# Non-analyticity in Holographic Complexity near Critical points

Uday Sood1, Martin Kruczenski1,2
1 Dep. of Physics and Astronomy, and
2 Purdue Quantum Science and Engineering Institute
Purdue University, W. Lafayette, IN, USA E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The region near a critical point is studied using holographic models of
second-order phase transitions. In a previous paper, we argued that the
quantum circuit complexity of the vacuum ($C_{0}$) is the largest at the
critical point. When deforming away from the critical point by a term $\int
d^{d}x\,\tau\,{\cal O}_{\Delta}$ the complexity $C(\tau)$ has a piece non-
analytic in $\tau$, namely
$C_{0}-C(\tau)\sim|\tau-\tau_{c}|^{\nu(d-1)}+\mathrm{analytic}$. Here, as
usual, $\nu=\frac{1}{d-\Delta}$ and $\xi$ is the correlation length
$\xi\sim|\tau-\tau_{c}|^{-\nu}$ and there are possible logarithmic corrections
to this expression. That was derived using numerical results for the Bose-
Hubbard model and general scaling considerations. In this paper, we show that
the same is valid in the case of holographic complexity providing evidence
that the results are universal, and at the same time providing evidence for
holographic computations of complexity.

###### Contents

  1. 1 Introduction
  2. 2 Bulk Gravity Setup
  3. 3 Non-analyticity near the critical point
  4. 4 $\cal N$ $=1$ Gapped Flow
  5. 5 Holographic Complexity and Ambiguities
    1. 5.1 Choice of cutoff
    2. 5.2 Null-null joint at the cutoff surfaces
    3. 5.3 Null-normal normalization and parametrization
    4. 5.4 Functional redefinitions of the null surface
    5. 5.5 Scalar boundary term
    6. 5.6 Some comments on ambiguity in the field-theoretic complexity
  6. 6 Conclusions
  7. 7 Acknowledgements
  8. A A Power Series solution near the boundary
  9. B Volumes in some toy models of RG Flows
  10. C Non-analytic piece as a bulk term in the action

##  1 Introduction

In a previous work [1], we studied the circuit complexity of the ground state
of systems near quantum critical points using numerical and field-theoretical
methods. The notion of complexity used in that work followed from identifying
optimal circuits with geodesics on the space of circuits similar to [2, 3, 4]
in the context of quantum computing and [5, 6, 7, 8, 9, 10, 11, 12] in the
context of quantum field theory. Related calculations of complexity in many
body systems near quantum phase transitions can be found in the recent works
[13, 14] for the LMG (Lipkin-Meshkov-Glick) model, [15] for the Proca theory
and the closely related work [16] on the N-site Bose-Hubbard model.

In this work, we continue our study of complexity this time through the lens
of known holographic conjectures. The first such conjecture is that the
complexity should be dual to the volume of the extremal codimension-one bulk
hypersurface which meets the asymptotic boundary on the time slice where the
boundary state is defined [17]. The second conjecture states that complexity
should be dual to the gravitational action evaluated on the WDW patch of the
spacetime [18, 19]. The WDW patch is the region of spacetime enclosed by past
and future light sheets sent into the bulk from the time slice on the
boundary. These proposals have led to numerous insights, for ex. see [17, 20,
21, 22, 23]. Using these conjectures, we study the critical behavior of the
complexity in holographic RG flows [24, 25].
A nice feature of the holographic correspondence is that it allows us to study
RG flows of strongly coupled field theories using weakly coupled dual
gravitational theories. The geometries that will be of interest to us are
asymptotically AdS geometries. The interpretation in the field theory is that
of a perturbed CFT which undergoes a renormalization group flow. In the
holographic correspondence, the radial coordinate of the AdS geometry is
interpreted in terms of the energy scale in the field theory, therefore a
dependence of a bulk field on the radial coordinate represents an RG flow. On
the field theory side, perturbations are introduced by adding a source term to
the CFT Lagrangian or by giving a vacuum expectation value to a certain
operator. These two types of perturbations correspond to the non-normalizable
and normalizable modes of the dual bulk fields, respectively [26].
In our previous study, we found that the complexity is the largest at the
critical point $\tau=\tau_{c}$ and, as $\tau\rightarrow\tau_{c}$, has a non-
analytic piece that behaves as $|\tau-\tau_{c}|^{\nu(d-1)}$ for a
d-dimensional spacetime field theory. We also found that this term is
independent of the UV cutoff. We find all these features in both the volume
and the action calculations near holographic critical points. This is the main
result of this paper. In addition, we found that the analytic terms in
$C(\tau)$ were all regularization-dependent and hence ambiguous even after
subtracting $C_{0}$. This continues to hold in the case of holographic
complexity.
The organization of the paper is as follows: we start with an explanation of
the general gravitational setup that we use for RG flow geometries in Section
2. We include expressions for the general form that the volume and action
calculations take for these geometries. In Section 3, we show that all such
expressions contain a term that is in general non-analytic in the deformation
coupling $\tau$ by deriving a scheme to extract such a term which we denote by
$v_{0}$ and $i_{0}$ for the volume and action respectively. Section 4 looks at
an example where the volume and action can be computed analytically. This is
the case of the ${\cal N}=1$ flow geometry. Next, in Section 5, we show that
the complexity defined using the volume or action conjectures has various
ambiguities as it is regulator-dependent. We look at a few such ambiguities in
the volume and action. We find that the non-analytic term discussed previously
is universal and independent of the cutoff. This is similar to the nature of
complexity found in [1] where we found a non-universal complexity but a
universal non-analytic term in the field theory calculation for the complexity
near the $O(2)$ fixed point in a Gaussian approximation. Finally, we summarize
our results with the discussion in Section 6.

##  2 Bulk Gravity Setup

We want to study holographic complexity near a critical point that defines a
UV CFT dual to AdS. Upon deforming it by an IR-relevant operator the theory
may flow in the IR to another CFT or a gapped phase. In the dual gravitational
theory, we consider an asymptotically AdS background of the form

$\displaystyle
ds^{2}=\frac{L^{2}}{z^{2}}(\eta_{\mu\nu}dx^{\mu}dx^{\nu}+\frac{dz^{2}}{f(z)})$
(2.1)

We require that the function $f(z)$ have a scale $\xi$ and be of the form
$f(z/\xi)$ such that $f(z)\simeq 1$ when $z\ll\xi$. Then, $\xi$ defines a
correlation length such that for length scales smaller than $\xi$ the theory
is described by a CFT. For larger length scales we can have a flow to another
CFT or to a gapped phase based on how $f(z)$ behaves in the region $z>\xi$.
The boundary deformation by an IR-relevant operator ${\cal O}$ is modeled by
turning on a bulk scalar field $\Phi$ with a mass related to the conformal
dimension of ${\cal O}$ in the bulk. Consider then the bulk action

$\displaystyle I_{bulk}$ $\displaystyle=\frac{1}{16\pi G_{N}}\int
d^{D}x\sqrt{-g}\left(R-\frac{1}{2}g^{AB}\partial_{A}\Phi\partial_{B}\Phi-V(\Phi)\right)$
(2.2)

where $g_{AB}$ is the spacetime metric and $V$ is the scalar potential which
has a critical point at $\Phi=0$, satisfying $V(0)=-\frac{d(d-1)}{L^{2}}$.
When we have $f(z)\rightarrow\frac{L^{2}}{L^{2}_{0}}>1$ as
$z\rightarrow\infty$, the geometry again becomes AdS with radius $L_{0}$. For
consistency, the potential again has a critical point at the
$z\rightarrow\infty$ value of the scalars $V_{0}=-\frac{d(d-1)}{L_{0}^{2}}$.
Such geometries are known as domain wall geometries in the literature. [27,
28]
On the other hand, flows to a gapped phase involve an $f(z)$ that keeps on
growing in the interior and typically blows up with some positive power of $z$
as $z\rightarrow\infty$ [29]. The spacetime is singular here but typically
becomes non-singular when considering extra dimensions and generically the
space-time caps off at a finite proper distance from the boundary.
In addition to how the metric behaves in the interior, the RG flow geometries
are also distinguished based on whether the leading deformation is source-like
or vev-like. The leading behavior in the scalar dual to the relevant
deformation ($\Delta<d$) of the field theory which we call $\Phi$ determines
whether we have a source-like deformation

$\displaystyle\Phi=\Phi_{(s)}z^{d-\Delta}+....$ (2.3)

or a vev-like deformation

$\displaystyle\Phi=\Phi_{(v)}z^{\Delta}+....$ (2.4)

in the standard quantization framework.
The equations of motion following from the action in Eq. 2.2 are

$\displaystyle R_{AB}$
$\displaystyle=\frac{1}{2}\partial_{A}\Phi\partial_{B}\Phi+\frac{1}{d-1}g_{AB}V(\Phi)$
(2.5) $\displaystyle\frac{1}{\sqrt{-g}}$
$\displaystyle\partial_{A}(\sqrt{-g}g^{AB}\partial_{B}\Phi)-\frac{\delta
V}{\delta\Phi}=0$ (2.6)

Near the boundary, the equation for $R_{zz}+R_{tt}$ gives,

$\displaystyle
zf^{\prime}(z)=\frac{(d-\Delta)^{2}\Phi_{(s)}^{2}}{d-1}z^{2(d-\Delta)}$ (2.7)

Thus the leading correction to $f(z)$ is positive.

$\displaystyle
f(z)=1+\frac{(d-\Delta)\Phi_{s}^{2}}{2(d-1)}z^{2(d-\Delta)}+...$ (2.8)

The critical exponent $\nu$ determines the scaling of $\Phi_{(s)}$ with the
correlation length in the dual field theory 111$\nu$ is defined in the
standard way via $\xi\sim\Phi_{(s)}^{-\nu}$. Since $\int
d^{d}x\Phi_{(s)}O_{\Delta}$ is a term in the action,
$\frac{1}{\nu}=d-\Delta$.. Thus, the above equation is of the form
$f(z)=1+(z/\xi)^{2(d-\Delta)}+...$ 222If $\Delta$ = d/2, then the leading
order term in f(z) is $(z/\xi)^{d}(\log(z/\xi))^{2}$. This also follows from a
near boundary analysis of Einstein equations and the fact that even the
asymptotic leading order behavior of the scalar has a $z^{d/2}\log(z)$ term..
A similar near-boundary analysis can be done in the case of vev deformations
which yields an asymptotic form of $f(z)$ given by

$\displaystyle f(z)=1+\frac{\Delta}{2(d-1)}z^{2\Delta}\,\Phi_{(v)}^{2}+...$
(2.9)

The positivity of the leading correction to $f(z)$ near the boundary follows
more generally from the null energy theorem and Einstein equations which imply
that $f(z)$ is monotonically increasing ${\it i.e.}$ $\partial_{z}f>0$. [28,
30].

The maximal-volume computation is straightforward for the class of metrics in
eq. 2.1. The maximal volume slices are fixed-$t$ slices and the maximal volume
is invariant under boundary time translations. Thus we can fix the boundary
time at which to evaluate the maximal volume to be $t=0$. Then, the maximal
slice satisfies $t=0$ for all $z$. The volume of such a slice is

$\displaystyle
V_{\Sigma}[f]=\sigma_{d-1}L^{d}\int_{z_{0}}^{Z_{0}}dz\frac{1}{z^{d}\sqrt{f(z)}}$
(2.10)

The volume of these slices would be infinite for $z_{0}\rightarrow 0$ because
proper distances near $z=0$ diverge and these slices extend to $z=0$. These UV
divergences are expected in the field theory definitions of complexity as well
[1, 5]. So, we use a regulated version of AdS space with a cutoff surface at
$z=z_{0}$. Also, $\sigma_{d-1}$ is the volume of the spatial field theory
background which is finite after making the spatial coordinates of the
boundary theory periodic. In principle, such periodic identification is
singular at the Poincare horizon $z\rightarrow\infty$ which can be avoided by
using a large-z cutoff $Z_{0}$. However, this is unnecessary here since the
complexity we compute is not singular in the limit $Z_{0}\rightarrow\infty$.
It would be useful to note that for $d\geq 2$ a fixed-$t$ AdS slice has a
volume given by

$\displaystyle
V_{\Sigma}(AdS_{d+1})=\frac{\sigma_{d-1}L^{d}}{(d-1)\epsilon^{d-1}}$ (2.11)

where $z_{0}=\epsilon$.
We also compute the action of the scalar-gravity system on the WDW patch of
spacetimes given by the class of metrics in eq. 2.1. This is yet another
coordinate-invariant object in the bulk. The full action on the WDW requires
an accounting of boundary terms as well along with $I_{bulk}$ [31]. These
terms come from co-dimension 1 boundary segments as well as co-dimension 2
joints formed by the intersection of these segments. Any spacelike/timelike
segments require the addition of the Gibbons-Hawking-York term [32, 33] and
any joints formed by these surfaces require additional terms [34, 35] for the
variational principle to be well defined. Similarly, null segments require a
boundary term [36, 37] and so do the joints formed by the intersection of null
segments with spacelike/timelike boundary segments.

Figure 1: WDW region for AdS spacetime

$\displaystyle I=I_{bulk}(W)$ $\displaystyle+I_{GHY}(\partial
W_{1})+I_{null}(\partial W_{2})+I_{jnts}(J)$ (2.12)

In the above equation, $\partial W_{1}$ refers to any spacelike/timelike
boundary segments of the WDW patch, $\partial W_{2}$ to null boundary
segments, and $J$ to any joints formed between boundary segments. Naively, it
may seem that the definition of the WDW patch should imply that one only has
null boundary segments. However, this may not be the case as some
regularization schemes may introduce non-null surfaces Fig, 1.
The precise formula for $I_{bulk}$ involves evaluating eq. 2.2 for the
specific choice of functions $f$ and $\Phi$ on the regulated WDW patch. The
second term is the usual Gibbons-Hawking-York term for spacelike/timelike
boundaries $\partial W_{1}$. It is

$\displaystyle I_{GHY}(\partial W_{1})=\frac{1}{8\pi G_{N}}\int
d^{d}x\sqrt{|h|}K$ (2.13)

with $h$ being the induced metric on $\partial W_{1}$ and K being the trace of
the extrinsic curvature. 333All normals are taken to point outwards w.r.t. W
The third term is the contribution from null boundaries $\partial W_{2}$. It
is given by

$\displaystyle I_{null}(\partial W_{2})=-\frac{1}{8\pi G_{N}}\int d\lambda
d^{d-1}\theta\sqrt{|\gamma|}\kappa$ (2.14)

This term is required for the variational principle to be well-defined
whenever a spacetime has null boundaries. This piece depends on the
parametrization 444 The variation and the equations of motion do not. through
$\kappa$ which measures the non-affinity in $\lambda$. $\gamma$ is the
$(d-1)-$dimensional metric on the $\theta$ coordinates. A choice of $\lambda$
gives a null normal $k^{A}$ to the surface which satisfies

$\displaystyle k^{A}\gradient_{A}k_{B}=\kappa k_{B}$ (2.15)

A choice of $\lambda$ can set this term to zero and we make this choice in
this section and the next. In Sec 5, we consider a different choice and check
whether that changes this term in the action. Finally, the term evaluated on
joints is

$\displaystyle I_{jnts}(J)=\frac{1}{8\pi G_{N}}\int d^{d-1}x\sqrt{\sigma}a$
(2.16)

where for timelike-null joints
$a=-\mathrm{sign}(k.s)\mathrm{sign}(k.\hat{t})\log|k.s|$ where k is the null
normal and s is the one form spacelike unit normal to the timelike surface.
$\hat{t}$ is a tangent vector in the tangent space of the timelike surface,
orthogonal to the junction and again pointing outwards as shown in fig. 2(a).

(a) Timelike-null joints

(b) Null-null joints

Figure 2: Joints in AdS WdW spacetime

We compute the action for $AdS_{D}$ as a warmup. Any scalars are turned off in
this case and therefore we only have the cosmological constant term from the
scalar. For the $AdS_{D}$ space, we have $R=-\frac{d(d+1)}{L^{2}}$ which gives

$\displaystyle
I_{bulk}(AdS_{D})=-\frac{dL^{d-1}}{\kappa^{2}}\int_{WDW}d^{D}x\frac{1}{z^{d+1}}$
(2.17)

In this case, the WDW patch is enclosed by $t=\pm z$ light sheets. Including
regulators at small $z=\epsilon$ and large $z=Z_{0}$, we have

$\displaystyle I_{bulk}(AdS_{D})=-\frac{d\sigma_{d-1}L^{d-1}}{(d-1)4\pi
G_{N}}\left(\frac{1}{\epsilon^{d-1}}-\frac{1}{Z_{0}^{d-1}}\right)$ (2.18)

We find that $I_{bulk}(AdS_{D})$ is negative and that we can take the
$Z_{0}\rightarrow\infty$ limit. For the GHY term, we find that the surface at
$z=Z_{0}$ again gives a vanishing contribution as $Z_{0}\rightarrow\infty$.
For general $d\geq 2$, one obtains

$\displaystyle I_{GHY}(AdS_{D})=\frac{d\sigma_{d-1}L^{d-1}}{4\pi
G_{N}}\left(\frac{1}{\epsilon^{d-1}}-\frac{1}{Z_{0}^{d-1}}\right)$ (2.19)

As mentioned above, we can affinely parametrize the null generators and make
any contributions from the null boundaries vanish. The WDW patch for vacuum
AdS has four joints all of which are null-timelike joints. The outward null
normals corresponding to affine parametrizations are $dt-dz$ and $-dt-dz$
respectively. The joint contributions give

$\displaystyle I_{jnts}(AdS_{D})=-\frac{L^{d-1}\sigma_{d-1}}{4\pi
G_{N}}\left(\frac{\log{\epsilon/L}}{\epsilon^{d-1}}-\frac{\log{Z_{0}/L}}{Z_{0}^{d-1}}\right)$
(2.20)

We can again drop the $Z_{0}$ terms. Adding up all these contributions, the
full action for $AdS_{d+1}$ is

$\displaystyle I(AdS_{d+1})=\frac{\sigma_{d-1}L^{d-1}}{4\pi
G_{N}\epsilon^{d-1}}\left(\frac{d(d-2)}{d-1}+\log{L/\epsilon}\right)$ (2.21)

We find that the total action is positive here, even though the bulk
contribution was negative. Hence, we find that the boundary terms are
important for the action to be a valid measure of complexity.

##  3 Non-analyticity near the critical point

This section shows that near the critical point, the volume and action have a
piece non-analytic in the deformation coupling $\tau$ and we derive the
general form for this piece.
The volume of interest is $V_{\Sigma}[f]$. To isolate the non-analytic piece,
we use the variables $u=z/\xi$ and also take $Z_{0}\rightarrow\infty$ as we
did in the previous section.

$\displaystyle
V_{\Sigma}[f]=\frac{\sigma_{d-1}L^{d}}{\xi^{d-1}}\int_{z_{0}/\xi}^{\infty}du\frac{1}{u^{d}\sqrt{f(u)}}$
(3.1)

where, as discussed later in sec.5, we introduce a value for $z_{0}$ defined
through

$\epsilon=\xi\int_{0}^{z_{0}/\xi}\\!\\!\\!\frac{du}{\sqrt{f(u)}}$ (3.2)

with $\epsilon$ a fixed UV cut-off. The function $f(u)$ is increasing with $u$
and $f(0)=1$. We start with some general considerations to show that the
complexity from the volume prescription decreases as we move away from the
critical point, i.e. $dV_{\Sigma}[f]/d\tau<0$ for $\tau>0$. Indeed

$\frac{dV_{\Sigma}[f]}{d\xi}=\sigma_{d-1}L^{d}\left\\{-(d-1)\frac{1}{\xi^{d}}\int_{z_{0}/\xi}^{\infty}du\frac{1}{u^{d}\sqrt{f(u)}}+\frac{\epsilon}{z_{0}^{d}\xi}\right\\}$
(3.3)

where we used, from eq. 3.2, that

$\frac{d}{d\xi}\left(\frac{z_{0}}{\xi}\right)=-\frac{\epsilon}{\xi^{2}}\sqrt{f(z_{0}/\xi)}$
(3.4)

Eq. 3.3 can be rewritten as

$\frac{dV_{\Sigma}[f]}{d\xi}=\sigma_{d-1}L^{d}\frac{(d-1)}{\xi^{d}}\int_{z_{0}/\xi}^{\infty}du\,\left(\frac{\epsilon}{z_{0}}-\frac{1}{\sqrt{f(u)}}\right)$
(3.5)

Now, due to the fact that $f(u)$ is increasing with $u$ we obtain from eq.3.2

$\frac{\epsilon}{z_{0}}>\frac{1}{\sqrt{f(z_{0}/\xi)}}>\frac{1}{\sqrt{f(u)}},\
\ \ \mbox{if}\ \ \ u>\frac{z_{0}}{\xi}$ (3.6)

implying that $\frac{dV_{\Sigma}[f]}{d\xi}>0$, and, in view of
$\xi\sim\tau^{-\nu}$ with $\nu>0$:

$\frac{dV_{\Sigma}[f]}{d\tau}<0$ (3.7)

showing that the complexity from the volume prescription indeed has a peak at
the transition $\tau=0$. To show that this peak has a non-analytic part we
start by recalling that near the boundary $z\ll\xi$,

$\displaystyle
f(z)=1+\sum_{m=2}^{\infty}c_{m}\left(\frac{z}{\xi}\right)^{m\alpha}+{\cal
O}\left((z/\xi)^{2\Delta}\right)$ (3.8)

where $\alpha=d-\Delta$ for a source deformation and $c_{2}=1$ [38, 30]. We
consider only source-like deformations here but the calculation in this
section can be generalized to the vev case as well. To see a power series
solution for the scalar-gravity action of this form, see Appendix A. The
second series of terms come from the subleading scalar terms discussed in the
previous section. Next, we introduce a parameter $\delta$ which is small
$\delta<<1$, and use it to break up the integral.

$\displaystyle
V_{\Sigma}[f]=\frac{\sigma_{d-1}L^{d}}{\xi^{d-1}}\left[\int_{z_{0}/\xi}^{\delta}du\frac{1}{u^{d}\sqrt{f(u)}}+\int_{\delta}^{\infty}du\frac{1}{u^{d}\sqrt{f(u)}}\right]$
(3.9)

Since $\delta$ is small, we can expand the function $f$ using the small $u$
expansion in the first integral.

$\displaystyle
f(u)^{-1/2}=1-\frac{1}{2}\sum_{m=2}^{\infty}\tilde{c}_{m}u^{m\alpha}+{\cal
O}\left(u^{2\Delta}\right)$ (3.10)

where $\tilde{c}_{2}=1$ and since we are interested in separating the
divergences, the last ${\cal O}(u^{2\Delta})$ term can be ignored because it
gives a finite contribution to the integral because $2\Delta>d$. After
performing the first integral and replacing $\xi=\tau^{-\nu}$, we obtain

$\displaystyle
V_{\Sigma}[f]=\sigma_{d-1}L^{d}\left[\frac{1}{(d-1)z_{0}^{d-1}}+\frac{1}{2}\sum_{m=2}^{\infty}\tilde{c}_{m}\frac{z_{0}^{m\alpha-d+1}}{m\alpha-d+1}\tau^{m}+\ldots+\tau^{\nu(d-1)}v_{0}\right]$
(3.11)

with $v_{0}$ given by

$\displaystyle v_{0}=\lim_{\delta\rightarrow
0}\left(\int^{\infty}_{\delta}du\frac{1}{u^{d}\sqrt{f(u)}}-\frac{1}{(d-1)\delta^{d-1}}-\frac{1}{2}\sum_{m=2}^{\infty}\tilde{c}_{m}\frac{\delta^{m\alpha-d+1}}{m\alpha-d+1}+\ldots\right)$
(3.12)

This limit is finite since we subtracted all the infinite pieces. The ellipsis
($\ldots$) in eq. 3.11 refer to other analytic terms similar to $\tau^{m}$ and
in eq. 3.12 refer to other divergent pieces in the limit $\delta\rightarrow
0$. The result (3.11) shows that the divergent part is analytic in $\tau$ but
there is a non-analytic contribution proportional to $\tau^{\nu(d-1)}$
independent of the cut-off $z_{0}$. Notice that, if for some integer $m_{0}$
we have $m_{0}\alpha=(d-1)$ then the complexity will have a (non-analytic)
logarithmic term $\ln\xi$. On the other hand, we will have
$\tau^{\nu(d-1)}=t^{2m_{0}}$, and that contribution will be analytic but still
independent of the cut-off. For some toy model calculations of $v_{0}$, see
Appendix B.

A similar analysis gives the non-analytic piece $i_{0}$ for the CA
prescription of the complexity 555The action has a purely gravitational part
considered here and a part coming from the scalar. The sum of both should have
a peak at $\tau=0$ as we show later in particular examples.. This term comes
from the bulk part of the action. Since the gravitational terms are always
present in the bulk action, we restrict to these terms here and show that they
contain a piece non-analytic in $\tau$.

$\displaystyle I_{bulk}$ $\displaystyle=\frac{1}{16\pi G_{N}}\int
d^{D}x\sqrt{-g}\left(R-V(0)\right)$ (3.13)
$\displaystyle=\frac{dL^{d-1}\sigma_{d-1}}{8\pi
G_{N}}\int_{z_{0}}^{\infty}\frac{dz}{z^{d+1}\sqrt{f(z)}}\big{(}d-1+zf^{\prime}(z)-(d+1)f(z)\big{)}\int_{0}^{z}\frac{dy}{\sqrt{f(y)}}$

Changing variables to $u=z/\xi$ and $w=y/\xi$ so that we have

$\displaystyle I_{bulk}=\frac{dL^{d-1}\sigma_{d-1}}{8\pi
G_{N}\xi^{d-1}}\int_{z_{0}/\xi}^{\infty}\frac{du}{u^{d+1}\sqrt{f(u)}}\big{(}d-1+uf^{\prime}(u)-(d+1)f(u)\big{)}\int_{0}^{u}\frac{dw}{\sqrt{f(w)}}$
(3.14)

By again introducing a scale $\delta$ between $z_{0}/\xi$ and $\infty$ with
$\delta<<1$ as before,

$\displaystyle I_{bulk}=-\frac{dL^{d-1}\sigma_{d-1}}{8\pi G_{N}}$
$\displaystyle\left(\frac{2}{(d-1)z_{0}^{d-1}}+\sum_{m=2}^{\infty}f_{m}\tau^{m}z_{0}^{m\alpha-d+1}\right)+L^{d-1}\sigma_{d-1}\tau^{\nu(d-1)}i_{0}$
(3.15)

with $i_{0}$ given by

$\displaystyle i_{0}=\frac{d}{8\pi G_{N}}\lim_{\delta\rightarrow 0}$
$\displaystyle\left(\int_{\delta}^{\infty}\frac{du}{u^{d+1}\sqrt{f(u)}}\big{(}d-1+uf^{\prime}(u)-(d+1)f(u)\big{)}\int_{0}^{u}\frac{dw}{\sqrt{f(w)}}\right.$
(3.16)
$\displaystyle\left.+\frac{2}{(d-1)\delta^{d-1}}+\sum_{m=2}^{\infty}f_{m}\delta^{m\alpha-d+1}\right)$

Here, the coefficients $f_{m}$ can be obtained from the coefficients $c_{m}$
and, the $I_{GHY},I_{null},I_{J}$ parts of the gravitational action do not
give any contributions to $i_{0}$ since they are analytic. We show this in
Appendix C.

##  4 $\cal N$ $=1$ Gapped Flow

The metric is known analytically for the flow of $\mathcal{N}=4$ SYM theory to
a confining theory under a mass-like source deformation with dimension
$\Delta=3$ in the UV [29]. The metric is asymptotically $AdS_{5}$ with $f(z)$
and $\Phi(z)$ given by 666Note that $\xi$ differs here from definition in
previous sections by a constant factor of $\sqrt{2}$ for ease of notation.

$\displaystyle f(z)=(1+z^{2}/\xi^{2})^{2}$ (4.1)
$\displaystyle\Phi(z)=\frac{\sqrt{3}}{2}\log{\frac{\sqrt{\xi^{2}+z^{2}}+z}{\sqrt{\xi^{2}+z^{2}}-z}}$
(4.2)

The potential $V(\Phi)$ in the action for the scalar is

$\displaystyle
V(\Phi)=-\frac{3}{2L^{2}}(3+4\cosh{(2\Phi/\sqrt{3})}+\cosh{(2\Phi/\sqrt{3})}^{2})$
(4.3)

The near boundary asymptotics $z<<\xi$ for the scalar and metric are
consistent with a source deformation with $\nu=1$.

$\displaystyle f(z)=1+\frac{2z^{2}}{\xi^{2}}+...$ (4.4)
$\displaystyle\Phi(z)=\frac{\sqrt{3}z}{\xi}+...$ (4.5)

Note here that since $\nu(d-1)=3$ takes a special value i.e an integer,
therefore $\tau^{\nu(d-1)}=\tau^{3}$ is analytic in $\tau$. However, this term
is still distinguished because it is the only term in the series with an odd
power of $\tau$. In that sense it is natural to use a variable
$\tilde{\tau}=\sqrt{\tau}$ in which case $\tilde{\tau}^{\frac{3}{2}}$ would be
the non-analytic term near $\tau=0$.
Of course, the $\mathcal{N}=1$ geometry just becomes the AdS geometry when
$\xi\rightarrow\infty$ or when the scalar source is turned off. Therefore, we
can expand the difference in maximal volumes or actions in powers $z_{0}/\xi$.
$V_{\Sigma}$ can be computed analytically and is given by

$\displaystyle\frac{V_{\Sigma}(\xi)}{\sigma_{3}L^{4}}$
$\displaystyle=\frac{1}{3z_{0}^{3}}-\frac{1}{\xi^{2}z_{0}}+\frac{\pi}{2\xi^{3}}-\frac{1}{\xi^{3}}\tan^{-1}{\frac{z_{0}}{\xi}}.$
(4.6)

Following up on the discussion above eq.4.4, the function on the right-hand
side is even under $\xi\rightarrow-\xi$ except for the term
$\frac{\pi}{2\xi^{3}}$ that leads to the $\tau^{3}$ term that we interpret as
the universal non-analytic term in this case. For $z_{0}/\xi$ small, replacing
$\xi=\tau^{-1}$ and writing explicitly the terms that do not vanish as
$z_{0}\rightarrow 0$ we obtain

$\displaystyle\frac{V_{\Sigma}(\tau)}{\sigma_{3}L^{4}}$
$\displaystyle=\frac{1}{3z_{0}^{3}}-\frac{\tau^{2}}{z_{0}}+\frac{\pi}{2}\,\tau^{3}+{\cal
O}(z_{0})$ (4.7)

We find regulator-dependent leading terms, a regulator-independent sub-leading
term in $V_{\Sigma}$ that contains the only odd power of $\tau$, and then
terms that go to 0 as $z_{0}\rightarrow 0$ and that only contain even powers
of $\tau$.
For the WDW action in the $\mathcal{N}=1$ flow case, the potential evaluated
on the solution of the scalar is

$\displaystyle V=-\frac{6}{L^{2}}\left(2+3u^{2}+u^{4}\right)$ (4.8)

with $u=z/\xi$. As in the vacuum AdS case, we have to regulate the geometry
with a small z cutoff $z_{0}$ and a large z cutoff at $Z_{0}$ to regulate the
singularity at $z\rightarrow\infty$. To label the WDW region, it is useful to
define light-cone coordinates $v_{1}$ and $v_{2}$ using

$\displaystyle dv_{1}$ $\displaystyle=dt-dz/\sqrt{f}$ (4.9) $\displaystyle
dv_{2}$ $\displaystyle=-(dt+dz/\sqrt{f})$ (4.10)

Integrating, one has $v_{1}=t-\xi\tan^{-1}z/\xi$ and
$v_{2}=-(t+\xi\tan^{-1}z/\xi)$. Then the region enclosed by $v_{1}=0$ and
$v_{2}=0$ in the regulated geometry is the WDW patch. These coordinates are
also useful as 4.9, 4.10 gives null normals $k_{1}=k_{1\mu}dx^{\mu}=dv_{1}$
and $k_{2}=k_{2\mu}dx^{\mu}=dv_{2}$ that satisfy the geodesic equation with
$\kappa=0$ and point outwards. In terms of the familiar Poincare coordinates,
the WDW patch is bounded between the rays $t=\pm\xi\arctan{z/\xi}$ on the
$t-z$ plane. For this spacetime,

$\displaystyle I_{bulk}=\frac{\sigma_{3}L^{3}}{8\pi
G_{N}\xi^{3}}\int_{z_{0}/\xi}^{Z_{0}/\xi}du\frac{\tan^{-1}{u}}{u^{5}}\left(\frac{u^{2}}{2}-8\right)$
(4.11)

We can again take the $Z_{0}\rightarrow\infty$ in this integral without
encountering any difficulties. The integral gives

$\displaystyle I_{bulk}=\frac{\sigma_{3}L^{3}}{8\pi
G_{N}\xi^{3}}\left(-\frac{2}{3u_{0}^{3}}-\frac{1}{2u_{0}^{4}}(1-\frac{u_{0}^{2}}{8}-\frac{9u_{0}^{2}}{2})\tan^{-1}u_{0}+\frac{9}{4u_{0}}-\frac{9\pi}{8}\right)$
(4.12)

with $u_{0}=z_{0}/\xi$. The terms in the bracket can be expanded for small
$u_{0}$ to get the leading behavior in the deformation away from the critical
point,

$\displaystyle I_{bulk}(\tau)=\frac{\sigma_{3}L^{3}}{8\pi
G_{N}}\left(-\frac{8}{3z_{0}^{3}}+\frac{19}{6z_{0}}\,\tau^{2}-\frac{9\pi}{8}\,\tau^{3}+{\cal
O}(z_{0})\right)$

The first term in the expression is the only one that survives when
$\tau\rightarrow 0$ and is the familiar AdS contribution. The sub-leading
universal piece again goes like $\tau^{3}$, the only odd power of $\tau$ in
the expansion. The coefficients of this term are different for the two
different prescriptions which we called $v_{0}$ and $i_{0}$ in Sec 3.
We next compute the GHY term for the timelike surfaces at $z=z_{0}$ and
$z=Z_{0}$. These give

$\displaystyle I_{GHY}=\frac{L^{3}\sigma_{3}}{\pi
G_{N}}\frac{\xi\tan^{-1}{z_{0}/\xi}}{z_{0}^{4}}\left(1+z_{0}^{2}/\xi^{2}\right)$
(4.13)

as the $z=Z_{0}$ gives zero as we take $Z_{0}$ to the Poincare horizon. For
small $u_{0}$

$\displaystyle I_{GHY}(\tau)=\frac{L^{3}\sigma_{3}}{\pi
G_{N}}\left(\frac{1}{z_{0}^{3}}+\frac{2}{3z_{0}}\,\tau^{2}+{\cal
O}(z_{0})\right)$ (4.14)

Here we find as expected that the $GHY$ piece does not contribute to the
universal piece in the WDW action. Next, we compute the contributions from the
joints in the WDW region. As in the AdS case, there are four joints each of
them formed by null-timelike intersections. The two joints connected to the
$z=Z_{0}$ do not contribute while the $z=z_{0}$ do contribute

$\displaystyle I_{joints}=-\frac{\sigma_{3}L^{3}}{4\pi
G_{N}z_{0}^{3}}\log{z_{0}/L}$ (4.15)

This quantity also does not contribute to the universal term in the action.
Adding up all the action contributions, we get

$\displaystyle I(\tau)=\frac{\sigma_{3}L^{3}}{8\pi
G_{N}}\left(\frac{1}{3z_{0}^{3}}(16+6\log{L/z_{0}})+\frac{17}{2z_{0}}\,\tau^{2}-\frac{9\pi}{8}\,\tau^{3}+{\cal
O}(z_{0})\right)$ (4.16)

Here again, the total action is positive as in the AdS case. Thus, the
coefficients $v_{0}$ and $i_{0}$ for this flow are

$\displaystyle v_{0}$ $\displaystyle=\frac{\pi}{2}$ (4.17) $\displaystyle
i_{0}$ $\displaystyle=-\frac{9}{64G_{N}}$ (4.18)

##  5 Holographic Complexity and Ambiguities

In general, the holographic complexity defined using either the CV or the CA
prescriptions is an ambiguous quantity even after choosing one prescription
(see e.g. [31]). However, in this section, by considering the various types of
ambiguities we show that the non-analytic term calculated in the previous
sections is free from such ambiguities and therefore continues to be
meaningful. Still, there is a difference in the coefficient of the universal
term as computed by the two prescriptions, and how that matches different
definitions of complexity in the field theory remains to be seen.
The quantities calculated in the previous section $V_{\Sigma}$ and $I$ have
been interpreted as representing the complexity of the quantum state dual to
the geometry according to

$\displaystyle C_{V}$ $\displaystyle=\frac{V_{\Sigma}}{G_{N}l}$ (5.1)

for CV and

$\displaystyle C_{A}$ $\displaystyle=\frac{I}{\pi\hbar}$ (5.2)

for the CA conjecture 777The reference state wrt which these quantities should
be considered as complexities and the length scale $l$ in the CV prescription
are both ambiguous. We take $l=L$ in this paper..
In this way, the previous sections allow us to compute $C_{V}$ and $C_{A}$ for
the ${\cal N}=1$ flow geometry. We see that both $C_{V}$ and $C_{A}$ for this
geometry start with the AdS contributions and then have corrections coming
from the deformation $\tau$. So we define the subtracted quantities $\delta
V_{\Sigma}$ and $\delta I$ for the $\mathcal{N}=1$ geometry by looking at the
difference

$\displaystyle\delta V_{\Sigma}$
$\displaystyle=V_{\Sigma}(AdS_{5})-V_{\Sigma}(\mathcal{N}=1)$ (5.3)
$\displaystyle\delta I$ $\displaystyle=I(AdS_{5})-I({\cal N}=1)$ (5.4)

These are the analogs of the complexity of formation for black hole spacetimes
where the deformation is in the temperature. In earlier holographic
calculations, this quantity was found to be free of several ambiguities that
may arise in the calculation of the volume or the action [31]. In this
section, we are comparing the action and volumes in two different
asymptotically AdS geometries. We use a systematic way of applying the cutoff
in the Fefferman-Graham coordinates $y=\epsilon$ where the metric is of the
form

$\displaystyle
ds^{2}=\frac{L^{2}}{y^{2}}(dy^{2}+g_{\mu\nu}(x,y)dx^{\mu}dx^{\nu})$ (5.5)

The quantity $z_{0}(\epsilon)$ then has an expansion for general $f(z)$ of the
form 3.8

$\displaystyle
z_{0}(\epsilon)=\epsilon\big{(}1+\sum_{m=2}^{\infty}\frac{d_{m}}{2(m\alpha+1)}\big{(}\frac{\epsilon}{\xi}\big{)}^{m\alpha}+{\cal
O}\left((\epsilon/\xi)^{2\Delta}\right)\big{)}$ (5.6)

with $d_{2}=1$. We find that $z_{0}$ is analytic in $\tau$ up to the ${\cal
O}\left((\epsilon/\xi)^{2\Delta}\right)$ terms in the above expansion. The
small-z cutoff in the case of the ${\cal N}=1$ flow is

$\displaystyle z_{0}(\epsilon)=\frac{\epsilon}{\sqrt{1-\tau^{2}\epsilon^{2}}}$
(5.7)

The pure AdS cutoff and the $\mathcal{N}=1$ flow cutoff differ by

$\displaystyle
z_{0}(\epsilon)-\epsilon=\frac{\tau^{2}\epsilon^{3}}{2}+O(\epsilon^{5})$ (5.8)

Using this result, we write down the expressions for $\delta C_{V}$ and
$\delta C_{A}$

$\displaystyle\delta
C_{V}=\frac{\sigma_{3}L^{3}}{G_{N}}\left(\frac{3\tau^{2}}{2\epsilon}-\frac{\pi\tau^{3}}{2}\right)$
(5.9)

and

$\displaystyle\delta
C_{A}=\frac{\sigma_{3}L^{3}}{16\pi^{2}G_{N}\hbar}\left(\frac{\tau^{2}}{\epsilon}(1+6\log{L/\epsilon})+\frac{9\pi\tau^{3}}{4}\right)$
(5.10)

Here in both the prescriptions, we find that $\delta C>0$ for $\tau>0$ ${\it
i.e.}$ the complexity is the largest at the critical point and decreases as we
move away. We showed that this holds generally in sec. 3 for the volume case,
but here we find that this also true for the action prescription in this
particular example.

###  5.1 Choice of cutoff

We find that the subtracted quantity $\delta C$ defined using either the CV or
the CA proposal is still dependent on the choice of the cutoff $\epsilon$.
This is the first ambiguity that we encounter.

###  5.2 Null-null joint at the cutoff surfaces

Instead of regulating the patch as we did in the previous section, one could
regulate the patch in such a way that the null normals meet at the $z=z_{0}$
surface as in Fig 2(b). Then instead of two null-timelike joints, we have one
null-null joint. This procedure also shifts the null normals infinitesimally
so that they are now,

$\displaystyle t=\xi\tan^{-1}z/\xi-\xi\tan^{-1}z_{0}/\xi$ (5.11)
$\displaystyle t=\xi\tan^{-1}z_{0}/\xi-\xi\tan^{-1}z/\xi$ (5.12)

for the $\mathcal{N}=1$ case and

$\displaystyle t=z-\epsilon$ (5.13) $\displaystyle t=\epsilon-z$ (5.14)

for the AdS case. Due to this infinitesimal shift, $I_{bulk}$ changes. With
this choice, the new $I_{bulk}$ expressions are

$\displaystyle I_{bulk}(AdS_{5})$ $\displaystyle=-\frac{\sigma_{3}L^{3}}{12\pi
G_{N}\epsilon^{3}}$ (5.15) $\displaystyle I_{bulk}(\mathcal{N}=1)$
$\displaystyle=I_{bulk}(AdS_{5})+\frac{\sigma_{3}L^{3}}{8\pi
G_{N}}\left(\frac{13}{4\epsilon}\,\tau^{2}-\frac{9\pi}{8}\,\tau^{3}\right)$
(5.16)

up to terms that vanish when $\epsilon\rightarrow 0$. In this new scheme, the
Gibbons-Hawking-York terms are 0 for both geometries because the timelike
surface at the cutoff is no longer present. The null piece can also be taken
to be zero in both cases by choosing $\kappa=0$. The character of the joint
terms changes as we now have a null-null joint. For this joint,
$a=-\mathrm{sign}(k.\tilde{k})\mathrm{sign}(\hat{k}.\tilde{k})\log{(k.\tilde{k}/2)}$
where $\hat{k}$ is in the tangent space of the boundary region which has the
normal $k$ $(\hat{k}.k=0)$, null and pointing outwards and away from the joint
while $\tilde{k}$ is the normal of the other null region. With this
prescription, the AdS joint term is

$\displaystyle I_{J}(AdS_{5})=-\frac{\sigma_{3}L^{3}}{4\pi
G_{N}\epsilon^{3}}\log{\epsilon/L}$ (5.17)

and the $\mathcal{N}=1$ term is

$\displaystyle I_{J}(\mathcal{N}=1)=-\frac{\sigma_{3}L^{3}}{4\pi
G_{N}z_{0}^{3}}\log{z_{0}/L}$ (5.18)

which is the same as the joint term in the case with joints as in Fig. 2(a).
Thus, the difference $\delta I_{J}$ is the same. However, $\delta I_{bulk}$
and $\delta I_{GHY}$ are different. The quantity $\delta C_{A}$ still
continues to be positive and is given by

$\displaystyle\delta
C_{A}=\frac{3\sigma_{3}L^{3}}{16\pi^{2}G_{N}\hbar}\left(\frac{\tau^{2}}{\epsilon}(2\log{L/\epsilon}-3/2)+\frac{3\pi}{4}\,\tau^{3}+{\cal
O}(\epsilon)\right)$ (5.19)

###  5.3 Null-normal normalization and parametrization

Another ambiguity that can be kept track of is in the normalization of the
null normals. We introduce $c>0$ so that $k_{1}.\hat{t}=c$ instead of 1. In
this case,

$\displaystyle I_{J}(AdS_{5})$ $\displaystyle=-\frac{\sigma_{3}L^{3}}{4\pi
G_{N}\epsilon^{3}}\log{(c\epsilon/L)}$ (5.20) $\displaystyle
I_{J}(\mathcal{N}=1)$ $\displaystyle=-\frac{\sigma_{3}L^{3}}{4\pi
G_{N}z_{0}(\epsilon)^{3}}\log{(cz_{0}(\epsilon)/L)}$ (5.21)
$\displaystyle\delta I_{J}$ $\displaystyle=\frac{\sigma_{3}L^{3}\tau^{2}}{8\pi
G_{N}\epsilon}\left(1+3\log{L/c\epsilon}\right)$ (5.22)

We find that $\delta I$ and thus $\delta C_{A}$ has some $c$ dependence in the
leading term.
Next, we consider a different parametrization $\tilde{\lambda}(\lambda)$ from
the previously chosen affine case $\lambda$ for the null surfaces. This
induces new normals $\tilde{k_{1}}$ and $\tilde{k_{2}}$ and hence we have to
recalculate both the null piece and the joint piece in the action. For
simplicity, we consider whether $\delta I$ changes when $\kappa$ changes from
zero to a constant not equal to zero. A different $\delta C_{A}$ would suggest
that the complexity has another ambiguity that comes from the parametrization
chosen for the boundary regions and therefore the existing definition requires
an additional choice of parametrization to be specified. An affine
parametrization $\lambda$ satisfies

$\displaystyle d\lambda=-\frac{L^{2}}{z^{2}}\frac{dz}{\sqrt{f}}$ (5.23)

This gives the null normals $k_{1}$ and $k_{2}$ discussed earlier that satisfy
$\kappa=0$. We want to make a change
$\lambda\rightarrow\tilde{\lambda}(\lambda)$ such that $\tilde{\kappa}$ is a
constant different from zero. We consider a change of parametrization of the
form

$\displaystyle\frac{d\tilde{\lambda}}{d\lambda}=e^{-\beta}$ (5.24)

Then, the new normals $\tilde{k}^{A}$ satisfy $\tilde{k}^{A}=e^{\beta}k^{A}$.
Then $\tilde{\kappa}=\frac{d}{d\lambda}e^{\beta}$ or equivalently
$\tilde{\kappa}=\frac{d}{d\tilde{\lambda}}\beta$ so that
$e^{\beta}=1+\tilde{\kappa}\lambda$. The quantity in the integrand
$\tilde{\kappa}d\tilde{\lambda}=d\beta=\frac{\tilde{\kappa}d\lambda}{1+\tilde{\kappa}\lambda}$
can be written as an integral over $z$ using eq. 5.23. For the AdS case,

$\displaystyle e^{\beta}=1+\frac{\tilde{\kappa}L^{2}}{z}$ (5.25)

while for the $\mathcal{N}=1$ case,

$\displaystyle
e^{\beta}=1+\frac{\tilde{\kappa}L^{2}}{z}+\frac{\tilde{\kappa}L^{2}}{\xi}\tan^{-1}z/\xi$
(5.26)

The integrals for $I_{null}$ are

$\displaystyle I_{null}(AdS_{5})$
$\displaystyle=\frac{\tilde{\kappa}\sigma_{3}L^{5}}{4\pi
G_{N}}\int_{\epsilon}^{\infty}\frac{dz}{z^{4}(z+\tilde{\kappa}L^{2})}$ (5.27)
$\displaystyle I_{null}(\mathcal{N}=1)$
$\displaystyle=\frac{\tilde{\kappa}\sigma_{3}L^{5}\xi^{2}}{4\pi
G_{N}}\int_{z_{0}}^{\infty}\frac{dz}{z^{5}(z^{2}+\xi^{2})(1+\tilde{\kappa}L^{2}/z+(\tilde{\kappa}L^{2}/\xi)\tan^{-1}z/\xi)}$
(5.28)

Using the notation $\tilde{\kappa}L^{2}=k$, the AdS integral is given by

$\displaystyle I_{null}(AdS_{5})=\frac{\sigma_{3}L^{3}}{4\pi
G_{N}}\left(\frac{1}{3\epsilon^{3}}-\frac{1}{2k\epsilon^{2}}+\frac{1}{k^{2}\epsilon}+\frac{1}{k^{3}}\log{\epsilon/k}\right)+{\cal
O}(\epsilon)$ (5.29)

Keeping terms upto $\tau^{5}$, the ${\cal N}=1$ integral gives

$\displaystyle I_{null}({\cal N}=1)=\frac{\sigma_{3}L^{3}}{4\pi
G_{N}}\left(A_{0}+A_{2}\tau^{2}+A_{4}\tau^{4}+A_{5}\tau^{5}+{\cal
O}(\tau^{6})\right)$ (5.30)

with

$\displaystyle A_{0}$
$\displaystyle=\frac{1}{3\epsilon^{3}}-\frac{1}{2k\epsilon^{2}}+\frac{1}{k^{2}\epsilon}+\frac{1}{k^{3}}\log{\epsilon/k}$
(5.31) $\displaystyle A_{2}$
$\displaystyle=-\frac{5}{2\epsilon}-\frac{1}{2k}-\frac{3}{k}\log{\epsilon/k}$
(5.32) $\displaystyle A_{4}$
$\displaystyle=\frac{k}{6}\left(-11+6\log{tk}\right)$ (5.33) $\displaystyle
A_{5}$ $\displaystyle=\frac{k^{2}}{12}\left(-22+\pi+6\pi\log{2}\right)$ (5.34)

where we again drop terms that go to $0$ as $\epsilon\rightarrow 0$.

The sign in the new $I_{J}$ is unchanged since $e^{\beta}$ is positive
definite, so only the term in the $\log$ is modified. The modification is

$\displaystyle I_{J}(AdS_{5})$ $\displaystyle=-\frac{\sigma_{3}L^{3}}{4\pi
G_{N}\epsilon^{3}}\log{((\epsilon+k)/L)}$ (5.35) $\displaystyle
I_{J}(\mathcal{N}=1)$ $\displaystyle=-\frac{\sigma_{3}L^{3}}{4\pi
G_{N}z_{0}^{3}}\log{((z_{0}+k+\tau z_{0}k\tan^{-1}\tau z_{0})/L)}$ (5.36)

so that

$\displaystyle\delta I_{null}$ $\displaystyle=\frac{\sigma_{3}L^{3}}{4\pi
G_{N}}\left(A_{2}\tau^{2}+A_{4}\tau^{4}+A_{5}\tau^{5}+{\cal
O}(\tau^{6})\right)$ (5.37) $\displaystyle\delta I_{J}$
$\displaystyle=\frac{3\sigma_{3}L^{3}\tau^{2}}{16\pi
G_{N}\epsilon}\left(1+2\log{L/2\epsilon}\right)$ (5.38)

###  5.4 Functional redefinitions of the null surface

Finally, we have functional redefinitions of the null surface. We look at
shifts in the quantity $a$ by constant $a_{0}$ at all the joints.

$\displaystyle I_{J}(AdS_{5})=\frac{\sigma_{3}L^{3}}{4\pi
G_{N}\epsilon^{3}}(a_{0}-\log{\epsilon/L})$ (5.39) $\displaystyle
I_{J}(\mathcal{N}=1)=\frac{\sigma_{3}L^{3}}{4\pi
G_{N}z_{0}^{3}}(a_{0}-\log{z_{0}/L})$ (5.40) $\displaystyle\delta
I_{J}=\frac{\sigma_{3}L^{3}\tau^{2}}{8\pi
G_{N}\epsilon}(1+3(a_{0}-\log{\epsilon/L}))$ (5.41)

###  5.5 Scalar boundary term

Following [39], we consider the case when the scalar action is modified by an
extra term

$\displaystyle I_{s}(\partial W_{1})=\frac{1}{16\pi G_{N}}\int
d^{d}x\sqrt{|h|}\frac{1}{2}\Phi s^{A}\partial_{A}\Phi$ (5.42)

Here, the unit normal $s$ and the region $\partial W_{1}$ are defined in sec.
2. We find that just as in the case of boundary terms for the gravitational
action, this term is analytic in $\tau$ for the $N=1$ case and does not affect
the universal piece of the action complexity calculation. Since it is zero for
the pure AdS case when $\tau=0$, the quantity $\delta I_{s}(\partial W_{1})$
is given by

$\displaystyle\delta I_{s}(\partial W_{1})=\frac{3\sigma_{3}L^{3}}{16\pi
G_{N}}\left(\frac{\tau^{2}}{\epsilon}+{\cal O}(\epsilon)\right)$ (5.43)

Thus we find that the term that goes like $\xi^{-3}$ or $\tau^{3}$ is
independent of $\epsilon$ and therefore universal ${\it i.e.}$ it does not
change under the ambiguities discussed above, while the $\epsilon$-dependent
terms are not universal. Analogous quantities to $C_{V}$ and $C_{A}$ can also
be computed for situations when the temperature T is the only deformation away
from the critical point. $\delta V$ or $\delta I$ is then proportional to the
complexity of formation for the thermal state based on whether we use the CV
or the CA proposal. For the black hole geometry dual to the thermal state, one
finds that in both these cases, the quantity $\delta C(T)=C(T)-C(T=0)$ is
independent of the regularization $\epsilon$. Not only is the complexity of
formation free from the ambiguity of being regulator-dependent, but it is also
free from other ambiguities that we looked at above [31]. Therefore, these
complexities of formation $\delta C$ appear to be meaningful in this case but
we find that this does not carry over to this RG flow example.

###  5.6 Some comments on ambiguity in the field-theoretic complexity

Thus we see that any holographic dual to the complexity should also
accommodate such ambiguities in RG flow computations of complexity in field
theories. Such a possibility may exist as it can be shown that the quantity
$\delta C$ can depend on the cutoff in existing definitions of complexity for
discretized Gaussian field theories. As an example, consider
$\kappa$-complexities for an $h$-dimensional spatial lattice [5]

$C_{\kappa}=\frac{1}{2^{\kappa-1}}\sum_{p}\left|\ln\left(\frac{\sqrt{p^{2}+m^{2}}}{\omega_{0}}\right)\right|^{\kappa}=\frac{1}{2^{\kappa-1}}\frac{V\Omega_{h-1}}{(2\pi)^{h}}\int_{0}^{\Lambda}p^{h-1}dp\left|\ln\left(\frac{\sqrt{p^{2}+m^{2}}}{\omega_{0}}\right)\right|^{\kappa}$
(5.44)

where $V$ is the volume of the system and
$\Omega_{h-1}=\frac{2\pi^{h/2}}{\Gamma(h/2)}$ is the volume of the unit sphere
$S^{h-1}$ with momentum cut-off $\Lambda=\sqrt{\omega_{0}^{2}-m^{2}}$ where
the logarithm vanishes. For $\omega_{0}\gg m$ we can take
$\omega_{0}\sim\frac{1}{a}$ where $a$ is the lattice spacing. In the quantum
field theory, we take $\omega_{0}$ as a UV cut-off. We get

$c_{\kappa}=\frac{C_{\kappa}}{V}=\frac{1}{2^{\kappa-1}}\frac{\Omega_{h-1}}{(2\pi)^{h}}\int_{0}^{\sqrt{\omega_{0}^{2}-m^{2}}}p^{h-1}dp\left|\ln\left(\frac{\sqrt{p^{2}+m^{2}}}{\omega_{0}}\right)\right|^{\kappa}$
(5.45)

A direct computation of $c_{1}$ gives

$\displaystyle
c_{1}=\frac{\Omega_{h-1}}{(2\pi)^{h}m^{2}}\frac{(\omega_{0}^{2}-m^{2})^{1+h/2}}{h(h+2)}{}_{2}F_{1}[1,1+h/2,2+h/2,1-\frac{\omega_{0}^{2}}{m^{2}}]$
(5.46)

For $h=2$:

$\displaystyle c^{h=2}_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi}\left\\{m^{2}\ln\frac{m^{2}}{\omega_{0}^{2}}-m^{2}+\omega_{0}^{2}\right\\}$
(5.47) $\displaystyle c^{h=2}_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi}\left\\{-\frac{m^{2}}{2}\ln^{2}\frac{m^{2}}{\omega_{0}^{2}}+m^{2}\ln\frac{m^{2}}{\omega_{0}^{2}}-m^{2}+\omega_{0}^{2}\right\\}$
(5.48)

This suggests that the quantity $\delta c=c(m=0)-c(m)$ still depends on the UV
cutoff. For example,

$\delta c_{1}^{h=2}=\frac{1}{4\pi}\log{\omega_{0}/m}+\frac{m^{2}}{8\pi}$
(5.49)

Thus here again in the context of a free theory near a Gaussian fixed point
$(m=0)$, we find that $\delta c$ exhibits regulator dependence. How the other
ambiguities should be accommodated or interpreted on the field theory side is
still unclear. However, once the cost function or $\kappa$ and the reference
state are fixed, the term $\tau^{\nu h}$ is again independent of the cutoff
and therefore universal in that sense.

##  6 Conclusions

In a previous study, we showed that there is a non-analytic behavior in field-
theoretic complexity up to logarithmic terms in the vicinity of critical
points. We demonstrated this using explicit lattice calculations and also
using general scaling arguments. This term scales like $\tau^{\nu(d-1)}$ where
$\nu$ is the standard critical exponent for the correlation length $\xi$ as a
function of the reduced coupling $\tau$: $\xi\sim\tau^{-\nu}$. In this work,
we show both, for a generic renormalization group flow geometry and also in a
specific known example, that this is also true when one studies holographic
complexity using the volume and action prescriptions. We also find that even
though holographic complexity, like field-theoretic circuit complexity, has
various ambiguities, the non-analytic term is free from such ambiguities.

##  7 Acknowledgements

We are very grateful to Chen-Lung Hung, Sergei Khlebnikov, Nima Lashkari and
Qi Zhou for comments and discussions. In addition, we are very grateful to the
DOE that supported in part this work through grant DE-SC0007884 and the DOE
QuantISED program of the theory consortium “Intersections of QIS and
Theoretical Particle Physics” at Fermilab, as well as to the Keck Foundation
that also provided partial support for this work.

##  Appendix A A Power Series solution near the boundary

Here we give an example of a power series solution near the boundary similar
to the form used in the eq. 3.8 for a particular choice of the potential. We
get two independent equations involving the functions $\Phi=\Phi(z)$ and
$f(z)$ from eq. 2.5

$\displaystyle\frac{(d-1)f(z)}{zf^{\prime}(z)}=(\Phi^{\prime}(z))^{2}$ (A.1)
$\displaystyle
z^{2}f(z)\left(\Phi^{\prime\prime}(z)+\frac{1}{2}\Phi^{\prime}(z)\frac{f^{\prime}(z)}{f(z)}-\frac{d-1}{z}\Phi^{\prime}(z)\right)-L^{2}\frac{\delta
V(\Phi)}{\delta\Phi}=0$ (A.2)

We look at power series solutions when $z\ll\xi$ of the form

$\displaystyle\Phi(z)$ $\displaystyle=\tau z^{\alpha}+\sum_{k=2}\phi_{k}(\tau
z^{\alpha})^{k}$ (A.3) $\displaystyle f(z)$
$\displaystyle=1+\sum_{k=1}f_{k}(\tau z^{\alpha})^{k}$ (A.4)

with $\tau\sim\xi^{-\alpha}$ and $\alpha=d-\Delta$ for source deformations.
Then, for the choice of the potential

$V(\Phi)=\frac{-d(d-1)}{L^{2}}+\frac{1}{2}m^{2}\Phi^{2}+\frac{1}{6L^{2}}g_{3}\Phi^{3}$
(A.5)

the coefficients $\phi_{k}$ and $f_{k}$ for some small k are

$\displaystyle f_{1}$ $\displaystyle=0$ (A.6) $\displaystyle f_{2}$
$\displaystyle=\frac{\alpha}{2(d-1)}$ $\displaystyle\phi_{2}$
$\displaystyle=\frac{-\kappa}{2\alpha(d-3\alpha)}$ $\displaystyle f_{3}$
$\displaystyle=\frac{-2\kappa}{3\alpha(d-3\alpha)(d-1)}$
$\displaystyle\phi_{3}$
$\displaystyle=\frac{\alpha(2\alpha-d)}{4(d-1)(d-4\alpha)}+\frac{\kappa^{2}}{4\alpha^{2}(d-4\alpha)(d-3\alpha)}$
$\displaystyle\dots$

##  Appendix B Volumes in some toy models of RG Flows

In this appendix, we consider some toy examples of deformations by postulating
functions $f(z)$ in the metric $(2.2)$. Consider the class of metrics

$\displaystyle f_{\alpha}(z)$ $\displaystyle=(1+(z/\xi)^{\gamma})^{\alpha}$
(B.1)

where $\alpha>0$ is a parameter and $\gamma$ is given by the dimension of the
relevant deformation from the critical point for source deformations or by the
dimension of the operator getting a vev in the vev deformation. $\xi$ is
related to the coupling in the case of the source deformation via the exponent
$\nu$ or it is related to the vev of an operator in the vev case. These
choices have the correct asymptotic form and, we explore the co-dimension one
volume in these cases. The ${\cal N}=1$ flow is a specific case in this class
with $\alpha=2$ and $\gamma=2$ in $d=4$.
For the $d=3$ case with $\alpha=2$,

$\displaystyle V(\xi)$
$\displaystyle=\sigma_{2}L^{3}\int_{z_{0}(\epsilon)}^{Z_{0}}\frac{dz}{z^{3}}\frac{1}{1+z^{\gamma}/\xi^{\gamma}}$
$\displaystyle=\frac{\sigma_{2}L^{3}}{(\gamma+2)z_{0}^{2}}\left(\frac{\xi}{z_{0}}\right)^{\gamma}{}_{2}F_{1}[1,1+\frac{2}{\gamma},2+\frac{2}{\gamma},-\frac{\xi^{\gamma}}{z_{0}^{\gamma}}]$
(B.2)

Using a mathematical identity that is true for $\gamma=2$, we find that

$\displaystyle V(\xi)$
$\displaystyle=\sigma_{2}L^{3}\left(\frac{1}{2z_{0}^{2}}-\frac{1}{2\xi^{2}}\log[1+\frac{\xi^{2}}{z_{0}^{2}}]\right)$

$\gamma=2$ holds for $\Delta=2$ source deformations and $\Delta=1$ vev
deformations in $d=3$. In fact when we generalize $\alpha$ to be arbitrary but
positive, for $d\geq 2$ one can analytically compute the quantity $V(\xi)$. It
is given by

$\displaystyle\frac{V(\xi)}{\sigma_{d-1}L^{d}}=\frac{z_{0}^{1-d-\alpha\gamma/2}\xi^{\alpha\gamma/2}}{(d-1+\alpha\gamma/2)}{}_{2}F_{1}[\frac{\alpha}{2},\frac{\alpha}{2}+\frac{d-1}{\gamma},1+\frac{\alpha}{2}+\frac{d-1}{\gamma},-\xi^{\gamma}/z_{0}^{\gamma}]$
(B.3)

Using the identities

$\displaystyle{}_{2}F_{1}[a,b;c;z]$
$\displaystyle=(1-z)^{-a}{}_{2}F_{1}[a,c-b;c;\frac{z}{z-1}]$ (B.4)
$\displaystyle{}_{2}F_{1}[a,b;c;z]$
$\displaystyle=\frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}{}_{2}F_{1}[a,b;a+b+1-c;1-z]$
(B.5)
$\displaystyle+\frac{\Gamma(c)\Gamma(a+b-c)}{\Gamma(a)\Gamma(b)}(1-z)^{c-a-b}{}_{2}F_{1}[c-a,c-b;1+c-a-b;1-z]$
$\displaystyle{}_{2}F_{1}[a,b;a;z]$ $\displaystyle=(1-z)^{-b}$ (B.6)

the above expression can be expanded in a series of the form

$\displaystyle\frac{V(\xi)}{\sigma_{d-1}L^{d}}$
$\displaystyle=\frac{1}{(d-1)z_{0}^{d-1}}-\frac{\alpha}{(d-1-\gamma)z_{0}^{d-1-\gamma}\xi^{\gamma}}+{\cal
O}(1/z_{0}^{d-1-2\gamma}\xi^{2\gamma})$ (B.7)
$\displaystyle+\frac{\Gamma{(\frac{\alpha}{2}+\frac{d-1}{\gamma})}\Gamma{((1-d)/\gamma)}}{\gamma\Gamma{(\alpha/2)}\xi^{d-1}}$

Thus we find a series of terms in powers of $\xi^{-\gamma}$ or equivalently
$\tau^{2}$ and a universal term going like $\xi^{-(d-1)}$ or
$\tau^{\nu(d-1)}$. The first series is analytic in the coupling $\tau$ while
the universal term is non-analytic in $\tau$ in general. This matches our
expectation from the general results of sec. 3. The exact form of $v_{0}$ for
this class of metrics is then

$\displaystyle
v_{0}=\frac{\Gamma{(\frac{\alpha}{2}+\frac{d-1}{\gamma})}\Gamma{((1-d)/\gamma)}}{\gamma\Gamma{(\alpha/2)}}$
(B.8)

For toy models that have AdS domain walls i.e a flow from one AdS space to
another, we look at the class of metrics with

$\displaystyle
f_{\alpha}(z)=1+\frac{\alpha(z/\xi)^{\gamma}}{1+(z/\xi)^{\gamma}}$ (B.9)

Here $\alpha$ is related to the $L_{IR}$ by $\alpha=L^{2}/L_{IR}^{2}-1$ and
$\gamma$ is related to the operator dimension of the UV deformation. This can
be seen by expanding $f$ close to the boundary $z<<\xi$. We analytically
compute the volumes in the special case when $\alpha\rightarrow 0$ and find
that

$\displaystyle\frac{V(\xi)}{\sigma_{d-1}L^{d}}$
$\displaystyle=\frac{1}{(d-1)z_{0}^{d-1}}+\frac{\alpha}{2\xi^{d-1}}\left(\frac{\pi}{\gamma}\csc{\pi(1-d)/\gamma}\right.$
$\displaystyle+\left.\frac{z_{0}^{\gamma+(1-d)}}{(\gamma+1-d)(\xi^{\gamma}+z_{0}^{\gamma})^{1+\frac{1-d}{\gamma}}}{}_{2}F_{1}[1+\frac{1-d}{\gamma},1+\frac{1-d}{\gamma},2+\frac{1-d}{\gamma},\frac{z_{0}^{\gamma}}{\xi^{\gamma}+z_{0}^{\gamma}}]\right)$
(B.10)

which again gives a series of terms analytic in the coupling $\tau$ and a non-
analytic term of the form $\tau^{\nu(d-1)}$.

$\displaystyle\frac{V(\xi)}{\sigma_{d-1}L^{d}}$
$\displaystyle=\frac{1}{(d-1)z_{0}^{d-1}}-\frac{\alpha}{2(d-1-\gamma)}\left(\frac{1}{z_{0}^{d-1-\gamma}\xi^{\gamma}}-\frac{(\gamma+1-d)}{(2\gamma+1-d)}\frac{1}{z_{0}^{d-1-2\gamma}\xi^{2\gamma}}\right.$
(B.11) $\displaystyle+$ $\displaystyle\left.{\cal
O}(\frac{1}{z_{0}^{d-1-3\gamma}\xi^{3\gamma}})+\frac{\pi\alpha}{2\gamma\xi^{d-1}}\csc{\frac{\pi(1-d)}{\gamma}}\right)$

We thus have for this class of toy models,

$\displaystyle v_{0}=\frac{\pi\alpha}{2\gamma}\csc{\frac{\pi(1-d)}{\gamma}}$
(B.12)

##  Appendix C Non-analytic piece as a bulk term in the action

In this appendix, we show that if there is any universal term in the WDW
action, then it necessarily comes from the bulk piece. We do this by computing
the boundary action terms for a general $f(z)$ and showing that they are
analytic in $\tau$. Using the affine parametrization for the null sheets, we
find that $I_{null}=0$. We look at a more general parametrization for the
specific example of the ${\cal N}=1$ flow in Section 5. $I_{GHY}$ and $I_{J}$
are both evaluated on the cutoff surface $z=z_{0}$ and we study these for
small $z_{0}$.
We find that

$\displaystyle I_{GHY}=\frac{d\sigma_{d-1}L^{d-1}\sqrt{f(z_{0})}}{4\pi
G_{N}z_{0}^{d}}\int_{0}^{z_{0}}\frac{dz}{\sqrt{f(z)}}$ (C.1)

We see that using the near-boundary expansion for $f(z)$ in 3.8, the above
expression only contains integer powers of $\tau$ and hence is analytic in the
coupling.
For $I_{J}$, we find that it is independent of $f(z)$ with no $\xi$
dependence.

$I_{J}=\frac{\sigma_{d-1}L^{d-1}}{4\pi G_{N}z_{0}^{d-1}}\log{L/z_{0}}$ (C.2)

In Section 5, we compare different renormalization geometries along the RG
flow and look at differences in the action $\delta I$ for two geometries. The
cutoff $z_{0}$ picks up an $\xi$ dependence because the cutoff is kept the
same in Fefferman-Graham coordinates for the different geometries. In such
cases, we still find that the quantities $\delta I_{GHY}$ and $\delta I_{J}$
are analytic in the reduced coupling $\tau$.

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