#  Following the flow for large N and large charge

Domenico Orlando★✩, Susanne Reffert✩ and Tim Schmidt✩ ★ INFN sezione di
Torino, via Pietro Giuria 1, 10125 Turin, Italy ✩ Albert Einstein Center for
Fundamental Physics Institute for Theoretical Physics, University of Bern,
Sidlerstrasse 5, CH-3012 Bern, Switzerland

###### Contents

  1. 1 Introduction
  2. 2 The effective action from large N
    1. 2.1 One-loop result at fixed charge
    2. 2.2 The effective actions
  3. 3 The $\varphi^{4}$ model on flat space
    1. 3.1 The effective potential on flat space
    2. 3.2 The free energy density on flat space
    3. 3.3 Nonperturbative finite size corrections on flat space
  4. 4 The effective potential on the sphere
  5. 5 Conclusions

##  1 Introduction

Working in a sector of fixed and large global charge leads to numerous
simplifications in strongly coupled models (for a review, see [1] and
references therein). While most of the large-charge literature focuses on cft,
the approach is in principle applicable to any quantum field theory.

The O(2n) vector model in three dimensions has been studied in great detail at
large charge [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. While these works all study
the conformal ir (ir) wf (wf) fixed point, we use here the combined power of
working at large charge and large N to study the model away from the fixed
point for generic values of the quadratic and quartic couplings.

While working at large N gives a lot of control, working in the double-scaling
limit of large charge at fixed $Q/N$ provides us with essential extra
structure. From the grand potential (which is a function of the chemical
potential dual to the fixed charge), computed at leading order in N, we can
directly read off an effective nlsm (nlsm) action. This large-$Q$ effective
action captures the system at large charge, it describes the spontaneous
symmetry breaking due to the fixed charge and contains the same type of terms
as the nlsm at finite N, see _e.g._ the result for the O(2) model [2]. The
large-N behavior of the model can be described through an effective action
which takes the form of a lsm (lsm). It can be studied both in the limit of
large and small charge, and captures the flow from the uv (uv) to the ir. The
two actions apply in general to different ranges of the parameters $N$ and $Q$
but must agree in the double scaling limit of $Q$, $N$ large and $Q/N$ fixed.

Using this agreement we find that the grand potential is related to the
effective potential by a Legendre transform in the vev (vev) of the collective
field, $m^{2}$. This observation is crucial. The resulting effective potential
is exact in the sense that it corresponds to a resummed expression which
includes infinitely many Feynman diagrams, each computing the leading-$N$
contribution to a monomial in the expansion of the effective potential. This
is due to the large-N limit which captures directly the quantum effects in
terms of a semiclassical action. In the massless case we indeed reproduce a
known result found diagrammatically in [13].

Applied to the case of the $\varphi^{4}$ model on flat space the grand
potential at leading order in N is given by two terms only and leads to a
closed expression for the effective potential. Depending on whether the
coefficient $r$ of the quadratic term in the tree-level potential is bigger,
equal, or smaller than zero we are physically in the unbroken phase, at
criticality or in the broken phase. The power of the Legendre transform shows
itself in the case of the broken phase where the double-well tree-level
potential is replaced by a convex effective potential with a flat line between
the original minima.

All these results are obtained in a very simple and transparent fashion and
without the need of evaluating and resumming Feynman diagrams. Also the free
energy in the different phases can be computed straightforwardly via another
Legendre transform, this time in $m$.

Using the same methodology that is introduced here, the O(2N) vector model is
studied in various dimensions in [14], where the criterion of convexity allows
us to distinguish between unitary and non-unitary theories.

The plan of this paper is as follows. In Section 2.1, we extend the large N
results of [8] to non-critial values of the quadratic coupling $r$ and of the
quartic coupling $u$ and compute both the expressions for the free energy and
for the grand potential at large charge. In Section 2.2 we observe that the
effective potential at leading order in N is given by the Legendre transform
of the grand potential. It is important to use the definition of the Legendre
transform that includes taking the supremum. After the general remarks of
Section sec:largeN, we explicitly study the $\varphi^{4}$ model on flat space
in Section sec:phi4. Depending on the value of the coefficient $r$ of the
quadratic term, we distinguish three cases in Section 3.1: the unbroken phase
($r>0$), the critical point ($r=0$) and the broken phase ($r<0$). For each of
them, we compute the effective potential and observe that unlike the tree-
level potential, it is always convex. In Section 3.2, we give the free energy
density for the three cases. In Section 3.3 we derive the first
nonperturbative correction to the grand potential and its Legendre transform,
the effective potential. In Section sec:sphereEFT we treat the critical theory
on the sphere. Since we only have a series expansion for the zeta function, we
do not get a closed expression for the effective potential, but give the first
few terms of $V$ as an expansion in the curvature. In Section sec:Conclusions,
we give conclusions and an outlook.

##  2 The effective action from large N

We want to discuss the Landau–Ginzburg model for $2N$ real scalar fields in
the vector representation of $O(2N)$ in three dimensions. When fine-tuned,
this model flows to the wf fixed point in the ir. In the following we will
rely on the same ideas as [8] and compute the one-loop results at fixed charge
in the double scaling limit $N\to\infty$, $Q\to\infty$ with $Q/N$ fixed, while
keeping the coupling to the $\absolutevalue{\phi}^{4}$ term finite.

###  2.1 One-loop result at fixed charge

We work in Euclidean signature on $S^{1}_{\beta}\times\Sigma$, where $\Sigma$
is a Riemann surface. In the following we will specialize to the cases of flat
space (the torus $T^{2}$) and the sphere $S^{2}$. We start with the action for
$N$ complex scalar fields $\phi_{i}$,

$\displaystyle
S[\phi_{i}]=-\int\differential{\tau}\differential{\Sigma}\left[g^{\mu\nu}(\mathop{\mathrm{{}\partial}}\mathopen{}^{i}_{\mu}\phi_{i})^{\dagger}(\mathop{\mathrm{{}\partial}}\mathopen{}^{i}_{\nu}\phi_{i})-r(\phi_{i}^{\dagger}\phi_{i})-\frac{u}{2N}(\phi_{i}^{\dagger}\phi_{i})^{2}\right].$
(2.1)

For $r=R/8$, where $R$ is the scalar curvature of $\Sigma$, the model flows to
a non-trivial conformal fixed point in the ir where $u$ diverges. Here we want
to explore the regime where $u$ is finite.

After a Stratonovich transformation we have an action in terms of $\phi_{i}$
and the field $\lambda$ [15, 16, 17],

$\displaystyle
S[\phi_{i},\lambda]=-\int\differential{\tau}\differential{\Sigma}\left[g^{\mu\nu}(\mathop{\mathrm{{}\partial}}\mathopen{}^{i}_{\mu}\phi_{i})^{\dagger}(\mathop{\mathrm{{}\partial}}\mathopen{}^{i}_{\nu}\phi_{i})-(r+\lambda)(\phi_{i}^{\dagger}\phi_{i})-\frac{\lambda^{2}}{2u}\right].$
(2.2)

We want to study this system at fixed charge, where the canonical partition
function is given by

$\displaystyle Z(Q_{1},\dots,Q_{N})$ $\displaystyle=\Tr[e^{-\beta
H}\prod_{i=1}^{N}\delta(\hat{Q}_{i}-Q_{i})]=\int_{-\pi}^{\pi}\prod_{i=1}^{N}\frac{\differential{\theta_{i}}}{2\pi}\prod_{i=1}^{N}e^{i\theta_{i}Q_{i}}\Tr[e^{-\beta
H}\prod_{i=1}^{N}e^{-i\theta_{i}\hat{Q}_{i}}]$ (2.3)
$\displaystyle=\int_{-\pi}^{\pi}\prod_{i=1}^{N}\frac{\differential{\theta_{i}}}{2\pi}\prod_{i=1}^{N}e^{i\theta_{i}Q_{i}}Z_{gc}(i\theta_{1},\dots,i\theta_{N}),$

where $\hat{Q}_{i}$ are the Noether charges corresponding to the Cartan
generators of the global $O(2N)$ symmetry and $\theta_{i}$ are the dual
imaginary chemical potentials in the grand canonical description.

It is convenient to introduce the covariant derivative $D_{\mu}$, defined as

$\displaystyle
D^{i}_{\mu}\phi_{i}:=\begin{cases}\left(\partial_{0}+i\frac{\theta_{i}}{\beta}\right)\phi_{i},&\mu=0,\\\
\partial_{i}\phi_{i},&\text{otherwise,}\end{cases}$ (2.4)

so that the chemical potentials are realized by substituting
$\mathop{\mathrm{{}\partial}}\mathopen{}_{\mu}$ with $D_{\mu}$ in the action.
As shown in [3], the path integral localizes around a trajectory that only
depends on the sum of the charges $Q=Q_{1}+\dots+Q_{N}$.

We will decompose the fields $\phi_{i}$ into a constant part $A_{i}/\sqrt{2}$
and a fluctuating part $u_{i}$. The same applies to the field $\lambda$ where
we choose the constant part to be $m^{2}-r$ and the fluctuating part
$\hat{\lambda}$. Up to a total derivative the action becomes

$\displaystyle S_{Q}[u_{i},\hat{\lambda}]={}$
$\displaystyle\sum_{i=1}^{N}\biggl{[}-i\theta_{i}Q_{i}+\int\differential{\tau}\differential{\Sigma}\biggl{[}(D^{i}_{\mu}u_{i})^{\dagger}(D_{i}^{\mu}u_{i})+\frac{\theta_{i}^{2}A_{i}^{2}}{2\beta^{2}}$
(2.5)
$\displaystyle+(m^{2}+\hat{\lambda})\left(\frac{A_{i}^{2}}{2}+\frac{A_{i}}{\sqrt{2}}(u_{i}^{\dagger}+u_{i})+u_{i}^{\dagger}u_{i}\right)\biggr{]}-\left(\frac{(m^{2}-r)^{2}}{2u}+\frac{\hat{\lambda^{2}}}{2u}+\frac{(m^{2}-r)\hat{\lambda}}{u}\right)\biggr{]}.$

Since the $u_{i}$ appear at most quadratically we can integrate them out and
get

$\displaystyle
Z=\int\prod_{i=1}^{N}\frac{d\theta_{i}}{2\pi}\mathinner{\mathcal{D}u_{i}}\mathinner{\mathcal{D}\hat{\lambda}}e^{-S_{Q}[u_{i},\hat{\lambda}]}=\int\prod_{i=1}^{N}\frac{d\theta_{i}}{2\pi}\mathcal{D}\hat{\lambda}e^{i\theta_{i}Q_{i}}e^{-S_{\theta}^{\textrm{eff}}[\hat{\lambda}]},$
(2.6)

where

$\displaystyle S_{\theta}^{\textrm{eff}}[\hat{\lambda}]={}$
$\displaystyle\sum_{i=1}^{N}\biggl{[}V_{\Sigma}\beta\left(\frac{\theta_{i}^{2}}{\beta^{2}}+m^{2}-\frac{(m^{2}-r)^{2}}{uA_{i}^{2}}\right)\frac{A_{i}^{2}}{2}+\Tr\log(-D_{\mu}D^{\mu}+m^{2}+\hat{\lambda})$
(2.7)
$\displaystyle-\frac{A_{i}^{2}}{2}\Tr(\hat{\lambda}\Delta^{i}\hat{\lambda})-\frac{1}{u}\int\differential{\tau}\differential{\Sigma}\left(\frac{1}{2}\hat{\lambda}^{2}+(m^{2}-r)\hat{\lambda}\right)\biggr{]},$

and $V_{\Sigma}$ is the volume of $\Sigma$. In the expression above we have
introduced the propagator $\Delta^{i}$ which fulfills

$\displaystyle(-D^{i}_{\mu}D^{i}_{\mu}+m^{2})\Delta^{i}(x)=\frac{1}{\sqrt{\det(g)}}\delta(x).$
(2.8)

We will proceed in the same way as in [8] and compute the saddle-point
equations where we set the fluctuations $\hat{\lambda}$ to zero. Thus we get

$\displaystyle\evaluated{\partial_{m^{2}}S_{Q}}_{\hat{\lambda}=0}$
$\displaystyle=\sum_{i=1}^{N}\left[V_{\Sigma}\beta\frac{A_{i}^{2}}{2}\left(1-\frac{2(m^{2}-r)}{uA_{i}^{2}}\right)+\partial_{m^{2}}\Tr\log(-D_{\mu}D^{\mu}+m^{2})\right]=0,$
(2.9) $\displaystyle\evaluated{\partial_{\theta_{j}}S_{Q}}_{\hat{\lambda}=0}$
$\displaystyle=\theta_{j}\frac{V_{\Sigma}A_{j}^{2}}{\beta}+\partial_{\theta_{j}}\Tr\log(-D_{\mu}D^{\mu}+m^{2})-iQ_{j}=0,$
(2.10) $\displaystyle\evaluated{\partial_{A_{j}}S_{Q}}_{\hat{\lambda}=0}$
$\displaystyle=V_{\Sigma}\beta
A_{j}\left(\frac{\theta_{j}^{2}}{\beta^{2}}+m^{2}\right)=0.$ (2.11)

We want to consider the limit $\beta\to\infty$. The functional integral can be
expressed in terms of zeta functions:

$\displaystyle\partial_{m^{2}}\Tr\log(-D_{\mu}D^{\mu}+m^{2})=\zeta(1|\theta,\Sigma,m),$
(2.12)

where $\zeta(s|\theta,\Sigma,m)$ is

$\displaystyle\zeta(s|\theta,\Sigma,m)=\sum_{n\in\mathbb{Z}}\sum_{p}\left(\left(\frac{2\pi
n}{\beta}+\frac{\theta}{\beta}\right)^{2}+E(p)^{2}+m^{2}\right)^{-s}$ (2.13)

and $E(p)^{2}$ are the eigenvalues of the Laplacian on the manifold $\Sigma$.
In the $\beta\to\infty$ limit,

$\zeta(1|\theta,\Sigma,m)\underset{\beta\rightarrow\infty}{\longrightarrow}\frac{\beta}{2}\zeta(\nicefrac{{1}}{{2}}|\Sigma,m).$
(2.14)

From the second and the third saddle-point equations we get

$\displaystyle\theta_{j}$ $\displaystyle=\theta=im\beta,$ $\displaystyle
A^{2}_{j}=\frac{Q_{j}}{mV_{\Sigma}}.$ (2.15)

In terms of the charge $Q=\sum_{i=1}^{N}Q_{i}$ the first line becomes

$\frac{Q}{N}=m\left(\frac{2V_{\Sigma}}{u}(m^{2}-r)-\zeta(\nicefrac{{1}}{{2}}|\Sigma,m)\right).$
(2.16)

To compute the free energy at the saddle we use

$\Tr\log(-D_{\mu}D^{\mu}+m^{2})=-\evaluated{\derivative{s}\zeta(s|\theta,\Sigma,m)}_{s=0}\underset{\beta\rightarrow\infty}{\longrightarrow}\beta\zeta(-\nicefrac{{1}}{{2}}|\Sigma,m)$
(2.17)

and find

$\displaystyle F(Q)$
$\displaystyle=-\frac{1}{\beta}\sum_{i=1}^{N}\left[i\theta_{i}Q_{i}+\beta\frac{V_{\Sigma}(m^{2}-r)^{2}}{2u}-\beta\zeta(-\nicefrac{{1}}{{2}}|\Sigma,m)\right]$
(2.18)
$\displaystyle=mQ-\frac{NV_{\Sigma}(m^{2}-r)^{2}}{2u}+N\zeta(-\nicefrac{{1}}{{2}}|\Sigma,m),$

where $m$ is to be taken at the saddle point.

The large-N limit is the thermodynamical limit and the saddle-point equations
realize the standard thermodynamical relations. We can therefore introduce the
grand potential density

$\omega(m)=\lim_{\beta\to\infty}-\frac{1}{\beta
V_{\Sigma}}\log(Z_{gc}(m))=2N\left(-\frac{1}{2V_{\Sigma}}\zeta(-\nicefrac{{1}}{{2}}|\Sigma,m)+\frac{(m^{2}-r)^{2}}{4u}\right),$
(2.19)

so that the free energy density $f=F/V_{\Sigma}$ is the Legendre transform of
$\omega$,

$\begin{cases}f=m\rho-\omega(m),\\\ \rho=\derivative{\omega}{m}.\end{cases}$
(2.20)

###  2.2 The effective actions

Having computed the one-loop determinant at fixed charge allows us to write an
effective potential thanks to the extra structure that fixing the charge has
introduced.

We can construct the 1-loop effective action using thermodynamical reasoning
[18]. If a physical system is described by a grand potential
$\omega=\omega(m)$, one can write an effective microscopic description in
terms of a field $\chi$ using the Lagrangian

$\mathcal{L}_{\text{NLSM}}=\omega(\absolutevalue{\mathop{\mathrm{{}\partial}}\mathopen{}_{\mu}\chi\mathop{\mathrm{{}\partial}}\mathopen{}^{\mu}\chi}^{1/2}).$
(2.21)

In our case, we use the grand potential computed in Eq. (2.19) and find a nlsm
only in terms of the Goldstone field $\chi$. By construction, the energy of
the ground state $\chi=-im\tau$ of this eft (eft) reproduces the free energy
given in Eq. (2.18) computed at large N.111The imaginary units in $\chi$ and
later in $\arg{\varphi_{1}}$ are due to the Wick rotation to Euclidean space.
The Lagrangian is to be understood as an expansion around the ground state, so
possible fractional powers pose no issue. The Lagrangian (2.21) has the
drawback that, apart from containing fractional powers of the fields, it
breaks the $O(2N)$ symmetry, since it only describes a subsector of the
theory.

If we want to describe the full large-N behavior of the system we can write an
effective action in terms of N complex fields $\varphi_{i}$. Imposing
$O(2N)$-invariance, it must have the form

$\mathcal{L}_{\text{LSM}}=\sum_{i=1}^{N}\mathop{\mathrm{{}\partial}}\mathopen{}_{\mu}\varphi_{i}^{*}\mathop{\mathrm{{}\partial}}\mathopen{}^{\mu}\varphi_{i}-V(\absolutevalue{\varphi}).$
(2.22)

This Lagrangian must also reproduce the large-charge behavior, and it turns
out that this condition is sufficient to fix the form of
$V(\absolutevalue{\varphi})$. In other words, the (effective) potential $V$ is
determined by requiring that if we fix one of the U(1) charges, we must
recover the initial nlsm when eliminating the radial mode. By construction,
the effective potential preserves the global O(2N) symmetry, _i.e._ it depends
on $\varphi^{\dagger}\varphi=\absolutevalue{\varphi^{2}}$, so we can choose to
fix any of the $U(1)$ charges, say the one that rotates the field
$\varphi_{1}$. We make the fixed-charge ansatz

$\displaystyle\absolutevalue{\varphi_{1}}$ $\displaystyle=\Phi,$
$\displaystyle\arg(\varphi_{1})$ $\displaystyle=-im\tau.$ (2.23)

The Lagrangian evaluated on this ansatz is given by

$\mathcal{L}_{\text{LSM}}=\Phi^{2}m^{2}-V(\Phi)$ (2.24)

and the eom (eom) for the radial mode is

$m^{2}-\derivative{(\Phi^{2})}V=0.$ (2.25)

Its solution is a function $\Phi=\Phi(m^{2})$. Plugging this back into the
Lagrangian (2.24) we must recover

$\evaluated{\Phi^{2}m^{2}-V(\Phi)}_{\Phi=\Phi(m^{2})}=\omega(m).$ (2.26)

The last two equations define the Legendre transform of the effective
potential $V$ seen as a function of $\Phi^{2}$. In the following we will need,
however, to use a more general definition of the Legendre transform to take
into account the fact that some of these functions are only defined for
positive values of the parameters.222The same generalization is used to take
into account possible cusps in the effective potential [19]. For a more
detailed discussion of this definition, see also [14]. If we indicate the
Legendre transform as

$f^{*}(y)=\sup_{x}(xy-f(x)),$ (2.27)

and introduce the notation

$\displaystyle x$ $\displaystyle=\varphi_{i}^{*}\varphi_{i},$
$\displaystyle\Upsilon(x)$ $\displaystyle=V(\sqrt{x}),$ $\displaystyle y$
$\displaystyle=m^{2},$ $\displaystyle\varpi(y)$
$\displaystyle=\omega(\sqrt{y}),$ (2.28)

the two effective actions are related by

$V(\absolutevalue{\varphi})=\Upsilon(\absolutevalue{\varphi}^{2})\to\Upsilon^{*}(m^{2})=\varpi(m^{2})=\omega(m).$
(2.29)

The Legendre transform preserves the property of homogeneity in the variables,
so that for any constant $a$,

$\displaystyle f(x)$ $\displaystyle=ag(x)\implies f^{*}(y)=ag^{*}(y/a),$
(2.30) $\displaystyle f(x)$ $\displaystyle=g(ax)\implies f^{*}(y)=g^{*}(y/a).$
(2.31)

In particular, the free energy density as a function of the charge density has
the same functional form as the free energy as a function of the charge.

##  3 The $\varphi^{4}$ model on flat space

###  3.1 The effective potential on flat space

We have collected all the ingredients to write down the effective potential at
leading order in N. In the case of the torus, $\Sigma=T^{2}$, the grand
potential is given by Eq. (2.19) with the zeta function [8]

$\zeta(-1/2|T^{2},m)=-\frac{V_{\Sigma}}{6\pi}m^{3},$ (3.1)

resulting in

$\omega(m)=(2N)\left[\frac{m^{3}}{12\pi}+\frac{(m^{2}-r)^{2}}{4u}\right].$
(3.2)

In principle, this function contains the full information about the model. To
make the information more transparent, we follow the construction described in
Section 2.2 and define

$\displaystyle\varpi(y)$
$\displaystyle=(2N)\left[\frac{y^{3/2}}{12\pi}+\frac{(y-r)^{2}}{4u}\right],$
(3.3) $\displaystyle\Upsilon(x)$
$\displaystyle=\varpi^{*}(x)=\sup_{y\in\mathbb{R}^{+}}(xy-\varpi(y)),$ (3.4)

so that the effective potential is given by

$V(\absolutevalue{\varphi})=\Upsilon(\absolutevalue{\varphi}^{2}).$ (3.5)

The function $\varpi(y)$ is convex, so its Legendre transform is well defined.
To find the supremum, we derive w.r.t $y$ at fixed $x$,

$x-\varpi^{\prime}(y)=x-N\left[\frac{y^{1/2}}{4\pi}+\frac{y-r}{u}\right]=0.$
(3.6)

The variable $x$ (which we will identify with $\absolutevalue{\varphi}^{2}$)
is defined for $x\in\mathbb{R}^{+}$. There are two possibilities (see Figure
1):

  * •

if $r\geq 0$, the above equation always admits one solution, namely

$y(x)=\pqty{\frac{u}{8\pi}}^{2}\pqty{1+\sqrt{1+\eta}}^{2},$ (3.7)

where

$\eta=64\pi^{2}\pqty{\frac{x}{Nu}+\frac{r}{u^{2}}}.$ (3.8)

Then

$\Upsilon(x)=\evaluated{xy-\varpi(y)}_{y=y(x)}$ (3.9)

and finally

$V(\absolutevalue{\varphi})=\frac{Nu^{3}}{3\cdot
2^{10}\pi^{4}}\left[1+\frac{3}{2}\eta+\frac{3}{8}\eta^{2}-(1+\eta)^{3/2}\right]-\frac{Nr^{2}}{2u}=:\hat{V}(\absolutevalue{\varphi}).$
(3.10)

  * •

If $r<0$, Eq. (3.6) admits a solution only for $x>-\frac{Nr}{u}>0$. For
smaller values of $x$, the supremum is obtained for the value of $y$ which
minimizes $\varpi(y)$, namely $y=0$:

$\Upsilon(x)=\begin{cases}\evaluated{xy-\varpi(y)}_{y=y(x)}&\text{for
$x>-\frac{rN}{u}$}\\\ \evaluated{xy-\varpi(y)}_{y=0}&\text{for
$0<x<-\frac{rN}{u}$.}\end{cases}$ (3.11)

For small values of $x$ the potential is constant, while for big values we
recover the same form as above:

$V(\absolutevalue{\varphi})=\begin{cases}-\frac{Nr^{2}}{2u}&\text{for
$0<\absolutevalue{\varphi}<\sqrt{-\frac{rN}{u}}$}\\\
\hat{V}(\absolutevalue{\varphi})&\text{for
$\absolutevalue{\varphi}>\sqrt{-\frac{rN}{u}}$.}\end{cases}$ (3.12)

[mode=buildnew]OmegaAlone

Figure 1: The function $\varpi(y)=\omega(\sqrt{y})$ for different values of
$r$. Observe that for $r<0$ the derivative $\varpi^{\prime}(y)$ is never
smaller than the value reached of $y=0$ (_viz._ $\varpi^{\prime}(0)=-Nr/u$).

The regions $r<0$ and $r>0$ are precisely those that correspond to the broken
und unbroken phase of the tree-level potential

$V_{0}(\absolutevalue{\phi})=r\absolutevalue{\phi}^{2}+\frac{u}{2N}\absolutevalue{\phi}^{4}.$
(3.13)

These phases are separated by the critical point at $r=0$. We will now study
these three cases separately to see how the quantum effects (which have been
resummed at leading order in $N$) modify this picture.

#### The unbroken phase, $r>0$.

In the unbroken phase, the behavior of $V$ is qualitatively the same as in the
naive tree-level approximation $V_{0}$ (see Figure 2). The potential is
modified to $V(\absolutevalue{\varphi})=\hat{V}(\absolutevalue{\varphi})$ and
is given for small values of the field (up to an irrelevant constant) by

$V(\absolutevalue{\varphi})=c+\left[r+\frac{u^{2}}{32\pi^{2}}\left(1-\sqrt{1+\frac{64\pi^{2}r}{u^{2}}}\right)\right]\absolutevalue{\varphi}^{2}+\frac{u}{2N}\left[1-\frac{1}{\sqrt{1+\frac{64\pi^{2}r}{u^{2}}}}\right]\absolutevalue{\varphi}^{4}+\dots$
(3.14)

We can interpret the two coefficients as the values of the renormalized
couplings,

$\displaystyle r$ $\displaystyle\to
r\pqty{1+\frac{u^{2}}{32\pi^{2}r}\pqty{1-\sqrt{1+\frac{64\pi^{2}r}{u^{2}}}}},$
(3.15) $\displaystyle u$ $\displaystyle\to
u\pqty{1-\frac{1}{\sqrt{1+\frac{64\pi^{2}r}{u^{2}}}}}.$ (3.16)

More in general we can describe the rg (rg) flow of the model in terms of
Callan–Symanzik equations satisfied by the effective potential.

[mode=buildnew]UnbrokenAlone

Figure 2: Tree level (dashed line) and effective potential in the unbroken
phase.

#### The critical point $r=0$.

If we fine tune $r=0$, we are at the critical point (see Figure 3). The
effective potential is given by

$V(\absolutevalue{\varphi})=\frac{Nu^{2}}{3\cdot
2^{10}\pi^{4}}\left[1+\frac{3}{2}\times
64\pi^{2}\frac{\absolutevalue{\varphi}^{2}}{Nu}+\frac{3}{8}\pqty{64\pi^{2}\frac{\absolutevalue{\varphi}^{2}}{Nu}}^{2}-\left(1+64\pi^{2}\frac{\absolutevalue{\varphi}^{2}}{Nu}\right)^{3/2}\right],$
(3.17)

agreeing with the result of [13] based on the resummation of an infinite
number of Feynman diagrams, each corresponding to the leading-$N$ contribution
to the coefficient of $\absolutevalue{\varphi}^{n}$ in the expansion of the
full potential. We can think of this potential as describing the flow that
goes from the Gaussian uv fixed point ($u\to 0$) to the strongly coupled wf
fixed point for $u\to\infty$. For $u\to 0$ we can expand $V$ in inverse powers
of the field,

$V(\absolutevalue{\varphi})=\frac{u}{2N}\absolutevalue{\varphi}^{4}\left[1-\frac{3\pi\sqrt{uN}}{\absolutevalue{\varphi}}+\frac{uN}{16\pi^{2}\absolutevalue{\varphi}^{2}}+\dots\right],$
(3.18)

where each term is again associated to a given Feynman diagram [13].

For $u\to\infty$, we have an expansion in positive powers of
$\absolutevalue{\varphi}$,

$V(\absolutevalue{\varphi})=\frac{16\pi^{2}}{3N^{2}}\absolutevalue{\varphi}^{6}\left[1-\frac{24\pi^{2}\absolutevalue{\varphi}^{2}}{uN}+\frac{4}{3}\pqty{\frac{24\pi^{2}\absolutevalue{\varphi}^{2}}{uN}}^{2}+\dots\right].$
(3.19)

That the leading term of the potential goes like $\absolutevalue{\varphi}^{6}$
for $u\to\infty$ was to be expected since we are describing a cft in terms of
an ir field of dimension $1/2$. Since in a cft, there are no dimensionful
couplings (indeed, $r=0$ and $u\to\infty$), the only way to construct a
dimension-three operator is $\absolutevalue{\varphi}^{6}$. This is precisely
the effective potential obtained in the large-charge approximation [2] and
describes the large-charge sector also for finite values of $N$. We will study
it in more detail in the following using the state-operator correspondence and
computing also the leading non-perturbative corrections in the large-$N$
approximation.

[mode=buildnew]CriticalAlone

Figure 3: Tree level (dashed line) and effective potential in the critical
phase. The potential interpolates between the conformal regime (IR) for large
values of $u$ where $V\underset{u\to\infty}{\sim}\absolutevalue{\varphi}^{6}$,
and the free UV regime for small values of $u$ where $V\underset{u\to
0}{\sim}\absolutevalue{\varphi}^{4}$.

#### The broken phase, $r<0$.

The form of the tree-level potential (3.13) suggests that for $r<0$, we are in
a broken phase realized by the minima of the Mexican hat potential,

$\absolutevalue{\varphi}^{2}=\varphi_{\text{min}}^{2}=\frac{\absolutevalue{r}N}{u}.$
(3.20)

We know however that this picture is changed by quantum corrections: the tree-
level potential has a flex, but it is well-known that the effective potential
is always a convex function, even in finite volume [20, 21]. This is precisely
what we have found in our calculation above based on the Legendre transform
preserving convexity.

Qualitatively, the effective potential $V(\absolutevalue{\varphi})$ is close
to $V_{0}(\absolutevalue{\varphi})$ in the region of large
$\absolutevalue{\varphi}$, but constant in the region between the original
minima, $\absolutevalue{\varphi}^{2}<\frac{\absolutevalue{r}N}{u}$, see Figure
4.

[mode=buildnew]BrokenAlone

Figure 4: Tree level (dashed) and effective potential in the broken phase. The
effective potential is constant in the region between the two minima of the
tree-level potential as demanded by convexity.

At the minimum, $V(\absolutevalue{\varphi})$ has the expansion

$V(\absolutevalue{\varphi})=-\frac{r^{2}N}{2u}+\frac{128\pi^{2}}{3\sqrt{N}}\left(\frac{\absolutevalue{r}}{u}\right)^{3/2}\left(\absolutevalue{\varphi}-\sqrt{\frac{\absolutevalue{r}N}{u}}\right)^{3}+\dots$
(3.21)

and has a cusp in the second derivative.

It is natural to wonder what is the actual vacuum of the theory, given that
the effective potential is constant for
$\absolutevalue{\varphi}^{2}\leq\frac{\absolutevalue{r}N}{u}$ and there seems
to be no energy argument to prefer any choice in this range. Moreover, it is
very unlikely that this degeneracy can be lifted by higher-order corrections
in $N$, as convexity is a nonperturbative property and will in general lead to
the same qualitative behavior.

A strong argument for the actual vacuum of the theory to be at the location of
the minimum of the tree-level potential (at the extremal value
$\absolutevalue{\varphi}=\varphi_{\text{min}}$) can be made using cluster
decomposition [19]. To illustrate this, consider a simplified version of our
model with a single real scalar field, where there is no Goldstone mode and
the system has exactly two vacua $\ket{\pm}$, such that333It can be useful to
think of the two states $\ket{\pm}$ as two thermodynamic phases which coexist
in the intermediate region (Maxwell’s rule).

$\expectationvalue{\varphi}{\pm}=\pm\varphi_{\text{min}}.$ (3.22)

Consider now a linear superposition

$\ket{\theta}=\cos(\theta)\ket{+}+\sin(\theta)\ket{-}.$ (3.23)

The vev of the field on $\ket{\theta}$ covers all the values between $+v$ and
$-v$,

$\expectationvalue{\varphi}{\theta}=\cos(2\theta)\varphi_{\text{min}}.$ (3.24)

Since $\ket{\theta}$ has zero energy this implies that
$V(\absolutevalue{\varphi})$ is constant for
$-\varphi_{\text{min}}\leq\varphi\leq\varphi_{\text{min}}$ [22]. The states
$\ket{\theta}$ are, however, not in general acceptable vacua. To see this we
can compute the connected two-point function,

$\expectationvalue{\varphi(x_{1})\varphi(x_{2})}{\theta}_{c}=\expectationvalue{\varphi(x_{1})\varphi(x_{2})}{\theta}-\expectationvalue{\varphi(x_{1})}{\theta}\expectationvalue{\varphi(x_{2})}{\theta}.$
(3.25)

In the limit $|x_{1}-x_{2}|\to\infty$ we find

$\expectationvalue{\varphi(x_{1})\varphi(x_{2})}{\theta}_{c}=2\varphi_{\text{min}}^{2}\sin(\theta)\cos(\theta)+e^{-|x_{1}-x_{2}|}$
(3.26)

which vanishes, as it should, only for $\cos(\theta)=0$ or $\sin(\theta)=0$,
_i.e._ for one of the two original minima.

###  3.2 The free energy density on flat space

The grand potential $\omega(m)$ was originally introduced in [8] to compute
the free energy density (at fixed $U(1)$ charge) at the critical point. Since
we have now the expression for $\omega(m)$ in the general case, we can repeat
the computation, which amounts to a Legendre transform of $\omega(m)$, this
time with respect to the variable $m$:

$f(\rho)=\omega^{*}(\rho)=\sup_{m\in\mathbb{R}^{+}}(m\rho-\omega(m)).$ (3.27)

Note that there is no convexity constraint on $\omega(m)$ and in fact,

$\displaystyle\omega^{\prime\prime}(m_{\text{fl}})$ $\displaystyle=0,$ for
$m_{\text{fl}}=\frac{u}{12\pi}\pqty{\sqrt{1+\frac{48\pi^{2}r}{u^{2}}}-1}$.
(3.28)

For $r\leq 0$, _i.e._ in the broken phase and at the critical point, $\omega$
is convex, while there is a flex for positive values of $m$ in the massive
$r>0$ case.

The maximization condition $\rho=\omega^{\prime}(m)$ admits a solution for any
real positive value of $\rho$, independently of the sign of $r$. What changes
is that in the massive case, the region $m<m_{\text{fl}}$ does not contribute
to the Legendre transform. In other words, if we look at the inverse Legendre
transform, there are regions of $m$ that cannot be reached starting from a
fixed-charge description. This is not surprising: in general we expect the
fixed-chemical potential and fixed-charge regimes to be different (see _e.g._
the discussion in the appendix of [1]).

Explicitly, we can solve the maximization in terms of trigonometric functions:

$m(\rho)=\frac{u}{12\pi}\pqty{2\sqrt{1+\frac{48\pi^{2}r}{u^{2}}}\cos(\frac{\theta}{3})-1},$
(3.29)

where

$\cos(\theta)=\pqty{1+\frac{48\pi^{2}r}{u^{2}}}^{-3/2}\pqty{\frac{432\pi^{3}\rho}{Nu^{2}}-\pqty{1+\frac{72\pi^{2}r}{u^{2}}}}.$
(3.30)

Note that the function is real and positive for any value of $\rho$ since the
function $\cos(\frac{1}{3}\arccos(\theta))$ can be continued analytically for
$\theta>1$. The resulting free energy density is

$f(\rho)=\frac{Nu^{3}}{2^{9}3^{4}\pi^{4}}\Bigg{[}4\pqty{1+\frac{144\pi^{2}r}{u^{2}}}+3\pqty{1+\frac{48\pi^{2}r}{u^{2}}}^{2}\bqty{-1+4\cos(\frac{2\theta}{3})+2\cos(\frac{4\theta}{3})}\\\
-8\pqty{1+\frac{48\pi^{2}r}{u^{2}}}^{3/2}\cos(\theta)\Bigg{]}.$ (3.31)

To understand the actual physics it is convenient to expand the free energy
density for small values of the charge, using the identities

$\displaystyle f(0)$ $\displaystyle=-\omega(m_{\text{min}}),$ $\displaystyle
f^{\prime}(0)$ $\displaystyle=m_{\text{min}},$ $\displaystyle
f^{\prime\prime}(0)$
$\displaystyle=\frac{1}{\omega^{\prime\prime}(m_{\text{min}})}.$ (3.32)

In this way we can identify typical behaviors that characterize the possible
phases.

#### Broken phase.

In the broken phase ($r<0$), we find

$f_{\text{br}}(\rho)=-\frac{Nr^{2}}{2u}+\frac{u}{4N\absolutevalue{r}}\rho^{2}+\frac{u^{3}}{48N^{2}\pi
r^{3}}\rho^{3}-\frac{u^{3}}{32N^{3}r^{4}}\pqty{1+\frac{u^{2}}{8\pi^{2}r}}\rho^{4}+\order{\rho^{5}}.$
(3.33)

Apart from the unphysical constant, at leading order the energy depends on the
_square of the charge density_ , as in the simple mechanics example of the
rigid rotor.

#### Unbroken phase.

In the unbroken phase ($r>0$) we find

$f_{\text{unb}}(\rho)=\frac{Nu^{3}}{3\times
2^{10}\pi^{4}}\left(1+\frac{96\pi^{2}r}{u^{2}}-\left(1+\frac{64\pi^{2}r}{u^{2}}\right)^{3/2}\right)+\frac{u}{8\pi}\pqty{\sqrt{1+\frac{64\pi^{2}r}{u^{2}}}-1}\rho\\\
+\frac{16\pi^{2}}{Nu}\frac{1}{1+\frac{64\pi^{2}r}{u^{2}}-\sqrt{1+\frac{64\pi^{2}r}{u^{2}}}}\rho^{2}+\order{\rho^{3}},$
(3.34)

which at lowest order is _linear in the charge density_ and consistent with a
particle of mass

$m^{2}=r\pqty{1+\frac{u^{2}}{32\pi^{2}r}\pqty{1-\sqrt{1+\frac{64\pi^{2}r}{u^{2}}}}},$
(3.35)

which is the coupling of $\varphi^{2}$ in the effective potential in Eq.
(3.14).

#### Critical phase.

For $r=0$ we find

$f_{\text{crit}}(\rho)=\frac{2\sqrt{2\pi}\rho^{3/2}}{3\sqrt{N}}-\frac{2\pi^{2}\rho^{2}}{Nu}+\frac{8\sqrt{2}\pi^{7/2}\rho^{5/2}}{N^{3/2}u^{2}}-\frac{256\pi^{5}\rho^{3}}{3\left(N^{2}u^{3}\right)}+\frac{528\sqrt{2}\pi^{13/2}\rho^{7/2}}{N^{5/2}u^{4}}+\order{\frac{\rho^{4}}{u^{5}}}.$
(3.36)

In particular, at the critical point $u\to\infty$ the free energy is
proportional to $\rho^{3/2}$, consistently with the predictions of the large-
charge expansion in [2, 23]. In this case the system has an intermediate
behavior, midway between the rotor ($\propto\rho^{2}$) and the massive
particle ($\propto\rho$). The simplest explanation is that in the critical
limit scale invariance fixes the relationship between the energy density
(dimension three) and the charge density (dimension two).

###  3.3 Nonperturbative finite size corrections on flat space

Up to this point we have neglected non-perturbative corrections. On the torus,
we can however do better as the zeta function is known exactly. The
exponential terms in the zeta function can be understood as exponentially
suppressed finite-size effects contributing to the effective potential.

The expression that we have used for the zeta function on the torus receives
exponentially suppressed corrections in $m$. They have been computed _e.g._ in
[9] and we can use them to write down the exact expression of the grand
potential $\omega(m)$ at leading order in $N$. For a torus of side $L$, we
have

$\displaystyle\frac{\omega(m)}{2N}$
$\displaystyle=-\frac{1}{2L^{2}}\zeta(-\nicefrac{{1}}{{2}}|T^{2},m)+\frac{\pqty{m^{2}-r}^{2}}{4u}$
(3.37)
$\displaystyle=\frac{m^{3}}{12\pi}\pqty{1+\sideset{}{{}^{\prime}}{\sum}_{\mathbf{k}}\frac{e^{-\norm{\mathbf{k}}mL}}{\norm{\mathbf{k}}^{2}m^{2}L^{2}}\pqty{1+\frac{1}{\norm{\mathbf{k}}mL}}}+\frac{\pqty{m^{2}-r}^{2}}{4u},$

where the sum runs over a square lattice of length $1$ minus the origin:
$\mathbf{k}=\set{(k_{1},k_{2})|k_{1}=0,1,\dots;k_{2}=0,1,\dots;(k_{1},k_{2})\neq(0,0)}$.
The exponential terms have a geometric interpretation and can be understood in
terms of massive particles propagating along the cycles of the torus [9].

One can in principle take into account all the exponential corrections, but
for ease of exposition we will keep only the first one:

$\frac{\omega(m)}{2N}=\frac{m^{3}}{12\pi}\pqty{1+2\frac{e^{-mL}}{m^{2}L^{2}}\pqty{1+\frac{1}{mL}}+\order{e^{-\sqrt{2}mL}}}+\frac{\pqty{m^{2}-r}^{2}}{4u}.$
(3.38)

It is convenient to think of this this expression as a trans-series and
compute the Legendre transform of $\varpi(y)$ separately for the exponentially
suppressed term.

The final result is that the leading finite-size correction to the $\order{1}$
potential that we have computed in Section 3.1 is given by

$V(\varphi)=\hat{V}(\varphi)-\frac{N}{3\pi
L^{3}}e^{-\frac{uL}{8\pi}(1+\sqrt{1+\eta})}\pqty{1+\frac{4u\left(3+\eta+3\sqrt{1+\eta}\right)L+u^{2}\left(2+\eta+2\sqrt{1+\eta}\right)L^{2}}{32\pi^{2}\left(2+\sqrt{1+\eta}\right)}},$
(3.39)

where again

$\eta=64\pi^{2}\pqty{\frac{\absolutevalue{\varphi}^{2}}{Nu}+\frac{r}{u^{2}}}.$
(3.40)

This correction preserves the convexity of the potential.

##  4 The effective potential on the sphere

We can use the same approach as before for constructing an effective action in
curved space. We specialize in this section to the sphere as this allows us to
invoke the state-operator correspondence at the conformal point and directly
read off the scaling dimension of the lowest operator of a given charge $Q$
from the free energy $F(Q)$.

In this case the dimensionful parameter is the curvature of the sphere. There
is an important technical difference _w.r.t._ flat space since the large-
charge nlsm is only known perturbatively as an expansion in $R/\rho$. Also,
the effective potential can only be written perturbatively and it is not clear
whether it is possible to resum it in terms of elementary functions. The nlsm
Lagrangian at the critical point $r=1/4$, $u\to\infty$ on a sphere of radius
$r_{0}$ is given by

$\mathcal{L}_{\text{NLSM}}=\frac{1}{3}\absolutevalue{\mathop{\mathrm{{}\partial}}\mathopen{}_{\mu}\chi\mathop{\mathrm{{}\partial}}\mathopen{}^{\mu}\chi}^{3/2}-\frac{1}{6r_{0}^{2}}\absolutevalue{\mathop{\mathrm{{}\partial}}\mathopen{}_{\mu}\chi\mathop{\mathrm{{}\partial}}\mathopen{}^{\mu}\chi}^{1/2}+\frac{1}{60r_{0}^{4}}\absolutevalue{\mathop{\mathrm{{}\partial}}\mathopen{}_{\mu}\chi\mathop{\mathrm{{}\partial}}\mathopen{}^{\mu}\chi}^{-1/2}+\frac{17}{15360r_{0}^{6}}\absolutevalue{\mathop{\mathrm{{}\partial}}\mathopen{}_{\mu}\chi\mathop{\mathrm{{}\partial}}\mathopen{}^{\mu}\chi}^{-3/2}+\dots$
(4.1)

We can think of it as a perturbative expansion in $1/r_{0}^{2}$, or
equivalently in terms of the curvature $R=2/r_{0}^{2}$. Similarly, we can
think of the effective potential as an expansion

$V(\varphi)=(\varphi_{i}^{*}\varphi_{i})^{3}\sum_{l=0}\gamma_{l}y^{l}$ (4.2)

in terms of the dimensionless parameter

$y=\frac{RN^{2}}{96\pi^{2}(\varphi_{i}^{*}\varphi_{i})^{2}}.$ (4.3)

By performing the Legendre transform term by term,

$V(\varphi)=\mathcal{L}_{\text{NLSM}}^{*}(\varphi_{i}^{*}\varphi_{i}),$ (4.4)

we find

$\displaystyle V(\varphi)$
$\displaystyle=\frac{16\pi^{2}(\varphi_{i}^{*}\varphi_{i})^{3}}{3N^{2}}\bqty{1+3y-\frac{3}{20}y^{2}+\frac{53}{2\,560}y^{3}+\dots}$
(4.5)
$\displaystyle=\frac{16\pi^{2}}{3N^{2}}(\varphi_{i}^{*}\varphi_{i})^{3}+\frac{R}{6}\varphi_{i}^{*}\varphi_{i}-\frac{N^{2}R^{2}}{11\,520\pi^{2}}(\varphi_{i}^{*}\varphi_{i})^{-1}+\frac{53N^{4}R^{3}}{424\,673\,280\pi^{4}}(\varphi_{i}^{*}\varphi_{i})^{-3}+\dots$

The leading term is the same as in flat space since it is curvature-
independent. The linear coupling to the curvature may be surprising as it is
not the usual conformal coupling of $R/8$. This is, however, the value found
when expanding around the Gaussian fixed point. Here, we expand around a
strongly coupled fixed point. Since the value of this coupling is not
protected by any symmetry, there is no reason to suppose that the value does
not change when expanding around different points.

The free energy, which at the critical point directly corresponds to the
conformal dimension of the lowest operator of a given charge, was computed
using the large-$N$ techniques in [8].

##  5 Conclusions

In this work we have computed the large-N effective potential of the $O(2N)$
vector model in three dimensions in the presence of quadratic and quartic
interactions. To do so, we have taken advantage of the double-scaling limit
$N\to\infty$, $Q\to\infty$, $Q/N$ fixed, in which we can compute the grand
potential, which is related via a Legendre transform to the effective
potential to leading order in N. This potential is independent of the charge
fixing and describes the full theory and not just a subsector. It is
equivalent to the resummation of an infinite number of Feynman diagrams.
Depending on the sign of the coefficient of the quadratic term in the uv
action, the model can be in different phases –unbroken, critical point, broken
– but the effective potential always remains convex thanks to the properties
of the Legendre transform.
Our results can be naturally extended to a general number of dimensions. This
is done in the companion paper [14], where the convexity $\omega$ serves as a
criterion to identify non-unitary theories.

One limitation of our analysis is its reliance on the double scaling limit
$N\to\infty$, $Q\to\infty$ with $Q/N$ fixed. It would be very interesting to
generalize it to finite values of N, possibly starting from a nlsm whose form
is dictated by the symmetries of the problem and justifying a semiclassical
analysis using the large-charge limit. In this sense, the grand potential that
we have derived here could be the starting point for the exploration of the
phase space of more general systems.

### Acknowledgements

We would like to thank Luis Álvarez-Gaumé, Simeon Hellerman, Igor Pesando and
Uwe–Jens Wiese for enlightening discussions and Rafael Moser for collaboration
on a related project. The work of S.R. is supported by the Swiss National
Science Foundation under grant number 200021 192137. D.O. acknowledges partial
support by the nccr 51nf40–141869 “The Mathematics of Physics” (Swissmap).

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  *[ir]:
  *[wf]:
  *[nlsm]:
  *[lsm]:
  *[uv]:
  *[vev]:
  *[eft]:
  *[eom]:
  *[rg]:
  *[cft]: