# Universality in driven open quantum matter

Lukas M. Sieberer  Institute for Theoretical Physics, University of Innsbruck,
6020 Innsbruck, Austria  Michael Buchhold  Institute for Theoretical Physics,
University of Cologne, Zülpicher Straße 77, 50937 Cologne, Germany  Jamir
Marino  Institute for Physics, Johannes Gutenberg University of Mainz, 55099
Mainz, Germany  Sebastian Diehl  Institute for Theoretical Physics, University
of Cologne, Zülpicher Straße 77, 50937 Cologne, Germany

(August 28, 2024)

###### Abstract

Universality is a powerful concept, which enables making qualitative and
quantitative predictions in systems with extensively many degrees of freedom.
It finds realizations in almost all branches of physics, including in the
realm of nonequilibrium systems. Our focus here is on its manifestations
within a specific class of nonequilibrium stationary states: driven open
quantum matter. Progress in this field is fueled by a number of uprising
platforms ranging from light-driven quantum materials over synthetic quantum
systems like cold atomic gases to the functional devices of the noisy
intermediate scale quantum era. These systems share in common that, on the
microscopic scale, they obey the laws of quantum mechanics, while detailed
balance underlying thermodynamic equilibrium is broken due to the simultaneous
presence of Hamiltonian unitary dynamics and nonunitary drive and dissipation.
The challenge is then to connect this microscopic physics to macroscopic
observables, and to identify universal collective phenomena that uniquely
witness the breaking of equilibrium conditions, thus having no equilibrium
counterparts. In the framework of a Lindblad-Keldysh field theory, we discuss
on the one hand the principles delimiting thermodynamic equilibrium from
driven open stationary states, and on the other hand show how unifying
concepts such as symmetries, the purity of states, and scaling arguments are
implemented. We then present instances of universal behavior structured into
three classes: new realizations of paradigmatic nonequilibrium phenomena,
including a survey of first experimental realizations; novel instances of
nonequilibrium universality found in these systems made of quantum
ingredients; and genuinely quantum phenomena out of equilibrium, including in
fermionic systems. We also discuss perspectives for future research on driven
open quantum matter.

###### Contents

  1. I Introduction
    1. I.1 Universality
    2. I.2 Driven Open Quantum Matter
    3. I.3 Synopsis
  2. II Description of driven open systems: from Lindblad to Keldysh
    1. II.1 Driven open Bose-Einstein condensation
    2. II.2 Keldysh field theory for driven open systems
      1. II.2.1 Keldysh partition function and Lindblad-Keldysh action
      2. II.2.2 Keldysh rotation: classical and quantum fields
      3. II.2.3 Retarded, advanced, and Keldysh Green’s functions
      4. II.2.4 From a single to many modes
  3. III Equilibrium vs. nonequilibrium stationary states
    1. III.1 Thermal equilibrium and the fluctuation-dissipation theorem
    2. III.2 Thermal equilibrium as a symmetry of the Keldysh action
    3. III.3 Nonequilibrium and emergent equilibrium
  4. IV Overarching principles in- and out of equilibrium
    1. IV.1 Classical and quantum symmetries of the Keldysh action: symmetry breaking vs. conservation laws
      1. IV.1.1 Classical and quantum symmetries
      2. IV.1.2 Equivalence to weak and strong symmetries
      3. IV.1.3 Quantum symmetries, conservation laws, and slow hydrodynamic modes
      4. IV.1.4 Spontaneous breaking of classical symmetries and dissipative Goldstone theorem
    2. IV.2 Mixed vs. pure states, classical vs. quantum scaling
      1. IV.2.1 Spectral gap and criticality
      2. IV.2.2 Noise gap: classical and quantum scaling
    3. IV.3 Scaling arguments: deterministic and semiclassical limit
      1. IV.3.1 Deterministic limit and relation to non-Hermitian physics
      2. IV.3.2 Canonical power counting and semiclassical limit
  5. V Absorbing state phase transitions and directed percolation
    1. V.1 Rydberg atom lattice in the facilitation regime
    2. V.2 Field theory for directed percolation
    3. V.3 Universality and symmetries
    4. V.4 The quest for quantum directed percolation
  6. VI Self-organized criticality and Rydberg experiments
    1. VI.1 Field theory for self-organized criticality
    2. VI.2 Self-organized criticality in Rydberg atom ensembles
    3. VI.3 Alternative sources of scale invariance without fine-tuning in cold atom dynamics
  7. VII Driven open condensates in low spatial dimensions
    1. VII.1 Long-wavelength theory of driven open condensates
      1. VII.1.1 Incoherent pumping scheme and mapping to the compact isotropic KPZ equation
      2. VII.1.2 Coherent pumping scheme and mapping to the compact anisotropic KPZ equation
    2. VII.2 KPZ universality in the absence of topological defects
      1. VII.2.1 RG flow of the KPZ equation in 1D, 2D, and 3D
      2. VII.2.2 RG flow of the anisotropic KPZ equation in 2D
    3. VII.3 Vortex unbinding in two-dimensional driven open condensates
      1. VII.3.1 Electrodynamic duality
      2. VII.3.2 Noise-activated vortex unbinding in isotropic systems
      3. VII.3.3 Stabilization of the ordered phase through strong spatial anisotropy
      4. VII.3.4 Observability of KPZ universality in two-dimensional driven open condensates
    4. VII.4 One-dimensional driven open condensates
      1. VII.4.1 KPZ universality in one-dimensional driven open condensates
      2. VII.4.2 Destruction of KPZ scaling through noise activated space-time vortices
      3. VII.4.3 Transition to space-time vortex turbulence
  8. VIII Driven open criticality
    1. VIII.1 Bosonic driven open criticality
    2. VIII.2 Driven-dissipative criticality with competing order parameters
  9. IX Slowly and rapidly driven open systems
    1. IX.1 Slowly driven systems: activating critical exponents
      1. IX.1.1 Kibble-Zurek mechanism
      2. IX.1.2 Exact RG approach to slowly driven systems
      3. IX.1.3 Observable consequences
    2. IX.2 Rapidly driven systems: open Floquet criticality
      1. IX.2.1 Basic physical picture
      2. IX.2.2 Keldysh-Floquet model
      3. IX.2.3 Renormalization of the open Floquet problem
      4. IX.2.4 Relation to the slowly driven limit
  10. X Nonequilibrium first order phase transitions
    1. X.1 General phenomenology of first order phase transitions
    2. X.2 First order phase transitions out of equilibrium
    3. X.3 Dark state bistability
    4. X.4 Field theory of first order dark state phase transitions
    5. X.5 Renormalization group flow
  11. XI Quantum criticality in driven open systems
    1. XI.1 Quantum critical scaling of noise-driven Luttinger liquids
    2. XI.2 Quantum criticality of the 1D driven open Bose gas
      1. XI.2.1 Implementation of diffusion noise
      2. XI.2.2 Nonequilibrium quantum scaling
      3. XI.2.3 Properties of the nonequilibrium quantum fixed point
      4. XI.2.4 Comparison with noise-driven Luttinger Liquids
  12. XII Universality in dissipative quantum impurities
    1. XII.1 Solvable models of dissipative impurities and the quantum Zeno effect
    2. XII.2 Local dissipation in interacting quantum wires:  the dissipative Kane-Fisher problem
      1. XII.2.1 Qualitative overview: Canonical power counting
      2. XII.2.2 Keldysh field theory of the dissipative Kane-Fisher problem
  13. XIII Universality in fermion systems
    1. XIII.1 Topological phase transitions of fermions out of equilibrium: pure vs. mixed states
      1. XIII.1.1 Lindbladians with a topological dark state
      2. XIII.1.2 Phase transitions
    2. XIII.2 Universality of topological response out of equilibrium
      1. XIII.2.1 Universality of topological response: equilibrium vs. nonequilibrium dynamics
      2. XIII.2.2 Universality of topological response: pure vs. mixed states
    3. XIII.3 Symmetry classification of open fermion matter
  14. XIV Perspectives
    1. XIV.1 Realizations of paradigmatic nonequilibrium universality
    2. XIV.2 Novel nonequilibrium universality
    3. XIV.3 Nonequilibrium quantum phenomena
  15. A Derivation of the Lindblad-Keldysh action
    1. A.1 Derivation from a system-bath setting
    2. A.2 Derivation from the quantum master equation
    3. A.3 Temporal order and causality
  16. B Green’s functions in the Keldysh formalism
    1. B.1 From operators to fields
    2. B.2 Equal time arguments and Hermiticity
    3. B.3 Causality structure
  17. C Gaussian states and Gaussian actions
  18. D Dark states of Lindblad dynamics

##  I Introduction

Universality underlies the possibility to perform the transition from micro-
to macrophysics quantitatively by systematically discarding irrelevant
information. Historically, the concept of universality has guided and enabled
progress in many-particle physics in thermodynamic equilibrium. But it is
operative more generally in Nature, in systems with an extensive number of
degrees of freedom. This work surveys expressions of universality in driven
open quantum matter. Systems that belong to this class are defined by the
appearance of coherent and driven-dissipative dynamics on an equal footing,
placing them far from thermodynamic equilibrium, even if they reach a
stationary state.

###  I.1 Universality

The most dramatic expression of universality occurs close to the critical
point of a second order phase transition: There, the loss of memory about the
microscopic physics is so strong, that the long-wavelength physics is fully
determined by the dimensionality, symmetries and conservation laws, as well as
the range of interactions. A system with extensively many degrees of freedom
is then characterized by few universal critical exponents defining the
universality class. Only a handful of such universality classes are found in
Nature, despite the plethora of materials and tailor-made platforms that make
up the world around us [237, 642, 286].

Universality, however, manifests in a variety of forms. In its weakest form,
universality may be viewed as the fact that Nature is organized
hierarchically. Focusing on universality more quantitatively, a suitable
starting point is a model of a system which applies on length and time scales
that can be considered as microscopic. The goal is then to bridge to the
macroscopic physics emerging on scales that are a few (but not many) decades
larger. In that setting, universality often results in cases of degeneracies,
giving rise to gapless phases of matter that can exist without fine-tuning to
a critical point across an extended parameter range. Such situations are
characterized by large-scale fluctuations dominating the macroscopic behavior.
Mechanisms enabling degeneracies are manifold. One case in point is the
spontaneous breakdown of continuous global symmetries, leading to gapless
Goldstone modes. Examples include phonons in solids or phase modes in atomic
Bose-Einstein condensates. Additionally, soft modes can occur in situations
where symmetry breaking is prohibited due to low dimensionality. Universal
effects are even enhanced due to the restricted available phase space, as seen
in the Kosterlitz-Thouless critical phase in two dimensions. Another class of
stable gapless modes appears in the form of hydrodynamics, associated with
global conservation laws. Examples include particle and heat diffusion in
number and energy-conserving systems, as well as the response of topological
insulators.

All the examples mentioned above are consistent with thermodynamic equilibrium
conditions, but their fundamental characteristics are not limited to this
context. Both criticality and weaker realizations of universal behavior have
been established in nonequilibrium statistical mechanics. A paradigmatic
example of a nonequilibrium critical point occurs in directed percolation, a
variant of the percolation problem that violates detailed balance [520, 251,
248, 115, 297, 284, 451]. Universal nonequilibrium phases facilitated by
robust gapless modes are exemplified by the Kardar-Parisi-Zhang equation [316,
343, 269, 561]. It was introduced to describe the roughening of driven
interfaces observed in phenomena like the spreading of fire fronts [412] or
the growth of bacterial colonies [12]. The phenomenon of self-organized
criticality stands in between these cornerstones: A system self-tunes to
criticality, and due to this mechanism exhibits stable universal behavior in
extended parameter regimes [39, 38, 37, 453, 176, 256, 601, 29, 488]

###  I.2 Driven Open Quantum Matter

This article focuses on universality in driven open quantum matter,
representing a novel class of nonequilibrium many-body systems. The
distinguishing feature of systems in this class is the simultaneous presence
of coherent quantum dynamics, external drive, and dissipation. This
circumstance is typically realized when matter is strongly coupled to external
light fields, like lasers. The ability to drive thermodynamically large
systems with such fields without compromising their intricate many-body
behavior due to noise and heating has emerged relatively recently. Yet it
already includes a broad spectrum of platforms for atoms, light, and solids.
Among these are exciton-polariton systems in semiconductor heterostructures
[120], atoms [425] or solids [291] immersed into cavities, photonic platforms
and microcavity arrays [275, 447, 448, 456], circuit quantum electrodynamics
[593, 66], ultracold atoms and ions [435, 273], Rydberg gases [83, 509, 429],
and light-induced superconductors [424, 122]. Driven open quantum matter also
encompasses functionalized matter, such as noisy intermediate-scale quantum
(NISQ) [481] devices of superconducting circuits [514] and Rydberg tweezer
arrays [533]. The need to conceptualize these systems as instances of driven
open quantum matter is clear; this need also starts to get recognized more
widely for systems such as cold atoms in optical lattices, specifically when
long time scales are being considered [467].

All such systems in the class of driven open quantum matter have two common
characteristics:

##### Nonequilibrium conditions.

The combination of external driving and the coupling to external reservoirs,
inducing an open system character, pushes driven open quantum matter out of
equilibrium. Without external drive, open systems may still reach
thermodynamic equilibrium with their surroundings, and thus obey the principle
of detailed balance. The study of such open systems in thermodynamic
equilibrium has been pioneered by Caldeira and Leggett [104, 105, 611],
building on the influence functional techniques introduced by Feynman and
Vernon [197]. However, adding a drive—for example, in the form of a laser
field—generically induces stationary fluxes of quantities such as energy,
particle number, or entropy, and, therefore, leads to a violation of detailed
balance between system and environment. The presence of nonequilibrium
conditions even in the stationary state distinguishes driven open systems from
closed and undriven many-body systems. Even though the latter can exhibit
nonequilibrium behavior in their time evolution, they generically reach a
stationary state of thermodynamic equilibrium.

##### Quantum dynamics.

On the microscopic scale, instances of driven open quantum matter need to be
described as quantum systems—quantum mechanical effects such as phase
coherence and entanglement cannot be discarded. However, given the delicate
nature of quantum mechanical correlations, this does not imply that a quantum
mechanical effects necessarily persist up to macroscopic scales. In many
instances, an effective (semi-)classical description of the macroscopic
behavior is possible—in parallel to finite temperature quantum systems.
However, drive and dissipation need not act destructively on quantum many-body
correlations, and can even be harnessed to induce them [166, 585]. Identifying
and describing universal effects of this type is both a unique challenge and
an opportunity offered by driven open quantum systems. This holds the
potential to spark a new field of nonequilibrium quantum statistical
mechanics.

Realizations of paradigmatic | Novel nonequilibrium universality | Quantum nonequilibrium phenomena
---|---|---
nonequilibrium universality
V. Absorbing state phase transitions | VIII. Driven open criticality | XI. Quantum criticality in driven open systems
and directed percolation
VI. Self-organized criticality and | IX. Slowly and rapidly driven open systems | XII. Universality in dissipative quantum impurities
Rydberg experiments
| X. Nonequilibrium first order phase transitions | XIII. Universality in fermion systems
VII. Driven open condensates in low spatial dimensions |
Table 1: Manifestations of universality in driven open quantum matter are
grouped into three main directions in this review.

###  I.3 Synopsis

This review provides both the conceptual framework to characterize
universality in driven open quantum matter, and highlights key instances of
universality in various physical platforms. It is organized as follows.

##### Principles of universality in driven open quantum matter (Secs. II–IV).

Microscopically, the systems described above can be modelled in terms of a
Markovian quantum master equation in Lindblad form. Yet, this is not the ideal
language to distill universality. We thus first introduce an overarching
framework that enables discarding irrelevant information while keeping the
relevant one (Secs. II–IV). This is achieved by reformulating the quantum
master equation in terms of an equivalent but more flexible Lindblad-Keldysh
field theory (Sec. II). In particular, this allows us to identify three
principles to preserve the relevant information, and to guide the analysis of
the driven open quantum many-body problem:

  * (1)

Equilibrium vs. nonequilibrium stationary states (Sec. III): Detailed balance
characteristic of a system in thermodynamic equilibrium is indicated by the
presence of a discrete symmetry of the Keldysh action. This equips us with a
criterion to distinguish equilibrium from nonequilibrium conditions in
practice. By comparing the magnitudes of coupling constants which are
incompatible with that symmetry to those that are allowed, one can assess
quantitatively how far a system is away from thermodynamic equilibrium.

  * (2)

Symmetries and conservation laws (Sec. IV): Symmetries of the Keldysh action
come with a fine structure of “classical” and “quantum” (or “weak” and
“strong”) symmetries, while in equilibrium Euclidean field theories these
types of symmetries coincide. Both forms of symmetries, if continuous, are
tied to the existence of gapless modes, and are thus key to the identification
of universal phenomena: The spontaneous breakdown of weak continuous
symmetries is accompanied by the formation of gapless Goldstone modes. Strong
symmetries imply conservation laws, and are related to the existence of slow
hydrodynamic modes.

  * (3)

Mixed and pure states, classical and quantum scaling (Sec. IV): Intuitively,
one might expect that Markovian noise acts similarly to a finite temperature.
In cases where the stationary state is mixed, this is generally true, and
allows us to simplify the Keldysh action by taking the semiclassical limit.
Critical problems are then described by scaling forms of the Keldysh action
analogous to finite-temperature classical phase transitions. However, under
specific conditions, also pure stationary states can emerge, similar to the
fine-tuning of temperature to zero in systems in thermodynamic equilibrium.
Critical problems of this class exhibit scaling solutions which parallel
quantum phase transitions.

We then focus on universal phenomena, grouped into three main directions as
illustrated in Tab. 1:

##### Realizations of paradigmatic nonequilibrium universality (Secs. V–VII).

Driven open quantum systems enable the realization of paradigmatic
nonequilibrium scenarios, which hitherto were difficult to implement. This
includes the above mentioned directed percolation (Sec. V) and self-organized
criticality (Sec. VI), as well as Kardar-Parisi-Zhang (KPZ) universality (Sec.
VII). The challenge to theory is to extract these instances of universality
from concrete microscopic platforms made of quantum ingredients. This is
achieved by systematic coarse graining all the way to the macroscale. We also
describe recent experiments in polariton systems observing KPZ physics, and
driven Rydberg gases exhibiting directed percolation as well as self-organized
criticality. This highlights driven open quantum platforms as controllable
laboratories for nonequilibrium statistical mechanics.

##### Novel nonequilibrium universality (Secs. VII–X).

We next discuss universal phenomena which are unique to driven open quantum
systems and have not previously surfaced in nonequilibrium statistical
mechanics. Among these, by analyzing the impact of nonequilibrium conditions
on topological vortex defects, we assess the fate of one of the cornerstones
of low-dimensional statistical mechanics, the Kosterlitz-Thouless transition,
including in the case of strong spatial anisotropy. We furthermore show that
novel kinds of topological phase transitions exist out of equilibrium, such as
vortex turbulence—those transitions occur upon increasing the strength of
nonequilibrium conditions (Sec. VII). Then, we discuss novel universality in
the phase transitions in driven open quantum matter (Sec. VIII). A first
instance is the identification of a new independent critical exponent for the
Bose condensation transition, which distinguishes equilibrium from driven open
criticality. A novel nonequilibrium fixed point in nonreciprocally coupled
Ising models is also discussed. We then turn to the critical behavior in
slowly and rapidly driven open quantum systems (Sec. IX). In the slowly driven
limit, the paradigmatic Kibble-Zurek scenario is revealed as the tip of the
iceberg of a more general phenomenology, accessible in a renormalization group
approach, which gives experimental access to the full spectrum of critical
exponents by suitable driving protocols. The opposite limit of fast, yet not
infinitely fast, driving is realized in open Floquet systems, and is shown to
exhibit a similar phenomenology as its slowly driven counterpart, although
with a different physical origin. Next, we discuss universal aspects of first
order transitions far from equilibrium (Sec. X). Here nonthermal noise
modifies the formation of droplets—the universal mechanism behind first order
phase transitions. The most drastic modification occurs at first order dark
state phase transitions: The system realizes a bistability between a mixed
phase, displaying statistical fluctuations similar to systems at nonzero
temperature, and a dark state phase, represented by a single, pure quantum
state. All these phenomena are predicted for concrete platforms such as
polariton systems, driven optical lattices for ultracold atoms, or Rydberg
tweezer arrays, but yet await observation. We hope that our exposition may
foster an agenda for future experiments.

##### Nonequilibrium quantum phenomena (Secs. XI–XIII).

This final part reviews progress on one of the unique hallmarks of driven open
quantum systems: the potential for showing collective quantum behavior at the
largest distances. One example of the strong universality encountered near
phase transitions concerns an analog of quantum criticality at
equilibrium—Markovian quantum criticality (Sec. XI). Another one focuses on a
class of problems where nonequilibrium perturbations occur via impurities, in
systems otherwise kept in their quantum ground states. The interplay of
gapless many-body modes with the nonequilibrium impurity gives rise to
universal phenomena like the fluctuation induced quantum Zeno effect (Sec.
XII). We also present results on driven open systems composed of fermions,
which do not possess a simple classical limit (Sec. XIII). Here we highlight
how fermion systems can be cooled into pure topological states, and discuss
their topological phase transitions. We also demonstrate that topology
provides a strong principle of universality: the topological response of
fermion systems is identical, irrespective to their dynamics proceeding in- or
out of equilibrium. We also point out a dynamical fine structure in the
symmetry classification of interacting fermion matter, which distinguishes
equilibrium from nonequilibrium conditions. Results on quantum nonequilibrium
phenomena are just fledging here; further research will shape the field of
nonequilibrium quantum statistical mechanics.

With its focus on universal phenomena in driven open quantum matter, this
review complements related works: Several surveys focus on physical
realizations of driven open quantum systems, which can be structured into
platforms made of photons [275, 447, 448, 456], atoms [435, 425, 83], and
solids [120, 593, 66, 291]. On the methodological side, recent progress
regarding numerical techniques and simulation methods for open quantum many-
body systems has been reviewed in Refs. [151, 607], while the field theory
approach has been surveyed in Refs. [541, 568].

There are also several instances of universal nonequilibrium behavior in
quantum many-body systems which are not touched upon here. A first important
delineation concerns open vs. closed nonequilibrium systems. Clearly,
nonthermal stationary or quasi-stationary states can also exist in closed
quantum systems. One paradigmatic example is the absence of thermalization in
disordered interacting systems, known as many-body localization [1, 440, 11,
4]. Further examples in this class are provided by certain gauge theories
[80], fragmentation [430, 511], and scar states [579, 517]. Furthermore,
various instances of closed systems governed by unitary dynamics without
energy conservation have been studied. This concerns, for example, Floquet
systems with periodically time dependent Hamiltonians, which can be trapped in
long-lived states before generically heating up at long times [3, 428, 347, 2,
183, 426, 272, 531, 640]. Such systems can show collective phenomena, such as
the formation of time crystals [324, 629]. Also the study of unitary dynamics
in random quantum circuits has attracted attention. Circuit dynamics serve as
models exhibiting generic aspects of thermalization, entanglement dynamics, as
well as information spreading and scrambling [437, 550, 372, 203, 267, 323,
325].

We also do not touch upon the time evolution of many-body systems—our notion
of “nonequilibrium” refers to the nonthermal nature of the stationary states
of driven open quantum systems. Of course, the temporal evolution of closed or
open systems following, e.g., a quench of system parameters, realizes a
different kind of nonequilibrium condition. Quenches, or more generally
nonthermal initial states, have been studied intensely for closed systems (see
[477, 184, 103], and some of the instances in the preceding paragraph). Beyond
the thermalization expected at asymptotically long times [150], interesting
universal aspects of the transient temporal regimes such as turbulence [205],
prethermalized plateaus [58], or nonthermal fixed points [60, 187, 489, 403]
have been revealed. The dynamics of driven open quantum systems have so far
received far less attention, and we will come back to this point in the
Perspectives in Sec. XIV.

##  II Description of driven open systems: from Lindblad to Keldysh

The state of a driven open quantum system is specified by its density matrix
$\hat{\rho}$, and the dynamics are described by a quantum master equation in
Lindblad form [379, 245, 81, 213],

$\partial_{t}\hat{\rho}=\mathcal{L}\hat{\rho}=-i\left[\hat{H},\hat{\rho}\right]+\mathcal{D}\hat{\rho}.$
(1)

Intrinsic dynamics of the system are captured by the commutator of
$\hat{\rho}$ with the system Hamiltonian $\hat{H}$, and the dissipator
$\mathcal{D}$ encodes the coupling of the system to external reservoirs. The
dissipator takes the characteristic Lindblad form,
$\mathcal{D}\hat{\rho}=\hat{L}\hat{\rho}\hat{L}^{\dagger}-\frac{1}{2}\left\\{\hat{L}^{\dagger}\hat{L},\hat{\rho}\right\\}$,
where $\hat{L}$ is called a Lindblad or quantum jump operator. For a many-body
system, there are typically several decay channels—e.g., when the system is
coupled to several baths. Then, the dissipator contains a sum over different
types of jump operators. The Lindblad form of the dissipator ensures that the
time evolution generated by the Liouville superoperator $\mathcal{L}$, which
is also known as the Liouvillian or the Lindbladian, is Hermiticity- and
trace-preserving, and completely positive.

The focus of this review is on universal phenomena that emerge at large length
scales in driven open quantum many-body systems. In addressing this issue, we
take the quantum master equation (1) as the microscopic starting point of our
analysis. For derivations of master equations in the many-body context and a
discussion of the relevant approximations, we refer to the literature [166,
467, 287, 396, 570, 99, 239].

###  II.1 Driven open Bose-Einstein condensation

To illustrate key concepts in the theory of driven open quantum many-body
systems, we shall often refer to a model of short-range interacting bosons in
a $d$-dimensional spatial continuum, which are subjected to single-particle
pumping and loss as well as two-body loss. This model describes driven open
Bose-Einstein condensation [543, 544], e.g., of exciton-polaritons [120],
which is discussed in detail in Sec. VII. Its pedagogical value stands in the
direct comparison with the equilibrium BEC transition; furthermore, upon
including minimal modifications (change of dimensionality, or of jump
operators involved) we can explore several of the distinct notions of driven-
open criticality discussed in this review.

The Hamiltonian of the model reads

$\hat{H}=\int
d\mathbf{x}\left[\hat{\psi}^{\dagger}(\mathbf{x})\left(-\frac{\nabla^{2}}{2m}\right)\hat{\psi}(\mathbf{x})+u_{c}\hat{\psi}^{\dagger}(\mathbf{x})^{2}\hat{\psi}(\mathbf{x})^{2}\right],$
(2)

where $\hat{\psi}(\mathbf{x})$ and $\hat{\psi}^{\dagger}(\mathbf{x})$ are
annihilation and creation operators for bosons at position
$\mathbf{x}\in\mathbb{R}^{d}$. These bosons have mass $m$ and interact with
strength $u_{c}$. Single-particle incoherent pumping and loss and two-body
loss are described by the jump operators
$\hat{L}_{p}(\mathbf{x})=\sqrt{\gamma_{p}}\hat{\psi}^{\dagger}(\mathbf{x})$,
$\hat{L}_{l}(\mathbf{x})=\sqrt{\gamma_{l}}\hat{\psi}(\mathbf{x})$, and
$\hat{L}_{t}(\mathbf{x})=\sqrt{2u_{d}}\hat{\psi}(\mathbf{x})^{2}$,
respectively. Consequently, the dissipator takes the form

$\mathcal{D}\hat{\rho}=\int
d\mathbf{x}\left[\gamma_{p}\left(\hat{\psi}^{\dagger}(\mathbf{x})\hat{\rho}\hat{\psi}(\mathbf{x})-\frac{1}{2}\left\\{\hat{\psi}(\mathbf{x})\hat{\psi}^{\dagger}(\mathbf{x}),\hat{\rho}\right\\}\right)\right.\\\
+\gamma_{l}\left(\hat{\psi}(\mathbf{x})\hat{\rho}\hat{\psi}^{\dagger}(\mathbf{x})-\frac{1}{2}\left\\{\hat{\psi}^{\dagger}(\mathbf{x})\hat{\psi}(\mathbf{x}),\hat{\rho}\right\\}\right)\\\
\left.+2u_{d}\left(\hat{\psi}(\mathbf{x})^{2}\hat{\rho}\hat{\psi}^{\dagger}(\mathbf{x})^{2}-\frac{1}{2}\left\\{\hat{\psi}^{\dagger}(\mathbf{x})^{2}\hat{\psi}(\mathbf{x})^{2},\hat{\rho}\right\\}\right)\right].$
(3)

Other concrete models of bosonic and fermionic driven open quantum many-body
systems are introduced in Secs. V–XIII below. Condensation of the field
$\hat{\psi}(\mathbf{x})$ is indicated by a finite value of the condensate
amplitude $\psi(t,\mathbf{x})=\langle\hat{\psi}(t,\mathbf{x})\rangle$ in the
steady state. Here, the time-dependent expectation value is defined as
$\langle\hat{\psi}(t,\mathbf{x})\rangle=\mathop{\mathrm{tr}}\\!\left(\hat{\psi}(\mathbf{x})\hat{\rho}(t)\right)$,
where $\hat{\rho}(t)=e^{\mathcal{L}\left(t-t_{0}\right)}\hat{\rho}(t_{0})$ is
the formal solution of the master equation. To understand the mechanism that
underlies the driven open condensation transition, we consider a mean-field
description, which we derive in two steps: First, we obtain the equation of
motion of the condensate amplitude by taking the temporal derivative of the
relation
$\psi(t,\mathbf{x})=\mathop{\mathrm{tr}}\\!\left(\hat{\psi}(\mathbf{x})\hat{\rho}(t)\right)$;
and second, we implement a mean-field approximation by replacing field
operators by the classical condensate field $\psi(t,\mathbf{x})$ according to
$\hat{\psi}(t,\mathbf{x})\to\psi(t,\mathbf{x})$ and
$\hat{\psi}^{\dagger}(t,\mathbf{x})\to\psi^{*}(t,\mathbf{x})$. This leads to

$i\partial_{t}\psi=\left[-\frac{\nabla^{2}}{2m}+2u_{c}\left\lvert\psi\right\rvert^{2}+\frac{i}{2}\left(\gamma_{p}-\gamma_{l}-4u_{d}\left\lvert\psi\right\rvert^{2}\right)\right]\psi,$
(4)

which, for a closed system with $\gamma_{p}=\gamma_{l}=u_{d}=0$, reduces to
the Gross-Pitaevskii equation [469]. Instead, for a driven open system with
$\gamma_{p}>\gamma_{l}$, the above equation is solved by
$\psi(t,\mathbf{x})=\psi_{0}e^{-i\mu t}$ with
$\left\lvert\psi_{0}\right\rvert^{2}=(\gamma_{p}-\gamma_{l})/(4u_{d})$ and
$\mu=2u_{c}\left\lvert\psi_{0}\right\rvert^{2}$. Physically, this solution
describes the formation of an oscillating condensate when net single-particle
gain for $\gamma_{p}-\gamma_{l}>0$ is balanced by two-body loss with density-
dependent loss rate $4u_{d}\left\lvert\psi_{0}\right\rvert^{2}$.

Both in closed and open systems, a theoretical description of universal
emergent phenomena such as critical behavior at the driven open condensation
transition can be given in terms of collective variables and by employing
renormalization group (RG) techniques. The formal framework for introducing
these concepts is provided by quantum field theory—more specifically, for
systems driven out of thermal equilibrium, by Keldysh field theory [525, 526,
393, 40, 41, 321, 311, 16, 541, 568]. In the following, we describe how to
perform the transition from a description of the dynamics in terms of quantum
master equation (1) to an equivalent formulation using Keldysh field theory.

###  II.2 Keldysh field theory for driven open systems

Keldysh field theory provides a functional integral representation of the
quantum master equation (1). The starting point for the construction of the
functional integral representation is the formal solution of the master
equation,
$\hat{\rho}(t)=e^{\mathcal{L}\left(t-t_{0}\right)}\hat{\rho}(t_{0})$. A
crucial difference between the Hamiltonian evolution of a pure state
$\lvert\Psi(t)\rangle$ and the Lindbladian evolution of a mixed state
$\hat{\rho}(t)$ lies in the fact that the superoperator $\mathcal{L}$ acts on
the density matrix simultaneously from the left- and right-hand sides. This
necessitates a modification of the usual construction of the functional
integral for the time evolution operator
$\hat{U}(t,t_{0})=e^{-i\hat{H}\left(t-t_{0}\right)}$. For concreteness, we
consider a single bosonic or fermionic field mode, described by annihilation
and creation operators $\hat{\psi}$ and $\hat{\psi}^{\dagger}$, respectively.
These operators obey the canonical commutation relation
$\left[\hat{\psi},\hat{\psi}^{\dagger}\right]_{\zeta}=\hat{\psi}\hat{\psi}^{\dagger}-\zeta\hat{\psi}^{\dagger}\hat{\psi}=1$,
where $\zeta=+1$ for a bosonic and $\zeta=-1$ for a fermionic mode. Then, the
time evolution operator for a pure state can be represented as a functional
integral over a single complex or Grassmann field $\psi$. The required
modification to describe the dynamics of the density matrix is the doubling of
fields $\psi\to\psi_{\pm}$, which is best understood by considering first the
case without dissipation, i.e., when $\mathcal{D}=0$ in Eq. (1): Then
$\hat{\rho}(t)=e^{\mathcal{L}\left(t-t_{0}\right)}\hat{\rho}(t_{0})=\hat{U}(t,t_{0})\hat{\rho}(t_{0})\hat{U}(t,t_{0})^{\dagger}$,
where the operator $\hat{U}(t,t_{0})^{\dagger}=\hat{U}(t_{0},t)$, if applied
to a state vector, would describe evolution that proceeds backward in time.
For this reason, the time evolution of $\hat{\rho}(t)$ is commonly visualized
as two branches emanating from $\hat{\rho}(t_{0})$: A forward branch that
corresponds to $\hat{U}(t,t_{0})$ and is described, in the functional integral
to be specified below, by integration over the field $\psi_{+}$; and a
backward branch that corresponds to $\hat{U}(t,t_{0})^{\dagger}$ and is
described by integration over the field $\psi_{-}$. In the Keldysh partition
function, which is defined as the trace of the density matrix,
$Z=\mathop{\mathrm{tr}}(\hat{\rho}(t))=\mathop{\mathrm{tr}}(\hat{\rho}(t_{0}))=1$,
the two branches are connected at time $t$ to form a closed contour.

####  II.2.1 Keldysh partition function and Lindblad-Keldysh action

The derivation of the functional integral representation of the Keldysh
partition function is carried out in detail in Appendix A. For a single
bosonic or fermionic mode, which at time $t_{0}$ is in the state
$\hat{\rho}(t_{0})$, and the dynamics of which include dissipation with a
single jump operator $\hat{L}$, we obtain the following expression for the
Keldysh partition function:

$Z=\int
D[\psi_{+}^{*},\psi_{+},\psi_{-}^{*},\psi_{-}]\,e^{iS}\langle\psi_{+}(t_{0})|\hat{\rho}(t_{0})|\zeta\psi_{-}(t_{0})\rangle.$
(5)

The integration is over complex and Grassmann fields for bosons and fermions,
respectively. Grassmann fields on the backward branch are defined with a sign
$\zeta=-1$ which is required to represent the trace in the Keldysh partition
function in terms of coherent states as is discussed in Appendix A. The
Lindblad-Keldysh action is given by

$S=\int_{t_{0}}^{t}dt^{\prime}\left(\psi_{+}^{*}(t^{\prime})i\partial_{t}\psi_{+}(t^{\prime})-\psi_{-}^{*}(t^{\prime})i\partial_{t}\psi_{-}(t^{\prime})-i\mathcal{L}_{+-}(t^{\prime})\right).$
(6)

As detailed below, to describe steady states of driven open systems, we will
typically consider the limit $t_{0}\to-\infty$ and $t\to\infty$. The
Lindbladian that generates the dynamics of the system is encoded in

$\mathcal{L}_{+-}(t)=-i\left(H_{+}(t)-H_{-}(t)\right)+L_{+}(t)L_{-}^{\dagger}(t)\\\
-\frac{1}{2}\left(L^{\dagger}_{+}(t)L_{+}(t_{-})+L^{\dagger}_{-}(t)L_{-}(t_{+})\right),$
(7)

where the functions $H_{\pm}=H(\psi_{\pm}^{*},\psi_{\pm})$,
$L_{+}=L(\psi_{+}^{*},\psi_{+})$, $L_{-}=L(\zeta\psi_{-}^{*},\zeta\psi_{-})$,
$L_{+}^{\dagger}=L^{\dagger}(\psi_{+}^{*},\psi_{+})$, and
$L_{-}^{\dagger}=L^{\dagger}(\zeta\psi_{-}^{*},\zeta\psi_{-})$ can be obtained
from the respective operators $\hat{H}$, $\hat{L}$, and $\hat{L}^{\dagger}$,
which we assume to be normal ordered, simply by replacing the creation
operators by $\psi_{\pm}^{*}$, and the annihilation operators by $\psi_{\pm}$,
with an additional sign $\zeta=-1$ for Grassmann fields on the backward
branch. Infinitesimal time shifts are introduced in $t_{\pm}\to t\pm 0^{+}$ to
keep track of the original order of operators as detailed in Appendix A. These
shifts are required only to regularize certain classes of integrals that occur
in diagrammatic expansions, and can safely be neglected otherwise.

From Eq. (7), we can read off the following simple rule that allows us to
straightforwardly translate from the Lindblad master equation formalism to
Keldysh field theory: In the master equation (1), operators that act on the
density matrix from the left- and right-hand side lead to contributions on the
forward and backward branches, respectively, with an additional sign
$\zeta=-1$ for Grassmann fields on the backward branch. Indeed, this rule does
not only apply to the action, but also to expectation values as follows
directly from the construction of the Keldysh field integral described in
Appendix A. When there is no dissipation, the Keldysh action is the sum of two
independent contributions that correspond to the forward and backward
branches, and the branches are coupled only at $t_{0}$ through the matrix
element of the initial state in Eq. (5), and at $t$ through taking the trace.
In the presence of dissipation, the forward and backward branches are coupled
at all times through the term $L_{+}L_{-}^{\dagger}$.

####  II.2.2 Keldysh rotation: classical and quantum fields

For finite initial and final times $t_{0}$ and $t$, respectively, the Keldysh
functional integral Eq. (5) describes the full dynamics of the system and is
thus fully equivalent to the Lindblad equation (1). Expectation values can be
evaluated as functional integrals. For example, the expectation value
$\langle\hat{\psi}(t)\rangle=\mathop{\mathrm{tr}}(\hat{\psi}\hat{\rho}(t))$ of
a bosonic field $\hat{\psi}$ at time $t$ is given by

$\langle\psi_{+}(t)\rangle=\int
D[\psi_{+}^{*},\psi_{+},\psi_{-}^{*},\psi_{-}]\,\psi_{+}(t)e^{iS}\langle\psi_{+}(t_{0})|\hat{\rho}(t_{0})|\psi_{-}(t_{0})\rangle,$
(8)

where, according to the rule that is formulated at the end of the previous
section, the operator $\hat{\psi}$ that is acting on $\hat{\rho}(t)$ from the
left translates to the field $\psi_{+}(t)$ on the forward branch. Applying
this rule to both sides of the equation
$\mathop{\mathrm{tr}}(\hat{\psi}\hat{\rho}(t))=\mathop{\mathrm{tr}}(\hat{\rho}(t)\hat{\psi})$,
which follows from the cyclic property of the trace, we find
$\langle\psi_{+}(t)\rangle=\langle\psi_{-}(t)\rangle$. That is, expectation
values of bosonic fields on the forward and backward branches are identical.
This redundancy of the Keldysh formalism is eliminated by working with
symmetric and antisymmetric superpositions of the fields $\psi_{\pm}$, which
are referred to as classical and quantum fields, respectively, and defined
through the Keldysh rotation:

$\psi_{c}=\frac{1}{\sqrt{2}}\left(\psi_{+}+\psi_{-}\right),\qquad\psi_{q}=\frac{1}{\sqrt{2}}\left(\psi_{+}-\psi_{-}\right).$
(9)

A finite expectation value
$\langle\hat{\psi}(t)\rangle=\langle\psi_{c}(t)\rangle/\sqrt{2}$ of a bosonic
field, which can signify, e.g., Bose-Einstein condensation in the model
described by Eqs. (2) and (3), is captured by the classical field; this
justifies a posteriori the nomenclature: condensation is a mean-field
behaviour with subleading quantum fluctuations. In contrast, the quantum field
describes fluctuations, and its expectation value vanishes by definition,
$\langle\psi_{q}(t)\rangle=0$. Also fermionic fields cannot acquire a finite
expectation value—in other words, Grassmann variables are never classical.
Nevertheless, also for fermions it is meaningful to perform a Keldysh rotation
to classical and quantum fields. This is because the abovementioned redundancy
of the Keldysh formalism manifests also for even-order correlation functions
that are in general nonzero for both bosons and fermions. In particular, such
correlation functions vanish when the largest time argument belongs to a
quantum field as we discuss in more detail in Appendix B for the example of
the Green’s functions introduced below in Sec. II.2.3. However, while for
bosons the fields $\psi_{\pm}$ and $\psi_{\pm}^{*}$ are related by complex
conjugation, the corresponding Grassmann fields for fermions are independent.
Therefore, they can be transformed in an arbitrary manner. This freedom is
often exploited to introduce different forms of the Keldysh rotation for
bosons and fermions [311, 16]. Here, instead, we opt to keep the presentation
symmetric by defining $\psi_{c}^{*}=(\psi_{+}^{*}+\psi_{-}^{*})/\sqrt{2}$ and
$\psi_{q}^{*}=(\psi_{+}^{*}-\psi_{-}^{*})/\sqrt{2}$ also for fermions.

We have argued that the cyclic property of the trace motivates introducing
classical and quantum fields. More fundamentally, the cyclic property of the
trace is related to the general property that the Keldysh action vanishes when
$\psi_{+}=\psi_{-}$ and thus $\psi_{q}=0$. This property is part of what is
known as the causality structure of the Keldysh action [311]. To see how it
comes about, we start from the conservation of probability that is expressed
through $\partial_{t}\mathop{\mathrm{tr}}(\hat{\rho}(t))=0$, and which follows
indeed from using $\partial_{t}\hat{\rho}=\mathcal{L}\hat{\rho}$ and the
cyclic property of the trace. But $\mathop{\mathrm{tr}}(\hat{\rho}(t))=Z$ is
just the Keldysh partition function. Therefore, taking the temporal derivative
of Eq. (5) and expressing the Keldysh action Eq. (6) as
$S=\int_{t_{0}}^{t}dt^{\prime}\,s(\psi_{+}(t^{\prime}),\psi_{-}(t^{\prime}))$,
conservation of probability implies that $0=\partial_{t}Z=i\langle
s(\psi_{+}(t),\psi_{-}(t))\rangle=i\langle s(\psi_{+}(t),\psi_{+}(t))\rangle$,
where in the last equality we have used the redundancy of the Keldysh
formalism explained above. As can be seen in the explicit form of the Keldysh
action in Eq. (6), conservation of probability as expressed through
$\partial_{t}Z=0$ is ensured by the property of the Keldysh action
$s(\psi_{+},\psi_{+})=0$. In other words, the Keldysh action vanishes when
$\psi_{q}=0$.

####  II.2.3 Retarded, advanced, and Keldysh Green’s functions

The two-point functions of classical and quantum fields, which are the
retarded, advanced, and Keldysh Green’s functions, are basic elements of
Keldysh field theory:

$\begin{split}G^{R}(t,t^{\prime})&=-i\langle\psi_{c}(t)\psi_{q}^{*}(t^{\prime})\rangle=-i\theta(t-t^{\prime})\langle[\hat{\psi}(t),\hat{\psi}^{\dagger}(t^{\prime})]_{\zeta}\rangle,\\\
G^{A}(t,t^{\prime})&=-i\langle\psi_{q}(t)\psi_{c}^{*}(t^{\prime})\rangle=i\theta(t^{\prime}-t)\langle[\hat{\psi}(t),\hat{\psi}^{\dagger}(t^{\prime})]_{\zeta}\rangle,\\\
G^{K}(t,t^{\prime})&=-i\langle\psi_{c}(t)\psi_{c}^{*}(t^{\prime})\rangle=-i\langle[\hat{\psi}(t),\hat{\psi}^{\dagger}(t^{\prime})]_{-\zeta}\rangle,\end{split}$
(10)

where the Heaviside step function is defined by $\theta(t)=1$ for $t>0$ and
$\theta(t)=0$ for $t<0$, and
$[\hat{A},\hat{B}]_{\zeta}=\hat{A}\hat{B}-\zeta\hat{B}\hat{A}$. For driven
open systems, two-time expectation values in the operator formulation are
defined through the quantum regression theorem [214]. The equivalence between
the expressions for Green’s functions in the operator and Keldysh formalisms
is established in Appendix B.

In general, the Green’s functions depend on both time arguments $t$ and
$t^{\prime}$. However, in the applications of Keldysh field theory discussed
in this review, our main interest is in systems that reach a steady state due
to coupling to external reservoirs, and in properties of the steady state. A
straightforward way to access these properties is to shift the initial time
$t_{0}$ to the distant past, i.e., take the limit $t_{0}\to-\infty$. In the
steady state, there is no memory left of the initial state, and, therefore,
the matrix element
$\langle\psi_{+}(t_{0})|\hat{\rho}(t_{0})|\zeta\psi_{-}(t_{0})\rangle$ in Eq.
(5) can be omitted. Consequently, the Green’s functions depend only on the
difference $t-t^{\prime}$. To account for the fact that this difference can
become arbitrarily large, it is convenient to consider the limit of an
infinite final time $t\to\infty$ in Eq. (5).

####  II.2.4 From a single to many modes

So far, we have considered only a single field mode. The generalization to
many modes in a spatially extended many-body system is straightforward. For
many-body system in a spatial continuum or on a lattice, the Keldysh action in
Eq. (6) includes an integration over spatial coordinates or a summation over
lattice sites, respectively. As a specific example, the Keldysh action for the
Hamiltonian in Eq. (2) and the dissipator in Eq. (3) is the sum of two terms,
$S=S_{H}+S_{D}$, which encode coherent and dissipative contributions to the
dynamics, respectively:

$S_{H}=\sum_{\sigma=\pm}\sigma\int_{t,\mathbf{x}}\left[\psi_{\sigma}^{*}\left(i\partial_{t}+\frac{\nabla^{2}}{2m}\right)\psi_{\sigma}-u_{c}\left(\psi_{\sigma}^{*}\psi_{\sigma}\right)^{2}\right],$
(11)

where $\int_{t,\mathbf{x}}=\int_{-\infty}^{\infty}dt\int d\mathbf{x}$, and

$S_{D}=-i\int_{t,\mathbf{x}}\left(\gamma_{p}\left[\psi_{+}^{*}\psi_{-}-\frac{1}{2}\left(\psi_{+}^{*}\psi_{+}+\psi_{-}^{*}\psi_{-}\right)\right]\right.\\\
+\gamma_{l}\left[\psi_{+}\psi_{-}^{*}-\frac{1}{2}\left(\psi_{+}^{*}\psi_{+}+\psi_{-}^{*}\psi_{-}\right)\right]\\\
\left.+2u_{d}\left\\{\psi_{+}^{2}\psi_{-}^{*2}-\frac{1}{2}\left[\left(\psi_{+}^{*}\psi_{+}\right)^{2}+\left(\psi_{-}^{*}\psi_{-}\right)^{2}\right]\right\\}\right).$
(12)

##  III Equilibrium vs. nonequilibrium stationary states

A defining signature of thermodynamic equilibrium is that the state of a
system in equilibrium does not change when the system is isolated from its
environment [182]. This condition is clearly violated for a quantum many-body
system that is subjected to a time-periodic classical driving field and
coupled to a dissipative environment: Absorption of energy from the drive
causes the system to heat up indefinitely, and dissipation to the environment
is necessary to stabilize a nontrivial state. Microscopically, the systems
that are the focus of this review are exactly such periodically driven system
coupled to a bath, but in the limit of high-frequency driving. This limit is
captured by the rotating wave approximation, which amounts to taking the
average over the fast driving frequency. Consequently, the explicit time
dependence disappears, and time-translation symmetry is restored on the
temporally coarse grained scale described by the Lindblad evolution in Eq.
(1). For such systems, it is, therefore, not clear a priori whether they will
reach thermodynamic equilibrium, or a nonequilibrium stationary state. As we
discuss in the following, a clear-cut identification of the violation of
equilibrium conditions is possible through a symmetry criterion for the
Keldysh action [542, 26, 14, 263, 264, 265, 145, 235, 266].

###  III.1 Thermal equilibrium and the fluctuation-dissipation theorem

How can one decide whether a quantum system is in thermal equilibrium or in a
nonequilibrium steady state? Making this distinction is not possible by
focusing solely on static properties of the steady state, such as expectation
values of observables or equal-time correlation functions. Instead, it is
necessary to consider also dynamics in the steady state in the form of
unequal-time correlation functions. Thermal equilibrium is achieved when the
dynamics are unitary and governed by the same time-independent Hamiltonian
$\hat{H}$ that defines also the thermal Gibbs state
$\hat{\rho}_{\beta}=\left.e^{-\beta\hat{H}}\middle/\mathop{\mathrm{tr}}\\!\left(e^{-\beta\hat{H}}\right)\right.$
at inverse temperature $\beta=1/T$. More generally, thermal equilibrium
requires that the _generator of the dynamics_ also determines the _statistical
weights_ of the state. Otherwise nonequilibrium conditions are expected.

To formalize the distinction between thermal equilibrium and nonequilibrium
stationary states, one may first note that any positive semidefinite Hermitian
operator $\hat{\rho}$ can be written in the form of $\hat{\rho}_{\beta}$ for
some Hamiltonian $\hat{H}$ that is defined as the logarithm of $\hat{\rho}$.
This applies, in particular, to the steady states of the dynamics described by
a master equation in Lindblad form Eq. (1). However, thermal equilibrium
requires the dynamics to be generated by the same Hamiltonian $\hat{H}$ that
also determines the state. To test whether the dynamics obey thermal
equilibrium conditions, it is sufficient to consider the linear response of
the system to weak time-dependent perturbations. This response is encoded in
unequal-time correlation functions in the unperturbed state. For the example
of a single bosonic or fermionic mode $\hat{\psi}$, we consider two-point
functions such as $\langle\hat{\psi}^{\dagger}(t)\hat{\psi}(0)\rangle$, where,
for a system in thermal equilibrium,
$\hat{\psi}(t)=e^{i\hat{H}t}\hat{\psi}e^{-i\hat{H}t}$ and
$\langle\dotsc\rangle=\mathop{\mathrm{tr}}(\dotsc\hat{\rho}_{\beta})$. Thermal
equilibrium implies the validity of the fluctuation-dissipation theorem (FDT),
which is thus a necessary requirement for equilibrium conditions:

$G^{K}(\omega)=\left(1+2\zeta
n_{\zeta}(\omega)\right)\left(G^{R}(\omega)-G^{A}(\omega)\right),$ (13)

with the Bose or Fermi distribution function
$n_{\zeta}(\omega)=\left.1\middle/\left(e^{\beta\omega}-\zeta\right)\right.$,
where as above $\zeta=1$ for bosons and $\zeta=-1$ for fermions. In a
stationary state, the Green’s functions in Eq. (10) depend only on the
difference of time arguments. Then, the Fourier transform, e.g., of the
Keldysh Green’s function, is defined as
$G^{K}(\omega)=\int_{-\infty}^{\infty}dt\,e^{i\omega t}G^{K}(t)$.

Studying the response to perturbations that couple to products of bosonic or
fermionic operators leads to generalizations of the FDT for higher-order
correlation functions, and the validity of the full hierarchy of generalized
FDTs can be taken as the defining property of thermal equilibrium. Clearly,
this definition is not very practical and one may rely instead on an
equivalent symmetry criterion.

###  III.2 Thermal equilibrium as a symmetry of the Keldysh action

In field theories, exact relationships between correlation functions can often
be attributed to the symmetries of the action. This is also the case for the
FDT. In fact, the FDT can be seen as a consequence of the Ward-Takahashi
identity which is associated with the symmetry of the Keldysh action under a
specific field transformation on the forward and backward branches of the
Keldysh contour:

$\mathcal{T}_{\beta}\psi_{l,\pm}(t)=\left(\pm
i\right)^{(1-\zeta)/2}\sum_{l^{\prime}}\psi_{l^{\prime},\pm}^{*}(-t\pm
i\beta/2)U_{l^{\prime},l}.$ (14)

We consider here a multicomponent bosonic or fermionic field $\hat{\psi}_{l}$,
where $l$ is the site index in a lattice system or may be a combined index
including, e.g., spin or sublattice degrees of freedom. For any two-time
function, invariance under $\mathcal{T}_{\beta}$ is equivalent to the
combination of the Kubo-Martin-Schwinger relation [344, 406] with time
reversal $\hat{\mathcal{T}}$ [296, 542], and, therefore, implies the validity
of the FDT. The time reversal transformation $\hat{\mathcal{T}}$ is
antiunitary, $\hat{\mathcal{T}}i\hat{\mathcal{T}}^{-1}=-i$, and defined by its
action on the fields,
$\hat{\mathcal{T}}\hat{\psi}_{l}\hat{\mathcal{T}}^{-1}=\sum_{l^{\prime}}U_{l,l^{\prime}}^{\dagger}\hat{\psi}_{l^{\prime}}$,
where for generality we include a unitary matrix $U$ that obeys
$UU^{*}=U^{*}U=\pm\mathbbm{1}$ [510, 386]. If we further assume that the
Hamiltonian is invariant under time reversal,
$\hat{\mathcal{T}}\hat{H}\hat{\mathcal{T}}^{-1}=\hat{H}$, then invariance of
the Keldysh action under $\mathcal{T}_{\beta}$ implies a comprehensive
hierarchy of generalized FDTs for higher-order functions. This illustrates
that the _thermal symmetry_ of the Keldysh action, characterized by its
invariance under the transformation $\mathcal{T}_{\beta}$, can be considered
as the definition of thermal equilibrium conditions.

For systems that do not have time reversal symmetry, higher-order FDTs relate
correlation functions of the original system to those of its time-reversed
partner, in the sense that all external parameters are conjugated under time
reversal. For example, the sign of a magnetic field is reversed. This is
analogous to Onsager relations which, when time reversal symmetry is broken,
are replaced by Onsager-Casimir relations [454, 121]. Therefore, the validity
of FDTs and the obedience of equilibrium conditions do not require invariance
of the Hamiltonian under time reversal.

As a concrete example, we consider a single bosonic or fermionic mode with
Hamiltonian $\hat{H}=\omega_{s}\hat{\psi}^{\dagger}\hat{\psi}$, which has time
reversal symmetry with $U=1$. A thermal state of this mode is described by the
Keldysh action

$S=\int_{\omega}\left(\psi_{c}^{*}(\omega),\psi_{q}^{*}(\omega)\right)\\\
\begin{pmatrix}0&\omega-\omega_{s}-i\delta\\\
\omega-\omega_{s}+i\delta&i2\delta\left(1+2\zeta
n_{\zeta}(\omega)\right)\end{pmatrix}\begin{pmatrix}\psi_{c}(\omega)\\\
\psi_{q}(\omega)\end{pmatrix},$ (15)

where the shorthand notation
$\int_{\omega}=\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}$ is used. This
action is invariant under the transformation $\mathcal{T}_{\beta}$ for
arbitrary values of the parameter $\delta$. The Green’s functions that
correspond to a thermal state are obtained by taking the limit $\delta\to
0^{+}$ after calculating the functional integrals in Eq. (10) [311, 16].

The arguments above rely only on symmetry properties of the Keldysh action,
and, therefore, they apply equally to noninteracting and interacting theories.
An alternative formulation of the symmetry which is based on a different
implementation of time reversal in the Keldysh formalism is discussed for
bosons in Ref. [542] and for multicomponent fermions in Ref. [14]. The
generalization from thermal equilibrium to chemical equilibrium is readily
implemented [542, 26].

###  III.3 Nonequilibrium and emergent equilibrium

Having identified thermal symmetry of the Keldysh action as the defining
property of thermal equilibrium conditions, we define a nonequilibrium
stationary state as being described by an action that is not invariant under
$\mathcal{T}_{\beta}$. Different physical mechanisms can stabilize a
nonequilibrium stationary state, including the combination of a time-dependent
classical drive and the coupling to a heat bath, or the coupling to several
baths at different temperatures or chemical potentials. As a concrete example,
consider the driven open condensate described by in Eqs. (11) and (12). For
direct comparison with Eq. (15) for a system in thermal equilibrium, we focus
on the Gaussian or quadratic part of the Keldysh action (cf. Appendix C for a
general discussion of Gaussian Keldysh actions). After the Keldysh rotation
Eq. (9), the Gaussian action reads

$S_{\mathrm{Gaussian}}=\int_{\omega,\mathbf{q}}\left(\psi_{c}^{*},\psi_{q}^{*}\right)\begin{pmatrix}0&P^{A}(\omega,\mathbf{q})\\\
P^{R}(\omega,\mathbf{q})&P^{K}\end{pmatrix}\begin{pmatrix}\psi_{c}\\\
\psi_{q}\end{pmatrix},$ (16)

where
$P^{R}(\omega,\mathbf{q})=P^{A}(\omega,\mathbf{q})^{*}=\omega-q^{2}/(2m)+ir_{d}$
with $q=\left\lvert\mathbf{q}\right\rvert$ and
$r_{d}=(\gamma_{l}-\gamma_{p})/2$. The key difference to Eq. (15) is that the
Keldysh component $P^{K}=i2\gamma$ with $\gamma=(\gamma_{l}+\gamma_{p})/2$ is
frequency-independent. However, the nontrivial frequency dependence of the
distribution function in the Keldysh component of Eq. (15) is required for
invariance under $\mathcal{T}_{\beta}$. Therefore, the thermal symmetry
criterion establishes formally that the driven open condensate is in a
nonequilibrium stationary state. It is important to note that this analysis
applies to the _microscopic_ action and does not rule out that thermal
equilibrium conditions emerge upon coarse graining to obtain an effective
long-wavelength and low-frequency description. This is indeed the case for the
driven open condensate in sufficiently high spatial dimension, as discussed in
Sec. VIII.1.

##  IV Overarching principles in- and out of equilibrium

Field theories are ideally suited to describe emergent phenomena on long
distance and time scales, and, in particular, critical behavior at continuous
phase transitions. Key organizing principles that are fundamental to our
understanding of universal emergent behavior in thermal equilibrium, including
symmetries and the distinction between classical and quantum criticality,
retain their validity in systems that are driven out of equilibrium.
Consequently, their formulation can be adapted to the Keldysh contour
formalism, where they will likewise act as powerful guiding principles for
theoretical analysis.

###  IV.1 Classical and quantum symmetries of the Keldysh action: symmetry
breaking vs. conservation laws

A global unitary continuous symmetry of a closed system entails the existence
of a conserved charge via the Noether theorem [642]. When the closed system
consists of two subsystems, i.e., the actual system of interest and a bath,
there is a fundamental distinction as to whether Noether charge can or cannot
be transferred between the system of interest and the bath. In the Keldysh
formalism, after integrating out the bath, the former case is referred to as a
classical symmetry, while the latter case realizes a quantum symmetry. The
distinction between these two types of symmetry becomes important whenever one
deals with an open system, irrespective of whether it is driven or not.

Both cases can be illustrated using the example of a single bosonic or
fermionic mode coupled to a bath, as described by the Hamiltonian
$\hat{H}=\hat{H}_{s}+\hat{H}_{sb}+\hat{H}_{b}$, where the system Hamiltonian
reads $\hat{H}_{s}=\omega_{s}\hat{\psi}^{\dagger}\hat{\psi}$, and the system-
bath coupling and the bath Hamiltonian are given by

$\hat{H}_{sb}=\sum_{\mu}g_{\mu}\left(\hat{L}^{\dagger}\hat{\phi}_{\mu}+\hat{\phi}_{\mu}^{\dagger}\hat{L}\right),\quad\hat{H}_{b}=\sum_{\mu}\omega_{\mu}\hat{\phi}_{\mu}^{\dagger}\hat{\phi}_{\mu},$
(17)

with $\hat{L}=\hat{\psi}$. The Hamiltonian $\hat{H}$ commutes with the total
number of excitations of the system and the bath
$\hat{N}=\hat{N}_{s}+\hat{N}_{b}$ where
$\hat{N}_{s}=\hat{\psi}^{\dagger}\hat{\psi}$ and
$\hat{N}_{b}=\sum_{\mu}\hat{\phi}_{\mu}^{\dagger}\hat{\phi}_{\mu}$. Therefore,
$\hat{N}$ is conserved. Formally, we can regard $\hat{N}$ as the Noether
charge that is associated with the symmetry of the Hamiltonian under the
unitary transformation $\hat{G}_{\alpha}=e^{i\alpha\hat{N}}$. For a finite
system-bath coupling $g_{\mu}\neq 0$, excitations, and, therefore, Noether
charge, can be transferred between the system and the bath. In contrast, when
$g_{\mu}=0$, the numbers of excitations in the system and in the bath,
$\hat{N}_{s}$ and $\hat{N}_{b}$, respectively, are conserved individually.
This is reflected in the invariance of the Hamiltonian under both
$\hat{G}_{s,\alpha}=e^{i\alpha\hat{N}_{s}}$ and
$\hat{G}_{b,\alpha}=e^{i\alpha\hat{N}_{b}}$. The associated Noether charges
$\hat{N}_{s}$ and $\hat{N}_{b}$ pertain to the system and the bath,
respectively, and cannot be transferred between them.

These different types of symmetries—under only $\hat{G}_{\alpha}$ or under
both $\hat{G}_{s,\alpha}$ and $\hat{G}_{b,\alpha}$, which implies also
symmetry under $\hat{G}_{\alpha}=\hat{G}_{s,\alpha}\hat{G}_{b,\alpha}$—are
reflected in the Keldysh action even after integration over the bath. Only the
latter case of a global continuous quantum symmetry is associated with a
conserved Noether charge that pertains to the system alone. But also the
former case of a classical symmetry has important physical consequences. For
example, the breaking of global continuous classical symmetries leads to the
existence of Goldstone modes. The implications of quantum and classical
symmetries are further discussed in Secs. IV.1.3 and IV.1.4. Their origin in
the structure of the Keldysh formalism, and their equivalence to the notions
of weak and strong symmetries of Lindbladians [86] is elucidated below.

####  IV.1.1 Classical and quantum symmetries

The doubling of degrees of freedom in Keldysh field theory entails a
corresponding doubling of the symmetry group. This can be illustrated with the
functional integral representation of the time evolution operator
$\hat{U}(t,t_{0})=e^{-i\hat{H}\left(t-t_{0}\right)}$ for pure states. The
corresponding action is obtained by keeping only the part of the Keldysh-
Lindblad action Eq. (6) that describes Hamiltonian time evolution on the
forward branch,

$S=\int_{t_{0}}^{t}dt^{\prime}\left(\psi^{*}(t^{\prime})i\partial_{t}\psi(t^{\prime})-H(\psi^{*}(t^{\prime}),\psi(t^{\prime}))\right).$
(18)

For concreteness, consider the Hamiltonian of the bosonic many-body system
given in Eq. (2). The invariance of this Hamiltonian under the $\mathrm{U}(1)$
phase-rotation transformation of field operators
$\hat{\psi}(\mathbf{x})\mapsto\hat{G}_{\alpha}^{\dagger}\hat{\psi}(\mathbf{x})\hat{G}_{\alpha}=e^{i\alpha}\hat{\psi}(\mathbf{x})$
with $\hat{G}_{\alpha}=e^{i\alpha\hat{N}}$ where
$\hat{N}=\int_{\mathbf{x}}\hat{\psi}^{\dagger}(\mathbf{x})\hat{\psi}(\mathbf{x})$
is associated with the conservation of the number of particles $\hat{N}$. It
leads to invariance of the action $S$ in Eq. (18) under the transformation of
fields $\psi(t,\mathbf{x})\mapsto e^{i\alpha}\psi(t,\mathbf{x})$. How does
this symmetry transfer to the Keldysh action in Eq. (6)? For a closed system,
the Keldysh action reduces to the sum of two copies that correspond to the
forward and backward branches of the Keldysh contour, and one can perform
independent transformations of the fields on each branch,
$\psi_{\pm}(t,\mathbf{x})\mapsto e^{i\alpha_{\pm}}\psi_{\pm}(t,\mathbf{x})$.
The symmetry group of the action is thus enlarged to
$\mathrm{U}_{+}(1)\times\mathrm{U}_{-}(1)$ (branch indices are added for
clarity). Analogous arguments apply to symmetry groups other than
$\mathrm{U}(1)$, including discrete symmetries.

For closed systems, for which the contributions to the Keldysh action from the
forward and backward branches are independent and identical, the enlarged
symmetry group $\mathrm{U}_{+}(1)\times\mathrm{U}_{-}(1)$ does not yield
information beyond the original $\mathrm{U}(1)$ symmetry of the Hamiltonian.
For example, the explicit breaking of $\mathrm{U}_{+}(1)$ by adding terms to
the Hamiltonian implies that also $\mathrm{U}_{-}(1)$ is broken, since
modifications of the Hamiltonian affect the forward and the backward branches
in the same way. However, the distinction between symmetry transformations on
the forward and backward branches becomes meaningful and important for open
systems, when dissipative processes couple the forward and backward evolution.
Then, a fruitful perspective is opened up by distinguishing between classical
and quantum symmetries. In analogy to the distinction between classical and
quantum fields in the Keldysh rotation in Eq. (9), in Ref. [541], the
generators of continuous classical and quantum symmetries are defined,
respectively, as symmetric and antisymmetric superpositions of the generators
of symmetry transformations on the forward and backward branches. For the case
of $\mathrm{U}(1)$ phase rotations $\psi_{\pm}(t,\mathbf{x})\to
e^{i\alpha_{\pm}}\psi_{\pm}(t,\mathbf{x})$, this definition of classical and
quantum symmetries corresponds to the choices $\alpha_{+}=\alpha_{-}$ and
$\alpha_{+}=-\alpha_{-}$. To include also the case of discrete symmetry
groups, we employ here a more general definition of classical and quantum
symmetries. Let us consider a many-body system with multicomponent field
operators $\hat{\psi}$, where we leave the indices pertaining to internal and
external degrees of freedom implicit. A unitary transformation $\hat{G}$ of
operators can be represented through a unitary matrix $G$ according to
$\hat{G}\hat{\psi}\hat{G}^{\dagger}=G\hat{\psi}$. The transformation of
operators induces a transformation of complex or Grassmann fields, which is
also described by $G$. In particular, for the functional integral
representation of the propagator
$\hat{U}(t,t_{0})=e^{-i\hat{H}\left(t-t_{0}\right)}$ which is governed by the
action $S$ in Eq. (18), the fields transform as $\psi\mapsto G\psi$. Extending
this transformation to the Keldysh contour, we define a Keldysh action as
having a classical symmetry if it is invariant under $\psi_{\pm}\mapsto
G\psi_{\pm}$. That is, the fields on the forward and backward branches are
transformed in the same way. In contrast, we define a quantum symmetry as
requiring invariance of the Keldysh action under independent transformations
of the fields on the forward and backward branches. Clearly, this includes the
classical symmetry transformation $\psi_{\pm}\mapsto G\psi_{\pm}$, but also
two further possibilities: $\psi_{+}\mapsto G\psi_{+}$ and
$\psi_{-}\mapsto\psi_{-}$ as well as $\psi_{+}\mapsto\psi_{+}$ and
$\psi_{-}\mapsto G\psi_{-}$.

In our discussion of classical and quantum symmetries below, we will often
refer to the model of driven open Bose-Einstein condensation defined in terms
of the Hamiltonian and dissipation given in Eq. (2) and Eq. (3), respectively,
with Keldysh action given in Eqs. (11) and (12). In this model, particle
number conservation is broken explicitly by incoherent pumping and losses.
When any of the parameters $\gamma_{p}$, $\gamma_{l}$, and $u_{d}$ is nonzero,
the symmetry group $\mathrm{U}_{+}(1)\times\mathrm{U}_{-}(1)$ of the closed
system is reduced to $\mathrm{U}_{c}(1)$, which we define as symmetry under
classical phase rotations $\psi_{\pm}(t,\mathbf{x})\mapsto
e^{i\alpha}\psi_{\pm}(t,\mathbf{x})$. Symmetry under the full group of quantum
transformations, $\mathrm{U}_{q}=\mathrm{U}_{+}(1)\times\mathrm{U}_{-}(1)$
corresponding to $\psi_{\pm}(t,\mathbf{x})\mapsto
e^{i\alpha_{\pm}}\psi_{\pm}(t,\mathbf{x})$ with arbitrary values of
$\alpha_{\pm}$, is broken explicitly.

####  IV.1.2 Equivalence to weak and strong symmetries

Global continuous classical and quantum symmetries have a clear physical
interpretation in terms of whether Noether charge can be transferred between a
system and its environment. Formally, classical and quantum transformations of
fields correspond to, respectively, identical and independent transformations
of operators that act from the left and right on the density matrix in the
master equation (1). Therefore, classical and quantum symmetries are
equivalent to weak and strong symmetries as defined in Ref. [86, 9, 376].

In more detail, we first show that classical symmetries are equivalent to weak
symmetries. In Sec. II, we have derived the general rule that fields on the
forward and backward branches correspond to operators in the Lindbladian that
act on the density matrix from the left- and right-hand side, respectively.
Accordingly, the Keldysh action with fields transformed as $\psi_{\pm}\mapsto
G\psi_{\pm}$ corresponds to the following Lindbladian,

$\mathcal{L}_{\hat{G},c}\hat{\rho}=-i\left[\hat{H}_{\hat{G}},\hat{\rho}\right]+\hat{L}_{\hat{G}}\hat{\rho}\hat{L}_{\hat{G}}^{\dagger}-\frac{1}{2}\left\\{\hat{L}_{\hat{G}}^{\dagger}\hat{L}_{\hat{G}},\hat{\rho}\right\\},$
(19)

where for simplicity we consider a single jump operator $\hat{L}$, and we
write $\hat{H}_{\hat{G}}=\hat{G}^{\dagger}\hat{H}\hat{G}$, and
$\hat{L}_{\hat{G}}=\hat{G}^{\dagger}\hat{L}\hat{G}$. The classical symmetry of
the Keldysh action implies that also the transformed Lindbladian should be
identical to the original Lindbladian, $\mathcal{L}_{\hat{G},c}=\mathcal{L}$,
or, equivalently,

$\mathcal{L}(\hat{G}\hat{\rho}\hat{G}^{\dagger})=\hat{G}\mathcal{L}(\hat{\rho})\hat{G}^{\dagger}.$
(20)

This is the defining relation for a weak symmetry.

Next, we show the equivalence between quantum and strong symmetries. Applying
the same logic as above, one can see that invariance of the Keldysh action
under $\psi_{+}\mapsto G\psi_{+}$ and $\psi_{-}\mapsto\psi_{-}$, which is one
particular choice of the various possibilities to transform the fields
$\psi_{\pm}$ independently, implies that the transformed Lindbladian,

$\mathcal{L}_{\hat{G},q}\hat{\rho}=-i\left(\hat{H}_{\hat{G}}\hat{\rho}-\hat{\rho}\hat{H}\right)+\hat{L}_{\hat{G}}\hat{\rho}\hat{L}^{\dagger}-\frac{1}{2}\left(\hat{L}_{\hat{G}}^{\dagger}\hat{L}_{\hat{G}}\hat{\rho}+\hat{\rho}\hat{L}^{\dagger}\hat{L}\right),$
(21)

has to be identical to the original Lindbladian,
$\mathcal{L}_{\hat{G},q}=\mathcal{L}$. Notice that in Eq. (21) only the
operators that act on the density matrix from the left are transformed under
the symmetry $\hat{G}$. For generic choices of $\hat{H}$ and $\hat{L}$, this
implies that

$\left[\hat{G},\hat{H}\right]=0,\qquad\left[\hat{G},\hat{L}\right]=0.$ (22)

These are the defining equations for a strong symmetry. Returning to the
example of driven open Bose-Einstein condensation and $\mathrm{U}_{q}$
symmetry, we note that the second equality is clearly violated for the jump
operators in Eq. (3) that describe particle gain and loss.

####  IV.1.3 Quantum symmetries, conservation laws, and slow hydrodynamic
modes

Global continuous quantum symmetries of both open and closed systems are
associated, through the Noether theorem, with conservation laws [541]. An
important example is again given by $\mathrm{U}_{q}$ phase rotation symmetry,
which implies conservation of the number of particles. Clearly, incoherent
pumping and losses as in Eq. (3) break particle number conservation. Formally,
the absence of particle number conservation becomes manifest as the absence of
$\mathrm{U}_{q}$ symmetry of the Keldysh action in Eq. (12).

Here we discuss two examples, where quantum symmetries are realized in driven
open quantum matter:

##### Heating to infinite temperature.

One important example of particle number conserving dissipation, which is
relevant, in particular, for experiments with cold atoms in optical lattices
[467], is given by dephasing with
$\hat{L}(\mathbf{x})=\sqrt{\gamma}\hat{\psi}^{\dagger}(\mathbf{x})\hat{\psi}(\mathbf{x})$
[404, 102, 474, 473, 201, 369]. Notably, the jump operator that describes
dephasing is Hermitian, $\hat{L}(\mathbf{x})=\hat{L}^{\dagger}(\mathbf{x})$,
which generically leads to heating to infinite temperature. For Hermitian jump
operators, the dissipator can be rewritten as a double commutator,
$\mathcal{D}\hat{\rho}=\hat{L}\hat{\rho}\hat{L}^{\dagger}-\frac{1}{2}\left\\{\hat{L}^{\dagger}\hat{L},\hat{\rho}\right\\}=-\frac{1}{2}\left[\hat{L},\left[\hat{L},\hat{\rho}\right]\right]$.
Consequently, the infinite temperature state
$\hat{\rho}_{\infty}=\hat{1}/\mathop{\mathrm{tr}}(\hat{1})$ is a steady state
of the master equation (1). Further, when $\left[\hat{H},\hat{L}\right]\neq
0$, which is the case for dephasing and the generic bosonic Hamiltonian given
in Eq. (2), $\hat{\rho}_{\infty}$ is typically also the unique steady state.

As key implications of the conservation law, several works have reported
algebraic behavior in the dynamics approaching the infinite temperature state.
For example, Refs. [474, 473] find instances of anomalous diffusion in the
Fock space of bosonic modes with a fixed total particle number. Further, Ref.
[102] reports algebraic behavior in the decoherence of an XXZ magnet with a
universal power law $\sim t^{-1.58}$, which can be traced back to
$\mathrm{U}_{q}$ symmetry under rotations about the $z$-axis.

##### Cooling to pure states.

Heating to infinite temperature can be replaced by the opposite behavior of
cooling into a pure state by means of dissipation. This is accomplished
through bilocal, yet also particle number conserving Lindblad operators in the
absence of a Hamiltonian. For example, for a bosonic system on a one-
dimensional (1D) lattice with lattice sites $l\in\\{1,\dotsc,L\\}$,
$\mathrm{U}_{q}$ symmetric and thus number conserving cooling into a perfectly
phase-coherent condensate is obtained through purely dissipative evolution
with Lindblad operators
$\hat{L}_{l}=\sqrt{\gamma}\left(\hat{\psi}_{l}^{\dagger}+\hat{\psi}_{l+1}^{\dagger}\right)\left(\hat{\psi}_{l}-\hat{\psi}_{l+1}\right)$
[166]. The pure state reached as the dynamical fixed point of this evolution
for $N$ particles, $\lvert
D\rangle\sim\left(\sum_{l}\hat{\psi}^{\dagger}_{l}\right)^{N}\lvert 0\rangle$
where $\lvert 0\rangle$ is the vacuum state, is an example of a dark state
(see Appendix D). Further examples will be encountered in Secs. XI.2, and
XIII.1.

Also in this case, the conservation law has observable physical consequences.
For example, Ref. [293] has found diffusive behavior on top of dissipatively
stabilized, number conserving superfluid of fermions.

Most of the results reported here have been established numerically. Finally,
we briefly touch upon the fate and impact of continuous global quantum
symmetries in long-wavelength effective theories. In spatially extended
systems, local conservation of the Noether charge associated with a quantum
symmetry is expressed through a continuity equation and entails the existence
of a gapless hydrodynamic mode. (Notable exceptions occur when the associated
transport coefficient vanishes, as is the case for (topological) insulators at
zero temperature, or in the presence of anomalies, which are associated to
conservation laws of subsystems on the bare level of description [466, 209]).
Crucially, charge conservation is a consequence of quantum symmetry of the
action on the microscopic level. A long-wavelength effective theory is derived
through integrating out short-wavelength modes, which can act as a bath for
the long-wavelength modes. In particular, Noether charge can be transferred
between the short- and long-wavelength modes. Therefore, integrating out
short-wavelength modes generates dissipative contributions to the action that
break the microscopic quantum symmetry. Indeed, the paradigmatic classical
models of critical dynamics with conserved quantities do not exhibit such a
symmetry [286]. However, the continuity equation remains preserved as an exact
property of the theory, even if its transport coefficients receive RG
corrections, including a possible emergent irreversibility of the
hydrodynamics [263, 264, 265, 145, 235, 266].

####  IV.1.4 Spontaneous breaking of classical symmetries and dissipative
Goldstone theorem

While the $\mathrm{U}_{q}$ symmetry of the Keldysh action specified by Eqs.
(11) and (12) is broken explicitly by particle gain and loss, the remaining
$\mathrm{U}_{c}(1)$ symmetry can be broken spontaneously through the formation
of a condensate in the steady state, corresponding to a nonvanishing
expectation value of the classical field,
$\langle\psi_{c}(t,\mathbf{x})\rangle\neq 0$. In general, the spontaneous
breaking of a global continuous classical symmetry leads to the occurrence of
a massless mode, i.e., a mode whose dispersion relation $\omega(\mathbf{q})$
obeys $\omega(\mathbf{q})\to 0$ when the momentum goes to zero, $\mathbf{q}\to
0$. The existence of such a Goldstone mode is an exact property of the Keldysh
theory [541]. In particular, it serves as an important benchmark for
approximate calculations of the excitation spectrum. An explicit example is
provided by the following analysis of fluctuations around a mean-field
condensate.

Evaluating the Keldysh functional integral for a driven open condensate in a
mean-field approximation corresponds to neglecting fluctuations around the
solutions to the classical field equations given by $\delta
S/\delta\psi_{c}^{*}=0$ and $\delta S/\delta\psi_{q}^{*}=0$. The former is
solved by $\psi_{q}(t,\mathbf{x})=0$, and the latter reproduces Eq. (4) for
the operator expectation value
$\psi(t,\mathbf{x})=\langle\hat{\psi}(t,\mathbf{x})\rangle$, the only
difference being a factor $\sqrt{2}$ in the definition of the condensate
amplitude, which is introduced in the bosonic Keldysh rotation Eq. (9). We
thus find $\psi_{c}(t,\mathbf{x})=\psi_{0}(t,\mathbf{x})=\psi_{0}e^{-i\mu t}$,
where, for $\gamma_{p}>\gamma_{l}$, the mean-field condensate amplitude is
given by $\psi_{0}=\sqrt{-r_{d}/u_{d}}$ with $r_{d}=(\gamma_{l}-\gamma_{p})/2$
and, $\mu=u_{c}\psi_{0}^{2}$. Here, without loss of generality, we choose
$\psi_{0}$ to be real and positive. In contrast, for $\gamma_{p}<\gamma_{l}$,
there is no condensation in the steady state, $\psi_{0}=0$. To study
fluctuations around the mean-field condensate, we parameterize the classical
and quantum fields as
$\psi_{c}(t,\mathbf{x})=\left(\psi_{0}+\delta\psi_{c,1}(t,\mathbf{x})+i\delta\psi_{c,2}(t,\mathbf{x})\right)e^{-i\mu
t}$ and
$\psi_{q}(t,\mathbf{x})=\delta\psi_{q,1}(t,\mathbf{x})+i\delta\psi_{q,2}(t,\mathbf{x})$,
and we expand the action to second order in $\delta\psi_{\nu,i}$ for $\nu=c,q$
and $i=1,2$. We find that fluctuations of $\delta\psi_{c,1}$ are gapped, which
implies that terms that involve derivatives of this field can be neglected as
they do not affect the low-frequency and low-momentum form of the theory
qualitatively. Then, after integrating out $\delta\psi_{c,1}$ and
$\delta\psi_{q,1}$, the Keldysh partition function takes the form

$Z=\int D[\delta\psi_{c,2},\delta\psi_{q,2}]\,e^{iS^{\prime}},$ (23)

where the action is given by

$S^{\prime}=2\psi_{0}\int_{\omega,\mathbf{q}}\delta\psi_{q,2}(\omega,\mathbf{q})\left[\left(\omega+iDq^{2}\right)\delta\psi_{c,2}(\omega,\mathbf{q})\right.\\\
\left.\vphantom{\left(\omega+iDq^{2}\right)}+2\psi_{0}\Delta\delta\psi_{q,2}(\omega,\mathbf{q})\right].$
(24)

We write $q=\left\lvert\mathbf{q}\right\rvert$, and for the integration over
frequencies and $d$-dimensional momentum space, we abbreviate
$\int_{\omega,\mathbf{q}}=\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\int_{\mathbb{R}^{d}}\frac{d\mathbf{q}}{\left(2\pi\right)^{d}}$.
The parameters in the action are

$D=\frac{u_{c}}{2mu_{d}},\quad\Delta=\frac{\gamma+2u_{d}\psi_{0}^{2}}{2\psi_{0}^{2}}\left(1+\frac{u_{c}^{2}}{u_{d}^{2}}\right),$
(25)

where $\gamma=(\gamma_{l}+\gamma_{p})/2$. From the action $S^{\prime}$, we can
read off the retarded Green’s function:

$G^{R}(\omega,\mathbf{q})=\langle\delta\psi_{c,2}(\omega,\mathbf{q})\delta\psi_{q,2}(\omega,\mathbf{q})\rangle=\frac{1}{2\psi_{0}\left(\omega+iDq^{2}\right)}.$
(26)

The pole of the retarded Green’s function at $\omega=-iDq^{2}$ encodes the
dispersion relation $\omega=\omega(\mathbf{q})$ of elementary excitations. As
anticipated, $\omega(\mathbf{q})$ describes a Goldstone mode with
$\omega(\mathbf{q})\to 0$ for $\mathbf{q}\to 0$. The Goldstone mode
corresponds to fluctuations of $\delta\psi_{c,2}$ and, therefore, to
fluctuations of the phase of the condensate. However, since phase fluctuations
are gapless, they should not be assumed to be small, i.e., the expansion in
$\delta\psi_{c,2}$ is, in fact, not justified. Instead, to obtain a consistent
theory of fluctuations around the mean-field condensate, we must set
$\psi_{c}=\sqrt{\rho}e^{i\left(\theta-\mu t\right)}$ where
$\theta\in[0,2\pi)$, and only fluctuations of $\rho$ around the mean field
$\rho_{0}=\psi_{0}^{2}$ can safely be assumed to be small. A full treatment of
nonlinear fluctuations of the condensate phase is given in Ref. [541], and
leads to the Kardar-Parisi-Zhang (KPZ) action, or equivalently, the famous KPZ
equation [311, 566]. Originally, the KPZ equation was used to describe the
dynamics of growing interfaces, where the dynamical variable is the noncompact
height of the interface [316]. The consequences of an emergent KPZ action for
the compact phase variable $\theta$ are discussed in Sec. VII.

The Goldstone mode that is associated with the breaking of the
$\mathrm{U}_{c}(1)$ symmetry of a driven and open condensate propagates
diffusively with diffusion constant $D$. In contrast, the condensation
transition in a closed system which has $\mathrm{U}_{q}$ symmetry leads to the
appearance of coherently propagating sound waves with dispersion relation
$\omega(\mathbf{q})=\pm cq$, where $c$ is the speed of sound. In the models
for critical dynamics with conservation laws in thermal equilibrium, the
existence of such coherently propagating hydrodynamic modes is taken into
account explicitly by introducing the corresponding conserved densities as
dynamical variables [286]. While the existence of such modes is a consequence
of an underlying $\mathrm{U}_{q}$ symmetry, this symmetry is no longer visible
in the mesoscopic models used to analyze critical behavior as discussed at the
end of the previous section.

###  IV.2 Mixed vs. pure states, classical vs. quantum scaling

Closed quantum many-body systems in thermal equilibrium can exhibit both
classical and quantum critical behavior—the former occurring in mixed thermal
states at finite temperature $T>0$, and the latter in pure ground states at
$T=0$ [508]. As we explain in the following, the concepts of classical and
quantum criticality retain their validity in driven and open systems. We base
our discussion on an analysis of canonical scaling, first of the propagators
of the Gaussian theory, and then, in Sec. IV.3 below, of interaction vertices.
Thereby, we highlight the analysis of canonical scaling as an important tool
both in and out of equilibrium.

For concreteness, we consider the driven and open condensate, whose Gaussian
action is given in Eq. (16). In the retarded component, the difference between
loss and pump rates, $r_{d}=(\gamma_{l}-\gamma_{p})/2$, measures the distance
from the mean-field condensation transition that occurs at
$\gamma_{l}=\gamma_{p}$. In contrast, in the Keldysh component, the noise
caused by incoherent pumping and gain adds up,
$\gamma=(\gamma_{l}+\gamma_{p})/2$. More generally, by considering the
retarded and Keldysh or noise components of the inverse propagator (i.e., the
matrix in the quadratic action Eq. (16)), we are lead to distinguish two types
of gaps, the spectral gap and the noise gap, whose significance we discuss in
the following.

####  IV.2.1 Spectral gap and criticality

On the level of the Gaussian theory for a single field component, the
dispersion relation of elementary excitations is obtained by solving
$P^{R}(\omega,\mathbf{q})=0$ for $\omega=\omega(\mathbf{q})$. Propagators are
matrix-valued for multicomponent fields, and the equation to be solved in this
case is $\det\bigl{(}P^{R}(\omega,\mathbf{q})\bigr{)}=0$. The spectral or
dissipative gap is defined by

$\Delta_{p}=\min_{\mathbf{q}}\left(-\mathop{\mathrm{Im}}(\omega(\mathbf{q}))\right).$
(27)

That is, the dissipative gap is the minimal distance of the dispersion
relation from the real axis in the complex $\omega$ plane. Instead, for a
closed system with a purely real dispersion relation, the spectral gap should
be defined in terms of the minimal distance from the imaginary axis [15], and
it is just the usual mass term that should be tuned to zero to access
criticality [112, 642, 311]. For our example of a driven and open condensate,
on the Gaussian level, the spectral gap is thus given by $r_{d}$ in the
symmetric phase, and the mean-field critical point at $r_{d}=0$ corresponds to
a zero of the spectral gap. The calculation of corrections to mean-field
theory in a loop expansion involves integrals over products of Green’s
functions [642]. At the mean-field critical point $r_{d}=0$, the retarded and
advanced Green’s functions exhibit canonical critical scaling,
$G^{R}(\omega,\mathbf{q})=G^{A}(\omega,\mathbf{q})^{*}=1/P^{R}(\omega,\mathbf{q})=1/[\omega-q^{2}/(2m)]\sim
1/q^{2}$, which can lead to infrared (IR) divergencies, i.e., divergencies of
momentum integrals for $q\to 0$. In the framework of the RG, such IR
divergencies drive the RG flow at criticality towards the universal scaling
regime which encodes corrections to mean-field critical exponents [112]. In
contrast, a finite spectral gap implies the absence of zero modes of $P^{R}$,
i.e., $\left\lvert\omega(\mathbf{q})\right\rvert\neq 0$ for all momenta
$\mathbf{q}$, and, consequently, the absence of infrared divergencies.

####  IV.2.2 Noise gap: classical and quantum scaling

While a nonvanishing spectral gap results from the absence of zero modes of
$P^{R}(\omega,\mathbf{q})$, a nonvanishing noise gap signifies the absence of
zero modes of $P^{K}(\mathbf{q})$. For the sake of generality, we consider
here a momentum-dependent Keldysh component. However, for Markovian
dissipation, $P^{K}(\mathbf{q})$ does not depend on frequency. Denoting the
eigenvalues of $P^{K}(\mathbf{q})$ by $i2\gamma(\mathbf{q})$, we define the
noise gap as

$\Delta_{n}=\min_{\mathbf{q}}\left\lvert\gamma(\mathbf{q})\right\rvert.$ (28)

A finite or vanishing value of the noise gap decides whether the criticality
that is induced by a vanishing spectral gap is classical or quantum,
respectively.

To see that this is the case, it is instructive to compare the noise component
$P^{K}=i2\gamma$ of the driven and open condensate to the one of a bosonic
system coupled to a bath in thermal equilibrium,
$P^{K}(\omega)=i2\gamma\omega\coth(\beta\omega/2)$ [311]. In contrast to a
system that is subjected to Markovian dissipation, the Keldysh component
$P^{K}(\omega)$ of a system in thermal equilibrium depends on the frequency
$\omega$. Depending on the temperature, $P^{K}(\omega)$ exhibits different
behaviors:

##### Classical scaling.

In the high-temperature limit, we obtain $P^{K}(\omega)\sim i4\gamma T\sim
q^{0}$. That is, as in the case of the driven open condensate, the Keldysh
component is frequency- and momentum-independent and exhibits a finite noise
gap. If, under these conditions, a phase transition is induced by closing the
spectral gap, the system will exhibit classical criticality. Therefore, we
refer to $P^{K}(\omega)\sim q^{0}$ as classical scaling. Indeed, as explained
above, critical behavior is driven by fluctuations on small frequency and
momentum scales, $\omega,q\to 0$. For the system in thermal equilibrium, the
excitation energies of such fluctuations are much smaller than $T$. Therefore,
these fluctuations are effectively exposed to a high temperature and behave
classically. As we show explicitly in Sec. IV.3.2 below, the same applies to
systems out of equilibrium, with temperature replaced by Markovian noise
level.

##### Quantum scaling.

In the limit of zero temperature, $P^{K}(\omega)\sim
i2\gamma\left\lvert\omega\right\rvert$ is frequency-dependent and vanishes for
$\omega\to 0$. We refer to the scaling $P^{K}(\omega)\sim\omega\sim q^{z}$ as
quantum scaling, where for a system with parabolic dispersion relation the
dynamical exponent $z$ takes the value $z=2$. At $T=0$, the system in thermal
equilibrium is in its pure ground state, and tuning the spectral gap to zero
induces a quantum phase transition with associated quantum critical behavior.
A concrete example of such quantum criticality in a driven and open system is
discussed in Sec. XI.

The above argument establishes the connection between quantum scaling and zero
temperature. As we show in the following on the level of the Gaussian theory
in Eq. (16) for which the canonical scaling is developed, quantum scaling is
also directly related to the purity of the state. To that end, we consider the
Hermitian covariance matrix [51], given by the equal-time Keldysh Green’s
function,

$\Gamma(\mathbf{x},\mathbf{x}^{\prime})=\langle\\{\hat{\psi}(\mathbf{x}),\hat{\psi}^{\dagger}(\mathbf{x}^{\prime})\\}\rangle=iG^{K}(t,\mathbf{x},t,\mathbf{x}^{\prime}).$
(29)

For a mixed state bosonic state, the eigenvalues $\xi$ of the Hermitian matrix
$\Gamma$ satisfy $\left\lvert\xi\right\rvert\geq 1$; in contrast, for a pure
state, the eigenvalues $\xi$ have unit modulus,
$\left\lvert\xi\right\rvert=1$. Further, the covariance matrix squares to the
identity, $\Gamma^{2}=\mathbbm{1}$, which is equivalent to
$\mathop{\mathrm{tr}}\\!\left(\hat{\rho}^{2}\right)=1$. The covariance matrix
of a system with translational invariance obeys
$\Gamma(\mathbf{x},\mathbf{x}^{\prime})=\Gamma(\mathbf{x}-\mathbf{x}^{\prime})$
and is, therefore, diagonal in momentum space,

$\Gamma(\mathbf{q})=\int_{\mathbf{x}}e^{-i\mathbf{q}\cdot\mathbf{x}}\Gamma(\mathbf{x})=i\int_{\omega}G^{K}(\omega,\mathbf{q}).$
(30)

According to the abovementioned properties of the covariance matrix, the
absolute value of $\Gamma(\mathbf{q})$ is constant for a pure state,
$\left\lvert\Gamma(\mathbf{q})\right\rvert=1$. This condition is compatible
only with quantum scaling of $P^{K}$. To see that this is the case, note that
$G^{K}(\omega,\mathbf{q})=-G^{R}(\omega,\mathbf{q})P^{K}(\omega)G^{A}(\omega,\mathbf{q}),$
where
$G^{R}(\omega,\mathbf{q})=1/P^{R}(\omega,\mathbf{q})=G^{A}(\omega,\mathbf{q})^{*}$.
Therefore, canonical critical scaling $G^{R,A}(\omega,\mathbf{q})\sim q^{-2}$
implies $G^{K}(\omega,\mathbf{q})\sim q^{-4}$ and
$G^{K}(\omega,\mathbf{q})\sim q^{-2}$ for classical and quantum scaling of
$P^{K}(\omega)$, respectively. With $d\omega\sim q^{2}$, we find that
classical scaling leads to $\Gamma(\mathbf{q})\sim q^{-2}$, while quantum
scaling results in $\Gamma(\mathbf{q})\sim q^{0}$. In the classical case,
momentum modes are strongly occupied at low momenta in a bosonic system. But
this is not the case for pure states: As claimed above, only quantum scaling
is compatible with a pure state with
$\left\lvert\Gamma(\mathbf{q})\right\rvert=1$.

For simplicity, we have considered here a scalar bosonic field, and we have
assumed the system to be invariant under continuous spatial translations.
However, the argument generalizes to vector fields, and to fields defined on a
lattice. Gaussian states and the covariance matrix for these cases, both for
bosons and fermions, are discussed in Appendix C. Fermionic Gaussian states
will play a key role in our discussion of topological phase transitions out of
equilibrium in Sec. XIII.1. There, in Eq. (112), we will introduce yet another
type of gap, the purity gap, which measures the distance of the eigenvalues of
the covariance matrix from zero, and distinguishes pure from mixed states.
This definition of the purity gap applies only to fermions, for which the
eigenvalues of the covariance matrix are bounded between $-1$ and $+1$.

###  IV.3 Scaling arguments: deterministic and semiclassical limit

Scaling arguments are fundamental tools in both equilibrium and nonequilibrium
statistical physics. They help assess the relevance of operators at long
wavelengths and enable controlled approximations, such as the $1/N$ expansion
in $\mathrm{O}(N)$ models. In driven open quantum systems, the Keldysh field
integral treats quantum and statistical fluctuations equally, resulting in a
complex microscopic description. Here, scaling arguments are systematically
applied to reduce complexity and derive an effective long-wavelength theory
that governs the dynamics in relevant limiting cases.

There are two important limits in which subleading fluctuations can be
disregarded systematically: the deterministic limit and the semiclassical
limit. The _deterministic limit_ corresponds to an expansion around a
macroscopically occupied field configuration such as a condensate wave
function, using an $N^{-1/2}$ expansion, where $N$ is the total number of
particles. This limit intentionally omits both quantum and statistical
fluctuations, resulting in a non-Hermitian evolution equation equivalent to a
mean-field approximation of the quantum master equation. Instead, the
_semiclassical limit_ employs canonical power counting to neglect subleading
quantum fluctuations in systems in which statistical fluctuations are
dominant. This occurs, e.g., near critical points at finite noise level, and
both in and out of equilibrium as discussed in Sec. IV.2.

####  IV.3.1 Deterministic limit and relation to non-Hermitian physics

The Keldysh partition function
$Z=\int\mathcal{D}[\psi_{c}^{*},\psi_{c},\psi_{q}^{*},\psi_{q}]\,e^{iS}$ is an
integral over field configurations, whose respective importance is weighted by
the exponential $e^{iS}$. In contrast, classical field theories are controlled
by a single deterministic field configuration that extremizes the action $S$.
Such _classical_ configurations often yield a dominant contribution to the
Keldysh field integral. The deterministic limit focuses exclusively on this
classical configuration and discards any fluctuations. This limit arises as a
_scaling limit_ and is particularly relevant in the presence of condensation
phenomena like Bose-Einstein condensation or spontaneous magnetization, when a
single quantum state is macroscopically occupied by bosonic degrees of freedom
in the presence of weak noise and weak interactions.

For concreteness, consider the driven open condensate from Sec. II.1.
Rewriting its action $S=S_{H}+S_{D}$ in Eqs. (11) and (12) in Keldysh
coordinates yields

$S=\int_{\omega,\mathbf{q}}\left(\psi_{c}^{*},\psi_{q}^{*}\right)\begin{pmatrix}0&P^{A}(\omega,\mathbf{q})\\\
P^{R}(\omega,\mathbf{q})&P^{K}\end{pmatrix}\begin{pmatrix}\psi_{c}\\\
\psi_{q}\end{pmatrix}\\\
+\int_{t,\mathbf{x}}\left\\{4iu_{d}\left\lvert\psi_{c}\psi_{q}\right\rvert^{2}-\left[\left(u_{c}-iu_{d}\right)\psi_{q}^{\ast}\psi_{c}^{\ast}\left(\psi_{c}^{2}+\psi_{q}^{2}\right)+\mathrm{c.c.}\right]\right\\}.$
(31)

The quadratic part, including the propagators $P^{R,A,K}$, is specified in Eq.
(16). Consider now the case of weak nonlinearities and a macroscopically
occupied mode, i.e., a finite expectation value of the classical field
$\langle\psi_{c}\rangle$. An important example is Bose-Einstein condensation,
$\langle\psi_{c}\rangle\sim\sqrt{N/V}$, with $N$ the particle number and $V$
the system volume. In momentum space

$\displaystyle\psi_{c}(t,{\bf q}=0)\sim\langle\psi_{c}(t,{\bf
q}=0)\rangle=\sqrt{V}\langle\psi_{c}\rangle\sim\sqrt{N},$ (32)
$\displaystyle\quad\psi_{c}(t,{\bf q}\neq 0)\sim N^{0},\quad\psi_{q}(t,{\bf
q})\sim N^{0}.$ (33)

The quantum field cannot condense, hence its scaling. The condensed mode is
singled out according to $\psi_{c}(t,{\bf
q})\equiv\sqrt{V}\langle\psi_{c}\rangle\delta({\bf q})+\delta\psi_{c}(t,{\bf
q})$. In real space the Fourier transform

$\displaystyle\psi_{c}(t,{\bf
x})=\frac{1}{\sqrt{V}}\int_{\mathbf{\mathbf{q}}}e^{i\mathbf{q}\cdot\mathbf{x}}\psi_{c}(t,{\bf
q})=\langle\psi_{c}\rangle+\delta\psi_{c}(t,{\bf x})$ (34)

reveals the scaling of the condensate $\langle\psi_{c}\rangle\sim N^{0}$,
while the fluctuations and the quantum fields scale as $\delta\psi_{c}(t,{\bf
x}),\psi_{q}(t,{\bf x})\sim N^{-1/2}$. This consideration is readily
generalized to inhomogeneous condensate fields
$\langle\psi_{c}\rangle\to\langle\psi_{c}(t,{\bf x})\rangle$ that vary
smoothly in space-time on scales much larger than the microscopic length and
time scales.

Drawing the thermodynamic limit $V\to\infty$ at constant density $N/V$, both
$\psi_{q}$ and $\delta\psi_{c}$ yield subleading contributions. Expanding to
leading order in $N^{-1/2}$ yields an action

$\displaystyle
S=\int_{t,\mathbf{x}}\Big{\\{}\psi_{q}^{\ast}\big{[}P^{R}-(u_{c}-iu_{d})|\langle\psi_{c}\rangle|^{2}\big{]}\langle\psi_{c}\rangle+\text{c.c.}\Big{\\}},$
(35)

linear in the quantum field. Integration over $\psi_{q},\psi_{q}^{\ast}$ is
then performed exactly: The resulting $\delta$-functional constrains the
dynamics of the condensate to the saddle point equation $0=\delta
S/\delta\psi_{q}^{\ast}(x)$, reminiscent of a classical, i.e., fluctuationless
field theory. For the driven open condensate, this reproduces the dissipative
Gross-Pitaevskii equation (4), i.e., the mean-field approximation. Quantum
_and_ statistical fluctuations are discarded, yielding a non-Hermitian
equation of motion.

The field integral expresses the approximation as a controlled expansion
around a macroscopic condensate. It provides justification for scenarios where
a macroscopically large order parameter meets weak nonlinearities, such as,
e.g., realized in atomic condensates at low temperatures [120], or in photonic
non-Hermitian systems [456]. It also provides criteria for when the
approximation breaks down: An important case is a second order phase
transition where the macroscopic order parameter vanishes,
$\langle\psi_{c}\rangle\to 0$. Then, the expansion is no longer justified and
quantum and statistical fluctuations need to be incorporated.

Another common but different approximation that neglects statistical
fluctuations starts from the Lindblad equation (1) and discards the jump terms
by setting $\hat{L}\rho\hat{L}^{\dagger}\to 0$. This yields the evolution
equation

$\displaystyle\partial_{t}\hat{\rho}=-i\left(\hat{H}_{\text{eff}}^{\phantom{\dagger}}\hat{\rho}-\hat{\rho}\hat{H}_{\text{eff}}^{\dagger}\right),$
(36)

with an _effective_ non-Hermitian Hamiltonian
$\hat{H}_{\text{eff}}=\hat{H}-\frac{i}{2}\hat{L}^{\dagger}\hat{L}$. In this
case, the evolution neglects any feedback from the environment on the state,
and the system approaches a pure state
$\hat{\rho}\to|\psi_{0}\rangle\langle\psi_{0}|$, corresponding to the
eigenstate $\hat{H}_{\text{eff}}|\psi_{0}\rangle=\varepsilon|\psi_{0}\rangle$
with the largest imaginary part of $\varepsilon$. The corresponding equation
of motion, however, is not probability conserving,
$\partial_{t}\mathop{\mathrm{tr}}\left(\hat{\rho}\right)\neq 0$. In Keldysh
field theory, this manifests in a violation of the causality condition
$\left.S_{\text{eff}}[\psi_{+},\psi_{-}]\right|_{\psi_{+}\to\psi_{-}}\neq 0$
for the action associated with the non-Hermitian Hamiltonian
$\hat{H}_{\text{eff}}$.

The non-Hermitian evolution generated by $\hat{H}_{\mathrm{eff}}$ thus does
not reflect a single particular field configuration $\psi_{c},\psi_{q}$ and it
does not extremize the action. Instead of describing the probabilistically
most likely evolution, i.e., the largest contribution to the partition
function $Z$, the non-Hermitian Hamiltonian evolution rather selects a
trajectory which results from a _rare_ sequence (zero jumps) of system-bath
interactions. This interpretation is particularly transparent when modeling
the environment as performing measurements (or general positive operator-
valued measures, POVMs [443]) on the system. Then the effective Hamiltonian
yields a measurement trajectory in which measurement outcomes of a particular
type have been discarded, e.g., via postselection [578, 244, 644, 215]. Yet,
analyzing $\hat{H}_{\text{eff}}$ may still provide valuable information
regarding the response of the system or on potential dynamical instabilities.
This has been successfully exploited to explore the non-Hermitian band
structure in topological systems of photons [62, 30, 407, 456]. Care, however,
needs to be exerted regarding the interpretation of the effective Hamiltonian.
Consider, for instance, bosons or fermions with single-particle loss and pump
from Sec. II.1, i.e.,
$\hat{L}_{p}=\sqrt{\gamma_{p}}\hat{\psi}^{\dagger},\hat{L}_{l}=\sqrt{\gamma_{l}}\hat{\psi}$
and $\hat{H}_{0}=\omega_{0}\hat{\psi}^{\dagger}\hat{\psi}$. Then disregarding
the jump terms in the evolution yields

$\begin{split}\hat{H}_{\mathrm{eff}}&=\omega_{0}\hat{\psi}^{\dagger}\hat{\psi}-i\frac{\gamma_{l}}{2}\hat{\psi}^{\dagger}\hat{\psi}-i\frac{\gamma_{p}}{2}\hat{\psi}\hat{\psi}^{\dagger}\\\
&=\left[\omega_{0}-\frac{i}{2}\left(\gamma_{l}+\zeta\gamma_{p}\right)\right]\hat{\psi}^{\dagger}\hat{\psi}-\frac{i\gamma_{p}}{2}.\end{split}$
(37)

Using the effective Hamiltonian to define a retarded Green’s function in an
analogous way to Hamiltonian systems in equilibrium, i.e.,
$G^{R}_{\text{eff}}\equiv\left(\omega-\omega_{\text{eff}}+i0^{+}\right)^{-1}$,
where $\omega_{\text{eff}}$ are the eigenvalues of $\hat{H}_{\text{eff}}$,
does not provide the correct retarded Green’s function. The latter is instead
$G^{R}=\left[\omega-\omega_{0}+i\left(\gamma_{l}-\zeta\gamma_{p}\right)\right]^{-1}$,
see Appendix C. One observes a qualitative difference through the flipped sign
in front of the pumping term $\sim\gamma_{p}$. For nonlinear or interacting
systems, the effective Hamiltonian approximation may become further
uncontrolled as rare trajectories may in general not reflect the behavior of
the average ensemble.

####  IV.3.2 Canonical power counting and semiclassical limit

In Sec. IV.2, we have concluded that the driven open condensation transition
exhibits classical critical behavior. This conclusion was based on a
comparison of the scaling behavior of the noise component of the inverse
propagator for a driven open condensate and for a system in equilibrium at
high temperature. As we discuss next, a more formal and unifying argument that
corroborates this conclusion can be given in terms of canonical power
counting. In general, canonical scaling dimensions of the couplings that
appear in an action determine the RG relevance of the respective couplings for
the low-frequency and long-wavelength dynamics of the system.

##### Semiclassical limit of the driven open condensation transition.

The canonical momentum scaling dimensions of the fields $\psi_{c}$ and
$\psi_{q}$ and, therefore, of all couplings in the action, can be inferred
from the scaling of the inverse propagator and the condition that the action
is dimensionless, i.e., it does not scale with momentum. In particular, at the
mean-field critical point of the driven open condensate, which is determined
by the vanishing of the spectral gap $r_{d}$, the retarded component scales as
$P^{R}(\omega,\mathbf{q})=\omega-q^{2}/(2m)\sim q^{2}$. In contrast, the noise
component is frequency- and momentum-independent, and, therefore, does not
scale, $P^{K}=i2\gamma\sim q^{0}$. This leads to $\psi_{c}\sim q^{(d-2)/2}$
and $\psi_{q}\sim q^{(d+2)/2}$. While the canonical scaling dimensions of the
fields are determined by the Gaussian part of the action in Eq. (16), they in
turn determine the relevance of interaction vertices that are not part of the
Gaussian theory. Therefore, power counting has strong implications beyond the
analysis of the Gaussian theory in the previous sections. In particular, the
canonical scaling dimensions of the fields imply that local vertices with more
than two quantum fields are irrelevant in spatial dimensions $d>2$ [541].
Dropping these terms is equivalent to taking the semiclassical limit [311, 16]
and results in

$S=\int_{t,\mathbf{x}}\left\\{\psi_{q}^{*}\left[i\partial_{t}+\left(K_{c}-iK_{d}\right)\nabla^{2}-r_{c}+ir_{d}\right]\psi_{c}+\mathrm{c.c.}\right.\\\
\left.-\left[\left(u_{c}-iu_{d}\right)\psi_{q}^{*}\psi_{c}^{*}\psi_{c}^{2}+\mathrm{c.c.}\right]+i2\gamma\psi_{q}^{*}\psi_{q}\right\\},$
(38)

where $K_{c}=1/(2m)$ and the term that is proportional to $r_{c}$ corresponds
to the transformation to a rotating frame, $\psi_{\nu}(t,\mathbf{x})\mapsto
e^{ir_{c}t}\psi_{\nu}(t,\mathbf{x})$ for $\nu=c,q$, and is included here for
the sake of generality. Further, the diffusion term with coefficient $K_{d}$
is absent in the microscopic model, but is added here as it will inevitably be
generated upon integrating out short-scale fluctuations [543, 544, 541]. The
action in the semiclassical limit is equivalent to a stochastic equation of
motion for the condensate field [541],

$i\partial_{t}\psi=\left[-\left(K_{c}-iK_{d}\right)\nabla^{2}+r_{c}-ir_{d}+\left(u_{c}-iu_{d}\right)\left\lvert\psi\right\rvert^{2}\right]\psi+\xi,$
(39)

with a Gaussian noise source $\xi$ that vanishes on average,
$\langle\xi(t,\mathbf{x})\rangle=0$, and whose fluctuations are determined by
the noise component of the inverse propagator,
$\langle\xi(t,\mathbf{x})\xi^{*}(t^{\prime},\mathbf{x}^{\prime})\rangle=2\gamma\delta(t-t^{\prime})\delta(\mathbf{x}-\mathbf{x}^{\prime})$.
Here and in the following, we omit the subscript “$c$” of the field that
appears in the Langevin equation, with the understanding that such equations
always apply to classical fields. Compared to the deterministic saddle point
equation discussed in the last subsection, the Langevin differs by the
addition of noise. This random force is the element that allows the system to
explore configurations beyond the deterministic path. The description in terms
of a Langevin equation establishes the formal connection to the paradigmatic
models of classical dynamical critical behavior, which are determined by
systems of coupled Langevin equations for the order parameter field and slow
hydrodynamic modes that are associated with conservation laws [286]. However,
a fundamental difference between the models of Ref. [286] and Eq. (39) lies in
the fact that the former correspond to systems in thermal equilibrium whereas
the latter describes a condensation transition in a nonequilibrium steady
state. Formally, this can be seen by noting that the Keldysh action Eq. (38)
is, in general, not invariant under the semiclassical limit of the
transformation $\mathcal{T}_{\beta}$ Eq. (14). The latter is a symmetry of the
Keldysh action only when $K_{c}/K_{d}=u_{c}/u_{d}$. We note that the analogous
condition $K_{c}/K_{d}=r_{c}/r_{d}$ for the two mass scales $r_{c}$ and
$r_{d}$ can always be satisfied in a suitably chosen rotating frame, i.e., by
modifying $r_{c}$ through a transformation
$\psi_{c}(t,\mathbf{x})\mapsto\psi_{c}(t,\mathbf{x})e^{-i\omega t}$. A
violation of the condition $K_{c}/K_{d}=u_{c}/u_{d}$ leads to a modification
of critical behavior at the condensation transition [543, 567, 544, 541]. This
is discussed in more detail in Sec. VIII.1.

##### Absence of a semiclassical limit for models with quantum scaling.

As discussed above, quantum criticality can be realized for a system that is
coupled to a thermal bath at zero temperature, such that $P^{K}\sim\omega\sim
q^{2}$, which leads to $\psi_{c}\sim\psi_{q}\sim q^{d/2}$. Consequently,
interaction vertices with different numbers of classical and quantum fields
but an equal total number of fields are equally relevant, and a semiclassical
description is not applicable. The simultaneous vanishing of the spectral and
noise gaps corresponds to a double fine-tuning that is required to realize
nonequilibrium quantum criticality, see Sec. XI. This is analogous to quantum
phase transitions in thermal equilibrium, where one needs to fine-tune not
only a system parameter to its critical value, but also the noise gap to zero,
via $T\to 0$.

##  V Absorbing state phase transitions and directed percolation

We now start our overview of instances of universality in driven open systems,
starting with realizations of paradigmatic classes of nonequilibrium
universality using quantum simulators (cf. Tab. 1). An important class of
genuine nonequilibrium phase transitions are absorbing state phase
transitions. They appear in a variety of classical dynamical systems, e.g., in
population models [236], in epidemics [459] or in chemical reactions [451].
Here, the _absorbing state_ corresponds to a macroscopic configuration which
may be reached during the time evolution but which can never be left. A
drastic example is the extinction of an entire species in population models.

An absorbing state phase transitions is a transition in the structure of the
dynamics: an absorbing or inactive phase is separated from an active phase. In
the former, the absorbing state is dynamically stable and attractive. It will
be reached from any initial state in finite time, leading the dynamics to
always terminate in this particular state. In contrast, in the active phase,
the absorbing state remains a stationary solution of the evolution, _but_ it
becomes dynamically unstable. The system traverses a large part of the
accessible configuration space before reaching the absorbing state. This leads
to the persistence of nontrivial dynamics up to exponentially large time
scales $t\sim\exp(N)$, where $N$ is the number of degrees of freedom, e.g.,
the sites of a lattice, and $\exp(N)$ is the size of the configuration space
explored by these degrees of freedom. In the thermodynamic limit,
$N\rightarrow\infty$, the absorbing state thus becomes unreachable and the
dynamics proceed indefinitely.

An absorbing state breaks detailed balance, which becomes crucial close to the
absorbing state phase transition. It invalidates the description of the
critical dynamics in terms of an equilibrium ensemble. This may be illustrated
in a configuration space picture: when approaching the absorbing state, say
the extinction of a species, the accessible configuration space (the number of
individuals), becomes smaller and smaller until it shrinks down to a single
point. In such a scenario, not only the order parameter (the average number of
individuals) approaches zero, but also the fluctuations of the order
parameter, i.e., its noise, vanishes continuously. The noise thus scales
proportional to the order parameter itself (or powers thereof). In many cases,
this causes the noise gap discussed in Sec. IV.2.2 to become proportional to
the order parameter, $P^{K}\sim\phi$, when approaching the transition, and to
vanish at the critical point.

This poses a challenge for the field theory approach to absorbing state
transitions: an order parameter-dependent noise, which vanishes at the
transition and obeys scaling in the vicinity of the critical point is
fundamentally different from the noise at equilibrium phase transitions. For
the latter, noise is either governed by thermal or quantum fluctuations, and
thus does not change its structure when tuning across the critical point, see
Sec. IV.2.2. This leads to characteristic and significant modifications of the
dynamical action and the scaling dimensions of the fields, which are discussed
in the following.

Though the majority of known absorbing state phase transitions appear in
classical systems, the concept is general and can be extended to driven open
quantum many-body dynamics. Consider a system of $N$ degrees of freedom, e.g.,
spins, bosons or fermions. Its quantum state $\hat{\rho}$ shall follow a
Lindblad master equation of the form

$\partial_{t}\hat{\rho}=\mathcal{L}_{1}(\hat{\rho})+\gamma\mathcal{L}_{2}(\hat{\rho}).$
(40)

Here $\mathcal{L}_{1,2}$ are two many-body Lindbladians and $\gamma$ is a
dimensionless tuning parameter. An absorbing state $\hat{\rho}_{D}$ has zero
statistical fluctuations, i.e., is a represented by a pure state
$\hat{\rho}_{D}=|D\rangle\langle D|$, with wave function $|D\rangle$. It is
the dark state of both Lindbladians, $\mathcal{L}_{1,2}(\hat{\rho}_{D})=0$,
and thus the stationary solution of Eq. (40), independently of the value of
$\gamma$ (cf. Appendix D). In order to realize a dynamical transition, one may
assume that $\hat{\rho}_{D}$ is a repulsive or unstable fixed point of
$\mathcal{L}_{1}$, while being an attractive or stable fixed point of
$\mathcal{L}_{2}$. Thus $\mathcal{L}_{2}$ pushes any initial state
$\hat{\rho}$ toward $\hat{\rho}_{D}$, while $\mathcal{L}_{1}$ pushes any
initial state $\hat{\rho}\neq\hat{\rho}_{D}$ away from the dark state—an
intuitive example is $\mathcal{L}_{1}(\hat{\rho})=-i[\hat{H},\hat{\rho}]$ for
some Hamiltonian $\hat{H}$, and $\hat{\rho}_{D}$ being one of the eigenstates
of the Hamiltonian.

The absorbing state phase transition roots in the competition of
$\mathcal{L}_{1}$ and $\mathcal{L}_{2}$, tuned by the parameter $\gamma$, and
deciding whether the dark state $\hat{\rho}_{D}$ will be reached or not in the
thermodynamic limit. A possible way to construct an order parameter is to take
a set of operators $\\{\hat{O}_{l}\\}$ to which $|D\rangle$ is an eigenstate,
$\hat{O}_{l}|D\rangle=o_{l}|D\rangle$. Then
$\hat{\varphi}=\tfrac{1}{N}\sum_{l}(\hat{O}_{l}-o_{l})$ is zero and
fluctuationless in the stationary state of the absorbing phase, while it is
nonzero and fluctuating in the active phase.

An elementary, paradigmatic model for absorbing state phase transitions is
_directed percolation_ (DP); a review of general absorbing state phase
transitions is provided in Refs. [284, 451]. DP describes general absorbing
state phase transitions of a real scalar order parameter field without any
particular symmetries, disorder or long range couplings. Despite its
theoretical simplicity, it has been experimentally observed only in certain
classical, turbulent systems [562, 364] and in quantum simulators with
ultracold Rydberg atoms [262, 282, 331], see below.

###  V.1 Rydberg atom lattice in the facilitation regime

Directed percolation, and variants of it, can be realized in driven-
dissipative Rydberg atom ensembles. In order to illustrate the microscopic
origin of the phenomenon, one may consider a $d$-dimensional array of atoms,
e.g., created with optical tweezers or in an optical lattice, where
neighboring atoms are kept at a fixed relative distance $R$, and their
motional degree is frozen. The atoms are then optically driven into an excited
state with a high principal quantum number (see, e.g., Refs. [509, 5, 383] for
reviews on the microscopic physics of Rydberg atoms). This generates a lattice
of two level systems (ground and excited states), where the excited atoms
interact with a dipole-induced van der Waals interaction. In a pseudo-spin
representation of the atomic ground $\lvert\downarrow\rangle$ and excited
$\lvert\uparrow\rangle$ states, where
$\hat{\sigma}^{x}_{l}=\lvert\uparrow\rangle\langle\downarrow\rvert_{l}+\lvert\downarrow\rangle\langle\uparrow\rvert_{l}$
is the Pauli matrix and
${\hat{n}_{l}=\lvert\uparrow\rangle\langle\uparrow\rvert_{l}}$ is the
projector onto the excited state acting on the $l$-th atom, this is described
by the quantum Ising Hamiltonian

$\displaystyle\hat{H}=\sum_{l}\left(-\Delta\hat{n}_{l}+\sum_{m}V_{l,m}\hat{n}_{l}\hat{n}_{m}+\frac{\Omega}{2}\hat{\sigma}^{x}_{l}\right).$
(41)

The parameters are the detuning $\Delta$ of the drive frequency from the
atomic resonance and the Rabi frequency $\Omega$, set by the driving field
intensity. Excited states repel each other with the van der Waals interaction
$V_{l,m}=\frac{C_{6}}{|\mathbf{x}_{l}-\mathbf{x}_{m}|^{6}}$, where $C_{6}$ is
the van der Waals coefficient of the atom and $\mathbf{x}_{l}$ is the position
of atom $l$ in the lattice [603].

The collective behavior of the Rydberg atoms can be modified by adjusting the
detuning $\Delta$. When $|\Delta|\gg|V_{l,m}|$, the Hamiltonian suppresses the
simultaneous excitation of neighboring lattice atoms and the array is in the
blockade regime [133, 516]. By tuning $\Delta$ ($\Omega$) the ground state
undergoes a classical (quantum), $\mathds{Z}_{s}$-symmetry breaking phase
transition from a paramagnet to an ordered state (the integer $s$ is set by
the range of the blockade) [64, 349].

An intriguing dynamical regime, the _facilitation regime_ is realized in the
opposite case when the detuning $\Delta$ is large compared to the Rabi
frequency $|\Delta|\gg\Omega$. Then single atom excitations are energetically
unfavorable, i.e., significantly suppressed by a factor $\Omega^{2}/\Delta$.
Atoms can only transfer between the excited and the ground state when the
mutual interaction $\Delta=V_{l,m}$ induced through neighboring excited states
compensates the detuning. This regime is known as anti-blockade or
facilitation regime since excited atoms which push their neighbors into
resonance act as seeds that _facilitate_ the spreading of excitations [31,
518, 394, 580, 548, 362, 365, 119, 262, 283, 282, 238, 399, 363, 384, 280,
608, 23, 219, 549, 191, 366, 581, 490, 92, 497, 582]. On time scales
$t\lesssim|\Delta/\Omega^{2}|$, off-resonant processes can be perturbatively
eliminated, yielding the effective Hamiltonian

$\displaystyle\hat{H}=\frac{\Omega}{2}\sum_{\langle
l,m\rangle}\hat{\sigma}_{l}^{x}\hat{n}_{m},$ (42)

where the sum runs over nearest neighbors $l,m$. The Hamiltonian (42) contains
some simplifications, e.g., it neglects the effect of multiply occupied
neighbors, which can cause geometric frustration effects, see Refs. [497,
189]. However, it captures the essence of the dynamics: (i) it decouples the
total spin down state
$\lvert\downarrow\downarrow\downarrow\cdots\downarrow\rangle$ from all excited
states and (ii) it leads to the spreading of excitations from individual seeds
(spin-up particles).

In addition to the Hamiltonian dynamics, excited atoms spontaneously emit
optical photons when they decay back into the ground state. This happens at a
rate $\gamma$ and is modeled by the master equation for the density matrix
$\hat{\rho}$:

$\displaystyle\partial_{t}\hat{\rho}=-i\left[\hat{H},\hat{\rho}\right]+\gamma\sum_{l}\left(2\hat{L}_{l}\hat{\rho}\hat{L}^{\dagger}_{l}-\left\\{\hat{L}^{\dagger}_{l}\hat{L}_{l},\hat{\rho}\right\\}\right),$
(43)

with Lindblad jump operators
$\hat{L}_{l}=\hat{\sigma}^{-}_{l}=\lvert\downarrow\rangle\langle\uparrow\rvert_{l}$.
The master equation (43) always has a stationary solution
$\hat{\rho}_{D}=|D\rangle\langle D|$ with zero excitations $\lvert
D\rangle=\lvert\downarrow\downarrow\downarrow\cdots\downarrow\rangle$, as
follows from $\hat{H}\hat{\rho}_{D}=\hat{L}_{l}\hat{\rho}_{D}=0$. This is the
absorbing state. For strong $\Omega>\gamma$, the rapid facilitation prevents
the system from reaching this stationary state. Instead, the dynamics explore
the full $2^{N}$-dimensional Hilbert space of $N$ pseudo-spins. The absorbing
state is approached only at exponentially long times $t\sim\exp(N)$, and the
system is in the active phase.

###  V.2 Field theory for directed percolation

The order parameter for the transition between the active and the absorbing
state is the excitation density $\langle\hat{n}_{l}\rangle$. In order to study
its dynamics in a field theory framework, one may either derive the Keldysh
field integral for the master equation (43), see, e.g., [92], or construct the
Heisenberg-Langevin equation for the operators $\hat{n}_{l}$ and then treat
them in a classical Martin-Siggia-Rose functional approach [497, 328]. While
both paths are equivalent, the latter is outlined below.

The dynamics of any pseudo-spin operator
$\hat{O}_{\alpha}=\hat{\sigma}^{x}_{l},\hat{\sigma}^{y}_{l},\hat{n}_{l}$ are
governed by an operator-valued Heisenberg-Langevin equation
$\partial_{t}\hat{O}_{\alpha}=\mathcal{L}^{\dagger}(\hat{O}_{\alpha})+{\hat{\xi}}_{\alpha}$.
The operator $\mathcal{L}^{\dagger}(\hat{O}_{\alpha})$ is deterministic and
$\hat{\xi}_{\alpha}$ represents a _quantum noise_ [528].
$\mathcal{L}^{\dagger}(\hat{O}_{\alpha})$ describes the action of the adjoint
master equation,

$\displaystyle\mathcal{L}^{\dagger}(\hat{O}_{\alpha})=i\left[\hat{H},\hat{O}_{\alpha}\right]+\gamma\sum_{l}\left(2\hat{L}^{\dagger}_{l}\hat{O}_{\alpha}\hat{L}_{l}-\left\\{\hat{L}^{\dagger}_{l}\hat{L}_{l},\hat{O}_{\alpha}\right\\}\right).$
(44)

Since the time evolution is not unitary, $\hat{O}_{\alpha}$ experiences
statistical fluctuations, i.e.,
$\mathcal{L}^{\dagger}(\hat{O}_{\alpha}\hat{O}_{\beta})\neq\hat{O}_{\alpha}\mathcal{L}^{\dagger}(\hat{O}_{\beta})+\mathcal{L}^{\dagger}(\hat{O}_{\alpha})\hat{O}_{\beta}$.
The fluctuations are encoded in the quantum noise $\hat{\xi}_{\alpha}$. The
noise has zero mean and an operator-valued variance
$\hat{\chi}_{\alpha\beta}=\hat{\xi}_{\alpha}\hat{\xi}_{\beta}$, determined by
the Einstein relation

$\hat{\chi}_{\alpha\beta}=\partial_{t}\left(\hat{O}_{\alpha}\hat{O}_{\beta}\right)-\hat{O}_{\alpha}\mathcal{L}^{\dagger}(\hat{O}_{\beta})-\mathcal{L}^{\dagger}(\hat{O}_{\alpha})\hat{O}_{\beta}.$
(45)

The pseudo-spin coherences $\hat{\sigma}_{l}^{x,y}$ are exponentially
suppressed by the spontaneous emission. Eliminating them in second order
perturbation theory (adiabatic elimination) yields the Heisenberg-Langevin
equation

$\displaystyle\partial_{t}\hat{n}_{l}=-\gamma\hat{n}_{l}+\frac{\Omega^{2}}{\gamma}\left(1-2\hat{n}_{l}\right)\left(\sum_{\langle
m,l\rangle}\hat{n}_{m}\right)^{2}+\hat{\xi}_{l},$ (46)

where the sum is restricted to nearest neighbors of $l$ and the noise has
variance $\hat{\xi}_{l}\hat{\xi}_{m}=\gamma\delta_{l,m}\hat{n}_{l}$.
Importantly, the noise depends on the operator $\hat{n}_{l}$, representing
multiplicative quantum noise. The square over the sum of the nearest neighbor
excitations is of purely quantum mechanical origin. It enhances the effective
Rabi frequency $\Omega$ by a factor $\sqrt{N}$, where $N$ is the number atoms
in a coherent superposition of excited states, known as the Rydberg superatom
[603]. This quantum mechanical, nonlinear enhancement of the facilitation
probability can result in a first order dark state phase transition [397],
which is discussed in Sec. X.3.

The evolution equation for the excited state densities $\hat{n}_{l}$ describes
the competition between the local decay of excitations with rate $\gamma$ and
the spreading of excitations to nearby lattice sites with rate
$\Omega^{2}/\gamma$. It is an operator-valued version of the classical
_contact process_ [385]. However, all operators commute, and thus one obtains
effective classical dynamics from Eq. (46). Indeed, the absorbing state phase
transition is in the directed percolation universality class. The key
ingredient for an absorbing state is that the variance $\hat{\chi}$ is
multiplicative, proportional to the order parameter.

The effective long-wavelength theory for the contact process is derived by
coarse graining, i.e., by averaging the density $\hat{n}_{l}$ over a suitable
number of lattice sites. In the operator-valued version, one needs to also
take the quantum mechanical average, accompanied by a mean-field decoupling of
products
$\langle\hat{n}_{l}\hat{n}_{m}\rangle\rightarrow\langle\hat{n}_{l}\rangle\langle\hat{n}_{m}\rangle$,
which is perturbatively controlled by the low excitation density close to the
dark state [397, 92, 328]. Introducing the coarse-grained order parameter
field,

$\phi(t,{\bf x})=\frac{1}{V_{{\bf x}}}\sum_{l\in V_{\bf
x}}\langle\hat{n}_{l}(t)\rangle,$ (47)

as the average density of excitations in a small volume $V_{\bf x}$ centered
at the continuum coordinate ${\bf x}$, yields the Langevin equation (we set
$\phi\equiv\phi(t,{\bf x})$)

$\partial_{t}\phi=\left(D\nabla^{2}-m\right)\phi-\kappa_{2}\phi^{2}-\kappa_{3}\phi^{3}+\eta.$
(48)

The noise $\eta=\eta(t,\mathbf{x})$ is normal distributed with zero mean and
variance $\langle\eta(t,{\bf x})\eta(t^{\prime},{\bf
x}^{\prime})\rangle=\delta(t-t^{\prime})\delta({\bf x}-{\bf
x}^{\prime})\gamma\phi(t,{\bf x})$, i.e., the noise gap is set by the order
parameter. The precise values of $\kappa_{2}$, $\kappa_{3}$ and $m$ depend on
the microscopic details, such as, e.g., dephasing [282, 283] and the
dimension.

When the strength of the leading-order nonlinearity is positive
$\kappa_{2}>0$, it dominates the relaxation close to the transition. The cubic
term $\sim\phi^{3}$ is then subleading for small values of $\phi$ and can be
neglected $\kappa_{3}\rightarrow 0$, i.e., it does not modify the universal
scaling at the transition. The corresponding Langevin equation describes the
field theory for _directed percolation_ (DP). On the mean-field level, it
undergoes a continuous phase transition from an absorbing phase ($\phi=0,m>0$)
to an active phase ($\phi>0,m<0$) at $m=0$ (approximately at $\Omega=\gamma$).
In contrast, if $\kappa_{2}<0$, the field $\phi$ experiences a bistability and
the absorbing state phase transition becomes first order on the mean-field
level [497, 92, 398]. At $\kappa_{2}=0$, mean-field theory predicts a first
order phase transition, which is separated from the continuous directed
percolation transition by a tricritial point. The universality class of this
point, and the field theory describing it, is known as _tricritical directed
percolation_ (TDP) [248, 452, 385].

###  V.3 Universality and symmetries

As it is common for continuous phase transitions, the universality class of
the transition is determined by symmetries (or the absence thereof), and a
unique set of critical exponents. The peculiar universal behavior of DP and
TDP, and the manifestation of an absorbing state, may be better understood in
a nonequilibrium field integral framework. From the Langevin Eq. (48), one can
derive the equivalent Keldysh action from the Martin-Siggia-Rose construction
[92] (see also Sec. IV.3.2):

$\displaystyle S=\int
d\mathbf{x}\,dt\,\phi_{q}\left(\partial_{t}-D\nabla^{2}+m+\kappa_{2}\phi_{c}+\kappa_{3}\phi^{2}_{c}-\gamma\phi_{q}\right)\phi_{c}.$

For $\kappa_{2}>0$, the cubic nonlinearity can be set to zero as argued above,
i.e., $\kappa_{3}=0$. Upon rescaling, the action is invariant under the
transformation

$\displaystyle\left(\phi_{c}(t),\phi_{q}(t)\right)\mapsto-\left(\phi_{q}(-t),\phi_{c}(-t)\right).$
(49)

This invariance is known as rapidity-inversion symmetry and is characteristic
for DP [300]. It has important consequences for the critical behavior. For
instance, it implies that the classical field $\phi_{c}$, i.e., the order
parameter, and the quantum field $\phi_{q}$, possess the same scaling
dimension $\sigma$ ($\sigma=-d/2$ at the Gaussian fixed point). This is
unconventional at a classical phase transition, and reminiscent of quantum
critical behavior (see Sec. XI.2.2). At the same time, the term quadratic in