Emergent Supersymmetry in Local Equilibrium Systems

# Emergent Supersymmetry in Local Equilibrium Systems

Ping Gao Center for the Fundamental Laws of Nature, Harvard University, Cambridge, MA 02138    Hong Liu Center for Theoretical Physics,
Massachusetts Institute of Technology, Cambridge, MA 02139
###### Abstract

Many physical processes we observe in nature involve variations of macroscopic quantities over spatial and temporal scales much larger than microscopic molecular collision scales and can be considered as in local thermal equilibrium. In this paper we show that any classical statistical system in local thermal equilibrium has an emergent supersymmetry at low energies. We use the framework of non-equilibrium effective field theory for quantum many-body systems defined on a closed time path contour and consider its classical limit. Unitarity of time evolution requires introducing anti-commuting degrees of freedom and BRST symmetry which survive in the classical limit. The local equilibrium is realized through a dynamical KMS symmetry. We show that supersymmetry is equivalent to the combination of BRST and a specific consequence of the dynamical KMS symmetry, to which we refer as the special dynamical KMS condition. In particular, we prove a theorem stating that a system satisfying the special dynamical KMS condition is always supersymmetrizable. We discuss a number of examples explicitly, including model A for dynamical critical phenomena, a hydrodynamic theory of nonlinear diffusion, and fluctuating hydrodynamics for relativistic charged fluids.

preprint: MIT-CTP/4861

July 12, 2019

## I Introduction

The goal of many-body physics is to explain and predict macroscopic phenomena. Except for some very simple systems, however, it is rarely possible to compute macroscopic behavior of a system directly from its microscopic description. For static properties of equilibrium systems, we have the extremely successful Laudau-Ginsburg-Wilson paradigm, which provides an effective field theory (EFT) description of long-distance (IR) physics

 Z[ϕ]=e−βF[ϕ]=Tre−βH=∫Dχe−Seff[χ;ϕ] . (1)

In (1), denotes collectively external sources, denotes collectively gapless modes, and is the low energy effective action of the gapless modes obtained by integrating out gapped degrees of freedom. While in practice such direct integrations are almost always impossible, one can deduce the general form of on physical ground. Two elements are needed for this purpose: (i) choice of the IR dynamical variables which best capture gapless (collective) degrees of freedom; (ii) symmetries of . One can then write down as the most general local field theory consistent with the symmetries.

For a non-equilibrium system or dynamical quantities of an equilibrium system, partition function is inadequate. A large class of non-equilibrium observables can be extracted from the generating functional defined on a closed time path (CTP) schwinger (); keldysh (); Feynman:1963fq ()

 eW[ϕ1,ϕ2] =Tr(U(+∞,−∞;ϕ1)ρ0U†(+∞,−∞;ϕ2)) (2) =∫ρ0Dψ1Dψ2eiS0[ψ1,ϕ1]−iS0[ψ2;ϕ2] (3)

where denotes the state (density matrix) of the system, and is the evolution operator of the system from to in the presence of external sources denoted by . The sources are taken to be slowly varying functions and there are two copies of them, one for each leg of the CTP contour. The second line (3) is the “microscopic” path integral description, with denoting microscopic dynamical variables for the two copies of spacetime of the CTP and the microscopic action. Whereas in (1) derivatives with respect to sources give thermodynamic quantities of a system, in (2) derivatives with respect to give dynamical properties of a (non)equilibrium system such as (nonlinear) response and fluctuating functions.

As in (1) we can consider integrating out short-lived degrees of freedom in (3) to obtain a non-equilibrium EFT for slow modes (denoting them collectively by and there are now two copies of them)

 eW[ϕ1,ϕ2]=∫Dχ1Dχ2eiIeff[χ1,ϕ1;χ2,ϕ2;ρ0] . (4)

Again to write down the general form of one needs to specify appropriate dynamical variables and the symmetries satisfied by the low energy effective action , although these tasks normally become significantly more challenging in non-equilibrium situations. In (4) is also encoded in the couplings of (below for notational simplicity we will suppress in ). In general does not have the factorized form of (3), and is complex. It is often convenient to introduce the so-called variables Chou:1984es (); Wang:1998wg ()

 χr=12(χ1+χ2),χa=χ1−χ2,ϕr=12(ϕ1+ϕ2),ϕa=ϕ1−ϕ2 (5)

where as usual correspond to physical quantities while can be interpreted as noises.

The functional integral (4) defines a “bare” theory at some short distance (time) cutoff scale.111The cutoff is chosen so that it is much larger than all microscopic scales, but much smaller than macroscopic scales of questions of interests. Physics at larger distance and time scales is obtained by further applying renormalization group procedure. While in principle contains an infinite number of terms with increasingly higher number of derivatives, in practice to describe macroscopic phenomena one only needs to keep track of a finite number of relevant interactions.

Non-equilibrium EFTs provide powerful tools for dealing with dynamical questions and non-equilibrium systems. The effective action incorporates dissipations and retardation effects from the bath of short-lived degrees of freedom (which have been integrated out) in a medium. Its general structure has recently been used to derive from first principle the local second law of thermodynamics GL (), and a new formulation of fluctuating hydrodynamics has been proposed in terms of such an EFT CGL (); CGL1 () (see also Grozdanov:2013dba (); Kovtun:2014hpa (); Haehl:2014zda (); Harder:2015nxa (); Haehl:2015foa (); Haehl:2015uoc ()). See also Sieberer1 () for a review of applications to driven open systems. When an is truncated to quadratic order in noises (i.e. -variables) the path integral (4) reduces to a so-called Martin-Siggia-Rose-De Dominicis-Janssen msr (); Dedo (); janssen1 () functional integral which is in turn equivalent to a stochastic Langevin equations (for a review see Kamenev ()).

Compared to EFTs for equilibrium systems, there are new elements in identifying both dynamical variables and symmetries for a non-equilibrium EFT (4). The unitarity of time evolution in (2) implies that the action should in addition satisfy the following conditions (see e.g. CGL (); GL () for more details)

 I∗eff[χr,ϕr;χa,ϕa]=−Ieff[χr,ϕr;−χa,−ϕa] (6) ImIeff≥0 (7) Ieff[χ,ϕ;χ,ϕ]=0,orIeff[χr=χ,ϕr=ϕ;χa=0,ϕa=0]=0 , (8)

where for definiteness we have taken and sources to be real. These conditions are, however, enough only for performing the functional integrals of (4) at tree level. With loops included one also has to worry about defining the integration measure precisely.

To see this, in (2) taking , we then find that

 Tr(U(+∞,−∞;ϕ)ρ0U†(+∞,−∞;ϕ))=Tr(ρ0)=1⇒W[ϕ,ϕ]=0 . (9)

While equation (8) leads to (9) at tree-level, this is no longer so when including loops and one has to include an additional integration measure factor. See e.g. Sec. I E of CGL () for an explicit discussion. To ensure (9) at loop level can use the standard trick of parameterizing integration measures by introducing an anti-commuting partner for each bosonic variable, i.e. for respectively, and requiring the action to be invariant under the following BRST-type fermionic transformation zinnjustin ()

 δχr=ϵcr,δca=ϵχa . (10)

Here is an anti-commuting constant. To show that (10) is enough to ensure (9) at loop level is quite simple and is reproduced in Appendix A for completeness. In particular, the BRST invariance automatically leads to (8) for the bosonic part of the action. are anti-commuting but transform the same as their bosonic partners under spacetime rotations. They will be subsequently referred to as ghost variables following standard terminology. The need for ghosts and BRST symmetry can also be anticipated from results on the functional integral forms of stochastic equations zinnjustin (), and has been emphasized recently CGL (); Haehl:2015foa () in the context of fluctuating hydrodynamics.222See also Haehl:2016pec (). Refs Haehl:2015foa (); Haehl:2016pec () appear to require two BRST generators while zinnjustin () and CGL () require only one.

There are three different regimes for (4). The first is the full quantum regime where path integrations describe both quantum and classical statistical fluctuations. The second is the classical regime with . In the limit the path integrals survive and describe classical statistical fluctuations. The third is the level of equations of motion which corresponds to the thermodynamic limit with all classical and quantum fluctuations neglected. Since the constraints (6)–(8) concern only with the general structure of the action, they remain in the classical limit. Similarly the requirement also survives the classical limit, and so do ghost variables and the corresponding BRST symmetry. It is striking that a classical statistical system is significantly constrained by these remnants from quantum unitarity.

For many physical processes in nature, macroscopic physical quantities of interests typically vary over spatial and temporal scales much larger than microscopic molecular collision scales (or any microscopic interaction scales). Such a system is considered as in local equilibrium, for which an additional symmetry should be imposed on  GL (); CGL1 ().

A subclass of local equilibrium systems correspond to thermal systems perturbed by slowly varying external sources, and in this case the need for this symmetry can be readily understood as follows. For , the generating functional (2) satisfies an additional constraint coming from combining the Kubo-Martin-Schwinger (KMS) condition kubo57 (); mart59 (); Kadanoff () with time reversal invariance,

 W[ϕ1(x),ϕ2(x)]=W[~ϕ1(x);~ϕ2(x)] (11)

where denotes and

 ~ϕ1(x)=ϕ1(−t+iθ,−→x),~ϕ2(x)=ϕ2(−t−i(β0−θ),−→x) . (12)

for arbitrary . Below we will simply refer to (11) as the KMS condition, but it should be kept in mind it also encodes consequences of microscopic time-reversal symmetry.333The KMS condition itself only relates to a time-reversed one, and so does the time reversal symmetry of the microscopic theory. Only the combination of them leads to a nontrivial constraint on itself Chou:1984es (); Wang:1998wg (). See CGL () for a detailed discussion. Depending on circumstances one could combine KMS with or or . For definiteness here we follow CGL () to combine it with . It is simple to adapt (12) for a system with only invariance by simply removing the minus signs before .

Additional condition(s) then need to be imposed on for (4) to satisfy (11). For variables associated with non-conserved quantities, the required symmetry is well known, probably since 70’s janssen1 (); janssen2 (); Sieberer2 (). In this case the couplings between and external sources are the standard ones

 ∫ddx(χ1iϕ1i−χ2iϕ2i)=∫ddx(χriϕai+χaiϕri) (13)

which then immediately implies that for (4) to satisfy (11), the action should satisfy444One can readily check that the requirements (6) and (14) are compatible.

 Ieff[χ1,ϕ1;χ2,ϕ2]=Ieff[~χ1,~ϕ1;~χ2,~ϕ2] (14)

with

 ~χ1(x)=χ1(−t+iθ,−→x),~χ2(x)=χ2(−t−i(β0−θ),−→x) . (15)

Following GL (); CGL1 () we will refer to (14) as the dynamical KMS condition. In the absence of external sources it becomes a symmetry (dynamical KMS symmetry) of the action. We will refer to transformations (15) on dynamical variables as dynamical KMS transformations.

The story for variables associated with conserved quantities (hydrodynamical variables) is more complicated since the couplings to external sources are more intricate making it more difficult to deduce the needed transformations on dynamical variables. In CGL () a shortcut was proposed which imposes (11) on a contact-term action which in turn constrains the action for dynamical variables through the special structure of the couplings between dynamical variables and external sources. It was termed as the local KMS condition. Only very recently were dynamical KMS transformations on hydrodynamical variables finally found in CGL1 ().

We emphasize that a system in local equilibrium is not restricted to a thermal density matrix in the presence of slowly varying external sources. Such a system can, for example, be in a pure state. In these more general cases while there is no such requirement as (14), invariance of an action under dynamical KMS transformations ensures the system is in a local equilibrium.555In essence, the dynamical KMS condition is “local”, i.e. operating at the scale of local inverse temperature, and thus will not care about the global structure of a state, be it a thermal state or a pure state. For example, in the classical limit dynamical KMS condition ensures that first law and second law of thermodynamics, as well as fluctuation-dissipation and Onsager relations are all satisfied locally GL (); CGL1 ().

With dynamical KMS transformations for bosonic variables understood, in this paper we consider the extensions to ghost variables, which are needed to have a complete formulation of a non-equilibrium EFT. For example, to ensure (11) at loop level we need dynamical KMS transformations on all variables.

Furthermore, it has been long known in the context of functional representation for linear stochastic systems that there is an emergent supersymmetry as a consequence of fluctuation-dissipation relations parisi (); feigelman (); Gozzi:1983rk (); Mallick:2010su (); zinnjustin (). More recently, it was found in CGL () that after imposing the local KMS condition and BRST symmetry there is also an emergent supersymmetry for a hydrodynamic theory of nonlinear diffusion. We would like to understand the precise origin and the full extent of this emergent supersymmetry. In particular, we would like to extend the discussion to a general non-equilibrium EFT including full fluctuating hydrodynamics.666In Haehl:2015foa (); Haehl:2015uoc () a certain superalgebra was assumed as a basic input for constructing fluctuating hydrodynamics and an attempt was made to write down the action using superspace. See also Haehl:2016pec (); Haehl:2016uah ().

We will restrict our discussion to the classical level with . At the classical level, the dynamical KMS transformations dramatically simplify. For example, equations (12) and (15) become

 ~ϕr(−x)=ϕr(x),~ϕa(−x)=ϕa(x)+iβ0∂0ϕr(x), (16) ~χr(−x)=χr(x),~χa(−x)=χa(x)+iβ0∂0χr(x), (17)

which are local transformations combined with a spacetime reflection. We stress that these are finite transformations. The dynamical KMS transformations for hydrodynamical variable, although more involved, have a similar structure (see Sec. V). The quantum regime has a number of additional complications and will not be pursued here (see Sec. VI for a brief discussion).

We will show that any system in local equilibrium has an emergent supersymmetry at low energies. With increasing complications and generality we consider three classes of systems depending on whether or not a system has conserved quantities or dynamical temperature: (i) no conserved quantities with a fixed background temperature; (ii) with conserved quantities and a fixed background temperature; (iii) with conserved quantities and dynamical temperature. Clearly the third class includes all systems. As an example for class (i) we consider model A of critical dynamics hohenberg (), for class (ii) a theory of nonlinear diffusion, and for class (iii) a fluctuating hydrodynamics for charged fluids proposed in CGL (); CGL1 (). It turns out when expressed in terms of the right sets of variables, all three classes have essentially the same structure. Here is a summary of the main results:

1. We show that there is essentially a unique extension of dynamical KMS transformations to ghost variables which is self-consistent. The dynamical KMS transformation on ghost variables turn out to be a operation, but is still a operation of the action.

2. For any action the combination of BRST symmetry and dynamical KMS symmetry leads to an emergent fermionic symmetry which together with the BRST symmetry forms a supersymmetric algebra.

3. Starting with a supersymmetric action one can always construct an action which is both BRST and dynamical KMS invariant.

4. Supersymmetry does not impose the full dynamical KMS invariance, only a particular consequence of the dynamical KMS symmetry, to which we refer as the special dynamical KMS condition. Conversely we prove a theorem stating that any bosonic action satisfying the special dynamical KMS condition is always supersymmetrizable.

5. For a system for which temperature is non-dynamical (i.e. with a fixed constant temperature), one finds a global supersymmetry. For a system for which temperature is dynamical, such as a fluctuating hydrodynamics, one finds a local supersymmetry.

6. Supplementing the bosonic story of fluctuating hydrodynamics proposed in CGL (); CGL1 () with dynamics of ghosts, this paper finally gives a complete formulation of fluctuating hydrodynamics in the classical regime.

The plan of the paper is as follows. In next section we present a general discussion of emergence of supersymmetry from BRST and dynamical KMS symmetries. In Sec. IIIV we discuss three classes of examples. We conclude in Sec. VI with a discussion of future directions. In Appendix A we give further argument for the need of BRST symmetry. Appendix B contains details of a proof for a supersymmetrizability theorem.

While this paper is in preparation we learned that overlapping results have been obtained by Kristan Jensen, Natalia Pinzani-Fokeeva, and Amos Yarom yarom ().

## Ii Emergent supersymmetry: general structure

In this section we present a general discussion of emergence of supersymmetry from BRST and dynamical KMS symmetries.

### ii.1 General case

Consider an action with which denotes collectively bosonic source and dynamical fields, and denotes collectively anti-commuting source and dynamical fields (ghost variables here). To make our equations compact we will use index to denote both field species and spacetime points. We assume that the action is invariant under a BRST-type fermionic symmetry, i.e.

 δFi=ϵQFi,QFiδIδFi=0 (18)

where is an anti-commuting constant, and is an anti-commuting operator satisfying

 Q2Fi=0,i.e.(QFj)δQFiδFj=0 . (19)

Now let us suppose that is invariant under another bosonic symmetry

 Fi→KαFi,KαI[Fi]≡I[KαFi]=I[Fi] (20)

where is an invertible bosonic operator (i.e. maps bosons to bosons and ghosts to ghosts) and index denotes different elements of the symmetry group. Note that while acts as a derivation, acts as a finite transformation. Acting on a product, transforms all factors at the same time.

Clearly the action is also invariant under the combined operations ,

 QαFi=KαQK−1αFi=[(QFj)∂K−1αFi∂Fj]Fi→KαFi (21)

where the notation on the right hand side means after evaluating replace all by the corresponding . More explicitly

 0=KαQI[Fi]=KαQK−1αKαI[Fi]=QαI[Fi] . (22)

By definition

 Q2α=0 . (23)

Thus we find that for each symmetry transformation there is an emergent fermionic symmetry . Note that the collection also includes the original as includes the identity element. Note that

 {Qα,Qβ}=QαQβ+QβQα=Kα{Qα−1β,Q}K−1α,Kα−1β≡K−1αKβ . (24)

Suppose we have an action which is not invariant under a -transformation. Then it follows immediately from our definition that

 QI0[F]=0⟺QαIα[F]=0 (25)

where

 Iα[Fi]≡KαI0[Fi]=I0[KαFi] . (26)

### ii.2 A special case

Now let us specialize to a situation which will be relevant for the rest of this paper, with being a set of transformations satisfying

 K2bi=bi,K2fi=−fi . (27)

In this case for any action (which is not necessarily invariant under ) we have

 K2I0[Fi]=I0[Fi] (28)

as an action is always even in the number of ghosts variables. Also note that and , and thus the independent ’s are and .

From (26) and (28) we then have for any action

 Q~I0[Fi]=0⟺¯QI0[Fi]=0 (29)

where

 ~I0[Fi]≡KI0[Fi]=K−1I0[Fi] . (30)

Now suppose is BRST invariant, i.e. . We can construct a -invariant action as

 I=12(I0+~I0) . (31)

But this action is in general not BRST invariant as does not have to be zero. From (29) we conclude that for to be both BRST and -invariant, the sufficient and necessary condition is that should in addition be invariant under .

### ii.3 Strategy for extending dynamical KMS transformations to ghosts

Since ghost variables are introduced to give the correct integration measure and do not directly couple to external sources, there is no obvious principle to determine how they should transform under dynamical KMS symmetry. Our strategy is based on the following non-trivial self-consistency requirement: BRST and dynamical KMS invariance of the full action does not put further constraint on the pure bosonic part of the action. More explicitly, with the full action written in a form

 I[bi,fi]=Ib[bi]+If[bi,fi] (32)

then the pure bosonic part should coincide with the most general action one can construct based (6)–(8) and the bosonic dynamical KMS invariance. This requirement is due to that the bosonic action already provides a complete formulation of tree-level physics, thus extension of dynamical KMS symmetry to the ghost sector should not change that physics. The requirement is highly nontrivial mathematically as dynamical KMS invariance constrains the ghost part of the action which in turn constrains the bosonic part via BRST symmetry.

Our discussion contains the following elements:

1. Applying the consistency requirement at quadratic level in dynamical variables uniquely determines the dynamical KMS transformation for ghost variables at linear level.

2. As a simplest possibility we postulate the linear transformation deduced from the quadratic action is the full transformation. Including both bosons and ghosts, the dynamical KMS transformations have the structure discussed around (27). We then construct explicitly from and using (21). One finds that form a supersymmetric algebra. In other words, a non-equilibrium EFT must be supersymmetric invariant.

3. We provide a strong support for the postulate of item 2 by proving that the self-consistency requirement is indeed satisfied for the full nonlinear action.

From the above discussion and that around (31) we conclude that one can obtain a BRST and dynamical KMS invariant action by first writing down a most general supersymmetric action and then impose (31).

### ii.4 Special dynamical KMS symmetry and a theorem on supersymmtrizability

In this subsection we elaborate a bit further on the self-consistency requirement of the previous subsection.

Consider a most general action of bosonic variables which satisfies (6)–(8) and the dynamical KMS condition (14). Now adding a ghost partner for each bosonic variable to obtain a full action which is BRST and dynamical KMS invariant. From our discussion above we learned that this full action must be supersymmetric. The self-consistency requirement requires that the bosonic part of should coincide with the original . This in turn requires that be supersymmetrizable. Conversely, if is supersymmetrizable, then we can construct a BRST and dynamical KMS invariant action with the same bosonic part by first constructing a supersymmetric extension of and then using (31). Thus for to be supersymmetriable is both sufficient and necessary for constructing a full BRST and dynamical KMS invariant action. So the self-consistent condition boils down to the statement: a bosonic action which satisfies (6)–(8) and the bosonic dynamical KMS condition (14) should be supersymmetrizable under the supersymmetry generated by and .

We will be able to prove that this is indeed the case. In fact we will be able to prove a stronger statement which was first observed in CGL () for a theory of nonlinear diffusion at cubic level. To describe the statement, we need to be a bit more specific on the general structure of dynamical KMS condition.

Equations (6) and (8) imply that the bosonic Lagrangian density can be expanded in -fields as

 (33)

where we use to denote collectively - and -fields respectively. Note that the sum starts with as term is not allowed by (8). In the classical limit , the dynamical KMS transformation on bosonic variables is a transformation which can be schematically written as777Clearly (16)–(17) are the of the form (34). Those for hydrodynamical variables are given explicitly in Sec. IV and Sec. V.

 ~Λr(−x)=Λr(x),~Φa(−x)=Φa(x)+iΦr(x) (34)

where is a product of bosonic -variables with altogether one derivative. The dynamical KMS condition (14) can then be written as

 ~Lb=Lb+∂μVμ (35)

where is obtained by plugging (34) into (33) and taking , i.e.

 ~Lb=∞∑n=1˜L(n)b=∞∑n=1iηnf(n)∗[Λr](Φa+iΦr)n=∞∑k=0(~Lb)k . (36)

is obtained from by flipping the signs of all derivatives and denotes terms in with factors of . Note that the -sum starts with zero. Equating equation (35) order by order in the expansion of we then find an infinite number of conditions

 (~Lb)0=∂μVμ0 (37)

and

 (~Lb)k=L(k)b+∂μVμk,k≥1 (38)

where denotes terms containing factors of .

Alternatively we can also impose the dynamical KMS condition as follows. Take a Lagrangian density of the form (33). Due to nature of the transformation, then

 Lb=12(L0+~L0), (39)

automatically satisfies (14). But as in (36) contains terms with no , and we must require that such terms in vanish, which is precisely (37). Thus it is enough to impose (37) and (39) as all the conditions (38) with are automatically taken care of by (39). We will refer to (37) as the special dynamical KMS condition.

From (29)–(31) and that BRST symmetry implies (8) for the bosonic part, it then follows that supersymmetry of ensures the special dynamical KMS condition for its bosonic part. Conversely, we will be able to prove the following supersymmetrizability theorem:

Any local bosonic Lagrangian which satisfies (8) and the special dynamical KMS condition (37) is supersymmetrizable.

Comparing with the discussion around (31), we see the procedure of imposing (39) commutes with supersymmetrization. One could either do it before or after.

## Iii With no conserved quantities at a fixed temperature: model A

In this and next two sections we consider the extension of dynamical KMS transformations to ghosts and the associated supersymmetry for some explicit examples of non-equilibrium EFTs. In this section we will consider systems with no conserved quantities at a fixed background temperature, i.e. temperature is not a dynamical variable. In this case the story is technically much simpler, but captures all the essential elements.

As an illustration of a system with no conservation laws, we consider the critical dynamics of a -component real order parameter at a fixed inverse temperature  (i.e. model A hohenberg (); Folk ()). The dynamical variables in (4) are then and the action should be invariant under an symmetry which rotates simultaneously888The boundary condition for CTP requires that at , thus any global symmetry must rotate together.. The dynamical KMS transformation for bosonic variables is the same as (17)

 ~χri(x)=χri(−x),~χai(−x)=χai(x)+iβ0∂0χri(x) . (40)

In this case the couplings (13) to external sources are rather trivial, so we will suppress the sources below. At quadratic order in (but to all orders in derivatives) the bosonic part of the Lagrangian can be written as

 Lb=χaiGraχri+i2χaiGaaχai (41)

where are are some differential operators. Note that by definition satisfies where denotes the operator obtained from by taking all to . Imposing the dynamical KMS condition leads to the condition

 Gra−G∗ra=−β0∂0Gaa . (42)

As discussed in the Introduction we should also introduce anti-commuting partners for respectively, and require the action to be invariant under the following BRST transformations

 δχri≡ϵQχri=ϵcri,δcai≡ϵQcai=ϵχai,Qχai=Qcri=0 . (43)

At quadratic level the most general Lagrangian invariant under (43) can be written as999There cannot be a term as it is incompatible with BRST symmetry.

 L=χaiGraχri+i2χaiGaaχai−caiGracri+criGrrcri (44)

where is an arbitrary differential operator satisfying from anti-commuting nature of . At quadratic level the dynamical transformation on must be linear and requiring no further constraints on we find that the only possibility is to require Lagrangian be invariant under101010One can in fact consider and for any real , but such an can be absorbed by redefining .

 cai→~cai(x)=cri(−x),cri→~cri(x)=−cai(−x) (45)

which in turn requires . We thus propose (45) as the dynamical KMS transformation for ghosts.

Combining (40) and (45) we find the structure discussed around equation (27) with

 Kχri(x)=χri(−x),Kχai(x)=χai(−x)+iβ0∂0χri(−x), (46) Kcai(x)=cri(−x),Kcri(x)=−cai(−x) . (47)

Now applying (21) to (46)–(47) and (43), we find that

 ¯δχri≡¯ϵ¯Qχri=−¯ϵcai,¯δχai=i¯ϵβ0∂0cai,¯δcri=¯ϵ(χai+iβ0∂0χri),¯δcai=0 . (48)

It can also be readily checked that

 {Q,¯Q}=iβ0∂0 (49)

i.e. form a supersymmetric algebra. It can also be checked explicitly that invariance of (44) under and indeed leads to (42).

### iii.1 Superspace

To impose supersymmetry, it is convenient to use superspace formalism Bagger1992 (). We introduce two Grassmannian coordinates and the superfield

 Ψi=χri+θcri+cai¯θ+θ¯θχai . (50)

can then be written in terms of the following differential operators

 Q=∂θ,¯Q=∂¯θ−iθβ0∂0 (51)

with (43) and (48) given by

 δΨi=(ϵQ+¯ϵ¯Q)Ψi . (52)

Note that as usual acting on superfields

 {Q,¯Q}=−iβ0∂0 (53)

with an opposite sign from (49).

The corresponding covariant derivatives are

 ¯D=∂¯θ,D=∂θ+iβ0¯θ∂0 (54)

which satisfy

 D2=¯D2=0,{D,¯D}=iβ0∂0,{Q,D}={Q,¯D}={¯Q,D}={¯Q,¯D}=0 . (55)

Note that

 ¯DΨi =−cai−θχai, DΨi =cri+¯θ~χai(−x)−iβ0θ¯θ∂0cri (56) ¯DDΨi =~χai(−x)+iβ0θ∂0cri, D¯DΨi =−χai−iβ0¯θ∂0cai+iβ0θ¯θ∂0χai (57)

A general Lagrangian which is invariant under (43) and (48) can then be written as

 L=∫d¯θdθF[Ψi,D,¯D,∂μ] (58)

where is a local expression constructed out of , and their covariant and ordinary derivatives.

### iii.2 Proof of the supersymmetrizability theorem

We now present a proof of the supersymmetrizability theorem stated at the end of Sec. II.4. Here we will discuss the main steps. There is a key step whose proof is rather contrived, which we will leave to Appendix B.

Consider a general bosonic action , which satisfies (6), (8), and the special dynamical KMS condition (37). Since the dynamical KMS transformation (40) is linear in fields, it does not change the total number of fields in a given term. In other words, suppose we expand in terms of the power of dynamical variables

 Lb=∞∑n=2Ln (59)

where contains altogether factors of , then different ’s do not mix under dynamical KMS transformations. It is then enough to prove the theorem for a general .

can be written schematically in a form

 Lb=n∑m=1iηmf(m,n−m)χmaχn−mr (60)

where each term should be understood as

 f(m,k)χmaχkr=f(m,k)I1⋯ImJ1⋯JkχI1a⋯χImaχJ1r⋯χJkr (61)

and the indices include both species indices and indices for all possible derivatives on them. It is easy to write them in momentum space, for example,

 aijk(k1,k2,k3)χai(k1)χrj(k2)χrk(k3)≡aIJ1J2χIaχJ1rχJ2r (62)

with and similarly for . is then symmetric among the first and last indices.

Now take any term in (60) with , choose a factor and replace it by . The first term resulted from the replacement has the form

 f(m,n−m)χm−1a~χa(−x)χn−mr (63)

which can be supersymmetrized as

 ∫d¯θdθf(m,n−m)(−¯DΨ)(−D¯DΨ)m−2DΨΨn−m (64)

where we have used (56)–(57). Note that in (64) the only pure bosonic term is (63). The second term resulted from the replacement can be regrouped into terms with ’s. Continuing this procedure we will then be left with terms with one factor which we will denote as

 Ib=gJ1⋯JnχJ1aχJ2r⋯χJnr, (65)

where is symmetric in indices.

Note that under a dynamical KMS transformation, a term of the form (63) will always contain at least one factor of due to the factor there. Thus the special dynamical KMS condition (37) will only involve (65) which can be written as

 iβ0gJ1⋯Jn∂0χJ1rχJ2r⋯χJnr=∂μVμ0 (66)

and in momentum space

 ω1gJ1⋯JnχJ1rχJ2r⋯χJnr=0 . (67)

Recall that index include both species indices and momenta, i.e. with , and momentum conservation implies that .

From properties of symmetric polynomials one can show that (66) implies that can be written as (the proof of which is a bit involved and we leave it to Appendix B)

 Ib=I(s)b+I(a)b (68)

where

1. in the corresponding is fully symmetric under exchanges of its indices, for which using (50) can be supersymmetrized as

 ∫d¯θdθgJ1⋯JnΨJ1⋯ΨJn . (69)
2. can be written in a form

 I(a)b=hIJ(χr)χIa∂0χJr,hIJ=−hJI . (70)

Using (56)–(57) such a term can be supersymmetrized as

 iβ−10∫d¯θdθhIJ(Ψ)¯DΨIDΨJ . (71)

We thus have shown all terms in can be supersymmetrized, which concludes the proof. To conclude this subsection let us note that the proof does not depend on the nature of the species index , which can be generalized to any kinds of indices including spacetime indices, say for a tensor field. In particular we will see the proof applies also to the examples of next two sections.

### iii.3 Full formulation

To complete the formulation of the model A EFT, now let us consider the generalization of (6) to the full action. A natural generalization is

 I∗eff[χr,cr;χa,ca]=−Ieff[χr,ηrcr;−χa,ηaca] (72)

with . It can be readily checked that only the choice and is compatible with the BRST symmetry, and thus we should have

 I∗eff[χr,cr;−χa,−ca]=−Ieff[χr,cr;χa,ca] . (73)

To see that (73) is also compatible with and operations, let us define an operation as

 ^SFr≡F∗r,^SFa≡−F∗a,^SI[Fr,Fa]≡I∗[Fr,−Fa] (74)

where denote respectively any and -type variables (including sources, bosonic and ghost dynamical variables). Equation (73) can then be written as

 ^SIeff=−Ieff (75)

Now it can be readily checked that

 [Q,^S]=0,{¯Q,^S}=0 (76)

and thus supersymmetry is preserved by . Also note that acting on bosonic fields and acting on ghost fields, and thus acting on action commutes with due to the fact that an action must contain even number of ghost fields. This shows (73) is also compatible with dynamical KMS condition.

We also need to check the self-consistency: (73) should not put further constraints on the bosonic action. This amounts to showing the full action obtained by supersymmetrizing a bosonic action satisfying (6) satisfies . Note that from the discussion of last section any term in the bosonic action can be supersymmetrized to a single term in terms of superfields. We thus only need to show that any term in the superspace has a definite eigenvalue under (then this eigenvalue must agree with that of the bosonic part). To see this note that the superfield (50) has the following structure under transformation of : , , , and where denotes the type of fields with eigenvalue under . Note that does not contain . Thus any term consisting of products of such superfields will be the proportional to , where only one survives. It has a definite eigenvalue of . Finally given that acting on action, the step (31) does not change the eigenvalue of .

We can now present the full procedure for constructing the EFT for model A using supersymmetry:

1. Construct a most general supersymmetric action , which satisfies (73) (and of course whatever other symmetries of the system).

2. Construct the full action using (31).

3. The bosonic part of the action should further be constrained by (7).

Instead of using supersymmetry one can of course directly impose BRST symmetry and the special dynamical KMS condition. With the powerful formalism of superspace, supersymmetry should in general be a faster route.

Finally we note that in a most general supersymmetric action there can be terms which are not related to the pure bosonic action, i.e. terms involving ghosts transform among themselves under supersymmetric transformations. Whether one should include such terms requires further consideration.

## Iv With conserved quantities at a fixed temperature: nonlinear diffusion

In this section as an example of systems with conserved quantities at a fixed temperature we consider the hydrodynamic theory for nonlinear diffusion developed in CGL (). For slow variables associated with conserved quantities, couplings to external sources play an important role in the formulation of the theory. So in this section we will turn on external sources from the beginning. We will see that the same structure as that of model A emerges. The discussion here generalizes and systemizes some previous observations in CGL () regarding BRST invariance, KMS conditions and supersymmetry.

We consider the theory of diffusion mode associated with a conserved current at a fixed inverse temperature , ignoring possible couplings between the diffusion mode and other hydrodynamical modes. The dynamical variables are with interpreted as the diffusion mode and the corresponding noise variable. The background sources are and which couple to conserved currents and respectively. The bosonic action satisfy the following conditions:

1. must always be acted on by at least one derivatives. We will thus count as having zeroth derivative. In the presence of background fields , the action should depend only on the combinations

 Brμ=Arμ+∂μφr,Baμ=Aaμ+∂μφa (77)

i.e.

 Ib[φr,Arμ;φa,Aaμ]=Ib[Brμ,Baμ] . (78)

The local chemical potential is given by with giving the dynamical part, and it is often convenient to use and .

2. The action is invariant under

 φr→φr−λ(σi),φa→φa . (79)

The dynamical KMS transformation on bosonic variables are

 ~φr(x)=−φr(−x),~φa(x)=−φa(−x)−iβ0∂0φr(−x) (80)

and when including background fields

 ~Brμ(−x)=Brμ(x),~Baμ(−x)=Baμ(x)+iβ0∂0Brμ . (81)

We now introduce ghost partners for respectively, and require the action (in the absence of background fields) to be invariant under transformation

 δφr=ϵcr,δca=ϵφa . (82)

In the presence of external sources it is convenient to introduce ghost partners for respectively and the action should be now be invariant under the combinations of (82) and111111See Appendix A for motivation for introducing ghost partners and BRST transformations for external sources.

 δArμ=ϵηrμ,δηaμ=ϵAaμ . (83)

Introducing

 Hrμ=ηrμ+∂μcr,Haμ=ηaμ+∂μca . (84)

then (82)–(83) can be written in a unified way as