converts such a judgement into the standard statement reduction judgement. The
rules for the separating conjunction and the accessibility predicate (when the
receiver and permission are well-defined) are shown in Fig. 11. In the case of
the accessibility predicate, there is an additional rule where the inhale
fails if $e$ or $e_{p}$ are not well-defined, which is not shown in the
figure.

The accessibility predicate rule inh-acc shown in Fig. 11 expresses that if
the added permission is negative then the operation fails. If the permission
is nonnegative, then the operation succeeds if (1) the receiver is non-null if
$p>0$, and (2) the added permission does not yield an inconsistent state (i.e.
does not result in more than 1 permission for $(r,f)$). Otherwise, the
operation stops (denoted by outcome M). If the operation succeeds, then the
new state additionally contains the added permission $p$ at location $(r,f)$.

##  Appendix B Another Simulation Rule Example

$\displaystyle{tensy\vbox{\hbox spread0.0pt{\hskip 0.0pt plus
0.0001fil\hbox{$\displaystyle\penalty
1{\begin{array}[]{c}{\begin{array}[]{c}\forall\sigma_{v}.\;\textsf{wfSim}{}_{\Gamma_{b}}(\hat{R}(\sigma_{v}),\hat{R_{A}}(\sigma_{v}),[e,e_{p}],\gamma,\gamma_{1})\quad\text{(subexpression
well-definedness)}\\\ \forall
r,p.\;\textsf{sim}{}_{\Gamma_{b}}(R_{A},R_{B}(r,p),\mathit{Succ}_{A}(r,p),\mathit{Fail}_{A}(r,p),\gamma_{1},\gamma_{2})\quad\text{(non-
failure check)}\end{array}}\\\ \forall
r,p.\;\textsf{sim}{}_{\Gamma_{b}}(R_{B}(r,p),R^{\prime},\mathit{Succ}_{B}(r,p),(\lambda\\_.\;\bot),\gamma_{2},\gamma^{\prime})\quad\text{(state
update)}\end{array}}$}\hskip 0.0pt plus 0.0001fil}\hbox{\hbox{\kern
0.0pt\vrule height=0.25002pt,depth=0.25002pt,width=322.80193pt\hbox{\kern
3.00003pt${(\textsc{racc-sim})}$}}}\hbox{\kern
96.06738pt\hbox{$\displaystyle\textsf{rcSim}{}_{\Gamma_{b}}(R,R^{\prime},{\color[rgb]{0.25,0.41,0.88}\definecolor[named]{pgfstrokecolor}{rgb}{0.25,0.41,0.88}\texttt{acc}}(e.f,e_{p}),\gamma,\gamma^{\prime})$}}}}$
$\displaystyle\hat{R}(\sigma_{v})\triangleq{}\lambda\sigma_{v}^{0}\;\sigma_{b}.\;R((\sigma_{v}^{0},\sigma_{v}),\sigma_{b})\quad\hat{R_{A}}(\sigma_{v})\triangleq{}\lambda\sigma_{v}^{0}\;\sigma_{b}.\;R_{A}((\sigma_{v}^{0},\sigma_{v}),\sigma_{b})$
$\displaystyle\mathit{Succ}_{A}(r,p)\triangleq{}\left(\lambda(\sigma_{v}^{0},\sigma_{v})\;(\sigma_{v}^{1},\sigma_{v}^{\prime}).\;\begin{array}[]{l}{\color[rgb]{0.71,0.4,0.11}\definecolor[named]{pgfstrokecolor}{rgb}{0.71,0.4,0.11}\textsf{exhAccSucc}(r,p,\sigma_{v})}\wedge{}(\sigma_{v}^{0},\sigma_{v})=(\sigma_{v}^{1},\sigma_{v}^{\prime})\wedge{}\\\
{\color[rgb]{0.58,0.0,0.83}\definecolor[named]{pgfstrokecolor}{rgb}{0.58,0.0,0.83}\textsf{wfAccSucc}(e,e_{p},r,p,\sigma_{v}^{0})}\end{array}\right)$
$\displaystyle\mathit{Fail}_{A}(r,p)\triangleq{}\lambda(\sigma_{v}^{0},\sigma_{v}).\;\neg{\color[rgb]{0.71,0.4,0.11}\definecolor[named]{pgfstrokecolor}{rgb}{0.71,0.4,0.11}\textsf{exhAccSucc}(r,p,\sigma_{v})}\wedge{}{\color[rgb]{0.58,0.0,0.83}\definecolor[named]{pgfstrokecolor}{rgb}{0.58,0.0,0.83}\textsf{wfAccSucc}(e,e_{p},r,p,\sigma_{v}^{0})}$
$\displaystyle\mathit{Succ}_{B}(r,p)\triangleq{}\left(\lambda(\sigma_{v}^{0},\sigma_{v})\;(\sigma_{v}^{1},\sigma_{v}^{\prime}).\;\begin{array}[]{l}\sigma_{v}^{\prime}=\textsf{rem}(\sigma_{v},r,f,p)\wedge{}\sigma_{v}^{0}=\sigma_{v}^{1}\wedge{}\\\
{\color[rgb]{0.71,0.4,0.11}\definecolor[named]{pgfstrokecolor}{rgb}{0.71,0.4,0.11}\textsf{exhAccSucc}(r,p,\sigma_{v})}\wedge{}{\color[rgb]{0.58,0.0,0.83}\definecolor[named]{pgfstrokecolor}{rgb}{0.58,0.0,0.83}\textsf{wfAccSucc}(e,e_{p},r,p,\sigma_{v}^{0})}\end{array}\right)$
$\displaystyle{\color[rgb]{0.58,0.0,0.83}\definecolor[named]{pgfstrokecolor}{rgb}{0.58,0.0,0.83}\textsf{wfAccSucc}(e,e_{p},r,p,\sigma_{v}^{0})}\triangleq{}\langle
e,\sigma_{v}^{0}\rangle\Downarrow\textsf{V}(r)\wedge\langle
e_{p},\sigma_{v}^{0}\rangle\Downarrow\textsf{V}(p)$ Figure 12. Rule for the
simulation of
${\color[rgb]{0.25,0.41,0.88}\definecolor[named]{pgfstrokecolor}{rgb}{0.25,0.41,0.88}\texttt{remcheck}}\;{\color[rgb]{0.25,0.41,0.88}\definecolor[named]{pgfstrokecolor}{rgb}{0.25,0.41,0.88}\texttt{acc}}(e.f,e_{p})$.
The definition of  exhAccSucc is given in Fig. 2.

Consider the rule racc-sim in Fig. 12, which decomposes the simulation of
${\color[rgb]{0.25,0.41,0.88}\definecolor[named]{pgfstrokecolor}{rgb}{0.25,0.41,0.88}\texttt{remcheck}}\;{\color[rgb]{0.25,0.41,0.88}\definecolor[named]{pgfstrokecolor}{rgb}{0.25,0.41,0.88}\texttt{acc}}(e.f,e_{p})$
(ignore the universal quantifiers for now) into the simulation of three
separate Viper effects: (1) the check of well-definedness of the receiver $e$
and permission expression $e_{p}$ (via the wfSim instantiation from Fig. 4),
(2) a check  exhAccSucc ensuring that the operation will not fail (from the
semantics; see Fig. 2), and (3) the actual update of the Viper state, which
removes the permission.

The second premise includes contextual information, namely the conjunct
wfAccSucc expressing that $e$ and $e_{p}$ are well-defined (which is ensured
by the first premise) and evaluate to the reference value $r$ and permission
value $p$. The third premise modelling the removal of the permission includes
the same conjunct and that the operation will succeed ( exhAccSucc). Without
the latter, we could in general not prove that the resulting Boogie state
satisfies crucial invariants, for instance, that none of the permissions
stored in the Boogie state are negative. Again, we are agnostic as to
syntactically _how_ this is achieved by this check: our rule does _not_
require the Boogie program to emit an explicit Boogie assert command checking
that the permission is nonnegative. This is important, since the
implementation omits such a command, for example, if the permission is the
literal 1.

The first universal quantifier is technically motivated: it expresses that the
simulation must hold for _any_ reduction state. The other quantifiers over
reference values $r$ and permission values $p$ make the rule more powerful and
reusable. They permit the relation $R_{B}$ to directly talk about the values
that $e$ and $e_{p}$ evaluate to as specified by the success and failure
predicates. This is particularly useful for justifying cases where the
simulation of the non-failure check establishes a property on $r$ or $p$,
which is then used in the simulation of the state update. For example, the
Viper-to-Boogie translation stores $p$ into an auxiliary variable that is used
for both the non-failure check and the state update.

##  Appendix C Instantiation-Independent Consequence Rule

tensy  $\displaystyle\penalty
1{\begin{array}[]{c}{\begin{array}[]{c}{\begin{array}[]{c}{\begin{array}[]{c}\textsf{sim}{}_{\Gamma_{b}}(R_{0}^{\prime},R_{1}^{\prime},S^{\prime},F^{\prime},\gamma,\gamma^{\prime})\\\
\forall\sigma_{v}\;\sigma_{b}.\;R_{0}(\sigma_{v},\sigma_{b})\Rightarrow
R_{0}^{\prime}(\sigma_{v},\sigma_{b})\quad\text{(weaker input state
relation)}\end{array}}\\\
\forall\sigma_{v}\;\sigma_{v}^{\prime}.\;S(\sigma_{v},\sigma_{v}^{\prime})\Rightarrow
S^{\prime}(\sigma_{v},\sigma_{v}^{\prime})\quad{\text{(weaker success
predicate)}}\end{array}}\\\ \forall\sigma_{v}.\;F(\sigma_{v})\Rightarrow
F^{\prime}(\sigma_{v})\quad\text{(weaker failure predicate)}\end{array}}\\\
\forall\sigma_{v}\;\sigma_{v}^{\prime}\;\sigma_{b}^{\prime}.\;(\exists\sigma_{b}.\;R_{0}(\sigma_{v},\sigma_{b}))\Rightarrow
R_{1}^{\prime}(\sigma_{v}^{\prime},\sigma_{b}^{\prime})\Rightarrow
S(\sigma_{v},\sigma_{v}^{\prime})\Rightarrow
R_{1}^{\prime\prime}(\sigma_{v}^{\prime},\sigma_{b}^{\prime})\end{array}}$
${(\textsc{cons})}$
$\displaystyle\textsf{sim}{}_{\Gamma_{b}}(R_{0},R_{1}^{\prime\prime},S,F,\gamma,\gamma^{\prime})$

Figure 13. An instantiation-independent consequence rule for the generic
forward simulation judgement.

The rule cons in Fig. 13 shows an instantiation-independent consequence rule
for the generic forward simulation judgement. The rule enables proving a
simulation judgement by proving a different simulation judgement in the
premise where the input state relation, success predicate, and failure
predicate are weakened. The output state relation is potentially adjusted in
the premise as well. Additionally, the final premise expresses a condition
ensuring that the output state relation $R_{1}^{\prime\prime}$ in the
conclusion will be established for the output states if the initial state
relation holds for the initial states, the adjusted output relation holds on
the output states, and the Viper execution is successful.

##  Appendix D Detailed Results of the Evaluation

Table 3. Detailed results of our evaluation for the files from the test suite of Gobra. File | Methods | Viper | Boogie | Isabelle | Proof Check
---|---|---|---|---|---
| no. | Total [LoC] | Total [LoC] | Total [LoC] | Total [s]
concurrency | 2 | 24 | 164 | 1153 | 25.3
defer-simple-01 | 6 | 142 | 639 | 3344 | 49.6
defer-simple-02 | 9 | 211 | 853 | 4717 | 60.6
perm-fail1 | 15 | 165 | 661 | 6392 | 66.5
perm-simple1 | 9 | 131 | 622 | 4221 | 50.7
fail1 | 3 | 44 | 283 | 1574 | 31.6
fail3 | 2 | 19 | 116 | 1044 | 23.6
simple1 | 2 | 30 | 237 | 1210 | 28.4
simple2 | 1 | 10 | 90 | 672 | 21.4
simple3 | 1 | 17 | 186 | 801 | 24.6
global-const-8 | 6 | 49 | 206 | 2510 | 33.8
pointer-identity | 1 | 30 | 158 | 731 | 23.0
pointer-identity | 1 | 30 | 158 | 731 | 23.1
000008 | 1 | 10 | 85 | 672 | 21.4
000009 | 1 | 16 | 98 | 679 | 21.3
000039 | 3 | 49 | 178 | 1410 | 26.8
000155 | 2 | 39 | 152 | 1075 | 24.3
Table 4. Detailed results of our evaluation for the files from the test suite of MPP. File | Methods | Viper | Boogie | Isabelle | Proof Check
---|---|---|---|---|---
| no. | Total [LoC] | Total [LoC] | Total [LoC] | Total [s]
banerjee | 8 | 414 | 2014 | 9545 | 242.4
darvas | 2 | 91 | 582 | 2800 | 38.4
kusters | 3 | 112 | 583 | 3146 | 46.2
Table 5. Detailed results of our evaluation for the files from the test suite of VerCors. File | Methods | Viper | Boogie | Isabelle | Proof Check
---|---|---|---|---|---
| no. | Total [LoC] | Total [LoC] | Total [LoC] | Total [s]
BasicAssert-e1 | 6 | 41 | 197 | 2589 | 35.0
BasicAssert | 6 | 41 | 193 | 2589 | 35.0
DafnyIncr | 8 | 60 | 265 | 3419 | 41.6
DafnyIncrE1 | 8 | 57 | 220 | 3340 | 40.2
permissions | 5 | 39 | 208 | 2270 | 33.1
inv-test-fail1 | 5 | 90 | 510 | 2589 | 55.5
inv-test-fail2 | 5 | 92 | 514 | 2596 | 56.5
inv-test | 5 | 90 | 510 | 2589 | 55.1
SwapIntegerFail | 8 | 79 | 429 | 3645 | 49.8
SwapIntegerPass | 8 | 81 | 469 | 3688 | 53.0
SwapLong | 6 | 57 | 277 | 2725 | 36.7
SwapLongTwice | 8 | 81 | 469 | 3688 | 52.1
SwapLongWrong | 8 | 79 | 429 | 3645 | 48.9
frame-error-1 | 5 | 35 | 173 | 2191 | 32.8
refute3 | 6 | 49 | 246 | 2662 | 34.5
refute4 | 6 | 54 | 258 | 2676 | 35.9
refute5 | 6 | 50 | 253 | 2662 | 35.9
demo1 | 7 | 60 | 347 | 3185 | 44.6
Table 6. Detailed results of our evaluation for the files from the test suite of Viper. File | Methods | Viper | Boogie | Isabelle | Proof Check
---|---|---|---|---|---
| no. | Total [LoC] | Total [LoC] | Total [LoC] | Total [s]
0004 | 1 | 6 | 100 | 729 | 21.7
0004-CPG1 | 1 | 6 | 95 | 704 | 21.6
0005 | 1 | 4 | 78 | 665 | 21.1
0008 | 2 | 12 | 241 | 1396 | 26.8
0011 | 5 | 63 | 902 | 3284 | 55.7
0015 | 1 | 6 | 92 | 709 | 21.4
0052 | 1 | 7 | 100 | 719 | 21.5
0063 | 6 | 34 | 180 | 2595 | 35.2
0072 | 1 | 8 | 112 | 770 | 22.4
0073 | 1 | 10 | 132 | 737 | 22.2
0088-1 | 1 | 9 | 115 | 751 | 21.9
0094 | 1 | 6 | 91 | 679 | 21.2
0152 | 2 | 14 | 139 | 1137 | 24.5
0157 | 8 | 47 | 354 | 3508 | 45.1
0159 | 2 | 13 | 120 | 1083 | 23.8
0170 | 1 | 8 | 84 | 665 | 21.1
0177-1 | 1 | 10 | 102 | 665 | 21.4
0222 | 2 | 13 | 118 | 1054 | 23.9
0227 | 1 | 5 | 85 | 683 | 21.4
0324 | 1 | 7 | 104 | 704 | 21.2
0345 | 3 | 21 | 165 | 1463 | 24.4
0384 | 1 | 11 | 127 | 709 | 22.0
assert | 1 | 7 | 92 | 693 | 21.5
negative-amounts | 3 | 21 | 155 | 1517 | 27.4
old | 6 | 38 | 318 | 2805 | 37.9
swap | 2 | 16 | 177 | 1239 | 25.6
test | 1 | 6 | 81 | 663 | 20.9
testHistoryProcesses | 13 | 205 | 1711 | 7035 | 126.3
testHistoryProcessesPVL | 13 | 204 | 1711 | 7035 | 116.3
testHistoryProcessesPVL-CPG1 | 4 | 56 | 490 | 2304 | 46.1
testHistoryThreadsProcessesPVL | 4 | 56 | 490 | 2304 | 45.7
test-example1 | 4 | 57 | 374 | 2152 | 37.0
test-example3 | 5 | 74 | 430 | 2634 | 39.3
test-example4 | 5 | 71 | 451 | 2645 | 42.7

The detailed results of the evaluation are shown separately for each of the
files in Tab. 3 (Gobra), Tab. 4 (MPP), Tab. 5 (VerCors), and Tab. 6 (Viper).


# Polarization Rotation of Chiral Fermions in Vortical Fluid

Defu Hou  Institute of Particle Physics (IOPP) and Key Laboratory of Quark and
Lepton Physics (MOE), Central China Normal University, Wuhan 430079, China
Shu Lin  School of Physics and Astronomy, Sun Yat-Sen University, Zhuhai
519082, China

###### Abstract

The rotation of polarization occurs for light interacting with chiral
materials. It requires the light states with opposite chiralities interact
differently with the materials. We demonstrate analogous rotation of
polarization also exists for chiral fermions interacting with quantum
electrodynamics plasma with vorticity using chiral kinetic theory. We find
that the rotation of polarization is perpendicular both to vorticity and
fermion momentum. The effect also exists for chiral fermions in quantum
chromodynamics plasma with vorticity. It could lead to generation of a vector
current when the probe fermions contain momentum anisotropy.

##  I Introduction

It is known that polarized light interacting with stereoisomers can lead to
rotation of polarization LANDAU and LIFSHITZ (1984). The polarization rotation
effect has received much attention in different fields including optics Plum
_et al._ (2009), condensed matter physics Zhong  _et al._ (2016), cosmology Ni
(2008) etc. The mechanism of rotation of polarization is that light with
opposite circular polarizations interact differently with the chiral
materials, leading to circular birefringence. The polarization dependent
interaction is not particular for light. A natural question to ask is whether
analogous effect exist for chiral fermions?

In this letter, we give one such example with chiral fermions interacting with
polarized medium. Our medium is polarized by vorticity of fluid. On general
ground, the spin polarization of chiral fermion in local rest frame of the
fluid can be decomposed as follows

$\displaystyle{\cal
P}^{i}=A_{1}\hat{p}^{i}+A_{2}\omega^{i}+A_{3}\epsilon^{ijk}\hat{p}_{j}\omega_{k},$
(1)

with $\hat{p}^{i}$ and $\omega^{i}$ being the direction of momentum and fluid
vorticity. By parity, $A_{1}$ and $A_{3}$ are pseudoscalar functions and
$A_{2}$ is a scalar function. In fact, $A_{1}$ exists in free theory due to
spin-momentum locking for chiral fermion, and $A_{2}$ is a manifestation of
spin-orbit coupling giving rise to chiral vortical effect Vilenkin (1980a);
Erdmenger  _et al._ (2009); Banerjee  _et al._ (2011); Neiman and Oz (2011);
Landsteiner  _et al._ (2011). $A_{3}$ is a new contribution we will focus on.

The new contribution leads to net vector current in systems with momentum
anisotropy. To see that, we note momentum integration of ${\cal P}^{i}$ gives
rise to axial current, which receives opposite contribution from left and
right-handed components of chiral fermion due to the odd parity of $A_{3}$. It
follows immediately that a net vector current is generated as

$\displaystyle J_{V}^{i}\sim\epsilon^{ijk}\xi_{j}\omega_{k},$ (2)

with $\xi_{j}$ being the axis characterizing the momentum anisotropy. This is
analog of Lorentz force due to vorticity.

##  II Kinetic theory for chiral fermions

We illustrate this effect in weakly coupled quantum electrodynamics (QED)
plasma using kinetic theory. We will generalize to quantum chromodynamics
(QCD) plasma later. In high temperature limit of the two cases, electron/quark
are approximately chiral. While the spin averaged kinetic theory has been
widely used in describing transport coefficients of weakly coupled plasma
Blaizot and Iancu (2002); Arnold  _et al._ (2003a, 2000, b), its construction
limits its application in spin dependent phenomenon, such as chiral magnetic
effect Vilenkin (1980b); Kharzeev (2006); Kharzeev and Zhitnitsky (2007);
Fukushima  _et al._ (2008) and chiral vortical effect Vilenkin (1980a);
Erdmenger  _et al._ (2009); Banerjee  _et al._ (2011); Neiman and Oz (2011);
Landsteiner  _et al._ (2011). Spin dependent kinetic theory has been developed
in recent years under the names of chiral kinetic theory Son and Spivak
(2013); Son and Yamamoto (2012, 2013); Stephanov and Yin (2012); Pu  _et al._
(2011); Chen  _et al._ (2013); Hidaka  _et al._ (2017); Manuel and Torres-
Rincon (2014a, b); Huang  _et al._ (2018); Liu  _et al._ (2019); Lin and
Shukla (2019); Sheng  _et al._ (2018); Lin and Yang (2020) and spin kinetic
theory Hattori  _et al._ (2019); Wang  _et al._ (2019); Gao and Liang (2019);
Yang  _et al._ (2020); Weickgenannt  _et al._ (2019); Gao  _et al._ (2020a);
Liu  _et al._ (2020); Gao  _et al._ (2020b); Liu and Huang (2020), in which a
scalar-like distribution function is used. In this paper, we retain the spinor
structure and work with spinor equations. We start with the Kadanoff-Baym
equation (KBE) Blaizot and Iancu (2002)

$\displaystyle\frac{i}{2}\not{D}S^{<}(X,P)+\not{P}S^{<}(X,P)=\frac{i}{2}\left(\Sigma^{>}(X,P)S^{<}(X,P)-\Sigma^{<}(X,P)S^{>}(X,P)\right),$
(3)

where $\not{D}=\not{\partial}_{X}+ie\not{A}$. $S^{</>}$ are the Wigner
transform of the off-equilibrium lesser and greater fermion correlators
111$S^{<}(X,P)$ is related to the usual Wigner function $W(X,P)$ by
$S^{<}=-\frac{1}{(2\pi)^{4}}W$

$\displaystyle S_{\alpha\beta}^{>}(X,P)=\int
d^{4}(x-y)e^{iP\cdot(x-y)}\langle\psi_{\alpha}(x){\bar{\psi}}_{\beta}(y)\rangle,$
$\displaystyle S_{\alpha\beta}^{<}(X,P)=-\int
d^{4}(x-y)e^{iP\cdot(x-y)}\langle{\bar{\psi}}_{\beta}(y)\psi_{\alpha}(x)\rangle,$
(4)

with $X=\frac{x+y}{2}$ and similarly $\Sigma^{</>}$ for lesser and greater
self-energy correlators.

In the absence of collisional term on the right hand side (RHS), (3) is easily
solved in a gradient expansion assuming $P\gg\partial_{X}$. Denoting zeroth
order and first order solutions by $S^{<(0)}$ and $S^{<(1)}$, we obtain the
following equation

$\displaystyle\frac{i}{2}\not{\partial}S^{<(0)}+\not{P}S^{<(1)}=0.$ (5)

The zeroth order solution $S^{<(0)}$ is given by propagator in local
equilibrium with vortical fluid

$\displaystyle S^{<(0)}=-(2\pi)\not{P}\delta(P^{2})\epsilon(P\cdot u)f(P\cdot
u),$ (6)

where $\epsilon(P\cdot u)$ is the sign function, $u$ is the fluid velocity and
$f$ is Fermi-Dirac distribution function

$\displaystyle f(P\cdot u)=\frac{1}{e^{(P\cdot u-\mu)/T}+1},$ (7)

with temperature $T$ and chemical potential $\mu$ taken to be constants so
that the plasma has vorticity only 222If the system has net charge density,
magnetic field can be induced by charge current. The effect of the induced
magnetic field on the dynamics of fermions is suppressed by an additional
power of $e^{2}$ compared to vortical effect considered here.. With the
vorticity counted as first order in gradient, (5) can be solved by Gao  _et
al._ (2019); Fang  _et al._ (2016)

$\displaystyle
S^{<(1)}=-(2\pi)\frac{1}{2}\not{\tilde{P}}\gamma^{5}\delta(P^{2})\epsilon(P\cdot
u)f^{\prime}(P\cdot u),$ (8)

with $\tilde{P}_{\mu}=P^{\lambda}\tilde{\Omega}_{\lambda\mu}$ and
$\tilde{\Omega}^{\mu\nu}=\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}\partial_{\rho}u_{\sigma}$.
$\tilde{\Omega}^{\mu\nu}$ can be decomposed into vorticity
$\omega^{\mu}=\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}u_{\nu}\partial_{\rho}u_{\sigma}$
and acceleration
$\varepsilon^{\mu}=\frac{1}{2}u^{\lambda}\partial_{\lambda}u^{\mu}$ as

$\displaystyle\tilde{\Omega}^{\mu\nu}=\omega^{\mu}u^{\nu}-\omega^{\nu}u^{\mu}+\epsilon^{\mu\nu\rho\sigma}\varepsilon_{\rho}u_{\sigma}.$
(9)

We restrict ourselves to the case with only static vorticity in the local rest
frame of the fluid. In this case, $\not{\partial}=\gamma^{i}\partial_{i}$ in
(3). (8) is the off-equilibrium correction to propagator due to fluid
vorticity. The factor $\delta(P^{2})$ indicates that the on-shell condition is
not changed. We can infer the change of polarization due to vorticity. In
local rest frame of the fluid, the unintegrated polarization is given by

$\displaystyle{\cal
P}^{i}(X,\vec{p})=\int\frac{dp_{0}}{2\pi}\text{tr}\gamma^{i}\gamma^{5}S^{<}.$
(10)

(6) corresponds to an unpolarized fluid. (8) leads to a net polarization along
the vorticity: ${\cal P}^{i}\sim f^{\prime}(p)\omega_{i}$ for both fermions
and antifermions in neutral fluid.

Now we turn to the collisional term on the RHS. Recent works incorporating
collisional term in spin-dependent theories include Yang  _et al._ (2020);
Weickgenannt  _et al._ (2020, 2019); Sheng  _et al._ (2021); Carignano  _et
al._ (2020); Li and Yee (2019); Yamamoto and Yang (2020); Wang  _et al._
(2020). We use the following representation for the fermion self-energy
Blaizot and Iancu (1999)

$\displaystyle\Sigma^{>}(X,P)=e^{2}\int_{Q}\gamma^{\mu}S^{>}(X,P+Q)\gamma^{\nu}D_{\nu\mu}^{<}(X,Q)$
$\displaystyle\simeq-e^{2}\int_{Q}\gamma^{\mu}S^{>}(P+Q)\gamma^{\nu}D_{\nu\alpha}^{R}(Q)\Pi^{\alpha\beta<}(Q)D_{\beta\mu}^{A}(Q),$
(11)

with $\int_{Q}\equiv\int\frac{d^{4}Q}{(2\pi)^{4}}$. We have suppressed the
dependence on $X$ in $S^{>}$, $D^{R/A}$ and $\Pi^{<}$ in the last line for
notational simplicity. The representation is valid off-equilibrium, with the
second equality holds to the leading order in gradient expansion, which
requires $Q\gg\partial_{X}$. A similar representation exists for
$\Sigma^{<}(X,P)$ with the exchange of $<$ and $>$ in (II). The off-
equilibrium photon self-energy $\Pi^{\alpha\beta}$ can be expressed in terms
of fermion propagators as follows

$\displaystyle\Pi^{\alpha\beta<}(X,Q)=e^{2}\int_{K}tr\left[\gamma^{\alpha}S^{<}(X,K+Q)\gamma^{\beta}S^{>}(X,K)\right].$
(12)

In general, the KBE (3), and the representations for self-energies (II) and
(12) do not form a closed set of equations as they also involve photon
propagators, for which a separate kinetic theory for photons is needed. On the
other hand, it is known that the RHS contains possible IR divergence Pisarski
(1993); Blaizot and Iancu (1996, 1997a, 1997b). If we keep only the leading IR
divergence on the RHS, the kinetic theory for photons decouples for the
following reason: we know the divergence comes from Coulomb scattering with
soft photon exchange. The self-energy of soft photon $\Pi^{\alpha\beta}$ as
well as propagators $D_{\nu\alpha}^{R}$ and $D_{\beta\mu}^{A}$ are entirely
determined by hard fermion, which is governed by the kinetic theory. It will
not be true if we wish to go beyond the leading IR divergence, for which
Compton scattering is also needed 333This can be seen from the following power
counting. In Coulomb and Compton scatterings, the exchanged particles are
photon and fermion respectively. The former contains $1/Q^{4}$ from two
propagators of the exchanged photon while the latter contains only $1/Q^{2}$
from propagators of fermion. Moreover, the Coulomb scattering has an
additional $1/Q$ due to the Bose enhancement from the photon self-energy..
This allows us to use kinetic theory for fermion only to study the leading IR
divergence. However, we will see that similar simplification is not possible
in QCD where the presence of three-gluon vertex necessitates the inclusion of
gluon kinetic theory for the same IR divergence.

It is known that correction to the $S^{>/<}$ and $\Sigma^{>/<}$ corresponds to
correction to hard fermion distribution, while correction to $D^{R/A}$
corresponds to correction to scattering amplitude Blaizot and Iancu (1999). We
are interested in linear response to vorticity, thus the vortical correction
can enter only one case. The vortical correction to $D^{R/A}$ is irrelevant
because $S^{>/<}$ and $\Sigma^{>/<}$ arises from equilibrium fermion
distribution and the collision term vanishes identically by detailed balance
independent of the interaction.

##  III Probe fermions in vortical fluid

Now we introduce probe fermions as a perturbation to the vortical fluid and
study its spin polarization by solving the kinetic equation. We denote the
perturbation to $S^{<}$ and $S^{>}$ by $\Delta S^{<}$ and $\Delta S^{>}$
respectively. In the quasi-particle approximation, we have
$S^{>}(X,P)-S^{<}(X,P)=\rho(X,P)=2\pi\epsilon(P\cdot
u(X))\not{P}\delta(P^{2})$ Blaizot and Iancu (2002). The RHS is the local
spectral density, which depends on local temperature and fluid velocity only,
but not on the perturbation. It follows that $\Delta S^{>}(X,P)-\Delta
S^{<}(X,P)=0$. Below we will use $\Delta S$ to denote both $\Delta S^{<}$ and
$\Delta S^{>}$ and assume the on-shell condition is not changed, which will be
verified by the explicit solution.

Now we work out the RHS of (3) up to first order in vorticity. At zeroth
order, the RHS of (3) can be written as

$\displaystyle-\frac{i}{2}e^{2}\int_{Q}\big{[}\gamma^{\mu}S^{>(0)}(P+Q)\gamma^{\nu}D_{\nu\mu}^{<(0)}(Q)-\gamma^{\mu}S^{<(0)}(P+Q)\gamma^{\nu}D_{\nu\mu}^{>(0)}(Q)\big{]}\Delta
S(P).$ (13)

Note that the leading IR divergence comes from exchange of soft photon with
$Q\ll P\sim T$. We can then approximate the equilibrium photon propagators as
$D_{\nu\mu}^{<}(Q)\simeq D_{\nu\mu}^{>}(Q)=\frac{T}{Q\cdot
u}(u_{\mu}u_{\nu}\rho_{L}+P_{\mu\nu}^{T}\rho_{T})$ in Coulomb gauge with
$u_{\mu}u_{\nu}$ and $P_{\mu\nu}^{T}$ being the longitudinal and transverse
projection operators, and $\rho_{L/T}$ being longitudinal and transverse
spectral densities. We can then simplify the RHS as

$\displaystyle-\frac{i}{2}e^{2}\int_{Q}\frac{T}{Q\cdot
u}\gamma^{\mu}\rho(P+Q)\gamma^{\nu}D_{\nu\mu}^{<(0)}(Q)\Delta S(P),$ (14)

Using $Q\ll P$ and $\Delta S\propto\delta(P^{2})$ and contracting the gamma
matrices using

$\displaystyle\gamma^{\mu}\gamma^{\lambda}\gamma^{\nu}=g^{\mu\lambda}\gamma^{\nu}-g^{\mu\nu}\gamma^{\lambda}+g^{\mu\lambda}\gamma^{\nu}+i\epsilon^{\mu\lambda\nu\sigma}\gamma_{\sigma}\gamma^{5},$
(15)

we obtain the RHS in local rest frame of the plasma as

$\displaystyle-\frac{i}{2}e^{2}\int_{Q}$
$\displaystyle\frac{T}{q_{0}}\left(\frac{2p_{0}}{q^{2}}(Q^{2}\gamma^{0}-q_{0}\not{Q})\rho_{T}+2p_{0}\gamma^{0}\rho_{L}\right)\delta(2P\cdot
Q)\Delta S(P)$ (16)

We can perform the angular integration in the momentum integral to arrive at

$\displaystyle-\frac{i}{2}e^{2}\gamma^{0}\int\frac{dq_{0}qdq}{(2\pi)^{2}}\frac{T}{q_{0}}\left(\frac{q_{0}^{2}-q^{2}}{q^{2}}\rho_{T}-\rho_{L}\right)\Delta
S(P),$ (17)

where we have used the on-shell condition $\delta(P^{2})$. The $q_{0}$
integral can be performed by using the sum rule Valle Basagoiti (2002), but it
is not necessary as we only need the leading divergence. Note that the
longitudinal and transverse components correspond to electric and magnetic
interactions respectively. The former is dynamically screened by the plasma to
give finite contribution and the latter is only partially screened with
divergent contribution. The leading divergence is from the kinematic regime
$q_{0}\ll q$. We can approximate the retarded soft transverse correlator
$\Delta_{T}$ and spectral density as Bellac (2011).

$\displaystyle\Delta_{T}(q_{0}\ll q)\simeq\frac{1}{q^{2}-i(\pi
q_{0}/4q)m_{D}^{2}},$ $\displaystyle\rho_{T}\simeq
2\text{Im}\Delta_{T}=\frac{1}{q^{4}+(\pi q_{0}/4q)^{2}m_{D}^{4}}(\pi
q_{0}/2q)m_{D}^{2}.$ (18)

Keeping only the leading divergence, we obtain from (17)

$\displaystyle-\frac{i}{2}\gamma^{0}\frac{e^{2}T}{2\pi}\ln\frac{m_{D}}{q_{\text{IR}}}\Delta
S\equiv-i\gamma^{0}\Gamma_{0}\Delta S.$ (19)

Here $q_{\text{IR}}$ is an IR cutoff of momentum $q$. A resummation can be
used to render the result finite Blaizot and Iancu (1996). We will not attempt
it here as the IR regularized result is sufficient to illustrate the effect we
are after. Clearly the zeroth order contribution (19) is independent of the
spin as expected.

Now we turn to first order vortical correction to the collisional term, for
which spin-dependent kinetic theory must be used. We first derive vortical
correction to the collisional term (13), which can enter either through
$S^{>/<}$ or $D_{\nu\mu}^{</>}$. The former and the latter can be regarded as
vortical correction to fermion and photon in the fermion self-energy loop
respectively, or in language of kinetic theory, the former corresponds to the
final state of the probe fermion and the latter corresponds to initial and
final state of the medium fermion. We will show that the former vanishes
identically and the latter give similar type of divergence as (19).

Let us work out the basic elements we need. We already have $S^{<(1)}$. We can
solve for $S^{>(1)}$ from the following collisionless kinetic theory for
$S^{>}$

$\displaystyle\frac{i}{2}\not{\partial}S^{>}+\not{P}S^{>}=0,$ (20)

with the zeroth order solution

$\displaystyle S^{>(0)}=-(2\pi)\not{P}\delta(P^{2})\epsilon(P\cdot u)(f(P\cdot
u)-1).$ (21)

Since $S^{<}$ and $S^{>}$ satisfy the same equation and the zeroth order
solutions are related by the replacement $f\to f-1$, we easily obtain
$S^{>(1)}=S^{<(1)}$ by analogy of (8). We now work out the vortical correction
to the photon propagator $D_{\nu\mu}^{<(1)}$($D_{\nu\mu}^{>(1)}$). As we
already show before, vortical correction to $D^{R}_{\nu\alpha}$ and
$D^{A}_{\beta\mu}$ are not relevant as they lead to vanishing collisional
term, thus we only need to consider vortical correction to self-energy
$\Pi^{\alpha\beta<(1)}$, which is easily constructed using $S^{<}$ and $S^{>}$
as

$\displaystyle\Pi^{\alpha\beta<(1)}(Q)=e^{2}\int_{K}tr\big{[}\gamma^{\alpha}S^{<(1)}(K+Q)\gamma^{\beta}S^{>(0)}(K)+\gamma^{\alpha}S^{<(0)}(K+Q)\gamma^{\beta}S^{>(1)}(K)\big{]}.$
(22)

Using (6) and (8), we obtain

$\displaystyle\Pi^{\alpha\beta<(1)}(Q)\simeq
2i(2\pi)^{2}e^{2}\epsilon^{\alpha\nu\beta\lambda}\int_{K}K^{\mu}\tilde{\Omega}_{\mu\nu}K_{\lambda}\delta(K^{2})\delta(2K\cdot
Q)f^{\prime}(K\cdot u),$ (23)

where we have used for soft photon momentum $Q\ll K$, $\epsilon((K+Q)\cdot
u)\simeq\epsilon(K\cdot u)$ and $\delta((K+Q)^{2})\simeq\delta(2K\cdot Q)$.
Note that $\Pi^{\alpha\beta<(1)}$ is antisymmetric in the indices. We consider
$\Pi^{\alpha\beta<(1)}$ in a fluid with vorticity but no acceleration, i.e.
$\tilde{\Omega}_{\lambda\rho}=\omega_{\lambda}u_{\rho}-\omega_{\rho}u_{\lambda}$.
Completing the momentum integral, we obtain in local rest frame of the fluid
the following nonvanishing components for the vortical correction to the
photon self-energy:

$\displaystyle\Pi^{ij<(1)}$ $\displaystyle=-\frac{ie^{2}}{4\pi
q}\chi\left(-\epsilon^{ijk}\hat{q}_{k}\hat{q}\cdot\omega\frac{q_{0}^{2}}{q^{2}}-\frac{1}{2}\epsilon^{ijk}P_{kl}^{T}\omega_{l}\frac{q^{2}-q_{0}^{2}}{q^{2}}+\epsilon^{ijk}\omega_{k}\right),$
$\displaystyle\Pi^{0i<(1)}$ $\displaystyle=-\frac{ie^{2}}{4\pi
q}\chi\left(-\epsilon^{ijk}\hat{q}_{j}\omega_{k}\frac{q_{0}}{q}\right),$ (24)

with $\chi=-\int dkk^{2}\sum_{k_{0}=\pm
k}f^{\prime}(k_{0})=\frac{\pi^{2}T^{2}}{3}+\mu^{2}$. Note that
$\Pi^{\alpha\beta<(1)}\sim O(1/q)$. This is reminiscent of Bose-enhancement in
$\Pi^{\alpha\beta<(0)}$. The counterpart of $\Pi^{\alpha\beta>(1)}$ can be
obtained by the exchange of $>$ and $<$, which leads to
$\Pi^{\alpha\beta>(1)}=-\Pi^{\alpha\beta<(1)}$, and thus
$D_{\nu\mu}^{<(1)}=-D_{\nu\mu}^{>(1)}$.

The properties $D_{\nu\mu}^{<(1)}=-D_{\nu\mu}^{>(1)}$ and $S^{>(1)}=S^{<(1)}$
allow us to simplify the vortical correction to RHS of (3) as

$\displaystyle\frac{i}{2}e^{2}\int_{Q}\big{[}\gamma^{\mu}S^{(1)}(P+Q)\gamma^{\nu}\rho_{\nu\mu}(Q)-\gamma^{\mu}\left(S^{<}(P+Q)+S^{>}(P+Q)\right)\gamma^{\nu}D_{\nu\mu}^{<(1)}\big{]}\Delta
S(X,P).$ (25)

The two terms correspond to vortical correction to final state of the probe
fermion and to initial and final state of the medium fermion respectively. Let
us first show the former contribution vanishes. To see that, we first perform
the contraction of indices to obtain

$\displaystyle\gamma^{\mu}S^{(1)}(P+Q)\gamma^{\nu}\rho_{\nu\mu}(Q)=\delta(2P\cdot
Q)\big{[}-\not{\tilde{P}}\rho_{L}+2\tilde{P}_{0}\gamma^{0}\rho_{L}-\frac{2\tilde{P}_{0}}{q^{2}}(Q^{2}\gamma^{0}-q_{0}\not{Q})\rho_{T}-\frac{2\tilde{P}\cdot
Q}{q^{2}}\left(\not{Q}-q_{0}\gamma^{0}\right)\rho_{T}\big{]}.$ (26)

To proceed, we perform partial angular integration as

$\displaystyle\int_{Q}\delta(2P\cdot Q)=\int\frac{dq_{0}q^{2}dqd\cos\theta
d\phi}{(2\pi)^{4}}\delta(2p_{0}q_{0}-2pq\cos\theta)$
$\displaystyle=\int\frac{dq_{0}dqd\phi}{(2\pi)^{4}}\frac{q}{2p}.$ (27)

The terms proportional to $\rho_{L}$ in (26) by integration of $q_{0}$ because
$\rho_{L}(q_{0})$ is an odd function. The term proportional to $\gamma^{0}$ in
the first bracket of $\rho_{T}$ vanishes for the same reason. For the second
term in (26), we expand $\not{Q}$ as

$\displaystyle\not{Q}=\gamma^{0}q_{0}-\gamma_{\parallel}q_{\parallel}-\vec{\gamma}_{\perp}\cdot\vec{q}_{\perp}=\gamma^{0}q_{0}-\gamma_{\parallel}q_{0}p_{0}/p-\vec{\gamma}_{\perp}\cdot{\vec{q}}_{\perp}.$
(28)

Combining with the $q_{0}$ outside, we find the integrand is either odd in
$q_{0}$ or in $\vec{q}_{\perp}$, which vanishes upon integration of $q_{0}$ or
$\phi$. The remaining second bracket vanishes for similar reasons. Therefore
vortical correction to final state of the probe fermion vanishes. The
vanishing of this contribution is a consequence of the special kinematics
$Q\ll P$, in which case the momenta of both probe fermion and medium fermion
are almost unchanged in Coulomb scattering. It follows that in the absence of
vortical correction, the spin angular momentum is unchanged for probe and
medium fermions. The vortical correction to the final state of probe fermion
modifies the spin, thus is not allowed by angular mmentum conservation.

Now we move on to vortical correction to medium fermions, which is not
forbidden by the angular momentum conservation because vortical correction can
enter both the initial and final states of medium fermions. Using (III) and
the following representation for $D^{R}_{\mu\nu}$ and $D^{A}_{\mu\nu}$ in
Coulomb gauge:

$\displaystyle
D^{R}_{\mu\nu}=u_{\mu}u_{\nu}\Delta_{L}+P^{T}_{\mu\nu}\Delta_{T},\quad
D^{A}_{\mu\nu}=D^{R}_{\mu\nu}{}^{*},$ (29)

we obtain the following components of photon correlator relevant in our case

$\displaystyle P_{ik}^{T}\Pi^{kl<(1)}P_{lj}^{T}=\frac{-ie^{2}}{4\pi
q}\chi\big{[}\left(1-\frac{q_{0}^{2}}{q^{2}}\right)P_{im}^{T}P_{jn}^{T}\epsilon^{mnk}\omega_{k}\big{]},$
$\displaystyle\Pi^{0l<(1)}P_{li}^{T}=\frac{-ie^{2}}{4\pi
q}\chi\left(-\epsilon^{ijk}\hat{q}_{j}\omega_{k}\frac{q_{0}}{q}\right).$ (30)

After completing angular integration in a similar way as above, we obtain

$\displaystyle\frac{i}{2}e^{2}\int_{Q}\big{[}-\gamma^{\mu}\left(S^{<}(P+Q)+S^{>}(P+Q)\right)\gamma^{\nu}D_{\nu\mu}^{<(1)}\big{]}\Delta
S(X,P)$
$\displaystyle\simeq\frac{ie^{4}}{4p}\epsilon(p_{0})(2f(p_{0})-1)\chi\int\frac{dq_{0}dq}{(2\pi)^{3}}\big{[}\left(\gamma^{i}\gamma^{5}p_{0}-\gamma^{0}\gamma^{5}p_{i}\right)\frac{q_{0}^{2}-q^{2}}{q^{2}}\left(\frac{q_{0}^{2}}{q^{2}}\omega_{i}^{\parallel}+\frac{q^{2}-q_{0}^{2}}{2q^{2}}\omega_{i}^{\perp}\right)|\Delta_{T}|^{2}$
$\displaystyle+\gamma^{i}\gamma^{5}\frac{q_{0}^{2}}{q^{2}}\left(p\omega_{i}-p_{i}\omega_{\parallel}\right)\left(\Delta_{T}\Delta_{L}{}^{*}+\Delta_{L}\Delta_{T}{}^{*}\right)\big{]}.$
(31)

We are only interested in the leading divergence from the magnetic
interaction, i.e. the $|\Delta_{T}|^{2}$ term. Further noting the divergence
comes from the regime $q_{0}\ll q$, we can use (III) and
$\vec{\omega}^{\perp}\cdot\vec{p}=0$ to simplify (III) as

$\displaystyle\frac{ie^{4}}{8p}\epsilon(p_{0})(2f(p_{0})-1)\chi\int\frac{dq_{0}dq}{(2\pi)^{3}}\gamma^{i}\gamma^{5}p_{0}\omega_{i}^{\perp}|\Delta_{T}|^{2}\Delta
S$
$\displaystyle=\frac{ie^{4}}{8p}\epsilon(p_{0})(2f(p_{0})-1)\frac{1}{(2\pi)^{3}}\frac{4\chi}{m_{D}^{2}}\ln\frac{m_{D}}{q_{\text{IR}}}\gamma^{i}\gamma^{5}p_{0}\omega_{i}^{\perp}\Delta
S$
$\displaystyle\equiv-i\Gamma_{1}\gamma^{k}\gamma^{5}p_{0}\omega_{k}^{\perp}\Delta
S.$ (32)

We have used the same IR cutoff $q_{\text{IR}}$ as in (19). Using
$m_{D}^{2}=\frac{e^{2}}{\pi^{2}}\chi$ for QED, we obtain

$\displaystyle\Gamma_{1}=-\frac{e^{2}}{8p}\epsilon(p_{0})(2f(p_{0})-1)\frac{1}{2\pi}\ln\frac{m_{D}}{q_{\text{IR}}}.$
(33)

Note that at $\mu=0$, $\Gamma_{1}>0$ and is invariant under $p_{0}\to-p_{0}$.
This is consistent with the charge conjugation symmetry in neutral plasma.

Now we are ready to solve the full kinetic equation with the RHS given by the
sum of (19) and (III)

$\displaystyle\frac{i}{2}\gamma^{0}\partial_{t}\Delta
S+\frac{i}{2}\gamma^{i}\partial_{i}\Delta S+\not{P}\Delta
S=-i\gamma^{0}\Gamma_{0}\Delta
S-i\Gamma_{1}\omega_{i}^{\perp}\gamma^{i}\gamma^{5}p_{0}\Delta S.$ (34)

We have splitted $\not{\partial}$ into temporal and spatial parts, with
vortical correction to the LHS entering only through the spatial parts. In the
absence of vortical correction, (34) reduces to

$\displaystyle\frac{i}{2}\gamma^{0}\partial_{t}\Delta S+\not{P}\Delta
S=-i\gamma^{0}\Gamma_{0}\Delta S.$ (35)

It adopts the following solution

$\displaystyle\Delta
S_{0}=e^{-2\Gamma_{0}t}(-2\pi)\left(f_{V}\not{P}+f_{A}\gamma^{5}\not{P}\right)\epsilon(p_{0})\delta(P^{2}).$
(36)

Here $f_{V}$ and $f_{A}$ are distribution function for vector and axial
components of the probe fermions, to be distinguished from distribution
function $f$ for the medium fermions. In fact, $f_{V}$ and $f_{A}$ are allowed
to be aribitrary functions of momenta at this stage. The exponential factor
$e^{-2\Gamma_{0}t}$ indicates damping of probe fermion due to collision with
medium fermions. The vortical correction to $\Delta S$ can be splitted into
two parts $\Delta S_{1}$ and $\Delta S_{2}$, which satisfy respectively

$\displaystyle\frac{i}{2}\gamma^{0}\partial_{t}\Delta
S_{1}+\frac{i}{2}\gamma^{i}\partial_{i}\Delta S_{0}+\not{P}\Delta
S_{1}=-i\gamma^{0}\Gamma_{0}\Delta S_{1},$
$\displaystyle\frac{i}{2}\gamma^{0}\partial_{t}\Delta S_{2}+\not{P}\Delta
S_{2}=-i\gamma^{0}\Gamma_{0}\Delta
S_{2}-i\Gamma_{1}\omega_{i}^{\perp}\gamma^{i}\gamma^{5}p_{0}\Delta S_{0}.$
(37)

$\Delta S_{1}$ and $\Delta S_{2}$ arise due to spin-vorticity coupling and
vortical correction to collision term respectively. The solution to the first
equation of (III) is a simple generalization of (8)

$\displaystyle\Delta
S_{1}=e^{-2\Gamma_{0}t}(-2\pi)\left(\frac{1}{2}f_{V}^{\prime}\not{\tilde{P}}\gamma^{5}-\frac{1}{2}f_{A}^{\prime}\not{\tilde{P}}\right)\epsilon(p_{0})\delta(P^{2}).$
(38)

Note that the above solution arise from spatial derivative on $\Delta S_{0}$.
Its validity relies on $f_{V}$ and $f_{A}$ being functions of $P\cdot u$, i.e.
in local equilbirum. The second equation of (III) adopts the following
solution

$\displaystyle\Delta
S_{2}=e^{-2\Gamma_{0}t}(-2\pi)\left(\not{V}+\gamma^{5}\not{A}\right)\epsilon(p_{0})\delta(P^{2}),$
(39)

with

$\displaystyle
V^{\mu}=-\Gamma_{1}\epsilon^{\mu\nu\rho\sigma}\omega_{\nu}^{\perp}P_{\rho}u_{\sigma}f_{V},\;\;A^{\mu}=-\Gamma_{1}\epsilon^{\mu\nu\rho\sigma}\omega_{\nu}^{\perp}P_{\rho}u_{\sigma}f_{A}.$
(40)

Since the equation involves no spatial derivative on $\Delta S_{0}$, the
validity of (39) only requires medium fermions to be in local equilibrium. The
full solution is given by

$\displaystyle\Delta
S=e^{-2\Gamma_{0}t}(-2\pi)\epsilon(p_{0})\delta(P^{2})\big{[}$ $\displaystyle
f_{V}\not{P}-f_{A}^{\prime}\not{\tilde{P}}-f_{V}\Gamma_{1}\epsilon_{\mu\nu\rho\sigma}\gamma^{\mu}\omega_{\perp}^{\nu}P^{\rho}u^{\sigma}$
$\displaystyle+$ $\displaystyle
f_{A}\gamma^{5}\not{P}-f_{V}^{\prime}\gamma^{5}\not{\tilde{P}}-f_{A}\Gamma_{1}\epsilon_{\mu\nu\rho\sigma}\gamma^{5}\gamma^{\mu}\omega_{\perp}^{\nu}P^{\rho}u^{\sigma}\big{]}.$
(41)

The corresponding spin polarization is given by

$\displaystyle{\cal P}^{i}=e^{-2\Gamma_{0}t}\sum_{p_{0}=\pm
p}\big{[}f_{A}\frac{2\epsilon(p_{0})p_{i}}{p}-f_{V}^{\prime}\omega_{i}+f_{A}\Gamma_{1}\frac{2\epsilon(p_{0})\epsilon^{ijk}\omega_{j}p_{k}}{p}\big{]}.$
(42)

(42) has the expected structure in (1). Note that $f_{V}$ and $f_{A}$ are
scalar and pseudoscalar functions respectively, which is consistent with the
parity of $A_{1}$, $A_{2}$ and $A_{3}$ in (1). The pseudoscalar function
$f_{A}$ implies that the new contribution is opposite for left and right-
handed fermions.

##  IV Generalization to QCD plasma

The polarization rotation effect can be generalized to QCD plasma by similar
derivations. The new ingredients to take into account are the color factors
and scattering processes not present in QED. The color structure modifies the
KB equation only marginally: both the quark and gluon propagators are diagonal
in color space:

$\displaystyle S^{<}_{IJ}\propto\delta_{IJ},\quad
D_{\mu\nu}^{ab<}\propto\delta_{ab},$ (43)

with $I,J$ and $a,b$ being indices in fundamental and adjoint representations
of color group. The color sum in the quark self-energy reads
$\sum_{a}t_{IJ}^{a}t_{KI}^{a}=C_{F}\delta_{JK}=\frac{N_{c}^{2}-1}{2N_{c}}\delta_{JK}$.
This amounts to the replacement $e^{2}\to g^{2}C_{F}$ in (II) for quark self-
energy.

The gluon self-energy contains both quark loop and gluon loop contributions.
The former involves a replacement $e^{2}\to g^{2}N_{f}/2$ in (12) for $N_{f}$
flavor of quarks. The gluon loop contribution has no analog in QED. For the
self-energy $\Pi^{</>}$, the four-gluon vertex is excluded. A further
simplification can be made by noting that in Coulomb gauge contribution from
unphysical polarizations and ghosts cancel each other in self-energy in
equilirium. We assume it is still true for the off-equilibrium self-energy in
the vortical plasma, hence we keep only transverse polarizations of gluons in
the loop.

To proceed, we need the off-equilbrium propagator of gluon in vortical plasma.
This is provided by solution of KB equation for gluons in Coulomb gauge
Blaizot and Iancu (2002)

$\displaystyle\big{[}\left(-i\partial\cdot
P-P^{2}\right)g^{\mu\nu}-P^{\mu}P^{\nu}+\frac{i}{2}\left(\partial^{\mu}P^{\nu}+\partial^{\nu}P^{\mu}\right)\big{]}D^{ab<}_{\nu\rho}=0,$
(44)

with the Coulomb gauge condition

$\displaystyle
P^{\mu\alpha}\left(\frac{1}{2}\partial_{\alpha}-iP_{\alpha}\right)D^{ab<}_{\mu\nu}=0.$
(45)

Here $P^{\mu\nu}=-g^{\mu\nu}+u^{\mu}u^{\nu}$. The zeroth order solution to
(44) is simply the transversely polarized gluon propagator

$\displaystyle D^{ab<(0)}_{\nu\rho}(X,P)=2\pi
P^{T}_{\nu\rho}\delta(P^{2})\epsilon(P\cdot u)f_{b}(P\cdot u)\delta^{ab},$
(46)

with $f_{b}$ being the Bose-Einstein distribution function. The first order
correction is given by Huang  _et al._ (2020); Hattori  _et al._ (2021)

$\displaystyle
D^{ab<(1)}_{\mu\rho}(X,P)=-i(2\pi)P_{\mu\lambda}P_{\rho\eta}\frac{P^{\lambda}\partial^{\eta}-P^{\eta}\partial^{\lambda}}{2(P\cdot
u)^{2}}\epsilon(P\cdot u)f_{b}(P\cdot u)\delta(P^{2})\delta^{ab}.$ (47)

They give the following correction to the gluon self-energy through the gluon
loop

$\displaystyle\Pi^{ab,\alpha\beta<(1)}(Q)=-\frac{1}{2}g^{2}\int_{K}\left(D_{\mu\rho}^{ce<(1)}(K+Q)D^{df>(0)}_{\sigma\nu}(K)+D_{\mu\rho}^{ce<(0)}(K+Q)D^{df>(1)}_{\sigma\nu}(K)\right)$
$\displaystyle\times
f^{acd}[g^{\mu\alpha}(-K-2Q)^{\nu}+g^{\alpha\nu}(Q-K)^{\mu}+g^{\nu\mu}(2K+Q)^{\alpha}]f^{bef}[g^{\rho\beta}(K+2Q)^{\sigma}+g^{\beta\sigma}(K-Q)^{\rho}+g^{\sigma\rho}(-2K-Q)^{\beta}].$
(48)

Let us focus on the contribution from the term proportional to
$D_{\mu\rho}^{ce<(1)}D^{df>(0)}_{\sigma\nu}$ for the moment. We can use
$D^{ce>(0)}_{\sigma\nu}\propto P^{T}_{\sigma\nu}$ and transverse property of
$P_{\sigma\nu}^{T}$ to simplify the vertices. As before we only keep $Q$ in
$\delta((K+Q)^{2})\simeq\delta(2K\cdot Q)$ to arrive at

$\displaystyle\Pi^{ab,\alpha\beta<(1)}_{10}(Q)$
$\displaystyle=-\frac{1}{2}g^{2}\int_{K}D_{\mu\rho}^{ce<(1)}(K+Q)D^{df>(0)}_{\sigma\nu}(K)f^{acd}[-g^{\alpha\nu}K^{\mu}+2g^{\nu\mu}K^{\alpha}]f^{bef}[g^{\beta\sigma}K^{\rho}-2g^{\sigma\rho}K^{\beta}].$
(49)

The subscript $10$ indicates it from the term
$D_{\mu\rho}^{ce<(1)}D^{df>(0)}_{\sigma\nu}$. The color factor is evaluated
using $f^{acd}f^{bef}\delta^{ce}\delta^{df}=N_{c}\delta^{ab}$. The contraction
of Lorentz indices reads

$\displaystyle
P_{\sigma\nu}^{T}P_{\mu\lambda}P_{\rho\eta}[-g^{\alpha\nu}K^{\mu}+2g^{\nu\mu}K^{\alpha}][g^{\beta\sigma}(K)^{\rho}-2g^{\sigma\rho}K^{\beta}]$
$\displaystyle=$
$\displaystyle-P^{T,\alpha\beta}\bar{K}_{\lambda}\bar{K}_{\eta}-4P^{T}_{\lambda\eta}K^{\alpha}K^{\beta}+2P^{T,\beta}_{\lambda}K^{\alpha}\bar{K}_{\eta}+2P^{T,\alpha}_{\eta}K^{\beta}\bar{K}_{\lambda},$
(50)

with $\bar{K}_{\mu}=-P_{\mu\nu}K^{\nu}$. The first two terms vanish upon
contracting with $K^{\lambda}\partial^{\eta}-K^{\eta}\partial^{\lambda}$ in
$D_{\mu\rho}^{ce<(1)}(K+Q)\simeq D_{\mu\rho}^{ce<(1)}(K)$. The last two terms
give

$\displaystyle\Pi^{ab,\alpha\beta<(1)}_{10}(Q)$
$\displaystyle=-\frac{i(2\pi)^{2}}{2}g^{2}N_{c}\delta_{ab}\int_{K}[-2K_{T}^{\beta}K^{\alpha}\bar{K}\cdot\partial
f_{b}(K\cdot u)+2\bar{K}^{2}K^{\beta}\partial_{T}^{\alpha}f_{b}(K\cdot
u)-(\alpha\leftrightarrow\beta)]\frac{1+f_{b}(K\cdot u)}{2(K\cdot
u)^{2}}\delta(K^{2})\delta(2K\cdot Q)$
$\displaystyle=-\frac{i(2\pi)^{2}}{2}g^{2}N_{c}\delta_{ab}\int_{K}[2\bar{K}^{2}K^{\beta}\partial_{T}^{\alpha}f_{b}(K\cdot
u)-(\alpha\leftrightarrow\beta)]\frac{1+f_{b}(K\cdot u)}{2(K\cdot
u)^{2}}\delta(K^{2})\delta(2K\cdot Q),$ (51)

where we have defined $K_{T}^{\mu}=-P_{T}^{\mu\nu}\bar{K}_{\nu}$ and
$\partial_{T}^{\mu}=-P_{T}^{\mu\nu}\partial_{\nu}$. In the second line we have
used $\bar{K}\cdot\partial
f_{b}\propto\bar{K}^{\mu}K^{\nu}\partial_{\mu}u_{\nu}=0$ for fluid with
vorticity only.

The evaluation of the term proportional to
$D_{\mu\rho}^{ce<(0)}D^{df>(1)}_{\sigma\nu}$ is similar by noting that
$D^{df>(1)}_{\sigma\nu}=D^{df<(1)}_{\sigma\nu}$. It gives a contribution
identical to (IV) except for the replacement $1+f_{b}(K\cdot u)\to-
f_{b}(K\cdot u)$. Hence we obtain for their sum

$\displaystyle\Pi^{ab,\alpha\beta<(1)}(Q)$
$\displaystyle=-\frac{i(2\pi)^{2}}{2}g^{2}N_{c}\delta_{ab}\int_{K}[2\bar{K}^{2}K^{\beta}\partial_{T}^{\alpha}f_{b}(K\cdot
u)-(\alpha\leftrightarrow\beta)]\frac{1}{2(K\cdot
u)^{2}}\delta(K^{2})\delta(2K\cdot Q).$ (52)

The vortical correction to the gluon self-energy from gluon loop is manifestly
anti-symmetric in Lorentz indices as in quark loop counterpart. The explicit
components are given by

$\displaystyle\Pi^{ab,0i<(1)}$
$\displaystyle=-\frac{i(2\pi)^{2}g^{2}N_{c}\delta_{ab}}{2}\int_{K}k^{2}\frac{q_{0}}{q}P_{T}^{im}\epsilon^{mnk}\hat{q}_{n}\omega_{k}f_{b}^{\prime}(k_{0})\delta(K^{2})\delta(2K\cdot
Q),$ $\displaystyle\Pi^{ab,ij<(1)}$
$\displaystyle=\frac{-i(2\pi)^{2}g^{2}N_{c}\delta_{ab}}{2}\int_{K}\big{[}k^{2}\hat{q}_{j}P_{T}^{im}\epsilon^{mnk}\hat{q}_{n}\omega_{k}\frac{q_{0}^{2}}{q^{2}}+P_{T}^{im}P_{T}^{jn}\epsilon^{mnk}\omega_{k}\frac{k^{2}}{2}\left(1-\frac{q_{0}^{2}}{q^{2}}\right)\big{]}f_{b}^{\prime}(k_{0})-(i\leftrightarrow
j).$ (53)

Similar to the fermion loop case, only $\Pi^{ab,ij<(1)}$ enters the leading
divergence we are after. Completing the momenta integral, we obtain the
following combination

$\displaystyle
P^{T}_{im}P^{T}_{nj}\Pi^{ab,mn<(1)}=-\frac{ig^{2}N_{c}\delta_{ab}}{8\pi
q}\chi_{0}\left(1-\frac{q_{0}^{2}}{q^{2}}\right)P_{T}^{im}P_{T}^{jn}\epsilon^{mnk}\omega_{k},$
(54)

where $\chi_{0}=\frac{\pi^{2}T^{2}}{3}=\chi(\mu=0)$. Comparing this with
(III), we find it is obtainable from QED case with $e^{2}\to g^{2}N_{c}/2$ and
$\chi\to\chi_{0}$.

Now we are ready to combine the contribution from quark and gluon loops to
give the following $\Gamma_{1}$ for QCD

$\displaystyle\Gamma_{1}=-g^{4}C_{F}\frac{\epsilon(p_{0})(2f(p_{0})-1)}{8p}\frac{1}{(2\pi)^{3}}\frac{2(N_{c}\chi_{0}+N_{f}\chi)}{m_{D}^{2}}\ln\frac{m_{D}}{q_{\text{IR}}}.$
(55)

For QCD, the Debye mass is given by

$\displaystyle m_{D}^{2}=\frac{N_{c}\chi_{0}+N_{f}\chi/2}{\pi^{2}},$ (56)

which allows us to express (55) in a more suggestive form

$\displaystyle\Gamma_{1}=-g^{2}C_{F}\frac{\epsilon(p_{0})(2f(p_{0})-1)}{8p}\frac{1}{4\pi}\frac{N_{c}\chi_{0}+N_{f}\chi}{N_{c}\chi_{0}+N_{f}\chi/2}\ln\frac{m_{D}}{q_{\text{IR}}}.$
(57)

Note that the color and flavor factors do not cancel out. This has a simple
explanation that the vorticity couples to quarks and gluons differently.

Let us apply the spin polarization (42) to early stage heavy ion collisions,
where a significant vorticity is formed Adamczyk  _et al._ (2017). If the
system is isotropic, (42) vanishes upon momentum integration. However for
anisotropic system, which is also realizable at early stage of heavy ion
collisions Kovchegov (2009), it could lead to an observable effect. As we
emphasized above, $f_{V/A}$ can be arbitrary functions of momenta in (39).
This could then lead to net current from the probe fermion with momentum
anisotropy. For example, in a system with $f_{A}=0$ and anisotropic $f_{V}$, a
vector current can be generated from the new contribution in $\Delta S$ as:

$\displaystyle\Delta J_{V}^{i}=\int_{P}\text{tr}[\gamma^{i}\Delta
S]\sim\epsilon^{ijk}\xi_{j}\omega_{k},$ (58)

with $\xi$ being the weighted average of momentum. A charge asymmetry for the
probe quarks is needed such that the quark contribution to the current is not
entirely canceled by the anti-quark counterpart.

##  V Conclusions and Outlook

We have found a new contribution to spin polarization due to interaction of
chiral fermion with medium polarized by fluid vorticity. This is analogous to
rotation of polarization in light interacting with polarized medium. We have
considered the leading IR divergent part to the new contribution coming from
Coulomb scattering between probe fermions and medium particles. The effect
could arise due to vortical correction of final state of probe fermions or the
counterpart of initial and final state of medium particles. We found the
former contribution vanishes kinematically due to spin angular momentum
conservation. For the latter contribution, the medium particles are fermions
for QED plasma and are quarks and gluons for QCD plasma. The effect are
qualitatively similar for the two cases. The fact that the vorticity couples
differently to quarks and gluons is visible in a non-trivial color and flavor
factor for the QCD case.

The effect we found is opposite for left and right-handed fermions, which
cancels out in unpolarized probe fermions. Observation of this effect might be
possible in polarized probe fermions. It is also desirable to generalize the
current study to the case of massive fermions, where connection to spin
polarization in heavy ion collisions can be made Becattini and Lisa (2020). We
expect the general structure in (1) remains unchanged.

The new contribution can also lead to net current in a system with momentum
anisotropy. The current is perpendicular to both vorticity and axis
characterizing the momentum anisotropy, which can be realized at early stage
of heavy ion colisions. We defer more quantative analysis for future studies.

###### Acknowledgements.

We are grateful to Jianhua Gao, Hai-cang Ren, Qun Wang, Di-Lun Yang, Ho-Ung
Yee, Yi Yin and Pengfei Zhuang for useful discussions. This work is in part
supported by NSFC under Grant Nos 11675274 and 11735007.

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# Long cycles in percolated expanders

Maurício Collares 111Institute of Discrete Mathematics, Graz University of
Technology, Steyrergasse 30, 8010 Graz, Austria. Emails:
<EMAIL_ADDRESS><EMAIL_ADDRESS>Sahar Diskin 222School of
Mathematical Sciences, Tel Aviv University, Tel Aviv 6997801, Israel. Emails:
<EMAIL_ADDRESS><EMAIL_ADDRESS>Joshua Erde
††footnotemark:  Michael Krivelevich ††footnotemark:

###### Abstract

Given a graph $G$ and probability $p$, we form the random subgraph $G_{p}$ by
retaining each edge of $G$ independently with probability $p$. Given
$d\in\mathbb{N}$ and constants $0<c<1,\varepsilon>0$, we show that if every
subset $S\subseteq V(G)$ of size exactly $\frac{c|V(G)|}{d}$ satisfies
$|N(S)|\geq d|S|$ and $p=\frac{1+\varepsilon}{d}$, then the probability that
$G_{p}$ does not contain a cycle of length
$\Omega(\varepsilon^{2}c^{2}|V(G)|)$ is exponentially small in $|V(G)|$. As an
intermediate step, we also show that given $k,d\in\mathbb{N}$ and a constant
$\varepsilon>0$, if every subset $S\subseteq V(G)$ of size exactly $k$
satisfies $|N(S)|\geq kd$ and $p=\frac{1+\varepsilon}{d}$, then the
probability that $G_{p}$ does not contain a path of length
$\Omega(\varepsilon^{2}kd)$ is exponentially small. We further discuss
applications of these results to $K_{s,t}$-free graphs of maximal density.

##  1 Introduction

The binomial random graph $G(n,p)$ can be seen as a model of a random subgraph
of the complete graph $K_{n}$, obtained by retaining each edge of $K_{n}$
independently with probability $p$. A particularly interesting feature of this
model, first observed by Erdős and Rényi [8], is the phase transition that it
undergoes with respect to its component structure when $p$ is around
$\frac{1}{n}$. In [8] they showed that when $p\leq\frac{1-\varepsilon}{n}$,
where $\varepsilon>0$ is some arbitrary constant, then with high probability
(whp) every component of the graph is _small_ , of logarithmic order, whereas
when $p\geq\frac{1+\varepsilon}{n}$, whp there is a unique giant component of
linear order. Later, Ajtai, Komlós, and Szemerédi [1] showed that when
$p\geq\frac{1+\varepsilon}{n}$, whp $G(n,p)$ contains a cycle of length
$\Theta(n)$, settling a long-standing conjecture of Erdős. For more background
on the theory of random graphs, see [5, 9, 11].

More recently, the emergence of similar structures in generalisations of this
model has been studied. Given a host graph $G$, consider the random subgraph
$G_{p}$ obtained by retaining each edge of $G$ independently with probability
$p$. What (minimal) structural assumptions on the host graph $G$ are
sufficient to guarantee the likely emergence of particular structures past
certain “natural” thresholds?

Traditionally, these assumptions have been “local”, in terms of the degree
sequence of the graph, and in particular its minimum degree. Indeed, a
standard coupling argument with a Galton–Watson branching process implies that
for any graph $G$ with minimum degree $\delta(G)\geq d$ and $\varepsilon>0$,
whp $G_{p}$ will contain a component of order $\Omega(d)$ when
$p\geq\frac{1+\varepsilon}{d}$. The existence of long paths in this model and
regime is less obvious. Through an analysis of the Depth-First Search
algorithm, Krivelevich and Sudakov [16] showed that if $\delta(G)\geq d$ and
$\varepsilon>0$, whp $G_{p}$ will contain a path of length $\Omega(d)$ when
$p\geq\frac{1+\varepsilon}{d}$. However, in these general models, where the
girth of $G$ could be much larger than the minimum degree, it is much less
clear how to show the typical existence of a long cycle. Using more delicate
methods, this was eventually proven by Krivelevich and Samotij [14] (see also
[17]), who further conjectured that similar statements should hold when the
bound on the minimum degree is replaced with a bound on the average degree.

Recent work of Diskin, Erde, Krivelevich, and Kang [6] on the connection
between the phase transition in $G_{p}$ and the isoperimetric properties of
the host graph $G$ suggests another way of thinking about the quantitative
aspects of the phase transition.

######  Theorem 1 ([6, Theorem 4]).

Let $k=\omega(1),d\leq k$ and let $G$ be a graph on more than $k$ vertices,
such that every $S\subseteq V(G)$ with $|S|\leq k$ satisfies
$e(S,V(G)\setminus S)\geq d|S|$. Let $\varepsilon>0$ be a small constant and
let $p=\frac{1+\varepsilon}{d}$. Then, with probability tending to $1$ as $k$
tends to infinity, $G_{p}$ contains a component of order at least
$\frac{k}{2}$.

In fact, with slightly more assumptions on the graph, and a slightly weaker
bound on the probability, a quantitatively similar statement to Theorem 1
holds when the “global” assumption of expansion at all small scales is
replaced by the assumption that sets of size _exactly_ $k$ expand well, see
[6, Theorem 3.1].

One can think of Theorem 1 as giving an alternative heuristic for the nature
of the phase transition. Indeed, here the point of criticality is controlled
by the expansion ratio of subsets (that is, $d$), whereas the quantitative
aspects of the component structure above the critical point are controlled by
the _scale_ at which this level of expansion holds (that is, $k$).

Returning to the conjecture of Krivelevich and Samotij [16], let us note that
the assumption that sets of size exactly $k$ expand by a factor of $d$ is, in
a sense, a strengthening of the assumption of having an average degree $d$
(indeed it implies the average degree is at least $d$). Nevertheless, for
large $k$ this assumption is still somewhat weaker than a constraint on the
minimum degree (in particular, it implies that there are at most $k$ vertices
of degree less than $d$). It is thus perhaps tempting to conjecture that the
threshold for the appearance of long paths and cycles might also be controlled
by the expansion ratio of the host graph, with a lower bound on their size
being determined by the scale on which this expansion holds. Perhaps somewhat
surprisingly, as noted in [6, Remark 3.3], such a result cannot hold, even
before percolation. Indeed, if we take our host graph to be a very unbalanced
bipartite graph, for example, $G=K_{d,d^{10}}$, then whilst the graph has an
expansion factor of at least $d$ for every set of size up to $d^{9}$, there is
no path in $G$ of length longer than $2d$, since each path must have half of
its vertices in the smaller partition class.

This suggests that one should perhaps consider the _vertex-expansion_ of the
host graph, instead of the edge-expansion. Indeed, our first result shows the
typical emergence of long paths after percolation under the assumption of
vertex-expansion at a fixed scale.

######  Theorem 2.

Let $k,d\in\mathbb{N}$, and let $G$ be a graph on at least $k$ vertices such
that every $S\subseteq V(G)$ with $|S|=k$ satisfies $|N(S)|\geq kd$. Let
$\varepsilon>0$ be a sufficiently small constant and let
$p=\frac{1+\varepsilon}{d}$. Then $G$ contains a path of length at least
$\frac{\varepsilon^{2}kd}{10}$ with probability at least
$1-\exp\left(-\Omega_{\varepsilon}\left(kd\right)\right)$.

Note that for $k=1$ and $G=K_{d+1}$, the above recovers the result of Ajtai,
Komlós, and Szemerédi [1]. In fact, when $G$ is a clique of size $k(d+1)$, the
above can be seen to be tight up to a multiplicative factor in the bound of
the length of the path.

However, as is the case with arbitrary host graphs of large minimum degree, it
is not immediately obvious how this result can be strengthened to find a long
cycle. In the deterministic setting, that is, when considering a graph $G$ on
at least $k$ vertices such that every $S\subseteq V(G)$ with $|S|=k$ satisfies
$|N(S)|\geq kd$, it is known that $G$ contains a cycle of length $\Omega(kd)$
[13]. In the specific case where $kd$ is linear in $n$, our main result shows
that this holds after percolation as well.

######  Theorem 3.

Let $0<c<1$, $d\in\mathbb{N}$, and let $G$ be a graph on $n$ vertices such
that every $S\subseteq V(G)$ with $|S|=\frac{cn}{d}$ satisfies $|N(S)|\geq
d|S|$. Let $\varepsilon\coloneqq\varepsilon(c)>0$ be a sufficiently small
constant and let $p=\frac{1+\varepsilon}{d}$. Then $G_{p}$ contains a cycle of
order $\Omega\left(\varepsilon^{2}c^{2}n\right)$ with probability at least
$1-\exp\left(-\Omega_{\varepsilon,c}\left(n\right)\right)$.

As an application, we consider “optimal” $K_{s,t}$-free graphs, that is,
graphs not containing a copy of $K_{s,t}$ (for $2\leq s\leq t$) and having
maximal density. Tightly related to the Zarankiewicz problem, it is known that
the maximal number of edges in an $n$-vertex $K_{s,t}$-free graph is
$O(n^{2-\frac{1}{s}})$, and for $t$ sufficiently larger than $s$ (as well as
for some fixed small values of $s$ and $t$), this has been established as the
correct order of magnitude (see, for example, [3] and references therein).
When a graph $G$ is an optimal $K_{s,t}$-free graph, it has sufficiently good
vertex expansion properties [15], which allows one to derive that it has a
cycle of length linear in $|V(G)|$ (utilising the connections between local
expansion and long cycles as in [13]).

Let us show that in fact one can apply Theorem 3 to optimal $K_{s,t}$-free
graphs. Indeed, let $\alpha$ be a positive constant, and let $G=(V,E)$ be a
$K_{s,t}$-free graph with $|E|\geq\alpha n^{2-\frac{1}{s}}$. By a standard
argument, there is a subgraph $G_{0}\subseteq G$ with minimum degree
$\delta(G_{0})\geq\alpha n^{1-\frac{1}{s}}$, and we note that
$|V(G_{0})|=\Theta(n)$ (indeed, otherwise $G_{0}$ will be too dense,
contradicting it being $K_{s,t}$-free). Recalling that any $K_{s,t}$-free
graph $H$ has $O\left(|V(H)|^{2-\frac{1}{s}}\right)$ edges, for every
$X\subseteq V(G_{0})$ with $|X|=n^{1/s}$, we have $e(X,V\setminus
X)\geq\frac{\alpha}{2}n=\frac{\alpha n^{1-1/s}}{2}|X|$. Hence, by [15, Lemma
7.2], for every $X\subseteq V(G_{0})$ with $|X|=n^{1/s}$ we have
$|N(X)|\geq\frac{\alpha n^{1-1/s}}{2t}|X|=\frac{\alpha}{2t}\cdot n$. We may
thus apply Theorem 3, and obtain the following corollary:

######  Corollary 4.

Let $\alpha$ be a positive constant, and let $s,t\in\mathbb{N}$ with $2\leq
s\leq t$. Let $G$ be a $K_{s,t}$-free graph on $n$ vertices satisfying
$|E(G)|\geq\alpha n^{2-\frac{1}{s}}$. Then, there exists a sufficiently large
constant $K\coloneqq K(\alpha,t)$ such that for any
$p\geq\frac{K}{n^{1-1/s}}$, with high probability $G_{p}$ contains a cycle of
length $\Omega(n)$.

The paper is structured as follows. In Section 2 we collect several
definitions and results which will be useful throughout the paper. In Section
3 we prove Theorem 2, and in Section 4 we prove Theorem 3. Finally, in Section
5 we discuss our results and present avenues for future research.

##  2 Preliminaries

###  2.1 Notation

Given subsets $A,B\subseteq V(G)$ with $A\cap B=\varnothing$ and a subgraph
$H\subseteq G$, we denote by $e_{H}(A,B)$ the number of edges in $H$ with one
endpoint in $A$ and the other endpoint in $B$. When $H=G$, we may omit the
subscript. Furthermore, we denote by $N(A)$ the external neighbourhood of $A$
in $G$, that is, $N(A)\coloneqq\\{v\in V(G)\setminus A\colon\exists u\in
A,\\{v,u\\}\in E(G)\\}$. For the sake of clarity we omit all rounding signs.

Given $I\subseteq\mathbb{N}$, we say that a graph $G$ is an $(I,d)$-expander
if for every set $X$ with $|X|\in I$, it holds that $|N(X)|\geq d|X|$. If $G$
is a $(\\{k\\},d)$-expander we may also simply say $G$ is a $(k,d)$-expander.
We refer the reader to the surveys [10, 12] for a comprehensive study of
expander graphs and their applications.

###  2.2 Depth-First Search (DFS)

The Depth-First Search (DFS) algorithm explores the components of a graph
using a “last-in-first-out” discipline. The algorithm receives as input a
graph $G=(V,E)$ and an ordering $\sigma$ of the vertex set $V$. During the
algorithm, we maintain three sets of vertices: $W$, the set of _processed_
vertices; $A$, the set of _active_ vertices, which we treat as a stack; and
$U$, the set of _unvisited_ vertices.

We initialise $W=A=\emptyset$ and $U=V$. At each step, if the stack $A$ is
non-empty, then we consider the most recently added $a\in A$, and query the
edges from $a$ to $U$, according to the order $\sigma$, until a vertex $u\in
U$ is discovered with $(a,u)\in E(G)$. If no vertex is found, $a$ is moved
from the stack $A$ to $W$. Otherwise, we add the newly discovered vertex $u$
to the top of the stack $A$. If the stack $A$ is empty, we move the first
vertex in $U$, according to $\sigma$, into $A$.

We note some elementary facts about this process:

  1. (P0)

at each stage of the algorithm the stack $A$ spans a path in $G$; and,

  2. (P0)

at each stage of the algorithm there are no edges in $G$ between $U$ and $W$.

To analyse the DFS algorithm on a percolated graph $G_{p}$ we will utilise the
_principle of deferred decisions_. That is, we will sample a sequence
$(X_{i})_{i=1}^{|E(G)|}$ of i.i.d Bernoulli$(p)$ random variables, which we
can think of as representing a positive (with probability $p$) or negative
(with probability $1-p$) answer to a query in the algorithm. When the
algorithm makes the $i$-th query (about an edge of $E(G)$), it receives a
positive answer if and only if $X_{i}=1$. Let us denote by $W(i),A(i),U(i)$
the sets of vertices in $W,A,U$, respectively, after the $i$-th query. To
complete the exploration of the graph, once $U$ is empty we make the algorithm
query all the remaining edges in $G$ not queried before. It is then clear that
the obtained graph has the same distribution as $G_{p}$.

###  2.3 Concentration inequalities

We will make use of a typical Chernoff-type tail bound on the binomial
distribution (see, for example, [4, Appendix A]).

######  Lemma 5.

Let $n\in\mathbb{N}$, let $p\in[0,1]$ and let $X\sim\text{Bin}(n,p)$. Then,
for every $0\leq t\leq\frac{np}{2}$,

$\mathbb{P}\left[\left|X-np\right|>t\right]<2\exp\left(-\frac{t^{2}}{3np}\right).$

##  3 Long paths: Proof of Theorem 2

As discussed in the introduction, an assumption on edge-expansion alone does
not suffice to prove Theorem 2. Nevertheless, the proof here is heavily
inspired by that of [16].

We explore $G_{p}$ via the DFS algorithm described in Section 2.2 until it
performs $N_{1}=\frac{\varepsilon kd^{2}}{2(1+\varepsilon)}$ queries (indeed,
$|E(G)|\geq kd^{2}/2>N_{1}$). The expected number of positive queries by this
point is $\frac{\varepsilon kd}{2}$, and hence by Lemma 5 there have been at
least $\left(1-\frac{\varepsilon}{5}\right)\frac{\varepsilon kd}{2}$ positive
queries with probability at least
$1-\exp\left(-\Omega\left(\varepsilon^{3}kd\right)\right)$. We will assume in
what follows that this occurs deterministically.

Since either $U(N_{1})=\emptyset$ (and $|V(G)|\geq kd$) or each positive query
corresponds to a vertex which moves from $U$ to $A$ (which may later move to
$W$), it follows that $|A(N_{1})\cup
W(N_{1})|\geq\left(1-\frac{\varepsilon}{5}\right)\frac{\varepsilon kd}{2}$.
Let $N_{2}$ be the first time such that

$|A(N_{2})\cup W(N_{2})|=\left(1-\frac{\varepsilon}{5}\right)\frac{\varepsilon
kd}{2},$ (1)

noting that $N_{2}\leq N_{1}$. Suppose towards a contradiction that $G_{p}$
does not contain a path of length $\frac{\varepsilon^{2}kd}{10}$. By 1,
$|A(N_{2})|\leq\frac{\varepsilon^{2}kd}{10}$, and therefore by (1),
$|W(N_{2})|\geq\left(1-\frac{2\varepsilon}{5}\right)\frac{\varepsilon kd}{2}$.
Let us fix disjoint sets $W_{1},W_{2},\ldots,W_{r}\subseteq W(N_{2})$ of size
$k$, where $r\geq\left(1-\frac{2\varepsilon}{5}\right)\frac{\varepsilon
d}{2}$. Since $G$ is a $(k,d)$-expander, $|N(W_{i})|\geq kd$ for each
$i\in[r]$, and so by (1)

$e(W_{i},U(N_{2}))\geq|N(W_{i})|-|A(N_{2})\cup W(N_{2})|\geq kd-|A(N_{2})\cup
W(N_{2})|=\left(1-\frac{\varepsilon}{2}+\frac{\varepsilon^{2}}{10}\right)kd.$

Since the sets $W_{i}$ are disjoint, it follows that

$e(W(N_{2}),U(N_{2}))\geq
r\left(1-\frac{\varepsilon}{2}+\frac{\varepsilon^{2}}{10}\right)kd\geq\left(1-\frac{\varepsilon}{2}\right)\left(1-\frac{2\varepsilon}{5}\right)\frac{\varepsilon
kd^{2}}{2}$

However, by 2 every edge between $U(N_{2})$ and $W(N_{2})$ has already been
queried and hence

$\left(1-\frac{\varepsilon}{2}\right)\left(1-\frac{2\varepsilon}{5}\right)\frac{\varepsilon
kd^{2}}{2}\leq e(W(N_{2}),U(N_{2}))\leq N_{2}\leq N_{1}=\frac{\varepsilon
kd^{2}}{2(1+\varepsilon)},$

or equivalently
$(1+\varepsilon)\left(1-\frac{\varepsilon}{2}\right)\left(1-\frac{2\varepsilon}{5}\right)\leq
1,$ which is a contradiction for $\varepsilon$ sufficiently small. ∎

##  4 Long cycles: Proof of Theorem 3

The proof will proceed via a two-round exposure process, using
$p_{1}=\frac{1+\frac{\varepsilon}{2}}{d}$ and
$p_{2}=\frac{p-p_{1}}{1-p_{1}}\geq\frac{\varepsilon}{2d}$, noting that $G_{p}$
has the same distribution as $G_{p_{1}}\cup G_{p_{2}}$, since
$1-p=(1-p_{1})(1-p_{2})$.

Let
$\alpha\coloneqq\frac{c(\varepsilon/2)^{2}}{10}=\frac{c\varepsilon^{2}}{40}$.
By Theorem 2, $G_{p_{1}}$ contains a path $P$ of length $\alpha n$ with
probability at least $1-\exp\left(-\Omega_{\varepsilon}\left(n\right)\right)$.
We continue assuming that this holds deterministically. Let us fix
$r\coloneqq\frac{\alpha d}{c}$ disjoint subpaths $P_{1}\cup\ldots\cup P_{r}$
of $P$, which we call _blocks_ , each of length $\frac{cn}{d}$. We may assume
without loss of generality that these blocks are ordered according to the
order they appear in the path $P$.

For each $v\in V(G)\setminus V(P)$, let $B_{v}=\\{i\in[r]\colon N(v)\cap
V(P_{i})\neq\varnothing\\}$ be set of indices of blocks to which $v$ is
adjacent (in $G$). We call $v$ _good_ if
$b_{v}\coloneqq|B_{v}|\geq\frac{\alpha d}{2}$, and denote the number of good
vertices by $g$.

By our assumption on $G$, every block $P_{i}$ has at least
$d\cdot|P_{i}|-|P|=\left(c-\alpha\right)n$ neighbours outside $P$, and in
total there are $r=\frac{\alpha d}{c}$ blocks. Hence, by a double-counting
argument

$\left(c-\alpha\right)n\cdot\frac{\alpha d}{c}\leq\sum_{v\in V\setminus
P}b_{v}\leq g\cdot\frac{\alpha d}{c}+n\cdot\frac{\alpha d}{2}.$

In particular, $g\geq\left(\frac{c}{2}-\alpha\right)n\geq\frac{cn}{3}$, since
$\varepsilon$ is sufficiently small.

We say a good vertex is _successful_ if there is at least one edge of
$G_{p_{2}}$ between $v$ and a vertex $v_{f}$ of the first $\frac{b_{v}}{3}$
blocks of $B_{v}$ and one edge between $v$ and a vertex $v_{\ell}$ of the last
$\frac{b_{v}}{3}$ blocks of $B_{v}$. Note that a successful vertex $v$ lies in
a cycle in $G_{p_{1}}\cup G_{p_{2}}$ of length at least $\frac{\alpha cn}{6}$.
Indeed, by assumption there are $i$ and $j$ with $v_{f}\in P_{i}$,
$v_{\ell}\in P_{j}$, $\\{\\{v,v_{f}\\},\\{v,v_{\ell}\\}\\}\subseteq
E(G_{p_{2}})$ and $|i-j|-1\geq\frac{b_{v}}{3}\geq\frac{\alpha d}{6}$. Since
$v\not\in V(P)$, the vertex $v$ together with (part of) $P$ forms a cycle in
$G_{p_{1}}\cup G_{p_{2}}$ of length at least $\frac{\alpha
d}{6}\cdot\frac{cn}{d}=\frac{\alpha cn}{6}=\Omega(\varepsilon^{2}c^{2}n)$.

Each good vertex $v$ satisfies $b_{v}\geq\frac{\alpha d}{2}$ by definition,
and hence

$p_{\text{s}}\coloneqq\mathbb{P}[v\text{ is
successful}]\geq\left(1-(1-p_{2})^{\frac{b_{v}}{3}}\right)^{2}\geq\left(1-\exp\left(-\frac{\varepsilon\alpha}{12}\right)\right)^{2}=\Omega(c^{2}\varepsilon^{6}).$

There are $g\geq\frac{cn}{3}$ good vertices, and each good vertex is
successful independently of the others. Therefore, the probability that no
good vertex is successful is
$(1-p_{\text{s}})^{g}=\exp\left(-\Omega_{\varepsilon,c}\left(n\right)\right)$.
We conclude that with probability at least
$1-\exp\left(-\Omega_{\varepsilon}\left(n\right)\right)-\exp\left(-\Omega_{\varepsilon,c}\left(n\right)\right)=1-\exp\left(-\Omega_{\varepsilon,c}\left(n\right)\right)$,
there is a cycle of length at least $\Omega(\varepsilon^{2}c^{2}n)$ in
$G_{p_{1}}\cup G_{p_{2}}$.∎

##  5 Discussion

Whilst Theorem 3 requires that the graph expands at an almost optimal scale,
that is $d$-vertex-expansion for sets of size $\Omega(n/d)$, we conjecture
that the conclusion should hold for any $(k,d)$-expander.

######  Conjecture 6.

Let $k,d\in\mathbb{N}$ and let $G$ be a graph on at least $k$ vertices such
that every $S\subseteq V(G)$ with $|S|=k$ satisfies $|N(S)|\geq kd$. Let
$\varepsilon>0$ be sufficiently small and let $p=\frac{1+\varepsilon}{d}$.
Then with probability tending to one as $k$ tends to infinity, $G_{p}$
contains a cycle of length $\Omega_{\varepsilon}(kd)$.

Note that Conjecture 6 would be a strengthening of Theorem 2, where we show
that typically $G_{p}$ contains a path of length $\Omega_{\varepsilon}(kd)$.

As noted in the introduction, while assuming that the minimum degree is at
least $d$ suffices to show that $G_{p}$ contains a cycle of length linear in
$d$ [14], an assumption on the edge-expansion alone does not suffice to prove
Theorem 2. Such an assumption is tightly related to an assumption on
$\lambda_{2}$, the second largest eigenvalue of the adjacency matrix of the
graph. In contrast, it is known that an appropriate assumption on
$\lambda=\min\\{|\lambda_{2}|,|\lambda_{|V(G)|}|\\}$ does suffice. Indeed, for
$(n,d,\lambda)$-graphs $G$ with growing degree $d$, when
$p=\frac{1+\varepsilon}{d}$ and $\lambda\leq\varepsilon^{4}d$, Diskin and
Krivelevich [7] showed that whp $G_{p}$ contains a cycle of length linear in
$n$.

Considering constant $d$, Alon and Bachmat [2] showed that given a
$(d+1)$-regular graph $G$ on $n$ vertices, for any fixed $\varepsilon>0$, if
$p=\frac{1+\varepsilon}{d}$, then with probability at least $1-o_{n}(1)$ the
random subgraph $G_{p}$ contains a cycle. Theorems 2 and 3 can be seen as a
quantitative strengthening of this result, showing what additional
requirements are sufficient to show the typical existence of paths or cycles
of some given length.

## Acknowledgement

The first and third authors were supported in part by the Austrian Science
Fund (FWF) [10.55776/P36131]. Part of this work was done while the first and
the third authors were visiting Tel Aviv University, and they would like to
thank the university for its hospitality. For the purpose of open access, the
authors have applied a CC-BY public copyright licence to any Author Accepted
Manuscript version arising from this submission.

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