# Colossal stability of antiferromagnetically exchange coupled nanomagnets

Kuntal Roy<EMAIL_ADDRESS>Department of Electrical Engineering and
Computer Science, Indian Institute of Science Education and Research (IISER)
Bhopal, Bhopal, Madhya Pradesh 462066, India

###### Abstract

Bistable nanomagnets store a binary bit of information. Exchange coupled
nanomagnets can increase the thermal stability at low dimensions. Here we show
that the antiferromagnetically (AFM) coupled nanomagnets can be highly stable
at low dimensions than that of the ferromagnetically (FM) coupled nanomagnets.
By solving stochastic Landau-Lifshitz-Gilbert equation of magnetization
dynamics at room temperature, we analyze the stability of the exchange coupled
nanomagnets in the presence of correlated, uncorrelated, and anti-correlated
noise. The results show that the correlated noise can make the stability of
the AFM coupled nanomagnets very high. Such finding will lead to very high-
density non-volatile storage and logic devices in our future information
processing systems.

††preprint: AIP/123-QED

The invention and development of charge-based transistor electronics has been
a story of great success Shockley, Bardeen, and Brattain (1956); Kilby (2000)
and it has been materialized by miniaturization, i.e., cramming more and more
transistors on a chip following the so-called Moore’s law purely based on
observation Moore (1998). However, excessive energy dissipation in the
transistors limits the further improvement of the charge-based electronics
Borkar (1999); Landauer (1961); Keyes and Landauer (1970); Roy (2014a).
Electron’s spin-based counterpart, so-called spintronics has profound
potential Fert (2007); Grünberg (2007) to be the replacement of current
transistor-based technology in our future energy-efficient information
processing systems Chappert, Fert, and Dau (2007); Kajiwara  _et al._ (2010);
Ikeda  _et al._ (2010); Roy, Bandyopadhyay, and Atulasimha (2011); Roy (2016,
2014b); Ramaswamy  _et al._ (2018); Tokura, Yasuda, and Tsukazaki (2019);
Puebla  _et al._ (2020). Switching the magnetization direction between the two
symmetry-equivalent stable states allows us to implement _non-volatile_
storage and process the information Roy (2013, 2017a); Kajiwara  _et al._
(2010).

There has been enormous progress in the field of spintronics and nanomagnetics
in recent years with the advent of new materials and phenomena Roy (2017b),
however, it is a formidable challenge to minimize the dimension of the
nanomagnets Roy (2015a) to incur less and less area on a chip. This is due to
the reason that the thermal stability of the magnetization state depends on
the energy barrier between the two stable states and that is proportional to
the volume of the nanomagnet. An energy barrier of about 40 _kT_ (_k_ is the
Boltzmann constant and _T_ is the temperature) at room temperature (T = 300 K)
is required to maintain the stability of the magnetic bits for 10 years
Sharrock (1994); Weller and Moser (1999).

With an eye to achieve higher and higher areal density of magnetic bits, the
magnetic recording has made transition from longitudinal to perpendicular
magnetic anisotropy Richter (2007), and also there has been demonstration of
perpendicular interface anisotropy Ikeda  _et al._ (2010); Sato  _et al._
(2014). However, at small lateral dimensions ($\sim$10 nm), the energy barrier
is not sufficient to hold the information bits for long as per standard
requirement. The thermal stability at low dimensions remained a primary
concern as it goes down to $20\,kT$ at room temperature for nanomagnets of
$\sim$10 nm lateral dimension Sato  _et al._ (2014). There have been studies
on whether a coupled composite system of magnetic bits can increase the energy
barrier and the thermal stability Margulies  _et al._ (2004); Suess  _et al._
(2005, 2007); Richter, Girt, and Zhou (2002); Richter and Dobin (2006); Choo
_et al._ (2007); Bertram and Lengsfield (2007); Hauet  _et al._ (2009); Nolan,
Valcu, and Richter (2011); Nogués  _et al._ (2005); Wang, Shen, and Bai
(2005); Wang, Shen, and Hong (2007); Victora and Shen (2005a, b, 2008);
Fullerton  _et al._ (2000); Tudosa  _et al._ (2010); Yulaev  _et al._ (2011);
Cuchet  _et al._ (2015); Clément  _et al._ (2015); Roy (2015b).

Here we analyze the thermal stability of exchange coupled nanomagnets using
the stochastic Landau-Lifshitz-Gilbert (LLG) equation of magnetization
dynamics Landau and Lifshitz (1935); Gilbert (2004) in the presence of room
temperature thermal fluctuations Brown (1963). We consider both the
antiferromagnetically (AFM) and ferromagnetically (FM) exchange coupled
nanomagnets. We analyze the critical magnetization dynamics in the presence of
room temperature thermal fluctuations when the noise applied on the two
nanomagnets are correlated, uncorrelated, and anti-correlated. We find that
the AFM coupled nanomagnets in the presence of _correlated_ noise can be very
effective in increasing the thermal stability of the nanomagnets. It makes a
perfect sense in assuming _correlated_ noise experienced by the two
nanomagnets due to their proximity while in idle mode (but the magnetizations
fluctuate at their respective positions due to thermal agitations). While the
FM coupled nanomagnets with very high exchange strength would just act like a
composite system, the AFM coupled nanomagnets are more crucial to understand
and here the insights have been provided using the LLG equation of
magnetization dynamics.

Figure 1:  Exchange coupled nanomagnets and the mean deflections of
magnetizations in the presence of room temperature (300 K) thermal
fluctuations, determined from stochastic LLG simulations. (a) Two nanomagnets
NM1 and NM2 are _antiferromagnetically_ (AFM) exchange coupled via an exchange
coupling layer (ECL) with exchange coupling strength $|J_{ex}|$. (b) Two
nanomagnets are _ferromagnetically_ (FM) coupled via an ECL with exchange
coupling strength $J_{ex}$. Here, $J_{ex}$ is positive while $J_{ex}$ is
negative for the AFM coupled case. (c) Axes assignment in the standard
spherical coordinate system with $\theta$ as polar angle and $\phi$ as
azimuthal angle. For the AFM coupled case, NM1 and NM2 are pointing towards
$\theta=180^{\circ}$ and $\theta=0^{\circ}$, respectively. For the FM-coupled
case, both NM1 and NM2 are pointing towards $\theta=180^{\circ}$. (d) Mean
deflections of the magnetizations in the AFM and FM coupled systems in the
presence of room temperature thermal noise (correlated, uncorrelated, and
anti-correlated) for $|J_{ex}|=1K_{u}V$ ($K_{u}V$ is the uniaxial anisotropy
energy barrier).

Figures 1(a) and (b) show the AFM and FM exchange coupled nanomagnets (NM1 and
NM2), respectively separated by an exchange coupling layer (ECL) e.g.,
Ruthenium Parkin, More, and Roche (1990); Parkin (1991) with strength
$|J_{ex}|$. The magnetizations $\mathbf{M_{1(2)}}$ of the two coupled
nanomagnets have a constant magnitude but a variable direction and thus we can
represent it by a vector of unit norm
$\mathbf{n_{m,1(2)}}=\mathbf{M_{1(2)}}/|\mathbf{M_{1(2)}}|=\mathbf{\hat{e}_{r}}$,
where $\mathbf{\hat{e}_{r}}$ is the unit vector in the radial direction in
standard spherical coordinate system and the other two unit vectors are
$\mathbf{\hat{e}_{\theta}}$ and $\mathbf{\hat{e}_{\phi}}$ for $\theta$ and
$\phi$ rotations, respectively [Figure 1(c)]. The potential energy $E_{pot}$
of the nanomagnets can be expressed as the sum of the uniaxial anisotropy
energy ($E_{uni}$) and the exchange energy ($E_{ex}$) as

$\displaystyle E_{pot,1(2)}(\theta_{1},\phi_{1},\theta_{2},\phi_{2})$
$\displaystyle\,=E_{uni,1(2)}(\theta_{1(2)})+E_{ex}(\theta_{1},\phi_{1},\theta_{2},\phi_{2}),$
(1)

where

$E_{uni,1(2)}(\theta_{1(2)})=K_{u,1(2)}V_{1(2)}sin^{2}\theta_{1(2)},$ (2)
$E_{ex}(\theta_{1},\phi_{1},\theta_{2},\phi_{2})=J_{ex}\mathbf{M_{1}}\cdot\mathbf{M_{2}},$
(3)

$K_{u,1(2)}=(1/2)M_{s,1(2)}H_{k,1(2)}$, $K_{u,1(2)}$, $M_{s,1(2)}$,
$H_{k,1(2)}$, and $V_{1(2)}$ are the uniaxial anisotropy, the saturation
magnetization, the anisotropy field for switching of magnetization, and the
volume of the nanomagnets. In the above and onwards, the subscripts $1$ and
$2$ denote the cases for the nanomagnets NM1 and NM2, respectively.

The random thermal fluctuations experienced by the nanomagnets are accounted
by a random magnetic field

$\mathbf{h}(t)=h_{x}(t)\mathbf{\hat{e}_{x}}+h_{y}(t)\mathbf{\hat{e}_{y}}+h_{z}(t)\mathbf{\hat{e}_{z}},$
(4)

where $h_{i}(t)$ ($i\in x,y,z$) are the three components of the random thermal
field in Cartesian coordinates. We assume the properties of the random field
$\mathbf{h}(t)$ as described in Ref. Brown (1963). The random thermal field
can be written as Brown (1963)

$h_{i}(t)=\sqrt{\frac{2\alpha kT}{|\gamma|M\Delta t}}\;G_{(0,1)}(t)\quad(i\in
x,y,z)$ (5)

where $\gamma$ is the gyromagnetic ratio for electrons, $\alpha$ is the
dimensionless phenomenological Gilbert damping parameter Landau and Lifshitz
(1935); Gilbert (2004), $M=\mu_{0}M_{s}V$, $1/\Delta t$ is proportional to the
attempt frequency of the thermal field, and $G_{(0,1)}(t)$ is a Gaussian
distribution with zero mean and unit variance.

We denote _correlated_ and _anti-correlated_ noises when
$h_{i,1}(t)=h_{i,2}(t)$ and $h_{i,1}(t)=-h_{i,2}(t)$, respectively for $i\in
x,y,z$. When there is no correlation between the noises experienced by the two
nanomagnets, we term it _uncorrelated_.

Figure 2:  Different motions associated with the AFM and FM coupled
nanomagnets. (a) For the AFM coupled case, correlated noise rotates the
magnetizations in the same direction (see ${d\phi_{1,n}}/{dt}$ and
${d\phi_{2,n}}/{dt}$, ${d\theta_{1,n}}/{dt}$ and ${d\theta_{2,n}}/{dt}$),
which deflects NM1 away from its stable orientation and attracts NM1 towards
its stable position. While ${d\phi_{1,n}}/{dt}$ (${d\theta_{1,n}}/{dt}$)
counters the motion due to potential energy ${d\phi_{1,pot}}/{dt}$
(${d\theta_{1,pot}}/{dt}$) for NM1, ${d\phi_{2,n}}/{dt}$
(${d\theta_{2,n}}/{dt}$) aids ${d\phi_{2,pot}}/{dt}$
(${d\theta_{2,pot}}/{dt}$) for NM2. (b) For the FM coupled case, correlated
noise rotates the magnetizations in the same direction (see
${d\phi_{1,n}}/{dt}$ and ${d\phi_{2,n}}/{dt}$, ${d\theta_{1,n}}/{dt}$ and
${d\theta_{2,n}}/{dt}$), which deflects both the magnetizations NM1 and NM2
away from their stable orientations. Therefore, both ${d\phi_{1,n}}/{dt}$
(${d\theta_{1,n}}/{dt}$) and ${d\phi_{2,n}}/{dt}$ (${d\theta_{2,n}}/{dt}$)
counter the motion due to potential energy ${d\phi_{1,pot}}/{dt}$
(${d\theta_{1,pot}}/{dt}$) and ${d\phi_{2,pot}}/{dt}$
(${d\theta_{2,pot}}/{dt}$), respectively.

The magnetization dynamics under the action of the torques due to potential
energy and thermal noise ($\mathbf{T_{pot}}(t)$ and $\mathbf{T_{n}}(t)$,
respectively) is described by the stochastic Landau-Lifshitz-Gilbert (LLG)
equation Landau and Lifshitz (1935); Gilbert (2004); Brown (1963) as

$\frac{d\mathbf{n_{m}}}{dt}-\alpha\left(\mathbf{n_{m}}\times\frac{d\mathbf{n_{m}}}{dt}\right)=-\frac{|\gamma|}{M}\left[\mathbf{T_{pot}}+\mathbf{T_{n}}\right].$
(6)

We can numerically solve the above equation (using subscripts 1 and 2 for NM1
and NM2, respectively) for $d\theta_{1(2)}/dt$ and $d\phi_{1(2)}/dt$. The
dynamics of the rotational angles ($\theta_{1(2)}$,$\phi_{1(2)}$) of NM1 (NM2)
are

$\displaystyle\frac{d\theta_{1(2)}}{dt}$
$\displaystyle=\left(\frac{d\theta_{{1(2)},uni}}{dt}+\frac{d\theta_{{1(2)},ex}}{dt}\right)+\frac{d\theta_{{1(2)},n}}{dt}$
$\displaystyle=\frac{d\theta_{{1(2)},pot}}{dt}+\frac{d\theta_{{1(2)},n}}{dt},$
(7) $\displaystyle\frac{d\phi_{1(2)}}{dt}$
$\displaystyle=\left(\frac{d\phi_{{1(2)},uni}}{dt}+\frac{d\phi_{{1(2)},ex}}{dt}\right)+\frac{d\phi_{{1(2)},n}}{dt}$
$\displaystyle=\frac{d\phi_{{1(2)},pot}}{dt}+\frac{d\phi_{{1(2)},n}}{dt},$ (8)

where the three components in the two expressions above denote the rotations
due to energy barrier caused by uniaxial anisotropy, exchange energy, and the
thermal noise, respectively. The first two terms in the expressions can be
summed up to denote the rotations $d\theta_{1(2),pot}/dt$ and
$d\phi_{1(2),pot}/dt$ due to potential energy of the nanomagnets.

Figure 1(d) shows the mean deflections of magnetizations for the AFM and FM
coupled nanomagnets when $|J_{ex}|=1K_{u}V$. The magnetizations are not
positioned exactly along the $+z$ axis or $-z$ axis, rather they fluctuate due
to thermal agitations. Stochastic LLG equation of magnetization dynamics in
the presence of room temperature (300 K) thermal noise is solved to generate
these results. Three cases are considered for the characteristics of the
thermal noise while applying to the two exchange coupled nanomagnets: (1)
Correlated, (2) Uncorrelated, and (3) Anti-correlated. For the AFM coupled
system, with the _correlated_ noise, the mean deflections of the
magnetizations is _less_ (i.e., the _stability_ is _high_) than that of the
other two cases of noise. For the FM coupled nanomagnets, it shows just the
opposite trend. For _uncorrelated_ noise, AFM and FM coupled systems act
similarly, as depicted by the corresponding mean deflections of the
magnetizations.

Figure 2 depicts the explanation behind having _lower_ mean deflections of the
magnetizations (i.e., _higher_ stability) with _correlated_ noise for the case
of AFM exchange coupled nanomagnets compared to the FM coupled case. The
dynamics of the rotations ${d\theta_{1,pot}}/{dt}$, ${d\theta_{1,n}}/{dt}$,
${d\phi_{1,pot}}/{dt}$, ${d\phi_{1,n}}/{dt}$ for NM1 and
${d\theta_{2,pot}}/{dt}$, ${d\theta_{2,n}}/{dt}$, ${d\phi_{2,pot}}/{dt}$,
${d\phi_{2,n}}/{dt}$ for NM2 with _correlated_ noise are shown in the Fig. 2.

Figure 3:  Potential energy landscapes and the stochastic LLG simulation
results for the two AFM exchange coupled nanomagnets with _correlated_ noise.
(a,b,c) Potential energy landscapes when $|J_{ex}|=0,1\,K_{u}V,$ and
$2\,K_{u}V$, where $K_{u}V$ is the energy barrier of the individual
nanomagnets. The magnetizations are fluctuating at their respective potential
minima due to thermal agitations. (d,e,f) The distributions of the two
magnetizations and the fitting with Boltzmann distributions of energy barrier
$\sim$20 _kT_ ($|J_{ex}|=0$), $\sim$60 _kT_ ($|J_{ex}|=1\,K_{u}V$), and
$\sim$120 _kT_ ($|J_{ex}|=2\,K_{u}V$), respectively at room temperature (300
K). The mean deflections of the magnetizations are $7.8^{o}$ ($|J_{ex}|=0$),
$4.3^{o}$ ($|J_{ex}|=1\,K_{u}V$), and $3^{o}$ ($|J_{ex}|=2\,K_{u}V$),
respectively.

_Correlated noise applied to AFM coupled NM1 and NM2._ Figure 2(a) depicts the
rotational motions of the magnetizations NM1 and NM2. NM1 points downward
(i.e., $\theta_{1}=180^{\circ}$), while NM2 points upward (i.e.,
$\theta_{2}=0^{\circ}$). Therefore, when the magnetizations are deflected from
their positions, the rotational motions $d\phi_{1,pot}/dt$ and
$d\phi_{2,pot}/dt$ are in the opposite directions. The other rotational
motions $d\theta_{1,pot}/dt$ and $d\theta_{2,pot}/dt$ are in the opposite
directions as well so that the magnetizations are attracted towards their
respective energy minima.

Now, at a particular time step, let us assume that thermal noise is attempting
to deflect the magnetization NM1 from its stable position, i.e.,
$d\phi_{1,n}/dt$ due to noise is in the _opposite_ direction to
$d\phi_{1,pot}/dt$ ($d\theta_{1,n}/dt$ is in the _opposite_ direction to
$d\theta_{1,pot}/dt$ as well). If the magnetization NM2 faces a _correlated_
noise, NM2 experiences the same torque i.e., $d\phi_{2,n}/dt$ due to noise is
in the _same_ direction to $d\phi_{2,pot}/dt$ ($d\theta_{2,n}/dt$ is in the
_same_ direction to $d\theta_{2,pot}/dt$ as well). Therefore, NM2 goes toward
its stable position (i.e., $\theta_{2}=0^{\circ}$) rather than deflecting
away. Such _counteraction_ makes the the AFM exchange coupled system _more_
stable with _correlated_ noise than that of the case of _uncorrelated_ noise.

If we assume _anti-correlated_ noise, $d\phi_{2,n}/dt$ and $d\theta_{2,n}/dt$
would be in the opposite direction than what is shown in the Fig. 2(a) and
this will allow both the magnetizations NM1 and NM2 to deflect away from their
stable positions. Such trend is depicted in the stochastic LLG simulation
results for AFM coupled nanomagnets as presented in the Fig. 1(d).

_Correlated noise applied to FM coupled NM1 and NM2._ For the FM coupled
nanomagnets, as depicted in the Fig. 2(b), if thermal noise attempts to
deflect the magnetization NM1 away from its stable position, it also deflects
magnetization NM2 from the stable position for the case of _correlated_ noise
(see ${d\phi_{1,n}}/{dt}$ and ${d\phi_{2,n}}/{dt}$, ${d\theta_{1,n}}/{dt}$ and
${d\theta_{2,n}}/{dt}$). If the noise is strong enough at some time step,
magnetizations can deflect away from their stable orientation ($-z$ axis).
Hence, _no counteraction_ happens as for the AFM coupled case. Therefore, for
the FM coupled nanomagnets, the mean deflections of the magnetizations is
_higher_ for _correlated_ noise than that of case of the _uncorrelated_ noise.

Note that the _anti-correlated_ noise for the FM coupled nanomagnets
conceptually acts in the very same way as the _correlated_ noise acts on the
AFM coupled nanomagnets. Such trend is depicted in the stochastic LLG
simulation results for FM coupled nanomagnets as presented in the Fig. 1(d).

Figure 4:  Comparison and understanding of the thermal stability with _random_
noise field and _deterministic_ magnetic field. (a) Mean deflections of the
AFM exchange coupled magnetizations with the exchange coupling strength
$-J_{ex}$ when _correlated_ noise acting on the nanomagnets. (b) The magnetic
field $H$ required to switch the AFM exchange coupled nanomagnets with the
exchange coupling strength $-J_{ex}$. The field $H$ is applied to one
nanomagnet only.

Figure 3 shows the potential landscapes of the AFM exchange coupled
nanomagnets and the distributions of magnetizations due to thermal
fluctuations at their potential energy minima. Without any coupling [Figures
3(a) and (d)], the nanomagnets act like isolated ones. Figure 3(d) shows the
fitting with Boltzmann distribution of $\sim$20 _kT_ at room temperature (300
K). This confirms the validity of the simulation results since the nanomagnets
chosen have the energy barrier of $\sim$20 _kT_ at 300 K ($M_{s}=1\,T$,
$H_{k}=2.5\,T$, $\alpha=0.01$, 10 nm diameter with 1 nm thickness). As the
exchange coupling strength is increased, the magnetizations become more
confined at their potential energy minima [Figures 3(b) and (c)]. With
exchange coupling strengths $|J_{ex}|=1\,K_{u}V$ and $|J_{ex}|=2\,K_{u}V$, the
distributions of both the nanomagnets correspond to Boltzmann distributions of
$\sim$60 _kT_ and $\sim$120 _kT_ at room temperature (300 K), respectively.
The energy barriers in the fitted Boltzmann distributions can be derived from
the mean deflections of the magnetizations $\theta_{mean}$ as
$E_{b}=(1/3)\,(1/sin^{2}\theta_{mean})$ in the units of _kT_. The exchange
coupling strength $|J_{ex}|=1\,K_{u}V$ corresponds to $\sim$1 erg/cm2, which
can be achieved experimentally Parkin, More, and Roche (1990); Yulaev  _et
al._ (2011). Therefore, the results clearly demonstrate that the AFM coupled
nanomagnets can be highly stable with _correlated_ noise.

Figure 4(a) depicts that as the exchange coupling strength $|J_{ex}|$
increases, the mean deflection of the magnetizations goes _lower_ , i.e., the
stability becomes _higher_. As explained earlier, the _inherent_ magnetization
dynamics of the AFM coupled nanomagnet counters the motion of the
magnetizations with _correlated_ noise and such counteraction (i.e.,
stability) increases with $|J_{ex}|$.

Figure 4(b) shows that as the exchange coupling strength $|J_{ex}|$ increases,
the magnetic field $H$ required to switch the AFM exchange coupled nanomagnets
does also increase, which is obvious. The required field tends to saturate
towards $2\,H_{k}$, where $H_{k}$ is the individual switching field for the
nanomagnets. Note that a _deterministic_ $H$ field is applied at each time
step contrary to the case in the Figure 4(a), where a _random_ noise field
with mean zero is applied on the nanomagnets. This makes the difference
between the switching using a $H$ field and the field due to random thermal
noise. It is the noise field (not the $H$ field) that attempts to deflect
magnetizations from their stable orientations in real time and the _inherent_
dynamics of the AFM exchange coupled nanomagnets determines the stability of
the system with _correlated_ noise.

To summarize, we have shown by solving stochastic LLG of magnetization
dynamics that AFM exchange coupled nanomagnets with _correlated_ noise can
lead to very high thermal stability at low-dimensions, i.e., the magnetic bits
can be scaled down further to achieve higher areal density. With the
experimentally feasible exchange strengths, we have shown that the magnetic
bits can be highly stable as per the standard requirement of storage. Such
finding would allay our concern on the thermal stability of magnetic bits at
low-dimensions and would pave the pathway towards ultra high-density non-
volatile information storage and logic systems.

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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11footnotetext: These authors contributed equally.

# Lift force in odd compressible fluids

Ruben Lier a<EMAIL_ADDRESS>Max Planck Institute for the Physics of Complex
Systems, 01187 Dresden, Germany  Würzburg-Dresden Cluster of Excellence
ct.qmat, 01187 Dresden, Germany  Charlie Duclut a<EMAIL_ADDRESS>Laboratoire Physico-Chimie Curie, UMR 168, Institut Curie, PSL Research
University, CNRS, Sorbonne Université, 75005 Paris, France  Université Paris
Cité, Laboratoire Matière et Systèmes Complexes (MSC), UMR 7057 CNRS, F-75205
Paris, France  Max Planck Institute for the Physics of Complex Systems, 01187
Dresden, Germany  Stefano Bo<EMAIL_ADDRESS>Max Planck Institute for
the Physics of Complex Systems, 01187 Dresden, Germany  Department of Physics,
King’s College London, London WC2R 2LS, U.K.  Jay Armas<EMAIL_ADDRESS>Institute for Theoretical Physics, University of Amsterdam, 1090 GL Amsterdam,
The Netherlands  Dutch Institute for Emergent Phenomena (DIEP), University of
Amsterdam, 1090 GL Amsterdam, The Netherlands  Frank Jülicher
<EMAIL_ADDRESS>Max Planck Institute for the Physics of Complex Systems,
01187 Dresden, Germany  Center for Systems Biology Dresden, Pfotenhauerstrasse
108, 01307 Dresden, Germany  Cluster of Excellence Physics of Life, TU
Dresden, 01062 Dresden, Germany  Piotr Surówka<EMAIL_ADDRESS>Institute of Theoretical Physics, Wrocław University of Science and
Technology, 50-370 Wrocław, Poland  Institute for Theoretical Physics,
University of Amsterdam, 1090 GL Amsterdam, The Netherlands  Dutch Institute
for Emergent Phenomena (DIEP), University of Amsterdam, 1090 GL Amsterdam, The
Netherlands

###### Abstract

When a body moves through a fluid, it can experience a force orthogonal to its
movement, called lift force. Odd viscous fluids break parity and time-reversal
symmetry, suggesting the existence of an odd lift force on tracer particles,
even at vanishing Reynolds numbers and for symmetric geometries. It was
previously found that an incompressible odd fluid cannot induce lift force on
a tracer particle with no-slip boundary conditions, making signatures of odd
viscosity in the two-dimensional bulk elusive. By computing the response
matrix for a tracer particle, we show that an odd compressible fluid can
produce an odd lift force. Using _shell localization_ , we provide analytic
expressions for the drag and odd lift forces acting on the tracer particle in
a steady state and also at finite frequency. Importantly, we find that the
existence of an odd lift force in a steady state requires taking into account
the non-conservation of the fluid mass density due to the coupling between the
two-dimensional surface and the three-dimensional bulk fluid.

Introduction..— Odd materials are characterized by the breaking of parity
symmetry, which manifest itself in viscous and elastic tensor contributions
that are odd under index exchange. Breaking this symmetry results in the
emergence of novel phenomena, endowing odd materials with fascinating
properties that are interesting for various fields of physics, including
electron fluids [1, 2, 3], topological waves [4, 5, 6, 7], fluid dynamics [8,
9, 10, 11], complex materials [12, 13, 14, 15, 16], soft active matter,
statistical physics and biological physics [17, 18, 19, 20, 21, 22, 23, 24,
25, 26, 27, 28]. Notably, these materials are now within experimental reach
and their properties can be measured, validated and further explored [3, 22,
27].

The simplest examples of odd materials are odd fluids, which are characterized
by odd viscosity. Odd viscosity is a transport coefficient in two dimensions
breaking parity and time-reversal symmetry, which can occur in passive fluids
subject to a background magnetic field [29, 30], as well as in active chiral
systems [26, 8].

The signatures of odd viscosity in fluids have been explored in various
contexts (see e.g. [31, 32, 33, 34, 8, 10, 35, 36, 9, 37, 22, 24, 38, 21, 39,
11, 40, 41, 20]). The experimental realization of an active odd fluid in Ref.
[22] showed that the strongest signatures of odd behavior, such as edge flow
or the rotation of asymmetric droplets, are found at interfaces. Inserting a
tracer or probe particle in an odd fluid naturally introduces a boundary,
making it an ideal candidate to probe the odd properties of a fluid, and has
been the subject of several numerical and theoretical studies [23, 25, 33, 10,
38, 39]. In particular, due to the parity-breaking nature of odd viscosity,
symmetry allows a fluid with a constant velocity at infinity not only to
induce a drag force on a tracer particle, but also a lift force, orthogonal to
the movement of the tracer. This odd lift force is allowed at vanishing
Reynolds number and in a symmetric geometry. This illustrates its different
physical origin compared to the lift force observed for instance in
aeronautics, that requires a nonvanishing Reynolds number or a symmetry-
breaking mechanism such as the shape of the wing [42].

Surprisingly, such a lift force is absent111A nonvanishing odd lift force, on
a tracer in an incompressible fluid, assuming no-slip boundary conditions, was
obtained in Ref. [33], but was contradicted in Ref. [10]. The discrepancy can
be traced to the computation of the force on the probe, which in Ref. [33]
used an incorrect pressure field. in incompressible odd fluids [10], and the
motion of a tracer particle cannot be used to detect signatures of odd
properties in these systems.

This brings us to a variant of the more-than-a-century-old question: _how much
force does a tracer particle in a fluid experience?_ Answering this question
typically requires finding a smooth and regular solution for the velocity
profile of the fluid flows satisfying appropriate boundary conditions near the
tracer particle and far away from it. However, when one tries this for two-
dimensional fluids one encounters a problem, commonly known as the Stokes
paradox, which prevents a solution to the Stokes equation for a disk moving
through a two-dimensional fluid with infinitely large system size [43]. The
Stokes paradox can be circumvented by adding a scale to the problem which
“regularizes” the paradox. One way to do this is, is a through the Oseen
approximation [44], which introduces the far field velocity through an inertia
term. The Oseen approximation can be improved in an iterative way, which is
called asymptotic expansion [45, 46, 47, 48, 49, 50]. Another way that the
Stokes paradox is evaded, is by assuming that the two-dimensional fluid is in
contact with a three-dimensional bulk, to which momentum is relaxed [51, 52,
53]. This is what will be considered in this work.

In this Letter, we show that a tracer particle in an odd compressible fluid
experiences a lift force proportional to the odd viscosity coefficient, and
that compressibility is a necessary condition for the existence of an odd lift
force in two dimensions. As commonly done when studying the motion of tracers
in fluids and to make direct contact with the incompressible case studied in
Ref. [10], we consider no-slip boundary conditions on the surface of the
tracer. Lifting the incompressibility constraint dramatically complicates the
two-dimensional fluid equations, the description of the fluid velocity
requires, in addition to the stream function, a second scalar field. The
differential equations of these two scalar fields are coupled due through odd
viscosity. In addition, one also needs to account for the non-trivial role of
density. To tackle these difficulties, we avoid computing the fluid profile
and instead use the “shell localization” approach [54, 55, 56] to analytically
compute the drag and lift forces on a tracer particle in two different
situations: a fluid in a steady-state configuration, and a fluid excited by an
external force with finite driving frequency.

Crucially —and this point was overlooked in previous studies on this subject
in which an instantaneous density relaxation was considered [38]— we show that
lift force only persists in a steady state in systems for which the density is
not conserved. Non-conservation of density is generic in active systems as a
consequence of birth and death processes, for instance in “Malthusian flocks”
[57, 58], cellular tissues [59], and in chemotactic systems [60]. Furthermore,
absence of mass density conservation in two dimensions can arise from
exchanges with a three-dimensional fluid bulk [61, 62]. This is for instance
the case if the odd properties of the fluid stem from the activity of chiral
particles, such as bacteria [63] or spermatozoa [64] that swim in a three-
dimensional fluid and can accumulate at a surface.

As a further step, we also investigate the response of a probe excited
periodically. At finite frequency, we show that an odd lift force can be
measured in compressible fluids even if the mass density is conserved. This
paves the way towards measurements of odd transport coefficients using
frequency-dependent micro-rheology.

Compressible odd fluid.— We consider a thin layer of an odd compressible
viscous fluid at the interface between two bulk (even) fluids, for instance
water and air. For simplicity, we consider this layer to be flat and
infinitely thin, such that the odd fluid can be described effectively as two-
dimensional. The stress tensor associated with the mechanical properties of
the odd fluid with velocity field $v_{i}$ reads

$\displaystyle\sigma_{ij}=2\eta_{\rm s}\partial_{\langle
i}v_{j\rangle}+2\eta_{\rm o}\partial_{\\{i}v_{j\\}}+\left(\eta_{\rm
b}\partial_{k}v_{k}-P\right)\delta_{ij}\,,$ (1)

where $i,j$ denote two-dimensional Cartesian coordinates and where summation
over repeated indices is implied. For an arbitrary tensor $A_{ij}$, we have
introduced the notation $A_{\langle
ij\rangle}=(A_{ij}+A_{ji})/2-A_{kk}\delta_{ij}/2$ for its traceless symmetric
part, such that $\partial_{\langle i}v_{j\rangle}$ is the fluid shear rate. We
have also introduced the odd tensor contraction
$A_{\\{ij\\}}=(\varepsilon_{ik}A_{kj}+\varepsilon_{ik}A_{jk}+\varepsilon_{jk}A_{ki}+\varepsilon_{jk}A_{ik})/4$,
where $\varepsilon_{ij}$ denotes the fully antisymmetric tensor in two
dimensions with $\varepsilon_{12}=-\varepsilon_{21}=1$. Finally, we denote by
$\eta_{\rm s,b,o}$ the shear, bulk and odd viscosities of the fluid, and by
$P$ its pressure field.

The divergence of the stress tensor (1) then allows us to write the momentum
balance equation, which corresponds to the odd version of the Navier–Stokes
equation. It reads:

$\displaystyle\begin{split}\partial_{t}\pi_{i}&+v_{k}\partial_{k}\pi_{i}=\eta_{\rm
s}\partial_{k}\partial_{k}v_{i}+\eta_{\rm
b}\partial_{i}\partial_{k}v_{k}-\partial_{i}P\\\ &\quad+\eta_{\rm
o}\varepsilon_{ij}\partial_{k}\partial_{k}v_{j}-\frac{1}{\tau}\pi_{i}+f_{i}\,.\end{split}$
(2)

where $\pi_{i}=\rho v_{i}$ is the fluid momentum density with $\rho$ the local
mass density. The first line of Eq. (2) is the usual isotropic Navier–Stokes
equation, while the first term in the second line is the signature of odd two-
dimensional fluids. In addition, we have included in Eq. (2) a momentum
relaxation process with timescale $\tau$. This process accounts for linear
friction between the two-dimensional fluid and the three-dimensional bulk
which can generically exists in our geometry. The last term $f_{i}$ in Eq. (2)
is an external force density acting on the fluid, which will prove convenient
to compute the drag and lift coefficients of a probe immersed in the fluid.

As will become clear below, the compressibility of the odd fluid layer is a
necessary condition to observe a nonvanishing lift force. A compressible fluid
can be described by providing an equation of state for the pressure field $P$
written as a series expansion in powers of the fluid density $\rho$. For a
weakly compressible fluid that we consider here, we keep only the first
nontrivial order and write:

$\displaystyle P(\rho)=P_{0}+\chi\frac{(\rho-\rho_{0})}{\rho_{0}}\,,$ (3)

where $\chi^{-1}$ is the compressibility, and $P_{0}$ and $\rho_{0}$ are the
reference pressure and density, respectively. Finally, the mass density obeys
the balance equation

$\displaystyle\partial_{t}\rho+\partial_{k}(\rho v_{k})$
$\displaystyle=-\frac{1}{\kappa}(\rho-\rho_{0})\,,$ (4)

where we have included a mass exchange process with timescale $\kappa$ to
account for particle exchange with the bulk of the fluid [62]. Note that
linear terms proportional to the density in Eq. (3) and in Eq. (4) would also
be allowed in an active fluid layer [65], such as a cell epithelium. In this
specific case, $\chi(\rho-\rho_{0})/\rho$ would correspond to an active
isotropic stress and $(\rho-\rho_{0})/\kappa$ would account for cell divisions
and extrusions. Finally, we emphasize that the case of a momentum-conserving,
mass-conserving, or incompressible fluid can be easily recovered by taking
respectively the limit $\tau\to\infty$, $\kappa\to\infty$, or $\chi\to\infty$
in Eqs. (2)-(4). These coupled equations thus provides the ideal starting
point for studying odd effects in two-dimensional fluid layers.

To simplify the system of coupled nonlinear differential equations (2)-(4), we
linearize it to first order in $v_{i}$ and $\delta\rho=\rho-\rho_{0}$ near a
vanishing velocity and homogeneous reference state. The balance equations then
take the form

$\displaystyle\begin{split}\rho_{0}\partial_{t}v_{i}&=\eta_{\rm
s}\partial_{k}\partial_{k}v_{i}+\eta_{\rm b}\partial_{i}\partial_{k}v_{k}\\\
&+\eta_{\rm
o}\varepsilon_{ij}\partial_{k}\partial_{k}v_{j}-\partial_{i}P-\frac{\rho_{0}}{\tau}v_{i}+f_{i}\,,\end{split}$
(5a) $\displaystyle\partial_{t}\delta\rho+\rho_{0}\partial_{k}v_{k}$
$\displaystyle=-\frac{1}{\kappa}\delta\rho\,.$ (5b)

We will use these equations to compute the response of a probe to an external
force in an odd fluid.

Shell localization.— Having defined the equations of motion, we move to
Fourier space with the convention

$\displaystyle g(t,x_{i})=\frac{1}{(2\pi)^{3}}\int{\rm d}\omega{\rm
d}^{2}k\,g(\omega,k_{i})e^{-{\rm i}\omega t+{\rm i}k_{j}x_{j}}~{}~{},$ (6)

for some function $g(t,x_{i})$ so that Eq. (5) can be written in matrix form
$\mathcal{G}_{ij}v_{j}=f_{i}$ with $\mathcal{G}_{ij}$ given by

$\displaystyle\begin{split}\mathcal{G}_{ij}=\hat{k}_{i}\hat{k}_{j}\left[\frac{\rho_{0}}{\tau}-{\rm
i}\omega\rho_{0}+\left(\eta_{\rm s}+\eta_{\rm b}+\frac{\chi\kappa}{1-{\rm
i}\omega\kappa}\right)k^{2}\right]\\\
+(\delta_{ij}-\hat{k}_{i}\hat{k}_{j})\left[\frac{\rho_{0}}{\tau}-{\rm
i}\omega\rho_{0}+\eta_{\rm s}k^{2}\right]+\varepsilon_{ij}\eta_{\rm
o}k^{2}~{}~{},\end{split}$ (7)

where $k=\sqrt{k_{i}k_{i}}$ and $\hat{k}_{i}=k_{i}/k$. This relation can be
inverted as

$\displaystyle
v_{i}(k_{i},\omega)=\mathcal{M}_{ij}(k_{i},\omega)f_{j}(k_{i},\omega)~{}~{},$
(8)

where we have defined $\mathcal{M}_{ij}=\mathcal{G}^{-1}_{ij}$. Equaqtion (8)
yields the velocity induced by a force distribution. Specifically, we consider
the force applied on a tracer particle, which is a rigid disk of radius $a$
located at the origin. Due to the rotational symmetry of the disk, we can
decompose the force density as
$f_{j}(k_{i},\omega)=L(k)\mathcal{F}_{j}(\omega)$. The shell localization
method consists in considering that the force density is located in real space
according to [54, 55, 66]:

$\displaystyle L(x)=\frac{1}{2\pi a}\delta(|x|-a)~{}~{}.$ (9)

Equation (9) enforces the force density exerted by the disk on the fluid to be
uniformly distributed along the entire edge of the disk. The disk is coupled
to the fluid through a no-slip boundary condition, which equates the velocity
of the tracer particle to the fluid velocity at the edge of the tracer
particle. Fourier transforming Eq. (9) yields $L(k)=J_{0}(ak)$ with $J_{n}(z)$
the $n^{\rm th}$ Bessel function of the first kind. The velocity of the disk
located at $|x|=0$ is then directly given by the inverse Fourier transform at
the origin

$\displaystyle
v_{i}(|x|=0,\omega)=\mathbb{M}_{ij}(\omega)\mathcal{F}_{j}(\omega)~{}~{},$
(10)

where the “response matrix” is

$\displaystyle\mathbb{M}_{ij}(\omega)=\frac{1}{(2\pi)^{2}}\int_{0}^{2\pi}\\!\\!{\rm
d}\theta\int_{0}^{\infty}\\!\\!{\rm
d}k\,kL(k)\mathcal{M}_{ij}(k_{i},\omega)\,.$ (11)

The response matrix $\mathbb{M}_{ij}(\omega)$ encodes the velocity of a rigid
probe immersed in an odd fluid as a function of the applied (frequency-
dependent) force $\mathcal{F}_{j}(\omega)$. Using the disk radius $a$ we can
introduce the dimensionless coefficients

$\displaystyle\begin{split}z_{i}&=ak_{i}\,,\,\,\tilde{\omega}=\omega
a^{2}\rho_{0}/\eta_{\rm s}\,,\,\,\tilde{\eta}_{\rm o}=\eta_{\rm o}/\eta_{\rm
s}\,,\,\,\tilde{\eta}_{\rm b}=\eta_{\rm b}/\eta_{\rm s}\\\
\tilde{\tau}&=\tau\eta_{\rm
s}/(\rho_{0}a^{2})\,,\,\,\tilde{\chi}=\chi\rho_{0}a^{2}/\eta_{\rm
s}^{2}\,,\,\,\tilde{\kappa}=\kappa\eta_{\rm s}/(\rho_{0}a^{2})\,,\end{split}$
(12)

so that Eq. (7) turns into

$\displaystyle\begin{split}\mathcal{G}_{ij}=\frac{\eta_{\rm
s}}{a^{2}}\Big{\\{}\hat{z}_{i}\hat{z}_{j}\left[\frac{1}{\tilde{\tau}}-{\rm
i}\tilde{\omega}+\\!\left(1+\tilde{\eta}_{\rm
b}+\frac{\tilde{\chi}\tilde{\kappa}}{1-{\rm
i}\tilde{\omega}\tilde{\kappa}}\right)\\!z^{2}\right]\\\
+(\delta_{ij}-\hat{z}_{i}\hat{z}_{j})\left[\frac{1}{\tilde{\tau}}-{\rm
i}\tilde{\omega}+z^{2}\right]+\varepsilon_{ij}\tilde{\eta}_{\rm
o}z^{2}\Big{\\}}\,,\end{split}$ (13)

where $z=\sqrt{z_{i}z_{i}}$ and $\hat{z}_{i}=z_{i}/z$. Before considering the
most general case of a compressible fluid, where a lift force can arise, we
first discuss the limiting case of an incompressible odd fluid. This
corresponds to the limit $\tilde{\chi}\rightarrow\infty$, for which the matrix
$\mathcal{M}$ reads

$\displaystyle\lim_{\chi\rightarrow\infty}\mathcal{M}_{ij}(z_{k},\tilde{\omega})=\frac{a^{2}}{\eta_{\rm
s}}\frac{\delta_{ij}-\hat{z}_{i}\hat{z}_{j}}{z^{2}+\tilde{\tau}^{-1}-{\rm
i}\tilde{\omega}}~{}~{}.$ (14)

It may be observed that this matrix is transverse to the wave-vector,
indicating the absence of an odd lift force as expected for an incompressible
odd fluid [10]. In addition, the odd viscosity transport coefficient is
absent, indicating that the response of the tracer particle in the case of an
odd incompressible fluid is identical to the response in the case of an even
incompressible fluid. In App. A, we verify that in the incompressible case the
shell localization gives a drag force that is consistent with results found by
explicitly solving the boundary value problem in two instances. Specifically,
we recover the result for two-dimensional oscillatory drag [67, 68] as well as
the result for the drag force found in the Saffman-Delbrück model [51, 52],
provided we appropriately match the relaxation time to the coefficients of
this model [53].

Odd lift force.— We now address the general case of a compressible fluid. In
this setting, the response matrix can written as

$\displaystyle\mathbb{M}_{ij}(\omega)=\frac{1}{\eta_{\rm s}}(M_{\rm
d}\delta_{ij}-M_{\rm l}\varepsilon_{ij})\,,$ (15)

where $M_{\rm d}$ and $M_{\rm l}$ are respectively the dimensionless response
functions for drag force, and for lift force, specific to compressible odd
fluids.

Figure 1: Steady-state drag coefficient $M_{\rm d}$ (A) and lift coefficient
$M_{\rm l}$ (B) as a function of the dimensionless inverse compressibility
$\Theta=(\tilde{\chi}\tilde{\kappa})^{-1}$ for different values of the
relaxation time $\tilde{\tau}$ and for $\tilde{\eta}_{\rm b}=\tilde{\eta}_{\rm
o}=1$.

Steady-state odd lift force.— We first consider the steady-state case
$\tilde{\omega}\rightarrow 0$ with a non-vanishing relaxation rate
$\tilde{\tau}^{-1}\neq 0$. The drag and lift are obtained by computing the
momentum integrals

$\displaystyle M_{\rm d}$ $\displaystyle=\frac{1}{4\pi}\int{\rm
d}z\,J_{0}(z)\frac{D(z)}{Q(z)}\,,$ (16a) $\displaystyle M_{\rm l}$
$\displaystyle=\frac{1}{2\pi}\int{\rm d}z\,J_{0}(z)\frac{L(z)}{Q(z)}\,,$ (16b)

where we have defined:

$\displaystyle\begin{split}Q(z)&=\tilde{\tau}^{2}z^{4}\left(\tilde{\eta}_{\rm
b}+\tilde{\eta}_{\rm o}^{2}+\Theta^{-1}+1\right)\\\
&\,\,\,\,\,+\tilde{\tau}z^{2}(\tilde{\eta}_{\rm
b}+\Theta^{-1}+2)+1\,,\end{split}$ (17a) $\displaystyle D(z)$
$\displaystyle=\tilde{\tau}z\left(\tilde{\tau}z^{2}(\tilde{\eta}_{\rm
b}+\Theta^{-1}+2)+2\right)\,,\,\,L(z)=\tilde{\eta}_{\rm
o}\tilde{\tau}^{2}z^{3}\,,$ (17b)

and where $\Theta^{-1}=\tilde{\kappa}\tilde{\chi}$. As advertised in the
introduction, we note that the odd lift force vanishes for $\tilde{\eta}_{\rm
o}\rightarrow 0$, which is expected as it can only be induced by a parity-odd
coefficient. Furthermore, $M_{\rm l}$ is only non-vanishing when
$\tilde{\kappa}^{-1}$ is non-vanishing, since in the steady case the limit
$\tilde{\kappa}\rightarrow\infty$ is equivalent to the incompressible limit
for which was shown in Eq. (14) that the lift force vanishes. This means that
in a steady state there can only be lift forces when density is not conserved,
for instance if exchanges between the surface and three-dimensional fluid,
parameterized by the relaxation time $\kappa$, take place.

As we detail in App. B, where we use Ref. [69], the momentum integrals can be
computed analytically using residues but their expression can become lengthy.
For the purpose of clarity, we consider a series expansion in powers of the
odd viscosity $\tilde{\eta}_{\rm o}$ and keep the first non-vanishing
contribution. Specifically, we find

$\displaystyle M_{\rm d}$
$\displaystyle=\frac{K_{0}(\tilde{\tau}^{-1/2})+K_{0}[(\Xi\tilde{\tau})^{-1/2}]/\Xi}{4\pi}+\mathcal{O}(\tilde{\eta}_{\rm
o}^{2})\,,$ (18a) $\displaystyle M_{\rm l}$
$\displaystyle=\frac{\tilde{\eta}_{\rm
o}\left[K_{0}(\tilde{\tau}^{-1/2})-K_{0}[(\Xi\tilde{\tau})^{-1/2}]/\Xi\right]}{2\pi(\Xi-1)}+\mathcal{O}(\tilde{\eta}_{\rm
o}^{2})\,,$ (18b)

with $\Xi=1+\tilde{\eta}_{\rm b}+\Theta^{-1}$ and where $K_{n}(x)$ is the
$n^{\rm th}$ modified Bessel function of the second kind. In the
incompressible fluid limit or for a compressible fluid without mass density
relaxation ($\Theta\to 0$), we find $M_{\rm l}=0$ and $M_{\rm
d}=K_{0}(\tilde{\tau}^{-1/2})/(4\pi)$.

We now evaluate $M_{\rm d}$ and $M_{\rm l}$ from Eq. (16) as a function of
$\Theta$ and provide the result in Fig. 1. We take $\tilde{\eta}_{\rm
b}=\tilde{\eta}_{\rm o}=1$ for the dimensionless viscosities. We observe in
Fig. 1 A that the drag force is significantly affected by the momentum
relaxation time $\tilde{\tau}$ but only weakly depends on the compressibility
parameter $\Theta$. On the other hand, Fig. 1 B shows the crucial role of the
compressibility in the magnitude of the lift force, which vanishes in the
incompressible limit $\Theta\rightarrow 0$.

We also consider the limit $\Theta\rightarrow\infty$ which corresponds to an
infinitely compressible fluid ($\tilde{\chi}=0$), or to a fluid with an
instantaneous density relaxation $\kappa=0$). In this limit, any deviation
from the reference density $\rho_{0}$ is instantly relaxed to the bulk, such
that pressure is constant and plays no role in the response matrix. In this
case, our equations reduce to the ones considered in Ref. [38] where numerical
expressions for the response function are computed.

Lastly, we note that the odd lift coefficient $M_{\rm 1}$ can become negative
for small values of $\tilde{\tau}$ and large values of $\Theta$. However, this
regime in parameter space for which $\tau\ll 1$ is precisely the regime in
which momentum relaxation dominates and the system given by Eqs. (5) no longer
provides an accurate description of two-dimensional fluid flows.

Figure 2: Real (A,C) and imaginary parts (B,D) of the complex drag and lift
coefficients $M_{\rm d,l}$ as a function of the dimensionless frequency
$\tilde{\omega}$ for different values of the inverse compressibility
$\tilde{\chi}$ and for $\tilde{\eta}_{\rm b}=\tilde{\eta}_{\rm o}=1$.

Frequency-dependent lift force.— We now consider the system in the absence of
relaxation processes ($\tilde{\tau}^{-1}\to 0$ and $\tilde{\kappa}^{-1}\to 0$)
to focus on the frequency-dependent response of the tracer. In Fig. 2, we
display the real and imaginary parts of the drag coefficient $M_{\rm
d}(\tilde{\omega})$ and odd lift coefficient $M_{\rm l}(\tilde{\omega})$ as a
function of the dimensionless frequency $\tilde{\omega}$ and for different
values of the inverse compressibility $\tilde{\chi}^{-1}$. The drag
coefficient $M_{\rm d}(\tilde{\omega})$ diverges as $\tilde{\omega}\to 0$,
which is a signature of the Stokes paradox, see Figs. 2 A and B.

On the other hand, the lift coefficient $M_{\rm l}(\tilde{\omega})$ vanishes
at steady state, see Figs. 2 C and D. At finite excitation frequency and
compressibility, a nonvanishing odd response can be measured. Note that both
the drag and lift responses vanish at large frequencies, as expected for a
fluid.

Additionally, a simple analytic expression for the drag and odd lift
coefficient $M_{\rm d,l}$ can be obtained by expanding Eq. (11) in the absence
of relaxation processes ($\tilde{\tau}^{-1}\to 0$ and $\tilde{\kappa}^{-1}\to
0$) and at leading order in the inverse compressibility $\tilde{\chi}^{-1}$.
One obtains

$\displaystyle M_{\rm d}$
$\displaystyle=\frac{1}{4\pi}K_{0}\left(\sqrt{\tilde{\omega}/{\rm
i}}\right)+\mathcal{O}(\tilde{\chi}^{-1})\,,$ (19a) $\displaystyle M_{\rm l}$
$\displaystyle=\frac{-{\rm i}\tilde{\omega}\tilde{\eta}_{\rm
o}}{2\pi\tilde{\chi}}K_{0}\left(\sqrt{\tilde{\omega}/{\rm
i}}\right)+\mathcal{O}(\tilde{\chi}^{-2})\,.$ (19b)

The drag and lift coefficients have a completely different behavior in the
limit of small frequencies. Indeed, we have the expansion222Note that because
the drag coefficient diverges in the limit $\tilde{\omega}\to 0$, the
expansion in a series of $\tilde{\omega}$ must be performed after computing
the momentum integral over $z$.

$\displaystyle M_{\rm d}$
$\displaystyle=-\frac{1}{8\pi}\left(\\!\log\frac{\tilde{\omega}}{4}+2\gamma_{\rm
EM}-\frac{{\rm
i}\pi}{2}\\!\right)+\mathcal{O}(\tilde{\chi}^{-1},\tilde{\omega})\,,$ (20a)
$\displaystyle M_{\rm l}$ $\displaystyle=\frac{{\rm
i}\tilde{\omega}\tilde{\eta}_{\rm
o}}{4\pi\tilde{\chi}}\left(\\!\log\frac{\tilde{\omega}}{4}+2\gamma_{\rm
EM}-\frac{{\rm
i}\pi}{2}\\!\right)+\mathcal{O}(\tilde{\chi}^{-2},\tilde{\omega}^{2})\,,$
(20b)

which shows a $\log\tilde{\omega}$ divergence of the drag, as expected from
the Stokes paradox, while the odd lift coefficient vanishes as
$\tilde{\omega}\log\tilde{\omega}$. This difference in the small
$\tilde{\omega}$ behavior is clearly visible in Fig. 2.

Discussion.— In this Letter we obtained analytical expressions for the drag
and lift coefficients of a disk in a two-dimensional odd compressible fluid.
We used a shell localization approach [54, 55] to study the probe response
both at steady-state and at finite frequency. In the incompressible limit, we
confirmed the absence of odd effects on the tracer with no-slip boundary
conditions [10]. Having in mind a two-dimensional system embedded in a three-
dimensional bulk, we have considered a finite momentum relaxation due to
friction, which remedies the Stokes paradox. We found that in order for lift
force to be non-vanishing in the steady case, an additional density relaxation
due to exchanges with the bulk is required333Note that in Ref. [38], the odd
lift force was computed in the limit of a vanishing density relaxation time
($\kappa\rightarrow 0$) using the Lorentz reciprocal theorem [70]. However,
this theorem relies on the index exchange symmetry $\eta_{ijkl}=\eta_{klij}$
of the viscosity tensor, which does not hold for an odd fluid. After
completing this work, a work appeared where a modified version of the Lorentz
reciprocal theorem is introduced which can accommodate for the anti-reciprocal
odd viscosity [71] and can therefore be used to overcome this problem..

The shell localization approach has also been used to compute drag force for
the incompressible Oseen equation [72]. An interesting question is whether it
is possible to also apply this computation for the case where the Oseen
approximation is applied to odd compressible fluids. Furthermore, it would be
interesting to see whether “effective boundary conditions” [73] accounting for
a small finite compressibility can be used to capture the odd lift force on
the probe while using an incompressible model in the bulk. Finally, when the
tracer is excited at finite frequency $\omega$, we found that an odd lift
response exists at finite frequency, and vanishes as $\omega\log(\omega)$ in
the limit of small frequency. For comparison, the drag response diverges in
the same limit as $\log(\omega)$, a signature of Stokes paradox [67, 68].
These results suggest that active micro-rheology could be used to measure the
properties of odd viscoelastic materials.

Acknowledgements.— J.A. is partly supported by the Nederlandse Organizatie
voor Wetenschappelijk Onderzoek (NWO) through the NWA Startimpuls funding
scheme and by the Dutch Institute for Emergent Phenomena (DIEP) cluster at the
University of Amsterdam. R.L. was supported, in part, by the cluster of
excellence ct.qmat (EXC 2147, project-id 39085490). P.S. acknowledges the
support of the Narodowe Centrum Nauki (NCN) Sonata Bis Grant No.
2019/34/E/ST3/00405 and NWO Klein grant via NWA route 2. C.D. acknowledges the
support of the LabEx “Who Am I?” (ANR-11-LABX-0071) and of the Université
Paris Cité IdEx (ANR-18-IDEX-0001) funded by the French Government through its
“Investments for the Future” program. The authors wish to thank Alexandre G.
Abanov for insightful comments on the manuscript.

##  Appendix A Incompressible drag force

In this Appendix, we explicitly compute the response matrix in two simple
incompressible scenarios. In the absence of relaxation ($\tilde{\tau}^{-1}\to
0$), the incompressible response matrix is given by

$\displaystyle\begin{split}\mathbb{M}_{ij}(\tilde{\omega})\\!&=\frac{\delta_{ij}}{4\pi\eta_{\rm
s}}\int{\rm d}z\,\frac{zJ_{0}(z)}{z^{2}-{\rm i}\tilde{\omega}}\\\
&\\!=-\frac{\delta_{ij}}{8\pi\eta_{\rm
s}}\\!\left[\log(\frac{\tilde{\omega}}{4})+2\gamma_{\rm EM}\\!-{\rm
i}\frac{\pi}{2}\\!\right]\\!+\mathcal{O}(\tilde{\omega})\,,\end{split}$ (21)

where $\gamma_{\rm EM}$ is the Euler-Mascheroni constant. We see that this
result is divergent in the steady-state limit ($\tilde{\omega}\rightarrow 0$),
which is a signature of the Stokes paradox [47]. Note that the shell
localization result given in Eq. (21) matches the drag force that one would
obtain from solving explicitly the Stokes equation with no-slip boundary
conditions [67, 68].

A second scenario is the steady-steady case $\tilde{\omega}\rightarrow 0$ with
a finite relaxation rate $\tilde{\tau}^{-1}\neq 0$. The incompressible
response matrix now reads

$\displaystyle\begin{split}\mathbb{M}_{ij}(0)&=\frac{\delta_{ij}}{4\pi\eta_{\rm
s}}\int{\rm d}z\,\frac{zJ_{0}(z)}{z^{2}+1/\tilde{\tau}}\\\
&=\frac{\delta_{ij}}{4\pi\eta_{\rm
s}}\left[\log(2\sqrt{\tilde{\tau}})-\gamma_{\rm
EM}\right]\\!+\mathcal{O}(\tilde{\tau}^{-1})~{}.\end{split}$ (22)

In this case, the response matrix is non-divergent thanks to the momentum
relaxation circumventing the Stokes paradox [74, 53, 75, 76]. The result in
Eq. (22) can be compared to the result from works of Saffman and Delbrück [51,
52], if one matches the relaxation $\tilde{\tau}$ as [53]

$\displaystyle\tilde{\tau}=\left(\frac{\eta_{\rm s}}{2a\eta^{\prime}_{\rm
s}}\right)^{2},$ (23)

where $\eta^{\prime}_{\rm s}$ is the shear viscosity of the surrounding bulk
fluid that is tied to the substrate in Refs. [51, 52]444Note in Refs. [51, 52]
the shear viscosity in the substrate $\eta^{(SD)}_{\rm s}$ is three-
dimensional and therefore it has different units from the $\eta_{\rm s}$
appearing in this letter. In Eq. (23) the two viscosities are related by
taking $\eta^{(SD)}_{\rm s}\rightarrow\eta_{\rm s}/h$, with $h$ being the
height of the substrate.. We thus find that in these two instances, the shell
localization approach yields the same results as in previous works where the
fluid velocity profile is computed over the entire two-dimensional surface
[51, 52].

##  Appendix B Analytical computation of the response matrix

In this Appendix, we show how the integrals performed throughout this Letter
can be performed using the method of residues. For a compressible fluid as
described in the main text, the response coefficients are obtained by
performing momentum integrals that take the form

$\displaystyle I[R]=\int_{0}^{\infty}{\rm d}z\,R(z)J_{0}(z)\,,$ (24)

where $R(z)=A(z)/B(z)$ is an odd function of $z$, and where $A$ and $B$ are
polynomials in $z$. We call $z_{n}$ the $n^{\rm th}$ root of $B(z)$, such that
$B(z_{n})=0$. Following Ref. [77], the integral $I[R]$ can be computed
analytically in terms of the Hankel functions of the first kind
$H^{(1)}_{\nu}$ and the Bessel functions of the second kind $Y_{\nu}$. It
reads:

$\displaystyle\begin{split}I[R]&={\rm
i}\pi\sum_{z_{n}\in\mathbb{C}^{+}\backslash\mathbb{R}}{\rm
Res}\left(R(z)H^{(1)}_{0}(z),z_{n}\right)\\\
&\quad-\pi\sum_{z_{n}\in\mathbb{R}^{+}}{\rm
Res}\left(R(z)Y_{0}(z),z_{n}\right)\,,\end{split}$ (25)

where the first sum is over the roots of $B(z)$ whose imaginary part is
strictly positive, and the second one is over the positive real roots of
$B(z)$. We denote by ${\rm Res}(f(z),z_{n})$ the residue of $f$ at point
$z_{n}$.

As an illustration, we consider the oscillatory incompressible case in the
absence of relaxation ($\tilde{\tau}^{-1}\to 0$), for which one has

$\displaystyle R(z)=\frac{z}{z^{2}-{\rm i}\tilde{\omega}}\,,$ (26)

and thus for which $A(z)=z$ and $B(z)=z^{2}-{\rm i}\tilde{\omega}$ with the
roots $z_{1,2}=\pm\sqrt{{\rm i}\tilde{\omega}}$. In this case, only the first
term in the right-hand side of Eq. (25) contributes, and since the Hankel
function $H^{(1)}_{0}$ has no pole in $z_{1}=\sqrt{{\rm i}\tilde{\omega}}$, it
yields

$\displaystyle I\left[z/(z^{2}-{\rm i}\tilde{\omega})\right]=\frac{{\rm
i}\pi}{2}H^{(1)}_{0}(\sqrt{{\rm i}\tilde{\omega}})=K_{0}(-{\rm i}\sqrt{{\rm
i}\tilde{\omega}})~{}~{},$ (27)

where $K_{\nu}(x)$ is the $\nu^{\rm th}$ modified Bessel function of the
second kind. An expansion of Eq. (27) in series of $\tilde{\omega}$ yields the
result given in Eq. (21).

The same procedure can be applied for the compressible case, and was used to
obtain Eqs. (18) and (19) of the main text.

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