JHEP07(2020)200 Springer July 4, 2020 July 28, 2020 : June 30, 2020 : March 11, 2020 : : Revised Accepted Published Received Published for SISSA by https://doi.org/10.1007/JHEP07(2020)200 . 3 1912.12301 The Authors. c Effective Field Theories, Quantum Dissipative Systems, Thermal Field

We propose a systematic coset construction of non-equilibrium effective field , [email protected] E-mail: Department of Physics, Center538W for 120th Theoretical Physics, Street, New Columbia York, University, NY, 10027, U.S.A. Open Access Article funded by SCOAP ArXiv ePrint: that can be used to study many-body systems outKeywords: of thermal equilibrium. Theory in an effort to makeconstruction, this we paper successfully reproduce self-contained. the Totemperature. known demonstrate EFTs the Then, for legitimacy to fluids of and demonstratesolids, this superfluids its coset and at utility, finite four we construct phasesthe novel of non-equilibrium EFTs liquid for effective crystals, solids, action all super- and at the finite coset temperature. construction to We create thereby a combine powerful tool significant advances beyond more standardof coset recently-developed techniques, constructions which in allowactions that the that it formulation account takes of for advantage quantum non-equilibriumthese and effective systems thermal fluctuations exist as atSchwinger-Keldysh well contour. as finite Since dissipation. temperature, the Because coset theactions construction may EFTs and be the live non-equilibrium unfamiliar on effective to the many readers, closed-time-path we of include the brief introductions to these topics Abstract: theories (EFTs) governing the long-distancetemperature and condensed late-time matter dynamics systems. of relativistic, finite- Our non-equilibrium coset construction makes Michael J. Landry The coset construction for non-equilibrium systems JHEP07(2020)200 3 33 37 5 15 20 8 7 11 23 34 – i – 11 37 32 36 24 9 27 35 12 27 29 29 5 22 18 1 31 16 20 7.2.1 Phase7.2.2 A Phase7.2.3 B Phase C C.2 Unbroken inverse Higgs C.3 More on unbroken inverse Higgs C.1 Thermal inverse Higgs 6.2 Supersolids 7.1 Nematic liquid7.2 crystals Smectic liquid crystals 5.1 Fluids 5.2 Superfluids 6.1 Solids 4.1 Goldstones at4.2 finite temperature The method of cosets 2.1 Inverse Higgs 2.2 Zero-temperature superfluids: a simple example 3.1 Rules for3.2 constructing non-equilibrium EFTs The fluid worldvolume D Fluids and volume-preserving diffeomorphisms A Emergent gauge symmetries B tricks St¨uckelberg and the Maurer-Cartan form C Explicit inverse Higgs computations 8 Conclusion 7 Liquid crystals at finite temperature 6 Solids and supersolids at finite temperature 4 The non-equilibrium coset construction 5 Fluids and superfluids at finite temperature 3 Non-equilibrium effective actions: a review Contents 1 Introduction 2 The zero-temperature coset construction: a review JHEP07(2020)200 Once this 1 ]. The reasons for doing 29 – 2 input could resolve some of these issues. – 1 – only ]. Then, the long-distance and late-time dynamics are 10 ]. This construction takes the symmetry-breaking pattern — which is 34 – 30 ]. Even though such an approach is heavily based in symmetry considerations it 1 From the effective field theory (EFT) perspective, many-body systems can often be In order to make the approach to hydrodynamics — and many-body physics as a whole Near-equilibrium thermodynamics in the usual formulation is ill-defined in the sense that thermody- 1 described entirely by Goldstoneis bosons. a For powerful our andconstruction purposes, [ systematic this way is to very good construct news: EFTs of there Goldstonesnamic parameters known like as temperature and the chemical potential coset are only rigorously defined in true equilibrium. tools of analysis. understood as systems that spontaneouslypossible break to classify spacetime the symmetries. state In ofbreaking matter fact, (SSB) of it a pattern is many-body often system alone by [ its spontaneous symmetry alone. Once theaction symmetries by writing and down field ametries. content linear combination are Thus, of specified, all there one termsexpress is local that constructs thermodynamic are no the quantities consistent guesswork like with effective temperaturethe when the and fields, sym- formulating chemical yielding potential effective in a terms actions. moremodynamics. of rigorous Applying One understanding these of can methods the to then concept condensed of matter systems near-equilibrium has ther- provided powerful — more systematic, over thehydrodynamics last from decade the or point so ofso there view are has of been as an a follows. effectiveenergy great action Effective effort and [ actions to particle reformulate provide powerful physics tools because for they describing can systems be in high- constructed from symmetry principles clear what are the rulesone of might the wonder game if for constructing therenamics hydrodynamic are that theories. additional one For constraints example, must beyondmethodically impose. the account second for Additionally, law thermalapproach of a fluctuations that thermody- way or utilizes to symmetries quantum extend as effects hydrodynamic the is theories not to at all clear. An it relies on therelations somewhat among ill-defined the notion conserved quantities of andis ‘near-equilibrium’ local done, to thermodynamic the parameters. construct second constitutive lawon of various coefficients. thermodynamics must While be thisabout method imposed hydrodynamic has by systems, led hand, there to putting remain very constraints powerful some and difficulties. general In statements particular, it is not at all most important guiding principlesand in their corresponding constructing Noether such currents.dynamics effective For comes example theories the from are standard the formulation symmetries number of equations [ hydro- for conservationis of not energy, very momentum, systematic and as particle other non-symmetry constraints must be imposed. In particular, When trying to understandall a of many-body the system, degreesonly the on of goal quantities freedom; is thatone there often persist course-grains are over not over long simply to the distances keep tooan microscopic, and track effective many. or extended of theory ultraviolet time Instead, of (UV) scales. the degrees the To aim of macroscopic, do freedom is this, or to to infrared obtain focus (IR) observables. Often, one of the 1 Introduction JHEP07(2020)200 ]. 12 ] in which a new 24 – non-equilibrium coset 19 ] and [ 8 – 2 We term the result the ] to build simple EFTs on the SK contour; however their – 2 – 2 25 ]. 2 ]. These gauge symmetries lead to diffusive behavior. 9 give quick reviews of these topics. Readers already familiar with these 3 . and 2 To demonstrate the validity of this new coset construction, we use it to formulate Our approach is as follows: we start from the observation that at finite temperature, Nevertheless, despite significant advances, in the current literature there exist very few Historically, the EFT approach to many-body physics has been unable to account for The coset construction was recently used in [ 2 ism may be unfamiliarsections to some readers. Totopics make can this skip paper these as sections self-contained entirely. as possible, approach only dealt with internal symmetries. ture. Then, to demonstrate thesupersolids, utility of and our four approach, we phasesexpand construct the of novel number EFTs liquid of for crystals, known solids, to non-equilibrium all EFTs. construct at EFTs Moreover, finite our for formalism essentially temperature.Goldstone-like) can any excitations. be state We used The of thus coset matter vastly construction that and is the describable non-equilibrium by EFT Goldstone formal- (or We use these cosetsthat to transform construct covariantly building-blocks under both fortype the the gauge global non-equilibrium symmetries. symmetries effective as actions well as the chemical shift- actions for already known EFTs, namely those of fluids and superfluids at finite tempera- erators behave very differently thancorresponding those to unbroken arising generators from have SSB.the infinitely many chemical In gauge shift particular symmetries symmetries the analogousSince to Goldstones of the [ non-equilibrium effective actiondoubled. is defined This on requires the the SK introduction contour, of the two field cosets; content is one for each leg of the SK contour. body systems and theconstruction coset construction. there exist Goldstone-like excitations correspondingis to not each spontaneously symmetry broken. generator However, even the if Goldstones corresponding it to the unbroken gen- systems for which the non-equilibriumunderstanding EFTs are of known. the The reasonpattern field is has that yet content, been no emergent overarching providedproblems for gauge non-equilibrium by systems. symmetries, combining In two or this powerful paper, tools: symmetry-breaking we address the these non-equilibrium EFT approach to many- kind of non-equilibrium action wasKeldysh formulated using (SK) the contour. in-in Using formalismcan symmetry on principles account the Schwinger- alone for these bothare new statistical constructed non-equilibrium on fluctuations EFTs the assymmetry SK well group, contour, as there the exist dissipation. field emergent contentof gauge is Because which symmetries doubled. remain these at poorly In EFTs finite understood addition temperature, [ to the origins the global has been used to formulate numerous EFTs describing variousstatistical states fluctuations of and matter dissipation. [ less, The conservative reason systems is and that are therefore ordinarypative incapable actions dynamics. of describe accounting However, for noise- this stochastic changed and with dissi- recent work [ specified by the state of matter of the system — as the only input. The coset construction JHEP07(2020)200 (2.1) , there . While ]. ]. It will H G ); instead / 32 µ 10 – G P 30 . Then the IR , which includes H G , we represent the symmetry , to the subgroup , G → H G , is linearly realized on the fields, the action H . and internal symmetry generators [ – 3 – A +). µ , T P ] and for spontaneously broken spacetime symme- + , ]. Importantly, we do not require that the unbroken 33 + , both refer to unbroken generators, they will play very 10 − A may be some combination of internal and spacetime gen- = unbroken translations = other unbroken generators = broken generators T A µ α A ¯ T τ P T and = diag( is nonlinearly realized. In general, therefore, generic symmetry µ and ¯ P µν H α η / τ G ]. be the original Poincar´etranslation generators (represented by 34 µ ¯ P that is generated exclusively by The EFT of the Goldstones must be invariant under the full symmetry group The coset construction is an especially powerful technique since it provides a systematic Often, the only gapless modes in the system are the Goldstones. If we are only con- Throughout this paper we will use the ‘mostly plus convention’ so the Minkowski ⊂ H 0 the action of the unbroken symmetry subgroup of the broken coset transformations will act inwriting a down the highly most non-trivialchallenging general unless manner effective we on action use the the consistent construction Goldstones. with that symmetries follows. As can be a rather result generators they can be someturn linear out combination of that although different roles in the cosetH construction. Therefore, it is convenient to define the subgroup where the generators erators and we have assumedremain that unbroken. there In exist this somenotions way, notions states of of can energy spacetime still and translations be momentum that classified [ according to the corresponding only input. Supposing the symmetry-breakinggenerators patten is by time symmetries. Forbroken internal in-depth symmetries, discussions consult of [ tries, consult the [ coset construction for spontaneously method for generating effective actions of Goldstone bosons using the SSB pattern as the cerned with the deep IRan dynamics, we effective can action integrate for out allwriting the non-Goldstone down Goldstones. Goldstone modes effective to It actions obtain turns knowngive as out a the that brief coset construction. there review is of This section a the will coset systematic construction method for for spontaneously broken internal and space- the system spontaneously breaks thedynamics of symmetry the group system arespontaneously described broken, by then Goldstone modes. for everyis If broken a only symmetry internal corresponding symmetries generator Goldstone are inbroken, boson. the there However, are coset if often spacetime fewer symmetries Goldstones are than spontaneously broken symmetry generators [ metric takes the form 2 The zero-temperature cosetConsider construction: a relativistic quantum a field review theoryboth whose internal full symmetry symmetries group as is well as the Poincar´egroup. Suppose that the ground state of JHEP07(2020)200 . . π dg 1 (2.4) (2.5) (2.2) (2.3) − as the g G ) can be , we have g 2.3 It is convenient to 3 . 0 H / . -invariant combinations of G
H A transforms covariantly under , ] T . As a result, for any element α in the coset. But in any case it turns A ν 0 π . , meaning that the coset of sym- B µ . µ ¯ H A P α ; and higher-order-derivative terms µ T τ + ∇ α x, π, g ) ix A µ ( x α π e ( 0 τ B µ α i h α need not be contracted in the usual way are transformed coordinates and fields. · ∇ iπ π + ] e , the terms on the r.h.s. of ( 0 ν 0 µ g ν π ¯ coordinates P ∇ µ, ν ∂ , π µ 0 ν µ + ) x ix 1 ( – 4 – and e ν and , where repeated indices are summed over. γ − ¯ P 0 2 , it is possible to express any element of π ) E 0 ), etc. Then, the symmetry-invariant terms of the x ] = ν µ α α ] = H π π / i = ( iE ρ ), we can construct the Maurer-Cartan one-form ). In particular, under the transformation by x, π G and ∇ ( x, π H µ ∇ = ( ( γ ) correspond to the Goldstones. 2.2 0 ], so we have 2.3 ∇ g γ x H ν · µ ( H ∂ 34 α ∇ and ( g , it is then easy to identify the covariant building-blocks. At , 1 π g H µ ; we therefore will refer to it as the covariant derivative of 2 − µ 33 ) g G ∇ ∂ α 1 π ), − 0 α g ∇ π ν . 0 ∇ transforms like a connection (i.e. gauge field) and we can therefore use it π serves as a vierbein; in particular, the invariant integration measure is ( A H µ ν ν µ → B ∇ E π ) has the correct symmetry properties for our purposes. is some element of . It should be noted that the indices 2.2 0 and E h 0 , we may write x After computing Using the parameterization ( By the definition of the coset We are interested in ensuring that the effective action remain invariant under all sym- det This is a somewhat hand-wavy justification for including 3 x → ∈ G 4 effective action are simplythese constructed covariant by terms. taking If manifestly invariant terms boosts are are ( broken but rotations are not, thenout that examples ( of such d if the Lorentz group is broken. leading order in the derivativeare expansion we given have by Additionally, to compute higher-order covariant derivates by And finally, the symmetry generators [ Using the symmetry algebra alone,ator it in is the possible above tothe expression. compute full the It symmetry coefficients can of group each be gener- checked that therefore read off the transformations ofsymmetry the transformation coordinates from and ( Goldstones underx an arbitrary This one-form has the property that it can always be expressed as a linear combination of g where It is important toexplicitly computed note if that we for know the given commutation relations among the generators. We can parameterize this coset by Then, up to normalization, product of an element of the coset and an element of metry transformations. The symmetriesthat that act act linearly non-linearly are presentgenerators easy act more to non-linearly deal of on with, aunbroken the but translations fields challenge. those act while non-linearly unbroken By onmetry generators definition, the act generators linearly. spontaneously that However, broken have some sort of non-linear action is JHEP07(2020)200 Q to ¯ P (2.6) -Goldstone, τ , we will take a contains another 0 5.1 τ do not belong to the , 0 τ ρµ J σν and η is the charge associated with τ + Q ρν J -Goldstone in the direction of τ σµ η are the Lorentz generators. From the − , µν νσ ρ J and a broken generator J -Goldstone to derivatives of the P 0 ]. Physically, ¯ µρ τ P µσ η – 5 – η 37 , − − 36 σ µσ . Suppose further that is the chemical potential. Additionally, since every P J τ . Then it turns out that it is consistent with symmetry µ 0 , νρ µρ ⊃ η η H ] 0 -Goldstone. The setting of this covariant derivative to zero ] = 0 ] = ] = 0 ν ¯ τ P , τ ρσ ρσ ,P (i.e. time translation) are spontaneously broken but a diagonal ,J ,J is preserved [ µ 0 µ P µν [ P P i [ J µQ [ i i + , that is [ ]. Essentially, there are two possibilities. Firstly, sometimes when space- and 0 τ P Q 35 , ≡ 32 0 , ¯ P are the translation generators and 11 µ P The physical reasons for imposing these inverse Higgs constraints have been investi- Pragmatically, the rules of the game are as follows: suppose that the commutator EFT perspective, a superfluidsuch is that defined both as asubgroup, system that has aparticle-number conserved conservation U(1) and charge condensed matter system (including superfluids) has a zero-momentum frame, boosts are where 2.2 Zero-temperature superfluids:Thus a far, simple our example discussionwill of focus the on coset a constructionsuperfluid has simple EFT. been concrete First, rather since example: ourour abstract. theory the ‘mostly is This plus’ coset relativistic, convention, section construction it the ought of Poincar´ealgebra is to the be In zero-temperature Poincar´e-invariant. From this perspective, the inverseGoldstones. Higgs For constraints the correspond purposes of topurely this integrating pragmatic paper, out approach with gapped and the always exception imposeremaining of inverse agnostic section Higgs about constraints whenever the possible, precise reasons for doing so. independent fluctuations. As a result,that there all can correspond be multiple toconstraints Goldstone the can field be same configurations understood physicalwhen as state. a a Goldstone convenient From choice can thisif of be view we gauge-fixing removed point, did condition. via the include Secondly, inversewe the inverse Higgs are Goldstone Higgs constraints, in often it the only is effective interested often action, in the it gapless case would excitations, that have an so energy we gap. may integrate But out these fields. allowing the removal of the is known as an inverse Higgs constraint. gated in [ time symmetries are spontaneously broken, the resulting Goldstones do not correspond to between an unbroken translation generator broken generator same irreducible multiplet under transformations to set the covariantzero. derivative This of gives the a constraint that relates the We mentioned earlier that whenstones only internal exactly symmetries matches are the broken,metries number the are of number of broken broken, Gold- generators. thisfrom need However, the not when perspective be spacetime of the sym- the case. coset construction. In this section, we will see how this works 2.1 Inverse Higgs JHEP07(2020)200 . . 3 π , jk 0 2 J , (2.7) (2.8) (2.9) ∇ (2.10) (2.12) (2.13) (2.14) (2.11) ijk  = 1 1 2 i ≡ and Lorentz i for J Q i , so the leading- P x 4 = d i ¯ P ,
i J and i ν , µQ + Ω jk can be thought of as the spin- i + . i Λ] K k µ 0 i ν µ. , K , Ω P i η ) ∂ . 0 0 µ ν x 1 − , ψ. ( ) ψ ijk ψ i = − ψ. µ ∇ Λ] i µδ 0  ψ y iη µ ∂ ν 0 ∂ ( ∂ µ i [Λ e ¯ + ∂ − P µ 1 − Q ψ∂ ) , we find that ijk ψ∂ ψ − µ x P x  π µ ν ( = πQ 4 ∂ ν µ 0 ∂ ∂ [Λ = Λ ν ) d iπ η ν µ ν µ − 1 ∇ – 6 – e − ∇ ) ) π . Here, − i µ 1 1 Z , p ¯ π P p + ν − − E ∇ µ ( = µ tanh ν ≡ + E E = ix ¯ 1 2 i P S e y η η π 0 = µt 0 ν = = Λ = ( = ( µ = i ν i µ ∇ µ π ≡ g iE η µ Ω E µ , meaning that we may impose inverse Higgs constraints to = ψ ∇ ∇ g µQ µ ij ∂ and 1 iδ − ν g ⊃ µ ] 3 directions, yielding ] i j , K 2 i ) into the expression for , iη ,K i e . The unbroken translations are ¯ [ i P 2.11 = 1 0 J i = 1, we have that the invariant integration measure is just ≡ ≡ ν E µ i is just a constant, we find that at leading order in derivatives, the zero-temperature K µ Now, notice that the commutator between unbroken spatial translations and broken along the Since det order action is By plugging ( Since EFT must be built out of an arbitrary function of tives of the U(1) Goldstones Using these relations, we can removeThus, the at boost leading Goldstones order as in dynamical the degrees derivative of expansion, freedom. the only covariant building-block is remove the boost Goldstones.π In particular, we may set to zero the covariant derivative of These constraints can be solved to give a relation between the boost Goldstones and deriva- where Λ connection (or at least therotations. spatial components of it) and transforms as a gaugeboosts field yields under [ By explicit computation, the Maurer-Cartan form is where boots and the remaining unbrokenWith generators this are symmetry-breaking the pattern, spatial the rotation coset generators is then necessarily broken. As a result, the broken generators are the U(1) charge JHEP07(2020)200 , i µ for (3.2) (3.3) (3.1) J . This to denote I ). We are −∞ 3.2 = as well as Ω t , ] i 2 represent the UV η J ; µ ir 2 uv ∇ i ψ ,ψ ) 2 uv 2 4 J ψ ; [ iS , − ] −∞ , 0 1 ] ¯ , in the presence of source P 2 J ; 0 t ,J is the unbroken time-translation β ∞ ir 1 2 − 0 ,ψ [Ψ e to (+ ¯ P † uv 1 0 iS t ψ [ − to denote an ordinary action and ] indicates that field configurations are ρU 1 = tr iS ) S e ,J 1 ρ 1 J and then returns again to uv 2 ; [Ψ ψ iS ∞ D e – 7 – −∞ 2 uv , 1 Ψ ψ = + ,Z ∞ D D 0 t 1 ¯ ρ P (+ ]. In this formalism, the sources are doubled. Letting Ψ 0 Z Z β U D 39 h − = ρ e ] tr Z 2 = goes to ,J ≡ ≡ 1 ρ ] J ; represent the IR degrees of freedom and 2 ), and the subscript ir 2 ,J ir −∞ 1 ,ψ J [ ψ ir 1 = ψ W → ∞ [ t e t ( EFT 2 iI e , where } ). But to keep things as streamlined as possible, in this paper we will only uv 2.5 ) = Ψ is the inverse equilibrium temperature and is some function that is determined by the equation of state. Terms that are , ψ ) be the time-evolution operator from time 0 ir P β ψ ,J 0 → ∞ { Consider the path-integral representation of the generating functional ( It is possible to use this formalism to construct an effective action of the IR degrees of t Here as well as in the remainder of the paper, we use ( t, t 4 ( 1 degrees of freedom. Define the effective action for the IR fields by an action defined on the SK contour. interested in the meaningIn of typical the Wilsonian low-frequency, long-wavelength fashion,tion dynamics we for of integrate the the out slow system. Ψ modes. the = fast Suppose modes the to fields obtain can an be effective divided ac- up into IR and UV fields by time that starts at contour is often referred to as a closed timefreedom path on (CTP). the CTP, knownfunctional as depends on the two non-equilibrium copies of effectivedoubled the action. field fields, content. the Since non-equilibrium the effective generating action will have where inΨ the pathweighted integral by representation, the thermalare we two density time-evolution operators require matrix such thatward functional that one in in evolves time, forward in the in we time infinite can the and conceive the past. other of distant back- the Because SK future, there path integral as existing on a closed contour in some field Ψ, the generating functional is where generator. In order to describeor correlation Schwinger-Keldysh functions in formalism time, [ weU must use the so-called in-in At zero temperature, theuum. equilibrium At state finite is temperature, describedequilibrium however, state by no must a such be pure pure described state, equilibrium by namely state a the mixed-state exists. vac- thermal Instead, density the matrix of the form higher order inwhich the plays derivative the expansion role canthe of form be a ( constructed connection using construct and leading-order actions. can be used to create3 covariant derivatives of Non-equilibrium effective actions: a review where JHEP07(2020)200 , . ] 0 2 ¯ P J 0 EFT ]. ; β (3.5) (3.4) I 2 2 − e [Ψ S ∝ ρ − ] 1 J must transform ; 1 ir 2 ] ψ [Ψ 1 2 , S 1 ,J 2 ,J and J 2 , ; ir ir 1 1 , except for time-reversing ir 1 , ψ ψ ] are purely real, the coeffi- ) 2 so we do not need to include , ψ J EFT , ~x ; ir 2 ) I 2 θ ψ [ EFT [Ψ I − . ) S for any 0 EFT β I − ( EFT , . iθ, ~x i ] I − 0 1 − + J ≥ ; t t ] = ] ] = 0 1 ( ( . Notice that the information contained in ] 2 2 2 2 ir ir 1 2 J ], but there are several important constraints [Ψ ,J ,J ,J ψ ψ 2 1 S 1 1 – 8 – = ] obeys what are known as the KMS conditions. J takes the form of a thermal matrix, Θ Θ is a symmetry of ; J J 2 1 ; ; ir 2 ρ J S → → ,J ir ir 2 2 ; ,ψ 1 ) ) . Finally we require that the Green functions of ir ir 1 2 J 2 , ψ , ψ x x [ , ψ ψ ( ( [ ir 1 ir ir 1 1 . We outline some important features of this effective ] may be complex. There are three important constraints ir ir 1 2 W ψ ψ ψ 2 = [ [ ψ ψ EFT EFT ir ,J iI 1 ]. I e ψ ir 2 ∗ ] EFT EFT 2 [ I I ψ ir 2 ; can be absorbed in the low-frequency limit by a finite number of 1 Im , ψ EFT I ρ ir ,J 1 ir ψ 1 is absorbed into the coefficients of [ at any given order in the derivative and field expansions. It has been ψ ρ [ D R EFT EFT I I = ] 2 ,J 1 J [ W reversing symmetry Θ;simultaneous at charge, a parity, and minimum,the time the dynamical inversion. KMS UV symmetries Then, theory act setting on will the the sources be fields to by invariant zero, under a If the equilibrium densitythen matrix the partition function These KMS conditions for thedynamical partition KMS function symmetries can of beis the as used effective follows. to action. derive Suppose the that The so-called the way UV these theory symmetries possesses act some kind of anti-unitary time- of the effective actionthe allows 1-and the 2-contours production must be ofsimultaneously equal entropy. in under Because the any distant the global future, fieldcopy symmetry values of transformation. on the global Thus, symmetry there group. is just one These conditions help to ensure that GreenAny functions symmetry are of path-ordered. the UVsymmetries. action The fact that these time-reversing transformations are not symmetries cients of that come from unitarity, namely The UV action describing theThe system of effective interest action, is factorized however,will by does exist not terms that admit couple a 1-and factorized 2-fieldsNotice form. in that, while In the general, coefficients there of e • • • • Following the usual EFT philosophy, it isand our information hope about that all ofparameters the complicated in UV dynamics demonstrated that this isthat in must fact the be case imposedaction [ below upon without proof [ the density matrix a density matrix for theare IR path-ordered fields on the SK contour; the path-ordered3.1 Green functions are Rules given for in [ constructing non-equilibrium EFTs then JHEP07(2020)200 . . 2 θ , sµν − (3.8) (3.9) (3.6) (3.7) g 0 = 1 (3.10) β ] must s . More µν 2 sµν and g , g symmetries. ) for θ x µν 2 . Thus, when ]. Consider a ( 1 2 Z ir r µ g s [ ζ ψ , i W ) . µν 2 ir 2 , g ] ψ ; .  MN − ) 2 2 indicates on which leg of x , −∞ ir 1 ( ,G , , ] ψ ir r ∞ ψ = 1 MN ≡ t 1 MN 2 s ∂ G (+ ζ 2 ir [ a 0 † β , g  EFT ρU Θ ) iI i . Now, we can use the trick St¨uckelberg MN 1 e ζ 1 µν ν s N 1 , where g 2 [ g ) ) + ∂ζ ∂φ X ; x x , ψ sµν W ( ( )) D
g 1 ir ir φ r a – 9 – ir 2 ( −∞ ψ ψ X ] = s ψ , ζ Θ Θ D ( µν + ∞ 2 is proportional to the derivative of Z → → ir 1 sµν , g (+ ) ) g ψ = ir a U x x µν µ s ] M ( ( 1 h 3 denote diffeomorphism transformations of ψ 1 2 g , as contributing to the same order. ∂ζ ir ir a r µν [ ∂φ 2 2 ≡ . As a result, they are non-local transformations. This non- ψ ψ , ir r ,g 0 W = tr 1 ≡ ψ ir r , ] µν t ) iβ 1 ψ ∂ g µν φ [ − 2 ( ]. It can be checked that these transformations are their own ) is the time-evolution operator in the presence of source 0 = 0 ,g W e µν and , β s sµν 1 ζ sMN g g [0 [ g ir a ; ) such that the generating functional becomes W ψ φ M,N ∈ e ( µ s −∞ θ , for X ∞ s (+ ζ sMN U g Notice that the changewriting in down terms ofto the consider effective action in the derivative expansion, it is natural Then the classical dynamical KMS symmetry transformations are these symmetries become local; oneFurther, need in only the perform a classical Taylor limitit series they in is become convenient to exactly perform local. a To change take of the field classical basis limit, by inverse, meaning that theThese dynamical symmetries KMS involve temporal symmetries translationsbecause are along the discrete the thermal imaginary-time density matrix directions by can imaginary be time interpreted as alocality time-translation may operator seem rather odd, but at any given order in the derivative expansion, for any explicitly, we have and promote the gaugein’ transformations the to fields dynamical fields. In particular, we ‘integrate be invariant under two independent diffeomorphismrepresent transformations. two Let different diffeomorphisms. Then we have where where Since the stress-energy tensor is conserved, the generating functional At finite temperature theInstead, effective we action must introduce is the oftenis notion not so, of defined the we on so-calledfluid will the fluid with reproduce worldvolume. physical no To spacetime. the conserved seestress-energy derivation currents why tensor other this of are than the the the fluidthe stress-energy metric SK action tensor. tensors contour presented the The metrics in sources live. for [ Then, the the SK generating functional takes the form 3.2 The fluid worldvolume JHEP07(2020)200 are µ s (3.13) (3.11) (3.12) (3.14) (3.15) 3. The X , 2 , = 1 , I ) I for , φ 0 I 0 as ‘fluid coordinates’ φ φ ( M τ φ 0 ∂f ∂φ ) can be succinctly encapsu- ν s → N ) 3.14 I ∂φ ∂X , , , ) )) , φ ) ) I 0 φ I J ( φ φ fluid coordinates φ φ ( s ( ) and ( ( ( 0 I X M 0 0 ( ξ φ φ 3. We will refer to 3.12 + , sµν → → spatial g 2 M , 0 µ I s – 10 – 1 M φ φ φ ] is diffeomorpism invariant. However, there is no , τ , ) ∂X ∂φ I µν → 2 = 1, in which case we have the residual symmetry = 0 , φ ≡ M , g τ 0 φ φ M µν ( 1 f g sMN [ for G = W 0 M 0 φ φ for a discussion of their relation to local thermodynamic properties → 0 A ) plays the role of the worldline einbein for the fluid volume element 3. φ I , 2 3. , , φ , 0 2 , φ ( = 1 ) are arbitrary functions of the τ I ) are the dynamical fields and describe the embedding of the fluid worldvolume = 1 . We can then gauge-fix φ φ I ( I I,J ( φ µ s M ξ X at for reparameterization invariance, where for Treating each volume element as a point particle, we require independent world-line Fluid elements are indistinguishable fromrelabel one them another, in so we a should time-independent be manner. able We to therefore freely require invariance under The partial diffeomorphism symmetries ( • • justifications provided above forincluded to imposing build these the readersarise, diffeomorphism intuition; only symmetries it that is are they notcan merely are understood consult appendix necessary from to first principles describeof how fluid systems they phase. in fluid phase. Interested readers lated by the transformation law where in fluid phase,symmetries: we must require that the fluid coordinates enjoy partial diffeomorphism into the physical spacetime. fieldsSt¨uckelberg In as long a as certain sense,guarantee it that is the always resulting possible EFT willmatrix to be ‘integrate is non-trivial; in’ for a example, these pure, ifpure zero-temperature the equilibrium gauge ground density and state hence without any have SSB no then dynamics. the fields It turns out that to ensure the system exists are pull-back metrics.introduce the Notice coordinates thatand when the performing corresponding manifold thefields on trick, St¨uckelberg which we they had live as to the ‘fluid worldvolume.’ Then the where JHEP07(2020)200 . as- (4.1) (4.2) gives 3 and , 2 A , unbroken 1 , ] = 0 2 by [ M,N M φ be the unbroken generators ]. While it is not fully un- A , 38 T A 2 indicates on which leg of the , T , ) 9 I , , φ 4 ( ) = 1 – non-equilibrium coset construction I A 2 φ s iλ ( e A M T ξ ) symmetry regardless of whether it is broken φ + ( A s M i – 11 – e φ every → → ). We use the convention that A I M T ) φ φ φ ( ( A A s λ i e and those corresponding to unbroken generators as symmetry. As a result, we expect that the non-equilibrium effective every ) be the unbroken Goldstones. Here, φ 3. Additionally, the effective action will be invariant under a certain subgroup . ( , that these gauge symmetries act as follows. Let A s 2  , broken Goldstones = 1 It turns out that there is an important distinction between broken and unbroken Gold- of diffeomorphisms that act on the fluid worldvolume coordinates which we interpret as additionalsymmetries gauge arise, symmetries. it Since is it is conceivable unknown that how they exactly may these not hold in all situations. However, we CPT the fields live. Then we have the gauge redundancies for arbitrary spatial functions I,J gauge symmetries. Forderstood specific from examples, first consult principlesa where [ these pragmatic gauge explanation symmetries for comesumption why from, they appendix should exist.and let We will take as a well-motivated action should consist of Goldstonesbroken. as if We every will symmetry oftors refer the as to theory the were spontaneously GoldstonesGoldstones corresponding to spontaneously broken genera- stones. In particular, in every known case, unbroken Goldstones possess infinitely many derstood from a semi-classical perspective.pragmatic tool Semi-classically, the that density is matrixof used is the to a system. account purely forcorresponds Essentially our to every classical a (classical) ignorance highly micro-stateneously of chaotic in break the classical a field true thermal configuration, micro-state which statistical will ensemble in general sponta- ken symmetry, there is aare corresponding imposed). Goldstone However, at mode finite (unlesslong temperature, inverse distances there Higgs are and constraints other extendedthere excitations are time that in survive scales, fact over Goldstone-like whichor fields resemble for unbroken Goldstones. (unless inverse Our Higgs-type claim constraints is are that imposed). This claim is best un- non-equilibrium effective actions, which we term the 4.1 Goldstones atAt finite finite temperature temperature, SSBequilibrium occurs density when matrix. a Goldstone’s symmetry theorem generator tells fails us to that for commute every with spontaneously the bro- Before we are able towith state the the procedure method for constructingmatter of non-equilibrium effective cosets, systems actions we at mustexcitations finite first at temperature. finite understand temperature thefield It are IR content turns a field is out bit content established, different that of than we Goldstone condensed at will and zero outline Goldstone-like temperature. the procedure After the for using cosets to construct 4 The non-equilibrium coset construction JHEP07(2020)200 2 = . , A T (4.4) (4.5) (4.3) M = 1 s for for M s φ 6 g , , A T ) , φ construction. Nevertheless we will use ( , )) A s , I be the subgroup generated by ) have been promoted to dynamical i A φ e T ( . ) coset α 2.2 I ) are in fact the correct symmetries for τ M ) φ To this end, we will treat the coefficients ( ⊂ H ξ φ ( A 4.2 7 0 3. α s Since unbroken Goldstones enjoy a gauge + , iλ ). H iπ 2 I e , e M ) 5 φ µ ( φ φ ¯ P ( ( λ = 1 – 12 – ) s s ) and ( ], gauge symmetries are treated as redundancies and only φ g g ( 34 µ s 4.1 I,J → → iX ) and ) ) e I φ φ φ ( ( ( 3 and s s , = broken generators = unbroken translations = other unbroken generators ) = g g in the Maurer-Cartan form as gauge-fields that have the 2 M φ , µ α A ξ ( 0 1 ¯ τ s P T , g symmetry generators of the theory are as follows: H as well as under the gauge transformations = 0 G global M,N ]. The be the full symmetry group of the theory (including spacetime symmetries), 2 indicates on which leg of the SK contour the fields live. Each , 10 ) in the non-equilibrium coset construction. These fields serve to embed the G be the unbroken subgroup, and let φ ( = 1 ], gauge transformations are treated as if they are physical symmetries and generators for each µ s s 3. Parameterize an arbitrary group element by X 40 , , 2 We are interested in building-blocks for the non-equilibrium effective action that can We will construct our theory on the fluid worldvolume with coordinates When using the coset construction for theories with gauge symmetries, there are two main approaches. Once again, we use Since we are parameterizing the full symmetry group as opposed to merely the non-linearly realized H ⊂ G , that appear in the ordinary coset construction ( 12 6 7 5 1 µ , the term ‘coset construction’ for the sake ofIn linguistic [ continuity. gauge transformation appear inglobal the symmetry coset. generators In appear [ second in approach the as it coset. is Both far simpler. methods yield correct results, but we will use the for arbitrary spatial functions of unbroken generators of coset, it is not technically correct to call our construction a be constructed from the Maurer-Cartan one-formglobal and symmetry that group transform covariantly under the where transforms under the same globalx symmetry action. Notice thatfields the spacetime coordinates fluid worldvolume into the physical spacetime. let 0 Finally, let imposed). As a result,group in of the the coset theory constructionsymmetry we whereas with must broken Goldstone parameterize Goldstones modes. the doof Goldstones. full not, symmetry we Additionally, we must assume distinguishgenerators that between [ there the exist some two sorts types of unbroken translation a wide range of physical systems. 4.2 The methodAs of discussed cosets in thecorresponding previous to subsection, each at symmetry finite of temperature the there theory is (unless a inverse Goldstone Higgs-type mode constraints are will see in the following sections that ( JHEP07(2020)200 M A 2 (4.9) (4.6) (4.7) (4.8) (4.10) − B ), and to M A 1 . and covariant 4.5 N ν 2 G ≡ B E µν A aM η B M µ 1 E , and These objects transform A , T 8 M ν 0 ) h 1 A sM I , − 2 ∂ B A i · E T ( 1 ). The Maurer-Cartan one-form is + A rI − 0 . M
µ B 1 i 4.5 α ih , ν , sN all transform as scalars, whereas the transform as E τ + α E s − 0 µ sM A sM A sI A s π I 0 µν E B µ B ∂ B h η I I · ∇ 0 0 M M = µ A sM + ∂φ ∂φ H I T ∂φ ∂φ E µ · 3. To contract coordinate indices, we use the , and – 13 – ∇ transform as connections. Under the fluid dif- ¯ , P 1 = α s → → 2 , which transform covariantly under ( − 0 π , A sI α h s µ µ A sI sI -gauge transformations, the vierbeins, the covariant µ B sM π A sI B ∇ E A are the covariant derivatives of the broken Goldstones, µ sMN = 1 T iE , G ∇ α s ) are invariant under the global symmetry group 0 , I µ 2 s behave like gauge connections. π = → B g 0 µ E · s ∂ A , which transform covariantly. Finally, we have terms that g . 1 ∇ ∂φ A sM 0 T − 1 and A s M B B g ) are as follows: first, there are the building-blocks from the ; that is, A sI B ∂
). A 1 B I I T A 2 − 4.5 s ) φ I g B ( and φ ( M transform linearly, whereas + 0 A ξ µ s I 0 iλ A 1 A s E + e B B M ≡ 1 2 φ ) are the vierbeins, I = ≡ ) as follows. Under the φ ( M A rI µ sM 0 0 4.5 h B φ E After computing these building-blocks, inverse Higgs-type constraints can be imposed The building-blocks that transform covariantly under both the global symmetries and Notice that terms from log( 8 one set of Goldstonessubsequent in examples favor that of there derivatives are of three another kinds set of of inverseunder Higgs-type Goldstones. the constraints. gauge We transformations, will yetis they see that are only not in building-blocks used that asof arise building-blocks which from of is the the Maurer-Cartan provided effective one-form in action. are appendix permissible, Our claim the demonstration transform covariantly and we can contract coordinate indices with to remove extraneous Goldstoneis modes. that The we basic mayand idea set gauge behind to symmetries inverse zero Higgs as any constraints long objects as that the transform resulting covariantly under equations the can symmetries be algebraically solved for metrics Second, there arefreedom, new namely building-blocks thatinvolve involve combinations the of 1-and unbroken 2-fields. Goldstone Notice degrees that of take higher-order covariant derivatives we can use where where the gauge symmetries ( usual coset construction, namely spatial component of the vierbeins and the connections transform as derivatives, and where feomorphism transformations, where and certain components of under ( partial gauge-invariance defined by the second line of ( JHEP07(2020)200 0 . ¯ ¯ P P ⊃ ] 0 ¯ P -Goldstone contains an T, τ and a broken 0 . Then, it is -Goldstone to . 0 ¯ ¯ 0 P P τ H C.3 – . Further, suppose τ 3 are coordinate in- , for unbroken trans- , . Then we may set to C.2 , that is [ B 2 ⊃ ¯ ¯ P , P ] 1 0 ⊃ , ⊃ ] ] ¯ 0 P ¯ P , τ -Goldstone. ¯ P = 0 0 A, τ -Goldstone. τ, τ M -Goldstone in terms of derivatives τ . This gives an equation that can be ¯ , that is [ P τ -Goldstone, allowing the removal of the , that is [ . Because the set of unbroken generators , where ¯ P ¯ P B M ν ) 1 . These conditions give constraints that re- − 0 2 and ¯ . Suppose that under the dynamical KMS sym- P E – 14 – 0 ( . Then, we may also set to zero the components A ¯ P µ M µν ) and an unbroken spacetime translation generator ˜ and 1 A 9 E T and ¯ ( P → ¯ P in the direction of Suppose that at finite temperature, the commutator be- ≡ µ µν 0 ν Suppose that at finite temperature, the commutator between A E µ 3 are Lorentz indices. Then we may set to zero the components and the unbroken time-translation generator , A 2 τ , 1 , to zero. This gives a constraint that relates the building-blocks. However they do remove the unbroken Goldstones entirely from ¯ P = 0 -Goldstone, allowing the removal of the and some generators 10 τ ¯ P -Goldstone to derivatives of the do not belong to the same irreducible multiplet under µ, ν T -Goldstone. This allows the removal of the 0 covariant in the directions of in the directions of contains another unbroken generator ¯ τ P building-blocks and hence from the effective action; see appendices 0 µν τ µν ˜ A A . In this appendix, to keep things as concrete as possible, we perform an in-depth analysis of and -Goldstone. of late the T contains another unbroken spacetimeConsider translation the generator matrix dices and of metry transformation, zero the component of algebraically solved to yield anof expression the for the Unbroken inverse Higgs. tween an unbroken generator Thermal inverse Higgs. a broken generator unbroken spacetime translation generator C τ One might wonder if the aforementioned inverse Higgs constraints are the only possi- Notice that the ordinary inverse Higgs constraints only allow for the removal of certain There are the usual inverse Higgs constraints that exist for zero-temperature systems. To get a better understanding of why these newIt inverse turns Higgs out constraints that the can unbroken be inverse Higgs imposed, constraints do see not ap- necessarily remove the unbroken Gold- • • invariant 9 10 pendix inverse Higgs-type constraints for fluids without conserved charge. stones from the the imposed any time there iswhich can a be covariant object algebraically that solvedlevel when for of generality, one set it set to is hard of zero,constraints to Goldstones results are say in related in if to terms we an have commutator of equation been relationslation another. exhaustive. of generator the With However, form most this inverse [ Higgs ble such constraints. From a very general stand-point, an inverse Higgs constraint can be are possible; the rules are as follows: derivatives of the broken Goldstones by relating themwonder to if derivatives of there other arerelating broken Goldstones. other them One possibilities. to might derivativesconstraints Perhaps of that we allow unbroken us could Goldstones; to remove or remove broken unbroken perhaps Goldstones Goldstones. we by It could turns specify out that some both of these Suppose that the commutators betweengenerator an unbroken translation generator that consistent with symmetry transformations to setin the covariant derivative the of the direction of JHEP07(2020)200 is B ), the (4.11) describe , but such 4.11 α B are broken, π µ B ∇ 11 and unbroken, we have ] — can be thought A B 10 , unbroken. Every other ) 3 a A χ ( O ] + are unbroken, we have unbroken 2 χ B is broken and [ A EFT and S ) counts as higher order in the derivative 3 a A − to be broken yet χ ] ( 1 B O χ [ – 15 – describe the unbroken Goldstones. For the sake of EFT , and A S 0 2 B χ ] = ]. Thus, we will not study them here. For all other states − a 1 and , χ 13 χ , µ 0 r χ E 9 ≡ [ , 2 a χ EFT I ), 2 χ + 1 χ ( 1 2 ≡ r χ It should be noted that the primary reason for defining effective actions on the SK Once inverse Higgs constraints have been imposed and an effective action has been Strictly speaking, the thermal and unbroken inverse-Higgs constraints make the assumption that 11 studied in great detail inof [ matter, however, we will include brief discussions ofan the unbroken resulting translation equations generator. ofinverse Higgs motion. It constraints is will possible not to be conceive relevant of for more any constructions general in possibilities this for paper. section gives correct results,superfluids we at will reproduce finite theframids temperature. — known systems effective for Along actions whichof for only the as fluids. boosts fluids way, are In and we particular, spontaneously atgapped broken will finite [ and temperature, find the can boost therefore that Goldstonesthe automatically be finite-temperature become equations integrated out, of resulting motion in for the fluids ordinary fluid and EFT. superfluids Finally, from the EFT perspective have been dissipation. We leave this important work of computing higher-order terms for future study. 5 Fluids and superfluidsTo at demonstrate finite that temperature our non-equilibrium coset construction presented in the previous contour is to accountleading-order for physics dissipation. is When fullyTherefore, an captured the action by majority an factorizes of ordinary accordingfor the dissipative action to effective effects. and ( actions However, is our constructed cosetscription thus in construction for non-dissipative. this gives constructing a paper higher-order clear, terms, will almost which mechanical not if pre- included account in the EFT, will account for the broken Goldstones and simplicity, in the subsequent sections,at we leading will order in construct the EFTswith derivative for expansion. just various As one states a copy of result, of matter we the will fields construct ordinary whenever actions possible. where expansion. The onlynematic exceptions and to smectic C this phases rulewith of just that liquid one crystals. appear copy When in of this the this rule fields. paper holds, Thus, it are at allows the leading us order, to EFTs the work for building-blocks constructed, it isleading then order necessary in toworking the impose in derivative the the expansion subsequent examples) dynamical (whichbe the is to KMS only mandate effect the symmetry. that of only the the Often, effective order dynamical action KMS to at factorize symmetry which as will we will be combination of broken and unbroken,then however, we is have permitted. ordinary If inversethermal both Higgs inverse constraints; Higgs if constraints;inverse and Higgs if constraints. both assume Thus, that our all list such of constraints must inverse be Higgs related constraints to is commutor exhaustive relations. if we forms a closed algebra, it is impossible for JHEP07(2020)200 . In i (5.4) (5.5) (5.6) (5.7) (5.1) (5.2) (5.3) η as well as , meaning for boosts. 0 G P i 0 ij J for translations iδ µ ≡ P i ] = j K . ,K i , P C.1 i , J , i M . i = 0 , jk Ω J i ) ji )] M i φ , ( ) R + R i . j 1 )) , 0 ,
iθ ji i I − 2 i (Λ i t e J = 0, from which we find that φ ) i R Λ] E i ( K I j M X X ( i K µ φ 0 µ 0 ) ∂ M ( η M , . Because spatial rotations are unbroken, 1 φ i ∂ ∂ Λ ∂ Λ ). Notice that [ ξ µ ν ( µ jM 1 µ i − I iλ ) ν µ µ − 0 ). Therefore we may fix ] ] ∇ − 1 e + φ iη Λ i X ( · 1 e R E 0 i = K + [Λ ∂φ µ 4.5 ( M ) ∂X − ) ∂ λ P η [Λ µ φ φ φ M µ ) R ( – 16 – ν ( ( − ) [ i P = φ 1 ( g g iη i X 0 µ − M j ijk e µ M )  ). The unbroken generators are E tanh M → → E iX 1 ) and 1 2 i ∂ e I ) ) − iE 2 = [ η η 2.6 φ φ φ E = ( = = ν ( ( ( 0 = = ( µ i g g ) = M g µ η i M M φ ξ iM µ ( Ω E ) M g 1 ∇ ∂ 1 E and Λ ( − ij g ] i J ) φ ( for spatial rotations; the broken generators are i iθ . For a more detailed calculation, see appendix e jk , so we may set i k J η i transforms covariantly under both the global symmetry group P = [ η ijk i 0  ij ijk p 2 1 E is invertible, this is equivalent to R i In order to do this, we must recall that the field-content is doubled. Notice that ≡ ij ≡ 12 η ] = R i j J Now we can impose unbroken inverse Higgs constraints to remove the rotation Gold- At this level in the derivative expansion, it is not actually necessary to solve the unbroken inverse Higgs ,J i 12 P [ constraints, but we will doHiggs so just constraints to in demonstrate later that sections, it such can calculations be done. will be When imposing omitted. the unbroken inverse where stones. Since for arbitrary spatial functions that we may impose theparticular thermal inverse Higgs constraints,the allowing gauge the and romoval diffeomorphism of symmetries ( such that the action must be invariant under the transformations The resulting Maurer-Cartan one-form is where the vierbein, covariant derivative, and spin-connection are given by Consider a fluidgroup, without whose a algebra conserved isand charge; given the by ( symmetryTherefore, group the is most general just group the element Poincar´e is 5.1 Fluids JHEP07(2020)200 (5.8) (5.9) as an (5.11) (5.12) (5.10) r ~ θ ]. . 2 j and is given by µ ) 2 1 ) . This way it is clear t t 0 with no derivatives (Λ E i ( ν µ 2 1 η can be removed at the − can only appear in the r ∂X ∂X ~ θ = r iν ~ θ ) , Thus at finite temperature, 00 1 ν G N − 2 15 . ) ∂φ ∂X (Λ ]. Thus, we can interpret a finite- ) is sufficient to remove 00 µν ≡ 10 G η ( . ij µ 5.10 M r 0 , = 0, then we could use it as a covariant ~ M a X have been successfully removed from the i ∂X ∂φ 0 GP a covariant derivative. Then, performing ∂φ ~ ∂ X a E − = ~ θ ) implies that our action must be invariant 14 √ not ν N ~ φ ~ ∇ × ∇ × 5.4 4 E – 17 – = 0 to indicate time-like components of Lorentz vectors and d , = = 1 µν r µ has an energy gap. a 0 − r η , we replace the Lorentz index with a ~ θ Z ), we have ~ θ 0 i ~ θ 0 . µ M η . As a result, ( ∂ E ∂φ = 3.6 0 E , meaning that terms involving C.3 ≡ ) agrees exactly with the pull-pack fluid metrics of [ ··· EFT /∂φ r ·M·M instead of S ~ θ + t ), meaning that at linear order, T 5.11 MN -basis ( ∂ i I η = G φ M ( r, a µ λ ~ ⊃ √ i 0 Defining the metric by = E ) + 1 13 φ . − 2 ( for more detailed calculations. To get an intuitive sense of what is ν 0 r R ~ θ E · 1 µν C.2 → η R µ 0 represents the ordinary curl and ) E φ ( − r has one raised Lorentz index and one lowered coordinate index. Whenever we feel that there may ~ ~ θ ∇× p t 0 Notice that the above system has the same symmetry-breaking pattern as a framid Thus the only covariant building-block at leading order in the derivative expansion is E The metric tensor definedAt in finite ( temperature, the notion of an energy gap is ill-defined. Really, what we mean is that the To avoid ambiguity, instead of writing = 14 15 13 t 0 that be an ambiguity, weand will tensors. use propagator will die offaction takes in the space form and of an time ordinary exponentially action fast. with an But energy gap, since we the use leading-order the term hydrodynamic ‘energy gap’. temperature framid as a fluid.Higgs With constraint this is interpretation, as the meaning follows.building-block. of If the we thermal But did inverse notmust set exist in the action. Therefore except for the fact that it exists at finite temperature [ E we find that the leading-order action is a generic function of effective action in theindependent form degree of freedom.effective Thus, action, so no rotationalnon-linearized Goldstones level, remain. see appendix To see how But notice that theunder linearized version of ( from the 1,2-basis to the where the (classical) dynamical KMS transformation on both sides gives See appendix happening, let us expand the above equation to linear order in the fields. Transforming which gives JHEP07(2020)200 . , 0 ]. ¯ P ψ 4 0 – µQ ∂ 2 Only (5.17) (5.18) (5.13) (5.14) (5.15) (5.16) + = 0 16 . P t 0 i ) and the P = µE 5.3 0 . = 0 ¯ P + are the broken i ¯ P ¯ also transforms 0 i 0 P β 0 t 0 − J e E and exists at finite ≡ B and = tr by imposing thermal , / 0 i i 0 i Q J ¯ B P K η such that µQ 0 i M β . . + − Q Ω i i e i reflects a choice since any linear 0 J J and ) ) P = . φ φ + ( ( ) Q ρ µQ i i = Q 00 iθ iθ + 0 e e ¯ M 0 i i , P ,G P B , K K by imposing unbroken inverse Higgs Q ], but it can be shown that they are 0 i ) ) ) 9 i I φ φ t = M B ( ( θ φ + ( i i ( 0 ¯
P iη iη Q µE i λ e e e K Q Q GP − i ) ) · η φ φ − ) ψ ( ( µ φ √ iπ iπ M ( ∇ – 18 – φ e e ∂ g transforms as a scalar. Since 4 µ µ + ¯ ¯ d P P = 0 ) ) → µ φ φ B ¯ ( ( ) Z P M µ µ ). As before, we may remove φ B I ( = µ iX iX M φ g e e ( Q iE EFT λ ) are precisely the leading-order actions presented in [ ), and we may remove ) = ) = = S ). We are interested in constructing an action that is invariant φ φ are unbroken, so defining g . are the unbroken generators and ( ( φ 5.5 ( g g Q 5.17 M D jk π ∂ J 1 is unbroken. But it is the most natural choice given that this definition of − and ) + ijk g Q 0  φ P ( 1 2 t ]. The most general group element is ) and ( and = ). 0 13 i µX , 5.12 P J 5.7 ≡ 10 ) ) as well as the chemical shift gauge symmetry φ ( ), such a system must also have a conserved U(1) charge , and 5.4 i ψ P 2.6 Notice that the U(1) gauge field Now suppose that the fluid carries a conserved U(1) charge Notice that both = 16 i ¯ generators [ combination of allows us to express the equilibrium density matrix in the usual form 5.2 Superfluids Consider a superfluid attry ( finite temperature.P In addition to possessing Poincar´esymme- Notice that ( They appear to differ fromequivalent; see the appendix actions presented in [ inverse Higgs constraints ( constraints ( as a scalar, itThe is effective convenient action to at define lowest the order invariant in building-block derivatives is therefore under ( for arbitrary spatial function where the vierbein, covariantU(1) derivative, gauge and field spin-connection is given are by given by ( where boosts are spontaneously broken and the most general group element is The resulting Maurer-Cartan one-form is be noted, however, thatgap at may be sufficiently quite low small.we temperatures, are In the interested this exclusively case framid in it Goldstones’ the would energy very only deep be IR appropriatechemical behavior to potential. of integrate Then the them the out system. unbroken if translations are framid Goldstones necessarily develop a gap and can therefore be integrated out. It should JHEP07(2020)200 = ). π µ (5.24) (5.20) (5.22) (5.23) (5.25) (5.19) (5.21) ∇ 2.14 µ 0 , and the ν E µ ] i ], but can be K ) 13 φ ( i iη e , i J ) = [ . These constraints give i M µ φ ( Ω i , giving our first symmetry- X ν . µ µ , ) M + , ∂ jk − 00
i ji . )] ψ ≡ K R ,G ψ ψ R N i , j 0 ψ. 0 0 µ η µ M M M (Λ , from which we find that N µ e ∂ ∂ ψ∂ Λ] µδ ν 0 ψ M ∇ Y,B M i M 0 M ψ∂ ∂ M ( E ) ) M . − , ∂ 1 ∂ + 1 1 ∂ µ M 0 1 µν − ψ − − ν ∂ η ] − M µ e e Λ ∂ψ MN GP ∂φ ) ( ( µ M 0 πQ 1 , where R 1 ∂ [Λ G µ − − E MN − ψ − ≡ [Λ − M µ M µ e ∇ R √ – 19 – G ν ) ) [ 0 M = ( = ), meaning that it only depends on the physical- φ 1 1 ∂ p − 1 + . Further, in the zero-temperature limit, the effec- φ B 4 Y X . To remove the boost Goldstones, impose inverse η − − ijk ( 00 d M ρ µ  ij p = µ M ) D E E ¯ ] ), reproducing the standard superfluid EFT ( G = P 1 2 ∂ 1 k Z π X ≡ J µ − t ) tanh t = = ( = ( = e µ M ] = φ 2.13 ∇ ( i Y ( i i η π R η k µ η M i M ρ iE µ ), the Lorentz boost matrix is Λ µ iθ Ω E φ e EFT ∇ = ∇ ( S = Λ g π π M i ) = [ )+ ∂ φ , namely ( φ 1 ∇ ( ( − Y 0 ij g R µX ≡ ) tells us that [Λ ) φ ( , yielding our second symmetry-invariant building-block 5.21 ψ ψ 0 ∂ ] µ Y − This effective action looksshown somewhat to different be from equivalent; the seetive one appendix action presented loses in its [ spacetime dependence version on of in the previous section,rotation we Goldstones. can Thus, impose the effective unbrokenis, action inverse describing to a Higgs lowest superfluid constraints order at in to finite temperature derivatives, remove the And just as in the fluid case, we have as our third symmetry-invariant building-block. Finally, following the same calculation as Equation ( [1 Using this relation, we haveinvariant that building-block such that rotation matrix is Higgs constraints 0 = a relation between the boost Goldstones and the U(1) Goldstones where And the resulting Maurer-Cartan one-form is JHEP07(2020)200 . is µ are ~π and X (6.3) (6.4) (6.5) (6.1) (6.2) , and 0 jk M ν ~ ∇ × J P ∂ µ ] i ijk ≡ = K  ) 1 2 0 φ µ M ¯ ( P e . Thus i = µ iη i ∇ e [ J Let ≡ , 17 )
i φ , and J ( i i ν 0 θ µ . J , µ i J jk ∇ ) = φ )] i ( ij + i ) R i K 1 iθ , e − K (Λ i i , , i a j ( K ji η Q M ) j µ ψ ∂ φ R = 0, respectively. ψ ( 1 j N i , ∇ 0 − ~π i ∂ µ M iη i δ Λ + ∂ e Λ] 1 i ψ i M t − M − Q ~ ∇× ). We therefore have our first set of invariant , ) Q M ) i ∂ i 1 µ R φ ∂ 1 [ ( ψ ν π − i ] − µ e . To remove the boost and rotation Goldstones, 5.11 M iπ ijk ( – 20 – R MN ∂ [Λ  ∇ e ij ] − µ G [Λ ¯ M µ M µ M µ k + P ν ) ) ) J ) = 0 and = ) 1 1 1 ≡ µ φ i φ X ( ¯ η − − − ( P π µ ij k t M E E E iθ Y ∂ iX µ ∇ M e e ), the Lorentz boost matrix is Λ [ tanh = ( = = ( = ( φ iE ( j i i i ≡ i η θ µ η η M ) = π = ) π µ µ 3 that commute with each other. We then take µ φ . Imposing the second inverse Higgs constraint tells us that E φ , g ( j ( ]. The most general group element is ∇ ∇ ∇ 2 g ), such a system must also have internal translation symmetry ψ M ij ) + , 10 ∂ , where φ M R 1 ( 2.6 ij ∂ i − δ = 1 g M i X i ) − 1 ≡ − ij is the inverse of the metric ( ) ) e for . 2 ( to be the unbroken generators such that φ i / ~π ( 1 i i MN Q ≡ should be thought of as the spatial components of the covariant derivative Q Y ψ G ~ ∇ ij + a = ( i ) P j is truly symmetry-invariant and not merely covariant. We have the additional invariant the curl of Here, π = i ( 17 ij i ¯ such that building-blocks. Notice that becauseY there is no notion of unbroken rotational symmetry, not where ∇ the rotation matrix is impose inverse Higgs constraints Then by imposing the first inverse Higgs constraint, we find that such that where P the broken generators [ And the resulting Maurer-Cartan one-form is 6.1 Solids Consider a chargelessPoincar´esymmetry crystalline ( solid atgenerators finite temperature. In addition to possessing Now that we haveknown established that results, the we non-equilibrium turnphysical coset systems our construction for attention can which the to reproduce solids non-equilibrium and the effective supersolids. construction actions are In ofEFTs as this novel for of section, these EFTs. yet we unknown will states are construct of The the matter. simplest leading-order non-equilibrium 6 Solids and supersolids at finite temperature JHEP07(2020)200 . ψ 0 ∂ (6.9) (6.6) (6.7) (6.8) (6.13) (6.12) (6.10) (6.11) ≡ 0 = 0. The B µν T ν ). As when we µ 0 ∂ φ ( ∂φ , ∂X π ν u 00 , as our final building- µ . ) + G u 00 1 Q φ ) − ( G φ . + 0 ( ) √ ), . iπ X µν ) 00 e = η i 00 J µ ,G ) 5.11 ≡ 0 φ ) = ( ,G ) with the addition of a term in- i i φ µν ( ,B iθ j i e ψ ,Z 6.2 ∆ i ψ , ij ν ,Z K j ) ∂ Y ij for i ψ φ . ( ( , namely µ i i ψ Y 0 ij ∂ ( P, u t µ M iη i ∂ e Y ∂ψ ∂φ ψ − i GP , ij µE µ Q − GP ) ≡ ∂ µν 00 – 21 – φ ∂P − i r . The most general group element is − ( √ ∂Y i i ≡ ∂P φ Z ψ , from which we extract our next set of invariant √ ∂G + iπ 4 Q j = ij φ e d M 00 ψ 4 y µ + µν ∂ 0 ¯ d µν P ]. G i Z ) ∂ ∆ r 2 ) P φ = p 12 Z ( = − ij µ . We see therefore that the zero-temperature EFT only ) = + 0 M = 1 i = iX µ B ¯ − be the corresponding generator. Then, the unbroken transla- e P u EFT Y /∂φ µ S Q ( EFT ) S φ ) = εu and ( − φ µ ( = ij g δ µQ µν ∂X , we find the conservation of the stress-energy tensor µ P, ε + T = ( 0 X = i P π p ], we compute the equations of motion. If we vary the chargeless action µ = ∇ 0 45 ¯ µ 0 – P E 41 To connect with other formulations of the finite-temperature hydrodynamics of Often, solids possess an additional unbroken U(1) symmetry corresponding to particle- are the thermodynamic pressure, energy density, and four-velocity, respectively and with respect to stress-energy tensor is given by where which agrees with the results of [ solids [ As a consistency check, letall us dependence take the on zero-temperaturedepends limit. on This the amounts physical-spacetime to version removing of volving the U(1) gauge-field added a conserved charge toThus the fluids, effective we action now describingtemperature have a the is, crystalline additional to solid building-block lowest with order conserved U(1) in charge derivatives, at finite number conservarion. Let tions are The corresponding Maurer-Cartan one-form is just ( lowest order in derivatives, building-blocks Lastly, we have the usual time-like pieceblock. of the Thus fluid metric the ( effective action describing a crystalline solid at finite temperature is, to objects JHEP07(2020)200 ) 0 /β ) = 2 6.14 φ / (6.16) (6.14) (6.18) (6.15) (6.17) ( 1 ν are the − µ ) , µ i 00 Q ψ G + − µ P = ]. The most general , . µ 10 i  ¯ P . ) νλ ∂P ∂Z ν J . µ νλ u µν ↔ θ J , ) µ + µ , φ ( νλ ( ν µ ∇ ij δ 1 2 µν Λ] − θ ∂P − ν i 2 M ∂Y + ν e ψ j ∂ ν 1 µ ψ ψ M , − Q Q µ ) ∂ M µ ν ∂ φ ∂ [Λ ν ( π M ρ 2 µ ) µ Λ M µ M µ 1 ν . Just like the additional equations of motion, ) ) iπ ≡ ]. The additional equations of motion ( ∇ i − 1 1 e X e µ Z − − – 22 – iµ 46 ( + ¯ P M µ ) E E µ ρ ∂ φ are the broken generators [ ¯ ( P µ = = ( = (  µν = Λ iX ν µ J ) and the Lorentz transformation matrix is Λ M µ M e νλ ] π φ ν θ µ E ( is the equilibrium inverse temperature. However, in our iE µ π ,J µ ∇ µ 0 ) = [ and ∇ π = β φ ∇ = 0 ( g µ g Q M ) + iµ ∂ φ J 0 = 1 ( µ µ − ∂ g . To remove the Lorentz Goldstones, impose inverse Higgs constraints X 3 that commute with each other such that ν , µ ] 2 k , J 1 ) = ) ), such a system must also have internal translation symmetry generators , φ φ ( ( k µ 2.6 iθ ψ = 0 e respectively, where · µ i ij K ) Y φ for ( i µ iη e such that [ where group element is And the resulting Maurer-Cartan one-form is 6.2 Supersolids Consider an anisotropic supersolid at finitesymmetry temperature. ( In addition to possessingQ Poincar´e unbroken generators and formalism, there is an additionalthis quantity quantity owes its existence towhen the presence the of phonon second gas sound. flowsdetermine In with how particular, respect it to to is remove the non-zero second solid sound lattice. from We solid leave EFTs. it as further work to like longitudinal phonon canaccount propagate for [ thephonon fact gas. that the Further,on solid in the degrees temperature the of and standardand a freedom formulation, symmetric can the tensor, move thermoelastic which independently variables we of depend can the identify with ( second sound modes,ordinary analogous sound to modes those ofof found solids the consist lattice. in of finite-temperature However,lattice transverse if structure superfluids. and a and longitudinal Umklapp finite-temperature scatteringof The vibrational crystalline events solid modes thermalized can has be solid a ignored, phonons sufficiently then pristine that there will behave be as a bath a gas through which an additional fluid- However, we have the additional equations of motion that come from varying These new equations indicateaffecting that the the stress-energy solid tensor. degrees The of physical freedom interpretation can is be that excited our without solid exhibits is the elastic stress-tensor. So far, our theory appears to agree with standard results. JHEP07(2020)200 ). be ), we = 0. 5.17 (6.21) (6.23) (6.19) (6.20) (6.22) ·M µν T 6.14 J ν ) or ( ∂ 5.12 ) and ( 6.11 = 0 and µν T ν . ∂ ) 00 . ,G µ µν ν Y ψ ,Z 3 in equations ( . Our next set of invariant building- , N , = ν ν µν ∂ 2 ψ µ , ψ . Y 0 λν ρ 1 , namely ( ψ µ ) 0 ∂ , ∂ 2 ) µ µν M / µ ν ∂φ 1 ∂ψ ∂ ) ψ Y 1 GP = 0 ρ Y − ( ≡ ∂ − µ – 23 – MN have been suppressed. Using this result, we have Y µ ρλ √ ≡ ( G η Z φ µν for 4 − µν µρ = η d ) y µ µ ν building-block. Thus the effective action describing a 2 ]. δ / µν ψ Z 1 00 12 Y Y = = ( G ( µ , where , the inverse Higgs constraints merely require that Λ . Thus at zero temperature, the effective action can only π ν 0 EFT 3 with µν ν , (a) spontaneously breaks all translations and rotations, corre- ψ , S η ∇ 2 1 M ν 0 , /∂φ − ∂ ) E ). As the temperature rises, symmetries are restored until at high φ µν M µ ( = 1 ) ) for a graphical representation of various phases of liquid crystals. At µ 2 1 6.9 i / − 1 1 e ∂X ( Y for ≡ i ) or ( = ( ψ ) µν 6.7 ν π M µ ( ∇ Notice that in figure Finally, the equations of motion are very similar to those of solids. In particular if we are spontaneously broken; theThe corresponding aim of EFT this is sectionbreak will therefore be the given to ISO(3) construct by group EFTs ( of for spatial the intermediate translations phases and that rotations. sponding partially to crystalline solid phase. Parenthetically, the profile of smectic liquid crystal in liquid crystal phases: nematicand liquid C. crystals See and figure low smectic temperatures, liquid the system crystals exists inall in phases the spatial crystalline A, translational solid B, phases,given and spontaneously by rotational breaking ( symmetries;temperatures, the the corresponding system EFT exists is in fluid therefore phase in which no symmetries other than boosts 7 Liquid crystals atWe finite now temperature turn our attentionnamely to those the of construction liquidthe crystals. of sake EFTs There of for are brevity more myriad (and exotic distinct the states phases reader’s of of patience) liquid matter, we crystals, will so focus for on four of the most common which agrees with the results of [ replace the find the finite-temperature supersolid equations of motion are As in thedependence solid on EFT case,depend taking on the the physical-spacetime zero-temperature version of limit amounts to removing all And finally, we have thecrystalline usual supersolid at finite temperature is, to lowest order in derivatives, Additionally, we have blocks is therefore a symmetric matrix, where factorsthat of are our first set of symmetry-invariant building-blocks and we use the convention that Letting JHEP07(2020)200 . We therefore see j ν δ , pointing parallel to i µ . To extend this order ~n δ ~n ij Q → − ≡ ~n µν Q , the long axes of the molecules ~n spatial rotations whereas (c) does not symmetry all 2 Z – 24 – . Therefore, it is common to use the order parameter ~n − ]. However, for any choice of 48 , which is invariant under the ij δ 1 3 − j . The figure above depicts the microscopic appearance of various phases of liquid crystal n i n ≡ ij the long axes of theis molecules. broken As to a SO(2) resultcould be the [ SO(3) equally well symmetry specified groupQ by of spatial rotations parameter to the relativistic case, it is natural to define Liquid crystals in nematic phaseof are an composed ordinary of oblong fluid, moleculesture; bounce that, however, around like on chaotically the average molecules and theBecause therefore long cannot no axes form lattice of a structure thealigned lattice molecules can long struc- align, axes form, thereby of spacetime breaking theorder translations isotropy. molecules parameter remain spontaneously associated break unbroken, with rotations. but broken Non-relativistically, the rotations the is a unit vector liquid crystal phase. Finally,rotations (e) and does is not therefore spontaneously in break isotropic any fluid spatial phase. translations7.1 or Nematic liquid crystals neously break translations intherefore the both vertical exist direction,states in but of not smectic matter is in liquid that thebreak crystal (b) rotations horizontal spontaneously about phase. directions; breaks the verticalA. direction; The thus Phase difference (b) (d) depicts between does phasespontaneously C these not break and spontaneously two rotations (c) break about depicts spatial phase the translations horizontal axes, in meaning any that direction, it but exists does in nematic i.e. fluid phase.phase Notice spontaneously that breaks (a) fewer and spontaneously few breaks symmetries the until most (e) symmetries only breaks andphase boosts. each B subsequent appears almostlayers identical of to molecules that are of allowed the to solid; slide however past in each phase other. B, the Phases horizontal (b) and (c) both sponta- Figure 1 in order from lowest tocrystal highest in temperature. phase C, The (c) phases smectic are liquid (a) crystal crystalline in phase solid, A, (b) (d) smectic nematic liquid liquid crystal, and (e) isotropic, JHEP07(2020)200 (7.5) (7.6) (7.7) (7.1) (7.2) (7.3) (7.4) sym- 2 Z are the bro- i and let indices K z , = ˆ 3 and J i ~n A h 3 sM , . Define the vector J . 3 3 s Ω 3 θ i B J ) )] φ + s ( 3
s R i iθ s e K . i s , , A (Λ , η J M j ji s , ) ) µ M ji s i t s s φ 1 R , ∂ ( ∇ R = 0. At this order in the derivative j 1 − 2 X X A s j 12 . We can impose the thermal inverse 0 0 − s 0 SO(2) sµ 3 a E iθ + ] )] ij ∂ ∂ ( e Λ s ] s Λ A i A × 1 3 − Λ i M µ s R 0 J J K , . Additionally, we can impose the unbroken − s 4 ) ) s 3 s µ A M 1 s φ = R R 00 iθ are invertible. Then, as in the ordinary fluid ν ( θ [ ∂φ ∂ s ] ∂X E (Λ i s e s 3 1 µ s ( η G ij s A → iη – 25 – s − M ∇ R J = e AB − R ∂ ijk s A s µ  [Λ 0  1 + i P s √ iθ 1 2 [Λ − s ) µ M µ M tanh e µ E φ ν s ) ) [ Λ ( = i s s P 1 1 1 ≡ µ s X η η 0 − − s s − s ≡ t s k a iX M ISO(1,3) 0 = E E are the unbroken generators and R µ E e sM A ∂ ij s . Without loss of generality choose 3 ~n R iE = = ( = ( = [ J ) = i s = 0 because A s = φ η θ µ i sM 3 sM ( s µ → − µ s and and g Ω E sµ g ∇ are broken so we may not impose such constraints. We therefore ∇ ~n ν µ M Λ µ ∂ P µ s ] A 1 i J X − s K 0 . Once again, are not to be confused with the indices indicating unbroken symmetry generators that i s g i s ∂ iη η e Then i s [ η A, B 18 ≡ p represents spacetime translations. We also have the unbroken discrete 2. ν ≡ , µ s 4 s R η = 1 In the case of ordinary fluids, we had the additional unbroken inverse Higgs constraints = 0. But now The indices 18 A a brevity we will not. A we used in previous sections. The unbroken inverse Higgsexpansion constraint it is is then not necessary to solve this inverse Higgs constraint, so in the interest of case, we have where inverse Higgs constraint, which removes the rotation Goldstone where Λ Higgs constraints which reduce to The resulting Maurer-Cartan one-forms are such that ken generators. It willthe turn leading-order out action that forelements we by this need phase to use of two matter. copies of Parameterize the the fields most to general construct group the Poincar´egroup is where metry associated with A, B that boosts are spontaneously broken as well. Then, relativistically, the SSB pattern of JHEP07(2020)200 A r θ ∇ (7.8) (7.9) (7.13) (7.11) (7.12) (7.14) (7.10) . For s ~n , 2 . )) 2 i s ) → − 2 ~n  A s , ) s
θ  , A s k r ~n 2 A θ ) 00 n ~ ∇× 3 19 s ∇ k ( A a a . Next, define ∇ )( A A a × X to count at the same i )( s contain mixtures of 1- A 00 )( ∂ s ~n A s 00 00 s θ G r − 0 ( , but we will see that the G = ( not i a ∂ 2 G ( 2 2 ( ) 3 ) K X ,G A s k K A s and hence is the usual material M A s θ + ∂ θ t θ 0 0 + k r i 3 2 3 β . φ ) ∇ n . ] ∇ ∇ A r ( B -basis, the contribution to the non- + s θ 1 2 θ mix A a ∇ I A , [ − 0 r, a A 2 β i a + ∇ A r )) . n indicate in which direction the elongated Θ 2 θ s )( i A r S , ~n θ 3 ∇  00 i s ) A s + s
− θ j r R 00 G A a 1 ~ ∇× n r B ( – 26 – ( j S r A 1 ≡ · G ∇ ˙ s ( Θ X K i s = i -invariant combinations of the above terms. The ~n n 2 ∂ M → Z ) + − = ( − EFT A a ×  2 i 00 r I r ) s A ˙ , X A s G G j θ A r ( . Note that our EFT is invariant under − ∂ θ r A j r P ~n p ∇ 2 n ∇ h ( φ 4 1 2 2 1 . Thus, at leading order in the derivative expansion, the effective d 0 , s ν s − , where 2 Z ) 2 G E i r s ˙ S n − µν ~n = ,  involves a time-like derivative of η − A a p 0 A ~ r ∇· µ s 1 = φ A θ mix in the derivative expansion. In the 4 S E I A a ∇ d = ( A s A ≡ 2 θ Z ) A r 2 i ] is the equilibrium inverse temperature and the relationship between the coeffi- θ 00 ∂ B s s = 0 θ ∇ s β G A [ S To compare the above action built with the non-equilibrium coset construction to more ∇ Notice that ( 19 where the dot (˙) indicates differentiationderivative with when respect to acting on limit, we have and standard results, let themolecules unit of vector the nematic are oriented. Then, working in the classical, non-relativistic where cients of the two termshigher-order is corrections, fixed the by full the non-equilibrium (classical) effective dynamical action is KMS given symmetry. by Then, up to Notice that we couldnematic have degrees included of a freedomorder term are as involving diffusive, ( soequilibrium we effective will action consider that contains mixtures of 1-and 2-fields is where action is constructedcontribution from to SO(2) the non-equilibriumand effective 2-fields action is that does implicitly as follows. Under a dynamical KMS transformation, we have The full set of covariant building-blocks at this order in derivatives is have a covariant building-block that mixes 1-and 2-fields, namely JHEP07(2020)200 3 , 2 , (7.16) (7.17) (7.15) are the = 1 3 direction. direction. i Q , z z  2 for )) i . Further, the . ~n K , and A i mix K ) I K ~ ∇× φ ( ( , A A , and × J iη ~n . e M ( ) 3 3 , J φ ) ( K that is spontaneously broken φ P π ( is unbroken. Throughout this 3 direction exists, it is necessary + 3 2 3 Q z iθ ) + )) + Q φ A ~n ( + J 20 ) 0 3 φ ) except ( ~ P ∇× A so that the inverse Higgs constraints are easier ( µX · 3 are evaluated on the equilibrium value iθ ≡ g , ~n 7.12 + ≡ 3 2 ( 3 be the corresponding unbroken Goldstone. ¯ , 2 ) P . K ) φ K π ( φ ( = 1 + 3 2 – 27 – i , we have the additional invariant building-block ) iη e ~n 3 . Let a priori for 6.1 . Q can be interpreted as the relaxation rate times the 0 ~ i Q ∇· ) ( φ B K ( 1 F as proportional to the standard splay, twist, and bend 3 , ψ K 1 0 , the dispersion relation for the nematic degrees of freedom iπ are the unbroken generators and M  3 e A r x and and 3 K ∂ψ µ ∂φ θ 3 ) because of the inclusion of the term ¯ 2 to indicate spatial indices perpendicular to the ˆ J P d , ) 00 M φ ≡ — and therefore the resulting diffusive dispersion relation — is ( G Z and section 0 µ 4.11 = 1 2 1 , and B iX 2 , and mix 5.1 , indicating diffusion, which is an intrinsically dissipative effect. It e 3 I 2 K Q F ≡ , A, B ik 3 µ 1 ) = δ K φ ∝ , ( + ]. We have thus recovered the standard nematic equations of motion. g F ω µ δ~n δ 47 P 1 M = − to be the spontaneously broken translation generator orthogonal to the layers. To µ ¯ P . We therefore see that 3 = ˙ P 00 ~n Finally, we often expect particle number to be conserved, requiring the inclusion of Notice that, linearizing in We use a somewhat non-standard parameterization of G 20 broken generators. The most general group element is to solve. such that the diagonal subgroupsection, generated we by will use 7.2.1 Phase A In phase A, the longThus, axes of the molecules are, on the average, aligned with the ˆ Smectic liquid crystals are composedone spatial of direction, oblong the molecules liquid thattake crystal form has layers a such periodic that structure.ensure along that Without some loss notion of of generality, unbrokento translations introduce along an the internal translation ˆ symmetry generated by As a result, theare effective now action functions is of both identical to ( 7.2 Smectic liquid crystals an additional unbroken U(1)Then charge just as in section is worth noting thatfactorized this according dissipation to owesnecessity entirely ( of to including the factan that automatic the consequence action ofnot cannot the need be to commutation assume relations diffusive of behavior the Poincar´ealgebra; we did of free energy, and respectively [ is of the form just on the nematicGoldstones degrees can be of written freedom. in the Then, form the equations of motion for the rotation where it is understood that simplicity, assume that the nematic and hydrodynamic modes decouple so we can focus JHEP07(2020)200 (7.23) (7.24) (7.26) (7.21) (7.22) (7.25) (7.18) (7.19) (7.20) . Begin ν µ ] 3 J 3 , iθ 3 + J A J 3 M A and tell us that Ω i iθ , 3 + i η 3 + 2 ) K
3 i 3 3 π B iη and K 3 i e [ ∂ . η A ( ) , Λ)] µ θ i 3 1 . Now impose thermal inverse ≡ L 0 00 , ∇ ( ν N ν M − t A . µ M E , + 3 Λ)] µ µ − A 3 L ∂ L ψ L A µ µν 1 2 L ( ψ µ ) , η J µ N − , Y,Z,G 3 N Λ] M ∂ 3 µ A ( 12 L X µ M and X π ∂ 3 δ ∂ L θ 1 0 . 0 3 [ 1 A E ν µ ψ ∂ − 3 0 ∂ µ − ψ ∂ Λ)] µ − 0 ] ∇ ( M GP , 3 = [Λ − L L 1 M A ∂φ ∂ψ ∂ µ ( 3 1 ∂φ − ψ + ∂ ∂X ν K = − M M 3 √ A ≡ M AB – 28 – Λ] ∂ MN = MN ⊥ φ ∂  [Λ iη − Q 1 MN 4 L η Z = 0, which remove e 3 G [ 2 G A − [ 0 M µ M µ M µ d G ) π ν 3 ) ) ) L 3 E p µ 1 1 1 π ≡ 1 ≡ Z X π − − − . And once again, solving these unbroken inverse Higgs A ∇ − ( ˙ ν tanh = 3 M 0 E E E µ = Y ∇ θ 3 0 = ∂ + A ⊥ M π η η µ = given by h 3 = = ( = ( = ( = [Λ ¯ P 1 2 3 , the second of which is just EFT i 3 ∇ A A 3 µ π η S M 3 M η π − θ t π µ µ M µ Ω E µ µ ∇ = ∇ ∇ iE ∇ ∇ . This allows us to construct our final two building-blocks µ 0 (2) = A E η L g and we have Λ A 3 M η ∂ π and 1 p + ν 0 − 3 g E X ≡ µν η ≡ µ ⊥ 0 3 η E ψ = 00 To compare with knownfields. results, For simplicity, let we will us neglectphysical the now spacetime, fluid the degrees expand quadratic of Lagrangian freedom. this describing Then, the EFT transforming behavior to of to the the quadratic smectic mode order is in the Thus, the leading-order action for smectic liquid crystals in phase-A is where G where the metric isHiggs given constrains to as remove usual by Then, we have yielding the first building-block by imposing unbroken inverse Higgsto constraints; remove the as rotation in Goldstone theconstraints is previous not subsection, necessary at they this serve inverse order Higgs in the constraints derivative expansion. Next, impose ordinary where such that The resulting Maurer-Cartan one-form is JHEP07(2020)200 . z ), - y 6.1 7.16 (7.29) (7.27) (7.28) , which and 3 a z A - x , 33 . ) — or if it contains ,Y 2 ) 6.7 6.1 23 , we find that the effective Y ( − 7.2.1 33 , 3 Y in the combinations π , 2 22 3 ) direction to be perpendicular to the ij . Finally, the equations of motion are ). The only difference is that to allow ∂ 3 i Y . Physically, these symmetries indicate 3 0 Y ψ z 3 Z ( 6.9 ≡ . M M ψ A 0 2 f B + plane independently without changing the 3 + is now spontaneously broken. Thus the only ) with five invariant additions. First, we have y π , b - A 3 2 2 A ). The addition of the building-block x – 29 – ) J ψ ∂ = 0 in the solid equations of motion in section 7.25 13 1 0 7.7 2 → Y M M ψ ( A = − ψ = . Further, this Goldstone cannot be removed with inverse 3 1 3 33 ]. For readers interested in the full equations of motion, ¨ = 3 component of π ψ θ Y . The resulting equations of motion, i 3 49 11 , M Y 45 ≡ 3 are now merely labels and do not need to be contracted in any ) may now depend on , -component of ( 1 is an arbitrary function of 2 z , and , A 1 7.25 denoted by 1 f , , b 3 M J ij direction, meaning that , in ( 0 = 0 Y z P M 2 and µ , det ]. As a result, the effective action is almost identical to ( ≡ = 1 , which is the ˆ b 45 3 a A A , where Going through a similar procedure to the one in section Finally, if particle number is conserved, we include the additional building-block ( 3 θ µ ∇ particular way since thehave entire Lorentz group ismixes 1-and spontaneously 2-fields, broken. means thatthe Additionally, the difference we non-equilibrium of effective ordinary action actions. cannot factorize into align with the ˆ difference between phases Abroken and generator C is thatHiggs-type constraints. phase C has a Goldstone associatedaction with has the the same building-blocks as ( and it can only dependjust on a the special case of the solid equations7.2.3 of motion given Phase in C section Phase C is much like phase A except now the long axes of the molecules do not on average macroscopic state of themean that system. the effective At action the can level only depend of on the EFT, these additional symmetries the layers to slide past each other, we must impose the symmetries where that we can translate each layer in the around freely without forminga a given lattice layer structure. arestacked In locked layers, phase then in B, phase place. B however,shears the is [ If essentially molecules a we in solid takean that the unbroken cannot U(1) ˆ sustain charge uniform it is almost identical to ( meaning that 7.2.2 Phase B Smectic liquid crystals in phasecan B, slide like past in each phase other. A, In consist phase of A, vertically stacked the layers molecules that within a given layer are able to move agree with the resultsthey of can be [ obtained by setting for constants JHEP07(2020)200 3 θ is, are 3 θ 3 (7.30) (7.31) (7.33) (7.34) (7.32) π ). Then and 7.16 µ , . X   3 contain mixtures s 2 ) θ j 3 a A ∇ not 3 s )( θ i 00 r ∇ ) G ( are merely labels. 00 0 s j M ,G is symmetric under exchange of 0 s i and β ), except the coefficient functions , as in the nematic phase, leads to . ij . i 3 r ,Z + θ s M mix mix . Y 3 a 7.33 I ∇ I 3 ( 0 to be the same order in the derivative A θ ij 3 r β j + 3 θ ∂ 2 Θ θ i M i 2 ∇ i S ∂ 1 2 ) ∂ 0 + ij − 00 3 a r 1 M ) + M – 30 – A S G and 00 . ( Θ s = 0 0 = 3 3 θ B ]. ˙ M → ,G θ 0 . Then, the equations of motion for s ∂ − 45 3 a 3 EFT  I π A ,Z r s G Y are summed over and ( − and P p i, j  , where µ φ 2 s 4 X S d G is purely for notational convenience as − − Z j 1 p S = φ 4 and may now depend on implicitly as follows. Under a dynamical KMS transformation, we have d i mix 0 3 r I Z θ M ∇ Just as in the case of nematic phase, we will see that the rotation Goldstone, = -basis, the contribution to the non-equilibrium effective action that contains s 21 S r, a . , and j ij It is worth noting that the presence of the term If particle number is conserved, we have the additional building-block ( The full equations of motion for smectic C are quite complicated and therefore not Define The contribution to the non-equilibrium effective action that does The sum over M is diffusive, so we will consider , agreeing with standard results [ 21 , and 3 3 a diffusive dispersion relationof and hence liquid dissipation. crystal are Thusdissipation nematic the at and leading only smectic order two C in states the phases of derivative expansion. matter considered in this paper that exhibit Thus, smectic C phase looksθ just like smectic A phase with the addition of a diffusive mode terribly enlightening. Thus,decouple for from the those sake of identical to of those simplicity of assume theignoring smectic that the A complications the phase. of dynamics hydrodynamical And fluctuations, of the linearized equation of motion for the non-equilibrium effective actionP is identical to ( As a result, up to higher-order corrections, the full non-equilibrium effective action is In the mixtures of 1-and 2-fields is i θ expansion. of 1-and 2-fields is such that repeated indices JHEP07(2020)200 By 22 – 31 – ]. We leave it as future work to construct EFTs for 46 ] was to treat these infinitely many symmetries as if they were 12 ] for a general discussion of how gauge symmetries can be treated as genuine symmetries ], thereby circumventing the need to introduce infinitely many symmetry 40 34 ] can be used to construct pseudo-Goldstone EFTs for spontaneously broken ] only constructed actions to leading order in the derivative expansion, it was not actually 33 12 The non-equilibrium coset construction admits generalizations and applications in The approach of [ Since [ 22 necessary to introduce allhigher of orders the using infinitely theirGoldstones; many method see symmetry [ would generators. requirein the However, the extending addition coset to of construction. arbitrarily infinitely many symmetry generators and liquid crystal phases exhibitof second matter, sound. it While isthese second states exceedingly sound of rare matter can without [ existexcitations second in sound. that such Fourth, survive some phases over condensedinterpreted long matter as distances systems Goldstone have and modes; extended forextending time example, the scales non-equilibrium such coset but modes construction exist that to near cannot allow critical for be couplings points. between Thus, Goldstone sented in [ approximate symmetries. Itstruction will to be cases for of which approximate interestthere symmetries to exist. ought to extend By the be the philosophytries associated of non-equilibrium this are pseudo-Goldstones coset paper, spontaneously whether con- or broken. not Third, the approximate the symme- EFTs we constructed for solids and smectic many directions. First, in theto interest lowest of order simplicity, in the the EFTssmectic in derivative C expansion. this phases paper As of were liquid aWe computed crystal result, leave admit all it nothing but as of thehigher statistical future EFTs fluctuations orders for work and nematic in to dissipation. and ‘turn the the derivative expansion. crank’ and Second, extend the the actions ordinary presented coset here construction to pre- one to formulate non-equilibriumthe actions coset in construction full for generality,only whereas systems be previous with used attempts spontaneously to at fluctuations formulate broken ordinary and spacetime dissipation. actions symmetries that can are incapable of accounting for statistical true symmetries of theits underlying own theory. generator As and acontrast, result, Goldstone our each approach to was of parameterize to thesethe the treat symmetries style these coset required of additional of [ symmetriesgenerators broken as and symmetries. gauge Goldstones. redundancies Further, in the coset construction presented in this paper allows generators behave rather differently thanerators. those In corresponding particular, to the unbrokenwhereas unbroken symmetry the Goldstones gen- broken possess Goldstones, infinitely like many ordinarygauge gauge Goldstones symmetries. at symmetries, zero temperature, have no such In this paper, we defined ait systematic to coset formulate construction of both non-equilibriummatter known EFTs and and systems. used heretofore We unknown postulated non-equilibriumcan be EFTs that characterized for IR by condensed Goldstones dynamics almostneously of as broken. if thermal However, every the systems symmetry Goldstones corresponding of out to the spontaneously of system broken were equilibrium symmetry sponta- 8 Conclusion JHEP07(2020)200 , 3 (A.1) (A.2) ) and [ 5.17 )–( 5.15 ). It is pragmatic in is the equilibrium in- 4.2 0 β )–( 4.1 . , where µ 23 s ) corresponding to unbroken sym- M , M x ( 00 ∂X ∂φ 0) A 0 0 , G A s µ -type fields correspond to classical field β 0 B r − , 0 = √ , 0 µ s M + β ), the A 0 ∂X ∂φ µ = ( – 32 – ] do not necessitate a long-wavelength expansion. 3.6 M 2 β M ) = β x ≡ ( ) s A are the equilibrium chemical potentials of the unbroken x µ ( A 0 µ s -basis of ( µ β -type fields describe quantum and statistical fluctuations. As r, a a ) and 4.6 ) as well as the chemical potentials x ( µ β . From the EFT perspective, we posit that the fluid manifold is a space on which is given in ( A 0 A s T B In the ordinary construction of the hydrodynamic equations of motion it is usually sup- It may not be immediately obvious that this combination of fields should be identified with the chemical ]. 23 symmetries. Working in the configurations, whereas the potential. For some insight into38 why this should be so, see the discussion surrounding ( Additionally, we can identify the chemical potential with where verse temperature. Then, in thefour-vectors physical — spacetime, one we for can each define leg the inverse-temperature of the SK contour — via the push-forwards posed that the systembroken exists Goldstones, the in state ‘local of the equilibrium.’four-vector system can This be essentially specified by meansmetries the that, local inverse ignoring temperature the fluid four-vector is fixed, namely In this section,equilibrium effective we actions offer to possesthe a the sense gauge that somewhat symmetries it should ( pragmatic convincesituations, the explanation but reader that it these for is symmetries by are why necessary no in we means ordinary a expect derivation of non- the symmetries from first principles. mentorship as well as Austin JoyceThis and Noah work Bittermann was for partially many supported insightful conversations. by the US Department of Energy grant DE-SC0011941. A Emergent gauge symmetries Acknowledgments I am pleasedColumbia to University. I dedicate would like this to thank Alberto paper Nicolis and to Lam Hui Dorian for their Goldfeld, wonderful Professor of Mathematics at freedom in the non-equilibrium EFTs ofHowever [ our non-equilibrium coset constructionis — designed like to the generate ordinary covariantour coset terms construction coset for — construction an action towith in allow much a for broader derivative non-local applicability, expansion. e.g. terms Extending actions would describing allow low-temperature one systems. to formulate actions and non-Goldstone modes is of significant interest. Finally, the hydrodynamic degrees of JHEP07(2020)200 ) ) = µ, ν ] to 4.6 2 x (B.1) (A.3) ( ) with ). To x ( sµν A.3 A g µ ) and . x ( µ β be the set of unbroken ⊂ H 0 H H .  , , ) and physical spacetime coordinate )] x ( x ( A 2 µ 2 , alone. The external sources are as µ m, n β 3 to denote Lorentz indices and , A 2 , T ) + , ) + As before, let x 1 x ( , ( A 1 µ 24 1 µ β  [ = 0 ]. 1 2 1 2 51 ≡ , – 33 – ≡ m, n ) ) 50 x x . The metric tensors are then given by ( ( m µ A r r ¯ µ β P = unbroken translations = other unbroken generators = broken generators ) = ) = x x α ), these emergent symmetries cannot change the state of the be the gauge fields (or spin connections if the Lorentz group A m ( ( τ ¯ x T P µ be the gauge fields (or spin connections) corresponding to the ( A A β T µ A r α ) ). Further, since we expect that the (classical) state of the system µ the subgroup generated by τ x ) ( that the correct way to construct covariant building-blocks for non- x 4.2 ( , where we now use A sµ 4 . This amounts to introducing external gauge fields. Let ). G ⊂ H α sµ x )–( G . In this section, we will use a trick St¨uckelberg inspired by [ ) and c ( 0 x G ≡ A ( n sν 4.1 ≡ H µ ) ε r ) be the vierbeins, which can be thought of as the gauge fields correspond- ) x β x x ( mn ( ( η sµ ) m sµ sµ because the trick St¨uckelberg requires that we gauge all symmetries including Lorentz. c ε A x ( m sµ µ, ν is involved) corresponding to the unbroken symmetriesLet of broken symmetries. Let ing to unbroken translations ε Let We begin by introducing sources for the Noether currents corresponding to each sym- It is worth mentioning that beyond these heuristic arguments, there has been some It is now necessary to distinguish between Lorentz indices • • • 24 indices be the generators of to denote physical spacetime coordinatesymmetries indices. and follows: metry generator of We claimed in section equilibrium effective actions was to— construct one two for distinct each Maurer-Cartan legsymmetry one-forms of group ( the SKdemonstrate contour that — using that two transform copies under of a the single Maurer-Cartan copy form of is the the global correct approach. success in derivingAdS/CFT chemical duality shift from first symmetries principles for [ unbroken U(1)B Goldstones using tricks St¨uckelberg the and the Maurer-Cartan form of motion onlyensure depend that on this the isthe unbroken the symmetries ( Goldstone case, we fields mustis as specified require they by that appear thesystem effective in and action ( therefore be ought invariant to under be considered as mere gauge redundancies. If we wish to reproduce ordinary hydrodynamics, then we must require that the equations a result, we ought to identify the physical thermodynamic quantities JHEP07(2020)200 , sµ θ = 0, , re- m, n ) for i (B.6) (B.2) (B.3) (B.4) (B.5) θ sµ α sµ θ c ; 0 ) we have , = x A t, t ( ) defined on T A sµ ( 2 ) ζ φ to refer to the A U ( 4.6 A s µ ¯ i P e . and i α ) and τ ) m µ ) x µ δ φ ( 2 . , ( ] 1 θ α s A ζ ; M T 2 iπ ) = ) e Θ x , x ( ( −∞ ≡ M , 1 m sµ A sµ ) ε [Θ ∞ φ . A ( ] i s µ 2 (+ EFT ζ 2 † + iI , θ α e , allowing us to use τ in the presence of external sources µ 1 ρU ) ζ 1 ) A s t θ x  µ , γ we can identify the Lorentz indices [ ] must be invariant under two independent ( 1 s ). Moreover because these fields live on the µ, ν µ D θ to γ 2 m α sµ µ W ; α s  0 δ 4.3 t π ic , θ M µ and the unbroken rotations Goldstones D ] = ∂ 1 + −∞ – 34 – µ i s θ µ , ) = ∂φ [ 2 η m X x ¯ ∞ ( P , θ + W D ) µ m sµ x 1 µ s (+ ε M ( θ Z [ U ]; that is, for gauge parameters m sµ h ∂X ∂φ ≡ 2 W iε ] )) µ 2 φ . = tr ( ,θ G ) = s ] µ 1 x µ 3 are the fluid worldvolume coordinates. Now we can implicitly X θ 2 ( , [ ( ,θ 2 are included for later convenience. Now, letting sµ W µ , sµ θ 1 e 1 i θ θ [ ,  · W are nothing other than the Maurer-Cartan one-forms ( e 1 = 0 − s sM γ M ≡ ) for φ couple to conserved currents, ( M sµ φ 2 be the time evolution operator from sM θ , Θ = 1 the thermal and unbroken inverseremoving Higgs the constraints broken in the boostspectively. case Goldstones But of before ordinary we fluids, solveit thereby these is new inverse first Higgs helpfulinvestigate constraints these to at inverse full understand Higgs nonlinear the constraints order, expanded motivations to for linear imposing order them. in the To fields. do this, we will C Explicit inverse Higgs computations Since the thermal andsee unbroken how inverse they Higgs can be constrains solved are explicitly. new, This it appendix may explains be how helpful to to algebraically solve each leg of thewith SK the contour. physical spacetime Since unbroken coordinate translation generators indices as we didSK in ( contour, their values mustthere match were up two copies in of thethe the global infinite gauge symmetry future, group fields meaning and that gauge even symmetries, though there is only one copy of Notice that if we remove thewe external source find fields that by fixing Θ where define the non-equilibrium effective action by We can therefore ‘integrate trick.St¨uckelberg in’ In both particular, the define broken and unbroken Goldstone fields using the Since copies of the gauge symmetries [ where the factorss of the generating functional for the conserved currents is On each leg of the SK contour, we can combine these fields into a single object, JHEP07(2020)200 ). )– and 5.5 i (C.7) (C.4) (C.5) (C.6) (C.1) (C.2) (C.3) ) and C.5 0 s φ θ ( µ s = ε . However, i s i s 3 transform η + -Goldstones. θ , i s , µ 2 η , φ ··· = 1 + ) = φ i ( aji µ s θ 2 for X − 0 i . s . i , aj . Thus, the constraints ( j µ s E ε r ε i Λ ~ θ ··· ∂ j 0 0 . ∂ enjoys a chemical shift-type gauge + − ∂φ ∂X r µ ai ··· , ~ θ ε r + j sM ~ε , + i , θ ∂ 0 a t that transform covariantly and that can i i s ∂ ~ε 0 ε Λ s − = 0 t θ 0 ∂ µ µ s sM ε M i − − E ~ ~ ). Since ) ∇ × ∇ × ∂φ ∂X 1 i s M ij 2 – 35 – ). This allows the removal of the = ε ∂ − = = 2 θ = 0 and derivatives of i s r a E ∂ i + η + 5.6 ( ~ θ µ ~ θ µ -Goldstones entirely from the (quadratic) effective . Solving the above equation gives the constraint 0 ). Then, acting with the (classical) dynamical KMS = s ij 1 µ M Λ ∂ ij 2 ~ θ Mj sM θ δ 0 µ θ 1 0 ( θ i 5.8 s 1 2 E = E − ) represent all Lorentz Goldstones such that ∂φ ∂X φ − ≡ ij 1 ( µ sM θ ij r M j µν s E can only appear in the from ) θ 0 = ≡ θ 1 r − ~ θ 2 ij a θ E and (
i 2 Mi and ε 1 i 2 E + ε i 1 − ε 2 indicates on which leg of the SK contour the fields are defined. We are 0 = , which transforms covariantly under all symmetries and gauge symmetries. . Then at linear order in the fields, the vierbeins are given by i 1 , 1 2 ε M j jk s ) θ ≡ ≡ 1 = 1 is invertible, these constraints give − 2 i r i a ijk s ε ε  E R 1 2 ( ) are sufficient to remove the Removal of the rotational Goldstones is a bit trickier. There are no terms in a single To remove the broken boost Goldstones, notice that Consider the case of a fluid without conserved charge. Let = Mi 1 i s C.6 C.1 Thermal inverse Higgs Recall that for ordinarySince fluids, the thermal inverse Higgs constraints are given by ( where symmetry, at liner order, ( action. which is the linearizedsymmetry version on of both ( sides gives Setting the antisymmetric part to zero gives the relation where which is just the linearized version of ( vierbein that transform covariantly and, whenwe set to have zero, two vierbeins allow the — removalE of one for each leg of the SK contour — thus we consider the object Thus setting the above expression to zero gives the linearized constraint interested in setting togive zero an components algebraic of relation between covariantly under all symmetries and gauge symmetries and are given by let the antisymmetric field θ where JHEP07(2020)200 ). 5.7 (C.8) (C.9) (C.12) (C.13) (C.14) (C.15) (C.16) (C.10) (C.11) , ), so it can be li ) . 1 2 C.5 ) R ( l be the 3-vector with k η, ·M l ~ 1 , M − 2 ν k j sinh R sρ ) -type rotational Goldstones η · 1 r Λ 1 − 2 ρ s = ˆ M , , R R and let l ~ , 1 2 ( i − ∂X ∂φ M R j = ( ≡ j · ) ≡ T 1 η, T M ) . Then unbroken inverse Higgs con- l, ν ~ − s l k 1 = Λ · ) a µ ) becomes − 2 , that is, ( 1 ) 1 j sM ·M j 1 i η, − 2 R ·M·M i tanh − ) · T λ 1 a C.7 i λ . Rearranging this equation we have that T ( 1 (Λ ·M η 1 1 η R k ν − µ − 2 s 1 − j s T R M cosh − ) R ( = = R 1 ∂X ∂X ˆ η · – 36 – √ M 1 = − 2 ν t , a T k = ( ⊗ = ~ R √ i t = i R X ) X ( 0 η 1 0 1 sj X X T s M i = li ∂ ∂ − 2 0 0 − 2 ) ) R R 1 1 ∂ ∂ j R ) + ˆ ·M· − s R · ˆ η ), from which it is immediate that ·M ( T 2 sM E 1 l 1 ·M·M = (Λ ) ( a ⊗ − 2 R ) R 1 M . Now equation ( l 1 ˆ η ) − 2 R = 1 − 2 · M 3 matrix equation R a − ) ~η/η R 1 · 1 µ sM · × a 1 R ≡ ( 1 = ( ( E . Then, we have that R η j = (1 R ij M ] k i . Then we have M λ ) i J ) 3 matrix with components 1 t ) = ( 1 ·M· and ˆ φ − 2 -type rotational Goldstones. To remove the ( − 2 ) × i a s a ~η 1 ( E = Λ iθ · − 2 ( ·M e i ~η l R ≡ T ) yield the 3 iM · √ l ) = [ 1 k 1 M 5.7 ≡ ij s R E ( M be the 3 R η = ( λ T as desired. Noticeused that to the remove abovebeyond expression, the when linearized linearized, level requiresthe gives a next ( bit subsection. more work; we will carry out the computation in M Finally, we have that ( straints ( where we have used the fact that which gives us where where as desired. C.2 Unbroken inverse Higgs Recall that for ordinaryExpanding fluids, out the the unbroken vierbeins, inverse we Higgs find constraints that are given by ( which simplifies to components where Let JHEP07(2020)200 ) 1 − -type (D.1) C.16 r ·M (C.17) (C.18) T ) that we ]. In this covariant M 13 √ 5.25 ). It can be ≡ -derivatives only C.6 a 0 φ , M 2 R · . This expedient comes 0 r r M 0 ∂ · , in which case we can use ( T 1 ) respectively, are equivalent to Then, by applying the (classical) )) T 2 , R x r R ( . · 25 I = s 1 5.17 M φ 0 0 R R ( r ) to remove Ω ∂ can appear without M Ω Θ ). It should be noted that the s ξ ) to remove this combination of rotational R iβ C.18 ) and ( ) + . It turns out that there are two possibilities: − C.18 ⇒ s x a attached to them that cannot be removed with C.16 ( = R 5.12 – 37 – s ), we have M M R φ Θ -type spin connection. The above equation is our r r ) and ( C.16 → → M ) 0 . a x r are contracted in such a way that they cancel completely, ∂ appear in the form ( C.16 M s M s building-block, = M R ) and they enjoy the gauge symmetry R φ T 2 ) is the x ( R appears in an invariant building-block, the only other package M · 2 M s 0 φ ), which is defined on the fluid worldvolume. Perform a change of r R Ω + Ω invariant · 5.12 , in which case we can use ( M 1 . But then we can use ( 0 1 r R T 2 (Ω R ) as a definition of ]. Additionally, the action for finite-temperature superfluids ( 1 2 · 9 1 ≡ C.17 R building-blocks using ( rM to remove them. The additional factors of in which case we have no remainingThe factors additional of factors of Think of ( • • 25 coordinates so that it isof now defined freedom on are the the physical fields spacetime; then the dynamical degrees charge that we constructedthose with given in cosets, [ ( constructed with cosets turnssection, out we to will be see equivalent toconsider how the these the action seemingly action ( different presented actions in are [ equivalent. For definiteness, inverse Higgs-type constraints, but this isare not relevant an for issue constructing as EFTs. only the invariant building-blocks D Fluids and volume-preservingIt diffeomorphisms turns out that despite appearances, the effective actions for fluids with and without Therefore, no matter what,invariant the rotational Goldstonesbuilding-blocks can may still be have entirely factors of removed from the at the price of introducing more factors of where Ω second unbroken inverse Higgschecked that constraint in every andin when the linearized, form givesGoldstones. ( Further, if it can come in is Ω where Θ is a time-reversing symmetrydynamical of KMS the UV transformations theory. to ( Now we will see that therotational unbroken inverse Goldstones Higgs beyond constraints allow linear theand order removal suppose of in the that the under fields. the (classical) First, dynamical KMS define symmetries, we have C.3 More on unbroken inverse Higgs JHEP07(2020)200 0 ≡ ), we (D.6) (D.7) (D.2) (D.3) (D.4) (D.5) µ J µ D.1 ∂ .  3 . Since 3. The resulting  , dφ ) 2 , T ∧ ( 0 2 ]. Notice that now the P 9 = 1 dφ µ b J ∧  ) can be solved algebraically 1 µ . ) is equivalent to the effective I,J ) is equivalent to the action ∂ dφ D.5  = 1 5.17 ? 5.25 I J , ≡ , 3 and . ) is 0 = ) ∂g , ∂φ  ]. , b T 2 ) 0 ) ( ( , I φ 13 T 1 ( φ det , 0 ( x F x P P f 4 4 b 1 d d = 0  ] and that ( ,J ) = 9 µ Z µ Z ∂ – 38 – T J µ , which enjoy a volume-preserving diffeomorphism ( = ). Plugging this solution back into the expression µ , = 0 I ) b J J M,N ( φ P J − b 1 T φ EFT ( EFT 0 = I S p S = g ). Using the diffeomorphism gauge symmetry ( ≡ I T → φ ( . I f 00 φ ) gives /G 1 D.2 , that is − , we have that b , b µ . The equation of motion for such that = )). But this is precisely the action given in [ µ J 0 b b 2 0 J ( φ T ) = 1. With this gauge-fixing condition, ( φ T I µ . As always, we use ≡ ( φ ∂ ( M µ P µ ξ f u u ≡ ≡ ) b T ( F as a function of 26 By almost identical procedures, it can be shown that ( T It turns out that 26 action describing charged fluidsdescribing finite-temperature given superfluids in given [ in [ where action only depends on the threegauge fields symmetry for some arbitrary function can gauge-fix for for the effective action ( By the definition of Now integrate out identically, this becomes action defined on the physical spacetime is where for arbitrary JHEP07(2020)200 ] , ]. , DOI , ]. , 09 ]. SPIRE (2012) 046 SPIRE IN IN ][ ]. 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