# Superfluid Edge Dislocation: Transverse Quantum Fluid

Leo Radzihovsky  Department of Physics and Center for Theory of Quantum
Matter, University of Colorado, Boulder, CO 80309  Anatoly Kuklov  Department
of Physics & Astronomy, College of Staten Island and the Graduate Center of
CUNY, Staten Island, NY 10314  Nikolay Prokof’ev  Department of Physics,
University of Massachusetts, Amherst, MA 01003, USA  Boris Svistunov
Department of Physics, University of Massachusetts, Amherst, MA 01003, USA
Wilczek Quantum Center, School of Physics and Astronomy and T. D. Lee
Institute, Shanghai Jiao Tong University, Shanghai 200240, China

###### Abstract

Recently, it has been argued [Kuklov et al., Phys. Rev. Lett. 128, 255301
(2022)] that unusual features associated with the superflow-through-solid
effect observed in solid 4He can be explained by unique properties of dilute
distribution of superfluid edge dislocations. We demonstrate that stability of
supercurrents controlled by quantum phase slips (instantons), and other exotic
infrared properties of the superfluid dislocations readily follow from a one-
dimensional quantum liquid distinguished by an effectively infinite
compressibility (in the absence of Peierls potential) associated with the edge
dislocation’s ability to climb. This establishes a new class of quasi-one-
dimensional superfluid states that remain stable and long-range ordered
despite their low dimensionality. We propose an experiment to test our mass-
current–pressure characteristic prediction.

Introduction. About a decade ago the superflow-through-solid (STS) effect in a
structurally imperfect crystal of 4He Hallock ; Hallock2012 ; Hallock2019 ;
Beamish ; Moses ; Moses2019 ; Moses2020 ; Moses2021 , along with the striking
companion effect of anomalous isochoric compressibility (also known as the
syringe effect) Hallock  have been attributed—by means of ab initio
simulations—to the properties of superfluid edge dislocations (SED) sclimb .
However, two apparently unrelated experimental features, namely (i) an
exponentially strong suppression of the flow by a moderate increase in
pressure and (ii) an enigmatic temperature dependence of the flow rate,
remained unexplained until very recently, when both dependencies were argued
to be linked and accounted for by the highly unusual properties of isolated
SED Kuklov2022 . The arguments of Ref. Kuklov2022  rest on the self-consistent
assumption that a SED can support stable supercurrents despite being quasi-
one-dimensional and featuring the spectrum of elementary excitations in
violation of the Landau criterion (see the Supplemental Material SM  for the
discussion of the Landau criterion in a non-Galilean superfluid system).

An outstanding (apparent) inconsistency between the scenario of Ref.
Kuklov2022  and the experimental data has been the observation of mass-
current–pressure (I-V) characteristic reminiscent of a Luttinger liquid (LL)
Hallock2012 , while the temperature dependence of the flux was incompatible
with the LL physics.

In this Letter, we put the theory of SED on a solid theoretical basis by
examining the consequences of the key feature distinguishing a SED from a LL:
SED is characterized by a (nearly) divergent linear compressibility imposed by
the approximate translational invariance of the dislocation transverse to its
Burgers vector and its axis, as illustrated in Fig. 1. The I-V characteristic
that we find resolves the above-mentioned inconsistency with the experimental
data and suggests a simple experiment to confirm our theory.

Transverse Quantum Fluid model. The minimal model—which we refer to as
Transverse Quantum Fluid (TQF)—proposed in Ref. sclimb  for SED in an ideal
crystal of 4He with an average superflow velocity $\sim v_{0}$ along its
$x$-directed core, is given by a 1D bosonic Hamiltonian,
$H[\phi,n]=\int{\mathcal{H}}\,dx$, with

${\mathcal{H}}={1\over 2\kappa}({\partial}_{x}n)^{2}+{n_{s}\over
2}(v_{0}+{\partial}_{x}\phi)^{2}\,.$ (1)

The 1D boson density $n$ is canonically conjugate to the superfluid phase
$\phi$, and the second term is the kinetic energy of the flow, with the
superfluid stiffness, $n_{s}$, exponentially sensitive to the local pressure.
The parameter $\kappa$ is determined by the inverse shear modulus of the
crystal sclimb . As shown in Ref. Kuklov2022 , namely this feature accounts
for the unusual dependence of the critical current on temperature.

Figure 1: (Color online) Sketch of the superfluid edge dislocation marked by
the bold (red) wavy line. Its Burgers vector $\bf b$ (along the C-axis) is
perpendicular to the planes of atoms—two complete layers are shown as two
parallel planes with straight edges and the incomplete one is limited by the
dislocation line. The transverse translation invariance of $h(x)$ (continuous
in the absence of the pinning Peierls potential, and discrete otherwise) is
responsible for infinite compressibility of the bosonic fluid confined to the
dislocation core.

The key feature distinguishing TQF from LL is the absence of the
compressibility term $\chi^{-1}n^{2}$, namely, TQF is characterized by a
divergent compressibility $\chi$. The condition $\chi^{-1}=0$ is enforced by
the translation invariance of the dislocation motion transverse to its core
axis and its Burgers vector (Peierls potential effects that violate this
symmetry and lead to finite compressibility are discussed below). This feature
is illustrated in Fig. 1: The dislocation climb displacement $\delta h(x)$
corresponds to the 1D boson density change,

$\delta n(x)=\delta h(x)/a^{2},$ (2)

along the dislocation, where $a$ is a lattice constant that we set to unity
hereafter. The associated spectrum $\omega_{k}$ of elementary excitations (at
$v_{0}=0$) is straightforwardly found to be quadratic in the wavevector $k$
along the dislocation,

$\omega_{k}=Dk^{2},\qquad D=\sqrt{n_{s}/\kappa},$ (3)

as a direct consequence of the aforementioned translation invariance, that
leads to $n\to n+{\rm const}$ invariance of (1).

We note that the quadratic dispersion (3) appears to violate the Landau
criterion for the critical velocity, $v_{0}<\\{\omega_{k}/k\\}_{\rm min}$, for
arbitrary small $v_{0}$. However, as we discuss in the Supplemental Material
(SM) SM , the Landau criterion for instability relies on the existence of the
term $\delta n(v_{0}+{\partial}_{x}\phi)^{2}=2v_{0}\delta
n{\partial}_{x}\phi+\dots$, which is forbidden for SED by the climb
translation invariance, allowing only density derivatives, $\partial_{x}n$, in
the TQF Hamiltonian. Thus, at sufficiently small $v_{0}$, SED (described by
TQF) does not develop the Landau instability, and we must consider superflow
relaxation via quantum phase slips, i.e., instantons (space-time vortices) in
the phase field $\phi(x,\tau)$ configuration ColemanInstantons .

The Euclidean Lagrangian density of TQF, corresponding to (1) can be obtained
after eliminating the density $n$,

${\mathcal{L}}={\kappa\over
2}({\partial}_{x}^{-1}{\partial}_{\tau}\phi)^{2}+{n_{s}\over
2}({\partial}_{x}\phi)^{2}+n_{s}v_{0}{\partial}_{x}\phi,$ (4)

where the long-range operator ${\partial}_{x}^{-1}$ is defined by its Fourier
transform, namely, ${\partial}_{x}^{-1}\to-ik^{-1}$. The last (boundary) term
in (4) is important only in the presence of instantons, and we have omitted an
irrelevant constant.

Goldstone modes and long-range order. Mean-squared fluctuations of the
superfluid phase $\phi$, computable within the Gaussian approximation,

$\langle\phi^{2}\rangle=\frac{1}{\kappa}\int^{2\pi/a}\frac{d\omega
dk}{(2\pi)^{2}}\frac{k^{2}}{\omega^{2}+D^{2}k^{4}}\approx\frac{1}{a\sqrt{\kappa
n_{s}}}\,,$ (5)

are finite, and TQF thus features a long-range order in the superfluid field
$\psi=e^{i\phi}$ at zero temperature ($T$), despite being one-dimensional
ODLRO . Given the exact mathematical symmetry between $\phi$ and $n$ in (1),
the field $\eta=e^{i2\pi n}$ also exhibits long-range order at zero-
temperature.

At this point we recall that SED is a subject to a Peierls potential,
$U=U_{0}\cos(2\pi n)$, [utilizing the density $n$ – displacement $h$ relation,
(2)], omitted in the formulation (1). This sets certain limitations on the
applicability of the TQF as an asymptotic model for SED. Long-range order in
$\eta$ implies that $U$ is a relevant perturbation, which suppresses the
dislocation climb motion and induces a crossover of the excitation spectrum to
a conventional (LL-type) linear form at small momenta, $k/(2\pi)<\xi^{-1}$,
with the length scale $\xi$ diverging with crystal’s vanishing shear modulus
$\sim\kappa^{-1}$ MAX . (In the opposite limit, the LL superfluid undergoes a
phase transition to a Mott-Insulator MAX .) Concomitantly, at length scales
longer than $\xi$, the zero-temperature long-range order of SED changes to the
algebraic quasi-long-range one, typical for LL. Such a crossover was observed
in model simulations of Refs. MAX ; ODLRO  through a finite-size scaling of
compressibility $\chi$ for a pinned dislocation with $h(0)=h(L)=0$: from TQF’s
$\chi\sim L^{2}$ to a constant in the $L\to\infty$ limit, as $T\sim 1/L\to 0$.

In what follows, we will focus on the exotic TQF regime on scales below $\xi$,
where SED is quantum rough and the associated 1D superfluid is long-range
ordered. Equivalently, one can view TQF as the asymptotic limit
($\xi\to\infty$) of SED. We demonstrate that such a simplification is
sufficient for explaining the experimental data with high accuracy.

Confinement of instantons. We now analyze the stability of this novel one-
dimensional superfluidity to superflow-induced instantons ColemanInstantons .
To account for these, we consider non-single-valued configurations of the
superfluid phase $\phi(x,\tau)$, corresponding to its space-time vortex
configurations. To this end, we introduce the velocity field
$v_{\mu}\equiv{\partial}_{\mu}\phi$ in the $(1+1)$-dimensional space-time
$x_{\mu}=(x,\tau)$. The instantons have the form of point-vortex singularities
in the otherwise regular field $v_{\mu}$:

${\partial}\times
v=q(x_{\mu})=\sum_{j}\,q_{j}\,\delta^{2}(x_{\mu}-x_{\mu,j}),$ (6)

where ${\partial}\times v\equiv\epsilon_{\mu\nu}{\partial}_{\mu}v_{\nu}$ is
short-hand notation for $(1+1)$ space-time curl of $v_{\mu}$ and $q_{j}$ and
$x_{\mu,j}$ are, respectively, the “charge” (an integer multiple of $2\pi$)
and space-time position of the $j$-th instanton.

With these ingredients we can now straightforwardly derive the instanton
action $S[q(x_{\mu})]$ from the Lagrangian density (4) for $\phi(x_{\mu})$.
The most direct way to obtain this (with alternative derivations presented in
Ref. SM ) is to enforce the topological constraint (6) using a functional
delta function in a path-integral for a partition function over $v_{\mu}$,

$Z=\int[dv_{\mu}][dq][d\lambda]e^{-\int
dxd\tau{\mathcal{L}}[v_{\mu},\lambda,q]}\,,$ (7)

with the Lagrangian density (for $v_{0}=0$)

$\displaystyle{\mathcal{L}}$ $\displaystyle=$
$\displaystyle\frac{\kappa}{2}({\partial}_{x}^{-1}v_{\tau})^{2}+\frac{n_{s}}{2}(v_{x})^{2}+i\lambda({\partial}\times
v-q)\,.$ (8)

In ${\mathcal{L}}$ we implemented the vorticity constraint (6) via a
functional delta function as an integral over the auxiliary field
$\lambda(x_{\mu})$. Performing the Gaussian integration over fields
$v_{\mu}(x_{\mu})$ and $\lambda(x_{\mu})$, gives the instanton action,

$\displaystyle S$ $\displaystyle=$ $\displaystyle{n_{s}\over 2}\int{d\omega
dk\over(2\pi)^{2}}\,{|q_{\omega,k}|^{2}\over\omega^{2}+D^{2}k^{4}}$ (9)
$\displaystyle=$
$\displaystyle{\frac{1}{2}}\int_{x_{\mu},x^{\prime}_{\mu}}q(x_{\mu})V(x_{\mu}-x^{\prime}_{\mu})q(x^{\prime}_{\mu})\;,$
(10)

where $V(x,\tau)$ is the space-time instanton-instanton interaction, which
after subtracting the self action $V(0,0)$, becomes

$\displaystyle V(x,\tau)$ $\displaystyle=$ $\displaystyle
n_{s}\int\frac{d\omega
dk}{(2\pi)^{2}}\,\frac{e^{i(\omega\tau+kx)}-1}{\omega^{2}+D^{2}k^{4}}$ (13)
$\displaystyle\approx$
$\displaystyle\left\\{\begin{array}[]{ll}-\frac{n_{s}\sqrt{|\tau|}}{2\sqrt{\pi
D}},\;\;\mbox{ for $x^{2}\ll D\tau$},\\\
-\;\frac{n_{s}|x|}{4D},\;\;\;\;\;\mbox{ for $x^{2}\gg
D\tau$.}\end{array}\right.$

This result can be complementarily obtained (see Ref. SM ) through the saddle-
point equation for $v_{\mu}$ together with the Fourier-transformed instanton
constraint, Eq. (6), finding

$v_{\tau}\,=\,{-iD^{2}k^{3}q\over D^{2}k^{4}+\omega^{2}}\,,\qquad
v_{x}\,=\,{i\omega q\over D^{2}k^{4}+\omega^{2}}\,.$ (14)

The interaction kernel in (13) is reminiscent (but misses a factor of
$-{\partial}_{x}^{2}$ in the numerator) of interaction between dislocations in
the 2D classical smectic TonerNelson81 . This difference is expected as the 2D
smectic elasticity is “softer” than in the XY model, while here it is
“stiffer” than in the XY model because of the extra ${\partial}_{x}^{-1}$
factor in ${\mathcal{L}}$ of (4). The latter effect is directly related to
divergent compressibility, which allows stronger density fluctuations and
thereby “stiffens” the canonically conjugate superfluid phase $\phi$. As a
result, in contrast to the 2D smectic (where interaction is weak and
dislocations are deconfined) TonerNelson81 ; QSmGaugeLR , the TQF instantons
with opposite “charges” are always confined by the interaction $V(\tau,x)$ in
(13), featuring a power-law growth with spatial and temporal separation. Note
that this behavior is consistent with the 1D long-range superfluid order at
$T=0$ discussed above ODLRO .

Metastability of the superflow and nonlinear I-V response. We now demonstrate
that instanton confinement, (13), leads to the exponential in $1/v_{0}$
metastability of the superflow. This is readily seen by examining the
contribution of the $v_{0}$-dependent term to the Langragian density (4).
Since this is a boundary term, its presence does not change the conditions (6)
and action in (13), leaving the solution (14) intact. Further simplification
comes from the fact that the metastable regime corresponds to appropriately
small values of $v_{0}$, when the destabilization channel is associated with a
well-isolated instanton–anti-instanton pair characterized by space-time
coordinates $(x_{+},\tau_{+})$ and $(x_{-},\tau_{-})$. Straightforward
integration in the $v_{0}$-dependent term in (4), $\int
dx\,\partial_{x}\phi(x,\tau)=\phi(+\infty,\tau)-\phi(-\infty,\tau)$, gives its
contribution to the instanton pair action (for the pair with “charges” $\pm
2\pi$)

$\displaystyle S_{v_{0}}\,=\,v_{0}n_{s}\iint d\tau dxv_{x}\,$ $\displaystyle=$
$\displaystyle\,2\pi v_{0}n_{s}\int_{\tau_{+}}^{\tau_{-}}d\tau$ (15)
$\displaystyle=$ $\displaystyle\,2\pi v_{0}n_{s}(\tau_{+}-\tau_{-})\,.\qquad$

We see that, in a close similarity with the vortex–anti-vortex pair in a 2D
superfluid, the superflow generates a transverse (in space-time) “force”
pulling the pair apart along the imaginary-time direction.

Combining this with the competing $v_{0}$-independent part of the pair action
at short distances, $S_{\rm inst}^{(0)}\approx V(x=0,\tau)$, given by
(10)–(13), we find the total instanton-pair action,

$\displaystyle S_{\rm inst}$ $\displaystyle\approx$
$\displaystyle\frac{2\pi^{3/2}n_{s}}{\sqrt{D}}\sqrt{|\tau|}-2\pi
n_{s}|v_{0}\tau|\,,$ (16)

where $(x,\tau)\equiv(x_{+}-x_{-},\tau_{+}-\tau_{-})$. The maximum-action
instanton-pair configuration controlling the dissipation is reached at
$\sqrt{\tau^{*}(v_{0})}=\sqrt{\pi}/(2\sqrt{D}v_{0})$, and the corresponding
action is given by

$S_{\rm inst}^{*}\,\equiv\,S_{\rm
inst}(\tau^{*})\,\approx\,\frac{\pi^{2}n_{s}}{2Dv_{0}}\,.$ (17)

In the exponential approximation ColemanInstantons , this defines the
probability of the instanton nucleation:

${\cal P}\sim e^{-v_{c}/v_{0}},\qquad
v_{c}=\frac{\pi^{2}n_{s}}{2D}=\frac{\pi^{2}\sqrt{n_{s}\kappa}}{2},$ (18)

in sharp contrast with the LL’s power-law dependence ${\cal
P}\sim(v_{0})^{g}$, with $g>0$.

Another qualitative difference with the LL physics comes from kinematic
considerations regarding the ultimate decay of the superflow. In the
translation-invariant LL, the decay of the supercurrent is kinematically
forbidden because it is impossible to simultaneously satisfy the conservation
of energy and momentum under generic conditions. One needs either impurities
and/or commensurate external potential to absorb the momentum released by the
supercurrent when the phase winding number changes by $\pm 1$ due to a quantum
phase slip. In TQF, the decay of the supercurrent into elementary excitations
is kinematically allowed due to the quadratic dispersion, Eq. (3). However, at
small values of $v_{0}$, this involves a large number of elementary
excitations. Indeed, the energy and momentum released by a supercurrent in a
single phase slip are (respectively): $\Delta E=2\pi n_{s}v_{0}$ and $\Delta
P=2\pi n_{s}$. Suppose these are absorbed by ${\cal N}$ quasiparticles with
momentum $k_{*}=\Delta P/{\cal N}$. From the energy conservation, $\Delta
E={\cal N}D\Delta P^{2}/{\cal N}^{2}=D\Delta P^{2}/{\cal N}$, we then readily
obtain ${\cal N}=D\Delta P^{2}/\Delta E=2\pi n_{s}D/v_{0}$ and
$k_{*}=v_{0}/D$. In what follows we will not be pursuing further the analysis
of specific details of the decay and assume that the exponential factor in
Eq.(18) controls the rate of the phase slips.

Figure 2: (Color online) The I-V characteristic master curve (solid line) of
the TQF , Eq. (20) (units are arbitrary). Symbols are the experimental data
from Fig. 5(a) of Ref. Moses2019  [presented also in Fig.1 of Ref. SM ]
collected at different $T$ and interpreted within the sub-Ohmic dependence
(19). The corresponding values of $T$ and $\alpha$ (in parenthesis) are shown
in the legend. The data points from a set at a given $T$ are shifted without
changing their log-log slopes to achieve the collapse onto the master curve
(see the text below).The dashed line, as an example, corresponds to the power
law (19) with $\alpha=0.16$.

Experimental implications. The results obtained above allow us to resolve an
apparent disagreement between the scenario advocated in Ref. Kuklov2022  and
the experimental data on the I-V characteristic of the STS effect.
Experimentally, the flow rate $F$ (the mass current, proportional to $v_{0}$)
as a function of the chemical potential bias (the “voltage”) $\Delta\mu$ was
found to be consistent with the sublinear power-law Hallock2012 ; Moses2019 :

$F=A(T)(\Delta\mu)^{\alpha},\qquad(\alpha<1).$ (19)

The authors of Ref. Hallock2012  reported $\alpha\approx 0.3\pm 0.1$,
independent of $T$ up to $T\approx 0.5$K. This value agrees well with
$\alpha\approx 0.24$, observed at $T<0.2$K in Ref. Moses2019 . However, as
$T\to 1$K, the value of $\alpha(T)$ was found to cross over to $\alpha\approx
0.5$ Moses2019 . If it were not for the drastic temperature dependence of the
flux amplitude, $A(T)$, as well as the less shocking but still unexpected
temperature dependence of $\alpha(T)$, it would be natural to interpret (19)
as the manifestation of the LL behavior Hallock2012 .

By suggesting a solution for the temperature dependence of the amplitude
$A(T)$, Ref. Kuklov2022  questioned the LL origin of the dependence (19),
without providing a detailed mechanism. Present analysis offers a concrete
alternative to the LL interpretation based on the TQF instanton mechanism of
the phase slips described above by Eqs. (17)–(18). If the probability of phase
slips is controlled by the TQF instanton action (17) rather than by the matrix
element of the transition from the initial to the final state of the system,
the I-V characteristic of the superfluid edge dislocation can be obtained
within an elementary hydrodynamics approach (see Supplemental Material SM ).
Accordingly, the resulting I-V curve can be described by the relation ${\cal
P}_{0}v_{0}{\cal P}=\Delta\mu$ where ${\cal P}$ is given in Eq.18 and ${\cal
P}_{0}$ is a constant that includes the details of the mechanisms for the
transfer of the total momentum and energy from the current to the excitations.
Then, introducing dimensionless variables
$\hat{v}=v_{0}/v_{c},\,\,\mu=\Delta\mu/({\cal P}_{0}v_{c})$, the normalized
flow velocity $\hat{v}$ (in units of critical velocity, $v_{c}$), obey the
relation

$\hat{v}\,{\rm e}^{-1/\hat{v}}=\mu\,,$ (20)

with the flux $F$ through a sample containing many dislocations being
$F\propto\hat{v}$. While the overall curve $\hat{v}$ vs $\mu$ is obviously
inconsistent with the dependence (19), its significant parts can be well fit
by this dependence, provided the interval of $\mu$ variation does not exceed
two orders of magnitude – as demonstrated by the dashed line in Fig. 2.

In accordance with the large-fluctuation scenario of Ref. Kuklov2022 , the
crossover between different parts of the curve takes place with increasing
temperature when at fixed $\Delta\mu$, the dimensionless bias
$\mu\sim\Delta\mu/\sqrt{n_{s}(T)}$ increases exponentially, shifting the I-V
characteristic from $\alpha\approx$ 0.24 at low $T$ towards ${\hat{v}}\sim 1$,
where $\alpha\approx 0.5\div 0.6$. This behavior is demonstrated by the good
collapse of the data $F$ vs $\mu$ Moses2019  on the master curve in Fig. 20.
[See more details in Ref. SM ].

Summary and outlook. Motivated by a recent suggestion that phase slips in a
superfluid edge dislocation are expected to be qualitatively distinct from
those in a LL and allow for a metastable superflow in 1D, we presented and
analyzed the TQF model as the natural asymptotic limit of SED. We demonstrated
that quantum Lagrangian of TQF predicts confinement of quantum phase slips,
implying the exponential nonlinear I-V characteristic, along with other
special properties that have no analogs in known one-dimensional systems.

This confinement of instantons, accompanied by the quadratic dispersion of
elementary excitations—the hallmark of the TQF—follows directly from its
divergent compressibility in the absence of the Peierls potential. Despite
superficial similarity with the ideal Bose gas, the physics here is
qualitatively distinct because (i) SED lacks Galilean invariance and (ii)
translational invariance in the climb direction prohibits terms in the TQF
Hamiltonian that are responsible for the Landau-type instability—a vanishing
critical current – for a parabolic dispersion in the ideal Bose gas.

Our theory is consistent with the existing experimental data on the superflow-
through-solid effect thus offering a resolution of a vexing controversy in the
data interpretation. The exponential I-V characteristic that we predict is in
a stark contrast with the LL power-law, that requires a variable exponent to
fit the experimental data. This naturally brings up a proposal for the
compelling experimental test: detailed measurements of the I-V characteristic
at low temperature and different external pressures and then collapsing the
data on the master curve predicted by Eq. (20) and illustrated in Fig. 2.
Specifically, (i) extending the range of the biases $\Delta\mu$ should
demonstrate the deviation from the LL dependence (19), (ii) applying external
pressure at small $T$, should decrease $n_{s}$ and, accordingly, shift the
effective $\alpha$ from 0.24 to higher values, (iii) decreasing temperature
below 45mK (the lowest studied in Ref. Moses2019 ) should either decrease
$\alpha$ to even smaller values—as demonstrated by the dashed line in Fig.
2—or eventually bring the dislocation to the LL regime, where $\alpha$ is
determined by the emerging finite compressibility.

Acknowledgents. This work was supported by the National Science Foundation
under the grants DMR-2032136 and DMR-2032077. LR acknowledges support by the
Simons Investigator Award from the Simons Foundation. We thank Robert Hallock
and Moses Chan for providing us with their data and for discussion of our
results.

Supplemental Material

##  I Instanton Action: Extremal Configuration of the Velocity Field

Given that TQF action $S=\int{\cal L}\,dxd\tau$ is bi-linear with respect to
$\phi$ (apart from the linear in $\phi$ term for non-zero $v_{0}$) and that
the dependence on $\phi$ is through the velocity components: $S[\phi]\equiv
S[v_{x},v_{\tau}]$ [see Eq. (8) of the main text], the problem of the
instanton action can be reduced to the problem of finding the velocity field
minimizing the action at given instanton positions. In this section, we use a
straightforward approach to the problem. In the next two sections, we will
present several complementary analyses.

Let $\delta\varphi$ be a small single-valued variation of $\phi$ and
$B[\phi]\equiv B[v_{x},v_{\tau}]$ the variational derivative of the action
with respect to $\delta\varphi$:

$\delta S[\phi]\,=\,\int dxd\tau B[v_{x},v_{\tau}]\,\delta\varphi\,,$
$B[v_{x},v_{\tau}]\,=\,G^{-1}\partial_{x}^{-2}\partial_{\tau}v_{\tau}-n_{s}\partial_{x}v_{x}\,.$

For the extremal configuration of the velocity field, we have
$B[v_{x},v_{\tau}]\equiv 0$ corresponding to

$\partial_{\tau}v_{\tau}-D^{2}\partial_{x}^{3}v_{x}\,=\,0\,.$ (21)

Combining (21) with the condition [Eq. (6) of the main text]

$\epsilon_{\mu\nu}{\partial}_{\mu}v_{\nu}=q\,,$

we obtain a system of two equations relating the fields $v_{x}$ and $v_{\tau}$
to $q=2\pi p,\,\,p=\pm 1,\pm 2,...$. Fourier-transforming these equations,

$\omega v_{\tau}\,+\,D^{2}k^{3}v_{x}\,=\,0\,,$ (22) $kv_{\tau}-\omega
v_{x}\,=\,-iq\,.$ (23)

This readily gives the Fourier transforms of $v_{x}$ and $v_{\tau}$ as
functions of the Fourier transforms of $q$, thereby arriving at Eq. (12) of
the main text.

##  II Euler-Lagrange equation in a generic gauge

Here we utilize the fact that the field $\phi$ minimizing the action at fixed
positions of instantons can be represented—in a continuum of different ways—as
the sum of the smooth single-valued part $\phi_{s}$ and the part $\phi_{0}$
accounting for the instantons:

$\phi=\phi_{s}+\phi_{0},\qquad\quad\epsilon_{\mu\nu}{\partial}_{\mu}{\partial}_{\nu}\phi_{0}=q,$
(24)

with a particular choice of $\phi_{0}$ corresponding to a particular “gauge.”
By substituting this decomposition inside the Lagrangian, and minimizing it
over $\phi_{s}$, we obtain the Euler-Lagrange equation of the form

$(D^{2}{\partial}_{x}^{4}-{\partial}^{2}_{\tau})\phi_{s}={\partial}_{\tau}v^{(0)}_{\tau}-D^{2}{\partial}_{x}^{3}v^{(0)}_{x}\,,$
(25)

where $v^{(0)}_{\mu}={\partial}_{\mu}\phi_{0}$, so that

$\epsilon_{\mu\nu}{\partial}_{\mu}v^{(0)}_{\nu}=q\,.$ (26)

Gauge freedom amounts to the choice of $v^{(0)}_{\mu}$ satisfying the
condition (26).

There is a special gauge reproducing the theory of the previous section. In
this gauge, we require that the right-hand side of (25) be identically zero,
which, on one hand, leads to Eq. (21) for $v^{(0)}_{\mu}$ and, on the other
hand, implies the trivial solution $\phi_{s}\equiv 0$, meaning $v_{\mu}\equiv
v^{(0)}_{\mu}$.

Alternatively, we can use a simple gauge that nullifies one of the
$v^{(0)}_{\mu}$ components—for definiteness, let it be $v^{(0)}_{x}$:

$v^{(0)}_{x}=0\,,\qquad\quad v^{(0)}_{\tau}={\partial}_{x}^{-1}q\,.$ (27)

Substitutinq this into the r.h.s. of (25) we find $\phi_{s}$ (in the operator
form):

$\phi_{s}={{\partial}_{x}^{-1}{\partial}_{\tau}q\over
D^{2}{\partial}_{x}^{4}-{\partial}^{2}_{\tau}}\,,$ (28)

which then yields the expressions for the two velocity components,

$v_{x}=v^{(0)}_{x}+{\partial}_{x}\phi_{s}={{\partial}_{\tau}q\over
D^{2}{\partial}_{x}^{4}-{\partial}^{2}_{\tau}}\,,$ (29)
$v_{\tau}=v^{(0)}_{\tau}+{\partial}_{\tau}\phi_{s}={D^{2}{\partial}_{x}^{3}q\over
D^{2}{\partial}_{x}^{4}-{\partial}^{2}_{\tau}}\,,$ (30)

reproducing result (12) in the main text.

##  III Constrained Euler-Lagrange equations for components of the velocity
field

Here we observe that since the two components of the velocity field,

$v_{x}=\partial_{x}\phi,\qquad v_{\tau}=\partial_{\tau}\phi,$ (31)

are allowed to be singular and thereby have both longitudinal and transverse
parts, they can be conveniently treated as independent fields subject to the
constraint

$\partial_{x}v_{\tau}-\partial_{\tau}v_{x}=q.$ (32)

In contrast to the phase $\phi$, its spatial and temporal derivatives are
single-valued functions (with isolated singular points corresponding to
instantons), and this is one of the advantages of parametrization (31)–(32).
The role played by the constraint (32) is two-fold. On the one hand, it takes
into account the fact that the two components are not independent, so that
there is only one independent physical degree of freedom. On the other hand,
it provides a simple way of including the positions of instantons into the
action in a “gauge-invariant” form.

In its Fourier representation, the constraint is given by Eq. (23), so that
the Lagrangian of corresponding extended action reads

${\cal L}^{\prime}={\cal L}+[\lambda^{*}(kv_{\tau}-\omega v_{x})+{\rm c.c.}],$
(33)

where

${\cal L}={\kappa\over 2k^{2}}|v_{\tau}|^{2}+{n_{s}\over 2}|v_{x}|^{2},$ (34)

is the unconstrained part of the Lagrangian and $\lambda$ is the Fourier
transform of the Lagrange multiplier.

The Euler-Lagrange equations immediately allow us to relate each of the two
velocity components to $\lambda$,

$v_{\tau}=-2\kappa^{-1}k^{3}\lambda,\qquad v_{x}={2\omega\over n_{s}}\lambda,$
(35)

that leads to

${\cal L}={2\over n_{s}}(D^{2}k^{4}+\omega^{2})|\lambda|^{2}.$ (36)

Substituting $v_{x}$ and $v_{\tau}$ from (35) into (23) to express $\lambda$
in terms of $q$, we obtain

$\lambda={i\over 2}{n_{s}q\over D^{2}k^{4}+\omega^{2}}\,$ (37)

and find the Lagrangian (12) of the main text.

##  IV Duality transformation

Here we start with the action (3) of the main text for $v_{0}=0$, and
introduce the Hubbard-Stratonovich field $j_{\mu}=(n,j_{x})$ in the density-
phase $(n,\phi)$ path integral to obtain

$\displaystyle{\mathcal{L}}$ $\displaystyle=$ $\displaystyle
in{\partial}_{\tau}\phi+ij_{x}{\partial}_{x}\phi+{\frac{1}{2}}\kappa^{-1}({\partial}_{x}n)^{2}+{\frac{1}{2}}n_{s}^{-1}j_{x}^{2},\qquad$
(38) $\displaystyle=$ $\displaystyle
ij_{\mu}v_{\mu}+{\frac{1}{2}}\kappa^{-1}({\partial}_{x}n)^{2}+{\frac{1}{2}}n_{s}^{-1}j_{x}^{2}.$
(39)

By integrating out the smooth field $\phi_{s}$, we arrive at the charge-
conservation constraint, ${\partial}_{\mu}j_{\mu}=0$, i.e.,

$\displaystyle{\partial}_{\tau}n+{\partial}_{x}j_{x}=0\;.$ (40)

This is solved by
$j_{\mu}=\epsilon_{\mu\nu}{\partial}_{\nu}\theta\equiv({\partial}\times\theta)_{\mu}$,
with

$n={\partial}_{x}\theta\;,\qquad j_{x}=-{\partial}_{\tau}\theta\;.$ (41)

By substituting this solution into the Lagrangian, integrating the first term
by parts, and using the definition ${\partial}\times j=q$, we find,

$\displaystyle{\mathcal{L}}$ $\displaystyle=$ $\displaystyle
iq\theta+E_{c}q^{2}+\frac{1}{2\kappa}({\partial}^{2}_{x}\theta)^{2}+\frac{1}{2n_{s}}({\partial}_{\tau}\theta)^{2},$
(42)

where to account for lattice scale physics in this continuum treatment we
added the instanton core energy, $E_{c}$. For $E_{c}=0$, integration over
$\theta$ reproduces our result for the instanton action (9),(10) of the main
text.

##  V TQF as the asymptotic limit model for SED

As explained in the main text, the TQF is the asymptotic model for SED – that
is, at scales shorter than $\xi$. In the SED at scales longer than $\xi$, the
pinning effects of the Peierls potential change the denominator of the
solution in Eq.(12) of the main text from $\omega^{2}+D^{2}k^{4}$ to
$\omega^{2}+v_{s}^{2}k^{2}$ for $k\lesssim\xi^{-1}$, with
$v_{s}=\sqrt{n_{s}/\chi}$ the speed of sound of the emerging LL and $\chi$
being the emerging finite compressibility MAX . This changes the asymptotic
behavior of $V(\tau,x)$ to the conventional LL logarithmic growth at scales
longer than $\xi$, asymptotically resulting in the standard (1+1)-dimensional
XY-model phenomenology.

##  VI Irrelevance of Landau criterion

In its original formulation, the Landau criterion of the stability of the
superflow (a detailed discussion can be found in, e.g., Ref. book ) relies on
the system being Galilean invariant. Since our system is not Galilean
invariant, it is instructive to see how the criterion changes when the system
features an asymptotic—in the long-wavelength limit—translation invariance in
the absence of Galilean invariance. We are concerned only with potential
violation of the criterion in the infra-red limit. Therefore, it is sufficient
to perform a general analysis at the level of the classical-field bilinearized
hydrodynamic Hamiltonian, $H_{\rm hd}[\eta,\phi]$, written in terms of two
canonically conjugate fields $\eta,\phi$. In the case of the LL, these are
small deviation $\eta({\bf r})$ of the coarse-grained value of matter density
$n$ from its equilibrium value $n_{0}$ and small deviation $\phi({\bf r})$ of
the field of the superfluid phase from its equilibrium configuration
$\phi_{0}({\bf r})=m\,{\bf v}_{0}\cdot{\bf r}$. Here ${\bf v}_{0}$ is the
velocity of the stationary superflow and $m$ is the bare atomic mass.

In the absence of the Galilean invariance, the ground-state energy density of
the system, ${\cal E}(n,v_{s})$, is a function of the density, $n$, and the
magnitude of the superfluid velocity, $v_{s}$, with the following relations
playing key hydrodynamic role book :

${\partial{\cal E}(n,v_{s})\over\partial
n}\,=\,\mu\,,\qquad\quad{\partial{\cal E}(n,v_{s})\over\partial
v_{s}}\,=\,mj\,.\quad$ (43)

Here $\mu$ is the chemical potential, and $j$ is the magnitude of the
supercurrent density, which is related to $v_{s}$ as $j=n_{s}v_{s}$, with
$n_{s}$ the superfluid stiffness.

Performing an expansion of $\cal E$ around $n=n_{0},v_{s}=v_{0}$, the
Hamiltonian density ${\cal H}_{\rm hd}[\eta,\phi]$ follows—up to the lowest-
order in $(\nabla\phi,\eta)$—as

${\cal H}_{\rm hd}(\eta,\phi)\,=\,{\tilde{n}_{s}\over
2m}\,(\nabla\phi)^{2}\,+\,{\eta^{2}\over 2\chi}\,\,+\,\lambda_{s}\eta\,{\bf
v}_{0}\cdot\nabla\phi\,,$ (44)

where

$\tilde{n}_{s}\,=\,{1\over m}\frac{\partial^{2}\cal E}{\partial
v_{0}^{2}}\,=\,\frac{\partial j}{\partial v_{0}}\,=\,n_{s}+v_{0}\frac{\partial
n_{s}(n_{0},v_{0})}{\partial v_{0}}\,,$ (45)

and

$\chi\,=\,{\partial
n(\mu,v_{0})\over\partial\mu}\,,\qquad\quad\lambda_{s}\,=\,{\partial
n_{s}(n_{0},v_{0})\over\partial n_{0}}\,,$ (46)

where $\chi$ the compressibility. Note that in the absence of the Galilean
invariance, $n_{s}$, can be a nontrivial function of $(n,v_{s})$.

The dispersion of the normal modes governed by the Hamiltonian (44) is

$\omega({\bf k})\,=\,v_{*}k\,+\,\lambda_{s}{\bf v}_{0}\cdot{\bf
k}\,,\qquad\quad v_{*}\,=\,\sqrt{\tilde{n}_{s}\over\chi}\,.\quad$ (47)

The critical value, $v_{0}^{(c)}$, of the magnitude of the superflow velocity
corresponds to vanishing of $\omega({\bf k})$ at the critical speed

$v_{0}^{(c)}\,=\,v_{*}/\lambda_{s}\,,$ (48)

so that $\omega({\bf k})$ can become negative for a range of
$v_{0}>v_{0}^{(c)}$. In a Galilean invariant superfluid,
$\tilde{n}_{s}=n_{s}=n$, implying $\lambda_{s}=1$ and the critical velocity
coincides with the velocity of sound: $v_{0}^{(c)}=v_{*}$. In a general case,
$\lambda_{s}$ can change not only its value but even the sign. For example, in
a system of hard-core lattice bosons, $\lambda_{s}$ is close to unity at small
fillings and approaches negative unity when the filling factor approaches
unity. By continuity, this means that there is a special value of $n$ at which
$\lambda_{s}=0$ and the generic Landau-type mechanism of instability becomes
irrelevant. This does not exclude system-specific alternative mechanisms, but
does imply that these have do deal with essentially finite wavevectors.

In the TQF the compressibility is infinite, $\chi=\infty$, and the role of the
density $n$ is played by the dislocation climbing displacement $h$ (see Eq.(2)
of the main text setting a relation between the dislocation displacement
$\delta h$ and the density $\delta n$ variations), with the potential-energy
density determined by the elastic deformation of the dislocation
$\kappa^{-1}({\partial}_{x}h)^{2}/2$. Furthermore, due to the translational
invariance, $n_{s}$ can only depend on the curvature
$\sim{\partial}^{2}_{xx}h$ of the dislocation and not on the position $\sim n$
of the line Kuklov2022 . Accordingly, the spectrum $\omega\sim k^{2}$ at
finite $v_{0}$ acquires the term $\sim v_{0}k^{3}$. In the $k\to 0$ limit,
this correction becomes irrelevant (equivalently, we can put it as
$\lambda_{s}\equiv 0$), and there is no room for the Landau-type hydrodynamic
instability.

##  VII I-V characteristic

Figure 3: (Color online) Raw data from Ref. Moses  for the flux dependence on
the chemical potential difference (in arbitrary units) at a range of
temperatures. The lines are guides to an eye. Symbols are the same as in Fig.
2 of the main text and correspond to the same temperatures.  Figure 4: (Color
online) Scaling factors $F_{0}(T)$ and $\mu_{0}(T)$ (symbols) used in
processing the data from Fig.5 of Ref.Moses . Solid lines are guides to the
eye.

The instanton mechanism of the superflow relaxation by phase slips is
described here in the leading exponential approximation when $S^{*}_{\rm
inst}>>1$, see Eq.(15) of the main text. Without knowing the nature of the
pre-exponential factor in the probability (16) (in the main text), we use the
Newtonian approach in deriving the I-V characteristics – Eq. (18) in the main
text. Phase slip events conclude with a jet of normal excitations carrying
away the superflow momentum—they produce a friction force $f=\tau^{-1}2\pi
n_{s}$ with the relaxation rate $\tau^{-1}\propto{\cal P}$, where ${\cal P}$
is given in Eq.(16) of the main text. This force balances the force $f_{\rm
bias}\approx n_{s}\Delta\mu$ imposed by the bias, and the stationary flow
corresponds to $\Delta\mu=\tau^{-1}2\pi$.

In order to compare this model with the experiment, an assumption should be
made about the proportionality coefficient in the relation
$\tau^{-1}\propto{\cal P}$ as a function of the superflow velocity $v_{0}$. We
will follow the model of liquid friction, that is,
$\tau^{-1}=(v_{0}/l_{0}){\cal P}\equiv(v_{0}/l_{0})\exp(-v_{c}/|v_{0}|)$,
where $l_{0}$ is some length scale $l_{0}$ independent of $v_{0}$. This form
respects the reflection symmetry $\Delta\mu\to-\Delta\mu$ and $v_{0}\to-v_{0}$
together with the breaking of the time reversal due to the dissipation caused
by the phase slips. Thus, the stationary flow velocity obeys

$(v_{0}/l_{0}){\rm e}^{-v_{c}/|v_{0}|}\approx\Delta\mu.$ (49)

In the dimensionless units ${\hat{v}}=v_{0}/v_{c}$ and $\mu=\delta\mu
l_{0}/v_{c}$ this leads to Eq. (18) of the main text, which plays the role of
universal master curve presented in Fig 2 of the main text. Its parts over
about two orders of magnitude in $\mu$ and a factor of 4 in $\hat{v}$ can be
approximated by the power law (see Eq.(17) of the main text) – similarly to
the experiments Hallock ; Moses . One can define the effective exponent
$\alpha$ using the log-derivative of the master curve, Eq.(18), of the main
text: $\alpha^{-1}=d\ln\mu/d\ln{\hat{v}}$, or,
$\alpha={\hat{v}}/(1+{\hat{v}})$. By comparing this value with the exponent
observed for each data set taken at temperature $T$, see Figs.5(a) and (b) of
Ref. Moses  (presented above in Fig. 3 in arbitrary units ) , we determine the
rescaling parameters $F_{0}(T),\mu_{0}(T)$ used to plot the experimental data
together with the master curve, Fig. 2 of the main text. More specifically,
each data set in the temperature range from $T=0.045$K to $T=0.75$K
(characterized by the corresponding value of $\alpha(T)$) from Fig.5 of
Ref.Moses  has been rescaled as $F\to F_{0}(T)F$ and
$\Delta\mu\to\mu_{0}(T)\Delta\mu$ by some parameters in order to achieve the
best fit by the dependence (18) of the main text. On the log-log graph, Fig.2
of the main text, this implies that the data at a given $T$ was shifted
vertically and horizontally (without changing its slope).

Temperature dependence of the scaling factors is shown in Fig. 4. It can be
interpreted in terms of exponential suppression of the superfluid density
$n_{s}$ by rare thermal fluctuations of the dislocation shape Kuklov2022 .
This mechanism leads to the stretched exponential dependence $\ln
n_{s}(T)\propto-T^{5/4}$ on temperature with weak links providing nucleation
centers for efficient phase slips. Since the instanton action is proportional
to $\sqrt{n_{s}}$ (see Eq.(16) of the main text), this results in the growth
of $\mu_{0}(T)$ and $\alpha(T)$ with increasing $T$.

## References

  * (1) M. W. Ray and R. B. Hallock, Phys. Rev. Lett. 100, 235301 (2008); Phys. Rev. B 79, 224302 (2009); M. W. Ray and R. B. Hallock, Phys. Rev. B 81, 214523 (2010).
  * (2) Ye. Vekhov and R. B. Hallock, Phys. Rev. Lett. 109, 045303 (2012).
  * (3) R.B. Hallock, J. Low Temp. Phys. 197, 167 (2019)
  * (4) Z. G. Cheng, J. Beamish, A. D. Fefferman, F. Souris, S. Balibar, and V. Dauvois, Phys. Rev. Lett. 114, 165301 (2015); Z. G. Cheng and J. Beamish, Phys. Rev. Lett. 117, 025301 (2016).
  * (5) J. Shin, D. Y. Kim, A. Haziot, and M. H. W. Chan, Phys. Rev. Lett. 118, 235301 (2017).
  * (6) J. Shin and M. H. W. Chan, Phys. Rev. B 99, 140502(R) (2019).
  * (7) J. Shin and M. H. W. Chan, Phys. Rev. B 101, 014507 (2020).
  * (8) M.H.W. Chan, J. Low Temp. Phys. 205, 235 (2021).
  * (9) S. G. Söyler, A. B. Kuklov, L. Pollet, N. V. Prokof’ev, and B. V. Svistunov, Phys. Rev. Lett. 103, 175301(2009).
  * (10) A. B. Kuklov, L. Pollet, N. V. Prokof’ev, and B. V. Svistunov, Phys. Rev. Lett. 128, 255301 (2022).
  * (11) Supplemental material.
  * (12) S. Coleman, Aspects of Symmetry (CUP, NY, 1985).
  * (13) L. Liu and A. B. Kuklov, Phys. Rev. B 97, 104510 (2018).
  * (14) M. Yarmolinsky and A. B. Kuklov, Phys. Rev. B 96, 024505 (2017).
  * (15) J. Toner and D. R. Nelson, Phys. Rev. B 23, 316 (1981).
  * (16) L. Radzihovsky, Phys. Rev. Lett. 125, 267601 (2020); Z. Zhai, L. Radzihovsky, Annals of Physics 435, 168509 (2021).
  * (17) B. Svistunov, E. Babaev, and N. Prokof’ev, Superfluid States of Matter, Taylor & Francis, 2015.


# Bayesian Game Formulation of Power Allocation in Multiple Access Wiretap
Channel with Incomplete CSI

Basharat Rashid1, Majed Haddad 2, and Shahid M Shah1  1Dept. of ECE, National
Institute of Technology Srinagar, J&K, India Email<EMAIL_ADDRESS><EMAIL_ADDRESS>2CERI/LIA, University of Avignon, Avignon, France
Email<EMAIL_ADDRESS>
###### Abstract

In this paper, we address the problem of distributed power allocation in a $K$
user fading multiple access wiretap channel, where global channel state
information is limited, i.e., each user has knowledge of their own channel
state with respect to Bob and Eve but only knows the distribution of other
users’ channel states. We model this problem as a Bayesian game, where each
user is assumed to selfishly maximize his average _secrecy capacity_ with
partial channel state information. In this work, we first prove that there is
a unique Bayesian equilibrium in the proposed game. Additionally, the price of
anarchy is calculated to measure the efficiency of the equilibrium solution.
We also propose a fast convergent iterative algorithm for power allocation.
Finally, the results are validated using simulation results.

###### Index Terms:

Game theory, Bayesian game, Physical layer security, Secrecy capacity,
Multiple access wiretap channel.

##  I Introduction

Fading Multiple Access Channel (F-MAC) is well studied in wireless
communication as it models the uplink scenario in a cellular system, where $K$
users are trying to communicate with a base station. Further, optimal resource
allocation techniques in an F-MAC are extensively established in the
literature. Wyner and Shamai first studied F-MAC, assuming that only the
receiver has access to channel state information (CSI) [1]. This work was
later extended in [2] to include the assumption that transmitters have global
CSI, with the authors utilizing the polymatroidal structure of the capacity
region to derive power control policies. In [3], the problem of resource
allocation was framed as a complete information game, with transmitters having
global CSI. Power allocation in F-MAC was first approached as an incomplete
information game in [4], with the assumption of individual CSI and global
knowledge of the channel state distribution. The same setup was extended to a
two-tier cellular network in [5].

In the literature, the concept of F-MAC has been extended to the fading
multiple access wiretap channel (F-MAC-WT), where an eavesdropper attempts to
intercept information in addition to a legitimate receiver. In [6], F-MAC-WT
with global CSI at the transmitters is studied. In [7], the authors proposed a
power control scheme for F-MAC-WT without any knowledge of the channel state
of the eavesdropper. In [8] the authors assumed local channel state
information, where each user is assumed to know his channel state with respect
to Bob and Eve, but the transmitter relies on low-rate feedback from the
receiver. In [9], authors employed a zero-sum game to maximize the _secrecy
rate_ of the network.

In this study, we employ Bayesian game theory to explore the optimal
distributed power allocation in a single input single output (SISO) F-MAC-WT
with limited global CSI. Specifically, we consider a $K$ user uplink scenario,
where each user possesses knowledge of his channel state in relation to Bob
and Eve but only knows the distribution of other users channel states. The
Bayesian game proved to be a useful tool to investigate this incomplete
information scenario, where each rational user aims to maximize their
achievable _secrecy rate_ with restricted knowledge of fading channel gains.
Further, we use the concept of diagonal concavity [10] to demonstrate the
existence of a unique Bayesian equilibrium (BE) in this context and propose an
iterative algorithm to characterize it. Finally, the price of anarchy (POA) is
calculated as a metric to measure the efficiency of the equilibrium state.

As cellular technologies continue to advance, networks are becoming denser
(for example 5G/6G communication systems), resulting in an increased demand
for high data rates and decentralized cellular systems [11]. In this regard,
the proposed Bayesian game theoretical framework aligns with the concept of
secure “self-organizing networks”, which reduces operational complexity and
eliminates the need for global CSI.

The rest of the paper is organized as follows: In Section II, we introduce the
channel model, and in Section III, we develop the Bayesian game setup and
prove the uniqueness of BE. Subsequently, in Section IV, we characterize the
BE and evaluate its efficiency. Finally, in Section V, we provide simulation
results to establish our theoretical claims.

##  II Fading MAC-WT Model

We consider a Gaussian wiretap channel for an uplink scenario where $K$ users
are attempting to communicate with a legitimate receiver (Bob) while being
intercepted by an adversary (Eve). The signal transmitted by the $k^{th}$ user
is represented as $X_{k}$, while the channel gain of the $k^{th}$ user in
relation to Bob and Eve is represented by $\widetilde{H}_{k}$ and
$\widetilde{G}_{k}$ respectively. The received signals by Bob and Eve are
represented by $Y$ and $Z$, respectively, and are given as:

$\displaystyle
Y=\sum_{k=1}^{K}\widetilde{H}_{k}X_{k}+N_{k}^{b}\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\
Z=\sum_{k=1}^{K}\widetilde{G}_{k}X_{k}+N_{k}^{e},$ (1)

where $N_{k}^{b}$ and $N_{k}^{e}$ are zero-mean additive white Gaussian noise
with variance $\sigma^{2}$. It is assumed that the noise $N_{k}^{b}$ is
independent of channel gain $\widetilde{H}_{k}$, also the channel realization
$\widetilde{H}_{k}$ is independent of $\widetilde{H}_{l},\leavevmode\nobreak\
\forall l\neq k$. Similar assumptions hold for the channel gain
$\widetilde{G}_{k}$ and the noise $N_{k}^{e}$. Additionally,
$\widetilde{H}_{k}$ is assumed to be independent of
$\widetilde{G}_{k},\leavevmode\nobreak\ \forall k$. In this paper, we examine
the transmission of wireless signals in a fast-fading environment, where the
channel’s coherence time is relatively small compared to the time delay. It is
further assumed that the channel is stationary and ergodic during the
transmission. For a degraded Gaussian wiretap channel where the signal
received by Bob is stronger than Eve, the achievable rate in this scenario for
secure communication is characterized by _secrecy capacity_[Leung1978Gaussian]
and for the assumed fast fading setup we use the notion of _ergodic secrecy
capacity_ , which is defined as:

$\displaystyle
C^{s}_{k}=\mathbb{E}_{\textbf{H},\textbf{G}}\Bigg{[}\Bigg{(}C_{k}^{b}-C_{k}^{e}\Bigg{)}^{+}\Bigg{]},$
(2)

where $\textbf{H}=\\{H_{1},\dots,H_{K}\\}$,
$\textbf{G}=\\{G_{1},\dots,G_{K}\\}$ and the notation is defined as
$(x)^{+}=\max(0,x)$. Further, $C^{b}_{k}$ and $C^{e}_{k}$ are instantaneous
_Shannon capacity_ for Bob and Eve, respectively, and for non-successive
decoding at the receiver, they are given by

$\displaystyle
C_{k}^{b}=\log_{2}\Bigg{(}1+\frac{\mathit{H}_{k}P_{k}}{\sigma^{2}+\sum\limits_{l\neq
k,l=1}^{K}\mathit{H}_{l}P_{l}}\Bigg{)}$ (3) $\displaystyle
C_{k}^{e}=\log_{2}\Bigg{(}1+\frac{\mathit{G}_{k}P_{k}}{\sigma^{2}+\sum\limits_{l\neq
k,l=1}^{K}\mathit{G}_{l}P_{l}}\Bigg{)},$ (4)

where $H_{k}=\widetilde{H}_{k}^{2}$, $G_{k}=\widetilde{G}_{k}^{2}$ and $P_{k}$
is the transmitted power.

##  III Bayesian game setup for F-MAC-WT

Consider an incomplete information scenario, where each transmitter can
perfectly estimate his channel gain $H_{k}$ and $G_{k}$ with Bob and Eve,
respectively, but does not know the value of the channel state of other users,
i.e., $\textbf{H}_{-k}=\\{H_{1},\dots,H_{k-1},H_{k+1},\dots,H_{K}\\}$ and
$\textbf{G}_{-k}=\\{G_{1},\dots,G_{k-1},G_{k+1},\dots,G_{K}\\}$ are unknown at
$k^{th}$ transmitter. However, the distribution of each $H_{l}$ and $G_{l}$,
$\forall l\neq k$ is known at the $k^{th}$ transmitter. In such a scenario,
each user aims to maximize his _ergodic secrecy capacity_ subject to the
average power constraint $\overline{P}_{k}$. Hence, the following optimization
problem is sought for each user

$\displaystyle\max_{P_{k}}\leavevmode\nobreak\
\mathbb{E}_{\textbf{H},\textbf{G}}\Bigg{[}\Bigg{(}C_{k}^{b}-C_{k}^{e}\Bigg{)}^{+}\Bigg{]},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $ (5)
$\displaystyle\text{\it{s.t.}}\leavevmode\nobreak\
\mathbb{E}_{H_{k},G_{k}}[P_{k}(H_{k},G_{k})]\leq\overline{P}_{k},\leavevmode\nobreak\
P_{k}(H_{k},G_{k})\geq 0$

For the given setup we define

$\displaystyle
C_{k}^{b}=\log_{2}\Bigg{(}1+\frac{\mathit{H}_{k}P_{k}(H_{k},G_{k})}{\sigma^{2}+\sum\limits_{l\neq
k,l=1}^{K}\mathit{H}_{l}P_{l}(H_{l},G_{l})}\Bigg{)}$ (6) $\displaystyle
C_{k}^{e}=\log_{2}\Bigg{(}1+\frac{\mathit{G}_{k}P_{k}(H_{k},G_{k})}{\sigma^{2}+\sum\limits_{l\neq
k,l=1}^{K}\mathit{G}_{l}P_{l}(H_{l},G_{l})}\Bigg{)},$ (7)

where $P_{k}(H_{k},G_{k})$ is the transmitted power, which is the function of
the user’s own channel gains with Bob and Eve. However, the optimization
problem (5) is dependent on the power strategies of all the transmitting
users, which is not common knowledge. Thus, in order to achieve optimal power
allocation, each user needs to adapt his power output depending on his guess
of the strategies of all other users. Bayesian game theory is a well-suited
framework to address the problem of incomplete information, as it provides a
structured approach for users to modify their strategies based on their
assumptions about other users.

Consider each user’s channel gains are i.i.d, discrete with $L$ states i.e,
$H_{k}=\\{h_{1},\dots,h_{L}\\}$ and $G_{k}=\\{g_{1},\dots,g_{L}\\}$, where
$h_{i}$ and $g_{j}$, $i,j\in[1,L]$ are the gains over a coherence time.
Therefore, the $K-$ user Bayesian game for F-MAC-WT can be completely
characterized as:

$\displaystyle\mathcal{G}_{F-MAC-
WT}\triangleq\left(\mathcal{K},\mathcal{T},\mathcal{P},\mathcal{Q},\mathcal{U}\right),$
(8)

where each element of a tuple is defined as:

  * •

Player set: $\mathcal{K}$ is the set of users, $\mathcal{K}=\\{1,\dots,K\\}$.

  * •

Type set: $\mathcal{T}=\mathcal{T}_{1}\times\dots\times\mathcal{T}_{K}$
(‘$\times$’ defines Cartesian product), where $\mathcal{T}_{k}=H_{k}\times
G_{k}$ is the set of tuples. A player’s type is defined by its channel gains,
that is $(h_{i},g_{j})\in\mathcal{T}_{k}$.

  * •

Action set: $\mathcal{P}=\mathcal{P}_{1}\times\dots\times\mathcal{P}_{K}$,
where $\mathcal{P}_{k}=[0,P_{k}^{max}]$. A player’s action is defined as his
transmit power, that is, $P_{k}\in\mathcal{P}_{k}$.

  * •

Probability set:
$\mathcal{Q}=\mathcal{Q}_{1}\times\dots\times\mathcal{Q}_{K}$, where
$\mathcal{Q}_{k}=A_{k}\times B_{k}$ with
$A_{k}=\\{\alpha_{1},\dots,\alpha_{L}\\}$ and
$B_{k}=\\{\beta_{1},\dots,\beta_{L}\\}$, where $\alpha_{i}\triangleq
Pr\\{H_{k}=h_{i}\\}$ and $\beta_{j}\triangleq Pr\\{G_{k}=g_{j}\\}$.

  * •

Utility function set: $\mathcal{U}=\\{U_{1},\ldots,U_{K}\\}$, where $U_{k}$ is
$k^{th}$ user’s achievable rate.

$\displaystyle
U_{k}(P_{1},\ldots,P_{K})=\Big{(}C^{b}_{k}-C^{e}_{k}\Big{)}^{+}$ (9)

######  Definition 1 (Bayesian equilibrium).

The strategy profile $P^{*}(\cdot)=\\{P_{k}^{*}(\cdot)\\}_{k\in\mathcal{K}}$
is a pure strategy Bayesian equilibrium, if for all $k\in\mathcal{K}$, and for
all $P_{k}(\cdot)\in\mathcal{P}_{k}$ and
$\mathbf{P}_{-k}(\cdot)\in\mathcal{P}_{-k}$, we have:

$\displaystyle\overline{U}_{k}(P^{*}_{k},\mathbf{P}_{-k}^{*})\geq\overline{U}_{k}(P_{k},\mathbf{P}_{-k}^{*}),$
(10)

where $\overline{U}_{k}=\mathbb{E}_{\textbf{H},\textbf{G}}[U_{k}]$ is the
average utility.

###  III-A Existence of unique Bayesian equilibrium

######  Theorem 1.

For a degraded fading multiple access wiretap channel, there exists a unique
Bayesian equilibrium for $K$ user game $\mathcal{G}_{F-MAC-WT}$.

_Proof._ Since the strategy space $P_{k}$ is convex, compact, and non-empty
for each user $k$, the utility function $U_{k}$ is continuous with respect to
both $P_{k}$ and $\textbf{P}_{-k}$. Additionally $U_{k}$ is concave in $P_{k}$
for any $\textbf{P}_{-k}$[12].

Based on the above conditions, according to [10]: a non-cooperative game has
unique equilibrium if the non-negative weighted sum of the utility functions
is diagonally strictly concave. Before we proceed further, we will state the
following definition.

######  Definition 2 (Diagonally strictly concave).

To ensure that a non-negative sum function
$f(\textbf{X},\lambda)=\sum_{i=1}^{N}\lambda_{i}\phi_{i}(\textbf{X})$ is
diagonally strictly concave for a fixed vector $\lambda\in\mathbb{R}^{N\times
1}_{+}$ and any vector
$\textbf{X}:=\prod\limits_{i=1}^{N}X_{i}\in\mathbb{R}^{\sum m_{i}}$, where
$X_{i}\in\mathbb{R}^{m_{i}}$, a sufficient condition is that the symmetric
matrix $[J(\textbf{X},\lambda)+J^{T}(\textbf{X},\lambda)]$ is negative
definite, Where $J(\textbf{X},\lambda)$ is the Jacobian of
$\triangledown(\textbf{X},\lambda)$ with respect to X and $J^{T}$ is transpose
of $J$. $\triangledown(\textbf{X},\lambda)$ is called the pseudo gradient of
$f(\textbf{X},\lambda)$ and is defined as:

$\displaystyle\triangledown(\mathbf{X},\lambda)\triangleq\begin{bmatrix}\lambda_{1}\triangledown_{1}\phi_{1}(\textbf{X})\\\
\vdots\\\ \lambda_{N}\triangledown_{N}\phi_{N}(\textbf{X})\end{bmatrix},$ (11)

where $\triangledown_{i}\phi(\textbf{X})$ is the gradient of
$\phi(\textbf{X})$ with respect to $X_{i}$.

######  Lemma 1.

The weighted non-negative sum of the average utilities $\overline{U}_{k}$ is
diagonally strictly concave for $\lambda=c^{+}\mathbf{1}$, where $c^{+}$ is a
positive scalar, $\mathbf{1}$ is a vector whose every entry is 1.

_Proof._ For the sake of clarity, we examine a two-user scenario, i.e., $K=2$.
We emphasize that this is not a restriction of the proposed model. Let us
first define $P_{k}\in\mathbb{R}^{M}$ as the strategy vector of $k^{th}$ user
and $\textbf{P}:=\prod\limits_{k=1}^{2}P_{k}\in\mathbb{R}^{2M}$ as the action
profile for the game. Furthermore, it is considered that $P_{ki},i=1,\dots,M$,
is the $i^{th}$ strategy of the $k^{th}$ user, which corresponds to the value
of channel states $H_{k}$ and $G_{k}$. For the assumed setup, we can define
the average utility as follows:

$\displaystyle\overline{U}_{k}(\mathbf{P})=$
$\displaystyle\sum_{j}\sum_{i}{\gamma_{j}}\gamma_{i}\Bigg{[}\log\Bigg{(}1+\frac{H_{k}P_{ki}(H_{k},G_{k})}{\sigma^{2}+H_{l}P_{lj}(H_{l},G_{l})}\Bigg{)}$
$\displaystyle-\log\Bigg{(}1+\frac{G_{k}P_{ki}(H_{k},G_{k})}{\sigma^{2}+G_{l}P_{lj}(H_{l},G_{l})}\Bigg{)}\Bigg{]}^{+},\leavevmode\nobreak\
k\neq l$ (12)

where $\gamma_{i}$ is the joint probability of different combinations of
channel gains $H_{k}$ and $G_{k}$ and is related to the value of $\alpha_{i}$
and $\beta_{i}$. Similarly, $\gamma_{j}$ is defined for $H_{l}$ and $G_{l}$.
Further, we will define the weighted, non-negative sum of the average payoffs
as:

$\displaystyle
f^{u}(\mathbf{P},\lambda)\triangleq\sum_{k=1}^{2}\lambda_{k}\overline{U}_{k}(\mathbf{P})$
(13)

Similar to (11), we define the pseudo gradient as:

$\displaystyle\triangledown^{U}(\mathbf{P},\lambda)=\begin{bmatrix}c^{+}\frac{\partial\overline{U}_{1}(\textbf{P})}{\partial
P_{11}}\\\ \vdots\\\ c^{+}\frac{\partial\overline{U}_{1}(\textbf{P})}{\partial
P_{1M}}\\\\[6.0pt] c^{+}\frac{\partial\overline{U}_{2}(\textbf{P})}{\partial
P_{21}}\\\ \vdots\\\ c^{+}\frac{\partial\overline{U}_{2}(\textbf{P})}{\partial
P_{2M}}\end{bmatrix}$ (14)

In order to evaluate each element of the column vector (14), we have $i^{th}$
utility for the $k^{th}$ user given as:

$\displaystyle U(P_{ki})=$
$\displaystyle\Bigg{[}\log\Bigg{(}1+\zeta^{b}_{lj}P_{ki}(H_{k},G_{k})\Bigg{)}$
$\displaystyle-\log\Bigg{(}1+\zeta^{e}_{lj}P_{ki}(H_{k},G_{k})\Bigg{)}\Bigg{]}^{+},$
(15)

where,

$\displaystyle\zeta^{b}_{lj}=\frac{H_{k}}{\sigma^{2}+H_{l}P_{lj}(H_{l},G_{l})},\leavevmode\nobreak\
\leavevmode\nobreak\
\zeta^{e}_{lj}=\frac{G_{k}}{\sigma^{2}+G_{l}P_{lj}(H_{l},G_{l})}\leavevmode\nobreak\
\leavevmode\nobreak\ $ (16)

Throughout, we will assume that the channel is degraded, i.e.,
$\zeta^{b}_{lj}>\zeta^{e}_{lj}$, this inequality captures the fact that the
signal received by Bob is better than Eve. Otherwise, the achievable _secrecy
capacity_ is zero. Therefore, the partial derivative of the average utility
function, whenever it exists, is given by

$\displaystyle\frac{\partial\overline{U}_{k}(\textbf{P})}{\partial P_{ki}}=$
$\displaystyle\sum_{j}\gamma_{j}\gamma_{i}\Bigg{(}\frac{\zeta^{b}_{lj}}{1+\zeta^{b}_{lj}P_{ki}(H_{k},G_{k})}$
$\displaystyle-\frac{\zeta^{e}_{lj}}{1+\zeta^{e}_{lj}P_{ki}(H_{k},G_{k})}\Bigg{)}$
(17)

Therefore the pseudo gradient can be written as:

$\displaystyle\triangledown^{U}(\mathbf{P},\lambda)=$
$\displaystyle\begin{bmatrix}c^{+}\sum\limits_{j}\gamma_{j}\gamma_{1}\Bigg{(}\frac{\zeta^{b}_{2j}}{1+\zeta^{b}_{2j}P_{11}(H_{1},G_{1})}-\frac{\zeta^{e}_{2j}}{1+\zeta^{e}_{2j}P_{11}(H_{1},G_{1})}\Bigg{)}\\\
\vdots\\\
c^{+}\sum\limits_{j}\gamma_{j}\gamma_{M}\Bigg{(}\frac{\zeta^{b}_{2j}}{1+\zeta^{b}_{2j}P_{1M}(H_{1},G_{1})}-\frac{\zeta^{e}_{2j}}{1+\zeta^{e}_{2l}P_{1M}(H_{1},G_{1})}\Bigg{)}\\\
c^{+}\sum\limits_{i}\gamma_{1}\gamma_{i}\Bigg{(}\frac{\zeta^{b}_{1i}}{1+\zeta^{b}_{1i}P_{21}(H_{2},G_{2})}-\frac{\zeta^{e}_{1i}}{1+\zeta^{e}_{1i}P_{21}(H_{2},G_{2})}\Bigg{)}\\\
\vdots\\\
c^{+}\sum\limits_{i}\gamma_{M}\gamma_{i}\Bigg{(}\frac{\zeta^{b}_{1i}}{1+\zeta^{b}_{1i}P_{2M}(H_{2},G_{2})}-\frac{\zeta^{e}_{1i}}{1+\zeta^{e}_{1i}P_{2M}(H_{2},G_{2})}\Bigg{)}\end{bmatrix}$

The second order partial derivatives of $\overline{U}_{k}(\textbf{P})$ are
given in (18)-(20).

$\displaystyle\frac{\partial\overline{U}_{k}^{2}(\textbf{P})}{\partial
P_{ki}^{2}}=c^{+}\sum_{j}\gamma_{j}\gamma_{i}\Bigg{[}\Bigg{(}\frac{\zeta^{e}_{lj}}{1+\zeta^{e}_{lj}P_{ki}(H_{k},G_{k})}\Bigg{)}^{2}-\Bigg{(}\frac{\zeta^{b}_{lj}}{1+\zeta^{b}_{lj}P_{ki}(H_{k},G_{k})}\Bigg{)}^{2}\Bigg{]},\forall
i$ (18)

$\displaystyle\frac{\partial\overline{U}_{k}^{2}(\textbf{P})}{\partial
P_{lj}P_{ki}}=c^{+}\gamma_{j}\gamma_{i}\Bigg{(}\frac{\zeta^{e}_{lj}\zeta^{e}_{ki}}{(1+\zeta^{e}_{lj}P_{ki}(H_{k},G_{k}))(1+\zeta^{e}_{ki}P_{lj}(H_{l},G_{l}))}-\frac{\zeta^{b}_{lj}\zeta^{b}_{ki}}{(1+\zeta^{b}_{lj}P_{ki}(H_{k},G_{k})(1+\zeta^{b}_{ki}P_{lj}(H_{l},G_{l})}\Bigg{)},\forall
i,j$ (19)

$\displaystyle\frac{\partial\overline{U}_{k}^{2}(\textbf{P})}{\partial
P_{ki^{{}^{\prime}}}\partial P_{ki}}=0,\leavevmode\nobreak\
\leavevmode\nobreak\ \forall i^{{}^{\prime}}\neq i$ (20)

Now, the Jacobian of $\triangledown^{U}(\mathbf{P},\lambda)$ with respect to P
is defined as:

$\displaystyle J(\textbf{P},\lambda)=$
$\displaystyle\begin{bmatrix}\frac{\partial\overline{U}_{1}^{2}(\textbf{P})}{\partial
P_{11}^{2}}&\dots&\frac{\partial\overline{U}_{1}^{2}(\textbf{P})}{\partial
P_{1M}\partial
P_{11}}&\frac{\partial\overline{U}_{1}^{2}(\textbf{P})}{\partial
P_{21}\partial
P_{11}}&\dots&\frac{\partial\overline{U}_{1}^{2}(\textbf{P})}{\partial
P_{2M}\partial P_{11}}\\\ \vdots&\ddots&\ddots&\ddots&\ddots&\vdots\\\
\frac{\partial\overline{U}_{2}^{2}(\textbf{P})}{\partial P_{11}\partial
P_{2M}}&\dots&\frac{\partial\overline{U}_{2}^{2}(\textbf{P})}{\partial
P_{1M}\partial
P_{2M}}&\frac{\partial\overline{2}_{1}^{2}(\textbf{P})}{\partial
P_{21}\partial
P_{2M}}&\dots&\frac{\partial\overline{U}_{2}^{2}(\textbf{P})}{\partial
P_{2M}^{2}}\end{bmatrix}$

From (16) its evident that $\zeta_{lj}^{b}$ and $\zeta_{lj}^{e}$ are positive,
now for degraded channel condition, i.e., $\zeta_{lj}^{b}>\zeta_{lj}^{e}$ from
(18) we have:

$\displaystyle\Bigg{(}\frac{\zeta^{e}_{lj}}{1+\zeta^{e}_{lj}P_{ki}(H_{k},G_{k})}\Bigg{)}^{2}-\Bigg{(}\frac{\zeta^{b}_{lj}}{1+\zeta^{b}_{lj}P_{ki}(H_{k},G_{k})}\Bigg{)}^{2}$
$\displaystyle=\frac{(\zeta^{e}_{lj})^{2}-(\zeta^{b}_{lj})^{2}+\zeta^{b}_{lj}\zeta^{e}_{lj}P_{ki}(H_{k},G_{k})(\zeta^{e}_{lj}-\zeta^{b}_{lj})}{(1+\zeta^{e}_{lj}P_{ki}(H_{k},G_{k}))^{2}(1+\zeta^{b}_{ki}P_{lj}(H_{l},G_{l}))^{2}}<0$
(21)

Similarly, we can prove (19) is also negative. Therefore, all the non-zero
terms in $J(\textbf{P},\lambda)$ are negative, which proves that the symmetric
matrix is negative definite, i.e.,
$\textbf{P}^{T}[J(\textbf{P},\lambda)+J^{T}(\textbf{P},\lambda)]\textbf{P}<0$.

Therefore, from definition (2), $f^{u}(\mathbf{P},\lambda)$ is diagonally
strictly concave, which proves lemma (1). As we have proved
$f^{u}(\mathbf{P},\lambda)$ is diagonally strictly concave, it follows from
[10] that there exists a unique Bayesian equilibrium for the game
$\mathcal{G}_{F-MAC-WT}$. Although we have proven our results for a two-user
scenario, it is evident that the same can be extended to any $K$.

##  IV Characterization of Bayesian Equilibrium

For the two-user setup of the proposed game $\mathcal{G}_{F-MAC-WT}$, let
$P_{ij}^{k}=P^{k}(h_{i},g_{j})$ be the power allocated by the $k^{th}$ user
corresponding to the specific channel gain $h_{i}$ and $g_{j}$. Considering
that each user observes the same set of $L$ channel states with Eve and Bob,
the power allocation vector for $k^{th}$ user is
$\textbf{P}^{k}=\\{P_{11}^{k},\dots
P_{1L}^{k},\dots,P_{L1}^{k},\dots,P_{LL}^{k}\\}$ with the average power
constraint $\overline{P}^{k}$. Further, for the degraded channel condition,
the average utility of each player can be specified as:

$\displaystyle\overline{U}_{k}(\textbf{P}^{k},\textbf{P}^{l})=$
$\displaystyle\sum_{m=1}^{L}\sum_{n=1}^{{L}}\sum_{j=1}^{L}\sum_{i=1}^{L}\beta_{m}\alpha_{n}\beta_{j}\alpha_{i}\Bigg{[}\log\Bigg{(}1+\frac{h_{i}P_{ij}^{k}}{\sigma^{2}+h_{n}P_{nm}^{l}}\Bigg{)}$
$\displaystyle\quad\quad\quad-\log\Bigg{(}1+\frac{g_{j}P_{ij}^{k}}{\sigma^{2}+g_{m}P_{nm}^{l}}\Bigg{)}\Bigg{]},\leavevmode\nobreak\
l\neq k$ (22)

To find the best-response strategy of the $k^{th}$ player given the strategy
profile $\textbf{P}^{l}$ of another player, we need to solve the following
maximization problem

$\displaystyle\max_{\textbf{P}^{k}}\leavevmode\nobreak\
\overline{U}_{k}(\textbf{P}^{k},\textbf{P}^{l}),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ (23)
$\displaystyle\text{\it{s.t.}}\leavevmode\nobreak\
\mathbb{E}_{H_{k},G_{k}}[P_{ij}^{k}]\leq\overline{P}^{k},\leavevmode\nobreak\
\leavevmode\nobreak\ P_{ij}^{k}\geq 0$

In this problem, the objective function $\overline{U}_{k}$ is a concave
function in $\textbf{P}^{k}$, and the set of constraints is convex. Hence, the
optimization problem can be classified as convex optimization, and the
corresponding Lagrangian $\mathcal{L}$ is given by

$\displaystyle\mathcal{L}=$
$\displaystyle\overline{U}_{k}(\textbf{P}^{k},\textbf{P}^{l})+\lambda^{k}\Bigg{(}\sum_{i=1}^{L}\sum_{j=1}^{L}\beta_{j}\alpha_{i}P_{ij}^{k}-\overline{P}^{k}\Bigg{)}-\sum_{i=1}^{L}\sum_{j=1}^{L}\emph{v}_{ij}^{k}P_{ij}^{k}$

where $\lambda^{k}>0$ and $\nu_{ij}^{k}>0,\forall\leavevmode\nobreak\ k,i,j$
are the Lagrange multipliers associated with inequality constraints.
Therefore, the Karush-Kuhn-Tucker (KKT) conditions for optimization are
necessary and sufficient for optimality and are given in (24) - (25).

$\displaystyle\sum_{m=1}^{L}\sum_{n=1}^{{L}}\beta_{m}\alpha_{n}\beta_{j}\alpha_{i}\Bigg{(}\frac{h_{i}}{\sigma^{2}+h_{n}P_{nm}^{l}+h_{i}P_{ij}^{k}}-\frac{g_{i}}{\sigma^{2}+g_{n}P_{nm}^{l}+g_{i}P_{ij}^{k}}\Bigg{)}+\lambda^{k}\beta_{j}\alpha_{i}-\nu_{ij}^{k}=0,\leavevmode\nobreak\
\forall i,j$ (24)

$\displaystyle\lambda^{(k)}\Bigg{(}\sum_{i=1}^{L}\sum_{j=1}^{L}\beta_{j}\alpha_{i}P_{ij}^{k}-\overline{P}^{k}\Bigg{)}=0,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \&\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \nu_{ij}^{k}P_{ij}^{k}=0,\leavevmode\nobreak\
\leavevmode\nobreak\ \forall i,j$ (25)

For the degraded channel from (24), it is easy to verify that $\lambda^{k}>0$.
Therefore, from (25) we have

$\displaystyle\sum_{i=1}^{L}\sum_{j=1}^{L}\beta_{j}\alpha_{i}P_{ij}^{k}=\overline{P}^{k}\vspace{-0.2cm}$
(26)

The above equation signifies that at Bayesian equilibrium, each user
distributes his power across all the channel realizations depending on their
distributions. Based on equation (24), the optimal solution for each user is
dependent on the transmission power of other user’s, which is not universally
known. Therefore, in order to achieve the optimal power allocation, each user
must individually optimize his transmission power based on his estimate of the
power allocation of the other user. In equation (22), we define
$\psi_{nm}^{b,k}=\sigma^{2}+h_{n}P_{nm}^{l}$ and
$\psi_{nm}^{e,k}=\sigma^{2}+g_{n}P_{nm}^{l}$ as the aggregate interference
observed by $k^{th}$ user in relation to Bob and Eve, respectively. Hence, for
distributed power allocation by $k^{th}$ user, we propose an Algorithm (1).

Algorithm 1 Distributed Iterative Power Allocation

1:

2:Input: Set of channel states $H_{k}=\\{h_{1},\dots,h_{L}\\}$ and
$G_{k}=\\{g_{1},\dots,g_{L}\\}$

3:Initialize $t=0$ , $P_{ij}^{k(0)}=0\leavevmode\nobreak\ \leavevmode\nobreak\
\forall\leavevmode\nobreak\ k$ and $\forall\leavevmode\nobreak\ i,j$

4:Repeat: Until converge

5:$t\leftarrow t+1$

6:for $k\leftarrow 1$ to $2$ do

7: for $n\leftarrow 1$ to $L$ do

8: for $m\leftarrow 1$ to $L$ do

9: $\psi_{nm}^{b,k(t)}=\sigma^{2}+h_{n}P_{nm}^{l(t)}$

10: $\psi_{nm}^{e,k(t)}=\sigma^{2}+g_{n}P_{nm}^{l(t)}$

11: end for

12: end for

13:
$\displaystyle[P_{11}^{k},\dots,P_{1L}^{k},\dots,P_{L1}^{k},\dots,P_{LL}^{k}]\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ $ $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
=\underset{\mathbf{E}_{H_{k},G_{k}}[P_{ij}^{k}]\leq\overline{P}^{k},\leavevmode\nobreak\
P_{ij}^{k}\geq 0}{argmax}\leavevmode\nobreak\
\overline{U}_{k}(\textbf{P}^{k},\textbf{P}^{l})$

14:end for

This algorithm assumes that the players will play the same game multiple
times, without any memory of past games or knowledge of future events. In
every round, each player will choose their own optimal strategy by solving KKT
conditions which are determined by the game’s current state. Hence, if each
player’s power allocation $P_{ij}^{k}$ is determined by applying the single-
player KKT conditions while treating other player’s signals as noise, the set
$\textbf{P}^{k}$ is guaranteed to be a BE of the original game $\mathcal{G}$.
Therefore, we can infer that the Algorithm (1) converges through iteration to
a BE point.

###  IV-A Efficiency of Bayesian equilibrium

In order to evaluate the system’s efficiency when operating at the equilibrium
state we calculate PoA as a metric. We first define social welfare as the sum
of individual _ergodic secrecy_ rates

$\displaystyle
U_{\emph{sum}}(\textbf{P})=\sum_{k=1}^{2}\overline{U}_{k}(\textbf{P}),$ (27)

where $\textbf{P}:=\prod\limits_{k=1}^{2}\textbf{P}^{k}$ and
$\overline{U}_{k}(\cdot)$ is given in (22). We define $R_{\emph{sum}}^{BE}$ as
the sum rate at the BE, where each user optimizes his rate in a decentralized
manner by solving (23) based on the estimate of other user’s strategies.
Consider $C_{\emph{sum}}^{OPT}$ as the social optimal that can be achieved
with centralized control and feedback signaling between the base station and
the users.

######  Definition 3 (Price of Anarchy).

The Price of Anarchy (PoA) is the ratio between the network sum rate at unique
Bayesian equilibrium and the social optimal.

$\displaystyle PoA=\frac{R_{\emph{sum}}^{BE}}{C_{\emph{sum}}^{OPT}}$ (28)

The problem of maximizing the overall network rate in a centralized approach
can be expressed as:

$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\max_{\textbf{P}}\leavevmode\nobreak\ {U}_{\emph{sum}}(\textbf{P}),$ (29) s.t.
$\displaystyle\leavevmode\nobreak\
\mathbb{E}_{H_{k},G_{k}}[P_{ij}^{k}]\leq\overline{P}^{k}\leavevmode\nobreak\
\forall\leavevmode\nobreak\ k,\leavevmode\nobreak\ P_{ij}^{k}\geq 0$

Solving the above optimization problem is challenging because the objective
function is non-convex with respect to P. However, we can address this problem
by utilizing the lower bound given in [13], namely

$\displaystyle\omega\log(z)+\Omega\leq\log(1+z),$ (30)

where the bound is said to be tight for a chosen $z_{o}$ if

$\displaystyle\omega=\frac{z_{o}}{1+z_{o}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\Omega=\log(1+z_{o})-\frac{z_{o}}{1+z_{o}}\log(z_{o})$ (31)

Applying relaxation (30) to (22) with the transformation of variable
$\tilde{P}_{ij}^{k}=\log(P_{ij}^{k})$, we can convert the optimization problem
(29) into a Difference of Convex (DC) problem, namely

$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \min_{\tilde{\textbf{P}}}\leavevmode\nobreak\
\sum_{k=1}^{2}\mathcal{F}_{k}(\tilde{\textbf{P}})-\mathcal{H}_{k}(\tilde{\textbf{P}}),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $ (32)
$\displaystyle\text{\it{s.t.}}\leavevmode\nobreak\
\mathbb{E}_{H_{k},G_{k}}[2^{\tilde{P}_{ij}^{k}}]\leq\overline{P}^{k}\leavevmode\nobreak\
\leavevmode\nobreak\ \forall\leavevmode\nobreak\ k,\leavevmode\nobreak\
\leavevmode\nobreak\ \tilde{P}_{ij}^{k}\geq 0$

where,

$\displaystyle\mathcal{F}_{k}(\tilde{\textbf{P}})=$
$\displaystyle-\sum_{m=1}^{L}\sum_{n=1}^{{L}}\sum_{j=1}^{L}\sum_{i=1}^{L}\beta_{m}\alpha_{n}\beta_{j}\alpha_{i}\Bigg{\\{}\omega_{ij,nm}^{b,k}\Bigg{(}\log(h_{i})$
$\displaystyle+\tilde{P}_{ij}^{k}-\log(\sigma^{2}+h_{n}2^{\tilde{P}_{nm}^{l}})\Bigg{)}+\Omega_{ij,nm}^{b,k}\Bigg{\\}}$
(33) $\displaystyle\mathcal{H}_{k}(\tilde{\textbf{P}})=$
$\displaystyle-\sum_{m=1}^{L}\sum_{n=1}^{{L}}\sum_{j=1}^{L}\sum_{i=1}^{L}\beta_{m}\alpha_{n}\beta_{j}\alpha_{i}\Bigg{\\{}\omega_{ij,nm}^{e,k}\Bigg{(}\log(g_{i})$
$\displaystyle+\tilde{P}_{ij}^{k}-\log(\sigma^{2}+g_{n}2^{\tilde{P}_{nm}^{l}})\Bigg{)}+\Omega_{ij,nm}^{e,k}\Bigg{\\}}$
(34)

The above two equations are clearly convex since the log-sum-exp function is
convex. Hence, we can solve (32) using the generic DCA algorithm [14].
However, as we are optimizing the lower bound, we can use the iterative
procedure given in [13] to improve the bound, and finally do the reverse
transformation $P_{ij}^{k}=2^{\tilde{P}_{ij}^{k}}$ to get the value of actual
power.

##  V Simulations

###  V-A Convergence of Ergodic rate

(a)

(b)

Figure 1: Convergence of Algorithm (1) for different SNR values:
(a) $SNR=1$. (b) $SNR=5$.

To ensure a clear understanding, we consider a two-user scenario with two-
channel states concerning Eve and Bob i.e., $L=2$, with system parameters
$\alpha_{i}=0.5$ and $\beta_{j}=0.5$,$\forall i,j$ also
$H_{k}=\\{2,3.5\\},G_{k}=\\{0.2,0.3\\}$ and $\sigma^{2}=1$ for both users. For
an SNR of 1 and 5, figures (1a) and (1b) above show the rapid convergence of
_ergodic secrecy capacity_ to a unique solution by using the proposed
Algorithm (1). Here, SNR is defined as the ratio of average power
$(\bar{P}^{k})$ and noise power $(\sigma^{2})$. Numerical simulations confirm
the convergence of the Algorithm (1). However, it is observed that the
convergence rate slows down at higher SNR levels.

###  V-B Ergodic sum-rate efficiency

(a)

(b)

(c)

Figure 2: (a) Sum rate vs $SNR$ for: (a) $H_{k}=\\{2,3.5\\}$ and
$G_{k}=\\{0.2,0.3\\}$. (b) $H_{k}=\\{5,7\\}$ and $G_{k}=\\{0.5,0.7\\}$. (c)
Performance of Bayesian Equilibrium

To assess the efficiency of the Bayesian equilibrium in terms of the network
sum rate, in figure (2), we plot the _ergodic secrecy sum rate_ and POA as a
function of SNR. In figure (2a) we consider the same system parameters as
specified above while in figure (2b) we have $H_{k}=\\{5,7\\}$ and
$G_{k}=\\{0.5,0.7\\}$ while rest of the parameters remain unchanged. The
“Uniform” curve illustrates the sum rate when the user lacks channel state
information, resulting in a uniform power allocation across all channel states
by each mobile user. The “Bayesian” curve represents the sum rate achieved at
BE, where each user allocates power based on Algorithm (1). Lastly, the
“Central” curve depicts the sum rate obtained when a centralized approach is
employed to maximize the network sum rate, as explained in Section (IV-A).
This curve serves as an upper bound for the BE. Finally, in figure (2c) we
have plotted POA w.r.t to SNR for specified channel realizations. The
objective is to achieve a POA value of one, indicating an efficient system at
equilibrium. However, the figures clearly demonstrate the inefficiency of the
system at equilibrium, which becomes more pronounced at higher SNR values.

Moreover, using a centralized approach for power allocation necessitates
global CSI, which can be challenging to obtain, particularly in the presence
of adversaries. Therefore, the significance of the Bayesian solution lies in
its ability to develop secure self-organizing networks that reduce operational
complexity and eliminate the requirement for global CSI.

##  VI Conclusion

We have addressed the problem of distributed power allocation in a multiple
access wiretap channel, where global channel state information is limited. The
problem is modeled as a Bayesian game, where each user is assumed to selfishly
maximize his average _secrecy capacity_ with partial channel state
information. We have proved that there exists a unique BE in the proposed game
and developed a distributed iterative algorithm to characterize the BE.
However, the network sum rate at BE exhibits inefficiency when compared to the
centralized policy, especially in the high SNR regime. Hence, future direction
involves exploring the implementation of pricing strategies or enabling
strategic interactions among users to enhance the achievable _secrecy rate_ at
BE.

## References

  * [1] S. Shamai and A. D. Wyner, “Information-theoretic considerations for symmetric, cellular, multiple-access fading channels.i,” _IEEE Transactions on Information Theory_ , vol. 43, no. 6, p. 1877–1894, 1997.
  * [2] D. N. C. Tse and S. V. Hanly, “Multiaccess fading channels. i. poly-matroid structure, optimal resource allocation and throughput capacitie,” _IEEE Transactions on Information Theory_ , vol. 44, no. 7, p. 796–2815, 1998.
  * [3] L. Lai and H. E. Gamal, “The water-filling game in fading multiple access channels,” _IEEE Transactions on Information Theory_ , vol. 54, no. 5, pp. 2110 – 2122, 2008.
  * [4] G. He, M. Debbah, and E. Altman, “A bayesian game-theoretic approach for distributed resource allocation in fading multiple access channels,” _EURASIP Journal on Wireless Communications and Networking_ , 2010.
  * [5] N. Duy Duong, A. S. Madhukumar, and D. Niyato, “Stackelberg bayesian game for power allocation in two-tier networks,” _IEEE Transactions on Vehicular Technology_ , vol. 65, no. 4, pp. 2341 – 2354, 2015.
  * [6] E. Tekin and A. Yener, “Secrecy sum-rates for the multiple-access wire-tap channel with ergodic block fading,” in _45th Annual Allerton Conference on Communication, Control and Computing_ , 2007, pp. 856–863.
  * [7] S. M. Shah, V. Kumar, and V. Sharma, “Achievable secrecy sum-rate in a fading mac-wt with power control and without csi of eavesdropper,” in _2012 International Conference on Signal Processing and Communications (SPCOM)_. IEEE, 2012, pp. 1–5.
  * [8] S. M. Shah, A. K. Chaitanya, and V. Sharma, “Resource allocation in fading multiple access wiretap channel via game theoretic learning,” in _2016 Information Theory and Applications Workshop (ITA)_. IEEE, 2016, pp. 1–7.
  * [9] Z. Xu and M. Baykal-Gürsoy, “A friendly interference game in wireless secret communication networks,” _10th International Conference, NetGCooP_ , 2021\.
  * [10] J. B. Rosen, “Existence and uniqueness of equilibrium points for concave n-person games,” _Econometrica: Journal of the Econometric Society_ , p. 520–534, 1965.
  * [11] B. T. Tinh, L. D. Nguyen, H. H. Kha, and T. Q. Duong, “Practical optimization and game theory for 6g ultra-dense networks: Overview and research challenges,” _IEEE Access_ , 2022.
  * [12] S. P. Boyd and L. Vandenberghe, _Convex optimization_. Cambridge university press, 2004.
  * [13] J. Papandriopoulos and J. S. Evans, “Distributed algorithms for spectrum balancing in multi-user dsl networks,” _IEEE International Conference on Communications_ , 2006.
  * [14] L. An and P. Tao, “The dc (difference of convex functions) programming and dca revisited with dc models of real world nonconvex optimization problems,” _Annals of Operations Research_ , vol. 133, pp. 23–46, 2005.