Some Calculable Contributions to Entanglement Entropy

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by  Mark P. Hertzberg

Some Calculable Contributions to Entanglement Entropy
Mark P. Hertzberg1,2∗ and Frank Wilczek1
1Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
2KIPAC and SITP, Stanford University, Stanford, CA 94305, USA
Entanglemententropyappearsasacentralpropertyofquantumsystemsinbroadareasofphysics.
However, its precise value is often sensitive to unknown microphysics, rendering it incalculable. By
consideringparametricdependenceoncorrelationlength,weextractfinite,calculablecontributions
to the entanglement entropy for a scalar field between the interior and exterior of a spatial domain
of arbitrary shape. The leading term is proportional to the area of the dividing boundary; we
also extract finite subleading contributions for a field defined in the bulk interior of a waveguide in
3+1dimensions,includingtermsproportionaltothewaveguide’scross-sectionalgeometry;itsarea,
1 perimeter length, and integrated curvature. We also consider related quantities at criticality and
1 suggest a class of systems for which these contributions might be measurable.
0
2

n Introduction: The quantum nature of matter is rarely cutoff-independentcontributionstotheentanglementen-
a evident on macroscopic scales, often due to the decoher- tropy in d≥2 dimensions.
J ence of excited states toward classical states. However, In this Letter we do just that, albeit in the very spe-

1 for certain states, such as ground/vacuum states, their cial case of free field theory. We place the system in its
3 quantumnaturecanappear,inprinciple,onmacroscopic groundstateinordertoisolatetheentanglemententropy.

scales. One of the most dramatic properties of quan- Ofcourserealsystemsarenormallyatfinitetemperature,
]
h tum matter is entanglement and its associated entropy, whichleadstoavolumecontributiontoentropy,butthis
t which, if observed on mesoscopic or macroscopic scales, is not our focus. Instead our focus is toward gaining
-
p would be of broad interest. To define this entropy, con- insight into novel quantum phenomena at zero tempera-
e sideraquantumsystemwhosedegreesoffreedomcanbe ture,suchasquantumphasetransitions. Forafreescalar
h
divided into two parts in space A, A (see Fig. 1). The fieldind+1-dimensionsatfinitecorrelationlengthξ (i.e.,
[
geometric or entanglement entropy is defined by the von massµ=1/ξ),weshowthatinadditiontothedivergent

2 Neumann formula S =−Tr (ρ lnρ ), where ρ =Tr ρ terms, such as S ∼A /(cid:15)d−1, there is also a finite area
A A A A A d−1
v is the reduced density matrix of the subsystem A. This law contribution for general smooth geometries
3 quantity has appeared in recent investigations in several
9 A ξ
domains including quantum field theory, condensed mat- ∆S =γ d−1 ln , for d odd,
9 ter physics, quantum computing, and black hole physics. d ξd−1 (cid:15)
0
It is a measure of one’s ignorance of the full system due A
7. toquantumentanglementbetweenthedegreesoffreedom ∆S =γd ξdd−−11, for d even, (1)
0
in the subsystem A and its complement A.
0
1 For d+1-dimensional systems with local dynamics the where γd ≡ (−1)d−21[6(4π)d−21((d−1)/2)!]−1 for d odd
: entanglement entropy typically obeys an area law S ∼ andγd ≡(−1)d/2[12(2π)(d−2)/2(d−1)!!]−1fordeven. For
v A /(cid:15)d−1 ford≥2,whereA isthed−1-dimensional awaveguidegeometrywithspecifiedboundaryconditions
i d−1 d−1
X area of the boundary dividing the subsystem from its (see Fig. 1, left panel) we define and calculate additional
complement[1]. Variousdiscussionsofthisarealawhave power law corrections using heat kernel methods. Those
r
a been made in the literature, including extensive study methods allow us to express the entropy as an expansion
of bosonic systems in Ref. [2] and fermionic systems in in terms of the geometric properties of the waveguide’s
Ref.[3](thelattercaninvolveextralogarithmicfactors). cross-section. We also consider a massless field (ξ →∞)
Withoutfurtherrefinementtheconstantofproportional- anddefineandcalculateunambiguousfinitetermsforthe
ity is usually ill-defined, as it depends sensitively on an interval in a waveguide (see Fig. 1, right panel).
ultraviolet cutoff (cid:15). By contrast, the entanglement en- Thearealawforgeneralsmoothgeometriesinthemas-
tropy of 1+1-dimensional systems is well-defined, since sivecase,aswellasthewaveguideexpansionforboththe
the (cid:15) dependence is only logarithmic. For example, the massiveandmasslesscases,extendtheresultsofRef.[4].
entanglement entropy between a pair of half-spaces of Heat Kernel Method: The replica trick is a powerful
a 1+1-dimensional conformal field theory at correlation method for computing the entanglement entropy. Since
length ξ was shown to be S = c/6 ln ξ/(cid:15), where c is thelogarithmthatappearsinthedefinitionofthatquan-
the central charge of the conformal field theory [4]; here tityisawkwardtocomputedirectly,oneexploitstheiden-
rescaling of (cid:15) does not alter the coefficient of lnξ. This tityS =−Tr (ρ lnρ )=(cid:0)− d +1(cid:1)lnTrρn| andthe
A A A dn A n=1
raisesthechallenge,todefineandcalculateunambiguous, strategy to compute Trρn for integer n and use analytic
A
2
continuation. (A similar analytic continuation is known (cid:45) (cid:45) (cid:45)
A A A A A
to fail for spin glasses [5], and we suspect that it can fail
for the entanglement entropy in complicated quantum
field theories, but it should be safe for the very simple
Ξ L
field theories considered here.) Consider, for example,
a field theory in 1 spatial dimension. In that case the
quantity Trρn is a trace over an n-sheeted Riemann sur-
A
face with cut along the subsystem of interest A. If the
subsystem A is a half-space, then as explained in Ref. [4]
FIG. 1: Waveguide geometry in d = 3. Left: Region A is a
Trρn can be identified with the partition function Z on
A δ half-space at finite correlation length ξ. Right: Region A is
aconeofdeficitangleδ =2π(1−n),andtheentropycan
an interval of length L at criticality.
be recast as
(cid:18) d (cid:19) (cid:12)
S = 2π +1 lnZ (cid:12) (2)
dδ δ(cid:12)δ=0 wherethedotsrepresenttermsthateitherareannihilated
bythe2π d +1| operatororelsevanishintheR→∞
dδ δ=0
which can be calculated analytically. limit,andthereforedonotcontributetotheentropy. We
NowconsiderthewaveguidegeometryshowninFig.1, thereby obtain
leftpanel. Thefieldlivesinthebulkinteriorofthewaveg-
uide, satisfying boundary conditions on its surface. For S = 1 (cid:90) ∞ dtζ (t)e−t/ξ2 (6)
this geometry we formulate a Euclidean field theory on 12 t d−1
0
thespaceC ×M ,whereC isa2-dimensionalconeof
δ d−1 δ
fortheentanglemententropyforwaveguidegeometryind
radius R (infrared cutoff) and deficit angle δ, and M
d−1
spatial dimensions traced over half-space. Evidently the
is the (d−1)-dimensional cross-section of the waveguide.
entropy is determined by the geometry of the waveguide
The cone’s radius (R → ∞) corresponds to the physi-
cross-section, through its heat kernel ζ (t).
cal region in space we are tracing over and the angular d−1
Waveguide Cross-Section: The heat kernel for a closed
direction is associated with a geometric “temperature”
domain satisfying either Dirichlet (η =−1) or Neumann
(imaginary time) direction in the Euclidean path inte-
(η = +1) boundary conditions in dimensions 0, 1, and 2
gral. Let Z be the partition function for a field in its
δ
has the following small t expansion [7]:
ground state defined on this space. For a free scalar field
of inverse mass ξ, the partition function is Gaussian
ζ (t)=1,
0
lnZ =−1lndet(cid:0)−∆+ξ−2(cid:1), (3) ζ (t)= √a + η +...,
δ 2 1 2 πt 2
where ∆ is the Laplacian satisfying the appropriate A ηP
ζ (t)= + √ +χ+.... (7)
boundary conditions on Cδ×Md−1. 2 4πt 8 πt
Now let us introduce the heat kernel for the Laplacian
operator ζ(t) ≡ tr(cid:0)et∆(cid:1). The trace is defined by im- Here a is the cross-sectional length of a waveguide in 2-
posingDirichletorNeumannboundaryconditionsonthe dimensions, while A, P, and χ are the cross-sectional
waveguide ∂M and Dirichlet boundary conditions on area, perimeter length, and integrated curvature of a
d−1
∂C . This allows us to rewrite the partition function Z waveguide in 3-dimensions, respectively. (This expan-
δ δ
and hence the entropy S in terms of the heat kernel: sion is also of use in computations of the Casimir ef-
fect between two partitions in a waveguide, see Ref. [9].)
1(cid:90) ∞ dt(cid:18) d (cid:19) (cid:12)
S = 2π +1 ζ(t)e−t/ξ2(cid:12) . (4) The curvature term for an arbitrary piecewise smooth
2 0 t dδ (cid:12)δ=0 2-dimensional cross-section is given by
waSviengcueidtheeismaadniirfoecldtfporrotdhuectECucδl×idMeand−fi1e,ldthtehehoeraytfkoerrtnheel χ=(cid:88)214(cid:18)απ − απi(cid:19)+(cid:88)121π (cid:90) κ(γj)dγj, (8)
factorizes as ζ(t)=ζδ(t)ζd−1(t). Thus our problem sim- i i j γj
plifies into that of obtaining expansions for two separate
where α is the interior angle of any sharp corners and
heat kernels: One for the 2-dimensional cone ζ (t); and i
δ
κ(γ )isthecurvatureofanysmoothpieces. Forexample,
the other for the (d−1)-dimensional cross-section of the j
χ = 1/6 for any smooth shape (such as a circle) and
waveguide ζ (t). The heat kernel for the cone has the
d−1
χ=(n−1)/(n−2)/6 for any n-sided polygonal (so χ=
form [6]
1/4 for a square). This result differs from Ref. [8] where
(cid:18) (cid:19)
1 2π 2π−δ the curvature piece was argued to be proportional to the
ζ (t)= − +... (5)
δ 12 2π−δ 2π number of corners in an arbitrary shape.
3
Regularization - Finite Terms: Direct insertion of the length. Thethirdtermistopologicalford=3. Notethat
heat kernel expansion into eq. (6) for the entanglement by taking the appropriate number of anti-derivatives it
entropy leads to divergences as t → 0+. These diver- is straightforward to isolate cutoff-independent contribu-
gences are associated with the behavior of the theory at tions to the entropy itself. We have also computed some
arbitrarily short distances. As is known, these lead to exact results. In d = 2 and for a square cross-section of
infinities that cannot be renormalized away: logarithmic width a in d = 3, ζ (t) is known exactly. The result
d−1
in 1 dimension, linear in 2 dimensions, and quadratic in for S is
ξ
3 dimensions [1]. There are various ways to regulate the
divergences. For instance, we could impose a hard cutoff acoth(a/ξ) η
onthetintegralandintegratefromt=tc =(cid:15)2 tot=∞, Sξ = 24ξ + 24, for d=2
and find terms that only diverge in the (cid:15)→0 limit. An-
other procedure is to use Pauli-Villars regularization by S = a2 (cid:104)1+2a(cid:88)(cid:48)f K (2f a/ξ)(cid:105)
subtracting off terms with µ replaced by Λ and taking ξ 48πξ2 ξ n,m 1 n,m
n,m
Λ large. This is perhaps more appealing as it respects (cid:20) (cid:21)
ηa a 1
theunderlyinggeometry. However,eitherapproachgives + coth(a/ξ)+ csch(a/ξ) + , for d=3
48ξ ξ 48
results containing the cutoff parameters (cid:15) or Λ.
Fortunately,byreturningtoeq.(6)wecanidentifycut-
√
offindependentdependenceoftheentropyontheinverse where f ≡ n2+m2, the primed summation means
n,m
correlation length µ=1/ξ. In general, the leading order {n,m} ∈ Z2/{0,0}, and K is the modified Bessel func-
1
behavior of the heat kernel as t→0 is tion of the second kind of order 1. For a(cid:29)ξ we recover
α eq. (12) plus exponentially small corrections. (Note that
ζd−1(t)= t(d−1)/2 +... (9) for the square A=a2, P =4a, χ=1/4.)
GeneralGeometries: Althoughthesubleadingtermsin
where α=A /(4π)(d−1)/2 is a constant. Inserting this
d−1 eq.(12)arespecifictoawaveguidegeometry,theleading
into eq. (6) reveals that the integrand has the leading
terms have a meaning for arbitrary geometries. In par-
order behavior ∼1/t(d+1)/2; giving a divergence of order
ticular, for any boundary in 1 dimension we pick up a
d−1 as t → 0+ with respect to a cutoff, say (cid:15), defined
contribution of 1/12 to S , as is known [10]. For closed
through t = (cid:15)2. (For d−1 = 0 there is a logarithmic ξ
c geometries in 2 dimensions, the leading contribution is
divergence.) This singularity can be regulated by taking
S = P/(24ξ), where P is the perimeter length. For
ξ
some number of partial derivatives of the entropy with
closed geometries in 3 dimensions, the leading contribu-
respecttothecorrelationlengthξ,asthatprocedurepulls tion is S = A/(48πξ2). By integrating up these results,
down factors of t from the exponential exp(−t/ξ2). In ξ
we recover the d = 1,2,3 cases in eq. (1). Furthermore,
particular by taking
using the heat kernel in arbitrary dimensions (9) we re-
(cid:20) (cid:21) cover the general result for arbitrary dimensions.
d+1
k ≡Floor (10)
This general result differs from estimates made in Sec-
2
tion 7 of Ref. [11], where the corresponding term in the
derivatives of S with respect to ξ−2 gives a manifestly entropy did not appear. We have checked our result nu-
finite integral whose value is independent of any cutoff merically for the cases of spheres and cylinders, finding
(note k = 1,1,2 for d = 1,2,3). Hence we define a di- excellent agreement. In fact our numerics suggests that
mensionless, cutoff-independent quantity through the area term is the only polynomial contribution to S
ξ
for large A/ξ2. We can understand that heuristically, as
∂kS
S ≡(−ξ−2)k . (11) follows: In the regime ξ (cid:28)a, where a is a typical length
ξ ∂(ξ−2)k scale of curvature of the boundary, the correlations re-
quiredtofeelthecurvatureareexponentiallysuppressed.
Using eqs. (6) and (7) and integrating t, we obtain
Ontheotherhandiftheboundarycontainssharpcorners,
1 we expect power law corrections to appear. (For related
S = , for d=1,
ξ 12 discussion, see [12].) We have verified this numerically
1 a η for squares.
S = + +..., for d=2,
ξ 24ξ 24 Massless Case - Finite Interval: The previous expan-
1 A η P χ sion requires the field theory to be massive. Let us turn
Sξ = 48πξ2 + 192 ξ + 12 +..., for d=3. (12) now to the critical case (ξ → ∞). To use our strategy
to define finite entropy quantities we need a length scale,
Ford=1thisresultisexact[4]. Ford≥2thisexpansion which will now come from considering a finite interval of
isonlyvalidfora(cid:29)ξ,whereaisatypicalcross-sectional length L, as in Fig. 1, right panel. We can define the
4
cutoff-independent quantity: independent piece emerging after taking k = Floor[(d+
1)/2] derivatives of S with respect to ξ−2. For a waveg-
dS
uide geometry we used our construction to obtain an
S ≡L . (13)
L dL asymptotic expansion of the entropy for small values of
the correlation length to cross-section width ratio. For
The small t heat kernel expansion of eq. (5) is insuf-
arbitrary smooth manifolds the leading order area law
ficient here because we must know the form of ζ (t) not
δ should be applicable. In contrast to the 1-dimensional
only for t (cid:46) L2, but also for t (cid:38) L2 where t is large.
case, with S =1/12, these higher-dimensional entropies
In general the full form of ζ (t) is difficult to calculate. ξ
δ can be large numerically. It would be interesting to ex-
However, we do not need ζ (t) for arbitrary δ, but only
δ tend our results to fields involving alternative dispersion
the specific limit indicated in eq. (4). There are powerful
relations, fermions, and interacting field theories.
toolsavailableforthis,aswenowexplain. Thederivative
of the entanglement entropy can be written in terms of (ii) Measurement of entanglement entropy of the kind
an object defined for 2d conformal field theories known discussed above, in the massive, or non-critical, case, re-
as the c-function, denoted C. It is related to the inverse quires changing the correlation length ξ in such a way
Laplace transform of 1 (2π d +1)ζ (t)| . Convolving that the microphysics is only weakly affected, and mea-
12 dδ δ δ=0
with the transverse density of states, we have suring the corresponding change in entropy ∆S (leav-
ing the cutoff-dependent pieces, such as A /(cid:15)d−1, un-
(cid:90) ∞ √ d−1
(cid:0) (cid:1) affected). Though fluctuations of the vacuum state of
S = dEC L E ρ (E). (14)
L d−1
a relativistic QFT may not be directly measurable [14],
0
we can turn to condensed matter systems. Consider, for
Thec-functionC hasbeenstudiedintensely,seeRef.[13].
example, magnetic media. In the absence of an exter-
ItisknownthatC(0)=1/3andthatC ismonotonically
nal magnetic field, there is a massless mode; but for an
decreasing. Using the heat kernel expansion (7), we can
externallyappliedB-field,φacquiresanadjustableeffec-
inverse Laplace transform to obtain an expansion for the tive mass µ=1/ξ. In the regime: (cid:15)2 (cid:28)ξ2 (cid:28)A, where (cid:15)
density of states ρ (E). The quantity S can then be
d−1 L is the inter-spin spacing, the area law should be an ade-
expressed in terms of a few integrals of C, which have
quate description. Another possibility would be to work
been computed numerically. We find
near, butnottoonear, aquantumphasetransition; then
thecorrelationlengthcouldbevariedinacontrolledway.
1
S = , for d=1,
L 3 (iii) A proposal for obtaining the entanglement en-
a η tropy experimentally by measuring current fluctuations
S =k + +..., for d=2,
L 1L 6 in 1-dimensional electron systems has been presented
S =k A + ηk1P + χ +..., for d=3. (15) in Ref. [15]. Extending such proposals to 3-dimensions
L 2L2 4 L 3 would be of great interest.
Here k ≡ 1 (cid:82)∞dxC(x) and k ≡ 1 (cid:82)∞dxxC(x). The Acknowledgments: WewouldliketothankCarlosSan-
1 π 0 2 2π 0
numerical values are: k ≈ 0.04 and k ≈ 0.01 [11]. For tana, Brian Swingle, and Erik Tonni for helpful discus-
1 2
d≥2 this expansion is valid for a(cid:29)L, analogous to the sions. We thank the Department of Energy (D.O.E.) for
expansion in (12) which was valid for a(cid:29)ξ. supportundercooperativeresearchagreementDE-FC02-
Discussion: (i) We have shown that arbitrary shaped 94ER40818 and M.P.H thanks the Kavli Foundation.
domainshaveanareaterm,givenbyeq.(1),withacutoff ∗Electronic address: [email protected]
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