Global Anomalies, Discrete Symmetries and Hydrodynamic Effective Actions

Global anomalies, discrete symmetries and hydrodynamic effective actions

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As Published https://doi.org/10.1007/JHEP01(2019)043

Publisher Springer Berlin Heidelberg

Version Final published version

Citable link http://hdl.handle.net/1721.1/120114

Detailed Terms https://creativecommons.org/licenses/by/4.0/ JHEP01(2019)043 Springer January 4, 2019 : December 19, 2018 November 29, 2018 : : Published Received Accepted [email protected] b , Published for SISSA by https://doi.org/10.1007/JHEP01(2019)043 [email protected] hong , and Srivatsan Rajagopal b . 3 Hong Liu 1710.03768 The Authors. a c Anomalies in Field and String Theories, Discrete Symmetries, Eﬀective Field

We derive effective actions for parity-violating fluids in both (3+1) and (2+1) , [email protected] Kadanoff Centre for Theoretical Physics,933 University East of 56th Chicago, Street, Chicago,Center IL for 60637, Theoretical U.S.A. Physics, Massachusetts77 Institute Massachusetts of Avenue, Technology, Cambridge, MA 02139,E-mail: U.S.A. b a Open Access Article funded by SCOAP Theories, Space-Time Symmetries ArXiv ePrint: (3+1) dimensions we elucidateglobal connections anomalies, between and clarify anomalous a transport previousanomalies. puzzle coefficients concerning transports and and local gravitational Keywords: Abstract: dimensions, including those withconstitutive anomalies. relations for As such systems ain derived corollary detail previously connections using we between other confirm parity-odd methods. the transport We and most discuss underlying general discrete symmetries. In Paolo Glorioso, Global anomalies, discrete symmetrieshydrodynamic and effective actions JHEP01(2019)043 26 27 29 16 16 17 18 ], it 15 – 13 19 ]). Effects of 12 16 21 ], phenomenological 7 29 – 5 5 20 8 24 ], holographic duality [ – 1 – ], and equilibrium partition functions [ 4 ].) These anomalous transports could be relevant µ o – j ˆ 12 1 28 – – 8 21 and 16 µ o q in physical spacetime 12 30 13 hydro I ]. (See also [ 6 23 20 T CPT PT – 17 1 T TP CPT 3.3.1 Θ3.3.2 = Θ3.3.3 = Θ = 6.1 Θ = 6.2 Θ = 6.3 Θ = 4.1 Equilibrium partition4.2 function from effective Parity-odd action equilibrium partition function and global anomalies 3.3 Dynamical KMS condition 3.4 Explicit expressions for 2.1 General setup 2.2 Formulation of 3.1 Generating functional 3.2 Parity odd action has been recognized that systems within quantum the anomalies exhibit presence novel of transportanomalies behavior rotation on or transport in in superfluids, abeen superconductors magnetic discussed and field in topological [ (for insulators have ain also recent a review wide see range [ of physical contexts: from the study of quark-gluon plasma at subnuclear 1 Introduction Through studies of freearguments based field on theories entropy current [ [ A Explicit expressions for various discreteB transformations Some useful formulae 6 Parity-violating action in 2 + 1-dimension 4 Equilibrium partition function and global gravitational anomalies 5 Entropy current 3 Effective action for parity-violating systems in (3+1)-dimension Contents 1 Introduction 2 Review of hydrodynamical action in physical spacetime JHEP01(2019)043 ≡ ] µ ]. (1.2) (1.3) (1.4) (1.5) (1.1) B 13 , 40 – 9 , , 8 38 0 p 0 + n 0 3 T 3 a 2 ) as follows [ of charge current can be − gauge transformations of 1.1 ) µ o α J 2 − small ], and astrophysical phenomena , (1 0 2 p αβ . 35 0 , T F µ 2 + n a 34 B 0 µν F 0 B 2 p ), the Euclidean partition function of the ξ 0 ) + T are local chemical potential, temperature, α 2 + + 1.1 µn µναβ a 0 0 , µ − ] µ , p ω ~ − – 2 – ˆ 4 0 J c ω (1 ) . Suppose the symmetry becomes anomalous in ≡ 30 ξ µ , α , ˆ = α 0 J for = µT − µ is the local velocity field). This is called the chiral 29 1 is not invariant under , µ ˆ µ o J a µ (1 . Due to ( J A µ µ 9 , u T A ( A ∇ 8 µ, T, n 1 + 2 a ρ u α λ ]. In addition, there have been various experimental searches 3 2 ∂ ) + 2 ν α u − 37 , ) as a local U(1) anomaly, in contrast to a global anomaly in which 1 − receive contributions from local anomaly ( 36 µνλρ 1.1 (2 2 B ξ µ µ ~ ~ ≡ c c 3 3 µ and − − ω ], to cosmology, where the dynamics of primordial magnetic fields plays an , which is often referred to as the chiral magnetic effect (CME). The transport are constants, and ω ξ = = 3 33 αβ , – 2 is the field strength for F , ω B 1 ν ξ ξ 29 a u F gauge transformations when the system is put on a topologically nontrivial manifold. Consider a parity-violating relativistic system in (3 + 1)-dimension with a global U(1) Given their importance, it is of primary interest to incorporate anomalous transports µναβ . We will refer to ( 1 2 with where charge density, energy density and pressure respectively. coefficients The first term impliesthe a vorticity contribution tovortical the effect (CVE). current The that second term is is induced proportional to by the and magnetic field parallel strength to, case the partition functionlarge is invariant under small gaugeTo first transformations, order but in not thewritten under derivative in expansion, the the Landau parity-odd frame part as [ where system in the presence ofA source new insights which we will discuss momentarily. symmetry whose conserved currentthe is presence of an external source has a number of advantages.hydrodynamic fluctuations Firstly, can an be effective systematically fieldfor incorporated, theory new thus physical provides enabling effects a one due to frameworktive to search where action fluctuations approach in provides parity-violating a systems. first-principleautomatically derivation Secondly, incorporates of the the effec- all constitutive relationsreproduces the which fully phenomenological the constraints. constitutive relations Indeed of previous our approaches. derivation It also highlights some important role in thesuch early as pulsar stage kicks offor [ the the signatures universe of [ anomalies on transports in condensedin matter an systems, effective see field [ theory framework, which is the goal of this paper. Such a formulation scales [ JHEP01(2019)043 ] ] 2 – is a 53 48 14 T , (1.6) (1.7) ]. 13 , , such as 56 9 , , T ] for earlier 8 62 50 = 0, there can , – c with 60 ], whose physical 48 We find: , = 0. If only = 0, i.e. no chiral P ] that the transport 1 3 , 45 18 c , a C 55 13 ] (see [ , = 59 1 54 – a ), i.e. with ). In particular, we offer an 57 1.1 = 0 and 1.4 3 , 2 , )–( is allowed [ αλρ 1 ) have only one derivative. Further- β a 2 ). Arguments have been made in [ 1.3 R a 1.2 ]. ]. In (2+1)-dimension the story is much λ . 1.7 13 βµν 2 α 63 π , R 32 is conserved, then 14 ) as the most general constitutive relation for − as in ( , – 3 – µνλρ of the local mixed gravitational anomalies π 13 = 1.4 ) contains four derivatives and thus should modify λ λ 2 a CPT )–( are allowed. Thus detection of possible existence of violations. 1.6 itself, or any combinations of = c µ 1.2 T ˆ is conserved, then J µ CPT and ∇ PT 3 , ) is violated for systems with gravitinos [ 2 , 1 1.7 a ) should apply at least to field theory systems smoothly connected ) which reconciles various different perspectives. 1.7 ), when 1.7 1.4 )–( should be considered as being directly related to global mixed gravitational ) is puzzling from the perspective of anomaly matching in a low energy effec- 1.2 2 can be used to test a 3 1.7 ] , a 1 47 – conserved, then all a sitively depends on the underlyingports discrete can symmetries. be used Henceform to hydrodynamic probe trans- ( microscopic discrete symmetries.vortical or For example, magnetic given effects. the If . As a corollary we confirm ( In both (3 + 1) and (2 + 1) dimensions, possible parity-odd transport behavior sen- In this paper we work out effective actions for parity-violating fluids in both (2 + 1) It is curious that even in the absence of local anomaly ( While these points follow naturally from our discussion, some aspects could have been realized before 41 only at the third derivative order while terms in ( 1. 1 ] which show that ( µ using the approaches already discusseddiscussed in the below literature. could have For example, been the read connection from with the global anomalies results of [ constitutive relations obtained earlier in [ richer, containing six independentThe functions rest of of local the temperaturehighlight paper and a is chemical couple devoted potential. of tointerpretation conceptual detailed for ( points derivations of related the to effective ( actions. Here we and (3 + 1) dimensions followingattempts the at approach an developed effective in actionlevel [ for the anomalous transports). system WeHere has assume Θ an that at can underlying microscopic beCPT discrete the symmetry time Θ reversal a which parity-violating includes system in time (3 reversal. + 1)-dimensions, and in (2 + 1)-dimension we confirm the and subsequently explicitly worked outcoefficient in various examples inanomalies when [ putting the systemknown on that a topologically relation nontrivial like manifold. ( It has also been J more, matching with constitutivewill relations not or lead partition to functions any52 as multiplicative factor done in [ to free theories through continuous parameter(s). Alternatively, it has been hinted in [ as [ Relation ( tive theory, as the right hand side of ( still be chiral vortical andpointed magnetic out effects, that determined for uporigin to a has three CTP generated constants. much invariant It recent theory,appears has interest. only to been be From holography related and to free the theory coefficient examples, JHEP01(2019)043 2 2 ]. a S = 0. × (1.8) (1.9) 55 , (1.10) (1.11) (1.12) 1 c S 54 i dx i denoting the ), i.e. b , under a large 1 T 2 1.1 S S ) + i denotes directions i dx Z g i x i v as well. ∈ ∂ , with . 2 − τ + S , i i , then under a large gauge dτ b b × ) aries from time diffeomor- 2 ( CPT 0 1 S → → is real; recall that the presence 1.9 S A i i Z 1 × = a ) is the stationary gauge transfor- ∈ are nonzero, the partition function 1 Z, m, n T µ 3 S , 2 dx 2 1.10 , µ 1 a q as πna has a magnetic flux along 2 µ . Equation ( Z, r i i r τ ) are associated with global anomalies, re- A while ( 3 v a ,A + 3 2 j – 4 – 1.4 we find that 1 we find that the partition function transforms as − 2 e 1 dx 1 is broken and we are at a finite temperature. i )–( 2 S ma S q 2 → dx 1.3 f, b π i ij Z 8 ∂ a CPT is also known in the literature as Kaluza-Klein U(1). has a magnetic flux along along along in ( − f + i , b i i i i b 2 3 v v , v v exp ) does not matter at this derivative order. 2 i , 1 1.6 → → → a dx are independent of and i i i i Z v v v g b . Similarly when only − . Let us suppose there is no local gauge anomaly ( and . It turns out, however, when = 4. But here dτ 1 µ is preserved for all relativistic local field theories, searching for its pos- CPT f S d A 00 × g is the minimal U(1) charge of the system. The term proportional to ) is fully consistent with the discussion of various examples in [ 2 ) the term in the exponent proportional to q = CTP S breaks 2 1.11 1.11 1 ds a which is again real. The standardanomaly lore in is that there can be no pure global gravitational in ( In ( of gauge transformation of transformation of where phism along the Euclidean timemation circle for is only invariant under transformations whichbut are not smoothly invariant connected under to large the identity, gaugeMore transformations. explicitly, suppose where both with all components toalong be independent ofThen Euclidean to time first derivativefollowing order, two the U(1) partition transformations function should be invariant under the explicitly, consider the partition functionat of the a system finite on temperature, aEuclidean i.e. spatial time the manifold direction full along manifold whichturn is we on put the thermal external boundary metric conditions. and source We also sible violations throughsystems transports exhibit could emergent be relativisticbe interesting. symmetries, potentially and used to Some transport probe behavior condensed whether can matter there then is emergent spectively with pure gauge, mixed gauge, and pure gravitational anomalies. More While All three constants The gravitational anomaly ( The transformation associated to 2. 2 3 JHEP01(2019)043 ) 3 – – a we we 57 57 with 1.11 4 3 and kq 1 a + 2 a ) should not be → ) and the global 1.7 2 ) to global gravita- 1.6 a 1.4 in ( )–( , i.e. λ 2 ) in a phase, so the global a 1.3 1.11 ): equation ( ) nor the global anomaly in ( 1.7 1.4 we briefly review the formalism of [ )–( 2 1.3 and global anomalies described above are we discuss the entropy current for (3 + 1)- appears in ( 3 5 , 2 2 , – 5 – 1 a a -related transports in ( 2 a ] can be considered as establishing (b) for field theory ). Nevertheless, when UV physics is taken into consid- 51 ) which has been known to be valid for some class of sys- 1.6 – ) which is a universal low energy relation; 48 1.11 1.11 we repeat the analysis for (2+1)-dimensional parity-violating systems, ) are real. As a result the global anomalies associated with them are ) only captures the “fractional” part of 6 1.12 ] for other discussions of action formulation. 1.11 ) which from the light of the above discussion may be interpreted as the 76 )–( – does not change the phase. In contrast, the factors associated with 1.7 64 mixed anomaly ( tems. This relation goes beyond low energy physics. tional anomaly ( Z a relation between local mixed anomaly coefficient the connection between 1.11 ∈ (a) (b) has anything to doeration, with they ( are controlledlight by the the same discussion number ofsystems in [ smoothly a connected large to class free of theories systems. through In continuous parameter(s). this This resolves the twoviewed puzzles as mentioned a below low energy ( field relation. theory, neither Indeed, transport from behavior in the ( perspective of low energy effective The relations between coefficients universal relations which cantheory, be without deduced any solely knowledgetion at of ( the UV level physics. ofcombination low of Now the energy let following: effective us come back to the rela- anomalies of aanomaly system. ( Note that k in ( fully equivalent to the corresponding transport coefficients. We thus see measuring parity-violating transports can also be used to probe global 4 The plan of the paper is as follows. In section See also [ 4 ] to set up the notations and formalism for deriving anomalous transports in later sec- ] to set up the notations and the rules for derivations of later sections. In section In this section, we review59 the formulation of the hydrodynamicaltions. action introduced in [ obtaining the effective action,included partition a function number of and appendices the for entropy technical current. details. We have also 2 Review of hydrodynamical action in physical spacetime 59 obtain the effective action ofdiscuss a the parity-violating connection fluid in betweenconnection (3 the with + global 1)-dimension. effective anomalies. In actionsystems. In section and In section section thermal partition function, and JHEP01(2019)043 in to 0 1 t (2.8) (2.1) (2.2) (2.3) (2.4) ρ Equa- 5 . i µ ] should be ) . , ξ 2 ) in terms of µ ) ) 2 ] for a system x x 2 ,A (sources for the ( ( 2.1 2 ,A µ µ 1 2 µ g ,A 1 J J ; 2 µν 1 2 1 2 g A ] (2.6) 0 g g g ; 2 ρ ξ 2 1 ; − − ,A is the field strength of g,A

1 . √ √ 2 (2.7) g µ ,A ,A 2 [ µν µ , 2 1 J −∞ 1 ξ 2 = = g A , [ W g F and independent diﬀeomor- = ) ) ; = 1 ∞ = 1 ] (2.5) x x 2 W ] ξ 1 2 ( ( and external sources which we µ χ µ µ [ g,A (+ 2 0 ,A

1 2 † ,A ,A δW δW , and ρ 1 µ A , s 1 1 1 U ξ 1 ν EFT s J A δA δA µν 0 I A g = 1 J i [ [ i ~ ρ g = ) are obtained by µ − µ 1 W W sµν µ µ 1 A F A Dχ e generated by a vector ﬁeld ,A ] = ] = = 2 2 Z µν and depends on 1 ν sµ dλ ,A ,J g = T 2 g, A and an external vector ﬁeld ; – 6 – µν ] + g 2 g sν g,A ; EFT 2

, , i µν 1 ,A I ) ) ∇ 1 = −∞ 2 A µν x x g 2 g , ; ; ( ( ,A 2 T 1 1 µν g ∞ 2 g µν µν 1 2 ,A ; [ g = 1 , 1 T T g (+ [ 1 2 of the stress tensor and the U(1) current in the state W = g g dλ U g,A W µ = 0

h − − e µν + µ s 1 ,J µν √ √ 1 1 Tr g J , i.e. 1 2 1 2 T µν 2 sµ degrees of freedom of the system ≡ ,A ) denotes the quantum evolution operator of the system from T = = = 1 ] and external background ∇ µ ,A g µ 1 [ ) ) 2 2 µν g x x µν of any gravitational and U(1) anomalies, ,A slow ( ( T g W ,A µν µν µν 2 µν δW δW are all assumed to vanish at spatial and time inﬁnities. 1 2 g 1 and ; ) in turn ensure that g 2 µ , denote diﬀeomorphisms of δg δg 1 ; 1 µ 1 i i 1 ξ 2.6 A ξ ,A absence 1 − , t denotes setting g is the covariant derivative associated with µν 2 )–( 1 ,A t and collectively denotes slow variables of the system which in general also come in two 1 g ξ [ 2 g,A g

∇ χ 2.5 W , λ e 1 For slowly varying sources, we can express the generating functional ( In the . Similarly for quantities with subscript 2. λ 5 µ in the presence of spacetime metric 1 2 where copies. The low energyhave effective suppressed, action and is assumed to be local. A path integrals over where tions ( where invariant under independent gauge transformationsphisms of of where The expectation values an external metric where U( t U(1) current). The sourcesWe for introduce two the legs “on-shell” of stress the tensors CTP and contour currents are for taken each to leg be as independent. Consider the closed timewith path a (CTP) U(1) generating symmetry functional in some state specified by the density matrix 2.1 General setup JHEP01(2019)043 ) ), A sµν ) to and σ EFT g i 2.6 ( (2.9) I 2 2 (2.10) (2.12) , , , σ µ µ 1 1 0 )–( σ X X )) (2.11) . ( 2.5 2 ) σ , 7 ( µ 1 in two copies σ s ( X s A X ( σ ϕ in two copies of µ s A is their “internal” f ∂ A 0 σ σ . The slow variables ) = are equivalent to the )) + σ 2 1 ( , σ 2): 1 µ ( , 0 s s ) are coordinates for a “fluid ϕ , i is mapped by X )) i ( = 1 , σ σ σ 0 . s ,X ( sµ and ] σ ) s β s A 2 Physical spacetime , X ; µ ). Furthermore, the form of the µ 1 s = ( X ( A 2 ( s X A λ 2.6 ∂σ sλ ,B ∂X σ 2 A h )–( )+ µ to be a local action of pullbacks of ; λ s 0 s σ . We will limit ourselves to the generic ) = 1 ]. ( ), i.e. hydrodynamical modes, with 2.5 σ s ) which can be interpreted as U(1) phase ( ∂X 76 ∂X ,B ϕ A , 2.7 1 hydro σ hydro sA h 59 I respectively. See figure ( I [ – 7 – 2 ) = ) = , 0 2 s 1 , σ respectively. µ 1 ( X ϕ 2 0 s , ( Fluid spacetime hydro X µ 1 I ,B 0 sµ X ν are invariant under independent diffeomorphisms and s = B 2 , ∂σ ,A 1 ∂X ) B s 2 to the fluid spacetime )) hydro , I σ X ( , ϕ ( ) s s and = 1 X sλρ ( X 2 s g ) immediately implies ( , ( 1 , ν s ρ s 0 s s h sµν λ are interpreted as labels of each fluid element while ϕ g µ 2.10 i µ ∂X µ ∂ s ∂X A Physical spacetime σ ∂ ) describe motion of a continuum of fluid elements labelled by µ ) which gives the local inverse temperature in fluid spacetime. − λ s A 0 s ∂σ + ∂X A σ sµ ( σ 2 ∂X sµ ( ∂X , A µ 1 ) implies that the equations of motion of β describing a medium in local equilibrium, generically the only slow modes are The slow variables associated with the stress tensor can be chosen to be A ) = = X σ 0 6 ) = = ( . 0 ρ s 0 sµ 2.10 A X sµ ( sAB are the Stuckelberg fields for diffeomorphisms and gauge transformations ( B By construction For h The discussion can be readily generalized to systems suchNote as that near there a critical is point only where one one temperature should field also rather than two copies. 2 6 7 0 sµν , 1 g which along with ( action ( include the corresponding order parameter(s). See [ gauge transformations of the two legs of the CTP contour ( i.e. scalar field ϕ and we require theand hydrodynamical action the corresponding hydrodynamical action situation. which describe motions ofphysical a spacetimes continuum with of coordinates fluidassociated elements with labelled the by U(1)rotations associated currents for are each fluid elements. It is also convenient to introduce an additional time. The redphysical straight spacetime line trajectories in (also the in fluid red) spacetime of the with corresponding constant fluid element. those associated with conserved quantities ( Figure 1 of physical spacetimes with coordinates spacetime”, where JHEP01(2019)043 - ) ]. r X 59 ( . In – µ ) and A (2.14) (2.15) (2.16) (2.17) (2.18) (2.19) (2.13) x σ 57 X ( ν a , β aµ expansion. X ) , µ a A x . ~ ( -variables. ϕ 2 ~ ∇ should satisfy ) r , ν 2 ~ x , ϕ ) − ( ), where we have ) A x ), i.e. use µ − µ 2 x x ( ˆ ( A + ( J µ A ϕ 1 hydro -expansion [ 2 ˆ σ µ A I J aµ ~ g ( = = -variables correspond σ 1 A µ a g − 2 µ ν ,A 2 X ) − √ , ∇ x ν should be invariant under a √ ( aµ X ,A X ≡ ≡ − aµν . ν ) ) aµ , ϕ )+ hydro , g ∇ x x a I A x ) ( ( ··· ( ϕ 2 + ~ µ µ x a 1 2 ( hydro hydro 2 ~ + ϕ µ aν + δI δI µ + µ δA δA X ∂ ,A µ A ϕ ) (1) hydro x ∇ = = I )+ ( ). Invariance under these transformations ~ 1 x µ + ( 1 µν + g is naturally formulated in the fluid spacetime aµ 2.12 as aµν -variables and those without as g A – 8 – a ~ ,A interpreted as the corresponding position noises. )–( , ϕ = = (0) hydro , µ hydro a as I µ ) µ a aµν I , µν ) apply to a general quantum system. At sufficiently x X g a A ) X g 2.11 = ( a 2 2 ~ ~ x a ( ) survive and describe classical statistical averages. We 2.8 X µν X 2 − − ˆ L T µν L 1 µ 2.8 2 ˆ hydro µν in physical spacetime T g + ) is interpreted as mapping fluid spacetime into the physical I g X 1 )+ A 1 ~ − g x = = σ ( aµν − √ ( a µ 2 g µ 2 1 µν √ ϕ hydro 2 ≡ µ 1 2 I X ) ∂ can be expanded in x ≡ − ≡ ( , g -fields (including both background and dynamical variables) must ap- )+ ,X ) ) to emphasize they are now just coordinates for physical spacetime. The a µ x x x a aµν ( aµν ( ( µ X g hydro x G aµ I µν µν hydro hydro 2 2 ~ ~ 2 1 A δI δI by + + ), while the background fields are δg δg ≡ µ µ x µν ( X g X aµ a C = = , ϕ implies that pear through the combinations itational and charged current anomalies,physical the spacetime action version of ( ) µ 1 Spacetime diffeomorphisms and gauge transformations. In the absence of any grav- µν While the hydrodynamical action As defined the path integrals ( x 1 ( X , one can also formulate it in physical spacetime by inverting 1. g µ a A when formulated in theThey can physical be spacetime separated to into leading the following order categories: in the physical spacetime formulation hasconnects the more advantage directly of with being the more traditional phenomenological physically2.2 approach. intuitive and Formulation of We now list various symmetries and consistency requirements which σ as dynamical variables andthe express physical all spacetime other formulation, the variables dynamicalX accordingly variables as are then functions of replaced will refer to variablesvariables with can subscript be consideredto as noises. describing For physical example, quantitiesspacetime while (now only one copy) with and the action In this limit the path integrals ( high temperatures it is oftenFor this enough purpose to we consider decompose the leading order in a small conservations of the “off-shell” hydrodynamical stress tensors and currents defined as JHEP01(2019)043 (2.24) (2.29) (2.30) (2.20) (2.21) (2.22) (2.23) (2.25) (2.26) (2.27) (2.28) a fluid. We µ . define A ), 0 i µ ν ν 0 σ σ ∂ ) B ). 2.1 1 ) , − − x → → ] ( 2.9 i ν 0 a K µ a ( A Λ β . -variables which can appear µ 0 ϕ µ r ) − ∂ ) defines a normal fluid. For a = 1 µν , is not invariant under diagonal → r − = µ ∆ µ a ) . [Λ K 2.23 µ x B ) ( defined by ( , B x µ ( β µν ν 1. µν g u ∂ ϕ hydro − I µ , , − , ϕ − ∂ ) = − 0 -variables including both dynamical and = )) ν x = a + ( µ µ ) involve only dynamical variables, yet they ≥ B 2 ˆ µ x µ u ] = µ ( µ a i , σ ∂ A ) u , µ, F 2.23 σ Λ i – 9 – ( = 0] = 0 , µ = hydro = - and r λ I , r a , σ )–( , b µ ) 0 µν µ 0 Λ [Λ − x B ) σ , Im ( , u ( 1 r ϕ ) , σ µ 2.21 ) implies that the only invariant which can be con- f − ) x u i [Λ ∗ ( hydro ) → K = σ I β is the velocity field x ( only. Invariance under ( ( ,F 2.23 ( i 0 ϕ b 1 0 0 i µ be invariant under A β hydro σ σ σ σ B I )–( = . By definition = µ µ ν → → ∂ µ u µ u i u hydro 0 2.21 β µ I ≡ ≡ σ σ u A must appear through µ µ , but + ϕ K ϕ µν collectively denote all g ) of is now unconstrained. and = is a function of r,a µ µ β λ µν 2.23 A where Λ background fields. where are invariant. To summarize, theare only combinations of It is often convenient to combine the first three variables further into and ∆ shift ( transformations are deemed physically equivalent. Invariance under ( structed from where superfluid where the U(1)be symmetry dropped. is The spontaneously symmetries broken ( should this be symmetry viewed should as “global gauge symmetries,” i.e. configurations related by such Furthermore we require the action be invariant under the diagonal shift The above variables are the physical spacetime version of ( require the action while Classical remnants of constraints from quantum unitarity of ( Spatial and time diffeomorphisms in the fluid spacetime which 3. 2. JHEP01(2019)043 ,Θ hydro (2.32) P I . For (2.34) (2.35) or ηx The fact C 8 . T W . Θ can be time . Θ can also be is invariant under the ) (2.33) ) as ~ x ( W CPT ( O G , ] (2.31) . Similarly for ) + a should be understood as i . G r Λ ] , , x G aµ r time reversal 0 η ΘΦ x i , such as [Λ ,A − T ) is more general, applicable also − , aµν ) ) leads to Onsager relations, local a g ) can be written schematically as = ( hydro ; , and I µ ηx 2.31 µ G ( with , since Θ does not take the generating 2.31 ,A 2.31 dynamical KMS symmetry G ] = = ΘΛ P µν 2 G a , g -variables with total one derivative. More a η [ Z ˜ 9 hydro Λ C r , ηx ˜ Λ I , ) W ≡ i r ] that ( ) ˜ A Λ [ x ] = − 59 – 10 – ( , , – aµ G ) 0 ˜ A ~ hydro 57 , ) to itself, the generating functional A Θ ( I µ O 2.1 aµν , such discrete symmetries should be imposed on A ≡ = ( g ; ˜ + µ = µ CP r ˜ A is given by a thermal density matrix, while neither Θ nor the KMS A 1) one for each component for transformation which is a combination of Θ and the Kubo- G , contains dissipative terms, thus it cannot be invariant under Θ alone. A hydro 0 or ˜ 2 impose either Θ or KMS separately, but should impose the combination I η ρ µν g Z P [˜ as we will discuss in the next item. , the tilde operation in ( = ΘΛ and with some other internal discrete operations. Unlike ~ hydro not W r I ) we denoted Θ transformation of a tensor to be invariant under a T ˜ T Λ be imposed directly on . It should be understood as a mathematical characterization of a ], when hydro ) plays the dual role of imposing microscopic time-reversibility and I to itself, but to a time reversed generating functional 57 2.33 , parity not one can hydro C I itself, or any combinations of 2.31 W denotes certain combination of in local equilibrium. The prototype of such a state is the thermal density T r hydro I 0 ρ ). is invariant under a discrete symmetry Θ containing 2.31 0 a collection of phasesexample, for ( Θ = where Φ explicitly, in ( where we have suppressed tensor indices for to pure states. Itfirst was law, local found second in law, [ andTo local leading fluctuation-dissipation order relations. in Martin-Schwinger (KMS) transformation. Equation ( local equilibrium state matrix in slowly varying external sources, but ( where tilde denotes a a combination of by itself can functional that the underlying Hamiltonian is invariantconstraints under on Θ nevertheless leads to important conjugation and they can be imposed straightforwardly as usual. ρ reversal We require We assume the microscopic Hamiltonian underlying the macroscopic many-body state Discrete spacetime symmetries. If the microscopic system is invariant under charge As emphasized in [ This is quite intuitive as 6. 5. 4. 9 8 operation takes the generating functionalcombination ( of them i.e. Accordingly in of them ( JHEP01(2019)043 1 L L and -field ). (2.41) (2.42) (2.43) (2.36) (2.37) (2.38) (2.39) (2.40) a ··· , 2.31 -variables ) to order consistent + r L ··· , aµν 2.41 g ) aµ g + µ . − C β ) = ν √ aρP µν x ∇ = 2 d G F . aµν d ν + ) i β G aµd ν R x aνN β − µ − = G , ˆ µ ∇ 0 ,G µ ( x ) i aµM ∇ − hydro ( ··· G for various tensors are given in I i give the standard hydrodynamic µ, d ) + contains first derivative terms in + = ( a x 3 2 ( µ ) + = ( L , ϕ -fields and the number of derivatives . Given that L . x CPT µ a a ( aµν , + Θ X µνρ,MNP i aµ 2 G aµM 10 -variables can be written explicitly as L Y C − PT ,M G ]. a while 1 8 , ) , ηx + = ) = µ . The explicit expressions for ( ) 58 T ˆ 1 x i i + as the “off-shell” hydrodynamic stress tensor ) = J , ( , x µM L ) A Θ µ – 11 – ) for ( ˆ . They are separately invariant under ( x µν T ˆ − µν µ J ( = g n aνN , ˆ µν,MN T B µ ϕ 0 n G β 2.33 µ µ A L β W ∂ L and = ( i µ − L i =1 aµM ∞ X n iβ µν G µM ) contains one derivative, the dynamical KMS transforma- = ( ) + ˆ T ˆ = ) + x T µ ( ) + x -variables and derivatives. The first few terms in the L ( A x 2.33 a are covariant tensors constructed out of ( , for conformal fluids. A aµν µν,MN aµ a ) η 3 ϕ G C in ( L aµ W T r i , which implies that terms in the action which have the same value 4 as the sum of the number of C ··· 2 n ) = ) = ) = , , n CPT + x x x ( ( ( and covariant derivatives on a , we identify . aµν aµ ϕ } aµν aµM G ˜ A C µν,MN ··· Θ ˜ ˜ µν G G Θ = ( + Θ ∆ ,W , µM contains all terms with given ) preserves ˆ aµ T µν n µM 1 2 A aµM ˆ L T The explicit transformations forappendix Θ = The second set of equations in ( Since Θ contains while for Θ = G 2.31 and zeroth derivative terms in ˆ = µ, F transform separately among themselves. We can thus write the action as = If we introduce It is straightforward to write down the most general , They are given to order for a parity-preserving fluid are given in [ L n µ 10 µM aµ 2 β ˆ where contains only zeroth derivativeT term in L tion ( of C and U(1) current, and theequations. equations of motion of in a term, then since Φ where we have introduced notation and { in terms of theexpansion number can of be written schematically as with the above prescriptions. We can expand the corresponding Lagrangian density JHEP01(2019)043 (3.1) (3.3) (2.44) (2.45) ) (3.2) 2 ). It can F ∧ ) should be 2.33 gravitational 2 F mixed gravita- 2.5 2 λ − global local 1 F ∧ 1 F ··· , equation ( 1 2 , have any λ + 1 , ( as introduced in ( µ A µ 1 r ˆ Z -fields. The entropy current can V J not , and the second term on the right is the fully antisymmetric tensor a c ˆ µ λρ + λρ ) the consistent currents introduced -dependence so as to be clear about ]+ − F dA µ F 0 2 ~ ν 3.2 0 (2.46) V µν µν µνλρ = β , g F F 2 ν ≥ = µν A factor on the left hand side of various equations ˆ ) implies that µ T dx ; R factors of µνλρ 1 ~ µνλρ ∧ − = k , g µ ~ ~ 2.31 4 µ 1 µ c 1 – 12 – c 4 ˆ V dx A S [ ,V = µ = µν − µ µ ∇ µ F µ iW V 0 ˆ J 2 1 J µ − µ V µ contains ∇ ≡ ∇ ∇ = µ k ] = + F µ V , 2 S L , there should be a dλ = ~ ) we have made manifest replaced by the corresponding Φ 12 − ˜ L 2 ], and a 3.1 r ˜ Λ ,A 2 Θ g depends on specific systems. , ; . In ( a 1 with Λ c g ˜ Λ − 1 dλ µ 1 -expansion at which the corresponding anomalous transports appear in the √ [Θ ~ − V ) now satisfy L 1 ]. Dynamical KMS invariance ( 11 = is ). ), under independent local transformations of ) remains. Note that 2.3 59 = is a local non-negative expression. ,A µ 1 1 2.3 ˜ 3.1 ˆ g 2.6 L V R )–( [ 0123 )–( We now give a brief review of the derivation of the entropy current, whose details are 2.2 We emphasize that here we consider only smallNote gauge transformations that and when diffeomorphisms, restoring i.e. those iW 2.2 11 12 − vanish at spatial and time infinities and smoothlyin connected ( to the identity. while ( hand side is independent ofin metrics. ( Indeed, from ( 3.1 Generating functional From ( replaced by with the order in hydrodynamical action. We assume thattional the anomalies. system does We will seeanomalies that which the are system closely can connected nevertheless to possess certain novel transports. In this section we applysystems the formalism which reviewed break in parity, the including previous those section with to a four-dimensional local U(1) anomaly where constant where 3 Effective action for parity-violating systems in (3+1)-dimension where be shown upon using equations of motion given in [ where then be defined as JHEP01(2019)043 ) as was µ anom (3.9) (3.4) (3.5) (3.6) (3.7) (3.8) 2.33 ) and I (3.12) (3.10) (3.11) J even I 2.16 is invariant . )–( ) 2 ), while inv , ~ can be further o ( . 2.15 I 2.9 O γδ odd F I ) + a αβ . ), and F F ∧ 3.2 even I αβγδ λF + µ A introduced in ( , ) becomes (see ( + 2 inv ~ . , 2 γδ o ρλ c 4 , F ). I 1 3.2 F F ) is independent of the metric). To odd ν − ∧ B = αβ odd L A ν I 3.2 anom 2.43 F F g J I a inv + can be treated independently. is invariant under gauge transformations . − λ µν + ( µνρλ ν √ αβγδ F J ~ x odd even Z inv ~ even c , = 4 I I 2. It should be understood there are two copies I o µν ~ c we also have d ,I 4 , ν I c – 13 – 3 + F = ) nor the dynamical KMS transformations ( J µ Z = ν W = = and only through J anom ) = ∇ ) as = µ µ 2.23 I 2 2 , µ as defined in ( ν = hydro odd F 1 J I I + A 2 T 3.4 even µ µ ϕ odd ∧ ν I L must be expressible in terms of covariant current I J − ∇ 2 inv ∇ I ν F µν 2 J ) below. Defining the covariant current as = ]. Here we focus on λ T ) and ( µν 3.7 − F 58 , 3.3 1 to order )–( = hydro F are parity even and odd parts respectively. 57 I µ ∧ 3.5 odd ν 1 L T odd F -expansion, the anomalous piece in ( ν I 1 ~ ∇ λ should depend on ( ) and ) we have suppressed indices 1 is responsible for generating the anomalous term in ( Z 2 inv I c λ 3.4 − even anom I 1 )–( I λ should be gauge invariant (the last term in ( Since neither the diagonal shift ( 3.3 = µν a discussed in detail in [ and will construct Note that does not have to. mix parity even and odd parts, decomposed as where under gauge transformations.we Given can that also write We now construct the hydrodynamicanomaly. action for We a can parity-violating write system the with action a as local U(1) where 3.2 Parity odd action Note that the equationT for leading order in λ we can write equations ( In ( of them and so are ( and from diffeomorphism invariance of JHEP01(2019)043 and s (3.17) (3.20) (3.15) (3.22) (3.18) (3.21) (3.14) (3.16) ϕ ). Again 2.14 -fields (order a ), we take the )–( 3.2 should also not be )] (3.13) 2 2.13 X inv ). Under a diagonal , ( o ~ ρλ 2 ( , we then have I F F ) (3.19) respectively, which then O ν a ), . Note that under gauge inv ∧ , 2 I B ) µν X 2 1 γδ ∧ F ). To see this, for two terms 2.23 F F . X µνρλ ( , 2 aµ αβ ϕF ~ 3.13 and F γδ F C c ). At linear order in F ∧ F 2 µ 1 o F a is invariant under gauge transforma- ˆ , the equations of motion of ϕ + 2 + J is of order F X X 2 , αβ ) µ αβγδ i − 3.21 F , 1 L + ˆ µ F inv J σ ) ˆ µ B γδ ∧ J ( 1 + . ) from ( A F λ anom aµν a ν X F = ~ I αβγδ ( ˆ a F G αβ 3.2 1 ~ c J 4 1 Z ϕ ~ ρλ F F ( ~ = µν µν c o − F c 4 ˆ – 14 – 3 a T F ∧ ν x ν 4 1 2 ) ˆ is the pull-back of αβγδ A J 1 = = d dC = 2 ~ = µν µ µ X Z = ( ν c 4 ˆ AB ) to be invariant under ( F µνρλ 1 J 1 ˆ ~ a T inv µ , F c anom F ν = = ~ F o 1 c ∇ , 3.10 depends only on δI µ µ ∇ L = ϕ [ ˆ ν A ). Note that 1 ~ J + 1 ~ . To match with the anomalous term in ( ˆ σ µ T inv µ Z ν ˆ I ∇ J 2 and each equation should be understood to have two copies. ) transforms as c anom 2.19 and rewriting the resulting expressions in physical spacetime we ∇ , anom I ) makes it manifest that = I ~ = 1 ~ µ 3.19 = 1 ) we precisely recover ( ˆ 3.16 J s anom I 2.12 1 ~ ) and that ) becomes are functions of 3.13 is defined as the off-shell currents corresponding to 3.13 2 ), equation ( , was defined in ( µ 1 µ ) one changes the integration variables to inv ˆ X J aµ 2.23 C Expanding in small Given ( Let us first look at lead to )) we can write 3.13 a µ s ( In order for the fullinvariant odd and action its ( variation shouldO precisely cancel ( and shift ( find that ( where The last equality of ( tions. Defining the covariant off-shell currents as where where the off-shell stress tensors andwe consistent have currents suppressed are defined in ( in ( become dummy variables. X anomalous action as (written in fluid spacetime) where transformations ( JHEP01(2019)043 ). aµ 2 (3.28) (3.29) (3.25) (3.26) (3.27) (3.32) (3.30) (3.31) (3.23) (3.24) C a should ( ). Using µ o ρ O j ˆ will never B ] for details) 3.28 νλ F 58 aµν ), ( G µνλρ λρ 3.24 λρ λρ ) and thus ~ F c and derivatives. F F ν u ~ − µν µν 3.21 F µ F o a a j ˆ )) to zeroth order in deriva- µνλρ ϕ , ϕ 2 a 1 2 µν o + ( µνλρ µνλρ O ≡ µ o aµν + Σ j ~ ˆ ~ µ ) c G c 4 4 B ν ) o + µ should be diagonal shift invariant by o µ µ ν o q q aνN ρ + + q µ B B 0 ( µν µ 2 B G 2 o p . ( u aµ aµ g h 0 T , u ). µ νλ o C + C n ) j ˆ x + + F ), aµM ( 0 λ + 2 ). µ µ ρ µ ρ = u ) + G -fields (order x are given respectively by ( ρ ( ω ω can be written as µν a A A − µνλρ 1 1 2.23 µ µ – 15 – o aλρ µ g ∆ . Since terms proportional to h odd νλ νλ µ j ˆ o o ~ ˆ F µ j ˆ odd J F , − ∇ F c p = = u ) and ˆ µν,MN µ o o L ρ µν − ≡ x µ q o µ + o in the Laudau frame (see section VI of [ u ( q ) and ˆ W j ˆ µ o λ µνλρ ν µνλρ = β ϕF x i ` 4 u ( ∇ µ ) under ( ~ ~ o odd µ ( ˆ β c c J u + 2 L ν o ), and − u − ε which as usual can be decomposed as 3.21 which can be written as λρ µ µ o o F ` j ˆ ν = o µν 3.20 o q µνλρ ˆ µν T ) we thus find that µν o F + 1 2 = ˆ a T ϕ ≡ 3.16 ( are transverse to aµν odd µ L G ω ) µν o ν o µνλρ are respectively zeroth order energy, pressure and charge densities. is parity odd and is diagonal shift invariant. Such a term does not exist at q should then be transverse and can be written as 0 µ ~ ( µ are some functions of o c 4 u is defined by ( j ˆ , n 2 0 are some functions of and Σ = = µν,MN , h o 2 , p aλρ given by ν o 1 can be written as 0 q F h W , g µ o µ odd 1 o ` ˆ g Collecting the above expressions, Now let us consider quadratic terms in As discussed above to first derivative order since there is no diagonal shift invariant J Let us first consider L 1 ~ with where where field redefinitions one can write where zero derivative order so we conclude there are no new parity-odd terms at order where tive, which should have the form be invariant. From ( scalar term and where the variation of the first term under diagonal shift cancels ( non-vanishing quantity is where where generate a term of theitself. form At ( zeroth derivative order there is no such term. At first derivative order the only and the terms on the right hand side may be further expanded in JHEP01(2019)043 2 B , h 1 ). In , h ). We (3.38) (3.33) (3.34) (3.35) (3.37) ~ 2 ( , g O 1 φ 3.10 g α 13 ∂ α ). β . ρσ 2.33 ) F x )) . , note that is of order µν )( F i o ) are adequate. F ˆ iβ inv J , µ L o − 2.33 I B anom , µνρσ + β I 0 o ˆ 1 ~ a L J ~ 4 µ F o c ) are such that even if ˆ ), ( J ) to be a total derivative. More − (3.36) ∧ . ) = ( + 2.33 3.3 ) ) i are undetermined at the moment. F 2 µν αµ , x 3.38 ϕF = 0 g 0 ∧ F c β , h ) on the parity-odd action ( x ) α 1 L + 2 β − F = ( , h · µν F 2 o 2.31 µ + 2 o ˆ ˜ ˆ ). Thus in our discussion below it is enough β h T J ˆ ∧ µ ( ~ , g 1 2 ( µ and will see that they lead to very different 1 ϕ = F ∂ g ( O ) ) corrections in ( 2 d )( ~ – 16 – g ϕ x ( , ρ µ 4 ν ) ) terms the leading terms in ( iβ O = d A CPT Z β a x B L ( , µ 1 β ~ νλ )( O Z h ) while i T o L + ∇ F ic i , q ~ µ ν o o a ( = ˆ q J − + ϕ µ 1 , O + 2 µνλρ PT + 0 g u o (( q on the right hand side of ( inv ~ , ~ − µν o ) is Z c ), we find after some algebraic manipulations (see appendix ~ g ~ I anom ~ β − I c ) = ( ~ odd for useful formulae) L = i µ 3.26 o . For dynamical KMS transformation of 3.28 L ) then requires j ˆ = = ν µν B o ( , x 2 a ˆ inv 0 ( T )–( , ~ dx − − o x 1 2 O ˜ I ~ µν − = anom ( 3.24 ˜ F I µ x o µ ˜ 4 q β odd d T PT L , at first derivative order for ≡ 0 Z 1 ~ ) we find that under dynamical KMS transformation, the anomalous action ~ i F to be invariant, we need the second term of ( − · 3.19 the first term in ( β odd odd ˜ Due to the presence of I inv L In fact one can check that the structure of , o 1 ~ 13 I 1 ~ for useful formulae) are of order For explicitly, using ( where We then find that becomes (see appendix 3.3.2 Θ = From ( KMS invariance at We will later argue that theyto should also consider be the leading order terms in dynamical3.3.1 KMS transformations ( Θ = We find in this case We now impose the dynamicalwill KMS consider condition respectively ( Θresults. = 3.3 Dynamical KMS condition JHEP01(2019)043 2 , g 1 (3.44) (3.45) (3.46) (3.39) (3.40) (3.41) (3.50) (3.49) (3.42) (3.43) , g 2 , , h du 1 . 1 ) . Similarly h ) ∧ 1 h 3 F µ u a H (ˆ = ∧ . 2 ∧ 1 + ) ) h H x 2 ˆ µB µ − ( ˆ µ (ˆ a dβ ) (3.47) 2 µ 1 d a x g ( + B ~ ) = ) du . c + ˆ − µ , ∂ ˆ µν µ 2 , 2 2 1 2 ∧ ˆ µ F is fully determined up to 1 ˆ − g a µ β − ˆ d µ, β β ( u a 1 2 1 − 2 1 L a − β h du i ( + 2 odd H = 2 + 2 I + ∧ 2 dQ, h 2 +( β + ) (3.48) 3 ˆ ˆ µ µ , h ) + ˆ ) x µ du ~ ~ H F x Z ( F c c µ ~ β ( β ) + i µ (ˆ ∧ c 6 3 2 1 ∧ 1 x B u g − − − ( = h ) aµν u µ and satisfy the following relations ∧ 2 , ∂ + 2 F ) = = = ω 2 2 H ) µ ˆ F h H µ, β 1 2 2 ) = dβ B ˆ µ ∧ + ,H − ) = 2 = = ) can be written as β ( 1 ˆ g µ, β u x − 2 2 F L ( β H g − ∧ − − 1 , we now have µ 2 o ), H ( ( ∧ ) ˆ 3.41 ˆ J µ 1 ˆ µ ) level requires – 17 – ω inv , g d h , A a ) + · 2 aµν + o 3 1 ( )–( x 3.41 , h ˜ I β a h a − ( F ) O ( µν µ ρ µ 2 + )–( g , ∂ β, g (ˆ g +( ω 3.40 B + 2 2 1 2 β ) g g F g + F ˆ L µ νλ , 2 3.40 ∧ ∧ = = F ,H, h µν a ˆ du o µ, β µν 2 2 ˆ 1 1 ) = F T ∧ − F a a ˆ µ ˆ ( 1 2 µB H + 2 u µνλρ 1 is very similar. Note that 2 ~ − ˆ + + µ 3 2 g c ( ∧ ~ ~ ) = ˆ ˆ ˆ 2 µ µ µ 2 β ) − c c x 1 1 1 x , h . With ( 4 − F 2 a a a 1 · − d 2 CPT a ( +3 H = , g β ) = ) = β β , but that constant can be absorbed in the definition of 2 ) ( Z + 2 + 2 + 2 x x µν 1 ) again applies. For = β µ 2 i Q ˜ + 2 3 2 2 F (ˆ − − 1 h βh . For the above expression to be a total derivative we find that β ˆ ˆ ˆ ( ( β 1 µ µ µ g ˆ µ are constants. Thus to first derivative order µ 2 µ µ g o ~ ~ ~ o + ~ , 3.36 ˜ ˜ g is defined only up to a closed three-form as such an addition will not q c c c 3 J c = − 1 CPT dx 2 3 3 6 g ). 1 µ , a Z Q − − − − 2 − u H i is a constant. Note that one could add a constant to the right hand side of ) = , a = = = ˆ = = 3.42 µ ≡ 1 = 1 2 1 1 a 2 1 − a u g h ( h H H 1 The most general solutions to ( β g ∂ and the dynamical KMS condition at The analysis for Θ = and equation ( where three constants. 3.3.3 Θ = Note that change ( equation 2 with equation where where must arise from derivatives of two functions JHEP01(2019)043 ) ). to ] did 3.50 3.31 (3.54) (3.57) (3.51) (3.52) (3.53) (3.55) (3.56) 13 CPT , µ 2 ω a ) or ( to zero. 3 3 a + 3 β 2 a , 2 2 ). Ref. [ 3.30 µ ˆ 0 µ β + = p B ~ 0 c 2 3.57 + 3 + n a µ 0 µ − B )–( = ω 3 3 1 = + ˆ µ β ) apply. Imposing ( a a β 2 2 µ 3.55 can be obtained by setting ω ) − 3.43 µ α to be used in ( β ) = 0. , g 2ˆ µ α o µ − 3 )–( ˆ µ . j ˆ 2 2 a CPT ω ], confirming that these methods µ 2 2 a (1 0 − ω a = β 13 1 p . 3.40 2 2 0 a and β 1 0 (1 + 2 a β 2 + 3 + n a p µ o 2 2 3 ) β = 0, and thus 0 0 q a β 0 ˆ + + µ + n 3 µ p ~ 3 a 0 3 c ˆ B µ a µ µ β + 2 0 ) + 2 = 2 2 B + 0 B n α − ) a β 1 − µ µ ( o α a + . − j ˆ = ω β − – 18 – µ − ˆ ) is then given by µ µ ˆ 1 ) µ β (1 2 ω ≡ B α ˆ a µ ˆ µ β 0 2 can be obtained by setting 1 α and p 3.32 − ˆ µ β a + , g + (2 ( 2 0 2 1 2 µ o 3 ˆ + β 1 µ n β a µ (1 µ o q ˆ µ a ~ PT β 2 0 ` 1 β + β c 2 ~ a β ˆ µω c 6 . The expressions for Θ = + 2 − + − − T α ) we find that 3 µ 2 α µ 3 ) reproduce previous results in the literature obtained from en- = = ) + ω µ 2 is fully determined up to a single constant. B ) α 1 2 − B 3.46 α − µ 1 2 − 3.57 + 3 ) we find that odd 1 B )–( + 2 I ) terms is the same as before and ( − µ ,H , h B (2 2 2 0 )–( µ ξ 2 2 invariant theories one should set ω ˆ ˆ a µ µ β β (1 ] and equilibrium partition function [ ω a a 3.46 ( µ 3.44 ˆ ~ ~ + µ β β ˆ µ β 2ˆ 9 c c O 3.55 + + , ~ µ 2 2 )–( 3 3 c 8 2 2 2 a β ω 3 CPT − − β ˆ ˆ µ µ β 2 2 ˆ ω µ β ξ − ~ ~ ˆ + µ β 3.45 ~ = = c c c ~ 3 3 = = ω c 3 B ξ = 0, while those for Θ = ξ − − µ o − − ` 3 a = = = = Equations ( From ( 1 2 µ µ o o = h j q ˆ H 1 tropy current [ indeed give the complete answerpinpoint for the the exact current discrete problem. symmetrypoint However, a those out system methods for should did have not for ( where we have introduced and The frame independent quantity We can now write down theIt explicit is expressions enough for to doa it for Θ = Thus for a macroscopicfirst system derivative whose order underlying Hamiltonian is invariant under 3.4 Explicit expressions for on the solutions ( The analysis for JHEP01(2019)043 )– ) in (3.64) (3.59) (3.60) (3.61) (3.63) (3.65) (3.62) 3.53 3.55 . ) as α in ( )–( ˆ 2 µ a 3 3.58 , , a − 3.53 . µ 2 ) dv ) ) from respective w α , a λ ∧ 3 µ 1 b − a v β ρ a w 3 2 3.25 (1 a α anomalies. When the is defined only up to a 1 . = 0 (3.58) 3 ˆ µ a − ∇ ~ + a ) + µ Q 2 ρ µ v b db . µ λ B − ) + 2 ) ∧ global ∇ α µ + ( v µ 2 w µ ν − a ˆ B v µ w , b (2 ˆ µ α µ + µ − ˆ µ (ˆ β , u ~ i µ µνλρ − 2 µ 6= 0. We now present ( db c µ β a µ B 3 = w 1 2 µ ∧ ( B 2 w − µ B + b ) ) a β = 1 µ α α µ = which were defined in ( a u + µ = , terms proportional to B µ − − B µ + µ B ˆ µ B T B B 1 , , v (2 (1 β – 19 – 1 a µ µ , ξ dv a 2 − ω 2 should be proportional to ) can be rewritten as µv α , a µ ∧ 3 3 , ) ˆ , µ b w + ˆ 2 a + ) + λ , µ 2 ) + 2 ˆ 2 µ µ v 1 3.55 µ ˆ µ µ µ w ρ a B 2 B − µ B B − 3 3 α ) ) β 2 B = found in the last section to obtain the parity-odd part of α α . But note that − ∇ − db = − B µ ν − ρ − ξ ) and ( µ − 1 ∧ µ B B v µ odd w λ µ ν ω + (1 w I (1 h 2 µ µb 1 ∇ ) can be written more transparently in the basis of ( µ µ 3.54 (ˆ invaraince, only a global mixed gravitational anomaly is present. a ˆ 2ˆ µ ( a (ˆ and w 2 ~ ˆ ν µ = ∆ + c ˆ )–( w µ µ ~ v ~ β 3.43 ξ µ c + 2 ~ u c b c 3 α − CPT = 3 = 3.53 µνλρ of ( = = 3 µ o − = = 0, and ` 1 2 µ µ o o 1 Q µ Q j q ˆ b = µ 2 v ˆ µ µ ~ w c ) are respectively associated with global U(1), mixed gravitational, and gravitational Similarly We presented our results in terms of = 3 w = 0, all the parity-odd transport terms are connected to ξ 3.57 c underlying theory is only( invariant under anomalies. With This connection also implies that 4 Equilibrium partition functionIn and this global section gravitational wehydrodynamical anomalies first explain effective how action. toapply obtain the We the equilibrium procedures discuss partition to the two function equilibrium different from partition ways the function. of doing We will it. see that We in then the absence local anomalies, i.e. where we have dropped anclosed exact three-form. three-form as mentioned earlier with and Then equations ( Note that “field strengths” of a slightly different basis which makes their expressions a bit more transparent. Introduce and JHEP01(2019)043 ] 14 as 57 (4.6) (4.3) (4.4) (4.5) (4.1) (4.2) W and their . i i 0 φ b g 00 g = − i √ ) becomes identically = b, u 2.1 − to Euclidean signature, = ) to time-independent field ˜ W of ( 0 ··· b, b . 0 2.34 W β as follows. One first obtains the ] + )] i , ) to the following equilibrium values 2 ~x , , u = ( -field expansion we can write 0 ~x φ )] i [ ( a i ~x A φ hydro [ ( ˜ 0 I i W hydro β i δφ ˜ , φ W )] is a functional defined on the spatial I δ ~x F − = , β [Θ ( 0 ] i β i ˆ satisfying the combined Θ and KMS trans- µ = 0) ˜ ~ as 1 φ W − , [ φ φ e [ i Z W ˜ , (1 ) W – 20 – µ ˜ = b 1 labelling different operators/components. In [ . W )] = i , ~x A ), i F ~x Z 2 = ( 0 i a = β µ ) should have a local expansion in terms of external ( ] = = 0 φ i µ [ ˜ 2 O W ω ˜ ( W i , φ ) is given by the thermal density matrix with an inverse i with 1 ,B hO , u φ ) implies that ˜ 2.1 [ aµ W with index ], the equilibrium partition function can be extracted from the A − i W = 0 4.1 in ( from the contact terms in O 59 µ expansion. a = (and thus – 0 X ~ ˜ ρ W aµ 57 F = a ϕ ,C by setting the dynamical fields in for an explicit example of the continuation. ) can be “factorized” in the stationary limit. That is, when the sources . By definition the generating functional s = 0 aµν I β 4.2 g ϕ 2.32 denotes terms of order = ) we can identify is the free energy, and doing analytic continuation of are time independent, to leading order in the ··· F aµν 4.3 i ] we can obtain 2 G The free energy For notational simplicity we will now denote the sources collectively by See section 57 , φ i 14 1 which give sources, as the equilibrium partitionEuclidean function manifold can with be a computed periodic byin putting [ time the circle, system which on generatessource a action a finite gap. As discussed where from ( Writing the equilibrium partition function manifold of the spacetime, and satisfies where Θ here shouldconfigurations. be Equation understood ( as the extension of ( formation ( φ where effective action with theleading order help in of small the dynamical KMS condition.corresponding operators We will againit was work shown to that a generating functional We will now describe two methodseffective of action obtaining the when equilibrium partitiontemperature function from the zero when we setalready indicated the in external [ fields for the two legs to be the same. Nevertheless, as 4.1 Equilibrium partition function from effective action JHEP01(2019)043 ]. ; to even 58 I (4.7) (4.8) (4.9) -field T (4.12) (4.10) a ], we can ) and local to equilib- 59 . Using the 0 i 0 2.31 dv V dx ∧ i v u ] (4.11) 2 3 0 ) as shown in [ a ≡ du β ∧ + 2.31 , u i u 1 db dx ). H i ∧ A v + F following the procedures dis- 0 0 2 3.50 to satisfy the combination of Θ and ≡ β a β there is no parity-odd contribu- s dA − I odd can be further expanded in terms A + I ∧ ··· = µ u db PT 2 V Z ∧ ] + H µ 2 b ··· is factorizable at this order is warranted )). φ V 1 + [ µ s a + I ˜ ∂ = log W µ + 0 dA ] it was shown the dynamical KMS ( 2.33 + eq vanishes in the stationary limit. − 58

∧ iV ] 0 0 L 1 0 – 21 – 1 dv manifold with ˆ A = V φ = 0 together with ( = ) [ ∧ gV µ 1 ). That 3 ] (see ( ˜ b , ˜ L a 2 − r V W 2 0 1 a ˜ Λ a √ ( A + = was imposed by requiring ) we immediately obtain that Θ x spatial 15 O ˆ µ s − , 1 ~ I a ) implies that − c d ˜ db Λ 2 hydro 4.10 ) can be written more transparently as ] and the dynamical KMS condition ( d I ∧ − [Θ 2.31 57 b Z [( L 0 4.11 A Z = 2 ), and ( ˜ L = 0 -ﬁelds only. From the discussion of the entropy current in [ Z β r 3.43 ~ one needs to take c ). log − ), ( ), equation ( 4.5 denotes the expression obtained by setting dynamical ﬁelds in 16 CPT Z 3.36 denotes terms of order contains eq

3.65 ]. = is defined as 0 ] the KMS condition on µ 0 0 ˜ Z V ··· L V 57 14 , From ( The equivalence of the two methods can be considered as a consequence of equivalence There is also an alternative way to obtain the equilibrium free energy as follows. The -fields as In [ See equation (3.14) there. The second term log 13 a 15 16 KMS, dubbed the localKMS KMS conditions are condition equivalent. there. In [ where the integration isbasis over of the ( the same answers. Thedescribe second here. approach is Recall significantly thattion simpler from to technically, our the which partition analysis we function for will obtain at Θ Θ first = = derivative order. The results below are for Θ = in [ 4.2 Parity-odd equilibriumWe partition now function obtain and the globalcussed parity-odd anomalies in partition the function previous from subsection. It can be readily checked that the two approaches give where rium values ( of local KMS conditionOne of [ can readily checkthe two that methods applied indeed to give the the same parity answers even and are part equivalent of to the the results effective discussed action where then identify dynamical KMS condition ( where of expansion where by the dynamical KMS condition. All external fields are taken to be time independent. Then to leading order in the JHEP01(2019)043 1 a ) we . We (4.22) (4.20) (4.13) (4.15) (4.16) (4.17) (4.18) (4.19) (4.21) (4.14) L ), under . 4.20 ~x ( f dv + i has size t ∧ . 1 v 0 dx → 2 i 3 0 S A a i t β A v , − + i + ), and the background b 0 i = 0. Then in ( dt db ). The background fields β 0 A c → , where ∧ 2 A j i = , we get the replacements v S 4.12 ) under which i 0 2 = dx ~x iτ i a β × ( µ i 1 λ dx i 0 S dx λ . − Z ∂ ij i → − µ f, b a ]. i , b ∂ iA ∈ t + db ∂ , i.e. i i + 13 0 0 2 00 ) as flavor U(1). + ∧ → g g 2 S i A A ) b b 0 i − 1 i + 4.16 ,A db . → a i j → dx = i , n dx i ∧ i i + A i v A dx b i A local anomalies, i.e. q πn − → ,A 2 – 22 – Z i dv + dx on a circle with period i 1 no ij iv dτ = ∧ a τ , b ( , v a dτ i b i 0 v 2 0 00 db + g A dx 2 A → − 2 i S → b i = = i − Z (with i v f, A i 2 v µ = ∂ dx db iτ i ds dx v ) is preserved by time reparameterizations − ∧ µ ) as time U(1) and ( i b − A v 0 → − 4.14 , b A dt 4.15 t i → 2 i dx v 00 0 i g β v ~ ) becomes = = ic invariant terms become pure imaginary while the terms proportional to real. 2 ) precisely agrees with that given in [ v − to have a monopole configuration on ds 4.12 0. Thus, under the analytic continutaion b 4.12 Z 0. Note that ( CPT > = < ) are those for a stationary Lorentzian manifold with remain 00 g 3 Z 00 a g Now let us consider a system with The thermal partition function is usually calculated by analytically continuing to Eu- Let us now explore a bit further physical implications of ( 4.12 log Let us take the spatialcan manifold choose to have the topology of have three Chern-Simons terms, respectively,and for time flavor U(1). U(1), mixedunder A time “large” defining and gauge flavor feature transformations U(1), of i.e.for those Chern-Simons example are terms the not flavor is connected U(1) that to Chern-Simons the they term identity. are Consider not invariant Note that and Here, after which ( fields are taken so thatand they gauge are field real to in Euclidean be signature. of form We take the Euclidean metric Below we will refer to ( clidean signature with and which and time-independent U(1) transformations Equation ( in ( where JHEP01(2019)043 , 2 )– S (5.1) (4.28) (4.23) (4.24) (4.25) (4.26) (4.27) 3.61 on i v ) does not have 3.19 . , ] in some free theory models. CPT 55 ··· , with mixed global gravitational Z Z + 2 54 ) we have Z ∈ a ∈ ) the latter has the form (now also breaks aµ ∈ a is 3 C ( 4.25 a x Z. µ 0 Z. Z O ˆ 2 v J 1 a kl or π 3 q + 2 , k , m a 2 1 mna , l q 2 0 a 2 0 6= 0, then the transport coefficients in ( − π ink L aµν 8 e πm c kβ lβ qL is the circle direction) is − e 2 – 23 – are terms, being real, are not mixed with local G e x = + → )) ( 3 µν + → x → ˆ Z x x T v , a dv couples to matter as a U(1) gauge field with minimal b b Z 1 1 2 Z 2 3.11 v a S → → = Z x x v b inv ] and shown explicitly in [ ) we need to consider a monopole configuration for L 53 1 ~ 4.20 ) the partition function transforms by a real number rather than ] 4.28 ). One thing to notice is that the anomalous action ( 78 , 77 2.45 ) and ( ,thus a large gauge transformation of 0 π β 2 4.24 is the minimal charge under U(1). q ) are then mixed among local and global anomalies. The same thing happens to the Possible connections of the term proportional to In the presence of a local anomaly, i.e. Thus we find in the absence of local anomaly, all the anomalous transports are asso- For the last term in ( Under Kaluza-Klein reduction, A large gauge transformation of 3.63 by applying ( the same structure ofincluding the the rest parity-even part, of see the ( action. At anomaly was first hinted in [ 5 Entropy current In this section we obtain the entropy current for a (3+1)-dimensional parity-violating fluid four-manifold with a thermal time circle. ( partition function. Butanomalies. note that Note in ( a phase. As mentioned earlier non-vanishing ciated with global gauge or gravitational anomalies for putting the system on a Euclidean and then under just a large gauge transformation ( we have “charge” we then have [ where JHEP01(2019)043 ) 5.3 (5.5) (5.6) (5.7) (5.2) (5.4) ) µ w , giving the there. µ Q 2 )ˆ . Equation ( Q α αβ 17 αβ − F µ F µ ν ν (1 ω ω A 1 A − H µ ) µ ω α + µναβ αβ 1 B µναβ ) µ F 1 H 2 1 ν α a B 2 β a 2 1 2 + B has a different structure does (1 + 2 H µ + 3 ]. It was denoted as + = 0. Furthermore, there is also B a µ + ((1 + µ 2 58 µναβ 3 B 2 anom is simply the dual of B a + H 2 ˆ αβ a I µ 0 (5.3) ˆ a µ µ ~ F 0 + = ) 2 c ν ≥ V µ 2 α ) + 2 A + 2 + αβ µ β β αa a B F 2 ˆ 2 µ µ w even ν 1 ˆ a ˆ = µ J (1 + a µναβ R A ˆ 2 µ 2 1 − β . 1 a + 2 a – 24 – 1) + = 2 − a µ µ + µ µ µναβ + − ν µ B B w S β α 3 ω 1 α µ + 1) 2 a a 3 ˆ )(3 ∂ µ µν (2 µ o a 1 ˆ ]. The fact that − α j T β ˆ 2 ] when a ˆ 2 ˆ + 3 µ 8 µ 2 α is KMS invariant by itself. We can then simply apply + 59 − − 1 ( β + 3 which will generate an entropy current with non-negative ρ µ o − a ˆ b µ µ ˆ 0 µ (1 1 λ = 0. After dropping duals of exact three forms, the vector ν 2 2 ) + ˆ V 2 µ` inv a ∂ β a ˆ µ α I 1 anom ν ) β ) + − ~ = I b ν a o α c − q + 2 µ = + 2 µ ) we find that µ ( − + S 2 ) to (1 µ ), for the parity-odd part, o µνλρ u ˆ ρ w µ 2 S 2 (1 ) b 1 1 ˆ µ 2.45 λ 3 a α − a . The entropy current in the Landau frame is then given by β ~ 2.45 ] when 3.43 ∂ ˆ µ c c 2 9 ν ~ 3 = = = β b c )–( µ o 2 S (1 + 2 + 3 µνλρ 3.42 to be divergence of the entropy current of the parity-even part. a = 1 a is given explicitly in equation (5.40) of section V C of [ + µ o S even = R even µ o From ( Now applying ( R S 17 6 Parity-violating action inLet 2 us + now 1-dimension consider the actionare for parity-violating exactly terms parallel in to 2+1-dimension. those The of procedures the 3 + 1-dimensional story. So we will be brief, only giving agreement with [ above can be written in the new basis introduced here as The parts of theframe expression entropy which current involve the given anomaly in coefficient [ agree with the Landau which gives where we have dropped ais term independent which of is dual to an exact 3-form. Note that this expression following odd-parity contribution to the entropy and with means that parity-odd part does not contribute to entropy dissipation. not cause athe problem, procedure as of ( divergence. which is the form assumed in [ JHEP01(2019)043 18 (6.7) (6.9) (6.1) (6.2) (6.3) (6.4) (6.5) (6.6) (6.15) (6.11) (6.10) (6.12) (6.13) (6.14) ˆ µ 2 λ ) ∂ ν ) (6.8) u α 2 ). The results u ] from entropy s a<µν> 2 α µνλ . f 63 λ G . ∇ ( 2.43 aν ˆ t µ = + ∂u λρ C ν µν o µ 4 aµν 1 + g u β, s σ G o 1 − . aα aµα f η µνλ µν λ C G , , β, t u α = = − ∆ α λ ρ γδ u µν µν o u o ∂ = F ∇ µν o ∆ ν aµν ρ µν ν aµν u Σ u γδ + u G + Σ ∆ λ G ν as defined in ( ρ ) C ν − ∇ 12 u ν µνλ o u 2 u r µ q αβ λ ˆ µ µ L µ u νλρ ν ( = ∆ ∇ u aµ are functions of ) as +2 ( ν 12 u ∇ µ 3 1 13 µ C s o , n ν s aν µ νρ 1 2 o 1 2 6.3 µ − ˆ , η + 2 s J C − ∆ νλ 4 d , t 2 ∆ − , ] using stationary partition function. )–( λ 1 3 h F aµ , 2 + µν , and ∆ 2 µλ as ) − − µ µ i 2 C v aαβ , t ρ ∆ + 14 ) u β ∆ 6.2 1 i ν , µ o o u . For this purpose let us list all the parity-odd l G 1 aµν µν ˆ – 25 – u aµν p , l J µ αβ s = ≡ λρ 13 µ ( 4 o G µνλ ∆ 1 u , αβ 4 G F =1 j + µ 3 i C h X ν , µνλ 22 2 µν µν ∆ 1 2 o µν 2 ν u r , + and ˆ T u 1 = = β ν aµ µ = = µ + 1 2 (∆ o µνλ µ o C u ∆ , k 2 µν µ 2 u o o 2 µ 22 o α µ ˆ , T s ε n aνβ = u 1 . 1 4 , s G 23 , f λρ = = λ β, v s 2 = ∆ ρ aλρ + F , , t u ν µ o 1 u i 2 + aµα λ ν ˆ µν G ∂ o ) J 1 ρ 2 νλρ ν ˆ ρλ µν T G , h ∇ v ) , p ) h µ u 2 µ ν µ ν 2 , µ aµν C µ s aµ 1 u ∆ u u aν are transverse to 2 1 g ν ν G ∆ , j C ( C g ν 1 2 u − µ i µ λ µ µν o µνλ µνλ t u β u i σ + u αβ µ , σ ( k 1 µ u − = = = ∆ 33 s ( µνλ 4 1 µ 1 s µ 1 ∇ =1 11 i t s 11 µν g X o µ 1 4 and Σ r r s σ ∂u u ) the hydro Lagrangian has terms 1 4 = = µ + + + o a = ≡ o ( µ o , j = ε ) the complete action at zero derivative order is q ν µ o ∂ O 2 1 q a (2) v ( L At i Note the identities O tensors : scalars : vectors : 18 − Note that only thesymmetric last traceless term part is of parity-odd. the corresponding Here, tensor, angular e.g., brackets deonte the transverse where all coefficients At We can then expand various quantities in ( where we have introduced scalars, vectors, and tensors which are diagonal shift invariant at first derivative order and as usual we can decompose where below are fully consistentcurrent analysis with and the those constitutive presented relations in [ presented in [ the main results. We will again work to the level of JHEP01(2019)043 . ρ u which (6.16) (6.17) (6.19) (6.25) (6.21) (6.22) (6.23) (6.24) (6.18) (6.20) ) were aλρ 1 G l aν 6.13 C . µ and )–( ρ u ) does not lead 2 u k 6.11 µνλ aλρ 6.17 . G µ o i r ρ q ] for details) 1 aν 0 v + C p 58 ν µ 0 2 aµ + v u n C µ 0 ) ε u . ), leads to various constraints µνλ µ o 0 j aν − µνρ C i r ≥ + µ o µ 2.29 ) again depends very much on the ν µ 2 j µ + 2 12 ∆ u ˆ rβ µJ r . = o µ o aµ 2.31 ` − 2 n µ o − − 4 ) are zero, except for r C βr . + ν µ 1 1 2 22 + ( t β ≥ ) = r 6.10 2 L = µν aµν 11 βr 2 µα 12 aµν , ` 1 T r G F = 0 l as (see section VI of [ )–( + G α ˜ ) − µν − n – 26 – L = β defined by 6.9 ∂ µ ∆ µν o odd 2 22 o aµν o , r µ o − k r L n θ can then be written as 0 pβ G ˆ µ , ` ) 11 1 2 − 2 + Σ o − µ r µ 2 > = θ ) t ∂ L 0 is nonzero, the second inequality of the above becomes = ν o ( ν µ p 22 q µ u o r ε S ˆ µ ∂ J , r ( the parity-odd action can be written as odd o + u . The constraints on the parity even part ( 2 11 ε + L ν 2 r t L ν (2) − µ + 2 β L o u µ ( p ]. Among other constraints we have µν ∇ 4 = ∆ 57 βr can be defined away using field redefinitions, so ( o µν o o p − ˆ r θ T + = ), there is no parity-odd contribution to the thermal partition function to are respectively zeroth order energy, pressure and charge densities. Note ν u 0 odd µ 6.23 u L , n T o 0 ε ( , p 1 2 0 = Due to ( The outcome of the dynamical KMS condition ( To summarize, to level odd first derivative order. The entropy current is given by The above Lagrangian satisfies which can be seen by noting the relation satisfy the relation The full parity-odd action to level to new constraints on transport coefficients. choice Θ, which we will discuss separately. 6.1 Θ = In this case, we find all coefficients in ( where that the coefficient Using field redefinitions one can write with frame independent quantities L among the coefficients of analyzed in detail in [ When the parity-odd coefficient Non-negativity of the imaginary part of the action, eq. ( JHEP01(2019)043 and (6.39) (6.36) (6.37) (6.38) (6.26) (6.27) (6.31) (6.33) (6.34) (6.35) (6.28) (6.29) (6.30) µ . µ 4 . 4 ) Y t ) should satisfy 4 ˆ µ βl ∂ βk = 1 β ( 6.10 β 2 ∂ f + such that )–( µ 3 − νρ − Y F such that ) 6.9 µ Y t 0 3 β V µνρ X ρ ∂ µ βk , 1 X. ( 1 3 β ˆ v µ ∇ 2 ˆ µ Y ν ) (6.32) 1 ∂ , l βl ∂ 2 3 + ˆ . µ v + k µν o = µ 2 ∂ ) = µ µ = = = ∂Y 2 . , 0 t σ µ 4 u β µ 4 t 2 o 2 2 4 µ V Y αµ l t ( ) 2 = η k µ ) ˆ µ v F + Y 4 µνρ + ∂ ∇ 0 α + Y + i 4 p µ 1 2 ) β − 3 ) we then have k t 0 = ν − o 2 = 1 + q n + l 3 rβ X µ 0 X ˆ µ ( ˆ µ 6.10 βl ˆ + µ + ∂ u µ odd , βl – 27 – ∂ ( µ − ∂ ( )–( ( 1 u β 2 ) + µ + 2 even = 1 o ∂β ∂Y 4 − L β ˆ , g J + 6.9 ν k X, β R µ 4 2 2 1 u β = µ 3 + + v µ odd β ∂ = βk 3 ) can be further written as Y s ( µ 3 ˜ ν u L ˆ µ Xt µ β β = βl = β ∂ ∂ S µ Xt 3 2 ∂ 1 µ 6.30 β k − ∇ f 1 , k β ∂ 2 ∇ 1 Y s − 2 β µν o s 1 β ) implies that there exists a function = β ˆ ) , l T = 0 ∂ ) µ 0 X 3 r + ) implies that there exists a function + ˆ µ V 6.30 , µ 2 βl 1 ∂ = t 3 ( 1 ˆ 2 µ l − 6.31 . Xs h βk ∂ o β + also include the parity-even part, and Y , η ∂ ( = = µ 1 µ 2 t 1 1 ) = β β 1 1 1 4 , l g h is the parity-even expression. Note that the second term in the right hand ,J k 1 ) vanishes by ideal fluid equation of motion TP βl , l ( µν = = = 1 β are unconstrained. Thus there are altogether six independent functions of ˆ even 6.26 µ µ o ∂ o ˆ µν q o o R J p, T ˆ T , η Applying the above relations to ( 2 X, Y, k , l : 1 with which gives It can be checked that the above expressions satisfy which upon using ( k β while the second equation of ( The first equation of ( 6.2 Θ = The dynamical KMS condition implies that the coefficients in ( where side of ( where JHEP01(2019)043 (6.47) (6.48) (6.49) (6.43) (6.44) (6.45) (6.46) (6.50) (6.51) (6.52) (6.53) (6.40) (6.41) (6.42) ρ u ) which makes µ λρ ˆ µJ F j 6.27 ν b i − u ∂ ν ij ) can be written as β µνλ , ) µν Y Y T 6.20 = + − µ 2 − t X νρ Y − ˆ µ 0 F µ 4 ∂ t ( A dx 1 . 0 1 ) µνρ β − β E 0 x 3 . β ( 5 p 5 χ 2 − Y x ˜ 0 Y ν ˜ Ω ν ˆ ≡ n µ ˆ f T µ + ˜ X + ∂ ∂ even ∂ ∂ µ 1 T 0 B , µ 4 ˜ k R p + t E ) j + + 2 0 ) Z v 1 3 M Y i n = Y ˜ ˜ ν ν ∂ ∂ β + 0 µ – 28 – − X , ij p − 2 β Ω S + , = 2 0 → → s µ ∂ X µ 3 0 + n 4 T 1 3 1 t X X, Y ˆ B µ β 1 M β ν ˜ ˜ 0 ∇ ν ν ˆ µ β β 2 ∂ T χ ∂ ∂ ( ∂ ˜ β χ = = ˜ ( ) 0 0 0 g + 0 p p p 1 β 1 − Y 0 0 p − X 0 + s ), the thermal partition function is obtained from the zeroth ). The above expression of the partition function agrees + n µ + 1 0 ∂ ∂ √ Ω + l ˜ 0 n E µ 3 x χ − − 0 0 t 2 p E ) 4.14 4.13 0 ( d = = = χ + n β + X o 2 Ω ], note that we need to first add the total derivative with zero diver- B β Z + ˜ 0 Y θ − χ ∂ χ β µ 2 ( 63 = Y 1 − β ) to their expression of the entropy current. This has the consequence ] are reproduced if we make the identifications ˜ β∂ σt ˆ µ 5 Z 2 ˜ − l − ∂ ν with dynamical fields set to their equilibrium values. We find that 63 + ρ µ µ log u = = = = 0 V µ o ˜ T σ E pβ ` µνρ ˜ ˜ χ χ − ]. = ( equivalent. As a result the number of independent functions reduce to four: . ν is as defined in ( µ o 14 µ 2 , i S t b −∇ 13 ˜ σ, η Finally let us note that the frame independent quantities ( For stationary sources ( To compare with [ The entropy current can then be obtained as and µ 1 Note that in thet above expressions we haveX,Y, used ideal fluid equation ( and where with [ component of in their expressions. Furtherand comparing eq. corresponding (3.24) terms, in we [ find eq. (3.22),eq. (3.23) gence of redefining with JHEP01(2019)043 , µ µ S (6.58) (6.57) (6.54) (6.55) (6.56) should be , Y ) µ, β and . (ˆ . µ µ 4 E βr t χ ) Y + ˜ be even functions of ˆ ) + ) should be modified to r − o 0 p 0 µ, β X 6.35 (ˆ ˆ µ ν + 1 , r, η 1 ∂ βr, l βn ) the covariant derivative of 2 0 ( v − − ρ 2 , f 2 1 1 β v + = 6.41 µ , g 1 ) = 1 + u l 4 k µ 3 2 , l − 2 ˆ µ, β µνρ 2 Xt 0 , l − 2 β p 3 ( 0 ∂ 1 rβ , k + 2 n 1 1 β 0 + k – 29 – + µ 2 should be an even function of ˆ , even , l t + ) µ ) ) are unchanged while ( R 1 X l βr 0 = ) of various tensors. For notational simplicity we have µ, β p 6.36 0 (ˆ µ − n + S 1 2 βr l 2.40 µ 0 ∇ )–( + ( ) + ) still apply except that for ( , k − ) and ( µ 1 2 t l 1 2.38 µ, β = 0 k 6.46 (ˆ 6.34 = 2 2 ) are unmodified, while = h k ˜ σ )–( µ o = be odd functions of ˆ − q 6.52 ), except that now ) and ( ) = 1 1 6.39 h )–( , f 2 ˆ µ, β 6.33 2.33 CPT , g − 6.47 3 ( ). )–( 2 , l k 4 k − 6.27 6.29 Eqs. ( Equations ( In this appendixmetries. we list They transformations are offormations important ( various for tensors obtaining undersuppressed the various the explicit discrete transformations forms of sym- the of the first dynamical arguments line KMS of of each trans- all table. the functions, which are given in DE-SC0012567. P. G. was supported by a Leo Kadanoff Fellowship. A Explicit expressions for various discrete transformations We would like tosteiner, thank Domingo Gallegos Umut Pazos, Gursoy, Andrey Kristan Sadofyev,Wang, Savdeep Jensen, Xiao-Gang Sethi, Dam Wen, Zohar Thanh and Komargodski, Son, AmosOffice Juven Karl of Yarom High for Land- Energy discussions. Physics of This U.S. work Department is of Energy supported under by grant Contract the Number Acknowledgments with the second termtion on ( right hand side again vanishing from ideal fluid equation of mo- now yields and ( odd. Thus equations ( Dynamical KMS invariance requireswhile that 6.3 Θ = JHEP01(2019)043 ) µ ) ω ) 22 o 3.36 (B.1) 22 ,σ ,g 11 o 11 ,σ ,g 12 o 12 σ g − ) , − , 2 ) ) ) ) 02 o 2 2 2 ) 1 02 ) ) 2 α σ ) i ) ) ,A ) i g 2 γδ i 2 i ,x 2 ,∂ 1 ,v ) ,t 2 α − ) u 1 v i 1 1 − F t 1 i , ,A 1 α , ω A , x , , ∂ x ∂ v t β B − 0 − 0 − 0 0 01 , o , − , , u , V 01 − − − − ∂ A ω x have the same transfor- 1 , 0 1 0 2 , , , ˆ , 1 α µ ∂ ( ( ( ( 0 ,σ µν 0 ,g 0 0 B u 1 0 α µα ,u ,v ,t ϕ g − − − ( − − ( − CPT 2 x A ∂ v t 00 0 o 0 2 00 ˆ ∂ µ s 0 α G aµν 1 − u − − − v g t − σ CPT ( ( ( ( − − ( ( ( s − ( ( ( , g αβγδ µν ) ) ) ) ) i i 2 ) i 22 o i ) ,T i ) B i 22 but with an overall minus sign. , ω , , x ,σ − , ∂ 0 0 0 0 ,A ,g , u 11 o ∂ B ω x 0 γδ 0 µν ϕ ∂ ( ( ( ( 11 µν A u g ,σ G ˆ − g µ − − − ( − ( − PT ,g β 12 o , while 12 V σ µ g − µα A , − , ) F ) ) ) ) 02 o 2 2 2 2 1 2 – 30 – 2 02 ) σ ) ) ) g 2 ,x A ,∂ ,v ,v 2 α 2 α − u 1 ij 1 1 1 − t t 1 2 , αβγδ − , x ∂ v v − , − − , g 01 , o ) 1 2 i , , 01 − ) ) − − − i ) ) 1 ) 0 i i , , , , 1 α 1 α i i i ,σ − g 0 ,g 0 0 0 A 1 2 ,A ω u ,u ,t ,t x ∂ v v 0 , x − , ∂ 00 0 o − 00 − −B ∂ 0 α 0 α − = 0 , 0 1 2 , − u A − − − g t t σ , , , ˆ ∂ x PT ( ( ( ( − µ ( ( ( s s ( ( ( 0 0 0 transform as 0 00 ϕ ∂ γδ ). g − B A ω u − ˆ µ − ( µ − ( ( ( ( ( ( T G ) 1.2 ij ,J o αβ transforms the same as ) Discrete transformations in 2+1-dimension Discrete transformations in 3+1-dimension σ F aµ ij − , C µν ,g i µ ) i 0 o σ i ) ) ) ) ) ) ∂ ) 0 αβγδ i i i i i 1 2 g µ i α i α A ,σ u u ,x µ ,∂ ,v ,v − ,t ,t − − 00 0 o , 0 0 0 1 2 , 0 α 0 α V , 1 2 = x ∂ v v t t σ 0 µ µ µ 0 µ µ µ 00 µν ∂ s s , and ˆ − u A − − − g − − − ϕ g µ ∂ ∂ B A ω u x ˆ T ( ( ( ( − µ ( ( ( − − ( ( ( µν g 3 4 ). , , µ are defined below ( ∂ µ = 1 = 2 µ u 3.39 B We first note an identity in (3 + 1)-dimension ,α ,α = µ µν µ µ o µ µ µ 1 2 µν 1 2 µ α µ α ˆ x u A ∂ ∂ µ v v g s s t t σ In this appendix weand give ( some useful formulae used in deriving equations such as ( mations as and B Some useful formulae Note that in all cases JHEP01(2019)043 (B.7) (B.8) (B.9) (B.2) (B.3) (B.4) (B.12) (B.10) (B.11) . ) F F 2 ∧ g F + 2 ξ 2 )) (B.5) du βh 1 F γ g · ( + F. W W µ o v ∧ β ( q λρ ∧ ∧ du F. F V ) u w ∧ β V ∧ V ∧ ∧ ∧ µα µ u ∧ νλρ u ∧ G B ϕF ) ) dβ ∧ ( ν ∧ ∧ d G µ − ∇ ) ) G ) αβγ − · · µ ∆ F F B w Z ) ξ 2 2 · and a vector field ξ β F 1 2 · ( (B.6) ) is a one-form. As an example, given ( dλ . h ) L β − du λ β ∧ F · ( − = F · − 2 + 2 + V 1 ξ u Z ρ h βγ λρ h G β ∧ + ( G F u ~ ∧ F − du G v (( + c F ∧ ) 1 w ∧ α λρ λ 2 ) + h g µB ξ ∧ V νλρ w V ∧ ( (ˆ − V du du ∧ L F ν L − ) 1 d · ν u B ∧ 2 ∧ u – 31 – h = µ αβγ √ h ) β ( W ∧ ∧ u Z ( µ x ∆ · ϕ F ) = µ F ) ) 2 µνλ + ∧ 4 ~ ∧ 1 2 B · λ g ξ c d F µ F W u β · Z ξ 2 · · v B du + ( = = L 2 ξ ∧ Z = β 1 = ( ( , we then have ρ β β − ρ − λ − ) h ). d L µ du B ( u = G βγ + F F = = ∂u = ( ∧ are two-forms, and · ∧ Z λρ a two-form. Here are two examples: νλ βu ˆ G ∧ µ ν ∧ µν 3.36 F F G α d u u F β w g = = ν ( G du β W ∧ W ∧ ∧ u ∧ + ˆ µ ) µ µF β L µνλ F,G Z 2 ˆ 1 µνλρ β µ ˆ B µ V g ~ µν d Z µνλ αβγ β o d · βF ϕF − g VF c ( ~ v L ξ · 2 µ − c ( µ o g T L ξ V Z Z − Z j ˆ √ , and − g Z x = = = = 3 = √ du 4 − ), we note that: d x √ 4 = . x d Z 4 3.39 w du ~ d , Z c is a one-form and µ = is a vector field, 2 1 − Z dx ξ w W µ u For (2 + 1)-dimension we have To see ( ≡ where or in differential form where which can be used to derive ( It is also useful to recall that for a differential form It then follows that for some vector where u which can be written in differential forms as JHEP01(2019)043 , , 03 D D ]. (1980) 103 JHEP ] , SPIRE ]. D 22 IN Phys. Rev. (2016) 2617 (2012) 046 , Phys. Rev. ][ , 09 SPIRE ]. B 47 IN He-A Towards 3 arXiv:1106.0277 ][ , Phys. Rev. JHEP , Phys. Rev. Lett. SPIRE , , Constraints on anomalous arXiv:0809.2596 IN (2012) 101601 [ [ Holographic thermal helicity 109 ]. arXiv:1012.1958 [ ]. 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