# Relativistic meson spectra on ion-trap quantum simulators

Johannes Knaute<EMAIL_ADDRESS>Max Planck Institute for
Gravitational Physics (Albert Einstein Institute), 14476 Potsdam-Golm, Germany
Department of Physics, Freie Universität Berlin, 14195 Berlin, Germany
Philipp Hauke<EMAIL_ADDRESS>INO-CNR BEC Center and Physics
Department of Trento University, Via Sommarive 14, I-38123 Povo (Trento),
Italy

###### Abstract

The recent rapid experimental advancement in the engineering of quantum many-
body systems opens the avenue to controlled studies of fundamental physics
problems via digital or analog quantum simulations. Here, we systematically
analyze the capability of analog ion traps to explore relativistic meson
spectra on current devices. We focus on the E8 quantum field theory regime,
which arises due to longitudinal perturbations at the critical point of the
transverse-field Ising model. As we show through exact numerics, for
sufficiently strong long-range suppression in experimentally accessible spin
chain models, absorption spectroscopy allows for the identification of the
low-lying meson excitations with a good degree of accuracy even for small
system sizes. Our proposal thus opens a way for probing salient features of
quantum many-body systems reminiscent of meson properties in high-energy
physics.

##  I Introduction

Emergent phenomena of quantum many-body (QMB) systems play a major role in
condensed matter and particle physics [1, 2, 3]. The recent progress of
quantum simulation technologies [4, 5, 6] in controllable platforms such as
ion traps [7, 8, 9, 10] has opened the prospect of treating fundamental
effects and systems beyond the capability of classical computers. Various
trapped-ion experiments have already unveiled static and dynamical properties
of quantum matter [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23] as well
as lattice gauge theories [24, 25].

In this work, we are interested in using trapped-ion devices to study mesons,
which are non-perturbative bound states consisting of two subparticles or
charges. They appear prominently in Quantum Chromodynamics (QCD), the theory
of strong interactions within the standard model of particle physics, where a
quark-antiquark pair is confined by a flux tube. Their properties and
phenomenology is of key importance for the understanding of heavy-ion
collisions, which provide an experimental way of studying far-from equilibrium
dynamics relevant to the physics of the early universe [26]. Beyond particle
physics, mesons exist also in condensed matter systems, in particular Ising
spin chain models, where symmetry breaking longitudinal fields [27] or long-
range interactions [28, 29] can confine domain walls into mesons. The
existence of mesons in the spectrum has severe consequences for both static
and dynamical properties of the QMB system at zero and finite temperature.
Some of the diverse implications for entanglement, correlations, and
thermalization are theoretically studied in [30, 31, 32, 33, 34, 35, 36, 37,
38, 39]. In long-range models, the existence or absence of meson states has
also profound implications on the emergence of anomalous cusps in dynamical
quantum phase transitions [40, 41, 42, 43, 44].

Analog quantum simulations can implement such spin Hamiltonians and therefore
provide access to meson features. Experimental evidence of dynamically induced
magnetic domain wall confinement was first provided in [45] and [46] by
analyzing the meson impact on correlation and entanglement spreading after
quantum quenches. Recently, the authors of [47] proposed a protocol to measure
fluctuations and dissipations in quantum simulators (see also [48]), and
illustrated a way of obtaining spectral information of the meson system from
their relations. An improved error mitigation technique for the extraction of
meson masses on quantum simulators was discussed in [49]. While these systems
are currently most developed in (1+1)-dimensional simulations, their
phenomenology can provide important insights that are relevant across
dimensions. For example, the recent papers [50, 51, 52] explored the
capabilities of quantum simulations for real-time string breaking and meson
scattering 111See also [109] for a QED study of meson scattering with tensor
network simulations.. All these studies focused on parameter regimes where
either a semiclassical interpretation of mesons in terms of domain walls is
possible or a formulation as a simple gauge theory is amenable 222As discussed
in detail in [32], the semiclassical regime is valid deep in the ferromagnetic
phase, i.e. $h<J$ in (1) and (3), when the system is not close to criticality.
In the case of nearest-neighbor interactions, isolated spin domains of
magnetization opposite to the longitudinal field give rise to a confining
potential linear in its spatial extension. Mesonic excitations and their
properties can then be estimated from the Bohr–Sommerfeld quantization
condition. In a similar way, using a two-kink model, it was shown in [28] that
long-range interactions give rise to an effective confining potential when
restricting to the Hilbert space of two domain-wall states. Details on a
reformulation of the nearest-neighbor Ising model at very weak transverse
field as a nonrelativistic $\mathds{Z}_{2}$ gauge theory are discussed, e.g.,
in [52]..

Alternatively, meson states can occur also far away from semiclassical
regimes, e.g., close to quantum critical points (QCPs), where an effective
(i.e. relativistic) quantum field theory (QFT) description is available.
Zamolodchikov’s E8 model [55] is such an example of an interacting QFT that
emerges through longitudinal perturbations at the Ising critical point. The
theoretically predicted E8 meson spectrum was first experimentally observed in
[56] and found recently renewed interest in [57, 58, 59]. These experiments
were based on neutron scattering measurements and spectroscopic methods in
solid-state crystals.

Here, we instead propose controlled measurements of the E8 meson spectrum on
an ion-trap quantum simulator using absorption spectroscopy [14, 16]. For that
purpose, we numerically explore the capabilities of experimentally realizable
small Ising spin systems to identify the lowest E8 meson states. We show that
for sufficiently strong long-range suppression in Ising models, the energy
absorption spectrum, which is accessible in the linear response framework, is
in close correspondence with the analytical expectation of the E8 QFT. We
corroborate these findings by a fidelity analysis, which suggest that small
systems retain the nature of the first meson across all interaction ranges
considered, while the thermodynamic limit may have a transition at a spatial
power-law interaction $\sim 1/r$. Due to the promising experimental [24, 25]
and theoretical [60, 61, 62, 63, 64, 65, 66] efforts to implement and study
gauge theories with ion trap quantum simulations, we see, as an implication of
our study, the potential to probe meson physics also in relativistic gauge
theories with these technologies.

##  II Ising models and QFTs

The transverse field Ising model is a famous example of a many-body system
exhibiting a quantum phase transition [67]. An additional longitudinal field
can break the integrability of the system and introduces interesting new
features, in particular mesons, appearing as non-perturbative bound states in
the spectrum of the model. The prototype is the nearest-neighbor (NN) Ising
model, defined in terms of Pauli matrices $\sigma^{x,z}_{j}$ by the
Hamiltonian

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

where the overall energy scale is set by the unit $J$. The transverse and
longitudinal fields are quantified by the parameters $h$ and $g$,
respectively. The Hamiltonian (1) is written for $N$ spins at positions $j$
assuming open boundary conditions (obc). Analogously, one can assume periodic
boundary conditions (pbc), defined by $\sigma_{N+1}=\sigma_{1}$ for a system
on a circle, by adding the interaction term $-J\sigma^{z}_{N}\sigma^{z}_{1}$.

In a proper continuum limit, the IR regime of $H_{\rm NN}$ is described by a
Majorana fermion QFT with Hamiltonian [68]

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

Here, ${\cal C}\approx 0.062$ is a numerical constant, and $M_{h}\equiv
2J|1-h|$ and $M_{g}\equiv\,{\cal D}J\,|g|^{8/15}$ with ${\cal D}\approx 5.416$
are mass scales in the transverse and longitudinal direction [68, 69]. The QCP
at $\\{J=h=1,g=0\\}$ translates into $M_{h}=M_{g}=0$, in which case the IR is
governed by the Ising CFT with central charge $c=1/2$ and scalar primary
operators $\epsilon=i\bar{\psi}\psi$ and $\sigma$ of dimensions
$\Delta_{\epsilon}=1$ and $\Delta_{\sigma}=1/8$. For longitudinal relevant
perturbations of the Ising CFT, i.e. $M_{h}=0,M_{g}\neq 0$, it is a remarkable
prediction of Zamolodchikov that the resulting interacting E8 QFT is also
integrable and governed by the exceptional simple Lie algebra of rank 8 [55].
This QFT contains 8 stable mesons – fermionic non-perturbative bound states –
whose masses are known as tabulated in table 1 in units of the lightest meson
mass $M_{1}\equiv M_{g}$.

Table 1:  Ratios of meson masses for the integrable interacting E8 QFT [55], which provides an effective description of the nearest-neighbor Ising model along longitudinal perturbations at its critical point. | $M_{2}/M_{1}$ | $M_{3}/M_{1}$ | $M_{4}/M_{1}$ | $M_{5}/M_{1}$ | $M_{6}/M_{1}$ | $M_{7}/M_{1}$ | $M_{8}/M_{1}$
---|---|---|---|---|---|---|---
analytical | $2\cos{\frac{\pi}{5}}$ | $2\cos{\frac{\pi}{30}}$ | $4\cos{\frac{7\pi}{30}}\cos{\frac{\pi}{5}}$ | $4\cos{\frac{2\pi}{15}}\cos{\frac{\pi}{5}}$ | $4\cos{\frac{\pi}{30}}\cos{\frac{\pi}{5}}$ | $8\cos^{2}{\frac{\pi}{5}}\cos{\frac{7\pi}{30}}$ | $8\cos^{2}{\frac{\pi}{5}}\cos{\frac{2\pi}{15}}$
numerical | 1.6180 | 1.9890 | 2.4049 | 2.9563 | 3.2183 | 3.8912 | 4.7834

On ion-trap quantum simulators, it is experimentally possible to implement a
long-range (LR) Ising model, defined by the Hamiltonian [12, 13, 15]

$H_{\mathrm{LR}}=-J\,\left(\sum_{i<j}^{N}\frac{1}{|i-j|^{\alpha}}\sigma^{z}_{i}\sigma^{z}_{j}+h\sum_{j=1}^{N}\sigma^{x}_{j}+g\sum_{j=1}^{N}\sigma^{z}_{j}\right),$
(3)

where the coefficient $\alpha$ quantifies the LR interaction of two spins at
position $i$ and $j$ 333Physical consequences of algebraic long-range
interactions in QMB systems for eigenstate thermalization and symmetry
properties are studied, e.g., in [85].. Similarly to the NN model, one can
consider the system for obc and pbc, where in the latter case we assume that
two spins at positions $i$ and $j$ interact along their minimal distance on
the ring. For $\alpha\to\infty$, one recovers the NN Hamiltonian (1). While
experimentally the range $0\leq\alpha\leq 3$ is in principle accessible [71,
72], it was observed, e.g., in [73] that already for $\alpha\approx 3$, the
physics of the system can resemble closely the NN model.

##  III Energy and absorption spectra

In what follows, we compare the ideal NN model with the LR model based on
numerical diagonalization, to characterize in how far the E8 meson spectrum
survives in presence of power-law interactions and for the relatively small
systems of few dozens sites to which current experiments on trapped ions are
restricted [9, 10]. The basis for the observability in small systems is that
the longitudinal field is chosen large enough such that the associated length
scale of the first meson $L\sim 1/M_{1}\sim|g|^{-8/15}$ is sufficiently small
to be captured by the finite size chain. As already observed in [74] for a
realistic model of a solid state crystal, even relatively large longitudinal
field values are able to reproduce the E8 spectrum, indicating the strong
impact of the QFT regime on the physics of the model.

###  III.1 Energy levels

Figure 1: (Color online) Numerical energy spectra for the NN (a) and LR (b)
Ising model with obc (blue circles) and pbc (orange triangles). The normalized
mass gaps $m_{n}/m_{1}$ of the lowest excited states with level
$n\in\mathds{N}$ are shown for the longitudinal field strength $g=3$ in a
chain of $N=12$ sites. The transverse field strength is fixed to the critical
value $h=1$ throughout this work. Grey dashed lines represent the analytical
E8 meson mass ratios $M_{n}/M_{1}$ (cf. table 1). The continuum threshold is
at $2M_{1}$. Grey dotted lines correspond to multiparticle states with masses
$M_{1}+M_{2}$, $M_{1}+M_{3}$, and $2M_{2}$ (in ascending order). While for obc
some deviations from the ideal result appear, for pbc even such a small system
reproduces well the expected low-lying mass spectrum, for NN as well as for
algebraic interactions.

In Fig. 1, the mass gaps $m_{n}/m_{1}$ for the lowest $n=1,\ldots,400$ excited
eigenstates, normalized to the lowest excited numerical state, are shown for a
chain of $N=12$ spins with an exemplary longitudinal field $g=3$ in the NN (a)
and LR (b) Ising model 444The mass gap $m_{n}$ of level $n$ is defined as the
energy difference to the groundstate, i.e. $m_{n}=E_{n}-E_{0}$.. In the finite
size system, energy levels appear as bands in the spectrum. In the ideal NN
model, pbc (shown as orange triangles) allow for a clean identification of the
first 6 meson levels. Apart from an underestimation of the fourth level, the
mass ratios agree well with the E8 theory (shown as grey dashed lines). The
first $n=1,\ldots,N$ eigenvalues can be associated to the the first meson
level and follow the momentum dispersion relation in the first Brillouin zone.
In contrast, while obc (blue circles) match particularly some of the higher
meson levels, they do not satisfy the ratio of the first E8 masses. We
therefore focus in the following on a finite system with pbc on a ring and
compare the results to the E8 theory on an infinite line (cf. table 1), i.e.
we neglect finite volume corrections given by Lüscher’s formula [76].

At energies above $2M_{1}$, multiparticle states exist and form a continuum.
Although we do not have a continuum in a finite system, we can nevertheless
identify the mass sum $M_{1}+M_{2}$ (shown as the lowest grey dotted line).
Higher order mass sums are very close to some of the analytical E8 mass
ratios. The LR results for $\alpha=3$ (right panel) resemble the NN profile
nearly identically for pbc and slightly smeared-out for obc, and therefore
similarly allow one to identify the analytical meson mass ratios.

###  III.2 Absorption spectra

In recent years, methods have been developed to reveal spectra of interacting
spin systems in trapped ions akin to neutron-scattering in the solid state
[14, 16]. Such experimentally measurable absorption spectra can be computed
within the framework of linear response theory 555See e.g. [93] for an
introduction.. Specifically, the mean energy absorption rate
$\overline{Q}=\overline{\langle\partial H/\partial t\rangle}$ is proportional
to the imaginary (dissipative) part
$\chi^{\prime\prime}(\omega)\equiv\chi^{\prime\prime}_{AA}(\omega)$ of the
susceptibility, which is given in general in the Lehmann representation by

$\chi^{\prime\prime}_{OA}(\omega)=\pi\sum_{n,m=0}^{2^{N}-1}\braket{n}{A}{m}\braket{m}{O}{n}\left(p_{n}-p_{m}\right)\delta[\omega-(E_{m}-E_{n})].$
(4)

Here, the double sum is taken over all eigenstates $\ket{n}$ and $\ket{m}$ of
the system, $A$ is an operator that perturbs the Hamiltonian in the time
domain, and $O$ is an operator whose response in the system is considered. The
delta function in eq. (4) expresses the fact that there is only a contribution
to the result when the perturbation frequency $\omega$ equals the energy
differences $E_{m}-E_{n}$. For general thermal states, the population factors
take the form $p_{n}=\mathrm{e}^{-\beta E_{n}}/Z$, where
$Z=\sum_{n}\mathrm{e}^{-\beta E_{n}}$ is the finite temperature partition
function. For our studies, we are interested in the zero temperature case
where absorption energies are measured with respect to the ground state
$\ket{0}$ with energy $E_{0}$, and $p_{0}=1$ and $p_{n}=0$ for $n>0$.

In the following, we find that the salient features of the spectrum become
accessible with the following straightforwardly measurable operator

$O=A=\sum_{i=1}^{N}\sigma_{i}^{z}\cos(kr_{i}),$ (5)

where $k\in[-\pi,\pi]$ is the quasi-momentum and $r_{i}=ai\equiv i$ the
lattice position for unit lattice spacing. For the special case of $k=0$, the
imaginary part dynamic susceptibility simplifies (in dimensionless units) to

$\chi^{\prime\prime}(\omega,k=0)=\pi
J\sum_{n=0}^{2^{N}-1}\sum_{i=1}^{N}\left|\braket{0}{\sigma_{i}^{z}}{n}\right|^{2}\left\\{\delta[\omega-(E_{n}-E_{0})]-\delta[\omega+(E_{n}-E_{0})]\right\\}.$
(6)

In a realistic situation, the energy resolution is restricted by the
accessible experimental observation time $t_{\rm obs}$. According to the
Wiener–Khintchine theorem [78, 79], the delta function is then approximated by
a Lorentzian

$\delta[\omega-(E_{n}-E_{0})]\approx\frac{\Gamma}{[\omega-(E_{n}-E_{0})]^{2}+\Gamma^{2}},$
(7)

with width $\Gamma=1/t_{\rm obs}$.

Figure 2: (Color online) Energy absorption spectrum of the LR model with pbc
in dependence on the power-law coefficient $\alpha$. The data are scaled to
the maximum of the spectrum. Black dashed lines represent the analytical E8
meson mass ratios (cf. table 1). Grey dotted lines correspond to multiparticle
states with masses $M_{1}+M_{2}$ and $M_{1}+M_{3}$. For the entire range of
$\alpha$, a strong peak appears at the lowest meson mass. With increasing
$\alpha$, more features become discernible that agree with the analytic E8
meson spectrum of the QFT at the critical point of the NN Ising model.
Numerical parameters: $N=18$ (pbc), $\Gamma/J=0.1$, $g=3$.

We numerically calculate the energy absorption spectrum according to Eqs. (6)
and (7) for the realistic value $\Gamma/J=0.1$ (see Sec. VI) on a chain of
$N=18$ sites, which is the largest system size that we can achieve by
iterative eigensolvers for sparse matrices, while keeping a large portion of
the spectrum 666We refer to the appendices for further details on the finite
size, long-range and longitudinal field dependence of the energy and
absorption spectra discussed in this section.. Figure 2 shows the energy
absorption spectrum in the LR model as a function of the frequency in
dependence of the coefficient $\alpha$. For low $\alpha$, only the first meson
mass can unequivocally be discerned. As $\alpha$ is increased, peaks at the
analytical E8 meson mass ratios are formed, whereby the first meson retains
the largest spectral density. The continuum threshold at $2M_{1}$ overlaps
with the third meson peak. Above, also the mass sum $M_{1}+M_{2}$ is
identifiable while the fifth meson peak overlaps with the mass sum
$M_{1}+M_{3}$.

In Fig. 3, the energy absorption spectrum in the LR model with $\alpha=3$
(green dash-dotted curve) is compared to the NN model (orange solid curve) for
one selected value of the longitudinal field. The analogon of this spectral
density in the E8 QFT is the dynamical structure function, which has been
calculated recently in [81]. The corresponding spectrum is shown as the blue
dotted curve for a similar frequency broadening. In the Ising model data, the
first 5 meson states and the mass sum $M_{1}+M_{2}$ are visible as peaks with
(apart from the 4th level) good quantitative agreement to the analytical mass
ratios. While in the exact E8 spectrum the meson peak heights are continuously
decreasing, the finite size data are not able to reproduce this feature above
the continuum threshold. However, the ratio of the first to the second meson
peak height is even in good quantitative agreement with the analytical
prediction.

Figure 3: (Color online) Comparison of the energy absorption spectrum in the
NN and LR model with the analytical E8 dynamical structure function from [81].
The data are scaled to the maximum of the spectrum. Grey dashed lines
represent the analytical E8 meson mass ratios (cf. table 1). Grey dotted lines
correspond to multiparticle states with masses $M_{1}+M_{2}$ and
$M_{1}+M_{3}$. From the numerical absorption spectra, meson peaks can be
identified very close to their expected analytical ratios. The LR model allows
one to resolve the quantitative ratio of the first to the second meson peak
height of the QFT prediction with nearly the same precision as the NN model.
Numerical parameters: $N=18$ (pbc), $\Gamma/J=0.1$, $g=3$, $\alpha=3$.

##  IV Fidelity analysis

The existence of a clear band structure in the energy spectrum of the LR Ising
model (cf. Fig. 1) as well as peaks in the absorption spectrum (cf. Fig. 2)
even for small values of $\alpha$ raises the question whether the underlying
quantum states still can be interpreted as mesons and whether they resemble
their counterparts in the E8 regime of the NN Ising model (corresponding to
$\alpha=\infty$) for finite values of $\alpha$ even on the deeper level of
quantum information measures. Regarding the first point, it is well known [28,
29] that LR interactions confine domain walls in the Ising model. It is
therefore in principle justified to interpret the existence of discrete band
structures as meson states. In the present case, we have additionally also the
effect of the longitudinal field. We address the resemblance with the E8
regime of the NN model for this case using the fidelity $F(\alpha)$ and
fidelity susceptibility $\chi_{F}(\alpha)$. These quantities have been used
previously for ground [82, 83] and excited states [84, 85] as a theoretical
framework to identify and characterize quantum phase transitions. Here, we use
them to detect if there is a fundamental change in the meson structure as the
LR coefficient $\alpha$ is varied.

We consider the system with pbc, in which the first meson band consists of the
first $n=1,\ldots,N$ excited eigenstates of the Hamiltonian. Since they have
different degeneracies in the LR and NN model, the overlap of some of these
states is not well-defined and hence numerically not unique. In fact, only the
first excited state ($n=1$) is nondegenerate in all cases and allows us to
define the fidelity as

$F(\alpha)=|\\!\braket{\phi_{1}(\alpha)}{\phi_{1}(\alpha=\infty)}\\!|,$ (8)

where $\phi_{1}$ denotes the first excited state in the LR and NN model,
respectively. Furthermore, following [82, 83] the fidelity susceptibility is
defined as

$\chi_{F}(\alpha)=-\frac{\partial^{2}F(\alpha,\delta\alpha)}{\partial(\delta\alpha)^{2}}\bigg{|}_{\delta\alpha=0}=\lim_{\delta\alpha\to
0}\frac{-2\ln F(\alpha,\delta\alpha)}{(\delta\alpha)^{2}},$ (9)

where
$F(\alpha,\delta\alpha)=|\\!\braket{\phi_{1}(\alpha)}{\phi_{1}(\alpha+\delta\alpha)}\\!|$.
In our numerics, we use the second relation with the numerical value
$\delta\alpha=0.01$ and probe the range $0\leq\alpha\leq 3$.

The results for the fidelity per site $f(\alpha)\equiv F(\alpha)^{1/N}$ and
the fidelity susceptibility $\chi_{F}(\alpha)$ are shown in Fig. 4 for several
chain lengths at the longitudinal field value $g=3$. For all finite system
sizes under consideration (colored solid curves), which are within
experimental scope, for $\alpha\gtrsim 2$, $f(\alpha)$ lies close to the
maximal value of 1, and even at all-to-all LR interactions ($\alpha=0$, lowest
solid curve) the fidelity per site decreases at most by 1%. These findings
indicate that the quantum nature of the first excited state in the LR model
resembles very closely its counterpart in the NN case, at least for finite
system sizes.

Figure 4: (Color online) Dependence of the fidelity (a), defined in (8), and
fidelity susceptibility (b), defined in (9), on the LR coefficient $\alpha$.
(a): In finite-size systems (colored solid lines, increasing $N$ from top to
bottom), the first meson state retains a large overlap to the one of the NN
model ($\alpha=\infty$). An extrapolation to $N\to\infty$ (black dashed line)
indicates a transition in the nature of the meson state in the thermodynamic
limit, occurring at some value of $\alpha\lesssim 1.5$. (b): In agreement with
this finding, the fidelity susceptibility shows a peak that becomes sharper
with system size (colored solid lines, increasing $N$ from bottom to top).
Assuming a scaling with $N^{-1}$, we obtain a peak position of $\alpha_{\rm
max}\approx 1.07\pm 0.02$ in the thermodynamic limit.

In addition to the finite-size results, we extrapolate the data to the
thermodynamic limit $N\to\infty$ by making the scaling ansatz
$f(\alpha)=f_{\infty}(\alpha)+c(\alpha){N^{-b(\alpha)}}$. We leave the
exponent $b(\alpha)$ together with $c(\alpha)$ and $f_{\infty}(\alpha)$ as
free fit parameter. The result for $f_{\infty}(\alpha)$, representing the
prediction for the thermodynamic limit, is shown as the black dashed curve in
the left panel of Fig. 4. While it shows fast convergence for large $\alpha$,
it decreases rapidly for $\alpha\lesssim 1.5$, indicating a transition in the
nature of the meson state for large system sizes. Similarly, the fidelity
susceptibility, shown in the right panel, exhibits a peak at small values of
$\alpha$ and then decreases towards 0 for strong LR suppression. Such a peak
is suggestive of a transition in the first excited meson state occurring at
some intermediate value of $\alpha$. As the system size increases, the peak
position $\alpha_{\rm max}$ moves towards larger values of $\alpha$. Assuming
a scaling with $N^{-1}$, we can extract the value $\alpha_{\rm max}\approx
1.07\pm 0.02$ for $N\to\infty$ in the thermodynamic limit. This range seems to
agree with the rapid decrease of the fidelity in the left panel.

Thus, while finite size systems retain the same physics across all considered
values of $\alpha$, the scaling analysis suggest the appearance of interesting
new physics for the first meson state in the LR versus NN model, which would
be worth to study on its own. This finding agrees with the fact that the Ising
model with variable-range interactions in a transverse field shows a
transition in its quench dynamics at $\alpha=1$ [73, 86]. Besides hinting at
interesting physics in the excited states, this result suggests that the
mesons for at least $\alpha\gtrsim 1$ remain smoothly connected to the NN
meson even at large system sizes.

##  V Meson mass identifications

For the previous discussions, one specific longitudinal field value was
chosen. In this section, we extract the meson masses from the energy
absorption spectrum in dependence of $g$ 777Further details on the underlying
spectra are discussed in appendix C.. The results are presented in Fig. 5.
Individual meson masses $\widetilde{M}_{n}$ are obtained from a Gaussian fit
to each peak in units of the mass gap $m_{1}$ of the first excited state with
an uncertainty corresponding to its full width at half maximum (shown in panel
(a)). Since the individual energy of an eigenstate is experimentally not
accessible, we additionally express the results with respect to the extracted
mass $\widetilde{M}_{1}$ of the first meson by propagating its uncertainty
(shown in panel (b)). With increasing value of $g$, the uncertainty of the
meson mass decreases, allowing for a more precise identification of the
analytical E8 mass ratios $M_{n}/M_{1}$ up to the fifth level for both the NN
(solid errorbars) and the LR model (dotted errorbars). The fourth meson is
constantly underestimated except for the largest considered longitudinal field
strength. Overall, the numerical data of the finite size system are closest to
the E8 QFT in the range $3\leq g\leq 4$, with an even smaller uncertainty for
the LR model. Moreover, one can observe that the measurable ratio in panel (b)
even allows for a slightly better consistency with the analytical E8 mass
ratios. The resulting uncertainties (peak widths) follow as a property of the
spectrum in combination with the chosen inverse observation time
$\Gamma/J=0.1$.

Figure 5: (Color online) Extracted meson mass ratios $\widetilde{M}_{n}/m_{1}$
and $\widetilde{M}_{n}/\widetilde{M}_{1}$ from the energy absorption spectra
in dependence of the longitudinal field g. The results are expressed in units
of the mass gap $m_{1}$ (a) and the first extracted meson mass
$\widetilde{M}_{1}$ (b). The level $n=1,2,\ldots,5$ increases from bottom to
top. Solid errorbars are for the NN model, dotted ones for the LR model (shown
slightly displaced for graphical purposes). Grey dashed lines represent the
the analytical E8 meson mass ratios $M_{n}/M_{1}$ (cf. table 1). Numerical
parameters: $N=18$ (pbc), $\Gamma/J=0.1$, $\alpha=3$. Once $g$ is sufficiently
large, the meson mass ratios can be reliably extracted even in small systems.

##  VI Quantum simulation in trapped ions

Figure 6: (Color online) Comparison of the energy absorption spectra for
adiabatically prepared groundstates with different time durations $t_{ramp}$
of a linear ramp (colored dashed curves) to the exact ground state case in the
LR model (black solid curve). Background lines are as in Fig. 3. Even for a
relatively fast ramp of $t_{ramp}J=1$, the qualitative agreement of the
spectrum with the exact case is very good and allows for the identification of
E8 meson mass ratios.

In trapped-ion quantum simulators, effective magnetic models are routinely
realized by encoding the basis states $\ket{\uparrow}$ and $\ket{\downarrow}$
of spins 1/2 in two long-lived hyperfine states and inducing effective
spin–spin interactions $\sim J$ through a phonon bus, e.g., using a
Moelmer–Soerensen-type laser or microwave beam [88]. Effective magnetic fields
$\sim h,g$ can be realized by a detuning of the Moelmer–Soerensen beams [13,
15, 16, 89, 90] or by additional lasers that are tuned off-resonantly to the
carrier transition [17, 23].

An experimental protocol to measure the E8 spectrum in such a system is as
follows. First, the effective spins are prepared in the electronic ground
state, corresponding to the fully polarized state
$\ket{\uparrow,..,\uparrow}$, the ground state at $g=\infty$. By slowly
decreasing $g$ and turning on $J$ and $h$, the system is adiabatically
transferred to the ground state at the desired parameter values. Such a
procedure can produce considerable excitations when crossing a quantum phase
transition [13]. In the present scenario, instead, even though we are
interested in the QFT regime emanating from the critical point, we find the
final value of the transverse field $g$ can be large, ensuring a large many-
body gap on the order of $M_{g}\equiv\,{\cal D}J\,|g|^{8/15}$. For $g=3$, we
have $M_{g}=9.7J$ and thus the initial state preparation can occur
adiabatically in times much shorter than $\hbar/J$, which in turn are much
shorter than typical coherence times. Alternatively, ground states in trapped-
ion quantum computers can be prepared to good precision using variational
algorithms [25].

After initialization, the system is perturbed with a time-dependent magnetic
field, which again can be realized by periodically modulating the detuning of
the Moelmer–Soerensen beams or by a time-modulated AC-Stark shift. Using
single-site addressing, site dependent AC-Stark shifts with switching times
much faster than the timescales of the internal dynamics (on the order of
$\hbar/J$) have already been demonstrated experimentally [17, 23]. It is thus
possible to perturb the effective spin system with an operator of the type
defined in Eq. (5).

Two spectroscopy protocols are thinkable. Either, the perturbation is turned
on abruptly, e.g., as a step function and the subsequent time evolution of the
same observable is tracked, which for local observables of the type $O$ can be
done by standard fluorescence measurements [91, 10, 92]. A Fourier transform
then yields the desired imaginary part of the dynamic susceptibility,
$\chi^{\prime\prime}(\omega)$, defined in Eq. (6) [93]. Alternatively, the
perturbation can be modulated temporally with a $\cos(\omega t)$. By tracking
the absorbed energy per unit time, which amounts to the measurement of few-
body correlators and which has already been demonstrated experimentally [25],
again $\chi^{\prime\prime}(\omega)$ is obtained.

Typical trapped-ion experiments on many-body spin systems generate long-range
interactions [12, 13, 14, 15, 16]. In linear chains with open boundary
conditions, for not too large systems these approximate a spatial power-law
decay to good precision [94, 72], and any deviations from the desired power-
law interactions can be mitigated by shaping of the interactions, e.g., by
additional laser beams [95, 90, 63], periodic driving [96, 97], or trap-
shaping techniques [98, 99, 61, 100]. It is nowadays also possible to prepare
ions in ring conformations, thus enabling the realization of periodic boundary
conditions [101, 102, 103]. While the theoretical range of power-law decay
exponents is $0\leq\alpha\leq 3$ [71], the experimentally most favorable
power-law decays are at $\alpha=0$ when working with the axial center-of-mass
mode or in an intermediate range when interactions are transmitted by the
radial phonon modes. For example, in [15] the range of $0.75\leq\alpha\leq
1.75$ has been accessed, which – as the spectra and fidelity analysis reported
in the previous sections show – enables access to meson spectra that closely
approach the physics of the ideal NN model.

The robustness of the adiabatic ground state preparation and subsequent
spectroscopic analysis is explicitly demonstrated in Fig. 6 for the LR model.
Here, we consider the previously discussed parameters $\alpha=g=3$ for a small
chain of $N=12$ sites. Starting from the fully polarized state, the ground
state in the E8 regime is adiabatically prepared by increasing the transverse
field from $h=0$ up to $h=1$ using a linear ramp profile of duration
$t_{ramp}$. We assume $J\mathrm{d}t=0.01$, from which $\mathrm{d}h$ follows
through the relation $\mathrm{d}h/\mathrm{d}t=1/t_{ramp}$. The figure shows
the resulting absorption spectra for several values of $t_{ramp}\lesssim
M_{g}$ (colored dashed curves) in comparison to the case when the exact ground
state is calculated and employed in eq. (6) (black solid curve). The curves
for $t_{ramp}J=\\{4,8\\}$ are nearly coinciding with the exact result. Only
for a fast ramp with $t_{ramp}J=1$ the peak heights are marginally
overestimated but still allow for a precise meson identification.

##  VII Summary and outlook

In this letter, we have demonstrated that the relativistic E8 QFT can be
identified experimentally on ion-trap quantum simulators. Surprisingly small
systems of only 12 to 18 sites with pbc, which implement the LR quantum Ising
model at the experimentally largest possible LR suppression, resemble the NN
Ising model closely and allow for the identification of E8 meson states. For
longer-ranged interactions, while most meson states disappear from the
spectrum, the lowest meson remains a strong feature. We have calculated the
energy absorption spectrum based on linear response theory and showed that it
shares qualitative and quantitative features with its QFT counterpart. Single
and multiparticle meson states appear as peaks in the energy absorption
spectrum, which allow for a precise extraction of analytically predicted E8
meson mass ratios even for large longitudinal field values. As a fidelity
analysis shows, for small systems the nature of the first meson changes only
insignificantly across all values of $\alpha$ considered, while we find
indications for a transition in the meson state in the thermodynamic limit at
a critical value around $\alpha_{c}\approx 1$. We have also discussed an
protocol adapted to existing trapped-ion technologies to experimentally access
the meson spectra. While we have focused on the zero momentum case, this
procedure can be extended to derive also relativistic dispersion relations at
finite momenta, which would enable, e.g., to extract the meson dispersion
relation.

We have focused in our study on the E8 regime, which appears in a parameter
region of the simple Ising model (longitudinal perturbations at the QCP) and
has been experimentally verified previously in solid state crystals. Using
ion-trap based quantum simulation technologies opens, however, a new avenue to
address relativistic meson physics also in more complicated gauge theories,
see, e.g., [104] for an overview of recent progress in the field. Furthermore,
ion-trap quantum simulations allow for studies of finite-temperature systems
[105, 106], which offers, for example, the possibility to study the rich
physics of meson melting [107], a process for which currently no complete
microscopic understanding is available 888See [110] for a related study of
this phenomenon from a tensor network perspective in quantum spin chains..

###### Acknowledgements.

The Gravity, Quantum Fields and Information group at AEI is generously
supported by the Alexander von Humboldt Foundation and the Federal Ministry
for Education and Research through the Sofja Kovalevskaja Award. JK is
partially supported by the International Max Planck Research School for
Mathematical and Physical Aspects of Gravitation, Cosmology and Quantum Field
Theory. The work of JK is supported in part by a fellowship from the
Studienstiftung des deutschen Volkes (German Academic Scholarship Foundation).
PH is supported by Provincia Autonoma di Trento, the ERC Starting Grant
StrEnQTh (project ID 804305), the Google Research Scholar Award ProGauge, and
Q@TN — Quantum Science and Technology in Trento.

##  Appendix A Finite size effects

In this appendix we elaborate on some further details about the finite size
dependence of the results presented in the main text.

Figure 7: (Color online) Numerical energy spectra (normalized mass gaps of
excited states) for the NN (a,b) and LR (c,d) Ising model with obc (blue
circles) and pbc (orange triangles). The left column is for a chain of size
$N=12$, the right for $N=18$. Grey dashed lines represent the the analytical
E8 meson mass ratios (cf. table 1). Grey dotted lines correspond to
multiparticle states with masses $M_{1}+M_{2}$, $M_{1}+M_{3}$ and $2M_{2}$ (in
ascending order).

In Fig. 7, the normalized mass gaps of a chain of size $N=12$ (left column
(a,c) as in Fig. 1 of the main text) are compared to a chain with $N=18$ sites
(right column (b,d)). The longitudinal field $g=3$ and the largest
experimentally accessible decay parameter $\alpha=3$ are considered. Due to
the exponential difference in the total number of eigenstates, different
portions of the spectrum are available for a comparable number of excited
states. For both obc and pbc, the effect of the finite size difference seems
to be very mild in the energy spectrum. Observe also that for obc, higher
bands in the LR model seem to resemble a continuous branch. There are only
mild differences between the NN model (top row (a,b)) and the LR model at
$\alpha=3$ (bottom row (c,d)).

The underlying eigenstates give rise to the energy absorption spectra shown in
Fig. 8. There are nearly no visible differences for the first two meson peaks.
Only above the continuum threshold, differences in multiparticle states occur.
We therefore conclude that the quantitative agreement with the analytical E8
result for the dynamical structure function, which is described in the main
text, is a stable feature for both the NN and LR model.

Figure 8: (Color online) Comparison of the energy absorption spectrum in the
NN (a) and LR Ising model (b) for different chain sizes $N$. The data are
scaled to the maximum of the spectrum. Grey dashed lines represent the the
analytical E8 meson mass ratios (cf. table 1). Grey dotted lines correspond to
multiparticle states with masses $M_{1}+M_{2}$ and $M_{1}+M_{3}$.

##  Appendix B Long-range dependence

In this appendix, we analyze the effect of the LR coefficient $\alpha$ on the
physics discussed in the main text.

Figure 9 displays the energy spectrum as a function of $\alpha$. We vary the
parameter in the range $\alpha=0$ (all-to-all LR interactions) up to the
previously used $\alpha=3$ (strong LR suppression). At $\alpha=0$, three
identical degenerate branches are visible within the considered portion of the
spectrum, for obc and pbc. For increasing values of $\alpha$, semi-continuous
branches in the obc spectrum are more and more split into discrete bands and
new bands appear in the case of pbc. Already for $\alpha\geq 2$, the bands in
the pbc spectrum resemble the E8 lines.

Figure 9: (Color online) Effect of the LR coefficient $\alpha$ on the
numerical energy spectrum (normalized mass gaps of excited states) with obc
(blue circles) and pbc (orange triangles). Background lines are as in Fig. 7.

Figure 10: (Color online) Comparison of the absorption spectrum in the NN (a)
and LR Ising model (b). In the LR model, the coefficient $\alpha$ is varied
over the whole experimentally accessible range of values. Background lines are
as in Fig. 8.

The resulting energy absorption spectra are shown in Fig. 10 (b) in absolute
units for a quantitative comparison to the NN model (a). As $\alpha$ is
increased, the intensities of the individual peaks are increased and the peak
positions resemble the E8 mass ratios. Observe that there is only a very small
difference in the absolute height of the first peak for the NN versus the LR
model at $\alpha=3$.

##  Appendix C Longitudinal field dependence

In this appendix we provide further details about the longitudinal field
dependence of the absorption spectra.

Figure 11 shows the energy spectra of the LR model at $\alpha=3$ for several
longitudinal field values. With increasing field strength, it becomes visible
that a large semi-continuous band breaks apart into several discrete bands,
which flatten out at the expected analytical E8 mass ratios for pbc.

Figure 11: (Color online) Effect of the longitudinal field value $g$ on the
numerical energy spectrum (normalized mass gaps of excited states) in the LR
model with obc (blue circles) and pbc (orange triangles). Background lines are
as in Fig. 7.

The resulting energy absorption spectra are compared to the NN model in Fig.
12. For a better visual presentation, we show both the scaled spectra in the
top panels (a,b) as well in absolute units in the bottom panels (c,d) for a
quantitative comparison. With increasing longitudinal field, the peaks get
narrower and allow for the identification of the proper meson mass ratios and
sums. For the system size under consideration, $N=18$, one can infer that a
longitudinal field $g\geq 2$ is necessary to capture all associated of the
low-lying meson length scales in the system. The NN spectrum differs
qualitatively from the LR model only at the smallest depicted field value
$g=1$ (red curves). At larger values, the quantitative differences in the
absolute scale are very small.

Figure 12: (Color online) Energy absorption spectrum of the NN (a,c) and LR
model (b,d) with pbc in dependence on the longitudinal field value $g$. In
(a,b) the data are scaled to the maximum of the spectrum, in (c,d) in absolute
units. Dashed lines represent the analytical E8 meson mass ratios (cf. table
1), dotted lines correspond to multiparticle states with masses $M_{1}+M_{2}$
and $M_{1}+M_{3}$. Numerical parameters: $N=18$ (pbc), $\Gamma/J=0.1$,
$\alpha=3$.

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