# Meson content of entanglement spectra after integrable and nonintegrable
quantum quenches

Johannes Knaute<EMAIL_ADDRESS>Racah Institute of Physics,
The Hebrew University of Jerusalem, Jerusalem 91904, Givat Ram, Israel

###### Abstract

We use tensor network simulations to calculate the time evolution of the lower
part of the entanglement spectrum and return rate functions after global
quantum quenches in the Ising model. We consider ground state quenches towards
mesonic parameter ranges with confined fermion pairs as nonperturbative bound
states in a semiclassical regime and the relativistic E8 theory. We find that
in both cases only the dominant eigenvalue of the modular Hamiltonian fully
encodes the meson content of the quantum many-body system or quantum field
theory, giving rise to nearly identical entanglement oscillations in the
entanglement entropy. When the initial state is prepared in the paramagnetic
phase, the return rate density exhibits regular cusps at unequally spaced
positions, signaling the appearance of dynamical quantum phase transitions, at
which the entanglement spectrum remains gapped. Our analyses provide a deeper
understanding on the role of quantum information quantities for the dynamics
of emergent phenomena reminiscent to systems in high-energy physics.

##  I Introduction and motivation

Properties of dynamical systems are of major importance for the understanding
of emergent phenomena in condensed matter, gravitational and particle physics.
In this line of research, quantum information concepts became increasingly
relevant for the study of entanglement properties in strongly-coupled quantum
many-body (QMB) systems and quantum field theories (QFTs) in and out of
equilibrium [1, 2, 3]. While entanglement entropy is the most popular measure
to quantify the amount of entanglement in pure states, to extract universal
information, or to use it as an order parameter in (quantum) phase transitions
(see e.g. the review [4]), the seminal paper [5] introduced the more general
entanglement spectrum, which allows to characterize the complete entanglement
structure of a physical system.

Consider a pure state density operator $\rho$ and a spatial bipartition into a
subsystem $A$ and its complement $B$. The modular (or entanglement)
Hamiltonian $\mathcal{H}_{mod}$ [6] is then defined from the reduced density
matrix $\rho_{A}$ of the subsystem via

$\rho_{A}=\operatorname{Tr}_{B}\rho\equiv\operatorname{\mathrm{e}}^{-\mathcal{H}_{mod}}.$
(1)

The corresponding set of eigenvalues is denoted as the entanglement spectrum,
from which the entanglement entropy and Rényi entropies can be calculated (cf.
section III). While this concept was originally employed to detect topological
order [5, 7], it found enormous amount of attention across different fields in
physics (see e.g. [8] for a partial review). In particular, it has been
studied for lattice models [9, 10, 11, 12, 13, 14, 15, 16, 17] and fermionic
systems [18, 19, 20, 21, 22]. Calculations of $\mathcal{H}_{mod}$ in QFTs, and
especially conformal field theories (CFTs), are based on the
Bisognano–Wichmann theorem [23, 24], which allowed to find some explicit forms
[25, 26, 27, 28]. The modular Hamiltonian and its spectrum have also been
studied using tensor networks [29, 30, 31, 32, 33] and via holography in
connection to further quantum information measures [25, 34, 35].

In this letter, we are interested in studying the impact of meson confinement
on the dynamics of entanglement spectra after quantum quenches. Mesons are
nonperturbative bound states, which appear in quantum chromodynamics (QCD) as
flux tube confined quark-antiquark pairs that are important for the physics of
the early universe after the big bang and heavy-ion collisions in nuclear
accelerators [36, 37, 38]. The phenomenology of meson confinement, however, is
not exclusive to QCD. Mesonic bound states exist also as confined fermion
pairs (domain walls) in the spectrum of the quantum Ising model with
longitudinal field [39] or long-range interactions [40, 41]. The seminal paper
[42] initiated the study of their impact on the entanglement dynamics.
Specifically, it was found that mesons give rise to entanglement oscillations,
i.e. an oscillating behavior of the entanglement entropy after quantum
quenches, which bounds the overall entanglement growth if the quench is
performed within the ferromagnetic phase and mesons are produced at rest.
While analyses of quantum quenches towards critical regimes revealed that the
entanglement spectrum carries universal information in form of the operator
scaling dimensions of the underlying boundary CFT [27, 43, 44], comparable
studies in mesonic models have not yet been pursued. We fill this gap in this
article using tensor network simulations [45, 46] for both nonintegrable
semiclassical and integrable relativistic regimes of the Ising model at early
and intermediate time scales.

We are particularly also interested in differences between quenches within the
ferromagnetic phase versus crossings from the paramagnetic one. It hence
becomes insightful to discuss our analyses in connection with dynamical
quantum phase transitions (DQPTs). These are non-equilibrium phase
transitions, which occur in the time domain after quenches, showing up as
nonanalyticities (cusps) in return rate functions. (For reviews on that topic
see [47, 48].) Originally discovered through regular cusps in [49] for
quenches across the critical point of the transverse field Ising model, it was
realized that DQPTs exist also for phase crossings in the longitudinal field
[50] and long-range Ising model [51, 52], i.e. in models where mesons can
exist. Their appearance was experimentally confirmed in [53, 54]. Moreover, it
was shown in [52] that anomalous DQPTs can even exist for quenches within the
ferromagnetic phase. (For further explorations see [55, 56, 57, 58, 59].)
Connections between DQPTs and the dynamics of the entanglement spectrum have
been pioneered in [60, 61, 62]. While the necessity of meson states for
anomalous DQPTs has been explored in [56, 58], their explicit role in the
entanglement spectrum, however, has not yet been addressed.

Our analyses are also strongly motivated by significant advances of quantum
simulation technologies for the study of fundamental physics problems [63, 64,
65, 66]. Recently, the impact of confinement and mesons on quantum
correlations, entanglement dynamics and related properties has been studied
experimentally [67, 68] and theoretically [69, 70, 71, 72, 73, 74, 75, 76,
77]. On the other hand, not only DQPTs became accessible in quantum
simulations [53, 54], but also the spectrum of the modular Hamiltonian via
entanglement tomography [78, 79, 80]. It therefore is a very timely problem to
address the impact of meson confinement also in the latter context.

##  II Model

The quantum Ising model represents an important QMB system, which gives rise
to a nontrivial QFT close to its critical point [81]. It is defined for
nearest-neighbor interactions in one spatial dimension by the Hamiltonian

$H=-J\,\left(\sum_{j=1}^{N-1}\sigma^{z}_{j}\sigma^{z}_{j+1}+h\sum_{j=1}^{N}\sigma^{x}_{j}+g\sum_{j=1}^{N}\sigma^{z}_{j}\right),$
(2)

where $\sigma^{\alpha}_{j}$ ($\alpha=\\{x,z\\}$) are Pauli matrices at lattice
position $j$ within an open chain of $N$ sites. The unit $J\equiv 1$ sets the
overall lattice energy scale, and the transverse and longitudinal field
perturbations w.r.t. the first interaction term are quantified by the
parameters $h$ and $g$, respectively. The transverse model ($g=0$) exhibits a
quantum critical point at $J\\!=\\!h\\!=\\!1$, at which a quantum phase
transition from a disordered paramagnetic phase ($h>1$) towards an ordered
ferromagnetic phase ($h<1$) occurs.

In the thermodynamics limit ($N\to\infty$), there exists a scaling limit, in
which the infrared regime is described by a Majorana fermion QFT, given by the
Hamiltonian [82]

$H_{\text{IR}}=\int_{-\infty}^{\infty}\mathrm{d}x\,\left\\{\frac{i}{4\pi}\left(\psi\partial_{x}\psi-\bar{\psi}\partial_{x}\bar{\psi}\right)-\frac{iM_{h}}{2\pi}\bar{\psi}\psi+{\cal
C}M_{g}^{15/8}\,\sigma\right\\}.$ (3)

Here, $M_{h}\\!\equiv\\!2J|1-h|$ is the free fermion mass, $M_{g}\equiv\,{\cal
D}J\,|g|^{8/15}$ is a longitudinal mass scale, and ${\cal C}\approx
0.062,{\cal D}\approx 5.416$ are numerical constants [82, 83].

At criticality, i.e. for $M_{h}=M_{g}=0$, the Hamiltonian (3) describes the
Ising conformal field theory (CFT) of central charge $c=1/2$, which possesses
two scalar primary operators, $\epsilon=i\bar{\psi}\psi$ and $\sigma$ with
scaling dimensions $\Delta_{\epsilon}=1$ and $\Delta_{\sigma}=1/8$. Transverse
perturbations of the Ising CFT ($M_{h}>0$, $M_{g}=0$) result in an integrable
massive free fermion regime. Longitudinal perturbations confine domain walls
as elementary excitations in the ferromagnetic phase into nonperturbative
meson bound states [39]. In particular, pure longitudinal perturbations
($M_{h}=0$, $M_{g}>0$) give rise to the integrable and interacting E8 QFT
[84], whose 8 stable meson masses $M_{n}$ are analytically known as ratios to
the lightest mass $M_{1}\equiv M_{g}$. Combined transverse and longitudinal
perturbations ($M_{h}>0$, $M_{g}>0$) result in a nonintegrable interacting QFT
with both stable and unstable mesonic bound states [85, 86, 87].

##  III Setup

In the present letter, we study real-time properties of entanglement spectra
and return rate functions after global quantum quenches in both the integrable
E8 QFT as well as in the nonintegrable meson regime. For this purpose, we
employ well established ab initio tensor network simulations, which directly
give access to the quantities of interest in the thermodynamic limit of a
translational invariant spin chain. In particular, based on the matrix product
state (MPS) ansatz [88, 89, 90, 91], we use the infinite time-evolving block
decimation (iTEBD) algorithm [92] to construct a MPS approximation to (gapped)
ground states
$\ket{\psi_{0}}=\lim_{\beta\to\infty}\operatorname{\mathrm{e}}^{-\beta H_{0}}$
w.r.t. an Ising model Hamiltonian $H_{0}$ of the form (2) via imaginary time
evolution. We then use the same iTEBD algorithm to calculate its real-time
evolution $\ket{\psi(t)}=\operatorname{\mathrm{e}}^{-itH_{1}}\ket{\psi_{0}}$
under a different Hamiltonian $H_{1}$. In nontrivial cases, the state
$\ket{\psi_{0}}$ is not an eigenstate of $H_{1}$, such that this quench
protocol drives the QMB system instantaneously out-of-equilibrium (at time
$t=0$) and causes the emergent phenomena.

Figure 1: Overview of the considered quench protocols in the transverse ($h$)
vs. longitudinal ($g$) field plane. Ground states are prepared in the
ferromagnetic and paramagnetic phase of the purely transverse field Ising
model (indicated by green dots) and quenched towards a nonintegrable
semiclassical meson regime [types (1,2)] and the integrable E8 QFT regime
(indicated by the grey dotted line) [types (3,4)].

We consider the specific quench protocols illustrated in Fig. 1. We choose two
distinct pre-quench parameter points in the free fermion ferromagnetic and
paramagnetic phase (shown as green dots) for the parameters $h=0.25$ and
$h=1.75$, respectively. Protocols 1 and 2 quench towards a nonintegrable meson
regime for which we exemplarily choose $\\{h=0.25,g=0.1\\}$ (indicated by the
left cross). This quench point is far away from criticality, i.e. a QFT
description is not amenable but instead a semiclassical approximation based on
the Bohr–Sommerfeld quantization condition can be used to determine four meson
states and their masses (see [42] for detailed discussions). Protocols 3 and
4, on the other hand, quench to the integrable E8 QFT regime from the
different pre-quench phases. The post-quench parameter point is given for
$\\{h=1,g=0.48\\}$. 111The values are chosen such that both final quench
points have an identical mass gap, given by the first meson mass
$M_{1}/J\approx 3.66$. To compare the resulting properties to a non-mesonic
case, we analyze in protocols 5 and 6 quenches from the paramagnetic phase to
the critical point and towards the ferromagnetic phase in the free fermion
regime. 222Note that in type (5) analytical CFT results are available, while
in type (6) it is well understood that a regular DQPT occurs. We refer to App.
A in the supplemental material (including ref. [107]), where we revisit and
analyze these cases.

We analyze real-time entanglement properties of the state $\ket{\psi(t)}$ for
a semi-infinite bipartition of the Ising chain. A Schmidt decomposition across
this cut takes the form
$\ket{\psi(t)}=\sum_{r}\sqrt{\lambda_{r}}\ket{\psi^{A}_{r}}\otimes\ket{\psi^{B}_{r}}$,
where the Schmidt values $\lambda_{0}\geq\lambda_{1}\geq\lambda_{2}\geq\ldots$
are directly related to the eigenvalues $\xi_{r}$ of the entanglement spectrum
via $\lambda_{r}\equiv\operatorname{\mathrm{e}}^{-\xi_{r}}$. A relevant
quantity of interest are the entanglement gaps
$g_{r}\equiv\log\lambda_{0}-\log\lambda_{r}=\xi_{r}-\xi_{0}$. The entanglement
entropy is given by
$S_{1}(\rho_{A})=-\operatorname{Tr}_{A}[\rho_{A}\log\rho_{A}]=-\sum_{r}\lambda_{r}\log\lambda_{r}$
and the 2-Rényi entropy follows as
$S_{2}(\rho_{A})=-\log\operatorname{Tr}_{A}\rho_{A}^{2}=-\log\sum_{r}\lambda_{r}^{2}$.

The central quantity to identify DQPTs is the Loschmidt amplitude
$G(t)\equiv\braket{\psi(0)}{\psi(t)}=\braket{\psi(0)}{\operatorname{\mathrm{e}}^{-itH_{1}}}{\psi(0)}$,
from which the return rate density is defined as

$r_{1}(t)=-\lim_{N\to\infty}\frac{1}{N}\log|G(t)|^{2}.$ (4)

The latter can be interpreted as an analogue of the free energy density in
equilibrium, such that nonanalyticities in $r_{1}(t)$ indicate the appearance
of DQPTs as dynamical analogues of equilibrium phase transitions [49, 47, 48].
As discussed in [55], this definition can be generalized to the rate functions

$r_{i}(t)=-2\log|\epsilon_{i}(t)|.$ (5)

Here, $\epsilon_{i}$ are the eigenvalues (in decreasing order) of the mixed
MPS transfer matrix between $\ket{\psi_{0}}$ and $\ket{\psi(t)}$. Cusps or
kinks in $r_{1}(t)$ hence correspond to level crossings between $r_{1}$ and
$r_{i>1}$.

##  IV Quenches to nonintegrable semiclassical meson regimes

1

2

Figure 2: Time dependence of physical quantities in quench protocols (1) (a-d)
and (2) (e-h) to a semiclassical meson regime. From top to bottom:
entanglement entropy $S_{1}$ and 2-Rényi entropy $S_{2}$ (a,e), eigenvalues
$\lambda_{r}$ of the entanglement spectrum (b,f), entanglement gap ratios
$g_{r\geq 2}/g_{1}$ (c,g), return rate functions $r_{i}$ (d,h). See text for
detailed discussions.

1

2

Figure 3: Results of the Prony signal analysis of $S_{1}$ (a,c) and
$\xi_{0}=-\log(\lambda_{0})$ (b,d) under quench type (1) (left column) and (2)
(right column). Grey vertical lines indicate meson masses $M_{i}$ obtained
from a semiclassical approximation, green vertical lines show all possible
mass differences $m_{ij}\equiv|M_{i}-M_{j}|$ between them. The results
demonstrate that the meson content of entanglement oscillations is fully
encoded in the dominant eigenvalue of the modular Hamiltonian.

Fig. 2 shows the simulation results for quenches from the ferromagnetic (type
1, left column) and paramagnetic phase (type 2, right column) into the
nonintegrable semiclassical meson regime. The time evolution of $S_{1}$ and
$S_{2}$ for quench 1 (panel (a)) within the ferromagnetic phase exhibits a
bounded oscillatory behavior, representing the known entanglement oscillations
induced through meson confinement [42]. 333We have chosen the parameters in
quench protocol (1) identical to one of the cases in [42] and hence reproduce
the form of $S_{1}$ here. On the other side, $S_{1}$ and $S_{2}$ show a very
large entanglement growth under quench 2 (panel (e)), which are superimposed
with oscillations. The latter are, in contrast, unbounded in the available
simulation times. 444By monitoring the truncation error in the iTEBD
simulations for sufficiently high bond dimensions $D$, we estimate the
reachable time scales in the different quench scenarios. More concretely, we
use a dynamical truncation, where singular values below machine precision are
discarded. For quench type (1) with bounded entanglement growth, $D=16$ then
suffices. In all other protocols, we use the maximal value $D=500$, while the
truncation error never exceeds $10^{-7}$.

Panels (b) and (f) show the first eigenvalues $\xi_{0},\ldots,\xi_{6}$ of the
corresponding entanglement spectra. One can observe that the dominant
eigenvalue $\xi_{0}=-\log(\lambda_{0})$ in quench type 1, shown as the blue
dashed curve in panel (b), oscillates on a much smaller magnitude (w.r.t. the
left axis) than the remaining eigenvalues, i.e. the entanglement spectrum is
largely gapped. The shape of $\xi_{0}$ follows nearly identically the time
evolution of $S_{1}$ and hence seems to encode the entanglement oscillations
(cf. the quantitative analyses below). In contrast, in all higher eigenvalues,
many level crossings appear, indicated by nonanalyticities (cusps) of any
single level. The same findings hold also for quench protocol 2 (panel (f))
with the difference that $\xi_{0}$ is of the same scale as $\xi_{r\geq 1}$.
Only at very early times after the quench, at $tJ\approx 0.9$, the
entanglement spectrum becomes gapless, corresponding to a singularity in
$g_{r}/g_{1}$ (cf. panel (g)).

The time dependence of the gap ratios $g_{r}/g_{1}$ is shown in panels (c) and
(g) for $r=2,\ldots,6$. We want to contrast their behavior to CFT expectations
in case of quenches to the critical point (cf. App. A). In the latter case,
the ratios assume the constant values $g_{r}/g_{1}=\Delta_{r}/\Delta_{1}$,
where $\Delta_{r}$ are the conformal dimensions of primary fields and their
descendants in the boundary CFT [27]. In the ferromagnetic meson quench in
panel (c), all shown values are instead oscillating at later times around
values smaller than the lowest integer CFT value $g_{2}/g_{1}=3$ (indicated by
the grey dashed line). In particular, $g_{2}/g_{1}$ (green curve) exhibits
multiple nonanalytic cusps, when the gap between $\xi_{1}$ and $\xi_{2}$
closes and the ratio hence assumes the value one as the lower bound. The same
features exist also under quench 2 (panel (g)). Here, the oscillations display
a larger amplitude and assume higher values (shown on a logarithmic scale),
while a single singularity appears immediately after the quench. 555We refer
to App. B for a comparison of the ratio $g_{2}/g_{1}$ in all quench examples.
Since also higher order ratios exhibit cusps, when the gap between other
eigenvalues closes, these gap ratios do not contain information on meson
masses (and obviously also no other universal information in type 2).

The behavior of the first five return rate functions $r_{i}$ is visible in
panels (d) and (h). For quench protocol 1, $r_{1}$ is on a much smaller scale
than all higher order ones. It exhibits regular oscillations, which carry the
meson content of the post-quench Hamiltonian (cf. the discussions below). On
the other side, all $r_{i}$ in quench type 2 exhibit multiple level crossings.
Since the first cusp in $r_{1}$ appears before the first minimum, we can
identify them as regular ones according to the nomenclature in [52]. 666While
we do not observe any nonanalyticities in quench type (1), the existence of
anomalous cusps [52] is in principle possible for other quench parameters
within the ferromagnetic phase. In contrast to the DQPT regime in the
transverse Ising model (cf. Fig. 6 in App. A), which is also characterized by
regular cusps, they are, however, not equally spaced in time. Moreover, while
regular cusp positions coincide in the previous case with times when the
entanglement spectrum becomes gapless [61], this is not a necessary
consequence in the mesonic regime under consideration, i.e. the modular
Hamiltonian remains gapped at these points in time, apart from the single
exception at early times.

To analyze the meson content of the entanglement spectrum quantitatively, we
show in Fig. 3 the results of a Prony signal analysis in the complex frequency
plane. 777Prony methods [108] are based on the representation of a function
$f(t)$ as a sum of complex exponentials,
$f(t)=\sum_{k=1}^{K}c_{k}\operatorname{\mathrm{e}}^{-i\,\omega_{k}\,t}$. The
complex coefficients $c_{k}$ and frequencies $\omega_{k}$ can be determined
independently through the linearity of the ansatz ($K$ is the (variable) total
number of modes). This allows to capture both oscillations (real part of
$\omega_{k}$) and exponential decay/growth (imaginary part of $\omega_{k}$).
Here, we employ a signal analysis technique developed in [109], for which the
Prony method is applied on a finite analysis window that is sequentially
shifted towards later times. Identified modes are visualized as poles in the
complex frequency plane (on a color scale from blue to red). Stable modes can
be interpreted as discrete frequencies, while streaks (a sequence of poles in
different time windows) can be associated with branch cuts. The first row
displays the analysis of $S_{1}$ in comparison to $\xi_{0}$ in the second row.
In quench type 1 (panels (a,b)) within the ferromagnetic phase, both
quantities allow the clear detection of four meson states $M_{i}$, which are
consistent with a semiclassical approximation [42] (shown as grey vertical
lines). Additionally, meson mass differences $m_{ij}$ (shown as green vertical
lines) and the continuum threshold at $2M_{1}$ can be identified. When the
initial state is in the paramagnetic phase, i.e. for type 2 (panels (c,d)),
remnants of the meson states are still visible, but less clear due to the
large entanglement growth. In both quenches, one can observe that $\xi_{0}$
even allows for a clearer extraction of meson poles in the Prony analyses than
$S_{1}$. These analyses show that the meson content of the post-quench
Hamiltonian, giving rise to entanglement oscillations, is fully encoded in the
dominant eigenvalue of the modular Hamiltonian. 888All higher order
eigenvalues and gaps exhibit nonanalytic cusps in their profile and hence do
not allow to define oscillation frequencies, which could be quantitatively
analyzed. Interestingly, $r_{1}$ in quench type 1 equally encodes the meson
masses in the frequency pattern, but in contrast to $\xi_{0}$, neither mass
differences, nor the continuum threshold are appearing.

##  V Quenches to the integrable E8 QFT regime

3

4

Figure 4: Time dependence of physical quantities in quench protocols (3) (a-d)
and (4) (e-h) to the integrable E8 QFT regime. Legends and quantities are the
same as in Fig. 2. See text for detailed discussions.

3

4

Figure 5: Fourier spectra of $\xi_{0}$ and $S_{1}$ (a,c), and their time
derivatives (b,d) under quench type (3) (left column) and (4) (right column).
Green background lines mark the following mass differences: $m_{23}$,
$m_{34}$, $m_{12}$ (ascending). Red vertical lines indicate the following mass
sums $M_{ij}\equiv M_{i}+M_{j}$: $M_{12}$, $M_{13}$, $M_{14}$, $M_{23}$
(ascending). The results allow to identify several meson states equally
accurate from entanglement oscillations in $S_{1}$ and $\xi_{0}$.

We now consider protocols 3 and 4, which quench towards the integrable E8 QFT
regime with 8 stable meson states. Fig. 4 shows the simulation results. In
type 3 (panel (a)), $S_{1}$ and $S_{2}$ show entanglement oscillations, which,
in contrast to 1, are not bounded but superimposed with a linear growth. As
discussed e.g. in [101], such a behavior can be explained in a quasiparticle
picture by mesons produced at finite velocity (due to a large quench
magnitude), which are able to spread entanglement and quantum correlations
faster. As in the previous section, the entanglement oscillations in the same
quantities are much less pronounced under quench type 4 (panel (e)), when the
initial state is in the paramagnetic phase.

The corresponding entanglement spectra (panels (b,f)) are gapped. As in the
semiclassical regime, $\xi_{0}$ (blue curves) shares the qualitative behavior
of $S_{1}$ in both quenches. Similarly, multiple level crossing appear in all
higher order eigenvalues. The gap ratios $g_{r}/g_{1}$ (panels (c,g)) are
oscillating around lower values than the constant CFT value. For
$g_{2}/g_{1}$, the oscillations are broken by several cusps at the minimal
lower value. While the return rate density $r_{1}$ in type 3 shows
oscillations, which are given by the E8 meson masses, only higher order rate
functions $r_{i>1}$ exhibit level crossings (panel (d)). 999In the calculation
of $r_{i}$, some numerical exception errors occur at late times. The
corresponding data points are left blank in panel (d). The same quantity in
quench type 4 (panel (h)) instead has numerous regular cusps at unequally
spaced positions, indicating the appearance of DQPTs.

Fig. 5 shows a Fourier analysis of $S_{1}$ (blue curves) and $\xi_{0}$ (purple
curves) for both quenches (panels (a,c)). Due to the dominating linear
entanglement growth, the Fourier spectra are decreasing towards larger
frequencies and overall relatively flat with only small peak structures. For
that reason we evaluate in panels (b,d) also their time derivatives, which
allow to identify the oscillating contributions more clearly. Several peaks
become discernible that match the analytical E8 meson mass ratios as well as
some mass differences and sums. For quench 4 from the paramagnetic phase,
these features are much less pronounced. There are only mild differences
between the behavior of $S_{1}$ and $\xi_{0}$.

The discussions of this section exemplify that the previously found
conclusions in the semiclassical meson regime hold equally also in the
relativistic E8 QFT. That is, the dominant eigenvalue of the entanglement
spectrum fully encodes the meson content of the QMB system or QFT. The
appearance of regulars cusps at irregular positions, indicating the appearance
of DQPTs, does not imply that the entanglement spectrum becomes gapless at
these points in time.

##  VI Discussion and outlook

In this letter we have studied the time evolution of the lowest eigenvalues in
the entanglement spectrum and other entanglement quantities after global
quantum quenches into mesonic parameter regimes of the Ising model. As post-
quench points we considered both a semiclassical meson regime (transverse and
longitudinal fields far away from criticality) and the relativistic E8 model
(purely longitudinal perturbations along the Ising critical point). In both
cases, we found that the dominant eigenvalue $\xi_{0}$ of the entanglement
spectrum fully encodes the meson content of the QMB system or QFT. That is,
the nonperturbative meson bound states, giving rise to entanglement
oscillations in the entanglement entropy, appear nearly identically in
$\xi_{0}$ as oscillation frequencies at their respective masses. In contrast,
all higher order eigenvalues exhibit level crossings, signaled by cusps, i.e.
nonanalyticities, in their time evolution. When the initial state is prepared
in the paramagnetic phase, the entanglement growth is always unbounded in the
studied parameter examples and the meson mass frequencies are less pronounced
in the entanglement spectrum than for quenches with initial states in the
ferromagnetic phase.

We combined the entanglement analyses in this letter with the calculation of
return rate functions $r_{i}$, which provide insights into the appearance of
DQPTs and their properties. For quenches initiated in the ferromagnetic phase,
we observe an oscillating behavior of $r_{1}$, dominated by the meson mass
frequencies. For quenches originating in the paramagnetic phase, regular cusps
appear at unequally spaced points in the time evolution of the return rate
density. In contrast to regular DQPTs in the transverse Ising model (i.e.
without meson states), the entanglement spectrum remains gapped at these
times.

Our analyses contribute to a deeper understanding of entanglement properties
of emergent phenomena in QMB systems and QFTs. The study of meson confinement
and DQPT properties in the (1+1)-dimensional Ising QFT is a first step to more
complex systems akin to QCD in particle physics. This necessarily involves the
consideration of gauge theories, where the existence of DQPTs has been
predicted in [103, 104] and further investigated in [105, 106]. Very recently,
its first experimental observation was realized on a quantum computer and
simultaneously discussed with entanglement tomography [80].

###### Acknowledgements.

We would like to thank Mari Carmen Bañuls, Jens Eisert, Jad C. Halimeh, Michal
P. Heller, Jacopo Surace and Luca Tagliacozzo for useful discussions and
correspondence. JK is supported by the Israel Academy of Sciences and
Humanities & Council for Higher Education Excellence Fellowship Program for
International Postdoctoral Researchers.

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  * Note [1] The values are chosen such that both final quench points have an identical mass gap, given by the first meson mass $M_{1}/J\approx 3.66$.
  * Note [2] Note that in type (5) analytical CFT results are available, while in type (6) it is well understood that a regular DQPT occurs. We refer to App.+.1667emA in the supplemental material (including ref.+.1667em[107]), where we revisit and analyze these cases.
  * Note [3] We have chosen the parameters in quench protocol (1) identical to one of the cases in [42] and hence reproduce the form of $S_{1}$ here.
  * Note [4] By monitoring the truncation error in the iTEBD simulations for sufficiently high bond dimensions $D$, we estimate the reachable time scales in the different quench scenarios. More concretely, we use a dynamical truncation, where singular values below machine precision are discarded. For quench type (1) with bounded entanglement growth, $D=16$ then suffices. In all other protocols, we use the maximal value $D=500$, while the truncation error never exceeds $10^{-7}$.
  * Note [5] We refer to App.+.1667emB for a comparison of the ratio $g_{2}/g_{1}$ in all quench examples.
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Supplemental Material to
“Meson content of entanglement spectra after integrable and nonintegrable
quantum quenches”

Johannes Knaute

This supplemental material contains two appendices, in which we discuss some
further background material of our analyses.

##  Appendix A Quenches at criticality and in the massive free fermion regime

5

6

Figure 6: Time dependence of physical quantities in quench protocol (5) (a-d)
to the critical point and (6) (e-h) to the massive free fermion regime regime.
Legends and quantities are the same as in Fig. 2 of the main text.

In this appendix we revisit quenches in the transverse Ising model as a
comparison to the mesonic cases studied in the main text. Fig. 6 shows the
simulation results of quench protocol 5 from the paramagnetic phase to the
critical point, and protocol 6 to the ferromagnetic phase (i.e. in the massive
free fermion regime; cf. Fig. 1). In type 5, $S_{1}$ and $S_{2}$ (panel (a))
exhibit an linear growth (after a short initial quench phase), which is
consistent with expectations of quasiparticle model interpretations [107]. As
alluded in the main text, the entanglement gap ratios $g_{r}/g_{1}$ in panel
(c), following from the entanglement spectrum in panel (b), are expected to
carry universal information by assuming integer values related to the
conformal dimensions in the boundary CFT. This observation was made originally
in [27] for a semi-infinite bipartition, as we consider it here, but holds
also for finite size blocks [43]. One can observe that the lowest two ratios
assume their corresponding values (shown as gray dashed lines) very accurately
already at early times. Higher order eigenvalues deviate more and tend to
converge to the analytical ratios towards later times. Note that these curves
are valid for the particular initial state constructed in the main text (with
a transverse field value $h=1.75$). In general, one can observe that the
analytical ratios are approached more accurately, the more the initial state
is in the paramagnetic phase, i.e. for $h\gg 1$. There is a gap in panel (d)
between the monotonously increasing return rate density $r_{1}$ and higher
order rate functions $r_{i>1}$.

In contrast, under quench type 6, many level crossing appear in the rate
functions $r_{i}$, indicated by equally spaced regular cusps appearing in the
time evolution of $r_{1}$; cf. panel (h). At these points in time, the
entanglement spectrum becomes gapless (cf. panel (f)), such that the gap
ratios $g_{r}/g_{1}$ diverge (cf. panel (g) on a logarithmic scale).

##  Appendix B Entanglement gap ratios in mesonic quenches

Figure 7: Entanglement gap ratio $g_{2}/g_{1}$ for all mesonic quench types
(1-4) of Fig. 1. The gray dashed background line denotes the corresponding CFT
value.

In Fig. 7, the entanglement gap ratios $g_{2}/g_{1}$ are compared for all
mesonic quench types. The values are decaying at early times after the initial
quench phase and oscillate around some mean value below the corresponding CFT
value, which is shown as a reference scale by the gray dashed line. The
oscillations are broken in all cases by cusps at the lower bound
$g_{2}/g_{1}=1$. Similar nonanalyticities exist also in higher order ratios
and eigenvalues of the modular Hamiltonian. For that reason, we concluded in
the main text that the meson content of the QMB system or QFT is solely and
fully contained in the dominant eigenvalue $\xi_{0}$, which gives rise to the
phenomenon of entanglement oscillations.