# Hohenberg-Mermin-Wagner-type theorems and dipole symmetry

Anton Kapustin<EMAIL_ADDRESS>California Institute of
Technology, Pasadena, CA 91125, United States  Lev Spodyneiko<EMAIL_ADDRESS>Massachusetts Institute of Technology, Cambridge, MA 02139, United States

###### Abstract

We study the possibility of spontaneous symmetry breaking in systems with both
charge and dipole symmetries. For $d$-dimensional systems at a positive
temperature, we show that charge symmetry cannot be spontaneously broken for
$d\leq 4$, while dipole symmetry cannot be spontaneously broken for $d\leq 2$.
For $T=0$, we show that charge symmetry cannot be spontaneously broken for
$d\leq 2$ if the compressibility is finite. We also show that continuum
systems with a dipole symmetry have infinite inertial mass density.

##  I Introduction

The concepts of spontaneous symmetry breaking and long-range order play a
central role in the theory of phase transitions. R. E. Peierls Peierls (1934,
1935) and L. D. Landau Landau (1937) were the first to realize that in
sufficiently low dimensions thermal fluctuations of the order parameter can
preclude genuine long-range order. The impossibility of long-range order in
one and two dimensions at positive temperatures has been proved by P. C.
Hohenberg Hohenberg (1967) in the case of superfluids and superconductors and
by N. D. Mermin and H. Wagner Mermin and Wagner (1966) for (anti)ferromagnets.
The relevant symmetries in these cases are either $U(1)$ or $SU(2)$. Analogous
theorems for one-dimensional systems at zero temperature were proved by L.
Pitaevskii and S. Stringari Pitaevskii and Stringari (1991). At $T=0$ it is
quantum fluctuations that destroy the order.

The impossibility of symmetry breaking in a particular system does not
guarantee the absence of continuous phase transitions. A textbook example is
the BKT phase transition Berezinskiǐ (1971, 1972); Kosterlitz and Thouless
(1973). In this case fluctuations of the order parameter are qualitatively
different on the two sides of the transition, even though symmetry is not
broken.

The main purpose of this paper is to generalize the above results to
$U(1)$-invariant systems which conserve not only the charge but also the
dipole moment. The additional symmetry possessed by such models is called the
dipole symmetry. A simple non-interacting field theory with a dipole symmetry
is the Lifshitz model which has a single real scalar field
$\phi(t,{\mathbf{x}})$ and an action

$S=\frac{1}{2}\int
dt\,d^{d}x\left({\dot{\phi}}^{2}-C(\partial^{2}\phi)^{2}\right).$ (1)

Here $C$ is a positive constant and $\partial^{2}=\partial_{j}\partial_{j}$.
This model has both a charge symmetry $\phi\mapsto\phi+a$ and a dipole
symmetry $\phi\mapsto\phi+{\mathbf{b}}\cdot{\mathbf{x}}$, where ${\mathbf{b}}$
is a constant vector. An example of a lattice system with a dipole symmetry
dubbed the dipolar Bose-Hubbard model was recently studied by E. Lake et al.
Lake  _et al._ . Its Hamiltonian is given by

$\displaystyle\begin{split}H=\sum_{i}&\Bigg{(}-t\sum_{a}\left[{b^{\dagger}_{i}}^{2}b_{i+a}b_{i-a}+b_{i-a}^{\dagger}b_{i+a}^{\dagger}{b_{i}}^{2}\right]\\\
&-t^{\prime}\sum_{a,b,b\neq
a}\left[b^{\dagger}_{i}b^{\dagger}_{i+a+b}b_{i+a}b_{i+b}+b^{\dagger}_{i+b}b^{\dagger}_{i+a}b_{i+a+b}b_{i}\right]+\frac{U}{2}n_{i}(n_{i}-1)-\mu
n_{i}\Bigg{)},\end{split}$ (2)

where $b_{i}$ are boson creation operators and $n_{i}=b^{\dagger}_{i}b_{i}$,
$i$ goes over square lattice sites and $a,b$ over basis vectors. The dipole
symmetries are generated by $\sum_{j}j_{a}n_{j}$ where $j_{a}$ is the
$a$-coordinate of the site $j$. The usual hoping term is forbidden by the
dipole symmetry and particles can only hop in pairs. Ref. Lake  _et al._ shows
that within the mean-field approximation the model has several different
phases characterized by spontaneous breaking of dipole and/or charge
symmetries.

Our interest in such models stems from a connection between dipole symmetry
and systems where the kinetic energy of charged quasi-particles is suppressed
relative to the potential interaction energy, leading to strongly correlated
physics. Whenever a system is effectively described by weakly interacting
quasi-particles, suppression of the kinetic energy is tantamount to the
flatness of the relevant band. For systems with strongly interacting quasi-
particles, the concept of a flat band is problematic. A possible way out is to
notice that an idealized exactly flat band has an additional symmetry of
shifts by a constant momentum, or equivalently a Galilean boost symmetry.
Invariance of the Hamiltonian under boost can be used as a definition of a
“flat band” which does not rely on the weakness of the interaction. Although
in the experimentally realized systems the band is never exactly flat (i.e.
the commutator of the boost generator and the Hamiltonian is never exactly
zero), the study of such idealized models is a good starting point to tackle
the theoretical challenges of strong interactions in these materials.

To explain the connection to the dipole symmetry, consider a system of
particles with mass $m$ and density-density interactions. The generator of a
shift in momentum space is given by

$\displaystyle{\bf K}=\int d^{d}x\left(t\frac{{\bf p}({\bf x})}{m}-{\bf
x}\rho({\bf x})\right)$ (3)

where ${\bf p}({\bf x}),\rho({\bf x})$ are the momentum and particle densities
and we divided the boost generator by $m$ since we want it to shift momentum
instead of velocity. In the limit $m\rightarrow\infty$ the kinetic energy
become quenched and ${\bf K}$ starts to commute with the Hamiltonian, i.e.
becomes a true symmetry instead of being part of the spectrum-generating
algebra. In this limit, the generator (3) becomes the total dipole moment of
the particles.

Dipole moment conservation on top of charge conservation leads to restricted
mobility of charged particles but does not obstruct the propagation of neutral
bound states. Similar phenomena appear in theories of fractons Haah (2011);
Vijay  _et al._ (2015); Nandkishore and Hermele (2019) and their connection to
dipole symmetry is known Pretko (2017, 2018); Gorantla  _et al._ (2022).
Therefore models with dipole symmetries and fractonic behavior can be used as
a playground for studying systems with vanishing kinetic energy. One can hope
that their exotic features can also manifest themselves in more realistic
“flat-band” systems.

The issue of spontaneous breaking of dipole symmetry is related to the
question of whether systems that are invariant under boost symmetry in the
above sense can nevertheless have quasi-particles with a nonzero dispersion.
To see that this might be possible, consider the famous Landau relation
between the bare mass $m$ and the quasi-particle mass $m^{*}$ in a Fermi
liquid Abrikosov  _et al._ (1975):

$\displaystyle\frac{1}{m^{*}}=\frac{1}{m}-\frac{2p_{f}}{(2\pi)^{3}}\int
d\Omega\,f(\theta)\cos\theta.$ (4)

This relation follows from Galilean invariance. One can see that in the limit
$m\rightarrow\infty$ (and assuming that the system stays in the Fermi liquid
phase) the effective mass of quasiparticles can be generated by the
interaction and remain finite. Since the Fermi surface is not invariant under
translations in momentum space, for this mechanism to work the dipole symmetry
of the $m=\infty$ model must be spontaneously broken.

Within the Lifshitz model, the effects of quantum and thermal fluctuations on
the existence of phases with broken charge or dipole symmetries have been
discussed in Yuan  _et al._ (2020); Stahl  _et al._ (2022) and are reviewed in
Section II. This type of analysis assumes that the only low-energy degrees of
freedom are Goldstone bosons. However, spontaneous breaking of dipole symmetry
(or more generally any symmetry that has a non-trivial commutator with
translations) does not necessarily lead to a Goldstone mode Watanabe and
Vishwanath (2014); Alberte and Nicolis (2020). For example, some Fermi liquids
spontaneously break rotational and/or Galilean symmetries but their low energy
excitation spectrum consists of the particle-hole continuum without any
bosonic collective modes. A further complication is that the choice of the
fields describing the Goldstone modes is not unique. For example, in the
traditional approach to crystalline solids, one does not introduce a separate
field for the rotational symmetry, even though both rotational and
translational symmetries are broken to a discrete subgroup. But the same low-
energy physics can also be described by a field theory with additional fields
which play the role of the Goldstone bosons for rotations Radzihovsky and
Hermele (2020); Radzihovsky (2020). Thus it is desirable to prove theorems of
Hohenberg-Mermin-Wagner (HMW) type without assuming anything about the low-
energy degrees of freedom.

In Section III, we consider a general $d$-dimensional lattice system with a
finite range Hamiltonian that commutes with both charge and dipole moment. We
show that for $T>0$ the dipole symmetry cannot be broken for $d\leq 2$, while
the charge symmetry cannot be broken for $d\leq 4$. Thus the mere presence of
dipole symmetry allows one to strengthen the conclusions of the HMW theorem.
At $T=0$ we show that the charge symmetry cannot be broken for $d\leq 2$ if
the charge compressibility does not diverge. The theorem does not put any
restriction on the dipole symmetry at zero temperature because the dielectric
constant diverges for a general system with a dipole symmetry. Our results put
on firm ground the analysis of Yuan  _et al._ (2020); Stahl  _et al._ (2022)
based on the Lifshitz model.

In Section IV we prove HMW-type theorems for continuum models with a dipole
symmetry. Unlike lattice systems, such models have a conserved momentum
${\mathbf{P}}$, and thus one may also consider a generalized Gibbs ensemble
with a nonzero velocity ${\mathbf{v}}$. We show that the inertial mass density
in continuum models with a dipole symmetry is always infinite. This is a kind
of converse to the statement that dipole symmetry emerges as the infinite-mass
limit of the Galilean symmetry. Section V contains concluding remarks.

We thank Xiaoyang Huang, Ethan Lake, Leo Radzihovsky, and Senthil Todadri for
discussions. AK was supported in part by the U.S. Department of Energy, Office
of Science, Office of High Energy Physics, under Award Number DE-SC0011632. AK
was also supported by the Simons Investigator Award. LS was supported by the
Simons Collaboration on Ultra-Quantum Matter, which is a grant (651446) from
the Simons Foundation.

##  II The Lifshitz model

The Lifshitz model (1) in $d$ spatial dimensions is the simplest field theory
which conserves both charge and its dipole moment. The charge symmetry is
generated by a Hermitian operator $Q$ satisfying $[Q,\phi]=-i$. The dipole
symmetry is generated by a vector-valued Hermitian operator ${\bf D}$
satisfying $[{\bf D},\phi(t,{\mathbf{x}})]=-i{\mathbf{x}}$. Naively, the
Lifshitz model describes a phase where both charge and dipole symmetries are
completely broken, with $\phi$ being the order parameter. Indeed, if
$\langle\cdot\rangle$ denotes averaging over a ground state or a thermal
state, then we have $\langle[Q,\phi(t,{\mathbf{x}})]\rangle=-i\neq 0$ and
$\langle[{\bf D},\phi(t,{\mathbf{x}})]\rangle=-i{\mathbf{x}}\neq 0$. By
definition, this means that symmetries generated by both $Q$ and ${\bf D}$ are
spontaneously broken. But it is well known that quantum and thermal
fluctuations may invalidate this conclusion by making $\phi$ ill-defined. Let
us examine this issue in more detail.

For $T=0$ and $d>2$ the connected two-point function of $\phi$ is

$\langle\phi(0,{\mathbf{x}})\phi(0,{\mathbf{y}})\rangle_{c}\sim\frac{1}{|{\mathbf{x}}-{\mathbf{y}}|^{d-2}}.$
(5)

This is a well-defined two-point function that satisfies the cluster
decomposition. Thus $\phi$ is a well-defined local field that may serve as an
order parameter for both charge and dipole symmetry breaking. In contrast, for
$d=2$ the connected two-point function of $\phi$ is not well-defined, because
its spatial Fourier transform

$\langle\phi(0)_{{\mathbf{k}}}\phi(0)_{{\mathbf{k}}}\rangle_{c}\sim\frac{1}{k^{2}}$
(6)

has a non-integrable singularity at ${\mathbf{k}}=0$.111If one cuts off the
integral at $k=\Lambda_{\rm IR}$, one finds a logarithmically growing two-
point function. Less formally, the average of $\phi(0,{\mathbf{x}})$ over a
large region of diameter $L$ has a standard deviation that grows like $(\log
L)^{1/2}$, indicating that it cannot serve as an order parameter.

On the other hand, the two-point function of the vector-valued operator ${\bf
u}=\nabla\phi$ has an integrable singularity in momentum space, and the two-
point function in coordinate space is well-defined and satisfies the cluster
decomposition:

$\langle
u_{j}(0,{\mathbf{x}})u_{l}(0,{\mathbf{y}})\rangle_{c}\sim\frac{(x_{j}-y_{j})(x_{l}-y_{l})}{|{\mathbf{x}}-{\mathbf{y}}|^{4}}$
(7)

Since $\bf u$ commutes with $Q$, it cannot serve as an order parameter for
charge symmetry breaking. Neither there are any other local fields whose
commutator with $Q$ has a non-zero expectation value. Thus charge symmetry is
unbroken in the $d=2$ Lifshitz model at zero temperature. But since
$\langle[D_{j},u_{l}]\rangle=-i\delta_{jl}$, the dipole symmetry is
spontaneously broken.

The situation for $d=1$ is similar. The two-point function of the field $\phi$
in momentum space again has a non-integrable singularity (6). The standard
deviation of the spatial average of $\phi$ grows like $\sqrt{L}$ for large
$L$. Since one lacks any order parameter for the charge symmetry breaking, it
remains unbroken. But the two-point function of the field $u=\partial_{x}\phi$
is constant in momentum space and thus we have

$\langle u(0,x)u(0,y)\rangle_{c}\sim\delta(x-y).$ (8)

This is a well-defined (although somewhat unusual) answer, and $u$ can serve
as an order parameter for the dipole symmetry breaking. Thus the dipole
symmetry in the $d=1$ Lifshitz model is broken at $T=0$, as noted in Stahl
_et al._ (2022).222Spontaneous breaking of a continuous symmetry in a 1d
system might seem surprising, but it does not contradict any known general
theorems. Another example is a system of non-relativistic fermions in 1d which
spontaneously breaks boost invariance by forming a Luttinger liquid.

For $T>0$ the static two-point function of the Lifshitz field $\phi$ in
momentum space is

$\langle\phi(0)_{{\mathbf{k}}}\phi(0)_{{\mathbf{k}}}\rangle_{c}\sim\frac{1}{k^{4}}$
(9)

For $d>4$ it is integrable, in which case the connected thermal two-point
function of $\phi$ in coordinate space is well-defined. Since $\phi$ is well-
defined and shifts under charge symmetry, the charge symmetry (and therefore
also the dipole symmetry) are spontaneously broken for $d>4$. For $d\leq 4$,
the field $\phi$ is not well-defined, and there are no other order parameters
for charge symmetry breaking. Thus the charge symmetry is not broken for
$d\leq 4$. As for dipole symmetry, for $d>2$ it is spontaneously broken, while
for $d\leq 2$ it is also unbroken. Indeed, for $d>2$ we can take
$u_{j}=\partial_{j}\phi$ as an order parameter, since its connected thermal
two-point function has an integrable singularity $\sim 1/k^{2}$ in momentum
space and decays as $1/x^{d-2}$ in coordinate space.

##  III Lattice systems

###  III.1 Definitions

We will consider a lattice system with discrete translation symmetry and
finite-range interactions. More formally, let $\Lambda$ be a lattice in
$\mathbb{R}^{d}$ with an action of translation group $\mathbb{Z}^{d}$. The
Hilbert space is the product of an infinite number of copies of a finite-
dimensional Hilbert space of a single site. Local operators are operators
which act non-trivially only on a finite number of sites. We can formally
combine all sites in a single unit cell into a single site and reduce the
lattice $\Lambda$ to $\mathbb{Z}^{d}$ without a loss of generality. If the
translational symmetry is broken to a sublattice symmetry (as in
antiferromagnets), we can formally combine the resulting unit cell to one cell
and again assume that the lattice is $\mathbb{Z}^{d}$.

The Hamiltonian density operator $H_{\bf n}$ is assumed to be translationally
invariant and finite-range. The integer-valued index ${\bf n}$ enumerates the
lattice $\mathbb{Z}^{d}$ sites. Moreover, we assume that the system has a
charge symmetry and an associated dipole symmetry. Namely, there exist charge
density operators $Q_{\bf n}$ which are finite-range, translationally
invariant, and satisfy

$\displaystyle[Q_{\bf n},Q_{\bf m}]$ $\displaystyle=0,$ (10)
$\displaystyle\sum_{{\bf n}\in\mathbb{Z}^{d}}[Q_{\bf n},H_{\bf m}]$
$\displaystyle=0,$ (11) $\displaystyle\sum_{{\bf
n}\in\mathbb{Z}^{d}}n_{i}[Q_{\bf n},H_{\bf m}]$
$\displaystyle=0,\quad\text{for all $i=1,\dots,d$},$ (12)

where $n_{i}$ is $i$-th component of the ${\bf n}$ in $\mathbb{Z}^{d}$. The
latter condition means the that dipole moment is conserved. We will define the
dipole charge density as $D^{i}_{\bf n}=n_{i}Q_{\bf n}$ and the dipole charge
as $D^{i}=\sum_{{\bf n}\in\mathbb{Z}^{d}}D_{\bf n}^{i}$. The operators $Q_{\bf
n},H_{\bf n}$ are assumed to be local operators supported on balls of diameter
$d_{\rm int}$ centered at ${\bf n}$.

We also assume that the state of the system has a clustering property. More
precisely, following Martin (1982); Momoi (1996), we assume that for any two
local operators $A,B$ one has

$\displaystyle|\langle AT_{\bf n}(B)\rangle-\langle A\rangle\langle
B\rangle|\leq\frac{C_{AB}}{|{\bf n}|^{\delta_{AB}}}\quad\text{as}\quad|{\bf
n}|\rightarrow\infty,$ (13)

where $T_{\bf n}$ is the translation operator and $\delta_{AB},C_{AB}$ are
positive numbers. In reality, we only use the clustering property when $A=B$
and $A$ is a local operator which is a candidate for the order parameter. Some
sort of clustering is also implicit in Hohenberg (1967); Mermin and Wagner
(1966), since it is assumed there that the Fourier transform of a two-point
correlator is a well-defined integrable function rather than a distribution.

###  III.2 HMW theorem for charge symmetry at a non-zero temperature

We repeat the argument of Momoi (1996); Martin (1982) making only small
adjustments. Suppose the charge symmetry is spontaneously broken, i.e. there
exists a local operator $A_{0}$ such that

$\langle[Q,A_{0}]\rangle\neq 0,$ (14)

where brackets denote the thermal average. Without a loss of generality, we
may assume that $A_{0}$ is Hermitian, has zero average $\langle
A_{0}\rangle=0$ and is localized around the point ${\bf n}=0$.

We will consider a spatially-averaged version of $A_{0}$ defined as

$\displaystyle A_{R}=\frac{1}{{\rm Vol}(B^{d}_{R})}\sum_{|{\bf n}|\leq
R}T_{\bf n}(A_{0}),$ (15)

where $T_{\bf n}$ is the translation operator, $R$ is some number, and ${\rm
Vol}(B^{d}_{R})$ is the number of lattice sites in a $d$-dimensional ball of
radius $R$. We assume that the full $\mathbb{Z}^{d}$ translational symmetry is
preserved and thus

$\langle[Q,A_{0}]\rangle=\langle[Q,A]\rangle.$ (16)

By increasing $d_{\rm int}$ if necessary, we may assume that $A_{0}$ is
supported on a ball of diameter $d_{\rm int}$ with a center at $0$. Then
$A_{R}$ is supported on a ball of diameter $2R+d_{\rm int}$ centered at $0$.

Without changing the commutator, we can also replace $Q$ in eq. (16) with
$Q(f)=\sum_{{\bf n}\in\mathbb{Z}^{d}}f({\bf n})Q_{\bf n}$, where $f({\bf n})$
is any compactly supported function equal to $1$ whenever ${\bf n}$ belongs to
a ball of diameter $2R+3d_{\rm int}$ centered at $0$.

We will use the Bogoliubov inequality valid for any two bounded operators
$A,B$ and a thermal (Kubo-Martin-Schwinger) state at a nonzero temperature
$T$:

$\displaystyle|\langle[A,B]\rangle|^{2}\leq\frac{1}{2T}\langle[B,[H,B^{\dagger}]]\rangle\langle
A^{\dagger}A+AA^{\dagger}\rangle.$ (17)

We will apply it to $A=A_{R}$ and $B=Q(f)$.

####  III.2.1 Estimates

We want to estimate $\langle[Q(f),[Q(f),H]]\rangle$. We can rewrite the
commutators as

$\displaystyle\begin{split}[Q(f),H]&=\sum_{{\bf m},{\bf k}}f({\bf m})[Q_{\bf
m},H_{\bf k}]=\sum_{{\bf m},{\bf k}}(f({\bf m})-f({\bf k}))[Q_{\bf m},H_{\bf
k}],\\\ [Q(f),[Q(f),H]]&=\sum_{{\bf n},{\bf m},{\bf k}}f({\bf n})(f({\bf
m})-f({\bf k}))[Q_{\bf n},[Q_{\bf m},H_{\bf k}]]\\\ &=\sum_{{\bf n},{\bf
m},{\bf k}}(f({\bf n})-f({\bf k}))(f({\bf m})-f({\bf k}))[Q_{\bf n},[Q_{\bf
m},H_{\bf k}]],\end{split}$ (18)

The commutator $\langle[Q_{\bf n},[Q_{\bf m},H_{\bf k}]]\rangle$ is non-zero
only if the supports of $H_{\bf k}$, $Q_{\bf n}$ and $Q_{\bf m}$ overlap. The
expectation value can be estimated as

$\displaystyle\left|\langle[Q_{\bf n},[Q_{\bf m},H_{\bf k}]]\rangle\right|\leq
4\|Q_{0}\|^{2}\|H_{0}\|,$ (19)

where the operator norm is defined as
$\|A\|=\max_{v\in\mathcal{H}}\frac{|Av|}{|v|}$.

We can expand the function $f$ in the Taylor series around $\bf k$ with the
reminder in the Lagrange form

$\displaystyle f({\bf n})=f({\bf k})+(({\bf n}-{\bf k})\cdot\nabla)f({\bf
k})+\frac{1}{2}(({\bf n}-{\bf k})\cdot\nabla)^{2}f(\xi),$ (20)

where $({\bf n}-{\bf k})\cdot\nabla$ is derivative in the direction of the
vector ${\bf n}-{\bf k}$ and $\xi$ is point on the straight interval from
${\bf k}$ to ${\bf n}$. Due to dipole symmetry linear terms proportional to
$(({\bf n}-{\bf k})\cdot\nabla)f({\bf k})$ and $(({\bf m}-{\bf
k})\cdot\nabla)f({\bf k})$ in (18) will be zero and we can estimate

$\displaystyle\left|\langle[Q(f),[Q(f),H]]\rangle\right|\leq
C_{d}R_{f}^{d}d_{\rm
int}^{4}\|Q_{0}\|^{2}\|H_{0}\|\big{(}\max|\nabla^{2}f|\big{)}^{2},$ (21)

where $R_{f}$ is half of the diameter of the support of the function $f$ and
$C_{d}$ is a constant that depends only on $d$.

Choosing the function $f({\bf r})$ to be $1$ for $r\leq R+\frac{3}{2}d_{\rm
int}$, $0$ for $r>2R$ and smoothly interpolating between these two values with
the second derivatives of order $O(\frac{1}{R^{2}})$, we find

$\displaystyle\langle[Q(f),[Q(f),H]]\rangle=O(R^{d-4}).$ (22)

Using the clustering property and letting $\delta=\delta_{A_{0}A_{0}}$, we can
estimate

$\displaystyle\begin{split}\langle
A_{R}^{\dagger}A_{R}+A_{R}A_{R}^{\dagger}\rangle=\frac{2}{{{\rm
Vol}(B^{d}_{R})}^{2}}\sum_{|{\bf n}|\leq R,|{\bf m}|\leq R}\langle T_{\bf
n}(A_{0})T_{\bf m}(A_{0})\rangle\\\ <\frac{2C_{A_{0}A_{0}}}{{{\rm
Vol}(B^{d}_{R})}}\sum_{|{\bf n}|<2R}\frac{1}{|{\bf
n}|^{\delta}}=\begin{cases}O\left(\frac{1}{R^{\delta}}\right)&\text{if}\quad\delta<d,\\\
O\left(\frac{\ln R}{R^{d}}\right)&\text{if}\quad\delta=d,\\\
O\left(\frac{1}{R^{d}}\right)&\text{if}\quad\delta>d.\end{cases}\end{split}$
(23)

Combining the two estimates and substituting them into the Bogoliubov
inequality we find

$\displaystyle|\langle[Q,A_{0}]\rangle|^{2}=|\langle[Q(f),A_{R}]\rangle|^{2}=O(R^{d-4-\delta}).$
(24)

Since $R$ is arbitrary, this means that for $d\leq 4$ and any local $A_{0}$ we
have $\langle[Q,A_{0}]\rangle=0$. Thus the charge symmetry cannot be
spontaneously broken for $d\leq 4$ if the temperature is non-zero.

###  III.3 HMW theorem for the dipole symmetry at a non-zero temperature

We can use the estimates from the previous section to constrain dipole
symmetry breaking.

Suppose dipole symmetry is spontaneously broken, with a local order parameter
$A_{0}$:

$\langle[D^{i},A_{0}]\rangle\neq 0.$ (25)

We will assume that the charge symmetry is not broken333If the charge symmetry
is broken then the dipole symmetry is broken as well.. This will allow us to
act with a translation operator on the left-hand side of (25) without
affecting the right-hand side. We will consider a spatially-averaged $A_{0}$
given by (15). We can replace $D^{i}$ in this equation with
$D^{i}(g)=\sum_{{\bf n}\in\mathbb{Z}^{d}}g({\bf n})D^{i}_{\bf n}$, where
$g({\bf n})$ is any compactly supported function equal to $1$ whenever ${\bf
n}$ belongs to a ball of diameter $2R+3d_{\rm int}$ centered at $0$.

We will use the Bogoliubov inequality (17) with $A=A_{R}$ and $B=D^{i}(g)$.

####  III.3.1 Estimates

We now want to estimate

$\displaystyle\langle[D^{i}(g),[D^{i}(g),H]]\rangle$ $\displaystyle=\sum_{{\bf
n},{\bf m}}g({\bf n})g({\bf m})\langle[D_{\bf n}^{i},[D^{i}_{\bf
m},H]]\rangle$ $\displaystyle=\sum_{{\bf n},{\bf m}}n^{i}g({\bf n})m^{i}g({\bf
m})\langle[Q_{\bf n},[Q_{\bf m},H]]\rangle.$

The considerations of the previous section did not use a specific form of the
function $f$. Therefore, we can use the same estimate (21) with the function
$f({\bf n})=n^{i}g({\bf n})$:

$\displaystyle|\langle[D^{i}(g),[D^{i}(g),H]]\rangle|\leq
C_{d}(R_{f})^{d}d_{\rm
int}^{4}\|Q_{0}\|^{2}\|H_{0}\|\big{(}\max|\nabla^{2}f|\big{)}^{2}.$ (26)

Since $g({\bf r})$ must be $1$ for $r\leq R+\frac{3}{2}d_{\rm int}$ the
function $f({\bf r})$ must be equal to $x^{i}$ for $r\leq R+\frac{3}{2}d_{\rm
int}$. We can smoothly interpolate it to zero for $r>2R$ while keeping the
second derivatives bounded by $O(\frac{1}{R})$. Then we get

$\displaystyle|\langle[D^{i}(f),[D^{i}(f),H]]\rangle|=O(R^{d-2}).$ (27)

Using this estimate, the Bogoliubov inequality, and the estimate (23), we find

$\displaystyle|\langle[D^{i},A_{0}]\rangle|^{2}=|\langle[D^{i}(f),A]\rangle|^{2}=O(R^{d-2-\delta}).$
(28)

Since $R$ is arbitrary, the dipole symmetry cannot be spontaneously broken in
$d\leq 2$ at non-zero temperatures.

###  III.4 HMW-type theorems at zero temperature

At zero temperatures instead of the Bogoliubov inequality we use the
uncertainty relation Pitaevskii and Stringari (1991):

$\displaystyle|\langle[A,B]\rangle|^{2}\leq\langle
A^{\dagger}A+AA^{\dagger}\rangle\langle B^{\dagger}B+BB^{\dagger}\rangle,$
(29)

where brackets denote the average over any state and we assumed that the
operators $A,B$ have zero averages. We will apply it to the case when
$A=A_{R}-\langle A_{R}\rangle$, $B=Q(f)-\langle Q(f)\rangle$, and the state is
the ground state of $H$. We can upper-bound the right-hand side of this
inequality in terms of more physical quantities as follows. Recall that for
any two local operators $B,B^{\prime}$ one defines the structure factor
$S_{BB^{\prime}}(\omega)$ as

$S_{BB^{\prime}}(\omega)=\int_{-\infty}^{\infty}\left[\langle
B(t)B^{\prime}\rangle-\langle B\rangle\langle
B^{\prime}\rangle\right]e^{i\omega t}dt.$ (30)

Since the average is over the ground state, $S_{BB^{\prime}}(\omega)=0$ for
$\omega<0$. If $B^{\prime}=B^{\dagger}$, then $S_{BB^{\dagger}}(\omega)$ is
positive and is known as the spectral density of $B$.

Another useful quantity is the static susceptibility $\chi_{BB^{\prime}}$
defined as the change in the expectation value of $B$ under an infinitesimal
variation of the Hamiltonian by $-\epsilon B^{\prime}$:

$\langle B\rangle_{H-\epsilon B^{\prime}}=\langle
B\rangle_{H}+\epsilon\chi_{BB^{\prime}}+O(\epsilon^{2}).$ (31)

Here it is assumed that $B,B^{\prime}$ are Hermitian, but one can formally
extend $\chi_{BB^{\prime}}$ to general operators by linearity. In linear
response theory, it is shown that $\chi_{BB^{\prime}}$ can be expressed
through $S_{BB^{\prime}}(\omega)$. At $T=0$ the formula looks as follows:

$\chi_{BB^{\prime}}=\int_{0}^{\infty}\frac{S_{BB^{\prime}}(\omega)+S_{B^{\prime}B}(\omega)}{2\omega}\frac{d\omega}{\pi}.$
(32)

The static susceptibility is not always well-defined because of a possible
non-integrable singularity at $\omega=0$. Assuming it exists for
$B=B^{\prime}=Q(f)-\langle Q(f)\rangle$ and using the Cauchy-Schwarz
inequality, we get an estimate:

$\begin{split}\langle B^{\dagger}B+BB^{\dagger}\rangle=2\langle
B(t)B\rangle\Big{|}_{t=0}=\int_{0}^{\infty}S_{BB}(\omega)\frac{d\omega}{\pi}\\\
\leq\sqrt{\int_{0}^{\infty}\omega
S_{BB}(\omega)\frac{d\omega}{\pi}\int_{0}^{\infty}\frac{S_{BB}(\omega)}{\omega}\frac{d\omega}{\pi}}=\sqrt{|\langle[B,[H,B]]\rangle|}\sqrt{\chi_{BB}}.\end{split}$
(33)

The quantity $\chi_{BB}$ depends on the function $f$ and thus also on $R$, but
with the choice of $f$ described at the end section III.2.1 it can be upper-
bounded by a number of order $R^{d}$ times an $R$-independent quantity with a
simple thermodynamic interpretation:

$\displaystyle\chi_{BB}=\sum_{{\bf n},{\bf m}}f({\bf n})f({\bf m})\chi_{Q_{\bf
n}Q_{\bf m}}\leq C_{d}R^{d}\sum_{\bf n}\chi_{Q_{0}Q_{\bf
n}}+o(R^{d})=C_{d}R^{d}\frac{\partial\langle
Q_{0}\rangle}{\partial\mu}+o(R^{d}).$ (34)

Here $\mu$ is the chemical potential. The non-negative quantity
$\partial\langle Q_{0}\rangle/\partial\mu$ is the derivative of the charge
density with respect to the chemical potential, also known as charge
compressibility. We assumed here that it is finite.

Combining (22, 23,29, 34) we find

$\displaystyle|\langle[Q,A]\rangle|^{2}\leq\sqrt{\frac{\partial\langle
Q_{0}\rangle}{\partial\mu}}\times O(R^{d-2-\delta}).$ (35)

Therefore if charge compressibility is finite and the ground state is
clustering, charge symmetry cannot be broken for $d\leq 2$. Note that in the
absence of dipole symmetry one can only show that the charge symmetry cannot
be broken for $d\leq 1$.

We cannot use the same method to exclude dipole symmetry breaking in $d=1$.
The main issue is that the analog of susceptibility for the dipole symmetry is

$\displaystyle\chi_{D^{i}(f)D^{i}(f)}=\sum_{{\bf n},{\bf m}}n^{i}m^{i}f({\bf
n})f({\bf m})\chi_{Q_{\bf n},Q_{\bf m}},$ (36)

Even if we assume that $\chi_{Q_{\bf n},Q_{\bf m}}$ is essentially non-zero
only if ${\bf n},{\bf m}$ are close to each other, the sum behaves as
$O(R^{d+2})$. This is not enough to exclude symmetry breaking in $d=1$. In
fact, the Lifshitz model shows that dipole symmetry can be spontaneously
broken in $d=1$ for $T=0$.

###  III.5 The dipolar Bose-Hubbard model

The above theorems apply, in particular, to lattice models of fermions with a
dipole symmetry. They cannot be directly applied to the dipolar Bose-Hubbard
model because the on-site Hilbert space is infinite-dimensional, and the
operators $Q_{\bf n},H_{\bf n}$ are unbounded. However, after a slight
modification, the proofs work for this model too, if local charge fluctuations
in the relevant equilibrium state are not too wild. Indeed, we can write

$H_{\bf n}=H_{\bf n}^{(1)}+H_{\bf n}^{(2)},\quad$ (37)

where $H_{\bf n}^{(1)}$ is the dipole hopping term and $H_{\bf n}^{(2)}$ is a
function of only $Q_{\bf m}$ on the neighboring sites. Then

$[Q_{\bf m},H_{\bf k}]=[Q_{\bf m},H_{\bf k}^{(1)}]=\epsilon_{{\bf m}-{\bf
k}}H^{(1)}_{\bf k},$ (38)

where $\epsilon_{{\bf m}-{\bf k}}$ is $\pm 2$ if ${\bf m}={\bf k}$, $\pm 1$ if
${\bf m},{\bf k}$ are nearest neighbors, and $0$ otherwise. Therefore the
estimate (19) can be replaced with

$\left|\langle[Q_{\bf n},[Q_{\bf m},H_{\bf k}]]\rangle\right|\leq
C_{d}\left|\langle H_{0}\rangle\right|,$ (39)

where $C_{d}$ depends only on $d$. Further, it is easy to see that

$\left|\langle H_{0}\rangle\right|\leq C^{\prime}_{d}|t|\left|\langle Q_{\bf
n}^{2}\rangle\right|,$ (40)

where $C^{\prime}_{d}$ is another $d$-dependent constant. Thus if we assume
that fluctuations of the charge density have a finite standard deviation, we
can still bound the double commutator by a (state-dependent) constant. Note
also that while the local operator $A_{0}$ need not be bounded (for example,
it could be $b_{0}$), the clustering property implies that it has a finite
standard deviation. Then we still get the estimates (24), (28) and (35).

Similar arguments can be used for more general systems with unbounded $Q_{\bf
n}$ and $H_{\bf n}$. As long as the fluctuations of the constituents are not
too wild one can bound the expectation value on the left-hand side of (19).

##  IV Continuum theories

###  IV.1 HMW-type theorems for continuum theories

For continuum systems, the momentum density operator $p_{j}({\bf x})$ is an
obvious candidate for an order parameter for the dipole symmetry. Indeed, if
we choose the dipole charge density to be $d_{j}({\bf x})=x_{j}\rho({\bf x})$,
where $\rho({\bf x})$ is the charge density, we find

$\displaystyle[D_{j},p_{k}({\bf x})]=i\delta_{jk}\rho({\bf x}).$ (41)

Thus, as long as the average charge density is not zero, the dipole symmetry
appears to be broken. This statement does not depend on either temperature of
dimensionality or space and suggests that the theorem of section III.3 cannot
be extended to continuum models with a dipole symmetry.

In this section using a specific Hamiltonian as an example, we will show that
the arguments of previous sections can be generalized without essential change
to the continuum case. We discuss how the apparent contradiction is avoided.

Consider a bosonic field $\psi$ with canonical commutation relations

$\displaystyle[\psi({\bf x}),\psi^{\dagger}({\bf y})]$
$\displaystyle=\delta({\bf x}-{\bf y}),$ (42) $\displaystyle[\psi({\bf
x}),\psi({\bf y})]$ $\displaystyle=[\psi^{\dagger}({\bf
x}),\psi^{\dagger}({\bf y})]=0,$ (43)

and the Hamiltonian

$\displaystyle H$ $\displaystyle=H_{\rm kin}+H_{\rm pot},$ (44)

where the kinetic energy is

$\displaystyle H_{\rm kin}$ $\displaystyle=\int
d^{d}x\Big{[}\frac{K_{1}}{2}\Big{|}\partial_{j}\psi\partial_{k}\psi-\psi\partial_{j}\partial_{k}\psi\Big{|}^{2}+\frac{K_{2}}{2}(\partial_{i}\psi^{\dagger}\partial_{j}\psi^{\dagger}-\psi^{\dagger}\partial_{j}\partial_{k}\psi^{\dagger})(\partial_{j}\psi\partial_{k}\psi-\psi\partial_{j}\partial_{k}\psi)\Big{]},$

and the potential energy depends only on the density $\rho=\psi^{\dagger}\psi$
but is arbitrary otherwise and can include long-range interactions. This
Hamiltonian is invariant under charge and dipole transformations
$\psi\rightarrow\exp(ia+ib_{j}x_{j})\psi$. Similar Hamiltonians were studied
in Yuan  _et al._ (2020); Jensen and Raz .

The charge conservation equation has the form

$\displaystyle\dot{\rho}$ $\displaystyle=\partial_{j}\partial_{k}j_{jk},$ (45)

where we defined

$\displaystyle j_{jk}$
$\displaystyle=i\frac{K_{1}}{2}\,\big{(}\partial_{j}\psi^{\dagger}\partial_{k}\psi^{\dagger}-\psi^{\dagger}\partial_{j}\partial_{k}\psi^{\dagger}\big{)}\psi^{2}+i\frac{K_{2}}{2}\,\delta_{jk}\big{(}\partial_{l}\psi^{\dagger}\partial_{l}\psi^{\dagger}-\psi^{\dagger}\partial^{2}\psi^{\dagger}\big{)}\psi^{2}+h.c.$
(46)

The commutator of the charge density with the current is

$\displaystyle[\rho({\bf x}),j_{jk}({\bf
y})]=-iK_{1}\partial_{j}\partial_{k}\delta({\bf x}-{\bf
y}){\psi^{\dagger}}^{2}\psi^{2}({\bf
y})-iK_{2}\delta_{jk}\partial^{2}\delta({\bf x}-{\bf
y}){\psi^{\dagger}}^{2}\psi^{2}({\bf y}).$ (47)

Using this commutation relation, we can do manipulations analogous to (18):

$\displaystyle\begin{split}\int d^{d}xd^{d}y\,f({\bf x})f({\bf
y})\langle[\rho({\bf x}),[\rho({\bf y}),H]]\rangle=i\int d^{d}xd^{d}y\,f({\bf
x})f({\bf y})\langle[\rho({\bf x}),\partial_{j}\partial_{k}j_{jk}({\bf
y})]\rangle\\\ =\int
d^{d}xd^{d}y\,\Big{(}K_{1}\partial_{j}\partial_{k}f(x)\partial_{j}\partial_{k}f({\bf
y})+K_{2}\partial^{2}f(x)\partial^{2}f({\bf
y})\Big{)}\langle{\psi^{\dagger}}^{2}\psi^{2}({\bf y})\rangle.\end{split}$
(48)

We can again apply the Bogoliubov inequality (17) to $A=A_{R}=\dfrac{1}{{\rm
Vol}\,B_{R}^{d}}\int_{B_{R}^{d}}d^{d}x\,\Big{(}a({\bf x})-\langle a({\bf
x})\rangle\Big{)}$ and $B=D_{j}(g)=\int d^{d}x\,x_{j}g({\bf x})\rho({\bf x})$,
where $a({\bf x})$ is order parameter, $B_{R}^{d}$ is a $d$-dimensional ball
of radius $R$ centered at $0$, and the function $g({\bf x})$ is chosen to be
$1$ inside of this ball and interpolates to 0 outside of $|{\bf x}|>2R$ so
that second derivatives of $f({\bf x})=x_{j}g({\bf x})$ are bounded by
$O(\frac{1}{R})$. We find

$\displaystyle\Big{|}\langle[D,a(0)]\rangle\Big{|}^{2}\leq\langle[\rho(f),[H,\rho(f)]]\rangle\langle
A_{R}^{\dagger}A_{R}+A_{R}A_{R}^{\dagger}\rangle$ (49)

We can use (48) to show that

$\displaystyle\langle[\rho(f),[H,\rho(f)]]\rangle=O(R^{d-2}).$ (50)

Using the estimate (23), we find

$\displaystyle\Big{|}\langle[D,a(0)]\rangle\Big{|}^{2}\leq O(R^{d-2-\delta}).$
(51)

Similarly, one can generalize the arguments for the charge symmetry at zero or
nonzero temperature leading to the same results as for lattice systems.

There are several ways in which the HMW theorem for the dipole symmetry and
the commutation relation (41) can be reconciled. Most obviously, the momentum
density operator may fail to satisfy the clustering property. For example,
take the Hamiltonian (44) and choose the potential term to have a minimum at
$|\rho|=\rho_{0}$. Then we can ignore the fluctuation of the modulus of $\psi$
and write approximately $\psi=\rho_{0}\exp(i\phi)$. The effective action for
$\phi$ is

$\displaystyle\mathcal{S}_{\rm eff}=\frac{k}{2}\int
d^{d}xdt\,\Big{[}\dot{\phi}^{2}-C_{1}(\partial^{2}\phi)^{2}\Big{]}.$ (52)

This is the Lifshitz model discussed in Section II. The charge density
operator in this approximation becomes $k\dot{\phi}$. Thus in a state with a
constant charge density we have $\phi=\mu t+\eta$, where $\mu$ is the chemical
potential and $\eta$ is the fluctuating field with zero expectation value. The
momentum density operator is $p_{j}=k\dot{\phi}\partial_{j}\phi$. To leading
order in $\eta$ we get $p_{j}=k\mu\partial_{j}\eta$. As discussed in Section
II, the two-point function of $\partial_{j}\eta$ grows with distance if $T>0$
and $d=1,2$. Therefore the same is true for $p_{j}$ provided $\mu\neq 0$. Note
that if the two-point function of a local operator grows with distance,
fluctuations of its spatial average increase as the averaging region becomes
larger. This is why it is natural to require an operator serving as an order
parameter to satisfy clustering.

Another possible loophole in the HMW theorem is that the spatially-smeared
momentum density operator may fail to be bounded, and thus the expression
$\langle A_{R}^{\dagger}A_{R}+A_{R}A_{R}^{\dagger}\rangle$ may be infinite.
For this to happen, $\langle p_{j}({\bf x})p_{k}({\bf y})\rangle$ must have a
non-integrable singularity at ${\bf x}={\bf y}$. However, this is a short-
distance issue that can be fixed by replacing the momentum density operator
with a suitably regularized expression.

###  IV.2 Inertial mass density

In Section I we motivated the dipole symmetry by arguing that it emerges in
the limit of infinite particle mass. For a large class of continuous systems
with dipole symmetry, one can be more precise and prove that they have
infinite inertial mass density.

It is well-known that the Bogolyubov inequality can be strengthened as
follows:

$\left|\langle[A,B]\rangle\right|^{2}\leq\chi_{AA^{\dagger}}\,\langle[B,[H,B^{\dagger}]]\rangle.$
(53)

This is simply the Cauchy-Schwarz inequality for the Hermitian inner product
$(A,C)=\chi_{AC^{\dagger}}$ applied to $C=i[H,B]$. Let us apply it to
Hermitian operators $A=\dfrac{1}{{\rm
vol}\,B_{R}^{d}}\int_{B^{d}_{R}}p_{j}({\bf x})d^{d}x$ and $B=D_{j}(g)$, as
above. Assuming that translational symmetry is unbroken, we get

$\langle\rho(0)\rangle^{2}\leq\chi_{AA}\,\langle[\rho(f),[H,\rho(f)]]\rangle.$
(54)

Here $f({\mathbf{x}})=x_{j}g({\mathbf{x}})$ as before. For the model
considered in Section IV.1, or for any continuum theory with a dipole symmetry
where the commutator of $\rho$ and $j_{jk}$ has the form similar to (47), we
have the estimate (50). Hence

$\chi_{AA}\geq\langle\rho(0)\rangle^{2}\times O(R^{2-d}).$ (55)

To relate $\chi_{AA}$ to inertial mass density, recall that whenever momentum
is conserved, one can define a generalized Gibbs ensemble by replacing the
Hamiltonian $H$ with $H-{\bf v}\cdot{\bf P}$, where ${\bf P}$ is the total
momentum. The chemical potential for ${\bf P}$ is the velocity ${\bf v}$. It
is natural to define the inertial mass density as the tensor

$m_{jk}=\frac{\partial\langle p_{j}(0)\rangle}{\partial
v_{k}}=\int\chi_{p_{j}(0)p_{k}({\bf x})}d^{d}x.$ (56)

If $m_{jk}$ is finite, then $\chi_{AA}\leq m_{jj}\times O(R^{-d})$, which
contradicts the inequality (55) for any nonzero $\langle\rho(0)\rangle$.

##  V Discussion

In this paper, we generalized the Hohenberg-Mermin-Wagner theorem to the
dipole symmetry. We showed that for systems with finite-range interactions and
clustering, dipole symmetry cannot be broken if $d=1,2$ and $T>0$. Thus a
system of fermions with a microscopic dipole symmetry cannot flow to a Fermi
liquid or any phase with a pronounced Fermi surface. Any phase of such a
system must be rather exotic.

We also showed that the mere presence of a dipole symmetry allows one to
strengthen the conclusions of the HMW and Pitaevskii-Stringari theorems as
applied to charge symmetry. Namely, with the same assumptions, the charge
symmetry cannot be broken if $d\leq 4,T>0$ and for $d\leq 2,T=0$ and finite
compressibility.

Systems with a dipole symmetry provide a counter-example to the widespread
belief that spontaneous breaking of charge symmetry implies superconductivity
(i.e. infinite d.c. conductivity). Indeed, for any system with a dipole
symmetry the current operator vanishes at zero wave-vector and thus
conductivity vanishes at all frequencies. On the other hand, the example of
the $d=3$ Lifshitz model at $T=0$ shows that systems with a dipole symmetry
can break charge symmetry.

Charge symmetry and continuous translational symmetry are alike in some
respects. For example, HMW-type theorems for them are essentially the same.
Superficially, dipole symmetry should be similar to a continuous rotational
symmetry: the dipole moment is an integral of charge density times a linear
function of coordinates, just like the angular momentum is the integral of
momentum density times a linear function of coordinates. Therefore it might
seem surprising that rotational symmetry can be spontaneously broken in two
dimensions for $T>0$ Mermin (1967), while dipole symmetry cannot. The
difference arises from the different group structure of the full symmetry
group and the corresponding constraints on the local conservation laws. Dipole
symmetry requires the charge current to be a total derivative:
$j_{j}=-\partial_{k}j_{jk}$. In contrast, rotational symmetry does not require
the momentum current (i.e. the stress tensor) to be a total derivative, it
only requires it to be symmetric. Consequently, the bounds resulting from the
Bogolyubov inequality do not preclude the spontaneous breaking of rotational
symmetry in two dimensions. This difference is also responsible for the
different structure of the effective action for the Goldstone bosons.

Finally, it would be interesting to consider similar issues for systems of
fermions in a magnetic field projected to the lowest Landau level. They have
an infinite-dimensional GMP symmetry Girvin  _et al._ (1986) which contains a
1d dipole symmetry.

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# OGLE-2017-BLG-1434Lb: Confirmation of a Cold Super-Earth using Keck Adaptive
Optics

J.W. Blackman School of Natural Sciences, University of Tasmania, Private Bag
37 Hobart, Tasmania 7001 Australia  Sorbonne Universités, UPMC Université
Paris 6 et CNRS, UMR 7095, Institut d’Astrophysique de Paris, 98 bis bd Arago,
75014 Paris, France  J.-P. Beaulieu School of Natural Sciences, University of
Tasmania, Private Bag 37 Hobart, Tasmania 7001 Australia  Sorbonne
Universités, UPMC Université Paris 6 et CNRS, UMR 7095, Institut
d’Astrophysique de Paris, 98 bis bd Arago, 75014 Paris, France  A.A. Cole
School of Natural Sciences, University of Tasmania, Private Bag 37 Hobart,
Tasmania 7001 Australia  N. Koshimoto  Laboratory for Exoplanets and Stellar
Astrophysics, NASA/Goddard Space Flight Center, Greenbelt, MD 20771, USA
Department of Astronomy, Graduate School of Science, The University of Tokyo,
7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan  A. Vandorou School of Natural
Sciences, University of Tasmania, Private Bag 37 Hobart, Tasmania 7001
Australia  A. Bhattacharya  Laboratory for Exoplanets and Stellar
Astrophysics, NASA/Goddard Space Flight Center, Greenbelt, MD 20771, USA
J.-B. Marquette Associated with Sorbonne Universités, UPMC Université Paris 6
et CNRS, UMR 7095, Institut d’Astrophysique de Paris, 98 bis bd Arago, 75014
Paris, France  Laboratoire d’astrophysique de Bordeaux, Univ. Bordeaux, CNRS,
B18N, allée Geoffroy Saint-Hilaire, 33615 Pessac, France  D.P. Bennett
Laboratory for Exoplanets and Stellar Astrophysics, NASA/Goddard Space Flight
Center, Greenbelt, MD 20771, USA

###### Abstract

The microlensing event OGLE-2017-BLG-1434 features a cold super-Earth planet
which is one of eleven microlensing planets with a planet-host star mass ratio
$q<1\times 10^{-4}$. We provide an additional mass-distance constraint on the
lens host using near-infrared adaptive optics photometry from Keck/NIRC2. We
are able to determine a flux excess of $K_{L}=16.96\pm 0.11$ which most likely
comes entirely from the lens star. Combining this with constraints from the
large Einstein ring radius, $\theta_{E}=1.40\pm
0.09\;$\mathrm{m}\mathrm{a}\mathrm{s}$$ and OGLE parallax we confirm this
event as a super-Earth with mass $m_{p}=4.43\pm 0.25M_{\earth}$. This system
lies at a distance of $D_{L}=0.86\pm 0.05\,$\mathrm{k}\mathrm{p}\mathrm{c}$$
from Earth and the lens star has a mass of $M_{L}=0.234\pm 0.012M_{\odot}$. We
confirm that with a star-planet mass ratio of $q=0.57\times 10^{-4}$,
OGLE-2017-BLG-1434 lies near the inflexion point of the planet-host mass-ratio
power law.

exoplanets, gravitational lensing

tablenum SIXtablenum

##  1 Introduction

The core accretion theory of planet formation (Pollack et al., 1996) predicts
a planetary desert at intermediate planet/host mass ratios of $1<q/10^{-4}<4$
(Suzuki et al., 2016). This is due to runaway gas accretion which involves the
rapid accumulation of hydrogen and helium gas onto protoplanetary cores as
they reach masses of $\sim 10M_{\earth}$. This results in a dearth of
intermediate-mass giant planets between Saturn mass planets at
$\sim$95$M_{\earth}$ and failed gas giants at $\sim$10$M_{\earth}$. The
predicted lack of planets with these masses is however in conflict with planet
demographics determined from microlensing observations (Suzuki et al., 2016,
2018).
Suzuki et al. (2016) compiled a sample of 30 planetary microlensing events
from the MOA (Microlensing Observations in Astrophysics) survey observed
between 2007 and 2012 and compared that with population synthesis models. They
show that these models underestimate the number of planets with
$1<q/10^{-4}<4$ by a factor of ten. They estimate the mass ratio distribution
of exoplanets to be a broken power law with a break occurring at
$q_{br}=1.7\times 10^{-4}$. A subsequent study used 15 low mass-ratio
microlensing events and estimated this break to instead be at $q_{br}\sim
0.55\times 10^{-4}$ (Jung et al., 2019). While the distribution at higher mass
ratios is a well-sampled as a decreasing power-law, there are only eleven
planets with mass ratios with $q<\times 10^{-4}$, which is why arriving at
robust statistical conclusions in this region of parameter space is difficult.
In this paper we discuss the planetary microlensing event OGLE-2017-BLG-1434
(Udalski et al., 2018), a cold super Earth with mass ratio $q=0.57\times
10^{-4}$. A measurement of the microlens parallax meant that the physical
parameters are comparatively well measured, with the mass of the planet and
its stellar host determined to be $m_{p}=4.4\pm 0.5M_{\earth}$ and
$M_{L}=0.23\pm 0.03M_{\odot}$, respectively. However, as is typical in
microlensing event analysis, a Bayesian analysis with constraints from the
Einstein radius crossing time and measurable secondary effects (for example,
due to the finite size of the source) are required in combination with a
galactic model (Sumi et al., 2011; Bennett et al., 2014) to obtain estimates
of these physical parameters. Without secondary constraints these estimates
usually have an accuracy of 30-40%.
The first of these constraints can be derived from the sharp light curve
features of many binary microlensing events. This enables one to measure the
finite angular source radius (Beaulieu, 2018) and hence a relationship between
the lens mass and distance:

$\displaystyle M_{L}=\frac{\theta_{E}^{2}}{\kappa\pi_{rel}}$ (1)

where $\theta_{E}$ is the Einstein radius, $\pi_{rel}$ the relative parallax
and
$\kappa=4G/{c^{2}\mathrm{AU}=8.144\;$\mathrm{m}\mathrm{a}\mathrm{s}$\;M_{\odot}^{-1}}$.
A second constraint on the lens mass and distance can be found by measuring
the microlensing parallax, $\pi_{E}$. This can be determined via detection of
the Earth’s orbital motion or between two spatially separated observing sites:

$\displaystyle M_{L}=\frac{\pi_{rel}}{\kappa\pi_{\mathrm{E}}^{2}}$ (2)

Space-based parallax from observatories such as Spitzer are ideal in
constraining $\pi_{E}$, though systematic errors may be a problem for both
orbital and space-based parallax (Koshimoto & Bennett, 2020; Gould et al.,
2020; Dang et al., 2020; Poleski et al., 2016). Finally, a third mass-distance
relation can be obtained by measuring the flux of the lensing system using
high-angular resolution observations from 8-10 metre class telescopes. This
makes it possible to decouple the source and lens contributions from that of
other blend stars. The relation that follows is:

$m_{L}(\lambda)=10+5log(D_{L}/1\>$\mathrm{k}\mathrm{p}\mathrm{c}$)+A_{L}(\lambda)\\\
+M_{iso}(\lambda,M_{L},age,[Fe/H])$ (3)

where $m_{L}(\lambda)$ is the apparent magnitude of the lens, $D_{L}$ is the
distance to the lens, $A_{L}(\lambda)$ is the extinction to the lens in band
$\lambda$, and $M_{iso}$ is an absolute magnitude derived from stellar
isochrones. It is this relationship which we use in this paper to better
quantify the mass and distance of OGLE-2017-BLG-1434.

Figure 1: K-band images of OGLE-2017-BLG-1434 obtained on August 7, 2018 with
Keck/NIRC2. On the far left is an image taken with the wide camera with a 40
arcsec field of view. On the right are two enlargements of this frame showing
the position of the source/lens blend, indicated with a white line.

We supplement the discovery paper of this event with the addition of high-
angular resolution data obtained using the NIRC2 imager on the Keck II
telescope on Mauna Kea, Hawaii. We obtained follow-up photometry in August
2018 as part of a NASA Keck Key Strategic Mission Support (KSMS) proposal in
support of the Nancy Grace Roman Space Telescope (formerly WFIRST, Spergel et
al. (2015)). This program is designed to acquire adaptive optics data of
microlensing events in order to constrain the lens flux and/or the source-lens
relative proper motion, and hence determine accurate estimates of the planet
and host masses. Results from this program can be found in a number of studies
(Beaulieu et al., 2016; Bhattacharya et al., 2018; Beaulieu et al., 2018;
Vandorou et al., 2020; Bennett et al., 2020; Bhattacharya et al., 2020; Terry
et al., 2021). We use this photometry to measure the lens apparent magnitude
$m_{L}(\lambda)$ from Eq. 3 and hence determine a relationship between $D_{L}$
and $M_{L}$. The finite source size, ground parallax and this newly obtained
adaptive optics data, combined with theoretical isochrones, allows us to
refine the estimate of the mass-distance relationship and confirm this event
as a cold super Earth, $m_{p}=4.43\pm 0.25M_{\earth}$, very nearby for a
microlensing planet at a distance of $D_{L}=0.86\pm
0.05\;$\mathrm{k}\mathrm{p}\mathrm{c}$$.

##  2 The cold super-Earth OGLE-2017-BLG-1434

OGLE-2017-BLG-1434 was detected by the OGLE Early Warning System at UT19:33 on
25 Jul 2017 using the OGLE 1.3m telescope at Las Campanas. Located at
$(RA,Dec)_{J2000}=(17^{h}53^{m}07s.29^{s},-30^{\circ}14^{\prime}44.6^{\prime\prime})$,
complementary follow-up data was taken with telescopes from the Korea
Microlensing Telescope Network (KMTNet, Kim et al. (2016)) and the MiNDSTEp
collaboration at the 1.54m Danish Telescope at La Silla, Chile. The majority
of the observations were taken in I-band with V-band data obtained to
determine source colors. The resultant light curve was similar to that of a
traditional point-lens with the addition of deviations near the event peak.
Udalski et al. (2018) determined the source brightness from the fitted light
curve and used red clump extinction estimates from Bensby et al. (2013) and
Nataf et al. (2013) to derive an angular source size of $\theta_{*}=0.657\pm
0.041\;$\mu\mathrm{a}\mathrm{s}$$. This results in an Einstein ring radius of
$\theta_{E}=1.40\pm 0.09\;$\mathrm{m}\mathrm{a}\mathrm{s}$$ and a relatively
large lens-source relative proper motion, $\mu=8.1\pm
0.5\;$\mathrm{m}\mathrm{a}\mathrm{s}$\;$\mathrm{y}\mathrm{r}^{-1}$$. This
large Einstein ring radius indicates that the lens must either be very close
or very massive. The long timescale of the event ($t_{E}\sim 63$ days)
suggests the presence of microlens parallax, though Udalski et al. (2018)’s
models were unable to decouple parallax from orbital motion. Udalski et al.
(2018) finds two degenerate ($\pm u_{0}$) models with parameters which are
statistically indistinguishable within $1\sigma$.
With a year having elapsed since the peak magnification of OGLE-2017-BLG-1434,
the relative source-lens proper motion is such that we would not expect to be
able to resolve these blended components with the best achievable resolution
of Keck/NIRC2 of $\sim 50\>$\mathrm{m}\mathrm{a}\mathrm{s}$$. We can, however,
compare the predicted source magnitude from the light-curve model with the
measured flux of the object at the position of the source, and attempt to
decouple and characterize the excess flux.

###  2.1 Keck Observations

We observed OGLE-2017-BLG-1434 with Keck/NIRC2 on August 7, 2018 (HJD:
2458337.80080). 29 images were obtained in the short $K_{s}$ band using the
wide camera. For simplicity in the rest of this paper we will drop the
subscript and refer to the magnitudes simply as $K$. The wide camera has a
plate scale of 0.03968 arcsec pixel-1 and a field of view of 40 arcseconds.
The best 15 of these images were stacked using SWARP (Bertin & Emmanuel, 2010)
and calibration performed by cross-matching with the VVV catalog, following
the process described in Blackman et al. (2020) and Vandorou et al. (2020).
The mean full-width half-maximum (FWHM) is 120 mas along the North axis and 90
mas along the east, indicating an elongation of the point spread function
(PSF). This elongation is not severe enough to affect our photometry. The
resulting stacked image can be seen in the left panel of Fig. 1. To determine
the location of the source we use precise astrometry determined from the OGLE-
III survey image. The OGLE image coordinates measured during magnification
were (X,Y) = (1829.25, 3196.66) (Private communication, Udalski, 5/12/18) with
the source having an OGLE ID of 198963. This places the object at
$(RA,Dec)_{J2000}=(17^{h}53^{m}07s.312^{s},-30^{\circ}14^{\prime}44.37^{\prime\prime})$
in our stacked image (Fig. 1). Photometry was performed on this image using
SExtractor (Bertin & Arnouts, 1996). We find the magnitude of the object at
the location of the source to be:

$\displaystyle K_{blend}=16.61\pm 0.07$ (4)

where the blend is defined here as the total flux of the object.

###  2.2 Extinction Estimates

Udalski et al. (2018) determines the intrinsic source color and magnitude to
be: $[(V-I)_{S0},I_{S0}]=(0.732,18.45)\pm(0.025,0.063).$ Following the color
relations of Bessell & Brett (1988) we can interpolate to find the V-K color,
$(V-K)_{S0}=1.59^{+0.05}_{-0.06}$, and hence the predicted intrinsic
K-magnitude of the source, $K_{S0}=17.59\pm 0.09$. In order to compare this
with our measured K magnitude, we must re-redden this using an estimate of the
extinction on the path towards the source. We find $A_{K}=0.259\pm 0.021$
using the OGLE extinction calculator111http://ogle.astrouw.edu.pl/cgi-
ogle/getext.py to estimate the K-band extinction at galactic coordinates
$(l,b)=(0.28,2.07)$. This value is derived from a natural neighbour
interpolation of good points from Table 3 in Nataf et al. (2013) and assuming
$E(J-K_{s})$ measurements from Gonzalez et al. (2012). We use the values
$E(V-I)=1.521\pm 0.125$, $R_{JKVI}=0.3195$ and the extinction law from
Nishiyama et al. (2009) with the relationship $E(J-K_{s})=R_{JKVI}E(V-I)$. The
I-band extinction towards this part of the galactic bulge is estimated as
$A_{I}=1.801.$
To check our $A_{K}$ estimate we calculate the extinction directly from the
OGLE-III field (Fig. 2). Comparing this to the intrinsic brightness of the red
clump, $[(V-I)_{RC0},I_{RC0}]=(1.06,14.46)$ (Nataf et al., 2013) gives
$E(V-I)=1.55$ which is well within the error given by the OGLE extinction
calculator. Using the value of $A_{K}=0.259\pm 0.021$ we hence find a
predicted source magnitude of $K_{\mathrm{predict}}=17.85\pm 0.09$.
Subtracting this from Eq. 4 we find an excess flux of

Figure 2: OGLE-III calibrated color-magnitude diagram in V and I. The red
circle marks the centroid of the red giant clump while the blue circle
indicates the position of the source. The centroid of the red clump is located
at $[(V-I)_{RC},I_{RC}]=(2.41,16.09)$.

$\displaystyle K_{\mathrm{excess}}=17.03\pm 0.11$ (5)

We now re-correct for extinction but now only to the distance of the lens. The
lens in Udalski et al. (2018) is predicted to be at a $0.86\pm
0.09\>$\mathrm{k}\mathrm{p}\mathrm{c}$$, in front of more than half of the
extinction along the 8 kpc line of sight towards the galactic bulge. We follow
the procedure as in Bennett et al. (2015) and Beaulieu et al. (2018), using
the relationship

$\displaystyle
A_{\mathrm{K}_{\mathrm{L}}}=(1-e^{-D_{\mathrm{L}/\tau_{\mathrm{dust}}}})/(1-e^{-D_{\mathrm{S}/\tau_{\mathrm{dust}}}})A_{\mathrm{K}_{\mathrm{S}}}$
(6)

where the scale height of the dust towards the galactic bulge is
$\tau_{dust}=(0.10\pm 0.02\>$\mathrm{k}\mathrm{p}\mathrm{c}$)\sin b$ and
$b=2.07$ the galactic longitude. In our case we calculate
$A_{\mathrm{K}_{\mathrm{L}}}=0.073$ adopting a source distance of
$D_{S}=8.0\pm 0.5\>$\mathrm{k}\mathrm{p}\mathrm{c}$$ as predicted by the OGLE
extinction calculator. If the light from this excess is entirely from the
lens, or from a combination of objects at the same distance of
$0.86\>$\mathrm{k}\mathrm{p}\mathrm{c}$$, we find a excess flux of

$\displaystyle K_{\mathrm{0,excess}}=16.96\pm 0.11$ (7)

Figure 3: Mass-distance diagram showing constraints from the Einstein Ring Radius ($\theta_{E}$, in blue), from OGLE parallax (in red) and our Keck measurement and K-band isochrones (Girardi et al., 2002), the solid and dotted blue lines. The original estimate from Udalski et al. (2018) is shown with the grey cross at the intersection of the $\theta_{E}$ and OGLE parallax constraints. This paper adds the constraint from lens flux + isochrones. Our estimate is plotted as the black cross. Table 1: Physical Parameters of OGLE-2017-BLG-1434 Parameter | Value | with Keck AO | Udalski et al. (2018)
---|---|---|---
Lens Distance |  $D_{L}$ (kpc) | $0.86\pm 0.05$ | $0.86\pm 0.09$
Lens Mass | $M_{L}(M_{\odot})$ | $0.234\pm 0.012$ | $0.234\pm 0.026$
Planet Mass | $m_{p}(M_{\earth})$ | $4.43\pm 0.25$ | $4.4\pm 0.5$
Instantaneous 2D star-planet separation | $a_{\perp}(\mathrm{AU})$ | $1.18\pm 0.10$ | $1.18\pm 0.14$

###  2.3 Lens Properties and Bayesian analysis

We detect an excess flux aligned to the source to better than the best 90mas
FWHM of our final swarped Keck image. In order to determine whether this light
is (a) entirely from the lens, (b) from a companion to the lens, (c) a
companion to the source, or (d) from an ambient star unrelated to either the
lens or the source, we plot a mass-distance diagram assuming all the light is
from the lens. Fig. 3 combines constraints from the Einstein Ring Radius,
$\theta_{E}$, OGLE parallax and our flux measurement of $K_{L}=16.96\pm 0.11$
combined with theoretical isochrones or, in other words, Eq. 1, 2 and 3. We
find values for the lens mass and distance consistent with that determined by
Udalski et al. (2018), but with smaller uncertainties. The agreement between
the model and our additional lens flux constraints is such that this excess
flux is most likely entirely from the lens.
To test this we perform a Bayesian analysis as described in Koshimoto et al.
(2020). Using the galactic model prior from Bennett et al. (2014) and
constraints from the large $\theta_{E}$, the observed $t_{E}$ and the measured
flux excess of $K=16.61\pm 0.07$, we find the excess to very likely be from
the lens with a probability of 0.96 (Fig. LABEL:fig:bayesiana). In this
calculation we deliberately exclude priors from parallax
$\boldsymbol{\pi}_{\mathrm{E}}=\left(\pi_{\mathrm{E},N},\pi_{\mathrm{E},E}\right)$.
In this case the $\pi_{E,N}$ determined by the Udalski et al. (2018) models
lie in the $3\sigma$ boundary while $\pi_{E,E}$ is slightly more likely and
sits within the $2\sigma$. When using the $\theta_{E}$ and $t_{E}$ constraints
only without our Keck measurements, the large parallax determined by Udalski
et al. (2018) is even less likely. In this case both components lie in the
$3\sigma$ range.
The parallax estimates are hence consistent with the large $\theta_{E}$ and
our Keck measurements, but not very likely. These measurements rely on the
measuring the microlensing parallax, $\boldsymbol{\pi}_{\mathrm{E}}$, most
often from the orbital motion of the Earth (Penny et al., 2016). The component
of parallax perpendicular to the Earth’s motion ($\pi_{\mathrm{E},N}$), is
degenerate with blending and the orbital motion of the lens and source (the
effect on the light curve from these qualities mimics that from orbital
parallax, meaning that they are difficult to disentangle from one another;
Bennett et al. (2010); Batista et al. (2011)). Caution, therefore, must be had
when trusting $\boldsymbol{\pi}_{\mathrm{E}}$ measurements, in particular for
close targets. A new reduction of the MOA data using the photometry code of
(Bond et al., 2017), for example, indicates that the large parallax signal
reported in the event MOA-2007-BLG-192 (Bennett et al., 2008; Kubas et al.,
2012) was incorrect. This was due to a photometric error due to color
dependent differential refraction, first noted to be significant in Bennett et
al. (2012). Penny et al. (2016) meanwhile notes an unexpectedly large number
of planets at close distances and cites six planetary events with
$D_{L}<2\>$\mathrm{k}\mathrm{p}\mathrm{c}$$, of which two could be plausibly
be moved to farther distances: OGLE-2013-BLG-0341 and OGLE-2013-BLG-0723.
Since that publication OGLE-0723 (Udalski et al., 2015; Han et al., 2016) has
been revealed to be a binary event with no evidence of a planet, while
variations in the baseline photometry of OGLE-0341 has been contradicted by
data taken after Gould et al. (2014) was published, and whose large $\pi_{E}$
value could be caused by large systematic photometry errors. In the case of
MOA-2007-BLG-192 (Kubas et al., 2012), the detection of a source excess using
AO imaging is also consistent with an unrelated source contaminant, rather
than solely the lens detection that was claimed.
Our Bayesian analysis predicts the system to be a $m_{p}=4.43\pm
0.25\;M_{\earth}$ planet orbiting a $M_{L}=0.234\pm 0.012\;M_{\odot}$ star at
a distance of $0.86\pm 0.05\;$\mathrm{k}\mathrm{p}\mathrm{c}$$. The updated
physical parameters of the lens and its companion calculated using this extra
constraint are shown in Table 1. The contaminant analysis of Koshimoto et al.
(2020) employed here shows no tension between the reported parallax of Udalski
et al. (2018) and our Keck measurements, however the values of parallax
determined from the light-curve model (eg.
$\boldsymbol{\pi}_{\mathrm{E}}=(\pi_{\mathrm{E},N},\pi_{\mathrm{E},E})=(-0.586\pm
0.081,0.472\pm 0.013$ for a parallax+orbital motion fit where $u_{0}>0$) are
unlikely and could be the results of systematic errors in the photometry.
Excluding the parallax constraints we find that the excess flux at the
position of the source is almost certainly the lens (with a probability of
0.96, Fig. LABEL:fig:bayesian, panel (a)). These probabilities were calculated
using the $\theta_{E}$ and the Keck excess flux ($K_{ex,obs}$) constraint. If
we also include the parallax (Fig. LABEL:fig:bayesian, panel (b)) the excess
flux is still very likely to be the lens ($K_{L}({<}f_{L}{>}=0.93$).

##  3 Discussion & Conclusion

We obtain Keck follow-up photometry of the microlensing event
OGLE-2017-BLG-1434 which is consistent with the physical interpretation of
Udalski et al. (2018) that the system is a super-Earth planet orbiting an M4V
dwarf star. When applying the additional constraint on the lens mass and
distance from our lens flux measurement we reduced the uncertainty in the lens
parameters (mass, distance, planetary mass) by half. As such we can now
describe the system as a super Earth at a distance of $0.86\pm
0.05\>$\mathrm{k}\mathrm{p}\mathrm{c}$$,. This is quite nearby for a
microlensing planet but farther away than the majority of radial velocity and
transit-detected planets. The planet and its host star had an instantaneous 2D
separation of $a_{\perp}=1.18\pm 0.10$, which places a lower limit on its
perihelion distance. Calculating this using the relationship
$a_{\perp}=s\theta_{E}D_{L}$ results in a value in agreement with Udalski et
al. (2018), but with an error reduced by 28%. Comparing this to the snow line,
defined as $a_{snow}=2.7$\mathrm{A}\mathrm{U}$\;(M/M_{\odot})$, the planet
lies at $1.9\pm 0.2\;\mathrm{a_{snow}}$, while the stellar lens host has a
mass of $0.234\pm 0.012\>$\mathrm{M}_{\odot}$$.

###  3.1 Galactic Environment

This planet joins a list of only 5 other planets detected by microlensing at a
distance of less than 1 kpc, assuming no systematic errors in the photometry
(see Section 2.3). Of the 100 or so planets so far detected using
microlensing, the median distance of these systems, almost all in the
direction of the galactic centre, is $\sim
5.7\;$\mathrm{k}\mathrm{p}\mathrm{c}$$222https://exoplanetarchive.ipac.caltech.edu/.
One consequence of this distance distribution is that the typical
microlensing-detected planet host is expected to be significantly more metal-
rich than planetary host stars in the Solar neighborhood, owing to the radial
metallicity gradient in the Galactic disk. The inner galaxy (RGC $\lesssim$ 3
kpc) is expected to contain the most metal-rich stars in the Milky Way (e.g.,
Larson, 1976); surveys have found this to be the case (e.g., Zoccali et al.,
2008, and references therein), with the kinematically cold population near the
disk plane home to proportionally more stars of super-Solar metallicity,
reaching up to [Fe/H] $\sim+0.4-0.6$ (McWilliam & Rich, 1994; Ness et al.,
2013).
Thin disk stars within $\approx$3–5 kpc of the Galactic centre have a
relatively narrow peak in their metallicity distribution at [Fe/H] $\approx$
$+$0.3 (e.g., Hayden et al., 2015), with a tail to lower values and little
evidence for radial gradients (Hayden et al., 2014). By contrast, in the Solar
neighborhood the radial metallicity gradient amounts to $-$0.09 dex/kpc
(Frinchaboy et al., 2013; Hayden et al., 2014). The mean metallicity of dwarfs
and subgiants in the Solar neighborhood is [Fe/H] = $-$0.04, with a dispersion
of $\sigma$ = 0.25 dex and a roughly Gaussian shape (Buder et al., 2019).
The stellar population structure of the Galactic disk leads us to expect a
metallicity difference of a factor of 1.5–2 between typical planetary hosts in
the inner Galaxy and the Solar neighborhood, which must be reckoned with when
comparing planet occurrence as a function of host mass and orbital period
(e.g. Sousa et al., 2019) across samples from different Galactic environments.
For microlensing surveys, continuing to build exoplanet samples at a wide
range of distances will allow for a statistical exploration of the frequency
of cold planets as a function of host star metallicity. This is highly
complementary to transit and Doppler exoplanet searches in the Solar
neighborhood, which are accessible to direct spectroscopic probes of stellar
metallicity but are biased towards the detection of hot planets around more
intrinsically luminous stars.

###  3.2 Mass-Ratio Function

OGLE-2017-BLG-1434 is one of five microlensing planets with a mass ratio of
$q\leq 0.6\times 10^{-4}$ and one of 11 planets with a mass between
$1-10M_{\earth}$. It lies right on the break of the exoplanet mass-ratio
power-law according to Jung et al. (2019), which is smooth and decreasing at
mass ratios higher than this inflection point, and increasing below. The slope
at these smaller mass ratios is not well constrained with Jung et al. (2019)
claiming a sharp break at the inflection point of the mass-ratio function.
They give a best fit value of $q=0.55\times 10^{-4}$. The lack of statistics
in this region is a significant limitation in attempts to determine the slope
and shape of the low-$q$ mass-ratio function. This is why lens flux
measurements such as those presented here are important. Even though it
requires high-resolution imaging from the largest ground based telescopes, or
from the Hubble Space Telescope, the time costs for each event is small
(usually 30-40 mins excluding calibration overhead).
Of the eleven planets with mass ratios $q<1\times 10^{-4}$, four of them have
reliable host mass measurements: OGLE-2005-BLG-169Lb (Batista et al., 2015),
MOA-2009-BLG-266Lb (Muraki et al., 2011), KMT-2018-BLG-0029Lb (Gould et al.,
2020) and OGLE-2017-BLG-1434 (this study), while a number of others have
estimates computed using a Bayesian analysis that are highly dependent on the
priors: OGLE-2005-BLG-390Lb (Beaulieu et al., 2006; Kubas et al., 2008),
OGLE-2007-BLG-368Lb (Sumi et al., 2010), OGLE-2017-BLG-0173Lb (Hwang et al.,
2017), OGLE-2018-0677Lb (Herrera-Martín et al., 2020) and KMT-2019-BLG-0842Lb
(Jung et al., 2020). Especially for events with limited secondary light-curve
effects and physical parameters only determined to 30-40%, single band near
infrared follow-up photometry as shown here is a time-cheap way of tightening
mass and distance constraints on microlensing systems, and – eventually – of
characterising the break in the planetary mass-ratio function.

This work was supported by the University of Tasmania through the UTAS
Foundation and the endowed Warren Chair in Astronomy. It was also supported by
ANR COLD-WORLDS (ANR-18-CE31-0002) at Le Centre National de la Recherche
Scientifique (CNRS) in Paris and the Laboratoire d’astrophysique de Bordeaux.
AAC and JWB are supported by funding from the Australian Research Council
through the Discovery Project grant scheme (DP200101909). DPB and AB were
supported by NASA through grant NASA-80NSSC18K0274. Data presented in this
work was obtained at the W. M. Keck Observatory from telescope time allocated
to the National Aeronautics and Space Administration through the agencies
scientific partnership with the California Institute of Technology and the
University of California. These Keck Telescope observations and analysis were
supported by a NASA Keck PI Data Award, administered by the NASA Exoplanet
Science Institute. The Observatory was made possible by the generous financial
support of the W. M. Keck Foundation. This research has made use of the NASA
Exoplanet Archive, which is operated by the California Institute of
Technology, under contract with the National Aeronautics and Space
Administration under the Exoplanet Exploration Program.

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