Carrollian Fluids and Spontaneous Breaking of Boost Symmetry
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Carrollian Fluids and Spontaneous Breaking of Boost Symmetry Jay Armas1,2,and Emil Have3,4,? 1Institute for Theoretical Physics and Dutch Institute for Emergent Phenomena, University of Amsterdam, 1090 GL Amsterdam, The Netherlands 2Institute for Advanced Study, University of Amsterdam, Oude Turfmarkt 147, 1012 GC Amsterdam, The Netherlands 3School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, Peter Guthrie Tait road, Edinburgh EH9 3FD, United Kingdom 4Niels Bohr International Academy, Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen ?, Denmark (Received 28 September 2023; revised 16 January 2024; accepted 29 February 2024; published 17 April 2024) In the hydrodynamic regime, field theories typically have their boost symmetry spontaneously broken due to the presence of a thermal rest frame although the associated Goldstone field does not acquireindependent dynamics. We show that this is not the case for Carrollian field theories where the boost Goldstone field plays a central role. This allows us to give a first-principles derivation of the equilibrium currents and dissipative effects of Carrollian fluids. We also demonstrate that the limit of vanishing speed oflight of relativistic fluids is a special case of this class of Carrollian fluids. Our results shine light on the thermodynamic properties and thermal partition functions of Carrollian field theories. DOI: 10.1103/PhysRevLett.132.161606 Introduction. —In the past few years Carrollian physics, emerging by taking the limit of the vanishing speed of light, has been found useful for describing a variety of phenomena in contexts ranging from black holes [1–4], cosmology [5,6], gravity [3,7–16], to hydrodynamics [4,5,17 –24]. Concretely, Carrollian fluids can be used to describe Bjorken flow, which is relevant for models of the quark-gluon plasma, cf. [23] (and its conformal generalization, Gubser flow [25]). Carrollian fluids also model dark energy in inflationary models [5]. Furthermore, Carrollian symmetries are expected to have a role to play in exotic phases of matter (e.g., viaCarroll-fracton dualities [26–29]and in superconducting twisted bilayer graphene [30]). Many of the properties encountered in this Carrollian limit are expected to be explained from underlying quantum field theories with inherent Carrollian sym-metries. Indeed, if conformal symmetry is present in addition, such theories would be putative holographic duals to flat space gravity [31–41].H o w e v e r ,w h e n attempting to formulate such Carrollian field theories, several issues have been pointed out including violations of causality, lack of well-defined thermodynamics, and ill-defined partition functions [5,24] . Our goal is to show that the lack of well-defined thermodynamics in Carrollian field theories is expected in the hydrodynamic regime, butthat this issue can be cured when carefully accounting for the Carrollian symmetries. The approach we take is to consider the hydrodynamic regime of such putative Carrollian field theories and showhow to construct their equilibrium partition function andnear-equilibrium dynamics. In particular, we will show thatthere is no proper notion of temperature in Carrollian fluids unless the Goldstone field of spontaneous broken boost symmetry is taken into account. This allows us to constructa well-defined hydrodynamic theory of Carrollian fluids(similar to framids in the language of [42,43] ). In the process, we show that seemingly different approaches toCarrollian hydrodynamics previously pursued in the liter-ature [5,17 –24]are in fact equivalent and special cases of the Carrollian fluids we derive. The fact that boost symmetry is spontaneously broken in hydrodynamics is not unexpected. Thermal states break the boost symmetry spontaneously due to thepresence of a preferred rest frame aligned with the thermal vector [43,44] . In the context of hydrodynamics the thermal vector is the combination u μ=Tof the unit normalized fluid velocity uμand temperature T. On the other hand, the Goldstone field associated with the break- ing of boost symmetry does not typically feature in the low energy spectrum of the theory because it is determinedin terms of the other dynamical fields (see, e.g., [42]). This is easy to show for relativistic fluids. Consider a ( d?1)- dimensional spacetime metric g μν?EAμEμ Awhere EAμis the set of vielbeins, μ?0;…;dare spacetime indices, and A?0;…;dare internal Lorentz indices. The Goldstone field associated to spontaneous breaking of Lorentz boostPublished by the American Physical Society under the terms of theCreative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article ’s title, journal citation, and DOI. Funded by SCOAP3.PHYSICAL REVIEW LETTERS 132, 161606 (2024) 0031-9007 =24=132(16) =161606(7) 161606-1 Published by the American Physical Societysymmetry is the vector lA?Λ0 Awhere Λ0 Ais the Lorentz boost matrix [42], and acquires an expectation value hlAi?δ0 Ain the ground state. In thermal equilibrium we can construct an equilibrium partition function S?Rdd?1x???????gpP?T;uμlμ?where Pis the fluid pressure. The temperature is given by T?T0=jKjwith T0a global constant temperature, and the fluid velocity is uμ? Kμ=jKjsatisfying uμuμ??1and defined in terms of the Killing (thermal) vector Kμwith modulus jKj2? ?gμνKμKν. We furthermore defined lμ?EAμlAsatisfying lμlμ??1. The Goldstone equation of motion obtained from varying the partition function Swith respect to lAis[42] ?δμ ν?lμlν?δS δlμ?0: ?1? This equation implies uμ??lνuνlμfor arbitrary thermo- dynamic coefficients, and can only be satisfied if lμ?uμ. Indeed, we see that the dynamics of the Goldstone field lμis determined by the dynamics of the fluid velocity and hencecan be removed from the hydrodynamic description. Thesame conclusion is reached for the case of spontaneousbreaking of Galilean (Bargmann) boost symmetry [45](see also[44]). However, in the case of Carrollian symmetry, as we will show, the boost Goldstone acquires its ownindependent dynamics. Below we introduce Carrolliangeometry and use it to show that, naively, there is nowell-defined notion of temperature. Carrollian geometry and the lack of temperature. —A weak Carrollian geometry on a ( d?1)-dimensional mani- fold Mis defined by a Carrollian structure ?v μ;hμν? consisting of the nowhere-vanishing Carrollian vector field vμ, and the corank-1 symmetric tensor hμν, the “ruler,” satisfying hμνvν?0. It is useful to define inverses ?τμ;hμν? satisfying vμτμ??1, and τμhμν?0, as well as the completeness relation ?vμτν?hμρhρν?δμ ν. It is also useful to introduce the spatial vielbeins eaμand their inverses eμ awhich can be used to write hμν?eaμeaν. Under Carrollian boosts, the inverses transform as δCτμ?λμ;δChμν?2λρhρ?μvν?; ?2? corresponding to δCeμ a?vμλa, where we have used thatvμλμ?0.A strong Carrollian geometry is a weak Carrollian geometry together with an affine connection. We focus mainly on weak Carrollian geometry but discussstrong Carrollian geometry in Appendix D of the Supple-mental Material [46]. Given the Carroll geometry and the existence of a thermal vector in equilibrium, namely the spacetimeKilling vector k μ, one can proceed as for other (non)- Lorentzian field theories [54–65]and construct an equi- librium partition function by identifying the invariantscalars under all local symmetries which the pressure Pcan depend on, as above Eq. (1)for the relativistic case. A more thorough construction of the partition function will be given in a later section. Here we note that for non-relativistic theories the temperature Tis given by the scalar T?T 0=?kμτμ?with T0a constant global temperature. However, in the Carrollian case Tis not invariant under boost transformations since δC?kμτμ??kμλμ≠0. Indeed, there is no well-defined notion of temperature for arbitrary observers [66]. This is rooted in the fact that it is not possible to impose a timelike normalization condition onspacetime vectors such as the fluid velocity since u μτμis not boost invariant. This argument does not rely on any specific model of Carrollian (quantum) field theory sincethese statements are valid in the hydrodynamic regime.This suggests that well-defined thermodynamic limits ofsuch putative theories are subtle. Below we introduce the boost Goldstone and use it to show that it can be used to define an appropriate notion of temperature. The Carroll boost Goldstone. —We define the boost Goldstone as the vector θ μwhich transforms under Carrollian boosts λμas δCθμ??hμνλν; ?3? where λμvμ?0. This implies that only the spatial part of θμ is physical, which we can enforce by endowing the Goldstone with a timelike Stueckelberg symmetry of theform δ Sθμ?χvμ; ?4? where χis an arbitrary function [67]. With this we can build the boost and Stueckelberg invariant vielbeine ?τμ?τμ?hμνθν; ?eμ a?eμ a?vμθμeμa; ?5? which lead to the following invariant ruler: ?hμν?δab?eμ a?eν b?hμν?vμvν?θ2?2τρθρ??2v?μθν?;?6? where θ2?hμνθμθν. Together with vμandhμν, these form anAristotelian structure [68], ?hμν?τν?0;vμ?τμ??1; ?hμρhρν?δμ ν?vμ?τν??hμ ν;?7? that is partly dynamical due to the Goldstone θμ. The low- energy effective action for the Carrollian boost Goldstone isa two-derivative Ho ˇrava–Lifshitz type action as we show in Appendix A of [46]. If coupled to Carrollian gravity the resultant action would be derivable from the limit of the vanishing speed of light of the Einstein-Aether theory [69]. Before showing how the Goldstone allows one to define anotion of temperature, we first discuss the currents andconservation laws.PHYSICAL REVIEW LETTERS 132, 161606 (2024) 161606-2Currents and conservation laws. —We now consider an arbitrary fluid functional (or free energy) S?τμ;hμν;θν/C138for a Carrollian fluid with spontaneously broken boosts. The variation of this functional is δS?Z dd?1xeh ?Tμδτμ?1 2Tμνδhμν?Kμδθμi ;?8? where Tμis the energy current, Tμνthe stress-momentum tensor, and Kμthe response to the Goldstone field. In particular Kμ?0gives the analogous equation of motion for the Goldstone as in (1). The measure is defined as e?det?τμ;eaμ??det??τμ;eaμ?. The Ward identities for the Stueckelberg and boost symmetries are, respectively, vμKμ?0;Tνhνμ?Kμ: ?9? The equation of motion for the Goldstone, Kμ?0, imposes the condition Tνhνμ?0. In other words, the boost Ward identity now becomes the equation of motion for the Goldstone. The momentum-stress tensor is not boost invari-ant. In fact, computing the second variation, which mustvanish δ C?δCS?? 0, we find that δCTμ?δCKμ?0and δCTμν?2T?μhν?ρλρ. The associated energy-momentum tensor (EMT) Tμ ν??τνTμ?Tμρhρνis also not boost invariant and transforms as δCTμ ν?Kνhμρλρ; ?10? where we used (9). Doing the same for the Stueckelberg symmetry, the condition δS?δSS?? 0implies that δSTμ ν? ?χvμKν. Hence, the EMT is both boost and Stueckelberg invariant if Kμ?0. The diffeomorphism Ward identity reads as e?1?μ?eTμ ρ??Tμ?ρτμ?1 2Tμν?ρhμν?0; ?11? wherewe used that Kμ?0. It is possible to obtain manifestly boost invariant currents, including the EMT, by formulating the action in terms of the effective Aristotelian structure (5), as we show in Appendix B of [46]. Equilibrium partition function and Carrollian fluids. — To derive the currents of Carrollian fluids, we consider the equilibrium partition function construction. An equilibriumCarrollian background consists of a set of symmetry parameters K??k μ;λμ K;χK?, where kμis a Killing vector and λμ Kis a boost symmetry parameter, while χKis a Stueckelberg symmetry parameter. The various structures transform according toδKvμ??kvμ?0;δKτμ??kτμ?λKμ?0; δKhμν??khμν?0; δKhμν??khμν?2λKρhρ?μvν??0; δKθμ??kθμ?hμνλKμ?χKvμ?0: ?12? The boost and Stueckelberg symmetry parameters trans- form as δλKμ??ξλKμ??kλμ;δχK??ξχK??kχ; ?13? under infinitesimal diffeomorphisms generated by ξμ, infinitesimal Carrollian boosts λμand Stueckelberg trans- formations χ. As we show in Appendix B of [46],λKμand χKwill not play a role in the effective fluid description. Before enumerating the possible invariant scalars, we must provide a gradient ordering. As usual we take the geometry itself to be of ideal order, that is, τμ?hμν? vμ?hμν?O?1?. Since θμenters the definition of ?τμit must have the same ordering, θμ?O?1?. Gradients of these structures are O???and hence suppressed in a hydrodynamic expansion. Given this gradient scheme the only two idealorder invariants are T?T 0=?τμkμ; ?u2?hμνuμuν; ?14? where uμ?kμ=?τρkρ, which satisfies ?τμuμ?1. We note that we can now define a notion of temperature Tthat is invariant for all observers. The scalar ?u2is the modulus of the spatial fluid velocity. Generically the fluid velocity can be decomposed as uμ??vμ? ?uμ, where ?uμ??hμ νuνwith ?hμ ν? ?hμρhρν. We furthermore define ?uμ?hμνuν?hμν ?uν, such that ?uμ??hμ νuν,b u t ?uμ≠hμν?uν. Note in particular thatuμdecomposes as follows relative to the Carrollian structure: uμ??vμ?1?θν?uν??hμ νuν: ?15? The hydrostatic partition function at ideal order is given by S?Rdd?1xeP?T; ?u2?. Using the general actionvariation (8) together with the “variational calculi ”δhμν?2v?μhν?ρδτρ? hμρhνσδhρσandδvμ?vμvνδτν?hμνvρδhρνwe obtain the ideal order currents: Tμ ?0??Pvμ?sTuμ?m ?u2uμ; Tμν ?0??Phμν?muμuν?2?sT?m ?u2?u?μθν?; K?0?μ??sT?m ?u2? ?uμ; ?16? where the subscript (0) indicates that the currents are of ideal orderO?1?and the entropy sand mass density mare defined viadP?sdT ?md ?u2. The associated EMT is given byPHYSICAL REVIEW LETTERS 132, 161606 (2024) 161606-3Tμ ?0?ν?Pδμ ν?muμ?uν??sT?m ?u2??uμ?τν?θμ?uν?;?17? which transforms as in (10). The equation of motion for the Goldstone K?0?μ?0, which is equivalent to the boost Ward identity, gives a constraint on the dynamics ?sT?m ?u2? ?uμ?0; ?18? and can be viewed as a framid condition for Carrollian fluids. Defining the energy density as E??τμTμ ?0?, the Goldstone equation has two branches of solutions: either E?P? sT?m ?u2?0or ?uμ?0. Neither of them allows for the elimination of the Goldstone θμfrom the low-energy description. The constraint (18)was derived in equilibrium, but we show in Appendix B of [46] that it also holds off equilibrium, although it receives corrections due to dissipa- tive effects. As such, together with (11), it provides the ideal order dynamics for Carrollian fluids. Equations (16)–(18)are a central result of this work as they provide a well-defined notion of Carrollian fluids. Below we show that the c→0 limit of relativistic fluids gives rise to a Carrollian fluid with ?uμ?0. Thec→0limit of a relativistic fluid. —Thec→0limit of relativistic fluids was considered in [17,18,20] (see also [23]) for a specific class of metrics. The same limit was taken in [24], where it was referred to as a “timelike fluid.”Here, we demonstrate that these notions coincide and correspond to the special case of the Carrollian fluid we introduced above with ?uμ?0, and that the emergence of the Goldstone can be understood from the ultralocal expansion of the Lorentzian geometry. The relativistic EMT is given by Tμν??E??P c2UμUν??Pδμ ν; ?19? where UμUνgμν??c2, and where the “hat”indicates relativistic thermodynamic quantities. To take the limit, we first consider the metric and its inverse in “pre-ultra- local (PUL) variables ”[3] gμν??c2TμTν?Πμν;gμν??1 c2VμVν?Πμν;?20? where TμVμ??1,TμΠμν?VμΠμν?0,ΠμρΠρν? δμ ν?VμTν. The leading order components of the PUL variables correspond to the fields that make up the Carrollian structure, e.g., Vμ?vμ?O?c2?. We write the expansion of the relativistic fluid velocity relative to the PUL variables as Uμ??Vμ?c2uμ; for some uμ. Crucially, Uμis invariant under local Lorentz boosts while δCVμ?c2hμνλν?O?c4?, implying thatδCuμ??hμνλν?O?c2?. This shows that uμcannot be identified with a fluid velocity in the Carrollian limit.Indeed, we may identify the spatial part of the leading order term in the c 2expansion of uμwith the spatial part of the boost Goldstone Πμνuν?hμνθν?O?c2?? ?θμ?O?c2?: ?21? Using this, together with Uμ?c2?τμ?O?c4?the EMT becomes Tμ ν??E?P?vμ?τν?Pδμ ν?O?c2?; ?22? where EandPare the leading order contributions of ?Eand ?P, respectively, satisfying the Euler relation E?P?sT.T h i s is exactly the “timelike ”fluid of [24], corresponding to the ?uμ?0branch of the Carrollian fluid we described above [70]. It is instructive to take the c→0limit of the relativistic equation of motion b?μTμ ν?0, where b?is the Levi-Civita connection of the spacetime metric gμν. Deferring the details to Appendix C of [46], we note here that the equations of motion in the limit c→0can be expressed as vμ?μE??E?P?K; hμν?νP???φμ?E?P???E?P?Khμν ?θν ?hμνvρe?ρ??E?P? ?θν/C138; ?23? where K?hμνKμν??1 2hμν?vhμνis the trace of the intrinsic torsion of the Carrollian structure [71], and is sometimes referred to as the “Carrollian expansion, ”while ?φλ?2hλμvν??ντμ/C138?hλμhνσ ?θσKμν. The first term in ?φλis sometimes referred to as the “Carrollian acceleration, ”while the second term comes from the c2expansion of the Levi- Civita connection. The covariant derivative e?is a Carroll compatible connection that arises in the O?1?piece of the c2 expansion of the Levi-Civita connection, which we discuss further in Appendix C of [46]. These equations are fully covariant and reduce to the special case of the equations of motion obtained in [17,18,20,23] when restricted to space- time metrics that admit a Randers-Papapetrou parametriza-tion. Furthermore, these equations can be obtained byprojecting the conservation law (11) along the time and spatial directions using (17) with ?u μ?0. We thus have shown that the “timelike ”fluid of [24] is the same as the Carrollian fluid of [17,18,23] , and both are a special case of the Carrollian fluid derived here. Dissipation and modes. —In Appendix B of [46] we show that at order O???the class of Carrollian fluids we introduced is characterized by two hydrostatic coefficientsand ten dissipative coefficients. Here we study the effect of specific coefficients in the linear spectrum of fluctuations. We consider flat Carrollian space with τ μ?δtμ,vμ??δμ t,PHYSICAL REVIEW LETTERS 132, 161606 (2024) 161606-4hμν?δiμδiν, and hμν?δμ iδν i(see Appendix B of [46] for more details). We then fluctuate the conservation equa- tions (11) and the boost Ward identity (9)around an equilibrium state with constant temperature T0, fluid velocity vi 0, and Goldstone field θi 0, such that, e.g., θi?θi 0?δθi. Using plane wave perturbations with fre- quency ωand wave vector ?kwe find a distinguishing feature of these Carrollian fluids: the mode structurestrongly depends on whether the equilibrium state carries nonzero velocity v i 0.I fvi 0?0, the linearized equations only admit a nontrivial solution if θi 0≠0,T0?0, and δvi?0. Denoting the angle between the momentum kiandθi 0by?, this leads to a single linear mode: ω??1 j ?θ0jcos?j ?kj; ?24? where we assumed that the value of the entropy density sin equilibrium s0remains finite and nonvanishing when T0→0; otherwise there is no mode. Interestingly, this mode is not affected by any of the 12 transport coefficientsentering at order O???[72]. This spectrum corresponds to the branch of solutions with ?u μ?0, and hence it is the expected spectrum arising from the c→0limit of an ideal relativistic fluid. On the other hand if vi 0≠0butθi 0?0a more interesting spectrum can be obtained. For simplicity we only consider the effect of a bulk viscosity s3and one anisotropic viscosity s2. Besides a gapped mode, we find for d?2 a single diffusive mode of the form ω?vi 0ki??iΓ1 2?εijvi 0kj?2; ?25? where εijis the two-dimensional Levi-Civita symbol, and Γ1?s2;0?T2 0χTT?jv0j2?3T0χTu?2jv0j2χuu?/C138 s0T0?T2 0χTT?jv0j2?2T0χTu?jv0j2χuu?/C138 ?2s3;0 s0T0; where s3;0is thevalue of s3in equilibrium, ditto s2,a n dw h e r e we defined χTT???2P=?T?T?0,χuu???2P=? ?u2? ?u2?0, and χTu???2P=?T? ?u2?0. The left-hand side of (25)is character- istic of a fluid without boost symmetry [63]while the right- hand side is typical of a diffusive mode. A salient signature ofCarrollian fluids is that the spectrum is only nontrivial for states with nonzero equilibrium velocity v i 0. Discussion. —We have given a first-principles derivation of Carrollian fluids based on symmetries, showing that thespontaneous breaking of boost symmetry plays a crucial role in defining equilibrium partition functions of Carrollianfield theories in the hydrodynamic regime. It is interesting to speculate whether this peculiar feature of Carrollianhydrodynamics can shed light on how to construct well- defined partition functions using specific microscopic mod- els of Carrollian field theories along the lines of [24,44] . Different approaches to Carrollian hydrodynamics have ap- peared in the literature in the past few years [5,17 –19,21 –24]. Revisiting the c→0limit of relativistic fluids we showed that there are subtleties regarding the interpretation of the dynamical variables that appear in the limit of the equations of motion. In particular, we showed that what naivelyappeared to be a fluid velocity was in fact a Goldstone fieldassociated to the spontaneous breaking of boost symmetry.This allowed us to show that the different approaches are not only equivalent to each other but also special cases of the more general Carrollian fluids we introduced here. Webelieve it could be interesting to revisit the black holemembrane paradigm [2]in light of this new understanding. The effective field theory geometry becomes Aristotelian once taking the Goldstone field into account. This allowed us to easily understand the dissipative structure of such fluids using earlier results [63,73] . The spectrum of exci- tations for certain classes of equilibrium states shares certainsimilarities with the spectrum of excitations of p-wave fracton superfluids in which the Goldstone field associated to the spontaneous breaking of dipole symmetry plays an analogous role to the boost Goldstone field [45,64,65] , albeit in the Carroll case a nonvanishing fluid velocity is needed.We believe that this relation can be made clearer if weconsider strong Carrollian geometries as we describe in Appendix D of [46]. Finally, it would be interesting to consider the addition of conformal symmetry as this could shine light on thermo- dynamic properties of holographic dual theories of flat space gravity. If we impose this symmetry the equation of state for the branch ?u μ?0becomes E?dPwhile for the branch sT?m ?u2?0it imposes the relation m????d?1?= ?u2/C138P. In addition, the number of first order transport coefficients reduces from 12 to 8 (see Appendix B of [46]). 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