Flowing holographic anyonic superfluid

We investigate the flow of a strongly coupled anyonic superfluid based on the holographic D3-D7' probe brane model. By analyzing the spectrum of fluctuations, we find the critical superfluid velocity, as a function of the temperature, at which the flow stops being dissipationless when flowing past a barrier. We find that at a larger velocity the flow becomes unstable even in the absence of a barrier.

speed of sound v s , the fluid can not exchange energy and momentum with the object; this is the reason the flow is dissipationless. Above such a critical velocity, energy and momentum can be exchanged, and the flow is no longer without dissipation. 1 This critical velocity can be understood by looking at the excitation spectrum of the flowing superfluid. At zero temperature, a moving superfluid can be obtained simply by a Lorentz boost of a static superfluid. The excitation spectrum of the flowing superfluid is likewise just the Lorentz transform of the zero-velocity spectrum. As the superfluid flows faster, the velocity of phonons in the opposite direction decreases.
When v f > v s , antiparallel phonons have negative energy at small momentum, signaling that the energy of the superfluid can be lowered by exciting them. If there are objects, such as impurities or the walls of the capillary, that can excite these modes, then the flow stops being dissipationless. This criterion for superfluid stability is called the Landau criterion [3].
At nonzero temperature, Lorentz symmetry is broken, and so the fluctuation spectra must actually be computed. There is still a critical superfluid velocity v crit above which the gapless phonons develop a negative dispersion and dissipation can occur. However, in general, this is less than the speed of sound; i.e. v crit < v s . In the holographic model we have an infinite, homogeneous superfluid without any barriers or impurities, but we can still compute the critical velocity by a fluctuation analysis. It turns out that there are three important velocity thresholds for the superfluid. First, there is the Landau critical velocity v crit , which is the velocity of the superfluid above which the energy of the backward directed phonons becomes negative. This would signal an instability towards dissipative flow if there were an object which could excite these modes. However, we also find another, larger velocity above which the phonon dispersion acquires a positive imaginary part, signaling a spontaneous instability of the flow, even if no object is present. We label this v complex . Finally, there is v max , the velocity above which the solution of the equations of motion ceases to exist. Interestingly, we find that at zero temperature v max is universal and equal to the speed of light, irrespective of the mass of the ambient fermions. We compute these velocities as functions of the temperature and comment on their physical meanings.
A similar analysis for a conventional holographic superfluid was performed in [4], where it was found that, when the velocity of the phonons became negative, these modes also became tachyonic. In contrast, we find v crit < v complex . In addition, while the model of [4] could only be analyzed at relatively high temperature, the special nature of our probe brane model allows trustworthy analysis all the way to zero temperature. This paper is organized as follows. We begin in Sec. 2 by briefly reviewing the D3-D7' model, the proxy of a fractional quantum Hall state, at nonzero background magnetic field and charge density, and in the presence of an electric field. In Sec. 3, we perform the SL(2, Z) transformation to map the system to an anyon superfluid phase with zero effective electric and magnetic fields but possessing a nonvanishing current. We present an analysis of the fluctuations of the flowing superfluid in Sec. 4 and discuss the physical meaning of the result. And finally, we summarize our results and discuss open questions in Sec. 5.

The model
The D3-D7' model is constructed by embedding a probe D7-brane in the background generated by a stack of N D3-branes in such a way that the intersection is (2+1)dimensional [5] (see also [6]). As described in [2], this system is a model for the fractional quantum Hall effect. For a specific ratio of the charge density to magnetic field and low enough temperature, the D7-brane smoothly ends outside the horizon at some r 0 . This Minkowski embedding holographically corresponds to a quantum Hall state.
Minkowski embeddings ordinarily have a gap for charged fluctuations. In the bulk, charges are sourced by open strings stretching from the horizon to the tip of the brane; the charge gap is given by the masses of these strings, which is proportional to r 0 . In [7] it was shown that this embedding also has a gap for neutral excitations.
However, we showed in [1] that the D3-D7' model can be generalized by considering alternative quantizations of the D7-brane gauge field, and for one particular choice, the neutral gap closes, giving a superfluid. The change of the gauge field boundary conditions is implemented by an SL(2, Z) electromagnetic transformation. On the boundary, this SL(2, Z) action maps from one CFT to another, mixing the charged current and the magnetic field and changing the statistics of the particles.
Here, we aim to study the flowing anyon superfluid. We will start with a quantum Hall state in a background electric field. Under the SL(2, Z) transformation, this electric field will map to a current.

Action and equations of motion
The metric of the thermal D3-brane background reads: where the blackening factor h = 1 − r T r 4 corresponds to a temperature T = r T /(πL) and the radius of curvature is related to the 't Hooft coupling: λα . We write the metric on the internal sphere as: with the ranges for angles: ψ ∈ [0, π/2]; θ, α ∈ [0, π]; and φ, β ∈ [0, 2π]. The Ramond-Ramond four-form potential is C We embed a flavor D7-brane probe such that it spans t, x, y, and r, wraps both of the S 2 's, and will therefore have a profile ψ = ψ(r) and z = z(r). Such an embedding is inherently non-supersymmetric. Indeed, in the decoupling limit, this is signaled by the presence of a tachyon in the open string spectrum. However, the mass of the tachyon can be lifted above the Breitenlohner-Freedman bound by turning on a sufficiently large magnetic flux on the internal manifold of the D7brane worldvolume [2,8], thus rendering the model perturbatively stable. The same mechanism has also been applied in the T-dual system to stabilize a probe D8-brane in the D2-background [9], where only an infinitesimally small internal flux is required. 2 We therefore introduce worldvolume flux on one of the internal spheres: We are concerned here only with Minkowski embeddings, and so we will not turn any flux on the other two-sphere. This also implies the embedding function z will be constant.
In order to construct a quantum Hall state with a background magnetic field, electric field, and charge density, we turn on the following additional components of the worldvolume gauge field: where the prime represents derivation with respect to r. Furthermore, the electric field will generate a current, so we also include We expect that there will be a Hall current in the y-direction dual to a y . But, because the quantum Hall state has vanishing longitudinal conductivity [2], there will not be a current in the x-direction and we will indeed find In these coordinates, the action of the D7-brane, which consists of a Dirac-Born-Infeld term and a Chern-Simons term, reads [1]: where N = 8π 2 T 7 V 3 L 5 , V 3 is the volume of spacetime, and The function c(r), essentially representing the axion, is the pullback of the RR fourform potential onto the worldvolume and is given by where the asymptotic embedding angle ψ ∞ = lim r→∞ ψ(r) is related to the internal flux: In addition a boundary term at the tip of the D7 brane has to be added. This can be seen from either requiring gauge invariance under shifts of the RR four-form potential [2] or by consistency of the variation principle. In our case, this boundary term takes the form where r 0 is the smallest r of the D7-brane embedding. In the action (10), a t and a y are cyclic variables; the associated conserved quantities are the charge density d ≡ j t and Hall current j y . 3 The radial displacement field isd(r) = d − 2c(r)b. While d gives the total charge density on the boundary,d(r) measures how much of that charge is due to sources in the bulk located at radial positions below r. Similarly, j y is the total Hall current, andj y (r) = j y − 2c(r)e is the current due sources below r.
With an eye toward the fluctuation analysis, we will consider Cartesian-like coordinates (R, ρ) instead of the polar coordinates (r, ψ): The embedding is now described by ρ = ρ(R), where R is the new worldvolume coordinate. We will still write r explicitly in the equations to follow, but it should be read as r = ρ(R) 2 + R 2 . Until now prime has denoted derivative with respect to r; from now on, it will instead indicate a derivative with respect to R.
Performing an appropriate Legendre transformation to eliminate the cyclic variables, we obtain the following action (including the appropriately mapped boundary term) for the embedding field ρ(R): where The solutions for the gauge fields are: We obtain a complete solution by first numerically solving the equation of motion for ρ(R) derived from (17) and then using this to numerically integrate (19) and (20).
The mass 4 of the fermions is extracted from the leading UV behavior of the embedding: 3 The physical charge density and currents, defined by the variation of the on-shell action with respect to the boundary values of A t and A i , are Here, m is related to the physical mass by M = 2πα m.
where the corresponding operator dimensions are In this paper, we fix f = 2 3 , which yields ψ ∞ = 1 2 arccos 1 3 . This choice leads to zero anomalous mass dimension for the fermions; this is, ∆ + = −1. We do not expect to find qualitatively different results for different values of f .

Minkowski embeddings
Probe brane embeddings can be classified into two categories. Generically, probe branes cross the horizon; these are black hole embeddings. They are interesting in myriad ways [11][12][13]. However, in special cases, probe branes can end smoothly at some r 0 above the horizon as one of the wrapped S 2 's shrinks to zero size, yielding Minkowski (MN) embeddings, which are what we focus on here.
There are constraints coming from the requirement that the embeddings are of MN type. Essentially, we are demanding that there are no sources at the tip of the D7-brane. Due to the effects of the Chern-Simons term, this does not mean that we need to require the charge density to vanish. Rather, via a mechanism revealed in [2], we require that the charge density be locked with the magnetic field in such a way that the radial displacement field is forced to vanish at the tip:d(R = 0) = 0. A similar argument holds for the currents: j x =j y (R = 0) = 0. These conditions yield: of which the former dictates the filling fraction ν, which ultimately follows from the amount of flux f we turned on. As expected, the only nonzero current is the Hall current, and the conductivities are precisely those of a quantum Hall state [2]:

Rescaled variables
We have the freedom to scale out some parameters. Since the system is pretty robust against temperature variations, it is better to scale out the magnetic field as follows: In terms of the rescaled variables, the action (17) has the same form, but with tildes added to all quantities and with the overall normalization: We are then left with four independent parameters; these are the reduced temperature, the electric field, and the mass of the fermions: along with the internal flux f . As was mentioned in Sec. 2.1, different values of f yield qualitatively similar results, and we will therefore fix f = 2 3 . Moreover, it turns out that differentm do not induce qualitative changes either; we therefore fixm = −8 in what follows. In practice, we thus have a two-dimensional parameter space (r T ,ẽ) to explore. For any CFT in 2+1 dimensions with a conserved U (1) charge, there are two natural operations which can transform it into another CFT: adding a Chern-Simons term for an external vector field and making an external vector field dynamical. Together, these operations generate an SL(2, Z) action transforming one CFT into another [14,15]. Holographically, this mapping corresponds to changing the boundary conditions of the bulk gauge field and thereby imposing alternative quantization.
In [1], we described the D3-D7' model with alternative quantization of the bulk gauge field. Standard quantization corresponds to Dirichlet boundary conditions for the gauge field A µ . The variation of the renormalized on-shell action is just a boundary term: where J µ = δS D δA µ is the conserved U (1) current in the CFT. This implies that A µ must be kept fixed at the boundary; that is, δA µ = 0. To implement mixed boundary conditions, we can add a general boundary term to the action. Defining, up to gauge transformations, a vector V µ such that the most general boundary term we can add takes the form for arbitrary a 1 , a 2 , and a 3 . The variation of the on-shell action can be written as where and where Note that (37) implies that a s d s − b s c s = 1. The parametrization of (35) makes clear that and we recognize this change of boundary conditions as an SL(2, R) transformation mapping the original boundary theory into a new one. The new boundary condition fixes B * µ , and the new conserved current is J * µ . These are related to the original variables by an SL(2, R) transformation: Because charges in the bulk theory are quantized, we are, in fact, restricted to transformations in the subgroup SL(2, Z).
An anyonic superfluid state can be obtained by a judicious SL(2, Z) transformation from a quantum Hall state in standard quantization. A quantum Hall state with filling fraction ν = 2πD B and background electric field E x has a Hall current J y = D B E x . A general SL(2, Z) transformation of such a quantum Hall state gives, via (39), To end up with a superfluid, we need a final state with no electric or magnetic field. We therefore choose a transformation with which implies B * = 0. Since we started with a quantum Hall state with J y = D B E x , this choice implies E * x = 0, as well. The current and charge density in the new state are The new state has a persistent current in the absence of a driving electric field, as would be expected for a superfluid. In [1], we also showed that, in the case with e = 0, this SL(2, Z) mapping produced a superfluid with the requisite gapless mode. We will further investigate the dispersion of this mode in Sec. 4.
At nonzero temperature, a superfluid can be described as a mixture of two components, the superfluid and an ordinary fluid of thermally excited phonons. Both components contribute to the current J, and the velocity v ave = J/D gives a weighted average of the superfluid and normal fluid velocities.
At zero temperature, the normal fluid is absent. In this case the superfluid velocity is The velocity of a conventional superfluid is given by the gradient of the order parameter. In holographic superfluids [17,18], the superfluid velocity is the dual source for the current J i and is therefore given by the boundary value of the bulk gauge field A i . For a superfluid flowing in the y-direction, the velocity is Anyonic superfluids are not characterized by a local order parameter. However, it seems natural that the superfluid velocity is still given by Z) transformed gauge field. In our case we have a nonzero J y and a nonzero J t , so we can pick a gauge where V y = V t = 0. We can also choose a gauge where the electric and magnetic field come from A x . In this case we can then compute The boundary values of a t and a y can be found by integrating the expressions (19) and (20) for a t and a y : which, for a given embedding ρ(R), can be integrated numerically. One subtlety with expressions (47) and (48) is the IR boundary condition at R = 0. For a MN embedding, the value of the gauge field at the tip is not fixed. For example, as was seen in various contexts [2,9,19,20], the charge density in a MN embedding is completely independent of the chemical potential. In order to obtain the correct zero-temperature limit (44), we must choose which fixes the IR boundary terms in (47) and (48).
Thus the expression (46), using our chosen boundary conditions, reduces at zero temperature to v f = e b =ẽ, as desired. More generally, from (47) and (48) and the fact that h ≤ 1, we find that v f ≥ẽ. We can thus parametrize our flowing solutions by v f or byẽ.

Superflowing solutions
To find solutions corresponding to flowing superfluids, we numerically solve the equations for the D7-brane embedding and gauge fields in the presence of worldvolume electric and magnetic fields. We focus here only on MN embeddings, so we need to impose the IR boundary conditions discussed in Sec. 2.2.
Note that, because different quantizations only differ by boundary terms, the bulk equations of motion are independent of the choice of quantization. After an SL(2, Z) transformation, the solutions of the equations of motion are therefore the same, but their physical interpretation is different. In standard quantization,ẽ is a background electric field, while in the alternative quantization appropriate to the anyonic superfluid, it is the average fluid velocity v ave .
was also shown to be perturbatively stable. We believe that this thermodynamic argument holds atẽ > 0. From Fig. 1, it is evident that for a given choice ofm, whenẽ becomes sufficiently large, there are no solutions (apart from an unstable branch of solutions for whichr 0 2). This can be seen even more clearly in Fig. 2 which showsr 0 as a function ofẽ for a fixedm. For each temperature, there is a maximumẽ such that there are no relevant MN solutions forẽ >ẽ max . In the standard quantization,ẽ max is the maximum electric field. For the anyonic superfluid, the physical interpretation is that there is a maximum velocity v max beyond which no relevant solutions exist.
For a given MN solution, the superfluid velocity can be computed numerically via (46). We find that, in general, v f is slightly larger thanẽ. We plot the difference v f −ẽ for two fixed temperatures in Fig. 3. The difference is largest asẽ approaches e max . At zero temperatureẽ max = v max = 1, which simply means that the superfluid can not flow faster than the speed of light. In this case, as the superfluid velocity increases to v max , the charge mass gap decreases, i.e.r 0 → 0 for allm, asẽ = v f → 1; see Fig. 2. At nonzero temperatures, bothẽ max < 1 and v max < 1, and both depend onm. In addition, at nonzero temperature the charge mass gap does not close as e →ẽ max . Instead, another branch of solutions with smallerr 0 appears and merges with the thermodynamically preferred stable solution atẽ =ẽ max .
We will see by analyzing the fluctuation spectrum, however, that the anyonic superfluid becomes unstable before the maximum superfluid velocity is reached.

Fluctuations
We now compute the spectrum of collective excitations of the anyon superfluid flowing with velocity v f . The SL(2, Z) transformation used in Sec. 3.1 to generate the superfluid state acts in the bulk to change the boundary conditions, and thereby, the quantization of the gauge field. We must, therefore, analyze the fluctuations of all fields around the MN background, imposing alternative boundary conditions on the bulk gauge fields, as previously described in [1].

Set up
In the case of a nonflowing, isotropic superfluid [1], one could use rotational symmetry in the x−y plane to align the fluctuation with, say, the x-axis. However, the superfluid flowing in the y-direction breaks this symmetry. The excitation frequency will in fact depend on the relative angle between the superflow and the fluctuation.
We impose the following wavelike ansatz on the fluctuations: 6 The rescaled frequency and momenta are It is preferable, however, to work with the gauge-invariant field strength perturbations: δẽ y =ωδã y +k y δã t (54) δb =k x δã y −k y δã x .
With this ansatz, we expand the D7-brane action (17) to second order and derive the equations of motion for δẽ x , δẽ y , δb, and δρ. This system of coupled, linear ordinary differential equations is extremely long and not particularly illuminating; we will therefore not reproduce it here.

Alternative boundary conditions
The general mixed boundary conditions for fluctuations of the gauge field can be written as: where indices µ, ν, and ρ are (2+1)-dimensional boundary coordinates, raised and lowered by the flat metric η µν , and the inverse radial coordinate u = 1/r. The boundary is therefore located at u = 0. The parameter n indicates the particular choice of quantization (see also [16] for more discussion on this). The standard quantization with Dirichlet boundary conditions corresponds to n = 0, and n = ∞ gives Neumann boundary conditions.
The parameter n is related to the SL(2, Z) parametrization in Sec. 3.1. From (39), we see that the new boundary condition, δB * = 0, implies mixed fluctuations in terms of the original charge and magnetic field. The charge d and magnetic field b are related to the physical charge D and magnetic field B by and the charge is related to the boundary value of the bulk gauge field via Writing (39) in terms of the bulk gauge field and comparing the result to (56) gives As explained in Sec. 3.1, the superfluid phase is obtained from the quantum Hall phase by an SL(2, Z) transformation for which d s /c s = ν, where the filling fraction ν is given in terms of ψ ∞ by (23). In the superfluid phase, therefore, the gauge field fluctuations obey boundary conditions with In order to write these boundary conditions entirely in terms of gauge-invariant quantities, we can use the gauge constraint coming from the equation of motion for δa u which, for u → 0, reads The boundary condition (56) can then be written as follows: Now we need to put the boundary conditions (62), (63), and (64) in terms of the rescaled radial coordinateR and the other rescaled variables for use in the numerical calculations: Using the fluctuation analysis techniques developed in [21,22] and used in [1,7], we search for normal modes by looking for pairs (ω,k) for which there is a solution to the fluctuation equations with the boundary conditions (65).

Phonon dispersion
In this section we analyze the spectrum of collective excitations of the anyonic superfluid with nonzero current. We consider a current in the y-direction and, as we do not expect rotational invariance, consider collective excitations with a general (k x ,k y ). The mode we are most interested in is the massless phonon. We compute its dispersion as a function of the angle in the (x, y)-plane in which it is directed.
When the superfluid is at rest, the phonon has an isotropic linear dispersion, for smallk in any direction [1]. For v f > 0, the phonon velocity becomes anisotropic, as shown in Fig. 4, increasing in the direction of the superfluid flow and decreasing in the opposite direction. Fig. 5 shows the phonon dispersion in the y-direction for various superfluid velocities. Similar dispersions have been found in [4]. At a critical superfluid velocity v crit , the phonon velocity vanishes, and for v f > v crit , the frequency of backward-directed phonons becomes negative. We show in Fig. 7 the temperature dependence of both v crit and the sound speed in the static superfluid v s .
At T = 0, the spectrum of excitations for the flowing superfluid is just given by a Lorentz transformation of the spectrum of a static superfluid. The nonzero current can be obtained by boosting an observer by v f in the negative y-direction; the frequency and wave number of fluctuations transform as: where the Lorentz factor γ = 1 √ Plugging in the linear phonon dispersion (66) into (67) gives the dispersion at nonzero v f . In particular, the critical velocity v crit , at whichω = 0 for negativek y , is exactly v s . To the accuracy of our numerical computations, this dispersion matches the numerical result shown in Fig. 5.
According to the Landau criterion, v crit is the largest current velocity for which the anyonic fluid remains a stable superfluid. 7 Indeed, as one goes to v f > v crit , there is a negative energy mode which signals an instability towards a different configuration. However, we wish to emphasize that the frequency of this mode continues to be real: Imω = 0. The remaining configuration should just be a superfluid with a lower velocity. At zero temperature, the critical velocity for the anyonic superfluid is found to be exactly the speed of sound when the fluid is at rest, i.e. v crit = v s . At nonzero temperature, the critical superfluid velocity is smaller than the speed of sound at rest; that is, v crit (T ) < v s (T ). When one tries to give the current a velocity above v crit , the phonon velocity becomes negative. If the fluid passes any barrier, it can excite modes with a negative energy, which is just the statement that the fluid flow is no longer dissipationless.
However, if there is no barrier, the fluid flow is still stable, and the existence of the negative velocity does not make the flow unstable. This is clear from the zero-temperature case where the flowing superfluid is just the stable static superfluid in a boosted reference frame. On the other hand, at nonzero temperature, the usual description of a superfluid consists of a superfluid component and some regular fluid component. If this is the case, then one might expect that relative velocities between the two components could induce interactions that would excite the negative-frequency mode and make the flow unstable. Indeed, we find that at nonzero temperature, there is a velocity v complex at which an instability occurs.
In Fig. 6, we show a typical phonon dispersion corresponding to v f > v complex . For excitations with smallk, we find a positive imaginary frequency, signifying an instability. We interpret v complex as the velocity at which the flow becomes unstable due to interactions, with the normal component making it possible to excite the negative-frequency mode.
The temperature dependence of v complex is shown in Fig. 7. In general, v complex > v crit , though the difference shrinks with temperature. At a sufficiently high temperature, we find v complex = v crit = 0. This is therefore the critical temperature above which the static superfluid is unstable.
At zero temperature, v complex = 1, the speed of light. This is in accord with our previous argument that at T = 0, the flowing superfluid is just a static superfluid   Figure 7: The temperature dependence of various velocities: (solid curves from bottom) v crit (black), v complex (red), and v max (brown). In addition, the sound speed v s at zero superfluid velocity is shown as a dashed blue curve. Atr T = 0, v crit = v s as expected from the Landau argument. Also atr T = 0, v max = v complex = 1 which is the speed of light. Atr T = 1.10, v crit = v complex = v s = 0; above this temperature, the nonflowing superfluid is unstable. which has been Lorentz boosted. The maximum v f obtainable by a boost is, of course, the speed of light, so the T = 0 superfluid should be stable for any v f < 1.
Note that this is a bit different than the results found in [4], where the authors found that v crit = v complex . We believe this is due to the relative high temperature at which they were working, where we find the two velocities become very close. On general grounds, however, v crit = v complex since at T = 0 the dispersion is fixed by Lorentz invariance.
As discussed in Sec. 3.2, at even higher velocities, we encounter a v max , the maximal superfluid velocity above which no MN solution exists on the stable branch, and whose temperature dependence is also shown in Fig. 7. Interestingly, v max (T = 0) = 1 for all values of the mass. The embedding geometry somehow knows that the highest superfluid velocity possible in the boundary is the speed of light.

Discussion
We have presented a holographic model of a flowing, strongly-coupled anyonic superfluid. A particularly elegant feature of this model is that, because it is based on a probe brane taking a MN embedding, there is no difficulty considering the zerotemperature limit. By analyzing the fluctuations, we found the critical superfluid velocity v crit at which the phonons can begin dissipating energy and showed that at zero temperature this critical velocity was equal to the phonon velocity v s , as argued by Landau. We further found that at an even higher velocity v complex , the superfluid is in fact unstable.
A interesting open question is what actually happens to the anyonic superfluid when v f > v complex ? For v f > v crit , the negative energy modes, if they are excited, simply act to slow down the superfluid until it is back to the critical velocity. However, for v f > v complex , the outcome is less clear. At sufficiently high temperature it is possible that the stable configuration is a BH embedding, corresponding to a metallic, conducting state rather than superfluid. This black hole embedding should obey the same boundary conditions as the flowing superfluid phase, which are E * = B * = 0. The only such BH solutions are those with B and D the same as in the superfluid solution but with E x = E y = 0. However, such solutions only exist at high enough temperature; for instance, form = −8 such BH solutions only exist forr T > 0.982, so at lower temperatures at least, this can not be the end point. Another option is that since the unstable modes occur also at nonzero momentum, perhaps the stable ground state is spatially modulated. An upcoming work [24] will investigate more generally the instabilities of the alternatively quantized system, and in another [25] we will solve for the inhomogeneous ground state to which these instabilities lead.