# Charge and spin gaps of the ionic Hubbard model with density-dependent
hopping

O. A. Moreno Segura  Centro Atómico Bariloche and Instituto Balseiro, 8400
Bariloche, Argentina  K. Hallberg  Instituto de Nanociencia y Nanotecnología
CNEA-CONICET, Centro Atómico Bariloche and Instituto Balseiro, 8400 Bariloche,
Argentina  A. A. Aligia  Instituto de Nanociencia y Nanotecnología CNEA-
CONICET, GAIDI, Centro Atómico Bariloche and Instituto Balseiro, 8400
Bariloche, Argentina

###### Abstract

We calculate the charge gap $\Delta E_{C}$ and the spin gap $\Delta E_{S}$ of
the ionic Hubbard chain including electron-hole symmetric density-dependent
hopping. The vanishing of $\Delta E_{C}$ ($\Delta E_{S}$) signals a quantum
critical point (QCP) in the charge (spin) sector. Between both critical
points, the system is a fully gapped spontaneously dimerized insulator (SDI).
We focus our study in this region. Including alternation in the hopping, it is
possible to perform an adiabatic Thouless pump of one charge per cycle, but
with a velocity limited by the size of the gaps.

##  I Introduction

The ionic Hubbard model (IHM) consists of the usual Hubbard model with on-site
Coulomb repulsion $U$ supplemented by an alternating one-particle potential
$\Delta$. It has been used to study the neutral-to-ionic transition in organic
charge-transfer salts naga ; hub  and the ferroelectric transition egam . More
recent studies have established that the chain has three different
thermodynamic phases, and different gaps, correlation functions and other
properties have been studied sdi ; phihm ; manih ; rihm ; exac ; sdi3 ; tinca
; abol1 ; abol2 . Here we assume half filling. The unit cell consists of two
sites with on-site energies $\pm\Delta$. Chains with larger unit cells have
also been studied abn ; stenzel .

The model has three phases, the band insulating (BI), the Mott insulating (MI)
and a narrow spontaneously dimerized insulating (SDI) phase in between. An
intuitive understanding of the first two phases is provided by the zero-
hopping limit, in which the occupancies of the different sites are like 2020…
(BI phase) for $\Delta>U/2$ and 1111… (MI phase) for $\Delta<U/2$. For finite
hopping the SDI phase appears in between, as first shown by bosonization sdi ,
and described in more detail later using an approximate mapping to an SU(3)
Heisenberg model exac ; sdi3 .

The phase diagram of the model has been constructed in Ref. phihm,  using the
method of crossings of excited energy levels (MCEL) based on conformal field
theory nomu ; naka ; naka1 ; naka2 ; som . For this model (including also
density-dependent hopping) the method also coincides with that of jumps of
charge and spin Berry phases used in Ref. phtopo, . The basis of the MCEL is
that in one dimension, the dominant correlations at large distances correspond
to the smallest excitation energies. Thus, the crossings of excited levels in
appropriate symmetry sectors therefore correspond to phase transitions.
Lanczos using total wave vector, inversion symmetry note  and time-reversal
symmetry has been used in order to separate the different symmetry sectors,
limiting the maximum size to 16 sites. The results were obtained extrapolating
to the thermodynamic limit phihm . Open-shell boundary conditions (OSBC) were
used, which correspond to periodic BC for a number of sites $L$ multiple of 4,
and antiperiodic BC for even $L$ not multiple of 4.

For fixed $U$, small $\Delta$, and half-filling as assumed here, the system is
in the MI phase with zero spin gap. Increasing $\Delta$, at the point
$\Delta=\Delta_{s}$, a spin gap $\Delta E_{S}$ opens signaling the transition
to the SDI phase. the transition is of the Kosterlitz-Thouless type sdi .
Although the spin gap is exponentially small near the transition, the MCEL
allows to identify it unambiguously and accurately from the crossing of the
even singlet with lowest energy and the odd triplet of lowest energy (both
states have higher energy than the ground state)- At $\Delta=\Delta_{s}$, also
the spin Berry phase $\gamma_{s}$ phihm ; phtopo  jumps from $\pi$ to 0
mod($2\pi$). Further increasing $\Delta$, rather soon, at the point
$\Delta=\Delta_{c}$ a charge transition from the SDI to the BI phase takes
place in which the charge reorders. At this point, there is a crossing of the
two singlets of lowest energy with opposite parity under inversion. In the BI
phase the ground state is the singlet even under inversion, while it is the
odd singlet in the other two phases. All these states have wave vector 0 for
$\Delta\neq 0$. In turn, this crossing leads to a jump in the charge Berry
phase $\gamma_{c}$ from $\pi$ to 0 mod($2\pi$). As explained above, for
$\Delta=\Delta_{c}$ and using OSBC, the charge gap $\Delta E_{C}$ defined as
the absolute value of difference in energy between the ground state and the
first excited state at half filling (in other works called exciton gap manih
or internal gap eric ) vanishes at the charge transition.

Changes in $\gamma_{c}$ are proportional to changes in the polarization.
Actually, calculations of the charge Berry phase form the basis of the modern
theory of polarization pola1 ; pola2 ; pola3 ; bradlyn ; om ; resor ; oc ;
song ; zup . A jump in $\pi$ in $\gamma_{c}$ is consistent with a displacement
of an electronic charge per unit cell in half a unit cell (to the next site)
on average. This is the change of polarization that corresponds to the change
in site occupancies from 1111.. to 2020… The IHM in a ring has inversion
symmetry with center at any site note , and as a consequence $\gamma_{c}$ and
$\gamma_{s}$ can only be 0 or $\pi$ mod($2\pi$). In other words, they are
$Z_{2}$ topological numbers protected by inversion symmetry zup .

Figure 1: Schematic representation of the pump trajectory. Dashed, dotted, and
solid lines indicate the BI, SDI and MI phases of the IHM (at $\delta=0$),
respectively.

If a modulation of the hopping $\delta$ is introduced, the inversion symmetry
is lost, and $\gamma_{c}$ can change continuously. This permits to transfer
one charge to the next unit cell in a Thouless pump cycle in the
$(\Delta,\delta)$ plane (see Fig. 1). This can be understood as follows.
Starting at a point $(\Delta_{1},0)$ with $\Delta_{1}>\Delta_{c}$,
$\gamma_{c}=0$. Then introducing a finite $\delta$, with the appropriate sign,
$\gamma_{c}$ increases continuously with increasing $|\delta|$. Decreasing
$\Delta$ to a value $\Delta_{2}<\Delta_{c}$ and returning $\delta$ to zero,
the point $(\Delta_{2},0)$ is reached where $\gamma_{c}=\pi$. Continuing the
cycle with the opposite sign of $\delta$, $\gamma_{c}$ continues to increase
and reaches the value $\gamma_{c}=2\pi$ at the end of the cycle at
$(\Delta_{1},0)$. This corresponds to the displacement of one unit charge by
one unit cell according to the modern theory of polarization. The values of
the Berry phases in the cycle and time-dependent calculations of the charge
transferred have been presented in Ref. eric, . Moreover, this pumping
procedure has been realized experimentally recently expe , allowing to study
the effects of interactions in the field of quantized topological charge
pumping in driven systems, which is of great interest in the last years citro
; konrad .

A problem with the pumping cycle mentioned above is that it usually crosses
the MI segment between the points $(\Delta_{s},0)$ and $(-\Delta_{s},0)$ at
which the spin gap vanishes. Since unavoidably this segment is traversed at a
finite speed, spin excitations are created, leading to the loss of adiabatic
quantized pumping eric ; expe  (we note that introducing $\delta$ in the MI
phase, a spin gap $\Delta E_{S}$ opens proportional to $|\delta|^{2/3}$ for
small $\delta$ eric ). To avoid this problem, one might choose the crossing
point $\Delta_{2}$ inside the SDI phase, that is
$\Delta_{s}<\Delta_{2}<\Delta_{c}$ (as it is shown in Fig. 1), and then the
system is fully gapped in the whole trajectory. However, at
$\Delta=\Delta_{2}$ both gaps $\Delta E_{C}$ and $\Delta E_{S}$ are small, and
their magnitude is not known. Previous calculations of $\Delta E_{S}$ were
affected by strong finite-size effects in the SDI region and were limited to
very large values of $U$ manih . On the other hand, it has been recently shown
that the SDI phase is enlarged at small values of $U$ if density dependent
hopping is introduced tab . A density-dependent hopping can be experimentally
engineered by near-resonant Floquet modulation ma ; meine ; go1 ; messer ;
goerg .

In this work, we calculate both gaps, $\Delta E_{C}$ and $\Delta E_{S}$ inside
and near the SDI phase, and explore the optimum value of $\Delta_{2}$ for
which the smallest gap is maximum. We use density-matrix renormalization group
(DMRG) S.white ; karen_rev ; Uli ; Uli2 ; banuls , as described in Section
III. We find that to calculate $\Delta E_{S}$ open BC are more convenient,
while to calculate $\Delta E_{C}$, a ring with OSBC leads to the optimum
results, improving previous estimates and allowing to calculate the charge gap
within the SDI phase with unprecedented accuracy.

The paper is organized as follows. In Section II we briefly explain the model.
In Section III, we describe the methods used to calculate the gaps. The
results are presented in Section IV. Section V contains a summary and
discussion.

##  II Model

The model we study here is the IHM with density-dependent hopping (DDH). It is
the version without alternation of the hopping ($\delta=0$) of the interacting
Rice-Mele model rice  including DDH. Because of its relevance for quantized
charge pumping, we describe the full Hamiltonian including also $\delta$ below

$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{j\sigma}\left[-1+\delta\;(-1)^{j}\right]\left(c_{j\sigma}^{\dagger}c_{j+1\sigma}+\text{H.c.}\right)$
(1)
$\displaystyle\times[t_{AA}(1-n_{j\bar{\sigma}})(1-n_{j+1\bar{\sigma}})+t_{BB}n_{j\bar{\sigma}}n_{j+1\bar{\sigma}}$
$\displaystyle+t_{AB}(n_{j\bar{\sigma}}+n_{j+1\bar{\sigma}}-2n_{j\bar{\sigma}}n_{j+1\bar{\sigma}})]$
$\displaystyle+\Delta\sum_{j\sigma}(-1)^{j}n_{j\sigma}+U\sum_{j}n_{j\uparrow}n_{j\downarrow}.$

The first term is the DDH, which is alternating for $\delta\neq 0$. The
amplitudes $t_{AA}$, $t_{AB}$ and $t_{BB}$ correspond to hopping of a particle
with a given spin, when the total occupancy of both sites for particles with
the opposite spin is 0, 1 and 2 respectively. In the following we assume the
electron-hole symmetric case $t_{BB}=t_{AA}$, which is the one implemented
experimentally with cold atoms ma ; meine ; go1 ; messer ; goerg . $\Delta$ is
the alternating on-site energy and $U$ is the on-site Coulomb repulsion.

The model with $\Delta=\delta=0$ has been derived and studied in two
dimensions as an effective model for cuprate superconductors fedro ; brt ;
hir2 ; lili . In one dimension also superconductivity is favored for some
parameters inco ; dobry ; ari ; chen ; jiang . Our interest in DDH here is
that for $t_{AB}$ larger than the other two, the fully gapped SDI phase is
favored tab ; naka2 ; phtopo ; jaka ; bosolili . This is important for fully
adiabatic quantized charge pumping of one charge. So far, charge pumping has
been studied in the interacting Rice-Mele model in absence of DDH
($t_{AA}=t_{AB}=t_{BB}$) eric ; expe ; konrad ; nakag .

##  III Methods

To perform the energy level calculations, we have used the DMRG method with a
code that relies on the ITensors library for Julia itensors . Conveniently
setting $S_{z}$ sectors, we have calculated ground and excited states with a
fixed bond dimension of 900. The truncation error is, in the worst case, on
the order of $10^{-6}$ for periodic BC (PBC), and $10^{-10}$ for open BC
(OBC).

In general, it is convenient to use OBC rather than PBC, because the
entanglement is lower in the former case, leading to more accurate results in
less amount of time. In turn, this fact permits to reach larger systems. This
is particularly important for the spin gap $\Delta E_{S}$, because even in
regions of parameters for which $\Delta E_{S}=0$ in the thermodynamic limit,
it is finite for finite systems, scaling as $1/L$ for increasing system size
$L$ nomu . We have calculated the spin gap by extrapolating the results for
different system sizes using a quadratic function in $1/L$. The calculations
were done for systems between $L=40$ and $L=100$, except in the case of
$U=10$, where we have used sizes up to $L=64$.

Figure 2: (Color online) Ground state and first excited state as a function of
$\Delta$ for 32 sites and PBC.

For the charge gap $\Delta E_{C}$, which is the difference of energies between
the first excited state and the ground state in the singlet sector, the
situation is different. For OBC we find similar difficulties as those found
before manih  for calculating the gap in the SDI phase and particularly near
the transition to the BI phase, where it should vanish in the thermodynamic
limit. The reason is the following. As it is clear using the MCEL method
mentioned in Section I, the ground state and the first excited state have
opposite parity under inversion, being the even state the one of lowest energy
in the BI phase, and both states cross at the BI-SDI transition. For a chain
with OBC and an integer number of unit cells, the inversion symmetry is lost,
the crossing becomes an anticrossing, and extrapolation to the thermodynamic
limit becomes problematic. Therefore, we change the method using a ring with
OSBC as described below.

The Lanczos method used in the MCEL has divided the Hilbert space in different
symmetry sectors, but the method is limited to 16 sites at half filling phihm
; naka2 ; tab . Our method allows us to use larger system sizes, but we do not
have access to the different symmetry sectors. In any case, just plotting the
energy of the ground state and first excited state as a function of $\Delta$
in a ring, both energy levels and the crossing can be clearly identified. This
is illustrated in Fig. 2 for a typical case. $t_{AB}=1$ is chosen as the unit
of energy. We find that extrapolating the energies of $L$ multiple of 4 for a
ring with PBC (which coincide with OSBC for the chosen $L$) between $L=12$ and
$L=32$ using a quadratic function in $1/L$, an accurate and reliable result
for $\Delta E_{C}$ in the thermodynamic limit is obtained. An example of the
extrapolation is presented in Fig. 3

Figure 3: Difference of energy between the even and odd states of lowest
energy as a function of the inverse of the systems size $L$ for all $L$
multiple of 4 in the range $12\leq L\leq 32$ with PBC for several values of
$\Delta$. The transition is calculated to be at $\Delta_{c}=0.978$.

Noting that the slopes of the gap are different at both sides of the
transition, we find that the difference between odd and even states can be
well fitted by the following function with three parameters

$E_{\mathrm{odd}}-E_{\mathrm{even}}=(\Delta-\Delta_{c})\left[A+\text{tanh}\left(\frac{\Delta-\Delta_{c}}{B}\right)\right].$
(2)

Examples will be shown in the next Section.

Comparing with previous results using the MCEL in smaller systems tab , we
have also found that using OSBC, the crossing between the first excited state
in the sector with total spin projection $S_{z}=0$ (corresponding to the even
singlet tab ) and the lowest-energy state in the sector with $S_{z}=1$ (an odd
triplet tab ) corresponds to the crossing at $\Delta=\Delta_{s}$ that signals
the opening of the spin gap $\Delta E_{S}$, and the SDI-MI transition, as
explained in Section I.

Therefore, our methods might be also used to improve the accuracy of phase
diagrams calculated with the MCEL, extending the results to larger systems.

We have found empirically, that sufficiently far from $\Delta_{s}$ and in the
SDI phase, or in the BI phase near the SDI-BI transition at
$\Delta=\Delta_{c}$, the dependence on $\Delta$ of the spin gap is well
described by the expression

$\Delta E_{S}=A_{s}\text{exp}\left[B_{s}\left(\Delta-C_{s}\right)\right].$ (3)

##  IV Results

Figure 4: (Color online) Charge gap (blue circles) and spin gap (red squares)
as a function of $\Delta$ for two values of $U$ and $t_{AA}=t_{BB}=t_{AB}=1$.
Blue solid [red dashed] line is a fit using Eq. (2) [Eq. (3)]. Vertical lines
in the top figure separate the different phases of the IHM.

In Fig. 4, we show the gaps for the model without DDH and two values of $U$.
The maximum difference between any excited state and the ground state in the
SDI phase is denoted by $\Delta E_{M}$. This value is obtained at the crossing
between both studied gaps. The value of $\Delta$ at this crossing is denoted
as $\Delta_{M}$. We take $t_{AB}=1$ as the unit of energy.

From the figure, one can see that the spread of the SDI phase
$\Delta_{c}-\Delta_{s}$ and also $\Delta E_{M}$ are larger for larger values
of $U$ than for moderate ones. The former fact is in agreement with
calculations of the phase diagram using up to 16 sites phihm , although
$\Delta_{c}-\Delta_{s}$ is a little bit smaller in our case. Our values should
be more accurate since we have calculated $\Delta_{c}$ and $\Delta_{s}$ using
up to 32 and 28 sites, respectively.

Figure 5: (Color online) Same as Fig. 4 for $t_{AA}=t_{BB}=0.5$ and
$t_{AB}=1$.

In Fig. 5 we analyze the effect of DDH, decreasing $t_{AA}=t_{BB}$ to half the
value of $t_{AB}=1$, for two extreme values of $U$, leaving the intermediate
values of $U$ for Fig. 6. For $U=10$, the maximum value of the gap $\Delta
E_{M}$ increases slightly. This effect is rather surprising, because one
naively expects that reducing the average value of the hopping, both $\Delta
E_{M}$ and the amplitude of the SDI phase should decrease. Therefore, the
effect of introducing DDH overcomes the effect of reducing the average hopping
regarding $\Delta E_{M}$. Instead, the amplitude of the SDI phase
$\Delta_{c}-\Delta_{s}$ decreases slightly.

As discussed earlier tab , for small values of $U$, the amplitude of the SDI
phase increases strongly, since it continues to exist even for $\Delta=0$.
However, the magnitude of the maximum gap $\Delta E_{M}$ is reduced by 25%
when $U$ is reduced from 10 to 1 in units of $t_{AB}$.

In order to look for the largest possible value of $\Delta E_{M}$ in presence
of DDH, we have calculated the gaps for intermediate values of $U$. The result
is shown in Fig. 6. While qualitatively, the results for $U=3$, 4 and 5 are
similar, the maximum gap $\Delta E_{M}=0.0188$ is obtained for $U=4$.

Figure 6: (Color online) Same as Fig. 5 for intermediate values of $U$.

##  V Summary and discussion

We have calculated the charge and spin gaps of the spontaneously dimerized
insulating (SDI) phase of the ionic Hubbard model, including electron-hole
symmetric density-dependent hopping. We have developed a new method using DMRG
to calculate the charge gap, which presents advantages with respect to
previously used ones, leading to substantially more accurate values. In
addition, phase diagrams constructed by the method of crossing of energy
levels might be calculated more accurately than using only Lanczos methods, if
they are combined with DMRG (the former can be used to identify the symmetry
sectors).

The results might be useful to present experiments with cold atoms in which
quantized Thouless pumping of one charge is observed, when a pump cycle in the
two-dimensional space $(\Delta,\delta)$ enclosing the point $(\Delta_{c},0)$
is performed in a realization of the interacting Rice-Mele model [Eq. (1)],
where $\Delta_{c}$ is the value of $\Delta$ at the transition between the SDI
and the band insulating (BI) phase. A fully adiabatic pump is possible if the
Mott insulating (MI) phase is avoided. This phase lies at the segment between
the points $(\Delta_{s},0)$ and $(-\Delta_{s},0)$, where $\pm\Delta_{s}$ are
the points of the MI-SDI phase transitions. For this purpose, the SDI phase
should be traversed.

Fixing $t_{AB}=1$ we find that the maximum gap inside the SDI phase is about
0.019. This is a rather small value, which by simple estimates seems to
require a velocity about 10 times smaller than that used in available
experiments expe  to guarantee adiabatic pumping in crossing the point
$(\Delta_{M},0)$. However introducing $\delta$ the gap increases quickly (as
$|\delta|^{2/3}$ in the MI phase). A time-dependent calculation, possibly
decreasing the velocity near $(\Delta_{M},0)$ would be useful to check this
procedure.

The effect of density-dependent hopping, reducing $t_{AA}=t_{BB}$ and keeping
$t_{AB}=1$ is moderate in increasing the gap, although it is important if the
average hopping is kept at the same value. Its main effect is that for small
$U$, the extension of the fully gapped SDI phase is strongly increased.

## Acknowledgments

AAA (KH) acknowledges financial support provided by PICT 2017-2726 and PICT
2018-01546 (PICT 2018-01546) of the ANPCyT, Argentina. KH acknowledges support
from ICTP through the Associates Programs.

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# Asymptotically flat galactic rotation curves in gravity theory

Sandipan Sengupta<EMAIL_ADDRESS>Department of Physics, Indian
Institute of Technology Kharagpur, Kharagpur-721302, INDIA

###### Abstract

We present a new set of four-geometries exhibiting asymptotically flat
galactic rotation curves. These are found as explicit solutions to 5D vacuum
Hilbert-Palatini theory, where the fifth dimension has vanishing proper
length. In the emergent 4D dynamics, governed by the condition that the Ricci
scalar must vanish (upto a cosmological constant), these correspond to
anisotropic effective pressure. The enhancement in the deflection angle of a
light ray penetrating the halo is obtained, which could provide a realistic
testing ground for the model as a purely geometric alternative to ‘dark
matter’. For very large halo radii, the leading nonbaryonic contribution to
the bending angle is predicted to be $\frac{3\pi v^{2}}{2c^{2}}$ ($v$ being
the asymptotic rotational velocity), a constant that is different from the
result for an isothermal CDM halo.

##  I Introduction

Perhaps, a dynamical explanation of asymptotically flat rotation curves in
spiral galaxies solely through spacetime geometry is far from being out of
question. Despite the astonishing success enjoyed by Einstein’s gravity theory
within solar system scales, its apparent failure in this context stems from
the fact that the predicted Newtonian fall-off for the circular velocity
$v(r)$ of a test particle at a radial distance $r$ far away from the luminous
mass distribution happens to be contrary to what is in fact observed [1,
*rubin1, 3].

While one could either continue exploring new forms of (‘dark’) matter in the
energy-momentum tensor of Einstein gravity or invoke arbitrary (geometric)
modications of gravity Lagrangian itself, we shall adopt a rather conservative
outlook here. We shall be concerned with the general question as to whether it
is possible to construct a purely geometric theory of gravity, where it must:
a) obey 4D general covariance, b) follow from a gravity action linear in
curvature tensor so that the field equations contain no higher than second
order derivatives, c) involve no matter-coupling, d) admit galactic spacetimes
with asymptotically flat rotation curves as explicit solutions and e) can
reproduce Einstein gravity in vacuum or with some restriction on the energy-
momentum tensor in a suitable limit.

The quest for a modified theory of gravity towards an explanation of the
asymptotic behaviour of rotation curves has a long history though. For
instance, MOND [4, *milgrom1, 6] and conformal gravity [7, *mannheim1]
represent two of the most extensively studied approaches in four dimensions.
However, finding a generally covariant action principle with metric as the
only gravitational degree of freedom is not possible in the first case [9,
10]. In the latter, the issues of defining the test mass through a conformally
invariant geodesic equation and of obtaining frame-independent results turn
out to be quite subtle [11, 12]. Other approaches, such as Braneworld gravity
[13, 14] and $f(R)$ or more generalized extensions of the action principle
[15, *mendoza] etc. have also been explored in this context. In any case, none
of these proposals do follow from the principles laid out in the previous
paragraph.

Here, in this context, we focus on the recent formulation of five dimensional
gravity in vacuum based on an extra dimension of vanishing proper length [17,
18]. The four dimensional effective theory is characterized by the vanishing
of the Ricci scalar with torsion (upto an effective cosmological constant).
Restricting to the case of vanishing four-torsion, the most general static
spherically symmetric solutions are obtained. Remarkably, we find that the
class where the metric exhibits a small deviation from the Newtonian form
implies asymptotically flat rotation curves. Interpreted in terms of an
effective energy-momentum tensor in the 4D emergent Einstein equation, these
solutions are associated with anisotropic effective pressure. The other
(Newtonian) class reproduces a generalized version of Einstein-Maxwell-(A)dS
geometry, with a ‘charge’ that is purely geometric though.

Next, we study the deflection of light passing through the galactic spacetime
solution so obtained. Both cases, involving a finite and infinite halo radius
respectively, are presented along with testable results. We conclude with a
summary and a few relevant remarks.

##  II Field equations and galactic spacetime solutions

The fundamental theory is defined by the following action principle:

$\displaystyle{\cal
L}(\hat{e},\hat{w})=\frac{1}{L^{3}}\epsilon^{\mu\nu\alpha\beta\rho}\epsilon_{IJKLM}\hat{e}_{\mu}^{I}\hat{e}_{\nu}^{J}\hat{e}_{\alpha}^{K}\hat{R}_{\beta\rho}^{~{}~{}LM}(\hat{w}),$

where $L$ is the (fundamental) five-dimensional Planck-length and
$\hat{R}_{\beta\rho}^{~{}~{}LM}(\hat{w})=\partial_{[\beta}\hat{w}_{\rho]}^{~{}LM}+\hat{w}_{[\beta}^{~{}LK}\hat{w}_{\rho]K}^{~{}~{}~{}M}$
is the field-strength. The internal metric is defined as
$\eta_{IJ}=[-1,1,1,1,\sigma]$, where $\sigma=\pm 1$ can take either of these
values. The resulting field equations are given by:

$\displaystyle\epsilon^{\mu\nu\alpha\beta\rho}\epsilon_{IJKLM}\hat{e}_{\mu}^{I}\hat{e}_{\nu}^{J}\hat{D}_{\alpha}(\hat{w})\hat{e}_{\beta}^{K}=0,$
(1a)
$\displaystyle\epsilon^{\mu\nu\alpha\beta\rho}\epsilon_{IJKLM}\hat{e}_{\mu}^{I}\hat{e}_{\nu}^{J}\hat{R}_{\alpha\beta}^{~{}KL}(\hat{w})=0,$
(1b)

where $\hat{D}_{\mu}$ is the gauge-covariant derivative with respect to the
five dimensional connection $\hat{w}_{\mu}^{~{}IJ}$. For a vielbein of the
form:

$\displaystyle\hat{e}_{\mu}^{I}=\left[\begin{array}[]{cc}\hat{e}_{a}^{i}\equiv
e_{a}^{i}&\hat{e}_{v}^{i}=0\\\ \hat{e}_{a}^{5}=0&\hat{e}_{5}^{5}=0,\\\
\end{array}\right]$

the full set of general solutions, already presented in [17], could be
summarized as follows. The connection equations are solved as:

$\displaystyle\hat{w}_{v}^{~{}IJ}$ $\displaystyle=$ $\displaystyle
0,~{}\hat{w}_{a}^{~{}4i}=M^{ij}e_{aj}~{}[M^{kl}=M^{lk}],$ (3)
$\displaystyle\hat{w}_{a}^{~{}ij}$ $\displaystyle=$
$\displaystyle\bar{w}_{a}^{~{}ij}(e)+K_{a}^{~{}ij}$ (4)

where $M^{kl}$ are arbitrary spacetime fields and $K_{a}^{~{}ij}$ denote the
four dimensional contortion satisfying $e^{a}_{j}K_{a}^{~{}ij}=0$. The
vielbein equations of motion imply:

$\displaystyle\hat{D}_{v}M^{kl}$ $\displaystyle=$ $\displaystyle
0,~{}\left[\delta^{a}_{b}\delta_{kl}-e^{a}_{k}e_{bl}\right]D_{a}M^{kl}=0,$
$\displaystyle e^{a}_{i}e^{b}_{j}R_{ab}^{~{}~{}ij}(\hat{w})$ $\displaystyle=$
$\displaystyle\bar{R}(\bar{w})+e^{a}_{i}e^{b}_{j}[D_{[a}(\bar{w})K_{b]}^{~{}ij}+K_{[a}^{~{}ik}K_{b]k}^{~{}~{}~{}j}-\sigma
M^{i}_{~{}k}M^{j}_{~{}l}e_{[a}^{k}e_{b]}^{l}]=0$ (5)

where $\bar{R}(\bar{w})$ denotes the torsionless Ricci scalar depending only
upon the tetrads $e_{a}^{i}$ through $\bar{w}(e)$.

Here, we shall consider the simplest possible class of solutions to eq.(II)
with trivial four-torsion:

$\displaystyle K_{a}^{~{}ij}=0,~{}M^{ij}=\lambda\eta^{ij}$ (6)

where $\eta_{ij}=diag[-1,1,1,1]$ is the internal four-metric. The constant
$\lambda$ above defines the effective cosmological constant
$\chi=-12\sigma\lambda^{2}$ in the 4D emergent field equation, obtained from
the last one among the set (II) as:

$\displaystyle\bar{R}(\bar{w})+\chi=0$ (7)

The general solution to the above is:
$\bar{R}_{ab}(\bar{w})=\bar{t}_{ab}-\frac{1}{4}\chi g_{ab}$, where
$\bar{t}_{ab}$ is an arbitrary symmetric traceless tensor subject to
$\nabla_{a}\bar{t}^{ab}=0$ owing to the Bianchi identities.

Next, we shall obtain the spherically symmetric static solutions to the
equations of motion. Assuming the four-geometry to be of the standard form
($G=1=c$):

$\displaystyle ds^{2}=-f(r)dt^{2}+g(r)dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$

the equation of motion (7) translates to:

$\displaystyle\frac{f^{\prime\prime}}{f}-\frac{f^{\prime}}{2f}\left(\frac{f^{\prime}}{f}+\frac{g^{\prime}}{g}\right)+\frac{2}{r}\left(\frac{f^{\prime}}{f}-\frac{g^{\prime}}{g}\right)-\frac{2}{r^{2}}(g-1)-\chi
g=0.$ (8)

In the following, we shall consider two classes of vacuum solutions of this
full nonlinear equation above.

A) Newtonian solutions: $f(r)g(r)=1$

In this case, the solution to eq.(8) is given by:

$\displaystyle
f(r)=1+\frac{B_{1}}{r}+\frac{B_{2}}{r^{2}}+\frac{\chi}{12}r^{2},$ (9)

where $B_{1,2}$ are arbitrary integration constants. This is a generalization
of the Einstein-Maxwell-(A)dS solution of Einstein’s theory, in the sense that
the charge $B_{2}$ could have either sign [19]. Note that the associated
(effective) pressure originates purely due to the spacetime geometry in our
case.

B) Non-Newtonian solutions: $f(r)g(r)=1+\delta(r)$ where $|\delta|<<1$

To be specific, we parametrize the behaviour above as:

$\displaystyle f(r)g(r)=\left(\frac{r}{R}\right)^{2\alpha},$ (10)

where $R$ is some length scale and $\alpha<<1$ is a small parameter to be
interpreted later. The explicit solution in this case reads:

$\displaystyle f(r)$ $\displaystyle=$
$\displaystyle\left(\frac{r}{R}\right)^{2\alpha}\left[\frac{1}{1+\alpha+\alpha^{2}}+\frac{\chi
r^{2}}{2(6+4\alpha+\alpha^{2})}\right]+C_{1}\left(\frac{r}{R}\right)^{\gamma_{+}}+C_{2}\left(\frac{r}{R}\right)^{\gamma_{-}};$
$\displaystyle g^{-1}(r)$ $\displaystyle=$
$\displaystyle\frac{1}{1+\alpha+\alpha^{2}}+\frac{\chi
r^{2}}{2(6+4\alpha+\alpha^{2})}+C_{1}\left(\frac{r}{R}\right)^{\gamma_{+}-2\alpha}+C_{2}\left(\frac{r}{R}\right)^{\gamma_{-}-2\alpha},$
(11)

where
$\gamma_{\pm}=\frac{-(3-\alpha)\pm(1+10\alpha+\alpha^{2})^{\frac{1}{2}}}{2}$
and $C_{1,2}$ are constants.

### Asymptotically flat rotation curves

To understand the implications of the vacuum solutions (II), which have no
analogue in Einstein gravity with (at least) standard matter couplings, let us
explore if these could be used to model the galactic halo which is assumed to
be spherically symmetric. In particular, the intriguing question is, if these
exact solutions could reproduce the non-Newtonian behaviour of rotation curves
for spiral galaxies.

The circular velocity $v^{2}(r)=\frac{rf^{\prime}}{2f}$ [20] of a massive test
particle moving in this geometry (at the equatorial plane
$\theta=\frac{\pi}{2}$), under the practical assumption that the effect of the
cosmological constant is negligible within the halo ($\chi\approx 0$), is
found to be:

$\displaystyle
v^{2}(r)=\frac{\frac{\alpha}{1+\alpha+\alpha^{2}}+\frac{1}{2}C_{1}\gamma_{+}\left(\frac{r}{R}\right)^{\gamma_{+}-2\alpha}+\frac{1}{2}C_{2}\gamma_{-}\left(\frac{r}{R}\right)^{\gamma_{-}-2\alpha}}{\frac{1}{1+\alpha+\alpha^{2}}+C_{1}\left(\frac{r}{R}\right)^{\gamma_{+}-2\alpha}+C_{2}\left(\frac{r}{R}\right)^{\gamma_{-}-2\alpha}}$

Noting that
$\gamma_{+}=-1+3\alpha+o(\alpha^{2}),~{}\gamma_{-}=-2-2\alpha+o(\alpha^{2})$,
the expression above implies: $v^{2}(r)\rightarrow\alpha$ as
$r\rightarrow\infty$. This corresponds precisely to an asymptotically flat
rotation curve, where the small parameter $\alpha$ in the halo metric should
be interpreted as the limiting rotation velocity of the test particle at the
flat region.

### Effective density, pressure and the mass function

It is worthwhile studying what the solution (II) implies from the perspective
of an emergent 4D Einstein gravity. Defining an energy-momentum tensor in an
effective sense as $\bar{R}_{ab}-\frac{1}{2}g_{ab}\bar{R}=T^{(eff)}_{ab}$, we
find the effective density and pressures as:

$\displaystyle 8\pi\rho^{(eff)}$ $\displaystyle=$
$\displaystyle\frac{\alpha(1+\alpha)}{1+\alpha+\alpha^{2}}\frac{1}{r^{2}}-(1+\gamma_{+}-2\alpha)\frac{C_{1}}{R^{2}}\left(\frac{r}{R}\right)^{-2+\gamma_{+}-2\alpha}$
$\displaystyle-$
$\displaystyle(1+\gamma_{-}-2\alpha)\frac{C_{2}}{R^{2}}\left(\frac{r}{R}\right)^{-2+\gamma_{-}-2\alpha},$
$\displaystyle 8\pi P^{(eff)}_{r}$ $\displaystyle=$
$\displaystyle\frac{\alpha(1-\alpha)}{1+\alpha+\alpha^{2}}\frac{1}{r^{2}}+(1+\gamma_{+})\frac{C_{1}}{R^{2}}\left(\frac{r}{R}\right)^{-2+\gamma_{+}-2\alpha}$
$\displaystyle+$
$\displaystyle(1+\gamma_{-})\frac{C_{2}}{R^{2}}\left(\frac{r}{R}\right)^{-2+\gamma_{-}-2\alpha},$
$\displaystyle 8\pi P_{\theta}^{(eff)}$ $\displaystyle=$
$\displaystyle\frac{\alpha^{2}}{1+\alpha+\alpha^{2}}\frac{1}{r^{2}}-(1+\gamma_{+}-\alpha)\frac{C_{1}}{R^{2}}\left(\frac{r}{R}\right)^{-2+\gamma_{+}-2\alpha}$
(12) $\displaystyle-$
$\displaystyle(1+\gamma_{-}-\alpha)\frac{C_{2}}{R^{2}}\left(\frac{r}{R}\right)^{-2+\gamma_{-}-2\alpha}~{}=~{}8\pi
P^{(eff)}_{\phi}$

Thus, the halo metric corresponds to anisotropic (effective) pressure. In
fact, it could be shown in this context that a solution with asymptotically
flat rotation curve cannot exhibit an isotropic (effective) pressure, since
the (Schwarzschild) metric (9) with $B_{2}=0$ turns out to be the unique
solution of eq.(8) under the demand $P_{r}=P_{\theta}=P_{\phi}$.

If we define a spatially averaged pressure as
$\bar{P}=\frac{1}{3}(P_{r}+P_{\theta}+P_{\phi})$, then we find:
$\bar{P}=\frac{1}{3}\rho$. This equation of state correctly captures the
traceless nature of $T^{(eff)}_{ab}$. The effective mass function is obtained
below by integrating the density over 3-space:

$\displaystyle m(r)$ $\displaystyle=$
$\displaystyle\frac{\alpha(1+\alpha)}{2(1+\alpha+\alpha^{2})}r-\frac{C_{1}R}{2}\left(\frac{r}{R}\right)^{1+\gamma_{+}-2\alpha}-\frac{C_{2}R}{2}\left(\frac{r}{R}\right)^{1+\gamma_{-}-2\alpha}~{}+~{}m_{0},$
(13)

where the constant $m_{0}$ depends on the lowest value of $r$ till which the
solution (II) is acceptable. Since the metric must be joined to some interior
solution at this small radius, the apparent divergence of $m(r)$ at $r=0$ is
not relevant. Note that the second and third terms go as $r^{\alpha}$ and
$r^{-1-4\alpha}$ respectively (upto $o(\alpha^{2})$), leaving the linear term
as the leading one at sufficiently large distances. This precisely reflects
the behaviour expected for flat rotation curves.

##  III Observational predictions: Deflection of light

In reality, the galactic halo does not extend to infinity. It would be more
practical to join the halo metric to an asymptotically flat geometry, such
that both these metrics are vacuum solutions to the fundamental equations of
motion (7). We choose the vacuum metric beyond the halo to be Schwarzschild,
obtained by setting $B_{1}=-2M,~{}B_{2}=0=\chi$ in (9) where $M$ is the total
mass enclosed within the halo radius $R_{H}$.

Since the term involving $C_{2}$ in the halo metric (II) has the fastest fall-
off, it is reasonable to assume $C_{2}=0$ (along with $\chi=0$) in what
follows next. Demanding continuity at the boundary of the halo fixes the two
constants $C_{1}$ and $R$ as:

$\displaystyle
C_{1}=-\frac{2M}{R_{H}}+\frac{\alpha(1+\alpha)}{1+\alpha+\alpha^{2}},~{}R=R_{H}$
(14)

From the null geodesic equation obeyed by a light ray at the equatorial plane
($\theta=\frac{\pi}{2}$) of the halo, the angular distance covered in going
from the radial distance ($r_{0}$) at closest approach from the centre of the
halo to infinity is given by [20]:

$\displaystyle\Delta\phi$ $\displaystyle=$
$\displaystyle\int_{r_{0}}^{\infty}dr~{}\frac{d\phi}{dr}=\int_{r_{0}}^{\infty}\frac{dr}{r}~{}\left[\frac{E^{2}}{L^{2}}\frac{r^{2}}{fg}-\frac{1}{g}\right]^{-\frac{1}{2}}$
(15) $\displaystyle=$ $\displaystyle\Delta\phi_{1}+\Delta\phi_{2}$

where $E,L$ are the conserved quantities associated with the $t$ and $\phi$
motions, respectively, satisfying
$\frac{E^{2}}{L^{2}}=\frac{f(r_{0})}{r_{0}^{2}}=\frac{f(r_{*})}{r_{*}^{2}}$
($r_{*}$ denotes the distance at closest approach in the Schwarzschild
geometry). The last line in (15) reflects the fact that the integral is to be
evaluated in two parts, for the two regions $r_{0}\leq r\leq R_{H}$ and
$R_{H}\leq r<\infty$, respectively. Under the weak-field approximation, we
have the condition:
$r_{0}\approx(1-\alpha+\frac{2M}{r_{*}})^{\frac{1}{2}}r_{*}$, which is
equivalent to $r_{0}\approx r_{*}$ for $r_{0}>>M$ ($2M$ being the
Schwarzschild radius of the mass within the halo) at the leading order. The
total angle of deflection of the ray is then given by:
$\delta=|2\Delta\phi-\pi|$.

Next, using the identity:

$\displaystyle\frac{E^{2}}{L^{2}}\frac{r^{2}}{fg}-\frac{1}{g}=\frac{1}{1+\alpha+\alpha^{2}}\left[\left(\frac{r}{r_{0}}\right)^{2-2\alpha}-1\right]\times$
$\displaystyle\left[1+(1+\alpha+\alpha^{2})C_{1}\left(\frac{r}{R}\right)^{-1+\alpha}\frac{\left(\frac{r}{r_{0}}\right)^{2-2\alpha}+\left(\frac{r}{r_{0}}\right)^{1-\alpha}+1}{\left(\frac{r}{r_{0}}\right)^{1-\alpha}+1}\right],$
(16)

the result for $\Delta\phi$ for the passage of the light ray within the halo
becomes:

$\displaystyle\Delta\phi_{1}\approx\int_{r_{0}}^{R_{H}}\frac{dr}{r}~{}(1+\alpha)^{\frac{1}{2}}\left[\left(\frac{r}{r_{0}}\right)^{2-2\alpha}-1\right]^{-\frac{1}{2}}\times$
$\displaystyle\left[1-\frac{(1+\alpha+\alpha^{2})C_{1}}{2}\left(\frac{r}{R}\right)^{-1+\alpha}\frac{\left(\frac{r}{r_{0}}\right)^{2-2\alpha}+\left(\frac{r}{r_{0}}\right)^{1-\alpha}+1}{\left(\frac{r}{r_{0}}\right)^{1-\alpha}+1}\right]$
$\displaystyle=~{}\frac{(1+\alpha+\alpha^{2})^{\frac{1}{2}}}{1-\alpha}\tan^{-1}\left[\left(\frac{R_{H}}{r_{0}}\right)^{2-2\alpha}-1\right]^{\frac{1}{2}}$
$\displaystyle-~{}\frac{(1+\alpha+\alpha^{2})^{\frac{3}{2}}C_{1}}{2(1-\alpha)}\left[\left(\frac{R_{H}}{r_{0}}\right)^{2-2\alpha}-1\right]^{\frac{1}{2}}\frac{2\left(\frac{R_{H}}{r_{0}}\right)^{1-\alpha}+1}{\left(\frac{R_{H}}{r_{0}}\right)^{1-\alpha}+1},$
(17)

In writing the first line above, we have used the weak-field approximation for
the term involving $C_{1}$, since such a limit is applicable to galactic
spacetimes. The remaining contribution to the deflection from the halo
boundary to infinity, upon using the expansion for a weak Schwarzschild
potential ($\frac{2M}{r}<<1$), is evaluated as:

$\displaystyle\Delta\phi_{2}$ $\displaystyle\approx$
$\displaystyle\int_{R_{H}}^{\infty}\frac{dr}{r}\left(\frac{r^{2}}{r_{*}^{2}}-1\right)^{-\frac{1}{2}}\left[1+\frac{M}{r}+\frac{Mr}{r_{*}(r+r_{*})}\right]$
(18) $\displaystyle=$
$\displaystyle\frac{\pi}{2}+\frac{2M}{r_{*}}-\tan^{-1}\left[\left(\frac{R_{H}}{r_{*}}\right)^{2}-1\right]^{\frac{1}{2}}-\frac{M}{R_{H}}\left[\left(\frac{R_{H}}{r_{*}}\right)^{2}-1\right]^{\frac{1}{2}}\left(\frac{\frac{2R_{H}}{r_{*}}+1}{\frac{R_{H}}{r_{*}}+1}\right),$

As a result, the total deflection angle upon using the expression for $C_{1}$
reads:

$\displaystyle\delta$ $\displaystyle=$
$\displaystyle\frac{4M}{r_{*}}+\frac{2(1+\alpha+\alpha^{2})^{\frac{1}{2}}}{1-\alpha}\tan^{-1}\left[\left(\frac{R_{H}}{r_{0}}\right)^{2-2\alpha}-1\right]^{\frac{1}{2}}$
(19) $\displaystyle+$
$\displaystyle\frac{(1+\alpha+\alpha^{2})^{\frac{3}{2}}}{(1-\alpha)}\left(\frac{2M}{R_{H}}-\frac{\alpha(1+\alpha)}{1+\alpha+\alpha^{2}}\right)\times\left[\left(\frac{R_{H}}{r_{0}}\right)^{2-2\alpha}-1\right]^{\frac{1}{2}}\frac{2\left(\frac{R_{H}}{r_{0}}\right)^{1-\alpha}+1}{\left(\frac{R_{H}}{r_{0}}\right)^{1-\alpha}+1}$
$\displaystyle-$ $\displaystyle
2\tan^{-1}\left[\left(\frac{R_{H}}{r_{*}}\right)^{2}-1\right]^{\frac{1}{2}}-\frac{2M}{R_{H}}\left[\left(\frac{R_{H}}{r_{*}}\right)^{2}-1\right]^{\frac{1}{2}}\left(\frac{\frac{2R_{H}}{r_{*}}+1}{\frac{R_{H}}{r_{*}}+1}\right)$

Note that for $r_{0}=r_{*}=R_{H}$, which implies that the light ray passes
only through the Schwarzschild geometry without entering the halo, we recover
the standard result: $\delta=\frac{4M}{r_{*}}$. Evidently, the presence of the
halo leads to an enhancement in the deflection compared to the Schwarzschild
bending. This effect is characterized completely by the ratio
$\frac{R_{H}}{r_{0}}\approx\frac{R_{H}}{r_{*}}$, given the asymptotic rotation
velocity. For $C_{1}=0$, reflecting a scenario where the light ray enters only
the outer region of the halo where the rotation velocity becomes flat, the
total deflection reads:

$\displaystyle\delta$ $\displaystyle=$
$\displaystyle\frac{4M}{r_{*}}+\frac{2(1+\alpha+\alpha^{2})^{\frac{1}{2}}}{1-\alpha}\tan^{-1}\left[\left(\frac{R_{H}}{r_{0}}\right)^{2-2\alpha}-1\right]^{\frac{1}{2}}~{}-~{}2\tan^{-1}\left[\left(\frac{R_{H}}{r_{*}}\right)^{2}-1\right]^{\frac{1}{2}}$
(20) $\displaystyle-$
$\displaystyle\frac{2M}{R_{H}}\left[\left(\frac{R_{H}}{r_{*}}\right)^{2}-1\right]^{\frac{1}{2}}\left(\frac{\frac{2R_{H}}{r_{*}}+1}{\frac{R_{H}}{r_{*}}+1}\right)$

We observe that upon using $r_{0}=r_{*}$ this matches the $o(\alpha)$ result
presented in ref.[20], which considers only the asymptotic part of halo
geometry.

### Infinitely extended halo

Next, we consider the limit of very large galactic halo radii preserving the
earlier assumptions. Using the fact that $\alpha<<1$ (the rotational
velocities being nonrelativistic), the term involving $C_{1}$ in the halo
metric (II) may be identified as the baryonic mass term with a small radial
variation:

$\displaystyle m(r)\approx-\frac{C_{1}R}{2}\left(\frac{r}{R}\right)^{\alpha}$
(21)

Next, using the general expression (15) for $\Delta\phi$ along with the
identity (III) and performing the integration from $r_{0}$ to infinity for the
halo metric, we obtain the deflection angle in this case as:

$\displaystyle\delta~{}\approx~{}\left[\frac{(1+\alpha)^{\frac{1}{2}}}{1-\alpha}-1\right]\pi+\frac{4(1+\alpha)^{\frac{3}{2}}m(r_{0})}{(1-\alpha)r_{0}}~{}\approx~{}\frac{4m(r_{0})}{r_{0}}+\frac{3}{2}\pi\alpha+\frac{10\alpha
m(r_{0})}{r_{0}},$ (22)

ignoring $o(\alpha^{2})$ corrections. Thus, for a very large halo radius, the
model predicts a constant leading correction $\frac{3}{2}\pi\alpha$ to the
Einsteinian bending encoded by the first term. Notably, this value is lesser
than the nonbaryonic contribution ($2\pi\alpha$) predicted for a singular
isothermal halo within cold ‘dark matter’ model [21].

To emphasize, the results in (19) and (22) make the formulation here amenable
to lensing observations, and also to comparisons with standard ‘dark matter’
scenarios as well as with other extra dimensional formulations such as the
Braneworld model [22, *pal].

##  IV Conclusions

We have shown that asymptotically flat rotation curves emerge as exact
dynamical solutions of a purely geometric theory, which reflects a number of
attractive features compared to some other gravitational alternatives to ‘dark
matter’. This theory is based on a recent formulation of five dimensional
gravity, where the fifth dimension has vanishing proper length. This property
is encoded through a degenerate vielbein (with one zero eigenvalue).

Among the static spherically symmetric solutions, the ones corresponding to
asymptotically flat rotation curves are associated with anisotropic effective
pressure. Though in a different context, the possibility of ‘dark matter’
models with nontrivial stresses have been considered in the literature [24].
We note that the spatially averaged equation of state is
$\frac{\bar{P}}{\rho}=\frac{1}{3}$. The Newtonian class, on the other hand,
corresponds to the generalized Einstein-Maxwell-de Sitter geometries as the
unique solution, providing a connect with Einstein gravity in presence of a
traceless energy-momentum tensor with cosmological constant.

We observe that the associated effective density within the halo, apart from
the $r^{-}{2}$ behaviour required for the flatness of rotation curves at
sufficiently large distance, also exhibits $r^{-3}$ and $r^{-4}$ contributions
(upto relativistic corrections) which should dominate at scales significantly
smaller than the halo radii. These encode the substructure of the rotation
curves. Notably, these resemble the large distance behaviour of two well-known
density profiles motivated from CDM simulation results, e.g. NFW $(r^{-3})$
[25] and Hernquist ($r^{-4}$) [26], respectively. A deeper study in this
regard could be pursued elsewhere.

Finally, we obtain the deflection angle of a light ray as it passes through
the halo metric joined to an asymptotically flat spacetime. The correction
with respect to the standard Schwarzschild result is obtained as a function of
$\alpha=\frac{v^{2}}{c^{2}}$ and $\frac{r_{0}}{R_{H}}$, the ratio of the
distance of closest approach to the halo radius. Since $\alpha$ could be
determined empirically from the rotation curve, this provides the hope that
lensing observations for a given $\frac{r_{0}}{R_{H}}$ could be used to test
the model set up here. In the case of very large halo radii, we obtain the
leading correction to the Einstein bending to be $\frac{3}{2}\pi\alpha$. This
provides an intriguing contrast with the predicted value for the standard CDM
halo modelled as an isothermal sphere. Furthermore, the full correction
including the subleading ones could be used to distinguish this formulation of
five-dimensional gravity built upon a noninvertible vielbein from other
existing extra dimensional scenarios, in particular from those which could
lead to similar 4D emergent theories.

###### Acknowledgements.

This work is supported by the MATRICS grant MTR/2021/000008, SERB, Govt. of
India. The author enjoyed many helpful conversations with Somnath Bharadwaj
and gratefully acknowledges a discussion on lensing with Sayan Kar.

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