the configuration with smaller noise is the more stable one. For the case of
state-dependent fluctuations, one can define a modified free energy landscape
$W(\phi)$ if and only if $W$ solves the equation
$\partial_{\phi}W(\phi)=-F(\phi)/\chi(\phi)$ for the entire configuration
space of $\phi$. If such a solution exists, then the system fulfills an
effective detailed balance with respect to $W(\phi)$. If no such solution
exists, then detailed balance is truly broken. This is the case when
$\chi(\phi)^{-1}$ is singular for some field configuration $\phi^{\prime}$,
e.g., when $\chi(\phi^{\prime})=0$. In particular, this is the case when a
bistability occurs between an active and an absorbing phase, i.e., a dark
state for which $\chi(\phi)=0$ at one of the bistable states [397, 497]. A
setup where this scenario occurs has been introduced for ultracold Rydberg
gases in the facilitation regime discussed in Sec. V.2. Here, the quantum
mechanical origin of the facilitation mechanism introduces a bistability
between a dark state and an active state at high atom density. This scenario
has been highlighted in Ref. [398, 92] and is reviewed in Sec. X.3 below.

###  X.3 Dark state bistability

For a bistability separating an ordered phase from a dark state, the first
order phase transition is significantly modified. As discussed in Sec. V.2, a
dark state $\hat{\rho}_{D}=|D\rangle\langle D|$ is a pure quantum state, in
which the order parameter $\varphi$ and its fluctuations vanish (cf. Appendix
D). In contrast, the ordered (_active_) phase yields a nonzero value for the
order parameter and its fluctuations, and resides in a mixed state.

A first order phase transition between an active and a dark state thus readily
dictates a set of modifications:

(1) _Coexistence:_ The dark state has a local product state structure, e.g.,
${\hat{\rho}_{D}=\prod_{l}|\downarrow\rangle\langle\downarrow|_{l}}$ for
Rydberg atoms (cf. Sec. V.1). Coexistence between an active phase and a dark
state implies that the system hosts an extensive volume $V_{D}$ which is in
the dark state. Inside this volume, the order parameter and thus the
fluctuations vanish. Hence, the state of the total system is well approximated
by a product
$\hat{\rho}=(\hat{\rho}_{\text{active}})_{\bar{V}_{D}}\otimes(\hat{\rho}_{D})_{V_{D}}$
between a pure state $\hat{\rho}_{D}$, constrained to the volume $V_{D}$, and
a mixed state $\hat{\rho}_{\text{active}}$ on its complement $\bar{V}_{D}$.

Figure 10: RG flow of the potential $V_{k}$ and noise kernel $\chi_{k}$ at a
first order phase transition in $d=3$ dimensions. The microscopic model is
given by the Langevin equation (88) with initial conditions
$V_{\Lambda}=\tfrac{1}{2}\phi^{2}+\tfrac{1}{4}\phi^{4}-\tfrac{2\alpha}{3}\phi^{3}$
and noise kernel $\chi_{\Lambda}=\tfrac{1}{10}\phi$. (a) $\alpha=3.48$, (b)
and (c) $\alpha=3.5$. The flow at large momenta $k\sim\Lambda$ yields small
corrections to the potential landscape: The minimum at $\phi>0$ moves slightly
and becomes nearly degenerate to the dark state at $\phi=0$ in (a) and (b).
(d) The flow of $V^{\prime\prime}(0)$ provides an estimate for the change of
the potential toward a convex function. At the droplet scale
$k_{D}=\xi_{D}^{-1}$, it undergoes a sudden, sharp drop. (c) At this scale
$\chi_{k}$ increases by three orders of magnitude. It assumes a bimodal form
indicating large droplet fluctuations. At the largest distances fluctuations
are suppressed again, confirmed by the inset. Figure adapted from Ref. [497].

(2) _Droplet nucleation:_ Fluctuations are absent in the dark state volume
$V_{D}$. This prohibits noise activation from the dark state to the active
phase. Droplets appear exclusively in the active phase volume $\bar{V}_{D}$,
This leads to a unidirectional fluctuation pattern where local noise-induced
transitions can only follow the path
$\hat{\rho}_{\text{active}}\to\hat{\rho}_{D}$. Hence, one finds a maximally
extended hysteresis loop: once the global dark state has been reached, there
is no return to the active phase.

(3) _Domain wall motion:_ The domain walls experience a similar unidirectional
fluctuation pattern. The presence (absence) of fluctuations in the active
(inactive) phase makes the domain walls favor the dark state and extend the
volume $\bar{V}_{D}$. This bias needs to be compensated by a nonzero
difference in the potential energy. Hence, the noise $\chi$ and the potential
$V$ can no longer be treated independently.

Items (1)–(3) are relevant for the field theory approach to a first order dark
state phase transition. In particular, the interplay between the potential and
the noise for both the domain wall motion and the droplet formation requires a
treatment of the potential and the fluctuations on equal footing, i.e., an RG-
approach in the Keldysh field integral framework. This approach provides an
unambiguous identification of the three dynamical regimes (i)–(iii) in Sec.
X.1, and their modifications (1)–(3) at a dark state transition [497].

###  X.4 Field theory of first order dark state phase transitions

The dynamics at a dark state bistability with a scalar order parameter field
$\phi\equiv\phi(t,\mathbf{x})$ and with a potential form can be examined in a
Langevin framework as in Eq. (86). The dark state at $\phi=0$ is implemented
by a modified noise kernel [497, 109] as in Eq. (87):

$\partial_{t}\phi=D\nabla^{2}\phi-V^{\prime}(\phi)+\eta.$ (88)

Here $V$ is as defined above and the Gaussian noise $\eta$ has zero mean and
variance $\langle\eta(t,{\bf x})\eta(t^{\prime},{\bf
x}^{\prime})\rangle=\chi(\phi)\delta(t-t^{\prime})\delta(\mathbf{x}-\mathbf{x}^{\prime})$.
The noise kernel $\chi(\phi)$ is a positive, continuous function of $\phi$,
which vanishes in the dark state, $\chi(\phi=0)=0$. For concreteness, one may
assume $\chi(\phi)=\kappa\phi$ for some $\kappa,\phi>0$. This form of $\chi$
and $V$ is inspired by the Rydberg setup discussed in Sec. V.2, which yields
Eq. (88) in the high density regime [398, 92].

The Langevin equation (88) describes the evolution of the field $\phi$ on
short distances. A conventional, perturbative RG approach yielding the
effective field theory on large distances is not applicable at a first order
phase transition, due to the presence of a length scale and the absence of
universal scaling behavior. Nonperturbative approaches based on the functional
RG [612, 109, 108, 180] are more promising, and have been successfully applied
[497, 61]. For a nonequilibrium bistability as described by Eq. (88), a
convenient implementation of the functional RG is performed in momentum space.
The general formulation for the RG transformations in the nonequilibrium
Keldysh (or Martin-Siggia-Rose) framework can be found, e.g., in Refs. [541,
109, 497]. At RG scale $k$, it yields an effective theory for the field $\phi$
defined over momenta $q<k$ by integrating out short distance fluctuations with
momenta $q>k$.

The first order transition can be approached in a Langevin truncation, which
was introduced in Ref. [497]. In this framework, each RG transformation, i.e.,
each integration over short distance fluctuations, modifies the potential $V$
and the noise kernel $\chi$. At RG scale $k$, the dynamics are described by
the Langevin equation (88) with

$V\to V_{k},\ \chi\to\chi_{k}.$ (89)

The potential $V_{k}$ and the noise kernel $\chi_{k}$ transform under each RG
step. A truncation of this form ensures that the evolution of the distribution
function of the field $\phi$ at each momentum scale $k$ follows a regular
Fokker-Planck equation, and thus remains a well-defined, nonnegative
probability distribution under coarse graining. In general, one might also
include a scale-dependent wave function renormalization, $\sim
Z_{k}\partial_{t}\phi$, and diffusion $\sim D_{k}\nabla^{2}\phi$, but their
evolution is expected to be negligible due to the finite correlation length
(verified in Ref. [497] for Eq. (88)).

The Keldysh action for the effective Langevin equation is

$S_{k}=\int_{t,{\bf
x}}\phi_{q}\left[\left(\partial_{t}-D\nabla^{2}\right)\phi_{c}+V^{\prime}_{k}(\phi_{c})-\chi_{k}(\phi_{c})\phi_{q}\right].$
(90)

This imposes a polynomial dependence on the quantum (response) field
$\phi_{q}$, but leaves the functional dependence on the classical field
$\phi_{c}$ general. As mentioned above, such a truncation ensures that the
theory at each RG step can be mapped to an appropriate Langevin or Fokker-
Planck equation. Loosening this restriction, one may allow a more general
dependence on the $\phi_{q}$-fields as, e.g., discussed in Refs. [109, 108].
For the bistability in Eq. (88), such modifications remain, however,
negligibly small [497]. One may expect this behavior to be general for
semiclassical theories whose distribution function is governed by a Fokker-
Planck equation at all relevant scales.

For the Langevin truncation, the RG equations for the (unrescaled) potential
and noise vertex are [497, 108] (setting the diffusion constant $D=1$ and
dropping the field dependence in the argument)

$\begin{split}\partial_{k}V^{(1)}_{k}&=U_{k,d}V_{k}^{(3)},\\\
\partial_{k}\chi_{k}&=U_{k,d}\left(\chi^{(2)}_{k}-4\frac{\chi^{(1)}_{k}V^{(3)}_{k}}{\left(k^{2}+V^{(2)}_{k}\right)}+\frac{\chi_{k}(V^{(3)}_{k})^{2}}{2\left(k^{2}+V^{(2)}_{k}\right)^{2}}\right),\end{split}$
(91)

with
$U_{k,d}=-k^{d+1}\chi_{k}\left[2^{d-1}\pi^{\frac{d}{2}}\Gamma\left(\frac{d}{2}\right)d\left(k^{2}+V^{(2)}_{k}\right)^{2}\right]^{-1}$.
At a first order phase transition, the fields $\phi_{c},\phi_{q}$ do not obey
canonical power counting. Thus a polynomial expansion of the potential and the
noise kernel cannot be justified. It is then necessary to keep the general
functional form of $V_{k}$ and $\chi_{k}$, and to solve the flow equations
numerically.

###  X.5 Renormalization group flow

The phenomenology of the first order phase transition, including the
modifications from the dark state, is reflected in the numerical solution of
the flow equations (91) for the bistability equation (88) [497]. Close to a
first order transition, the RG flow of the potential $V_{k}$ and noise
$\chi_{k}$ reveal regimes (i)–(iii) from Sec. X.1:

(i) At short distances, i.e., at large momenta $k\gg\xi_{D}^{-1}$, where
$k=\Lambda\gg\xi_{D}^{-1}$ is the UV cutoff, the potential $V_{k}$ is weakly
renormalized (flattened) due to integrating over short wavelength fluctuations
(see initial stage of the RG flow for $k/\Lambda\geq 0.1$ in Fig. 10).

(ii) At a sharp momentum scale $k_{D}\Lambda=\xi_{D}^{-1}\Lambda\approx 0.1$,
both the potential $V_{k}$ and the noise kernel $\chi_{k}$ experience a
sudden, strong renormalization (witnessed by $V_{k}^{\prime\prime}$ in Fig.
10(d)). The potential approaches a nearly degenerate bistability,
$V_{k}(\phi=0)\sim V_{k}(\phi=\varphi_{f})$. The noise kernel $\chi_{k}$
assumes a bimodal structure with peaks at $\phi=0,\varphi_{f}$, see Fig.
10(c). This indicates strong fluctuations of the field between the two minima,
i.e., the onset of droplet nucleation at the scale $k_{D}=\xi_{D}^{-1}$.

(iii) At momenta $k\ll k_{D}$, on distances much larger than the droplet scale
$\xi_{D}$, the noise kernel $\chi_{k}$ becomes flat $\chi_{k}(\phi)\approx 0$
for $\phi\in[0,\varphi_{f})$. The corresponding Langevin equation Eq. (88) is
noiseless and describes deterministic domain wall motion. Asymptotically
($k\to 0$) the potential assumes a linear form $V_{k}(\phi)\sim\phi$ and
vanishes at the bistability $V_{k}(\phi)=\chi_{k}(\phi)=0$ for
$0<\phi<\varphi_{f}$.

This reconciles the field theory with the general phenomenology of first order
phase transitions, and in particular with a dark state transition. Phase
coexistence between a dark state and an active state requires both the
deterministic dynamics generated by $V_{k}$ _and_ the noise $\sim\chi_{k}$ to
vanish. The renormalization group flow of the noise kernel $\chi_{k}$ and the
interplay between $V_{k}$ and $\chi_{k}$ is crucial to resolve this behavior.

Whether first order dark state phase transitions can be observed in driven
quantum systems is a topic of ongoing research. Several possible scenarios
have been proposed [186], ranging from Rydberg atoms in the facilitation
regime [397, 497, 92, 594, 368, 309], to cellular automata [230], or between
two different dark states [118]. However, first order phase transitions are
often unstable against fluctuations in low dimensions [497, 385, 284]. Thus,
they are restricted to higher dimensions $d\geq 2$, where numerical
simulations of quantum dynamics are challenging. This is at the moment a major
obstacle in identifying first order dark state transitions.

##  XI Quantum criticality in driven open systems

The previous sections have covered driven open quantum systems whose scaling
descriptions fall into extensions of thermal universality classes, or
constitute novel nonequilibrium fixed points _per se_. In each of the cases
presented, the microscopic quantum nature of the problem is levelled out on
long wavelengths and becomes inconsequential for the effective field theory
description. Here, instead, we will cover two instances of the quantum scaling
limit mentioned in Sec. IV.3.2, marking the commencement of the third and last
thematic part of this review (cf. Tab. 1) . We will focus on noise-driven one-
dimensional (1D) bosonic systems whose quantum features can persist over long
length and time scales. Specifically, we will consider a Luttinger liquid
driven by $1/f$ noise (Sec. XI.1), and a driven open condensate subjected to
diffusion noise (Sec. XI.2), i.e., Markovian noise whose variance scale
quadratically with momentum. Both models illustrate how the formation of
nonequilibrium quantum critical states is determined by the interplay between
dissipation and nonlinearities. In the former case (XI.1), scaling manifests
in nonuniversal exponents controlled by the noise strength, while in the
latter case (XI.2), scaling is governed by a quantum fixed point without
classical counterpart, leading to universal scaling exponents that define a
novel universality class.

###  XI.1 Quantum critical scaling of noise-driven Luttinger liquids

We follow Refs. [152, 153, 154] and consider a 1D bosonic wire, described as a
Luttinger liquid with periodic potential of strength $g$, and driven by a
noise field $f(t)$ that couples to fluctuations of the density. The
Hamiltonian reads

$\hat{H}(t)=\int\frac{dx}{u}\left[K\left(u\pi\partial_{x}\hat{\theta}(x)\right)^{2}+\frac{1}{K}\left(u\partial_{x}\hat{\phi}(x)\right)^{2}\right.\\\
\left.\vphantom{\left(u\partial_{x}\hat{\phi}(x)\right)^{2}}-g\cos\\!\left(2\hat{\phi}(x)\right)\right]-\frac{1}{\pi}\int\frac{dx}{u}f(t)\partial_{x}\hat{\phi}(x)+H_{\mathrm{bath}}(\hat{\phi}(x)),$
(92)

where $\hat{\theta}(x)$ and $\hat{\phi}(x)$ are Hermitian operators encoding,
respectively, the phase of the bosons and their long-wavelength density
fluctuations, satisfying canonical commutation relations [225]. $K$ is the
Luttinger parameter, and $u$ is the speed of sound. In Eq. (92), we have
included the Hamiltonian of a zero-temperature Ohmic bath,
$H_{\mathrm{bath}}(\small{\hat{\phi}(x)})$, linearly coupled to
$\partial_{x}\hat{\phi}(x)$ and with friction $\eta$, which serves to damp
heating induced by the noise term $f(t)$. The noise is assumed to be Gaussian
and with zero average. It has, however, a nontrivial dependence on frequency,
with fluctuations scaling as $\langle f^{*}(\omega)f(\omega)\rangle=F/\omega$,
on top of a subleading contribution in the $\omega\to 0$ limit proportional to
$\eta i\omega$ (standard zero temperature noise of Caldeira-Leggett bath
[16]). This should be contrasted with the noise used for the driven open Bose
gas in Sec. XI.2 below, which is also Gaussian but flat in frequency (i.e.,
Markovian), and with nontrivial dependence on momentum.

We first consider the model in the absence of nonlinearities ($g=0$). Then,
correlation ($G^{K}$) and response functions ($G^{R}$) of the Luttinger liquid
obey scaling behavior that explicitly violates the thermal symmetry of Sec.
III.2. For instance, on long length and time scales, the correlation function
of the crystalline order parameter, $\hat{O}(x)=\cos(2\hat{\phi}(x))$, is
described by the scaling form

$G^{K}(t-t^{\prime},x-x^{\prime})\propto\left((x-x^{\prime})^{2}-u^{2}(t-t^{\prime})^{2}\right)^{-K\left(1+\bar{F}\right)},$
(93)

with $\bar{F}=F/(u^{2}\pi^{2}\eta)$. The scaling of $G^{K}$ in Eq. (93) is
governed by a nonuniversal exponent, and is a consequence of the number-
conserving nature of the noise drive ($f(t)\partial_{x}\hat{\phi}(x)$). This
form of the drive preserves the sound mode of the Luttinger liquid, which
therefore remains gapless. In this respect, the noise does not fundamentally
alter the nature of the zero-temperature Luttinger Liquid fixed point, but
only its scaling exponents [165]. To derive Eq. (93), one has to first take
the limit $\eta\to 0$ and, concomitantly, $F\to 0$, with $\bar{F}$ fixed. In
fact, for finite friction $\eta\neq 0$, the quantum scaling of Eq. (93) would
be valid up to time and length scales smaller than $1/\eta$, where
correlations start to decay exponentially.

In the case $g=0$ we have considered so far, the model Eq. (92) is quadratic,
and one can assign an independent temperature to each field mode.
Consequently, the state of the system is nonthermal, in a way reminiscent of
the generalized Gibbs ensemble in integrable systems [352, 353]. In contrast,
when $g\neq 0$, the nonlinearities encoded in the periodic potential enable
the redistribution of the energy injected by the noise among the different
modes. As a result, the modes reach a common effective temperature
$T_{\textsl{eff}}\propto g^{2}$. By the fluctuation-dissipation theorem,
thermalization is accompanied by a damping term which cuts the power-law
scaling of $G^{K}$ in Eq. (93). This is similar to the effect of temperature
on equilibrium quantum critical points, where a thermal de Broglie length
delimits the scaling regime [508].

###  XI.2 Quantum criticality of the 1D driven open Bose gas

We now present the quantum scaling regime and universal properties of the
driven open Bose gas. First, we discuss the implementation of the Markovian
diffusion noise employed to access the novel fixed point (Sec. XI.2.1). We
then focus on the properties of the scaling limit from an RG perspective (Sec.
XI.2.2), and discuss its universal features (Sec. XI.2.3) in comparison to the
3D driven open Bose gas of Sec. II.1. Finally, we compare the 1D driven open
Bose gas with the noise-driven Luttinger Liquid discussed above (Sec. XI.2.4).

####  XI.2.1 Implementation of diffusion noise

We consider a 1D variant of the driven open Bose gas introduced in Eqs. (2)
and (3), augmented by an additional jump operator in the dissipator:

$\hat{L}(x)=\sqrt{\gamma_{d}}\partial_{x}\hat{\psi}(x).$ (94)

This term adds a diffusion contribution (quadratic in momentum) to the kinetic
coefficient and to the Markovian noise of the Bose gas in Eq. (38). Such
quadratic-in-momentum scaling of the noise is reminiscent of model B in the
Hohenberg-Halperin classification [286]. However, the analogy stops at this
level, since the dynamics of model B is order-parameter conserving, in
contrast to the driven open Bose gas considered here.

A possible realization of the jump operator in Eq. (94) is offered by an array
of microwave cavities coupled to superconducting qubits [66]. This
implementation relies on coupling each qubit to an anti-symmetric combination
of the photonic modes of its nearby cavities. Integrating out the qubit
dynamics leads to a loss term for the photons which is proportional to such an
anti-symmetric combination, and is converted to a spatial gradient of a
bosonic field in the continuum limit [402, 396].

For $q\to 0$, the infrared modes asymptotically decouple from this noise,
since it scales to zero with momentum, $\sim q^{2}$. In quantum optics
language, this circumstance is described by saying that the jump operator of
Eq. (94) supports dark states around $q=0$ (see App. D and Ref. [396]). The
presence of such dark states is the physical reason for the appearance of
quantum critical scaling and of a universal regime without asymptotic
decoherence in an otherwise driven open system. In the following, we formalize
this picture on the grounds of Keldysh field theory and an RG analysis.

####  XI.2.2 Nonequilibrium quantum scaling

When translated into the Keldysh action, the jump operators in Eq. (94) result
in the following contribution to the Keldysh component of the Gaussian action
(cf. Eq. (16)):

$P^{K}=i\left(2\gamma+\gamma_{d}q^{2}+\dotsb\right).$ (95)

As in Eq. (16), $\gamma$ is given by the sum of one-body loss ($\gamma_{l}$)
and pumping ($\gamma_{p}$) rates, $\gamma=(\gamma_{l}+\gamma_{p})/2$.
Similarly, the retarded component of the Gaussian action reads

$P^{R}=Z\omega-\left(K_{c}-i\gamma_{d}\right)q^{2}+ir_{d},$ (96)

with $r_{d}=(\gamma_{l}-\gamma_{p})/2$. The dynamical critical exponent here
is $z=2$. Indeed, as discussed in Sec. IV.1.4, by tuning $r_{d}\to 0$ the
condensation transition is accompanied by the onset of a diffusive critical
mode, with canonical dynamical exponent $z=2$. In Eq. (96) we have also
introduced the wave function renormalization coefficient, $Z$, for future
purposes [112]. Notice that $Z=1$ in the microscopic theory, cf. Sec. II.1.

The form of $P^{K}$ in Eq. (95) is the key to unveiling nonequilibrium quantum
critical behavior in the system. As we tune $\gamma_{p}\to\gamma_{l}$, the
spectral gap of the driven open Bose gas closes ($r_{d}\to 0$). If we tune
simultaneously $\gamma_{p,l}\to 0$, we enter a scaling regime with both
$P^{R}\sim q^{2}$ and $P^{K}\sim q^{2}$ scaling quadratically. This realizes
precisely the quantum scaling regime of Sec. IV.3.2. Although formally
analogous to the scaling of a zero temperature quantum critical point, there
are remarkable differences as we will discuss in the following.

When the $R/A/K$ components of the Gaussian action scale with the same power
of momentum, $P^{R/A/K}\sim q^{2}$, both classical and quantum fields have the
same canonical scaling dimension, $\psi_{c}\sim\psi_{q}\sim q^{d/2}$. By
following the canonical power counting of Sec. IV.3, one readily deduces that
any quartic term, regardless of the number of quantum fields, scales with
momentum as $\sim q^{2-d}$. These terms are not restricted to obey the thermal
symmetry of Sec. III.2, and their precise nature will be discussed in the next
subsection. This discussion sets $d=1$ as the natural context to explore the
Wilson-Fisher fixed point associated with nonequilibrium quantum scaling. The
canonical power counting would have been completely different had the system
supported a sound mode ($z=1$), as is the case for the noise driven Luttinger
Liquid discussed above. Indeed, in contrast to this latter case, sextic
operators are marginal in $d=1$, with higher-order nonlinearities being RG
irrelevant.

Figure 11: Diffusion noise, $\gamma_{d}q^{2}$ (blue line), and noise gap,
$\gamma$ (red line), as a function of momentum. For momentum scales
$q\gtrsim\Lambda_{G}$, the renormalization group flow is dominated by
nonuniversal effects, while below $\Lambda_{G}$ it is controlled by the
nonequilibrium quantum fixed point. For $q\lesssim\Lambda_{M}$, the noise gap
$\gamma$ dominates over the diffusion noise, and the quantum scaling regime is
left. Figure adapted from Ref. [402].

What is the extent of the quantum scaling regime in momentum space? Quantum
scaling can formally be accessed by setting the noise gap to zero, $\gamma\to
0$, or equivalently by working with momenta above the Markov scale
$q\gg\sqrt{\gamma/\gamma_{d}}\equiv\Lambda_{M}$, where diffusion noise
dominates. Accordingly, the quantum regime will be established between the two
momentum scales, $\Lambda_{M}\lesssim q\lesssim\Lambda_{G}$, where
$\Lambda_{G}$ is the Ginzburg scale [642]: for $q\lesssim\Lambda_{G}$
corrections to Gaussian scaling become dominant, and the scaling behavior of
the system is governed by the Wilson-Fisher fixed point. This situation is
sketched in Fig. 11. It is analogous to the conditions for observability of
quantum scaling in equilibrium at finite temperature. There, the relevant
range of momenta is $\Lambda_{dB}\lesssim q\lesssim\Lambda_{G}$, with
$\Lambda_{dB}\sim T^{1/2}$ the de Broglie momentum scale. Temperature,
however, is a fixed energy scale, protected against renormalization by the
thermal symmetry. In contrast, in our model, $\gamma$ can acquire sizable
corrections under RG, which alters the bare value of the Markov scale
estimate, $\Lambda_{M}$. The scales $\Lambda_{M,G}$ are nonuniversal and need
to be determined for a given microscopic model. Crucially, even after RG
corrections are taken into account, $\Lambda_{M}$ remains one order of
magnitude smaller than $\Lambda_{G}$ in the model specified above [401].

As indicated in the previous paragraphs, the quantum scaling regime discussed
here pertains to a nonequilibrium Bose condensation transition in 1D. However,
the following argument shows that the effective dimension of the system in the
quantum scaling regime is, in fact, $D=3$, enabling a Bose condensation and
spontaneous symmetry breaking phase transition at the point where the sign of
$(\gamma_{l}-\gamma_{p})/(\gamma_{l}+\gamma_{p})$ changes even though the
spatial dimension of the system is $d=1$. For illustration, we consider the
correction to the dissipative spectral gap, $\Delta r_{d}$, but the result
would remain unaltered if we were to focus on other couplings. At one-loop
order, the correction is given by the tadpole diagram [402]

$\Delta r_{d}\propto(u_{c}+iu_{d})\int d\omega\,dq\leavevmode\nobreak\
\frac{P^{K}(\omega,q)}{P^{R}(\omega,q)P^{A}(\omega,q)},$ (97)

where $u_{c}$ and $u_{d}$ are the strength of interactions and two-body loss,
respectively (cf. Sec. II.1). In the quantum scaling regime, $P^{K}$ scales
quadratically, enlarging the momentum phase space by $q^{2}$ in Eq. (97), and
mimicking an integration measure in three dimensions which would also bring an
extra quadratic factor ($d^{3}q\sim q^{2}dq$). As a result, the dimensionality
of the problem is effectively enlarged by the dynamical critical exponent,
$D=d+z=3$. The lack of particle number conservation is crucial: A closed
system would have $z=1$, which would still prohibit a condensation transition
in $d=1$.

When $\gamma\gg\gamma_{d}q^{2}$, this effective dimensional enlargement does
not hold anymore, since $P^{K}$ and $P^{R/A}$ will not scale alike ($P^{K}\sim
q^{0}$, $P^{R/A}\sim q^{2}$; classical scaling of Sec. IV.2) and the effective
dimensionality of the system will shrink ($D\to d$). In our setup, this
corresponds to $D\to 1$, where no continuous symmetry breaking or quantum
criticality can occur. Therefore, for momenta $q\lesssim\Lambda_{M}$ (or
length scales $l>\Lambda^{-1}_{M}$), the quantum scaling regime is left, as
marked in Fig. 11.

####  XI.2.3 Properties of the nonequilibrium quantum fixed point

At the microscopic scale, our model has the same nonlinear terms as the driven
open Bose gas of Eqs. (2) and (3): short-range contact interactions ($u_{c}$)
and two-body losses ($u_{d}$). As mentioned above, any quartic combination of
fields which respects the $\mathrm{U}_{c}(1)$ symmetry (cf. Sec. IV.1) will be
generated in the course of renormalization, irrespective of the number of
quantum fields involved. This is different from the semiclassical RG flow of
the 3D driven open Bose gas, which retains only the vertex
$\left(u_{c}-iu_{d}\right)\psi^{*}_{c}\psi^{*}_{q}\psi^{2}_{c}$ (and its
complex conjugate) as RG-relevant nonlinearity. The extra terms include the
quantum counterpart of this vertex ($\psi^{*}_{c}\psi^{*}_{q}\psi^{2}_{q}$),
multiplicative Markovian noise ($\psi^{*}_{q}\psi_{q}\psi^{*}_{c}\psi_{c}$),
and non-Gaussian Markovian noise fluctuations $(\psi^{*}_{q}\psi_{q})^{2}$. As
mentioned above, since the system is open ($z=2$) and the upper critical
dimension is $d_{c}=2$, the hierarchy of RG-relevant nonlinearities is
truncated at fourth order. This is in contrast to isolated, number conserving
($U_{q}$ restored), low-dimensional quantum systems with $z=1$, where all
higher-order nonlinearities are equally RG-relevant [225].

The scaling solution of the RG flow in the quantum critical regime is governed
by a Wilson-Fisher fixed point with two repulsive directions, associated with
simultaneously tuning the spectral gap and the noise gap to zero [402]. This
is analogous to thermal equilibrium, where both spectral gap and temperature
have to be tuned to zero in order to reach a quantum critical point. The
nonequilibrium quantum fixed point is characterized by three key properties:

(1) _Asymptotically broken thermal symmetry:_ The fluctuation-dissipation
relation is not restored at infrared scales, since quartic nonlinearities
breaking the thermal symmetry of Sec. III.2 have nonzero fixed-point values.
This is in contrast to several instances of driven open criticality, where an
emergent effective temperature rules the occupation of soft modes [423, 575,
391], cf. also Sec. IV.2 and VIII. We have already encountered in this review
instances of criticality where the effective temperature is momentum-dependent
at infrared scales and therefore the system spoils fluctuation-dissipation
relations (Sec. VIII.2 and [400]). However, in these cases the fixed point is
classical (associated to a classical scaling solution).

(2) _Absence of decoherence and RG limit cycle:_ Coherent (spectral gap,
interactions, etc.) and dissipative couplings (noise gap, two-body losses,
etc.) scale with the same anomalous dimension in the infrared, with their
ratios approaching a constant. This signals the absence of decoherence at the
Wilson-Fisher fixed point in sharp contrast to the dissipative nature of the
semiclassical fixed point, where coherent couplings vanish in the infrared,
determining universal decoherence (cf. Sec. VIII and Fig. 12). This is a
signature of the dark states supported by the jump operator, which are
decoupled from noise in the infrared. The simultaneous presence of coherent
and dissipative couplings at the fixed point is further mirrored by slowly
damped oscillations in the RG flow of the wave function renormalization $Z$.
Similar limit cycle phenomenology is found in coupled Langevin-Ising models
(Sec. VIII.2), and in cavity QED with engineered dissipation [530, 529]).

Figure 12: Comparison between the nonequilibrum quantum and classical fixed
points (cf. Fig. 6). In the former case ($d=1$), the coherent and dissipative
couplings flow under RG to a finite value, which indicates that the system has
not fully decohered at the critical point. This is in contrast to the
semiclassical fixed point in $d=3$, which displays decoherence at long-
wavelengths, with only the dissipative couplings nonvanishing at the fixed
point. Figure adapted from Ref. [402].

(3) _Absence of quantum-to-classical correspondence:_ The quantum fixed point
is not only without equilibrium analog, but it also does not belong to the
same universality class as its 3D driven open counterpart, as follows from the
discussion above. This violation of the quantum-to-classical correspondence
between low-dimensional ($d=1$) and high-dimensional ($d=3$) criticality is
rationalized by noting that the set of RG relevant operators at the Wilson-
Fisher fixed point is markedly different in the two cases. Another way to
appreciate such sharp discrepancy with a cornerstone of equilibrium
criticality [508] is to notice that the equilibrium symmetry is not restored
in the quantum case, in contrast to asymptotic thermalization of the
semiclassical fixed point. This mismatch in the symmetry properties leads to
different universality classes.

####  XI.2.4 Comparison with noise-driven Luttinger Liquids

The one dimensional driven-open Bose gas has similarities and important
differences to the noise-driven Luttinger Liquid discussed in Sec. XI.1. The
key difference is that by engineering Markovian noise which scales
quadratically with momentum, one can target the quantum scaling limit. The
$1/f$ noise used for the Luttinger Liquid is not tailored to access a novel
fixed point, and in fact the exponents are non universal in this latter case.

In the model of the driven open Bose gas, particle number conservation is
broken by the presence of incoherent gain and loss of particles (the $U_{q}$
symmetry of Sec. IV.1 is broken). These processes do not occur in the
Luttinger Liquid of Eq. (92), which evolves under a time-dependent, number-
conserving Hamiltonian. The different symmetry properties of the two models
result in different soft-mode dispersion relations at criticality (linear for
the Luttinger liquid, and quadratic for the open Bose gas), and assign the two
systems to different universality classes.

However, what the systems have in common is that the scaling regime of the
driven open Bose gas is also cut asymptotically by an emergent effective
thermal length ($\Lambda_{M}$), which is generated by the interplay of
nonlinearities and dissipation.

##  XII Universality in dissipative quantum impurities

This section is devoted to the interplay of local dissipative sources and
long-wavelength modes in low-dimensional quantum systems. The problem of a
quantum impurity, a situation where a few degrees of freedom are locally
coupled to an extended interacting quantum systems, is central to the field of
strongly correlated systems in equilibrium [225, 392]. It has been key to
access prototypical instances of strong-coupling phenomena, which would have
been hard to solve in generic conditions, and has provided guidance for
decades in solid state and AMO experiments [498].

In a nutshell, distinguishing a few degrees of freedom (the impurity) from the
remainder of the system allows us to find nonperturbative phenomena where
strong correlations play a role, on the midway between a few-body problem and
the full many-particle one. The many-body ‘environment’ serves as a feed of
correlations, to which the impurity degrees of freedom couple, leading to a
strong intertwining of the two.

In the realm of strongly correlated systems in equilibrium, examples range
from the disruptive effects of a local potential embedded in a Fermi sea
(Anderson catastrophe [392]), to the strong coupling of magnetic impurities to
fermionic or bosonic baths (Kondo effects [491]), and encompass the dressing
of static or moving particles in Fermi or Bose-Einstein condensed environments
(polarons [258]). This raises the question about the nonequilibrium
counterparts of these iconic cases of strongly correlated physics. Studying
dissipative quantum impurities elucidates the interplay of dissipation,
interactions, and correlations, which is at core of this review. Such
impurities can be, for example, local losses, incoherent pumping, or phase
noise, embedded in an extended interacting quantum many-body system.

We first consider solvable instances of dissipative impurities (Sec. XII.1).
Then, we present a Keldysh approach to treat strongly correlated regimes of
interacting wires with local losses (Sec. XII.2).

###  XII.1 Solvable models of dissipative impurities and the quantum Zeno
effect

A phenomenon that is central to the following sections is the quantum Zeno
effect [335, 190]. We will start by reviewing its manifestation in solvable
systems of dissipative impurities.

Let us consider a tight-binding model of fermions hopping along a one-
dimensional chain with one-body loss at the center of the chain, for reference
at the lattice site $l=0$ (for analogous setups in spin chains, see Refs.
[643, 57, 486, 485, 487, 583, 484, 564]). The corresponding Hamiltonian and
jump operator for the dissipative impurity site read

$\hat{H}=J\sum^{L}_{l=-L}\left(\hat{\psi}^{\dagger}_{l}\hat{\psi}_{l+1}+\text{H.c.}\right),\qquad\hat{L}_{0}=\sqrt{\gamma}\hat{\psi}_{0}.$
(98)

In the dynamics starting from the ground state of the closed tight-binding
model, i.e., a filled Fermi sea, the effect of switching on the loss
propagates along the chain, reaching distances $d$ from the impurity on
timescales $d/J$ [207]. On smaller distances, a nonequilibrium steady-state is
formed. The rate at which fermions leave the system at the impurity site is
$r(\gamma)$. It scales as $r(\gamma)\propto\gamma$ for small loss rate
($\gamma\ll J$), while, for strong dissipation $\gamma\gg J$, the rate falls
off to zero as $r(\gamma)\sim\gamma^{-1}$ [339, 340, 564]. Due to particle-
hole symmetry of the Hamiltonian, analogous results apply when local loss is
replaced by gain.

In order to rationalize the counterintuitive decay of $r(\gamma)$ for
$\gamma\gg J$, we employ a perturbative argument. The large energy scale set
by the loss rate splits the Hilbert space into fast and slow sectors [420,
295]. All states with a particle on the lossy site decay fast with a rate
$\sim\gamma$. On the other hand, states with zero occupancy on the dissipative
site form the slow sector of the Hilbert space. In second order perturbation
theory in the hopping $J$ that couples the slow and fast subspaces, one finds
the effective decay rate of states in the slow sector to be $r\sim
J^{2}/\gamma$. Another way to rephrase this result is to notice that for
$\gamma\gtrsim J$, the speed of particles replenishing the lossy site is
smaller than the rate at which they are lost into the environment, leading to
a smaller net current flowing toward the dissipative impurity. The dynamical
decoupling induced by dissipation finds numerous applications in the
engineering of quantum simulators based on ultracold gases, and it is known as
quantum Zeno effect [200, 270, 277, 212, 639]. Fig. 13 shows a qualitative
dependence of $r(\gamma)$. The RG flow shown in the figure will be discussed
in the next section.

To grasp the generality of the phenomenon, one could also consider a chain of
spins or fermions subject to a local incoherent scatterer: for instance, the
Hamiltonian in Eq. (98) supplemented by a local time-dependent potential
$V=\xi(t)\hat{\psi}_{0}^{\dagger}\hat{\psi}_{0}$ with $\xi(t)$ Gaussian white
noise [172, 572]. The presence of a fast scale (i.e., dissipation rate), no
matter if it is related to loss/pumping or dephasing, will induce the same
nonmonotonous behavior. In Ref. [172], the system is initialized in the ground
state of Eq. (98) at fixed filling and then the noise is switched on. After
the local nonequilibrium state is formed around the impurity, one can observe
that the fraction of fermions left inside the Fermi sea grows and then
collapses to zero upon increasing the noise strength. This is analogous to the
dependence on $\gamma$ discussed above for the case of local losses. The
quantum Zeno effect can also manifest in the spreading of correlations or in
entanglement dynamics, as reported in studies involving integrable 1D systems
[126, 7, 6, 100, 146].

Experimentally, the quantum Zeno effect at dissipative impurities has been
demonstrated in a wire of ultracold bosons with an electron beam focused on a
small region of space to induce localized loss. In this setup, nonmonotonous
behavior of the particle current toward the region of localized dissipation
has been observed [79, 49, 348]. There are a number of theory works modelling
this scenario. They mostly resort to a semiclassical description of dynamics
or Bogoliubov theory, which are appropriate in the presence of a condensate
[614, 532, 600]. However, in a wire of interacting fermions, strong
correlations could play a decisive role by intertwining with the effect of
local dissipation, leading to nontrivial renormalization effects. We will
discuss this case in detail in the next section.

Fermionic dissipative impurities can be realized in ultracold atomic wires by
mimicking the two-leads scenario typical of mesoscopic physics (experiments
[77, 141, 361, 289]; theory [589, 588]). A tightly focused beam at the center
of the chain leads to spin-dependent particle losses, with the reservoirs of
different spin species used to probe transport across the impurity by imposing
a chemical potential difference.

###  XII.2 Local dissipation in interacting quantum wires:
the dissipative Kane-Fisher problem

Our presentation of the interplay of local dissipation with the quantum
fluctuations of a 1D interacting fermionic system will be based on a close
analogy with the Kane-Fisher phenomenon. In the latter, a local coherent
scatterer acts as a relevant or irrelevant perturbation depending on the
repulsive or attractive character of the interactions in the wire prepared in
the ground state [225, 478, 312, 314, 313]. Under renormalization, the
strength of the impurity grows for repulsive interactions, and the effective
long-wavelength description corresponds to a wire split into two decoupled
segments. Instead, for attractive interactions, the strength of the impurity
diminishes, and the system ‘heals’ itself from the defect. Hence, the
conductivity vanishes in the former case and becomes perfect in the latter.

####  XII.2.1 Qualitative overview: Canonical power counting

The starting, yet crucial, observation is that, after bosonization (cf. also
Sec. XII.2.2), the scaling dimensions of a coherent and a dissipative impurity
in an interacting fermionic wire are the same. One would then expect, at least
on the level of canonical power counting, the same physical picture of the
Kane-Fisher problem. Indeed, the RG flow of the rescaled one-body loss rate,
$\bar{\gamma}=\gamma/\Lambda$, reads [206]

$\frac{d\bar{\gamma}}{d\ell}=\left(1-g\right)\bar{\gamma},$ (99)

where $g$ is the Luttinger parameter of the wire. From this flow equation we
retrieve the physical picture discussed above. For attractive interactions in
the wire ($g>1$) dissipation is renormalized to zero and the impurity becomes
transparent; in contrast, for repulsive interactions ($g<1$), the dissipation
strength flows to infinity, invalidating perturbative RG. However, in this
case, we can approach the problem from a complementary angle, which leads to
an RG equation valid for the inverse of $\gamma$ [206]. Its rescaled value is
defined as $\bar{D}=\Lambda/\gamma$ and obeys the RG flow

$\frac{d\bar{D}}{d\ell}=\left(1-g^{-1}\right)\bar{D}.$ (100)

In this second case, one assumes a strong dissipative impurity, treating the
remainder of the chain as a perturbation. This is analogous to a similar
approach in the RG treatment of the original Kane-Fisher problem [16, 315],
and to the perturbative argument outlined for the Zeno effect in Sec. XII.1.
The RG flow in Eq. (100) implies that for repulsive interactions ($g<1$), the
weak link is cut $\bar{D}\to 0$, while for attractive interactions, $\bar{D}$
is renormalized to infinity, invalidating the RG treatment. This discussion is
summarized in Fig. 13. In the following, we will refer to the two fixed points
$\gamma\to\infty$ and $\gamma\to 0$ as fluctuation-induced quantum Zeno (FIQZ)
effect and fluctuation-induced transparency (FIT), respectively. This
nomenclature is due to the fact that no fine-tuning of driving parameters is
required to reach these two fixed point; it rather roots in the presence of
gapless modes in the wire. These new universal regimes are purely fluctuation
driven: it is the nature of interactions in the chain ($g$ smaller or larger
than one), which determines the onset or lack of the Zeno effect.

Figure 13:  Dependence of the escape rate $r$ on the microscopic loss rate at
the impurity site $\gamma$. When $\gamma$ is large, $r$ drops to zero, which
is a manifestation of the quantum Zeno effect. The red and blue arrows
qualitatively portray the RG flow of an interacting fermionic wire with
localized loss, leading to the two fixed points achieved for repulsive and
attractive interactions, respectively. Figure adapted from Ref. [206].

####  XII.2.2 Keldysh field theory of the dissipative Kane-Fisher problem

We now turn our attention to the interplay of fluctuations and localized
dissipation, going beyond canonical power counting. The key new ingredient
here is that, similar to the equilibrium impurity problems, the interplay
between interactions and a continuum of gapless modes strongly renormalizes
the couplings. The couplings still flow to either of the two fixed points of
Fig. 13 (FIT or FIQZ). However, now the system may also enter a thermalizing
phase where it heats up and no fluctuation-induced phenomena can establish.
This is the physical picture encapsulated in Fig. 14, which we substantiate in
the following.

This case of study is, indeed, a clear example of emergent nonequilibrium
physics in the quantum regime resulting from the interplay of long-wavelength
modes, dissipation, and strong many-body interactions, which can be addressed
with the nonequilibrium RG approach widely employed in this review. We
consider the continuum Luttinger liquid description of a fermionic wire [207]

$\hat{H}=\frac{v}{2\pi}\int_{x}\left[g\left(\partial_{x}\hat{\phi}\right)^{2}+g^{-1}\left(\partial_{x}\hat{\theta}\right)^{2}\right],$
(101)

with $\hat{\theta}$ and $\hat{\phi}$ describing density and phase
fluctuations, and $v$ the velocity of sound. At the center of the wire at
$x=0$, there is a single-body fermion loss modeled by the Lindblad jump
operator, $\hat{L}(x=0)=\sqrt{\gamma}\hat{\psi}(x=0)$, which in bosonization
language is given by

$\hat{\psi}(x=0)\approx
e^{i(\hat{\phi}+\hat{\theta})|_{x=0}}+e^{i(\hat{\phi}-\hat{\theta})|_{x=0}}.$
(102)

Here we have retained only leading harmonics. Analogously to the treatment of
the usual Kane-Fisher problem [313], one can integrate out the chain at all
points $x\neq 0$ to obtain an effective action for the impurity, reducing the
extended system to a $0+1$-dimensional problem. This is feasible analytically
due to the quadratic nature of the Luttinger liquid Hamiltonian (101).
Physically, the remainder of the chain acts as a bath for the impurity degrees
of freedom, which can then be treated in an RG approach. We thus obtain a
contribution to the Keldysh action at $x=0$ in the form of an effective
Caldeira-Leggett bath [105] which describes the effect of the remainder of the
chain on the impurity as an environment:

${S^{B}}|_{x=0}=\frac{i}{\pi}\int_{\omega}\left(\theta^{*}_{c}\leavevmode\nobreak\
\theta^{*}_{q}\right)\begin{pmatrix}0&-\kappa\omega\\\
\kappa\omega&2\kappa_{0}|\omega|+4\kappa
T_{\text{eff}}\end{pmatrix}\begin{pmatrix}\theta_{c}\\\
\theta_{q}\end{pmatrix}.$ (103)

In the expression above, the damping term in the retarded and advanced sectors
is proportional to $\kappa=1/g$, while $T_{\text{eff}}$ accounts for the
possibility of generating an effectively temperature under RG transformations
(cf. Sec. XI.1), which is the noise gap in the language of Sec. IV.2. While
the damping coefficient $\kappa$ in the retarded and advanced sectors can grow
under renormalization, the corresponding term in the Keldysh sector will
remain at its microscopic value ($\kappa_{0}$) since perturbative corrections
cannot generate a nonanalytic function of the frequency ($\propto|\omega|$).

The master equation with jump operator Eq. (102) adds to the free Keldysh
action in Eq. (103) the nonlinear term

$S^{I}|_{x=0}=-2i\gamma\int_{t,x}\delta(x)\left[\left(e^{i\sqrt{2}\phi_{q}}-\cos\\!\left(\sqrt{2}\theta_{q}\right)\cos\\!\left(\sqrt{2}\theta_{c}\right)\right)\right.\\\
\left.+\left(e^{i\sqrt{2}\phi_{q}}\cos\\!\left(\sqrt{2}\theta_{q}\right)-1\right)\right].$
(104)

Therefore, the total Keldysh action of the problem at the impurity site
$S|_{x=0}$ consists of the sum of the nonlinear term coming from local
dissipation, and of the environment action,
$S|_{x=0}=S^{I}|_{x=0}+S^{B}|_{x=0}$.

Figure 14:  Phase diagram for a dissipative impurity in an interacting fermion
wire from second order RG equations, for different values of the microscopic
dissipation strength $\bar{\gamma}_{0}$ ($\bar{D}_{0}$ in the dual regime) and
of the Luttinger parameter $g$. The blue-shaded area in the lower-right corner
corresponds to values where the fluctuation-induced transparency (FIT) is
visible, while the upper-left one corresponds to values where the fluctuation-
induced quantum Zeno effect (FIQZ) is visible. The gray-shaded areas
correspond to values where the effective temperature suppresses these effects.
The white regions correspond to values where the perturbative RG breaks down.
Figure adapted from Ref. [207].

Performing an RG analysis up to second order in $\gamma$ on the total action
$S|_{x=0}$, one finds a non-trivial RG flow of $\gamma$, $\kappa$ and
$T_{\mathrm{eff}}$ (the RG flow is initialized at zero effective temperature;
an analogous procedure can be carried out for the dual theory at large
$\gamma$). The first order calculation reproduces the discussion above in Sec.
XII.2.1. At second order, fluctuations lead to a flow of $\kappa$ and,
importantly, to the emergence of a finite effective temperature
$T_{\mathrm{eff}}$, corrupting the zero temperature scaling solution (recall
that the system is out of equilibrium, therefore the thermal symmetry does not
protect temperature from renormalization, cf. Secs. III.2, VIII, XI). This
allows one to study quantitatively how the interplay of local heating and
nonlinearity destroy the two fluctuation-induced regimes, the FIT and FIQZ.
This analysis [207] is analogous to the comparison of the Ginzburg and de
Broglie scales in the context of quantum criticality, discussed in detail in
Sec. XI. Namely, for certain microscopic values of $g$ and $\gamma$ (gray
regions in Fig. 14), the length scale at which thermal fluctuations are
dominant is smaller than the typical length where one can observe the
renormalization of the dissipation strength; this implies that the FIT and
FIQZ are not observable in those parameter windows. (In addition, there are
the regions of instability of RG which match the result at the level of
canonical power counting, white in Fig. 14.) A simple physical picture
emerges: Interactions have to be strong ($|g|$ large) to make these two
effects visible in experiment [207, 434]. In these regions, the picture of
Fig. 13 remains qualitatively correct, yet quantitatively altered by higher-
order fluctuations.

A real-space RG calculation of transmission and reflection coefficients,
similarly to the traditional Kane-Fisher problem [411, 625], leads to
conclusions consistent with the momentum-space RG discussed above. The real-
space RG, working directly with fermionic degrees of freedom, has the
advantage of being more suitable for comparison with numerical approaches to
the problem [207, 617].

##  XIII Universality in fermion systems

In the previous section, we have made a first encounter with fermions in the
context of impurity problems, where the nonequilibrium perturbation was
restricted to a single point in space. Here, we continue our discussion of
fermions, but we return to setups of driven open quantum matter as discussed
in earlier sections, where out of equilibrium dynamics occurs everywhere in
space. Quantum mechanics must be expected to play a crucial role, as by the
fermionic exclusion principle there is no simple classical limit. We review
elements of universality in driven open fermion systems, and connect them to
the basic principles that underlie this review: pure vs. mixed states, the
impact of equilibrium vs. nonequilibrium generators of dynamics, and the
presence of strong symmetries.

A key additional ingredient, encountered only tangentially in the form of
vortex defects in Secs. VII.3 and VII.4.3 so far, is topology. In particular,
we will highlight that topology enables universality, in that it leads to
macroscopic behavior that is highly insensitive to the microscopic generator
of dynamics, and robust with respect to the state of the system being pure or
mixed in a sense made precise below.

To this end, we first discuss a scenario to create out-of-equilibrium analogs
of fermionic ground- and finite temperature states, which are equipped with a
nontrivial topology (Sec. XIII.1), and thus give rise to nonequilibrium
analogs of topological insulators and superconductors [276, 493, 130]. This is
achieved by a suitable design of the jump operators. Pure stationary states
can be generated as dark states of Lindblad dynamics. When different sets of
Lindblad operators are put into competition, topological phase transitions
ensue, which can proceed in pure or mixed states. Although a comprehensive
picture of such phase transitions—and more generally, nonequilibrium phase
transitions of fermions—is still outstanding, one particular aspect becomes
clear already: Critical fermions can exist only in pure states, analogous to
thermodynamic equilibrium [508]. We then continue to overview in Sec. XIII.2
that topology (1) ‘beats dynamics’, i.e., it is insensitive to whether
generators of dynamics describe in- or out-of-equilibrium situations, and that
it also (2) ‘beats mixedness’, i.e., topological quantization persists in
suitable mixed state observables. In both cases, a universal topological gauge
theory emerges, which describes physically observable quantized (albeit
generally nonlinear) responses, and entails a bulk-boundary correspondence. We
close with a discussion of fundamental discrete symmetry operations such as
time reversal and particle-hole transformations, pointing out a dynamical fine
structure in their implementation, depending exclusively on whether the
generator of dynamics obeys the thermal symmetry of Sec. III.2 or not (Sec.
XIII.3).

###  XIII.1 Topological phase transitions of fermions out of equilibrium: pure
vs. mixed states

The nonequilibrium analog of ground states of Hamiltonians are dark states of
Lindbladians: Both scenarios feature pure states, but are prepared by dynamics
respecting or violating equilibrium conditions, respectively, as per the
discussion in Sec. III. We have encountered instances of dark states already
in Secs. X and XI.2 (see also Appendix D for a more detailed discussion).
These states, represented by a pure density matrix $\hat{\rho}_{D}=\lvert
D\rangle\langle D\rvert$, are annihilated by the Lindbladian in Eq. (1),
$\mathcal{L}\hat{\rho}_{D}=0$, via satisfying the two conditions

$\hat{H}\lvert D\rangle=E\lvert D\rangle\quad\text{and}\quad\hat{L}_{l}\lvert
D\rangle=0\quad\forall l.$ (105)

Here, $l$ is a multi-index, which will be made concrete below; we will
consider the case $\hat{H}=0$ for simplicity. Once such a state is reached,
time evolution stops and the system becomes stationary; when the dark state is
unique and no other stationary solutions exist, the dark state will be reached
from an arbitrary initial (and possibly mixed) state (cf. Appendix D). This
circumstance can be used for state preparation in quantum optics [479] and
many-body physics [166, 585], including for topological states [167]. Below we
will consider a class of Lindbladians featuring a dark state with such
topological properties. Topological phase transitions result from the
competition between Lindbladians that stabilize topologically distinct dark
states. These transitions are nonequilibrium analogs of quantum and classical
topological phase transitions, proceeding in pure and mixed states,
respectively.

####  XIII.1.1 Lindbladians with a topological dark state

For concreteness, we focus here on dark states representing two-banded
topological insulators in one dimension in a translationly invariant setting;
such models have been introduced in Refs. [239, 539]. This scenario can be
generalized to arbitrary dimension [290]. It also comprises topological
superfluids [169, 622, 167, 47, 46, 287], see [46] for a review. Such
topological dark states are stabilized by a Lindbladian Eq. (1) with Lindblad
operators of the form

$\hat{L}_{a,1,i}=\hat{\psi}^{\dagger}_{a,i}\hat{l}_{1,i},\quad\hat{L}_{a,2,i}=\hat{\psi}_{a,i}\hat{l}^{\dagger}_{2,i},\quad\hat{l}_{a,i}=\sum_{b,j}U_{ab,ij}\hat{\psi}_{b,j}.$
(106)

Here, $a,b\in\\{1,2\\}$ label the band index, and $i,j$ are lattice site
indices. The operators $\hat{l}_{a,i}$ are superpositions of the local
annihilation operators $\hat{\psi}_{a,i}$. They are related to the latter by
the unitary, canonical transformation $U$, which encodes the topological
properties, see below. When the transformation $U$ is not exactly local, the
dark state carries real-space entanglement, unlike the state considered in
Sec. V.1. The operators in Eq. (106) uniquely [341] stabilize the Gaussian,
half filled dark state

$\displaystyle\lvert D\rangle=\prod_{i}\hat{l}_{2,i}^{\dagger}\lvert
0\rangle,\quad\hat{L}_{a,1,i}\lvert D\rangle=\hat{L}_{a,2,i}\lvert
D\rangle=0\quad\forall\quad a,i.$ (107)

The annihilation of the dark state by the Lindblad operators happens due to
distinct mechanisms (see Fig. 15(a)): $\hat{L}_{a,1}$ annihilates the dark
state since there are no particles in superposition $\hat{l}^{\dagger}_{1,i}$,
and $\hat{L}_{a,2,i}$ due to Pauli blocking for particles in superposition
$\hat{l}^{\dagger}_{2,i}$. Once the state is annihilated by the
$\hat{l}_{1,i}$ and $\hat{l}^{\dagger}_{2,i}$ pieces in the Lindblad
operators, the evolution stops. If the state is not annihilated, it is
recycled into the evolution until it is annihilated, by the uniqueness of the
dark state and the absence of other stationary solutions in the present
problem.

Figure 15: (a) Action of Lindblad operators: $\hat{L}_{a,1}$ annihilate
particles in the upper band to either recreate them in the lower band or
redistribute them in the upper band. Similarly, $\hat{L}_{a,2}$ create
particles in the lower band by transfer from the upper or redistribution in
the lower band (adapted from Ref. [571]). (b) Winding number and purity gap
closing: Pure states are characterized by
$|\mathbf{d}_{q}|=1\Leftrightarrow\Gamma_{q}^{2}=\mathbbm{1}$ for all $q$.
Then, $\mathbf{d}_{q}$ lies on a grand circle of the Bloch sphere in the
presence of a chiral symmetry
($\mathbf{s}\perp\mathbf{d}_{q}\Leftrightarrow\\{\Sigma,\Gamma_{q}\\}=0$ for
all $q$, red line). For mixed states, $|\mathbf{d}_{q}|<1$ (blue line). The
closing of a purity gap then corresponds to a zero in $|\mathbf{d}_{q}|$ for
at least one $q$ (black straight line). At such a point, the geometry of an
ellipse, for which a winding number is well defined, collapses to a line. No
winding can then be defined, which signals a topological phase transition.
(c),(d) Dissipative ($\Delta_{d}$), noise ($\Delta_{n}$), and purity gaps
($\Delta_{p}$) for topological quantum (c) and classical (d) phase
transitions. $\Delta_{d}$ and $\Delta_{n}$ are measured in units of
$\bar{\gamma}$, $\Delta_{p}$ is dimensionless. Discussion, see text.

The jump operators Eq. (106) conserve the total particle number,
$[\hat{L}_{a,1/2,i},\hat{N}]=0$, as it should be for operators that stabilize
an insulating state (there is a strong $\mathrm{U}_{q}(1)$ symmetry in the
associated field theory). Insisting on this property makes the Lindbladian
necessarily quartic in the field operators, and thus interacting. An
alternative version with Lindblad operators $\hat{L}_{a,1,i}=\hat{l}_{1,i}$
and $\hat{L}_{a,2,i}=\hat{l}^{\dagger}_{2,i}$ stabilizes the same dark state,
but does not conserve particle number, although it still exhibits a weak
$\mathrm{U}_{c}(1)$ symmetry.

The number nonconserving operators can also be viewed as a mean-field
approximation to the full interacting Lindblad dynamics, valid at late times
when the dynamics can be linearized around the exactly known dark state. Mean-
field theory can be implemented within the operator formalism [46] or in the
functional integral representation [571]. The latter lends itself to
extracting beyond mean-field physics, see Sec. XIII.2 below. In a functional
integral representation, mean-field theory has the status of a one-loop self-
consistent Born approximation [16]. It results in a simple replacement of the
overall decay rate $\gamma\to\bar{\gamma}=2n\gamma$, where $n=1/2$ for half
filling as in Eq. (107). The effective mean-field Gaussian action with
$l=(l_{1},l_{2})^{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\intercal$}}{\raisebox{0.0pt}{$\textstyle\intercal$}}{\raisebox{0.0pt}{$\scriptstyle\intercal$}}{\raisebox{0.0pt}{$\scriptscriptstyle\intercal$}}}}$
and in the frequency-momentum domain then reads

$\begin{gathered}S=\int_{\omega,q}\left(l_{c}^{*{\mathchoice{\raisebox{0.0pt}{$\displaystyle\intercal$}}{\raisebox{0.0pt}{$\textstyle\intercal$}}{\raisebox{0.0pt}{$\scriptstyle\intercal$}}{\raisebox{0.0pt}{$\scriptscriptstyle\intercal$}}}},l_{q}^{*{\mathchoice{\raisebox{0.0pt}{$\displaystyle\intercal$}}{\raisebox{0.0pt}{$\textstyle\intercal$}}{\raisebox{0.0pt}{$\scriptstyle\intercal$}}{\raisebox{0.0pt}{$\scriptscriptstyle\intercal$}}}}\right)\left(\begin{array}[]{cc}0&P^{A}(\omega,q)\\\
P^{R}(\omega,q)&P^{K}(q)\end{array}\right)\left(\begin{array}[]{c}l_{c}\\\
l_{q}\end{array}\right),\\\
P^{R}(\omega,q)=\left(\omega+i\bar{\gamma}_{q}\right)\mathbbm{1},\quad
P^{K}({q})=2i\bar{\gamma}_{q}\sigma_{z},\end{gathered}$ (108)

with damping rate $\bar{\gamma}_{q}\geq 0$. Due to the weak
$\mathrm{U}_{c}(1)$ symmetry of the number nonconserving mean-field theory,
the action does not contain pairing terms such as $l_{1,c}l_{1,q}$, which are
included in the general Nambu structure of Gaussian actions discussed in
Appendix C. The action Eq. (108) will allow us to discuss the basic properties
of nonequilibrium topological phase transitions. The noise matrix $P^{K}$ is
not positive definite—unlike for bosons, $P^{K}>0$ is not required for
convergence of the fermionic functional integral. The positive eigenvalue of
$P^{K}$ indicates that the modes $l_{1}$ are empty; the negative eigenvalue
indicates that the modes $l_{2}$ are occupied. This results from the different
ordering of $l_{1,q}^{*}l_{1,q}$ vs. $l_{2,q}l_{2,q}^{*}=-l_{2,q}^{*}l_{2,q}$.
Using $l(\omega,q)=U_{q}\psi(\omega,q)$ with
$\psi=(\psi_{1},\psi_{2})^{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\intercal$}}{\raisebox{0.0pt}{$\textstyle\intercal$}}{\raisebox{0.0pt}{$\scriptstyle\intercal$}}{\raisebox{0.0pt}{$\scriptscriptstyle\intercal$}}}}$
in the Fourier representation (cf. Eq. 106), the action can be expressed in
terms of the original lattice fermion fields. In this basis,
$P^{R/A}(\omega,q)$ continue to be unit matrices, but $P^{K}(q)\to
2i\bar{\gamma}U_{q}^{\dagger}\sigma_{z}U_{q}$ becomes a general anti-Hermitian
matrix, with the structure of a free topological Hamiltonian up to the
imaginary prefactor. The topology is thus encoded in the noise matrix in
dissipatively stabilized topological states of fermions.

The above discussion applies in a translationally invariant setting and
discards edge mode effects, which lead to topological degeneracy and higher
dimensional dark spaces. We refer to Refs. [46, 14, 513, 220, 221] for a
discussion of these aspects.

####  XIII.1.2 Phase transitions

Topological phase transitions are defined by an abrupt jump of an integer-
valued topological invariant as a function of parameter changes. We consider
here a simple model with a stationary state of Su-Schrieffer-Heeger type
[560], equipped with a chiral symmetry, and topologically characterized by a
chiral winding number invariant. The invariant can be constructed based on the
covariance matrix collecting the single-particle correlation functions
$\Gamma_{{q}}=i\int\frac{d\omega}{2\pi}G^{K}(\omega,{q})$, where
$G^{K}=-G^{R}P^{K}G^{A}$ can be read off Eq. (108). The winding number is then
defined as

$W=\int_{-\pi}^{\pi}\frac{dq}{2\pi}\mathop{\mathrm{tr}}\\!\left(\Sigma\Gamma^{-1}_{q}i\partial_{q}\Gamma_{q}\right),$
(109)

where the chiral symmetry is expressed by the constraint that there exist a
Hermitian matrix $\Sigma$ such that $\\{\Sigma,\Gamma_{q}\\}=0$ for all $q$.
The winding is illustrated in Fig. 15(b), where we represent the matrices in
terms of vectors according to
$\Gamma_{q}=\mathbf{d}_{q}\cdot\boldsymbol{\sigma},\Sigma=\mathbf{s}\cdot\boldsymbol{\sigma}$
($\boldsymbol{\sigma}$ collects the Pauli matrices). A winding number of
$W=+1$ corresponds to $\mathbf{d}_{q}$ encircling $\mathbf{s}$ once in
counterclockwise direction upon $q=-\pi\to\pi$. For example, a model defined
with quasi-local operators

$\begin{pmatrix}\hat{l}_{1,i}\\\
\hat{l}_{2,i}\end{pmatrix}=\frac{1}{\sqrt{2}}\begin{pmatrix}\hat{\psi}_{1,i+1}+\hat{\psi}_{2,i}\\\
-\hat{\psi}_{1,i+1}+\hat{\psi}_{2,i}\\\ \end{pmatrix},$ (110)

has the covariance matrix $\Gamma_{q}=e^{-iq}\sigma^{+}+e^{iq}\sigma^{-}$,
$\Sigma=\sigma^{z}$, and the winding $W=+1$. The opposite winding $W=-1$
results from operators $\hat{r}_{1,i}$ and $\hat{r}_{2,i}$, obtained upon
exchanging $\hat{\psi}_{1,i+1}\to\hat{\psi}_{1,i}$ and
$\hat{\psi}_{2,i}\to\hat{\psi}_{2,i+1}$ in the above equation.

Topological phase transitions with a jump $W=+1\to-1$ can be generated by
putting the operators $\hat{l}$ and $\hat{r}$ into competition. To understand
the basic properties of these transitions, it is sufficient to study the
linearized model, described by the Gaussian action Eq. (108). The abrupt
change of the topological invariant can occur in two ways: A first option is a
closing of the dissipative gap, which is analogous to the closing of the
energy gap in the ground state of Hamiltonian systems. A second one is the
closing of the noise and purity gaps. The latter case is illustrated in Fig.
15(b), straight black line inside the Bloch sphere. Indeed there are two
qualitatively distinct transitions reflecting these two options, which occur
in pure or mixed states, respectively:

(i) Topological quantum phase transition (in a pure dark state). We set the
operators $\hat{l}$ and $\hat{r}$ into competition by means of a coherent
superposition of operators,

$\hat{x}_{a,i}=\cos(\theta)\hat{l}_{a,i}+\sin(\theta)\hat{r}_{a,i},$ (111)

with the limiting cases $\hat{x}=\hat{l}$ for $\theta=0$ and $\hat{x}=\hat{r}$
for $\theta=\pi/2$. For any value of $\theta\in[0,\pi/2]$, the Lindbladian
with jump operators $\hat{x}$ has a dark state given by $\lvert
D\rangle=\prod_{i}\hat{x}^{\dagger}_{2,i}\lvert 0\rangle$. Upon tuning
$\theta=0\to\pi/2$, a topological phase transition is crossed, where the
winding number jumps from $W=+1\to-1$. The dissipative gap $\Delta_{d}$ and
the noise gap $\Delta_{n}$ (cf. Sec. IV.2), which are determined by
$P^{R}(\omega,q)$ and $P^{K}(q)$ given in Eq. (108) directly in the diagonal
basis, are shown in Fig. 15(c). At the transition at $\theta=\pi/4$, the gaps
close, $\Delta_{d}=\Delta_{n}=0$. The fact that the steady state is a pure
dark state for all values of $\theta$ is reflected in the constant value
$\Delta_{p}=1$ of the purity gap, which is defined in terms of the eigenvalues
$\lambda_{a,q}$ of the covariance matrix,

$\Delta_{p}=\min_{a,q}\left\lvert\lambda_{a,q}\right\rvert.$ (112)

This definition implies that a state with a vanishing purity gap features at
least one fermionic mode in a totally mixed (infinite temperature) state.

(ii) Topological classical phase transition (in a mixed state). Now we
consider a competition by means of an incoherent sum of operators, in terms of
a Lindblad generator

$\mathcal{L}=\cos(\theta)\mathcal{L}_{l}+\sin(\theta)\mathcal{L}_{r},$ (113)

where $\mathcal{L}_{l}$ features Lindblad operators involving $\hat{l}$, and
analogously for $\mathcal{L}_{r}$; again we set $\hat{H}=0$. In this case, the
stationary state is known exactly only in the cases $\theta=0,\pi/2$, where
the steady state is given by the respective dark states of opposite winding
number. In between, the stationary state is mixed. However, the chiral
symmetry is preserved [14], and there must be a topological phase transition.
This behavior is captured by an effective linear model, with a generator of
dynamics which is a weighted direct sum of the $l$ and $r$ models [290]. The
dissipative, noise, and purity gaps are shown in Fig. 15(d). The dissipative
gap is finite for $0\leq\theta\leq\pi/2$. However, both noise and purity gaps
close at $\theta=\pi/4$. At this point, the momentum modes $q=\pm\pi/2$ are in
a fully mixed state. The winding number evaluates to $W=+1$ for
$0\leq\theta<\pi/4$ and to $W=-1$ for $\pi/4<\theta\leq\pi/2$. In these
regions, the purity gap is open. The integrity of a winding number defined in
Eq. (109) is verified straightforwardly under this condition. Indeed,
intuitively, the number of windings of the vector $\mathbf{d}_{q}$ is well-
defined until the circle is deformed to a line or point, which can only happen
once the purity gap closes. The collapse of the purity gap is illustrated in
Fig. 15(b), as already mentioned above. In summary, the transition happens
when the noise and, consequently, the purity gap close, while the dissipative
gap remains open. This yields a topological phase transition without
thermodynamic signatures, such as divergent length or time scales.

For fermions, the topological (i) quantum and (ii) classical phase transitions
are the only types of transitions that can occur: Since the spectrum of
$\Gamma$ is bounded between $-1$ and $1$ by the Pauli principle, there cannot
be a transition where the dissipative gap closes, while the noise gap remains
open. Transitions with divergent length and time scales for the fermion
degrees of freedom are possible only in pure states. Topological phase
transitions are possible in mixed states, but they proceed without
thermodynamic signatures. This is a general consequence of Fermi statistics,
applies in and out of equilibrium, and includes interacting systems. The
absence of critical fermions in mixed states is seen perhaps more
straightforwardly at thermodynamic equilibrium: In a Euclidean functional
integral formulation, the fermionic Matsubara frequencies are characterized by
odd integers, $\omega_{n}=\left(2n+1\right)\pi/\beta$ with $n\in\mathbb{Z}$
[642], and do not allow for zero modes of the inverse fermion Green’s function
at nonzero temperature.

The concept of purity gaps and their closing as enabling topological
transitions is not bound to nonequilibrium problems. However, at thermal
equilibrium, the circumstances under which a purity gap can close are more
restrictive, since all modes experience the same temperature by the principle
of detailed balance: The covariance matrix (for a Gaussian problem for
definiteness) is $\Gamma=\tanh(\beta H)$, with inverse temperature $\beta$ and
(first quantized) Hamiltonian $H$. Zero modes of $\Gamma$, signalling the
purity gap closing, can only occur if there are zero modes of $H$ for finite
temperature, or for infinite temperature $\beta=0$ in the absence of zero
modes of $H$.

We close with a comment on the nature of the pure state phase transitions in
the present models. Due to the specific choice of Gaussian dark states, all
static correlation functions in stationary state can be computed exactly,
including at the critical point. The corresponding static exponents are thus
Gaussian. However, since the generator of dynamics is interacting, this does
not imply that the dynamic exponent is Gaussian as well (one example where a
non-Gaussian dynamic and Gaussian static exponents are realized is provided by
the KPZ equation in one dimension, cf. [566, 311]). A general analysis of out-
of-equilibrium fermion criticality is outstanding.

###  XIII.2 Universality of topological response out of equilibrium

In equilibrium ground states, topology is encoded in the ground-state wave
function, i.e., in the state of the system. A fundamental question is then,
how topological structures generalize once the realm of ground states is left.
This may concern (i) the breaking of equilibrium conditions while preserving
the pure state nature analogously to a ground state, and (ii) giving up
purity, in- and out of equilibrium.

An overarching framework to address this question is topological field theory
[635, 381, 338, 492]. Based on the interplay of topology and gauge structures,
effective theories provide a bridge between microphysics and observable
macrophysics [635, 381, 495, 504]. Where gauge principles exist, they show a
high level of robustness, including in the presence of interactions [504] or
translational symmetry breaking [13]. Indeed, general adiabatic gauge
principles have been identified for nonequilibrium Lindblad dynamics in a
series of seminal works [32, 34, 33, 8]. Here, we specialize to the case of
$\mathrm{U}(1)$ gauge theory, which also allows one to directly study the
physical consequences of the gauge structure in many-body systems.

####  XIII.2.1 Universality of topological response: equilibrium vs.
nonequilibrium dynamics

We consider the nonequilibrium analog of a Chern insulator in two dimensions,
described by a wave function $\lvert D\rangle$. As above, this scenario is
realized by choosing Lindblad operators such that $\lvert D\rangle$ is their
dark state. Then the stationary state $\lvert D\rangle$ is equivalent to the
equilibrium ground state of a suitably chosen Hamiltonian, while the dynamics
of the system differs from the Hamiltonian evolution. More precisely, we
require the following: $\lvert D\rangle$ should

(i) be unique, such that $\hat{\rho}_{D}=\lvert D\rangle\langle D\rvert$ is a
pure state;

(ii) be stable, so that local perturbations to the steady state relax at a
finite minimal rate, i.e., there is a dissipative (many-body) gap;

(iii) result as the stationary state of a particle number conserving dynamics,
implemented by the presence of a strong $\mathrm{U}_{q}(1)$ symmetry.

This implements the conditions required by Laughlin’s gauge argument [358]:
the threading of a quantum Hall annulus by a time varying magnetic flux can be
adiabatic only if the bulk state is nondegenerate and has a (many-body)
spectral gap. Then, the insertion of flux quanta will lead to the transfer of
an integer number of charges from one edge to the other, provided these
charges cannot be lost (e.g., to a bath).

The above requirements are met by a choice of Lindblad operators of the same
structure as considered in the preceding section (working in the spatial
continuum here, and with overall damping rate $\gamma$):
$\hat{L}_{a,1}=\hat{\psi}_{a}\hat{l}_{1}$ and
$\hat{L}_{a,2}=\hat{\psi}_{a}\hat{l}^{\dagger}_{2}$, where in momentum space
($\hat{l}=(\hat{l}_{1},\hat{l}_{2})^{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\intercal$}}{\raisebox{0.0pt}{$\textstyle\intercal$}}{\raisebox{0.0pt}{$\scriptstyle\intercal$}}{\raisebox{0.0pt}{$\scriptscriptstyle\intercal$}}}}$)

$\hat{l}_{\mathbf{q}}=V_{\mathbf{q}}\hat{\psi}_{\mathbf{q}},\quad
V_{\mathbf{q}}=q_{x}\mathbbm{1}+iq_{y}\sigma_{z}+im\sigma_{y}\equiv
d^{1/2}_{\mathbf{q}}U_{\mathbf{q}}.$ (114)

The transformation $V$ is local in real space,
$V_{\mathbf{x}}=-i\partial_{x}\mathbbm{1}-\partial_{y}\sigma_{z}+im\sigma_{y}$,
such that the Lindblad operators $\hat{L}$ share this property, and
$d_{\mathbf{q}}=q^{2}+m^{2}$ with $q=\left\lvert\mathbf{q}\right\rvert$ is a
normalization factor introduced here to ensure that $U_{\mathbf{q}}$ is
unitary. The dark state targeted by these Lindblad operators coincides with
the ground state of the Hamiltonian
$\hat{H}=\int_{\mathbf{q}}\hat{\psi}^{\dagger}_{\mathbf{q}}\mathbf{d}_{\mathbf{q}}\cdot\boldsymbol{\sigma}\hat{\psi}_{\mathbf{q}}=\int_{\mathbf{q}}\hat{l}^{\dagger}_{\mathbf{q}}\sigma_{z}\hat{l}_{\mathbf{q}}$,
with
$\mathbf{d}_{\mathbf{q}}=(2mq_{x},2mq_{y},-m^{2}+q^{2})^{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\intercal$}}{\raisebox{0.0pt}{$\textstyle\intercal$}}{\raisebox{0.0pt}{$\scriptstyle\intercal$}}{\raisebox{0.0pt}{$\scriptscriptstyle\intercal$}}}}$
such that $d_{\mathbf{q}}=\left\lvert\mathbf{d}_{\mathbf{q}}\right\rvert$.
Similar to the discussion in Sec. XIII.1, the winding of the map
$\mathbf{q}\mapsto\mathbf{d}_{\mathbf{q}}$ defines a Chern number $\theta=-1$
for any $m\neq 0$.

A gauge field is introduced by minimal coupling in the field theory framework:
This is enabled by the strong $\mathrm{U}_{q}(1)$ symmetry associated with
particle number conservation, which translates to independent transformations
$\psi_{\pm}(t,\mathbf{x})\mapsto e^{i\theta_{\pm}}\psi_{\pm}(t,\mathbf{x})$ on
each contour. Promoting global to local gauge invariance under transformations
$\theta_{\pm}(t,\mathbf{x})$ then requires us to introduce gauge fields
appearing in covariant derivatives
$\partial_{\mu}\to\partial_{\mu}-iA_{\pm,\mu}$, where $\mu\in\\{t,x,y\\}$. Up
to terms that are topologically trivial, this replaces the Lindblad operators
by, e.g.,

$L_{1,1}\to
L_{1,1}+\psi^{\dagger}_{1}\left(a_{i}l\right)_{1}A_{\pm,i}+\dotsb,\quad
a_{i}=\left(i\partial_{q_{i}}U_{\mathbf{q}}\right)U_{\mathbf{q}}^{\dagger},$
(115)

where $a_{i}$ ($i=x,y$) is a Berry connection represented in momentum space on
the right, encoding the topology of the problem. Indeed, in a mean-field
theory for the fermions as described above, followed by a gradient expansion
of the fermionic Gaussian functional integral to lowest order, a Chern-Simons
theory on the Keldysh contour emerges,

$\displaystyle S[A_{\pm}]$
$\displaystyle=\frac{\theta}{4\pi}\int_{t,\mathbf{x}}\left[\epsilon^{\mu\nu\rho}\left(A_{+,\mu+}\partial_{\nu}A_{+,\rho}-A_{-,\mu}\partial_{\nu}A_{-,\rho}\right)+\dotsb\right],$
$\displaystyle\theta$
$\displaystyle=-\frac{\pi}{2}\int_{\mathbf{q}}\mathop{\mathrm{tr}}\\!\left(\sigma_{z}\left(\partial_{q_{x}}a_{y}-\partial_{q_{y}}a_{x}\right)\right)=-1,$
(116)

where $\epsilon^{\mu\nu\rho}$ is the Levi-Civita symbol. Here,
$\partial_{q_{x}}a_{y}-\partial_{q_{y}}a_{x}$ is the Berry curvature
associated with $a_{i}$. Only the strong $\mathrm{U}_{q}(1)$ symmetry allows
for the existence of two independent contour fields $A_{\pm}$, underlining the
importance of particle number conservation. The prefactor of the effective
action is the Chern number, describing a quantized topological response on top
of a purely dissipative bulk. With the topological gauge theory at hand, the
bulk-boundary correspondence follows from standard arguments [559, 209]. It
entails the perhaps counterintuitive results that reversible, underdamped
chiral modes exist in a dark subspace at an edge of a bulk undergoing purely
irreversible, dissipative dynamics.

The robustness of this result—i.e., the universality of the topological
response, irrespective to the nonequilibrium nature of the generator of
dynamics—relies on the dissipative gap. Here, the dissipative gap for single
fermion excitations is finite, and evaluates to $2\gamma nm$ within mean-field
theory. The problem also has a gap for local particle-hole excitations, since
the particle numbers of upper and lower band are not separately conserved
[571, 449, 389].

####  XIII.2.2 Universality of topological response: pure vs. mixed states

We now give up the constraint on the purity of the state, and study to what
extent topological structures persist in mixed stationary quantum states,
which occur as dynamical fixed points of equilibrium or nonequilibrium
dynamics. Based on the understanding of the dissipative gap as the counterpart
of the energy gap in closed systems, and the universality of the topological
response for pure states, we now choose a symmetry-based approach to the
problem. An adiabatic long-wavelength topological $\mathrm{U}(1)$ gauge action
can be constructed in any dimension based on the following prerequisites
[290], generalizing those of the previous section:

(i) The dynamics converge to a form dubbed Dirac stationary state:
$\hat{\rho}_{s}\sim e^{-\hat{G}}$, where $\hat{G}$ represents a
(dimensionless) Dirac operator in second quantized formulation. For example,
in equilibrium, $\hat{G}=\beta\hat{H}$, with $\hat{H}=\int
d\mathbf{x}\sum_{a,b}\hat{\psi}^{\dagger}_{a}(\mathbf{x})H_{ab}\hat{\psi}_{b}(\mathbf{x})$.
The matrix $H$ is defined as

$H=\mathbf{d}\cdot\boldsymbol{\alpha},\text{ with
}\mathbf{d}=(-i\partial_{x_{1}},\dots,-i\partial_{x_{2}},\dots,-i\partial_{x_{d}},m),$
(117)

and the matrices $\alpha^{i}$ satisfy the Clifford algebra
$\\{\alpha^{i},\alpha^{j}\\}=2\delta^{ij}$. This form is generic near phase
transitions of weakly correlated systems including systems with dark states
[290], and general enough to cover the universal topological properties of
symmetry protected quantum matter [505, 131, 15]. We can thus work with the
parameterization $\hat{G}=\beta\hat{H}$ both in and out of equilibrium.

(ii) There exist a fast microscopic time scale (finite spectral gap, realized
as an energy gap in equilibrium and as a dissipative gap in Lindblad dynamics)
and a finite purity gap.

(iii) The charge conservation of the underlying dynamics ensures the existence
of a $\mathrm{U}(1)$ real-time response theory.

(iv) The charge quantization ensures large gauge invariance [178].

As above, gauge fields are then introduced by minimal coupling. In Ref. [290],
an effective gauge action is derived in odd space-time dimension from
combining the above ingredients with the Atiyah-Singer index theorem for Dirac
operators. The result is a mixed-state generalization of Chern-Simons
theories. The even-dimensional cases are constructed via the bulk-boundary
correspondence resulting in mixed-state $\theta$-terms. We focus here on the
example of $2+1$ dimensions (see also [161, 177] for a calculation in the
Euclidean equilibrium functional integral for this special case),

$\begin{split}S[A]&=\frac{\theta}{\pi}\,\int
d\mathbf{x}\,\mathcal{I}\\!\left(\int
dt\,A_{q,t}\right)\epsilon^{ij}\partial_{i}A_{c,j},\\\
\mathcal{I}(a)&=2\arctan\\!\left(\tanh\\!\left(\beta\left\lvert
m\right\rvert\\!/2\right)\tan\\!\left(a/2\right)\right).\end{split}$ (118)

The classical and quantum components of the gauge field in this Keldysh gauge
action are defined as $A_{c}=(A_{+}+A_{-})/2$ and $A_{q}=A_{+}-A_{-}$.
Further, $\tanh(\beta\left\lvert m\right\rvert\\!/2)$ is the purity gap of the
stationary density matrix parameterized by the effective Dirac Hamiltonian Eq.
(117), and $\theta$ is the Chern number.333For Dirac models, the actual
calculation yields a half-integer Chern number, but this can be cured by a
suitable UV renormalization. An example for such regularization is the term
$\sim q^{2}$ in $\mathbf{d}_{\mathbf{q}}$ of Eq. (114). A further detail is
that Eq. (118) is incomplete, unless $A_{q,t}(t,\mathbf{x})=A_{q,t}(t)$ only,
cf. Ref. [290] for the full action. In the pure state limit $\beta\to\infty$,
the function $\mathcal{I}$ becomes the identity, $\mathcal{I}(a)\to a$. The
action Eq. (118) then coincides with the one in Eq. (XIII.2.1), if a
projection of $A_{q,t}(t,\mathbf{x})$ onto the zero-frequency limit is
performed in the latter ($A_{q,t}(t,\mathbf{x})\to\int
dt\,A_{q,t}(t,\mathbf{x})/T$, $T=\int dt$)—which is sufficient to reproduce
the adiabatic physical response. Conversely, for mixed states described by
finite $\beta$, the action still has a large gauge invariance: under a
transformation $A_{q,t}(t,\mathbf{x})\to A_{q,t}(t,\mathbf{x})+2\pi n/T$ with
integer $n$ in Eq. (118), the action transforms as $S\to S+2\pi n\,\theta$.
Thus, $e^{iS}$ remains invariant, taking the integer quantization of the Chern
number $\theta$ into account. This is how topology is still encoded for mixed
states. The large gauge invariance physically reflects the quantization of
particle number [571, 290]. This rationalizes the finding of a quantized
nonlinear (in $A_{q,t}$) response, described by Eq. (118): particle number
quantization persists including for mixed states. We note that the linear
response, obtained by expanding $\mathcal{I}$ to first order in $A_{q,t}$,
ceases to be quantized for finite $\beta<\infty$, in accordance with
expectation [178, 598]. Nevertheless, there are interferometric physical
observables witnessing topological quantization in mixed states, such as the
ensemble geometric phase [48]. In the framework of the above topological gauge
theory, they are represented as nonlinear responses.

Finally, we note that Eq. (118) features an equilibrium symmetry, which is
emergent (cf. Sec. III.3). This is rationalized by the simple dependence of
the mixed state gauge action on the purity gap, induced by a gap of $\hat{G}$,
alone.

###  XIII.3 Symmetry classification of open fermion matter

We finally leave the ground of topology, and turn to symmetry. Specifically,
we ask: How universal is the paradigmatic symmetry classification of fermion
matter [17], when the realm of thermal equilibrium is left? Here we point out
a dynamical fine structure for the fundamental symmetry operations, which
universally distinguishes equilibrium and nonequilibrium generators of
dynamics.

For equilibrium ground states, Altland and Zirnbauer have provided a tenfold
symmetry classification of free Hamiltonians, in terms of the basic discrete
symmetry operations that are time reversal, particle-hole transformation, and
their concatenation, the chiral transformation [17]. This symmetry
classification has been harnessed for a topological classification of free
fermion systems [523, 327, 505].

The classified object in this case is the first quantized, Hermitian
Hamiltonian matrix. This setting has been generalized to non-Hermitian
matrices, leading to a 38-fold symmetry classification based on the absence of
Hermiticity [63, 318, 637]. However, this does not yet incorporate fundamental
constraints of physical evolution such as probability conservation, complete
positivity, particle statistics, and the equilibrium vs. nonequilibrium nature
of the generator of dynamics. A first step in this direction was taken in Ref.
[377] considering single-particle fermion systems, pointing at a reduction
from 38 classes down to 10\. In Ref. [15], starting from the Fock space
representation of interacting open fermion dynamics and requiring the
invariance of the equation of motion for the density matrix, it was
demonstrated that in fact there is a fine-structure relating to whether the
generator of dynamics obeys detailed balance or not. In either case, there are
10 classes. The classes are distinct from each other in the cases where time-
reversal is involved, due to a different implementation of this transformation
in the time domain in either case (the full quantum mechanical time reversal
transformation consists of a Fock space transformation, and an operation on
the time parameter). This concerns 7 out of 10 classes. The path of keeping
the first quantized approach working with matrices—which can be done on the
basis of exponentially large matrices—but building in physical constraints was
followed in Refs. [506, 317]. These works point at a further enrichment of the
10 classes upon incorporating the distinction between weak and strong
symmetries.

These results are not at variance with the universal topological response in-
and out of equilibrium discussed above: In that case, there is no time
reversal symmetry involved. Whether and how more generally topology can level
out the dynamical fine structure of discrete symmetries is an intriguing open
question for future research.

##  XIV Perspectives

We have reviewed universal physics in driven open quantum matter, which
witnesses the breaking of detailed balance on the microscopic level in
macroscopic observables. Significant advances have been made in both
experiment and theory. Yet, the field presents a number of challenges and
opportunities for future research. In addition, there are intriguing links to
other active areas of physics. This concerns all of the major directions
reported in this work.

###  XIV.1 Realizations of paradigmatic nonequilibrium universality

Above, we have focused on distilling universal long-wavelength phenomena
starting from a given microscopic physics. This procedure has deepened our
understanding of the microscopic ingredients required to obtain a desired
macroscopic behavior. The time seems right to invert the logic, and move down
in wavelength again, to identify new experimental platforms for the
exploration of nonequilibrium universality. To give a concrete example: The
challenge of observing KPZ physics in two spatial dimensions is largely open.
As we have seen, the basic ingredients for macroscopic KPZ behavior in driven
open quantum matter are minimal and natural. Required are a broken phase
rotation symmetry, absence of particle number conservation, and the breaking
of detailed balance. Such a situation could be realized in various solid state
or cold atom setups. For example, in superfluid films sandwiched between
particle reservoirs at different chemical potentials [77, 342]. More
generally, by a suitable engineering of “nonequilibrium simulators”—controlled
setups for the quantitative study of paradigmatic phenomena—novel platforms
for nonequilibrium universality could be created.

A case in point are NISQ platforms. These are operated under conditions of
drive and dissipation, and could be geared into nonequilibrium simulators. For
example, Kibble-Zurek scaling was observed on a Rydberg quantum simulator
[320], directed percolation has been implemented in a trapped ion quantum
computer [129], and 1D KPZ physics was reported on a noisy coupled transmon
device [319]. An arena for realizing the phenomena of epidemic spreading and
self-organized criticality has been established with ultracold Rydberg atoms
[282, 615, 170, 75].

There is also great potential for condensed matter platforms. These realize
open systems due to the simultaneous presence of electronic, spin, and
phononic degrees of freedom, with vastly different characteristic time scales.
Generically, such systems relax quickly to thermal equilibrium due to
scattering processes, which overwrite an external weak drive by lasers or
terahertz radiation. However, there might be a generic way out of this
scenario: For soft modes, a weak breaking of equilibrium conditions can be
sufficient to induce strong nonequilibrium universal effects. One prominent
example is again KPZ physics in low dimensional systems, which ensues at large
scales for an infinitesimal violation of detailed balance. But there are more
examples, like the activation of Goldstone mode limit cycles by an arbitrarily
weak driving of an equilibrium ferrimagnet [633]. An intriguing general
question emerges from these examples: Under which conditions are equilibrium
RG fixed points destabilized and left in favor of nonequilibrium fixed
points—of known, but also novel kind? The answer would pave the way toward
robust nonequilibrium universality induced by a weak drive only, as required
for solid state systems.

More broadly, searching for such scenarios in a targeted way may create an
intriguing link to the physics of active matter [395, 494]: In fact, the
overarching trait of such systems is a breaking of detailed balance in an
extensive way, i.e., at every point in space (as opposed to more familiar
solid state nonequilibrium settings, such as voltage biases at the
boundaries). This is precisely the setting discussed throughout this
review—many-body Lindbladians describe active quantum matter.

###  XIV.2 Novel nonequilibrium universality

The quest for new universal structures keeps inspiring the field. Such
structures may be found in stationary state, but also in the dynamics of
driven open quantum systems.

##### Non-Hermitian physics and universality near exceptional points.

One promising direction for finding universality in stationary states is
provided by the field of non-Hermitian physics (see also Sec. IV.3.1). A key
phenomenon are exceptional points, giving rise to degeneracies in the complex
excitation spectra of generators of dynamics, where reversible and
irreversible processes appear on an equal footing [30, 62]. Exceptional points
have fueled a stream of research in condensed matter, atomic condensates, and
optics: On the one hand, they hold promises for applications, such as sensing
due to an enhanced response to external perturbations in their vicinity [285,
128, 475, 30]. At the same time, on the conceptual side, such points host
novel topological phenomena, such as nodal topological phases with open Fermi
surfaces, or an anomalous bulk-boundary correspondence [634, 638, 606, 621,
281, 123, 62].

Much of the focus has so far been devoted to effective single-particle physics
in the absence of noise. This can be a good approximation deep in stable
phases of matter. Yet recently, important connections to the field of
nonlinear dynamics [144] and active matter [208, 610] have been made.
Transitions of first and second order have been identified on the mean-field
level. The time seems mature to add the final layer of complexity, namely to
include the interplay of nonlinearities and of noise fluctuations. As has been
emphasized throughout this review, this becomes indispensable once gapless
modes are present, or when a system is brought close to a critical point with
long-wavelength fluctuations. Recently, first such studies have appeared,
addressing the coalescence of an exceptional- with a critical point [271,
633]. Such points host giant fluctuations, which ultimately render the phase
transition weakly first order [633]. This finding holds universally for
critical exceptional points in the class of nonequilibrium $\mathrm{O}(N)$
models, and is protected by Goldstone’s theorem in this case. Yet, other
realizations of critical exceptionality are conceivable, for example, in
magnonic systems [599]. A systematic analysis of their universal properties
presents a challenge for future research.

##### Phases with limit cycles.

The physics of limit cycles, or, more generally, spatiotemporal pattern
formation in nonlinear dynamics [144], stands in between stationary states and
nonequilibrium time evolution. Recently, the related effect of time crystals
in closed quantum systems has received a lot of attention [324, 629]. These
phenomena are intimately connected to the driven nature of the system. They
can be rationalized as a transmutation of scales: Time-periodic drive on the
microscopic scale can surface as periodicity in time on the macroscopic level.
This remains true even if at an intermediate scale of modelling (e.g., by a
Lindblad equation), the system appears to be invariant under continuous
translations in time. A paradigmatic example is provided by the van der Pol
oscillator, originally realized in electrical circuits with active nonlinear
elements [472]. Limit cycles have recently been discussed for systems in the
quantum realm [596, 597, 294, 181, 88, 85, 569, 87, 56, 84]. First experiments
in cavity quantum electrodynamics realize this phenomenology [171, 175], and
might be scaled up to include an extensive number of spatial degrees of
freedom in the future [636, 425, 333]. A comprehensive understanding of the
phase transitions in such systems, which in part pass through exceptional
points [208, 610, 633], remains yet to be established. Conceptually, this
touches upon the breaking of time translation symmetry present on the level of
the Lindblad equation, and the nature of the associated Goldstone modes.

##### Dynamics of driven open quantum systems.

The identification of universal regimes in the time evolution of driven open
quantum matter has so far received relatively little attention, and has mainly
focused on one-dimensional atomic systems. An example concerns the heating
dynamics of atoms in optical lattices subject to phase noise from the lattice
laser [467, 474, 473, 102, 90, 101, 89, 330]. Here particle number is
conserved, rationalizing the existence of algebraically slow and universal
scaling regimes. Another example is provided by the critical behavior of
topological pumping phenomena at dynamical phase transitions [515, 555, 556].

Beyond ultracold atomic systems, motivation for addressing dynamics comes from
pump-probe experiments in solid state physics [52, 626, 574, 573, 380, 54,
414]: A system is photo-excited beyond the linear response regime, such that
its occupation number distribution becomes highly nonthermal, and then relaxes
back to its equilibrium state. As mentioned above, electronic, spin, and
phonon degrees of freedom relax on very different time scales, which enables a
description in terms of driven and open subsystems. These subsystem degrees of
freedom can experience various sources of universal behavior: Conservation
laws due to the presence of strong symmetries, gapless modes via the
spontaneous breakdown of weak symmetries, and (topological) defects [646].

The impact of conservation laws on nonequilibrium time evolution has been
studied extensively in the context of closed systems. They give rise to quasi-
stationary turbulent behavior in terms of the energy or particle number
cascade [205]. There are even more exotic phenomena, such as nonthermal fixed
points: Here, a dynamically evolving system gets trapped for a long time
before eventually reaching a thermal state, and time itself acts as a scaling
variable [60, 187, 489, 496]. While driven open systems violate the
conservation of energy, other quantities such as the number of particles can
be conserved, making room for possibly modified yet universal dynamical
scenarios.

Gapless modes due to the spontaneous breaking of weak symmetries are likewise
expected to strongly influence dynamics. Promising first steps have been taken
in Refs. [173, 645, 351], studying the recovery dynamics in pump-probe
settings, where an order parameter is coupled to a gapless mode continuum.
When the occupation number distribution function has not yet reached its
stationary value, there is a potential for universal physics beyond linear
response. Indeed, these works point out novel dynamical scaling laws due to
the bottleneck to equilibration provided by Goldstone modes.

Finally, defects—surfaces on which the order parameter vanishes—can also give
rise to slow dynamics. This has been explored extensively in the field of
phase ordering kinetics [78]. Defects can be present when the relevant
symmetry group is small enough: For example, for the group $\mathrm{O}(N)$,
the defect core defines a surface of dimension $d-N$, such as domain walls in
the Ising model ($N=1$) in arbitrary dimensions, or vortices and vortex lines
in the XY model ($N=2$) in $d=2$ and $d=3$. Such defects often have a long-
ranged shape. Therefore, they affect the long-distance equilibration dynamics
even when statistical fluctuations are neglected. So far, the focus has been
mainly on generators of dynamics which are compatible with thermodynamic
equilibrium. A novel ingredient specific to driven systems connects to the
above mentioned limit cycles: Out of equilibrium, defects with explicit time
dependence can be activated, such as Archimedean screws or other structures in
complex magnets [535, 536].

Disentangling the interplay of these distinct sources of dynamical
universality theoretically, and exploring them in experiment, is a challenging
and rewarding direction of research.

###  XIV.3 Nonequilibrium quantum phenomena

##### Nonequilibrium fermion criticality.

Fermionic systems do not possess a simple classical limit, and potentially
host novel instances of nonequilibrium universality in the quantum regime.
While we have discussed weak forms of fermionic nonequilibrium universality
here—such as the robustness of topological phenomena out of equilibrium—a
scenario of nonequilibrium fermionic criticality is still outstanding. Its key
ingredients are visible though—as discussed in Sec. XIII.1, fermionic
criticality is confined to pure states both in and out of equilibrium. One
route to preserve the pure state critical scaling of a fermionic Green’s
function is then to address the short-time dynamics after a quench, when the
fermion distribution is in a prethermal regime, giving rise to a new fixed
point with critical fermions [306]. In stationary states, a case in point are
the topological transitions proceeding in pure states far from equilibrium
[46, 14]. The infrared enhancement at exceptional points in interacting
systems [527, 143] tuned to criticality mentioned in Sec. XIV.2) above might
give rise to yet more exotic forms of fermion criticality without equilibrium
counterpart.

##### Scaling near pure states.

Other grounds to search for quantum effects out of equilibrium are provided by
scaling phenomena near pure states, beyond the fermionic systems described
above. One example is the quest for quantum generalizations of the directed
percolation universality class discussed in Sec. V.4 (see also below for a
different angle of approach), where there is a pure absorbing state on one
side of the phase transition; efforts concentrate on designing models where
coherence effects are nonnegligible near the critical point [117, 227, 118,
308, 94]. Another instance is the decay of a system toward the vacuum. When
the decay is nonlinear (e.g., two- or three-body losses), such decay dynamics
proceed algebraically slowly. This scenario was explored in the context of
classical reaction-diffusion systems, revealing non-Gaussian scaling behavior
despite the steady state being void of particles [113, 114]. Recently, first
studies of quantum mechanical systems described by Lindblad equations with
decay appeared, both in the reaction-limited (weak losses) [288, 465] and
diffusion limited (strong losses, Zeno regime) [501, 500, 499, 224] regimes,
in part showing structurally different scaling behavior from their classical
counterparts.

More broadly, we point out that once the ground of thermodynamic equilibrium
is left, the borders between “quantum” and “classical” necessarily get
blurred. The reason is the absence of a universal distribution function
imposed by thermal symmetry, and thus of fixed scaling regimes associated with
ground ($T=0$) and mixed ($T>0$) states for bosons or fermions, respectively.
In fact, the freedom of shaping nonequilibrium distribution functions could
enable new scaling regimes. At the same time, this discussion suggests a less
strained focus on such borders.

##### Monitored quantum systems.

A recent frontier in the study of universal nonequilibrium phenomena involves
monitored quantum systems [371, 372, 550]. In this context, _monitoring_
refers to a scenario where quantum measurements are not merely employed to
gather information from a system, but also to dynamically evolve the quantum
mechanical wave function over time. The measurement projection introduces an
irreversible, nonunitary evolution, potentially competing with the unitary
elements generated by a Hamiltonian or unitary gates. In this framework,
measurements can be harnessed both to build quantum correlations and
entanglement among different subsystems, or to dismantle them by extracting
local information. Regardless, the measurement-induced evolution yields
individual, nonunitarily evolved, and pure quantum states, representing
physical realizations of quantum trajectories.

A unique and universal phenomenon in monitored quantum systems are
measurement-induced phase transitions (MIPTs). These occur in ensembles of
pure state wave functions exposed to combined unitary- and measurement
dynamics in the thermodynamic limit, and result from the competition resp.
noncommutativity of distinct quantum operations. MIPTs surface, for example,
in a characteristic change of the growth of the entanglement entropy in the
system. In this sense, they are analogous to quantum phase transitions. But
there is also a major difference to the latter: Due to the randomness of the
recorded measurement outcomes, trajectory averages which are linear in the
quantum state projector—and thus any standard quantum mechanical
observable—behave as if evaluated in an infinite temperature state. Such
averages thus do not carry physical information on the trajectory ensemble any
more. A way out is presented by state-dependent observables, which are
nonlinear in the quantum state, such as the entanglement entropy.

MIPTs appear to be generic in one-dimensional monitored quantum circuits
composed of unitary gates and Pauli measurements [550, 375, 132, 259]; while
some results in two and more dimensions draw a similar picture [360, 546].
Very roughly, these are transitions from an information scrambling phase in
which trajectories feature (sub-)extensive entanglement scaling, e.g., a
volume law, into a quantum Zeno phase, where trajectories obey an area law.
The phases are connected by a second order phase transition with an emergent
conformal invariance [628]. In the limit of large onsite Hilbert space, where
each lattice site hosts a “qudit”, the circuit dynamics has been mapped to
statistical mechanics models that undergo a percolation transition [436, 43,
304, 609]. The numerical results for circuits composed of local qubits are,
however, partially at variance with this prediction [627] both for Haar random
circuits and Clifford circuits. The detailed understanding of the phase
transition therefore remains a topic of active research. Another class of
measurement-induced transitions occurs in monitored systems, for which the
unitary part of the dynamics is generated by a Hamiltonian. Particular
attention has been paid to free fermion systems. Depending on their
symmetries, numerical simulations predict critical weak-measurement regimes
where the entanglement entropy grows $S\sim L^{d-1}\ln(L)$ with system size
$L$ in $d$ dimensions [10, 578, 192, 44, 124, 470, 110], separated from
strong-measurement regimes featuring area law scaling $S\sim L^{d-1}$.
Analytical mappings predict a variety of different universality classes,
ranging from localization [329, 470, 124] to weak localization [471], weak
anti-localization [192] and a KT scenario [93]. The analytical approaches
indicate an intriguing connection between monitored quantum systems in $d$
dimensions and Anderson localization in $d+1$-dimensional systems, i.e., a
potential link between nonequilibrium quantum systems in $d$ dimensions and
the equilibrium _quantum_ statistical mechanics in $d+1$ dimension—a new
perspective that yet needs to be explored.

This scenario opens up a number of challenges for the physics of driven open
quantum systems. First, the proper description of trajectory resolved
monitored systems requires the development of new theoretical tools: The
measurement process is nonlinear in the state, and in addition requires
replica approaches able to produce the relevant trajectory averages. In
addition to the statistical mechanics mappings mentioned above, first
systematic field theory approaches have been constructed to this end [93, 44,
50], and exactly solvable models are emerging [438, 624, 419, 329]. These
represent promising steps toward a better understanding of the universality
classes of monitored quantum systems. A second challenge is to make these
transitions visible in experiment. For Clifford circuits, exploiting their
classical simulability, both postselection [332] and feedback [446] reveal a
transition in experiment. For monitored fermions, a preselection strategy has
been proposed recently [94]. It pulls the phase transition to the observable
level by performing active feedback chosen such that the universal properties
of the underlying measurement-induced transition are left unmodified. This has
revealed intriguing connections to quantum absorbing state transitions [94,
450, 547, 292, 468], beyond the quantum directed percolation scenario touched
upon above. This sparks the hope that novel classes of such transitions
between pure and mixed states with distinct entanglement properties can be
constructed. There are also striking parallels of postselection strategies to
reveal measurement-induced transitions in experiments, and the physics of
large deviations usually considered in the classical context [217, 216], which
remain to be explored. Another route for detecting measurement-induced
transitions in experiments is based on employing cross correlations, e.g., the
linear cross entropy, between the monitored system and a classically simulable
counterpart [374, 218, 260]. Finally, the insight that measurement-induced
transitions can be associated with the maximum threshold at which error
correction schemes fail [132, 373], might prove useful in transferring
concepts from the (field) theory of driven open quantum systems to quantum
information, and vice versa.

More broadly, real world monitored quantum systems present a fresh perspective
on driven open quantum systems. They encompass both reversible dynamics,
driven by Hamiltonian evolution and unitary gates, and irreversible dynamics,
involving measurements, dissipative environments, as well as combinations of
reversible and irreversible operations through feedforward mechanisms. This
inclusive framework covers all fundamental aspects of quantum mechanical
evolution. The theoretical description of such systems faces new challenges,
and Keldysh field theory is one promising tool to address them. At the same
time, this new arena unlocks possibilities for realizing unconventional forms
of universal behavior in driven open quantum matter outside of previously
anticipated paradigms.

###### Acknowledgements.

We would like to acknowledge interactions and collaborations with numerous
colleagues and students who have generously shared insights that appear in
this review. We would particularly like to thank our collaborators on the
lines of research reviewed here, O. Alberton, A. Altland, E. Altman, C.-E.
Bardyn, J. C. Budich, K. Chahine, A. Chiocchetta, E. Dalla Torre, R. Daviet,
O. Diessel, P. Dolgirev, R. Fazio, M. Fleischhauer, H. Fröml, A. Gambassi, M.
Gievers, L. He, M. Heyl, H. Hosseinabadi, D. Hsieh, Z.-M. Huang, K. Klocke, C.
Kollath, B. Ladewig, J. Lang, I. Lesanovsky, M. Lukin, M. Marcuzzi, S. Mathey,
Y. Minoguchi, Th. Müller, P. Rabl, A. Rosch, D. Roscher, S. Sachdev, M.
Scherer, P. Strack, X. Sun, M. H. Szymańska, A. Silva, U. C. Täuber, J. Toner,
F. Tonielli, G. Wachtel, S. Whitlock, C. Zelle and P. Zoller. We thank E.
Santos for help with the figures. L.S. acknowledges support from the Austrian
Science Fund (FWF) through the project P 33741-N. M.B. and S.D. acknowledge
support from the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) under Germany’s Excellence Strategy Cluster of Excellence Matter
and Light for Quantum Computing (ML4Q) EXC 2004/1 390534769, and by the DFG
Collaborative Research Center (CRC) 183 Project No. 277101999 - project B02,
and S.D. additionally by the DFG Collaborative Research Center (CRC) 1238
Project No. 277146847 - projects C03, C04. J.M. acknowledges support by the
Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through
Project-ID 429529648, TRR 306 QuCoLiMa (“Quantum Cooperativity of Light and
Matter”), through the grant HADEQUAM-MA7003/3-1, and by the Dynamics and
Topology Centre funded by the State of Rhineland-Palatinate.

##  Appendix A Derivation of the Lindblad-Keldysh action

Here we provide a detailed derivation of the Keldysh partition function for a
driven open system, focusing on the example of a single bosonic or fermionic
field mode $\hat{\psi}$. We pay special attention to the subtle issue of
operator ordering. As we argue in the following and have anticipated in Eq.
(7), the proper way to keep track of operator ordering is to introduce a
temporal regularization of the Keldysh action in the form of infinitesimal
time shifts [544]. The origin of the time shifts is understood most
transparently by performing the steps that lead to the master equation (1),
starting from a Hamiltonian system-bath setting, in the Keldysh formalism
[544]. In the following, we first briefly summarize this derivation and
generalize it to fermionic fields. Then, we present a derivation of the
Keldysh partition function starting directly from the master equation (1).

###  A.1 Derivation from a system-bath setting

The Hamiltonian that describes the dynamics of a system coupled to a bath is
of the form $\hat{H}=\hat{H}_{s}+\hat{H}_{sb}+\hat{H}_{b}$. We assume that the
system Hamiltonian $\hat{H}_{s}$ is explicitly time-dependent due to a
classical driving field with frequency $\omega_{0}$, but becomes time-
independent after a transformation to a suitably chosen rotating frame,
generated by an operator $\hat{H}_{0}$. A paradigmatic example is given by a
two-level system undergoing Rabi oscillations. The system-bath coupling
$\hat{H}_{sb}$ and the bath Hamiltonian $\hat{H}_{b}$ are given in Eq. (17),
where the bath modes $\hat{\phi}_{\mu}$ with frequencies $\omega_{\mu}$ are
bosonic when the system operator $\hat{L}$ in $\hat{H}_{sb}$ is bosonic or
even in fermionic operators; in contrast, when $\hat{L}$ is odd in fermionic
operators, the bath modes are fermionic.

We perform a transformation to a rotating frame described by the time-
dependent unitary operator
$\hat{U}=e^{i\left(\hat{H}_{0}+\hat{H}_{b}\right)t}$. By assumption, the
system Hamiltonian becomes time-independent in the rotating frame; the bath
Hamiltonian is removed by the transformation, and the system-bath coupling
becomes

$\hat{H}_{sb}^{\prime}=\sum_{\mu}g_{\mu}\left(\hat{L}^{\dagger}\hat{\phi}_{\mu}e^{-i\left(\omega_{\mu}-\omega_{0}\right)t}+\hat{\phi}_{\mu}^{\dagger}\hat{L}e^{i\left(\omega_{\mu}-\omega_{0}\right)t}\right).$
(119)

We note that a more general form of the system-bath coupling contains also the
terms $\hat{L}\hat{\phi}_{\mu}$ and
$\hat{L}^{\dagger}\hat{\phi}_{\mu}^{\dagger}$. In the rotating frame, such
terms oscillate with frequency $\omega_{\mu}+\omega_{0}$, and are dropped in a
rotating wave approximation.

After bringing the system Hamiltonian as well as $\hat{L}$ and
$\hat{L}^{\dagger}$ to normal ordered form, we can construct the usual Keldysh
functional integral for the system-bath setup without ambiguity [16, 311]. We
introduce an additional sign $\zeta=-1$ for fermionic fields on the backward
branch. This sign results from the representation of a trace in fermionic Fock
space in the basis of coherent states, and allows us to obtain a Keldysh
action that satisfies the usual causality structure described for the action
in Sec. II.2.2 and for the Green’s functions in Appendix B. Integrating out
the bath yields the following contribution to the Keldysh action:

$\Delta
S=-\sum_{\mu}g_{\mu}^{2}\int_{t,t^{\prime}}\left(L^{\dagger}_{+}(t),\zeta_{b}L^{\dagger}_{-}(t)\right)e^{i\omega_{0}\left(t-t^{\prime}\right)}\\\
\times\hat{\sigma}_{z}\begin{pmatrix}G_{\mu}^{++}(t-t^{\prime})&G_{\mu}^{+-}(t-t^{\prime})\\\
G_{\mu}^{-+}(t-t^{\prime})&G_{\mu}^{--}(t-t^{\prime})\end{pmatrix}\hat{\sigma}_{z}\begin{pmatrix}L_{+}(t^{\prime})\\\
\zeta_{b}L_{-}(t^{\prime})\end{pmatrix},$ (120)

where the Green’s functions of the bath are given by

$\begin{split}G_{\mu}^{+-}(t)&=-i\zeta_{b}n_{\zeta_{b}}(\omega_{\mu})e^{-i\omega_{\mu}t},\\\
G_{\mu}^{-+}(t)&=-i\left(1+\zeta_{b}n_{\zeta_{b}}(\omega_{\mu})\right)e^{-i\omega_{\mu}t},\\\
G_{\mu}^{++}(t)&=\theta(-t)G_{\mu}^{+-}(t)+\theta(t)G_{\mu}^{-+}(t),\\\
G_{\mu}^{--}(t)&=\theta(t)G_{\mu}^{+-}(t)+\theta(-t)G_{\mu}^{-+}(t).\end{split}$
(121)

In the above equations, the sign $\zeta_{b}=\pm 1$ refers to the bath:
$\zeta_{b}=+1$ for a bosonic bath, and $\zeta_{b}=-1$ for a fermionic bath.
Accordingly, the distribution function is
$n_{\zeta_{b}}(\omega)=\left.1\middle/\left(e^{\beta\omega}-\zeta_{b}\right)\right.$.
A possible additional sign is contained in the field variables $L_{\pm}$ and
$L_{\pm}^{\dagger}$. For example, for $\hat{L}=\hat{\psi}$, we obtain
$L_{+}=\psi_{+}$, $L_{-}=\zeta\psi_{-}$, and $L^{\dagger}_{+}=\psi_{+}^{*}$,
$L^{\dagger}_{-}=\zeta\psi_{-}^{*}$.

We assume that the bath remains in thermal equilibrium at all times—this is
essentially the Born approximation, and relies on weak system-bath coupling.
Further, we wish to perform the Markov approximation, assuming that the
correlation time of the bath is short on typical time scales of evolution of
the system, such that we can replace $L_{\pm}(t^{\prime})\to L_{\pm}(t)$ in
Eq. (120). However, we have to keep track of the temporal order of
$L_{\pm}^{\dagger}(t)$ and $L_{\pm}(t^{\prime})$ that is encoded in the
Heaviside functions in Eq. (121) as can be seen by introducing a new
integration variable $\tau=t-t^{\prime}$ and rewriting Eq. (120) as

$\Delta
S=-\int_{t,\tau}\left[L_{+}^{\dagger}(t)\left(\theta(-\tau)\Gamma^{+}(\tau)+\theta(\tau)\Gamma^{-}(\tau)\right)L_{+}(t-\tau)\right.\\\
-L_{-}(t-\tau)\Gamma^{+}(\tau)L_{+}^{\dagger}(t)-L_{+}(t-\tau)\Gamma^{-}(\tau)L_{-}^{\dagger}(t)\\\
\left.+L_{-}^{\dagger}(t)\left(\theta(\tau)\Gamma^{+}(\tau)+\theta(-\tau)\Gamma^{-}(\tau)\right)L_{-}(t-\tau)\right],$
(122)

where

$\begin{pmatrix}\Gamma^{+}(\tau)\\\
\Gamma^{-}(\tau)\end{pmatrix}=\sum_{\mu}g_{\mu}^{2}e^{i\omega_{0}\tau}\begin{pmatrix}G_{\mu}^{+-}(\tau)\\\
G_{\mu}^{-+}(\tau)\end{pmatrix}.$ (123)

In the terms that are mixed in the branch index, we have exchanged the order
of $L_{\pm}^{\dagger}(t)$ and $L_{\pm}(t-\tau)$. If the jump operator is odd
in fermionic operators, this leads to an additional minus sign, which cancels
$\zeta_{b}$ in Eq. (120).

Next, we consider the continuum limit of dense bath frequencies, centered
around $\omega_{b}\approx\omega_{0}$ with a large bandwidth $\vartheta$. We
can then replace the sum over bath modes $\mu$ by an integral,

$\begin{pmatrix}\Gamma^{+}(\tau)\\\
\Gamma^{-}(\tau)\end{pmatrix}=-i\int_{0}^{\infty}d\omega\,\nu(\omega)g(\omega)^{2}e^{i\left(\omega_{0}-\omega\right)\tau}\begin{pmatrix}\zeta_{b}n_{\zeta_{b}}(\omega)\\\
1+\zeta_{b}n_{\zeta_{b}}(\omega)\end{pmatrix}.$ (124)

where $\nu(\omega)$ is the density of states of the bath. The key assumption
underlying the Markov approximation is that the product
$\nu(\omega)g(\omega)^{2}$ is a smoothly varying function of $\omega$ in the
vicinity of $\omega_{0}$. This implies that the bath correlation functions
$\Gamma^{\pm}(\tau)$ appearing in Eq. (122) are sharply peaked around
$\tau=0$. Therefore, keeping track of temporal order, we make the replacement
$\theta(\pm\tau)L_{\pm}(t-\tau)\to\theta(\pm\tau)L_{\pm}(t_{\mp})$ where
$t_{\pm}\to t\pm 0^{+}$. We can now perform the integration over $\tau$ using

$\begin{split}\int_{-\infty}^{\infty}d\tau\,e^{i\omega\tau}\theta(\pm\tau)=\pi\delta(\omega)\pm
i\mathcal{P}\frac{1}{\omega},\end{split}$ (125)

where $\mathcal{P}$ denotes the principal value. This leads to

$\begin{split}\int_{-\infty}^{\infty}d\tau\begin{pmatrix}\Gamma^{+}(\tau)\\\
\Gamma^{-}(\tau)\end{pmatrix}&=-i\gamma\begin{pmatrix}\zeta_{b}n_{\zeta_{b}}(\omega_{0})\\\
1+\zeta_{b}n_{\zeta_{b}}(\omega_{0})\end{pmatrix},\\\
\int_{-\infty}^{\infty}d\tau\,\theta(\pm\tau)\begin{pmatrix}\Gamma^{+}(\tau)\\\
\Gamma^{-}(\tau)\end{pmatrix}&=-i\frac{\gamma}{2}\begin{pmatrix}\zeta_{b}n_{\zeta_{b}}(\omega_{0})\\\
1+\zeta_{b}n_{\zeta_{b}}(\omega_{0})\end{pmatrix}\mp\begin{pmatrix}\Delta_{+}(\omega_{0})\\\
\Delta_{-}(\omega_{0})\end{pmatrix},\end{split}$ (126)

where $\gamma=2\pi\nu(\omega_{0})g(\omega_{0})^{2}$, and

$\begin{pmatrix}\Delta_{+}(\omega_{0})\\\
\Delta_{-}(\omega_{0})\end{pmatrix}=\mathcal{P}\int_{0}^{\infty}d\omega\,\frac{\nu(\omega)g(\omega)^{2}}{\omega_{0}-\omega}\begin{pmatrix}\zeta_{b}n_{\zeta_{b}}(\omega)\\\
1+\zeta_{b}n_{\zeta_{b}}(\omega)\end{pmatrix}.$ (127)

We thus find that $\Delta S=\Delta S_{d}+\Delta S_{L}$ is the sum of a
dissipative contribution and a Lamb shift, which are given by

$\Delta
S_{d}=-i\gamma\int_{t}\left\\{\left(1+\zeta_{b}n_{\zeta_{b}}(\omega_{0})\right)\vphantom{\frac{1}{2}}\right.\\\
\times\left[L_{+}(t)L_{-}^{\dagger}(t)-\frac{1}{2}\left(L_{+}^{\dagger}(t)L_{+}(t_{-})+L_{-}^{\dagger}(t)L_{-}(t_{+})\right)\right]\\\
\left.+n_{\zeta_{b}}(\omega_{0})\left[L_{+}^{\dagger}(t)L_{-}(t)-\frac{1}{2}\left(L_{+}(t_{+})L_{+}^{\dagger}(t)+L_{-}(t_{-})L_{-}^{\dagger}(t)\right)\right]\right\\},$
(128)

and

$\Delta
S_{L}=-\int_{t}\left[\Delta_{+}(\omega_{0})\left(L_{+}^{\dagger}(t)L_{+}(t_{+})-L_{-}^{\dagger}(t)L_{-}(t_{-})\right)\right.\\\
\left.-\Delta_{-}(\omega_{0})\left(L_{+}^{\dagger}(t)L_{+}(t_{-})-L_{-}^{\dagger}(t)L_{-}(t_{+})\right)\right].$
(129)

For a bath at zero temperature, we can set $n_{\zeta_{b}}(\omega_{0})=0$.
Then, after absorbing $\sqrt{\gamma}$ in a redefinition of $L_{\pm}$, Eq.
(128) yields Eq. (7). The Lamb shift can be included in the Hamiltonian
contribution to the action.

###  A.2 Derivation from the quantum master equation

The above derivation of the Keldysh action for a driven open system shows that
infinitesimal time shifts occur naturally in a Hamiltonian system-bath
setting. Moreover, above we have required normal order only of $\hat{H}$,
$\hat{L}$, and $\hat{L}^{\dagger}$, but not of the product
$\hat{L}^{\dagger}\hat{L}$ that also appears in the master equation (1). These
insights serve as a guideline for a direct derivation of the Keldysh action
from the master equation [541], which we generalize here to fermionic systems.

A key ingredient in the construction of the functional integral representation
are coherent states. For bosons, a coherent state with amplitude
$\psi\in\mathbb{C}$ is defined as
$\lvert\psi\rangle=e^{\psi\hat{\psi}^{\dagger}}\lvert 0\rangle$, where the
vacuum state $\lvert 0\rangle$ obeys $\hat{\psi}\lvert 0\rangle=0$. Defining
fermionic coherent states, on the other hand, requires us to employ Grassmann
numbers that form an anticommutative algebra. To unify the presentation for
bosons and fermions, we will denote also Grassmann numbers with $\psi$. Then,
a fermionic coherent state is given by
$\lvert\psi\rangle=e^{-\psi\hat{\psi}^{\dagger}}\lvert 0\rangle$. Both bosonic
and fermionic coherent states have the following properties: They are
eigenstates of the annihilation operator,
$\hat{\psi}\lvert\psi\rangle=\psi\lvert\psi\rangle$; the overlap of two
coherent states is
$\langle\psi|\psi^{\prime}\rangle=e^{\psi^{*}\psi^{\prime}}$; and coherent
states span the bosonic or fermionic Fock space, as expressed by the
completeness relation

$\hat{1}=\int\frac{d\psi^{*}d\psi}{\pi^{(1+\zeta)/2}}\,e^{-\psi^{*}\psi}\lvert\psi\rangle\langle\psi\rvert.$
(130)

However, there is a fundamental difference between bosons and fermions: For
bosons, $\psi^{*}$ denotes the complex conjugate of $\psi\in\mathbb{C}$. In
contrast, the notation $\psi^{*}$ for a Grassmann number $\psi$ is merely
symbolic. Indeed, for fermions, $\psi$ and $\psi^{*}$ are strictly independent
variables [604].

As anticipated in Sec. II.2, the starting point for the construction of the
functional integral representation is the formal solution of the master
equation,
$\hat{\rho}(t)=e^{\mathcal{L}\left(t-t_{0}\right)}\hat{\rho}(t_{0})$. We split
the evolution from $t_{0}$ to $t$ into $N$ discrete steps of duration $\Delta
t=\left(t-t_{0}\right)/N$,

$\hat{\rho}(t)=e^{\mathcal{L}\left(t-t_{0}\right)}\hat{\rho}(t_{0})=\left(e^{\mathcal{L}\Delta
t}\right)^{N}\hat{\rho}(t_{0})\approx\left(\mathcal{I}+\mathcal{L}\Delta
t\right)^{N}\hat{\rho}(t_{0}),$ (131)

where $\mathcal{I}$ is the identity superoperator. We denote the density
matrix at time $t_{n}=t_{0}+n\Delta t$ by $\hat{\rho}_{n}=\hat{\rho}(t_{n})$.
The evolution during one time step is thus given by

$\hat{\rho}_{n+1}\approx\left(\mathcal{I}+\mathcal{L}\Delta
t\right)\hat{\rho}_{n}.$ (132)

To keep track of the order of operators, we further split each time step into
two, leading to a total of $2N$ time steps. Details of this splitting are
given below. We construct a field integral representation of the time
evolution by inserting resolutions of the identity in terms of coherent states
Eq. (130) in between consecutive time steps and both to the left and to the
right of the density matrix. In particular, for the density matrix
$\hat{\rho}_{n}$ at time $t_{n}$, which is reached after $2n$ time steps, we
write

$\hat{\rho}_{n}=\int\frac{d\psi_{+,2n}^{*}d\psi_{+,2n}}{\pi^{(1+\zeta)/2}}\frac{d\psi_{-,2n}^{*}d\psi_{-,2n}}{\pi^{(1+\zeta)/2}}e^{-\psi_{+,2n}^{*}\psi_{+,2n}-\psi_{-,2n}^{*}\psi_{-,2n}}\\\
\times\langle\psi_{+,2n}|\hat{\rho}_{n}|\zeta\psi_{-,2n}\rangle\lvert\psi_{+,2n}\rangle\langle\zeta\psi_{-,2n}\rvert.$
(133)

As explained in Sec. A.1, we have to include a sign $\zeta=-1$ for Grassmann
variables on the backward branch.

To obtain a representation of the full time evolution, we have to relate the
matrix element
$\langle\psi_{+,2n+2}|\hat{\rho}_{n+1}|\zeta\psi_{-,2n+2}\rangle$, appearing
in the coherent state representation of $\hat{\rho}_{n+1}$, to the
corresponding matrix element of $\hat{\rho}_{n}$. This can be done by
inserting Eq. (133) in Eq. (132), leading us to consider the matrix element of
the superoperator $\mathcal{I}+\mathcal{L}\Delta t$ appearing in Eq. (132),
$\langle\psi_{+,2n+2}|\left(\mathcal{I}+\mathcal{L}\Delta
t\right)\\!\left(\lvert\psi_{+,2n}\rangle\langle\zeta\psi_{-,2n}\rvert\right)|\zeta\psi_{-,2n+2}\rangle$,
which contains, in particular, the terms
$\langle\psi_{+,2n+2}|\hat{L}^{\dagger}\hat{L}|\psi_{+,2n}\rangle$ and
$\langle\zeta\psi_{-,2n}|\hat{L}^{\dagger}\hat{L}|\zeta\psi_{-,2n+2}\rangle$.
When $\hat{L}$ and $\hat{L}^{\dagger}$ are normal ordered, their product
$\hat{L}^{\dagger}\hat{L}$ is in general not normal ordered. Therefore, we
insert additional resolutions of the identity in terms of
$\lvert\psi_{+,2n+1}\rangle$ and $\lvert\zeta\psi_{-,2n+1}\rangle$ between
$\hat{L}^{\dagger}$ and $\hat{L}$. In other matrix elements such as
$\langle\psi_{+,2n+2}|\hat{H}|\psi_{+,2n}\rangle$ we also insert these
additional resolutions of the identity. The position, e.g., before or after
$\hat{H}$, is arbitrary, and we make the choice of applying the Hamiltonian
$\hat{H}$ and the jump operator $\hat{L}$ on the forward and
$\hat{L}^{\dagger}$ on backward branch between steps $2n$ and $2n+1$. We can
then replace the matrix elements of operators by functions of the coherent
state amplitudes, both for the Hamiltonian,
$H(\psi^{*},\psi^{\prime})=\langle\psi|\hat{H}|\psi^{\prime}\rangle/\langle\psi|\psi^{\prime}\rangle$,
and the jump operator,
$L(\psi^{*},\psi^{\prime})=\langle\psi|\hat{L}|\psi^{\prime}\rangle/\langle\psi|\psi^{\prime}\rangle$,
and its Hermitian conjugate,
$L^{\dagger}(\psi^{*},\psi^{\prime})=\langle\psi|\hat{L}^{\dagger}|\psi^{\prime}\rangle/\langle\psi|\psi^{\prime}\rangle$.
Specifically, following our convention for inserting resolutions of the
identity, we obtain the following matrix elements:

$\displaystyle H_{+,n}$ $\displaystyle=H(\psi_{+,2n+1}^{*},\psi_{+,2n}),$
$\displaystyle H_{-,n}$
$\displaystyle=H(\zeta\psi_{-,2n}^{*},\zeta\psi_{-,2n+1}),$ (134)
$\displaystyle L_{+,n}$ $\displaystyle=L(\psi_{+,2n+1}^{*},\psi_{+,2n}),$
$\displaystyle L_{-,n}$
$\displaystyle=L(\zeta\psi_{-,2n+1}^{*},\zeta\psi_{-,2n+2}),$ $\displaystyle
L^{\dagger}_{+,n}$
$\displaystyle=L^{\dagger}(\psi_{+,2n+2}^{*},\psi_{+,2n+1}),$ $\displaystyle
L^{\dagger}_{-,n}$
$\displaystyle=L^{\dagger}(\zeta\psi_{-,2n}^{*},\zeta\psi_{-,2n+1}),$

in terms of which we can write the Lindbladian as

$\mathcal{L}_{+-,n}=-i\left(H_{+,n}-H_{-,n}\right)\\\
+L_{+,n}L^{\dagger}_{-,n}-\frac{1}{2}\left(L^{\dagger}_{+,n}L_{+,n}+L^{\dagger}_{-,n}L_{-,n}\right).$
(135)

We thus find the desired relation between the matrix elements of the density
matrix at times $t_{n+1}$ and $t_{n}$:

$\langle\psi_{+,2n+2}|\hat{\rho}_{n+1}|\zeta\psi_{-,2n+2}\rangle\approx\int\frac{d\psi_{+,2n+1}^{*}d\psi_{+,2n+1}}{\pi^{(1+\zeta)/2}}\\\
\times\frac{d\psi_{-,2n+1}^{*}d\psi_{-,2n+1}}{\pi^{(1+\zeta)/2}}\frac{d\psi_{+,2n}^{*}d\psi_{+,2n}}{\pi^{(1+\zeta)/2}}\frac{d\psi_{-,2n}^{*}d\psi_{-,2n}}{\pi^{(1+\zeta)/2}}\\\
\times
e^{\left(\psi_{+,2n+2}^{*}-\psi_{+,2n+1}^{*}\right)\psi_{+,2n+1}+\left(\psi_{+,2n+1}^{*}-\psi_{+,2n}^{*}\right)\psi_{+,2n}+\psi_{-,2n}^{*}\left(\psi_{-,2n+1}-\psi_{-,2n}\right)}\\\
\times
e^{\psi_{-,2n+1}^{*}\left(\psi_{-,2n+2}-\psi_{-,2n+1}\right)+\mathcal{L}_{+-,n}\Delta
t}\langle\psi_{+,2n}|\hat{\rho}_{n}|\zeta\psi_{-,2n}\rangle.$ (136)