# Bond order via cavity-mediated interactions

Titas Chanda  The Abdus Salam International Centre for Theoretical Physics
(ICTP), Strada Costiera 11, 34151 Trieste, Italy  Institute of Theoretical
Physics, Jagiellonian University in Krakow, Łojasiewicza 11, 30-348 Kraków,
Poland  Rebecca Kraus  Theoretical Physics, Department of Physics, Saarland
University, 66123 Saarbrücken, Germany  Jakub Zakrzewski  Institute of
Theoretical Physics, Jagiellonian University in Krakow, Łojasiewicza 11,
30-348 Kraków, Poland  Mark Kac Complex Systems Research Center, Jagiellonian
University in Kraków, Łojasiewicza 11, 30-348 Kraków, Poland  Giovanna Morigi
Theoretical Physics, Department of Physics, Saarland University, 66123
Saarbrücken, Germany

###### Abstract

We numerically study the phase diagram of bosons tightly trapped in the lowest
band of an optical lattice and dispersively coupled to a single-mode cavity
field. The dynamics is encompassed by an extended Bose-Hubbard model. Here,
the cavity-mediated interactions are described by a two-body potential term
with a global range and by a correlated tunneling term where the hopping
amplitude depends on a global observable. We determine the ground state
properties in one dimension by means of the density matrix renormalization
group algorithm, focusing on the effects due to the correlated tunneling. The
latter is responsible for the onset of bond orders, manifesting in one
insulating and two gapless bond ordered phases. We discuss the resulting
phases for different geometries that correspond to different relative
strengths of the correlated tunneling coefficient. We finally analyze the
scaling of the entanglement entropy in the gapless bond ordered phases that
appear entirely due to global interactions and determine the corresponding
central charges.

##  I Introduction

Ultracold atomic gases in optical lattices realize the strongly-correlated
dynamics of the Hubbard model with tunable interactions Fisher  _et al._
(1989); Jaksch  _et al._ (1998); Greiner  _et al._ (2002); Bloch  _et al._
(2008); Lewenstein  _et al._ (2012). The dispersive coupling with a high-
finesse resonator, additionally, allows one to design interactions whose range
can be tailored and whose strength can be tuned Gopalakrishnan  _et al._
(2011); Periwal  _et al._ (2021). One prominent example is the all-to-all
interaction in a quantum gas of bosons realized by coupling an electric dipole
transition with a single-mode resonator Landig  _et al._ (2016). Here, the
appearance of phases with density modulations was observed by tuning the
effective strength of the coupling with the cavity. These patterns support
coherent scattering into the cavity mode Landig  _et al._ (2016) and can be
either superfluid or incompressible. The experimentally measured phases are
captured by the ground state of an extended Bose-Hubbard model with global
interactions, as shown in Dogra  _et al._ (2016); Niederle  _et al._ (2016);
Flottat  _et al._ (2017); Himbert  _et al._ (2019). This model describes the
effect of the cavity-mediated potential by means of an interaction term
between pair of sites that depends on the onsite density and has a global
range.

Ab-initio derivations of the Bose-Hubbard model show, however, that two-body
interactions give rise to further terms describing correlated tunneling
Fernández-Vidal  _et al._ (2010); Maik  _et al._ (2013); Dutta  _et al._
(2015); Elliott and Mekhov (2016); Caballero-Benitez and Mekhov (2016); Chanda
_et al._ (2021). These contributions can be important in determining the phase
of superconductors Strack and Vollhardt (1993); Hirsch (1994); Amadon and
Hirsch (1996), frustrated quantum magnets Schmidt  _et al._ (2008, 2006), and
dipolar gases Sowiński  _et al._ (2012); Maik  _et al._ (2013); Dutta  _et
al._ (2015); Biedroń  _et al._ (2018); Kraus  _et al._ (2020). They can
interfere with single-particle hopping Kraus  _et al._ (2020); Suthar  _et
al._ (2020); Chanda  _et al._ (2021) and, in the cavity quantum
electrodynamics (QED) setup, give rise to an effective periodic modulation of
the bonds. Indeed, at half filling this interference is at the basis of the
emergence of self-organized topological insulating phases Chanda  _et al._
(2021).

In this work, we characterize the quantum ground state of the extended Bose-
Hubbard model of cavity QED taking into account both the cavity-induced
density-density and correlated hopping terms for different lattice geometries.
We consider a one-dimensional lattice and determine the quantum phases for
half and unit fillings using the density matrix renormalization group (DMRG)
algorithm White (1992, 1993); Schollwöck (2011); Orús (2014) extended to the
case of global interactions Chanda  _et al._ (2020, 2021). We show that
correlated tunneling gives rise to a bond order that can be either supersolid,
superfluid, or insulating. We analyze, in particular, the scaling of the
entanglement entropy at the bond superfluid and bond supersolid phases. Our
analysis complements and extends the study of Chanda  _et al._ (2021) by
investigating the phase diagram for generic geometries. These geometries were
also considered in Caballero-Benitez and Mekhov (2016) where the ground state
of atoms in small chains using exact diagonalization was discussed.

The paper is organized as follows. In Sec. II we review the extended Bose-
Hubbard model of cavity QED and discuss the dependence of its coefficients on
the cavity system parameters. In Sec. III, we present the ground state phase
diagrams calculated by means of the DMRG approach and analyze the nature of
the phase transitions. We then perform the scaling analysis of the
entanglement entropy in the gapless bond ordered phases that are stabilized
entirely by the cavity-mediated global terms. The conclusions are drawn in
Sec. IV. The Appendix provides details on the determination of the
coefficients of the extended Bose-Hubbard model.

##  II Extended Bose-Hubbard model with cavity-mediated interactions

The system we consider consists of $N$ bosons tightly bound in the lowest band
of an one dimensional optical lattice with $L$ sites. Let $\hat{a}_{i}$ and
$\hat{a}_{i}^{\dagger}$ denote the bosonic operators destroying and creating,
respectively, a boson at site $i=1,\ldots,L$ with
$\left[\hat{a}_{i},\hat{a}^{\dagger}_{j}\right]=\delta_{i,j}$, and
$\hat{n}_{i}=\hat{a}_{i}^{\dagger}\hat{a}_{i}$ being the corresponding
particle number operator. The Hamiltonian $\hat{H}_{\text{EBH}}$ determining
their dynamics is the one of the extended Bose-Hubbard model Habibian  _et
al._ (2013a); Caballero-Benitez and Mekhov (2016):

$\hat{H}_{\text{EBH}}=\hat{H}_{\text{BH}}+\hat{H}_{\text{BH}}^{C}\,,$ (1)

which is the sum of the standard Bose-Hubbard Hamiltonian
$\hat{H}_{\text{BH}}$,

$\displaystyle\hat{H}_{\text{BH}}=-t\sum_{j}\left(\hat{a}^{\dagger}_{j}\hat{a}_{j+1}+\text{H.c.}\right)+\frac{U}{2}\sum_{j}\hat{n}_{j}(\hat{n}_{j}-1)\,,$
(2)

and of the terms containing the cavity-mediated interactions Fernández-Vidal
_et al._ (2010); Caballero-Benitez and Mekhov (2016):

$\displaystyle\hat{H}_{\text{BH}}^{C}\,=\frac{U_{1}}{L}\left(z^{2}\hat{D}^{2}+zy\left(\hat{D}\hat{B}+\hat{B}\hat{D}\right)+y^{2}\hat{B}^{2}\right)\,.$
(3)

The details of the derivation of Hamiltonian (3) follow from Habibian  _et
al._ (2013a, b); Niederle  _et al._ (2016) and are reported in Chanda  _et
al._ (2021) (see also the Appendix A). The coefficients in Eq. (2) are
positive and are the nearest-neighbor hopping rate $t$ and the strength of the
onsite repulsion $U$. The coefficient $U_{1}$ scaling the cavity term in Eq.
(3) can be either positive or negative, the factor $1/L$ warrants extensivity
Fernández-Vidal  _et al._ (2010); Habibian  _et al._ (2013a). Operators
$\hat{D}$ and $\hat{B}$ depend on the onsite densities and hoppings,
respectively Caballero-Benitez and Mekhov (2016); Chanda  _et al._ (2021):

$\displaystyle\hat{D}=\sum_{j}(-1)^{j}\hat{n}_{j}\,,$ (4)
$\displaystyle\hat{B}=\sum_{i}(-1)^{i}\left(\hat{a}^{\dagger}_{i+1}\hat{a}_{i}+{\rm
H.c.}\right)\,,$ (5)

where the staggered sum emerges when the cavity wavelength is twice the
lattice periodicity Habibian  _et al._ (2013a); Niederle  _et al._ (2016). The
coefficients $y$ and $z$ are dimensionless parameters whose strength depends
on the setup’s geometry and are discussed in the following.

We note that the term $\hat{D}^{2}$ is a global density-density interaction,
that promotes the onset of a population imbalance between even sites (with
$j=2n$) and odd sites (with $j=2n+1$) Landig  _et al._ (2016); Dogra  _et al._
(2016); Niederle  _et al._ (2016). The two other terms,
$\hat{D}\hat{B}+\hat{B}\hat{D}$ and $\hat{B}^{2}$, describe correlated
tunneling processes induced by the cavity field.

###  II.1 Bose-Hubbard coefficients

The coefficients in Eq. (1) are numerically calculated from the overlap
integrals using the Wannier functions of the lowest band of the static optical
lattice. In our calculations they are varied taking into account that in the
experiment the control parameters are the depth of the optical lattice, the
$s$-wave scattering length, and the cavity interaction amplitude and its sign.
When we sweep across the phase diagram, we keep fixed the lattice depth at the
value $V_{0}=4E_{R}$, where $E_{R}=\hbar^{2}k^{2}/(2m)$ is the recoil energy
for atoms of mass $m$ and $k$ the lattice and cavity wave number. Therefore,
in our numerical calculations the tunneling coefficient is kept constant. The
ratio $t/U$ is varied by tuning $U$ via the $s$-wave scattering length. The
sign of the detuning between cavity and driving laser determines the sign of
the coefficient $U_{1}$. Moreover, the detuning and the strength of the cavity
field determine the magnitude of $|U_{1}|$Baumann  _et al._ (2010); Fernández-
Vidal  _et al._ (2010); Habibian  _et al._ (2013a, b). Thus, in our
calculations the ratio $U_{1}/U$ and $t/U$ are varied by simultaneously
changing $U_{1}$ and $U$.

The coefficients $y$ and $z$ in Eq. (3) are overlap integrals between the
Wannier functions $w_{j}(x)$ and the cavity mode function $\cos(kx+\phi)$ (see
Appendix A):

$\displaystyle z=\int_{0}^{aL}dx$ $\displaystyle\ w_{j}(x)^{2}\cos(kx+\phi)$
$\displaystyle y=\int_{0}^{aL}dx$ $\displaystyle\
w_{j}(x)w_{j+1}(x)\cos(kx+\phi)\ ,$ (6)

where $j$ denotes the lattice site about which the Wannier function is
centered, $a$ is the lattice periodicity, $a=\pi/k$, and $\phi$ is the phase
shift between the lattice and the cavity standing wave. The phase shift
$\phi=0$ and $\phi=\pi/2$ correspond to trapping the atoms at the antinodes
and at the nodes, respectively, of the cavity standing wave. This is realized
by either tuning the laser on the blue or on the red side of the cavity
resonance. In this paper we will also consider the case $\phi=\pi/4$, for
which both $y$ and $z$ are different from zero. We remark that $y\leq 0$ for
the parameter regimes we inspect.

Figure 1 displays the coefficients $z$ and $|y|$ as a function of $V_{0}$ for
the three different phase shifts $\phi=0,\pi/4,\pi/2$ considered in this
paper. We also display the tunneling rate $t$ and the on-site coefficient $U$
for comparison, keeping in mind that these quantities are independent of
$\phi$. The upper panel shows the parameters for $\phi=0$, where the $y$
coefficient vanishes within machine precision. For $\phi=\pi/4$ both $y$ and
$z$ are finite: the $z$ coefficient is almost independent of the lattice depth
$V_{0}$, while $|y|$ decreases monotonically with $V_{0}$. For $\phi=\pi/2$
(see lower panel) the $z$ coefficient is zero within machine precision. The
vertical dashed line in Fig. 1 indicates the value of $V_{0}$ considered in
this work. For $\phi=0$ the value of the overlap integrals are given by
$z=0.8279$ and $y=0$, thus we neglect the correlated tunneling. For
$\phi=\pi/2$ we find $y=-0.0658$ and $z=0$. In this case the cavity-mediated
interactions are solely described by the term proportional to $\hat{B}^{2}$.
Setting $\phi=\pi/4$ the overlap integrals are $z=0.5854$ and $y=-0.0465$ and
we expect to observe an interplay between the density-density potential and
the correlated tunneling.

Figure 1: (Color online.) The coefficients $z$, $|y|$, $t/E_{R}$ and
$U/(gE_{R})$ as a function of the lattice depth $V_{0}$, in units of the
recoil energy $E_{R}$, for $\phi=0$ (upper panel), $\phi=\pi/4$ (middle
panel), and $\phi=\pi/2$ (lower panel). The vertical dashed line corresponds
to the lattice depth of $V_{0}=4E_{R}$. The coefficient $g$ scaling the onsite
repulsion coefficient contains the physical variables including the scattering
length, therefore $U/(gE_{R})$ is the overlap integral solely depending on the
Wannier functions, which in turn are determined by $V_{0}$.

###  II.2 Observables

The ground state of Hamiltonian (1) is determined in one dimension and for a
fixed number of bosons on a finite lattice $L$ with open boundaries. The
numerical program we use is based on the DMRG algorithm, we refer the readers
to Chanda  _et al._ (2021) and to Appendix B for details on its
implementation. In what follows we introduce and describe the observables we
use in order to identify the quantum phases.

We identify superfluidity by a non-vanishing value of the single-particle
correlations. In turn, a phase is incompressible (insulator) when the single-
particle correlations vanish and there is a finite energy gap between the
ground and the excited states in the thermodynamic limit. In order to gain
information on the properties of the superfluid phase, we analyze the Fourier
transform of the single-particle correlations, the so-called single-particle
structure form factor, that is defined as Kraus  _et al._ (2020):

$\displaystyle
M_{1}(k)=\frac{1}{L^{2}}\sum_{i,j}e^{ik(i-j)}\left\langle\hat{a}_{i}^{\dagger}\hat{a}_{j}\right\rangle\,,$
(7)

where $\langle\cdot\rangle$ denotes the expectation value over the ground
state. This quantity can be experimentally revealed by means of time-of-flight
measurements Greiner  _et al._ (2002). Depending on the value of $k$ at which
$|M_{1}(k)|$ reveals a maximum, off-diagonal order can exhibit modulations in
the phase. In the absence of the cavity, the ground-state superfluid is
spatially homogeneous and characterized by a non-vanishing value of $M_{1}(k)$
at $k=0$.

The superfluid (SF) phase acquires further features in the presence of the
cavity field, depending on whether the expectation values
$\langle\hat{D}\rangle$ and/or $\langle\hat{B}\rangle$ (compare Eqs. (4) and
(5)) are different from zero in the thermodynamic limit. For this purpose we
identify the order parameters

$\displaystyle\mathcal{O}_{D}=\frac{1}{L}\left|\left\langle\hat{D}\right\rangle\right|,$
(8)
$\displaystyle\mathcal{O}_{B}=\frac{1}{2L}\left|\braket{\hat{B}}\right|\,,$
(9)

which can be measured by detecting the light at the cavity output Larson  _et
al._ (2008); Landig  _et al._ (2016); Baumann  _et al._ (2011); Sierant  _et
al._ (2019). The order parameter $\mathcal{O}_{D}$ signals the onset of
density modulation (even-odd population imbalance), while $\mathcal{O}_{B}$
signals the formation of dimers along the lattice Caballero-Benitez and Mekhov
(2016), namely, a so-called dimerized or bond ordered state Affleck  _et al._
(1987); Jürgensen and Lühmann (2014). In addition to the “normal” SF, the
emerging SF phases can be lattice Supersolid (SS) in the presence of diagonal
long-range order ($\mathcal{O}_{D}\neq 0$ and $\mathcal{O}_{B}=0$); Bond SF
(BSF) for homogeneous density and bond order ($\mathcal{O}_{D}=0$ and
$\mathcal{O}_{B}\neq 0$), or Bond Supersolid (BSS) when both order parameters
are non-vanishing. The phases and the corresponding order parameters are
summarized in Table 1. The onsets of these gapless phases, i.e., SS, BSF and
the BSS, occur due to a spontaneous breaking of a discrete $\mathbb{Z}_{2}$
lattice translational symmetry. Such a spontaneous discrete symmetry breaking
is captured by the two-fold ground state degeneracy in these phases (Sec.
III.5).

The insulating phases, having vanishing $M_{1}(k)$ in the thermodynamic limit,
are classified according to the values of the population imbalance and of the
bond order parameters. The Bond Insulator (BI) is characterized by
$\mathcal{O}_{B}\neq 0$, the Charge-Density Wave (CDW) by $\mathcal{O}_{D}\neq
0$, while in the Mott Insulator (MI) all order parameters here discussed
vanish, see Table 1. Similarly to the SS, BSF, or BSS phases, the insulating
BI and CDW phases are also $\mathbb{Z}_{2}$ symmetry broken phases. We remark
that we have also determined the parity and string order parameters Dalla
Torre  _et al._ (2006); Rossini and Fazio (2012) in the resulting phases: for
the parameter regimes considered we do not find signatures of the Haldane
insulator (c.f., Chanda  _et al._ (2021)). This is consistent with other
numerical studies on globally interacting systems Sicks, Johannes and Rieger,
Heiko (2020).

Phase |  Acronyms |  $\mathcal{O}_{D}$ |  $\mathcal{O}_{B}$ |  max$|M_{1}(k)|$
---|---|---|---|---
Mott-Insulator |  MI |  $0$ |  $0$ |  $0$
Density Wave |  CDW |  $\neq 0$ |  $0$ |  $0$
Bond Insulator |  BI |  $0$ |  $\neq 0$ |  $0$
Superfluid |  SF |  $0$ |  $0$ |  $M_{1}(0)$
Supersolid |  SS |  $\neq 0$ |  $0$ |  $M_{1}(0)$
Bond Superfluid |  BSF |  $0$ |  $\neq 0$ |  $M_{1}(\pm\frac{\pi}{2})$
Bond Supersolid |  BSS |  $\neq 0$ |  $\neq 0$ |  $M_{1}(\pm\frac{\pi}{2})$
Table 1: Table of the quantum phases of the ground state of Eq. (1), of their
acronyms, and of the corresponding behavior of the order parameters.

##  III Ground-state phase diagram

We determine the phase diagrams for fixed densities as a function of the
ratios $U_{1}/U$ and $t/U$ that we vary as previously specified. We consider
in particular the densities $\rho=1/2$ and $\rho=1$ since they are
commensurate with the long-range potential, thus they can give rise to
insulating phases in addition to the gapless ones. In our model the ratio
$U_{1}/U$ controls the onset of structures that support the buildup of an
intra-cavity field, while $t/U$ determines the strength of quantum
fluctuations. We sweep the ratio $U_{1}/U$ from positive to negative values
for different $\phi$. Depending on $\phi$ we rescale $U_{1}$ by the maximum
between the coefficients $z^{2}$ and $y^{2}$ (i.e., $\max(z^{2},y^{2})$), thus
giving the effective strength of the cavity-induced interaction. The ground
state phase diagram is calculated by means of the DMRG algorithm with open
boundary conditions.

In the following we present the phase diagram for a finite system of size
$L=60$ sites and identify the transition lines when the corresponding order
parameter exceeds a threshold value: The line separating the incompressible
and the compressible phases is set at the threshold value
$\max|M_{1}(k)|=0.1$. Bond and density-wave order is signaled by
$\mathcal{O}_{D}>0.02$ and $\mathcal{O}_{B}>0.02$, respectively. We also
analyze the order parameters across different transitions for different
system-sizes $L\in[40,120]$ in order to determine the nature of the phase
transitions and to verify the stability of the phase diagram with the varying
system-size. We finally determine the central charge of the bond ordered
gapless phases, BSF and BSS, that are due to the global interactions.

###  III.1 Phase diagrams for $\phi=0$

Figure 2: (Color online.) Color plots of the phase diagrams in the plane
$U_{1}/U$ and $t/U$ for $\phi=0$ and density $\rho=1$ (upper row) and
$\rho=1/2$ (lower row). The subplots show the maximum of $|M_{1}(k)|$ (left
panels) and the density-wave order parameter (right panels), the phases are
identified according to Table 1. In the CDW phase two neighboring sites
forming a unit cell have occupations $n_{i}=2\rho$, $n_{i+1}=0$, where
$\rho=1/2,1$. Note that the interaction strength $U_{1}$ in the plots is
scaled by $z^{2}$. The phase diagram are obtained using DMRG on a lattice with
size $L=60$ and open boundaries. Here, $t/U$ is varied from $0.01$ to $1$ in a
logarithmic scale in $101$ steps, while $U_{1}/U$ is varied from $-20$ to $20$
in uniform steps of width $0.1$.

For $\phi=0$ the cavity-induced interactions in the extended Bose-Hubbard
Hamiltonian consist solely of global density-density interactions. These
interactions are periodic, with a periodicity that is twice the lattice
periodicity. The corresponding phase diagram has been extensively studied in
the literature for attractive interactions, corresponding to negative values
of $U_{1}$ Landig  _et al._ (2016); Dogra  _et al._ (2016); Niederle  _et al._
(2016); Caballero-Benitez and Mekhov (2016); Flottat  _et al._ (2017); Himbert
_et al._ (2019). In this case the cavity potential favors the formation of
ordered structures which support photon scattering into the cavity. For
$U_{1}$ positive, on the other hand, the cavity-induced potential is repulsive
and the energy is minimized for uniform densities, at which $\mathcal{O}_{D}$
vanishes.

Figure 2 displays the maximum of $|M_{1}(k)|$, signaling superfluidity, and
the density-wave order parameter $\mathcal{O}_{D}$. For positive $U_{1}$ and
for half-filling the ground state remains in a SF phase for the whole $t/U$
parameter range, while at unit density we find the MI-SF transition.
Interestingly, the transition line slightly depends on the value of $U_{1}$
and in particular is shifted to larger values of $t/U$ as $U_{1}$ increases:
the repulsive cavity interaction tends to stabilize the incompressible phase.

The situation is different for attractive global interactions ($U_{1}<0$).
Here, we identify the transition line $U_{1}^{(c)}$ separating the homogeneous
phase from the density wave, which is a monotonously increasing function of
$t/U$. The transition line qualitatively agrees with the one found by means of
a mean-field ansatz for a grand-canonical ensemble Himbert  _et al._ (2019):
At half-filling it vanishes at $t/U=0$, $U_{1}^{(c)}(0)=0$, while at unit
density $U_{1}^{(c)}(0)<0$. A direct transition between SF and the
incompressible CDW is found at half filling and for $0>U_{1}^{(c)}\gtrsim-U$,
while for $U_{1}^{(c)}\lesssim-U$ a SS phase separates SF from CDW. At unit
density there is no direct CDW-SF transition: The two phases are always
separated either by a MI or by a SS phase. We also note that the area covered
by the SS phase in parameter space is larger at unit density than at half-
filling.

Figure 3:  (Color online.) The order parameters, $\mathcal{O}_{D}$ and $|\max
M_{1}(k)|$ as a function of $U_{1}/U$ for $\phi=0$, fixed $t/U$ (as specified
in the panels), and different system-sizes $L\in[40,120]$ (see legend in
subplot (a)). The upper (lower) panels give the observables’ behavior at
density $\rho=1$ ($\rho=1/2$) across different phase transitions. The red
vertical lines indicate the value of $U_{1}/U$ at which we identify a phase
transition, the corresponding phases are reported. For $\rho=1$ (a) the MI-CDW
transition is first-order discontinuous, while (b) the SF-SS transition is
continuous and the CDW-SS transition is either continuous or a crossover. In
case of $\rho=1/2$ (c) the SF-CDW and (d) the SF-SS transitions are
continuous, while the SS-CDW transition could be either continuous or a
crossover.

By inspecting the behavior of the order parameters across different phase
transitions (see Fig. 3), we deduce the nature of the transitions.
Interestingly, for $\rho=1/2$, the transitions SF-SS and SF-CDW are now
characterized by a smooth change in the density-wave order parameter (lower
panels of Fig. 3), signaling that these transitions are continuous. The CDW-SS
transition, instead, is either continuous or a crossover. In the case of
$\rho=1$ (upper panels in Fig. 3), a jump in the order parameter
$\mathcal{O}_{D}$ signals a discountinuous transition between the MI and the
CDW phase.

###  III.2 Phase diagrams for $\phi=\pi/2$

Figure 4: (Color online.) Same as Fig. 2 but for $\phi=\pi/2$. Here, the right
column displays the bond order parameter, $\mathcal{O}_{B}$. Note that the
interaction strength $U_{1}$ in the plots is now rescaled by $y^{2}$ and we
vary $U_{1}/U$ in the range $[-400,400]$ in steps of size $4$ to generate the
data.

We now discuss the case in which the cavity-mediated interactions are
described by a global correlated hopping term. This configuration can be
realized experimentally when the atoms are tightly confined at the nodes of
the cavity field. In our model, this case corresponds to the choice
$\phi=\pi/2$ in the cavity standing wave, resulting in $z=0$ in Eq. (3).

Figure 4 displays the phase diagrams for density $\rho=1$ (upper row) and
$\rho=1/2$ (lower row). For $U_{1}>0$ the ground state at half-filling is SF.
At unit density, the MI-SF transition line is visibly shifted to smaller
values of $t/U$ as $U_{1}$ increases: the size of the incompressible phase is
reduced because the weight of quantum fluctuations at small $t/U$ is enhanced
due to the contribution of the global hopping. This trend is also visible in
the color plot of the maximum of $M_{1}(k)$ for $\rho=1/2$.

Figure 5: (Color online.) The order parameters, $\mathcal{O}_{D}$ and $|\max
M_{1}(k)|$ as a function of $U_{1}/U$ for $\phi=\pi/2$, fixed $t/U$ (as
specified in the panels), and different system-sizes $L\in[40,120]$ (see
legend in subplot (a)). The upper (lower) panels give the observables’
behavior at density $\rho=1$ ($\rho=1/2$) across different phase transitions.
The red vertical lines indicate the value of $U_{1}/U$ at which we identify a
phase transition, the corresponding phases are reported. For $\rho=1$ (a) the
MI-BSF transition is continuous. Similarly, (c) for $\rho=1/2$, the
transitions SF-BI and BI-BSF are continuous. The SF-BSF transitions, on the
other hand, are discontinuous for both (b) $\rho=1$ and (d) $\rho=1/2$.

For $U_{1}<0$ we observe a transition from SF to bond order. Similar to the
case $\phi=0$, also here the transition shifts to larger values of $t/U$ as
$|U_{1}|$ increases: the cavity-induced correlated hopping tends to stabilize
bond order, as expected. Remarkably, at $U_{1}<0$ we do not find
incompressible phases for $\rho=1$: in the considered parameter region the
bond ordered phase is BSF. Figure 5 displays the behavior of the order
parameters at the transition between the BSF and the MI phase: both $M_{1}(k)$
and $\mathcal{O_{B}}$ display a continuous behavior signaling a continuous
transition. On the other hand, at higher $t/U$, the transition between the SF
and the BSF phase is of first order kind as visible in the discontinuity of
$\max M_{1}(k)$ and $\mathcal{O_{B}}$

The phase diagrams at $\rho=1/2$ differ from the ones at $\rho=1$ due to the
appearance of an insulating phase with bond order separating the BSF from the
homogeneous phase. This phase is found for sufficiently small values of $t/U$.
In Chanda  _et al._ (2021) we showed that this is a topological insulator,
which shares several analogies with the Su-Schrieffer–Heeger model. Inspection
into the behavior of the order parameters show that the transition SF-BI and
BI-BSF is continuous Fig. 5), while the direct transition SF-BSF is
discontinuous (c.f. Chanda  _et al._ (2021)).

###  III.3 Phase diagrams for $\phi=\pi/4$

Figure 6: (Color online.) Color plots of the phase diagrams in the plane
$U_{1}/U$ and $t/U$ at $\phi=\pi/4$ and for density $\rho=1$ (top) and
$\rho=1/2$ (bottom). The subplots show the maximum of $|M_{1}(k)|$ (left
panels), the density-wave order parameter (central panels) and the bond order
parameter (right panels); the phases are labeled according to Table 1. The
phase diagrams are calculated using DMRG on a lattice with $L=60$ sites and
open boundary conditions, where we have varied $U_{1}/U$ from $-200$ to $200$
in steps of size $2$. Note that the interaction strength $U_{1}$ is scaled by
$z^{2}$ in all subplots.

We now discuss the ground state phase diagrams emerging from the interplay of
the density-density attractive potential and the correlated tunneling. We
choose $\phi=\pi/4$ for which both $z$ and $y$ in Eq. (3) are different from
zero, and recall that $z\sim 10|y|$, see Fig. 1. Therefore, the coefficient
scaling correlated tunneling is one order of magnitude smaller than the
coefficient scaling the cavity-induced potential term.

Figure 7:  (Color online.) The order parameters, $\mathcal{O}_{D}$ and $|\max
M_{1}(k)|$ as a function of $U_{1}/U$ for $\phi=\pi/4$, fixed $t/U$ (as
specified in the panels), and different system-sizes $L\in[40,120]$ (see
legend in subplot (a)). The upper (lower) panels give the observables’
behavior at density $\rho=1$ ($\rho=1/2$) across different phase transitions.
The red vertical lines indicate the value of $U_{1}/U$ at which we identify a
phase transition, the corresponding phases are reported. For $\rho=1$ the MI
to the CDW+BI transition (a) is first-order discontinuous, while SF-SS
transition (b) is continuous. The CDW+BI to the BSS (a) transition is either
continuous or a crossover. In case of $\rho=1/2$, the SF to the CDW+BI (c) and
the SF-SS (d) transitions are continuous, while the transition SS to CDW+BI
(d) is either continuous or a crossover.

Figure 6 displays the order parameters for SF, density-wave and bond order for
unit density and for half-filling. For $U_{1}>0$ the behavior we observe is
essentially the same as for $\phi=0$. Instead, for $U_{1}<0$ the phase diagram
becomes richer. In the first place, at small values of $t/U$ we observe a
transition from the homogeneous phases to an insulating phase with both
density-wave and bond order. This transition occurs for both density $\rho=1$
and $1/2$. The new phase is an insulator of dimers with population imbalance
within the dimer. We dub this phase CDW+BI since both order parameters
$\mathcal{O}_{D}$ and $\mathcal{O}_{B}$ are non-zero. As visible in Fig. 7,
the transition from the SF/SS to the CDW+BI at unit density is continuous,
while the MI to the CDW+BI transition is discontinuous. For $\rho=1/2$,
instead, all the transitions are continuous or crossovers. At larger tunneling
rates $t/U$ the CDW+BI phase undergoes a transition to a SS phase for both
$\rho=1$ and $1/2$. The SS phase is signaled by the non-zero values of the
maximum of $M_{1}(k)$ and by the vanishing value of the bond order parameter
$\mathcal{O}_{B}$. The density-wave order parameter, instead, stays finite
across the transition. We note that the transition CDW+BI to SS is continuous
(or a crossover) for both unit density and half filling (Fig. 7). The SS phase
is then separated from the SF phase by a continuous transition for both
$\rho=1$ and $1/2$ (analogously to the $\phi=0$ geometry).

We now discuss the phases encountered keeping $t/U$ fixed and tuning $U_{1}/U$
to larger values along the negative axis. At unit density we observe a
transition from CDW+BI to a compressible phase that has both density-wave and
bond order. This phase is a SS phase exhibiting dimers – the BSS phase. Hence,
at unit density and for low tunneling rates, large cavity-mediated
interactions promote superfluidity, which is at first sight seems
counterintuitive. Interestingly, for the parameter window we have considered,
we do not find a BSS phase at half filling. However, from the pattern of
$M_{1}(k)$, we suspect that the BSS will also appear at half filling but at
larger negative values of $U_{1}/U$.

###  III.4 Discussion

The phase diagrams for the three geometries have been also analyzed in Ref.
Caballero-Benitez and Mekhov (2016) using exact diagonalization and small
chains. In this work, we refrained from making a systematic comparison of our
predictions with the results of Caballero-Benitez and Mekhov (2016). In fact,
our results are qualitatively and quantitatively different in most regions of
the phase diagram. We believe that the discrepancy is mainly due to the very
small size considered in Caballero-Benitez and Mekhov (2016). To give few
examples, in Caballero-Benitez and Mekhov (2016) and at half filling the
authors reported insulating phases for $U_{1}>0$, while instead the phase we
find is always SF. Other phases, such as the SFD dimers (that would here
correspond to a sort of BSF phase) are reported in regions of the phase
diagram where the DMRG predicts different ground state phases. A Gutzwiller
mean-field analysis confirms the DMRG result and often finds that the SFD (or
BSF) phase in those regions is a metastable, excited state.

Remarkably, the phase diagram in Fig. 3 at $\rho=1/2$ is in qualitative
agreement with the mean-field predictions for a grand-canonical ensemble Dogra
_et al._ (2016); Himbert  _et al._ (2019), despite the fact that in the
present paper it has been determined using DMRG in one dimension. There are
instead qualitative differences when comparing the phase diagram at unit
density. For $\phi=\pi/2$ an analysis based on a Gutzwiller mean-field for a
canonical ensemble qualitatively reproduces the DMRG phase diagram for
$\rho=1$. It does not capture, however, the BI phase at half filling, see Ref.
Chanda  _et al._ (2021).

###  III.5 Symmetry-broken gapless phases

We have shown that the attractive cavity potential stabilizes inhomogeneous
gapless phases, the SS, the BSF, and the BSS phase. These phases have long-
range diagonal order, signaled by the non-vanishing order parameters
$\mathcal{O}_{D}$ and/or $\mathcal{O}_{B}$, and off-diagonal correlations that
decay algebraically with the distance. The long-range order in these three
gapless phases manifests itself due to a spontaneous breaking of discrete
$\mathbb{Z}_{2}$ lattice translational symmetry by doubling of the unit cell.
A way to verify such a spontaneous symmetry breaking, is by checking the
ground state degeneracy for finite system-sizes. For spontaneous
$\mathbb{Z}_{2}$ symmetry breaking, the energy gap between the ground state
and the first excited state must fall to zero much faster than $\sim 1/L$
(ideally, the gap should diminish exponentially in the system-size), while the
spectral gap, as measured by the gap between the ground state and the second
excited state should vanish as $\sim 1/L$ in a gapless phase.

Figure 8: (Color online.) The energy gaps in the SS, the BSF, and the BSS
phase at unit filling. In the left panel, we plot the energy gap between the
ground and the first excited states ($E_{1}-E_{0}$) as a function of the
system-size $L$ confirming the ground state degeneracy in these phases. The
right panel shows the gap between the ground and the second excited states
($E_{2}-E_{0}$) that decays to zero in the thermodynamic limit as $\sim 1/L$
confirming the gapless character of the phases. Here we consider an even
number of sites for the SS phase and an odd number of sites for the BSF phase
with open boundary condition, while for BSS phase we impose periodic boundary
condition with even number of sites.

In the SS phase ($O_{D}\neq 0$), the order develops in the sites by
$\mathbb{Z}_{2}$ symmetry breaking, and therefore the ground state must be
two-fold degenerate for even number of sites for large enough system-sizes. In
the left panel of Fig. 8 we plot the degenerate gap, i.e., the gap between the
ground and the first excited states ($E_{1}-E_{0}$). The gap reaches the
numerical precision $(\lesssim 10^{-10})$ set by DMRG simulations as we
increase the system-size. The right panel shows the gap between the ground and
the second excited states ($E_{2}-E_{0}$) as a function of $1/L$. It decays to
zero as $\sim 1/L$, confirming the gapless nature of the phase. The number of
sites shall be odd when analyzing the BSF phase ($O_{B}\neq 0$), because the
order develops in the bonds. Therefore, even number of bonds (and hence odd
number of sites) are needed to observe the ground state degeneracy. In Fig. 8
we verify the ground state degeneracy along with gapless nature of the BSF
phase for odd number of sites ($L=41,61,81,...$). In the BSS phase ($O_{D}$
and $O_{B}$ both are non-zero), the order develops in both sites and bonds.
Therefore, we need to impose periodic boundary condition with even number of
sites/bonds to observe the degeneracy. We find that, even for system-size
$L=8$ with periodic boundary condition, the gap in the BSS phase reaches the
numerical precision (Fig. 8), thus we expect that it vanishes in the
thermodynamic limit.

###  III.6 Scaling of the entanglement entropy in the gapless bond ordered
phases

Let us now analyze the two bond superfluid phases we find, namely the BSF for
$\phi=\pi/2$ and the BSS for $\phi=\pi/4$ and at unit density. We determine in
particular the scaling of the entanglement entropy with the system size. The
entanglement entropy of a lattice partition comprising the sites
$\ell=1,\ldots,j$ is defined as

$\mathcal{S}_{j}=-\text{Tr}\left[\rho_{j}\ln\rho_{j}\right],$ (10)

where $j$ denotes the bond that separates the system into two parts, and
$\rho_{j}=\text{Tr}_{j+1,j+2,...,L}\ket{\psi}\bra{\psi}$ is the reduced
density matrix obtained from the ground state $\ket{\psi}$ by tracing out the
degrees of freedom of the second partition. In a gapless critical system with
open boundary conditions the entanglement entropy scales with the size of the
block of consecutive sites according to Callan and Wilczek (1994); Vidal  _et
al._ (2003); Calabrese and Cardy (2004)

$\mathcal{S}_{j}=\frac{c}{6}\ln\left[\frac{2L}{\pi}\sin\left(\pi
j/L\right)\right]+b^{\prime},$ (11)

where $c$ is the central charge of the corresponding conformal field theory
(CFT) that describes the criticality and $b^{\prime}$ is a non-universal
constant. In Bose-Hubbard models with short-range interaction, the gapless
critical phases, e.g., SF and SS phases, obey the entropy scaling formula (11)
with $c=1$ that corresponds to the CFT of free compactified bosons described
by Tomonaga-Luttinger liquid theory Cazalilla  _et al._ (2011). However, the
fate of such an entropy scaling in the presence of infinite-range global
interactions is still an open question. This question is particularly
intriguing when considering the BSF and BSS phases in our study, since these
gapless phases are due to the infinite-range interactions.

Figure 9: (Color online.) The scaling of entanglement entropy in the BSF (top
panel) and the BSS (bottom panel) phases according to the formula in Eq, (11).
For both cases $t=0.5U$, the other parameters are given in the legends. The
numerical fits yield $c=1.00(2)$ and $c=0.99(5)$ respectively. Note that for
the BSF we get two distinct straight lines, both having a slope of $\sim 1/6$,
corresponding to the entanglement entropy when the partition is cut at a
strong and at a weak bond respectively. In the BSS the fluctuations of the
entropy with respect to the mean slope of $1/6$ is due to the modulations of
densities and bonds across the system.

Figure 9 displays the entanglement entropy as a function of the chord length
$\ln\left[\frac{2L}{\pi}\sin\left(\pi j/L\right)\right]$ in the BSF and the
BSS phases for system sizes $L=60,80,100,120$. Interestingly, although these
phases appear when the global correlated tunneling and the potential dominate
over the short-ranged counterparts, the entropy still shows a linear growth
with respect to the chord length and hence a logarithmic divergence with
respect to the system size. Moreover, the corresponding central charges are
$c\simeq 1$, as in the cases of SF and SS phases of the Bose-Hubbard model
with short-range interactions.

##  IV Conclusions

In this work we have presented a numerical analysis of the quantum phases of
bosons in an extended Bose-Hubbard model with global density-density
interactions and global correlated tunneling. The quantum phases have been
studied for an one dimensional lattice and for configurations which are
consistent with the setups of cavity quantum electrodynamics, where the global
interactions are mediated by multiple scattered photons in the dispersive
optomechanical regime Landig  _et al._ (2016); Zupancic  _et al._ (2019). The
geometry we considered permits one to tune the relative strength of the
coefficients scaling the long-range potential and the correlated tunneling
terms in the extended Bose-Hubbard model

In the geometry where correlated tunneling vanishes, the cavity-mediated
potential induces density-wave order and the phase diagrams we obtain are
consistent with the ones predicted in higher dimensions Dogra  _et al._
(2016); Niederle  _et al._ (2016); Flottat  _et al._ (2017); Himbert  _et al._
(2019). In the geometries where correlated tunneling cannot be discarded, we
find phases with bond order. These phases are analogous to valence bond
solids/fluids in spin systems Affleck  _et al._ (1987). We remark that bond
ordered phases are also present in other systems like for instance in zig-zag
optical lattices with density-dependent gauge fields Mishra  _et al._ (2016),
in spin-1/2 dipolar Fermi gases in lattice potentials Di Dio  _et al._ (2014)
and in honeycomb lattices with anisotropic tunneling Jürgensen and Lühmann
(2014). In contrast to those models here the bond order is a self-organized
phase induced by photon-atom interactions in a cavity.

The interplay of the cavity-induced potential and correlated tunneling gives
rise to phases that can simultaneously exhibit density-wave and bond order.
Remarkably, at low tunneling and unit density the cavity interactions favor a
bond supersolid, where one would otherwise expect an incompressible phase. We
have analyzed the nature of these self-organized bond superfluid phases for
exemplary parameter regimes by means of the scaling of the entanglement
entropy and found that these phases have the same central charge as the
superfluid phases of short-ranged Bose-Hubbard models.

Bond and density wave order correspond to self-organized patterns which
scatter coherently into the cavity mode. Correspondingly, they are associated
with the onset of an intracavity field that is proportional to density-wave
and bond-order parameters and can be measured at the cavity output Baumann
_et al._ (2010); Fernández-Vidal  _et al._ (2010); Sierant  _et al._ (2019).
Superfluidity can be revealed by time-of-flight measurements Baumann  _et al._
(2010). The low energy gap could be determined by means of a pump-probe
experiment.

Bond order in these dynamics can be often understood in terms of interference
between single-particle hopping and correlated tunneling Kraus  _et al._
(2020); Chanda  _et al._ (2021). The very same interference is at the basis of
the phases observed in one dimension for correlated tunneling with short range
interactions, even though the resulting phases are different Biedroń  _et al._
(2018); Kraus  _et al._ (2020). In two dimensions the phases result from an
interesting interplay between frustration and quantum interference Suthar  _et
al._ (2020); Zhang  _et al._ (2021). The extension of these dynamics to global
interactions is non-trivial and will be subject of future works.

###### Acknowledgements.

R.K. and G.M. acknowledge the support of the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation), Project-ID 429529648 TRR 306 QuCoLiMa
(“Quantum Cooperativity of Light and Matter”) and Priority Program SPP1929
GiRyd (“Giant Interactions in Rydberd Systems”). They also acknowledge the
support of the German Ministry of Education and Research (BMBF) via the
QuantERA projects “QTFLAG” and “NAQUAS”. Project NAQUAS has received funding
from the QuantERA ERA-NET Cofund in QuantumTechnologies implemented within the
European Union’s Horizon 2020 Programme. T.C. and J.Z. acknowledge the support
of PL-Grid Infrastructure and the National Science Centre (Poland) under
project Opus 2019/35/B/ST2/00034 (J.Z.) and Unisono 2017/25/Z/ST2/03029 (T.C.)
realized within QuantERA ERA-NET QTFLAG collaboration. The DMRG simulations
have been performed using ITensor library Fishman  _et al._ .

##  Appendix A On the extended Bose-Hubbard model of cavity quantum
electrodynamics

The extended Bose-Hubbard model of Eq. (1) is found by performing the Wannier
expansion of the Hamiltonian $\hat{H}_{\text{eff}}=\hat{H}+\hat{H}_{\text{C}}$
as a function of the bosonic field operators $\Psi(\mathbf{r})$ with
$\left[\Psi(\mathbf{r}),\Psi^{\dagger}(\mathbf{r}^{\prime})\right]=\delta(\mathbf{r}-\mathbf{r}^{\prime})$.
The Hamiltonian term $\hat{H}$ consists of the kinetic energy, the potential
of the optical lattice and contact interaction between the atoms:

$\displaystyle\hat{H}=$ $\displaystyle\int
d^{3}\mathbf{r}\hat{\Psi}^{\dagger}(\mathbf{r})\left(-\frac{\hbar^{2}}{2m}\nabla^{2}+V_{\text{trap}}\right)\hat{\Psi}(\mathbf{r})$
(12) $\displaystyle+\frac{g}{2}\int
d^{3}\mathbf{r}\hat{\Psi}^{\dagger}(\mathbf{r})^{2}\hat{\Psi}(\mathbf{r})^{2}\,,$
(13)

while the effect of the atom-cavity coupling is described by the effective
long-range Hamiltonian Fernández-Vidal  _et al._ (2010); Habibian  _et al._
(2013b)

$\displaystyle\hat{H}_{C}=\frac{U_{1}}{L}\left(\int
d^{3}\mathbf{r}\hat{\Psi}^{\dagger}(\mathbf{r})h(\mathbf{r})\hat{\Psi}(\mathbf{r})\right)^{2}\,,$
(14)

where $h(\mathbf{r})$ is the spatial mode function of the cavity field, with
$\max|h(\mathbf{r})|=1$. Hamiltonian (14) is obtained by eliminating the
cavity field assuming that this evolves on a faster time-scale than the atomic
motion. The details of the derivation of Eq. (14) starting from the full atom-
photon master equation (that accounts for cavity losses) have been reported
and discussed in Larson  _et al._ (2008); Fernández-Vidal  _et al._ (2010);
Habibian  _et al._ (2013a); Sierant  _et al._ (2019), see also Ref. Sebby-
Strabley  _et al._ (2006) for a systematic semiclassical treatment. We note
that in Eq. (14) we have neglected a term resulting from the dynamical Stark
shift of the cavity field. We further remark that the coefficient $U_{1}$ is
proportional to the intensity of a transverse laser field, driving the atoms,
and that its sign is experimentally controlled by the sign of the detuning
between the cavity and laser frequency Fernández-Vidal  _et al._ (2010);
Habibian  _et al._ (2013a); Ritsch  _et al._ (2013).

In our work we assume a trapping potential of the form
$V_{\text{trap}}=V_{\text{lat}}(x)+\frac{m\omega^{2}}{2}\left(y^{2}+z^{2}\right)$
with $\omega$ the trap frequency and

$V_{\text{lat}}(x)=V_{0}\sin^{2}(kx).$

The trap frequency $\omega$ is chosen so that the atoms can be assumed to be
in the ground state of the transverse trapping potential. In the single-band
approximation the Wannier expansion of the bosonic field reads

$\displaystyle\Psi(\mathbf{r})=\sum_{j}w_{j}(x)\psi_{0}(y,z)\hat{a}_{j},$ (15)

where $w_{j}(x)$ are one dimensional Wannier functions centered on $j$-sites
along $x$ direction, $\psi_{0}(y,z)$ is the ground state wave-function of the
transverse harmonic trap, and $\hat{a}_{j}$ annihilates a bosons at site $j$
and in the ground state of the harmonic trap.

The cavity spatial mode function is here assumed to be given by its expression
in the paraxial approximation,

$h({\bf r})=\cos(kx+\phi)\,,$

assuming that the transverse size of the atomic gas is much smaller than the
mode waist.

By using Eq. (15) in Hamiltonian $\hat{H}$ and performing the integrals as
outline in Refs. Larson  _et al._ (2008); Fernández-Vidal  _et al._ (2010);
Habibian  _et al._ (2013b), one obtains the Bose-Hubbard Hamiltonian of Eq.
(1).

##  Appendix B Details about DMRG simulations

The DMRG algorithm White (1992, 1993) that we employ is based on matrix
product state (MPS) ansatz Schollwöck (2011); Orús (2014) to find the ground
state and low-lying excited states of the system. We employ the global $U(1)$
symmetry corresponding to the conservation of the total number of particles
Singh  _et al._ (2010, 2011). For that purpose, we use ITensor C++ library
Fishman  _et al._ where the MPO for the all connected long-range Hamiltonian
can be constructed exactly Crosswhite  _et al._ (2008); Pirvu  _et al._ (2010)
using AutoMPO class. In our Hamiltonian (1) (with (3)), when both $\hat{B}$
and $\hat{D}$ terms are non-zero the MPO bond dimension turns out to be $13$.
The maximum number bosons ($n_{0}$) per site has been truncated to 6.

We consider random entangled states,
$\ket{\psi_{\text{ini}}}=\frac{1}{\sqrt{50}}\sum_{i=0}^{49}\ket{\psi^{\text{rand}}_{i}}$,
where $\ket{\psi^{\text{rand}}_{i}}$ are random product states with fixed
density (either $\rho=1$ or $1/2$), as our initial states for the DMRG
algorithm. The maximum bond dimension of MPS has been restricted to
$\chi_{\max}=600$, so that the discarded weights of the singular values remain
below $10^{-10}$ even in the gapless phases for system-sizes upto $L\leq 120$.
It is to be noted that in case of small system-sizes or the gapped phases, the
final bond dimension may not reach $\chi=600$, as in our DMRG sweeps we
discard any singular values having weights below $10^{-12}$. We verify the
convergence of the DMRG algorithm by checking the deviations in energy in
successive DMRG sweeps. When the energy deviation in successive sweeps falls
below $10^{-12}$, we conclude that the resulting MPS is the ground state of
the system.

To obtain low-lying excited states, as in Fig. 8, we first shift the
Hamiltonian by a weight factor multiplied with the projector of the previously
found state. To be precise, for finding the $n^{th}$ excited state
$\ket{\psi_{n}}$, we search for the ground state of the shifted Hamiltonian,

$\hat{H}^{\prime}=\hat{H}+W\sum_{m=0}^{n-1}\ket{\psi_{m}}\bra{\psi_{m}},$ (16)

where $W$ should be guessed to be sufficiently larger than $E_{n}-E_{0}$.

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# Tubed embeddings

Anton Petrunin

###### Abstract

We consider the following question: When does a Riemannian manifold admit an
embedding with a uniformly thick tubular neighborhood in another Riemannian
manifold of large dimension?

## Introduction

### 1.

Let $M$ and $N$ be Riemannian manifolds; we will always assume that Riemannian
manifolds are complete connected, and $C^{\infty}$-smooth.

A smooth embedding $f\colon M\hookrightarrow N$ will be called tubed if the
image $f(M)\subset\nobreak N$ is a closed subset and it admits a tubular
$\varepsilon$-neighborhood for some $\varepsilon>0$; that is, the closest-
point projection from the $\varepsilon$-neighborhood of $f(M)$ to $f(M)$ is
uniquely defined. Equivalently, $f(M)$ is a set with positive reach [6, 2].

We say that $M$ has uniformly polynomial growth if there is a polynomial $p$
such that

$\operatorname{\rm vol}B(x,R)_{M}\leqslant p(R)$

for any point $x\in M$ and any radius $R\geqslant 0$; here $B(x,R)_{M}$
denotes open ball in $M$ of radius $R$ centered at $x$, and $\operatorname{\rm
vol}$ stands for volume. In this case, we say that $p$ is a growth polynomial
of $M$.

Observation. Suppose a complete Riemannian manifold $M$ admits an isometric
tubed embedding in a Euclidean space. Then $M$ has

  1. (i)

bounded sectional curvature,

  2. (ii)

positive injectivity radius, and

  3. (iii)

uniformly polynomial growth.

For example, the hyperbolic plane has exponential growth; therefore, it does
not admit an isometric tubed embedding into $\mathbb{R}^{d}$ for any integer
$d$.

### 2.

Further, the sectional curvature of $M$ will be denoted by $\mathrm{sec}_{M}$;
its injectivity radius at point $p\in M$ will be denoted by $\operatorname{\rm
inj}_{p}M$, and $\operatorname{\rm inj}M$ will denote the injectivity radius
of $M$; that is,

$\operatorname{\rm inj}M\mathrel{{:}{=}}\inf\left\\{\,\left.{\operatorname{\rm
inj}_{p}M}\vphantom{p\in M}\,\right|\,{p\in M}\,\right\\}.$

We say that $M$ has bounded geometry if $\operatorname{\rm inj}M>0$ and for
every $k\in\mathbb{N}$ there is a constant $C_{k}>0$ such that
$|\nabla^{k}\operatorname{\rm Rm}|\leqslant C_{k}$; here $\nabla$ denotes the
Levi-Civita connection, and $\operatorname{\rm Rm}$ is the curvature tensor on
$M$. Note that manifolds with bounded geometry have bounded sectional
curvature. (Some authors define bounded geometry as bounded sectional
curvature and injectivity radius — we do not follow this convention.)

Main theorem. Suppose $M$ is a Riemannian manifold with bounded geometry.
Then, $M$ has uniformly polynomial growth if and only if there is an isometric
tubed embedding $M\hookrightarrow\mathbb{R}^{d}$ for some integer $d$.

Moreover, the dimension $d$ can be found in terms of the dimension and a
growth polynomial of $M$.

### 3.Other ambient spaces.

Loosely speaking, the following statement says that there is no universal
space — no space can serve as a target of an isometric tubed embedding for any
$n$-manifolds with bounded geometry.

Proposition. Let $N$ be a Riemannian manifold with bounded geometry and
$n\geqslant 2$. Then there is an $n$-dimensional Riemannian manifold $M$ with
bounded geometry that does not admit an isometric tubed embedding in $N$.

### 4.About the proofs.

The observation in §1 is proved in §§5–6. It is done by means of elementary
differential geometry.

To prove the main theorem (§2), we approximate the manifold by a graph (§9)
and apply a slight modification of the theorem by Robert Krauthgamer and James
Lee [12, 13] (§8). It produces a map $f_{1}\colon M\to\mathbb{R}^{d_{1}}$ such
that $|f_{1}(x)-f_{1}(y)|$ is bounded away from zero for all pairs $x,y\in M$
on distance at least $R$ from each other for fixed $R>0$ (§10).

Further, in §11, we construct another map $f_{2}\colon M\to\mathbb{R}^{d_{2}}$
that is bi-Lipschitz in every ball of radius $R$. The direct sum $f_{1}\oplus
f_{2}$ is an embedding. We can choose $\varepsilon>0$ such that
$\varepsilon^{2}{\hskip 0.5pt\cdot\hskip 0.5pt}h<\tfrac{1}{2}{\hskip
0.5pt\cdot\hskip 0.5pt}g$, where $h$ is the induced Riemannian metric on $M$
by $f_{1}\oplus f_{2}$ and $g$ is the original Riemannian metric on $M$.
Applying Nash’s embedding theorem for metric $g-\varepsilon^{2}{\hskip
0.5pt\cdot\hskip 0.5pt}h$, we find another map $f_{3}\colon
M\to\mathbb{R}^{d_{3}}$ such that the direct sum $f=(\varepsilon{\hskip
0.5pt\cdot\hskip 0.5pt}f_{1})\oplus(\varepsilon{\hskip 0.5pt\cdot\hskip
0.5pt}f_{2})\oplus f_{3}$ is isometric.

We show that the constructions of $f_{1}$, $f_{2}$, and $f_{3}$ produce
uniformly smooth maps; see §10. In particular, the normal curvatures of $f$
are bounded. Then we use elementary differential geometry to show that $f$ is
tubed.

The proof of the proposition in §3 is a slight modification of the argument of
Florian Lehner [14, 1.2]; see also [17]. It states that no connected graph
with bounded degree contains a copy of any connected graph of degree 3; see
§§15–16.

## Observation

### 5.Lemma.

Suppose $M$ is a smooth submanifold in $\mathbb{R}^{d}$ with normal curvatures
at most 1. Then $-2\leqslant\mathrm{sec}_{M}(\sigma)\leqslant 1$ for any
sectional direction $\sigma$ of $M$.

Proof. Denote by $\mathrm{T}_{p}$ and $\mathrm{N}_{p}$ the tangent and normal
spaces at $p\in M$. Denote by $s$ the second fundamental form of $M$ at $p$;
it is a symmetric quadratic form on $\mathrm{T}_{p}$ with values in
$\mathrm{N}_{p}$.

By the assumptions, $|s(v,v)|\leqslant 1$ for any $|v|=1$. We need to show
that curvature of $M$ at $p$ in any sectional direction lies in the range
$[-2,1]$. Passing to a 2-dimensional subspace of $\mathrm{T}_{p}$, we may
assume that $\operatorname{\rm dim}M=2$.

Let $h\in\mathrm{N}_{p}$ and $\textsl{Ж}\in\mathbb{R}$ be the average values
of $s(u,u)$ and $|s(u,u)|^{2}$ for unit vectors $u\in\mathrm{T}_{p}$. Since
$\operatorname{\rm dim}M=2$, the Gauss formula [18, 2.1] can be written as

$K=3{\hskip 0.5pt\cdot\hskip 0.5pt}|h|^{2}-2{\hskip 0.5pt\cdot\hskip
0.5pt}\textsl{Ж};$

here $K$ is the Gauss curvature of $M$ at $p$. Note that
$|h|^{2}\leqslant\textsl{Ж}$. Since the normal curvatures are at most 1, we
have $\textsl{Ж}\leqslant 1$. Hence the lemma follows. ∎

It is straightforward to check that the inequality
$0\leqslant|h|^{2}\leqslant\textsl{Ж}$ is the only restriction on $|h|^{2}$
and Ж. Therefore, the obtained bounds are optimal.

### 6.

The first two parts of the the observation in §1 follow from a more general
result by Stephanie Alexander and Richard Bishop [1, 1.4]; but we include
proofs for completeness.

Proof of the observation in §1. Suppose a smooth $n$-dimensional submanifold
$M$ in $\mathbb{R}^{d}$ has a $\varepsilon$-thick tubular neighborhood $W$.

Note that normal curvatures of $M$ cannot exceed $\tfrac{1}{\varepsilon}$. By
§5,

$-\tfrac{2}{\varepsilon^{2}}\leqslant\mathrm{sec}_{M}\leqslant\tfrac{1}{\varepsilon^{2}}.$

Let us show that $\operatorname{\rm inj}M\geqslant\varepsilon$; suppose
$\operatorname{\rm inj}_{p}M<\varepsilon$ for some $p$. Since
$\mathrm{sec}_{M}\leqslant\tfrac{1}{\varepsilon^{2}}$, Klingenberg’s lemma [3,
5.6] implies that there is a geodesic loop $\gamma$ based at $p$ with length
$2{\hskip 0.5pt\cdot\hskip 0.5pt}\operatorname{\rm inj}_{p}M$. Let

$r=\max\left\\{\,\left.{|p-q|_{\mathbb{R}^{d}}}\vphantom{q\in\gamma}\,\right|\,{q\in\gamma}\,\right\\}.$

Note that $r<\operatorname{\rm inj}_{p}M$, and $\gamma$ has curvature at least
$\tfrac{1}{r}$ at $q$. It follows that the normal curvature of $M$ at $q$ in
the direction of $\gamma$ exceeds $\tfrac{1}{\varepsilon}$ — a contradiction.

Consider the closest point projection $\pi\colon W\to\nobreak M$. Note that
there is a constant $c=\nobreak c(\varepsilon,n,d)>0$ such that

$\operatorname{\rm vol}_{d}(\pi^{-1}X)\geqslant c{\hskip 0.5pt\cdot\hskip
0.5pt}\operatorname{\rm vol}_{n}X$

for any measurable set $X\subset M$.

Note that $B(x,R)_{M}\subset B(x,R)_{\mathbb{R}^{d}}$ for any $x\in M$; here
$B(x,R)_{M}$ and $B(x,R)_{\mathbb{R}^{d}}$ denote balls in the metric of $M$
and $\mathbb{R}^{d}$ respectively. Therefore

$\pi^{-1}[B(x,R)_{M}]\subset B(x,R+\varepsilon)_{\mathbb{R}^{d}}.$

Since $R\mapsto\operatorname{\rm vol}_{d}B(x,R+\varepsilon)_{\mathbb{R}^{d}}$
is a polynomial function, we get that $M$ has uniformly polynomial growth. ∎

## Graph embedding

### 7.

Let $\Gamma$ be a connected graph. Recall that $\Gamma$ comes with the
shortest-path distance on its vertex set $\mathop{\rm Vert}\Gamma$. So, we can
talk about ball $B(x,R)_{\Gamma}$ for any center $x\in\mathop{\rm Vert}\Gamma$
and radius $R>0$. The number of vertices in $B(x,R)_{\Gamma}$ will be denoted
by $|B(x,R)_{\Gamma}|$. We say that $\Gamma$ has uniformly polynomial growth
if there is a polynomial $p$ such that

$|B(x,R)_{\Gamma}|\leqslant p(R)$

for any point $x\in\Gamma$ and any radius $R\geqslant 0$. In this case, we say
that $p$ is a growth polynomial of $\Gamma$. Note that any graph with
uniformly polynomial growth has bounded degree.

We will write $v\sim w$ if two vertices $v,w\in\mathop{\rm Vert}\Gamma$ are
adjacent. Let us denote by $\mathbb{Z}^{d}_{\infty}$ the graph with the vertex
set formed by integer lattice $\mathbb{Z}^{d}\subset\mathbb{R}^{d}$ and

$v\sim w\iff\|v-w\|_{\infty}=1.$

The next statement follows from [13, Theorem 5.5].

Graph-embedding theorem. Suppose $\Gamma$ is a connected graph with uniformly
polynomial growth. Then, $\Gamma$ is isomorphic to a subgraph of
$\mathbb{Z}^{d}_{\infty}$ for some integer $d$. Moreover, the dimension $d$
can be found in terms of a growth polynomial of $\Gamma$.

### 8.

Let us denote by $\text{\rm Conv}X$ the convex hull of the subset
$X\subset\mathbb{R}^{d}$.

Corollary. Suppose $\Gamma$ is a connected graph with uniformly polynomial
growth. Then for some $\rho>0$ and integer $d>0$ there is a map
$\Phi\colon\mathop{\rm Vert}\Gamma\to\mathbb{R}^{d}$ such that

  1. (a)

if $v\sim w$, then $|\Phi(v)-\Phi(w)|\leqslant\rho$, and

  2. (b)

if $V$ and $W$ are disjoint cliques in $\Gamma$, then the minimal distance
from $\text{\rm Conv}[\Phi(V)]$ to $\text{\rm Conv}[\Phi(W)]$ is at least $1$.

Moreover, the dimension $d$ and the distance $\rho$ can be found in terms of a
growth polynomial of $\Gamma$.

Proof. By §7, it is sufficient to prove the corollary for
$\Gamma=\mathbb{Z}^{n}_{\infty}$.

Take $d=n+2^{n}$. Let us color vertices of $\mathbb{Z}^{n}_{\infty}$ in
$2^{n}$ colors so that adjacent vertices get different colors. Let
$e_{1},\dots s_{2^{n}}$ be the standard basis in $\mathbb{R}^{2^{n}}$.
Consider the map
$\Phi\colon\mathbb{Z}^{n}\to\mathbb{R}^{d}=\mathbb{R}^{n}\oplus\mathbb{R}^{2^{n}}$
defined by $\Phi\colon v\mapsto v\oplus e_{i(v)}$ where $i(v)$ denotes the
color of $v$. After an appropriate rescaling, the obtained map meets the
conditions. ∎

## Approximation by graph

### 9.Intersection graph.

Let $M$ be a complete connected Riemannian manifold. Choose $r>0$. Suppose
$V\subset M$ is a maximal set of points (with respect to inclusion) on
distance at least $r$ from each other. Given $v\in V$, set $B_{v}=B(v,r)_{M}$
and $\lambda{\hskip 0.5pt\cdot\hskip 0.5pt}B_{v}=B(v,\lambda{\hskip
0.5pt\cdot\hskip 0.5pt}r)_{M}$. Note that

  1. (i)

the balls $\\{\tfrac{1}{2}{\hskip 0.5pt\cdot\hskip 0.5pt}B_{v}\\}_{v\in V}$
are disjoint;

  2. (ii)

$\\{B_{v}\\}_{v\in V}$ is an open cover of $M$;

  3. (iii)

$\\{2{\hskip 0.5pt\cdot\hskip 0.5pt}B_{v}\\}_{v\in V}$ is an open cover of $M$
with Lebesgue number $r$.

  4. (iv)

If $M$ has bounded geometry, then for any positive real $\lambda$, there is an
integer $N_{\lambda}$ such that any $\lambda{\hskip 0.5pt\cdot\hskip
0.5pt}r$-ball in $M$ contains at most $N_{\lambda}$ points of $V$.

Let $\Gamma_{\lambda}$ be the intersection graph of the covering
$\\{\lambda{\hskip 0.5pt\cdot\hskip 0.5pt}B_{v}\\}$ for $\lambda\geqslant 1$;
that is, $\mathop{\rm Vert}\Gamma_{\lambda}=V$, and

$v\sim w\quad\iff\quad\lambda{\hskip 0.5pt\cdot\hskip
0.5pt}B_{v}\cap\lambda{\hskip 0.5pt\cdot\hskip 0.5pt}B_{w}\neq\varnothing;$

here $v\sim w$ means that $v$ is adjacent to $w$ in $\Gamma_{\lambda}$.

Assume $M$ has bounded geometry. By iv, $\Gamma_{\lambda}$ has degree at most
$N_{2{\hskip 0.5pt\cdot\hskip 0.5pt}\lambda}-1$. Note that

$\lambda{\hskip 0.5pt\cdot\hskip 0.5pt}r{\hskip 0.5pt\cdot\hskip
0.5pt}|v-w|_{\Gamma_{\lambda}}\geqslant|v-w|_{M}$ ➊

for any $v,w\in V$.

If $M$ has bounded geometry, there is $\delta>0$ such that $\operatorname{\rm
vol}(\tfrac{1}{2}{\hskip 0.5pt\cdot\hskip 0.5pt}B_{v})>\delta$ for any $v\in
V$. Therefore, i and ➊ ‣ 9 implies the following:

  1. (v)

If $M$ has bounded geometry and uniformly polynomial growth, then the graph
$\Gamma_{\lambda}$ has uniformly polynomial growth for any $\lambda\geqslant
1$.
Moreover, a growth polynomial of $\Gamma_{\lambda}$ can be found in terms of
$\lambda$, $r$ and a growth polynomial of $M$. Namely, if $x\mapsto p(x)$ is a
growth polynomial of $M$, then $x\mapsto p(\lambda{\hskip 0.5pt\cdot\hskip
0.5pt}r{\hskip 0.5pt\cdot\hskip 0.5pt}x)$ is a growth polynomial of
$\Gamma_{\lambda}$.

## Factors of embedding

### 10.

Let $M$ be a smooth Riemannian manifold. A map $f\colon M\to\mathbb{R}^{d}$ is
called uniformly smooth if for any integer $k\geqslant 1$ there is a constant
$C_{k}$ such that

$|\tfrac{d^{k}}{dt^{k}}(f\circ\gamma)(t)|\leqslant C_{k}$

for all unit-speed geodesics $\gamma$.

Proposition. Let $M$ be a complete $n$-dimensional Riemannian manifold with
bounded geometry and uniformly polynomial growth. Suppose
$|\mathrm{sec}_{M}|\leqslant\tfrac{1}{100}$ and $\operatorname{\rm
inj}M\geqslant 10$. Then for some integer $d_{1}$ there is a uniformly smooth
map $f_{1}\colon M\to\nobreak\mathbb{R}^{d_{1}}$ such that

$|x-y|_{M}\geqslant
1\quad\Longrightarrow\quad|f_{1}(x)-f_{1}(y)|_{\mathbb{R}^{d_{1}}}\geqslant
1.$

Moreover, the dimension $d_{1}$ can be found in terms of a growth polynomial
of $M$.

Proof. Let us apply the construction in §9 for $r=\tfrac{1}{4}$; we obtain a
subset $V\subset M$, balls $\\{\lambda{\hskip 0.5pt\cdot\hskip
0.5pt}B_{v}\\}_{v\in V}$, integer values $N_{\lambda}$ and connected graphs
$\Gamma_{\lambda}$ with $\mathop{\rm Vert}\Gamma_{\lambda}=V$ for all
$\lambda\geqslant 1$.

Now, let us apply the construction of partition of unity. Choose a
$C^{\infty}$-smooth function $\sigma\colon\mathbb{R}\to[0,1]$ such that
$\sigma(t)=1$ for $t\leqslant 1$ and $\sigma(t)=0$ for $t\geqslant 2$.
Further, let $\psi_{v}=\sigma(\tfrac{\operatorname{\rm dist}_{v}}{r})$, for
each $v\in V$; note that the functions $\varphi_{v}$ are uniformly smooth. Set

$\Psi=\sum_{v\in
V}\psi_{v}\quad\text{and}\quad\varphi_{v}=\frac{\psi_{v}}{\Psi}.$

The obtained functions $\varphi_{v}$ form a partition of unity subordinate to
$\\{2{\hskip 0.5pt\cdot\hskip 0.5pt}B_{v}\\}$. Note that
$1\leqslant\nobreak\Psi\leqslant\nobreak N_{2}$; therefore, the functions
$\varphi_{v}$ are uniformly smooth.

Given $x\in M$, consider the set

$V_{x}=\left\\{\,\left.{v\in
V}\vphantom{\varphi_{v}(x)>0}\,\right|\,{\varphi_{v}(x)>0}\,\right\\}.$

Note that $\varphi_{v}(x)>0$ implies that $2{\hskip 0.5pt\cdot\hskip
0.5pt}B_{v}\ni x$. Therefore $V_{x}$ is a clique in $\Gamma_{2}$. Moreover,

$|x-y|\geqslant 4{\hskip 0.5pt\cdot\hskip 0.5pt}r=1\qquad\Longrightarrow\qquad
V_{x}\cap V_{y}=\varnothing.$ ➋

Let $\Phi\colon V\to\mathbb{R}^{n}$ be the map constructed in §8. Given $x\in
M$, define $f_{1}(x)$ as the barycenter of points $\Psi(v)$ with masses
$\varphi_{v}(x)$ for each $v\in V_{x}$. By §8, $\operatorname{\rm
diam}V_{x}\leqslant\rho$, and since the functions $\varphi_{v}$ are uniformly
smooth, so is $f_{1}$. Note that $f_{1}(x)$ lies in the convex hull of
$\Phi(V_{x})$ for any $x\in M$. It remains to apply ➋ ‣ 10 and §8. ∎

### 11.Proposition.

Let $M$ be a complete $n$-dimensional Riemannian manifold with bounded
geometry. Suppose $|\mathrm{sec}_{M}|\leqslant\tfrac{1}{100}$ and
$\operatorname{\rm inj}M\geqslant 10$. Then for some integer $d_{2}$, there is
a uniformly smooth map $f_{2}\colon M\to\mathbb{R}^{d_{2}}$ such that $f_{2}$
is uniformly bi-Lipschitz in all unit balls.

Proof. Choose $r=1$ and apply the constructions in §9.

Let $\sigma\colon\mathbb{R}\to[0,1]$ be as in §10. For each $v\in V$, choose
an isometry $\iota_{v}\colon\mathrm{T}_{v}\to\nobreak\mathbb{R}^{n}$. Consider
the map $s_{v}\colon M\to\mathbb{R}^{n}$ defined by

$s_{v}\colon\exp_{v}x\mapsto\sigma(2{\hskip 0.5pt\cdot\hskip 0.5pt}|x|){\hskip
0.5pt\cdot\hskip 0.5pt}\iota_{v}(x).$

Since $\Gamma_{2}$ has degree at most $N_{2}-1$, the set $V$ can be subdivided
into $N_{2}$ subsets $V_{1},\dots,V_{N_{2}}$ such that for any $i$, the balls
$2{\hskip 0.5pt\cdot\hskip 0.5pt}B_{v}$ for $v\in V_{i}$ are disjoint.
Consider the map $S_{i}\colon M\to\mathbb{R}^{n}$ defined by

$S_{i}\mathrel{{:}{=}}\sum_{v\in V_{i}}s_{v}.$

Note that $S_{i}$ is uniformly smooth and it is uniformly bi-Lipschitz in
every ball $2{\hskip 0.5pt\cdot\hskip 0.5pt}B_{v}$ for $v\in V_{i}$. It
follows that $S_{1}\oplus\dots\oplus S_{N_{2}}\colon
M\to\mathbb{R}^{N_{2}{\hskip 0.5pt\cdot\hskip 0.5pt}n}$ is uniformly smooth
and bi-Lipschitz in $2{\hskip 0.5pt\cdot\hskip 0.5pt}B_{v}$ for any $v\in V$.
The proposition follows since $r=1$ is a Lebesgue number of the covering
$\\{2{\hskip 0.5pt\cdot\hskip 0.5pt}B_{v}\\}$. ∎

## Uniform Nash’s construction

### 12.Uniformly regular metrics.

Let $M$ be a smooth $n$-dimensional manifold with a chosen locally finite
atlas $\mathcal{A}$. We say that $\mathcal{A}$ is appropriate if it meets the
following conditions:

  * $\diamond$

Each chart $\sigma\in\mathcal{A}$ is given by a map from an open set
$U_{\sigma}\subset M$ to the unit ball $\mathbb{D}^{n}\subset\mathbb{R}^{n}$.

  * $\diamond$

The covering $\\{U_{\sigma}\\}$ of $M$ has bounded multiplicity.

  * $\diamond$

There exists $r\in(0,1)$ such that $\\{\sigma^{-1}(r{\hskip 0.5pt\cdot\hskip
0.5pt}\mathbb{D}^{n})\\}_{\sigma\in\mathcal{A}}$ is a cover for $M$.

  * $\diamond$

All the transition maps of $\mathcal{A}$ are uniformly bi-Lipschitz and for
any integer $k\geqslant 0$ they have uniformly bounded $C^{k}$-norms.

A metric tensor $g$ on $M$ will be called uniformly regular with respect to an
appropriate atlas $\mathcal{A}$ if for every integer $k$, all components of
the metric tensor in all charts of $\mathcal{A}$ have bounded $C^{k}$-norms,
and all the charts are uniformly bi-Lipschitz with respect to the metric
induced by $g$ on $M$.

Analogously, we can define a uniformly regular sequence of metrics on a fixed
manifold with a chosen appropriate atlas. (Note that a smooth metric on a
fixed compact manifold, say $\mathbb{S}^{n}$, is uniformly smooth for some
appropriate atlas, but a sequence of smooth metrics does not have to be
uniformly smooth.)

Let $M$ be a Riemannian manifold with bounded geometry. Applying rescaling if
necessary, we may assume $|\mathrm{sec}_{M}|\leqslant\tfrac{1}{100}$ and
$\operatorname{\rm inj}M\geqslant 10$.

Let us apply the construction in §9 for $r=\tfrac{1}{2}$. Given $v\in V$,
consider the normal chart $\sigma_{v}\colon 2{\hskip 0.5pt\cdot\hskip
0.5pt}B_{v}\to\mathbb{D}^{n}$ with the center at $v$; we will call such an
atlas standard. It is straightforward to check that this atlas is appropriate
and the metric tensor is uniformly regular with respect to this atlas. In
particular, any manifold of bounded geometry has a uniformly regular metric in
an appropriate atlas. The following statement from [5] implies that the
converse holds as well.

Proposition. A Riemannian manifold $M$ has bounded geometry if and only if its
metric is uniformly regular for some appropriate atlas.

Also note that if $M$ comes with a uniformly regular metric for some
appropriate atlas $\mathcal{A}$, then a map $f\colon M\to\mathbb{R}^{d}$ is
uniformly smooth if and only if for every integer $k\geqslant 0$, the
functions $f\circ\sigma-f\circ\sigma(0)\colon\mathbb{D}^{n}\to\mathbb{R}^{d}$
for all charts $\sigma\in\mathcal{A}$ have uniformly bounded $C^{k}$-norms.

### 13.Proposition.

Let $M$ be a complete $n$-dimensional Riemannian manifold with bounded
geometry. Then for some integer $d$, there is a uniformly smooth isometric
immersion $f_{3}\colon M\to\mathbb{R}^{d}$.

The statement follows from the proof of Nash’s theorem about regular
embedding; we will describe the needed changes in [16].

Suppose we have a sequence of Riemannian metrics $g_{1},g_{2},\dots$ on the
sphere $\mathbb{S}^{n}$. By Nash’s theorem, there are smooth embeddings
$s_{1},s_{2},\dots\nobreak\colon\mathbb{S}^{n}\to\mathbb{R}^{d}$ such that
each $s_{i}$ induces $g_{i}$ on $\mathbb{S}^{n}$; here
$d=\nobreak\tfrac{1}{2}{\hskip 0.5pt\cdot\hskip 0.5pt}n{\hskip
0.5pt\cdot\hskip 0.5pt}(3{\hskip 0.5pt\cdot\hskip 0.5pt}n+11)$.

Suppose that the sequence of metrics $g_{i}$ is uniformly regular with respect
to a finite appropriate atlas on $\mathbb{S}^{n}$. Then the construction in
[16, Part C] with the same choices of functions produces uniformly smooth
embeddings $s_{i}$. Namely, the functions $\psi^{r}$ in [16, (C2)] and
$\alpha_{r}$ in [16, (C10)] are uniformly smooth and we can assume that
$\lambda$ in [16, (C12)] is the same for all $g_{i}$.

Further, the so-called Nash’s reduction [16, Part D] reduces the existence of
an isometric immersion of $M$ to the existence of isometric immersions
$(\mathbb{S}^{n},g_{i})\to\mathbb{R}^{d}$ for a sequence of Riemannian metrics
$g_{i}$. Nash starts with an atlas with bounded multiplicity, and for each
chart, he produces a sequence of metrics $g_{i}$ on $\mathbb{S}^{n}$ with a
sequence of smooth maps $\varphi_{i}\colon M\to(\mathbb{S}^{n},g_{i})$ such
that the sum of metrics induced by $\varphi_{i}$ on $M$ coincides with $g$.
Then an isometric immersion constructed from embeddings $s_{i}$ as above.

Applying rescaling, we may assume that
$|\mathrm{sec}_{M}|\leqslant\tfrac{1}{100}$ and $\operatorname{\rm
inj}M\geqslant 10$. Let us choose a standard atlas on $M$ as in §12. Then
Nash’s construction produces a uniformly smooth sequence of maps $\varphi_{i}$
and a sequence of metrics satisfying the above condition. As a result, we get
a uniformly smooth isometric immersion $(M,g)\to\mathbb{R}^{d}$, where
$d=\tfrac{1}{2}{\hskip 0.5pt\cdot\hskip 0.5pt}N_{2}{\hskip 0.5pt\cdot\hskip
0.5pt}n{\hskip 0.5pt\cdot\hskip 0.5pt}(3{\hskip 0.5pt\cdot\hskip 0.5pt}n+11)$.

## Proof assembling

### 14.Proof of the main theorem (§2).

Let $f_{1}\colon M\to\mathbb{R}^{d_{1}}$ and $f_{2}\colon
M\to\mathbb{R}^{d_{2}}$ be the maps constructed in §10 and §11. Consider the
map $f_{1}\oplus f_{2}\colon M\to\mathbb{R}^{d_{1}+d_{2}}$ defined by
$p\mapsto(f_{1}(p),f_{2}(p))\in\mathbb{R}^{d_{1}}\oplus\mathbb{R}^{d_{2}}=\mathbb{R}^{d_{1}+d_{2}}$.

Let $h$ be the metric induced by $f_{1}\oplus f_{2}$ and $g$ be the original
metric on $M$.

Choose $\varepsilon>0$ such that $\varepsilon^{2}{\hskip 0.5pt\cdot\hskip
0.5pt}h<\tfrac{1}{2}{\hskip 0.5pt\cdot\hskip 0.5pt}g$. Note that
$g-\varepsilon^{2}{\hskip 0.5pt\cdot\hskip 0.5pt}h$ is a uniformly regular and
smooth metric on $M$ for a standard atlas; see §12. By §12,
$(M,g-\varepsilon^{2}{\hskip 0.5pt\cdot\hskip 0.5pt}h)$ has bounded geometry.
By §13, there is a uniformly smooth map $f_{3}\colon M\to\mathbb{R}^{d_{3}}$
that induces metric $g-\varepsilon^{2}{\hskip 0.5pt\cdot\hskip 0.5pt}h$ on
$M$.

Therefore, the map $f=(\varepsilon{\hskip 0.5pt\cdot\hskip
0.5pt}f_{1})\oplus(\varepsilon{\hskip 0.5pt\cdot\hskip 0.5pt}f_{2})\oplus
f_{3}\colon M\to\mathbb{R}^{d_{1}+d_{2}+d_{3}}$ is a uniformly smooth
isometric immersion. In particular, the image $f(M)$ has bounded normal
curvatures. By §10 and §11, $f_{1}\oplus f_{2}$ is an embedding; therefore so
is $f$. It remains to show that $f$ is tubed.

Assume the contrary; that is, for any $\varepsilon>0$ there is a point
$x\in\mathbb{R}^{d}$ that lies on distance smaller than $\varepsilon$ to
$f(M)$ such that the distance function from $x$ has at least two minimum
points on $f(M)$, say $f(p)$ and $f(q)$. Denote by $\gamma$ a minimizing
geodesic from $p$ to $q$ in $M$.

Suppose $\varepsilon<\tfrac{1}{2}$. The properties in §10 and §11 imply that

$\mathop{\rm length}\nolimits\gamma<L{\hskip 0.5pt\cdot\hskip
0.5pt}\varepsilon$

for some fixed $L\in\mathbb{R}$. Let $y\in f(\gamma)$ be a point that
maximizes the distance from $x$, and $r=|x-y|$. The triangle inequality
implies that $r\leqslant\varepsilon{\hskip 0.5pt\cdot\hskip 0.5pt}(1+L)$. It
follows that the curvature of $f(\gamma)$ at $y$ is at least $\tfrac{1}{r}$.
The latter is impossible if $\varepsilon$ is small. ∎

## Universal space

### 15.Theorem.

Let $\Gamma$ be a connected graph of bounded degree. Then there is a connected
graph $\Delta$ of degree $\leqslant 3$ such that there is no map
$f\colon\mathop{\rm Vert}\Delta\to\mathop{\rm Vert}\Gamma$ such that

  1. (i)

$|f^{-1}(h)|\leqslant K$ for any vertex $h$ in $\Gamma$, and

  2. (ii)

$|f(v)-f(x)|_{\Gamma}\leqslant K$ for any two adjacent vertices $v,w$ in
$\Delta$

for some constant $K$.

A slightly weaker statement was proved by Florian Lehner [14, 1.2] and
rediscovered by Fedor Petrov [17]; we will modify their proof
straightforwardly. The conditions are closely related to the so-called regular
maps; see [8] and the references therein.

Proof. Suppose $\Gamma$ is an infinite connected graph with bounded degree.
Note that $\Gamma$ has a countable set of vertices; so we can identify
$\mathop{\rm Vert}\Gamma$ with the set of natural numbers $\mathbb{N}$.

Choose a sequence $n_{1},n_{2},\dots$ of natural numbers such that each
$n\in\mathbb{N}$ appears infinitely many times in it; for example, we may take
the following sequence: $1,2,1,3,1,2,1,4,1,\dots$

Let us construct a graph $\Delta$ with degree at most $3$ such that no map $f$
meets the condition in the theorem for any $K$. In the following construction,
the set of vertices of $\Delta$ will be $\mathbb{N}$, and it will contain an
infinite path $P=1\sim 2\sim\dots$. There will be a fast growing sequence
$S_{1}<S_{2}<\dots$ such that if $i\sim j$ for some $i<j$, then $j-i=1$ or
$S_{k}<i<j\leqslant S_{k+1}$ for some $k$.

The subgraphs induced by $\\{1,\dots,S_{k}\\}$ will be denoted by
$\Delta_{k}$; let $P_{k}$ be the corresponding path $1\sim 2\sim\dots\sim
S_{k}$. The graphs $\Delta_{k}$ will be constructed recursively so that they
meet the following condition:

➌  There is no map $f_{k}\colon\mathop{\rm Vert}\Delta_{k}\to\mathop{\rm
Vert}\Gamma$ that meets the conditions for $K=k$ and such that
$f_{k}(1)=\nobreak n_{k}$.

Once it is done, the theorem follows.

We assume that $\Delta_{0}$ is the empty graph, and $S_{0}=0$. Assume
$\Delta_{k-1}$ is constructed, let us construct $\Delta_{k}$. To do so, we
need to choose large $S_{k}\gg\nobreak S_{k-1}$ and add diagonals to the path
$S_{k-1}+1\sim\dots\sim S_{k}$.

Suppose $L<(S_{k}-S_{k-1})/2$. Let $A$ ($B$) be the first (respectively, the
last) $L$ elements in the integer interval from $S_{k-1}+1$ to $S_{k}$. We may
choose an arbitrary bijection $A\leftrightarrow B$ and connect corresponding
vertices by edges. This way we get $L!$ different graphs; in particular, the
total number of graphs $\Delta_{k}$ has superexponential growth with respect
to $S_{k}$. More precisely, we count labeled graphs; that is, graphs
$\Delta_{k}$ with vertices labeled by $\\{1,\dots,S_{k}\\}$ and a path
$P_{k}=1\sim 2\sim\dots\sim S_{k}$.

How much of these graphs have the required maps $f\colon\mathop{\rm
Vert}\Delta_{k}\to\mathop{\rm Vert}\Gamma$? Suppose that the degree of
$\Gamma$ does not exceed $d-1$. Note that each $k$-ball in $\Gamma$ has at
most $d^{k}$ vertices. Therefore we have at most $d^{k{\hskip 0.5pt\cdot\hskip
0.5pt}S_{k+1}}$ different maps of the path $P_{k}$. Given one such map, there
are at most $k{\hskip 0.5pt\cdot\hskip 0.5pt}d^{k}$ ways to add a diagonal at
each vertex. Therefore the number of labeled graphs that admit a required map
$f\colon\mathop{\rm Vert}\Delta_{k}\to\mathop{\rm Vert}\Gamma$ for $K=k$ grows
exponentially with respect to $S_{k}$. It follows that we may choose a
sufficiently large $S_{k}$, so that there is a graph $\Delta_{k}$ that meets
15. ∎

### 16.Proof of the proposition in §3.

Let $N$ be a manifold with bounded geometry and $n\geqslant 2$; we need to
construct an $n$-dimensional manifold $M$ with bounded geometry that does not
admit an isometric tubed embedding into $N$.

Let $\Gamma_{2}$ be the intersection graph obtained by applying the
construction in §9 to $N$. Recall that $\Gamma_{2}$ has bounded degree. Let
$\Delta$ be the graph provided by §15 for $\Gamma_{2}$; it is a connected
graph with degree at most 3.

Let us choose a tube; that is, the cylinder $\mathbb{S}^{n-1}\times[0,1]$ with
smooth Riemannian metric such that a neighborhood of each boundary component
is isometric to a small annulus in the unit sphere $\mathbb{S}^{n}$, so it can
be used to connect spheres. Prepare a unit sphere $\mathbb{S}^{n}$ for every
vertex in $\Delta$. Connect two spheres by a tube if the corresponding
vertices are adjacent. Note that the obtained manifold $M$ has bounded
geometry.

Assume there is an isometric tubed embedding $\iota\colon M\hookrightarrow N$.
We can assume that $\Delta$ is embedded in $M$ so that each vertex lies on the
corresponding sphere. Consider the map $\Delta\to\Gamma_{2}$ that sends a
vertex $v\in\mathop{\rm Vert}\Delta$ to a vertex of $\Gamma_{2}$ that lies at
minimal distance from $\iota(v)$. Note that for some $K$, the obtained map
satisfies the conditions in the theorem in §15 — a contradiction. ∎

## Final remarks

### 17.

Let us say that a Riemannian manifold $M$ has $k$-bounded geometry if there is
a constant $C_{k}>0$ such that $|\nabla^{i}\operatorname{\rm Rm}|\leqslant
C_{k}$ for any nonnegative integer $i\leqslant k$. For example, $0$-bounded
geometry means positive injectivity radius and bounded curvature.

In the proof of the main theorem, one can assume that $M$ has only $3$-bounded
geometry; the proof requires only minor changes. Below we discuss $1$\- and
$0$-bounded geometry.

Let us generalize the definitions of appropriate atlas and uniformly regular
metric (see §12). Choose an integer $m\geqslant 1$ and $\alpha\in(0,1)$.
Suppose that in the definition of appropriate atlas, instead of bounded
$C^{k}$ norm of the transition maps for all $k$ we only require bounded
$C^{m+1,\alpha}$ norm. Let us call such atlas $C^{m+1,\alpha}$-appropriate.
Furthermore, in the definition of uniformly regular metric let us assume that
the atlas is only $C^{m+1,\alpha}$-appropriate, and instead of bounded $C^{k}$
norm for all $k$, we only require that metric has bounded $C^{m,\alpha}$ norm
in every chart. Let us call such metrics uniformly $C^{m,\alpha}$-regular.
Note that a $C^{\infty}$-smooth metric might fail to be uniformly
$C^{m,\alpha}$-regular even if injectivity radius is positive.

Recall that if metric is $C^{m,\alpha}$, then it remains to be $C^{m,\alpha}$
in the harmonic charts. This fundamental result was obtained by Idzhad Sabitov
and Samuil Shefel [19] and rediscoved by Dennis DeTurck and Jerry Kazdan [4].
Applying this technique, one could get that manifolds with $1$-bounded
geometry have uniformly $C^{2,\alpha}$-regular metric for any
$\alpha\in(0,1)$.

Using Günther’s version of the embedding theorem [7] in §13, we obtain the
following: for any $m\geqslant 2$ and $\alpha\in(0,1)$, any manifold with
uniformly $C^{m,\alpha}$-regular metric admits a uniformly
$C^{m,\alpha}$-smooth isometric immersion into Euclidean space; in particular,
such an immersion has bounded normal curvatures. Furthermore, following §16,
we get that for any $m\geqslant 2$ and $\alpha\in(0,1)$, any manifold with
uniformly $C^{m,\alpha}$-regular metric and uniformly polynomial growth admits
a uniformly $C^{m,\alpha}$-smooth tubed isometric embedding into a Euclidean
space of large dimension.

Note that uniformly $C^{1,1}$-regular metric has bounded sectional curvature
and positive injectivity radius. It is unknown if the converse holds. Also it
is unknown if a manifold with $C^{1,1}$-regular metric admits a
$C^{1,1}$-smooth isometric embedding into Euclidean space. These are two parts
of the question posed by Vitali Kapovitch and Alexander Lytchak [11, 1.10]. If
both questions have affirmative answers, then the presented argument should
produce a converse to the observation in §1.

If the smoothness is below $C^{1,1}$, then there is no hope for tubed
isometric embedding. However, it is easy to see that an embedding $f\colon
M\to\mathbb{R}^{d}$ with bounded normal curvature is tubed if and only if it
is uniformly graphical; that is, there is $r>0$ such that for any $p\in M$ the
set $B(f(p),r)\cap f(M)$ is a graph of a function
$\mathbb{R}^{n}\to\mathbb{R}^{d-n}$ for a suitable choice of coordinate system
in $\mathbb{R}^{d}$. Applying the result of Anders Källén [10], the argument
in §16 produces that for any $\alpha\in(0,1)$, any manifold with
$C^{1,\alpha}$-regular metric and uniformly polynomial growth admits a
uniformly $C^{1,\alpha}$-smooth uniformly graphical embedding into a Euclidean
space of large dimension. In other words, for arbitrary small positive
$\varepsilon$, the $C^{2,\varepsilon}$ and $C^{1,1-\varepsilon}$ versions of
the uniformly graphical question have affirmative answers, but $C^{1,1}$—the
most important case—remains open.

Question. Suppose $M$ is an $n$-dimensional Riemannian manifold of bounded
geometry. What is necessary and sufficient condition for the existence of
tubed isometric embeddings of $M$ into (1) hyperbolic space of sufficiently
large dimension or (2) product of sufficiently many hyperbolic planes?

Note that polynomial growth is a sufficient condition for both questions. It
follows from the main theorem; the second case is trivial and the first case
follows since a horosphere defines a tubed isometric embedding of a Euclidean
space into a hyperbolic space. Evidently this condition is not necessary.

The argument presented by David Hume and Alessandro Sisto [9, 1.1] can be used
to find some necessary conditions; for example, the product of the hyperbolic
plane and the real line does not admit a tubed embedding into hyperbolic space
of arbitrary dimension. Also, a manifold with infinite asymptotic dimension
does not admit a tubed embedding into a product of hyperbolic planes. The
latter assertion relies on the monotonicity of asymptotic dimension with
respect to maps between graphs satisfying the two conditions in §15. Namely,
if $f\colon\Delta\to\Gamma$ is such a map between graphs with bounded degrees,
then $\mathrm{asdim}\,\Delta\leqslant\mathrm{asdim}\,\Gamma$. (The proof is
left to the reader.)

Finally, it would be interesting to understand what happens if we do not
require $N$ to have bounded geometry in §3.

### 18.Acknowledgments.

This note is inspired by the question of Peter Michor [15]. I want to thank
aorq (user:1079 at MathOverflow), Michael Gromov, Nina Lebedeva, James Lee,
Alexander Lytchak, Fedor Petrov, and Danielel Semola, for help.

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