# Operator spectrum of nonrelativistic CFTs at large charge

Vito Pellizzani<EMAIL_ADDRESS>Albert Einstein Center for
Fundamental Physics, Institute for Theoretical Physics, University of Bern,
Switzerland

###### Abstract

We extend and clarify the large-charge expansion of the conformal dimension
$\Delta_{Q}$ of the lowest operator of charge $Q$ in nonrelativistic conformal
field theories using the state-operator correspondence. The latter requires
coupling the theory to an external harmonic trap that confines the particles
to a spherical cloud, at the edge of which the effective theory breaks down
and leads to divergences. Only recently has this issue been overcome by
constructing appropriate counterterms at the edge of the cloud
[arXiv:2010.07967]. In this paper, we extend these results by systematically
analyzing the degree of divergence of operators in the effective action and
show that there always exist appropriate edge counterterms that make the final
contributions to $\Delta_{Q}$ finite. On the other side of the correspondence,
this also provides new corrections to the Thomas-Fermi approximation of the
unitary Fermi gas, and we comment on their relevance for ultracold atom
physics.

##  I Introduction

The large-charge approach to strongly coupled systems with global symmetries
is a systematic way of deriving the spectrum of charged operators in an
expansion in inverse powers of the charge, as first discussed in [2] (see [3]
for a recent review), as well as certain correlation functions [4, 5, 6].
Sequels of this approach include the large-charge expansion in nonrelativistic
conformal field theories [1, 7, 8, 9, 10], the $O(N)$ model at large charge
[11], its double-scaling large-$N$ limit [12, 13] and the study of
nonperturbative corrections thereof using resurgence techniques [14], the
large $R$-charge limit [15, 16, 17, 18, 19, 20, 21] and the
$\epsilon$-expansion at large charge [22, 23, 24, 25, 26, 27], among others.
In most cases, the state-operator correspondence turns out to be extremely
powerful.

In this paper, we are concerned with nrcfts where the correspondence maps the
spectrum of conformal dimensions of (positively charged) local operators to
the energy spectrum of states in an external spherical harmonic trap

$A_{0}(\vec{x})=\frac{m\omega^{2}}{2\hbar}|\vec{x}|^{2},$ (1)

and vice-versa [28, 29, 30]. In particular, we focus on the conformal
dimension $\Delta_{Q}$ of the lowest operator of fixed charge $Q\gg 1$, which
can be accessed via the ground-state energy $E_{0}$ of the trapped system with
$Q$ particles confined to a spherical cloud. The argument is presented for
general spatial dimension $d$ in dimensionless units $\hbar=m=\omega=1$, in
which case we simply have $\Delta_{Q}=E_{0}$.

In order to derive the large-charge expansion of the ground-state energy, we
construct the effective field theory (eft) for the Goldstone boson $\chi$
associated with the broken $U(1)$ (i.e. particle number) symmetry with
appropriate dilaton dressing rules that guarantee conformal invariance. To
leading order, this description corresponds to the usual Thomas-Fermi
approximation, and the first subleading corrections were found by Son and
Wingate in a small momentum expansion for the nonlinear sigma model (nlsm)
[31]. While the power counting they used in this work might seem somewhat
arbitrary, it is in fact best understood from a large-charge perspective [9]
(see also [8]). However, this effective theory is known to break down close to
the edge of the cloud, where the particle density falls off and gives rise to
divergences even at the classical level [31, 9, 7]. Building upon the recent
work [1], we classify all types of edge divergences and explain how to restore
tree-level consistency by constructing appropriate counterterms in a procedure
that we refer to as $\delta_{\epsilon}$-layer regularization. We show that, in
general,

$\displaystyle\Delta_{Q}$
$\displaystyle=Q^{\frac{d+1}{d}}\left[a_{1}+a_{2}Q^{-\frac{2}{d}}+a_{3}Q^{-\frac{4}{d}}+\ldots\right]$
(2)
$\displaystyle+Q^{\frac{2d-1}{3d}}\left[b_{1}+b_{2}Q^{-\frac{2}{3d}}+b_{3}Q^{-\frac{4}{3d}}+\ldots\right]$
$\displaystyle+Q^{\frac{d-5}{3d}}\left[c_{1}+c_{2}Q^{-\frac{2}{3d}}+c_{3}Q^{-\frac{4}{3d}}+\ldots\right]+\ldots$

plus quantum corrections starting at $Q^{0}$. The first line is analogous to
the relativistic case, while the second and third lines are new structures
arising from edge effects. Note that some of the $b_{i}$’s contain a $\log
Q$-factor when $d$ is even, e.g.

$\displaystyle\Delta_{Q}^{(d=2)}=c_{1}Q^{\frac{3}{2}}$
$\displaystyle+c_{2}\sqrt{Q}\log Q+c_{3}\sqrt{Q}$ (3)
$\displaystyle+c_{4}Q^{\frac{1}{6}}-0.29416+\ldots,$

where the last term is the model-independent one-loop Casimir energy
calculated in [7], and further corrections scale with negative powers of $Q$.
Equation (3) was first derived in [1], based on [7, 8, 9], although we shall
clarify the origin of the $Q^{\frac{1}{2}}$ and $Q^{\frac{1}{6}}$ terms.
Similarly, we show that

$\displaystyle\Delta_{Q}^{(d=3)}=c_{1}Q^{\frac{4}{3}}$
$\displaystyle+c_{2}Q^{\frac{2}{3}}+c_{3}Q^{\frac{5}{9}}$ (4)
$\displaystyle+c_{4}Q^{\frac{1}{3}}+c_{5}Q^{\frac{1}{9}}+c_{6}Q^{0}+\ldots,$

and we argue that one should expect the presence of a universal $\log Q$-term
associated with the Casimir energy, which will be computed in an upcoming
publication [32]. Son & Wingate [31] derived the first two terms of Eq. (4)
and anticipated the presence of a divergent term that would scale like
$Q^{\frac{5}{9}}$ after $\delta_{\epsilon}$-layer regularization. In this
paper, we renormalize it for the first time and extend the expansion down to
$Q^{0}$.

Moreover, the description of the system in terms of the dilaton mode allows us
to explore the near-conformal regime of the theory by introducing a small
dilaton mass $m_{\sigma}$ _à la_ Coleman [33], as was done in [7] based on the
general expectation detailed in [34, 35] that a dilatonlike mode appears near
a smooth quantum phase transition. We overcome the issues faced in [7] related
to boundary divergences and find that the signature of this mass deformation
is an additional $\sqrt{Q}\log Q$ and $\sqrt{Q}$-contributions in $d=3$, while
the structure of the expansion is unaffected in $d=2$. Note that, while the
concept of conformal dimension becomes (softly) ill-defined in this scenario,
the corrections induced by the dilaton mass to the ground-state energy of the
trapped system are interesting _per se_. In this work, one can therefore think
of $\Delta_{Q}$ as the latter energy, which exactly corresponds to the
conformal dimension of the lowest operator of charge $Q$ only when
$m_{\sigma}=0$.

Finally, let us comment on the relevance for ultracold atom physics, as
trapped gases can be realized experimentally (see e.g. the beautiful reviews
[36, 37] and references therein). Typically, this is achieved for cold and
dilute atomic Fermi gases whose interaction strength is dominated by the
s-wave scattering and can be tuned using Feshbach resonances. Correspondingly,
the value and even the sign of the dimensionless scattering parameter
$\frac{1}{k_{F}a_{s}}$—where $k_{F}$ is the Fermi wave-vector and $a_{s}$ the
s-wave scattering length—can be changed. In the Bardeen-Cooper-Schrieffer
(bcs) regime $\frac{1}{k_{F}a_{s}}\ll-1$, the interaction is weakly attractive
and fermions form Cooper pairs, while for $\frac{1}{k_{F}a_{s}}\gg 1$, the
attraction is strong and binds pairs of fermions with opposite spin together.
The latter system is effectively described by a weakly interacting bosonic gas
of such molecules (also called dimers), i.e. a Bose-Einstein condensate (bec)
[37]. Both regimes are known to exhibit superfluidity and no phase transition
occurs in between, indicating a smooth crossover that preserves superfluidity
for all values of $k_{F}a_{s}$. This is particularly relevant for the
crossover region $\frac{1}{k_{F}a_{s}}\in[-1,1]$, centered around the resonant
case $\frac{1}{k_{F}a_{s}}=0$ known as the _unitary_ limit, where the system
is strongly interacting and an expansion in $k_{F}a_{s}$ is inappropriate.
While a complete description of the crossover is still lacking, the emergent
scale invariance at unitarity allows for an eft description of the cold Fermi
gas with a large number of trapped particles, as initiated by Son & Wingate
[31] and completed in the present work from a linear sigma model (lsm)
perspective, where the only massless, low-energy degree of freedom $\chi$
corresponds to the phase of the condensate. As already mentioned, this eft
goes beyond the Thomas-Fermi approximation, yielding corrections e.g. to the
ground-state energy, Eq. (4), or the doubly integrated density
$n(x_{3})=\iint\differential x_{1}\differential x_{2}\,\rho(\vec{x})$ measured
experimentally in [38] (where $\rho(\vec{x})$ is the charge density):

$\displaystyle n(x_{3})$ $\displaystyle=\frac{2\pi
g}{5}\left[\frac{2c_{\frac{4}{3}}\mu}{g}\left(1-\frac{x_{3}^{2}}{2\mu}\right)\right]^{\frac{5}{2}}$
$\displaystyle\times\left[1+\frac{45c_{\frac{2}{3}}}{32c_{\frac{4}{3}}\mu^{2}}\left\\{\frac{5}{\left(1-\frac{x_{3}^{2}}{2\mu}\right)^{2}}-\frac{1}{\left(1-\frac{x_{3}^{2}}{2\mu}\right)^{3}}\right\\}+\ldots\right].$

Here, $\mu$ is the chemical potential and $g$, $c_{\frac{4}{3}}$,
$c_{\frac{2}{3}}$ are Wilsonian parameters. The latter is associated with the
simplest subleading operator in the eft; without it, only the first line above
matters, which can be written as
$n(x_{3})=\frac{16}{5\pi}\frac{Q}{R_{cl}}\left(1-\frac{x_{3}^{2}}{R_{cl}^{2}}\right)^{\frac{5}{2}}$,
where $R_{cl}=2\mu$ is the classical radius of the cloud, thus matching the
known expression [37]. Further corrections can readily be computed. It would
also be interesting to investigate to what extent the somewhat naïve breaking
of conformal invariance caused by the introduction of a small dilaton mass
mentioned earlier allows for the exploration of the crossover region (e.g. in
the spirit of [39]).

Another interesting direction for future research in ultracold atom physics
concerns bec, for which the Gross-Pitaevskii theory predicts that the ground-
state energy be given by [36]

$E_{0}=\int\differential^{3}x\left[\frac{1}{2}(\mathop{\mathrm{{}\partial}}\mathopen{}_{i}a(\vec{x}))^{2}+A_{0}(\vec{x})a(\vec{x})^{2}+2\pi
a_{s}\cdot a(\vec{x})^{4}\right],$

where $a(\vec{x})$ is the radial mode of the condensate wave-function
$\Phi(t,\vec{x})=a(\vec{x})e^{-i\mu t}$ and $\mu$ is the chemical potential.
The first term in this expression is called _quantum pressure_ and is
neglected in the Thomas-Fermi approximation. However, it becomes important
close to the edge of the cloud, in a region sometimes called _effective
surface thickness_ in this context. Upon approximating the potential by a
linear ramp in this region, the authors of [40] (based on [41]) found
schematically

$E_{0}=d_{1}Q^{\frac{7}{5}}+d_{2}Q^{\frac{3}{5}}\log
Q+d_{3}Q^{\frac{3}{5}}+\ldots,$ (5)

where we did not keep track of $a_{s}$ for simplicity. It would therefore be
interesting to understand how much this result could be improved by including
edge counterterms similar to the ones presented in the present work.

* * *

This paper is organized as follows. In Sec. II.1 and II.2, we review the
construction of the leading-order effective action from the lsm perspective
using the dilaton dressing, following [7]. We then discuss in Sec. II.3 how to
adapt the previous dilaton dressing as we approach the edge of the cloud, as
first discussed in [1], and we give the recipe for the construction of edge
counterterms in Sec. II.4, based on the same reference. The core of our work
is presented in Sec. II.5, where we analyze the possible diverging behaviors
of operators due to boundary effects and show that the previously constructed
counterterms always match. This allows to complete the large-charge expansion
of $\Delta_{Q}$ in any dimension up to terms that scale with negative powers
of $Q$, in which case quantum corrections have to be taken into account. We
make some general observations and comment on quantum corrections in Sec.
II.6, and finally work out the $d=2$ and $d=3$ cases in Sec. III.1 and Sec.
III.2, respectively, including the aforementioned dilaton mass deformation.

##  II Effective action

###  II.1 Dilaton dressing and radial mode

Working in a sector of fixed charge spontaneously breaks the associated global
symmetry, as well as conformal invariance. Describing the eft in terms of the
Goldstone mode that accounts for the breaking of conformal invariance, namely
the dilaton $\sigma$, turns out to be convenient even though it may actually
be massive (see e.g. [42] for a discussion on gapped Goldstone bosons). A
simple construction is due to Coleman [33], who pointed out that it is
possible to promote Lorentz invariance of a given theory to a full conformal
invariance by dressing the operators with an appropriate factor involving the
dilaton. The very same game can be played with nonrelativistic theories [7,
43], where the nonrelativistic conformal symmetry group is usually referred to
as the _Schrödinger group_. Scale transformation
$(t,\vec{x})\to(e^{z\tau}t,e^{\tau}\vec{x})$—where $z=1$ in the relativistic
case and $z=2$ in the nonrelativistic one—acts on the dilaton as

$\sigma(t,\vec{x})\longrightarrow\sigma(t,\vec{x})+\frac{d+z-2}{2f},$ (6)

where the dimensionful parameter $f$ can be regarded as the (inverse) decay
constant of the dilaton [44].

If one considers a theory featuring a global $U(1)$ symmetry, as is the case
of the Schrödinger group, one may first construct the most general eft for the
Goldstone mode $\chi$ invariant under Galilean or Lorentz symmetry that
nonlinearly realizes the $U(1)$ symmetry, and then appropriately dress the
operators with the dilaton $\sigma$ so as to make them marginal. Note that in
general, these two fields can then be conveniently recast as

$\psi=\frac{1}{f}e^{-f\sigma-i\chi},$ (7)

and we shall therefore refer to $a\equiv|\psi|=\frac{1}{f}e^{-f\sigma}$ as the
_radial mode_. Of course, this construction would require an infinite number
of Wilsonian coefficients, but one can then organize them in a large-charge
expansion and truncate to any desired order. While this provides an explicit
recipe for the construction of the large-charge lsm of the effective theory
(which essentially works in the same way for the relativistic [3, 45] and the
nonrelativistic cases [7]), it should be pointed out that the radial mode
becomes massive under spontaneous breaking of the $U(1)$ symmetry and thus
decouples below the energy scale associated with the charge. Upon integrating
it out, one would recover an equivalent large-charge effective action for the
Goldstone $\chi$ alone in the form of a nlsm, which can be obtained using
different methods, e.g. the coset construction [4, 9]. In this paper, we shall
use the lsm description in view of including a small dilaton mass deformation.

###  II.2 Leading-order Lagrangian

As mentioned in the introduction, we aim to compute the conformal dimension of
the lowest operator at large charge using the nonrelativistic state-operator
correspondence. Accordingly, we consider the theory coupled to an external
trapping potential,

$A_{0}(r)=\frac{1}{2}r^{2},$ (8)

which restricts the support of the (classical) theory to a ball of finite
radius, i.e., a cloud—or droplet—of particles at the edge of which the
particle density rapidly falls off to zero. Unlike in the relativistic case
where the state-operator correspondence is realized on a fixed background, the
cloud is a dynamical object whose boundary undergoes quantum fluctuations. It
is a very reasonable question to wonder whether short-distance physics causes
any trouble close to the edge, and the answer is known to be positive [31, 9,
7]. This issue is already present at the classical level, and a sharp cutoff
procedure was discussed in these references, where the so-called
$\delta_{\epsilon}$-layer is removed at the edge in order to regularize the
theory. More recently, a thorough discussion of the possible counterterms
located at the edge of the droplet has been carried out in [1]. We aim at
translating and extending these results into the language of the lsm at large
charge.

The building block of a generic Galilean invariant theory for the Goldstone
mode $\chi$ in the trap is the operator

$U\equiv\dot{\chi}-A_{0}(r)-\frac{1}{2}(\mathop{\mathrm{{}\partial}}\mathopen{}_{i}\chi)^{2},$
(9)

where the presence of $A_{0}(r)$ requires some notion of general coordinate
invariance, as discussed in [31]. As we shall see later, derivatives of this
operator, as well as other operators featuring more derivatives of the
Goldstone mode $\chi$ contribute to the effective action but, for now, let us
focus on the power series in $U$,

$\mathcal{L}(\chi)=-k_{0}+\sum_{n=1}^{\infty}k_{n}U^{n},$ (10)

where the $k_{i}$’s are Wilsonian coefficients. Promoting such a Galilean-
invariant Lagrangian to a fully Schrödinger-invariant one is now an easy task
with the dilaton dressing. Since the dimension of the radial mode
$a=\frac{1}{f}e^{-f\sigma}$ is $[a]=\frac{d}{2}$ and $[U]=2$, we simply have

$\mathcal{L}(\chi,a)=-k_{0}a^{2+\frac{4}{d}}+a^{2+\frac{4}{d}}\sum_{n=1}^{\infty}k_{n}\cdot\left(\frac{U}{a^{\frac{4}{d}}}\right)^{n}.$
(11)

In the superfluid ground state, the vacuum expectation value (vev) of the
Goldstone mode is $\langle\chi\rangle=\mu\cdot t$, where $\mu$ is the chemical
potential. The $U(1)$ and conformal symmetries are spontaneously broken, and
the equation of motion for $a$ imposes that the ratio
$\frac{U}{a^{\frac{4}{d}}}$ is necessarily a constant. Correspondingly, their
vev are of the form

$\langle a\rangle^{\frac{4}{d}}\sim\langle
U\rangle=\mu\left(1-\frac{r^{2}}{2\mu}\right).$ (12)

From the form of the action, Eq. (11), one readily sees that $a$ acquires a
mass $m_{a}^{2}\sim\langle a\rangle^{\frac{4}{d}}\sim\mu$. Moreover, the
ground-state charge density $\rho\propto\langle a\rangle^{2}$ is supported on
the interval $r\in[0,R_{cl}]$ where $R_{cl}\equiv\sqrt{2\mu}$ defines the
radius of the cloud, i.e. the classical turning point, and sets an infrared
(ir) length-scale. Upon integrating the charge density over this region, one
finds that the total charge is

$Q\sim\mu^{d}.$ (13)

If one associates an ultraviolet (uv) length-scale
$R_{\mu}=\sqrt{\frac{2}{\mu}}$ with the mass of the radial mode, the effective
theory description is under perturbative control when there is a separation of
scales,

$R_{cl}\gg r\gg R_{\mu},$ (14)

which amounts to requiring that the controlling parameter
$\frac{R_{\mu}}{R_{cl}}=\frac{1}{\mu}\sim Q^{-\frac{1}{d}}$ be small. This, in
turn, is guaranteed by the large-charge condition $Q\gg 1$.

The drawback of keeping track of the massive mode $a$ in the low-energy
description is that we technically have to account for series of operators, as
in Eq. (11), that give the same contribution to observables to leading-order.
Roughly speaking, integrating the radial mode out in Eq. (11) gives a single
leading-order term $U^{1+\frac{2}{d}}$ in the nlsm, and trading
$a^{\frac{4}{d}}$ for $U$ is therefore unseen at the level of the nlsm. The
minimal Lagrangian that captures all the above properties is given by

$\mathcal{L}_{LO}(\chi,a)=c_{\frac{d+1}{d}}a^{2}U-\frac{d}{2(d+2)}ga^{2+\frac{4}{d}},$
(15)

where we renamed and rescaled the Wilsonian coefficients for future
convenience. Correspondingly, the ground-state energy—and therefore, the
conformal dimension of the lowest operator of charge $Q$ in the system without
trap—is given by

$\Delta_{Q}=\frac{d}{d+1}\zeta Q^{\frac{d+1}{d}},$ (16)

where $\zeta=\sqrt{\frac{g}{4\pi
c_{\frac{d+1}{d}}}}\left[\frac{2\Gamma(d)}{c_{\frac{d+1}{d}}\Gamma\left(\frac{d}{2}\right)}\right]^{\frac{1}{d}}$
is a constant, in accordance with the nlsm results [9]. The advantage of this
description, however, is that it allows for a rather straightforward analysis
of the subleading corrections to Eq. (16), as discussed in Sec. II.5.

In order to further simplify the argument, we introduce the dimensionless
coordinate $z\equiv 1-\frac{r^{2}}{R_{cl}^{2}}=1-\frac{r^{2}}{2\mu}$, which
measures the distance from the classical boundary of the cloud. Since
spherical symmetry is preserved by the superfluid ground state, it will be
convenient to express every vev as a function of $z$. Useful properties are

$\displaystyle(\mathop{\mathrm{{}\partial}}\mathopen{}_{i}f(\vec{x}))(\mathop{\mathrm{{}\partial}}\mathopen{}_{i}g(\vec{x}))=\frac{2(1-z)}{\mu}f^{\prime}(z)g^{\prime}(z),$
(17)
$\displaystyle\nabla^{2}f(\vec{x})=\frac{2}{\mu}\left[(1-z)f^{\prime\prime}(z)-\frac{d}{2}f^{\prime}(z)\right],$
$\displaystyle\int_{cloud}\differential^{d}x\,f(\vec{x})=\frac{(2\pi\mu)^{\frac{d}{2}}}{\Gamma\left(\frac{d}{2}\right)}\int_{0}^{1}\differential
z\,(1-z)^{\frac{d-2}{2}}f(z),$

where primes refer to derivatives with respect to $z$ and $f,g$ are
spherically invariant functions. Note that spatial derivatives of operators _a
priori_ make their contributions to the conformal dimension $\Delta_{Q}$
parametrically smaller due to the division by $\mu\sim Q^{\frac{1}{d}}$.

At this stage, let us point out that Eq. (15) in $d=3$ corresponds to the
Thomas-Fermi approximation of the unitary Fermi gas, and yields, among others,
the known expression for the doubly integrated density mentioned in the
introduction and measured experimentally in [38], namely

$\displaystyle n(x_{3})$ $\displaystyle=\iint\differential x_{1}\differential
x_{2}\,\rho(\vec{x})$ (18) $\displaystyle=\frac{2\pi
g}{5}\left[\frac{2c_{\frac{4}{3}}\mu}{g}\left(1-\frac{x_{3}^{2}}{2\mu}\right)\right]^{\frac{5}{2}},$

where the $x_{i}$’s take values in the cloud. At the end of this paper, we
discuss corrections to this expression.

###  II.3 Dressing rules

The presence of the dilaton field, via the radial mode
$a(t,\vec{x})=\frac{1}{f}e^{-f\sigma(t,\vec{x})}$, allows for the dressing of
operators to marginality, as discussed in the previous section. At the same
time, the breakdown of the effective theory near the edge of the cloud is
associated with the vanishing of the particle density and, hence, the
vanishing of $a$ [31, 9, 7]. From a large-charge perspective, this indicates
that the dressing rule based on powers of $a$ is only appropriate when edge
effects are negligible, i.e. in the _bulk_ of the cloud (to be defined later).
As we approach the boundary, the dressed theory fails to describe the system,
and another nonvanishing, nonsingular operator needs to take over as the new
appropriate dressing rule [1].

Concretely, a generic dressing operator can involve powers of $a$ and its
derivatives $(\mathop{\mathrm{{}\partial}}\mathopen{}_{i}a)^{2}$, so we
consider

$\mathcal{D}_{b,c}\equiv\left[a^{2b}(\mathop{\mathrm{{}\partial}}\mathopen{}_{i}a)^{2c}\right]^{\frac{2}{d\cdot(b+c)+2c}},$
(19)

where $b,c$ can be any positive numbers for now, and the overall power is
chosen such that its dimension is fixed:

$[\mathcal{D}_{b,c}]=2.$ (20)

Indeed, note that $[a^{2}]=d$ and
$[(\mathop{\mathrm{{}\partial}}\mathopen{}_{i}a)^{2}]=d+2$. It is
straightforward to see that the dressing rule associated with
$\mathcal{D}_{b,c}$ for an operator $\mathcal{O}$ with dimension
$[\mathcal{O}]$ is

$\mathcal{O}_{dressed}\equiv\mathcal{O}\cdot\mathcal{D}_{b,c}^{\frac{d+2-[\mathcal{O}]}{2}}.$
(21)

So how do we fix $b$ and $c$ in the bulk and at the edge? Since the dressing
operator has the same dimension for any pair $(b,c)$, the selection criteria
are rather simple [1]. To leading-order in $\mu$, the vev of the dressing
operator scales like

$\langle\mathcal{D}_{b,c}\rangle\sim\mu^{1-\frac{4c}{d\cdot(b+c)+2c}}\cdot
z^{1-\frac{6c}{d\cdot(b+c)+2c}},$ (22)

since $\langle a\rangle\sim(\mu\cdot z)^{\frac{d}{4}}$ and
$(\mathop{\mathrm{{}\partial}}\mathopen{}_{i}\langle
a\rangle)^{2}\sim\mu^{\frac{d-2}{2}}\cdot z^{\frac{d-4}{2}}$. In the bulk of
the cloud, $z$ is of order 1 and the dressing rule is associated with the
operator $\mathcal{D}_{b,c}$ that has the highest $\mu$-scaling, i.e. $c=0$.
This yields the natural dressing rule used in the previous section, namely

$\mathcal{D}_{bulk}\equiv a^{\frac{4}{d}}.$ (23)

At the edge, however, the dressing operator is required to be nonvanishing
(unlike $\mathcal{D}_{bulk}$), and nonsingular. In short, its leading-order
dependence on $\mu$ in the ground state should feature neither positive nor
negative powers of $z$, i.e. it is a constant. This happens when
$d\cdot(b+c)=4c$, and we get

$\mathcal{D}_{edge}\equiv\left[a^{\frac{8}{d}-2}(\mathop{\mathrm{{}\partial}}\mathopen{}_{i}a)^{2}\right]^{\frac{1}{3}}.$
(24)

This operator is proportional to
$\left|\mathop{\mathrm{{}\partial}}\mathopen{}_{i}\left(a^{\frac{4}{d}}\right)\right|^{\frac{2}{3}}$
which is equivalent to the edge dressing rule originally discussed in [1] upon
trading $a^{\frac{4}{d}}$ for $U$ at the level of the nlsm.

###  II.4 $\delta_{\epsilon}$-layer and edge counterterms

In order to account for the lack of control at the droplet edge, one can
effectively cut off a small layer close the classical boundary of the cloud,
as discussed in [31, 9, 7]. Following [1], this regularization prescription
can be made slightly more precise in terms of the dressing operators we have
just found. Indeed, we eventually want to renormalize the theory by
introducing counterterms in the region where
$\mathcal{D}_{bulk}\sim\mathcal{D}_{edge}$ and beyond which
$\mathcal{D}_{edge}$ is the only appropriate dressing operator. This condition
is equivalent to

$(\mathop{\mathrm{{}\partial}}\mathopen{}_{i}a)^{2}\sim a^{\frac{4}{d}+2},$
(25)

which, in the ground state, is satisfied when $z\sim\mu^{-\frac{2}{3}}$, and
we thus define the $\delta_{\epsilon}$-layer,

$\delta_{\epsilon}\equiv\frac{\epsilon}{\mu^{\frac{2}{3}}}\sim\mathcal{O}\left(Q^{-\frac{2}{3d}}\right),$
(26)

for an arbitrary constant $\epsilon\sim\mathcal{O}(1)$. We thereby define the
bulk of the cloud as the region covered by the interval
$\delta_{\epsilon}\lesssim z\leq 1$, in agreement with the literature.

In order to construct a counterterm at the edge, we use Eq. (24) to dress to
marginality an operator $\mathcal{O}$ of dimension $[\mathcal{O}]$ together
with an operator-valued Dirac $\delta$-function $\delta(\mathcal{D}_{bulk})$
of dimension $-2$ [1]:

$\displaystyle\mathcal{O}_{edge}$
$\displaystyle\equiv\mathcal{O}\cdot\delta(\mathcal{D}_{bulk})\cdot\mathcal{D}_{edge}^{\frac{d+4-[\mathcal{O}]}{2}}$
(27)
$\displaystyle=\mathcal{O}\cdot\delta(a^{\frac{4}{d}})\cdot\left[a^{\frac{8}{d}-2}(\mathop{\mathrm{{}\partial}}\mathopen{}_{i}a)^{2}\right]^{\frac{d+4-[\mathcal{O}]}{6}}.$

Given that the vev $\langle a\rangle\equiv v(z)$ of the radial mode is of the
form [cf. Eq. (12)]

$v(z)=v_{hom}\cdot z^{\frac{d}{4}}\cdot[1+\mathrm{subleading}],$ (28)

where $v_{hom}$ is the superfluid ground-state solution of Eq. (15) in the
homogeneous case (i.e. without trap), the Dirac $\delta$-function becomes

$\delta(v^{\frac{4}{d}})=\delta\left(\frac{v^{\frac{4}{d}}}{v_{hom}^{\frac{4}{d}}}\right)\frac{1}{v_{hom}^{\frac{4}{d}}}=\frac{\delta(z)}{v_{hom}^{\frac{4}{d}}}.$
(29)

This shows that, in the ground state, counterterms are indeed located at
$z=0$, that is, at the classical edge.

The renormalization procedure thus consists in (1) regularizing the divergent
integrals of operators dressed in the bulk by removing the
$\delta_{\epsilon}$-layer from the domain of integration, and (2) introducing
edge counterterms whose coefficients get renormalized so as to absorb the
resulting $\epsilon$-dependence. After the first step, the regulator
$\epsilon$ appears in logarithms and denominators, and thus serves as a
diagnosis of divergences. However, the limit $\epsilon\to 0$ is not implicitly
understood at any point, since no physical quantity depends on it.

At this stage, it is worth mentioning that edge counterterms have already been
discussed in the literature in the context of effective open strings [46] and,
while this construction is very clear from an effective point of view, it
would be interesting to put it on a more formal basis.

###  II.5 Subleading operators

We now make a simple argument that allows to push the large-charge expansion
of $\Delta_{Q}$ beyond all known results so far. From a Wilsonian perspective,
the effective action contains infinitely many operators. However, in a large-
charge regime, there is a way in which we can organize them. The action Eq.
(15) provides the leading-order contribution to the conformal dimension
$\Delta_{Q}$, Eq. (16), which is of order $Q^{\frac{d+1}{d}}$. Operators with
more derivatives yield corrections to $\Delta_{Q}$ that are parametrically
suppressed by inverse powers of the charge. If one truncates the large-charge
expansion of $\Delta_{Q}$ to a desired order in $Q$, the task is to identify
the operators that survive in the action. In turn, this means we have to
understand how a generic operator contributes to $\Delta_{Q}$.

To do so, let us consider an operator $\mathcal{O}$ with dimension
$[\mathcal{O}]$, which we dress to marginality in the bulk as

$\mathcal{O}_{bulk}\equiv\mathcal{O}\cdot a^{\frac{2}{d}(d+2-[\mathcal{O}])}.$
(30)

If the latter appears in the Hamiltonian density, its contribution to
$\Delta_{Q}$ is then obtained by integrating its vev over the volume of the
cloud. Spherical invariance being preserved by the superfluid ground state,
this computation simplifies if the vev is expressed as a function of the
dimensionless coordinate $z$, in which case it turns into an integration over
$z\in[0,1]$—as indicated in Eq. (17)—which may need to be regularized upon
removing the $\delta_{\epsilon}$-layer. More specifically, let
$\mu[\mathcal{O}]$ and $z[\mathcal{O}]$ be such that the vev of the operator
$\mathcal{O}$ to leading-order in $\mu$ takes the form

$\langle\mathcal{O}\rangle\sim\mu^{\mu[\mathcal{O}]}\cdot
z^{z[\mathcal{O}]}+\mathrm{(subleading)}.$ (31)

The dressed operator in the bulk, Eq. (30), then has a vev that scales to
leading order in $\mu$ as

$\langle\mathcal{O}_{bulk}\rangle\sim\mu^{\mu[\mathcal{O}]+\frac{d+2-[\mathcal{O}]}{2}}\cdot
z^{z[\mathcal{O}]+\frac{d+2-[\mathcal{O}]}{2}}.$ (32)

It is now straightforward to analyze the leading contribution of this operator
to $\Delta_{Q}$. Indeed, if

$z[\mathcal{O}_{bulk}]\equiv
z[\mathcal{O}]+\frac{d+2-[\mathcal{O}]}{2}\leq-1,$ (33)

a divergence occurs when integrating over $z\in[0,1]$. In particular, a
logarithmic divergence appears if $z[\mathcal{O}_{bulk}]=-1$. We thus
regularize these divergences by removing the $\delta_{\epsilon}$-layer and,
accounting for the factor of $\mu^{\frac{d}{2}}\sim\sqrt{Q}$ from the measure
[cf. Eq. (17)], we find

$\Delta_{Q}\ni\begin{cases}Q^{\frac{d+1}{d}-\frac{[\mathcal{O}]-2\mu[\mathcal{O}]}{2d}}&\text{if}\quad[\mathcal{O}]<d+4+2z[\mathcal{O}]\\\
Q^{\frac{d+1}{d}-\frac{[\mathcal{O}]-2\mu[\mathcal{O}]}{2d}}\cdot\log\frac{Q}{\epsilon^{3d/2}}&\text{if}\quad[\mathcal{O}]=d+4+2z[\mathcal{O}]\\\
\frac{Q^{\frac{2}{3}-([\mathcal{O}]+4z[\mathcal{O}]-6\mu[\mathcal{O}]-2)/(6d)}}{\epsilon^{([\mathcal{O}]-2z[\mathcal{O}]-d-4)/2}}&\text{if}\quad[\mathcal{O}]>d+4+2z[\mathcal{O}].\end{cases}$
(34)

The last two cases are $\epsilon$-dependent and thus need to be renormalized
using edge counterterms.

We address this issue in the following, but let us first answer a natural
question: now that we have identified all possible (classical) leading
contributions to the conformal dimension $\Delta_{Q}$, what sort of operator
can $\mathcal{O}$ actually be? As dictated by general coordinate invariance
[31], one possibility is given by

$Z\equiv\nabla^{2}A_{0}-\frac{1}{d^{2}}\left(\nabla^{2}\chi\right)^{2},$ (35)

whose vev is $\langle Z\rangle=d$, but any other operator with more
derivatives of $\chi$ actually has a vanishing vev. Therefore, $\mathcal{O}$
is a composite operator made out of integer powers of $U$,
$(\mathop{\mathrm{{}\partial}}\mathopen{}_{i}U)^{2}$, $a$,
$(\mathop{\mathrm{{}\partial}}\mathopen{}_{i}a)^{2}$ and $Z$. As already
mentioned, the massive radial mode $a$ would have to be integrated out and one
can thus effectively trade $U$ for $a^{\frac{4}{d}}$—and likewise for
$\mathop{\mathrm{{}\partial}}\mathopen{}_{i}U$ and
$\mathop{\mathrm{{}\partial}}\mathopen{}_{i}a$—without changing the low-energy
description. This is convenient because if $U$ only appears in the Lagrangian
density as given in Eq. (15), then there is no need to worry about Legendre
transforming $\mathcal{O}$: it simply enters the Hamiltonian density
$\mathcal{H}=\frac{\mathop{\mathrm{{}\partial}}\mathopen{}\mathcal{L}}{\mathop{\mathrm{{}\partial}}\mathopen{}U}\dot{\chi}-\mathcal{L}$
with the opposite sign. Finally, we can strip off powers of $a$ as they will
be restored appropriately upon dressing $\mathcal{O}$ in the bulk [cf. Eq.
(30)]. Hence, we only need to consider operators of the form

$\mathcal{O}^{(m,n)}\equiv(\mathop{\mathrm{{}\partial}}\mathopen{}_{i}a)^{2m}Z^{n},$
(36)

with $m,n$ two positive integers and $[\mathcal{O}^{(m,n)}]=(d+2)m+4n$,
$\mu[\mathcal{O}^{(m,n)}]=\frac{d-2}{2}m$,
$z[\mathcal{O}^{(m,n)}]=\frac{d-4}{2}m$. The corresponding bulk operator is
then given by

$\displaystyle\mathcal{O}^{(m,n)}_{bulk}$
$\displaystyle\equiv(\mathop{\mathrm{{}\partial}}\mathopen{}_{i}a)^{2m}Z^{n}\cdot
a^{\frac{2}{d}((d+2)(1-m)-4n)}$ (37)
$\displaystyle=\left(\frac{(\mathop{\mathrm{{}\partial}}\mathopen{}_{i}a)^{2}}{a^{\frac{4}{d}+2}}\right)^{m}\left(\frac{Z}{a^{\frac{8}{d}}}\right)^{n}a^{\frac{4}{d}+2},$

and its leading contribution to $\Delta_{Q}$ is classified as follows:

$\Delta_{Q}\ni\begin{cases}Q^{\frac{d+1-2(m+n)}{d}}&\text{if}\quad
6m+4n<d+4\\\
Q^{\frac{2d-1}{3d}-\frac{2n}{3d}}\cdot\log\frac{Q}{\epsilon^{3d/2}}&\text{if}\quad
6m+4n=d+4\\\
\frac{Q^{\frac{2d-1}{3d}-\frac{2n}{3d}}}{\epsilon^{\frac{1}{2}(6m+4n-d-4)}}&\text{if}\quad
6m+4n>d+4.\end{cases}$ (38)

Notice that there are infinitely many operators contributing to the same power
of $Q$ in the last category, as it is independent of $m$. This remains true at
the level of the nlsm. For instance, the set of operators
$(\mathop{\mathrm{{}\partial}}\mathopen{}_{i}a)^{2m}$ with $m\geq 2$ in $d=2$
all give $Q^{\frac{1}{2}}$-contributions after
$\delta_{\epsilon}$-regularization, but they have not yet appeared in the
literature so far. The same holds for
$(\mathop{\mathrm{{}\partial}}\mathopen{}_{i}a)^{2m}Z$ with $m\geq 1$ in
$d=2$, yielding $Q^{\frac{1}{6}}$-contributions, and similarly in $d=3$ where
sets of equally contributing operators give terms of order $Q^{\frac{5}{9}}$,
$Q^{\frac{1}{3}}$, $Q^{\frac{1}{9}}$, etc. In Sec. III, we illustrate this by
including one operator of each set.

Note that these terms, however, appear with different powers of $\epsilon$ and
can thus be distinguished and compensated for by a single edge counterterm. In
order to identify the latter, we repeat the same analysis as before and remark
that the only candidates to be dressed at the edge are of the form $Z^{n}$
(with $n$ a positive integer), since powers of $a$ and
$(\mathop{\mathrm{{}\partial}}\mathopen{}_{i}a)^{2}$ are appropriately
incorporated by the dressing rule Eq. (27), which reads in this case

$Z^{n}_{edge}\equiv
Z^{n}\cdot\delta(a^{\frac{4}{d}})\cdot\left[a^{\frac{8}{d}-2}(\mathop{\mathrm{{}\partial}}\mathopen{}_{i}a)^{2}\right]^{\frac{d+4(1-n)}{6}}.$
(39)

After integrating the vev of the latter over the volume of the cloud, the
contribution to $\Delta_{Q}$ turns out to be

$\Delta_{Q}\ni Q^{\frac{2d-1}{3d}-\frac{2n}{3d}},$ (40)

which matches exactly the regulator-dependent part (i.e. the last two
categories) of Eq. (38). This analysis of divergences and counterterms thus
provides us with a simple way of constructing the effective action that
describes $\Delta_{Q}$ to a given order in $Q$. In Sec. III, we carry out this
derivation in $d=2$ and $d=3$ up to corrections that scale with negative
powers of the charge, but we first make some general observations.

###  II.6 Properties

_Equation of motion_ (eom). Consider the leading-order Lagrangian Eq. (15).
The equation of motion (eom) with respect to the radial mode $a$ then simply
reads $(v(z)/v_{hom})^{\frac{4}{d}}=z$ in the superfluid ground state, where
$v_{hom}\equiv(2\mu c_{\frac{d+1}{d}}/g)^{\frac{d}{4}}$ is the ground-state
solution in the homogeneous case, i.e. when the trap is turned off. As we
complement the Lagrangian with operators of the form Eq. (37) and the
corresponding counterterms, Eq. (39), the eom gets more and more complicated,
although it can always be put in the form

$\left(\frac{v(z)}{v_{hom}}\right)^{\frac{4}{d}}=z[1+B(z,v,v^{\prime},v^{\prime\prime})]$
(41)

in the ground state. In the bulk, $B(z,v,v^{\prime},v^{\prime\prime})\ll 1$
and one can solve this equation order by order in an expansion in
$\frac{1}{\mu}$. For instance, adding the first subleading operator
$-\frac{c_{1}}{2}\mathcal{O}^{(1,0)}_{bulk}$ to the Lagrangian yields

$\displaystyle\left(\frac{v(z)}{v_{hom}}\right)^{\frac{4}{d}}$
$\displaystyle=z\left[1-\frac{d}{16}\frac{c_{1}}{c_{\frac{d+1}{d}}}\frac{(4-d)+(3d-4)z}{\mu^{2}z^{3}}\right.$
(42) $\displaystyle\hskip
31.29802pt\left.+\mathcal{O}\left(\frac{1}{\mu^{4}}\right)\right].$

_Chemical potential._ The chemical potential can then be expressed as a
function of the charge $Q$ by inverting

$Q=\int_{cloud}\differential^{d}x\,\rho(\vec{x}),$ (43)

where $\rho=c_{\frac{d+1}{d}}v(z)^{2}$ is the ground-state charge density.
Using Eq. (42) and removing the $\delta_{\epsilon}$-layer to regularize the
divergent part, we find

$Q=\left(\frac{\mu}{\zeta}\right)^{d}\left[1+\mathcal{O}\left(\mu^{-\frac{d+2}{3}}\right)\right],$
(44)

where

$\zeta\equiv\sqrt{\frac{g}{4\pi
c_{\frac{d+1}{d}}}}\left[\frac{2\Gamma(d)}{c_{\frac{d+1}{d}}\Gamma\left(\frac{d}{2}\right)}\right]^{\frac{1}{d}},$
(45)

as first discussed in [7]. Therefore,

$\mu=\zeta
Q^{\frac{1}{d}}\left[1+\mathcal{O}\left(Q^{-\frac{d+2}{3d}}\right)\right].$
(46)

It is tempting to give an explicit expression for the correction in the square
bracket based on the solution found above for $v(z)$, but some remarks are in
order. Pushing the expansion further in Eq. (42), one would actually face
terms of the form $\frac{c_{1}^{k}}{(\mu^{2}z^{3})^{k}}$ ($k\in\mathbb{N}$),
which all become of order one close to the edge (i.e. where
$z\approx\delta_{\epsilon}\sim\mu^{-\frac{2}{3}}$), and contribute to the
$Q^{-\frac{d+2}{3d}}$-correction in the chemical potential, which makes it
hard to express it in closed form. A reasonable choice, though, is to limit
ourselves from now on to linear order in the Wilsonian coefficients of
subleading operators, since this does not change the nature of the expansion,
but merely its coefficients. Similarly, the operator
$-\frac{c_{2}}{4}\mathcal{O}^{(2,0)}_{bulk}$ also ends up contributing to the
next-to-leading order in the chemical potential for exactly the same reason,
and so does any operator $\mathcal{O}^{(m,0)}_{bulk}$, although we will not
need to consider $m>2$. With this,

$\displaystyle\mu$ $\displaystyle=\zeta
Q^{\frac{1}{d}}\left[1+\frac{d^{2}\Gamma(d)}{8c_{\frac{d+1}{d}}\Gamma\left(\frac{d}{2}\right)^{2}}\frac{1}{Q^{\frac{d+2}{3d}}}\left\\{\frac{c_{1}}{\epsilon^{\frac{4-d}{2}}}\right.\right.$
(47) $\displaystyle\hskip
45.5244pt\left.\left.+\frac{3d^{2}}{32}\frac{c_{2}g}{c_{\frac{d+1}{d}}}\frac{1}{\epsilon^{\frac{10-d}{2}}}\right\\}+\mathcal{O}\left(Q^{-\frac{d+4}{3d}}\right)\right].$

Counterterms contributions would allow us to renormalize this expression and
get rid of the $\epsilon$-dependence, but for practical purposes, we shall do
this and fix the renormalized coefficients such that they cancel all
divergences only at the very end of the computation of $\Delta_{Q}$. This is
because the latter coefficients are only fixed up to a finite piece which we
will not keep track of. Renormalizing $\mu$ at this stage would therefore just
clutter the computations.

_Structure of the expansion_. We now elaborate on the structure of the
expansion of $\Delta_{Q}$ by first noting that the contribution of
$\mathcal{O}^{(m,n)}_{bulk}$ is itself an expansion. Indeed, using the
leading-order solution $v(z)\sim(\mu\cdot z)^{\frac{d}{4}}$ (corrections do
not change the argument) and Eq. (17), we have

$\displaystyle\int_{cloud}\differential^{d}x\,\langle\mathcal{O}^{(m,n)}_{bulk}\rangle$
$\displaystyle\sim\mu^{d+1-2(m+n)}\int_{0}^{1}\differential
z\,\frac{(1-z)^{\frac{d}{2}-1+m}}{z^{3m+2n-1-\frac{d}{2}}}.$ (48)

The integral on the right-hand side either converges and corresponds to the
first case of Eq. (38), or needs to be regularized by setting the lower bound
to $\delta_{\epsilon}$, yielding the last two cases of this classification. In
the latter situation, however, the upper bound of the integral always gives a
finite result, thus continuing the expansion in $Q^{-\frac{2}{d}}$ starting at
$Q^{\frac{d+1}{d}}$. For concreteness, consider

$\mathcal{O}^{(1,1)}_{bulk}=\frac{(\mathop{\mathrm{{}\partial}}\mathopen{}_{i}a)^{2}Z}{a^{\frac{8}{3}}}$
(49)

in $d=3$. We obtain

$\displaystyle\int_{cloud}\differential^{3}x\,\langle\mathcal{O}^{(1,1)}_{bulk}\rangle$
$\displaystyle\sim\int_{\delta_{\epsilon}}^{1}\differential
z\,\frac{(1-z)^{\frac{3}{2}}}{z^{\frac{5}{2}}}$
$\displaystyle=\pi+\frac{2}{3\delta_{\epsilon}^{\frac{3}{2}}}-\frac{3}{\sqrt{\delta_{\epsilon}}}+\mathcal{O}\left(\sqrt{\delta_{\epsilon}}\right)$
$\displaystyle=\frac{2\zeta}{3\epsilon^{\frac{3}{2}}}Q^{\frac{1}{3}}-\frac{3\zeta^{\frac{1}{3}}}{\sqrt{\epsilon}}Q^{\frac{1}{9}}+\pi+\mathcal{O}\left(Q^{-\frac{1}{9}}\right),$

where $\pi$ comes from the upper bound, and the rest is an expansion in
$\delta_{\epsilon}$ whose dependence on $\epsilon$ is cured by the
counterterms. All in all, the expansion of $\Delta_{Q}$ reads

$\displaystyle\Delta_{Q}$
$\displaystyle=Q^{\frac{d+1}{d}}\left[a_{1}+a_{2}Q^{-\frac{2}{d}}+a_{3}Q^{-\frac{4}{d}}+\ldots\right]$
(50)
$\displaystyle+Q^{\frac{2d-1}{3d}}\left[b_{1}+b_{2}Q^{-\frac{2}{3d}}+b_{3}Q^{-\frac{4}{3d}}+\ldots\right]$
$\displaystyle+Q^{\frac{d-5}{3d}}\left[c_{1}+c_{2}Q^{-\frac{2}{3d}}+c_{3}Q^{-\frac{4}{3d}}+\ldots\right]+\ldots$

The first line is completely analogous to the relativistic case, while the
rest is specific to nrcft. The last line arises when $\mu$ is replaced by $Q$,
according to Eq. (46). When $d$ is even, it can be absorbed in the second
line, where some of the $b_{i}$’s contain $\log Q$-terms [see Eq. (38)].

_Casimir energy_. The leading quantum correction is due to the one-loop
Casimir energy [1], given by the Coleman-Weinberg formula applied to the
spectrum of excited states, Eq. (54) below. It is model-independent and _a
priori_ enters the expansion of $\Delta_{Q}$ at order $Q^{0}$. However, it was
shown in the relativistic case that the Casimir energy in odd spatial
dimensions is divergent and yields instead a universal $Q^{0}\log Q$-term
after renormalization [6]. This is hinted at by the presence of a classical
$Q^{0}$-contribution that serves as a counterterm for this divergence, and the
same phenomenon is thus to be expected in nrcft when $d$ is odd. This is the
object of a future publication [32].

_Dilaton mass_. In addition to the operators discussed in the previous
section, we now include a small dilaton mass deformation, as originally
proposed by Coleman in [33] in the relativistic case and studied in the
context of nrcfts at large charge in [7]. In the latter case, this potential
is of the form

$U_{C}\equiv\left(\frac{d}{d+2}\right)^{2}\frac{m_{\sigma}^{2}}{4f^{2}}\left[(fa)^{2\frac{d+2}{d}}-2\frac{d+2}{d}\log(fa)-1\right],$
(51)

where $m_{\sigma}\ll f^{-\frac{2}{d}}$ is a small dilaton mass, as can be seen
from the fact that, to quadratic order,
$U_{C}\approx\frac{1}{2}m_{\sigma}^{2}\sigma^{2}$. Adding this term to the
Lagrangian softly breaks conformal invariance and should trigger some
signature in the ground-state energy of the trapped system.

_Nonlinear sigma model_. If one is not interested in such a deformation, it
might be more natural to work with the nlsm, where the radial mode is
integrated out. Upon trading $a^{\frac{4}{d}}$ for $U$, the bulk operators,
Eq. (37), become

$\displaystyle\mathcal{O}^{(m,n)}_{bulk}$
$\displaystyle\equiv(\mathop{\mathrm{{}\partial}}\mathopen{}_{i}U)^{2m}Z^{n}\cdot
U^{\frac{d}{2}+1-(3m+2n)}$ (52)
$\displaystyle=\left(\frac{(\mathop{\mathrm{{}\partial}}\mathopen{}_{i}U)^{2}}{U^{3}}\right)^{m}\left(\frac{Z}{U^{2}}\right)^{n}U^{\frac{d}{2}+1},$

and edge counterterms, Eq. (39), read

$Z^{n}_{edge}\equiv
Z^{n}\cdot\delta(U)\cdot(\mathop{\mathrm{{}\partial}}\mathopen{}_{i}U)^{\frac{d+4(1-n)}{3}}.$
(53)

_Collective excitations_. As a final comment, let us mention that the energy
spectrum of collective excitations above the ground state (see e.g. [47] in
$d=3$, or [9]),

$\varepsilon(n,l)=\sqrt{\frac{4n}{d}(n+l+d-1)+l},$ (54)

can be given corrections, as initiated in [7], thanks to the renormalization
procedure discussed here. Similarly, the spectrum of spinning operators at
large charge described in [10] can be refined. We leave this for future work.

##  III Examples

We are now going to see this machinery in action through two examples, so let
us repeat the recipe. We are interested in the conformal dimension of the
lowest operator of charge $Q\gg 1$ in the theory without trap, $\Delta_{Q}$,
which is given by the ground-state energy of the trapped system. After
choosing at which order in $Q$ we want to truncate the expansion of
$\Delta_{Q}$ (in what follows, we go up to $Q^{0}$), we construct the
Lagrangian density by complementing Eq. (15) with subleading operators, Eq.
(37), based on the classification given in Eq. (38). We also include the
corresponding counterterms, Eq. (39), and we account for the small dilaton
mass deformation introduced above.

We then compute the ground-state energy density and we integrate it over the
volume of the cloud, regularizing the integrals when needed. We also express
the chemical potential $\mu$ as a function of the charge $Q$ to write
$\Delta_{Q}$ as an expansion in powers of the charge and, finally, we
renormalize the couplings of the counterterms so as to absorb any dependence
on the regulator $\epsilon$. We do this for both $d=2$ and $d=3$.

###  III.1 The $d=2$ case

Typically, nrcfts in two spatial dimensions are relevant for the description
of anyons [48, 29], which themselves are at the origin of the Aharonov-Bohm
effect [49] and whose existence has very recently been proven in the context
of the fractional quantum Hall effect [50, 51]. Based on the previous
discussion, we include the subleading operators $\mathcal{O}^{(m,n)}_{bulk}$
of Eq. (37) for $(m,n)\in\\{(1,0),(0,1),(2,0),(1,1)\\}$, as well as the
counterterms $Z^{0}_{edge}$ and $Z^{1}_{edge}$ constructed in Eq. (39). The
Lagrangian thus reads

$\displaystyle\mathcal{L}$
$\displaystyle=c_{\frac{3}{2}}a^{2}U-\frac{g}{4}a^{4}-\frac{c_{\frac{3}{2}}m^{2}_{\sigma}}{16f^{2}}\left[(fa)^{4}-4\log(fa)-1\right]$
$\displaystyle-\frac{c_{\frac{1}{2}}}{2}(\mathop{\mathrm{{}\partial}}\mathopen{}_{i}a)^{2}-\frac{c^{\prime}_{\frac{1}{2}}}{2}Z-\frac{c^{\prime\prime}_{\frac{1}{2}}}{4}\frac{(\mathop{\mathrm{{}\partial}}\mathopen{}_{i}a)^{4}}{a^{4}}-\frac{c_{\frac{1}{6}}}{4}\frac{(\mathop{\mathrm{{}\partial}}\mathopen{}_{i}a)^{2}}{a^{4}}Z$
$\displaystyle+\delta(a^{2})\left[\kappa_{\frac{1}{2}}\frac{a^{2}(\mathop{\mathrm{{}\partial}}\mathopen{}_{i}a)^{2}}{2}+\frac{\kappa_{\frac{1}{6}}}{2}\left(\frac{a^{2}(\mathop{\mathrm{{}\partial}}\mathopen{}_{i}a)^{2}}{2}\right)^{\frac{1}{3}}Z\right],$

where the Wilsonian parameters of subleading operators
$c_{\frac{1}{2}},c^{\prime}_{\frac{1}{2}},c^{\prime\prime}_{\frac{1}{2}}$,
etc. are labeled according to the order at which they enter the expansion of
$\Delta_{Q}$ in the end, and they are normalized so as to slightly simplify
the expression of the ground-state energy density $\mathcal{H}_{0}$. Indeed,
using $\langle\chi\rangle=\mu\cdot t$, $\langle a\rangle\equiv v(z)$ and Eq.
(17), we get

$\displaystyle\mathcal{H}_{0}$
$\displaystyle=c_{\frac{3}{2}}v^{2}\mu(1-z)+\frac{\tilde{g}}{4}v^{4}-\frac{c_{\frac{3}{2}}m^{2}_{\sigma}}{16f^{2}}\left[4\log(fv)+1\right]$
(55) $\displaystyle+\frac{(1-z)v^{\prime
2}}{\mu}\left[c_{\frac{1}{2}}+\frac{c_{\frac{1}{6}}}{v^{4}}\right]+c^{\prime}_{\frac{1}{2}}+\frac{c^{\prime\prime}_{\frac{1}{2}}(1-z)^{2}}{\mu^{2}}\left(\frac{v^{\prime}}{v}\right)^{4}$
$\displaystyle-\delta(v^{2})\left[\frac{\kappa_{\frac{1}{2}}(1-z)}{\mu}(vv^{\prime})^{2}+\frac{\kappa_{\frac{1}{6}}(1-z)^{\frac{1}{3}}}{\mu^{\frac{1}{3}}}(vv^{\prime})^{\frac{2}{3}}\right].$

The vev $v(z)$ is the solution of the eom (cf. Eq. (41))

$\left(\frac{v(z)}{v_{hom}}\right)^{2}=z\left[1+B(z,v,v^{\prime},v^{\prime\prime})\right],$
(56)

with $v_{hom}\equiv\sqrt{2\mu c_{\frac{3}{2}}/\tilde{g}}$, and
$B(z,v,v^{\prime},v^{\prime\prime})$ is given by

$\displaystyle B(z,v,v^{\prime},v^{\prime\prime})$
$\displaystyle=\frac{m_{\sigma}^{2}}{8f^{2}\mu
z}\frac{1}{v^{2}}+\frac{c_{\frac{1}{2}}}{c_{\frac{3}{2}}\mu^{2}z}\frac{(1-z)v^{\prime\prime}-v^{\prime}}{v}$
$\displaystyle+\frac{6c^{\prime\prime}_{\frac{1}{2}}(1-z)}{c_{\frac{3}{2}}\mu^{3}z}\frac{{v^{\prime}}^{2}}{v^{4}}\frac{(1-z)(vv^{\prime\prime}-{v^{\prime}}^{2})-vv^{\prime}}{v^{2}}$
$\displaystyle+\frac{c_{\frac{1}{6}}}{c_{\frac{3}{2}}\mu^{2}z}\frac{(1-z)(vv^{\prime\prime}-2{v^{\prime}}^{2})-vv^{\prime}}{v^{6}}.$

in the bulk. Referring to the previous section, we find that the chemical
potential is related to the charge as

$\displaystyle\mu$
$\displaystyle=\zeta\sqrt{Q}\left[1+\frac{1}{2c_{\frac{3}{2}}}\frac{1}{Q^{\frac{2}{3}}}\left\\{\frac{c_{\frac{1}{2}}}{\epsilon}+\frac{3}{8}\frac{c^{\prime\prime}_{\frac{1}{2}}\tilde{g}}{c_{\frac{3}{2}}}\frac{1}{\epsilon^{4}}\right\\}+\mathcal{O}\left(Q^{-1}\right)\right],$

where $\zeta\equiv\sqrt{\tilde{g}/(2\pi c_{\frac{3}{2}}^{2})}$. Note that the
dilaton mass $m_{\sigma}$ modifies the expression of the chemical potential
only beyond next-to-leading order. We are now in position to integrate Eq.
(55), removing the $\delta_{\epsilon}$-layer when necessary. Working linearly
in the Wilsonian coefficients of subleading operators, we find

$\displaystyle\Delta_{Q}^{(d=2)}$ $\displaystyle=\frac{2}{3}\zeta
Q^{\frac{3}{2}}+\left[\frac{c_{\frac{1}{2}}}{6\zeta
c_{\frac{3}{2}}}-\frac{\pi\zeta
c_{\frac{3}{2}}m_{\sigma}^{2}}{8f^{2}}\right]\sqrt{Q}\log Q$ (57)
$\displaystyle-\frac{k_{\frac{1}{2}}^{ren.}}{2\zeta
c_{\frac{3}{2}}}\sqrt{Q}-\left(2\pi^{4}c_{\frac{3}{2}}\right)^{\frac{1}{3}}\zeta
k_{\frac{1}{6}}^{ren.}\cdot Q^{\frac{1}{6}}-0.29416,$

up to corrections scaling with negative powers of $Q$. The last term is
universal and given by the Casimir energy found in [7], while the renormalized
couplings are

$\displaystyle\kappa_{\frac{1}{2}}^{ren.}$
$\displaystyle=\kappa_{\frac{1}{2}}+c_{\frac{1}{2}}\log\epsilon-\frac{\gamma}{24}\frac{c^{\prime\prime}_{\frac{1}{2}}}{\epsilon^{3}}+\mathrm{(finite)},$
(58) $\displaystyle\kappa_{\frac{1}{6}}^{ren.}$
$\displaystyle=\kappa_{\frac{1}{6}}+\left[\frac{c^{\prime\prime}_{\frac{1}{2}}}{24\gamma^{\frac{1}{3}}}-\frac{\gamma^{\frac{2}{3}}}{8}c_{\frac{1}{6}}\right]\frac{1}{\epsilon^{2}}+\mathrm{(finite)},$

where $\gamma\equiv\tilde{g}/c_{\frac{3}{2}}$. Note that we have absorbed the
contribution of $c^{\prime}_{\frac{1}{2}}$, which is finite, as well as a
finite correction due to $m_{\sigma}$ into the finite part of
$\kappa_{\frac{1}{2}}^{ren.}$. We thus see that the effect of the small
dilaton mass deformation is a mere shift in the coefficients of the
$\sqrt{Q}\log Q$ and $\sqrt{Q}$-terms.

Let us mention that anyons are not invariant under parity and it would thus be
interesting to extend this study to the case of parity-violating theories (cf.
[9] for suggestions), in the spirit of [52] in the relativistic case.

###  III.2 The $d=3$ case

As mentioned in the introduction, the case of nrcfts in three spatial
dimensions is relevant for the description of the unitary Fermi gas. In order
to build the large-charge eft, we again use the leading-order Lagrangian Eq.
(15), to which we add Coleman’s potential Eq. (51), the operators given in Eq.
(37) with $(m,n)\in\\{(1,0),(0,1),(2,0),(1,1),(0,2)\\}$ and the edge
counterterms, Eq. (39), constructed from $Z^{0}$, $Z^{1}$, and $Z^{2}$. We
thus consider the following Lagrangian density:

$\displaystyle\mathcal{L}$
$\displaystyle=c_{\frac{4}{3}}a^{2}U-\frac{3\tilde{g}}{10}a^{\frac{10}{3}}+\frac{9c_{\frac{4}{3}}m^{2}_{\sigma}}{100f^{2}}\left[\frac{10}{3}\log(fa)+1\right]$
$\displaystyle-\frac{c_{\frac{2}{3}}}{2}(\mathop{\mathrm{{}\partial}}\mathopen{}_{i}a)^{2}-\frac{c^{\prime}_{\frac{2}{3}}}{3}a^{\frac{2}{3}}Z-\frac{c_{\frac{5}{9}}}{4}\frac{(\mathop{\mathrm{{}\partial}}\mathopen{}_{i}a)^{4}}{a^{\frac{10}{3}}}-\frac{c_{\frac{1}{3}}}{6}\frac{(\mathop{\mathrm{{}\partial}}\mathopen{}_{i}a)^{2}}{a^{\frac{8}{3}}}Z$
$\displaystyle-\frac{c_{\frac{1}{9}}}{9}\frac{Z^{2}}{a^{2}}+\delta(a^{\frac{4}{3}})\left[\kappa_{\frac{5}{9}}\left(\frac{a^{\frac{2}{3}}(\mathop{\mathrm{{}\partial}}\mathopen{}_{i}a)^{2}}{2}\right)^{\frac{7}{6}}\right.$
$\displaystyle\left.+\frac{\kappa_{\frac{1}{3}}}{3}\left(\frac{a^{\frac{2}{3}}(\mathop{\mathrm{{}\partial}}\mathopen{}_{i}a)^{2}}{2}\right)^{\frac{1}{2}}Z+\frac{\kappa_{\frac{1}{9}}}{9}\left(\frac{2}{a^{\frac{2}{3}}(\mathop{\mathrm{{}\partial}}\mathopen{}_{i}a)^{2}}\right)^{\frac{1}{6}}Z^{2}\right],$

where $\tilde{g}\equiv
g\left(1+3c_{\frac{4}{3}}m_{\sigma}^{2}f^{\frac{4}{3}}/(10g)\right)$. The
ground-state energy density then reads

$\displaystyle\mathcal{H}_{0}$ $\displaystyle=c_{\frac{4}{3}}\mu
v^{2}(1-z)+\frac{3\tilde{g}}{10}v^{\frac{10}{3}}-\frac{9c_{\frac{4}{3}}m^{2}_{\sigma}}{100f^{2}}\left[\frac{10}{3}\log(fv)+1\right]$
(59)
$\displaystyle+\frac{{v^{\prime}}^{2}(1-z)}{\mu}\left[c_{\frac{2}{3}}+\frac{c_{\frac{1}{3}}}{v^{\frac{8}{3}}}\right]+c^{\prime}_{\frac{2}{3}}v^{\frac{2}{3}}+\frac{c_{\frac{5}{9}}(1-z)^{2}}{\mu^{2}}\frac{{v^{\prime}}^{4}}{v^{\frac{10}{3}}}$
$\displaystyle+\frac{c_{\frac{1}{9}}}{v^{2}}-\delta(v^{\frac{4}{3}})\left[\frac{\kappa_{\frac{5}{9}}(1-z)^{\frac{7}{6}}}{\mu^{\frac{7}{6}}}v^{\frac{7}{9}}{v^{\prime}}^{\frac{7}{3}}+\frac{\kappa_{\frac{1}{3}}\sqrt{z}}{\sqrt{z}}v^{\frac{1}{3}}v^{\prime}\right.$
$\displaystyle\hskip
128.0374pt\left.+\frac{\kappa_{\frac{1}{9}}\mu^{\frac{1}{6}}}{(1-z)^{\frac{1}{6}}}\frac{1}{v^{\frac{1}{9}}{v^{\prime}}^{\frac{1}{3}}}\right].$

Moreover, the vev of the radial mode $v(z)$ satisfies the equation of motion
Eq. (41) with $v_{hom}\equiv(2\mu c_{\frac{4}{3}}/\tilde{g})^{\frac{3}{4}}$
and

$\displaystyle B(z,v,v^{\prime},v^{\prime\prime})$
$\displaystyle=\frac{3m_{\sigma}^{2}}{20f^{2}\mu
z}\frac{1}{v^{2}}+\frac{c_{\frac{2}{3}}}{2c_{\frac{4}{3}}\mu^{2}z}\frac{2(1-z)v^{\prime\prime}-3v^{\prime}}{v}$
$\displaystyle-\frac{c^{\prime}_{\frac{2}{3}}}{3c_{\frac{4}{3}}\mu
z}\frac{1}{v^{\frac{4}{3}}}+\frac{c_{\frac{1}{9}}}{c_{\frac{4}{3}}\mu
z}\frac{1}{v^{4}}$
$\displaystyle+\frac{c_{\frac{5}{9}}(1-z)}{c_{\frac{4}{3}}\mu^{3}z}\frac{{v^{\prime}}^{2}}{v^{\frac{10}{3}}}\frac{(1-z)(6vv^{\prime\prime}-5{v^{\prime}}^{2})-9vv^{\prime}}{v^{2}}$
$\displaystyle+\frac{c_{\frac{1}{3}}}{6c_{\frac{4}{3}}\mu^{2}z}\frac{(1-z)(6vv^{\prime\prime}-8{v^{\prime}}^{2})-9vv^{\prime}}{v^{\frac{14}{3}}}$

in the bulk. Consequently, the chemical potential reads

$\displaystyle\mu$ $\displaystyle=\zeta Q^{\frac{1}{3}}\left[1+\frac{9}{\pi
c_{\frac{4}{3}}}\frac{1}{Q^{\frac{5}{9}}}\left\\{\frac{c_{\frac{2}{3}}}{\sqrt{\epsilon}}+\frac{27c_{\frac{5}{9}}g}{32c_{\frac{4}{3}}}\frac{1}{\epsilon^{\frac{7}{2}}}\right\\}+\mathcal{O}\left(Q^{-\frac{7}{9}}\right)\right].$

Proceeding as before, we finally find

$\displaystyle\Delta_{Q}^{(d=3)}$ $\displaystyle=\frac{3}{4}\zeta
Q^{\frac{4}{3}}+\left[\frac{27c_{\frac{2}{3}}}{8\zeta
c_{\frac{4}{3}}}+\frac{\pi^{\frac{4}{3}}\zeta
c^{\prime}_{\frac{2}{3}}}{c_{\frac{4}{3}}^{\frac{1}{3}}}\right]\cdot
Q^{\frac{2}{3}}$ (60)
$\displaystyle-\left[\frac{3^{42}}{2^{15}\pi^{14}c_{\frac{4}{3}}^{16}}\right]^{\frac{1}{18}}\frac{\kappa_{\frac{5}{9}}^{ren.}}{\zeta}\cdot
Q^{\frac{5}{9}}$ $\displaystyle-\frac{\sqrt{2}\pi
m_{\sigma}^{2}\zeta^{\frac{3}{2}}c_{\frac{4}{3}}}{5f^{2}}\cdot\sqrt{Q}\left[\log
Q+\left(\log\frac{512f^{4}}{\pi^{2}\zeta^{3}c_{\frac{4}{3}}^{2}}-\frac{34}{5}\right)\right]$
$\displaystyle-3\sqrt{2}\pi\zeta k_{\frac{1}{3}}^{ren.}\cdot
Q^{\frac{1}{3}}-\left[\frac{2^{33}\pi^{50}c_{\frac{4}{3}}^{16}}{3^{6}}\right]^{\frac{1}{18}}\zeta^{3}\kappa_{\frac{1}{9}}^{ren.}\cdot
Q^{\frac{1}{9}}$
$\displaystyle-\frac{\sqrt{2}\pi^{2}}{2}\left[\frac{3^{5}}{16\sqrt{\gamma}}c_{\frac{5}{9}}-\frac{3\sqrt{\gamma}}{4}c_{\frac{1}{3}}+\frac{\gamma^{\frac{3}{2}}}{27}c_{\frac{1}{9}}\right],$

where $\gamma\equiv 18\tilde{g}/c_{\frac{4}{3}}$ is a convenient parameter to
express the renormalized couplings, which are given by

$\displaystyle\kappa_{\frac{5}{9}}^{ren.}$
$\displaystyle=\kappa_{\frac{5}{9}}-\frac{\gamma^{\frac{5}{6}}}{80}\frac{c_{\frac{5}{9}}}{\epsilon^{\frac{5}{2}}}+\mathrm{(finite)}$
(61) $\displaystyle\kappa_{\frac{1}{3}}^{ren.}$
$\displaystyle=\kappa_{\frac{1}{3}}+\left[\frac{135c_{\frac{5}{9}}}{8\sqrt{\gamma}}-\frac{\sqrt{2}\gamma}{18}c_{\frac{1}{3}}\right]\frac{1}{4\epsilon^{\frac{3}{2}}}+\mathrm{(finite)}$
$\displaystyle\kappa_{\frac{1}{9}}^{ren.}$
$\displaystyle=\kappa_{\frac{1}{9}}+\left[\frac{3^{5}}{2^{3}}\frac{c_{\frac{1}{3}}}{\gamma^{\frac{5}{6}}}-\frac{3^{8}\cdot
5}{2^{7}}\frac{c_{\frac{5}{9}}}{\gamma^{\frac{11}{6}}}-\gamma^{\frac{1}{6}}c_{\frac{1}{9}}\right]\frac{1}{\sqrt{\epsilon}}+\mathrm{(finite)}.$

The dilaton mass deformation $m_{\sigma}$ is responsible for the presence of
the $\sqrt{Q}\log Q$ and $\sqrt{Q}$-terms, which is the signature of the soft
breaking of conformal invariance. Moreover, there is a mixed
$Q^{0}$-contribution that serves as a counterterm for the divergent one-loop
Casimir energy, and a universal $Q^{0}\log Q$-term is expected to arise as a
consequence of it. This will be computed in an upcoming article [32].

Among the many quantities that can be given corrections based on this
construction (cf. also [31]), let us come back to the doubly integrated
density mentioned in the introduction, whose leading-order expression is given
in Eq. (18). We solve the equation of motion as in Eq. (42), but we only
account for the corrections caused $c_{\frac{2}{3}}$ for simplicity. We then
integrate the charge density $\rho(z)=c_{\frac{4}{3}}v(z)^{2}$ over $x_{1}$
and $x_{2}$ to obtain

$\displaystyle n(x_{3})$ $\displaystyle=\frac{2\pi
g}{5}\left[\frac{2c_{\frac{4}{3}}\mu}{g}\left(1-\frac{x_{3}^{2}}{2\mu}\right)\right]^{\frac{5}{2}}$
(62)
$\displaystyle\times\left[1+\frac{45c_{\frac{2}{3}}}{32c_{\frac{4}{3}}\mu^{2}}\left\\{\frac{5}{\left(1-\frac{x_{3}^{2}}{2\mu}\right)^{2}}-\frac{1}{\left(1-\frac{x_{3}^{2}}{2\mu}\right)^{3}}\right\\}+\ldots\right].$

This expression is valid for $x_{3}\in[0,R_{cl}\sqrt{1-\delta_{\epsilon}}]$
and can in principle improve the fitting to experimental data.

##  IV Conclusion

Via the nonrelativistic state-operator map, charged operators correspond to
finite density states in a harmonic trap. In this work, we investigated the
class of nrcfts whose large-charge sector is effectively described by a
superfluid state in the trap, which is particularly relevant for the case of
the unitary Fermi gas. Specifically, we exploited this effective description
to extend all known results about the expansion of the conformal dimension of
the lightest charged operator, up to quantum corrections entering at order
$Q^{0}$, see Eq. (50), thereby uncovering a rich structure of logarithmic
contributions. This is based on the recent treatment of divergences at the
edge of the physical cloud of trapped particles [1], which we reviewed and
extended. We also accounted for a small dilaton mass deformation in order to
explore the near-conformal regime, and illustrated the full procedure in the
$d=2$ and $d=3$ cases, see Sec. III.

Whenever we found it appropriate, we commented on the connections with the
ultracold atom literature (cf. in particular the introduction) and computed
some new corrections, as in Eq. (62) for the doubly integrated density. This
fruitful direction of research remains to be explored systematically. See also
[31, 39].

Let us finally mention the seminal works [53, 54] paving the way toward a
gravity/nrcft correspondence (see also [55] for a recent proposal). It would
be fascinating to understand whether the large-charge sector of certain nrcfts
can be described in a dual picture and how this would relate to the effective
construction used here.

### Acknowledgments

Enlightening discussions with Simeon Hellerman, Domenico Orlando, Susanne
Reffert and Ian Swanson are gratefully acknowledged. This work is supported by
the Swiss National Science Foundation under grant No. 200021 192137.

## References

  * [1] S. Hellerman and I. Swanson, “Droplet-Edge Operators in Nonrelativistic Conformal Field Theories,” arXiv:2010.07967 [hep-th].
  * [2] S. Hellerman, D. Orlando, S. Reffert, and M. Watanabe, “On the CFT Operator Spectrum at Large Global Charge,” JHEP 12 (2015) 071, arXiv:1505.01537 [hep-th].
  * [3] L. A. Gaumé, D. Orlando, and S. Reffert, “Selected Topics in the Large Quantum Number Expansion,” arXiv:2008.03308 [hep-th].
  * [4] A. Monin, D. Pirtskhalava, R. Rattazzi, and F. K. Seibold, “Semiclassics, Goldstone Bosons and CFT data,” JHEP 06 (2017) 011, arXiv:1611.02912 [hep-th].
  * [5] G. Cuomo, “The OPE meets semiclassics,” arXiv:2103.01331 [hep-th].
  * [6] G. Cuomo, “A note on the large charge expansion in 4d CFT,” Phys. Lett. B 812 (2021) 136014, arXiv:2010.00407 [hep-th].
  * [7] D. Orlando, V. Pellizzani, and S. Reffert, “Near-Schrödinger dynamics at large charge,” arXiv:2010.07942 [hep-th].
  * [8] S. Favrod, D. Orlando, and S. Reffert, “The large-charge expansion for Schrödinger systems,” JHEP 12 (2018) 052, arXiv:1809.06371 [hep-th].
  * [9] S. M. Kravec and S. Pal, “Nonrelativistic Conformal Field Theories in the Large Charge Sector,” JHEP 02 (2019) 008, arXiv:1809.08188 [hep-th].
  * [10] S. M. Kravec and S. Pal, “The Spinful Large Charge Sector of Non-Relativistic CFTs: From Phonons to Vortex Crystals,” JHEP 05 (2019) 194, arXiv:1904.05462 [hep-th].
  * [11] L. Alvarez-Gaume, O. Loukas, D. Orlando, and S. Reffert, “Compensating strong coupling with large charge,” JHEP 04 (2017) 059, arXiv:1610.04495 [hep-th].
  * [12] L. Alvarez-Gaume, D. Orlando, and S. Reffert, “Large charge at large N,” JHEP 12 (2019) 142, arXiv:1909.02571 [hep-th].
  * [13] S. Giombi and J. Hyman, “On the Large Charge Sector in the Critical $O(N)$ Model at Large $N$,” arXiv:2011.11622 [hep-th].
  * [14] N. Dondi, I. Kalogerakis, D. Orlando, and S. Reffert, “Resurgence of the large-charge expansion,” arXiv:2102.12488 [hep-th].
  * [15] S. Hellerman, S. Maeda, and M. Watanabe, “Operator Dimensions from Moduli,” JHEP 10 (2017) 089, arXiv:1706.05743 [hep-th].
  * [16] S. Hellerman and S. Maeda, “On the Large $R$-charge Expansion in ${\mathcal{N}}=2$ Superconformal Field Theories,” JHEP 12 (2017) 135, arXiv:1710.07336 [hep-th].
  * [17] A. Bourget, D. Rodriguez-Gomez, and J. G. Russo, “A limit for large $R$-charge correlators in $\mathcal{N}=2$ theories,” JHEP 05 (2018) 074, arXiv:1803.00580 [hep-th].
  * [18] S. Hellerman, S. Maeda, D. Orlando, S. Reffert, and M. Watanabe, “Universal correlation functions in rank 1 SCFTs,” JHEP 12 (2019) 047, arXiv:1804.01535 [hep-th].
  * [19] M. Beccaria, “On the large R-charge $\mathcal{N}=2$ chiral correlators and the Toda equation,” arXiv:1809.06280 [hep-th].
  * [20] M. Beccaria, F. Galvagno, and A. Hasan, “$\mathcal{N}=2$ conformal gauge theories at large R-charge: the $SU(N)$ case,” arXiv:2001.06645 [hep-th].
  * [21] S. Hellerman, S. Maeda, D. Orlando, S. Reffert, and M. Watanabe, “S-duality and correlation functions at large R-charge,” arXiv:2005.03021 [hep-th].
  * [22] G. Badel, G. Cuomo, A. Monin, and R. Rattazzi, “The Epsilon Expansion Meets Semiclassics,” arXiv:1909.01269 [hep-th].
  * [23] G. Arias-Tamargo, D. Rodriguez-Gomez, and J. G. Russo, “The large charge limit of scalar field theories and the Wilson-Fisher fixed point at $\epsilon=0$,” arXiv:1908.11347 [hep-th].
  * [24] M. Watanabe, “Accessing Large Global Charge via the $\epsilon$-Expansion,” arXiv:1909.01337 [hep-th].
  * [25] O. Antipin, J. Bersini, F. Sannino, Z.-W. Wang, and C. Zhang, “Charging the $O(N)$ model,” Phys. Rev. D 102 no. 4, (2020) 045011, arXiv:2003.13121 [hep-th].
  * [26] O. Antipin, J. Bersini, F. Sannino, Z.-W. Wang, and C. Zhang, “Charging non-Abelian Higgs theories,” Phys. Rev. D 102 no. 12, (2020) 125033, arXiv:2006.10078 [hep-th].
  * [27] O. Antipin, J. Bersini, F. Sannino, Z.-W. Wang, and C. Zhang, “Untangling scaling dimensions of fixed charge operators in Higgs theories,” Phys. Rev. D 103 no. 12, (2021) 125024, arXiv:2102.04390 [hep-th].
  * [28] F. Werner and Y. Castin, “Unitary gas in an isotropic harmonic trap: Symmetry properties and applications,” Phys. Rev. A 74 (Nov, 2006) 053604. https://link.aps.org/doi/10.1103/PhysRevA.74.053604.
  * [29] Y. Nishida and D. T. Son, “Nonrelativistic conformal field theories,” Phys. Rev. D76 (2007) 086004, arXiv:0706.3746 [hep-th].
  * [30] W. D. Goldberger, Z. U. Khandker, and S. Prabhu, “OPE convergence in non-relativistic conformal field theories,” JHEP 12 (2015) 048, arXiv:1412.8507 [hep-th].
  * [31] D. T. Son and M. Wingate, “General coordinate invariance and conformal invariance in nonrelativistic physics: Unitary Fermi gas,” Annals Phys. 321 (2006) 197–224, arXiv:cond-mat/0509786 [cond-mat].
  * [32] S. Hellerman, D. Orlando, V. Pellizzani, S. Reffert, and I. Swanson, “Nonrelativistic CFTs at Large Charge: Casimir Energy and Logarithmic Enhancements,” arXiv:2111.12094 [hep-th].
  * [33] S. Coleman, Aspects Of Symmetry.  Cambridge University Press, 1988.
  * [34] D. Orlando, S. Reffert, and F. Sannino, “Charging the conformal window,” Phys. Rev. D 103 (May, 2021) 105026. https://link.aps.org/doi/10.1103/PhysRevD.103.105026.
  * [35] D. Orlando, S. Reffert, and F. Sannino, “Near-conformal dynamics at large charge,” Phys. Rev. D 101 (Mar, 2020) 065018. https://link.aps.org/doi/10.1103/PhysRevD.101.065018.
  * [36] F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of bose-einstein condensation in trapped gases,” Reviews of Modern Physics 71 no. 3, (Apr, 1999) 463–512. http://dx.doi.org/10.1103/RevModPhys.71.463.
  * [37] S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of ultracold atomic fermi gases,” Rev. Mod. Phys. 80 (Oct, 2008) 1215–1274. https://link.aps.org/doi/10.1103/RevModPhys.80.1215.
  * [38] M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, C. Chin, J. H. Denschlag, and R. Grimm, “Crossover from a molecular bose-einstein condensate to a degenerate fermi gas,” Phys. Rev. Lett. 92 (Mar, 2004) 120401. https://link.aps.org/doi/10.1103/PhysRevLett.92.120401.
  * [39] M. A. Escobedo, M. Mannarelli, and C. Manuel, “Bulk viscosities for cold Fermi superfluids close to the unitary limit,” Phys. Rev. A 79 (2009) 063623, arXiv:0904.3023 [cond-mat.quant-gas].
  * [40] A. L. Fetter and D. L. Feder, “Beyond the thomas-fermi approximation for a trapped condensed bose-einstein gas,” Phys. Rev. A 58 (Oct, 1998) 3185–3194. https://link.aps.org/doi/10.1103/PhysRevA.58.3185.
  * [41] F. Dalfovo, L. Pitaevskii, and S. Stringari, “Order parameter at the boundary of a trapped bose gas,” Phys. Rev. A 54 (Nov, 1996) 4213–4217. https://link.aps.org/doi/10.1103/PhysRevA.54.4213.
  * [42] T. Brauner and H. Watanabe, “Spontaneous breaking of spacetime symmetries and the inverse Higgs effect,” Phys. Rev. D 89 no. 8, (2014) 085004, arXiv:1401.5596 [hep-ph].
  * [43] I. Arav, I. Hason, and Y. Oz, “Spontaneous Breaking of Non-Relativistic Scale Symmetry,” JHEP 10 (2017) 063, arXiv:1702.00690 [hep-th].
  * [44] Z. Komargodski and A. Schwimmer, “On Renormalization Group Flows in Four Dimensions,” JHEP 1112 (2011) 099, arXiv:1107.3987 [hep-th].
  * [45] D. Orlando, S. Reffert, and F. Sannino, “Near-Conformal Dynamics at Large Charge,” Phys. Rev. D 101 no. 6, (2020) 065018, arXiv:1909.08642 [hep-th].
  * [46] S. Hellerman and I. Swanson, “Boundary Operators in Effective String Theory,” JHEP 04 (2017) 085, arXiv:1609.01736 [hep-th].
  * [47] M. A. Baranov and D. S. Petrov, “Low-energy collective excitations in a superfluid trapped fermi gas,” Phys. Rev. A 62 (Sep, 2000) 041601(R). https://link.aps.org/doi/10.1103/PhysRevA.62.041601.
  * [48] F. Wilczek, “Quantum Mechanics of Fractional Spin Particles,” Phys. Rev. Lett. 49 (1982) 957–959.
  * [49] O. Bergman and G. Lozano, “Aharonov-Bohm scattering, contact interactions and scale invariance,” Annals Phys. 229 (1994) 416–427, arXiv:hep-th/9302116 [hep-th].
  * [50] H. Bartolomei, M. Kumar, R. Bisognin, A. Marguerite, J.-M. Berroir, E. Bocquillon, B. Plaçais, A. Cavanna, Q. Dong, U. Gennser, and et al., “Fractional statistics in anyon collisions,” Science 368 no. 6487, (Apr, 2020) 173–177. http://dx.doi.org/10.1126/science.aaz5601.
  * [51] J. Nakamura, S. Liang, G. C. Gardner, and M. J. Manfra, “Direct observation of anyonic braiding statistics at the $\nu$=1/3 fractional quantum hall state,” 2020\.
  * [52] G. Cuomo, L. V. Delacretaz, and U. Mehta, “Large Charge Sector of 3d Parity-Violating CFTs,” JHEP 05 (2021) 115, arXiv:2102.05046 [hep-th].
  * [53] D. T. Son, “Toward an AdS/cold atoms correspondence: A Geometric realization of the Schrodinger symmetry,” Phys. Rev. D 78 (2008) 046003, arXiv:0804.3972 [hep-th].
  * [54] K. Balasubramanian and J. McGreevy, “Gravity duals for non-relativistic CFTs,” Phys. Rev. Lett. 101 (2008) 061601, arXiv:0804.4053 [hep-th].
  * [55] X. Bekaert, E. Meunier, and S. Moroz, “Towards a gravity dual of the unitary fermi gas,” Phys. Rev. D 85 (May, 2012) 106001. https://link.aps.org/doi/10.1103/PhysRevD.85.106001.

  *[nrcfts]: nonrelativistic conformal field theory
  *[eft]: effective field theory
  *[nlsm]: nonlinear sigma model
  *[bcs]: Bardeen-Cooper-Schrieffer
  *[bec]: Bose-Einstein condensate
  *[lsm]: linear sigma model
  *[vev]: vacuum expectation value
  *[ir]: infrared
  *[uv]: ultraviolet
  *[eom]: equation of motion
  *[nrcft]: nonrelativistic conformal field theory