Non-conformal entanglement entropy

Non-conformal entanglement entropy

Marika Taylor and William Woodhead
Mathematical Sciences and STAG Research Centre, University of Southampton,
Highfield, Southampton, SO17 1BJ, UK.
E-mail:
m.m.taylor@soton.ac.uk; w.woodhead@soton.ac.uk
Abstract:

We explore the behaviour of renormalized entanglement entropy in a variety of holographic models: non-conformal branes; the Witten model for QCD; UV conformal RG flows driven by explicit and spontaneous symmetry breaking and Schrödinger geometries. Focussing on slab entangling regions, we find that the renormalized entanglement entropy captures features of the previously defined entropic c-function but also captures deep IR behaviour that is not seen by the c-function. In particular, in theories with symmetry breaking, the renormalized entanglement entropy saturates for large entangling regions to values that are controlled by the symmetry breaking parameters.

1 Introduction

Entanglement entropy is widely used in condensed matter physics, quantum information theory and, more recently, in high energy physics and black holes. Consider a reduced density matrix , obtained from tracing out certain degrees of freedom from a quantum system. The associated entanglement entropy is then the von Neumann entropy:

(1)

Throughout this paper we will be interested in the case for which a quantum system is subdivided into two, via partitioning space. In such a case is a spatial region, with boundary .

The entanglement entropy characterizes the nature of the quantum state of a system. For example, in the ground state of a quantum critical system in spatial dimensions:

(2)

where , and are dimensionless; is a characteristic scale of the region and is an UV cutoff. Logarithmic terms arise when is odd, and their coefficients are related to the anomalies of the stress energy tensor. More generally, the famous area law leading term characterizes the ground state of a system and can be used to test trial ground state wavefunctions. Entanglement entropy can also be used to distinguish between different phases of a system, such as the confining/deconfining phase transition [1].

Continuum quantum field theory (with a cutoff) is often used as a tool to describe discrete condensed matter systems. In this context, the cutoff appearing in (2) is related to the underlying physical lattice scale in the discrete system and the coefficients of power law terms such as capture the leading physical contributions to the entanglement entropy. From a quantum field theory perspective, the expansion in (2) implicitly assumes the use of a direct energy cutoff as a regulator. Different methods of regularisation result in different regulated divergences and thus the power law divergences are often called non-universal. By contrast logarithmic divergences are often denoted as universal as their coefficients are related to the anomalies of the theory.

In even spatial dimensions, the logarithmic term in (2) is absent but the constant term is believed to be related to the number of degrees of freedom of the system. However, is manifestly dependent on the choice of the cutoff. In two spatial dimensions, if

(3)

for a spatial region with boundary of length , then changing the cutoff as

(4)

for any choice of the dimensionless constant gives

(5)

so the constant term in the entanglement entropy clearly depends on the choice of regulator.

If one is interested in isolating finite contributions to the entanglement entropy, one can evade the issue of regulator dependence. For example, if the entangling region is a strip of width and regulated length , then the divergent contributions in (3) cannot, by locality of the quantum field theory, depend on the width of slab, so

(6)

is finite as [2, 3, 4]. However, such an approach has several drawbacks. The regularisation is specific to the shape of the geometry (a slab) and a modified prescription is needed for curved entangling region boundaries such as spheres, for which the scale of the entangling region is related to the local curvature of the entangling region boundary (see proposals in [5]). Any such prescription depends explicitly on the UV behaviour of the theory. More generally, extraction of finite terms by differentiation obscures scheme dependence: there is no connection with the renormalization scheme used for other QFT quantities such as the partition function and correlation functions.

From a quantum field theory perspective, as opposed to a condensed matter perspective, it is very unnatural to work with a regulated rather than a renormalized quantity. In previous papers [6, 7], we introduced a systematic renormalization procedure for entanglement entropy, in which the counterterms are inherited directly from the partition function counterterms. As we review in section 2, such renormalization guarantees that the counterterms depend only on the quantum field theory sources (non-normalizable modes in holographic gravity realisations) and not on the state of the quantum field theory (normalizable modes in holographic gravity realisations).

The renormalized entanglement entropy expressed as a function of a characteristic scale of the entangling region implicitly captures the behaviour of the theory under an RG flow: small entangling regions probe the UV of the theory, while larger regions probe the IR. In this paper we will establish how these finite contributions to entanglement entropy behave in a variety of theories, using holographic models.


The plan of this paper is as follows. In section 2 we review the definition of renormalized entanglement entropy introduced in [6]. In section 3 we calculate the renormalized entanglement entropy for a slab region in anti-de Sitter (in general dimensions). The latter is relevant for the non-conformal branes discussed in section 4, as the latter can be viewed as dimensional reductions of anti-de Sitter theories in general dimensions. In section 4 we also compute the renormalized entanglement entropy for a slab region in the Witten holographic model for QCD. Section 5 explores renormalized entanglement entropy for operator and driven holographic RG flows, which are UV conformal. In section 6 we consider renormalized entanglement entropy in holographic Schrödinger geometries. In section 7 we summarise the main features of the renormalized entanglement entropy, using both our holographic results and earlier perturbative/lattice calculations. We conclude in section 8.

2 Renormalized entanglement entropy

Entanglement entropy is usually calculated using the replica trick. The Rényi entropies are defined as

(7)

where is the partition function and is the partition function on the replica space obtained by gluing copies of the geometry together along the boundary of the entangling region. The entanglement entropy is obtained as the limit

(8)

Note that this limit implicitly assumes that the Rényi entropies are analytic in .

Both sides of (7) are UV divergent. In a local quantum field theory, the UV divergences of cancel with those of except at the boundary of the entangling region; therefore the divergences of scale with the area of this boundary.

We can formally define the renormalized entanglement entropy as [6]

(9)

with . Here the renormalized partition functions are defined with any suitable choice of renormalization scheme.

The replica space matches the original space, except at the boundary of the entangling region where there is a conical singularity. To define the renormalization on the replica space it is therefore natural to work within a renormalization method that works for generic curvature backgrounds for the quantum field theory.

2.1 Direct cutoff: field theory

Consider for example a Euclidean free massive scalar field theory on a background geometry of dimension and let be the partition function in the ground state. Using locality of the quantum field theory and dimensional analysis, the UV divergences in the partition function behave as

(10)

where is the UV cutoff, is the volume of the background (Euclidean) geometry, is the mass and is the Ricci scalar. The coefficients are dimensionless and in the above expressions we ignore boundaries of .

The divergences of the partition function on the replica space have exactly the same structure and coefficients. However, the curvature of the replica space has an additional term from the conical singularity [8]

(11)

where is localised on a constant time hypersurface, on the boundary of the entangling region. (Here and in what follows we consider only static situations.) Therefore, when we use the replica formula (8) the leading divergences of the partition functions cancel so that the leading divergence in the entanglement entropy behaves as

(12)

Such a divergence can clearly be cancelled by the counterterm

(13)

which is covariantly expressed in terms of the geometry of the entangling region.

2.2 Holographic renormalization

In gauge-gravity duality, the defining relation is [9, 10]

(14)

where is the onshell action for the bulk theory dual to the field theory. In the supergravity limit this is given by the onshell Euclidean Einstein-Hilbert action together with appropriate matter terms i.e.

(15)

where the latter is the usual Gibbons-Hawking-York boundary term. The volume divergences of the bulk gravity action correspond to UV divergences of the dual quantum field theory; these divergences can be removed by appropriately covariant counterterms at the conformal boundary.

For example, in the case of asymptotically locally anti-de Sitter solutions of Einstein gravity the action counterterms are

(16)

where the ellipses denote terms of higher order in the curvature and logarithmic counterterms arise for even.

Applying the replica formula to the bulk terms in the action, as discussed in [11], and using the analogue of (11) for the bulk curvature, namely,

(17)

gives the Ryu-Takayanagi functional [12] for the entanglement functional:

(18)

Applying the replica formula to the counterterms gives

(19)

with the leading counterterm being proportional to the regulated area of the entangling surface boundary. Analogous expressions for higher derivative gravity and gravity coupled to scalars can be found in [6].

Using a radial cutoff to regulate is perhaps the most geometrically natural way to renormalize the area of the minimal surface but it is not the only holographic renormalization scheme. Dimensional renormalization for holography was developed in [13] and this method could also be used to renormalize the holographic entanglement entropy.

3 AdS entanglement entropy in general dimensions

In this section we derive the renormalized entanglement entropy for a slab domain in Anti-de Sitter in general dimensions. This quantity in relevant to the non-conformal brane backgrounds discussed in the next section, as the latter can be understood in terms of parent Anti-de Sitter theories, and also relevant for the Schrödinger backgrounds discussed in section 6.

Let us parameterise as

(20)

The entangling functional is

(21)

We now consider an entangling region in the boundary of width in the direction, on a constant time hypersurface, longitudinal to the other coordinates . The bulk entangling surface is then specified by the hypersurface minimising

(22)

where . The equation of motion admits the first integral

(23)

where is the turning point of the surface, related to via

(24)

or equivalently

(25)

The regulated onshell value of the entangling functional is then

(26)

where is the regulated volume of the directions. For the only contributing counterterm is the regulated area of the boundary i.e.

(27)

(where we assume that ) and therefore

(28)

which can be rewritten in terms of dimensionless quantities as

(29)

This can be evaluated to give

(30)

and hence

(31)

As we discuss later, this quantity is closely related to the entropic function for slabs in anti-de Sitter computed in [3]. In the case of () the entangling functional is logarithmically divergent, and the renormalized entanglement entropy depends explicitly on the renormalization scale: for a single interval

(32)

where is the (dimensionless) renormalization scale.

4 Non-conformal branes

In this section we will consider entangling surfaces in Dp-brane and fundamental string backgrounds. It is convenient to express these backgrounds in the so-called dual frame in ten dimensions as [14]

(33)

where the constants are given below for Dp-branes and fundamental strings respectively. (Note that it is convenient to express the field strength magnetically, so for we use .)

The field equations admit solutions with a linear dilaton. The field equations following from the action above can be reduced over a sphere, truncating to a -dimensional metric and scalar. The resulting action is then

(34)

where and the constants depend on the type of brane under consideration.

For Dp-branes

(35)

where

(36)

and is the dimensionful coupling of the dual field theory, which is related to the string coupling as

(37)

At any length scale there is an effective dimensionless coupling constant

(38)

For the fundamental string

(39)

and the dimensionful coupling is

(40)

so

(41)

In all cases, the dual frame is chosen such that the equations of motion admit an solution:

(42)

where the constant again depends on the case of interest: for Dp-branes

(43)

while for fundamental strings. In general the equations admit an AdS solution with linear dilaton provided that the parameters are related as

(44)

For further discussion of this point, see [15].

The non-conformal branes are formally related to AdS gravity in the following way [16]. Let us define a parameter as

(45)

Now we consider -dimensional gravity with cosmological constant , so that the action is

(46)

Reducing on a -dimensional torus with coordinates via a diagonal reduction ansatz

(47)

results in the action (34) where

(48)

with the volume of the compactification torus.

4.1 Entanglement functional and surfaces

The entanglement functional follows from the replica trick: in the dual frame

(49)

The equations for the entangling surface can be expressed geometrically as

(50)

where is the background metric, is the induced metric on the entangling surface, specifies the embedding of the entangling surface into the background and are the associated traces of the extrinsic curvatures.

The dual frame entanglement functional follows directly from the reduction of the pure gravity entanglement functional

(51)

when one again uses the diagonal reduction ansatz (47), and assumes that the entangling surface wraps the torus and that the shape of the surface does not vary along the torus directions. In the upstairs picture the entangling surface satisfies

(52)

where the background metric is now denoted and denotes the traces of the extrinsic curvatures. Thus, any AdS entangling surface which factorises as will give an entangling surface for non-conformal branes; moreover, the non-conformal brane surface will inherit its renormalized entanglement entropy from the upstairs entangling surface.

As an example, let us consider slab entangling regions, characterised by a width . The bulk entangling surface is specified as and in the background (42) the entangling functional is

(53)

which is indeed precisely the functional obtained in (22), identifying . The renormalized entanglement entropy can then be expressed as

(54)

The renormalized entanglement entropy for a strip in the F1 background can be expressed as

(55)

where the effective coupling is expressed as . The expression for the renormalized entanglement entropy of a strip in the D1 background is analogous:

(56)

4.2 Witten model

The Witten [17] holographic model for YM can be expressed in terms of the following six-dimensional background:

(57)

where

(58)

Regularity of the geometry requires that the circle direction must have periodicity

(59)

This model originates from D4-branes wrapping the circle with anti-periodic boundary conditions for the fermions. which breaks the supersymmetry. At low energies the model resembles a four-dimensional gauge theory, with the gauge coupling being . The gravity solution captures the behaviour of this theory in the limit of large ’t Hooft coupling .

One of the main applications of this model is in the context of flavour physics: Sakai and Sugimoto [18, 19] introduced D8-branes wrapped around the on which the theory is reduced from ten to six dimensions. These D8-branes model chiral flavours in the dual gauge theory and the resulting Witten-Sakai-Sugimoto model has been used extensively as a simple holographic model of a non-supersymmetric gauge theory with flavours.

The operator content of the dual theory captured by the metric and scalar field is the four-dimensional stress energy tensor , a scalar operator corresponding to the component of the five-dimensional stress energy tensor and the gluon operator corresponding to the bulk scalar field. These operators satisfy a Ward identity [15]

(60)

and their expectation values can be extracted from the above geometry. For example, the condensate of the gluon operator

(61)

and therefore controls the QCD scale of the theory.


Next we can consider a slab entangling region, wrapping the circle direction , characterised by a width . Entanglement entropy in this theory was previously discussed in [1], with the confinement transition being associated with a discontinuity in the derivative of the entanglement entropy with respect to . The bare entanglement functional is

(62)

where is the volume of the two-dimensional cross-section of the slab. The entanglement can then be written as

(63)

where is the turning point of the surface, related to the width of the entangling region as

(64)

The entanglement entropy can be renormalized as before, with the counterterm contributions being

(65)

For large entangling regions, the only possible entangling surface is the disconnected configuration, for which the renormalized entanglement entropy is

For small entangling regions the condensate is negligible and the renormalized entanglement entropy is controlled by the conformal structure

(67)

The renormalized entanglement entropy is plotted in Figure 1. As discussed in [1] there is a discontinuity in the derivative of the entanglement entropy for slab widths around . For larger values of the entanglement entropy saturates at a constant value.

Figure 1: The renormalized entanglement entropy for the Witten model. The solid blue line and solid orange lines indicate the renormalized entropy for the two possible connected minimal surfaces.

5 Renormalized entanglement entropy for RG flows

In this section we will consider holographic entanglement entropy in geometries dual to RG flows. We work in Euclidean signature with a bulk action

(68)

Holographic RG flows with flat radial slices can be expressed as

(69)

where the warp factor is related to a radial scalar field profile via the equations of motion

(70)

These equations can always be expressed as first order equations [20]

(71)

where the (fake) superpotential is related to the potential as

(72)

Near the conformal boundary the potential can be expanded in powers of the scalar field as

(73)

and hence the superpotential can be written as

(74)

where . The higher order terms in the superpotential are not unique, as different choices are associated with different RG flows.

Note that for flat domain walls, a single counterterm (in addition to the usual Gibbons-Hawking term) is sufficient

(75)

although the derivation of the entanglement entropy counterterms requires knowledge of the counterterms for a curved background (since the replica space is curved).


The entanglement entropy for a slab region in the RG flow geometry is

(76)

where is the number of spatial directions in the dual theory, is the regulated volume of the longitudinal directions and defines the entangling surface. Then

(77)

where at the turning point of the surface . The regulated onshell action is

(78)

with the cutoff being .

The entanglement entropy counterterms for RG flows driven by relevant deformations were discussed in [6], working perturbatively in the deformation. Here we will analyse both spontaneous and explicit symmetry breaking, using exact supergravity solutions.

5.1 Spontaneous symmetry breaking: Coulomb branch of Sym

In this section we consider the case of VEV driven flow, i.e. spontaneous symmetry breaking. In such a situation, the scalar field has only normalizable modes and thus asymptotically the scalar field behaves as

(79)

where is related to the operator expectation value as

(80)

From (71) and (74), one can immediately read off the asymptotic form of the warp factor:

(81)

Substituting into the regulated action, we then obtain

(82)

The second term vanishes as for , and is logarithmically divergent for . (The latter case does not however arise holographically, as when the lower bound on the conformal dimension is saturated the operator automatically obeys free field equations.) Therefore, for VEV driven flows the only counterterm required is the regulated area of the boundary of the entangling surface:

(83)

Note that one can derive the same result from the bulk action counterterms, using the replica trick; see below for the case of the Coulomb branch of SYM. Thus the renormalized entanglement entropy for slabs in VEV driven flows is

(84)

Now let us consider the general structure of the renormalized entropy. In the vacuum of the conformal field theory, the renormalized entropy must behave as

(85)

with a dimensionless constant on dimensional grounds: the entropy scales with the longitudinal volume and the width of the entangling region is the only other dimensionful scale in the problem. The value of in holographic theories is given in (31).

Now working perturbatively in the operator expectation value the renormalized entropy must behave as

(86)

where is dimensionless and we work in a limit in which

(87)

i.e. the width of the entangling region is much smaller than the length scale set by the condensate.

5.1.1 Coulomb branch disk distribution

We now analyse a specific example: the renormalized entanglement entropy of slab domains on the Coulomb branch of SYM.

We consider the case of a disk distribution of branes preserving symmetry, for which the equations of motion follow from (68), with the superpotential being [21]

(88)

The metric in five-dimensional gauged supergravity is then

(89)

with

(90)

Here the coordinate at the conformal boundary and characterises the expectation value of the dual scalar operator. The scalar field can be expressed by the relation

(91)

Using the standard Fefferman-Graham coordinates near the conformal boundary:

(92)

We can then read off the expectation values of the dual stress energy tensor and scalar operator, following [22, 23]:

(93)

where we use the standard relation between the Newton constant and the rank of the dual gauge theory:

(94)

The vanishing of the dual stress energy tensor is required given the supersymmetry but careful holographic renormalization is required to derive this answer.


The regulated entanglement entropy of a slab domain in this geometry can be written as

(95)

Using the first integral of the equations of motion the width of the entangling region can be expressed in terms of the turning point of the surface as

(96)

where is an integration constant and satisfies

(97)

The regulated entanglement entropy is then

(98)

and the required counterterm is expressed in terms of the regulated area of the boundary of the entangling surface i.e. there are counterterm contributions

(99)

at each side of the slab. (The total contribution is therefore twice this value.) Note that the counterterms in this case clearly contribute both divergent and finite parts: expanding in powers of the cutoff

(100)

It is then convenient to write the entanglement entropy in terms of dimensionless quantities as

(101)

where is a rescaled dimensionless cutoff. Implicitly this expression assumes that and is the turning point of the surface. Then

(102)

These integrals can be computed numerically. There is a maximal value of (for fixed ) for which a connected entangling surface exists: the critical value of is such that

(103)

For there is no connected entangling surface but the disconnected entangling surface consisting of two components and still exists. For the latter one can straightforwardly calculate the renormalized entanglement entropy as

(104)

The renormalized entanglement entropy is plotted in Figure 2: its first derivative is discontinuous at . For small values of , the analytic expressions (86) is valid:

(105)

and the constant can be determined as:

(106)
Figure 2: The renormalized entropy for Coulomb branch disk distribution. The blue line shows the numerical results for a cut-off of , the dotted red line shows the small fit of equation (105), the dashed yellow line shows the value of the renormalized entropy for disconnected surfaces.

5.1.2 Coulomb branch spherical distribution

We now consider the renormalized entanglement entropy of slab domains on the Coulomb branch of SYM for the case of a spherical distribution of branes, preserving symmetry. The equations of motion follow from (68), with the superpotential being [21]

(107)

The metric in five-dimensional gauged supergravity is then

(108)

with

(109)

Here the coordinate at the conformal boundary and characterises the expectation value of the dual scalar operator. The scalar field can be expressed by the relation

(110)

Using the standard Fefferman-Graham coordinates near the conformal boundary:

(111)

We can then read off the expectation values of the dual stress energy tensor and scalar operator, following [22, 23]:

(112)

where we use the standard relation between the Newton constant and the rank of the dual gauge theory:

(113)

The vanishing of the dual stress energy tensor is required given the supersymmetry but again careful holographic renormalization is required to derive this answer.


The regulated entanglement entropy is then

(114)

and the required counterterm is expressed in terms of the regulated area of the boundary of the entangling surface i.e. there are counterterm contributions

(115)

at each side of the slab. (The total contribution is therefore twice this value.) Note that the counterterms in this case clearly contribute both divergent and finite parts: expanding in powers of the cutoff

(116)

It is then convenient to write the entanglement entropy in terms of dimensionless quantities as

(117)

where is a rescaled dimensionless cutoff. Implicitly this expression assumes that and is the turning point of the surface. Then