# Enhancement of spin-orbit coupling in Dirac semimetal Cd3As2 films by Sb-
doping

Yusuke Nakazawa  Department of Applied Physics and Quantum-Phase Electronics
Center (QPEC), the University of Tokyo, Tokyo 113-8656, Japan  Masaki Uchida
<EMAIL_ADDRESS>Department of Applied Physics and Quantum-Phase
Electronics Center (QPEC), the University of Tokyo, Tokyo 113-8656, Japan
PRESTO, Japan Science and Technology Agency (JST), Tokyo 102-0076, Japan
Shinichi Nishihaya  Department of Applied Physics and Quantum-Phase
Electronics Center (QPEC), the University of Tokyo, Tokyo 113-8656, Japan
Mizuki Ohno  Department of Applied Physics and Quantum-Phase Electronics
Center (QPEC), the University of Tokyo, Tokyo 113-8656, Japan  Shin Sato
Department of Applied Physics and Quantum-Phase Electronics Center (QPEC), the
University of Tokyo, Tokyo 113-8656, Japan  Masashi Kawasaki  Department of
Applied Physics and Quantum-Phase Electronics Center (QPEC), the University of
Tokyo, Tokyo 113-8656, Japan  RIKEN Center for Emergent Matter Science (CEMS),
Wako 351-0198, Japan

###### Abstract

We present a study on magnetotransport in films of the topological Dirac
semimetal Cd3As2 doped with Sb grown by molecular beam epitaxy. In our weak
antilocalization analysis, we find a significant enhancement of the spin-orbit
scattering rate, indicating that Sb doping leads to a strong increase of the
pristine band-inversion energy. We discuss possible origins of this large
enhancement by comparing Sb-doped Cd3As2 with other compound semiconductors.
Sb-doped Cd3As2 will be a suitable system for further investigations and
functionalization of topological Dirac semimetals.

###### pacs:

Topologically non-trivial phases are commonly characterized by band inversion
induced by large spin-orbit coupling RMP2010Hasan ; RMP2011Qi ;
RMP2018Armitage . As seen in the field of topological insulators, tuning the
band inversion through modulation of the spin-orbit coupling has advanced the
understanding such as of topological phase transitions PRB2011Guo ;
Science2011Xu . A topological Dirac semimetal, which is characterized by
stable pairs of bulk Dirac dispersions in the presence of band inversion, is
regarded as an ideal platform to study a variety of topological phases
PRL2012Young ; PRB2012Wang ; PRB2013Wang ; NatCom2014Yang . In particular,
tuning the band inversion in topological Dirac semimetals does not only serve
as a source of various topological phase transitions but also allows to
control the separation of the Dirac points in the momentum space, leading to
exotic quantum transport phenomena. Moreover, the quantum spin Hall insulator
phase in quantum confinement PRB2013Wang  can be stabilized by increasing the
band-inversion energy. These few examples already suggest that gaining control
over the tunability of the spin-orbit coupling is strongly desired for
investigations of topological Dirac semimetals.

Cd3As2 is prototypical topological Dirac semimetal NatMat2014Liu ;
NatMat2014Jeon ; Nature2016Moll ; NatCom2017Uchida ; NatCom2017Zhang ;
PRL2019Lin , exhibiting a simple band structure, where only Cd 5$s$ and As
4$p$ bands, which form the Dirac dispersions, exist over a wide energy range
around the Fermi level, as shown in Fig. 1(a). As an example of exotic quantum
transport, quantum Hall states mediated by both the surface and bulk states
Nature2019Zhang ; NatCom2019Nishihaya  have been reported in this compound.
Although chemical doping has been considered effective for tuning the band
inversion as in the cases of topological insulators, it has been limited to
Zn- and P-doping because of the high volatility of Cd3As2 itself SciRep2017Lu
; PRB2018Thirupathaiah . Also in high-crystallinity Cd3As2 films obtained by
solid-phase epitaxy NatCom2017Uchida ; NatCom2019Nishihaya ;
SciRep2018Nakazawa ; PRB2019Uchida , Zn-doping has been demonstrated to
successfully modulate the quantum transport and to induce a topological phase
transition to a trivial insulator simply by reducing the spin-orbit coupling
NatCom2019Nishihaya ; PRB2018Nishihaya ; SciAdv2018Nishihaya . Unfortunately,
however, solid-phase epitaxy does not work on other dopant elements, and so
far, chemical doping efficiently enhancing the spin-orbit coupling has not
been reported for Cd3As2.

Doping Sb which exhibiting a large spin-orbit coupling is promising to
overcome this issue. From the band-gap change measured in P-doped Cd3As2, 10 %
Sb-doping is expected to increase the valence-band-maximum energy by 80 meV,
judged from the band alignment of conventional compound semiconductors
PRB2018Thirupathaiah ; PRB2014Hinuma . In addition to enhancement of the band-
inversion energy, lowering of the Fermi level is also expected, as shown in
Fig. 1(a). This is because the main origin of the $n$-type carriers in Cd3As2
is the deficiency at the anion site (As and Sb), and since Sb is less volatile
than As, a decrease of the carrier density can be expected. Considering that
it is difficult to dope especially volatile atoms into Cd3As2 under thermal
equilibrium, molecular beam epitaxy, which is thermally a non-equilibrium
process as compared to solid-phase epitaxy, is expected more suitable to dope
Sb into Cd3As2 effectively. For undoped Cd3As2, high-quality films have been
grown also by molecular-beam epitaxy, and a quantum Hall state has been
observed for the films with two-dimensional Fermi surface PRL2018Schumann .
Quantum Hall states have also been observed for the films with thickness of
three-dimensional Fermi surface, where involved magnetic orbits can be
described neither by a simple two-dimensional nor by a three-dimensional
electronic state APLMat2019Nakazawa . In this letter, we report transport
signatures of an enhancement of spin-orbit coupling in single-crystalline
Cd3(As1-ySby)2 films grown by molecular-beam epitaxy. A systematic analysis of
weak antilocalization in films with various Sb concentrations reveals that the
spin-orbit scattering rate is dramatically enhanced by Sb doping.

Cd3(As1-ySby)2 films with $y$ ranging from 0.005 to 0.24 were grown by
molecular-beam epitaxy, following the same procedure described in Ref.
APLMat2019Nakazawa . The single-crystalline (1 1 1)A CdTe substrate was heated
at 160 ∘C during the growth. The molecular beams were simultaneously provided
from conventional Knudsen cells containing 6N Cd and 6N Sb and a valved
cracker source containing 7N5 As. The beam equivalent pressures were set to
1$\times$10-4 Pa for Cd and As4 and 0.05 - 5$\times$10-5 Pa for Sb. Transport
measurements were performed by using the conventional four-terminal method on
film samples of 5 mm $\times$ 1 mm size.

Figure 1(b) shows a cross-sectional scanning transmission electron microscopy
(STEM) image taken on a Cd3(As1-ySby)2 film with $y$ = 0.15. Cd3(As1-ySby)2
layers are clearly formed from the heterointerface to the CdTe substrate. An
x-ray diffraction (XRD) $\theta$-2$\theta$ scan for the $y$ = 0.15 film in
Fig. 1(c) shows the reflections from the {1 1 2} lattice planes without any
impurity phases. As shown in Fig. 1(d), XRD scans show a systematic peak shift
with increasing $y$, indicating that the lattice constant systematically
increases upon Sb doping. The film thicknesses determined by atomic force
microscopy are 186 nm, 156 nm, 82 nm, 52 nm, and 83 nm for $y$ = 0.005, 0.07,
0.12, 0.15, and 0.24 samples, respectively. These thickness variation is
caused by difficulty of estimating accurate growth rate for each sample due to
Sb doping effects. However, transport properties of the Cd3(As1-ySby)2 samples
show systematic dependence on the Sb concentration as presented below,
indicating that the thickness variation does not have a considerable effect on
discussions.

Figure 2(a) shows longitudinal resistance $R_{\mathrm{xx}}$ and Hall
resistance $R_{\mathrm{yx}}$, taken on the $y$ = 0.15 film with sweeping the
magnetic field at 2 K. $R_{\mathrm{yx}}$ has a positive slope and is slightly
bent at low fields. This indicates that the conduction is dominated by holes
($p$ $\sim$ 1018 cm-3) and that there is a small number of remnant electrons
($n$ $\sim$ 1015 cm-3). The low-temperature carrier mobility and carrier
density obtained for all the Cd3(As1-ySby)2 films are summarized in Figs. 2(b)
and 2(c). As also confirmed from the Hall resistance in Figs. 2(d) - 2(i), the
electron carrier density is systematically reduced with Sb doping. For
$y=0.12$, the Fermi energy measured from the Dirac points is estimated from
its carrier density to be about 40 meV, and the saddle point energy is
estimated to be about 100 meV PRB2018Thirupathaiah ; PRB2014Hinuma ,
suggesting that the Fermi energy is located inside the band-inversion region.
At $y$ = 0.15, the hole carriers become dominant. This indicates that the
deficiency at As sites is greatly suppressed by Sb doping as expected, and
eventually the system becomes $p$ type. As commonly observed in III-V and II-
VI compound semiconductors, the valence band level is higher in heavier anions
PRB2014Hinuma ; Nature2003VanDeWalle . Based on these observations in the
conventional compound semiconductors, the valence band level of Cd3As2 is also
expected to be shifted higher by Sb doping, and hole carriers are considered
to become easier to be induced as a result of the shallower valence band
level.

Magnetoresistivity ratio
$\mathrm{MRR}\equiv[\rho(B)-\rho(0)]/\rho(0)=\Delta\rho(B)/\rho(0)$ of the
Cd3(As1-ySby)2 films at 2 K are shown in Figs. 2(j) - 2(n). Due to Sb doping,
the shape of the MRR curves changes from parabolic-like in the lightly doped
region to cusp-like in the heavily-doped region. This change indicates that
the Sb doping suppresses the classical magnetoresistivity and enhances weak
antilocalization effects as discussed below. The magnitude of the classical
magnetoresistivity originating from the Lorentz force is expressed by

$\frac{\Delta\rho_{\mathrm{orb}}(B)}{\rho(0)}\approx(\mu B)^{\mathrm{2}},$ (1)

with the carrier mobility $\mu$ MinM1989Pippard , and, therefore, its
contribution decreases when the carrier mobility is reduced.

The observed magnetoresistivity $\rho_{\mathrm{total}}$ can be expressed as
$\rho_{\mathrm{total}}=\rho_{\mathrm{orb}}+\rho_{\mathrm{WAL}}$, where
$\rho_{\mathrm{orb}}$ is the classical magnetoresistivity and
$\rho_{\mathrm{WAL}}$ is a quantum correction to the magnetoresistivity due to
weak antilocalization. According to Kohler’s rule PhysC2002Luo , Eq. (1) can
be rewritten as

$\frac{\Delta\rho_{\mathrm{orb}}(B)}{\rho(0)}=k\left(\frac{B}{\rho(0)}\right)^{2}$
(2)

with a temperature-independent constant $k$, as long as the carrier density is
independent of temperature. In the present case, $k$ can be obtained by
fitting the total magnetoresistivity at high temperatures ($T$ $\sim$ 50 K),
above which only $\Delta\rho_{\mathrm{orb}}(B)$ is observed. By using this $k$
value, $\Delta\rho_{\mathrm{WAL}}(B)$ at low temperatures can be derived by
subtracting $\Delta\rho_{\mathrm{orb}}(B)$ from
$\Delta\rho_{\mathrm{total}}(B)$. Details about the subtraction of the
classical magnetoresistivity can be found in the Supplementary Materials
Supple .

Figure 3(a) shows
$\Delta\rho_{\mathrm{WAL}}(B)\equiv\Delta\rho_{\mathrm{total}}(B)-\Delta\rho_{\mathrm{orb}}(B)$
at 2 K for all examined Cd3(As1-ySby)2 films. Before moving on to a detailed
discussion of the weak antilocalization, we first check the dimensionality of
the Cd3(As1-ySby)2 films. The dimensionality of weak localization and weak
antilocalization is determined by comparing the film thickness $t$ to the
dephasing length $L_{\phi}=\sqrt{D\tau_{\phi}}$, with the diffusion
coefficient $D$ and the dephasing scattering time $\tau_{\phi}$. The dephasing
process is dominated by inelastic scatterings including electron-phonon and
electron-electron scattering JPCM2002Lin . When $L_{\phi}$ is smaller than
$t$, the quantum interference process causing the weak antilocalization is
treated as three-dimensional (Fig. 3(b)), and otherwise, as two-dimensional
(Fig. 3(c)). In the case of our Cd3(As1-ySby)2 films, we employed both two-
dimensional and three-dimensional models to obtain $L_{\phi}$ and compare it
with $t$ to determine the dimensionality of the weak antilocalization. For
$y=0.07-0.24$, the experimental data can be well fitted by the three-
dimensional weak antilocalization model JPCM2002Lin ; JPSJ1981Fukuyama ;
JPSJ1981Maekawa ; JPhys1989Baxter  as presented in Figs. 3(d) - 3(g), and
$L_{\phi}$ obtained from this model is smaller than $t$. This suggests that
the weak antilocalization observed in these films can be treated as three-
dimensional. On the other hand, for $y=0.005$, a narrow dip observed at low
fields ($B<40$ mT) cannot be described by the three-dimensional model.
Instead, it is well fitted by the two-dimensional weak antilocalization model
PTP1980Hikami ; PhysRep1984Bergmann , and $L_{\phi}$ for the $y=0.005$ film
obtained from the two-dimensional model satisfies $L_{\phi}>t$, suggesting
that the weak antilocalization observed in this film is of two-dimensional
nature. Two-dimensional weak antilocalization has also been reported for
undoped Cd3As2 films previously SciRepZhao2016 . In the following discussion,
we focus on the three-dimensional weak antilocalization observed in
$y=0.07-0.24$ films. See also the Supplementary Materials for further
discussion on the dimensionality and the fitting of the $y=0.005$ film Supple
.

Three-dimensional weak localization and weak antilocalization including the
effect of the spin-orbit coupling are expressed as JPCM2002Lin ;
JPSJ1981Fukuyama ; JPSJ1981Maekawa ; JPhys1989Baxter ; approx

$\displaystyle\frac{\Delta\rho_{\mathrm{WAL}}(B)}{\rho^{2}(0)}$
$\displaystyle=$
$\displaystyle\frac{e^{2}}{2\pi^{2}\hbar}\sqrt{\frac{eB}{\hbar}}\Biggl{(}\frac{1}{2\sqrt{1-\gamma}}\biggl{[}f_{3}\left(\frac{B}{B_{-}}\right)-f_{3}\left(\frac{B}{B_{+}}\right)\biggr{]}-f_{3}\left(\frac{B}{B_{2}}\right)$
(3) $\displaystyle-$
$\displaystyle\sqrt{\frac{4B_{\mathrm{SO}}}{3B}}\left[\frac{1}{\sqrt{1-\gamma}}\left(\sqrt{t_{+}}-\sqrt{t_{-}}\right)+\sqrt{t}-\sqrt{t+1}\right]\Biggr{)},$

where

$\displaystyle\gamma=\left[\frac{3g^{*}\mu_{B}B}{4eD(2B_{\mathrm{SO}}-B_{0})}\right]^{2},$
$\displaystyle
B_{\pm}=B_{\phi}+\frac{1}{3}\left(2B_{\mathrm{SO}}-B_{\mathrm{0}}\right)\left(1\pm\sqrt{1-\gamma}\right),$
$\displaystyle
B_{\phi}=B_{i}+B_{\mathrm{0}},\;\;\;B_{2}=B_{i}+\frac{1}{3}B_{\mathrm{0}}+\frac{4}{3}B_{\mathrm{SO}},$
$\displaystyle
t=\frac{3B_{\phi}}{2\left(B_{\mathrm{SO}}-B_{0}\right)},\;\;\;t_{\pm}=t+\frac{1}{2}\left(1\pm\sqrt{1-\gamma}\right),$
$\displaystyle
f_{3}(z)=\sum_{N=0}^{\infty}\left[2\left(\sqrt{N+1+z}-\sqrt{N+z}\right)-\frac{1}{\sqrt{N+\frac{1}{2}+z}}\right].$

Here, $g^{\ast}$ is the Landé factor, $\mu_{\mathrm{B}}$ is the Bohr magneton,
and $D=\mu k_{\mathrm{B}}T/e$ is the diffusion constant. The characteristic
fields $B_{j}$ ($j$ = $i$, SO, and 0) are related to the characteristic
scattering times through $B_{\mathrm{j}}=\hbar/4eD\tau_{\mathrm{j}}$, where
the indices denote inelastic, spin-orbit, and residual scattering,
respectively. The dephasing scattering time $\tau_{\phi}$ is defined by

$\frac{1}{\tau_{\phi}(T)}=\frac{1}{\tau_{0}}+\frac{1}{\tau_{i}(T)},$ (4)

where its temperature dependence is governed by $\tau_{i}$, and its low-
temperature limit is determined by temperature-independent $\tau_{0}$
JPCM2002Lin . In the spin-orbit scattering process, on the other hand, not
only the momentum but also the spin direction of the scattered electrons is
changed due to the spin-orbit interaction JETP1962Abrikosov . In the limit of
$1/\tau_{i}\gg 1/\tau_{\mathrm{SO}}$, Eq. (3) describes weak localization, and
in the opposite limit, $1/\tau_{\mathrm{SO}}\gg 1/\tau_{i}$, it describes weak
antilocalization.

Temperature dependences of $1/\tau_{\phi}$ and $1/\tau_{\mathrm{SO}}$ are
shown in Figs. 4(a) and 4(b). An exponential temperature dependence of
$1/\tau_{\phi}$ is confirmed for all films. The extracted exponents
$p=1.9-3.1$ are quite typical for three-dimensional systems in the diffusive
regime. Similar exponents $p\sim 2-4$ have been previously reported for
conventional compound semiconductors JPCM2002Lin ; PRB2002Oszwaldowski ;
PRB2009Peres .

$1/\tau_{\mathrm{SO}}$ is significantly enhanced by Sb doping, as shown in
Fig. 4(b). The spin-orbit scattering cross section of an impurity
$\sigma_{\mathrm{SO}}$ has been previously studied theoretically and
experimentally PRL1992Geier ; PRL1992Papanikolaou ; JdePhys1982Monod ;
PRB2009Fedorov , and it is expressed as

$\sigma_{\mathrm{SO}}\sim\frac{1}{\tau_{\mathrm{SO}}}\sim\sum_{l\geqq
0}\frac{l(l+1)}{2l+1}\xi_{l}^{2}n_{l}^{2}(E_{F}),$ (5)

where $l$ is the azimuthal quantum number for impurity atoms, $\xi_{l}$ is
half of the spin-orbit constant PRL1992Papanikolaou , and $n_{l}(E_{F})$ is
the angular momentum resolved local density of states of the impurity atoms at
the Fermi level. Eq. (5) suggests that $1/\tau_{\mathrm{SO}}$ largely depends
on the local density of $p$ states ($l=1$) of impurity at the Fermi level
($n_{l=1}(E_{F})$) PRL1992Papanikolaou , and, therefore, the observed
enhancement of $1/\tau_{\mathrm{SO}}$ in Cd3(As1-ySby)2 is considered a result
of a larger $n_{l=1}(E_{F})$ with an increased impurity density by Sb doping.

The relation between $1/\tau_{\mathrm{SO}}$ and $1/\tau_{\phi}$ obtained for
the Cd3(As1-ySby)2 films is summarized in Fig. 4(c), and compared with
previously reported data on both three-dimensional and two-dimensional systems
PRB2002Oszwaldowski ; PRB2009Peres ; PRB2003Studenikin ; PRB2008Grbic ;
PRB2009Kallaher ; PRB2006Thillosen ; APL2006Thillosen . In particular, a large
enhancement of $1/\tau_{\mathrm{SO}}$ has been observed for the $p$-type
Cd3(As1-ySby)2 films, and this may suggest that a similar enhancement can be
also expected for conventional compound semiconductors where the following two
conditions are satisfied: (i) the doping takes place at the anion sites and
(ii) the Fermi level is tuned into the valence band. For the compound
semiconductors shown in Fig. 4(c), at least one of these conditions is not
satisfied. It is further suggested that a large enhancement of
$1/\tau_{\mathrm{SO}}$ can be expected, for example, for $p$-type GaAs1-xSbx
and $p$-type InP1-xAsx, which satisfy the aforementioned two conditions but
these samples have not been examined yet.

In summary, we have systematically investigated magnetotransport of
Cd3(As1-ySby)2 films with varying Sb concentrations $y=0.005-0.24$. Weak
antilocalization is commonly observed in the Cd3(As1-ySby)2 films and has been
analyzed using a three-dimensional model. In the analysis,
$1/\tau_{\mathrm{SO}}$ is greatly enhanced upon increasing the Sb
concentration. In particular, the Fermi energy of the $y=0.12$ film is
estimated to be located inside the band-inversion region, based on the band
alignment of conventional compound semiconductors. Our demonstration of a
large enhancement of the spin-orbit coupling in Cd3(As1-ySby)2 films paves the
way for further investigations and functionalization of topological Dirac
semimetals, including the realization of a robust quantum spin Hall insulator
phase under confinement conditions as in HgTe/CdTe quantum wells PRB2013Wang ;
Science2006Bernevig ; Science2007Konig . Tuning a confinement-induced sub-band
position and the Fermi energy inside the band-inversion region is considered
more feasible with the increased band inversion in Cd3(As1-ySby)2.

We thank N. Nagaosa and M. Tokunaga for fruitful discussions. We also thank M.
Kriener for proofreading of the manuscript. This work was supported by JST
PRESTO Grant No. JPMJPR18L2 and CREST Grant No. JPMJCR16F1, Japan, and by
Grant-in-Aids for Scientific Research (B) No. JP18H01866 from MEXT, Japan.

## References

  * (1) M. Z. Hasan and C. L. Kane, Colloquium : Topological insulators, Rev. Mod. Phys. 82, 3045 (2010).
  * (2) X.-L. Qi and S.-C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys. 83, 1057 (2011).
  * (3) N. P. Armitage, E. J. Mele, and A. Vishwanath, Weyl and Dirac semimetals in three-dimensional solids, Rev. Mod. Phys. 90, 015001 (2018).
  * (4) H. Guo, K. Sugawara, A. Takayama, S. Souma, T. Sato, T. Satoh, A. Ohnishi, M. Kitaura, M. Sasaki, Q.-K. Xue, and T. Takahashi, Evolution of surface states in Bi1-xSbx alloys across the topological phase transition, Phys. Rev. B 83, 201104(R) (2011).
  * (5) S.-Y. Xu, Y. Xia., L. A. Wray, S. Jia, F. Meier, J. H. Dil, J. Osterwalder, B. Slomski, A. Bansil, H. Lin, R. J. Cava, and M. Z. Hasan, Topological Phase Transition and Texture Inversion in a Tunable Topological Insulator, Science 332, 560 (2011).
  * (6) S. M. Young, S. Zaheer, J. C. Teo, C. L. Kane, E. J. Mele, and A. M. Rappe, Dirac Semimetal in Three Dimensions, Phys. Rev. Lett. 108, 140405 (2012).
  * (7) Z. Wang, Y. Sun, X.-Q. Chen, C. Franchini, G. Xu, H. Weng, X. Dai, and Z. Fang, Dirac semimetal and topological phase transitions in A3Bi (A = Na, K, Rb), Phys. Rev. B 85, 195320 (2012).
  * (8) Z. Wang, H. Weng, Q. Wu, X. Dai, and Z. Fang, Three-dimensional Dirac semimetal and quantum transport in Cd3As2, Phys. Rev. B 88, 125427 (2013).
  * (9) B.-J. Yang and N. Nagaosa, Classification of stable three-dimensional Dirac semimetals with nontrivial topology, Nat. Commun. 5, 4898 (2014).
  * (10) Z. K. Liu, J. Jiang, B. Zhou, Z. J. Wang, Y. Zhang, H. M. Weng, D. Prabhakaran, S.-K. Mo, H. Peng, P. Dudin, T. Kim, M. Hoesch, Z. Fang, X. Dai, Z. X. Shen, D. L. Feng, Z. Hussain, and Y. L. Chen, A stable three-dimensional topological Dirac semimetal Cd3As2, Nat. Mater. 13, 677 (2014).
  * (11) S. Jeon, B. B. Zhou, A. Gyenis, B. E. Feldman, I. Kimchi, A. C. Potter, Q. D. Gibson, R. J. Cava, A. Vishwanath, and A. Yazdani, Landau quantization and quasiparticle interference in the three-dimensional Dirac semimetal Cd3As2, Nat. Mater. 13, 851 (2014).
  * (12) P. J. W. Moll, N. L. Nair, T. Helm, A. C. Potter, I. Kimchi, A. Vishwanath, and J. G. Analytis, Transport evidence for Fermi-arc-mediated chirality transfer in the Dirac semimetal Cd3As2, Nature 535, 266 (2016).
  * (13) M. Uchida, Y. Nakazawa, S. Nishihaya, K. Akiba, M. Kriener, Y. Kozuka, A. Miyake, Y. Taguchi, M. Tokunaga, N. Nagaosa, Y. Tokura, and M. Kawasaki, Quantum Hall states observed in thin films of Dirac semimetal Cd3As2, Nat. Commun. 8, 2274 (2017).
  * (14) C. Zhang, A. Narayan, S. Lu, J. Zhang, H. Zhang, Z. Ni, X. Yuan, Y. Liu, J.-H. Park, E. Zhang, W. Wang, S. Liu, L. Cheng, L. Pi, Z. Sheng, S. Sanvito, and F. Xiu, Evolution of Weyl orbit and quantum Hall effect in Dirac semimetal Cd3As2, Nat. Commun. 8, 1272 (2017).
  * (15) B.-C. Lin, S. Wang, S. Wiedmann, J.-M. Lu, W.-Z. Zheng, D. Yu, and Z.-M. Liao, Observation of an Odd-Integer Quantum Hall Effect from Topological Surface States in Cd3As2, Phys. Rev. Lett. 122, 036602 (2019).
  * (16) C. Zhang, Y. Zhang, X. Yuan, S. Lu, J. Zhang, A. Narayan, Y. Liu, H. Zhang, Z. Ni, R. Liu, E. S. Choi, A. Suslov, S. Sanvito, L. Pi, H.-Z. Lu, and A. C, Quantum Hall effect based on Weyl orbits in Cd3As2, Nature 565, 331 (2019).
  * (17) S. Nishihaya, M. Uchida, Y. Nakazawa, R. Kurihara, K. Akiba, M. Kriener, A. Miyake, Y. Taguchi, M. Tokunaga, and M. Kawasaki, Quantized surface transport in topological Dirac semimetal films, Nat. Commun. 10, 2564 (2019).
  * (18) H. Lu, X. Zhang, Y. Bian, and S. Jia, Topological Phase Transition in Single Crystals of (Cd1-xZnx)3As2, Sci. Rep. 7, 3148 (2017).
  * (19) S. Thirupathaiah, I. Morozov, Y. Kushnirenko, A. V. Fedorov, E. Haubold, T. K. Kim, G. Shipunov, A. Maksutova, O. Kataeva, S. Aswartham, B. Büchner, and S. V. Borisenko, Spectroscopic evidence of topological phase transition in the three-dimensional Dirac semimetal Cd3(As1-xPx)2, Phys. Rev. B 98, 085145 (2018).
  * (20) Y. Nakazawa, M. Uchida, S. Nishihaya, M. Kriener, Y. Kozuka, Y. Taguchi, and M. Kawasaki, Structural characterisation of high-mobility Cd3As2 films crystallised on SrTiO3, Sci. Rep. 8, 2244 (2018).
  * (21) M. Uchida, T. Koretsune, S. Sato, M. Kriener, Y. Nakazawa, S. Nishihaya, Y. Taguchi, R. Arita, and M. Kawasaki, Ferromagnetic state above room temperature in a proximitized topological Dirac semimetal, Phys. Rev. B 100, 245148 (2019).
  * (22) S. Nishihaya, M. Uchida, Y. Nakazawa, K. Akiba, M. Kriener, Y. Kozuka, A. Miyake, Y. Taguchi, M. Tokunaga, and M. Kawasaki, Negative magnetoresistance suppressed through a topological phase transition in (Cd1-xZnx)3As2 thin films, Phys. Rev. B 97, 245103 (2018).
  * (23) S. Nishihaya, M. Uchida, Y. Nakazawa, M. Kriener, Y. Kozuka, Y. Taguchi, and M. Kawasaki, Gate-tuned quantum Hall states in Dirac semimetal (Cd1-xZnx)3As2, Sci. Adv. 4, eaar5668 (2018).
  * (24) Y. Hinuma, A. Grüneis, G. Kresse, and F. Oba, Band alignment of semiconductors from density-functional theory and many-body perturbation theory, Phys. Rev. B 90, 155405 (2014).
  * (25) T. Schumann, L. Galletti, D. A. Kealhofer, H. Kim, M. Goyal, and S. Stemmer, Observation of the Quantum Hall Effect in Confined Films of the Three-Dimensional Dirac Semimetal Cd3As2, Phys. Rev. Lett. 120, 016801 (2018).
  * (26) Y. Nakazawa, M. Uchida, S. Nishihaya, S. Sato, A. Nakao, J. Matsuno, and M. Kawasaki, Molecular beam epitaxy of three-dimensionally thick Dirac semimetal Cd3As2 films, APL Mater. 7, 071109 (2019).
  * (27) Chris G. Van de Walle and J. Neugebauer, Universal alignment of hydrogen levels in semiconductors, insulators and solutions, Nature 423, 626 (2003).
  * (28) A. B. Pippard, Magnetoresistance in Metals (Cambridge University Press, 1989).
  * (29) N. Luo and G. H. Miley, Kohler’s rule and relaxation rates in high-Tc superconductors, Physica C 371, 259 (2002).
  * (30) See Supplementary Materials for more information about the analysis of magnetoresistance.
  * (31) J. J. Lin and J. P. Bird, Recent experimental studies of electron dephasing in metal and semiconductor mesoscopic structures, J. Phys.: Condens. Matter. 14, R501 (2002).
  * (32) H. Fukuyama and K. Hoshino, Effect of Spin-Orbit Interaction on Magnetoresistance in the Weakly Localized Regime of Three-Dimensional Disordered Systems, J. Phys. Soc. Jpn. 50, 2131 (1981).
  * (33) S. Maekawa and H. Fukuyama, Magnetoresistance in Two-Dimensional Disordered Systems : Effects of Zeeman Splitting and Spin-Orbit Scattering, J. Phys. Soc. Jpn. 50, 2516 (1981).
  * (34) D. V. Baxter, R. Richter, M. L. Trudeau, R. W. Cochrane, and J. O. Strom-Olsen, Fitting to magnetoresistance under weak localization in three dimensions, J. Physique 50, 1673 (1989).
  * (35) G. Bergmann, Weak localization in thin films : a time-of-flight experiment with conduction electrons, Physics Reports 107, 1 (1984).
  * (36) S. Hikami, A. I. Larkin, and Y. Nagaoka, Spin-Orbit Interaction and Magnetoresistance in the Two Dimensional Random System, Prog. Theor. Phys. 63, 707 (1980).
  * (37) B. Zhao, P. Cheng, H. Pan, S. Zhang, B. Wang, G. Wang, F. Xiu, and F. Song, Weak antilocalization in Cd3As2 thin films, Sci. Rep. 6, 22377 (2016).
  * (38) The summation function $f_{3}$ can be approximated by the formula
$f_{3}(z)\approx
2\left(\sqrt{2+\frac{1}{z}}-\sqrt{\frac{1}{z}}\right)-\left[\left(\frac{1}{2}+\frac{1}{z}\right)^{-\frac{1}{2}}+\left(\frac{3}{2}+\frac{1}{z}\right)^{-\frac{1}{2}}\right]+\frac{1}{48}\left(2.03+\frac{1}{z}\right)^{-\frac{3}{2}}$
with sufficient accuracy. We used this approximation in the present analysis.

  * (39) A. A. Abrikosov and L. P. Gor’kov, Spin-orbit interaction and the Knight shift in superconductors, JETP 15, 752 (1962).
  * (40) M. Oszwaldowski and T. Berus, Weak localization in InSb thin films heavily doped with lead, Phys. Rev. B 65, 235418 (2002).
  * (41) M. L. Peres, V. A. Chitta, N. F. Oliveira, Jr., D. K. Maude, P. H. O. Rappl, A. Y. Ueta, and E. Abramof, Antilocalization of hole carriers in Pb1-xEuxTe alloys in the metallic regime, Phys. Rev. B 79, 085309 (2009).
  * (42) N. Papanikolaou, N. Stefanou, P. H. Dederichs, S. Geier, and G. Bergmann, First-principles calculations of the spin-orbit scattering cross section of $sp$ impurities in Mg, Phys. Rev. Lett. 69, 2110 (1992).
  * (43) S. Geier and G. Bergmann, Residual atomic behavior of ($sp$)-impurities in a Mg host, Phys. Rev. Lett. 68, 2520 (1992).
  * (44) P. Monod and S. Schultz, Conduction electron spin-flip scattering by impurities in copper, Journal de Physique 43, 393 (1982).
  * (45) D. V. Fedorov, P. Zahn, M. Gradhand, and I. Mertig, First-principles calculations of spin relaxation times of conduction electrons in Cu with nonmagnetic impurities, Phys. Rev. B 79, 059901(E) (2009).
  * (46) S. A. Studenikin, P. T. Coleridge, N. Ahmed, P. J. Poole, and A. Sachrajda, Experimental study of weak antilocalization effects in a high-mobility InxGa1-xAs/InP quantum well, Phys. Rev. B 68, 035317 (2003).
  * (47) B. Grbić, R. Leturcq, T. Ihn, K. Ensslin, D. Reuter, and A. D. Wieck, Strong spin-orbit interactions and weak antilocalization in carbon-doped p-type GaAs/AlxGa1-xAs, Phys. Rev. B 77, 125312 (2008).
  * (48) R. L. Kallaher and J. J. Heremans, Spin and phase coherence measured by antilocalization in $n$-InSb thin films, Phys. Rev. B 79, 075322 (2009).
  * (49) N. Thillosen, S. Cabanas, N. Kaluza, V. A. Guzenko, H. Hardtdegen, and Th. Schäpers, Weak antilocalization in gate-controlled AlxGa1-xN/GaN two-dimensional electron gases, Phys. Rev. B 73, 241311(R) (2006).
  * (50) N. Thillosen, Th. Schäpers, N. Kaluza, H. Hardtdegen, and V. A. Guzenko, Weak antilocalization in a polarization-doped AlxGa1-xN/GaN heterostructure with single subband occupation, Appl. Phys. Lett. 88, 0022111 (2006).
  * (51) B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells, Science 314, 1757 (2006).
  * (52) M. König, S. Wiedmann, C. Brüne, A. Roth, H. Buhmann, L. W. Molenkamp, X.-L. Qi, and S.-C. Zhang, Quantum Spin Hall Insulator State in HgTe Quantum Wells, Science 318, 766 (2007).

Figure 1:  (a) Schematic band-structure change induced by Sb doping in Cd3As2.
The band inversion is expected to be enhanced by Sb doping which exhibits a
larger spin-orbit coupling than As. Electron carriers caused by As and Sb
deficiency are also expected to be reduced. (b) Cross-section HAADF-STEM image
and (c) XRD $\theta-2\theta$ scan of a Cd3(As0.85Sb0.15)2 film grown on a CdTe
substrate. The asterisks denote peaks from the CdTe substrate. (d) XRD
$\theta-2\theta$ scans magnified around the (4 4 8) Cd3(As1-ySby)2 film peak
with different Sb concentrations $y$. The Sb concentrations are estimated
using Vegard’s law, which linearly interpolates the lattice constants between
$y$ = 0 and 0.15 films determined by EDX analyses.

Figure 2:  (a) Longitudinal resistance $R_{\mathrm{xx}}$ and Hall resistance
$R_{\mathrm{yx}}$ of Cd3(As0.85Sb0.15)2 film at 2 K. The magnetic field $B$ is
applied out of plane. Mobility vs carrier density summarized for our
Cd3(As1-ySby)2 films in (b) $n$-type and (c) $p$-type regions. (d)-(i) Hall
resistance times thickness $R_{\mathrm{yx}}$$t$ and (j)-(n) magnetoresistance
ratio for our Cd3(As1-ySby)2 films at 2 K. In (g) and (i), the curves are
multiplied by ten for clarity.

Figure 3:  (a) Normalized magnetoresistivity
$\Delta\rho_{\mathrm{WAL}}/\rho^{2}(0)=[\Delta\rho_{\mathrm{total}}-\Delta\rho_{\mathrm{orb}}]/\rho_{\mathrm{total}}^{2}(0)$
of the Cd3(As1-ySby)2 films at 2 K. Trajectories of the scattered electrons
for (b) three-dimensional and (c) two-dimensional weak antilocalization.
(d)-(g) Temperature dependence of the normalized magnetoresistivity for $y$ =
0.07, 0.12, 0.15, and 0.24. The red curves are fits to the three-dimensional
weak antilocalization model (Eq. (3)). The theory assumes that the magnetic
field is within the range satisfying $\gamma<1$ (see Eq. (3) in main text).
The magnetoresistivity below 7 K is not shown for $y=0.07$ because
$L_{\phi}<t$ is not satisfied at lower temperature (see also the Supplementary
Material).

Figure 4:  (a) Dephasing scattering rate $1/\tau_{\mathrm{\phi}}$ and (b)
spin-orbit scattering rate $1/\tau_{\mathrm{SO}}$ as a function of temperature
for $y$ = 0.07, 0.12, 0.15, and 0.24. The horizontal bars on the right axis in
(a) represent the temperature-independent residual scattering rate extracted
from the fit at 7 K for $y$ = 0.07 and at 2 K for $y$ = 0.12, 0.15, and 0.24.
(c) Spin-orbit scattering rate $1/\tau_{\mathrm{SO}}$ vs dephasing scattering
rate $1/\tau_{\phi}$ summarized for the Cd3(As1-ySby)2 films. Data taken at
different temperatures as shown in (a) and (b) are plotted. For comparison,
$1/\tau_{\mathrm{SO}}$ vs $1/\tau_{\phi}$ previously reported for three-
dimensional systems ($\bullet$ In1-xPbxSb PRB2002Oszwaldowski ,
$\blacktriangle$ Pb1-xEuxTe PRB2009Peres ) and two-dimensional systems
($\bigtriangledown$ Ga1-xInxAs PRB2003Studenikin , $\diamond$ GaAs/Al1-xGaxAs
PRB2008Grbic , $\bowtie$ InSb PRB2009Kallaher , $\Box$ Al1-xGaxN/GaN
PRB2006Thillosen ; APL2006Thillosen ) are also plotted. $1/\tau_{i}$
($1/\tau_{\phi}$ at $\tau_{0}\rightarrow\infty$) is shown for Pb1-xEuxTe
instead of $1/\tau_{\phi}$.


Holographic RG flows and Janus solutions from matter-coupled $N=4$ gauged
supergravity

Parinya Karndumri

String Theory and Supergravity Group, Department of Physics, Faculty of
Science, Chulalongkorn University, 254 Phayathai Road, Pathumwan, Bangkok
10330, Thailand

E-mail<EMAIL_ADDRESS>

###### Abstract

We study an $SO(2)\times SO(2)\times SO(2)\times SO(2)$ truncation of four-
dimensional $N=4$ gauged supergravity coupled to six vector multiplets with
$SO(4)\times SO(4)$ gauge group and find a new class of holographic RG flows
and supersymmetric Janus solutions. In this truncation, there is a unique
$N=4$ supersymmetric $AdS_{4}$ vacuum dual to an $N=4$ SCFT in three
dimensions. In the presence of the axion, the RG flows generally preserve
$N=2$ supersymmetry while the supersymmetry is enhanced to $N=4$ for vanishing
axion. We find solutions interpolating between the $AdS_{4}$ vacuum and
singular geometries with different residual symmetries. We also show that all
the singularities are physically acceptable within the framework of four-
dimensional gauged supergravity. Accordingly, the solutions are
holographically dual to RG flows from the $N=4$ SCFT to a number of non-
conformal phases in the IR. We also find $N=4$ and $N=2$ Janus solutions with
$SO(4)\times SO(4)$ and $SO(2)\times SO(2)\times SO(3)\times SO(2)$
symmetries, respectively. The former is obtained from a truncation of all
scalars from vector multiplets and can be regarded as a solution of pure $N=4$
gauged supergravity. On the other hand, the latter is a genuine solution of
the full matter-coupled theory. These solutions describe conformal interfaces
in the $N=4$ SCFT with $N=(4,0)$ and $N=(2,0)$ supersymmetries.

##  1 Introduction

Over the past twenty years, solutions of gauged supergravities in different
space-time dimensions have provided a large number of useful holographic
descriptions for strongly coupled dual field theories. Among various types of
these solutions, holographic RG flows and Janus solutions are of particular
interest since the original proposal of the AdS/CFT correspondence [1, 2, 3].
These explain deformations of the dual conformal field theories (CFT) and the
presence of conformal interfaces or defects within the parent CFTs. The most
useful results along this direction arise when there is some amount of
unbroken supersymmetry in which many properties at strong coupling are
controllable.
Although there are AdS/CFT dualities in many space-time dimensions, AdS4/CFT3
correspondence attracts much attention due to the relevance in describing
world-volume dynamics of M2-branes, the fundamental degrees of freedom in
M-theory. In this case, four-dimensional gauged supergravities are very useful
in obtaining holographic solutions of interest which in some cases can be
uplifted to ten or eleven dimensions. A number of holographic RG flows and
Janus configurations have previously been found in gauged supergravities with
various gauge groups and different numbers of supersymmetries [4, 5, 6, 7, 8,
9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26], for
similar solutions with other space-time dimensions, see [27, 28, 29, 30, 31,
32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44] for an incomplete list.
In this paper, we will give a new class of holographic RG flows and
supersymmetric Janus solutions within matter-coupled $N=4$ gauged supergravity
with $SO(4)\times SO(4)$ gauge group. This gauged supergravity can be
constructed by coupling pure $N=4$ gauged supergravity to $n\geq 6$ vector
multiplets. We will consider the simplest case of $n=6$ here. In this gauged
supergravity, a number of supersymmetric $AdS_{4}$ vacua and RG flows between
them have recently appeared in [17]. In that case, the truncation to
$SO(3)\times SO(3)$ singlet scalars have been considered and solutions with
only the dilaton and the axion in the gravity multiplet or solutions with
vanishing axion are consistent with supersymmetry. In addition, all solutions
given in [17] preserve $N=4$ supersymmetry. In the present work, we will find
more general solutions namely solutions with both the axion and vector
multiplet scalars non-vanishing and breaking the original $N=4$ supersymmetry
to $N=2$.
We will also find a new class of supersymmetric Janus solutions in $N=4$
gauged supergravity which has not been considered in [17]. To the best of the
author’s knowledge, only Janus solutions found in [15] and [16] are obtained
within the framework of $N=4$ gauged supergravity. All the solutions given in
[15] are the so-called singular Janus in the sense that they interpolate
between non-conformal phases rather than between conformal phases, so these
are holographically expected to describe conformal interfaces within non-
conformal field theories in three dimensions. On the other hand, regular Janus
solutions given in [16] only involve the dilaton and axion from the gravity
multiplet. There is also a singular Janus solution found in [16] that involve
only scalars from vector multiplets. Therefore, we believe that the $N=2$
Janus solution obtained in this paper is the first regular Janus solution in
the context of matter-coupled $N=4$ gauged supergravity that involves scalars
from both gravity and vector multiplets.
The paper is organized as follows. In section 2, we review the general
structure of $N=4$ gauged supergravity coupled to vector multiplets in the
embedding tensor formalism and at the end focus on the case of six vector
multiplets with $SO(4)\times SO(4)$ gauge group. The truncation to
$SO(2)\times SO(2)\times SO(2)\times SO(2)$ singlet scalars, $AdS_{4}$ vacua
and general structure of relevant BPS equations in both RG flow and Janus
solutions are also given in detail. This sets up the framework for finding
solutions in subsequent sections. In sections 3 and 4, a number of $N=4$ and
$N=2$ holographic RG flow and Janus solutions are given, respectively. We end
the paper by giving some conclusions and comments on the results in section 5.

##  2 Matter-coupled $N=4$ gauged supergravity

In four dimensions, $N=4$ supersymmetry allows two types of supermultiplets,
the gravity and vector multiplets. The former contains the following field
content

$(e^{\hat{\mu}}_{\mu},\psi^{i}_{\mu},A_{\mu}^{m},\chi^{i},\tau)$ (1)

which are given by the graviton $e^{\hat{\mu}}_{\mu}$, four gravitini
$\psi^{i}_{\mu}$, six vectors $A_{\mu}^{m}$, four spin-$\frac{1}{2}$ fields
$\chi^{i}$ and one complex scalar $\tau$. The scalar consists of the dilaton
$\phi$ and the axion $\chi$ and can be parametrized by
$SL(2,\mathbb{R})/SO(2)$ coset. Indices $\mu,\nu,\ldots=0,1,2,3$ and
$\hat{\mu},\hat{\nu},\ldots=0,1,2,3$ are respectively space-time and tangent
space indices while $m,n=1,\ldots,6$ and $i,j=1,2,3,4$ indices describe
fundamental representations of $SO(6)_{R}$ and $SU(4)_{R}$ R-symmetry.
The component fields in a vector multiplet are given by a vector field
$A_{\mu}$, four gaugini $\lambda^{i}$ and six scalars $\phi^{m}$. In general,
the gravity multiplet can couple to an arbitrary number $n$ of vector
multiplets. These vector multiplets will be labeled by indices
$a,b=1,\ldots,n$, so all fields in the vector multiplets will carry an
additional index in the form

$(A^{a}_{\mu},\lambda^{ia},\phi^{ma}).$ (2)

Similar to the dilaton and the axion in the gravity multiplet, the $6n$ scalar
fields $\phi^{ma}$ can be parametrized by $SO(6,n)/SO(6)\times SO(n)$ coset.
Fermionic fields and supersymmetry parameters transforming in fundamental
representation of $SU(4)_{R}\sim SO(6)_{R}$ are subject to the chirality
projections

$\gamma_{5}\psi^{i}_{\mu}=\psi^{i}_{\mu},\qquad\gamma_{5}\chi^{i}=-\chi^{i},\qquad\gamma_{5}\lambda^{i}=\lambda^{i}$
(3)

while those transforming in anti-fundamental representation of $SU(4)_{R}$
satisfy

$\gamma_{5}\psi_{\mu i}=-\psi_{\mu
i},\qquad\gamma_{5}\chi_{i}=\chi_{i},\qquad\gamma_{5}\lambda_{i}=-\lambda_{i}\,.$
(4)

In general, all possible gaugings of a supergravity theory can be described by
the embedding tensor. For the matter-coupled $N=4$ supergravity, supersymmetry
requires that the embedding tensor can have only two non-vanishing components
denoted by $\xi^{\alpha M}$ and $f_{\alpha MNP}$. Indices $\alpha=(+,-)$ and
$M,N=(m,a)=1,\ldots,n+6$ describe fundamental representations of the global
$SL(2,\mathbb{R})$ and $SO(6,n)$ symmetries, respectively. The electric vector
fields $A^{+M}=(A^{m}_{\mu},A^{a}_{\mu})$, appearing in the ungauged
Lagrangian with the usual Yang-Mills kinetic term, together with their
magnetic dual $A^{-M}$ form a doublet under $SL(2,\mathbb{R})$ denoted by
$A^{\alpha M}$. In addition, the embedding tensor needs to satisfy the
quadratic constraint in order to define a consistent gauging. This ensures
that the resulting gauge generators form a closed subalgebra of
$SL(2,\mathbb{R})\times SO(6,n)$. We will not discuss all these details here
but refer to the original construction in [45], see also [46, 47, 48] for an
earlier construction.
Consistent gauge groups can generally be embedded in both $SL(2,\mathbb{R})$
and $SO(6,n)$ factors, and the magnetic vector fields can also participate in
the gauging. However, each magnetic vector field must be accompanied by an
auxiliary two-form field in order to remove the extra degrees of freedom. The
analysis of the existence of maximally supersymmetric $AdS_{4}$ vacua given in
[49], see also [47, 48] for an earlier result, requires the gaugings to
involve both electric and magnetic vector fields with the corresponding gauge
groups embedded solely in $SO(6,n)$. This implies that both electric and
magnetic components of $f_{\alpha MNP}$ must be non-vanishing and $\xi^{\alpha
M}=0$. We also note that the magnetic components $f_{-MNP}$ are related to the
duality phases first introduced in [47, 48].
In this paper, we are interested in supersymmetric solutions that are
asymptotic to $AdS_{4}$ vacua and involve only the metric and scalar fields.
Accordingly, from now on, we will set all the other fields including
$\xi^{\alpha M}$ to zero. The scalar coset $SL(2,\mathbb{R})/SO(2)\times
SO(6,n)/SO(6)\times SO(n)$ can be described by the coset representative
$\mathcal{V}_{\alpha}$ and $\mathcal{V}_{M}^{\phantom{M}A}$, respectively.
With the definition

$\tau=\chi+ie^{\phi},$ (5)

we will choose the explicit form of $\mathcal{V}_{\alpha}$ to be

$\mathcal{V}_{\alpha}=e^{\frac{\phi}{2}}\left(\begin{array}[]{c}\chi+ie^{\phi}\\\
1\\\ \end{array}\right).$ (6)

The coset representative $\mathcal{V}_{M}^{\phantom{M}A}$ transforms under the
global $SO(6,n)$ and local $SO(6)\times SO(n)$ by left and right
multiplications, respectively. This implies the splitting of the index
$A=(m,a)$. We can then write the $SO(6,n)/SO(6)\times SO(n)$ coset
representative as

$\mathcal{V}_{M}^{\phantom{M}A}=(\mathcal{V}_{M}^{\phantom{M}m},\mathcal{V}_{M}^{\phantom{M}a}).$
(7)

Furthermore, as an element of $SO(6,n)$, the matrix
$\mathcal{V}_{M}^{\phantom{M}A}$ also satisfies the relation

$\eta_{MN}=-\mathcal{V}_{M}^{\phantom{M}m}\mathcal{V}_{N}^{\phantom{M}m}+\mathcal{V}_{M}^{\phantom{M}a}\mathcal{V}_{N}^{\phantom{M}a}$
(8)

with $\eta_{MN}=\textrm{diag}(-1,-1,-1,-1,-1,-1,1,\ldots,1)$ being the
$SO(6,n)$ invariant tensor.
With $\xi^{\alpha M}=0$ and only the metric and scalars non-vanishing, the
bosonic Lagrangian can simply be written as

$e^{-1}\mathcal{L}=\frac{1}{2}R+\frac{1}{16}\partial_{\mu}M_{MN}\partial^{\mu}M^{MN}-\frac{1}{4(\textrm{Im}\tau)^{2}}\partial_{\mu}\tau\partial^{\mu}\tau^{*}-V$
(9)

where $e=\sqrt{-g}$ is the vielbein determinant. The scalar potential is given
in terms of the scalar coset representative and the embedding tensor by

$\displaystyle V$ $\displaystyle=$ $\displaystyle\frac{1}{16}\left[f_{\alpha
MNP}f_{\beta
QRS}M^{\alpha\beta}\left[\frac{1}{3}M^{MQ}M^{NR}M^{PS}+\left(\frac{2}{3}\eta^{MQ}-M^{MQ}\right)\eta^{NR}\eta^{PS}\right]\right.$
(10) $\displaystyle\left.-\frac{4}{9}f_{\alpha MNP}f_{\beta
QRS}\epsilon^{\alpha\beta}M^{MNPQRS}\right].$

At this point, we should note that the components $f_{\alpha MNP}$ of the
embedding tensor include the gauge coupling constants.
The $SO(6)\times SO(n)$ invariant and symmetric matrix $M_{MN}$ is defined by

$M_{MN}=\mathcal{V}_{M}^{\phantom{M}m}\mathcal{V}_{N}^{\phantom{M}m}+\mathcal{V}_{M}^{\phantom{M}a}\mathcal{V}_{N}^{\phantom{M}a}$
(11)

with its inverse denoted by $M^{MN}$. The tensor $M^{MNPQRS}$ is obtained from
raising indices of $M_{MNPQRS}$ given by

$M_{MNPQRS}=\epsilon_{mnpqrs}\mathcal{V}_{M}^{\phantom{M}m}\mathcal{V}_{N}^{\phantom{M}n}\mathcal{V}_{P}^{\phantom{M}p}\mathcal{V}_{Q}^{\phantom{M}q}\mathcal{V}_{R}^{\phantom{M}r}\mathcal{V}_{S}^{\phantom{M}s}\,.$
(12)

Similarly, $M^{\alpha\beta}$ is the inverse of the symmetric $2\times 2$
matrix $M_{\alpha\beta}$ defined by

$M_{\alpha\beta}=\textrm{Re}(\mathcal{V}_{\alpha}\mathcal{V}^{*}_{\beta}).$
(13)

Fermionic supersymmetry transformations are given by

$\displaystyle\delta\psi^{i}_{\mu}$ $\displaystyle=$ $\displaystyle
2D_{\mu}\epsilon^{i}-\frac{2}{3}gA^{ij}_{1}\gamma_{\mu}\epsilon_{j},$ (14)
$\displaystyle\delta\chi^{i}$ $\displaystyle=$
$\displaystyle-\epsilon^{\alpha\beta}\mathcal{V}_{\alpha}D_{\mu}\mathcal{V}_{\beta}\gamma^{\mu}\epsilon^{i}-\frac{4}{3}igA_{2}^{ij}\epsilon_{j},$
(15) $\displaystyle\delta\lambda^{i}_{a}$ $\displaystyle=$ $\displaystyle
2i\mathcal{V}_{a}^{\phantom{a}M}D_{\mu}\mathcal{V}_{M}^{\phantom{M}ij}\gamma^{\mu}\epsilon_{j}-2igA_{2aj}^{\phantom{2aj}i}\epsilon^{j}\,.$
(16)

The fermion shift matrices are in turn defined by

$\displaystyle A_{1}^{ij}$ $\displaystyle=$
$\displaystyle\epsilon^{\alpha\beta}(\mathcal{V}_{\alpha})^{*}\mathcal{V}_{kl}^{\phantom{kl}M}\mathcal{V}_{N}^{\phantom{N}ik}\mathcal{V}_{P}^{\phantom{P}jl}f_{\beta
M}^{\phantom{\beta M}NP},$ $\displaystyle A_{2}^{ij}$ $\displaystyle=$
$\displaystyle\epsilon^{\alpha\beta}\mathcal{V}_{\alpha}\mathcal{V}_{kl}^{\phantom{kl}M}\mathcal{V}_{N}^{\phantom{N}ik}\mathcal{V}_{P}^{\phantom{P}jl}f_{\beta
M}^{\phantom{\beta M}NP},$ $\displaystyle A_{2ai}^{\phantom{2ai}j}$
$\displaystyle=$
$\displaystyle\epsilon^{\alpha\beta}\mathcal{V}_{\alpha}{\mathcal{V}_{a}}^{M}{\mathcal{V}_{ik}}^{N}\mathcal{V}_{P}^{\phantom{P}jk}f_{\beta
MN}^{\phantom{\beta MN}P}$ (17)

where $\mathcal{V}_{M}^{\phantom{M}ij}$ is defined in terms of the ’t Hooft
symbols $G^{ij}_{m}$ and $\mathcal{V}_{M}^{\phantom{M}m}$ as

$\mathcal{V}_{M}^{\phantom{M}ij}=\frac{1}{2}\mathcal{V}_{M}^{\phantom{M}m}G^{ij}_{m}\,.$
(18)

Similarly, the inverse elements ${\mathcal{V}_{ij}}^{M}$ can be written as

${\mathcal{V}_{ij}}^{M}=-\frac{1}{2}{\mathcal{V}_{m}}^{M}(G^{ij}_{m})^{*}\,.$
(19)

$G^{ij}_{m}$ satisfy the relations

$G_{mij}=(G^{ij}_{m})^{*}=\frac{1}{2}\epsilon_{ijkl}G^{kl}_{m}\,.$ (20)

An explicit representation of these matrices can be chosen as

$\displaystyle G_{1}^{ij}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cccc}0&1&0&0\\\ -1&0&0&0\\\ 0&0&0&1\\\
0&0&-1&0\\\ \end{array}\right],\qquad
G_{2}^{ij}=\left[\begin{array}[]{cccc}0&0&1&0\\\ 0&0&0&-1\\\ -1&0&0&0\\\
0&1&0&0\\\ \end{array}\right],$ (29) $\displaystyle G_{3}^{ij}$
$\displaystyle=$ $\displaystyle\left[\begin{array}[]{cccc}0&0&0&1\\\
0&0&1&0\\\ 0&-1&0&0\\\ -1&0&0&0\\\ \end{array}\right],\qquad
G_{4}^{ij}=\left[\begin{array}[]{cccc}0&i&0&0\\\ -i&0&0&0\\\ 0&0&0&-i\\\
0&0&i&0\\\ \end{array}\right],$ (38) $\displaystyle G_{5}^{ij}$
$\displaystyle=$ $\displaystyle\left[\begin{array}[]{cccc}0&0&i&0\\\
0&0&0&i\\\ -i&0&0&0\\\ 0&-i&0&0\\\ \end{array}\right],\qquad
G_{6}^{ij}=\left[\begin{array}[]{cccc}0&0&0&i\\\ 0&0&-i&0\\\ 0&i&0&0\\\
-i&0&0&0\\\ \end{array}\right].$ (47)

Finally, we also note that the scalar potential can be written in terms of the
fermion shift matrices $A_{1}$ and $A_{2}$ as

$V=-\frac{1}{3}A^{ij}_{1}A_{1ij}+\frac{1}{9}A^{ij}_{2}A_{2ij}+\frac{1}{2}A_{2ai}^{\phantom{2ai}j}A_{2a\phantom{i}j}^{\phantom{2a}i}\,.$
(48)

It is useful to remark here that upper and lower $i,j,\ldots$ indices are
related by complex conjugation.

###  2.1 $SO(4)\times SO(4)$ gauge group and $SO(2)\times SO(2)\times
SO(2)\times SO(2)$ truncation

We now consider the case of $n=6$ vector multiplets with $SO(4)\times SO(4)$
gauge group as constructed in [50]. With a slightly different notation for
various gauge coupling constants, the corresponding embedding tensor has the
following non-vanishing components

$\displaystyle f_{+\hat{m}\hat{n}\hat{p}}$ $\displaystyle=$ $\displaystyle
g_{1}\epsilon_{\hat{m}\hat{n}\hat{p}},\qquad
f_{+\hat{a}\hat{b}\hat{c}}=\tilde{g}_{1}\epsilon_{\hat{a}\hat{b}\hat{c}},$
$\displaystyle f_{-\tilde{m}\tilde{n}\tilde{p}}$ $\displaystyle=$
$\displaystyle g_{2}\epsilon_{\tilde{m}\tilde{n}\tilde{p}},\qquad
f_{-\tilde{a}\tilde{b}\tilde{c}}=\tilde{g}_{2}\epsilon_{\tilde{a}\tilde{b}\tilde{c}},$
(49)

in which we have used the convention on the $SO(6,6)$ fundamental index as
$M=(m,a)=(\hat{m},\tilde{m},\hat{a},\tilde{a})$ with $\hat{m}=1,2,3$,
$\tilde{m}=4,5,6$, $\hat{a}=7,8,9$ and $\tilde{a}=10,11,12$. We also note that
the two $SO(4)$ factors are electrically and magnetically embedded in
$SO(6,6)$ and will be denoted by $SO(4)_{+}\times SO(4)_{-}$. In terms of the
$SO(3)$ factors given by the embedding tensor in (49), we will write the gauge
group as $SO(3)_{+}\times SO(3)_{-}\times SO(3)_{+}\times SO(3)_{-}$ with the
first two factors embedded in the $SU(4)_{R}\sim SO(6)_{R}$.
To parametrize the $SO(6,6)/SO(6)\times SO(6)$ coset representative, we first
define $SO(6,n)$ generators in the fundamental representation by

$(t_{MN})_{P}^{\phantom{P}Q}=2\delta^{Q}_{[M}\eta_{N]P}\,.$ (50)

We then identify the $SO(6,6)$ non-compact generators as

$Y_{ma}=t_{m,a+6}\,.$ (51)

The truncation of this gauged supergravity to $SO(4)_{\textrm{diag}}\sim
SO(3)_{\textrm{diag}}\times SO(3)_{\textrm{diag}}$ singlet scalars has already
been studied in [17] in which a number of supersymmetric $AdS_{4}$ critical
points and domain walls interpolating between them have been given.
In the present case, we are interested in another truncation to $SO(2)\times
SO(2)\times SO(2)\times SO(2)$ invariant scalars. For later convenience, we
also note that the gauge generators are given by

$X_{\alpha M}={f_{\alpha M}}^{NP}t_{NP}\,.$ (52)

Each $SO(2)$ factor is embedded in the $SO(3)$ factor of $SO(3)_{+}\times
SO(3)_{-}\times SO(3)_{+}\times SO(3)_{-}\sim SO(4)_{+}\times SO(4)_{-}$ as
$\mathbf{3}\rightarrow\mathbf{2}+\mathbf{1}$. To identify these singlets, we
first consider the transformation of the scalars under the $SO(2)\times
SO(2)\times SO(2)\times SO(2)$ subgroup. The $36$ scalars transform under the
compact subgroup $SO(6)\times SO(6)$ as $(\mathbf{6},\mathbf{6})$. The
fundamental representation $\mathbf{6}$ of $SO(6)$ is in turn decomposed as
$(\mathbf{3},\mathbf{1})+(\mathbf{1},\mathbf{3})$ under $SO(3)\times SO(3)$.
With the aforementioned embedding of $SO(2)$ in $SO(3)$, we find the
transformation of all scalars under $SO(2)\times SO(2)\times SO(2)\times
SO(2)$

$(\mathbf{6},\mathbf{6})\rightarrow(\mathbf{2}+\mathbf{1},\mathbf{2}+\mathbf{1},\mathbf{1},\mathbf{1})+(\mathbf{2}+\mathbf{1},\mathbf{1},\mathbf{1},\mathbf{2}+\mathbf{1})+(\mathbf{1},\mathbf{2}+\mathbf{1},\mathbf{2}+\mathbf{1},\mathbf{1})+(\mathbf{1},\mathbf{1},\mathbf{2}+\mathbf{1},\mathbf{2}+\mathbf{1})$
(53)

from which we can easily see that there are four singlets corresponding to the
representation $(\mathbf{1},\mathbf{1},\mathbf{1},\mathbf{1})$. With the gauge
generators obtained from the embedding tensor given in (49), we can choose the
$SO(2)\times SO(2)\times SO(2)\times SO(2)$ generators to be $X_{+3}$,
$X_{-6}$, $X_{+9}$ and $X_{-12}$. We then identify non-compact generators
corresponding to these singlets as $Y_{33}$, $Y_{36}$, $Y_{63}$ and $Y_{66}$
in term of which the coset representative can be written as

$\mathcal{V}=e^{\phi_{1}Y_{33}}e^{\phi_{2}Y_{36}}e^{\phi_{3}Y_{63}}e^{\phi_{4}Y_{66}}\,.$
(54)

Together with the dilaton and axion from the gravity multiplet which are
$SO(4)\times SO(4)$ singlets, there are six scalars in the $SO(2)\times
SO(2)\times SO(2)\times SO(2)$ sector.
By a straightforward computation, we find a very simple scalar potential

$V=-\frac{1}{2}e^{-\phi}(g_{1}^{2}+e^{2\phi}g_{2}^{2}+g_{2}^{2}\chi^{2})-2g_{1}g_{2}\cosh\phi_{1}\cosh\phi_{2}\cosh\phi_{3}\cosh\phi_{4}\,.$
(55)

It can be readily verified that this potential admits a unique $AdS_{4}$
critical point at

$\phi=\ln\left[\frac{g_{1}}{g_{2}}\right]\qquad\textrm{and}\qquad\phi_{1}=\phi_{2}=\phi_{3}=\phi_{4}=\chi=0\,.$
(56)

The cosmological constant and $AdS_{4}$ radius are given by

$V_{0}=-3g_{1}g_{2}\qquad\textrm{and}\qquad
L=\sqrt{-\frac{3}{V_{0}}}=\frac{1}{\sqrt{g_{1}g_{2}}}\,.$ (57)

This is the maximally $N=4$ supersymmetric $AdS_{4}$ vacuum preserving the
full $SO(4)\times SO(4)$ gauge symmetry. By shifting the dilaton $\phi$ by a
constant, we can choose $g_{2}=g_{1}=g$ which make the dilaton vanish at the
$AdS_{4}$ critical point. It is useful to note that all scalars have mass
$m^{2}L^{2}=-2$ at this vacuum and are dual to operators of dimensions
$\Delta=1,2$ in the dual $N=4$ SCFT, see more detail in [17].

###  2.2 Superpotential and BPS equations

We are interested in two types of supersymmetric solutions in the forms of
holographic RG flows and Janus configurations. The former is described by the
standard (flat) domain wall with the metric

$ds^{2}=e^{2A(r)}dx^{2}_{1,2}+dr^{2}$ (58)

while the latter takes the form of a curved domain wall with an
$AdS_{3}$-sliced world-volume corresponding to the metric

$ds^{2}=e^{2A(r)}(e^{\frac{2\xi}{\ell}}dx^{2}_{1,1}+d\xi^{2})+dr^{2}\,.$ (59)

It should be noted that the Janus ansatz (59) is obtained from the domain wall
metric (58) by replacing the three-dimensional Minkowski metric $dx^{2}_{1,2}$
by the metric on $AdS_{3}$ with radius $\ell$. The flat domain wall is in turn
recovered from the Janus metric (59) in the limit $\ell\rightarrow\infty$.
To preserve the isometry of $dx_{1,2}^{2}$ or the $AdS_{3}$ metric, scalar
fields can depend only on the radial coordinate $r$. In the $SO(2)\times
SO(2)\times SO(2)\times SO(2)$ truncation, the kinetic term for all the six
scalars $\Phi^{r}=(\phi,\chi,\phi_{1},\phi_{2},\phi_{3},\phi_{4})$, with
$r,s=1,2,\ldots,6$, takes the form

$\displaystyle\mathcal{L}_{\textrm{kin}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}G_{rs}{\Phi^{r}}^{\prime}{\Phi^{s}}^{\prime}$ (60)
$\displaystyle=$ $\displaystyle-\frac{1}{4}(\phi^{\prime
2}+e^{-2\phi}\chi^{\prime 2})-\frac{1}{16}\left[6+\cosh
2(\phi_{2}-\phi_{3})\right.$ $\displaystyle\left.+\cosh
2(\phi_{2}+\phi_{3})+2\cosh 2\phi_{4}(\cosh 2\phi_{2}\cosh
2\phi_{3}-1)\right]\phi^{\prime 2}_{1}$
$\displaystyle-\cosh\phi_{2}\cosh\phi_{4}\sinh\phi_{3}\sinh\phi_{4}\phi^{\prime}_{1}\phi^{\prime}_{2}-\cosh\phi_{3}\cosh\phi_{4}\sinh\phi_{2}\sinh\phi_{4}\phi^{\prime}_{1}\phi^{\prime}_{3}$
$\displaystyle+\sinh\phi_{2}\sinh\phi_{3}\phi^{\prime}_{1}\phi^{\prime}_{4}-\frac{1}{2}\cosh^{2}\phi_{4}\phi^{\prime
2}_{2}-\frac{1}{2}\cosh^{2}\phi_{4}\phi^{\prime 2}_{3}-\frac{1}{2}\phi^{\prime
2}_{4}$

in which we have introduced a symmetric matrix $G_{rs}$. Throughout the paper,
we will denote the $r$-derivative by ′.
We now analyze the BPS equations obtained by setting fermionic supersymmetry
transformations to zero. We use Majorana representation for space-time gamma
matrices with all $\gamma^{\hat{\mu}}$ real and $\gamma_{5}$ purely imaginary.
This implies that left and right chiralities of fermions are simply related to
each other by complex conjugation. To proceed, we first note that for both
types of solutions, the variations $\delta\chi^{i}$ and
$\delta\lambda^{i}_{a}$ contain $\gamma^{\hat{r}}$ matrix due to the
$r$-dependence of scalar fields. We then impose the following projector

$\gamma^{\hat{r}}\epsilon^{i}=e^{i\Lambda}\epsilon_{i}$ (61)

with an $r$-dependent phase $\Lambda$. This projector relates the two
chiralities of $\epsilon^{i}$ and breaks half of the supersymmetry.
To complete the analysis, we consider the variation of the gravitini
$\delta\psi^{i}_{\mu}$. With the coset representative (54), we obtain the
$A_{1}^{ij}$ tensor of the form

$A^{ij}_{1}=\textrm{diag}(\alpha_{+},\alpha_{-},\alpha_{-},\alpha_{+})$ (62)

with

$\displaystyle\alpha_{\pm}$ $\displaystyle=$
$\displaystyle\frac{3}{4}e^{-\frac{\phi}{2}}\left[(g_{2}e^{\phi}\cosh\phi_{3}\pm
ig_{1}\sinh\phi_{1}\sinh\phi_{3})\cosh\phi_{4}\right.$ (63)
$\displaystyle+g_{1}\cosh\phi_{1}(\cosh\phi_{2}\pm
i\sinh\phi_{2}\sinh\phi_{4})+$
$\displaystyle\left.+ig_{2}\chi\cosh\phi_{3}\cosh\phi_{4}\right].$

For $\chi=0$, $\alpha_{+}$ and $\alpha_{-}$ are complex conjugate of each
other. The eigenvalue of $A_{1}^{ij}$ tensor corresponding to unbroken
supersymmetry will give a superpotential in term of which the scalar potential
can be written. From the structure of $A_{1}^{ij}$ given above, we find that
in general, solutions within the $SO(2)\times SO(2)\times SO(2)\times SO(2)$
truncation will preserve at most $N=2$ supersymmetry corresponding to
$\epsilon^{1,4}$ or $\epsilon^{2,3}$ depending on which eigenvalue among
$\alpha_{+}$ and $\alpha_{-}$ is chosen to be the superpotential.
By defining the superpotential $\mathcal{W}$ as

$\mathcal{W}=\frac{2}{3}\alpha$ (64)

and taking into account the projector (61), we can write the variations
$\delta\psi^{i}_{\mu}$ for $\mu=0,1,2$ in the RG flow case as

$A^{\prime}e^{i\Lambda}-\mathcal{W}=0\,.$ (65)

This condition is satisfied along $\epsilon^{1,4}$ and $\epsilon^{2,3}$ for
$\mathcal{W}=\mathcal{W}_{\pm}=\frac{2}{3}\alpha_{\pm}$, respectively. In the
following analysis, we will consider only $\mathcal{W}=\mathcal{W}_{+}$. The
final results show that the other choice $\mathcal{W}=\mathcal{W}_{-}$ is
related to this by a sign change in $\chi$, so the two options are equivalent.
Equation (65) implies that

$A^{\prime}=\pm W=\pm|\mathcal{W}|\qquad\textrm{and}\qquad
e^{i\Lambda}=\pm\frac{\mathcal{W}}{W}\,.$ (66)

Using the explicit phase factor $e^{i\Lambda}$ in term of $\mathcal{W}$ in
$\delta\chi^{i}$ and $\delta\lambda^{i}_{a}$ equations, we obtain the BPS
equations for scalar fields. In what follows, we will choose the upper sign
choice in order to relate the $AdS_{4}$ vacuum to the limit
$r\rightarrow\infty$.
Finally, the variation $\delta\psi_{r}^{i}$ leads to

$2{\epsilon^{\hat{i}}}^{\prime}-\mathcal{W}\gamma_{\hat{r}}\epsilon_{\hat{i}}=0,\qquad\hat{i}=1,4$
(67)

which, by the projector (61) and equation (65), gives the Killing spinors of
the form

$\epsilon^{\hat{i}}=e^{\frac{A}{2}}\epsilon^{\hat{i}}_{0}$ (68)

for constant spinors $\epsilon_{0}^{\hat{i}}$ satisfying (61).
For Janus solutions, we can perform a similar analysis with some
modifications. The variations $\delta\psi^{\hat{i}}_{0,1}$ and
$\delta\psi^{\hat{i}}_{\xi}$ lead to the following conditions

$\displaystyle\left(A^{\prime}\gamma^{\hat{r}}+\frac{e^{-A}}{\ell}\gamma^{\hat{\xi}}\right)\epsilon^{\hat{i}}-\mathcal{W}\epsilon_{\hat{i}}$
$\displaystyle=$ $\displaystyle 0,$ (69) $\displaystyle
2\partial_{\xi}\epsilon^{\hat{i}}+A^{\prime}e^{A}\gamma^{\hat{\xi}}\gamma^{\hat{r}}\epsilon^{\hat{i}}-e^{A}\mathcal{W}\gamma^{\hat{\xi}}\epsilon_{\hat{i}}$
$\displaystyle=$ $\displaystyle 0\,.$ (70)

The analysis closely follows that given in [19] to which we refer for more
detail.
Equation (69) implies the integrability condition

$A^{\prime 2}+\frac{e^{-2A}}{\ell^{2}}=W^{2}\,.$ (71)

Using equation (69) in (70), we find

$2\partial_{\xi}\epsilon^{\hat{i}}=\frac{1}{\ell}\epsilon^{\hat{i}}$ (72)

which gives

$\epsilon^{\hat{i}}=e^{\frac{\xi}{2\ell}}\tilde{\epsilon}^{\hat{i}}$ (73)

with $\tilde{\epsilon}^{\hat{i}}$ being $\xi$-independent spinors.
It should be noted that condition (69) takes the form of a projector on
$\epsilon^{\hat{i}}$ which must be compatible with the projector (61) used in
the scalar flow equations. Following [19], we impose the projector

$\gamma^{\hat{\xi}}\epsilon^{\hat{i}}=i\kappa e^{i\Lambda}\epsilon_{\hat{i}}$
(74)

with $\kappa^{2}=1$. Using equations (61) and (74) in (69), we arrive at

$\left(A^{\prime}+\frac{i\kappa}{\ell}e^{-A}\right)e^{i\Lambda}=\mathcal{W}$
(75)

which can be used to determine the phase $e^{i\Lambda}$ in terms of
$\mathcal{W}$ and $A$. We note again that in the limit
$\ell\rightarrow\infty$, the phase factor reduces to that given in (66).
Finally, with all the previous results, the condition
$\delta\psi^{\hat{i}}_{r}=0$ gives

$2\partial_{r}\epsilon^{\hat{i}}=A^{\prime}\epsilon^{\hat{i}}+\frac{i\kappa}{\ell}e^{-A}\epsilon^{\hat{i}}\,.$
(76)

By redefining the Killing spinors with the phase factor
$e^{\frac{i\Lambda}{2}}$, this condition implies the following form of the
Killing spinors

$\epsilon^{\hat{i}}=e^{\frac{A}{2}+\frac{r}{2\ell}+\frac{i\Lambda}{2}}\epsilon_{0}^{\hat{i}}$
(77)

in which $\epsilon_{0}^{\hat{i}}$ can include an $r$-dependent phase and
satisfy the $\gamma^{\hat{r}}$ and $\gamma^{\hat{\xi}}$ projectors without the
phase $e^{i\Lambda}$

$\gamma^{\hat{r}}\epsilon^{\hat{i}}_{0}=\epsilon_{0\hat{i}}\qquad\textrm{and}\qquad\gamma^{\hat{\xi}}\epsilon^{\hat{i}}_{0}=i\kappa\epsilon_{0\hat{i}}\,.$
(78)

We also note that the constant $\kappa=\pm 1$ is related to the chirality of
the Killing spinors on the two-dimensional interface dual to the $AdS_{3}$
slice.
With all these results, we will look for supersymmetric RG flow and Janus
solutions within the $SO(2)\times SO(2)\times SO(2)\times SO(2)$ truncation of
the $SO(4)\times SO(4)$ $N=4$ gauged supergravity.

##  3 Holographic RG flows

We are now in a position to present an explicit form of the BPS equations
given in the previous section and look for possible solutions. In this
section, we consider RG flow solutions. Since there is only one supersymmetric
$AdS_{4}$ critical point dual to a single conformal fixed point, the solutions
will describe RG flows from the dual $N=4$ SCFT to non-conformal phases in the
IR. This type of solutions in the case of pure $N=4$ gauged supergravity, with
all scalars but the dilaton and axion vanishing, has already been studied in
[17]. In this paper, we will find solutions in the matter-coupled $N=4$ gauged
supergravity.
As previously mentioned, the flow equations for scalars can be obtained by
using the projector (61) and the phase $e^{i\Lambda}$ given in (66). Within
the $SO(2)\times SO(2)\times SO(2)\times SO(2)$ truncation, the variation
$\delta\chi^{i}$ reduces to a single complex equation which can be
straightforwardly solved for $\phi^{\prime}$ and $\chi^{\prime}$. However, it
turns out that consistency of the equations from the variation
$\delta\lambda^{i}_{a}$ requires some scalars to vanish. This is similar to
the $SO(3)\times SO(3)$ truncation studied in [17] in which the BPS equations
from $\delta\lambda^{i}_{a}$ imply $\chi=0$, and non-vanishing $\chi$ is
possible only when all scalars from the vector multiplets are set to zero. In
that case, the axion and vector multiplet scalars cannot be turned on
simultaneously.
In the present case, the constraint is somewhat weaker. We find that a
consistent set of BPS equations that are compatible with the second-order
field equations can be obtained only for two of the scalars $\phi_{i}$,
$i=1,2,3,4$ vanishing. This leads to $N=2$ supersymmetric solutions. In
addition, if the axion $\chi$ vanishes, supersymmetry is enhanced to $N=4$. As
expected, truncating out all scalars from the vector multiplets also leads to
$N=4$ supersymmetry. We then see that in the presence of both axion and vector
multiplet scalars, the solutions break some supersymmetry. We also point out
that this is not possible in the $SO(3)\times SO(3)$ truncation considered in
[17] since in that case the $A^{ij}_{1}$ tensor is proportional to the
identity matrix. Therefore, partial supersymmetry breaking is not possible
there.

###  3.1 $N=4$ holographic RG flow with $SO(2)\times SO(3)\times SO(2)\times
SO(2)$ symmetry

We first consider $N=4$ solutions with $\chi=0$ and $SO(2)\times SO(3)\times
SO(2)\times SO(2)$ symmetry. In this case, the $SO(4)$ subgroup of the $SO(6)$
R-symmetry is broken to $SO(2)\times SO(3)$ while the remaining $SO(2)\times
SO(2)$ factor is a subgroup of $SO(3)\times SO(3)\sim SO(4)$ symmetry of the
vector multiplets. There are two possibilities namely setting
$\phi_{1}=\phi_{2}=0$ or $\phi_{3}=\phi_{4}=0$. The two choices are related to
each other by interchanging $e^{\phi}$ with $e^{-\phi}$ factors. Therefore, we
will consider only the solution with $\phi_{3}=\phi_{4}=0$.
The superpotential is real and given by

$W=W_{+}=W_{-}=\frac{1}{2}(g_{2}e^{\frac{\phi}{2}}+g_{1}e^{-\frac{\phi}{2}}\cosh\phi_{1}\cosh\phi_{2}).$
(79)

The resulting BPS equations read

$\displaystyle\phi^{\prime}$ $\displaystyle=$ $\displaystyle-4\frac{\partial
W}{\partial\phi}=g_{1}e^{-\frac{\phi}{2}}\cosh\phi_{1}\cosh\phi_{2}-g_{2}e^{\frac{\phi}{2}},$
(80) $\displaystyle\phi_{1}^{\prime}$ $\displaystyle=$
$\displaystyle-2\textrm{sech}^{2}\phi_{2}\frac{\partial
W}{\partial\phi_{1}}=-g_{1}e^{-\frac{\phi}{2}}\textrm{sech}\phi_{2}\sinh\phi_{1},$
(81) $\displaystyle\phi_{2}^{\prime}$ $\displaystyle=$
$\displaystyle-2\frac{\partial
W}{\partial\phi_{2}}=-g_{1}e^{-\frac{\phi}{2}}\cosh\phi_{1}\sinh\phi_{2},$
(82) $\displaystyle A^{\prime}$ $\displaystyle=$ $\displaystyle
W=\frac{1}{2}(g_{2}e^{\frac{\phi}{2}}+g_{1}e^{-\frac{\phi}{2}}\cosh\phi_{1}\cosh\phi_{2}).$
(83)

To find the solution, we first take the following combination

$\frac{d\phi_{1}}{d\phi_{2}}=\textrm{csch}\phi_{2}\textrm{sech}\phi_{2}\tanh\phi_{1}$
(84)

with the solution

$\sinh\phi_{1}=\tan\varphi_{0}\tanh\phi_{2}$ (85)

for a constant $\varphi_{0}$. Similarly, taking the combination between
$\phi^{\prime}$ and $\phi_{2}^{\prime}$ equations together with this result
leads to

$\frac{d\phi}{d\phi_{2}}=\frac{g_{2}e^{\phi}\textrm{csch}\phi_{2}}{g_{1}\sqrt{1+\tan\varphi_{0}^{2}\tanh^{2}\phi_{2}}}-\coth\phi_{2}\,.$
(86)

The solution to this equation is given by

$\phi=\ln\left[\frac{2\sqrt{2}g_{1}\cos^{2}\varphi_{0}\textrm{csch}\phi_{2}(\cos\varphi_{0}\textrm{csch}\phi_{2}\sqrt{\cos
2\varphi_{0}+\cosh 2\phi_{2}}-C)}{g_{2}[1-2C^{2}+\cos
2\varphi_{0}+2\cos^{2}\varphi_{0}(\coth^{2}\phi_{2}+\cos
2\varphi_{0}\textrm{csch}^{2}\phi_{2})]}\right].$ (87)

Using these results and redefining the radial coordinate to $\rho$ given by
$\frac{d\rho}{dr}=e^{-\frac{\phi}{2}}$, we obtain the solution for
$\phi_{2}(\rho)$ of the form

$\cosh
2\phi_{2}=2\cos^{2}\varphi_{0}\tanh\left[g_{1}\sec\varphi_{0}(\rho-\rho_{0})\right]-\cos
2\varphi_{0}$ (88)

with an integration constant $\rho_{0}$.
Finally, taking a linear combination $2A^{\prime}+\phi^{\prime}$, we find

$2\frac{dA}{d\phi_{2}}+\frac{d\phi}{d\phi_{2}}=-\frac{2(e^{2\phi_{2}}+1)}{e^{2\phi_{2}}-1}$
(89)

with the solution

$A=\phi_{2}-\frac{1}{2}\phi-\ln(1-e^{2\phi_{2}}).$ (90)

We have neglected an additive integration constant in $A$ since this can be
removed by rescaling coordinates on $dx^{2}_{1,2}$.
We now look at asymptotic behaviors of the solution. For convenience, we first
set $g_{2}=g_{1}$ to bring the $AdS_{4}$ critical point to the origin of the
scalar manifold. As $r\rightarrow\infty$, we find

$\phi\sim\phi_{1}\sim\phi_{2}\sim e^{-g_{1}r}\sim e^{-\frac{r}{L}},\qquad
A\sim g_{1}r\sim\frac{r}{L}\,.$ (91)

This gives the supersymmetric $AdS_{4}$ vacuum and indicates that all scalars
are dual to relevant operators of dimensions $\Delta=1,2$.
As usual for a flow to non-conformal phases, the solution is singular for a
finite value of $\rho$. We find that, for $C<0$, the above solution is
singular at $\rho=\rho_{0}+\rho_{*}$ with

$\rho_{*}=\frac{1}{g_{1}}\coth^{-1}\left[\sec\varphi_{0}\sqrt{\frac{\cos
2\varphi_{0}+\cosh\left(2\coth^{-1}\sqrt{1-\sec^{2}\varphi_{0}+\frac{1}{2}C^{2}\sec^{4}\varphi_{0}}\right)}{2}}\right].$
(92)

Near the singularity, we find

$\displaystyle\phi_{2}=\phi_{*}=\coth^{-1}\sqrt{1-\sec^{2}\varphi_{0}+\frac{1}{2}C^{2}\sec^{4}\varphi_{0}},$
$\displaystyle\phi_{1}=\sinh^{-1}(\tan\varphi_{0}\tanh\phi_{*}),\qquad\phi\sim-\ln(\phi_{2}-\phi_{*}),$
$\displaystyle A\sim-\frac{\phi}{2}\sim\frac{1}{2}\ln(\phi_{2}-\phi_{*})$ (93)

and

$V\sim-g_{2}^{2}e^{\phi}\rightarrow-\infty\,.$ (94)

We see that near the singularity both $\phi_{1}$ and $\phi_{2}$ are constant,
and the singularity is physical by the criterion given in [51]. Therefore, the
solution describes an RG flow from $N=4$ SCFT to a non-conformal phase in the
IR with the $N=4$ superconformal symmetry broken to $N=4$ Poincare
supersymmetry in three dimensions. The $SO(4)\times SO(4)$ symmetry is also
broken to $SO(2)\times SO(3)\times SO(2)\times SO(2)$ along the flows.

###  3.2 $N=4$ holographic RG flow with $SO(2)\times SO(2)\times SO(2)\times
SO(2)$ symmetry

Unlike the previous case, if we set $\phi_{2}=\phi_{3}=0$ or
$\phi_{1}=\phi_{4}=0$ along with $\chi=0$, we obtain $N=4$ solutions with only
$SO(2)\times SO(2)\times SO(2)\times SO(2)$ symmetry. The two choices give
rise to equivalent results with scalars $(\phi_{1},\phi_{4})$ and
$(\phi_{2},\phi_{3})$ interchanged. We will consider only the
$\phi_{2}=\phi_{3}=0$ case for definiteness.
Setting $\chi=\phi_{2}=\phi_{3}=0$, we again find a real superpotential

$\mathcal{W}=\frac{1}{2}(g_{1}e^{-\frac{\phi}{2}}\cosh\phi_{1}+g_{2}e^{\frac{\phi}{2}}\cosh\phi_{4}).$
(95)

By repeating the same procedure as in the previous case, we find the following
BPS equations

$\displaystyle A^{\prime}$ $\displaystyle=$ $\displaystyle
W=\frac{1}{2}(g_{1}e^{-\frac{\phi}{2}}\cosh\phi_{1}+g_{2}e^{\frac{\phi}{2}}\cosh\phi_{4}),$
(96) $\displaystyle\phi^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{\partial
W}{\partial\phi}=g_{1}e^{-\frac{\phi}{2}}\cosh\phi_{1}-g_{2}e^{\frac{\phi}{2}}\cosh\phi_{4},$
(97) $\displaystyle\phi_{1}^{\prime}$ $\displaystyle=$
$\displaystyle-2\frac{\partial
W}{\partial\phi_{1}}=-g_{1}e^{-\frac{\phi}{2}}\sinh\phi_{1},$ (98)
$\displaystyle\phi_{4}^{\prime}$ $\displaystyle=$
$\displaystyle-2\frac{\partial
W}{\partial\phi_{4}}=-g_{2}e^{\frac{\phi}{2}}\sinh\phi_{4}\,.$ (99)

Unlike the previous case, solving these equations is slightly more complicated
due to the different dilaton prefactors in $\phi_{1}^{\prime}$ and
$\phi_{4}^{\prime}$ equations. However, with some effort, we can find an
analytic solution to these equations.
We first treat $\phi_{4}$ as an independent variable and take a combination

$\frac{d\phi_{1}}{d\phi_{4}}=\frac{g_{1}}{g_{2}}e^{-\phi}\textrm{csch}\phi_{4}\sinh\phi_{1}\,.$
(100)

By changing to a new variable $\varphi$ defined by

$\phi_{1}=\ln\sinh\frac{\varphi}{2}-\ln\cosh\frac{\varphi}{2},$ (101)

we can rewrite the above equation as

$\frac{d\varphi}{d\phi_{4}}=\frac{g_{1}}{g_{2}}e^{-\phi}\textrm{csch}\phi_{4}\,.$
(102)

We then combine $\phi^{\prime}$ and $\phi_{4}^{\prime}$ equations with the
$e^{-\phi}$ factor obtained from (102) and arrive at

$\frac{d\phi}{d\phi_{4}}=\coth\phi_{4}+\coth\varphi\frac{d\varphi}{d\phi_{4}}$
(103)

with the solution

$\phi=\phi_{0}+\ln\sinh\phi_{4}+\ln\sinh\varphi\,.$ (104)

We can now use this solution in (102) and find the solution

$\cosh\varphi=\frac{g_{1}}{g_{2}}e^{-\phi_{0}}\coth\phi_{4}\,.$ (105)

Taking a linear combination $2A^{\prime}-\phi^{\prime}$ gives

$2\frac{dA}{d\phi_{4}}-\frac{d\phi}{d\phi_{4}}=-2\coth\phi_{4}$ (106)

with the solution

$A=\frac{1}{2}\phi-\ln\sinh\phi_{4}\,.$ (107)

Finally, changing $r$ to a new coordinate $\rho$ given by
$\frac{d\rho}{dr}=e^{\frac{\phi}{2}}$, we can readily find $\phi_{4}(\rho)$
solution

$\phi_{4}=\ln(1+e^{-g_{2}(\rho-\rho_{0})})-\ln(1-e^{-g_{2}(\rho-\rho_{0})}).$
(108)

As in the previous case, for $r\rightarrow\infty$, we find

$\phi\sim\phi_{1}\sim\phi_{4}\sim e^{-g_{1}r}\sim e^{-\frac{r}{L}}$ (109)

in which we have taken $g_{2}=g_{1}$ for convenience. This is an asymptotic
$AdS_{4}$ geometry.
As $\rho\rightarrow\rho_{0}$, the solution is singular with

$\displaystyle\phi_{1}\sim\textrm{constant},\qquad\phi_{4}\sim\ln(\rho-\rho_{0}),$
$\displaystyle\phi\sim-\ln(\rho-\rho_{0}),\qquad
A\sim\frac{1}{2}\ln(\rho-\rho_{0}).$ (110)

Unlike the previous case, the scalars from vector multiplets are not all
constant near the singularity. Using the scalar potential given in (55), we
find that, as $\rho\rightarrow\rho_{0}$,

$V\sim-\frac{g_{1}^{2}}{(\rho-\rho_{0})}\rightarrow-\infty\,.$ (111)

The singularity is again physically acceptable, and the solution could be
interpreted as an RG flow from the $N=4$ SCFT to a non-conformal phase in the
IR driven by relevant operators of dimensions $\Delta=1,2$.

###  3.3 $N=2$ holographic RG flow with $SO(2)\times SO(2)\times SO(3)\times
SO(2)$ symmetry

We now consider the most general solutions in the $SO(2)\times SO(2)\times
SO(2)\times SO(2)$ truncation with non-vanishing $\chi$ and
$\phi_{1}=\phi_{3}=0$. In this case, the $SO(4)\subset SO(6)_{R}$ is broken to
$SO(2)\times SO(2)$ while the $SO(4)$ subgroup of the $SO(6)$ symmetry of the
vector multiplets is broken to $SO(3)\times SO(2)$, or $SO(2)\times SO(3)$ if
we choose $\phi_{2}=\phi_{4}=0$ rather than $\phi_{1}=\phi_{3}=0$.
With $\phi_{1}=\phi_{3}=0$, the superpotential is given by

$\mathcal{W}=\mathcal{W}_{+}=\frac{1}{2}\left[g_{1}e^{-\frac{\phi}{2}}(\cosh\phi_{2}+i\sinh\phi_{2}\sinh\phi_{4})+g_{2}e^{\frac{\phi}{2}}\cosh\phi_{4}+ig_{2}e^{-\frac{\phi}{2}}\chi\cosh\phi_{4}\right]$
(112)

in which for definiteness, we have chosen $\mathcal{W}_{+}$ to be the
superpotential with $\epsilon^{1}$ and $\epsilon^{4}$ being the corresponding
Killing spinors. If we choose $\mathcal{W}=\mathcal{W}_{-}$, the unbroken
supersymmetry will correspond to $\epsilon^{2}$ and $\epsilon^{3}$.
With the same analysis as in the previous cases, we find the following BPS
equations

$\displaystyle A^{\prime}$ $\displaystyle=$ $\displaystyle W$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sqrt{(g_{1}e^{-\frac{\phi}{2}}\cosh\phi_{2}+g_{2}e^{\frac{\phi}{2}}\cosh\phi_{4})^{2}+e^{-\phi}(g_{1}\sinh\phi_{2}\sinh\phi_{4}+g_{2}\cosh\phi_{4}\chi)^{2}},$
$\displaystyle\phi^{\prime}$ $\displaystyle=$ $\displaystyle-4\frac{\partial
W}{\partial\phi}$ (114) $\displaystyle=$
$\displaystyle\frac{1}{8W}e^{-\phi}\left[g_{1}^{2}(3+\cosh 2\phi_{2}+2\cosh
2\phi_{4}\sinh^{2}\phi_{2})-4g_{2}^{2}e^{2\phi}\cosh^{2}\phi_{4}\right.$
$\displaystyle\left.+4g_{2}\chi(g_{1}\sinh\phi_{2}\sinh
2\phi_{4}+g_{2}\chi\cosh^{2}\phi_{4})\right],$ $\displaystyle\chi^{\prime}$
$\displaystyle=$ $\displaystyle-4e^{2\phi}\frac{\partial
W}{\partial\chi}=-\frac{e^{\phi}}{W}g_{2}\cosh\phi_{4}(g_{1}\sinh\phi_{2}\sinh\phi_{4}+g_{2}\cosh\phi_{4}\chi),$
(115) $\displaystyle\phi_{2}^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{2}{\cosh^{2}\phi_{4}}\frac{\partial W}{\partial\phi_{2}}$
(116) $\displaystyle=$
$\displaystyle-\frac{1}{2W}e^{-\phi}g_{1}\left[\sinh\phi_{2}(g_{1}\cosh\phi_{2}+g_{2}e^{\phi}\textrm{sech}\phi_{4})+g_{2}\chi\cosh\phi_{2}\tanh\phi_{4}\right],\quad\,\,\,$
$\displaystyle\phi_{4}^{\prime}$ $\displaystyle=$
$\displaystyle-2\frac{\partial W}{\partial\phi_{4}}$ (117) $\displaystyle=$
$\displaystyle-\frac{1}{4W}e^{-\phi}\left[\sinh
2\phi_{4}(g_{2}^{2}e^{2\phi}+g_{2}^{2}\chi^{2}+g_{1}^{2}\sinh^{2}\phi_{2})\
\right.$
$\displaystyle\left.+2g_{1}g_{2}e^{\phi}\cosh\phi_{2}\sinh\phi_{4}+2g_{1}g_{2}\chi\cosh
2\phi_{4}\sinh\phi_{2}\right].$

In this case, we are not able to find analytic solutions, so we will instead
perform a numerical analysis for solving these equations. By choosing the
numerical values of the coupling constants $g_{1}=g_{2}=\sqrt{2}g_{0}$, we
obtain examples of numerical RG flow solutions for suitable boundary
conditions and different values of the parameter $g_{0}$ as shown in figures
1, 2 and 3. From these figures, we see some patterns namely, in all of these
solutions, $\phi_{2}$ is constant near the singularities as in the previous
cases, and all the singularities are physically acceptable. The former indeed
seems to be a general feature at least among all the solutions we have found.
The latter follows directly from the scalar potential (55) for $g_{2}=g_{1}$,

$V=-\frac{1}{2}g_{1}^{2}e^{-\phi}(1+e^{2\phi}+\chi^{2}+4e^{\phi}\cosh\phi_{2}\cosh\phi_{4})$
(118)

from which we immediately see that $V\rightarrow-\infty$ for any diverging
behaviors of all the scalar fields. This implies that all possible
singularities in this $SO(2)\times SO(2)\times SO(3)\times SO(2)$ truncation
are physical. Accordingly, the corresponding solutions describe RG flows from
the dual $N=4$ SCFT to various non-conformal phases in the IR.

(a) $\phi(r)$ solution

(b) $\chi(r)$ solution

(c) $\phi_{2}(r)$ solution

(d) $\phi_{4}(r)$ solution

(e) $A(r)$ solution

(f) Scalar potential

Figure 1: An $N=2$ RG flow from the $N=4$ SCFT in the UV to a non-conformal
phase in the IR for $g_{1}=g_{2}=\frac{1}{\sqrt{2}}$.

(a) $\phi(r)$ solution

(b) $\chi(r)$ solution

(c) $\phi_{2}(r)$ solution

(d) $\phi_{4}(r)$ solution

(e) $A(r)$ solution

(f) Scalar potential

Figure 2: An $N=2$ RG flow from the $N=4$ SCFT in the UV to a non-conformal
phase in the IR for $g_{1}=g_{2}=\frac{3\sqrt{2}}{2}$.

(a) $\phi(r)$ solution

(b) $\chi(r)$ solution

(c) $\phi_{2}(r)$ solution

(d) $\phi_{4}(r)$ solution

(e) $A(r)$ solution

(f) Scalar potential

Figure 3: An $N=2$ RG flow from the $N=4$ SCFT in the UV to a non-conformal
phase in the IR for $g_{1}=g_{2}=\frac{5\sqrt{2}}{2}$.

##  4 Supersymmetric Janus solutions

In this section, we look at supersymmetric Janus solutions with $N=4$ and
$N=2$ supersymmetries. In this case, the resulting BPS equations are much more
complicated than those in the RG flow case, and we are not able to obtain any
analytic solutions.

###  4.1 $N=4$ Janus solutions with $SO(4)\times SO(4)$ symmetry

We first consider solutions with all $\phi_{i}=0$ that can be regarded as
solutions of pure $N=4$ gauged supergravity with $SO(4)\sim SO(3)\times SO(3)$
gauge group. In this case, we find a simple superpotential

$\mathcal{W}=\frac{1}{2}e^{-\frac{\phi}{2}}[g_{1}+g_{2}(e^{\phi}+i\chi)].$
(119)

Using the prescription given in section 2, we obtain the following BPS
equations

$\displaystyle\phi^{\prime}$ $\displaystyle=$
$\displaystyle-4\frac{A^{\prime}}{W}\frac{\partial
W}{\partial\phi}-4\kappa\frac{e^{-A}}{\ell W}e^{\phi}\frac{\partial
W}{\partial\chi}$ (120) $\displaystyle=$ $\displaystyle\frac{2\ell
A^{\prime}(g_{1}^{2}-g_{2}^{2}e^{2\phi}+g_{2}^{2}\chi^{2})-4\kappa
g_{2}^{2}\chi e^{\phi-A}}{\ell[(g_{1}+g_{2}e^{\phi})^{2}+g_{2}^{2}\chi^{2}]},$
$\displaystyle\chi^{\prime}$ $\displaystyle=$
$\displaystyle-4e^{2\phi}\frac{A^{\prime}}{W}\frac{\partial
W}{\partial\chi}+4\kappa\frac{e^{-A}}{\ell W}e^{\phi}\frac{\partial
W}{\partial\phi}$ (121) $\displaystyle=$ $\displaystyle-\frac{4\ell
g_{2}^{2}A^{\prime}\chi e^{2\phi}-2\kappa
e^{\phi-A}(g_{1}^{2}-g_{2}^{2}e^{2\phi}+g_{2}^{2}\chi^{2})}{\ell[(g_{1}+g_{2}e^{\phi})^{2}+g_{2}^{2}\chi^{2}]},$
$\displaystyle A^{\prime 2}$ $\displaystyle=$
$\displaystyle\frac{1}{4}e^{-\phi}[(g_{1}+g_{2}e^{\phi})^{2}+g_{2}^{2}\chi^{2}]-\frac{e^{-2A}}{\ell^{2}}\,.$
(122)

As expected, in the limit $\ell\rightarrow\infty$, these equations reduce to
those of the RG flows studied in [17] after a redefinition of the scalars and
coupling constants. It should also be noted that setting $\chi=0$ is not
consistent unless $\ell\rightarrow\infty$. Therefore, Janus solutions are
possible only for non-vanishing pseudoscalars needed to support the curvature
of the $AdS_{3}$-sliced world-volume. This has also been pointed out in [19].
We now perform a numerical analysis and give some examples of supersymmetric
Janus solutions as shown in figure 4. These solutions preserve half of the
original $N=4$ supersymmetry, or eight supercharges, with all $\epsilon^{i}$
non-vanishing and subject to the projector (61). On the conformal interfaces,
the Killing spinors can have left or right chiralities depending on whether
the values of $\kappa=1$ or $\kappa=-1$. The two-dimensional interfaces then
preserve $(4,0)$ or $(0,4)$ supersymmetries. In the numerical analysis, we
have conventionally set $\kappa=1$.

(a) $\phi(r)$ solutions

(b) $\chi(r)$ solutions

(c) $A(r)$ solutions

Figure 4: $N=4$ supersymmetric Janus solutions from $N=4$ $SO(4)\times SO(4)$
gauged supergravity for $g_{1}=g_{2}=2\sqrt{2}$ and $\ell=\kappa=1$ with
different boundary conditions.

###  4.2 $N=2$ Janus solutions with $SO(2)\times SO(2)\times SO(3)\times
SO(2)$ symmetry

We now look at more complicated solutions in matter-coupled $N=4$ gauged
supergravity. We first note that Janus solutions are not possible for all the
subtruncations with $\chi=0$. We then consider the case of $N=2$ supersymmetry
and $SO(2)\times SO(2)\times SO(3)\times SO(2)$ symmetry with
$\phi_{1}=\phi_{3}=0$ and $\epsilon^{2}=\epsilon^{3}=0$. The other equivalent
cases can be obtained by replacing $(\phi_{2},\phi_{4})$ and/or
$(\epsilon^{1},\epsilon^{4})$ by $(\phi_{1},\phi_{3})$ and/or
$(\epsilon^{2},\epsilon^{3})$.
With the superpotential (112) and the same procedure as in the previous case,
we find the following BPS equations

$\displaystyle\phi_{2}^{\prime}$ $\displaystyle=$
$\displaystyle-2\textrm{sech}^{2}\phi_{4}\frac{A^{\prime}}{W}\frac{\partial
W}{\partial\phi_{2}}-2\textrm{sech}\phi_{4}\frac{\kappa e^{-A}}{\ell
W}\frac{\partial W}{\partial\phi_{4}},$ (123) $\displaystyle\phi_{4}^{\prime}$
$\displaystyle=$ $\displaystyle-\frac{A^{\prime}}{W}\frac{\partial
W}{\partial\phi_{4}}+2\textrm{sech}\phi_{4}\frac{\kappa e^{-A}}{\ell
W}\frac{\partial W}{\partial\phi_{2}},$ (124) $\displaystyle\phi^{\prime}$
$\displaystyle=$ $\displaystyle-4\frac{A^{\prime}}{W}\frac{\partial
W}{\partial\phi}-4e^{\phi}\frac{\kappa e^{-A}}{\ell W}\frac{\partial
W}{\partial\chi},$ (125) $\displaystyle\chi^{\prime}$ $\displaystyle=$
$\displaystyle-4e^{2\phi}\frac{A^{\prime}}{W}\frac{\partial
W}{\partial\chi}+4e^{\phi}\frac{\kappa e^{-A}}{\ell W}\frac{\partial
W}{\partial\phi},$ (126) $\displaystyle A^{\prime 2}$ $\displaystyle=$
$\displaystyle W^{2}-\frac{e^{-2A}}{\ell^{2}}$ (127)

with

$\displaystyle\frac{1}{W}\frac{\partial W}{\partial\phi_{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{4W^{2}}g_{1}e^{-\phi}\cosh\phi_{4}\left[\sinh\phi_{2}(g_{2}e^{\phi}+g_{1}\cosh\phi_{2}\cosh\phi_{4})\right.$
(128) $\displaystyle\left.+g_{2}\chi\cosh\phi_{2}\sinh\phi_{4}\right],$
$\displaystyle\frac{1}{W}\frac{\partial W}{\partial\phi_{4}}$ $\displaystyle=$
$\displaystyle\frac{1}{4W^{2}}\left[e^{-\phi}(g_{1}\sinh\phi_{2}\sinh\phi_{4}\right.$
(129)
$\displaystyle+g_{2}\chi\cosh\phi_{4})(g_{1}\cosh\phi_{4}\sinh\phi_{2}+g_{2}\chi\sinh\phi_{4})$
$\displaystyle\left.g_{2}\sinh\phi_{4}(g_{1}\cosh\phi_{2}+g_{2}e^{\phi}\cosh\phi_{4})\right],$
$\displaystyle\frac{1}{W}\frac{\partial W}{\partial\phi}$ $\displaystyle=$
$\displaystyle\frac{1}{8W^{2}}\left[2g_{2}\cosh\phi_{4}(g_{1}\cosh\phi_{2}+g_{2}e^{\phi}\cosh\phi_{4})\right.$
(130)
$\displaystyle-e^{-\phi}(g_{1}\sinh\phi_{2}\sinh\phi_{4}+g_{2}\chi\cosh\phi_{4})^{2}$
$\displaystyle\left.-e^{-\phi}(g_{1}\cosh\phi_{2}+g_{2}e^{\phi}\cosh\phi_{4})^{2}\right],$
$\displaystyle\frac{1}{W}\frac{\partial W}{\partial\chi}$ $\displaystyle=$
$\displaystyle\frac{1}{4W^{2}}g_{2}e^{-\phi}\cosh\phi_{4}(g_{1}\sinh\phi_{2}\sinh\phi_{4}+g_{2}\chi\cosh\phi_{4})\,.$
(131)

Examples of numerical solutions are given in figure 5. In the figure, the
solutions for the warped factor $A$ are very close to each other. In this
case, the solutions describe conformal interfaces with $N=(2,0)$ or $N=(0,2)$
supersymmetries corresponding respectively to $\kappa=1$ or $\kappa=-1$ and
could provide a holographic description of deformations by position-dependent
operators within the dual $N=4$ SCFT.

(a) $\phi(r)$ solutions

(b) $\chi(r)$ solutions

(c) $\phi_{2}(r)$ solutions

(d) $\phi_{4}(r)$ solutions

(e) $A(r)$ solutions

Figure 5: $N=2$ supersymmetric Janus solutions from $N=4$ $SO(4)\times SO(4)$
gauged supergravity for $g_{1}=g_{2}=2\sqrt{2}$, $\ell=2$ and $\kappa=1$ with
different boundary conditions.

##  5 Conclusions and discussions

In this paper, we have studied a number of holographic RG flows from matter-
coupled $N=4$ gauged supergravity with $SO(4)\times SO(4)$ gauge group by
truncating to $SO(2)\times SO(2)\times SO(2)\times SO(2)$ singlet scalars. For
vanishing axion, there exist $N=4$ supersymmetric flow solutions with
$SO(2)\times SO(3)\times SO(2)\times SO(2)$ and $SO(2)\times SO(2)\times
SO(2)\times SO(2)$ symmetries. In the presence of axion, the solutions
preserve only $N=2$ supersymmetry and $SO(2)\times SO(2)\times SO(3)\times
SO(2)$ symmetry. All of these solutions are singular at finite values of the
radial coordinate, but all the singularities turn out to be physically
acceptable according to the criterion given in [51]. The solutions can be
interpreted as RG flows from the $N=4$ SCFT, dual to the unique $N=4$
supersymmetric $AdS_{4}$ vacuum within this truncation, to non-conformal
phases in the IR with different global symmetries. All of these solutions
could hopefully be useful in studying various mass deformations of strongly
coupled $N=4$ SCFT via AdS/CFT holography.
We have also found $N=4$ and $N=2$ supersymmetric Janus solutions with
$SO(4)\times SO(4)$ and $SO(2)\times SO(2)\times SO(3)\times SO(2)$
symmetries. The former is obtained by truncating out all scalars from the
vector multiplets resulting in the solutions of pure $N=4$ gauged supergravity
while the latter is a genuine solution of the matter-coupled theory. These
solutions holographically describe two-dimensional conformal interfaces in the
dual $N=4$ SCFT with unbroken $N=(4,0)$ or $N=(2,0)$ supersymmetries on the
interfaces. Generally, these configurations are identified with deformations
of the dual SCFT by operators or vacuum expectation values that depend on the
coordinate transverse to the interfaces.
It would be interesting to explicitly identify the dual $N=4$ SCFT together
with relevant deformations dual to the solutions found here. Uplifting the
$SO(4)\times SO(4)$ gauged supergravity considered here to higher dimensions
will allow to embed these solutions in ten or eleven dimensions giving rise to
new examples of AdS/CFT duality in the context of string/M-theory. In
particular, it could be interesting to check whether the singularities allowed
by the criterion of [51] are physical in string/M-theory by using the
criterion of [52]. If this is indeed the case, identifying relevant M-brane or
D-brane configurations related to these four-dimensional solutions also
deserves further study. Finally, finding more general holographic solutions of
$N=4$ gauged supergravities in other truncations or in other gauge groups
could be interesting as well. These solutions might include holographic
solutions describing spatially varying mass terms given in [53] and [54]. We
leave all these issues for future investigation.
Acknowledgement
This work is supported by The Thailand Research Fund (TRF) under grant
RSA6280022. It is a pleasure to thank Patharadanai Nuchino for reading the
manuscript.

## References

  * [1] J. M. Maldacena, “The large $N$ limit of superconformal field theories and supergravity”, Adv. Theor. Math. Phys. 2 (1998) 231-252, arXiv: hep-th/9711200.
  * [2] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, “Gauge Theory Correlators from Non-Critical String Theory”, Phys. Lett. B428 (1998) 105-114, arXiv: hep-th/9802109.
  * [3] E. Witten, “Anti De Sitter Space and holography”, Adv. Theor. Math. Phys. 2 (1998) 253-291, arXiv: 9802150.
  * [4] R. Corrado, K. Pilch and N. P. Warner, “An $N=2$ supersymmetric membrane flow”, Nucl. Phys. B629 (2002) 74-96, arXiv: hep-th/0107220.
  * [5] C. N. Gowdigere and N. P. Warner, “Flowing with Eight Supersymmetries in M-Theory and F-theory”, JHEP 12 (2003) 048, arXiv: hep-th/0212190.
  * [6] K. Pilch, A. Tyukov and N. P. Warner, “Flowing to Higher Dimensions: A New Strongly-Coupled Phase on M2 Branes”, JHEP 11 (2015) 170, arXiv: 1506.01045.
  * [7] C. Ahn and K. Woo, “Supersymmetric Domain Wall and RG Flow from 4-Dimensional Gauged $N=8$ Supergravity”, Nucl. Phys. B599 (2001) 83-118, arXiv: hep-th/0011121.
  * [8] C. Ahn and T. Itoh, “An $N=1$ Supersymmetric $G_{2}$-invariant Flow in M-theory”, Nucl. Phys. B627 (2002) 45-65, arXiv: hep-th/0112010.
  * [9] N. Bobev, N. Halmagyi, K. Pilch and N. P. Warner, “Holographic, $N=1$ Supersymmetric RG Flows on M2 Branes”, JHEP 09 (2009) 043, arXiv: 0901.2376.
  * [10] A. Guarino, “On new maximal supergravity and its BPS domain-walls”, JHEP 02 (2014) 026, arXiv: 1311.0785.
  * [11] J. Tarrio and O. Varela, “Electric/magnetic duality and RG flows in $AdS_{4}/CFT_{3}$”, JHEP 01 (2014) 071, arXiv: 1311.2933.
  * [12] Y. Pang, C. N. Pope and J. Rong, “Holographic RG Flow in a New $SO(3)\times SO(3)$ Sector of $\omega$-Deformed $SO(8)$ Gauged $N=8$ Supergravity”, JHEP 08 (2015) 122, arXiv: 1506.04270.
  * [13] P. Karndumri, “Holographic RG flows in $N=3$ Chern-Simons-Matter theory from $N=3$ 4D gauged supergravity”, Phys. Rev. D94 (2016) 045006, arXiv: 1601.05703.
  * [14] P. Karndumri and K. Upathambhakul, “Gaugings of four-dimensional $N=3$ supergravity and AdS4/CFT3 holography”, Phys. Rev. D93 (2016) 125017 arXiv: 1602.02254.
  * [15] P. Karndumri, “Supersymmetric deformations of 3D SCFTs from tri-sasakian truncation”, Eur. Phys. J. C (2017) 77, 130, arXiv: 1610.07983.
  * [16] P. Karndumri and K. Upathambhakul, “Supersymmetric RG flows and Janus from type II orbifold compactification”, Eur. Phys. J. C (2017) 77, 455, arXiv: 1704.00538.
  * [17] P. Karndumri and K. Upathambhakul, ”Holographic RG flows in N = 4 SCFTs from half-maximal gauged supergravity”, Eur. Phys. J. C78 (2018) 626, arXiv:hep-th/1806.01819.
  * [18] P. Karndumri and C. Maneerat, “Supersymmetric solutions from $N=5$ gauged supergravity”, Phys. Rev. D101 (2020) 126015, arXiv: 2003.05889.
  * [19] N. Bobev, K. Pilchand N. P. Warner, “Supersymmetric Janus Solutions in Four Dimensions”, JHEP 1406 (2014) 058, arXiv: 1311.4883.
  * [20] P. Karndumri, “Supersymmetric Janus solutions in four-dimensional $N=3$ gauged supergravity”, Phys. Rev. D93 (2016) 125012, arXiv: 1604.06007.
  * [21] M. Suh, “Supersymmetric Janus solutions of dyonic $ISO(7)$-gauged $N=8$ supergravity”, JHEP 04 (2018) 109, arXiv: 1803.00041.
  * [22] N. Kim and S. J. Kim, “Re-visiting Supersymmetric Janus Solutions: A Perturbative Construction”, arXiv: 2001.06789.
  * [23] A. Guarino, “On new maximal supergravity and its BPS domain-walls”, JHEP 02 (2014) 026, arXiv: 1311.0785.
  * [24] J. Tarrio and O. Varela, “Electric/magnetic duality and RG flows in AdS4/CFT3”, JHEP 10 (2014) 071, arXiv: 1311.2933.
  * [25] P. Karndumri and J. Seeyangnok, “Supersymmetric solutions from $N=6$ gauged supergravity”, arXiv: 2012.10978.
  * [26] P. Karndumri and C. Maneerat, “Supersymmetric Janus solutions in $\omega$-deformed $N=8$ gauged supergravity”, arXiv: 2012.15763.
  * [27] P. Karndumri, “RG flows in 6D $N=(1,0)$ SCFT from $SO(4)$ half-maximal 7D gauged supergravity”, JHEP 06 (2014) 101, arXiv: 1404.0183.
  * [28] P. Karndumri, “Holographic RG flows in six dimensional F(4) gauged supergravity”, JHEP 01 (2013) 134, erratum ibid JHEP 06 (2015) 165, arXiv: 1210.8064.
  * [29] P. Karndumri, “Gravity duals of 5D $N=2$ SYM from F(4) gauged supergravity”, Phys. Rev. D90 (2014) 086009, arXiv: 1403.1150.
  * [30] G. Bruno De Luca, A. Gnecchi, G. Lo Monaco and A. Tomasiello, “Holographic duals of 6d RG flows”, JHEP 03 (2019) 035, arXiv: 1810.10013.
  * [31] L. Girardello, M. Petrini, M. Porrati, and A. Zaffaroni, “Novel local CFT and exact results on perturbations of $N=4$ super Yang Mills from AdS dynamics, JHEP 12 (1998) 022, arXiv: hep-th/9810126.
  * [32] D. Z. Freedman, S. S. Gubser, K. Pilch, and N. P. Warner, “Renormalization group flows from holography supersymmetry and a c theorem”, Adv. Theor. Math. Phys. 3 (1999) 363–417, arXiv: hep-th/9904017.
  * [33] D. Cassani, G. Dall’Agata and A. F. Faedo, “BPS domain walls in $N=4$ supergravity and dual flows”, JHEP 03 (2013) 007, arXiv: 1210.8125.
  * [34] N. Bobev, D. Cassani and H. Triendl, “Holographic RG flows for four-dimensional $N=2$ SCFTs”, JHEP 06 (2018) 086, arXiv: 1804.03276.
  * [35] H. L. Dao and P. Karndumri, “Holographic RG flows and $AdS_{5}$ black strings from 5D half-maximal gauged supergravity”, Eur. Phys. J. C79 (2019) 137, arXiv: 1811.01608.
  * [36] C. Bachas, J. de Boer, R. Dijkgraaf, and H. Ooguri, “Permeable conformal walls and holography”, JHEP 06 (2002) 027, arXiv:hep-th/0111210.
  * [37] C. Bachas and M. Petropoulos, “Anti-de-Sitter D-branes”, JHEP 02 (2001) 025, arXiv:hep-th/0012234.
  * [38] D. Bak, M. Gutperle and S. Hirano, “Three dimensional Janus and time-dependent black holes”, JHEP 02 (2007) 068, arXiv: hep-th/0701108.
  * [39] M. Chiodaroli, M. Gutperle and D. Krym, “Half-BPS Solutions locally asymptotic to $AdS_{3}\times S^{3}$ and interface conformal field theories”, JHEP 02 (2010) 066, arXiv: 0910.0466.
  * [40] M. Chiodaroli, E. D’Hoker, Y, Guo and M. Gutperle, “Exact half-BPS string-junction solutions in six-dimensional supergravity”, JHEP 12 (2011) 086, arXiv: 1107.1722.
  * [41] D. Bak and H. Min, “Multi-faced Black Janus and Entanglement”, JHEP 03 (2014) 046, arXiv: 1311.5259.
  * [42] E. D’Hoker, J. Estes, M. Gutperle and D. Krym, “Janus solutions in M-theory”, JHEP 06 (2009) 018, arXiv: 0904.3313.
  * [43] M. Gutperle, J. Kaidi and H. Raj, “Janus solutions in six-dimensional gauged supergravity”, JHEP 12 (2017) 018, arXiv: 1709.09204.
  * [44] K. Chen and M. Gutperle “Janus solutions in three-dimensional $N=8$ gauged supergravity”, arXiv: 2011.10154.
  * [45] J. Schon and M. Weidner, “Gauged $N=4$ supergravities”, JHEP 05 (2006) 034, arXiv: hep-th/0602024.
  * [46] E. Bergshoeff, I. G. Koh and E. Sezgin, “Coupling of Yang-Mills to $N=4$, $d=4$ supergravity”, Phys. Lett. B155 (1985) 71-75.
  * [47] M. de Roo and P. Wagemans, “Gauged matter coupling in $N=4$ supergravity”, Nucl. Phys. B262 (1985) 644-660.
  * [48] P. Wagemans, “Breaking of $N=4$ supergravity to $N=1$, $N=2$ at $\Lambda=0$, Phys. Lett. B206 (1988) 241.
  * [49] J. Louis and H. Triendl, “Maximally supersymmetric $AdS_{4}$ vacua in $N=4$ supergravity”, JHEP 10 (2014) 007, arXiv:1406.3363.
  * [50] D. Roest and J. Rosseel, “De Sitter in Extended Supergravity”, Phys. Lett. B685 (2010) 201-207, arXiv: 0912.4440.
  * [51] S. S. Gubser, “Curvature singularities: the good, the bad and the naked”, Adv. Theor. Math. Phys. 4 (2000) 679-745, arXiv: hep-th/0002160.
  * [52] J. Maldacena and C. Nunez, “Supergravity description of field theories on curved manifolds and a no go theorem”, Int. J. Mod. Phys. A16 (2001) 822, arXiv: hep-th/0007018.
  * [53] J. P. Gauntlett and C. Rosen, “Susy Q and spatially modulated deformations of ABJM theory”, JHEP 10 (2018) 066, arXiv: 1808.02488.
  * [54] I. Arav, J. P. Gauntlett, M. Roberts, C. Rosen, “Spatially modulated and supersymmetric deformations of ABJM theory”, JHEP 04 (2019) 099, arXiv: 1812.11159.