# Scrambling in the Quantum Lifshitz Model

Eugeniu Plamadeala and Eduardo Fradkin  Department of Physics and Institute
for Condensed Matter Theory, University of Illinois at Urbana-Champaign, 1110
West Green Street, Urbana, Illinois 61801-3080, USA

###### Abstract

We study signatures of chaos in the quantum Lifshitz model through out-of-time
ordered correlators (OTOC) of current operators. This model is a free scalar
field theory with dynamical critical exponent $z=2$. It describes the quantum
phase transition in 2D systems, such as quantum dimer models, between a phase
with an uniform ground state to another one with a spontaneously translation
invariance. At the lowest temperatures the chaotic dynamics are dominated by a
marginally irrelevant operator which induces a temperature dependent stiffness
term. The numerical computations of OTOC exhibit a non-zero Lyapunov exponent
(LE) in a wide range of temperatures and interaction strengths. The LE (in
units of temperature) is a weakly temperature-dependent function; it vanishes
at weak interaction and saturates for strong interaction. The Butterfly
velocity increases monotonically with interaction strength in the studied
region while remaining smaller than the interaction-induced
velocity/stiffness.

##  1 Introduction

Scrambling is the delocalization of initially local quantum information over
the entire system. It implies that information about the initial state cannot
be deduced by any local measurements of the final state. This notion is
closely related the notion of quantum chaos in many-body systems[1]. The so-
called out-of-time-order correlators were invented to capture the failure of
semiclassical methods in a study of superconductivity.[2] More recently it has
been realized that they are a way to quantify chaos and information scrambling
in black holes [3, 4, 5]. Specifically, for generic local Hermitean operators
$W$ and $V$, the signature of chaos is a regime of exponential growth of the
OTOC $C(t)\sim e^{\lambda t}$ after local relaxation has occured but before
the scrambling time when $C(t)\sim 1$. The exponent $\lambda$ is called a
“Lyapunov exponent” by analogy with the Lyapunov exponents of classical
dynamical systems:[6, 7] in both classical and quantum cases the system looses
memory of its initial state exponentially fast. It is generally conjectured
that systems with this behavior are “chaotic”. In the classical setting this
means that the system over very long times eventually explores all its
available phase state with probability 1. Although the precise definition of
chaos is lacking (or rather there is an abundance of them[1]), in the quantum
setting of a chaotic system it is believed that, at times long compared with
the scrambling time, the system becomes described by a density matrix and in
some sense has thermalized.[8] The out-of-time ordered correlator $C(t)$
(OTOC) of two hermitian operators $V$ and $W$ is

$\displaystyle C(t)$ $\displaystyle=\tr\\{\rho[W(t),V]^{\dagger}[W(t),V]\\}$
(1.1)

Numerous works have recently studied quantum chaos both at strong[9, 3] and
weak coupling,[10, 11] in gapped phases, in rational CFTs,[12, 13] and in
strongly disordered phases.[14, 15, 16]

In this paper we will compute the chaos (Lyapunov) exponents and the Butterfly
velocity for a system close to a quantum critical point. In general, this is a
challenging problem since most quantum critical points are associated to a
non-trivial UV fixed point of a quantum field theory. However in 2+1
dimensions there is a quantum critical theory, the quantum Lifshitz model [17]
(qLM), which is a free (compactified) scalar field theory with dynamical
exponent $z=2$. Since this fixed point theory is a free field, it is not
chaotic. In spite of its simplicity, the qLM is a non trivial theory with a
rich spectrum of operators, and has been studied quite extensively.[17, 18,
19] One aspect of this theory that is useful in this context is that it has a
marginally irrelevant operator[18, 20, 21] which, if included in the
Lagrangian, spoils the integrability of the fixed point theory. This is thus a
controlled setting to examine the behavior of the OTOCs of this theory within
a perturbative approach.

The imaginary-time action of the qLM in 2+1-dimensions is:

$\displaystyle S_{0}$ $\displaystyle=\frac{1}{2}\int d\tau
d^{2}x\left(\left(\partial_{\tau}\phi\right)^{2}+\kappa\left(\bigtriangledown^{2}\phi\right)^{2}\right)$
(1.2)

There are several variations of this model: the compactified and non-
compactified real scalar, as well as the complex scalar. The leading
irrelevant operator at all these critical points is the same -
$\left(\bigtriangledown\phi\right)^{4}$, and for this reason we believe the
physics concerning chaos will be qualitatively the same. For simplicity, here
we choose to study the case of the non-compactified real scalar theory.

Since this QCP is described by a quadratic action, it is integrable and cannot
thermalize. Lack of integrability is not sufficient to guarantee
thermalization.[22] The $\left(\bigtriangledown\phi\right)^{4}$ interaction is
meant to both break integrability[23] and lead to thermalization, although we
have no proof it does this. It is marginally irrelevant at the QCP and,
barring $\cos(n\phi)$ terms only allowed (but less relevant at $\kappa\ll 1$)
in the compactified theory, it is the the leading perturbation to the scaling
form of the action. As with other irrelevant operators, its main effect is to
shift the location of the QCP by inducing a renormalized stiffness, the
coefficient of the relevant operator $(\bigtriangledown\phi)^{2}$, whose
renormalized value is tuned to zero at $T=0$ (but not at $T>0$) as the
definition of the QCP. Thus, our approach consists of deforming $S_{0}$ by the
following:

$\displaystyle S^{\prime}$ $\displaystyle=\int d\tau
d^{2}x\left[r_{qc}\left(\bigtriangledown\phi\right)^{2}+\frac{u}{4}\left(\bigtriangledown\phi\right)^{4}\right]$
(1.3)

where the stiffness $r_{qc}=r_{qc}(u)$ is chosen such that we hit the QCP as
$T\rightarrow 0$ keeping the bare value of $u$ fixed. The field $\phi$ is non-
compact.

In order to perform a controlled calculation we choose to consider an
$N$-flavor version of this model:

$\displaystyle{\cal
L}_{QLM,N}=\sum_{a}\frac{1}{2}\left[(\partial_{\tau}\phi_{a})^{2}+r_{qc}(\bigtriangledown\phi_{a})^{2}+\kappa(\bigtriangledown^{2}\phi_{a})^{2}\right]+\frac{u}{4N}\sum_{a,b}(\bigtriangledown\phi_{a})^{2}(\bigtriangledown\phi_{b})^{2}$
(1.4)

Furthermore it is helpful to rewrite the $(\bigtriangledown\phi)^{4}$ term by
introducing an auxiliary field $\sigma$ through a Hubbard-Stratonovich
transformation:

$Z_{E}=\int{\cal D}\phi e^{-S_{QLM,N}[\phi]}=\int{\cal D}\phi{\cal D}\sigma
e^{-S_{N}[\phi,\sigma]}$ (1.5)

In this form the (Euclidean) Lagrangian is

$\displaystyle{\cal
L}_{N}[\phi,\sigma]=\sum_{a}\frac{1}{2}\left[(\partial_{\tau}\phi_{a})^{2}+r_{qc}(\bigtriangledown\phi_{a})^{2}+\kappa(\bigtriangledown^{2}\phi_{a})^{2}\right]-\frac{\sigma^{2}}{4u}+\frac{\sigma}{2\sqrt{N}}\sum_{a}\left(\bigtriangledown\phi_{a}\right)^{2}$
(1.6)

As discussed in Appendix A, at $N=\infty$ we can solve the theory most
naturally by introducting a new coupling:

$r_{\rm
eff}=\frac{\langle\sigma\rangle}{\sqrt{N}}+r_{qc}\quad\textrm{where}\quad
r_{\rm eff}=\frac{uT}{4\pi\kappa}\frac{\ln\left(\frac{\sqrt{\kappa}T}{r_{\rm
eff}}\right)}{1+\frac{u}{16\pi\kappa^{3/2}}\left[\ln\left(\frac{4\kappa\Lambda^{2}}{r_{\rm
eff}}\right)\right]}$ (1.7)

We note that the expectation value of the auxiliary field
$\langle\sigma\rangle$ induces a stiffness (“velocity”) for the $\phi$ field
(instead of a “thermal mass” as in $\phi^{4}$ theory). We now discuss
solutions of the equation above.
At fixed interaction strength $u$ and sufficiently low temperatures the
logarithm dominates the denominator and we obtain (see discussion around
Eqn1.12)

$\displaystyle r_{\rm eff}$ $\displaystyle\approx
4T\sqrt{\kappa}\frac{\ln\ln\left(\frac{\Lambda^{2}}{T\sqrt{\kappa}}\right)}{\ln\left(\frac{\Lambda^{2}}{T\sqrt{\kappa}}\right)}$
(1.8)

which is consistent with our assumptions at low $T$. The marginality of the
$(\bigtriangledown\phi)^{4}$ term has induced log corrections that violate
$z=2$ scaling. Note that $r_{\rm eff}$ appears to be independent of the
interaction strength $u$ at low $T$. This is only true when the approximation
we made in the denominator holds:
$\frac{u}{16\pi\kappa^{3/2}}\left[\ln\left(\frac{4\kappa\Lambda^{2}}{r_{\rm
eff}}\right)\right]\gg 1$. The assumption we began with is clearly valid. We
will call this the low temperature regime.
On the other hand at higher temperatures or small $u$ the opposite is true

$\displaystyle\frac{u}{16\pi\kappa^{3/2}}\left[\ln\left(\frac{4\kappa\Lambda^{2}}{r_{\rm
eff}}\right)\right]\ll 1$ (1.9)

and using similar methods we obtain a different saddle-point equation

$\displaystyle r_{\rm
eff}=\frac{uT}{4\pi\kappa}{\ln\left(\frac{\sqrt{\kappa}T}{r_{\rm
eff}}\right)}\quad\textrm{with solution}\quad r_{\rm
eff}=\sqrt{\kappa}T\left(\frac{u}{4\pi\kappa^{3/2}}\right)\ln\left(\frac{4\pi\kappa^{3/2}}{u}\right)$
(1.10)

We call this the weak interaction regime. We will study both regimes
simultaneously by definig $\alpha(u,T)=r_{\rm eff}(u,T)/T$. Since $\alpha$ has
very weak temperature dependence in both regimes, we will often treat $\alpha$
as an indepedent parameter (in a wide but finite range of temperatures).

The $1/N$ corrections are then captured by the following action (see B) where
the $\lambda$ field is the deviation from the saddle-point value of $\sigma$,
i.e. $\lambda=\sigma-\langle\sigma\rangle_{N=\infty}$,

$\displaystyle{\cal
L}_{\phi,\lambda,N}=\sum_{a}\frac{1}{2}\left[(\partial_{\tau}\phi_{a})^{2}+r_{\rm
eff}(\bigtriangledown\phi_{a})^{2}+\kappa(\bigtriangledown^{2}\phi_{a})^{2}\right]-\frac{\lambda^{2}}{4u}+\frac{\lambda}{2\sqrt{N}}\sum_{a}\left(\bigtriangledown\phi_{a}\right)^{2}$
(1.11)

At leading order in $N$ the $\phi_{a}$ fields decouple from $\lambda$ and each
other, and there is no chaos. Therefore, the chaos (Lyapunov) exponent is at
most of order $1/N$ at large $N$ and we must study $1/N$ corrections using the
theory of Eq.1.11.

The building blocks of our perturbative expansions are the imaginary-time
$\phi_{a}$ propagator

$\displaystyle{\cal G}(\tau,{\bm{x}})\delta_{a,b}$ $\displaystyle=\langle
T_{\tau}\phi_{a}(\tau,{\bm{x}})\phi_{b}(0,0)\rangle$ (1.12)
$\displaystyle{\cal G}(i\omega_{n},{\bm{q}})$
$\displaystyle=\frac{1}{\omega_{n}^{2}+\epsilon_{\bm{q}}^{2}},\quad\textrm{
where }\quad\epsilon_{\bm{q}}^{2}=r_{\rm
eff}(u,T){\bm{q}}^{2}+\kappa{\bm{q}}^{4}$ (1.13)

The retarded $\phi_{a}$ propagator ${\cal G}_{R}(\omega,{\bm{k}})$ is obtained
as usual by analytic continuation $i\omega_{n}\rightarrow\omega+i0$. The
spectral function is

$\displaystyle A(\omega,{\bm{k}})$ $\displaystyle=-2\Im{\cal
G}_{R}(\omega,{\bm{k}})=\frac{\pi}{\epsilon_{\bm{k}}}\left[\delta(\omega-\epsilon_{\bm{k}})-\delta(\omega+\epsilon_{\bm{k}})\right]$
(1.14)

We will also make use of the symmetrized Wightman propagator of the $\phi$
fields, defined as

$\displaystyle{\cal G}_{W}(t,{\bm{x}})\delta_{a,b}$
$\displaystyle=\tr\\{\rho^{1/}\phi_{a}({\bm{x}},t)\rho^{1/2}\phi_{b}(0)\\}$
(1.15)

By manipulations similar to those in Appendix C of Ref.[[11]] the Wightman
propagator can be related to the spectral function through

$\displaystyle{\cal G}_{W}(\omega,{\bm{k}})$
$\displaystyle=\frac{A(\omega,{\bm{k}})}{2\sinh\frac{\beta\omega}{2}}$ (1.16)

We will shortly need the $\phi$ and $\lambda$ propagators dressed to leading
order. We denote the bare $\lambda$ field propagator as ${\cal
G}^{0}_{\lambda}(\tau,{\bm{x}})=-2u$. It picks up a $O(N^{0})$ correction
through its self-energy $\Pi(i\omega_{n},{\bm{q}})$ to give the dressed
$\lambda$ propagator

$\displaystyle{\cal
G}_{\lambda}(i\omega_{n},{\bm{q}})=\frac{1}{\frac{1}{-2u}-\Pi(i\omega_{n},{\bm{q}})}$
(1.17)

from which one can derive the other functions:

the retarded propagator $\displaystyle\quad{\cal
G}_{R,\lambda}(\omega,{\bm{q}})$ $\displaystyle=-{\cal
G}_{\lambda}(i\omega_{n}\rightarrow\omega+i0,{\bm{q}})$ (1.18) the spectral
function $\displaystyle\quad A_{\lambda}(\omega,{\bm{q}})$
$\displaystyle=-2\Im{\cal G}_{R,\lambda}(\omega,{\bm{q}})$ (1.19) the Wightman
propagator $\displaystyle\quad{\cal G}_{W,\lambda}(\omega,{\bm{q}})$
$\displaystyle=\frac{A_{\lambda}(\omega,{\bm{q}})}{2\sinh\frac{\beta\omega}{2}}$
(1.20)

The Feynman rule for the $\phi-\lambda$ vertex follows from

$\displaystyle-{\cal
L}_{E}\propto\frac{1}{2\sqrt{N}}\int_{\omega_{n},\omega_{m}}\int_{{\bm{k}},{\bm{q}}}(-{\bm{k}}\cdot{\bm{q}})\lambda(-i\omega_{n}-i\omega_{m},-{\bm{k}}-{\bm{q}})\phi_{a}(i\omega_{n},{\bm{k}})\phi_{a}(i\omega_{m},{\bm{q}})$
(1.21)

Here and below we use the notation
$\int_{\bm{p}}=\int\frac{d^{2}p}{(2\pi)^{2}}$ and
$\int_{\omega_{n}}=T\sum_{\omega_{n}}$. The vertex then equals
$\frac{-{\bm{k}}_{1}\cdot{\bm{k}}_{2}}{\sqrt{N}}\delta(\sum_{i}{\bm{k}}_{i})$,
where the dotted momenta belong to the $\phi$ fields.

Figure 1: The self-energy correction of the $\lambda$ field at leading order
in $1/N$.

The one-loop self-energy $\Pi(i\omega_{n},{\bm{q}})$ of the $\lambda$ field
comes from the Dyson equation for the field $\lambda$, the Feynman diagram
shown in Fig.1 and equals

$\displaystyle\Pi(i\omega_{n},{\bm{q}})$
$\displaystyle=\frac{T}{2}\sum_{i\nu_{n}}\int_{\bm{k}}^{\Lambda}\frac{\left({\bm{k}}\cdot({\bm{k}}+{\bm{q}})\right)^{2}}{\left(\nu_{n}+\omega_{n}\right)^{2}+\epsilon^{2}_{{\bm{k}}+{\bm{q}}}}\frac{1}{\nu_{n}^{2}+\epsilon^{2}_{\bm{k}}}$
(1.22)

where the factor of $1/2$ comes from combinatorics ($1/2$ from second order of
exponential series, and two factors from vertex; on top there are two ways to
contract the $\phi$’s, and two to contract the $\lambda$’s). In D we discuss
the computation of this object, which must ultimately be done numerically.

Similarly, the one-loop self-energy of $\phi$ is given by the Feynman
diagrams:

$\Sigma(i\omega_{n},{\bm{q}})=$ (1.23)

The second diagram in Eq.(1.23) induces only a real shift of the pole and
plays no role in our calculation.

$\displaystyle\Sigma\left(i\omega_{n},{\bm{q}}\right)$
$\displaystyle=\frac{T}{N}\sum_{i\nu_{n}}\int_{\bm{k}}^{\Lambda}\left({\bm{q}}\cdot({\bm{q}}+{\bm{k}})\right)^{2}{\cal
G}(i\omega_{n}+i\nu_{n},{\bm{q}}+{\bm{k}}){\cal
G}_{\lambda}(i\nu_{n},{\bm{k}})$ (1.24)

where ${\cal G}_{\lambda}$ is already one-loop corrected. This object is
computed in E. Note that the second diagram also contributes at order $1/N$
but as it induces only a real shift of the pole we drop it with the philosophy
that the location of the pole is a physical quantity that we choose to keep
fixed with a suitable counterterm.

##  2 Perturbative expansion of OTOC

Instead of studying the OTOC of Eq.1.1 we will study a closely related object
with the same growth properties. Following the approach employed for matrix
$\phi^{4}$ theory in the large $N$ limit by Stanford[10], we shift half the
fields half-way along the thermal circle.[11] This removes spurious short-
distance divergences of coincident operators, without modifying the
exponential growth. The resulting operators insertions in complex time lie on
the contour shown in Fig. 2.

Figure 2: Complex time contour of the operators insertions of the regulated
OTOC. The blue (red) dot corresponds to the $V(0)$ ($W(t)$) operator.

Specifically we choose to study the following (regulated) OTOC

$\displaystyle C(t_{1}-t_{2},{\bm{x}}_{1}-{\bm{x}}_{2})=$
$\displaystyle\qquad-\frac{1}{N^{2}}\sum_{a,b}{\rm
Tr}\\{\sqrt{\rho}\left[\bigtriangledown\phi_{a}({\bm{x}}_{1},t_{1}),\bigtriangledown\phi_{b}({\bm{x}}_{2},t_{2})\right]\sqrt{\rho}\left[\bigtriangledown\phi_{a}({\bm{x}}_{1},t_{1}),\bigtriangledown\phi_{b}({\bm{x}}_{2},t_{2})\right]\\}$
(2.1)

Note that gradients are dotted into each other within each commutator. The
OTOC can be expanded in powers of the interaction vertex.

We derive the rules in C (they are almost identical to those in Chowdhury and
Swingle [11]), and simply present them here:

  1. 1.

Vertex insertions can occur on either time-fold. Each comes with a factor of
$\frac{-i{\bm{k}}_{1}\cdot{\bm{k}}_{2}}{2\sqrt{N}}\delta_{ab}$ (coming from
$\phi_{a},\phi_{b}$). Contractions inside each time fold represent self-energy
corrections and are counted separately.

  2. 2.

Horizontal lines correspond to retarded propagators $i{\cal G}_{R},i{\cal
G}_{R,\lambda}$, vertical lines to Wightman propagators $G_{W},G_{W,\lambda}$.

  3. 3.

Each line must be directed, left to right or top to down. This determines
whether a momentum/frequency is incoming or outgoing. The sum of incoming must
equal sum of outgoing at each vertex.

Figure 3: The simplest ladder diagram with rungs of type 1,
$C_{1}(\nu,{\bm{q}}=0)$ defined in Eq.2.2.

There are two classes of diagrams then that are most important, both involve
contractions between the different time folds. The first class involve ladder
rungs of $\lambda$ propagators (we will call them type 1). The simplest
example of a type 1 diagram is shown in Fig.3.

Its value at ${\bm{q}}=0$ momentum transfer is

$\displaystyle
C_{1}(\nu,{\bm{q}}=0)=\frac{1}{N^{2}}\int_{{\bm{p}},{\bm{p}}^{\prime}}\left({\bm{p}}\cdot{\bm{p}}^{\prime}\right)^{4}\int_{\omega,\omega^{\prime}}{\cal
G}_{R}({\bm{p}},\omega)$ $\displaystyle{\cal G}_{R}(-{\bm{p}},\nu-\omega){\cal
G}^{W}_{\lambda}({\bm{p}}-{\bm{p}}^{\prime},\omega-\omega^{\prime})$ (2.2)
$\displaystyle\times{\cal G}_{R}({\bm{p}}^{\prime},\omega^{\prime}){\cal
G}_{R}(-{\bm{p}}^{\prime},\nu-\omega^{\prime})$

For clarity and conciseness, in Eq.2.2 we present the expression at
${\bm{q}}=0$. The full expression for $C_{1}(\nu,{\bm{q}})$ is worked out in
C.

An example of the second class of diagrams is shown in Fig.4. These involve
ladder rungs of $\phi$ propagators (we will call them type 2). The expression
for the simplest such diagram is

$\displaystyle
C_{2}(\nu,{\bm{q}}=0)=\frac{1}{N^{2}}\int_{{\bm{p}},{\bm{p}}^{\prime}}\left({\bm{p}}\cdot{\bm{p}}^{\prime}\right)^{2}\int_{\omega,\omega^{\prime}}\,$
$\displaystyle{\cal G}_{R}(\nu-\omega,-{\bm{p}}){\cal
G}_{R}(\omega,{\bm{p}}){\cal G}_{\rm
eff}(\nu,{\bm{q}};\omega^{\prime},\omega,{\bm{p}}^{\prime},{\bm{p}})$ (2.3)
$\displaystyle\times{\cal G}_{R}(\nu-\omega^{\prime},-{\bm{p}}^{\prime}){\cal
G}_{R}(\omega^{\prime},{\bm{p}}^{\prime})$

where ${\cal G}_{\rm eff}$ above represents the loop of four propagators that
form the inner rectangle in Fig 4. It is given by

$\displaystyle{\cal G}_{\rm
eff}(\nu,{\bm{q}}=0;\omega^{\prime},\omega,{\bm{p}}^{\prime},{\bm{p}})=\int_{\omega^{\prime\prime},{\bm{p}}^{\prime\prime}}$
$\displaystyle\left({\bm{p}}\cdot({\bm{p}}^{\prime\prime}-{\bm{p}})\right)^{2}\left({\bm{p}}^{\prime}\cdot({\bm{p}}^{\prime}-{\bm{p}}^{\prime\prime})\right)^{2}$
(2.4) $\displaystyle\times{\cal
G}_{W}(\omega^{\prime\prime}-\omega,{\bm{p}}^{\prime\prime}-{\bm{p}}){\cal
G}_{W}(\omega^{\prime}-\omega^{\prime\prime},{\bm{p}}^{\prime}-{\bm{p}}^{\prime\prime})$
$\displaystyle\times{\cal
G}_{R,\lambda}(\nu-\omega^{\prime\prime},-{\bm{p}}^{\prime\prime}){\cal
G}_{R,\lambda}(\omega^{\prime\prime},{\bm{p}}^{\prime\prime})$

The functions ${\cal G}_{W}$ and ${\cal G}_{R,\lambda}$ are defined in Eqns
1.16 and 1.18. In the sequel, we will typically suppress the $\nu$ and
${\bm{q}}$ dependence of ${\cal G}_{\rm eff}$ and consider it implicit.

Figure 4: The simplest ladder diagram with rungs of type 2,
$C_{2}(\nu,{\bm{q}}=0)$ defined in Eq.2.3.

At late times both $C_{1}$ and $C_{2}$ contribute at order $t/N^{2}$. A
general diagram with $n_{1}$ rungs of type 1 and $n_{2}$ rungs of type 2 is of
order $1/N^{n_{1}+n_{2}+1}$, so we expect their sum to be of order $1/N$. We
ignored crossed diagrams because they are parametrically smaller than the
ladder diagrams (see C.1).

##  3 Computation of the OTOC and the Bethe-Salpeter Equation

Before we compute the OTOC, ${\cal C}(t)=\int d^{2}x\;{\cal C}(t,{\bm{x}})$,
we will briefly describe the structure of the calculation.
Instead of aiming directly for ${\cal C}(t,{\bm{x}})$ we will obtain ${\cal
C}(\nu,{\bm{q}})$, which is its Laplace transform in frequency and the Fourier
transform in momentum. The ${\bm{q}}=0$ value is sufficient to compute the
Lyapunov exponent $\lambda_{L}$, while the ${\bm{q}}\neq 0$ value is necessary
to obtain the Butterfly velocity (and a semi-independent verification of
$\lambda_{L}$).

The function ${\cal C}(\nu,{\bm{q}})$ has a diagrammatic expansion in powers
of $1/N$. The first two terms are $C_{1}(\nu,{\bm{q}})$ and
$C_{2}(\nu,{\bm{q}})$ previously defined in Eqns 2.2 and 2.3 corresponding to
diagrams with rungs of type 1 and 2 respectively. We will set up and solve the
Bethe-Salpeter equation for ${\cal C}(\nu,{\bm{q}})$ in terms of an auxiliary
function $g(\nu,{\bm{q}};\omega,{\bm{p}})$ defined by

$\displaystyle{\cal C}(\nu,{\bm{q}})$
$\displaystyle=\frac{1}{N}\int\frac{d\omega}{2\pi}\int_{\bm{p}}{\bm{p}}^{2}g(\nu,{\bm{q}};\omega,{\bm{p}}),$
(3.1)

The Bethe-Salpeter equation will be finally recast to an integral equation,
which we will need to solve numerically. To do that we will discretize and
turn it into a matrix eigenvalue problem. Finally the eigenvalue with the
largest positive real part will dominate the exponential growth of the OTOC in
time.
We study the ${\bm{q}}=0$ case first. We subsequently suppress the ${\bm{q}}$
index (when zero) and write ${\cal C}(\nu)$ in place of ${\cal
C}(\nu,{\bm{q}})$. To compute the OTOC ${\cal C}(\nu)$ we now proceed with an
evaluation of the sum of ladder diagrams of Fig.5.

Figure 5: Top left is a general uncrossed diagram. The bottom represents the
complete sum ${\cal C}(\nu)$.

To this end we set up a Bethe-Salpeter equation, which diagrammatically is
represented in figure Fig.6.

Figure 6: A diagrammatic definition of the Bethe-Salpeter equation for ${\cal
C}(t)$.

We write down the value of the first few diagrams with rungs of type 1:

$\displaystyle{\cal C}(\nu)$
$\displaystyle=\frac{1}{N}\int_{\omega,{\bm{p}}}{\bm{p}}^{4}{\cal
G}_{R}(\nu-\omega,-{\bm{p}}){\cal G}_{R}(\omega,{\bm{p}})$
$\displaystyle+\frac{1}{N^{2}}\int_{\omega,\omega^{\prime},{\bm{p}},{\bm{p}}^{\prime}}{\bm{p}}^{2}{\cal
G}_{R}(\nu-\omega,-{\bm{p}}){\cal
G}_{R}(\omega,{\bm{p}})\left({\bm{p}}\cdot{\bm{p}}^{\prime}\right)^{2}{\cal
G}_{W,\lambda}(\omega^{\prime}-\omega,{\bm{p}}^{\prime}-{\bm{p}})$
$\displaystyle\qquad\qquad\times{\cal
G}_{R}(\nu-\omega^{\prime},-{\bm{p}}^{\prime}){\cal
G}_{R}(\omega^{\prime},{\bm{p}}^{\prime}){\bm{p}}^{\prime 2}$ (3.3)
$\displaystyle+\frac{1}{N^{3}}\int{\bm{p}}^{2}\left({\bm{p}}\cdot{\bm{p}}^{\prime}\right)^{2}\left({\bm{p}}^{\prime}\cdot{\bm{p}}^{\prime\prime}\right)^{2}({\bm{p}}^{\prime\prime})^{2}G^{2}_{\bm{p}}{\cal
G}_{W,\lambda}G^{2}_{{\bm{p}}^{\prime}}{\cal
G}_{W,\lambda}G^{2}_{{\bm{p}}^{\prime\prime}}+\cdots$ (3.4)

where the last line is written schematically. This allows us to infer the
Bethe-Salpeter equation for ${\cal C}(\nu)$, written in terms of the auxiliary
function $g$:

$\displaystyle g(\nu;\omega,{\bm{p}})=$ $\displaystyle{\cal
G}_{R}(\nu-\omega,-{\bm{p}}){\cal G}_{R}(\omega,{\bm{p}})$ (3.5)
$\displaystyle\times\left[{\bm{p}}^{2}+\frac{1}{N}\int_{\omega^{\prime},{\bm{p}}^{\prime}}\left({\bm{p}}\cdot{\bm{p}}^{\prime}\right)^{2}{\cal
G}_{W,\lambda}(\omega^{\prime}-\omega,{\bm{p}}^{\prime}-{\bm{p}})g(\nu;\omega^{\prime},{\bm{p}}^{\prime})\right]$

We can also include the type 2 rungs by making the following replacement in
the expression given above

$\displaystyle\left({\bm{p}}\cdot{\bm{p}}^{\prime}\right)^{2}{\cal
G}_{W,\lambda}(\omega^{\prime}-\omega,{\bm{p}}^{\prime}-{\bm{p}})\rightarrow\left({\bm{p}}\cdot{\bm{p}}^{\prime}\right)^{2}{\cal
G}_{W,\lambda}(\omega^{\prime}-\omega,{\bm{p}}^{\prime}-{\bm{p}})+{\cal
G}_{\rm
eff}(\nu,{\bm{q}}=0;\omega^{\prime},\omega;{\bm{p}}^{\prime},{\bm{p}})$

where ${\cal G}_{\rm eff}$ is given by Eq.2.4.

To make progress and extract the leading time-dependence we must simplify the
object ${\cal G}_{R}(\nu-\omega,-{\bm{p}}){\cal G}_{R}(\omega,{\bm{p}})$. We
do this by the following replacement:

$\displaystyle{\cal G}_{R}(\nu-\omega,-{\bm{p}}){\cal G}_{R}(\omega,{\bm{p}})$
$\displaystyle\rightarrow\frac{\pi
i}{\epsilon_{\bm{p}}}\frac{\delta(\omega^{2}-\epsilon_{\bm{p}}^{2})}{\nu+i2\Gamma_{\bm{p}}}$
(3.6)

As explained in C.2 this replacement achieves three things: it simplifies the
expression, it captures the pole structure that leads to leading late-time
dependence in ${\cal C}(t)$, and finally it also captures the scattering rate,
$\Gamma_{\bm{p}}$, of the $\phi$ fields (see discussion around Eqn.C.2).

Since the product of the two Green functions ${\cal G}_{R}$ has an on-shell
delta function condition, we choose the following ansatz

$\displaystyle g(\nu;\omega,{\bm{p}})$
$\displaystyle=g(\nu;{\bm{p}})\delta(\omega^{2}-\epsilon_{\bm{p}}^{2})$ (3.7)

This simplifies the Bethe-Salpeter equation 3.5 to the following

$\displaystyle-i\nu\,g(\nu;{\bm{p}})$
$\displaystyle=\frac{\pi{\bm{p}}^{2}}{\epsilon_{\bm{p}}}+\frac{1}{N}\int_{{\bm{l}}}\left({\cal
R}_{1}({\bm{l}},{\bm{p}})+{\cal
R}_{2}({\bm{l}},{\bm{p}})-2N\Gamma_{\bm{p}}\,\delta^{(2)}({\bm{l}}-{\bm{p}})\right)g(\nu;{\bm{l}})$
(3.8)

where we introduce auxiliary functions ${\cal R}_{1},{\cal R}_{2}$ to
compactify the expression. They are given by

$\displaystyle{\cal R}_{1}({\bm{l}},{\bm{p}})$ $\displaystyle={\cal
R}_{1,+}({\bm{l}},{\bm{p}})+{\cal R}_{1,-}({\bm{l}},{\bm{p}})$ (3.9)
$\displaystyle{\cal R}_{1,\pm}({\bm{l}},{\bm{p}})$
$\displaystyle=\frac{\left({\bm{l}}\cdot{\bm{p}}\right)^{2}}{4\epsilon_{\bm{l}}\epsilon_{\bm{p}}}{{\cal
G}_{W,\lambda}(\pm\epsilon_{\bm{l}}-\epsilon_{\bm{p}},{\bm{l}}-{\bm{p}})}$
(3.10) $\displaystyle{\cal R}_{2}({\bm{l}},{\bm{p}})$ $\displaystyle={\cal
R}_{2,+}({\bm{l}},{\bm{p}})+{\cal R}_{2,-}({\bm{l}},{\bm{p}})$ (3.11)
$\displaystyle{\cal R}_{2,\pm}({\bm{l}},{\bm{p}})$
$\displaystyle=\frac{1}{4\epsilon_{\bm{l}}\epsilon_{\bm{p}}}{\cal G}_{\rm
eff}\left(\pm\epsilon_{\bm{l}},\epsilon_{\bm{p}};{\bm{l}}-{\bm{p}},\frac{{\bm{l}}+{\bm{p}}}{2}\right)$
(3.12)

Note that ${\cal R}_{2,\pm}({\bm{l}},{\bm{p}})$ actually depends on
$\nu,\kappa,u,T$ in addition to ${\bm{l}},{\bm{p}}$. Similarly ${\cal R}_{1}$
depends on $\kappa,u,T$. We convinced ourselves numerically that indeed
$R_{1}({\bm{l}},{\bm{l}})$ vanishes. Both $R_{1}({\bm{l}},{\bm{p}})$ and
$R_{2}({\bm{l}},{\bm{p}})$ also vanish when either
${\bm{l}},{\bm{p}}\rightarrow 0$ due to the factors of momentum in the
numerator.

##  4 Lyapunov Exponent and Butterfly velocity

We note that 3.8 can be made dimensionless by the replacement
$\nu\rightarrow\nu T,{\bm{p}}\rightarrow{\bm{p}}\sqrt{T},r_{\rm
eff}(u,T)=\alpha(u,T)T$, with $[g]=[1/T]=2$. Then, by dimensional
considerations and inspection of $r_{\rm eff}$, the Lyapunov exponent
$\lambda_{L}$ will take the form

$\lambda_{L}=n(u,\alpha(u,T))\frac{T}{N}$ (4.1)

where $n(u,\alpha(u,T))$ is a dimensionless function with the following
behaviors

  * •

In the weak interaction regime $n(u,T)=n(u)$ is temperature independent with
$n(u=0)=0$ (since at $u=0$ the model becomes integrable).

  * •

In the low temperature regime, $n(u,T)$ inherits a weak temperature dependence
from $\alpha(T)$.

We solve the integral equation of Eq.3.8 numerically. Given the eigenfunction
$g_{\eta}(\nu;{\bm{p}})$ of the right-hand-side integral with eigenvalue
$\eta$ we obtain the following final form for 3.8:

$\displaystyle-i\nu\,g_{\eta}(\nu;{\bm{p}})$
$\displaystyle=\frac{\pi{\bm{p}}^{2}}{\epsilon_{\bm{p}}}+\frac{\eta}{N}g_{\eta}(\nu;{\bm{p}})$
(4.2)

All the eigenvalues $\eta$ we obtained numerically were complex; most had a
negative real part which leads to exponential decay. We typically found only a
handful of eigenvalues with a positive real part, and it is these that lead to
exponential growth of chaotic correlations. An inverse Laplace transform
${\cal C}(t)=\int_{\nu}e^{-i\nu t}{\cal C}(\nu)$ then gives

$\displaystyle{\cal C}(t)\sim e^{\eta tT/N}$ (4.3)

which allows us to identify the eigenvalue $\eta$ as the dimensionless
function $n(u,T)$ introduced earlier in Eq.(4.1).

We remark that while maximal chaos is often associated with the absence of
“quasiparticles”[24, 25] (so presumably large scattering rate $\Gamma_{p}$),
in equation 3.8 the scattering rate appears to decrease the chaos exponent in
an apparent contradiction. There is however no contradiction as ${\cal
R}_{1}({\bm{l}},{\bm{p}})$ which is present in Eq.3.8 and contributes to chaos
also contributes to $\Gamma_{\bm{p}}$ (see Eq.5.5).

Despite its $z=2$ scaling, the quantum Lifshitz critical point has apriori a
velocity scale $\sqrt{\kappa\Lambda}$. This is consistent with Lieb-Robinson
type bounds that show the existence of a light cone even in non-relativistic
systems.[26] This is explicitly a UV sensitive quantity. As we argued earlier
a “thermal velocity”, $v=\sqrt{r(T)}\sim\sqrt{\alpha(u,T)T}$, is induced in
our model close to the QCP.

There is growing evidence that OTOCs like ours propagate ballistically in
systems that exhibit a non-zero chaos exponent.[27, 28]. Roberts and
Swingle[29] have argued that the associated velocity is an effective Lieb-
Robinson velocity in the IR, and is UV insensitive (although this depends on
the considered operators, see supplemental material of Ref [29]). Multiple
calculations support this conjecture[30].

In the qLM we find numerical evidence that $C(t,{\bm{k}})$ grows exponentially
with an exponent

$\lambda({\bm{k}})=\lambda_{L}-D_{L}{\bm{k}}^{2}+\cdots$ (4.4)

where $\lambda_{L}$ denotes the Lyapunov exponent. With some additional
assumptions, that will be spelled out later, it follows that

$\displaystyle C(t,{\bm{x}})$
$\displaystyle\sim\int_{\bm{k}}e^{i{\bm{k}}\cdot{\bm{x}}}e^{\left(\lambda_{L}-D_{L}{\bm{k}}^{2}\right)t}\sim
e^{\lambda_{L}t-\frac{{\bm{x}}^{2}}{4D_{L}t}}$ (4.5)

and the wavefront propagates ballistically with (by definition) Butterfly
velocity $v_{B}=\sqrt{4D_{L}\lambda_{L}}$.

Mirroring the discussion for ${\cal C}(t)$, for ${\bm{q}}\neq 0$ we define the
auxiliary function $g(\nu,{\bm{q}};\omega,{\bm{p}})$

$\displaystyle{\cal C}(\nu)$
$\displaystyle=\frac{1}{N}\int\frac{d\omega}{2\pi}\int_{\bm{p}}{\bm{p}}\cdot({\bm{p}}-{\bm{q}})g(\nu,{\bm{q}};\omega,{\bm{p}})$
(4.6)

and obtain the new Bethe-Salpeter equation for Type 1 ladders:

$\displaystyle g(\nu,{\bm{q}};\omega,{\bm{p}})={\cal
G}_{R}(\nu-\omega,{\bm{q}}-{\bm{p}}){\cal G}_{R}(\omega,{\bm{p}})$ (4.7)
$\displaystyle\times\left[{\bm{p}}\cdot({\bm{p}}-{\bm{q}})+\frac{1}{N}\int_{\omega^{\prime},{\bm{p}}^{\prime}}\left({\bm{p}}\cdot{\bm{p}}^{\prime}\right)({\bm{q}}-{\bm{p}})\cdot({\bm{q}}-{\bm{p}}^{\prime}){\cal
G}_{W,\lambda}(\omega^{\prime}-\omega,{\bm{p}}^{\prime}-{\bm{p}})g(\nu,{\bm{q}};\omega^{\prime},{\bm{p}}^{\prime})\right]$

The Type 2 rungs are included by shifting

$\displaystyle\left({\bm{p}}\cdot{\bm{p}}^{\prime}\right)$
$\displaystyle({\bm{q}}-{\bm{p}})\cdot({\bm{q}}-{\bm{p}}^{\prime}){\cal
G}_{W,\lambda}(\omega^{\prime}-\omega,{\bm{p}}^{\prime}-{\bm{p}})$ (4.8)
$\displaystyle\rightarrow\left({\bm{p}}\cdot{\bm{p}}^{\prime}\right)({\bm{q}}-{\bm{p}})\cdot({\bm{q}}-{\bm{p}}^{\prime}){\cal
G}_{W,\lambda}(\omega^{\prime}-\omega,{\bm{p}}^{\prime}-{\bm{p}})+{\cal
G}_{\rm eff}(\nu,{\bm{q}};\omega,\omega^{\prime},{\bm{p}},{\bm{p}}^{\prime})$

where

$\displaystyle{\cal G}_{\rm
eff}(\nu,{\bm{q}};\omega,\omega^{\prime},{\bm{p}},{\bm{p}}^{\prime})=$
$\displaystyle\quad\int_{\omega^{\prime\prime},{\bm{p}}^{\prime\prime}}({\bm{q}}-{\bm{p}})\cdot({\bm{p}}^{\prime\prime}-{\bm{p}}){\bm{p}}\cdot({\bm{p}}^{\prime\prime}-{\bm{p}})({\bm{q}}-{\bm{p}}^{\prime})\cdot({\bm{p}}^{\prime}-{\bm{p}}^{\prime\prime})({\bm{p}}^{\prime}-{\bm{p}}^{\prime\prime})\cdot{\bm{p}}^{\prime}$
$\displaystyle\quad\times{\cal
G}_{W}(\omega^{\prime\prime}-\omega,{\bm{p}}^{\prime\prime}-{\bm{p}}){\cal
G}_{W}(\omega^{\prime}-\omega^{\prime\prime},{\bm{p}}^{\prime}-{\bm{p}}^{\prime\prime}){\cal
G}_{R,\lambda}(\nu-\omega^{\prime\prime},{\bm{q}}-{\bm{p}}^{\prime\prime}){\cal
G}_{R,\lambda}(\omega^{\prime\prime},{\bm{p}}^{\prime\prime})$ (4.9)

A simplified expression for ${\cal G}_{\rm
eff}(\nu,{\bm{q}};\omega,\omega^{\prime},{\bm{p}},{\bm{p}}^{\prime})$ that is
more amenable to numerical evaluation is provided in Eqn C.3.

Adopting the ansatz

$\displaystyle g(\nu,{\bm{q}};\omega,{\bm{p}})$
$\displaystyle=\frac{1}{2\epsilon_{\bm{p}}}\left[g_{+}({\bm{p}})\delta(\omega-\epsilon_{\bm{p}})+g_{-}({\bm{p}})\delta(\omega+\epsilon_{\bm{p}})\right]$
(4.10)

we arrive at the following system of integral equations

$\displaystyle\left(-i\nu+i\delta\epsilon_{\bm{q}}+2\Gamma_{\bm{p}}\right)g_{+}({\bm{p}})=$
$\displaystyle\frac{\pi}{\epsilon_{{\bm{q}}-{\bm{p}}}}{\bm{p}}\cdot({\bm{q}}-{\bm{p}})$
(4.11) $\displaystyle+\frac{1}{N}\int_{{\bm{p}}^{\prime}}{\cal
R}_{+}({\bm{p}}^{\prime},{\bm{p}})g_{+}({\bm{p}}^{\prime})+{\cal
R}_{-}({\bm{p}}^{\prime},{\bm{p}})g_{-}({\bm{p}}^{\prime})$
$\displaystyle\left(-i\nu-i\delta\epsilon_{\bm{q}}+2\Gamma_{\bm{p}}\right)g_{-}({\bm{p}})=$
$\displaystyle\frac{\pi}{\epsilon_{{\bm{q}}-{\bm{p}}}}{\bm{p}}\cdot({\bm{q}}-{\bm{p}})$
(4.12) $\displaystyle+\frac{1}{N}\int_{{\bm{p}}^{\prime}}{\cal
R}_{-}({\bm{p}}^{\prime},{\bm{p}})g_{+}({\bm{p}}^{\prime})+{\cal
R}_{+}({\bm{p}}^{\prime},{\bm{p}})g_{-}({\bm{p}}^{\prime})$

The integral kernels are almost identical to the earlier versions:

$\displaystyle{\cal R}_{\pm}(\nu,{\bm{q}};{\bm{p}}^{\prime},{\bm{p}})$
$\displaystyle=R_{1,\pm}(\nu,{\bm{q}};{\bm{p}}^{\prime},{\bm{p}})+R_{2,\pm}(\nu,{\bm{q}};{\bm{p}}^{\prime},{\bm{p}})$
(4.13) $\displaystyle{\cal R}_{1,\pm}(\nu,{\bm{q}};{\bm{l}},{\bm{p}})$
$\displaystyle=\frac{({\bm{q}}-{\bm{l}})\cdot({\bm{q}}-{\bm{p}})\,\,{\bm{l}}\cdot{\bm{p}}}{4\epsilon_{\bm{l}}\epsilon_{\bm{p}}}{\cal
G}_{W,\lambda}(\pm\epsilon_{\bm{l}}-\epsilon_{\bm{p}},{\bm{l}}-{\bm{p}})$
(4.14) $\displaystyle{\cal R}_{2,\pm}(\nu,{\bm{q}};{\bm{l}},{\bm{p}})$
$\displaystyle=\frac{1}{4\epsilon_{\bm{l}}\epsilon_{\bm{p}}}{\cal G}_{\rm
eff}\left(\nu,{\bm{q}};\pm\epsilon_{\bm{l}},\epsilon_{\bm{p}};{\bm{l}}-{\bm{p}},\frac{{\bm{l}}+{\bm{p}}}{2}\right)$
(4.15)

We have made the following approximations in reaching the above expressions:
$\epsilon_{{\bm{p}}^{\prime}}\epsilon_{{\bm{q}}-{\bm{p}}}\approx\epsilon_{{\bm{p}}^{\prime}}\epsilon_{\bm{p}}$,
$\epsilon_{\bm{p}}-\epsilon_{{\bm{q}}-{\bm{p}}}\equiv\delta\epsilon_{\bm{q}}\approx{\bm{q}}\cdot
v_{\bm{p}}$. We also used the fact that the Wightman functions are even in
frequency and rotationally invariant. As noted in Ref [11, 10], Eq.4.11 has
the form of a kinetic theory Boltzmann equation. This analogy inspires the
ansatz that “particle” density at $-{\bm{p}}$ is equals “hole” density at
${\bm{p}}$, that is $g_{+}(-{\bm{p}})=g_{-}({\bm{p}})$. This finally allows us
to decouple the two equations; we arrive at the following (dropping the
homogeneous term) final expression:

$\displaystyle\left(-i\nu+i{\bm{q}}\cdot v_{\bm{p}}\right)g_{+}({\bm{p}})=$
$\displaystyle\qquad\qquad\qquad\frac{1}{N}\int_{{\bm{p}}^{\prime}}\left[{\cal
R}_{+}({\bm{p}}^{\prime},{\bm{p}})+{\cal
R}_{-}(-{\bm{p}}^{\prime},{\bm{p}})-2N\Gamma_{{\bm{p}}}\delta^{(2)}({\bm{p}}^{\prime}-{\bm{p}})\right]g_{+}({\bm{p}}^{\prime})$
(4.16)

Our identification of the Butterfly velocity following Eqn 4.5 holds assuming
the eigenvectors $g_{+}({\bm{p}})$ of the equation above are weakly ${\bm{q}}$
dependent.

##  5 Numerical Results

We now describe the solution of 3.8 and 4. It is achieved by discretizing the
momentum integral on the right hand side into a sum. This turns the integral
kernel into a matrix and the function $g(\nu,{\bm{q}};{\bm{p}}^{\prime})$ into
a vector. We diagonalize this matrix. Using eigenfunctions we simplify to the
form 4.2 (or similar for ${\bm{q}}\neq 0$). Finally an inverse Laplace
transform takes us to real time.

The numerical calculations were carried out as described below for
$\alpha=10^{-1},10^{-2},10^{-5}$:

  * •

We were able to eliminate the Dirac delta functions present in
$\Im\,\Pi_{\lambda,R}(\nu,{\bm{q}})$ by solving the quartic polynomial in the
argument. This produced four integrals over two-dimensional momentum that
nevertheless still vanish in some regions.

  * •

We computed $\Im\,\Pi_{\lambda,R}(\nu,{\bm{q}})$ at fixed ${\bm{q}}$ for many
values of $\nu$, particularly densely around
$\nu_{*}({\bm{q}})=\sqrt{\alpha{\bm{q}}^{2}+\kappa{\bm{q}}^{4}}$.

  * •

We rescaled the $\nu$ coordinate of these slices by $\nu_{*}({\bm{q}})$ and
interpolated linearly between them (in the ${\bm{q}}$ direction).

  * •

We performed the Kramers-Kronig transform as outlined in D.

  * •

We computed $A_{\lambda}$ from ${\cal G}_{R,\lambda}$ and from this obtained
the ${\cal R}_{1}$ functions.

  * •

To obtain ${\cal R}_{2}$ we performed yet another momentum integral with the
same delta function solution as earlier. This function depends on four
independent real parameters (frequency, two momentum magnitudes, a relative
angle) and therefore cannot be precomputed and stored for later fast lookup.

  * •

For the ${\bm{q}}=0$ case we discretized the integral equation of Eq.3.8 and
turned it into an eigenvalue equation on an adaptively spaced grid of at least
60 x 60, or as fine as necessary to achieve grid spacing independence of the
result.

  * •

For ${\bm{q}}\neq 0$ due to the lack of rotational invariance of Eq. 4 we are
forced to use the separate components of the momentum, and which lead to much
smaller grid sizes around 25 x 25 (still leading to matrices of roughly 8000 x
8000).

The total duration of our calculations was several thousand CPU hours. All our
computations were performed with $\kappa=1$. Furthermore, we chose to study
OTO correlations only of current operators, $\bigtriangledown\phi$. The main
result of this paper is the existence of a finite chaotic Lyapunov exponent
for such operators; we express it in the form $\lambda_{L}=n(u,T)\frac{T}{N}$,
where $n(u,T)$ is a dimensionless function that is either weakly dependent on
or completely independent of temperature. We estimate an error margin of 50%
on our results for $n(u,T)$. Recall that the ratio $\alpha=r_{\rm eff}(u,T)/T$
is uniquely determined by $T,\kappa,u,\Lambda$. However, in light of its very
weak dependence on temperature, we consider it an effective independent
parameter in wide temperature ranges.

Figure 7: Lyapunov exponent at different temperatures as a function of
interaction strength.

In the low temperature regime our data (Fig.7) suggests that $\lambda_{L}$ is
weakly dependent on $\alpha$ as it goes to zero, which corresponds to
decreasing temperature. At large values of the interaction strength there does
not appear to be a monotonic dependence of the Lyapunov exponent on $\alpha$;
our numerical error estimates preclude us from resolving this.

This same figure can be interpreted instead in the weak interaction regime
when $\alpha\sim u\log(u)$ is temperature independent.

Figure 8: Lyapunov exponent at different temperatures at small interaction
strength. The non-monotonicities are due to the inaccuracy of our numerical
algorithm.

As expected chaos vanishes as $u\rightarrow 0$. The saturation of the chaotic
exponent at large interaction strenghts can be understood from the dressed
propagator of the auxiliary field $\lambda$, (suppressing indices) of the form

$\displaystyle{\cal G}_{\lambda}$ $\displaystyle=\frac{1}{\frac{-1}{2u}-\Pi}$
(5.1)

This is the only place in the calculation where the interaction strength
enters explicitly in this regime (the self-energy $\Pi$ is interaction
independent). The saturation of the chaos exponent then follows from the fact
that fluctuations, captured by $\Pi$, dominate the bare propagator. An
intuitive way to understand the saturation is to realize that as
$u\rightarrow\infty$, field configurations with non-zero values of the current
$\bigtriangledown\phi$ become energetically suppressed and therefore the
dynamics become more constrained.

Figure 9: Lyapunov exponent $\lambda({\bm{k}})$ vs momentum, in
$C(t,{\bm{k}})\sim\int d{\bm{k}}\exp\left(\lambda({\bm{k}})t\right)$, at
$\alpha=0.1$ and $u=1$. The fit gives
$\lambda({\bm{k}})=0.054-0.008k-0.045k^{2}$, from which it follows that
$v_{B}=2\sqrt{0.054\times 0.045}\sqrt{T}=0.1\sqrt{T}$.

The two independent ways of computing $\lambda_{L}$ (with ${\bm{q}}=0$ and
${\bm{q}}\geq 0$) were within 50% of each other. Based on this we estimate our
Butterfly velocities to have an error margin of about 50%. It is worth noting
that the “thermal velocity” $\sqrt{r(T)}=\sqrt{\alpha(T)T}$ equals
$\sqrt{0.1T}\approx 0.3\sqrt{T}$, which is larger than largest value of the
Butterfly velocity of $0.23\sqrt{T}$ we obtained at $\alpha=1,u=2$. We did not
compute the Butterfly velocity for values other than $\alpha=0.1$ because the
computations required more accuracy for small values of $\alpha$ and we
estimated their running time to be impractically long.

Figure 10: Butterfly velocity as a function of interaction strength at
$\alpha=0.1$.

##  6 Conclusions

We performed a perturbative calculation of a out-of-time-order correlator of
current operators in the non-compact quantum Lifshitz model. We extracted the
numerical values of the Lyapunov exponent and Butterfly velocity for a wide
range of temperatures and/or interaction strengths. Our results indicate that
the $(\bigtriangledown\phi)^{4}$ term is sufficient to generate chaos and is
its dominant cause at sufficiently low energies. We observe that the Lyapunov
exponent has a monotonic dependence on the interaction strength. In the small
temperature regime it saturates at large values of interaction (which
correspons to the non-linear sigma model version of the quantum Lifshitz). In
the weak interaction regime the Lyapunov exponent vanishes as the interaction
strength vanishes. It is expected that for sufficiently generic interactions
the Lyapunov exponent does not depend on the choice of operators in the OTO
correlator. It would be interesting to verify this for our model, and for
vertex operators in the compact version of qLM.

We thank Yuxuan Wang, Hart Goldman, Cristian Gaidau, Thomas Scaffidi, and Juan
Maldacena for discussions. We also thank Debanjan Chowdhury and Brian Swingle
for making available their MATLAB code for the O(N) model [11]. This work was
supported in part by the Gordon and Betty Moore Foundation EPiQS Initiative
through Grant No. GBMF4305 (EP) and by the National Science Foundation through
the grant DMR 1725401 (EF).

##  Appendix A Saddle-point solution at $N=\infty$

We rewrite Eq.1.11 in a form convenient for integrating out $\phi$

$\displaystyle S_{\phi,\lambda}=\frac{1}{2}\int
d\tau_{1}d^{2}{\bm{r}}_{1}d\tau_{2}d^{2}{\bm{r}}_{2}$
$\displaystyle\phi(\tau_{1},x_{1})G^{-1}(\tau_{1},{\bm{r}}_{1};\tau_{2},{\bm{r}}_{2})\phi(\tau_{2},x_{2})$
(1.1) $\displaystyle+\int d\tau d^{2}{\bm{r}}\frac{\lambda^{2}(\tau,x)}{4u}$

where now

$\displaystyle G^{-1}$
$\displaystyle(\tau_{1},x_{1};\tau_{2},x_{2})=\delta(\tau_{1}-\tau_{2})\delta^{(2)}({\bm{r}}_{1}-{\bm{r}}_{2})$
(1.2)
$\displaystyle\times\left(-\partial_{\tau_{1}}^{2}-r_{qc}\bigtriangledown^{2}_{{\bm{r}}_{1}}+\kappa\bigtriangledown^{4}_{{\bm{r}}_{1}}-\frac{1}{\sqrt{N}}\bigtriangledown\lambda(\tau_{1},{\bm{r}}_{1})\cdot\bigtriangledown-\frac{1}{\sqrt{N}}\lambda(\tau_{1},{\bm{r}}_{1})\bigtriangledown^{2}\right)$

From now on we will write ${\bf i}$ for the vector $(\tau_{i},{\bm{r}}_{i})$
in order to make things more readable.

We divide the $\phi_{a}$ into $N-1$ components $\phi_{\perp}$ and $\pi$ and we
choose to integrate out $\phi_{\perp}$ and subsequently use the $\pi\pi$
correlations as a proxy for the $\phi\phi$ correlations of the original
theory. At large $N$ the difference between $N$ and $N-1$ is unimportant.

$\displaystyle S_{\rm eff}(\pi,\lambda)$ $\displaystyle=\int d\tau
d^{2}{\bm{r}}\frac{1}{2}\left[\left(\partial_{\tau}\pi\right)^{2}+r_{qc}\left(\bigtriangledown\pi\right)^{2}+\kappa\left(\bigtriangledown^{2}\pi\right)^{2}+\frac{\lambda(\tau,{\bm{r}})}{\sqrt{N}}(\bigtriangledown\pi)^{2}\right]$
(1.3) $\displaystyle-\int d\tau
d^{2}{\bm{r}}\frac{\lambda^{2}(\tau,x)}{4u}+\frac{N-1}{2}\ln\det G^{-1}$

The saddle-point equation for a spatially uniform $\lambda$ is easiest found
assuming all components of $\phi_{a}$ have been integrated out

$\displaystyle\frac{\delta S_{\rm eff}}{\delta\lambda}$ $\displaystyle=0$
(1.4) $\displaystyle=-\beta
L^{2}\frac{2\lambda}{4u}+\frac{N}{2}\frac{\delta\ln\det
G^{-1}}{\delta\lambda}$ (1.5) $\displaystyle=-\beta
L^{2}\frac{2\lambda}{4u}+\frac{\sqrt{N}}{2}\beta
L^{2}\frac{1}{\beta}\sum_{\omega_{n}}\int\frac{d^{2}{\bm{q}}}{(2\pi)^{2}}\frac{{\bm{q}}^{2}}{\omega_{n}^{2}+(r_{\rm
qc}+\lambda/\sqrt{N}){\bm{q}}^{2}+\kappa{\bm{q}}^{4}}$ (1.6)

Hence,

$\lambda={u\sqrt{N}}\frac{1}{\beta}\sum_{\omega_{n}}\int\frac{d^{2}{\bm{q}}}{(2\pi)^{2}}\frac{1}{\omega_{n}^{2}+(r_{\rm
qc}+\lambda/\sqrt{N}){\bm{q}}^{2}+\kappa{\bm{q}}^{4}}$ (1.7)

It will be convenient to define $r_{\rm eff}=r_{qc}+\lambda/\sqrt{N}$ and
replace $\lambda$ in terms of it:

$\displaystyle r_{\rm eff}-r_{qc}$
$\displaystyle=u\frac{1}{\beta}\sum_{\omega_{n}}\int\frac{d^{2}{\bm{q}}}{(2\pi)^{2}}\frac{{\bm{q}}^{2}}{\omega_{n}^{2}+r_{\rm
eff}^{2}{\bm{q}}^{2}+\kappa{\bm{q}}^{4}}$ (1.8)

Using the identity

$\displaystyle\frac{1}{\beta}\sum_{\omega_{n}}\frac{1}{\omega_{n}^{2}+a^{2}}$
$\displaystyle=\frac{1}{a}\left[\frac{1}{2}+\frac{1}{e^{\beta a}-1}\right]$
(1.9)

we arrive at the final form of this self-consistent equation

$\displaystyle r_{\rm eff}-r_{qc}$
$\displaystyle=u\int\frac{d^{2}{\bm{q}}}{(2\pi)^{2}}\frac{{\bm{q}}^{2}}{\sqrt{r_{\rm
eff}{\bm{q}}^{2}+\kappa{\bm{q}}^{4}}}\left[\frac{1}{2}+\frac{1}{\exp\left(\beta\sqrt{r_{\rm
eff}{\bm{q}}^{2}+\kappa{\bm{q}}^{4}}\right)-1}\right]$ (1.10)

The physics is now most transparent. $r_{\rm eff}$ is the effective
stifness/velocity of the theory, and must vanish at the zero-temperature QCP.
This implies $r_{qc}$ must have the value

$\displaystyle-r_{qc}$
$\displaystyle=u\frac{1}{\beta}\sum_{\omega_{n}}\int\frac{d^{2}{\bm{q}}}{(2\pi)^{2}}\frac{{\bm{q}}^{2}}{\omega_{n}^{2}+\kappa{\bm{q}}^{4}}=\frac{u}{8}\int_{\Lambda}\frac{d^{2}{\bm{q}}}{(2\pi)^{2}}\frac{{\bm{q}}^{2}}{\sqrt{\kappa{\bm{q}}^{4}}}$
(1.11)

which needs to be regularized with a UV cutoff.

It is difficult to solve Eq.1.10 directly, but it can be guessed that $r_{\rm
eff}$ vanishes faster than $T$ due to log corrections, and therefore in the
regime where $r_{\rm eff}\ll\sqrt{\kappa}T$ the equation becomes

$r_{\rm eff}=\frac{uT}{4\pi\kappa}\frac{\ln\left(\frac{\sqrt{\kappa}T}{r_{\rm
eff}}\right)}{1+\frac{u}{16\pi\kappa^{3/2}}\left[\ln\left(\frac{4\kappa\Lambda^{2}}{r_{\rm
eff}}\right)\right]}$ (1.12)

When the log in the denominator dominates, and after rescaling $\bar{r}=r_{\rm
eff}/\sqrt{\kappa}T$, the equation simplifies to

$\bar{r}=4\frac{\ln\left(\frac{1}{\bar{r}}\right)}{\ln\left(\frac{4\Lambda^{2}}{\bar{r}\sqrt{\kappa}T}\right)}=4\frac{\ln\left(\frac{1}{\bar{r}}\right)}{\ln\left(\frac{4\Lambda^{2}}{\sqrt{\kappa}T}\right)-\ln\bar{r}}$
(1.13)

The condition that in Eqn 1.12 the denominator is dominated by the log implies
in particular that $\bar{r}\ll\Lambda^{2}/\sqrt{\kappa}T$, which implies

$\bar{r}\approx
4\frac{\ln\left(\frac{1}{\bar{r}}\right)}{\ln\left(\frac{4\Lambda^{2}}{\sqrt{\kappa}T}\right)}$
(1.14)

Ignoring the log in the numerator we see that a putative solution to the
equation is $\bar{r}=4/\ln\left(\frac{4\Lambda^{2}}{\sqrt{\kappa}T}\right)$.
We substitute it into the original equation and find that it is wrong by a
prefactor $\ln\ln\left(\frac{4\Lambda^{2}}{\sqrt{\kappa}T}\right)$ which we
incorporate into the new trial solution $\bar{r}$. Another substitution shows
that the solution is now correct up to factors of
$\left[\ln\ln\left(\frac{4\Lambda^{2}}{\sqrt{\kappa}T}\right)\right]^{2}/\ln\left(\frac{4\Lambda^{2}}{\sqrt{\kappa}T}\right)^{2}\approx\bar{r}^{2}$
which is guaranteed to be small because $\bar{r}\ll 1$.

##  Appendix B 1/N corrections

Rewrite Eq.1.3 with the saddle-point solution substituted and simultaneously
denote deviations from it by $\lambda$:

$\displaystyle\frac{\lambda_{\rm sp}}{\sqrt{N}}$ $\displaystyle=r_{\rm
eff}-r_{qc}$ $\displaystyle S_{\rm eff}(\pi,\lambda_{\rm sp}+\lambda)$
$\displaystyle=\int d\tau
d^{2}{\bm{r}}\frac{1}{2}\left[\left(\partial_{\tau}\pi\right)^{2}+r_{\rm
eff}\left(\bigtriangledown\pi\right)^{2}+\kappa\left(\bigtriangledown^{2}\pi\right)^{2}+\frac{\lambda(\tau,{\bm{r}})}{\sqrt{N}}(\bigtriangledown\pi)^{2}\right]$
$\displaystyle-\int d\tau d^{2}{\bm{r}}\frac{\left(\lambda_{\rm
sp}+\lambda\right)^{2}(\tau,x)}{4u}+\frac{N-1}{2}\ln\det G^{-1}$

where now

$\displaystyle G^{-1}(1,2)$
$\displaystyle=\delta(1-2)\left(-\partial_{1}^{2}-r_{\rm
eff}\bigtriangledown^{2}_{1}+\kappa\bigtriangledown^{4}_{1}-\frac{1}{\sqrt{N}}\bigtriangledown\lambda(1)\cdot\bigtriangledown_{1}-\frac{1}{\sqrt{N}}\lambda(1)\bigtriangledown_{1}^{2}\right)$
(2.1)

Now expand $\ln\det G^{-1}$ to second order in $\lambda(\tau,{\bm{x}})$ around
the saddle-point value.

$\displaystyle\ln\det G^{-1}$ $\displaystyle=\tr\ln G^{-1}$ (2.2)
$\displaystyle=\tr\ln G_{0}^{-1}\left(1+G_{0}B\right)=\tr\ln
G_{0}^{-1}+\tr\ln\left(1+G_{0}B\right)$ (2.3) $\displaystyle=\tr\ln
G_{0}^{-1}+\tr G_{0}B-\frac{1}{2}\tr(G_{0}B)^{2}+\cdots$ (2.4)

The first order term will vanish when combined with the variation of the rest
of the action due to the stationary phase condition. Finally, the second order
term gives

$\displaystyle G_{0}^{-1}(1,2)$
$\displaystyle=\delta(1-2)\left(-\partial_{1}^{2}-r_{\rm
eff}\bigtriangledown^{2}_{1}+\kappa\bigtriangledown^{4}_{1}\right)$ (2.5)
$\displaystyle B(1,2)$
$\displaystyle=\frac{-1}{\sqrt{N}}\delta(1-2)\left[\bigtriangledown\lambda(1)\cdot\bigtriangledown+\lambda(1)\bigtriangledown^{2}\right]$
(2.6) $\displaystyle\tr(G_{0}B)^{2}$ $\displaystyle=\frac{1}{N}\int
d1\,d2\,d3\,d4\,G_{0}(1,2)B(2,3)G_{0}(3,4)B(4,1)$ (2.7)
$\displaystyle=\frac{1}{N}\int
d1\,d2\,\lambda(1)\lambda(2)\bigtriangledown_{1}\bigtriangledown_{2}G_{0}(1,2)\bigtriangledown_{1}\bigtriangledown_{2}G_{0}(2,1)$
(2.8)

with which the effective action for $\lambda$ at order $O(N^{0})$ becomes

$\displaystyle S_{\rm eff}(\pi,\lambda)$ $\displaystyle=\int d\tau
d^{2}{\bm{r}}\frac{1}{2}\left[\left(\partial_{\tau}\pi\right)^{2}+r_{\rm
eff}\left(\bigtriangledown\pi\right)^{2}+\kappa\left(\bigtriangledown^{2}\pi\right)^{2}+\frac{\lambda(\tau,{\bm{r}})}{\sqrt{N}}(\bigtriangledown\pi)^{2}\right]$
$\displaystyle-\frac{1}{\beta}\sum_{\omega_{n}}\int\frac{d^{2}{\bm{q}}}{(2\pi)^{2}}\lambda(\omega_{n},{\bm{q}})\lambda(-\omega_{n},-{\bm{q}})G_{1}^{-1}(i\omega_{n},{\bm{q}})$
$\displaystyle G_{1}^{-1}(i\omega_{n},{\bm{q}})$
$\displaystyle=\left[-\frac{1}{4u}+\frac{1}{\beta}\sum_{\omega^{\prime}_{n}}\int\frac{d^{2}{\bf
p}}{(2\pi)^{2}}\left[({\bm{p}}+{\bm{q}})\cdot{\bm{p}}\right]^{2}G_{0}(\omega^{\prime}_{n}+\omega_{n},{\bm{p}}+{\bm{q}})G_{0}(\omega^{\prime}_{n},{\bm{p}})\right]$

##  Appendix C Derivation of OTO rules

We work in the interacting picture, where the time-evolution operator is

$\displaystyle U_{I}$
$\displaystyle=\mathcal{T}\exp\left(\frac{i}{2\sqrt{N}}\sum_{a}\int_{0}^{t}ds\int_{\bm{x}}\lambda_{0}({\bm{x}},s)\left(\bigtriangledown\phi_{a}({\bm{x}},s\right)^{2}\right)$
(3.1)

Its expansion looks like

$\displaystyle U_{I}=1+\frac{i}{2\sqrt{N}}$
$\displaystyle\sum_{a}\int_{s_{1}}\int_{\bm{x}}\lambda_{0}({\bm{x}},s)\left(\bigtriangledown\phi_{a}({\bm{x}},s\right)^{2}$
(3.2)
$\displaystyle+\left(\frac{i}{2\sqrt{N}}\right)^{2}\sum_{a,b}\int^{t}ds_{1}\int^{s_{1}}ds_{2}\int_{{\bm{y}}_{1},{\bm{y}}_{2}}\lambda_{0}({\bm{y}}_{1},s_{1})\left(\bigtriangledown\phi_{a}({\bm{y}}_{1},s_{1}\right)^{2}$
$\displaystyle\qquad\qquad\qquad\times\lambda_{0}({\bm{y}}_{2},s_{2})\left(\bigtriangledown\phi_{b}({\bm{y}}_{2},s_{2}\right)^{2}$

We will need an expansion of $U_{I}^{\dagger}\phi_{0}({\bm{x}},t)U_{I}$, for
which it will be useful to first establish a few identities:

$\displaystyle\left[\lambda_{0},\phi_{0}\right]$ $\displaystyle=0$ (3.3)
$\displaystyle\left[\phi_{0,a}({\bm{x}},t),\phi_{0,b}(0)\right]$
$\displaystyle=i{\cal G}_{R}^{ab}({\bm{x}},t)$ (3.4)
$\displaystyle\left[\phi_{0,a}({\bm{x}},t),\bigtriangledown\phi_{0,b}({\bm{y}})\right]$
$\displaystyle=\bigtriangledown_{y}i{\cal G}_{R}^{ab}({\bm{x}}-{\bm{y}},t)$
(3.5)
$\displaystyle\left[\bigtriangledown\phi_{0,a}({\bm{x}},t),\bigtriangledown\phi_{0,b}({\bm{y}})\right]$
$\displaystyle=\bigtriangledown_{x}\bigtriangledown_{y}i{\cal
G}_{R}^{ab}({\bm{x}}-{\bm{y}},t)$ (3.6)
$\displaystyle\left[\phi_{0,a}({\bm{x}},t),\left(\bigtriangledown\phi_{0,b}({\bm{y}})\right)^{2}\right]$
$\displaystyle=2\bigtriangledown_{y}i{\cal
G}_{R}^{ab}({\bm{x}}-{\bm{y}},t)\cdot\bigtriangledown_{y}\phi_{0,b}({\bm{y}})$
(3.7)
$\displaystyle\left[\bigtriangledown\phi_{0,a}({\bm{x}},t),\left(\bigtriangledown\phi_{0,b}({\bm{y}})\right)^{2}\right]$
$\displaystyle=2\bigtriangledown_{x}\bigtriangledown_{y}i{\cal
G}_{R}^{ab}({\bm{x}}-{\bm{y}},t)\cdot\bigtriangledown_{y}\phi_{0,b}({\bm{y}})$
(3.8)

Finally, we got back to $U_{I}^{\dagger}\phi_{0}({\bm{x}},t)U_{I}$:

$\displaystyle U_{I}^{\dagger}\phi_{0}({\bm{x}},t)U_{I}$
$\displaystyle=\phi_{0}({\bm{x}},t)+\frac{i}{2\sqrt{N}}\sum_{a}\int^{t}ds\int_{\bm{y}}\left[\bigtriangledown\phi_{0}({\bm{x}},t),\lambda_{0}({\bm{y}},s)\left(\bigtriangledown\phi_{0,a}({\bm{y}},s)\right)^{2}\right]+\cdots$

Which means the first-order term in $C(t,{\bm{x}})$ is

$\displaystyle C_{1}(t,{\bm{x}}_{1}-{\bm{x}}_{2})$
$\displaystyle=\frac{-1}{N^{2}}\left(\frac{i}{2\sqrt{N}}\right)^{2}\sum_{a,b,c,d}\int_{s_{1},s_{2}}\int_{{\bm{x}}_{1},{\bm{x}}_{2}}$
$\displaystyle{\rm
Tr}\\{\sqrt{\rho}\left[\left[\bigtriangledown\phi_{a}({\bm{x}}_{1},t),\lambda_{0}({\bm{y}}_{1},s_{1})\left(\bigtriangledown\phi_{0,c}({\bm{y}}_{1},s_{1})\right)^{2}\right],\bigtriangledown\phi_{b}({\bm{x}}_{2})\right]$
$\displaystyle\sqrt{\rho}\left[\left[\bigtriangledown\phi_{a}({\bm{x}}_{1},t),\lambda_{0}({\bm{y}}_{2},s_{2})\left(\bigtriangledown\phi_{0,c}({\bm{y}}_{2},s_{2})\right)^{2}\right],\bigtriangledown\phi_{b}({\bm{x}}_{2})\right]\\}$
$\displaystyle=\frac{-1}{N^{2}}\left(\frac{i}{2\sqrt{N}}\right)^{2}\sum_{a,b,c,d}\int_{s_{1},s_{2}}\int_{{\bm{x}}_{1},{\bm{x}}_{2}}$
$\displaystyle{\rm
Tr}\\{\rho\lambda_{0}({\bm{y}}_{1},s_{1})\lambda_{0}({\bm{y}}_{2},s_{2})\\}\,2\bigtriangledown_{x}\bigtriangledown_{y}i{\cal
G}_{R}^{ac}({\bm{x}}_{1}-{\bm{y}}_{1},t-s_{1})\cdot\bigtriangledown_{y}\bigtriangledown_{x}i{\cal
G}^{cb}({\bm{y}}_{1}-{\bm{x}}_{2},s_{1})$
$\displaystyle\,2\bigtriangledown_{x}\bigtriangledown_{y}i{\cal
G}_{R}^{ad}({\bm{x}}_{1}-{\bm{y}}_{2},t-s_{2})\cdot\bigtriangledown_{y}\bigtriangledown_{x}i{\cal
G}^{db}({\bm{y}}_{2}-{\bm{x}}_{2},s_{2})$
$\displaystyle=\frac{-1}{N^{2}}\left(\frac{i}{2\sqrt{N}}\right)^{2}\sum_{a,b,c,d}\int_{s_{1},s_{2}}\int_{{\bm{x}}_{1},{\bm{x}}_{2}}{\cal
G}^{W}_{\lambda}({\bm{y}}_{1}-{\bm{y}}_{2},s_{1}-s_{2})$
$\displaystyle\,2\bigtriangledown_{x}\bigtriangledown_{y}i{\cal
G}_{R}^{ac}({\bm{x}}_{1}-{\bm{y}}_{1},t-s_{1})\cdot\bigtriangledown_{y}\bigtriangledown_{x}i{\cal
G}^{cb}({\bm{y}}_{1}-{\bm{x}}_{2},s_{1})$
$\displaystyle\,2\bigtriangledown_{x}\bigtriangledown_{y}i{\cal
G}_{R}^{ad}({\bm{x}}_{1}-{\bm{y}}_{2},t-s_{2})\cdot\bigtriangledown_{y}\bigtriangledown_{x}i{\cal
G}^{db}({\bm{y}}_{2}-{\bm{x}}_{2},s_{2})$

Note that above the $\bigtriangledown_{x}$ and $\bigtriangledown_{y}$ are
dotted with themselves. Another observation is that

$\displaystyle\frac{1}{N^{3}}\sum_{abcd}\delta_{ac}\delta_{cb}\delta_{ad}\delta_{db}=\frac{1}{N^{3}}\sum_{ab}\delta_{ab}=\frac{1}{N^{2}}$
(3.10)

Finally, we Fourier transform and obtain:

$\displaystyle C_{1}(\nu)$
$\displaystyle=\int_{t}\int_{{\bm{x}}_{1}-{\bm{x}}_{2}}C_{1}(t,{\bm{x}}_{1}-{\bm{x}}_{2})$
(3.11)
$\displaystyle=\frac{1}{N^{2}}\int_{{\bm{p}},{\bm{p}}^{\prime}}\left({\bm{p}}\cdot{\bm{p}}^{\prime}\right)^{4}\int_{\omega,\omega^{\prime}}{\cal
G}_{R}({\bm{p}},\omega){\cal G}_{R}(-{\bm{p}},\nu-\omega)$
$\displaystyle\qquad\qquad\times{\cal
G}^{W}_{\lambda}({\bm{p}}-{\bm{p}}^{\prime},\omega-\omega^{\prime}){\cal
G}_{R}({\bm{p}}^{\prime},\omega^{\prime}){\cal
G}_{R}(-{\bm{p}}^{\prime},\nu-\omega^{\prime})$

Had we chosen to instead dot product the gradient terms of
$\bigtriangledown_{x_{1}}$ (and $\bigtriangledown_{x_{2}}$) with
$\bigtriangledown_{x_{1}}$ (and $\bigtriangledown_{x_{2}}$) from other
commutator, we would have had a different momentum-dependent numerator:
$\left({\bm{p}}\cdot{\bm{p}}^{\prime}\right)^{2}{\bm{p}}^{2}{\bm{p}}^{\prime
2}$:

$\displaystyle C_{1}(\nu)$
$\displaystyle=\frac{1}{N^{2}}\int_{{\bm{p}},{\bm{p}}^{\prime}}{\bm{p}}^{2}\left({\bm{p}}\cdot{\bm{p}}^{\prime}\right)^{2}{\bm{p}}^{\prime
2}\int_{\omega,\omega^{\prime}}{\cal G}_{R}({\bm{p}},\omega){\cal
G}_{R}(-{\bm{p}},\nu-\omega){\cal
G}^{W}_{\lambda}({\bm{p}}-{\bm{p}}^{\prime},\omega-\omega^{\prime})$ (3.12)
$\displaystyle\qquad\qquad\times{\cal
G}_{R}({\bm{p}}^{\prime},\omega^{\prime}){\cal
G}_{R}(-{\bm{p}}^{\prime},\nu-\omega^{\prime})$

Or generalized:

$\displaystyle C_{1(ijij)}(\nu,{\bm{q}})$
$\displaystyle=\frac{1}{N^{2}}\int_{{\bm{p}},{\bm{p}}^{\prime}}{\bm{p}}\cdot({\bm{q}}-{\bm{p}})({\bm{q}}-{\bm{p}})\cdot({\bm{q}}-{\bm{p}}^{\prime})({\bm{q}}-{\bm{p}}^{\prime})\cdot{\bm{p}}^{\prime}{\bm{p}}\cdot{\bm{p}}^{\prime}$
$\displaystyle\times\int_{\omega,\omega^{\prime}}{\cal
G}_{R}({\bm{p}},\omega){\cal G}_{R}(-{\bm{p}},\nu-\omega){\cal
G}^{W}_{\lambda}({\bm{p}}-{\bm{p}}^{\prime},\omega-\omega^{\prime})$
$\displaystyle\qquad\qquad\times{\cal
G}_{R}({\bm{p}}^{\prime},\omega^{\prime}){\cal
G}_{R}(-{\bm{p}}^{\prime},\nu-\omega^{\prime})$ (3.14)

This gives a diagram with one rung of Type 1. We can read off the Feynman
rules from the above. They are:

  * •

Draw a diagram

  * •

Each line must be directed: left to right, and top to down. This determines
whether a momentum/frequency is incomming or outgoing. The sum of incomming
must equal sum of outgoing.

  * •

Assign a value of
$\frac{i{\bm{k}}_{1}\cdot{\bm{k}}_{2}}{\sqrt{N}}\delta_{a,b}$ to each vertex,
where the momenta belong to $\phi$.

  * •

Multiply overall diagram by $\left({\bm{k}}_{i}\cdot{\bm{k}}_{f}\right)^{2}$
for $C_{iijj}$ or ${\bm{k}}_{i}^{2}{\bm{k}}_{f}^{2}$ for $C_{ijij}$, where the
momenta ${\bm{k}}_{i},{\bm{k}}_{f}$ emmanate from ${\bm{x}}_{1}$ and
${\bm{x}}_{2}$ correspondingly.

  * •

Retarded propagators for horizontal lines, Wightman for vertical.

To verify the rules for diagrams with Type 2 rungs we must expand
$U_{I}^{\dagger}\phi U_{I}$ to second order.

$\displaystyle\left[U_{I}^{\dagger}\phi_{0}(x,t)U_{I}\right]_{2}$
$\displaystyle=\left(\frac{i}{2\sqrt{N}}\right)^{2}\sum_{bc}\int_{0}^{t}ds_{1}\int_{0}^{s_{1}}ds_{2}\int_{{\bm{y}}_{1},{\bm{y}}_{2}}$
$\displaystyle\left[\left[\phi_{0}({\bm{x}},t),\lambda_{0}(s_{1},{\bm{y}}_{1})\left(\bigtriangledown\phi_{0,b}(s_{1},{\bm{y}}_{1})\right)^{2}\right],\lambda_{0}(s_{2},{\bm{y}}_{2})\left(\bigtriangledown\phi_{0,c}(s_{2},{\bm{y}}_{2})\right)^{2}\right]$

The commutator in the term above becomes

$\displaystyle[[,],]$ $\displaystyle=2\bigtriangledown_{y_{1}}i{\cal
G}_{R}({\bm{x}}-{\bm{y}}_{1},t-s_{1})\cdot\bigtriangledown_{y_{1}}$
$\displaystyle\qquad\qquad\times\bigtriangledown_{y_{2}}2i{\cal
G}_{R}({\bm{y}}_{1}-{\bm{y}}_{2},s_{1}-s_{2})\cdot\bigtriangledown_{y_{2}}\phi({\bm{y}}_{2},s_{2})\lambda(s_{1})\lambda(s_{2})$
$\displaystyle+2\bigtriangledown_{y_{1}}i{\cal
G}_{R}({\bm{x}}-{\bm{y}}_{1},t-s_{1})i{\cal
G}_{R,\lambda}({\bm{y}}_{1}-{\bm{y}}_{2},s_{1}-s_{2})\left(\bigtriangledown_{y}\phi({\bm{y}}_{2},s_{2})\right)^{2}\bigtriangledown_{y_{1}}\phi({\bm{y}}_{1})$

where the $\bigtriangledown$’s of ${\bm{y}}_{1}$ are contracted amonst
themselves, similarly for ${\bm{y}}_{2}$.

Finally, we must actually consider
$\left[\left[U_{I}^{\dagger}\phi_{0}(x_{1},t)U_{I}\right]_{2},\bigtriangledown_{x_{2}}\phi(x_{2},0)\right]$
on each time fold.

$\displaystyle\left[[[,],],\bigtriangledown_{x_{2}}\phi(x_{2},0)\right]$
$\displaystyle=2\bigtriangledown_{y_{1}}i{\cal
G}_{R}({\bm{x}}-{\bm{y}}_{1},t-s_{1})\cdot\bigtriangledown_{y_{1}}$
$\displaystyle\qquad\times\bigtriangledown_{y_{2}}2i{\cal
G}_{R}({\bm{y}}_{1}-{\bm{y}}_{2},s_{1}-s_{2})\cdot\bigtriangledown_{y_{2}}$
$\displaystyle\qquad\qquad\times\bigtriangledown_{x_{2}}i{\cal
G}_{R}({\bm{y}}_{2}-{\bm{x}}_{2},s_{2})\lambda(s_{1})\lambda(s_{2})$
$\displaystyle+2\bigtriangledown_{y_{1}}i{\cal
G}_{R}({\bm{x}}-{\bm{y}}_{1},t-s_{1})i{\cal
G}_{R,\lambda}({\bm{y}}_{1}-{\bm{y}}_{2},s_{1}-s_{2})$
$\displaystyle\quad\qquad\times\left[\left(\bigtriangledown_{y}\phi({\bm{y}}_{2},s_{2})\right)^{2}\bigtriangledown_{y_{1}}\phi({\bm{y}}_{1}),\bigtriangledown_{x_{2}}\phi(x_{2},0)\right]$
(3.16)

where

$\displaystyle\left[\left(\bigtriangledown_{y}\phi({\bm{y}}_{2},s_{2})\right)^{2}\bigtriangledown_{y_{1}}\phi({\bm{y}}_{1}),\bigtriangledown_{x_{2}}\phi(x_{2},0)\right]=\bigtriangledown_{x}\bigtriangledown_{y_{1}}i{\cal
G}_{R}({\bm{y}}_{1}-{\bm{x}}_{2},s_{1})\left(\bigtriangledown_{y_{2}}\phi\right)^{2}$
$\displaystyle+2\bigtriangledown_{x}\bigtriangledown_{y_{2}}i{\cal
G}_{R}({\bm{y}}_{2}-{\bm{x}}_{2},s_{2})\cdot\bigtriangledown_{y_{2}}\phi({\bm{y}}_{2})\bigtriangledown_{y_{1}}\phi({\bm{y}}_{1})$

The first line in Eq.C gives the diagram with two rungs of Type 1 (if the
$\lambda$’s are contracted between the time folds), or a correction to the
self-energy of $\phi$. The second line can be verified to give a diagram with
a single Type 2 rung, and a correction to the vertex (which is $1/N$
suppressed) in a Type 1 diagram. Furthermore, there are two types of Type 2
diagrams, ladder type, and crossed type. This latter is argued to be
kinematically suppressed and is subsequently ignored.

Putting it all together, for a diagram with a single uncrossed Type 2 rung:

$\displaystyle C_{2}(t,{\bm{x}}_{1}-{\bm{x}}_{2})$
$\displaystyle=\frac{-1}{N^{2}}\left(\frac{i}{2\sqrt{N}}\right)^{4}\sum_{abcdef}\int_{s_{1},s_{2},s_{1}^{\prime},s_{2}^{\prime}}\int_{{\bm{y}}_{1},{\bm{y}}_{2},{\bm{y}}_{1}^{\prime},{\bm{y}}_{2}^{\prime}}$
$\displaystyle\times 2\bigtriangledown_{x_{1}}\bigtriangledown_{y_{1}}i{\cal
G}_{R}({\bm{x}}_{1}-{\bm{y}}_{1},t-s_{1})i{\cal
G}_{R,\lambda}({\bm{y}}_{1}-{\bm{y}}_{2},s_{1}-s_{2})$ $\displaystyle\times
2\bigtriangledown_{x_{2}}\bigtriangledown_{y_{2}}i{\cal
G}_{R}({\bm{y}}_{2}-{\bm{x}}_{2},s_{2})2\bigtriangledown_{x_{1}}\bigtriangledown_{y_{1}^{\prime}}i{\cal
G}_{R}({\bm{x}}_{1}-{\bm{y}}_{1}^{\prime},t-s_{1}^{\prime})$
$\displaystyle\times i{\cal
G}_{R,\lambda}({\bm{y}}_{1}-{\bm{y}}_{2}^{\prime},s_{1}^{\prime}-s_{2}^{\prime})2\bigtriangledown_{x_{2}}\bigtriangledown_{y_{2}^{\prime}}i{\cal
G}_{R}({\bm{y}}_{2}^{\prime}-{\bm{x}}_{2},s_{2}^{\prime})$
$\displaystyle\times\bigtriangledown_{y_{1}}\bigtriangledown_{y_{1}^{\prime}}{\cal
G}_{W}({\bm{y}}_{1}-{\bm{y}}_{1}^{\prime},s_{1}-s_{1}^{\prime})\bigtriangledown_{y_{2}}\bigtriangledown_{y_{2}^{\prime}}{\cal
G}_{W}({\bm{y}}_{2}-{\bm{y}}_{2}^{\prime},s_{2}-s_{2}^{\prime})$

which in the end gives (with all factors of $i,2,N$ upfront):

$\displaystyle C_{2}(t,{\bm{x}}_{1}-{\bm{x}}_{2})$
$\displaystyle=\frac{1}{N^{2}}\int_{s_{1},s_{2},s_{1}^{\prime},s_{2}^{\prime}}\int_{{\bm{y}}_{1},{\bm{y}}_{2},{\bm{y}}_{1}^{\prime},{\bm{y}}_{2}^{\prime}}$
$\displaystyle\times\bigtriangledown_{x_{1}}\bigtriangledown_{y_{1}}{\cal
G}_{R}({\bm{x}}_{1}-{\bm{y}}_{1},t-s_{1}){\cal
G}_{R,\lambda}({\bm{y}}_{1}-{\bm{y}}_{2},s_{1}-s_{2})$
$\displaystyle\times\bigtriangledown_{x_{2}}\bigtriangledown_{y_{2}}{\cal
G}_{R}({\bm{y}}_{2}-{\bm{x}}_{2},s_{2})\bigtriangledown_{x_{1}}\bigtriangledown_{y_{1}^{\prime}}{\cal
G}_{R}({\bm{x}}_{1}-{\bm{y}}_{1}^{\prime},t-s_{1}^{\prime})$
$\displaystyle\times{\cal
G}_{R,\lambda}({\bm{y}}^{\prime}_{1}-{\bm{y}}_{2}^{\prime},s_{1}^{\prime}-s_{2}^{\prime})\bigtriangledown_{x_{2}}\bigtriangledown_{y_{2}^{\prime}}{\cal
G}_{R}({\bm{y}}_{2}^{\prime}-{\bm{x}}_{2},s_{2}^{\prime})$
$\displaystyle\bigtriangledown_{y_{1}}\bigtriangledown_{y_{1}^{\prime}}{\cal
G}_{W}({\bm{y}}_{1}-{\bm{y}}_{1}^{\prime},s_{1}-s_{1}^{\prime})\bigtriangledown_{y_{2}}\bigtriangledown_{y_{2}^{\prime}}{\cal
G}_{W}({\bm{y}}_{2}-{\bm{y}}_{2}^{\prime},s_{2}-s_{2}^{\prime})$

which in momentum space becomes

$\displaystyle C_{2}(\nu)$
$\displaystyle=\frac{1}{N^{2}}\int_{{\bm{p}},{\bm{p}}^{\prime},{\bm{p}}^{\prime\prime}}\int_{\omega,\omega^{\prime},\omega^{\prime\prime}}\left({\bm{p}}\cdot{\bm{p}}^{\prime}\right)^{2}\left({\bm{p}}\cdot({\bm{p}}^{\prime\prime}-{\bm{p}})\right)^{2}\left({\bm{p}}^{\prime}\cdot({\bm{p}}^{\prime}-{\bm{p}}^{\prime\prime})\right)^{2}$
$\displaystyle\qquad\times{\cal G}_{R}(\nu-\omega,-{\bm{p}}){\cal
G}_{R}(\omega,{\bm{p}}){\cal
G}_{W}(\omega^{\prime\prime}-\omega,{\bm{p}}^{\prime\prime}-{\bm{p}})$
$\displaystyle\qquad\times{\cal
G}_{R,\lambda}(\nu-\omega^{\prime\prime},-{\bm{p}}^{\prime\prime}){\cal
G}_{R,\lambda}(\omega^{\prime\prime},{\bm{p}}^{\prime\prime}){\cal
G}_{R}(\nu-\omega^{\prime},-{\bm{p}}^{\prime}){\cal
G}_{R}(\omega^{\prime},{\bm{p}}^{\prime})$

The $1/N$ counting follows from

$\displaystyle\frac{1}{N^{2}}\left(\frac{1}{\sqrt{N}}\right)^{4}\sum_{abcdef}\delta_{ac}\delta_{cd}\delta_{da}\delta_{be}\delta_{ef}\delta_{fb}=\frac{1}{N^{2}}$
(3.17)

such that we can write the diagram with one Type 2 rung as

$\displaystyle C_{2(iijj)}(\nu)$
$\displaystyle=\frac{1}{N^{2}}\int_{{\bm{p}},{\bm{p}}^{\prime}}\left({\bm{p}}\cdot{\bm{p}}^{\prime}\right)^{2}\int_{\omega,\omega^{\prime}}\,{\cal
G}_{R}(\nu-\omega,-{\bm{p}}){\cal G}_{R}(\omega,{\bm{p}}){\cal G}_{\rm
eff}(\omega^{\prime},\omega,{\bm{p}}^{\prime},{\bm{p}})$ (3.18)
$\displaystyle\times{\cal G}_{R}(\nu-\omega^{\prime},-{\bm{p}}^{\prime}){\cal
G}_{R}(\omega^{\prime},{\bm{p}}^{\prime})$

Compare it with

$\displaystyle C_{1(iijj)}(\nu)$
$\displaystyle=\frac{1}{N^{2}}\int_{{\bm{p}},{\bm{p}}^{\prime}}\left({\bm{p}}\cdot{\bm{p}}^{\prime}\right)^{4}\int_{\omega,\omega^{\prime}}{\cal
G}_{R}({\bm{p}},\omega){\cal G}_{R}(-{\bm{p}},\nu-\omega)$ (3.19)
$\displaystyle\times{\cal
G}^{W}_{\lambda}({\bm{p}}-{\bm{p}}^{\prime},\omega-\omega^{\prime}){\cal
G}_{R}({\bm{p}}^{\prime},\omega^{\prime}){\cal
G}_{R}(-{\bm{p}}^{\prime},\nu-\omega^{\prime})$

we notice that replacing $\left({\bm{p}}\cdot{\bm{p}}^{\prime}\right)^{2}{\cal
G}^{W}_{\lambda}({\bm{p}}^{\prime}-{\bm{p}},\omega^{\prime}-\omega)$ in
$C_{1}$ by $\left({\bm{p}}\cdot{\bm{p}}^{\prime}\right)^{2}{\cal
G}^{W}_{\lambda}({\bm{p}}^{\prime}-{\bm{p}},\omega^{\prime}-\omega)+{\cal
G}_{\rm eff}(\omega^{\prime},\omega,{\bm{p}}^{\prime},{\bm{p}})$ allows us to
treat both one-rung diagrams simultaneously.

Recall that in defining the OTO correlator we had a choice in how to contract
the spatial derivatives:

$\displaystyle C_{ijkl}(t_{1}-t_{2},{\bm{x}}_{1}-{\bm{x}}_{2})=$
$\displaystyle\qquad\frac{1}{N^{2}}\sum_{a,b}{\rm
Tr}\\{\sqrt{\rho}\left[\partial_{i}\phi_{a}({\bm{x}}_{1},t_{1}),\partial_{j}\phi_{b}({\bm{x}}_{2},t_{2})\right]^{\dagger}\sqrt{\rho}\left[\partial_{k}\phi_{a}({\bm{x}}_{1},t_{1}),\partial_{l}\phi_{b}({\bm{x}}_{2},t_{2})\right]\\}$
(3.20)

N.B. So far we have worked with the contraction $C=\sum_{i,k}C_{iikk}$, but
another choice is $C=\sum_{i,j}C_{ijij}$. I will work with the latter from now
on due to certain simplifications.

Using the $ijij$ contraction:

$\displaystyle C_{2(ijij)}(\nu)$
$\displaystyle=\frac{1}{N^{2}}\int_{{\bm{p}},{\bm{p}}^{\prime}}{\bm{p}}^{2}\cdot{\bm{p}}^{\prime
2}\int_{\omega,\omega^{\prime}}\,{\cal G}_{R}(\nu-\omega,-{\bm{p}}){\cal
G}_{R}(\omega,{\bm{p}})$ (3.21) $\displaystyle\qquad\qquad\times{\cal G}_{\rm
eff}(\omega^{\prime},\omega,{\bm{p}}^{\prime},{\bm{p}}){\cal
G}_{R}(\nu-\omega^{\prime},-{\bm{p}}^{\prime}){\cal
G}_{R}(\omega^{\prime},{\bm{p}}^{\prime})$

and

$\displaystyle
C_{1(ijij)}(\nu)=\frac{1}{N^{2}}\int_{{\bm{p}},{\bm{p}}^{\prime}}{\bm{p}}^{2}\left({\bm{p}}\cdot{\bm{p}}^{\prime}\right)^{2}{\bm{p}}^{\prime
2}\int_{\omega,\omega^{\prime}}{\cal G}_{R}({\bm{p}},\omega){\cal
G}_{R}(-{\bm{p}},\nu-\omega)$ $\displaystyle\times{\cal
G}^{W}_{\lambda}({\bm{p}}-{\bm{p}}^{\prime},\omega-\omega^{\prime}){\cal
G}_{R}({\bm{p}}^{\prime},\omega^{\prime}){\cal
G}_{R}(-{\bm{p}}^{\prime},\nu-\omega^{\prime})$ (3.22)

Or, in full generality:

$\displaystyle C_{1(ijij)}(\nu,{\bm{q}})$
$\displaystyle=\frac{1}{N^{2}}\int_{{\bm{p}},{\bm{p}}^{\prime}}({\bm{q}}-{\bm{p}})\cdot{\bm{p}}({\bm{q}}-{\bm{p}}^{\prime})\cdot{\bm{p}}^{\prime}\int_{\omega,\omega^{\prime}}({\bm{q}}-{\bm{p}})\cdot({\bm{q}}-{\bm{p}}^{\prime}){\bm{p}}\cdot{\bm{p}}^{\prime}$
$\displaystyle\times{\cal G}_{R}({\bm{p}},\omega){\cal
G}_{R}(-{\bm{p}},\nu-\omega){\cal
G}^{W}_{\lambda}({\bm{p}}-{\bm{p}}^{\prime},\omega-\omega^{\prime}){\cal
G}_{R}({\bm{p}}^{\prime},\omega^{\prime}){\cal
G}_{R}(-{\bm{p}}^{\prime},\nu-\omega^{\prime})$ $\displaystyle
C_{2(ijij)}(\nu)$
$\displaystyle=\frac{1}{N^{2}}\int_{{\bm{p}},{\bm{p}}^{\prime}}({\bm{q}}-{\bm{p}})\cdot{\bm{p}}\,\,({\bm{q}}-{\bm{p}}^{\prime})\cdot{\bm{p}}^{\prime}\int_{\omega,\omega^{\prime}}\,{\cal
G}_{R}(\nu-\omega,-{\bm{p}}){\cal G}_{R}(\omega,{\bm{p}})$ (3.24)
$\displaystyle{\cal G}_{\rm
eff}(\omega^{\prime},\omega,{\bm{p}}^{\prime},{\bm{p}};{\bm{q}}){\cal
G}_{R}(\nu-\omega^{\prime},-{\bm{p}}^{\prime}){\cal
G}_{R}(\omega^{\prime},{\bm{p}}^{\prime})$

where now

$\displaystyle{\cal G}_{\rm
eff}(\omega^{\prime},\omega,{\bm{p}}^{\prime},{\bm{p}};\nu,{\bm{q}})=$
$\displaystyle\quad\int_{{\bm{p}}^{\prime\prime},\omega^{\prime\prime}}({\bm{p}}^{\prime\prime}-{\bm{p}})\cdot{\bm{p}}\,\,({\bm{q}}-{\bm{p}})\cdot({\bm{p}}^{\prime\prime}-{\bm{p}})\,\,({\bm{q}}-{\bm{p}}^{\prime})\cdot({\bm{p}}^{\prime}-{\bm{p}}^{\prime\prime})\,\,{\bm{p}}^{\prime}\cdot({\bm{p}}^{\prime}-{\bm{p}}^{\prime\prime})$
$\displaystyle\quad\times{\cal
G}_{W}(\omega^{\prime\prime}-\omega,{\bm{p}}^{\prime\prime}-{\bm{p}}){\cal
G}_{W}(\omega^{\prime}-\omega^{\prime\prime},{\bm{p}}^{\prime}-{\bm{p}}^{\prime\prime}){\cal
G}_{R,\lambda}(\nu-\omega^{\prime\prime},{\bm{q}}-{\bm{p}}^{\prime\prime}){\cal
G}_{R,\lambda}(\omega^{\prime\prime},{\bm{p}}^{\prime\prime})$ (3.25)

NB. It really doesn’t matter whether ${\cal
G}^{W}_{\lambda}({\bm{p}}^{\prime}-{\bm{p}})$ or ${\cal
G}^{W}_{\lambda}({\bm{p}}-{\bm{p}}^{\prime})$ is used. It amounts to a
redefinition of momenta that keeps all other terms in $C_{1}$ the same.
Therefore, it must be that ${\cal
G}^{W}_{\lambda}({\bm{p}}^{\prime}-{\bm{p}})={\cal
G}^{W}_{\lambda}({\bm{p}}-{\bm{p}}^{\prime})$

###  C.1 Crossed Diagrams

The first important observation comes from the Fourier space form of a
Wightman correlator:

$\displaystyle{\cal
G}_{W}(\omega,{\bm{q}})=\frac{A(\omega,{\bm{q}})}{2\sinh\left(\frac{\beta\omega}{2}\right)}$
(3.26)

We see that in real time, ${\cal G}_{W}(t,{\bm{r}})$ has significant support
only for $t\sim\beta^{-1}$. Now consider a ladder diagram of type 1 with two
rungs. Its value is, schematically:

$\displaystyle\chi_{\rm ladder}$ $\displaystyle=\int
s_{1}s_{2}s_{1}^{\prime}s_{2}^{\prime}{\cal G}_{R}(t-s_{1}){\cal
G}_{R}(t-s_{1}^{\prime}){\cal G}_{R}(s_{1}-s_{2}){\cal
G}_{R}(s_{1}^{\prime}-s_{2}^{\prime})$ (3.27) $\displaystyle\times{\cal
G}_{W,\lambda}(s_{1}-s_{1}^{\prime}){\cal
G}_{W,\lambda}(s_{2}-s_{2}^{\prime})\ {\cal G}_{R}(s_{2}){\cal
G}_{R}(s_{2}^{\prime})$

The retarded Green functions impose, in particular,
$\theta(s_{1}-s_{2})\theta(s_{1}^{\prime}-s_{2}^{\prime})\theta(s_{2})\theta(s_{2}^{\prime})$,
while the Wightman function effectively imposes $s_{1}-s_{1}^{\prime}\approx
0,s_{2}-s_{2}^{\prime}\approx 0$. Putting them together the integration phase
space is restricted to $\theta(s_{1}-s_{2})\theta(s_{2})$.

Now compare this with the case of a crossed diagram

$\displaystyle\chi_{\rm crossed}$ $\displaystyle=\int
s_{1}s_{2}s_{1}^{\prime}s_{2}^{\prime}{\cal G}_{R}(t-s_{1}){\cal
G}_{R}(t-s_{1}^{\prime}){\cal G}_{R}(s_{1}-s_{2}){\cal
G}_{R}(s_{1}^{\prime}-s_{2}^{\prime})$ (3.28) $\displaystyle\times{\cal
G}_{W,\lambda}(s_{1}-s_{2}^{\prime}){\cal
G}_{W,\lambda}(s_{1}^{\prime}-s_{2})\ {\cal G}_{R}(s_{2}){\cal
G}_{R}(s_{2}^{\prime})$

where the constraints are
$\theta(s_{1}-s_{2})\theta(s_{1}^{\prime}-s_{2}^{\prime})\theta(s_{2})\theta(s_{2}^{\prime})$,
coming from the Green functions, and $s_{1}-s_{2}^{\prime}\approx
0,s_{2}-s_{1}^{\prime}\approx 0$ from the Wightman function. Together they
require $\theta(s_{1}-s_{2})\theta(s_{2}-s_{1})$, to within $\beta^{-1}$.

That is why we conclude that the crossed diagrams are parametrically
suppressed compared to the ladder diagrams, a fact important at late times.

###  C.2 Product of retarded Green functions

Let us extract the part of Eq.C that contributes to the leading time
dependence of $C(t)$. Define $f(\omega)={\cal
G}^{W}_{\lambda}({\bm{p}}^{\prime}-{\bm{p}},\omega^{\prime}-\omega)$. We first
work with the free expressions for ${\cal G}_{R}$.

$\displaystyle{\cal G}_{R}(\omega,{\bm{k}})$
$\displaystyle=\frac{1}{(\omega+i0)^{2}+\epsilon_{\bm{k}}^{2}}=\frac{1}{2\epsilon_{\bm{k}}}\left[\frac{1}{\omega-\epsilon_{\bm{k}}+i0}-\frac{1}{\omega+\epsilon_{\bm{k}}+i0}\right]$
$\displaystyle{\cal G}_{R}(\nu-\omega,-{\bm{p}}){\cal G}_{R}(\omega,{\bm{p}})$
$\displaystyle=\frac{1}{4\epsilon_{\bm{p}}^{2}}\left[\frac{1}{\nu-\omega-\epsilon_{\bm{p}}+i0}-\frac{1}{\nu-\omega+\epsilon_{\bm{p}}+i0}\right]$
$\displaystyle\qquad\times\left[\frac{1}{\omega-\epsilon_{\bm{p}}+i0}-\frac{1}{\omega+\epsilon_{\bm{p}}+i0}\right]$
$\displaystyle C_{1}(\nu)$ $\displaystyle\propto
I(\omega^{\prime})=\int\frac{d\omega}{2\pi}{\cal G}_{R}({\bm{p}},\omega){\cal
G}_{R}(-{\bm{p}},\nu-\omega)f(\omega)$ $\displaystyle I(\omega^{\prime})$
$\displaystyle=\frac{-i\,f(\omega=\nu-\epsilon_{\bm{p}})}{4\epsilon_{\bm{p}}^{2}}\left[\frac{1}{\nu-2\epsilon_{\bm{p}}+i0}-\frac{1}{\nu+i0}\right]$
$\displaystyle+\frac{-i\,f(\omega=\nu+\epsilon_{\bm{p}})}{4\epsilon_{\bm{p}}^{2}}\left[\frac{1}{\nu+2\epsilon_{\bm{p}}+i0}-\frac{1}{\nu+i0}\right]$

NB. It is important that above ${\cal G}_{W,\lambda}(\omega^{\prime}-\omega)$
not have any singularities in $\omega$, otherwise we might get extra poles, or
worse.

Now doing the $\omega^{\prime}$ integral picking up the other poles, at
$\omega^{\prime}=\nu\pm\epsilon_{\bm{p}}+i0$:

$\displaystyle I_{2}(\nu,{\bm{p}},{\bm{p}}^{\prime})$
$\displaystyle=\int\frac{d\omega^{\prime}}{2\pi}{\cal
G}_{R}({\bm{p}}^{\prime},\omega^{\prime}){\cal
G}_{R}(-{\bm{p}}^{\prime},\nu-\omega^{\prime})I(\omega^{\prime})$
$\displaystyle=\frac{(-i)^{2}{\cal
G}_{W,\lambda}({\bm{p}}^{\prime}-{\bm{p}},\nu-\epsilon_{{\bm{p}}^{\prime}}-\nu+\epsilon_{\bm{p}})}{16\epsilon_{\bm{p}}^{2}\epsilon_{{\bm{p}}^{\prime}}^{2}}$
$\displaystyle\times\left[\frac{1}{\nu-2\epsilon_{\bm{p}}+i0}-\frac{1}{\nu+i0}\right]\left[\frac{1}{\nu-2\epsilon_{{\bm{p}}^{\prime}}+i0}-\frac{1}{\nu+i0}\right]$
$\displaystyle+\frac{(-i)^{2}{\cal
G}_{W,\lambda}({\bm{p}}^{\prime}-{\bm{p}},\nu+\epsilon_{{\bm{p}}^{\prime}}-\nu-\epsilon_{\bm{p}})}{16\epsilon_{\bm{p}}^{2}\epsilon_{{\bm{p}}^{\prime}}^{2}}$
$\displaystyle\times\left[\frac{1}{\nu+2\epsilon_{\bm{p}}+i0}-\frac{1}{\nu+i0}\right]\left[\frac{1}{\nu+2\epsilon_{{\bm{p}}^{\prime}}+i0}-\frac{1}{\nu+i0}\right]$
(3.30)

We must now consider the three kinds of poles in $\nu$ that are important for
the inverse Laplace transform to $C(t)$: $\nu^{-2},\nu^{-1}(\nu\pm
2\epsilon_{\bm{p}})^{-1},(\nu\pm 2\epsilon_{\bm{p}})^{-1}(\nu\pm
2\epsilon_{{\bm{p}}^{\prime}})^{-1}$. The $\nu$ integral will be done above
all singularities of $I_{2}(\nu,{\bm{p}},{\bm{p}}^{\prime})$ in the complex
plane

$\displaystyle C(t>0)_{(1)}$
$\displaystyle=\frac{-1}{N^{2}}\int\frac{d\nu}{2\pi}e^{-i\nu
t}\frac{1}{(\nu+i0)^{2}}\int_{{\bm{p}},{\bm{p}}^{\prime}}\left({\bm{p}}\cdot{\bm{p}}^{\prime}\right)^{4}$
$\displaystyle\qquad\times\frac{{\cal
G}_{W,\lambda}({\bm{p}}^{\prime}-{\bm{p}},\epsilon_{{\bm{p}}^{\prime}}-\epsilon_{\bm{p}})+{\cal
G}_{W,\lambda}({\bm{p}}^{\prime}-{\bm{p}},\epsilon_{\bm{p}}-\epsilon_{{\bm{p}}^{\prime}})}{16\epsilon_{\bm{p}}^{2}\epsilon_{{\bm{p}}^{\prime}}^{2}}$
$\displaystyle=\frac{t}{N^{2}}\int_{{\bm{p}},{\bm{p}}^{\prime}}\left({\bm{p}}\cdot{\bm{p}}^{\prime}\right)^{4}\frac{{\cal
G}_{W,\lambda}({\bm{p}}^{\prime}-{\bm{p}},\epsilon_{{\bm{p}}^{\prime}}-\epsilon_{\bm{p}})+{\cal
G}_{W,\lambda}({\bm{p}}^{\prime}-{\bm{p}},\epsilon_{\bm{p}}-\epsilon_{{\bm{p}}^{\prime}})}{16\epsilon_{\bm{p}}^{2}\epsilon_{{\bm{p}}^{\prime}}^{2}}$

and

$\displaystyle C(t>0)_{(2)}$
$\displaystyle=\frac{1}{N^{2}}\int_{{\bm{p}},{\bm{p}}^{\prime}}\int\frac{d\nu}{2\pi}e^{-i\nu
t}\frac{1}{(\nu+i0)(\nu-2\epsilon_{\bm{p}}+i0)}\left({\bm{p}}\cdot{\bm{p}}^{\prime}\right)^{4}$
$\displaystyle\times\frac{{\cal
G}_{W,\lambda}({\bm{p}}^{\prime}-{\bm{p}},\epsilon_{\bm{p}}-\epsilon_{{\bm{p}}^{\prime}})}{16\epsilon_{\bm{p}}^{2}\epsilon_{{\bm{p}}^{\prime}}^{2}}+\cdots$
$\displaystyle=\frac{i}{N^{2}}\int_{{\bm{p}},{\bm{p}}^{\prime}}\frac{1+e^{-i2\epsilon_{\bm{p}}t}}{2\epsilon_{\bm{p}}}\left({\bm{p}}\cdot{\bm{p}}^{\prime}\right)^{4}\frac{{\cal
G}_{W,\lambda}({\bm{p}}^{\prime}-{\bm{p}},\epsilon_{\bm{p}}-\epsilon_{{\bm{p}}^{\prime}})}{16\epsilon_{\bm{p}}^{2}\epsilon_{{\bm{p}}^{\prime}}^{2}}+\cdots$

Clearly all poles besides $\nu^{-2}$ give subleading time dependence at late-
times, and therefore we will ignore them, and use the simplifying form

$\displaystyle{\cal G}_{R}(\nu-\omega,-{\bm{p}}){\cal G}_{R}(\omega,{\bm{p}})$
$\displaystyle=\frac{\pi
i}{2\epsilon_{\bm{p}}^{2}}\left(\frac{\delta(\omega-\nu+\epsilon_{\bm{p}})}{\nu+i0}+\frac{\delta(\omega-\nu-\epsilon_{\bm{p}})}{\nu+i0}\right)$
(3.31)

or because of the two delta functions, might as well

$\displaystyle{\cal G}_{R}(\nu-\omega,-{\bm{p}}){\cal G}_{R}(\omega,{\bm{p}})$
$\displaystyle=\frac{\pi
i}{2\epsilon_{\bm{p}}^{2}}\left(\frac{\delta(\omega+\epsilon_{\bm{p}})}{\nu+i0}+\frac{\delta(\omega-\epsilon_{\bm{p}})}{\nu+i0}\right)$
(3.32) $\displaystyle=\frac{\pi
i}{\epsilon_{\bm{p}}}\frac{\delta(\omega^{2}-\epsilon_{\bm{p}}^{2})}{\nu+i0}$
(3.33)

Finally, anticipating a self-energy induced finite lifetime
$\Gamma_{\bm{p}}=\Im\left[\Sigma_{R}(\epsilon_{\bm{p}},{\bm{p}})\right]/2\epsilon_{\bm{p}}$
(see Eqn 5.5):

$\displaystyle{\cal G}_{R}(\nu-\omega,-{\bm{p}}){\cal G}_{R}(\omega,{\bm{p}})$
$\displaystyle=\frac{\pi
i}{\epsilon_{\bm{p}}}\frac{\delta(\omega^{2}-\epsilon_{\bm{p}}^{2})}{\nu+i2\Gamma_{\bm{p}}}$
(3.34)

###  C.3 Simplifying ${\cal G}_{\rm eff}$

To include the contribution of the type 2 rungs we must simplify ${\cal
G}_{\rm eff}$ first. In what follows we use the free $\phi$ field expressions
for ${\cal G}_{W}$ based on the observation that the self-energy
$\Sigma_{R}\propto\frac{1}{N}$. In can be verified that:

$\displaystyle{\cal G}_{W}(\omega,{\bm{k}})$ $\displaystyle={\cal
Q}(\omega)A(\omega,{\bm{k}})$ (3.35) $\displaystyle{\cal Q}(\omega)$
$\displaystyle=\frac{1}{2\sinh\left(\frac{\beta\omega}{2}\right)}$ (3.36)
$\displaystyle{\cal G}_{W}(\omega,{\bm{k}})$
$\displaystyle=Q(\omega)\frac{\pi}{\epsilon_{\bm{k}}}\left(\delta(\omega-\epsilon_{\bm{k}})-\delta(\omega+\epsilon_{\bm{k}})\right)$
(3.37)

Then

$\displaystyle{\cal G}_{W}^{2}$ $\displaystyle\equiv{\cal
G}_{W}(\omega^{\prime\prime}-\omega,{\bm{p}}^{\prime\prime}-{\bm{p}}){\cal
G}_{W}(\omega^{\prime}-\omega^{\prime\prime},{\bm{p}}^{\prime}-{\bm{p}}^{\prime\prime})$
$\displaystyle=\frac{\pi^{2}{\cal Q}(\omega^{\prime\prime}-\omega){\cal
Q}(\omega^{\prime}-\omega^{\prime\prime})}{\epsilon_{{\bm{p}}^{\prime\prime}-{\bm{p}}}\epsilon_{{\bm{p}}^{\prime}-{\bm{p}}^{\prime\prime}}}\left[\delta(\omega^{\prime\prime}-\omega-\epsilon_{{\bm{p}}^{\prime\prime}-{\bm{p}}})-\delta(\omega^{\prime\prime}-\omega+\epsilon_{{\bm{p}}^{\prime\prime}-{\bm{p}}})\right]$
$\displaystyle\qquad\times\left[\delta(\omega^{\prime}-\omega^{\prime\prime}-\epsilon_{{\bm{p}}^{\prime}-{\bm{p}}^{\prime\prime}})-\delta(\omega^{\prime}-\omega^{\prime\prime}+\epsilon_{{\bm{p}}^{\prime}-{\bm{p}}^{\prime\prime}})\right]$

And after performing the $\omega^{\prime\prime}$ integral and changing
coordinates to
$\bar{{\bm{p}}}={\bm{p}}^{\prime}-{\bm{p}},{\bm{P}}=({\bm{p}}+{\bm{p}}^{\prime})/2,\bar{\omega}=\omega^{\prime}-\omega$…
we arrive at

$\displaystyle{\cal G}_{\rm
eff}(\omega^{\prime},\omega,\bar{{\bm{p}}},\bar{{\bm{P}}})=\frac{1}{2}\int\frac{d^{2}{\bm{p}}^{\prime\prime}}{(2\pi)^{2}}\left[({\bm{P}}-\frac{{\bar{\bm{p}}}}{2})\cdot({\bm{p}}^{\prime\prime}+\frac{{\bar{\bm{p}}}}{2})\,\,({\bm{P}}+\frac{{\bar{\bm{p}}}}{2})\cdot({\bm{p}}^{\prime\prime}-\frac{{\bar{\bm{p}}}}{2})\right]^{2}\frac{\pi}{\epsilon_{+}\epsilon_{-}}$
(3.38) $\displaystyle\times\\{Q(\epsilon_{+})Q(\bar{\omega}-\epsilon_{+}){\cal
G}_{R,\lambda}(\nu-\omega-\epsilon_{+},-{\bm{p}}^{\prime\prime}-{\bm{P}}){\cal
G}_{R,\lambda}(\omega+\epsilon_{+},{\bm{p}}^{\prime\prime}+{\bm{P}})\left(\delta(\bar{\omega}-\epsilon_{+}-\epsilon_{-})-\delta(\bar{\omega}-\epsilon_{+}+\epsilon_{-})\right)$
$\displaystyle-Q(-\epsilon_{+})Q(\bar{\omega}+\epsilon_{+}){\cal
G}_{R,\lambda}(\nu-\omega+\epsilon_{+},-{\bm{p}}^{\prime\prime}-{\bm{P}}){\cal
G}_{R,\lambda}(\omega-\epsilon_{+},{\bm{p}}^{\prime\prime}+{\bm{P}})\left(\delta(\bar{\omega}-\epsilon_{+}-\epsilon_{-})-\delta(\bar{\omega}+\epsilon_{+}+\epsilon_{-})\right)\\}$

Let $\theta$ ($\phi$) be the angle between ${\bm{P}}$
(${\bm{p}}^{\prime\prime}$) and ${\bar{\bm{p}}}$, then

$\displaystyle\left[({\bm{P}}-\frac{{\bar{\bm{p}}}}{2})\cdot({\bm{p}}^{\prime\prime}+\frac{{\bar{\bm{p}}}}{2})\,\,({\bm{P}}+\frac{{\bar{\bm{p}}}}{2})\cdot({\bm{p}}^{\prime\prime}-\frac{{\bar{\bm{p}}}}{2})\right]^{2}$
$\displaystyle\qquad\qquad=\left[\right(Pp^{\prime\prime}\cos(\theta-\phi)-\frac{\bar{p}^{2}}{4}\left){}^{2}-\left(\frac{P\bar{p}}{2}\cos\theta-\frac{\bar{p}p^{\prime\prime}}{2}\cos\phi\right)^{2}\right]^{2}$
(3.39)

For ${\bm{q}}\neq 0$, we can simplify to

$\displaystyle{\cal G}_{\rm
eff}(\nu,{\bm{q}};\omega,\omega^{\prime},{\bm{p}},{\bm{p}}^{\prime})=\frac{1}{2}\int\frac{d^{2}{\bm{p}}^{\prime\prime}}{(2\pi)^{2}}\left[({\bm{q}}-{\bm{P}}+\frac{{\bar{\bm{p}}}}{2})\cdot({\bm{p}}^{\prime\prime}+\frac{{\bar{\bm{p}}}}{2})\,\,({\bm{q}}-{\bm{P}}-\frac{{\bar{\bm{p}}}}{2})\cdot({\bm{p}}^{\prime\prime}-\frac{{\bar{\bm{p}}}}{2})\right]$
$\displaystyle\times\left[({\bm{P}}-\frac{{\bar{\bm{p}}}}{2})\cdot({\bm{p}}^{\prime\prime}+\frac{{\bar{\bm{p}}}}{2})\,\,({\bm{P}}+\frac{{\bar{\bm{p}}}}{2})\cdot({\bm{p}}^{\prime\prime}-\frac{{\bar{\bm{p}}}}{2})\right]\frac{\pi}{\epsilon_{+}\epsilon_{-}}$
$\displaystyle\times\\{Q(\epsilon_{+})Q(\bar{\omega}-\epsilon_{+}){\cal
G}_{R,\lambda}(\nu-\omega-\epsilon_{+},{\bm{q}}-{\bm{p}}^{\prime\prime}-{\bm{P}}){\cal
G}_{R,\lambda}(\omega+\epsilon_{+},{\bm{p}}^{\prime\prime}+{\bm{P}})\left(\delta(\bar{\omega}-\epsilon_{+}-\epsilon_{-})-\delta(\bar{\omega}-\epsilon_{+}+\epsilon_{-})\right)$
$\displaystyle-Q(-\epsilon_{+})Q(\bar{\omega}+\epsilon_{+}){\cal
G}_{R,\lambda}(\nu-\omega+\epsilon_{+},{\bm{q}}-{\bm{p}}^{\prime\prime}-{\bm{P}}){\cal
G}_{R,\lambda}(\omega-\epsilon_{+},{\bm{p}}^{\prime\prime}+{\bm{P}})\left(\delta(\bar{\omega}-\epsilon_{+}-\epsilon_{-})-\delta(\bar{\omega}+\epsilon_{+}+\epsilon_{-})\right)\\}$

##  Appendix D $\lambda$ self-energy

###  D.1 Zero temperature

$\displaystyle\Pi(i\omega_{n},{\bm{q}})$
$\displaystyle=\frac{1}{2}\int\frac{d\nu}{2\pi}\int_{\bm{k}}^{\Lambda}\frac{\left({\bm{k}}\cdot({\bm{k}}+{\bm{q}})\right)^{2}}{\left(\omega_{n}+\nu\right)^{2}+\epsilon^{2}_{{\bm{k}}+{\bm{q}}}}\frac{1}{\nu^{2}+\epsilon^{2}_{\bm{k}}}$
(4.1)
$\displaystyle=\frac{1}{2}\int_{\bm{k}}\left({\bm{k}}\cdot({\bm{k}}+{\bm{q}})\right)^{2}\frac{\epsilon_{\bm{k}}+\epsilon_{{\bm{k}}+{\bm{q}}}}{2\epsilon_{{\bm{k}}}\epsilon_{{\bm{k}}+{\bm{q}}}\left((\epsilon_{\bm{k}}+\epsilon_{{\bm{k}}+{\bm{q}}})^{2}+\omega_{n}^{2}\right)}$
(4.2) $\displaystyle\Pi_{R}(\omega,{\bm{q}})$
$\displaystyle=\Pi(i\omega=\omega+i0,{\bm{q}})$ (4.3)
$\displaystyle=\frac{1}{2}\int_{\bm{k}}\left({\bm{k}}\cdot({\bm{k}}+{\bm{q}})\right)^{2}\frac{1}{4\epsilon_{{\bm{k}}}\epsilon_{{\bm{k}}+{\bm{q}}}}\left[\frac{1}{(\omega+\epsilon_{\bm{k}}+\epsilon_{{\bm{q}}+{\bm{k}}}+i\epsilon)}-\frac{1}{(\omega-\epsilon_{\bm{k}}-\epsilon_{{\bm{q}}+{\bm{k}}}+i\epsilon)}\right]$
(4.4) $\displaystyle\mathrm{Im}\,\Pi_{R}(\omega,{\bm{q}})$
$\displaystyle=\frac{1}{8}\int_{\bm{k}}\frac{\left({\bm{k}}\cdot({\bm{k}}+{\bm{q}})\right)^{2}}{\epsilon_{{\bm{k}}}\epsilon_{{\bm{k}}+{\bm{q}}}}\left[-\pi\delta(\omega+\epsilon_{\bm{k}}+\epsilon_{{\bm{q}}+{\bm{k}}})+\pi\delta(\omega-\epsilon_{\bm{k}}-\epsilon_{{\bm{q}}+{\bm{k}}})\right]$
$\displaystyle=\frac{1}{8}\int_{\bm{k}}\frac{\left(({\bm{k}}-{\bm{q}}/2)\cdot({\bm{k}}+{\bm{q}}/2)\right)^{2}}{\epsilon_{{\bm{k}}-{\bm{q}}/2}\epsilon_{{\bm{k}}+{\bm{q}}/2}}$
$\displaystyle\times\left[-\pi\delta(\omega+\epsilon_{{\bm{k}}-{\bm{q}}/2}+\epsilon_{{\bm{k}}+{\bm{q}}/2})+\pi\delta(\omega-\epsilon_{{\bm{k}}-{\bm{q}}/2}-\epsilon_{{\bm{k}}+{\bm{q}}/2})\right]$
(4.6)

where in the last line we shifted ${\bm{k}}\rightarrow{\bm{k}}-{\bm{q}}/2$.
Now, some simplifications. We work first with
$\epsilon^{2}_{{\bm{k}}}=\sqrt{\kappa}{\bm{k}}^{4}$.

$\displaystyle\epsilon_{{\bm{k}}+{\bm{q}}/2}\epsilon_{{\bm{k}}-{\bm{q}}/2}$
$\displaystyle=\kappa\left({\bm{k}}^{2}+\frac{{\bm{q}}^{2}}{4}+kq\cos\phi\right)\left({\bm{k}}^{2}+\frac{{\bm{q}}^{2}}{4}-kq\cos\phi\right)$
(4.7)
$\displaystyle=\kappa\left[\left({\bm{k}}^{2}+\frac{{\bm{q}}^{2}}{4}\right)^{2}-{\bm{k}}^{2}{\bm{q}}^{2}\cos^{2}\phi\right]$
(4.8)
$\displaystyle\epsilon_{{\bm{k}}-{\bm{q}}/2}+\epsilon_{{\bm{k}}+{\bm{q}}/2}$
$\displaystyle=\sqrt{\kappa}\left(2{\bm{k}}^{2}+\frac{{\bm{q}}^{2}}{2}\right)$
(4.9)
$\displaystyle\left(({\bm{k}}-{\bm{q}}/2)\cdot({\bm{k}}+{\bm{q}}/2)\right)^{2}$
$\displaystyle=\left({\bm{k}}^{2}-\frac{{\bm{q}}^{2}}{4}\right)^{2}$ (4.10)

Since the imaginary part of $\Pi_{R}$ is easily seen to be odd in $\omega$, we
choose to evaluate the $\omega>0$ part.

$\displaystyle\mathrm{Im}\,\Pi_{R}(\omega>0,{\bm{q}})$
$\displaystyle=\frac{1}{8}\int_{\bm{k}}\frac{\left(({\bm{k}}-{\bm{q}}/2)\cdot({\bm{k}}+{\bm{q}}/2)\right)^{2}}{\epsilon_{{\bm{k}}-{\bm{q}}/2}\epsilon_{{\bm{k}}+{\bm{q}}/2}}\pi\delta(\omega-\epsilon_{{\bm{k}}-{\bm{q}}/2}-\epsilon_{{\bm{k}}+{\bm{q}}/2})$
(4.11) $\displaystyle=\frac{\pi}{8}\int\frac{d\phi
kdk}{(2\pi)^{2}}\frac{\left({\bm{k}}^{2}-\frac{{\bm{q}}^{2}}{4}\right)^{2}\delta(\omega-\sqrt{\kappa}\left(2{\bm{k}}^{2}+\frac{{\bm{q}}^{2}}{2}\right))}{\kappa\left[\left({\bm{k}}^{2}+\frac{{\bm{q}}^{2}}{4}\right)^{2}-{\bm{k}}^{2}{\bm{q}}^{2}\cos^{2}\phi\right]}$
(4.12)

For $\omega\geq\sqrt{\kappa}q^{2}/2$ we have the (physical) root

$k_{r}=+\sqrt{\frac{\omega-\sqrt{\kappa}q^{2}/2}{2\sqrt{\kappa}}}=\sqrt{\frac{\omega}{2\sqrt{\kappa}}-\frac{q^{2}}{4}}$

Putting it all together, and recalling that $|{\bm{k}}|\leq\Lambda$:

$\displaystyle\mathrm{Im}\,\Pi_{R}(\omega>0,{\bm{q}})$
$\displaystyle=\frac{\pi}{8}\int\frac{d\phi}{(2\pi)^{2}}\frac{\left(k_{r}^{2}-\frac{q^{2}}{4}\right)^{2}}{\kappa\left[\left(k_{r}^{2}+\frac{q^{2}}{4}\right)^{2}-k_{r}^{2}q^{2}\cos^{2}\phi\right]}\frac{k_{r}}{|-4\sqrt{\kappa}k_{r}|}$
(4.13) $\displaystyle\times\theta\left(\Lambda-
k_{r}\right)\theta\left(\omega-\frac{\sqrt{\kappa}q^{2}}{2}\right)$
$\displaystyle=\frac{\pi}{8}\int\frac{d\phi}{(2\pi)^{2}}\frac{\frac{1}{2^{2}}\left(\frac{\omega}{\sqrt{\kappa}}-q^{2}\right)^{2}}{\kappa\left[\left(k_{r}^{2}+\frac{q^{2}}{4}\right)^{2}-k_{r}^{2}q^{2}\cos^{2}\phi\right]}\frac{1}{4\sqrt{\kappa}}$
(4.14)
$\displaystyle=\frac{\pi}{2^{7}\kappa^{3/2}}\left(\frac{\omega}{\sqrt{\kappa}}-q^{2}\right)^{2}\int\frac{d\phi}{(2\pi)^{2}}\frac{1}{\left[\left(k_{r}^{2}+\frac{q^{2}}{4}\right)^{2}-k_{r}^{2}q^{2}\cos^{2}\phi\right]}$
(4.15)

Doing the $\phi$ integral, we have

$\displaystyle\mathrm{Im}\,\Pi_{R}(\omega>0,{\bm{q}})$
$\displaystyle=\frac{\pi}{32\kappa^{3/2}}\frac{1}{2^{2}}\frac{\left(\frac{\omega}{\sqrt{\kappa}}-q^{2}\right)^{2}}{(2\pi)^{2}}\frac{2\pi\theta\left(\Lambda-
k_{r}\right)\theta\left(\omega-\frac{\sqrt{\kappa}q^{2}}{2}\right)}{\sqrt{\left(k_{r}^{2}+\frac{q^{2}}{4}\right)^{2}\left[\left(k_{r}^{2}+\frac{q^{2}}{4}\right)^{2}-k_{r}^{2}q^{2}\right]}}$
(4.16)
$\displaystyle=\frac{\pi}{32\kappa^{3/2}}\frac{1}{2^{2}}\left(\frac{\omega}{\sqrt{\kappa}}-q^{2}\right)^{2}\frac{1}{(2\pi)^{2}}\frac{32\pi}{q^{4}-16k_{r}^{4}}$
(4.17)
$\displaystyle=\frac{\pi}{32\kappa^{3/2}}\frac{1}{2^{2}}\left(\frac{\omega}{\sqrt{\kappa}}-q^{2}\right)^{2}\frac{1}{(2\pi)^{2}}\frac{8\pi\kappa}{\omega\left(\omega-\sqrt{\kappa}q^{2}\right)}$
(4.18)
$\displaystyle=\frac{\pi}{32\kappa^{3/2}}\frac{1}{2^{2}}\frac{1}{(2\pi)^{2}}\frac{8\pi}{\omega}\left(\omega-\sqrt{\kappa}q^{2}\right)$
(4.19)

and finally

$\displaystyle\mathrm{Im}\,\Pi_{R}(\omega>0,{\bm{q}})$
$\displaystyle=\frac{1}{64\kappa^{3/2}}\frac{\left(\omega-\sqrt{\kappa}q^{2}\right)}{\omega}\theta\left(\omega-\frac{\sqrt{\kappa}q^{2}}{2}\right)\theta\left(2\sqrt{\kappa}(\Lambda^{2}+\frac{q^{2}}{4})-\omega\right)$
(4.21)

###  D.2 Finite temperature

At $T>0$, we have expressions similar to Appendix B of Chowdhury:

$\displaystyle\Im\left[\Pi_{R}(\nu+i0,q)\right]_{\nu>0}$
$\displaystyle=\frac{1}{2}\int_{\bm{k}}^{\Lambda}\frac{\pi\left({\bm{k}}\cdot({\bm{k}}+{\bm{q}})\right)^{2}}{4\epsilon_{\bm{k}}\epsilon_{{\bm{k}}+{\bm{q}}}}2\left[b(\epsilon_{{\bm{k}}+{\bm{q}}})-b(\epsilon_{\bm{k}})\right]\delta(\nu+\epsilon_{{\bm{k}}+{\bm{q}}}-\epsilon_{\bm{k}})$
$\displaystyle+\frac{1}{2}\int_{\bm{k}}^{\Lambda}$
$\displaystyle\frac{\pi\left({\bm{k}}\cdot({\bm{k}}+{\bm{q}})\right)^{2}}{4\epsilon_{\bm{k}}\epsilon_{{\bm{k}}+{\bm{q}}}}\left[b(\epsilon_{{\bm{k}}+{\bm{q}}})-b(-\epsilon_{\bm{k}})\right]\delta(\epsilon_{{\bm{k}}+{\bm{q}}}+\epsilon_{\bm{k}}-\nu)$
(4.22)

So far this expression is general, and $\epsilon_{\bm{k}}$ could be the one-
loop corrected dispersion. Note that if we set ${\bm{q}}=0$ above we obtain:

$\displaystyle\Im\left[\Pi_{R}(\nu+i0,q=0)\right]_{\nu>0}$
$\displaystyle=\frac{1}{2}\int_{\bm{k}}\frac{\pi\left({\bm{k}}\right)^{4}}{4\epsilon_{\bm{k}}^{2}}\left[b(\epsilon_{{\bm{k}}})-b(-\epsilon_{\bm{k}})\right]\delta(2\epsilon_{{\bm{k}}}-\nu)$
(4.23)

This can be finally simplified to:

$\displaystyle\Im\left[\Pi_{R}(\nu+i0,q=0)\right]_{\nu>0}$
$\displaystyle=\frac{b(\nu/2)-b(-\nu/2)}{16\nu\sqrt{r^{2}+\kappa\nu^{2}}}\left(\frac{-r+\sqrt{r^{2}+\kappa\nu^{2}}}{2\kappa}\right)^{2}\theta\left(2\sqrt{\kappa}\Lambda^{2}-\nu\right)$

Now, at $T=0$ we must set $r=0$ and we obtain

$\displaystyle\Im\left[\Pi_{R}(\nu+i0,q)\right]_{\nu>0}$
$\displaystyle=\frac{1}{64\kappa^{3/2}}\left(2\sqrt{\kappa}\Lambda^{2}-\nu\right)$
(4.24)

Sending $\nu\rightarrow 0$ instead we have

$\displaystyle\Im\left[\Pi_{R}(\nu,q=0)\right]$
$\displaystyle=\frac{T}{16\nu^{2}r}\frac{\nu^{4}}{16r^{2}}=T\frac{\nu^{2}}{2^{8}r^{3}}$
(4.25)

which implies that ${\cal G}_{W,\lambda}(\nu,q=0)\rightarrow 0$ as
$\nu\rightarrow 0$ because the $1/\sinh(\beta\nu)$ is overcome.

Aftering shifting ${\bm{k}}\rightarrow{\bm{k}}-{\bm{q}}/2$, and
$\epsilon_{\pm}=\epsilon_{{\bm{k}}+\pm{\bm{q}}/2}$ in (D.2):

$\displaystyle\Im\left[\Pi_{R}(\nu+i0,q)\right]_{\nu>0}$
$\displaystyle=\frac{1}{2}\int_{\bm{k}}\frac{\pi\left(k^{2}-q^{2}/4\right)^{2}}{4\epsilon_{+}\epsilon_{-}}2\left[b(\epsilon_{+})-b(\epsilon_{-})\right]\delta(\nu+\epsilon_{+}-\epsilon_{-})$
$\displaystyle+\frac{1}{2}\int_{\bm{k}}\frac{\pi\left(k^{2}-q^{2}/4\right)^{2}}{4\epsilon_{+}\epsilon_{-}}\left[b(\epsilon_{+})-b(-\epsilon_{-})\right]\delta(\epsilon_{+}+\epsilon_{-}-\nu)$
$\displaystyle=\frac{\pi}{8}\int\frac{d\phi}{(2\pi)^{2}}\int
k\,dk\,\frac{\left(k^{2}-q^{2}/4\right)^{2}}{\epsilon_{+}\epsilon_{-}}2\left[b(\epsilon_{+})-b(\epsilon_{-})\right]\delta(\nu+\epsilon_{+}-\epsilon_{-})$
$\displaystyle+\frac{\pi}{8}\int\frac{d\phi}{(2\pi)^{2}}\int
k\,dk\,\frac{\left(k^{2}-q^{2}/4\right)^{2}}{\epsilon_{+}\epsilon_{-}}\left[b(\epsilon_{+})-b(-\epsilon_{-})\right]\delta(\epsilon_{+}+\epsilon_{-}-\nu)$
$\displaystyle=\frac{\pi}{8}\frac{1}{(2\pi)^{2}}\left(I_{1}+I_{2}\right)$
(4.27)

At finite temperature it is imperative that we use the full expression for
$\epsilon_{\pm}^{2}=r(T)k_{\pm}^{2}+\kappa k_{\pm}^{4}$. Let us scale
somethings out:

$\displaystyle A(\nu,q,r,\kappa,T)$
$\displaystyle=\Im\Pi_{R}(\nu,q,r,\kappa,T)$ (4.28)
$\displaystyle=B(\frac{\nu}{T},\frac{q}{\sqrt{T}},\frac{r}{T},\kappa)$ (4.29)

Since $r(T)=\alpha(T)T$, where $\alpha$ is dimensionless and very weakly
dependent on $T$, we will just replace $r/T=\alpha\approx const$, but keeping
in mind it is actually a function of temperature. We can do this once more
with dimensionless $\kappa$:

$\displaystyle A(\nu,q,r,\kappa,T)$
$\displaystyle=B(\frac{\nu}{T},\frac{q}{\sqrt{T}},\alpha,\kappa)$ (4.30)
$\displaystyle=\frac{1}{\kappa^{3/2}}C(\frac{\nu}{T},\frac{q\kappa^{\frac{1}{4}}}{\sqrt{T}},\frac{\alpha}{\sqrt{\kappa}})\frac{\pi}{8}\frac{1}{(2\pi)^{2}}$
(4.31)

where finally

$\displaystyle C(\nu,q,\alpha)$
$\displaystyle=I_{1}(\nu,q,\alpha)+I_{2}(\nu,q,\alpha)$ (4.32)

So we need to evaluate two integrals where implicitly we have set
$\kappa=1,T=1$ and $\alpha\ll 1$ a constant.

$\displaystyle I_{1}(\nu,q,\alpha)$
$\displaystyle=2\int_{0}^{2\pi}d\phi\int_{0}^{\infty}k\,dk\,\frac{\left(k^{2}-q^{2}/4\right)^{2}}{\epsilon_{+}\epsilon_{-}}\left[b(\epsilon_{+})-b(\epsilon_{-})\right]\delta(\nu+\epsilon_{+}-\epsilon_{-})$
(4.33) $\displaystyle I_{2}(\nu,q)$ $\displaystyle=\int d\phi\int
k\,dk\,\frac{\left(k^{2}-q^{2}/4\right)^{2}}{\epsilon_{+}\epsilon_{-}}\left[b(\epsilon_{+})-b(-\epsilon_{-})\right]\delta(\epsilon_{+}+\epsilon_{-}-\nu)$
(4.34)

Some observations that are useful for testing numerics: $I_{2}=0$ for
$q>q^{*}$ where $q^{*}$ is a root of $2\epsilon_{\bm{q}}=\nu$. Recall due the
the factor of $\sinh(\beta\omega)$ in ${\cal G}_{W,\lambda}$ we only care
about $\omega/T\sim 1$, and certainly not $\omega/T\gg 1$. We verified
numerically that $\Im\Pi_{R}(\nu,{\bm{q}})\rightarrow\frac{1}{64\kappa^{3/2}}$
for $\nu\gg T,\kappa{\bm{q}}^{2}$, which matches the $T=0,{\bm{q}}=0$ result.

###  D.3 $\Re\Pi_{R}$

The KK relation tells us that for a function analytic in the upper half plane
(which implies its real-time version vanishes for $t<0$), its real and
imaginary parts can be determined from each other.

$\displaystyle\Re\Pi_{R}(\nu,q)$
$\displaystyle=\frac{2}{\pi}P\int_{0}^{\infty}d\nu\frac{\omega\Im\Pi_{R}(\omega,q)-\nu\Im\Pi_{R}(\nu,q)}{\omega^{2}-\nu^{2}}$
(4.35)

Armed with $\Re\Pi_{R}(\nu,q)$ we have the $O(1)$ dressed $\lambda$
propagator:

$\displaystyle{\cal G}_{\lambda}(i\omega_{n},{\bm{q}})$
$\displaystyle=\frac{1}{\frac{-1}{4v}-\Pi(i\omega_{n},{\bm{q}})}$ (4.36)
$\displaystyle{\cal G}_{R,\lambda}(\omega+i0,{\bm{q}})$ $\displaystyle=-{\cal
G}_{\lambda}(i\omega_{n}=\omega+i0,{\bm{q}})$ (4.37)
$\displaystyle=\frac{1}{\frac{1}{4v}+\Pi_{R}(\omega,{\bm{q}})}$ (4.38)

From the $T=0$ expression for $\Im\Pi_{R}(\nu,{\bm{q}})$ we can see that since
$\Im\Pi_{R}(\nu,{\bm{q}})\rightarrow\frac{1}{64\kappa^{3/2}}$ at large $\nu$,
we expect a logarithmic divergence in the KK transform.

Numerically, it can be handled as follows. Find a number $c$, such that at
$\nu=c\nu_{*}({\bm{q}})$, $\Im\Pi_{R}$ is sufficiently close to its asymptotic
value. In our case, for $r(T)=10^{-5}T$, this turns out to be $c\approx 2500$.
Split the integral

$\displaystyle\Re\Pi_{R}(\omega,{\bm{q}})$
$\displaystyle=\frac{2}{\pi}\int_{0}^{2500\nu_{*}({\bm{q}})}d\nu\frac{\omega\Im\Pi_{R}(\omega,q)-\nu\Im\Pi_{R}(\nu,q)}{\omega^{2}-\nu^{2}}$
(4.39)
$\displaystyle+\frac{2}{\pi}\int_{2500\nu_{*}({\bm{q}})}^{E}d\nu\frac{\omega\Im\Pi_{R}(\omega,q)-\nu\Im\Pi_{R}(\nu,q)}{\omega^{2}-\nu^{2}}$
(4.40)

The first line can be done numerically, while the second can be done
analytically with the numerics-backed observation that the imaginary part of
$\Pi_{R}$ doesn’t change much.

$\displaystyle\Re\Pi_{R}(\omega,{\bm{q}})$
$\displaystyle=\frac{2}{\pi}\int_{0}^{2500\nu_{*}({\bm{q}})}d\nu\frac{\omega\Im\Pi_{R}(\omega,q)-\nu\Im\Pi_{R}(\nu,q)}{\omega^{2}-\nu^{2}}$
(4.41) $\displaystyle-$
$\displaystyle\frac{2}{\pi}\Im\Pi_{R}(\omega,{\bm{q}}){\rm
Arctanh}\left(\frac{\omega}{2500\nu_{*}({\bm{q}})}\right)$ (4.42)
$\displaystyle+\frac{1}{64\pi\kappa^{3/2}}\log\left|\frac{E^{2}-\omega^{2}}{(2500\nu_{*}({\bm{q}}))^{2}-\omega^{2}}\right|$
(4.43)

In subsequent computations we drop the term in the last line above.

##  Appendix E Scattering rate of $\phi$

We need to evaluate $G_{R}(\omega,{\bm{q}})$, and more specifically

$\displaystyle\Sigma\left(i\omega_{n},{\bm{q}}\right)$
$\displaystyle=\frac{T}{N}\sum_{i\nu_{n}}\int_{\bm{k}}^{\Lambda}\left({\bm{q}}\cdot({\bm{q}}+{\bm{k}})\right)^{2}{\cal
G}(i\omega_{n}+i\nu_{n},{\bm{q}}+{\bm{k}}){\cal
G}_{\lambda}(i\nu_{n},{\bm{k}})$ (5.1)

Following the Chowdhury[11] we derive

$\displaystyle\Im\Sigma_{R}(\omega+i0,{\bm{q}})$
$\displaystyle=\frac{1}{N}\int_{\bm{k}}^{\Lambda}\left({\bm{q}}\cdot{\bm{k}}\right)^{2}\frac{\sinh(\beta\omega/2)}{4\epsilon_{\bm{k}}\sinh(\beta\epsilon_{\bm{k}})/2}$
(5.2) $\displaystyle\qquad\times\left[{\cal
G}_{W,\lambda}(\epsilon_{\bm{k}}-\omega,{\bm{k}}-{\bm{q}})+{\cal
G}_{W,\lambda}(-\epsilon_{\bm{k}}-\omega,{\bm{k}}-{\bm{q}})\right]$

Written in rescaled variables

$\displaystyle\Im\Sigma_{R}(\omega T,{\bm{q}}\sqrt{T})$
$\displaystyle=\frac{T^{2}}{N}\int_{\bm{k}}^{\Lambda/\sqrt{T}}\left({\bm{q}}\cdot{\bm{k}}\right)^{2}\frac{\sinh(\omega/2)}{4\epsilon_{\bm{k}}\sinh(\epsilon_{\bm{k}})/2}$
(5.3) $\displaystyle\qquad\times\left[{\cal
G}_{W,\lambda}(\epsilon_{\bm{k}}-\omega,{\bm{k}}-{\bm{q}})+{\cal
G}_{W,\lambda}(-\epsilon_{\bm{k}}-\omega,{\bm{k}}-{\bm{q}})\right]$

Some observations: At $T=0$, $\Im\Pi_{R}$ is cutoff independent (see Eq.4.21),
but $\Re\Pi_{R}$ is clearly cutoff dependent and goes like $\log\Lambda$
because $\Im\Pi_{R}(\nu,{\bm{q}})\rightarrow\frac{1}{64\kappa^{3/2}}$ as
$\nu\rightarrow\infty$. Since the location of the pole is a physical quantity,
we imagine adding corresponding counterterms that keep the
$\nu=\sqrt{r(T){\bm{k}}^{2}+{\bm{k}}^{4}}$ dispersion for $\phi$ once $\Pi$ is
used to compute corrections to ${\cal G}$. Therefore, the theory can be fixed
such that ${\cal G}_{W,\lambda}$ is cutoff independent.

Proceeding further, we note that in Eq.5.3, the cutoff $\Lambda$ can be safely
sent to infinity.

Finally, assuming $\Sigma_{R}(\epsilon_{\bm{q}},{\bm{q}})$ is smaller than
$\epsilon_{\bm{q}}$ (always true for sufficiently large $N$) we define the
inverse lifetime

$\displaystyle\Gamma_{\bm{q}}$
$\displaystyle=\frac{\Im\Sigma_{R}(\epsilon_{\bm{q}},{\bm{q}})}{2\epsilon_{\bm{q}}}$
(5.4) $\displaystyle\Gamma({\bm{q}}\sqrt{T})$
$\displaystyle=\frac{T}{2N}\int_{\bm{k}}^{\Lambda/\sqrt{T}}\left({\bm{q}}\cdot{\bm{k}}\right)^{2}\frac{\sinh(\omega/2)}{4\epsilon_{\bm{q}}\epsilon_{\bm{k}}\sinh(\epsilon_{\bm{k}})/2}$
(5.5) $\displaystyle\qquad\times\left[{\cal
G}_{W,\lambda}(\epsilon_{\bm{k}}-\omega,{\bm{k}}-{\bm{q}})+{\cal
G}_{W,\lambda}(-\epsilon_{\bm{k}}-\omega,{\bm{k}}-{\bm{q}})\right]$

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