$ (46)

Pluggin this expression in (45), the integral over $(t_{5},t_{6})$ factorises
and gives a constant. If we replace $G_{B,ud}^{K}$ by its conformal expression
given in Appendix B, we finally obtain

$\displaystyle F^{(f)}_{c}(t_{1},t_{2})=\textrm{cte}\times
e^{\lambda_{L}\frac{t_{1}+t_{2}}{2}}H(t_{1}-t_{2})$ (47)

with

$\begin{split}H(t)&=\int\frac{d\omega}{2\pi}e^{-i\omega(t_{1}-t_{2})}G_{c}^{R,0}\Big{(}\omega+i\frac{\lambda_{L}}{2}\Big{)}\\\
&\times
G_{c}^{A,0}\Big{(}-\omega+i\frac{\lambda_{L}}{2}\Big{)}f_{c}(\omega),\end{split}$
(48)

with $f_{c}(\omega)\equiv\bigg{(}\frac{\pi}{\beta\cosh\big{(}\frac{\pi
t}{\beta}\big{)}}\bigg{)}^{1+h}$. One would find a similar expression for
$F_{c}^{(B)}$ and so we can conclude that $F_{c}$ grows exponentially
$F_{c}(t_{1},t_{2})\sim e^{\lambda_{L}\frac{t_{1}+t_{2}}{2}}$ witht the same
Lyapunov exponent $\lambda_{L}$ as the impurity fermions and the auxiliary
bosons. This result is particularly remarkable, since the conduction electrons
are essentially non-interacting degrees of freedom in their bulk and hence
non-chaotic (their Lyapunov exponent far away from the impurity should
vanish). However due to the strong coupling at the boundary with the impurity
spin they develop many-body chaos with a linear-in temperature Lyapunov
exponent. This suggests that the OTOC of the conduction electrons should
develop a non-trivial spatial dependence, analogue to the Kondo screening
cloud. We leave the investigation of this interesting feature to future work.
Next we focus on the OTOC and scrambling properties of the impurity spin.

Figure 7:  Feynman diagrams for the impurity spin OTOC. Resummation of ladder
diagrams for the OTOC of the impurity spin fermions. Dashed lines are
conduction electrons propagators while full red-lines are bosonic ones. The
resulting expression is given in Eq. 57.

##  V Impurity spin OTOC

In the previous sections we have computed the OTOC of auxiliary fermions,
bosons and conduction electrons. These types of correlators can be written as
two-particle Green’s functions in the doubled Keldysh contour and their
evaluation can be done, to large extent, following similar calculations for
the SYK model. In this section we tackle a more challenging OTOC correlator of
the impurity spin, which as discussed, is a composite object in our theory of
the MCK model. As we are going to show this results in a different
diagrammatic structure, summing simple ladder diagrams will not be enough.
Remarkably we will be able to extract the leading low temperature behavior of
the Lyapunov exponent and show that it saturates the bound on chaos.

Let us consider the regularised retarded OTOC for the impurity spin, defined
as

$\begin{split}C_{S}(t_{1},t_{2})&=\frac{\theta(t_{1})\theta(t_{2})}{N^{4}}\sum_{\alpha,\beta,\gamma,\delta=1}^{N}\textrm{Tr}\Big{[}\sqrt{\rho}\big{[}\hat{S}_{\alpha\beta}(t_{1}),\hat{S}_{\delta\gamma}(0)\big{]}\\\
&\times\sqrt{\rho}\big{[}\hat{S}_{\alpha\beta}(t_{2}),\hat{S}_{\delta\gamma}(0)\big{]}^{\dagger}\Big{]}\end{split}$
(49)

First we need to rewrite this correlator as an average over the two-fold
Keldysh action. We perform the Keldysh rotation for the spin degrees of
freedom and define

$S^{\sigma,cl}_{\alpha\beta}=\frac{S^{\sigma,+}_{\alpha\beta}+S^{\sigma,-}_{\alpha\beta}}{\sqrt{2}},\qquad
S^{\sigma,q}_{\alpha\beta}=\frac{S^{\sigma,+}_{\alpha\beta}-S^{\sigma,-}_{\alpha\beta}}{\sqrt{2}}$
(50)

with
$S_{\alpha\beta}^{\sigma,\pm}=\overline{f}_{\alpha}^{\sigma,\pm}f_{\beta}^{\sigma,\pm}$.
Then we can rewrite $C_{S}$ as

$C_{S}(t_{1},t_{2})=-\frac{1}{N^{4}}\sum_{\alpha,\beta,\gamma,\delta}\left\langle
S_{\alpha\beta}^{d,cl}(t_{1})S_{\delta\gamma}^{d,q}(0)S_{\beta\alpha}^{u,cl}(t_{2})S_{\gamma\delta}^{u,q}(0)\right\rangle\,.$
(51)

It will be convinient to express $S^{\sigma,cl}$ and $S^{\sigma,q}$ in terms
of $f^{\sigma,cl}$ and $f^{\sigma,q}$:

$S_{\alpha\beta}^{\sigma,cl}=\overline{\Psi}_{\alpha}^{\sigma}\gamma^{q}\Psi_{\beta}^{\sigma}\qquad
S_{\alpha\beta}^{\sigma,q}=\overline{\Psi}_{\alpha}^{\sigma}\gamma^{cl}\Psi_{\beta}^{\sigma}$
(52)

where

$\Psi^{\sigma}=\begin{pmatrix}f^{\sigma,cl}\\\
f^{\sigma,q}\end{pmatrix},\qquad\overline{\Psi}^{\sigma}=\begin{pmatrix}\overline{f}^{\sigma,cl}&\overline{f}^{\sigma,q}\end{pmatrix},$
(53)

and where

$\gamma^{cl}=\begin{pmatrix}1&0\\\
0&1\end{pmatrix},\qquad\gamma^{q}=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}$ (54)

The impurity spin OTOC now takes the form

$\begin{split}C_{S}(t_{1},t_{2})&=-\frac{1}{N^{4}}\sum_{\alpha,\beta,\gamma,\delta=1}^{N}\big{\langle}\overline{\Psi}_{\alpha}^{d}(t_{1})\gamma^{q}\Psi_{\beta}^{d}(t_{1})\overline{\Psi}_{\delta}^{d}(0)\gamma^{cl}\Psi_{\gamma}^{d}(0)\overline{\Psi}_{\beta}^{u}(t_{2})\gamma^{q}\Psi_{\alpha}^{u}(t_{2})\overline{\Psi}_{\gamma}^{u}(0)\gamma^{cl}\Psi_{\delta}^{u}(0)\big{\rangle}\\\
&=-\sum_{\begin{subarray}{c}\mu_{1}\cdots\mu_{4}\\\
\nu_{1}\cdots\nu_{4}\end{subarray}}\gamma^{q}_{\mu_{1}\mu_{2}}\gamma^{cl}_{\nu_{2}\nu_{1}}\gamma^{q}_{\mu_{4}\mu_{3}}\gamma^{cl}_{\nu_{3}\nu_{4}}C_{\mu_{1}\nu_{1};\mu_{2}\nu_{2}}^{\mu_{3}\nu_{3};\mu_{4}\nu_{4}}(t_{1},t_{2})\,.\end{split}$
(55)

with

$C_{\mu_{1}\nu_{1};\mu_{2}\nu_{2}}^{\mu_{3}\nu_{3};\mu_{4}\nu_{4}}(t_{1},t_{2})=\frac{1}{N^{4}}\sum_{\alpha,\beta,\gamma,\delta=1}^{N}\left\langle\overline{f}_{\alpha}^{d,\mu_{1}}(t_{1})f_{\beta}^{d,\mu_{2}}(t_{1})\overline{f}_{\delta}^{d,\nu_{2}}(0)f_{\gamma}^{d,\nu_{1}}(0)\overline{f}_{\beta}^{u,\mu_{4}}(t_{2})f_{\alpha}^{u,\mu_{3}}(t_{2})\overline{f}_{\gamma}^{u,\nu_{3}}(0)f_{\delta}^{u,\nu_{4}}(0)\right\rangle$
(56)

As previously, we expand the correlator
$C_{\mu_{1}\nu_{1};\mu_{2}\nu_{2}}^{\mu_{3}\nu_{3};\mu_{4}\nu_{4}}(t_{1},t_{2})$
in powers of the interaction vertex and keep only the leading diagrams in the
large $N$ and $K$ limit, which are shown on Fig. 7. Remarkably, these diagrams
can be factorized into the product of auxiliary fermions OTOC

$C_{\mu_{1}\nu_{1};\mu_{2}\nu_{2}}^{\mu_{3}\nu_{3};\mu_{4}\nu_{4}}(t_{1},t_{2})=\overline{F}_{\mu_{1}\nu_{1}}^{\mu_{3}\nu_{3}}(t_{1},t_{2})F_{\mu_{2}\nu_{2}}^{\mu_{4}\nu_{4}}(t_{1},t_{2})\,,$
(57)

where

$\displaystyle F_{\mu_{2}\nu_{2}}^{\mu_{4}\nu_{4}}(t_{1},t_{2})$
$\displaystyle=\frac{1}{N^{2}}\sum_{\beta,\delta}\big{\langle}f_{\beta}^{\mu_{2},d}(t_{1})\overline{f}_{\delta}^{\nu_{2},d}(0)\overline{f}_{\beta}^{\mu_{4},u}(t_{2})f_{\delta}^{\nu_{4},u}(0)\big{\rangle}$
(58)
$\displaystyle\overline{F}_{\mu_{1}\nu_{1}}^{\mu_{3}\nu_{3}}(t_{1},t_{2})$
$\displaystyle=\frac{1}{N^{2}}\sum_{\alpha,\gamma}\big{\langle}\overline{f}_{\alpha}^{\mu_{1},d}(t_{1})f_{\gamma}^{\nu_{1},d}(0)f_{\alpha}^{\mu_{3},u}(t_{2})\overline{f}_{\gamma}^{\nu_{3},u}(0)\big{\rangle}$
(59)

Using the expressions of the matrices $\gamma^{cl}$ and $\gamma^{q}$, the fact
that the correlator $\langle f^{q}\overline{f}^{cl}\rangle$ is zero and
symetries between the different OTOCs we can simplify the sum into

$\begin{split}C_{S}(t_{1},t_{2})&=-\Big{[}F_{++}(t_{1},t_{2})F_{--}(t_{1},t_{2})^{*}\\\
&-F_{+-}(t_{1},t_{2})F_{-+}(t_{1},t_{2})^{*}+\textrm{c.c}\Big{]}\end{split}$
(60)

with

$\begin{split}F_{ss^{\prime}}(t_{1},t_{2})&\propto\frac{1}{N^{2}}\sum_{\alpha,\beta=1}^{N}\textrm{Tr}\bigg{[}\sqrt{\rho}\Big{[}\hat{f}_{\alpha}(t_{1}),\hat{f}_{\beta}^{\dagger}(0)\Big{]}_{s}\\\
&\times\sqrt{\rho}\Big{[}\hat{f}_{\alpha}(t_{2}),\hat{f}_{\beta}^{\dagger}(0)\Big{]}_{s^{\prime}}^{\dagger}\bigg{]}\end{split}$
(61)

with $s,s^{\prime}=\pm$. The four possible combinations of product of
commutator and anticommutator $F_{ss^{\prime}}$ appear in the definition of
the OTOC. We have already computed the first one $F_{++}$ and one could be
tempted to write down a ladder equation for the other three and try to find
the scrambling rate by the same technique that we have used above. However the
Keldysh structure of the resulting Bethe-Salpeter equations makes it hard to
find a closed analytic solution. Since we are only interested in computing the
scrambling rate of the spin, we will take a different approach. To simplify
the discussion from now on we drop the spin and channel indices and we take
$t_{1}=t_{2}=t$. Then we can write

$C_{S}(t)=-4\,\theta(t)\,\textrm{Re}\Big{\\{}\textrm{Tr}\Big{[}\sqrt{\rho}\hat{f}(t)\hat{f}^{\dagger}(0)\sqrt{\rho}\hat{f}^{\dagger}(t)\hat{f}(0)\Big{]}^{2}\Big{\\}}+\cdots$
(62)

Here we have only shown the OTOC part of $C_{S}$ which is responsible for the
exponential chaotic growth. To see how it behaves, notice that

$\begin{split}\textrm{Tr}\Big{[}\sqrt{\rho}&\hat{f}(t)\hat{f}^{\dagger}(0)\sqrt{\rho}\hat{f}^{\dagger}(t)\hat{f}(0)\Big{]}\\\
&=\textrm{Tr}\Big{[}\rho^{\frac{1}{4}}\hat{f}\Big{(}t+i\frac{\beta}{4}\Big{)}\rho^{\frac{1}{4}}\hat{f}^{\dagger}(0)\rho^{\frac{1}{4}}\hat{f}^{\dagger}\Big{(}t+i\frac{\beta}{4}\Big{)}\rho^{\frac{1}{4}}\hat{f}(0)\Big{]}\\\
&=F_{1/4}\Big{(}t+i\frac{\beta}{4},t+i\frac{\beta}{4}\Big{)}\end{split}$ (63)

with

$F_{1/4}(t_{1},t_{2})\equiv\textrm{Tr}\Big{[}\rho^{\frac{1}{4}}\hat{f}(t_{1})\rho^{\frac{1}{4}}\hat{f}^{\dagger}(0)\rho^{\frac{1}{4}}\hat{f}^{\dagger}(t_{2})\rho^{\frac{1}{4}}\hat{f}(0)\Big{]}$
(64)

This function $F_{1/4}$, which corresponds to a different OTOC regularisation
Tsuji  _et al._ (2018); Romero-Bermúdez  _et al._ (2019), satisfies the same
ladder equation as the retarded OTOC

$\begin{split}F_{1/4}(t_{1},t_{2})&=F_{1/4}^{0}(t_{1},t_{2})+\int\,dt_{3}dt_{4}\,K(t_{1},t_{2},t_{3},t_{4})\\\
&\times F_{1/4}(t_{3},t_{4})\end{split}$ (65)

with $K(t_{1},t_{2},t_{3},t_{4})$ the kernel defined above. Thus
$F_{1/4}(t_{1},t_{2})$ is of the form

$F_{1/4}(t_{1},t_{2})=e^{i\frac{\alpha}{\beta}(t_{1}-t_{2})}f(t_{1}-t_{2})e^{\lambda_{L}\frac{t_{1}+t_{2}}{2}}$
(66)

with $f$ a real function (the expression is written above). Then we conclude
that $F_{1/4}$ grows exponentially with a Lyapunov exponent $\lambda_{L}$
equal to the one of the retarded OTOC correlator.

$F_{1/4}\Big{(}t+i\frac{\beta}{4},t+i\frac{\beta}{4}\Big{)}=f(0)\,e^{i\frac{\beta\lambda_{L}}{4}}e^{\lambda_{L}\frac{t_{1}+t_{2}}{2}}$
(67)

Using this result and the expression for the spin OTOC in Eq.(62) we obtain

$C_{S}(t)\sim
f(0)\cos\Big{(}\frac{\beta\lambda_{L}}{2}\Big{)}e^{2\lambda_{L}\frac{t_{1}+t_{2}}{2}}$
(68)

We conclude therefore that the impurity spin OTOC has a Lyapunov exponent
given by _twice_ the value of the fermionic and bosonic one. This factor of
two is crucial since it allows the impurity spin, a physical and gauge
invariant degree of freedom of our theory, to become maximally chaotic and to
saturates the bound on chaos for $\gamma=1$, i.e. when the size of the
impurity spin is equal to the number of channels. Right at this point bosons
and fermions, the composite degrees of freedom of the impurity spin are also
chaotic, but only acquire half of the maximal Lyapunov exponent, a sort of
equipartition of maximal chaos.

##  VI Discussion

The results obtained in the previous section show that the MCK model in the
infinite $N,K$ limit at fixed $K/N$ is maximally chaotic and saturates the
bound on chaos. This is a striking result from several standpoints that we
briefly highlight here. First, maximal chaos occurs in a model that, quite
differently from the SYK and its variations, does not present any random all-
to-all coupling but only boundary interactions between a large quantum spin,
strongly coupled to a large number of free fermionic channels. As we will
argue this has direct consequences on the physical origin of the fast
scrambling dynamics of the MCK model that sets it apart from other well known
examples.

A second reason for which this result is remarkable is from the point of view
of the properties of the MCK model. Infact these models are integrable
Tsvelick and Wiegmann (1985); Jerez  _et al._ (1998) at any finite $N,K$. For
example, for $N=K=2$ the model reduces to the two-channel Kondo model (2CK)
which is known to be integrable at low energy. In particular it is known from
CFT that the spectrum is made of a finite number of tower of states, where
within each tower is the levels are equally spaced by $\pi v_{F}/L$. Hence,
although the system is a non-Fermi Liquid, there is a quasiparticle
description which is revealed by CFT Affleck and Ludwig (1991); Sengupta and
Georges (1994); Georges and Sengupta (1995). Furthermore in this case at zero
temperature there is a residual entropy coming from a decoupled Majorana mode
Emery and Kivelson (1992). Recent work investigating the OTOC in the 2CK model
Dóra  _et al._ (2017) did not find any trace of exponential growth and
Lyapunov chaos, a result which is compatible with the fact that models with
finite local Hilbert space (i.e. spin chains) do not have an exponential
regime for the growth of OTOC.

To reconcile the known results at finite $N,K$ with our results in the limit
$N,K\rightarrow\infty$ is it crucial to understand how the finite-size
spectrum, residual entropy and OTOC evolve in the MCK as the large $N,K$ limit
is approached. The case of the SYK model in this respect is well understood:
the finite-size spectrum contains exponentially many states above the
groundstate, with exponentially small level spacing, leading to an extensive
residual zero temperature entropy Chowdhury  _et al._ (2022) if the large
volume limit is taken first with respect to the zero-temperature one. However
the structure of the spectrum for the finite $N,K$ MCK model is very different
from the case of SYK. In the 2CK model the residual entropy at $T=0$ arises
from high-energy, large quantum number states whose level spacing is still
regular and set by the inverse system size. In the thermodynamic limit, as the
level spacing becomes small, these high-energy states with high degeneracy
yield a residual entropy, when temperature is sent to zero after the taking
the thermodynamic limit Affleck and Ludwig (1991). A similar mechanism is at
play for any $K,N$ finite, with levels described by $SU_{K}(N)$. We can now
understand qualitatively how the structure of the spectrum of the MCK model
evolves as $N,K$ are increased: while the CFT tower with the $1/L$ spacing is
always there, the key is the behavior of the spectrum at large quantum numbers
and whether exponentially many-levels contribute. This interpretation is
indeed confirmed by looking at the residual entropy of the MCK for generic
$N,K$, computed in Ref. Parcollet  _et al._ (1998), which gives a result of
order $N$ in the large $N$ limit, where $N$ is the size of the spin. We
conjecture that the fast scrambling we find in our calculation is associated
with this manifold of states with high-degeneracies.

What are the consequences for the OTOC at finite but large $N,K$? The
exponential growth of the OTOC is expected to hold on time scales shorter than
the Ehrenfest time Maldacena  _et al._ (2016), that in the present case should
be identified with $t_{E}\sim\mbox{log}N$. In the case of the SYK the long-
time dynamics of the OTOC is controlled by Goldstone-mode fluctuations
associated to time-reparametrization. These fluctuations act as a restoring
force giving rise to a time scale above which a power-law decay of the OTOC is
observed Bagrets  _et al._ (2016, 2017). We can speculate that a mechanism
similar to this one will be at play in controlling the long time behavior of
the OTOC for $t\gg t_{E}$ at finite $N,K$.

##  VII Conclusions

Models that saturate the quantum bound on chaos are appealing for a wide range
of communities, from high-energy theory to condensed matter and statistical
mechanics. While few examples displaying maximal chaos exist, beyond the
celebrated SYK model, they typically involve extended systems with random all
to all interactions or conformal symmetries. In this work we exhibit arguably
the simplest model of fast scrambler: a quantum $SU(N)$ spin coupled to a
metallic bath with $K$ channels, in the large $N$ and $K$ limit, with fixed
$\gamma=K/N$. In this limit the quantum spin fractionalise due to strong
correlations into auxiliary fermionic and bosonic degrees of freedom, coupled
to the conduction electrons of the metallic bath. Using the generalised two-
fold Keldysh time contour, we have computed diagrammatically the OTOC for
auxiliary fermions and bosons as well as the conduction electrons. We have
demonstrated that for all these degrees of freedom the Lyapunov exponent is
linear in temperature as $T\rightarrow 0$, with a tunable prefactor maximum at
$\gamma=1$. Finally, we have shown that the OTOC of the physical impurity spin
factorizes in the large $N$ limit and can be deduced from the OTOC of the
auxiliary fermions. In particular for the value $\gamma=K/N=1$ the spin OTOC
saturates the bound on chaos.

The onset of maximal chaos raises the intriguing question of whether there is
a gravity analogue of the MCK model, i.e. whether the effective theory of the
soft mode associated to a broken reparametrization symmetry would be related
to some model of gravity. Remarkably, holographic realization of Multichannel
Kondo impurity models have been proposed in the context of the AdS/CFT
correspondence Mück (2011); Erdmenger  _et al._ (2013, 2017), even though
their chaos properties on the gravity side has not been investigated to the
best of our knowledge.

The field theory technique used in this work to evaluate the OTOC and Lyapunov
exponent of composite operators such as the impurity spin could find immediate
applications in the investigation of scrambling and chaos in models for doped
quantum spin glasses which display similar fractionalisation Joshi  _et al._
(2020); Christos  _et al._ (2022a, b). Finally, a natural extension of the
present work would be to investigate the non-equilibrium properties of this
model and compare it to the known results on the SYK model. In particular, it
could be interesting to see if the relaxation towards thermal equilibrium in
this model also displays a Planckian behaviour, with a thermalization rate
proportional to the temperature.

## Acknowledgments

This work was supported by the ANR grant ”NonEQuMat” (ANR-19-CE47-0001).

##  Appendix A Details on Double Contour

In this appendix we provide additional details on the two-fold Keldysh
technique used in the main text to compute the retarded regularized OTOCs,
like the $f$-OTOC

$C_{f}(t_{1},t_{2})=\frac{\theta(t_{1})\theta(t_{2})}{N^{2}}\sum_{\alpha,\beta=1}^{N}\textrm{Tr}\Big{[}\sqrt{\rho}\big{\\{}\hat{f}_{\alpha}(t_{1}),\hat{f}_{\beta}^{\dagger}(0)\big{\\}}\sqrt{\rho}\big{\\{}\hat{f}_{\alpha}(t_{2}),\hat{f}_{\beta}^{\dagger}(0)\big{\\}}^{\dagger}\Big{]}\,,$
(69)

where $\sqrt{\rho}=e^{-\beta\hat{H}/2}$ and
$\hat{f}_{\alpha}(t)=e^{i\hat{H}t}\hat{f}_{\alpha}e^{-i\hat{H}t}$. By
rearranging the exponential factors, $C_{f}(t_{1},t_{2})$ can be rewritten
like

$C_{f}(t_{1},t_{2})=\frac{\theta(t_{1})\theta(t_{2})}{N^{2}}\sum_{\alpha,\beta=1}^{N}\left\langle\Big{\\{}\hat{f}_{\alpha}\Big{(}t_{1}-i\frac{\beta}{2}\Big{)},\hat{f}_{\beta}^{\dagger}\Big{(}-i\frac{\beta}{2}\Big{)}\Big{\\}}\Big{\\{}\hat{f}_{\alpha}(t_{2}),\hat{f}_{\beta}^{\dagger}(0)\Big{\\}}^{\dagger}\right\rangle_{\beta}\,,$
(70)

with $\langle\cdots\rangle_{\beta}=\textrm{Tr}[\cdots e^{-\beta\hat{H}}]/Z$
the thermal average. The partition function of the model $Z$ along the two-
fold time contour of Fig. 2 is given by

$Z=\int\mathcal{D}[\overline{f},f,\overline{c},c,\overline{B},B,\mu]e^{i(S_{0}+S_{int})}$
(71)

with $S_{0}$ the non-interacting action and $S_{int}$ the interacting part

$S_{int}=-\frac{i}{\sqrt{N}}\int_{-\infty}^{+\infty}dt\sum_{\sigma=u,d}\sum_{s=+,-}\sum_{i\alpha}\Big{(}\overline{B}_{i}^{\sigma,s}(t)\overline{c}^{\sigma,s}_{i\alpha}(t)f_{\alpha}^{\sigma,s}(t)+B^{\sigma,s}_{i}(t)\overline{f}^{\sigma,s}_{\alpha}(t)c^{\sigma,s}_{i\alpha}(t)\Big{)}$
(72)

$\sigma=u,d$ denotes the upper and lower folds of the contour and $+,-$ the
forward and backward branch on the corresponding fold. Thermal expectation
values $\langle\cdots\rangle_{\beta}$ can then be represented as averages over
the action $S$ on the two-fold contour $\langle\cdots\rangle_{S}$, like for
instance

$\Big{\langle}\hat{f}\Big{(}t_{1}-i\frac{\beta}{2}\Big{)}\hat{f}^{\dagger}\Big{(}-i\frac{\beta}{2}\Big{)}\hat{f}^{\dagger}(t_{2})\hat{f}(0)\Big{\rangle}_{\beta}=\Big{\langle}f^{d,-}(t_{1})\overline{f}^{d,+}(0)\overline{f}^{u,-}(t_{2})f^{u,+}(0)\Big{\rangle}_{S}\,.$
(73)

The retarded OTOCs, like 70, are more easily represented by using the Keldysh
rotated fields on each time fold $\sigma$

$\displaystyle f_{\alpha}^{\sigma,cl/q}(t)$
$\displaystyle=\frac{f_{\alpha}^{\sigma,+}(t)\pm
f_{\alpha}^{\sigma,-}(t)}{\sqrt{2}},\qquad\qquad\overline{f}_{\alpha}^{\sigma,cl/q}(t)=\frac{\overline{f}_{\alpha}^{\sigma,+}(t)\mp\overline{f}_{\alpha}^{\sigma,-}(t)}{\sqrt{2}}$
(74) $\displaystyle B_{i}^{\sigma,cl/q}(t)$
$\displaystyle=\frac{B_{i}^{\sigma,+}(t)\pm
B_{i}^{\sigma,-}(t)}{\sqrt{2}},\qquad\qquad\overline{B}_{i}^{\sigma,cl/q}(t)=\frac{\overline{B}_{i}^{\sigma,+}(t)\pm\overline{B}_{i}^{\sigma,-}(t)}{\sqrt{2}}$
(75) $\displaystyle c_{i\alpha}^{\sigma,cl/q}(t)$
$\displaystyle=\frac{c_{\alpha}^{\sigma,+}(t)\pm
c_{i\alpha}^{\sigma,-}(t)}{\sqrt{2}},\qquad\qquad\overline{c}_{i\alpha}^{\sigma,cl/q}(t)=\frac{\overline{c}_{i\alpha}^{\sigma,+}(t)\mp\overline{c}_{i\alpha}^{\sigma,-}(t)}{\sqrt{2}}$
(76)

In this basis, the interacting part of the action on the two-folds Keldysh
contour is expressed as

$\begin{split}S_{int}=-\frac{i}{\sqrt{2N}}\int_{-\infty}^{+\infty}dt\sum_{i,\alpha,\sigma}\Big{[}&\overline{B}_{i}^{\sigma,cl}\big{(}\overline{c}_{i\alpha}^{\sigma,cl}f_{\alpha}^{\sigma,cl}+\overline{c}_{i\alpha}^{\sigma,q}f_{\alpha}^{\sigma,q}\big{)}+\overline{B}_{i}^{\sigma,q}\big{(}\overline{c}_{i\alpha}^{\sigma,cl}f_{\alpha}^{\sigma,q}+\overline{c}_{i\alpha}^{\sigma,q}f_{\alpha}^{\sigma,cl}\big{)}\\\
&+B_{i}^{\sigma,cl}\big{(}\overline{f}_{\alpha}^{\sigma,cl}c_{i\alpha}^{\sigma,cl}+\overline{f}_{\alpha}^{\sigma,q}c_{i\alpha}^{\sigma,q}\big{)}+B_{i}^{\sigma,q}\big{(}\overline{f}_{\alpha}^{\sigma,cl}c_{i\alpha}^{\sigma,q}+\overline{f}_{\alpha}^{\sigma,q}c_{i\alpha}^{\sigma,cl}\big{)}\Big{]}\end{split}$
(77)

The intra-fold two-point functions are the same as in the usual single-fold
Keldysh formalism and are identical on the upper and lower folds:

$\mathbf{G}_{f}^{\sigma\sigma}=\begin{pmatrix}\,G^{R}_{f}&\,G^{K}_{f}\\\
0&\,G^{A}_{f}\end{pmatrix}\qquad\mathbf{G}_{c}^{\sigma\sigma}=\begin{pmatrix}\,G^{R}_{c}&\,G^{K}_{c}\\\
0&\,G^{A}_{c}\end{pmatrix}\qquad\mathbf{G}_{B}^{\sigma\sigma}=\begin{pmatrix}\,G^{K}_{B}&\,G^{R}_{B}\\\
\,G^{A}_{B}&0\end{pmatrix}\qquad\sigma=u,d$ (78)

In these notations the matrix elements for a given field $\psi$ are
$[\textbf{G}^{\sigma\sigma^{\prime}}_{\psi}]^{ss^{\prime}}(t,t^{\prime})=-i\xi\langle\psi^{\sigma,s}(t)\,\overline{\psi}^{\sigma^{\prime},s^{\prime}}(t^{\prime})\rangle$
with $\sigma,\sigma^{\prime}=u,d$ and $s,s^{\prime}=cl,q$ and where $\chi=1$
(resp. $-1$) if $\psi$ is fermionic (resp. bosonic). We have omitted the spin
and channel indices to simplify notations. The retarded, advanced and Keldysh
Green’s functions are defined by

$\displaystyle iG_{\psi}^{R}(t,t^{\prime})$
$\displaystyle=\xi\theta(t-t^{\prime})\,\langle[\hat{\psi},\hat{\psi}^{\dagger}(t^{\prime})]_{\xi}\rangle$
(79) $\displaystyle iG_{\psi}^{A}(t,t^{\prime})$
$\displaystyle=-\xi\theta(t^{\prime}-t)\,\langle[\hat{\psi},\hat{\psi}^{\dagger}(t^{\prime})]_{\xi}\rangle$
(80) $\displaystyle iG_{\psi}^{K}(t,t^{\prime})$
$\displaystyle=\xi\,\langle[\hat{\psi},\hat{\psi}^{\dagger}(t^{\prime})]_{-\xi}\rangle$
(81)

where the average is over the thermal density matrix and $[A,B]_{\xi}=AB+\xi
BA$. For the inter-fold Green’s functions, only the Keldysh component is non-
zero

$\mathbf{G}_{f}^{\sigma\overline{\sigma}}=\begin{pmatrix}0&\,G^{K}_{f,\sigma\overline{\sigma}}(t,t^{\prime})\\\
0&0\end{pmatrix},\quad\mathbf{G}_{c}^{\sigma\overline{\sigma}}=\begin{pmatrix}0&\,G^{K}_{c,\sigma\overline{\sigma}}(t,t^{\prime})\\\
0&0\end{pmatrix},\quad\mathbf{G}_{B}^{\sigma\overline{\sigma}}=\begin{pmatrix}\,G^{K}_{B,\sigma\overline{\sigma}}(t,t^{\prime})&0\\\
0&0\end{pmatrix}$ (82)

where we used the notation $\overline{u}=d$ and reciprocally. These are the
so-called Wightman correlators, which are defined as the thermal averages

$iG^{K}_{\psi,ud}(t,t^{\prime})=2\xi\,\Big{\langle}\hat{\psi}\Big{(}t-i\frac{\beta}{2}\Big{)}\hat{\psi}^{\dagger}(t^{\prime})\Big{\rangle},\qquad
iG^{K}_{\psi,du}(t,t^{\prime})=-2\xi\,\Big{\langle}\hat{\psi}^{\dagger}\Big{(}t^{\prime}-i\frac{\beta}{2}\Big{)}\hat{\psi}(t)\Big{\rangle}$
(83)

To write the OTOC 70 as a four-point function on the two-fold contour 2, one
can develop the product of the two anti-commutators and use the identity

$\theta(t)\Big{(}\hat{f}(t)\hat{f}^{\dagger}(0)+\hat{f}^{\dagger}(0)\hat{f}(t)\Big{)}=\frac{1}{2}\Big{[}T\big{(}\hat{f}(t)\hat{f}^{\dagger}(0)\big{)}-\tilde{T}\big{(}\hat{f}(t)\hat{f}^{\dagger}(0)\big{)}+\hat{f}(t)\hat{f}^{\dagger}(0)+\hat{f}^{\dagger}(0)\hat{f}(t)\Big{]}\,,$
(84)

where $T$ and $\tilde{T}$ are the time-ordering and anti-time ordering
operations. Using identities like 73 and the definition of the rotated fields
74, after some rearrangements we finally get

$C_{f}(t_{1},t_{2})=\frac{1}{N^{2}}\sum_{\alpha,\beta=1}^{N}\big{\langle}f_{\alpha}^{d,cl}(t_{1})\overline{f}_{\beta}^{d,cl}(0)\overline{f}_{\alpha}^{u,q}(t_{2})f_{\beta}^{u,q}(0)\big{\rangle}$
(85)

##  Appendix B Real time Green’s function

For bosons and fermions, the real time retarded, advanced, and Wightman two-
point functions are related to the imaginary time Matsubara Green’s function
by analytic continuation

$\displaystyle G^{R}(t)$
$\displaystyle=i\theta(t)\Big{(}G(\tau=it+\epsilon)-G(\tau=it-\epsilon)\Big{)}\,,$
(86) $\displaystyle G^{A}(t)$
$\displaystyle=-i\theta(-t)\Big{(}G(\tau=it+\epsilon)-G(\tau=it-\epsilon)\Big{)}\,,$
(87) $\displaystyle G^{K}_{du}(t)$
$\displaystyle=2iG\Big{(}\tau=it+\frac{\beta}{2}\Big{)}\,,$ (88)
$\displaystyle G^{K}_{ud}(t)$
$\displaystyle=2iG\Big{(}\tau=it-\frac{\beta}{2}\Big{)}\,.$ (89)

By using the conformal expression of the Matsubara Green’s functions 11, one
finds after some simple manipulations for the auxiliary fermions $f$:

$\displaystyle G_{f}^{R}(t)$
$\displaystyle=-2iA_{f}\theta(t)\cos\Big{(}\pi\Delta_{f}-i\frac{\alpha}{2}\Big{)}\frac{e^{i\frac{\alpha}{\beta}t}}{\cosh\big{(}\frac{\alpha}{2}\big{)}}\bigg{(}\frac{\pi}{\beta\sinh\big{(}\frac{\pi
t}{\beta}\big{)}}\bigg{)}^{2\Delta{f}}$ (90) $\displaystyle G_{f}^{A}(t)$
$\displaystyle=2iA_{f}\theta(-t)\cos\Big{(}\pi\Delta_{f}+i\frac{\alpha}{2}\Big{)}\frac{e^{i\frac{\alpha}{\beta}t}}{\cosh\big{(}\frac{\alpha}{2}\big{)}}\bigg{(}\frac{\pi}{\beta\sinh\big{(}-\frac{\pi
t}{\beta}\big{)}}\bigg{)}^{2\Delta{f}}$ (91) $\displaystyle G_{f,du}^{K}(t)$
$\displaystyle=-2iA_{f}\frac{e^{i\frac{\alpha}{\beta}t}}{\cosh\big{(}\frac{\alpha}{2}\big{)}}\bigg{(}\frac{\pi}{\beta\cosh\big{(}\frac{\pi
t}{\beta}\big{)}}\bigg{)}^{2\Delta{f}}$ (92) $\displaystyle G_{f,ud}^{K}(t)$
$\displaystyle=2iA_{f}\frac{e^{i\frac{\alpha}{\beta}t}}{\cosh\big{(}\frac{\alpha}{2}\big{)}}\bigg{(}\frac{\pi}{\beta\cosh\big{(}\frac{\pi
t}{\beta}\big{)}}\bigg{)}^{2\Delta{f}}$ (93)

For the auxiliary bosons $B$, we have

$\displaystyle G_{B}^{R}(t)$
$\displaystyle=-2A_{B}\theta(t)\sin\Big{(}\pi\Delta_{B}-i\frac{\alpha}{2}\Big{)}\frac{e^{i\frac{\alpha}{\beta}t}}{\cosh\big{(}\frac{\alpha}{2}\big{)}}\bigg{(}\frac{\pi}{\beta\sinh\big{(}\frac{\pi
t}{\beta}\big{)}}\bigg{)}^{2\Delta{B}}$ (94) $\displaystyle G_{B}^{A}(t)$
$\displaystyle=-2A_{B}\theta(-t)\sin\Big{(}\pi\Delta_{B}+i\frac{\alpha}{2}\Big{)}\frac{e^{i\frac{\alpha}{\beta}t}}{\cosh\big{(}\frac{\alpha}{2}\big{)}}\bigg{(}\frac{\pi}{\beta\sinh\big{(}-\frac{\pi
t}{\beta}\big{)}}\bigg{)}^{2\Delta{B}}$ (95) $\displaystyle G_{B,du}^{K}(t)$
$\displaystyle=-2iA_{B}\frac{e^{i\frac{\alpha}{\beta}t}}{\cosh\big{(}\frac{\alpha}{2}\big{)}}\bigg{(}\frac{\pi}{\beta\cosh\big{(}\frac{\pi
t}{\beta}\big{)}}\bigg{)}^{2\Delta{B}}$ (96) $\displaystyle G_{B,ud}^{K}(t)$
$\displaystyle=-2iA_{B}\frac{e^{i\frac{\alpha}{\beta}t}}{\cosh\big{(}\frac{\alpha}{2}\big{)}}\bigg{(}\frac{\pi}{\beta\cosh\big{(}\frac{\pi
t}{\beta}\big{)}}\bigg{)}^{2\Delta{B}}$ (97)

Finally, we get for the conduction electrons

$\displaystyle G_{c,du}^{K}(t)$
$\displaystyle=-2i\rho_{0}\frac{\pi}{\beta\cosh\big{(}\frac{\pi
t}{\beta}\big{)}}$ (98) $\displaystyle G_{c,ud}^{K}(t)$
$\displaystyle=2i\rho_{0}\frac{\pi}{\beta\cosh\big{(}\frac{\pi
t}{\beta}\big{)}}$ (99)

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