Lifshitz entanglement entropy from holographic cMERA
Revised: May
Lifshitz entanglement entropy from holographic
Simon A. Gentle 0 1 2
Stefan Vandoren 0 1 2
0 Leiden University , 2333CA Leiden , The Netherlands
1 Utrecht University , 3508TD Utrecht , The Netherlands
2 Institute for Theoretical Physics and Centre for Extreme Matter and Emergent Phenomena
We study entanglement entropy in free Lifshitz scalar eld theories holographically by employing the metrics proposed by Nozaki, Ryu and Takayanagi in [1] obtained from a continuous multiscale entanglement renormalisation ansatz (cMERA). In these geometries we compute the minimal surface areas governing the entanglement entropy as functions of the dynamical exponent z and we exhibit a transition from an area law to a volume law analytically in the limit of large z. We move on to explore the e ects of a massive deformation, obtaining results for any z in arbitrary dimension. We then trigger a renormalisation group ow between a Lifshitz theory and a conformal theory and observe a monotonic decrease in entanglement entropy along this ow. We focus on strip regions but also consider a disc in the undeformed theory.
Conformal Field Theory; Field Theories in Lower Dimensions; Holography

3
Lifshitz entanglement entropy
Massless Lifshitz scalar
Disc geometry
Massive deformation
RG
1 Introduction
2 cMERA and holographic entanglement entropy
Metrics from cMERA
Consistency checks in relativistic theories
Lifshitz theories are characterised by a scaling symmetry under which space and time scale
~x !
~x; t !
zt ;
where z is referred to as the dynamical exponent. Such theories govern quantum critical
points in many condensed matter systems. As a simple example, consider the following
theory of a free massless scalar in d + 1 dimensions:
I =
2
2( r~z )2i :
Little is known about entanglement entropy in such theories. For the special case of
d = z = 2 one can map the ground state to a Euclidean conformal eld theory in 1 + 1
dimensions and compute some subleading universal terms in the entanglement entropy [2{
7], see also [
8, 9
]. Other work on z = 2 in d = 1 can be found in [10, 11]. This will be
useful for us to x the overall normalization factor in the entanglement entropy obtained
from cMERA for the case z = 2, as we discuss in section 3.1.
The discretised theory with d = 1 and arbitrary z, including a mass term, was studied
recently in [12, 13] and some partial results for d = 2 were obtained in [12]. In [13] an
analytical approach was also put forward based on the holographic cMERA technique [1]
by this metric, that ends on this region at the boundary.
However, the holographic dual of a Lifshitz theory has yet to be universally agreed
upon. Whilst a dual spacetime, termed Lifshitz spacetime, was proposed in [16, 17] and
has been studied intensively ever since (see [18] for a review), it is unclear whether the RT
prescription should be applied to this spacetime. Indeed, a study of various perspectives
on the holographic reconstruction of Lifshitz spacetime can be found in [19]. Other recent
work suggests that NewtonCartan geometry may provide a more natural bulk dual for a
HJEP07(218)3
nonrelativistic theory [20]. Regardless, holography typically computes the entanglement
entropy for stronglycoupled eld theories with large central charges, whereas here we focus
on a very di erent setting.
In this paper we use a method inspired by holography that is applicable to free
eld theories. In particular, NozakiRyuTakayanagi (NRT) proposed in [1] that a metric
emerges from a continuous version of the multiscale entanglement renormalisation ansatz
(cMERA) [
21
]. For a given theory, expectation values of the appropriate disentangler
operator determine various components of this metric. In some sense, the NRT proposal
geometrises the entanglement entropy of free elds. Our goal is to compute entanglement
entropies by applying the RT prescription to the cMERA metric for various Lifshitz
theories. As we explain later, our results should be viewed as predictions for eld theory
calculations. We should stress however that, while the NRT proposal is holography
inspired, it is not embedded within the AdS/CFT correspondence, since we apply it to a
single free scalar eld that is neither large N nor at strong coupling. It is well known
that free conformal elds do not have a gravity dual, at least not with a weakly coupled
gravity sector coupled to matter elds. Nevertheless, the NRT proposal is similar in avor
to the RT prescription at a technical level (extremisation of surface areas using metric
geometries), so we can make concrete calculations. The justi cation of this comes from the
MERA approach, and the expectation that a continuum version of it should exist.
We begin in the following section with a brief review of cMERA and the de nition of
the cMERA metric. As a crucial consistency check we rst compute entanglement entropy
in the ground state of a relativisitic free massive scalar theory and compare with known
results. We then turn to our three main calculations of Lifshitz entanglement entropy in
section 3. We consider the original Lifshitz theory (1.2) and two relevant deformations: a
mass term m2 2 and a relativistic term ( r~ )2. We conclude with a discussion in section 4.
2
cMERA and holographic entanglement entropy
The multiscale entanglement renormalisation ansatz (MERA) is a variational approach
based on the renormalisation group (RG) to construct (ground) states in quantum many
body systems and study their entanglement properties [22, 23]. A continuum version was
developed in [
21
] for free elds, which we now summarise. We follow the notation of [1].
{ 2 {
Choose a quantum eld theory in d + 1 dimensions and introduce a length cuto ".
Consider then a oneparameter family of states j (u)i living in the Hilbert space of this
theory. The dimensionless parameter u keeps track of the current length scale, with uUV = 0
and uIR =
1 in the ultraviolet and infrared, respectively. Focus initially on a reference
state j (uIR)i that has no entanglement. Run up the RG scale and generate entanglement
by acting with a unitary transformation based on a local operator K(u). Next act with
a scale transformation L to introduce new degrees of freedom at shorter length scales.
Repeat this process until the UV cuto is reached. The nal state we are interested in
can then be written as
j (uUV)i = U (uUV; uIR)j (uIR)i;
U (u1; u2)
P exp
du (K(u) + L)
(2.1)
i
Z u1
u2
and the variational principle can then be applied to minimise the energy of this state.
The variational parameters are encoded in the coe cients of the interactions in K(u).
Note that the states j (u)i are manifestly translationally invariant if these coe cients are
independent of ~x.
2.1
Metrics from cMERA
The authors of [1] associate a metric in d + 2 dimensions with a type of cMERA. In
particular, they argue that a cMERA yielding a translationally invariant ground state
should correspond to a metric of the form
ds2 = guu(u) du2 +
e2u
"2 d~xd2 + gtt(u) dt2 :
The metric in the RG direction parametrised by u is given by the HilbertSchmidt distance
between cMERA states at nearby scales u. It can be expressed in terms of the variance of
the operator K(u) in the state j (u)i:1
guu(u) = h (u)jK(u)2j (u)i
h (u)jK(u)j (u)i2 :
This metric component e ectively measures the density of disentanglers at the scale u. The
gtt component cannot be determined from the eld theory entanglement on a xedtime
slice and therefore plays no role in our discussion.
The explicit form of this metric can be calculated for the ground state of a free scalar
theory [1], given the choice of disentangler proposed in [
21
]. It depends purely on the
dispersion relation !(k), with k
j~kj, and will be used throughout this paper:
pguu(u) =
2 !
k=eu="
:
The cMERA metric constructed in this way captures how the quantum degrees of
freedom in the eld theory are entangled with each other at di erent RG scales. It has the
1Equation (2.3) is a simple rewrite of equation (90) in [1] using their equations (18) and (21), relating
j (u)i and K(u) with j (u)i and K^(u).
{ 3 {
(2.2)
(2.3)
(2.4)
HJEP07(218)3
avour of holography, but the precise connection with AdS/CFT is not understood since
we are neither at large N nor at strong coupling and no expression for gtt is provided.
For relativistic conformal eld theories, the holographic cMERA approach gives the AdS
metric [1]. And as we review below, applying the RT prescription to the cMERA metric
for massive scalar elds yields the correct answers for the entanglement entropy when the
correlation length is small. Based on this, we take a pragmatic approach and apply the
holographic cMERA techniques to nonrelativistic theories with Lifshitz scaling and with
mass deformations corresponding to small correlation lengths. The method yields
predictions for the Lifshitz entanglement entropy for general values of the dynamical exponent z.
2.2
Consistency checks in relativistic theories
Before embarking on our main calculations, in this section we illustrate the consistency
of this approach with a simple example: a free massive scalar eld in 1 + 1 dimensions.
The action for this theory is (we absorb a factor c2=~ in m such that m has dimension of
inverse seconds)
I =
2
2
and the dispersion relation is !(k)2 = c2k2 + m2. Computing guu via (2.4), we should
therefore associate the following metric with the ground state of this theory:
ds2 =
e2u
2(e2u + (m"=c)2)
2
du2 +
e2u
For m = 0 this reduces to the metric on AdS3 in Poincare coordinates with z = "e u
and gtt =
c2=z2, together with a simple rescaling of x and t. The AdS radius in this
normalisation is RAdS = 2, but one can rescale the overall metric to get any radius.
Our task now is to compute entanglement entropy from this metric using the RT
prescription. We will focus on an interval in the xdirection of width ` at t = 0. It is useful
to de ne the dimensionless quantities
m"
c
J1
and
J2
m`
c
:
We want to compute entanglement entropy as a function of J1 and J2. We require J1 <
1 and J2 > J1 to ensure that the cuto
" is the smallest length scale in the theory.
Furthermore, the regime J2 < 1 means that the correlation length
c=m is larger than
the subsystem size `, whereas J2 > 1 means that the correlation length is smaller than the
subsystem size. The entanglement entropy is known to be di erent in these two regimes [
24
]
 a result we now rederive.
First we change coordinates:
(2.5)
(2.6)
(2.7)
(2.8)
(2.9)
HJEP07(218)3
e2u =
"
2
r2
J
2
1
=)
ds2 =
dr2
4r2 +
"
2
r2
J
2
1
dx2
"2 + gtt dt2 :
The boundaries of the r coordinate are xed by the limits of the cMERA length scale u:
rUV(u = 0) =
"
p1 + J12
;
rIR(u !
1) =
=
=
:
"
J1
c
m
{ 4 {
0.2
0.1
0.1
end at mrUV=c = 1=p26 and the dashed line is the IR cuto
mrIR=c = 1.
connected geodesics and the pair of green lines is the disconnected geodesic. All three geodesics
HJEP07(218)3
We seek geodesics of this metric parametrised by x(r) that end at x(rUV) =
`=2. The
length of the shortest geodesic is proportional to the entanglement entropy. The steps to
nd the appropriate geodesics and compute their lengths are identical to those in [14, 15];
we adapt them to our setup and only present the results and a few intermediate steps.
Two types of geodesic are relevant for a nonzero mass. See gure 1 for examples.
The rst type connects the endpoints and is smooth at the point of deepest extent in r:
x(r?) = 0 and x0(r?) ! 1. The boundary condition for this type of geodesic relates r? to
the interval width `:
J2 = p
1
a2 tanh 1 qa2(1 + J12)
J12 ;
a
mr? :
c
At xed J1, the function on the righthand side is real and nonzero for J1(1 + J12) 1=2 <
a < 1, in accordance with (2.9). It is positive within this range and has a single maximum.
Thus, equation (2.10) cannot be satis ed for large J2, but may have two solutions for small
J2. The second type of geodesic exists for all J2 and consists of two disconnected straight
sections x(r)
`=2 that end at r = rIR.
We must be careful to identify the geodesic with the shortest length for a given J2.
The lengths of the connected and disconnected geodesics are given respectively by
LC = tanh 1
LD =
We demonstrate in
gure 2 that the disconnected geodesic is shorter than any connected
geodesic above a critical value of J2. This value is slightly below that for which a connected
geodesic no longer exists.
In the massless case (i.e. setting m = 0 from the beginning in (2.6)) the connected
geodesic is always the shortest. We can nd its length explicitly as a function of `=":
L = log 4 "
+
2
`
s
` 2
"
3
+ 15 = log
`
"
+ log 2 + O
" 2
`
:
(2.10)
(2.11)
(2.12)
(2.13)
1.0
0.8
0.6
0.4
0.2
0.00.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
J2
HJEP07(218)3
as a function of J2. The red, green and blue dots correspond to the curves plotted in
J2 = 1=2 and the black dot marks the critical value of J2 for this J1. The region to the left of the
vertical dashed line is unphysical since J2 < J1 therein.
In the last equation, we made an expansion in small "=`. The rst term is the dominant term
and leads to the area law, which in 1 + 1 dimensions is logarithmic in `. The entanglement
entropy is proportional to the length of the geodesic and the proportionality factor is known
to be related to the central charge [
25
]:
c
3
`
"
S =
log
+
;
(2.14)
with c = 1 for a real scalar eld.
Given that we have xed the overall normalisation, we can return to the massive case
and perform another consistency check for the case when the correlation length is smaller
than the size of the interval. This translates into large J2, for which the disconnected bulk
curve (2.12) is the shortest. Expanding for small J1 and using the same normalisation
factor c=3 as before, we nd the following leading term:
c
6
1
S =
log
+
=
log
+
;
(2.15)
where we recall that the correlation length is de ned by
c=m. This result for the
entanglement entropy matches perfectly with the universal result of Cardy and Calabrese
in [
24
] applied to the case of one interval with two boundary points.
In the opposite limit, i.e. for large correlation lengths compared to the length of the line
interval, the cMERA approach seems to not reproduce known results. Namely, it was shown
in [26, 27] that the entanglement entropy for a free massive scalar eld in 1+1 dimensions
in the regime `
contains a term proportional to log( log(m"=c)) = log( log J1) in
the limit of small J1. It is unclear to us how such a double logarithm would appear in the
cMERA approach (see also section 3.3 for the case of general z), and it would be interesting
to understand better where this apparent discrepancy comes from.
This concludes the consistency checks on the holographic cMERA techniques applied
to free relativistic scalar eld theories. The generalisation to higher dimensions can also be
done, but we include it in the general analysis of Lifshitz theories with arbitrary dynamical
exponent z. When we set z = 1 we recover the relativistic results.
{ 6 {
We now apply the holographic cMERA technique to compute entanglement entropy in free
Lifshitz scalar eld theories. This possibility was in fact already pointed out in [1] but not
worked out. In fact, in that reference, the case z = 1 treated in the previous section was
also not worked out in detail. The case of d = 1 was recently also considered in [13] which
we review and extend below.
As suggested in [1], all we need is the dispersion relation and the rest is computational.
In this section we perform all the calculations and provide results for arbitrary z and d,
treat the massive case as well, and compute the RG ow of the entanglement entropy from
kz: According to the prescription given in the previous section, we should associate
the following metric with its ground state:
z!1
lim L =
`
"
;
{ 7 {
This can be seen immediately at the level of the metric. Suppose we scale out the factor z2
in front of the metric. Then up to this overall rescaling, this is the z = 1 metric provided
we replace the cuto
" ! z" in (3.1). (Physically, we are not changing the cuto
" for
the Lifshitz theory, it is just a trick to get the answer. Equivalently, one can also rescale
x ! zx.) It is then clear that the length of geodesics on the t = 0 slice is just a rescaling
by z of that obtained from (2.6) at m = 0. This observation was already made in [13],
but there the result was only given for large `=z", which does not permit taking large
values of z.
what we nd:
Here, we also obtain a sensible large z limit. In a discrete model of a continuum
local theory, entanglement entropy is dominated by contributions from nearest neighbors
across the boundary of the entangling region. As z is increased, more and more lattice
sites are involved and we expect the full interval to contribute at in nite z. This is indeed
i.e. it becomes extensive with the region size, so we recover a volume law in the large z
limit. This agrees with the observations and results of [12]. It has to be noted though that
We wish to compute the entanglement entropy of an interval in this state, and for this
we calculate the length of the geodesics in this metric. Remarkably, the result is a simple
rescaling of the massless relativistic result (2.13):
ds2 =
du2 +
z
2
4
e2u
+
0
`
s
`
z"
2
1
There might still be a nonuniversal additive constant independent of `, just like the
coefcient c0 that is cuto dependent. We will leave it out of the discussion here.
The expansion around small values of `=z" gives small deviations from the volume law,
whereas expanding around the area law `=z"
1 gives deviations from the area law,
this limit is only natural in the continuum limit (where also the cuto " is sent to zero),
as long as z
l=".
The entanglement entropy S follows from L by a multiplicative factor. In AdS/CFT,
this factor comes from translating Newton's constant into the central charge of the theory,
which in 1+1 dimensions gives a multiplicative factor of c=3, leading to S = c=3 log(`= )+c0
for relativistic CFTs. The cMERA approach, however, does not x the overall
normalization. The best we can do is to replace the central charge c by an overall multiplicative
constant cz, independent of ` but which can depend on the dynamical exponent z. For a
relativistic scalar eld, we have cz=1 = 1. In general, we call cz the Lifshitz central charge.
Then we obtain the following entanglement entropy formula for general values of z:
In [13], the assumption was made that cz is independent of z. Recent numerical work,
however, shows that this assumption should be relaxed, as the work of [10, 11, 28] showed
that for z = 2, one has cz=2 = 3=4, such that the entanglement entropy for z = 2 starts
like S = 12 log( ` ) to leading order. This implies that cz does depend on z. But the fact
that we don't know cz does not prevent us from making predictions that could be checked
using numerics or other methods. Indeed for xed but arbirtray z, we can still check the
functional dependence on `=z inside the logarithm in (3.4) with numerical methods. Or
stated di erently, we can take the ratio of the entanglement entropies for large and small
values of `=z and compare this to lattice results. This ratio is independent of cz, so is
not sensitive to our ignorance of it. Admittedly, this issue needs to be better understood,
either from further numerical work, or perhaps analytically, by computing cz from the
replica trick or from Lifshitz scale anomaly coe cients. We leave this for further study,
and leave the coe cient cz undetermined in this paper. More conveniently, we will focus
on just the length or area of the minimal surfaces, which do not involve this factor cz.
We now generalise our treatment to higher dimensions.
We make the replacement
dx2
! d~xd2 with ~x = (x1; : : : ; xd) and also the change of coordinates e2u = r"22 in (3.1):
ds2 =
{ 8 {
(3.4)
(3.5)
(3.6)
(3.7)
directions and end at xd(") =
entanglement entropy.
extending to r = r?, respectively:
Our region of interest is now the strip ~x
at t = 0. Translational invariance
of the metric and strip in the additional spatial directions simpli es the problem
dramatically: we seek extremal surfaces of this metric parametrised by xd(r) that ll the other
`
xd
`=2. The area of the smallest surface is proportional to the
We nd the following expressions for the width of the strip and the area of the surface
`
"
= z
b
2d
A = z "d 1
Vd 1
1 + d
2d
" p
2d
"
p
1+2d
2d
1 d
2d
r?=" and regulated the in nite volume of the remaining directions
with Vd 1. Note that these results are again related to those of the relativistic case with a
rescaling by z: we scale " ! z" and multiply A ! zdA .
We deduce from these results that the entanglement entropy follows an area law for
nite z. The equation (3.8) relating ` and r? has a unique solution for given d and z. We
can invert this asymptotically at large `=" and nd the following expansion for the area:
A =
d
z
Vd 1
The rst term represents the area term, and the second term is nite and independent of
the cuto . (See also, for example, [15] for the case z = 1 using AdS/CFT and the RT
formula.) The entanglement entropy is proportional to A, with a proportionality factor
that is not universal for d > 1. In the cMERA approach, this normalisation inherits from
that of the disentangler operator K(u) and is usually
xed by comparing with known eld
theory results for the entanglement entropy. Because of the nonuniversality, we leave this
overall factor undetermined.
In the large z limit, we again obtain a volume law instead of an area law. We can
invert (3.8) asymptotically at large z, nding
b = 1 +
1
2
pd` !2
z"
+
2d
5
24
pd` !4
z"
We substitute this into the area expression (3.9) and nd
which is indeed proportional to the regulated volume ` Vd 1 of the strip.
3.2
Disc geometry
In this section we depart brie y from the strip to consider a region of nite size: the disc.
We study the same state of theory (1.2) but simply write the appropriate cMERA metric
z!1
lim A =
` Vd 1
"d
;
{ 9 {
in di erent coordinates:
The following function describes a surface lying on the constant time slice t = 0 that ends
at r = " on a disc of radius p = R:
ds2 =
This surface is smooth at p = 0, independent of d, minimises the area functional and has
area
z
d
2dd
A = Vol S
d 1
1 + x2
d=2
2F1
2
;
2
;
d d + 1 d + 2
2
;
1
1 + x2
;
x
: (3.15)
z"
2R
The area diverges as the cuto is taken much smaller than the disc radius. For nite z,
this divergence is proportional to the area of the disc:
The lowest subleading divergence is logarithmic when d is odd; in particular, we
recover (3.2) for d = 1 since the interval width satis es ` = 2R. For even d, the small
x expansion contains a constant term which is universal. All these results are
straightforward generalisations and rescalings of the z = 1 case described in, for example, [15].
It is interesting to compare our results with [2], where the entanglement entropy was
studied for d = 2 with z = 2. In particular it was found that for the disc subspace, no
logarithmic terms were found, see case (a) in
gure 1 in [2]. This is consistent with our
general result that for even d, no logarithms appear in the expansion of (3.15). More
interesting would be to study the geometries where logarithmic terms do arise, such as the
rectangular or half disc subspaces studied in [2] as well. To reproduce the coe cients in
front of the logterms using the cMERA approach, we would need to perform extremisation
of surfaces in geometries that end on these rectangles or half discs that have less symmetry.
This is much harder in d 2 but is an interesting problem for separate study. Given that we don't know proportionality coe cient between the area and the entanglement entropy, it is cumbersome to perform such a test at present.
In the large z limit we again obtain a volume law:
z!1
lim A =
Vol Sd 1
d
R
"
d
=
Vol BRd ;
"d
Here BRd is the ddimensional ball whose boundary is the (d
1)dimensional sphere of radius R. From now on we focus exclusively on strip regions. { 10 {
3.3
We now deform the theory (1.2) by adding the mass term m2 2. The dispersion relation
is modi ed to !(k)2 =
2k2z + m2, leading to a cMERA metric
ds2 =
z e2zu
2 (e2zu + (m"z= )2)
2
du2 +
e2u
"2 d~xd2 + gtt dt2 :
The dimensionless parameters that characterise the mass deformation and the region size
can be chosen
J1
"
and
J2
`
;
( =m)1=z ;
respectively. They reduce to (2.7) in the relativistic limit. We have written them in terms
of the correlation length , generalising the one from the relativistic case. We want to
compute entanglement entropy as a function of d, z, J1 and J2. Again we require J1 < 1
and J2 > J1 to ensure that the cuto " is the smallest length scale in the theory. As in the
relativistic case, the correlation length maybe larger or smaller than the interval length or
strip width `, corresponding to the regimes J2 < 1 or J2 > 1 respectively.
Just like the relativistic case covered in section 2.2, there is a competition between two
types of extremal surface: connected and disconnected. We analyse this case in a similar
fashion. We begin with a change of coordinates:
e2zu =
"2z
r2z
J 2z
1
=)
ds2 =
z2dr2
4r2
+
"2z
r2z
"2
1=z d~xd2 + gtt dt2 :
For a connected surface extending to r = r?, the width of the strip and the area of the
surface can be written
J2 = z
Z t?
J1
dt
(t=t?)d
(1 + t2z)p1
(t=t?)2d
AC = z J1(d 1) Vd 1 Z t?
dt
"d 1
J1
td (1 + t2z)p1
(t=t?)2d
;
J 2z
1
;
1
respectively, where it is convenient to parametrise the bulk depth via
a
r?
with t?
a2z
1
a2z
1
2z
:
We can evaluate these integrals exactly for various values of d and z; more generally, they
are straightforward to evaluate numerically. Equation (3.21) relates J2 and r?. We nd that
it may have zero, one or two solutions depending on the value of J2, just like the relativistic
case. We observe an area law divergence in AC that degenerates to a logarithmic divergence
for d = 1.
For example, we have obtained analytical expressions for d = 1, z 2 N:
J2 =
z
1 X
t? n=1
n
p1 + n=t?2
(3.18)
(3.19)
(3.20)
(3.21)
(3.22)
(3.23)
(3.24)
(3.25)
HJEP07(218)3
where the n satisfy
It is clear from the rst term in (3.25) that the length diverges like
log J1 as J1 ! 0. Note
that the correct formulae (2.10) and (2.11) are recovered for z = 1. In addition, analytical
expressions can be obtained for z = d, but these are lengthy and not very illuminating.
The disconnected surface exists for all J2. Its area is in fact independent of J2 and can
be evaluated in closed form:
AD = z J1(d 1) Vd 1 Z 1
"d 1
J1
1
dt td (1 + t2z)
=
d
z
This result is again proportional to the area of the region. For d = 1 we nd a simple
expression that diverges logarithmically as J1 ! 0:
LD =
We must now identify which surface has the least area for a given region size J2. For
xed d, z and J1 we nd that the disconnected surface has minimal area above a critical
value of J2. This value is slightly below that for which a connected surface no longer exists.
We observe the same qualitative behaviour as presented in gure 2 for the relativistic case
regardless of d and z. Besides yielding analytical solutions, the point z = d does not appear
to be special in this calculation.
For any given J1 it is straightforward to numerically nd the critical value of J2 at
which the two types of surface have equal area. In this way we can construct the phase
diagram of a given theory. Our results in d = 1 for various values of z are plotted in gure 3.
Note that the J2intercepts of the phase boundaries follow a roughly linear relationship:
J2(J1 = 0)
z=2. For large values of J2 the disconnected curve is always the shortest.
This is the limit in which the correlation length is smaller than the length of the interval:
"
`. For z = 1 this was the regime in which the result of Cardy and Calabrese
holds: cf. (2.15). For z > 1, the result obtained in (3.28) generalises the CardyCalabrese
result, and we obtain
where we have used the same normalisation factor cz=3 as before that relates the length of
the geodesic to the entanglement entropy for a real scalar eld.
For large correlation lengths
> `, so small J2 (and therefore small J1), we get
again logarithmically diverging terms
log J1. There are no known analytical or numerical
results in this case, but we expect similar discrepancies as for z = 1, where the leading
diverging term involves a double logarithm, log( log J1) in the limit of small J1.
1.2
1.0
(green) and z = 4 (yellow). To the left of each curve the shortest geodesic is connected, whereas
to the right the shortest geodesic is disconnected. The dashed lines mark the boundaries of the
physical region J1 < 1 and J2 > J1. The black dots are extracted directly at J1 = 0.
3.4
RG
In this section we begin with a massless free scalar eld in 1 + 1 dimensions and turn on
an irrelevant Lifshitz coupling:
I =
2
2
Our goal is to compute the entanglement entropy along the entire renormalisation group
ow from the UV Lifshitz theory to the IR CFT. For sublattice entanglement entropy,
such a study was carried out in [13], where it was found that the entanglement entropy
decreases along the renormalisation group
ow, for any starting value of z > 1. This
provides evidence for a generalisation of the ctheorem for entanglement entropy in the
relativistic case [29], applicable to
ows between two Lifshitz xed points. Our analysis
below will give further support for this.
The dispersion relation is given by !(k)2 = c2k2 + 2k2z. After a change of coordinates,
this results in a cMERA metric of the following form:
ds2 =
f (r)2
4r2 dr2 +
dx2
r2 + gtt dt2 :
The function f (r) interpolates between the two limits of the ow as the dimensionless
parameter K
=(c "z 1) is varied:
f (r) =
1 + zK2("=r)2(z 1)
1 + K2("=r)2(z 1)
!
( z; K
1; K
1
1
We seek geodesics of the form x(r) extending to r = r? that end at x(") =
`=2.
the UV. Left: length as a function of log10 K for `=" = 2; 5; 10; 20 (bottom to top). These curves
interpolate monotonically between the correct IR (left) and UV (right) limits. Right: length as a
function of `=" for K = 0:5; 5; 50 (bottom to top, solid). The dashed lines correspond to the IR
(black) and UV (red) limits.
For example, for z = 2 we nd that the interval width and the geodesic length can be
written respectively as
100 `="
HJEP07(218)3
`
"
= pb2
K2
1 + pb2 + K2
L = 2 log b + pb2
1
r?=". Note that the expression for `=" increases
monotonically with b (at xed K) so can be inverted uniquely. It is straightforward to check that
these results have the correct limits: (3.2) for large K and (2.13) for small K. These are
the UV and IR limits, respectively. We plot the geodesic length as a function of K or `="
in
gure 4. We see that the length decreases monotonically along this RG
ow from UV
to IR, as expected. (We have checked that qualitatively similar results can be obtained
numerically for other values of z.)
4
Discussion
An approach was put forward in [1] to geometrise entanglement entropy based on cMERA
techniques [
21
]. It is similar in spirit to AdS/CFT but is applicable to free
elds, i.e.
weak coupling and small N . We have illustrated in this paper that this approach leads to
concrete predictions for the entanglement entropy of free scalar elds. In the relativistic
case, it reproduces wellknown analytical eld theory results, as we showed in our analysis.2
In the Lifshitz case, it leads to new predictions, generalising and extending the observations
made in [13]. The overall normalisation for the entanglement entropy is however not xed
from the metrics introduced by [1]; rather, it depends on the choice and normalisation of
the disentangler operator. In AdS/CFT this normalisation is xed by the dictionary that
2With the exception of the massive case where the correlation length exceeds the length of the interval,
as discussed in section 2.2.
relates Newton's coupling constant to the central charge or number of colours. We have not
been able to determine this overall normalization, which we denoted by cz, and which can
depend on the value of the dynamical exponent z. It would be interesting to understand
if cz is related to Lifshitz scale anomaly coe cients. These anomalies have been studied
in [30{32], but also in earlier work for z = 2 [33, 34] in 2+1 and 3+1 dimensions, and z = 3
in 3+1 dimensions [35]. The results obtained in e.g. [2] show that central charges do appear
in certain logarithmic terms entanglement entropy for z = d = 2 (so 2+1 dimensions), but
it is not straightforward to see the relation to Lifshitz scale anomaly coe cients. Moreover,
for the strip and disc geometries we consider here, these logarithmic terms are absent. It
would be important progress to
nd such a relation if it exists.
In the massless case, our results for free Lifshitz scalars are obtained from a simple
rescaling of the relativistic case, but this is not true in the massive case. In the latter
case, we generalised the CalabreseCardy result to Lifshitz scalar elds with values of the
dynamical exponent z > 1. It would be interesting to repeat our analysis for fermions.
Clearly, it would be important to better understand the validity of the holographic
cMERA approach. From this point of view, our results are merely predictions rather than
solid results. It would be nice to reproduce some of our predictions for Lifshitz entanglement
based solely on eld theoretic techniques. For free Lifshitz scalar elds, one imagines this
should not be so di cult by attempting the replica method, but the presence of longer
range interactions for large z complicates this. In low dimensions, numerical and lattice
methods can also be used, as was illustrated in [12, 13]. We leave this topic for future study.
Acknowledgments
It is our pleasure to thank Temple He, Javier Magan, Ali Mollabashi, Aurelio
RomeroBermudez, Philippe SabellaGarnier and William WitczakKrempa for useful and enjoyable
discussions and correspondence. This work was supported in part by the DeltaInstitute for
Theoretical Physics (DITP) that is funded by the Dutch Ministry of Education, Culture
and Science (OCW).
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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