A refinement of entanglement entropy and the number of degrees of freedom

A refinement of entanglement entropy and the number of degrees of freedom

Hong Liu Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139    Márk Mezei Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139
Abstract

We introduce a “renormalized entanglement entropy” which is intrinsically UV finite and is most sensitive to the degrees of freedom at the scale of the size of the entangled region. We illustrated the power of this construction by showing that the qualitative behavior of the entanglement entropy for a non-Fermi liquid can be obtained by simple dimensional analysis. We argue that the functional dependence of the “renormalized entanglement entropy” on can be interpreted as describing the renormalization group flow of the entanglement entropy with distance scale. The corresponding quantity for a spherical region in the vacuum, has some particularly interesting properties. For a conformal field theory, it reduces to the previously proposed central charge in all dimensions, and for a general quantum field theory, it interpolates between the central charges of the UV and IR fixed points as is varied from zero to infinity. We conjecture that in three (spacetime) dimensions, it is always non-negative and monotonic, and provides a measure of the number of degrees of freedom of a system at scale . In four dimensions, however, we find examples in which it is neither monotonic nor non-negative.

preprint: MIT-CTP 4336

July 6, 2019

I Introduction

Quantum entanglement has been seen to play an increasingly important role in our understanding and characterization of many-body physics (see e.g. amico (); eisert ()). The entanglement entropy for a spatial region provides an important set of observables to probe such quantum correlations.

In spacetime dimensions higher than two, however, the entanglement entropy for a spatial region is dominated by contributions from non-universal, cutoff-scale physics Bombelli:1986rw (); Srednicki:1993im (). This implies that for a region characterized by a size , the entanglement entropy is sensitive to the physics from scale all the way down to the cutoff scale , no matter how large is. As a result the entanglement entropy is ill-defined in the continuum limit. The common practice is to subtract the UV divergent part by hand, a procedure which is not unique and often ambiguous, in particular in systems with more than one scales. Even with the UV divergent part removed, the resulting expression could still depend sensitively on physics at scales much smaller than the size of the entangled region. As a result, in the limit of taking to infinity, one often does not recover the expected behavior of the IR fixed point (see for example the case of a free massive scalar in Sec. VI).

Such a situation is clearly awkward both operationally and conceptually. We should be able to probe and characterize quantum entanglement at a given macroscopic scale without worrying about physics at much shorter distance scales.

In this paper we show that there is a simple fix of the problem.111See also Hertzberg:2010uv () for a discussion based on free theories. Consider a quantum field theory on which is renormalizable non-perturbatively, i.e., equipped with a well-defined UV fixed point. Suppose is the entanglement entropy in the vacuum across some smooth entangling surface characterized by a scalable size .222In this paper we will always consider to be a closed connected surface. Also note that not all closed surfaces have a scalable size. In Sec. II and Appendix A we make this more precise. We introduce the following function

(1)

In Sec. II we show that it has the following properties:

  1. It is UV finite in the continuum limit (i.e. when the short-distance cutoff is taken to zero).

  2. For a CFT it is given by a -independent constant .

  3. For a renormalizable quantum field theory, it interpolates between the values and of the UV and IR fixed points as is increased from zero to infinity.

  4. It is most sensitive to degrees of freedom at scale .

The differential operator in (1) plays the role of stripping from of short-distance correlations. The stripping includes also finite subtractions and is -dependent; it gets of rid of not only the UV divergences, but also contributions from degrees of freedom at scales much smaller than . can be also be used at a finite temperature or finite density where it is again UV finite in the continuum limit. In the small limit it reduces to the vacuum behavior while for large we expect it to go over to the thermal entropy.

may be considered as the “universal part” of the original entanglement entropy, a part which can be defined intrinsically in the continuum limit. Below we will sometimes refer to it as the “renormalized entanglement entropy,” although this name is clearly not perfect. We believe such a construction gives a powerful tool for understanding entanglement of a many-body system. As an illustration, in Sec. III we show that the entanglement entropy of a non-Fermi liquid also has a logarithmic enhancement just as that for a Fermi liquid by a simple dimensional analysis. We also predict the behavior of the entanglement entropy from higher co-dimensional Fermi surfaces.

In the rest of the paper we focus on the behavior of in the vacuum, studying its possible connections to renormalization group flow (RG) and the number of degrees of freedom along the flow. Items (2)–(4) above, especially (4), indicate that can be interpreted as characterizing entanglement correlations at scale . Thus in the continuum limit as we vary from zero to infinity, can be interpreted as describing the renormalization group (RG) flow of the “renormalized entanglement entropy” from short to large distances. In contrast to the usual discussion of RG using some auxiliary mass or length scale, here we have the flow of a physical observable with real physical distances. Its derivative

(2)

can then be interpreted as the “rate” of the flow. With the usual intuition that RG flow leads to a loss of short-distance degrees of freedom, it is natural to wonder whether it also leads to a loss of entanglement. In other words, could also track the number of degrees of freedom of a system at scale ? which would imply (2) should be negative, i.e. should be monotonically decreasing.

For ,333for which is given by two points and there is no need to have a superscript in . a previous result of Casini and Huerta Casini:2004bw () shows that is indeed monotonically decreasing for all Lorentz-invariant, unitary QFTs, which provides an alternative proof of Zamolodchikov’s -theorem Zamolodchikov:1986gt ().

In higher dimensions, the shape of also matters. We argue in Sec. V that has the best chance to be monotonic. At a fixed point, reduces to the previously proposed central charge in all dimensions.444That the entanglement entropy could provide a unified definition of central charge for all dimensions was recognized early on in Ryu:2006ef () and was made more specific in Myers:2010xs () including proof of a holographic -theorem. Its monotonicity would then establish the conjectured -theorems Cardy:1988cwa (); Myers:2010xs (); Jafferis:2011zi () for each . (For notational simplicity, from now on we will denote the corresponding quantities for a sphere simply as and without the superscript.)

In Sec. VI we consider a free massive scalar and Dirac field in , where available partial results again support that555A similar construction which involves the partition function (instead of entanglement entropy) has been used in Friedan:2003yc () in connection with the g-theorem for .

(3)

is monotonic.

In Sec. VII and VIII we turn to holographic systems whose gravity dual satisfies the null-energy condition. In Sec. VII, among other things we show that when the central charges of the UV and IR fixed points are sufficiently close, is always monotonic in all dimensions. Sec. VIII is devoted to numerical studies of various holographic systems in and . We find all the examples support the conjecture that: in , is always non-negative and monotonically decreasing with for Lorentz-invariant, unitary QFTs.

In , where

(4)

we find that while appears to have the tendency to be monotonically decreasing, there exist holographic systems where it, however, is not always monotonic and can become negative.

We conclude in Sec. IX with a summary and a discussion of future directions.

Note Added: When this paper is finalized, we became aware of  Myers:2012ed () which has some overlap with our study. After the first version of this preprint appeared Casini:2012ei () proved that is indeed monotonic, thereby proving the -theorem in three dimensions.

Ii A refinement of entanglement entropy

In our discussion below we will assume that the system under consideration is equipped with a bare short-distance cutoff , which is much smaller than all other physical scales of the system. The continuum limit is obtained by taking while keeping other scales fixed. The entanglement entropy for a spatial region is not a well-defined observable in the continuum limit as it diverges in the limit. The common practice is to subtract the UV divergent part by hand, a procedure which is often ambiguous. The goal of this section is to introduce a refinement of the entanglement entropy which is not only UV finite, but also is most sensitive to the entanglement correlations at the scale of the size of the entangled region.

ii.1 Structure of divergences in entanglement entropy

In this subsection we consider the structure of divergent terms in the entanglement entropy. We assume that the theory lives in flat and is rotationally invariant. The discussion below is motivated from that in tarun () which considers the general structure of local contributions to entanglement entropy in a gapped phase.666We thank Tarun Grover for discussions. We will mostly consider the vacuum state and will comment on the thermal (and finite chemical) state at the end.

Let us denote the divergent part of the entanglement entropy for a region enclosed by a surface as . Then should only depend on local physics at the cutoff scale near the entangling surface. For a smooth , one then expects that should be expressible in terms of local geometric invariants of , i.e.

(5)

where denotes coordinates on , is a sum of all possible local geometric invariants formed from the induced metric and extrinsic curvature of . Note that here we are considering a surface embedded in flat space, all intrinsic curvatures and their derivatives can be expressed in terms of and its tangential derivatives, thus all geometric invariants can be expressed in terms of the extrinsic curvature and its tangential derivatives. The proposal (5) is natural as should not depend on the spacetime geometry away from the surface nor how we parametrize the surface. Thus when the geometry is smooth, the right hand side is the only thing one could get after integrating out the short-distance degrees of freedom. In particular, the normal derivatives of cannot appear as they depend on how we extend into a family of surfaces, so is not intrinsically defined for the surface itself.

Here we are considering a pure spatial entangled region in a flat spacetime, for which the extrinsic curvature in the time direction is identically zero. Thus in (5) we only have for the spatial normal direction. In more general situations, say if the region is not on a spatial hypersurface or in a more general spacetime, then should be considered as a co-dimensional two surface in the full spacetime and in (5) we will have with running over two normal directions.

Given (5), now an important point is that in the vacuum (or any pure state),

(6)

where denotes the entanglement entropy for the region outside , and in particular

(7)

Recall that is defined as the normal derivative of the induced metric and is odd under changing the orientation of , i.e., in it enters with an opposite sign. Thus (5) and (7) imply that should be an even function of . In a Lorentz invariant theory, there is also an alternative argument777We thank R. Myers for pointing this out to us. which does not use (6) or (7). Consider a more general situation with both as mentioned above. The index has to be contracted which implies that must be even in . Then for a purely spatial surface we can just set the time component of to zero, and is still even for the remaining .

As a result one can show that for a smooth and scalable surface of size , the divergent terms can only contain the following dependence on

(8)

See Appendix A for a precise definition of scalable surfaces. Heuristically speaking, these are surfaces whose shape does not change with their size , i.e. they are specified by a single dimensional parameter plus possible other dimensionless parameters describing the shape. For such a surface, one can readily show that various quantities scale with as (see Appendix A for more details)

(9)

where denotes covariant derivative on the surface. As a result, any fully contracted quantity which is even in , such as in (5), can only give rise to terms proportional to with a non-negative integer, which then leads to (8). Below we restrict our discussion to scalable surfaces.

Now let us consider a scale invariant theory in the vacuum. On dimensional ground, the only other scale can appear in (8) is the short-distance cutoff . We should then have

(10)

and so on. For odd , the term is not among those in (8) and thus should be finite. For even , there can be a term at the order and should come with in order to have to the right dimension. We thus conclude that for a scale invariant theory, the entanglement entropy across a scalable surface in the vacuum should have the form

(11)

where for notational simplicity we have suppressed the coefficients of non-universal terms. It is important to emphasize that does not contain any divergent terms with negative powers of in the limit . The form (11) was first predicted from holographic calculations in Ryu:2006ef () for CFTs with a gravity dual. is an -independent constant which gives the universal part of the entanglement entropy. The sign factors before in (11) are chosen for later convenience. As indicated by the superscript, in general depends on the shape of the surface.

For a general QFT, there could be other mass scales, which we will denote collectively as . Now the coefficients in (8) can also depend on , e.g., we can write as

(12)

and similarly for other coefficients. Note that by definition of , we always have and can be expanded in a power series of . Now for a renormalizable theory, the dependence on must come with a non-negative power, as when taking , should not be singular and should recover the behavior of the UV fixed point. In other words, for a renormalizable theory, the scale(s) arises from some relevant operator at the UV fixed point, which implies that should always come with a non-negative power in the limit . This implies that the UV divergences of should be no worse than those in (10). In particular, there cannot be divergent terms with negative powers of for even , and for odd the divergence should stop at order . These expectations will be confirmed by our study of holographic systems in Sec. VII.3 and VII.4 (see e.g. (82)), where we will find that has the expansion , where with the UV dimension of the leading relevant perturbation at the UV fixed point.

So far we have been considering the vacuum. The discussion of the structure of divergences should work also for systems at finite temperature or finite chemical potential. In such a mixed state, while (6) no longer holds, equation (7) should still apply as the short-distance physics should be insensitive to the presence of temperature or chemical potential. Also recall that for a Lorentz invariant system, there is an alternative argument for (8) which does not use (7).

ii.2 Properties of

Given the structure of divergent terms in discussed in the previous subsection, one can then readily check that when acting on with the differential operator in (1), all the UV divergent terms disappear and the resulting is finite in the continuum limit . In fact, what the differential operator does is to eliminate any term (including finite ones) in which has the same -dependence as the terms in (8). We believe, for the purpose of extracting long range correlations, it is sensible to also eliminate possible finite terms with the same -dependence, as they are “contaminated” by short-distance correlations. In particular, in the continuum limit this makes invariant under any redefinitions of the UV cutoff which do not involve .888Since is the scale at which we probe the system, reparameterizations of the short-distance cutoff should not involve . With a finite , does depend on , but only very weakly, through inverses powers of . This will be important in our discussion below.

In the rest of this section we show that the resulting is not only UV finite, but also have various desirable features. In this subsection we discuss its behavior in the vacuum, while in Sec. II.3 discuss its properties at a finite temperature and chemical potential.

For a scale invariant theory, from (11) we find that for all

(13)

is -independent. The sign factors in (11) were chosen so that there is no sign factor in (13). Note that if we make a redefinition of the form where is some mass scale, for odd the UV finite term in (11) is modified. But , and as defined from (13), is independent of this redefinition.

Let us now look at properties of for a general renormalizable QFT (i.e. with a well-defined UV fixed point). Below we will find it convenient to introduce a floating cutoff , which we can adjust depending on scales of interests. At the new cutoff , the system is described by the Wilsonian effective action , which is obtained by integrating out degrees of freedom from the bare cutoff to . The entanglement entropy calculated from with cutoff should be independent of choice of . So should the resulting . Below we will consider the continuum limit, i.e. with bare cutoff .

First consider the small limit, i.e. is much smaller than any other length scale of the system. Clearly as , these other scales should not affect , which should be given by its expression at the UV fixed point. Accordingly, also reduces to that of the UV fixed point, i.e.

(14)

As we will see in Sec. VII.5, studies of holographic systems (with given by a sphere) predict that the leading small correction to (14) is given by

(15)

where with the UV dimension of the leading relevant scalar perturbation. Equation (15) has a simple interpretation that the leading contribution from a relevant operator comes at two-point level. We believe it can be derived in general, but will not pursue it here.

The story is more tricky in the large limit, as all degrees of freedom at scales between the bare UV cutoff and could contribute to the entanglement entropy in this regime. Nevertheless, one can argue that

(16)

as follows. When becomes much larger than all other length scales of the system, we can choose a floating cutoff to be also much larger than all length scales of the system while still much smaller than , i.e.

(17)

where denote possible mass parameters of the system. Now the physics between and is controlled by the IR fixed point, i.e. we should be able to write again as (11), but with replaced by , and by . Then equation (16) immediately follows. In other words, while in terms of the bare cutoff , the entanglement entropy could be very complicated in the large regime, involving many different scales, there must exist a redefinition of short-distance cutoff , in terms of which reduces to the standard form (11) with replaced by , and by . In fact, higher order terms in (11) with negative powers of also imply that generically we should expect the leading large corrections to (16) to have the form

(18)

This expectation is supported by theories of free massive scalar and Dirac fields as we will see in Sec. VI, and by holographic systems as we will see in Sec. VII.5. Holographic systems also predict an exception to (18) which happens when the flow away from the IR fixed point toward UV is generated by an irrelevant operator with IR dimension sufficiently close to , for which we have instead (see Sec. VII.5)999The expression below is derived in Sec. VII.5 for given by a sphere and closely separated UV/IR fixed points. We believe the result should be more general, applicable to generic systems and smooth , but will not pursue a general proof here.

(19)

where .

By adjusting the floating cutoff , one can also argue that should be most sensitive to contributions from degrees of freedom around . Consider e.g. a length scale which is much smaller than . In computing , we can choose a floating short-distance cutoff which satisfies

(20)

As discussed at the beginning of this subsection, by design is insensitive to short-distance cutoff when .101010Of course ultimately as mentioned earlier should be independent of choice , when one includes all possible dependence on including those in coupling constants. Here we are emphasizing that even explicit dependence on should be suppressed by negative powers of . We thus conclude that should be insensitive to contributions of from d.o.f around .

While our above discussion around and after (16) assumes a conformal IR fixed point, the discussion also applies to when the IR fixed point is a gapped phase, where there are some differences depending on the spacetime dimension. For odd , using as an illustration, the entanglement entropy for a smooth surface in a gapped phase has the form (see e.g. also tarun ())

(21)

where is the topological entanglement entropy wenx (); Kita (). We then have

(22)

In gapped phases without topological order, . Thus a nonzero signals the system has long range entanglement, i.e. the system is either gapless or topological-ordered in the IR. The two cases can be distinguished in that for a topological ordered phase should be shape-independent, but in a gapless case, in (14) is shape-dependent.

For even , in a gapped phase we expect that does not have a term proportional to for large , and thus we should have

(23)

Nevertheless, it has been argued in tarun () that the size-independent part of the entanglement entropy contains topological entanglement entropy. Such a topological term could not be captured by , as all terms in (1) contain derivatives with respect to for even . This is not surprising, as in even , the -independent part of the entanglement entropy also contains a finite non-universal local part, as is clear from the discussion around (8). Thus it is not possible to separate the topological from the non-universal contribution using a single connected entangling surface, and one has to resort to constructions like those in wenx (); Kita () to consider combination of certain regions in such a way that the local part cancels while the topological part remains tarun ().

ii.3 Finite temperature and chemical potential

As discussed at the end of Sec. II.1, we expect should also be UV finite in the continuum limit at a finite temperature or chemical potential. Here we briefly discuss its properties, and for simplicity will restrict to a scale invariant theory.111111See also sw2 () regarding scaling behavior of the entanglement entropy at finite . In ref. sw2 () considered the entanglement entropy itself with UV part subtracted manually.

For a scale invariant system at a finite temperature , since there is no other scale in the system, must have a scaling form, i.e.

(24)

In particular, in the high temperature limit, i.e. it must be dominated by thermal entropy at leading order, while in the low temperature limit , it should reduce to . For a scale invariant theory, the thermal entropy has the form , where is volume of the spatial region enclosed by and is some constant. Thus we should have

(25)

More explicitly, for we expect when ,

(26)

where the first term is simply the thermal entropy, denotes terms with negative powers of . The second term is a constant. It would be interesting to compute this constant for some explicitly examples to see whether some physical interpretation (or significance) can be attached to it.

Similarly, with a nonzero chemical potential , as a generalization of (24) we expect that

(27)

Iii Entanglement entropy of a (non)-Fermi liquid

In this section we show that the entanglement entropy of a (non)-Fermi liquids can be obtained by simple dimensional analysis.121212See also a recent discussion in sw2 () based on finite temperature scaling and crossover. See also Ogawa:2011bz (); lizaetal (); Shaghoulian:2011aa (); Iizuka:2012iv () for recent discussion of logarithmic enhancement of holographic “non-Fermi liquids.” Consider a -dimensional system of a finite fermions density whose ground state is described by a Fermi surface of radius . We have in mind a Fermi liquid, or a non-Fermi liquid described by the Fermi surface coupled to some gapless bosons (as e.g. in sungsik ()). In either case, the low energy dynamics of the system involves fermionic excitations locally in momentum space near the Fermi surface, and different patches of the Fermi surface whose velocities are not parallel or anti-parallel to each other essentially decouple. In particular, drops out of the low energy effective action. We thus expect in the large limit, the “renormalized entanglement entropy” should be proportional to the area of the Fermi surface , which can be considered as the “volume” of the available phase space. Since there is no other scale in the system than , should then have the form

(28)

where denotes the area of the entangling surface . In other words, our “renormalized entanglement entropy” should satisfy a “area law.” Using (1) one can readily see that the area law (28) translates into the well-known behavior in the original entanglement entropy FS1 (); FS2 () (see also sw1 ())

(29)

where in the logarithm is added on dimensional ground and denotes other non-universal parts. We note that this result does not depend on whether the Fermi surface has quasi-particles or not, i.e. whether it is a Fermi or non-Fermi liquid, only depends on the expectation that is proportional to the area of the Fermi surface.

This analysis can also be immediately generalized to predict the qualitative behavior of the entanglement entropy of higher co-dimensional Fermi surfaces. For a co-dimensional Fermi surface we should have131313We define the co-dimension with respect to the full spacetime dimension .

(30)

which implies that in the entanglement entropy itself141414These results were also obtained by B. Swingle (unpublished).

(31)

Thus we find that there is a factor only for even co-dimensional Fermi surfaces. These results are again independent of whether there are quasi-particles. Note that for a Fermi point where , equation (31) in consistent with one’s expectation that for massless fermions there is a universal term only for even . For general , at least for free fermions, the alternating behavior of logarithmic enhancement in (31) may also be understood (by generalizing an argument of sw1 ()) as follows: at each point of a co-dimensional Fermi surface, there is an -dimensional free fermion CFT. The appearance in (31) is then consistent with the fact that for an -dimensional CFT, there is a universal piece only for even.

It would be interesting to see whether our discussion may also be used to understand the logarithmic enhancement in the entanglement entropy of the critical spin liquids in zhang () which are described by a projected Fermi sea state.

Iv Renormalized Rényi entropies

In addition to the entanglement entropy, other important measures of entanglement properties of quantum states include Rényi entropies, which are defined as

(32)

The entanglement entropy can be obtained from them by analytic continuation in :

(33)

The discussions of Sec. II–Sec. III for the entanglement entropy can be applied almost without any change to Rényi entropies. The main results include:

  1. The divergent pieces of should be expressible in terms of the local geometric invariants as in (5).

  2. For a pure state, equations (6)–(7) apply to Rényi entropies. As a result the renormalized Rényi entropies , obtained by acting the differential operators in (1) to , are UV finite.

  3. For a CFT, the Rényi entropies have the same structure as (11), i.e.

    (34)

    with

    (35)
  4. For a general (renormalizable) QFT interpolate between the values of the UV and IR fixed point

    (36)

    and are most sensitive to the degrees of freedom at the scale .

  5. For a scale invariant theory at finite temperature and chemical potential, should take the scaling form

    (37)

    Unlike for entanglement entropy we do not expect a simple relation with the thermal entropy in the high temperature limit.

  6. All Rényi entropies contain logarithmic violations of the area law for a (non)-Fermi liquid

    (38)

    This generalizes a previous result for the free Fermi gas calaetal ().

The key difference between entanglement entropy and the Rényi entropies is that strong subadditivity does not hold for the latter. In the following sections we discuss how entanglement entropy is related to the number of degrees of freedom. These relations do not appear to have obvious generalization to Rényi entropies.

V Entanglement entropy as measure of number of degrees of freedom

For the rest of this paper we will restrict our discussion to the renormalized entanglement entropy in the vacuum. In Sec. II we showed that in the vacuum introduced in (1) has various desirable features:

  1. It has a well-defined continuum limit.

  2. For a CFT, it is independent of and given by the universal part of the entanglement entropy (11)

    (39)
  3. For a renormalizable quantum field theory, it interpolates between the values of UV and IR fixed points as is increased from zero to infinity.

    (40)

    It should be understood that in (40) if the IR fixed point is described by a gapped phase, then is either given by the topological entanglement entropy (for odd ) or zero (for even ).

  4. It is most sensitive to degrees of freedom at scale .

Thus provides a set of observables which can be used to directly probe and characterize quantum entanglement at a given scale . As discussed in the Introduction, these properties also imply that we may interpret the dependence on as a RG flow. A natural question which then arises is whether could also provide a scale-dependent measure of the number of degrees of freedom of a system. Given the physical intuition that RG leads to a loss of degrees of freedom, a necessary condition for this interpretation is then

(41)

which in turn requires (given (40))

(42)

Note that (42) alone is enough to establish as a measure of the number of degrees of freedom for CFTs, while establishing as a measure of degrees of freedom for general QFTs requires a much stronger condition (41).

For , the entangled region becomes an interval (there is no shape difference) and equation (1) reduces to151515The function (43) has also been discussed e.g. in Casini:2009sr (); Hertzberg:2010uv () as the universal part of the entanglement entropy in .

(43)

which for a CFT then gives Callan:1994py (); Holzhey:1994we (); Calabrese:2004eu ()

(44)

where is the central charge. In this case, Zamolodchikov’s -theorem Zamolodchikov:1986gt () ensures (42) and there exists a beautiful proof by Casini and Huerta Casini:2004bw () showing that is indeed monotonically decreasing for all Lorentz-invariant, unitary QFTs. Note that while there already exist an infinite number of -functions Cappelli:1990yc () including Zamolodchikov’s original one, has some special appeal, given that it also characterizes the entanglement of a system. We would like to propose that it gives a “preferred” -function which best characterizes the number of d.o.f. of a system at scale .

In higher dimensions, the shape of also matters. Could (41) and (42) apply to generic or only certain shapes? For this purpose, consider first the weaker condition (42).

For even , since appears as the coefficient of the divergent term in (11), it can be expressed in terms of integrals of the geometric invariants associated with  (recall (5)), and in particular related to trace anomaly Ryu:2006ef (); solodu (). For example, for  Ryu:2006ef (); solodu ()

(45)

where and are coefficients of the trace anomaly161616We use the convention (46) with the Weyl tensor and the Euler density. The relation to and of solodu () is , . Also note there is a minus sign in the definition of in (11). In comparing with solodu () we have also set the extrinsic curvature in the time direction to zero. and (below )

(47)
(48)

In (47) is the intrinsic curvature on and in the second equality of (47) we have used the Gauss-Codacci relation in flat space. is the Euler density for and is a topological invariant with value for a surface with spherical topology. is a Weyl invariant and is zero for a sphere. For a sphere we then have

(49)

while for other shapes, will be a linear combination of and . More than twenty years ago, Cardy conjectured that Cardy:1988cwa () and its higher dimensional generalizations obey the analogue of -theorem. Only very recently was it proven for  Komargodski:2011vj (). In addition, there are strong indications any combination of and (including ) will not satisfy such a condition Komargodski:2011vj (). Thus for only for could the condition (42) be satisfied. For higher even dimensions the situation is less clear, but one again has Ryu:2006ef (); Myers:2010xs (); Casini:2011kv ()171717In addition, the structure of (45) persists in higher even dimensions except that there are more Weyl invariants Myers:2010xs (); hung2 ().

(50)

For odd , does not arise from local terms in (11), thus we do not expect that it can be expressed in terms of local geometric invariants on . This is in contrast to the local shape dependence in the even dimensional case. It would be interesting to understand how depends on the shape of the entangling surface. It was found in Casini:2011kv () that for a sphere

(51)

where is the finite part of the Euclidean partition for the CFT on . Some support has been found that (equivalently ) satisfies the condition (42Myers:2010xs (); Jafferis:2011zi ().

To summarize, for given by a sphere, there are (strong) indications that could satisfy (42) and thus provide a measure of the number of degrees of freedom for CFTs in all dimension (including both odd and even) Myers:2010xs (); Casini:2011kv (). Below we will simply call the central charge of a CFT. For , other shapes could still provide a similar measure, which will be left for future investigation. For the rest of the paper we study the stronger condition (41) for in and . For notational simplicity, we will drop the superscript “sphere” in various quantities and denote them simply as and .

Vi Free massive scalar and Dirac fermions in

In this section we consider for a free massive scalar and Dirac field. For a free massive field, we expect that should approach that for a massless field as , and as . We would like to see whether it is monotonic and positive in between. Recently, in the limit of , it was found in Hertzberg:2010uv (); Huerta:2011qi () that ( is a short-distance cutoff)

(52)
(53)

From (3) we thus find that

(54)
(55)

which are indeed monotonically decreasing with . Note that the fall-off in the above expressions is also consistent with our earlier expectation (18).

We emphasize that if one simply subtracts the divergent part in (52), then the resulting

(56)

approaches minus infinity linearly as and thus does not have a good asymptotic limit. The presence of such a linear term can be understood as a finite renormalization between the short distance cutoffs of the UV and IR fixed points, as discussed in Sec. II. In contrast, approaches zero as as one would expect of a system with a mass gap.

We have also calculated numerically for a massive scalar field181818Compared to that of a scalar, the computation for a Dirac fermion requires significantly more computer time to achieve the same accuracy. We will leave it for future investigation. for a range of as shown in Fig. 1 (see Appendix B for details for the numerical calculation). The numerical result is consistent with our expectation of the limiting values of in the small and large limits, and also suggests that it is monotonic in between.

Figure 1: for a free massive scalar field: The red point is the value for . The black dashed line is the result of the asymptotic expansion (54). The numerical results are computed following the method of Srednicki:1993im () with a radial lattice discretization. We choose the system size to be , where is the lattice spacing. To avoid boundary effects the restriction to was made. To extend the range of we obtained the results for . In the plots, the orange dots are data points for , the blue ones are for , and the green ones are for . As expected all our data points collapse into one curve as can only depend on in the continuum limit. For more details see Appendix B.

Vii for Holographic flow systems

In this section we discuss properties of (defined for a sphere) for systems with a gravity dual using the proposal of Ryu:2006bv (); Ryu:2006ef () (see Nishioka:2009un () for a review), which relates the entanglement entropy to the area of a minimal surface. Other recent discussion of entanglement entropy in holographic RG flow systems include Myers:2010xs (); deboer (); jimliu (); sinha (); Paulos (); Albash:2011nq ()

We will restrict our discussion to . After a brief discussion of the general set-up, we derive a relation between and the undetermined constant in the asymptotic expansion of the minimal surface. We then show that when the central charges of the IR and UV fixed points are close, for all dimensions, is always monotonically deceasing with at leading order in the expansion ot difference of central charges of the UV and IR fixed points. Thus for flows between two sufficiently closely separated fixed points, appears to provide a good central function.

vii.1 Gravity set-up

We consider a bulk spacetime which describes a renormalization group flow in the boundary theory. We assume that the system is Lorentz invariant. The flow can be induced either by turning on the source for some relevant scalar operators or by certain scalar operators developing vacuum expectation values (without a source). Below we will refer to them as source and vev deformation respectively. We denote the corresponding bulk fields by .

The bulk action can be written as

(57)

where is some positive-definite metric on the space of scalar fields. The spacetime metric can be written in a form

(58)

We assume that has a critical point at with , which corresponds to the UV fixed point. Near the boundary ,

(59)

and the spacetime metric is that of AdS with curvature radius . Einstein equations and positive-definiteness of the kinetic term coefficients require that the evolution of with should satisfy Freedman:1999gp ()

(60)

i.e. is a monotonically increasing function. More generally, equation (60) follows from the null energy condition and Einstein equations, regardless of specific form of the scalar action.

At small , can be expanded as

(61)

where is some mass scale and some positive constant. For a source deformation, with the UV dimension of the leading relevant perturbating operator (i.e. the one with the smallest ).191919Note that when for which , we should replace the second term in (61) by . For a vev deformation we have .202020The above description is for the standard quantization. In the alternative quantization (which applies to ), we have instead for a source deformation and for a vev deformation.

As , we can have the following two possibilities:

  1. Flow to an IR conformal fixed point. In this case approaches a neighboring critical point with , and , . The flow solution then describes a domain wall with the metric (58) interpolating between two AdS with curvature radius given by and respectively, i.e.

    (62)

    Near the IR fixed point, i.e. , can be expanded as

    (63)

    where with the dimension of the leading irrelevant perturbing operator at the IR fixed point and is a mass scale characterizing irrelevant perturbations.

  2. Flow to a “gapped” phase: since has to increase with , instead of approaching a constant as in (62), can blow up as , e.g.

    (64)

    The spacetime then becomes singular at . Given , the singularity in fact sits at a finite proper distance away. From the standard IR/UV connection we expect that the system should be described by a gapped phase in the IR. Explicit examples include the GPPZ Girardello:1999 () and Coulomb branch flow Freedman:1999gk (); Brandhuber:1999hb () which we will discuss in more detail in next section. While one should be wary of such singular spacetimes, they appear to give sensible answers for correlation functions consistent with the interpretation of a gapped phase (see e.g. Bianchi:2001de ()).212121In Coulomb branch flow Freedman:1999gk (); Brandhuber:1999hb () which describes a Higgs phase of the SYM theory, there is a single Goldstone mode corresponding to spontaneous breaking of conformal symmetry. In the large limit, the effect of such a gapleess mode on observables like entanglement entropy can be neglected. We will thus still call it a ”gapped” phase. In this paper we will assume such singular geometries make sense.

vii.2 Holographic entanglement entropy

From the prescription of Ryu:2006bv (), the entanglement entropy for a spherical region of radius is obtained by

(65)

where is the area of a unit -dimensional sphere and is obtained by minimizing the surface area

(66)

with the boundary condition at infinity

(67)

Depending on the spacetime metric, there can be two kinds of minimal surfaces as indicated in Fig. 2. For the disk type, the minimal surface ends at a finite with

(68)

The cylinder type solution extends all the way to with