momentum is conserved since the tension along each bond is cancelled by its
corresponding color.

To show that working with a microscopic system directly can lead to the
explicit breaking of additional symmetries, let us add a next-nearest-neighbor
odd spring constant with distinct parity-even and parity-odd coefficients
($k^{\,}_{2}$ and $k_{2}^{\circ}$, respectively). Naïvely, an identical
argument to the above goes through, and the system should still be PT-
symmetric. However, an explicit microscopic calculation Scheibner  _et al._
(2020) shows that theories with nearest- and next-nearest-neighbor odd springs
does not obey (263). What happens is that the stationary distribution,
captured by $\Phi$, _explicitly breaks parity_ :

$\Phi=\frac{1}{2}\int\mathrm{d}^{2}x\left[c^{\vphantom{\prime}}_{1}(\nabla\times
u)^{2}+(c_{1}+c_{2})(\nabla\cdot u)^{2}+c^{\vphantom{\prime}}_{3}(\nabla\cdot
u)(\nabla\times u)\right]\,,~{}~{}$ (265)

where the P-odd coefficient $c_{3}$ depends on the details of the microscopic
spring constants. Repeating the calculation above, we find that

$\displaystyle B$ $\displaystyle=Q_{1}c_{2}+2Q_{2}c_{3},$ (266a)
$\displaystyle\mu$ $\displaystyle=Q_{1}c_{1}-Q_{2}c_{3},$ (266b)
$\displaystyle K^{\circ}$ $\displaystyle=Q_{1}c_{3}+Q_{2}c_{1},$ (266c)
$\displaystyle A$ $\displaystyle=Q_{2}c_{2}-2Q_{1}c_{3}.$ (266d)

Thus the ratio (263) is broken and we get four distinct elastic coefficients,
due to the explicit breaking of parity. Observe that it is possible for
$\mu<0$ or $B<0$ if $Q_{2}$ is large (which is not incompatible with the
stability of the theory); however, $B+2\mu\geq 0$ since $Q_{1},c_{1},c_{2}\geq
0$ are all required by stability of the fixed point.

Interestingly, we can explicitly break parity, meaning that $c_{3}\neq 0$, but
restore time-reversal symmetry, meaning that $Q_{2}=0$. We still in this case
find that the two new odd elastic moduli do not vanish, implying that time-
reversal symmetry breaking was not essential to see odd elastic coefficients
Lier  _et al._ (2022). In this case, we do see that $B,\mu>0$ directly, while
$A=-2K^{\circ}$.

Lastly, let us return to a point remarked at the beginning of this subsection:
the choice that $\Phi$ had a shift symmetry has led to direct physical
consequences. Notice that the Lagrangian (259), or more formally the generator
of the Fokker-Planck equation, has no derivatives acting on $\pi$. This means
that when we write down the noisy equations of motion:
$\partial_{t}u_{i}=\cdots+\xi_{i}$, the variance of the noise will be

$\langle\xi_{i}(x,t)\xi_{j}(x^{\prime},t^{\prime})\rangle\sim\delta(x-x^{\prime})\delta(t-t^{\prime}).$
(267)

In other words, this noise will _translate_ the solid in space. This theory
therefore corresponds to an odd solid that is pinned to some substrate, and
therefore can slide around over time. In most experiments, this is indeed
likely to be the case.

####  7.2.2 Translationally pinned potential

However, it is also reasonable to postulate—at least theoretically—the
existence of an intrinsically odd solid, which itself forms a closed system,
and where the stochastic dynamics has only arisen due to integrating out short
wavelength fluctuations of the solid itself. Along the lines of the above
discussion, let us now turn to discuss the case where the solid itself is
taken to be an isolated system, that cannot slide itself spontaneously in the
plane. Then, the quantities

$\displaystyle X_{i}$ $\displaystyle=\int\mathrm{d}^{2}x\,u_{x},$ (268a)
$\displaystyle Y_{i}$ $\displaystyle=\int\mathrm{d}^{2}x\,u_{y}\,,$ (268b)

are conserved. Noether’s theorem then implies that the Lagrangian must be
invariant under $\pi_{x,y}\rightarrow\pi_{x,y}+1$, and therefore (259) must be
modified to

$\mathcal{L}=\pi_{i}\partial_{t}u_{i}+\mathrm{i}\partial_{i}\pi_{j}Q_{ijkl}\partial_{k}\left(\pi_{l}-\mathrm{i}\mu_{l}\right).$
(269)

Due to the addition of two extra derivatives on $\pi_{i}$, if we take the same
choice of $\Phi$ as before, evidently the elastic equations of motion will be
fourth order. On the other hand, since $X_{i}$ and $Y_{i}$ are conserved, it
need not be the case that $\Phi$ depend on $u_{i}$ only via derivatives: we
are _not_ just as likely to find the solid in all configurations translated by
a constant. So now, let us instead postulate

$\Phi=\frac{1}{2}\int\mathrm{d}^{2}x\;u_{i}u_{i}$ (270)

For simplicity we have chosen units so there is no constant prefactor in
$\Phi$ above. In this case $\mu_{i}=u_{i}$, and the equation of motion is

$\partial_{t}u_{i}=Q_{ijkl}\partial_{j}\partial_{k}u_{l}+\xi_{i},$ (271)

where

$\langle\xi_{j}(x,t)\xi_{l}(x^{\prime},t^{\prime})\rangle=-2Q_{ijkl}\partial_{k}\partial_{l}\delta^{(2}(x-x^{\prime})\delta(t-t^{\prime}).$
(272)

Now we want to know what components are allowed in the fourth rank tensor
$Q_{ijkl}$. We consider the basis of $2\times 2$ matrices, $\\{\omega^{i}\\}$
such that

$\omega^{0}=\delta_{ij},\quad\omega^{1}=\epsilon_{ij},\quad\omega^{2}=\sigma_{ij}^{z},\quad\omega^{3}=\sigma_{ij}^{x}$
(273)

In this basis, we express the fourth rank tensor $Q_{ijkl}$ as a $4\times 4$
matrix;

$Q^{\alpha\beta}=\left(\omega^{\alpha}\right)_{ij}Q_{ijkl}\left(\omega^{\beta}\right)_{kl}$
(274)

First, we assume that the solid is isotropic for simplicity, as before. This
means that $\omega^{2,3}$ lie in the same real-valued representation of the
rotation group distinct from $\omega^{0,1}$. In order to be invariant under
rotations,

$Q^{\alpha\beta}=\begin{pmatrix}Q^{00}&Q^{01}&0&0\\\ Q^{10}&Q^{11}&0&0\\\
0&0&Q^{22}&Q^{23}\\\ 0&0&-Q^{23}&Q^{22}\end{pmatrix}$ (275)

Table 2 describes how the components of $Q^{\alpha\beta}$ transform under
parity, inversion, and time-reversal symmetry.

Component | Parity | Inversion | Time
---|---|---|---
$\delta\otimes\delta$ | Even | Even | Even
$\delta\otimes\epsilon+\epsilon\otimes\delta$ | Odd | Even | Even
$\delta\otimes\epsilon-\epsilon\otimes\delta$ | Odd | Even | Odd
$\epsilon\otimes\epsilon$ | Even | Even | Even
$\sigma_{z}\otimes\sigma_{z}$ | Even | Even | Even
$\sigma_{z}\otimes\sigma_{x}-\sigma_{x}\otimes\sigma_{z}$ | Odd | Even | Odd
$\sigma_{x}\otimes\sigma_{x}$ | Even | Even | Even
Table 2: Tensor transformations for $Q_{ijkl}$ under parity, inversion, and
time-reversal.

While we will not consider all possible combinations exhaustively, let us note
that if we wish to have PT-symmetry only, the only further constraint is that
$Q^{10}=-Q^{01}$.

As above, we can calculate the elastic moduli. Assuming PT-symmetry, we find:

$\displaystyle B$ $\displaystyle=Q^{00}+Q^{22},$ (276a) $\displaystyle\mu$
$\displaystyle=Q^{11}+Q^{22},$ (276b) $\displaystyle K^{\circ}$
$\displaystyle=Q^{01}+\frac{1}{2}Q^{23},$ (276c) $\displaystyle A$
$\displaystyle=0.$ (276d)

Note that because $\Phi$ is explicitly parity invariant, $A$ and $K^{\circ}$
are now necessarily T-odd coefficients.

A system with $A=0$ and $K^{\circ},B,\mu\neq 0$ arises in an active solid on a
two-dimensional honeycomb lattice. At vertices of the lattice, we place hinges
between the edges, which are active components that deform according to the
deformation of its nearest clockwise neighbor Fruchart  _et al._ (2022). We
further assume that $B$ and $\mu$ are nonzero so that the solid resists
changes in volume and shape (this can be realized by adding next-nearest-
neighbor (NNN) Hookean springs). Each NNN bond responds to the strain of its
counterclockwise neighbor in the unit cell. Reversing time would imply that
each NNN bond responds to the tension of its clockwise neighbor which is
tantamount to a parity (P) transformation. We therefore expect this system to
have PT symmetry. Our effective theory shows that angular momentum must be
conserved so long as $Q^{11}=0$, which is always the case when (as in this
microscopic model) the system is invariant under a global rotation
$(u_{i}=\epsilon_{ij}x_{j}$). In this microscopic model we can indeed confirm
that angular momentum is conserved, and therefore $A=0$: all forces from
Hookean springs are radial so momentum is conserved pair by pair, while the
active hinges exert equal and opposite torques on the two bonds they connect.

###  7.3 Chiral fluid

Now, let us turn to the most complex example of fluid we will consider in this
paper: a two-dimensional chiral fluid with both density and momentum
conservation. There are three possible “odd” transport coefficients we should
expect: odd diffusivity Hargus  _et al._ (2021) (which we have already
discussed), odd viscosity Banerjee  _et al._ (2017), and odd stress Han  _et
al._ (2021). While the former two have a close analogy with dissipationless
Hall transport in equilibrium systems, the last one appears to us unique to
active phases: in particular, it violates the thermal KMS condition in the
presence of a generic background geometry Huang and Lucas (2022).

####  7.3.1 Odd viscosity

Let us begin by discussing odd viscosity. For simplicity, we’ll momentarily
neglect fluctuations of density, and assume that the most likely fluid
configuration has zero velocity. Then for small fluctuations around
equilibrium, the stationary distribution is given by the nonrelativistic
kinetic energy (with background mass density / momentum susceptibility
Hartnoll  _et al._ (2018) $\rho$):

$\displaystyle\Phi=\beta_{0}\int\differential[d]{x}\frac{\rho}{2}u_{i}u_{i}.$
(277)

As usual, we introduce ‘chemical potentials’ as

$\displaystyle\mu_{i}=\functionalderivative{\Phi}{u_{i}}=\rho u_{i}.$ (278)

In the usual nonactive case, the effective Lagrangian should satisfy $T$ and
$P$ and also be consistent with conservation of momentum,
$\mathbf{P}=\int\differential[d]{x}\rho\mathbf{u}$. The minimal Lagrangian
therefore features two gradients of $\mathbf{u}$:

$\displaystyle\mathcal{L}(u,\pi)=\pi_{i}\partial_{t}u_{i}+\mathrm{i}\beta_{0}^{-1}\rho^{-2}\eta_{ijkl}\partial_{i}\pi_{j}\partial_{k}(\pi_{l}-\mathrm{i}\mu_{l}).$
(279)

The constant object $\eta_{ijkl}$ is known as the viscosity tensor. In the Itô
scheme of the MSR path integral, integrating out $\pi_{i}$ leads to a
fluctuating diffusion equation for the velocity:

$\displaystyle\rho\partial_{t}u_{i}=\eta_{ijkl}\partial_{j}\partial_{k}u_{l}+\xi_{i},$
(280)

where the statistics of the white noise $\xi_{i}(\mathbf{x},t)$ are such that

$\displaystyle\langle\xi_{i}(\mathbf{x},t)\xi_{j}(\mathbf{x}^{\prime},t^{\prime})\rangle=\beta_{0}^{-1}(\eta_{ikjl}+\eta_{jlik})\partial_{k}\partial_{l}\delta(\mathbf{x}-\mathbf{x}^{\prime})\delta(t-t^{\prime}).$
(281)

Thus viscous hydrodynamics naturally emerges as the simplest $P$,
$T$-invariant effective field theory for conserved momentum. We will discuss
the pressure and stress, so far omitted, in the next subsection.

However, in a chiral fluid composed, for example, of spinning particles, P and
Tmay be violated such that only PT is conserved. In that case, we may
decompose

$\displaystyle\eta_{ijkl}=\eta^{\mathrm{even}}_{ijkl}+\eta^{\mathrm{odd}}_{ijkl},$
(282)

where $\eta^{\mathrm{even}}_{ijkl}=\eta^{\mathrm{even}}_{klij}$,
$\eta^{\mathrm{odd}}_{ijkl}=-\eta^{\mathrm{odd}}_{klij}$. The latter is called
the odd viscosity Fruchart  _et al._ (2022). Note that $\eta^{\mathrm{even}}$,
but not $\eta^{\mathrm{odd}}$, contribute to the variance of the statistical
fluctuations; hence $\eta^{\mathrm{even}}$ must a positive semidefinite
tensor.

To find odd viscosity in our effective field theory, we simply need to enforce
PT instead of T and P separately. The minimal Lagrangian becomes:

$\displaystyle\mathcal{L}(u,\pi)=\pi_{i}\partial_{t}u_{i}+\beta_{0}^{-1}\rho^{-2}\eta^{\mathrm{odd}}_{ijkl}\partial_{i}\pi_{j}\partial_{k}\mu_{l}+\mathrm{i}\beta_{0}^{-1}\rho^{-2}\eta^{\mathrm{even}}_{ijkl}\partial_{i}\pi_{j}\partial_{k}(\pi_{l}-\mathrm{i}\mu_{l}),$
(283)

where, as in the case of odd diffusivity, the antisymmetry of
$\eta^{\mathrm{odd}}_{ijkl}$ means this coefficient enters only at linear
order in $\pi_{i}$. Accordingly, $\eta^{\mathrm{odd}}_{ijkl}$ is not related
to noise—the statistics of the noise remain (281). The resulting equation of
motion is in agreement with (280).

Our approach differs from Han  _et al._ (2021); in this work, the authors
attributed the odd viscosity to an “antisymmetric” non-Markovian noise where
$\langle\xi(t)\xi(t^{\prime})\sim f(t-t^{\prime})$ with $f$ an odd, sharply
localized function. In our discussion, odd viscosity is allowed due to the gT
invariance of the effective Lagrangian, and it is independent of noise as a
dissipationless force in the effective theory. Odd viscosity, like odd
diffusivity, therefore emerges naturally from the simplest PT-invariant
effective field theory of a dissipative incompressible fluid. The fact that we
obtain the same hydrodynamic universality class as Han  _et al._ (2021)
therefore explains the emergence of an FDT and a second law of thermodynamics
in more microscopically active models as well. Like in the discussion of odd
elasticity, our approach does not give an unambiguous definition of the stress
tensor, so certain global correlation functions of stress will not be
universally captured by our EFT.

####  7.3.2 Fluid with spinning particles

Let us now work in a regime where both $n$ and $\mathbf{u}$ are nonzero. We
also consider an additional spin density $s$, as slowly varying fields. The
spin density $s$ could correspond to a local angular momentum density of
spinning particles, and we will first assume that $s$ is a conserved quantity.
The minimal steady-state distribution is given by

$\displaystyle\Phi=\beta_{0}\int\mathrm{d}^{d}x~{}\left[\frac{\rho}{2}(u_{i})^{2}+\frac{1}{2\chi}(\delta
n)^{2}+\frac{1}{2\chi_{s}}(\delta s)^{2}\right],$ (284)

where $\chi_{s}$ is the spin susceptibility. Spinning motions are P-odd and
T-odd, but PT-even.

Under PT symmetry, the leading Lagrangian is given by

$\displaystyle\mathcal{L}=$
$\displaystyle\pi_{i}\partial_{t}u_{i}+\pi^{n}\partial_{t}\delta
n+\pi^{s}\partial_{t}\delta
s-n_{0}u_{i}\partial_{i}\pi^{n}-\frac{n_{0}}{\chi\rho}\partial_{i}\pi_{i}\delta
n-s_{0}u_{i}\partial_{i}\pi^{s}-\frac{s_{0}}{\chi_{s}\rho}\partial_{i}\pi_{i}\delta
s$
$\displaystyle+\mathrm{i}\beta_{0}^{-1}\rho^{-2}\eta_{ijkl}^{\mathrm{even}}\partial_{i}\pi_{j}\partial_{k}(\pi_{l}-\mathrm{i}\mu_{l})+\sum_{\alpha,\beta=n,s}\mathrm{i}\beta_{0}^{-1}D^{\alpha\beta,\mathrm{even}}\partial_{i}\pi^{\alpha}\partial_{i}(\pi^{\beta}-\mathrm{i}\mu^{\beta})$
$\displaystyle+\beta_{0}^{-1}\rho^{-2}\eta^{\mathrm{odd}}_{ijkl}\partial_{i}\pi_{j}\partial_{k}\mu_{l}+\sum_{\alpha,\beta=n,s}\beta_{0}^{-1}D^{\alpha\beta,\mathrm{odd}}\epsilon_{ij}\partial_{i}\pi^{\alpha}\partial_{j}\mu^{\beta},$
(285)

where $D^{ns}=D^{sn}$. Importantly, we do not include the following invariant
term

$\displaystyle\mathcal{L}\sim\sum_{\alpha=n,s}\frac{\mathrm{i}}{2}\beta_{0}^{-1}K^{\alpha}\epsilon^{ij}(\partial_{i}\pi_{j}(\pi^{\alpha}-\mathrm{i}\mu^{\alpha})+\partial_{i}(\pi_{j}-\mathrm{i}\mu_{j})\pi^{\alpha}).$
(286)

This is because to fulfill the positivity constraint of the noise invariance
matrix, one has to also include terms with zero derivative order:

$\displaystyle\mathcal{L}\sim\sum_{\alpha=n,s}\mathrm{i}\beta_{0}^{-1}\Gamma^{\alpha}\pi^{\alpha}(\pi^{\alpha}-\mathrm{i}\mu^{\alpha}).$
(287)

Moreover, this zeroth order term is forbidden by the shift symmetry
$\pi^{\alpha}\to\pi^{\alpha}+1$ corresponding to the conservation of charge
and spin. Indeed, (287) will lead to equations of motion:
$\partial_{t}n\sim-\Gamma^{n}n$ and $\partial_{t}s\sim-\Gamma^{s}s$ that decay
to zero. Without them, the stochastic equations of motion corresponding to
(7.3.2) are given by

$\displaystyle\partial_{t}\delta
n+n_{0}\partial_{i}u_{i}-\chi^{-1}D^{n,\mathrm{even}}\partial_{i}^{2}\delta
n-\chi_{s}^{-1}D^{ns,\mathrm{even}}\partial_{i}^{2}\delta s$
$\displaystyle=\xi^{n},$ (288a) $\displaystyle\partial_{t}\delta
s+s_{0}\partial_{i}u_{i}-\chi_{s}^{-1}D^{s,\mathrm{even}}\partial_{i}^{2}\delta
s-\chi^{-1}D^{ns,\mathrm{even}}\partial_{i}^{2}\delta n$
$\displaystyle=\xi^{s},$ (288b)
$\displaystyle\rho\partial_{t}u_{i}+\chi^{-1}n_{0}\partial_{i}\delta
n+\chi_{s}^{-1}s_{0}\partial_{i}\delta
s-\eta_{jikl}\partial_{j}\partial_{k}u_{l}$ $\displaystyle=\xi_{i},$ (288c)

with the noise variance

$\displaystyle\expectationvalue{\xi^{\alpha}(\mathbf{x},t)\xi^{\beta}(\mathbf{x}^{\prime},t^{\prime})}$
$\displaystyle=2\beta_{0}^{-1}D^{\alpha\beta,\mathrm{even}}\,\partial_{i}^{2}\delta(\mathbf{x}-\mathbf{x}^{\prime})\,\delta(t-t^{\prime}),$
(289a)
$\displaystyle\expectationvalue{\xi_{i}(\mathbf{x},t)\xi_{j}(\mathbf{x}^{\prime},t^{\prime})}$
$\displaystyle=2\beta_{0}^{-1}\eta_{ikjl}^{\mathrm{even}}\,\partial_{k}\partial_{l}\delta(\mathbf{x}-\mathbf{x}^{\prime})\,\delta(t-t^{\prime}),$
(289b)

for $\alpha,\beta=n,s$. Therefore, we find that it describes the odd fluid
dynamics, including the odd diffusivity and the odd viscosity that we
discussed above, with two species of charge. In particular, to the leading
order, the equation of state is given by

$\displaystyle\mathrm{d}p\approx\chi^{-1}n_{0}\mathrm{d}n+\chi_{s}^{-1}s_{0}\mathrm{d}s+\cdots,$
(290)

where $p$ is the thermodynamic pressure, and there exist no odd stress.

However, if we were allowed to include (286), it will actually lead to the
desirable odd stress. To see it, we vary (286) with respect to $\pi_{i}$,
which gives

$\displaystyle\sigma_{ij}^{\mathrm{P-odd}}=\sum_{\alpha=n,s}\frac{1}{2}\beta_{0}^{-1}\rho
K^{\alpha}\epsilon^{ij}\mu^{\alpha},$ (291)

where $\sigma_{ij}^{\mathrm{P-odd}}$ is the odd stress that is defined by
$\rho\partial_{t}u_{i}+\partial_{j}\sigma_{ij}=\cdots$ and antisymmetric. To
avoid that $\mu^{\alpha}\to 0$ as we argued above, we can introduce an active
torque $\tau$ into the equations of motion for $\delta s$: Banerjee  _et al._
(2017)

$\displaystyle\partial_{t}\delta s\sim\tau-\Gamma^{s}s,$ (292)

such that a spinning steady state with $s_{0}\equiv\tau/\Gamma^{s}$ can exist.
Now, the odd stress is given by

$\displaystyle\sigma_{ij}^{\mathrm{P-odd}}=\frac{1}{2}\rho\chi_{s}^{-1}K^{s}\epsilon^{ij}s,$
(293)

in agreement with Banerjee  _et al._ (2017). Several remarks follow. First,
the odd stress strength $K^{s}$ is constrained by the positivity of the noise
invariance: $(K^{s})^{2}\leq\eta\Gamma^{s}$. Second, the torque $\tau$ is not
an ingredient that can be captured in the EFT – $\mathcal{L}\sim\pi^{s}\tau$
is not invariant under PT – but the resulting steady state with $s_{0}\neq 0$
is a valid starting point for our EFT and we have seen that it leads to the
odd stress. Let us turn to the relevant changes of the equations of motion due
to this odd stress. We have, to the leading order,

$\displaystyle\partial_{t}\delta n+n_{0}\partial_{i}u_{i}$ $\displaystyle=0,$
(294a) $\displaystyle\partial_{t}\delta
s+s_{0}\partial_{i}u_{i}+\frac{1}{2}\rho K^{s}\epsilon^{ij}\partial_{i}u_{j}$
$\displaystyle=-\Gamma^{s}\delta s,$ (294b)
$\displaystyle\rho\partial_{t}u_{i}+\chi^{-1}n_{0}\partial_{i}\delta
n+\chi_{s}^{-1}s_{0}\partial_{i}\delta
s+\frac{1}{2}\rho\chi_{s}^{-1}K^{s}\epsilon^{ij}\partial_{j}\delta s$
$\displaystyle=0.$ (294c)

Solving the above equations to the linear order in $O(k)$, we find,
schematically, a longitudinal sound mode $\omega\approx\pm v_{0}k$ with
$v_{0}=n_{0}/\sqrt{\chi\rho}$, a decaying spin density mode
$\omega\approx-\mathrm{i}\Gamma^{s}$, and a transverse diffusion mode
$\omega\sim-\mathrm{i}k^{2}$.

There is an intrinsic spin density $\delta s=\epsilon^{ij}\partial_{i}u_{j}$
corresponding to the angular momentum of the fluid. It also contributes to the
odd stress if the rotational symmetry is broken explicitly. Specifically,
there exists a rotational viscosity Cook and Lucas (2019, 2021); Rao and
Bradlyn (2023)

$\displaystyle\eta^{\mathrm{even}}_{ijkl}\sim\eta_{\circ}\epsilon^{ij}\epsilon^{kl},$
(295)

and plugging it in (288c), we obtain

$\displaystyle\sigma_{ij}^{\mathrm{odd}}=-\eta_{\circ}\epsilon^{ij}\epsilon^{kl}\partial_{k}u_{l}.$
(296)

Notice that this is typically a higher derivative contribution to the odd
stress, and that this contribution is P-even and T-even separately.161616The
“odd” stress refers to the fact that
$\sigma_{ij}^{\mathrm{odd}}\sim\epsilon^{ij}$, but the tensor structure of
(295) renders it even under P. Recently, this relaxation of intrinsic angular
momentum has been argued to relax the momentum along the helical axis of a
fluid with helical symmetry in thermal equilibrium Farrell  _et al._ (2022).

####  7.3.3 Odd stress and PT-odd charge

Lastly, suppose that we add to our fluid a P-odd but T-even charge, which we
will denote $w$. A possible candidate for this P-odd and T-even charge is to
consider a fluid consisting of two kinds of chiral particles (which cannot be
created or destroyed), with $n_{\mathrm{L,R}}$ the densities of left- and
right-handed particles respectively. Since parity will exchange left/right-
handed particles, we would then obtain both a P-even and P-odd charge:
$n=n_{\mathrm{L}}+n_{\mathrm{R}}$ and $w=n_{\mathrm{L}}-n_{\mathrm{R}}$. Both
of them, however, remain T-even.

We take similarly a minimal steady-state distribution

$\displaystyle\Phi=\beta_{0}\int\mathrm{d}^{d}x~{}\left[\frac{\rho}{2}(u_{i})^{2}+\frac{1}{2\chi}(\delta
n)^{2}+\frac{1}{2\chi_{w}}(\delta w)^{2}\right],$ (297)

where $\chi_{w}$ is the susceptibility associated with $w$. Then, the leading
order Lagrangian that is invariant under PT symmetry is given by

$\displaystyle\mathcal{L}=$
$\displaystyle\pi_{i}\partial_{t}u_{i}+\pi^{n}\partial_{t}\delta
n+\pi^{w}\partial_{t}\delta
w-n_{0}u_{i}\partial_{i}\pi^{n}-\frac{n_{0}}{\chi\rho}\partial_{i}\pi_{i}\delta
n-w_{0}\epsilon^{ij}u_{i}\partial_{j}\pi^{w}+\frac{w_{0}}{\chi_{w}\rho}\epsilon^{ij}\partial_{i}\pi_{j}\delta
w$
$\displaystyle+\mathrm{i}\beta_{0}^{-1}\rho^{-2}\eta_{ijkl}^{\mathrm{even}}\partial_{i}\pi_{j}\partial_{k}(\pi_{l}-\mathrm{i}\mu_{l})+\sum_{\alpha=n,w}\mathrm{i}\beta_{0}^{-1}\chi_{\alpha}D^{\alpha,\mathrm{even}}\partial_{i}\pi^{\alpha}\partial_{i}(\pi^{\alpha}-\mathrm{i}\mu^{\alpha})$
$\displaystyle+\beta_{0}^{-1}\rho^{-2}\eta^{\mathrm{odd}}_{ijkl}\partial_{i}\pi_{j}\partial_{k}\mu_{l}+\sum_{\alpha=n,w}\beta_{0}^{-1}\chi_{\alpha}D^{\alpha,\mathrm{odd}}\epsilon_{ij}\partial_{i}\pi^{\alpha}\partial_{j}\mu^{\alpha}$
$\displaystyle+\beta_{0}^{-1}D_{\mathrm{mix},\mathrm{odd}}\left(\partial_{i}\pi^{n}\partial_{i}\mu^{w}-\partial_{i}\pi^{w}\partial_{i}\mu^{n}\right),$
(298)

where the subscript “odd” indicates the term is T-odd. In the above equation,
we did not include

$\displaystyle\mathcal{L}\sim\mathrm{i}\frac{1}{2}\beta_{0}^{-1}D_{\mathrm{mix},\mathrm{even}}\epsilon^{ij}\left(\partial_{i}\pi^{n}\partial_{j}(\pi^{w}-\mathrm{i}\mu^{w})+\partial_{i}(\pi^{n}-\mathrm{i}\mu^{n})\partial_{j}\pi^{w}\right),$
(299)

which is a total derivative, and the invariant terms

$\displaystyle\mathcal{L}\sim\frac{\mathrm{i}}{2}\beta_{0}^{-1}K^{n}\epsilon^{ij}(\partial_{i}\pi_{j}(\pi^{n}-\mathrm{i}\mu^{n})+\partial_{i}(\pi_{j}-\mathrm{i}\mu_{j})\pi^{n})+\frac{\mathrm{i}}{2}\beta_{0}^{-1}K^{w}(\partial_{i}\pi_{i}(\pi^{w}-\mathrm{i}\mu^{w})+\partial_{i}(\pi_{i}-\mathrm{i}\mu_{i})\pi^{w})$
(300)

for the same reason as in Sec. 7.3.2. Hence, the stochastic equations of
motion is

$\displaystyle\partial_{t}\delta
n+n_{0}\partial_{i}u_{i}-D^{n,\mathrm{even}}\partial_{i}^{2}\delta
n-\chi_{s}^{-1}D^{\mathrm{mix},\mathrm{odd}}\partial_{i}^{2}\delta w$
$\displaystyle=\xi^{n},$ (301a) $\displaystyle\partial_{t}\delta
w-w_{0}\epsilon^{ij}\partial_{i}u_{j}-D^{w,\mathrm{even}}\partial_{i}^{2}\delta
w+\chi^{-1}D^{\mathrm{mix},\mathrm{odd}}\partial_{i}^{2}\delta n$
$\displaystyle=\xi^{w},$ (301b)
$\displaystyle\rho\partial_{t}u_{i}+\chi^{-1}n_{0}\partial_{i}\delta
n+\chi_{w}^{-1}w_{0}\epsilon^{ij}\partial_{j}\delta
w-\eta_{jikl}\partial_{j}\partial_{k}u_{l}$ $\displaystyle=\xi_{i},$ (301c)

with the noise variance

$\displaystyle\expectationvalue{\xi^{\alpha}(\mathbf{x},t)\xi^{\alpha}(\mathbf{x}^{\prime},t^{\prime})}$
$\displaystyle=2\beta_{0}^{-1}\chi_{\alpha}D^{\alpha,\mathrm{even}}\,\partial_{i}^{2}\delta(\mathbf{x}-\mathbf{x}^{\prime})\,\delta(t-t^{\prime}),$
(302a)
$\displaystyle\expectationvalue{\xi_{i}(\mathbf{x},t)\xi_{j}(\mathbf{x}^{\prime},t^{\prime})}$
$\displaystyle=2\beta_{0}^{-1}\eta_{ikjl}^{\mathrm{even}}\,\partial_{k}\partial_{l}\delta(\mathbf{x}-\mathbf{x}^{\prime})\,\delta(t-t^{\prime}),$
(302b)

for $\alpha=n,w$. A crucial difference from Sec. 7.3.2 is that there exists an
odd stress (pressure) obeying

$\displaystyle\mathrm{d}p_{\mathrm{P-odd}}\approx\chi_{w}^{-1}w_{0}\mathrm{d}w.$
(303)

The odd stress results in a transversal sound mode, which is absent in the
normal hydrodynamics. To see it, we compute the quasinormal modes
corresponding to (301) to the leading (ideal) order. This leads to

$\displaystyle\omega=\pm\frac{n_{0}}{\sqrt{\chi\rho}}k,\quad\omega=\pm\frac{w_{0}}{\sqrt{\chi_{w}\rho}}k,$
(304)

where the former is the usual longitudinal sound mode, and the latter is the
transversal sound mode due to the odd stress. It would be interesting if this
effect can be discovered in active chiral matter.

###  7.4 Fracton hydrodynamics

“Fracton hydrodynamics” is the name for the universality classes of
hydrodynamics associated to theories with multipolar conservation laws (e.g.
charge and dipole conservation together) Gromov  _et al._ (2020); Feldmeier
_et al._ (2020); Morningstar  _et al._ (2020); Hart  _et al._ (2022); Glorioso
_et al._ (2022); Grosvenor  _et al._ (2021); Burchards  _et al._ (2022); Guo
_et al._ (2022). These exotic conservation laws require a careful formulation
of hydrodynamic EFTs Guo  _et al._ (2022), including in nonthermal systems. As
mentioned in Sec. 1.2 for example, certian terms naively allowed in equations
of motion can be forbidden by an EFT. Using the weak noise limit formalism
described in Section 2.7, Guo  _et al._ (2022) has studied theories of fracton
hydrodynamics without time-reversal symmetry. To quickly re-cap why this
formalism is very helpful, notice that if

$0=\frac{\mathrm{d}}{\mathrm{d}t}\int\mathrm{d}^{d}x\;(a+b_{i}x_{i})n$ (305)

for any $a$ and $b_{i}$, and $n(x)$ is a conserved charge density, we must
write down a theory invariant under $\pi\rightarrow\pi+a+b_{i}x_{i}$. Hence,
the lowest derivative motif of $\pi$ that can show up in $\mathcal{L}$,
outside of $\pi\partial_{t}n$, becomes $\partial_{i}\partial_{j}\pi$. So long
as the time-reversed theory remains local, one can show that in generic
dimensions171717In one dimension only, one can also add a third-derivative
term which contributes a total derivative after applying time-reversal., the
most general Lagrangian at lowest order in derivatives is

$\mathcal{L}=\pi\partial^{\vphantom{\prime}}_{t}n+\mathrm{i}D_{ijkl}(n)\partial^{\vphantom{\prime}}_{i}\partial^{\vphantom{\prime}}_{j}\pi\partial^{\vphantom{\prime}}_{k}\partial^{\vphantom{\prime}}_{l}\left(\pi-\mathrm{i}\mu\right).$
(306)

If we take

$\Phi\,=\,\int\mathrm{d}^{d}x\,\frac{n^{2}}{2\chi}\,,~{}~{}$ (307)

for constant $\chi$, then Eq. (306) predicts subdiffusive with
$\partial^{\vphantom{\prime}}_{*}{t}n\sim\partial^{4}_{x}n$, in agreement with
Ref. 14; 21, and resolving the puzzle of Sec. 1.2.

Note that the above choice of $\Phi$ (307) is only reasonable if $n<0$ makes
sense—e.g., there are both positive and negative charges in the theory. If we
wish to consider only positive charges, and we wish to consider nonlinear
dynamics close to $n=0$, an appropriate, alternative choice of $\Phi$ is

$\displaystyle\Phi=\int\mathrm{d}x~{}n\log n,$ (308)

which is inspired by the formula for Shannon entropy. Keeping the Lagrangian
(306), and working in $d=1$ spatial dimensions for simplicity, a short
calculation reveals the equation of motion

$\displaystyle\partial_{t}\delta
n+\partial_{x}^{2}\left(D(n)\partial_{x}^{2}\mu\right)=0.$ (309)

Using (308) we find

$\displaystyle\mu=\frac{\partial\Phi}{\partial\delta n}=\log n-1.$ (310)

In order for the equation of motion to be regular, we demand

$D(n)=D_{0}n^{2}+\cdots$ (311)

as $n\rightarrow 0$. The resulting nonlinear equation of motion reproduces the
predictions of a recent paper Han  _et al._ (2023) in the absence of noise.
This effective theory approach conveniently streamlines what is otherwise a
somewhat complicated microscopic calculation of the equations of motion.
Dynamics conserving higher multipole moments can be similarly derived
following Guo  _et al._ (2022).

One further generalization considered in Han  _et al._ (2023), which again
immediately follows from our effective field theory, is to consider quantum
particles (bosons or fermions) subject to the same dipole-conserving dynamics.
In our EFT, we can take the _same_ Lagrangian (306), but simply modify $\Phi$
to

$\displaystyle\Phi_{s}=\int\mathrm{d}x\left(n\log n-s(1+sn)\log(1+sn)\right)$
(312)

where we take $s=1$ for bosons and $s=-1$ for fermions. Calculating

$\displaystyle\mu_{s}=\frac{\partial\Phi_{s}}{\partial\delta
n}=\log\frac{n}{1+sn}$ (313)

along with $D_{s}(n)=Dn^{2}(1+sn)^{2}$, which is the minimal scaling in which
the equation of motion is not singular, we find the same equations of motion
as Han  _et al._ (2023).

Although this theory is nonlinear, it is straightforward to convert the
discussion above into the operator formalism, where we maintain an exact
implementation of time-reversal symmetry using regularizations analogous to
App. D.

##  8 Lattice model for active phase separation

In this final section, we describe a lattice discretization of active phase
separation between two interacting species. Such models have been of recent
interest Saha  _et al._ (2020); You  _et al._ (2020). Here, we show how to
readily build such a model consistently coupled to noise—as usual, in our
construction, we will have explicit control over the thermodynamic (time-
reversal even part of the) phase diagram of the model. While we leave an
exhaustive analysis of this model to future work, we include the construction
here to emphasize the versatility of our methods.

Let us begin by overviewing how an active Cahn-Hilliard model for phase
separation has been derived in the literature Saha  _et al._ (2020). Let
$\phi_{\alpha,i}$ denote the concentration of species $\alpha\in\\{1,2\\}$ at
site $i$. We assume that the net concentration of each species is conserved in
the dynamics as a whole. A nonactive dynamical system can be obtained on a
discretized lattice by considering the noisy stochastic equation

$\dot{\phi}_{\alpha,i}=-\Gamma\sum_{j}\Lambda_{ij}\frac{\partial\Phi}{\partial\phi_{\alpha,j}}+\text{noise}$
(314)

where, as usual in this paper, $\Phi$ denotes the logarithm of the stationary
distribution:181818Note that the $\lambda$ term is different here vs. e.g.
Ref. Saha  _et al._ (2020); however our construction is not too sensitive to
the choice of $\Phi$.

$\Phi=\beta\left[\sum_{\alpha=1}^{2}\left[\sum_{i\sim
j}\frac{1}{2}(\phi_{\alpha,i}-\phi_{\alpha,j})^{2}+\sum_{i}\left(\frac{\phi_{\alpha,i}^{4}}{4}-\frac{\phi_{\alpha,i}^{2}}{2}\right)\right]+\lambda\sum_{i}(\phi_{1,i}+\phi_{2,i})^{2}\right],$
(315)

where $i\sim j$ denotes nearest-neighbor sites on a two-dimensional square
lattice; the parameter $\lambda$ denotes an interaction between the two
species. Here we have also defined $\Lambda_{ij}$ to be the graph Laplacian:

$\Lambda_{ij}=\left\\{\begin{array}[]{ll}-1&\ i,j\text{ adjacent}\\\ 4&\
i=j\end{array}\right..$ (316)

The noise in (314) conserves the net concentration of each species, and will
be explicitly defined in the Fokker-Planck formalism shortly. Note that
$(\nabla^{2})_{ij}\approx-\Lambda_{ij}$. If $\beta$ is very large, we expect
to see $\phi_{\alpha,i}\approx\pm 1$, so that the first term in $\Phi$ is
small. When $\lambda\gg 1$, we would anticipate phase separation, where
$\phi_{1,i}\approx-\phi_{2,i}$. To add activity to this passive model, Ref.
Saha  _et al._ (2020) added a nonreciprocal diffusion term to the equation of
motion:

$\dot{\phi}_{\alpha,i}=-\Gamma\sum_{j}\Lambda_{ij}\frac{\partial\Phi}{\partial\phi_{\alpha,j}}+\kappa\sum_{\beta,j}\epsilon_{\alpha\beta}\Lambda_{ij}\phi_{\beta
j}+\text{noise}.$ (317)

It is deliberately the point of Ref. Saha  _et al._ (2020) that the
$\kappa$-dependent term above does not obviously follow from considering
derivatives of $\Phi$ with respect to $\phi_{\alpha i}$ – this term will thus
break detailed balance. The perspective of our paper is that a minimal model
for active phase separation can instead demand that $\Phi$ remains the
stationary distribution _even out of equilibrium_ ; therefore, we elect to
incorporate the nonreciprocal diffusion ($\kappa$ term) consistent with
stationarity. We can achieve this as follows. With Gaussian noise and detailed
balance, the passive model we would write down becomes a lattice
discretization of the passive model of Saha  _et al._ (2020):

$W=\sum_{\alpha,i,j}\Gamma\Lambda_{ij}\partial_{\alpha,i}\left(\partial_{\alpha,j}+\mu_{\alpha,j}\right).$
(318)

where $\Gamma$ denotes the noise strength. To write down active terms
consistent with this stationary distribution, we can invoke (67) and look for
an antisymmetric matrix $V_{\alpha i,\beta j}$ that leads to the desired
terms. Our strategy will be to look for $V$ that give us a desired “active
term”, up to presumably irrelevant nonlinear corrections (with the caveat that
we will not perform an exhaustive renormalization group analysis in this
paper). The key observation is that even in equilibrium configurations where
$\mu_{\alpha i}=0$, we can still find active terms in the equations of motion
of the form $[\partial_{\beta j}V_{\alpha i,\beta j}]$, and so we will choose
a $V$ so that this divergence gives us a discretization of the terms of
interest. Consider the following choice of $V$:

$V_{\alpha i,\beta
j}=-\kappa\epsilon_{\alpha\beta}\Lambda_{ij}\left(\frac{\phi_{\alpha
i}^{2}+\phi_{\beta j}^{2}}{2}-1\right)$ (319)

where $\epsilon_{12}=-\epsilon_{21}=1$ is the two-dimensional Levi-Civita
tensor. This choice of $V$ is chosen to be small near the global minima of
$\Phi$ (where $\phi^{2}=1$); hence, it will make the “unwanted” terms added to
the equations of motion to enforce stationarity as “small” as possible.
Explicit calculation gives us

$\displaystyle\sum_{\beta,j}\frac{\partial V_{\alpha i,\beta
j}}{\partial\phi_{\beta j}}$
$\displaystyle=-\sum_{\beta,j}\kappa\epsilon_{\alpha\beta}\Lambda_{ij}\phi_{\beta
j}$ (320a) $\displaystyle\sum_{\beta,j}V_{\alpha i,\beta j}\mu_{\beta j}$
$\displaystyle=-\sum_{\beta,j}\kappa\beta\epsilon_{\alpha\beta}\left(\frac{\phi_{\alpha
i}^{2}+\phi_{\beta
j}^{2}}{2}-1\right)\Lambda_{ij}\left[\sum_{k}\Lambda_{jk}\phi_{\beta
k}-\phi_{\beta j}+\phi_{\beta j}^{3}+2\lambda(\phi_{1,j}+\phi_{2,j})\right]$
(320b)

Notice that the continuum limit of the first line is precisely the active term
considered in Saha  _et al._ (2020), and which appeared in (317) –
$\kappa\epsilon_{\alpha\beta}\nabla^{2}\phi_{\beta}$ – while the second line
is an irrelevant correction to the standard passive dynamics, if we expand
around a static solution $\phi_{1i}=-\phi_{2i}=1$: the $V\mu$ term is
_quadratic_ in fluctuations while the $\partial V$ term is _linear_ in
fluctuations, with each term having at least two spatial derivatives in the
continuum limit. It is thus reasonable to speculate that this lattice model
ought to lie in the same universality class as the active scalar mixture
model, in which we simply do not include the $V\mu$ term in the equations of
motion.

In our construction, we know precisely that the stationary distribution of the
dynamics is independent of $\kappa$. Thus while $\kappa$ may lead to
nonreciprocal phases, as we saw in the Kuramoto model, these nonreciprocal
phases will provably obey all the laws of equilibrium statistical mechanics –
in particular, in one dimension, this model cannot exhibit any long-range
order.

This construction is curious in that it uses precisely the regularization
subtlety which App. D aims to avoid, in order to construct equations of motion
with desired active terms (at leading order in a long wavelength expansion).
This means that the construction of this section cannot be reproduced by any
MSR field theory in the continuum in which time-reversal takes the simple form
(97). We leave a more detailed investigation of such methods to future work
Lew-Smith  _et al._ (to appear).

##  9 Outlook

In this paper, we have proposed an effective theory formalism for
nonequilibrium systems that extends the approaches recently pioneered for
thermal systems Harder  _et al._ (2015); Crossley  _et al._ (2017); Glorioso
_et al._ (2017); Haehl  _et al._ (2016); Jensen  _et al._ (2018). Numerous
examples and applications demonstrate the broad applicability of our approach,
along with its relative ease of use; see Tab. 3 for a summary. Many of the
theories described in this manuscript were first derived and studied
elsewhere, but our effective theory approach provides an instructive
derivation of these theories, and can provide broader context into how one
particular model fits into a broader paradigm of nonequilibrium matter.

Our methods can elucidate subtleties that are easily overlooked when only
considering equations of motion, such as the subtle nature of parity and
generalized time-reversal symmetries (and their breaking) in recently
developed theories of active odd elastic solids. In other cases, our methods
give a clear framework within which existing models can be generalized: e.g.,
Kuramoto models of chiral rotors on higher-dimensional spheres.

Perhaps most importantly, our conceptual starting point for building models of
active matter is not to begin by breaking detailed balance or a second law of
thermodynamics. Instead, we begin by understanding the typical states in the
stationary distribution (i.e., we deduce $\Phi$), and in turn, understand how
it is possible for the system to “flow” around phase space. From our
perspective, a second law can often emerge, albeit with an entropy that may
differ from equilibrium analogues (see App. B).

Theory | Equation of motion | $\Phi$ | $V_{ab}$ | MSR-compatible
---|---|---|---|---
Overdamped harmonic oscillator | (126) | (124) | N/A | const. noise
Finite-mass harmonic oscillator | (130) | (127) | symplectic | const. noise
Vortex in superfluid thin films | (137) | (131) | symplectic | const. noise
Landau-Lifshitz-Gilbert spin | (138) | (146) | symplectic | const. noise
Odd harmonic oscillator | (152) | (127) | (149) | const. noise
Non-reciprocal Kuramoto model | (178) | (171) | (176) | const. noise
Wealth | (185) | (180) | N/A | const. noise
Rigid body | (246)(250) | (228) | (247) | const. noise
Odd diffusion | (254) | (251) | (256) | yes
Odd elasticity | (260)(271) | (257)(265)(270) | (259)(269) | yes
Odd viscosity | (280) | (277) | (283) | yes
Odd stress | (294) (301) | (284) (297) | (286) (7.3.3) | yes
Fracton hydrodynamics | (309) | (308) (312) | N/A | yes
Phase separation | (317) | (315) | (319) | break Itô
Table 3: Summary of the various example systems considered in this paper. If
the stationary state is given by some Hamiltonian $H$, then $\Phi=\beta H$
corresponds to thermal equilibrium. The symplectic $V_{ab}$ refers to the
Hamiltonian dynamics in Sec. 3.1. The last column indicates whether the theory
is compatible with MSR for different regularizations: “const. noise” means to
be compatible with Sec. 2.7 the noise must not be multiplicative; “yes” means
that MSR path integrals are exact upon applying the spatial regularization of
App. D, MSR is applicable; “break Itô” means the MSR formalism of Sec. 2.7 is
not applicable. A formal resolution to the case of “break Itô” is provided in
App. C.

###  9.1 Generalizations and comparisons to previous literature

Looking forward, there are a number of open directions for extending this
work, which we now enumerate. First, this paper has focused on active
_classical_ matter; some of us will propose a notion of active _quantum_
matter in a follow-up paper Guo  _et al._ (to appear). We also plan to develop
a more systematic classification for active systems with spontaneously broken
symmetries, including spacetime symmetries, which may lead to new universality
classes with “flocking” phenomena Lew-Smith  _et al._ (to appear).

It is also important to understand the extent to which our framework does
_not_ crisply capture known phases of active matter. The key challenge is
that, in our formalism, it is important to know the stationary distribution
$\exp[-\Phi]$ of the dynamics; in many currently studied systems of active
matter, this is not easily determined. Moreover, it is possible that $\Phi$ is
not expressible within a Wilsonian (i.e., derivative, few-body) expansion. For
example, the Toner-Tu model violates the equilibrium Mermin-Wagner theorem
Toner and Tu (1998), meaning that $\Phi$ must be nonlocal. It would be
interesting to understand if there is a systematic means by which to construct
a nonlocal $\Phi$, while maintaining strict locality in the equations of
motion and noise correlations.

On an optimisitc note, it is sometimes the case that _equilibrium_ quantum
phase transitions once believed to be beyond Landau’s paradigm are now argued
to be compatible therewith McGreevy (2022) (once one considers, e.g., higher-
form symmetries). So it might be the case that our EFT approach to active
matter can classify all known phases, once suitable generalized symmetries
and/or constructions of $\Phi$ are identified. We leave these important open
questions to future work.

The thermal EFTs for fluids with conserved charges and energy/momentum neatly
couple to background gauge fields and geometry, respectively. In _quantum_
field theory, it seems crucial to have these background couplings to ensure
that a theory is not anomalous; however, we have not found that such couplings
are necessarily natural within our formalism. We anticipate that it is not
possible, in general, to couple to background geometry in the same way as in a
thermal fluid—a simple obstruction we have already encountered is the
existence of odd stress in certain active chiral fluids, since this effect is
forbidden by KMS symmetry Huang and Lucas (2022). The lattice model of Qi  _et
al._ (2023) demonstrates that such anomalous effects, forbidden in thermal
systems, are readily possible in microscopic classical models of active
systems. However, our formalism additionally allows for time-reversal-breaking
terms are possible that are not compatible with any known anomaly in quantum
field theory. A simple example is a biased diffusion equation
$\partial^{\vphantom{\prime}}_{*}{t}\rho+\partial^{\vphantom{\prime}}_{*}{x}v(\rho)-D\partial^{2}_{x}\rho+\cdots=0$—for
generic $v(\rho)$ and $\Phi=\int\mathrm{d}x\,\rho^{2}$, one can show that this
theory is consistent with stationarity, and analyze it using the EFT framework
we develop. However, the apparent anomaly coefficient is nonlinear and field
dependent, rendering it beyond the existing classification of anomalies in
quantum field theory. As far as we can tell, one must use our MSR framework to
describe such a theory, rather than a “geometrically inspired” approach.

###  9.2 Towards classifying active phases of matter

Finally, conclude by posing the question, _when can a phase only be realized
out of equilibrium_? Certain phases of matter may arise in active matter,
where the microscopic degrees of freedom are driven by, e.g., internal
batteries, and yet the long-wavelength description of such phases may
reproduce a thermal theory (such as a diffusion equation). In what phases can
we show that _no thermal system_ can realize an active phase of matter?

A first proposal is that any system that breaks detailed balance (time-
reversal) is active. For example, such fluids can have odd viscosity and
diffusion. However, such odd transport coefficients can be reproduced by PT-
symmetric thermal systems Huang and Lucas (2022). An experimental
“realization” of such a system—at least at the level of linear response—is an
electron fluid in a background magnetic field Berdyugin  _et al._ (2019),
albeit strictly speaking, momenta is only conserved if the fluid is overall
charge neutral, as can be realized in graphene Crossno  _et al._ (2016); Lucas
and Fong (2018). Since such fluids can have odd viscosity, this time-reversal-
odd transport coefficient is not unique to active fluids.

An example of a system that appears to be truly nonequilibrium—yet also
describable fully in our framework—is the chiral phase of the nonreciprocal
Kuramoto model Fruchart  _et al._ (2021). To understand why, notice that if we
have a Hamiltonian system that spontaneously breaks time-reversal symmetry, it
must be the case that the minima of $\Phi$ (where the system is most likely to
be found) are _not_ extrema of $H$ (as there is no deterministic dynamics if
the gradient of $H$ vanishes). This means that $\Phi\neq\beta H$ is required
to have any mathematical “Hamiltonian description” of a nonreciprocal phase
with spontaneous T-breaking. At best, therefore, one relaxes to a strongly
nonthermal stationary distribution, so it makes sense to us to call this phase
active.

That active matter can both realize genuinely new phases, or simply reproduce
physics possible in thermal systems, may have interesting implications for
frameworks in which the amount of entropy production in the steady-state is
used as a diagnostic for “activity” Schnakenberg (1976); Tomé and de Oliveira
(2012). Such definitions can lead to entropy production from a single particle
spinning in a cyclotron orbit in a magnetic field (a system which has PT, but
not T, symmetry). On the other hand, at least in hydrodynamics, there is a
natural notion of entropy production in hydrodynamics that vanishes for a
fluid in thermal equilibrium, even when placed in a background magnetic field
Hartnoll  _et al._ (2007); Son and Surówka (2009).

Our proposal that nonreciprocal phases where a generalized time-reversal
symmetry is spontaneously broken can only be realized in active matter could
be a first step towards identifying phases unique to active matter, yet surely
there are other possibilities as well. We expect that the tools, and simply
the perspective, advocated in this manuscript can play an important role in
comprehensively understanding what is possible in active systems constrained
not only by thermal equilibrium, but mere statistical stationarity.

## Acknowledgments

We thank Jay Armas, Michel Fruchart, Jinkang Guo, Oliver Hart, Akash Jain,
Marvin Qi, Leo Radzihovsky, Yi-Zhuang You, and especially Vincenzo Vitelli,
for useful discussions. This work was supported in part by the Alfred P. Sloan
Foundation through Grant FG-2020-13795 (AL), the National Science Foundation
through CAREER Grant DMR-2145544 (XH, IZ, AL) and CAREER Award DMR-2045181
(PG), through the Gordon and Betty Moore Foundation’s EPiQS Initiative via
Grant GBMF10279 (XH, JHF, AL), by the Laboratory for Physical Sciences through
the Condensed Matter Theory Center of the University of Maryland, College Park
(PG), by the Department of Energy through Award DE-SC0019380 (PG), and by the
Simons Foundation through Award No. 620869 (PG) and a Simons Investigator
Award via Leo Radzihovsky (AJF). AL thanks NORDITA and KITP for hospitality
while this manuscript was written. KITP is supported by the National Science
Foundation through Grant PHY-1748958.

##  Appendix A Topological exceptions to the form of dissipationless terms

Are expressions of the form (67) are the _only_ solutions to the condition
(70)? In fact, there exists a single, special case that is not captured by
constructions of the form (67). Suppose that the (phase-space) coordinates
$\\{q^{\,}_{a}\\}$ belong to an $N$-dimensional manifold $M$; for simplicity
suppose that $M$ is compact. In this case, one can use de Rham cohomology to
classify all possible solutions to (67) by expressing the equation in the
language of differential forms:

$0\,=\,\partial^{\vphantom{\prime}}_{a}\left(v_{a}\mathrm{e}^{-\Phi}\right)=\mathrm{d}*\left(v\mathrm{e}^{-\Phi}\right)\,.~{}~{}$
(A.321)

In the right equation above, we wrote $v=v_{a}\mathrm{d}x_{a}$ as a 1-form;
the Hodge star $*$ converts
$*\mathrm{d}x_{1}=\mathrm{d}x_{2}\wedge\cdots\wedge\mathrm{d}x_{N}$, e.g., and
allows us to express divergences in terms of the exterior derivative
$\mathrm{d}$ acting on forms. The solutions captured by (67) in terms of the
antisymmetric matrix $V^{\,}_{ab}$ in (57) are those in which $*v$ is an exact
form:

$*v=\mathrm{d}(*V)$ (A.322)

where $V$ is a 2-form, which can be thought of as an antisymmetric matrix. The
de Rham cohomology ensures that these are _all_ solutions to (A.321) up to
certain topological solutions that can exist when the manifold $M$ has
noncontractible cycles (a nontrivial $\mathrm{H}^{1}(M)$ cohomology group).
For example, if we look at $M=\mathrm{T}^{N}$, where $\mathrm{T}^{N}$ is the
$N$-dimensional torus with coordinates $(x_{1},\ldots,x_{N})$, each of which
obeys $x_{i}\sim x_{i}+1$, we find $N$ linearly independent topological
solutions to (A.321) corresponding to

$v\mathrm{e}^{-\Phi}=\mathrm{d}x^{\vphantom{\prime}}_{i}$ (A.323)

for any $i$. These solutions correspond to flows that cycle around one
direction of the torus persistently. In Sec. 5.1, we will be particularly
interested in the existence of such cycles on nontrivial manifolds $M$.

##  Appendix B Second law of thermodynamics

In this appendix, we describe settings in which the second law of
thermodynamics arises naturally in our formalism, even in nonequilibrium
matter.

###  B.1 Linear response

In Sec. 2.7, we discussed an analogy between our formalism and recent EFTs for
dissipative thermal systems. In thermal field theories for dissipative
systems, there is an elegant way to construct an entropy current whose
divergence is nonnegative Glorioso  _et al._ (2017); Glorioso and Liu (2016,
2018). The existence of this entropy current underpins the Landau formulation
of hydrodynamics. Let us now show that – subject to two key assumptions – the
prescription of Sec. 2.5 will reproduce the thermal EFT derivation of an
entropy current and extend it to active fluids.

We proceed by using the Lagrangian formalism, closely following Glorioso  _et
al._ (2017); Glorioso and Liu (2016, 2018) to derive an entropy current. Then,
we will connect to the formalism in Sec. 2.5. Before starting the derivation,
let us emphasize that the two key assumptions we will be using. Firstly, we
will use that we work only in the linear response regime (we hope to
generalize this elsewhere). Secondly, we will assume Gaussian noise; recall
that the construction of Sec. 2.5 is also restricted to this case. With these
assumptions, gT is equivalent to the KMS-like transformation (97), which
simplifies the resulting analysis greatly.

Let us begin by considering the general (and most interesting) limit of a
field theory (Section 7). Under gT symmetry, $\mathcal{L}$ will transform to
$\tilde{\mathcal{L}}$ where tilde represents gT transformations, and we have

$\displaystyle\tilde{\mathcal{L}}-\mathcal{L}=\partial_{t}V^{t}+\partial_{i}V^{i}=\partial_{\mu}V^{\mu}.$
(B.324)

For this appendix only, we will use an Einstein summation convention on
spacetime $\mu$ indices. Expanding
$V^{\mu}=\mathrm{i}V^{\mu}_{(0)}+V^{\mu}_{(1)}+\ldots$, where
$V^{\mu}_{(k)}\sim O(\pi^{k})$, we find that if using the manifestly gT/KMS-
invariant $\mathcal{L}$ in Sec. 2.7, there may be only one contribution
$V^{\mu}_{(0)}\neq 0$ and $V^{\mu}_{(k)}=0,\;k>0$. To build the entropy
current explicitly, it is therefore convenient to integrate by parts and write
$\mathcal{L}$ as, schematically,

$\displaystyle\mathcal{L}=E_{a}[q]\pi_{a}+\frac{\mathrm{i}}{2}\pi_{a}F_{ab}[q,\overleftrightarrow{\partial}]\pi_{b},$
(B.325)

where $E_{a}[q],F_{ab}[q,\overleftrightarrow{\partial}]$ are local functions
except that $F_{ab}[q,\overleftrightarrow{\partial}]=F_{ba}$ contains
derivatives acting symmetrically on the $\pi_{a}$, while $E_{a}\pi_{a}$ has no
derivatives acting on $\pi_{a}$. For example, for the theory of charge
diffusion, we would write

$\mathcal{L}=\pi\partial_{t}\rho+\mathrm{i}\sigma\partial_{i}\pi\partial_{i}\left(\pi-\mathrm{i}\mu\right)\rightarrow\pi\partial_{t}\rho-\pi\sigma\partial_{i}\partial_{i}\mu+\mathrm{i}\sigma\partial_{i}\pi\partial_{i}\pi.$
(B.326)

Using this Lagrangian, we can have nontrivial $V^{\mu}_{(1)}$ as well, which
is important to have entropy production out of equilibrium. Defining a
diagonal matrix $\eta_{ab}$ whose $\pm 1$ entries encode whether the variable
$q_{a}$ is even or odd under T, we can explicitly use the definition (B.324)
to obtain

$\displaystyle\partial_{\mu}V^{\mu}_{(0)}$
$\displaystyle=\eta_{ab}\tilde{E}_{a}\mu_{b}-\frac{1}{2}\mu_{a}\tilde{F}_{ab}\mu_{b},$
(B.327a) $\displaystyle\partial_{\mu}V^{\mu}_{(1)}$
$\displaystyle=-\eta_{ab}\tilde{E}_{a}\pi_{b}-E_{a}\pi_{a}+\frac{1}{2}\mu_{a}\tilde{F}_{ab}\pi_{b}+\frac{1}{2}\pi_{a}\tilde{F}_{ab}\mu_{b},$
(B.327b)

where we used $\pi_{a}\tilde{F}_{ab}\pi_{b}=\pi_{a}F_{ab}\pi_{b}$. Let us
define $\hat{V}^{\mu}_{(1)}$ by replacing the $\pi$ by its corresponding
conjugate chemical potential $\mu$ in $V^{\mu}_{(1)}$, and then define the
entropy current as

$\displaystyle s^{\mu}\equiv V^{\mu}_{(0)}+\hat{V}^{\mu}_{(1)}.$ (B.328)

Hence, we find

$\displaystyle\partial_{\mu}s^{\mu}$
$\displaystyle=-E_{a}\mu_{a}+\frac{1}{2}\mu_{a}\tilde{F}_{ab}\mu_{b}=\frac{1}{2}\mu_{a}F_{ab}\mu_{b}\geq
0,$ (B.329)

where we used $E_{a}=0$ as the equation of motion, and positive
semidefiniteness comes from the fact that distribution of fluctuations is
bounded, and thus $\mathrm{Im}~{}\mathcal{L}\geq 0$. When there is no
dissipation, we find a conserved entropy current $\partial_{\mu}s^{\mu}=0$,
and, from the construction in Sec. 2.7, we can immediately see that the
entropy is equal to the steady-state distribution

$\displaystyle\Phi=-\int\mathrm{d}^{d}xV^{0}_{(0)}=-\int\mathrm{d}^{d}xs^{0}.$
(B.330)

As an illustrative example, consider a simple chiral fluid with degree of
freedom $u_{i}$ and $n$. From (7.3.2) with the spin turned off, we can extract
the $V^{\mu}$ as

$\displaystyle\partial_{\mu}V^{\mu}_{0}$
$\displaystyle=-\partial_{t}u_{i}\mu_{i}-\partial_{t}\delta
n\mu^{n}-n_{0}\partial_{i}(u_{i}\mu^{n})-\beta_{0}^{-1}\rho^{-2}\eta^{\mathrm{even}}_{jikl}\partial_{j}(\mu_{i}\partial_{k}\mu_{l})-\beta_{0}^{-1}D^{\mathrm{even}}_{ij}\partial_{i}(\mu^{n}\partial_{j}\mu^{n})$
$\displaystyle+\beta_{0}^{-1}\rho^{-2}\eta^{\mathrm{odd}}_{jikl}\mu_{i}\partial_{j}\partial_{k}\mu_{l}+\beta_{0}^{-1}D^{\mathrm{odd}}_{ij}\mu^{n}\partial_{i}\partial_{j}\mu^{n},$
(B.331a) $\displaystyle\partial_{\mu}\hat{V}^{\mu}_{1}$
$\displaystyle=2\beta_{0}^{-1}\rho^{-2}\eta^{\mathrm{even}}_{jikl}\partial_{j}(\mu_{i}\partial_{k}\mu_{l})+2\beta_{0}^{-1}D^{\mathrm{even}}_{ij}\partial_{i}(\mu^{n}\partial_{j}\mu^{n}).$
(B.331b)

This leads to the entropy current

$\displaystyle s^{0}$
$\displaystyle=-\beta_{0}\frac{\rho}{2}u_{i}^{2}-\beta_{0}\frac{1}{2\chi}\delta
n^{2},$ (B.332a) $\displaystyle s^{i}$
$\displaystyle=-\beta_{0}\frac{n_{0}}{\chi}\delta
nu_{i}+\beta_{0}\frac{1}{\chi}D_{ij}\delta n\partial_{j}\delta
n+\beta_{0}\eta_{ijkl}u_{j}\partial_{k}u_{l},$ (B.332b)

where $D_{ij}$ and $\eta_{ijkl}$ include both T-even and T-odd terms. Notice
that (B.330) is satisfied. Upon using equations of motion, we arrive at

$\displaystyle\partial_{\mu}s^{\mu}=\beta_{0}\frac{1}{\chi}D^{\mathrm{even}}_{ij}\partial_{i}\delta
n\partial_{j}\delta
n+\beta_{0}\eta^{\mathrm{even}}_{ijkl}\partial_{i}u_{j}\partial_{k}u_{l}\geq
0.$ (B.333)

While this goes against the common expectations in the literature as far as we
can tell, we showed how this arises naturally at least in simple settings. We
leave a careful generalization of this argument to arbitrary nonlinear
systems, using our operator formalism, to future work.

Next, let us connect back to the discussion of Sec. 2.5. For simplicity of
notation in the following two paragraphs, we describe systems with a finite-
dimensional phase space, but the conclusions are general. As before, we are
focused in this appendix on the linear response limit. In this limit, we can
take $P(q_{a},t)$ to be a perturbed Gaussian distribution with variance
$\sigma_{a}^{2}$ and mean zero:

$\displaystyle
P(q_{a},t)\propto\exp\left[-\sum_{a}\frac{(q_{a}-\hat{q}_{a}(t))^{2}}{2\sigma_{a}^{2}}\right],$
(B.334)

where $\hat{q}_{a}(t)$ represent the degrees of freedom of hydrodynamics
(e.g., in a nonfluctuating limit we would have above equation of motion
$E_{a}(\hat{q}_{a})=0$) – i.e. they are expectation values of $q_{a}$
instantaneously in $P(t)$. We have, without loss of generality in the linear
response regime, further rotated the variables $q_{a}$ so that the genuine
stationary state has

$\Phi=\sum_{a}\frac{q_{a}^{2}}{2\sigma_{a}^{2}}.$ (B.335)

Plugging in (75), with $P^{\rm ss}=P(q_{a};\hat{q}_{a}=0)$, we find

$\displaystyle
D(\hat{q}_{a})=-\sum_{a}\frac{\hat{q}_{a}^{2}}{2\sigma_{a}^{2}}=\Phi(\hat{q}_{a}).$
(B.336)

We see that the entropy (B.330) agrees with the Kullback-Leibler divergence
(75).

As a further sanity check, it is helpful to explicitly confirm that the FPE of
(B.334) precisely reproduces the naive hydrodynamic equation for $\hat{q}_{a}$
alone. First we evaluate the FPE directly:

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}P=-WP=\partial_{a}Q_{ab}(\partial_{b}+\mu_{b})P=-(\mu_{a}-\hat{\mu}_{a})Q_{ab}\hat{\mu}_{b}P.$
(B.337)

where $\hat{\mu}_{a}=\hat{q}_{a}/\sigma_{a}^{2}$. Then, we can evaluate the
time-derivative by simply assuming the form (B.334), and checking when the two
answers are identical:

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}P=\left(\mu_{a}-\hat{\mu}_{a}\right)\frac{\mathrm{d}\hat{\mu}_{a}}{\mathrm{d}t}P.$
(B.338)

So the FPE (B.337) agrees with (B.338) so long as we enforce the noise-free
equation of motion

$\displaystyle
E_{a}(\hat{\mu}_{a})=-\partial_{t}\hat{\mu}_{a}+Q_{ab}\hat{\mu}_{b}=0.$
(B.339)

This short calculation explains why our notion of second law always exists
within linear response: the full FPE is mathematically equivalent to an
equation for the averages $\hat{q}_{a}$ alone Banks and Lucas (2019).

###  B.2 A nonequilibrium system

Lastly, let us describe how such an entropy current arises even in a driven
nonequilibrium system that breaks time-reversal symmetry: a diffusing charge
between two reservoirs held at different densities. The degrees of freedom are
$\rho(x)$ for $|x|\leq a$, and we take the boundary conditions to be fixed:

$\rho(\pm a)=\pm a\alpha,$ (B.340)

for some constant $\alpha$. The steady-state solution to the linear diffusion
equation $\partial_{t}\rho=D\partial_{x}^{2}\rho$ is

$\rho_{\mathrm{ss}}(x)=\alpha x.$ (B.341)

In a thermal fluid which relaxes to a uniform density, the entropy production
rate is locally proportional to $D(\partial_{x}\rho)^{2}$, so one could
compute the entropy production in this diffusive theory as

$\dot{S}_{\mathrm{ss}}\sim\int\limits_{-a}^{a}\mathrm{d}xD(\partial_{x}\rho_{\mathrm{ss}})^{2}=2Da\alpha^{2}>0,$
(B.342)

and thus deduce that this system is out of equilibrium as entropy is locally
produced for all times (and leaks through the boundaries).

In our framework, this situation is understood a bit differently (and we hope
that the contrast clarifies the sense in which nonequilibrium systems
nonetheless can obey a second law). The steady state of the dynamics is
readily found by making the elementary observation that if we write

$\rho(x,t)=\rho_{\mathrm{ss}}(x)+\tilde{\rho}(x,t),$ (B.343)

then (fixing the charge susceptibility to be 1, i.e. $\mu=\tilde{\rho}$) we
see that

$\Phi\sim\frac{1}{2}\int\mathrm{d}x\;\tilde{\rho}^{2}$ (B.344)

is the same $\Phi$ that would arise for a theory of conserved charge with
ordinary “equilibrium boundary conditions” ($\tilde{\rho}(\pm a)=0$). From our
perspective, the entropy current becomes

$\displaystyle s^{0}$
$\displaystyle=-\Phi=-\frac{1}{2}\tilde{\rho}^{2}=-\frac{1}{2}(\rho-\alpha
x)^{2},$ (B.345a) $\displaystyle s^{x}$
$\displaystyle=-D\tilde{\rho}\partial_{x}\tilde{\rho}=-D(\rho-\alpha
x)(\partial_{x}\rho-\alpha).$ (B.345b)

We have written the equations in terms of both the original $\rho$, and the
“nicer” $\tilde{\rho}$, to emphasize that the entropy current naturally
defined by our formalism becomes space-dependent in the “physical” coordinates
due to the nonequilibrium boundary conditions. One can straightforwardly check
that this entropy current obeys the second law of thermodynamics for the
nonequilibrium steady state. We thus put forth the point of view that
restoring the second law by _changing the definition of entropy_ is an
alternative perspective to the common notion of defining nonequilibrium
systems as those in which entropy is produced. While our perspective then
loses a physically motivated definition for nonequilibrium, we gain the power
of the second law and the fluctuation-dissipation theorem in return.

Finally, we emphasize that in this simple example, our proposal that entropy
production vanishes in the nonequilibrium steady state is similar to previous
hydrodynamic theories in background fields Hartnoll  _et al._ (2007); Son and
Surówka (2009). In this setting, one may elect to define the entropy _density_
differently as

$s^{0}=-\frac{1}{2}\rho^{2},$ (B.346)

instead of (B.345a), although a quick calculation confirms that
$\partial_{t}\int\mathrm{d}xs^{0}$ is the same with both definitions in this
theory. Beyond this convention, and the normalization of the entropy current,
our results then agree with the hydrodynamic theories of Hartnoll  _et al._
(2007); Son and Surówka (2009).

##  Appendix C Derivation of the MSR path integral

To compute the Jacobian in (92), we adopt the $\alpha$-regularization Lau and
Lubensky (2007) as a generalization of the Itô ($\alpha=0$) and Stratonovinch
($\alpha=1/2$) regularizations. The discretization of the path integral
depends on the choice of $\alpha$ as follows. By dividing the time to $N+1$
intervals $\Delta_{t}$, the matrix elements of the Jacobian become
$\mathcal{J}^{(ij)}_{ab}=\partial\hat{N}^{(j)}_{b}/\partial q^{(i)}_{a}$, with
$q^{(i)}_{a}\equiv q_{a}(t_{i})$ and

$\displaystyle\hat{N}^{(j)}_{b}=q_{b}^{(j)}-q_{b}^{(j-1)}-\Delta_{t}\left[f_{b}\left(\alpha
q^{(j)}+(1-\alpha)q^{(j-1)}\right)+b_{bc}\left(\alpha
q^{(j)}+(1-\alpha)q^{(j-1)}\right)\xi_{c}^{(j-1)}\right]\,,~{}~{}$ (C.347)

and we see that $\mathcal{J}_{ab}^{(ij)}$ is a lower-triangular matrix in
$(ij)$ with diagonal elements

$\displaystyle\mathcal{J}_{ab}^{(ii)}=\delta_{ab}-\alpha\Delta_{t}\partial_{a}f_{b}-\alpha\Delta_{t}\partial_{a}b_{bc}\xi_{c}^{(i-1)}.$
(C.348)

Using $\det A=\exp\Tr\ln A$, we obtain the Jacobian
$\mathcal{J}=\prod_{i}\mathcal{J}^{(ii)}$, where

$\displaystyle\mathcal{J}^{(ii)}[\rho;\xi]=1-\alpha\Delta_{t}\partial_{a}f_{a}-\alpha\Delta_{t}\partial_{a}b_{ab}\xi_{b}^{(i-1)}+\frac{(\alpha\Delta_{t})^{2}}{2}\left(\partial_{a}b_{ac}\partial_{b}b_{bc^{\prime}}-\partial_{a}b_{bc}\partial_{b}b_{ac^{\prime}}\right)\xi_{c}^{(i-1)}\xi_{c^{\prime}}^{(i-1)}+\ldots.$
(C.349)

In the above equation, we have only kept terms at leading order
$\mathcal{O}(\Delta_{t})$ and used $\xi^{(i)}\sim(\Delta_{t})^{-1/2}$. Since
the third term scales as $(\Delta_{t})^{1/2}$, direct exponentiation does not
work. Therefore, we first replace every noise term in $\mathcal{J}^{(ii)}$ by
$\xi_{a}^{(i-1)}\Delta_{t}=b^{-1}_{ab}(q_{b}^{(i)}-q_{b}^{(i-1)}-\Delta_{t}f_{b})$
according to the equation of motion, then perform the noise averaging. With
the help of an auxiliary field $\pi_{\alpha}$, we obtain

$\displaystyle Z$
$\displaystyle=\int\mathrm{D}\xi~{}Z[\xi]\mathrm{e}^{-\frac{1}{2}\int\mathrm{d}t~{}\xi^{2}}=\int\mathrm{D}\rho\prod_{i}\mathcal{J}^{(ii)}[\rho]\int\mathrm{D}\pi~{}\mathrm{e}^{\mathrm{i}\int\mathrm{d}t\left[\pi_{a}\partial_{t}q_{a}-f_{a}\pi_{a}+\frac{\mathrm{i}}{2}\pi_{a}Q_{ab}\pi_{b}\right]}.$
(C.350)

After applying the trick

$\displaystyle\left(q_{b}^{(i)}-q_{b}^{(i-1)}-\Delta_{t}f_{b}\right)\mathrm{e}^{\mathrm{i}\int\mathrm{d}t\left[\pi_{a}\partial_{t}q_{a}-f_{a}\pi_{a}\right]}=-\mathrm{i}\frac{\partial}{\partial\pi_{b}^{(i)}}\mathrm{e}^{\mathrm{i}\int\mathrm{d}t\left[\pi_{a}\partial_{t}q_{a}-f_{a}\pi_{a}\right]},$
(C.351)

and integrating by parts, we arrive at

$\displaystyle
Z=\int\mathrm{D}q\mathrm{D}\pi\prod_{i}\left(1-\alpha\Delta_{t}\partial_{a}f_{a}+\mathrm{i}\alpha\Delta_{t}\partial_{a}b_{ab}b_{cb}\pi_{c}+\frac{\alpha^{2}\Delta
t}{2}\left(\partial_{a}b_{ac}\partial_{b}b_{bc}-\partial_{a}b_{bc}\partial_{b}b_{ac}\right)\right)~{}\mathrm{e}^{\mathrm{i}\int\mathrm{d}t\left[\pi_{a}\partial_{t}q_{a}-f_{a}\pi_{a}+\frac{\mathrm{i}}{2}\pi_{a}Q_{ab}\pi_{b}\right]}.$
(C.352)

Now we can exponentiate the Jacobian and obtain
$Z=\int\mathrm{D}q\mathrm{D}\pi\mathrm{e}^{\mathrm{i}\int\mathrm{d}tL}$ with
the MSR Lagrangian given by

$\displaystyle L(q,\pi)=\pi_{a}\partial_{t}q_{a}-\left(f_{a}-\alpha
b_{ac}\partial_{b}b_{bc}\right)\pi_{a}+\mathrm{i}\alpha\partial_{a}f_{a}-\frac{\mathrm{i}\alpha^{2}}{2}\left(\partial_{a}b_{ac}\partial_{b}b_{bc}-\partial_{a}b_{bc}\partial_{b}b_{ac}\right)+\frac{\mathrm{i}}{2}\pi_{a}Q_{ab}\pi_{b}.$
(C.353)

There are two problems to address The first problem is the existence of the
$O(\pi^{0})$ terms. In thermal equilibrium, one can use causality or
retardness of response function to argue that these $O(\pi^{0})$ terms must
vanish Gao  _et al._ (2020), but this is not known for systems out-of-
equilibrium. We will leave this problem open, although we find it might be
possible to argue that the loop diagrams that violate the causality would
exactly cancel. In fact, these $O(\pi^{0})$ terms does not show up once we
convert (C.353) into the FPE, which is given by Lau and Lubensky (2007)

$\displaystyle\partial_{t}P=-\partial_{a}\left[(f_{a}+\alpha\partial_{b}b_{ac}b_{bc})P-\frac{1}{2}\partial_{b}(Q_{ab}P)\right].$
(C.354)

The second problem is how to implement the T symmetry in (C.353). We wish to
have the minimal transformation: $\pi\to-\pi+\mathrm{i}\mu$ as in (97). Let
$f_{a}=G_{a}+v_{a}$ with $G_{a}/v_{a}$ T-even/odd. The minimal transformation
requires the T-even force to be

$\displaystyle G_{a}-\alpha
b_{ac}\partial_{b}b_{bc}=-\frac{1}{2}Q_{ab}\mu_{b}.$ (C.355)

On the other hand, in order for (C.354) to support a steady-state distribution
up to a T-odd force, we find

$\displaystyle
G_{a}+(\alpha-\frac{1}{2})\partial_{b}b_{ac}b_{bc}-\frac{1}{2}b_{ac}\partial_{b}b_{bc}=-\frac{1}{2}Q_{ab}\mu_{b}.$
(C.356)

To satisfy both (C.355) and (C.356), we must choose $\alpha=1/2$. Therefore,
to implement (97) as our T symmetry, we must take $\alpha=1/2$ in the
Lagrangian (C.353). Note that in the main text, we can deal with the Itô
scheme $\alpha=0$ when we assume the weak noise limit, such that
$G_{a}\approx-\frac{1}{2}Q_{ab}\mu_{b}$ for both (C.355) and (C.356), or we
use the regularization of App. D in the continuum limit.

Stratonovinch regularization $\alpha=1/2$ is also the most natural choice to
make the T-odd force invariant under the generalized T symmetry (gT). Taking
$\alpha=1/2$, the T-odd terms in (C.353) are

$\displaystyle
H^{\mathrm{odd}}=\pi_{a}v_{a}-\mathrm{i}\frac{1}{2}\partial_{a}v_{a}.$ (C.357)

One can show that only under the constraint (70) can $H^{\mathrm{odd}}$ be gT-
invariant. This generalizes the weak noise limit in the main text. Moreover,
if the system only contains T-odd terms, i.e. dissipationless, we can also let
the gT symmetry to act on an arbitrary regularization $\alpha$ but, at the
same time, shift $\alpha\to 1-\alpha$. For example,
$H^{\mathrm{odd}}_{\alpha=0}=\pi_{a}v_{a}$ is invariant under gT by
transforming to
$H^{\mathrm{odd}}_{\alpha=1}=\pi_{a}v_{a}-\mathrm{i}\partial_{a}v_{a}$. This
thus explains why the naive KMS symmetry discussed at the end of Sec. 3.1 for
Hamiltonian dynamics with a state-dependent symplectic form will not know the
shift in $\Phi$ (118) – the gT symmetry includes both the KMS transformation
and the shift in $\alpha$.

Lastly, we note that it is challenging to straightforwardly construct higher-
order terms in (C.353), whereas such a generalization would be straightforward
in the Itô path integral, so long as time-reversal is implemented by (97). For
this reason, we believe that these MSR path integrals will be most valuable in
models where the method of App. D can be employed to neglect subtleties with
regularization.

###  C.1 Correlation functions

An advantage of the above Lagrangian is that, although reversibility is not
manifest, it does allows us to compute response/correlation functions using a
more familiar diagrammatic approach. As one example, let us show how the FDT
(63) is derived using the T symmetry. We perturb the steady-state distribution
by $\Phi\to\Phi-h_{a}(t)q_{a}(t)$. Under such perturbation, the forces will
shift in the Lagrangian, so we obtain (note that we take $\alpha=1/2$)

$\displaystyle-\frac{\mathrm{i}}{2}\langle
q_{a}(t)Q_{bc}\pi_{c}(t^{\prime})\rangle_{0}-\mathrm{i}\langle
q_{a}(t)\frac{\delta v_{c}}{\delta
h_{b}}\pi_{c}(t^{\prime})\rangle_{0}-\frac{1}{2}\langle
q_{a}(t)\partial_{c}\frac{\delta v_{c}}{\delta
h_{b}}(t^{\prime})\rangle_{0}=\frac{\delta\langle q_{a}(t)\rangle}{\delta
h_{b}(t^{\prime})}|_{h_{a}=0},\quad\mathrm{for}\;\;t>t^{\prime}.$ (C.358)

According to (70), $\delta v_{a}$ satisfies

$\displaystyle\mu_{a}\frac{\delta v_{a}}{\delta
h_{b}}-\partial_{a}\frac{\delta v_{a}}{\delta h_{b}}=v_{b}.$ (C.359)

Consider acting T on the pieces in (C.358) that are proportional to $\delta
v_{a}$:

$\displaystyle\mathrm{T:}\;\;-\mathrm{i}\langle q_{a}(t)\frac{\delta
v_{c}}{\delta h_{b}}\pi_{c}(t^{\prime})\rangle_{0}-\frac{1}{2}\langle
q_{a}(t)\partial_{c}\frac{\delta v_{c}}{\delta h_{b}}(t^{\prime})\rangle_{0}$
$\displaystyle\to-\mathrm{i}\langle q_{a}(t^{\prime})\frac{\delta
v_{c}}{\delta h_{b}}\pi_{c}(t)\rangle_{0}-\frac{1}{2}\langle
q_{a}(t^{\prime})\partial_{c}\frac{\delta v_{c}}{\delta h_{b}}(t)\rangle_{0}$
$\displaystyle\;\;\;\;\;-\langle q_{a}(t^{\prime})\frac{\delta v_{c}}{\delta
h_{b}}\mu_{c}(t)\rangle_{0}+\langle q_{a}(t^{\prime})\partial_{c}\frac{\delta
v_{c}}{\delta h_{b}}(t)\rangle_{0}.$

Using the T-invariance of it and (C.359), we obtain, for $t>t^{\prime}$,

$\displaystyle-\mathrm{i}\langle q_{a}(t)\frac{\delta v_{c}}{\delta
h_{b}}\pi_{c}(t^{\prime})\rangle_{0}-\frac{1}{2}\langle
q_{a}(t)\partial_{c}\frac{\delta v_{c}}{\delta
h_{b}}(t^{\prime})\rangle_{0}=\langle q_{a}(t)v_{b}(t^{\prime})\rangle_{0}.$
(C.360)

Plugging (C.360) in (C.358) and using (97), we obtain

$\displaystyle\chi_{ab}(t,t^{\prime})\equiv\frac{\delta\langle
q_{a}(t)\rangle}{\delta h_{b}(t^{\prime})}|_{h=0}=\frac{1}{2}\langle
q_{a}(t)Q_{bc}\mu_{c}(t^{\prime})\rangle_{0}+\langle
q_{a}(t)v_{b}(t^{\prime})\rangle_{0},\quad\mathrm{for}\;\;t>t^{\prime}.$
(C.361)

From (C.358) and employing the time-translation symmetry, $\partial_{t}\langle
q_{a}(t)q_{b}(t^{\prime})\rangle_{0}=-\partial_{t^{\prime}}\langle
q_{a}(t)q_{b}(t^{\prime})\rangle_{0}$, we arrive at our generalized FDT:

$\displaystyle\partial_{t}\langle
q_{a}(t)q_{b}(t^{\prime})\rangle_{0}=-\chi_{ab}(t,t^{\prime})\Theta(t-t^{\prime}).$
(C.362)

Remarkably, both the T-odd force $v_{a}$ and the multiplicative noise are
encoded in the response function $\chi_{ab}$ which can be probed by perturbing
the steady-state distribution. We note that (C.362) generalizes the result in
Lau and Lubensky (2007) to $v_{a}\neq 0$.

##  Appendix D Convenient regularization of a nonlinear diffusion equation

In this appendix191919We thank Akash Jain for discussions that led to the
calculation of this appendix. we further elaborate on subtleties for
continuum, hydrodynamic field theories, associated with the apparent mismatch
between the KMS transformation and time-reversal or detailed balance in our
stochastic framework. We will see that (at least in this simple example) it is
possible to choose a discretization in which, _in the long wavelength limit_ ,
time-reversal symmetry and the KMS transformation exactly agree.

We consider a theory of nonlinear diffusion which, in the continuum, takes the
form

$\partial_{t}\rho=\partial_{x}\left(D(\rho)\partial_{x}\rho+\text{noise}\right).$
(D.363)

To apply our Fokker-Planck formalism precisely, it is necessary to discretize
this equation _in space_ : $\rho_{n}(t)$ (for $n\in\mathbb{Z}$) become our
degrees of freedom, which model $\rho(x_{n},t)$ for some discretization
$\\{x_{n}\\}$ of the real line. For convenience in what follows, we take the
lattice spacing to be 1, so $x_{n}=n$, so that in the long-wavelength limit,

$(\partial_{x}\rho)_{n}\approx\rho_{n+1}-\rho_{n}\approx\rho_{n}-\rho_{n-1}\approx\frac{\rho_{n+1}-\rho_{n-1}}{2}\approx\cdots.$
(D.364)

We will use this relative freedom to our advantage, in what follows. In order
to discretize the continuum nonlinear diffusion theory, we wish to build a
Fokker-Planck generator $W=-\partial_{a}Q_{ab}(\partial_{b}+\mu_{b})$ (with
$\mu_{b}=\rho_{b}$ for simplicity). One choice is:

$Q^{(1)}_{n,m}=\displaystyle\left\\{\begin{array}[]{ll}\displaystyle
D\left(\dfrac{\rho_{n}+\rho_{n-1}}{2}\right)+D\left(\dfrac{\rho_{n}+\rho_{n+1}}{2}\right)&\
n=m\\\ \displaystyle-D\left(\dfrac{\rho_{n}+\rho_{m}}{2}\right)&\ |n-m|=1\\\
0&\ \text{otherwise}\end{array}\right..$ (D.365)

Note that $Q^{(1)}$ has a null vector $\rho_{b}=1$, compatible with Noether’s
theorem from Sec. 2.6. Another logical choice is

$Q^{(2)}_{n,m}=\displaystyle\left\\{\begin{array}[]{ll}\displaystyle\dfrac{2D(\rho_{n})+D(\rho_{n+1})+D(\rho_{n-1})}{2}&\
n=m\\\ \displaystyle-\dfrac{D(\rho_{n})+D(\rho_{m})}{2}&\ |n-m|=1\\\ 0&\
\text{otherwise}\end{array}\right.$ (D.366)

which also has the desired null vector encoding charge conservation. A simple
calculation shows that

$\partial_{m}Q^{(1)}_{nm}=0$ (D.367)

while

$\partial_{m}Q^{(2)}_{nm}\neq 0.$ (D.368)

Therefore, so long as we have in mind the continuum KMS-invariant theory as
being discretized according to $Q^{(1)}$, the MSR Lagrangian exactly encodes
time-reversal symmetry, and the regularization subtleties are not important.

A similar method can be employed to study more complicated corrections, such
as

$\partial_{t}\rho=\partial_{x}\left(D(\rho)\partial_{x}\rho+F(\rho)(\partial_{x}\rho)^{3}+\cdots+\text{noise}\right).$
(D.369)

In this example, one could choose to add (this construction is not unique):

$Q^{(F)}_{n,m}=\displaystyle\left\\{\begin{array}[]{ll}\displaystyle
F\left(\dfrac{\rho_{n+2}+\rho_{n-1}}{2}\right)\left(\dfrac{\rho_{n+2}-\rho_{n-1}}{3}\right)^{2}+F\left(\dfrac{\rho_{n-2}+\rho_{n+1}}{2}\right)\left(\dfrac{\rho_{n-1}-\rho_{n}}{3}\right)^{2}&\
n=m\\\ \displaystyle-F\left(\dfrac{\rho_{n\pm 2}+\rho_{n\mp
1}}{2}\right)\left(\dfrac{\rho_{n\pm 2}-\rho_{n\mp 1}}{3}\right)^{2}&\ m=n\pm
1\\\ 0&\ \text{otherwise}\end{array}\right..$ (D.370)

This theory does, on long wavelengths, reduce to the same continuum theory as
desired, while obeying $\partial_{m}Q^{(F)}_{nm}=0$.

The deviations between KMS and time-reversal can always be removed up to an
arbitrary order in derivatives by choosing a suitable discretized
regularization of $Q$, as long as the derivative expansion is treated
perturbatively, as stated in the main text. This strategy will thus work so
long as we can approximate $\Phi$ to be a strictly local integral in space
(e.g. there are no Goldstone bosons), as in this case one can always choose a
“point splitting” of the continuum equations of motion that ensures that
$\partial_{b}Q_{ab}=0$, while $Q_{ab}\mu_{b}$ matches the continuum theory to
arbitrary orders in derivatives (by adding “counter terms” to cancel the
discrepancies that arise at lower orders). Thus one can safely use a KMS-
invariant path integral to describe a T-invariant continuum theory.

An interesting exception to the validity of this regularization, which we will
not explore in detil in this paper, is when the steady state of the model
possesses an intrinsic length scale that is shorter or comparable to the
hydrodynamic scales of interest, such as in the presence of a domain wall –
see e.g. Section 8. In these situations, the derivative expansion is not
strictly perturbative, and thus the above regularization is not directly
applicable.

##  Appendix E Breaking gT in a four-species Kuramoto model on the 2-sphere

We describe the case of $N=4$ species of particles on the 2-sphere
$\mathrm{S}^{2}$, and look for generalized Kuramoto models compatible with
spontaneous gT breaking. We begin by looking for dynamics with an
$\mathrm{S}_{4}$-invariant $\Phi$, where $\mathrm{S}_{4}$ is the permutation
group acting on four objects. There is no obvious physical application for
this dynamics; however, the symmetry group of $\Phi$ is sufficiently
complicated that one needs to think in group theoretic terms to make progress.
Hence, the group-theoretic arguments below are helpful in determining whether
gT symmetry can be spontaneously broken in models whose stationary state
$\Phi$ has a non-Abelian symmetry, and generalize beyond this particular
example.

First, we recall a few group theory facts about $\mathrm{S}_{4}$, following
the notation of Zee (2016). $\mathrm{S}_{4}$ has a normal subgroup isomorphic
to $\mathbb{Z}_{2}$, which corresponds to the sign of a permutation, with
alternating group $\mathrm{A}_{4}=\mathrm{S}_{4}/\mathbb{Z}_{2}$. There are 5
irreducible representations of $\mathrm{S}_{4}$, which we denote using the
high energy physics notation with $\mathbf{1}$, $\mathbf{1}^{\prime}$,
$\mathbf{2}$, $\mathbf{3}$, $\mathbf{3}^{\prime}$.

(a)

(b)  (c)

Figure 6: Positive and negative examples of spontaneous gT-breaking of 4
species of particles on $\mathrm{S}^{2}$, which coalesce into (a)
$\mathrm{A}_{4}$ symmetry: in this case, no spontaneous gT-breaking is
possible, as explained in the appendix. Geometrically, the lack of natural gT-
breaking corresponds to the high symmetry of the tetrahedron—there is no
preferred rotation axis. (b) Three blue rotors and one red with nonreciprocal
interactions (red pursues blue, blue avoids red). There is still no
spontaneous gT-breaking since there is no natural axis around which to rotate.
If the blue species have $\mathrm{S}_{3}$ symmetry, the only natural choice of
$\Omega_{IJ}$ is oriented in the direction of the red species. (c) Two “inert”
rotors an two others with nonreciprocal interactions. This time, spontaneous
gT-breaking is allowed since there is a clear axis around which to rotate,
given by (206).

With these facts in mind, we look for a theory in which the $\mathbb{Z}_{2}$
normal subgroup of $\mathrm{S}_{4}$ is spontaneously broken, while also
playing the role of the “g” in gT-symmetry. In other words, we wish to find an
$\Omega_{IJ}$ which flips sign if we apply a sign-changing permutation, but is
otherwise invariant. In order to build such an $\Omega_{IJ}$ in an
$\mathrm{S}_{4}$-covariant manner without explicitly breaking the SO(3)
rotational symmetry of $\mathrm{S}^{2}$, we need to ensure that the $IJ$
indices on $\Omega$ come from $n_{I}^{(a)}$ and $n_{J}^{(b)}$. Moreover, there
cannot be any dangling $(a)$ or $(b)$ indices, namely we must write
$\Omega_{IJ}$ in terms of an $\mathrm{S}_{4}$-covariant expression that flips
sign under permutation. To deduce a possible construction, we use the
following group theoretical facts. A “defining” representation of
$\mathrm{S}_{4}$ is the four-dimensional representation
$\mathbf{4}=\mathbf{1}\oplus\mathbf{3}$, on which the permutation group
elements are represented by permutation matrices with 1s and 0s only. Notice
that with respect to the $(a)$ index, $n^{(a)}_{I}$ transforms in the
$\mathbf{4}$. The antisymmetric $IJ$ indices on $\Omega$ must come from an
antisymmetric tensor product which contains representations

$\mathbf{4}\otimes_{\mathrm{A}}\mathbf{4}=\mathbf{3}\oplus\mathbf{3}^{\prime}.$
(E.371)

Under $\mathrm{S}_{4}$, we wish for $\Omega_{IJ}$ to be an element of the
signed $\mathbf{1}^{\prime}$ irreducible representation. Since
$\mathbf{3}\otimes\mathbf{1}^{\prime}=\mathbf{3}^{\prime}$, we wish to find a
scalar object with respect to SO(3) (no dangling $IJ$ indices) that transforms
in either the $\mathbf{3}$ or $\mathbf{3}^{\prime}$. An appropriate inner
product between these two motifs in three-dimensional representations of
$\mathrm{S}_{4}$ will give us the desired $\Omega_{IJ}$.

Actually, it is a little easier to look for objects in the
$\mathbf{4}^{\prime}=\mathbf{1}^{\prime}\oplus\mathbf{3}^{\prime}$ and
$\mathbf{4}$ representations, since permutations act more simply on these
representations. We have found that the following transforms in the
$\mathbf{4}^{\prime}$:

$\displaystyle
G^{(1)}_{IJ}=n^{(2)}_{[I}n^{(3)}_{J]}+n^{(3)}_{[I}n^{(4)}_{J]}+n^{(4)}_{[I}n^{(2)}_{J]},\;\;\;\;-G^{(2)}_{IJ}=n^{(3)}_{[I}n^{(4)}_{J]}+n^{(4)}_{[I}n^{(1)}_{J]}+n^{(1)}_{[I}n^{(3)}_{J]},$
$\displaystyle
G^{(3)}_{IJ}=n^{(4)}_{[I}n^{(1)}_{J]}+n^{(1)}_{[I}n^{(2)}_{J]}+n^{(2)}_{[I}n^{(4)}_{J]},\;\;\;\;-G^{(4)}_{IJ}=n^{(1)}_{[I}n^{(2)}_{J]}+n^{(2)}_{[I}n^{(3)}_{J]}+n^{(3)}_{[I}n^{(1)}_{J]},$
(E.372)

while in the $\mathbf{4}$ we have

$F^{(a)}=\left|-n^{(a)}_{I}+\sum_{b=1}^{4}n^{(b)}_{I}\right|^{2}.$ (E.373)

Hence,

$\Omega_{IJ}=\sum_{a=1}^{4}F^{(a)}G^{(a)}_{IJ}$ (E.374)

transforms in the $\mathbf{1}^{\prime}$ of $\mathrm{S}_{4}$, implying that the
resulting $W$ is gT-invariant. The resulting theory will spontaneously break
gT symmetry so long as $\Omega_{IJ}$ does not vanish in the minima of $\Phi$.
These minima of $\Phi$ must also spontaneously break $\mathrm{S}_{4}$ in a
generic configuration where $\Omega_{IJ}\neq 0$. We have numerically confirmed
that for generic choices of $n^{(a)}_{I}$, $\Omega_{IJ}$ given in (E.374) is
indeed nonvanishing while transforming in the $\mathbf{1}^{\prime}$.

In the simplest theory, we may attempt to choose $\Phi$ such that its minima
correspond to the $n^{(a)}_{I}$ equidistant on $\mathrm{S}^{2}$. The unique
such configuration is a tetrahedron, as shown in Fig. 6. Unfortunately, one
can check that (E.374) vanishes in such a configuration; intuitively this is
because there is no preferred axis to rotate around, when in such a highly
symmetric configuration we can perform an even permutation on the
configuration by a global rotation (which would in turn rotate $\Omega_{IJ}$).
A preferred axis can arise as one deviates from the tetrahedral configuration.

If the four particles form a tetrahedral configuration, but the interactions
between them are not $\mathrm{A}_{4}$-symmetric, then it is possible to have
nontrivial dynamics: see, e.g., Fig. 6 for a few different possibilities.

Finally, if the four particles form instead a square configuration on
$\mathrm{S}^{2}$—in this case, there are two sensible equilibrium
configurations with unique T-breaking dynamics. The first reasonable
equilibrium configuration would feature the 4 particles forming a square in
some planar slice of $\mathrm{S}^{2}$, for example, around the equator. Take
the axis perpendicular to the square to be the $3$ direction. In that case,
none of the permutations in $\mathrm{A}_{4}$ correspond to rotations outside
of the $1-2$ plane! Therefore the matrix $\epsilon_{IJ3}$ is invariant, and we
may include a term $\Omega_{IJ}\sim\epsilon_{IJ3}$, which is similar to the
model studied in (5.2.2).

## References

  * Wilson (1971a) Kenneth G. Wilson, “Renormalization group and critical phenomena. I. Renormalization group and the Kadanoff scaling picture,” Phys. Rev. B 4, 3174–3183 (1971a).
  * Wilson (1971b) Kenneth G. Wilson, “Renormalization group and critical phenomena. II. Phase-space cell analysis of critical behavior,” Phys. Rev. B 4, 3184–3205 (1971b).
  * Hohenberg and Halperin (1977) P. C. Hohenberg and B. I. Halperin, “Theory of dynamic critical phenomena,” Rev. Mod. Phys. 49, 435–479 (1977).
  * Weinberg (1979) Steven Weinberg, “Phenomenological Lagrangians,” Physica A: Stat. Mech. Appl. 96, 327–340 (1979).
  * Altland and Simons (2010) Alexander Altland and Ben D. Simons,  _Condensed Matter Field Theory_, 2nd ed. (Cambridge University Press, 2010).
  * Martin  _et al._ (1973) P. C. Martin, E. D. Siggia, and H. A. Rose, “Statistical dynamics of classical systems,” Phys. Rev. A 8, 423–437 (1973).
  * Toner and Tu (1998) John Toner and Yuhai Tu, “Flocks, herds, and schools: A quantitative theory of flocking,” Phys. Rev. E 58, 4828–4858 (1998).
  * Scheibner  _et al._ (2020) C. Scheibner, A. Souslov, D. Banerjee, and  _et al_ , “Odd elasticity,” Nature Phys. , 475–480 (2020).
  * Fruchart  _et al._ (2022) Michel Fruchart, Colin Scheibner, and Vincenzo Vitelli, “Odd viscosity and odd elasticity,”  (2022).
  * Huang and Lucas (2022) Xiaoyang Huang and Andrew Lucas, “Hydrodynamic effective field theories with discrete rotational symmetry,” JHEP 2022 (2022), 10.1007/jhep03(2022)082.
  * Friedman  _et al._ (2023) Aaron J. Friedman, Caleb Q. Cook, and Andrew Lucas, “Hydrodynamics with triangular point group,” SciPost Phys. 14, 137 (2023), arXiv:2202.08269 [cond-mat.str-el] .
  * Delacrétaz and Glorioso (2020) Luca V. Delacrétaz and Paolo Glorioso, “Breakdown of diffusion on chiral edges,” Phys. Rev. Lett. 124, 236802 (2020).
  * Qi  _et al._ (2023) Marvin Qi, Jinkang Guo, and Andrew Lucas, “Anomalous hydrodynamics with triangular point group in 2+1 dimensions,” Phys. Rev. B 107, 144305 (2023), arXiv:2209.08108 [cond-mat.stat-mech] .
  * Gromov  _et al._ (2020) Andrey Gromov, Andrew Lucas, and Rahul M. Nandkishore, “Fracton hydrodynamics,” Phys. Rev. Research 2, 033124 (2020).
  * Feldmeier  _et al._ (2020) Johannes Feldmeier, Pablo Sala, Giuseppe De Tomasi, Frank Pollmann, and Michael Knap, “Anomalous diffusion in dipole- and higher-moment-conserving systems,” Phys. Rev. Lett. 125, 245303 (2020).
  * Morningstar  _et al._ (2020) Alan Morningstar, Vedika Khemani, and David A. Huse, “Kinetically constrained freezing transition in a dipole-conserving system,” Phys. Rev. B 101, 214205 (2020).
  * Hart  _et al._ (2022) Oliver Hart, Andrew Lucas, and Rahul Nandkishore, “Hidden quasiconservation laws in fracton hydrodynamics,” Phys. Rev. E 105, 044103 (2022), arXiv:2110.08292 [cond-mat.stat-mech] .
  * Glorioso  _et al._ (2022) Paolo Glorioso, Jinkang Guo, Joaquin F. Rodriguez-Nieva, and Andrew Lucas, “Breakdown of hydrodynamics below four dimensions in a fracton fluid,” Nature Phys. 18, 912–917 (2022), arXiv:2105.13365 [cond-mat.str-el] .
  * Grosvenor  _et al._ (2021) Kevin T. Grosvenor, Carlos Hoyos, Francisco Peña Benítez, and Piotr Surówka, “Hydrodynamics of ideal fracton fluids,” Phys. Rev. Res. 3, 043186 (2021), arXiv:2105.01084 [cond-mat.str-el] .
  * Burchards  _et al._ (2022) Ansgar G. Burchards, Johannes Feldmeier, Alexander Schuckert, and Michael Knap, “Coupled hydrodynamics in dipole-conserving quantum systems,” Phys. Rev. B 105, 205127 (2022), arXiv:2201.08852 [cond-mat.quant-gas] .
  * Guo  _et al._ (2022) Jinkang Guo, Paolo Glorioso, and Andrew Lucas, “Fracton hydrodynamics without time-reversal symmetry,” Phys. Rev. Lett. 129, 150603 (2022), arXiv:2204.06006 [cond-mat.stat-mech] .
  * Harder  _et al._ (2015) Michael Harder, Pavel Kovtun, and Adam Ritz, “On thermal fluctuations and the generating functional in relativistic hydrodynamics,” Journal of High Energy Physics 2015, 1–25 (2015).
  * Crossley  _et al._ (2017) Michael Crossley, Paolo Glorioso, and Hong Liu, “Effective field theory of dissipative fluids,” JHEP 2017 (2017), 10.1007/jhep09(2017)095.
  * Glorioso  _et al._ (2017) Paolo Glorioso, Michael Crossley, and Hong Liu, “Effective field theory of dissipative fluids (II): Classical limit, dynamical KMS symmetry, and entropy current,” JHEP 2017 (2017), 10.1007/jhep09(2017)096.
  * Haehl  _et al._ (2016) Felix M. Haehl, R. Loganayagam, and Mukund Rangamani, “The fluid manifesto: Emergent symmetries, hydrodynamics, and black holes,” JHEP 01, 184 (2016), arXiv:1510.02494 [hep-th] .
  * Jensen  _et al._ (2018) Kristan Jensen, Natalia Pinzani-Fokeeva, and Amos Yarom, “Dissipative hydrodynamics in superspace,” JHEP 09, 127 (2018), arXiv:1701.07436 [hep-th] .
  * Kubo (1966) R Kubo, “The fluctuation-dissipation theorem,” Rep. Prog. Phys. 29, 255–284 (1966).
  * Martin and Schwinger (1959) Paul C. Martin and Julian Schwinger, “Theory of many-particle systems. I,” Phys. Rev. 115, 1342–1373 (1959).
  * Araki (1978) Huzihiro Araki, “On the Kubo-Martin-Schwinger boundary condition,” Prog. Theor. Phys. 64, 12–20 (1978).
  * Glorioso and Liu (2016) Paolo Glorioso and Hong Liu, “The second law of thermodynamics from symmetry and unitarity,” (2016), arXiv:1612.07705 [hep-th] .
  * Martin  _et al._ (2021) David Martin, Jérémy O’Byrne, Michael E. Cates, Étienne Fodor, Cesare Nardini, Julien Tailleur, and Frédéric van Wijland, “Statistical mechanics of active Ornstein-Uhlenbeck particles,” Phys. Rev. E 103, 032607 (2021).
  * Mermin and Wagner (1966) N. D. Mermin and H. Wagner, “Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic Heisenberg models,” Phys. Rev. Lett. 17, 1133–1136 (1966).
  * Landry (2023) M. J. Landry, “Active actions: effective field theory for active nematics,” (2023), arXiv:2309.15142 [cond-mat.soft] .
  * Tailleur and Cates (2008) J. Tailleur and M. E. Cates, “Statistical mechanics of interacting run-and-tumble bacteria,” Phys. Rev. Lett. 100, 218103 (2008).
  * Han  _et al._ (2021) Ming Han, Michel Fruchart, Colin Scheibner, Suriyanarayanan Vaikuntanathan, Juan J. de Pablo, and Vincenzo Vitelli, “Fluctuating hydrodynamics of chiral active fluids,” Nature Phys. 17, 1260–1269 (2021).
  * Fruchart  _et al._ (2021) Michel Fruchart, Ryo Hanai, Peter B. Littlewood, and Vincenzo Vitelli, “Non-reciprocal phase transitions,” Nature 592, 363–369 (2021).
  * Fokker (1914) A. D. Fokker, “Die mittlere Energie rotierender elektrischer Dipole im Strahlungsfeld,” Annalen der Physik 348, 810–820 (1914).
  * Planck (1917) M. Planck,  _Über einen Satz der statistischen Dynamik und seine Erweiterung in der Quantentheorie_, Sitzungsberichte der Königlich-Preussischen Akademie der Wissenschaften zu Berlin (Reimer, 1917).
  * Kolmogorov (1931) A. Kolmogorov, “Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung,” Mathematische Annalen 104, 415–458 (1931).
  * Kramers (1940) H.A. Kramers, “Brownian motion in a field of force and the diffusion model of chemical reactions,” Physica 7, 284–304 (1940).
  * Moyal (1949) J. E. Moyal, “Stochastic processes and statistical physics,” J. R. Stat. Soc. Ser. B Methodol. 11, 150–210 (1949).
  * Pawula (1967a) R. Pawula, “Generalizations and extensions of the Fokker-Planck-Kolmogorov equations,” IEEE Trans. Inf. Theory 13, 33–41 (1967a).
  * Pawula (1967b) R. F. Pawula, “Approximation of the linear Boltzmann equation by the Fokker-Planck equation,” Phys. Rev. 162, 186–188 (1967b).
  * Kadanoff (2000) L.P. Kadanoff,  _Statistical Physics: Statics, Dynamics and Renormalization_ (World Scientific, 2000).
  * Doi (1976a) M. Doi, “Second quantization representation for classical many-particle system,” J. Phys. A: Math. Gen. 9, 1465 (1976a).
  * Doi (1976b) M. Doi, “Stochastic theory of diffusion-controlled reaction,” J. Phys. A: Math. Gen. 9, 1479 (1976b).
  * Peliti, L. (1985) Peliti, L., “Path integral approach to birth-death processes on a lattice,” J. Phys. France 46, 1469–1483 (1985).
  * Peliti (1986) L. Peliti, “Renormalisation of fluctuation effects in the A+A to A reaction,” J. Phys. A: Math. Gen. 19, L365 (1986).
  * Langevin (1908) P. Langevin, “Sur la théorie du mouvement brownien,” C. R. Acad. Sci. (Paris) 146, 530–536 (1908).
  * Lemons and Gythiel (1997) Don S. Lemons and Anthony Gythiel, “Paul Langevin’s 1908 paper “On the Theory of Brownian Motion”,” Amer. J. Phys. 65, 1079–1081 (1997).
  * Kwon  _et al._ (2005) Chulan Kwon, Ping Ao, and David J. Thouless, “Structure of stochastic dynamics near fixed points,” PNAS 102, 13029–13033 (2005).
  * Prost  _et al._ (2009) J. Prost, J.-F. Joanny, and J. M. R. Parrondo, “Generalized fluctuation-dissipation theorem for steady-state systems,” Phys. Rev. Lett. 103, 090601 (2009).
  * Lau and Lubensky (2007) A. W. C. Lau and T. C. Lubensky, “State-dependent diffusion: Thermodynamic consistency and its path integral formulation,” Phys. Rev. E 76, 011123 (2007).
  * Schütz (2001) G.M. Schütz, “Exactly solvable models for many-body systems far from equilibrium,” in  _Phase Transitions and Critical Phenomena_, Vol. 19, edited by C. Domb and J.L. Lebowitz (Academic Press, 2001) pp. 1–251.
  * Kelly (1979) F.P. Kelly,  _Reversibility and Stochastic Networks_, Probability and Statistics Series (Wiley, 1979).
  * Kwon and Ao (2011) Chulan Kwon and Ping Ao, “Nonequilibrium steady state of a stochastic system driven by a nonlinear drift force,” Phys. Rev. E 84, 061106 (2011).
  * Schnakenberg (1976) J. Schnakenberg, “Network theory of microscopic and macroscopic behavior of master equation systems,” Rev. Mod. Phys. 48, 571–585 (1976).
  * Tomé and de Oliveira (2012) Tânia Tomé and Mário J. de Oliveira, “Entropy production in nonequilibrium systems at stationary states,” Phys. Rev. Lett. 108, 020601 (2012).
  * Cover and Thomas (2012) T.M. Cover and J.A. Thomas,  _Elements of Information Theory_ (Wiley, 2012).
  * Dubkov and Spagnol (2005) Alexander Dubkov and Bernardo Spagnol, “Generalized Wiener process and Kolmogorov’s equation for diffusion induced by non-Gaussian noise source,” (2005), arXiv:cond-mat/0505270 [cond-mat.stat-mech] .
  * Banks and Lucas (2019) Tom Banks and Andrew Lucas, “Emergent entropy production and hydrodynamics in quantum many-body systems,” Phys. Rev. E 99, 022105 (2019).
  * Glorioso and Liu (2018) Paolo Glorioso and Hong Liu, “Lectures on nonequilibrium effective field theories and fluctuating hydrodynamics,” (2018), arXiv:1805.09331 [hep-th] .
  * Mullins  _et al._ (2023) Nicki Mullins, Mauricio Hippert, Lorenzo Gavassino, and Jorge Noronha, “Relativistic hydrodynamic fluctuations from an effective action: causality, stability, and the information current,” (2023), arXiv:2309.00512 [hep-th] .
  * Kleinert (2009) H. Kleinert,  _Path integrals in quantum mechanics, statistics, polymer physics, and financial markets_ , 5th ed. (World Scientific, 2009).
  * Gao  _et al._ (2020) Ping Gao, Paolo Glorioso, and Hong Liu, “Ghostbusters: Unitarity and causality of non-equilibrium effective field theories,” JHEP 2020 (2020), 10.1007/jhep03(2020)040.
  * Delacretaz and Mishra (2023) Luca V. Delacretaz and Ruchira Mishra, “Nonlinear response in diffusive systems,” (2023), arXiv:2304.03236 [cond-mat.str-el] .
  * Onsager (1949) L. Onsager, “Statistical hydrodynamics,” Il Nuovo Cimento (1943-1954) 6, 279–287 (1949).
  * Newton (2011) Paul K. Newton,  _The N- vortex problem: Analytical Techniques_ (Springer, 2011).
  * (69) David Pines, Philippe Nozières, Philippe Nozières, and David Pines,  _The Theory of Quantum Liquids. 2: Superfluid Bose Liquids_ , 4th ed. (Addison-Wesley).
  * Reeves  _et al._ (2022) Matthew T. Reeves, Kwan Goddard-Lee, Guillaume Gauthier, Oliver R. Stockdale, Hayder Salman, Timothy Edmonds, Xiaoquan Yu, Ashton S. Bradley, Mark Baker, Halina Rubinsztein-Dunlop, Matthew J. Davis, and Tyler W. Neely, “Turbulent relaxation to equilibrium in a two-dimensional quantum vortex gas,” Phys. Rev. X 12, 011031 (2022).
  * Ambegaokar  _et al._ (1980) Vinay Ambegaokar, B. I. Halperin, David R. Nelson, and Eric D. Siggia, “Dynamics of superfluid films,” Phys. Rev. B 21, 1806–1826 (1980).
  * Qi and Lucas (2023) Marvin Qi and Andrew Lucas, “Kinematically constrained vortex dynamics in charge density waves,” (2023), arXiv:2310.00051 [cond-mat.stat-mech] .
  * Heisenberg (1928) W. Heisenberg, “Zur Theorie des Ferromagnetismus,” Zeitschrift für Physik 49, 619–636 (1928).
  * Girvin and Yang (2019) Steven M. Girvin and Kun Yang,  _Modern Condensed Matter Physics_ (Cambridge University Press, 2019).
  * Landau and Lifshitz (1935) L. D. Landau and E. M. Lifshitz, “On the theory of the dispersion of magnetic permeability in ferromagnetic bodies,” Physikalische Zeitschrift der Sowjetunion 8, 153–169 (1935).
  * Gilbert (1955) T. L. Gilbert, “A phenomenological theory of damping in ferromagnetic materials,” Journal of Applied Physics 26, 1101–1103 (1955).
  * Xiao  _et al._ (2010) Di Xiao, Ming-Che Chang, and Qian Niu, “Berry phase effects on electronic properties,” Rev. Mod. Phys. 82, 1959–2007 (2010).
  * Brandenbourger  _et al._ (2021) Martin Brandenbourger, Colin Scheibner, Jonas Veenstra, Vincenzo Vitelli, and Corentin Coulais, “Limit cycles turn active matter into robots,”  (2021).
  * Acebrón  _et al._ (2005) Juan A. Acebrón, L. L. Bonilla, Conrad J. Pérez Vicente, Félix Ritort, and Renato Spigler, “The Kuramoto model: A simple paradigm for synchronization phenomena,” Rev. Mod. Phys. 77, 137–185 (2005).
  * Kuramoto (1975) Y. Kuramoto, “Self-entrainment of a population of coupled non-linear oscillators,” in  _International Symposium on Mathematical Problems in Theoretical Physics_ (1975) pp. 420–422.
  * Kuramoto (1984) Y. Kuramoto,  _Chemical Oscillations, Waves, and Turbulence_ (Springer, 1984).
  * Strogatz (2000) S. H. Strogatz, “From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators,” Physica D: Nonlinear Phenomena 143, 1–20 (2000).
  * (83) M. Kalecki, “On the Gibrat distribution,” Econometrica 13, 161.
  * (84) J. Angle, “The surplus theory of social stratification and the size distribution of personal wealth,” Social Forces 65, 293.
  * (85) S. Ispolatov, P. L. Krapivsky, and S. Redner, “Wealth distributions in asset exchange models,” Eur. Phys. J. B2, 267.
  * Yakovenko and Rosser (2009) Victor M. Yakovenko and J. Barkley Rosser, “Colloquium: Statistical mechanics of money, wealth, and income,” Rev. Mod. Phys. 81, 1703–1725 (2009).
  * Newman (2005) MEJ Newman, “Power laws, Pareto distributions and zipf's law,” Contemporary Physics 46, 323–351 (2005).
  * Lew-Smith  _et al._ (to appear) E. Lew-Smith, J. H. Farrell, M. Qi, A. J. Friedman, and A. Lucas, (to appear).
  * José and Saletan (1998) Jorge V. José and Eugene J. Saletan,  _Classical Dynamics: A Contemporary Approach_ (Cambridge University Press, 1998).
  * McGreevy (2022) John McGreevy, “Generalized symmetries in condensed matter,”  (2022), 10.1146/annurev-conmatphys-040721-021029, arXiv:2204.03045 [cond-mat.str-el] .
  * Berg (1993) Howard C. Berg,  _Random Walks in Biology_ (Princeton University Press, 1993).
  * Binney and Tremaine (2011) James Binney and Scott Tremaine,  _Galactic Dynamics_ (Princeton University Press, 2011).
  * Gunn and Tinsley (1972) James E Gunn and Beatrice M Tinsley,  _The Physics of Galaxy Formation_ (Princeton University Press, 1972).
  * Hut (1980) Jean M. Hut, “Tidal evolution of close binary systems,” Astronomical Journal 85, 919–935 (1980).
  * Raston  _et al._ (2013) Paul L. Raston, Tao Liang, and Gary E. Douberly, “Anomalous $\lambda$-doubling in the infrared spectrum of the hydroxyl radical in helium nanodroplets,” J. Phys. Chem. A 117, 8103–8110 (2013).
  * Arnold and Khesin (1998) V. I. Arnold and B. A. Khesin,  _Topological Methods in Hydrodynamics_ (Springer, 1998).
  * de Groot and Mazur (1962) R. de Groot and P. Mazur,  _Non-Equilibrium Thermodynamics_ (Dover Publications, 1962).
  * Liebchen and Levis (2022) Benno Liebchen and Demian Levis, “Chiral active matter,” EPL 139, 67001–67001 (2022).
  * Banerjee  _et al._ (2017) Debarghya Banerjee, Anton Souslov, Alexander G. Abanov, and Vincenzo Vitelli, “Odd viscosity in chiral active fluids,” Nature Comm. 8 (2017), 10.1038/s41467-017-01378-7.
  * Hargus  _et al._ (2021) Cory Hargus, Jeffrey M. Epstein, and Kranthi K. Mandadapu, “Odd diffusivity of chiral random motion,” Phys. Rev. Lett. 127, 178001 (2021).
  * Nash  _et al._ (2015) Lisa M. Nash, Dustin Kleckner, Alismari Read, Vincenzo Vitelli, Ari M. Turner, and William T. M. Irvine, “Topological mechanics of gyroscopic metamaterials,” PNAS 112, 14495–14500 (2015).
  * Lier  _et al._ (2022) Ruben Lier, Jay Armas, Stefano Bo, Charlie Duclut, Frank Jülicher, and Piotr Surówka, “Passive odd viscoelasticity,” Phys. Rev. E 105, 054607 (2022).
  * Hartnoll  _et al._ (2018) Sean Hartnoll, Andrew Lucas, and Subir Sachdev,  _Holographic quantum matter_ (The Mit Press, 2018).
  * Cook and Lucas (2019) Caleb Q. Cook and Andrew Lucas, “Electron hydrodynamics with a polygonal fermi surface,” Phys. Rev. B 99, 235148 (2019).
  * Cook and Lucas (2021) Caleb Q. Cook and Andrew Lucas, “Viscometry of electron fluids from symmetry,” Phys. Rev. Lett. 127, 176603 (2021).
  * Rao and Bradlyn (2023) Pranav Rao and Barry Bradlyn, “Resolving Hall and dissipative viscosity ambiguities via boundary effects,” Phys. Rev. B 107, 075148 (2023).
  * Farrell  _et al._ (2022) Jack H. Farrell, Xiaoyang Huang, and Andrew Lucas, “Hydrodynamics with helical symmetry,” (2022), arXiv:2208.12269 [cond-mat.stat-mech] .
  * Han  _et al._ (2023) Jung Hoon Han, Ethan Lake, and Sunghan Ro, “Scaling and localization in kinetically constrained diffusion,” (2023), arXiv:2304.03276 [cond-mat.stat-mech] .
  * Saha  _et al._ (2020) Suropriya Saha, Jaime Agudo-Canalejo, and Ramin Golestanian, “Scalar active mixtures: The nonreciprocal Cahn-Hilliard model,” Phys. Rev. X 10, 041009 (2020).
  * You  _et al._ (2020) Zhihong You, Aparna Baskaran, and M. Cristina Marchetti, “Nonreciprocity as a generic route to traveling states,” Proceedings of the National Academy of Sciences 117, 19767–19772 (2020).
  * Guo  _et al._ (to appear) J. Guo, O. Hart, C-F. Chen, A. J. Friedman, and A. Lucas, “Active quantum matter,” (to appear).
  * Berdyugin  _et al._ (2019) A. I. Berdyugin, S. G. Xu, F. M. D. Pellegrino, R. Krishna Kumar, A. Principi, I. Torre, M. Ben Shalom, T. Taniguchi, K. Watanabe, I. V. Grigorieva, M. Polini, A. K. Geim, and D. A. Bandurin, “Measuring Hall viscosity of graphene’s electron fluid,” Science 364, 162–165 (2019).
  * Crossno  _et al._ (2016) Jesse Crossno, Jing K. Shi, Ke Wang, Xiaomeng Liu, Achim Harzheim, Andrew Lucas, Subir Sachdev, Philip Kim, Takashi Taniguchi, Kenji Watanabe, Thomas A. Ohki, and Kin Chung Fong, “Observation of the Dirac fluid and the breakdown of the Wiedemann-Franz law in graphene,” Science 351, 1058–1061 (2016).
  * Lucas and Fong (2018) Andrew Lucas and Kin Chung Fong, “Hydrodynamics of electrons in graphene,” J. Phys. Condens. Matter 30, 053001 (2018), arXiv:1710.08425 [cond-mat.str-el] .
  * Hartnoll  _et al._ (2007) Sean A. Hartnoll, Pavel K. Kovtun, Markus Müller, and Subir Sachdev, “Theory of the Nernst effect near quantum phase transitions in condensed matter and in dyonic black holes,” Phys. Rev. B 76, 144502 (2007).
  * Son and Surówka (2009) Dam T. Son and Piotr Surówka, “Hydrodynamics with triangle anomalies,” Phys. Rev. Lett. 103, 191601 (2009).
  * Zee (2016) A. Zee,  _Group Theory in a Nutshell for Physicists_ (Princeton Univ. Press, 2016).