Contents
Abstract

The strong coupling dynamics of a 2+1 dimensional U(1) gauge theory coupled to charged matter is holographically modeled via a top-down construction with intersecting D3- and D5-branes. 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 higher-dimensional bulk theory, as a diagnostic.

Monopole correlation functions and holographic phases of matter in 2+1 dimensions


T. Alho111 alho@hi.is, V. Giangreco M. Puletti222 vgmp@hi.is, R. Pourhasan333 razieh@hi.is, L. Thorlacius444 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

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 Yang-Mills theories in a large limit, which are rather exotic from the point of view of many-body 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+1-dimensional 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 anti-ferromagnetic order to valence bond solid in a gauge theory description of certain two-dimensional lattice anti-ferromagnets [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].111See [Chester:2015wao] for a study of monopole operators by means of expansion.

In a 2+1-dimensional 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 non-perturbative 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 D-branes, that we will be using was developed in [Iqbal:2014cga, Filev:2014mwa].222For other works related to holographic monopoles, see e.g. [Bolognesi:2010nb, Sutcliffe:2011sr, Rougemont:2015gia]. The starting point for the construction is a well-known top-down model for a 2+1-dimensional field theory living on the intersection of a single D5-brane and a large number of coincident D3-branes [DeWolfe:2001bt, Erdmenger:2002ed, Karch:2002sh, Skenderis:2002vf]. In this model, the D5-brane is treated as a probe brane in the background geometry sourced by the D3-branes. The embedding of the D5-brane into the D3-brane geometry is obtained by minimizing the DBI action of the D5-brane in an background (we review the calculation in Section 2.1). There exists a solution where the D5-brane 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+1-dimensional boundary theory. There are other solutions where the D5-brane embedding caps off at a finite radial coordinate, corresponding to a deformation away from criticality and a mass gap in the 2+1-dimensional theory.

Open strings stretching between the D3- and D5-branes give rise to matter fields in the fundamental representation of the SU() gauge group that are localised on the 2+1-dimensional 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+1-dimensional boundary corresponds to a bulk field carrying magnetic charge under the bulk gauge field [Sachdev:2012ks, Witten:2003ya]. In the top-down construction of [Iqbal:2014cga] the monopoles are realized as a probe D3-brane, oriented in such a way as to appear as a one-dimensional curve in , with the remaining world-volume coordinates filling (at most) half an in and ending on the wrapped by the D5-brane. A D-brane ending on a D-brane carries magnetic charge in the D world-volume [Strominger:1995ac] and thus the probe D3-brane 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 two-point function of boundary monopole operators is given by the on-shell D3 action,

(1.1)

The D3-brane 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 D3-brane, in a metric that depends on the D5-brane embedding, while the remaining term involves the integral of the magnetic dual of the world-volume 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 equal-time two-point function is expected to saturate with distance between the monopole insertion points [Sachdev:2012ks]. In a Fermi-liquid phase (a compressible phase with non-zero charge density and no broken symmetries), field theory computations become rather involved due to the non-perturbative nature of the monopoles but [Lee:2008cl] predicted a faster than power-law falloff for the monopole equal-time two-point 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 non-zero temperature affects monopole correlation functions. In particular, we wish to determine whether the spatial dependence of the monopole equal-time two-point 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 D3-brane 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 bottom-up models. It would be interesting to pursue this in future work but for now we will take advantage of the higher-dimensional geometric perspective provided by the specific top-down construction of [Iqbal:2014cga].

The paper is organized as follows. In Section 2 we review the D3/D5-brane construction at finite temperature. We then introduce a monopole D3-brane and compute the D3-brane action that gives the monopole two-point function. In Section 3 we turn to the Fermi-liquid phase at finite charge density. After briefly introducing the relevant background D3/D5-brane solutions, we proceed to the D3-brane 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 D-brane dynamics studied in the paper are collected in Appendix A. This is followed in Appendix B by a short discussion of the boundary counter-terms that are required for the regularisation of the D5-brane free energy. A detailed examination of the on-shell D3-brane 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 D5-brane in a black 3-brane background

Throughout the paper we consider probe D-branes in the finite temperature near-horizon geometry of D3-branes,

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 D5-brane with an -embedding.333D3/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 D5-brane world-volume coordinates are and , as indicated in Table 1. The D5-brane profile is described by two functions, and , that, due to translation and rotation symmetries in the world-volume directions, only depend on the radial coordinate . We set throughout and focus on the angle , which controls the size of the 2-sphere wrapped by the probe D5-brane.444In Section 3 we consider a D5-brane carrying non-vanishing 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 D5-brane 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.

D3-branes (background)
D5-brane (probe)
Table 1: Background D-brane construction.

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 D5-brane is given by

(2.5)

where the dot denotes a derivative with respect to . The Euclidean DBI action for the probe D5-brane (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 non-trivial profile depending on whether the probe D5-brane 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 so-called “Minkowski Embedding" (ME) solutions [Mateos:2006nu, Mateos:2007vn]. A one-parameter 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 non-singular 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 D5-brane 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 one-parameter 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 non-singular at .

Figure 1 shows two D5-brane 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 D5-brane 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 .

Figure 1: The embedding angle for the “Minkowski embedding” solution (dashed green curve) and the “black-hole embedding” solution (solid blue curve). The dashed black curve is the horizon at .

In order to determine the thermodynamically stable D5-brane 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 low-temperature phase, i.e. low , corresponds to a Minkowski embedding (or “gapped”) solution, where the two-sphere shrinks down at a finite distance away from the horizon () and the spectrum of “quark-antiquark” bound states has a mass gap [Mateos:2006nu, Mateos:2007vn, Albash:2006ew]. At high the D5-tension can no longer balance the gravitational attraction of the background D3-branes 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 right-hand plot in Figure 2 zooms in on the region near the critical temperature and reveals the characteristic swallow tail of a first-order transition. This is a universal feature of all D-D systems [Mateos:2006nu, Mateos:2007vn].

Figure 2: Regularised free energy for a D5-brane in D3-branes background. The dashed green (solid blue) curves correspond to Minkowski (black hole) embeddings. The figure on the right shows an expanded view of the region near the phase transition. The dotted vertical red line indicates the critical temperature .

The free energy shown in Figure 2 is obtained from numerical solutions of the field equations (2.7) and (2.10) at different temperatures. In Appendix D we consider its high and low temperature limits, where the field equations simplify and analytic results can be obtained.

2.2 Monopole two-point function

In order to calculate the two-point correlation function of monopole operators in the dual field theory we consider a probe D3-brane on top of the background D3-D5-brane system. The probe D3-brane ends on the D5-brane and thus appears as a magnetically charged object in the D5 worldvolume [Strominger:1995ac]. Furthermore, the D3-brane is embedded in in such a way that it wraps the same as the D5-brane does and extends along a one-dimensional 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 D5-brane, along in the parametrisation (2.1), from to where the D3-brane ends on the D5-brane. The probe D3-brane thus appears as a particle in and is magnetically charged under the D5 world-volume gauge field , i.e. it models a bulk magnetic monopole [Iqbal:2014cga].

The D3-brane 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 D5-brane is at and the D3-brane fills half of the . The curve connecting the insertion points on the boundary extends into the bulk, where the D5-brane generically moves away from and the D3-brane occupies a smaller fraction of the volume. In particular, if the curve extends to where the D5-brane caps off in a Minkowski embedding then the volume of the D3-brane shrinks to zero at that point.

Computing the two-point boundary monopole correlation function in the large -limit amounts to evaluating the corresponding on-shell D3-brane 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 two-point monopole correlator at finite temperature and background charge density, generalising the zero-temperature 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 D3-brane world-volume coordinates that match the coordinates we used for the D5-brane embedding in Section 2.1. Here parametrises the curve traced out by the probe D3-brane in the part of the background geometry (2.4). This curve is spacelike when we consider a two-point function of monopole operators inserted at equal time on the boundary.555In this case the probe D3-brane is strictly speaking a D3-instanton. The variable is restricted to the range , where corresponds to the intersection between the D5- and D3-brane worldvolumes.

In the charge neutral case, the action (A.20) for a probe D3-brane 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 pull-back of the ten-dimensional spacetime metric to the D3-brane world-volume, 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.666Note that our normalisation convention for differs from that of [Iqbal:2014cga] by a factor of . It follows that the dynamics of the probe D3-brane depends on the embedding of the D5-brane 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 non-zero 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 on-shell D3-brane 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 D3-brane action (2.14) can be re-expressed 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 D3-brane action and the endpoint separation as a function of for a given D5-brane embedding solution.

The location of the turning point, of a geodesic with depends on the D5-brane 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 D5-brane because goes to zero as the D5-brane 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 D5-brane caps off, depending on the type of D5-brane embedding. A pair of such vertical D3-branes turns out to be the thermodynamically favoured configuration at sufficiently large endpoint separation when the probe D5-brane 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 .

(a) Bulk monopole effective mass
(b) Regularised D3-brane action
Figure 3: (a) Bulk monopole effective mass for Minkowski (black hole) embeddings on the left (right) at different temperatures; (b) Regularised D3-brane action versus for Minkowski (black hole) embeddings on the left (right) at . The dashed vertical line in the left panel indicates the critical endpoint separation marking the transition from a single “hanging" geodesic to two disconnected vertical geodesics in a Minkowski embedding.

The D3-brane 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 cut-off at in (2.17) and subtracting the action of a geodesic in the limit,

(2.23)

where depends on the D5-brane embedding. For a black hole embedding it is at the horizon, , while for a Minkowski embedding it is where the D5-brane 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 D3-branes 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 zero-temperature 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 D5-brane. 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 D3-branes. 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 D5-brane 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.777Such a low momentum expansion is explicitly carried out for the more general case with non-zero background charge density in Appendix D.2.2.

Figure 4: Temperature dependence of , the critical separation between D3-brane endpoints in a Minkowski embedding, beyond which the thermodynamically favoured configuration consists of two separate segments extending vertically to where the D5-brane caps off. Above the critical temperature (indicated by the dotted vertical line) the D5 brane is in a black hole embedding where there is no disconnected D3-brane configuration.

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 D3-brane 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.888In a black hole embedding there is no analog of the disconnected branch, with separate D3-brane segments extending from the boundary to where the D5-brane caps off, that dominates at large endpoint separation in a Minkowski embedding. If a D3-brane were to end at the horizon its boundary would include an of finite area that is not inside the D5-brane world-volume. 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 D3-brane 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 that999A detailed analysis of these divergences, including the more general case at non-vanishing charge density, is presented in Appendix D.2.1.

(2.25)

The monopole two-point 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 D5-brane is in a Minkowski embedding and the favoured D3-brane configuration at large endpoint separation is a pair of disconnected vertical segments, for which the monopole two-point 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 low-temperature Minkowski embedding phase, has a weak temperature dependence shown in Figure 4.

3 Finite charge density phase

3.1 Thermodynamics of charged D5-branes

By turning on a gauge field on the D5-brane we can generalise the results of the previous section to study monopole correlation functions in a compressible Fermi-liquid phase at finite charge density. We begin by giving a quick overview of the resulting charged D5-brane thermodynamics before turning our attention to the monopole correlators. Our discussion of D5-brane thermodynamics parallels that of [Kobayashi:2006sb], which considered charged D7-branes in a D3-brane background.

The Euclidean action for D5-brane 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 D5-brane 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 charge-neutral 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 D5-brane 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.

(a) Low temperature
(b) High temperature
Figure 5: D5-brane solutions at constant charge density for different temperatures. The red line represents constant embedding while the blue lines describe the BH embeddings for a) and b) from top to bottom. The dashed black line indicates the horizon at .

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.101010The 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 zero-charge case in Section 2.1. The constant solution is also present and can be viewed as the high-temperature limit of a black hole embedding, as is apparent in Figure 5.

Figure 6: (a) Regularised free energy of charged D5-branes with (from left to right). (b) Left panel: A close-up of the phase transition at low charge density (). Right panel: Phase transition is absent at high charge density (). (c) Charge density vs. critical temperature of phase transition at .

In order to decipher the phase diagram, we compare the on-shell free energy density (3.1) on different branches of solutions. The UV divergence encountered as the D5-brane approaches the AdS boundary is regulated by introducing boundary counter-terms, as outlined in Appendix B. Numerical results for the regularised D5-brane 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 right-hand panels in Figure 6(b) showcase the different behaviour of the D5-brane 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 low-temperature behaviour can be seen on the left in Figure 5 even if the D5-brane profiles in the figure are for a value somewhat above .

A similar phase diagram, involving charged D7-branes in the finite temperature background of a black 3-brane, was worked out in [Kobayashi:2006sb]. There it was argued that the favoured low-temperature configuration at charge densities below the analog of in the D7-brane system may in fact be unstable. In the present paper we are mainly concerned with evaluating two-point correlation functions of monopole operators in the presence of the probe D5-brane 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 D5-brane solutions at very low . In what follows, we therefore restrict our attention to charged D5-branes 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 two-point function at finite charge density

We now proceed to compute the action of a probe D3-brane ending on the charged D5-brane, which, under holographic duality, determines the two-point correlation function of monopole operators in a compressible finite charge density phase of the 2+1-dimensional defect field theory [Iqbal:2014cga]. The calculation is a straightforward generalisation from the charge-neutral case that was presented in Section 2.2. The main new ingredient is the magnetic coupling between the probe D3-brane and the non-vanishing gauge field on the D5-brane world-volume. This means that the second term in the D3-brane action (A.20) in Appendix A comes into play and the spacelike curve traced out by the probe D3-brane 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 on-shell action of the probe D3-brane 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 equal-time two-point function as a function of spatial separation at different temperatures and charge densities.

The Euclidean action of a D3-brane ending on a charged D5-brane 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 D3-brane ending on a D5-brane 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 DBI-action of the D3-brane and reduces to (2.17) for the uncharged case. The second term, which arises from the magnetic coupling between the probe D3- and D5-branes, vanishes in the limit. A regularised D3-brane 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 two-point 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.

(a)
(b)
Figure 7: Euclidean D3-brane action against at (a) and (b) . Each curve corresponds to a given temperature as indicated. The unstable branch is indicated by dashed lines in the middle and bottom right plots, but omitted from the others to avoid clutter.

The graphs in figure 7 show our results for the regularised D3-brane 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 first-order transition between two D3-branes that have different values of . Both have and thus the turning point is outside the horizon on both branches.111111As in the charge-neutral 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 D3-brane configurations that we saw for D5-branes in Minkowski embedding in the charge-neutral 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 connected-disconnected transition.

Figure 8: The dashed red curve shows the critical endpoint separation for the first-order transition between two branches of solutions, at as a function of temperature. The dashed green curve (repeated from Figure 4 for reference) shows the endpoint separation at which the disconnected branch of D3-brane solutions becomes thermodynamically favoured in a Minkowski embedding at .

The graphs showing the regularised action of a connected D3-brane 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 short-distance behaviour. A detailed analysis carried out in Appendix D.2.1 shows that the regularised D3-brane action always depends quadratically on in the “magnetic regime" at sufficiently large . This, in turn, leads to Gaussian suppression of the equal-time two-point 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 two-point 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 two-point 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 two-point 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 equal-time two-point correlation functions of magnetic monopole operators in a strongly coupled 2+1-dimensional 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 top-down holographic construction involving intersecting D5- and D3-branes 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 two-point 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 D5-brane, on which the D3-brane representing the bulk monopole ends, caps off in a Minkowski embedding. We find that the saturation of the monopole two-point function at long distances persists at finite temperature, up to the critical temperature where the D5-brane 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 two-point 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 non-zero charge density implemented by introducing a gauge field on the D5-brane world-volume. On the one hand, the finite charge density forces the D5-brane into a black hole embedding at any non-zero temperature and on the other hand it gives rise to a direct coupling between the D3-brane and the magnetic dual of the world-volume gauge field on the D5-brane. Studying the monopole correlation function at finite temperature and charge density, we find that the transition to a disconnected D3-brane configuration at large separation found at vanishing charge density is replaced by a transition between different connected D3-brane configurations at low, but non-vanishing, 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 DBI-term in the D3-brane action. On the dual field theory side, this shows up in the dependence of the monopole operator two-point function on spatial distance. In particular, at finite charge density and low temperature the magnetic coupling contribution is the dominant one and the two-point function has a Gaussian falloff with distance. This is in line with the zero-temperature findings of [Iqbal:2014cga]. At high temperature and low charge density, on the other hand, the monopole two-point 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 two-point functions of monopole operators. An interesting and rather straightforward extension would be to add a non-zero 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 top-down holographic model used here in order to explore more general bulk monopole embeddings and the corresponding phase diagrams. Our treatment involved a probe D3-brane ending on a probe D5-brane in an background. An important question is whether including the back-reaction of the D3/D5-brane system on the background geometry would stabilise or wash out the features we have found. Modelling the back-reaction may require the added flexibility of a bottom-up approach, while retaining essential features of the top-down D-brane construction. At the same time, the study of monopole dynamics in phenomenologically motivated bottom-up models would be of considerable interest in its own right. Finally, there are other intersecting D-brane systems which can be used to model monopole operators in strongly coupled gauge theory. A D5/D7-brane model might for instance be a more natural setting to study non-Abelian 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 163419-051 and 163422-051, the Swedish Research Council under contract 621-2014-5838, and by grants from the University of Iceland Research Fund.

Appendix A Action functionals for probe D-branes

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 D5-brane action in the coordinate system used in the main text. This is standard material but is included here in order to have a self-contained presentation. We also obtain an explicit expression for the probe D3-brane action of a bulk monopole proposed by Iqbal in [Iqbal:2014cga] in a black 3-brane background.

The tension of a D-brane 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 D3-branes as follows [Maldacena:1997re],

(A.2)

a.1 D5-brane

The world-volume action for the probe D5-brane is

(A.3)

where denotes the Ramond-Ramond 4-form field sourced by the background D3-branes, the 2-form field strength of the D5-brane world-volume gauge field, and the induced D5 world-volume metric. The first term in (A.3) vanishes for the D5-brane configuration investigated in this work. To see this, we choose a gauge where the 4-form 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 2-form while the latter has vanishing pullback to the D5-brane world-volume.

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 D5-brane world-volume as follows,

(A.4)

After some straightforward algebra the D5-brane action reduces to

(A.5)

where is the number of background D3-branes, 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 D5-brane 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 D5-brane at charge density is given by the on-shell value of the Euclidean action (A.9). The boundary counter-terms needed to regularise the free energy are discussed in Appendix B and numerical results for the resulting regularised on-shell action are presented in Section 3.1. Switching off the charge density gives the free energy of a charge neutral D5-brane,

(A.11)

considered in Section (2.1).

a.2 D3-brane

The action for the probe D3-brane is

(A.12)

where is the field strength of the D3-brane world-volume gauge field and the induced D3-brane world-volume 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 3-form 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 3-form provides the magnetic coupling between the gauge field living on the D5-brane and the edge of the D3-brane. Indeed, adopting the same ansatz as in [Iqbal:2014cga],

(A.17)

with the volume form on the two-dimensional unit sphere that the D5 and D3-branes wrap around inside , the three-dimensional term in (A.14) reduces to an integral,

(A.18)

along the curve in traced out by the probe D3-brane.

The field strength of is the magnetic dual of the field strength of the D5-brane world-volume 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 DBI-term in (A.3) and the D5-brane world-volume 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 D3-brane embedding employed in our analysis, the first term in (A.12) containing the Ramond-Ramond potential vanishes and the remaining DBI term simplifies because there is no gauge field on the D3-brane worldvolume. As explained in Section 2.2, for equal-time correlation functions the curve spanned in by the probe D3-brane is spacelike and therefore the induced metric has Euclidean signature. The relevant terms in the Euclidean D3-brane action are thus [Iqbal:2014cga]

(A.20)

In the charge-neutral 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 on-shell D3-brane 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 non-vanishing 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 counter-terms for D5-brane

In the main text, we encountered several branches of D5-brane 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 on-shell Euclidean action of the D5-brane and compare between different branches of solutions. In the charge-neutral 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 D5-brane free energy (A.9) is UV-divergent and needs to be regularised by introducing appropriate counter-terms at the boundary. We use a well-established regularisation procedure for general D-D systems described by DBI actions [Karch:2005ms] and specialise to the system at hand. We take the UV cut-off surface to be at constant radial coordinate and find that the following counter-term action will cancel the UV-divergence of the bulk D5-brane 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 charge-neutral case, and evaluating the sum

(B.2)

before taking the limit. The ellipsis in (B.1) denotes sub-leading terms that give a vanishing contribution in the limit. We note that the presence of a gauge field on the D5-brane world-volume does not require any additional boundary counter-terms compared to the charge-neutral case.

The regularised free energy can now be calculated numerically as a function of temperature for both black hole and Minkowski embeddings.121212For 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 charge-neutral case and in Figure 6 for D5-branes at finite charge density.

Appendix C On-shell D3-brane action

In this Appendix we generalise the discussion of monopole two-point functions in Section 2.2 to finite charge density. We obtain integral expressions for the on-shell D3-brane 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.

Our starting point is the Euclidean D3-brane action (A.20). Using the D3-brane world-volume coordinates that were introduced in Section 2.2, the action reduces to