A Holographic Model of Quantum Hall Transition

We consider a phenomenological holographic model, inspired by the D3/D7 system with a 2+1 dimensional intersection, at finite chemical potential and magnetic field. At large 't Hooft coupling the system is unstable and needs regularization; the UV cutoff can be decoupled by considering a certain double scaling limit. At finite chemical potential the model exhibits a phase transition between states with filling fractions plus and minus one--half as the magnetic field is varied. By varying the parameters of the model, this phase transition can be made to happen at arbitrary values of the magnetic field.


Introduction and summary
In condensed matter physics, the quantum Hall effect (QHE) is a general feature of 2+1 dimensional, low-temperature electron systems subject to strong magnetic field B [1][2][3]. At zero temperature, by varying the magnetic field B, the transverse conductivity σ xy experiences sudden jumps between quantized values (plateaux) where ν is the filling fraction, defined as the ratio of the charge density to the magnetic field, and it can assume integer (IQHE) or fractional values (FQHE). Although the IQHE is well explained by considering localization-delocalization processes for free electrons moving in a random potential, a complete understanding of the fractional case, which relies on the strong interaction between electrons, is still lacking. Remarkably, in both cases experiments show the presence of scaling behaviour with respect to the temperature. Indeed, when the temperature T is increased, the profile of the transition between plateaux is smoothed out and it is described by a power law of the temperature while at the same critical value of magnetic field the longitudinal conductivity exhibits sharp spikes. Moreover, the width of the region in which the transition occurs (or, equivalently, in which the longitudinal resistivity is different from zero) scales with the temperature ∆B ∝ T κ . (1. 3) The exponent κ has been experimentally measured for different materials and between different pairs of plateaux (both in the integer and fractional case). Initially, the same value κ ∼ 0.42 had been found [4][5][6] and this was interpreted as a signal of universal behaviour.
However, further investigations suggested that the value of κ may be in general dependent on the experimental apparatus and the plateau transition considered [7,8] even if, concerning the IQHE, recent papers conjectured that the presence or absence of universality is affected by the range of the disorder potentials in the sample [8].
Due to the presence of strong interactions, it is difficult to understand the physics underneath the plateau transitions in the QHE. Therefore, it would be interesting to have a holographic model of this phenomenon and to investigate the finite temperature behaviour.
In this paper we focus on the phase transition at zero temperature, leaving the non-zero temperature analysis to future work. There is a wide literature concerning the QHE and its holographic description. Refs. [9,10] studied quantum Hall plateaux using holographic D-brane constructions where the fermions are represented by open strings living on the 2+1 dimensional intersection of D3 and D7 system. This approach was pursued further by several authors in various D-brane contexts [11][12][13]. Another interesting approach is based on the observation that some experimental results can be explained by a discrete duality group relating the different quantum Hall states. Refs. [14,15] and, more recently [16], considered a holographic model encoding this feature based on Einstein-Maxwell axiondilaton action. In this description, the quantum Hall states are represented by dyonic black holes and it is possible to capture the quantization of the Hall plateaux. Other work on holographic quantum Hall physics includes .
Although these attempts succeeded in explaining some of the features of QHE such as the presence of constant conductivity plateaux, the description of phase transitions between different quantum Hall plateaux remains elusive. In this paper we consider a holographic model that exhibits such a transition. We follow the approach of [49][50][51][52] where the physics of interacting three-dimensional fermions was argued to be holographically related to the physics of a tachyon field in the bulk of AdS space. The three-dimensional fermions coupled to four-dimensional N = 4 super Yang Mills are realized as a low energy theory of the D3/D7 branes configuration in which a small number of D7 branes intersects a large number of D3-branes along 2 + 1 dimensions. The holographic description involves finding a profile of the D7 brane propagating in the AdS 5 × S 5 space; there is a (below Breitenlohner-Freedman bound) tachyon mode which appears because the system is non-supersymmetric and unstable. To understand the physics of this system, [49] proposed introducing a cutoff in the radial direction of AdS. The model can be rendered renormalizable by taking cutoff to infinity and the tachyon mass to the BF value, while the physical scale remains fixed.
We study the consequences of having a finite chemical potential and a finite magnetic field in this system. The equations of motion, together with the regularity of the brane profile and the gauge field, give rise to two energetically inequivalent solutions. Therefore, the system undergoes a first order phase transition precisely at b = 0. By computing the conductivities via linear response we observe that the transition is between two different plateaux, characterized by filling fractions ν = ± 1 2 . To obtain the phase transition for a non-zero value of the magnetic field b c we can phenomenologically modify the action (this involves explicit breaking of parity in three space-time dimensions). Again, the phase transition is of the first order and it occurs between the two solutions of the equations of motion.
The remainder of the paper is organized as follows. In section 2 the D3/D7 model is reviewed. After discussing the Dirac-Born-Infeld action in the absence of gauge field, we consider the addition of the Chern-Simons term and turn on both the magnetic field and the charge density. Then, we analyse the scaling symmetry of the full action and derive the charge density from the holographic dictionary. Eventually we solve the equations of motion for different values of the magnetic field. In section 3 we show that the system undergoes a phase transition between states with filling fraction ν = ± 1 2 at b c = 0. We also show how to change the value of b c by modifying the action. The computation of the conductivities shows that the two phases exhibit two different values of the transverse conductivity. Finally, in section 4 we discuss the results and comment on future prospects.

Introduction
We consider a brane system consisting of N D3-branes intersecting a single D7-brane along a 2+1 dimensional defect. The branes are oriented as follows (Similar model has been used to analyse N = 4 SYM with gauge group SU (N ) coupled to an N = 2 superfield in the fundamental representation of SU (N ) [53].) The gauge field lives in the 3 + 1-dimensional Minkowski space-time labelled by the coordinates {t, x, y, z} while the fermions are located in the 2 + 1-dimensional defect z = 0. At strong coupling, the model is described by the D7-brane propagating in the background AdS 5 ×S 5 geometry generated by the stack of D3-branes. In the probe limit, the back-reaction of the D7-brane on the background geometry is negligible.
To introduce the features relevant to the present work, this section is devoted to a brief review of the D3/D7 system. The metric for the AdS 5 × S 5 geometry can be written as follows where L is the AdS radius, x represents the boundary space directions {x, y, z}, the foursphere is parametrized by the coordinates {x 4 , · · · , x 8 } and the following identities hold In this background the D7 probe brane wraps a four-sphere inside the S 5 and stretches along the t, x, y directions. The rest of the D7 world-volume is specified by a single embedding function x 9 = f (ρ), giving the following induced metric The Dirac-Born-Infeld (DBI) action, up to an overall constant, is (2.5) The action (2.5) becomes quadratic in the regime of small f (ρ); it is clear that f (ρ) is tachyonic with the tachyon mass below the Breitenlohner-Freedman bound [54]. This is the holographic manifestation of the fact the the field theory is unstable. As reviewed in detail in [49] this instability is related to the formation of the bi-fermion condensate at finite value of the 't Hooft coupling. At infinitely large coupling the theory develops a gap of the order of the UV cutoff. This can also be seen in holography by introducing the UV cutoff Λ in (2.5).
In principle, the theory can be rendered renormalizable by taking the tachyon mass to the BF value. However we will not be doing this here: we believe that the main features of the model relevant to the description of quantum Hall physics are not affected by this procedure. A convenient way to think about the cutoff Λ in (2.5) is to imagine that the model at that scale is modified and the UV physics lifts the tachyon mass above the BF bound. Again, the details of this physics should not affect the infrared observables that we are after. 1 The profile of f (ρ) is obtained by solving the equations of motion, supplemented with the following boundary conditions where the first one reflects the regularity of the brane at ρ = 0 and the second one sets the fermion bare mass to zero. Note that the scalar assumes a finite value at ρ = 0. It is useful to mention that the equations of motion derived from the action (2.5) preserve the scaling symmetry and this allows to chose the value of f (0) arbitrarily.
One can also use a different coordinate system, defined as ρ = r cos θ , x 9 = r sin θ . The D-brane profile is now described by θ(r) and the action (2.5) becomes the tachyon DBI action where the lower limit of the integral is defined as r 0 = f (0). The function V (θ(r)) = cos 4 θ(r) assumes the role of tachyon potential: the mass of the tachyon in the vacuum is obtained by expanding it up to second order in small θ that is, in our dimensionless units, m 2 = −4. As stated previously, the value of the tachyon mass is below the Breitenlohner-Freedman bound m 2 BF = −d 2 /4, with d number of boundary dimensions, therefore the scalar undergoes condensation.
The boundary conditions (2.6) are mapped to the following ones and where the ± reflects the fact that the action is invarian under the transformation θ → −θ and therefore both the positive and negative profiles ±θ(r) satify the equations of motion. The potential V (θ(r)) assumes monotonically the values from the maximum V (θ(Λ)) = The equation of motion for the action (2.9) reads r 2 cos 3 θ(r) (2. 13) or, equivalently, (2.14) A term-wise study of the equation (2.13) near the point r = r 0 shows that the asymptotic expansion of the field θ(r) reads In the following only the plus-solution will be considered and, thanks to the scaling symmetry (2.7), we choose r 0 = 1. The result (2.15) can also be obtained in a different coordinate system by solving the equations of motion for x 9 = f (ρ) and then by using the change of coordinates (2.8). The near ρ = 0 behaviour of the field x 9 (ρ) is described by which reduces to eq. (2.15) by means of the aforementioned map (2.8).
To solve the equation (2.14) numerically it is convenient to consider a different set of boundary conditions (2.17) The solutions in the two coordinate systems are shown in Fig. 1. The numerical value of UV cutoff Λ is obtained by requiring θ(Λ) = 0 and it reads Λ ∼ 4.7305, in perfect agreement with [49].

Adding magnetic field and charge density
In this section we present the complete D3/D7 model studied in this work. The orientation of the branes is the same as in the setup (2.1). The background near-horizon metric for the D3-branes system reads with L the AdS 5 radius and the five-sphere is parametrized as The coordinate θ can be defined in two different patches, covering each half a five-sphere: θ ∈ [0, π/2] and θ ∈ [−π/2, 0, ] . In the followings we will consider the patch θ ∈ [0, π/2], while the other case will be commented later in this section. The D7-brane extends along the t , x , y and r directions and wraps the S 4 : its embedding is encoded in the two scalar fields θ(r) and z(r). For our purposes we set z(r) = 0, while, as already stated in the previous sections, θ(r) vanishes at the UV cutoff r = Λ and assumes the value of π/2 at r = r 0 .
The background is supported by the following Ramond-Ramond (RR) five-form The RR four-form potential is defined as dC 4 = F 5 and in our conventions it reads The constant of integration c 1 is fixed by the requirement that c(θ)dΩ 4 is a well defined differential form on the patch, namely its norm is not divergent in the whole domain considered. The norm reads For θ ∈ [0, π/2], it is easy to see that in the neighbourhood of θ = π/2 the norm blows up as therefore, to cancel the divergence we fix the integration constant to be and eq. (2.23) becomes Thus, the norm of c(θ)dΩ 4 is defined everywhere for θ ∈ [0, π/2] and, in the same range of θ, the function c(θ) is always negative except at θ = π/2, where it vanishes. 2 It is useful to remind that the field θ assumes the value π/2 when r = r 0 The case of θ ∈ [−π/2, 0] is slightly different. Indeed the request of a well defined norm for c(θ)dΩ 4 fixes differently the integration constant in (2.23) and we have Therefore, the function c(θ) depends on the patch considered and note that the following relation holds As last ingredient, we introduce in the boundary theory a finite charge density and an external magnetic field b (directed along z) by means of the gauge field, expressed in the Landau gauge

Action
Three terms contribute to the full action that is, the Dirac-Born-Infeld action S DBI , the Chern-Simons action S CS and the boundary term S bdy . The DBI action is defined as where we have with g s the string coupling, related with the N = 4 Yang-Mills coupling by 4πg s = g 2

Y M
and N the number of D7-branes. The volume for the four angular coordinates φ i is while the induced metric G αβ on the probe D7-brane is The Chern-Simons contribution reads 3 which, after integrating by parts, generates two terms. We will refer to the first one as the Chern-Simons term The sign convention is the same of [11].
while the second is a boundary term where the relevant part of pullback of the RR four-form potential is obtained from eq. (2.22) where α 4 is a closed form dα 4 = 0. The full gauge invariant action can be written as and where, as in eq. (2.9), we introduce the tachyonic potential V (θ(r)) = cos 4 (θ(r)).

Equations of Motion
In this section we solve the equations of motion obtained from the action (2.42) for the scalar field θ(r) and the gauge field a 0 (r). Since the action depends on a 0 (r) only through its derivative a ′ 0 (r), the equation of motion for the gauge field can be written as The presence of the Chern-Simons term (2.38), as it will be explained in the following section, gives rise to a non-zero charge density. Therefore, we can consider a solution with a non-zero charge density.
We can solve algebraically eq. (2.44), obtaining In the same way we compute the equation of motion for θ(r). We have The equations of motion (2.48) and (2.47) show the following scaling symmetry

Computing the charge density
To find the favored state we have to compare the Gibbs free energies computed on the different solutions. We keep the chemical potential fixed and therefore we study the system in the grand-canonical ensemble by considering the Gibbs free energy. By the holographic dictionary, the Gibbs free energy is associated with the on-shell (euclidean) action S The chemical potential µ is defined, as usual, as the value of the temporal component of the gauge field (2.31) at the cutoff Λ while the charge density ρ is its conjugate variable To compute ρ we perform a variation at constant b of the Gibbs free energy (2.52) The first term evaluates as and it vanishes as in eq. (2.46). B noting that eq. (2.53) implies δµ = L 2 2πα ′ δa 0 (Λ) and by observing that δa 0 (0) = 0, eq. (2.55) reduces to namely, the complete action S depends on a 0 (Λ) only through the boundary term. Comparing eq. (2.57) with the first law of thermodynamics δΩ = −V 1,2 ρ δµ allows us to write where N is the number of D7-brane (which has been set to 1). From the charge density we can define the filling fraction ν as and therefore the filling fraction ν is invariant under the rescaling (2.49). Note that the filling fraction (plus or minus) one-half is consistent with the parity anomaly of a threedimensional fermion coupled to the gauge field, as already noticed in [10].

Rescaling and Normalizing the Free Energy
The computation of the Gibbs free energy requires the introduction of a UV cutoff Λ.
One can always use the scaling symmetry to make any solution θ(r) satisfy θ with µ 0 the external, fixed value for the chemical potential. Finally, since we are only interested in the difference between the Gibbs free energies, we choose to normalize Ω by subtracting the value Ω 0 , namely the Gibbs free energy at zero b. Therefore, the magnetic field and the Gibbs free energy after the rescaling and the normalization read (2.64)

Black-Hole embedding profiles
The solution we have considered so far is characterized by the fact that it stops at a certain value of the radial coordinate r = r 0 and, therefore, it does not enter the Poincaré horizon, which is defined in our coordinate system as the line r = 0 in the {r, θ} plane.
However, in general we could expect that the equations of motion can be satisfied also by profiles of θ(r) which go up to r = 0. We will refer to this kind of solution as black  At zero magnetic field, by comparing the Gibbs free energies Ω as functions of µ we find that for small values of the chemical potential the favored solution is the MN one.
For µ larger than a critical value µ c (but below the maximal value of µ discussed above), the BH solution becomes the favored one, while the θ = 0 solution never has the lowest energy. The phase diagram slightly changes when b is increased: for small µ the MN profile is always favored and, as before, after a critical value of µ the interpolating solution dominates. The difference with respect to the b = 0 case is that, as we stated previously, the constant θ = 0 profile does not exist anymore.These results are represented in Fig. 5. To summarize, we found that the the model we are considering admits profiles which enter the Poincaré horizon r = 0. However, in a certain range of µ (below a critical value µ c ) these new solutions have larger Gibbs free energy than the MN profile. Therefore we can limit ourselves to study only the MN solution.

Conductivities
To complete the analysis of the holographic model of the quantum Hall effect we compute the longitudinal and transverse component of the conductivity σ, defined as where J i is the electric current induced in the medium and E i is the external electric field.
To introduce in the model these new features, we modify the gauge field (2.31) as follows A = L 2 2πα ′ [a 0 (r)dt + (e t + a x (r)) dx + (b x + a y (r)) dy] , (2.69) namely we add a background electric field and the fluctuations along the longitudinal and transverse directions a x (r) and a y (r) respectively, As in the case of magnetic field explained in section 2.5, the physical electric field is defined as The action (2.42) is then modified as follows  The AdS/CFT dictionary prescribes that the physical currents J x and J y are computed by differentiating the action (2.71) with respect to the boundary values of the gauge field A x (Λ) = L 2 2πα ′ a x (Λ) and A y (Λ) = L 2 2πα ′ a y (Λ). As in the computation of the charge density, the only contribution to the action (2.71) of the boundary values of the gauge field is through the boundary term. Therefore, in both patches, the results for the currents read From eq. (2.68) and the expression of the filling fraction ν eq. (2.60), we obtain the longitudinal and Hall (transverse) conductivities The longitudinal conductivity vanishes while, since c(0) is constant, we see the emergence of the plateaux for the transverse conductivity. Note that, since we are in a zero-temperature regime, the transition is not smoothed out.

Tachyon Model and Phase Transitions
In this section we study the Gibbs free energies of the two sets of solutions we found previously, namely the one starting from θ = π/2 and the one starting from θ = −π/2. In particular, we analyse the competition between the two solutions to understand whether a phase transition occurs by varying the magnetic field, namely if the difference between the Gibbs free energies changes sign.  (c.f. eq. (2.10)) to be the same, up to second order To fulfill the previous requirements and, at the same time, to generate a new phase we modify the behaviour of the potential for θ ∈ [−π/2, 0].
In section. 2.2 we showed that both the tachyon potential and the c(θ) function are derived from the geometry of the five sphere S 5 . More precisely, the tachyon potential is related to the metric (2.19) while the RR four-form is determined by the volume element dΩ 5 (2.21). These two quantities can be expressed in terms of the single function, for instance the tachyon potential. Indeed we have and We consider the following V (θ) while c(θ) is given by The profiles are shown in Fig. 7. We note that with the choice made the value of c(θ) in θ = 0 is different from the original one (2.29). We then solve numerically the equations of motion in the two patches θ ∈ [−π/2, 0] and θ ∈ [0, π/2] for different values of b. In Fig. 8 we draw the corresponding results. Finally, we compute the Gibbs free energy Ω, as described in the previous sections.
We observe that the three contributions to the Gibbs free energy behave differently with increasing b (c.f. Fig. 9,

Discussion
In this paper we propose a holographic model for the quantum Hall plateau transition based on the non-supersymmetric D3/D7 system in the probe limit and at zero temperature. The full action is obtained by considering the tachyon-DBI action and a Chern-Simons term, which allows the system to have a non-zero charge density. The By playing with the holographic action [changing V (θ) and c(θ)], and breaking parity explicitly, we can also make phase transitions happening at finite values of the magnetic field. Note that our description requires introduction of the cutoff, but can be rendered renormalizable by taking the physical limit, where the cutoff Λ is taken to infinity and the tachyon mass to the BF value, while the physical scale remains fixed. We expect the physics of the phase transition described in this paper to not be significantly affected by this procedure. At finite Λ there can be multiple solutions with the same value of the cutoff which oscillate around θ = 0. We show that their (Gibbs) energy is always higher than those of the solutions we discuss in the paper and hence they are suppressed thermodynamically. All these solutions disappear in the physical limit.
We also considered another possibility to generate phase transitions phenomenologically. Suppose the tachyon potential has two minima, θ 1 and θ 2 , and both satisfy V (θ 1,2 ) = 0 condition, to assure the absence of external forces (finite energy density) at r = r 0 . Can we have two distinct solutions which interpolate between θ = 0 at r = Λ and θ = θ 1,2 at r = r 0 ? Then, there can be a competition between their energies, and, possibly, a phase transition. We show that at least within the set of examples we considered two solutions do not arise -the tachyon field always stops at the first minimum.
One of the motivations for using the language of a non-linear tachyon action to describe QHE has been its ability to model both the hard-wall and the soft-wall behaviour. The latter opens up a possibility of describing a crossover between different Hall plateaux at finite temperature (the class of the hard-wall models, to which all currently available holographic quantum Hall models belong, is not suitable for this purpose: we expect temperatures much lower than the hard wall scale to not affect the order of the phase transition). We leave investigation of the holographic quantum Hall in a soft-wall type model for future work.

Acknowledgments
We