# Certifying entanglement of spins on surfaces using ESR-STM

Y. del Castillo1,2, J. Fernández-Rossier1111On permanent leave from
Departamento de Física Aplicada, Universidad de Alicante, 03690 San Vicente
del Raspeig<EMAIL_ADDRESS>1International Iberian
Nanotechnology Laboratory (INL), Av. Mestre José Veiga, 4715-330 Braga,
Portugal  2Centro de Física das Universidades do Minho e do Porto,
Universidade do Minho, Campus de Gualtar, 4710-057 Braga, Portugal

###### Abstract

We propose a protocol to certify the presence of entanglement in artificial
on-surface atomic and molecular spin arrays using electron spin resonance
carried by scanning tunnel microscopes (ESR-STM). We first generalize the
theorem that relates global spin susceptibility as an entanglement witness to
the case of anisotropic Zeeman interactions, relevant for surfaces. We then
propose a method to measure the spin susceptibilities of surface-spin arrays
combining ESR-STM with atomic manipulation. Our calculations show that
entanglement can be certified in antiferromagnetically coupled spin dimers and
trimers with state of the art ESR-STM magnetometry.

Very much like electricity and magnetism, quantum entanglement is a naturally
occurring phenomena. Entanglement lies at the heart of the most intriguing
aspects of quantum mechanicsBell and Bell (2004), such as quantum
teleportationBennett  _et al._ (1993); Bouwmeester  _et al._ (1997); Boschi
_et al._ (1998), non-locality and the emergence of exotic quantum states of
matterKitaev and Preskill (2006). The notion that quantum entanglement is also
a resource that can be exploited is at the cornerstone of the fields of
quantum computingNielsen and Chuang (2002), quantum sensingDegen  _et al._
(2017) and quantum communicationsWootters (1998). In order to harness quantum
entanglement, it seems imperative to develop tools to probe itFriis  _et al._
(2019). This can be particularly challenging at the nanoscaleHeinrich  _et
al._ (2021).

Quantum entanglement is predicted to occur spontaneously in interacting spin
systemsArnesen  _et al._ (2001); Amico and Fazio (2009), going from small
clusters to crystals. In the last few decades, tools to create and probe
artificial spin arrays based on both magnetic atoms and molecules on surface
have been developedHirjibehedin  _et al._ (2006); Khajetoorians  _et al._
(2019); Choi  _et al._ (2019). There is now a growing interest in the
exploitation of quantum behaviour of this class of systemsDelgado and
Fernández-Rossier (2017); Chen  _et al._ (2022). Surface-spin artificial
arrays can display exotic quantum magnetism phenomena, including spin
fractionalization in $S=1$ Haldane chainsMishra  _et al._ (2021), resonant
valence bond states in spin plaquetesYang  _et al._ (2021) and quantum
criticalityToskovic  _et al._ (2016) that are known to host entangled states.
However, experimental protocols that unambiguously determine the presence of
entanglement in on-surface spins are missing.

Here we propose a protocol to certify the presence of entanglement on
artificial arrays of surface spins. The approach relies on two ideas. First,
it was both proposedWieśniak  _et al._ (2005); Brukner  _et al._ (2006);
Vedral (2008) and demonstrated experimentallyBrukner  _et al._ (2006); Sahling
_et al._ (2015) in bulk systems, that the trace of the spin susceptibility
matrix can be used as an entanglement witness. Second, the developmentBaumann
_et al._ (2015) of single-atom electron spin resonance using scanning
tunneling microscopy (ESR-STM) that, combined with atomic manipulation, has
made it possible to carry out absolute magnetometryNatterer  _et al._ (2017);
Choi  _et al._ (2017) of surface spins. As illustrated in the scheme of Figure
1, the magnetic field created by the different quantum states of the surface
spins can be measured by an ESR-STM active atomic spin sensor placed nearby.
The accurate determination of the height of the corresponding peaks and their
shift makes it possible to pull out both their occupation and magnetic moment,
which permits one to determine the spin susceptibility.

The global spin susceptibility is defined as the linear coefficient that
relates the external magnetic field to the average total magnetization

$\langle M_{\alpha}\rangle=\chi_{\alpha}B_{\alpha}$ (1)

where $\langle\rangle$ stands for statistical average in thermal equilibrium,
and

$M_{\alpha}=\sum_{i}m_{\alpha}^{i}=-\mu_{B}\sum_{i}g_{\alpha}S_{\alpha}^{i}$
(2)

where $\alpha=x,y,z$ labels the principal axis that diagonalize the
susceptibility tensor and $i$ labels the site in a given spin lattice.

Figure 1: Scheme of the proposed STM-ESR experimental protocol. (a)
Characterization of the bare resonance frequency $f_{0}$ of the sensor. (b)
Determination of the $g$ coefficients of a single unit (atom) of the on-
surface spin array. (c) ESR-STM readout of the bare sensor at two different
fields. (d) ESR-STM readouts of a sensor nearby antiferromagnetically coupled
dimer (top) and a trimer (bottom), with $J=124\mu$eV and $T=1$ K, using
different magnetic fields that lead to different occupations of the states.
(e) and (f) Energy spectra of the dimer and trimer as a function of $B$.

The interaction between the spins of interest and the external field is given
by: ${\cal H}_{1}=-\sum_{\alpha}M_{\alpha}B_{\alpha}$. For spin-rotational
invariant Hamiltonians that commute with the Zeeman operator, $[{\cal
H}_{0},{\cal H}_{1}]=0$, the spin susceptibility and the statistical variance
of the total magnetization, $\Delta M_{\alpha}^{2}$ are related by Wieśniak
_et al._ (2005)333See supplementary material for a derivation:

$\chi_{\alpha}=\beta\Delta M_{\alpha}^{2}=\beta\left(\sum_{i,j}\langle
m_{\alpha}^{i}m_{\alpha}^{j}\rangle-(\sum_{i}\langle
m_{\alpha}^{i}\rangle)^{2}\right)$ (3)

where $\beta=\frac{1}{k_{B}T}$ . The variance for an isolated spin satisfies
$\sum_{\alpha}g_{\alpha}^{2}\Delta S_{\alpha}^{2}\geq g_{\rm min}^{2}S$ where
$g_{\rm min}={\rm Min}(g_{x},g_{y},g_{z})$. For a multi-spin system it has
been shownHofmann and Takeuchi (2003) that, for non-entangled states, the
variance of total magnetization is at least as large as the sum over variances
of individual magnetizations. We thus can write the equation:

$\displaystyle{\rm
Tr}\,\chi=\sum_{\alpha}\chi_{\alpha}=\chi_{x}+\chi_{y}+\chi_{z}=$
$\displaystyle=\beta\sum_{\alpha}\left(\sum_{i,j}\langle
m_{\alpha}^{i}m_{\alpha}^{j}\rangle-(\sum_{i}\langle
m_{\alpha}^{i}\rangle)^{2}\right)\geq\frac{g_{\rm
min}^{2}\mu_{B}^{2}S}{k_{B}T}N$ (4)

where $N$ is the number of spins and $S$ is the spin quantum number
($S=\frac{1}{2},1,...$). Equation (4) generalizes the result derived by
Wiesnak et al.Wieśniak  _et al._ (2005) to the case of anisotropic $g$ factor,
relevant for surface spinsFerrón  _et al._ (2019), and constitutes the
starting point to our proposed entanglement certification method. If the spin-
susceptibilities are measured, and violate the inequality of eq. (4), the
presence of entanglement is certified. Equation (4) is a sufficient condition
for entanglement, but not necessary: entanglement may be present, but go un-
noticed, if the susceptibility satisfies eq. (4).

We discuss now how to certify entanglement using a single STM-ESR-active atom,
that will be used as a sensor, placed nearby the spin array of interest.
Specifically, we consider the case of dimersYang  _et al._ (2017); Kot  _et
al._ (2022) and trimersYang  _et al._ (2021) of antiferromagnetically coupled
$S=1/2$ spins on MgOYang  _et al._ (2021), the canonical surface for STM-ESR.
The choice of $S=1/2$ rules out single ion anisotropies. In the suppl.
material we show how the role of dipolar interactions can be neglected in most
cases. Examples of $S=1/2$ adsorbates deposited on MgO include TiH Yang  _et
al._ (2017, 2018, 2019); Seifert  _et al._ (2020); Steinbrecher  _et al._
(2021); Veldman  _et al._ (2021); Kot  _et al._ (2022), CuYang  _et al._
(2018), dimers of alkali atomsKovarik  _et al._ (2022) and even organic
moleculesLutz  _et al._ (2022).

The spectra of the dimer and trimer are shown in Figure 1(e,f) as a function
of the applied field. The dimer shows a singlet ground state with wave
function
$\frac{1}{\sqrt{2}}\left(|\uparrow\downarrow\rangle-|\downarrow\uparrow\rangle\right)$
and a triplet of excited states $(S=1,S_{z})$. The $S=1,\,S_{z}=\pm 1$ are
product states, whereas the $S=1,\,S_{z}=0$ is entangled. The trimer ground
state is a $S=1/2$ doublet with wave functions:

$|\frac{1}{2},+\frac{1}{2}\rangle=-\frac{1}{\sqrt{6}}\left(|\uparrow\uparrow\downarrow\rangle+|\downarrow\uparrow\uparrow\rangle\right)+\sqrt{\frac{2}{3}}|\uparrow\downarrow\uparrow\rangle$
(5)

and an analogous formula for $|S=\frac{1}{2},S_{z}=-\frac{1}{2}\rangle$.
Therefore, both structures, dimer and trimer, have entangled ground states.
The lowest energy excited state, with excitation energy $J$, is also a $S=1/2$
doublet, whereas the highest energy multiplet has $S=3/2$ and excitation
energy $\frac{3J}{2}$.

ExperimentsChoi  _et al._ (2017) show that the ESR spectrum, i.e., the
$I_{DC}(f)$ curve of the sensor adatom, under the influence of a nearby spin-
array, can be described by:

$\displaystyle I_{DC}(f)=\sum_{n}p_{n}L(f-f_{n})$ (6)

where the sum runs over the eigenstates of the Hamiltonian of the spin array,
$\left({\cal H}_{0}+{\cal H}_{1}\right)|n\rangle=E_{n}|n\rangle$, $L$ is a
lorentzian curve centered around the resonant frequency $f=f_{n}$, given by
the expression

$f_{n}(\vec{r})=\frac{\mu_{B}}{h}\sqrt{\sum_{\alpha}\left[g_{\alpha}^{\rm
sensor}(B_{\alpha}+b_{n,\alpha}(\vec{r}))\right]^{2}}$ (7)

Here, $h=2\pi\hbar$, $g_{\alpha}^{\rm sensor}$ are the components of the
gyromagnetic tensor of the sensor, $B_{\alpha}$ represents the the external
field and $b_{n,\alpha}(\vec{r})$ is the $\alpha$ component of the the stray
field generated by the spin-array in the state $n$. The vector is given by

$\vec{b}_{n}(\vec{r})=\frac{\mu_{0}}{4\pi}\sum_{i=1,N}\frac{3(\vec{m}^{i}(n)\cdot\vec{d}^{i})\vec{d}^{i}}{(d^{i})^{5}}-\frac{\vec{m}^{i}(n)}{(d^{i})^{3}}$
(8)

where $\vec{r}^{i}$ are the positions of the surface spins and $\vec{r}$ is
the position of the sensor, $\vec{d}^{i}=\vec{r}-\vec{r}^{i}$ and
$\vec{m}^{i}(n)$ is a vector given by with components by

$m_{\alpha}^{i}(n)=-g_{\alpha}\mu_{B}\langle n|S_{\alpha}i|n\rangle$ (9)

the encode the quantum-average magnetization of the atomic moments of the spin
array for state $n$. Equation 6 implicitly assumes the absence of resonant
spin-flip interactions between the sensor spin and the spin-array.

In Figure 1d we show the spectra for a sensor nearby a dimer and a trimer of
antiferromagnetically coupled spins, in the presence of a external field along
the $z$ direction, assuming a broadening $\Delta f=5\,\rm{MHz}$ Baumann  _et
al._ (2015). The frequency shift of the $n$ peak, with respect to the bare
sensor, are governed by the stray field $\vec{b}_{n}(\vec{r})$ generated by
that state. For the dimer, the ESR-spectrum shows a prominent peak, that has
contributions from both the $S=0,\,S_{z}=0$ and $S=1,\,S_{z}=0$ states, whose
stray field vanish. As a result, the prominent peak has the frequency of the
bare sensor. The two smaller peaks, symmetrically located around the central
peak, correspond to the $S=1,S_{z}=\pm 1$ states. In the case of the trimer up
to eight different peaks appear in our simulation, as the different doublets
generate states with different magnetization and stray field.

We now express the average magnetization and the susceptibility in terms of
the occupations $p_{n}$ and the magnetic moment of each state, the two
quantities that can be obtained from ESR-STM. For spin-rotational invariant
systems, we write eq. (1) as:

$\langle M_{\alpha}\rangle=\sum_{n}p_{n}M_{\alpha}(n)$ (10)

where $p_{n}=\frac{1}{Z}e^{-\beta E_{n}}$ is the equilibrium occupation of the
$n$ eigenstate and $M_{\alpha}(n)=-g_{\alpha}\mu_{B}S_{\alpha}(n)$, and
$S_{\alpha}(n)$ is the (half-)integer projection of the total spin operator of
the spin-array along the $\alpha$ direction. The susceptibility would be given
by

$\chi_{\alpha}=\sum_{n}\frac{dp_{n}}{dB_{\alpha}}M_{\alpha}(n).$ (11)

The experimental protocol to determine the $p_{n}$ is the followingChoi  _et
al._ (2017). For an ESR spectrum with $N_{p}$ visible peaks, we obtain the
ratios $r_{n}=\frac{I_{n}}{I_{0}}$ where $I_{0}$ corresponds to the largest
peak that arises from a single state. For trimers, $I_{0}$ is the peak that
arises from the ground state. For dimers, the ground state peak has also
contributions from the $S=1,S_{z}=0$ state and therefore we normalize with
respect to the $S=1,S_{z}=-1$ peak. It has been demonstrated Choi  _et al._
(2017) that the ratio of thermal occupations is the same than the ratio of
Boltzman factors $r_{n}=\frac{p_{n}}{p_{0}}=\frac{I_{n}}{I_{0}}$. Assuming
that the peaks exhaust the occupations of the states, we obtain $p_{n}$ out of
the ratios (see suppl. mat.)

The determination of the $M_{\alpha}(n)$ can be done in two ways. First, the
(half-)integer values of $S_{\alpha}$ could be assigned by inspection of the
ESR spectrum relying on the fact that the states with largest $S_{\alpha}$
have larger stray field. We have verifieddel Castillo and Fernández-Rossier
the feasibility of a second approach where the readout of the ESR spectra is
repeated with the sensor in different locations, mapping the stray field of
every state in real space, and pulling out the atomic magnetization averages,
inferring $M_{\alpha}(n)$ thereby.

The protocol to measure the spin susceptibility of a spin array has the
following steps, that have to be implemented for the three orientations
$\alpha=x,y,z$ of the external field: 1\. The bare resonant frequencies for
the sensor $f_{0\alpha}=\frac{\mu_{B}}{h}g_{\alpha}^{\rm sensor}B_{\alpha}$
(Figure 1a) are determined in the absence of the spin array.

2\. The $g_{\rm min}$ for the individual spins that form the array are
determined. If they are ESR-STM active, a conventional resonance experiment is
carried out, for the three directions of the field. Otherwise, the nearby ESR-
STM sensor atom can be used Choi  _et al._ (2017).

3\. The $M_{\alpha}(n)$ and $p_{n}$ are obtained as explained above, for two
values of $B_{\alpha}^{(1)}$ and $B_{\alpha}^{(2)}$. This permits us to obtain
$\langle M_{\alpha}(B_{\alpha}^{(1)})\rangle$ and $\langle
M_{\alpha}(B_{\alpha}^{(2)})\rangle$ for both fields and
$\chi_{\alpha}=\frac{\langle M_{\alpha}(B_{\alpha}^{(2)})\rangle-\langle
M_{\alpha}(B_{\alpha}^{(1)})\rangle}{B_{\alpha}^{(2)}-B_{\alpha}^{(1)}}$ and
compare to eq. (4) to establish the presence of entanglement.

In the top panels of figure 2 we show the spin susceptibility for dimers and
trimers as a function of temperature, together with the EW boundary, assuming
isotropic factors, $g_{\alpha}=2$. We define the temperature $T^{*}$, below
which the trace of the susceptibility matrix is smaller than the entanglement
witness limit, so that equation (4) is not satisfied, and therefore the
presence of entanglement is certified. The $\rm{Tr}\,\chi(T)$ curves are very
different for dimer and trimer. At low T, the dimer is in the singlet state
and its susceptibility vanish. In contrast, the trimer with $J>>k_{B}T$, the
excited states play no role and the spin susceptibility is governed by the two
states of the $S=1/2$ ground state doublet.

Figure 2:  Entanglement certification window (see text) for a dimer (a) and
trimer (b) with with $J=124\mu$eV, using a sensor located 0.9 nm away. Top
panels show $\rm{Tr}\,\chi(T)$ and the EW condition of eq. (4). Color map
inside the ECWs represents the largest experimental error $\Delta I/I_{0}$
that still makes entanglement certification possible, assuming $\eta=5$ in eq.
12.

The implementation of the entanglement certification protocol will only be
possible if the stray fields induce a shift larger than the spectral
resolution of ESR-STM. For ESR-STM , $\Delta f\simeq 4$MHzBaumann  _et al._
(2015), determined by the $T_{1},T_{2}$ times of the adsorbate in that system.
In the simulations of Figure 1 we have used $\Delta f=5MHz$. In addition, the
experiment has to meet four experimental conditions. Condition $I$:
temperature $T$ has to be smaller than $T^{*}$. Condition $II$:
$M_{\alpha}(B_{\alpha})$ has to be linear. We define $B^{*}$ as the upper
value of the field above which $\frac{dM_{\alpha}}{dB_{\alpha}}$ deviates from
its value for $B=0$ more than 5 percent.

Condition $III$: the relative error in the current of the reference peak
$\frac{\Delta I}{I_{0}}$ should be small enough to allow for the accurate
entanglement certification. Specifically, the error in the spin susceptibility
associated to $\frac{\Delta I}{I_{0}}$, should be smaller, by a factor
$\eta>>1$, than the difference between $\chi_{\alpha}$ and the EW condition.
In the suppl. mat. show this condition is

$\frac{\Delta I}{I_{0}}<\frac{B|{\rm Tr\,}\chi-\chi_{EW}|}{6\eta
p_{0}\sum_{n}\left(\left(2+r_{n}+(N_{p}-2)p_{0}\right)|M_{z}(n)|\right)}.$
(12)

Condition $IV$: the external field must not drive the sensor resonant
frequency above the excitation bandwidth of ESR-STM. The record, so far, is
$f_{0}=61GHz$ Kot  _et al._ (2022), that translates into $B\simeq 2.2$ T for
$S=1/2$ Ti-H adsorbates.

The independent satisfaction of conditions $I$ to $IV$ occurs only in a region
of the $(B,T)$ plane, defining four boundary lines, one per condition. We
define the entanglement certification window (ECW) as the area where the four
conditions are satisfied simultaneously. In Figure 2 we show the ECW, assuming
$\eta=5$ and imposing for condition $III$ that $\Delta I/I_{0}$ can not go
below $10^{-3}$, for the case of a dimer (a) and a trimer (b) with the same
$J$ reported by Yang et al.Yang  _et al._ (2021) $J=124\mu eV$ (30 GHz). The
colour code signals the maximal $\Delta I/I_{0}$ that satisfies eq. (12).

Figure 3: (a) $T^{*}(J)$ for different chain sizes.(b,c) Temperature and $J$
dependence of the spin susceptibility (b) and $B^{*}$ (c) of dimer (top) and
trimer (bottom).

We now discuss the effect of the value of $J$ on the size of the ECW. In
Figure 3a we show that $T^{*}$ is strictly linear with $J$ for spin chains
with $N$ sites, and the slope is an increasing function of the number of $N$.
Therefore, both long chains and large $J$ increase the crossover temperature.
As of condition two, our numerical simulations show that $B^{*}$ is an
increasing function of both $T$ and $J$: the magnetization vs field curves
remain linear as long as the Zeeman energy is smaller than the other two
relevant energy scales $J$ and $T$. Condition three is governed both by the
ratio $B/p_{0}(B,T)$ (see eq. 12 ) and the susceptibility. In Figure 3b we
show the $J$ dependence of the $\chi(T)$ curves. We see that reducing $J$
tends to increase $\chi$, at a fixed value of $T$. Again, this reflects the
competition between exchange and Zeeman energies. Therefore, increasing $J$
pushes $T^{*}$ up and pushes the $\chi(T)$ away from the EW boundary. The only
advantage of smaller values of $J/K_{B}T$ is the presence of more peaks in the
$I(f)$ spectra that makes the spectra clearly distinguishable from simpler
structures.

In conclusion, we propose a method to determine the presence of quantum
entanglement in on-surface spin arrays using ESR-STM. For that matter, the
trace of the spin susceptibility of the spin arrays has to be measured taking
advantage of the STM potential i) to manipulate atoms and build spin
structures, and ii) to simultaneously resolve the stray fields and occupation
probabilities of several quantum states of nearby spinsChoi  _et al._ (2017).
Our calculations show that the experiments can be carried out with the state
of the art ESR-STM instrumentation in terms of both spectral resolution
($\Delta f$) and relative error in the current $\Delta I/I_{0}$Choi  _et al._
(2017). The use of STM to certify the presence of entanglement would thus add
a new functionality to this versatile tool and would open a new venue in
quantum nanoscienceHeinrich  _et al._ (2021).

We acknowledge Antonio Costa for useful discussions. J.F.R. acknowledges
financial support from FCT (Grant No. PTDC/FIS-MAC/2045/2021), SNF Sinergia
(Grant Pimag), FEDER /Junta de Andalucía, (Grant No. P18-FR-4834), Generalitat
Valenciana funding Prometeo2021/017 and MFA/2022/045, and funding from MICIIN-
Spain (Grant No. PID2019-109539GB-C41). YDC acknowledges funding from FCT,
QPI, (Grant No. SFRH/BD/151311/2021) and thanks the hospitality of the
Departamento de Física Aplicada at the Universidad de Alicante.

## References

  * Bell and Bell (2004) J. S. Bell and J. S. Bell,  _Speakable and unspeakable in quantum mechanics: Collected papers on quantum philosophy_ (Cambridge university press, 2004).
  * Bennett  _et al._ (1993) C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, Physical Review Letters 70, 1895 (1993).
  * Bouwmeester  _et al._ (1997) D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, Nature 390, 575 (1997).
  * Boschi  _et al._ (1998) D. Boschi, S. Branca, F. De Martini, L. Hardy, and S. Popescu, Physical Review Letters 80, 1121 (1998).
  * Kitaev and Preskill (2006) A. Kitaev and J. Preskill, Physical Review Letters 96, 110404 (2006).
  * Nielsen and Chuang (2002) M. A. Nielsen and I. Chuang, “Quantum computation and quantum information,” (2002).
  * Degen  _et al._ (2017) C. L. Degen, F. Reinhard, and P. Cappellaro, Reviews of Modern Physics 89, 035002 (2017).
  * Wootters (1998) W. K. Wootters, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 356, 1717 (1998).
  * Friis  _et al._ (2019) N. Friis, G. Vitagliano, M. Malik, and M. Huber, Nature Reviews Physics 1, 72 (2019).
  * Heinrich  _et al._ (2021) A. J. Heinrich, W. D. Oliver, L. M. Vandersypen, A. Ardavan, R. Sessoli, D. Loss, A. B. Jayich, J. Fernandez-Rossier, A. Laucht, and A. Morello, Nature Nanotechnology 16, 1318 (2021).
  * Arnesen  _et al._ (2001) M. Arnesen, S. Bose, and V. Vedral, Physical Review Letters 87, 017901 (2001).
  * Amico and Fazio (2009) L. Amico and R. Fazio, Journal of Physics A: Mathematical and Theoretical 42, 504001 (2009).
  * Hirjibehedin  _et al._ (2006) C. F. Hirjibehedin, C. P. Lutz, and A. J. Heinrich, Science 312, 1021 (2006).
  * Khajetoorians  _et al._ (2019) A. A. Khajetoorians, D. Wegner, A. F. Otte, and I. Swart, Nature Reviews Physics 1, 703 (2019).
  * Choi  _et al._ (2019) D.-J. Choi, N. Lorente, J. Wiebe, K. Von Bergmann, A. F. Otte, and A. J. Heinrich, Reviews of Modern Physics 91, 041001 (2019).
  * Delgado and Fernández-Rossier (2017) F. Delgado and J. Fernández-Rossier, Progress in Surface Science 92, 40 (2017).
  * Chen  _et al._ (2022) Y. Chen, Y. Bae, and A. J. Heinrich, Advanced Materials , 2107534 (2022).
  * Mishra  _et al._ (2021) S. Mishra, G. Catarina, F. Wu, R. Ortiz, D. Jacob, K. Eimre, J. Ma, C. A. Pignedoli, X. Feng, P. Ruffieux, J. Fernandez-Rossier, and R. Fasel, Nature 598, 287 (2021).
  * Yang  _et al._ (2021) K. Yang, S.-H. Phark, Y. Bae, T. Esat, P. Willke, A. Ardavan, A. J. Heinrich, and C. P. Lutz, Nature Communications 12, 1 (2021).
  * Toskovic  _et al._ (2016) R. Toskovic, R. Van Den Berg, A. Spinelli, I. Eliens, B. Van Den Toorn, B. Bryant, J.-S. Caux, and A. Otte, Nature Physics 12, 656 (2016).
  * Wieśniak  _et al._ (2005) M. Wieśniak, V. Vedral, and Č. Brukner, New Journal of Physics 7, 258 (2005).
  * Brukner  _et al._ (2006) Č. Brukner, V. Vedral, and A. Zeilinger, Physical Review A 73, 012110 (2006).
  * Vedral (2008) V. Vedral, Nature 453, 1004 (2008).
  * Sahling  _et al._ (2015) S. Sahling, G. Remenyi, C. Paulsen, P. Monceau, V. Saligrama, C. Marin, A. Revcolevschi, L. Regnault, S. Raymond, and J. Lorenzo, Nature Physics 11, 255 (2015).
  * Baumann  _et al._ (2015) S. Baumann, W. Paul, T. Choi, C. P. Lutz, A. Ardavan, and A. J. Heinrich, Science 350, 417 (2015).
  * Natterer  _et al._ (2017) F. D. Natterer, K. Yang, W. Paul, P. Willke, T. Choi, T. Greber, A. J. Heinrich, and C. P. Lutz, Nature 543, 226 (2017).
  * Choi  _et al._ (2017) T. Choi, W. Paul, S. Rolf-Pissarczyk, A. J. Macdonald, F. D. Natterer, K. Yang, P. Willke, C. P. Lutz, and A. J. Heinrich, Nature Nanotechnology 12, 420 (2017).
  * Hofmann and Takeuchi (2003) H. F. Hofmann and S. Takeuchi, Physical Review A 68, 032103 (2003).
  * Ferrón  _et al._ (2019) A. Ferrón, S. A. Rodríguez, S. S. Gómez, J. L. Lado, and J. Fernández-Rossier, Physical Review Research 1, 033185 (2019).
  * Yang  _et al._ (2017) K. Yang, Y. Bae, W. Paul, F. D. Natterer, P. Willke, J. L. Lado, A. Ferrón, T. Choi, J. Fernández-Rossier, A. J. Heinrich, _et al._ , Physical Review Letters 119, 227206 (2017).
  * Kot  _et al._ (2022) P. Kot, M. Ismail, R. Drost, J. Siebrecht, H. Huang, and C. R. Ast, arXiv preprint arXiv:2209.10969 (2022).
  * Yang  _et al._ (2018) K. Yang, P. Willke, Y. Bae, A. Ferrón, J. L. Lado, A. Ardavan, J. Fernández-Rossier, A. J. Heinrich, and C. P. Lutz, Nature Nanotechnology 13, 1120 (2018).
  * Yang  _et al._ (2019) K. Yang, W. Paul, F. D. Natterer, J. L. Lado, Y. Bae, P. Willke, T. Choi, A. Ferrón, J. Fernández-Rossier, A. J. Heinrich, _et al._ , Physical Review Letters 122, 227203 (2019).
  * Seifert  _et al._ (2020) T. S. Seifert, S. Kovarik, D. M. Juraschek, N. A. Spaldin, P. Gambardella, and S. Stepanow, Science Advances 6, eabc5511 (2020).
  * Steinbrecher  _et al._ (2021) M. Steinbrecher, W. M. Van Weerdenburg, E. F. Walraven, N. P. Van Mullekom, J. W. Gerritsen, F. D. Natterer, D. I. Badrtdinov, A. N. Rudenko, V. V. Mazurenko, M. I. Katsnelson, _et al._ , Physical Review B 103, 155405 (2021).
  * Veldman  _et al._ (2021) L. M. Veldman, L. Farinacci, R. Rejali, R. Broekhoven, J. Gobeil, D. Coffey, M. Ternes, and A. F. Otte, Science 372, 964 (2021).
  * Kovarik  _et al._ (2022) S. Kovarik, R. Robles, R. Schlitz, T. S. Seifert, N. Lorente, P. Gambardella, and S. Stepanow, Nano Letters (2022).
  * Lutz  _et al._ (2022) C. Lutz, G. Czap, and M. Sherwood, Bulletin of the American Physical Society (2022).
  * (39) Y. del Castillo and J. Fernández-Rossier, In preparation .


# Probing the spin structure of the fractional quantum Hall magnetoroton with
polarized Raman scattering

Dung Xuan Nguyen<EMAIL_ADDRESS>Brown Theoretical Physics Center and
Department of Physics, Brown University, 182 Hope Street, Providence, Rhode
Island 02912, USA  Dam Thanh Son  Kadanoff Center for Theoretical Physics,
University of Chicago, 933 East 56th Street, Illinois 60637, USA

###### Abstract

Starting from the Luttinger model for the band structure of GaAs, we derive an
effective theory that describes the coupling of the fractional quantum Hall
(FQH) system with photons in resonant Raman scattering experiments. Our theory
is applicable in the regime when the energy of the photons $\omega_{0}$ is
close to the energy gap $E_{G}$, but $|\omega_{0}-E_{G}|$ is much larger than
the energy scales of the quantum Hall problem. In the literature, it is often
assumed that Raman scattering measures the dynamic structure factor
$S(\omega,\mathbf{k})$ of the FQH. However, in this paper, we find that the
light scattering spectrum measured in the experiments are proportional to the
spectral densities of a pair of operators which we identified with the spin-2
components of the kinetic part of the stress tensor. In contrast with the
dynamic structure factor, these spectral densities do not vanish in the long-
wavelength limit $k\to 0$. We show that Raman scattering with circularly
polarized light can measure the spin of the magnetoroton excitation in the FQH
system. We give an explicit expression for the kinetic stress tensor that
works on any Landau level and which can be used for numerical calculations of
the spectral densities that enter the Raman scattering amplitudes. We propose
that Raman scattering provides a way to probe the bulk of the $\nu=5/2$
quantum Hall state to determine its nature.

##  I Introduction

The fractional quantum Hall effect (FQHE) was discovered in experiment almost
forty years ago [1]. Fractional quantum Hall (FQH) systems support a host of
intriguing physical phenomenons; they are also a playground for many exotic
theoretical ideas ranging from anyons [2, 3, 4] to superconductivity [5],
skyrmion [6], and bimetric gravity [7], to name a few. Anyonic excitations in
a nonabelian FQH states such as the Moore-Read state at filling fraction 5/2
[8] may provide the building blocks for a topological quantum computer [9].
However, FQHE is still one of the most difficult and important unsolved
problems of modern physics.

In a classic paper [10], Girvin, MacDonald, and Platzman proposed a single
mode approximation for the FQHE, in which the only excitation of the FQH
system is a gapped charge-neutral mode called “magnetoroton.” In the original
treatment, the magnetoroton was interpreted as the charge density wave in the
lowest Landau level (LLL). The dispersion relation of this neutral mode has a
minimum at the wave length is of order the magnetic length $\ell_{B}$, which
imitate the behavior of the roton mode in superfluid 4He. Experiments have
confirmed the existence of the magnetoroton mode in light scattering
experiments [11, 12, 13] 111For an alternative interpretation of the mode, see
Ref. [14].. In the widely accepted theoretical interpretation of these
experiments, the light scattering intensity has been associated with the
dynamical structure factor $S(\omega,\mathbf{k})$ in the LLL [15]. However, in
the LLL limit, the dynamical structure factor goes to zero as $k^{4}$ [10] in
the limit where the momentum of the magnetoroton $k$ goes to zero. On the
other hand, the Raman signal seems to persist down to $k=0$, signaling that
the identification of the intenstity of Raman scattering with the dynamic
structure factor may not be correct.

In this paper, we provide a new theoretical treatment for Raman scattering
experiments. We first note that the problem of Raman scattering involves many
energy scales. The largest energy scale is that of the semiconductor gap
$E_{G}$ and the photon energy $\omega_{0}$. The next scale is the distance
between the Landau levels of the conduction-band electrons,
$\omega_{c}=eB/m^{*}c$, and the smallest energy scale is the Coulomb
interacting energy between these electrons, $\Delta$. We assume a hierarchy

$\Delta\ll\omega_{c}\ll E_{G}.$ (1)

Only the physics at the scale $\Delta$ is “hard,” i.e., nonperturbative or
strongly correlated, while the physics at the scales $E_{G}$ and and
$\omega_{c}$ are weakly coupled. To solve the Raman scattering problem
effectively, one needs to separate out the nonperturbative, strongly
correlated physics at the scale $\Delta$ from the perturbative, weak coupling
physics at the other scales. (This is similar to the philosophy of
“factorization” in quantum chromodynamics [16] where the perturbative physics
at the hadronic scale is separated from the perturbative physics of higher
energy scales.)

We perform this “factorization” procedure in two steps. The first is to
integrate out the energy scale $E_{G}$. Starting with the Luttinger’s
Hamiltonian for GaAs [17], we introduce a coupling between the lowest
conduction band and highest valence band due to the interaction with light
waves. We focus on the regime of resonant Raman scattering in which the
frequency of the incoming light is close to the semiconductor energy gap:
$\omega_{0}\approx E_{G}$. Under the assumption that the detuning between the
frequency of light and the gap, $|\omega_{0}-E_{G}|$, is larger than both
energy scales of the Hall effect $\omega_{c}$ and $\Delta$, we integrate out
valence bands to obtain the coupling of the conduction-band electon to the
photon. The second step is to do projection to a single Landau level. The
result is an effective coupling of the Raman photons to operators acting on a
single Landau level.

Our result differs drastically from that of Ref. [15]. We find that instead of
measuring the spectral density of the density operator (the dynamic structure
factor), the Raman scattering experiment measures the spectral densities of
the operators that can be called the “kinetic stress tensor” operators,
$T^{\text{kin}}_{ij}=\frac{1}{m^{*}}\partial_{i}\psi^{\dagger}\partial_{j}\psi$.
These operatores are the components of the stress tensor that arise from the
kinetic energy term in the many-body Hamiltonian. For simple model
Hamiltonians leading to exact zero-energy trial wave functions, the kinetic
stress tensor coincides with the full stress tensor, but that is not the case
for the general case, including that of Coulomb interaction. Moreover, in the
lowest Landau level limit, in the long-wavelength limit the only components of
the kinetic stress tensor that have nonvanishing spectral densities are the
two spin-2 components, $T^{\text{kin}}_{zz}$ and
$T^{\text{kin}}_{\bar{z}\bar{z}}$ (the spin-0 component,
$T^{\text{kin}}_{z\bar{z}}$ has vanishing spectral density in the limit $k\to
0$).

Recent theoretical works [18, 19, 20, 21] advance a new proposal on the
interpretation of the magnetoroton. According to this proposal, the
magnetoroton is the quantum a dynamical metric, and at the long-wavelength
regime $k\sim 0$ has an angular momentum equals 2 in the direction of the
applied magnetic field [19]. In order to determine the spin of the
magnetoroton, it has been suggested [19, 20, 21] that the spin of the
magnetoroton can be detected through polarized Raman scattering.

We show it this paper how Raman scattering with cicrularly polarized light can
indeed be used to confirm the spin of the magnetoroton, including the sign.
The basic idea is very simple: if angular momentum is exactly conserved, an
FQH state can absorb only one specific circular polarized photon to excite a
spin-2 magnetoroton mode and emit a photon with opposite circular
polarization. However, the real system does not have full rotational symmetry,
but only $C_{4}$, so spins 2 and $-2$ are not distinct from the point of view
of symmetry. In this paper we carefully analyze the resonant Raman scattering
using the Luttinger model of GaAs. We determine the magnitude of the effect of
nonconservation of angular momentum through the Luttinger parameters and show
that it is numerically small (of order $1/40$).

We organize the paper as follows. We start Section II by introducing our
theoretical model for Raman scattering of an FQH state. In Section III, we
present the calculation of the intensity of cirularly polarized light
scattering . We show that, in contrast with previous theoretical proposals,
the peaks in scattering intensity, at the long wavelength regime, was obtained
mainly due to the poles in the correlation functions of the kinetic part of
stress tensor $\langle T^{\text{kin}}T^{\text{kin}}\rangle$. We relate the
cross section of Raman scattering by circularly polarized light with the
spectral densities of the stress tensor. In Section IV, we derive the explicit
formulas for the stress tensor projected to a Landau level, which can be used
to evaluate numerically the spectral function. We then conclude the paper in
Section V. The Appendicies are devoted to the details of the calculation.

##  II Model

In GaAs the light hole and heavy hole bands make the main contribution to
Raman scattering process [15]. Thus the spin-off bands can be ignored in this
sense. We consider the effective Lagrangian, with only conduction band
$\psi^{\alpha}$ and $j=3/2$ valence bands (which is nothing but light hole and
heavy hole bands) $\chi_{i}^{\alpha}$. In the notation, $\alpha=1,2$
represents spinor index, and $i=1,2,3$ represents 3 components of $p$ wave
function. The (“Rarita-Schwinger”) constraint is imposed,

${(\sigma^{i})^{\alpha}}_{\beta}\chi^{\beta}_{i}=0,$ (2)

which projects out the $j=1/2$ part from $\chi^{\alpha}_{i}$. Consider the
Luttinger Hamiltonian for heavy hole and light hole (within approximation
$\mathbf{k}\approx 0$) [17]

$H=\frac{1}{m}\biggl{\\{}\left(\gamma_{1}+\frac{5\gamma_{2}}{2}\right)\frac{\mathbf{D}^{2}}{2}-\gamma_{2}\sum_{i}J_{i}^{2}D_{i}^{2}\\\
-2\gamma_{3}\left[\\{J_{x}J_{y}\\}\\{D_{x}D_{y}\\}+\\{J_{y}J_{z}\\}\\{D_{y}D_{z}\\}+\\{J_{z}J_{x}\\}\\{D_{z}D_{x}\\}\right]+\frac{e}{c}\kappa\mathbf{J}\cdot\mathbf{B}\biggr{\\}},$
(3)

where $\gamma_{1,2,3}$ are the Luttinger’s parameters, $m$ is the mass of
electron, $J_{i}$ are the SO(3) generators: $(J_{i})_{jk}=-i\epsilon_{ijk}$,
$D_{i}\chi=(\partial_{i}-i\frac{e}{c}A_{i})\chi$ is the covariant derivative,
$\\{\cdot\\}$ denotes symmetrization, e.g.,
$\\{D_{x}D_{y}\\}\equiv\frac{1}{2}(D_{x}D_{y}+D_{y}D_{x})$, and $e<0$ is the
electric charge of the electron. Using the equality

$\\{J_{i}J_{j}\\}\\{D_{i}D_{j}\\}=(\mathbf{J}\cdot\mathbf{D})^{2}-\frac{e}{2c}\mathbf{J}\cdot\mathbf{B},$
(4)

we can rewrite Hamiltonian (3) as

$H=\frac{1}{m}\biggl{[}\Bigl{(}\gamma_{1}+\frac{5\gamma_{2}}{2}\Bigr{)}\frac{\mathbf{D}^{2}}{2}+(\gamma_{3}-\gamma_{2})\sum_{i}J_{i}^{2}D_{i}^{2}-\gamma_{3}(\mathbf{J}\cdot\mathbf{D})^{2}+\frac{e}{c}\left(\kappa+\frac{\gamma_{3}}{2}\right)\mathbf{J}\cdot\mathbf{B}\biggr{]}.$
(5)

The Lagrangian for the hole band is

$\mathcal{L}_{v}=i\chi^{\dagger}_{i\alpha}\partial_{t}\chi^{\alpha}_{i}-\chi^{\dagger}_{i\lambda}H_{ij}\chi^{\lambda}_{j}+E_{G}\chi^{\dagger}_{i\alpha}\chi^{\alpha}_{i},$
(6)

where $E_{G}$ is energy gap and the covariant derivative is
$D_{i}\psi\equiv(\partial_{i}-i\frac{e}{c}A_{i})\psi$. We also have Lagrangian
of the conduction band

$\mathcal{L}_{c}=i\psi_{\alpha}^{\dagger}\partial_{t}\psi^{\alpha}-\frac{D_{i}\psi_{\alpha}^{\dagger}D_{i}\psi^{\alpha}}{2m^{*}}\,,$
(7)

and the coupling of the valence band and conduction band through interaction
with light

$\mathcal{L}_{i}=e(P^{*}\psi_{\alpha}^{\dagger}\chi^{\alpha}_{i}E_{i}+P\chi^{\dagger}_{i\alpha}\psi^{\alpha}E_{i}),$
(8)

where $P$ is the strength of the dipole transition between the conduction and
valence bands (it will be related to the parameter usually denoted as $E_{p}$
in the literature), $E_{i}$ is the electric field of electromagnetic wave.

We consider the regime of resonant Raman scattering, where the photon energy
be is close to the gap: $|\omega_{0}-E_{G}|\ll E_{G}$. Here we chose
$\omega_{0}=\frac{\omega_{L}+\omega_{S}}{2}$, with $\omega_{L}$ ($\omega_{S}$)
is the frequency of the incoming (scattered) photon. In this case we can write

$\displaystyle E_{i}$
$\displaystyle=\tilde{E}_{i}e^{-i\omega_{0}t}+\tilde{E}_{i}^{*}e^{i\omega_{0}t},$
(9) $\displaystyle\chi_{i}^{\alpha}$
$\displaystyle=\tilde{\chi}_{i}^{\alpha}e^{i\omega_{0}t},$ (10)

where $\tilde{E}_{i}$ and $\tilde{\chi}$ are slowly varying fields (e.g.,
fields that vary with frequencies much smaller than $\omega_{0}$). Subsituting
into the action and dropping the rapidly oscillating terms, the action can be
rewritten as (for notational simplicity we also drop the tildas in
$\tilde{E}_{i}$ and $\tilde{\chi}$)

$\mathcal{L}=i\psi_{\alpha}^{\dagger}\partial_{t}\psi^{\alpha}-\frac{D_{i}\psi_{\alpha}^{\dagger}D_{i}\psi^{\alpha}}{2m^{*}}+e(P^{*}\psi_{\alpha}^{\dagger}\chi^{\alpha}_{i}E_{i}+P\chi^{\dagger}_{i\alpha}\psi^{\alpha}E_{i}^{*})+i\chi^{\dagger}_{i\alpha}\partial_{t}\chi^{\alpha}_{i}+(E_{G}-\omega_{0})\chi^{\dagger}_{i\alpha}\chi^{\alpha}_{i}\\\
-\chi^{\dagger}_{i\lambda}H_{ij}\chi^{\lambda}_{j}+\lambda^{\dagger}_{\alpha}{(\sigma^{i})^{\alpha}}_{\beta}\chi^{\beta}_{i}+\chi^{\dagger}_{i\alpha}{(\sigma^{i})^{\alpha}}_{\beta}\lambda^{\beta}.$
(11)

The last two terms are Lagrange multiplier for constraints (2).

Integrating out $\chi$ and $\lambda$ is equivalent to solving the field
equations and the constraints

$\displaystyle
eP\psi^{\alpha}E_{i}^{*}+(E_{G}-\omega)\chi_{i}^{\alpha}+{(\sigma^{i})^{\alpha}}_{\beta}\lambda^{\beta}+i\partial_{t}\chi_{i}^{\alpha}-H_{ij}\chi^{\alpha}_{j}=0,$
(12) $\displaystyle{(\sigma^{i})^{\alpha}}_{\beta}\chi_{i}^{\beta}=0.$ (13)

We will focus on the regime where the photon energy is not too close to the
gap. More precisely, we will assume that the detuning $|\omega_{0}-E_{G}|$ is
still much larger than the distance between the Landau levels in the bands,

$\omega_{c}\equiv\frac{|e|B}{mc}\ll|\omega_{0}-E_{G}|\ll\omega_{0}.$ (14)

In the FQH regime that and holes the typical momentum scale is $1/\ell_{B}$,
one has

$H\sim\frac{1}{m}D^{2}\sim\frac{eB}{mc}\,.$ (15)

This means that one can solve Eq. (12), ignoring the $\partial_{t}$ and $H$
terms,

$\chi^{\alpha}_{i}=-\frac{eP}{3(E_{G}-\omega_{0})}\left[2\psi^{\alpha}E^{*}_{i}+i\epsilon^{ijk}{(\sigma^{j})^{\alpha}}_{\beta}\psi^{\beta}E_{k}^{*}\right]+O\left(\frac{\omega_{c}}{|E_{G}-\omega_{0}|}\right).$
(16)

Substituting this solution into the action, we then find the effective action
for $\psi$ alone. In fact since $\chi$ is the saddlie point, an error of order
$O(\omega_{c}/|E_{G}-\omega_{0}|)$ translated into an error
$O(\omega_{c}^{2}/|E_{G}-\omega_{0}|^{2})$ in the action; thus it is justified
to also keep terms the terms $\chi^{\dagger}\partial_{t}\chi$ and
$\chi^{\dagger}H\chi$ when we do the subsitution (16).

To simplify the result, we assume that the electrons are fully polarized with
spin $s_{z}=\frac{1}{2}$, so

$\psi^{\alpha}=\begin{pmatrix}\psi\\\ 0\end{pmatrix}.$ (17)

We assume the incoming and outgoing photons to have momenta along the $z$
direction, so $E_{i}$ are independent of $x$ and $y$. In this case, the
Lagrangian describing the interaction of the conduction-band electron with the
Raman photons have the form

$\mathcal{I}_{\text{int}}=V_{\alpha\beta}(E^{*})^{\alpha}E^{\beta},$ (18)

where $V_{\alpha\beta}$ is some operators quadratic over $\psi$.

The photon spin then points along or opposite to the $z$-axis, corresponding
to the two circular polarizations. We will distinguish processes in which the
direction of the spin of the photon flips from those in which the photon spin
does not change direction. Introducing the complex coordinates (in the quantum
Hall convention)

$z=x-iy,\quad\bar{z}=x+iy,\quad\partial_{z}=\frac{1}{2}(\partial_{x}+i\partial_{y}),\quad\partial_{\bar{z}}=\frac{1}{2}(\partial_{x}-i\partial_{y}),$
(19)

so

$E^{z}=E_{x}-iE_{y},\qquad E^{\bar{z}}=E_{x}+iE_{y},$ (20)

and

$D_{z}=\frac{1}{2}(D_{x}+iD_{y}),\qquad
D_{\bar{z}}=\frac{1}{2}(D_{x}-iD_{y}),$ (21)

the interaction Lagrangian can be written as

$\mathcal{I}_{\rm
int}=V_{zz}(E^{\bar{z}})^{*}E^{z}+V_{\bar{z}\bar{z}}(E^{z})^{*}E^{\bar{z}}+V_{z\bar{z}}(E^{\bar{z}})^{*}E^{\bar{z}}+V_{\bar{z}z}(E^{z})^{*}E^{z}.$
(22)

The terms responsible for scatterings with a switch in the photon helicity
contain $V_{zz}$ and $V_{\bar{z}\bar{z}}$. Direct calculation yields

$\displaystyle V_{\bar{z}\bar{z}}$
$\displaystyle=\frac{e^{2}|P|^{2}}{6(E_{G}-\omega_{0})^{2}}\left[(\gamma_{3}+\gamma_{2})D_{z}\psi^{\dagger}D_{z}\psi-(\gamma_{3}-\gamma_{2})D_{\bar{z}}\psi^{\dagger}D_{\bar{z}}\psi\right],$
(23) $\displaystyle V_{zz}$
$\displaystyle=\frac{e^{2}|P|^{2}}{6(E_{G}-\omega_{0})^{2}}\left[(\gamma_{3}+\gamma_{2})D_{\bar{z}}\psi^{\dagger}D_{\bar{z}}\psi-(\gamma_{3}-\gamma_{2})D_{z}\psi^{\dagger}D_{z}\psi\right].$
(24)

Terms proportional to $\gamma_{3}+\gamma_{2}$ in $V_{zz}$ and
$V_{\bar{z}\bar{z}}$ preserves rotational symmetry, while terms proportional
to $\gamma_{3}-\gamma_{2}$ breaks the angular momentum conservation by 4.

It is convenient to introduce the “kinetic” stress tensor

$T_{zz}^{\text{kin}}=\frac{1}{m^{*}}D_{z}\psi^{\dagger}D_{z}\psi,\quad
T_{\bar{z}\bar{z}}^{\text{kin}}=\frac{1}{m^{*}}D_{\bar{z}}\psi^{\dagger}D_{\bar{z}}\psi,\quad
T_{z\bar{z}}^{\text{kin}}=\frac{1}{2m^{*}}(D_{z}\psi^{\dagger}D_{\bar{z}}\psi+D_{\bar{z}}\psi^{\dagger}D_{z}\psi).$
(25)

These components are the variation of the kinetic-energy part of the
Hamiltonian over the external metric. This is different from the full stress
tensor which contains also the variation of the potential energy over the
metric. The effective operators coupled to the Raman photon (and flips the
direction of its spin) are

$\displaystyle V_{\bar{z}\bar{z}}$
$\displaystyle=\frac{e^{2}|P|^{2}m^{*}}{6(E_{G}-\omega_{0})^{2}}\left[(\gamma_{3}+\gamma_{2})T_{zz}-(\gamma_{3}-\gamma_{2})T_{\bar{z}\bar{z}}\right],$
(26) $\displaystyle V_{zz}$
$\displaystyle=\frac{e^{2}|P|^{2}m^{*}}{6(E_{G}-\omega_{0})^{2}}\left[(\gamma_{3}+\gamma_{2})T_{\bar{z}\bar{z}}-(\gamma_{3}-\gamma_{2})T_{zz}\right].$
(27)

For details of calculations see the Appendix A. For completeness, we also
write down the operators that do not flip the photon spin,

$\displaystyle V_{\bar{z}z}$
$\displaystyle=\frac{e^{2}|P|^{2}}{(E_{G}{-}\omega_{0})^{2}}\left[\left(\theta^{*}\omega_{c}{-}\frac{(E_{G}{-}\omega_{0})}{2}{+}\frac{\beta^{*}\omega_{c}}{4}\right)\rho+\left(2\alpha^{*}{-}\beta^{*}{+}\gamma^{\prime*}\right)T_{z\bar{z}}+\frac{i}{2}\psi^{\dagger}\partial_{t}\psi\right],$
(28a) $\displaystyle V_{z\bar{z}}$
$\displaystyle=\frac{e^{2}|P|^{2}}{9(E_{G}{-}\omega_{0})^{2}}\left[\left(-\theta^{*}\omega_{c}{-}\frac{9(E_{G}{-}\omega_{0})}{2}{-}\frac{\beta^{*}\omega_{c}}{4}\right)\rho+\left(6\alpha^{*}{-}5\beta^{*}{+}5\gamma^{\prime*}\right)T_{z\bar{z}}+\frac{9i}{2}\psi^{\dagger}\partial_{t}\psi\right],$
(28b)

where

$\displaystyle\alpha^{*}$
$\displaystyle=\frac{m^{*}}{2m}\left(\gamma_{1}+\frac{5\gamma_{2}}{2}\right),\quad\gamma^{\prime*}=\frac{m^{*}}{m}(\gamma_{3}-\gamma_{2}),$
(29) $\displaystyle\beta^{*}$
$\displaystyle=\frac{m^{*}}{m}\gamma_{3},\quad\theta^{*}=-\frac{m^{*}}{m}\left(\kappa+\frac{\gamma_{3}}{2}\right).$
(30)

##  III Scattering of circularly polarized light in the FQH regime

In this Section, we will calculate the Raman scattering on the fractional
quantum Hall state. The magnitude of the effect can be characterized by the
per-particle differential cross-ssection

$\frac{d\sigma_{\lambda\lambda^{\prime}}}{d\omega d\Omega}\,,$ (31)

where $\omega$ is the difference between the energy of the incoming photon
$\omega_{L}$ and the scattered photon $\omega_{S}$:
$\omega=\omega_{L}-\omega_{S}$, $d\Omega$ is the infinitesimal solid angle of
the scattered photon, and $\lambda$ and $\lambda^{\prime}$ are the indices
denoting the polarization of the incoming and scattered photons. For
simplicity, we consider the case when the incident and reflection light are
directed perpendicularly to the sample. The light can pass through the sample,
or, as depicted in Fig. 1, be reflected from the sample. We will assume that
both incident light and scattered lights have circular polarization, and and
$\lambda$ and $\lambda^{\prime}$ can be either $+$ and $-$ depending on the
projection of the proton spin on the $z$ axis. For example, for $\sigma_{++}$
the incident light is left-handed (in the “classical optics” convention, see,
e.g., Ref. [22]) and so the incident photons have spin pointing along the
direction of their momentum, and the scattered light is right-handed, as in
Fig. 1 (a)),

Figure 1: Setup experiment for circular polarized light scattering

we have the formula for cross section per electron [15, 23],

$\frac{d\sigma_{++}(\omega)}{d\omega
d\Omega}=\frac{1}{N_{e}}\frac{\omega_{S}}{\omega_{L}}\omega_{S}^{2}\omega_{L}^{2}\sum_{f}|\langle
f|V_{\bar{z}z}|i\rangle|^{2}\delta(\epsilon_{f}-\epsilon_{i}-\hbar\omega)\approx-\frac{1}{\bar{\rho}}\frac{\omega_{0}^{4}}{\pi}\mathop{\mathrm{Im}}\langle
V^{\dagger}_{\bar{z}z}V_{\bar{z}z}\rangle_{\omega,\mathbf{0}},$ (32)

with $N_{e}$ being the total electron number in the conductance band and
$\bar{\rho}$ is the electron density in the conductance band,
$\epsilon_{f},\epsilon_{i}$ are energies of final and initial states, and

$\langle
A^{\dagger}A\rangle_{\omega,\mathbf{k}}\equiv\int\\!dt\,d\mathbf{x}\,e^{i\omega
t-i\mathbf{k}\cdot\mathbf{x}}\langle
TA(t,\mathbf{x})A^{\dagger}(0,\mathbf{0})\rangle.$ (33)

Thus we need to calculate the spectral density of the operator
$V_{\bar{z}z}(\omega,0)$.

Similarly, in the case of setups in Figure 1 (b), (c), and (d), we have

$\displaystyle\frac{d\sigma_{--}(\omega)}{d\omega d\Omega}$
$\displaystyle=-\frac{1}{\bar{\rho}}\frac{\omega_{0}^{4}}{\pi}\mathop{\mathrm{Im}}\langle
V^{\dagger}_{z\bar{z}}V_{z\bar{z}}\rangle,$ (34)
$\displaystyle\frac{d\sigma_{+-}(\omega)}{d\omega d\Omega}$
$\displaystyle=-\frac{1}{\bar{\rho}}\frac{\omega_{0}^{4}}{\pi}\mathop{\mathrm{Im}}\langle
V^{\dagger}_{zz}V_{zz}\rangle,$ (35)
$\displaystyle\frac{d\sigma_{-+}(\omega)}{d\omega d\Omega}$
$\displaystyle=-\frac{1}{\bar{\rho}}\frac{\omega_{0}^{4}}{\pi}\mathop{\mathrm{Im}}\langle
V^{\dagger}_{\bar{z}\bar{z}}V_{\bar{z}\bar{z}}\rangle.$ (36)

The intensity of the Raman scattering in these channels are proportional to
the spectral densities of he operators $V_{z\bar{z}}$, $V_{zz}$, and
$V_{\bar{z}\bar{z}}$.

Let us now show that in the limit of negligible Landau level mixing, the
spectral densities of the operators $V_{z\bar{z}}$ and $V_{\bar{z}z}$ are
zero, implying that the processes depicted on Fig. 1 (a) and (b) do not
happen. For that, we note from Eqs. (28) that the integrals of $V_{z\bar{z}}$
and $V_{\bar{z}z}$ over space are linear combinations of

$\int\\!d\mathbf{x}\,\rho,\quad\int\\!d\mathbf{x}\,T_{z\bar{z}}^{\text{kin}},\quad\int\\!d\mathbf{x}\,i\psi^{\dagger}\partial_{t}\psi.$
(37)

The first integral is the total number of particles $N_{e}$. As this quantity
is conserved, it does not contribute to the spectral density. From Appendix D
we find the results for $N^{th}$ Landau level

$\displaystyle\int\\!d\mathbf{x}\,T_{z\bar{z}}^{\text{kin}}$
$\displaystyle=(N+\frac{1}{2})\frac{\omega_{c}}{2}N_{e},$ (38)
$\displaystyle\int\\!d\mathbf{x}\,i\psi^{\dagger}\partial_{t}\psi$
$\displaystyle=2E-(N+\frac{1}{2})\omega_{c}N_{e},$ (39)

where $E$ is the total energy. Both integrals reduce to conserved quantities.
Thus, the Raman processes that does not involve flipping the direction of the
photon spin are suppressed.

In previous experiments [11, 12], the momentum transfer to the electron gass
is rather small $kl_{B}\leq 0.15$. This implies that these experiments mainly
probe the transitions where the photon spin flips sign, and effectively
measures the spectral densities of the traceless components of the kinetic
stress tensor. The picture suggested here is different from the previous one
suggested in Ref. [15] where the main coupling of the Raman photon to the
electron liquids is through the $\psi^{\dagger}\psi A_{i}^{2}$ term in the
Lagrangian. This coupling would lead to a vanishing Raman scattering at $k=0$.

Let us introduce the short-hand notation for the spectral densities of the
off-diagonal components of the stress tensor,

$\displaystyle-\mathop{\mathrm{Im}}\left\langle
T_{zz}^{\text{kin}}T_{\bar{z}\bar{z}}^{\text{kin}}\right\rangle_{\omega,\mathbf{0}}$
$\displaystyle=I_{+}(\omega),$ (40)
$\displaystyle-\mathop{\mathrm{Im}}\left\langle
T_{\bar{z}\bar{z}}^{\text{kin}}T_{zz}^{\text{kin}}\right\rangle_{\omega,\mathbf{0}}$
$\displaystyle=I_{-}(\omega).$ (41)

These functions should be calculated numerically. The intensity of Raman
scatterings can now be expressed as

$\displaystyle\frac{d\sigma_{+-}}{d\omega d\Omega}=$
$\displaystyle\frac{1}{\pi\bar{\rho}}\left[\frac{e^{2}|P|^{2}\omega_{0}^{2}m^{*}}{6(E_{0}-\omega_{0})^{2}}\right]^{2}\left[(\gamma_{3}+\gamma_{2})^{2}I_{+}(\omega)+(\gamma_{3}-\gamma_{2})^{2}I_{-}(\omega)\right],$
(42) $\displaystyle\frac{d\sigma_{-+}}{d\omega d\Omega}=$
$\displaystyle\frac{1}{\pi\bar{\rho}}\left[\frac{e^{2}|P|^{2}\omega_{0}^{2}m^{*}}{6(E_{0}-\omega_{0})^{2}}\right]^{2}\left[(\gamma_{3}+\gamma_{2})^{2}I_{-}(\omega)+(\gamma_{3}-\gamma_{2})^{2}I_{+}(\omega)\right].$
(43)

In Ref. [24] it was proven that $I_{+}(\omega)=0$ for the trial ground states
of model Hamiltonians with contact interactions. While there is no argument
that $I_{+}$ should be zero for more general Hamiltonians, numerically it was
found that for Coulomb interaction $I_{+}$ is much smaller than $I_{-}$ for
the Laughlin $\nu=1/3$ state [20].

If one ignore $I_{+}$ compared to $I_{-}$, we find the ratio of scattered
light intensity of experiment setups 1 (c) and 1 (d)

$\frac{I_{-+}(\omega=\Delta)}{I_{+-}(\omega=\Delta)}=\frac{(\gamma_{3}+\gamma_{2})^{2}}{(\gamma_{3}-\gamma_{2})^{2}}\,.$
(44)

The ratio only depends on the Luttinger parameters. Moreover, the fact that
$I_{+-}$ will vanish if $\gamma_{3}-\gamma_{2}=0$ suggests that the signal of
$I_{+-}$ is due to rotational symmetry breaking. These results confirm that at
zero momentum ($\mathbf{k}=0$), the magneto-roton excitation has spin 2 in
$\hat{z}$ direction. However, in the case of finite momentum, the magneto-
roton excitation will be mixed of modes with spin $+2$ and spin $-2$ in
$\hat{z}$ direction, which was suggested in the previous work [19].

The numerical values for the Luttinger parameters of GaAs are [25]

$\gamma_{1}=6.9,\quad\gamma_{2}=2.1,\quad\gamma_{3}=2.9,\quad\kappa=1.2.$ (45)

Substituting these parameters in equation (44) yields the ratio of intensities

$\frac{I_{-+}(\omega=\Delta)}{I_{+-}(\omega=\Delta)}\approx 40.$ (46)

Note that this relies on the assumption that $I_{+}=0$, which is not expected
to hold exactly for the Coulomb interaction. However, if $I_{+}$ is small
compared to $I_{-}$, one still expect that $I_{-+}\ll I_{+-}$ for $\nu=1/3$
states. This is also expected for the Jain states $\nu=n/(2n+1)$, in which the
composite fermion theory implies that the magnetoroton has the same sign of
spin as in the $\nu=1/3$ state. In the particle-hole conjugate Jain states
$\nu=(n+1)/(2n+1)$, in contrast, one expects that $I_{-+}\gg I_{+-}$ [26, 27].

##  IV Stress tensor projected on a Landau level

As we have seen from the previous section, to obtain the cross section of
polarized Raman scattering, we need to calculate the spectral function of the
kinetic part of the stress tensor projected on a specific Landau level (the
fractionally filled one). To enable future numerical calculations of these
spectral functions we need the expressions for the operators
$T^{\text{kin}}_{ij}$ after the projection to a Landau level. In this section,
we will derive the explicit form of the projected kinetic stress tensor.

We summarize the result here. For a system of particles interacting through a
two-body isotropic potential $V(|\mathbf{x}-\mathbf{y}|)$ on the $N$th Landau
level, the kinetic part of the stress tensor (at zero momentum) can be written
as

$\int\\!d\mathbf{x}\,T_{ij}^{\text{kin}}=\int_{\mathbf{q}}\frac{q_{i}q_{j}}{q}\frac{\partial}{\partial
q}\left\\{e^{-x_{q}}[L_{N}(x_{q})]^{2}\right\\}V(q)\bar{\rho}(\mathbf{q})\bar{\rho}(-\mathbf{q}),$
(47)

where $\int_{\mathbf{q}}\equiv\int\\!d\mathbf{q}/(2\pi)^{2}$,

$x_{q}\equiv\frac{q^{2}\ell_{B}^{2}}{2}\,,$ (48)

$L_{N}(x)$ is the Laguerre polynomial, and $ij$ can be either $zz$ or
$\bar{z}\bar{z}$, and $\bar{\rho}(\mathbf{q})$ is the projected density
operator in momentum space [10].

The interpretation of the above equation is rather simple. Recall that the
projected Hamiltonian of the system is

$H=\int_{\mathbf{q}}e^{-x_{q}}[L_{N}(x_{q})]^{2}V(q)\bar{\rho}(\mathbf{q})\bar{\rho}(-\mathbf{q}),$
(49)

where the form-factor $e^{-x_{q}}[L_{N}(x_{q})]^{2}$ arises from the
projection to the $N$th Landau level. Polarized Raman scattering, as explained
above, has the effect of changing the effective metric in the kinetic term for
the electron (making the effective mass $m^{*}$ anisotropic). This makes the
Landau orbit on the $N$th Landau level anistropic, and the effect of that is
the operator $(q_{i}q_{j}/q)\partial_{q}$ acting on the form-factor.

For $N=0$, Eq. (47) reads

$\int\\!d\mathbf{x}\,T_{zz}^{\text{kin}}=-\ell_{B}^{2}\int_{\mathbf{q}}q_{z}^{2}e^{-q^{2}\ell_{B}^{2}/2}V(q)\bar{\rho}(\mathbf{q})\bar{\rho}(-\mathbf{q}),$
(50)

which is exactly the operator considered in Ref. [20]. Thus the spectral
densities computed in Ref. [20] are directly related to polarized Raman
scattering on FQH states on the LLL.

For the next-to-lowest Landau level $N=1$ we have

$\int\\!d\mathbf{x}\,T_{zz}^{\text{kin}}=-\ell_{B}^{2}\int_{\mathbf{q}}q_{z}^{2}e^{-x_{q}}(1-x_{q})(3-x_{q})V(q)\bar{\rho}(\mathbf{q})\bar{\rho}(-\mathbf{q}).$
(51)

The general expression for the kinetic stress tensor (47) has also been found
by Kun Yang [28]. In the rest of this Section, we provide a derivation of Eq.
(47).

###  IV.1 Preliminaries

We use the complex coordinates (19) and the symmetric gauge
$A_{x}=-\frac{1}{2}By$, $A_{y}=\frac{1}{2}Bx$. In the complex coordinates

$A_{z}=\frac{1}{2}(A_{x}+iA_{y})=i\frac{B}{4}\bar{z},\qquad
A_{\bar{z}}=\frac{1}{2}(A_{x}-iA_{y})=-i\frac{B}{4}z,$ (52)

Then in the symmetric gauge

$\displaystyle D_{z}$
$\displaystyle=\partial_{z}-\frac{ie}{c}A_{z}=\partial_{z}-\frac{\bar{z}}{4}\,,$
(53) $\displaystyle D_{\bar{z}}$
$\displaystyle=\partial_{\bar{z}}-\frac{ie}{c}A_{\bar{z}}=\partial_{\bar{z}}+\frac{z}{4}\,.$
(54)

Note that

$[D_{z},\,D_{\bar{z}}]=-\frac{eB}{2c}=\frac{1}{2\ell_{B}^{2}}\,.$ (55)

The (complex) guiding center coordinates are defined as

$\displaystyle Z$
$\displaystyle=z-2\ell_{B}^{2}D_{\bar{z}}=\frac{z}{2}-2\ell_{B}^{2}\partial_{\bar{z}},$
(56) $\displaystyle\bar{Z}$
$\displaystyle=\bar{z}+2\ell_{B}^{2}D_{z}=\frac{\bar{z}}{2}+2\ell_{B}^{2}\partial_{z}$
(57)

which satisfy

$[D_{z},\,Z]=[D_{z},\,\bar{Z}]=[D_{\bar{z}},\,Z]=[D_{\bar{z}},\,\bar{Z}]=0.$
(58)

and

$[\bar{Z},\,Z]=2\ell_{B}^{2}.$ (59)

We define another set of coordinates: the relative coordinates which describes
the motion around the guiding center,

$\displaystyle\zeta$
$\displaystyle=2\ell_{B}^{2}D_{\bar{z}}=\frac{z}{2}+2\ell_{B}^{2}\partial_{\bar{z}},$
(60) $\displaystyle\bar{\zeta}$
$\displaystyle=-2\ell_{B}^{2}D_{z}=\frac{\bar{z}}{2}-2\ell_{B}^{2}\partial_{z},$
(61)

which commute with $Z$ and $\bar{Z}$ [Eqs. (58)] and have the commutator

$[\bar{\zeta},\,\zeta]=-2\ell_{B}^{2}.$ (62)

Then $z=Z+\zeta$, $\bar{z}=\bar{Z}+\bar{\zeta}$. We denote the 2D vector whose
complex coordinates are $Z$ and $\bar{Z}$ as $\mathbf{R}$, and the vector with
complex coordinates $\zeta$ and $\bar{\zeta}$ as $\tilde{\mathbf{r}}$. That
means $\mathbf{x}=\mathbf{R}+\tilde{\mathbf{r}}$.

One defines two sets of creation and annihilation operators. One set moves
between different Landau levels

$a=\sqrt{2}\ell_{B}D_{\bar{z}}=\frac{\zeta}{\sqrt{2}\ell_{B}},\qquad
a^{\dagger}=-\sqrt{2}\ell_{B}D_{z}=\frac{\bar{\zeta}}{\sqrt{2}\ell_{B}}\,,$
(63)

and another set moves within a Landau level

$b=\frac{1}{\sqrt{2}\ell_{B}}\bar{Z},\qquad
b^{\dagger}=\frac{1}{\sqrt{2}\ell_{B}}Z.$ (64)

The orbitals are obtained by acting creation operator on the lowest state

$|M,m\rangle=\frac{1}{\sqrt{M!m!}}a^{\dagger M}b^{\dagger m}|0,0\rangle,$ (65)

where

$\langle\mathbf{x}|0,0\rangle\sim e^{-|z|^{2}/4\ell_{B}^{2}}.$ (66)

###  IV.2 The kinetic stress tensor on a Landau level

Our task is to find the expression for the kinetic part of the stress tensor
in the theory where the electrons live on one Landau level. This will be done
through a field-theory formalism. The action describing electrons on the $N$th
Landau level is

$S=\int\\!dt\,d\mathbf{x}\left[i\psi^{\dagger}\partial_{t}\psi+\chi^{\dagger}\left(2\ell_{B}^{2}D_{z}D_{\bar{z}}\psi+N\psi\right)+\left(2\ell_{B}^{2}D_{\bar{z}}D_{z}\psi^{\dagger}+N\psi^{\dagger}\right)\chi\right]\\\
-\frac{1}{2}\\!\int\\!dt\,d\mathbf{x}\,d\mathbf{x}^{\prime}\,V(\mathbf{x}-\mathbf{x}^{\prime})\psi^{\dagger}(\mathbf{x})\psi^{\dagger}(\mathbf{x}^{\prime})\psi(\mathbf{x}^{\prime})\psi(\mathbf{x}).$
(67)

The fields $\chi$ and $\chi^{\dagger}$ are simply the Lagrange multipliers
enforcing the constraint

$2\ell_{B}^{2}D_{z}D_{\bar{z}}\psi+N\psi=0,$ (68)

which is simply the condition that $\psi$ lies on the $N$th Landau level.

To find the stress tensor, we first rewrite the action by integration by part,

$S=\int\\!dt\,d\mathbf{x}\left[i\psi^{\dagger}\partial_{t}\psi-2\ell_{B}^{2}D_{z}\chi^{\dagger}D_{\bar{z}}\psi-2\ell_{B}^{2}D_{z}\psi^{\dagger}D_{\bar{z}}\chi+N(\chi^{\dagger}\psi+\psi^{\dagger}\chi)\right]\\\
-\frac{1}{2}\\!\int\\!dt\,d\mathbf{x}\,d\mathbf{x}^{\prime}\,V(\mathbf{x}-\mathbf{x}^{\prime})\psi^{\dagger}(\mathbf{x})\psi^{\dagger}(\mathbf{x}^{\prime})\psi(\mathbf{x}^{\prime})\psi(\mathbf{x}),$
(69)

then the kinetic part of the stress tensor can be calculated from Noether’s
theorem:

${T^{i}}_{j}=-\frac{\partial
L}{\partial(\partial_{i}\phi_{a})}\partial_{j}\phi_{a},$ (70)

where one sums over all fields $\phi_{a}$, which in our case encompass $\psi$,
$\phi^{\dagger}$, $\chi$, and $\chi^{\dagger}$. For the polarized Raman
experiment with perpendicularly incoming and outgoing photons, with a flipping
of the photon spin, one only needs the traceless part of the stress tensor,
integrated over space:

$\displaystyle\int\\!d\mathbf{x}\,T^{\text{kin}}_{zz}$
$\displaystyle=-\ell_{B}^{2}\int\\!d\mathbf{x}\,(\chi^{\dagger}D_{z}^{2}\psi+D_{z}^{2}\psi^{\dagger}\chi),$
(71) $\displaystyle\int\\!d\mathbf{x}\,T^{\text{kin}}_{\bar{z}\bar{z}}$
$\displaystyle=-\ell_{B}^{2}\int\\!d\mathbf{x}\,(\chi^{\dagger}D_{\bar{z}}^{2}\psi+D_{\bar{z}}^{2}\psi^{\dagger}\chi).$
(72)

We can expand $\chi$ as a sum over Landau levels:
$\chi=\chi_{0}+\chi_{1}+\chi_{2}+\cdots$. We recall that when acting on $\psi$
and $\chi$, $D_{z}$ raises and $D_{\bar{z}}$ lowers the Landau level index,
while when acting on $\psi^{\dagger}$ and $\chi^{\dagger}$ they switch roles.
Due to the orthogonality of wavefunctions on different Landau levels, only the
parts of $\chi$ that are on the $(N+2)$th and (if $N\geq 2$) $(N-2)$th Landau
level’s contribute to the integrals in Eqs. (71). We then have, for $N\geq 2$

$\displaystyle\int\\!d\mathbf{x}\,T^{\text{kin}}_{zz}$
$\displaystyle=-\ell_{B}^{2}\int\\!d\mathbf{x}\,(\chi_{N+2}^{\dagger}D_{z}^{2}\psi+D_{z}^{2}\psi^{\dagger}\chi_{N-2}),$
(73) $\displaystyle\int\\!d\mathbf{x}\,T^{\text{kin}}_{\bar{z}\bar{z}}$
$\displaystyle=-\ell_{B}^{2}\int\\!d\mathbf{x}\,(\chi_{N-2}^{\dagger}D_{\bar{z}}^{2}\psi+D_{\bar{z}}^{2}\psi^{\dagger}\chi_{N+2}),$
(74)

and for $N=0$ or 1

$\displaystyle\int\\!d\mathbf{x}\,T^{\text{kin}}_{zz}$
$\displaystyle=-\ell_{B}^{2}\int\\!d\mathbf{x}\,\chi_{N+2}^{\dagger}D_{z}^{2}\psi,$
(75) $\displaystyle\int\\!d\mathbf{x}\,T^{\text{kin}}_{\bar{z}\bar{z}}$
$\displaystyle=-\ell_{B}^{2}\int\\!d\mathbf{x}\,D_{\bar{z}}^{2}\psi^{\dagger}\chi_{N+2}.$
(76)

The equation determining $\chi$ is

$0=\frac{\delta
S}{\delta\psi^{\dagger}}=i\partial_{t}\psi+(2\ell_{B}^{2}D_{z}D_{\bar{z}}+N)\chi+W(\mathbf{x}),$
(77)

which, for $n\neq N$, implies

$\displaystyle\chi_{n}=\frac{W_{n}}{n-N}\,.$ (78)

In particular

$\displaystyle\chi_{N+2}$ $\displaystyle=\frac{1}{2}W_{N+2}\,,$ (79)
$\displaystyle\chi_{N-2}$ $\displaystyle=-\frac{1}{2}W_{N-2}\,\quad(N\geq 2),$
(80)

and therefore

$\displaystyle\int\\!d\mathbf{x}\,T^{\text{kin}}_{zz}$
$\displaystyle=-\frac{1}{2}\ell_{B}^{2}\int\\!d\mathbf{x}\,(W^{\dagger}D_{z}^{2}\psi-
D_{z}^{2}\psi^{\dagger}W),$ (81)
$\displaystyle\int\\!d\mathbf{x}\,T^{\text{kin}}_{\bar{z}\bar{z}}$
$\displaystyle=\frac{1}{2}\ell_{B}^{2}\int\\!d\mathbf{x}\,(W^{\dagger}D_{\bar{z}}^{2}\psi-
D_{\bar{z}}^{2}\psi^{\dagger}W),$ (82)

where we have used the orthogonality of the functions on different Landau
levels to replace $W_{N+2}$ and $W_{N-2}$ by simply $W$.

Using formulas of Appendix C, we then find

$\int\\!d\mathbf{x}\,T^{\text{kin}}_{zz}=-\frac{1}{2}\int_{\mathbf{q}}\ell_{B}^{2}q_{z}^{2}e^{-x_{q}}L_{N}(x_{q})\left[L_{N}^{2}(x_{q})-L_{N-2}^{2}(x_{q})\right]V(q)\bar{\rho}(\mathbf{q})\bar{\rho}(-\mathbf{q}),\quad
x_{q}\equiv\frac{q^{2}\ell_{B}^{2}}{2}\,,$ (83)

and a similar equation where $T_{zz}$ is replaced by $T_{\bar{z}\bar{z}}$ and
$q_{z}$ by $q_{\bar{z}}$. Here $L_{N}$ is the Laguerre polynomial and
$L_{N}^{2}$ is not the square of $L_{N}$ but the associate Laguerre polynomial
$L_{N}^{k}$ with $k=2$, and for the uniformity of the equation we have defined
$L^{2}_{-1}=L^{2}_{-2}=0$.

These equations can be brought to an alternative form by using the following
identities involving the associated Laguerre polynomials,

$L_{N}^{k}(x)=L_{N}^{k+1}(x)-L_{N-1}^{k+1}(x),\qquad\frac{d}{dx}L^{k}_{N}(x)=-L_{N-1}^{k+1}(x).$
(84)

One can show that, for $N\geq 2$

$L_{N}^{2}(x)-L_{N-2}^{2}(x)=L_{N}(x)-2\frac{d}{dx}L_{N}(x),$ (85)

while one can also check directly that

$L_{0}(x)\left[L_{0}(x)-2L^{\prime}_{0}(x)\right]=L_{0}(x)L_{0}^{2}(x),\qquad
L_{1}(x)\left[L_{1}(x)-2L^{\prime}_{1}(x)\right]=L_{1}(x)L_{1}^{2}(x).$ (86)

We then can rewrite the kinetic part of stress tensor for a general Landau
level $N$ as

$\int\\!d\mathbf{x}\,T^{\rm
kin}_{zz}=-\frac{\ell_{B}^{2}}{2}\sum_{\mathbf{q}}q_{z}^{2}e^{-x_{q}}L_{N}(x_{q})\left[L_{N}(x_{q})-2L^{\prime}_{N}(x_{q})\right]V(q)\bar{\rho}(\mathbf{q})\bar{\rho}(-\mathbf{q}),$
(87)

and another equation with the replacement $T_{zz}^{\text{kin}}\to
T_{\bar{z}\bar{z}}^{\text{kin}}$ and $q_{z}\to q_{\bar{z}}$. This can be
further transformed to Eq. (47).

Some remarks are in order. The kinetic part of stress tensor operators (87)
for the LLL share the same form as the spin-2 operators in Ref [20], in which
the authors calculated the normalized spectral functions. One can employ the
same approach to obtain the spectral density of the stress tensor for higher
Landau levels. The result will provide the estimation for Raman scattering
intensity of an FQH system at higher Landau levels in our theoretical model.
In Appendix E, we give the expression for the full stress tensor operators,
including the contribution from the interaction. This can be used calculate
the spectral function of LLL stress tensor and check the sum rules derived in
Ref. [19].

##  V Conclusions

In this paper, we have derived the coupling of the electrons in a single
Landau level with applied electromagnetic waves, which effectively captures
the essential physics of Raman scattering on FQH systems. We show that the
electron operator responsible for Raman scattering is not the density
operator, but the “kinetic stress tensor,” and we derive the expression of the
latter after projection to a single Landau level. We then show that, in the
long-wavelength regime, the light scattering intensity in Raman experiments
measures the spectral function of the kinetic part of stress tensors. Our
calculation explains the scattering intensity peaks at zero momentum without
relying on any momentum-noncoserving processes.

In addition, we proposed experimental setups to verify the spin-2 hypothesis
of magneto-roton mode in FQH systems using Raman scattering with circularly
polarized light. We show that, for a magnetoroton with a well-defined sign of
spin, the ratio between light scattering intensities of different
configurations of circularly polarized Raman experiments only depends on
Luttinger parameters, which are well known. Measuring those ratios can confirm
our theoretical model and unveil the spin of the magnetoroton excitations in a
FQH state.

Using the explicit form of the stress tensor operator derived in this paper,
one can perform the numerical calculation to obtain the stress tensors’
spectral function. One then use the numerical results to verify the LLL sum
rules proposed in Ref. [19], and to predict the result of Raman scattering on
states on higher Landau levels.

Raman scattering may help resolve the question about the nature of the
$\nu=5/2$ state. In a recent experiment [29], the thermal Hall conductance at
the edge of the $\nu=5/2$ state was determined to be consistent with the PH-
Pfaffian state [30], but not the Pfaffian [8], or the anti-Pfaffian state [31,
32], seemingly contradicting the results of numerical simulations [33].
Theoretical proposals aiming to explain this discrepancy include a disorder-
stabilized thermal metal phase which is adiabatically connected to the PH-
Pfaffian phase [34, 35] and an incomplete thermalization on the edge [36, 37,
38]. Raman scattering provides a way to probe directly the bulk of the
$\nu=5/2$ state. The magnetoroton in the Pfaffian (Moore-Read) state [8] must
have a spin of the same sign as in the $\nu=1/3$ Lauglin state, while in the
anti-Pfaffian state [31, 32] it must have the opposite sign. The PH-Pfaffian
state [30], in the absence of Landau-level mixing, is particle-hole symmetric,
hence the Raman scattering probabilities $I_{+-}$ and $I_{-+}$ must be the
same. However, it is not clear how significant the effect of Landau level
mixing would be in this case.

To derive the coupling of the Raman photons to FQH electron liquid, we have
assume that the detuning $|\omega_{0}-E_{G}|$ is much larger than the
cyclotron energy $\omega_{c}$. This allows us to perform the first step of our
“factorization” procedure—integrating out the holes—without having to think
about the effect of the magnetic field on the conduction-band electrons. We
suspect that our final result is valid under a weaker assumption—that the
detuning is larger than the energy scale of the FQHE, i.e., of the Coulomb
interaction between the conduction-band electrons. A derivation of this result
would need to be a one-step procedure—integrating out the valence bands and
the projecting to one Landau level at the same time. We defer this to future
work.

###### Acknowledgements.

The authors thank Duncan Haldane, Edward Rezayi, and Kun Yang for discussion.
This work is supported, in part, by by the U.S. DOE grant No. DE-
FG02-13ER41958, a Simons Investigator grant and by the Simons Collaboration on
Ultra-Quantum Matter, which is a grant from the Simons Foundation (651440,
DTS). DXN was supported by Brown Theoretical Physics Center.

##  Appendix A Detailed derivation of Raman scattering coupling

In this section, we present the detailed derivation the coupling of the FQH
system with photon. We define new parameters

$\displaystyle\alpha$
$\displaystyle=\frac{1}{2m}\left(\gamma_{1}+\frac{5\gamma_{2}}{2}\right),\quad\gamma^{\prime}=\frac{1}{m}(\gamma_{3}-\gamma_{2}),$
(88) $\displaystyle\beta$
$\displaystyle=\frac{1}{m}\gamma_{3},\quad\theta=-\frac{1}{m}\left(\kappa+\frac{\gamma_{3}}{2}\right).$
(89)

After integrating out fields in valence band, we derive the effective
Lagrangian for conduction band

$\mathcal{L}=i\psi_{\alpha}^{\dagger}\partial_{t}\psi^{\alpha}-\frac{D_{i}\psi_{\alpha}^{\dagger}D_{i}\psi^{\alpha}}{2m^{*}}+i\chi^{\dagger}_{i\alpha}\partial_{t}\chi^{\alpha}_{i}-(E_{G}-\omega_{0})\chi^{\dagger}_{i\alpha}\chi^{\alpha}_{i}\\\
-\chi^{\dagger}_{k^{\prime}\lambda}\biggl{[}\alpha\mathbf{D}^{2}+\gamma^{\prime}\sum_{i}J_{i}^{2}D_{i}^{2}-\beta(\mathbf{J}\cdot\mathbf{D})^{2}-\frac{e}{c}\theta\mathbf{J}\cdot\mathbf{B}\biggr{]}_{k^{\prime}k}\chi^{\lambda}_{k}.$
(90)

All terms which contain valence band field $\chi^{\alpha}_{i}$ can be
considered as coupling of conduction band field $\psi^{\alpha}$ with the
electric field through substitution (16). We define

$\displaystyle\mathcal{I}_{0}$
$\displaystyle=i\chi_{i\lambda}^{\dagger}\partial_{t}\chi_{i}^{\lambda},$ (91)
$\displaystyle\mathcal{I}_{1}$
$\displaystyle=-(E_{G}-\omega_{0})\chi^{\dagger}_{i\alpha}\chi^{\alpha}_{i},$
(92) $\displaystyle\mathcal{I}_{\alpha}$
$\displaystyle=-\alpha\chi^{\dagger}_{k\lambda}\mathbf{D}^{2}\chi^{\lambda}_{k},$
(93) $\displaystyle\mathcal{I}_{\beta}$
$\displaystyle=\beta\chi^{\dagger}_{k^{\prime}\lambda}\left[(\mathbf{J}\cdot\mathbf{D})^{2}\right]_{k^{\prime}k}\chi^{\lambda}_{k},$
(94) $\displaystyle\mathcal{I}_{\gamma^{\prime}}$
$\displaystyle=-\gamma^{\prime}\chi^{\dagger}_{k^{\prime}\lambda}\sum_{i}(J_{i}^{2})_{k^{\prime}k}D_{i}^{2}\chi^{\lambda}_{k},$
(95) $\displaystyle\mathcal{I}_{\theta}$
$\displaystyle=\theta\frac{e}{c}\sum_{k,k^{\prime},\lambda}\chi^{\dagger}_{k^{\prime}\lambda}(\mathbf{J}_{k^{\prime}k}\cdot\mathbf{B})\chi^{\lambda}_{k}.$
(96)

Consequently, the effective Lagrangian can be rewritten as

$\mathcal{L}_{\rm
eff}=i\psi^{\dagger}\partial_{t}\psi-\frac{(D_{i}\psi)^{\dagger}D_{i}\psi}{2m^{*}}+\mathcal{I}_{0}+\mathcal{I}_{1}+\mathcal{I}_{\alpha}+\mathcal{I}_{\beta}+\mathcal{I}_{\gamma^{\prime}}+\mathcal{I}_{\theta}.$
(97)

Substitution of equation (16) for $\chi^{\alpha}_{i}$ in equation (92) yields

$\mathcal{I}_{1}=-\frac{e^{2}|P|^{2}}{3(E_{G}-\omega_{0})}\left[2\psi^{\dagger}\psi|E|^{2}+i\epsilon^{ijk}\psi^{\dagger}\sigma^{j}\psi
E_{k}^{*}E_{i}\right].$ (98)

The first term in $\mathcal{I}_{1}$ is the interaction of light with charge
density, the second term is the interaction of light with spin density.
Considering that the electrons in the conduction band, under strong magnetic
field in $\hat{z}$ direction, only have the spin component
$s_{z}=\frac{1}{2}$, we can rewrite

$\mathcal{I}_{1}=-\frac{e^{2}|P|^{2}}{3(E_{G}-\omega_{0})}\left[2E_{i}E_{i}^{*}+i(E_{2}^{*}E_{1}-E_{1}^{*}E_{2})\right]\rho,$
(99)

where $\rho=\psi^{\dagger}\psi$. Since $E_{i}^{*}$ is a slow varying field
under the redefinition (9), the term with $\partial_{t}E_{i}^{*}$ in
$\mathcal{I}_{0}$ is small in comparison with $\mathcal{I}_{1}$. We then have

$\mathcal{I}_{0}=i\frac{e^{2}|P|^{2}}{3(E_{G}-\omega_{0})^{2}}\left[2E_{i}E_{i}^{*}+i(E_{2}^{*}E_{1}-E_{1}^{*}E_{2})\right]\psi^{\dagger}\partial_{t}\psi.$
(100)

To understand the next interaction terms (the Luttinger terms), we recall the
formula for the kinetic part of stress energy tensor

$T^{\rm
kin}_{ij}=\frac{(D_{i}\psi)^{\dagger}D_{j}\psi}{2m^{*}}+\frac{(D_{j}\psi)^{\dagger}D_{i}\psi}{2m^{*}}.$
(101)

Under above assumption of spin state of electrons in the conduction band and
$k_{3}=0$ (we consider 2D system in $xy$ plane, and the applied magnetic field
is in $\hat{z}$ direction), we can rewrite the Luttinger terms as

$\mathcal{I}_{\alpha}=\frac{\alpha^{*}e^{2}|P|^{2}}{3(E_{G}-\omega_{0})^{2}}\left[2E_{i}E_{i}^{*}+i(E_{2}^{*}E_{1}-E_{1}^{*}E_{2})\right](T^{\rm
kin}_{11}+T^{\rm kin}_{22}),$ (102) $\displaystyle\mathcal{I}_{\beta}=$
$\displaystyle-\frac{\beta^{*}e^{2}|P|^{2}}{9(E_{G}-\omega_{0})^{2}}\left\\{T^{\rm
kin}_{11}\left[5E_{2}E_{2}^{*}+5E_{3}E_{3}^{*}+2E_{1}E_{1}^{*}+i(E_{2}E_{1}^{*}-E_{1}E_{2}^{*})\right]\right.$
(103) $\displaystyle\left.+T^{\rm
kin}_{22}\left[5E_{1}E_{1}^{*}+5E_{3}E_{3}^{*}+2E_{2}E_{2}^{*}+i(E_{2}E_{1}^{*}-E_{1}E_{2}^{*})\right]\right.$
$\displaystyle\left.-3T^{\rm
kin}_{12}\left(E_{1}E_{2}^{*}+E_{2}E_{1}^{*}\right)+i\frac{\omega_{c}}{2c}\left[5(E_{2}E_{1}^{*}-E_{1}E_{2}^{*})-4iE_{a}E_{a}^{*}-2iE_{3}E_{3}^{*}\right]\rho\right\\},$
$\displaystyle\mathcal{I}_{\gamma^{\prime}}=$
$\displaystyle\frac{{\gamma^{\prime}}^{*}e^{2}|P|^{2}}{9(E_{G}-\omega_{0})^{2}}\left\\{T^{\rm
kin}_{11}\left[5E_{2}E_{2}^{*}+5E_{3}E_{3}^{*}+2E_{1}E_{1}^{*}+i(E_{2}E_{1}^{*}-E_{1}E_{2}^{*})\right]\right.$
$\displaystyle\left.+T^{\rm
kin}_{22}\left[5E_{1}E_{1}^{*}+5E_{3}E_{3}^{*}+2E_{2}E_{2}^{*}+i(E_{2}E_{1}^{*}-E_{1}E_{2}^{*})\right]\right\\},$
(104)
$\mathcal{I}_{\theta}=i\theta^{*}\omega_{c}\frac{e^{2}|P|^{2}}{9(E_{G}-\omega_{0})^{2}}\left[5(E_{1}E_{2}^{*}-E_{2}E_{1}^{*})+4iE_{a}E_{a}^{*}+2iE_{3}E_{3}^{*}\right]\rho,$
(105)

where we have defined the parameters

$\displaystyle\beta^{*}=\beta m^{*},\quad\alpha^{*}=\alpha m^{*},$ (106)
$\displaystyle\gamma^{\prime*}=\gamma^{\prime}m^{*},\quad\theta^{*}=\theta
m^{*},$ (107)

and the cyclotron frequency

$\omega_{c}=-\frac{eB}{cm^{*}}=\frac{|eB|}{cm^{*}}.$ (108)

The effective Lagrangian for the conduction band includes the coupling of the
electric field $E_{i}$ with charge density $\rho$ and the kinetic part of
stress energy tensor $T^{\rm kin}_{ij}$. We have the effective interaction of
conduction band with light through
$\mathcal{I}_{0},\mathcal{I}_{1},\mathcal{I}_{\alpha},\mathcal{I}_{\beta},\mathcal{I}_{\gamma^{\prime}},\mathcal{I}_{\theta}$.
We can rewrite the interaction term in the convenient form for circular
polarized light scattering experiment setup in the Figure 1. In this case, we
can consider $E_{3}=0,E_{3}^{*}=0$. Going to the complex coordinates (19) and
(20), in which

$T^{\rm kin}_{zz}=\frac{1}{4}(T^{\rm kin}_{xx}-T^{\rm kin}_{yy}+2iT^{\rm
kin}_{xy}),\quad T^{\rm kin}_{\bar{z}\bar{z}}=\frac{1}{4}(T^{\rm
kin}_{xx}-T^{\rm kin}_{yy}-2iT^{\rm kin}_{xy}),\quad T^{\rm
kin}_{z\bar{z}}=\frac{1}{4}(T^{\rm kin}_{xx}+T^{\rm kin}_{yy}),$ (109)

we can rewrite the interaction terms as

$\mathcal{I}_{0}=i\frac{e^{2}|P|^{2}}{6(E_{G}-\omega_{0})^{2}}\left[3E^{\bar{z}}(E^{\bar{z}})^{*}+E^{z}(E^{z})^{*}\right]\psi^{\dagger}\partial_{t}\psi,$
(110)
$\mathcal{I}_{1}=-\frac{e^{2}|P|^{2}}{6(E_{G}-\omega_{0})}\left[3E^{\bar{z}}(E^{\bar{z}})^{*}+E^{z}(E^{z})^{*}\right]\rho,$
(111)
$\mathcal{I}_{\alpha}=\alpha^{*}\frac{e^{2}|P|^{2}}{3(E_{G}-\omega_{0})^{2}}\left[6E^{\bar{z}}(E^{\bar{z}})^{*}+2E^{z}(E^{z})^{*}\right]T^{\rm
kin}_{z\bar{z}},$ (112)
$\mathcal{I}_{\beta}=-\frac{\beta^{*}e^{2}|P|^{2}}{9(E_{G}-\omega_{0})^{2}}\left\\{-3E^{z}(E^{\bar{z}})^{*}T^{\rm
kin}_{zz}-3E^{\bar{z}}(E^{z})^{*}T^{\rm
kin}_{\bar{z}\bar{z}}+(5E^{z}(E^{z})^{*}+9E^{\bar{z}}(E^{\bar{z}})^{*})T^{\rm
kin}_{z\bar{z}}\right.\\\
\left.+\frac{\omega_{c}}{4}(E^{z}(E^{z})^{*}-9E^{\bar{z}}(E^{\bar{z}})^{*})\rho\right\\},$
(113) $\displaystyle\mathcal{I}_{\gamma^{\prime}}=$
$\displaystyle\frac{\gamma^{\prime*}e^{2}|P|^{2}}{9(E_{G}-\omega_{0})^{2}}\left\\{-\frac{3}{2}\left[E^{z}(E^{\bar{z}})^{*}+E^{\bar{z}}(E^{z})^{*}\right](T^{\rm
kin}_{zz}+T^{\rm
kin}_{\bar{z}\bar{z}})+\left[5E^{z}(E^{z})^{*}+9E^{\bar{z}}(E^{\bar{z}})^{*}\right]T^{\rm
kin}_{z\bar{z}}\right\\},$ (114)
$\mathcal{I}_{\theta}=\theta^{*}\frac{\omega_{c}e^{2}|P|^{2}}{18(E_{G}-\omega_{0})^{2}}\left[E^{z}(E^{z})^{*}-9E^{\bar{z}}(E^{\bar{z}})^{*}\right]\rho.$
(115)

We can easily check that only $\mathcal{I}_{\gamma^{\prime}}$ violates
rotational invariance.

##  Appendix B The dipole-transition oefficient $P$

In this Appendix, we follow Ref. [39] to derive an expression for the dipole-
transition coefficient $P$ through the electron Bloch wave functions. The
first term of Eq. (8) absorbs a photon and creates a hole in the valence band
and adds an electron to the conductance band. We can rewrite this term in the
Hamiltonian as

$-\int\\!dx\,eP^{*}_{ij}\psi^{\alpha\dagger}\chi^{i}_{\alpha}E^{j},$ (116)

with $P_{ij}^{*}=P^{*}\delta_{ij}$. Comparing with Eq. (11.23) in Ref. [39],
we see that 222We use the Coulomb gauge $A_{0}=0$ in this section.

$-eP_{ij}^{*}i\omega_{ph}\left<n_{ph}{-}1|A^{j}|n_{ph}\right>=-\frac{e}{m}\left<\psi^{\alpha}_{\mathbf{k}},n_{ph}{-}1|A^{j}p_{j}|\chi_{i\alpha,\mathbf{k}},n_{ph}\right>,$
(117)

with $m$ being the free electron mass and $p_{j}$ is the free electron
momentum operator. Then we have

$P^{*}_{ij}=-\frac{i}{m\omega_{ph}}\left<\psi^{\alpha}_{\mathbf{k}}|p_{j}|\chi_{i\alpha,\mathbf{k}}\right>,$
(118)

where $\psi^{\alpha}_{\mathbf{k}}$ is the Bloch wavefunction of electron in
the conductance band ($s$-band)

$\psi^{\alpha}_{\mathbf{k}}=\frac{1}{\sqrt{N_{\rm
site}}}\sum_{a}e^{i\mathbf{k}\cdot\mathbf{x}}u^{\alpha}_{c}(\mathbf{x}-\mathbf{R}_{a}),$
(119)

and $\chi_{i\alpha,\mathbf{k}}$ is the Bloch wavefunction of electron in the
valence bands (p-band)

$\chi_{i\alpha,\mathbf{k}}=\frac{1}{\sqrt{N_{\rm
site}}}\sum_{a}e^{i\mathbf{k}\cdot\mathbf{x}}u^{i\alpha}_{v}(\mathbf{x}-\mathbf{R}_{a}).$
(120)

We have

$\left<\psi^{\alpha}_{\mathbf{k}}|p_{j}|\chi_{i\alpha,\mathbf{k}}\right>=\frac{1}{N_{\rm
site}}\sum_{a,b}\int\\!d^{3}\mathbf{x}\,[k_{j}(u^{\alpha})^{*}_{c}(\mathbf{x}-\mathbf{R}_{a})u^{i\alpha}_{v}(\mathbf{x}-\mathbf{R}_{b})-i(u^{\alpha})^{*}_{c}(\mathbf{x}-\mathbf{R}_{a})\partial_{j}u^{i\alpha}_{v}(\mathbf{x}-\mathbf{R}_{b})],$
(121)

The first term vanishes due to the orthogonality of LCAO. The second term can
be written as

$\left<\psi^{\alpha}_{\mathbf{k}}|p_{j}|\chi_{i\alpha,\mathbf{k}}\right>=-i\sum_{a,b}\int_{\text{unit
cell}}d^{3}\mathbf{x}\,(u^{\alpha})^{*}_{c}(\mathbf{x}-\mathbf{R}_{a})\partial_{j}u^{i\alpha}_{v}(\mathbf{x}-\mathbf{R}_{b}).$
(122)

Due to the angular momentum conservation, we have

$\left<\psi^{\alpha}_{\mathbf{k}}|p_{j}|\chi_{i\alpha,\mathbf{k}}\right>=-i\delta_{ij}\sum_{a,b}\int_{\text{unit
cell}}d^{3}\mathbf{x}\,(u^{\alpha})^{*}_{c}(\mathbf{x}-\mathbf{R}_{a})\partial_{i}u^{i\alpha}_{v}(\mathbf{x}-\mathbf{R}_{b}).$
(123)

Following Ref. [39], we obtain

$\left<\psi^{\alpha}_{\mathbf{k}}|p_{i}|\chi_{i\alpha,\mathbf{k}}\right>=|\hat{e}\cdot\vec{p}_{\rm
cv}|,$ (124)

where $\hat{e}$ is any unit vector and $p^{x}_{\rm cv}=p^{y}_{\rm
cv}=p^{z}_{\rm cv}$ with the definition

$p^{i}_{\rm cv}=-i\sum_{a,b}\int_{\text{unit
cell}}d^{3}\mathbf{x}\,(u^{\alpha})^{*}_{c}(\mathbf{x}-\mathbf{R}_{a})\partial_{i}u^{i\alpha}_{v}(\mathbf{x}-\mathbf{R}_{b}).$
(125)

We also have the relation between $|\hat{e}\cdot\vec{p}_{\rm cv}|$ and the
parameter $E_{p}$ often use in the photonics literature [39]

$|\hat{e}\cdot\vec{p}_{\rm cv}|^{2}=\frac{E_{p}m}{2}\,.$ (126)

We then obtain

$P^{*}=-\frac{i}{m\omega_{ph}}\sqrt{\frac{E_{p}m}{2}}\,.$ (127)

Note that the coupling depends on the frequency of photon. However, if we
rewrite the coupling with electric field to coupling with gauge potential,
there will be no dependence on the photon frequency. We can rewrite the
coupling with photon (8) as

$\mathcal{L}_{i}=e\sqrt{\frac{E_{p}}{2m}}(\psi_{\alpha}^{\dagger}\chi^{\alpha}_{i}A_{i}+\chi^{\dagger}_{i\alpha}\psi^{\alpha}A_{i}^{*}).$
(128)

##  Appendix C How to project to a Landau level

In this section, we provide the detail calculation for kinetic part of stress
tensor in a specific Landau level. The electron field can be expanded in
orbitals

$\psi(\mathbf{x})=\sum_{Mm}\langle\mathbf{x}|Mm\rangle c_{Mm},$ (129)

where $M$ labels the Landau level, and $m$ labels the states within the Landau
level. In order to obtain the kinetic part of stress tensor, we need to
compute

$\int\\!d\mathbf{x}\,W^{\dagger}(\mathbf{x})D_{z}^{2}\psi(\mathbf{x})=-\\!\int\\!d\mathbf{x}\,d\mathbf{x}^{\prime}\,\psi^{\dagger}(\mathbf{x})\psi^{\dagger}(\mathbf{x}^{\prime})V(\mathbf{x}-\mathbf{x}^{\prime})\psi(\mathbf{x}^{\prime})D_{z}^{2}\psi(\mathbf{x}).$
(130)

Inserting the expansion over modes, and limiting to the $N$th Landau level,
this becomes

$-\int\\!d\mathbf{x}\,d\mathbf{x}^{\prime}\sum_{mnm^{\prime}n^{\prime}}\langle
Nm|\mathbf{x}\rangle\langle Nm^{\prime}|\mathbf{x}^{\prime}\rangle
V(\mathbf{x}-\mathbf{x}^{\prime})\langle\mathbf{x}^{\prime}|Nn^{\prime}\rangle
D_{z}^{2}\langle\mathbf{x}|Nn\rangle
c^{\dagger}_{Nm}c^{\dagger}_{Nm^{\prime}}c_{Nn^{\prime}}c_{Nn}.$ (131)

Introducing the Fourier transform of the potential

$V(\mathbf{x}-\mathbf{x}^{\prime})=\int\\!\frac{d\mathbf{q}}{(2\pi)^{2}}\,e^{i\mathbf{q}\cdot(\mathbf{x}-\mathbf{x}^{\prime})}V(\mathbf{q}),$
(132)

the expression becomes

$-\\!\sum_{mnm^{\prime}n^{\prime}}\int_{\mathbf{q}}V(\mathbf{q})\\!\int\\!d\mathbf{x}\,\langle
Nm|\mathbf{x}\rangle
e^{i\mathbf{q}\cdot\mathbf{x}}D_{z}^{2}\langle\mathbf{x}|Nn\rangle\\!\int\\!d\mathbf{x}^{\prime}\,\langle
Nm^{\prime}|\mathbf{x}^{\prime}\rangle
e^{-i\mathbf{q}\cdot\mathbf{x}^{\prime}}\langle\mathbf{x}^{\prime}|Nn^{\prime}\rangle
c^{\dagger}_{m}c^{\dagger}_{m^{\prime}}c_{n^{\prime}}c_{n}.$ (133)

Now we have

$\int\\!d\mathbf{x}^{\prime}\,\langle Nm^{\prime}|\mathbf{x}^{\prime}\rangle
e^{-i\mathbf{q}\cdot\mathbf{x}^{\prime}}\langle\mathbf{x}^{\prime}|Nn^{\prime}\rangle=\langle
Nm^{\prime}|e^{-i\mathbf{q}\cdot\hat{\mathbf{x}}}|Nn^{\prime}\rangle=\langle
N|e^{-i\mathbf{q}\cdot\tilde{\mathbf{r}}}|N\rangle\langle
m^{\prime}|e^{-i\mathbf{q}\cdot\mathbf{R}}|n^{\prime}\rangle,$ (134)

but

$\langle N|e^{-i\mathbf{q}\cdot\tilde{\mathbf{r}}}|N\rangle=\langle
N|\exp\left[-\frac{i\ell_{B}}{\sqrt{2}}(q^{z}a^{\dagger}+q^{\bar{z}}a)\right]|N\rangle\\\
=e^{-x_{q}/2}\langle
N|\exp\left(-\frac{i\ell_{B}}{\sqrt{2}}q^{z}a^{\dagger}\right)\exp\left(-\frac{i\ell_{B}}{\sqrt{2}}q^{\bar{z}}a\right)|N\rangle=e^{-x_{q}/2}L_{N}(x_{q}),\quad
x_{q}=\frac{q^{2}\ell_{B}^{2}}{2}\,,$ (135)

therefore

$\sum_{m^{\prime}n^{\prime}}\\!\int\\!d\mathbf{x}^{\prime}\,\langle
Nm^{\prime}|\mathbf{x}^{\prime}\rangle
e^{-i\mathbf{q}\cdot\mathbf{x}^{\prime}}\langle\mathbf{x}^{\prime}|Nn^{\prime}\rangle
c^{\dagger}_{m^{\prime}}c_{n^{\prime}}=e^{-x_{q}/2}L_{N}(x_{q})\bar{\rho}(\mathbf{q}).$
(136)

Analogously

$\int\\!d\mathbf{x}\,\langle Nm|\mathbf{x}\rangle
e^{i\mathbf{q}\cdot\mathbf{x}}D_{z}^{2}\langle\mathbf{x}|Nn^{\prime}\rangle=\frac{1}{2}\langle
Nm|e^{i\mathbf{q}\cdot\hat{\mathbf{x}}}\frac{a^{\dagger
2}}{\ell_{B}^{2}}|Nn\rangle\\\
=\frac{1}{2\ell_{B}^{2}}\sqrt{(N+1)(N+2)}\langle
N|e^{i\mathbf{q}\cdot\tilde{\mathbf{r}}}|N+2\rangle\langle
m|e^{i\mathbf{q}\cdot\mathbf{R}}|n\rangle=-q_{z}^{2}e^{-x_{q}/2}L_{N}^{2}(x_{q})\langle
m|e^{i\mathbf{q}\cdot\mathbf{R}}|n\rangle,$ (137)

therefore

$\sum_{mn}\\!\int\\!d\mathbf{x}\,\langle Nm|\mathbf{x}\rangle
e^{-i\mathbf{q}\cdot\mathbf{x}}D_{z}^{2}\langle\mathbf{x}|Nn\rangle
c^{\dagger}_{m}c_{n}=-q_{z}^{2}e^{-x_{q}/2}L_{N}^{2}(x_{q})\bar{\rho}(-\mathbf{q}).$
(138)

Finally we obtain

$\int\\!d\mathbf{x}\,W^{\dagger}(\mathbf{x})D_{z}^{2}\psi(\mathbf{x})=\int\\!\frac{d\mathbf{q}}{(2\pi)^{2}}\,q_{z}^{2}e^{-x_{q}}L_{N}(x_{q})L_{N}^{2}(x_{q})V(q)\bar{\rho}(\mathbf{q})\bar{\rho}(-\mathbf{q}).$
(139)

Similarly, for $N\geq 2$

$\int\\!d\mathbf{x}\,W^{\dagger}(\mathbf{x})D_{\bar{z}}^{2}\psi(\mathbf{x})=\int\\!\frac{d\mathbf{q}}{(2\pi)^{2}}\,q_{\bar{z}}^{2}e^{-x_{q}}L_{N}(x_{q})L_{N-2}^{2}(x_{q})V(q)\bar{\rho}(\mathbf{q})\bar{\rho}(-\mathbf{q}),$
(140)

while for $N=0$ or 1 the expression is obviously zero due to the presence of
two lowering operators $D_{\bar{z}}$ acting on $\psi$. Equations (139) and
(140) are used in Sec. IV to obtain the explicit form of the kinetic part of
stress tensor on a specific Landau level.

##  Appendix D $i\psi^{\dagger}\partial_{t}\psi$ and $T^{\rm kin}_{z\bar{z}}$

In this Appendix, we derive the explicit form of
$i\psi^{\dagger}\partial_{t}\psi$ in the lowest Landau level. The field
equation reads

$i\partial_{t}\psi(\mathbf{x})=-\frac{1}{m^{*}}\left(D_{z}D_{\bar{z}}+D_{\bar{z}}D_{z}\right)\psi(\mathbf{x})+\int\\!d\mathbf{x}^{\prime}\,V(\mathbf{x}-\mathbf{x}^{\prime})\psi^{\dagger}(\mathbf{x}^{\prime})\psi(\mathbf{x}^{\prime})\psi(\mathbf{x}),$
(141)

we then use the constraint equation $2\ell_{B}^{2}D_{z}D_{\bar{z}}\psi=-N\psi$
and the commutator (55) from that we get

$i\\!\int\\!d\mathbf{x}\,\psi^{\dagger}\partial_{t}\psi=I_{0}+(N+\frac{1}{2})\omega_{c}N_{e},$
(142)

with $N_{e}$ being the total electron number in the conductance band and

$I_{0}=\int\\!d\mathbf{x}\,d\mathbf{x}^{\prime}\,V(\mathbf{x}-\mathbf{x}^{\prime})\psi^{\dagger}(\mathbf{x})\psi^{\dagger}(\mathbf{x}^{\prime})\psi(\mathbf{x}^{\prime})\psi(\mathbf{x})=\int_{\mathbf{q}}V(\mathbf{q})e^{-q^{2}\ell_{B}^{2}/2}\bar{\rho}(\mathbf{q})\bar{\rho}(-\mathbf{q}).$
(143)

The first term on the on the right hand side of Eq. (142) is twice the
interacting energy and second term is the kinetic energy of electrons333In
Ref. [24] we eliminate the second term in the LLL case by introducing the
Landré factor $\mathfrak{g}=2$ . We also have

$\displaystyle\int\\!d\mathbf{x}\,T^{\rm kin}_{z\bar{z}}$
$\displaystyle=\int\\!d\mathbf{x}\,\frac{1}{2m^{*}}\left(D_{z}\psi^{\dagger}D_{\bar{z}}\psi+D_{\bar{z}}\psi^{\dagger}D_{z}\psi\right)$
$\displaystyle=\int\\!d\mathbf{x}\,\frac{1}{2m^{*}}\left(-2\psi^{\dagger}D_{z}D_{\bar{z}}\psi-\frac{eB}{2c}\psi^{\dagger}\psi\right)$
$\displaystyle=\frac{\omega_{c}}{2}(N+\frac{1}{2})\\!\int\\!d\mathbf{x}\,\rho=(N+\frac{1}{2})\frac{N_{e}\omega_{c}}{2}.$
(144)

Where we used the contraint $2\ell_{B}^{2}D_{z}D_{\bar{z}}\psi=-N\psi$ to
obtain the last equality.

##  Appendix E Two sum rules

This Appendix is not directly related to Raman scattering, but contains some
exact sum rules. First we write down formulas for the full stress tensor. For
the model Hamiltonian, at the long wave length regime $\mathbf{k}\sim 0$, the
full stress tensor is the same as $T^{\rm kin}$ [24]. For the general case,
one needs to take into account the potential-energy term in the Lagrangian.
When the metric varies with time (but remains uniform in space), the potential
changes according to

$V(\mathbf{x})\to V(\sqrt{g_{ij}x^{i}x^{j}}),$ (145)

and so the Fourier transform changes as

$V(\mathbf{q})\to V(\sqrt{g^{ij}q_{i}q_{j}}).$ (146)

Given that the stress tensor is give by $\delta
S=\frac{1}{2}\int\\!dx\,T_{ij}\delta g_{ij}$, the potential part of the
stress-energy tensor is then

$\int\\!d\mathbf{x}\,T_{ij}^{\rm
pot}=\frac{1}{2}\int_{\mathbf{q}}\frac{q_{i}q_{j}}{q}V^{\prime}(q)\rho(\mathbf{q})\rho(-\mathbf{q}).$
(147)

This can be projected to the $N$th Landau level to become

$\int\\!d\mathbf{x}\,T_{ij}^{\rm
pot}=\frac{1}{2}\int_{\mathbf{q}}e^{-x_{q}}\left[L_{N}(x_{q})\right]^{2}\frac{q_{i}q_{j}}{q}V^{\prime}(q)\bar{\rho}(\mathbf{q})\bar{\rho}(-\mathbf{q}).$
(148)

It is interesting to compare the formula to to that of the kinetic stress
tensor, Eq. (47): the “stretching operator” $(q_{i}q_{j}/q)\partial_{q}$ now
acts on the potential $V(q)$ instead of acting on the form-factor. The full
stress tensor is the sum of the kinetic stress tensor, Eqs. (87), and the
potential stress tensor, Eq. (148). It can be written as

$\int\\!d\mathbf{x}\,T_{ij}^{\text{full}}=\frac{1}{2}\sum_{\mathbf{q}}\frac{q_{i}q_{j}}{q}\frac{\partial}{\partial
q}\left\\{e^{-x_{q}}[L_{N}(x_{q})]^{2}V(q)\right\\}\bar{\rho}(\mathbf{q})\bar{\rho}(-\mathbf{q}).$
(149)

We define the spectral densities [19]

$\displaystyle\rho_{T}(\omega)$ $\displaystyle=\frac{1}{N}\sum_{n}|\langle
n|\int\\!d\mathbf{x}\,T_{zz}|0\rangle|^{2}\delta(\omega-E_{n}),$ (150)
$\displaystyle\bar{\rho}_{T}(\omega)$
$\displaystyle=\frac{1}{N}\sum_{n}|\langle
n|\int\\!d\mathbf{x}\,T_{\bar{z}\bar{z}}|0\rangle|^{2}\delta(\omega-E_{n}),$
(151)

where $N$ is the total number of electrons, $|0\rangle$ is the ground state,
the sum is taken over all excited states $|n\rangle$ in the lowest Landau
level, and $E_{n}$ is the energy of the state $|n\rangle$. The two spectrum
densities satisfy the sum rules [19]

$\displaystyle\int_{0}^{\infty}\frac{d\omega}{\omega^{2}}\left[\rho_{T}(\omega)-\bar{\rho}_{T}(\omega)\right]$
$\displaystyle=\frac{\bar{s}}{4}\,,$ (152)
$\displaystyle\int_{0}^{\infty}\frac{d\omega}{\omega^{2}}\left[\rho_{T}(\omega)+\bar{\rho}_{T}(\omega)\right]$
$\displaystyle=S_{4},$ (153)

where $\bar{s}$ is the “guiding center spin” [18], which is equal to
$(\mathcal{S}-1)/2$ on the LLL where $\mathcal{S}$ is the shift, and $S_{4}$
is the coefficient in front of $(k\ell_{B})^{4}$ in the static structure
factor.

## References

  * Tsui  _et al._ [1982] D. C. Tsui, H. L. Stormer, and A. C. Gossard, Two-Dimensional Magnetotransport in the Extreme Quantum Limit, Phys. Rev. Lett. 48, 1559 (1982).
  * Laughlin [1983] R. Laughlin, Anomalous quantum Hall effect: An Incompressible quantum fluid with fractionallycharged excitations, Phys. Rev. Lett. 50, 1395 (1983).
  * Halperin [1984] B. I. Halperin, Statistics of Quasiparticles and the Hierarchy of Fractional Quantized Hall States, Phys. Rev. Lett. 52, 1583 (1984).
  * Wilczek [1982] F. Wilczek, Magnetic Flux, Angular Momentum, and Statistics, Phys. Rev. Lett. 48, 1144 (1982).
  * Read and Green [2000] N. Read and D. Green, Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect, Phys. Rev. B 61, 10267 (2000), cond-mat/9906453 .
  * Ezawa [1997] Z. F. Ezawa, Quantum coherence and Skyrmions in a bilayer quantum Hall system, Phys. Rev. B 55, 7771 (1997).
  * Gromov and Son [2017] A. Gromov and D. T. Son, Bimetric Theory of Fractional Quantum Hall States, Phys. Rev. X 7, 041032 (2017), arXiv:1705.06739 .
  * Moore and Read [1991] G. Moore and N. Read, Nonabelions in the fractional quantum Hall effect, Nucl. Phys. B 360, 362 (1991).
  * Nayak  _et al._ [2008] C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Non-Abelian anyons and topological quantum computation, Rev. Mod. Phys. 80, 1083 (2008), arXiv:0707.1889 .
  * Girvin  _et al._ [1986] S. M. Girvin, A. H. MacDonald, and P. M. Platzman, Magneto-roton theory of collective excitations in the fractional quantum Hall effect, Phys. Rev. B 33, 2481 (1986).
  * Pinczuk  _et al._ [1998] A. Pinczuk, B. Dennis, L. Pfeiffer, and K. West, Light scattering by collective excitations in the fractional quantum Hall regime, Physica B: Condens. Matter 249-251, 40 (1998).
  * Kang  _et al._ [2000] M. Kang, A. Pinczuk, B. S. Dennis, M. A. Eriksson, L. N. Pfeiffer, and K. W. West, Inelastic Light Scattering by Gap Excitations of Fractional Quantum Hall States at $1/3\leq\nu\leq 2/3$, Phys. Rev. Lett. 84, 546 (2000), cond-mat/9911350 .
  * Kukushkin  _et al._ [2009] I. V. Kukushkin, J. H. Smet, V. W. Scarola, V. Umansky, and K. von Klitzing, Dispersion of the Excitations of Fractional Quantum Hall States, Science 324, 1044 (2009).
  * Wiegmann [2018] P. Wiegmann, Inner Nonlinear Waves and Inelastic Light Scattering of Fractional Quantum Hall States as Evidence of the Gravitational Anomaly, Phys. Rev. Lett. 120, 086601 (2018), arXiv:1708.04282 .
  * Platzman and He [1994] P. M. Platzman and S. He, Resonant Raman scattering from mobile electrons in the fractional quantum Hall regime, Phys. Rev. B 49, 13674 (1994).
  * Collins  _et al._ [1989] J. C. Collins, D. E. Soper, and G. F. Sterman, Factorization of Hard Processes in QCD, Adv. Ser. Direct. High Energy Phys. 5, 1 (1989), hep-ph/0409313 .
  * Luttinger [1956] J. M. Luttinger, Quantum Theory of Cyclotron Resonance in Semiconductors: General Theory, Phys. Rev. 102, 1030 (1956).
  * Haldane [2011] F. D. M. Haldane, Geometrical Description of the Fractional Quantum Hall Effect, Phys. Rev. Lett. 107, 116801 (2011), arXiv:1106.3375 .
  * Golkar  _et al._ [2016a] S. Golkar, D. X. Nguyen, and D. T. Son, Spectral sum rules and magneto-roton as emergent graviton in fractional quantum Hall effect, J. High Energy Phys. 1601, 021, arXiv:1309.2638 .
  * Liou  _et al._ [2019] S.-F. Liou, F. D. M. Haldane, K. Yang, and E. H. Rezayi, Chiral Gravitons in Fractional Quantum Hall Liquids, Phys. Rev. Lett. 123, 146801 (2019), arXiv:1904.12231 .
  * Son [2019] D. T. Son, Chiral Metric Hydrodynamics, Kelvin Circulation Theorem, and the Fractional Quantum Hall Effect, (2019), arXiv:1907.07187 .
  * Barron [2004] L. D. Barron,  _Molecular Light Scattering and Optical Activity_ (Cambridge University Press, 2004).
  * Blum [1970] F. A. Blum, Inelastic Light Scattering from Semiconductor Plasmas in a Magnetic Field, Phys. Rev. B 1, 1125 (1970).
  * Nguyen  _et al._ [2014] D. X. Nguyen, D. T. Son, and C. Wu, Lowest Landau Level Stress Tensor and Structure Factor of Trial Quantum Hall Wave Functions, (2014), arXiv:1411.3316 .
  * Ruf [1998] T. Ruf,  _Phonon Raman Scattering in Semiconductors, Quantum Wells and Superlattices_ (Springer, Berlin, 1998).
  * Golkar  _et al._ [2016b] S. Golkar, D. X. Nguyen, M. M. Roberts, and D. T. Son, Higher-Spin Theory of the Magnetorotons, Phys. Rev. Lett. 117, 216403 (2016b), arXiv:1602.08499 .
  * Nguyen  _et al._ [2018] D. X. Nguyen, S. Golkar, M. M. Roberts, and D. T. Son, Particle-hole symmetry and composite fermions in fractional quantum Hall states, Phys. Rev. B 97, 195314 (2018), arXiv:1709.07885 .
  * [28] K. Yang, private communication.
  * Banerjee  _et al._ [2018] M. Banerjee, M. Heiblum, V. Umansky, D. E. Feldman, Y. Oreg, and A. Stern, Observation of half-integer thermal Hall conductance, Nature 559, 205 (2018), arXiv:1710.00492 .
  * Son [2015] D. T. Son, Is the Composite Fermion a Dirac Particle?, Phys. Rev. X 5, 031027 (2015), arXiv:1502.03446 .
  * Levin  _et al._ [2007] M. Levin, B. I. Halperin, and B. Rosenow, Particle-Hole Symmetry and the Pfaffian State, Phys. Rev. Lett. 99, 236806 (2007), cond-mat/0707.0483 .
  * Lee  _et al._ [2007] S.-S. Lee, S. Ryu, C. Nayak, and M. P. A. Fisher, Particle-Hole Symmetry and the $\nu=\frac{5}{2}$ Quantum Hall State, Phys. Rev. Lett. 99, 236807 (2007), cond-mat/0707.0478 .
  * Rezayi [2017] E. H. Rezayi, Landau Level Mixing and the Ground State of the $\nu$=5/2 Quantum Hall Effect, Phys. Rev. Lett. 119, 026801 (2017), arXiv:1704.03026 .
  * Mross  _et al._ [2018] D. F. Mross, Y. Oreg, A. Stern, G. Margalit, and M. Heiblum, Theory of Disorder-Induced Half-Integer Thermal Hall Conductance, Phys. Rev. Lett. 121, 026801 (2018), arXiv:1711.06278 .
  * Wang  _et al._ [2018] C. Wang, A. Vishwanath, and B. I. Halperin, Topological order from disorder and the quantized Hall thermal metal: Possible applications to the $\nu=5/2$ state, Phys. Rev. B 98, 045112 (2018), arXiv:1711.11557 .
  * Simon [2018] S. H. Simon, Interpretation of thermal conductance of the $\nu=5/2$ edge, Phys. Rev. B 97, 121406 (2018), arXiv:1801.09687 .
  * Ma and Feldman [2019] K. K. W. Ma and D. E. Feldman, Partial equilibration of integer and fractional edge channels in the thermal quantum Hall effect, Phys. Rev. B 99, 085309 (2019), arXiv:1809.05488 .
  * Simon and Rosenow [2020] S. H. Simon and B. Rosenow, Partial Equilibration of the Anti-Pfaffian Edge due to Majorana Disorder, Phys. Rev. Lett. 124, 126801 (2020), arXiv:1906.05294 .
  * Haus [2016] J. Haus, Nanophotonic devices, in  _Fundamentals and Applications of Nanophotonics_ (Elsevier, 2016) pp. 341–395.