Fragile fate of driven-dissipative XY phase in two dimensions
FFragile fate of driven-dissipative XY phase in two dimensions
Mohammad F. Maghrebi Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824, USA
Driven-dissipative systems define a broad class of non-equilibrium systems where an externaldrive (e.g. laser) competes with a dissipative environment. The steady state of dynamics is gener-ically distinct from a thermal state characteristic of equilibrium. As a representative example, adriven-dissipative system with a continuous symmetry is generically disordered in two dimensionsin contrast with the well-known algebraic order in equilibrium XY phases. In this paper, we studya 2D driven-dissipative model of weakly interacting bosons with a continuous U (1) symmetry. Ouraim is two-fold: First, we show that an effectively equilibrium XY phase emerges despite the drivennature of the model, and that it is protected by a natural Z symmetry of the dynamics. Second, weargue that this phase is unstable against symmetry-breaking perturbations as well as static disorder,whose mechanism in most cases has no analog in equilibrium. In the language of renormalizationgroup theory, we find that, outside equilibrium, there are more relevant directions away from theXY phase. PACS numbers:
I. INTRODUCTION
A time-dependent drive continuously pumps energyinto a driven system, and eventually heats it up to in-finite temperature. On the other hand, a driven systemcoupled to a dissipative bath approaches a nontrivial non-equilibrium steady state due the competition betweendissipation and external drive. In many-body driven-dissipative systems, the steady state of dynamics mayexhibit new, and inherently nonequilibrium, phases. Thelatter, however, pose a fundamental challenge to our un-derstanding of phases of matter.Non-equilibrium systems are, almost by definition, lessconstrained than their equilibrium counterparts. Thisimplies that, away from equilibrium, dynamics and fluc-tuations can explore a larger “phase space”. It is thennatural to expect non-equilibrium phases that are not ac-cessible in equilibrium. The converse of this statementcould also be true in the sense that a generic equilibriumphase may be non-generic far from equilibrium. A repre-sentative example is a driven-dissipative model with U (1)symmetry in low dimensions. This model is particularlyrelevant to driven-dissipative condensates consisting ofexciton polaritons in semiconductor quantum wells [1–6]. It has been argued that such driven-dissipative Bosesystems in two dimensions cannot exhibit algebraic order,characteristic of the equilibrium XY model, unless theyare strongly anisotropic [7]. This is partly due to theemergence of the Kardar-Parisi-Zhang (KPZ) equationthat describes a broad range of driven classical phenom-ena [8].This manuscript makes a case for the emergence ofthe XY phase in driven-dissipative systems on the basisof symmetry. We present a case study of a 2D driven-dissipative bosonic model with U (1) symmetry whichnevertheless gives rise to an XY phase. We further ar-gue that this is due to the underlying symmetries of themodel including an additional Z symmetry. Despite theemergence of the XY phase, the model is shown to be generically unstable to symmetry-breaking perturbationsas well as static disorder. We shall argue that, while U (1)-symmetry breaking perturbations find a descriptionsimilar to those in equilibrium, perturbations of the Z symmetry as well as static disorder are of a genuinelynon-equilibrium nature (see Fig. 1).The structure of this paper is as follows. In Sec. II, weintroduce the 2D driven-dissipative model of weakly in-teracting bosons, and argue on the basis of the Keldyshfunctional integral that an effectively classical equilib-rium XY phase emerges. In Sec. III, we undertake a de-tailed study of the symmetries of the model and the waythey constrain the emergent thermodynamic phase. Wefurther discuss perturbations away from symmetries aswell as static disorder. Finally, in Sec. IV, we summarizeour results and discuss future directions. II. MODEL
We consider a driven-dissipative model of weakly in-teracting bosons on a square lattice in two dimensions.This model is inspired by the spin model introduced inRef. [9] and its subsequent treatment in Ref. [10] wherespins were mapped to bosons. To define the model, westart from the quantum master equation ∂ t ρ = − i [ ˆ H, ρ ] + (cid:88) j (cid:18) ˆ L j ρ ˆ L † j −
12 ˆ L † j ˆ L j ρ − ρ ˆ L † j ˆ L j (cid:19) . (1)The first term on the right-hand side gives the usual co-herent evolution via the Hamiltonian ˆ H . The dissipa-tion is subsumed in the second term characterized by theLindblad operators L j s that describe the incoherent pro-cesses. We take the Hamiltonian asˆ H = J (cid:88) (cid:104) ij (cid:105) (cid:16) ˆ a i ˆ a j + ˆ a † i ˆ a † j (cid:17) + U (cid:88) j ˆ a † j ˆ a † j ˆ a j ˆ a j . (2) a r X i v : . [ c ond - m a t . qu a n t - g a s ] J u l The first term in the Hamiltonian describes anomalous hopping between nearest neighbors, while the secondterm is the on-site interaction. We can also considera “chemical-potential” term ∼ (cid:80) j a † j a j in the Hamilto-nian; the latter, however, does not alter our main con-clusions, and will be discussed from the point of view ofsymmetry in Sec. III. Furthermore, we consider weaklyinteracting bosons where the the interaction ( U ) can betreated perturbatively. Finally, the incoherent dynamicsis given by a single-particle lossˆ L j = √ Γ ˆ a j . (3)The Hamiltonian and the Lindblad operators should beunderstood in a rotating frame—determined by the fre-quency of the external drive—after making the rotatingwave approximation. The latter is an excellent approxi-mation provided that the drive frequency is much largercompared to other energy scales. We shall not provide amicroscopic time-dependent model ; however, we arguethat the driven nature of the dynamics is inherent in thequantum master equation. To this end, note that there isa competition between the Hamiltonian and dissipativedynamics. While the dissipation via L j s favors a statewith no particles, or a vacuum , the Hamiltonian producespairs of particles out of the vacuum state. The competi-tion between the two gives rise to a steady state at longtimes with a finite density of particles. This feature hasno analog in equilibrium, and is the defining character ofdriven-dissipative models.An important property of the model introduced hereis that it possesses a U (1) symmetry. To see this, letus consider the checkerboard sublattices A and B of thesquare lattice. The quantum master equation is invariantunder the following staggered U (1) transformationˆ a j ∈ A → e iθ ˆ a j ∈ A , ˆ a j ∈ B → e − iθ ˆ a j ∈ B , (4)where bosons on the two sublattices are “rotated” in op-posite directions. This is to ensure that the anomaloushopping in the Hamiltonian remains invariant; all theother terms in the master equation (including the Lind-blad terms) are acting on a single site, and respect thesymmetry as well. It is then natural to ask whether thecontinuous U (1) symmetry is broken in the steady state.A mean-field analysis would be a first step to this end(for the spin analog of this model, see Ref. [9]). How-ever, mean-field-type treatments are at best incompletesince they ignore fluctuations that are crucial to findingthe fate of ordered phases in low dimensions. Further-more, in a nonequilibrium setting, there is even a larger For notational convenience, J is defined two times that ofRefs. [9, 10]. The underlying time-dependent model is not unique, and its ex-plicit form is constrained by experimental feasibility rather thanphysical principles. phase space available to dynamics and fluctuations. In-stead, we shall follow a field-theory treatment based onthe Keldysh formalism. pert.pert. Disorder XY N o n - E q . Figure 1: Emergence of an effectively thermal XY phase asthe steady state of a non-equilibrium driven-dissipative modelwith Z × U (1) symmetry of the unit cell. The schematicplot shows relevant perturbations away from the XY phasewhich include symmetry-breaking perturbations as well asstatic disorder. While U (1) symmetry breaking finds an ef-fective equilibrium character, the corresponding mechanismsfor Z symmetry breaking and disorder are of a genuinelynon-equilibrium nature. (The highlighted plane representsthe subspace spanned by genuinely non-equilibrium pertur-bations.) A. Overview of Keldysh formalismand previous results
The Keldysh formalism adapts the functional-integraltechniques to density matrices where two time con-tours/branches represent the evolution of the bra andket states in the density matrix. In transitioning tothe functional integral, the operator ˆ a j is mapped tothe fields a j, ± ( t ) with the subscripts ± representing thetwo branches. The Keldysh functional integral gives aweighted sum (integral) over all configurations of a j, ± ( t ).The weight associated with each configuration is givenby the Keldysh action S K [ a j, ± ( t )], the form of which isdirectly determined from the quantum master equation(1). In a coherent-state representation, the Keldysh ac-tion can be cast as [11] S K = ˆ t (cid:88) j (cid:0) a ∗ j, + i∂ t a j, + − a ∗ j, − i∂ t a j, − (cid:1) − i L . (5) L contains information about dynamics, and is given by L = − i ( H + − H − ) (6)+ (cid:88) j (cid:20) L j, + L ∗ j, − − (cid:0) L ∗ j, + L j, + + L ∗ j, − L j, − (cid:1)(cid:21) where H ± as well as L ± contain fields on the ± contouronly. Clearly, the first line of this equation captures theunitary dynamics, while the second line describes the dis-sipative dynamics. The particular form of various termsare determined by the simple rule that a term of theform ˆ Oρ ˆ O (cid:48) in the quantum master equation translates to O + O (cid:48)− in the action [12, 13].It is often more convenient to work in the Keldysh basisdefined as [11, 14] a j,cl = a j, + + a j, − √ , a j,q = a j, + − a j, − √ . (7)This basis is more convenient in separating out the meanvalue (represented by a cl ) from the fluctuations aroundit (due to both a cl and a q which may be nevertheless ofdifferent nature). Next, we provide a summary of theprevious results obtained in the context of a spin model[9, 10] to the extent that it is relevant to our discus-sion. Along the way, we also give an overview of theby-now standard techniques and methods. Motivated bythe staggered U (1) symmetry (4), we allow the order pa-rameter to be different on the two sublattices, but assumethat it is uniform within each sublattice. With this as-sumption, one can take the continuum limit of the latticemodel. Following Ref. [10], we define the bosonic oper-ators on the two sublattices A and B in the continuumas ˆ a j ∈ A −→ ˆ a ( x ) , ˆ a j ∈ B −→ ˆ b ( x ) . (8)(With a slight abuse of notation, we have now used ˆ a ( x )to denote the bosonic operators corresponding to thesublattice A .) The corresponding quantum and classicalfields associated with the operators ˆ a ( x ) and ˆ b ( x ) shouldbe identified as a cl/q ( t, x ) and b cl/q ( t, x ). Subsequently,the Keldysh action can be written as a functional of thesefields. It was pointed out in Ref. [10] that one can makethe transformation ψ cl/q ( t, x ) = ∓ (cid:104) e ± iπ/ b cl/q ( t, x ) + e ∓ iπ/ a ∗ cl/q ( t, x ) (cid:105) ,χ cl/q ( t, x ) = e ∓ iπ/ b cl/q ( t, x ) + e ± iπ/ a ∗ cl/q ( t, x ) , (9)to bring the Keldysh action into a more transparent format or near the critical point to be further discussed be-low. This transformation casts the quadratic part of theKeldysh Lagrangian density (the integrand of the space-time integral in the action) as L (2) K = 12 (cid:110) ψ ∗ q (cid:2) − ∂ t + J ∇ − r (cid:3) ψ cl + c . c . + i Γ | ψ q | + χ ∗ q (cid:2) − ∂ t − R (cid:3) χ cl + c . c . + i Γ | χ q | (cid:111) , (10) with the constants r = Γ / − J, R = Γ / J. (11)Importantly, the constant r can be tuned to zero, or crit-icality , while R is always finite. Indeed we have used thisfact to drop the gradient term acting on χ cl at long wave-lengths. It should be then clear that the critical behavioris captured by ψ cl/q , while χ cl/q are non-critical. At thequadratic level, the two fields are decoupled, and χ cl/q can be simply dropped; however, interaction mixes thecritical and non-critical fields together. We shall not re-produce the interaction terms in the new basis, and referthe interested reader to Ref. [10]. Integrating out χ cl/q produces an effective interaction term of the form L int K = − u (cid:0) | ψ cl | ψ cl ψ ∗ q + c . c . (cid:1) . (12)The (real) coefficient u ∼ U /J is obtained via a second-order perturbation theory in the vicinity of the criticalpoint r = 0 or J = Γ /
8. Of course, a perturbative treat-ment is justified in the limit of weak coupling U (cid:28) J . Weremark that there are various nonlinear terms generatedin the second-order perturbation theory; however, a sim-ple scaling analysis renders nonlinear terms with higherpowers of the quantum field ψ q irrelevant in the sense ofrenormalization group (RG) theory. A first step of per-turbative RG is to determine scaling dimensions of thefields at the Gaussian fixed point corresponding to thequadratic part of the action. Demanding that the lattershould be scale-invariant at the critical point under thetransformation x → b x and t → b t (relative scaling ofspace and time coordinates follows from the diffusive na-ture of the dynamics) requires ψ cl → b ψ and ψ q → b ψ q in two dimensions. The corresponding scaling dimensionsare then [ ψ cl ] = 0 and [ ψ q ] = 2. The difference in thescaling dimensions is a consequence of the fact that ψ q is“gapped” in the sense that the action contains the termΓ | ψ q | whose coefficient, unlike that of rψ ∗ q ψ cl +c . c . , can-not be tuned to zero. The relevance of nonlinear termsat the Gaussian fixed point is determined by their RGflow; generally, terms containing higher powers of fieldsof a larger scaling dimension will be less relevant.Putting together the quadratic terms in the first lineof Eq. (10) with the interaction term in Eq. (12), wefind the effective Keldysh Lagrangian density obtainedby integrating out the non-critical fields. Incidentally,the latter can be written as L eff K = 12 (cid:110) ψ ∗ q (cid:2) − ∂ t ψ cl − δ H eff /δψ ∗ cl (cid:3) + c . c . + iT eff | ψ q | (cid:111) , (13)where T eff = Γ, and the functional H eff ≡ H eff [ ψ cl ] isgiven by H eff [ ψ ] = ˆ x J |∇ ψ | + r | ψ | + u | ψ | . (14)(A function(al) of ψ should be always interpreted as afunction(al) of both ψ and ψ ∗ .) The latter has the sameform as the Landau-Ginzburg free energy for a complex-valued field ψ . The description in terms of the effec-tive action and Hamiltonian can be equivalently cast asa stochastic equation, ∂ t ψ = − δ H eff δψ ∗ + ξ ( t, x ) , (15)where ξ represents a stochastic noise that is correlated as (cid:104) ξ ( t, x ) ξ ∗ ( t (cid:48) , x (cid:48) ) (cid:105) = 2 T eff δ ( t − t (cid:48) ) δ ( x − x (cid:48) ). Using standardtechniques [15], one can show that the asymptotic steadystate of the effective dynamics in Eq. (15) is given by theprobability distribution ( ψ cl → ψ ) P [ ψ ] ∼ exp (cid:18) − H eff [ ψ ] T eff (cid:19) , (16)which is nothing but a thermal distribution function.Despite the nonequilibrium dynamics at the microscopicscale, at long wavelengths, the model effectively behavesas if it is in equilibrium. Of course, the effective Hamilto-nian (free energy) and temperature are not in any directway related to those at the microscopic level. B. Emergence of XY phase
An effective classical and equilibrium behavior opensup the sophisticated toolbox of statistical mechanics. Inthe context of the model considered here, one can imme-diately draw intuition from the classical XY model in twodimensions. In particular, vortices should be properlytaken into account. To this end, let us take ψ = (cid:104) ψ cl (cid:105) ,and define K ≡ J | ψ | / Γ which is commonly known asspin stiffness. It is a classical result due to Kosterlitz andThouless that a quasi-long-range order with algebraic de-cay of correlations emerges when [15]
K > π . (17)In the opposite regime where this constraint is not sat-isfied, vortices proliferate destroying the algebraic orderand leading to an exponential decay of correlations.For the spin model introduced in Ref. [9], it was shownthat the constraint (17) cannot be satisfied [10], and theXY phase will not be realized.This is because ψ repre-sents the expectation value of a spin operator which maybe saturated, and consequently K ∼ | ψ | would not belarge enough. In contrast, in our bosonic model, | ψ | canbe arbitrarily large in favor of the constraint (17). Thesaddle-point approximation of Eq. (14) yields | ψ | ≈ | r | u , (18)when r <
0. Recalling that u ∼ U /J , the constraint(17) is easily satisfied in the weak-coupling regime U (cid:28) J . We thus conclude that an XY phase is realized in the driven-dissipative model of weakly interacting bosonsintroduced here.Our perturbative treatment still lacks an importantdiscussion. We have used second-order perturbation the-ory to show that the resulting action takes a form thatfinds a description in terms of an effective free energy. Itis important, however, to show that this is not an arti-fact of our approximation, but rather is protected by thesymmetries of the model. This is particularly importantin the context of a 2D driven-dissipative model with U (1)symmetry where the XY phase is shown to be genericallyunstable to KPZ-like physics [7]. In the next section, wediscuss the symmetries of the model, and argue that theXY phase is indeed protected by these symmetries. Fur-thermore, we show that relaxing these symmetries gener-ically tends to destroy the XY phase. III. ROLE OF SYMMETRY
In addition to the U (1) symmetry (Eq. (4)), the modeldefined in the previous section has a Z symmetry undersublattice exchange A ↔ B . Hence, in a unit cell con-sisting of two sites (one from each sublattice), we havean enlarged Z × U (1) symmetry. In the continuum, the Z symmetry interchanges the fields, ˆ a ( x ) ↔ ˆ b ( x ) , whichconstitutes a fundamental symmetry of the model be-yond any approximation or perturbation scheme. Thelatter directly translates to a transformation in terms ofthe fields, a cl/q ( t, x ) ↔ b cl/q ( t, x ), as a symmetry of theKeldysh action S K [ a cl/q , b cl/q ]. In the basis of the fields ψ cl/q and χ cl/q defined in Eq. (9), this symmetry simplyreads as complex conjugation, ψ cl/q ( t, x ) ↔ ψ ∗ cl/q ( t, x ) ,χ cl/q ( t, x ) ↔ χ ∗ cl/q ( t, x ) . (19)Being a fundamental symmetry of the model, this trans-formation should also be a symmetry of the effectiveKeldysh action S eff K [ ψ cl/q ] obtained by integrating out thenon-critical fields χ cl/q , S eff K [ ψ ∗ cl , ψ ∗ q ] = S eff K [ ψ cl , ψ q ] . (20)This equation imposes a strong constraint on the form ofthe Keldysh action. To fully exploit it, we also note thata general Keldysh action S K [ a cl , a q ] (with a cl/q repre-senting all the fields with quantum numbers suppressed)always satisfies S ∗ K [ a cl , a q ] = −S K [ a cl , − a q ] . This equa-tion follows from the causal structure of the Keldyshaction, and ensures that ( G R ) † = G A and ( G K ) † = − G K where G R,A,K are retarded, advanced, and KeldyshGreen’s functions, respectively. It then follows thatRe( S K [ a cl , a q ]) is odd in a q while Im( S K [ a cl , a q ]) is evenin a q . For future reference, we specialize the causalitystructure to the effective Keldysh action, (cid:0) S eff K [ ψ cl , ψ q ] (cid:1) ∗ = −S eff K [ ψ cl , − ψ q ] . (21)Using the Z × U (1) symmetry, we next expand theKeldysh action in classical and quantum fields, and onlykeep spatial and time derivatives to the lowest order.With the exception of the classical field, this is welljustified due to the corresponding scaling dimensions([ ∂ t ] = 2, [ ∂ x ] = 1, and [ ψ q ]=2). However, a similar scal-ing argument fails for the classical field since [ ψ cl ] = 0 atthe critical point. Moreover, we need to consider a pa-rameter regime at a finite distance away from the criticalpoint where Eq. (17) is satisfied and the classical fieldassumes a finite value. We will present a more generalargument below; for now, we simply expand the actionin both classical and quantum fields as S eff K = ˆ t, x ψ ∗ q (cid:104) − Z∂ t ψ cl − ˜ J ∇ ψ cl − ∂V /∂ψ ∗ cl (cid:105) + c . c . + i ˜Γ | ψ q | + · · · . (22) V ( ψ cl ) is a function of the modulus | ψ cl | , and can beexpanded as V ( ψ ) = ˜ r | ψ | + ˜ u | ψ | + · · · . Note that theaction is at most quadratic in the quantum field, but pos-sibly contains higher-order terms in the classical field.Also it is written in a way that explicitly respects the U (1) symmetry. Furthermore, the complex conjugationin the first line of the action is to ensure the reality of ψ q -odd terms that follows from Eq. (21). In general, thecoefficients Z, ˜ J, ˜ r, ˜ u, · · · can be complex-valued (˜Γ has tobe real on the basis of Eq. (21)). However, the symme-try constraint in Eq. (20) ensures that all the coefficientsare real. Therefore, the action can be written in a formconsistent with Eqs. (13, 14) with a (real-valued) Hamil-tonian, which is what we wanted to show.It is instructive to present a more general argumentwithin the XY phase. Let us first represent the classicaland quantum fields as ψ cl = √ ρ + π e iθ , ψ q = ζe iθ , (23)where ρ ≡ | ψ | = (cid:104)| ψ cl | (cid:105) is the average density in theordered phase, π characterizes density fluctuations, and ζ = ζ + iζ is a complex field representing the quantumfield. Here we have followed the notation in Ref. [11] infactoring out a common phase factor from both classicaland quantum fields. The symmetry constraint (20) inthe new basis reads S eff K [ π, θ, ζ , ζ ] = S eff K [ π, − θ, ζ , − ζ ] . (24)Moreover, the U (1) symmetry requires the action to beinvariant under θ → θ + const. With these constraints onthe form of the Keldysh action together with Eq. (21), The expansion starts at the linear order in quantum field since S K [ a cl , a q = 0] = 0 as a general property of the Keldysh action[14]. we can write the most general Keldysh action as S eff K = ˆ t, x ζ (cid:0) − Z (cid:48) ∂ t θ − J (cid:48) ∇ θ (cid:1) + J (cid:48)(cid:48) ζ ( ∇ θ ) − u (cid:48) ζ π + i Γ (cid:48) ζ + i Γ (cid:48)(cid:48) ζ + · · · , (25)where the ellipses represent irrelevant terms that con-tain higher powers of spatial and time derivatives or thefields π , ζ and ζ due to the corresponding scaling di-mensions ([ π ] = [ ζ ] = [ ζ ] = 2). All the coefficients( Z (cid:48) , K (cid:48) , K (cid:48)(cid:48) , u (cid:48) , Γ (cid:48) , Γ (cid:48)(cid:48) ) in the action are real as a conse-quence of the causality structure in Eq. (21). We notethat, unlike Eq. (22), we have not made an expansion inpowers of ψ cl which in the ordered phase can be possiblylarge. For the special case of the action in Eq. (22), wehave Z (cid:48) ∼ √ ρ Z , K (cid:48) ∼ K (cid:48)(cid:48) ∼ √ ρ ˜ J , u (cid:48) ∼ √ ρ u , andΓ (cid:48) = Γ (cid:48)(cid:48) = ˜Γ. Now note that the integration over π inEq. (25) gives a delta function that sets ζ = 0, and theKeldysh action can be simply written as S eff K = ˆ t, x ζ (cid:0) − Z (cid:48) ∂ t θ − K (cid:48) ∇ θ (cid:1) + i Γ (cid:48)(cid:48) ζ , (26)where irrelevant terms are simply dropped. It is thenstraightforward to see that the steady state is given as aneffective thermal distribution with the partition function ˆ Dθ exp (cid:20) − K ˆ ( ∇ θ ) (cid:21) , (27)corresponding to the XY Hamiltonian with the spin stiff-ness K = K (cid:48) Z (cid:48) / Γ (cid:48)(cid:48) .Having shown that a description in terms of the XYHamiltonian is guaranteed by the symmetries of themodel, we next turn our attention to symmetry-breakingperturbations. We shall see that the XY phase is gener-ically unstable to such perturbations (see Fig. 1). A. Perturbing U (1) symmetry There are a number of ways that U (1) symmetry canbe explicitly broken. A representative example is nearest-neighbor hopping, (cid:100) ∆ H = α (cid:88) (cid:104) ij (cid:105) ˆ a † i ˆ a j + H . c . (28)Since neighboring sites belong to different sublattices, the U (1) symmetry in Eq. (4) is explicitly broken. In the spinanalog of Ref. [9], this amounts to having J x ≈ − J y witha slightly different | J x | and | J y | of the corresponding XX Again, the expansion starts at the linear order in ζ representingthe quantum field. A linear term in ζ is also allowed, but nev-ertheless can be absorbed in a redefinition of π , or equivalentlya renormalization of the density ρ . and YY interactions. In the continuum, we have (cid:100) ∆ H ∝ α ´ x ˆ a † ( x )ˆ b ( x )+H . c . + · · · , and the corresponding term inthe action reads ∆ S ∝ α ´ t, x a ∗ q b cl + a ∗ cl b q + c . c . + · · · withthe ellipses indicating the less relevant terms. Writing thelatter in the basis of ψ cl/q and χ cl/q and integrating outthe non-critical fields χ cl/q , we find, to leading order in U/J ,∆ S eff K ∝ αUJ ˆ t, x (cid:2) ψ ∗ q (cid:0) ψ ∗ cl + | ψ cl | ψ ∗ cl + ψ cl (cid:1) + c . c . (cid:3) . (29)All the terms reported here explicitly break the U (1)symmetry, while those that simply renormalize terms al-ready present in the non-perturbed action as well as theless relevant terms are omitted. Also, we have not kepttrack of relative coefficients (of order 1) between differentterms. The first term under the integral in the effectiveaction ( ψ ∗ q ψ ∗ c + c . c . ) can be cast as a correction to theeffective Hamiltonian in Eq. (14) as∆ H eff1 ∝ αUJ ˆ x ψ + ( ψ ∗ ) , (30)with the subscript denoting the corresponding term. Thisexpression is nothing but a perturbation of the XY modelfamiliar in the context of statistical physics. In the or-dered phase, ψ cl ≈ √ ρ e iθ , the correction to the effectiveHamiltonian becomes∆ H eff1 ∝ αρ UJ ˆ x cos(2 θ ) . (31)Generally, a cosine perturbation of the from cos( pθ ) isirrelevant if K < p / (8 π ) [16]. However, the latter cannotbe satisfied due to the condition (17), and thus the cosineperturbation grows under RG pinning the value of θ . TheXY phase and its characteristic algebraic order will bethen destroyed at long wavelengths.On the other hand, the second and the third termsunder the integral in Eq. (29) cannot be derived froma Hamiltonian. To treat these terms, we resort to thedensity-phase representation of Eq. (23). In this repre-sentation, the U (1) perturbation leads to a correction tothe action of the form (relative and overall coefficientsare neglected)∆ S eff K ∝ ˆ t, x ζ cos(2 θ ) + ζ sin(2 θ ) . (32)This action contains the most relevant perturbations thatalso respect the Z symmetry (Eq. (24)). Now the firstterm under the integral simply drops since the functionalintegration over π sets ζ = 0. The second term perturbs A term in the Keldysh action of the form ψ ∗ q f ( ψ cl ) + c . c . can becast as ψ ∗ q δ H /δψ ∗ cl +c . c . —in a fashion similar to Eq. (13)—if thefunction f ( ψ ) satisfies the condition ∂f/∂ψ = ∂f ∗ /∂ψ ∗ . the U (1) symmetry, but can be similarly cast as a cor-rection to the effective Hamiltonian as in Eq. (31).In short, a perturbation of the U (1) symmetry of theform considered in this section can be simply consideredas a perturbation of the XY Hamiltonian, which, usingstandard techniques, can be shown to destroy the XYphase at long wavelengths. B. Perturbing Z symmetry In this section, we study the consequences of break-ing the Z sublattice symmetry. Naively, this symmetrycan be broken by adding to the Hamiltonian a staggeredchemical potential (with different chemical potentials onthe two sublattices), µ A (cid:88) j ∈ A ˆ a † j ˆ a j + µ B (cid:88) j ∈ B ˆ a † j ˆ a j . (33)However, one can remove the asymmetry by exploiting agauge transformation via the unitary operatorˆ U ( t ) = exp − iµt (cid:16) (cid:88) j ∈ A ˆ a † j ˆ a j − (cid:88) j ∈ B ˆ a † j ˆ a j (cid:17) . (34)In an appropriate rotating frame with µ = ( µ A − µ B ) / µ (cid:48) A = µ (cid:48) B . The latter satisfies all the symmetries( Z × U (1) as well as the translation symmetry) of themodel, and thus only slightly renormalizes the effectiveHamiltonian. It is instructive to view this argument inthe basis of the Keldysh action. The corresponding cor-rection to the effective Keldysh action is given by∆ S eff K = i µ A − µ B ˆ t, x (cid:0) ψ ∗ q ψ cl − c . c . (cid:1) . (35)Naively, this term breaks the Z symmetry since it isnot invariant under ψ cl,q ↔ ψ ∗ cl,q . However, the stochas-tic equation that follows from the Keldysh action reads( ψ cl → ψ ) ∂ t ψ = − δ H eff δψ ∗ + i µ A − µ B ψ + ξ ( t, x ) . (36)Making the transformation ψ → ψe it ( µ A − µ B ) / , we re-cover the unperturbed form of the stochastic equa-tion that can be equivalently described by an effectiveHamiltonian, consistent with the gauge transformationin Eq. (34).To explicitly break the Z sublattice symmetry, we takethe decay rates to be slightly different on the two sublat-tices as ˆ L j ∈ A = (cid:112) Γ A ˆ a j , ˆ L j ∈ B = (cid:112) Γ B ˆ a j , (37)with Γ A/B = Γ ± ∆Γ. We stress that any generic pertur-bation of the Z symmetry should also lead to the sameconclusions. Writing the corresponding Keldysh actionand integrating out the non-critical fields, we find an ef-fective Keldysh action, to the first nontrivial order in U/J , as∆ S eff K ∝ i ∆Γ UJ ˆ t, x (cid:0) ψ ∗ q ψ cl + ψ ∗ q | ψ cl | ψ cl − c . c . (cid:1) . (38)(We have not kept track of relative coefficients.) Thefirst term under the integral can be gauged away bygoing to a rotating frame similar to Eq. (35); how-ever, the second term cannot be dealt with in a similarfashion. Indeed adding the latter to the non-perturbedKeldysh action, we find a renormalized interaction term ∼ u ren ψ q | ψ cl | ψ + c . c . with a complex-valued coefficient u ren = u (cid:48) ren + iu (cid:48)(cid:48) ren . As shown in Refs. [7], this featuregenerically leads to the KPZ equation, and takes us out-side the effective equilibrium description. The emergenceof the KPZ equation can be generally argued on the basisof symmetry. In the absence of the Z symmetry, manynew terms are allowed in the Keldysh action. One suchterm is ζ ( ∇ θ ) , a term that was previously disalloweddue to the symmetry under the simultaneous transforma-tion ζ → − ζ and θ → − θ . The inclusion of the latterterm in Eq. (13) leads to the KPZ equation ∂ t θ = J ∇ θ + λ ( ∇ θ ) + η ( t, x ) , (39)where η represents a (real-valued) stochastic noise that iscorrelated as (cid:104) η ( t, x ) η ( t (cid:48) , x (cid:48) ) (cid:105) = (Γ / √ ρ ) δ ( t − t (cid:48) ) δ ( x − x (cid:48) ).The coefficient of the nonlinear term ( λ ) vanishes withthe perturbation ( ∼ ∆Γ) away from the Z symmetry.Notice that this term cannot be derived from a Hamil-tonian in a similar fashion as Eq. (15). At the Gaussianfixed point (ignoring the compact nature of θ ), the scalingdimension [ θ ] = 0, and the new term in the KPZ equa-tion is marginal [15]. To a higher order in perturbationtheory, the latter term can be shown to be marginally rel-evant, and leads to a stretched-exponential decay of thecorrelation function and the destruction of the XY phase[7]. The interested reader is referred to Refs. [6, 11, 17]for more details on the emergence of the KPZ equationin driven-dissipative condensates. C. Random disorder
In this section, we consider the effect of disorder on thebehavior of our model. In a disordered system, transla-tion symmetry is broken at a microscopic level, whichnevertheless is restored by an ensemble average over dis-order configurations. A generic example is a disorderedchemical potential (cid:100) ∆ H = (cid:88) j µ j ˆ a † j ˆ a j , (40) Other generated terms, ζ ∇ θ , ζ π , and ζ ζ , also lead to thesame qualitative behavior. where µ j on each site is a static random variable drawnfrom a Gaussian distribution. Unlike a staggered chem-ical potential (see the previous subsection), a disorderedchemical potential cannot be gauged away. Carrying outthe same steps of writing the perturbation in the con-tinuum, and integrating out noncritical fields, we find acorrection to the effective action, to the zeroth order in U/J , as ∆ S eff K = i ˆ t, x υ (cid:0) ψ ∗ q ψ cl − c . c . (cid:1) , (41)where υ ≡ υ ( x ) is correlated as (cid:104) υ ( x ) υ ( x (cid:48) ) (cid:105) = κ δ ( x − x (cid:48) )with κ the disorder strength. Disorder superficiallybreaks the Z symmetry (Eq. (20)); however, the integralover the Gaussian distribution restores this symmetry (inthe same way that translation symmetry is restored in adisordered system). This can be more precisely formu-lated as a modified symmetry under the transformationin Eq. (20) together with υ ( x ) → − υ ( x ). It is more con-venient to cast the above equation in the density-phaserepresentation of Eq. (23) to find ∆ S eff K = √ ρ ´ t, x ζ υ. This term satisfies the Z symmetry which, in this rep-resentation, is defined as the symmetry under θ → − θ , ζ → − ζ , and υ → − υ as a close analog of Eq. (24).Indeed this is the only relevant correction to the actionthat involves υ . This follows from the scaling dimensionof static disorder, [ υ ] = 1, determined from its Gaussiandistribution. Putting all the relevant terms together inthe Keldysh action after integrating out π and setting ζ = 0, one finds S eff K = ˆ t, x √ ρ ζ (cid:0) − ∂ t θ + J ∇ θ + υ (cid:1) + i Γ2 ζ . (42)Importantly the term depending on υ cannot be cast asthe functional derivative of a proper potential term (anaive guess υθ does not respect the gauge freedom θ → θ + const . ). It is instructive to write the correspondingLangevin equation, ∂ t θ = J ∇ θ + υ ( x ) + η ( t, x ) . (43)Note that η represents white noise, while υ denotes delta-function-correlated static disorder. Clearly, the lattercannot be gauged away by going to a rotating frame dueto its spatial dependence. Furthermore, static disorder,being perfectly correlated in time, should be expected todominate over white noise. This is indeed the case, andis easily seen on the basis of scaling analysis. A conve-nient way to see this is to obtain the disorder-averaged Note that the disorder average can be performed at the levelof the Keldysh functional integral since the partition function isnormalized to Z = 1 by construction. This allows us to directlycompare the terms in the action to the disorder distribution [14]. Keldysh action by integrating over υ , S eff K = ˆ t, x (cid:20) √ ρ ζ (cid:0) − ∂ t θ + J ∇ θ (cid:1) + i Γ2 ζ (cid:21) + iρ κ ˆ t,t (cid:48) , x ζ ( t, x ) ζ ( t (cid:48) , x ) , (44)where the double time integral in the last line runs from −∞ to + ∞ for both t and t (cid:48) . With [ ζ ] = 2 at the XYfixed point described by the first line of this equation,a simple power-counting analysis reveals that κ growsunder RG as d κ dl = 2 κ . (45)Therefore, static disorder takes the system into a disor-dered phase at long wavelengths.We stress that the instability to static disorder dis-cussed here is a purely non-equilibrium phenomenon. Al-ternatively, imagine that the effect of disorder could beabsorbed in a correction to the effective Hamiltonian ofthe form ∆ H eff ∼ ´ x ˜ υ | ψ | with ˜ υ ( x ) a static randompotential; in fact, the disorder potential in Eq. (40) pro-duces such a correction at a higher order in U/J . Nev-ertheless, various terms generated from the disorderedeffective Hamiltonian can be shown to either vanish orbecome irrelevant in the sense of RG. Crucially, it is thenon-equilibrium nature of the disorder potential that isresponsible for the destruction of the XY phase.
IV. SUMMARY AND OUTLOOK
In this paper, we have considered a driven-dissipativemodel of weakly interacting bosons with U (1) symmetry in two dimensions. We have shown that an effectivelyclassical equilibrium XY phase emerges as the steadystate despite the driven nature of the model. The emer-gence of the XY phase has been argued on the basis of anadditional Z symmetry due to the sublattice exchangeof the lattice model. Various perturbations of symmetryas well as static disorder have been considered, againstwhich the XY phase is shown to be unstable. It is furtherargued that Z symmetry-breaking perturbations as wellas static disorder are genuinely of nonequilibrium nature,perturbing the XY phase in directions that are not ac-cessible in equilibrium. More generally, nonequilibriumsystems allow for new types of dynamics and fluctuations,which should be properly taken into account in order todetermine the nature of phases and phase transitions inthe thermodynamic limit. A natural question for futurestudy is the fate of this model in the limit of strong cou-pling (large U ). The perturbative arguments presented inthis manuscript are not directly applicable in this limit.It is also interesting to consider other types of symme-try ( O ( n ) symmetry, for example), and to compare andcontrast the emergent behavior in and out of equilibriumon the basis of symmetry. It would be worthwhile toidentify additional symmetries, if any, that constrain thecorresponding driven-dissipative models to exhibit an ef-fectively equilibrium behavior. V. ACKNOWLEDGEMENTS
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