Also at ]CAMTP-Center for Applied Mathematics and Theoretical Physics,
University of Maribor, Mladinska 3, SI-2000 Maribor, Slovenia

Also at ]Instituto Carlos I de Física Teórica y Computacional, Universidad de
Granada, Fuentenueva s/n, 18071 Granada, Spain

Also at ]Instituto Carlos I de Física Teórica y Computacional, Universidad de
Granada, Fuentenueva s/n, 18071 Granada, Spain

Also at ]Instituto Carlos I de Física Teórica y Computacional, Universidad de
Granada, Fuentenueva s/n, 18071 Granada, Spain

# Degeneracy in excited state quantum phase transitions of two-level bosonic
models and its influence on system dynamics

J. Khalouf-Rivera  Departamento de Ciencias Integradas y Centro de Estudios
Avanzados en Física, Matemáticas y Computación, Universidad de Huelva, Huelva
21071, Spain  Qian Wang  Department of Physics, Zhejiang Normal University,
Jinhua 321004, China  [  Lea F. Santos  Department of Physics, University of
Connecticut, Storrs, CT, USA  J.E. García Ramos  Departamento de Ciencias
Integradas y Centro de Estudios Avanzados en Física, Matemáticas y
Computación, Universidad de Huelva, Huelva 21071, Spain  [  M. Carvajal
Departamento de Ciencias Integradas y Centro de Estudios Avanzados en Física,
Matemáticas y Computación, Universidad de Huelva, Huelva 21071, Spain  [  F.
Pérez-Bernal  Departamento de Ciencias Integradas y Centro de Estudios
Avanzados en Física, Matemáticas y Computación, Universidad de Huelva, Huelva
21071, Spain  [

###### Abstract

Excited state quantum phase transitions in collective many-body quantum
systems influence the system spectral properties due to changing degeneracy
patterns in different phases. We report a fundamental difference in the nature
of the degeneracy for boson two-level models, depending on the dimension of
the space where the models were defined. To illustrate the consequences, we
assess the validity of an out-of-time-order correlator as a possible order
operator for excited state quantum phase transitions in different models.

Quantum phase transitions (QPTs) are zero-temperature phase transitions driven
by quantum instead of thermal fluctuations. In QPTs the system’s ground state
undergoes abrupt changes once one or several Hamiltonian parameters (control
parameters) go through particular (critical) values [1]. In collective
systems, true discontinuities only happen in the large system size limit (also
called mean-field or thermodynamic limit) though QPT precursors can be found
in finite-size systems [2]. Particular attention has been paid to such
transitions in algebraic models of application to the study of nuclear
structure [3, *Casten2009, *Cejnar2010]. The QPT concept was later extended to
encompass the realm of excited states, introducing the excited state quantum
phase transition (ESQPT) concept [6, *Cejnar2008]. An extensive review paper
on ESQPTs has been recently published where the interested reader can find the
main developments in the field [[][, andreferencestherein.]Cejnar2021].

Two-level bosonic models have been extensively used as approximations to
widely different physical systems. In the present work we focus on two such
models: the Lipkin-Meshkov-Glick (LMG) model and the two-dimensional limit of
the vibron model (2DVM). We also extend our results to other two models, the
vibron model (VM) and the interacting boson model (IBM) in App. A. These four
models have a similar algebraic structure, with a unitary Lie algebra $u(n+1)$
as their dynamical algebra, where $n$ is the dimension of the problem under
study [9]. In the four cases, the dynamical algebra generators can be built as
bilinear products of creation and annihilation boson operators. In accordance
with the algebraic approach, the different subalgebra chains that start in the
dynamical algebra and end in the system’s symmetry algebra are called
dynamical symmetries and they provide solvable example of physical limits of
interest, as well as a basis to carry out calculations [10, 11]. The ground
state QPTs for the aforesaid four models, among others, were studied in Ref.
[12] considering a general system Hamiltonian including one- and two-body
interactions.

The LMG model is a one dimensional model ($n=1$) introduced in the 60’s as a
toy model for the study of the validity of approximate methods in nuclear
structure studies [13, *Meshkov1965, *Glick1965]. Besides its simplicity, due
to its rich physical content, the LMG model has been extensively used in the
study of QPTs [16, *Ribeiro2008, *Engelhardt2013, *Campbell2016, *Heyl2018,
21, 22] and ESQPTs [23, *PFernandez2009, *PFernandez2011, 26, 27, 28, 29, 30,
31]. The interest on the LMG model has been further fostered by the existence
of different realizations for this model [32, 33, 34, 35, 36, *Jurcevic2017,
*Muniz2020, 39, 40, 41, 42]. This model can be realized with a chain of
$N_{2}$ spins subject to all-to-all long range interactions. In this case,
introducing a collective pseudospin,
$\hat{J}_{\alpha}=\frac{1}{2}\sum_{i}^{N_{2}}\sigma_{i,\alpha}$, where
$\alpha=x,y,z$ and $\sigma_{i,\alpha}$ is the $\alpha$ Pauli spin matrix for
the $i$-th spin in the chain. The general LMG model has first-, second-, and
third-order ground-state QPTs [21, 22]. We can define a simplified LMG model
Hamiltonian

$\hat{H}_{LMG}=(1-\xi)\left(\frac{\hat{N}_{2}}{2}+\hat{J}_{z}\right)-\frac{4\xi}{N_{2}}\hat{J}_{x}^{2}~{},$
(1)

with a control parameter $\xi\in[0,1]$. This Hamiltonian is written in an
intensive form dividing the second term, a two-body term, by the system size,
$N_{2}$, to make possible to study the large size limit of the system. The
algebraic structure of this model is clearer in a bosonic realization,
introducing a scalar boson, $s^{\dagger}(s)$, and a pseudoscalar boson,
$t^{\dagger}(t)$. The four generators of the $u(n+1)=u(2)$ dynamical algebra
are the bilinear products
$\\{s^{\dagger}s,t^{\dagger}t,t^{\dagger}s,s^{\dagger}t\\}$ and, making use of
a Schwinger transform, the generators can be recast in the form of the
quasispin components with $\\{\hat{N},\hat{J}_{x},\hat{J}_{y},\hat{J}_{z}\\}$
[11]

$\displaystyle\hat{N}_{2}$ $\displaystyle=t^{\dagger}t+s^{\dagger}s~{},$
$\displaystyle\hat{J}_{x}$
$\displaystyle=\frac{1}{2}(t^{\dagger}s+s^{\dagger}t)~{},$
$\displaystyle\hat{J}_{y}$
$\displaystyle=\frac{1}{2i}(t^{\dagger}s-s^{\dagger}t)~{},$
$\displaystyle\hat{J}_{z}$
$\displaystyle=\frac{1}{2}(t^{\dagger}t-s^{\dagger}s)~{}.$

There are two possible dynamical symmetries starting from the $u(2)$ dynamical
algebra

$\displaystyle u(2)\supset u(1)$ $\displaystyle\mbox{Chain (LMG-I)}~{},$ (2)
$\displaystyle u(2)\supset so(2)$ $\displaystyle\mbox{Chain (LMG-II)}~{}.$ (3)

The $\hat{N}_{2}$ operator in this case is a constant and it is equal to the
total number of $s$ and $t$ bosons, denoting the totally symmetric $u(2)$
irrep that spans the system’s Hilbert space. The Hamiltonian Eq. (1) can be
recast as

$\hat{H}_{LMG}=(1-\xi)\hat{n}_{t}-\frac{4\xi}{N_{2}}\hat{P}_{t}~{},$ (4)

where $\hat{n}_{t}=t^{\dagger}t$ and
$\hat{P}_{t}=\frac{1}{4}(t^{\dagger}s+s^{\dagger}t)^{2}$. Hence, the model
Hamiltonian combines the first order Casimir operator of $u(1)$ and the second
order Casimir operator of $so(2)$. The LMG-I(II) dynamical symmetries are
recovered for $\xi=0(1)$. The simplified $\hat{H}_{LMG}$ can be split into
even and odd symmetry blocks, as it conserves the parity
$\hat{\Pi}=e^{\imath\pi\hat{n}_{t}}$. The model has a second-order ground-
state QPT for a critical value of the control parameter $\xi_{c}=0.2$ and an
associated ESQPT [23, 27]. The high density of states associated with the
ESQPT along the separatrix line can be clearly seen in the correlation energy
diagram shown in the panel (a) of Fig. 1 for a LMG model with a system size
$N_{2}=50$. It is also apparent how even and odd parity states (depicted with
full blue and dashed red lines, respectively) are degenerate in the broken-
symmetry region ($\xi>\xi_{c}$) for states below the critical energy marked by
the separatrix. This degeneracy is broken for states at energies above the
critical ESQPT energy. This fact has been used to define a constant of the
motion able to identify ESQPT phases in quantum collective models in one
dimension [43, 44].

Figure 1:  (Color online) Panel (a): Excitation energy scaled by the system
size ($\varepsilon=(E-E_{0})/N_{2}$) as a function of the control parameter
$\xi$ for the LMG model with model Hamiltonian Eq. (4) and system size
$N_{2}=50$. Full blue (dashed red) lines mark even (odd) parity levels. Panel
(b): Excitation energy scaled by the system size
($\varepsilon=(E-E_{0})/N_{3}$) as a function of the control parameter $\xi$
for the 2DVM model Hamiltonian Eq. (7) with a system size $N_{3}=50$. Full
blue (dashed red) lines mark levels with angular momentum $\ell=0(1)$. In both
panels the color-filled area marks the energy difference between selected
states with different parity (panel (a)) or angular momentum (panel (b)).
Panel (c): Energy difference between selected pairs of states of the LMG model
Hamiltonian Eq. (4) having different parity as a function of the control
parameter $\xi$. Panel (d): Energy difference between selected states of
Hamiltonian Eq. (7) with angular momentum $\ell=0$ and $1$ as a function of
the control parameter $\xi$. In both cases, the labels of the selected pairs
of levels are provided in the legend of the panels and the color used for each
pair of states is the same color used to fill the corresponding area in the
upper panel.

We compare the LMG model results with results obtained for the 2DVM, a model
that was first introduced for the study of vibrational bending degrees of
freedom in molecules [45]. Molecular bending vibrations are two-dimensional,
thus $n=2$ and the system dynamical algebra is the $u(3)$ Lie algebra. In this
approach, bending vibrations are treated as collective bosonic excitations and
the model building blocks are a scalar boson operator
$\sigma^{\dagger}(\sigma)$ and two circular bosons
$\tau_{i}^{\dagger}(\tau_{i})$ with $i=+,-$ [46]. As in the LMG model case,
the nine $u(3)$ generators are the bilinear product of creation and
annihilation operators [45, 46]. The 2DVM is the simplest two-level model with
nontrivial angular momentum, a fact that has made it a convenient model for
QPT and ESQPT studies [47, 48, 6, 46, 49, 50, 51, 52]. In fact, the first
experimental signatures of ESQPT precursors have been found in the molecular
bending spectrum of nonrigid molecules [53, *Larese2013, *KRivera2020].

As in the LMG model, the 2DVM has two dynamical symmetries that, due to
angular momentum conservation, converge in the system’s symmetry algebra,
$so(2)=\text{span}\left\\{\hat{\ell}=\tau_{+}^{\dagger}\tau_{+}-\tau_{-}^{\dagger}\tau_{-}\right\\}$,

$\displaystyle u(3)\supset u(2)\supset so(2)$ $\displaystyle\mbox{Chain
(2DVM-I)}~{},$ (5) $\displaystyle u(3)\supset so(3)\supset so(2)$
$\displaystyle\mbox{Chain (2DVM-II)}~{}.$ (6)

In this case, the total number of $\sigma$ and $\tau$ bosons is $N_{3}$, a
constant that denotes the totally symmetric $u(3)$ irrep that spans the
system’s Hilbert space. The dynamical symmetry in Eq. (5) is a convenient
approximation to model the bending degrees of freedom of a linear molecule,
while Eq. (6) dynamical symmetry is applied to the modeling of rigidly-bent
molecular species [45, 46, 55]. In the same spirit as we have done for the LMG
model, we introduce a simple model Hamiltonian that allows for the transition
between the two limiting cases associated with the dynamical symmetries, using
$\hat{n}_{\tau}$, the first order Casimir operator of the $u(2)$ subalgebra in
Eq. (5) and a pairing operator $\hat{P}_{\tau}$, built with the second order
Casimir operator of $so(3)$ in Eq. (6)

$\hat{H}_{2DVM}=(1-\xi)\hat{n}_{\tau}+\frac{\xi}{N_{3}}\hat{P}_{\tau}~{},$ (7)

with a control parameter $\xi\in[0,1]$,
$\hat{n}_{\tau}=\tau^{\dagger}_{+}\tau_{+}+\tau^{\dagger}_{-}\tau_{-}$, and
$\hat{P}_{\tau}=N_{3}^{2}-\hat{W}^{2}$ where $\hat{W}^{2}$ is the second order
Casimir operator of
$so(3)=\text{span}\left\\{\hat{D}_{\pm}=\sqrt{2}\left(\pm\tau_{\pm}^{\dagger}\sigma\mp\sigma^{\dagger}\tau_{\mp}\right),\hat{\ell}\right\\}$
[46]. This model Hamiltonian is block-diagonal for states belonging to
different irreps of the symmetry algebra $so(2)$, as expected considering the
conservation of the system angular momentum, $\ell$ 111We used the notation
$\ell$ for the $so(2)$ quantum number, that is commonly used in molecular
spectroscopy for the vibrational angular momentum in the doubly-degenerate
bending degree of freedom of linear species.. In molecular bending vibrations
this conserved quantity is identified either with the vibrational angular
momentum (in the $u(2)$ dynamical symmetry) or with the projection of the
angular momentum on the molecular figure axis (in the $so(3)$ dynamical
symmetry) [46, 55]. The correlation energy diagram for the 2DVM model is shown
in panel (b) of Fig. 1, where excitation energies for $\ell=0$ (full blue
lines) and $1$ (dashed red lines) states are plot as a function of the control
parameter $\xi$ for a system size $N_{3}=50$. At first sight, the results are
completely equivalent to the ones obtained for the LMG model and shown in
panel (a) of Fig. 1. As in the LMG model, the 2DVM has a second order ground
state QPT and an associated ESQPT [46]. In this case the degeneration in the
broken-symmetry phase occurs for eigenstates with different angular momentum
values with energies under the ESQPT critical energy, instead of states with
different parity. Again, the critical energy is marked by a separatrix with a
high density of states. States with angular momentum $\ell>1$ and energy under
the ESQPT critical energy also become degenerate and they have not been shown
in the figure for the sake of clarity.

Despite the different $n$ values of the models considered, their underlying
algebraic structure is very similar. Both models have a $u(n)$ dynamical
algebra and undergo a second-order ground-state QPT between $u(n-1)-so(n)$ and
its associated ESQPT. In fact, the same happens for the two models considered
in App. A, with $n=3,5$. From the energy functional obtained using the
coherent state formalism, the four cases considered reduce to the cusp
catastrophe [12]. Therefore, it could be expected that the ESQPT precursors
and their influence on system dynamics would be common to the different
models. This fact has been verified in some instances, e.g. the participation
ratio of the system’s eigenstates expressed in the basis associated with the
$u(n-1)$ dynamical symmetry has been proposed as a precursor of the ESQPT
obtaining similar results in the LMG model [27], the 2DVM [57], and the three
dimensional vibron model [58]. In the present work, we pay heed to a
particular aspect that implies a qualitative difference between the one-
dimensional LMG model and the 2DVM or other models in higher dimensions (see
results in App. A). From the (a) and (b) panels in Fig. 1, it is clear that
states under the critical energy of the ESQPT in the broken-symmetry region
are degenerate. This implies a degeneracy of states with even and odd parity
in the LMG model or states from different rotational bands in the 2DVM case.
However, the nature of this degeneracy in the LMG model is utterly different
to the degeneracy in the 2DVM (or in the models included in App. A). This fact
was already pointed out by Caprio, Cejnar, and Iachello in the ESQPT seminal
reference [6] (see Fig. 6 and the second paragraph in page 1120 of this
reference). The difference of energy between pairs of selected levels
belonging to different Hamiltonian matrix blocks is highlighted in panels (a)
and (b) of Fig. 1 by filling the energy interval with certain colors. As
expected, the width of the colored surface decreases as the system gets closer
to the critical ESQPT energy and it disappears once the pair of levels become
degenerate as they go under the corresponding ESQPT critical energy. This is
what has recently been called level kissing and experimentally shown using a
Kerr oscillator realized with circuit QED [59, 60]. The value of the energy
difference between the selected pairs of levels for the LMG model and the 2DVM
are depicted, in logarithmic scale and using the same colors, in panels (c)
and (d) of Fig. 1.

From panel (c) of Fig. 1, it is clear that once that a pair of states cross
the ESQPT separatrix in the LMG model they become degenerate up to the
numerical precision of our calculation even for finite-size systems. The
situation in the 2DVM is utterly different, as shown in panel (d) of Fig. 1.
In the 2DVM case, the degeneracy is not complete, the energy differences
decrease to a value close to $10^{-2}$, thus levels are not completely
degenerate until the control parameter $\xi$ is equal to unity. The comparison
of the results in panels (c) and (d) of Fig. 1 indicates a fundamental
difference between the models. Energy differences for adjacent states with
different parity in the LMG model are already zero in the finite size case,
while in the 2DVM we expect that the corresponding energy differences decrease
with the system size, and will be zero only in the thermodynamic limit.

The different nature of the degeneracy in the two models considered can be
understood from the panels in Fig. 2. In both panels energy differences
between the first four pair of states with different symmetry for a control
parameter $\xi=0.5$ as a function of the system size are shown. Panel (a)
includes results for the LMG model, computing the difference between even and
odd states. However, data shown in panel (b) are for the 2DVM corresponding
states with angular momentum $\ell=0$ and $\ell=1$. The abscissa axis is
linear in the case of the LMG model, while it is logarithmin in the 2DVM case.
This implies that the energy splitting between even and odd states in the LMG
model decreases exponentially, while in the 2DVM the splitting follows a power
law. In the LMG case, to achieve the required accuracy in the calculations,
the use library for real and complex floating-point arithmetic with arbitrary
precision has been required [61].

Figure 2:  (Color online) Panel (a): Differences of energy between even and
odd energy levels in the LMG model $\Delta E_{i}=E_{i}^{-}-E_{i}^{+}$ with
$i=0,1,2,3$ as a function of the system size, $N_{2}$, using log-lin axes.
Black lines are the result of fitting the depicted data to an exponential law.
Panel (b): Differences of energy between $\ell=0$ and $\ell=1$ energy levels
in the 2DVM model, $\Delta E_{i}=E_{i}^{\ell=1}-E_{i}^{\ell=0}$ with
$i=0,1,2,3$ as a function of the system size, $N_{3}$, using log-log axes.
Black lines are the result of fitting the depicted data to a power law. In
both cases the control parameter value is $\xi=0.5$.

Such a striking difference on the degeneracy patterns in the two models is
expected to have a noticeable influence over the system dynamics. In order to
illustrate this point, we consider the time evolution of an out-of-time-order
correlator (OTOC). An OTOC is a four-point temporal correlation function,
originally introduced in the context of superconductivity studies [62]. In
recent times, OTOCs have been in the limelight for two main reasons [63]. On
the first hand, they were suggested as an efficient quantum chaos probe as it
was shown that OTOCs grow exponentially, in accordance with the system
Lyapunov exponent value, at early times in non-integrable systems [64, 65, 66,
*Roberts2015, *Maldacena2016, *Akutagawa2020, *Zhao2021]. Also, it has been
found that in chaotic systems they approach their long-time limit saturation
value in a characteristic way [71, *Rammensee2018, *Fortes2019]. On the second
hand, they can be used to quantify information scrambling in quantum systems,
as they depend on the system entanglement spread [74, *Lewis2019, *Xu2019,
*Niknam2020]. Despite the fact that the unusual time ordering of its
constituents operators hinders the experimental access to OTOCs using local
operators, several approaches using different platforms have successfully
provided OTOC data [78, *Garttner2017, *Wei2018, *Landsman2019, *Pegahan2021,
*Green2022, *Braumuller2022]. OTOCs have also been recently used to
characterize QPTs [85, *Heyl2018, *Ceren2019, *Nie2020, *Lewis2020, *Bin2023].
Finally, it has been recently found that they are also valuable ESQPT probes
because the instability associated with ESQPTs’ critical points fosters an
exponential OTOC increase at short times, even in integrable systems [90,
*Pilatowsky2020, *Hashimoto2020, *Xu2020, 60].

Given any two operators, $\hat{W}$ and $\hat{V}$, where $\hat{W}(t)$ is the
operator $\hat{W}$ in the Heisenberg’s representation,
$\hat{W}(t)=e^{\imath\hat{H}t}\hat{W}e^{-\imath\hat{H}t}$, the spread of
$\hat{W(t)}$ with $\hat{V}$ can be obtained through the expectation value of
the squared commutator [63, 74, 66]

${\cal
C}_{w,v}(t)=\left\langle\left[\hat{W}(t),\hat{V}(0)\right]^{\dagger}\left[\hat{W}(t),\hat{V}(0)\right]\right\rangle~{}.$
(8)

The expectation value is usually computed in the canonical ensemble, however,
it can also be computed over the system eigenstates (microcanonical OTOC) [66,
92]. The squared commutator Eq. (8) can be expressed as ${\cal
C}_{w,v}(t)={\cal A}_{w,v}(t)-2\Re{[{\cal F}_{w,v}(t)]}$: the sum of a two-
point correlator, ${\cal
A}_{w,v}(t)=\left\langle\hat{W}^{\dagger}(t)\hat{V}^{\dagger}(0)\hat{V}(0)\hat{W}(t)\right\rangle+\left\langle\hat{V}^{\dagger}(0)\hat{W}^{\dagger}(t)\hat{W}(t)\hat{V}(0)\right\rangle$
plus a four-point correlator, ${\cal F}_{w,v}(t)$, where out-of-time order
effects take place

${\cal
F}_{w,v}(t)=\left\langle\hat{W}^{\dagger}(t)\hat{V}^{\dagger}(0)\hat{W}(t)\hat{V}(0)\right\rangle.$
(9)

Two of the authors (QW and FPB) have recently shown for the LMG model that the
long time average value of an OTOC operator is a valid ESQPT order operator
[29]. A similar OTOC has proved also a valid order parameter in an anharmonic
LMG model [94]. In both cases, the average value of the OTOC at the infinite
time limit is able to distinguish between phases below and above the ESQPT
critical energy. In the present work, our aim is to show how different are the
results obtained for the long time OTOC values depending on the model
dimensionality, something that stems from the different nature of the
degeneracy below the ESQPT critical energy shown in Fig. 1. For this purpose,
we derive the formulas needed to compute the long-time average value of the
microcanonical OTOC. Using Eq. (9) and the closure for the system eigenstates,
$|\psi_{k}\rangle$ with $k=1,\ldots,D$, the microcanonical OTOC for a system
$j$-th eigenstate, ${\cal F}^{(j)}_{VW}(t)$, can be expressed as follows

${\cal
F}^{(j)}_{VW}(t)=\sum_{j_{1},j_{2},j_{3}}^{D}e^{i\omega(j,j_{1},j_{2},j_{3})t}\langle\psi_{j}|\hat{W}^{\dagger}|\psi_{j_{1}}\rangle\langle\psi_{j_{1}}|\hat{V}^{\dagger}|\psi_{j_{2}}\rangle\langle\psi_{j_{2}}|\hat{W}|\psi_{j_{3}}\rangle\langle\psi_{j_{3}}|\hat{V}|\psi_{j}\rangle=\sum_{j_{1},j_{2},j_{3}}^{D}e^{i\omega(j,j_{1},j_{2},j_{3})t}N^{(j)}_{j_{1},j_{2},j_{3}}~{},$
(10)

where $\omega(j,j_{1},j_{2},j_{3})=E_{j}+E_{j_{2}}-E_{j_{1}}-E_{j_{3}}$.

The time-averaged value of ${\cal F}^{(j)}_{VW}(t)$ in the infinite time
limit, $\overline{{\cal F}^{(j)}_{VW}}$, is the equilibrium value of this
four-point correlator

$\displaystyle\overline{{\cal F}^{(j)}_{VW}}=$
$\displaystyle\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}{\cal
F}^{(j)}_{VW}(t)dt=\lim_{T\to\infty}\overline{{\cal F}^{(j)}_{VW}}(T)$
$\displaystyle=$
$\displaystyle\sum_{j_{1},j_{2},j_{3}}^{D}N^{(j)}_{j_{1},j_{2},j_{3}}\delta_{\omega(j,j_{1},j_{2},j_{3}),0}~{}.$
(11)

Figure 3:  (Color online) Panel (a): Time-averaged value of ${\cal
F}^{(j)}_{VW}(t)$ for the LMG model with $\hat{V}=\hat{W}=\hat{J}_{x}$ as a
function of the system’s excitation energy scaled by the system size
($E/N_{2}$) for even parity eigenstates of Eq. (1) Hamiltonian with a system
size $N_{2}=300$. Panel (b): Time-averaged value of ${\cal F}^{(j)}_{VW}(t)$
with $\hat{V}=\hat{D}_{-}$ and $\hat{W}=\hat{D}_{+}$ as a function of the
system’s excitation energy scaled by the system size ($(E-E_{gs})/N_{3}$) for
$\ell=0$ eigenstates of the 2DVM Hamiltonian in Eq. (7) with $N_{3}=300$. All
panels: The control parameter value is $\xi=0.6$ and the full red line is the
stationary value obtained with Eq. (11). Dashed lines are the result of
averaging for different time interval values (see panel legend). The inset in
the left panel is a zoom to the vicinity of the ESQPT critical energy. A
vertical dot-dashed pink line marks the ESQPT critical energy in the mean
field limit.

Taking Eq. (10) into consideration, it is clear that a nonzero average value
of ${\cal F}^{(j)}_{VW}(t)$ in the long-time limit implies that one or more
instances of the $\omega(j,j_{1},j_{2},j_{3})$ quantity should be equal to
zero, to avoid oscillations. In order to check this, we have performed
calculations for the LMG and 2DVM cases. Following Ref. [29], in the LMG model
we define $\hat{V}=\hat{W}=\hat{J}_{x}$. The use of Eq. (11) allows for a
direct calculation of the long-time average value of the correlator, avoiding
the evaluation of computationally expensive oscillatory time integrals. The
results obtained in the stationary $t\to\infty$ limit for the even parity
states of the LMG Hamiltonian (4) with a system size $N_{2}=300$ and a control
parameter $\xi=0.6$ are shown in panel (a) of Fig. 3 with a full red line.
This result is in agreement with the results obtained in Ref. [29]. If one
computes intermediate average values for finite times, depicted with dashed
blue lines in panel (a) of Fig. 3, one can see how the results tend to the
stationary solution as the averaging time increases. This is clearly seen in
the inset included in this panel with a zoom to the results in the vicinity of
the ESQPT critical energy. As already noticed in [29], only eigenstates whose
energy is less than the ESQPT critical energy have a nonzero value of
$\overline{{\cal F}^{(j)}_{VW}}$. This can be explained taking into account
that the operator $\hat{J}_{x}$ transforms even parity states into odd parity
ones (see App. B) and, from Eq. (10) for $\hat{V}=\hat{W}=\hat{J}_{x}$, it is
clear that levels $j$ and $j_{2}$ are even parity states while levels $j_{1}$
and $j_{3}$ are odd parity states. Hence, the $\omega(j,j_{1},j_{2},j_{3})=0$
condition is only fulfilled for even and odd parity eigenstates that are
degenerate. As shown in panel (c) of Fig. 1, this only occurs for LMG states
below the critical energy of the ESQPT, whose value in the mean field limit is
marked by a dot-dashed pink vertical line in panel (a) of Fig. 3.

In the 2DVM case, an equivalent choice for the OTOC operators would be
$\hat{V}=\hat{D}_{-}$ and $\hat{W}=\hat{D}_{+}$. These two generators enter
into the pairing operator and connect states with different values of the
vibrational angular momentum, $\ell$ (see App. B). If this is the case,
assuming we compute the microcanonical OTOC for $\ell=0$ states, the
fulfilment of the $\omega(j,j_{1},j_{2},j_{3})=0$ condition in Eq. (11)
implies that the sum of energies of the $j$-th and $j_{2}$-th $\ell=0$
eigenstates should be equal to the sum of energies for the $j_{1}$-th and
$j_{3}$-th $|\ell|=1$ eigenstates. From the results in panel (d) Fig. 1 it is
clear that 2DVM eigenstates belonging to different Hamiltonian blocks are not
degenerate for finite size systems though eigenstates with energies below the
ESQPT critical energy can be close in energy. Due to this lack of zero
$\omega(j,j_{1},j_{2},j_{3})$ terms, the stationary value of the four-point
correlator, $\overline{{\cal F}^{(j)}_{VW}}$, is zero for all possible values
of $j$ (see full red line in panel (b) of Fig. 3). The smaller energy
differences for eigenstates with energies below the ESQPT critical energy
imply that it takes longer for such states to reach the zero stationary limit
value of the microcanonical OTOC, compared to what happens for states with
energies above the ESQPT critical energy. This is clearly seen in panel (b) of
Fig. 3, where for shorter times (e.g. the dashed lighter green shade lines)
eigenstates above and below the critical energy can be distinguished. As in
the previous case, the critical ESQPT energy is marked by a dot-dashed pink
vertical line. As the time over which the average value is computed is larger,
$\overline{{\cal F}^{(j)}_{D_{-}D_{+}}}\to 0$ for all eigenstates. However,
the small energy gap between levels with different angular momentum below the
critical energy of the ESQPT indicates that $\overline{{\cal
F}^{(j)}_{D_{-}D_{+}}}$ could be considered as an approximate order parameter
for the ESQPT, computing the average time value of ${\cal F}^{(j)}_{VW}(t)$
for finite time values and taking profit of the much longer time needed for
states below the critical energy to reach the OTOC stationary value.

To check our previous results we have performed a second set of calculations,
shown in Fig. 4, where we display the $\overline{{\cal F}^{(j)}_{VW}}(T)$ OTOC
for the two models under study, fixing the averaging time to a constant value,
$T=1000$, and including results for different system sizes. As in the previous
figure, the results for the LMG model are shown in panel (a) with different
shades of blue and the results for the 2DVM model are shown in panel (b), with
different shades of green. Again, as in Fig. 3, we have marked the mean field
limit value of the critical energy with a vertical dot-dashed pink line. In
the LMG case, there is no dependence on the system size for eigenstates with
energies below the ESQPT critical energy, something that can be explained
because such states are degenerate up to the numerical precision used in this
calculation for all system sizes. In the panel (a) inset, with a zoom to the
critical energy region, it can be clearly appreciated how, as expected, for
larger system sizes, states go to zero closer to the expected mean field
critical value marked with a vertical dot-dashed pink line. The
$\overline{{\cal F}^{(j)}_{VW}}(T)$ OTOC value, for eigenstates with energies
larger than the ESQPT critical enery are closer to the expected stationary
limit zero value for lower system sizes. This can be easily explained because
we are performing the averaging of the four-point OTOC over a finite $T=1000$
value and this time is large enough to cancel the oscillatory integral for the
smaller system sizes, but not for the larger ones. The results for the 2DVM
case are shown in panel (b) of Fig. 4. In this case the smaller the system
size, the closest to the zero stationary result, for all eigenstates. The
difference with the stationary value increases for increasing system sizes, in
particular for energies smaller than the critical ESQPT energy. As in the LMG
case, as the system size grows, the microcanonical OTOC at energies less than
the critical energy goes to zero closer to the mean field critical energy. The
differences between the stationary limit are larger for states below the
critical energy than for states with energies above the critical energy, due
to the decreasing energy gap between states with different angular momentum
values in this zone that is evinced in Fig. 1. As shown in App. A, the results
obtained for $\overline{{\cal F}^{(j)}_{VW}}$ in the VM and IBM cases are akin
to the ones shown in panel (b) of Fig. 3) for the 2DVM.

Figure 4:  (Color online) Panel (a): Time-averaged value of ${\cal
F}^{(j)}_{VW}(t)$ for the LMG model with $\hat{V}=\hat{W}=\hat{J}_{x}$ as a
function of the system’s excitation energy scaled by the system size
($(E-E_{gs})/N_{2}$) for even parity eigenstates of Eq. (1) Hamiltonian for
various system size values. Panel (b): Time-averaged value of ${\cal
F}^{(j)}_{VW}(t)$ with $\hat{V}=\hat{D}_{-}$ and $\hat{W}=\hat{D}_{+}$ as a
function of the system’s excitation energy scaled by the system size
($(E-E_{gs})/N_{3}$) for $\ell=0$ eigenstates of the 2DVM Hamiltonian in Eq.
(7) for various system size values, $N_{3}$. All panels: All calculations were
carried out for a control parameter value $\xi=0.6$ and the time average is
performed over a time interval $T=1000$. Dashed lines are the result of
averaging for different system size values (see panel legend). A vertical dot-
dashed pink line marks the ESQPT critical energy in the mean field limit.

The main conclusion of the present work is that the degeneracy for eigenstates
belonging to different symmetries below the critical energy of the ESQPT is
different in systems defined in two-level bosonic models in one dimension
compared to models defined for higher dimensions. In fact, in one dimension
the states are degenerate even for finite size systems and the energy
splitting decreases exponentially with the system size, while in dimensions
above one the degeneracy decreases following a potential law and states are
only truly degenerate in the thermodynamic limit. We have checked the
consequences of this different degeneracy in the system dynamics, computing an
out-of-time-order correlator as a possible order parameter for the ESQPT. In
the LMG system (and similar 1D systems) $\overline{{\cal F}^{(j)}_{VW}}$, the
four-point OTOC component, can be used as an ESQPT order parameter, even for
finite size systems. This is not true for the ESQPT higher dimension models.
For such models the degeneracy is not complete for finite size systems.
However, in this last case, the microcanonical OTOC is an order parameter in
the mean field limit. We have also checked that, with an appropriate
combination of the system size and the averaging time, one could also define
with an OTOC an approximate order parameter for ESQPTs in models with
dimension larger than one.

###### Acknowledgements.

We wish to acknowledge useful discussions with José Miguel Arias, Pedro Pérez
Fernández, Jorge Dukelsky, Armando Relaño, Ángel López Corps, Jorge Hirsch,
and Saúl Pilatowski-Cameo. This project has received funding from the European
Union’s Horizon 2020 research and innovation program under the Marie
Skłodowska-Curie grant agreement No 872081 and from grant PID2019-104002GB-C21
funded by MCIN/AEI/ 10.13039/501100011033 and, as appropriate, by “ERDF A way
of making Europe”, by the “European Union” or by the “European Union
NextGenerationEU/PRTR”. This work has also been partially supported by the
Consejería de Conocimiento, Investigación y Universidad, Junta de Andalucía
and European Regional Development Fund (ERDF), UHU-1262561 (JKR and FPB) and
US-1380840 (JKR), and PY2000764, by the Ministerio de Ciencia, Innovación y
Universidades (ref.COOPB20364) (MC). JKR also thanks to Margarita Salas
Fellowship. Computing resources supporting this work were provided by the
CEAFMC and Universidad de Huelva High Performance Computer (HPC@UHU) located
in the Campus Universitario el Carmen and funded by FEDER/MINECO project
UNHU-15CE-2848.

##  Appendix A Extension to models in $3$ and $5$ dimensions

The vibron model (VM) was introduced in the 80s by Iachello [95], extending
the algebraic approach to the study of molecular structure [11, 96]. In
particular, ro-vibrational excitations for a diatomic molecule are treated as
bosonic collective excitations [95, 96]. Due to the three-dimensional nature
of the problem associated with the dipole degree of freedom in diatomic
molecules, this case has $n=3$ and a $u(4)$ Lie algebra as its dynamical
algebra. The bosonic operators requested to build the sixteen generators of
the dynamical algebra are a scalar boson operator $s^{\dagger}(s)$ and an
angular momentum one boson $p_{\mu}^{\dagger}(p_{\mu})$ with $\mu=\pm 1,0$. As
in the previous cases, the $u(4)$ generators are built as bilinear products of
creation and annihilation operators [11, 45, 96, 95]. As in the two previous
cases, the VM has two dynamical symmetries converging in $so(3)$, the system’s
symmetry algebra, associated with the conservation of the angular momentum

$\displaystyle u(4)\supset u(3)\supset so(3)$ $\displaystyle\mbox{Chain
(VM-I)}~{},$ (12) $\displaystyle u(4)\supset so(4)\supset so(3)$
$\displaystyle\mbox{Chain (VM-II)}~{}.$ (13)

In this case, the total number of $s$ and $p$ bosons is denoted as $N_{4}$ and
it denotes the totally symmetric $u(4)$ irrep that spans the system’s Hilbert
space. The dynamical symmetry in Eq. (12) is a convenient approximation to
model vibration of floppy, weakly-bent molecules and the Eq. (13) dynamical
symmetry provides a Morse-like spectrum and it has been applied to many
molecular species [96, 95]. A model Hamiltonian defined in the same way as in
the previous two cases can be built using the first order Casimir operator of
the $u(3)$ subalgebra in Eq. (12) and the $so(4)$ pairing operator, built with
the second order Casimir operator of $so(4)$ in Eq. (13)

$\hat{H}_{VM}=(1-\xi)\hat{n}_{p}+\frac{\xi}{N_{4}}\hat{P}_{p}~{},$ (14)

with a control parameter $\xi\in[0,1]$,
$\hat{n}_{\tau}=\sum_{\mu}p^{\dagger}_{\mu}p_{\mu}$, and
$\hat{P}_{p}=N_{4}^{2}-\hat{D}^{2}-\hat{J}^{2}$ where
$\hat{D}^{2}+\hat{J}^{2}$ is the second order Casimir operator of the $so(4)$
subalgebra [11, 45, 96, 10]. In this case the symmetry algebra is $so(3)$, due
to conservation of angular momentum, and Hamiltonian (7) is split into
different blocks, one for each angular momentum value, $J$ considered [96].

The model Hamiltonian (14) has a second-order ground-state QPT between the
$u(3)$ and $so(4)$ dynamical symmetries, and an associated ESQPT. The
correlation energy diagram is displayed in the panel (a) of Fig. 5 for levels
with angular momentum $J=0$ (full blue lines) and $J=1$ (dashed red lines).
The correlation energy diagram is akin than the correlation energy diagrams
for the LMG and the 2DVM shown in Fig. 1. Once again, the difference between
levels with different angular momentum is hightlighted. In the panel (c) of
Fig. 5 these differences are plotted using semi-log scale. As we observed in
the 2DVM, the order of the degeneracy doesn’t reach the numerical precision
until $\xi=1$.

Figure 5:  (Color online) Panel (a): Excitation energy scaled by the system
size ($\varepsilon=(E-E_{0})/N_{4}$) as a function of the control parameter
$\xi$ for the VM with model Hamiltonian Eq. (14) and system size $N_{4}=50$.
Full blue (dashed red) lines mark levels with angular momumtum $J=0(1)$. Panel
(b): Excitation energy scaled by the system size
($\varepsilon=(E-E_{0})/N_{6}$) as a function of the control parameter $\xi$
for the IBM model Hamiltonian Eq. (17) with a system size $N_{6}=50$. Full
blue (dashed red) lines mark levels with seniority
$\nu_{s}=\nu_{d}=0(\nu_{s}=\nu_{d}=1)$. In both panels the color-filled area
marks the energy difference between selected states with different angular
momentum (panel (a)) or seniority (panel (b)). Panel (c): Energy difference
between selected pairs of states of the VM Hamiltonian Eq. (17) having
different angular momentum as a function of the control parameter $\xi$. Panel
(d): Energy difference between selected states of Hamiltonian Eq. (17) with
different seniority as a function of the control parameter $\xi$. In both
cases, the labels of the selected pairs of levels are provided in the legend
of the panels and the color used for each pair of states is the same color
used to fill the corresponding area in the upper panel.

The IBM has a very successful history in the study of nuclear structure using
algebraic methods. This model was introduced in the 70’s of the XX-th century
by Arima and Iachello [97, 98, 99, 100]. Since then, it has become a standard
tool in the study of nuclear structure from a collective point of view [101],
and it has been enriched in many different ways, including an odd fermion,
IBFM [102], or considering explicitly the difference between protons and
neutrons, IBM-2 [103], so the original model is sometimes known as IBM-1. In
this case, $n=5$ due to the five dimensions that characterize the nuclear
collective problem and the model dynamical algebra is $u(6)$ [101]. The
nuclear excitations are treated as bosons that can be traced back to coupled
nucleons with angular momentum zero or two. Therefore the model building
blocks are a scalar boson, $s$ and a boson with angular momentum two,
$d_{\mu}$, with $\mu=\pm 2,\pm 1,0$ and the 36 generators of $u(6)$ are
expressed as the bilinear products of creation and annihilation boson
operators [101, 11, 10]. As in the previous case, the angular momentum is
conserved in this case and, therefore, the system’s symmetry algebra is
$so(3)$. In this case there are at least three relevant dynamical symmetries,
but we concentrate in this work on the two subalgebra chains that are
equivalent to the cases previously mentioned

$\displaystyle u(6)\supset u(5)\supset so(5)\supset so(3)$
$\displaystyle\mbox{Chain (IBM-I)}~{},$ (15) $\displaystyle u(6)\supset
so(6)\supset so(5)\supset so(3)$ $\displaystyle\mbox{Chain (IBM-II)}~{}.$ (16)

In this case, the total number of $s$ and $d$ bosons is denoted as $N_{6}$ and
it defines the totally symmetric $u(6)$ irrep that spans the system’s Hilbert
space. The dynamical symmetry in Eq. (15) is a convenient approximation to
model nuclear structure in spherically symmetric (vibrational) nuclides, while
the dynamical symmetry Eq. (13) provides a way to model the so called gamma-
unstable nuclei [11, 101]. Hence, we define a model Hamiltonian in the same
way as in the previous cases. The first term, $\hat{n}_{d}$, is the number
operator of $d$ bosons, which is the first order Casimir operator of $u(5)$ in
Eq. (15). The second term is the Casimir operator of the $so(6)$ subalgebra in
Eq. (13), $\hat{P}_{d}=2\left(N_{6}(N_{6}+4)-\left[d^{\dagger}\cdot
d^{\dagger}-s^{\dagger}s^{\dagger}\right]\left[\tilde{d}\cdot\tilde{d}-\tilde{s}\tilde{s}\right]\right)$,

$\hat{H}_{IBM}=(1-\xi)\hat{n}_{d}+\frac{\xi}{N_{6}}\hat{P}_{d}~{}.$ (17)

As in the three previous cases, the Hamiltonian has one control parameter,
$\xi\in[0,1]$ [11, 101, 10]. If we do not consider other possible dynamical
symmetries, the symmetry algebra is $so(5)$ instead of $so(3)$, and the
conserved quantity is seniority ($\nu_{s}$ and $\nu_{d}$), which is related
with the label $\tau$ of the irrep. This implies that Hamiltonian Eq. (17) can
be split into seniority blocks [11, 101, 10]. As in the previous cases, the
model Hamiltonian (17) has a second-order ground-state QPT between the $u(5)$
and $so(6)$ dynamical symmetries, and the corresponding ESQPT. The correlation
energy diagram is plotted in panel (b) of Fig. 5, for $\tau=0$ and $1$ levels.
On more time, a very similar spectra is obtained, where the eigenvalues seem
to be degenerate in the broken-symmetry phase. However, the highlighted
differences in the panel (b) are plotted using log scale in panel (d) of Fig.
5 versus the control parameter. As it was expected, the degeneracy is not
achieved until the system is in the dynamical symmetry $so(6)$.

##  Appendix B Matrix element of the relevant operators of the $u(2)$ and
$u(3)$ models

In this section we provide of the matrix elements that are necessaries to
develop the calculations presented in the present manuscript.

###  B.1 Matrix elements of the LMG model

The chain I of the LMG model, introduced in Eq. (2), provides of one label
$n_{t}$ to name the states of the basis
$\left\\{\left|[N_{2}]n_{t}\right\rangle,\;n_{t}=0,1...,N_{2}\right\\}$. The
elements of this basis conserve the parity,
$\hat{\Pi}\left|[N]n_{t}\right\rangle=(-1)^{n_{t}\%2}\left|[N]n_{t}\right\rangle$,
where the symbol $\%$ denotes the modulo-$2$ operation. The expected value of
the operator $\hat{J}_{x}$ in this basis is

$\displaystyle\left\langle[N]n_{t}^{\prime}\left|\hat{J}_{x}\right|[N]n_{t}\right\rangle=$
$\displaystyle\frac{1}{2}\sqrt{(N-n_{t})(n_{t}+1)}\delta_{n_{t}^{\prime},n_{t}+1}$
$\displaystyle+$
$\displaystyle\frac{1}{2}\sqrt{(N-n_{t}-1)n_{t}}\delta_{n_{t}^{\prime},n_{t}-1}~{}.$
(18)

From Eq. (B.1) it is trivial to realize that operator $\hat{J}_{x}$ mixes
elements with different parity, however $\hat{J}_{x}^{2}$ connects the state
$\left|[N]n_{t}\right\rangle$ with itself and with $\left|[N]n_{t}\pm
2\right\rangle$, both with the same parity as it.

###  B.2 Matrix element of the 2DVM

The relevant operators in this model are $\hat{n}$, $\hat{D}_{\pm}$,
$\hat{\ell}$, and $\hat{W}^{2}$, the latter can be expressed as
$\hat{W}^{2}=\frac{1}{2}\left(\hat{D}_{+}\hat{D}_{-}+\hat{D}_{-}\hat{D}_{+}\right)+\hat{\ell}^{2}$.
Let’s choose the basis provided by the chain I, Eq. (5). The element of this
basis can be labeled using the vibrational quantum number $n$ and the
vibrational angular momentum $\ell$ as
$\left\\{\left|[N]n,\ell\right\rangle\equiv\left|n^{\ell}\right\rangle\right\\}$
for $n=N,N-1,N-2,...,0$ and $\ell=\pm n,\pm(n-2),...,\pm(n\%2)$. The matrix
elements of the operators $\hat{n}$ and $\hat{\ell}$ are trivial, and the
matrix element of $\hat{D}_{\pm}$ are

$\displaystyle\left\langle
n^{\prime\ell^{\prime}}\left|\hat{D}_{\pm}\right|n^{\ell}\right\rangle=\pm\sqrt{(N-n)(n\pm\ell+2)}\delta_{\ell^{\prime},\ell\pm
1}\delta_{n^{\prime},n+1}~{}.$ (19)

From Eq. (19) it is understood how $\hat{D}_{\pm}$ connect a state with
angular momentum $\ell$ to another with $\ell^{\prime}=\ell\pm 1$. The matrix
element of $\hat{W}^{2}$ can be easily derived from Eq. (19).

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