Holographic plasma and anyonic fluids
We use alternative quantisation of the system to explore properties of a strongly coupled charged plasma and strongly coupled anyonic fluids. The -transform of the system is used as a model for charged matter interacting with a gauge field in the large coupling regime, and we compute the dispersion relationship of the propagating electromagnetic modes as the density and temperature are changed. A more general transformation gives a strongly interacting anyonic fluid, and we study its transport properties as we change the statistics of the anyons and the background magnetic field.
Much work has been done in recent years exploring strongly coupled CFTs with global symmetry and their associated conserved currents via the gauge/gravity relationship. These systems have a rich variety of physically relevant phenomena to condensed matter, such as superfluidity, non-Fermi liquids, fractional quantum Hall effect etc. Less explored are strongly coupled CFTs with a local gauge symmetry. Such theories can also be tackled using the gauge/gravity duality if they live in dimensions. In fact, as shown in Witten:2003ya (), given any CFT with a global charge there is a well defined procedure generated by an element of to turn it into a CFT with a local gauge symmetry, which also has a conserved current. An example of a phenomenologically relevant model with such a conformal gauge field is given by the large charge limit of zero temperature -dimensional QED333In condensed matter physics one application of QED and its cousins can be seen in the two-dimensional Heisenberg spin model (and variants) on a square lattice where the effective Hamiltonian turns out to be the fermionic term of lattice QEDPhysRevB.37.3774 ()..
Given a holographic description of a CFT with a global current one can find a holographic description of the transformed field theory and compute correlation functions of the transformed conserved current. Thus we can describe using holographic techniques field theories whose field content has some matter coupled to a dynamical gauge field (in the sense that the gauge field is integrated over in the path integral) but with no Maxwell term. We can however add a Chern-Simons’ term by an appropriate transformation. Using such gravitational duals we can begin to study properties of these theories at finite temperature and density, i.e. we can study strongly coupled “real” plasma physics in dimensions.
A second application of -dimensional gauge theories in condensed matter physics concerns the appearance of “anyons”, particles whose statistics interpolate between Bose-Einstein and Fermi-Dirac, which may be useful in understanding the quantum Hall effect. A simple model of such systems is given by a Chern-Simon’s gauge theory with charged matter. The sourced Chern-Simon’s equations of motion, with level , attach magnetic flux to an electrically charged particle. Hence a point particle with charge sourcing the current also carries units of magnetic flux. Moving one such point particle around another leads to an Aharanov-Bohm phase where is the exchange phase angle. We see that it is Chern-Simon’s level dependent and potentially neither integer nor half-integer. This leads to the unusual statistics of these anyonic particles PhysRevLett.49.957 (); Arovas:1985yb (). A holographic model for an anyonic superconductor which bears some resemblence to our system has recently been investigated Jokela:2013hta ().
1.1 transformation of a CFT
Let us start with a -dimensional CFT which has a conserved current that can couple to a background source . We can define another CFT also with a conserved current by an transformation of the original theory using the prescription we shall now review Witten:2003ya (); Leigh:2003ez (); Herzog:2007ij (). First note that is generated by an -transformation and a -transformation. The -transformed CFT is obtained by including the background field in the path integral, but with no kinetic term. A new conserved current in this CFT is , which can be coupled to some new background vector field , thus defining a new generating functional. A -transformation is induced by adding to the theory a Chern-Simon term for the background field. The -transformation and the -transformation do not commute. The current in the CFT and the Hodge dual of the background field strength transform as a doublet under the transformation.
Given the above we can readily determine the transformation of the two point function between the CFTs. Assuming rotational invariance, the general expression for the current-current correlator in a finite temperature CFT is
where the functions , and are scale invariants and the tensor structures displayed above are defined in terms of the Minkowski metric , spatial metric , momentum and by
The parity violating projector, , corresponds to a contact term ambiguity in the definition of two point functions that must be specified to completely define the -dimensional field theory of a conformal current Witten:1995gf ().
Under the -transformation the two point function has of course the same form,
but with new scalar functions: . These are given in terms of the original scalars by
Under a -transformation however only the parity violating scalar is altered,
where we have normalised our currents suitably, leaving the other functions untouched.
1.2 Bulk formalism
The procedure of using holographic techniques to describe a theory with a dynamical gauge field based on the holographic description of a theory without a dynamical gauge field is called alternative quantisation. The reason is that the bulk description of the theories is the same but the quantisation procedure (i.e which fluctuations we quantise and which we treat as sources) changes.
The usual boundary condition (Dirichlet) imposed on the gauge field in AdS stems from using a bulk action which after holographic renormalization and imposing the equation of motion has the property that
where is interpreted as a conserved current in the CFT. Consistency of the variational principle (equivalently requiring no flux through the time-like boundary at infinity) requires imposing the condition that is fixed at the boundary. However this “normal quantisation” procedure is not the only consistent boundary condition one can impose on a gauge field in AdS. Both independent solutions of the equation of motion for the bulk gauge field are normalisable modes, and one can consider the theory with other boundary conditions (which again guarantee that no information is lost through time like infinity)Ishibashi:2004wx (); Marolf:2006nd ().
Since one can write , where is defined up to gauge transformations. Also it is convenient to define . The most general boundary condition comes from the action
Now the variation of the action takes the form
Evidently and so form a matrix. The new boundary condition requires to be fixed, or in gauge invariant form
and the new current is just
The -transformation is given by setting and . The -transformation has and . It is sometimes easier to take derivatives with respect to the gauge invariant combination (10). The resulting objects are correlation functions of a gauge dependent quantity, but acting on it with an appropriate derivative operator gives the current correlation functions. Note that if the bulk theory has only integer electric and magnetic charges (which is what happens in string theory), for consistency the transformations must be restricted to a subset , acting on appropriately normalised quantities.
From equation (10) we see that the new current, , is a current of particles carrying (with respect to the original definition) for each unit of the original charge, units of the original magnetic flux. This is precisely the realisation of anyons described in the introduction.
In this paper we explore theories which are connected to part of the phase space of the system through an transformation. More specifically we look at this theory as the original theory at finite density and temperature and use a transformation to explore possible phenomena. For the pure -transformation we prefer to view the transformed theory as the original theory but coupled to a gauge field without a Maxwell term. We then holographically compute the field strength correlation function and interpret the quasi-normal modes as the spectrum of electromagnetic excitations (transverse) or plasmon excitations (longitudinal) in a finite temperature plasma of particles charged under a gauge field and also strongly interacting via an gauge field. For a more general transformation we prefer to view the transformed theory as just another finite temperature CFT with a conserved charge carried by some excitations. In this case we holographically compute the current correlation functions and extract from them the collective excitations, conductivites etc. As explained above, after the transformation, the charge carrying excitations are anyons. The new background charge density and magnetic field depend on the actual transformation we have used.
2 Holographic model
We will use as our bulk theory the brane system where the -branes are probes of the background given by -branes at finite temperature. The contribution of a -brane to the bulk action is
where are the embedding coordinates, the tension of the brane and the world-volume field strength. We have absorbed a factor of into the field strength compared to the usual definition and thus it is dimensionless. As the brane is treated as a probe we neglect its back-reaction upon the bulk metric which we must specify. We take the metric to be
where with the AdS radius. We now choose . The embedding we will consider is the usual massless black hole embedding with some background charge, carried by bosonic and fermionic excitations, also considered recently in Brattan:2012nb (). The embedding is determined by the gauge field configuration since the scalar profiles are all trivial. It is well understood and we simply record the relevant results here, namely, the bulk field strength is given by Kobayashi:2006sb (); Karch:2007br (),
where , , and
This embedding has been proposed as the thermodynamically preferred state of the system for all values of (see Chang:2012ek () for some controversy in the regard) and corresponds to the decoupling limit of the brane embedding displayed in table 1. The current displayed above does not have the correct length dimension to be a current due to our normalisation of in (12). Subsequently the physical charge density is .
Now we turn to gauge fluctuations about the background field in (14). The boundary theory corresponding to (13) and (14) has explicit spatial rotation invariance and so all choices of the direction of spatial momentum for our fluctuation are equivalent. As such we shall turn on momentum in the direction, which perturbatively breaks this , and then Fourier decompose our fluctuation ,
where our transform conventions are as displayed. The quadratic action in terms of these fluctuations, , and the resulting equations of motion are straightforward to obtain but unilluminating, so we will omit them. We will record the equation for , which in gauge is a constraint on the other bulk fluctuations,
With our choice of momentum, the gauge-invariant fluctuations are itself and also the bulk electric field (in Fourier space) Kovtun:2005ev ()
which is dual to the operator
In terms of these gauge-invariant fluctuations, using eq. (17) and performing an integration-by-parts, we can write the quadratic action as
where and are generically complex. The equations of motion that follow from are
These match the equations of motion in refs. Karch:2008fa (); Myers:2008me (); Davison:2011ek (); Brattan:2012nb (). Notice in particular that the bulk equations of motion for and are decoupled and degenerate to the same equation when is taken to zero (i.e. when the perturbation is homogeneous and thus respects spatial rotation invariance).
2.1 Alternative quantisation
Under a general transformation the appropriate quasi-normal mode boundary condition444This is a slight abuse of standards as quasi-normal mode typically refers to the normal quantisation condition. can be written in terms of the bulk fields and a single parameter, which we will label as , as
Of course there is an appropriate only for particular values of but we will compute as though it were a continuous parameter since the actual values corresponding to an parameter depend on the precise value of . The spectrum of the quasi-normal modes, which are the poles of the retarded Green’s function, are given by finding the normalisable modes relative to the boundary condition (23) which only occur for particular pairs . Thus the spectrum of the quasi normal modes depends only on the choice of and not on the precise transformation. If we wish to compute the current two-point function however we need to completely specify the actual transformation.
Turning now to correlation function in order to compute the current-current correlator one needs to take derivatives of the on shell action with respect to the boundary conditions
In the treatment of the fluctuation analysis this is actually computed by
However in the fluctuation analysis the boundary condition is given in terms of gauge invariant quantities .
Let be such that . Using the chain rule
and that with the kinematics , and are not independent, we get that
where we have defined . These equations allow us to compute the current two point function from the fluctuation analysis.
For low momentum and frequency it is often possible to solve (22a) and (22b) analytically. When this is not possible or unenlightening we shall resort to numerics. We now layout the numerical procedure Kaminski:2009dh () used to determine solutions to the bulk equations satisfying the mixed quantisation conditions555In this section we shall only display an explicit expression for the boundary source term and not the one-point functions. However when we reach section 4 we will show how the Green’s function, and thus the one point functions, are obtained from our procedure.. We shall employ the notation of Brattan:2012nb (). Eqs. (22a) and (22b) are second-order, hence for each field, and , we need two boundary conditions to specify a solution completely. On the black hole horizon, a solution for or looks like a linear combination of in-going and out-going waves, with some normalizations. The prescription for obtaining the retarded Green’s function requires that we choose our normalisations to remove any outgoing modes Son:2002sd (); Policastro:2002se (); Skenderis:2008dh (); vanRees:2009rw (). Let
and at large identify
to some value, denoted , at the boundary. The first component of is equal to and when multiplied by and respectively. For numerical purposes however it is preferable to fix all our boundary conditions at the future black hole horizon. The second boundary condition is then the normalisation of the ingoing wave at the horizon. The two ways to fix boundary conditions are related to each other by a change of basis transformation. The vector of near-horizon normalization factors, , is, when the temperature is non-zero,
Notice that is constant, independent of , , and . On the right-hand-side of eq. (30), the exponential factor is designed to cancel the exponential factor that represents an in-going wave at the future horizon.
We now pick two convenient values of and solve the equations for each of these choices. This provides us with a basis of solutions in terms of which we can write any solution. The typical choices we have used in our numerics are: and . Let us call the corresponding solutions and and use them to define a matrix by
Using this matrix we can write any solution to the equations of motion with initial condition at the horizon as
In terms of the bulk to boundary propagator and the near horizon vector we have
The limiting case of is the normal quantisation condition Brattan:2012nb () while is the -transform quantisation.
We call any solution to the bulk equations with and complex frequency a quasi-normal mode. For a non-trivial solution with and it must be the case that Kaminski:2009dh ()
which places a constraint on and yielding the dispersion relation of the mode666Note that the constraint for finding quasi-normal modes, as we have written it, is blind to poles at zero momentum and frequency.. If we are interested in only normal or alternate boundary conditions we can take the limiting value of this expression. Setting requires we set the determinant of to zero which is entirely standard. However, when we set , it is very important to include the matrix prefactor to . It ensures that is replaced by or when searching for the quasi-normal mode and cancels out an illusory light-like pole which would otherwise dominate the spectrum. Finally, we note that on setting the equations of motion degenerate into a single expression and thus finding the quasi-normal modes means solving a single equation equivalent to solving the boundary condition given by the upper or lower row of the above matrix.
We note that it will be useful throughout the remainder of the paper to set some conventions. A variable normalised by temperature will be denoted . The object is evaluated in normal quantisation while, the quantity is calculated using -transform quantisation. Finally if carries the sub- or superscript , e.g. , it is determined by imposing mixed quantisation conditions.
3 A strongly coupled -dimensional plasma
In this section we consider the effects of gauging the external source vector field, , in the absence of parity violation. We then employ this gauge field to probe properties of the finite density system. We compute various features from the two-point function of two gauge operators such as the penetration length and the Debye mass, as well as the dispersion of the electromagnetic waves propagating in the plasma.
The currents are related to the field strength by . At zero temperature where , one finds that for a wave vector in the direction, the only poles are lightlike and in the and correlators with no poles in the correlator. This is what we expect in the vacuum for an electromagnetic wave. Given this identification, and also to agree with condensed matter literature, at non-zero temperature we shall refer to the longest lived quasi-particle excitations in the transverse electric field correlator as the photon. The longest lived mode in the longitudinal electric field correlator at finite temperature will be called the plasmon.
3.1 The photon
We shall begin by describing the lowest lying poles in the transverse electric field correlator, organised by increasing . In fig. 1 we display the lowest lying modes in this correlator at , and . We display both the case where is chosen to be real and the case where is chosen to be real. The first choice represents the dispersion of electromagnetic waves in the plasma in response to an incident wave or due to time periodic phenomena. The second choice gives the excitations that arise when a spatially periodic phenomena occurs in the plasma.
Let us describe the behaviour of the dominant mode for real values of . For all the hydrodynamical mode, the mode for which , is dominant for small or . As is increased there is a transition to a mode with both real and imaginary parts for , with and for large . This mode may look like a massive mode if is large enough. This can also be seen in the representaion using real .
The dispersion relation of the hydrodynamical mode can be solved for analytically in a small frequency and momentum analysis. This mode, for all values of is the dominant one for small and . It satisfies the dispersion relation
for sufficiently small and . An important physical quantity that can be obtained from this dispersion relation is the penetration length which measures how far the electromagnetic wave will penetrate the plasma. It is defined as where is chosen to be real. This is compared to numeric data in fig. 2. The analytic calculations are given in appendix A. We note that at low frequency the penetration depth decreases as for and is roughly constant for . The penetration depth at arbitrary frequency, in other words outside the low frequency regime, can be obtained numerically as the imaginary parts of the real frequency dispersion relations. Examples for and are given in fig. 1.
As described above eventually the hydrodynamical mode ceases to be dominant and is overtaken by a mode which is approximately linear in . We have tracked the crossing point in for as shown in fig. 2. We note at large the position of the crossover in grows as . We do not expect the numbers displayed in this expression to have any element of universality and have simply recorded them for posterity. Indeed, we should expect them to depend strongly on the matter content of the bulk theory. However, the existence of the cross-over behaviour above should be more general and indeed has been seen often in the normally quantised system Brattan:2012nb ().
3.2 Plasma oscillations
We now turn to the plasmon. This is an excitation which is a collective excitation of the plasma and thus vanishes at strictly zero temperature and zero density. The behavior of the leading excitation i.e. that with the smallest imaginary part, changes as we change . In figure 3 we give the quasi normal mode at zero momentum behavior as we change . As increases two purely imaginary modes (at ) come together and become a complex mode with a decreasing imaginary part and increasing real part as grows. At even larger two complex poles merge to become two other complex modes (at ) where the leading tends to having an almost constant real part as grows. In figure 4 we look at what happens to these poles as one changes . At small the purely imaginary mode comes together with another purely imaginary mode and becomes a complex mode with at large . At approximately there is only a complex mode for which again has at large , and an increasing mass with increasing . At an even larger , for small the dominant mode has an almost constant , but at larger the dominant mode switches to a mode with . As is increased the range of where the almost constant mode is the dominant one increases, but eventually at large enough another mode comes up from the complex plane to dominate which has .
3.2.1 Debye length
Now we turn to computing the Debye mass. This can be defined using the zero frequency electrostatic two point function. At zero frequency we find
and we note that the Fourier transform of this time component of the two point function will give the potential between two static point charges. Our system has conformal invariance at zero temperature and thus for . Hence the correlator begins as in vacuum but, as turning on temperature introduces a length scale, it becomes exponentially decaying at large for non-zero . We call the dependent length scale in the exponential the “Debye length”. In summary at finite temperature the potential between static charges for our system will be
The Debye length of our system is defined by where is the position of the lowest pole in the longitudinal correlator for real frequencies. This is displayed on the left of fig. 5. We have checked numerically that the pole is simple and paired (i.e. there exists a second imaginary pole with opposite sign). On the right of fig. 5 we also give the two point function of the zero component of the gauge field at zero frequency and real momenta.
4 Anyonic fluids
We now turn to the more general boundary condition of (23) labelled by . The conserved charge is just the anyon number. The quasi-normal mode computation is insensitive to the actual transformation since it only depends on . So the following results for the quasi-normal modes and their properties are true whenever has a particular value. However it is useful to have in mind a particular transformation when describing the results. So we will imagine that we are using the transformation , with ( and are integer)
and so our parameter in (23) takes the value . This means that the ground state of our system has and . As such our results are relevant for a thermal anyonic fluid at temperature and anyon density in a magnetic field whenever the filling fraction is equal to an integer . In particular when this fluid consists of an equal number of anyons and anti-anyons in a background magnetic field.
As we change from zero we will get different finite temperature anyonic fluids at the same density in the same background magnetic field. However the fermionic and bosonic excitations of the original system are changed into anyonic excitations by a phase . We will treat as a continuous parameter but of course it can only take certain values. Thus the change of properties as we change indicates how the behaviour of the fluid depends on statistics of the theory’s excitations. In order to avoid clutter we will label from now on .
4.1 Anyon correlator
The pole structure of the correlators will be dependent on and . Turning from small to large switches the pole structure from alternate quantisation () to normal quantisation () . The behaviour for various is displayed in fig. 6. Not displayed in this figure is small where there is little difference between and . In this ultra low regime two purely imaginary modes come together at and form a complex mode which asymptotes to having the light-like behaviour that governs the system at large . As such we shall only further describe intermediate () and high () values of .
At and the behaviour of the lowest lying poles is slightly more complicated than the case. There is a purely dissipative hydrodynamical mode, but instead of the purely imaginary mode below it, there is a complex mode starting at . As increases it separates into two purely imaginary modes. One of these remains imaginary and sinks lower into the complex plane for the values of examined. The other combines with the hydrodynamical mode to become a complex mode which then asymptotes to a light-like pole. As is increased the splitting of the complex mode starts at smaller and as the complex mode disappears and becomes two imaginary modes. All this is shown in the top of fig. 6.
At larger the change in behaviour with is different. At the low-lying spectrum consists of a purely imaginary (diffusive) pole that descends deep into the complex frequency plane with increasing and a massive complex mode. For some finite this complex pole, which asymptotes to a light-like behaviour at large , has smaller imaginary part than the diffusive pole and governs the late time behaviour. As is increased a region of finite extent in appears where the complex mode has a real part with gradient . This looks like a massive zero sound mode for the anyons. At larger there is a kink and the dispersion then becomes light-like. The range of for which the zero sound mode behaviour is seen increases with increasing and . For any given there is an small enough where this behaviour is absent and similarly for a given there is a small enough where no zero sound behaviour is seen, but rather just a lightlike pole. For a fixed large enough for zero sound to exist as we increase the mass of the zero sound mode decreases. For larger at small enough the lower mode is still complex. However at some small value of it splits into two imaginary modes, one of which sinks deeper into the complex plane while the other joins up with the hydrodynamical mode as increases. All this is seen in the bottom row of fig. 6.
As a final comment on the pole structure we note that the emergence of the massive pole at non-zero , fig. 1, is similar to what is seen in the normally quantised and systems at finite density and magnetic field Brattan:2012nb (); Jokela:2012vn (). For vanishing magnetic field in that system the collisionless mode (whether it be zero sound at large or light-like at small ) has a massless dispersion relation after the cross-over. When the magnetic field is non-zero it acquires a small mass but the system has sufficient thermal energy to excite this mode regardless so we still see the purely imaginary diffusion pole connect up with a pole from deeper in the complex plane to become complex. As magnetic field increases further the mass of the mode becomes too large to be overcome by thermal effects and the diffusion poles ceases to connect with another pole and simply sinks lower into the complex plane as increases. Now at some value of the collisionless pole has a smaller imaginary part than the diffusion pole and instead governs the late time behaviour of the system. See Brattan:2012nb (); Jokela:2012vn () for further discussion.
4.1.1 Diffusion constant
A feature that is common to perturbations of interacting thermal systems by conserved current operators is the existence of a diffusion regime at sufficiently long-times and low momenta. This regime is governed by the behaviour of poles close to the origin.
The diffusion constant of the anyon current as a function of is computed in appendix A, and the result is
Note that for a given it has a maximum value at and thus anyons are much less efficient at depositing charge into the ground state. Here we have an infinite class of diffusion constants for differing types of anyonic excitations. The analytic expression for the diffusion constant at various choices of against numerical data is depicted in fig. 7. Notice also that the above behaviour with makes the normal quantisation diffusion something of a denegerate case. For sufficiently large , except when , the diffusion constant tends to the -transform diffusion constant.
At zero temperature, density and magnetic field the current correlator obtained from gauging the external source when the original theory contains a non-zero Chern-Simon’s number takes the form
We see that in the large limit the correlator is vanishing unless we rescale with . This scaling reproduces the two point function of the currents in normal quantisation with Chern-Simon’s level . In the numerics, at large , and , we expect our anyon correlators to have vanishing residues but a non-trivial pole structure. In particular, the large frequency AC conductivities should tend to zero for increasing .
To extract the AC conductivities of our theory we will need to numerically compute the Green’s function. The two point function is given in terms of the matrix by
If the reader would prefer components with as opposed to they need only replace the explicit factors of by . The AC conductivities are related to the mixed Green function by
where the transverse and longitudinal conductivities are equal due to rotation invariance. The DC conductivity is given by the limit of the above expression.
As a cross-check of the numeric results obtained from this expression, given the transformation formulae of appendix A, we can obtain the conductivities of the -transformed system from the original system. The blue “analytic” lines of fig. 8 are obtained via this transformation of the original system AC conductivities. The DC conductivities of the transformed system can be obtained analytically and they are