# Zeroth-order considerations on an accelerator-based gravitational wave
amplifier

Velizar Miltchev<EMAIL_ADDRESS>
###### Abstract

The operation of an accelerator-based gravitational wave amplifier has been
studied, taking into account the interaction between beam and gravitational
wave and neglecting all effects related to the emission of - and interaction
with electromagnetic radiation. The small-gain operation mode has been
considered. The gravitational counterpart of the Madey theorem as well some
basic amplifier quantities as resonance wavelength, amplifier gain and gain
length have been derived.

##  1 Introduction

The set-up of the considered gravitational wave amplifier is identical to a
free-electron laser (FEL) as shown in Fig.1. It consists of an arrangement of
permanent magnetic dipoles called undulator and a relativistic charged
particle beam, which propagates along the axis ($z$-axis) of the undulator.

Figure 1: Sketch of the gravitational wave amplifier set-up.

The polarity of undulator dipoles alternates along the z-axis with a period
$\mathrm{\lambda_{u}}$ so that they create a static magnetic filed, which
ideally is given by $\mathrm{\textbf{
B}=\left(0,B\sin\left(k_{u}z\right),0\right)}$. Here
$\mathrm{k_{u}}=2\pi/\lambda_{u}$ and the non-zero component of the magnetic
field points along the $y$-axis. For to facilitate the following discussion it
is helpful to write explicitly the electromagnetic tensor for the undulator
field:

$\displaystyle\begin{bmatrix}{F^{\mu}}_{\nu}\end{bmatrix}=\begin{bmatrix}0&0&0&0\\\
0&0&0&-B\sin\left(k_{u}z\right)\\\ 0&0&0&0\\\
0&B\sin\left(k_{u}z\right)&0&0\end{bmatrix}$ (1)

The equation of motion in the rest frame of the undulator for a beam particle
of charge $q$, Lorentz factor $\gamma$ and rest mass $m$ is:

$\displaystyle\gamma m\frac{d}{dt}\bm{u}^{\mu}=q{F^{\mu}}_{\nu}\bm{u}^{\nu}$
(2)

here $\mu,\nu=0,1,2,3$ and
$[\bm{u}^{\mu}]=\gamma\left(c,v_{x},v_{y},v_{z}\right)$ is the 4-velocity of
the particle. A solution of (2) for $\mu=1,2,3$ can be found elsewhere [1]:

$\displaystyle\bm{u}^{1}=cK\cos\left(k_{u}z\right)$ (3)
$\displaystyle\bm{u}^{3}=\gamma\bar{\beta}c-\frac{cK^{2}}{4\gamma}\cos\left(2\bar{\beta}ck_{u}t\right)$
(4)

where the following abbreviations have been introduced:

$\displaystyle K=\frac{qB}{mck_{u}}$ (5)

$\displaystyle\bar{\beta}=1-\frac{1}{2\gamma^{2}}\left(1+\frac{K^{2}}{2}\right)$
(6)

The particle motion along the $y$-axis is unaffected by the undulator field.
Therefore, for simplicity, the vertical velocity $\bm{u}^{2}$ and position
will be assumed zero. For $\mu=0$ equation (2) yields $\gamma=\mathrm{const}$,

$\displaystyle mc\gamma\frac{d\gamma}{dt}=0$ (7)

exactly as expected in the absence of electric field.
Essential for the amplifier operation is that massive particles undergo
transverse motion. Therefore in the following, in order to be able to
distinguish the effects related to the gravitational wave the particles will
be assumed with no electric charge and thus all effects related to the
emission of - and interaction with electromagnetic radiation as well as space
charge effects are going to be neglected.

##  2 Energy exchange between particles and gravitational wave

In the following, a plane linearly polarized gravitational wave traveling
along the $z$-axis in the same direction as the particle beam will be
considered. The gravitational wave of wavelength $\lambda$ has the form:

$\displaystyle h_{\mu\nu}=A_{\mu\nu}\cos\left(kz-\omega t\right)$ (8)

where $k=2\pi/\lambda$ and the amplitude tensor is:

$\displaystyle\left[A_{\mu\nu}\right]=\mathrm{diag}\left(0,-a,a,0\right)$ (9)

with $\lvert a\rvert\ \ll 1$. Even though $a$ may vary along the undulator,
the rate of change is very small and therefore it will be considered constant
in the first order. The metric tensor then becomes:

$\displaystyle g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ (10)

where $\left[\eta_{\mu\nu}\right]=\mathrm{diag}\left(1,-1,-1,-1\right)$. In
the new metric, equation (7) changes to:

$\displaystyle
mc\gamma\frac{d\gamma}{dt}+m{\Gamma^{0}}_{\mu\nu}\bm{u}^{\mu}\bm{u}^{\nu}=0$
(11)

The metric connection is given by:

$\displaystyle{\Gamma^{\sigma}}_{\mu\nu}=\frac{1}{2}\eta^{\sigma\rho}\left(\partial_{\nu}h_{\rho\mu}+\partial_{\mu}h_{\rho\nu}-\partial_{\rho}h_{\mu\nu}\right)$
(12)

Considering (9), in the case for $\sigma=0$ there are only two nonzero
connection coefficients:

$\displaystyle{\Gamma^{0}}_{11}=-\frac{1}{2}\partial_{0}h_{11}=-\frac{1}{2}\frac{\partial}{\partial\left(ct\right)}\left(-a\cos\left(kz-\omega
t\right)\right)=\frac{a\omega}{2c}\sin\left(kz-\omega t\right)$ (13)

and

$\displaystyle{\Gamma^{0}}_{22}=-\frac{1}{2}\partial_{0}h_{22}=-\frac{1}{2}\frac{\partial}{\partial\left(ct\right)}\left(a\cos\left(kz-\omega
t\right)\right)=-\frac{a\omega}{2c}\sin\left(kz-\omega t\right)$ (14)

The gravitational wave has very small effect on the particle motion. Therefore
it is safe to assume that equations (3) and (4) will excellently approximate
the velocities when the particles co-propagate with the gravitational wave.
Taking into account that $\bm{u}^{2}=0$ and plugging (3) and (13) into (11)
yields:

$\displaystyle\frac{d\gamma}{dt}=-\frac{a\omega
K^{2}}{2\gamma}\sin\left(kz-\omega t\right)\cos^{2}\left(k_{u}z\right)$ (15)

Equation (15) governs the energy exchange between a particle and the
gravitational wave.

##  3 Resonance condition

A sustained energy transfer from or to the gravitational wave can happen only
when the right-hand side of (15) on average does not vanish along the
undulator. The necessary conditions can be obtained by recasting the term in a
different form:

$\displaystyle\frac{a\omega K^{2}}{2\gamma}\sin\left(kz-\omega
t\right)\cos^{2}\left(k_{u}z\right)=\frac{a\omega
K^{2}}{8\gamma}\left[\sin\left(\Psi\right)+\sin\left(\Xi\right)+2\sin\left(\alpha\right)\right]$
(16)

where the following phases have been defined:

$\displaystyle\Psi=\left(k+2k_{u}\right)z-\omega t$ (17)
$\displaystyle\Xi=\left(k-2k_{u}\right)z-\omega t=\Psi-4k_{u}z$ (18)
$\displaystyle\alpha=kz-\omega t=\Psi-2k_{u}z$ (19)

The phase $\Psi$ is the analogue of the ponderomotive phase known from the
FEL-amplifiers theory [1]. The condition $\Psi=\mathrm{const}$ implies:

$\displaystyle\frac{d}{dt}\Psi=\left(k+2k_{u}\right)\frac{d}{dt}z-\omega=\left(\frac{2\pi}{\lambda}+2\frac{2\pi}{\lambda_{u}}\right)\bar{\beta}c-\frac{2\pi}{\lambda}c=0$
(20)

In the last line the fast-oscillating term in equation (4) has been neglected,
assuming $z(t)=\bar{\beta}ct$. After plugging $\bar{\beta}$ from (6) and
solving for $\lambda$ one obtains the required resonance condition for the
gravitational wavelength:

$\displaystyle\lambda=\frac{\lambda_{u}}{4\gamma^{2}}\left(1+\frac{K^{2}}{2}\right)$
(21)

where an ultra-relativistic ($\gamma\gg K$) particle beam has been assumed.
If $\Psi=\mathrm{const}$, the terms with $\Xi$ and $\alpha$ in (16) oscillate
four and two times per undulator period respectively and on average do not
contribute to the overall energy transfer. One can also show that only the
phase $\Psi$ can remain constant i.e. the equations $d\Xi/dt=0$ and
$d\alpha/dt=0$ do not have physically meaningful solution.
The resonance energy $\gamma_{r}$ is the particle energy at which equation
(21) exactly holds for a given $\lambda$:

$\displaystyle\gamma_{r}=\sqrt{\frac{\lambda_{u}}{4\lambda}\left(1+\frac{K^{2}}{2}\right)}$
(22)

Finally, it is convenient to rewrite (15) in terms of $\Psi$ and $z$:

$\displaystyle\frac{d\gamma}{dz}=-\frac{a\omega K^{2}}{8\gamma
c}\sin\left(\Psi\right)$ (23)

##  4 Longitudinal phase-space oscillations

As discussed above, when $\gamma=\gamma_{r}$ the resonance condition is
exactly fulfilled, $\Psi$ is constant and hence:

$\displaystyle\frac{d\Psi}{dz}(\gamma_{r})=\left(k+2k_{u}\right)-\frac{\omega}{\bar{\beta}(\gamma_{r})c}=0$
(24)

In the last line $\bar{\beta}$ has been explicitly written as a function of
$\gamma$ and the substitution $t=z/(\bar{\beta}c)$ has been made. If a
particle has a slightly different energy $\gamma=\gamma_{r}+\Delta\gamma$,
then to first order in $\Delta\gamma$ the rate of change of $\Psi$ is given
by:

$\displaystyle\begin{split}\frac{d\Psi}{dz}(\gamma_{r}+\Delta\gamma)&\approx\frac{d\Psi}{dz}(\gamma_{r})+\left.\Delta\gamma\frac{d}{d\gamma}\left(\frac{d\Psi}{dz}\right)\right|_{\gamma=\gamma_{r}}\\\
&=\left.\Delta\gamma\left(\frac{\omega}{{\bar{\beta}}^{2}c}\frac{d\bar{\beta}}{d\gamma}\right)\right|_{\gamma=\gamma_{r}}\end{split}$
(25)

which on using (6), (21) and assuming again $\bar{\beta}\approx 1$ transforms
to

$\displaystyle\frac{d\Psi}{dz}(\gamma_{r}+\Delta\gamma)=4k_{u}\eta$ (26)

where the relative energy deviation has been defined as
$\eta=\left(\gamma-\gamma_{r}\right)/{\gamma_{r}}$. Equation (26) together
with (23) forms a system of coupled differential equations.

$\displaystyle\begin{split}\frac{d\eta}{dz}&=-\frac{a\omega
K^{2}}{8\gamma_{r}^{2}c}\sin\left(\Psi\right)\\\
\frac{d\Psi}{dz}&=4k_{u}\eta\end{split}$ (27)

On combining them one obtains a second order differential equation, which is
formally identical to the pendulum equation occurring in the low-gain FEL
theory [1, 2]:

$\displaystyle\frac{d^{2}\Psi}{dz^{2}}+\Omega_{g}^{2}\sin(\Psi)=0\leavevmode\nobreak\
\leavevmode\nobreak\ \mathrm{with}\leavevmode\nobreak\ \leavevmode\nobreak\
\Omega_{g}^{2}=\frac{a\omega K^{2}k_{u}}{2c\gamma_{r}^{2}}$ (28)

##  5 Gravitational analogue of Maday theorem

The conventional electromagnetic FEL-amplifiers operating in the low-gain
regime are typically installed at storage rings inside an optical cavity. The
electromagnetic wave remains captured in the cavity and gets amplified over
many revolutions. Thus the output field amplitude exceeds the initial one by
many orders of magnitude even though the gain per passage might be a few
percents only. If, as for the gravitational waves, suitable mirrors are not
available, then one can think of high-gain operation mode, in which the output
amplitude increases exponentially with $z$ and just one passage is required.
Nevertheless, in the following a gravitational wave cavity-amplifier operating
in the low-gain regime is considered. The amplifier gain $\Gamma_{g}$ is
defined as the relative energy transfer from the beam particles to the
gravitational wave over a single pass through the undulator:

$\displaystyle\Gamma_{g}=-\frac{Nmc^{2}\Delta\gamma}{V\rho_{g}}=-\frac{nmc^{2}\Delta\gamma}{\rho_{g}}\leavevmode\nobreak\
\leavevmode\nobreak\ \mathrm{with}\leavevmode\nobreak\ \leavevmode\nobreak\
\rho_{g}=\frac{c^{2}a^{2}\omega^{2}}{32\pi G}$ (29)

with $N$ \- the total number of beam particles, $n$ \- particle density, $V$
\- volume occupied by the gravitational wave, $\rho_{g}$ \- energy density of
the gravitational wave and $G$ \- gravitational constant. For simplicity it is
assumed that $V$ coincides with the volume occupied by the beam. Following the
arguments in [2], the total energy loss $\Delta\gamma$ for one particle can be
calculated from (27) and (28) applying second order perturbation theory,

$\displaystyle\Delta\gamma=\frac{\gamma_{r}L^{3}\Omega_{g}^{4}}{32k_{u}}\frac{d}{d\xi}\left(\frac{\sin\xi}{\xi}\right)^{2}$
(30)

where $L$ is the undulator length and $\xi=2k_{u}L\eta$. Substituting the
relations (30) and (5) in (29) one obtains111for simplicity’s sake, higher-
order effects related to variation of longitudinal velocity are neglected
here.

$\displaystyle\Gamma_{g}=-\frac{Gnq^{4}L^{3}B^{4}\lambda_{u}^{3}}{32\pi^{2}m^{3}c^{6}\gamma_{r}^{3}}\frac{d}{d\xi}\left(\frac{\sin\xi}{\xi}\right)^{2}$
(31)

It is convenient to introduce the the power gain length $L_{g}$

$\displaystyle
L_{g}=\frac{1}{\sqrt{3}}\left(\frac{8c^{6}m^{3}\pi^{2}\gamma_{r}^{3}}{B^{4}Gnq^{4}\lambda_{u}^{3}}\right)^{1/3}$
(32)

Thus (31) can be recast in the form

$\displaystyle\Gamma_{g}=-\frac{1}{12\sqrt{3}}\left(\frac{L}{L_{g}}\right)^{3}\frac{d}{d\xi}\left(\frac{\sin\xi}{\xi}\right)^{2}$
(33)

Equation (31) is the gravitational counterpart of the Madey theorem, which was
first derived for electromagnetic wave amplifiers [3]. The corresponding
energy gain for an electromagnetic wave is given by:

$\displaystyle\Gamma_{em}=-\frac{nq^{4}B^{2}L^{3}\lambda_{u}}{16\pi
c^{4}\epsilon_{0}m^{3}\gamma_{r}^{3}}\frac{d}{d\xi}\left(\frac{\sin\xi}{\xi}\right)^{2}$
(34)

and hence

$\displaystyle\frac{\Gamma_{g}}{\Gamma_{em}}=\frac{G\epsilon_{0}}{2\pi
c^{2}}\left(B\lambda_{u}\right)^{2}$ (35)

The gain ratio in (35) is independent on the particles type and depends only
on physical constants and undulator properties. This gives a possibility to
roughly estimate the gain length $L_{g}$ of the gravitational wave amplifier
using the FEL gain length $L_{em}$:

$\displaystyle L_{g}\propto L_{em}\sqrt[\scriptstyle
3]{\frac{\Gamma_{em}}{\Gamma_{g}}}=L_{em}\sqrt[\scriptstyle 3]{\frac{2\pi
c^{2}}{G\epsilon_{0}B^{2}\lambda_{u}^{2}}}$ (36)

Equation (36) bears interpretation. On the one hand, $L_{em}\propto m$, so one
expects lighter particles to produce shorter gain lengths, what makes
$e^{-}$-accelerators more feasible. On the other hand, for to minimize EM-
radiation effects, one needs heavy particles such as $p^{+}$ or even heavy
ions.
As a numerical example one can take a peak field $B$ = 2 T, undulator period
$\lambda_{u}$ = 0.2 m and FEL-gain length $L_{em}$ = 1 m. In this case Eq.(36)
yields $L_{g}\sim$ 10 light-hours.
Obviously such effective amplifier length can only be realized in a storage-
ring, if at all. For example, similar to conventional ring lasers, one can
imagine a storage-ring amplifier, but with the classical mirrors replaced by
gain guiding.

##  6 Summary

The operation of an FEL-like gravitational wave amplifier has been studied in
the framework of a simplified model, in which the effects related to the
emission of - and interaction with electromagnetic radiation have been
ignored. This made possible to derive the gravitational counterparts of some
basic FEL-quantities like ponderomotive phase, resonance wavelength, amplifier
gain and gain length.

## References

  * [1] Schmüser P., Dohlus M., Rossbach J.: Ultraviolet and Soft X-Ray Free-Electron Lasers, Springer 2009
  * [2] Wiedemann H.: Particle Accelerator Physics, Springer, 2007
  * [3] Madey, J.M.J.: Relationship between mean radiated energy, mean squared radiated energy and spontaneous power spectrum in a power series expansion of the equation of motion in a free electron laser. Nuovo Cimento 50B, 64 (1979) 33


2^{2n+1}n\Gamma(n+1)\Gamma\left(n+1+\frac{N}{2}\right)}{\Gamma\left(\frac{N}{2}+1\right)}\left(1-\varepsilon\frac{54n+7N-16}{6(N+8)}\mathcal{A}_{p}+O(\varepsilon^{2})\right)~{}.$
(B.32)

Dividing this by the normalizations of the operators we find the third line of
(3.4).

With no extra complications, we can also compute the one-loop correction to
the three-point function with $S_{n+1}$. The final result reads:

$\displaystyle C_{W_{1}W_{n}S_{n+1}}$
$\displaystyle=\frac{1}{N}{4^{1+n}(6n+N+2)\Gamma(n+2)\left(\frac{N}{2}\right)_{n+1}}{}$
$\displaystyle\times\left(1-\varepsilon\frac{n(54n+17N+28)+3(N+2)}{(N+8)(6n+N+2)}\mathcal{A}_{p}+O(\varepsilon^{2})\right)~{}.$
(B.33)

Dividing this by the normalizations of the operators we find the fourth line
of (3.4).

###  B.4 Bulk two-point functions

In this section, we collect some perturbative results for the bulk two-point
function of $\Phi$ and its composites $(\Phi\cdot\Phi)^{n}$. It will be
convenient to work with the connected two-point function, defined for a
generic operator $O$ as

$\displaystyle\langle O(x_{1})O(x_{2})\rangle_{c}=\langle
O(x_{1})O(x_{2})\rangle-|x_{12}|^{-2\Delta_{O}}~{}.$ (B.34)

####  B.4.1 Two-point function of $\Phi$ near the short-range end

We start from the bulk two-point function of $\Phi=\phi$ near the short-range
end. At tree-level, this is the bulk-defect crossing symmetric solution with
$b_{0}^{\phi,+}=0$ – see eq. (2.15) –, and the connected correlator is
therefore completely specified. In order to extract $a_{\phi^{2}}$ we can take
$x_{1}\to x_{2}$ (meaning $\hat{r}\to 1$) to get

$\displaystyle|y|^{2\Delta_{\phi}}\langle\phi(x)\phi(x)\rangle_{c}^{(0)}$
$\displaystyle=(b_{0}^{\phi,-})^{2}{}_{2}F_{1}(p-\Delta_{\phi},p/2;1-\Delta_{\phi}+p/2;1)-{}_{2}F_{1}(\Delta_{\phi},p/2;1+\Delta_{\phi}-p/2;1)$
$\displaystyle=\begin{cases}-\frac{7}{8}+\delta,&p=2\\\
-0.575+0.749\delta,&p=3\end{cases},$ (B.35)

matching (2.16). Turning on interactions, the first correction to the result
above comes from two $\sigma\hat{\chi}$ insertions and gives

$\displaystyle\langle\phi(x_{1})\phi(x_{2})\rangle^{(2)}_{c}$
$\displaystyle=\frac{g^{2}(b_{0}^{\phi,-})^{2}}{2}\int\frac{d^{p}\tau_{3}d^{p}\tau_{4}}{|x_{13}|^{2\Delta^{*}_{\sigma}}|\tau_{34}|^{2\Delta_{\chi}}|x_{24}|^{2\Delta^{*}_{\sigma}}}$
$\displaystyle=\frac{g^{2}(b_{0}^{\phi,-})^{2}}{2}\frac{\pi^{p}\Gamma\left(\frac{p}{2}-\Delta^{*}_{\sigma}\right)}{|y_{1}|^{\Delta^{*}_{\sigma}+\Delta_{\chi}}|y_{2}|^{\Delta^{*}_{\sigma}+\Delta_{\chi}}|\tau_{12}|^{2\Delta^{*}_{\sigma}-2p}\Gamma\left(\Delta_{\chi}\right)^{2}\Gamma\left(\Delta^{*}_{\sigma}\right)}$
(B.36) $\displaystyle\times\int_{-i\infty}^{+i\infty}\frac{dtdu}{(2\pi
i)^{2}}\frac{\Gamma\left(\frac{p}{2}-t\right)\Gamma\left(\frac{p}{2}-u\right)\Gamma(\Delta_{\chi}-t)\Gamma(\Delta_{\chi}-u)\Gamma(-p+t+u+\Delta^{*}_{\sigma})}{\Gamma\left(-\Delta^{*}_{\sigma}+\frac{3p}{2}-t-u\right)|\tau_{12}|^{2(t+u)}|y_{1}|^{-2t}|y_{2}|^{-2u}}~{}.$

To get to the second line we have used (B.4). To compute this integral we
deform the contours such that poles for positive $t,u$ and negative $t,u$ are
separated, and then close to the left, picking up the poles with negative
$t,u$. We can now take the residue at $u=p-\Delta_{\sigma}^{*}-t$ and use the
first Barnes’ lemma to conclude:

$\displaystyle|y|^{2\Delta_{\phi}}\langle\phi(x)\phi(x)\rangle_{c}^{(2)}=\frac{g^{2}(b_{0}^{\phi,-})^{2}}{2}\frac{\Gamma(\tfrac{p}{2}-\Delta^{*}_{\sigma})\Gamma(\Delta^{*}_{\sigma}+2\Delta_{\chi}-p)\Gamma(\Delta^{*}_{\sigma}+\Delta_{\chi}-\tfrac{p}{2})^{2}}{\Gamma(\tfrac{p}{2})\Gamma(2\Delta^{*}_{\sigma}+2\Delta_{\chi}-p)\Gamma(\Delta_{\chi})^{2}}~{}.$
(B.37)

It is now a simple matter to plug in the fixed point and the numerical value
for $\Delta_{\sigma}^{*}$. Doing so and adding (B.35) recovers the result
(3.20) originally obtained from the OPE relations.

####  B.4.2 Two-point function of $\Phi$ near the mean-field end

We perform a similar computation near the mean-field end. Here the tree-level
contribution is just the correlator in the presence of a trivial defect of co-
dimension $q=2-\mathfrak{s}$, which has $b_{0}^{\phi,+}=1$ and
$b_{0}^{\phi,-}=0$. The first non-trivial correction to the connected
correlator comes at $O(\lambda^{2})$, and reads:

$\displaystyle\langle\phi^{I}(x_{1})\phi^{J}(x_{2})\rangle^{(2)}_{c}$
$\displaystyle=2(N+2)\delta^{IJ}\lambda^{2}\int\frac{d^{p}\tau_{3}d^{p}\tau_{4}}{|x_{13}|^{2\Delta_{\phi}}|\tau_{34}|^{6\Delta_{\phi}}|x_{24}|^{2\Delta_{\phi}}}$
(B.38)
$\displaystyle=2(N+2)\delta^{IJ}\lambda^{2}\frac{\pi^{p}\Gamma\left(\frac{p}{2}-3\Delta_{\phi}\right)}{|y_{1}|^{2\Delta_{\phi}}|y_{2}|^{2\Delta_{\phi}}|\tau_{12}|^{6\Delta_{\phi}-2p}\Gamma\left(\Delta_{\phi}\right)^{2}\Gamma\left(3\Delta_{\phi}\right)}$
$\displaystyle\times\int_{-i\infty}^{+i\infty}\frac{dtdu}{(2\pi
i)^{2}}\frac{\Gamma\left(\frac{p}{2}-t\right)\Gamma\left(\frac{p}{2}-u\right)\Gamma(\Delta_{\phi}-t)\Gamma(\Delta_{\phi}-u)\Gamma(-p+t+u+3\Delta_{\phi})}{\Gamma\left(-3\Delta_{\phi}+\frac{3p}{2}-t-u\right)|\tau_{12}|^{2(t+u)}|y_{1}|^{-2t}|y_{2}|^{-2u}}~{}.$

The computation is very similar to what we did before except now the defect
will have three $\hat{\phi}$ propagators instead of one $\hat{\chi}$
propagator. We deform the contours again such that poles for positive $t,u$
and negative $t,u$ are separated, and then close to the left, picking up the
poles with negative $t,u$. This yields a result that can be decomposed into
only the $s=0$ blocks of eq. (2.4) because there is no dependence on the
transverse angle as expected. Adding back the tree-level contribution, this
function is:

$\displaystyle\langle\phi^{I}(x_{1})\phi^{J}(x_{2})\rangle=\frac{\delta^{IJ}}{(|y_{1}||y_{2}|)^{p/2}}\left((b_{0}^{\phi,+})^{2}{\hat{g}}^{(+)}_{0}({\hat{r}},\hat{\eta})+(b_{0}^{\phi,-})^{2}{\hat{g}}^{(-)}_{0}({\hat{r}},\hat{\eta})\right)+O(\varepsilon^{3})~{},$
(B.39)

where

$\displaystyle(b_{0}^{\phi,+})^{2}$
$\displaystyle=1+\frac{(N+2)\Gamma\left(-\frac{p}{4}\right)\Gamma\left(\frac{p}{2}\right)^{2}}{2(N+8)^{2}\Gamma\left(\frac{3p}{4}\right)}\varepsilon^{2},\quad(b_{0}^{\phi,-})^{2}=\frac{(N+2)\Gamma\left(1-\frac{p}{4}\right)^{2}\Gamma\left(\frac{p}{2}\right)^{2}}{2(N+8)^{2}\Gamma(\tfrac{p}{4}+1)^{2}}\varepsilon^{2}~{}.$
(B.40)

The ratio

$\displaystyle(b_{0}^{\phi,-})^{2}/(b_{0}^{\phi,+})^{2}=\frac{(N+2)\Gamma\left(1-\frac{p}{4}\right)^{2}\Gamma\left(\frac{p}{2}\right)^{2}}{2(N+8)^{2}\Gamma\left(\frac{p}{4}+1\right)^{2}}\varepsilon^{2}+O(\varepsilon^{3})~{},$
(B.41)

can then be used to see that we get precisely (3.34) to $O(\varepsilon^{2})$.
Another way to extract $a_{\phi^{2}}$ quickly is to use the fact that it is
the coefficient of $|y|^{-2\Delta_{\phi}}$ in the connected correlator when we
set $y_{1}=y_{2}=y$ and take $\tau_{12}\to 0$. In this limit, only the
residues with $t+u=p-3\Delta_{\phi}$ contribute.

####  B.4.3 Composite operators

Finally, we will show how anomalous dimensions for defect operators can also
be obtained by perturbing two-point functions in the bulk. The idea is to use
an expansion very similar to (4.13) for a generic bulk operator $O$, namely:

$\displaystyle(|y_{1}||y_{2}|)^{\Delta_{O}}\left<O(x_{1})O(x_{2})\right>=\sum_{\Delta_{\mathcal{O}},s_{\mathcal{O}}}\left[\omega^{(0)}_{\mathcal{O}}+\varepsilon\left(\omega^{(1)}_{\mathcal{O}}+\omega^{(0)}_{\mathcal{O}}\hat{\gamma}^{(1)}_{\mathcal{O}}\partial_{\Delta}\right)+O(\varepsilon^{2})\right]\hat{g}_{\Delta_{\mathcal{O}}^{(0)},s_{\mathcal{O}}}(\hat{r},\hat{\eta})~{},$
(B.42)

where the squared bulk-defect OPE coefficient has been written as

$\displaystyle(b^{O}_{\mathcal{O}})^{2}\equiv\omega_{\mathcal{O}}=\omega_{\mathcal{O}}^{(0)}+\varepsilon\omega_{\mathcal{O}}^{(1)}+O(\varepsilon^{2})~{}.$
(B.43)

Since the bulk is free, $O$ must be a composite operator for unprotected
defect operators to appear in (B.42). We will consider the important example
of $\mathcal{S}_{n}(x)=(\phi\cdot\phi)^{n}(x)/\sqrt{4^{n}n!(N/2)_{n}}$ which
is the bulk version of $\mathcal{S}_{n}(\tau)$ considered in section 3. This
will provide an alternative proof of the fact that a double-twist operator
$[\hat{\phi}\hat{\phi}]_{n,\ell}$ only has a one-loop anomalous dimension when
$n=0$ (the need for $\ell=0$ was more clear).

As appropriate for the mean-field end of a long-range $O(N)$ model, our
starting point will be the two-point function in the presence of a trivial
defect

$\displaystyle\langle\mathcal{S}_{n}(x_{1})\mathcal{S}_{n}(x_{2})\rangle^{(0)}=\frac{1}{(x_{12}^{2})^{2n\Delta_{\phi}}}\equiv\frac{1}{(x_{12}^{2})^{\Delta_{O}}}~{}.$
(B.44)

This has a decomposition into defect primaries of the form

$\displaystyle\psi_{m,s}$
$\displaystyle\sim{(\partial^{2}_{\perp})}^{m}\partial^{i_{1}}_{\perp}\dots\partial^{i_{s}}_{\perp}S_{n}-\text{traces}~{},$
(B.45) $\displaystyle\Delta^{(0)}_{m,s}$ $\displaystyle=\Delta_{O}+2m+s~{},$

whose coefficients were found in [11]

$\displaystyle\omega^{(0)}_{m,s}$
$\displaystyle=\frac{2^{s}\Gamma\left(\frac{q}{2}+s\right)\Gamma\left(-\frac{d}{2}+m+\Delta_{O}+1\right)\Gamma(2m+s+\Delta_{O})\Gamma\left(m+s+\Delta_{O}-\frac{p}{2}\right)}{m!s!\Gamma(\Delta_{O})\Gamma\left(-\frac{d}{2}+\Delta_{O}+1\right)\Gamma\left(m+\frac{q}{2}+s\right)\Gamma\left(2m+s+\Delta_{O}-\frac{p}{2}\right)}~{}.$
(B.46)

These coefficients are positive for $\Delta_{O}$ above the unitarity bound and
integer $s$, as they should be. As a check, when
$\Delta_{O}=\Delta_{\phi}=d/2-1$, all but $\omega^{(0)}_{0,s}$ vanish, and we
recover (2.9). Note that for $\Delta_{O}\neq d/2-1$ there are infinitely many
$s=0$ defect blocks that contribute.

The first correction appears at $O(\lambda)$ and is given by:

$\displaystyle\langle\mathcal{S}_{n}(x_{1})\mathcal{S}_{n}(x_{2})\rangle$
$\displaystyle=\langle\mathcal{S}_{n}(x_{1})\mathcal{S}_{n}(x_{2})\rangle^{(0)}-\lambda
n(6n+N-4)\frac{1}{|x_{12}|^{4(n-1)\Delta_{\phi}}}\int\frac{d^{p}\tau_{3}}{|x_{13}|^{4\Delta_{\phi}}|x_{23}|^{4\Delta_{\phi}}}~{}.$
(B.47)

Note that there is no wave-function renormalization for $\mathcal{S}_{n}$.
From the integral alone we get:

$\displaystyle\frac{1}{|x_{12}|^{4(n-1)\Delta_{\phi}}}\sum_{i,j=0}^{\infty}\left[\frac{A_{ij}(\Delta_{\phi})}{\left(\tau_{12}^{2}\right)^{2\Delta_{\phi}+i+j}}(|y_{1}|^{2j}|y_{2}|^{-4\Delta_{\phi}+2i+p}+(1\leftrightarrow
2))+\frac{B_{ij}(\Delta_{\phi})|y_{2}|^{2i}|y_{1}|^{2j}}{\left(\tau_{12}^{2}\right)^{(4\Delta_{\phi}+i+j-p/2)}}\right]~{},$
(B.48)

with

$\displaystyle A_{ij}(\Delta_{\phi})$
$\displaystyle=\frac{\pi^{p/2}(-1)^{i+j}\Gamma(i+j+2\Delta_{\phi})\Gamma\left(-i+2\Delta_{\phi}-\frac{p}{2}\right)\Gamma\left(-j+\frac{p}{2}-2\Delta_{\phi}\right)}{i!j!\Gamma(2\Delta_{\phi})^{2}\Gamma\left(-i-j+\frac{p}{2}-2\Delta_{\phi}\right)}~{},$
$\displaystyle B_{ij}(\Delta_{\phi})$
$\displaystyle=\frac{\pi^{p/2}(-1)^{i+j}\Gamma\left(-i+\frac{p}{2}-2\Delta_{\phi}\right)\Gamma\left(-j+\frac{p}{2}-2\Delta_{\phi}\right)\Gamma\left(i+j+4\Delta_{\phi}-\frac{p}{2}\right)}{i!j!\Gamma(2\Delta_{\phi})^{2}\Gamma(-i-j+p-4\Delta_{\phi})}~{}.$
(B.49)

Upon including all the prefactors, evaluating at the IR fixed point, and
expanding in $\varepsilon$, we get

$\displaystyle\frac{\langle\mathcal{S}_{n}(x_{1})\mathcal{S}_{n}(x_{2})\rangle^{(1)}}{n(6n+N-4)}=-\frac{\varepsilon}{N+8}{\Gamma\left(\frac{p}{2}\right)}\frac{\log(\zeta)\,\zeta^{-\frac{p}{2}}\,\,_{2}F_{1}\left(\frac{1}{2},\frac{p}{2};1;-\frac{4}{\zeta}\right)}{|y|^{np}(2-2\hat{\eta}+\zeta)^{\frac{1}{2}(n-1)p}}$
$\displaystyle+\frac{2\varepsilon}{N+8}\sum_{k=0}^{\infty}\frac{(-1)^{k}\Gamma\left(2k\right)\Gamma\left(k+\frac{p}{2}\right)}{k^{2}\Gamma(k)^{3}\Gamma\left(\frac{p}{2}\right)}\left[\psi\left(k+\frac{1}{2}\right)-2\psi\left(k+1\right)+\psi\left(k+\frac{p}{2}\right)+\log
4\right]$
$\displaystyle\times|y|^{-np}\zeta^{-k-\frac{p}{2}}\left(2-2\hat{\eta}+\zeta\right)^{-\frac{1}{2}(n-1)p}+O(\varepsilon^{2})~{},$
(B.50)

where we have set $|y_{2}|=|y_{1}|=y$ for simplicity, $\hat{\eta}$ is defined
in (2.5) and
$\zeta=|\tau_{12}|^{2}/|y|^{2}=\hat{\chi}-2=(1-\hat{r})^{2}/\hat{r}$
($\hat{\chi}$ is defined in (2.6)).

From the coefficient of the logarithmic piece in the first line above we
obtain the $O(\varepsilon)$ correction to the scaling dimensions. This is
because (B.42), with the defect blocks (2.4) plugged in, instructs us to look
for the coefficient of $\log\hat{r}$ which is the same as the coefficient of
$-\log\zeta$. As a warm-up, one can start with $n=1$ which is especially
simple. Eq. (B.4.3) does not depend on the transverse angle for $n=1$, which
means that it can be expanded into scalar defect blocks only. The anomalous
dimensions of the defect operators, as well as their OPE coefficients, turn
out to be

$\displaystyle\hat{\gamma}_{m,s}^{(1)}=\frac{N+2}{N+8}\delta_{m,0}\delta_{s,0}~{},\quad\omega^{(1)}_{m,s}=\hat{\gamma}_{0,0}^{(1)}\left[\psi\left(\frac{p}{2}\right)+\psi(1)\right]\delta_{m,0}\delta_{s,0}~{},$
(B.51)

which can now be applied to the long-range $O(N)$ model.181818The bulk channel
expansion is even simpler. It only features the bulk identity, as well as the
operator $\mathcal{S}_{2}$ with coefficient  $\displaystyle
a_{\mathcal{S}_{2}}\sqrt{2\frac{N+2}{N}}=-\frac{\varepsilon\hat{\gamma}_{0,0}^{(1)}}{2}\frac{\Gamma\left(\frac{p}{2}\right)^{2}}{\Gamma(p)}+O(\varepsilon^{2})~{}.$
(B.52) This immediately shows why we had
$\gamma^{(1)}_{n,\ell}\propto\delta_{n,0}$ for double-twist operators on the
defect. Non-trivial powers of $\partial_{i}\partial^{i}$ cannot appear in
operators that renormalize at one loop and these can be traded for
$\partial_{a}\partial^{a}$ with Laplace’s equation.

There is no reason to stop at $n=1$. For general $\mathcal{S}_{n}$, this
procedure will also yield non-trivial CFT data in the extension of the long-
range $O(N)$ model which includes transverse spin. We find:

$\displaystyle\hat{\gamma}_{m,s}^{(1)}$
$\displaystyle=f_{s,m}\frac{(n-1)(6n+N-4)\Gamma\left(\frac{1}{4}(2n-1)p\right)\Gamma\left(\frac{np}{2}+1\right)\Gamma\left(m+\frac{np+s}{2}-\frac{p}{2}\right)}{2^{4m}m!(N+8)\Gamma\left(m+\frac{1}{4}(2n-1)p\right)\Gamma\left(\frac{np-p}{2}+1\right)\Gamma\left(2m+\frac{np}{2}+s\right)}~{},$
(B.53)

where $f_{s,m}$ are complicated polynomials of $(n,p,s)$, which for $m\leq 3$
are found to be:

$\displaystyle f_{s,0}$ $\displaystyle=1~{},$ $\displaystyle f_{s,1}$
$\displaystyle=4n^{2}p^{2}+np(8s+8-6p)+p^{2}(4s+8-p)-12p(s+1)~{},$
$\displaystyle f_{s,2}$ $\displaystyle=32n^{4}p^{4}+32n^{3}p^{3}(4s+12-3p)$
$\displaystyle-8n^{2}p^{2}\left[2p^{3}-p^{2}(8s+33)+12p(4s+11)-16(s^{2}+7s+11)\right]$
$\displaystyle+8np(3p-4s-12)\left[p^{3}-4p^{2}(s+3)+6p(2s+5)-8(s+2)\right]$
$\displaystyle+p(p-4)\left[16(p^{2}-4p+6)s^{2}-8(p(p^{2}-12p+44)-60)s+(p-6)^{2}(p-4)^{2}\right]~{},$
$\displaystyle f_{s,3}$ $\displaystyle=384n^{6}p^{6}-576n^{5}p^{5}(3p-4(s+5))$
$\displaystyle-96n^{4}p^{4}\left(3p^{3}-3p^{2}(4s+25)+36p(3s+14)-16(3s(s+11)+85)\right)$
$\displaystyle+48n^{3}p^{3}(3p-4(s+5))\left(6p^{3}-3p^{2}(8s+35)+24p(4s+17)-16(s(s+16)+45)\right)$
$\displaystyle+12n^{2}p^{2}\left[3p^{6}-6p^{5}(4s+25)+12p^{4}\left(4s^{2}+70s+231\right)-24p^{3}(s(44s+423)+994)\right.$
$\displaystyle+\left.96p^{2}(s(s(4s+89)+553)+1035)-1152p(s(s(s+18)+97)+162)+512(3s(s(s+14)+63)+274)\right]$
$\displaystyle-6np(3p-4(s+5))\left[3p^{6}-24p^{5}(s+4)+12p^{4}(2s+9)(2s+11)\right.$
$\displaystyle-\left.48p^{3}(2s+9)(4s+17)+96p^{2}(s(15s+122)+245)-768p(s(3s+23)+43)+1024(s+3)(s+4)\right]$
$\displaystyle-(p-4)p\left[p^{7}-4p^{6}(3s+11)+4p^{5}(3s(4s+33)+220)-16p^{4}(2s(s(2s+33)+181)+641)\right.$
$\displaystyle+\left.16p^{3}\left(s\left(44s^{2}+630s+2941\right)+4483\right)-128p^{2}(s(s(31s+390)+1625)+2248)\right.$
$\displaystyle+\left.128p(s+4)(s+5)(71s+237)-9216(s+3)(s+4)(s+5)\right]~{}.$
(B.54)

From the last two lines of (B.4.3) we obtain $O(\varepsilon)$ corrections to
the (squared) tree-level bulk-defect OPE coefficients. For the first few
operators, these are

$\displaystyle\omega^{(1)}_{0,s}$
$\displaystyle=\frac{2^{s}n(6n+N-4)\left(\frac{1}{2}(n-1)p\right)_{s}}{(N+8)s!}[\psi(\tfrac{p}{2})+\psi(1)]~{},$
$\displaystyle\omega^{(1)}_{1,s}$
$\displaystyle=\frac{2^{s-1}n(6n+N-4)\left(\frac{1}{2}(n-1)p\right)_{s+1}[\psi(\tfrac{p}{2})+\psi(1)]}{(N+8)s!(4s+4-p)(np-p+2s+2)}\left[f_{s,1}+\frac{g_{s,1}}{(np-p+2s+2)[\psi(\tfrac{p}{2})+\psi(1)]}\right]~{},$
$\displaystyle\omega^{(1)}_{2,s}$
$\displaystyle=\frac{2^{s-4}n(6n+N-4)\left(\frac{1}{2}(n-1)p\right)_{s+2}[\psi(\tfrac{p}{2})+\psi(1)]}{(N+8)s!(p-4(s+1))(p-4(s+2))((n-1)p+2(s+2))((n-1)p+2(s+3))}$
$\displaystyle\times\left[f_{s,2}+\frac{g_{s,2}}{2(np-p+2s+4)(np-p+2s+6)[\psi(\tfrac{p}{2})+\psi(1)]}\right]~{},$
(B.55)

with $f_{s,m}$ given in (B.4.3) and

$\displaystyle g_{s,1}$ $\displaystyle=(p-2)p(p-4s-4)(np-p+2s+4)~{},$
$\displaystyle g_{s,2}$
$\displaystyle=(p-2)p(p-4s-8)(np-p+2s+8)\left[3(n-1)p^{4}\right.$
$\displaystyle\left.+2p^{3}(-n(8n(2n-5)+6s+41)+9s+24)\right.$
$\displaystyle\left.-4p^{2}\left(16n^{2}(2s+5)-2n(27s+64)+6s^{2}+45s+69\right)\right.$
$\displaystyle\left.-16p\left(n\left(8s^{2}+42s+58\right)-s(7s+33)-42\right)-64(s+3)^{2}\right]~{}.$
(B.56)

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#  Why Scanning Tunneling Microscopy on Sr2RuO4 sometimes doesn’t see the
superconducting gap

Adrian Valadkhani<EMAIL_ADDRESS>Institut für Theoretische
Physik, Goethe-Universität, 60438 Frankfurt am Main, Germany  Jonas B. Profe
Institut für Theoretische Physik, Goethe-Universität, 60438 Frankfurt am Main,
Germany  Andreas Kreisel  Niels Bohr Institute, University of Copenhagen,
DK-2200 Copenhagen, Denmark  P.J. Hirschfeld  Department of Physics,
University of Florida, Gainesville, Florida 32611, USA  Roser Valentí
<EMAIL_ADDRESS>Institut für Theoretische Physik, Goethe-
Universität, 60438 Frankfurt am Main, Germany  Institut für Theoretische
Physik, Goethe-Universität, 60438 Frankfurt am Main, Germany  Institut für
Theoretische Physik, Goethe-Universität, 60438 Frankfurt am Main, Germany
Niels Bohr Institute, University of Copenhagen, DK-2200 Copenhagen, Denmark
Department of Physics, University of Florida, Gainesville, Florida 32611, USA
Institut für Theoretische Physik, Goethe-Universität, 60438 Frankfurt am Main,
Germany

###### Abstract

Scanning tunneling microscopy (STM) is perhaps the most promising way to
detect the superconducting gap size and structure in the canonical
unconventional superconductor Sr2RuO4 directly. However, in many cases,
researchers have reported being unable to detect the gap at all in simple STM
conductance measurements. Recently, an investigation of this issue on various
local topographic structures on a Sr-terminated surface found that
superconducting spectra appeared only in the region of small nanoscale
canyons, corresponding to the removal of one RuO surface layer. Here, we
analyze the electronic structure of various possible surface structures using
first principles methods, and argue that bulk conditions favorable for
superconductivity can be achieved when removal of the RuO layer suppresses the
RuO4 octahedral rotation locally. We further propose alternative terminations
to the most frequently reported Sr termination where superconductivity
surfaces should be observed.

## Introduction

The superconducting order parameter of Sr2RuO4 (Mackenzie_RMP_2003, ;
Maeno_JPSJ_2012, ; Kallin_IOP_2016, ; Mackenzie_npj_2017, ) is thought to be
of unconventional nature, but has proven unexpectedly difficult to identify.
Soon after its discovery in 1994, it was proposed as a promising candidate for
a chiral $p$-wave, spin triplet superconductor by analogy to superfluid 3He-A,
based in particular on early evidence from NMR (Ishida_Nature_1998, ). Muon-
spin rotation (Luke_Nature_1998, ) and Kerr effect (Xia_PRL_2006, )
measurements suggested intrinsic time-reversal symmetry (TRS) breaking
(Rice_1995, ) below $T_{c}$, consistent with this proposal. Thermodynamic
measurements provided clear evidence for low-energy quasiparticle states,
however, suggesting the existence of gap nodes or deep minima Bonalde_PRL_2000
; Mao_JPSJ_2000 ; Maeno_PRL_2004 ; Hassinger_PRX_2017 .

In 2019, the authors of Ref. Pustogow_Nature_2019  challenged the chiral
$p$-wave paradigm with in-plane 17O nuclear magnetic resonance measurements
that found a significant decrease in the Knight shift below $T_{\mathrm{c}}$
Pustogow_Nature_2019 ; Ishida_correct . Spin-triplet pairing was then
definitively ruled out by comparison to the change of the entropy from earlier
specific heat experiments (Brown_2020, ). These measurements were accompanied
by observations of shifts in the elastic constants Benhabib2021 ;
ghosh2020thermodynamic  together with experiments under strain Grinenko2021
suggesting a two component nature of the order parameter. All these results
led to renewed theoretical attempts to calculate the superconducting ground
state of Sr2RuO4 within a spin singlet picture (Astrid_PRL_2019, ;
Scaffidi_PRR_2019, ; Gingras_PRL_2019, ; suh2019, ; Kaba19, ; Ramires2019, ;
Acharya_CommPhys_2019, ; Kallin_PRB_2020, ; Romer2020, ;
romer2020fluctuationdriven, ; Romer2021, ; kivelson_npj, ; clepkens2021, ;
willa2021, ; Astrid3D2022, ; Merce2022, ; Henrik2022, ; palle2023constraints,
; hauck2023competition, ), leading to a variety of proposals, including the
even-parity 1D irreducible representation $B_{1g}(d_{x^{2}-y^{2}})$, multi-
component orders such as $d_{x^{2}-y^{2}}+ig_{xy(x^{2}-y^{2})}$ and
$s^{\prime}+id_{xy}$, as well as the 2D irreducible representation
$E_{1g}(d_{xz}+id_{yz})$. A two-component state is thought to be required to
explain ultrasound measurements Benhabib2021 ; ghosh2020thermodynamic  and
recent $\mu$SR experiments under strain Grinenko2021  (see however Ref.
Andersen2024 ; palle2023constraints ), but other recent measurements provide
evidence for a single order parameter component Li2021 ; Li2022 .

While the experiments cited above and many others provide indirect evidence in
support of one superconducting pairing channel or another, the community’s
ability to definitively identify the order parameter is severely hindered by
the difficulty of making direct measurements of the superconducting gap over
the Fermi surface. The extremely small gap ($|\Delta|\leq 350\,\mu$eV)
(Firmo2013, ; Madhavan_PNAS_2020, ) of the superconducting order parameter
$\Delta(\mathbf{k})$ in Sr2RuO4 means that angle-resolved photoemission
(ARPES) experiments do not currently have the fine energy and momentum
resolution to detect spectral features reflecting the gap. Scanning tunneling
spectroscopy experiments that detect Bogoliubov quasiparticle interference
(BQPI) provide, on the other hand, good momentum resolution and finer energy
resolution in the best circumstances. Recently, the BQPI technique was used
for Sr2RuO4 (Madhavan_PNAS_2020, ). This analysis, based on comparison with a
simple lattice calculation of the joint density of states, suggested a
$d_{x^{2}-y^{2}}$ superconducting gap symmetry for Sr2RuO4. A somewhat more
sophisticated calculation involving first-principles surface Wannier functions
was also compared to the same data Bhattacharyya2023 , but more than one gap
function appeared to fit nearly equally well. Nevertheless, BQPI appears to be
the best possibility of “directly” measuring the superconducting gap in this
canonical unconventional superconductor.

There is however one enduring, poorly understood puzzle regarding scanning
tunneling microscopy (STM) studies of superconducting Sr2RuO4. While some STM
measurements have reported signatures of superconductivity on the surface of
Sr2RuO4 for many years Suderow2009 ; Firmo2013 ; Madhavan_PNAS_2020 , others
on apparently equivalent surfaces under similar conditions are unable to
detect any gap at all kambara_scanning_2006 ; lupien_mk-stm_2011 ; Marques2021
; Madhavan_private . Recognizing that this question was an important one to
resolve in order to properly interpret STM data on the Sr2RuO4 surface, the
authors of Ref. Olivares2022 ; Mueller2023  performed a systematic study of
local STM conductance spectra at several distinct types of local topographic
structures on the surface with different termination layers. They found that
superconducting spectra with coherence peaks appeared only in the region of
small nanoscale “pits” on the surface, corresponding to the removal of one RuO
surface layer. Since the Sr2RuO4 surface is thought to be reconstructed in a
pattern of RuO6 octahedral rotation similar to that of bulk Sr3Ru2O7 and
calcium doped bulk Sr2RuO4 Nakatsuji2000 ; Matzdorf2002 ; Marques2021 ;
Kreisel2021 , it is natural to ask if the atomic layer on the surface might be
electronically different to that placed immediately underneath it, such that
conditions favorable to superconductivity are “masked”. In other words, is it
possible that superconductivity in Sr2RuO4 can never be observed on a
hypothetical perfect but reconstructed Sr2RuO4 surface, but is only revealed
when such pits form? Here we investigate the plausibility of such a scenario
by performing first-principles-based electronic structure simulations.

## Results

Figure 1:  Electronic structure of Sr Monolayer. (a) Band structure along the
high symmetry path of the folded Brillouin zone (BZ) in Sr2RuO4 and the
density of states (DOS) for the rotations $0^{\circ}$ (lightgray), $5^{\circ}$
(gray) and the optimal one (black). (b) Fermi surfaces in the folded BZ for
the same three rotations. We observe a Lifshits transition happening between
the $5^{\circ}$ (gray) and the optimal rotation (black). (c) $d$ orbitally
resolved DOS plotted with different shades of blue. The $d_{xy}$ orbitals are
the dominant orbitals at the Fermi level $E_{F}$.

In the following we examine how the surface termination and rotation angles of
the RuO6 octahedra in Sr2RuO4 affect its electronic structure employing
density functional theory (DFT). We determine the optimal rotation angle
depending on the type of termination, and perform large-scale canyon structure
simulations to investigate the above scenario predicting that pits in the
surface allow to observe signatures of bulk physics.

To understand better the changes induced by the octahedral rotation in Sr2RuO4
we first focus on a Sr-terminated monolayer for which we consider different
RuO6 rotation angles. The effect of the rotation on the band structure, Fermi
surface, total density of states (DOS) and projected DOS is visualized in Fig.
1 and is in agreement with previous reported results
chandrasekaran_engineering_2023 . As shown in Fig. 1, the octahedral rotation
pushes the van-Hove singularity (vHs) – located in the $d_{xy}$ orbital – from
a position above the Fermi level at $0^{\circ}$ rotation, as in the bulk case
tamai_high-resolution_2019 ; Gingras_PRL_2019 ; romer2020fluctuationdriven ,
to a position below the Fermi level at the optimal octahedral rotation value.
As a consequence of this, the $d_{xy}$-orbital dominated Fermi surface section
near the $\bar{M}$-point vanishes (see the band structure and Fermi surface
plotted in Fig. 1 (a)). Furthermore, a pocket at the $\bar{\Gamma}$ point in
the $d_{xy}$ orbital opens up at approximately $6^{\circ}$ octahedral
rotation. This feature has been identified as a source of suppressed
superconductivity in a recent theoretical investigation by some of the authors
Jonas2024  by performing functional renormalization group (FRG) calculations
for wannierized tight-binding models for the surface of Sr2RuO4.

In the same spirit and motivated by recent strain dependent experiments
Steppke_Science_2017 ; Watson_2018_strain ; Grinenko2021 ; Jerzembeck2022 ;
li2022elastocaloric  indicating a strong effect of the vHs position on
$T_{c}$, we investigate different possible realizations of surfaces with the
aim to recover bulk-like behavior in the surface layer, which would support
the scenario that the detection of superconductivity on the surface is
directly related to the surface having similar electronic properties as the
bulk. By cleaving such surfaces, surface sensitive probes, like STS, as
mentioned in the introduction, can then be utilized to image the
superconducting order parameter.

Figure 2: Slab terminations considered in this work. (a-b) depict the surface
and subsurface layers used for our slab calculations where the shaded gray
region on top marks the position of the void for the calculations. The labels
on the left (Sr-O or Ru-O) denote the layer composition. Panel (a) shows the
Sr-terminated, Ru-terminated and 2Sr-terminated slabs. The rotation of RuO6
octahedra is depicted on top of the Sr-termination figure. (b) Canyon-like
structure with 2Sr termination in the pit (2Sr Canyon). This canyon-like
structure has been reported in STM experiments Olivares2022 . The dark blue
and red horizontal bars are color indicators for Fig. 4. (c) Definition of the
RuO6 octahedra rotation angle.

We consider four different charge neutral surface configurations in Sr2RuO4
including the experimentally realized Sr termination Marques2021 , a Ru
termination, and a 2Sr termination (see Fig. 2 (a)). While the latter two
terminations, to our knowledge, have not yet been reported as a clean surface,
our findings indicate that trying to cleave or grow these is highly desirable
in order to restore bulk physics on the surface. As a fourth configuration, we
examine a canyon-like structure (Fig. 2 (b)) similar to the suggested pit in
Olivares2022  and investigate how the electronic structure of Sr2RuO4 is
affected thereby.

Figure 3:  Octahedral rotation $\boldsymbol{\theta}$ (see Fig. 2 (c) for the
angle definition) as a function of number of layers within the slab for both,
surface (continuous line) and subsurface (dashed line) for all three
terminations in Fig. 2 (a). Note that only as of a three-layer slab a
subsurface layer exists. We find a much stronger surface reconstruction in the
case of Sr termination (green solid curve) than for the Ru termination (blue,
in inset) and the 2Sr termination (red, inset). All subsurface layers show
negligible rotations. The exact values of the relaxation can be found in
Tables SI, SII and SIII.

The optimized RuO6 rotation angles for slabs with Sr surface termination (Fig.
2 (a), left panel) are summarized in Fig. 3 as a function of number of layers
$n$ (see also Tab. SI). We find that the octahedral rotation in both surface
and subsurface show rapid convergence to stable values with the number of
layers considered in the slab calculation, indicating the presence of small
interlayer couplings, as expected from resistivity measurements hussey_normal-
state_1998 . Incorporating spin-orbit coupling (SOC) in the structural slab
relaxations is not found to play a crucial role for the rotation angles, in
contrast to the role it plays for the electronic properties near the Fermi
surface tamai_high-resolution_2019  where it was found to be essential. In
other words, even though to reproduce the experimental Fermi-surface shape
spin-orbit coupling is essential, it has negligible effect on the structural
relaxation which is the central focus of this study. The optimal rotation
angle of $7.17^{\circ}$ at the surface layer obtained in non-relativistic
calculations is in agreement with experiments Matzdorf2000 ; Matzdorf2002  and
earlier theoretical investigations Veenstra2013 ;
chandrasekaran_engineering_2023 . Therefore, for the structure optimization we
neglect the effect of SOC in multilayered slabs. Furthermore, we checked that
structure optimization using different setups, like surface and subsurface
optimization or structural optimization of the full slab with different
depths, leads to changes in the rotation angle of the RuO6 octahedra of at
most $0.1^{\circ}$.

We next consider slabs with Ru termination (Fig. 2 (a) middle panel). In this
case the surface RuO6 octahedra don’t have top apex oxygens and the surface
reconstruction after relaxation is less severe. The optimized rotation angles
are therefore much smaller than in the previous case as summarized in Fig. 3
and in Tab. SII. The slabs exhibit once again a fast convergence in rotation
angles with number of layers considered within the slab (see inset of Fig. 3).

Figure 4:  DOS for the Ru atoms of the 2Sr canyon plotted in three setups. The
colors here correspond to the colors of the layers in Fig. 2 (b) for the
canyon. (a) shows the DOS of the top layer (blue, dashed) of the canyon for
all $d$ orbitals. (b) It is the same as (a) but for the subsurface layer (red,
solid). (c) Displays the comparison of the most dominant Ru orbital, $d_{xy}$,
at the Fermi level for the surface and subsurface layer. The dos are
normalized to one Ru atom.

We also performed relaxations by considering a 2Sr termination (Fig. 2 (a)
right panel), where the surface is assumed to be terminated at a SrO layer,
two layers away from the RuO plane. In this case we observe, similarly to the
Ru termination, a strong suppression of the RuO6 octahedral rotation in the
most outer RuO surface (see Fig. 3 and Table SII). The rotation angles again
converge quickly with the number of layers within the slab. As expected from
the electronic structure analysis of the effect of RuO6 octahedral rotations
in Sr2RuO4 (Fig. 1), we find bulk-like behavior in the absence of the
octahedral rotation for both the 2Sr and Ru terminations.

We proceed now with a more complex slab termination. As reported in Ref.
Olivares2022  imperfections in the cleaving process of the crystal, which
create small canyon-like structures, enable the observation of a
superconducting state in STM. Here, we perform DFT calculations for one such
case, a 2Sr canyon termination (Fig. 2 (b)). For that we choose a trade-off
between numerical feasibility and complexity of the structure. The canyon-like
structure is built from previously relaxed Sr-terminated slabs. After removing
the atoms in order to resemble the structures suggested in Ref. Olivares2022 ,
we perform a relaxation of the whole structure. We find that near the edges of
the canyon, a substantial displacement of the atoms occurs - while at the
bottom of the pit, the RuO6 octahedral rotation is nearly zero. Fig. 4 shows
the orbitally resolved DOS for the 2Sr canyon (Fig. 2 (b)) where the
contribution of the surface Ru is shown in shades of dashed blue (Fig. 4 (a))
and the contribution coming from subsurface Ru is shown in shades of solid red
(Fig. 4 (b)). Fig. 4 (c) depicts the comparison of the $d_{xy}$ orbital
projected DOS for the surface and subsurface. By this direct comparison it is
visible that the subsurface layer has a clear bulk-like feature, with a clear
peak close to the Fermi level (compare, for instance, with Ref. Marques2021 ,
and with Fig. 1 (c) left panel where the electronic structure of bulk Sr2RuO4
is emulated). As can be seen in Fig. 4 the influence of the RuO6 octahedral
rotation is quite drastic - the electronic structure of the subsurface Ru
layer retains the position of the bulk system’s van-Hove singularity while the
vHs of the surface Ru is shifted below the Fermi level, again resembling what
we observed in the Sr-terminated slab (Fig. 1). This indicates that the
subsurface electronic structure is the same as for bulk Sr2RuO4, since it is
not affected by the octahedra rotation. Therefore, bulk-like behavior is
expected, i.e. the layer should show superconductivity when the tip is placed
in the area of the canyon, as indicated by STM measurements Olivares2022 .

##  Discussion

In this work, we investigated via DFT calculations the surface reconstruction
of different charge-neutral surfaces of Sr2RuO4, including experimentally
observed canyon structures. Our motivation was to understand the microscopic
origin of contradictory reports of detection of superconductivity in STM
measurements of Sr2RuO4. The results of our calculations are condensed into
two main quantities which are (i) the structural properties in terms of the
behavior of the RuO6 octahedra at the surface and layers below and (ii) the
(partial) density of states of the Ru-d states in the corresponding layers. To
examine whether to expect a structure to have bulk-like behavior, we compare
the bulk and relaxed surface electronic structures. From experiments, we know
that the bulk system is superconducting Mackenzie_RMP_2003  while clean
surface samples show no superconductivity kambara_scanning_2006 ; lupien_mk-
stm_2011 ; Marques2021 ; Madhavan_private . In particular, we focus on the DOS
of the $d_{xy}$ orbital since it is the main contributor to the DOS near the
Fermi level, and to the van-Hove singularity driving the value of the $T_{c}$
as examined by strain experiments Steppke_Science_2017 ; Watson_2018_strain ;
Grinenko2021 ; Jerzembeck2022 ; li2022elastocaloric . It should be noted that
at large octahedral rotation ($\theta$ $\geq$ 6∘), which is the case for
experimentally reported Sr-terminated structures and found in our simulations,
the van-Hove singularity lies below the Fermi level (EF) and the DOS at EF is
predominantly provided by the new $d_{xy}$ pocket opening near the $\Gamma$
point (see Fig. 1 (b) right panel), in contrast to the bulk-like electronic
structure (no RuO6 rotation, Fig. 1 (b) left panel) where this pocket is
absent. Recent FRG calculations suggest that while carrying significant DOS,
this new pocket is irrelevant for the pairing Jonas2024 , and this might
explain the absence of superconductivity at such surfaces.

Using this qualitative understanding of how superconductivity is influenced by
the structural state of the Sr2RuO4 and by examining the signatures of the
density of states, we can indirectly connect the tendency towards
superconducting pairing to the structural state of the RuO6 octahedra. We find
that the density of states in the layers without rotation exhibits similar
positions of the van Hove peaks as that of an unreconstructed (bulk) system;
in contrast to layers with significant octahedra rotation, thus connecting the
electronic structures in the corresponding layers to the bulk electronic
structure. Our calculations corroborate that for Sr-terminated surfaces, the
RuO6 octahedra show a rotation of about $7^{\circ}$, in agreement with
previous investigations Matzdorf2000 ; Matzdorf2002 ; Veenstra2013 ;
chandrasekaran_engineering_2023 . However, for both 2Sr- and Ru-terminated
surfaces, no significant octahedral rotation was found.

This suggests that if these latter terminations could be successfully grown or
cleaved, superconducting surfaces should be observed. This in turn would allow
for direct measurements of the superconducting order parameter utilizing
Bogoliubov quasiparticle interference.

This suggestion is supported by our 2Sr canyon slab simulations resembling
reported surface imperfections in Ref. Olivares2022  where we found that the
octahedral rotation in the subsurface is essentially absent. Accordingly, the
electronic structure does not differ substantially from bulk - explaining why
within these canyons a superconducting gap is observed in STM measurements
Olivares2022 . Therefore, we argue that STM (or other surface) experiments do
not see the superconducting gap if they are located with their tip on the Sr-
terminated surface with a reconstructed surface that contains octahedral
rotation.

While so far there has been no report of a clean 2Sr or Ru surface
termination, our results suggest that fabricating such a termination could
help significantly in determining the gap structure in Sr2RuO4. Furthermore,
this could settle the longstanding debate of whether there is a second order
parameter or not ghosh2020thermodynamic ; Benhabib2021 ; Grinenko2021 ;
li2022elastocaloric ; palle2023constraints ; mueller2023constraints ; Li2021 .
While a relatively clean surface might be required to obtain Bogoliubov
quasiparticle interference patterns, the observation of a STM gap in canyon
defects already allows for a more thorough investigation of the gap structure.

## Methods

First principles calculations

We performed ab initio electronic structure calculations within density
functional theory (DFT) Hohenberg1964 ; Kohn1965  by using the Vienna Ab
initio Simulation Package (VASP) Kresse1993 ; Kresse1996 ; Kresse1996_2
within the pseudo-potential augmented plane-waveBloechl1994 ; Kresse1999
(PAW) basis set. The calculations were performed with the
Perdew–Burke–Ernzerhof (PBE) Perdew1996  exchange correlation functional as a
generalized gradient approximation (GGA), and a plane-wave cutoff of $800$ eV
was chosen. First the bulk structure was relaxed on a $16\times 16\times 4$
$k$-point mesh using the conventional unit cell of Sr2RuO4. The relaxed unit
cell was then transformed to a tetragonal $\sqrt{2}\times\sqrt{2}\times 1$
unit cell in order to have two inequivalent Ru sites and four inequivalent O
sites in the $ab$ plane. The convergence criterion was chosen to be $2\cdot
10^{-3}eV/$Å for these slabs and $1\cdot 10^{-3}eV/$Å for the bulk.

From the optimized bulk unit cell we generated slabs with up to 5 Ru layers
with Sr, 2Sr or Ru termination on both ends. A void of $15$ Å was set on top
of the surfaces. In these slab unit cells we rotated the RuO6 octahedra
starting form their initial position, i. e. $180^{\circ}$ between two
neighboring Ru atoms and the O in between, by adding a displacement
$\textbf{d}_{\text{O}}$ for the O atoms in the Ru plane dependent on the
rotation angle $\theta$

$\textbf{d}_{\text{O}}(\theta)=\begin{pmatrix}0&\tan(\theta)&0\\\
-\tan(\theta)&0&0\\\ 0&0&0\end{pmatrix}\cdot\textbf{r}_{\text{O}},$ (1)

where $\textbf{r}_{\text{O}}$ is the initial positions of the O atoms in the
Ru plane. For all slabs we optimized every pair of Ru layers symmetrically.
Crosscheck for non symmetric setups were done, however not found to be
optimal.

The scheme of optimizing was done carefully for all terminations as follows:
(i) We first started with one Ru layer. (ii) This was optimized first with a
low resolution energy landscape. (iii) Then we relocated and zoomed in to
restart with a higher resolution. (iv) We repeated this procedure until we had
a sufficient amount of resolution. (v) As a last step of refinement we let
VASP internally relax, to printout the forces and confirm that we are in a
minimum. (vi) Restart with an additional Ru layer, and use the previous
optimal rotation angle as new starting rotation angle. The $k$ mesh for
optimizing these structures was $6\times 6\times 1$.
For the 2Sr canyon structure, we started from the relaxed 3 layer Sr-
terminated slab and expanded the unit cell by $\sqrt{2}\times\sqrt{2}\times
1$. This supercell was then extended in one direction $3\times 1\times 1$.
Finally we dug a hole on both surfaces and started a relaxation. The forces
were reduced to $5\cdot 10^{-3}eV/$Å with a $k$-mesh of $2\times 6\times 1$.

## Acknowledgments

The authors are grateful to S. Mukherjee for valuable discussions in the early
stages of this project and V. Madhavan for providing valuable information
concerning STM measurements of Sr2RuO4. R.V., J.B.P. and A.V. gratefully
acknowledge support by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) for funding through Project No. 411289067 (VA117/15-1)
and TRR 288 — 422213477 (project A05). A.K. acknowledges support by the Danish
National Committee for Research Infrastructure (NUFI) through the ESS-
Lighthouse Q-MAT. P.J.H. acknowledges support from NSF-DMR-2231821.

#### Author Contributions

All authors contributed to the data analysis, discussions and writing of the
manuscript. A.V. performed the DFT simulations.

#### Data availability

All simulation data is available upon reasonable request.

## References

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Supplementary Information:
Why STM on Sr2RuO4 (sometimes) doesn’t see the gap

Supplementary Information:
Why STM on Sr2RuO4 (sometimes) doesn’t see the gap Adrian Valadkhani Jonas
Profe

Andreas Kreisel

Peter Hirschfeld

Roser Valentí

### Supplementary Note 1: Tables

No. Ru | type | RuL1 | RuL2 | RuL3 | RuL4 | RuL5
---|---|---|---|---|---|---
1 | all | 10.2837 |  |  |  |
2 | all | 7.3645 | 7.3644 |  |  |
3 | sur | 7.2825 | 0 | 7.2845 |  |
3 | all | 7.2672 | 0.1171 | 7.2662 |  |
4 | sur | 7.1673 | 0 | 0 | 7.1721 |
4 | all | 7.2929 | 0.0916 | 0.091 | 7.2789 |
5 | sur | 7.1709 | 0 | 0 | 0 | 7.1719
5 | sub | 7.2518 | 0.0959 | 0 | 0.0961 | 7.2519
5 | all | 7.2626 | 0.0734 | 0.0742 | 0.0735 | 7.2632
Table SI:  Here we list the rotation angles in ∘ in dependence of the number of Ru layers (No. Ru) for 1 layer (RuL1) to 5 layers (RuL5) with Sr termination. Each layer has several rows, which correspond to either a full relaxation (all), only surface (sur), surface and subsurface (sub).  No. Ru | type | RuL1 | RuL2 | RuL3 | RuL4 | RuL5
---|---|---|---|---|---|---
1 | all | 0 |  |  |  |
2 | all | 0 | 0 |  |  |
3 | sur | 0 | 0 | 0 |  |
3 | all | 0 | 0.0618 | 0 |  |
4 | sur | 0 | 0 | 0 | 0 |
4 | all | 0 | 0.0378 | 0.0375 | 0 |
5 | sur | 0 | 0 | 0 | 0 | 0
5 | sub | 0 | 0.0699 | 0 | 0.0696 | 0
5 | all | 0 | 0.0702 | 0.0001 | 0.0704 | 0
Table SII:  Here we list the rotation angles in ∘ in dependence of the number of Ru layers (No. Ru) for 1 layer (RuL1) to 5 layers (RuL5) with Ru termination. Each layer has several rows, which correspond to either a full relaxation (all), only surface (sur), surface and subsurface (sub).  No. Ru | type | RuL1 | RuL2 | RuL3 | RuL4 | RuL5
---|---|---|---|---|---|---
1 | all | 0.0292 |  |  |  |
2 | all | 0.106 | 0.1067 |  |  |
3 | sur | 0.0883 | 0 | 0.0881 |  |
3 | all | 0.0957 | 0.0002 | 0.0958 |  |
4 | sur | 0.0993 | 0 | 0 | 0.0993 |
4 | all | 0.0954 | 0.0045 | 0.0051 | 0.0956 |
5 | sur | 0.0882 | 0 | 0 | 0 | 0.0881
5 | sub | 0.0964 | 0.0009 | 0 | 0.0007 | 0.0961
5 | all | 0.0959 | 0.0069 | 0.0001 | 0.0069 | 0.096
Table SIII:  Here we list the rotation angles in ∘ in dependence of the number
of Ru layers (No. Ru) for 1 layer (RuL1) to 5 layers (RuL5) with 2Sr
termination. Each layer has several rows, which correspond to either a full
relaxation (all), only surface (sur), surface and subsurface (sub).  Figure
S1:  Rotation angle in dependence of freedom during the relaxation in DFT for
all 3 terminations of the 5 layer slabs.

Here we list all tables containing the optimized rotation angles from our DFT
calculations with VASP. The Sr termination is in SI, the Ru termination in SII
and the 2Sr termination in SIII. Fig. S1 shows the dependence of the rotation
angle on the freedom during the relaxation. This plot is generated for the 5
layer case and benchmarks the rotation angle of the RuO6 octahedra for the
surface, by letting the structure relax only at the surface (labeled as:
surface), using subsurface and surface (labeled as: subsurface) and also by
letting the full structure relax (labeled as: full). The rotation angle
converges very fast, showing very little deviation between surface and the
other relaxation types.

Comparison of bulk, monolayer, and canyon

Figure S2:  Comparison of the DOS of bulk, Sr-termination monolayer and 2Sr-
canyon for the Ru $d_{xy}$ orbital to verify, that all of them have the same
bulk like feature.

Here we show that the subsurface of the 2Sr canyon DOS has bulk-like features.
In Fig. S2 we visualize the DOS of the $d_{xy}$ Ru orbital. For all three
cases it shows a significant peak between 0.025 to 0.1 eV, which corresponds
to the vHs in the $d_{xy}$ orbital.

Ru canyon electronic structure

(a)

(b)

Figure S3: Results for Ru canyon (a) Comparison between surface and
subsurface, and (b) the Ru canyon.

A second pit proposed by the experiments in Olivares2022  is depicted in Fig.
S3. Instead of a 2Sr termination in the pit one has a Ru termination. Similar
to the 2Sr canyon (Fig. 4 main text) we decomposed the DOS in surface and
subsurface layer. The DOS of the surface with the Ru octahedra without the top
apex oxygen is very low. One the other hand one can see a nice agreement of
the DOS for the subsurface layer and the bulk, which has much higher DOS
values. Only the subsurface layer shows the expected bulk-like DOS.
Unfortunately, the DOS of the subsurface might be screened by the Ru layer in
the pit, therefore, visualizing the contributions of this canyon
experimentally might be more challenging.


11institutetext: Indian Institute of Science Education and Research (IISER)
Tirupati, India

# Elliptic flow of strange and multi-strange hadrons in isobar collisions at
$\sqrt{s_{\mathrm{NN}}}$ = 200 GeV at RHIC

Priyanshi Sinha(for the STAR Collaboration) 11
<EMAIL_ADDRESS>
###### Abstract

We report the elliptic flow ($v_{2}$) of $K_{s}^{0}$, $\Lambda$,
$\bar{\Lambda}$, $\phi$, $\Xi^{-}$, and $\overline{\Xi}^{+}$ at mid-rapidity
for Ru+Ru and Zr+Zr collisions at $\sqrt{s_{\mathrm{NN}}}$ = 200 GeV. The
transverse momentum ($p_{T}$) and centrality dependence of $v_{2}$ have been
studied. The number of constituent quark (NCQ) scaling for all the
(multi-)strange hadrons has also been tested. The dependence in the ratio of
average $v_{2}$ ($\langle$$v_{2}$$\rangle$) on centrality shows a deviation
from unity indicating a difference in nuclear structure and deformation
between the isobars. The results show a systematic size dependence when
compared to Cu+Cu, Au+Au, and U+U collisions at similar beam energies.

##  1 Introduction

A dedicated isobar collision run of ${}^{96}_{44}$Ru+${}^{96}_{44}$Ru and
${}^{96}_{40}$Zr+${}^{96}_{40}$Zr at $\sqrt{s_{\mathrm{NN}}}$ = 200 GeV was
successfully carried out at RHIC in 2018 to measure the charge separation
along the magnetic field, called the Chiral Magnetic Effect (CME) isobarData .
These collisions are considered to be an effective way to minimize the flow-
driven background contribution in search for the possibly small CME signal
expProp ; ReviewCME . The recent studies also show probing of nuclear
structures via $v_{2}$ ratios as well as the $v_{2}$-[$p_{T}$] correlations in
isobar collisions betaDeform ; v2ptcorrelation . The deformation parameters
are different between the two species and flow measurements are highly
sensitive to them. Strange and multi-strange hadrons have a small hadronic
cross-section compared to light hadrons, making their elliptic flow an
excellent probe for understanding the initial state anisotropies of these
isobar collisions.

##  2 Analysis details

In these proceedings, we report strange and multi-strange hadron $v_{2}$ in
Ru+Ru and Zr+Zr collisions at $\sqrt{s_{\mathrm{NN}}}$ = 200 GeV using the
data collected by the STAR experiment. A total of nearly 650M events have been
analysed for both the isobar collisions. $\phi$-mesons have been reconstructed
using the invariant mass technique through its hadronic decay channel:
$\phi\rightarrow K^{+}K^{-}$ ($48.9\pm 0.5\%$) in midrapidity $\left|y\right|$
$<$ 0.5 pdg . Event mixing technique is used for combinatorial background
estimation. The weakly decaying neutral strange particles $K_{s}^{0}$ and
$\Lambda(\bar{\Lambda})$ are reconstructed using invariant mass technique and
their weak-decay (V0) topology through the decay channel:
$K_{s}^{0}\rightarrow\pi^{+}+\pi^{-}$ ($69.20\pm 0.05\%$) and
$\Lambda\rightarrow p+\pi^{-}$ ($63.9\pm 0.5\%$), respectively pdg ; Strange .
The multi-strange particle $\Xi^{-}(\overline{\Xi}^{+})$ decays into a charged
daughter and a neutral V0 particle ($\Lambda$), which in turn decays into two
charged particles. Their reconstruction involves finding two secondary
vertices and the various topological selections. The combinatorial background
for the weakly decaying particles is constructed using rotational background
method multiStrange . The $\eta$-sub event plane method with an $\eta$ gap of
0.1 has been used to calculate $v_{2}$ of these (multi-)strange hadrons
Strange . The maximum EP resolution of nearly 48% is achieved for both the
collision systems.

##  3 Results

The left panels of Figs. 1 and 2 show the $v_{2}$ of strange and multi-strange
hadrons as a function of $p_{T}$ for minimum bias Ru+Ru and Zr+Zr collisions
at $\sqrt{s_{\mathrm{NN}}}$ = 200 GeV, respectively. An approximate mass
ordering at low $p_{T}$ and a baryon-meson splitting at intermediate $p_{T}$
have been observed. All particles and anti-particles tend to follow the number
of constituent quark (NCQ) scaling within 10% shown in the right panel of Fig.
1 and 2, indicating the partonic collectivity as well as domination of quark
coalescence mechanism during hadronization at intermediate $p_{T}$-region.

|
---|---
Figure 1: Left panel: $v_{2}$ as a function of $p_{T}$ of (multi-)strange hadrons; Right panel: NCQ-scaled $v_{2}$ as a function of transverse kinetic energy for Ru+Ru collisions at $\sqrt{s_{\mathrm{NN}}}$ = 200 GeV. The vertical lines and shaded boxes denote statistical and systematic uncertainties, respectively. |
---|---
Figure 2: Left panel: $v_{2}$ as a function of $p_{T}$ of strange hadrons;
Right panel: NCQ-scaled $v_{2}$ as a function of transverse kinetic energy for
Zr+Zr collisions at $\sqrt{s_{\mathrm{NN}}}$ = 200 GeV. The vertical lines and
shaded boxes denote statistical and systematic uncertainties, respectively.

A clear centrality dependence of $v_{2}$ has been observed for $K_{s}^{0}$,
$\Lambda$ and $\Xi^{-}$ as shown in Fig. 3 for the isobar collision systems.

|
---|---
|
|
Figure 3: Left panel: Centrality dependence of $v_{2}$ of $\Lambda$ as a
function of $p_{T}$ in Ru+Ru collisions; Right Panel: Same for Zr+Zr
collisions at $\sqrt{s_{\mathrm{NN}}}$ = 200 GeV. The vertical lines and
shaded boxes denote statistical and systematic uncertainties, respectively.

We also study the $p_{T}$-integrated $v_{2}$ for strange hadrons as a function
of the collision centrality in the isobar collisions as shown in Fig. 4. The
ratios of $v_{2}$ between the two isobar collisions for $K_{s}^{0}$,
$\Lambda$, and $\bar{\Lambda}$ are comparable with charged hadron data and
show a deviation of nearly 2% from unity in mid-central collisions, indicating
a difference in nuclear structure and shape isobarData .

Figure 4: $p_{T}$-integrated $v_{2}$ as a function of centrality for
$K_{s}^{0}$, $\Lambda$, and $\bar{\Lambda}$ in Ru+Ru and Zr+Zr collisions at
$\sqrt{s_{\mathrm{NN}}}$ = 200 GeV. The vertical lines on the ratio includes
statistical and systematic uncertainties. The dotted lines denotes the fitting
with a constant. Figure 5: $v_{2}$ of strange hadrons in minimum bias Cu+Cu,
Ru+Ru, Zr+Zr, Au+Au collisions at $\sqrt{s_{\mathrm{NN}}}$ = 200 GeV and U+U
collisions at $\sqrt{s_{\mathrm{NN}}}$ = 193 GeV CuData ; AuData ; UData .

We studied the system size evolution of $v_{2}$ by comparing the minimum bias
${}^{63}_{29}$Cu+${}^{63}_{29}$Cu, ${}^{96}_{44}$Ru+${}^{96}_{44}$Ru,
${}^{96}_{40}$Zr+${}^{96}_{40}$Zr, ${}^{197}_{79}$Au+${}^{197}_{79}$Au
collisions at $\sqrt{s_{\mathrm{NN}}}$ = 200 GeV, and
${}^{238}_{92}$U+${}^{238}_{92}$U collisions at $\sqrt{s_{\mathrm{NN}}}$ = 193
GeV CuData ; AuData ; UData . Figure 5 shows an approximate system size
dependence of $v_{2}$ for $p_{T}$ $>$ 1.8 GeV/$c$, based on the nuclear size.
$v_{2}$ in U+U and Au+Au is observed to be higher, whereas that in Cu+Cu is
slightly lower than those in isobar collisions.

##  4 Summary

In summary, we have presented the elliptic flow of $K_{s}^{0}$, $\Lambda$,
$\bar{\Lambda}$, $\phi$, $\Xi^{-}$, and $\overline{\Xi}^{+}$ in Ru+Ru and
Zr+Zr collisions at $\sqrt{s_{\mathrm{NN}}}$ = 200 GeV. We observed a mass
ordering at low $p_{T}$ and a baryon-meson splitting at intermediate $p_{T}$
in both isobar species. All the strange particles and anti-particles follow
the NCQ scaling indicative of the partonic degrees of freedom and coalescence
picture for hadronization. The $p_{T}$-integrated $v_{2}$ ratio of strange
hadrons shows a deviation of about 2% from unity. We also observed a system
size dependence of $v_{2}$ when comparing to different collision systems at
similar beam energy. These measurements provide further constraint on the
deformation and nuclear density difference in the two isobar nuclei.

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