On the surface of superfluids

Journal of High Energy Physics, Jun 2017

Developing on a recent work on localized bubbles of ordinary relativistic fluids, we study the comparatively richer leading order surface physics of relativistic superfluids, coupled to an arbitrary stationary background metric and gauge field in 3 + 1 and 2 + 1 dimensions. The analysis is performed with the help of a Euclidean effective action in one lower dimension, written in terms of the superfluid Goldstone mode, the shape-field (characterizing the surface of the superfluid bubble) and the background fields. We find new terms in the ideal order constitutive relations of the superfluid surface, in both the parity-even and parity-odd sectors, with the corresponding transport coefficients entirely fixed in terms of the first order bulk transport coefficients. Some bulk transport coefficients even enter and modify the surface thermodynamics. In the process, we also evaluate the stationary first order parity-odd bulk currents in 2 + 1 dimensions, which follows from four independent terms in the superfluid effective action in that sector. In the second part of the paper, we extend our analysis to stationary surfaces in 3 + 1 dimensional Galilean superfluids via the null reduction of null superfluids in 4 + 1 dimensions. The ideal order constitutive relations in the Galilean case also exhibit some new terms similar to their relativistic counterparts. Finally, in the relativistic context, we turn on slow but arbitrary time dependence and answer some of the key questions regarding the time-dependent dynamics of the shape-field using the second law of thermodynamics. A linearized fluctuation analysis in 2 + 1 dimensions about a toy equilibrium configuration reveals some new surface modes, including parity-odd ones. Our framework can be easily applied to model more general interfaces between distinct fluid-phases.

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On the surface of superfluids

Received: April On the surface of superfluids Jay Armas 0 1 4 Jyotirmoy Bhattacharya 0 1 2 Akash Jain 0 1 2 Nilay Kundu 0 1 3 Campus Plaine CP 0 1 B- 0 1 Brussels 0 1 Belgium 0 1 South Road 0 1 Durham DH 0 1 LE 0 1 U.K. 0 1 0 Open Access , c The Authors 1 Kitashirakawa Oiwakecho , Sakyo-ku, Kyoto 606-8502 , Japan 2 Centre for Particle Theory & Department of Mathematical Sciences, Durham University 3 Center for Gravitational Physics, Yukawa Institute for Theoretical Physics (YITP) 4 Physique Th ́eorique et Math ́ematique Universit ́e Libre de Bruxelles Developing on a recent work on localized bubbles of ordinary relativistic fluids, we study the comparatively richer leading order surface physics of relativistic superfluids, coupled to an arbitrary stationary background metric and gauge field in 3 + 1 and 2 + 1 dimensions. The analysis is performed with the help of a Euclidean effective action in one lower dimension, written in terms of the superfluid Goldstone mode, the shape-field (characterizing the surface of the superfluid bubble) and the background fields. We find new terms in the ideal order constitutive relations of the superfluid surface, in both the parity-even and parity-odd sectors, with the corresponding transport coefficients entirely fixed in terms of the first order bulk transport coefficients. Some bulk transport coefficients even enter and modify the surface thermodynamics. In the process, we also evaluate the stationary first order parity-odd bulk currents in 2 + 1 dimensions, which follows from four independent terms in the superfluid effective action in that sector. In the second part of the paper, we extend our analysis to stationary surfaces in 3 + 1 dimensional Galilean superfluids via the null reduction of null superfluids in 4 + 1 dimensions. The ideal order constitutive relations in the Galilean case also exhibit some new terms similar to their relativistic counterparts. Finally, in the relativistic context, we turn on slow but arbitrary time dependence and answer some of the key questions regarding the time-dependent dynamics of the shape-field using the second law of thermodynamics. A linearized fluctuation analysis in 2 + 1 dimensions about a toy equilibrium configuration reveals some new surface modes, including parity-odd ones. Our framework can be easily applied to model more general interfaces between distinct fluid-phases. Effective Field Theories; Spontaneous Symmetry Breaking; Holography and 1 Introduction and summary Stationary superfluid bubbles Non-relativistic stationary superfluid bubbles Time dependent fluctuations of the surface Stationary superfluid bubbles in 3+1 dimensions Perfect superfluid bubbles (d+1 dimensions, d ≥ 3) First order corrections away from the interface Stationary superfluid bubbles in 2+1 dimensions Parity-odd effects for perfect superfluid bubbles First order corrections away from the interface Constraints on the bulk parity-odd constitutive relations Bulk parity-odd effects on the surface currents Surface currents and thermodynamics Galilean stationary superfluid bubbles in 3+1 dimensions Surface dynamics Surface entropy current analysis at zero derivative order Surface entropy current for ordinary fluids Surface entropy current for 3+1 dimensional superfluids Surface entropy current for 2+1 dimensional superfluids Ripples on the surface A Surface thermodynamics in 2+1 dimensions B Equation of motion for the shape-field and the Young-Laplace equation C Useful notations and formulae Introduction and summary Matter in the universe exists in diverse forms and very often its collective behaviour is so complex that its detailed microscopic description becomes intractable. Fortunately, in many situations of interest, the low-energy collective behaviour can be captured by an effective theory with a few degrees of freedom. A prominent example of such a finite temperature effective theory is hydrodynamics, where the description is provided in terms of a few fluid variables in the long-wavelength approximation. In this effective description, the relevant microscopic information is conveniently packaged into the parameters of the theory, referred to as the transport coefficients. The universal nature of this description has lead to its applications in a diverse range of physical situations, ranging from neutron-stars, quark-gluon plasma to numerous condensed matter systems. Hence, this subject has had a long history and has been extremely well studied in the past. However, quite recently there has been a renewed interest in this area, particularly following the realization that there are some important lacunae in the structural aspects of the fluid equations that have been considered so far. It was understood that new transport coefficients must be incorporated in the effective theory in order to adequately describe certain physical situations.1 In fact, one of the most interesting aspects of some of these newly discovered coefficients is their parity-odd nature — a possibility that has been largely ignored in the rich and classic literature on the subject.2 In regimes where the hydrodynamic approximation is applicable, it is often observed that the same underlying microscopic theory can exist in distinct macroscopic phases. In situations where two such phases coexist, they are separated by a dynamical interface (or surface). If we wish to provide an effective description for such scenarios, then the hydrodynamic description must be appropriately generalized in order to include the effects specific to such surfaces. Our main goal in this paper is to explore new surface properties, especially in the context of superfluids, focusing on the parity-odd effects. For the case of ordinary relativistic space-filling fluids, the degrees of freedom include the fluid velocity uµ , temperature T and chemical potential(s) µ corresponding to any global symmetries that the fluid may enjoy. In this paper, we will assume this global symmetry to be a U(1) symmetry. The equation of motion for these fluid fields are simply the conservation of the energy-momentum tensor and charge current, which in turn are expressed in terms of the fluid variables subjected to constitutive relations. The structure of the constitutive relations is determined based on symmetry principles and is severely constrained by the second law of thermodynamics [6]. In the case of superfluids, the U(1) symmetry is spontaneously broken and the phase in the low-energy effective description in addition to the ordinary fluid fields. In order to the basics of superfluid dynamics). In the case of space-filling superfluids, the most general constitutive relations consistent with the second law of thermodynamics up to first order in the derivative expansion have been worked out more recently in [4]. If we wish to provide a unified description of two (super)fluid phases separated by a dynamical surface, we need to include a new field f in the hydrodynamic description, 1For instance, in situations where a symmetry of the underlying theory suffers from an anomaly. This was first realized in the study of the map between long-wavelength fluctuation of black holes and relativistic hydrodynamics [1, 2]. Subsequently, the existence of such new coefficients has been inferred employing pure hydrodynamic techniques [3]. 2See [4, 5] for recent discussions on such parity odd transport properties. which keeps track of the shape of the surface. The surface is considered to be located of superfluids. In fact, f may be considered to be the Goldstone boson corresponding to the spontaneous breaking of translational invariance in the direction normal to the fluid surface.3 The guiding symmetry principle for incorporating this shape-field into the constitutive relations is the reparametrization invariance, i.e. the fluid must be invariant under arbitrary redefinitions of f as long as its zeroes are unchanged. This essentially implies that the dependence of the fluid currents on f happens primarily4 through nµ , the normal vector to the surface, and its derivatives. Now, for a superfluid bubble placed inside an ordinary fluid, there is a rich interplay bubble. In this paper, we study these surface effects and work out the ideal order surface currents for a superfluid. This paper is organized as follows: in the remaining of this section we will give a detailed summary of the main points and techniques used in this paper. In section 2 we discuss stationary superfluid bubbles suspended in ordinary fluids in 3 + 1 dimensions, and extend it to 2 + 1 dimensions in section 3 (see the summary in section 1.1). Then in section 4, we discuss stationary Galilean superfluid bubbles using the technique of null superfluids [10], and use it to understand about the non-relativistic limit of surface phenomenon in superfluids (see the summary in section 1.2). Later in section 5, we turn on slow but arbitrary time dependence and study time-dependent dynamics of the shape-field f using the second law of thermodynamics as well as linearized fluctuations about an equilibrium configuration. We finish with some discussion in section 6. The paper has three appendices. In appendix A we discuss surface thermodynamics for 2 + 1 dimensional superfluid bubbles. Then in appendix B, we give a generic derivation of the Young-Laplace equation for stationary superfluid bubbles, that determines the shape of the surface. Finally, in appendix C we collect some useful formulae and notations. Stationary superfluid bubbles To begin with, following [8], we shall mainly focus on stationary relativistic superfluid bubbles, which will enable us to employ the partition function techniques discussed in [11– 13]. Our main objective here is to write down a Euclidean effective action for the Goldstone be easily read off using a variational principle. One of our primary focuses in this analysis will be the parity violating terms. Therefore, we will separately discuss the cases of 3 + 1 and 2 + 1 dimensions,5 which have significantly different parity-odd structures. 3See [7, 8] for a relevant recent discussion in the stationary case and [9] for an application of similar ideas to the study of polarization effects on surface currents in the context of magnetohydrodynamics. 4As we shall explain in more detail below, another way in which f may enter the constitutive relations 5Note that there is a subtlety in the discussion of finite temperature superfluidity in 2 + 1 dimensions. At finite temperature, the low-energy physics is blind to the time-like direction and therefore the dynamics is effectively two dimensional. In our context, this is clearly reflected by the fact that in 2 + 1 dimensions we write down a two dimensional Euclidean action for the massless Goldstone boson. This brings us within the purview of the Mermin-Wagner theorem implying that superfluidity in these dimensions may be destroyed We will consider stationary bubbles of a superfluid in the most general background ds2 = Gµν dxµ dxν = −e2σ(~x) dt + ai (~x) dxi 2 + gij (~x) dxidxj , A = A0 (~x) dt + Ai (~x) dxi . Here, the i-index runs over the spatial coordinates. We will denote the covariant derivative define respective surface derivatives by ∇˜ µ (· · · ) = 1/√ since we wish to provide a finite temperature partition function6 description of our system, we will Wick-rotate to Euclidean time and compactify this direction, with an inverse radius T0. Thus, the set of all background data comprises of (see [8, 11] for more details) Apart from T0, there is another length scale in the problem corresponding to the chemical potential µ 0. However, it is always possible to absorb this into the time component of the arbitrary gauge field A0. Therefore, we will not make µ 0 explicit in our discussions. In addition to the background data (1.2), there are two fields which must be included in the partition function if we wish to describe superfluid bubbles. One of them is the phase of the scalar operator responsible for the spontaneous breaking of the U(1) symmetry, which charged fluid (see [8] for more details). In the superfluid description, the first derivative As has been explained in detail in [8, 11], the partition function must be constructed in terms of quantities that are invariant under spatial diffeomorphisms, Kaluza-Klein (KK) gauge transformations (redefinitions of time, t → t+ϑt(~x)) and U(1) gauge transformations. Therefore, following [11], we first define a KK invariant gauge field A = A0dt + Aidxi, A0 ≡ A0, Ai ≡ Ai − A0ai . of arbitrary background sources. by strong quantum fluctuations. However, this conclusion is rendered invalid in the large-N limit. In fact, much of our discussions here might be relevant for 3 + 1 dimensional hairy black holes in AdS, via the AdS/CFT correspondence [14]. Also, our discussion in 2 + 1 dimensions may be relevant for other microscopic mechanisms for 2 + 1 dimensional superfluidity like the BKT transition. It would be definitely interesting to make this connection more precise. 6Here by partition function we refer to (the exponential of) the Euclidean effective action in the presence 7Here we follow the conventions of [12]. See also [4, 15] for an out of equilibrium discussion of relativistic In the context of superfluids, it is convenient to redefine the spatial components of the superfluid velocity so that they are invariant under both the U(1) and KK gauge transformations (the time component is automatically invariant) [12] The dependence of the partition function on the shape-field f (~x) follows exactly the same form as described in [8]. This dependence is primarily constrained by the reparametrization invariant building block made out of f is the normal vector to the surface nµ , which for stationary configurations takes the form ∂µ f nµ = − √ = {0, ni}, ni = − pDj f Dj f ∂if f enter the partition function. In fact following the analogy, we will consider the normal Now, we wish to describe a stationary bubble of a superfluid inside an ordinary charged fluid. The entire set of data which constitutes the building blocks of the partition function for ordinary charged fluids away from the interface are B˜ = {σ (~x) , ai (~x) , gij (~x) , A0 (~x) , Ai (~x) , T0} . On the superfluid side, away from the interface, there is an additional ingredient On the surface, this set must also include the normal vector to the interface B = B˜ B(s) = B ∪ {ni(~x)} . The structure of the Euclidean effective action for a bubble of a superfluid inside a charged fluid will take the most general form W = {d~x}√g θ(f ) S(b) (B, ∂B, . . . )+δ˜(f ) S(s) B(s), . . . +θ(−f ) S(e) B˜, ∂B˜, . . . where S(b) and S(e) are the partition functions of space-filling superfluids and ordinary namics. In addition, one must also consider terms containing reparametrization invariant in fact, two dimensionless small parameters in the effective theories studied in this paper. tuations), which allows us to make the usual derivative expansion in fluid dynamics. The rameter. Thus, (1.9) should be thought of as a double expansion in both these parameters. The energy-momentum tensor and charge current that follows from the partition function (2.1), have the structural form = θ(f ) T (µνb) + δ˜(f ) T (µνs) + θ(−f ) T (µνe) + . . . , µ = θ(f ) J (µb) + δ˜(f ) J (µs) + θ(−f ) J (µe) + . . . , bulk of the superfluid bubble, S(e) as the exterior and S(s) as the surface. Correspondingly, surface currents. The former two have been well explored in the literature (see e.g. [11, 12]), so our main focus here will be on the surface currents, and how the bulk/exterior of the bubble affects the surface. The conservation of energy-momentum tensor and charge current in (1.10), serve as the fluid equation of motion. In this paper, we will obtain the surface currents in (1.10) in a special hydrodynamic frame,8 which is the frame that follows directly from equilibrium partition functions. In usual ordinary fluid variables are given by to all orders in the derivative expansion. We will refer to this frame as the partition function frame. Furthermore, the surface equations of motion that follow from the conservation of the currents (1.10) can be thought of as equations which constrain the boundary conditions terms can be neglected, then the exercise of finding new configurations reduces to a boundary value problem from the bulk point of view. This problem should be solved with the boundary conditions themselves being determined by the surface conservation equations.9 Before summarizing our results regarding the detailed structure of the partition functhe Landau-Ginzburg paradigm. Here, we are describing an interface between two phases, distinguished by the status of a U(1) symmetry, which is spontaneously broken in one phase, while is intact in the other. The two phases, therefore, are distinguished by an order parameter, with the help of which it is possible to write down a Landau-Ginzburg 8See [8] for a detailed and complete description of issues on the choice of frames. 9We would like to emphasize that using solutions of the surface equations as boundary conditions for the bulk equations is clearly consistent at least in equilibrium, where there exists a continuous solution (1.11) of the fluid variables for the combined set of equations, following from the conservation of currents in (1.10) (see [8] for more detailed discussion of this issue). In section 5, we also show that this method may be applied in time-dependent situations as well at leading order in derivatives. ALG = superfluid phase and is 0 outside, and smoothly interpolates between 1 and 0 on the interface separating the two phases. The hydrodynamic degrees of freedom can be seen as small referred to as the superfluid velocity, enters the superfluid dynamics. We would like to point and goes to zero with the onset of the ordinary charged fluid. This implies that it is possible of the interface partition function S(s) on the superfluid velocity. In this context, it is worthof the superfluid velocity, as well as on its component along the direction normal to the surface, both being Lorentz scalars from the interface point of view. Following the analogy with ordinary fluids,10 it is tempting to anticipate that the component of the superfluid velocity normal to the surface should vanish in the stationary case. However, we were unable to obtain any rigorous justification why this should be the case, and hence we will perform all our analyses keeping this component non-zero and arbitrary. In fact, an entropy current analysis at leading order, performed in section 5.1.2 for situations away from equilibrium, also allows for a non-zero component of the superfluid velocity normal to the surface. It may also be noted that, while describing superfluids where there is a normal fluid constituted out of the more fundamental degrees of freedom. In such situations, we may In 2+1 dimensions, in particular, it is possible to write down such an interesting parity-odd interaction term of the form ALG = independent context, in the reduced language, this would imply that in general the surface important and non-trivial consequence on the surface thermodynamics of 2 + 1 dimensional superfluid bubbles. 10For ordinary fluids, the normal component of the fluid velocity at the surface vanishes in equilibrium, i.e. uμnμ|f=0 = 0. See [8] and section 5 for further details. The construction of S(b), up to first order in derivative expansion in 3 + 1 dimensions was presented in [12] and is given by11 W = Weven + Wodd, where Weven = Wodd = d3x√g d3x√g θ(f ) T0α4 ǫijkζi∂j ak + α5 ǫijkζi∂j Ak . In this paper, we also construct S(b) in 2 + 1 dimensions up to first order in derivatives in section 3. The parity-even sector is identical to that of (1.15), while the parity-odd sector is richer than its 3 + 1 dimensional counterpart13 Wodd = d2x√g The bulk currents that follow from (1.16) have not yet been analyzed in the literature, to the best of our knowledge. We perform this exercise in section 3.1. We find that there is a total of 35 relations among transport coefficients that are determined in terms of the four coefficients in (1.16) (in addition to the parity-even terms). Since we are only considering terms up to the first order in derivatives on both the sides far away from the surface, it suffices to only consider a zeroth order term at the surface for S(s). This is the surface tension term which was considered in [8]. Since we will be dealing with superfluids on one side of the interface, the surface tension can now also depend on the superfluid velocity. In 3 + 1 dimensions, we work out the ideal order surface currents in (2.17) by varying the partition function, with associated surface thermodynamics given in (2.20). Later in section 3, we work out the analogous surface currents in 2 + 1 dimensions in (3.36), with respective thermodynamics given in (3.37), which also includes parity-odd effects. One of the most interesting features of our equilibrium analysis is the fact that the equation of motion for the shape-field f (the Young-Laplace equation) is identical to the normal component of the energy-momentum conservation equation at the surface. We rigorously argue in appendix B that this must continue to hold at all orders in the derivative expansion. → −ǫij , in 2 + 1 and 3 + 1 dimensions respectively. of the surface partition function. sional superfluid bubble. 13Note that these terms are also the parity-odd first order corrections on the surface of the 3 + 1 dimenIn order to obtain an understanding of the non-relativistic limits of superfluid surface currents, in section 4 we study Galilean14 superfluids in 3 + 1 dimensions. For this analysis, we use the technique of null (super)fluids developed in [10, 16, 17], where it was realized that transport properties of a Galilean (super)fluid are in one-to-one correspondence with that of a relativistic system - null (super)fluid in one higher dimension. Here, the basic idea is that in order to obtain the most generic Galilean (super)fluid currents in 3+1 dimensions, we can start with a null (super)fluid on a null background15 in 4+1 dimensions, and then perform a null reduction on it [18, 19] (also see [20, 21] for some earlier application of null reductions in the context of fluid dynamics). The null reduction reduces the underlying Poincar´e symmetry algebra of a null (super)fluid to the Bargmann symmetry algebra (Galilean algebra with a central extension with the mass operator) of a Galilean (super)fluid. Though we find the null-reduction prescription more useful for our purposes, it is worth mentioning that these Galilean results can also be obtained directly in a 3 + 1 dimensional NewtonCartan setting following [22, 23] (see also [24]). The equilibrium currents of a null (super)fluid can be obtained from a partition function written in terms of the background fields, Goldstone boson and the shape-field, in very much the same way as for the relativistic fluids discussed in section 1.1. There is however, one crucial new ingredient for null backgrounds: in addition to the time-like Killing vector K˜ as in (1.1), null backgrounds also have a null Killing vector V . Choosing a set of coordinates {xM } = { configurations respecting both Killing vectors are given as ds2 = GMN dxM dxM = −2eσ(~x) dt + ai(~x)dxi dx− − B0(~x)dt − Bi(~x)dxi + gij (~x)dxidxj , A = −dx− + A0(~x)dt + Ai(~x)dxi, However, while writing an equilibrium partition function, we will not require our background to be torsionless and will only impose it at the end of the computation (see [16] for details). As in the relativistic case, we would like to construct the partition function in terms of all the background data which are manifestly invariant under diffeomorphisms on the null background and gauge transformations. In order to do so, we need to consider the following invariant combinations (we refer the reader to [10] for more details regarding the transformation properties) Bi = Bi − aiB0, B0 = B0, Ai = Ai − aiA0 − Bi, A0 = A0. 14There is a subtle difference between Galilean and non-relativistic systems. As we will explain in more detail below, in the context of fluid dynamics, non-relativistic fluids are only a special class of the Galilean ones. Moreover, there can be other non-relativistic systems, such as Lifshitz systems with a dynamical exponent z 6= 1, 2, which are not Galilean. 15A null background is one which admits a null Killing vector V M such that a component of the gauge field is fixed as V M Compared to [10], the additional ingredient in our discussion is the shape-field f , since eventually we are interested in the non-relativistic limit of the superfluid surface. The surface of the null superfluid needs to respect both the Killing vectors V and K˜ , rendering function in a reparametrization invariant fashion, the primary dependence on f comes The background data invariant under all the required symmetries, in terms of which the partition function for bubbles of a null superfluid should be constructed, is given by (1/T0 is the radius of the Euclidean time circle) B(s) = B ∪ {ni(~x)}. (1.19) Note that in this case the background data is clearly larger compared to the relativistic case, leading to more terms in the partition function at any given derivative order. This in turn implies that the Galilean fluid obtained after null reduction will in general have more transport coefficients than its relativistic counterpart. This is to be expected for a non-relativistic fluid as well, e.g. in the non-relativistic limit the energy of a relativistic fluid splits into a rest mass density part and the residual internal energy, hence increasing the count. Though this counting accounts for the extra coefficients at ideal order, there is no reason to believe that at higher orders as well such splitting will account for all the extra transport coefficients of a Galilean fluid.16 Therefore, the most generic non-relativistic fluid is, at best, a subset of the Galilean fluid discussed in this paper, exploration of which we leave for future work. Finally, the equilibrium partition function for a 4+1 dimensional null superfluid bubble immersed in an ordinary fluid, up to first derivative order in the bulk and ideal order on P(b) − f1ζi∂iσ + e−σf2ζi∂iA0 + e−σf3ζi∂iB0 the surface, can be written as W = Note that there are no possible first order terms that we can write on the ordinary fluid side outside the bubble. All the transport coefficients are functions of the zeroth order in the relativistic case. Note that in writing (1.20), we have ignored a total derivative term 16Furthermore, transport coefficients of a Galilean fluid have dependence on an extra scalar as opposed to a relativistic fluid, namely, the mass chemical potential. However, for any non-relativistic fluid obtained as a limit of a relativistic fluid, this dependence must be trivial. in the bulk, which can be absorbed in the surface term and, similarly to the relativistic case, Using the partition function (1.20) and the variational formulae (C.4), we can work out the currents for a 4 + 1 dimensional null superfluid bubble, which we report in (4.8). Given this, it is straightforward to exploit the null isometry to perform a null reduction and get the surface currents for a Galilean superfluid (4.9). Even in this case, we find that the ideal order surface currents receive contributions from the bulk transport coefficients leading to different thermodynamics compared to the bulk. Time dependent fluctuations of the surface Having understood the nature of the surface currents in equilibrium, we proceeded and introduced a slow but arbitrary time dependence. Away from equilibrium, there is no variational principle that can help us in deducing the structure of surface currents.17 Therefore, we have to resort back to the second law of thermodynamics in order to constrain the transport coefficients. The surface of the fluid interacts freely with the bulk. In order to account for this exchange of degrees of freedom between the bulk and the surface, the local form of the second law at the surface needs to be suitably modified. This modification takes the following form ∇˜ µ J (µs)ent − nµ J (µb)ent ≥ 0 , where J (µs)ent and J (µb)ent represent the local surface and bulk entropy currents respectively. entropy current which is of the form18 J eµnt = J (µb)entθ(f ) + J (µs)entδ˜(f ) + . . . . There are a few important aspects of out of equilibrium dynamics that are a priori unclear, even in the context of ordinary fluid surfaces. One of the key aspects that needs to be understood is the nature of the normal component of the fluid velocity uµ nµ at the surface. In equilibrium, uµ nµ vanishes by construction but once the location of the surface becomes time dependent, this component may become non-trivial. Drawing from observed that the Josephson equation, even at ideal order, followed from the second law of 17Given some of the latest developments in writing down actions in terms of fluid variables in nonequilibrium situations [25, 26], it would be interesting to understand if this setup can be suitably generalized to describe out of equilibrium fluid surfaces as well. 18The reader may wonder, since the second law is expressed as an inequality for the divergence of the total entropy current, whether it is legitimate to implement the inequality separately for terms proportional entropy current is divergence free and the second law inequality must be valid for all fluid configurations. in equilibrium. This leads to the interpretation of the Josephson condition as the equation frame which is the appropriate generalization of the partition function frame in (1.11). In equilibrium, it reduces to the equation of motion of f (or equivalently to the Young-Laplace equation, which is the component of the energy-momentum conservation equation normal to the surface). Also, it is noteworthy that in out of equilibrium situations, the equation determine two scalar degrees of freedom at the boundary: uµ nµ and f , the former of which turns out to be trivial in equilibrium. Proceeding to the superfluid case in out of equilibrium scenarios, we tackle the corresponding problem for the normal component of the superfluid velocity at the surface the entropy density is identical to what is obtained from the equilibrium partition function Also, as explained previously, the surface equations may also be interpreted as determining the possible set of boundary conditions that are allowed for the bulk fluid equations. Clearly, in the equilibrium case, there are consistent solutions to the full set of bulk and surface equations. In the partition function frame, such a solution corresponded to the one where the fluid velocity is aligned with a Killing vector field of the background. However, away from equilibrium, even with a judicious choice of frame, such a solution may be considerably complicated. In order to obtain some idea of the nature of such solutions in time-dependent cases, we study the linearized fluctuations around a toy equilibrium configuration, only considering the perfect fluid equations of motion. In section 5.2, we work with 2+1 dimensional ordinary fluids in flat space and consider the background equilibrium configuration to be one in which a static fluid fills half space. At first, we set the surface entropy to zero, recovering the standard dispersion relation larger than the surface thickness, then ignoring the surface degrees of freedom is a perfectly legitimate approximation. However, as soon as we allow the surface tension to be a function of T , thus introducing some non-trivial surface entropy, our surface equations predict a such sound-like boundary conditions. This new kind of surface sound wave for ordinary fluids is expected to be visible if the amplitude of the waves is comparable or less than the surface thickness. These waves are very reminiscent of the third sound mode for superfluids. We perform a similar analysis for 2 + 1 dimensional superfluids, for which the leading order surface equations contain parity-odd terms. We find that parity violation leaves its imprint its partner under a parity transformation k → −k is absent. Stationary superfluid bubbles in 3+1 dimensions In this section we study stationary bubbles of a 3 + 1 dimensional relativistic superfluid immersed in an ordinary fluid. We work out the respective constitutive relations up to first derivative order in the bulk and ideal order at the interface using equilibrium partition A discussion of the surface properties in perfect superfluids was initiated in [8]. Here we will elaborate and extend upon that discussion. As explained in section 1.1, the equilibrium partition function for superfluids takes the form given in (1.9). If the partition function does not contain any derivatives, the respective superfluid is called a perfect superfluid. It is of course a fictitious simplified system just like a perfect fluid, nevertheless it is an instructive toy system to study before moving to more complicated generalizations. For a perfect superfluid bubble with an ordinary charged fluid outside, the most generic partition function takes the form W = d3x√g θ(f )P(b) (T, µ, χ ) + δ˜(f )C (T, µ, χ˜, λ) + θ(−f )P(e) (T, µ ) , − nµ nν )ξν and χ˜ = −ξ˜µ ξ˜µ = µ 2 see later, P(b) and P(e) are the bulk and external pressures while C will be identified as the negative of surface tension. The discussion in this subsection is immediately applicable to perfect superfluid bubbles in all dimensions, except in 2+1 dimensions19 where there can be parity-odd effects at ideal order, and will be treated separately in section 3. We start by varying the partition function (2.1) with respect to the Goldstone boson ∂P(b) ∂C = 0 , = 0 , = 0 , 19Even in 2+1 dimensions, if we restrict only to the parity even sector, then the discussion of this section where Di denotes the spatial covariant derivative associated with gij , while D˜ i denotes the spatial covariant derivative on the surface defined in section 1.1. The last line of this equation is particularly interesting, as it tells us that on-shell, the boundary function C is The first line in (2.2) is a non-linear second order differential equation, which yields the superfluid velocity can be taken to be small, this equation may be linearized and converted into a second order linear partial differential equation. This equation must be solved with suitable boundary conditions at the interface, which are provided by the solutions to the the interface, while the second equation provides the initial condition necessary to evolve the first equation away from the interface. Note that as we move to higher orders, we will have an additional condition at the surface, and correspondingly, the order of the first differential equation will increase by one. Varying the partition function (2.1) and using the variational formulae (C.3), we can read out the bulk and boundary currents. The form of the energy-momentum tensor and charge current inside the bubble takes the usual perfect superfluid form and has been thoroughly discussed in [12], while outside the bubble it is just an ordinary perfect charged fluid. The new ingredients in our discussion however are the currents at the interface, there is energy and charge transport along the superfluid velocity, also on the surface. It is further instructive to write down the equation for the shape-field f that follows P(b) − P(e) + T Di = 0 . This is the modified Young-Laplace equation in the present case. As argued in appendix B, this equation is simply the normal component of the energy-momentum conservation equation on the surface. Let us now study the implications of this analysis on the covariant form of the charge current and energy-momentum tensor. We would like to work in a hydrodynamic frame most suitable for the analysis using the partition function. It is a frame where we have −C +T J(s)0 = −e ∂C +µ ∂C , ∂C +2ξ02 ∂∂χC˜ , T(s)i0 = ξ02 ∂∂χC˜ ζ˜i , T(s) 20Since the surface tension function C is given a priory, derived from the microscopics, the condition everywhere to all derivative orders, including at the interface. Such frame choice should be always possible to make as long as we are in equilibrium. The most general ideal order − nµ nν ) + F ξ˜µ ξ˜ν + 2λ1u(µ ξ˜ν) + 2λ2ξ(µ n¯ν) + 2U u(µ n¯ν) , reducing (2.6) on the time circle and comparing it with (2.3), we obtain22 E = −C + T ∂C + µ ∂C Y = −C, Q = ∂C F = −F ′ = 2 ∂C S = ∂C , the surface energy, charge and entropy densities, while F is the surface superfluid density. We will see, however, that the coefficients U , V get non-zero values when we introduce first order terms in the bulk. The coefficient S introduced in (2.7) is the surface entropy density recover the Euler relation and the Gibbs-Duhem relation of thermodynamics respectively E − Y = T S + µ Q , dY = −SdT − Qdµ − 2 F dχ˜ . The first law of thermodynamics trivially follows from here as These thermodynamic relations are exactly the same as their bulk counterparts. However, as we will show in the next subsection, the surface thermodynamics will modify upon including first order corrections in the bulk. First order corrections away from the interface Since the surface currents sit on a boundary separating two phases of a fluid, transport coefficients at a particular derivative order in the bulk can affect the surface currents at lower orders via an “inflow” (via a differentiation by parts in the partition function language). Therefore, we expect the ideal order surface currents to get contributions from first order terms in the bulk. In order to do so, we consider first order corrections to the bulk superfluid partition function (discussed in [12]) W (1) = We(v1e)n + Wo(d1)d , 21The tangentiality conditions on the surface energy-momentum tensor and currents are a direct consequence of their conservation equations to leading order in the surface thickness. If thickness corrections are currents) then these tangentiality conditions will be modified by extra terms on the right hand side [28–30]. as in classical literature. We(v1e)n = Wo(d1)d = d3x√g d3x√g θ(f ) α1ζi∂iT + α2ζi∂iν − α3Di θ(f ) T0α4ǫijkζi∂j ak + α5ǫijkζi∂j Ak . As discussed in section 1.1, while working up to first order in derivatives in the bulk of the superfluid, it is consistent to consider only the ideal order surface tension term at the surface, which was considered in (2.1). Also, far outside the superfluid bubble, the ordinary charged fluid does not receive any first order corrections, as there are no possible terms that can be written in the partition function. Consequently, W (1) in (2.11) constitutes the entire first order corrections to the perfect fluid partition function in (2.1). The bulk energy-momentum tensor and charge current that follow form (2.11) have been thoroughly examined in [12]. In particular, it was pointed out in [12] that the term surface quantities. However, at the level of the partition function for instance, we can always redefine the surface tension to absorb these terms and ignore any higher order terms. The surface energy-momentum tensor and charge current, in addition to (2.3), will T(s)00 = e2σλT α1, T(s)i0 = eσ (T α4 − µα 5) n¯i, T(s) ij = 0 , +2eσ ∂α4 ζiǫajkζa∂j ak − eσα4ǫijk∂j ak + = 0, ζ ∂j ν − α2nj ∂j ν + 2λT0 ∂χ ∂α5 ǫajkζa∂j Ak − α5ǫijkni∂j Ak = 0, ∂C = 0 , while the modified f equation of motion (Young-Laplace equation) is given as (see appendix B for a detailed discussion on Young-Laplace equations in the generic case) P(b) − P(e) + α1ζi∂iT + α2ζi∂iν + T0α4ǫijkζi∂j ak + α5ǫijkζi∂j Ak = 0 . (2.14) It is worth pointing out that, instead of the partition function W in (2.11), we could have We(v1e)n = Wo(d1)d = 4 √ 4 √ Comparing it to (2.11), we can simply read out the respective coefficients Now, the covariant form of the energy-momentum tensor and charge current, after contributions from the first order transport coefficients and we have E = −C + T Q = ∂C + µ Y = −C , U = g1 , S = V = g2 . ∂C + (f1 − µf 2) , Here we have defined S as the surface entropy density with the respective entropy current J (µs)ent = Suµ + U − µ V) n¯µ . The identification (2.18) leads to the Euler relation and a modified Gibbs-Duhem relation of thermodynamics at the surface24 E − Y = T S + µ Q , dY = − S − T (f1 − µf 2) dT − Q − λf2 dµ − 2 F dχ˜ . (2.20) 24In (2.20), f1 and f2 should be thought of as first order bulk transport coefficients (see [12]) rather than parameters of the partition function. We clearly see that the thermodynamics is different from usual.25 The respective modified first law of thermodynamics now takes the form = T d S − T (f1 − µf 2) This modification can be interpreted as follows. The surface densities E , Q, S and F have, in general, two contributions: from the thermodynamics on the surface and from the thermodynamics (2.20)–(2.21). Note that the parity-odd ideal order surface transport coefficients U and V (or correspondingly g1 and g2) do not enter the thermodynamics (2.20)–(2.21). However, since all the first order bulk transport coefficients f1, f2, g1, g2 do modify the ideal order surface transport, they can be measured by carefully designing experiments which probe the ideal order surface properties of superfluids. Stationary superfluid bubbles in 2+1 dimensions In this section, we study stationary superfluid bubbles in 2 + 1 dimensions and particularly focus on the parity-odd sector, where there is a significant difference compared to the 3 + 1 dimensional case. In fact, an exhaustive analysis of the first order parity-odd terms in the bulk of 2 + 1 dimensional superfluids has not been executed so far, to the best of our knowledge. Therefore, we also evaluate the stationary bulk currents following from the parity-odd first order bulk partition function in section 3.1 before analyzing their surface effects. Parity-odd effects for perfect superfluid bubbles We have discussed perfect superfluids in general dimensions in section 2.1. However, as explained in section 1.1, in 2 + 1 dimensions there can be parity-odd terms which may have a non-trivial effect on the surface tension. Hence, before going into the details of the surface effects of first order corrections in the bulk of 2 + 1 dimensional superfluids, we will revisit the zeroth order case once more. possible presence of a term like (1.13) in a Landau-Ginzburg effective theory, the surface 25Note that the difference here is not just a mere matter of definition of S and Q. We could have simply Gibbs-Duhem relation, which would then become different from usual. Note that our definition of the charge Q, for instance, corresponds to the quantity which is the zeroth order value of the surface charge current, projected along the direction of the surface fluid velocity. In the usual case of bulk thermodynamics, these two definitions of charge density would coincide, but not for the surface thermodynamics. This is because, the surface current contains additional zeroth order terms proportional to bulk first order transport coefficients. 1 ∂C ni 2 ∂P(b) i ∂P(b) ∂C = 0 , = 0 , = 0 , where again, D˜ i denotes the covariant derivative on the surface. The energy-momentum tensor and charge current far away from the interface is exactly the same as in the 3 + 1 dimensional case. At the interface however, we can use the formulae in appendix C in order to determine the energy-momentum tensor and charge current as following (after imposing −C + T J(s)0 = −e ∂C + µ ∂C ij = Chij W = d2x√g θ(f )P(b) (T, µ, χ ) + δ˜(f )C T, µ, λ, λ¯ + θ(−f )P(e) (T, µ ) . where hij = gij parity-odd and parity-even contributions and is given by (3.2). We can also easily obtain the equation of motion for the shape-field f , which now involves parity-odd pieces as well, P(b) − P(e) + Di = 0 . We again choose to work in a hydrodynamic frame suitable for the partition function redundant, since it is no longer an independent variable and is given by This is due to the fact that, on the interface there are only two independent components of the superfluid to the relation (3.1), the surface term in (3.3) may also be written as C T, µ, λ, λ¯ = C0 (T, µ, χ˜, λ) + λ¯C1 (T, µ, χ˜, λ) , however in our discussion we choose to proceed with the form (3.3). that the most generic form of the constitutive relations at ideal order (transverse to nM ) the energy-momentum tensor, making a total of 12 independent terms. The equilibrium partition function fixes these 12 coefficients in terms of a boundary function C and 6 first order bulk coefficients fi, gi. Finally, upon performing the null reduction, the leading order surface currents and densities for a 3 + 1 dimensional Galilean superfluid can be obtained as ti(js) = Rnuiuj − Yhij + Rsξ˜iξ˜j + 2g1u(i n¯j), ǫ(s) = E + Rsµ˜s + E − Y + 2 Rnukuk + 2 Rsξ˜kξ˜k + g1 n¯iui, 2 Rnukuk + g1 n¯iuj q(s) = Q − Rs, It is interesting to contrast these results with those in the bulk, as reported by [10]. Not only there are new terms in the leading order Galilean constitutive relations, but some of them are parity-odd as well. Furthermore, all these new terms are completely determined in terms of the first order bulk transport coefficients. In fact, since all the first order stationary bulk coefficients appear in the surface constitutive relations, they can, in principle, be measured by performing carefully designed experiments on the surface of the superfluid. Surface dynamics In this section, we study the consequences of a non-trivial time dependence of the shapefield on the surface. Once we relax the assumption of stationarity, we cannot deduce the constitutive relations of a (super)fluid through an equilibrium partition function, as we did in section 2 and section 3. Therefore, we have to resort to the second law of thermodynamics to constrain and understand the full time-dependent dynamics. Hence, we first analyze the surface entropy current at ideal order in section 5.1, to understand the structure of the equations governing the surface dynamics. With this understanding, in section 5.2 we study linearized fluctuations on the surface and its relation with the fluctuations in the bulk, both for an ordinary fluid and a superfluid. Surface entropy current for ordinary fluids Before proceeding to the superfluid case, we study the entropy current and the consequences of the second law of thermodynamics for ordinary fluids in the presence of a surface. Once we give up the assumption of stationarity, the first aspect of surface dynamics we would like to understand is what determines the normal component of the fluid velocity uµ nµ at Killing vector field.30 The second aspect of surface dynamics we would like to understand is what determines the equation of motion for the shape-field f , since it is not clear a priori if the normal component of the surface energy-momentum conservation continues to serve as a proxy for the equation of motion of f in non-equilibrium situations. In this section, we will try to answer both these questions and demonstrate that they are interrelated. As mentioned above, in the analysis of equilibrium partition functions, uµ nµ was zero by construction. In fact, this condition served as one of the boundary conditions for solving the bulk fluid equations (see section 1 and [8] for more details). However, as we move away from stationarity, the status of uµ nµ is not clear a priori and we need a principle to determine it. In order to address this problem, it is extremely useful to remember the of a spontaneously broken symmetry. Momentarily, if we take this analogy seriously then independent variable in superfluid dynamics. In fact, it is given by the chemical potential µ [6] at leading order and receives further corrections at higher orders, as determined by the second law of thermodynamics [4]. As noted in [27], the generalized Josephson observed in [27] that in equilibrium, and in a hydrodynamic frame chosen appropriately for equilibrium. This gives us an important clue for the case of the shape-field: uµ nµ should also be determined by the second law of thermodynamics in terms of other fluid variables, and the respective determining relation should be the equation of motion for f outside equilibrium. For this purpose, let us define relation among these variables. Later in this section, we will show that the second law of thermodynamics treated as an independent thermodynamic variable at the surface, as was done in [9]. not correspond to the more standard frame choices like the Landau frame, neither would it be a generalization of the equilibrium frame defined in section 1. Let us now proceed to analyze the structure of the divergence of the surface entropy current. The bulk energy-momentum tensor and entropy current have the well known form On the other hand, the ideal order surface currents are given by31 where the Y, E , S are the surface tension, energy density and entropy density on the surface tions will not play any significant role in our discussion below, but we retain them for completeness. The surface conservation equation projected along the fluid velocity takes µν µν uν ∇˜ µ T(s) − uν nµ T(b) = −uµ ∂µ E − (E − Y) ∇˜ µ uµ ∇µ (Ynµ ) + E = 0 . Now, the divergence of the entropy current on the boundary, including the possible entropy exchange with the bulk, must be positive semi-definite. This condition upon using the equation of motion (5.4) simplifies to ∇µ (Ynµ ) − P + ∇˜ µ Υµ(s)new − nµ Υµ(b)new ≥ 0 , the surface, (5.5) implies that ∇µ (Ynµ ) − P ≥ 0 . energy-momentum tensor and entropy current, since we wish to derive such tangentiality conditions at leading order from the entropy current analysis. Furthermore, note that in (5.3) we have not considered into account but for clarity of presentation we have not introduced them. In any case, the second law of The condition (5.6) must hold for an arbitrary fluid configuration, including the ones for which the term inside the bracket may have a negative sign. This implies that at leading order uµ nµ must vanish, that is This is the first important conclusion of this section. As we move to higher orders, other terms in (5.5) become important for this analysis. An important noteworthy structural feature in (5.5) is the fact that the only term which to the surface transport coefficients. An interesting observation can be made, if we focus which has a solution ∇µ (Ynµ ) − P ≥ 0 , motion away from equilibrium is then Laplace equation, defined as the normal component of the surface energy-momentum conservation equation is −T (µνs) Kµν = T YK − T immediately follows. However, under the assumption of perfect fluid bubbles, for which (5.8) applies, one may use the fact that, on-shell, the normal component of the vector bulk equation of fluids, even away from equilibrium.32 When we include first order terms in the bulk, i.e. respectively, Young-Laplace equation modifies as YK − T 32This equivalence holds on-shell but not off-shell in the sense of [32]. On the other hand, the f equation of motion in (5.8) remains unchanged, since J (µb)new is known to be zero at first order for ordinary fluids. Hence for onshell configurations, we can rewrite (5.12) as uµ nµ = γdiss = −ς (ησµν nµ nµ + ζΘ) + O γdiss, ∂γdiss . 2 rium. We would like to note that, upon including further higher order corrections, either in the bulk or at the surface, and hence moving further away from the simplified case of perfect fluid bubbles, we might expect (5.8) as well as (5.11), to be modified. Surface entropy current for 3+1 dimensional superfluids Having understood the behaviour of uµ nµ for neutral fluids, in this subsection we will explore the similar entropy current analysis for superfluids with a surface. We will demonstrate that the first law of thermodynamics in 3 + 1 dimensions modifies like (2.20), and interface, in contrast to the normal component of the fluid velocity uµ nµ which is set to zero at ideal order. For superfluids, the bulk currents take the well known form ST + µQ, dP = SdT + Qdµ +µ 12 F dχ. In our analysis here, the first order corrections Υµ(b)new = ∇ν c1u[µ ξν] + c2ǫµνρσ uρξσ μ 33At first order, the only contribution to Υ(b)new comes from the equilibrium sector and is obtained as follows [27]: write down the most general scalar L made out of first order data that survives in equilibrium (it can be thought of as a covariant version of the partition function), and perform a variation keeping the fluid variables constant √1 δ(√GL) = 21 TLμνδGμν + JLμδAμ + KLδϕ + YLδϕ + ∇μΘμL, In fact, TLμν, JL, KL and YL are the first order equilibrium energy-momentum tensor, charge current, φ μ variation and f variation respectively in the bulk, gained via the equilibrium partition function. 2 Fρσ + uσT ∂ρν where ζµ = ξµ + (uν ξν )uµ and we have defined an operator Oχ(·)µν for clarity. Note that the first term here is a total derivative, i.e. its divergence trivially vanishes, and hence is not important in the bulk. However it might have some non-trivial boundary effects. Following (2.6), the surface currents have the general form34 − nµ nν ) + F ξ˜µ ξ˜ν + 2U u(µ n¯ν) + Πµ(νs) , J (µs) = Q uµ − F ξ˜µ + V n¯µ + Υµ(s) , conservation equations µν µν uν ∇˜ µ T(s) − uν nµ T(b) − uν F νµ Jµ (s) = −uµ ∂µ E − (E − Y) ∇˜ µ uµ − 2 F uν ∂ν χ˜ − T ∇˜ µ µν µν + uν ∇˜ µ Π(s) − nµ Π(b) − uν F = 0 . Now, it is possible to show that the divergence of the entropy current conservation at the surface reduces to ∂µ E −T ∂µ S −µ∂ µ Q+ 2 F ∂µ χ˜ − T (E −Y +T S +µ Q) ∇˜ µ uµ ∇µ (Ynµ − (nν ξν )F ξ˜µ ) + E − T S − µQ − T Υ(s) (T ∇µ ν + uν Fνµ ) − 2U u(µ n¯ν)∇µ − T V n¯µ (T ∇µ ν + uν Fνµ ) + ∇˜ µ Note that we have not imposed the thermodynamics yet, as there are first derivative terms in J (µb)new which might modify it. Restricting ourselves to first order in the bulk and ideal momentum tensor and currents. However, such terms will be ultimately set to zero by the entropy current analysis and hence we did not consider them here for clarity of presentation. order at the boundary, this equation modifies to − T − T +∇˜ µ S − T (f1 − µf 2) − µ∂ µ E − Y + T S + µ Q) ∇˜ µ uµ + ≥ 0 , Eφ = ∇˜ µ (F ξ˜µ ) − (nµ ξµ )F − Oχ(f1)µν n − Oχ(g1)µ αnµ ǫανρσuν ∂ρuσ − 2 Oχ(g2)µ αnµ ǫανρσuν Fρσ , Ef = ∇µ + g1ǫµνρσ uµ ζν ∂ρuσ + g2 1 ǫµνρσ uµ ζν Fρσ . The condition of positive semi-definiteness implies the surface thermodynamics = T d S − T (f1 − µf 2) E − Y = T S + µ Q , and the relations U = g1 , V = g2 , which are exactly the same as the ones found using the equilibrium partition function. The second law also implies the corrections to the entropy current35 After imposing all of these, the second law of thermodynamics will turn into U − µ V − T c2) n¯µ + c1u[ν ξµ ]nν . f ≥ 0 , which will admit a general solution uµ ξµ − µ = αEφ + (β + β′)Ef , 35Note that, we can always modify the entropy currents as without changing the second law, hence the entropy currents always have this ambiguity. Interestingly, using this ambiguity we can get rid of both the c1 and c2 contributions from the theory. the Josephson and Young-Laplace equation respectively Ef = 0 , which are same as the ones derived using an equilibrium partition function. It is worthwhile noting that outside equilibrium, contrary to the ordinary fluid case discussed in the previous section, the equation of motion of f is not the Young-Laplace equation. Surface entropy current for 2+1 dimensional superfluids In this subsection we will give the entropy current analysis for 2+1 dimensional superfluids with a surface. We will only focus on the boundary computation here, for simplicity. As pointed out in the previous section, the only way in which the bulk interacts with the parity-even sector, but is quite different in the parity-sector. It is given by36 Υµ(b)new = ∇ν c1u[µ ξν] + c2ǫµνρ uρ + c3ǫµνρ ξρ κ1 ∂(mω − mBν) ζµ ǫανρuα∂ν uρ 2 Fνρ + uρT ∂ν ν On the other hand, the most generic surface currents are given as37 − nµ nν ) + F ξ˜µ ξ˜ν + 2U u(µ n¯ν) + Πµ(νs) , J (µs) = Q uµ − F ξ˜µ + V n¯µ + Υµ(s) , currents for clarity of presentation. 36We do not know of any reference which discusses generic first order corrections to entropy current for − T +∇˜ µ − T (E − Y + T S + µ Q) ∇˜ µ uµ + − T bulk and ideal order at the boundary, the second law (5.19) takes the form Υµ(s)new − T Uξn¯µ +νVξn¯µ +c1u[µ ξν]nν +c2ǫµνρ nν uρ +c3ǫµνρ nν ξρ ≥ 0 , Eφ = ∇˜ µ (F ξ˜µ ) − (nµ ξµ )F − Oχ(f1)µν n − Oχ(κ3)µ ν nµ ǫνρσuρ T ∂σT − Oχ(κ4)µ ν nµ ǫνρσuρT ∂σν , Ef = ∇µ T ∂ν T − f2ξν T ∂ν ν − κ2ǫµνρ u − κ1ǫµνρ uµ ∂ν uρ − κ3ǫµνρ ζµ uν Demanding positive definiteness, we can read out the surface thermodynamics dC = − S − T E − Y = T S + µ Q . (f1 − µf 2) + − d Q − λf2 + λ¯κ4 dµ − F (µ dµ − λ¯dλ¯) , and the constraints which are exactly the same as found using the equilibrium partition function. The respective first law of thermodynamics has been discussed in appendix A. Furthermore, we get the correction to the entropy current38 T (U − µ V)n¯µ − c1u[µ ξν]nν − c2ǫµνρ nν uρ − c3ǫµνρ nν ξρ . After implementing all of these constraints, the second law takes the form f ≥ 0 , 38We can remove the c1, c2, c3 dependence of the system, by using the entropy current ambiguity (see which can be solved, just like in the 3 + 1 dimensional case, by uµ ξµ − µ = αEφ + (β + β′)Ef , recover the equilibrium version of the Josephson and Young-Laplace equations respectively Ef = 0 , which are same as the ones derived using an equilibrium partition function. Ripples on the surface After studying the structure of the leading order surface equations away from equilibrium, in this section we shall study the nature of linearized fluctuations about an equilibrium configuration. For simplicity, we shall confine ourselves to the discussion in 2+1 dimensions. Ordinary fluids. Let us first consider the case of ordinary charged fluids in flat spacetime. We choose the coordinates { xµ } = {t, x, y} with the flat Minkowski metric variables take the form T (t, x, y) = T0 , uµ (t, x, y) = (1, 0, 0) , f (t, x, y) = y . pressure must be uniform everywhere. Also, since the extrinsic curvature of the line Now, let us consider linearized fluctuations about this configuration uµ = (1, ǫ δux, ǫ δuy) + O(ǫ2) , transition temperature T0. Note that in (5.40), uµ −1 + O(ǫ2). remains unit normalized up to the relevant order, i.e. The linearized equations in the bulk, which follow from the 39Note that the vanishing of the extrinsic curvature only implies that the pressure difference at the surface vanishes. If we consider a scenario similar to the one in [33], where a plasma fluid is separated from the vacuum by a surface, then the surface pressure and hence the equilibrium pressure everywhere in the bulk for the configuration (5.39) must vanish. This may be achieved if the equation of state is of the form conservation of the leading order energy-momentum tensor in (5.2), are given by, = 0 , = 0 . As we have argued in 5.1.1, nµ uµ at leading order must vanish due to the second law, variable at the surface. In the linearized approximation this equation is given by Using this and the leading order surface energy-momentum tensor (5.3), the surface conservation laws take the form S(T0) ∂xδux + S′(T0) ∂tδT = 0 , = 0 , E(T0) ∂t2δf − Y(T0) ∂x2δf = S(T0) δT . Now, the procedure for solving these equations as outlined in section 1 includes first solving the solutions as a boundary condition for solving the remaining 3 bulk equations (5.41) for In the classical computation of capillary waves [34], the surface entropy S is considered to be zero, or equivalently, a constant surface tension is assumed. In this limit, (5.43a) and (5.43b) are automatically satisfied. This implies that the set of allowed boundary conditions is less constrained compared to the more general case. Thus, the bulk equations, in that case, may be solved with partially arbitrary boundary conditions, as long as (5.43c) and (5.42) are ensured to be satisfied. In order to obtain the dispersion relation of capillary waves, in the absence of any external gravitational field, the equations (5.41b), (5.41c), (5.42) and (5.43c) are solved by cos (kxx + ωt) e−κy , δuy(t, x, y) = δf0 ω sin (kxx + ωt) e−κy , linearized fluctuation. The remaining equation (5.41a) provides a condition for determining relation of the form ω ≈ ±kx3/2pY/E. However, if we take into account a non-zero surface entropy, then the boundary conditions for solving the bulk equations must satisfy all the equations in (5.43) and (5.42). This completely determines the possible set of boundary conditions. In fact, (5.43) and (5.42) admits a sinusoidal solution with the following dispersion relations ∂E /∂T We see that there are two sound-like modes on the surface. We can solve the bulk the full bulk solution corresponding the to first dispersion relation in (5.46) takes the form P ′(T0) E (T0) − 1 and the dispersion is, ω = ±kx It can be easily checked that (5.47) solves both the bulk (5.41) and surface equations (5.43) simultanuously. There also exists a similar sinusoidal solution corresponding to the second dispersion in (5.46). Note that it should be possible to have both, the capillary waves in (5.44), as well as the tiny ripples (5.47) on the surface of the same fluid. If the amplitude of the waves is large compared to the thickness of the surface, then neglecting the surface entropy would be a legitimate approximation. Hence, in that case, we shall have capillary waves as in (5.44). On the other hand, if the amplitude of the surface waves is small or comparable to the surface thickness, then waves like (5.47) would be generated.40 We now move on to surface linear fluctuations in a 2 + 1 dimensional superfluid. To start with, we will consider an equilibrium configuration similar to (5.39), with the superfluid phase filling half spacetime y ≥ 0 T (t, x, y) = T0 , µ (t, x, y) = µ 0 , φ(t, x, y) = φ0 , ξµ (t, x, y) = (−µ 0, 0, 0) , uµ (t, x, y) = (1, 0, 0) , f (t, x, y) = y , 40In this sense, the linearized solution (5.47) is similar to the third sound mode on superfluid surfaces [35]. We consider the following linearized fluctuations about this equilibrium configuration µ = µ 0 − ǫ∂tδφ + O(ǫ2) , uµ = (1, ǫδux, ǫδuy) + O(ǫ2) , ξµ = {−µ 0 + ǫ∂tδφ, −ǫ∂xδφ, −ǫ∂yδφ} + O(ǫ2) , λ¯ = −ǫ(∂xδφ − µ 0δux) + O(ǫ2) , the ansatz (5.49), up to the relevant order. The surface equations are given by conservation of currents in (3.10), which includes parity-odd effects. With the most general analysis of the fluctuation equations, we found that a system face. These modes can further be used as boundary conditions to solve the bulk equations. For simplicity, however, here we consider a simplified equation of state where Y1, Y2 are constants. With this ansatz, the linearized surface conservation equations following from the leading order currents (5.14) and (3.10), together with the condition T0Y1∂xδux + µ 0Y2∂tδux = µ 0F (∂yδφ + µ 0∂tδf ) , −Y1 (∂xδT + T0∂tδux) − Y2 2µ 0∂xδux − ∂t2δφ = 0 , Y1T0∂x2δf + 2µ 0Y2∂t∂xδf = −SδT + (Q + µ 0F )∂tδφ , Y2∂tδux = F0 (∂yδφ + µ 0∂tδf ) , µ 0ω2 (SY2ω − Y1(Q + µ 0F )kx) − ℓkx2Y1 (2µ 0Y2ω + T0Y1kx) = 0 , of state41), one of which is a sound-like mode. We observe that none of these three modes on the spectrum of linearized fluctuations. this limit, the second clearly breaks it, as is expected for a system with no parity invariance. 41For instance, in ordinary fluids, the sound modes disappear if we consider an equation of state where the pressure is linear in temperature. This is because the velocity of sound becomes infinite in this limit. In this paper, we have worked out the leading order surface energy-momentum tensor and charge current for a finite bubble of superfluid, both in equilibrium and slightly away from it. In equilibrium, we were able to write down the most general Euclidean effective action for the Goldstone boson and the shape-field (in one lower dimension), coupled to arbitrary slowly varying background fields. By appropriately varying this action, we obtained all surface currents. Away from equilibrium, we used the second law of thermodynamics, implemented via an entropy current with a positive semi-definite divergence. Our near equilibrium results reduce to those obtained from the effective action, upon restricting to the stationary sector. The ideal order surface currents contain new terms, compared to their bulk counterparts, which are entirely determined by the first order bulk transport coefficients. This exercise has revealed new parity-even and parity-odd terms in the ideal order surface currents. In the case of the parity-odd terms, we have shown that they leave an imprint in the spectrum of linearized fluctuations. Such terms are also present in the surface currents of Galilean superfluids, which we have obtained by a null-reduction of 4 + 1 dimensional null superfluids. Hence, such new effects should also be relevant in realistic non-relativistic The parity-odd surface effects that we discussed here are relevant for theories with microscopic parity violation,42 but they may also be present as an emergent parity odd phenomenon. In order to better understand the nature of the physical systems in which our results would play an important role, it would be interesting to write down Kubo-like formulae for the first order parity-odd superfluid coefficients, along the lines of [37]. The results found here are extremely relevant in the context of black holes via the AdS/CFT correspondence. In this holographic context, the space-filling configurations of the boundary fluid have a one-to-one correspondence with slowly varying black brane configurations in the bulk [38]. It is also possible to generalize such maps to the context where the plasma of the deconfined phase fills the space partially while the rest of space is occupied by the confined phase [33, 39]. In the large N limit, such situations may be described by a plasma fluid separated from the vacuum by a surface in the hydrodynamic approximation. The holographic dual of such fluid configurations is a combination of black branes and the AdS-soliton patched up in a suitable fashion to account for the fluid surface at the boundary [7, 40–42]. Similarly, the holographic dual of the space filling superfluid phase are AdS hairy black holes [15]. It would be extremely interesting to construct the holographic duals of the superfluid bubbles discussed in this paper, along the lines of [33]. Such hairy black holes, besides being new and interesting solutions of the Einstein equations, may provide a suitable microscopic setting for a better understanding of the functional dependence of the surface tension on its arguments. 42For instance, in theories with anomalies, there may be additional terms in the first order bulk currents that are entirely determined by the anomaly coefficient [2, 3, 36]. Although we have refrained from discussing such (non-gauge invariant) terms in this paper, we hope to return to this question in a later work. We would like to thank Felix Haehl for many useful discussions and collaboration during the initial stages of this project. We would also like to thank Nabamita Banerjee, Sayantani Bhattacharyya, Suvankar Dutta, Arthur Lipstein, Mukund Rangamani, Simon Ross, Tadashi Takayanagi and Amos Yarom for useful correspondences and comments. We are grateful to Suvankar Dutta, Arthur Lipstein and Simon Ross for valuable comments on the draft of this manuscript. JA would like to thank NBI for hospitality during the course of this project. JB would like to acknowledge local hospitality at YITP during the long term workshop on “Quantum Information in String Theory and Many-body Systems” where a part of this work was done. JB would also like to thank IISER Pune and TIFR for hospitality during the final stages of the project. NK would like to acknowledge the support received from his previous affiliation HRI, Allahabad, during the course of this project. AJ is funded by the Durham Doctoral Scholarship offered by Durham University. JA acknowledges the current support of the ERC Starting Grant 335146 HoloBHC. JB is supported by the STFC Consolidated Grant ST/L000407/1. Research of NK is supported by the JSPS Grant-in-Aid for Scientific Research (A) No.16H02182. Surface thermodynamics in 2+1 dimensions In section 3 we derived the surface Euler relation and Gibbs-Duhem relation for a superfluid (f1 − µf 2) + Q − λf2 + λ¯κ4 dµ + F (µ dµ − λ¯dλ¯) , bubble, which take the form −dY = S − T E − Y = T S + µ Q . Though these relations are correct, they mix parity-even and parity-odd sectors. In this appendix, we will write down mutually independent thermodynamics for parity-even and parity-odd sectors on the surface, and derive the respective first law of thermodynamics. into a parity-even and a parity-odd sector Y(T, µ, λ¯) = Y+(T, µ, χ˜) + λ¯Y−(T, µ, χ˜). inflow from the bulk, we define mutually independent Gibbs-Duhem and Euler relations in −dY+ = Comparing these to the parity-mixed expressions, we can read out the parity splitting of energy, charge, entropy and superfluid density respectively µ 2 E− − µ 2 F− + Q− − µ 2 µ 2 Using (A.3), it is easy to derive the first law of thermodynamics for parity-even and parityodd sectors respectively = T d S+ − T (f1 − µf 2) + µ d Q+ − λf2 − 2 F+dχ˜, = T d S− + Equation of motion for the shape-field and the Young-Laplace equaIn this appendix, we rigorously show that in the stationary case, the Young-Laplace equation that follows by projecting the surface conservation equation along nµ , is identical to the equation of motion of f which follows from the equilibrium partition function, up to all orders in derivatives. Let us start with the most generic partition function variation 4 √ 4 √ 4 √ δ˜(n)(f ) = (−)n+1(nµ ∂µ )n+1θ(f ). p∇µ f ∇µ f know that W is a gauge invariant scalar, so it must be invariant under a diffeomorphism δX Aµ = ∇µ (Λϑ + Aµ ϑµ ) + ϑν Fνµ , δX f = ϑµ ∂µ f = −p∇µ f ∇µ f ϑµ nµ . This leads to a set of identities = 0, = 0, νρJρ(s) − nµ (T (µνb) − T (µνe) ) + nνY δ˜(f ) ∇µ J (µs) − nµ (J (µb) − J (µe)) = 0, = 0 . is the nµ component of the surface energy-momentum conservation equation This equation can be thought of as extremizing the partition function W under a restricted and Aµ along ϑµ = ϑnµ keeping f fixed. It might be beneficial to see this explicitly. Let the partition function have the form W = We use the facts that the bulk Lagrangians L(b), L(e) do not have any dependence on the shape-field f , and the dependence of L(s) only comes via the reparametrization invariant nµ . For the sake of simplicity, we further assume that L(s) is only dependent on nµ and not on its derivatives, which is true for our analysis in the bulk of the paper. We can perform a f variation of W to get p∇µ f ∇µ f This allows us to write the Young-Laplace equation directly as L(b) − L(e) + ∇µ ∂nµ = 0. fixed one can check that we get θ(f )L(b) + θ(−f )L(e) + δ˜(f )L(s) G +θ(f )ϑnµ ∂µ L(b) + θ(−f )ϑnµ ∂µ L(e) + δ˜(f )ϑnµ ∂µ L(s) − δ˜(f ) ∂L(s) δf nµ . ∂nµ We have used the fact that L(b), L(e) and L(s) are scalars and transform accordingly. Note however that L(s) also contains f which we are supposed to keep constant. To balance this we subtract the last term in (B.9). We can simplify this expression as ∂nµ which leads to the same Young-Laplace equation (B.8). Metric & cov. derivative Gauge field & strength Temperature & chemical pot. Superfluid phase & velocity Superfluid potential Shape-field & normal vector Surface metric & derivative Surface fluid velocity Surface superfluid velocity Even surface scalars Distribution fn. & derivatives Odd surface velocity Odd surface scalar (2 + 1) Energy-momentum tensor Derivative corrections Energy-mom. conservation Local second law Young-Laplace equation f equation of motion Background Quantities Superfluid Quantities J(b)ent (bulk), J(s)ent (sur.), J(e)ent (ext.), Jeμnt (full) μ μ μ Equations of Motion ∇μT(μbν/e) = F νρJρ(b/e), ∇˜ μT(μsν) = F νρJρ(s) + nμ(T(μbν) − T(μeν) ) ∇μJ(μs)ent − nμ(J(μb)ent − J(e)ent) ≥ 0 μ nν ∇˜μT(μsν) = nν F νρJρ(s) + nμnν (T(μbν) − T(μeν) ) Useful notations and formulae In this appendix, we recollect some useful notations and formulae used throughout this Relativistic superfluids. We have given a list of useful definitions and relations for relativistic superfluids in tables 3 and 4. In equilibrium, the metric and gauge field can be Constitutive Relations T(μbν) = (E + P )uμuν + P Gμν + F ξμξν + Π(μbν), T(μsν) = (E−C)uμuν − CG˜μν + F ξ˜μξ˜ν + 2U u(μn¯ν) + Π(μsν), μ μ J(s)ent = Suμ + T1 (U −µ V)n¯μ + Υ(s)ent Pressure and surface tension First order even bulk coeff. First order odd bulk coeff. Surface inflow densities (3+1) Bulk first law Surface first law (3+1) Surface first law (2+1) Y = −C (surface) E (energy), Q (charge), S (entropy), F (superfluid den.) E (energy), Q (charge), S (entropy), F (superfluid den.) g1, g2 (3+1), U = g1, V = g2 (3+1), E + P = T S + µQ (bulk), E − Y = T S + µ Q (surface) d(E−λf1) = T d(S− Tλ (f1−µf 2)) + µ d(Q−λf2) − 21 F dχ˜ d(E − λf1 + λ¯κ4) = T d(S − Tλ (f1 − µf 2) + Tλ¯ (κ3 − µκ 4)) dimensionally reduced in a Kaluza-Klein framework as Aµ = −a Let W be a partition function for relativistic superfluids with a surface, written as a shape-field f . Varying it, we can read out the components of the energy-momentum tensor and charge current as, T i0 = J i = √g Here T0 is the inverse radius of the Euclidean time circle. Null/Galilean superfluids. In a similar spirit (see [10] for details), let W be a partition various components of the energy-momentum tensor and charge current respectively as T i− = − √g T0− = − √g T i0 = J i = √g , T ij = √g Note that the components T00 and J0 is not determined by the partition function. In fact, these two components are “unphysical” as they do not enter the respective conservation laws. The formulae (C.4) can also be recasted directly into a null reduced Galilean language ǫ = − √g , ǫi = qi = √g √g tij = √g Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. holes, JHEP 01 (2009) 055 [arXiv:0809.2488] [INSPIRE]. 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Jay Armas, Jyotirmoy Bhattacharya, Akash Jain, Nilay Kundu. On the surface of superfluids, Journal of High Energy Physics, 2017, 1-52, DOI: 10.1007/JHEP06(2017)090