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The Reflected Entanglement Spectrum for Free Fermions

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We consider the reflected entropy and the associated entanglement spectrum for free fermions reduced to two intervals in 1+1 dimensions. Working directly in the continuum theory the reflected entropy can be extracted from the spectrum of a singular integral equation whose kernel is determined by the known free fermion modular evolved correlation function. We find the spectrum numerically and analytically in certain limits. For intervals that almost touch the reflected entanglement spectrum approaches the spectrum of the thermal density matrix. This suggests that the reflected entanglement spectrum is well suited to the task of extracting physical data of the theory directly from the ground state wave function.
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Prepared for submission to JHEP
The Reflected Entanglement Spectrum for Free
Fermions
Souvik Dutta,1,2Thomas Faulkner,1Simon Lin1
1Department of Physics, University of Illinois, Urbana-Champaign
1110 W. Green St., Urbana IL 61801, USA.
2Veritas Technologies, 2625 Augustine Drive, Santa Clara, CA 95054, USA
E-mail: sdutta9@illinois.edu,tomf@illinois.edu,shanlin3@illinois.edu
Abstract: We consider the reflected entropy and the associated entanglement spectrum for
free fermions reduced to two intervals in 1+1 dimensions. Working directly in the continuum
theory the reflected entropy can be extracted from the spectrum of a singular integral
equation whose kernel is determined by the known free fermion modular evolved correlation
function. We find the spectrum numerically and analytically in certain limits. For intervals
that almost touch the reflected entanglement spectrum approaches the spectrum of the
thermal density matrix. This suggests that the reflected entanglement spectrum is well
suited to the task of extracting physical data of the theory directly from the ground state
wave function.
arXiv:2211.17255v1 [hep-th] 30 Nov 2022
Contents
1 Introduction and summary 1
2 Setup 5
3 The x0limit 8
4 The x1limit 10
4.1 Reflected entropy 10
4.2 Deflected entropy 13
4.3 Next order corrections 14
5 Discussion 20
A Deflected entropy for holographic CFTs 21
B Next order eigenvalue correction to deflected entropy 24
1 Introduction and summary
Entanglement entropy is now a central topic in the study of QFT [1,2]. Entanglement en-
tropy itself is UV divergent, so many of the derivations/proofs of important results involving
entanglement entropy must play a delicate game of UV regularization [3]. We would like
to study quantities insensitive to this regularization procedure. One approach is to study
new quantities, different from entanglement entropy, that are well defined in the continuum
limit. In favorable circumstances these UV finite quantities can be thought of as approach-
ing a regularized version of entanglement entropy in certain limits. A prominent example
[3] is half the mutual information I(A:B)/2of two spatial regions A, B on a fixed Cauchy
slice that are almost complement to each other, but leave a small finite corridor between the
entangling surfaces. The appropriate limit sends the size of the corridor to zero. Another
quantity, the focus of this paper, is half the reflected entropy SR(A:B)/2which uses this
same geometric setup for A, B [4].
An advantage of reflected entropy compared to the mutual information is that the
reflected entropy is an actual von Neumann entropy of some density matrix - the reflected
density matrix. Thus we can use the same regulator to study the reflected entanglement
spectrum as a proxy for the regular entanglement spectrum. For a 2d (non-chiral) CFT
the entanglement spectrum for Aa single interval of length Lwas computed by Calabrese-
Lafevre [5]. The result reads:
D(λ) = δ(λλmax) + ln λ1
max
λf(ln λ1
max ln λmax)f(t) = I1(2t1/2)t1/2θ(t)(1.1)
1
where λmax = (L/)c/6and I1is the Bessel function. This reproduces the expected Renyi
entropies:
exp(Sn(A)/(n1)) = ZdλλnD(λ)(1.2)
computed in [6,7]. There are several undesirable features to this formula. The spectrum
depends on the UV cutoff and so will depend on the regularization procedure. The
spectrum is continuous (aside from the single delta function at the edge of the continuum).
It is also universal, only depending on the central charge of the CFT and none of the other
CFT data such as the operator spectrum and OPE coefficients. In contrast the reflected
entanglement spectrum is UV insensitive, discrete and depends on the operator spectrum
and OPE coefficients of the CFT.
We now give a brief introduction to reflected entropy and the associated spectrum,
see [4] for further details. Given a density matrix ρacting on a finite-dimensional Hilbert
space H, one can form the canonical purification |ρiby interpreting ρas a state in the
doubled Hilbert space
|ρi End(H) = H H,(1.3)
where His the dual of H. This doubled Hilbert space is equipped with the inner product
hρ|σi=Tr(ρσ). In the case where ρAB HA HBis a bipartite density matrix, the
canonical purification lives in the space (HA HA)(HB HB) HAABBand one
defines the reflected entropy as the von Neumann entropy
SR(A:B) = S(AA)|ρAB i=SvN(ρAA),(1.4)
where ρAA= trBB|ρABihρAB|is the reduced density matrix obtained by tracing over
HBB. The reflected entanglement spectrum is simply the spectrum of ρAA?and we claim
this is discrete even in the continuum limit.
Taking the continuum limit proceeds as follows. As we send BAc, the complement
region to A, the reflected entropy reduces to twice the entanglement entropy which is now
divergent. This divergent behavior can be understood as the non-existence of a tensor
factorization H 6=HAHAcof the global Hilbert space without introducing a cutoff. This
is an intrinsic property of type-III von Neumann algebras AAthat govern the local bounded
operators associated to region A[8]. Keeping a finite gap between Aand Bhowever allows
reflected entropy to be used as a regulated version of entanglement entropy. In particular
for two disjoint regions AB, the “split property” [9,10] guarantees the existence of at
least one type-I factor Nsplitting of the local algebras
AA N A0
B.(1.5)
The canonical purification introduced in (1.3), in the algebraic language, corresponds to
the state induced on a canonical type-I splitting factor, which can be written algebraically
as [11]
N=AAJABAAJAB,(1.6)
where JAB is the anti-unitary Tomita-Takesaki modular conjugation operator associated to
AAB and the vacuum state. In this language the reflected entropy is defined as the von
2
Neumann entropy of this type-I factor. In particular, the density matrix of Nis trace-class,
with a well-defined and discrete spectrum [12], allowing one to make sense of the entropy
and spectrum directly in the continuum.
It was shown in [4] that in the AdS/CFT setting, the reflected entropy is dual to the
area of entanglement wedge cross-section [13]. Since the entanglement wedge cross-section
is typically a deep bulk probe of the emergent geometry, it is interesting to study the
reflected entropy in a more general class of QFTs. Indeed, there has been many previous
works computing the reflected entropy on different quantum systems, such as free fermions
[14], free scalars [15], CFT in arbitrary dimensions [16], 3D Chern-Simons [17], holographic
tensor networks [18] and JT gravity with EOW branes [19].
In this note we will focus on free fermions in 1 + 1 dimensions. Reflected entropy in
free fermion systems was also already studied in [14,20]. In this paper we will make a few
new observations. In particular compared to [14] we will work directly in the continuum
theory bypassing the need of discretization. Some of our results will be numerical although
we will also make some new analytic predictions in various limits.
Consider two intervals Aand Bon some equal time slice in vacuum and define the
cross-ratio of the end points of these intervals as x. We will give analytic predictions for
SRin the limit where the Aand Bintervals are far separated x0or nearly touching
x1. In the former case the reflected entropy behaves as:
SR=α(xln x) + β+. . . (1.7)
where we give a simple integral expression (3.6) for αand a numerical prediction (3.7) for
β. This is in agreement with our numerics as well as the numerics in [14] and qualitative
agreement with [16]. In the later case we show that to the leading order
SR=1
6(ln(1 x) + ln 4) + . . . (1.8)
in agreement with the universal behavior of 2D CFTs [4]. In addition we give the next order
correction to SR, agreeing well with our numerics. We summarize the analytical predictions
and numerics in Figure 1.
We also derive analytically the entanglement spectrum of ρAA?as x1. It takes
the simple form of that of the spectrum of the thermal density matrix for a free chiral
fermion on the circle in the NS sector with inverse temperature to circle length ratio β/L
1/ln(1 x). By studying perturbative corrections to this later spectrum we give evidence
that the reflected density matrix approaches rapidly the thermal density matrix. This agrees
well with the eigenvalues obtained from numerical method, see Figure 2.
In this paper we do not make use of the replica trick, as in previous work [14], but we use
the correlation matrix technique [22] that is based on the fact that a many-body Gaussian
state is entirely determined by it’s two point correlation function. And so the many-body
entropy can also be extracted from this correlation function. It is sufficient to know certain
modular flowed two point functions in order to construct the relevant correlation matrix
that computes the reflected entropy. These modular flow correlators were computed in the
continuum in [2325].
3

+
 ( )


     






 -
Figure 1. Numerical result of free fermion reflected entropy. We plot here the markov gap [21]
SRMI(A:B)(where MI is the mutual information) of the free Fermion. In contrast to the
reflected entropy which is divergent as x1, the markov gap is finite and approaches 1/6 ln 4 as
x1, as implied from the general behavior of 2D CFT [4]. This asymptotic value is shown as the
gray dashed line in the figure. The orange dots are obtained from numerical approximations of the
correlation kernel (2.11). The asymptotic formula at x0is obtained from perturbation theory
around small x(3.4). The asymptotic formula at x1is the entropy calculated from thermal
distribution (4.25) plus next order corrections (4.56). We have also included the reflected entropy
of a holographic CFT (divided by 2cfor comparison), which undergoes a phase transition in which
its value jumps from O(c0)to O(c)at x= 1/2.
Our analytic computations proceed as follows. We setup a systematic expansion for
the eigenfunctions of the correlation matrix in the limit x1via a certain matching
procedure, between the endpoints of A. This is similar to a QM scattering problem. In this
limit one starts out with a continuum of Rindler eigenfunctions near each Aendpoint and
the discrete spectrum arises from a matching condition in-between. The discreteness of the
spectrum is important for the finite-ness of the reflected entropy.
The plan of this paper is as follows. We setup the singular integral equations in Sec-
tion 2. In Section 3and Section 4we discuss the x0limit and x1limits respectively.
In the later limit we make a more careful study of the spectrum, computing sub-leading
corrections to the thermal spectrum in Section 4.3. We also introduce a s-modular flowed
version of reflected entropy, that we call deflected entropy in Section 4.2 - this is a natu-
ral one parameter generalization of reflected entropy. We compute this quantity for free
fermions and also in AdS3/CFT2using the methods of [26] - the dual roughly corresponds
to a reflected geodesic that tracks the entanglement wedge cross section but picks up a
boosts of rapidity sat the Ryu-Takayanagi surface. In Section 5we discuss our results,
and make some conjectures about more general CFTs. Numerical results will be presented
throughout the paper but are summarized in Figure 1.
4
=
=
=
=
---






λ
Figure 2. Eigenvalues λmof the correlation kernel obtained by numerical method (dots) versus
the spectrum (4.21) of a thermal partition function with appropriate temperature (solid lines) for
various x1. Only the first few eigenvalues are shown. λmare symmetric across m= 1/2line.
Although not shown in this figure, the discrepancy between analytics and numerics is well resolved
by including the second order corrections, see Figure 5.
2 Setup
a1
b1
a2b2
φ
A
B
z
z
Figure 3. Our setup for calculating the reflected entropy for a free fermion on a circle. Global
conformal symmetry allows us to fix three of the four the interval endpoints A= [a1, b1] = [0, π]
and B= [a2, b2] = [3π/2φ, 3π/2 + φ]. The angle φis related to the conformal cross-ratio by
x= 2 sin φ/(1 + sin φ). The modular conjugation for region Atakes z¯z, or equivilently θ θ
after circular identification.
We would like to compute the reflected entropy for 2d free chiral fermions ψand two
intervals A, B on a circle. As discussed in [14], the reflected density matrix is still Gaussian
(that is the Fermion fields satisfy Wick’s theorem) so that the reduced density matrix is
5
completely determined by the two point correlation function of fundamental fermions. We
can then infer the entropy by solving the singular integral equation of this correlation kernel
[22]. The entire system consists of Fermions in AB and in the doubled system (AB)?. This
later system is governed by Fermionic operators b
ψ(x) = i˜
JABψ(x)˜
J
AB for xAB. Since
Fermions anti-commute, we need a generalization of the Tomita-Takesaki theory to a graded
algebra [14,20,27]. Denoting Γ = (1)Fwhere Fis the Fermion number operator, we
can form the Klein operator Z=1iΓ
1i, which is a unitary operator that acts as 1on even
states and ion odd states. The modular conjugation operator ˜
JAB for the graded system
is then related to the vacuum modular conjugation of Tomita-Takesaki theory (for the full
fermionic algebra of operators in AB)JAB by ˜
JAB =ZJAB =JABZ. The conjugated
modes b
ψ(x)then satisfy the canonical anti-commutation relations amongst themselves as
well as with the original AB system. We are interested in the entanglement entropy of AA?
so we may restrict to correlation functions of ψ|Aand b
ψ|A.
In order to understand the smooth B limit we will choose to geometrize the A?
reflected fermions by using instead the following description:
e
ψ(x)˜
JAB ˜
JAψ(x)˜
J
A˜
J
AB =JABJAψ(x)JAJAB , x Ac(2.1)
which is a fermion that now lives on Acwhere Acis the complementary region to A. The
˜
JAoperator acts as CPT conjugation with a reflection across the Aentangling surface, thus
these still represent the same modes as b
ψ|A. The system AA?has now been mapped to the
entire circle: S1=AAc. If we additionally define e
ψ(x)|Aψ(x)then e
ψis the new Fermion
that lives on this circle: it satisfies the standard canonical anti-commutation relations. We
thus, simply need to work out the state of the e
ψfermion on this circle. As discussed above
this is determined by the correlation function of e
ψ. Note that we expect this to be an
appropriately smooth state since near A one can check that JAB acts geometrically like
the Rindler reflection or JA, so the two modular operators cancel out.
Notice that the B now reproduces the Fermion on a circle with the vacuum state.
This is as expected and will give zero reflected entropy. The free fermion correlator on the
cylinder in the NS vacuum |iis the following distribution:
h|ψ(x)ψ(y)|i=Pz1/2w1/2
π(zw)+1
2δ(xy)z=eix, w =eiy (2.2)
where 0< x, y 2πon the cylinder. We have the canonical anti-commutation relations
{ψ(x), ψ(y)}=δ(xy)(with all other anti-commutators vanishing). Consider a basis
of normalized functions on the circle that is anti-periodic: ei(m+1/2)x(2π)1/2. We can
consider the modes:
ψm=1
2πIdz
iz zm+1/2ψ(x)mZ(2.3)
which gives {ψ
n, ψm}=δm,n from the canonical anti-commutation relations.
The Fourier modes of this correlator (m, n Z) can then be calculated:
Cmn =Idz
zi2πzm+1/2Idw
wi2πw(n+1/2) h|ψ(x)ψ(y)|i= Θmδmn (2.4)
6
where Θm= 1 if m0and 0otherwise. Thus we find that the correlation function/matrix
becomes a projector with eigenvalues 0and 1.
Let us now compute the correlation function when B6=:
TrAAcρAAce
ψ(x)e
ψ(y)=
h|ψ(x)ψ(y)|ix, y Aor x, y Ac
∂y
∂yJ1/2h|ψ(x)∆1/2
ABψ(yJ)|ixA , y Ac
∂x
∂xJ1/2h|ψ(y)∆1/2
ABψ(xJ)|iyA , x Ac
(2.5)
where xJis given by the geometric action of ˜
JAon x: if xAthen xJAc. We have
used Tomita-Takesaki theory [28] to replace the modular conjugation operator JAB with the
modular operator AB for the vacuum state. The phase of the square root can be fixed by
demanding we reproduce the vacuum correlator as B . This correlation function along
with the Gaussian nature of ρAAcwill allow us to compute its spectrum. The spectrum of
ρAAcis the same as the spectrum of the reflected density matrix ρAA?or equivilently the
density matrix on the canonical type-I factor N.
The correlation functions can be computed from the formula for modular flow for
fermions on the plane [2325]:
Dψ(z)∆is
ABψ(w)E=Dψ(z)∆is
ABψ(w)E=eπs
2πi(zw)11e2πs
Qe2πs (2.6)
Q=(za1)(za2)(wb1)(wb2)
(zb1)(zb2)(wa1)(wa2)(2.7)
where A= [a1, b1]and B= [a2, b2]and we may consider the boundaries of these intervals
to be complex coordinates in the plane.
For Fermions on the circle we are free to fix a1= 1, b1=1and a2=ie, b2=ie
and set z=eix and w=eiy. Here we may consider xas a holomorphic coordinate on the
cylinder with 0Re x2π(not to be confused with the cross-ratio that we will introduce
later) and in these coordinates A= [0, π]and B= [3π/2φ, 3π/2 + φ](see figure 3). We
use:
Dψ(x)∆is
ABψ(y)EDψ(x)∆is
AB ψ(y)Ecyl =∂z
∂x 1/2∂w
∂y 1/2Dψ(x)∆is
AB ψ(y)E
(2.8)
and xJ=x. If we subtract the vacuum correlator we find:
De
ψ(x)e
ψ(y)EρAAcDψ(x)ψ(y)E(2.9)
=1
4πi sinxy
2θA(x)θAc(y) sin(x) sin(y) sin(φ)
(sin2(xy
2)sin φsin2(x+y
2) + cos φsinxy
2cosx+y
2)
(xy)
where θA(x)(θAc(x)) is a unit step for xA(Ac). Define the Fourier transform of the
correlation kernel
Cmn =Z2π
0
dx
2πeix(m+1/2) Z2π
0
dy
2πeiy(n+1/2) De
ψ(x)e
ψ(y)EρAA?Dψ(x)ψ(y)E
(2.10)
7
So that we have
Cmn =Zπ
0
dx Z2π
π
dy 1
4π2sinxy
2sin(x(m+ 1/2) y(n+ 1/2)) sin(x) sin(y) sin(φ)
(sin2(xy
2)sin φsin2(x+y
2) + cos φsinxy
2cosx+y
2)
(2.11)
from which it is clear that Cmn = Cnm (from sending x2πy). Furthermore
xπxand y3πygives:
Cmn = (1)mnCmn (2.12)
which implies that Cmn vanishes unless mnis even. Also it is clear that Cm,n =
C1m,1n.
The actual correlation matrix is:
Cmn = Cmn +δmnΘm(2.13)
with Θmdefined in (2.4). The second term comes from the vacuum correlator (this is the
projector which gave zero entanglement when Cmn = 0.) The reflected entropy can be
computed via [22]
SR=tr(Cmn ln Cmn)tr((1 Cmn) ln(1 Cmn )) (2.14)
After truncation of the modes we can numerically compute Cmn and use this to
extract the reflected entropy (Figure 1,4,6,7) and entanglement spectrum (Figure 2,5). The
relevant cross ratio is:
x=2 sin φ
1 + sin φ(2.15)
We truncate the matrix at some finite value |m|,|n| Lso that the resulted entropy
converge to prescribed tolerance. 1
Starting from the next section we will study the reflected entropy under various limits.
Clearly the limit φ0(x0) is most naturally studied on the cylinder; whereas the
limit φπ/2(x1) is most appropriately studied using Rindler space, since this is the
limit where we recover the usual (divergent) entanglement entropy.
3 The x0limit
Let us first consider the case when φ0. We will apply degenerate perturbation theory
to the correlation matrix, which allows us to extract the φand φln φterms of the reflected
entropy. Taylor expand (2.11) to the first order in φwe have:
Cmn φRe Zπ
0
dx
π
eixm sin(x)
1 + ieix Z2π
π
dy
π
eiyn sin(y)
1 + ieiy
1
2 sin2(xy
2)(3.1)
= (1)nφRe Zπ
0
dx
πZπ
0
dy
π
fm(x)fn(y)?
2 cos2(xy
2), fm(x) = eixm sin(x)
1 + ieix (3.2)
1The appropriate value of Lwhere we see convergence is highly dependent on the cross-ratio xas well
as desired precision. In general we need higher Lwhen x1. For the plots in this note the value of L
ranges from 20 to 500.
8
Since the eigenvalue is degenerate at zeroth order, we need to diagonalize the two
matrices on these degenerate subspaces:
A= lim
φ0
1
φΘ∆CΘB= lim
φ0
1
φ(1 Θ)∆C(1 Θ) (3.3)
where we defined the projector Θmn =δmnΘm. Define the eigenvalues of these matrices as
ak, bkrespectively such that ak, bk0. Then the eigenvalues of Cmn are approximately
(1 φak)and φbkto first order in φ. The leading correction to the reflected entropy is:
SRX
k
(φak(1 ln (φak)) + φbk(1 ln (φbk))) (3.4)
=φln φX
k
(ak+bk) + φX
k
(ak+bkakln akbkln bk)(3.5)
For the φln φterm the sum of eigenvalues is just the trace of the respective matrices which
we can compute explicitly:
X
k
(ak+bk) = 1
4π2Zπ
0
dx Zπ
0
dy sin(x) sin(y)
cos3(xy
2)(cosxy
2+ sinx+y
2).149 (3.6)
It seems hard to find the coefficient of the term linear to φusing analytical method. Instead
we have from the direct eigenvalue decomposition of matrices Aand Bthat
X
k
(ak+bkakln akbkln bk).560 (3.7)
These coefficients agree with our numerical results, see figure 4.
Fermion
Asymptotics (small x)
0.0 0.1 0.2 0.3 0.4 0.5
0.00
0.05
0.10
0.15
0.20
0.25
x
Rferm
Figure 4. The reflected entropy for the analytical prediction (3.4) versus the numerics. Numerical
fitting of the data points with x < 0.1to SR=αφ ln φ+βφ gives {α, β}={0.149,0.563}, agreeing
well with the expression we obtained in (3.6) and (3.7).
9
4 The x1limit
We move on to studying the reflected entropy in the limit x1(or φπ/2) in this section.
We will setup a systematic expansion for the eigenfunctions of the correlation matrix via a
certain matching procedure between the endpoints of A. Similar to QM scattering, solving
for these matching conditions discretizes the eigenvalue spectrum and we obtain a finite
reflected entropy.
4.1 Reflected entropy
We need to reformulate our calculations adapted to Abeing a half space cut. We pick
a1= 0, b1=and a2=1/b, b2=b. The cross ratio of these points is:
x=(a1b1)(a2b2)
(a1a2)(b1b2)= 1 b2(4.1)
where we restrict to 0<b<1. We also have:
Q=z(z+ 1/b)(wJ+b)
wJ(z+b)(wJ+ 1/b)(4.2)
where we now used directly (z, w)(rather than (x, y)) as coordinates for the reflected
fermion. Also wJ=w. Taking the limit b0we find the correlator for the reflected
fermion approaches (note that Q1):
lim
b0De
ψ(z)e
ψ(w)EρAAc
=ΘA(zA(w)+ΘAc(zAc(w)
2πi(zwi)(4.3)
The eigenfunctions of this operator are Rindler modes with a continuous spectrum. This
continuum leads to a divergent entanglement (since the operator is no longer trace class).
These eigenfunctions are [23]:
fA
ν(z) = 1
2πθ(z)z1/2 fAc
ν(z) = 1
2πθ(z)(z)1/2 (4.4)
and these satisfy:
Z
0
dz 1
2πi(zwi)fA
ν(z) = 1
2(1 tanh(πν))fA
ν(w), w A(4.5)
and Z0
−∞
dz 1
2πi(zwi)fAc
ν(z) = 1
2(1 + tanh(πν))fAc
ν(w), w Ac(4.6)
To resolve this continuum we need to take the limit b0more carefully. We hold
|z|/b, |w|/b fixed as we send b0(with fixed ratio |z/w|). This zooms in on the region
near the entangling surface. Inside this region the correlator is:
δDe
ψ(z)e
ψ(w)EρAAc1
2πi(zw2wz/b)ΘA(zAc(w)(zw)(4.7)
10
Overall we can approximate the correlator as:
C(z, w)De
ψ(z)e
ψ(w)EρAAc1
2πi(zw2wz/b(θ(w)θ(z)θ(w)θ(z)) (4.8)
We can write this correlator as:
C(z, w) = ez
∂z 1/2ew
∂w 1/21
2πi(ezewi)(4.9)
where
ez(z) = (z) + z
12z/b θ(z),ew(w) = (w) + w
12w/b θ(w)(4.10)
and the new coordinate satisfies b/2ez . So the negative axis gets compactified.
On this half space we know the Rindler eigenfunctions:
e
fν(ez) = 1
2πθ(ez+b)(ez+b)1/2 (4.11)
and these are the eigenfunctions of the integral Kernel (ezewi)1. Taking into account
the conformal scale factors the would be eigenfunctions of C(z, w)are:
fν(z)e
fν(ez)ez
∂z 1/2
=1
2π(z+b/2)1/2 θ(z)+(b/2)2(b/2z)1/2+ θ(z)
(4.12)
with eigenvalue (1 tanh(πν))/2. This eigenfunction behaves badly at z . This is to
be expected since at this point (when z1/b) our scaling limit breaks down. Instead we
must match onto a new solution here. If we again go back to the full correlator (determined
by (4.2)) and hold fixed |z|band |w|bas we send b0then we find:
δDe
ψ(z)e
ψ(w)EρAAc1
2πi(zw+ 2/b)ΘA(zAc(w)(zw)(4.13)
Overall we can approximate the correlator here as:
D(z, w)De
ψ(z)e
ψ(w)EρAAc1
2πi(zw+ 2/b(θ(w)θ(z)θ(w)θ(z)) (4.14)
which we can write as:
D(z, w) = 1
2πi(bzbwi)(4.15)
where
bz(z) = (z)+(z2/b)θ(z),bw(w) = (w)+(w2/b)θ(w)(4.16)
the new coordinate satisfies 0bz and −∞ bz 2/b (which is a domain that wraps
around .) This can be described via 1/bz b/2. On this domain we use the Rindler
eigenfunctions:
b
fν(bz) = 1
2πθ(1/bz+b/2)(bz)1(1/bz+b/2)1/2+ (4.17)
11
which has the same eigenvalue, for the kernel (bzbw+i)1as we used in the previous
patch. The eigenfunction for the integral equation of interested, in this new scaling limit
must then be:
fν(z)1
2πθ(z)z1(1/z +b/2)1/2+ +θ(z)z1(b/2)2 (1/z +b/2)1/2
(4.18)
The small zlimit of this expression should match onto the large zlimit of the previous
expression (4.12) up to an overall scaling:
κθ(z)z1(1/z)1/2+ +θ(z)z1(b/2)2 (1/z)1/2
(z)1/2 θ(z)+(b/2)2(z)1/2+ θ(z)(4.19)
which gives κ= 1 and (b/2)4 =1. This is our quantization condition. More explicitly
the two limits of the eigenfunction given above match smoothly if this condition is satisfied.
We have the following allowed values of ν:
νm=π(m1
2)
2(ln(b/2)) (4.20)
where mZ. These give rise to eigenvalues of Cas:
Cm=1tanh(πνm)
2= (1 + e2πνm)1(4.21)
The reflected entropy is:
SR=
X
m=−∞
(Cmln Cm(1 Cm) ln(1 Cm)) (4.22)
Since the spacing is small as b0, we can approximate this by an integral with density:
2Z1
0
dC
dC
dm
m(Cln C) = 2(ln(b/2))
π2Z1
0
dC 1
C(1 C)(Cln C) = (ln(b/2))
3
(4.23)
Writing this in terms of the cross-ratio we have:
SR1
6(ln(1 x) + ln 4) + . . . (4.24)
which was as predicted by the replica trick [4]. In principle we can compute corrections to
this as an expansion in (1 x)and 1/ln(1 x).
In fact we can give a more general expression that is leading in (1 x)but one that
re-sums all 1/ln(1 x)effects. Firstly we notice that (4.22) is the entropy deriving from a
partition function:
SR=β2ββ1ln Z(β)Z(β) =
Y
m=0
(1 + qm+1/2)2(4.25)
12
where q=ei2πτ and the inverse temperature is:
τ=iπ
2(ln((1 x)/4)) (4.26)
We learn that the spectrum of the reflected density matrix approaches that of a free fermion
in the NS sector. So the reflected entropy is simply given by the appropriate Jacobi theta
function.
And indeed, as is often the case, the universal term in (4.24) arises simply via the Cardy
formula. We would speculate that this is a more general result: the reflected entanglement
spectrum approaches that of the thermal partition function of the CFT under considera-
tion, with temperature given by (4.26). There seems to be some connection between the
relfected entropy and the computable cross norm negativity [29] (at least for some Renyi
generalization of reflected entropy) which could possibly be used to give a more general
proof of this fact.
4.2 Deflected entropy
We now study a generalization of the reflected entropy that we call deflected entropy. It is
given by applying a modular flow on the correlation function (2.5) by an additional amount
of sR:
D˜
ψ(x)˜
ψ(y)EρAAc
=
ψ(x)ψ(y), x, y Aor x, y Ac
∂yJ
∂y 1/2Dψ(x)∆1/2+is
AB ψ(yJ)E, x A, y Ac
∂xJ
∂x 1/2Dψ(xJ)∆1/2is
AB ψ(y)E, y A, x Ac
(4.27)
When s6= 0 this correlator is not continuous across the entangling surface. To fix
this discontinuity we replace the geometric reflection map by another vacuum modular flow
(with respect to region A) with an amount s+i/2. In our current coordinate settings
this action is simply
xxS=e2πsx(4.28)
This extra vacuum modular flow for the A region leaves the entropy invariant since it is
generated by an operator that acts solely in AA?(working in the original frame of ABA?B?)
- the easiest way to see this is by using the modular operator for the split state, which agrees
with the vacuum flow inside A. When s= 0 we simply get back the geometric reflection;
whereas when s6= 0 this additional flow smooths out the jump discontinuity across the
entanglement surface. Algebraically one can think of this construction as performing a
Connes cocycle flow [3032] defined with the AB algebra and for the vacuum and split
state. See for example [30].
13
The analysis of the previous subsection still carries over as long as we assume b<eπs
when we scale b0. We can approximate the new correlation function as
C(x, y)'
1
2πi
1
xy2xy
beπs cosh(πs)(θ(x)θ(y)θ(x)θ(y)) ,|x|,|y| eπs
1
2πi
1
xy+2
beπs cosh(πs)(θ(x)θ(y)θ(x)θ(y)) ,|x|,|y| eπs
(4.29)
The effect of the Connes cocyle flow is to shift bin both regimes accordingly and the
matching point from 1to eπs. One can show the spectrum is now determined by:
νm=π(m1/2)
2 ln(2 cosh(πs)/b)(4.30)
such that the reflected entropy becomes:
SR1
6(ln(1 x) + ln 4 + 2ln(cosh(πs))) + . . . (4.31)
This result also agrees well with our numerics, see Figure 7.
In the Appendix Awe perform a different computation of the deflected entropy in
AdS3/CFT2. We use this to compare to the free fermion computations. We find:
SR,holographic =c
6ln
1 + w
1w+O(c0), w =x(1 + e2πs)2
(x+e2πs)(1 + xe2πs)(4.32)
From which we get the following asymptotic when x1:
SR,holographic =c
6(ln(1 x) + ln 4 + ln(cosh(πs))) + ··· (4.33)
The overall factor of c(where in holographic theories cis large) accounts for the extra
degrees of freedom. Comparing to (4.31) we see similar behavior aside from the factor of 2.
There is of course no reason to expect agreement. This difference persists away from x1
as can be seen in Figure 7.
4.3 Next order corrections
In this subsection we systematically improve on our results for the spectrum as x1. This
also serves to convince the reader that our results are under control.
We firstly setup the leading order answer more carefully. We will construct approximate
eigenfunctions of the integral equation, valid as b0(that is x1). This also fills in
some holes in the original discussion in Section 4.1. This subsection is a bit technical so
the reader might prefer to skip to the conclusions.
Let us see that the functions we wrote down are really approximate eigenfunctions as
b0.Define the function:
gν(z)(z+b/2)1/2 θ(z)+(b/2)2(b/2z)1/2+ θ(z)θ(1 |z|)
+θ(z)z1(1/z +b/2)1/2+ +θ(z)z1(b/2)2 (1/z +b/2)1/2 θ(|z| 1)
(4.34)
14
where we have designated an arbitrary matching point of |z|= 1. This function has a
non-uniform expansion as b0. For |z| bthen |gν(z)| b1/2, for |z| 1/b then
|gν(z)| b1/2and finally for |z| 1then gν(z)1.
Consider, for w > 0the following integral:
Z
−∞
dzC(z, w)gν(z)(4.35)
We need to check this integral in the various regimes. We first consider wb. One can see
that the integral is dominated by the regime 1/w < z < 1/w giving an answer O(b1/2):
Z
−∞
dzC(z, w)gν(z)λνgν(w)(4.36)
Similarly for w1/b the integral is dominated in the region z > 1/w and z < 1/w
giving an answer O(b1/2)with the same form as above. Finally for w1we only get a
contribution from away from the scaling regimes, and z > 0(since the kernel is otherwise
suppressed by O(b)for z < 0when w > 0) where:
Z
−∞
dzC(z, w)gν(z)Z
0
dz 1
2πi(zwi)z1/2 =λνw1/2 λνgν(w)(4.37)
Thus, at least for w > 0it looks like gνis an approximate eigenfunction with eigenvalue
λν.
A similar analysis applies to the case w < 0, where the O(1) regime is now:
Z
−∞
dzC(z, w)gν(z)Z0
−∞
dz (z)1/2+
2πi(zwi)(b/2)2 θ(1 + z)(b/2)2 θ((z+ 1))
=λν(b/2)2 (w)1/2+ + ((b/2)2 + (b/2)2 )Z0
1
dz (z)1/2+
2πi(zwi)(4.38)
λνgν(w) + [(b/2)2 + (b/2)2 ] λνθ(1 + w)(w)1/2+ +Z0
1
dz (z)1/2+
2πi(zwi)!
(4.39)
So all together:
Z
−∞
dz(C(z, w)λνδ(zw))gν(z)[(b/2)2 + (b/2)2 ]Gν(w)(4.40)
where Gν(w)is θ(w)times the function above and satisfies Gν(w) O(1) if |w|
1and O(1) for |w| band O(b1)if |w| 1/b. Thus we find that gνis only
an approximate eigenfunction upon imposing the quantization condition discussed in the
previous subsection.
We can now prove approximate orthogonality for distinct eigenvalues:
0Z
−∞
dw Z
−∞
dzg?
νm(w)(C(z, w)λνnδ(zw))gνn(z)(λνmλνn)Z
−∞
dwg?
νm(w)gνn(w)
(4.41)
15
and we can compute the normalization by moving slightly away from the quantization
condition for one of the ν’s above:
(λνmλν)Z
−∞
dwg?
νm(w)gν(w)[(b/2)2 + (b/2)2 ]Z0
−∞
dwg?
νm(w)Gν(w)(4.42)
One can see that the later integral I=R0
−∞ dwg?
νm(w)Gν(w)is dominated in the order one
regime where it evaluates to:
I=e(m1/2) Z0
−∞
dw λνθ(1 + w)(w)1/2+ +Z0
1
dz (z)1/2+
2πi(zwi)!(w)1/2m
(4.43)
This integral is most efficiently evaluated by taking νcomplex and so that the wintegral
can be done first (the integral is well defined as stated above)
I=e(m1/2) λνλνm
i(ννm)(4.44)
Thus: Z
−∞
dwg?
νm(w)gνn(w)δm,n4(ln(b/2)) (4.45)
This gives (at least formally) the completeness relation:
X
m=−∞
g?
νm(w)gνm(w0)1
4(ln(b/2))δ(ww0)(4.46)
We can now attempt to find corrections to these eigenfunctions and eigenvalues using
perturbation theory. We write:
hm(z) = gm(z) +
X
n=−∞
δµmngn(z) + . . . (4.47)
where δµmn =O(b). Plugging this into the eigenfunction equation:
Z
−∞
dzC(z, w)hm(z)=(λm+δλm)hm(w)(4.48)
To first order in bwe have:
4(ln(b/2))δλm=Z
−∞
dw λmδ(gm(w)) + δZ
−∞
dzC(z, w)gm(z)g?
m(w)
=Z
−∞
dw Z
−∞
dzδ(C(z, w))gm(z)g?
m(w)
(4.49)
which does not depend on dmn or δνm. One sees the usual simplicity of first order pertur-
bation theory.
We now seek the leading term in (4.49). We expect an expansion of the form:
δλm=
X
k=1
bk`k(ln b)(4.50)
16
where `kare functions of ln b. We will aim for the order k= 1 term.
Since the integrand in (4.49) has a non-uniform expansion in bwe must expand it in
various regions z, w b, 1, b1. We expand only in C(z, w), keeping the leading terms
in the wave-functions and the measure dzdw (which gets bdependence after scaling into
these regions.) This procedure goes under the name of matched asymptotic expansion [33
35]. One often finds power law divergences after isolating the various regions using scaling
arguments. The general rule is that, since the total integral is well defined, then any power
law divergences will cancel amongst the various regions, that is after a re-arrangement
of the order in the bexpansion (these divergences typically lead to enhancements in the
bexpansion that mix between the orders.) In some instances we can proceed by using
dimensional regularization. We give νa small imaginary part and this removes any power
law diverges (log’s would show up as poles, but we do not find any.)
The diagonal regions (z1, w 1),(zb, w b)and (z1/b, w 1/b)all give rise
to O(b2)corrections. Since the pole/singularity in the kernel lives in the diagonal regions,
we have the symmetry δ(C(z, w)) = δ(C(w, z)) = δ(C(w, z))?which allows us to restrict
the integral to z > w:
4(ln(b/2))δλm=Zz>w
dwdzδ(C(z, w))gm(z)g?
m(w) + Zz<w
dwdzδ(C(z, w))gm(z)g?
m(w)
= 2Re Zz>w
dwdzδ(C(z, w))gm(z)g?
m(w)(4.51)
after re-labeling wzin the second term.
The non-diagonal regions with one z1or w1naively give b1/2contribution, but
they evaluate to simple power laws that do not contribute after following the above rules.
We are left with the crossed regions (zb, w 1/b)and (z1/b, w b). Note that the
correlator satisfies C(z, w) = C(z1, w1)/(zw)under inversion. Also gm(1/z) = g
m(z)/z,
one finds that the contribution from the two crossed regions are equal. 2Therefore we only
need to study one crossed region, say (z1/b, w b):
δC(z, w) = 1
2πiz θ(zw) + (w/b + 1)(zb 1) + w/b
(w/b + 1)(zb 1) w/b θ(z)θ(w)
+(w/b 1)(zb + 1) z/b
(w/b 1)(zb + 1) + z/b θ(z)θ(w)(4.52)
The wave function in this region is
gm(z)g?
m(w)i()m(1 + 2w/b)1/2+mθ(w) + (1 2w/b)1/2mθ(w)
×(2/(zb))(1 + 2/(zb))1/2+mθ(z) + (2/(zb))(1 2/(zb))1/2mθ(z)
(4.53)
It is convenient to use the scaled coordinates (related to ˆz, ezdefined above):
x= (1 + 2/(bz)) , y = (1 + 2w/b)θ(w) + (1 2w/b))1θ(w)(4.54)
2This follows from the scaling behavior of the 2D free Fermions.
17
where we have assumed z > 0. In the scaled region we cover 0< y < and 1< x < .
After rescaling to x, y coordinates we find:
4(ln(b/2))δλm=b()m
πRe Z
1
dxx1/2+mZ1
0
dyy1/2+m1 + xy
y(x+y)
+Z
1
dxx1/2+mZ
1
dyy1/2+m(4.55)
These later integrals converge if we analytically continue the various appearances of νm
differently. While this is rather crude, it does the job of extract the non-power law diver-
gences.
The final result is:
δλm=b
2(ln b/2)
()msech(πνm)
(1 + 4ν2
m)+O(b2)(4.56)
This leads to a shift in the inferred spectrum:
δm =b
2
()mcosh(πνm)
π2(1/4 + ν2
m)+. . . (4.57)
Which competes with the leading order answer when m2
π2(ln b/2)2. While we don’t
expect perturbation theory to break down here (since δλmremains small), it will become
difficult to extract the spectrum at this order of m, since the density matrix has an ex-
ponentially small depends on these energy shifts. These shifts that we found analytically
agrees surprisingly well with our numerics, see Figure 5:
With the eigenvalue corrections at hand, one can check that the correlation matrix
(hence also the density matrix) converges to the thermal answer (4.21) by examining the
distance between two matrices. Given spectrum of two operators one can define the spectral
distance by minimizing over the sum of the differences between two set of eigenvalues. The
spectral distance is known to be equivalent to the distance between unitary orbits of two
operators [36]. In our setting this is simply
X
m|δλm|=X
m
b
2(ln b/2)
sech(πνm)
(1 + 4ν2
m)(4.58)
It is easy to see that the sum is upper bounded by
X
m|δλm| b
(ln b/2) Z
1/2
sech(πνm)dm + sech(πν1)!= 2πb +O(b/ ln(b)) (4.59)
which approaches zero as b0.
Repeating the same analysis for deflected entropy correlator Cs(z, w)with non-zero s,
one gets (For details see Appendix B):
δλs
m=bs
2(ln bs/2)
()msech(πνs
m)
(1 + 4νs
m)2cosh(πs) cos(2πs
m)sinh(πs) sin(2πs
m)
2νs
m+. . .
(4.60)
where bs=b/ cosh(πs)and νs
m=π(m1/2)
2 ln(bs/2) is the effective spectrum parameters for s6= 0.
We see that it reduces to the unflowed case when s= 0. It gives a prediction of deflected
entropy that agrees well with our numerics (Figure 7).
18
---
-
-



δλ
---
-
-



δλ
Figure 5. The difference δλmbetween the numerical eigenvalues and the leading order analytics,
plotted with the predictions from next order corrections (4.56). The cross-ratio in these figures are
x= 0.914 (top) and 0.9987 (bottom), respectively.

+


   







 -
Figure 6. The Markov gap SRMI of the analytics versus results obtained from numerics at large
conformal cross-ratio x. The analytical prediction is based on the leading order thermal prediction
(4.21), plus the corrections (4.56) obtained in this subsection. This figure is the magnified version
of Figure 1at x1.
19

+

      
-
-
-
-
-

Δ

+

      






Figure 7. (Top) The free fermion deflected entropy Rferm for fixed s= 0.2with the asymptotic
term 1/6(ln(1 x)+ 2 ln(cosh πs)) subtracted, plotted against the thermal prediction with correc-
tions (4.60) and holographic CFTs. (Bottom) The deflected entropy for fixed cross-ratio x= 0.981
and variable s, plotted against the same analytical prediction and holographic CFTs.
5 Discussion
In this note we calculated the reflected entropy of a chiral free fermion both numerically
and analytically, and we give asymptotic formulas for SRin both the limits x0and
x1. In particular we have shown that in the latter case, the reflected spectrum of a free
chiral fermion in the NS sector approaches a thermal spectrum with inverse temperature
to length ratio β/L 1/ln(1 x). Indeed this temperature coincides with the expected
answer one would have obtained in AdS/CFT. This thermal spectrum is theory dependent,
as opposed to the universal spectrum of entanglement entropy in 2d CFTs [37] which one
can regard as being in the infinite temperature limit.
As a practical matter one can then use the reflected entropy to extract the spectrum of
operator dimensions directly from the vacuum wave-function. In this regard, the reflected
20
entropy can be used as a diagnostic of different topological orders [38]. However, due to
the perturbative corrections and Boltzman suppression, the weights extracted in this way
are only perturbatively reliable for a window of operator weights satisfying:
h.1
2π2(ln(1 x))2(5.1)
It is natural to conjecture that the above statement are more general statements about the
reflected entanglement spectrum for any 2d CFT, with likely some modification of (5.1) to
include a varying central charge.
Acknowledgments
This work is partially supported by the Air Force Office of Scientific Research under
award number FA9550-19-1-0360 and the Department of Energy under award number DE-
SC0019183.
A Deflected entropy for holographic CFTs
In this appendix we present a quick derivation for the holographic deflected entropy using
a replica trick. We work in the same Rindler coordinate as in Section 4.1. Consider the
vacuum reduced density matrix ρAB where we define A= [a1, b1]and B= [a2, b2]. The
deflected entropy Rsfor this state is defined by the entanglement entropy of the following
density matrix
ρAA=TrBB|ρ1/2+is
AB ihρ1/2+is
AB |(A.1)
We will construct this density matrix by first calculating the holographic entanglement
entropy of the following replica state 3
ρm,n
AA=TrBB|ρn/2+mi hρn/2m|, n 2Z+, m Z(A.2)
and then take the analytic continuation n1, m is.
In general we can write down the canonical purified density matrix TrBB|ρn/2
AB ihρn/2
AB |
as a path integral on some Riemann surface. See [4] for detailed construction. In order
to apply the holographic RT formula we also need to construct a bulk solution whose
conformal boundary limits to the aforementioned surface geometry. Note that all the 3-
manifolds of constant negative curvature can be expressed as a global quotient of AdS3by
some discrete subgroup ΣP SL(2; C). These quotient group actions, when taken limit at
the conformal boundary of AdS3, descends to conformal isometries on the Riemann sphere.
Therefore, if one can find a conformal mapping which maps our Riemann surface on which
the path integral is defined to a single Riemann sphere Cwith quotient induced by some
conformal isometric group Σ0, the bulk solution is then easily obtained by extending action
of Σ0back into AdS3. The technology for finding such a mapping goes under the name of
Schottky uniformization. We will omit the details about how the uniformization mapping
21
Figure 8. (left) The Schottky domain (gray) of the first replica. The intervals Aand Bare mapped
to [0, xS]and [1,]. The dual intervals Aand Bare mapped to [0, e2πi/n xS]and [e2πi/nxS,].
xSis a complicated function of the conformal cross-ratio x=(a1b1)(a2b2)
(a1a2)(b1b2). It’s exact form does not
matter to us as we only need the fact that xSxwhen n1. The two identifying circular arcs
corresponds to the complementary segments ABin the original replica. The angle between the
arcs and the radial axes is π/n. (right) The full Schottky domain of n= 6, produced by replicating
the single Schottky domain 6 times and glue accordingly. Two adjacent arcs of different colors are
identified in the same way.
is constructed and only show the image of the map (Figure 8) in this appendix. For a more
complete review please refer to [26] and also [39,40].
The bulk solution for the Schottky domain is obtained by extending the circular arcs
to hemispheres in the bulk. We can then use the RT formula to find the entanglement
entropy for region AA. The holonomy condition now allows the endpoints of the minimal
surface to freely move on the hemispheres, see Figure 9. The length of the RT surface can
Figure 9. The bulk solution for the Schottky domain. Only a slice directly below the boundary real
line is shown. The green line is the RT surface for region AA. To achieve the minimal condition
the green line must meet both the circles at a perpendicular angle.
3The change of sign from is min the bra comes from the fact hρn/2+is|=hρn/2|ρis.
22
be readily found through a minimization procedure. It is
L= ln
1 + w
1w(A.3)
where w= (4xS)/(1+xS)2is a conformal cross-ratio for the four boundary points {−1,xS, xS,1}
and xSis some complicated function of the endpoints of the regions. We will not need the
detailed form of xS. What we only need is the result that xSxas n1, the actual
cross-ratio for the region {A, B}on the original geometry. We then obtain the formula of
the reflected entropy for a holographic CFT
Rf=c
3ln
1 + x
1x(A.4)
Now let’s see what happens when we turn on m. With nonzero and integer mthe
picture in figure 8is almost the same, as mdoes not change the total replica number.
Its only effect is to shift the dual regions Aand Bin such a way that they now meet
with their counterpart at a non-flat angle. In terms of coordinates we have the following
Figure 10. The Schottky domain for n= 6 and m= 1. Note the shift of the dual regions Aand
B.
expression
A= [0, xS], B = [xS,], A= [0,xSe2πim/n], B = [e2πim/n,](A.5)
After an analytical continuation mis and n1they become
A= [0, xS], B = [xS,], A= [0,xe2πis], B = [e2πis,](A.6)
They all lie on the real line of the boundary after the analytical continuation. Our previous
formula for the minimal surface still applies. We only need to replace wby the cross-ratio
for the new points. The result is
Rf(s) = c
6ln
1 + w
1w, w =x(1 + e2πis)2
(x+e2πis)(1 + xe2πis)(A.7)
This expression is invariant under the reflection s s.
23
B Next order eigenvalue correction to deflected entropy
In this appendix we give a quick derivation of (4.60). The deflected correlator is (assuming
z > 0and w < 0)
Cs(z, w) = 1
2πi(z+e2πs w)
Qs1
Qs+e2πs (B.1)
Qs=z(z+ 1/b)
(z+b)
e2πswb
e2πsw(e2πsw1/b)(B.2)
For z < 0and w > 0we use Cs(w, z ) = Cs(z, w). For zw > 0we have Cs(w, z) = 1
2πi(zw).
This correlator undergoes a change s safter inversion Cs(z, w) = Cs(1/z, 1/w)/(zw).
A direct consequence of this is that the contribution from two different crossed regions
(zb, w 1/b)and (z1/b, w b)are no longer equal but evaluates to s s. This
ensures that the eigenvalues (and hence the entropy) of Csare invariant as we invert the
sign of s, as expected from the symmetry of modular flow. It also greatly simplifies our
work since we only need to evaluate one scaling region. The eigenfunctions are related to
the s= 0 case
gs
ν(z) = gνs(eπsz, b b/ cosh(πs)) (B.3)
It has a new normalization
Zgs
ν(z)gs
ν0(z)dz =4δν,ν0eπs ln b
2 cosh(πs)(B.4)
Therefore the first order correction to eigenvalues are given by
4 ln b
2 cosh(πs)δλ =eπs ZdzdwδCs(z, w)gs
ν(z)gs
ν(w)(B.5)
We have factored eπs to RHS as it restores the s ssymmetry of the integral, as we
will see later. Again there are multiple scaling regions to consider, but as in the s= 0 case
only the crossed region z |b|, w |1/b|and z |1/b|, w |b|contributes.
Consider z > 0and w < 0and zb, w 1/b. The correlator is
δCs(z, w) = 1
2πiw
(z/b + 1)(e2πsbw 1) + z/b
(z/b + 1)(e2πsbw 1) e2πs z/b (B.6)
The relavent integral is
()mb
2πZΛ
1
dx ZΛ
1
dyx1/2 y1/2+ 1
e2πs +xy eπs tanh(πs) + x+y
2 cosh πs (B.7)
with x= 1 + 2eπs zcosh(πs)/b and y=2 cosh(πs)/(eπswb)+1. For the same sign but for
z1/b, w bwe have the integral
()mb
2πZΛ
1
dx ZΛ
1
dyx1/2 y1/2+ 1
e2πs +xy eπs tanh(πs) + x+y
2 cosh(πs)(B.8)
24
with y= 1 2eπswcosh(πs)/b and x= 2 cosh(πs)/(eπszb) + 1. One can see the aformen-
tioned symmetry of the correlator with s s.
These integrals are composed of two different pieces. The first one is absent in s= 0:
I1ZΛ
1
dx ZΛ0
1
dyx1/2 y1/2+ eπs
e2πs +xy + (s s)
=2 tanh(πs)
νsech(πν) sin(2πsν) + divergent terms.
(B.9)
The second integral is similar to the s= 0 case but with modified bounds:
I2ZΛ
1
dx ZΛ0
1
dyx1/2 y1/2+ x+y
e2πs +xy + (s s)
=2 sech(πν)
1+4ν2(cos(2πsν )+2νsin(2πsν) tanh(πs)) + divergent terms.
(B.10)
Note that we have only kept the terms that is constant as we scale Λ,Λ0 since we know
the divergent terms must cancel across different regions from the discussion in Section 4.3.
We conclude that we can write the correction for finite sto be
4 ln b
2 cosh(πs)δλs= ()mb(I1+I2)
= ()mbsech(πν)
1+4ν22 cos(2πsν ) + tanh(πs) sin(2πsν)
ν
(B.11)
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