Generalized global symmetries and dissipative magnetohydrodynamics

Generalized global symmetries and dissipative magnetohydrodynamics

Sašo Grozdanov grozdanov@lorentz.leidenuniv.nl Instituut-Lorentz for Theoretical Physics, Leiden University,
Niels Bohrweg 2, Leiden 2333 CA, The Netherlands
   Diego M. Hofman d.m.hofman@uva.nl    Nabil Iqbal n.iqbal@uva.nl Institute for Theoretical Physics, University of Amsterdam, Science Park 904, Postbus 94485, 1090 GL Amsterdam, The Netherlands
Abstract

The conserved magnetic flux of electrodynamics coupled to matter in four dimensions is associated with a generalized global symmetry. We study the realization of such a symmetry at finite temperature and develop the hydrodynamic theory describing fluctuations of a conserved 2-form current around thermal equilibrium. This can be thought of as a systematic derivation of relativistic magnetohydrodynamics, constrained only by symmetries and effective field theory. We construct the entropy current and show that at first order in derivatives, there are seven dissipative transport coefficients. We present a universal definition of resistivity in a theory of dynamical electromagnetism and derive a direct Kubo formula for the resistivity in terms of correlation functions of the electric field operator. We also study fluctuations and collective modes, deriving novel expressions for the dissipative widths of magnetosonic and Alfvén modes. Finally, we demonstrate that a non-trivial truncation of the theory can be performed at low temperatures compared to the magnetic field: this theory has an emergent Lorentz invariance along magnetic field lines, and hydrodynamic fluctuations are now parametrized by a fluid tensor rather than a fluid velocity. Throughout, no assumption is made of weak electromagnetic coupling. Thus, our theory may have phenomenological relevance for dense electromagnetic plasmas.

I Introduction

Hydrodynamics is the effective theory describing the long-distance fluctuations of conserved charges around a state of thermal equilibrium. Despite its universal utility in everyday physics and its pedigreed history, its theoretical development continues to be an active area of research even today. In particular, the new laboratory provided by gauge/gravity duality has stimulated developments in hydrodynamics alone, including an understanding of universal effects in anomalous hydrodynamics Son:2009tf (); Neiman:2010zi (), potentially fundamental bounds on dissipation Kovtun:2004de (); Son:2007vk (), a refined understanding of higher-order transport Baier:2007ix (); Bhattacharyya:2008jc (); Romatschke:2009kr (); Bhattacharyya:2012nq (); Jensen:2012jh (); Grozdanov:2015kqa (); Haehl:2014zda (), and path-integral (action principle) formulations of dissipative hydrodynamics Dubovsky:2011sj (); Endlich:2012vt (); Grozdanov:2013dba (); Kovtun:2014hpa (); Harder:2015nxa (); Crossley:2015evo (); Haehl:2015foa (); Haehl:2015uoc (); Torrieri:2016dko (); see e.g. Son:2007vk (); Rangamani:2009xk (); Kovtun:2012rj () for reviews of hydrodynamics from the point of view afforded by holography.

It is well-understood that the structure of a hydrodynamic theory is completely determined by the conserved currents and the realization of such symmetries in the thermal equilibrium state of the system. In this paper we would like to apply such a symmetry-based approach to the study of magnetohydrodynamics, i.e. the long-distance limit of Maxwell electromagnetism coupled to light charged matter at finite temperature and magnetic field.

To that end, we first ask a question with a seemingly obvious answer: what are the symmetries of electrodynamics coupled to charged matter? One might be tempted to say that there is a current associated with electric charge. There is indeed such a divergenceless object, related to the electric field-strength by Maxwell’s equations:

(1)

However, the symmetry associated with this current is a gauge symmetry. Gauge symmetries are merely redundancies of the description, and thus are presumably not useful for organizing universal physics.

The true global symmetry of electrodynamics is actually something different. Consider the following antisymmetric tensor

(2)

It is immediately clear from the Bianchi identity (i.e. the absence of magnetic monopoles) that . This is not related to the conservation of electric charge, but rather states that magnetic field lines cannot end.

What is the symmetry principle behind such a conservation law? It has recently been stressed in Gaiotto:2014kfa () that just as a normal -form current is associated with a global symmetry, higher-form symmetries such as are associated with generalized global symmetries, and should be treated on precisely the same footing. We first review the physics of a conventional global symmetry, which we call a -form symmetry in the notation of Gaiotto:2014kfa (): with every -form symmetry comes a divergenceless -form current , whose Hodge dual we integrate over a codimension- manifold to obtain a conserved charge. If this codimension- manifold is taken to be a time slice, then the conserved charge can be conveniently thought of as counting a conserved particle number: intuitively, since particle world-lines cannot end in time, we can “catch” all the particles by integrating over a time slice. The objects that are charged under -form symmetries are local operators which create and destroy particles, and the symmetry acts (in the case) by multiplication of the operator by a -form phase that is weighted by the charge of the operator : .

Consider now the less familiar but directly analogous case of a -form symmetry. A -form symmetry comes with a divergenceless -form current , whose Hodge star we integrate over a codimension- surface to obtain a conserved charge . This conserved charge should be thought of as counting a string number: as strings do not end in space or in time, an integral over a codimension- surface is enough to “catch” all the strings111Note that the dynamics of string-like degrees of freedom has been discussed in the context of superfluid hydrodynamics in the interesting recent paper Horn:2015zna (). In that case strings arise as solitons and, unlike in our work, interact through long range forces., as shown in Figure 1. The objects that are charged under -form symmetries are -dimensional objects such as Wilson or ’t Hooft lines. These 1d objects create and destroy strings, and the symmetry acts (in the -form case) by multiplication by a -form phase integrated along the contour of the ’t Hooft line: .

Figure 1: Integration over a codimension- surface counts the number of strings that cross it at a given time.

In the case of electromagnetism, the -form current is given by (2), and the strings that are being counted are magnetic field lines. We could also consider the dual current itself, which would count electric flux lines: however from (1) we see that is not conserved in the presence of light electrically charged matter, because electric field lines can now end on charges. Thus, electrodynamics coupled to charged matter has only a single conserved -form current. This is the universal feature that distinguishes theories of electromagnetism from other theories, and the manner in which the symmetry is realized should be the starting point for further discussion of the phases of electrodynamics.222In electrodynamics in dimensions this point of view is somewhat more familiar, as the analog of is a conventional -form “topological” current . For example, this symmetry is spontaneously broken in the usual Coulomb phase (where the gapless photon is the associated Goldstone boson), and is unbroken in the superconducting phase (where magnetic flux tubes are gapped). We refer the reader to Gaiotto:2014kfa () for a detailed discussion of these issues.

In this paper, we discuss the long-distance physics of this conserved current near thermal equilibrium, applying the conventional machinery of hydrodynamics to a theory with a conserved -form current and conserved energy-momentum. We are thus constructing a generalization of the (very well-studied) theory that is usually called relativistic magnetohydrodynamics. To the best of our knowledge, most discussions of MHD separate the matter sector from the electrodynamic sector. It seems to us that this separation makes sense only at weak coupling, and may often not be justified: for example, the plasma coupling constant , defined as the ratio of potential to kinetic energies for a typical particle, is known to attain values up to in various astrophysical and laboratory plasmas RevModPhys.54.1017 (). Experimental estimates of the ratio of shear viscosity to entropy density (where a small value is widely understood as being a universal measure of interaction strength Kovtun:2004de ()) in such plasmas at high obtain minimum values that are PhysRevLett.111.125004 (). These suggest the presence of strong electromagnetic correlations.

Our discussion will not make any assumptions of weak coupling and should therefore be valid for any value of : we will be guided purely by symmetries and the principles of the effective field theory of hydrodynamics. Beyond the (global) symmetries, the construction of the hydrodynamic gradient expansions also requires us to choose relevant hydrodynamic fields (degrees of freedom), which, as we will discuss, crucially depend on the symmetry breaking pattern in the physical system at hand. In particular, in addition to conventional hydrodynamics at finite temperature, we will also study a variant of magnetohydrodynamics at at very low temperatures. This theory has an emergent Lorentz invariance associated with boosts along the background magnetic field lines, and the parametrization of hydrodynamic fluctuations is considerably different. Interestingly, at leading-order corrections to ideal hydrodynamics only enter at second order, thus showing the direct relevance of higher-order hydrodynamics (see e.g. Baier:2007ix (); Bhattacharyya:2012nq (); Romatschke:2009kr (); Grozdanov:2015kqa ()). While this treatment does not include the typical light modes that emerge at , it does capture a universal self-contained sector of magnetohydrodynamics.

We now describe an outline of the rest of this paper. In Section II we discuss the construction of ideal hydrodynamic theory at finite temperature. In Section III we move beyond ideal hydrodynamics: we work to first order in derivatives, and demonstrate that there are seven transport coefficients that are consistent with entropy production, describing also how they may be computed through Kubo formulas. In Section IV we study linear fluctuations around the equilibrium solution and derive the dispersion relations and dissipative widths of gapless magnetohydrodynamic collective modes. In Section V we study the simple extension of the theory associated with adding an extra conserved -form current (e.g. baryon number). In Section VI we turn to the theory at strictly zero temperature, where we discuss novel phenomena that can be understood as arising from a hydrodynamic equilibrium state with extra unbroken symmetries. We conclude with a brief discussion and possible future applications in Section VII.

While this work was being written up, we came to learn of the interesting paper Schubring:2014iwa (), which also studies a dissipative theory of strings and makes the connection to MHD. Though the details of some derivations differ, there is overlap between that work and our Sections II and III.

Note added: In the original version of this work on the arXiv there was an inaccurate count of transport coefficients; we thank the authors of Hernandez:2017mch () for bringing this issue to our attention.

Ii Ideal magnetohydrodynamics

Our hydrodynamic theory will describe the dynamics of the slowly evolving conserved charges, which in our case are the stress-energy tensor and the antisymmetric current .

ii.1 Coupling external sources

For what follows, it will be very useful to couple the system to external sources. The external source for the stress-energy tensor is a background metric , and we also couple the antisymmetric current to an external -form gauge field source by deforming the microscopic on-shell action by a source term:

(3)

The currents are defined in terms of the total action as

(4)
(5)

Demanding invariance of this action under the gauge symmetry with a -form gauge-parameter results in

(6)

Similarly, demanding invariance under an infinitesimal diffeomorphism that acts on the sources as a Lie derivative , , gives us the (non)-conservation of the stress-energy tensor in the presence of a source:

(7)

where . The term on the right-hand side of the equation states that an external source can perform work on the system.

We now discuss the physical significance of the -field source. A term should be thought of as a chemical potential for the charge , i.e. a string oriented in the -th spatial direction.

For our purposes we can obtain some intuition by considering the theory of electrodynamics coupled to such an external source, i.e. consider using (2) to write the current as

(8)

with the familiar gauge potential from electrodynamics.333We choose conventions whereby . Then the coupling (3) becomes after an integration by parts:

(9)

The field strength associated to can be interpreted as an external background electric charge density to which the system responds.

For example, consider a cylindrical region of space that has a nonzero value for the chemical potential in the direction:

(10)

where is if and is otherwise. Then from (9) we see that we have

(11)

i.e. we have an effective electric current running in a delta-function sheet in the direction along the outside of the cylinder. Thus the chemical potential for producing a magnetic field line poking through a system is an electrical current running around the edge of the system, as one would expect from textbook electrodynamics. In our formalism the actual magnetic field created by this chemical potential is controlled by a thermodynamic function, the susceptibility for the conserved charge density .

We will sometimes return to the interpretation of as charge source to build intuition: however we stress that in general when there are light electrically charged degrees of freedom present the defined in (8) does not have a local effective action and is not a useful quantity to consider.

ii.2 Hydrodynamic stress-energy tensor and current

We now turn to ideal hydrodynamics at non-zero temperature. We first discuss the equilibrium state. Recall that the analog of a conserved charge for our 2-form current is its integral over a codimension- spacelike surface with no boundaries, as shown in Figure 1.

(12)

counts the number of field lines crossing at any instant of time and is thus unaltered by deformations of in both space and time. A thermal equilibrium density matrix is then given (for a particular choice of ) by

(13)

where is the chemical potential associated with the -form charge. This density matrix can be generated by a Euclidean path integral with an appropriate component of turned on, e.g. the is the plane then we would use .

Elementary arguments, which we spell out in detail in Appendix A, then give us the form of the stress-energy tensor and the conserved higher rank current in thermal equilibrium444Equilibrium thermodynamics in the presence of magnetic fields has also recently been studied in Kovtun:2016lfw (); that work differs from ours in that the magnetic fields there are fixed external sources for a conventional 1-form current, whereas in our case the magnetic fields are themselves the fluctuating degrees of freedom of a 2-form current.:

(14)

satisfying the conservation equations in the ideal limit

(15)

We have labeled this expression with a subscript , as this will be only the zeroth order term in an expansion in derivatives. Here is the fluid velocity as in conventional hydrodynamics. is the direction along the field lines, and we impose the following constraints:

(16)

It will also often be useful to use the projector onto the two dimensional subspace orthogonal to both and :

(17)

with trace . In (14), is the conserved flux density, and is the pressure. There is no mixed term, as this can be removed with no loss of generality by a Lorentz boost in the plane.555We note that the form of the stress-energy tensor (14), including constraints (16), is precisely that of anisotropic ideal hydrodynamics with different longitudinal and transverse pressures (with respect to some vector) Ryblewski:2008fx (); Florkowski:2008ag (). In that case, measures the difference between the two pressures. The role of this additional vector is now played by .

Note the presence of the term in the stress-energy tensor, representing the tension in the field lines. Its coefficient in equilibrium is . It is a bit curious from the effective field theory perspective that this coefficient is fixed and is not given by an equation of state, like , for example. There is a quick thermodynamic argument to explain this fact. Consider the variation of the internal energy for a system containing field lines running perpendicularly to a cross section of area , with an associated tension and a conserved charge given by the flux through the section:

(18)

where is the length of the system perpendicular to . Because is a charge defined by an area integral, it is given by and the factor of in front of is the correct scaling with the height of the system. Now perform a Legendre transform to the Landau grand potential:

(19)
(20)

where is the entropy density. Notice that is the quantity naturally calculated by the on-shell action and we expect it to scale with volume in local Quantum Field Theory. This scaling is spoiled by the term proportional to unless . This condition is, therefore, enforced by extensivity.

The thermodynamics is, thus, completely specified by a single equation of state, i.e. by the pressure as a function of temperature and chemical potential . The relevant thermodynamic relations are

(21)

with the entropy density. Here we have made use of the volume scaling assumption.

The microscopic symmetry properties of do not actually determine those of and , only that of their product. In this work we assume the charge assignments in Table 1, which are consistent with magnetohydrodynamical intuition and are particularly convenient. Note that that all scalar quantities (such as and ) are taken to have even parity under all discrete symmetries, and charge conjugation is taken to flip the sign of . These symmetries will play a useful role later on in restricting corrections to the entropy current.

Table 1: Charges under discrete symmetries of 2-form current and hydrodynamical degrees of freedom.

Hydrodynamics is a theory that describes systems that are in local thermal equilibrium but can globally be far from equilibrium, in which case the thermodynamic degrees of freedom become space-time dependent hydrodynamic fields. Thus the degrees of freedom are the two vectors and two thermodynamic scalars which can be taken to be and , leading to seven degrees of freedom. The equations of motion are the conservation equations (7) and (6). As is antisymmetric, one of the equations for the conservation of does not include a time derivative and is a constraint on initial data. This constraint is consistently propagated by the remaining equations of motion, thus leaving effectively six equations for six variables, and the system is closed.

We now demonstrate that the equations of motion of ideal hydrodynamics result in a conserved entropy current. Consider dotting the velocity into the conservation equation for the stress-energy tensor (7). Using the thermodynamic identities (21) we find

(22)

We now project the conservation equation for along :

(23)

Inserting this into (22) and using to rearrange derivatives we find

(24)

We thus see that the local entropy current is conserved, as we expect in ideal hydrodynamics.

We now turn to the interpretation of the other components of the hydrodynamic equations. The projections of (7) along and , respectively, are

(25)
(26)

These are the components of the Euler equation for fluid motion in the direction parallel and perpendicular to the background field.

Similarly, the evolution of the magnetic field is given by the projection of the conservation equation for along in (23) and along below:

(27)

The equation states that the transverse part of the magnetic field is Lie dragged by the fluid velocity.

This is the most general system that has the symmetries of Maxwell electrodynamics coupled to charged matter. In particular, unlike conventional treatments of MHD, we have made no assumption that the gauge coupling is weak. Indeed it appears nowhere in our equations: in theories with light charged matter, the fact that runs means that it does not have a universal significance and will not appear as a fundamental object in hydrodynamic equations.

To make contact with the traditional treatments of MHD, consider expanding the pressure in powers of , e.g.

(28)

Here should be thought of as the pressure of the matter sector alone. The expansion is given in powers of , as the sign of is not physical666In this theory, the sign of the magnetic field is carried by the direction of the vector.. If we stop at this order and then further assume that the coefficient of the term is independent of temperature , then the theory of ideal hydrodynamics arising from this particular equation of state is entirely equivalent to traditional relativistic MHD with gauge coupling given by . From our point of view, this is then a weak-magnetic-field limit of our more general theory. Note that this weak-field limit is entirely different from the hydrodynamic limit that we are taking throughout this paper, and there is an entirely consistent effective theory even if we do not take the weak-field limit. We discuss some physical consequences of keeping higher order terms in this expansion (which will be generically present in any interacting theory, even if their coefficient may be small under particular circumstances) later on in this paper.

Nevertheless, if we truncate the expansion for the pressure as in (28) then we find from (21): and with . The ideal hydrodynamic theory of our -form current is now entirely equivalent to conventional treatments of ideal MHD, as presented in e.g. PhysRevD.18.1809 (). As , the -independence of and thus of the -dependent piece of the pressure essentially means that the magnetic field degrees of freedom carry no entropy.

Iii First-order hydrodynamics

Hydrodynamics is an effective theory, and thus (14) are only the zeroth order terms in a derivative expansion. We now move on to first order in derivatives: to be more precise, the full stress-energy tensor is given by

(29)
(30)

where the zeroth order term is given by the ideal MHD expressions in (14), and our task now is to determine the first-order corrections as a function of the fluid variables such as the velocity and magnetic field. The numbers that parametrize these corrections are the transport coefficients such as viscosity and resistivity. The physics of dissipation and entropy increase enter at first order in the derivative expansion: as usual in hydrodynamics, the possible tensor structures that can appear (and thus the number of independent transport coefficients) are greatly constrained by the requirement that entropy always increases.

iii.1 Transport coefficients

We follow the standard procedure to determine these corrections landau1987fluid (). We begin by writing down the most general form for the first-order terms:

(31)

Here , and are transverse vectors (i.e. orthogonal to both and ), is a transverse, traceless and symmetric tensor, and is a transverse, antisymmetric tensor.

Next, we exploit the possibility to change the hydrodynamical frame. In hydrodynamics, there is no intrinsic microscopic definition of the fluid variables . Each field can therefore be infinitesimally redefined, as e.g. . The microscopic currents and the stress-energy tensor must remain invariant under this operation, and thus the redefinition alters the functional form of the relationship between the currents and the fluid variables. In conventional hydrodynamics of a charged fluid this freedom is often used to set (Landau frame) or (Eckart frame). We will use the scalar redefinitions of and to set and the vector redefinitions of and to set . We now have the simpler expansion:

(32)
(33)

Our task now is to determine the form of the reduced set in terms of derivatives of the fluid variables.

To proceed, we require an expression for the non-equilibrium entropy current . The textbook approach to this problem is to postulate a standard “canonical” form for this entropy current, motivated by promoting the thermodynamic relation to the following covariant expression:

(34)

Up to first order in derivatives, this is equivalent to

(35)

We will take this to be our entropy current. As in conventional hydrodynamics Bhattacharya:2011eea (), one can show that it is invariant under frame redefinitions of the sort described above.

Next, we directly evaluate the divergence . Using the contraction of the conservation equations (7) and (6) with , we find after some straightforward algebra:

(36)

We see that entropy is no longer conserved, as one expects for a dissipative theory. The second law of thermodynamics in its local form states that entropy should always increase. Thus the right-hand side of Eq. (III.1) should be a positive definite quadratic form for all conceivable fluid flows. For the vector and tensor dissipative terms, positivity implies that the right-hand side is simply a sum of squares, requiring that the dissipative corrections take the following form:

(37)
(38)
(39)
(40)

where the four transport coefficients and must all be positive.

In the bulk channel parametrized by and mixing is possible. The most general allowed form that is consistent with positivity is parametrized by three transport coefficients :

(41)
(42)

Note that this mixing matrix is symmetric, in that the mixing term is the same for and . This follows from an Onsager relation on mixed correlation functions, as we explain in Section III.2 below.777In the first version of this paper on the arXiv, the possibility of a nonzero was not taken into account, leading to an incorrect count of transport coefficients. This inaccuracy was pointed out to us by the authors of Hernandez:2017mch (), and we thank them for bringing this to our attention.

Further demanding that the right-hand side of (III.1) be a positive-definite quadratic form imposes two constraints on the bulk viscosities, which may be written as

(43)

There are no further constraints that we know of. At first order we thus have seven transport coefficients , and . If we were to allow all coefficients permitted by symmetries, we would instead have concluded that there were eleven independent transport coefficients consistent with the parity assignments under , illustrating the constraints enforced by the second law of thermodynamics.

We now turn to the interpretation of these transport coefficients. It is clear that and are anisotropic bulk and shear viscosities respectively: for a charged fluid in a fixed external magnetic field one finds instead seven independent viscosities Huang:2011dc (), where the difference in counting arises from the fact that we have imposed a charge conjugation symmetry .

The transport coefficients can be interpreted as the conventional electrical resistivity parallel and perpendicular to the magnetic field. To understand this, first note that the familiar electric field is defined in terms of the electromagnetic field strength as . Using (2) we find

(44)

where the ellipsis indicates further higher-order corrections. Note that a nonzero electric field enters only at first order in hydro: an electric field is not a low-energy object, as the medium is attempting to screen it.

Next, we note that a resistivity is conventionally defined as the electric 1-form current response to an applied external electric field. However, our formalism instead naturally studies the converse object, i.e. the 2-form current response in a field theory with a total action deformed by a fixed external -field source (which can be interpreted as an external electric current via (9)). Thus, we need to perform a Legendre transform to find the analog of the quantum effective action , which is a function of a specified 2-form current :

(45)

Here, is defined to be the on-shell action in the presence of the -field source, and is implicitly determined by the condition that , i.e. that the stationary points of the action coincide with the specified value for . We now write in terms of a vector potential using (8) and define the electrical 1-form current response via

(46)

Note the sign difference with respect to the external fixed source defined in (9). This arises from the Legendre transform and is the difference between having a fixed external source and a current response.

We now need to determine the relationship between the electric field (44) and the response 1-form current (46). Consider a static and homogenous fluid flow with

(47)

in the presence of a homogenous but time-dependent field source , . From (46), in the fixed ensemble, this -field can be interpreted as an electrical current response Now inserting the expansion (39) and (40) into (44) and neglecting the fluid gradient terms, we find that the electric field created by this current source is

(48)

Thus, are indeed anisotropic resistivities as claimed.

Finally, we discuss a technical point: our starting point for the discussion of dissipation was the canonical form for the non-equilibrium entropy current (35). It is now well-understood that this form for the entropy current is not unique: for example, in the hydrodynamics of fluids with anomalous global symmetries (and thus with parity violation), the second law requires that extra terms must be added to the entropy current, resulting eventually in extra transport coefficients corresponding to the chiral magnetic and vortical effects Son:2009tf (); Neiman:2010zi (). It was however shown in Bhattacharya:2011tra () that for a parity-preserving fluid with a conserved -form current all ambiguities in the entropy current can be fixed by demanding that entropy production on an arbitrary curved background be positive. We have performed a similar analysis for the -form current. Here, charge-conjugation invariance acts as , and this symmetry together with positivity of entropy production on curved backgrounds is sufficient to show that the form of the entropy current exhibited in (35) is unique.

iii.2 Kubo formulae

We now derive Kubo formulae—i.e. expressions in terms of real-time correlation functions—for these transport coefficients. We follow an approach described in Son:2007vk () which we briefly review below.

It is a standard result in linear response theory that in the presence of a perturbation out of equilibrium by an infinitesimal source, the response is given of the system is given by the retarded correlator of the operator coupled to the source. For example, if we turn on a small -field source, we find:

(49)

where is the retarded correlator of .

However, above we saw that in the presence of an infinitesimal perturbation around a static flow (47) by a time-varying but spatially homogenous -field source , , the response within the hydrodynamic theory was

(50)

Equating these two relations we find the following Kubo formulas for the parallel and perpendicular resistivities:

(51)

We will return to the physical interpretation of this formula shortly. First, we derive Kubo formulas for the viscosities. To do this, we consider perturbing the spatial part of the background metric slightly away from flat space:

(52)

where . The response of the stress-energy tensor to such a perturbation is given in linear response theory by

(53)

The hydrodynamic response to such a source is given by (37) to (42) where the full contribution comes from the Christoffel symbol

(54)

Matching the response in each tensor channel just as above we find the following set of Kubo relations:

(55)
(56)
(57)

These are a straightforward anisotropic generalization of the usual formulas for the bulk and shear viscosity. Our normalization for the anisotropic bulk viscosity has been chosen so that no dimension-dependent factors enter into the Kubo formula; however this is not the standard normalization. Note that we present two equivalent formulas for the mixed bulk viscosity ; the equality of these two correlation functions is guaranteed by the Onsager relations for off-diagonal correlation functions. Indeed, it is this Onsager relation that sets to zero a possible antisymmetric transport coefficient in (41)–(42).888The Kubo formulae (51) and (55)–(57) agree with those presented in Hernandez:2017mch (). We thank Pavel Kovtun for discussions regarding these matters.

We now turn to a discussion of the resistivity formula (51). Unlike the hydrodynamics of a conventional -form current where we generally obtain a Kubo formula for the conductivity, here we find a Kubo formula directly for its inverse, the resistivity, in terms of correlators of the components of the antisymmetric tensor current corresponding to the electric field. The resistivity is the natural object here: in a theory of dynamical electromagnetism, we examine how an electric field responds to an external current flow, not the other way around.

To the best of our knowledge, such a Kubo formula for the resistivity in terms of electric field correlations is novel. Traditionally, in order to compute a resistivity one instead computes the conductivity of the -form global current that is being gauged, and then takes the inverse of the resulting number “by hand”. This procedure—which essentially treats a gauge symmetry as a global one—is probably only physically reasonable at weak gauge coupling. On the other hand, the Kubo formula above permits a precise universal definition for the resistivity in a dynamical gauge theory, independently of the strength of the gauge coupling. It is interesting to study its implications.

For example, we might see whether it agrees with the traditional prescription. Consider a weakly coupled gauge theory with action

(58)

where is a -form current that is built out of other matter fields (schematically denoted by ), that has been weakly gauged. The considerations here do not involve the background magnetic field and so we turn it to zero. Within this theory we may compute the finite-temperature correlator of the electric field to compute the resistivity through (51).

Figure 2: Sum over current-current insertions to compute electrical resistivity.

One first attempt to do so might involve summing the series of diagrams shown in Figure 2. The geometric sum leads to an answer of the schematic form

(59)

where is the free photon propagator for spatial polarizations and is the correlation function of the electrical current. The photon propagator at zero spatial momentum has a pole at : at low frequencies we now zoom in on this pole to find for the resistivity :

(60)

where we have used the standard Kubo formula for the -form global conductivity . Thus, within this approximation scheme, it is indeed true that the resistivity (defined via our Kubo formula) is equal to the inverse of the conductivity of the current that is being gauged.999Here we have been somewhat cavalier with details. To make these considerations precise, one should imagine performing the sum over bubbles in Euclidean signature, then analytically continuing to the retarded propagator at frequency via before taking the small frequency limit. We have assumed here that no subtleties arise in this continuation.

Figure 3: Example of new diagram that contributes to electrical resistivity.

Note, however, that this class of diagrams is not the only set of diagrams that one should include. One might also imagine diagrams of the form Figure 3: computationally they arise from the fact that the photon is now dynamical, and thus the classification of diagrams as “one-particle-irreducible” has changed. Such diagrams will contribute to (51): as they simply do not exist in the theory of the global -form current , they will necessarily modify the conclusion above, changing away from . We have not attempted a systematic study of such diagrams, but it would be very interesting to understand their effect. It seems likely that they can be suppressed at weak gauge coupling, justifying the approximation scheme above, but it is an important open issue to demonstrate precisely when this is possible.

Iv Application: Dissipative Alfvén and magnetosonic waves

In this section, we study the collective modes of the relativistic MHD theory constructed above. We will linearly perturb the background solution and determine the dispersion relations of the resulting modes. We organize the fluctuations in the following way: without loss of generality, we fix the direction of the background magnetic field by setting the field to point in the -direction, (note that its size is fixed by the normalization of ). Furthermore, we can use a residual symmetry to fix the 4-momentum as

(61)

so that measures the angle between the direction of the background magnetic field and momentum of the hydrodynamic waves. The background velocity field is fixed to at rest and the background temperature and chemical potential are kept general and space-time independent. We then linearly perturb , , and as

(62)
(63)
(64)
(65)

Note that linearized constraints (16) impose that

(66)

For a background source without curvature, i.e. , the fluctuations can be organized into two classes:

  • Transverse Alfvén waves with

    (67)
    (68)
    (69)

    Note that the fluid displacement is perpendicular to the background magnetic field; thus, they can be thought of as the usual vibrational modes that travel down a string with tension. These modes were first discovered in the magnetohydrodynamic context by Alfvén in 1942Natur.150..405A (). For an introductory treatment, see e.g. bellan2008fundamentals ().

  • Magnetosonic waves with and contained in the space spanned by . These are more closely related to the usual sound mode in a finite temperature plasma. We will see that there are two branches of this kind: “fast” and “slow”.

We first study Alfvén waves. Solving the conservation equations (6) and (7), we find the dispersion relation for Alfvén waves to to be

(70)

where the parameter that enters the Alfvén phase velocity is

(71)

The expression for the speed of the wave is standard. Recall that is the tension in the field lines; in the nonrelativistic limit is dominated by the rest mass, and this becomes the textbook formula for the speed of wave propagation down a string. We are not however aware of much previous discussion of dissipative corrections to Alfv́en waves; Jedamzik:1996wp () studied a dissipative fluid perturbatively coupled to electrodynamics, and our expression reduces to their angle-independent result if we assume an isotropic shear viscosity and no resistivity.

When the magnetic field is perpendicular to the direction of momentum, i.e. , the Alfvén wave ceases to propagate and becomes entirely diffusive, as is usually the case for transverse excitations in standard hydrodynamics. Note that the width of the mode depends on the momentum perpendicular to the strings; elementary treatments of MHD often assume that the Alfvén wave has no dependence on the perpendicular momentum at all, which is sometimes taken as license to make it arbitrarily high, allowing Alfvén waves that are arbitrarily well-localized in the plane perpendicular to the field (see e.g. bellan2008fundamentals ()). Here, we see that this is an artefact of the ideal hydrodynamic limit.

Turning now to the magnetosonic waves, a straightforward but somewhat tedious calculation shows that the dispersion relations for the two magnetosonic waves are given by

(72)

where

(73)

Note that “fast” magnetosonic waves have a sign before the square-root in Eq. (73) and “slow” magnetosonic waves have a sign. Above, we have defined the following quantities:

(74)
(75)
(76)

and the susceptibilities:

(77)

It is easy to see that the formulae above predict generically the existence of a two fully dissipative modes at , namely the “slow” magnetosonic mode and the Alfvén mode. We can interpret as the speed of the “fast” magetosonic mode at , a kind of speed of sound for the system. At , on the other hand, one magnetosonic mode has the same speed as the Alfvén mode while the other one has velocity . We plot these velocities as a function of the angle for some interesting examples below.

The dissipative parts of these modes can be calculated in a straightforward manner by going to one higher order in derivatives using the formalism above. Unfortunately, explicit expressions are rather cumbersome to write in print. We quote below only the values for at and , where we indicate which mode the width applies to by specifying the value of the phase velocity at that angle101010Note that depending on the equation of state and the specific values of and (which determine the relative numerical magnitudes of and ) it can be either the fast or the slow magnetosonic mode that has phase velocity coinciding with the Alfvén wave at , as can be seen explicitly in Figures 3(a) and 3(b).:

(78)
(79)
(80)
(81)

While the coefficient enters into the dispersion relations of magnetosonic waves, its coefficient is proportional to , which implies that the magnetosonic dispersion relations have neither any dependence on the bulk viscosity at nor at . Notice that the dissipative part (78) coincides exactly with the limit of (IV). This is expected, as in this limit there is an enhanced rotational symmetry around the shared axis of background magnetic field and momentum, relating the modes in question. As a result of this coincidence the results presented allow the measurement of only 5 of the 7 dissipative coefficients. As it turns out, if we allow measurements at arbitrary angles, then can be determined, but the value of can’t be measured from the study of dissipation of linear modes alone. By introducing sources, one can of course use the Kubo formulae previously discussed to determine all transport coefficients.

iv.1 Magnetohydrodynamics at weak field

In order to recover the familiar results from standard magnetohydrodynamics, we can take the small chemical potential limit, which corresponds to weak magnetic fields. This is the regime in which the standard treatment is valid.

In the weak field limit, we can expand the equation of state as (cf. (28))

(82)

where and are temperature-dependent functions that control the leading order behavior. In this limit, to leading order,

(83)
(84)
(85)

This agrees with the standard treatment (for a relativistic discussion, see e.g. Jedamzik:1996wp ()). Notice that the slow magnetosonic mode and the Alfvén wave are indistinguishable to this order. If we want to separate them we need to go to higher order in the expansion. One nice example when one can do this and obtain concrete expressions is in the case where is much larger than any other scale in the problem (while still being much smaller than ). In this case, we have no other scale and the expansion of the equation of state to the necessary order is:

(86)

where , and are dimensionless constants. We find the leading effects on the velocities of modes to be

(87)
(88)
(89)
(90)

In each of the expressions, we have kept only the first non-trivial term to illustrate the angular dependence. The factor of in the leading order expression for is characteristic of the sound mode of conformal fluids in 4 dimensions. The fact that sound is the fastest mode is in agreement with our expectations at high temperatures where propagation is by nature diffusive. Note that both the Alfvén and the slow magnetosonic wave speeds start at , which is the small expansion parameter in this limit. Thus, they propagate very slowly indeed. We present some illustrative plots of these dispersion relations in Figure 3(a).

iv.2 Magnetohydrodynamics at strong field

The situation is quite different for a fluid in which magnetic fields are strong. Here, our formalism can make concrete predictions away from the weak coupling limit. For concreteness let us assume, similarly as in the previous discussion, that is much larger than any other scale in the problem, while still much smaller than . In that case we can write the equation of state in a small temperature expansion (strong magnetic field) as:

(91)

where , and are dimensionless constants. The expansion above is shown to the second subleading order to highlight that this expansion is, despite similarities, indeed different from (86). The fact that the leading order terms agree (in form, but not numerical coefficients) between the two expansions is a coincidence due to our working in dimensions.

From the above equation of state, we can calculate the mode velocities to first non-trivial order in temperature corrections: