Max Planck Institute for the Structure and Dynamics of Matter and Center for
Free-Electron Laser Science, Luruper Chaussee 149, 22761 Hamburg, Germany The
Hamburg Center for Ultrafast Imaging, Luruper Chaussee 149, 22761 Hamburg,
Germany The Hamburg Center for Ultrafast Imaging, Luruper Chaussee 149, 22761
Hamburg, Germany The Hamburg Center for Ultrafast Imaging, Luruper Chaussee
149, 22761 Hamburg, Germany The Hamburg Center for Ultrafast Imaging, Luruper
Chaussee 149, 22761 Hamburg, Germany Center for Computational Quantum Physics,
Flatiron Institute, 162 5th Avenue, New York, NY 10010, USA Nano-Bio
Spectroscopy Group, University of the Basque Country (UPV/EHU), 20018 San
Sebastián, Spain

# Unraveling a cavity induced molecular polarization mechanism from collective
vibrational strong coupling

Dominik Sidler<EMAIL_ADDRESS>Laboratory for Materials Simulations,
Paul Scherrer Institute, 5232 Villigen PSI, Switzerland  Thomas Schnappinger
Department of Physics, Stockholm University, AlbaNova University Center,
SE-106 91 Stockholm, Sweden  Anatoly Obzhirov  Max Planck Institute for the
Structure and Dynamics of Matter and Center for Free-Electron Laser Science,
Luruper Chaussee 149, 22761 Hamburg, Germany  Michael Ruggenthaler  Max Planck
Institute for the Structure and Dynamics of Matter and Center for Free-
Electron Laser Science, Luruper Chaussee 149, 22761 Hamburg, Germany  Markus
Kowalewski<EMAIL_ADDRESS>Department of Physics, Stockholm
University, AlbaNova University Center, SE-106 91 Stockholm, Sweden  Angel
Rubio<EMAIL_ADDRESS>Max Planck Institute for the Structure and
Dynamics of Matter and Center for Free-Electron Laser Science, Luruper
Chaussee 149, 22761 Hamburg, Germany

###### Abstract

We demonstrate that collective vibrational strong coupling of molecules in
thermal equilibrium can give rise to significant local electronic
polarizations in the thermodynamic limit. We do so by first showing that the
full non-relativistic Pauli-Fierz problem of an ensemble of strongly-coupled
molecules in the dilute-gas limit reduces in the cavity Born-Oppenheimer
approximation to a cavity-Hartree equation for the electronic structure.
Consequently, each individual molecule experiences a self-consistent coupling
to the dipoles of all other molecules, which amount to non-negligible values
in the thermodynamic limit (large ensembles). Thus collective vibrational
strong coupling can alter individual molecules strongly for localized
”hotspots” within the ensemble. Moreover, the discovered cavity-induced
polarization pattern possesses a zero net polarization, which resembles a
continuous form of a spin glass (or better polarization glass). Our findings
suggest that the thorough understanding of polaritonic chemistry, requires a
self-consistent treatment of dressed electronic structure, which can give rise
to numerous, so far overlooked, physical mechanisms.

Polaritonic chemistry and materials science is a rapidly growing research
field evidenced by a large number of recent review articles. 1, 2, 3, 4, 5, 6,
7, 8, 9, 10, 11 The strong coupling of matter and light within optical
cavities offers a novel way not only to alter and design matter properties,
but also to shape the (quantum) properties of light in various ways. For
example, magnetic12 or metal-to-insulator13 phase transitions can be altered.
Furthermore, cavities can also cause the breakdown of topological protection
as reported for the integer quantum Hall effect.14 In chemistry, the
electronic strong coupling, the quantum yield of emissions 15 or inter-system
crossings 16 can be modified, and photo-chemical reactions can be influenced
17, 18, 19, 20, 21, 22. For vibrational strong coupling even ground-state
(thermally-driven) chemical reactions can be affected 23, 24, 25, 26 However,
despite a plethora of suggested applications and observed novel effects, we
still lack a fundamental understanding of all the relevant underlying
microscopic/macroscopic physical mechanisms, specifically in the context of
vibrational strong coupling effects. 6, 27, 28 One of the main reason for this
deficiency is the complexity of the full description, which a priori requires
a holistic approach combining the expertise from different fields of physics
and chemistry such as quantum optics, electronic-structure theory, (quantum)
statistical mechanics, quantum electrodynamics, molecular and solid state
physics.11 Besides questions concerning the observed resonance conditions 29,
30, 31, 32, currently one of the most pressing, unresolved issue in the field
is how individual molecules can experience cavity-induced modifications under
collective strong coupling. 6, 27, 28, 11 Theoretical attempts to determine
how the coupling of the cavity to the ensemble of molecules can modify the
chemistry of individual molecules in the thermodynamic limit have so far only
been able to describe certain aspects. 33, 34, 35, 36 While there have been
theoretical suggestions that collective strong coupling can lead to local
changes once impurities or (thermally-induced) disorder are introduced in an
ensemble 37, 38 the existence and nature of such effects for a large ensemble
of molecules remained elusive. In this letter we close this important gap by
demonstrating numerically that the cavity can indeed induce local polarization
effects akin to those observed for small molecular ensembles 38 for collective
coupling in the thermodynamic limit, when treating the many-molecule problem
self-consistently within the cavity Born-Oppenheimer approximation of the full
Pauli-Fierz theory.

We consider a dilute gas-phase ensemble of $N$ molecules coupled to a photonic
environment with confined modes $\alpha$. Each molecule consists of $N_{e}$
electrons and $N_{n}$ nuclei/ions such that in the long-wavelength limit the
Pauli-Fierz Hamiltonian becomes 39, 40, 41

$\displaystyle\hat{H}$ $\displaystyle=$ $\displaystyle\hat{H}_{\rm
m}+\frac{1}{2}\sum_{\alpha=1}^{M}\bigg{[}\hat{p}_{\alpha}^{2}+\omega_{\alpha}^{2}\Big{(}\hat{q}_{\alpha}-\frac{\hat{X}_{\alpha}+\hat{x}_{\alpha}}{\omega_{\alpha}}\Big{)}^{2}\bigg{]}$
(1)

with $\hat{H}_{\rm m}$ the usual cavity-free/bare matter Hamiltonian
consisting of $N$ molecules. The coupled polarization operators are defined as
$\hat{X}_{\alpha}:=\sum_{n=1}^{N}\sum_{i_{n}=1}^{N_{n}}Z_{i_{n}}\boldsymbol{\lambda}_{\alpha}\cdot\hat{\boldsymbol{R}}_{i_{n}}$
and
$\hat{x}_{\alpha}:=-\sum_{n=1}^{N}\sum_{i_{n}=1}^{N_{e}}\boldsymbol{\lambda}_{\alpha}\cdot\hat{\boldsymbol{r}}_{i_{n}}$,
where $Z_{i_{n}}$ is the nuclear charge and $\hat{\boldsymbol{R}}_{i_{n}}$ is
the coordinate of the $i$-th nucleus/ion of the $n$-th molecule and
accordingly for the electrons $\hat{\mathbf{r}}_{i_{n}}$. The coupling
strength and polarization to the canonical displacement field operators
$\hat{q}_{\alpha},\hat{p}_{\alpha}$ with mode frequency $\omega_{\alpha}$ is
defined by $\boldsymbol{\lambda}_{\alpha}$ and can be obtained from, e.g.,
macroscopic quantum electrodynamics. 42, 43

In the next step we perform the cavity Born-Oppenheimer approximation
(cBOA),44, 45, 46 i.e., we treat the electrons of the ensemble as a
conditional many-body wave function of all the nuclear degrees of freedom
$\underline{\mathbf{R}}$ and all the displacement field coordinates
$\underline{q}$. 12 We subsequently assume the dilute-gas limit and thus the
overlaps of local many-electron ground-state wave functions $|\chi_{n}\rangle$
of different molecules is considered negligible, and thus a mean-field ansatz
for the ensemble electronic wave function

$\displaystyle|\Psi\rangle=|\chi_{1}\rangle\otimes\dots\otimes|\chi_{N}\rangle$
(2)

becomes accurate (see Supporting Information). We note that this ansatz leads
to the same set of equations as a Slater determinant of all electrons, where
we assume that the individual electronic wave functions of different molecules
do not overlap. This leads to a set of coupled equations, where the local
electronic structure of the $n$-th molecule depends self-consistently on all
the $N-1$ other molecules. Disregarding bare molecular interaction in the
dilute limit, we then have to find the lowest electronic energy state for the
following cavity-Hartree equations

$\displaystyle\left(\hat{H}_{n}(\underline{\mathbf{R}}_{n})+\sum_{\alpha=1}^{M}\bigg{[}\frac{p_{\alpha}^{2}}{2}+\frac{\omega_{\alpha}^{2}}{2}\Big{(}q_{\alpha}-\frac{X_{\alpha}}{\omega_{\alpha}}\Big{)}^{2}\\!\\!\\!+\Big{(}X_{\alpha}-q_{\alpha}\omega_{\alpha}\\!+\\!\\!\sum_{m\neq
n}^{N}\langle\chi_{m}|\hat{x}_{m,\alpha}|\chi_{m}\rangle\Big{)}\hat{x}_{n,\alpha}\right.$
$\displaystyle\qquad+\left.\frac{\hat{x}_{n,\alpha}^{2}}{2}\bigg{]}\right)\chi_{n}(\mathbf{z}_{1},...,\mathbf{z}_{N_{e}})=\epsilon_{n}\chi_{n}(\mathbf{z}_{1},...,\mathbf{z}_{N_{e}})$
(3)

for all $N$ molecules simultaneously, i.e., by a self-consistent solution. Eq.
(Unraveling a cavity induced molecular polarization mechanism from collective
vibrational strong coupling) can be solved analytically for an ensemble of $N$
simple harmonic (model) molecules, which is discussed in Ref. 47. The bare
matter Hamiltonian of a single molecule is defined as $\hat{H}_{n}$ from
$\hat{H}_{\rm
m}(\underline{\mathbf{R}})=\sum_{n=1}^{N}\hat{H}_{n}(\underline{\mathbf{R}}_{n})$
within the dilute-limit approximation and
$\hat{x}_{n,\alpha}=-\sum_{i_{n}=1}^{N_{e}}\boldsymbol{\lambda}_{\alpha}\cdot\hat{\boldsymbol{r}}_{i_{n}}$
is the electronic polarization operator of the $n$-th molecule and
$\mathbf{z}=\mathbf{r}\sigma$ the space-spin variable of an individual
electron. Consequently, the cavity induces an inter-molecular dipole-dipole
energy term

$\displaystyle V_{\rm
dd}=\sum_{\alpha=1}^{M}\sum_{n=1}^{N}\langle\chi_{n}|\hat{x}_{n,\alpha}|\chi_{n}\rangle\sum_{m\neq
n}^{N}\langle\chi_{m}|\hat{x}_{m,\alpha}|\chi_{m}\rangle$ (4)

that scales with $N(N-1)$ over the entire ensemble size. This macroscopic
scaling is crucial for molecular ensembles, since it counteracts the usual
$1/\sqrt{N}$ scaling law of the coupling terms $\boldsymbol{\lambda}_{\alpha}$
for a fixed Rabi splitting, as we show subsequently. Indeed, the inter-
molecular dipole-dipole interaction $V_{\rm dd}$ is physically straightforward
to understand. Because we work in the length gauge, the electric field
operator is48

$\displaystyle\hat{\boldsymbol{E}}=4\pi(\hat{\boldsymbol{D}}-\hat{\boldsymbol{P}}).$
(5)

Notice that the effect of the macroscopic polarization
$\hat{\boldsymbol{P}}=-\sum_{\alpha=1}^{M}\boldsymbol{\lambda}_{\alpha}(\hat{X}_{\alpha}+\hat{x}_{\alpha})/(4\pi)$
on the microscopic constituents is captured by the dipole self-interaction and
scales as $\boldsymbol{\lambda}_{\alpha}^{2}$. That these self-interaction
terms are important to properly describe the optical response of a material
system has been pointed out earlier. 49 In addition, it is also established
that disregarding this term (as often done in model calculations) and only
keeping the cavity-mediated displacement field
$\hat{\boldsymbol{D}}=\sum_{\alpha=1}^{M}\boldsymbol{\lambda}_{\alpha}\omega_{\alpha}\hat{q}_{\alpha}/(4\pi)$
introduces severe theoretical inconsistencies for ab-initio simulations. 50,
48 In the following, we will investigate the chemical relevance of treating
$\hat{\boldsymbol{P}}$ self-consistently on macroscopic and microscopic scale.

From the self-consistent solution of Eq. (Unraveling a cavity induced
molecular polarization mechanism from collective vibrational strong coupling)
for all $N$ molecules, we obtain the classical forces for the nuclei/ions and
the displacement-field coordinates. In more detail, we can perform an ab-
initio molecular dynamics simulation on the polarization-dressed ground-state
potential energy surface. To account for classical canonical equilibrium
conditions at temperature $T$, which are relevant for many ground-state
chemical processes, the classical Langevin equations of motion are propagated,
i.e., 51, 6

$\displaystyle M_{i_{n}}\ddot{\boldsymbol{R}}_{i_{n}}$ $\displaystyle=$
$\displaystyle-\partial_{\boldsymbol{R}_{i_{n}}}H_{n}^{\rm
n}-\langle\partial_{\boldsymbol{R}_{i_{n}}}\hat{H}_{n}^{\rm
e}\rangle_{0}+\sum_{\alpha=1}^{M}Z_{i_{n}}\boldsymbol{\lambda}_{\alpha}\Big{(}\omega_{\alpha}q_{\alpha}-X_{\alpha}-\langle\hat{x}_{\alpha}\rangle_{0}\Big{)}$
(6) $\displaystyle-\gamma
M_{i_{n}}\dot{\boldsymbol{R}}_{i_{n}}+\sqrt{2M_{i_{n}}\gamma
k_{B}T}\boldsymbol{S}_{i_{n}}$ $\displaystyle\ddot{q}_{\alpha}$
$\displaystyle=$
$\displaystyle-\omega_{\alpha}^{2}q_{\alpha}+\omega_{\alpha}X_{\alpha}+\omega_{\alpha}\langle\hat{x}_{\alpha}\rangle_{0}-\gamma\dot{q}_{\alpha}+\sqrt{2\gamma
k_{B}T}S_{\alpha}.$ (7)

These equations implicitly assume that the Hellmann-Feynman theorem applies,
i.e., that the cavity Hartree equations are not only globally but also locally
solved using a variational method. The bare matter Hamiltonian $\hat{H}_{n}$
is separated into a classical part, describing nuclear interactions
$H_{n}^{\rm n}$, and the quantized electronic part $\hat{H}_{n}^{\rm e}$ that
parametrically depends on the nuclear positions. Furthermore, we have
introduced nuclear masses $M_{i_{n}}$, friction constant $\gamma$ and
component-wise delta-correlated Gaussian noise terms, i.e., $\langle
S(t)\rangle=0$, $\langle S(t)S(t^{\prime})\rangle=\delta(t-t^{\prime})$. Each
degree of freedom possesses its individual independent stochastic noise term
indexed by $i_{n}$ and $\alpha$ respectively. We note here that treating the
displacement coordinates classically with thermal noise means that we consider
photonic excitations due to free charges to be in a classical thermal state
48.

In the first step, the collective Rabi splitting is calculated for a model
system consisting of $N=900$ randomly-oriented and slowly rotating Shin-Metiu
molecules 52 strongly coupled to a single cavity mode $\omega_{\alpha}$ at
$T=158$ K, yielding a clear lower and upper polaritonic resonance, as depicted
by the dotted line in Fig. 1(a) (see SI for further details). Notice, the
differently aligned molecules in the dilute limit can also be re-interpreted
as aligned molecules coupled to a spatially modulated displacement field,
which makes our results more generally applicable. Taking into account the
self-consistent treatment of the cavity-induced molecular polarization
$\hat{\boldsymbol{P}}$ leads to a detuning of the cavity towards lower
frequencies, which manifests itself in the asymmetric splitting with respect
to the bare cavity mode $\omega_{\alpha}$, indicated by the black vertical
line. The redshift of the cavity frequency can directly be related to the
refractive indesx of the ensemble within the harmonic approximation.47, 53
Simulations of a few less polarizable hydrogen fluoride (HF) dimers54 show the
same tendency as demonstrated in Fig. 1(b) and Fig. S3, but of significantly
smaller magnitude. Qualitatively similar results can also be found for
perfectly aligned Shin-Metiu molecules (see Fig. S1 in the SI). The observed
cavity induced de-tuning resembles the dipole-dipole interaction induced
Lorentz redshift within dense atomic ensembles (in absence of a cavity), which
depends strongly on the microscopic polarizability of the media, similar to
our result.55, 56 Notice that when tuning the cavity to much lower frequencies
(e.g. ro-vibrational regime), the presence (back-action) of permanent
molecular dipoles is expected to significantly contribute to the red-shift
alongside with the molecular polarizability. This dynamic re-orientation
contribution is neglected in our simulations. The observed collectively
induced redshift of a filled cavity with respect to a bare one has also been
seen in experiments 57 and may in principle be simply approximated by a
suitably chosen refractive index of the ensemble 6, with perfect agreement for
harmonic molecules as discussed in Ref. 47. We also note, that here we get the
redshift directly from the simulation, where we calculate the self-consistent
polarization of the ensemble of molecules. That is, we calculate implicitly
the ensemble polarizability and its back-action on the cavity mode. In the
case of free-space modes, this is the standard way to determine the refractive
index of a material. 49, 58, 59 Now the question arises whether the accurate
self-consistent and microscopic treatment of the polarization can additionally
induce local field effects that cannot be disregarded in the thermodynamic
limit ($N\gg 1$) and that are not captured by a simple refractive index
picture? Earlier evidence for collective electronic strong coupling for a few
molecules indeed indicates that there might be such an effect, 38 yet the
existence of similar local polarization effects for a thermal ensemble under
vibrational strong coupling in the large-$N$ limit remained unclear. As the
first local observable, we analyze the local molecular dipole vibrations for
individual Shin-Metiu molecules, which reveals a (locally) populated lower
polariton (solid line in Fig. 1(a)) and a strongly populated dark state at
$\omega=\omega_{\alpha}$. A local upper polariton could not be identified,
i.e., may be too weakly populated to overcome the thermal broadening for the
given system. Simulations show that the usual $\sqrt{N}$-collective Rabi split
scaling law of the Tavis-Cumming model remains preserved collectively as well
as locally, when including local polarization effects self-consistently (see
Fig. S1 of the Supporting Information). As the second local observable, we
propagate the system self-consistently and measure at every time step, i.e.,
for every realized classical configuration $(\underline{R},q_{\alpha})$, the
difference between the exact solution of Eq. (Unraveling a cavity induced
molecular polarization mechanism from collective vibrational strong coupling)
and the electronic bare matter problem by monitoring $\Delta
r_{n}(t)=\langle\hat{r}_{n}\rangle_{\lambda}-\langle\hat{r}_{n}\rangle_{\lambda=0}$
in the electronic ground-state. This allows to measure cavity-induced local
polarization effects in thermal equilibrium, since the full electronic problem
reduces to the bare local matter problem in the thermodynamic limit if only
the displacement field is considered instead. Our simulation results in Fig. 2
reveal a nonvanishing cavity-induced local ensemble polarization, i.e.,
$\langle|\Delta r_{n}|\rangle\neq 0$, that persist even in the large-$N$
limit. At the same time, the total electronic polarization of the ensemble
remains zero, i.e., $\langle\Delta r_{n}\rangle=0$, as expected from the
symmetry of Eq. (1). Consequently, our numerical results show that the
chemical properties of individual molecules can be locally modified by
collective strong vibrational coupling to the cavity. Fundamentally speaking,
our observation of a continuous distribution of cavity induced molecular
polarizations with zero net polarization resembles a continuous form of a
spin-glass60 (or better polarization-glass). The continuous distribution
automatically implies the existence of hotspots within the molecular ensemble,
where the collective coupling can strongly polarize single molecules and thus
significantly alter their chemical properties. Hoever, the average cavity-
induced polarization remains rather small, which seems in agreement with
recent NMR experiments.61 Notice this collectively induced local mechanism
occurs without external driving, i.e. the sole presence of a thermal bath is
sufficient. Analogous results hold also for perfectly aligned molecules, as
shown in Fig. S2 of the Supporting Information. Physically, the appearance of
local strong-coupling effects can be understood by interpreting the local
polarization in a dipole picture, as previously done for electronic strong
coupling of a few nitrogen dimers 38. While the total (macroscopic)
polarization is zero, non-trivial local dipole modifications are possible for
heterogeneous systems that can still cancel each other, i.e., as seen from the
sum $m\neq n$ in Eq. (Unraveling a cavity induced molecular polarization
mechanism from collective vibrational strong coupling). This local
polarization induces a mirror dipole in the rest of the ensemble. At this
point, we would like to highlight the relevance of (random) disorder in the
ensemble (temperature and/or different molecular orientations, vibrational
states and polarizabilities), which enables a heterogeneous structure of
modified local polarizations that can cancel collectively in analogy to a spin
glass62. For atoms, which do not have a static dipole moment, no local effect
is expected, as can be confirmed by simulating a small ensemble of up to five
neon atoms (see Fig S4 in the Supporting Information). In other words, having
spherically symmetric systems without (different) internal nuclear degrees of
freedom, all local dipole contributions will be equivalent and thus the local
polarization needs to be zero, in order to have a zero macroscopic
polarization. Furthermore, the simple harmonic model considerations in Ref. 47
demonstrate another important ingredient for the formation of polarization
glass, which is the presence of a complex electronic structure, i.e.,
anharmonic electron interactions (e.g. Coulomb) are mandatory. Overall, our
results do not contradict well-established knowledge from quantum-optical
models for atomic systems. However, they show that for molecular ensembles,
the formalism becomes more involved, and the self-consistent (!) treatment of
the local polarization may become decisive in capturing all relevant aspects
of polaritonic chemistry. We also note that the free-space mode structure of
the electromagnetic field, which is homogeneous and isotropic, is not able to
test for disorder in the same way as cavity modes do by having preferred
polarization directions and frequencies. This breaking of symmetry explains
why similar effects are not expected for coupling to free-space modes.

To conclude, we have reformulated the computationally inaccessible many-
molecule Pauli-Fierz problem of polaritonic chemistry in terms of an efficient
cavity-Hartree many-molecule problem, within the dilute-gas limit and the
cBOA. Simulating the corresponding Langevin equations of motion under
vibrational strong coupling in thermal equilibrium reveals that solving the
cavity-Hartree equations self-consistently, and thus including dipole-dipole
interactions between molecules, can be decisive to capture all relevant
aspects of polaritonic chemistry. The reason is that non-trivial local (on the
individual-molecule level) polarization distributions can arise with zero net
polarization, which can persist in the thermodynamic limit and thus may be
regarded as a polarization-glass phase. The continuous distribution implies
the existence of molecular hotspots, where chemistry is locally altered
significantly by the cavity. Furthermore, our self-consistent accounting for
ensemble polarization effects lead to a detuning of the cavity towards lower
frequencies, which is in line with experimental evidence57 and shows that the
dipole self-interaction term is a necessary ingredient to capture the basic
effect of a changed refractive index.

The present result may have far-reaching consequences for the theoretical
description of polaritonic chemistry and materials science, since they provide
a so far overlooked, yet simple and intuitive physical mechanism that can
induce local molecular changes in the thermodynamic limit. This local
mechanism may be the missing piece to settle current discrepancies between
existing simplified models for a macroscopic ensemble of molecules and
experiments. Furthermore, our cavity-Hartree equations are well suited to be
included in existing computational methods, 63, 64, 45, 40, 65, 66, 67, 68
which will enable the efficient exploration of the large chemical space with a
multitude of observables. Particularly, large ensemble sizes under self-
consistent vibrational strong coupling should become accessible by established
ab-initio molecular dynamics codes 64, 65, 66, 69 and potentially with the
help of self-consistent embedding schemes 70. Last but not least, the
existence of a macroscopically induced microscopic polarization mechanism
opens many interesting fundamental physical questions. For instance, can we
efficiently control microscopic (quantum) properties of individual molecules
via a thermal macroscopic field, or are the experimentally observed
modifications of chemical reactions purely due to change in the statistics? On
the more theoretical side, can our results be generalized to the liquid or
even solid phase under collective strong coupling conditions? What are the
physical properties of the suggested cavity induced polarization glass phase
and its relation to a spin glass? Can thus the thermodynamics of molecules
under VSC be affected akin to frustration60, 71 in a spin glass? All these
aspects open many interesting questions that lie at the boundaries between
physics and chemistry and need the combination of various different viewpoints
and methods 2.

(a)

(b)

Figure 1: (a) Vibrational absorption spectra $\alpha(\omega)$ for 900 randomly
oriented Shin-Metiu molecules under collective vibrational strong coupling in
canonical equilibrium at $k_{B}T=0.5$ [mH]. The cavity is tuned to the first
vibrational mode of the uncoupled molecule at $\omega_{\alpha}=6.27$ [mH]
(black vertical line) with $\lambda_{\alpha}=0.0085$. The collective Rabi
splitting (dotted line) is calculated from the fluctuations of the total
ensemble dipole moment (see SI) and shows an asymmetric splitting (red-shifted
cavity). In addition, local molecular vibrations (bold line) are monitored in
a similar way (see SI), which reveals a significant fraction of individual
molecules that are locally strongly coupled, i.e. that vibrate at the
frequency of the lower polariton. Furthermore, the local spectrum also
indicates that the dark states at $\omega=\omega_{\alpha}$ are strongly
populated. In contrast, no local populations of the upper polaritonic states
could be detected at the given temperature. (b) Relative red-shifted cavity
frequencies $\Delta\Delta\omega=|\Delta\omega(N)-\Delta\omega(1)|$ with
respect to the single molecule shift $\Delta\omega(1)$ for a few perfectly
parallelly aligned HF molecules. The collective Rabi splitting was kept
constant with respect to $N$ for each chosen $\lambda_{1}$ by re-scaling
$\lambda=\lambda_{1}/\sqrt{N}$ throughout the computations. The cavity is
tuned to the first vibrational mode of the uncoupled HF at
$\omega_{\alpha}=20.35$ [mH] (see SI for further details). Notice the de-
tuning is about two orders of magnitude smaller for the HF molecule than for
the Shin-Metiu molecule. However, the results agree qualitatively, since they
suggests a similar finite collective detuning in the large $N$ limit. The
overall very small shift is a consequence of the very low polarizability $0.8$
[$\text{\AA}^{3}$] of the HF molecules.72 Figure 2: Statistical evaluation of
cavity-induced local electronic changes $\Delta
r_{n}(t)=\langle\hat{r}_{n}\rangle_{\lambda}-\langle\hat{r}_{n}\rangle_{\lambda=0}$
of the $n$-th molecule with respect to the bare Shin-Metiu molecule in
canonical equilibrium at $k_{B}T=0.5$ [mH] for randomly-oriented molecules
(see Supporting Information for details). The Rabi-splitting was kept constant
when increasing the number of molecules by choosing a re-scaled
$\lambda_{\alpha}(N)=0.256/\sqrt{N}$. By monitoring $|\Delta r_{n}|$ (red
dots), we observe a non-zero saturation of the cavity induced local
polarizations in the large-$N$ limit, where the standard deviations with
respect to different molecules are indicated by vertical red lines. At the
same time, the total polarization of the ensemble, which is related to $\Delta
r_{n}$ (black triangles), quickly approaches zero, since the cavity cannot
induce a non-zero polarization of the ensemble. Consequently, our simulations
suggest that cavity-induced local strong coupling effects persist in the
thermodynamic limit $(N\gg 1)$ of a molecular ensemble under collective
vibrational strong coupling. In other words, the self-consistent treatment of
Eq. (Unraveling a cavity induced molecular polarization mechanism from
collective vibrational strong coupling) is decisive to describe ground-state
polaritonic chemistry accurately for realistic molecular ensembles.

We thank I-Te Lu, Simone Latini, Abraham Nitzan and Gerrit Groenhof for
inspiring discussions and helpful comments. This work was made possible
through the support of the RouTe Project (13N14839), financed by the Federal
Ministry of Education and Research (Bundesministerium für Bildung und
Forschung (BMBF)) and supported by the European Research Council
(ERC-2015-AdG694097), the Cluster of Excellence “CUI: Advanced Imaging of
Matter” of the Deutsche Forschungsgemeinschaft (DFG), EXC 2056, project ID
390715994 and the Grupos Consolidados (IT1249-19). The Flatiron Institute is a
division of the Simons Foundation. M.K. acknowledges funding from the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation program (grant agreement no. 852286).

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# Fundamental role of nonlocal orders in 1D Extended Bose-Hubbard Model

Nitya Cuzzuol<EMAIL_ADDRESS>Arianna Montorsi
<EMAIL_ADDRESS>Institute for Condensed Matter Physics and Complex
Systems, DISAT, Politecnico di Torino, I-10129 Torino, Italy

###### Abstract

Nonlocal order parameters capture the presence of correlated fluctuations
between specific degrees of freedom, in otherwise disordered quantum matter.
Here we provide a further example of their fundamental role, deriving the
ground state phase diagram of the filling one extended Bose Hubbard model
exclusively in terms of their ordering. By means of a density matrix
renormalization group numerical analysis, we show that besides the (even)
parity order characteristic of the Mott insulating phase, and the string order
non vanishing in the Haldane insulator, the recently proposed odd parity order
completes the picture, becoming nonvanishing at the transition from the normal
superfluid to the paired superfluid phase. The above three nonlocal parameters
capture all the distinct phases, including the density wave phase which local
order is seen as the simultaneous presence of correlated fluctuations in
different channels. They provide a unique tool for the experimental
observation of the full phase diagram of strongly correlated quantum matter,
by means of local density measurements.

###### pacs:

05.30.Jp, 03.65.Aa, 67.85.Hj, 05.10.Cc

> The full phase diagram of the one dimensional three body constrained
> extended Bose Hubbard model is re-derived here in terms of solely three
> nonlocal order parameters, identified with the expectation values of
> appropriate disorder operators: the string operator, and the even and odd
> parity operators. Their finiteness unveils the emergence of distinct
> correlated density fluctuations for each disordered phase, which can persist
> at non-zero temperatures thanks to their nonlocal nature. These parameters
> can be detected by simple local density measurements in real experiments
> with quantum matter.

##  I Introduction

The (1D) Hubbard model is the simplest paradigmatic model capturing the
complex physics of strongly correlated particles (fermions or bosons) on a
lattice. The model has been thoroughly studied in the last decades. In
particular, at integer fillings and low temperatures, for repulsive on site
density-density interaction $U$ a Mott insulating (MI) phase is observed,
which escapes the Landau classification of spontaneous symmetry breaking (SSB)
local orders Landau (1937); Landau and Lifshitz (1980). Indeed, it has been
shown that the phase is characterized by a nonlocal order, in 1D known as
charge parity order, in both the bosonic Berg  _et al._ (2008) and the
fermionic Montorsi and Roncaglia (2012) cases. Whereas in 2D it is captured by
its generalization, namely the brane parity order Fazzini, Becca, and Montorsi
(2017). Despite their nonlocal nature, such orders are observed by local
density measurements in atomic matter trapped onto optical lattices Endres
_et al._ (2011); Wei  _et al._ (2023); Hur  _et al._ (2024). In fact, since
they do not break any continuous symmetry, these order persist at sufficiently
low temperatures.
In the 1D fermionic extended Hubbard modelSengupta, Sandvik, and Campbell
(2002) the above MI phase is challenged by other insulating phases, which turn
out to be associated to other types of nonlocal ordersDalla Torre, Berg, and
Altman (2006); Barbiero, Montorsi, and Roncaglia (2013), ultimately capturing
the fundamental role of correlated quantum fluctuations in each of the
possibly disordered distinct phases. The mathematical framework for nonlocal
orders is provided by the concept of disorder operatorsFradkin (2017), which
is not peculiar of insulating phases. Examples are spin parity order, observed
in the Luther Emery liquid phase of 1D fermionsBarbiero, Montorsi, and
Roncaglia (2013), or the brane spin parity, which is non vanishing in the
superconducting phase of the 2D caseTocchio, Becca, and Montorsi (2019). Most
notably, besides parity orders, also string nonlocal orders appear in these
systems, capturing symmetry protected topological (SPT) phases with non
trivial topological featuresGu and Wen (2009). This is the case of the Haldane
phase, which can be observed in insulating (HI) phases of bosonsDalla Torre,
Berg, and Altman (2006) and fermionsBarbiero, Montorsi, and Roncaglia (2013),
as well as in conducting phasesFazzini, Barbiero, and Montorsi (2019);
Montorsi, Fazzini, and Barbiero (2020).

Figure 1:  Phase diagram obtained by analyzing the nonlocal operators in
eqs.(2),(3) for 3-body constrained 1D EBH model (left panel). Our results are
represented by filled symbols, while the unfilled symbols correspond to the
findings in Ejima, Lange, and Fehske (2014) for repulsive onsite interaction
and Dalmonte  _et al._ (2011) for attractive onsite interaction. The full
symbols of different colors signal that the expectation value of the
corresponding nonlocal operator is different from zero (green for
$O_{P}^{(e)}$, red for $O_{S}$, blue for $O_{P}^{(o)}$), separating phases
with different nonlocal orders. The red dotted line close to the origin is the
SF border predicted in Ejima, Lange, and Fehske (2014). The cartoons to the
right schematically represent the correlated density fluctuations underlying
the possible hidden nonlocal orders. The red ellipses highlight the two kind
of pairs (holon-doublon or boson-boson) with finite correlation length which
correspond to parity order (even or odd respectively). The empty rectangles
highlight the alternation of holons and doublons in the background of single
bosons characteristic of string order.

On more general grounds -at least for correlated spinful fermions in one
dimension Montorsi  _et al._ (2017) described in the low energy limit by
decoupled sine Gordon models- the framework of SPT phases can be exploited to
classify all distinct low temperature phases by means of appropriate nonlocal
order parameters. In fact, taking advantage of such observation also for
bosonic models, very recently it has been shown that another type of charge
parity order (named odd charge parity) is finite in the paired superfluid
(PSF) phase of the Bose-Hubbard model Cuzzuol, Barbiero, and Montorsi (2024).

Here we will explore the crucial role that nonlocal orders play in the full
phase diagram of the Extended Bose Hubbard (EBH) model. Based on the
theoretical framework, we will provide numerical evidence that the different
possible correlated fluctuations shown in the right panel of Fig.(1) are in
fact associated to three different nonlocal order parameters. Their finiteness
uniquely identifies the disordered MI, HI, and PSF phases, as well as the
locally ordered density wave (DW) phase. Each phase boundary is then
identified by the vanishing of one of these operators, as shown in Fig.(1).
There for completeness also the phase separated (PS) region is represented,
not investigated here.

Our results, besides reinforcing evidence on the fundamental role of
correlated fluctuations in disordered quantum matter, provide an immediate
tool for the observation of these distinct phases by local density
measurements in experiments at sufficiently low temperature.

##  II Model

We will focus onto the study of the 1D Extended Bose-Hubbard model, that can
be described by the Hamiltonian operator

$\begin{split}H=&-t\sum_{i}(b_{i}^{\dagger}b_{i+1}+b_{i+1}^{\dagger}b_{i})+\frac{U}{2}\sum_{i}n_{i}(n_{i}-1)+V\sum_{i}n_{i}n_{i+1}\end{split}$
(1)

where $b_{i}$, $b_{i}^{\dagger}$ are the bosonic creation and annihilation
operators, with algebra $[b_{i},b^{\dagger}_{j}]=\delta_{ij}$,
$[b_{i},b_{j}]=0=[b_{i}^{\dagger},b_{j}^{\dagger}]$,
$n_{i}=b^{\dagger}_{i}b_{i}$ is the number of particles at site $i$, $t$ is
the hopping matrix element, and $U$ is the on-site interaction. In the
following we will set $t=1$, and fix the average filling $\bar{n}=1$.
Moreover, we assume the three body constraint $(b_{i}^{\dagger})^{3}=0$ to
hold, which amounts to a truncation of the Hilbert space of each site to the
three lowest occupation states $0,1,2$.

The model, besides capturing the SF-MI transition Fisher  _et al._ (1989)
characteristic of these systems for $U\gtrsim 0$, gained a lot of attentions
because for non vanishing nearest neighbor density-density interaction $V>0$
it also describes an HI phase, with non-trivial topological properties Dalla
Torre, Berg, and Altman (2006); Berg  _et al._ (2008) and a finite nonlocal
string order.

For higher values of $V>0$ also a DW phase appearsKühner, White, and Monien
(2000); Rossini and Fazio (2012). For moderately attractive values of
$V\lesssim 0$ instead, a SF phase is expected, which was observed to turn into
a PSF phase in case of appropriate attractive $U$. The ground state phase
diagram is reported in Fig.(1).

The low energy effective-field theory developed in the constrained case Berg
_et al._ (2008), correctly captures the three insulating phases at filling
$\bar{n}=1$ in the repulsive interaction regime $U,V>0$, as well as a SF phase
for weakly attractive $U$. The Hamiltonian $H$ is first mapped into a spin-$1$
model, then each spin-$1$ variable is written as a sum of two spin-$1/2$,
which are mapped onto two spinless fermions via Jordan Wigner transforms. A
standard bosonization analysis can be applied to the resulting fermionic
model, which associates bosonic fields to the two fermionic degrees of freedom
(DOF) and ends up in mapping the fermionic model into two sine Gordon
Hamiltonians –decoupled in the symmetric and anti-symmetric combinations of
the bosonic fields– and a coupling term negligible in the low energy limit. We
refer to Berg  _et al._ (2008) for the details of the calculation and of the
subsequent renormalization group analysis. In fact, in case of fermions the
above approach was exploited exhaustively to derive a general framework for
associating nonlocal string and parity order parameters to each bosonic
fieldBarbiero, Montorsi, and Roncaglia (2013); Montorsi  _et al._ (2017). More
recentlyCuzzuol, Barbiero, and Montorsi (2024), it was noticed that such
procedure, when applied to the Bose Hubbard case, provides evidence of the
transition to the PSF phase and of its characterization though a further
nonlocal order parameter, named odd charge parity. Here we will prove that
within the same theoretical framework, the full phase diagram of the EBH model
can in fact be derived also at $V\geq 0$ through the appropriate nonlocal
order parameters.

##  III Nonlocal Order Parameters

The bosonization approach to constrained EBH modelBerg  _et al._ (2008)
associated to the symmetric (here even) bosonic field a local representations
through the parity and string operators, which can be expressed in terms of
local densities $n_{i}$. The MI and HI phases were identified with the pinning
of such field to two possible distinct values, implying a finite expectation
value of the even parity and string operators respectively. The previous
mathematical property in fact signals the emergence of an hidden order, i.e.
the ordering of a subset of DOF in the disordered background of the remaining
DOF. In the present case of $3$ DOF per site, the different possible hidden
orders are shown in the left panel of Fig.(1). The bosonization analysisBerg
_et al._ (2008); Cuzzuol, Barbiero, and Montorsi (2024) proves that these
correspond to finite expectation values of one or more of the following
nonlocal operators:

$O_{P}^{(\nu)}(j)=\prod_{i=0}^{j-1}\exp\left[i\pi\delta_{\nu}n_{i}\right],\quad\nu=e,o$
(2) $O_{S}(j)=\delta n_{j}\prod_{i=0}^{j-1}\exp\left[i\pi\delta n_{i}\right]$
(3)

known as parity ($O_{P}$) and string ($O_{S}$) operators. The further index
$\nu$ in the parity operator, which can be even ($e$) or odd ($o$), was
introduced in Cuzzuol, Barbiero, and Montorsi (2024), and corresponds to a
different type of density fluctuation $\delta_{\nu}n_{i}$. Specifically,

$\begin{split}&\delta_{e}n_{i}=\delta n_{i}\doteq n_{i}-1\\\
&\delta_{o}n_{i}=n_{i}\quad.\end{split}$ (4)

Let’s discuss first the parity operator. It is evident from the definition of
eq.(2) that the exponential factors assume either value $+1$ or $-1$ depending
on the $\delta n_{i}$ on site i-th. Then the even parity operator
$O_{P}^{(e)}$ will maintain on average a finite value only when the
fluctuations with respect to the background of singly occupied sites, occur in
the form of correlated nearby empty (holon) and doubly occupied (doublon)
pairs. Reversely, for odd parity $O_{P}^{(o)}$ now the background amounts to
disordered holons and doublons, and fluctuations must occur in pairs of nearby
singly occupied sites to maintain its expectation value finite. This is shown
in the right panel of Fig.(1).
Moreover, a non vanishing average value of the string operator $O_{S}$ in
eq.(3) is obtained when holons and doublons are alternated though diluted in
the disordered single site background.

Figure 2: The analysis of the phase diagram depicted in Fig.(1) for $U>0$ and
varying $V>0$ reveals four different possible phases: SF, MI, HI, DW. The
three distinct scenarios are: (a) Plot of the case $U=9$, here there is a
transition from a region dominated by even parity order (MI) to one where both
string and odd parity are present (DW). (b) Plot of the case $U=3$, here
between the two previous phases there is a region where only the string is
finite (HI). (c) Plot of the case $U=0.5$, here there is first a region
without any order (SF), then initially only the string is finite and finally
also the odd parity becomes greater than zero.

The results are summarized in Tab.(1). Beyond the SF phase, four possible
distinct phases can be obtained at filling $\bar{n}=1$ from the ordering of
the above nonlocal order parameters, according to the following table. Phases
having just one non vanishing nonlocal order (MI, HI, PSF) do not break any
symmetry. The nonvanishing parameters become two in case of the DW phase: the
simultaneous presence of two nonlocal orders ends up into the genuine local
order of the DW phase, which breaks the translational symmetry.

NLOP | SF | MI | HI | PSF | DW
---|---|---|---|---|---
$O_{P}^{(e)}$ | 0 | $\neq 0$ | 0 | 0 | 0
$O_{S}$ | 0 | 0 | $\neq 0$ | 0 | $\neq 0$
$O_{P}^{(o)}$ | 0 | 0 | 0 | $\neq 0$ | $\neq 0$
Table 1: Summary of the expectation values of the nonlocal order parameters
(NLOP) in eqs.(2)(3) for each phase according to bosonization analysis.

Also, in correspondence to each non vanishing nonlocal order parameter,
similarly to what happens in the fermionic case, one or more excitation gaps
are finite. In particular, we will be interested in

$\begin{split}&\Delta_{n}=E^{(1)}(N;L)-E^{(0)}(N;L)\\\
&\Delta_{\alpha=1,2}(L)=E(N+\alpha;L)+E(N-\alpha;L)-2E(N;L)\quad,\\\
\end{split}$ (5)

namely the neutral gap, and the single and double particle excitation gaps for
$N$ bosons on $L$ sites. These were already used to discuss the phase diagram
at $U>0$Ejima, Lange, and Fehske (2014); Rossini and Fazio (2012) and at
$U<0$Dalmonte  _et al._ (2011). As shall be clarified, they behave differently
in the different phases but unlike the nonlocal order parameters their
finiteness cannot be used to identify uniquely all distinct phases.

##  IV Results

This section presents the outcomes of our analysis of the extended Bose-
Hubbard model in one dimension (1D). As mentioned earlier, the comprehensive
phase diagram of this model is here inferred by evaluating the expectation
values of the nonlocal operators described in previous section. To accomplish
this, we represented the state using the matrix product state (MPS) Fannes,
Nachtergaele, and Werner (1992); Klümper, Schadschneider, and Zittartz (1992);
Östlund and Rommer (1995); Vidal (2003) formalism and applied the Density
Matrix Renormalization Group (DMRG) algorithm to ascertain the ground state
White (1992, 1993); Östlund and Rommer (1995); Dukelsky  _et al._ (1998);
Schollwöck (2005, 2011).
In particular, we employed the infinite DMRG algorithm McCulloch (2008);
Hauschild and Pollmann (2018) to approximate an infinite-size chain in our
simulations. This approach enables us to circumvent significant issues related
to boundary effects Degli Esposti Boschi, Montorsi, and Roncaglia (2016) when
evaluating expectation of equations (2),(3). Moreover there is no need for
extrapolation to the thermodynamic limit, requiring fewer DMRG states for
obtaining the ground state; we specifically set a maximal bond dimension of
$\chi=200$. This is enough to obtain a converged expectation value inside the
phase characterized by the corresponding nonlocal order. The situation differs
in computing the gaps, where the finite size algorithm is necessary.
To derive our results, we imposed a maximum limit of 2 bosons per site to
prevent the collapse of bosons onto a single site. While this restriction
alters the model’s delocalization of bosons in the lattice, the various phases
identified in the literature for $U>0$ remain qualitatively consistent, as
illustrated for larger maximal occupation numbers ($n_{\text{max}}=3$ in
Rossini and Fazio (2012) and $n_{\text{max}}=8$ in Deng  _et al._ (2013)). In
fact, it has been demonstratedKühner, White, and Monien (2000) that a maximum
of $n_{\text{max}}=5$ provides a reasonable approximation of the system (in
$U>0$ regime) without a specific limit on boson numbers.
The resulting phase diagram is presented in Fig.(1). Similar markers indicate
equivalent transitions, with filled symbols representing our findings and
empty symbols denoting results from the literature for the extended Bose-
Hubbard model with the same three-body constraint. Besides a region of PS in
which empty and doubly occupied regions coexists, five distinct phases emerge:
MI, HI, DW, SF, PSF. Our transition points are identified where the
expectation value of the operators become greater than zero (actually we
assume the expectation is different from zero only when our estimate becomes
greater then $3\times 10^{-3}$ for $j=200$ in eqs.(2),(3)). While for the
empty symbols we are referring to Ejima, Lange, and Fehske (2014) in the
positive interaction regimes, and for negative onsite interaction the data are
taken from Dalmonte  _et al._ (2011).

###  IV.1 Repulsive U Regime

Figure 3: The analysis of the phase diagram depicted in Fig.(1) for $U<0$ and
varying $V>0$ reveals four different possible phases: SF, PSF, HI, DW. The two
distinct scenarios are: (a) Plot of the case $U=-5$, here there is an initial
absence of order (SF). Then the first finite operator is the string (HI) and
subsequently also the odd parity becomes non-zero (DW). (b) Plot of the case
$U=-6.5$, here the odd parity is already greater than zero (PSF), and the
transition from PSF to DW is signalled by $O_{S}$.

In the region of $U>0$, it is possible to locate the following phases: SF, MI,
HI, DW.
In order to show the different behaviours of the operators defined in eqs.(2)
and (3), it is interesting to have a look at some sections of the phase
diagram. In particular we will examine three different lines in the phase
diagram, namely $U=9,3,0.5$.
For large onsite repulsion ($U=9$), by increasing $V$ from zero, one goes,
through a first order phase transition, from an initial phase where the
dominant order is captured by the even parity (MI) to one where both the odd
parity and the string are greater than zero (DW), as presented in Fig.(2)(a).
When the onsite interaction is reduced, for instance to $U=3$ as in
Fig.(2)(b), at sufficiently weak nearest-neighbor interaction the even parity
is the only operator different from zero: this is the MI phase. In this
scenario, by further increasing the nearest-neighbor interaction the system
first enters –through a Gaussian type phase transition– a phase where the
string operator is finite (HI), and then, upon an Ising type phase transition,
also the odd parity becomes non vanishing in the DW phase.
Further decreasing $U$ towards the origin, for instance in the case $U=0.5$
shown in Fig.(2)(c), $O_{P}^{(e)}$ becomes vanishing. At weak $V$ there is
neither local or nonlocal order and the system is in a normal SF phase. By
increasing $V$ the string parameter slowly becomes nonvanishing at the SF-HI
transition. Then, by further increasing $V$, also the odd parity becomes
different from zero and we are again in the DW phase. Even if both transition
are continuous they belong to different universality classes. The SF-HI is a
BKT transition while the HI-DW transition is an Ising one.
In the phase diagram Fig.(1), we compared our findings with those reported in
Ejima, Lange, and Fehske (2014) by determining the central charge Giamarchi
(2003). Their results closely align with ours, and the two phase diagrams
exhibit a significant overlap.
We also investigated the behavior of the gaps defined in equation (5) in this
region. The results are summarized in the subsequent Tab.(2), for the
repulsive $U$ region as well.

Gap | SF | PSF | MI | HI | DW
---|---|---|---|---|---
$\Delta_{n}$ | 0 | 0 | $\neq 0$ | $\neq 0$ | $\neq 0$
$\Delta_{1}$ | 0 | $\neq 0$ | $\neq 0$ | $\neq 0$ | $\neq 0$
$\Delta_{2}$ | 0 | 0 | $\neq 0$ | $\neq 0$ | $\neq 0$
Table 2: Gaps introduced in eq.(5) for each phase. As explained in the text,
the opening (or temporary vanishing) of the gaps supports the results obtained
by the nonlocal order parameters. A null gap is intended in the thermodynamic
limit.

Since the SF phase is the only one where all gaps are zero, both SF-MI and SF-
HI are signaled by any of the three gaps becoming finite. The scenario differs
for other phase transitions in the region of $U>0$, since all MI, HI, DW
phases possess a finite gap. Indeed the transition between the distinct MI and
HI phases is signaled by a temporary vanishing of both the charge
($\Delta_{1}$ and $\Delta_{2}$) and the neutral ($\Delta_{n}$) gaps, since the
two phases are protected by different symmetries. While the transition to the
latter is signaled solely by the vanishing of the neutral gap.
In the repulsive $U$ region of the phase diagram the gap $\Delta_{2}$ provides
no additional information.

###  IV.2 Attractive U Regime

Switching to the attractive interaction regime $U<0$, it is possible to find
the following phases: SF, PSF, HI, DW and the PS region.
As before let’s have a look to the different behaviours of the three nonlocal
operators defined in eqs.(2)(3). Contrarily to the case of $U>0$, now the
$O_{P}^{(e)}$ will be always zero and the correspondent order is absent. Thus
the phase diagram can be discussed here through the behavior of $O_{P}^{(o)}$
and $O_{S}$ solely. We can consider two representative lines of the phase
diagram, $U=-5,-6.5$, which are located above and below the transition line
from SF to PSF phase. These are shown in Fig.(3).
In particular, in (a) we consider $U=-5$: there, increasing $V$ from zero, the
system initially remains in the phase where all nonlocal orders are zero (SF).
Subsequently, similarly to the weakly repulsive $U$ regime, first the string
$O_{S}$ becomes different from zero (HI) and then also the odd parity (DW).
Moving down to stronger onsite attraction between bosons, in Fig.(3)(b) we
consider the case $U=-6.5$. There also at weakly attractive $V$ a PSF phase
characterised by non-null odd parity is present. Contrary to the SF phase,
this phase now is partly gapped and characterized by the emergence of the
hidden order identified in Cuzzuol, Barbiero, and Montorsi (2024). By further
increasing $V$, one enters directly the region in which both string and odd
parity orders are finite (DW).
Also for the $U<0$ region we can compare our findings with existing results
Dalmonte  _et al._ (2011), obtaining a good agreement.
Moreover, also in this region we investigated the behavior of the different
gaps defined in eq.(5). The results were anticipated in Tab.(2). For
sufficiently weakly attractive $U$, the scenario is the same as for weakly
repulsive $U$, in agreement with the phase diagram of Fig.(1). Entering the
PSF phase instead, just the single particle gap opens, to signal the cost in
energy of adding an unpaired single particle fluctuationCuzzuol, Barbiero, and
Montorsi (2024). On the contrary, the neutral gap remains zero (for an
infinite chain), since the PSF phase is still superfluid. So that the
transition from PSF to DW is in fact signaled by $\Delta_{n}$ becoming
different from zero. The same happens if instead one moves from PSF to PS.

##  V Conclusions

In summary, we presented an alternative numerical derivation of the full phase
diagram of the extended Bose Hubbard model in 1D, which is based solely on the
behavior of three nonlocal order parameters. These amount, besides the string
and (even) parity disorder operators introduced in Berg  _et al._ (2008) to
describe the MI and HI insulator phases, also to the odd parity operator
introduced in Cuzzuol, Barbiero, and Montorsi (2024) to describe the PSF
phase. The latter captures the correlated fluctuations of pairs of single
particles in a disordered background of holons and doublons. Besides
describing the PSF phase, odd parity is here found to be non vanishing also in
presence of a finite string order, describing the alternation of holons and
doublons. In this case, a true local order appears, and the system enters a
SSB DW phase.
The phase diagram obtained by the interplay of the three above disorder
operators is shown in Fig.(1) and is in full accordance with previous results
in literature Ejima, Lange, and Fehske (2014); Dalmonte  _et al._ (2011). Thus
our analysis is capable of characterizing distinct conducting and insulating
phases of these systems by the appropriate order or disorder operator, at the
same time identifying all the transition lines and the type of transition.
Notably, since the presence of nonlocal orders does not violate Mermin Wagner
theorem, the MI, HI, and PSF nonlocal orders could persist even at non zero
temperature, and can thus be observed in experiments with quantum matter.
Another significant advantage of the identified nonlocal orders discussed here
is that their measurement can be gained by just local density measures.
Indeed, in case of the MI phase, the (even) parity order was already measured
in this way both in 1D and in 2DEndres  _et al._ (2011); Wei  _et al._ (2023);
Hur  _et al._ (2024). We expect that a similar result could be obtained also
in case of the odd parity characteristic of the PSF phase, which in principle
can be generalized also to 2D as a odd brane parity orderFazzini, Becca, and
Montorsi (2017).

## ACKNOWLEDGMENTS

AM sincerely thanks David Campbell for introducing her several years ago to
the rich physics of the extended Hubbard model and for stimulating her with
his profound insights to delve into its hidden orders.
We offer our thanks for the financial support from the ICSC – Centro Nazionale
di Ricerca in High Performance Computing, Big Data and Quantum Computing,
funded by European Union – NextGenerationEU (Grant number CN00000013).
Computational resources were provided by HPC@POLITO
(http://www.hpc.polito.it). Calculations were performed using the TeNPy
Library (version 0.10.0)Hauschild and Pollmann (2018).

## Author Declaration

### Data Availability Statement

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

### Conflict of Interest Statement

The authors have no conflicts to disclose.

### Author Contributions

Nitya: Formal Analysis, Data Curation, Writing – original draft, review and
editing
Arianna: Funding Acquisition, Conceptualization, Supervision, Writing – review
and editing

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# Integral solutions of certain Diophantine equation in quadratic fields

Richa Sharma  Richa Sharma @Kerala School of Mathematics, Kozhikode-673571,
Kerala, India<EMAIL_ADDRESS>
###### Abstract.

Let $K=\mathbf{Q}(\sqrt{d})$ be a quadratic field and $\mathcal{O}_{K}$ be its
ring of integers. We study the solvability of the Diophantine equation
$r+s+t=rst=2$ in $\mathcal{O}_{K}$. We prove that except for $d=-7,-1,17$ and
$101$ this system is not solvable in the ring of integers of other quadratic
fields.

###### Key words and phrases:

Elliptic curves, Diophantine equation

###### 2010 Mathematics Subject Classification:

11D25, 11D41, 11G05

##  1\. Introduction

In 1960, Cassels [4] proved that the system of equations

$r+s+t=rst=1,$ (1.1)

is not solvable in rationals $r,s$ and $t$. Later in 1982, Small [3] studied
the solutions of (1.1) in the rings $\mathbb{Z}/m\mathbb{Z}$ and in the finite
fields $F_{q}$ where $q=p^{n}$ with $p$ a prime and $n\geq 1$. Further in
1987, Mollin et al. [8] considered (1.1) in the ring of integers of
$K=\mathbf{Q}(\sqrt{d})$ and proved that solutions exist if and only if
$d=-1,2$ or $5$, where $x,y$ and $z$ are units in $\mathcal{O}_{K}$. Bremner
[1, 2] in a series of two papers determined all cubic and quartic fields whose
ring of integers contain a solution to (1.1). Later in 1999, Chakraborty et
al. [6] also studied (1.1) in the ring of integers of quadratic fields
reproducing the findings of Mollin et al. [8] for the original system by
adopting a different technique.

Extending the study further, we consider the equation

$r+s+t=rst=2.$ (1.2)

The sum and product of numbers equals $1$ has natural interest where as sum
and product equals other naturals is a curious question. The method adopted
here may not be suitable to consider a general $n$ instead of $2$ as for each
particular $n$ the system give rise to a particular elliptic curve which may
have different ‘torsion’ and ‘rank’ respectively. The next case, i.e. when the
sum and product equals to $3$ is discussed in the last section.

To begin with we perform suitable change of variables and transform (1.2) to
an elliptic curve with the Weierstrass form

$E_{297}:Y^{2}=X^{3}+135X+297$ (1.3)

and then study $E_{297}$ in the ring of integers of $K=\mathbb{Q}(\sqrt{d})$.

######  Remark 1.

We transform (1.2) into an elliptic curve (1.3) to show that one of the
$(r,s,t)$ has to belong to $\mathbb{Q}$ (shown in §3).

System (1.2) give rise to the quadratic equation

$x^{2}-(2-r)x+\frac{2}{r}=0,~{}r\neq 0,$

with discriminant

$\Delta=\frac{r(r^{3}-4r^{2}+4r-8)}{r}.$ (1.4)

At hindsight there are infinitely many choices for the quadratic fields
contributed by each $r$ of the above form where the system could have
solutions. The main result of this article is that the only possibilities are
$r=\pm 1,2$ and $-8$. Thus (1.2) is solvable only in $K=\mathbf{Q}(\sqrt{d})$
with $d=-7,-1,17$ and $101$. Also the solutions are explicitly given.
Throughout this article we denote ‘the point at infinity’ of an elliptic curve
by ${\mathcal{O}}$. Now we state the main result of the paper.

######  Theorem 1.1.

Let $K=\mathbb{Q}(\sqrt{d})$ be a quadratic field and $\mathcal{O}_{K}$ denote
its ring of integers. Then the system

$r+s+t=rst=2$

has no solution in $\mathcal{O}_{K}$ except for $d=-7,-1,17$ and $101$.

In §4 we discuss the rank of $E_{297}$ in $\mathbb{Q}$ and in quadratic fields
of our interest.

##  2\. Preliminaries

In this section we mention some results which are needed for the proof of the
Theorem 1.1. First we state a basic result from algebraic number theory.

######  Theorem 2.1.

Let $K=\mathbb{Q}(\sqrt{d})$ with $d$ a square-free integer, then

$\mathcal{O}_{K}=\begin{cases}\mathbb{Z}[\frac{1+\sqrt{d}}{2}]{\ \text{ if }\
d\equiv 1\pmod{4},}\\\ \mathbb{Z}[\sqrt{d}]~{}~{}{\ \text{ if }\ d\equiv
2,3\pmod{4}.}\end{cases}$

We study the solutions of a family of elliptic curves defined over
$\mathbb{Q}$ in the ring of integers of a quadratic field.

Let $K=\mathbb{Q}(\sqrt{d})$ and $s^{\prime}$ be the conjugate of an element
$s\in K$ over $\mathbb{Q}$. Further $R=\mathcal{O}_{K}[S^{-1}]$, where $S$ is
some finite set of primes in $\mathcal{O}_{K}$. Thus
$\mathcal{O}_{K}\subset\mathcal{O}_{K}[S^{-1}]\subset K$.
Laska [7] considered the equation (for $r\in\mathbb{Z}$ and $r\neq 0$)

$\Gamma_{r}:~{}y^{2}=x^{3}-r,$

which defines the Weierstrass form of an elliptic curve (call it $E_{r}$) over
$\mathbb{Q}$. For an elliptic curve $E$ over $\mathbb{Q}$ the “trace map”

$\sigma:E(K)\longrightarrow E(\mathbb{Q})$

is given by

$\sigma(\mathcal{P})=\mathcal{P}\oplus\mathcal{P}^{\prime}.$

Here $\mathcal{P}^{\prime}$ is the conjugate of the element $\mathcal{P}\in
E(K)$ arising from the conjugation in $K$ and $\oplus$ is the usual elliptic
curve addition. Laska considered the “trace map” for $\Gamma_{r}$ and calls it

$\sigma_{r,R}:\Gamma_{r}(R)\rightarrow\Gamma_{r}(\mathbb{Q})\cup\\{\mathcal{O}\\}.$

If $r,R$ are fixed we simply write $\sigma$ instead of $\sigma_{r,R}$. The aim
was to study $\sigma^{-1}(P)$ for a given
$P\in\Gamma_{r}(\mathbb{Q})\cup\\{\mathcal{O}\\}$. He divided this inverse
image set into two parts and called them ‘exceptional’ and ‘non-exceptional’
respectively. These two sets consist of:
$\bullet$ an element $\mathcal{P}=(s,t)\in\Gamma_{r}(R)$ with $s\neq
s^{\prime}$ is called an exceptional solution of $\Gamma_{r}$ in $R$,
$\bullet$ non-exceptional solutions of $\Gamma_{r}$ in $R$ are contained in
$\sigma^{-1}(P)$ for a given
$P\in\Gamma_{r}(\mathbb{Q})\cup\\{\mathcal{O}\\}$.

If $\mathcal{P}=(s,t)\in\Gamma_{r}(R)$ is a non-exceptional solution then
$s=s^{\prime}$ and from the Weierstrass equation that implies $t=\pm
t^{\prime}$. Thus if $P=\mathcal{O}$, then all candidates in $\sigma^{-1}(P)$
are non-exceptional. More precisely, one can show that,

$\displaystyle\sigma^{-1}(\mathcal{O})=\\{(z^{-1}p,z^{-2}q\theta):(p,q)\in\Gamma_{z^{3}r}(R\cap\mathbb{Q}),$
$\displaystyle\hskip 56.9055ptp\in z(R\cap\mathbb{Q}),q\in
z^{2}(R\cap\mathbb{Q})\\}$ (2.1)

where for simplicity it is assumed that $\theta^{-1}\in R$.

If $P\neq\mathcal{O}$, then the non-exceptional solutions $\mathcal{P}$
contained in $\sigma^{-1}(P)$ are exactly given by the conditions

$\mathcal{P}\in\Gamma_{r}(R\cap\mathbb{Q}),\mathcal{P}\oplus\mathcal{P}=P.$

Thus the non-exceptional solutions of $\Gamma_{r}$ in $R$ which are contained
in $\sigma^{-1}(P)$ are obtained by solving the equation $\Gamma_{z^{3}r}$
respectively $\Gamma_{r}$ in $R\cap\mathbb{Q}$. Hence either
$\mathcal{P}\in\Gamma_{r}(\mathbb{Q})$ or
$\mathcal{P}\oplus\bar{\mathcal{P}}=\mathcal{O}$.

Here we substitute $\Gamma_{r}$ by $E_{297}$ and $R$ by $\mathcal{O}_{K}$ to
study the solutions of (1.2) in $\mathcal{O}_{K}$ by pulling back the elements
of $E_{297}(\mathbb{Q})$ using the above mentioned technique.

##  3\. Proof of Theorem 1.1

###### Proof.

Let us first transform the system (1.2) to the desired Weierstrass form
$E_{297}$. The system is

$r+s+t=rst=2.$

Putting the value of $t$ upon simplification it becomes

$s+r+\frac{2}{rs}=2.$

Now substituting $r=-2/x$ and $s=-y/x$ in the last equation give

$y^{2}+2y+2xy=x^{3}.$ (3.1)

Putting $x=x_{1}$ \- $1/2$ and $y=\frac{y_{1}}{2}-x_{1}-\frac{1}{2}$ in (3.1)
rids it from the $xy$ term

$y_{1}^{2}=4x_{1}^{3}-2x_{1}^{2}+7x_{1}+\frac{1}{2}.$ (3.2)

Further substituting $x_{1}=x_{2}+\frac{1}{6}$ and $y_{1}=y_{2}$ rids (3.2)
the $x_{1}^{2}$ term

$y_{2}^{2}=4x_{2}^{3}+\frac{20}{3}x_{2}+\frac{44}{27}.$ (3.3)

Now again putting $y_{2}=Y_{1}/27$ and $x_{2}=X_{1}/9$ give

$Y_{1}^{2}=4X_{1}^{3}+540X_{1}+1188.$ (3.4)

Finally substituting $Y_{1}=2Y$ and $X_{1}=X$ get us the required Weierstrass
form

$E_{297}:Y^{2}=X^{3}+135X+297.$ (3.5)

Here with the help of SAGE [9] we can conclude that

$E_{297}(\mathbb{Q})_{tors}\cong\mathbb{Z}_{3}=\left\\{{\mathcal{O},(3,\pm{27})}\right\\}$

and the $\mathbb{Q}$-rank of $E_{297}$ is zero (we give a mathematical proof
of this fact in §4).

Thus $\mathcal{O}$ and $(3,\pm{27})$ are the only $\mathbb{Q}$ rational points
of (3.5). It is not difficult to see that the inverse transformation

$r=18/(3-X)~{}~{}\mbox{and}~{}~{}s=(Y-3X-18)/(3(3-X))$

allows us to pass from (3.5) to (1.2). Before proceeding to final leg of the
proof of Theorem 1.1, we make a couple of claims and give their proofs.

Claim 1: One of the $(r,s,t)$ satisfying (1.2) must belong to $\mathbb{Q}$.

###### Proof.

(Proof of claim 1) Two cases needed to be considered.

Case I: $\mathcal{P}$ is non-exceptional. In this case either $\mathcal{P}\in
E_{297}(\mathbb{Q})$ or $\mathcal{P}\oplus\bar{\mathcal{P}}=\mathcal{O}$. If
$\mathcal{P}\in E_{297}(\mathbb{Q})$ then this implies $r\in\mathbb{Q}$.

Let $\mathcal{P}\oplus\bar{\mathcal{P}}=\mathcal{O}$ and
$\mathcal{P}=(a+b\sqrt{d},k+l\sqrt{d})$. As $\mathcal{P}$ is non-exceptional,
$b=0$ and $k=0$. Thus $\mathcal{P}=(a,l\sqrt{d})$ and in this case too

$r=18/(3-a)\in\mathbb{Q}.$

Case II: Now if $\mathcal{P}$ is exceptional then
$\mathcal{P}+\bar{\mathcal{P}}=(3,\pm 27)$. The curve (3.5) has exactly three
elements over $\mathbb{Q}$ and the non-trivial elements are of order $3$. Lets
call them $\mathcal{O},\Omega$ and $2\Omega$. If
$\mathcal{P}+\bar{\mathcal{P}}=\Omega$, then clearly $\mathcal{P}+\Omega$ is
non-exceptional, since

$(\overline{\mathcal{P}+\Omega})+\mathcal{P}+\Omega=\overline{\mathcal{P}}+\mathcal{P}+2\Omega=3\Omega=\mathcal{O}.$

Now if $\mathcal{P}+\overline{\mathcal{P}}=2\Omega$, as before we can show
that $\mathcal{P}+2\Omega$ is also non-exceptional. As the claim is valid for
non-exceptional elements, it is true for $\mathcal{P}+\Omega$ and
$\mathcal{P}+2\Omega$. Hence it is true for $\mathcal{P}$ itself. ∎

Hence without loss of generality we assume that $r\in\mathbb{Q}$.

Claim 2 The only possibilities for $r$ satisfying the system

$r+s+t=rst=2$

in $\mathcal{O}_{K}$ are $\pm 1,2$ and $-8$.

###### Proof.

(Proof of claim 2) As $r\in\mathbb{Q}$ and we are looking for solutions in
$\mathcal{O}_{K}$, this would imply that $r\in\mathbb{Z}$. Three possibilities
needed to be considered:

  * •

If $r=\pm 1$. In this case solutions exist.

  * •

If $r$ is odd. In this case the denominator of (1.4) will be multiple of an
odd number but in $\mathcal{O}_{K}$ denominator is only $2$ or $1$ by Theorem
2.1. So in this case there do not exist any solution in $\mathcal{O}_{K}$.

  * •

If $r$ is even. In this case save for $r=2$ and $-8$, the denominator of (1.4)
will be multiple of $2$ and in some other cases denominator is $1$ but
$d\equiv 1\mod 4$. Thus again by Theorem 2.1, in this case too we don’t
(except for $r=2,-8$) get any solution in $\mathcal{O}_{K}$.

Thus except these values of $r=\pm 1,2$ and $-8$, this system of equation is
not solvable in the ring of integers of other quadratic fields.

∎

We are now in a position to complete the proof of Theorem 1.1. We deal with
all the four possibilities of $r$ separately.

When $r=1$, from (3) we have $s+t=1$ and $st=2$. Thus we get

$(s,t)=(\frac{1-\sqrt{-7}}{2},\frac{1+\sqrt{-7}}{2})~{}\mbox{and}~{}(\frac{1+\sqrt{-7}}{2},\frac{1-\sqrt{-7}}{2}).$

When $r=-1$, we have $s+t=3$ and $st=-2$. In this case

$(s,t)=(\frac{3-\sqrt{17}}{2},\frac{3+\sqrt{17}}{2})~{}\mbox{and}~{}(\frac{3+\sqrt{17}}{2},\frac{3-\sqrt{17}}{2}).$

Similarly when $r=2$ and $-8$, we get

$(s,t)=(i,-i)(-i,i),(\frac{10+\sqrt{101}}{2},\frac{10-\sqrt{101}}{2})$
$\mbox{and}~{}(\frac{10-\sqrt{101}}{2},\frac{10+\sqrt{101}}{2}).$

To conclude

$(1,\frac{1-\sqrt{-7}}{2},\frac{1+\sqrt{-7}}{2}),(1,\frac{1+\sqrt{-7}}{2},\frac{1-\sqrt{-7}}{2}),$
$(-1,\frac{3-\sqrt{17}}{2},\frac{3+\sqrt{17}}{2}),(-1,\frac{3+\sqrt{17}}{2},\frac{3-\sqrt{17}}{2})$
$(-8,\frac{10+\sqrt{101}}{2},\frac{10-\sqrt{101}}{2})(-8,\frac{10-\sqrt{101}}{2},\frac{10+\sqrt{101}}{2})$
$(2,i,-i)~{}\mbox{and}~{}(2,-i,i).$

are the only solutions of (3) in $\mathcal{O}_{K}$. ∎

##  4\. Rank of $E_{297}$

In this section we discuss the rank of $E_{297}$ over $\mathbb{Q}$ and over
$\mathbb{Q}(\sqrt{d})$ for $d=-7,-1,17$ and $101$.

######  Lemma 4.1.

The rank of $E_{297}(\mathbb{Q})$ is zero.

###### Proof.

If possible let rank $E_{297}(\mathbb{Q})\neq 0$. This would imply that (3.5)
have rational solutions. Let $(X,Y)=(\frac{x_{1}}{x_{2}},\frac{y_{1}}{y_{2}})$
with $(x_{1},x_{2})=(y_{1},y_{2})=1$, be one such solution. Now putting the
values of $X$ and $Y$ in (3.5), we obtain

$y_{1}^{2}x_{2}^{3}=x_{1}^{3}y_{2}^{2}+135x_{1}y_{2}^{2}x_{2}^{2}+297y_{2}^{2}x_{2}^{3}.$
(4.1)

Let $p$ be a prime such that $y_{2}=p^{\alpha}y_{22}$ where
$\alpha\in\mathbb{Z},\alpha\geq 1$ and $(p,y_{22})=1$. Putting the value of
$y_{2}$ in (4.1), gives that right hand side of (4.1) is divisible by $p$.
Therefore $p\mid y_{1}x_{2}.$ Which implies either $p\mid y_{1}$ or $p\mid
x_{2}$. Suppose $p\mid y_{1}$ then since $y_{2}=p^{\alpha}y_{22}$, we get a
contradiction to the fact that $(y_{1},y_{2})$ = $1$. Thus $p\mid x_{2}$ and
we write $x_{2}=p^{\beta}x_{22}$ where $\beta\in\mathbb{Z},\beta\geq 1$ and
$(p,x_{22})=1$. Now putting this $y_{2}$ and $x_{2}$ in (4.1), gives

$p^{3\beta}x_{22}^{3}y_{1}^{2}=p^{2\alpha}y_{22}^{2}x_{1}^{3}+135x_{1}p^{2\alpha+2\beta}y_{22}^{2}x_{22}^{2}+297p^{2\alpha+3\beta}y_{22}^{2}x_{22}^{3}.$
(4.2)

Three cases can occur.
Case I. Suppose $3\beta>2\alpha$. In this case,

$p^{3\beta-2\alpha}x_{22}^{3}y_{1}^{2}=y_{22}^{2}x_{1}^{3}+135x_{1}p^{2\beta}y_{22}^{2}x_{22}^{2}+297p^{3\beta}y_{22}^{2}x_{22}^{3}.$
(4.3)

Further from (4.3)

$p\mid y_{22}x_{1}$

and forces $p\mid x_{1}$ as $(p,y_{22})=1$, which is a contradiction to the
fact that $(x_{1},x_{2})=1$.
Case II. Suppose $3\beta>2\alpha$. In that case $p\mid y_{1}$, which is also
contradiction to the fact that $(y_{1},y_{2})=1$.
Case III. This case is analogous to Case I.

Hence (4.3) has no solution in $\mathbb{Z}$ and that in turn implies (3.5) has
no solution in $\mathbb{Q}$. Thus the rank of $E_{297}(\mathbb{Q})$ defined by
the equation (3.5) is zero. ∎

######  Lemma 4.2.

The rank of $E_{297}(\mathbb{Q}(\sqrt{d}))$ for $d=-7,17$is $1$ and for
$d=101$ it is $2$.

###### Proof.

Let $E/K$ be an elliptic curve and $d\in K^{*}$ be such that $L=K(\sqrt{d})$
is a quadratic extension. Let $E_{d}/K$ be the twist of $E/K$, then by [5],

$\text{rank}~{}E(L)=\text{rank}~{}E(K)+\text{rank}~{}E_{d}(K).$ (4.4)

We conclude using SAGE [9] that (already we have noted that rank
$E_{297}(\mathbb{Q})=0$) rank $E_{-7\cdot 297}(\mathbb{Q})=1$, rank
$E_{-1\cdot 297}(\mathbb{Q})=1$, rank $E_{17\cdot 297}(\mathbb{Q})=1$ and rank
$E_{101\cdot 297}(\mathbb{Q})=2$. Now using (4.4) we have,

$\text{rank}~{}(E(\mathbb{Q}(\sqrt{-7}))=\text{rank}~{}(E(\mathbb{Q}(\sqrt{-1}))=\text{rank}~{}(E(\mathbb{Q}(\sqrt{17}))=1.$
$\mbox{and}~{}\text{rank}~{}(E(\mathbb{Q}(\sqrt{101}))=2$

∎

##  5\. Concluding remarks

We showed that the system

$r+s+t=rst=2$

has no solution in $\mathcal{O}_{K}$ except for $d=-7,-1,17$ and $101$ and the
solutions are explicitly given.

It would of interest to consider the next case, i.e,

$r+s+t=rst=3.$ (5.1)

Suitable change of variables transform (5.1) to an elliptic curve with
Weierstrass form

$E_{13122}:y^{2}=x^{3}+3645x-13122.$

Torsion of $E_{13122}$ over $\mathbb{Q}$ is isomorphic to $\mathbb{Z}_{3}$ and
it’s rank is zero (using SAGE [9]).

We can conclude that except for $d=-2,-1,7,10$ and $13$, the system (5.1) has
no solutions in the ring of integers of $K$ and the solutions can be
explicitly given (following analogous argument). Analogously other individual
cases can be treated.

##  6\. Acknowledgement

A part of this manuscript was completed while the author was visiting Prof.
Sanoli Gun at the Institute of Mathematical Sciences (IMSc), Chennai. She is
grateful to Prof. Sanoli for her support and encouragement. It is a pleasure
to Prof. Michel Waldschmidt for his valuable comments and suggestions during
the visit to IMSc. Last but not the least, the serene ambience of Kerala
School of Mathematics (KSoM) is a constant energy booster and is of great help
to research.

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