Statistical Field Theory and Effective Action Method for scalar Active Matter

We employ Statistical Field Theory techniques for coarse-graining the steady-state properties of Active Ornstein-Uhlenbeck particles. The computation is carried on in the framework of the Unified Colored Noise approximation that allows an effective equilibrium picture. We thus develop a mean-field theory that allows to describe in a unified framework the phenomenology of scalar Active Matter. In particular, we are able to describe through spontaneous symmetry breaking mechanism two peculiar features of Active Systems that are (i) The accumulation of active particles at the boundaries of a confining container, and (ii) Motility-Induced Phase Separation (MIPS). We develop a mean-field theory for MIPS and for Active Lennard-Jones (ALJ) fluid. We thus discuss the universality class of MIPS and ALJ. Finally, we show that spherical active particles fall into the Ising universality class, i. e., a $\varphi^4$ field theory. Within this framework, we compute analytically the critical line $T_c(\tau)$ for both models. In the case of MIPS, $T_c(\tau)$ gives rise to a reentrant phase diagram compatible with an inverse transition from liquid to gas as the strength of the noise decreases. In the case of asymmetric particles, e. g., rod-shaped swimmers, the field theory acquires a $\varphi^3$ term that explicitly breaks $\varphi \to -\varphi$ symmetry. In this case, the critical point might be destroyed and substituted by a first-order phase transition.


I. INTRODUCTION
In nature, there are many and diverse examples of Living Materials [1] ranging from epithelial monolayers [2], bacterial colonies [3], or dense drops of ants [4]. Even though the elementary units composing such materials are complex biological objects, many of the emerging collective behaviors can be described using concepts of Condensed Matter and, in particular, through the analytical and numerical tools developed during the last decades in Active Matter [5][6][7][8][9]. Active Systems are defined as a class of nonequilibrium systems consisting of interacting entities that individually dissipate energy to generate forces and motion and exhibit self-organized behavior at large scales.
Active systems can develop complex patterns that change dynamically, as in the case of flocking [9]. Flocking has been observed at very different scales ranging from birds [10] to epithelial monolayers [11]. However, Active Systems can also give rise to pattern formation and condensed phases with structural properties remarkably similar to those of ordinary materials, e. g., gasliquid phase transition [12][13][14][15], glassy or jamming states [16][17][18], polar order [19], and nematic order [20]. Pattern formation in Active Matter is driven by out-of- * Matteo.Paoluzzi@roma1.infn.it equilibrium dynamics and thus these condensed phases are emergent properties of steady-state configurations that, in general, can not be described by a Boltzmann distribution. In particular, the dependency on the microscopic dynamics makes hard to establish possible universality classes. Understanding how condensed phases in Active Systems are related to those at equilibrium plays an important role in both, basic and applied science. In basic science, it would allow to gain insight into the concept of universality in non-equilibrium systems. In applied science, for instance, it would allow to design living and synthetic materials with desired structural properties.
In this paper, we present a study on universal properties of a specific class of Active Matter system that is described on large-scale by a scalar field theory. We start from a microscopic model where particles are selfpropelled through a persistent noise [21][22][23] and we perform a coarse-graining using the machinery of Statistical Field Theory. We show that, within the framework of Unified Colored Noise (UCN) approximation [24,25], the structural properties of the system can be described through an opportune Effective Action [26]. Active field theories based on the dynamical evolution of opportune set of order parameters have been largely employed for capturing the large scale behavior of active systems [27][28][29][30][31][32][33]. In our work, we show that some peculiar behaviors of Active Systems can be captured through an Equilibrium Statistical Field Theory approach.

A. Summary of Results
We aim to develop field theoretical description of Active Systems using the machinery of equilibrium Statistical Physics. As a main result, we obtain that the Active System can be described by an Effective Action that counts of non-local terms that can be systematically studied at mean-field level.
Focusing our attention on mean-field computations, we show that: (i) The Effective Action Method allows to describe the steady-state properties of Scalar Active Matter.
(ii) We can put into a unified theoretical framework two peculiar phenomena of Active Matter: the accumulation of active particles at the boundaries of a container, and Motility-Induced Phase Separation (MIPS) [34]. In particular, both phenomena can be interpreted in term of a spontaneous symmetry breaking ϕ → −ϕ.
(iii) We can discuss in a simple picture the universality class of MIPS and Active Lennard-Jones Fluids (ALJ). In particular, In both cases, we can compute analytically the critical line T c (τ ). Moreover, the mean-field theory suggests that both MIPS, and ALJ fall into Ising universality class.
(iv) In the case of MIPS, the curve T c (τ ) develops a reentrance in the phase diagram indicating that the system undergoes an inverse transition from liquid to gas as the effective temperature T is decreased above a threshold value τ th . This means that the condensed MIPS phase evaporates as the effective temperature decreases.
(v) We show that the Effective Action explicitly breaks the symmetry ϕ → −ϕ through a ϕ 3 terms that vanish for spherical active particles and it is non zero for rod-shaped swimmers. Moreover, the presence of this term suggests that the MIPS critical point can be hidden by a first-order phase transition.
The paper is organized as follows. In Sec (II) we introduce the theoretical framework we consider for obtaining an Effective Field theory. In Sec. (III) we introduce the microscopical model. In Sec. (IV) we employ the theoretical set-up for studying the one-body problems in Active Matter. In particular, we show how the accumulation of active particles at the boundaries of a container can be interpreted as a spontaneous symmetry breaking in the Effective Action. In Sec. (V) we address the many-body problem and discuss the general features of the theory. In Sec. (VI) we discuss a unified mean-field theory of Active Systems. In particular, we study the mean-field phase diagram of purely repulsive potentials and Lennard-Jones potentials. In sec. (VII) we discuss the effect of anisotropic interaction on MIPS. Finally, in Sec. (VIII) we present our conclusions.

II. THEORETICAL SET-UP
We start our discussion with introducing a formalism in equilibrium Statistical Mechanics that allows to perform the coarse-graining of generic n−body interactions. We consider a system composed by N particles in d spatial dimensions confined in a box of side L and volume V = L d . To keep the presentation simple, we indicate with [r i ] or (r i ) a generic particle configuration (r 1 , ..., r N ) where r i is a d−dimensional vector representing the position of the particle i. For the sake of completeness, we consider a hamiltonian system composed of classical particles whose degrees of freedom are canonical coordinates and conjugated momenta. In the next section, we will apply the formalism for computing configurational integrals in the case Active Systems, and thus we will not have generalized momenta.
Denoting p i the momentum of the particle i, we assume that the mechanical properties of the system are fully specified through the hamiltonian function H[p, r] that is where the configurational part H[r] takes into account 1−body, 2−body and k−body interactions, with k ≥ 3. We write The thermodynamic is obtained through the computation of the partition function Z [35] that is with f (β) the density of free energy. In Eq. (3), the thermodynamic limit N, V → ∞ is performed maintaining fixed the mean density ρ = N/V . We have introduced the thermal wavelength λ = h 2 2πmk B T . β the inverse temperature, i.e, β = 1/k B T , h is the Planck constant, and µ is the chemical potential. Working in natural unit, one has k B = 1 = h and thus β = T −1 .
For studying the behavior of the system on large scales, we perform a coarse-graining based on the local density field ψ(r) that is Using standard manipulations [26,36], one can enforce the field ψ(r) into Eqs. (3) through a delta functional and thus we can write where the auxiliary fieldψ has been introduced for representing the delta functional introduced in Eq. (5). The details of the computation are provided in the appendix (A). The functional G takes the form Performing a shift to the fieldψ − b →ψ, the thermodynamics can be recasted in the following form As one can appreciate, the auxiliary fieldψ in Eq. (9) plays the role of external source in a quantum field theory [36].
and . . . C indicates a connected correlation function.
A. Stationary points, mean-field approximation and fluctuations around the mean-field solution It is well known that mean-field theories neglect fluctuations [37]. In particular, in a mean-field approximation, one usually replaces the value of the order parameter in a given point of the space with its mean-value in the same point, i.e., ψ(r) → ψ(r) and ψ(r)ψ(r ) → ψ(r) ψ(r ) . The latter replacement holds whenever [ψ(r) − ψ(r)] 2 ∼ V −1/2 ∼ N −1/2 that vanishes in the thermodynamic limit. Through Eqs. (9) we define the functional F [ψ] that is Since the exponent in the functional integral that defines our theory is of order N , it makes sense to perform a saddle-point approximation for evaluating the functional integral and then compute systematically the stability of the stationary pointψ SP against fluctuations. Considering field configurationsψ =ψ SP + ∆ψ, we can thus write where the kernel in Eqs. (13) is and we have used the fact that δF δψ(r) SP = 0 (15) In term of the generating functional W , we can define Neglecting terms that scales with 1/N , we can rewrite the partition function that turns to be factorized as follows where we have performed the change of variable ∆ψ(r) = i ds G −1 (r, s)ϕ(s) (computation details are provided in appendix (B)).
Using the saddle-point relation Eqs. (15), we can write ρ(r) = N e −ψ(r)+b(r) z (18) meaning that we can writeψ SP as a function of ρ(r) obtaining in this way the following expression for the thermodynamics in the mean-field approximation In Eqs. (17), G is the full propagator of the theory that takes contribution from both, the two-body potential φ 2 and the k−body potentials φ k .
The equations of motion for AOUPs are [16,21,22,38,39,43] Here we indicate with x i a vector in d spatial dimensions that determines the position of the active particle i. The self-propulsive force is f a i , τ is the persistence time of the self-propulsion. The total mechanical force acting on the particle i is f m i . The term f ext i represents forces due to external conservative fields. Finally, µ is the mobility. The noise η i satisfies η α i = 0 and η α i (t)η β j (s) = 2T µδ ij δ αβ δ(t − s), where greek symbols indicate cartesian components. According to Eqs. (20), the self-propulsion is exponentially correlated in time, in particular one has f a,α i (t)f a,β j (s) = 2T µ τ e −|t−s|/τ . Let us introduce the steady-state distribution Adopting the Unified Colored Noise approximation [24,25], it has been shown [21,[40][41][42]44] that the approximate solution for P ss takes the form with Z β fixed by the normalization condition i dx i P ss (x i ) = 1. The hamiltonian H 0 is responsible for both, the mechanical interactions and the interactions with external fields, i. e., f m i + f ext As one can see, P ss (x i ) takes the form of an equilibrium distribution where the hamiltonian H 0 is replaced by an effective one that we named H U CN [x i ]. In this way, the structural properties of the system on large-scale can be computed through an equilibrium Statistical Mechanics theory based on H U CN [x i ]. In the present paper, we will apply the theoretical machinery introduced in the previous section for coarse-graining the equilibrium-like model [45].
The effective hamiltonian H U CN [x i ] is composed of three contributions. The term H 0 is the mechanical energy of the system at equilibrium. The terms H 1 and H 2 introduce non-local many-body interactions among particles that disappear in the limit τ → 0. The computation of det M in d spatial dimensions requires the diagonalization of a dN × dN matrix. Moreover, Eq. (21) requires that M must be a positive definite matrix. Here we will consider a small τ expansion based on the approximation det(δ αγ ij + τ D αγ ij ) = 1 + τ T rD αγ ij + o(τ 2 ) (see appendix E ), with D αγ ij the Hessian matrix. It is worth noting that the small τ breaks when the hessian develops negative eigenvalues of order τ −1 . In this work, we will restrict our computation in cases where D αγ ij is positive definite.

IV. ONE-BODY PROBLEMS
Since active particles break the fluctuation-dissipation theorem at single particle level [46][47][48], they show intriguing non-equilibrium phenomena even at the level of gas of non-interacting particles. In particular, when active particles are confined by a container or immersed into a confining potential, i. e., a central field that tends to confine particles into a region of space, the steadystate distribution strongly deviates from the Boltzmann distribution showing a double-peaked structure at high persistence time [5,[49][50][51]. The double-peaks signal the accumulation of active particles at the boundary of the confining potential instead in the center, i. e., where the potential is zero. This effect is due to the fact that active particles remain trapped in regions of space where the external field exerts forces that balance the self-propulsion force rather than in a region where the potential is zero. The reason why active particles accumulate at the boundaries of a confining potential has been discussed in detail by Tailleur and Cates in Ref. [49] in the case of run-andtumble particles. Accumulation at the boundaries has been also observed in experiments with E. coli bacterial confined into drops [52].
The phenomenology is easy to understand in the case of particles driven by an external force that represent the limit of active particles with infinite persistence time, i. e., τ → ∞. In that case, the mechanical equilibrium is reached where the force exerted by the potential balances the driving force. At finite persistence time, particles change their swimming direction with a rate τ −1 . However, active particles tend to spend more time in regions where external forces balance self-propulsion forces [21]. In this section, we will see that this non-equilibrium localization phenomenon can be interpreted as a spontaneous symmetry breaking at zero temperature. In particular, while a system at equilibrium develops condensation at the bottom of the external potential, active particles condensate away from the bottom: this can be rationalized through our formalism in term of an effective potential that develops a double-well structure as soon as τ > 0. The condensation away from the center of the trapping potential can be thus interpreted as a spontaneous symmetry breaking in an effective equilibrium picture. As a benchmark for the formalism, in appendix Eq. C we report the case of gas at equilibrium in one spatial dimension embedded into an external potential where no accumulation at the boundaries take place and localization appears at the bottom of the potential.

A. Gas of Active Particles in external potentials
We consider a gas of active particles embedded into an external potential A(x) in one spatial dimension. As we showed in the previous section, the steady-state properties of the system can be obtained through the computation of the following partition function where the effective hamiltonian we can thus write In terms of the density field ψ( We can approximate the integral using the saddle-point approximation where we have indicated with n(x) the field configuration that solves δG δψ(x) ψ=n = 0 .
The steady-state density profile reads At small temperatures the density profile is dominated by the minima x 0 of B(x) and thus n(x) is concentrated around that minima, i. e., n( We specialize our computation in the case of harmonic and anharmonic trapping, the latter due to a quartic confining potential, i. e., α = 2.

B. Soft confining potentials in Active Matter
Self-propulsion naturally introduces a typical scale for the forces, i. e., the self-propulsion force f s , that competes with the other force fields interacting with the active particles. Due to this fact, even smooth potential fields can give rise to dramatic confining effects. Now we consider a generic confining potential of the form with α > 0. The effective potential B(x) reads and the derivative is

Harmonic trapping
The harmonic trapping is recovered for α = 1. As it has been proved experimentally and verified in simulations [48], harmonic potentials lead to a generalization of the equipartition theorem. This is due to the fact that the effective energy takes the simple form that is the energy of an harmonic oscillator where the natural frequency ω 0 frequency results shifted from ω 0 = 1 to ω 0 = 1 + τ . This result means that no accumulation at the boundaries occurs in the case of AOU particles trapped through a harmonic potential.

Anharmonic trapping
Now we are going to show that the effective equilibrium picture reproduces the accumulations of active particles at the boundary of a confining container. For sake of simplicity, we consider the case α = 2, i. e., a soft anharmonic confining potential instead of a container with hard boundaries. The effective potential B(x) reads Now a density profile n(x) that is peaked around x 0 = turns to be unstable. For rationalizing that we compute the second derivative of B(x) that is as one can immediately check, the configuration x 0 = 0 turns to be unstable since B (0) = − 6τ β < 0, i. e., as soon as τ = 0, x 0 = 0 is not a minimum anymore. In terms of the effective potential B(x), the emerging phenomenology can be interpreted as a spontaneous symmetry breaking. To prove that we consider the small τ expansion of Eq. (34) that is As one can see, if we think at the potential A(x) in terms of a Landau theory, the potential corresponds to a meanfield theory a 2 x 2 + x 4 4 at the critical point, i. e., a = 0. Now, for τ > 0, the system is described by an effective Landau energy that is given by Eq. (36). If we interpret the coefficient of the x 2 term as a mass it turns to be unphysical since it is negative. This means that the original vacuum of the theory, i. e., x 0 = 0, is not a minimum anymore and thus B(x) spontaneously breaks the symmetry x → −x that was satisfied by A(x). Now the new stable configuration of lower energy is x 1,2 A finite but small temperature T = 1/β introduces a finite variance in the two distributions that remain peaked around ±v. Expanding around the new minima x = v +δ at the linear order in τ one has As one can see, fluctuations δx around the minimum v that spontaneously break the symmetry x → −x acquire a mass that is linear in τ . The emerging phenomenology is shown in the upper panel of Fig n(x) ∼ exp [−B(x)] for τ = 0, and τ = 0.05, (dashed red, and dashed green, respectively). As one can appreciate, for τ = 0, one has the peak of the distribution at x 0 = 0. For τ = 0.05, the distribution become double peaked. Moreover, the theoretical picture provides also approximation schemes for obtaining quantitative predictions. A comparison between numerical simulations of a gas of AOUPs embedded into a anharmonic x 4 /4 potential is shown in the lower panel of Fig. (1). The red circles represent the histogram P (x) computed from numerical data, considering N = 4000 independent runners. P ef f (x) has been computed considering P ef f (x) = N e −B(x)/T with N a normalization constant that guaranties dx P ef f (x) = 1. Here, the computation has been performed considering the expression of B(x) given in Eq. (34), i.e., without performing any small τ expansion. Now we will address the many-body problem. With this aim, we perform the coarse-grained computation of Eq. (21) in absence of external fields, i. e., φ 1 = 0. For sake of simplicity we indicate the two-body potential φ 2 (x, y) = φ(x, y). Moreover, for the computation of the determinant that appears in Eq. (21), we recover to a small τ expansion. In term of the density field ρ(x), using the theoretical framework introduced in Eq. (II), we can write the following mean-field model (details of the computation are provided in appendix E ) As one can see, the effective action S ef f results from three contributions: (i) a two-body interaction with kernel A(x, y), (ii) a three-body interaction whose kernel is B(x, y, z), and (iii) an entropic term that has combinatorial origin [53,54]. It is worth noting that, as we will see in details in the next sections, the two-body interaction, that is responsible also for the mass term in the corresponding field theory, can change the sign because of the competition between the original interaction φ and quadratic terms originated by the hessian contribution. Moreover, the cubic term breaks the symmetry ρ(x) → −ρ(x), meaning that, when the system undergoes phase separation, it promotes one of the two coexisting phases.

A. Central pair potentials
Eq. (38) defines the thermodynamic of a wide class of scalar active systems. In order to make quantitative progresses, we have to define the form of the interacting potential. Let us focus our attention in the case of central pair potentials that take the form φ( Indicating with the prime the derivative with respect r in φ(r), the diagonal part of the hessian matrix reads In terms of the local density field ψ(x) we can thus rewrite H 2 as follows The contribution of H 1 becomes At the saddle point we have ψ(x) = ρ(x) and thus we can finally write the effective action for central potentials that is where the functions A(r) and B(r, s) now reads

B. Motility-Induced Phase Separation
The action S ef f in Eq. (42) describes the phenomenology of Motility-Induced Phase Separation (MIPS) [8,15,34,[55][56][57], i. e., the ability of active systems to undergo a spinodal decomposition similar to gas-liquid phase separation that happens in absence of any attractive forces.
For rationalizing that, let us consider the stability of a homogeneous density profileρ. As we will see in details in the next sections, considering homogeneous solutions, at the saddle point we can write S ef f [ρ] = V g(ρ), where the function g(ρ) defines the intensive Gibbs free energy. The stability ofρ that minimizes g(ρ) depends on the sign of the second derivative of g with respect ρ computed in ρ. In particular, in the case of van der Waals theory, the negative sign of the coefficient ρ 2 ensures the possibility that a homogeneous density profile may become unstable in a certain region of the phase diagram [53]. If one neglects attractive interactions, the coefficient of the quadratic term turns to be positive and thus there is not hope to observe a spinodal decomposition.
In the case of active particles, A(r) takes two contributions. The first one is due to the two-body central potential φ(r). This is the standard term that one has in the equilibrium theory and it is attractive or repulsive depending on the type of potential. The second one is given by the function f (r) and is linear in τ , i. e., thus disappears for τ = 0. The important thing is that the function f (r) can be always seen as an attractive potential. The intensity of the attraction is tuned by τ . However, as we will discuss in the next section, in the case of repulsive potentials, there is a threshold value of τ for observing MIPS. This fact implies a reentrant phase diagram and thus, at small enough T , i. e., since T = v 2 τ d , and thus at small enough self-propulsion velocities, MIPS disappears, meaning that the system remains in the gas state.

VI. MEAN-FIELD THEORY FOR SCALAR ACTIVE SYSTEMS
In this section, we will discuss the mean-field solutions of Eq. (42) assuming homogeneous density profiles. In this way, we describe the phase coexistence in active fluids through a van der Waals-like equation. We discuss the general case of active particles interacting via a central potential. We thus specialize our computation in both cases, repulsive potentials that give rise to MIPS, and Lennard-Jones potentials. Since the effective equilibrium theory represented by Eq. (42) depends on spatial derivatives of the interacting potential φ(r), it is important that φ(r) is repulsive on short distances but also a smooth function, in order to have well defined first and second derivatives with respect r.
With this aim, we consider the following central potential that results from two contributions, the first one represents a hard-core repulsion that provides well defined excluded volume effects, the second one is a smooth function that ensures well-defined derivatives. We consider the following function where sigma is the particle radius. The smooth part φ smooth (r) is a continuous function of class C ∞ , as it is shown in Fig. (2). We can thus write and φ smooth (r) = 0 r ≤ σ φ(r) r > σ .
Homogeneous density profiles ρ are described by the mean-density of the system that is ρ(x) = ρ = N/V . We can thus rewrite the partition function in Eq. 42 as follows where the free energy g(ρ) reads In the case of homogeneous density profiles due to spherical particles, the coefficient b = 0 for symmetry reasons.
The theory reduces to an effective van der Waals model defined by the following free energy − g(ρ) = α(τ, β)ρ 2 + ρ ln 1 − ρ ρ + 1 . As one can see, Eq. (49) is precisely the Van der Waals free energy [54] where the effect of interaction and motility are reabsorbed into the coefficient α(τ, β). Homogeneous density profiles turn to be stable whenever α(τ, β) ≥ 0. It is worth noting that, in the case of equilibrium systems, the coefficient is always positive in the case of purely repulsive potentials and thus there is no hope to observe spinodal decompositions at equilibrium without attractive forces.

A. Repulsive potentials: MIPS critical point and inverse melting
For a purely repulsive potential as in the case of φ smooth (r) = ( σ r ) 12 , the coefficient α reads with , and Γ(x) the Gamma function. As we said in the previous section, α is always positive for τ → 0 indicating that there is not way to observe spinodal decomposition in equilibrium systems. For d < 12, the negative and positive values of α are bounded by the curve We can write the pressure P (ρ) that is .
The phase diagram can be obtained considering the solution of ∂ ρ P = 0 that provides the coexistence curve τ (T, ρ). The computation in d spatial dimensions and σ = 1 brings to The phase diagram in two spatial dimensions is shown in Fig. (3), . As one can see, we obtain the same critical density of van der Waals theory.
We can also compute the critical temperature as a function of the correlation time of the noise The behavior of T c (τ ) for d = 2 is shown in the right panel of Fig. (3). It is worth noting that there is a threshold value τ th of τ for having spinodal decomposition. Below τ th , i. e., for τ < τ th , the system does not undergo a phase transition. Moreover, the critical line T c (τ ) shows a reentrance meaning that, decreasing T at fixed τ , the system undergoes an order-to-disorder transition at T 1 = T c (τ 1 ) and then, as T decreases below T 2 = T c (τ 2 ) ≤ T 1 , the system goes back to a disordered phase passing from liquid to gas. The reentrance in the phase diagram is highlighted in the inset of Fig. (3), right panel. case of Lennard-Jones potentials, one has Again, for τ = 0 the system reduces to a Lennard-Jones fluid in equilibrium at temperature T . In equilibrium, one recovers the standard van der Waals theory that lo-cates the critical point at ρ c = 1/3, T c = 8α/27. Also in the case of LJ active fluids , for τ = 0, the critical density remains the same while the critical temperature becomes a function of τ , i .e., T c = T c (τ ). Considering the solutions of ∂ ρ P (ρ c ) = 0, we compute the critical temperature that turns to be Since LJ fluids undergoes gas-liquid coexistence at equilibrium, one has T c (0) = 0. The resulting phase diagram is shown in Fig. (4). The solid lines in the left panel represent the spinodal lines, different colors refer to different values of T . The dashed purple line is the critical density. In the right panel it is shown the behavior of the critical temperature as a function of τ , i.e., T c (τ ).

C. Universality class of Motility-Induced Phase Separation
For investigating the universality class of MIPS within the framework presented here, we have to expand the effective action S ef f around the critical density ρ c and thus we write ρ(x) = ρ c + ϕ(x), where ϕ(x) represents a density fluctuation near the transition. In doing that, let us introduce the Gibbs free energy G[ρ] = S ef f [ρ]. Moreover, we have to introduce the chemical potential µ that guarantees density fluctuations around the critical point, i.e., we have to consider the Legendre transform of G[ρ] → G[ρ] − µρ. In this way, expanding G in power of ϕ(x), one has the cancellation of the linear term in ϕ(x) that is balanced by the chemical potential term µϕ(x). When b = 0, the leading terms in the expansion G = l 1 l! a l ϕ l contains only even power of ϕ. and thus the expansion has the form G 2l! a 2l (ϕ(x)) 2l . As a result, the effective theory reduces to a Landau-Ginzburg ϕ 4 theory that puts MIPS in the universality class of the Ising model.
However, if we consider elongated particles or density fluctuations that are not isotropic, the term bϕ 3 might be different from zero. As a consequence, we might observe deviations from the Ising universality class. This prediction of the mean-field theory has strong consequences in several experimental situations. For instance, in the case of Myxococcus xanthus where a MIPS-like phase separation has been observed [58]. In the next section, we will discuss predictions about the theory for b = 0.

VII. MIPS IN PRESENCE OF ANISOTROPIC INTERACTIONS
When we have performed a mean-field approximation on the partition function Eq. (42), we observed that, in the case central pair potential, the integral in front of the contribution ρ 3 in Eq. (48) vanishes identically for symmetry reasons. It is worth noting that, in the case of elongated and rod-shaped particles, the integral might assume a finite value. In this section, we use again Eq. (48) as a starting point, i. e., we consider homogeneous solutions of the saddle-point equations. We thus consider a phenomenological coarse-grained theory where b can be tuned as an external control parameter that tunes the degree of anisotropy in the two-body interactions. In this way, we can provide a qualitative estimate of the effect of particle asymmetries on spindally decomposed Active Systems. We will start by focusing our attention in the case of a system undergoing MIPS and thus described by α(β, τ ) given by Eq. (50). Looking at the solution of ∂ ρ P (ρ) = 0, we can compute the mean-field coexis- tence curves τ (ρ, T, b) that now depends also on b that is an external tunable parameter of our theory. Considering d = 2, we obtain the following expression for the coexistence curve that reduces to the previous results for b = 0. The behavior of Eq. (57) is shown in Fig. (5), upper panel. As one can appreciate, the coexistence region for b > 0 (blue curve) occupies a smaller area with respect the case of b = 0. This effect can be rationalized looking at the free energy Eq. (48). The term bρ 3 increases the energy and thus the system tends to minimize the value of ρ. This is made evident near the critical point where we can write down an effective ϕ 3 theory for describing the system at criticality. As we discussed before, near the critical point the scalar field ϕ represents density fluctuations around δρ ≡ ρ L − ρ G = 0. In the case of a ϕ 3 theory where one has an explicit breaking of ϕ → −ϕ symmetry, one of the two phases become energetically preferred. This is signaled by the presence of a metastable minimum in the corresponding Landau-Ginzburg free energy. The details are given in the Appendix (G 0 b). It is worth noting that, depending on the value assumed by b with respect to α, the ϕ 3 term might destroy the second-order transition that would be replaced by a first-order one. Using Eq. (57), we can compute numerically the critical line T c (τ, b), i. e., the location of the critical point as a function of the remaining control parameters. The result is shown in Fig. (5), lower panel, for b = 0.00, 2.475, 4.975. Again, the critical curve remains reentrant, however, the asymmetric interaction tends to move the transition at higher τ values. This effect is made evident looking at the shift of the threshold value τ th as a function of b, as it is shown in Fig. (6).

VIII. DISCUSSION AND CONCLUSIONS
In this paper, using the machinery of Statistical Field Theory, we have developed a unified theoretical framework for studying peculiar non-equilibrium phenomena in scalar Active Matter, i. e., Active System described on large scales by a scalar order parameter ϕ. We have derived an effective equilibrium action that contains nonlocal terms responsible for pattern formation in Active Systems. The non-local terms can be tackled using a mean-field approximation that, in the presented framework, turns to be well defined, in the sense that we do not introduce uncontrolled approximation. The meanfield picture is obtained considering the saddle-point solution of the corresponding field theory. Finally, looking at homogeneous solutions and at the stability of homogeneous configurations, we are able to compute the phase diagram of the system. We have focused our attention on three phenomena that concern the condensation of active particles: (i) Accumulation of active particles at the boundaries of a container.
(iii) Gas-liquid phase transitions in Lennard-Jones Active Fluids.
It turns out that the phenomenology of scalar Active Matter is captured by the effective equilibrium field theory based on UCN approximation. Non-equilibrium phase transitions as MIPS are due to non-local manybody contributions in the effective action. As a general result, looking the Active System at criticality, scalar Active Matter is described by the following Landau-Ginzburg free energy In the case of repulsive potential, the coefficient a(T, τ ) is positive for τ = 0, i. e., at equilibrium. In the case of Lennard-Jones systems, a(T, τ ) changes sign at T c at equilibrium (τ = 0). For τ > 0, a(T, τ ) vanishes along a critical line T c (τ ). The shape of the critical line depends on the microscopic details of the system. In the case of purely repulsive systems, a(T, τ ) = 0 at the MIPS critical point that, at the mean-field level, turns to be in the Ising universality class. According to this picture, in the case b(τ ) = 0, MIPS results from the spontaneously symmetry breaking of ϕ → −ϕ symmetry. We obtained that MIPS takes place above a threshold value of persistence time τ th . Moreover, MIPS is characterized by a reentrant phase diagram in the T vs τ plane, i. e., above τ th . This means that the system evaporates into a gas state by decreasing the driving force. On the other hand, when b(τ ) = 0, f Active LG (ϕ) predicts also the presence of a ϕ 3 term that explicitly breaks the ϕ → −ϕ symmetry. We showed that the cubic term has to be taken into account when active particles are anisotropic, i. e., in the case of elongated swimmers. This prediction suggests that MIPS critical point can be destroyed in the case of elongated active particles. In this case, it might be substituted by a first-order phase transition where a metastable MIPS state is nucleated at a higher effective temperature. This state will become eventually the stable one at smaller effective temperature. Looking at the properties of the system inside the coexistence region, the system is described by a van der Waals-like equation in both cases, MIPS and Active Lennard-Jones fluid.
In the case of non-interacting active particles, we showed that the accumulation of particles away from the minimum of the trapping potential can be rationalized in term of a spontaneous symmetry breaking mechanism. In particular, performing a zero temperature approximation, the density distribution ρ(x) ∼ δ(x − x α ) turns to be concentrated at x α = 0 as soon as τ = 0.
In conclusion, we have shown that effective equilibrium field theories are suitable for gaining insight into collective behaviors typical of Active Systems. In this Appendix, we discuss the theoretical framework that we have employed for performing the coarse-graining of the UCN hamiltonian defined in Eq. (21). Let start with the more general problem of n−body interactions in hamiltonian systems, i. e., the hamiltonian given in Eq. (2). We focus our attention to system described by a scalar field, i. e., the density ρ(r). As a standard starting point [54], we consider the identity and thus we can rewrite the interactions in Eq. (2) as follows It is worth noting that N = i dr δ(r − r i ) = dr i δ(r − r i ). The density field ρ(r) is provided by the following relation In order to perform a coarse-graining of the microscopic model, we introduce the field ψ(r) through the identity we can thus rewrite the partition function in terms of the coarse-grained scalar field ψ(r) as follows ..dr k βφ k (r 1 , ..., r k )ψ(r 1 )...ψ(r k ) .

Now, using standard techniques in Statistical Field
Theory [26,36], we are going to employ the following representation for the delta distribution We can thus represent the delta-functional in Z introducing an auxiliary fieldψ(r) and writing Now the partition function reads The fieldψ(r) can be shifted in a way thatψ − b →ψ and thus the partition function becomes For making further progresses, we introduce the following generating functional W [ψ] that is where we have introduced the normalization N ≡ Dψ e −S [ψ] . As one can appreciate, the auxiliary fieldψ in Eq. (A10) plays the role of external source in a quantum field theory. It is worth noting that W [ψ] = O(N ). We can now introduce W [ψ] in the expression of Z obtaining Finally, we define the functional F [ψ] through the relation

Appendix B: Fluctuations
In this section, we compute the fluctuations around the saddle-point configuration ψ SP . Letψ SP (r) be the field configuration that makes the action F [ψ] stationary, we can writeψ (r) =ψ SP (r) + ∆ψ(r) . (B1) With the field configurationψ SP (r) the solution of the saddle-point equation We can now expand the effective action F [ψ] up to the second order obtaining where the quadratic part is defined as follows It is worth noting that the computation of the stationary configurations brings to the following self-consistency equation Now we compute the Gaussian fluctuations around the stationary solution. In order to perform the computation we notice that δF δψ(r) = δW δψ(r) where we have defined z ≡ dr λ e −ψ(r)+b(r) .
The functional derivatives of z with respect the fieldψ(r) gives the following relations δz δψ(r) = e −ψ(r)+b(r) (B8) Now we define ρ(r) and G(r, s) as follows δW δψ(r) SP ≡ ρ(r) (B9) and thus the quadratic part in Eq. (B4) can be rewritten as After introducing G(r, s) and ρ(r), we can finally write the partition function in terms of two contributions: the first one is due to the saddle-point configurationψ SP of the field, the second one due to fluctuation around it. We can thus write = dsds ϕ(s) dr G −1 (r, s)ρ(r) × dr G −1 (r, s )ρ(r) ϕ(s ) .
It is worth noting that the last term in Eq. (B10) gives subd-leading contribution since it is multiplied by 1/N . Finally, we have to take into account the Jacobian determinant of the transformationψ → ϕ. Putting all together we arrive to the following expression for the partition function Now we consider a gas composed by non-interacting particles in one spatial dimension that are embedded into an external potential A(x). The external potential is a smooth and continuous function of class C ∞ . The hamiltonian H reads The partition function is in terms of the number density field ψ(x) and the auxiliary fieldψ(x) we can write Indicating with n(x) ≡ ψ(x) SP andn(x) ≡ψ(x) SP the filed configurations that satisfy the self-consistency equations, we have and we can thus writê Since dx n(x) = N we can finally write and the corresponding self-consistency equation is with c = const. > 0. At small temperature β → ∞ the density profile is dominated by the minima of A(x) and thus n(x) ∼ α δ(x − x α 0 ) with x α 0 given by a. Soft confining potentials As a case study we are going to consider a gas at equilibrium confined through a potential the corresponding density profile is At high temperatures β → 0 and then n(x) = c. The constant is fixed by the normalization, considering the particles free to move in a segment of length L centered around the origin, one has that has x 0 = 0 as unique and stable solution ∀ α > 0.

Appendix D: Mean-Field Theory of Two-body Interactions
Here we start with a specific problem that is the computation of the partition function Z β in the case of a particle system interacting through a pairwise potential. The hamiltonian reads and the partition function Z β is where we have defined the configurational integral In order to perform a coarse-graining of the microscopic dynamics, we introduce the density field ρ(r) that is we can then rewrite the two-body interaction Φ ≡ i<j φ(r i , r j ) in the following way The density field can be forced into the partition function using a delta functional that brings to the following expression for the configurational integral The delta functional can be expressed using an auxiliary fieldψ(r) = Dψ(r) e − drψ(r)ψ(r)+ drψ(r) i δ(r−ri) and the configurational integral becomes Let us introduce the one-body partition function z[ψ] that is and then we can write drdr ψ(r)φ(r, r )ψ(r ) .
Using the first equation, we can write and thusρ (r) = ln ρ(r) + ln z − ln N .
We can rewrite the free energy at the saddle-point as a functional of the density field ρ(r). Now we plug Eq. (D14) into Eq. (D11) for obtaining and finally where we have used the constraint drρ(r) = N . In order to study the stability of a solution (ρ(r),ρ(r)), one ha to compute the hessian matrix H with components H α,β = δ 2 G δρα(r)δρ β (r ) , where the greek indices takes the values α = 1, 2 with ρ 1 ≡ ρ and ρ 2 ≡ρ. the stability condition is det In the case of a two-body potential one has to consider the eigenvalues of the matrix The two-body potential φ(x, y) is translationally invariant, we can thus write φ(x, y) = φ(x − y) and the matrix becomes Since the continuous matrix M (x − y) is translationalinvariant, it is diagonal in Fourier space. Now we study the stability of a homogeneous density profileρ. Using the expressions , and the sums run over wave vectors k = 2π L n. We obtain that homogeneous configurations are stable if Since we are interested on the large scale behavior of the system, we look at k → 0 and thuŝ where Ω(d) results form the integration on the solid angle in d spatial dimensions. As we will see in the next section, for a Van der Waals gas the integral is negative and thus there is a critical density above that the homogeneous solution is not stable anymore.

Stability of homogeneous density profiles
Differently from the van der Waals theory, here we did not consider explicitly excluded volume effects. As a consequence, we can not observe a spinodal decomposition between a liquid and a gas phase. However, we can still study the stability of homogeneous density profiles. For homogeneous solutions ρ(r) = Const. = ρ, assuming translational invariant interactions, we can write The pressure P (ρ) can be computed as and thus we have If we require the thermodynamical stability of the solution ρ, we have to compute the second derivative of g(ρ) that is For purely repulsive potentials, α > 0 and thus ∂ 2 g ∂ρ 2 ≥ 0, ∀ρ ∈ [0, 1], i.e., homogeneous density profiles are always stable in equilibrium systems when attracting forces between particles are negligible.
Considering an equilibrium system where particles interact through a potential that is attractive on short distances, the integral in the third of Eqs. (D23) turns to assume negative values, i. e., α < 0. Let us write α = −|α|, now the stability of homogeneous profiles is related to that changes sign at ρ = ρ c = 1 β|α| . The location of the critical point can computed considering the equations The system of equations (D29) does not have solutions and thus, in the model we have considered, we do not have any critical point.

Including excluded volume effects in the mean-field theory
For obtaining the van der Waals theory it results convenient to do study the system discretized on a lattice. We follow a standard procedure that can be found in Refs. [53,54,59]. We illustrate the method considering an equilibrium system composed by N particles in a volume V in d spatial dimensions. We perform a coarsegraining dividing the systme into a lattice that defines a set of occupation number N i that must satisfy the constraint i N i = N . Each cell occupies a volume ∆ and thus V = N ∆, each particle occupies a volume δ. We consider the potential φ(r i , r j ) composed by two parts where φ δ HS indicates a hard core potential, i. e., each particle is a hard spheres that occupies a volume δ. The second part φ smooth (r) is a smooth function of r. The hard core potential causes a contraction of the phase space ω(N i ) that now is ω(N i ) = (∆ − N i δ) Ni . As we have done before when we have defined the local density filed ψ(r), through the occupation numbers N i the energy can be written as with φ i,j ≡ φ smooth (r i , r j ) The configurational integral X N becomes At the saddle-point one has to find the solution of the set of equations where the Lagrangian multiplier µ guaranties i N i = N . Considering the uniform solution of the saddle-point equations that has the form N i = ρ∆, ∀i with ρ = N/V . Setting δ = 1, the free energy becomes Focusing our attention on repulsive potential, we can write α = −|α| and compute the van der Waals equation of state that is The critical point is determined through Eqs. (D27) that brings to β c = 27 4α and ρ c = 1 3 . In order to make in contact Eq. (D34) with Eq. (D36) we perform the continuous limit ∆ → 0 that brings to −G[ρ] = dr ρ(r) ln 1 − ρ(r) ρ(r) + 1 + (D36) + β 2 drdr φ smooth (r, r )ρ(r)ρ(r ) .
As one can appreciate, Eq. (D36) has the form of a Density Functional Theory (DFT). For obtaining the kinetic term ∇ρ, it is convenient to go back to the discretized action and look at interaction term using the identity the interaction term can be written as Let ρ i be the density in the box i that is ρ i = Ni ∆ . We can thus rewrite In the continuum limit, i.e., ∆ → 0, we get The first term in the last equation is nothing else than a mass term in a field theory.
H 1 term.-This term brings to non-linear and nonlocal interactions in the effective equilibrium theory. In particular, it is responsible for a ψ 3 term that explicitly breaks the symmetry ψ → −ψ. This term can be rewritten as follows Where we have introduced the notation ∂ z φ(z, x) ≡ ∂φ(z,x) ∂z . H 2 term.-In order to perform the coarse-graining of the interactions due to H 2 , we have to deal with the determinant of a dN × dN matrix. For doing that, tet us assume a small τ limit, in this case, indicating with H the hessian matrix and with 1 the identity matrix, one has det(1 + τ H) 1 + τ T r H (E7) In this limit we can write where we have defined ∂ 2 x,y φ(x, y) ≡ ∂ 2 φ(x,y) ∂x∂y . For proving that, we write the determinant of a generic matrix M of elements M ij as where we have introduced two sets of conjugated Grassmann variables θ i andθ i [36], respectively, that satisfy the anti-commutation rules Since θ i andθ i are Grassmann variables, one has the following rules for the Grassmann integration A(x, y) ≡ φ(x, y) − 2τ T ∂ 2 x,y φ(x, y) B(x, y, z) ≡ ∂ z φ(z, x)∂ z φ(z, y) .
To obtain the thermodynamics of the model we have to compute the following partition function (E15) where we have introduced the "single-particle" partition function z ≡ dx eψ (x) . Let us rewrite the action as follows The action (E20) reduces to the equilibrium case for τ → 0. Moreover, for τ = 0 the theory include a ρ 3 term. are α and τ b/2 → b. In particular, we are interested in evaluating the impact that the cubic term has on spinodal decomposition. For this reason we will consider the situation α < 0. The coexistence region in the T vs ρ plane, obtained considering the solution of ∂ ρ P = 0, is that has been obtained considering α = −1. The curves T (ρ, b) are shown in Fig. (3) (left panel) for different values of the asymmetry parameter b. As b increases we obtain a contraction of the coexistence region. We can now compute the location of the critical point that is obtained considering the set of equations ∂ ρ P = ∂ 2 ρ P = 0. Once we get the critical density ρ c (b), that is shown in Fig. (3), central panel, we can also compute how the critical temperature changes with b (same figure, right panel). It turns that both, the critical density and the critical temperature, are decreasing function of b.
b. Landau-Ginzburg ϕ 3 Theory In this section, we provide a brief discussion about ϕ 3 field theories in Statistical Physics. In particular, we consider both interactions, ϕ 3 and ϕ 4 in order to have a well-defined Landau-Ginzburg energy functional. For b > 0, the critical behavior of the system can be studied through the following Landau-Ginzburg free energy (G5) where the order parameter ϕ(x) represents fluctuations around the critical density ρ c , i. e., ρ(x) = ρ c + ϕ(x). Without loss of generality, we consider the homogeneous case ϕ(x) = ϕ. We immediately realize that the cubic term, breaking the symmetry ϕ → −ϕ, promotes one phase with respect to the other [54]. Minimizing Eq. (G5), one obtains three configurations ϕ 0,1,2 that are ϕ 0 = 0 (G6) For sake of simplicity, let us fix c = 1. We consider the coefficients a and b as two independent and tunable external parameters of our coarse-grained model. For b = 0, the coefficient a changes sign at the MIPS critical point. For b = 0 the value a = 0 is not necessary the MIPS critical point. In particular, the coefficient b tunes the intensity of asymmetry ϕ → −ϕ. For b = 0, we recover the standard ϕ 4 theory, and thus the symmetry ϕ → −ϕ is preserved. For b > 0 and b 2 < 4a, at a > 0 the only real and stable solution is ϕ 0 that becomes marginal at a = 0 and eventually unstable for a < 0. In the latter case, ϕ 1,2 become reals and the solution ϕ 1 is the new minimum. When b 2 ≥ 4a, ϕ 1,2 are reals for a > 0, meaning that the system develops a metastable state. The two situations are showed in Fig. (8).