Holographic Entanglement Entropy and Renormalization Group Flow
Using holography, we study the entanglement entropy of strongly coupled field theories perturbed by operators that trigger an RG flow from a conformal field theory in the ultraviolet (UV) to a new theory in the infrared (IR). The holographic duals of such flows involve a geometry that has the UV and IR regions separated by a transitional structure in the form of a domain wall. We address the question of how the geometric approach to computing the entanglement entropy organizes the field theory data, exposing key features as the change in degrees of freedom across the flow, how the domain wall acts as a UV region for the IR theory, and a new area law controlled by the domain wall. Using a simple but robust model we uncover this organization, and expect much of it to persist in a wide range of holographic RG flow examples. We test our formulae in two known examples of RG flow in 3+1 and 2+1 dimensions that connect non–trivial fixed points.
Holographic Entanglement Entropy
Renormalization Group Flow
Tameem Albash, Clifford V. Johnson
Department of Physics and Astronomy
University of Southern California
Los Angeles, CA 90089-0484, U.S.A.
talbash, johnson1, [at] usc.edu
A useful probe of the properties of various field theories that has received increased interest in recent times is the entanglement entropy, with applications being pursued in diverse areas such as condensed matter physics, quantum information, and quantum gravity. One of the main motivators, in the context of strongly coupled field theories (perhaps modeling novel new phases of matter), is that the entanglement entropy may well act as a diagnostic of important phenomena such as phase transitions, in cases where traditional order parameters may not be available.
Within a system of interest, consider a region or subsystem and call it , with the remaining part of the system denoted by . A definition of the entanglement entropy of with is given by:
where is the reduced density matrix of given by tracing over the degrees of freedom of , , where is the density matrix of the system. When the system is in a pure state, i.e., , the entanglement entropy is a measure of the entanglement between the degrees of freedom in with those in .
It is of interest to find ways of computing the entanglement entropy in various strongly coupled systems, in diverse dimensions, and under a variety of perturbations, such as the switching on of external fields, or deformations by relevant operators. A powerful tool for studying such strongly coupled situations is gauge/gravity duality, which emerged from studies in string theory and M-theory. The best understood examples are the conjectured AdS/CFT correspondence and its numerous deformations [1, 2, 3, 4] (See e.g., ref. for an early, but still very useful, review.) There has been a great deal of activity for over a decade now, applying these tools to strongly coupled situations of potential interest in condensed matter and nuclear physics, for example. Fortunately, there has been an elegant proposal[6, 7] for how to compute the entanglement entropy in systems with an Einstein gravity dual (or, more generally, a string or M–theory dual in the large limit and large t’ Hooft limit), which provides a new way to calculate the entanglement entropy using geometrical techniques (for a review see ref.). In an asymptotically Anti–de Sitter (AdS) geometry, consider a slice at constant AdS radial coordinate . Recall that this defines the dual field theory (with one dimension fewer) as essentially residing on that slice in the presence of a UV cutoff set by the position of the slice. Sending the slice to the AdS boundary at infinity removes the cutoff (see ref.  for a review). On our slice, consider a region . Now find the minimal–area surface bounded by the perimeter of and that extends into the bulk of the geometry. (Figure 1 shows examples of the arrangement we will consider in this paper.) Then the entanglement entropy of region with is given by:
where is Newton’s constant in the dual gravity theory.
This prescription for the entropy coincides nicely with various low dimensional computations of the entanglement entropy, and has a natural generalization to higher dimensional theories. Note that there is no formal derivation of the prescription. Steps have been made, such as in refs. [9, 10], but they are not complete. However, there is a lot of evidence for the proposal. See e.g., refs.[11, 12, 13, 14, 15, 16]. A review of several of the issues can be found in ref.. Further progress has been made recently in ref. .
In this paper we shall assume that this holographic prescription does give the correct result for the entanglement entropy in systems with gravity duals, and proceed to examine the interesting question of how the entanglement entropy behaves when a system is perturbed by an operator that triggers a Renormalization Group (RG) flow. For simplicity, we will work with flows that connects two conformal field theories, and we will consider (for concreteness) a four dimensional example and one in three dimensions. Such examples are extremely natural to study using holographic duality since (at large ) it is possible to find geometries that represent the full flow from the maximally supersymmetric theory to theories with fewer super symmetries. (This was first proposed in refs.[18, 19], and several examples have since been found.) Flow between field theory fixed points correspond to flows between fixed points of the supergravity scalar potential. The examples we will study begin with the four dimensional case of the flow[20, 21, 22] to the Leigh–Strassler point[23, 24], which results from giving a mass to one of the chiral multiplets that make up the Yang Mills gauge multiplet. We then continue with the three dimensional generalization of it discussed in ref. . The gravity dual of the four dimensional flow connects AdS at the extreme of a radial coordinate to AdS at , where the space results from squashing the along the flow. There are two of the 42 supergravity scalars switched on at the latter endpoint, and correspondingly the characteristic radius of the AdS in the IR is larger than that of the UV theory: The gravity dual for the three dimensional flow has related structures, this time connecting an AdS UV geometry to an AdS in the IR, where results from squashing the along the flow.
Before studying the specific examples, however, we step back and try to anticipate some of the key physics that we should expect from the entanglement entropy in this type of situation, more generally. Generically, holographic RG flow involves a flow from one dual geometry in the UV to another in the IR, separated by an interpolating region that can be thought of as a domain wall separating the two regions. The key to understanding the behaviour of the content of the holographic entanglement entropy formula is to then understand how the computation incorporates the structure of the domain wall, and how the field theory quantities it extracts are encoded. To anticipate how to mine this information, we do an analytic computation of the proposed entanglement entropy (2) in an idealized geometry given by a sharp domain wall separating two AdS regions with different values for the cosmological constant. Working in various dimensions (AdS, AdS, and AdS, pertaining to flows in four, three, and two dimensional field theories), we find a fascinating and satisfying structure, seeing how the entanglement entropy tracks the change in degrees of freedom under the flow, and several other features. We expect that these features will be present in a wide range of examples, and we confirm our results in the examples mentioned above.
The outline of this paper is as follows. In section 2 we carry out the study of the entanglement entropy in the presence of the idealized (i.e., sharp domain wall) holographic RG flow model, and discover how the physics is organized in the results. Then, ready to study examples, we review the four dimensional Leigh–Strassler RG flow of interest, and its dual AdS flow geometry in section 3. We explicitly solve (numerically) the non–linear equations that define the geometry and scalars in the interpolating dual supergravity flow. We then compute the entanglement entropy and extract the physics, comparing to our predictions from section 2. Section 4 presents the analogous studies for the three dimensional field theory, with the AdS dual flow geometry. We end with a discussion in section 5.
2 Entanglement Entropy and a Sharp Domain Wall Model
As mentioned in the introduction, the generic holographic RG flow involves a flow from one dual geometry in the UV to another in the IR, separated by an interpotating domain wall. In all examples, understanding the behaviour of holographic entanglement entropy, as proposed in equation (2), requires us to understand how the area formula incorporates the structure of the domain wall in terms of field theory quantities. So we start by doing an analytic computation in an idealized geometry given by a sharp domain wall in AdS. In general, the location of the wall, and its thickness, are determined by field theory parameters corresponding to the details of the relevant operator - for example, in the case of the Leigh–Strassler flow and its generalization we later study, the detail in question is the bare value of the mass given to the chiral multiplet. A sharp domain wall is of course not a supergravity solution, and falls somewhat outside the usual supergravity duality to any (large ) theory, but nevertheless is a clean place to start to capture how the physics is organized. We expect it to capture a great deal of the key physics of holographic RG flow, as regards how the entanglement entropy formula works.
We use the following background metric:
Here , and is either four, three, or two coordinates (the spatial coordinates of the dual field theory), depending upon whether we are in AdS, AdS, or AdS, the cases we will consider. Also, . The length scale of AdS on either aide of the wall is set by in the UV at and in the IR at .
2.1 The Ball and AdS.
We begin by studying a region in the three spatial dimensions which is a round ball of radius . Using a radial coordinate in the spatial dimensions, the area of the surface, , that extends into the bulk is given by:
where the function defines the enbedding. We can calculate the equations of motion that result from minimizing this “action,” and we find that the solution is given by:
where is the position of the domain wall in the AdS radial direction, and is the spatial radial position where and is given by:
Note that, rather than integrating out to , we integrate out to large positive radius , defining our UV cutoff, with small defined by:
Note that with the solution given for , we are assuming that is larger than a critical radius such that our surface extends past the doman wall into the second AdS region. The critical radius is given by setting in the above:
Substituting the solution back into equation (5), we can analytically calculate the area of our minimal surface, , and hence the entanglement entropy via equation (2). This gives a long expression that we will not display here. For our purposes it is enough to first expand the area for small :
We find it useful to rewite it in a suggestive way:
There are a number of notable features of this expression. First, we see the results from pure AdS in the first line. There, we see the usual UV divergent terms and the –independent constant that results from the fact that the ball preserves some of the conformal invariance of AdS. Second, the terms that have as coefficients (the last two lines) always have appearing in the combination:
We are tempted to interpret this as the effective ball radius as seen in the IR, as opposed to the simple seen in the UV. Furthermore, the combination:
appears in a manner analogous to how the UV cut-off appears. (This might not be clear in our expansion of the above equation. One way to see that it does appear as does is to look at equation (6)). Now is not necessarily small, but we will see that it is useful to think of it as the cut–off in the IR theory. With these observations in mind, we rewrite the last three lines of equation (11) as follows:
Let us focus on the terms proportional to . If we expand these terms assuming that , i.e., the effective length in the putative IR theory is larger than the IR cutoff, which also means that the length is such that the surface extends very far past the wall into the IR AdS space, we get:
Pleasingly, this is exactly the result we would have obtained if we were purely in the IR theory!
So far therefore, we have seen how the entanglement entropy formula encodes key behaviours of both the UV and the IR theories, in terms of the appropriate scales, and . The boundary of AdS at is the UV region and the quantities of the UV theory appear accordingly. From the point of view of the IR theory, the domain wall acts (for small) as the effective UV region, with acting as the effective regulator.
We are left with understanding the first two terms in equation (15). These two terms mix the properties of the UV and the IR regions, and are more subtle. We associate them with the region around the domain wall, which connects the UV and IR regions (through and abrupt change in our idealized example). It is prudent to try to understand the role of these terms toward the end of the flow, and so we do a large expansion of them, giving:
So we see that these terms give contributions very analogous to our UV and IR results, where here the reference scale is played by . (Note that the constant term is actually different than the UV and IR constant terms’ form.) At fixed or , we may think of this as a new set of divergences.
Now that we have an understanding of the contributions of the various pieces to the area, we combine everything together again and consider the large (and small ) expansion:
The first key result here is that we no longer have a scaling associated with the UV theory. The remaining dependence has a coefficient that is only associated with the IR theory and that is independent of the domain wall. In a non–RG flow scenario, the coefficient of such a term is determined by the central charge of the theory (see e.g., refs.[7, 26]), but here we see that the coefficient has shifted from its UV value (associated with the UV central charge) to its IR value (associated with the IR central charge). The second thing to note is that the area law associated with the UV cut–off (the first term) is joined by a second area law. Its coefficient is sourced by the details of the domain wall. For clarity, we display this term here:
We expect this new area law to be a robust feature of RG flow geometries, but anticipate that the coefficient’s precise form will be different as we move away from the thin wall limit we are in here. The above result predicts that the coefficient grows more positive as is pushed to the UV. In realistic RG flows, while the domain wall position and sharpness cannot be varied arbitrarily, it is expected to get thinner toward the UV and so at least in that regime we should recover positivity. Finally, the constant terms in the last line of equation (18) are a mixture of both the UV, IR, and domain wall physics.
2.2 The Disc and AdS.
We can repeat the same procedure for AdS, pertaining to RG flows in dimensional theories. As our system we consider a circular disc of radius . The solution for the surface embedding are exactly as in equation (6). We can calculate the minimal area and expand for small to get:
We see the reappearance of many of the key players that we saw in the AdS case, such as , and , appearing in similar types of term. For , we recover the pure UV result (proportional to ) and also the constant , the constant ensured by the fact that the disc preserves some conformal invariance, as expected. For large , we have:
So in the AdS case, the constant term shifts from its UV result to its IR result . Again, in addition to the usual UV area law (proportional to ), we have a new area law controlled by the domain wall:
which should be compared to the example from AdS in equation (19). The same comments we made for the new area law there apply here: It is not necessarily positive, but we expect it to get more positive as the domain wall is sent to the UV, where generically it gets thinner.
2.3 The Case of AdS.
Next we consider the case of AdS, pertaining to flows in dimensions. We use a spatial interval of length for our region . The area is given by:
In the large limit, this gives
where and are defined in equations (12) and (14) respectively. So again we see that the universal coefficient (in front of the natural logarithm) becomes the IR factor in the large limit. The IR cutoff replaces the UV cutoff just as observed before.
2.4 The Strip and AdS.
We next consider an area that is a strip in AdS, to compare our results for the disc. We take the strip to be of finite width in the direction, and of length in the remaining direction, which will be taken to be large, making an infinite strip. The area is given by:
Since there is no explicit dependence on , there is a constant of motion in the dynamical problem associated to minimizing the area. However, we must be careful since the constant of motion on either side of the domain wall is not the same:
On the IR side, the constant is simply given by , which occurs at a radial position we will denote as . The constant on the UV side is determined by asking that when , which is the critical situation before our embedding enters the IR AdS. We can in turn calculate the area and length in terms of and expand for small :
Here, we see the appearance of the Gauss hypergeometric function:
The large limit corresponds to taking , which gives us:
So we see that that the constant term here is exactly the new area law’s coefficient that we saw in the disc case, in equation (22). Again, far enough in the UV, for large enough mass, our analysis suggests that this coefficient is positive.
2.5 The Box and AdS.
Returning to AdS, we consider for region a box in AdS, in order to compare to the round ball we studied before. Here the finite width is again and the two other sides are of length , which we again take to be large. The computation proceeds in a similar way. The area gives:
and taking the limit gives:
Again, we have that the constant term has the same coefficient as the new area law term for the ball case, as seen in equation (19).
3 The Four Dimensional Holographic RG Flow
3.1 The Holographic Dual Gravity Background
In field theory terms, the RG flow is defined by an supersymmetric deformation of the supersymmetric Yang Mills theory given by introducing a mass term for one of the chiral multiplets.This relevant deformation causes the theory to flow to an fixed point in the IR called the Leigh–Strassler fixed point [22, 23, 24]. For the theory at large , there is an holographic dual of this physics , represented by a flow between two five dimensional anti–de Sitter (AdS) fixed points of gauged supergravity in five dimensions. One point has the maximal symmetry, and the other has , global symmetries of the dual field theories. The relevant five dimensional gauged supergravity action is [20, 21, 27]:
where the superpotential is given by:
with . The scalar field is dual to an operator of dimension three in the field theory while the scalar field is dual to a dimension two operator:
where , . Here () are the six scalars in the multiplet, and the (three of that adjoint multiplet’s four fermions) are partners of the , forming the three chiral multiplets. This combination of operators is exactly what is needed to reproduce the deformation. The geometry in five dimensions, of domain wall form, can be parametrised in the following manner:
The supergravity equations of motion yield the following flow equations:
In these coordinates, the UV is at and the IR is at , as in earlier sections. In either limit, the right hand side of the first two equations vanish, and the scalars run to specific values ( at one end, at the other), while becomes in the UV and in the IR, defining an AdS in each case, and hence a conformally invariant dual field theory at each end. This is a fat, smooth version of our simple thin domain wall model of the previous sections. Here and .
To study the UV behavior of the fields, we find it convenient to define a coordinate given by:
and we find the asymptotic behavior of the fields near (UV AdS boundary):
The constant is related to the mass of the multiplet via :
To study the IR behavior of the fields, we define a coordinate given by:
where . The asymptotic (near ) behavior in the IR is given by:
3.2 Numerically Solving for the Flow
To solve the flow numerically (as we will need to do in order to compute the entanglement entropy), it is convenient to work with a coordinate:
and employ a shooting method to solve the equations. To shoot from the IR we take , and towards the UV we take , where we use . To get good numerical stability, we define new fields ,
such that near the AdS boundary, the leading behavior of these fields is given by:
In the IR, we use the results in equation (3.1) (up to ) as our shooting conditions (we have the freedom of choosing the parameter , which is always less than zero). We can then extract the values of and at the AdS boundary for our solution. Note that the choice of will determine the choice of . We can choose to eliminate the constant by appropriately rescaling our coordinate . In particular, if we solve our equations with , we can extract the constant , and then simply perform the following transformation to eliminate it from our metric:
Finally, as indicated in ref. , there are constants of the motion regardless of the choice of (and subsequently, the choice of ), given by: