# Modular Hamiltonian of the scalar in the semi infinite line: dimensional
reduction for spherically symmetric regions

Marina Huerta111e-mail<EMAIL_ADDRESS>Centro Atómico
Bariloche, 8400-S.C. de Bariloche, Río Negro, Argentina  Guido van der
Velde222e-mail<EMAIL_ADDRESS>Centro Atómico Bariloche,
8400-S.C. de Bariloche, Río Negro, Argentina

###### Abstract

We focus our attention on the one dimensional scalar theories that result from
dimensionally reducing the free scalar field theory in arbitrary $d$
dimensions. As is well known, after integrating out the angular coordinates,
the free scalar theory can be expressed as an infinite sum of theories living
in the semi-infinite line, labeled by the angular modes $\\{\ell,\vec{m}\\}$.
We show that their modular Hamiltonian in an interval attached to the origin
is, in turn, the one obtained from the dimensional reduction of the modular
Hamiltonian of the conformal parent theory in a sphere. Remarkably, this is a
local expression in the energy density, as happens in the conformal case,
although the resulting one-dimensional theories are clearly not conformal. We
support this result by analyzing the symmetries of these theories, which turn
out to be a portion of the original conformal group, and proving that the
reduced modular Hamiltonian is in fact the operator generating the modular
flow in the interval. By studying the spectrum of these modular Hamiltonians,
we also provide an analytic expression for the associated entanglement
entropy. Finally, extending the radial regularization scheme originally
introduced by Srednicki, we sum over the angular modes to successfully recover
the conformal anomaly in the entropy logarithmic coefficient in even
dimensions, as well as the universal constant $F$ term in $d=3$.

##  1 Introduction: Modular flow and modular Hamiltonian

The successful application of information theory tools to quantum field theory
(QFT) along the last decades, has given place to the solid current consensus
that these tools must be definitively incorporated into the usual QFT
machinery. In this context, the study of quantities related to different
information measures for quantum field theories gains relevance and with them,
the study of states reduced to a region. These states are described by reduced
(local) density matrices that live in the core of the definition of all the
information measures referenced to spatial regions $R$. From the quantum field
algebraic perspective [1], each region $R$ is attached to the algebra of the
degrees of freedom localized in $R$. The reduced state to a local algebra of
operators in a region can be expressed, in presence of a cutoff, as a density
matrix

$\rho=\frac{e^{-K}}{\textrm{tr}e^{-K}}\,,$ (1)

where the exponent $K$ is the modular Hamiltonian operator. This convenient
way of encoding the reduced state admits an interesting interpretation of the
entanglement entropy as the thermodynamic entropy of a system in equilibrium
at temperature $1$, but with respect to the modular Hamiltonian $K$. Moreover,
there is a time notion associated to the state through the modular
Hamiltonian, whose evolution is implemented by the unitary operator in the
algebra

$U(\tau)=\rho^{i\tau}\sim e^{-i\tau K}\,.$ (2)

The induced evolution of operators $O(\tau)=U(\tau)OU(-\tau)$ is called the
modular flow. This is a purely quantum transformation, which becomes trivial
in the classical limit.

Historically, the earliest recognition of the structural importance of modular
flows can be found in the algebraic formulation of QFT [2, 3] and more
recently, in the framework of the study of different information measures and
statistical properties of reduced states in QFT [4, 5, 6]. The modular
Hamiltonian is a fundamental constitutive part of the relative entropy and
plays an essential role in the entropy bounds formulations and proof of
several energy conditions [7, 8, 9, 10, 11, 12]. Besides, profiting that
entanglement and relative entropy have well established geometric duals for
holographic QFT [13, 14, 15], modular Hamiltonians have also been used to
clarify localization properties of degrees of freedom in quantum gravity [16,
17, 18].

Currently, our knowledge of the explicit form of modular Hamiltonians reduces
mostly to some examples where the modular flow is local, and it is primarily
determined by spacetime symmetries.

This is the case for the Rindler wedge $x^{1}>|t|$ in Minkowski space and any
QFT. Choosing the causal region to be the half spatial plane $x^{1}>0$ and
$t=0$ then, the rotational symmetry of the euclidean theory allows us to
express the reduced density matrix corresponding to the vacuum state in terms
of the energy density $T_{00}$

$\rho=k\,e^{-2\pi\int_{x^{1}>0}d^{d-1}x\,x^{1}T_{00}(x)}\,.$ (3)

The above expression manifestly reveals a non trivial connection between
entanglement in vacuum and energy density. Moreover, in equation (3), the
exponent corresponds to the modular Hamiltonian for half space which results
to be an integral of a local operator. $K$ is in fact $2\pi$ times the
generator of boosts restricted to act only on the right Rindler wedge

$K=-2\pi\int_{x^{1}>0}d^{d-1}x\,x^{1}T_{00}(x)\,.$ (4)

The modular flow $\rho^{i\tau}$ moves operators locally following the orbits
of the one parameter group of boost transformations. On the other hand, it is
interesting to note that from equation (3), the vacuum state in half space
corresponds to a thermal state of inverse temperature $2\pi$ with respect to
the boost operator. This is directly connected to the Unruh’s effect [19]
according to which accelerated observers see the vacuum as a thermally excited
state. For an observer following a trajectory given by a boost orbit, the
state looks like a thermal state with respect to the proper time
$\tilde{\tau}$. For these trajectories, the proper time and the boost
parameter $s$ are proportional $s=a\tilde{\tau}$ with $a$ the proper
acceleration of the observer, constant along boost orbits. In turn, this
implies there is a relation $K=\tilde{H}/a$ between the boost operator and the
proper time Hamiltonian $\tilde{H}$ of the accelerated observer. For such an
observer there is a thermal bath at (proper time) temperature
$T=\frac{2\pi}{a}$.

The other very well known example where symmetries again facilitate the
derivation of the exact modular Hamiltonian is the case of conformal field
theories (CFT) for spheres in any dimensions. For a CFT, Poincare symmetries
are enlarged to the conformal group. These theories are characterized by
having a traceless, symmetric and conserved stress tensor. This enlarges the
number of conserved currents related to space-time symmetries which in general
can be written as

$j_{\mu}=a^{\nu}\,T_{\nu\mu}+b^{\alpha\nu}\,x_{\alpha}\,T_{\nu\mu}+c\,x^{\nu}\,T_{\nu\mu}+d_{\alpha}\,(x^{2}g^{\alpha\nu}-2\,x^{\alpha}x^{\nu})\,T_{\nu\mu}\,.$
(5)

The corresponding conserved charges depend on parameters $a^{\mu}$,
determining translations, the antisymmetric $b^{\mu\nu}$, giving Lorentz
transformations, $c$, related to dilatations, and $d^{\mu}$, for the so called
special conformal transformations.

Since there is a conformal transformation that maps the Rindler wedge to
causal regions with spherical boundary, and the same transformation leaves the
vacuum invariant for a CFT, then, the modular Hamiltonian is just the
transformed Rindler modular Hamiltonian. It is easy to get

$K=2\pi\int_{|\vec{x}|<R}d^{d-1}x\,\frac{R^{2}-r^{2}}{2R}\,T_{00}(\vec{x})\,.$
(6)

In this example, $K$ is again local and proportional to $T_{00}$, with a
proportionality weight function $\beta(r)\equiv\frac{R^{2}-r^{2}}{2R}$.

Except for the two examples discussed above, the vacuum of a QFT in the
Rindler wedge and the vacuum of a CFT in the sphere, there are only some other
few known modular Hamiltonians, either local or not. The local ones in general
are derived profiting from symmetry transformations that leave the state
invariant. This is for example the case of the modular Hamiltonian for CFTs in
$1+1$ dimensions in presence of a global or local quench [20, 21, 22, 23, 24,
25, 26]. However, on general grounds, from the point of view of quantum
information we do not expect locality to hold. In general, $K$ will be given
by a non local and non linear combination of the field operators at different
positions inside the region.

An example of a non local modular Hamiltonian which has been explicitly
computed is the one for the vacuum state of the free massless fermion in $d=2$
for several disjoint intervals [27, 28, 29]. In this case $K$ has a local term
proportional to the energy density and an additional non local part given by a
quadratic expression in the fermion field that connects in a very particular
way points located in different intervals.

In this paper we calculate the modular Hamiltonian for the vacuum state of non
conformal $(1+1)$ dimensional theories in the interval $(0,R)$. These theories
are defined in the semi infinite line, and result from the dimensional
reduction of the $d$ dimensional free massless scalar. Our strategy is to
calculate the modular Hamiltonian of the reduced system by profiting of the
known modular Hamiltonian of CFTs in spheres in any dimension.

The free massless scalar in $d$ space time dimensions can be dimensionally
reduced to a sum of one dimensional theories, one for each angular mode. Since
the reduction is obtained by integrating over the angular coordinates, these
systems live in the semi infinite line. From the algebraic point of view, this
is convenient when studying algebras assigned to spherical regions to
calculate, for example, the entanglement entropy. In these coordinates, the
local algebra assigned to the region can be easily written in terms of fields
$\phi(r,\Omega)$ with nice localization properties. For example, points in the
semi infinite line correspond to shells in the original space and intervals
connected to the origin, to $d$-spheres (see figure 1).

Figure 1: The sphere of radius $R$ corresponds to intervals of length $R$ with
one edge in the origin in the radial semi infinite line.

Concretely, in the radial coordinate, the canonical Hamiltonian for the
massless free scalar decomposes as a sum over angular modes $H_{\ell\vec{m}}$

$H=\sum_{\ell\vec{m}}H_{\ell\vec{m}}\,.$ (7)

with $(\ell\vec{m})$ the angular mode label. In fact, there is a family of one
dimensional Hamiltonian $H_{\ell\vec{m}}$ for each dimension. In turn, the
same decomposition occurs for the modular Hamiltonian (6)

$K=\sum_{\ell\vec{m}}K_{\ell\vec{m}}\,.$ (8)

Taking into account that the vacuum state for a system composed by independent
subsystems is a product of density matrices, here
$\rho=\otimes\rho_{\ell\vec{m}}$, then it is immediate to identify the modular
Hamiltonian mode $K_{\ell\vec{m}}$ with the modular Hamiltonian of the one
dimensional reduced system $H_{\ell\vec{m}}$. The Hamiltonian
$H_{\ell\vec{m}}$ does not correspond to a conformal relativistic theory due
to an extra quadratic term proportional to $1/r^{2}$, whose proportionality
constant depends on the dimension of the original problem and the angular mode
$\ell$333For a different context in which a free scalar, albeit conformal, is
obtained from dimensional reduction, see [30].. Surprisingly, we find that
$K_{\ell\vec{m}}$ is still local and proportional to the energy density
$T_{00}$444Since translational invariance is lost, there is no conserved
energy momentum tensor. The notation for the energy density is just a matter
of convention., with the same weight function $\beta(r)$ that characterizes
the modular Hamiltonian for CFTs in spheres. Our analytic results coincide
with the suggested continuum limit of the entanglement Hamiltonian of blocks
of consecutive sites in massless harmonic chains, recently studied in [31].

This article is organized as follows. In section 2 we explicitly carry out the
dimensional reduction. We write the scalar field in a basis of hyper-spherical
harmonics, and after integrating out the angular coordinates we are left with
a Hamiltonian for the reduced systems $H_{\ell\vec{m}}$ of the form

$H_{\ell\vec{m}}=\frac{1}{2}\int
dr\left[\widetilde{\pi}_{\ell\vec{m}}^{2}+(\partial_{r}\widetilde{\phi}_{\ell\vec{m}})^{2}+\frac{\mu_{d}(\ell)}{r^{2}}\widetilde{\phi}_{\ell\vec{m}}^{2}\right],$
(9)

with

$\mu_{d}(\ell)=\frac{(d-4)(d-2)}{4}+\ell(\ell+d-3).$ (10)

In section 3 the same procedure is followed to find the modular Hamiltonian

$K_{\ell,\vec{m}}=2\pi\int_{|\vec{x}|<R}dr\,\frac{R^{2}-r^{2}}{2R}\,T_{00}^{\ell,\vec{m}}(\vec{x})\,.$
(11)

In some way, the reduced theory, manifestly invariant under dilatations but
non conformal, keeps the memory of the conformal symmetry of the parent
$d$-dimensional theory [32, 33], with the same local modular Hamiltonian as
that representing the vacuum of a CFT in a sphere. We delve into this in
section 4, where we show that the reduced theories preserve an
$SL(2,\mathbb{R})$ symmetry, and that the modular transformation belongs to
this subgroup. The modular Hamiltonian (11) written as a Noether charge can be
correctly interpreted as the local operator implementing the modular flow.

In section 5 we solve the spectrum of the modular Hamiltonian (11) and compute
the entanglement entropy in a segment connected to the origin. We find the
analytic expression

$S(\ell,d)=\frac{1}{6}\log{\frac{R}{\epsilon}}-\frac{i\pi}{2}\int_{0}^{\infty}ds\frac{s}{\sinh^{2}(\pi
s)}\log{\left(\frac{4^{is}\Gamma\left[is\right]\Gamma\left[-1+d/2+\ell-
is\right]}{\Gamma\left[-is\right]\Gamma\left[-1+d/2+\ell+is\right]}\right)},$
(12)

which is logarithmically divergent, with coefficient $1/6$ as expected for
$(1+1)$ theories, and has a constant term that depends both on the mode $\ell$
and the space time dimensions $d$ of the original theory. Although the above
integral cannot in general be solved analytically, we make some useful
approximations to extract relevant information out of it. Moreover, by summing
over $\ell$ we are able to recover the conformal anomaly in the logarithmic
coefficient for the free scalar field in even dimensions, as well as the
constant universal $F$ term in $d=3$. In doing the sum over the angular modes
$\ell$, we introduce a novel regularization implemented by a damping
exponential $\exp[-\ell\epsilon/R]$, with the same cutoff $\epsilon$ that
regularizes the radial coordinate $r$. This procedure generalizes the radial
regularization scheme introduced by Srednicki in [34], where it is explicitly
stated that for $d\geqslant 4$ regularization by a radial lattice turns out to
be insufficient and the sum over partial waves does not converge. We end the
discussion with some concluding remarks.

##  2 Spherical coordinates

The free scalar action in spherical coordinates reads

$S=\frac{1}{2}\int
dtdrr^{d-2}d\Omega\left[-(\partial_{0}\phi)^{2}+(\partial_{r}\phi)^{2}-\frac{\phi}{r^{2}}\Delta_{S^{d-2}}\phi\right].$
(13)

With the aim of reducing the above to a single integral in the radial
direction, we Fourier transform the scalar field in the angular coordinates,
using the real hyper-spherical harmonics as basis functions,

$\phi(\vec{r})=\sum_{\ell m_{1}...m_{d-3}}\phi_{\ell
m_{1}...m_{d-3}}(r)Y_{\ell}^{m_{1}...m_{d-3}}(\hat{r}),$ (14)

with

$\Delta_{S^{d-2}}Y_{\ell}^{m_{1}...m_{d-3}}(\hat{r})=-\ell(\ell+d-3)Y_{\ell}^{m_{1}...m_{d-3}}(\hat{r}),$
(15) $\int_{S^{d-2}}d\Omega
Y_{\ell}^{m_{1}...m_{d-3}}(\hat{r}){Y}_{\ell^{\prime}}^{m_{1}^{\prime}...m_{d-3}^{\prime}}(\hat{r})=\delta_{\ell\ell^{\prime}}\delta_{m_{1}m_{1}^{\prime}}...\delta_{m_{d-3}m_{d-3}^{\prime}}.$
(16)

After integrating the angular coordinates, we are left with

$S=\frac{1}{2}\sum_{\ell\vec{m}}\int
dtdrr^{d-2}\left[-(\partial_{0}\phi_{\ell\vec{m}})^{2}+(\partial_{r}\phi_{\ell\vec{m}})^{2}+\frac{\ell(\ell+d-3)}{r^{2}}\phi_{\ell\vec{m}}^{2}\right].$
(17)

However, the theory looks simpler when defined in terms of the rescaled field
$\widetilde{\phi}_{\ell\vec{m}}=r^{\frac{d-2}{2}}\phi_{\ell\vec{m}}$, whose
canonically conjugated momentum is
$\widetilde{\pi}_{\ell\vec{m}}\equiv\partial_{0}\widetilde{\phi}_{\ell\vec{m}}$,

$S=\frac{1}{2}\sum_{\ell\vec{m}}\int
dtdr\left[-(\partial_{0}\widetilde{\phi}_{\ell\vec{m}})^{2}+r^{d-2}\left(\partial_{r}\left(\frac{\widetilde{\phi}_{\ell\vec{m}}}{r^{\frac{d-2}{2}}}\right)\right)^{2}+\frac{\ell(\ell+d-3)}{r^{2}}\widetilde{\phi}_{\ell\vec{m}}^{2}\right].$
(18)

Functional variation with respect to the field leads to the equation of
motion. Nevertheless, in order for the variational problem to be well posed we
should impose specific boundary conditions at $r=0$. In fact,

$\begin{split}\delta S=\sum_{\ell\vec{m}}&\left\\{\int
dtdr\left[\partial_{0}^{2}\widetilde{\phi}_{\ell\vec{m}}-\frac{1}{r^{\frac{d-2}{2}}}\partial_{r}\left(r^{d-2}\partial_{r}\left(\frac{\widetilde{\phi}_{\ell\vec{m}}}{r^{\frac{d-2}{2}}}\right)\right)+\frac{\ell(\ell+d-3)}{r^{2}}\widetilde{\phi}_{\ell\vec{m}}\right]\delta{\widetilde{\phi}}_{\ell\vec{m}}\right.\\\
&\left.+\int
dt\left[r^{\frac{d-2}{2}}\partial_{r}\left(\frac{\widetilde{\phi}_{\ell\vec{m}}}{r^{\frac{d-2}{2}}}\right)\delta{\widetilde{\phi}}_{\ell\vec{m}}\right]\Biggr{\rvert}_{0}^{\infty}\right\\}\,,\end{split}$
(19)

which requires either $\delta\widetilde{\phi}_{\ell\vec{m}}(r=0,t)=0$
(Dirichlet boundary conditions) or
$r^{\frac{d-2}{2}}\partial_{r}\left(\frac{\widetilde{\phi}_{\ell\vec{m}}}{r^{\frac{d-2}{2}}}\right)\rightarrow
0$ (analogous to the ordinary Neumann boundary conditions). In the following
we will adopt the former.

The second term in (19) can be further simplified, which leads to the saddle
point

$\partial_{0}^{2}\widetilde{\phi}_{\ell\vec{m}}-\partial_{r}^{2}\widetilde{\phi}_{\ell\vec{m}}+\frac{\mu_{d}(\ell)}{r^{2}}\widetilde{\phi}_{\ell\vec{m}}=0,$
(20)

with

$\mu_{d}(\ell)=\frac{(d-4)(d-2)}{4}+\ell(\ell+d-3).$ (21)

This partial differential equation can be solved by separation of variables,
and expressed in terms of the original field $\phi_{\ell\vec{m}}$, the radial
eigenfunction problem is a Bessel equation, with solution
$j_{\ell}(r)\equiv\frac{1}{r^{(d-3)/2}}J_{\ell+\frac{d-3}{2}}(kr)$. Therefore,
the solution is

$\widetilde{\phi}_{\ell\vec{m}}(t,r)=e^{\pm
ikt}\sqrt{kr}J_{\ell+\frac{d-3}{2}}(kr),$ (22)

which means that $\widetilde{\phi}_{\ell\vec{m}}\sim r^{\ell+d/2-1}$ near
$kr\sim 0$, in agreement with the boundary conditions.

Having stated that, it is also possible to rewrite the second term in (18) by
getting rid of a boundary term555We would be able to ignore the boundary term
provided $\widetilde{\phi}_{\ell\vec{m}}^{2}$ went to zero faster than $r$.
This is at least satisfied by the classical configuration (22).. More
explicitly,

$S=\sum_{\ell\vec{m}}S_{\ell\vec{m}}$ (23)

where

$S_{\ell\vec{m}}=\frac{1}{2}\int
dtdr\left[-(\partial_{0}\widetilde{\phi}_{\ell\vec{m}})^{2}+(\partial_{r}\widetilde{\phi}_{\ell\vec{m}})^{2}+\frac{\mu_{d}(\ell)}{r^{2}}\widetilde{\phi}_{\ell\vec{m}}^{2}\right]$
(24)

can be thought of as the action for a free scalar living in the half line,
satisfying Dirichlet boundary conditions at the origin. Note that, unlike the
theory we started with, this is not a CFT because of the last term.

The dimensional reduction of the free scalar Hamiltonian can be made following
the same steps. But we can alternatively calculate the conserved charge due to
time translations associated directly to the $1+1$ dimensional action (24),
yielding

$H=\frac{1}{2}\sum_{\ell\vec{m}}\int
dr\left[\widetilde{\pi}_{\ell\vec{m}}^{2}+(\partial_{r}\widetilde{\phi}_{\ell\vec{m}})^{2}+\frac{\mu_{d}(\ell)}{r^{2}}\widetilde{\phi}_{\ell\vec{m}}^{2}\right]$
(25)

Once again we stress that $\widetilde{\pi}_{\ell\vec{m}}$ and
$\widetilde{\phi}_{\ell\vec{m}}$ satisfy canonical commutation relations

$\left[\widetilde{\phi}_{\ell\vec{m}}(r),\widetilde{\pi}_{\ell^{\prime}\vec{m}^{\prime}}(r^{\prime})\right]=i\delta_{\ell,\ell^{\prime}}\delta_{\vec{m},\vec{m}^{\prime}}\delta(r-r^{\prime}).$
(26)

##  3 The sphere modular Hamiltonian

On the other hand, since the free scalar field theory in $d$ spacetime
dimensions is conformally invariant, when the whole system is in its ground
state the modular Hamiltonian of a sphere is

$K=\frac{1}{2}\int_{|x|<R}dx^{d-1}\left(\frac{R^{2}-r^{2}}{2R}\right)T_{00}.$
(27)

However, although the stress tensor involved in this expression must be
traceless, the canonical stress tensor of the free scalar field is

$T_{\mu\nu}^{(c)}=\partial_{\mu}\phi\partial_{\nu}\phi-\frac{1}{2}\eta_{\mu\nu}(\partial\phi)^{2},$
(28)

which has non vanishing trace
$T_{\mu}^{\mu}=\left(1-d/2\right)(\partial\phi)^{2}=\frac{(1-d/2)}{2}\partial^{2}(\phi^{2})$666This
identity holds on-shell.. Hence, it must be improved by adding a conserved
symmetric tensor. A possible choice is

$T_{\mu\nu}^{\prime}=T_{\mu\nu}^{(c)}-\frac{(1-d/2)}{2(1-d)}(\partial_{\mu}\partial_{\nu}-\eta_{\mu\nu}\partial^{2})\phi^{2}.$
(29)

Therefore,

$K=\frac{1}{2}\int_{|x|<R}dx^{d-1}\left(\frac{R^{2}-r^{2}}{2R}\right)\left[(\partial_{0}\phi)^{2}+(\partial_{i}\phi)^{2}-\frac{(1-d/2)}{(1-d)}\partial_{i}^{2}\phi^{2}\right].$
(30)

Using the following identities:

$(\partial_{i}\phi)^{2}=(\partial_{r}\phi)^{2}-\frac{\phi}{r^{2}}\Delta_{S^{d-2}}\phi,$
(31)

where we have partially integrated the angular piece, and

$\partial_{i}^{2}\phi^{2}=\frac{1}{r^{d-2}}\partial_{r}\left(r^{d-2}\partial_{r}\phi^{2}\right)+\frac{1}{r^{2}}\Delta_{S^{d-2}}\phi^{2},$
(32)

we arrive at

$\begin{split}K=\frac{1}{2}\sum_{\ell\vec{m}}\int
drr^{d-2}\left(\frac{R^{2}-r^{2}}{2R}\right)&\left\\{\pi_{\ell\vec{m}}^{2}+(\partial_{r}\phi_{\ell\vec{m}})^{2}+\frac{\ell(\ell+d-3)}{r^{2}}\phi_{\ell\vec{m}}^{2}-\right.\\\
&\left.-\frac{(1-d/2)}{(1-d)}\left[\partial_{r}^{2}\phi_{\ell\vec{m}}^{2}+\frac{d-2}{r}\partial_{r}\phi_{\ell\vec{m}}^{2}\right]\right\\},\end{split}$
(33)

In terms of the canonically conjugated operators,

$\begin{split}K=\frac{1}{2}\sum_{\ell\vec{m}}\int
dr\left(\frac{R^{2}-r^{2}}{2R}\right)&\left\\{\widetilde{\pi}_{\ell\vec{m}}^{2}+(\partial_{r}\widetilde{\phi}_{\ell\vec{m}})^{2}+\frac{\mu_{d}(\ell)}{r^{2}}\widetilde{\phi}_{\ell\vec{m}}^{2}-\right.\\\
&\left.-\frac{d-2}{2(d-1)}\left[3\partial_{r}\left(\frac{\widetilde{\phi}_{\ell\vec{m}}^{2}}{r}\right)+r\partial_{r}^{2}\left(\frac{\widetilde{\phi}_{\ell\vec{m}}^{2}}{r}\right)\right]\right\\},\end{split}$
(34)

Note that the second line of (34), together with the prefactor
$(R^{2}-r^{2})$, is a total derivative in disguise. Hence,

$\begin{split}K=\frac{1}{2}\sum_{\ell\vec{m}}&\left\\{\int_{0}^{R}dr\left(\frac{R^{2}-r^{2}}{2R}\right)\left[\widetilde{\pi}_{\ell\vec{m}}^{2}+(\partial_{r}\widetilde{\phi}_{\ell\vec{m}})^{2}+\frac{\mu_{d}(\ell)}{r^{2}}\widetilde{\phi}_{\ell\vec{m}}^{2}\right]-\right.\\\
&\left.-\frac{d-2}{2(d-1)}\left[\frac{(R^{2}-r^{2})}{2R}r\partial_{r}\left(\frac{\widetilde{\phi}_{\ell\vec{m}}^{2}}{r}\right)+R\left(\frac{\widetilde{\phi}_{\ell\vec{m}}^{2}}{r}\right)\right]\biggr{\rvert}_{0}^{R}\right\\},\end{split}$
(35)

The boundary terms (coming from the improving) can be interpreted in general,
as an ambiguity in the definition of modular Hamiltonian in a region, and
safely ignored as explained in [35]. Consequently, the modular Hamiltonian of
the $d$ dimensional free scalar is

$K=\sum_{\ell\vec{m}}K_{\ell\vec{m}},$ (36)

where

$K_{\ell\vec{m}}=\frac{1}{2}\int_{0}^{R}dr\left(\frac{R^{2}-r^{2}}{2R}\right)\left[\widetilde{\pi}_{\ell\vec{m}}^{2}+(\partial_{r}\widetilde{\phi}_{\ell\vec{m}})^{2}+\frac{\mu_{d}(\ell)}{r^{2}}\widetilde{\phi}_{\ell\vec{m}}^{2}\right]$
(37)

can be interpreted as the modular Hamiltonian for the vacuum of (24) in a
segment. This identification rests on the fact that the theory decomposes into
independent sectors, labeled by the angular modes, so the state must write as
the direct product of the states pertaining to each sector. But, most
remarkably, this modular Hamiltonian is still local in the energy density. In
other words, (37) agrees with the general expression (27) in spite of the
reduced one dimensional theories being non conformal. In the next section we
analyse this in detail, paying attention to the symmetries which survive the
dimensional reduction.

Provided that (37) defines the reduced state of a free field theory, Wick’s
theorem guarantees it can be expressed in terms of the two-point correlators.
In fact, for a Gaussian state with modular Hamiltonian

$K=\int_{V}d^{d-1}x_{1}d^{d-1}x_{2}\left[\phi(x_{1})M(x_{1},x_{2})\phi(x_{2})+\pi(x_{1})N(x_{1},x_{2})\pi(x_{2})\right],$
(38)

and correlators

$X=\langle\phi(x_{1})\phi(x_{2})\rangle\,,\quad
P=\langle\pi(x_{1})\pi(x_{2})\rangle\,,$ (39)

the following relation must be satisfied [36]777Here the product is a bi-local
function constructed as
$\left[M.X\right](x_{1},x_{2})\equiv\int_{V}dyM(x_{1},y)X(y,x_{2})$

$M.X=P.N$ (40)

In the case at hand,

$M(r,r^{\prime})=-2\pi\delta(r-r^{\prime})\left[\beta(r)\partial_{r}^{2}+\partial_{r}\beta(r)\partial_{r}-\beta(r)\frac{\mu}{r^{2}}\right]$
(41)

and

$N(r,r^{\prime})=2\pi\delta(r-r^{\prime})\beta(r)\,.$ (42)

Meanwhile, the explicit form of the correlators for the one dimensional theory
(24) is [37]

$X(r_{1},r_{2})=\frac{\Gamma\left[\ell+d/2-1\right]}{2\Gamma\left[\frac{1}{2}\right]\Gamma\left[\ell+\frac{d-1}{2}\right]}\left(\frac{r_{1}}{r_{2}}\right)^{\ell+\frac{d}{2}-1}{}_{2}F_{1}\left[\frac{1}{2},\ell+\frac{d}{2}-1;\ell+\frac{d-1}{2};\left(\frac{r_{1}}{r_{2}}\right)^{2}\right]\,,$
(43)
$\begin{split}P(r_{1},r_{2})&=\frac{2\Gamma(\ell+d/2)}{\Gamma\left[\frac{1}{2}\right]\Gamma\left[\ell+\frac{d-1}{2}\right](r_{2}^{2}-r_{1}^{2})}\left(\frac{r_{1}}{r_{2}}\right)^{\ell+\frac{d}{2}-1}\left(A\leavevmode\nobreak\
{}_{2}F_{1}\left[\frac{1}{2},\ell+\frac{d}{2};\ell+\frac{d-1}{2};\left(\frac{r_{1}}{r_{2}}\right)^{2}\right]\right.\\\
&\left.+B\leavevmode\nobreak\
{}_{2}F_{1}\left[-\frac{1}{2},\ell+\frac{d}{2};\ell+\frac{d-1}{2};\left(\frac{r_{1}}{r_{2}}\right)^{2}\right]\right)\,,\end{split}$
(44)

where $A=\left(\ell+\frac{d-1}{2}\right)\left(1-r_{1}^{2}/r_{2}^{2}\right)-1$,
$B=1-\ell-d/2$, and $r_{1}<r_{2}$. Using these concrete expressions it is
possible to check that (40) indeed holds.

##  4 Symmetries

The locality of (37) suggests the existence of a symmetry with a conserved
current such that the modular Hamiltonian is the corresponding Noether charge.
This has to be an endomorphism in the causal wedge of the region, and must
point in the time direction at $t=0$. For CFTs in spheres in any dimensions,
this is the conformal transformation that maps the spherical boundary in
itself. For an interval $(0,R)$ in the half line, whose causal wedge is a half
diamond, the symmetry transformation leaves the boundary point $r=R$ fixed.
The identification of this symmetry in the present case is the natural path to
justify the locality of (37). With this aim, we first discuss the symmetries
of the reduced theories with action (24).

The symmetries of (24) are a subgroup of the conformal transformations
inherited from higher dimensions, in particular those which involve only the
time and radial coordinates, and that map the line $r=0$ into itself. These
are

  * •

Time translations:

$t\rightarrow t+t_{0}$ (45)

  * •

Dilatations:

$(t,r)\rightarrow(\lambda t,\lambda r)$ (46)

  * •

Special conformal transformations with parameter
$b^{\mu}=\frac{\alpha}{R}\hat{e}^{\mu}_{t}$:

$(t,r)\rightarrow\left(\frac{tR^{2}+\alpha R(t^{2}-r^{2})}{R^{2}+2\alpha
Rt+\alpha^{2}(t^{2}-r^{2})},\frac{rR^{2}}{R^{2}+2\alpha
Rt+\alpha^{2}(t^{2}-r^{2})}\right).$ (47)

Infinitesimally, that is, if we set $\alpha=\epsilon<<1$, then

$(t,r)\rightarrow\left(t-\epsilon(t^{2}+r^{2})/R,r-2\epsilon tr/R\right).$
(48)

The generators of the transformations listed above are $P_{0}=i\partial_{t}$,
$D=i(t\partial_{t}+r\partial_{r})$, and
$K_{0}=i\left((t^{2}+r^{2})\partial_{t}+2tr\partial_{r}\right)$ respectively.
These close an $sl(2,\mathbb{R})$ algebra, which can be expressed in a more
suggestive way identifying $L_{-1}\equiv P_{0}$, $L_{0}\equiv D$, $L_{1}\equiv
K_{0}$, so that

$i\left[L_{m},L_{n}\right]_{LB}=(m-n)L_{m+n}.$ (49)

Just for completeness, we note that one would have expected the original
conformal group $SO(d,2)$ to break into $SO(2,2)\sim
SL(2,\mathbb{R})\otimes\overline{SL(2,\mathbb{R})}$ [32, 33], with six
generators. In fact, besides the three generators already mentioned, there are
three more that do not mix the angular coordinates with $(t,r)$, associated to

  * •

Translations in the radial direction:

$r\rightarrow r+r_{0}$ (50)

  * •

Boosts:

$(t,r)\rightarrow(t+\epsilon r,r+\epsilon t)$ (51)

  * •

Special conformal transformations with parameter
$b^{\mu}=\frac{\alpha}{R}\hat{e}^{\mu}_{r}$

$(t,r)\rightarrow\left(\frac{tR^{2}}{R^{2}-2\alpha
Rr-\alpha^{2}(t^{2}-r^{2})},\frac{rR^{2}+\alpha R(t^{2}-r^{2})}{R^{2}-2\alpha
Rr-\alpha^{2}(t^{2}-r^{2})}\right).$ (52)

These are $\hat{e}^{\mu}_{r}P_{\mu}$, $\hat{e}^{\mu}_{r}M_{0\mu}$ and
$\hat{e}^{\mu}_{r}K_{\mu}$, respectively. However, it is easy to see that they
fail to become symmetries of the dimensionally reduced theory.

Then, the modular symmetry of the reduced theories we are looking for must be
a particular composition of the identified symmetry transformations (45) -
(47).

On the other hand, we know that the modular symmetry for the parent conformal
theory is associated to the generator of the boosts as seen from the domain of
dependence of the ball [38],

$\zeta=\frac{\pi}{R}\left[(R^{2}-t^{2}-|\vec{x}|^{2})\partial_{t}-2tx^{i}\partial_{i}\right]\,.$
(53)

In fact, comparing with (45) and (47), we notice that this transformation in
the semi infinite line is the composition of a time translation of parameter
$\epsilon\pi R$ and a special conformal transformation of parameter
$\epsilon\frac{\pi}{R}$. Let us check this explicitly.

In spherical coordinates, the infinitesimal transformation reads

$\begin{split}&t\longrightarrow
t^{\prime}=t+\epsilon\frac{\pi}{R}(R^{2}-t^{2}-r^{2})\\\ &r\longrightarrow
r^{\prime}=r+\epsilon\frac{\pi}{R}(-2tr)\\\
&\Omega\longrightarrow\Omega^{\prime}=\Omega\end{split}$ (54)

Since the invariance of the kinetic term is guaranteed, we need only to check
the invariance of the quadratic term $dtdr/r^{2}$, which is less evident. On
the one hand, we have

$\displaystyle dtdr$ $\displaystyle=$ $\displaystyle
dt^{\prime}dr^{\prime}\begin{vmatrix}\frac{\partial t}{\partial
t^{\prime}}&\frac{\partial t}{\partial r^{\prime}}\\\ \frac{\partial
r}{\partial t^{\prime}}&\frac{\partial r}{\partial
r^{\prime}}\end{vmatrix}=dt^{\prime}dr^{\prime}\begin{vmatrix}1+2\pi\epsilon
t^{\prime}/R+\mathcal{O}(\epsilon^{2})&2\pi\epsilon
r^{\prime}/R+\mathcal{O}(\epsilon^{2})\\\ 2\pi\epsilon
r^{\prime}/R+\mathcal{O}(\epsilon^{2})&1+2\pi\epsilon
t^{\prime}/R+\mathcal{O}(\epsilon^{2})\end{vmatrix}$ (55) $\displaystyle\sim$
$\displaystyle dt^{\prime}dr^{\prime}(1+4\pi\epsilon t^{\prime}/R).$

On the other hand,

$\frac{1}{r^{2}}\sim\frac{1}{(r^{\prime}+2\pi\epsilon
t^{\prime}r^{\prime}/R)^{2}}=\frac{1}{r^{\prime 2}}(1-4\pi\epsilon
t^{\prime}/R+\mathcal{O}(\epsilon^{2})).$ (56)

Hence,

$\frac{dtdr}{r^{2}}=\frac{dt^{\prime}dr^{\prime}}{{r^{\prime}}^{2}}+\mathcal{O}(\epsilon^{2}).$
(57)

By Noether’s theorem, there must exist a conserved current associated to (54),
which is of the form888We remove the tildes and the angular mode labels to
avoid cluttering.

$j^{\mu}=\left(\frac{\delta\mathcal{L}}{\delta(\partial_{\mu}\phi)}\partial_{\nu}\phi-\mathcal{L}\delta^{\mu}_{\nu}\right)\zeta^{\nu},$
(58)

or, in components,

$j^{t}=\frac{1}{2}\left[(\partial_{t}\phi)^{2}+(\partial_{r}\phi)^{2}+\frac{\mu}{r^{2}}\phi^{2}\right]\frac{(R^{2}-t^{2}-r^{2})}{R}-2\frac{tr}{R}\partial_{r}\phi\partial_{t}\phi,$
(59)
$j^{r}=\frac{1}{2}\left[(\partial_{t}\phi)^{2}+(\partial_{r}\phi)^{2}-\frac{\mu}{r^{2}}\phi^{2}\right]2\frac{tr}{R}-\partial_{r}\phi\partial_{t}\phi\frac{(R^{2}-t^{2}-r^{2})}{R}.$
(60)

Finally, the current above corresponds to a modular Hamiltonian

$\begin{split}K_{\ell\vec{m}}&=\int_{0}^{R}drj_{0}(t=0,r)\\\
&=\frac{1}{2}\int_{0}^{R}dr\left(\frac{R^{2}-r^{2}}{2R}\right)\left[\widetilde{\pi}_{\ell\vec{m}}^{2}+(\partial_{r}\widetilde{\phi}_{\ell\vec{m}})^{2}+\frac{\mu_{d}(\ell)}{r^{2}}\widetilde{\phi}_{\ell\vec{m}}^{2}\right],\end{split}$
(61)

the same as (37) deduced in the previous section from different arguments.

##  5 Modular Hamiltonian and entropy

In this section we study the spectrum of the modular Hamiltonian (37). Solving
the eigenfunction problem allows us to compute the entanglement entropy for an
interval attached to the origin, as a function of the angular mode $\ell$ and
the original spacetime dimension $d$. Then we sum over the modes and compare
the result with the entanglement entropy of the $d$-sphere.

###  5.1 Eigenfunctions

In general, given a quadratic modular Hamiltonian of a region $V$, of the form

$K=\int_{V}d^{d-1}x\,d^{d-1}x^{\prime}\,\,\left(\phi(x)M(x,x^{\prime})\phi(x^{\prime})+\pi(x)N(x,x^{\prime})\pi(x^{\prime})\right),$
(62)

with $M$ and $N$ real symmetric operators, the eigenfunctions are those of the
right and left action of $M.N$, namely

$(N.M)u_{s}=s^{2}u_{s}$ (63) $(M.N)v_{s}=s^{2}v_{s}.$ (64)

This leads to the alternative way of writing $K$

$K=\int_{V}d^{d-1}x\int_{0}^{\infty}ds\leavevmode\nobreak\
u_{s}(x)\leavevmode\nobreak\ s\leavevmode\nobreak\ v_{s}^{*}(x).$ (65)

More concretely, the problem we are interested in is defined by (41) and (42),
so the eigenfunctions $u$ and $v$ satisfy the following hypergeometric
equations999For later convenience we renormalize the eigenvalues to absorb a
factor $1/(2\pi)^{2}$.

$\left[\beta^{2}\partial_{r}^{2}+\beta\partial_{r}\beta\partial_{r}-\beta^{2}\frac{\mu}{r^{2}}\right]u_{s}=-s^{2}u_{s}$
(66)

$\left[\beta^{2}\partial_{r}^{2}+3\beta\partial_{r}\beta\partial_{r}+\left(\beta\partial_{r}^{2}\beta+(\partial_{r}\beta)^{2}-\beta^{2}\frac{\mu}{r^{2}}\right)\right]v_{s}=-s^{2}v_{s}$
(67)

The solutions of these equations are101010There is an additional independent
solution, but we dismiss it because it does not go to zero at $r=0$, as
mandated by the boundary conditions.

$\begin{split}u_{s}(r)&=N_{u}\left(\frac{r}{R}\right)^{-1+\frac{d}{2}+\ell}\left(\frac{R^{2}-r^{2}}{R^{2}}\right)^{-is}{}_{2}F_{1}\left[\frac{1}{2}-is,-1+\frac{d}{2}+\ell-
is,\frac{d}{2}-\frac{1}{2}+\ell,\frac{r^{2}}{R^{2}}\right]\\\
v_{s}(r)&=N_{u}\frac{R}{\beta(r)}u_{s}(r),\end{split}$ (68)

where $N_{u}$ is a normalization constant.

Near $r\sim 0$ the solutions behave as,

$u_{s}(r)\sim v_{s}(r)\propto r^{-1+\frac{d}{2}+\ell}$ (69)

in agreement with the classical profile (22), whereas near $r\sim R$ they
behave as

$\begin{split}u_{s}(r)&\sim
N_{u}\left[\left(\frac{R-r}{R}\right)^{-is}\alpha(s)+c.c.\right]\\\
v_{s}(r)&\sim
N_{u}\left[\left(\frac{R-r}{R}\right)^{-1-is}\alpha(s)+c.c.\right],\end{split}$
(70)

with

$\alpha(s)=\frac{2^{-is}\Gamma\left[\frac{d-1}{2}+\ell\right]\Gamma\left[2is\right]}{\Gamma\left[is+\frac{1}{2}\right]\Gamma\left[is+\ell+\frac{d}{2}-1\right]}.$
(71)

It is very important to keep in mind that there is a branch point at $r=R$. In
fact, since the eigenfunctions must satisfy the orthogonality relation

$\int_{0}^{R}dru_{s}(r)v_{s^{\prime}}^{*}(r)=\delta(s-s^{\prime})\,,$ (72)

in order to find out the normalization factor $N_{u}$ we substitute in (72)
the leading terms in their Taylor series expansion (70), because only the
region near $r\sim R$ can contribute with a Dirac delta function. That results
in

$\int_{0}^{R}dru_{s}(r)v_{s^{\prime}}^{*}(r)\sim
2|N_{u}|^{2}\text{Re}\left[I(s-s^{\prime})\alpha(s)\alpha^{*}(s^{\prime})+I(s+s^{\prime})\alpha(s)\alpha(s^{\prime})\right]\,,$
(73)

where

$\begin{split}I(s)&\equiv\int_{0}^{R}dr\frac{R}{R-r}\exp{\left[-is\log{\left(\frac{R-r}{R}\right)}\right]}\\\
&=R\left[\frac{i}{s}+\pi\delta(s)\right].\end{split}$ (74)

Hence, neglecting the finite terms111111We also neglect a contribution of the
form $\delta(s+s^{\prime})$, coming from the second term in (73), because it
is non-zero only at $s=s^{\prime}=0$., we have that (72) holds provided that

$N_{u}=\frac{1}{\sqrt{2\pi R}|\alpha(s)|},$ (75)

save an overall phase that we set to one for convenience.

###  5.2 The entropy

As explained in [39] in the context of the free chiral scalar, we can take
advantage of the orthogonality relation to simplify the computation of the
entanglement entropy, which can be expressed as a regularized integral over a
small region behind the end point $r=R$, of the form

$\begin{split}S(\ell,d)&=\int_{0}^{R-\epsilon}dr\,\int_{0}^{\infty}ds\,u_{s}(r)g(s)v_{s}^{*}(r)\\\
&=-\,\lim\limits_{\delta s\rightarrow
0}\int_{R-\epsilon}^{R}dr\,\int_{0}^{\infty}ds\,u_{s}(r)g(s)v_{s+\delta
s}^{*}(r)\,,\end{split}$ (76)

with

$g(s)=\frac{1+\coth(\pi s)}{2}\log\left(\frac{1+\coth(\pi
s)}{2}\right)+\frac{1-\coth(\pi s)}{2}\log\left(\frac{\coth(\pi
s)-1}{2}\right)$ (77)

Note that since we expect the entanglement entropy of a QFT to diverge due to
the short range correlations between modes at both sides of the boundary, we
regularized it by introducing a small UV cutoff $\epsilon$. Furthermore, in
going from the first to the second line of (76) we shifted the $v$ sub index,
summing over slightly off diagonal elements. For fixed $\delta s\neq 0$ the
integral defined on the whole interval vanishes because of (72), leading to an
integral just behind the boundary. This trick allows us to substitute the
expansion (70), which is much easier to integrate than the original solutions
(68). Finally, we get

$S(\ell,d)=\frac{1}{6}\log{\frac{R}{\epsilon}}-\frac{1}{\pi}\int_{0}^{\infty}ds\,g^{\prime}(s)\text{Arg}(\alpha(s)),$
(78)

or, more explicitly,

$S(\ell,d)=\frac{1}{6}\log{\frac{R}{\epsilon}}-\frac{i\pi}{2}\int_{0}^{\infty}ds\frac{s}{\sinh^{2}(\pi
s)}\log{\left(\frac{4^{is}\Gamma\left[is\right]\Gamma\left[-1+d/2+\ell-
is\right]}{\Gamma\left[-is\right]\Gamma\left[-1+d/2+\ell+is\right]}\right)}$
(79)

The logarithmic coefficient $1/6$ is the expected result for a $(1+1)$
dimensional theory. Meanwhile, the constant term is expressed in terms of an
integral that cannot be solved explicitly. For later convenience, we write it
as a sum of two contributions, one that does not depend neither on the
dimension nor on the angular mode

$c\equiv-\frac{i\pi}{2}\int_{0}^{\infty}ds\frac{s}{\sinh^{2}(\pi
s)}\log{\left(\frac{4^{is}\Gamma\left[is\right]}{\Gamma\left[-is\right]}\right)}\,,$
(80)

and another which does depend on both parameters

$f(\ell,d)\equiv-\frac{i\pi}{2}\int_{0}^{\infty}ds\frac{s}{\sinh^{2}(\pi
s)}\log{\left(\frac{\Gamma\left[-1+d/2+\ell-
is\right]}{\Gamma\left[-1+d/2+\ell+is\right]}\right)}$ (81)

Figure 2: Constant term of the entropy at $d=3$, as a function of the angular
mode $\ell$. The red dots represent the exact numerical value of (81), for
$\ell=\\{1,2,5,10,15,20,30,40,100\\}$. The blue curve corresponds to the fit
$f(\ell,d=3)=c_{0}+c_{1}\log{\ell}$, with $c_{0}=1.345\times 10^{-5}$ and
$c_{1}=-0.1666$.

Although it is unfortunately impossible to find an analytic expression for the
integral, for sufficiently large modes we can make use of the Stirling’s
approximation

$\log{\Gamma(z)}\sim
z\log{z}-z+\frac{1}{2}\log{\frac{2\pi}{z}}+\sum_{n=1}^{N-1}\frac{B_{2n}}{2n(2n-1)z^{2n-1}},\quad|z|\rightarrow\infty$
(82)

to write

$\begin{split}&\log{\left(\frac{\Gamma\left[-1+d/2+\ell-
is\right]}{\Gamma\left[-1+d/2+\ell+is\right]}\right)}\sim-2is\log{\ell}+\sum_{k=2}^{\infty}\sum_{m=1}^{\lfloor\frac{k+1}{2}\rfloor}\frac{(k-2)!}{\ell^{k-1}}a_{k,m}(s)\\\
&+\frac{1}{2}\sum_{k=1}^{\infty}\sum_{m=1}^{\lfloor\frac{k+1}{2}\rfloor}\frac{(k-1)!}{\ell^{k}}a_{k,m}(s)+\sum_{n=1}^{\infty}\sum_{k=1}^{\infty}\sum_{m=1}^{\lfloor\frac{k+1}{2}\rfloor}\frac{B_{2n}(2n+k-2)!}{(2n)!\ell^{2n+k-1}}a_{k,m}(s),\quad\ell>>1,\end{split}$
(83)

where

$a_{k,m}(s)=\frac{2i(-1)^{k+m}}{(2m-1)!(k+1-2m)!}\left(-1+\frac{d}{2}\right)^{k+1-2m}s^{2m-1}.$
(84)

This means that the constant term grows logarithmically with the mode $\ell$,
with corrections that decay as positive powers of $1/\ell$. In fact,
performing the integration over the variable $s$ order by order in the
expansion, we can straightforwardly check that the first few leading terms
read

$f(\ell,d)\sim-\frac{1}{6}\log{\ell}+\frac{a_{1}}{\ell}+\frac{a_{2}}{\ell^{2}}+\frac{a_{3}}{\ell^{3}}+\frac{a_{4}}{\ell^{4}}+\mathcal{O}\left(\frac{1}{\ell^{5}}\right),$
(85)

with

$\displaystyle a_{1}$ $\displaystyle=$ $\displaystyle\frac{1}{4}-\frac{d}{12}$
(86) $\displaystyle a_{2}$ $\displaystyle=$
$\displaystyle\frac{7}{40}-\frac{d}{8}+\frac{d^{2}}{48}$ (87) $\displaystyle
a_{3}$ $\displaystyle=$
$\displaystyle\frac{3}{20}-\frac{7d}{40}+\frac{d^{2}}{16}-\frac{d^{3}}{144}$
(88) $\displaystyle a_{4}$ $\displaystyle=$
$\displaystyle\frac{73}{560}-\frac{9d}{40}+\frac{21d^{2}}{160}-\frac{d^{3}}{32}+\frac{d^{4}}{384}$
(89)

Quite surprisingly, the logarithmic term already approximates $f(\ell,d)$ at
$\ell\sim\mathcal{O}(1)$ very accurately, as shown in figure (2).

In figure (3) we compare the numerical value of (81) with the one obtained
from direct calculation in a radial lattice, again at $d=3$. Since the
constant term depends on the regularization scheme, we subtract the one
corresponding to $\ell=1$ and compare $\Delta f(\ell,3)\equiv
f(\ell,3)-f(\ell=1,3)$. Although it is very hard to achieve good precision in
the lattice121212Roughly speaking, the value of $\ell$ gives a lower bound for
the meaningful radios $R/\epsilon>>\ell$, we find reasonable agreement. For
example, for $\ell=10$, numerical integration yields $\Delta f=-0.3737$, while
the lattice computation gives $\Delta f=-0.3795$.

Figure 3: $\Delta f(\ell,3)\equiv f(\ell,3)-f(\ell=1,3)$. Blue: direct
numerical integration of (81). Orange: calculation with a radial lattice
regularization. $\ell=\\{1,2,5,10,20\\}$

###  5.3 Recovering the scalar entropy

As discussed in section 3, the modular Hamiltonian of the free scalar in the
sphere is equal to the sum over $\ell$ of the modular Hamiltonian pertaining
to each one dimensional theory in the segment. Consequently, we expect that
summing (79) must necessarily reproduce the general structure for the
entanglement entropy,

$S=\begin{cases}\\#\left(\frac{R}{\epsilon}\right)^{d-2}+...+c_{\text{log}}\log{\frac{R}{\epsilon}},&d\quad\text{even}\\\
\\#\left(\frac{R}{\epsilon}\right)^{d-2}+...+F,&d\quad\text{odd}\end{cases}$
(90)

that is, an infinite contribution controlled by the area term, and a universal
piece, either in the form of a logarithmic coefficient in even dimensions,
which is precisely the trace anomaly coefficient associated to the Euler
density [40, 41, 42], or a constant term in odd dimensions [43, 44, 45].

To show that this indeed holds, we need to introduce a cutoff in $\ell$ to
regularize the sum. More concretely, we introduce a damping exponential so
that

$S=\sum_{\ell=0}^{\infty}\lambda(\ell,d)S(\ell,d)e^{-\ell\epsilon/R}.$ (91)

where

$\lambda(\ell,d)=(2\ell+d-3)\frac{(\ell+d-4)!}{\ell!(d-3)!}$ (92)

is the density of states. Note that this grows as $\ell^{d-3}$ for $\ell>>1$.

Given the complicated expression of the constant term $f(\ell,d)$, we
approximate it by its large $\ell$ expansion, leading to

$S=\sum_{\ell=1}^{\infty}\lambda(\ell,d)\left(\frac{1}{6}\log{\frac{R}{\epsilon}}+c-\frac{1}{6}\log{\ell}+\sum_{j=1}^{j_{max}}\frac{a_{j}}{\ell^{j}}\right)e^{-\ell\epsilon/R}+\lambda(0,d)S(0,d)+\text{correction}.$
(93)

The correction above accounts for the error made when approximating
$f(\ell,d)$ by its series expansion, truncated at
$\mathcal{O}({\ell^{-j_{max}}})$.

It is straightforward to verify that (93) reproduces (90). For example, the
divergent pieces come from terms with the general structure

$\sum_{\ell=1}^{\infty}\ell^{p}\left(\log{\frac{R}{\epsilon}}-\log{\ell}\right)e^{-\ell\epsilon/R}=-\Gamma^{\prime}(p+1)\left(\frac{R}{\epsilon}\right)^{p+1}+\zeta(-p)\log{\frac{R}{\epsilon}}+\zeta^{\prime}(-p),$
(94)
$\sum_{\ell=1}^{\infty}\ell^{p}e^{-\ell\epsilon/R}=p!\left(\frac{R}{\epsilon}\right)^{p+1}+\zeta(-p),$
(95)

with $\\{p\mid p\in\mathbb{N}_{0}\wedge p\leq d-3\\}$, and

$\sum_{\ell=1}^{\infty}\frac{1}{\ell}e^{-\ell\epsilon/R}=\log{\frac{R}{\epsilon}}.$
(96)

Note that the logarithmic term, only present in even dimensions, stems from
(94) and (96). Based on this observation, it is worth pointing out that in
order to compute the logarithmic coefficient we only need to take into account
the first $d-1$ terms in the expansion of $f(\ell,d)$, that is, $j_{max}=d-2$.
Subleading corrections give finite contributions at most.

Just to explicitly address some relevant specific cases, at $d=6$ we get

$c_{\text{log}}(d=6)=\frac{29}{540}+a_{1}+\frac{13}{6}a_{2}+\frac{3}{2}a_{3}+\frac{1}{3}a_{4},$
(97)

and, substituting (86), (87), (88), (89),

$c_{\text{log}}(d=6)=\frac{1}{756},$ (98)

in agreement with the expected anomaly value. On the other hand, at $d=4$ we
get

$c_{\text{log}}(d=4)=\frac{1}{18}+a_{1}+2a_{2},$ (99)

which leads to the expected anomaly coefficient

$c_{\text{log}}(d=4)=-\frac{1}{90}.$ (100)

The case of $d=3$ is different from the ones discussed above in that it has no
logarithmic term and the universal piece in the entanglement entropy is
associated to the constant term $F$. In fact, direct calculation yields, using
(86)

$c_{\text{log}}(d=3)=2a_{1}=0.$ (101)

Regarding the constant $F$, the infinite tail in the $1/\ell$ expansion must
in principle be taken into account. For that reason, we regularize the sum
taking up to $j_{max}=2$ and then add a finite contribution which corrects the
approximation, giving the exact value for the constant term in the series of
$f(\ell,d)$. That is,

$\text{correction}=2\lim\limits_{\ell_{max}\rightarrow\infty}\sum_{\ell=1}^{\ell_{max}}\left(f(\ell,d=3)+\frac{1}{6}\log{\ell}-\frac{a_{2}}{\ell^{2}}\right)$
(102)

In figure (4) we plot the above correction as a function of $\ell_{max}$ and
show that it converges very fast to

$\text{correction}\sim 0.00519641$ (103)

Figure 4: Error made in the computation of the constant term $F$ when
approximating $f(\ell,3)$ by $-\frac{1}{6}\log{\ell}+\frac{a_{2}}{\ell^{2}}$.
$\ell_{max}$ is the greatest angular momentum that is summed over. We see that
the correction converges very fast to $\sim 0.00519641$

According to (93), another term which contributes is

$f(0,3)=-\frac{i\pi}{2}\int_{0}^{\infty}ds\frac{s}{\sinh^{2}(\pi
s)}\log{\left(\frac{\Gamma\left[1/2-is\right]}{\Gamma\left[1/2+is\right]}\right)}\sim
0.278435.$ (104)

Gathering all the pieces together, we finally get

$F=\frac{\pi^{2}}{3}a_{2}+\frac{\zeta^{\prime}(0)}{3}+f(0,3)+\text{correction}\sim-0.0638049,$
(105)

which is within $0.003\%$ of the exact value [43, 45]. Note that the constant
$c$ does not contribute at all to $F$.

##  6 Final remarks

In this article, we focused on theories in the semi infinite line constructed
from the dimensional reduction of a free scalar in $d$ dimensions. Given that
the decomposition of the parent theory $H$ into independent sectors
$H_{\ell\vec{m}}$, labeled by the angular modes, also holds for the vacuum
modular Hamiltonian in spheres $K=\sum_{\ell\vec{m}}K_{\ell\vec{m}}$, and
provided that the vacuum state of the system is the product
$\rho=\otimes\rho_{\ell\vec{m}}$, then, it is immediate to identify the
modular Hamiltonian mode $K_{\ell\vec{m}}$ with the modular Hamiltonian of the
one-dimensional reduced system $H_{\ell\vec{m}}$ in the interval $(0,R)$.
Remarkably, the resulting modular Hamiltonian is local and proportional to the
energy density, with the same weight function
$\beta(r)=\frac{R^{2}-r^{2}}{2R}$ as the one characteristic of a CFT in a
sphere $R$.

We complemented the previous analysis with the study of the symmetries
inherited from the $d$ dimensional conformal theory. This approach evidences
the fact that the symmetry behind the locality of the reduced modular
Hamiltonian is just the restriction to the semi infinite line of the original
modular symmetry in $d$ dimensions. We identified the conserved current
associated to this symmetry transformation and checked that the
$K_{\ell\vec{m}}$ found by dimensional reduction coincides with the Noether
charge.

On the other hand, the spectral decomposition of the modular Hamiltonian leads
to an analytic expression for the corresponding entanglement entropy (EE)
which in turn, after summing over the angular modes, allowed us to recover the
EE of the original $d$ dimensional theory in the sphere. To make sense of the
sum, we used a novel regularization implemented by a damping exponential
parametrized by the same cutoff $\epsilon$ that regularizes the radial
coordinate. As we mentioned in the introduction, in a way, this procedure
generalizes the one introduced by Srednicki in [34] and provides an additional
tool to calculate analytically the EE logarithmic coefficient in even
dimensions, the universal constant term in $d=3$, among others.

It would certainly be interesting to explore in the future the modular
Hamiltonian of non-conformal theories constructed from the dimensional
reduction of other free theories, fermions for example. We expect that the
decomposition of the parent theory into independent sectors must carry over
unaltered, as well as the symmetry arguments which justify the resulting
modular Hamiltonian is a conserved charge.

## Acknowledgments

We thank H.Casini, C.Fosco, E.Tonni and G.Torroba for discussions while this
work was being carried out. This work was supported by CONICET, CNEA and
Universidad Nacional de Cuyo, Instituto Balseiro, Argentina.

## References

  * [1] R.Haag, Local quantum physics: Fields, particles, algebras.  1992\.
  * [2] R. Brunetti, D. Guido, and R. Longo, “Modular structure and duality in conformal quantum field theory,” Communications in Mathematical Physics 156 no. 1, (1993) 201–219.
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