# Critical properties of the Floquet Time Crystal

Muath Natsheh1 Andrea Gambassi2,3 Aditi Mitra1 1Center for Quantum Phenomena,
Department of Physics, New York University, 726 Broadway, New York, NY, 10003,
USA
2SISSA - International School for Advanced Studies, via Bonomea 265, 34136
Trieste, Italy
3INFN - Sezione di Trieste, via Bonomea 265, 34136, Trieste, Italy

###### Abstract

The critical properties characterizing the formation of the Floquet time
crystal are investigated analytically in the periodically driven $O(N)$ model.
In particular, we focus on the critical line separating the trivial phase with
period synchronized dynamics and absence of long-range spatial order from the
non-trivial phase where long-range spatial order is accompanied by period-
doubling dynamics. In the vicinity of the critical line, with a combination of
dimensional expansion and exact solution for $N\to\infty$, we determine the
exponent $\nu$ that characterizes the divergence of the spatial correlation
length of the equal-time correlation functions, the exponent $\beta$
characterizing the growth of the amplitude of the order-parameter, as well as
the initial-slip exponent $\theta$ of the aging dynamics when a quench is
performed from deep in the trivial phase to the critical line. The exponents
$\nu,\beta,\theta$ are found to be identical to those in the absence of a
drive. In addition, the functional form of the aging is found to depend on
whether the system is probed at times that are small or large compared to the
drive period. The spatial structure of the two-point correlation functions,
obtained as a linear response to a perturbing potential in the vicinity of the
critical line, is found to show algebraic decays that are longer ranged than
in the absence of a drive, and besides being period-doubled, are also found to
oscillate in space at the wave-vector $\omega/(2v)$, $v$ being the velocity of
the quasiparticles, and $\omega$ being the drive frequency.

##  I Introduction

Floquet time crystals (FTCs) are systems that show spontaneous breaking of
discrete time-translation symmetry (TTS), accompanied by the breaking of
another internal symmetry of the system such as an Ising symmetry Sacha and
Zakrzewski (2017); Else  _et al._ (2020); Khemani  _et al._ (2019). Simply
broken Ising symmetry would imply the well known transition from a
paramagnetic to a ferromagnetic Ising phase Goldenfeld (1992). However, the
appearance of broken TTS adds a new flavor to this problem. Thus natural
questions that arise are: Is there any universality associated with the
transition from a trivial phase where there is no long-range Ising order, and
the dynamics is synchronized with the drive, to a FTC phase? And if yes, what
are the critical exponents of the transition? Are they related to those
encountered at the Ising transition in the absence of drive, or are they
different? This question is particularly relevant due to the numerous
microscopically different experimental platforms that have realized this
phenomenon Zhang  _et al._ (2017); Choi  _et al._ (2017); Rovny  _et al._
(2018); Autti  _et al._ (2018); Pal  _et al._ (2018); Smits  _et al._ (2018).

Large-$N$ methods have proven to be very useful in understanding diverse
systems ranging from impurity models Read and Newns (1983); Ratiani and Mitra
(2009); Mitra and Rosch (2011), Kondo lattice models Read  _et al._ (1984),
and strange metals Sachdev and Ye (1993); Kitaev (2015). In this paper, we
address the question of universality in the context of the transition to a FTC
by studying the periodically driven $O(N)$ model. In the absence of the drive,
and depending on the value of $N$, this model is a textbook example for
studying the transition to various broken symmetry phases both in equilibrium
Eyal  _et al._ (1996); Moshe and Zinn-Justin (2003), and out of equilibrium,
due to, e.g., a quantum quench Sotiriadis  _et al._ (2009); Sotiriadis and
Cardy (2010); Sciolla and Biroli (2011, 2013); Chandran  _et al._ (2013);
Gagel  _et al._ (2014, 2015); Chiocchetta  _et al._ (2015); Maraga  _et al._
(2015); Smacchia  _et al._ (2015); Maraga  _et al._ (2016); Chiocchetta  _et
al._ (2016); Lemonik and Mitra (2016); Chiocchetta  _et al._ (2017). In
addition, the model can be exactly solved in the limit $N\to\infty$, providing
access to equilibrium and non-equilibrium collective behaviors and critical
properties beyond perturbation theory Moshe and Zinn-Justin (2003); Maraga
_et al._ (2015); Lemonik and Mitra (2016).

This paper is organized as follows. In Section II we present the model,
outline the setup of the problem, and summarize the Gaussian results. In
Section III we present the perturbative one-loop calculation and discuss its
effect on the phase diagram and on the correlation length. In Sections IV and
V we determine the exact expressions of the exponents $\nu$ and $\beta$ in the
limit $N\to\infty$ and to $O(\epsilon)$ where $\epsilon=4-d$, $d$ being the
spatial dimension. In Section VI we present results for aging following a
quench, also in the limit $N\rightarrow\infty$ and to $O(\epsilon)$. In
Section VII we present results for the spatial structure of the correlation
functions that are obtained as a linear response to a perturbing potential. We
present our conclusions in Section VIII. Intermediate details of the
calculations are presented in several Appendices.

Figure 1:  Stability phase diagram of the periodically driven $O(N)$ model in
the $a$-$q$ plane where $a=4(r+k^{2})/\omega^{2}$ and $q=2r_{1}/\omega^{2}$.
Compared to the Gaussian approximation reported in the diagram on the left
Natsheh  _et al._ (rint), interactions shift the critical line from
$a_{1}\approx 1+q$ (black line between regions (2) and (b) on the left
diagram) to $a_{1}\approx 1+q-3\widetilde{B}_{4}/\omega^{2}$ (yellow line on
the right diagram), for $q\ll 1$, where $\widetilde{B}_{d}$ is defined in Eq.
(21) and given in Eq. (77) for $d=4$.

##  II Model and set-up

The periodically driven $O(N)$ model we consider is

$\displaystyle H=\sum_{i=1}^{N}\int d^{d}x\frac{1}{2}\bigl{[}$
$\displaystyle(r-r_{1}\cos{(\omega
t)})\phi_{i}^{2}(\textbf{x})+(\vec{\nabla}\phi_{i})^{2}$
$\displaystyle+\Pi_{i}^{2}(\textbf{x})\bigr{]}+V,$ (1)

where $\phi_{i}$ and $\Pi_{i}$ are $N$-component bosonic fields that obey the
canonical commutation relation

$[\phi_{j}(\textbf{x}),\Pi_{l}(\textbf{y})]=i\delta_{jl}\delta^{(d)}(\textbf{x}-\textbf{y}).$
(2)

$V$ is the interaction term

$\displaystyle V=\frac{u}{4N}\int
d^{d}x\biggl{(}\sum_{i=1}^{N}\phi_{i}^{2}\biggr{)}^{2}.$ (3)

In Eq. (1), $\omega=2\pi/T$ is the frequency of the drive with period $T$,
$r_{1}$ is the drive amplitude, and we also define the dimensionless drive
amplitude

$\displaystyle q=\frac{2r_{1}}{\omega^{2}}.$ (4)

In addition, $r$ is a detuning parameter which, for negative values, causes an
instability in the free, undriven model (i.e., with $V=r_{1}=0$). Note that
the Hamiltonian in Eq. (1) has a ${\mathbb{Z}}_{2}$ symmetry under
$\phi\rightarrow-\phi$ and it also has a discrete TTS, i.e., $H(t+T)=H(t)$.
While a conventional Ising ferromagnet corresponds to broken
${\mathbb{Z}}_{2}$ symmetry, we are interested in the FTC phase where both
${\mathbb{Z}}_{2}$ and the discrete TTS are spontaneously broken.

For later convenience, we represent the fields
$O_{i}\in\\{\phi_{i},\Pi_{i}\\}$ in momentum space according to

$\displaystyle
O_{i}(\textbf{x})=\int^{\Lambda}\\!\frac{d^{d}k}{(2\pi)^{d}}e^{i\textbf{k}.\textbf{x}}O_{i,\textbf{k}},$
(5)

where $\Lambda$ is the large-momentum cut-off of the model, which will be
implemented as specified further below. In the absence of interactions, i.e.,
within the Gaussian approximation $V=0$, each mode $k$ is independent of the
others and its dynamics is given by the Mathieu equations Oliver  _et al._
(2010); McLachlan (1947); Richards (1983). The solutions of these equations is
characterized by the dimensionless drive amplitude $q$ introduced in Eq. (4)
and another dimensionless quantity

$\displaystyle a=4\frac{r+k^{2}}{\omega^{2}}.$ (6)

The solutions of the Mathieu equation result in the stability diagram reported
in Fig. 1, consisting of allowed bands (green regions) and band-gaps (red
regions). In the limit of weak drive $q\rightarrow 0$, the band edges are
determined by the condition of parametric resonance for the longest wavelength
mode Natsheh  _et al._ (rint), i.e., integer multiples of the drive frequency
$\omega$ should equal the energy for creating a pair of excitations at $k=0$,
i.e., $n\omega=2\sqrt{r}$, where $n$ is an integer and $\sqrt{r+k^{2}}$ is the
excitation energy of a mode of wavelength $k$ in the undriven system. For
generic values of $q$, the band edges are given by the Mathieu characteristic
values Oliver  _et al._ (2010); McLachlan (1947); Richards (1983) $a_{n}(q)$
and $b_{n}(q)$, where the former corresponds to the upper boundaries and the
latter to the boundaries of the unstable red regions of Fig. 1. Note that
$a_{n}(q\rightarrow 0)=n^{2}$ with $a_{1}(q)=4r_{c}/\omega^{2}$, $r_{c}$ being
the critical value of $r$ that tunes the system to the upper edge of the
unstable region (b) in Fig. 1.

In the presence of interactions, the phase-diagram of the model, before the
onset of heating (which is controlled by $N$ and $\Lambda$), was discussed in
Refs. Chandran and Sondhi (2016); Natsheh  _et al._ (rint). In the $q$-$a$
plane, it comprises of a trivial phase with no spatial long-range order and no
period doubling, separated by a series of critical lines to either period
synchronized ferromagnetic phases (regions (a) and (c) in Fig. 1, for $V=0$),
or to period-doubled FTC phases (region (b), for $V=0$). Here we discuss the
critical properties in the vicinity of one of the FTC critical lines (line
separating regions (2) and (b)) which corresponds to the Mathieu
characteristic value $a_{1}(q)$ when $V=0$.

With drive, the emerging critical properties of Eq. (1) was discussed in
Natsheh  _et al._ (rint) for $V=0$, i.e., within the Gaussian approximation.
It was shown that algebraic behaviors may appear in two-point correlation
functions as a prelude to universality. Here we explore the effect of
interactions on them. In particular we combine a perturbative approach with an
exact solution in the limit $N\to\infty$, finding that these algebraic
behaviors are robust, that the associated exponents are indeed universal and
that they turn out to be identical to those which emerge in the undriven model
when tuned to its critical point.

For a weak drive $q\ll 1$, and within the Gaussian approximation, the value
$r_{c}(q)$ of the parameter $r$ corresponding to the critical line is given by
$r_{c}=(\omega^{2}/4)a_{1}(q)$ with $a_{1}(q)\approx 1+q$ Natsheh  _et al._
(rint), and it is therefore convenient to introduce the parameter

$\displaystyle y=r-r_{c}\approx
r-\left(\frac{\omega}{2}\right)^{2}\left(1+q\right),$ (7)

which controls the detuning away from criticality.

In our analysis, we consider a quantum quench Calabrese and Cardy (2007);
Mitra (2018) in which the initial state is the thermal equilibrium state of
the undriven $O(N)$ model (i.e., $r_{1}=0$) with a positive initial value
$r_{0}>0$ of the detuning $r$ from the corresponding critical line. This state
is subsequently evolved under the periodically driven $O(N)$ model in Eq. (1).
We choose the initial value $r_{0}\gg r>0$ so that the initial state is deep
in the paramagnetic phase with short-range spatial correlations. We also
choose $r_{0}\gg u\langle\phi^{2}\rangle$, where $\langle\phi^{2}\rangle$ is
the average in the initial state. This condition ensures that interactions
hardly modify this initial state.

Central objects in our study are the two-point correlation functions of the
$\phi$ fields, defined by the retarded and Keldysh Green’s functions Kamenev
(2011) $G_{R}$ and $G_{K}$, respectively,

$\displaystyle\delta_{lj}\delta_{\textbf{k},-\textbf{q}}iG_{K}(k,t,t^{\prime})$
$\displaystyle=\langle\left\\{\phi_{l,\textbf{k}}(t),\phi_{j,\textbf{q}}(t^{\prime})\right\\}\rangle,$
(8)
$\displaystyle\delta_{lj}\delta_{\textbf{k},-\textbf{q}}iG_{R}(k,t,t^{\prime})$
$\displaystyle=\theta(t-t^{\prime})\langle\left[\phi_{l,\textbf{k}}(t),\phi_{j,\textbf{q}}(t^{\prime})\right]\rangle,$
(9)

where $\theta(t)$ is the step function which is non-vanishing only for $t>0$
and equals one. Above, since we will be approaching the critical line from the
symmetric phase where all the $N$ components of the order-parameter behave
equivalently, for notational brevity, we do not indicate the field components
which $G_{K,R}$ refer to. The Green’s functions obtained within the Gaussian
approximation will be denoted below by $G_{0K}$ and $G_{0R}$.

It is instructive to write the Keldysh path integral in terms of the
$N$-component classical ($\phi_{i,c}$) and quantum fields ($\phi_{i,q}$). In
terms of the vectors
$\bm{\phi}_{b}=\left(\phi_{1,b},\phi_{2,b},...,\phi_{N,b}\right)$ with
$b\in\\{c,q\\}$, the Keldysh action $\mathcal{S}$ turns out to be

$\displaystyle\mathcal{S}$ $\displaystyle=S_{\rm
initial}+\int_{\mathbf{x},t>0}\biggl{\\{}\dot{\bm{\phi}}_{q}\cdot\dot{\bm{\phi}}_{c}-\vec{\nabla}\bm{\phi}_{q}\cdot\vec{\nabla}\bm{\phi}_{c}$
$\displaystyle\quad\quad-\left[r-r_{1}\cos(\omega
t)\right]\bm{\phi}_{q}\cdot\bm{\phi}_{c}\biggr{\\}}+S_{\rm int},$ (10)

where $\int_{\mathbf{x},t}\equiv\int d^{d}xdt$ and all information about the
pre-quench Hamiltonian enters in $S_{\rm initial}$. Since the quench takes
place at $t=0$, all time integrals run over $t>0$. The contribution $S_{\rm
int}$ of the interactions is

$S_{\rm
int}=-\frac{u}{2N}\int_{\mathbf{x},t>0}\\!\\!\\!\bm{\phi}_{c}\cdot\bm{\phi}_{q}\left(|\bm{\phi}_{c}|^{2}+|\bm{\phi}_{q}|^{2}\right).$
(11)

###  II.1 Correlations in the Gaussian approximation

Under the initial conditions mentioned above, and for the Gaussian theory in
which the fields at different wavevectors decouple, the Green’s functions are
given by Natsheh  _et al._ (rint)

$\displaystyle iG_{0K}(k,t,t^{\prime})=$ $\displaystyle
q^{2}\frac{\sqrt{r_{0}}}{2\overline{\omega_{k}}^{2}}\cos\left(\frac{\omega}{2}t\right)\cos\left(\frac{\omega}{2}t^{\prime}\right)$
$\displaystyle\times[\cos(\overline{\omega_{k}}(t-t^{\prime}))-\cos(\overline{\omega_{k}}(t+t^{\prime}))],$
(12) $\displaystyle G_{0R}(k,t,t^{\prime})=$
$\displaystyle-\theta(t-t^{\prime})q\cos\left(\frac{\omega}{2}t\right)\cos\left(\frac{\omega}{2}t^{\prime}\right)$
$\displaystyle\times\frac{\sin(\overline{\omega_{k}}(t-t^{\prime}))}{\overline{\omega_{k}}},$
(13)

where

$\displaystyle\overline{\omega_{k}}=\sqrt{\bar{k}^{2}+qy/2},\quad{\rm
with}\quad\bar{k}=\sqrt{q/2}\,k.$ (14)

As expected, on approaching the FTC phase, the unequal time correlators
$G_{0R,0K}$ above show period doubling due to the $\cos(\omega t/2)$
prefactors. On the critical line $y=0$, algebraic prefactors $\bar{k}^{-1}$
and $\bar{k}^{-2}$ appear in $G_{0R}$ and $G_{0K}$, respectively, which imply
an emerging light-cone with quasiparticle velocity $v=\sqrt{q/2}$.

The spatial Fourier transform of $G_{R,K}$ is

$\displaystyle G_{R,K}(x,t,t^{\prime})=\frac{1}{(2\pi)^{d/2}x^{d/2-1}}$
$\displaystyle\quad\quad\quad\times\int_{0}^{\Lambda}\\!\\!dk\,k^{d/2}J_{d/2-1}(kx)\
G_{R,K}(k,t,t^{\prime}),$ (15)

where $J_{\alpha}$ is the Bessel function of the first kind which emerges
because of the rotational invariance of the integrand around the direction of
$\vec{x}$. Eq. (15) reveals that in spatial dimension $d=4$, $G_{0K}(x,t,t)$
has the following behavior Natsheh  _et al._ (rint)

$\displaystyle iG_{0K}(x\gg 2vt)$ $\displaystyle\approx 0,$ (16a)
$\displaystyle iG_{0K}(x=2vt)$
$\displaystyle\propto\cos^{2}\left(\frac{\omega}{2}t\right)\frac{1}{x^{3/2}},$
(16b) $\displaystyle iG_{0K}(x\ll 2vt)$
$\displaystyle\propto\cos^{2}\left(\frac{\omega}{2}t\right)\frac{1}{x^{2}}.$
(16c)

Accordingly, $G_{0K}$ decays algebraically as $1/x^{3/2}$ upon increasing $x$
on the light-cone $x=2vt$, as $1/x^{2}$ inside the light-cone $2vt\gg x$,
while it rapidly vanishes outside the light-cone $x\gg 2vt$. Note also that
$G_{0K}(k,t,t^{\prime})$ at equal times $t=t^{\prime}$ does not show period
doubling, but it is synchronized with the drive.

The response function $G_{0R}(x,t,t^{\prime})$, on the other hand, has an
almost delta-function weight on the light-cone Natsheh  _et al._ (rint), as it
vanishes both inside and outside the light-cone while it features an algebraic
decay upon increasing $x$ along the light cone $x=v|t-t^{\prime}|$:

$\displaystyle G_{0R}(x\gg v|t-t^{\prime}|)$ $\displaystyle\approx 0,$ (17a)
$\displaystyle G_{0R}(x=v|t-t^{\prime}|)$
$\displaystyle\propto\cos\left(\frac{\omega}{2}t\right)\cos\left(\frac{\omega}{2}t^{\prime}\right)\frac{1}{x^{3/2}},$
(17b) $\displaystyle G_{0R}(x\ll v|t-t^{\prime}|)$ $\displaystyle\approx 0.$
(17c)

##  III Perturbative correction at one loop

In this section we determine the perturbative one-loop correction to $G_{R,K}$
and discuss its effect on the phase diagram of the model and on its exponent
$\nu$.

In particular, the predictions presented here are derived at the lowest order
in the coupling constant $u$ and they can be easily extended to account for a
generic value of the number $N$ of the components of the field. However, we
will be eventually interested in the limit $N\to\infty$ and therefore we focus
directly on this case. As the fixed-point value $u^{*}$ of the coupling
constant $u$ is expected to be of order $\epsilon=4-d$ for spatial
dimensionality $d\lesssim 4$, the final expression of the quantities analyzed
here will eventually take the form of a dimensional expansion in $\epsilon$,
where terms of order $u^{2}$ and $u\epsilon$ are neglected. In order to
simplify the notation we will explicitly indicate the order of approximation
only in the final expressions.

The one-loop corrections $\delta G_{R,K}$ to $G_{R,K}$ can be written as
Chiocchetta  _et al._ (2015, 2016)

$\displaystyle\delta
G_{R}(k,t,t^{\prime})=\int_{0}^{\infty}d\tau\,G_{0R}(k,t,\tau)\;iT(\tau)\;G_{0R}(k,\tau,t^{\prime}),$
(18a) $\displaystyle\delta iG_{K}(k,t,t^{\prime})$
$\displaystyle=\int_{0}^{\infty}d\tau\,G_{0R}(k,t,\tau)\;iT(\tau)\;iG_{0K}(k,\tau,t^{\prime})$
$\displaystyle\quad\quad\quad\quad+(t\leftrightarrow t^{\prime}),$ (18b)

where $iT(t)$ is the tadpole integral given by

$\displaystyle iT(t)$
$\displaystyle=u\langle\phi^{2}\rangle=\frac{u}{2}iG_{0K}(x=0,t,t)$
$\displaystyle=\frac{u}{2}\int\frac{d^{d}k}{(2\pi)^{d}}\,iG_{0K}(k,t,t)f_{c}\left(k/\Lambda\right),$
(19)

for $N\rightarrow\infty$. In the expression above $f_{c}(x)$ is a cut-off
function with $f_{c}(0)=1$. This function is also assumed to be smooth and to
vanish for $x\gg 1$ in order to reduce the effects of microscopic oscillations
in the dynamical quantities and expose possible underlying universal behaviors
Chiocchetta  _et al._ (2015); Maraga  _et al._ (2015); Chiocchetta  _et al._
(2016). In Appendix A we show that the tadpole in Eq. (19) takes the form

$\displaystyle iT(t)$ $\displaystyle=\widetilde{B}_{d}\cos^{2}(\omega
t/2)+iT^{\prime}(t),$ (20)

where $\widetilde{B}_{d}$ is the constant determined by the leading behavior
$iT(t)\propto\cos^{2}(\omega t/2)$ for $t\to\infty$ while $iT^{\prime}$ is the
time-dependent transient part of the tadpole, which vanishes as $t$ increases.
In $d$ spatial dimension, $\widetilde{B}_{d}$ was reported in Eq. (76) in
Appendix A, which we report here for convenience

$\displaystyle\widetilde{B}_{d}(y)=a_{d}\frac{qu}{2}\sqrt{r_{0}}\int_{0}^{\infty}dk\,k^{d-1}\frac{f_{c}\left(k/\Lambda\right)}{k^{2}+y},$
(21)

where $a_{d}=2/[(4\pi)^{d/2}\Gamma(d/2)]$. For convenience we choose
$f_{c}(x)=e^{-x}$ but a different choice does not affect the large-distance
behavior as long as $f_{c}(0)=1$ and $f_{c}(x)$ vanishes sufficiently fast as
$x$ increases. As the tadpole above is $\propto u$ — the fixed-point value
$u^{*}$ of which is, in turn, proportional to $\epsilon=4-d$ — the rest of the
expression can be actually evaluated directly at $d=4$: the transient part of
the tadpole then turns out to be

$\displaystyle iT^{\prime}(t)\approx\frac{8\theta}{3qt^{2}}\cos^{2}(\omega
t/2)\quad{\rm with}\quad\theta=\frac{3qu\sqrt{r_{0}}}{(16\pi)^{2}},$ (22)

at times $t\gg\bar{\Lambda}^{-1}$ which are longer than the microscopic scale
set by $\bar{\Lambda}=\sqrt{q/2}\Lambda$.

Before considering the effects of the transient contribution to the tadpole on
the resulting $G_{R,K}$, we discuss below the effect of the long-time part
$\propto{\widetilde{B}}_{d}$ in Eq. (20). In particular, in Sec. III.1 we
discuss the perturbative correction it provides to the critical line while in
Sec. III.2 we focus on the correction to the correlation length $\xi$ which
characterize the spatial decay of the equal-time correlation functions at long
times.

###  III.1 Perturbative correction to the critical line

The Keldysh action of the model is given by Eq. (II) and, in the absence of
interactions, i.e., with $S_{\rm int}\mapsto 0$, the part of it which is
proportional to the fields $\bm{\phi}_{q}\cdot\bm{\phi}_{c}$ is
$[r-r_{1}\cos(\omega t)]\bm{\phi}_{q}\cdot\bm{\phi}_{c}$. In the presence of
$S_{\rm int}$, this term is corrected as follows at one-loop

$\displaystyle\left[r-r_{1}\cos(\omega
t)\right]\bm{\phi}_{q}\cdot\bm{\phi}_{c}\mapsto$
$\displaystyle\quad\left[r-r_{1}\cos(\omega
t)+u\langle\phi^{2}\rangle\right]\bm{\phi}_{q}\cdot\bm{\phi}_{c}.$ (23)

Using Eqs. (19) and (20) this expression implies the following correction at
long times

$\displaystyle\left[r-r_{1}\cos(\omega
t)\right]\bm{\phi}_{q}\cdot\bm{\phi}_{c}\mapsto$
$\displaystyle\quad\left[r-r_{1}\cos(\omega
t)+\widetilde{B}_{4}\cos^{2}(\omega
t/2)\right]\bm{\phi}_{q}\cdot\bm{\phi}_{c}.$ (24)

Writing $\cos^{2}(\omega t/2)=[1+\cos(\omega t)]/2$, it is clear that the
effect of the long-time part of the tadpole is to shift

$\displaystyle r\mapsto r+\frac{\widetilde{B}_{4}}{2}\quad\mbox{and}\quad
r_{1}\mapsto r_{1}-\frac{\widetilde{B}_{4}}{2},$ (25)

with the resulting effective action, after these shifts, being still Gaussian
at this order in perturbation theory.

Recall that the critical line for the Gaussian model is given by the condition
$y=0$ in Eq. (7), i.e., the critical value $r_{c}$ of $r$ for $q\ll 1$ is
determined by

$\displaystyle r_{c}\approx\frac{\omega^{2}}{4}+\frac{r_{1}}{2}.$ (26)

The shifts in Eq. (25) imply that the critical value $r_{c}$ of $r$ is now
determined by the condition above imposed on the shifted parameters, i.e.,

$\displaystyle
r_{c}+\frac{\widetilde{B}_{4}}{2}=\frac{\omega^{2}}{4}+\frac{r_{1}}{2}-\frac{\widetilde{B}_{4}}{4}.$
(27)

Accordingly, in terms of the parameters $r$ and $r_{1}$ of the original model
and of the associated dimensionless quantities $a$ and $q$ (see Eqs. (4) and
(6)), the critical line is shifted compared to that in the absence of
interaction (i.e., with $\widetilde{B}_{4}\mapsto 0$) as

$\displaystyle
a_{1}(q)=\frac{4r_{c}}{\omega^{2}}=1+q-3\frac{\widetilde{B}_{4}}{\omega^{2}}.$
(28)

The resulting shift of the critical line is therefore downwards, i.e., the
stable green disordered region, due to the interaction, widens at the expense
of the red unstable (ordered) region, as usually occurs also in equilibrium.
This fact is shown schematically in Fig. 1.

###  III.2 Perturbative correction to the exponent $\nu$

We now discuss the perturbative correction to the correlation length $\xi$
which characterizes the spatial decay of the equal-time correlation function
at long times. This then allows us to determine perturbatively the associated
critical exponent $\nu$ to $O(u)$, which will eventually yield an estimate at
the lowest order in the dimensional expansion $\epsilon=4-d$. In Section IV we
will then determine the exact dependence of $\nu$ on the dimensionality $d$
for $N\to\infty$ beyond the perturbative result presented here.

The shifts in Eq. (25) caused by the long-time limit of the tadpole imply a
shift

$y\mapsto y+3\widetilde{B}_{4}(y)/4$ (29)

in the detuning from the critical line introduced in Eq. (7). Based on the
dependence of $G_{0R}$ on $k$ in Eq. (13) (which, is not altered by the
contribution of the tadpole beyond the shifts in the parameters in Eq. (25)),
one can easily identify the spatial correlation length $\xi$ of the field as
being determined by $y=\xi^{-2}$. Accordingly, the correlation length
accounting for the one loop correction is (see Eq. (29))

$\displaystyle\xi^{-2}=y+\frac{3}{4}\widetilde{B}_{4}(y).$ (30)

Let us denote by $y_{c}$ the critical value of $y$, which is shifted by the
interaction and is determined by the condition that, correspondingly, $\xi$
diverges, i.e., from Eq. (30),

$\displaystyle 0=y_{c}+\frac{3}{4}\widetilde{B}_{4}(y_{c}).$ (31)

Subtracting Eq. (31) from Eq. (30) one finds

$\displaystyle\xi^{-2}$
$\displaystyle=y-y_{c}+\frac{3}{4}\left[\widetilde{B}_{4}(y)-\widetilde{B}_{4}(y_{c})\right]$
$\displaystyle=y-y_{c}+3\frac{u}{8}a_{4}q\sqrt{r_{0}}$
$\displaystyle\times\int_{0}^{\infty}dk\,k^{3}\biggl{[}\frac{1}{k^{2}+y}-\frac{1}{k^{2}+y_{c}}\biggr{]}f_{c}\left(k/\Lambda\right),$
(32)

where, in the last equality, we used Eq. (21). Denoting by $\delta y=y-y_{c}$
the distance from the actual critical line, the previous equation can be
written as

$\displaystyle\xi^{-2}$ $\displaystyle=\delta
y\biggl{[}1-3\frac{u}{8}a_{4}q\sqrt{r_{0}}$
$\displaystyle\quad\quad\times\int_{0}^{\infty}dkk^{3}\frac{f_{c}\left(k/\Lambda\right)}{\left(k^{2}+\delta
y+y_{c}\right)\left(k^{2}+y_{c}\right)}\biggr{]}.$ (33)

Note that, from Eq. (31), we expect $y_{c}$ to vanish in perturbation theory
as the interaction strength $u$ vanishes and therefore we may set $y_{c}=0$ in
the denominator of the integrand in Eq. (33) as the prefactor of the integral
is already of order $u$ and the calculation is done at the lowest order in
perturbation theory. By choosing the exponential cut-off function introduced
after Eq. (21) (see also Eq. (79)) and after performing the momentum integral,
one finds

$\displaystyle\xi^{-2}$ $\displaystyle=\delta
y\biggl{[}1-3\frac{u}{16}a_{4}q\sqrt{r_{0}}\ln\biggl{|}\frac{\Lambda^{2}}{\delta
y}\biggr{|}+O(u^{2})\biggr{]},$ (34)

for $\delta y\ll\Lambda^{2}$. At this order in the perturbative expansion,
this logarithmic correction can be exponentiated in order to obtain

$\displaystyle\xi^{-2}$ $\displaystyle=\delta
y\biggl{[}1+quA\ln\biggl{|}\frac{\delta
y}{\Lambda^{2}}\biggr{|}+O(u^{2})\biggr{]}$
$\displaystyle\approx\Lambda^{2}\biggl{|}\frac{\delta
y}{\Lambda^{2}}\biggr{|}^{1+Aqu}+O(u^{2}),$ (35)

where we introduced

$\displaystyle
A\equiv\frac{3}{16}a_{4}\sqrt{r_{0}}=\frac{3}{128\pi^{2}}\sqrt{r_{0}}.$ (36)

This analysis reveals that the exponent $\nu$, defined by the algebraic
singularity of $\xi\approx|\delta y|^{-\nu}$, is modified by the interaction
compared to its Gaussian value $\nu=1/2$. The latter is recovered here by
setting $u=0$. Although in Eq. (35) the coupling constant $u$ is determined by
the microscopic value entering the model, in order to determine the leading
scaling behavior close to the critical line, $u$ is effectively replaced by
its renormalization-group fixed-point value $u^{*}$; we later argue that in
the limit $N\to\infty$ this value can be determined and is given by, c.f., Eq.
(54). Accordingly, from Eq. (35), we conclude that the resulting exponent
$\nu$ takes the value

$\displaystyle\nu=\frac{1+Aqu^{*}}{2}=\frac{1}{2}\left[1+\frac{\epsilon}{2}+O(\epsilon^{2})\right],$
(37)

where $\epsilon=4-d$.

So far we have considered the effects of introducing the interactions within a
perturbative expansion at the lowest order in the coupling constant $u$,
eventually leading to expressions which are perturbative in the actual
expansion parameter $\epsilon$. While the expansion in $u$ can be easily
generalized to the case of finite $N$, and may also be used also to analyze
the effects of the transient part of the tadpole, the fixed-point value
$u^{*}$ of $u$ is easily determined only for $N\to\infty$, as shown below.
This is the reason why we primarily focus on limit $N\rightarrow\infty$. In
the next section, we determine the exact exponents $\nu$ and $\beta$ for
$N\to\infty$ as functions of the spatial dimensionality $d$. In particular,
the expansion of the expression of $\nu$ for $d=4-\epsilon$ at the first order
in $\epsilon$ renders, as expected, Eq. (37).

##  IV Exact solution for $\nu$

In the $N\rightarrow\infty$ limit, the evolution equation of $G_{K}$ derived
from the action in Eq. (II) becomes exact and can be solved self-consistently.
Noting that $\langle\phi_{q}^{2}\rangle=0$ Kamenev (2011) we obtain

$\begin{split}\big{[}\partial_{t}^{2}+k^{2}+r-r_{1}\cos(\omega
t)&+\frac{u}{2}iG_{K}(x=0,t,t)\big{]}\\\ &\times
iG_{K}(k,t,t^{\prime})=0.\end{split}$ (38)

Motivated by Eqs. (19) and (20), which show that
$iG_{0K}(x=0,t,t)\propto\cos^{2}(\omega t/2)$, we make the ansatz that in the
equation above one has

$\displaystyle r-r_{1}\cos(\omega t)+\frac{u}{2}iG_{K}(x=0,t,t)$
$\displaystyle=r+\delta r-(r_{1}+\delta r_{1})\cos(\omega t),$ (39)

where $iG_{K}$ is the Keldysh function of the Gaussian model ($u=0$) but
calculated with the renormalized parameters $r+\delta r$ and $r_{1}+\delta
r_{1}$ instead of $r$ and $r_{1}$, respectively. These new parameters also
imply that the effectively Gaussian detuning from the critical line,
determined from Eq. (7), takes the form

$\displaystyle y=r+\delta r-\frac{r_{1}+\delta
r_{1}}{2}-\frac{\omega^{2}}{4}.$ (40)

Substituting Eq. (12) in Eq. (39), and using the long-time limit of the
tadpole in Eq. (20), we find that the self-consistency condition requires

$\delta r=\widetilde{B}_{d}/2\quad\mbox{and}\quad\delta
r_{1}=-\widetilde{B}_{d}/2,$ (41)

where $\widetilde{B}_{d}$ is given in Eq. (21). Accordingly, the detuning in
Eq. (40) becomes

$\displaystyle
y=r-\frac{r_{1}}{2}-\frac{\omega^{2}}{4}+\frac{3}{4}\widetilde{B}_{d}(y).$
(42)

This equation provide the implicit relationship between the Gaussian detuning
$y$ and the control parameters $r$ and $r_{1}$. We also note here that within
this effective Gaussian model with renormalized parameters, the spatial
correlation length $\xi$ is still related to $y$ as $y=\xi^{-2}$. Since the
critical line corresponds to $y=0$, the critical values $r_{c}$ and $r_{1c}$
of the parameters $r$ and $r_{1}$, respectively, satisfy Eq. (42) with $y=0$
(equivalently $\xi=+\infty$), i.e.,
$0=r_{c}-r_{1c}/2-\omega^{2}/4+(3/4)\widetilde{B}_{d}(0)$. Subtracting this
condition from Eq. (42), and using Eq. (21), we find the self-consistency
condition in terms of the correlation length $\xi$, i.e.,

$\displaystyle\xi^{-2}=r-r_{c}-\frac{r_{1}-r_{1c}}{2}+g\Lambda^{d-2}\biggl{[}\omega_{d}\left(\Lambda^{-1}/\xi\right)-\omega_{d}(0)\biggr{]},$
(43)

where we introduced

$\displaystyle g=\frac{3}{8}qu\sqrt{r_{0}},$ (44)

and the function $\omega_{d}(z)$ which captures the scaling behavior of
$\widetilde{B}_{d}(\xi^{-2})$ as

$\int\frac{d^{d}k}{(2\pi)^{d}}\frac{f_{c}\left(k/\Lambda\right)}{k^{2}+\xi^{-2}}=\Lambda^{d-2}\omega_{d}\left(\Lambda^{-1}/\xi\right).$
(45)

We emphasize here that Eq. (43) turns out to be independent of the driving
frequency $\omega$ which influences, instead, the location of the critical
line. In fact, Eq. (43) has the same structure as the equation which controls
the correlation length $\xi$ in the undriven model, both after the quench
Chiocchetta  _et al._ (2016) and in equilibrium Moshe and Zinn-Justin (2003).
In the latter context it was shown in Ref. Moshe and Zinn-Justin (2003) that

$\displaystyle\omega_{d}(z)=\omega_{d}(0)-K_{d}z^{d-2}+c_{d}z^{2}+\mbox{h.o.},$
(46)

where $c_{d}$ is a constant that depends on the cut-off function $f_{c}$ in
Eq. (45), while $K_{d}$ is a universal constant given by

$\displaystyle
K_{d}=a_{d}\int_{0}^{\infty}dz\frac{z^{d-1}}{z^{2}(1+z^{2})}=-a_{d}\frac{\pi/2}{\sin(\pi
d/2)},$ (47)

with $K_{d}>0$ for $2<d<4$. When $d<4$, the contribution from $c_{d}z^{2}$ is
subleading compared to $K_{d}z^{d-2}$ for $z\to 0$. Accordingly, for
$\xi\gg\Lambda^{-1}$, the term in brackets on r.h.s. of Eq. (43) is more
relevant than that on the l.h.s. and the equation implies that

$\xi\propto\left(r-r_{c}-\frac{r_{1}-r_{1c}}{2}\right)^{-\nu},$ (48)

with

$\displaystyle\nu=\frac{1}{d-2}.$ (49)

This exponent $\nu$ controls the divergence of the spatial correlation length
of the fluctuations within the system as $r-r_{1}/2\to r_{c}-r_{1c}/2$, i.e.,
as the critical line in Fig. 1(b) is approached. For $d>4$, instead, the term
$\propto z^{2}$ in Eq. (46) is the dominant one for $z\to 0$ and therefore in
Eq. (43) the contributions $\propto\xi^{-2}$ on the l.h.s. and r.h.s. are the
relevant ones. Accordingly, one finds the same expression as in Eq. (48) with

$\displaystyle\nu=\frac{1}{2}.$ (50)

The predictions presented above for the leading scaling behavior and for the
exponent $\nu$ are actually independent of the coupling constant $g$ of the
model which can take any (positive) value. However, among these values, there
is a specific one $g^{*}$ for which the leading corrections to the scaling
behavior discussed above do vanish. In the renormalization-group framework,
$g^{*}$ correspond to the fixed-point value of $g$, see, e.g., Ref. Moshe and
Zinn-Justin (2003). In particular, for $2<d<4$, the corrections to the scaling
in Eq. (48) come from the term $\propto z^{2}$ in Eq. (46) and from the
analogous one on the l.h.s. of Eq. (43). Taking them into account, the
consistency condition in Eq. (43) can then be written as

$\displaystyle\xi^{-2}\biggl{[}1-gc_{d}\Lambda^{d-4}\biggr{]}=r-r_{c}-\frac{r_{1}-r_{1c}}{2}-gK_{d}\xi^{2-d}.$
(51)

The expression of $\xi$ which can be derived from this equation matches the
scaling behavior in Eq. (48), i.e., with no leading corrections to scaling,
only if the l.h.s. of the equation vanishes, i.e., for $g=g^{*}$, where

$g^{*}=\Lambda^{4-d}/c_{d}.$ (52)

While $c_{d}$ is a non-universal constant which depends on the cutoff function
$f_{c}$ (see Eqs. (46) and (45)), it turns out Moshe and Zinn-Justin (2003)
that its limit for $d\to 4$, i.e., $\epsilon\to 0$, is universal and given by
$c_{4-\epsilon}=1/(8\pi^{2}\epsilon)+O(\epsilon^{0})$ which renders

$\displaystyle g^{*}=8\pi^{2}\epsilon+O(\epsilon^{2}),$ (53)

in agreement with what is expected from renormalization-groups arguments. This
fixed-point value $g^{*}$ of the coupling constant $g$ in the dimensional
expansion can now be used in the perturbative expansions discussed in Sec. III
to set the fixed-point value of the coupling constant $u$ which, using Eq.
(44), is given by

$\displaystyle
u^{*}=\frac{(8\pi)^{2}}{3q\sqrt{r_{0}}}\epsilon+O(\epsilon^{2}).$ (54)

In particular, as anticipated in Sec. III.2, the perturbative prediction for
$\nu$ can be recovered from the expression in Eq. (35) if we substitute $u$ by
its fixed point value $u^{*}$, and assume $\epsilon\ll 1$. This leads to Eq.
(37), which indeed coincides with the dimensional expansion of Eq. (49) for
$d=4-\epsilon$ at the first order in $\epsilon$. This is also similar to what
happens in equilibrium.

In Sec. VI the perturbative fixed-point value $u^{*}$ of $u$ determined here
will be used to provide a prediction for the scaling exponents which emerge in
perturbation theory when studying the short-time behavior of $G_{R,K}$.

##  V Exact Solution for $\beta$

In order to extract the exponent $\beta$ that governs the behavior of the
order parameter close to criticality, we assume that the symmetry of the model
is (spontaneously) broken along one spatial direction, i.e., that
$\langle\phi_{i}(t)\rangle=\sqrt{N}M(t)\delta_{i,1}$ for $i=1,\ldots,N$. Then
for $N\rightarrow\infty$, the equations of motion for $M$ and $G_{K}$ are as
follows Chiocchetta  _et al._ (2016); Chandran and Sondhi (2016):

$\displaystyle\biggl{[}\partial_{t}^{2}+r-r_{1}\cos(\omega t)+uM^{2}(t)$
$\displaystyle\qquad+\frac{u}{2}iG_{K}(x=0,t,t)\biggr{]}M(t)=0,$ (55a)
$\displaystyle\biggl{[}\partial_{t}^{2}+k^{2}+r-r_{1}\cos(\omega t)+uM^{2}(t)$
$\displaystyle\qquad+\frac{u}{2}iG_{K}(x=0,t,t)\biggr{]}iG_{K}(k,t,t^{\prime})=0.$
(55b)

Assuming that the magnetization $M$ shows period doubling, we make the ansatz
$M(t)=M_{0}\cos(\omega t/2)$ (see Appendix B for details) and find that the
self-consistent equations written above imply that

$|M_{0}|\propto\sqrt{r_{c}-r+r_{1c}/2-r_{1}/2},$ (56)

and therefore

$\displaystyle\beta=1/2,$ (57)

as for the model in equilibrium Moshe and Zinn-Justin (2003).

##  VI Aging following a quench

In this section we discuss how the correlation functions behave in the
transient regime following a quantum quench. For this, we consider the
transient part of the tadpole given in Eq. (22) and we substitute it into Eqs.
(18a) and (18), which are derived at the lowest order in perturbation theory.
We recall that in Secs. III.2, IV, and V we focused on the long-time behavior
by neglecting this transient. A key quantity that turns out to characterize
this early-time regime is the constant $\theta$, defined perturbatively on the
basis of the transient behavior $iT^{\prime}(t)$ of the tadpole in Eq. (22).
As discussed below, this constant is eventually the exponent which appears in
the scaling of $G_{R,K}$. By using the perturbative fixed-point value $u^{*}$
of the coupling $u$ in Eq. (54), $\theta$ in Eq. (22) takes the value

$\displaystyle\theta^{*}=\frac{\epsilon}{4}+O(\epsilon^{2}).$ (58)

This exponent $\theta^{*}$ is identical to the one obtained for the undriven
model Chiocchetta  _et al._ (2015); Maraga  _et al._ (2015); Chiocchetta  _et
al._ (2016), in which the higher-order corrections $O(\epsilon^{2})$ turn out
to vanish as $N\rightarrow\infty$ for $2<d<4$, providing an exact exponent
Maraga  _et al._ (2015). In this section we discuss how $\theta$ appears in
the driven problem.

In the transient regime, for $k=0$ and $t\gg t^{\prime}$, but with both times
longer than the drive period $T=2\pi/\omega$, we find that
$G_{R}=G_{0R}+\delta G_{R}$ becomes (see Appendix C for details)

$\displaystyle G_{R}(k=0,t\gg t^{\prime},\omega t\gg\omega t^{\prime}\gg 1)$
$\displaystyle\quad=-qt\cos(\omega t/2)\cos(\omega
t^{\prime}/2)\left[1-\theta\ln\bigl{(}t/t^{\prime}\bigr{)}\right]$
$\displaystyle\quad\approx-qt\cos(\omega t/2)\cos(\omega
t^{\prime}/2)\bigl{(}t^{\prime}/t\bigr{)}^{\theta}.$ (59)

Using Eq. (58), this implies that at the fixed point, the universal exponent
$\theta^{*}$ governs the aging dynamics, which manifests itself via the
dependence of $G_{R}$ on $(t^{\prime}/t)^{\theta^{*}}$. On the other hand for
$t\gg t^{\prime}$, but with one time being smaller than the drive period $T$,
and the other longer, i.e., for $\omega t\gg 1\gg\omega t^{\prime}$, one finds
(see Appendix C for details)

$\displaystyle G_{R}(k=0,t\gg t^{\prime},\omega t\gg 1\gg\omega
t^{\prime})=-qt\cos(\omega t/2)$ $\displaystyle\qquad\quad\times\cos(\omega
t^{\prime}/2)\biggl{[}1-\theta\ln{(\omega t)}+\frac{8}{3}\theta\ln{(\omega
t^{\prime})}\biggr{]}$ $\displaystyle\qquad\approx-qt\cos(\omega
t/2)\cos(\omega t^{\prime}/2)(\omega t^{\prime})^{8\theta/3}/(\omega
t)^{\theta}.$ (60)

Equations (59) and (60) show that, due to the appearance of two exponents,
$\theta$ and $8\theta/3$, the functional form of the observed aging, and the
exponents which control it, are modified depending on whether the system is
probed at longer or shorter times compared to the drive period. The
resummation of the leading logarithmic behavior done in the previous equation,
leading to an algebraic dependence, is consistent within perturbation theory
but it certainly needs to be put on a firmer ground by an analytic solution or
a full-fledged renormalization-group analysis of the problem, as was done in
the absence of driving Maraga  _et al._ (2015); Chiocchetta  _et al._ (2016,
2015).

In a similar manner, the Keldysh Green’s function in perturbation theory at
one-loop, when probed at times longer than the drive period, turns out to be
(see Appendix D for details)

$\displaystyle iG_{K}(k=0,\omega t\gg 1,\omega t^{\prime}\gg 1)$
$\displaystyle\quad=q^{2}\sqrt{r_{0}}\cos(\omega t/2)\cos(\omega
t^{\prime}/2)\;tt^{\prime}\biggl{[}1-\theta\ln\bigl{(}\bar{\Lambda}^{2}tt^{\prime}\bigr{)}\biggr{]}$
$\displaystyle\quad\propto q^{2}\sqrt{r_{0}}\cos(\omega t/2)\cos(\omega
t^{\prime}/2)\bigl{(}\bar{\Lambda}^{2}tt^{\prime}\bigr{)}^{1-\theta}.$ (61)

On the other hand, for $\omega t\gg 1\gg\omega t^{\prime}$, one finds (see
Appendix D)

$\displaystyle iG_{K}(k=0,\omega t\gg 1\gg\omega
t^{\prime})=q^{2}\sqrt{r_{0}}\cos(\omega t/2)$ $\displaystyle\times\cos(\omega
t^{\prime}/2)tt^{\prime}\biggl{[}1-\theta\ln(\bar{\Lambda}t)-\frac{8}{3}\theta\ln(\bar{\Lambda}t^{\prime})\biggr{]}$
$\displaystyle\propto q^{2}\sqrt{r_{0}}\cos(\omega t/2)\cos(\omega
t^{\prime}/2)(\bar{\Lambda}t)^{1-\theta}(\bar{\Lambda}t^{\prime})^{1-8\theta/3}.$
(62)

Thus here too — with the same proviso as the one spelled out after Eq. (60) on
the resummation of the logarithmic dependence — the perturbative expressions
indicate that the aging behavior changes depending on whether the time is
small or large compared to the period of the drive.

Now we discuss the behavior of $iG_{K}(k,t,t)$ for $kt\gg 1$. Within the
Gaussian approximation in Eq. (12) this quantity decays as $1/\bar{k}^{2}$
upon increasing $\bar{k}$. The one loop corrections slow down this decay to
$1/\bar{k}^{2-3\theta}$. In particular we have (see Appendix E for details)

$\displaystyle iG_{K}(k,t,t)\approx q^{2}\cos^{2}(\omega
t/2)\frac{\sqrt{r_{0}}}{\bar{k}^{2-2\theta}}\sin^{2}(\bar{k}t).$ (63)

The expressions above indicate that, for a quantum quench to the FTC critical
line, the correlators at long wavelengths $\bar{k}\ll\omega$ and with one of
the times longer than the drive period, i.e., $\omega t\gg 1$ can be cast in
the following scaling forms, valid within the prethermal regime and
reminiscent of what was found in the undriven model Chiocchetta  _et al._
(2015); Maraga  _et al._ (2015); Chiocchetta  _et al._ (2016),

$\displaystyle G_{K}(k,t,t^{\prime})$ $\displaystyle=\cos(\omega
t/2)\cos(\omega t^{\prime}/2)\frac{1}{k^{2-2\theta}}{\cal
G}_{K}(kt,kt^{\prime},\omega t^{\prime}),$ (64) $\displaystyle
G_{R}(k,t,t^{\prime})$ $\displaystyle=\cos(\omega t/2)\cos(\omega
t^{\prime}/2)\frac{\bigl{(}t^{\prime}/t\bigr{)}^{\theta}}{k}{\cal
G}_{R}(kt,kt^{\prime},\omega t^{\prime}).$ (65)

In order for these expressions to render the behaviors discussed above, we
assume ${\cal G}_{R,K}(x,y,z)\sim 1$ for $x$, $y$, $z\gg 1$. For $x$, $y\ll
1$, instead, ${\cal G}_{K}(x,y,z)\sim(xy)^{1-\theta}f^{-1}(z)$, with $f(z)\sim
1$ for $z\gg 1$ and $f(z)\sim z^{5\theta/3}$ for $z\ll 1$. In the remaining
case $y\ll x\ll 1$, one has ${\cal G}_{R}(x,y,z)\sim xf(z)$.

Figure 2:  One-loop correction $G_{R}^{(1)}(x,\tau+t^{\prime},t^{\prime})$
arising from the perturbation in Eq. (66) in $d=4$, in the $(\tau,x)$ plane
for fixed value of $t^{\prime}=0.1$, $r_{0}=1$, and dimensionless drive
amplitude $q=0.22$. We take the strength of the perturbation to be $a_{p}=1$,
while $\tau$ and $x$ are measured in units of the drive period $T$. The dashed
line indicates the light-cone with quasiparticle velocity $v=\sqrt{q/2}\simeq
0.33$ while the dot-dashed line corresponds to $v=1$ Natsheh  _et al._ (rint),
which characterizes the light-cone in the absence of the drive. The spatial
oscillations occur with a characteristic wave-vector $\omega/(2v)$, while the
asymptotic decay of $G_{R}^{(1)}(x,\tau+t^{\prime},t^{\prime})$ at large
distances is described by Eq. (69).

##  VII Spatial structure due to resonances

We now discuss the effects on the dynamics of adding a perturbing potential of
the form

$\displaystyle V_{p}(t)=a_{p}\left[1-2\cos(\omega t)\right],$ (66)

with $a_{p}/\omega^{2}\ll 1$. This particular choice of $V_{p}$ does not
change the value of $r-r_{1}/2$ compared to the unperturbed case $V_{p}=0$ and
thus, from Eq. (7), the detuning from the critical line is not affected.
Accordingly, the Floquet quasi-modes for $a_{p}\neq 0$ will lead to a behavior
which is still described by the Gaussian correlators in Eqs. (12) and (13).
However, here we address a different question: assuming that the perturbation
$V_{p}$ was switched on suddenly at time $t=0$, what is the linear response to
this perturbation of a state which is initially at the critical line and
therefore described by the correlators in Eqs. (12), (13) with $y=0$?

Treating $V_{p}$ perturbatively, the one-loop correction to the correlators,
given by Eqs. (18a) and (18), with $iT(\tau)\mapsto V_{p}(\tau)$, are found to
be

$\displaystyle G_{R}^{(1)}(k,t,t^{\prime})$ $\displaystyle=-\cos(\omega
t/2)\cos(\omega
t^{\prime}/2)\frac{q^{2}a_{p}}{8\bar{k}^{2}}\\!\\!\\!\sum_{m=\pm 1,\pm
2}\\!\\!\\!I^{R}_{m\omega},$ (67a) $\displaystyle iG_{K}^{(1)}(k,t,t)$
$\displaystyle=\cos^{2}(\omega
t/2)\frac{q^{3}\sqrt{r_{0}}a_{p}}{8\bar{k}^{3}}\\!\\!\sum_{m=\pm 1,\pm
2}I^{K}_{m\omega},$ (67b)

where $G^{(1)}_{R,K}$ are the one-loop corrections to the Gaussian parts
already reported in Eqs. (12), (13), and

$\displaystyle
I^{R}_{m\omega}(k,t,t^{\prime})=\frac{\cos(m\omega(t+t^{\prime})/2)\sin((\bar{k}-m\omega/2)(t-t^{\prime}))}{\bar{k}-m\omega/2}$
$\displaystyle\qquad\quad-\cos(\bar{k}(t-t^{\prime}))\biggl{[}\frac{\sin(m\omega
t)-\sin(m\omega t^{\prime})}{m\omega}\biggr{]},$ (68a) $\displaystyle
I^{K}_{m\omega}(k,t)=\frac{[1+\cos(m\omega
t)][1-\cos(2\bar{k}t)]}{2\bar{k}-m\omega}$
$\displaystyle\qquad\quad-\sin(2\bar{k}t)\sin(m\omega
t)\biggl{[}\frac{1}{m\omega}+\frac{1}{2\bar{k}-m\omega}\biggr{]}.$ (68b)

Although $G^{(1)}_{R,K}$ have the same form as the Gaussian correlators in
Eqs. (12) and (13), for long wavelengths $\bar{k}/\omega\ll 1$, the spatial
Fourier transform in Eq. (15), which we perform for $d=4$, is sensitive to the
presence of resonances for $k=m\omega/(2v)$, with $m=\pm 1,\pm 2$. These
resonances modify drastically the dependence of $G^{(1)}_{R,K}$ on space, as
shown in Figs. 2 and 3. As expected, a light-cone is visible in
$G^{(1)}_{R}(x,t,t^{\prime})$ for $x=v|t-t^{\prime}|$ and in the equal-time
correlator $iG^{(1)}_{K}(x,t,t)$ for $x=2vt$. However, the one-loop
corrections to $G^{(1)}_{R,K}$ upon increasing the distance $x$ are
characterized by a decay in space which is slower compared to the Gaussian
correlators. Moreover, $G^{(1)}_{R,K}$ are found to oscillate in space with
the wave-vector $m\omega/2v$ with $m=1,2$. The spatial oscillations are
clearly visible in Figs. 2 and 3 for $G_{R}^{(1)}$ and $G_{K}^{(1)}$,
respectively.

The asymptotic form of $G_{R,K}^{(1)}$ on the light cone and inside it turn
out to be (see Appendix F for details)

$\displaystyle G_{R}^{(1)}(x\leq v|t-t^{\prime}|)\propto\cos(\omega
t/2)\cos(\omega t^{\prime}/2)$
$\displaystyle\quad\times\sum_{m=1,2}\cos\biggl{(}m\frac{\omega(t+t^{\prime})}{2}\biggr{)}\frac{\sin(m\omega
x/(2v)+\delta)}{x^{3/2}},$ (69a) $\displaystyle G^{(1)}_{K}(x\leq
2vt,t,t)\propto\cos^{2}(\omega t/2)\sum_{m=1,2}\frac{\cos(m\omega
x/2v+\delta^{\prime})}{x^{3/2}},$ (69b)

where $\delta$ and $\delta^{\prime}$ are phase-shifts. These expressions
should be compared with the corresponding ones in the unperturbed state (i.e.,
with $a_{p}=0$), which are given in Eqs. (16) and (17). In particular, on the
light-cone one finds that the correlators $G^{(1)}_{R,K}$ decay as $x^{-3/2}$
upon increasing $x$, which is also the case for the Gaussian correlators on
the light-cone. However, the spatial decay of $G^{(1)}_{R,K}$ is also
accompanied by oscillations in space, as is clearly visible in Figs. 2 and 3
when looking along the light-cone (dashed lines).

Inside the light-cone, the difference between the behavior of $G^{(1)}_{R,K}$
and the Gaussian correlators $G_{0R,K}$, are more pronounced. For example
while $G_{0R}$ vanishes inside the light-cone, the one loop correction to it,
$G^{(1)}_{R}$, is non-zero, and decays as $x^{-3/2}$. In a similar manner
while $G_{0K}$ decays as $x^{-2}$ inside the light-cone, the one-loop
correction to it, $G^{(1)}_{K}$ decays more slowly, as $x^{-3/2}$. All these
decays are also accompanied by spatial oscillations at the wave-vectors
$m\omega/(2v)$, with $m=1,2$.

Figure 3:  One-loop correction $G_{K}^{(1)}(x,t,t)$ in the $(t,x)$ plane, for
the same values of the parameters as in Fig. 2. Both space $x$ and time $t$
are expressed in units of the period $T$ of the drive. The light-cone (dashed
line) is located at $x=2vt$, with $v=\sqrt{q/2}\simeq 0.33$. For comparison,
we also report the line corresponding to $v=1$ (dot-dashed line), which
characterizes the light-cone in the absence of the driving. The correlator
decays as $x^{-3/2}$ upon increasing $x$ both on and inside the light-cone and
it oscillates in space with typical wave-vector $\omega/(2v)$. This
correlator, like all the equal-time correlators does not show period doubling.

##  VIII Conclusions

In this paper we have studied the physics of a Floquet time crystal (FTC) in
the vicinity of the critical line that separates it from a trivial phase. The
results are derived for the periodically driven $O(N)$ model with
$N\to\infty$. Exact results for the exponents $\nu$ and $\beta$ are derived in
the long-time (but still prethermal) regime. Results are also obtained in the
transient regime following a quantum quench where the system shows aging
dynamics controlled by an exponent $\theta$. The results in this transient
regime are obtained to the leading order in $\epsilon=4-d$.

The critical exponents $\nu$ (see Eq. (49)), $\beta$ (see Eq. (57)), and
$\theta$ (see Eq. (58)) are found to be the same as those of the undriven
problem, provided that the system is probed at times longer than the drive
period $T$ (see Eqs. (59), (61)). At times shorter than $T$, a perturbative
treatment reveals that a different scaling behavior possibly emerges (see Eqs.
(60), (62)).

FTCs always arise when the drive is resonant with some microscopic scale of
the undriven problem. We showed that this leads to certain peculiarities such
as the linear response of the system to a time-dependent perturbing potential
can lead to spatial oscillations controlled by the drive frequency and the
quasiparticle velocity.

Our study leaves a number of open questions for further investigations.
Solving numerically the dynamics of the system near the FTC critical line and
extracting from these solutions the exponents would be a useful exercise.
Carrying out a complete renormalization-group analysis which not only renders
the exponents predicted here, but generalizes them to finite $N$ is a
direction worth pursuing together with an exact analysis of the short-time
behavior for $N\to\infty$.

Finally, exploring other universality classes of FTCs, especially those where
the exponents $\nu$, $\beta$, and $\theta$ may be different from the undriven
situation, is an exciting direction of research. Exploring critical properties
even of undriven systems Wilczek (2012); Shapere and Wilczek (2012); Li  _et
al._ (2012) which can nevertheless show time-crystal behavior Iemini  _et al._
(2018); Prokof’ev and Svistunov (2020); Else  _et al._ (2017, 2020); Khemani
_et al._ (2019); Kozin and Kyriienko (2019); Khemani  _et al._ (2020); Kozin
and Kyriienko (2020); Öhberg and Wright (2019); Syrwid  _et al._ (2020);
Öhberg and Wright (2020) despite no-go theorems Bruno (2013a, b, c); Watanabe
and Oshikawa (2015), is also an important open question.

Acknowledgements: This work was supported by the US National Science
Foundation Grant NSF-DMR 2018358.

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##  Appendix A Tadpole correction

In this section we outline the derivation of Eq. (20). Note that the Green’s
functions within the Gaussian approximation, in the long wavelength limit
$k\ll\sqrt{q}\omega\ll\omega$, and along the critical line $y=0$ are obtained
from Eqs. (12) and (13) by setting $\overline{\omega_{k}}=\bar{k}$, see Eq.
(14),

$\displaystyle iG_{0K}(k,t,t^{\prime})$
$\displaystyle=q^{2}\frac{\sqrt{r_{0}}}{2\bar{k}^{2}}\cos\left(\frac{\omega}{2}t\right)\cos\left(\frac{\omega}{2}t^{\prime}\right)$
$\displaystyle\times[\cos(\bar{k}(t-t^{\prime}))-\cos(\bar{k}(t+t^{\prime}))],$
(70) $\displaystyle G_{0R}(k,t,t^{\prime})$
$\displaystyle=-\theta(t-t^{\prime})q\cos\left(\frac{\omega}{2}t\right)\cos\left(\frac{\omega}{2}t^{\prime}\right)$
$\displaystyle\times\frac{\sin(\bar{k}(t-t^{\prime}))}{\bar{k}}.$ (71)

It is interesting to note that the above Green’s functions can be written in
terms of the Green’s functions of the undriven problem following a quantum
quench Natsheh  _et al._ (rint), as follows

$\displaystyle G_{0R}(k,t,t^{\prime})$ $\displaystyle=q\cos(\omega
t/2)\cos(\omega t^{\prime}/2)\ G_{0R,u}(\bar{k},t,t^{\prime}),$ (72a)
$\displaystyle iG_{0K}(k,t,t^{\prime})$ $\displaystyle=q^{2}\cos(\omega
t/2)\cos(\omega t^{\prime}/2)\ G_{0K,u}(\bar{k},t,t^{\prime}),$ (72b)

where Chiocchetta  _et al._ (2015, 2016) we denote the undriven Green’s
functions following a quantum quench as $G_{0R,u},G_{0K,u}$

$\displaystyle G_{0R,u}(k,t,t^{\prime})$
$\displaystyle=-\theta(t-t^{\prime})\frac{\sin(k(t-t^{\prime}))}{k},$ (73a)
$\displaystyle iG_{0K,u}(k,t,t^{\prime})$
$\displaystyle=\frac{\sqrt{r_{0}}}{2k^{2}}\bigl{[}\cos(k(t-t^{\prime}))-\cos(k(t+t^{\prime})\bigr{]}.$
(73b)

Using Eq. (70), the tadpole in Eq. (20) can be written as

$\displaystyle iT(t)$
$\displaystyle=\frac{u}{2}\int\frac{d^{d}k}{(2\pi)^{d}}iG_{0K}(k,t,t)f_{c}\left(k/\Lambda\right)$
$\displaystyle=q^{2}\cos^{2}(\omega t/2)\frac{u}{2}\sqrt{r_{0}}$
$\displaystyle\times\int\frac{d^{d}k}{(2\pi)^{d}}\frac{f_{c}\left(k/{\Lambda}\right)}{2\bar{k}^{2}}\biggl{[}1-\cos(2\bar{k}t)\biggr{]}$
(74) $\displaystyle=\widetilde{B}_{d}\cos^{2}{(\omega t/2)}+iT^{\prime}(t).$
(75)

The first term in the last line corresponds to the long-time behavior of the
tadpole

$\displaystyle\widetilde{B}_{d}(y)=a_{d}\frac{qu}{2}\sqrt{r_{0}}\int_{0}^{\infty}dk\,k^{d-1}\frac{f_{c}\left(k/\Lambda\right)}{k^{2}+y},$
(76)

with $a_{d}=2/[(4\pi)^{d/2}\Gamma(d/2)]$. The second term $iT^{\prime}(t)$ in
Eq. (75) denotes the transient behavior of the tadpole.

We discuss the long-time behavior first. For spatial dimension $d=4$, and
restoring a small detuning $y$, the coefficient $\widetilde{B}_{4}$ in Eq.
(76) is given by

$\displaystyle\widetilde{B}_{4}(y)=a_{4}\frac{qu}{2}\sqrt{r_{0}}\int_{0}^{\infty}dk\,k^{3}\frac{f_{c}\left(k/\Lambda\right)}{k^{2}+y},$
(77)

where $a_{4}=1/(8\pi^{2})$.

Now we turn to the leading transient behavior. Removing the contribution from
the long-time part, and keeping track of the cut-off $\Lambda$, we have

$\displaystyle iT^{\prime}(t;\Lambda)$
$\displaystyle=\frac{u}{2}\int\frac{d^{d}k}{(2\pi)^{d}}iG_{0K}(k,t,t)f_{c}\left(k/\Lambda\right)$
$\displaystyle\qquad\qquad-\widetilde{B}_{d}\cos^{2}(\omega t/2).$ (78)

We use an exponential cut-off function,

$\displaystyle f_{c}(x)=e^{-x}.$ (79)

Note that the long time and long distance behavior is independent of the
precise details of the cut-off function. Using Eq. (72b), and at spatial
dimension $d=4$, we obtain

$\displaystyle iT^{\prime}(t;\Lambda)$
$\displaystyle=-\widetilde{B}_{4}\cos^{2}(\omega t/2)+q^{2}\cos^{2}(\omega
t/2)\frac{u}{16\pi^{2}}$
$\displaystyle\times\left(\frac{4}{q^{2}}\right)\biggl{[}\int_{0}^{\infty}d\bar{k}\bar{k}^{3}\exp(-\bar{k}/\bar{\Lambda})iG_{0K,u}(\bar{k},t,t)\biggr{]}$
$\displaystyle=\frac{8\theta}{3q}\cos^{2}(\omega
t/2)\biggl{[}(2\bar{\Lambda})^{2}\frac{(2\bar{\Lambda}t)^{2}-1}{[(2\bar{\Lambda}t)^{2}+1]^{2}}\biggr{]},$
(80)

where

$\displaystyle\theta=\frac{3qu\sqrt{r_{0}}}{256\pi^{2}}.$ (81)

In the long-time limit $\bar{\Lambda}t\gg 1$, one obtains

$iT^{\prime}(t,\Lambda)\xrightarrow{\bar{\Lambda}t\rightarrow\infty}\frac{8\theta}{3qt^{2}}\cos^{2}(\omega
t/2),$ (82)

i.e, Eq. (22) in the main text.

##  Appendix B Exact calculation of $\beta$

Let us denote the non-zero magnetization along a certain direction $i$ to be
$\langle\phi_{i}\rangle=\sqrt{N}M$. The equations of motion in the
$N\rightarrow\infty$ limit are in Eq. (55). The self-consistent equations are
equivalent to the coefficient before the $\bm{\phi}_{q}\cdot\bm{\phi}_{c}$
term in the Keldysh action Eq. (II) having the following form

$\displaystyle r-r_{1}\cos(\omega t)+uM^{2}(t)+\frac{u}{2}iG_{K}(x=0,t,t).$
(83)

Let us make the ansatz for period doubling

$\displaystyle M(t)=M_{0}\cos(\omega t/2)\Rightarrow
M^{2}(t)=\frac{M_{0}^{2}}{2}\left[1+\cos(\omega t)\right].$ (84)

In addition, as for the case of $M=0$ discussed in Section IV, we assume that
$iG_{K}$ is the Keldysh function of the Gaussian model ($u=0$) but calculated
with renormalized parameters $r+\delta r$ and $r_{1}+\delta r_{1}$. Thus,
substituting for $iG_{K}$ from Eq. (20), and for $M^{2}$ from Eq. (84), we
obtain the self-consistent equation

$\displaystyle r-r_{1}\cos(\omega t)+\frac{uM_{0}^{2}}{2}\left[1+\cos(\omega
t)\right]+\frac{\widetilde{B}_{d}}{2}\left[1+\cos(\omega t)\right]$
$\displaystyle=r+\delta r-(r_{1}+\delta r_{1})\cos(\omega t).$ (85)

Matching the coefficients on both sides gives

$\displaystyle r+\frac{1}{2}\widetilde{B}_{d}(y)+\frac{uM_{0}^{2}}{2}$
$\displaystyle=r+\delta r,$ (86) $\displaystyle
r_{1}-\frac{1}{2}\widetilde{B}_{d}(y)-\frac{uM_{0}^{2}}{2}$
$\displaystyle=r_{1}+\delta r_{1}.$ (87)

The self-consistent equation for the detuning $y$ follows from Eq. (40):

$\displaystyle
y=r-\frac{r_{1}}{2}-\frac{\omega^{2}}{4}+\frac{3}{4}\widetilde{B}_{d}(y)+\frac{3}{4}uM_{0}^{2}.$
(88)

When $y\rightarrow 0$, the above becomes

$\displaystyle
0=r-\frac{r_{1}}{2}-\frac{\omega^{2}}{4}+\frac{3}{4}\widetilde{B}_{d}(0)+\frac{3}{4}uM_{0}^{2}.$
(89)

Recal that the critical couplings $r_{c},r_{1c}$ are such that $y=0$ when
$M_{0}=0$, corresponding to the condition

$\displaystyle-\frac{\omega^{2}}{4}+\frac{3}{4}\widetilde{B}_{d}(0)=\frac{r_{1c}}{2}-r_{c}.$
(90)

Substituting the above condition in Eq. (89), we obtain

$\displaystyle
0=r-\frac{r_{1}}{2}+\frac{r_{1c}}{2}-r_{c}+\frac{3}{4}uM_{0}^{2}.$ (91)

On solving for $M_{0}$, we arrive at Eq. (56) in the main text,

$\displaystyle|M_{0}|\propto\sqrt{r_{c}-r-\biggl{(}\frac{r_{1c}-r_{1}}{2}\biggr{)}}\Rightarrow\beta=1/2.$
(92)

##  Appendix C Retarded Green’s Function: Transient behavior

We now study the one-loop correction to the retarded Green’s function in Eq.
(18a) taking into account the transient part of the tadpole Eq. (80). Using
the relation between the driven and the undriven Green’s functions Eq. (72),
the one-loop contribution to $\delta G_{R}$ is given by

$\displaystyle\delta G_{R}(k,t,t^{\prime})$ $\displaystyle=q^{2}\cos(\omega
t/2)\cos(\omega t^{\prime}/2)\int_{0}^{\infty}d\tau\cos^{2}(\omega\tau/2)$
$\displaystyle\times
G_{0R,u}(\bar{k},t,\tau)\left[iT^{\prime}(\tau)\right]G_{0R,u}(\bar{k},\tau,t^{\prime}).$
(93)

We will discuss the behavior in the limit $\bar{k}t,\bar{k}t^{\prime}\ll 1$,
but with both times long as compared to the cutoff
$\bar{\Lambda}t,\bar{\Lambda}t^{\prime}\gg 1$. In order to access these
limits, we set $k=0$, and use the expression Eq. (82) for the tadpole. We also
use the identity,
$\cos^{4}(\omega\tau/2)=3/8+\cos(\omega\tau)/2+\cos(2\omega\tau)/8$. This
leads to

$\displaystyle\delta G_{R}(k=0,t,t^{\prime})=\frac{8q}{3}\cos(\omega
t/2)\cos(\omega t^{\prime}/2)$
$\displaystyle\times\int_{t^{\prime}}^{t}d\tau\biggl{[}\frac{3}{8}+\frac{1}{2}\cos(\omega\tau)+\frac{1}{8}\cos(2\omega\tau)\biggr{]}(t-\tau)(\tau-t^{\prime})\frac{\theta}{\tau^{2}}.$
(94)

Now we show that logarithmic corrections begin to emerge when $t\gg
t^{\prime}$. In this limit, the dominant term corresponds to approximating
$\tau-t^{\prime}\rightarrow\tau$. Following this, of the two remaining terms,
the one that dominates at long times is

$\displaystyle\delta G_{R}(k=0,t\gg t^{\prime})=\frac{8\theta}{3}q\cos(\omega
t/2)\cos(\omega t^{\prime}/2)$ $\displaystyle\times
t\int_{t^{\prime}}^{t}d\tau\frac{1}{\tau}\biggl{[}\frac{3}{8}+\frac{1}{2}\cos(\omega\tau)+\frac{1}{8}\cos(2\omega\tau)\biggr{]}$
$\displaystyle=-G_{0R}(k=0,t\gg
t^{\prime})\frac{8\theta}{3}\biggl{\\{}\frac{3}{8}\ln\left(t/t^{\prime}\right)$
$\displaystyle+\frac{1}{2}\biggl{[}{\rm Ci}(\omega t)-{\rm Ci}(\omega
t^{\prime})\biggr{]}+\frac{1}{8}\biggl{[}{\rm Ci}(2\omega t)-{\rm Ci}(2\omega
t^{\prime})\biggr{]}\biggr{\\}},$ (95)

where ${\rm Ci}(x)=-\int_{x}^{\infty}dt\frac{\cos{t}}{t}$ is the cosine
integral, and we have used that $G_{0R}(k=0,t\gg t^{\prime})=-qt\cos(\omega
t/2)\cos(\omega t^{\prime}/2)$. Note that ${\rm
Ci}(x)\rightarrow\gamma+\ln{x}+O(x)$ for $x\ll 1$ where $\gamma$ is Euler’s
constant, an observation which will be used below. At long-times $\omega t\gg
1$, two different limits arise depending on whether $\omega t^{\prime}\ll 1$
or $\omega t^{\prime}\gg 1$. In fact for $\omega t^{\prime}\gg 1$, the terms
in Eq. (94) containing the cosine integral ${\rm Ci}(x)$ do not contribute to
any logarithmic corrections, and we find

$\begin{split}&\delta G_{R}(k=0,t\gg t^{\prime},\omega t\gg\omega
t^{\prime}\gg 1)=-G_{0R}\\\\[5.69054pt]
&\qquad\qquad\times\left[\theta\ln\left(t/t^{\prime}\right)+{\rm
finite}\right],\end{split}$ (96)

implying that the leading logarithmic singularity of the correction modifies
the full Green’s function as

$\displaystyle G_{R}(k=0,t\gg t^{\prime},\omega t\gg\omega t^{\prime}\gg 1)=$
$\displaystyle-qt\cos(\omega t/2)\cos(\omega
t^{\prime}/2)\biggl{[}1-\theta\ln\bigl{(}t/t^{\prime}\bigr{)}\biggr{]}$
$\displaystyle\approx$ $\displaystyle-qt\cos(\omega t/2)\cos(\omega
t^{\prime}/2)\bigl{(}t^{\prime}/t\bigr{)}^{\theta}.$ (97)

The above corresponds to Eq. (59) in the main text.

On the other hand for $\omega t^{\prime}\ll 1$, but with still $\omega t\gg
1$, the cosine-integral terms in Eq. (95) do provide logarithmic contributions
and therefore give

$\displaystyle\delta G_{R}(k=0,t\gg t^{\prime},\omega t\gg 1\gg\omega
t^{\prime})=-G_{0R}(k=0,t\gg t^{\prime})$
$\displaystyle\times\frac{8\theta}{3}\biggl{[}\frac{3}{8}\ln\left(t/t^{\prime}\right)-\frac{1}{2}\ln(\omega
t^{\prime})-\frac{1}{8}\ln(2\omega t^{\prime})\biggr{]},$ (98)

implying that the leading logarithmic singularity of the correction modifies
the full Green’s function as

$\displaystyle G_{R}(k=0,t\gg t^{\prime},\omega t\gg 1\gg\omega t^{\prime})=$
$\displaystyle\qquad-qt\cos(\omega t/2)\cos(\omega t^{\prime}/2)$
$\displaystyle\qquad\qquad\times\biggl{[}1-\theta\ln{(\omega
t)}+\frac{8\theta}{3}\ln{(\omega t^{\prime})}\biggr{]}$ $\displaystyle\approx$
$\displaystyle-qt\cos(\omega t/2)\cos(\omega t^{\prime}/2)(\omega
t^{\prime})^{8\theta/3}/(\omega t)^{\theta}.$ (99)

The above corresponds to Eq. (60) in the main text.

##  Appendix D Keldysh Green’s Function: Transient behavior

Now we discuss the one loop correction to the Keldysh Green’s function Eq.
(18) in the transient regime. Using Eq. (72) and Eq. (80), we may write the
one loop correction Eq. (18) as

$\displaystyle\delta iG_{K}(k,t,t^{\prime})=q^{3}\cos(\omega t/2)\cos(\omega
t^{\prime}/2)$
$\displaystyle\times\int_{0}^{\infty}d\tau\cos^{2}(\omega\tau/2)G_{0R,u}(\bar{k},t,\tau)\left[iT^{\prime}(\tau)\right]\
iG_{0K,u}(\bar{k},\tau,t^{\prime})$ $\displaystyle+\left(t\leftrightarrow
t^{\prime}\right)$ (100)
$\displaystyle=-\frac{8\theta}{3}q^{2}\frac{\sqrt{r_{0}}}{\bar{k}^{3}}\cos(\omega
t/2)\cos(\omega t^{\prime}/2)$
$\displaystyle\times\int_{0}^{t}d\tau\cos^{4}(\omega\tau/2)\sin(\bar{k}(t-\tau))(2\bar{\Lambda})^{2}\frac{(2\bar{\Lambda}\tau)^{2}-1}{\left[(2\bar{\Lambda}\tau)^{2}+1\right]^{2}}$
$\displaystyle\times\sin(\bar{k}t^{\prime})\sin(\bar{k}\tau)$
$\displaystyle+\left(t\leftrightarrow t^{\prime}\right),$ (101)

where $\left(t\leftrightarrow t^{\prime}\right)$ indicate the same expression
but with $t$ and $t^{\prime}$ exchanged.

We will now use the relation

$\displaystyle(2\bar{\Lambda})^{2}\frac{(2\bar{\Lambda}\tau)^{2}-1}{\left[(2\bar{\Lambda}\tau)^{2}+1\right]^{2}}=-\frac{d}{d\tau}\biggl{[}\frac{4\bar{\Lambda}^{2}\tau}{1+(2\bar{\Lambda}\tau)^{2}}\biggr{]},$
(102)

in order to integrate by parts, further below, in the integral above. Thus,

$\displaystyle\delta
iG_{K}(k,t,t^{\prime})=\frac{8\theta}{3}q^{2}\frac{\sqrt{r_{0}}}{\bar{k}^{3}}\cos(\omega
t/2)\cos(\omega t^{\prime}/2)$
$\displaystyle\times\int_{0}^{t}d\tau\cos^{4}(\omega\tau/2)\sin(\bar{k}(t-\tau))\sin(\bar{k}t^{\prime})\sin(\bar{k}\tau)$
$\displaystyle\times\frac{d}{d\tau}\biggl{[}\frac{4\bar{\Lambda}^{2}\tau}{1+(2\bar{\Lambda}\tau)^{2}}\biggr{]}$
$\displaystyle+\left(t\leftrightarrow t^{\prime}\right).$ (103)

As we did for the one loop correction to the retarded Green’s function, we
first consider $k=0$ as we are interested in the behavior at
$\bar{k}t,\bar{k}t^{\prime}\ll 1$. Further, we will explore the same two
conditions considered for the retarded Green’s functions, i.e, either $\omega
t,\omega t^{\prime}\gg 1$, or $\omega t\gg 1\gg\omega t^{\prime}$. We use the
identity $\cos^{4}(\omega\tau/2)=3/8+\cos(\omega\tau)/2+\cos(2\omega\tau)/8$
to write,

$\displaystyle\delta
iG_{K}(k=0,t,t^{\prime})=\frac{8\theta}{3}q^{2}\sqrt{r_{0}}\cos(\omega
t/2)\cos(\omega t^{\prime}/2)$
$\displaystyle\times\int_{0}^{t}d\tau\biggl{[}\frac{3}{8}+\frac{1}{2}\cos(\omega\tau)+\frac{1}{8}\cos(2\omega\tau)\biggr{]}(t-\tau)t^{\prime}\tau$
$\displaystyle\times\frac{d}{d\tau}\biggl{[}\frac{4\bar{\Lambda}^{2}\tau}{1+(2\bar{\Lambda}\tau)^{2}}\biggr{]}$
$\displaystyle+\left(t\leftrightarrow t^{\prime}\right).$ (104)

Integrating by parts, we obtain

$\displaystyle\delta
iG_{K}(k=0,t,t^{\prime})=-\frac{8\theta}{3}q^{2}\sqrt{r_{0}}\cos(\omega
t/2)\cos(\omega t^{\prime}/2)$
$\displaystyle\times\int_{0}^{t}d\tau\biggl{[}\frac{3}{8}+\frac{1}{2}\cos(\omega\tau)+\frac{1}{8}\cos(2\omega\tau)\biggr{]}(tt^{\prime}-2t^{\prime}\tau)$
$\displaystyle\times\frac{4\bar{\Lambda}^{2}\tau}{1+(2\bar{\Lambda}\tau)^{2}}+\left(t\leftrightarrow
t^{\prime}\right)$
$\displaystyle+\frac{8\theta}{3}q^{2}\sqrt{r_{0}}\cos(\omega t/2)\cos(\omega
t^{\prime}/2)$
$\displaystyle\times\int_{0}^{t}d\tau\biggl{[}\frac{\omega}{2}\sin(\omega\tau)+\frac{\omega}{4}\sin(2\omega\tau)\biggr{]}(tt^{\prime}\tau-t^{\prime}\tau^{2})\frac{4\bar{\Lambda}^{2}\tau}{1+(2\bar{\Lambda}\tau)^{2}}$
$\displaystyle+\left(t\leftrightarrow t^{\prime}\right).$ (105)

At times which are large as compared to $\bar{\Lambda}^{-1}$, the dominant
terms are those which give logarithmic corrections. These in particular arise
from the first and second lines above. Among them, it is the second term in
the numerator $\propto 2t^{\prime}\tau,2t\tau$ that dominates. Keeping only
these dominant terms, and approximating
$\frac{4\bar{\Lambda}^{2}\tau}{1+(2\bar{\Lambda}\tau)^{2}}\approx 1/\tau^{2}$
we obtain

$\displaystyle\delta
iG_{K}(k=0,t,t^{\prime})\approx-\frac{8\theta}{3}q^{2}\sqrt{r_{0}}tt^{\prime}\cos(\omega
t/2)\cos(\omega t^{\prime}/2)$
$\displaystyle\times\int_{\bar{\Lambda}^{-1}}^{t}d\tau\biggl{[}\frac{3}{8}+\frac{1}{2}\cos(\omega\tau)+\frac{1}{8}\cos(2\omega\tau)\biggr{]}\frac{1}{\tau}+\left(t\leftrightarrow
t^{\prime}\right).$ (106)

Performing the integrals, one obtains

$\displaystyle\delta
iG_{K}(k=0,t,t^{\prime})\approx-\frac{8\theta}{3}q^{2}\sqrt{r_{0}}tt^{\prime}\cos(\omega
t/2)\cos(\omega t^{\prime}/2)$
$\displaystyle\times\biggl{\\{}\frac{3}{8}\ln\bigl{(}tt^{\prime}\bar{\Lambda}^{2}\bigr{)}+\frac{1}{2}\biggl{[}{\rm
Ci}(\omega t)-{\rm Ci}(\omega/\bar{\Lambda})\biggr{]}$
$\displaystyle+\frac{1}{8}\biggl{[}{\rm Ci}(2\omega t)-{\rm
Ci}(2\omega/\bar{\Lambda})\biggr{]}+\frac{1}{2}\biggl{[}{\rm Ci}(\omega
t^{\prime})-{\rm Ci}(\omega/\bar{\Lambda})\biggr{]}$
$\displaystyle+\frac{1}{8}\biggl{[}{\rm Ci}(2\omega t^{\prime})-{\rm
Ci}(2\omega/\bar{\Lambda})\biggr{]}\biggr{\\}}.$ (107)

For $\omega t\gg 1,\omega t^{\prime}\gg 1$ the terms involving the cosine-
integral ${\rm Ci}(x)$ do not give any logarithmic corrections. Noting that
the Green’s function at the Gaussian level is

$\displaystyle iG_{0K}(k=0,t,t^{\prime})=q^{2}\sqrt{r_{0}}\cos(\omega
t/2)\cos(\omega t^{\prime}/2)\,\,tt^{\prime},$ (108)

the leading logarithmic singularities and the correction discussed above
modify the interacting Green’s function at one-loop as

$\displaystyle\\!\\!iG_{K}(k=0,\omega t\gg 1,\omega t^{\prime}\gg
1)\\!\\!=q^{2}\sqrt{r_{0}}\cos(\omega t/2)\cos(\omega t^{\prime}/2)$
$\displaystyle\times
tt^{\prime}\biggl{[}1-\theta\ln\bigl{(}\bar{\Lambda}^{2}tt^{\prime}\bigr{)}\biggr{]}$
(109) $\displaystyle\propto q^{2}\sqrt{r_{0}}\cos(\omega t/2)\cos(\omega
t^{\prime}/2)\bigl{(}\bar{\Lambda}^{2}tt^{\prime}\bigr{)}^{1-\theta}.$ (110)

The above is Eq. (61) in the main text.

On the other hand if $\omega t\gg 1\gg\omega t^{\prime}$, then, some of the
cosine-integrals do contribute with a logarithm singularity, giving

$\displaystyle\delta iG_{K}(k=0,\omega t\gg 1\gg\omega
t^{\prime})\approx-\frac{8\theta}{3}q^{2}\sqrt{r_{0}}tt^{\prime}$
$\displaystyle\times\cos(\omega t/2)\cos(\omega
t^{\prime}/2)\biggl{[}\frac{3}{8}\ln(\bar{\Lambda}t)+\ln(\bar{\Lambda}t^{\prime})\biggr{]},$
(111)

where we have used that ${\rm
Ci}(\omega/\bar{\Lambda})\approx\ln(\omega/\bar{\Lambda})$ for
$\omega\ll\bar{\Lambda}$. Thus

$\displaystyle iG_{K}(k=0,\omega t\gg 1\gg\omega
t^{\prime})=q^{2}\sqrt{r_{0}}\cos(\omega t/2)\cos(\omega t^{\prime}/2)$
$\displaystyle\times
tt^{\prime}\biggl{[}1-\theta\ln(\bar{\Lambda}t)-\frac{8\theta}{3}\ln(\bar{\Lambda}t^{\prime})\biggr{]}$
(112) $\displaystyle\approx q^{2}\sqrt{r_{0}}\cos(\omega t/2)\cos(\omega
t^{\prime}/2)(\bar{\Lambda}t)^{1-\theta}(\bar{\Lambda}t^{\prime})^{1-8\theta/3}.$
(113)

The above is Eq. (62) in the main text.

##  Appendix E Keldysh Green’s function: Long-time behavior

At the Gaussian level, the equal-time Keldysh Green’s function is

$\displaystyle iG_{0K}(k,t,t)=q^{2}\cos^{2}(\omega
t/2)\frac{\sqrt{r_{0}}}{\bar{k}^{2}}\sin^{2}(\bar{k}t).$ (114)

We will now study the one-loop correction to this quantity in the limit of
$\bar{k}t\gg 1$. Using Eq. (18) and Eq. (80) along with the identity Eq.
(102), we obtain

$\displaystyle\delta
iG_{K}(k,t,t)=\frac{16\theta}{3}q^{2}\frac{\sqrt{r_{0}}}{\bar{k}^{3}}\cos^{2}(\omega
t/2)$
$\displaystyle\times\int_{0}^{t}d\tau\cos^{4}(\omega\tau/2)\sin(\bar{k}(t-\tau))\sin(\bar{k}t)\sin(\bar{k}\tau)$
$\displaystyle\times\frac{d}{d\tau}\biggl{[}\frac{4\bar{\Lambda}^{2}\tau}{1+(2\bar{\Lambda}\tau)^{2}}\biggr{]}.$
(115)

Above when we expand $\sin(\bar{k}(t-\tau))$ we only keep the
$\sin(\bar{k}t)\cos(\bar{k}\tau)$ term as this is the term that is
proportional to the Gaussian Green’s function $G_{0K}$. Thus

$\displaystyle\delta
iG_{K}(k,t,t)\approx\frac{8\theta}{3}q^{2}\frac{\sqrt{r_{0}}}{\bar{k}^{3}}\sin^{2}(\bar{k}t)\cos^{2}(\omega
t/2)$
$\displaystyle\times\int_{0}^{t}d\tau\cos^{4}(\omega\tau/2)\sin(2\bar{k}\tau)\frac{d}{d\tau}\biggl{[}\frac{4\bar{\Lambda}^{2}\tau}{1+(2\bar{\Lambda}\tau)^{2}}\biggr{]}$
$\displaystyle=\frac{8\theta}{3}q^{2}\frac{\sqrt{r_{0}}}{\bar{k}^{3}}\sin^{2}(\bar{k}t)\cos^{2}(\omega
t/2)$ $\displaystyle\times\biggl{[}\cos^{4}(\omega
t/2)\sin(2\bar{k}t)\frac{4\bar{\Lambda}^{2}t}{1+(2\bar{\Lambda}t)^{2}}$
$\displaystyle-2\bar{k}\int_{0}^{t}d\tau\cos^{4}(\omega\tau/2)\cos(2\bar{k}\tau)\frac{4\bar{\Lambda}^{2}\tau}{1+(2\bar{\Lambda}\tau)^{2}}$
$\displaystyle+2\omega\int_{0}^{t}d\tau\cos^{3}(\omega\tau/2)\sin(\omega\tau/2)\cos(2\bar{k}\tau)\frac{4\bar{\Lambda}^{2}\tau}{1+(2\bar{\Lambda}\tau)^{2}}\biggr{]}.$
(116)

For $\bar{\Lambda}t\gg 1$, i.e., for times that are long as compared to the
microscopic scale $\bar{\Lambda}^{-1}$, the first term falls off at long times
as $1/t$, and therefore we drop it. Collecting the remaining terms, we obtain

$\displaystyle\delta
iG_{K}(k,t,t)\approx\frac{8\theta}{3}q^{2}\frac{\sqrt{r_{0}}}{\bar{k}^{3}}\sin^{2}(\bar{k}t)\cos^{2}(\omega
t/2)$
$\displaystyle\times\biggl{\\{}-2\bar{k}\int_{{\bar{\Lambda}^{-1}}}^{t}d\tau\biggl{[}\frac{3}{8}+\frac{1}{2}\cos(\omega\tau)+\frac{1}{8}\cos(2\omega\tau)\biggr{]}\cos(2\bar{k}\tau)\frac{1}{\tau}$
$\displaystyle+\omega\int_{{\bar{\Lambda}^{-1}}}^{t}d\tau\biggl{[}\frac{1}{2}+\frac{1}{2}\cos(\omega\tau)\biggr{]}\sin(\omega\tau)\cos(2\bar{k}\tau)\frac{1}{\tau}\biggr{\\}}.$
(117)

Note that the second term does not give any logarithmic contributions. We then
define $T(\alpha)$ for $\alpha\ll\bar{\Lambda}$

$\displaystyle T(\alpha)$
$\displaystyle=\int_{{\bar{\Lambda}^{-1}}}^{t}d\tau\frac{\cos(\alpha\tau)}{\tau}={\rm
Ci}(\alpha t)-{\rm Ci}(\alpha/\bar{\Lambda})$
$\displaystyle\approx-\gamma-\ln(\alpha/\bar{\Lambda})+{\rm Ci}(\alpha t),$
(118)

to write

$\displaystyle\delta
iG_{K}(k,t,t)\approx-\frac{16\theta}{3}q^{2}\frac{\sqrt{r_{0}}}{\bar{k}^{2}}\sin^{2}(\bar{k}t)\cos^{2}(\omega
t/2)$
$\displaystyle\times\biggl{[}\frac{3}{8}T(2\bar{k})+\frac{1}{4}T(2\bar{k}+\omega)+\frac{1}{4}T(2\bar{k}-\omega)$
$\displaystyle+\frac{1}{16}T(2\bar{k}+2\omega)+\frac{1}{16}T(2\bar{k}-2\omega)\biggr{]}.$
(119)

Now at times longer than $\alpha^{-1}$, where
$\alpha=2\bar{k},\bigl{|}2\bar{k}\pm\omega\bigr{|}$, and $\bigl{|}2\bar{k}\pm
2\omega\bigr{|}$, the ${\rm Ci}(\alpha t)$ terms can be dropped because they
decrease as a power-law in $\alpha t$. Thus at long-times we obtain

$\displaystyle\delta iG_{K}(k,t,t)\approx
iG_{0K}(k,t,t)\frac{16\theta}{3}\biggl{[}\frac{3}{8}\ln(\bar{k}/\bar{\Lambda})$
$\displaystyle+\frac{1}{4}\ln((2\bar{k}+\omega)/\bar{\Lambda})+\frac{1}{4}\ln((2\bar{k}-\omega)/\bar{\Lambda})$
$\displaystyle+\frac{1}{16}\ln((2\bar{k}+2\omega)/\bar{\Lambda})+\frac{1}{16}\ln((2\bar{k}-2\omega)/\bar{\Lambda})\biggr{]}.$
(120)

In the long wavelength limit $\bar{k}\ll\omega$, only the first logarithm
dominates in the correction above. Therefore the resulting $G_{K}$ can be
written as

$\displaystyle iG_{K}(k,t,t;\bar{k}\ll\omega)\approx
q^{2}\frac{\sqrt{r_{0}}}{\bar{k}^{2-2\theta}}\sin^{2}(\bar{k}t)\cos^{2}(\omega
t/2).$ (121)

The above is Eq. (63) in the main text.

##  Appendix F Derivation of Eq. (69)

In order to determine the asymptotic behavior of the Green’s functions
discussed in Sec. VII, it is sufficient to study the terms with $m=\pm 1$ in
Eq. (68), as the behavior for $m=\pm 2$ follows in a straightforward manner.
In addition, Eqs. (68) have terms that are resonant due to their denominators
being proportional to $\bar{k}-m\omega/2$. In doing the Fourier transform in
space for highlighting the structure of the spatial correlations, these
resonant terms dominate over the non-resonant terms. Accordingly, the analysis
below focuses only on the resonant contributions.

Using Eq. (15), the spatial Fourier transform in $d=4$ requires computing

$\displaystyle{\cal
I}_{R,K}(x,t,t^{\prime})=\frac{1}{4\pi^{2}x}\int_{0}^{\Lambda}dkk^{2}J_{1}(kx)I_{R,K}(k,t,t^{\prime}),$
(122)

where ${\cal I}_{R,K}(x,t,t^{\prime})$ contribute to the one loop corrections
$G_{R,K}^{(1)}$ and

$\displaystyle I_{R}(k,t,t^{\prime})$ $\displaystyle=\sum_{m=\pm
1}\cos(m\omega(t+t^{\prime})/2)$
$\displaystyle\quad\times\frac{\sin((k-m\omega/2)(t-t^{\prime}))}{k^{2}(k-m\omega/2)},$
(123) $\displaystyle I_{K}(k,t,t)$ $\displaystyle=\sum_{m=\pm
1}\frac{1}{k^{3}(2k-m\omega)}$
$\displaystyle\quad\times\biggl{\\{}[1+\cos(m\omega t)][1-\cos(2kt)]$
$\displaystyle\quad-\sin(2kt)\sin(m\omega t)\biggr{\\}}.$ (124)

Above we have kept only the resonant terms in Eq. (68). For notational
convenience, we have also set $v=1$, so that $\bar{k}=k$.

###  F.1 Evaluation of ${\cal I}_{R}$

In order to evaluate ${\cal I}_{R}$, we perform a shift of variables
$k-m\omega/2\rightarrow k$ and obtain

$\displaystyle{\cal I}_{R}(x,t,t^{\prime})$
$\displaystyle=\cos(\omega(t+t^{\prime})/2)\frac{1}{4\pi^{2}x}$
$\displaystyle\times\biggl{[}\int_{-\omega/2}^{\Lambda-\omega/2}dk\frac{\sin(k(t-t^{\prime}))}{k}J_{1}(kx+\omega
x/2)$
$\displaystyle+\int_{\omega/2}^{\Lambda+\omega/2}dk\frac{\sin(k(t-t^{\prime}))}{k}J_{1}(kx-\omega
x/2)\biggr{]}$
$\displaystyle\approx\cos(\omega(t+t^{\prime})/2)\frac{1}{4\pi^{2}x}$
$\displaystyle\times\biggl{\\{}\int_{-\omega/2}^{0}dk\frac{\sin(k(t-t^{\prime}))}{k}J_{1}(kx+\omega
x/2)$
$\displaystyle+\int_{\omega/2}^{0}dk\frac{\sin(k(t-t^{\prime}))}{k}J_{1}(kx-\omega
x/2)$
$\displaystyle+\int_{0}^{\Lambda}dk\frac{\sin(k(t-t^{\prime}))}{k}\biggl{[}J_{1}(kx+\omega
x/2)$ $\displaystyle+J_{1}(kx-\omega x/2)\biggr{]}\biggr{\\}},$ (125)

where in the last term we have used that $\Lambda\gg\omega$. In the first term
of Eq. (125), we perform the transformation $k\rightarrow-k$ and use that
$J_{1}(-x)=-J_{1}(x)$ to write

$\displaystyle{\cal
I}_{R}(x,t,t^{\prime})=\cos(\omega(t+t^{\prime})/2)\frac{1}{4\pi^{2}x}$
$\displaystyle\times\biggl{\\{}-2\int^{\omega/2}_{0}dk\frac{\sin(k(t-t^{\prime}))}{k}J_{1}(kx-\omega
x/2)$
$\displaystyle+\int_{0}^{\Lambda}dk\frac{\sin(k(t-t^{\prime}))}{k}\biggl{[}J_{1}(kx+\omega
x/2)$ $\displaystyle+J_{1}(kx-\omega x/2)\biggr{]}\biggr{\\}}.$ (126)

Following this, we use that $J_{1}$ may be replaced by its asymptotic value
$J_{1}(kx+\omega x/2)=-\sqrt{\frac{2}{\pi(kx+\omega x/2)}}\cos(kx+\omega
x/2+\pi/4)$ for $\omega x\gg 1$. This gives

$\displaystyle{\cal
I}_{R}(x,t,t^{\prime})\approx\cos(\omega(t+t^{\prime})/2)\sqrt{\frac{2}{\pi}}\frac{1}{4\pi^{2}x^{3/2}}$
$\displaystyle\times\biggl{\\{}2\int^{\omega/2}_{0}dk\frac{\sin(k(t-t^{\prime}))}{k}\frac{\cos(kx-\omega
x/2+\pi/4)}{\sqrt{k-\omega/2}}$
$\displaystyle-\int_{0}^{\Lambda}dk\frac{\sin(k(t-t^{\prime}))}{k}\biggl{[}\frac{\cos(kx+\omega
x/2+\pi/4)}{\sqrt{k+\omega/2}}$ $\displaystyle+\frac{\cos(kx-\omega
x/2+\pi/4)}{\sqrt{k-\omega/2}}\biggr{]}\biggr{\\}}.$ (127)

We now discuss the two non-trivial cases, i.e., the behavior on the light-cone
and the behavior inside it.

On the light-cone, since $x=t-t^{\prime}$, (and $t>t^{\prime}$ due to the
causal structure of the retarded Green’s function) we write

$\displaystyle{\cal
I}_{R}(x=t-t^{\prime})\approx\cos(\omega(t+t^{\prime})/2)\sqrt{\frac{2}{\pi}}\frac{1}{4\pi^{2}x^{3/2}}$
$\displaystyle\times\biggl{\\{}2\int^{\omega/2}_{0}dk\frac{\sin(kx)}{k}\frac{\cos(kx-\omega
x/2+\pi/4)}{\sqrt{k-\omega/2}}$
$\displaystyle-\int_{0}^{\Lambda}dk\frac{\sin(kx)}{k}\biggl{[}\frac{\cos(kx+\omega
x/2+\pi/4)}{\sqrt{k+\omega/2}}$ $\displaystyle+\frac{\cos(kx-\omega
x/2+\pi/4)}{\sqrt{k-\omega/2}}\biggr{]}\biggr{\\}}$
$\displaystyle\approx\cos(\omega(t+t^{\prime})/2)\sqrt{\frac{2}{\pi}}\frac{1}{4\pi^{2}x^{3/2}}$
$\displaystyle\times\biggl{\\{}2\sin(\omega
x/2-\pi/4)\int^{\omega/2}_{0}dk\frac{\sin^{2}(kx)}{k}\frac{1}{\sqrt{k-\omega/2}}$
$\displaystyle-\int_{0}^{\Lambda}dk\frac{\sin^{2}(kx)}{k}\biggl{[}-\frac{\sin(\omega
x/2+\pi/4)}{\sqrt{k+\omega/2}}$ $\displaystyle+\frac{\sin(\omega
x/2-\pi/4)}{\sqrt{k-\omega/2}}\biggr{]}\biggr{\\}}.$ (128)

In the last equality of the previous equation we have only kept terms that
oscillate in phase. Since this expression is dominated by $k$ away from
$\pm\omega/2$ we may write

$\displaystyle{\cal
I}_{R}(x=t-t^{\prime})\approx\cos(\omega(t+t^{\prime})/2)\sqrt{\frac{2}{\pi}}\frac{x^{1/2}}{4\pi^{2}x^{3/2}}$
$\displaystyle\times\biggl{\\{}2\sin(\omega x/2-\pi/4)\int^{\omega
x/2}_{0}dy\frac{\sin^{2}(y)}{y}\frac{1}{\sqrt{y}}$
$\displaystyle-\int_{0}^{\Lambda
x}dy\frac{\sin^{2}(y)}{y}\biggl{[}-\frac{\sin(\omega x/2+\pi/4)}{\sqrt{y}}$
$\displaystyle+\frac{\sin(\omega x/2-\pi/4)}{\sqrt{y}}\biggr{]}\biggr{\\}}.$
(129)

Performing the $y$-integral we find

$\displaystyle{\cal I}_{R}(x=t-t^{\prime})$
$\displaystyle\propto\cos(\omega(t+t^{\prime})/2)\sin(\omega
x/2+\delta)\frac{1}{x^{3/2}},$ (130)

where $\delta$ stands for a constant phase shift that originates from the
$\pi/4$ phase in the asymptotic expansion of the Bessel function.

For studying the behavior of ${\cal I}_{R}$ at points away from the light
cone, let us introduce $y=k(t-t^{\prime})$. Then Eq. (127) becomes

$\displaystyle{\cal
I}_{R}(x,t,t^{\prime})=\cos(\omega(t+t^{\prime})/2)\frac{1}{4\pi^{2}x}$
$\displaystyle\times\biggl{\\{}-2\int^{\omega(t-t^{\prime})/2}_{0}dy\frac{\sin(y)}{y}J_{1}(yx/(t-t^{\prime})-\omega
x/2)$
$\displaystyle+\int_{0}^{\Lambda(t-t^{\prime})}dy\frac{\sin(y)}{y}\biggl{[}J_{1}(yx/(t-t^{\prime})+\omega
x/2)$ $\displaystyle+J_{1}(yx/(t-t^{\prime})-\omega x/2)\biggr{]}\biggr{\\}}.$
(131)

When $x\ll t-t^{\prime}$, and since
$\omega(t-t^{\prime}),\Lambda(t-t^{\prime})\gg 1$, the upper limits of
integration in the expression above can be set to $\infty$. We can also
approximate $J_{1}(yx/(t-t^{\prime})+\omega x/2)\approx J_{1}(\omega x/2)$.
Moreover, in the second term above, the two Bessel functions cancel each other
giving

$\displaystyle{\cal I}_{R}(x\ll t-t^{\prime})$
$\displaystyle=\cos(\omega(t+t^{\prime})/2)\frac{1}{4\pi^{2}x}$
$\displaystyle\times J_{1}(\omega x/2)\biggl{(}2\int^{\infty}_{0}dy\frac{\sin
y}{y}\biggr{)}.$ (132)

Accordingly, inside the light-cone, we obtain

$\displaystyle{\cal I}_{R}(x\ll t-t^{\prime})$
$\displaystyle=\cos(\omega(t+t^{\prime})/2)\frac{1}{4\pi x}J_{1}(\omega x/2)$
$\displaystyle\propto\cos(\omega(t+t^{\prime})/2)\sin(\omega
x/2+\widetilde{\delta})\frac{1}{x^{3/2}},$ (133)

where in the last line we have used the asymptotic form of $J_{1}(\omega x/2)$
and $\widetilde{\delta}$ denotes a constant phase, which as before, originates
from the $\pi/4$ phase coming from the asymptotic expansion of the Bessel
function. Thus, together with Eq. (130) and Eq. (133), we have recovered the
asymptotic form reported in Eq. (69) for the retarded Green’s function.

###  F.2 Evaluation of ${\cal I}_{K}$

We now derive the asymptotic behavior of the equal-time Keldysh Green’s
function at one loop. Using Eq. (F) and assuming $\Lambda\gg\omega$, we obtain

$\displaystyle{\cal I}_{K}(x,t,t)$
$\displaystyle=\frac{1}{4\pi^{2}x}\int_{-\omega/2}^{\Lambda}dk\frac{1}{2k}\biggl{\\{}\sin(2kt)\sin(\omega
t)$ $\displaystyle+[1-\cos(2kt)][1+\cos(\omega
t)]\biggr{\\}}\frac{J_{1}(kx+\omega x/2)}{k+\omega/2}$
$\displaystyle+\frac{1}{4\pi^{2}x}\int_{\omega/2}^{\Lambda}dk\frac{1}{2k}\biggl{\\{}-\sin(2kt)\sin(\omega
t)$ $\displaystyle+[1-\cos(2kt)][1+\cos(\omega
t)]\biggr{\\}}\frac{J_{1}(kx-\omega x/2)}{k-\omega/2}.$ (134)

As before, we split the integral as follows:

$\displaystyle{\cal
I}_{K}(x,t,t)=\frac{1}{4\pi^{2}x}\int_{-\omega/2}^{0}dk\frac{1}{2k}\biggl{\\{}\sin(2kt)\sin(\omega
t)$ $\displaystyle+[1-\cos(2kt)][1+\cos(\omega
t)]\biggr{\\}}\frac{J_{1}(kx+\omega x/2)}{k+\omega/2}$
$\displaystyle+\frac{1}{4\pi^{2}x}\int_{0}^{\omega/2}dk\frac{1}{2k}\biggl{\\{}\sin(2kt)\sin(\omega
t)$ $\displaystyle-[1-\cos(2kt)][1+\cos(\omega
t)]\biggr{\\}}\frac{J_{1}(kx-\omega x/2)}{k-\omega/2}$
$\displaystyle+\frac{1}{4\pi^{2}x}\int_{0}^{\Lambda}dk\frac{1}{2k}\biggl{\\{}\frac{J_{1}(kx+\omega
x/2)}{k+\omega/2}\biggl{[}\sin(2kt)\sin(\omega t)$
$\displaystyle+(1-\cos(2kt))(1+\cos(\omega t))\biggr{]}$
$\displaystyle+\frac{J_{1}(kx-\omega
x/2)}{k-\omega/2}\biggl{[}-\sin(2kt)\sin(\omega t)$
$\displaystyle+(1-\cos(2kt))(1+\cos(\omega t))\biggr{]}\biggr{\\}}.$ (135)

Performing the transformation $k\rightarrow-k$ in the first term and noting
that $J_{1}(-x)=-J_{1}(x)$, we obtain

$\displaystyle{\cal
I}_{K}(x,t,t)=\frac{1}{2\pi^{2}x}\int_{0}^{\omega/2}dk\frac{1}{2k}\biggl{\\{}\sin(2kt)\sin(\omega
t)$ $\displaystyle-[1-\cos(2kt)][1+\cos(\omega
t)]\biggr{\\}}\frac{J_{1}(kx-\omega x/2)}{k-\omega/2}$
$\displaystyle+\frac{1}{4\pi^{2}x}\int_{0}^{\Lambda}dk\frac{1}{2k}\biggl{\\{}\frac{J_{1}(kx+\omega
x/2)}{k+\omega/2}\biggl{[}\sin(2kt)\sin(\omega t)$
$\displaystyle+(1-\cos(2kt))(1+\cos(\omega t))\biggr{]}$
$\displaystyle+\frac{J_{1}(kx-\omega
x/2)}{k-\omega/2}\biggl{[}-\sin(2kt)\sin(\omega t)$
$\displaystyle+(1-\cos(2kt))(1+\cos(\omega t))\biggr{]}\biggr{\\}}.$ (136)

We now separately discuss the behavior on the light-cone and inside the light-
cone. On the light-cone, since $x=2t$, we write Eq. (136) as

$\displaystyle{\cal I}_{K}(x=2t)=\frac{1}{2\pi^{2}}\int_{0}^{\omega
t}dy\frac{1}{2y}\biggl{\\{}\sin y\sin(\omega t)$ $\displaystyle-(1-\cos
y)[1+\cos(\omega t)]\biggr{\\}}\frac{J_{1}(y-\omega x/2)}{y-\omega x/2}$
$\displaystyle+\frac{1}{4\pi^{2}}\int_{0}^{\Lambda
x}dy\frac{1}{2y}\biggl{\\{}\frac{J_{1}(y+\omega x/2)}{y+\omega
x/2}\biggl{[}\sin y\sin(\omega t)$ $\displaystyle+(1-\cos y)(1+\cos(\omega
t))\biggr{]}$ $\displaystyle+\frac{J_{1}(y-\omega x/2)}{y-\omega
x/2}\biggl{[}-\sin y\sin(\omega t)$ $\displaystyle+(1-\cos y)(1+\cos(\omega
t))\biggr{]}\biggr{\\}}.$ (137)

Replacing $J_{1}$ by its leading asymptotic form we obtain

$\displaystyle{\cal
I}_{K}(x=2t)=-\frac{1}{2\pi^{2}}\sqrt{\frac{2}{\pi}}\int_{0}^{\omega
t}dy\frac{1}{2y}\biggl{\\{}\sin y\sin(\omega t)$ $\displaystyle-(1-\cos
y)[1+\cos(\omega t)]\biggr{\\}}\frac{\cos(y-\omega x/2+\pi/4)}{(y-\omega
x/2)^{3/2}}$
$\displaystyle-\frac{1}{4\pi^{2}}\sqrt{\frac{2}{\pi}}\int_{0}^{\Lambda
x}dy\frac{1}{2y}\biggl{\\{}\frac{\cos(y+\omega x/2+\pi/4)}{(y+\omega
x/2)^{3/2}}\biggl{[}\sin y\sin(\omega t)$ $\displaystyle+(1-\cos
y)(1+\cos(\omega t))\biggr{]}$ $\displaystyle+\frac{\cos(y-\omega
x/2+\pi/4)}{(y-\omega x/2)^{3/2}}\biggl{[}-\sin y\sin(\omega t)$
$\displaystyle+(1-\cos y)(1+\cos(\omega t))\biggr{]}\biggr{\\}}.$ (138)

This equation implies

$\displaystyle{\cal I}_{K}(x=2t)$ $\displaystyle\propto\frac{\cos(\omega
x/2+\alpha^{\prime\prime})}{x^{3/2}},$ (139)

where $\alpha^{\prime\prime}$ is a constant phase-shift originating from the
$\pi/4$ phase in the asymptotic expansion of the Bessel function.

Now we discuss the behavior inside the light-cone, corresponding to having
$2t\gg x$. Here we may replace $1-\cos(2kt)\approx 1$, obtaining

$\displaystyle{\cal I}_{K}(2t\gg x)$
$\displaystyle\approx\frac{1}{4\pi^{2}}\int_{1}^{\Lambda
x}dy\frac{1}{2y}\biggl{[}\frac{J_{1}(y+\omega x/2)}{y+\omega x/2}$
$\displaystyle+\frac{J_{1}(y-\omega x/2)}{y-\omega x/2}\biggr{]}$
$\displaystyle-\frac{1}{2\pi^{2}}\int_{1}^{\omega
x/2}dy\frac{1}{2y}\frac{J_{1}(y-\omega x/2)}{y-\omega x/2}.$ (140)

Let us assume $\omega x\gg 1$, so that we can set both the upper-limits of
integration $\Lambda x=\omega x=\infty$. Then

$\displaystyle{\cal I}_{K}(2t\gg x)$
$\displaystyle\approx\frac{1}{4\pi^{2}}\int_{1}^{\infty}dy\frac{1}{2y}\biggl{[}\frac{J_{1}(y+\omega
x/2)}{y+\omega x/2}$ $\displaystyle-\frac{J_{1}(y-\omega x/2)}{y-\omega
x/2}\biggr{]}$ $\displaystyle\propto\frac{\cos(\omega
x/2+\alpha^{\prime\prime\prime})}{x^{3/2}}.$ (141)

Above $\alpha^{\prime\prime\prime}$ is a constant phase-shift, which as before
can be traced back to the $\pi/4$ phase in the asymptotic form of the Bessel
function. Equations (139) and (141) together summarize the asymptotic behavior
of the Keldysh Green’s function at one loop, and yield Eq. (69) anticipated in
the main text.