Abstract
The strong coupling dynamics of a 2+1 dimensional U(1) gauge theory coupled to charged matter is holographically modeled via a topdown construction with intersecting D3 and D5branes. We explore the resulting phase diagram at finite temperature and charge density using correlation functions of monopole operators, dual to magnetically charged particles in the higherdimensional bulk theory, as a diagnostic.
Monopole correlation functions and holographic phases of matter in 2+1 dimensions
T. Alho^{1}^{1}1 alho@hi.is, V. Giangreco M. Puletti^{2}^{2}2 vgmp@hi.is, R. Pourhasan^{3}^{3}3 razieh@hi.is, L. Thorlacius^{4}^{4}4 lth@hi.is,
University of Iceland, Science Institute, Dunhaga 3, 107 Reykjavik, Iceland The Oskar Klein Centre for Cosmoparticle Physics, Department of Physics, Stockholm University, AlbaNova University Centre, 10691 Stockholm, Sweden
Contents
1 Introduction
Gauge/gravity duality provides an interesting setting for the study of compressible quantum phases, where strongly correlated quantum dynamics is encoded into spacetime geometry in a gravitational dual description. The best understood cases all involve supersymmetric YangMills theories in a large limit, which are rather exotic from the point of view of manybody physics. Emergent gauge fields are known to arise in various quantum critical systems but these are almost exclusively fields and do not immediately lend themselves to a large treatment. Gauge/gravity duality does, however, offer a rare glimpse into strongly coupled dynamics in a setting where explicit computations are relatively straightforward and some aspects of the dynamics, in particular at finite temperature and density, may be generic to more general strongly coupled field theories.
Motivated by recent work of N. Iqbal [Iqbal:2014cga], we apply the formalism of gauge/gravity duality to map out the phase diagram of a 2+1dimensional many body system with a conserved current at finite temperature and charge density. We use correlation functions of suitably defined magnetic monopole operators to probe the relevant physics [Iqbal:2014cga, Faulkner:2012gt]. The fact that magnetic monopoles can strongly influence the infrared behavior of gauge theories is well known. For instance, the key role of monopoles in precipitating confinement in 2+1 dimensional gauge dynamics was emphasized in the pioneering work of Polyakov [Polyakov:1975rs, Polyakov:1976fu]. In a condensed matter context, monopoles provide an order parameter for the transition from antiferromagnetic order to valence bond solid in a gauge theory description of certain twodimensional lattice antiferromagnets [Read:1990zza, Read:1989aa, Murthy:1989ps]. The phase transition is continuous and described by a model with monopoles condensing at the critical point, which has motivated the computation of monopole correlation functions in the model in a expansion [Pufu:2013vpa, Pufu:2013eda, Dyer:2015zha].^{1}^{1}1See [Chester:2015wao] for a study of monopole operators by means of expansion.
In a 2+1dimensional gauge theory, a magnetic monopole operator, , corresponds to a localized defect where a magnetic flux is inserted. Such operators belong to a more general class of topological disorder operators [Borokhov:2002ib]. Their construction in terms of singular boundary conditions in a path integral formalism is outlined in [Iqbal:2014cga]. Due to flux quantization, monopole operators are intrinsically nonperturbative and difficult to handle using conventional field theory techniques. In holography, on the other hand, correlation functions of monopole operators have a straightforward geometric representation and can be numerically evaluated using relatively simple methods.
The holographic description of magnetic monopole operators, in terms of intersecting Dbranes, that we will be using was developed in [Iqbal:2014cga, Filev:2014mwa].^{2}^{2}2For other works related to holographic monopoles, see e.g. [Bolognesi:2010nb, Sutcliffe:2011sr, Rougemont:2015gia]. The starting point for the construction is a wellknown topdown model for a 2+1dimensional field theory living on the intersection of a single D5brane and a large number of coincident D3branes [DeWolfe:2001bt, Erdmenger:2002ed, Karch:2002sh, Skenderis:2002vf]. In this model, the D5brane is treated as a probe brane in the background geometry sourced by the D3branes. The embedding of the D5brane into the D3brane geometry is obtained by minimizing the DBI action of the D5brane in an background (we review the calculation in Section 2.1). There exists a solution where the D5brane wraps an S of fixed radius inside the S and extends along an subspace of the . This corresponds to a conformally invariant state in the dual 2+1dimensional boundary theory. There are other solutions where the D5brane embedding caps off at a finite radial coordinate, corresponding to a deformation away from criticality and a mass gap in the 2+1dimensional theory.
Open strings stretching between the D3 and D5branes give rise to matter fields in the fundamental representation of the SU() gauge group that are localised on the 2+1dimensional intersection. The boundary theory also has a conserved global current, which corresponds under gauge/gravity duality to a bulk gauge field living in . In general, a monopole operator inserted at the 2+1dimensional boundary corresponds to a bulk field carrying magnetic charge under the bulk gauge field [Sachdev:2012ks, Witten:2003ya]. In the topdown construction of [Iqbal:2014cga] the monopoles are realized as a probe D3brane, oriented in such a way as to appear as a onedimensional curve in , with the remaining worldvolume coordinates filling (at most) half an in and ending on the wrapped by the D5brane. A Dbrane ending on a Dbrane carries magnetic charge in the D worldvolume [Strominger:1995ac] and thus the probe D3brane represents a magnetically charged particle in . If the D3 curve reaches the boundary at a point , it corresponds to an insertion of a magnetic flux at that point, i.e. a boundary monopole operator. We review the construction in more detail in Section 2.2 and extend it to finite temperature backgrounds.
In the large limit, the twopoint function of boundary monopole operators is given by the onshell D3 action,
(1.1) 
The D3brane action consists of the usual DBI term and a magnetic coupling term. The DBI term is proportional to the length of the curve in traced out by the D3brane, in a metric that depends on the D5brane embedding, while the remaining term involves the integral of the magnetic dual of the worldvolume gauge field along the same curve. The magnetic coupling will play a key role when we consider backgrounds at finite charge density.
In a charge gapped phase monopoles are expected to condense at large enough separation, that is their equaltime twopoint function is expected to saturate with distance between the monopole insertion points [Sachdev:2012ks]. In a Fermiliquid phase (a compressible phase with nonzero charge density and no broken symmetries), field theory computations become rather involved due to the nonperturbative nature of the monopoles but [Lee:2008cl] predicted a faster than powerlaw falloff for the monopole equaltime twopoint function. This behaviour was indeed found in the holographic computation in [Iqbal:2014cga], which gave a constant value for the monopole correlation as a function of distance in a charge gapped case, and a Gaussian falloff at large separation in a compressible phase.
Our goal is to understand how turning on a nonzero temperature affects monopole correlation functions. In particular, we wish to determine whether the spatial dependence of the monopole equaltime twopoint function can still serve as an order parameter for phase transitions at finite . Our starting point is the holographic model employed in [Iqbal:2014cga], except now the D3brane background is an Schwarzschild black brane, and we investigate the behavior of monopole correlation functions across the rather rich phase diagram spanned by temperature and charge density. Similar questions can in principle be addressed in other holographic models, including various phenomenologically motivated bottomup models. It would be interesting to pursue this in future work but for now we will take advantage of the higherdimensional geometric perspective provided by the specific topdown construction of [Iqbal:2014cga].
The paper is organized as follows. In Section 2 we review the D3/D5brane construction at finite temperature. We then introduce a monopole D3brane and compute the D3brane action that gives the monopole twopoint function. In Section 3 we turn to the Fermiliquid phase at finite charge density. After briefly introducing the relevant background D3/D5brane solutions, we proceed to the D3brane action, and map out the corresponding phase diagram. We conclude with a brief discussion in Section 4. Our conventions and definitions of the action functionals governing the probe Dbrane dynamics studied in the paper are collected in Appendix A. This is followed in Appendix B by a short discussion of the boundary counterterms that are required for the regularisation of the D5brane free energy. A detailed examination of the onshell D3brane action at finite charge density and temperature is carried out in Appendix C and referred to in the main text. In Appendix D we consider asymptotic limits of model parameters, where analytic results can be obtained. This complements the numerical investigation in the rest of the paper and provides a useful check on the numerics.
2 Monopole correlators at finite temperature
2.1 Probe D5brane in a black 3brane background
Throughout the paper we consider probe Dbranes in the finite temperature nearhorizon geometry of D3branes,
where , , , and is a characteristic length scale. In these coordinates there is an event horizon at and the asymptotic AdS boundary is at . The Hawking temperature is
(2.2) 
and we note that a rescaling of can be absorbed by a rescaling of the coordinate. We work in the supergravity limit, so in particular at very large , and insert a probe D5brane with an embedding.^{3}^{3}3D3/D5 systems at finite temperature and at finite chemical potential have been widely employed in applied holography [Evans:2008nf, Filev:2009ai, Jensen:2010ga, Evans:2010hi, Grignani:2012jh]. For reviews see [CasalderreySolana:2011us, Erdmenger:2007cm]. In the static gauge the D5brane worldvolume coordinates are and , as indicated in Table 1. The D5brane profile is described by two functions, and , that, due to translation and rotation symmetries in the worldvolume directions, only depend on the radial coordinate . We set throughout and focus on the angle , which controls the size of the 2sphere wrapped by the probe D5brane.^{4}^{4}4In Section 3 we consider a D5brane carrying nonvanishing charge density. The ansatz remains consistent in this case as well, as long as the charge density is uniform. The stability of the configuration was studied in [Karch:2001cw]. The D5brane introduces matter fields in the fundamental representation of , charged under a global (baryon number), and localized in (2+1)dimensions. This is the field theory we have in mind throughout the paper.
D3branes (background)  
D5brane (probe) 
For numerical computations, we find it convenient to introduce a dimensionless radial coordinate as follows [Babington:2003vm]:
(2.3) 
The horizon is at and the background metric (2.1) becomes
(2.4) 
where and . The AdS boundary is at .
Writing , the induced metric on the D5brane is given by
(2.5) 
where the dot denotes a derivative with respect to . The Euclidean DBI action for the probe D5brane (see Appendix A) reduces to
(2.6) 
with the constant given in (A.10).
The field equation for ,
(2.7) 
can be solved numerically using standard methods. There are two classes of solutions with a nontrivial profile depending on whether the probe D5brane extends all the way to the horizon at or caps off outside the horizon. The former are referred to as “Black Hole Embedding" (BHE) solutions and the latter are socalled “Minkowski Embedding" (ME) solutions [Mateos:2006nu, Mateos:2007vn]. A oneparameter family of black hole embedding solutions, with , is obtained by numerically integrating the field equation (2.7) from outwards using the initial values , . The condition on comes from requiring the field equation to be nonsingular at the horizon.
For the Minkowski embedding solutions, the numerical evaluation is streamlined by a further change of variables. By viewing and as polar coordinates, the metric on may be rewritten as follows,
(2.8)  
with
(2.9) 
In the new coordinates the D5brane profile is described by a function and a Minkowski embedding solution caps off at . The field equation (2.7) becomes
(2.10) 
where prime denotes a derivative with respect to . We obtain a oneparameter family of Minkowski embedding solutions by integrating (2.10) using the initial values and . The initial condition on comes from requiring the field equation to be nonsingular at .
Figure 1 shows two D5brane profiles. One is a Minkowski embedding that ends at with , while the other is a black hole embedding that extends to the horizon at . From the figure it is clear that there exists a borderline solution that belongs to both embedding classes. It can either be viewed as a black hole embedding solution with that enters the horizon at a vanishing angle, or equivalently as a Minkowski embedding solution with that caps off at the horizon.
Each D5brane solution is characterised by two constants, and , that can be read off from the asymptotic behavior at the boundary,
(2.11)  
They represent the boundary mass and the condensate of flavour charged degrees of freedom in the dual field theory [Mateos:2006nu, Mateos:2007vn, Kobayashi:2006sb],
(2.12) 
Note that our definition of the boundary mass differs from that in [Iqbal:2014cga] by a factor of . By using (2.2) and (A.2), we see that the mass parameter read off from our numerical solutions is proportional to the scale invariant ratio of the boundary mass and the temperature,
(2.13) 
In the following, we will use as a measure of the (inverse) temperature at fixed . Note that the trivial constant profile, (), is a solution of the field equation (2.7) at any temperature and corresponds to .
In order to determine the thermodynamically stable D5brane solution at a given temperature, we compare the regularised free energy of the different solutions. We review the main points of the regularisation procedure worked out in [Karch:2005ms] in Appendix B and the resulting free energies are shown in Figure 2. The lowtemperature phase, i.e. low , corresponds to a Minkowski embedding (or “gapped”) solution, where the twosphere shrinks down at a finite distance away from the horizon () and the spectrum of “quarkantiquark” bound states has a mass gap [Mateos:2006nu, Mateos:2007vn, Albash:2006ew]. At high the D5tension can no longer balance the gravitational attraction of the background D3branes and the favored solution is a black hole embedding solution, dual to a gapless meson spectrum in the boundary field theory. There is a phase transition between the two types of embeddings. The righthand plot in Figure 2 zooms in on the region near the critical temperature and reveals the characteristic swallow tail of a firstorder transition. This is a universal feature of all DD systems [Mateos:2006nu, Mateos:2007vn].
2.2 Monopole twopoint function
In order to calculate the twopoint correlation function of monopole operators in the dual field theory we consider a probe D3brane on top of the background D3D5brane system. The probe D3brane ends on the D5brane and thus appears as a magnetically charged object in the D5 worldvolume [Strominger:1995ac]. Furthermore, the D3brane is embedded in in such a way that it wraps the same as the D5brane does and extends along a onedimensional curve in with endpoints at the boundary. This accounts for three out of four of the D3 worldvolume directions. The remaining worldvolume direction is transverse to the D5brane, along in the parametrisation (2.1), from to where the D3brane ends on the D5brane. The probe D3brane thus appears as a particle in and is magnetically charged under the D5 worldvolume gauge field , i.e. it models a bulk magnetic monopole [Iqbal:2014cga].
The D3brane fills part of the described by in (2.1). The fraction of the volume that is filled depends on the D5 embedding. At the boundary, the D5brane is at and the D3brane fills half of the . The curve connecting the insertion points on the boundary extends into the bulk, where the D5brane generically moves away from and the D3brane occupies a smaller fraction of the volume. In particular, if the curve extends to where the D5brane caps off in a Minkowski embedding then the volume of the D3brane shrinks to zero at that point.
Computing the twopoint boundary monopole correlation function in the large limit amounts to evaluating the corresponding onshell D3brane action as a function of the separation between the brane endpoints on the boundary. In Section 3 we present results from a numerical evaluation of the equal time twopoint monopole correlator at finite temperature and background charge density, generalising the zerotemperature results obtained in [Iqbal:2014cga]. We begin, however, with the simpler case of vanishing charge density at finite temperature.
We find it convenient to use D3brane worldvolume coordinates that match the coordinates we used for the D5brane embedding in Section 2.1. Here parametrises the curve traced out by the probe D3brane in the part of the background geometry (2.4). This curve is spacelike when we consider a twopoint function of monopole operators inserted at equal time on the boundary.^{5}^{5}5In this case the probe D3brane is strictly speaking a D3instanton. The variable is restricted to the range , where corresponds to the intersection between the D5 and D3brane worldvolumes.
In the charge neutral case, the action (A.20) for a probe D3brane only contains the DBI term. Upon integrating over the coordinates the DBI action reduces to that of a point particle,
(2.14) 
where is the pullback of the tendimensional spacetime metric to the D3brane worldvolume, a dot indicates a derivative with respect to , and is a position dependent mass given by
(2.15) 
We refer to the dimensionless quantity as the effective mass of the bulk monopole.^{6}^{6}6Note that our normalisation convention for differs from that of [Iqbal:2014cga] by a factor of . It follows that the dynamics of the probe D3brane depends on the embedding of the D5brane it ends on. In a Minkowski embedding shrinks to zero at the point where the D5 caps off, while in a black hole embedding remains nonzero all the way to the horizon. This is clearly visible in Figure 3(a), which shows as a function of position at different temperatures.
For the actual computation, it is convenient to absorb the effective mass into the induced metric as a conformal factor and define a rescaled metric [Iqbal:2014cga],
(2.16) 
The onshell D3brane action is then given by the length of a geodesic in the rescaled metric connecting the monopole operator insertion points at the boundary, which can without loss of generality be assumed to lie on the axis. The geodesic extends along and intersects the boundary at , . It has a turning point at , , where .
As shown in Appendix C, the D3brane action (2.14) can be reexpressed as
(2.17) 
where is the conserved charge associated with translation invariance along the spatial direction, and is the corresponding dimensionless variable,
The separation of the D3 endpoints at the boundary is given by
(2.18) 
which can in turn be expressed in terms of dimensionless quantities as
(2.19) 
In our numerical computation, is an input parameter and we evaluate both the D3brane action and the endpoint separation as a function of for a given D5brane embedding solution.
The location of the turning point, of a geodesic with depends on the D5brane embedding. The condition for having a turning point is
(2.20) 
or equivalently
(2.21) 
A real valued solution requires . In a Minkowski embedding, this condition is always satisfied for some value of on the D5brane because goes to zero as the D5brane caps off. In addition to geodesics with turning points, the Minkowski embedding supports a geodesic that extends “vertically" from the boundary to the point where the D5brane caps off, depending on the type of D5brane embedding. A pair of such vertical D3branes turns out to be the thermodynamically favoured configuration at sufficiently large endpoint separation when the probe D5brane is in a Minkowski embedding.
In a black hole embedding, on the other hand, the geodesic may reach the horizon at before the turning point condition (2.21) is satisfied. In this case, the geodesic instead turns around at the horizon. This is not immediately apparent in the coordinate, because the coordinate transformation (2.3) is degenerate at the horizon, but by going back to the original coordinate it is straightforward to show that the geodesic is quadratic in near the horizon,
(2.22) 
when , where is evaluated at the horizon . This behaviour is also apparent in numerical solutions of the geodesic equation at sufficiently low .


The D3brane action is in fact divergent for all the geodesic curves we have described. The divergence comes from the region near the boundary . We regularise it by introducing an upper cutoff at in (2.17) and subtracting the action of a geodesic in the limit,
(2.23) 
where depends on the D5brane embedding. For a black hole embedding it is at the horizon, , while for a Minkowski embedding it is where the D5brane caps off. In this case, we can use the coordinates introduced in (2.9) and set the lower limit of the radial variable in the integral to . By this convention, a disconnected configuration in Minkowski embedding, where two separate vertical D3branes extend from the boundary, has vanishing regularised action.
In Figure 3(b) we plot the regularised action,
(2.24) 
obtained by numerically evaluating the integrals in (2.17) and (2.23) for different values of the dimensionless parameter , against the endpoint separation obtained at the same value of . For a Minkowski embedding the system undergoes a first order phase transition, similar to the zerotemperature case studied in [Iqbal:2014cga], at a critical value of the endpoint separation, indicated by a dashed vertical line in the figure. At small values of the endpoint separation , the thermodynamically stable branch consists of connected solutions with large , for which the turning point is located far from the cap of the D5brane. Following this branch towards smaller , the turning point moves deeper into the bulk geometry while both the endpoint separation and the regularised free energy (2.24) increase. Eventually the free energy becomes positive and this branch is disfavoured compared to a disconnected branch with two separate vertical D3branes. The critical endpoint separation is indicated by a dashed vertical line in the figure.
Following the (now unstable) connected branch to smaller , the endpoint separation increases towards a local maximum, which marks the endpoint of this branch of solutions. Even smaller values of give rise to a third branch of solutions where the endpoint separation decreases from its local maximum and eventually approaches zero in the limit of vanishing . On this branch the turning point continues to move deeper into the bulk as is decreased and touches the cap of the D5brane precisely when . The small branch of solutions is always disfavoured compared to the other two branches. This is apparent from our numerical results but can also be seen by expanding the integrands in (2.17) and (2.19) in powers of for very low . The endpoint separation is a linear function of , while the regularised action is a quadratic function of , and thus of . It follows that the regularised action is always positive on this branch.^{7}^{7}7Such a low momentum expansion is explicitly carried out for the more general case with nonzero background charge density in Appendix D.2.2.
For a black hole embedding the story is rather different. In this case the turning point of a “hanging" geodesic can be either outside the black hole or exactly at the horizon . The two possibilities give rise to two branches of D3brane solutions and the branch that is thermodynamically stable at all values of the endpoint separation turns out to be the one where the turning point is outside the horizon.^{8}^{8}8In a black hole embedding there is no analog of the disconnected branch, with separate D3brane segments extending from the boundary to where the D5brane caps off, that dominates at large endpoint separation in a Minkowski embedding. If a D3brane were to end at the horizon its boundary would include an of finite area that is not inside the D5brane worldvolume. We thank N. Iqbal for pointing out an error in a previous version of the paper concerning this issue.
The right panel in Figure 3(b) shows our numerical results for the regularised D3brane action as a function of the endpoint separation in a black hole embedding. Following the stable branch towards larger one finds a critical value of the conserved charge for which the geodesic touches the horizon at the turning point. At the critical value of the term in (2.22) vanishes. The unstable branch corresponds to below the critical value and geodesics that turn around at the horizon.
Both the stable and unstable branches extend to infinite endpoint separation. To see this, consider the integrals in (2.17) and (2.19) precisely at the critical value . It is straightforward to establish that both integrals diverge logarithmically in this case, with the divergence coming from the low end of the integration, near the horizon. The divergence can, for instance, be regulated by introducing a cutoff at and in the limit of one easily finds that^{9}^{9}9A detailed analysis of these divergences, including the more general case at nonvanishing charge density, is presented in Appendix D.2.1.
(2.25) 
The monopole twopoint function (1.1) at high temperature is thus exponentially suppressed at large spatial separation with a characteristic length scale that scales inversely with temperature. This is the expected behaviour of a thermally screened system. At low temperatures, on the other hand, the D5brane is in a Minkowski embedding and the favoured D3brane configuration at large endpoint separation is a pair of disconnected vertical segments, for which the monopole twopoint function is a constant independent of . This signals the condensation of monopoles at low temperatures in this system. The critical endpoint separation, at which the disconnected configuration becomes dominant in the lowtemperature Minkowski embedding phase, has a weak temperature dependence shown in Figure 4.
3 Finite charge density phase
3.1 Thermodynamics of charged D5branes
By turning on a gauge field on the D5brane we can generalise the results of the previous section to study monopole correlation functions in a compressible Fermiliquid phase at finite charge density. We begin by giving a quick overview of the resulting charged D5brane thermodynamics before turning our attention to the monopole correlators. Our discussion of D5brane thermodynamics parallels that of [Kobayashi:2006sb], which considered charged D7branes in a D3brane background.
The Euclidean action for D5brane at finite charge density is worked out in Appendix A. For convenience, we repeat the final expression (A.9) here,
(3.1) 
The induced metric on the D5brane is still parametrised as in (2.5) and the finite charge density enters via the dimensionless parameter in the action (see Appendix A for details). The field equation for , obtained by varying (3.1), can be solved numerically using the same methods as employed for the chargeneutral case in Section 2.1. In the absence of explicit bulk sources, electric field lines emanating from the boundary have nowhere to end if the D5brane caps off before the horizon [Kobayashi:2006sb]. The Minkowski embedding solutions are therefore unphysical at finite charge density and the only consistent solutions are black hole embeddings.
The relevant variables when it comes to the physical interpretation and presentation of our results are the temperature and the charge density in the boundary theory.^{10}^{10}10The physical charge density is related to the temperature and the dimensionless parameter appearing in the action through (A.6). We fix the overall scale by working at a fixed boundary mass and express our results in terms of dimensionless combinations,
(3.2) 
The phase diagram of the model is mapped out by separately varying and . In particular, if we keep fixed and consider very high temperature we expect thermal effects to swamp any effect of the charge density while in the limit of low temperature the finite charge density should dominate. This is readily apparent in our numerical results, but we also demonstrate it explicitly by considering the different asymptotic limits of parameters in Appendix D.
Numerical solutions for are obtained by integrating the field equation outwards from the horizon, with the charge density and the boundary value at the horizon as dimensionless input parameters. For given values of the input parameters in the range and , the inverse temperature can be read off from the asymptotic behaviour of the numerical solution as in (2.11). The charge density is then easily determined using the relation . This procedure uniquely determines the physical variables and as functions of the numerical input parameters and . The inverse mapping is not single valued, however, and this leads to phase transitions as was already seen in the zerocharge case in Section 2.1. The constant solution is also present and can be viewed as the hightemperature limit of a black hole embedding, as is apparent in Figure 5.

In order to decipher the phase diagram, we compare the onshell free energy density (3.1) on different branches of solutions. The UV divergence encountered as the D5brane approaches the AdS boundary is regulated by introducing boundary counterterms, as outlined in Appendix B. Numerical results for the regularised D5brane free energy are shown in Figure 6(a) for different values of . At low charge densities, , we find a first order phase transition between two branches of black hole embedding solutions. The left and righthand panels in Figure 6(b) showcase the different behaviour of the D5brane free energy at and , respectively. Figure 6(c) plots against the critical temperature of the phase transition and shows how the critical line in the plane terminates at
The low phase transition connects to the phase transition between the black hole embedding and Minkowski embedding solutions at zero charge density. We note, in particular, that as the critical temperature of the phase transition between the different black hole embedding branches approaches , which is the critical temperature found in Section 2.1 at zero charge density. Furthermore, the stable black hole embedding solution at low temperature and low charge density approaches a Minkowski embedding solution. It almost caps off at a finite radial distance outside the black hole, leaving a narrow throat that extends all the way to the horizon to accommodate the electric field lines emanating from the black hole. In the limit the throat degenerates and the solution takes the form of a Minkowski embedding solution. The onset of this lowtemperature behaviour can be seen on the left in Figure 5 even if the D5brane profiles in the figure are for a value somewhat above .
A similar phase diagram, involving charged D7branes in the finite temperature background of a black 3brane, was worked out in [Kobayashi:2006sb]. There it was argued that the favoured lowtemperature configuration at charge densities below the analog of in the D7brane system may in fact be unstable. In the present paper we are mainly concerned with evaluating twopoint correlation functions of monopole operators in the presence of the probe D5brane and how they depend on the transverse spatial separation between the monopole insertions on the boundary. As it turns out, we can determine this spatial dependence without having to rely on D5brane solutions at very low . In what follows, we therefore restrict our attention to charged D5branes with , where there is only one branch of solutions and the question of an instability, analogous to the one discussed in [Kobayashi:2006sb], does not arise.
3.2 Monopole twopoint function at finite charge density
We now proceed to compute the action of a probe D3brane ending on the charged D5brane, which, under holographic duality, determines the twopoint correlation function of monopole operators in a compressible finite charge density phase of the 2+1dimensional defect field theory [Iqbal:2014cga]. The calculation is a straightforward generalisation from the chargeneutral case that was presented in Section 2.2. The main new ingredient is the magnetic coupling between the probe D3brane and the nonvanishing gauge field on the D5brane worldvolume. This means that the second term in the D3brane action (A.20) in Appendix A comes into play and the spacelike curve traced out by the probe D3brane in the part of the bulk geometry is no longer a geodesic in the rescaled metric (2.16). The endpoints at the boundary can still be taken to be at , , , but at intermediate points the curve extends in the direction and lies along . We refer the reader to Appendix C for the derivation of the shape of and the regularised onshell action of the probe D3brane at finite charge density. The main focus of the present section will instead be on presenting our numerical results and exploring the behaviour of the monopole equaltime twopoint function as a function of spatial separation at different temperatures and charge densities.
The Euclidean action of a D3brane ending on a charged D5brane is worked out in Appendix C below and is given by
(3.3) 
where
(3.4) 
and at the endpoints of at the boundary. The frequency , defined in (C.3), is the analog of a cyclotron frequency for a magnetic monopole in a background electric field. The curve has a turning point at at the radial coordinate . The dimensionless constant of integration is a measure of transverse momentum in the direction. It plays the same role as the parameter in Section 2.2 and it is straightforward to see that as (see Appendix C for details). In fact, the turning point analysis for a D3brane ending on a D5brane with a black hole embedding goes through unchanged, with replaced by . When , the curve turns around at some outside the horizon and returns to the boundary. On the other hand, for any , the curve turns around at the horizon.
The first term in (3.3) comes from the geometric DBIaction of the D3brane and reduces to (2.17) for the uncharged case. The second term, which arises from the magnetic coupling between the probe D3 and D5branes, vanishes in the limit. A regularised D3brane action is obtained as before, by subtracting the action (2.23) of a curve. This cancels the divergence coming from the near boundary region .
In order to determine how the monopole twopoint function behaves as a function of the endpoint separation, we plot the result of a numerical evaluation of the regularised action against for different values of at fixed temperature and charge density. In Appendix C we obtain the following expression for the endpoint separation in terms of dimensionless input parameters,
(3.5) 
which can easily be evaluated numerically.

The graphs in figure 7 show our results for the regularised D3brane action as a function of for several temperatures at two values of the charge density: in Figure 7(a) and in Figure 7(b). At low charge density () and low temperature ( at ) we see evidence for a firstorder transition between two D3branes that have different values of . Both have and thus the turning point is outside the horizon on both branches.^{11}^{11}11As in the chargeneutral case, there is also a branch with and a turning point on the horizon itself but this branch is never the most stable one. It is indicated with dashed lines in some of the graphs in Figure 7 but is left out of the others to avoid clutter. In the limit of vanishing charge density, this transition reduces to the transition between connected and disconnected D3brane configurations that we saw for D5branes in Minkowski embedding in the chargeneutral case in Section 2.2. This is evident in Figure 8, which plots the critical endpoint separation as a function of temperature at low charge density () and compares it to that of the connecteddisconnected transition.
The graphs showing the regularised action of a connected D3brane in Figure 7 share a common feature in that they all become convex at sufficiently large . This can be traced to the magnetic term being dominant over the geometric DBI term in (3.3) at large endpoint separation while the DBI term governs the shortdistance behaviour. A detailed analysis carried out in Appendix D.2.1 shows that the regularised D3brane action always depends quadratically on in the “magnetic regime" at sufficiently large . This, in turn, leads to Gaussian suppression of the equaltime twopoint correlation function of monopole operators as a function of the distance between the operator insertion points on the boundary.
The significance of the Gaussian suppression depends, however, on the value of the charge density in relation to the temperature. At high or low the Gaussian behaviour sets in at relatively short distances, as can for instance be seen in Figure 7(b). This matches the zero temperature results of [Iqbal:2014cga] where the monopole twopoint function was found to be Gaussian suppressed with distance at finite charge density. At low or high , on the other hand, the Gaussian behaviour only sets in at such long distances that the monopole twopoint function is already vanishingly small due to the exponential suppression from thermal screening discussed at the end of Section 2.2.
Our results for the monopole twopoint function at finite and thus interpolate nicely between Gaussian suppression obtained when the charge density dominates over the temperature and exponential thermal screening found at low charge density.
4 Discussion
We have computed equaltime twopoint correlation functions of magnetic monopole operators in a strongly coupled 2+1dimensional gauge theory and studied their spatial dependence. This provides information about the phase structure of the theory, including possible monopole condensation in a phase with a charge gap, and also allows us to probe a compressible phase at finite charge density. Our investigation employs a topdown holographic construction involving intersecting D5 and D3branes in spacetime, originally developed by Iqbal in [Iqbal:2014cga], and extends it to an black hole background in order to include thermal effects and explore monopole correlators across the phase diagram.
In Section 2 we focused on thermal effects on monopole correlators in a holographic phase with a charge gap at zero temperature. The analysis performed in [Iqbal:2014cga] showed that in this phase the holographic monopole twopoint function at zero temperature saturates to a constant value as the separation between the operator insertions is increased. This is the expected behaviour when the monopole operator has condensed. On the gravitational side of the holographic duality the condensation is attributed to the vanishing of the bulk monopole effective mass where the D5brane, on which the D3brane representing the bulk monopole ends, caps off in a Minkowski embedding. We find that the saturation of the monopole twopoint function at long distances persists at finite temperature, up to the critical temperature where the D5brane makes a transition from a Minkowski to a black hole embedding. Above the critical temperature, however, the monopole operator is no longer condensed and the twopoint function is exponentially suppressed at long distances due to thermal screening.
In Section 3 we turned our attention to thermal effects in a compressible phase in the presence of a nonzero charge density implemented by introducing a gauge field on the D5brane worldvolume. On the one hand, the finite charge density forces the D5brane into a black hole embedding at any nonzero temperature and on the other hand it gives rise to a direct coupling between the D3brane and the magnetic dual of the worldvolume gauge field on the D5brane. Studying the monopole correlation function at finite temperature and charge density, we find that the transition to a disconnected D3brane configuration at large separation found at vanishing charge density is replaced by a transition between different connected D3brane configurations at low, but nonvanishing, charge densities and relatively low temperature.
We also observe effects of the interplay at finite temperature and charge density between the magnetic coupling and the geometric DBIterm in the D3brane action. On the dual field theory side, this shows up in the dependence of the monopole operator twopoint function on spatial distance. In particular, at finite charge density and low temperature the magnetic coupling contribution is the dominant one and the twopoint function has a Gaussian falloff with distance. This is in line with the zerotemperature findings of [Iqbal:2014cga]. At high temperature and low charge density, on the other hand, the monopole twopoint function exhibits exponential falloff due to thermal screening, but eventually crosses over to Gaussian suppression at very long distances.
In this work we restricted our attention to the combined effects of finite charge density and temperature on twopoint functions of monopole operators. An interesting and rather straightforward extension would be to add a nonzero magnetic field, for instance along the lines of [Evans:2008nf, Filev:2009ai, Jensen:2010ga]. Another future direction would be to relax some of the constraints that are built into the particular topdown holographic model used here in order to explore more general bulk monopole embeddings and the corresponding phase diagrams. Our treatment involved a probe D3brane ending on a probe D5brane in an background. An important question is whether including the backreaction of the D3/D5brane system on the background geometry would stabilise or wash out the features we have found. Modelling the backreaction may require the added flexibility of a bottomup approach, while retaining essential features of the topdown Dbrane construction. At the same time, the study of monopole dynamics in phenomenologically motivated bottomup models would be of considerable interest in its own right. Finally, there are other intersecting Dbrane systems which can be used to model monopole operators in strongly coupled gauge theory. A D5/D7brane model might for instance be a more natural setting to study nonAbelian monopole correlators.
Acknowledgements
We thank N. Evans, G. Grignani, N. Iqbal, N. Jokela, M. Lippert, R. Myers, A. Peet, S. Ross, and T. Zingg for helpful discussions. We thank J. Á. Thorbjarnarson and P. R. Bryde for useful comments. R.P. thanks the Perimeter Institute for Theoretical Physics for hospitality during the completion of this work. This research was supported in part by the Icelandic Research Fund under contracts 163419051 and 163422051, the Swedish Research Council under contract 62120145838, and by grants from the University of Iceland Research Fund.
Appendix A Action functionals for probe Dbranes
In this Appendix we collect some formulae and expressions which are used in the main body of the paper and in later appendices. We work out the explicit form of the D5brane action in the coordinate system used in the main text. This is standard material but is included here in order to have a selfcontained presentation. We also obtain an explicit expression for the probe D3brane action of a bulk monopole proposed by Iqbal in [Iqbal:2014cga] in a black 3brane background.
The tension of a Dbrane is given by
(A.1) 
where is the string coupling constant, and the string length, which in turn are related to the ’t Hooft coupling constant , the AdS radius , and the number of background D3branes as follows [Maldacena:1997re],
(A.2) 
a.1 D5brane
The worldvolume action for the probe D5brane is
(A.3) 
where denotes the RamondRamond 4form field sourced by the background D3branes, the 2form field strength of the D5brane worldvolume gauge field, and the induced D5 worldvolume metric. The first term in (A.3) vanishes for the D5brane configuration investigated in this work. To see this, we choose a gauge where the 4form is a sum of two terms, one proportional to the volume form on and the other to the volume form on the product of the two factors inside . The former gives zero when wedged with the 2form while the latter has vanishing pullback to the D5brane worldvolume.
In order to evaluate the remaining DBI term in (A.3) we take to be the induced metric in static gauge (2.5) and parametrise the U(1) gauge potential on the D5brane worldvolume as follows,
(A.4) 
After some straightforward algebra the D5brane action reduces to
(A.5) 
where is the number of background D3branes, is the (infinite) volume from the integration over the variables, is the temperature of the background (2.2), and the dot denotes a derivative with respect to . Since the action (A.5) depends only on the derivative of the gauge potential , it is convenient to introduce a charge density,
(A.6) 
with
(A.7) 
The equation of motion of the gauge field implies radial conservation of the charge density,
(A.8) 
We can take advantage of this by performing a Legendre transform on (A.5) that trades the gauge potential for as the independent field variable. This leads to an action functional for that includes the conserved charge density as a parameter,
(A.9) 
In order to study D5brane thermodynamics, we have changed to Euclidean signature and taken Euclidean time to be periodic with period . The temperature dependence of the constant in front of the action is left explicit and
(A.10) 
with the transverse area coming from the integral over and .
The free energy of a D5brane at charge density is given by the onshell value of the Euclidean action (A.9). The boundary counterterms needed to regularise the free energy are discussed in Appendix B and numerical results for the resulting regularised onshell action are presented in Section 3.1. Switching off the charge density gives the free energy of a charge neutral D5brane,
(A.11) 
considered in Section (2.1).
a.2 D3brane
The action for the probe D3brane is
(A.12) 
where is the field strength of the D3brane worldvolume gauge field and the induced D3brane worldvolume metric. Gauge invariance of (A.3) and (A.12) with respect to the gauge transformation,
(A.13) 
requires the presence of additional terms,
(A.14) 
involving a 3form Lagrange multiplier that transforms as follows under the gauge transformation (A.13),
(A.15) 
Note that this fixes the value of the coupling constant to be
(A.16) 
The 3form provides the magnetic coupling between the gauge field living on the D5brane and the edge of the D3brane. Indeed, adopting the same ansatz as in [Iqbal:2014cga],
(A.17) 
with the volume form on the twodimensional unit sphere that the D5 and D3branes wrap around inside , the threedimensional term in (A.14) reduces to an integral,
(A.18) 
along the curve in traced out by the probe D3brane.
The field strength of is the magnetic dual of the field strength of the D5brane worldvolume gauge field [Iqbal:2014cga]. To see this, we vary the full action with respect to the field strength of . Only two terms contribute, i.e. the DBIterm in (A.3) and the D5brane worldvolume term in (A.14). Using the definitions (A.6) and (A.17), we obtain the following rather simple result,
(A.19) 
which is the magnetic dual of the radial electric field sourced by the charge density . It follows that we can choose a gauge where .
Finally, we collect the terms that contribute to the bulk monopole dynamics. Due to the specific D3brane embedding employed in our analysis, the first term in (A.12) containing the RamondRamond potential vanishes and the remaining DBI term simplifies because there is no gauge field on the D3brane worldvolume. As explained in Section 2.2, for equaltime correlation functions the curve spanned in by the probe D3brane is spacelike and therefore the induced metric has Euclidean signature. The relevant terms in the Euclidean D3brane action are thus [Iqbal:2014cga]
(A.20) 
In the chargeneutral case, considered in Section 2.2, the second term is absent and the DBI term reduces to the action for a point particle (2.14) with a mass that depends on the radial position in . In this case, the onshell D3brane action is simply given by the length of a geodesic in a rescaled metric where the position dependent mass has been absorbed as a conformal factor, as discussed in Section 2.2. At finite charge density, on the other hand, the magnetic term in (A.20) is nonvanishing and the curve will no longer be a geodesic in the rescaled metric. This case is considered in detail in Appendix C.
Appendix B Boundary counterterms for D5brane
In the main text, we encountered several branches of D5brane solutions. When two or more different solutions exist for the same values of physical parameters it is important to identify which solution is thermodynamically stable. For this, we need to evaluate the free energy given by the onshell Euclidean action of the D5brane and compare between different branches of solutions. In the chargeneutral case this involves a comparison between Minkowski and black hole embedding solutions, while at finite charge density we compare the free energy of different branches of black hole embedding solutions.
As it stands, the D5brane free energy (A.9) is UVdivergent and needs to be regularised by introducing appropriate counterterms at the boundary. We use a wellestablished regularisation procedure for general DD systems described by DBI actions [Karch:2005ms] and specialise to the system at hand. We take the UV cutoff surface to be at constant radial coordinate and find that the following counterterm action will cancel the UVdivergence of the bulk D5brane action,
(B.1) 
where are boundary coordinates and the induced metric at . A finite free energy is obtained by cutting off the integral at in (A.9), or in (A.11) in the chargeneutral case, and evaluating the sum
(B.2) 
before taking the limit. The ellipsis in (B.1) denotes subleading terms that give a vanishing contribution in the limit. We note that the presence of a gauge field on the D5brane worldvolume does not require any additional boundary counterterms compared to the chargeneutral case.
The regularised free energy can now be calculated numerically as a function of temperature for both black hole and Minkowski embeddings.^{12}^{12}12For efficient numerical evaluation of the free energy of a Minkowski embedding solution, it is convenient to change to the variables introduced in Section 2.1. Results are shown in Figure 2 for the chargeneutral case and in Figure 6 for D5branes at finite charge density.
Appendix C Onshell D3brane action
In this Appendix we generalise the discussion of monopole twopoint functions in Section 2.2 to finite charge density. We obtain integral expressions for the onshell D3brane action and for the endpoint separation on the boundary, in terms of dimensionless parameters that characterise the background charge density and the curve in that connects the two endpoints. Results from the numerical evaluation of these expressions are presented and discussed in Section 3.2 of the main text.