# The overtone level spacing of a black hole quasinormal frequencies: a
fingerprint of a local $SL(2,\mathbb{R})$ symmetry

Bernard Raffaelli<EMAIL_ADDRESS>Institut de Mathématiques
de Bourgogne, UMR 5584 CNRS, Université Bourgogne Franche-Comté, F-2100 Dijon,
France

###### Abstract

The imaginary part of the quasinormal frequencies spectrum for a static and
spherically symmetric black hole is analytically known to be equally spaced,
both for the highly damped and the weakly damped families of quasinormal
modes. Some interesting attempts have been made in the last twenty years to
understand in simple ways this level spacing for the only case of highly
damped quasinormal frequencies. Here, we show that the overtone level spacing,
for both the highly damped and weakly damped families of quasinormal modes,
can simply be understood as a fingerprint of a hidden local $SL(2,\mathbb{R})$
symmetry, near different regions of the black hole spacetime, i.e. the near-
horizon and the near-photon sphere regions.

##  I Introduction

In the context of small perturbations of a black hole (BH) spacetime, the
quasinormal modes (QNM), which are damped modes of perturbation which
propagate towards spatial infinity, are playing a fundamental role both in our
experimental and theoretical understanding of gravity and of BH in particular.
The QNM are related to complex quasinormal frequencies (QNF) at which they
occur. The real part of the QNF corresponds to the frequency of oscillation
and the imaginary part to the damping rate of the QNM. Analytically, for
example in the case of static and spherically symmetric (asymptotically flat)
BH perturbed by a massless scalar field, two families of QNM are known

  * •

the _highly damped QNM_ with the associated QNF (at first order in $n$)

$\omega_{n}=\frac{\kappa}{2\pi}\ln\left[1+2\cos\left(\frac{\pi(qD-2)}{2}\right)\right]-i\kappa\left(n+\frac{1}{2}\right)$
(1)

where $\kappa$ is the surface gravity of the BH, $D=d-2$ for $d$-dimensional
spacetimes and $q$ is a purely real power law exponent that characterizes the
BH metric near its singularity [1, 2],

  * •

and the _weakly damped QNM_ with the associated QNF (at first order in $\ell$
and $n$)

$\omega_{\ell,n}=\Omega_{c}\left(\ell+\frac{1}{2}\right)-i|\Lambda_{c}|\left(n+\frac{1}{2}\right)$
(2)

where $\Omega_{c}$ is the angular velocity of massless particles on unstable
circular orbits around the BH, and $|\Lambda_{c}|$ is the Lyapunov exponent
associated with these unstable circular orbits [3, 4].

The imaginary part of the QNF, in both cases, is interesting. It contains the
overtone number $n$ and is equally spaced. Some remarks about its amplitude
will also be given in this paper. Then, it would be natural to ask whether the
seemingly simple behavior of the imaginary part of the QNF in both cases may,
or may not, hide a “universal” feature of BH spacetimes. For static and
spherically symmetric spacetimes, such an imaginary part for the BH QNF, in
the only case of the highly damped QNM, has been discussed in [5], where it
has been obtained as poles of the scattering amplitude in the Born
approximation, and in [6] where it has been extracted via a Coulomb-like phase
shift.
In this paper, we would like to show that such a behavior for the imaginary
part of a BH QNF is in fact more “universal” and can simply be understood, in
both the highly and weakly damped regimes, as a fingerprint of a hidden local
$SL(2,\mathbb{R})$ symmetry. We will focus on static and spherically symmetric
BH endowed with a photon sphere (also known as “lightrings”, or “photon
rings”). The strategy is made simple as long as one can show that certain
particular regions of a BH spacetime admit a Rindler metric as an
approximation. Indeed, on the one hand, the resonant scattering of a massless
scalar field in a Rindler spacetime is shown to be deeply linked to the
resonant scattering problem of a scalar field by an inverted harmonic
oscillator (IHO) potential of the form $V(x)=-(\alpha^{2}/4)x^{2}$, where
$\alpha$ is the potential curvature, or potential strength. On the other hand,
the resonant scattering problem by an IHO potential can be algebraically
solved using the $SL(2,\mathbb{R})$ algebra for which the representations and
the eigenvalues spectrum are discrete and indexed by a non negative integer.
The general claim of this paper follows: in any BH geometry, if there exist
some regions where the BH metric can be locally approximated as a Rindler one,
in some set of coordinates, then the underlying algebra is $SL(2,\mathbb{R})$,
and the imaginary part of the BH QNF will be indexed by a non negative
integer, i.e. the overtone number, and it will be equally spaced. The
amplitude of the overtone level spacing is shown to be related to the IHO
potential curvature.

The paper is organized as follows. In section II, we give an
$SL(2,\mathbb{R})$ approach to the resonant scattering problem of a scalar
field by an IHO potential. This allows to highlight that the eigenvalues
spectrum of the corresponding Hamiltonian is discrete, purely imaginary and
indexed by a non negative integer. In section III, we focus first on the
motions of massless particles in Rindler spacetime, and then on the resonant
scattering problem of a massless scalar field in such a spacetime. We show
that massless dynamics in Rindler spacetime is deeply linked to the problem
treated in section II, and we apply the results we have obtained to the
Rindler case. Finally, in section IV, we apply the $SL(2,\mathbb{R})$ algebra
to the case of static and spherically symmetric BH endowed with a photon
sphere, and show that the imaginary part of the BH QNF in both the highly an
weakly damped regimes are equally spaced. In sections V and VI we discuss the
amplitude of the overtone level spacing and gives somes conclusions about the
results obtained in this paper.

##  II An $SL(2,\mathbb{R})$ approach to the resonant scattering by an
inverted harmonic oscillator potential

With a particular choice of “position” operator $X$ and conjugate “linear
momentum” operator $P$, acting both on a same Hilbert space, the Hamiltonian
of a particle in an IHO potential can always be written as

$H=P^{2}-\frac{\alpha^{2}}{4}X^{2},$ (3)

where $\alpha$ is the curvature of the IHO potential, and the operators, which
satisfy $[X,P]=i\mathbb{I}$, are defined, as usual, as $P=-i\frac{d}{dx}$ and
$X=x$, in an “$x$-representation”. In a system of units such that $h=c=1$, it
should be noted that if one wants $H$ to have dimension $L^{-1}$, $P$ and $X$
have to be defined here with dimensions $L^{-1/2}$ and $L^{1/2}$ respectively.
The IHO potential curvature $\alpha$ has dimension $L^{-1}$.

From the point of view of a scalar field theory, the Hamiltonian gives the
following equation for its eigenstates $\psi$ in the $x$-representation

$H\psi(x)=\left(-\frac{d^{2}}{dx^{2}}-\frac{\alpha^{2}}{4}x^{2}\right)\psi(x)=h\psi(x).$
(4)

Here $h$ is the Hamiltonian eigenvalue associated with the eigenstate $\psi$.
This equation is a Weber differential equation whose solutions are related to
parabolic cylinder functions [7]. They describe the scattering states of the
scalar field by an IHO potential. Amongst the scattering states, the system
presents a discrete set of eigenstates, i.e. the resonant modes, or QNM, which
can be obtained, for example, by looking at the poles of the corresponding
$S$-matrix in the complex energy plane.

An expression of $H$ in the form $(\ref{Hamiltonian})$ (or
$(\ref{ScalarFieldEq})$) plays an “universal” role for the resonant scattering
problem of a scalar field by the top of any potential barrier. Indeed, in such
case, the QNM are associated with energies close to the top of the potential.
For such energies, one can always approximate any potential barrier close to
its maximum by a second order Taylor series expansion which locally gives an
inversed parabola. With the right choice of $X$ and $P$ operators, the
corresponding Hamiltonian $H$ can then always be reduced to the form
$(\ref{Hamiltonian})$ (or $(\ref{ScalarFieldEq})$). This form of $H$ is not
only “universal” for a problem of resonant scattering by a potential barrier,
but it also permits to reveal a hidden (dynamical) conformal symmetry near the
top of the potential through an $SO(2,1)\sim SL(2,\mathbb{R})$ algebra which
allows to obtain the associated set of QNM from the algebra representations
and, especially for the aim of this paper, to obtain the overtone level
spacing in the imaginary part of the quasinormal frequencies. We briefly
recall below the outline of one of the possible proofs that mainly lies on the
introduction of “creation and annihilation operators” analogues in the
factorization of $H$, which allows to highlight the underlying
$SL(2,\mathbb{R})$ algebra (see [8] for more details).

We have seen that the time-independent, one-dimensional Hamiltonian of a
quantum particle scattered by an IHO potential reads in the $x$-representation

$H=-\frac{d^{2}}{dx^{2}}-\frac{\alpha^{2}}{4}x^{2}.$ (5)

As one usually proceeds in the context of the well-known harmonic oscillator
problem, let us begin by factorizing $H$. From the operators
$P=-i\frac{d}{dx}$ and $X=x$ (mentioned in the previous section), one can
introduce “creation and annihilation operators” analogues

$U_{\pm}=\pm\frac{1}{\sqrt{\alpha}}P+\frac{\sqrt{\alpha}}{2}X=\mp
i\frac{1}{\sqrt{\alpha}}\frac{d}{dx}+\frac{\sqrt{\alpha}}{2}x,$ (6)

with $(U_{\pm})^{\dagger}=U_{\pm}\neq U_{\mp}$ (with respect to the usual
scalar product) and $[U_{-},U_{+}]=i\mathbb{I}$. Let us note that in our
system of units, $U_{\pm}$ are dimensionless operators. The Hamiltonian $H$
can then be written in terms of $U_{\pm}$ as

$H=P^{2}-\frac{\alpha^{2}}{4}X^{2}=-\frac{\alpha}{2}\left(U_{+}U_{-}+U_{-}U_{+}\right).$
(7)

It is also possible to introduce the dilation operator $D$ and the operator
$S$, which is associated with special conformal transformations generator
$K=X^{2}$, as follows

$\displaystyle D$
$\displaystyle=\frac{1}{2}\left(PX+XP\right)=\frac{1}{2}\left(U_{+}^{2}-U_{-}^{2}\right)=-i\left(x\frac{d}{dx}+\frac{1}{2}\right),$
(8) $\displaystyle S$
$\displaystyle=P^{2}+\frac{\alpha^{2}}{4}X^{2}=\frac{\alpha}{2}\left(U_{+}^{2}+U_{-}^{2}\right)=-\frac{d^{2}}{dx^{2}}+\frac{\alpha}{4}x^{2}.$

Up to dimensional factors, it is important to note that the expressions of $H$
and $D$ are switched whether one is looking at their expressions in terms of
$X$ and $P$ operators, or in terms of $U_{\pm}$ operators. In other words, for
what concerns the IHO problem, $H$ can be seen as a dilation process or as a
scattering process by an IHO potential, depending on the choice of
“coordinates” ($x$ or $u_{\pm}$) and the associated conjugate momenta ($p$ or
$u_{\mp}$). It is interesting to note that the operator $S$ remains unchanged
in both “coordinates” systems, as a sum of squared operators, and that it
would play the role of the Hamiltonian in the usual harmonic oscillator
problem. It becomes then quite trivial to write the expressions of $H$, $D$
and $S$ as differential operators in “$u_{\pm}$”-representations, from their
expressions in the $x$-representation (see [9] and references therein, for a
nice exhaustive analysis of the physics of the IHO). For example, $H$ in the
$u_{\pm}$-representation reads simply

$H=\mp i\alpha\left(u_{\pm}\frac{d}{du_{\pm}}+\frac{1}{2}\right).$ (9)

In order to simplify the underlying algebra, one can finally define the
following three (dimensionless) operators

$\displaystyle J_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}D=-\frac{i}{2}\left(x\frac{d}{dx}+\frac{1}{2}\right),$
(10a) $\displaystyle J_{2}$ $\displaystyle=$
$\displaystyle-\frac{i}{2\alpha}S=\frac{i}{2\alpha}\left(\frac{d^{2}}{dx^{2}}-\frac{\alpha^{2}}{4}x^{2}\right),$
(10b) $\displaystyle J_{3}$ $\displaystyle=$
$\displaystyle\frac{i}{2\alpha}H=-\frac{i}{2\alpha}\left(\frac{d^{2}}{dx^{2}}+\frac{\alpha^{2}}{4}x^{2}\right).$
(10c)

Those operators satisfy the commutation relations of an $SO(2,1)$ algebra

$[J_{1},J_{2}]=-iJ_{3};\quad[J_{2},J_{3}]=iJ_{1};\quad[J_{3},J_{1}]=iJ_{2}.$
(11)

Finally, from $J_{1}$ and $J_{2}$, one can also introduce “ladder operators”.
For example

$J_{\pm}=\pm iJ_{1}-J_{2}=\frac{i}{2}U_{\pm}^{2},$ (12)

which, together with $J_{3}$, satisfy an $SL(2,\mathbb{R})$ algebra

$[J_{+},J_{-}]=-2J_{3};\quad[J_{3},J_{\pm}]=\pm J_{\pm}.$ (13)

It should be noted first that one could have obviously constructed $J_{3}$ and
$J_{\pm}$, which satisfy commutation relations
$(\ref{LadderOp_CommutationRelation})$, more directly from $H$ and
$U_{\pm}^{2}$, i.e. without any references to the operators $D$ and $S$.
Finally, let us also note that that the ladder operators $J_{\pm}$ are not
self-adjoint conjugate to each other. In our example here, one has instead
$\left(J_{\pm}\right)^{\dagger}=-J_{\pm}$, i.e. anti-self-adjoint operators.

Once the algebra is found, the computation of the eigenstates and eigenvalues
of $H$ follows. Indeed, as usually done in the harmonic oscillator case, one
can begin by looking for the ground states of the ladder operators $J_{\pm}$.
Then, by an application of $(J_{\pm})^{n}$ to the associated ground state (for
every non negative integer $n$), it is an easy task to construct the set of
eigenstates which belong to the associated rigged Hilbert space, that allows,
in a few words, states to be defined as distributions (see [10, 11] for more
details about the rigged Hilbert space formalism). The interested reader may
found a detailed construction of the set of eigenstates and eigenvalues of
$(J_{\pm})^{n}$ and $H$ in [8], in the context of a hidden $SL(2,\mathbb{R})$
symmetry on a BH photon sphere. Finally, one shows that the associated
spectrum of $H$ is discrete, purely imaginary, and indexed by an integer. The
eigenvalues $h_{n}^{(\pm)}$ of $H$, for every non negative integer $n$, can
finally be written as

$h_{n}^{(\pm)}=\mp i\alpha\left(n+\frac{1}{2}\right).$ (14)

These discrete and purely imaginary eigenvalues are shown to be related to the
frequencies of the resonant modes in the resonant scattering problem of a
scalar field by an IHO potential [9, 8]. The non negative integer index $n$
will be the key to understand the overtone level spacing of a BH quasinormal
frequencies. It should be noted that the curvature $\alpha$ of the IHO
potential appears to be the amplitude of the eigenvalues spacing of $H$.

In the following, we will first show that the resonant scattering problem of a
massless scalar field in a Rindler spacetime can precisely be reduced to the
study of a resonant scattering problem by an IHO potential with the
corresponding Hamiltonian $H$. From the above, this obviously allows its
algebraic treatment through the $SL(2,\mathbb{R})$ algebra with all the
related results. By a natural extension, it should be noted that every regions
of a black hole spacetime that locally admit the Rindler metric as an
approximation will be places where a similar treatment can be applied. In
particular, we will show in the following sections that the horizon (which is
the best known case) but also the photon sphere are both such regions.

##  III The Rindler spacetime case

###  III.1  Massless particles in Rindler spacetime

We consider the line element of a $(1+1)$-dimensional Rindler spacetime, in
Rindler’s coordinates $(t,x)$:

$ds^{2}=a^{2}x^{2}dt^{2}-dx^{2}.$ (15)

It is known to describe a line element seen by an uniformly accelerated
observer (with a constant acceleration $a$), in coordinates $(t,x)$. The study
can be restricted, for example, to the right Rindler wedge, for which
$t\in]-\infty,+\infty[$ and $x\in[0,+\infty[$. In the following, we will be
focusing on the motions of massless particles in those coordinates. With this
aim in mind, we introduce an affine parameter $\lambda$ to describe the
geodesics, and we decide to use a (quadratic) Lagrangian approach which is a
minimalist but efficient way to highlight our point. Let us consider the
following quadratic lagrangian:

$\mathcal{L}=\frac{1}{2}\frac{ds^{2}}{d\lambda^{2}}=\frac{1}{2}(a^{2}x^{2}\dot{t}^{2}-\dot{x}^{2})$
(16)

where $\dot{t}=dt/d\lambda$ and $\dot{x}=dx/d\lambda$. In units such that
$h=c=1$, here $x$ and $t$ have dimension $L$, $\lambda$ has dimension $L^{2}$,
while $a$ has dimension $[x]^{-1}=L^{-1}$.

From Euler-Lagrange equations and spacetime symmetries, the only integral of
motion (equivalently associated with the Killing vector $\partial/\partial t$)
is here the “energy” $E$ defined by

$E=\frac{\partial\mathcal{L}}{\partial\dot{t}}=a^{2}x^{2}\dot{t}.$ (17)

The “linear momentum” is defined as

$p=\frac{\partial\mathcal{L}}{\partial\dot{x}}=-\dot{x}.$ (18)

Inserting $(\ref{E})$ into the line element $(\ref{RindlerMetric})$ for a
massless particle, i.e. $ds^{2}=0$, one finds

$\dot{x}^{2}-\frac{E^{2}}{a^{2}x^{2}}=0$ (19)

or

$\left(\frac{dx}{dt}\right)^{2}-a^{2}x^{2}=0.$ (20)

This equation, whose solution grows exponentially with an associated Lyapunov
exponent $\Lambda=|a|$ (see [12] and references therein), describes the
trajectory of a massless particle in Rindler’s coordinates $(t,x)$, i.e. seen
by an uniformly accelerated observer. Equation $(\ref{MasslessMotion2})$
clearly shows that this motion, in $(t,x)$ coordinates, is actually a motion
in an IHO potential. This equation can then also be interpreted as describing
an instability near the (hyperbolic) point $x=0$ (the top of the IHO potential
barrier) in the phase space $(dx/dt,x)$. In other words, the Rindler horizon
corresponds to an unstable equilibrium point for massless particles, in the
phase space $(dx/dt,x)$ of an accelerated observer. Let us note that for
massless geodesics ($ds^{2}=0$), dividing $(\ref{RindlerMetric})$ by $dt^{2}$
would have given directly $(\ref{MasslessMotion2})$.

It should be noted that there exist a simple dispersion relation between the
energy $E$ and the linear momentum $p$ of the particle, obtained by inserting
$(\ref{p})$ in $(\ref{MasslessMotion1})$

$E=\pm axp.$ (21)

This dispersion relation is characteristic of a particle in a IHO potential.
Indeed, with a change of variables of the form

$\tilde{x}=x-\frac{2}{a}p\quad\textrm{and}\qquad\tilde{p}=\frac{a}{2}x+p,$
(22)

which is equivalent to $(\ref{CreationAnnihilationOperators})$, one can
rewrite $(\ref{RindlerDispersionRelation})$ as

$E=\pm\left(\tilde{p}^{2}-\frac{a^{2}}{4}\tilde{x}^{2}\right).$ (23)

where the IHO potential appears explicitly.

###  III.2  Scattering of a massless scalar field in a Rindler spacetime

The Klein Gordon equation for a massless scalar field $\Phi(x^{\mu})$ in a
spacetime with metric $g_{\mu\nu}$ reads

$\Box\Phi=g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\Phi=\frac{1}{\sqrt{-g}}\partial_{\mu}\left(\sqrt{-g}g^{\mu\nu}\partial_{\nu}\Phi\right)=0.$
(24)

In a Rindler spacetime with metric $(\ref{RindlerMetric})$, the Klein Gordon
equation becomes

$\left[-\partial_{t}^{2}+a^{2}x\partial_{x}\left(x\partial_{x}\right)\right]\Phi(t,x)=0.$
(25)

Let us assume a stationary scalar field with harmonic time dependence of the
form $\Phi(t,x)=\phi(x)e^{-i\omega t}$. The Klein-Gordon equation becomes

$\left[(-i\omega)^{2}-ax\partial_{x}\left(x\partial_{x}\right)\right]\Phi(t,x)=0,$
(26)

which can be easily factorized in the form

$\left[(-i\omega)+ax\partial_{x}\right]\left[(-i\omega)-ax\partial_{x}\right]\Phi(t,x)=0.$
(27)

It should be noted that a massless scalar plane wave, e.g.
$\Phi(t,x)=\Phi_{0}e^{-i(\omega t\mp kx)}$, solution of each term of
$(\ref{KG_factorized})$ gives a dispersion relation

$\omega=\pm axk$ (28)

which is equivalent to the relation $(\ref{RindlerDispersionRelation})$.

In order to establish explicitly the link between $(\ref{KG_factorized})$ and
$(\ref{H_Diff_Upm})$, we will have to introduce a “trick” that will highlight
an important aspect of thermality comparing the Rindler and the IHO case. Let
us begin to rewrite $(\ref{KG_factorized})$ as

$\left[-i\omega-\frac{a}{2}+\frac{a}{2}+ax\partial_{x}\right]\left[-i\omega+\frac{a}{2}-\frac{a}{2}-ax\partial_{x}\right]\Phi(t,x)=0.$
(29)

Most of the solutions $\Phi$ [13] can be reduce to a linear combination of
$\tilde{\Phi}^{(\pm)}$, such as

$\left\\{\begin{array}[]{ll}\left(ax\partial_{x}+\dfrac{a}{2}\right)\tilde{\Phi}^{(+)}=\left(i\omega+\dfrac{a}{2}\right)\tilde{\Phi}^{(+)}\\\
-\left(ax\partial_{x}+\dfrac{a}{2}\right)\tilde{\Phi}^{(-)}=\left(i\omega-\dfrac{a}{2}\right)\tilde{\Phi}^{(-)}.\end{array}\right.$
(30)

In other words, $\tilde{\Phi}^{(\pm)}$ both satify

$\pm
ia\left(x\partial_{x}+\frac{1}{2}\right)\tilde{\Phi}^{(\pm)}=-\left(\omega\mp\frac{ia}{2}\right)\tilde{\Phi}^{(\pm)},$
(31)

which reads

$H\tilde{\Phi}^{(\pm)}=\left(\omega\mp\frac{ia}{2}\right)\tilde{\Phi}^{(\pm)}$
(32)

where $H$ is the Hamiltonian of an IHO written in the form
$(\ref{H_Diff_Upm})$, i.e. as a dilation operator. In other words, the
equation of motion of a massless scalar field, of frequency $\omega$, seen by
an uniformly accelerated observer (with acceleration $a$) in Rindler
coordinates $(t,x)$, can simply be expressed in term of an IHO Hamiltonian. It
should be noted that the naive “trick” used in $(\ref{KG_factorized_2})$ that
leads to $(\ref{H_Rindler_IHO})$, could be understood as being related to the
thermal behavior of the number of quantized particles in the Unruh effect.
Indeed, on the one hand, the factorization of the massless Klein Gordon
equation for a scalar field $\Phi$ in a $(1+1)$-dimensional Rindler spacetime
with metric $(\ref{RindlerMetric})$ gives a system of two massless scalar
field $\tilde{\Phi}^{(\pm)}$ equations, the same way the usual
$(1+1)$-dimensional massless scalar wave equation can be factorized into a
system of two $(1+1)$-dimensional uncoupled Weyl-like equations. On the other
hand, it is well-known that, for a scalar field, the thermal behavior of the
number of quantized particles seen by an accelerated observer is given by a
Bose-Einstein factor $(e^{2\pi\omega/T}-1)^{-1}$ with Unruh temperature $T$,
whereas for a spin-$1/2$ field the thermal behavior is given by a Fermi-Dirac
factor $(e^{2\pi\omega/T}+1)^{-1}$. From this perspective, the “trick” in
$(\ref{KG_factorized_2})$ formally corresponds to the passage from one thermal
behavior, that originates from equation $(\ref{KleinGordon_Rindler})$, to the
other, that originates from equations $(\ref{Weyl_like_eqs1})$, with the
change $\omega\to\omega\mp ia/2$ in the computation of the related Unruh
effect [14].

We have explicitly shown in this section that the scattering by an IHO
potential is actually hidden in the scattering problem of a massless scalar
field in Rindler spacetime, in coordinates $(t,x)$. The resonant scattering in
Rindler spacetime is then simply given by the $SL(2,\mathbb{R})$ algebraic
approach presented in section II. Indeed, by combining $(\ref{HEigenvalues})$
and $(\ref{H_Rindler_IHO})$ gives a set of purely imaginary frequencies, for
every non negative integer $n$

$\omega_{n}^{(\pm)}=\mp ian.$ (33)

As anticipated previously, it should be noted that the discrete eigenvalues
spectrum of the IHO Hamiltonian for the associated resonant modes can be
understood as being at the origin of the regular level spacing in the QNF.
This level spacing appears clearly here, in the case of resonant scattering in
Rindler spacetime. Let us note that the amplitude of the level spacing is
nothing less than the IHO potential curvature, which is interpreted as a
constant acceleration in the Rindler case. This will appear in a very similar
manner in the QNF spectrum for BH, as long as the BH metric admits close to
certain regions of spacetime, the Rindler metric as an approximation.

##  IV The static and spherically symmetric black hole case

In the Schwarzschild coordinates $(t,r,\theta,\phi)$, we consider a static
spherically symmetric four-dimensional spacetime with metric

$ds^{2}=f(r)dt^{2}-\frac{dr^{2}}{f(r)}-r^{2}d\sigma^{2}.$ (34)

As usual, $d\sigma^{2}=d\theta^{2}+\sin^{2}\theta d\varphi^{2}$ is the line
element on the unit 2-sphere $S^{2}$, with $\theta\in[0,\pi]$ and
$\varphi\in[0,2\pi]$.
Moreover, let us consider the BH exterior with $r\in]r_{h},+\infty[$, where
$r=r_{h}$ is a simple root of $f(r)$ and defines the location of the BH event
horizon. Let us assume also that the background geometry is asymptotically
flat and the tortoise coordinate $r_{\ast}=r_{\ast}(r)$, defined as
$dr_{\ast}/dr=1/f(r)$, is a bijection from $]r_{h},+\infty[$ to
$]-\infty,+\infty[$.

Without loss of generality, we will consider motions on the equatorial plane
$\theta=\pi/2$. A free-falling massless particle moves along null geodesics
according to

$-f(r)\dot{t}^{2}+\frac{1}{f(r)}\dot{r}^{2}+r^{2}\dot{\varphi}^{2}=0,$ (35)

where, as done earlier in this paper, $\dot{t}=dt/d\lambda$,
$\dot{r}=dr/d\lambda$, $\dot{\varphi}=d\varphi/d\lambda$ and $\lambda$ is an
affine parameter which describes null geodesics.

From symmetries of the BH spacetime, one can define integrals of motion, i.e.
energy $E$ and angular momentum $L$ of the massless particle, associated
respectively with the Killing vectors $\partial/\partial t$ and
$\partial/\partial\varphi$:

$f(r)\dot{t}=E\quad;\quad r^{2}\dot{\varphi}=L.$ (36)

The equation of motion is easily deduced from (35) and reads

$\dot{r}^{2}+V_{\textrm{eff}}(r)=E^{2},$ (37)

where the effective potential $V_{\textrm{eff}}$ is defined as

$V_{\textrm{eff}}(r)=\frac{L^{2}}{r^{2}}f(r).$ (38)

A photon sphere (or “lightrings”, or “photon rings”) located at $r=r_{c}$
corresponds to a local maximum of $V_{\textrm{eff}}(r)$ at $r=r_{c}$ such as

$\displaystyle\left.\frac{d}{dr}V_{\textrm{eff}}(r)\right|_{r_{c}}=0$
$\displaystyle\Leftrightarrow$
$\displaystyle\frac{2}{r_{c}}f_{c}=f^{\prime}_{c},$ (39a)
$\displaystyle\left.\frac{d^{2}}{dr^{2}}V_{\textrm{eff}}(r)\right|_{r_{c}}<0$
$\displaystyle\Leftrightarrow$ $\displaystyle
f^{\prime\prime}_{c}-\frac{2}{r_{c}^{2}}f_{c}<0.$ (39b)

The subscript “$c$” means, above and in the following, that the quantity
considered is evaluated at $r=r_{c}$, and the superscripts “ ′ ” and “ ′′ ”
respectively mean the first and second derivatives with respect to $r$.

A massless particle reaches the photon sphere when the turning point of its
motion satisfies $E^{2}=V_{\textrm{eff,c}}$, i.e. $L/E=r_{c}/\sqrt{f_{c}}$,
where $r_{c}/\sqrt{f_{c}}=b_{c}$ is the critical impact parameter for massless
particles to reach tangentially the photon sphere, before circling the BH at
$r=r_{c}$. Moreover, at $r=r_{c}$ one also has

$V^{\prime\prime}_{\textrm{eff},c}=-2\eta_{c}^{2}\frac{L^{2}}{r_{c}^{4}},$
(40)

where

$\eta_{c}=\frac{1}{2}\sqrt{4f_{c}-2r_{c}^{2}f^{\prime\prime}_{c}}.$ (41)

The study of instability associated with the circular orbits of massless
particles on the photon sphere follows [3]. Indeed, writing down a second
order Taylor series expansion of $V_{\textrm{eff}}$ near the photon sphere at
$r=r_{c}$ gives

$V_{\textrm{eff}}(r)\simeq
V_{\textrm{eff},c}+\frac{1}{2}V^{\prime\prime}_{\textrm{eff},c}(r-r_{c})^{2},$
(42)

such as (37) becomes

$\dot{r}^{2}+V_{\textrm{eff},c}+\frac{1}{2}V^{\prime\prime}_{\textrm{eff},c}(r-r_{c})^{2}=E^{2}.$
(43)

If the turning point is close enough to $r=r_{c}$, one can consider that
$E^{2}\approx V_{\textrm{eff},c}$. The equation of motion then simplifies to

$\dot{r}^{2}+\frac{1}{2}V^{\prime\prime}_{\textrm{eff},c}(r-r_{c})^{2}=0,$
(44)

which can be written, in $(r,t)$ coordinates, as

$\left(\frac{dr}{dt}\right)^{2}+\frac{V^{\prime\prime}_{\textrm{eff},c}}{2\dot{t}^{2}}(r-r_{c})^{2}=0.$
(45)

Close to $r=r_{c}$, one has $\dot{t}=dt/d\lambda\approx E/f_{c}$, and the
equation of motion in the vicinity of the photon sphere simply reads

$\left(\frac{dr}{dt}\right)^{2}-\Lambda_{c}^{2}(r-r_{c})^{2}=0,$ (46)

where

$|\Lambda_{c}|=\eta_{c}\frac{\sqrt{f_{c}}}{r_{c}}$ (47)

is the Lyapunov exponent associated with the unstable circular motion of a
free-falling massless particle near the BH photon sphere.

###  IV.1 The near horizon region: the highly damped QNM

In this short section, we consider the well-known case of a purely radial
motion in the line element $(\ref{metric_BH})$

$ds^{2}=f(r)dt^{2}-\frac{dr^{2}}{f(r)}.$ (48)

In the near-horizon region, one can approximate $f(r)$ by
$f(r)=f^{\prime}_{h}(r-r_{h})$ at first order, with
$f^{\prime}_{h}=(df/dr)(r=r_{h})$. In a coordinate systems for which

$d\rho=\frac{dr}{\sqrt{f^{\prime}_{h}(r-r_{h})}}$ (49)

and introducing the surface gravity $\kappa=(1/2)f^{\prime}_{h}$, the Rindler
approximation of the metric $(\ref{metric_BH_radial})$ in the near horizon
region follows easily

$ds^{2}=\kappa^{2}\rho^{2}dt^{2}-d\rho^{2}$ (50)

The form $(\ref{RindlerMetric})$ is obviously recovered from which an equation
similar to $(\ref{MasslessMotion2})$ is derived. It should be noted that the
equation of motion of a free-falling massless particle can also be obtained
from $(\ref{EqMotion})$, with $L=0$ because $\varphi$ is zero for purely
radial motions towards the BH, which reads

$\dot{r}^{2}=E^{2}.$ (51)

With $(\ref{EL})$, one deduces

$\left(\frac{dr}{dt}\right)^{2}-f(r)^{2}=0.$ (52)

In the near horizon region, $f(r)=f^{\prime}_{h}(r-r_{h})$ at first order in
$r$, and with the change of variable $(\ref{rhoRindler})$ one obviously
recovers $(\ref{MasslessMotion2})$ in the form

$\left(\frac{d\rho}{dt}\right)^{2}-\kappa^{2}\rho^{2}=0.$ (53)

All the results of section III.2 can now be applied to the near horizon case
with the $SL(2,\mathbb{R})$ algebra approach. In particular, the highly damped
QNF are given by a relation like $(\ref{Rindler_QNF})$

$\omega_{n}^{(\pm)}\approx\mp i\kappa n,$ (54)

for which the imaginary part is equally spaced as expected. It should be noted
that this approach does not give the real part of the QNF that appears in
$(\ref{HighlyDampedQNF})$, which is related to the behavior of the BH metric
near the singularity $r\to 0$.

Of course, although almost everything has already been said in the literature
about the Hawking-Unruh effect in the near horizon region, this
$SL(2,\mathbb{R})$ algebraic approach is worth mentioning, because it gives
easily the equally spaced overtone level of the imaginary part of the highly
damped QNF. Moreover, an interpretation that is less known is that the BH
horizon appears also in this context as an unstable point, as for
$(\ref{MasslessMotion2})$, here for purely radial null geodesics, in the phase
space of an accelerated observer $(d\rho/dt,\rho)$. In this context, the
surface gravity plays a role of a Lyapunov exponent, i.e. an inverse
characteristic time, that characterizes the (in)stability of the BH horizon
for radial trajectories of massless particles seen from an accelerated
observer $(t,\rho)$ (see [12] and references therein for a very exhaustive
study of this interpretation).

###  IV.2 The near photon sphere region: the weakly damped QNM

####  IV.2.1 Geodesics near the photon sphere region

In this section, we reproduce the outline of the proof in [8] according to
which, in the ultrarelativistic limit, the near photon sphere limit of the
metric $(\ref{metric_BH})$ is also a Rindler metric. Then, from the above, all
the results of section III could also be easily applied.

We start by considering the motion of a free-falling test particle of mass $m$
following the geodesic line element $(\ref{metric_BH})$. For a massive test
particle to get very close to the photon sphere, we will need to consider the
ultrarelativistic limit of its motion.

Without loss of generality, let us focus again on the equatorial plane
$\theta=\pi/2$ of the metric $(\ref{metric_BH})$

$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\varphi^{2}.$ (55)

In the massive case, the integrals of motion read

$f(r)\left(\frac{dt}{d\tau}\right)=\frac{E}{m}\quad;\quad
r^{2}\left(\frac{d\varphi}{d\tau}\right)=\frac{L}{m}$ (56)

where $\tau$ is the masssive particle proper time. Using (56), the equation of
motion for the test particle reads

$m^{2}\left(\frac{dr}{d\tau}\right)^{2}+U_{\textrm{eff}}(r)=E^{2},$ (57)

where

$U_{\textrm{eff}}(r)=f(r)\left[\frac{L^{2}}{r^{2}}+m^{2}\right].$ (58)

The effective potential $U_{\textrm{eff}}(r)$ has extrema located at $r=r_{i}$
such that

$f^{\prime}(r_{i})-\frac{2}{r_{i}^{2}}f(r_{i})+\frac{m^{2}}{L^{2}}f^{\prime}(r_{i})=0.$
(59)

We simplify the study by assuming that the particle angular momentum $L$ value
is such that $U_{\textrm{eff}}(r)$ admits a local maximum. Let us call
$r_{0}(L)$ the location of this local maximum. In the limit $L\gg 1$,
$(\ref{r0})$ tends to $(\ref{rc})$, the local maximum coincides with the
location of the photon sphere, i.e. $r_{0}(L)$ tends to $r_{c}$ and
$U_{\textrm{eff}}(r_{0}(L))$ tends to $V_{\textrm{eff,c}}$.

In other words, a test particle (with energy $E$ and angular momentum $L$)
coming from infinity, gets very close to the photon sphere if at least

$\displaystyle m^{2}$ $\displaystyle<E^{2}\approx U_{\textrm{eff}}(r_{0}(L)),$
(60a) $\displaystyle L$ $\displaystyle\gg 1\quad\textrm{with}\quad
L/E\quad\textrm{finite}.$ (60b)

The condition (60a) implies that the turning point of the particle motion is
located in the vicinity of the maximum of $U_{\textrm{eff}}(r)$. The second
condition (60b) implies that the location of this maximum tends to the
location of the photon sphere. If the conditions (60) are both satisfied, then
one has $E^{2}\approx U_{\textrm{eff}}(r_{0}(L))\approx V_{\textrm{eff,c}}$,
i.e. $L/E\approx r_{c}/\sqrt{f_{c}}$, which corresponds to the limit of an
ultrarelativistic test particle. Then, in the ultrarelativistic limit, the
test particle has an impact parameter $b$ which tends, from above, to the
critical impact parameter $b_{c}$ associated with massless particles motion,
and the particle coming from infinity will get close to the photon sphere
before moving away, back to infinity.

Let us now focus on the near-photon sphere limit of
$(\ref{geodesics_equatorial_plane})$ to describe the motion of the test
particle in the ultrarelativistic limit. We first use the constants of motions
$(\ref{EL_massive})$ for geodesic motions [15], to restrict ourselves to an
equivalent effective geodesic line element in the $(t,r)$-plane. From
$(\ref{EL_massive})$, one can write

$r^{2}d\varphi^{2}=f(r)^{2}\frac{L^{2}}{E^{2}r^{2}}dt^{2}.$ (61)

This allows to transform $(\ref{geodesics_equatorial_plane})$ into a line
element, that would give the same radial equation of motion
$(\ref{EqMotionMassive})$,

$ds^{2}=f(r)\left(-1+\frac{V_{\textrm{eff}}(r)}{E^{2}}\right)dt^{2}+\frac{dr^{2}}{f(r)},$
(62)

where $V_{\textrm{eff}}(r)$ is still, from a pure formal point of view,
defined by expression $(\ref{EffPotential})$ but with $L$ being now the
angular momentum of the massive test particle. It should be noted that the
line elements (62) and (55) have the same magnitude for any geodesic motion,
i.e. for any given $E$ and $L$. Moreover, let us emphasize that (62) does not
describe a purely radial motion in the BH background (in such case $\varphi$
would have been constant, i.e. $L=0$, and the photon sphere would have had no
effect on the test particle motion), but rather describes the effective non-
radial geodesic motion of a test particle in the $(t,r)$-plane of a static and
spherically symmetric BH background, taking into account explicitly the effect
of the centrifugal potential barrier in its time component. Let us recall that
we will not consider the case where the particle gets trapped into the BH,
i.e. here one has $b\gtrsim b_{c}$.

The near-photon sphere limit of $(\ref{metric_BH2})$ requires the conditions
$(\ref{EL_conditions})$ to be satisfied, i.e. $E^{2}\approx
V_{\textrm{eff,c}}$, and is obtained from the lowest order Taylor series
expansion around $r=r_{c}$ which does not cancel the time component of
$(\ref{metric_BH2})$. Using $(\ref{Taylor_EffPotential})$, the effective line
element $(\ref{metric_BH2})$ then becomes in this limit

$ds^{2}\simeq-\frac{V^{\prime\prime}_{\textrm{eff,c}}}{2E^{2}}f_{c}(r-r_{c})^{2}dt^{2}+\frac{dr^{2}}{f_{c}}.$
(63)

Now, from $(\ref{VeffSecond})$ and $(\ref{Lyapunov})$, and introducing the
variable

$\rho=\frac{r-r_{c}}{\sqrt{f_{c}}}\Leftrightarrow
d\rho=\frac{dr}{\sqrt{f_{c}}},$ (64)

one finally obtains a Rindler form of the effective line element near the
photon sphere, which acts in this setting as an effective Rindler horizon

$ds^{2}\simeq-\Lambda_{c}^{2}\rho^{2}dt^{2}+d\rho^{2},$ (65)

where we have used $L^{2}/E^{2}\approx r_{c}^{2}/f_{c}$ because $E^{2}\approx
V_{\textrm{eff,c}}$.
The Lyapunov exponent $|\Lambda_{c}|$ associated with the massless particle
motions around the photon sphere plays the role of a constant proper
acceleration in the near-photon sphere limit of the line element (62),
describing the test particle effective geodesic motion in the $(t,r)$-plane, a
role analogue to the role played by the surface gravity in the near-horizon
limit.

####  IV.2.2 Massless scalar field in the BH metric

In this section, we look at the Klein-Gordon equation for a massless scalar
field $\Phi$ in a spacetime metric $(\ref{metric_BH})$. After separation of
variables, assuming a harmonic time dependence ($e^{-i\omega t}$) for $\Phi$
and the introduction of the radial partial wave functions
$\Phi_{\ell\omega}(r)$ with $\ell=0,1,2,\dots$, the Klein-Gordon equation
gives the well-known Regge-Wheeler equation

$\frac{d^{2}\Phi_{\ell\omega}}{dr_{\ast}^{2}}+\left[\omega^{2}-V_{\ell}(r_{\ast})\right]\Phi_{\ell\omega}=0.$
(66)

where $r_{\ast}=r_{\ast}(r)$ is the tortoise coordinate, introduced
previously, and $V_{\ell}(r)$ is the Regge-Wheeler potential defined as

$V_{\ell}(r)=f(r)\left[\frac{\ell(\ell+1)}{r^{2}}+\frac{1}{r}f^{\prime}(r)\right].$
(67)

It should be noted that, for every $\ell\in\mathbb{N}$, $V_{\ell}(r)$ admits a
local maximum at $r=r_{0}(\ell)$ which is close to the photon sphere located
at $r=r_{c}$. Moreover, in the limit $\ell\gg 1$, $r_{0}(\ell)\approx r_{c}$
and $V_{\ell}(r)\approx V_{\textrm{eff}}(r)$. Using the tortoise coordinate,
we will denote $(r_{\ast})_{0,\ell}=r_{\ast}(r_{0}(\ell))$ the location of the
maximum of $V_{\ell}(r)$.

Following [16], let us consider $V_{\ell}(r_{\ast})$ around the location of
its local extremum at $(r_{\ast})_{0,\ell}$, i.e. around a location as close
to the photon sphere as $\ell\gg 1$. A second order Taylor series expansion
gives

$V_{\ell}(r_{\ast})\approx
V_{0}(\ell)+\frac{1}{2}V^{(2)}_{0}(\ell)\left(r_{\ast}-(r_{\ast})_{0,\ell}\right)^{2},$
(68)

where we have used the notation

$V_{0}^{(2)}(\ell)=\left(\frac{d^{2}V_{\ell}(r_{\ast})}{dr_{\ast}^{2}}\right)_{(r_{\ast})_{0,\ell}}.$
(69)

Introducing $x$, $h(\ell,\omega)$ and $\alpha(\ell)$ such that

$\displaystyle x$ $\displaystyle=$
$\displaystyle\left(r_{\ast}-(r_{\ast})_{0,\ell}\right),$ $\displaystyle
h(\ell,\omega)$ $\displaystyle=$ $\displaystyle\omega^{2}-V_{0}(\ell),$
$\displaystyle\alpha(\ell)$ $\displaystyle=$
$\displaystyle\sqrt{-2V_{0}^{(2)}(\ell)},$ (70)

equation (66) reads

$H\tilde{\Phi}_{\ell\omega}(x)=h(\ell,\omega)\tilde{\Phi}_{\ell\omega}(x)$
(71)

where $\tilde{\Phi}_{\ell\omega}(x)=\Phi_{\ell\omega}(r_{\ast}(x))$ and

$H=-\frac{d^{2}}{dx^{2}}-\frac{\alpha(\ell)^{2}}{4}x^{2}$ (72)

is the time-independent “Hamiltonian” (here with dimension $L^{-2}$) governing
the massless scalar field dynamics in the near-photon sphere limit. In the
coordinate $x$, $H$ is in the form $(\ref{Hamiltonian})$, and then all the
eigenvalue problem can be solved via algrebraic $SO(2,1)\sim SL(2,\mathbb{R})$
method.

For the weakly damped BH QNF (i.e. $\ell\gg 1$ and $\ell\gg n$), it should be
noted that one can make the substitution $\ell(\ell+1)=(\ell+1/2)^{2}-1/4$ in
$(\ref{RWPotential})$ and look at $(\ell+1/2)$ (instead of $\ell$) as the
quantity in which one will compute a Taylor series expansion of the Regge-
Wheeler potential and its second derivative at $(r_{\ast})_{0,\ell}$, in the
limit $(\ell+1/2)\gg 1$. At lowest order in $\ell+1/2$, one has

$\displaystyle\alpha(\ell)$ $\displaystyle\approx$ $\displaystyle
2\eta_{c}\frac{f_{c}}{r_{c}^{2}}\left(\ell+\frac{1}{2}\right),$ (73)
$\displaystyle V_{0}(\ell)$ $\displaystyle\approx$
$\displaystyle\frac{f_{c}}{r_{c}^{2}}\left(\ell+\frac{1}{2}\right)^{2}.$ (74)

Equation $(\ref{HEigenvalues})$ then reads

$\omega^{2}=V_{0}(\ell)\pm i\alpha(\ell)\left(n+\frac{1}{2}\right),$ (75)

which gives, in the weakly damped regime ($(\ell+1/2)\gg 1$ and $\ell\gg n$),
a equally spaced imaginary part of the BH QNF

$\textrm{Im}\left(\omega_{\ell
n}^{(\pm)}\right)\approx-\frac{i}{2}\frac{\alpha(\ell)}{\sqrt{V_{0}(\ell)}}\left(n+\frac{1}{2}\right)=-i|\Lambda_{c}|\left(n+\frac{1}{2}\right),$
(76)

with $|\Lambda_{c}|=\eta_{c}\sqrt{f_{c}}/r_{c}=$ is the Lyapunov exponent that
characterizes the instability of null motions on the photon sphere.

##  V Remarks on the amplitude of the overtone spacing

A relevant fact is that the amplitude of the overtone level spacing is related
to a Lyapunov exponent that characterizes the (in)stability of the top of the
underlying IHO potential barrier. Rindler spacetime is an example of such
behavior: the surface gravity in $(\ref{RindlerSimple})$ is understood in this
framework as a Lyapunov exponent, and reciprocally, the Lyapunov exponent in
$(\ref{Rindler_metric})$ can be interpreted as a locally constant acceleration
for a certain family of observers. It should be noted that in both cases, the
Lyapunov exponent characterizes motions of massless particles that seem to
play a fundamental role in this description. Of course, as long as one has a
Rindler approximation of a BH metric, the thermal aspects follow directly (see
[12, 8] for more details). As a Lyapunov exponent is interpreted as the
inverse of a characteristic time, this allows to expect a fundamental link
between “time-thermality-instability”.

Another remark is about the real part of the QNF in the highly damped regimes.
This approach does not allow to access such a real part, because it has been
shown to be related to the behavior of the BH metric near its singularity
$r\to 0$, which is not considered here.

##  VI Conclusion

In this paper, we propose a simple way to understand the equally spaced
overtone level of BH QNF both in the highly and weakly damped regimes. To do
so, we have shown that the resonant scattering problem in Rindler spacetime is
deeply linked to the resonant scattering problem by an IHO potential, which in
turn can be completely described through the $SL(2,\mathbb{R})$ algebra. The
$SL(2,\mathbb{R})$ algebra allows to show that the spectrum of the
“Hamiltonian” is discrete, purely imaginary and indexed by a non negative
integer. This integer turns out to be simply the overtone of the QNF, and is
shown to be equally spaced. A general claim could follow: in any BH geometry,
at last for static and spherically symmetric BH, if there exist some regions
where the BH metric can be locally approximated as a Rindler spacetime, in
some set of coordinates, then the underlying algebra should be
$SL(2,\mathbb{R})$, and the imaginary part of the BH QNF, be it in the highly
damped or weakly damped regimes, will be indexed by a non negative integer,
i.e. the overtone number, which will be equally spaced. A very interesting
fact is that we explicitly obtained this result for both highly and weakly
damped QNM families, by looking at different regions of the BH spacetime. This
work can be easily extended to higher dimensional spacetime. A future look at
the Kerr BH is of interest.

###### Acknowledgements.

The author would like to thank J.P. Provost and J.L. Jaramillo for stimulating
discussions. The IMB receives support from the EIPHI Graduate School (contract
ANR-17-EURE-0002).

## References

  * Das and Shankaranarayanan [2005] S. Das and S. Shankaranarayanan, Class. Quantum Grav. 22, L7 (2005).
  * Ghosh  _et al._ [2006] A. Ghosh, S. Shankaranarayanan, and S. Das, Class. Quantum Grav. 23, 1851 (2006).
  * Cardoso  _et al._ [2009] V. Cardoso, A. S. Miranda, E. Berti, H. Witek, and V. T. Zanchin, Phys. Rev. D 79, 064016 (2009).
  * Décanini  _et al._ [2010] Y. Décanini, A. Folacci, and B. Raffaelli, Phys. Rev. D 81, 104039 (2010).
  * Padmanabhan [2004] T. Padmanabhan, Class. Quantum Grav. 21, L1 (2004).
  * Batic  _et al._ [2011] D. Batic, N. G. Kelkar, and M. Nowakowski, Eur. Phys. J. C 17, 1831 (2011).
  * Gradshteyn  _et al._ [2014] I. S. Gradshteyn, I. M. Ryzhik, D. Zwillinger, and V. Moll,  _Table of integrals, series, and products; 8th ed._ (Academic Press, Amsterdam, 2014).
  * Raffaelli [2022] B. Raffaelli, J. High Energ. Phys 125 (2022).
  * Subramanyan  _et al._ [2019] S. Subramanyan, H. Hedge, V. Vishveshwara, and B. Bradlyn, Phys. Rev. Lett. 123, 156802 (2019).
  * Böhm [1981] A. Böhm, J. Math. Phys. 22, 2813 (1981).
  * Civitarese and Gadella [2004] O. Civitarese and M. Gadella, Physics Reports 396, 41 (2004).
  * Dalui and Majhi [2020] S. Dalui and B. R. Majhi, Phys. Rev. D 102, 124047 (2020).
  * Moretti and Pinamonti [2003] V. Moretti and N. Pinamonti, arXiv:hep-th/0304102 (2003).
  * Alsing and Milonni [2004] P. M. Alsing and P. W. Milonni, Am. J. Phys. 72, 1524 (2004).
  * Rindler [2006] W. Rindler,  _Relativity: Special, General, and Cosmological_ (Oxford University Press, Oxford, 2006).
  * Schutz and Will [1985] B. F. Schutz and C. M. Will, Astrophys. J. 291, L33 (1985).


# Phase transition of four-dimensional lattice $\phi^{4}$ theory with tensor
renormalization group

Shinichiro Akiyama<EMAIL_ADDRESS>Graduate School of Pure and
Applied Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan
Yoshinobu Kuramashi<EMAIL_ADDRESS>Center for Computational
Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8577, Japan  Yusuke
Yoshimura<EMAIL_ADDRESS>Center for Computational Sciences,
University of Tsukuba, Tsukuba, Ibaraki 305-8577, Japan

###### Abstract

We investigate the phase transition of the four-dimensional single-component
$\phi^{4}$ theory on the lattice using the tensor renormalization group
method. We have examined the hopping parameter dependence of the bond energy
and the vacuum condensation of the scalar field $\braket{\phi}$ at a finite
quartic coupling $\lambda$ on large volumes up to $V=1024^{4}$ in order to
detect the spontaneous breaking of the $\mathbb{Z}_{2}$ symmetry. Our results
show that the system undergoes the weak first-order phase transition at a
certain critical value of the hopping parameter. We also make a comparative
study of the three-dimensional $\phi^{4}$ theory and find that the properties
of the phase transition are consistent with the universality class of the
three-dimensional Ising model.

††preprint: UTHEP-755, UTCCS-P-136

##  I Introduction

The issue of the triviality of the four-dimensional ($4d$) $\phi^{4}$ theory
has been a theoretical concern among particle physicists, because it is
related to the scalar sector in the standard model Wilson and Kogut (1974);
Aizenman (1981, 1982); Frohlich (1982); Dashen and Neuberger (1983); Lindner
(1986); Hasenfratz  _et al._ (1987); Lüscher and Weisz (1987, 1988, 1989);
Huang (1989); Frick  _et al._ (1990); Kribs  _et al._ (2007). The single-
component $\phi^{4}$ theory becomes equivalent to the Ising model in the
infinite limit of the quartic coupling $\lambda=\infty$ so that numerical
studies of the $4d$ Ising model have been performed as a nonperturbative test
of the triviality, assuming the universality Blöte and Swendsen (1980);
Sanchez-Velasco (1987); Kenna and Lang (1993); Bittner  _et al._ (2002); Kenna
(2004); Lundow and Markström (2009, 2011). 111In the standard model, we need
to consider the $\phi^{4}$ interaction as a part of a combined Higgs-Yukawa
sector, whose nonperturbative aspects were investigated with the lattice
simulations Lee  _et al._ (1990a, b). Also there are some recent studies to
discuss the triviality of $O(N)$ $\phi^{4}$ theory with the higher-loop beta
function Shrock (2014, 2016, 2017). So far, no Monte Carlo calculation has
confirmed the logarithmic correction to the mean-field exponents in the
scaling behavior of the specific heat, which is expected from the perturbative
renormalization group analysis Wegner and Riedel (1973). Moreover, a detailed
Monte Carlo study has found a serious finite-volume effect due to nontrivial
boundary effects in the $4d$ Ising model Lundow and Markström (2011).

Recently, the authors have investigated the phase transition of the $4d$ Ising
model with the higher-order tensor renormalization group (HOTRG) algorithm
Akiyama  _et al._ (2019). The tensor renormalization group (TRG) method,222In
this paper, the TRG method or the TRG approach refers to not only the original
numerical algorithm proposed by Levin and Nave Levin and Nave (2007) but also
its extensions Xie  _et al._ (2012); Adachi  _et al._ (2020); Kadoh and
Nakayama (2019); Shimizu and Kuramashi (2014a); Sakai  _et al._ (2017);
Akiyama  _et al._ (2020a). which contains the HOTRG algorithm, has several
superior features over the Monte Carlo method. (i) Since the TRG provides a
deterministic numerical method, it does not have the sign problem encountered
in stochastic methods, including the standard Monte Carlo simulation, as
confirmed in various studies of quantum field theories Shimizu and Kuramashi
(2014a, b, 2018); Takeda and Yoshimura (2015); Kadoh  _et al._ (2018, 2020);
Kuramashi and Yoshimura (2020); Akiyama  _et al._ (2020b). (ii) Its
computational cost depends on the system size only logarithmically. (iii) The
computational cost to simulate fermions is almost equivalent to that to bosons
because the TRG can directly manipulate the Grassmann variables Shimizu and
Kuramashi (2014a); Sakai  _et al._ (2017); Yoshimura  _et al._ (2018); Akiyama
_et al._ (2020a). (iv) We can obtain the partition function or the path-
integral itself. Thanks to the above feature (ii), we have been allowed to
enlarge the lattice volume up to $V=1024^{4}$, which is essentially identified
as the thermodynamic limit, and found finite jumps for the internal energy and
the magnetization as functions of temperature in the $4d$ Ising model Akiyama
_et al._ (2019). These are characteristic features of the first-order phase
transition. Having shown that the $4d$ Ising model undergoes the weak first-
order phase transition, our interest turns to the order of the phase
transition in the $4d$ single-component $\phi^{4}$ theory, which has the
global $\mathbb{Z}_{2}$ symmetry as with the Ising model. 333The scenario of
the weak first-order phase transition in the Ising model or the $\phi^{4}$
theory has been discussed phenomenologically in some recent studies Cea  _et
al._ (2019); Consoli and Cosmai (2020a, b, c).

In this paper, we investigate the phase transition of the $4d$ single-
component $\phi^{4}$ theory with the quartic coupling $\lambda$ and the
hopping parameter $\kappa$, employing the anisotropic TRG (ATRG) algorithm
Adachi  _et al._ (2020), which was proposed to reduce the computational cost
of the TRG method. The ATRG has been successfully applied to analyze the $4d$
complex $\phi^{4}$ theory at the finite density with parallel computation
Akiyama  _et al._ (2020b). Our main purpose is to determine the order of the
phase transition by examining the $\kappa$ dependence of the bond energy and
the vacuum condensation of the scalar field $\braket{\phi}$ around the
critical value of $\kappa_{\rm c}$ for the fixed $\lambda$, the latter of
which is an order parameter of the phase transition caused by the spontaneous
$\mathbb{Z}_{2}$ symmetry breaking. We study the model with a single choice of
$\lambda=40$, which is a finite-$\lambda$ generalization of the Ising model
study performed in Ref. Akiyama  _et al._ (2019), corresponding to
$\lambda=\infty$. The choice of $\lambda=40$ may also be helpful to avoid the
weak coupling region affected by the Gaussian fixed point at $\lambda=0$. For
comparison, we also make the same analysis of the $3d$ single-component
$\phi^{4}$ theory at $\lambda=40$, which is believed to belong to the
universality class of the 3$d$ Ising model. We discuss the differences between
the results of the 3$d$ and 4$d$ cases.

This paper is organized as follows. In Sec. II we explain the formulation of
the lattice $\phi^{4}$ theory and the ATRG algorithm. We present numerical
results for the 4$d$ and 3$d$ cases in Sec. III and discuss the properties of
the phase transition. Section IV is devoted to summary and outlook.

##  II Formulation and numerical algorithm

We use the following popular action for the $d$-dimensional single-component
$\phi^{4}$ theory on a lattice $\Gamma$:

$\displaystyle
S[\phi]=\sum_{n\in\Gamma}\left[-\kappa\sum_{\nu=1}^{d}\left(\phi_{n}\phi_{n+\hat{\nu}}+\phi_{n}\phi_{n-\hat{\nu}}\right)+\phi_{n}^{2}+\lambda\left(\phi_{n}^{2}-1\right)^{2}\right],$
(1)

where $\hat{\nu}$ is the unit vector of the $\nu$-direction. This formulation,
which is explicit about the relation to the Ising model, is equivalent to the
more conventional expression

$\displaystyle
S[\varphi]=\sum_{n\in\Gamma}\left[\frac{1}{2}\sum_{\nu=1}^{d}\left(\varphi_{n+\hat{\nu}}-\varphi_{n}\right)^{2}+\frac{1}{2}m_{0}^{2}\varphi_{n}^{2}+\frac{g_{0}}{4!}\varphi_{n}^{4}\right]$
(2)

with

$\displaystyle\varphi_{n}=\sqrt{2\kappa}\phi_{n},$ (3) $\displaystyle
m_{0}^{2}=\frac{1-2\lambda}{\kappa}-2d,$ (4) $\displaystyle
g_{0}=\frac{6\lambda}{\kappa^{2}}.$ (5)

The partition function is defined by

$\displaystyle Z=\int\mathcal{D}\phi~{}\mathrm{e}^{-S[\phi]}$ (6)

using the action of Eq. (1) with the path integral measure

$\displaystyle\int\mathcal{D}\phi=\prod_{n\in\Gamma}\int_{-\infty}^{\infty}\mathrm{d}\phi_{n}.$
(7)

We express the partition function as a tensor network in the similar way to
Ref. Akiyama  _et al._ (2020b). The continuous variables $\phi_{n}$ are
discretized by the $K$-point Gauss-Hermite quadrature rule as

$\displaystyle\int_{-\infty}^{\infty}\mathrm{d}\phi_{n}~{}\mathrm{e}^{-\phi_{n}^{2}}f(\phi_{n})\simeq\sum_{\alpha_{n}=1}^{K}\omega_{\alpha_{n}}f(\phi_{\alpha_{n}}),$
(8)

where $\phi_{\alpha}$ and $\omega_{\alpha}$ are the $\alpha$-th node and its
weight. The partition function is thus discretized as

$\displaystyle
Z(K)=\sum_{\\{\alpha\\}}\prod_{n,\nu}M_{\alpha_{n}\alpha_{n+\hat{\nu}}},$ (9)

where

$\displaystyle
M_{\alpha_{n}\alpha_{n+\hat{\nu}}}=\sqrt[2d]{\omega_{\alpha_{n}}\omega_{\alpha_{n+\hat{\nu}}}}\exp\left[2\kappa\phi_{\alpha_{n}}\phi_{\alpha_{n+\nu}}-\frac{\lambda}{2d}\left(\phi_{\alpha_{n}}^{2}-1\right)^{2}-\frac{\lambda}{2d}\left(\phi_{\alpha_{n+\hat{\nu}}}^{2}-1\right)^{2}\right].$
(10)

Each matrix $M$ is approximated by the singular value decomposition (SVD) with
a bond dimension $D$ as

$\displaystyle M_{\alpha\beta}\simeq\sum_{k=1}^{D}U_{\alpha
k}\sigma_{k}V_{\beta k},$ (11)

where $\sigma_{k}$ is the $k$-th singular value sorted in the descending
order, and $U,V$ are the orthogonal matrices composed of the singular vectors.
One finally obtains a tensor network representation for $Z(K)$ as

$\displaystyle
Z(K)=\sum_{\\{i_{1},\cdots,i_{d}\\}}\prod_{n\in\Gamma}T_{n;i_{1}\cdots
i_{d}i^{\prime}_{1}\cdots i^{\prime}_{d}},$ (12)

where

$\displaystyle T_{n;i_{1}\cdots i_{d}i^{\prime}_{1}\cdots
i^{\prime}_{d}}=\sum_{\alpha=1}^{K}\prod_{\nu=1}^{d}\sqrt{\sigma_{i_{\nu}}\sigma_{i^{\prime}_{\nu}}}U_{\alpha
i_{\nu}}V_{\alpha i^{\prime}_{\nu}},$ (13)

with the shorthand notations such as $i_{\nu}=i_{\nu,n}$ and
$i^{\prime}_{\nu}=i_{\nu,n-\hat{\nu}}$.

In this study, we employ the parallelized $d$-dimensional ATRG algorithm
developed in Refs. Akiyama  _et al._ (2020c, b). We keep the bond dimension
$D$ fixed throughout the ATRG procedure. For the swapping bond parts explained
in Refs. Adachi  _et al._ (2020); Oba (2020), the randomized SVD is applied
with the choice of $p=2D$ and $q=2D$, where $p$ is the oversampling parameter
and $q$ is the numbers of QR decomposition.

##  III Numerical results

###  III.1 4$d$ case

The partition function of Eq. (12) is evaluated using the ATRG algorithm on
lattices with the volume $V=L^{4}$ ($L=2^{m},m\in\mathbb{N}$) employing the
periodic boundary conditions for all the space-time directions. As explained
in the previous section, there are two important algorithmic parameters. One
is the number of nodes $K$ in the Gauss-Hermite quadrature method to
discretize the scalar field. The other is the bond dimension $D$. We check the
convergence behavior of the free energy as a function of $K$ and $D$ by
defining the following quantities:

$\displaystyle\delta_{K}=\left|\frac{\ln Z(K,D=50)-\ln Z(K=2000,D=50)}{\ln
Z(K=2000,D=50)}\right|$ (14)

and

$\displaystyle\delta_{D}=\left|\frac{\ln Z(K=2000,D)-\ln Z(K=2000,D=50)}{\ln
Z(K=2000,D=50)}\right|.$ (15)

Figure 1 shows the $K$ dependence of $\delta_{K}$ with $D=50$ on $V=1024^{4}$
at $\kappa=0.0763059$ and $0.0765000$, which are in the symmetric and broken
symmetry phases. Note that $\kappa=0.0763059$ is close to the transition point
$\kappa_{\rm c}$, as we will see below. We observe that $\delta_{K}$ decreases
monotonically as a function of $K$ and reaches the order of $10^{-7}$ around
$K=1500$. This shows that the Gauss-Hermite quadrature method is not affected
by whether the system is in the symmetric or broken symmetry phase. We also
plot the $D$ dependence of $\delta_{D}$ in Fig. 2, which shows the fluctuation
of free energy is suppressed as $\delta_{D}\approx 10^{-5}$ up to $D=50$.
Since the double-well potential in the $\phi^{4}$ theory becomes sharper for
larger $\lambda$, we take a large number of $K$ to achieve good convergence
for $\delta_{K}$. In the following, numerical results at $\lambda=40$ are
presented for $K=2000$ and $D=50$ which are large enough in this study.

Figure 1: Convergence behavior of free energy with $\delta_{K}$ of Eq. (14) at
$\kappa=0.0763059$ and $0.0765000$ as a function of $K$ on $V=1024^{4}$.
Figure 2: Same as Fig. 1 for $\delta_{D}$ of Eq. (15).

The phase transition point $\kappa_{\rm c}$ is determined by following the
method employed in the Ising case Akiyama  _et al._ (2019). Suppose we have
obtained a coarse-grained tensor
$T^{(m)}_{i_{1}i_{2}i_{3}i_{4}i^{\prime}_{1}i^{\prime}_{2}i^{\prime}_{3}i^{\prime}_{4}}$
after the $m$ times of coarse-graining. Defining a $D\times D$ matrix as

$\displaystyle
A^{(m)}_{i_{4}i^{\prime}_{4}}=\sum_{i_{1},i_{2},i_{3}}T^{(m)}_{i_{1}i_{2}i_{3}i_{4}i_{1}i_{2}i_{3}i^{\prime}_{4}},$
(16)

we calculate

$\displaystyle X^{(m)}=\frac{\left({\rm Tr}A^{(m)}\right)^{2}}{{\rm
Tr}\left(A^{(m)}\right)^{2}}.$ (17)

This quantity, introduced in Ref. Gu and Wen (2009), possibly counts the
number of the largest singular value of $A^{(m)}$. Therefore, it is expected
that $X^{(m)}=1$ holds for the symmetric phase and $X^{(m)}=2$ for the broken
symmetry phase. We may distinguish both phases by observing the plateau of
$X^{(m)}$ after sufficient coarse-graining iterations.

Figure 3: History of $X^{(m)}$ as a function of the coarse-graining step $m$
at $\kappa=0.089225$ (circle) and 0.089300 (diamond). Figure 4: Comparison of
$\kappa_{\rm c}$ at $\lambda=5$ obtained by various methods. All numerical
values except for the ATRG result are taken from Table III in Ref. Akerlund
_et al._ (2013). For details on the dynamical or effective mean field theory,
see Ref. Akerlund  _et al._ (2013). For Kikuchi’s method, see Ref. Kikuchi
(1951).

In order to check the applicability of the above method to determine the value
of $\kappa_{\rm c}$, we calculate $\kappa_{\rm c}$ at $\lambda=5$ and compare
it with the previous results obtained by various methods including the Monte
Carlo simulation Akerlund  _et al._ (2013). Since we have found that the
convergence of the free energy with respect to the bond dimension at
$\lambda=5$ becomes slightly slower than that at $\lambda=40$, we have taken
$D=55$ (and $K=2000$) to evaluate $\kappa_{\rm c}$ at $\lambda=5$. Up to
$D=55$, the relative error for the free energy is suppressed to $O(10^{-5})$.
Figure 3 shows the $m$ dependence of the value of $X^{(m)}$ at
$\kappa=0.089225$ and $0.089300$, whose difference $\Delta\kappa=7.5\times
10^{-5}$ is the finest resolution across the transition point. We find
$X^{(m)}=1$ for $m\gtrsim 30$ at $\kappa=0.089225$ and $X^{(m)}=2$ for
$m\gtrsim 25$ at $\kappa=0.089300$. Based on this observation, we determine
the critical kappa $\kappa_{\rm c}=0.0892625(375)$ on the $1024^{4}$ lattice,
whose error bar is provided by the resolution of $\kappa$. In Fig. 4 we find
that our result is comparable to the Monte Carlo result $\kappa_{\rm
c}=0.08893(20)$ in Ref. Akerlund  _et al._ (2013). Slight deviation from the
Monte Carlo result may be attributed to the finite size effect: our result is
obtained on the $1024^{4}$ lattice, while the previous one is on the $32^{4}$
lattice.

Having confirmed the validity of the method using $X^{(m)}$, we determine
$\kappa_{\rm c}$ at $\lambda=40$ with $D=50$ and $K=2000$. The result is
$\kappa_{\rm c}=0.076305975(25)$ on the $1024^{4}$ lattice, whose error bar is
provided by the resolution of $\kappa$. In Fig. 5 we check the $1/\lambda$
dependence of $\kappa_{\rm c}$ toward the Ising limit, where the result at
$\lambda=100$ is obtained in the same way as the $\lambda=40$ case with $D=50$
and $K=2000$. We observe that the value of $\kappa_{\rm c}$ seems
monotonically approaching that in the Ising case. The error bars are provided
by the resolution of $\kappa$ but they are all within symbols.

Figure 5: $\kappa_{\rm c}$ as a function of $1/\lambda$. $1/\lambda=0$
corresponds to the Ising model. Square symbol at $1/\lambda=0$ denotes the
result obtained by the HOTRG Akiyama  _et al._ (2019). All error bars are
within symbols.

We now turn to the investigation of the phase transition with the bond energy
defined by

$\displaystyle E_{\rm{b}}=-\frac{1}{2}\frac{\partial}{\partial\kappa}\frac{\ln
Z}{V}$ (18)

and the vacuum condensation of the scalar field $\braket{\phi}$. Both
quantities are evaluated with the impure tensor method. Figure 6 plots the
bond energy as a function of $\kappa$ on the $1024^{4}$ lattice. The
resolution of $\kappa$ becomes finer toward the transition point and the
finest one is $\Delta\kappa=5.0\times 10^{-8}$ around the transition point.
The phase transition point is consistent with $\kappa_{\rm c}$ (gray band)
determined by $X^{(m)}$. Inset graph in Fig. 6 shows an emergence of a finite
gap with mutual crossings of curves for different volumes, $m\geq 23$, around
$\kappa_{\rm c}$. These are characteristic features of the first-order phase
transition as discussed in Ref. Fukugita  _et al._ (1990). As the gap, we
obtain

$\displaystyle\Delta E_{\rm b}=0.001318(3),$ (19)

by the linear extrapolation toward the transition point both from the
symmetric and broken symmetry phases. In this extrapolation, we have used data
points in $[0.07630560,0.07630595]$ for the symmetric phase and
$[0.0763060,0.0763064]$ for the broken symmetry one. Note that we do not
extrapolate $\Delta E_{\rm b}$ to the $D\rightarrow\infty$ limit in this paper
because a systematic study of the $D$ dependence demands enormous
computational cost and the theoretical formula for the extrapolation is not
known so far. The value of $\Delta E_{\rm b}$ becomes smaller than the latent
heat $\Delta E=0.0034(5)$ found in the Ising case with the HOTRG Akiyama  _et
al._ (2019).

Figure 6: Bond energy as a function of $\kappa$ on $V=1024^{4}$. Inset graph
shows it for various lattice sizes and gray band denotes $\kappa_{\rm c}$
estimated by $X^{(m)}$ of Eq. (17).

Another quantity to detect the phase transition is the vacuum condensation of
the scalar field $\braket{\phi}$, which is the order parameter of spontaneous
breaking of the $\mathbb{Z}_{2}$ symmetry. We calculate $\braket{\phi}$ by
introducing the external fields of $h=1.0\times 10^{-10}$ and $2.0\times
10^{-10}$ at each $\kappa$. After taking the infinite volume limit, we
extrapolate the value of $\braket{\phi}$ at $h=0$. Figure 7 shows the $\kappa$
dependence of $\braket{\phi}_{h=0}$. The resolution of $\kappa$ is the same as
that in Fig. 6. We find that the value of $\kappa_{\rm c}$, where the vacuum
condensation sets in, is consistent with both estimates by $X^{(m)}$ and the
bond energy. A finite jump in $\braket{\phi}_{h=0}$ at $\kappa_{\rm c}$ is
another indication of the first-order phase transition. We find

$\displaystyle\Delta\braket{\phi}_{h=0}=0.0105(9),$ (20)

as the value of finite jump, where we have used data points in
$[0.07630560,0.07630595]$ for the symmetric phase and $[0.0763060,0.0763064]$
for the broken symmetry one, as in the case with the bond energy, to
extrapolate linearly the values of $\braket{\phi}_{h=0}$ toward the transition
point. Note that this quantity is estimated as $0.037(2)$ in the Ising case
with the HOTRG Akiyama  _et al._ (2019).

Figure 7: Vacuum condensation $\braket{\phi}_{h=0}$ as a function of $\kappa$
on $V=1024^{4}$. Gray band in inset graph shows $\kappa_{\rm c}$ estimated by
$X^{(m)}$ of Eq. (17).

###  III.2 3$d$ case

The 2$d$ single-component lattice $\phi^{4}$ theory is believed to belong to
the same universality class as the 2$d$ Ising model. The previous TRG
analysis, which was carried out by two of the authors and collaborators,
supports this ansatz Kadoh  _et al._ (2019). Although the 3$d$ case should
undergo the second-order phase transition belonging to the universality class
of the 3$d$ Ising model, the direct check with the TRG method has not been
performed so far. Here it must be instructive to repeat the same TRG
calculation for the 3$d$ case and compare the results between the 3$d$ and
4$d$ cases at $\lambda=40$.

We first show the convergence behavior of the free energy as a function of $K$
and $D$ by defining the relative error in the following way:

$\displaystyle\delta_{K}=\left|\frac{\ln Z(K,D=90)-\ln Z(K=2000,D=90)}{\ln
Z(K=2000,D=90)}\right|$ (21)

and

$\displaystyle\delta_{D}=\left|\frac{\ln Z(K=2000,D)-\ln Z(K=2000,D=90)}{\ln
Z(K=2000,D=90)}\right|.$ (22)

The $K$ dependence of $\delta_{K}$ with $D=90$ on $V=4096^{3}$ at
$\kappa=0.112859$ and $0.112920$ in Fig. 8. $\kappa=0.112859$ is near the
transition point in the symmetric phase, while $\kappa=0.112920$ is in the
broken symmetric phase. We observe a monotonic decrease of $\delta_{K}$ as a
function of $K$, which is quite similar to the 4$d$ case. Figure 9 shows the
$D$ dependence of $\delta_{D}$, where $\delta_{D}$ reaches the order of
$10^{-5}$ up to $D=90$. Notice that the achieved order of $\delta_{D}$ is
similar with the $4d$ case. In the following, we present the results at
$\lambda=40$ for $K=2000$ and $D=90$.

Figure 8: Convergence behavior of 3$d$ free energy with $\delta_{K}$ of Eq.
(21) at $\kappa=0.112859$ and $0.112920$ as a function of $K$ on $V=4096^{3}$.
Figure 9: Same as Fig. 8 for $\delta_{D}$ of Eq. (22).

Now let us discuss the results of the bond energy and the vacuum condensation
of the scalar field $\braket{\phi}$, which are calculated with the impure
tensor method as in the 4$d$ case. We plot the bond energy as a function of
$\kappa$ on the $4096^{3}$ lattice in Fig. 10, where the gray band with
$0.11285890\leq\kappa\leq 0.11285905$ in the inset indicates the location of
the phase transition point determined by $X^{(m)}$. Note that in the $3d$
case, $X^{(m)}$ is also given in the same way as Eq. (17), defining the three-
dimensional counterpart of Eq. (16). The value of the bond energy evaluated at
$\kappa=0.11285900$ is located within this gray band. This is due to the
situation that $X^{(m)}$ at $\kappa=0.11285900$ does not show any clear
plateau at $X^{(m)}=1$ or 2. We observe that the bond energy on all the
volumes smoothly varies as a function of $\kappa$ without generating any gap.
In addition, we find no mutual crossing of curves for different volumes around
the phase transition point: The curve of the bond energy monotonically
approaches that on the largest volume of $4096^{3}$. These behaviors, which
are in clear contrast to the 4$d$ case, are characteristics of the second-
order phase transition as discussed in Ref. Fukugita  _et al._ (1990).

Figure 10: 3$d$ bond energy as a function of $\kappa$ on $V=4096^{3}$
($m=36$). Inset graph shows it for various lattice sizes and gray band
restricts the location of $\kappa_{\rm c}$ by $X^{(m)}$.

In Fig. 11, we show the $\kappa$ dependence of $\braket{\phi}_{h=0}$, which is
calculated in the same way as in the 4$d$ case. The resolution of $\kappa$ is
the same as that in Fig. 10. In order to determine the transition point
$\kappa_{\rm c}$ and extract the critical exponent $\beta$, we make a fit of
$\braket{\phi}_{h=0}$ on $4096^{3}$ lattice, which is essentially in the
thermodynamic limit, employing the function of $A(\kappa-\kappa_{\rm
c})^{\beta}$ over the range of $\kappa\in[0.11285900,0.11300000]$ in the
broken symmetry phase. The fit results are $A=3.7(9)$, $\kappa_{\rm
c}=0.112859(6)$ and $\beta=0.32(2)$. The value of $\beta$ is consistent with
recent estimates of $\beta\approx 0.3295$ and 0.3264 for 3$d$ Ising model with
the HOTRG algorithm Xie  _et al._ (2012) and the Monte Carlo method Hasenbusch
(2010), respectively. Numerical results for the bond energy and
$\braket{\phi}_{h=0}$ show consistency with the second-order phase transition
in the universality class of the 3$d$ Ising model.

Figure 11: 3$d$ vacuum condensation $\braket{\phi}_{h=0}$ as a function of
$\kappa$ on $V=4096^{3}$. Inset graph also shows $\braket{\phi}_{h=0}$
together with the fitting result (dotted line) as a function of the reduced
parameter $|(\kappa-\kappa_{\rm c})/\kappa_{\rm c}|$ on a logarithmic scale.
Gray band indicates $\kappa_{\rm c}$ estimated by $X^{(m)}$.

##  IV Summary and outlook

We have investigated the phase transition of the $4d$ single-component
$\phi^{4}$ theory at $\lambda=40$ employing the bond energy and the vacuum
condensation of the scalar field. Both quantities show finite jumps at the
transition point on the extremely large lattice of $V=1024^{4}$, corresponding
to the thermodynamic limit, and they indicate the weak first-order phase
transition as found in the Ising limit Akiyama  _et al._ (2019). This means
that the single-component lattice $\phi^{4}$ theory does not have a continuum
limit. In the current ATRG calculation, the resulting latent heat $\Delta
E_{\rm b}$ and the gap $\Delta\braket{\phi}$ are smaller than those in the
Ising case obtained by the HOTRG with $D=13$. As a next step, it would be
interesting to investigate the phase transition of the O(4)-symmetric
$\phi^{4}$ theory, which is more relevant to the SU(2) Higgs model.

###### Acknowledgements.

Numerical calculation for the present work was carried out with the
supercomputer Fugaku provided by RIKEN (Project ID: hp200170) and also with
the Oakforest-PACS (OFP) and the Cygnus computers under the Interdisciplinary
Computational Science Program of Center for Computational Sciences, University
of Tsukuba. This work is supported in part by Grants-in-Aid for Scientific
Research from the Ministry of Education, Culture, Sports, Science and
Technology (MEXT) (No. 20H00148).

## References

  * Wilson and Kogut (1974) K. G. Wilson and J. B. Kogut, Phys. Rept. 12, 75 (1974).
  * Aizenman (1981) M. Aizenman, Phys. Rev. Lett. 47, 886 (1981).
  * Aizenman (1982) M. Aizenman, Commun. Math. Phys. 86, 1 (1982).
  * Frohlich (1982) J. Frohlich, Nucl. Phys. B 200, 281 (1982).
  * Dashen and Neuberger (1983) R. F. Dashen and H. Neuberger, Phys. Rev. Lett. 50, 1897 (1983).
  * Lindner (1986) M. Lindner, Z. Phys. C 31, 295 (1986).
  * Hasenfratz  _et al._ (1987) A. Hasenfratz, K. Jansen, C. B. Lang, T. Neuhaus, and H. Yoneyama, Phys. Lett. B 199, 531 (1987).
  * Lüscher and Weisz (1987) M. Lüscher and P. Weisz, Nucl. Phys. B290, 25 (1987).
  * Lüscher and Weisz (1988) M. Lüscher and P. Weisz, Nucl. Phys. B295, 65 (1988).
  * Lüscher and Weisz (1989) M. Lüscher and P. Weisz, Nucl. Phys. B318, 705 (1989).
  * Huang (1989) K. Huang, Int. J. Mod. Phys. A4, 1037 (1989).
  * Frick  _et al._ (1990) C. Frick, K. Jansen, J. Jersak, I. Montvay, P. Seuferling, and G. Munster, Nucl. Phys. B331, 515 (1990).
  * Kribs  _et al._ (2007) G. D. Kribs, T. Plehn, M. Spannowsky, and T. M. Tait, Phys. Rev. D 76, 075016 (2007), arXiv:0706.3718 [hep-ph] .
  * Blöte and Swendsen (1980) H. W. J. Blöte and R. H. Swendsen, Phys. Rev. B 22, 4481 (1980).
  * Sanchez-Velasco (1987) E. Sanchez-Velasco, J. Phys. A20, 5033 (1987).
  * Kenna and Lang (1993) R. Kenna and C. B. Lang, Nucl. Phys. B393, 461 (1993), [Erratum: Nucl. Phys. B411, 340(1994)], arXiv:hep-lat/9210009 [hep-lat] .
  * Bittner  _et al._ (2002) E. Bittner, W. Janke, and H. Markum, Phys. Rev. D66, 024008 (2002), arXiv:hep-lat/0205023 [hep-lat] .
  * Kenna (2004) R. Kenna, Nucl. Phys. B691, 292 (2004), arXiv:hep-lat/0405023 [hep-lat] .
  * Lundow and Markström (2009) P. H. Lundow and K. Markström, Phys. Rev. E 80, 031104 (2009).
  * Lundow and Markström (2011) P. H. Lundow and K. Markström, Nucl. Phys. B845, 120 (2011), arXiv:1010.5958 [cond-mat.stat-mech] .
  * Lee  _et al._ (1990a) I.-H. Lee, J. Shigemitsu, and R. E. Shrock, Nucl. Phys. B 330, 225 (1990a).
  * Lee  _et al._ (1990b) I.-H. Lee, J. Shigemitsu, and R. E. Shrock, Nucl. Phys. B 334, 265 (1990b).
  * Shrock (2014) R. Shrock, Phys. Rev. D 90, 065023 (2014), arXiv:1408.3141 [hep-th] .
  * Shrock (2016) R. Shrock, Phys. Rev. D 94, 125026 (2016), arXiv:1610.03733 [hep-th] .
  * Shrock (2017) R. Shrock, Phys. Rev. D 96, 056010 (2017), arXiv:1707.06248 [hep-th] .
  * Wegner and Riedel (1973) F. J. Wegner and E. K. Riedel, Phys. Rev. B 7, 248 (1973).
  * Akiyama  _et al._ (2019) S. Akiyama, Y. Kuramashi, T. Yamashita, and Y. Yoshimura, Phys. Rev. D100, 054510 (2019), arXiv:1906.06060 [hep-lat] .
  * Levin and Nave (2007) M. Levin and C. P. Nave, Phys. Rev. Lett. 99, 120601 (2007), arXiv:cond-mat/0611687 [cond-mat.stat-mech] .
  * Xie  _et al._ (2012) Z. Y. Xie, J. Chen, M. P. Qin, J. W. Zhu, L. P. Yang, and T. Xiang, Phys. Rev. B 86, 045139 (2012).
  * Adachi  _et al._ (2020) D. Adachi, T. Okubo, and S. Todo, Phys. Rev. B 102, 054432 (2020), arXiv:1906.02007 [cond-mat.stat-mech] .
  * Kadoh and Nakayama (2019) D. Kadoh and K. Nakayama, (2019), arXiv:1912.02414 [hep-lat] .
  * Shimizu and Kuramashi (2014a) Y. Shimizu and Y. Kuramashi, Phys. Rev. D90, 014508 (2014a), arXiv:1403.0642 [hep-lat] .
  * Sakai  _et al._ (2017) R. Sakai, S. Takeda, and Y. Yoshimura, PTEP 2017, 063B07 (2017), arXiv:1705.07764 [hep-lat] .
  * Akiyama  _et al._ (2020a) S. Akiyama, Y. Kuramashi, T. Yamashita, and Y. Yoshimura, (2020a), arXiv:2009.11583 [hep-lat] .
  * Shimizu and Kuramashi (2014b) Y. Shimizu and Y. Kuramashi, Phys. Rev. D90, 074503 (2014b), arXiv:1408.0897 [hep-lat] .
  * Shimizu and Kuramashi (2018) Y. Shimizu and Y. Kuramashi, Phys. Rev. D97, 034502 (2018), arXiv:1712.07808 [hep-lat] .
  * Takeda and Yoshimura (2015) S. Takeda and Y. Yoshimura, PTEP 2015, 043B01 (2015), arXiv:1412.7855 [hep-lat] .
  * Kadoh  _et al._ (2018) D. Kadoh, Y. Kuramashi, Y. Nakamura, R. Sakai, S. Takeda, and Y. Yoshimura, JHEP 03, 141 (2018), arXiv:1801.04183 [hep-lat] .
  * Kadoh  _et al._ (2020) D. Kadoh, Y. Kuramashi, Y. Nakamura, R. Sakai, S. Takeda, and Y. Yoshimura, JHEP 02, 161 (2020), arXiv:1912.13092 [hep-lat] .
  * Kuramashi and Yoshimura (2020) Y. Kuramashi and Y. Yoshimura, JHEP 04, 089 (2020), arXiv:1911.06480 [hep-lat] .
  * Akiyama  _et al._ (2020b) S. Akiyama, D. Kadoh, Y. Kuramashi, T. Yamashita, and Y. Yoshimura, JHEP 09, 177 (2020b), arXiv:2005.04645 [hep-lat] .
  * Yoshimura  _et al._ (2018) Y. Yoshimura, Y. Kuramashi, Y. Nakamura, S. Takeda, and R. Sakai, Phys. Rev. D97, 054511 (2018), arXiv:1711.08121 [hep-lat] .
  * Cea  _et al._ (2019) P. Cea, M. Consoli, and L. Cosmai, (2019), arXiv:1912.00849 [hep-ph] .
  * Consoli and Cosmai (2020a) M. Consoli and L. Cosmai, Int. J. Mod. Phys. A 35, 2050103 (2020a), arXiv:2006.15378 [hep-ph] .
  * Consoli and Cosmai (2020b) M. Consoli and L. Cosmai, (2020b), arXiv:2007.10837 [hep-ph] .
  * Consoli and Cosmai (2020c) M. Consoli and L. Cosmai, Symmetry 12, 2037 (2020c).
  * Akiyama  _et al._ (2020c) S. Akiyama, Y. Kuramashi, T. Yamashita, and Y. Yoshimura, PoS LATTICE2019, 138 (2020c).
  * Oba (2020) H. Oba, PTEP 2020, 013B02 (2020), arXiv:1908.07295 [cond-mat.stat-mech] .
  * Gu and Wen (2009) Z.-C. Gu and X.-G. Wen, Phys. Rev. B 80, 155131 (2009).
  * Akerlund  _et al._ (2013) O. Akerlund, P. de Forcrand, A. Georges, and P. Werner, Phys. Rev. D 88, 125006 (2013), arXiv:1305.7136 [hep-lat] .
  * Kikuchi (1951) R. Kikuchi, Phys. Rev. 81, 988 (1951).
  * Fukugita  _et al._ (1990) M. Fukugita, H. Mino, M. Okawa, and A. Ukawa, Journal of Statistical Physics 59, 1397 (1990).
  * Kadoh  _et al._ (2019) D. Kadoh, Y. Kuramashi, Y. Nakamura, R. Sakai, S. Takeda, and Y. Yoshimura, JHEP 05, 184 (2019), arXiv:1811.12376 [hep-lat] .
  * Hasenbusch (2010) M. Hasenbusch, Phys. Rev. B 82, 174433 (2010), arXiv:1004.4486 [cond-mat.stat-mech] .