# Dual chiral density wave induced oscillating Casimir effect

Daisuke Fujii ID<EMAIL_ADDRESS>Advanced Science Research Center,
Japan Atomic Energy Agency (JAEA), Tokai, 319-1195, Japan  Research Center for
Nuclear Physics, Osaka University, Ibaraki 567-0048, Japan  Katsumasa Nakayama
ID<EMAIL_ADDRESS>RIKEN Center for Computational Science, Kobe,
650-0047, Japan  Kei Suzuki ID<EMAIL_ADDRESS>Advanced
Science Research Center, Japan Atomic Energy Agency (JAEA), Tokai, 319-1195,
Japan

###### Abstract

The Casimir effect is known to be induced from photon fields confined by a
small volume, and also its fermionic counterpart has been predicted in a wide
range of quantum systems. Here, we investigate what types of Casimir effects
can occur from quark fields in dense and thin quark matter. In particular, in
the dual chiral density wave, which is a possible ground state of dense quark
matter, we find that the Casimir energy oscillates as a function of the
thickness of matter. This oscillating Casimir effect is regarded as an analog
of that in Weyl semimetals and is attributed to the Weyl points in the
momentum space of quark fields. In addition, we show that an oscillation is
also induced from the quark Fermi sea, and the total Casimir energy is
composed of multiple oscillations.

##  I Introduction

The Casimir effect, proposed by Casimir [1], is crucially important for
understanding small-volume physics in quantum field theory (see Refs. [2, 3,
4, 5, 6] for reviews). Casimir predicted that the decrease of the zero-point
energy of photon fields by two parallel plates would cause an attractive force
for the plates, which are the so-called Casimir energy and the Casimir force.
The Casimir force was experimentally verified about fifty years later [7, 8].

Beyond academic interest, the engineering application of the Casimir effect to
nanotechnology (Casimir engineering) has recently attracted much attention
[9]. The most typical feature of the Casimir effect is an attractive force. On
the other hand, when one tunes the permittivity and/or permeability of plates
and medium, a repulsive force can be also realized [10, 11, 12, 13]. In
contrast to such attractive or repulsive Casimir effects, it would be
interesting that the third type of Casimir effect, where “third” means that
attraction or repulsion is not fixed. For example, under a setup, the sign of
Casimir energy flips from attraction to repulsion as the separation distance
increases, which may be called the sign-flipping Casimir effect (e.g., see
Refs. [14, 15, 16, 17, 18, 19]). As another example, the value of Casimir
energy can oscillate as a function of distance, which may be called the
oscillating Casimir effect (e.g., see Refs. [20, 21, 22, 23, 24, 25, 26, 27,
28, 29, 30, 31, 32]). Among them, Ref. [31] found that it occurs inside Weyl
semimetals, where the origins of the oscillation are Weyl points (WPs) at
finite momenta in the dispersion relations of Weyl fermions. Such new types of
Casimir effects are not only of theoretical interest but also will be
important for Casimir engineering.

The Casimir effect is also important for elucidating quark and gluon dynamics
described by Quantum Chromodynamics (QCD) in a small volume. For example, (i)
as an ab initio method for solving QCD, numerical simulations of lattice QCD
are done in finite volume (e.g., in a box of a few fm size), and finite-volume
effects must be understood. Since the finite-volume effect for zero-point
energy is nothing but the Casimir effect, its understanding is helpful for
interpretations of results in small-volume simulations. (ii) In relativistic
heavy-ion collision experiments, quark-gluon plasma is produced as a fireball
with a size of a few fm. Therefore, we must well understand the contribution
of the Casimir effect to physics inside the fireball and near its boundary.
(iii) In the interiors of neutron stars, dense quark matter as well as nuclear
matter may exist. Depending on the microscopic density profile inside stars,
there may be small regions of quark matter.

Figure 1:  Schematics of the Casimir effect for quark fields inside thin quark
matter, where the quark matter is confined between boundary conditions with a
thickness $L_{z}$. The “outside vacuum” should be regarded as an arbitrary
vacuum consistent with the boundary conditions.

Under these motivations, this paper focuses on what types of Casimir effects
can occur in various phases of quark matter where the thickness of $z$
direction is extremely short (i.e., “thin” quark matter) as illustrated in
Fig. 1. In particular, in this paper, we propose a QCD counterpart of the
oscillating Casimir effect predicted in Weyl semimetals. This is realized for
fermion fields in the dual chiral density wave (DCDW) phase [33] (also see
Refs. [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51,
52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70,
71, 72, 73, 74, 75, 76, 77, 78, 79, 80] and Ref. [81] for a review). The DCDW
phase is a candidate for the ground state of quark/nuclear matter in the
density region of $\rho/\rho_{0}=3\sim 6$ with the normal nuclear density
$\rho_{0}=0.16$ ${\rm fm}^{-3}$ [46, 47], and complex dispersion relations of
fermions in this phase can be an origin of the oscillating Casimir effect.

In this paper, we aim to investigate types of Casimir effects induced from
quark fields in dense quark matter. To accomplish this, this paper is
organized as follows. In Sec. II, we formulate the zero-point energy and the
Casimir energy within the Nambu–Jona-Lasinio (NJL) model [82, 83] as an
effective model of QCD. In Sec. III, we show our numerical results and
classify the types of Casimir effects in three density regions. Section IV is
devoted to our conclusion and outlook.

##  II Formulation

###  II.1 Dual chiral density waves

A chiral density wave is a spatially inhomogeneous chiral condensate, which is
represented as a position-dependent Lorentz-scalar condensate made of the
Dirac fields, $\psi$ and $\bar{\psi}=\psi^{\dagger}\gamma_{0}$:
$\langle\bar{\psi}\psi\rangle\propto\cos(\vec{q}\cdot\vec{r})$ with a
wavenumber vector $\vec{q}$ and a position vector $\vec{r}$. Among various
types of chiral density waves, in this study, we focus on the DCDW, where both
the scalar and pseudoscalar condensates are position-dependent [33],

$\displaystyle\langle\bar{\psi}\psi\rangle=\Delta\cos(\vec{q}\cdot\vec{r}),\ \
\ \langle\bar{\psi}i\gamma_{5}\psi\rangle=\Delta\sin(\vec{q}\cdot\vec{r}),$
(1)

where $\Delta$ is the amplitude of the DCDW and is regarded as the radius of
the chiral circle,
$\langle\bar{\psi}\psi\rangle^{2}+\langle\bar{\psi}i\gamma_{5}\psi\rangle^{2}=\Delta^{2}$.

###  II.2 NJL model

To investigate the DCDW phase of quark matter, in this work, we use the NJL
model [82, 83] (see Refs. [84, 85, 86, 87] for reviews), which was done in
Refs. [41, 44, 46, 47] for early studies. The Lagrangian density of the two-
flavor NJL model is written as

$\displaystyle\mathcal{L}_{\rm NJL}=\bar{\psi}(i\partial\hbox
to0.0pt{\hss/}+\mu\gamma_{0})\psi+G[(\bar{\psi}\psi)^{2}+(\bar{\psi}i\gamma_{5}\vec{\tau}\psi)^{2}],$
(2)

where $\mu$ is the quark chemical potential, $G$ is the coupling constant of
the four-point interactions, and $\vec{\tau}=(\tau_{1},\tau_{2},\tau_{3})$ is
the Pauli matrix for the isospin. Here, we apply the mean-field ansatz for the
DCDW,

$\displaystyle\langle\bar{\psi}\psi\rangle=\Delta\cos(\vec{q}\cdot\vec{r}),$
$\displaystyle\langle\bar{\psi}i\gamma_{5}\tau_{3}\psi\rangle=\Delta\sin(\vec{q}\cdot\vec{r}),$
(3) $\displaystyle\langle\bar{\psi}i\gamma_{5}\tau_{1}\psi\rangle=0,$
$\displaystyle\langle\bar{\psi}i\gamma_{5}\tau_{2}\psi\rangle=0,$ (4)

where $\Delta$ is the amplitude of the DCDW, and $\vec{q}=(0,0,q)$ is the
wavenumber of the DCDW propagating in the $z$ direction. Using this ansatz,
the mean-field (MF) Lagrangian density is

$\displaystyle\mathcal{L}_{\rm MF}=$
$\displaystyle\bar{\psi}\big{[}i\partial\hbox
to0.0pt{\hss/}+\mu\gamma_{0}-M\big{(}\cos(qz)+i\gamma_{5}\tau_{3}\sin(qz)\big{)}\big{]}\psi$
(5) $\displaystyle-\frac{M^{2}}{4G},$

where $M=-2G\Delta$.

Using a local chiral transformation (which may be called a Weinberg
transformation), $\psi e^{i\gamma_{5}\tau_{3}qz/2}\to\psi_{\rm W}$,
$\bar{\psi}e^{i\gamma_{5}\tau_{3}qz/2}\to\bar{\psi}_{\rm W}$, the original
quark field $\psi$ is redefined as a new field $\psi_{\rm W}$, and the
position dependence of the original Lagrangian (5) is removed in the redefined
quark fields. Then, from the diagonalization of the inverse quark propagator
in momentum space, we obtain the four eigenvalues of (quasi-)quarks,
$\omega_{\pm}$ (the positive-energy modes if $\mu=0$) and
$\tilde{\omega}_{\pm}$ (the negative-energy modes):

$\displaystyle\omega_{\pm}=E_{s}-\mu,$ (6)
$\displaystyle\tilde{\omega}_{\pm}=-E_{s}-\mu,$ (7)

where

$\displaystyle
E_{s}=\sqrt{k^{2}+M^{2}+q^{2}/4\pm\sqrt{(qk_{z})^{2}+M^{2}q^{2}}},$ (8)

and $k^{2}=k_{x}^{2}+k_{y}^{2}+k_{z}^{2}$. When $q=0$, the dispersion
relations return to those of usual massive quarks with a mass $M$, where the
two modes with different spins are degenerate. When $q\neq 0$, the two modes
are split, which is labeled by $s=\pm$. In this work, we focus on the case of
$M<q/2$, which is a typical situation of the DCDW phase.111The case of $M\geq
q/2$ is also interesting, but the WP-induced oscillating Casimir effect does
not occur.

Using these eigenvalues, the thermodynamic potential (per a spatial volume
$V=L_{x}L_{y}L_{z}$) at temperature $T=1/\beta$ is written as

$\displaystyle\frac{\Omega(T)}{V}=$ $\displaystyle-
N_{f}N_{c}\int\frac{d^{3}k}{(2\pi)^{3}}\sum_{s=\pm}\left[E_{s}+\frac{1}{\beta}\ln\big{(}1+e^{-\beta(E_{s}-\mu)}\big{)}\right.$
$\displaystyle\left.+\frac{1}{\beta}\ln\big{(}1+e^{-\beta(E_{s}+\mu)}\big{)}\right]+\frac{M^{2}}{4G}.$
(9)

In this work we fix the number of flavors as $N_{f}=2$ and the number of
colors as $N_{c}=3$.

By taking the zero-temperature limit $T\rightarrow 0$ of the thermodynamic
potential (9), we obtain

$\displaystyle\frac{\Omega(T\to 0)}{V}\equiv\frac{E_{0}}{L_{z}}=$
$\displaystyle-
N_{f}N_{c}\int\frac{d^{3}k}{(2\pi)^{3}}\sum_{s=\pm}\left[E_{s}\right.$
$\displaystyle\left.+\left(\mu-E_{s}\right)\theta(\mu-
E_{s})\right]+\frac{M^{2}}{4G}.$ (10)

Here, the first term is regarded as the contribution from the Dirac sea, i.e.,
the negative zero-point energy from the fermion fields. The second term with
the step function is the contribution from the Fermi sea of quarks. The third
term is the positive energy from the order parameter $M$. To provide a
Casimir-effect-like picture also for the Fermi sea contribution (as well as
the Dirac sea), we have denoted Eq. (10) as $E_{0}/L_{z}$ where $E_{0}$ is the
zero-point energy per unit area.

Note that in the traditional treatment of the NJL model, we determine the
values of the order parameters $M$ and $q$ by minimizing Eq. (10) at a fixed
$\mu$. In a finite volume, the thermodynamic potential in a fixed volume is
minimized, and the values of $M$ and $q$ should be determined. This type of
analysis is important for studying the phase diagram in finite volume, but it
is not the purpose of this work. The purpose of this work is to investigate
the feasibility of the oscillating Casimir effect in dense quark matter.

###  II.3 Casimir energy with Lifshitz formula

The Casimir energy is defined as a finite-volume effect for the zero-point
energy (10). In a finite volume, the momentum integral with respect to the
three-dimensional momentum of quarks in Eq. (10) is replaced by a discrete
sum. In this paper, we impose the periodic boundary conditions (PBC) on quark
fields at $z=0$ and $z=L_{z}$, where the $z$ component of quark momentum is
discretized as $k_{z}\rightarrow 2n\pi/L_{z}$ where
$n=0,1,\cdots,\infty$.222The case with boundary conditions for the $x$ or $y$
direction is straightforward, but these boundaries do not induce the WP-
induced oscillating Casimir effect since now the DCDW is along the $z$ axis.
Note that for the PBC, there is no outside vacuum as in Fig. 1, but the
discretization of momentum (and corresponding eigenvalues) can induce a
nonzero Casimir energy.

The zero-point energy (10) as the infinite integral and the corresponding
infinite sum is divergent (unless an energy or a momentum cutoff is
introduced), but the Casimir energy should be finite after using a
mathematical technique. In this paper, we propose two approaches: (i) the
Lifshitz formula and (ii) lattice regularizations.

Using the first approach, at zero temperature $T=0$ and zero chemical
potential $\mu=0$, the analytical solution for the Casimir energy (per unit
area) from quark fields in the DCDW phase can be written as

$\displaystyle E_{\rm Cas}$
$\displaystyle=-4N_{f}N_{c}\sum_{s=\pm}\int_{0}^{\infty}\frac{d\xi}{2\pi}\int\frac{dk_{x}dk_{y}}{(2\pi)^{2}}\ln\big{[}1-e^{-L_{z}k_{z}^{[s]}}\big{]},$
$\displaystyle k^{[\pm]}_{z}$
$\displaystyle=\sqrt{k_{\perp}^{2}+M^{2}+\xi^{2}-\frac{q^{2}}{4}\mp
iq\sqrt{k_{\perp}^{2}+\xi^{2}}},$ (11)

where the first integral variable is the imaginary part $\xi$ of imaginary
frequency $\omega\equiv i\xi$, and $k_{\perp}^{2}\equiv k_{x}^{2}+k_{y}^{2}$.
The overall factor of $2N_{f}N_{c}$ means the degrees of freedom for particle-
antiparticle, flavors, and colors, and the spin degrees of freedom are labeled
as $s=\pm$. The overall factor of $-2$ and the factor of
$-e^{-L_{z}k_{z}^{[s]}}$ are a property of the PBC on fermion fields. For the
MIT bag boundary condition, see Appendix A.

Note that Eq. (11) is one of the main findings in this paper, which is an
analog of the Lifshitz formula [88] to calculate the Casimir effect for photon
fields. When we substitute $M=q=0$ and $q=0$, Eq. (11) returns to the formulas
for the usual massless and massive quarks, respectively.

Furthermore, we define a dimensionless quantity, which we call the Casimir
coefficient, by multiplying $L_{z}^{3}$,

$\displaystyle C^{[3]}_{\rm Cas}\equiv L_{z}^{3}E_{\rm Cas}.$ (12)

This is a convenient quantity for checking the $L_{z}$ dependence of the
Casimir energy.

Figure 2:  (a) Dispersion relations of the four eigenmodes, Eqs. (6) and (7),
at $(M/\Lambda,q/\Lambda,\mu/\Lambda)=(0.1,0.6,0,6)$. (b) Each contribution
from the Dirac sea (red line) or the Fermi sea (blue line) and their sum
(purple line).

###  II.4 Casimir energy with a lattice regularization

As another approach to calculate the Casimir energy, we use lattice
regularizations [89, 90, 28, 29, 91, 92, 30, 31, 93, 32, 94, 95, 96], which is
regarded as a generalization of the original definition by Casimir [1]. Then,
the Casimir energy (per unit area) on the three-dimensional lattice with a
lattice spacing $a$ is defined as

$\displaystyle E^{\rm Lat}_{\rm Cas}=$ $\displaystyle E^{\rm sum}_{0}-E^{\rm
int}_{0},$ (13) $\displaystyle E^{\rm sum}_{0}=$
$\displaystyle-\frac{N_{f}N_{c}}{a^{3}}\sum_{s=\pm}\int_{\rm
BZ}\frac{d(ak_{x})d(ak_{y})}{(2\pi)^{2}}$ $\displaystyle\times\sum^{\rm
BZ}_{n}\Big{[}\frac{1}{2}a|E^{\rm Lat}_{s,n}-\mu|+\frac{1}{2}a|E^{\rm
Lat}_{s,n}+\mu|\Big{]},$ (14) $\displaystyle E^{\rm int}_{0}=$
$\displaystyle-\frac{N_{f}N_{c}}{a^{3}}\sum_{s=\pm}\int_{\rm
BZ}\frac{d(ak_{x})d(ak_{y})d(ak_{z})}{(2\pi)^{3}}$ $\displaystyle\times
N_{z}\Big{[}\frac{1}{2}a|E^{\rm Lat}_{s}-\mu|+\frac{1}{2}a|E^{\rm
Lat}_{s}+\mu|\Big{]},$ (15)

where the dispersion relations $E^{\rm Lat}_{\pm}$ on the lattice is obtained
by replacing the original momenta $k^{2}_{i}$ ($i=x,y,z$) in Eqs. (6) and (7)
as

$\displaystyle k^{2}_{i}\rightarrow\frac{1}{a^{2}}(2-2\cos ak_{i}).$ (16)

The first term $E^{\rm sum}_{0}$ in Eq. (13) is the zero-point energy in
finite volume (and at finite chemical potential) with the momentum discretized
by a finite thickness $L_{z}=aN_{z}$, where $N_{z}$ is the number of lattice
cells. The second term $E^{\rm int}_{0}$ in Eq. (13) is the zero-point energy
with the continuous momentum in infinite volume. The Casimir energy $E^{\rm
Lat}_{\rm Cas}$ is defined as the difference between these two energies. Also,
as a result of lattice regularization, the momentum integral and the momentum
sum are taken within the first Brillouin zone (BZ). In the case of the PBC,
the sum is over $n=0,1,...,N_{z}-1$ (or equivalently $n=0,1,...,N_{z}$).

Finally, we also define the dimensionless Casimir coefficient on the lattice,

$\displaystyle C^{[3]{\rm Lat}}_{\rm Cas}\equiv L_{z}^{3}E^{\rm Lat}_{\rm
Cas}=a^{3}N_{z}^{3}E^{\rm Lat}_{\rm Cas}.$ (17)

The quantities on the lattice, $E^{\rm Lat}_{\rm Cas}$ and $C^{[3]{\rm
Lat}}_{\rm Cas}$, depend on the lattice spacing $a$, but its continuum limit
($a\to 0$) should coincide with Eqs. (11) and (12) in the continuum theory (if
the lattice regularization is appropriate):

$\displaystyle C^{[3]}_{\rm Cas}=\lim_{a\to 0}C^{[3]{\rm Lat}}_{\rm Cas}.$
(18)

###  II.5 Dispersion relations

We remark on the dispersion relations (6) and (7) for quark fields in the DCDW
phase. In Fig. 2(a), we show the four eigenmodes in the DCDW phase. Here, the
parameters are fixed as ($M/\Lambda,q/\Lambda,\mu/\Lambda,)=(0.1,0.6,0.6)$,
where each quantity is dimensionless by dividing by a dimensional parameter
$\Lambda$. In this figure, the two touching points of $\omega_{-}$ and
$\tilde{\omega}_{-}$ are regarded as Weyl points, and in addition,
$\omega_{-}$ intersects the Fermi level at the Fermi points (FPs).
Furthermore, $\omega_{+}$ also intersects the Fermi level. Now, since we set
$q=\mu$ for simplicity, the momenta of the Weyl points of $\omega_{-}$ and the
intersection between the Fermi level and $\omega_{+}$ coincide.

Figure 2(b) shows each contribution from the Dirac or Fermi sea and their sum.
The Dirac-sea and Fermi-sea contributions correspond to the absolute value of
the first and second terms of the integrand in Eq. (10), respectively. These
dispersion relations become flat bands at low momentum and bend twice in the
middle of the dispersion relation. Such non-differentiable points in the
dispersion relation generally lead to an oscillating Casimir effect.

In the next section, we will show plots as shown in Fig. 2(b) for intuitively
understanding the mechanism of the Casimir effect.

##  III Results

In this section, we show the results for the Casimir coefficients in the low-,
intermediate-, and high-density regions, with the PBC (see Appendix A for the
discussion with the MIT bag boundary condition). In the following, with
$\Lambda=860\ {\rm MeV}$, we fix the values of the order parameters, $M$ and
$q$, at three $\mu$ (low-, intermediate, and high-density regions) as those
obtained in Ref. [46], where the authors applied the mean-field approach in
the NJL model with the proper-time regularization.333Note that the values of
order parameters, $M$ and $q$, in the NJL model depend on regularization
schemes. The Casimir energy is a quantity independent of regularization
schemes. Therefore, once $M$ and $q$ are obtained by a regularization, we can
calculate the Casimir energy equivalently by any regularization (using the
fixed $M$ and $q$).

###  III.1 Oscillating Casimir effect (virtual parameters)

We will see in the following section that the oscillating Casimir effect is
attributed to the flat band effect in the total dispersion relation caused by
the presence of Weyl points. Therefore, in this subsection, we show how the
flat band induces the oscillating Casimir effect by using a virtual parameter
set that is not a solution to the gap equation of the NJL model but gives a
typical flat band.

As typical parameters, we consider $(M/\Lambda,q/\Lambda)=(0.1,0.6)$, as shown
in Fig. 2. Then, the Casimir coefficient defined as Eq. (12) is shown in Fig.
3. The obtained results show the oscillating Casimir effect. The Casimir
energy is obtained from the difference between the momentum integral of the
dispersion relation and the infinite sum of the momentum discretized by the
boundary conditions. The oscillation of Casimir energy arises from the
matching of the Weyl points and the discrete levels.

Figure 3: An example of the Casimir coefficient for the oscillating Casimir
effect from the Dirac-sea contribution, where we fix
$(M/\Lambda,q/\Lambda)=(0.1,0.6)$ as virtual parameters.

In order to intuitively understand the mechanism of oscillations, we provide a
graphical explanation. In the following explanation, as shown in Fig. 4, we
represent $E_{0}^{\rm sum}$ by the sum of the blue rectangular areas $S^{\rm
sum}_{n}$, while the corresponding $E_{0}^{\rm int}$ is the sum of the red
areas $S^{\rm int}_{n}$.

Figure 4:  Graphical explanation for the oscillating Casimir effect. The sum
of the blue rectangles $S^{\rm sum}_{n}$ and the integral over the red areas
$S^{\rm int}_{n}$ correspond to the zero-point energies, $E_{0}^{\rm sum}$ and
$E_{0}^{\rm int}$ in finite and infinite volumes, respectively. The Casimir
energy is defined as $E_{0}^{\rm sum}-E_{0}^{\rm int}$, which corresponds to
$\sum_{n}S^{\rm sum}_{n}-\sum_{n}S^{\rm int}_{n}$.

For example, in the low-momentum region of a massive particle, since the
dispersion relation is a downward convex function, then $S^{\rm
int}_{n}>S^{\rm sum}_{n}$ is satisfied, leading to a positive contribution to
$N_{z}(S^{\rm sum}_{n}-S^{\rm int}_{n})\propto E^{\rm Lat}_{\rm Cas}$.444This
is a simplified situation. As a more complex case, when a dispersion relation
is inverted by a Wely point or a Fermi point, the function in the high-
momentum side can be an upward convex function. On the other hand, in the
high-momentum region, since the dispersion relation is an upward convex
function due to the lattice regularization, $S^{\rm sum}_{n}>S^{\rm int}_{n}$,
which leads to a negative contribution to $E^{\rm Lat}_{\rm Cas}$. In a region
that can be regarded as a linear dispersion, we have $S^{\rm sum}_{n}=S^{\rm
int}_{n}$.

As a crucial situation in this paper, in the flat-band region, it is also
obvious that $S^{\rm sum}_{n}=S^{\rm int}_{n}$. The only exception is the case
containing a Weyl point, where the corresponding areas are denoted as $S^{\rm
sum}_{\rm W}$ and $S^{\rm int}_{\rm W}$. As the thickness $N_{z}$ increases,
the position of the Weyl point relative to a rectangle changes. The case where
the endpoint of a rectangle coincides with the Weyl point is shown in Fig.
4(a). In this case, if the dispersion relation around the Weyl point is
assumed to be linear, then $S^{\rm sum}_{n}=S^{\rm int}_{n}$ is exactly held.
Precisely speaking, it deviates somewhat from the linear dispersion and yields
minor contributions as discussed in the last paragraph. As $N_{z}$ increases,
the size and positions of rectangles change and reach the case shown in Fig.
4(b). In this case, we always have $S^{\rm int}_{n}>S^{\rm sum}_{n}$, and
$E^{\rm Lat}_{\rm Cas}$ increases with $N_{z}$. As we continue to increase
$N_{z}$, we eventually reach the case shown in Fig. 4(c), where the Weyl point
coincides with the midpoint of the rectangle’s edge. In this case, $S^{\rm
sum}_{\rm W}$ goes under the flat band, which maximizes $E^{\rm Lat}_{\rm
Cas}$. Furthermore, as $N_{z}$ increases, the rectangle and the Weyl point
reach the situation shown in Fig. 4(d). In this case, the excess of $S^{\rm
int}_{\rm W}$ decreases and eventually reaches zero. Further increasing
$N_{z}$, the position of the Weyl point for the rectangle returns to the case
of Fig. 4(a).

Because the process from (a) to (d) occurs periodically, so that the value of
$E^{\rm Lat}_{\rm Cas}$ oscillates with $L_{z}$. This is the origin of the
$L_{z}$-dependent oscillation of Casimir energy. Then, the oscillation period
(for the PBC) with respect to $L_{z}$ is written as [31]

$\displaystyle L_{z}^{\rm osc}=\frac{2\pi}{k_{\rm WP}},$ (19)

where $k_{\rm WP}$ is the momentum separation between the Weyl point and the
origin. For the current parameters, the period is estimated to be $L_{z}^{\rm
osc}\sim 5.1$ fm with $k_{\rm WP}/\Lambda\sim 0.283$, which is consistent with
the plot in Fig. 3. In addition, in the $3+1$ dimensions, the minor deviation
from this naive estimate is caused by the integral over the $k_{x}\neq 0$ and
$k_{y}\neq 0$.

###  III.2 Low-density region

First, we consider the Casimir effect in the low-chemical potential region,
where we fix $\mu/\Lambda=0.46$. In this region, we find a chiral condensed
phase with $M/\Lambda=0.47$, but the DCDW phase has not yet developed $(q=0)$.

Figure 5 shows the dispersion relation of the quark field along the momentum
$k_{z}$ on the lattice with $a\Lambda=1$, where the blue line represents the
contribution from the Dirac sea, the red line from the Fermi sea, and the
purple line their sum.

Figure 5:  Dispersion relations on the lattice with $a\Lambda=1$ in the low-
density region ($\mu/\Lambda=0.46$).

In this case, there is no mode crossing the Fermi level, which means that the
total dispersion relation has no non-differentiable point. Therefore, the four
dispersion relations are nothing but that of a massive Dirac fermion.

Figure 6 shows the results for the Casimir coefficient at
$(M/\Lambda,q/\Lambda,\mu/\Lambda)=(0.47,0,0.46)$, obtained from the two
approaches.

Figure 6:  Casimir coefficients in the low-density region. We denote
$a\Lambda$ as $a$ in the legends.

The resultant Casimir effect occurs only from the contribution of the Dirac
sea. On the other hand, there is no contribution to the Casimir effect from
the Fermi sea, as expected. In this figure, we find that $C^{[3]}_{\rm Cas}$
approaches zero in the long-thickness region. This behavior means that the
damping of the Casimir energy $E_{\rm Cas}$ is faster than $1/L_{z}^{3}$
($1/L_{z}^{3}$ is expected for the massless fields in the three-dimensional
space). Such a faster damping is well known for free massive fields [97, 98,
99].

We present results for the two cases of lattice spacing $a\Lambda=0.2,0.4$ in
Fig. 6. We show analytical solutions for only the Dirac-sea part of the
Casimir coefficient, obtained from the Lifshitz formula. We observe that in
the long-thickness region of about $aN_{z}=L_{z}>1.0\ {\rm fm}$, the lattice
regularization works well even for $a\Lambda=0.4$. On the other hand, in the
short-thickness region, the results for $a\Lambda=0.4$ deviate from the
analytical solution due to the enhanced ultraviolet lattice cutoff effect. On
the other hand, the results for $a\Lambda=0.2$ reproduce the analytical
solution well up to about $aN_{z}=L_{z}=0.5\ {\rm fm}$. This result guarantees
that the results from our lattice regularization are accurate up to a quite
short-thickness region.

###  III.3 Intermediate-density region

As the chemical potential increases, the contribution from the quark Fermi sea
causes a significant change in the behavior of the Casimir effect. Figure 7
shows the dispersion relation of the quark field along the momentum $k_{z}$
for fixed parameters $(M/\Lambda,q/\Lambda,\mu/\Lambda)=(0.46,0,0.485)$ and
$a\Lambda=1$. In this parameter set, the DCDW phase has not yet occurred, but
the $\omega_{\pm}$ with low momentum is below the Fermi sea. Then, there is an
additional contribution to the Casimir effect from the Fermi sea as well as
from the Dirac sea. Furthermore, the crossing points (Fermi points) of
$\omega_{\pm}$ and the Fermi level lead to non-differentiable points in the
total dispersion relation. Due to this, the oscillating Casimir effect occurs,
by a mechanism similar to that discussed in the subsection II.5.

Figure 7:  Dispersion relations on the lattice with $a\Lambda=1$ in the
intermediate-density region ($\mu/\Lambda=0.485$). Figure 8:  Casimir
coefficients in the intermediate-density region. We denote $a\Lambda$ as $a$
in the legends.

Figure 8 shows the results for the Casimir coefficient. In the short thickness
of Fig. 8, we find that the Casimir coefficient is dominated by the
contribution from the Dirac sea, which is qualitatively the same as the result
shown in the previous subsection III.2. On the other hand, in the long
thickness of Fig. 8, the Casimir coefficient is dominated by the contribution
from the Fermi sea. As a result, the sign of the total Casimir coefficient
flips around $aN_{z}=L_{z}\sim 2.3\ {\rm fm}$ and $8.8\ {\rm fm}$. At further
long-thickness, we expect an oscillation of the Casimir coefficient. However,
now we do not show the numerical results due to the limitations of
calculations with sufficient accuracy. By using Eq. (19), the period of
oscillation is estimated to be $L_{z}^{\rm osc}=2\pi/k_{\rm FP}\sim 9.4\ {\rm
fm}$, where $k_{\rm FP}/\Lambda\sim 0.154$ is the momentum of the Fermi point.

###  III.4 high-density region

When the chemical potential is large enough, the DCDW phase ($q\neq 0$) is
realized. Figure 9 shows the dispersion relation of the quark fields along the
momentum $k_{z}$ for fixed parameters
$(M/\Lambda,q/\Lambda,\mu/\Lambda)=(0.09,0.62,0.72)$ with $a\Lambda=1$.

Figure 9:  Dispersion relations on the lattice with $a\Lambda=1$ in the high-
density region ($\mu/\Lambda=0.72$).

In this case, the two touching points of $\omega_{-}$ and $\tilde{\omega}_{-}$
are regarded as the Weyl points. Furthermore, the $\omega_{+}$ and
$\omega_{-}$ cross the Fermi level. Thus, there are six non-differentiable
points. In the low-momentum region, the total dispersion relation becomes a
flat band due to the cancellations between $\omega_{-}$ and
$\tilde{\omega}_{-}$ and between $\omega_{+}$ and $\tilde{\omega}_{+}$, and
the edges of the flat band of the total dispersion relation are non-
differentiable. Furthermore, due to the four Fermi points, the total
dispersion relation is non-differentiable at the four momenta.

Figure 10 shows the results for the Casimir coefficient.

Figure 10:  Casimir coefficients in the high-density region. We denote
$a\Lambda$ as $a$ in the legends.

For the Dirac-sea effect, the Casimir effect shows oscillating behavior, as
expected from the Weyl points. On the other hand, for the Fermi-sea effect as
depicted by the red line, we see that the Fermi-sea part of Casimir
coefficient oscillates with higher frequency than that from the Dirac sea. Its
frequency is determined by the positions of the non-differentiable points on
the dispersion relation. By using Eq. (19), the oscillation periods are
estimated to be $L_{z}^{\rm osc}=2\pi/k_{\rm WP}\sim 4.9$ fm with $k_{\rm
WP}/\Lambda\sim 0.297$ (for the Dirac sea part) and $L_{z}^{\rm
osc}=2\pi/k_{\rm FP}\sim 1.4$ fm with $k_{\rm FP}/\Lambda\sim 1.03$ (for the
Fermi sea part of $\omega_{-}$), and $L_{z}^{\rm osc}=2\pi/k_{\rm FP}\sim 3.6$
fm with $k_{\rm FP}/\Lambda\sim 0.40$ (for the Fermi sea part of
$\omega_{+}$). The Dirac-sea and Fermi-sea contributions shown in Fig. 10 are
dominated by the periods of $L_{z}^{\rm osc}\sim 4.9$ fm and $L_{z}^{\rm
osc}\sim 1.4$ fm, respectively, and the total result is their superposition.

##  IV Conclusion and outlook

In this paper, we have discussed the Casimir effect from quark fields in some
phases of quark matter. In particular, we consider the QCD counterpart of a
new type of Casimir effect i.e., the oscillating Casimir effect. We have
classified the typical behaviors of the Casimir effect by focusing on the
three types of density regions:

  1. 1.

When the quark chemical potential is small enough, the quark-number density is
zero. Then, the Casimir effect for massive quark fields occurs, where the
damping of the Casimir energy with increasing thickness is characterized by
the dynamical masses of quarks.

  2. 2.

As the quark chemical potential increases, the quark number density becomes
nonzero. In this region, the contribution from the quark Fermi sea effect
leads to the oscillating Casimir effect.

  3. 3.

Further increasing the chemical potential eventually leads to a phase
transition into the DCDW phase. In this region, in addition to the Fermi sea
effect, the Weyl-point structure of quark fields in the DCDW phase results in
the oscillating Casimir effect. The total Casimir energy behaves as the
superposition of two types of oscillations.

To clarify terminology, one of our main findings may be called the DCDW-
induced oscillating Casimir effect, which is a kind of the WP-induced
oscillating Casimir effect predicted in Ref. [31]. On the other hand, the FP-
induced oscillating Casimir effect would be a common term for general
fermionic systems at finite density in finite volume.

In the following, we list some potential future studies.

  * •

Solving gap equation in finite volume—In this work, we input the values of
order parameters obtained from the gap equation in infinite volume, in order
to understand the typical behavior of the Casimir effect for typical
dispersion relations. As a more precise analysis, one can calculate the order
parameters by minimizing the thermodynamic or effective potential in finite
volume, and then one can understand the self-consistent relationship between
the volume-dependent order parameters and the Casimir effect. This type of
study would help elucidate the phase diagram of interacting fermions in finite
volume (e.g., see Refs. [100, 101, 102, 103, 104, 105, 106, 107]) from the
viewpoint of the Casimir effect.

  * •

Lattice simulations—Our predictions will be examined by future lattice NJL
(and also QCD) simulations. Since Monte Carlo simulations at large quark
chemical potential suffer from the sign-problem, sign-problem-free techniques
should be applied. In particular, numerical simulations of Casimir effects for
lattice gauge fields have been vigorously studied, such as the U(1) gauge
[108, 109, 110], the compact U(1) gauge [111, 112, 113, 114, 115], the SU(2)
gauge [116, 117], and the SU(3) gauge fields [118, 119], so that the interplay
between quark and gluon dynamics would be interesting.

  * •

Finite temperature—At finite temperature, thermal fluctuations (as well as
quantum fluctuations) also induce the Casimir effect, which is the so-called
thermal Casimir effect well established theoretically [88] and experimentally
[120]. The Casimir energy at finite temperature can be calculated by using a
straightforward modification of Eq. (11) or Eq. (13). Although the DCDW phase
disappears at sufficiently high temperature, in the low-temperature region the
oscillating Casimir effect can be realized.

  * •

Axion electrodynamics—In this work, we consider only the Casimir effect for
quark fields, which is induced by boundary conditions on quark fields. The
dynamics of photons in the DCDW phase may be described by modified Maxell
equations [63, 121, 76] which is the so-called axion electrodynamics [122,
123]. The Casimir energy for such modified photon fields can exhibit a sign-
flipping behavior [19, 124, 125, 126, 127, 95] when boundary conditions on
photon fields are imposed. It will be interesting whether or not such a
photonic Casimir effect occurs also in the DCDW phase.

  * •

Models with nucleons—In this work, we consider the Casimir effect for quark
degrees of freedom based on the NJL model. Apart from the NJL model, in the
low-density phase of QCD, relevant degrees of freedom are hadrons,
specifically nucleons. Therefore, it will be important to examine the
consistency between the quark Casimir effect in the NJL model and the nucleon
Casimir effect in an effective model such as the nucleon linear sigma model
which can also investigate the DCDW phase.

  * •

Other inhomogeneous chiral phases—We have shown the WP-induced oscillating
Casimir effect in the DCDW phase, which is particular to systems with
momentum-dependent Weyl points. It would be interesting what types of Casimir
effects can occur in other inhomogeneous chiral phases. For example, the real-
kink crystal phase is known as a possible ground state in the 3+1 dimension
NJL model [50]. In this phase, because $k_{z}$ is not a conserved quantity,
our approach formulated in spatial-momentum space cannot be applied, but the
Casimir energy should be calculated by approaches formulated with discrete
eigenenergies. Furthermore, apart from the dense quark matter, a magnetic
field can also induce the DCDW phase [53], which is the so-called magnetic
dual chiral density wave (MDCDW) (see Ref. [128] for a review). The Casimir
effect in such a phase is also interesting.

  * •

Lower-dimensional models of QCD—The WP-induced oscillating Casimir effect can
occur also in lower spatial dimensions (i.e., $1+1$ or $2+1$ dimensions) [31].
Therefore, investigation of the Casimir effect in possible inhomogeneous
phases of low-dimensional QCD-like models, such as the Gross-Neveu model [129,
130, 131], the chiral Gross-Neveu (NJL2) model [132], and the ’t Hooft model
(QCD2) [132, 133, 134], will be also important.

## ACKNOWLEDGMENTS

The authors thank Tsutomu Ishikawa for fruitful discussions. This work was
supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI
(Grant No. JP20K14476).

##  Appendix A Results with MIT bag boundary

In the main text, we applied the PBC to consider the Casimir effect. In this
Appendix, we consider the case with the MIT bag boundary condition, which is
more realistic when we consider quark matter in nature or experiments.

The MIT bag boundary condition [135] requires that the quark-current flux
toward normal to the boundary is zero. For the massless Dirac field on the MIT
bag boundaries, one can obtain the discretized momentum
$k_{z}\to(n+1/2)\pi/L_{z}$ as an analytic solution satisfying the Dirac
equation. On the other hand, for the massive case, one cannot obtain an
analytic form of momentum, so that one should apply an alternative approach
with no analytic form of $k_{z}$ (e.g., Refs. [136, 137, 138]). In this
Appendix, as an approximate estimate, we show the results using the massless
solution $k_{z}\to(n+1/2)\pi/L_{z}$, which may be called “MIT bag” boundary
conditions.

The Lifshitz formula for the Casimir energy from the Dirac sea with the “MIT
bag” boundaries is given by

$\displaystyle E_{\rm Cas}^{\rm``MIT"}$
$\displaystyle=-2N_{f}N_{c}\sum_{s=\pm}\int_{0}^{\infty}\frac{d\xi}{2\pi}\int\frac{d^{2}k_{\perp}}{(2\pi)^{2}}\ln\big{[}1+e^{-2L_{z}k_{z}^{[s]}}\big{]},$
$\displaystyle k^{[\pm]}_{z}$
$\displaystyle=\sqrt{k_{\perp}^{2}+M^{2}+\xi^{2}-\frac{q^{2}}{4}\mp
iq\sqrt{k_{\perp}^{2}+\xi^{2}}},$ (20)

where the first integral variable is the imaginary part $\xi$ of imaginary
frequency $\omega\equiv i\xi$, $k_{\perp}^{2}\equiv k_{x}^{2}+k_{y}^{2}$, and
$d^{2}k_{\perp}\equiv dk_{x}dk_{y}$. The overall factor of $2N_{f}N_{c}$ means
the degrees of freedom for particle-antiparticle, flavors, and colors, and the
spin degrees of freedom are labeled as $s=\pm$. The overall minus sign and the
factor of $+e^{-2L_{z}k_{z}^{[s]}}$ is a property of the “MIT bag” boundary on
fermion fields. In the lattice rugularization, we have to replace as
$\sum^{\rm BZ}_{n}\to\frac{1}{2}\sum^{\rm BZ}_{n}$ in Eq. (14), and the sum is
over $n=0,1,...,2N_{z}-1$.

Figure 11:  Casimir coefficients in the low-density region with the “MIT bag”
boundary conditions. We denote $a\Lambda$ as $a$ in the legends.

Figure 11 shows the results of the Casimir coefficients at low-,
intermediate-, and high-density regions. In the case of the “MIT bag”
boundary, the number of discretized levels is twice larger than that with the
PBC. As a result, the period of oscillation is half of that with the PBC:
analogous to Eq. (19),

$\displaystyle L_{z}^{\rm osc``MIT"}=\frac{\pi}{k_{\rm WP}}.$ (21)

Furthermore, from these results, we can also discuss the case with the
antiperiodic boundary conditions (APBCs). The discrete momentum,
$k_{z}\to(2n+1)\pi/L_{z}$, for the APBCs is shifted by a half period from
$k_{z}\to 2n\pi/L_{z}$ for the PBCs, and hence the sign of the Casimir energy
is flipped and also its absolute value changes slightly. The corresponding
Lifshitz formula is written as

$\displaystyle E_{\rm Cas}^{\rm APBC}$
$\displaystyle=-4N_{f}N_{c}\sum_{s=\pm}\int_{0}^{\infty}\frac{d\xi}{2\pi}\int\frac{d^{2}k_{\perp}}{(2\pi)^{2}}\ln\big{[}1+e^{-L_{z}k_{z}^{[s]}}\big{]}.$
(22)

In the $3+1$ dimensions, the Casimir energy with $k_{z}\to(2n+1)\pi/L_{z}$ is
$16$ times larger than that with $k_{z}\to(n+1/2)\pi/L_{z}$. As a result, by
multiplying the vertical axis obtained in Fig. 11 by $16$ and the horizontal
axis by $2$, we can obtain the exact results for the APBCs.

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Herding LLaMaS: Using LLMs as an OS Module

Aditya K Kamath
Both authors contributed equally to this research.

University of Washington

Sujay Yadalam
[1]

University of Wisconsin–Madison

§ INTRODUCTION

Computer systems are becoming increasingly heterogeneous with the emergence of new memory technologies and compute devices.
GPUs alongside CPUs have become commonplace and CXL is poised to be a mainstay of cloud systems.

The operating system is responsible for managing these hardware resources, requiring modification every time a new device is released.
Years of research and development are sunk into tuning the OS for high performance with each new heterogeneous device <cit.>.
With the recent explosion in memory technologies and domain-specific accelerators, it would be beneficial to have an OS that could provide high performance for new devices without significant effort.

We propose  which can adapt to new devices easily.  uses Large Language Models (LLMs) to extract the useful features of new devices from their textual description and uses these features to make operating system decisions at runtime. Adding support to  for a new device is as simple as describing the system and new device properties in plaintext.

reduces the burden on system administrators to enable easy integration of new devices into production systems.

Preliminary evaluation using ChatGPT <cit.> shows that LLMs are capable of extracting device features from text and make correct OS decisions based on those features.

* Brief background of LLMs. Papers where LLMs are used as decision making engines.

§ BURDEN OF HETEROGENEITY ON THE OS

* Many advances in technologies. Our focus is on two, memory and compute. Memory: NUMA, NVM, CXL, HBM. Compute: GPUs, accelerators, FPGA, TPUs.

* OS is responsible for the interface between the app and devices. As such, every new tech requires OS changes for support.

* Significant research effort spent on improving performance. List papers.

The end of Moore's law and Dennard scaling has made the use of heterogeneous systems necessary.
Modern high-performance systems are embracing heterogeneity in both memory and compute.
These systems combine the best properties of different memory technologies to optimize for latency, bandwidth, capacity, and cost.
For processors, the adoption of GPUs and other domain-specific accelerators (DSAs) has helped push the boundaries of compute.

Different applications exhibit different memory requirements, necessitating a diverse set of memory devices to satisfy all of them.
A modern HPC system could be connected to local DRAM and NVM, and have disaggregated memory over CXL.
Non-volatile memory (NVM) <cit.> provides high capacities, but experiences read/write asymmetry as well as reduced bandwidth.
Similarly, CXL provides greater memory capacity than on-board DRAM, at the expense of increased latencies and lower bandwidth <cit.>.

Data-intensive applications like machine learning or scientific computations require high throughput that is not met by conventional architectures.
This has led to the development of accelerators such as GPUs and specialized hardware.
In the face of this explosive growth of diverse DSAs each with their own unique API, there has been significant effort being put into unifying application development <cit.>.
The RISC-V group endeavors to provide a unified ISA that can support the unique attributes that these different accelerators require <cit.>.
On the compiler side, the MLIR project <cit.> is providing an intermediate layer that allows developers to code in their language of choice and then compile the source code into optimized binaries for a chosen processor.
In the face of these advancements, we envision a future where an application binary could be deployed on any processing device without programmer input.
The operating system (OS) would be tasked with selecting the optimal processing device for the application.

The operating system is the gatekeeper between applications and hardware devices.
Beyond providing minimal support for these devices, the OS must be aware of the different intricacies and characteristics under which the devices perform optimally, to remove the reliance on application programmers.

This requirement of OS modification leads to a significant amount of research effort being spent in trying to devise the best method for handling these devices.
For example, there has been significant work in page placement for NVM <cit.> and CXL <cit.>.
In addition, many works have explored techniques for managing data placement and replication for NUMA systems <cit.>. Similarly, we foresee that significant effort will need to be made in order to allow the OS to select the optimal processing device.

It would be beneficial to have an operating system that could adapt to any heterogeneous system quickly.
Such an operating system would reduce the burden of researchers and system administrators.
It would also reduce the effort required to integrate new devices into production systems.

§ OUR SYSTEM: HERDING LLAMAS

Our goal is to design an operating system that would be able to (1) adapt to new heterogeneous technologies while (2) requiring minimal intervention from the programmer/system administrator.

To this end, we propose using Large Language Models as an OS Module () for managing heterogeneous resources and devices (a.k.a., the herd)[It is worth noting that for our title "Herding ",  is responsible for managing the herd, and so is performing the herding, not being herded.].

Language models are a class of Natural Language Processing algorithms that aim to recognize, summarize, translate, predict and generate text.
Large language models or LLMs are models trained on very large datasets with billions to trillions of parameters.
OpenAI's GPT-3 has over 100 billion parameters and is trained on almost the entirety of the internet.
The recent success of LLMs is attributed to few-shot or zero-shot learning.
LLMs can solve various tasks by simply training the models on a few examples (few-shot) or by providing instructions describing the task (zero-shot).

takes advantage of LLMs' ability to perform zero-shot learning.  is able to flexibly adapt to new devices with quirky characteristics, requiring only a simple description of the system and hardware characteristics.

Figure <ref> describes how  functions.
There are two main components of : (1) frontend LLM and (2) backend prediction model.
The frontend LLM takes the system description as a text file along with the program binary and/or source code if available.
It processes them to identify the important features and patterns that could influence decisions during program execution and generates embeddings that contain these information.

The backend prediction model (BPM) is responsible for making runtime OS decisions. The embeddings generated by the frontend LLM guide the BPM on  which heuristics influence decisions and when the decisions need to be made. For instance, the LLM could inform the BPM to migrate data when the accesses per second to a page exceed a certain value.

As an example, the frontend LLM may be informed that a system contains two types of memory, repetitive access tech (RAT) which provides low latency for frequently accessed data but high latency for infrequently accessed data, and DRAM which provides predictable latencies for all data.
Through the embeddings, it will inform the BPM that frequently accessed data should be kept in RAT.
During an application's runtime, the hardware counters may indicate that data in DRAM is being frequently accessed, causing the BPM to move it to RAT.
Through this process, the OS teaches itself how to handle new technology with minimal external support.

LLaMaS system design.

* What we require from our system. Reiteration of problems and goals.

* Our proposal: Herding LLaMaS, and what it hopes to achieve.

* In-depth description of Figure <ref>.

§ FEASIBILITY STUDY

To study the feasibility of , we conducted experiments using ChatGPT <cit.>, a chatbot built on top of OpenAI's GPT-3.5 and GPT-4 language models.
The goal of the experiments was to understand the abilities of a language model to extract insights from a textual description of device characteristics.
The prompts to ChatGPT had 2 components: (1) descrition of a heterogeneous system configuration and device properties, and (2) OS decisions to be made.
Below we share the results of 3 such experiments.
We first show the prompt to ChatGPT then its response:

Experiment 1:  Here, we wanted to ensure that LLMs could understand system descriptions to extract useful features.
Prompt: "System you manage has CPU and GPU. CPU is connected to DRAM and GPU is connected to HBM. For the GPU matrix multiplication program shown below, identify variables that should be allocated on DRAM and those on HBM."

ChatGPT: "Variable h_A is used to store the input matrix on the host, so allocate in host memory. Variable d_A is used to store the input matrix on the device, so allocate in device memory."

ChatGPT was able to successfully understand the system description. Since ChatGPT had been trained on data involving CPUs and GPUs, we didn't have to describe the system to it.
It was able to understand the use of different variables and map them to the host or device memory.

Experiment 2:  While the above experiment proved that LLMs could work well on data it was trained on,  requires that the OS module adapts to new devices with unseen characteristics. Therefore, our second experiment imagined a heterogenous system with a futuristic device.

Prompt: "System you manage has CPU connected to DRAM and Repetitive Access Technology (RAT) memories. RAT is a new memory technology that provides low latency for frequently accessed data. Given below program, identify variables that should be allocated in RAT memory.

ChatGPT: "RAT memory should be used for variables that are accessed repeatedly such as arrays or matrices. DRAM should be used for variables that are accessed randomly and infrequently. These variables should be allocated in RAT: priceArr, avg, ..."

ChatGPT was able to conceptualize the new memory technology and capture its properties. It identified the important features to consider while making memory placement decisions.

Experiment 3: Task placement: Above experiments demonstrated that abilities of LLMs to aid in memory management. We then tried to understand the abilities of LLMs to perform task mapping to different computational platforms.

Prompt: You manage a system with a CPU of 24 cores and a GPU. For best performance, where would you run the following functions?

ChatGPT: Func_A involves matrix multiplication and is highly parallelizable. Therefore, we would prefer to run it on the GPU. On the other hand, Func_B involves pointer manipulation and memory operations, so we would prefer to run on the CPU.

ChatGPT was able to understand functions' goals and properties (parallelizable, memory access patterns) and match it with the properties of the underlying hardware.

What we foresee in the future.