# Double-trace deformation in Keldysh field theory

Xiangyi Meng<EMAIL_ADDRESS>Department of Physics, Boston University, Boston,
Massachusetts 02215, USA  Department of Physics, Northeastern University,
Boston, Massachusetts 02115, USA

###### Abstract

The Keldysh formalism is capable of describing driven-dissipative dynamics of
open quantum systems as nonunitary effective field theories that are not
necessarily thermodynamical, thus often exhibiting new physics. Here, we
introduce a general Keldysh action that maximally obeys Weinbergian
constraints, including locality, Poincaré invariance, and two “ _CPT_ ”
constraints: complete positivity and trace preserving as well as charge,
parity, and time reversal symmetry. We find that the perturbative Lindblad
term responsible for driven-dissipative dynamics introduced therein has the
natural form of a double-trace deformation $\mathcal{O}^{2}$, which, in the
large $N$ limit, possibly leads to a new nonthermal conformal fixed point.
This fixed point is IR when $\Delta<d/2$ or UV when $\Delta>d/2$ given $d$ the
dimensions of spacetime and $\Delta$ the scaling dimension of $\mathcal{O}$.
Such a UV fixed point being not forbidden by Weinbergian constraints may
suggest its existence and even completion of itself, in contrast to the common
sense that dissipation effects are always IR relevant. This observation
implies that driven-dissipative dynamics is much richer than thermodynamics,
differing in not only its noncompliance with thermodynamic symmetry (e.g., the
fluctuation-dissipation relation) but its UV/IR relevance as well. Examples
including a $(0+1)$-$d$ harmonic oscillator under continuous measurement and a
$(4-\epsilon)$-$d$ classic $O(N)$ vector model with quartic interactions are
studied.

##  I Introduction

Quantum mechanics, despite its extreme success in predicting and agreeing with
every experimental outcome, still bemuses everyone by “suspiciously”
postulating the existence of _two_ distinct but necessary time-evolution
mechanisms, i.e., reversible unitary dynamics versus irreversible wave
function collapse [1]. A better understanding of the origins and essentials of
why a physical system may evolve in such two different ways seems the key to
answering some of the most fundamental problems such as the black-hole
information paradox [2, 3, 4], quantum measurement problem [5, 6], or even
consistency of quantum mechanics itself [7]. Not only that, it has recently
been shown that the entanglement entropy for a quantum circuit undergoing the
competition of unitary evolution and random measurements may behave quite
distinctly by adjusting the rate of measurements [8], a better understanding
of which serves practical purposes such as universal quantum computing [9],
etc.

Even though the underlying physics remains unclear, the proper mathematical
framework to integrate the two distinct time-evolution mechanisms has been
well developed, such as the Choi-Kraus’s theorem [10, 11] and the Gorini-
Kossakowski-Sudarshan-Lindblad theorem [12, 13]. These theorems have further
become the basis for the study of open quantum systems (OQS) [14], which aims
to understand all possible forms of evolution of a quantum system that is not
closed, and hence, unitarity can be lost. This does not only include examples
such as how a system reaches equilibrium by being in contact with a _thermal_
reservoir (e.g., quantum Brownian motion [15]), but also deals with a much
more general scenario where, after tracing out part of the degrees of freedom
(d.o.f.) of the system, the rest behaves effectively as an OQS coupled to an
artificial environment consisting of the traced-out d.o.f. As for the latter
scenario, the OQS undergoes driving force and dissipation, both induced by the
environment, but is not necessarily thermalized. Led by driven-dissipative
dynamics, the OQS may reach a _nonthermal stationary state_ —a realizable
state of matter that can often be seen in many light-matter systems of Bose-
Einstein condensates or Rydberg ensembles [16]. Moreover, in the context of
decoherence theory [17, 18], the wave function collapse induced by quantum
measurement [19] may be framed as an environmental effect on the OQS as well,
the idea of which has eventually triggered the development of a number of
quantum collapse models [5], including the continuous spontaneous localization
(CSL) model [20].

Unfortunately, for _many-body_ OQS, there are much fewer theoretical tools
having been developed, despite the rich emergent and universal phenomena that
the system may possess at the macroscopic scale. Most commonly, a field-
theoretical tool is preferred, so that a number of well-developed techniques
such as renormalization group (RG) will be applicable. A complete field-
theoretical solution involves first the use of the Keldysh path integral [21,
22, 23], which formulates how to set up a path integral that governs the
evolution of not a wave function but a density matrix by _doubling_ the to-be-
integrated field $\phi$ into two separate fields in the ket ($+$) and bra
($-$) basis, respectively [24, 25]. Next, tracing out the environment’s d.o.f.
is carried out by the use of the Feynman-Vernon influential functional [26],
leaving an effective field theory (EFT) as a Keldysh path-integral functional
where only the bra and ket fields corresponding to the remaining d.o.f. are
kept. This promising method of describing many-body OQS in a field-theoretical
language has been actively discussed both in the nonrelativistic context [16]
and relativistic context [24, 25] and has led to new research directions such
as novel universality class in quantum phase transition [27], information loss
in EFT [28, 29], etc. Note that a Feynman-Vernon influential functional is
usually highly nonlocal. However, after applying the Born-Markov approximation
[14], a Markovian form can often be produced where the time-evolution
mechanisms are automatically local.

Inspired by this field-theoretical language, in this paper, we start directly
with a general action formalized as a Keldysh path integral that works as a
heuristic phenomenological model. Our propose is to study the driven-
dissipative dynamics originating from mixing of the reversible and
irreversible time-evolution mechanisms. We will discuss their implications
rather than arguing about the origins of the mechanisms. Our Keldysh action is
similar to the open-EFT Keldysh action introduced in Ref. [30] but more
general and only required to maximally obey “Weinbergian” constraints such as
locality and Poincaré invariance [31]. Note that unitarity, however, must be
revoked because of the existence of irreversible wave function collapse.

The main result we find for our Keldysh action is that, under appropriately
renormalized perturbation, the nonunitary terms responsible for driven-
dissipative dynamics have the natural form of a _double-trace deformation_
[32, 33, 34, 35] which has been very thoroughly studied—especially in the
literature of conformal field theory (CFT) [36, 37]. In particular, in the
large $N$ limit, it is argued that the existence of a double-trace deformation
guarantees and produces a healthy RG flow from infrared (IR) backwards to
ultraviolet (UV), suggesting that the theory is UV complete [38], not merely
an EFT. In fact, this argument also holds true here, and we further show that
this UV theory is free of ghosts and tachyons, hence a physical relativistic
theory. Yet, the scaling dimension of the dissipation effect observed at the
UV fixed point clearly deviates from what a thermodynamic _flutuation-
dissipation relation_ (FDR) would predict. Our finding thus demonstrates a key
difference between thermodynamics and driven-dissipative dynamics.

Note that although deviations of OQS from thermodynamics have been widely
studied [16] by explicit use of the Keldysh path integral, to our best
knowledge, no similar studies have been done using CFT tools. Our observation
implies that driven-dissipative dynamics is much richer than thermodynamics,
differing in not only its non-compliance with thermodynamic symmetry [39] but
its UV/IR relevance as well. In fact, although a UV-relevant dissipation
effect seems unphysical in thermodynamics, it has never, in theory, been
forbidden in driven-dissipative dynamics, which could remind us Gell-Mann’s
totalitarian principle: “ _Everything not forbidden is compulsory_ ” [40]. Our
results may shed light on field-theoretical relativistic collapse models where
the breakdown of unitarity may indeed happen at some nontrivial high energy
scale. Our results may also offer a better understanding of universal
dynamical phase transitions of condensed matters which are quasirelativistic
(the dynamical exponent $z\approx 1$) near the critical point. Finally, our
results pave a new path that brings the Keldysh formalism to the gravitational
side under the pronounced holographic AdS/CFT correspondence [41, 42, 43, 44,
45]. This may lead to a refreshing perspective of how a Keldysh CFT living on
the boundary corresponds to the bulk theory, which we will briefly discuss at
the end.

##  II Keldysh formalism

We start with a Keldysh path integral of the most general form [24, 25] for a
real scalar field $\phi$,

$\displaystyle\int_{\phi_{\pm}(t^{i},\mathbf{x})=\phi_{\pm}^{i}(\mathbf{x})}^{\phi_{\pm}(t^{f},\mathbf{x})=\phi_{\pm}^{f}(\mathbf{x})}D\phi_{+}D\phi_{-}e^{iS[\phi_{+},\phi_{-}]},$
(1)

which is identified as the twofold time-evolution amplitude

$\left\langle\phi_{+}^{f}(\mathbf{x})\right|\left[\mathcal{V}(t^{f},t^{i})(\left|\phi_{+}^{i}(\mathbf{x})\right\rangle\left\langle\phi_{-}^{i}(\mathbf{x})\right|)\right]\left|\phi_{-}^{f}(\mathbf{x})\right\rangle$

between the initial and final boundary conditions $\phi^{i}_{\pm}(\mathbf{x})$
and $\phi^{f}_{\pm}(\mathbf{x})$. The time-evolution superoperator
$\mathcal{V}(t^{f},t^{i})(\cdot)$ governs the dynamics of not bras or kets in
the Hilbert space, but operators (matrices). The Keldysh action
$S[\phi_{+},\phi_{-}]$ as a functional can be constructed given any specific
form of $\mathcal{V}$ following the standard path integral approach (Appendix
A).

Given an initial distribution $\rho(t^{i})$ as a functional of
$\phi^{i}_{\pm}(\mathbf{x})$, multipoint functions of the final distribution
$\rho(t^{f})=\mathcal{V}(t^{f},t^{i})(\rho(t^{i}))$ are well defined by Eq.
(1),

$\displaystyle\left\langle\phi_{+}(x_{1})\cdots\phi_{+}(x_{m})\phi_{-}(x_{1})\cdots\phi_{-}(x_{n})\right\rangle$
(2) $\displaystyle=$
$\displaystyle\left.\frac{\left(-i\right)^{m+n}\delta^{m+n}Z[J_{+},J_{-}]}{\delta
J_{+}(x_{1})\cdots\delta J_{+}(x_{m})\delta J_{-}(x_{1})\cdots\delta
J_{-}(x_{n})}\right|_{J_{+}=J_{-}=0}$

in terms of $\phi_{\pm}$, where the partition function

$\displaystyle
Z[J_{+},J_{-}]=\int\limits_{\rho(t^{i})}^{\phi^{f}_{+}(\mathbf{x})=\phi^{f}_{-}(\mathbf{x})}D\phi_{+}D\phi_{-}e^{i\left(S[\phi_{+},\phi_{-}]+J_{+}\phi_{+}+J_{-}\phi_{-}\right)}\quad$
(3)

is constructed by adding the source terms $J_{\pm}$. The multi-point function
in Eq. (2) is equal to

$\displaystyle\text{Tr}\left\\{\mathcal{T}\left\\{\phi(x_{1})\cdots\phi(x_{m})\right\\}\rho(t^{i})\mathcal{\bar{T}}\left\\{\phi(x_{1})\cdots\phi(x_{n})\right\\}\right\\}$
(4)

in the operator formalism (Appendix A), in which $\mathcal{V}$ is invoked in
$\phi(t,\mathbf{x})=\mathcal{V}^{\dagger}(t,t^{i})(\phi(t^{i},\mathbf{x}))$,
given $\mathcal{V}^{\dagger}$ the adjoint superoperator of $\mathcal{V}$ [14].
Note that $\rho(t^{i})$ is not necessarily a density matrix. Being so,
however, allows Eq. (4) to be interpreted as a positive and normalized
physical correlation function. Also, note that two independent time-ordering
and anti-time-ordering superoperators $\mathcal{T}$ and $\mathcal{\bar{T}}$
are present in Eq. (4). Thus, combinations of local operators do not
necessarily follow one single time order [30]—compared to traditional Feynman
path integrals where operators can only be combined in one time order. Indeed,
Eq. (4) is the only form of correlation functions that are directly measurable
[46] (in contrast to the out-of-time-order correlation functions which are not
[47]), indicating that any measurable quantum evolution can be completely
described by $S[\phi_{+},\phi_{-}]$.

Here, we write down an explicit functional form for the Keldysh action,
$S[\phi_{+},\phi_{-}]=\int dxL(\phi_{+},\phi_{-})$, where

$\displaystyle L(\phi_{+},\phi_{-})=L_{0}(\phi_{+})-L_{0}(\phi_{-})$ (5)
$\displaystyle-$ $\displaystyle
i\sum\nolimits_{i}\gamma_{i}\left(\mathcal{O}_{i,+}\mathcal{O}^{*}_{i,-}-\frac{1}{2}\mathcal{O}^{*}_{i,+}\mathcal{O}_{i,+}-\frac{1}{2}\mathcal{O}^{*}_{i,-}\mathcal{O}_{i,-}\right)\qquad$

is locally composed of an unperturbed unitary Lagrangian $L_{0}(\phi)$ and
_arbitrary_ complex scalar fields $\mathcal{O}_{i}$ as perturbations, and our
metric signature convention is $\left(+,-,-,-\right)$. We require Eq. (5) to
be Lorentz covariant; we also require that $\gamma_{i}\geq 0$ and is time
independent. As a result, the special form of Eq. (5) guarantees that our
$S[\phi_{+},\phi_{-}]$ is at least perturbatively a field theory that obeys
some Weinbergian-like constraints [31], as explained below.

First, we must reverse engineer the superoperator $\mathcal{V}$ from Eq. (5).
When $\mathcal{O}_{i}$ does not contain spacetime derivatives, one derives

$\mathcal{V}(t^{f},t^{i})=e^{\left(t^{f}-t^{i}\right)\mathcal{L}}$

where the superoperator $\mathcal{L}$ admits a Lindblad form [14]

$\displaystyle\mathcal{L}(\rho)=-i\left[H,\rho\right]$ (6) $\displaystyle+$
$\displaystyle\sum\nolimits_{i}\gamma_{i}\int
d\mathbf{x}\left(\mathcal{O}_{i}(\mathbf{x})\rho\mathcal{O}^{\dagger}_{i}(\mathbf{x})-\frac{1}{2}\\{\mathcal{O}^{\dagger}_{i}(\mathbf{x})\mathcal{O}_{i}(\mathbf{x}),\rho\\}\right),\hskip
14.22636pt$

given $H$ the corresponding Hamiltonian of $L_{0}$ by the Legendre
transformation (Appendix A). Note that in a proper Lindblad form,
$\mathcal{O}_{i}(\mathbf{x})$ are required to be trace-zero and orthogonal to
each other, but this can indeed be made so for arbitrary
$\mathcal{O}_{i}(\mathbf{x})$ by linear recombination [48]. This explains why
there is no constraint on $\mathcal{O}_{i}$ in Eq. (5).

When $\mathcal{O}_{i}$ contains spacetime derivatives (which is not forbidden
since unitarity is not concerned), $\mathcal{V}$ is intractable and may not be
Lindblad as wished. Nevertheless, _Matthews’s theorem_ [49] implies that the
naïve Lorentz-covariant Lagrangian path-integral formalism [Eq. (5)] should
yield at least perturbatively identical results compared to the correct
Hamiltonian path-integral formalism [Eq. (6)] even if $\mathcal{O}_{i}$
contains spacetime derivatives [50, 51]. Thus, Eq. (6) is still valid
perturbatively.

Note that our $\mathcal{V}$ is Markovian, i.e.,
$\mathcal{V}(t_{1}+t_{2},0)=\mathcal{V}(t_{1},0)\mathcal{V}(t_{2},0)$, for
$t_{1},t_{2}>0$, so that the evolution of the system in the future does not
depend on its history, as demanded by locality and translational invariance
[14]. Therefore, $\\{\mathcal{V}(t,0)\\}$ as a one-parameter family of $t$
forms a quantum dynamical semigroup. One may suspect if the semigroup can
still produce some sort of conservation law. The answer is yes [52]. Equation
(6) has been widely used to describe the dynamics of an OQS under the Born-
Markov approximation, which means that rich physics of nonlocality (non-
Markovianity [53]) such as decrease of quantum speed limit [54] or emergence
of multiple timescales [55] has to be ignored. In the language of quantum
measurement, this amounts to the assumption that the nonselective continuous
measurements the OQS undergoes are weak enough [14]. It is, however, unclear
to what extent non-Markovianity should be seriously considered in a
fundamental field theory of some kind, e.g., quantum collapse models [5],
where a violation of locality is perhaps unfavorable.

We stress that our $\mathcal{V}$ is known to obey the following two “ _CPT_ ”
constraints:

###  II.1 CPT (complete positivity and trace persevering)

Any physical $\mathcal{V}$ that maps density matrices to density matrices must
be complete positive and trace persevering [14]. The Choi-Kraus’s theorem [10,
11] states that an arbitrary superoperator $\mathcal{V}(\rho)$ is _CPT_ if and
only if it can be expressed as
$\mathcal{V}(\rho)=\sum\nolimits_{i}\Omega_{i}\rho\Omega_{i}^{\dagger}$ where
$\Omega_{i}$ is a bounded Kraus operator that satisfies
$\sum\nolimits_{i}\Omega_{i}^{\dagger}\Omega_{i}=1$. The Gorini-Kossakowski-
Sudarshan-Lindblad theorem [12, 13] then states that if
$\\{\mathcal{V}(t^{f},t^{i})\\}$ forms a dynamical semigroup, then it
satisfies the Choi-Kraus’s form if and only if it can be written as
$e^{\mathcal{L}\left(t^{f}-t^{i}\right)}$ with a Lindblad form
$\mathcal{L}$—which is the same as in Eq. (6). A caveat, however, lies in the
fact that operators in field theories are almost always unbounded, and hence,
they have to be properly regularized before the Choi-Kraus’s theorem being
applied.

###  II.2 CPT (charge, parity, and time reversal symmetry)

A _CPT_ transformation $\Theta$ may be defined as any one of the various
antiunitary transformations of a theory that has some global invariance (e.g.,
a Poincaré invariance) [24]. In fact, from the Choi-Kraus’s theorem
immediately comes an invariance under (antiunitary) complex conjugation, i.e.,
$\text{Tr}\\{\mathcal{V}(\rho)\\}^{*}=\text{Tr}\\{\sum\nolimits_{i}\Omega_{i}\rho^{\dagger}\Omega_{i}^{\dagger}\\}=\text{Tr}\\{\sum\nolimits_{i}\Omega_{i}\rho\Omega_{i}^{\dagger}\\}=\text{Tr}\\{\mathcal{V}(\rho)\\}$.
There are different ways to realize $\Theta$ in the path-integral formalism:
for example, when $\mathcal{O}^{*}_{i}=\mathcal{O}_{i}$, Eq. (5) explicitly
obeys a $C$ symmetry by remaining invariant under $L(\phi_{+},\phi_{-})\to
L^{\prime}(\phi_{+},\phi_{-})=L(\phi_{+}^{*},\phi_{-}^{*})$, and thus,
$\Theta$ can be realized as $Z[J_{+},J_{-}]\to
Z^{\prime}[J_{+},J_{-}]=Z^{*}[-J_{-}^{*},-J_{+}^{*}]$, the invariance of which
becomes a _CPT_ symmetry [24]. Note that under this realization, $\Theta$ does
not actually reverse the time label $t\to-t$ but rather just swap the bra-ket
subscripts, because $\\{\mathcal{V}(t,0)\\}$ cannot have a time-reversal
symmetry since it is only a semigroup, not a group.

##  III Double-trace deformation

In the following, w.l.o.g., we will restrict ourselves to only one $\gamma$
term. We will also demand $\mathcal{O}^{*}=\mathcal{O}$ and save the
discussion of $C$-symmetry-breaking terms for later. Instead of the bra-ket
basis, a so-called Keldysh basis is particularly useful by defining

$\displaystyle\mathcal{O}_{c}$ $\displaystyle=$
$\displaystyle\left(\mathcal{O}_{+}+\mathcal{O}_{-}\right)/\sqrt{2},\text{ and
}$ $\displaystyle\mathcal{O}_{q}$ $\displaystyle=$
$\displaystyle\left(\mathcal{O}_{+}-\mathcal{O}_{-}\right)/\sqrt{2}$

for arbitrary fields $\mathcal{O}_{\pm}$. The subscripts $c$ and $q$
correspond to “classical” and “quantum”, respectively, as they are
individually and closely related to the classical and quantum contributions to
the noise spectrum in the theory of quantum noise [19]. We will work directly
in this basis for matrix representations throughout the context.

After rotated into the Keldysh basis, the full Lagrangian Eq. (5) reads

$\displaystyle
L(\phi_{+},\phi_{-})=L_{0}(\phi_{+})-L_{0}(\phi_{-})+i\gamma\mathcal{O}_{q}^{2}.$
(7)

We let the coupling parameter scale as $\gamma\sim\Lambda^{d-2\Delta}$
assuming that $\mathcal{O}_{q}$ is a single-trace operator of scaling
dimension $\Delta$, thus identifying $\Lambda$ as the cutoff of the theory. As
a result, one immediately recognizes that Eq. (7) is nothing but a double-
trace deformation [32, 33, 34, 35] by $\mathcal{O}_{q}^{2}$ on the undeformed
part $L_{0}(\phi_{+})-L_{0}(\phi_{-})$. The only subtlety is that the double-
trace deformation in Eq. (7) must be considered as not a single scalar but a
multiplet, the coupling strength of which is a two-by-two matrix
${\mathbf{f}}=\begin{pmatrix}0&0\\\ 0&-2i\gamma\end{pmatrix}$ in the Keldysh
basis. Such a multiplet deformation gives rise to a recombination of different
operators (in our case, $\mathcal{O}_{c}$ and $\mathcal{O}_{q}$) that occurs
along the RG flow by tuning $\gamma$ [56]. This can be most effectively
understood in the large $N$ limit where all multipoint connected functions are
automatically suppressed by powers of $N$ [41]. Only left are the two-point
functions.

###  III.1 Two-point functions in general

Recall that given a general $S[\phi_{+},\phi_{-}]$, the two-point functions

$\left\langle{\mathbf{O}}(x){\mathbf{O}}^{\intercal}(0)\right\rangle=\int\frac{d^{d}k}{\left(2\pi\right)^{d}}\mathbf{G}(k)e^{-ikx}$

for an arbitrary $\mathcal{O}$ [with shorthand
${\mathbf{O}}=\left(\mathcal{O}_{c},\mathcal{O}_{q}\right)^{\intercal}$]
encode all possible second-order correlations in the form of Eq. (4), provided
that $\rho(t^{i})$ in Eq. (4) is identified as a stationary state $\rho_{0}$
of $S[\phi_{+},\phi_{-}]$ [16]. For a unitary QFT, with the infinitesimal Wick
rotation understood, $\rho_{0}$ is just the vacuum state. For nonunitary
dynamics, $\rho_{0}$ is not necessarily a thermal equilibrium state. In the
momentum space, one has

$\displaystyle\mathbf{G}(k)=\begin{pmatrix}G^{K}(k)&G^{R}(k)\\\
G^{A}(k)&0\end{pmatrix}.$

There is a redundancy reflected by the $\mathcal{O}_{q}$-$\mathcal{O}_{q}$
correlation being nilpotent [16]. In fact, any correlation function composed
of only $\mathcal{O}_{q}$ operators will be nilpotent, a reflection of the
conservation law of probability [24, 25]. Noticing the use of the Keldysh
basis and the cyclic property of trace in Eq. (4), one can proceed to see that
$G^{R/A}$ are exactly the _retarded/advanced functions_ that by definition
describe the linear response of the system. The _Keldysh function_ $G^{K}$, on
the other hand, describes the spectrum of the symmetric auto-correlation of
$\phi$ [16].

Would the underlying dynamics of a general Keldysh theory be _thermal_ , there
would be a universal relation between $G^{R/A}$ and $G^{K}$, namely, the FDR.
In that case,
$\left\langle{\mathbf{O}}(x){\mathbf{O}}^{\intercal}(0)\right\rangle$ could be
derived alternatively by the Matsubara formalism [57]: In short, one
introduces a Euclidean action $S^{E}[\phi;\beta]=-\int_{0}^{\beta}d\tau\int
d\mathbf{x}L_{0}(\phi)$ with periodicity $\beta$ (inverse temperature) in the
imaginary time direction $\tau=it$. From $S^{E}[\phi;\beta]$ a Euclidean
Green’s function $G^{E}(\omega_{n},\mathbf{k})$ defined at discrete Matsubara
frequencies $\omega_{n}=2\pi n/\beta$, $n\in\mathbb{Z}$ can be derived, which
is connected to the Minkowski-space $G^{R/A}$ through analytical continuation,

$\displaystyle G^{R}(\omega,\mathbf{k})$ $\displaystyle=$ $\displaystyle-
iG^{E}(-i\omega+\varepsilon,\mathbf{k}),$ $\displaystyle
G^{A}(\omega,\mathbf{k})$ $\displaystyle=$ $\displaystyle-
iG^{E}(-i\omega-\varepsilon,\mathbf{k}).$ (8)

Then, using the Kubo-Martin-Schwinger condition [39], a connection between the
fluctuation spectrum $G^{K}$ and the susceptibility $G^{R}-G^{A}$ is given by

$\displaystyle G^{K}$ $\displaystyle=$
$\displaystyle\coth\left(\beta\omega/2\right)\left(G^{R}-G^{A}\right)\text{
(only if thermal) }$ (9)

which is nothing but the FDR [16]. When $\beta\to\infty$, Eq. (9) reduces to
$G^{K}=\text{sign}(\omega)\left(G^{R}-G^{A}\right)$ which always holds for
unitary QFTs.

###  III.2 RG flow induced by deformation

Now we calculate $\mathbf{G}$ for our deformed Lagrangian [Eq. (7)]. Following
the large-$N$ approach, we find, in the large $N$ limit (Appendix B),

$\displaystyle\mathbf{G}\simeq\left(\mathbf{G}_{0}^{-1}+i\mathbf{f}\right)^{-1}\simeq\begin{pmatrix}G^{K}_{0}-2\gamma
G^{R}_{0}G^{A}_{0}&G^{R}_{0}\\\ G^{A}_{0}&0\end{pmatrix}$ (10)

where $G^{R/A}_{0}$ and $G^{K}_{0}$ are components of $\mathbf{G}_{0}$ for the
undeformed part $L_{0}(\phi_{+})-L_{0}(\phi_{-})$. We find that, in the large
$N$ limit, $G^{R/A}=G^{R/A}_{0}$ remains unchanged, suggesting that the
deformation term $i\gamma\mathcal{O}_{q}^{2}$ does not modify the
susceptibility of the system. The new Keldysh function,
$G^{K}=G^{K}_{0}+G^{K}_{\mathcal{L}}$, however, contains a deformation-induced
correction which we define as $G^{K}_{\mathcal{L}}=-2\gamma
G^{R}_{0}G^{A}_{0}$.

If we tune the cutoff $\Lambda$ from zero to infinity, then it not only
changes the dimensional coupling $\gamma$ but also drives the theory to flow
between the two ends of the RG trajectory, which we assume to be two different
CFTs. If we let $L_{0}$ be a unitary CFT, then $G^{R/A}_{0}$ and $G^{K}_{0}$
for the vacuum (i.e., $\beta=\infty$) can be derived from Eqs. (III.1) and
(9),

$\displaystyle G^{R/A}_{0}$
$\displaystyle=-iC\left[\mathbf{k}^{2}-\left(\omega\pm
i\varepsilon\right)^{2}\right]^{\Delta-d/2},\text{ and}$ $\displaystyle
G^{K}_{0}$
$\displaystyle=\begin{cases}2C\sin\left[(d/2-\Delta)\pi\right]\left(k^{2}\right)^{\Delta-d/2}&k^{2}>0\\\
0&k^{2}<0\end{cases},$

with $C=4^{d/2-\Delta}\pi^{d/2}\Gamma(d/2-\Delta)/\Gamma(\Delta)$ [38] given
$k^{2}=\omega^{2}-\mathbf{k}^{2}$. Condition $\Delta-d/2\geq-1$ should be
imposed as required by unitarity [36]. Special care should also be taken when
$\Delta-d/2\in\mathbb{Z}$ since the results above will have log corrections.
Apparently, $G^{R/A}_{0}\sim G^{K}_{0}\sim\left|k\right|^{2\Delta-d}$ should
have the same scaling behavior, as they come from the same unitary operator
$\mathcal{O}$. Note that the CFT $L_{0}(\phi_{+})-L_{0}(\phi_{-})$ is not just
a sum of two identical and independent Minkowski CFTs: twisted fields (since
there is an internal symmetry between $\phi_{\pm}$) that connect the two CFTs
through branch points should also exist [58], which should be responsible for
the nonzero $G^{K}_{0}$.

The RG flow is triggered after $\gamma$ is turned on. We find

$G^{K}_{\mathcal{L}}=2\gamma
C^{2}\left|k^{2}\right|^{2\left(\Delta-d/2\right)},$

which scales with $\left|k\right|$ but differently than $G^{K}_{0}$ does.
Hence, $G^{K}=G^{K}_{0}+G^{K}_{\mathcal{L}}$ cannot obey the FDR [Eq. (9)].
This is the main difference between a thermal field theory and a driven-
dissipative field theory: the former always obeys the FDR as guaranteed by
microscopic unitarity, yet the latter describes driven-dissipative dynamics
which usually exhibits nonthermodynamic characteristics [16]. We thus conclude
that $\gamma\to\infty$ leads to a new, nonunitary, and nonthermal CFT. The new
scaling dimension for $\mathcal{O}_{c}$ can be identified as
$\Delta_{c}|_{\gamma\to\infty}=2\Delta-d/2$.

Now we emphasize the key observation: at the $O(N^{0})$ level, the Keldysh
function $G^{K}$ remains positive (free of ghosts) and finite (free of
tachyons) and hence, _physical_ [38] if and only if $\gamma>0$, a condition
which nevertheless has been guaranteed by our general construction of the
Keldysh theory. Therefore, Eq. (10) defines a complete RG trajectory—along
which the theory works at _all_ energy scales and may flow in both way between
UV and IR fixed points [38]. In particular, when $\Delta>d/2$, Eq. (10) yields
a UV fixed point at $|k|/\Lambda\to\infty$ which the IR-irrelevant deformation
$\sim{\Lambda^{d-2\Delta}}\mathcal{O}_{q}^{2}$ will eventually flow backwards
to. This is to say that our irrelevantly perturbed Keldysh theory can be
rephrased into a new nonunitary field theory which is a relevant perturbation
of the UV fixed point. Such UV _completion_ has been well known and studied
for unitary QFTs, e.g., $O(N)$ vector model near $d=6$ [59], yet now we have
seen that a Lindblad-form deformation could also be UV completed in the same
manner. This observation is unexpected, since a dissipation-like perturbation
which breaks unitarity was always expected to be IR relevant, and thus,
_irreversibility_ should only be observed at the macroscopic scale [39]. This
IR-relevant behavior corresponds to $\Delta<d/2$ in our driven-dissipative
field theory [Eq. (7)]. Yet as we have seen, $\Delta>d/2$ neither contradicts
the Weinbergian requirements of the Keldysh theory nor denies its
renormalizability. Driven-dissipative quantum dynamics is hence much richer
than thermodynamics, differing not only in whether complying with the FDR but
in the UV/IR relevance as well.

The large-$N$ approach is also applicable to thermal CFTs. For $d=2$,
multipoint functions at finite temperature $\beta^{-1}$ can be calculated from
the vacuum by the Weyl transformation [60]. Putting exactly known results [61,
62] into Eqs. (III.1) and (9), we derive
$G^{K}_{0}=2{\Im}\left\\{f(k,\beta)\right\\}$ and
$G^{K}_{\mathcal{L}}=2\gamma\left|f(k,\beta)\right|^{2}$, where

$\displaystyle f(k,\beta)=$
$\displaystyle-C\left(\frac{\beta}{4\pi}\right)^{2-2\Delta}\frac{\Gamma(\frac{\Delta}{2}+i\frac{\beta\left(\omega+\left|\mathbf{k}\right|\right)}{4\pi})\Gamma(\frac{\Delta}{2}+i\frac{\beta\left(\omega-\left|\mathbf{k}\right|\right)}{4\pi})}{\Gamma(\frac{2-\Delta}{2}+i\frac{\beta\left(\omega+\left|\mathbf{k}\right|\right)}{4\pi})\Gamma(\frac{2-\Delta}{2}+i\frac{\beta\left(\omega-\left|\mathbf{k}\right|\right)}{4\pi})}.$

We compare $G^{K}_{0}$ and $G^{K}_{\mathcal{L}}$ for $\beta=\infty$ and
$\beta=5.0$ in Fig. 1, choosing $\Delta=1/8$ (which is the scaling dimension
of the $\mathbb{Z}_{2}$-odd field $\sigma$ in $d=2$ Ising CFT) and
$\gamma=0.5$. We clearly see that $G^{K}_{0}$ and $G^{K}_{\mathcal{L}}$ have
different scaling behaviors and thus $G^{K}_{\mathcal{L}}$ cannot obey the
FDR.

We finally discuss in brief what if $\mathcal{O}$ is charged: a $C$-symmetry-
breaking term

$i\gamma\left(\mathcal{O}_{c}\mathcal{O}^{*}_{q}-\mathcal{O}^{*}_{c}\mathcal{O}_{q}\right)/2$

will appear in Eq. (7), which will alter $G^{R/A}$ as well, by shifting the
poles in $G^{R/A}_{0}$ to both upper and lower half planes, causing the
retarded/advanced functions $G^{R/A}$ to become _unphysical_. This is because
in a relativistic scalar field theory, particles and antiparticles always come
in pairs so that the charge spectrum is not bounded below—and thus a
dissipation of charge will blow up in either time direction, which is a
crucial difference from previously studied nonrelativistic field theories
[16].

(a)

(b)

Figure 1: The Keldysh function $G^{K}_{0}+G^{K}_{\mathcal{L}}$ consists of two
parts: $G^{K}_{0}$ is the undeformed Keldysh function; $G^{K}_{\mathcal{L}}$
is the correction induced by double-trace deformation
$i\gamma\mathcal{O}^{2}$. Both $G^{K}_{0}$ and $G^{K}_{\mathcal{L}}$ are even
functions of $\omega$ and are exactly solvable in the large $N$ limit in two
dimensions for arbitrary temperature $\beta$: 1 $\beta=\infty$; 1 $\beta=5.0$.
Both have $\gamma=0.5$. The scaling dimension of $\mathcal{O}$ is
$\Delta=1/8$.

##  IV Examples

###  IV.1 $\left(0+1\right)$-dimensional massive scalar field

Our first example is simply quantum mechanical: a one-dimensional quantum
harmonic oscillator of intrinsic frequency $m$ under weak, continuous
nonselective position $q$ measurement with rate $\gamma$ [14]. This is the
simplest model that does not admit a quantum nondemolition measurement [19],
since the back action of measurement will affect the correlation functions of
the observable $q$. The Lagrangian deformed by
$\mathcal{O}_{q}^{2}=\left(q_{+}-q_{-}\right)^{2}/2$ is given by

$\displaystyle L(q_{+},q_{-})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\left(\partial_{t}q_{+}\right)^{2}-m^{2}q_{+}^{2}\right]-\frac{1}{2}\left[\left(\partial_{t}q_{-}\right)^{2}-m^{2}q_{-}^{2}\right]$
(11)
$\displaystyle-i\gamma\left(q_{+}q_{-}-\frac{1}{2}q_{+}q_{+}-\frac{1}{2}q_{-}q_{-}\right),$

which is quadratic. Thus, Eq. (10) is exact even when $N=1$ as in our example,
and therefore, $G^{R/A}=i/[(\omega\pm i\varepsilon)^{2}-m^{2}]$ remains
unchanged under the deformation. This can also be seen directly by a Gaussian
integral of Eq. (11). Meanwhile, we have $G^{K}_{\mathcal{L}}=-2\gamma
G^{R}G^{A}\sim\omega^{-4}$ by dimensional analysis. A more careful calculation
further yields an approximate relation (Appendix C),

$\displaystyle
G^{K}_{\mathcal{L}}\simeq\frac{\gamma}{m\varepsilon}\pi\delta\left(\omega^{2}-m^{2}\right)=\frac{\gamma}{2\omega\varepsilon}\left(G^{R}-G^{A}\right).$

This is not an exact FDR since we are not looking at a thermal theory, but it
looks very similar to Eq. (9). Hence, we can introduce a “sloppy” effective
inverse temperature $\beta_{\text{eff}}\simeq
4\varepsilon/\gamma-\left(8m\varepsilon^{2}/\gamma^{2}\right)\coth\left(\beta
m/2\right)+O(\varepsilon^{3})$ regulated by $\varepsilon$ and rewrite the
Keldysh two-point function as
$G^{K}\simeq\left[\coth\left(\beta\omega/2\right)+{\gamma}/{\left(2\omega\varepsilon\right)}\right]\left(G^{R}-G^{A}\right)\simeq\coth\left(\beta_{\text{eff}}\omega/2\right)\left(G^{R}-G^{A}\right)$.
Thus, $G^{K}$ effectively describes the fluctuation of an infinitely heated
harmonic oscillator, which is a stationary state of Eq. (11), in consistence
with the Hamiltonian dynamics (Appendix C). This steadily infinite heating is
also a well-known feature of collapse models [5]. Interestingly, we see that
in the Keldysh formalism, the role of the regulator is played by $\varepsilon$
instead of the spatial localization width $r_{C}$ commonly used in those
models [5].

###  IV.2 $\left(4-\epsilon\right)$-dimensional $O(N)$ massless scalar field

The second example we are interested in is described by

$\displaystyle
L(\phi^{\alpha}_{+},\phi^{\alpha}_{-})=L_{0}(\phi^{\alpha}_{+})-L_{0}(\phi^{\alpha}_{-})$
$\displaystyle-i\gamma\left[\left(\phi^{\alpha}\phi^{\alpha}\right)_{+}\left(\phi^{\alpha}\phi^{\alpha}\right)_{-}-\frac{1}{2}\left(\phi^{\alpha}\phi^{\alpha}\right)_{+}^{2}-\frac{1}{2}\left(\phi^{\alpha}\phi^{\alpha}\right)_{-}^{2}\right],\qquad$
(12)

where
$L_{0}(\phi^{\alpha})=\frac{1}{2}\partial_{\mu}\phi^{\alpha}\partial^{\mu}\phi^{\alpha}-\frac{\lambda}{4!}\left(\phi^{\alpha}\phi^{\alpha}\right)^{2}$
is an ordinary $O(N)$ massless scalar field theory. In the Keldysh basis, we
have
$2\mathcal{O}_{q}^{2}=\left(\mathopen{:}\phi^{\alpha}_{+}\phi^{\alpha}_{+}-\phi^{\alpha}_{-}\phi^{\alpha}_{-}\mathclose{:}\right)^{2}=4\left(\mathopen{:}\phi^{\alpha}_{c}\phi^{\alpha}_{q}\mathclose{:}\right)^{2}$
and
$2\mathcal{O}_{c}^{2}=\left(\mathopen{:}\phi^{\alpha}_{+}\phi^{\alpha}_{+}+\phi^{\alpha}_{-}\phi^{\alpha}_{-}\mathclose{:}\right)^{2}=\left(\mathopen{:}\phi^{\alpha}_{c}\phi^{\alpha}_{c}\mathclose{:}\right)^{2}$.
Note that $\mathcal{O}$ is normal ordered for being a single-trace operator,
which should always be understood below. If $\gamma$ remains zero, then the
real and positive coupling $\lambda$ will trigger an RG flow from the Gaussian
fixed point at $\lambda=0$ to the Wilson-Fisher (WF) fixed point at
$\lambda/4!\simeq\mu^{\epsilon}{2\pi^{2}\epsilon}/\left({N+8}\right)+O(\epsilon^{2})$
by expansion around $\epsilon=4-d$ at an energy scale $\mu$ [63, 64]. Note
that when $\epsilon>0$, the stationary states at both fixed points are stable.

Now we turn on $\gamma$ in Eq. (IV.2). Diagrammatic perturbative calculations
can be carried out in the same spirit of the usual counterterm renormalization
scheme. Yet, the diagram counting is more complex, since we have three free
propagators $\bm{\mathcal{P}}(k)$ in total in the Keldysh basis (Appendix D).
Such an analysis has been done for $N=1$ [30]. Here, we extend it to $O(N)$
and find that the beta functions for the dimensionless renormalized parameters
$\bar{\lambda}=\mu^{-\epsilon}\lambda$ and
$\bar{\gamma}=\mu^{-\epsilon}\gamma$ are given by (Appendix D)

$\displaystyle\beta_{\bar{\lambda}}$ $\displaystyle\simeq$
$\displaystyle-\epsilon\bar{\lambda}+\frac{1}{\left(4\pi\right)^{2}}\frac{N+8}{3}\bar{\lambda}^{2},$
$\displaystyle\beta_{\bar{\gamma}}$ $\displaystyle\simeq$
$\displaystyle-\epsilon\bar{\gamma}+\frac{1}{\left(4\pi\right)^{2}}\frac{2N+13}{3}\bar{\lambda}\bar{\gamma},$

which have _four_ fixed points in total: there are two fixed values for
$\bar{\lambda}$, which are $\bar{\lambda}_{*}=0$ and
$\bar{\lambda}_{*}/4!\simeq{2\pi^{2}\epsilon}/\left({N+8}\right)$; and two for
$\bar{\gamma}$, which are $\bar{\gamma}_{*}=0$ and $\bar{\gamma}_{*}=\infty$.
We see that both the Gaussian and the WF fixed points for $\bar{\lambda}$ for
the unitary theory ($\bar{\gamma}_{*}=0$) are recovered. Meanwhile,
$\beta_{\bar{\lambda}}$ and $\beta_{\bar{\gamma}}$ can reduce to and match the
$N=1$ result [30]. At the Gaussian fixed point, the scaling dimensions
corresponding to the couplings $\lambda$ and $\gamma$ are given by
$\Delta^{\text{Gauss}}_{\phi^{\alpha}_{c}\phi^{\alpha}_{c}\phi^{\beta}_{c}\phi^{\beta}_{q}}=\Delta^{\text{Gauss}}_{\left(\phi^{\alpha}_{c}\phi^{\alpha}_{q}\right)^{2}}=4-2\epsilon$,
indicating that $\mathcal{O}_{c}\mathcal{O}_{q}$ and $\mathcal{O}^{2}_{q}$ as
relevant operators when $\epsilon>0$ have the same trivial engineering
dimension. At the WF fixed point, however, we find
$\Delta^{\text{WF}}_{\phi_{c}^{\alpha}\phi_{c}^{\alpha}\phi_{c}^{\beta}\phi_{q}^{\beta}}\simeq
4+O(\epsilon^{2})$ and
$\Delta^{\text{WF}}_{\left(\phi_{c}^{\alpha}\phi_{q}^{\alpha}\right)^{2}}\simeq
4-{3\epsilon}/{\left(N+8\right)}+O(\epsilon^{2})$. Both operators are thus
irrelevant when $\epsilon>0$ and independent.

The operators of which the scaling dimensions we are mostly interested in are
$\mathcal{O}_{c}$ and $\mathcal{O}_{q}$. To this end, we add quadratic terms
$-m^{2}\phi^{\alpha}_{c}\phi^{\alpha}_{q}$ and
$ic\phi^{\alpha}_{c}\phi^{\alpha}_{c}/2$ to the free Lagrangian
$L_{0}(\phi^{\alpha}_{+})-L_{0}(\phi^{\alpha}_{-})$. The first term generates
a positive mass for the unitary theory. The second term is a pure vertex of
some power of $\phi_{c}$—the kind of which is known to be responsible for
breaking the Lindblad constraint [30]—hence, only used for bookkeeping and
should be taken to zero at the final step of calculation. Including the second
term adds to the free propagator $\bm{\mathcal{P}}$ a new term
$c\bm{\mathcal{P}}^{(1)}+O(c^{2})$ where
$\bm{\mathcal{P}}^{(1)}=-\begin{pmatrix}\left(\mathcal{P}^{K}\right)^{2}&\mathcal{P}^{R}\mathcal{P}^{K}\\\
\mathcal{P}^{A}\mathcal{P}^{K}&\mathcal{P}^{A}\mathcal{P}^{R}\end{pmatrix}$ is
the first-order correction. From the modified propagators we derive two
identical beta functions (Appendix D),

$\displaystyle\beta_{\bar{m}^{2}}$ $\displaystyle\simeq$
$\displaystyle-2\bar{m}^{2}+\frac{1}{\left(4\pi\right)^{2}}\frac{N+2}{3}\bar{\lambda}\bar{m}^{2},$
$\displaystyle\beta_{\bar{c}}$ $\displaystyle\simeq$
$\displaystyle-2\bar{c}+\frac{1}{\left(4\pi\right)^{2}}\frac{N+2}{3}\bar{\lambda}\bar{c},$

for $\bar{m}^{2}=\mu^{-2}m^{2}$ and $\bar{c}=\mu^{-2}c$, respectively. Again,
we recover the beta function of mass renormalization. Note that both scaling
dimensions are equal, given by
$\Delta^{\text{Gauss}}_{\phi^{\alpha}_{c}\phi^{\alpha}_{q}}=\Delta^{\text{Gauss}}_{\phi^{\alpha}_{c}\phi^{\alpha}_{c}}=2-\epsilon$
and
$\Delta^{\text{WF}}_{\phi^{\alpha}_{c}\phi^{\alpha}_{q}}=\Delta^{\text{WF}}_{\phi^{\alpha}_{c}\phi^{\alpha}_{c}}\simeq
2-6\epsilon/\left({N+8}\right)+O(\epsilon^{2})$. This makes sense since
$\mathcal{O}_{c}$ and $\mathcal{O}_{q}$ must have the same scaling dimension:
they are only linear combinations of bra and ket fields, which are completely
equivalent at $\bar{\gamma}_{*}=0$.

Now we can calculate the retarded/advanced functions, i.e., the two-point
functions between $\mathcal{O}_{c}$ and $\mathcal{O}_{q}$. When $N\to\infty$,
we find

$\displaystyle G^{R/A}_{\text{Gauss}}$
$\displaystyle=\left(2N\right)iC_{\epsilon}^{-1}\left[\mathbf{k}^{2}-\left(\omega\pm
i\varepsilon\right)^{2}\right]^{-\epsilon/2},$ (13) $\displaystyle
G^{R/A}_{\text{WF}}$
$\displaystyle=-\left(2N\right)^{-1}iC_{\epsilon}\left[\mathbf{k}^{2}-\left(\omega\pm
i\varepsilon\right)^{2}\right]^{\epsilon/2},$

at the Gaussian and the WF fixed points, respectively, with $C_{\epsilon}$ a
normalization coefficient (Appendix D). This result is derived by noticing
that the RG flow connecting the two unitary fixed points is also triggered by
a double-trace deformation by $\lambda$ [59], and hence, $G^{R/A}_{\text{WF}}$
can be easily calculated from $G^{R/A}_{\text{Gauss}}$ in the large $N$ limit
(following Appendix B). Note that we have dropped a contact term $\sim
1/\lambda\sim\mu^{-\epsilon}$ from $G^{R/A}_{\text{WF}}$ and then renormalized
it by a multiplicative factor $\lambda^{2}\sim\mu^{2\epsilon}$ in Eq. (13).

We move our focus to the nonunitary fixed points at $\bar{\gamma}_{*}=\infty$.
The crucial difference between the RG flows originating from the two unitary
fixed points is that the $i\gamma\mathcal{O}^{2}_{q}$ deformation is
_relevant_ at the Gaussian fixed point but _irrelevant_ at the WF fixed point
when $\epsilon>0$. This can be clearly seen from Eq. (13). When approaching
the nonunitary fixed points, we have

$\displaystyle G^{K}_{\text{Gauss}}\xrightarrow{\gamma\to\infty}$
$\displaystyle-2\gamma G^{R}_{\text{Gauss}}G^{A}_{\text{Gauss}}$
$\displaystyle\simeq
2\gamma\left(2N\right)^{2}C_{\epsilon}^{-2}\left|k^{2}\right|^{-\epsilon},$
$\displaystyle G^{K}_{\text{WF}}\xrightarrow{\gamma\to\infty}$
$\displaystyle-2\gamma G^{R}_{\text{WF}}G^{A}_{\text{WF}}$
$\displaystyle\simeq
2\gamma\left(2N\right)^{-2}C_{\epsilon}^{2}\left|k^{2}\right|^{\epsilon},$

in the large $N$ limit. We can write down the new scaling dimensions for
$\mathcal{O}_{c}$ at the two nonunitary fixed points:

$\displaystyle\Delta^{\text{Gauss}}_{\phi^{\alpha}_{c}\phi^{\alpha}_{c}}\xrightarrow{\gamma\to\infty}$
$\displaystyle
2\Delta^{\text{Gauss}}_{\phi^{\alpha}_{c}\phi^{\alpha}_{c}}-d/2$
$\displaystyle\simeq 2-3\epsilon/2,$
$\displaystyle\Delta^{\text{WF}}_{\phi^{\alpha}_{c}\phi^{\alpha}_{c}}\xrightarrow{\gamma\to\infty}$
$\displaystyle 2\Delta^{\text{WF}}_{\phi^{\alpha}_{c}\phi^{\alpha}_{c}}-d/2$
$\displaystyle\simeq 2+\epsilon/2.$

As a result, at the IR nonunitary fixed point which can be reached from the
Gaussian fixed point, one should expect to see a nontrivial Keldysh two-point
function $\left\langle\mathcal{O}_{c}(x)\mathcal{O}_{c}(0)\right\rangle\sim
x^{-\left(4-3\epsilon\right)}$; at the UV nonunitary fixed point which is
backshot from the WF fixed point, however, one should expect to see
$\left\langle\mathcal{O}_{c}(x)\mathcal{O}_{c}(0)\right\rangle\sim
x^{-\left(4+\epsilon\right)}$. Both two-point functions should be conformal
invariant.

##  V Discussion and conclusions

From a holographic perspective, two unitary IR and UV CFTs connected by a
double-trace deformation $\mathbf{O}^{\intercal}\mathbf{O}$ of large $N$ have
a natural explanation in the bulk AdS space: the only difference between them
amounts to a change of the boundary conditions [65] to the bulk equations of
motion (EOM) near $z\to 0$ in the Poincaré patch,

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{z^{2}}\left(dt^{2}-dz^{2}-\sum^{d-1}_{i=1}dx_{i}^{2}\right).$

In particular, the bulk fields
$\mathbf{\Phi}=\left(\Phi_{1},\Phi_{2},\cdots\right)^{\intercal}$ dual to the
boundary fields
$\mathbf{O}=\left(\mathcal{O}_{1},\mathcal{O}_{2},\cdots\right)^{\intercal}$
of dimension $\Delta_{1},\Delta_{2},\cdots$ can be expanded near
$z=\varepsilon\ll 1$ as
$\Phi_{i}(z,\mathbf{x})\sim\alpha_{i}(\mathbf{x})\varepsilon^{d-\Delta_{i}}+\beta_{i}(\mathbf{x})\varepsilon^{\Delta_{i}}$
[32]. Multipoint functions of $\mathcal{O}_{i}$ of the undeformed CFT can be
recovered by taking $\alpha_{i}(\mathbf{x})\to 0$. On the other hand, by
taking $\beta_{i}(\mathbf{x})\to 0$, one instead recovers multipoint functions
for the deformed CFT located at the other end of the RG flow. The roles of
$\alpha_{i}(\mathbf{x})$ and $\beta_{i}(\mathbf{x})$ are thus exchanged.

What is special for Lorentzian AdS space, however, is that the bulk EOM allows
not one but two independent smooth solutions that vanish exponentially when
$z\to\infty$ [66]. Therefore, unlike in Euclidean space where
$\Phi_{i}(z,\mathbf{x})$ could be completely fixed by the EOM with a boundary
condition
$\Phi_{i}(\varepsilon,\mathbf{x})=\varepsilon^{d-\Delta_{i}}\Phi_{i}^{0}(\mathbf{x})$
near $\varepsilon=0$, here $\Phi_{i}(z,\mathbf{x})$ cannot be fixed. The
ambiguity therein is nothing but exactly the time-ordering ambiguity of
propagators in the corresponding Minkowski CFT [66] and is usually fixed by
hand to match known propagators in the Euclidean AdS space by Wick rotation
[67]. However, there are no general rules (other than thermodynamic reasoning)
forbidding us to fix the ambiguity in other ways. Thus, under the Keldysh
formalism, $\mathcal{O}_{i,c}$ and $\mathcal{O}_{i,q}$ in the boundary theory
can still be coupled to the same boundary condition $\Phi_{i}^{0}(\mathbf{x})$
but mapped to two different solutions $\Phi_{i}(z,\mathbf{x})$ in the bulk
theory. Naturally, we want to know how we can reproduce our deformed Keldysh
theory on the boundary by imposing some boundary condition to the bulk theory,
a solution of which should require rewriting the bulk theory in the Keldysh
formalism. We notice that similar interests have already appeared in recent
ongoing studies [68, 69].

To conclude, we formalize a nonunitary Keldysh path integral that satisfies
two “ _CPT_ ” constraints. We find that the introduced Lindblad term, which is
responsible for driven-dissipative dynamics, has the natural form of a double-
trace deformation $\mathcal{O}^{2}$. A calculation in the large $N$ limit
shows the possible existence of a new, nonunitary, and nonthermal conformal
fixed point induced by the double-trace deformation. This fixed point is IR
when $\Delta<d/2$ but UV when $\Delta>d/2$, given $\Delta$ the scaling
dimension of $\mathcal{O}$. Interestingly, such a fixed point remains physical
in both IR and UV regimes in the large $N$ limit, suggesting that general
driven-dissipative dynamics may unexpectedly differ from thermodynamics in its
UV/IR relevance as well. The retarded/advanced functions and the Keldysh
function are also calculated at this nonunitary fixed point for general
$\Delta$, as well as for two specific examples of a
$\left(0+1\right)$-dimensional massive scalar field and a
$\left(4-\epsilon\right)$-dimensional $O(N)$ massless scalar field. For the
former example, the known feature of infinite heating in quantum collapse
models is successfully recovered; for the latter, the RG flows between the two
unitary Gaussian and WF fixed points and the two corresponding nonunitary
fixed points originated from them are studied. We look forward to a thorough
investigation of the possible AdS/CFT correspondence of our results in the
bulk theory using the Keldysh formalism.

###### Acknowledgements.

X. M. thanks the support of Naoki Hanzawa and Houtarou Oreki. This work is
supported by the NetSeed: Seedling Research Award of Northeastern University.

##  Appendix A CONSTRUCT THE KELDYSH ACTION $S[\phi_{+},\phi_{-}]$ FROM THE
TIME-EVOLUTION SUPEROPERATOR $\mathcal{V}(t^{f},t^{i})$

Construction of $S[\phi_{+},\phi_{-}]$ from $\mathcal{V}(t^{f},t^{i})$ is
similar to the traditional Feynman path integral approach. We stress that our
approach of constructing $S[\phi_{+},\phi_{-}]$ is designed for _relativistic_
dynamics where the spacetime derivatives of $\phi_{\pm}$ are Lorentz
covariant. This is different from the earlier coherent-state-based approach
for nonrelativistic dynamics [16]. We start by rewriting the time-evolution
superoperator $\mathcal{V}$ in a general exponential form,

$\displaystyle\mathcal{V}(t^{f},t^{i})=\mathcal{T}e^{\int_{t^{i}}^{t^{f}}dt\mathcal{L}(t)},$
(14)

where $\mathcal{T}$ is the anti-time-ordering superoperator. Dividing the time
integral into $N$ equal intervals of length $\Delta t=(t^{f}-t^{i})/N$ yields

$\displaystyle\mathcal{V}(t^{f},t^{i})\simeq e^{\Delta
t\mathcal{L}(t_{N-1})}e^{\Delta t\mathcal{L}(t_{N-2})}\cdots e^{\Delta
t\mathcal{L}(t_{0})}$ (15)

which is an exact equality when $N\to\infty$ by applying the
Baker–Campbell–Hausdorff formula. Here the $n$th step is labeled by
$t_{n}=t^{i}+n\Delta t$. Therefore, we can divide the Hilbert-space time
evolution into infinitesimal segments,

$\displaystyle\left\langle\phi_{+}^{f}(\mathbf{x})\right|\left[\mathcal{V}(t^{f},t^{i})(\left|\phi_{+}^{i}(\mathbf{x})\right\rangle\left\langle\phi_{-}^{i}(\mathbf{x})\right|)\right]\left|\phi_{-}^{f}(\mathbf{x})\right\rangle$
(16) $\displaystyle=$ $\displaystyle\int d\phi_{+,N-1}\int
d\phi_{-,N-1}\cdots\int d\phi_{+,1}\int d\phi_{-,1}$
$\displaystyle\left\langle\phi_{+}^{f}\right|\left[e^{\Delta
t\mathcal{L}(t_{N-1})}(\left|\phi_{+,N-1}\right\rangle\left\langle\phi_{-,N-1}\right|)\right]\left|\phi_{-}^{f}\right\rangle$
$\displaystyle\left\langle\phi_{+,N-1}\right|\left[e^{\Delta
t\mathcal{L}(t_{N-2})}(\left|\phi_{+,N-2}\right\rangle\left\langle\phi_{-,N-2}\right|)\right]\left|\phi_{-,N-1}\right\rangle$
$\displaystyle\cdots$
$\displaystyle\left\langle\phi_{+,2}\right|\left[e^{\Delta
t\mathcal{L}(t_{1})}(\left|\phi_{+,1}\right\rangle\left\langle\phi_{-,1}\right|)\right]\left|\phi_{-,2}\right\rangle$
$\displaystyle\left\langle\phi_{+,1}\right|\left[e^{\Delta
t\mathcal{L}(t_{0})}(\left|\phi_{+}^{i}\right\rangle\left\langle\phi_{-}^{i}\right|)\right]\left|\phi_{-,1}\right\rangle,$

which is valid provided that $\mathcal{L}(t_{n})(\cdot)$ is a linear
superoperator, i.e.,

$\displaystyle\int d\phi_{+,n}\int
d\phi_{-,n}f(\phi_{+,n},\phi_{-,n})\left[\mathcal{L}(t_{n})(\left|\phi_{+,n}\right\rangle\left\langle\phi_{-,n}\right|)\right]$
$\displaystyle=$ $\displaystyle\mathcal{L}(t_{n})(\int d\phi_{+,n}\int
d\phi_{-,n}f(\phi_{+,n},\phi_{-,n})\left|\phi_{+,n}\right\rangle\left\langle\phi_{-,n}\right|)$

for arbitrary $f$. Here, each $\int d\phi_{\pm,n}$ is a shorthand of
$\prod_{\mathbf{x}}\int d\phi_{\pm,n}(\mathbf{x})$, i.e., the product of
integrals for every $\phi_{\pm,n}(\mathbf{x})$ living at $\mathbf{x}$ over the
space (and same for $\int d\Pi_{\pm,n}$ below), and hence, $\int
d\phi_{\pm,n}\left|\phi_{\pm,n}\right\rangle\left\langle\phi_{\pm,n}\right|=1$.
The space label $\mathbf{x}$ is also omitted.

From Eq. (16), one can proceed to write down a path integral in terms of the
bra and ket fields $\phi_{\pm}$ in the limit of $N\to\infty$. Note that
complexification of $\phi_{\pm}$ can also be done by adding a new real scalar
field $\varphi_{\pm}$, which together establishes an $SO(2)$ symmetry between
$\phi_{\pm}$ and $\varphi_{\pm}$. In the following, we will look at both
unitary and nonunitary dynamics and proceed to derive the corresponding
Keldysh actions.

###  A.1 Unitary dynamics

For unitary QFTs which admit linear Hamiltonian dynamics, i.e.,
$\mathcal{V}(t^{f},t^{i})\rho(t^{i})=e^{-iH(t^{f}-t^{i})}\rho(t^{i})e^{iH(t^{f}-t^{i})}$,
the time evolution reads

$\displaystyle\left\langle\phi_{+}^{f}(\mathbf{x})\right|\left[e^{-iH(t^{f}-t^{i})}\left|\phi_{+}^{i}(\mathbf{x})\right\rangle\left\langle\phi_{-}^{i}(\mathbf{x})\right|e^{iH(t^{f}-t^{i})}\right]\left|\phi_{-}^{f}(\mathbf{x})\right\rangle$

which is nothing but a twofold of the time evolution amplitude in terms of a
single operator $\phi(\mathbf{x})$. We further assume that the time-
independent Hamiltonian $H$ can be written in terms of a local integral,
$H=\int g(\mathbf{x})\Pi^{2}(\mathbf{x})+v(\phi(\mathbf{x}))$ (for arbitrary
$g$ and $v$), where $\Pi(\mathbf{x})$ as the conjugate momentum operator of
$\phi(\mathbf{x})$ only appears in a quadratic form. Then, each infinitesimal
segment in Eq. (16) can be rewritten as

$\displaystyle\left\langle\phi_{+,n+1}\right|\left[e^{-iH\Delta
t}\left|\phi_{+,n}\right\rangle\left\langle\phi_{-,n}\right|e^{iH\Delta
t}\right]\left|\phi_{-,n+1}\right\rangle$ (17) $\displaystyle=$
$\displaystyle\int\frac{d\Pi_{+,n}}{2\pi}e^{-i\Pi_{+,n}\left(\phi_{+,n+1}-\phi_{+,n}\right)}e^{-iH_{+}\Delta
t}$
$\displaystyle\int\frac{d\Pi_{-,n}}{2\pi}e^{i\Pi_{-,n}\left(\phi_{-,n+1}-\phi_{-,n}\right)}e^{iH_{-}\Delta
t},$

where $H_{+}$ only depends on $\phi_{+}$ and $\Pi_{+}$ (similarly for
$H_{-}$). In the second step of Eq. (17), we have used
$\left\langle\Pi_{\pm}|\phi_{\pm}\right\rangle=\left(2\pi\right)^{1/2}e^{-i\Pi_{\pm}\phi_{\pm}}$.
Noticing that the integrand in Eq. (17) is an exponential of a quadratic term
in $\Pi_{\pm,n}$, we can calculate the Gaussian integral exactly and find that
Eq. (17) is proportional to

$\displaystyle
e^{i\left\\{\left(g^{-1}/4\right)\left[\left({\phi_{+,n+1}-\phi_{+,n}}\right)/\Delta
t\right]^{2}-v(\phi_{+,n})\right\\}\Delta t}$ (18)
$\displaystyle+e^{-i\left\\{\left(g^{-1}/4\right)\left[\left({\phi_{-,n+1}-\phi_{-,n}}\right)/\Delta
t\right]^{2}-v(\phi_{-,n})\right\\}\Delta t}$ $\displaystyle=$ $\displaystyle
e^{i\left(L(\phi_{+,n})-L(\phi_{-,n})\right)\Delta t}.$

It is obvious that the Lagrangian $L(\phi_{\pm})$ and the Hamiltonian
$H(\phi_{\pm})$ are connected by the Legendre transformation. From this
Lagrangian expression, one can go back to Eq. (16), take $N\to\infty$, and
construct the Keldysh action,

$\displaystyle S[\phi_{+},\phi_{-}]=S[\phi_{+}]-S[\phi_{-}],$ (19)

where $S[\phi_{\pm}]=\int_{t^{i}}^{t^{f}}dt\int d\mathbf{x}L(\phi_{\pm})$.

The unitary Keldysh action [Eq. (19)] is of special interest [24, 25]. We can
see that it is just two copies of the ordinary unitary action. Since there is
no crossing term between $\phi_{+}$ and $\phi_{-}$, it seems that $\phi_{+}$
and $\phi_{-}$ are completely independent. However, the correlations between
them [as defined by Eq. (3)] are not zero due to the upper-limit boundary
condition imposed on Eq. (3). To see this, we look at two-point functions for
the vacuum. The ground state $\left|0\right\rangle$ is picked up by taking
$t^{f}-t^{i}\to(1-i\varepsilon)\infty$ with an infinitesimal Wick rotation
applied. The initial condition can thus be safely taken as
$\rho(t^{i})=\left|0\rangle\langle 0\right|$, which no longer depends on
$t^{i}$, thus recovering the translational symmetry in Eq. (4). Noticing the
cyclic property of Eq. (4), we have

$\displaystyle\left\langle\phi_{-}(x)\phi_{+}(0)\right\rangle$
$\displaystyle=$ $\displaystyle\left\langle
0\right|\phi(x)\phi(0)\left|0\right\rangle,$
$\displaystyle\left\langle\phi_{+}(x)\phi_{-}(0)\right\rangle$
$\displaystyle=$ $\displaystyle\left\langle
0\right|\phi(0)\phi(x)\left|0\right\rangle,$
$\displaystyle\left\langle\phi_{+}(x)\phi_{+}(0)\right\rangle$
$\displaystyle=$ $\displaystyle\left\langle
0\right|\mathcal{T}\left\\{\phi(x)\phi(0)\right\\}\left|0\right\rangle,$
$\displaystyle\left\langle\phi_{-}(x)\phi_{-}(0)\right\rangle$
$\displaystyle=$ $\displaystyle\left\langle
0\right|\mathcal{\bar{T}}\left\\{\phi(x)\phi(0)\right\\}\left|0\right\rangle,$
(20)

which cover all possible time orders made of two local operators. Perturbative
diagrams thus can be drawn from these propagators. Through a Fourier
transform, we see that the correlation between $\phi_{+}$ and $\phi_{-}$ is
not zero only when the exchange of particles $\phi$-$\phi$ is on shell [30].
In addition, since there are no vertices mixing $\phi_{+}$ and $\phi_{-}$ in
the unitary dynamics $S[\phi_{+}]-S[\phi_{-}]$, the first two in Eq. (A.1) are
the only connection between $\phi_{+}$ and $\phi_{-}$ and are called _cut
propagators_ [30]—namely, any diagram under the Keldysh formalism will reduce
to two independent Feynman diagrams for the bra and ket fields, respectively,
after cutting these on-shell propagators.

By direct inspection, the sum of all correlation functions in Eq. (A.1) is
always equal to zero, a result of the conservation of probability [16]. This
is why, after some rearrangement, in the Keldysh basis
$\left\langle{\Phi}(x){\Phi}^{\intercal}(0)\right\rangle=\begin{pmatrix}\left\langle
0\right|\\{\phi(x),\phi(0)\\}\left|0\right\rangle&\theta(x)\left\langle
0\right|[\phi(x),\phi(0)]\left|0\right\rangle\\\ \theta(-x)\left\langle
0\right|[\phi(0),\phi(x)]\left|0\right\rangle&0\end{pmatrix}$, the lower right
corner is always equal to zero. Also, one sees clearly that the upper right
and the lower left corners are called the retarded and the advanced functions
because of the existence of the Heaviside step function $\theta(x)$.

###  A.2 Nonunitary dynamics

We pick the simplest nonunitary and time-independent $\mathcal{L}$ that admits
a Lindblad form [as in Eq. (6)] and satisfies the quantum master equation,

$\displaystyle\partial_{t}\rho=\mathcal{L}(\rho)=-i\left[H,\rho\right]-\left(\gamma/2\right)\left[\mathcal{O},\left[\mathcal{O},\rho\right]\right],$
(21)

given an arbitrary real operator $\mathcal{O}$. The corresponding adjoint
master equation [14] is

$\displaystyle\partial_{t}A$ $\displaystyle=$
$\displaystyle\mathcal{L}^{\dagger}(A)=-i\left[A,H\right]-\left(\gamma/2\right)\left[\mathcal{O},\left[\mathcal{O},A\right]\right]$
(22)

for an arbitrary operator $A$. Note that $\mathcal{L}$ is a linear
superoperator, since

$\displaystyle\left\langle\phi_{+,n+1}\right|\left[\mathcal{L}(\left|\phi_{+,n}\right\rangle\left\langle\phi_{-,n}\right|)\right]\left|\phi_{-,n+1}\right\rangle$
(23) $\displaystyle=$
$\displaystyle-i\left\langle\phi_{+,n+1}\right|H\left|\phi_{+,n}\right\rangle\left\langle\phi_{-,n}|\phi_{-,n+1}\right\rangle$
$\displaystyle+i\left\langle\phi_{+,n+1}|\phi_{+,n}\right\rangle\left\langle\phi_{-,n}\right|H\left|\phi_{-,n+1}\right\rangle$
$\displaystyle+\gamma\left\langle\phi_{+,n+1}\right|\mathcal{O}\left|\phi_{+,n}\right\rangle\left\langle\phi_{-,n}\right|\mathcal{O}\left|\phi_{-,n+1}\right\rangle$
$\displaystyle-\left(\gamma/2\right)\left\langle\phi_{+,n+1}\right|\mathcal{O}^{2}\left|\phi_{+,n}\right\rangle\left\langle\phi_{-,n}|\phi_{-,n+1}\right\rangle$
$\displaystyle-\left(\gamma/2\right)\left\langle\phi_{+,n+1}|\phi_{+,n}\right\rangle\left\langle\phi_{-,n}\right|\mathcal{O}^{2}\left|\phi_{-,n+1}\right\rangle.\quad$

If $\mathcal{O}$ does not contain spacetime derivatives of $\phi_{\pm}$, then
for the last three terms on the right-hand side of Eq. (23), we can safely
take

$\displaystyle\left\langle\phi_{\pm,n+1}\right|\mathcal{O}\left|\phi_{\pm,n}\right\rangle\approx\left\langle\phi_{\pm,n}\right|\mathcal{O}\left|\phi_{\pm,n}\right\rangle=\mathcal{O}_{\pm}(\phi_{\pm}),\qquad$
(24)

where $\mathcal{O}_{\pm}$ are the bra and ket fields corresponding to
$\mathcal{O}$. They appear in the following term:

$\displaystyle-i\int
dx\left(\mathcal{O}_{+}(\phi_{+})\mathcal{O}_{-}(\phi_{-})-\frac{1}{2}\mathcal{O}_{+}^{2}(\phi_{+})-\frac{1}{2}\mathcal{O}_{-}^{2}(\phi_{-})\right),$

which should be added to Eq. (19) to yield the final nonunitary
$S[\phi_{+},\phi_{-}]$ that matches Eq. (21).

If $\mathcal{O}$ contains spacetime derivatives of $\phi_{\pm}$, then we have
to take into account the nontrivial overlap between
$\left|\phi_{\pm,n+1}\right\rangle$ and $\left|\phi_{\pm,n}\right\rangle$ and
thus cannot invoke Eq. (24). However, Matthews’s theorem [49, 50, 51] implies
that we can safely neglect this complication and simply assume Eq. (24) is
valid for perturbatively calculating the multipoint functions. To be more
specific, Matthews’s theorem has established a one-to-one correspondence
between the Feynman path-integral formalism and the Hamiltonian dynamics in
the perturbative regime (even when the momentum dependence of the Hamiltonian
is higher than second order [51]). Since the Keldysh formalism is nothing but
just a double copy of the usual path-integral formalism, we expect that this
correspondence holds here too.

##  Appendix B DOUBLE-TRACE DEFORMATION

The usual approach of studying double-trace deformation [38, 33] of a
Minkowski QFT is to deform a Euclidean QFT first and then apply the Wick
rotation to clear up the time-ordering ambiguity. In the Keldysh formalism,
however, the deformation coupling strength parameter is already complex, which
encodes the time-ordering information. Therefore, one must work in the
Minkowski space directly.

###  B.1 Singlet

We start with double-trace deformation of a singlet that contains only one
double-trace scalar term $f\mathcal{O}^{2}$. In general, the partition
function $Z_{f}[J]$ with source $J$ that describes the deformed QFT (which may
not be unitary) is given by

$\displaystyle Z_{f}[J]$ $\displaystyle\equiv$ $\displaystyle
e^{W_{f}[J]}=\left\langle e^{i\int
d^{d}x\left(-\frac{f}{2}\mathcal{O}^{2}(x)+J(x)\mathcal{O}(x)\right)}\right\rangle_{0},\quad$
(25)

where $W_{f}[J]$ is the free energy, and
$\left\langle\cdots\right\rangle_{0}\equiv\int D\phi\left(\cdots\right)\exp
iS_{\text{0}}[\phi]$ is the path-integral measure for the undeformed QFT [33].
By applying the Hubbard-Stratonovich transformation, one introduces an
auxiliary field $\sigma$ to Eq. (25) and gets

$\displaystyle Z_{f}[J]$ $\displaystyle=$ $\displaystyle\int
D\sigma\left\langle e^{i\int
d^{d}x\left[\frac{1}{2f}\sigma^{2}(x)+\left(\sigma+J\right)(x)\mathcal{O}(x)\right]}\right\rangle_{0}$
(26) $\displaystyle=$ $\displaystyle\int D\sigma e^{i\int
d^{d}x\frac{1}{2f}\sigma^{2}(x)}e^{W_{0}[\sigma+J]},$

where $W_{0}\equiv W_{f=0}$ is the undeformed free energy. Such a
transformation is possible in the Minkowski space if and only if
$\Im\\{f\\}<0$. Note that under a change of variables
$\tilde{\sigma}=\sigma+J$ and $\tilde{J}=J/f$, one has

$\displaystyle e^{W_{\infty}[\tilde{J}]}$ $\displaystyle=$
$\displaystyle\lim\limits_{f\to\infty}\int D\tilde{\sigma}e^{i\int
d^{d}x\frac{1}{2f}\left(\tilde{\sigma}-f\tilde{J}\right)^{2}(x)}e^{W_{0}[\tilde{\sigma}]}$
(27) $\displaystyle\propto$ $\displaystyle\int D\tilde{\sigma}e^{-i\int
d^{d}x\tilde{\sigma}(x)\tilde{J}(x)+W_{0}[\tilde{\sigma}]},$

up to a divergent multiplicative coefficient (contact term) $\exp(if\int
d^{d}x\tilde{J}^{2}(x)/2)$. From Eq. (27), one immediately sees that the two
free energies at the UV and IR fixed points, $W_{\infty}$ and $W_{0}$, are
connected by the Legendre transformation [33].

If we assume that $\mathcal{O}$ after normalization is proportional to a
hidden factor of $N^{-1}$ [e.g., $\mathcal{O}=\phi^{\alpha}\phi^{\alpha}$ for
$O(N)$ model which after normalization will be divided by $N$], then $W_{0}$
admits a cumulant expansion,

$\displaystyle{W_{0}[\tilde{\sigma}]}$ $\displaystyle\simeq$
$\displaystyle-\frac{1}{2}\left\langle\left(\int
d^{d}x\tilde{\sigma}(x)\mathcal{O}(x)\right)^{2}\right\rangle_{0}+\cdots$
$\displaystyle=$ $\displaystyle-\frac{1}{2}\int d^{d}x\int
d^{d}y\tilde{\sigma}(x)\tilde{\sigma}(y)\left\langle\mathcal{O}(x)\mathcal{O}(y)\right\rangle_{0}+\cdots,$

where higher terms are suppressed by factors of $N$.

At the $O(N^{0})$ level, only the first term which is quadratic in $\sigma$ in
Eq. (B.1) is kept, where the undeformed two-point function is given by
$\left\langle\mathcal{O}(x)\mathcal{O}(y)\right\rangle_{0}=\int\frac{d^{d}k}{\left(2\pi\right)^{d}}e^{-ik\left(x-y\right)}G_{0}(k^{2})$,
with $G_{0}(k^{2})$ the free propagator in the momentum space. Putting Eq.
(B.1) into Eq. (26), applying the Fourier transform, and then integrating out
$\sigma(k)$ yield

$\displaystyle{W_{f}[J]}$ $\displaystyle=$
$\displaystyle\frac{i}{2}\int\frac{d^{d}k}{\left(2\pi\right)^{d}}J(k)\frac{iG_{0}(k^{2})}{1+ifG_{0}(k^{2})}J^{*}(k),$
(29)

plus some contact terms. Therefore, the full propagator is given by

$G(k^{2})=-2\delta^{2}W_{f}[J]/\left(\delta J(k)\delta
J^{*}(k)\right)=\left(G_{0}^{-1}(k^{2})+if\right)^{-1}.$

We see that $if$ is directly added to the two-point vertex function
$G_{0}^{-1}(k^{2})$ of the QFT. This is because in the large $N$ limit, a
double-trace deformation is plainly additive to the effective action [38].

###  B.2 Multiplet

Following similar procedures, Eq. (29) can be generalized to a multiplet where
the deformation coupling strength is given by a matrix $\mathbf{f}$ [56]. The
result is

$\displaystyle\mathbf{G}\simeq\left(\mathbf{G}_{0}^{-1}+i\mathbf{f}\right)^{-1}+O(N^{1}),$
(30)

which leads to Eq. (10).

##  Appendix C HAMILTONIAN DYNAMICS OF ONE-DIMENSIONAL QUANTUM HARMONIC
OSCILLATOR UNDER $q$ MEASUREMENT

Solving the adjoint master equation Eq. (22) with
$H=\left(p^{2}+m^{2}q^{2}\right)/2$, $\mathcal{O}=q$, and
$A=q,p,q^{2},p^{2},\cdots$, under the Gaussian approximation [70] we get
(given $[q,p]=i$)

$\displaystyle\partial_{t}\left\langle q(t)\right\rangle$ $\displaystyle=$
$\displaystyle\left\langle p(t)\right\rangle,$
$\displaystyle\partial_{t}\left\langle p(t)\right\rangle$ $\displaystyle=$
$\displaystyle-m^{2}\left\langle q(t)\right\rangle,$
$\displaystyle\partial_{t}\left\langle q^{2}(t)\right\rangle_{c}$
$\displaystyle=$ $\displaystyle\left\langle
q(t)p(t)+p(t)q(t)\right\rangle_{c},$ $\displaystyle\partial_{t}\left\langle
p^{2}(t)\right\rangle_{c}$ $\displaystyle=$
$\displaystyle\gamma-m^{2}\left\langle q(t)p(t)+p(t)q(t)\right\rangle_{c},$
$\displaystyle\partial_{t}\left\langle q(t)p(t)+p(t)q(t)\right\rangle_{c}$
$\displaystyle=$ $\displaystyle 2\left\langle
p^{2}(t)\right\rangle_{c}-2m^{2}\left\langle q^{2}(t)\right\rangle_{c},$

where $\left\langle AB\right\rangle_{c}=\left\langle
AB\right\rangle-\left\langle A\right\rangle\left\langle B\right\rangle$. We
see that the only direct contribution of $\gamma$ is to $\left\langle
p^{2}(t)\right\rangle_{c}$ [70]. This contribution generates momentum $p$
diffusion and heats the system at rate $\gamma$, leading to the long-time
limit $\left\langle p^{2}(t)\right\rangle_{c}\sim\left\langle
q^{2}(t)\right\rangle_{c}\sim\gamma t/2$ and leaving us an infinitely heated
“stationary” state.

We can also calculate the correlations for $q$,

$\displaystyle\left\langle q_{+}(t)q_{+}(0)\right\rangle$ $\displaystyle=$
$\displaystyle-\left\langle q_{-}(t)q_{-}(0)\right\rangle$ $\displaystyle=$
$\displaystyle
i\int\frac{d\omega}{2\pi}\frac{\omega^{2}-m^{2}-i\gamma}{\left(\omega^{2}-m^{2}\right)^{2}}e^{-i\omega
t},$ $\displaystyle\left\langle q_{+}(t)q_{-}(0)\right\rangle$
$\displaystyle=$ $\displaystyle\left\langle q_{-}(t)q_{+}(0)\right\rangle$
$\displaystyle=$ $\displaystyle
i\int\frac{d\omega}{2\pi}\frac{-i\gamma}{\left(\omega^{2}-m^{2}\right)^{2}}e^{-i\omega
t},$

from which we can immediately derive the linear response functions and see
that they are independent of $\gamma$.

As for the Keldysh function $G^{K}$, we may simply write
$G^{K}_{\mathcal{L}}=-2\gamma
G^{R}G^{A}=2\gamma/\left(\omega^{2}-m^{2}\right)^{2}$. But its Fourier
transform diverges when $t\to\pm\infty$. Instead, noticing that

$\displaystyle\int\frac{d\omega}{2\pi}\frac{2\gamma}{\left[\left(\omega+i\varepsilon\right)^{2}-m^{2}\right]\left[\left(\omega-i\varepsilon\right)^{2}-m^{2}\right]}e^{-i\omega
t}$ $\displaystyle=$
$\displaystyle\frac{\gamma}{4m\varepsilon}\left\\{\theta(t)\left[\frac{e^{-t\left(\varepsilon+im\right)}}{m-i\varepsilon}+\frac{e^{-t\left(\varepsilon-
im\right)}}{m+i\varepsilon}\right]+\left(t\leftrightarrow-t\right)\right\\}$
$\displaystyle\simeq$
$\displaystyle\frac{\gamma\cos(mt)}{2m^{2}\varepsilon}=\int\frac{d\omega}{2\pi}\frac{\gamma}{m\varepsilon}\pi\delta\left(\omega^{2}-m^{2}\right)e^{-i\omega
t},$

the true $G^{K}_{\mathcal{L}}$ should be given by
$\left({m\varepsilon}\right)^{-1}{\gamma}\pi\delta\left(\omega^{2}-m^{2}\right)$,
the Fourier transform of which does not diverge when $t\to\pm\infty$.

##  Appendix D DIAGRAMMATIC PERTURBATION FOR THE KELDYSH FORMALISM

As a path-integral formalism, the Keldysh formalism naturally admits
diagrammatic perturbative calculations. Here, we explain how to do the
calculations for our second example, i.e., a
$\left(4-\epsilon\right)$-dimensional $O(N)$ massless scalar field. The
Lagrangian Eq. (IV.2), after rewritten in the Keldysh basis, reads

$\displaystyle
L(\phi^{\alpha}_{c},\phi^{\alpha}_{q})=\partial_{\mu}\phi^{\alpha}_{c}\partial^{\mu}\phi^{\alpha}_{q}-m^{2}\phi^{\alpha}_{c}\phi^{\alpha}_{q}+\frac{ic}{2}\phi^{\alpha}_{c}\phi^{\alpha}_{c}$
(31) $\displaystyle-$ $\displaystyle\frac{\lambda}{2\cdot
3!}\left({\phi^{\alpha}_{c}\phi^{\alpha}_{c}\phi^{\beta}_{c}\phi^{\beta}_{q}}+{\phi^{\alpha}_{c}\phi^{\alpha}_{q}\phi^{\beta}_{q}\phi^{\beta}_{q}}\right)+2i\gamma\left(\phi^{\alpha}_{c}\phi^{\alpha}_{q}\right)^{2},\quad$

where we have already included the quadratic terms
$\phi^{\alpha}_{c}\phi^{\alpha}_{q}$ and $\phi^{\alpha}_{c}\phi^{\alpha}_{c}$.
Note that $c$ should always be zero if we want the Keldysh formalism to be
physical. At $c=0$, we have three propagators in total,

$\displaystyle\bm{\mathcal{P}}(k)=\begin{pmatrix}\mathcal{P}^{K}(k)&\mathcal{P}^{R}(k)\\\
\mathcal{P}^{A}(k)&0\end{pmatrix},$

given by

$\displaystyle\mathcal{P}^{R/A}(\omega,\mathbf{k})$ $\displaystyle=$
$\displaystyle\frac{i}{\left(\omega\pm
i\varepsilon\right)^{2}-\mathbf{k}^{2}-m^{2}},$
$\displaystyle\mathcal{P}^{K}(\omega,\mathbf{k})$ $\displaystyle=$
$\displaystyle 2\pi\delta(\omega^{2}-\mathbf{k}^{2}-m^{2})$ (32)

which can be drawn as shown in Fig. 2, following the same diagrammatic
convention as in Ref. [30]. Also, we see that the FDR
$\mathcal{P}^{K}=\text{sign}(\omega)\left(\mathcal{P}^{R}-\mathcal{P}^{A}\right)$
is satisfied.

Figure 2: Free propagators in the Keldysh basis.

(a)

(b)

Figure 3: Vertices from Eq. (31) in the Keldysh basis.

The vertices of which the beta functions we are interested in are shown in
Fig. 3. Near the unitary WF fixed point, the beta functions for
${\phi^{\alpha}_{c}\phi^{\alpha}_{c}\phi^{\beta}_{c}\phi^{\beta}_{q}}$ and
${\phi^{\alpha}_{c}\phi^{\alpha}_{q}\phi^{\beta}_{q}\phi^{\beta}_{q}}$ should
behave similarly, so we will only look at the renormalization of one of the
two vertices
(${\phi^{\alpha}_{c}\phi^{\alpha}_{c}\phi^{\beta}_{c}\phi^{\beta}_{q}}$ only).

(a)

(b)

Figure 4: Divergent loops in the Keldysh basis at $O(c^{0})$.

(a)

(b)

(c)

Figure 5: One-loop corrections for the vertex
${\phi^{\alpha}_{c}\phi^{\alpha}_{c}\phi^{\beta}_{c}\phi^{\beta}_{q}}$.

(a)

(b)

(c)

(d)

(e)

Figure 6: One-loop corrections for the vertex
$\left(\phi^{\alpha}_{c}\phi^{\alpha}_{q}\right)^{2}$.

We note that the most advantage of working in the Keldysh basis is that not
all loops are divergent. The only one-vertex and two-vertex loops that diverge
near $d=4-\epsilon$ are shown in Fig. 4, respectively. The first loop [Fig. 4]
diverges as

$\overset{\text{dim.~{}reg.}}{\sim}-\frac{m^{2}}{4\pi^{2}\epsilon},$

where dim. reg. stands for dimensional regularization, and the second loop
[Fig. 4] diverges as

$\overset{\text{dim.~{}reg.}}{\sim}-\frac{i}{8\pi^{2}\epsilon}.$

Note that they are different from the results in Ref. [30] by a factor of $2$,
because in Ref. [30] the Keldysh basis was defined differently from ours.

As a demonstration, here we explicitly calculate the second loop [Fig. 4] at
$m^{2}=0$. Introducing external legs with total momentum $p$ into the loop, we
have

$\displaystyle\int\frac{d^{d}k}{\left(2\pi\right)^{d}}\mathcal{P}^{R}(k)\mathcal{P}^{K}(p-k)$
(33) $\displaystyle=$
$\displaystyle\int\frac{d^{d}k}{\left(2\pi\right)^{d}}\frac{i}{\left(\omega+i\varepsilon\right)^{2}-\mathbf{k}^{2}}2\pi\delta(\left(p-k\right)^{2})$
$\displaystyle=$
$\displaystyle-i\int\frac{d^{d}k_{\text{E}}}{\left(2\pi\right)^{d}}\frac{1}{k_{\text{E}}^{2}}\frac{1}{\left(p_{\text{E}}-k_{\text{E}}\right)^{2}}$
$\displaystyle=$
$\displaystyle-i\frac{\Gamma(2-d/2)}{2^{d}\pi^{d/2}}\left(p_{\text{E}}^{2}\right)^{d/2-2}\int_{0}^{1}dx\left[x\left(1-x\right)\right]^{d/2-2}$
$\displaystyle=$ $\displaystyle
iC_{\epsilon}^{-1}\left(-p^{2}\right)^{-\epsilon/2}\sim-{i}/{\left(8\pi^{2}\epsilon\right)}$

where
$C_{\epsilon}=-2^{5-2\epsilon}\pi^{\left(1-\epsilon\right)/2}\Gamma(3/2-{\epsilon}/{2})\sin({\pi\epsilon}/{2})$.
In the third step, we have used both the FDR for Eq. (D) and the momentum-
space Wick rotation
$\int{d^{d}k}\mathcal{P}(k^{2})=i\int{d^{d}k_{\text{E}}}\mathcal{P}(-k_{\text{E}}^{2})$.

A direct by-product of Eq. (33) is the two-point function between the two
composite operators,
$\mathcal{O}_{c}=\phi^{\alpha}_{c}\phi^{\alpha}_{c}/\sqrt{2}$ and
$\mathcal{O}_{q}=\sqrt{2}\phi^{\alpha}_{c}\phi^{\alpha}_{q}$, which can be
calculated by

$\displaystyle\left\langle\phi^{\alpha}_{c}\phi^{\alpha}_{c}(x)\phi^{\beta}_{c}\phi^{\beta}_{q}(0)\right\rangle$
$\displaystyle=$ $\displaystyle
2N\left[\int\frac{d^{d}k}{\left(2\pi\right)^{d}}\mathcal{P}^{R}(k)e^{-ikx}\right]\left[\int\frac{d^{d}k}{\left(2\pi\right)^{d}}\mathcal{P}^{K}(k)e^{-ikx}\right]$
$\displaystyle=$ $\displaystyle
2N\int\frac{d^{d}k}{\left(2\pi\right)^{d}}\int\frac{d^{d}p}{\left(2\pi\right)^{d}}\mathcal{P}^{R}(p)\mathcal{P}^{K}(k-p)e^{-ikx}$
$\displaystyle=$
$\displaystyle\int\frac{d^{d}k}{\left(2\pi\right)^{d}}\left(2NiC_{\epsilon}^{-1}\right)\left[\mathbf{k}^{2}-\left(\omega+i\varepsilon\right)^{2}\right]^{-\epsilon/2}e^{-ikx}.$

It is clear that the interior of this integral is the retarded function
$G^{R}_{\text{Gauss}}$ at the Gaussian fixed point for $\mathcal{O}$.

Below, we calculate the renormalization flow by looking at the perturbation of
each vertex in details.

###  D.1
${\phi^{\alpha}_{c}\phi^{\alpha}_{c}\phi^{\beta}_{c}\phi^{\beta}_{q}}$

Combining the diagrams above, now we look at the perturbative corrections for
the vertex
${\phi^{\alpha}_{c}\phi^{\alpha}_{c}\phi^{\beta}_{c}\phi^{\beta}_{q}}$, which
are given by three diagrams (s, t, and u channels) in total at the one-loop
level, as shown in Fig. 5. The full correction to $\lambda$ is

$\left[-\frac{i}{2\cdot
3!}\left(3!\right)\right]^{-1}\frac{1}{2}\left(-\frac{i\lambda}{2\cdot
3!}\right)^{2}\cdot
3\left[72+8\left(N-1\right)\right]\cdot\frac{-i}{8\pi^{2}\epsilon},$

from which $\beta_{\bar{\lambda}}$ is derived and is given in the main text.

###  D.2 $\left(\phi^{\alpha}_{c}\phi^{\alpha}_{q}\right)^{2}$

Similarly, for $\left(\phi^{\alpha}_{c}\phi^{\alpha}_{q}\right)^{2}$, there
are five diagrams in total (Fig. 6) that contribute. The full correction to
$\gamma$ is

$\displaystyle\left[-2\left(2!2!\right)\right]^{-1}\left(-2\gamma\right)\left(-\frac{i\lambda}{2\cdot
3!}\right)$
$\displaystyle\cdot\left\\{24+4\left[24+4\left(N-1\right)\right]\right\\}\cdot\frac{-i}{8\pi^{2}\epsilon},$

from which $\beta_{\bar{\gamma}}$ is derived and is given in the main text.

Figure 7: One-loop correction for the vertex
$\phi^{\alpha}_{c}\phi^{\alpha}_{q}$.  Figure 8: New divergent loop in the
Keldysh basis at $O(c^{1})$.  Figure 9: One-loop correction for the vertex
$\phi^{\alpha}_{c}\phi^{\alpha}_{c}$.

###  D.3 $\phi^{\alpha}_{c}\phi^{\alpha}_{q}$

For $\phi^{\alpha}_{c}\phi^{\alpha}_{q}$, there is only one diagram (Fig. 7).
The full correction to $m^{2}$ is

$\left(-i\right)^{-1}\left(-\frac{i\lambda}{2\cdot
3!}\right)\cdot\left[3+\left(N-1\right)\right]\cdot\frac{-m^{2}}{4\pi^{2}\epsilon},$

from which $\beta_{\bar{m}^{2}}$ is derived.

###  D.4 $\phi^{\alpha}_{c}\phi^{\alpha}_{c}$

The vertex $\phi^{\alpha}_{c}\phi^{\alpha}_{c}$ needs special treatment, as we
have to let $c$ be nonzero and consider the perturbative correction to the
free propagators $\bm{\mathcal{P}}$. In particular, the new perturbed
retarded/advanced propagators are not $\mathcal{P}^{R/A}$, but
$\mathcal{P}^{R/A}-c\mathcal{P}^{R/A}\mathcal{P}^{K}+O(c^{2})$, the second
term of which now diverges in the same way as Fig. 4 does.

Up to $O(c^{1})$, we also have to include a new one-vertex loop (Fig. 8) which
diverges as

$\overset{\text{dim.~{}reg.}}{\sim}\frac{ic}{8\pi^{2}\epsilon}.$

Therefore, taken into account the new diagram (Fig. 9) at $O(c^{1})$, the full
correction to $c$ is

$\left[-\frac{1}{2}\left(2!\right)\right]^{-1}\left(-\frac{i\lambda}{2\cdot
3!}\right)\cdot\left[6+2\left(N-1\right)\right]\cdot\frac{ic}{8\pi^{2}\epsilon},$

from which $\beta_{\bar{c}}$ is derived.

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