1 Introduction and Results

MCTP-15-12

Exact results for corner contributions to the

[2.5mm] entanglement entropy and Rényi entropies

[2.5mm] of free bosons and fermions in 3d

Randall Laboratory of Physics, Department of Physics,

University of Michigan, Ann Arbor, MI 48109, USA

In the presence of a sharp corner in the boundary of the entanglement region, the entanglement entropy (EE) and Rényi entropies for 3d CFTs have a logarithmic term whose coefficient, the corner function, is scheme-independent. In the limit where the corner becomes smooth, the corner function vanishes quadratically with coefficient for the EE and for the Rényi entropies. For a free real scalar and a free Dirac fermion, we evaluate analytically the integral expressions of Casini, Huerta, and Leitao to derive exact results for and for all . The results for agree with a recent universality conjecture of Bueno, Myers, and Witczak-Krempa that in all 3d CFTs, where is the central charge. For the Rényi entropies, the ratios do not indicate similar universality. However, in the limit , the asymptotic values satisfy a simple relationship and equal times the asymptotic values of the free energy of free scalars/fermions on the -covered 3-sphere.

## 1 Introduction and Results

For a 3d conformal field theory (CFT) in the ground state, the entanglement entropy for a region whose boundary has a sharp corner with angle can be written as

 S=BLϵ−a(θ)ln(Lϵ)+O(1). (1.1)

Here is a length scale associated with the size of the entangling region, is a short distance cutoff, and is a non-universal constant. The corner contribution to the entanglement entropy is the scheme-independent positive function of the opening angle [1, 2, 3]. Since the entanglement entropy of the region equals that of the complement region, the corner contribution satisfies . If the curve bounding the entangling region is smooth, the logarithmic term is absent, hence must vanish in the limit and it does so quadratically as

 a(θ)=σ(θ−π)2+…    for    θ→π. (1.2)

The value of the corner coefficient depends on the theory.

For the theory of a free real scalar or a Dirac fermion, Casini, Huerta, and Leitao [4, 5, 2] derived expressions that give implicitly in terms of some rather involved integrals. In the limit, one can extract double-integral expressions for the corner coefficient in (1.2). These integrals have been evaluated numerically [5, 6] and the results indicate that the exact values are [6]

 σ(B)=1256    and    σ(F)=1128 (1.3)

for the free boson and free fermion, respectively.

Bueno, Myers, and Witczak-Krempa [6] conjectured that the ratio of the coefficient in (1.2) to the central charge is universal in 3d CFTs and that it takes the value

 conjecture \@@cite[cite]{[\@@bibref{}{Bueno:2015rda}{}% {}]}:        σCT=π224. (1.4)

The conjecture (1.4) has passed non-trivial holographic tests for gravity models with a family of higher derivative corrections [6, 7]. The central charge is defined as the coefficient of the vacuum 2-point function of the stress tensor (see eq. (3) in [6]). For free bosons and fermions, Osborn and Petkou [8] found that and in 3d. So with the values (1.3), the ratio is indeed for both free bosons and fermions.

In this paper, we evaluate analytically the integral expressions [6] of Casini, Huerta, and Leitao [4, 5, 2] for and and prove that their exact values are indeed those in (1.3). This verifies the universality conjecture (1.4) for the case of free bosons and fermions. One way of viewing the conjecture is simply as the statement that the corner coefficient in (1.2) does not contain independent information about the CFT, but is fixed in terms of the central charge .

Turning to the Rényi entropies , one can define a similar corner contribution which in the smooth limit goes to zero as for . (The limit of the Rényi entropy is the entanglement entropy.) It is not known if has any universal properties.

We calculate analytically for the free boson and free fermion using integral expressions for derived in [4, 5, 2].111We are grateful to Horacio Casini for sharing with us the integral expression for . For the free scalar we find

 σ(B)n=n−1∑k=1k(n−k)(n−2k)tan(kπn)24πn3(n−1). (1.5)

Note that when is even, the contribution from must be taken carefully using .

The result for the free fermion is

 σ(F)n=(n−1)/2∑k=−(n−1)/2k(n2−4k2)tan(kπn)24πn3(n−1), (1.6)

where sum is to be taken in integer steps from to .

For low values of , the finite sums of the trigonometric functions in (1.5) and (1.6) simplify quite nicely. The results for first nine values of are

 nσ(B)nσ(F)n2148π2 ≈ 0.00211086164π ≈ 0.0049735931108√3π ≈ 0.001701635216√3π ≈ 0.0042540848+3π1152π2 ≈ 0.001532551+6√2768π ≈ 0.003931335√25−2√51000π ≈ 0.00144219√425+58√52000π ≈ 0.00374840681+34√3π19440π2 ≈ 0.00138643261+20√325920π ≈ 0.0036306172cot(π14)+5cot(3π14)+5tan(π7)4116π ≈ 0.0013487413cot(π14)+22cot(3π14)+15tan(π7)8232π ≈ 0.00354841832+9π(1+2√2)10752π2 ≈ 0.001321611+6√2+4√274+17√27168π ≈ 0.00348777927√3+10cot(π18)+28tan(π9)+35tan(2π9)34992π ≈ 0.00130116135√3+68cot(π18)+77tan(π9)+130tan(2π9)69984π ≈ 0.0034411810125+6π√565+142√554000π2 ≈ 0.001285225+300√5+4√425+58√572000π ≈ 0.00340427

In the case of the scalar, the exact results were guessed by the authors of [6] based on their high precision numerical evaluation of the integrals.

Since the ratios of the central charges of free fermions and bosons differ only by a factor of 2, universality of the ratio would require that obeys some simple, possibly -dependent, relation. Based on our results above, there is no hint of such a simple relationship. Of course to fully exclude this, one would need values of for other 3d CFTs.

As a function of , the Rényi corner coefficient decreases monotonically, as shown on the left in figure 1. When is large, asymptotes to a constant value, which we calculate analytically:

 σ(B)∞=3ζ(3)32π4    and    σ(F)∞=ζ(3)4π4. (1.7)

The appearance of the Riemann zeta-function is intriguing since also shows up in the free energies and Rényi entropies for free scalars/fermions on a 3-sphere, as shown by Klebanov, Pufu, Sachdev, and Safdi [9]. Specifically, the free energy of a free real scalar or free fermion on an -covered 3-sphere behaves as222The authors of [9] work with a complex scalar, so the free energy there is twice that of a real scalar. for with

 F(B)∞=3ζ(3)8π2    and    F(F)∞=ζ(3)π2. (1.8)

Thus, for both free scalars and fermions we have

 σ(B/F)∞=14π2F(B/F)∞. (1.9)

For finite , there is no apparent relation between and , however there are some similarities in the subleading large- behaviors, as we discuss in section 4. The plot on the right in figure 1 shows the large- behaviors of the Rényi corner coefficients . A priori it is not clear if there is any relation at large between and , but it would be curious to test (1.9) in other examples.

The remainder of the paper details the derivations of the results summarized above. In section 2, we derive the results (1.3) for the entanglement entropy corner coefficient . We then evaluate the Rényi entropy corner coefficients in section 3. In section 4, we discuss the asymptotic behavior at large .

## 2 Evaluation of the EE integrals

In this section we describe the procedure for analytically evaluating the integrals for the coefficients and of the entanglement entropy. Our starting point is the integrals [4, 5, 2] presented in equations (B1)-(B3) of [6]. After a change of integration variable from to , the integrals take the form

 σ(B)= −12∫∞0dμ∫∞0dbμ2Ha(1−a)πcosh2(πb),σ(F)= −∫∞0dμ∫∞0db[μ2Ha(1−a)−μF4π]πsinh2(πb), (2.1)

where for the scalar and for the fermion. The functions and are defined as

 H=−T2(chX1+1cX2)+116πa(a−1),          F=−F1F2, (2.2)

with333We simplified the expression for in [6] by writing it in terms of .

 F1= 4πchHa(1−a)[(2a−1)2+μ2]−14ch2(μ2+1),F2= ch[(2a−1)2+μ2]22(2a−1)μ. (2.3)

The functions , , , , and are defined as follows:

 h = 2(μ2+(2a−1)2)sin2(πa)(μ2+1)(cos(2πa)+cosh(πμ)), c = 22aπa(1−a)sec(πa+iπμ2)Γ(32−a+iμ2)√μ2+1(Γ(2−a))2Γ(−12+a+iμ2), X1 = −Γ(−a)[πsinh(πμ2)+icosh(πμ2)(ψ(12+a+iμ2)−ψ(12+a−iμ2))]22a+1μΓ(a+1)Γ(12−a+iμ2)Γ(12−a−iμ2)(cos(2πa)+cosh(πμ)), X2 = (X1 with a replaced by (1−a)), T = 12√h[(h+1)(μ2+1)−4a(1−a)]. (2.4)

Here denotes the digamma function, .

Our first line of attack involves calculating the quantities and that appear in in (2.2). Beyond the immediate cancellations that occur in these ratios, one can perform further simplifications using identities involving gamma functions. Namely, one can use the recurrence relation

 Γ(1+z)=zΓ(z) (2.5)

and the reflection relation

 Γ(1−z)Γ(z)=πsin(πz). (2.6)

Surprisingly, all the gamma functions cancel after a series of such substitutions, giving

 (2.7)

It is suggestive that the pre-factors and the form of these two results are the same. We then proceed by adding them together as in (2.2). The linear combination of digamma functions that appears in the result can be simplified using properties easily derived from (2.5) and (2.6). In the form that is useful for our purpose, these identities are

 ψ(32−a±iμ2)=ψ(12−a±iμ2)+112−a±iμ2

and

 ψ(12+a±iμ2)−ψ(12−a∓iμ2)=πtan(πa±iπμ2).

Then the combination of and that appears in simplifies to

 chX1+1cX2=√μ2+14πa(1−a)⎛⎜ ⎜⎝πsin(πa)sinh(πμ2)μ[cos(2πa)+cosh(πμ)]−csc(πa)cosh(πμ2)(1−2a)2+μ2⎞⎟ ⎟⎠. (2.8)

The last ingredient we need to construct in (2.2) is . Using (2.4), it is

 T=  ⎷((1−2a)2+μ2)2sin2(πa)cosh2(πμ2)(μ2+1)(cos(2πa)+cosh(πμ))2. (2.9)

Further simplifications of depend on the nature of variable , as we will see when we specialize to the cases of the free scalar and the free fermion.

Free scalar. To proceed with the evaluation of the integral , we set as prescribed for the free scalar. It is furthermore convenient to change integration variable . Using that both and are positive, the integrand of simplifies dramatically and becomes

 (2.10)

Next, we integrate by parts. Writing

 σ(B)=164∫∞0dμ∫∞0db [∂∂μ(μ(μ2−b2)cosh(πb)−cosh(πμ))+b2−μ2cosh(πb)−cosh(πμ)], (2.11)

we see that the boundary term vanishes and we get

 σ(B)=1256∫+∞−∞dμ∫+∞−∞db b2−μ2cosh(πb)−cosh(πμ). (2.12)

We have extended the limits of integration to facilitate the change of integration variables

 μ=x−y    and   b=x+y. (2.13)

This separates the two integrations and reduces the expression to

 σ(B)=(18∫+∞−∞dx xsinh(πx))2=1256. (2.14)

This completes the derivation of the result (1.3) for the free scalar.

Free fermion. With given in terms of as in (2.3), we have already done most of the leg-work needed to compute . For the free fermion, we have to take and it is again convenient to change integration variable . After putting everything together, we have

 σ(F)=−132∫∞0dμ∫∞0db μ[π(μ2−b2−1)sinh(πμ)−2μcosh(πb)−2μcosh(πμ)][cosh(πb)+cosh(πμ)]2. (2.15)

We can express the integrand as a total derivative plus remaining terms as

 σ(F)=132∫∞0dμ∫∞0db [∂∂μ(μ(μ2−b2−1)cosh(πb)+cosh(πμ))+1−μ2+b2cosh(πb)+cosh(πμ)]. (2.16)

As before, the boundary term vanishes and we are left with the expression (after extending the limits of integration)

 σ(F)=1128∫+∞−∞dμdb 1−μ2+b2cosh(πb)+cosh(πμ)=1128∫+∞−∞dxdy1+4xycosh(πx)cosh(πy). (2.17)

In the last step, we changed integration variables using (2.13). Since is odd, that part of the integral vanishes and the result is therefore simply

 σ(F)=1128(∫+∞−∞dx 1cosh(πx))2=1128. (2.18)

Thus we have derived the result (1.3) for the free fermion.

## 3 Rényi entropies

We now proceed to calculate the corner coefficients for the Rényi entropies.

Free scalar. For the scalar field, the Rényi corner coefficient is given by the integral (B7) in [6]. We change of the integration variable to to write it as

 σ(B)n=−n−1∑k=1k(n−k)2n2(n−1)∫∞0dμ μ2Hk/n, (3.1)

where is in (2.2) with replaced by . With the simplified expression for from section 2, we get

 !σ(B)n=n−1∑k=1sin2(πkn)32πn2(n−1)∫∞0dμ μ[(n−2k)2+μ2n2]πsinh(πμ)−2μ2n2[cos(2πkn)+cosh(πμ)][cos(2πkn)+cosh(πμ)]2. (3.2)

As before, we write the integrand as a total derivative plus the remaining terms:

 !σ(B)n=−n−1∑k=1sin2(πkn)32πn2(n−1)∫∞0dμ⎡⎢ ⎢⎣∂∂μ⎛⎜ ⎜⎝μ[(n−2k)2+μ2n2]cos(2πkn)+cosh(πμ)⎞⎟ ⎟⎠−(n−2k)2+μ2n2cos(2πkn)+cosh(πμ)⎤⎥ ⎥⎦. (3.3)

The boundary term vanishes and the expression simplifies to

 σ(B)n=n−1∑k=1sin2(πkn)32πn2(n−1)∫∞0dμ (n−2k)2+μ2n2cos(2πkn)+cosh(πμ). (3.4)

The contribution of is easy to calculate and is equal to

 164π(n−1)∫∞0dμ μ2sinh2(πμ2) = 148π2(n−1). (3.5)

For , there are contributions from two integrals:

 I(1)n;k=∫∞0dμcos(2πkn)+cosh(πμ)=2tan−1(tan(πkn))πsin(2kπn)=2sin(2kπn)×{kn,kn/2 (3.6)

and

 I(2)n;k = ∫∞0μ2dμcos(2πkn)+cosh(πμ)=2i[Li3(−e2ikπn)−Li3(−e−2ikπn)]π3sin(2kπn) = −ilog(e2ikπn)[π2+log2(e2ikπn)]3π3sin(2kπn)=2sin(2kπn)×⎧⎪⎨⎪⎩k(n2−4k2)3n3,kn/2.

Above, we manipulated the tri-logarithm using the polylog identity

 Li3(z)−Li3(z−1)=−16log3(−z)−π26log(−z), (3.8)

which holds for .

Combining the results (3.6) and (3), we find that the result is the same for and , namely

 ∫∞0dμ (n−2k)2+μ2n2cos(2πkn)+cosh(πμ) = (n−2k)2I(1)n;k+n2I(2)n;k = 8k(n−k)(n−2k)3nsin(2kπn). (3.9)

Thus, having evaluated the integral in (3.4), we can write as the finite sum

 σ(B)n=124π n3(n−1)n−1∑k=1k(n−k)(n−2k)tan(πkn). (3.10)

Note that taking the limit as described below (1.5), the summand evaluates precisely to the special case (3.5). The expression (3.10) is the result for the Rényi corner coefficient presented in (1.5), so this completes our evaluation for the free scalar.

Free fermion. For the fermion field, the Rényi corner coefficient is given by the integral

 σ(F)n=−2n−112(n−1)∑k>0∫∞0dμ [a(1−a)μ2H−μF4π]a=k/n, (3.11)

where the sum is over from ( even) or ( odd) in integer steps to . Substituting the expressions for and obtained earlier gives

 (3.12)

We then use integration by parts to simplify the integral

 (3.13)

The boundary term integrates to zero and the expression simplifies to

 σ(F)n=12(n−1)∑k>0sin2(πkn)8π(n−1)∫∞0dμ 1−μ2−4k2n2cos(2πkn)+cosh(πμ). (3.14)

The result of the integral again involves a difference of two tri-logarithms and it can be simplified using equation (3.8). The result is even in and we can write the final answer as

 σ(F)n=124π n3(n−1)12(n−1)∑k=−12(n−1)k(n2−4k2)tan(πkn). (3.15)

This is the answer we presented in (1.6). Values for low were tabulated in section 1 for both and .

## 4 Asymptotic behavior of the Rényi’s

Let us now study the large behavior of the Rényi entropy corner coefficients . In particular, we evaluate analytically the value for the coefficients in the limit where . This is done by introducing a new variable and multiplying by . Then in the limit, the sum becomes an integral and we have

 σ(B)∞= 124π∫10dx x(x−1)(2x−1)tan(πx)=3ζ(3)32π4,σ(F)∞= 124π∫1/2−1/2dx x(1−4x2)tan(πx)=ζ(3)4π4. (4.1)

These values turn out to be proportional to the asymptotic values of the calculated on the -covered 3-sphere [9]; as noted in (1.9) we have .

On the right in figure 1, we illustrated the asymptotic behavior of the corner coefficient which we find to be

 σn=σ∞(1+b1n+b2n2+b3n3+…). (4.2)

Numerical fits show that , , and are for the free boson while is and in (4.2) for the free fermion. In fact, fitting up to , we find numerical evidence that for both the scalar and fermion. This indicates that a factor of can be factored out of the function in (4.2), so that

 σn=σ∞n+1n(1+b2n2+b4n4+b6n6+…). (4.3)

It is also interesting to study the ratios of the Rényi corner coefficients at large : based on numerical fits in the range to we find

 σ(B)nσ(F)n=38[1+π212ζ(3)1n2−0.938711491n4+O(1n5)]. (4.4)

The value of the -coefficient is guessed based on the numerics. Specifically, we fit to the function

 38(1+d1n+d2n2+d3n3+…), (4.5)

and find that , , , , etc. The vanishing of the odd powers in (4.5) is consistent with (4.3). Note also that we can now identify the number from the fit (4.2) of the free fermion Rényi entropy corner coefficient at large as ; this is the value given in the caption of figure 1.

Taking the Hurwitz zeta-function expressions for from [9] and using (4.5) to perform a similar fit at large in the range to , we find

 F(B)nF(F)n=38[1−π212ζ(3)1n2+0.9371065861n4+O(1n5)]. (4.6)

Again, the value of the -coefficient is guessed based on the numerics which give , , , , etc. The behaviors of individually is, however, very different that that of the Rényi corner coefficients. We find that while .

It is not clear whether the similarities observed at large between and have any significance or if it is a coincidence. Perhaps future investigations will clarify this.

## Acknowledgments

We are grateful to Horacio Casini for sharing with us the integral expression for the corner coefficient of the free fermion. We would like to thank Pablo Bueno, Finn Larsen, Rob Myers, Silviu Pufu, and William Witczak-Krempa for useful discussions. HE is supported in part by NSF CAREER Grant PHY-0953232 and she is a Cottrell Scholar of the Research Corporation for Science Advancement. MH is supported by a Fulbright Fellowship and by the Department of Physics at the University of Michigan.

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