# Nested Fermi surfaces and correlated electronic phases in hole-doped
semiconductor quantum wells

Tommy Li  Dahlem Center for Complex Quantum Systems and Fachbereich Physik,
Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany  Julian Ingham
Physics Department, Boston University, Commonwealth Avenue, Boston, MA 02215,
USA  Harley D. Scammell  School of Physics, University of New South Wales,
Sydney 2052, Australia  Australian Research Council Centre of Excellence in
Future Low-Energy Electronics Technologies, University of New South Wales,
Sydney 2052, Australia

###### Abstract

We demonstrate the existence of novel interaction effects in hole-doped
semiconductor quantum wells which are connected to dramatic changes in the
Fermi surface geometry occurring upon variation of the doping. We present band
structure calculations showing that quantum wells formed in $p$-type cubic
semiconductors develop perfectly nested Fermi surfaces at a critical hole
density $p\sim 1/d^{2}$ set by the width $d$ of the quantum well. Nesting
gives rise to competing superconducting and charge or spin density wave order,
which we analyze using the perturbative renormalization group method. The
correlated phases may be created or destroyed by tuning the hole density
towards or away from the critical density. Our results establish $p$-type
semiconductor quantum wells as a platform for novel correlated phases, which
may be precisely controlled using electrostatic gating and external magnetic
fields.

##  I Introduction

Two-dimensional (2D) hole-doped semiconductor systems, which host
spin-$\frac{3}{2}$ charge carriers, are known for having an extremely strong,
electrically controllable spin-orbit interaction which makes them ideal
candidates for the study of novel transport effects and a platform for
spintronics. While the effect of spin-orbit coupling on single-particle
physics in 2D semiconductor quantum wells has been studied in several recent
experimental and theoretical works Yeoh2014 ; Nichele2015 ; Li2016 ;
Marcellina2017 ; Marcellina2018 ; Bladwell2019 ; Cullen2021 , the evolution of
the Fermi surface geometry with hole density and its consequences for many-
body effects have remained unexplored. The recent observation of coexisting
density wave and superconductivity in transition metal dichalcogenides
Barrera2018 ; Lu2018 ; Lu2015 ; Yang2018 ; Ye2012 ; Song2021 , kagome metals
Ortiz2020 ; Zhu2021 ; Chen2021 ; Ortiz2021 ; Ni2021 ; Chenb2021 ; Liang2021 ,
and superconductivity with proximate correlated insulators in spin-orbit
coupled Moiré materials Wang2020 ; Jin2021 ; Xu2020 ; Huang2021 ; Miao2021 ;
Ghiotto2021 ; Shi2021  have stimulated an emergence of interest in the effect
of spin-orbit coupling on correlated phases, yet the possibility of many-body
instabilities originating from the spin-orbit coupling in 2D
spin-$\frac{3}{2}$ systems has not been considered. Hole-doped semiconductor
quantum wells therefore currently provide an intriguing opportunity for
investigation.

In this work we demonstrate the existence of correlated electronic states
which arise due to the spin-$\frac{3}{2}$ nature of the positively charged
carriers characteristic of hole-doped semiconductors. We perform band
structure calculations that show that, at certain values of the carrier
density, the 2D Fermi surfaces exhibit perfect nesting over a range of
materials, due to a transition in which the curvature of the Fermi surface
changes sign at high symmetry points in $k$-space. Accounting for the presence
of an arbitrarily weak Coulomb interaction, we find that competing
superconducting and charge (CDW) or spin (SDW) density wave instabilities
emerge, with either SDW or coexisting CDW and SDW orders present when the
Fermi surface is spin degenerate, and a pure CDW when the spin degeneracy is
lifted by either broken time reversal or spatial inversion symmetry. The
nested regions do not cover the entire Fermi surface, leading to a Fermi
surface reconstruction in which the nested portions are gapped and the
unnested portions form Fermi arcs, so that the ordered phases are metallic.
Nesting remains possible in an in-plane magnetic field directed along a high
symmetry axis, and the nesting densities exhibit a dependence on the magnetic
field, allowing the identification and study of the correlated ground state
via magnetotransport; in this case only CDW order is present. In diamond-
structure semiconductors in zero magnetic field, SDW order always emerges upon
tuning of the hole density close to the nesting density; thus electrical gates
provide a highly sensitive control over the magnetic ordering, a property
which could prove useful for spintronic applications Jungwirth2016 .

##  II Hole band structure

We consider a semiconductor quantum well, which consists of a semiconducting
layer typically of width $d\approx 20$ nm, sandwiched between two insulators,
with charge carriers confined to a 2D plane. Hole-doped heterostructures
formed in cubic semiconductors are described by the Luttinger Hamiltonian
Luttinger1955 ; Dresselhaus1956 ; Winkler2003

$\displaystyle
H=\frac{p^{2}}{2m}+T^{\mu\nu\alpha\beta}p_{\mu}p_{\nu}S_{\alpha}S_{\beta}+V(z)+$
$\displaystyle
C_{1}^{\mu\nu\alpha\beta}S_{\mu}p_{\nu}p_{\alpha}p_{\beta}+C^{\mu\nu\alpha\beta}_{2}p_{\mu}S_{\nu}S_{\alpha}S_{\beta}\
\ ,$ (1)

where $S_{\mu}$ are the spin-$\frac{3}{2}$ operators, $T,C_{1},C_{2}$ are
fourth-rank tensors originating from spin-orbit coupling, and $V(z)$ is a
confining potential. Taking a coordinate system aligned with the cubic axes,
the tensor $T$ has two independent nonvanishing components,
$T^{xxxx}=T^{yyyy}=T^{zzzz}<0$ and $T^{xyxy}=T^{yzyz}=T^{zxzx}<0$. The final
two terms in (1) are the Dresselhaus interactions, which appear only in
materials with broken bulk inversion symmetry. The tensors
$C=\\{C_{2},C_{3}\\}$ only have one independent nonvanishing component
$C^{xxyy}=C^{yyzz}=C^{zzxx}=-C^{xxzz}=-C^{yyxx}=-C^{zzyy}$.

Figure 1: Dispersion of 2D subbands, $\varepsilon_{n}(|\bm{k}|)$ in Ge (left)
and GaAs (right) semiconductor quantum wells of width 20 nm along the (100)
and (110) directions (solid and dashed lines respectively). The horizontal
blue dashed lines show the values of the Fermi energy for typical a 2D hole
density $p=10^{11}$ cm-2, while the horizontal solid red lines show the values
of the Fermi energy at the critical density at which the curvature of the
Fermi surface changes sign, which in Ge (left) is $p\approx 2.56\times
10^{11}$ cm-2 and in GaAs (right) are $p\approx 1.99\times
10^{11}$/$2.92\times 10^{11}$ cm-2 for the larger/smaller spin split Fermi
surfaces. Energy and momentum are plotted in units of meV and $\pi/d$
respectively.

We consider the case where a confining potential quantizes motion in the
$z\parallel(001)$ direction, leaving the charge carriers free to move in the
[001] plane. The single particle eigenstates possess definite 2D momentum
$\bm{k}=(k_{x},k_{y})$ and form a discrete series of bands corresponding to
the quantization of the transverse modes. We perform band structure
calculations for the case of an infinite square well of width $d=20$ nm. In
Fig. 1 we plot the 2D dispersion of Ge (left) and GaAs (right) quantum wells,
$\varepsilon_{n}(\bm{k})$, $n=1,2,\dots$ as a function of $|\bm{k}|$ along the
(100) (solid lines) and (110) (dashed lines) directions. We observe that, for
low densities, the dispersion of the lowest bands is parabolic and isotropic,
however as the momentum becomes of the order $k\sim\pi/d$, the dispersion
becomes both highly anisotropic and nonparabolic due to the level repulsion
between the lowest two pairs of 2D bands. The anisotropy of the Fermi surface
is due to the tensor $T^{\mu\nu\alpha\beta}$ appearing in the Luttinger
Hamiltonian (1) lacking spherical symmetry, $T^{xxxx}\neq T^{xyxy}$. The
horizontal lines indicate the Fermi energy for various hole densities $p$. The
dashed blue lines correspond to a typical doping $p=10^{11}$ cm-2; the solid
red line in the left panel corresponds to $p=2.56\times 10^{11}$ cm-2, and the
solid red lines in the right panel correspond to $p=1.99\times 10^{11}$ and
$2.92\times 10^{11}$ cm-2.

The level repulsion and anisotropy of the 2D dispersion in Ge and GaAs quantum
wells may be understood straightforwardly from the structure of the Luttinger
Hamiltonian (1). The spin-independent terms provide an isotropic contribution
to the dispersion that is quadratic in energy, thus only the spin-dependent
terms contribute to the effect. We note that, at small in-plane momenta, the
bands in GaAs are nearly twofold degenerate, which shows that the Dresselhaus
interaction is negligible for $k_{x}=k_{y}=0$, thus the splitting of the 2D
bands for small in-plane momentum is entirely due to the term
$T^{\mu\nu\alpha\beta}p_{\mu}p_{\nu}S_{\alpha}S_{\beta}\rightarrow
T^{zzzz}p_{z}^{2}S_{z}^{2}$ in (1), which for an infinite square well results
in an energy splitting $\Delta\varepsilon=2T^{zzzz}\pi^{2}/d^{2}$ between the
lowest two pairs of subbands at $k_{x}=k_{y}=0$. The bands have definite
$S_{z}^{2}$, with the lowest and second lowest pair possessing maximum
($S_{z}^{2}=\frac{9}{4}$) and minimum ($S_{z}^{2}=\frac{1}{4}$) out-of-plane
polarization along the growth axis respectively, since $T^{zzzz}<0$
Winkler2003 . As the in-plane momentum is increased, terms mixing states with
different $S_{z}^{2}$, i.e. those containing $S_{x},S_{y}$, become
significant. The lowest pair of bands disperses upwards, while the second
lowest disperses downwards, leading to an anticrossing at $|\bm{k}|\approx
k_{z}=\pi/d$ which is the wavevector related to transverse quantum
confinement.

Figure 2: Evolution of the Fermi surface of a Ge quantum well with increasing
hole doping, with hole densities (in units of $10^{11}$ cm-2) (_left_):
$p=5.13,1.03,1.54,2.05$, (_right_): $p=2.18,2.31,2.43,2.56$. These densities
correspond to $p=0.2p^{*},0.4p^{*},0.6p^{*},0.8p^{*}$ for the left and
$p^{*}=0.85p^{*},0.9p^{*},0.95p^{*},p^{*}$ where $p^{*}$ is the doping at
which the curvature of the sides of the Fermi surface changes sign (as defined
in the text).

At the anticrossing, the level repulsion between the two lowest pairs of bands
is considerably stronger in the (110) direction than in the (100) direction
(shown in the solid and dashed lines respectively in Fig 1). As the Fermi
energy is increased to the anticrossing, the Fermi surface becomes
progressively more distorted due to the anisotropy of the dispersion. In Fig.
2 we plot the hole Fermi surfaces in a 2D Ge quantum well at various hole
densities. We observe the evolution of the Fermi surface from almost perfectly
circular at low densities to progressively more square. As the hole density
approaches a critical value $p^{*}=2.56\times 10^{11}$ cm-2, the curvature of
the sides of the Fermi surface $v_{\perp}^{-1}\partial^{2}\varepsilon/\partial
k_{\parallel}^{2}$ changes sign (with $v_{\perp},k_{\parallel}$ being the
velocity and momenta transverse and longitudinal to the Fermi surface) and the
sides of the Fermi surface become perfectly flat. We observe that almost
perfect nesting of the Fermi surface occurs over a range of densities.

Figure 3: The spin-split Fermi surfaces in a GaAs quantum well in zero
magnetic field, with hole densities (a): $p=1.99\times 10^{11}$ cm-2, at which
the larger Fermi surface is nested, and (b) $p\approx 2.91\times 10^{11}$
cm-2, at which the smaller Fermi surface is nested.

In Fig. 3 we show the Fermi surfaces in a GaAs quantum well at the hole
densities (a) $p=1.99\times 10^{11}$ cm-2, (b) $p=2.91\times 10^{11}$ cm-2.
Due to bulk inversion asymmetry, the bands are no longer spin degenerate, but
the same transition between convex and concave Fermi surfaces occurs for each
band at different densities.

The nesting of the Fermi surfaces in both Ge and GaAs systems reflects shared
features of the 2D dispersion arising from the spin structure of the Luttinger
Hamiltonian 1 common to hole-doped cubic semiconductors. At small in-plane
momenta, the bands are approximately isotropic and parabolic, giving rise to
circular energy contours. The quantum states in the lowest band have
approximately definite $S_{z}^{2}=\frac{9}{4}$. As the momentum is
sufficiently increased, the terms in the Hamiltonian containing the in-plane
momenta $k_{x},k_{y}$ compete with the 2D quantization (corresponding to
wavevector $k_{z}=\pi/d$) which splits the energy of the
$S_{z}^{2}=\frac{9}{4}$ and $S_{z}^{2}=\frac{1}{4}$ states, and both the level
repulsion and cubic anisotropy become important and eventually distort the
energy contours into a four-lobed shape (Fig. 2b). For some density between
the circular and lobed shape, the energy contours become square, implying
there is always a transitional hole density $p\sim 1/d^{2}$ with exact
nesting. The terms in the Hamiltonian (1) responsible for this effect are
quadratic in the in-plane spin components,
$S_{x}^{2},S_{y}^{2},\\{S_{x},S_{y}\\}$, which are proportional to the
identity matrix for spin-$\frac{1}{2}$ systems but nontrivial for
spin-$\frac{3}{2}$. Thus we may directly connect the nesting phenomenon to the
combination of spin-$\frac{3}{2}$ nature of holes and 2D confinement. We
should expect this to occur generally among quantum wells formed in hole-doped
cubic semiconductors, and we indeed find nested Fermi surfaces for Si, InAs,
InSb, CdTe and ZnSe, for which we provide figures (Figs. S1–S5) in the
Supplementary Material. In all cases, assuming a well width of $d=20$ nm,
nesting occurs within an order of magnitude of $p\sim 1/d^{2}=2.5\times
10^{11}$ cm-2. The robustness of the existence of nesting densities may be
connected to the change in sign of the curvature of the sides of the Fermi
surface $v_{\perp}^{-1}\partial^{2}\varepsilon/\partial k_{\parallel}^{2}$,
which we show in Fig. S6 in the Supplementary Material.

Figure 4: Nested Fermi surfaces for a 20 nm Ge quantum well in a 1T magnetic
field applied along the (010) direction, at hole densities (a) $1.97\times
10^{11}$ cm-2, (b) $3.00\times 10^{11}$ cm-2. Portions of the Fermi surface on
which the normal momentum coordinate varies by less than 0.25% are highlighted
in red.

Due to the cubic symmetry of the bulk Hamiltonian, the Fermi surface is
symmetric under reflections about the (100) and (010) axes as well as fourfold
rotations, for confinement in the [001] plane. Thus, all four sides are
simultaneously nested at the critical densities. We may also consider the
situation in an in-plane magnetic field applied along a high symmetry axis,
which breaks the fourfold symmetry of the Fermi surface. For diamond
semiconductors, the Fermi surface possesses two mirror axes parallel and
perpendicular to the magnetic field, however fourfold rotation symmetry is
broken, and the twofold spin degeneracy is lifted. In Fig. 4 we show nested
Fermi surfaces of a Ge quantum well in a 1T magnetic field along the (010)
direction at two densities, corresponding to a situation when the larger and
smaller Fermi surfaces are nested. We find that, for the densities chosen, the
sides of the Fermi surface perpendicular to the magnetic field exhibit nesting
over a greater region than the sides parallel to the field. However, we also
found that at slightly different densities, the opposite scenario was also
possible.

Figure 5: Nested Fermi surfaces for a 20 nm GaAs quantum well in a 1T magnetic
field applied along the (010) direction, at densitie, (a) $p=1.94\times
10^{11}$ cm-2, (b) $2.83\times 10^{11}$ cm-2. Portions of the Fermi surface on
which the normal momentum coordinate varies by less than 0.25% are highlighted
in red.

In zincblende semiconductors, bulk inversion asymmetry results in a lack of
mirror symmetry about the axis transverse to the magnetic field, however
mirror symmetry about the direction parallel to the field remains. Among the
five zincblende semiconductors we studied (GaAs, InAs, InSb, CdTe, ZnSe), all
except InSb exhibited transitions at which the curvature of at least one pair
of opposite sides of the Fermi surface changed sign in a 1T magnetic field. In
the case of CdTe and ZnSe, significant nesting occurred for all four sides of
the Fermi surface, however in the cases of GaAs and InAs, densities were found
for which significant nesting occurred only for one pair of sides. In Fig. 5
we show the nested Fermi surface of a GaAs quantum well in a 1T magnetic field
applied along the (010) direction, at densities for which nesting is exhibited
in the larger/smaller Fermi surfaces in the left/right panels. Portions of the
Fermi surface on which the normal momentum coordinate varies by less than
0.25% are highlighted in red. We show the nested Fermi surfaces in a 1T
magnetic field for the remaining zincblende semiconductors in the
Supplementary Material.

So far we have presented results for the case where the confining potential
$V(z)$ is an infinite square well. We have observed qualitatively identical
behavior for generic quantum wells which are symmetric along the growth axis,
however nesting can be significantly reduced by the asymmetry of the quantum
well in zincblende structures, even at zero magnetic field. We present a
comparison of the effect of structural inversion asymmetry in Ge and GaAs
quantum wells in Fig. S7 in the Supplementary Material.

##  III RG analysis

Figure 6: One-loop corrections to the interaction vertex.

The existence of nested Fermi surfaces suggests a dramatic enhancement in the
role of interactions, as is well known from the theory of the fermionic
renormalization group Shankar1994 . We define an effective theory by
restricting the allowed momenta $\bm{k}$ to lie within a cut-off of width
$\Lambda$ about the two Fermi surfaces (which are degenerate when both
inversion and time-reversal symmetry are present but otherwise split), and
study instabilities of the Fermi liquid phase by analyzing the RG flow of the
effective couplings
$\Gamma^{\Lambda}_{\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{4}}(\bm{k}_{1},\bm{k}_{2},\bm{k}_{3},\bm{k}_{4})$
describing scattering between two holes with initial momenta
$\bm{k}_{1},\bm{k}_{2}$ and final momenta $\bm{k}_{3},\bm{k}_{4}$, and in the
initial/final Fermi surfaces indexed by
$\sigma_{1},\sigma_{2}$/$\sigma_{3},\sigma_{4}$. We calculate the RG flow of
the effective couplings by identifying logarithmic divergences in perturbative
corrections to the scattering vertex; many-body instabilities are signaled by
a divergence in the effective couplings at low energies ($\Lambda\rightarrow
0$). We begin by presenting a general argument why, irrespective of the
specific interacting model, nesting in our systems always leads to an
instability, due to the divergence of certain scattering amplitudes at low
energies, and later introduce a toy model which allows us to explore the phase
diagram.

Noting that the bare values of interactions involving large momentum transfer
are suppressed due to the $1/q$ momentum dependence of the Coulomb interaction
as well as the overlap between spin–$\frac{3}{2}$ wavefunctions in the initial
and final hole states, we might expect that these interactions grow slower
under the RG flow and thus the strongest instabilities are associated with
small momentum transfer processes with
$\sigma_{1}=\sigma_{3},\sigma_{2}=\sigma_{4}$. To second order in the
interactions, the RG flow is driven by corrections to the scattering vertex
associated with the virtual processes shown in Fig. 6, which involve
intermediate states consisting of two holes with momenta
$\bm{p},\bm{p}^{\prime}$ (Fig. 6a) or a hole and an electron with momenta
$\bm{p},-\bm{p}^{\prime}$ (Fig. 6b). Logarithmic divergences always appear in
the hole-hole channel when $\bm{p}+\bm{p}^{\prime}=0$ as is usual for Fermi
liquids, however in the nested situation, additional logarithmic divergences
appear when $\bm{p}$ and $\bm{p}^{\prime}$ lie on opposite nested sides and
$\bm{p}+\bm{p}^{\prime}\neq 0$. In the hole-electron channel, logarithmic
divergences appear only when nesting is present, with $\bm{p}$,
$\bm{p}^{\prime}$ required to lie on opposite nested sides of the Fermi
surface.

Virtual processes shown in Fig. 6a provide a negative correction to the
forward scattering vertex,
$\delta\Gamma^{\Lambda}_{\sigma_{1}\sigma_{2}\sigma_{1}\sigma_{2}}(\bm{k}_{1},\bm{k}_{2},\bm{k}_{1},\bm{k}_{2})=-\sum_{\bm{p};\sigma,\sigma^{\prime}}{|\Gamma^{\Lambda}_{\sigma_{1}\sigma_{2}\sigma\sigma^{\prime}}(\bm{k}_{1},\bm{k}_{2},\bm{p},\bm{p}^{\prime})|^{2}/|E_{hh}|}$
where momentum conservation implies
$\bm{p}^{\prime}=\bm{k}_{1}+\bm{k}_{2}-\bm{p}$, and $E_{hh}$ is the energy of
the intermediate two-hole pair. Thus, for the repulsive hole-hole interactions
which we consider, this diagram provides a suppression of forward scattering.
For the process shown in Fig. 6b, the correction is always positive,
$\delta\Gamma^{\Lambda}_{\sigma_{1}\sigma_{2}\sigma_{1}\sigma_{2}}(\bm{k}_{1},\bm{k}_{2},\bm{k}_{1},\bm{k}_{2})=+\sum_{\bm{p};\sigma,\sigma^{\prime}}{|\Gamma^{\Lambda}_{\sigma_{1}\sigma\sigma^{\prime}\sigma_{2}}(\bm{k}_{1},\bm{p},\bm{p}^{\prime},\bm{k}_{2})|^{2}/|E_{he}|}$
where momentum conservation implies
$\bm{p}^{\prime}=\bm{p}+\bm{k}_{2}-\bm{k}_{1}$ and $E_{he}$ is the energy of
the hole-electron pair; thus these virtual processes enhance forward
scattering.

For general momenta $\bm{k}_{1},\bm{k}_{2}$, both diagrams in Fig. 6
contribute, however a simple phase space argument shows that, at special
points on the Fermi surface, the contribution from Fig. 6a vanishes, resulting
in a positive definite RG flow and therefore a divergence in the running
coupling at a finite energy scale, indicating an instability. These scattering
processes correspond to $\bm{k}_{1},\bm{k}_{2}$ lying laterally opposite at
the edges of parallel nested portions of the Fermi surface, labelled A,B in
Fig. 7. Since $\bm{k}_{1}+\bm{k}_{2}\neq 0$, the diagram Fig. 6a diverges only
if the internal momenta $\bm{p},\bm{p}^{\prime}$ lie on nested regions of the
Fermi surface. Choosing the components of both $\bm{k}_{1}$ and $\bm{k}_{2}$
parallel to the Fermi surface to be equal to the maximum possible values for
momenta lying in the nested regions, the momentum conservation condition
$\bm{k}_{1}+\bm{k}_{2}=\bm{p}+\bm{p}^{\prime}$ can only be satisfied for a
vanishingly small momentum space volume, and there is no contribution to the
RG flow.

At the same time, we note that the contribution to the RG flow from Fig. 6b
arises from processes in which $\bm{p}$ can take any value along the nested
portion of the Fermi surface, since $\bm{k}_{2}-\bm{k}_{1}=\bm{Q}$ is a
nesting vector and $\bm{p}^{\prime}=\bm{p}+\bm{Q}$ lies laterally opposite to
$\bm{p}$ for any value of $\bm{p}$ on a straight section of the Fermi surface.
Since Fig. 6b always provides a positive correction and never vanishes, the
coupling grows to $+\infty$ under the RG flow. This allows us to establish
that the Fermi liquid phase is unstable, with one possible instability
involving the condensation of hole-electron pairs with total momentum
$\bm{Q}$, which gives rise to a density wave.

While we have established an instability via generic phase space volume
arguments, determining whether this instability is dominant requires analysis
of a more detailed and specific model. One alternative to the density wave
instability is superconductivity, which is driven by interactions for which
the corrections due to virtual scattering in the hole-hole channel (Fig. 6a)
rather than the hole-electron channel (Fig. 6b) dominate. Generic phase space
arguments allow us to identify one such interaction, which involves forward
scattering in the Cooper channel, $\bm{k}_{1}+\bm{k}_{2}=0$, with
$\bm{k}_{1},\bm{k}_{2}$ lying at the edges of the nested portions of the Fermi
surface (the points labelled A,C in Fig. 7). The logarithmically divergent
processes corresponding to Fig. 6a involve hole-hole pairs with
$\bm{p}^{\prime}=-\bm{p}$; when $\bm{p}$ lies in one nested region,
$\bm{p}^{\prime}$ automatically lies on the opposite nested region, thus the
phase space volume is maximized. However, the hole-electron processes in Fig.
6b provide a vanishing contribution to the RG flow, since
$\bm{k}_{2}-\bm{k}_{1}$ is a vector of maximum length connecting the nested
regions of opposite sides of the Fermi surface, and the momentum conservation
condition $\bm{p}^{\prime}-\bm{p}=\bm{k}_{2}-\bm{k}_{1}$ with
$\bm{p},\bm{p}^{\prime}$ in the nested regions can only be fulfilled within a
vanishingly small momentum space volume. Thus the correction is purely due to
Fig. 6a which is negative. If the correction itself vanishes at low energies,
then the interaction flows to zero; however if the correction remains finite,
the interaction will grow to $-\infty$ and lead to a superconducting
instability. The former situation occurs ordinarily in Fermi liquids, however,
when nesting is present, the second situation is generically true, due to the
fact that the individual effective vertices appearing in the second-order
processes are themselves dressed by both virtual hole-hole and hole-electron
processes.

We may introduce a minimal model possessing both density wave and
superconducting instabilities, involving only one pair of nested sides, shown
in Fig. 7a. We define interactions on four patches of the Fermi surface close
to the points A,B,C,D; we find that in these regions the velocity normal of
the Fermi surface is minimum, which enhances the interactions. These patches
play a similar role to the “hot-spots” in previously studied patch RG models
of the cuprates Schulz1987 ; Dzyaloshinskii1987 ; Furukawa1998 . Rather than
coupling functions, we introduce a discrete set of coupling constants
$g_{1}=\nu\Gamma_{\sigma\sigma\sigma\sigma}(ABAB),g_{2}=\nu\Gamma_{\sigma\sigma\sigma\sigma}(ACAC),g_{3}=\nu\Gamma_{\sigma\sigma\sigma\sigma}(ACDB)$,
neglecting backscattering and spin-flip scattering, with $\nu$ being the
single-band density of states averaged over a single patch. The scaling
relations are

$\displaystyle\frac{dg_{1}}{dl}=g_{1}^{2}+g_{3}^{2}\ \ ,$
$\displaystyle\frac{dg_{2}}{dl}=-g_{2}^{2}-g_{3}^{2}\ \ ,$
$\displaystyle\frac{dg_{3}}{dl}=2g_{1}g_{3}-2g_{2}g_{3}\ \ .$ (2)

For an initial value $g_{3}(l=0)=0$, we find that $g_{1}$ diverges at a scale
$l=\log(\Lambda_{0}/\Lambda)=1/g_{1}$, while $g_{2}\rightarrow 0$. However,
when $g_{3}(l=0)\neq 0$, we find that $g_{3}$ is monotonically enhanced by the
RG flow, so that $dg_{2}/dl<-g_{3}^{2}$ is strictly negative, leading to
$g_{2}\rightarrow-\infty$. The competition between the density wave and
superconducting phases may be resolved by determining which coupling diverges
the fastest, since the RG flow must be stopped when the couplings become
large. In Fig. 7b we plot the flow of the couplings for initial values
$g_{1}=g_{2}=0.3,g_{3}=0.1$. We find that, since $g_{2}$ must first decrease
from an initially positive value, it reaches strong coupling at a later RG
time than $g_{1}$ and thus the dominant instability is associated with
$g_{1}$, corresponding to a $\bm{Q}$-density wave.

While this model is illustrative, it is not predictive since it employs a very
small subset of the possible couplings that contribute to the RG flow. The
most significant modification to the scaling relations arises from scattering
in the Cooper channel ($\bm{k}_{1}+\bm{k}_{2}=\bm{k}_{3}+\bm{k}_{4}=0$)
involving large momentum transfer, which enhances the flow of $g_{2}$. While
the couplings are initially small, they may grow sufficiently rapidly under
the RG flow to allow $g_{2}$ to diverge faster than $g_{1}$, in which case the
superconducting instability becomes dominant.

Figure 7: (a) Coupling constants used in the patch RG model
$g_{1}=\nu\Gamma(ABAB)$, $g_{2}=\nu\Gamma(ACAC)$, $g_{3}=\nu\Gamma(ACDB)$. (b)
RG flow of the couplings $g_{1},g_{2},g_{3}$ plotted in the blue, green and
red lines respectively, for initial values $g_{1}=g_{2}=0.3,g_{3}=0.1$. Figure
8: The spatial modulation of the charge density in the CDW phase for the cases
where (a) four sides of the Fermi surface exhibit nesting equally, (b) the
left and right sides are nested over a greater region than the top and bottom
sides, and (c) only the left and right sides are nested. Darker/brighter
regions correspond to higher/lower density.

##  IV Properties of the ordered phases

When the Fermi surfaces are non-degenerate, as is the case when either a
magnetic field is present or bulk inversion symmetry is broken, the RG
analysis predicts a CDW ground state, corresponding to the condensation of
hole-electron pairs on opposite nested sides of the Fermi surface. In the
absence of a magnetic field, there are two perpendicular nesting vectors
$\bm{Q}_{1},\bm{Q}_{2}$ along the $\hat{\bm{x}},\hat{\bm{y}}$ directions, and
the CDW order forms a 2D checkerboard lattice. However, when a magnetic field
is present, nesting can be significantly suppressed or destroyed for one pair
of sides of the Fermi surface, leading to a striped CDW order. We illustrate
these cases in Fig. 8.

In the case of degenerate Fermi surfaces, both CDW and SDW ordering is
possible. In order to determine the ground state, we must minimize the Landau-
Ginzburg free energy simultaneously with respect to the CDW and SDW order
parameters, which we may collect into a four-dimensional complex vector
$\bm{\Phi}=(\varphi,M_{x},M_{y},M_{z})$ with $\varphi$ being the CDW order
parameter and $(M_{x},M_{y},M_{z})$ the SDW order parameter. The free energy
may be calculated by performing a Hubbard-Stratonovich transformation to
decompose the hole-hole interaction into a coupling between holes and the
bosonic order parameters. We divide the action into contributions involving
the left/right and top/bottom sides of the Fermi surface
$\mathcal{S}_{LR}+\mathcal{S}_{TB}$, and accounting for the $g_{1}$
interaction the Euclidean action for one pair of sides is

$\displaystyle\mathcal{S}_{LR}=\int{}d\tau\\{\frac{1}{g_{1}}\bm{\Phi}^{*}\cdot\bm{\Phi}+$
$\displaystyle\sum_{\bm{k}\in
R,L}{c^{\dagger}_{\bm{k}}\left[\partial_{\tau}+\varepsilon_{\bm{k}}\right]c_{\bm{k}}}+\sum_{\bm{k}\in
L}{\left[c^{\dagger}_{\bm{k}+\bm{Q}}\bm{\Phi}\cdot\bm{\sigma}c_{\bm{k}}+\text{H.c.}\right]}\\}$
(3)

where $c^{\dagger}_{\bm{k}}=(c^{\dagger}_{\bm{k}+},c^{\dagger}_{\bm{k}-})$ is
a two-component spinor and
$\bm{\sigma}=(\sigma_{0},\sigma_{x},\sigma_{y},\sigma_{z})$ with $\sigma_{0}$
being the identity matrix. Integration over the fermionic degrees of freedom
yields the Landau-Ginzburg free energy

$\displaystyle\mathcal{F}=\left(g_{1}^{-1}-a(T)\right)|(|\varphi|^{2}+|\bm{M}|^{2})+$
$\displaystyle
b(T)\\{|\varphi|^{4}+|\bm{M}|^{4}+|\bm{M}\times\bm{M}^{*}|^{2}+4|\varphi|^{2}|\bm{M}|^{2}+$
$\displaystyle(\varphi^{*})^{2}\bm{M}\cdot\bm{M}+\varphi^{2}\bm{M}^{*}\cdot\bm{M}^{*}\\}$
(4)

where we have ignored fluctuations of the order parameters, and

$\displaystyle a(T)=\frac{1}{2}\sum_{\omega_{n},\bm{p}\in
L}{\frac{1}{\omega_{n}^{2}+\varepsilon^{2}_{\bm{p}}}}\ ,\
b(T)=\frac{1}{16}\sum_{\omega_{n},\bm{p}\in
L}{\frac{1}{(\omega_{n}^{2}+\varepsilon_{\bm{p}}^{2})^{2}}}\ ,$ (5)

with $\omega_{n}$ being the fermionic Matsubara frequencies.

Minimizing the free energy yields degenerate CDW and SDW orders, with
$\varphi$ purely real and $\bm{M}$ purely imaginary, thus $(|\varphi|,\bm{M})$
is a real four-dimensional vector. The free energy possesses an SO(4) symmetry
and the system will spontaneously choose any orientation in the four-
dimensional order parameter manifold. However, this SO(4) symmetry is an
artifact of our restricted model in which only the $g_{1}$ coupling, which we
assume dominates at low energies, is present. The symmetry is broken as soon
as subleading interactions are considered. The most significant effect is due
to the spin dependence of the interactions, which are negligible for small
angle scattering but substantial for interactions involving large momentum
transfer.

An accurate treatment of the density wave orders accounting for the large
range of possible couplings is beyond the scope of this work, however, we may
form some generic conclusions by considering the spin structure of the
interactions, which is restricted by the presence of both inversion and time
reversal symmetry (since we are considering the case when the Fermi surfaces
are spin degenerate). In order to do this we define a basis of single-hole
states $|\bm{k},\sigma=\pm\rangle$ which are eigenstates of the combined
symmetry operation $IC_{2z}$ consisting of inversion and a twofold combined
spin and orbital rotation about the $z$–axis with eigenvalue
$e^{-\frac{3i\pi}{2}}\sigma$. States with opposite $\sigma$ are related by the
combination of time reversal and spatial inversion,
$|\bm{k},-\rangle=I\mathcal{T}|\bm{k},+\rangle$. The symmetries of the
Hamiltonian then imply that $\sigma_{1}\sigma_{2}=\sigma_{3}\sigma_{4}$, which
restricts the possible couplings to $\Gamma_{++++}=\Gamma_{----}^{*}$,
$\Gamma_{+-+-}=\Gamma_{-+-+}^{*}$, $\Gamma_{+--+}=\Gamma^{*}_{-++-}$,
$\Gamma_{++--}=\Gamma^{*}_{--++}$. Performing a Hubbard-Stratonovich
decomposition of these interactions leads to additional terms in the free
energy containing products $\delta
F=\Gamma_{0z}\varphi^{*}M_{z}+\Gamma_{xy}M_{x}^{*}M_{y}$ and their complex
conjugates. Noting that the quartic terms in the only contain even powers of
$\varphi$, the derivative of the quartic terms with respect to $\varphi$
vanish at $\varphi=0$, and thus

$\displaystyle\frac{\partial\mathcal{F}}{\partial\varphi^{*}}|_{\varphi=0}=\Gamma_{0z}M_{z}\
\ ,$ (6)

which implies that, at the minimum of the free energy, $\varphi$ cannot vanish
unless $M_{z}$ also vanishes. Therefore the ground state can either consist of
an in-plane SDW ($\varphi=M_{z}=0$), or coexisting CDW and out-of-plane SDW
orders ($\varphi,M_{z}\neq 0$).

The presence of CDW/SDW order leads to a Fermi surface reconstruction which
bears similarities to that observed in the cuprates Norman1998 . While the
nested portions of the Fermi surface are gapped by density wave ordering, the
corners do not couple to the order parameter and maintain a nonvanishing
density of states at the Fermi level. The reconstructed Fermi surface consists
of Fermi arcs formed from the unnested corner regions of the original Fermi
surface.

##  V Conclusion

We have demonstrated the existence of a generic nesting phenomenon in
semiconductor quantum wells formed in materials with a cubic lattice
structure, resulting in density wave instabilities, although our analysis does
not exclude the possibility of a competing superconducting ground state. In
the ordered phases we explore, the single-electron spectrum is partially
gapped, but the corner regions of the Fermi surface which are not nested
remain ungapped, allowing the ground state to remain metallic. The correlated
ground state bears striking similarities to the pseudogap phase observed in
the cuprates, which have been observed to exhibit Fermi arcs as well as
density wave order Norman1998 ; Vershinin2004 ; Comin2014 .

The realization of these features in semiconductor systems, which possess
extremely long mean free paths (of the order of microns Dobbie2012 ), presents
an important opportunity for further study to determine whether these systems
share other physical properties with the cuprates. The advantages of using
highly versatile nanofabricated semiconductor systems over high-$T_{c}$
superconductors include the ability to precisely tune the density _in situ_
via gates, which combined with detailed knowledge of the band structure
provides a setting in which the source of the interaction effects may be
definitively known, unlike in high-$T_{c}$ superconductors, in which the role
of nesting is still not conclusively established. Since the dimensionless
interactions depend on the velocity normal to the Fermi surface at the “hot-
spots” (the points A,B,C,D indicated in Fig. 7), which is controlled by the
width of the well $v_{\perp}\propto 1/d$, a crucial opportunity exists to
control the interaction strength.

The nesting vector is determined by the width of the quantum well
$|\bm{Q}|\sim 2\pi/d$ rather than the lattice spacing, and the ordered phases
exhibit a continuous symmetry breaking and gapless sliding modes which might
enhance the role of fluctuations. Novel interactions between holes may emerge,
enriching the phenomenology of the correlated ground state and allowing
further comparisons with the pseudogap phase in the high-$T_{c}$
superconductors. Our systems are also expected to exhibit a higher degree of
nesting than those possible in the cuprates as well as other interacting
systems including graphene Nandkishore2012 , kagome materials Ortiz2020 ;
Zhu2021 ; Chen2021 ; Ortiz2021 ; Ni2021 ; Chenb2021 ; Liang2021  and Moiré
superlattices Wang2020 ; Jin2021 ; Xu2020 ; Huang2021 ; Miao2021 ; Ghiotto2021
; Shi2021 , since in our case exact nesting emerges upon tuning of the density
to a transition from two distinct Fermi geometries which always exist for
doping in different regimes. The existence of a nesting density is thus
guaranteed in real systems across a wide range of materials. This is in
contrast to nested Fermi surfaces appearing in lattice models, in which
nesting can be destroyed e.g. by next-nearest-neighbor hopping in the case of
the square lattice and third neighbor hopping for honeycomb lattices.

In our analysis, realization of nested Fermi surfaces requires a number of
specific conditions: firstly, the cubic symmetry of the lattice must be
maintained in the heterostructure. This implies that the crystal growth axis
must be aligned along a high-symmetry direction. Furthermore, while for Ge and
Si quantum wells, the asymmetry of the transverse confining potential does not
prohibit the existence of nested Fermi surfaces, for zincblende materials such
as GaAs, nesting is completely destroyed when the asymmetry is significant
(see Fig. S7 in the Supplementary Material).

While we have used the perturbative RG method to demonstrate the existence of
ordered phases for weak coupling, our analysis suffers from the common issue
among 2D systems that the couplings may realistically be too large for
perturbative methods to be justified. Our analysis is applicable only for the
case when the superconducting (SC)/CDW/SDW gaps are exponentially smaller than
an ultraviolet scale $\Lambda_{0}$ which is nominally similar to the Fermi
energy, which at the nesting density is $\approx 5$ meV in Ge and $\approx
2.5$ meV in GaAs; the gap is $\Delta\sim\Lambda_{0}e^{-l}$ where $l\gg 1$ is
the RG scale at which the flows enter the strong coupling regime, with the
bare couplings being $\approx l^{-1}$. We have not attempted to determine a
value for $\Delta$, since the precise values of the bare interactions that
enter into our weak-coupling RG flow cannot be accurately predicted. We have
included terms in the scaling relations which contain the logarithmically
divergent contributions to the interaction vertex arising from nesting,
neglecting RPA screening effects as well as vertex corrections, which should
enter at the ultraviolet scale and present a significant modification of the
initial values entering the RG flow. Variation of the well width provides the
possibility of varying the interactions between the weak and strong coupling
regimes, which may reveal a more complex phase diagram than the one we have
established via perturbative methods.

Disorder also poses an important issue for the realization of the ordered
ground states predicted by our RG analysis. In order for the correlated phases
to be observable, the SC/CDW/SDW gaps must be larger than the variation of the
chemical potential due to disorder, i.e.
$\tau\Delta=\tau\varepsilon_{F}e^{-l}>1$. The typical mobilities $\mu\approx
10^{6}$ cm2V-1s-1 observed at densities $p\approx 10^{11}$cm-2 imply a
disorder lifetime $\varepsilon_{F}\tau\approx 200$ in typical experiments
Dobbie2012 . Assuming that $\varepsilon_{F}\tau$ takes similar values for the
higher dopings required to observe nesting (which require an increase in the
Fermi energy by approximately a factor of two), we would expect the correlated
phases to be observable unless the dimensionless initial couplings are very
small, $\tau\Delta<1\rightarrow l^{-1}<0.2$.

We also need to the address the question of whether the nesting densities are
attainable in real systems. While we have related nesting to a transition in
the Fermi surface geometry which occurs at a single density, we have also seen
that the Fermi surfaces exhibit perfect nesting over an extended range of
dopings (see Fig. 2b), thus extreme fine tuning is not required. The nesting
density predicted for GaAs lies within the range currently used for
experiments Marcellina2018 . This leaves open the possibility that the
correlated phases have already been created, although so far there have been
no specific studies to identify them.

##  VI Acknowledgements

H. D. Scammell acknowledges funding from ARC Centre of Excellence FLEET. TL
acknowledges support from the Deutsche Forschungsgemeinschaft.

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Supplementary Figures

Figure S1: Nesting in Si wells. Evolution of the Fermi surface with doping in
a Si quantum well with infinite square well confinement, in zero magnetic
field (a,b,c) and a 1T magnetic field along (010) (d,e,f). (a,d) The 2D
dispersion (in meV) for a $d=20$ quantum well. The red lines indicate the
Fermi energy at the nesting densities. (b,c,e,f): Fermi surfaces at various
hole densities (in units of $10^{10}$ cm-2) (b): $p=4.83,5.67,6.46,7.27$, (c):
$p=7.67,8.08,8.47,8.86$, (e) $p=5.49$, (f) $p=6.28$. Portions of the Fermi
surface on which the normal momentum coordinate varies by less than 0.25% are
highlighted in red. Units of wavevector are $\pi/d$. Figure S2: Nesting in
InAs wells. Nested Fermi surfaces in an InAs quantum well with infinite square
well confinement, in zero magnetic field (a,b,c) and a 1T magnetic field along
(010) (d,e,f). (a,d) The 2D dispersion (in meV) for a $d=20$ quantum well. The
red lines indicate the Fermi energy at the nesting densities. (b,c,e,f): The
Fermi surface for various hole densities (b) $p=2.64\times 10^{11}$ cm-2 (c)
$p=9.56\times 10^{11}$, (e) $p=2.67\times 10^{11}$ cm-2 and (f) $p=5.37\times
10^{11}$ cm-2. Portions of the Fermi surface on which the normal momentum
coordinate varies by less than 0.25% are highlighted in red. Units of
wavevector are $\pi/d$. Figure S3: Nesting in InSb wells. Nested Fermi
surfaces in an InSb quantum well with infinite square well confinement in zero
magnetic field. (a) The 2D dispersion (in meV) for a $d=20$ quantum well. The
red lines indicate the Fermi energy at the nesting densities. (b) The Fermi
surface at the hole density $p=1.42\times 10^{12}$ cm-2. In addition to two
large hole Fermi surfaces, four small electron pockets exist in the upper
band. Nesting occurs between the electron pockets rather than across the
exterior sides of the hole Fermi surfaces. The unnested portions of the
boundaries of the electron pockets are shown in green. No nesting densities
were found for a $B=$1T magnetic field along (010), unlike in the other
materials discussed in the Supplementary Material. Portions of the Fermi
surface on which the normal momentum coordinate varies by less than 0.25% are
highlighted in red. Units of wavevector are $\pi/d$. Figure S4: Nesting in
CdTe wells. Nested Fermi surfaces in a CdTe quantum well with infinite square
well confinement, in zero magnetic field (a,b) and a 1T magnetic field along
(010) (c,d). (a,c) The 2D dispersion (in meV) for a $d=20$ quantum well. The
red lines indicate the Fermi energy at the nesting densities. (b,d) The Fermi
surface for the hole density $p=1.98\times 10^{11}$ cm-2. Portions of the
Fermi surface on which the normal momentum coordinate varies by less than
0.25% are highlighted in red. Units of wavevector are $\pi/d$. Unlike the
other materials studied, no densities were found at which the smaller Fermi
surface was nested. This may be explained by the fact that, for $\bm{k}$ along
the (100) and (010) directions, the curvature
$v_{\perp}^{-1}\partial^{2}\varepsilon/\partial k_{\parallel}^{2}$ (where
$v_{\perp}$ and $k_{\parallel}$ are the velocity perpendicular and momentum
parallel to the equal-energy contours) never vanishes, as can be seen in Fig.
S6. Figure S5: Nesting in ZnSe wells. Nested Fermi surfaces in a ZnSe quantum
well with infinite square well confinement, in zero magnetic field (a,b,c) and
a 1T magnetic field along (010) (d,e,f). (a,d) The 2D dispersion (in meV) for
a $d=20$ quantum well. The red lines indicate the Fermi energy at the nesting
densities. (b,c,e,f): The Fermi surface for various hole densities (b)
$1.53\times 10^{11}$ cm-2 (c) $4.88\times 10^{11}$ cm-2, (e) $p=1.52\times
10^{11}$ cm-2 and (f) $4.86\times 10^{11}$ cm-2. Portions of the Fermi surface
on which the normal momentum coordinate varies by less than 0.25% are
highlighted in red. Units of wavevector are $\pi/d$. Figure S6: Origin of
nesting in cubic semiconductors. The transition from concave to convex Fermi
surfaces occurs as a result of the change of sign of the curvature of the
sides of the equal-energy contours
$v_{\perp}^{-1}\partial^{2}\varepsilon/\partial k_{\parallel}^{2}$ (where
$v_{\perp}$ and $k_{\parallel}$ are the velocity perpendicular and momentum
parallel to the contours). We plot $\partial^{2}\varepsilon/\partial
k_{y}^{2}$ for the (a) lowest and (b) second lowest 2D bands along the
$\bm{k}\parallel(100)$ direction in Si, Ge, InAs, InSb, GaAs, CdTe and ZnSe
quantum wells with infinite square well confinement. The existence of points
at which each curve crosses zero indicates the presence of a transition in the
Fermi surface geometry which is accompanied by nesting. Note that this
transition is absent for the second lowest 2D band in CdTe, indicating that
only one nesting density is present, at which the larger Fermi surface is
nested. Figure S7: Effect of structural inversion asymmetry on nesting in Ge
and GaAs wells. Nested Fermi surfaces for Ge and GaAs quantum wells with
increasing asymmetry, either in zero magnetic field or in a 1T magnetic field
along (010). Portions of the Fermi surface on which the normal momentum
coordinate varies by less than 0.25% are highlighted in red. The densities
chosen are optimal for the nesting of the highlighted pairs of sides. The top
row shows Fermi surfaces for a symmetric 20 nm well, the middle row for a 20
nm well with an electric field $E_{z}=10^{4}$ V m-1, and the bottom row for a
20 nm well with an electric field $E_{z}=2.5\times 10^{4}$ V m-1. The
rightmost figures show the confining potential (solid lines) and the
transverse quantization energy (dashed lines).


# Confidence Is All You Need for MI Attacks

Abhishek Sinha*†, Himanshi Tibrewal*†, Mansi Gupta*†, Nikhar Waghela*†,
Shivank Garg*†

###### Abstract

In this evolving era of machine learning security, membership inference
attacks have emerged as a potent threat to the confidentiality of sensitive
data. In this attack, adversaries aim to determine whether a particular point
was used during the training of a target model. This paper proposes a new
method to gauge a data point’s membership in a model’s training set. Instead
of correlating loss with membership, as is traditionally done, we have
leveraged the fact that training examples generally exhibit higher confidence
values when classified into their actual class. During training, the model is
essentially being ’fit’ to the training data and might face particular
difficulties in generalization to unseen data. This asymmetry leads to the
model achieving higher confidence on the training data as it exploits the
specific patterns and noise present in the training data. Our proposed
approach leverages the confidence values generated by the machine-learning
model. These confidence values provide a probabilistic measure of the model’s
certainty in its predictions and can further be used to infer the membership
of a given data point. Additionally, we also introduce another variant of our
method that allows us to carry out this attack without knowing the ground
truth(true class) of a given data point, thus offering an edge over existing
label-dependent attack methods.

## Introduction

In recent years, using MI (Membership Inference) attacks to predict whether a
specific training example was used as training data in a particular model has
garnered significant momentum. Currently, approaches rely predominantly on
loss-based techniques, rooted in the fundamental concept that data points in
the training set exhibit lower loss when processed by the target model. In
(Shokri et al. 2017), various strategies for executing MI attacks from first
principles are outlined. The LOSS attack (Yeom et al. 2018), which initially
showed promise with an accuracy of 60%, fails to identify members effectively
in CIFAR-10, rendering it impractical for membership inference. Following
this, the Likelihood Ratio Test, (Carlini et al. 2022), was introduced,
improving the attack’s success across various metrics, including accuracy and
AUC. This method essentially computes the loss on a given data point and
measures the likelihood of this loss under two worlds, one where a model is
trained on this data point and the other where it is not.
Building upon this concept, we propose using confidence values and measuring
the likelihood of the confidence values under the distribution of models
containing a given data point against the distribution of models that have
never seen the given data point. Observing that the training set samples
should ideally display a higher output confidence level is crucial. This
compels us to investigate an alternative classification method based on
confidence levels.This strategy is based on the premise that training examples
should exhibit a specific level of confidence, and deviations from this
pattern may indicate group membership or non-membership, as the model, through
training, is learning the characteristics of the training data. In addition,
traditional membership inference attacks rely on knowledge of the actual class
label(ground truth) of the data point whose membership we infer. We
investigate the possibility of removing the requirement of authentic
identifiers from our classification strategy. This can be achieved by
selecting the class with the highest predicted probability, effectively
indicating the model’s most confident prediction for the given input data
point. This can be used as a proxy for the confidence value, and further, the
attack can be carried out in the same way as before.

| Loss Value (Baseline) | Confidence Values | log(Confidence Values) | Argmax | log(Argmax)
---|---|---|---|---|---
Attack (Online) | 0.5753 | 0.5668 | 0.575 | 0.5464 | 0.5447
Attack (Online, Fixed Variance) | 0.5879 | 0.593 | 0.6009 | 0.5622 | 0.5602
Attack (Offline) | 0.5181 | 0.492 | 0.4721 | 0.478 | 0.4756
Attack (Offline, Fixed Variance) | 0.5184 | 0.4928 | 0.4804 | 0.4834 | 0.4815
Attack Global Threshold | 0.5448 | 0.5439 | 0.5469 | 0.5376 | 0.5377
Table 1: Using AUC as metric

## Methodology

Drawing reference from (Carlini et al. 2022), we commenced our process by
training 16 shadow models, for 21 epochs each, on random samples drawn from
the encompassing data distribution $D$. Precisely half of these models were
trained on a target point $(x,y)$, whose membership or non-membership we
sought to establish. The remaining models were trained without involvement
with this particular point (These are the IN and OUT models, respectively).
Then, we fit two Gaussian distributions to the confidence of the OUT and IN
models on $(x,y)$ (
$Q\stackrel{{\scriptstyle\smash{\raisebox{-3.01389pt}{\kern-2.2604ptout}}}}{{}}$
and
$Q\stackrel{{\scriptstyle\smash{\raisebox{-3.01389pt}{\kern-2.2604ptin}}}}{{}}$).
While earlier the assessment of ‘confidence’ was accomplished by employing a
logit scaled loss function (baseline), we used a more direct and intuitive
method that aligns more closely with the inherent properties of a model’s
prediction. Both of our proposed methods are based on the intuition that
training examples exhibit higher output confidence scores than out-of-training
data points. If we let $f(x)$ denote the evaluation of the model along with a
final softmax activation function, we initially used the confidence
probability associated with the actual class of the sample and studied the
results. It is important to note that f(x) is a c-dimensional column vector
where c denotes the number of classes and $y_{\text{true}}$ refers to the
actual class of the target point.

$\text{confidence}=\log\left(c=y_{\text{true}}(f(x))+1\times 10^{-45}\right)$
(1)

This is logically sound as a higher confidence value calculated using our
proposed metric, indicates a higher likelihood of belonging to either
$Q\stackrel{{\scriptstyle\smash{\raisebox{-3.01389pt}{\kern-2.2604ptout}}}}{{}}$
or
$Q\stackrel{{\scriptstyle\smash{\raisebox{-3.01389pt}{\kern-2.2604ptin}}}}{{}}$,
thus reinforcing the validity of our methodology. As an alternative path, we
leveraged the application of the argmax operation to the vector, $f(x)$, and
finally attributed the logit scaled, the maximum probabilistic value obtained
as ‘confidence’. This eliminates the need to know the actual labels of the
data point, thereby freeing us from label-dependent constraints.

$\text{confidence}=\log(\text{argmax}(f(x)))$ (2)

Subsequently, a parametric likelihood ratio test was employed as the
conclusive step in our methodology to infer the membership of the target
point. This Likelihood Ratio Test was used as it is proven more powerful at
low false positive rates, making the attack more versatile.

## Results

In (Carlini et al. 2022), the authors employed LOSS Values (baseline) for
fitting IN and OUT models. In reference to Table 1, the first column shows the
baseline, while the next four columns display our results. Our findings
demonstrate that we have achieved AUC metrics that are commensurate with the
anticipated performance levels.

As indicated above, our approach has led to successful results in various
evaluation metrics. A good AUC shows that our methods excel in distinguishing
between positive and negative classes across varying decision thresholds.
Further, due to the lack of availability of computational resources, our
results are restricted. They can additionally be improved to a great extent by
scaling up the number of shadow models and increasing the epoch count.

## Future Work

Modifications to the baseline LiRA ratio can be explored to enhance our
approach further. One avenue for improvement involves considering the
difference between confidence values obtained for the IN and OUT models.
Instead of employing logarithmic scaling for confidence values, we could
further investigate other transformations that may help us model a Gaussian
distribution from the confidence values. Our confidence methodology can also
be further applied to other membership inference (MI) attacks. Owing to
computational constraints, our model was trained for a limited duration of 21
epochs. Consequently, there exists the potential for further enhancements in
results through the allocation of additional computational resources.

Figure 1: :

Comparison of different methods using AUC as metric
(Color coding: Blue = Attack Global Threshold
Gold=Attack (offline,fixed Variance)
Grey= Attack (offline)
Orange = Attack (online,fixed variance)
Dark Blue= Attack (online) )

## Conclusion

The above table demonstrates that we have achieved results on par with those
derived from the LiRA baseline approach. The assurance of our attack’s success
remains steadfast, and our second methodology also allows us to carry out this
attack without knowing the actual labels of data points, making the use of
confidence values a viable metric in the case of MIA attacks.

## References

  * Carlini et al. (2022) Carlini, N.; Chien, S.; Nasr, M.; Song, S.; Terzis, A.; and Tramer, F. 2022.  Membership inference attacks from first principles.  In _2022 IEEE Symposium on Security and Privacy (SP)_ , 1897–1914. IEEE.
  * Shokri et al. (2017) Shokri, R.; Stronati, M.; Song, C.; and Shmatikov, V. 2017.  Membership inference attacks against machine learning models.  In _2017 IEEE symposium on security and privacy (SP)_ , 3–18. IEEE.
  * Yeom et al. (2018) Yeom, S.; Giacomelli, I.; Fredrikson, M.; and Jha, S. 2018.  Privacy risk in machine learning: Analyzing the connection to overfitting.  In _2018 IEEE 31st computer security foundations symposium (CSF)_ , 268–282. IEEE.


Consistent couplings between a massive spin-3/2 field and a partially massless
spin-2 field

Nicolas Boulanger, Guillaume Lhost and Sylvain Thomée111Research Fellow of the
F.R.S.-FNRS (Belgium).

Service de Physique de l’Univers, Champs et Gravitation,
Université de Mons – UMONS, 20 place du Parc, 7000 Mons, Belgium

<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
Abstract

We revisit the problem of constructing consistent interactions between a
massive spin-3/2 field and a partially massless graviton in four-dimensional
(A)dS spacetime. We use the Stueckelberg formulation of the action principle
for these fields and find two non-trivial cubic vertices with less than two
derivatives, when going to the unitary gauge. One of the vertices is
reminiscent of the minimal coupling of the massive spin-3/2 field to gravity,
except that now the graviton is partially massless.

##  1 Introduction

The use of the Stueckelberg formulation for the problem of constructing
consistent interactions between massive fields has proved very efficient,
mainly through the works of Zinoviev and collaborators, see e.g. [1, 2, 3] and
references therein. Some years ago in [4], Zinoviev constructed cubic
couplings between a massive spin-3/2 field and a massive spin-2 graviton
around (A)dS4 background, with the assumption that the couplings should not
bring more than one derivative. Then, the partially-massless limit for the
graviton was considered, bringing the conclusion that no cubic vertex with one
derivative survived in this limit.

We wish to revisit this question, this time taking the spin-2 field as
partially massless from the very beginning of the analysis. Indeed, the
operations of introducing interactions and taking a partially massless limit
do not commute, in general. In fact, in this Letter we report about a coupling
between a partially massless spin-2 field and a massive spin-3/2 field that
seems to have gone unnoticed in previous investigations, as far as we could
see.

In order to build consistent vertex involving massive fields (in the present
case, the massive spin-3/2 field), we use the method proposed in [5] that
combines the cohomological reformulation of the Noether method for gauge
systems [6, 7] with the Stueckelberg formulation for massive fields. The
advantage of the method [5] is that it exploits the gauge structure of
massless theories to describe interactions for massive fields. It proved
useful in showing that dRGT gravity (see e.g. [8] for a review) can be recast
in a frame where the Einstein-Hilbert structure disappears, leaving only a
Born-Infeld-like theory, with vertices obtained by contraction of products of
the manifestly gauge-invariant field strengths in the Stueckelberg formulation
of the free massive theory. In particular, in [5] the full list of cubic
vertices of massive dRGT gravity theory was recovered, showing the usefulness
of the method.

In this Letter, we use the cohomological method of [5] for the search of
consistent couplings between a massive spin-$3/2$ field and a partially
massless (PM) spin-$2$ field, also called PM graviton. The spacetime
backgrounds considered are the Anti-de Sitter (AdS) and the de Sitter (dS)
geometries where the cosmological constant $\Lambda$ is negative and positive,
respectively. In such spacetimes, a PM graviton possesses four propagating
degrees of freedom and is characterized by a well-suited mass directly related
to the cosmological constant $\Lambda\,$, see [9, 10, 11] and references
therein. In its PM phase, the graviton therefore propagates four degrees of
freedom, exactly like the massive spin-3/2 field that carries four degrees of
freedom. It is therefore natural to ask whether it is possible to elaborate a
consistent gauge theory in which a PM graviton $k_{\mu\nu}$ and a massive
spin-$3/2$ field $\psi_{\mu}$ are involved.

We study the problem in both dS ($\Lambda>0$) and AdS ($\Lambda<0$)
backgrounds at a stroke, through the use of a parameter $\sigma$ that takes
the value $+1$ in AdS and $-1$ in dS.

##  2 Main results

The main results we report in this Letter consist in the construction of two
vertices expressed in the Stueckelberg formulation for both the PM spin-2 and
the massive spin-3/2 fields. In the unitary gauge where the Stueckelberg
fields are set to zero, the first vertex is proportional to

$\ell^{(1)}=2\,\nabla_{[\mu}k_{\nu]\rho}\,{\overline{\psi}}{}^{\mu}\,\gamma^{\rho}\,\psi^{\nu}\;,$
(1)

where the spinor field $\psi_{\mu}$ satisfies the Majorana reality condition
and denotes the field for the massive spin-3/2 particle (the spinor indices
are left implicit), the Lorentz-covariant derivative for the background
geometry222We use conventions whereby the Lorentz-covariant derivative
satisfies
$[\nabla_{\mu},\nabla_{\nu}]V^{\rho}=-2\sigma\,\lambda^{2}\,\delta^{\rho}{}_{[\mu}\,V_{\nu]}\,$,
where $\sigma=\pm 1\,$. In other terms, the cosmological constant is
$\Lambda=-3\,\sigma\,\lambda^{2}\,$ in four dimensions, where $\sigma=-1$
corresponds to dS4 and $\sigma=1$ to AdS${}_{4}\,$. On a Dirac spinor
$\psi\,$, we have
$[\nabla_{\mu},\nabla_{\nu}]\psi=-\tfrac{1}{2}\,\sigma\,\lambda^{2}\,\gamma_{\mu\nu}\,\psi\,$.
is denoted by the symbol $\nabla_{\mu}\,$, and $k_{\mu\nu}$ represents the PM
spin-2 field. The components of the (A)dS background metric will be denoted
$\overline{g}_{\mu\nu}\,$. As usual, the four Dirac matrices are denoted by
$\gamma^{a}\,$, $a=0,1,2,3\,$, and
$\gamma^{\mu}:=\overline{e}^{\mu}{}_{a}\,\gamma^{a}\,$ featuring the
components of the background (A)dS vierbein.

The other coupling is more interesting. In the unitary gauge, it reads

$\displaystyle\ell^{(2)}=k_{\mu\nu}\,\mathcal{T}^{\mu\nu}\;,\quad\mathcal{T}^{\mu\nu}=$
$\displaystyle\;\omega\,\overline{\psi}_{\rho}~{}\gamma^{\rho(\mu}\,\psi^{\nu)}+\overline{\psi}_{\sigma}\,\gamma^{\rho\sigma(\mu}\,\nabla_{\rho}\psi^{\nu)}\;,\quad\omega:=\sqrt{m^{2}+\sigma\lambda^{2}}\,\;,$
(2)

where the real parameter $m$ is the mass of the spin-3/2 field in AdS, in the
sense that the limit $m\rightarrow 0$ is the limit where the spin-3/2 field
enjoys a gauge symmetry that removes the helicity $\pm 1/2$ degrees of
freedom, leaving only the helicity $\pm 3/2$ degrees of freedom on-shell. The
tensor $\mathcal{T}^{\mu\nu}$ is traceless and divergenceless on-shell:

$\overline{g}_{\mu\nu}\,\mathcal{T}^{\mu\nu}\approx
0\quad,\qquad\nabla_{\mu}\mathcal{T}^{\mu\nu}\approx 0\;,$ (3)

where a weak equality is an equality that holds on the solutions of the field
equations for the free theory. The above vertex
$\ell^{(2)}=k_{\mu\nu}\,\mathcal{T}^{\mu\nu}\,$ induces a deformation of the
gauge transformations on the physical fields $(k_{\mu\nu},\psi_{\mu})\,$ in
the unitary gauge, given by

$\displaystyle\delta_{1}\psi_{\mu}=-\psi^{\nu}\,\nabla_{\mu}\nabla_{\nu}\xi+\sigma\,\lambda^{2}\,\psi_{\mu}\,\xi\;,\qquad\delta_{1}k_{\mu\nu}=0\;,$
(4)

where we recall that the free, quadratic action $S_{0}[k,\psi]$ is invariant
under [9, 10, 11]

$\displaystyle\delta_{0}k_{\mu\nu}=\nabla_{\mu}\nabla_{\nu}\xi-\sigma\,\lambda^{2}\,\overline{g}_{\mu\nu}\,\xi\;,\qquad\delta_{0}\psi_{\mu}=0\;.$
(5)

From the knowledge of the quadratic and cubic actions $S_{0}[k,\psi]$ and
$S_{1}[k,\psi]=\int\\!d^{4}x\sqrt{\bar{g}}\,\ell^{(2)}\,$ in the unitary
gauge, one readily finds the consistency of the deformation reported above:

$\displaystyle\delta_{0}S_{1}[k,\psi]+\delta_{1}S_{0}[k,\psi]=0\;.$ (6)

As far as the deformation $\ell^{(2)}$ is concerned, notice from (4) and (5)
that the transformation of the massive spin-3/2 field can be written as
$\delta_{1}\psi_{\mu}=-\,\psi^{\nu}\,\delta_{0}k_{\mu\nu}\,$, from which it is
tempting to view the contravariant spinor $\psi^{\mu}$ as a gauge-invariant
quantity, defining the covariant field as
$\psi_{\mu}:=\psi^{\nu}(\overline{g}_{\mu\nu}-\kappa\,k_{\mu\nu}+{\cal
O}(\kappa^{2}))\,$, where $\kappa$ is the deformation parameter that we take
with units of length, that defines the perturbative expansion
$S[\phi]=S_{0}[\phi]+\kappa\,S_{1}[\phi]+{\cal O}(\kappa^{2})\,$,
$\delta\phi=\delta_{0}\phi+\kappa\,\delta_{1}\phi+{\cal O}(\kappa^{2})\,$ such
that $\delta S[\phi]=0+{\cal O}(\kappa^{2})\,$. In this sense, like in
Riemannian geometry, it would appear that a metric
$g_{\mu\nu}:=\overline{g}_{\mu\nu}-\kappa\,k_{\mu\nu}+{\cal O}(\kappa^{2})$
could be defined in terms of the (A)dS background metric and the PM spin-2
field $k_{\mu\nu}\,$.

In the following section, we present the two deformations reported above in
their Stueckelberg form, and explain how one can reach the unitary gauge at
first order in perturbation, thereby reproducing the main results presented
above.

##  3 Consistent couplings in the Stueckelberg formulation

In this section we first spell out the free model, then exhibit the first
order interactions we found and finally explain how one can reach the unitary
gauge at first order in deformation.

### The free model

We want to investigate the couplings between a massive spin-3/2 field and a PM
spin-2 field. Our starting point will be the Stueckelberg formulation for
these models. The Stueckelberg action for a massive spin-3/2 field and a PM
spin-2 field in (A)dS4 reads

$\displaystyle S_{0}[k_{\mu\nu},B_{\mu},\psi_{\mu},\chi]=\int
d^{4}x\,\sqrt{-\bar{g}}{}$
$\displaystyle\Big{(}-\frac{1}{2}\nabla_{\rho}k^{\mu\nu}\nabla^{\rho}k^{\mu\nu}+\nabla_{\rho}k^{\mu\nu}\nabla_{\mu}k^{\rho}_{~{}\nu}-\nabla_{\mu}k\nabla_{\nu}k^{\mu\nu}$
(7)
$\displaystyle+\frac{1}{2}\nabla_{\mu}k\nabla^{\mu}k+\frac{\sigma}{4}F_{\mu\nu}F^{\mu\nu}-2\sigma{\lambda}^{2}(k_{\mu\nu}k^{\mu\nu}-\frac{1}{4}k^{2})$
$\displaystyle+2{\lambda}~{}[k\nabla_{\mu}B^{\mu}-k^{\mu\nu}\nabla_{\mu}B_{\nu}]+3\lambda^{2}\,B^{\mu}B_{\mu}\Big{)}$
$\displaystyle+\Big{(}-\frac{1}{2}~{}\overline{\psi}_{\mu}~{}\gamma^{\mu\nu\rho}~{}\nabla_{\nu}\psi_{\rho}+\frac{\omega}{2}~{}\overline{\psi}_{\mu}~{}\gamma^{\mu\nu}~{}\psi_{\nu}-\frac{3\omega}{2}~{}\overline{\chi}\chi$
$\displaystyle-\frac{3}{4}~{}\overline{\chi}~{}\gamma^{\mu}~{}\nabla_{\mu}\chi-\frac{3m}{2}\,\overline{\psi}_{\mu}~{}\gamma^{\mu}~{}\chi\Big{)}\,,$

where $F_{\mu\nu}=2\,\nabla_{[\mu}B_{\nu]}\,$. For the spin-$2$ sector, we use
the conventions of [12]. The vector field $B_{\mu}$ and the Majorana spinor
$\chi$ are, respectively, the Stueckelberg companions of $k_{\mu\nu}$ and
$\psi_{\mu}\,$. The action (7) is invariant under the following Abelian gauge
transformations:

$\displaystyle\delta_{0}k_{\mu\nu}$
$\displaystyle=2\,\nabla_{(\mu}\varepsilon_{\nu)}+\lambda\,\bar{g}_{\mu\nu}\,\pi\;,\quad\delta_{0}B_{\mu}=\nabla_{\mu}\pi+2\sigma\lambda\,\varepsilon_{\mu}\;,$
(8) $\displaystyle\delta_{0}\psi_{\mu}$
$\displaystyle=\nabla_{\mu}\theta+\frac{\omega}{2}\,\gamma_{\mu}\,\theta\;,\quad\delta_{0}\chi=m\,\theta\;.$

It is useful to introduce the gauge-invariant quantities

$\displaystyle\mathbf{\Psi}_{\mu}$
$\displaystyle:=\psi_{\mu}-\tfrac{1}{m}\,\nabla_{\mu}\chi-\tfrac{\omega}{2m}\,\gamma_{\mu}\,\chi\;,$
(9) $\displaystyle K_{\mu\nu\rho}$
$\displaystyle:=2\,\nabla_{[\mu}k_{\nu]\rho}+2\,\lambda\,\bar{g}_{\rho[\mu}B_{\nu]}-\tfrac{\sigma}{2\lambda}\,\nabla_{\rho}F_{\mu\nu}\;,$

in terms of which the free action (7) can be written in a manifestly gauge
invariant way:

$\displaystyle S_{0}[k_{\mu\nu},B_{\mu},\psi_{\mu},\chi]=\int
d^{4}x\,\sqrt{-\bar{g}}\,L_{0}=\frac{1}{2}\,\int
d^{4}x\,\sqrt{-\bar{g}}\,\Big{(}$
$\displaystyle-\overline{\mathbf{\Psi}}_{\mu}\;\gamma^{\mu\nu\rho}~{}\nabla_{\nu}\mathbf{\Psi}_{\rho}+\omega\,\overline{\mathbf{\Psi}}_{\mu}\;\gamma^{\mu\nu}\;\mathbf{\Psi}_{\nu}$
(10)
$\displaystyle-\tfrac{1}{2}\;K_{\mu\nu\rho}K^{\mu\nu\rho}+K^{\mu}K_{\mu}\Big{)}\;,$

where $K^{\mu}=\bar{g}^{\alpha\beta}K^{\mu}{}_{\alpha\beta}\,$. The equations
of motion obtained by extremizing the above action with respect to the fields
$k_{\mu\nu}$ and $\psi_{\mu}$ are, respectively,

$\displaystyle\nabla_{\rho}K^{\rho(\mu\nu)}-\overline{g}^{\mu\nu}\nabla_{\rho}K^{\rho}+\nabla^{(\mu}K^{\nu)}\approx\;$
$\displaystyle 0\;,$ (11)
$\displaystyle\nabla_{\nu}\overline{\mathbf{\Psi}}_{\rho}\,\gamma^{\mu\nu\rho}+\omega\,\overline{\mathbf{\Psi}}_{\nu}\,\gamma^{\mu\nu}\approx\;$
$\displaystyle 0\;.$ (12)

These equations are useful in checking the consistency of the vertex
$\ell^{(2)}\,$.

### Interactions to first order

We now report our main findings obtained following the method proposed in [5]
for constructing interactions of massive fields in the Stueckelberg
formulation. For the sake of conciseness, in this Letter we refrain from
reviewing this method and instead spell out the results that can be checked
without referring to the formalism developed in [5].

  * •

In the Stueckelberg formulation, the deformation $L^{(1)}_{1}$ of the
Lagrangian that corresponds to the first vertex $\ell^{(1)}$ presented in the
Introduction, reads

$\displaystyle{L}^{(1)}_{1}$
$\displaystyle=\overline{\mathbf{\Psi}}_{\sigma}\,\gamma^{\rho}\,{\mathbf{\Psi}}_{\alpha}\,K^{\sigma\alpha}{}_{\rho}\;,$
(13)

up to trivial field redefinitions. The coefficient in front of it is, at this
stage, arbitrary.

  * •

In the Stueckelberg formulation, the deformation ${L}^{(2)}_{1}$ of the
Lagrangian that corresponds to the second vertex $\ell^{(2)}$ presented in the
Introduction reads

$\displaystyle{L}^{(2)}_{1}=$
$\displaystyle\omega\,\overline{\mathbf{\Psi}}_{\mu}\;\gamma^{\mu\nu}\;\mathbf{\Psi}^{\rho}\,k_{\rho\nu}-\tfrac{\omega^{2}}{m}\,\overline{\mathbf{\Psi}}^{\mu}\,\gamma^{\nu}\,\chi\,k_{\mu\nu}+\tfrac{\omega^{2}}{m}\,\overline{\mathbf{\Psi}}_{\mu}\,\gamma^{\mu}\,\chi\,k$
(14)
$\displaystyle\;+\overline{\mathbf{\Psi}}_{\sigma}\,\gamma^{\sigma\nu\rho}\,\nabla_{\rho}\mathbf{\Psi}^{\mu}\,k_{\mu\nu}-\tfrac{2\,\omega}{m}\,\nabla_{[\rho}\overline{\mathbf{\Psi}}_{\sigma]}\,\gamma^{\sigma\alpha}\chi\,k^{\rho}{}_{\alpha}-\tfrac{\omega}{m}\,\nabla_{\rho}\overline{\mathbf{\Psi}}_{\sigma}\,\gamma^{\rho\sigma}\chi\,k$
$\displaystyle\;-\tfrac{\sigma\,\omega}{2\lambda}\,\overline{\mathbf{\Psi}}_{\sigma}\gamma^{\rho\sigma}\mathbf{\Psi}^{\alpha}F_{\rho\alpha}-\tfrac{\sigma}{2\lambda}\,\overline{\mathbf{\Psi}}_{\sigma}\gamma^{\alpha\rho\sigma}\nabla_{\alpha}\mathbf{\Psi}^{\mu}F_{\rho\mu}\;.$

Contrarily to the free Stueckelberg theory where the flat limit is smooth, in
the interacting case one cannot take the limit $\lambda\rightarrow 0\,$, as
the vertex is non-analytical in the constant $\lambda\,$.

  * •

The above vertex induces a deformation of the gauge transformations given by

$\displaystyle\delta_{1}\psi_{\mu}=$
$\displaystyle\;-\omega\,k_{\mu\nu}\,\gamma^{\nu}\,\theta+\omega\,\gamma_{\mu}\,\psi^{\nu}\,\varepsilon_{\nu}-\tfrac{\omega\,\lambda}{m}\,\gamma_{\mu}\,\chi\,\pi-\lambda\,\psi_{\mu}\,\pi$
$\displaystyle-\tfrac{\omega}{m}\,\gamma^{\nu}\,\nabla_{\nu}\varepsilon_{\mu}\,\chi-\tfrac{\omega}{m}\,\gamma_{\mu}\,\nabla_{\nu}\chi\,\varepsilon^{\nu}+2\,\nabla_{\mu}\psi_{\nu}\,\varepsilon^{\nu}-\tfrac{\lambda}{m}\,\chi\,\nabla_{\mu}\pi$
$\displaystyle-\tfrac{2}{m}\,\nabla_{\mu}\nabla_{\nu}\chi\,\varepsilon^{\nu}\;,$
(15) $\displaystyle\delta_{1}\chi=$
$\displaystyle\;2\,m\,\psi_{\mu}\,\varepsilon^{\mu}-2\,\varepsilon^{\mu}\,\nabla_{\mu}\chi-\lambda\,\chi\,\pi\;,$
(16) $\displaystyle\delta_{1}k_{\mu\nu}=$
$\displaystyle\;0\;,\quad\delta_{1}B_{\mu}=0\;.$ (17)

The corresponding gauge algebra is

$\displaystyle[\delta_{\theta},\delta_{\varepsilon}]\varphi$
$\displaystyle=\delta_{\tilde{\theta}}\varphi\;,\qquad\tilde{\theta}=2\,\omega\,\gamma^{\mu}\theta\,\varepsilon_{\mu}\;,$
(18) $\displaystyle[\delta_{\theta},\delta_{\pi}]\varphi$
$\displaystyle=\delta_{\overline{\theta}}\varphi\;,\qquad\overline{\theta}=-2\,\lambda\,\theta\,\pi\;.$
(19)

The redefinition of the gauge parameters that trivializes the gauge algebra is

$\theta\quad\rightarrow\quad\theta-\kappa\,\frac{\omega}{m}\,\gamma^{\mu}\,\chi\,\varepsilon_{\mu}+\kappa\,\frac{\lambda}{m}\,\chi\,\pi\;.$
(20)

In order to express our results in the unitary gauge, we first need to explain
how to reach the unitary gauge, in perturbation.

### Reaching the unitary gauge at first order in deformation

The starting point is a free theory $S_{0}[\varphi^{i},\chi^{I}]$ with a
spectrum of fields $(\varphi^{i},\chi^{I})$ such that the latter are
Stueckelberg companions of the former. In other words, the action $S_{0}=\int
d^{n}x\,{\cal L}_{0}$ is invariant under the gauge transformations

$\displaystyle\delta_{0}\varphi^{i}$
$\displaystyle={R}_{0}{}^{i}{}_{I}\,\varepsilon^{I}+{R}_{0}{}^{i}{}_{\alpha}\,\epsilon^{\alpha}\;,$
(21) $\displaystyle\delta_{0}\chi^{I}$
$\displaystyle=m_{I}\,\varepsilon^{I}+{R}_{0}{}^{I}{}_{\alpha}\,\epsilon^{\alpha}\;,$
(22)

where we use De Witt’s condensed notation. The gauge invariance under the
Stueckelberg gauge parameters $\varepsilon^{I}$ imply the Noether identities

$\displaystyle\frac{\delta{\cal
L}_{0}}{\delta\chi^{I}}\equiv-\frac{1}{m_{I}}\,R_{0}^{+}{}^{i}{}_{I}\frac{\delta{\cal
L}_{0}}{\delta\varphi^{i}}\;,$ (23)

where the operator $R_{0}^{+}{}^{i}{}_{I}$ denotes the adjoint of
$R_{0}{}^{i}{}_{I}\,$.

We assume we also have a consistent, first order deformation of the action and
gauge transformations, i.e., a functional $S_{1}[\varphi^{i},\chi^{I}]=\int
d^{n}x\,{\cal L}_{1}$ and gauge transformation laws

$\displaystyle\delta_{1}\varphi^{i}$
$\displaystyle=R_{1}{}^{i}{}_{I}(\varphi,\chi)\,\varepsilon^{I}+R_{1}{}^{i}{}_{\alpha}(\varphi,\chi)\,\epsilon^{\alpha}\;,$
(24) $\displaystyle\delta_{1}\chi^{I}$
$\displaystyle=R_{1}{}^{I}{}_{J}(\varphi,\chi)\,\varepsilon^{J}+R_{1}{}^{I}{}_{\alpha}(\varphi,\chi)\,\epsilon^{\alpha}\;,$
(25)

such that

$\displaystyle\delta_{1}S_{0}[\varphi^{i},\chi^{I}]+\delta_{0}S_{1}[\varphi^{i},\chi^{I}]=0\;.$
(26)

Upon expanding the latter equation using (21)-(25) gives the following Noether
identity associated with the gauge parameters $\varepsilon^{I}\,$:

$\displaystyle\frac{\delta{\cal
L}_{1}}{\delta\chi^{I}}\equiv-\frac{1}{m_{I}}\,\Big{(}R_{0}^{+}{}^{i}{}_{I}\,\frac{\delta{\cal
L}_{1}}{\delta\varphi^{i}}+[R_{1}^{+}{}^{i}{}_{I}(\varphi,\chi)-\frac{1}{m_{J}}\,R_{1}^{+}{}^{J}{}_{I}(\varphi,\chi)R_{0}^{+}{}^{i}{}_{J}]\,\frac{\delta{\cal
L}_{0}}{\delta\varphi^{i}}\Big{)}\;.$ (27)

Inserting this expression for $\frac{\delta{\cal L}_{1}}{\delta\chi^{I}}$ in
the equation (26) yields

$\displaystyle 0=\int d^{n}x\,\epsilon^{\alpha}\,\Big{[}{\cal
R}^{+}_{0}{}^{i}{}_{\alpha}\frac{\delta{\cal L}_{1}}{\delta\varphi^{i}}+{\cal
R}^{+}_{1}{}^{i}{}_{\alpha}(\varphi,\chi)\,\frac{\delta{\cal
L}_{0}}{\delta\varphi^{i}}\Big{]}\;,$ (28)

where

$\displaystyle{\cal R}_{0}{}^{i}{}_{\alpha}$
$\displaystyle=R_{0}{}^{i}{}_{\alpha}-\frac{1}{m_{I}}\,R_{0}{}^{i}{}_{I}\,R_{0}{}^{I}{}_{\alpha}\;,$
(29) $\displaystyle{\cal R}_{1}{}^{i}{}_{\alpha}(\varphi,\chi)$
$\displaystyle=R_{1}{}^{i}{}_{\alpha}(\varphi,\chi)-\frac{1}{m_{I}}\,R_{1}{}^{i}{}_{I}(\varphi,\chi)\,R_{0}{}^{I}{}_{\alpha}-\frac{1}{m_{I}}\,R_{0}{}^{i}{}_{I}\,R_{1}{}^{I}{}_{\alpha}(\varphi,\chi)$
$\displaystyle\quad+\;\frac{1}{m_{I}m_{J}}\,R_{0}{}^{i}{}_{J}\,R_{1}{}^{J}{}_{I}(\varphi,\chi)R_{0}{}^{I}{}_{\alpha}\;.$
(30)

Equation (28) expresses the $\epsilon^{\alpha}$-gauge invariance of the action
$S[\varphi^{i},\chi^{I}]=S_{0}[\varphi^{i},\chi^{I}]+g\,S_{1}[\varphi^{i},\chi^{I}]\,$,
to first order in perturbation — here $g$ denotes the coupling constant used
in perturbation. This equation is valid for an arbitrary field configuration,
in particular it is valid when we set the Stueckelberg fields $\chi^{I}$ to
zero. Using the following obvious equality

$\displaystyle\frac{\delta{\cal
L}_{1}}{\delta\varphi^{i}}(\varphi,\chi=0)=\frac{\delta\check{{\cal
L}}_{1}}{\delta\varphi^{i}}(\varphi)\;,\qquad\check{{\cal L}}_{1}:=\left.{\cal
L}_{1}\right|_{\chi=0}\;,$ (31)

gives

$\displaystyle 0=\int d^{n}x\,\epsilon^{\alpha}\,$ $\displaystyle\Big{[}{\cal
R}^{+}_{0}{}^{i}{}_{\alpha}\frac{\delta\check{{\cal
L}}_{1}}{\delta\varphi^{i}}+\check{\cal
R}^{+}_{1}{}^{i}{}_{\alpha}(\varphi)\,\frac{\delta\check{{\cal
L}}_{0}}{\delta\varphi^{i}}\Big{]}\;,$ (32) $\displaystyle\check{{\cal
L}}_{0}:=\left.{\cal L}_{0}\right|_{\chi=0}\;,\quad\check{\cal
R}_{1}{}^{i}{}_{\alpha}(\varphi):={\cal
R}_{1}{}^{i}{}_{\alpha}(\varphi,\chi=0)\;.$

In its turn, the latter equation expresses the gauge invariance, to first
order in perturbation, of the reduced action
$\check{S}[\varphi^{i}]=S_{0}[\varphi^{i},\chi^{I}=0]+g\,S_{1}[\varphi^{i},\chi^{I}=0]$
under the gauge transformations

$\displaystyle\delta_{\epsilon}\varphi^{i}=[{\cal
R}_{0}{}^{i}{}_{\alpha}\,+g\,\check{\cal
R}_{1}{}^{i}{}_{\alpha}(\varphi)]\,\epsilon^{\alpha}+{\cal O}(g^{2})\;.$ (33)

In the particular case studied in this Letter where we have the physical
fields $\varphi^{i}=\\{k_{\mu\nu},\psi_{\mu}\\}$ and the Stueckelberg fields
$\chi^{I}=\\{B_{\mu},\chi\\}\,$ with the gauge transformations at zeroth and
first order given in (8) and (15)-(17), respectively, we find the equation
(6): the reduced action is invariant under (4)-(5) as announced in the
Introduction, where we renamed the scalar parameter $\pi$ into $\xi\,$,
absorbing in it the constant factor $-\frac{\sigma}{\lambda}\,$.

From the above-derived formula (33) for the gauge transformations
$\delta_{\epsilon}\varphi^{i}$ that leave invariant the reduced action, one
can make an observation on the corresponding transformations of the
Stueckelberg field strengths

$\displaystyle\Phi^{i}:=\varphi^{i}-\tfrac{1}{m_{J}}\,{R}_{0}{}^{i}{}_{J}\,\chi^{J}\;.$
(34)

As is well-known, these quantities are invariant under the Stueckelberg
transformations

$\displaystyle\delta^{\varepsilon}_{0}\varphi^{i}$
$\displaystyle={R}_{0}{}^{i}{}_{I}\,\varepsilon^{I}\;,\qquad\delta^{\varepsilon}_{0}\chi^{I}=m_{I}\,\varepsilon^{I}\;.$
(35)

Under the complete transformation laws (21), (22), (24) and (25), the
Stueckelberg field strengths $\Phi^{i}$ transform as

$\displaystyle\delta\Phi^{i}=$
$\displaystyle\;\mathcal{R}_{0}{}^{i}{}_{\alpha}\epsilon^{\alpha}+g\,\Bigl{(}R_{1}{}^{i}{}_{\alpha}(\varphi,\chi)\epsilon^{\alpha}+R_{1}{}^{i}{}_{I}(\varphi,\chi)\varepsilon^{I}$
(36)
$\displaystyle-\frac{1}{m_{I}}\,R_{0}{}^{i}{}_{I}\,R_{1}{}^{I}{}_{\alpha}(\varphi,\chi)\epsilon^{\alpha}-\frac{1}{m_{J}}\,R_{0}{}^{i}{}_{J}\,R_{1}{}^{J}{}_{I}(\varphi,\chi)\varepsilon^{I}\Bigr{)}+{\cal
O}(g^{2})\;.$

If, on the right-hand-side of the above formula, one sets $\chi^{I}=0$ and
$\varepsilon^{I}=-\frac{1}{m_{I}}{R}_{0}{}^{I}{}_{\alpha}\,\epsilon^{\alpha}$
for the residual $\varepsilon^{I}(\epsilon)$ parameters that preserve the
unitary gauge $\chi^{I}=0$ at zeroth order in perturbation, it turns out that
one exactly recovers the expression for the $\delta_{\epsilon}\varphi^{i}$
transformations (33).

In other words, one could turn the argument around and get a heuristic way of
producing the right-hand side of formula (33): by demanding that the
operations of setting the fields $\chi^{I}$ to zero and performing gauge
transformations commute on the Stueckelberg field strengths, i.e., imposing
$(\left.\delta\Phi^{i})\right|_{\chi=0,\varepsilon=\varepsilon(\epsilon)}=\delta_{\epsilon}(\left.\Phi^{i}\right|_{\chi=0})\,$.

##  4 Conclusions and outlook

The first vertex $L_{1}^{(1)}$ that we presented above in (13) does not deform
the Stueckelberg gauge transformations (8). It is exactly invariant under the
latter transformations. The second vertex $L_{1}^{(2)}$ given in (14) is more
interesting in the sense that it truly deforms the transformations (8). The
first term of on the right-hand side of (15) is reminiscent of the local
supersymmetry transformations in AdS background, giving the minimal
deformation of the mass-like term on the right-hand side of
$\delta_{0}\psi_{\mu}=\nabla_{\mu}\theta+\frac{\omega}{2}\,\gamma_{\mu}\,\theta\,$,
see (8). Correspondingly, the first term on the right-hand side of (14) is the
minimal deformation of the mass-like term for the spinor $\mathbf{\Psi}_{\mu}$
in the free action (10). However, contrarily to the situation in supergravity
theories, there is no deformation proportional to the linearised “spin-
connection” $\nabla_{[\mu}k_{\nu]\rho}\,$ in (14) or in (15).

It will be interesting to investigate the consistent interactions among the
fields of the enlarged spectra given in [13]. There it was shown that the
doublet $(k_{\mu\nu},\psi_{\mu})$ consisting of a PM spin-2 and a massive
spin-3/2 field studied in the present Letter must be completed with a
masseless spin-$(3/2,1)$ doublet in order to carry the action of
supersymmetry. We hope to report soon on the interactions among these four
fields. In the paper [14], on the other hand, it was found that the partially
massless doublet $(5/2,2)$ can be completed with a massless doublet $(2,3/2)$
in order to carry the action of supersymmetry. We intend to investigate the
consistent couplings among those fields in a future work.

## Acknowledgments

It is a pleasure to thank Yurii Zinoviev for useful comments and discussions.
The work of N.B. was partially supported by the FNRS grant No. T.0022.19
“Fundamental issues in extended gravity”. The work of S.T. was partially
supported by the FNRS ASP fellowship FC 54793 MassHighSpin. The work of G.L.
was partially supported by the FNRS grant F.4503.20.

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