\section{Introduction}

The emergence of bulk spacetimes with gravity from nongravitational boundary theories is a remarkable and fundamental discovery~\cite{Maldacena:1997re}.  A deeper understanding of how local bulk information is encoded in the boundary theory has been the goal of much  work.  It is believed that a subregion of the boundary theory describes a corresponding subregion of the bulk, the causal wedge~\cite{Bousso:2012sj,Czech:2012bh,Hubeny:2012wa}.  Recently, Almheiri, Dong, and Harlow have framed some sharp puzzles in this regard~\cite{Almheiri:2014lwa}.    They note a striking parallel between these and the ideas of quantum erasure correction and quantum secret sharing.

In this paper, we show that quantum error correction and quantum secret sharing appear automatically as a consequence of boundary gauge invariance.  The resulting construction differs from standard examples of error correction in that the precursors act in the full space of gauge-invariant states; it is not necessary to introduce a distinct code subspace.  We illustrate this in two simple models, a discrete model similar to one in Ref.~\cite{Almheiri:2014lwa}, and a free-field model of precursors.   This observation gives a deeper and more general understanding of the nonuniqueness of the precursor construction. While a local bulk operator is dual to many different CFT precursor operators, they are all equivalent when acting on gauge invariant states.

\section{A discrete model}

The puzzles of Ref.~\cite{Almheiri:2014lwa} deal with ``precursors''~\cite{BDHM,Balasubramanian:1998de,Polchinski:1999yd,Bena99,HKLL}, CFT operators dual to local bulk operators.  The first puzzle is that bulk locality requires a precursor to commute with all spacelike separated local CFT operators.  (To be precise, there are limitations to the commutativity due to gravitational dressing at higher order in $1/N$~\cite{Heemskerk:2012np,Kabat:2012hp,Kabat:2013wga}, which we will discuss later.)  The second puzzle is that one can cover the CFT by three patches $A$, $B$, $C$, such that the precursor can be reconstructed from any of $AB$, $BC$, or $CA$ though not from $A$, $B$, or $C$ separately.  Thus, the bulk information is stored in the CFT in a very nonlocal way.  As Ref.~\cite{Almheiri:2014lwa} notes, this is strongly reminiscent of quantum erasure correction and quantum secret sharing~\cite{QEC}.

\begin{figure}[tbp]
\centering
	\includegraphics[width=2in]{fig1.pdf}
\caption{Focusing on a single time slice of AdS, bulk microcausality requires the operator $\Phi(0)$ to commute with all spacelike-separated local operators.  By the holographic dictionary, the boundary limits of such bulk operators are local CFT operators $\mathcal{O}(x)$, which must thus commute with $\Phi(0)$.}
\label{fig1}
\end{figure}

It might appear challenging to construct CFT operators having such properties, but we exhibit here a simple model in which they are realized precisely, and in which gauge invariance plays an essential role.  As in Ref.~\cite{Almheiri:2014lwa}, we model the boundary theory by three sites as in Fig.~1.  The new ingredient is an $O(N)$ gauge symmetry, which for simplicity we take to depend on time but not on space, so that its only effect is to constrain the Hilbert space to invariant states.  In our example, each site has a single bosonic degree of freedom transforming in the $N$ of $O(N)$.  These are denoted $(\phi_a^i, \pi_a^i)$, where the subscript denotes the site and the superscript is the $O(N)$ index;  dot products will always be on the $O(N)$ indices.

We wish to construct a nontrivial precursor operator that commutes with the local gauge invariants, those that are restricted to any single site.  To do this, we note that $\phi \cdot \phi$, $\phi \cdot \pi$, and $\pi \cdot \pi$, which generate the complete set of $O(N)$ invariants on site $a$, form an $SL(2,R)$ algebra.  The doublet $(\phi_a^i, \pi_a^i)$ transforms as the fundamental representation of this algebra, so the antisymmetric combination
\begin{equation}
L^{ij}_a = \phi^i_a \pi^j_a - \phi^j_a \pi^i_a
\end{equation}
is a singlet with respect to the $SL(2,R)$ on site $a$, while obviously commuting with the invariants on all other sites.  Thus any linear combination of the operators
\begin{equation} \label{eq:P}
{\cal P}_{ab} = L^{ij}_a L^{ij}_b
\end{equation}
is a gauge-invariant operator satisfying the conditions.  These are nontrivial; for example ${\cal P}_{12}$ does not commute with $\phi_2 {\cdot} \phi_3$.  The bulk precursors can then be built out of the ${\cal P}_{ab}$.  For example, ${\cal P}_{12}
+ {\cal P}_{23} + {\cal P}_{31}$ should be $\Phi(0)$.

The ${\cal P}_{ab}$ appear to be double trace operators, while the bulk fields should be single-trace.  However, normal ordering will induce single-trace terms, and these are the leading terms in $1/N$; we will see an example in the next section.  Since the appropriate normal-ordering prescription will depend on the background, a background-independent precursor will have different single-trace parts in different backgrounds.

A better way to understand this construction is to note that  $L^{ij}_a$ is the generator of $O(N)$ transformations acting on site $a$, and so commutes with invariants on any single site.  Any invariant built out of the $L^{ij}_a$ thus commutes with all local invariants.  These are nontrivial in general, as only the total $O(N)$ generator $L^{ij} = \sum_a L_a^{ij}$ annihilates all physical states.
Note also that
\begin{equation}
{\cal P}_{12} = - {\cal P}_{22} - {\cal P}_{32}  + L^{ij} L_2^{ij} \,.
\end{equation}
It follows that on physical states ${\cal P}_{12}$ is equivalent to $- {\cal P}_{22} - {\cal P}_{32}$, which does not involve the site~1 at all.

For any operator ${\cal A}^{ij}$ in the adjoint, the combination $L^{ij} {\cal A}^{ij}$ annihilates all gauge-invariant states,
\begin{equation}
L^{ij} {\cal A}^{ij} \cong 0 \,. \label{gaugefree}
\end{equation}
Note that $L^{ij} {\cal A}^{ij} = {\cal A}^{ij} L^{ij}$.  Also, the commutator of the operator~(\ref{gaugefree}) with an  invariant,
\begin{equation}
[ L^{ij} {\cal A}^{ij} , {\cal S}] = L^{ij} [ {\cal A}^{ij} , {\cal S}] \,,
\end{equation}
need not vanish identically, but it vanishes on physical states because we can commute $L^{ij}$ through to the right.

The  gauge-invariant precursors thus have natural error-correction built in.  We can read messages encoded in any of the ${\cal P}_{ab}$ by looking at the state on any two sites, while observables on any single site contain no information about any ${\cal P}_{ab}$.  Note that in addition to erasing any single site, we can introduce arbitrary gauge-invariant one-site errors on the remaining sites without losing the message.  This may not fit precisely into the standard framework of quantum error correction~\cite{QEC}. The gauge-invariant subspace in which the precursors act is not a standard code subspace, in that not all states represent distinct messages: it also includes  error degrees of freedom from the one-site gauge invariant operators.\footnote{The code subspace in the model of Ref.~\cite{Almheiri:2014lwa} can also be characterized as a gauge-invariant space, under the $Z_3 \times Z_3$ where the first factor cyclicly permutes the sites and the second shifts all registers equally.  However, there are no analogs of the single-site gauge invariants, which correspond to local operators in the \mbox{CFT}.}

\section{Free-field precursors}

\begin{figure}[tbp]
\centering
	\includegraphics[width=2in]{fig6.pdf}
\caption{A bulk operator (black dot) can be represented as a CFT precursor consisting of bilocals (red arcs ending on x's) stretching over the entire boundary. A different construction represents the bulk operators in the right bulk Rindler wedge as a precursor with support on only the right half of the boundary. Thus, bilocals localized on the left half of the CFT, as well as bilocals stretching from the left to the right half, can be removed from the precursor. This is due to the additional freedom that comes with considering the action of the precursors only on gauge invariant states.}
\label{fig4}
\end{figure}

Now let us apply these lessons to a continuum model of precursors.  Essentially this is an infinite-site version of the preceding example.  We will take the CFT to be a free massless scalar $\phi^i$ in two dimensions; this will not have a fully local holographic dual, but at least at the level of the two-point function it provides a good model \cite{Jevicki,MP}.  The operator $\phi^2$ has dimension zero, so is dual to a massless $AdS_3$ scalar $\Phi(t,\rho,\theta)$.

By the usual precursor construction \cite{BDHM,Balasubramanian:1998de,Polchinski:1999yd,Bena99,HKLL}, one uses bulk equations of motion to express $\Phi(t,\rho,\theta)$ in terms of the CFT field $\phi^2$ smeared over space and time, and then uses CFT equations of motion to evolve this to an operator at a single time. We will work in the large $N$ limit, expanding $\Phi$ in terms of bulk creation operators $b_{\Delta l}^{\dagger}$ for the normal modes $j_{\Delta l}$,
where $\Delta,l$ are the global energy and angular momentum:
\begin{equation} \label{eq:pre11}
\Phi(t,\rho,\theta) = \sum_{\Delta l} j_{\Delta l}(\rho, \theta) b_{\Delta l}(t) + \text{h.c.}
\end{equation}
Taking the boundary limit of $\Phi$, we find that $b_{\Delta l}$ is the Fourier transform over space and time of $\phi^2$. Expanding $\phi$ in modes give the precursor
\begin{equation} \label{eq:balal}
b_{\Delta l} =\frac{1}{\sqrt{N}} \alpha_{\Delta_+} {\cdot} \tilde\alpha_{ \Delta_-} \,,\quad \Delta_{\pm} = \frac12(\Delta \pm l) \,.
\end{equation}
Here $\alpha_m$  for $m$ positive (negative) is the annihilation (creation) operator for $\phi$ for left-movers of momentum $m$, and similarly for $\tilde{\alpha}_m$ for right-movers (these are normalized $[\alpha_m,\alpha_n] = m \delta_{mn}$), so a bulk particle is dual to a pair of  left and right moving boundary ``particles". The  $1/\sqrt { N}$ normalization is to give the correct $O(1)$ two-point function. Thus (\ref{eq:pre11}) expresses the bulk field in terms of CFT bilinears.
The precursor can alternatively be expressed in position space,
\begin{eqnarray}
\Phi(t,\rho,\theta) &=& \int d\theta'd\theta'' \frac{1}{\sqrt N}\Bigl\{ f(\rho,\theta; \theta', \theta'') \phi(t,\theta') {\cdot} \phi(t,\theta'') \nonumber\\
&& \quad + \ g(\rho,\theta; \theta', \theta'') \pi(t,\theta') {\cdot} \pi(t,\theta'') \Bigr\} \,. \label{prec}
\end{eqnarray}

The integrals~(\ref{prec}) run over the full range $0 \leq \theta',\theta'' \leq 2\pi$, and so determine $\Phi$ in terms of CFT operators on a full Cauchy slice for the \mbox{CFT}.  However, it is expected that bulk physics can be reconstructed in terms of the state on a smaller region of the CFT, namely any one whose causal wedge contains the given bulk point.  One can make this explicit, for example, by separating the CFT into two halves: each  reconstructs a Rindler wedge of the bulk; see Fig.~2.  One can expand the fields in terms of Rindler modes, in a way analogous to (\ref{eq:pre11}). Using now the precursor for a Rindler mode, one has the analogous expression to (\ref{eq:pre11}), but now in terms of the continuous Rindler energy and momenta $\omega, k$ with $\omega_\pm = (\omega \pm k)/2$,
\begin{eqnarray}
\Phi(t,\rho,\theta) &=&
 \int_{ -\infty}^\infty   \frac{d^2\omega_\pm}{\sqrt {N}}h_{\omega_+ \omega_- }^R(\rho,\theta) \alpha_{\omega_+ }(t) {\cdot} \tilde\alpha_{\omega_-}(t)\,. \label{mnprecR}
\end{eqnarray}
The  $\rho,\theta$ are now functions of the AdS-Rindler coordinates, and $h_{\omega_+ \omega_-}^R$ is related to the AdS-Rindler modes.\footnote{There is no AdS-Rindler analog of the position space expression (\ref{prec})~\cite{HKLL}. This is a reflection of the existence of field configurations passing through the bulk wedge and leaving an arbitrarily small imprint on the boundary wedge \cite{BFLRZ}. To reconstruct the bulk on sub-AdS scales, one would need to measure the boundary data to exponential precision \cite{RR}.  While the significance of this on the form of the precursors remains to be understood, in our context the issue is sidestepped by working in momentum space; one could alternatively write a position space expression with complexified boundary coordinates \cite{HKLL} (see also \cite{Morrison:2014jha})}

The Rindler Fourier operators $\alpha_{\omega_\pm}$ (distinguished from the global CFT $\alpha_{\Delta_\pm}$ by their subscript) have support only within the Rindler patch of the CFT, which is half of the CFT.  On the other hand,   the global CFT operators $\alpha_{\Delta_\pm}$, or equivalently the functions~$f(t,\rho,\theta; \theta', \theta'')$ and  $g(t,\rho,\theta; \theta', \theta'')$,  take values in the full Cauchy slice, so that bilinears may lie in either half of the CFT, or stretch between them. So there appears to be a contradiction between the forms~(\ref{eq:pre11}) and~(\ref{mnprecR}).

The point is that there is freedom in the choice of $h_{\Delta_+ \Delta_-}= j_{\Delta l}$.  This is particularly simple in this large-$N$ limit where only the two-point functions of gauge-invariant operators remain. From the form (\ref{eq:balal}), we see that any additional contribution $\alpha_{\Delta_+} \cdot \tilde{\alpha}_{\Delta_-}$ for $\Delta_+\Delta_- <0$, annihilates the vacuum in both directions and so makes no contribution to the two-point function.  (Such an operator would be dual to a bulk mode with $\Delta < l$, which does not exist in global AdS.)  Therefore to this order we have freedom to shift
\begin{equation}
h_{\Delta_+\Delta_-}(\rho,\theta) \to h_{\Delta_+\Delta_-}(\rho,\theta) + \lambda_{\Delta_+\Delta_-}(\rho,\theta)  \label{hfree}
\end{equation}
for arbitrary $\lambda_{\Delta_+\Delta_-}$ restricted to $\Delta_+\Delta_-<0$.  In the Appendix we show explicitly that the Poincar\'e and Rindler precursors are equal up to the equivalence~(\ref{hfree}).

This freedom is well-known~\cite{Bena99}, but in light of the preceding section we can express it in a new and powerful way.  The generator of gauge transformations is
\begin{equation}
L^{ij} = \sum_{r=1}^\infty \frac{2}{r}(\alpha_{-r}^{[i} \alpha_r^{j]} + \tilde\alpha_{-r}^{[i} \tilde\alpha_r^{j]}) +  L_0^{ij}  \,,
\end{equation}
where the last term acts on the zero modes.  Then for $\Delta_+< 0$ and $\Delta_->0$, we have
\begin{equation}
L^{ij} \alpha^i_{\Delta_+} \tilde\alpha^j_{\Delta_-} = \alpha^i_{\Delta_+} L^{ij} \tilde\alpha^j_{\Delta_-} + (1-N) \alpha_{\Delta_+} {\cdot} \tilde\alpha_{\Delta_-}\,.  \label{normord}
\end{equation}
The first term on the right is normal ordered, and so of order $N$, while the second is of order $N^{3/2}$.  Thus the freedom~(\ref{hfree}) is a special case of the freedom~(\ref{gaugefree}), a surprising result.  What had seemed to be a somewhat obscure dynamical freedom is reinterpreted in terms of gauge invariance. Note too that for $\Delta_+ \Delta_- > 0$ there is no normal-ordering constant in $L^{ij} \alpha^i_{\Delta_+} \alpha^j_{\Delta_-}$, so these bilinears are nonvanishing as we would expect.  This interpretation of the operator identity~(\ref{hfree}) shows that it is not special to the leading large-$N$ behavior, nor to a given vacuum.

In fact, it is not possible for the precursors to commute exactly with all local operators because they cannot commute with the Hamiltonian, which is an integral over $T_{\mu\nu}$.  This corresponds to the need for gravitational dressing to define the bulk position in a coordinate invariant way~\cite{Heemskerk:2012np,Kabat:2012hp,Kabat:2013wga}.
This did not arise in the free-field model because it is of higher order in $1/N$.  Also at higher orders in $1/N$ there are other complications.  The free-field toy model will not be local in the bulk below the AdS scale.  And, in a realistic model the gauge symmetry will be a function of space as well as time, necessitating Wilson lines in the precursors; their commutators will enter at next order in $1/N$.  In the full-fledged duality, precursors can be constructed order by order in $1/N$ around a semiclassical background~\cite{Kabat:2011rz,Heemskerk:2012mn}, and by construction the localization properties noted above will hold in this expansion.  Extension of the construction beyond this expansion is an interesting question for the future.

The gravitational dressing can be such that the precursor fails to commute with local CFT operators at only one point~\cite{Kabat:2013wga}, for example by working in Fefferman-Graham gauge.  This may seem like too strong a requirement to satisfy, but the precursor construction guarantees it at least in the $1/N$ expansion around a semiclassical background.  It is sometimes argued that this would require the precursors to commute with nonlocal operators as well, because the latter can be expanded in an infinite series of local operators.  However, taking a commutator does not commute with summing such a series.  To see this, consider a free scalar, where
\begin{equation}
\phi(t,0) = e^{iHt} \phi(0,0) e^{-iHt} = \sum_{n=0}^\infty \frac{t^n}{n!} \partial^n_t \phi(0,0)
\end{equation}
The operator $\pi(0,x)$ commutes with every operator on the right for $x \neq 0$, but only commutes with the operator on the left for $|x|>|t|$.
It should be noted that taking a commutator also does not commute with the operator product expansion, nor with taking the continuum limit of a lattice theory.  The point is that a commutator is the zero-time limit of a difference of time-orderings, and this limit need not commute with the others.

\section{Conclusion}

Dualities only act on gauge-invariant quantities, and so the role of gauge symmetry is often deemphasized.  However, our work suggests a deep connection between gauge invariance in the boundary theory and the emergence of the dual spacetime, which deserves further exploration.   This connection shows up in the natural error-correction that allows for the localization of bulk information in different regions, and which seems to  capture the gauge/gravity structure more precisely than the standard error correction schemes.  This is seen both in the three-site model and in the continuum precursors.  The introduction of gauge symmetries into quantum information theory may be interesting to explore for other reasons as well.

\begin{acknowledgments}
We thank Ahmed Almheiri, Xi Dong and Daniel Harlow for useful discussions, and for comments on the manuscript. This work is supported by NSF Grants PHY11-25915 and PHY13-16748.

\end{acknowledgments}

\section{Appendix}

Here we show at leading order in $N$ the operator equality of the Rindler and Poincar\'e precursors, up to pure gauge terms.  We use the Poincar\'e description in place of the global description of the main text for ease of notation.

Let $b_{\omega k} = \alpha_{\omega_+} {\cdot} \tilde\alpha_{ \omega_-}/{\sqrt{N}} $ denote the bulk annihilation operator for a Rindler mode with frequency $\omega$ and momentum $k$.
We do a Bogoliubov transformation to the Minkowski mode operators\footnote{Our notation is such that ${\Gamma}_{\omega\nu}$ includes both the positive and negative frequency Bogoliubov coefficients, depending  on whether $\nu$ is positive or negative.}
\begin{equation} \label{eq:bBoug}
\alpha_{\omega} = \int \frac{d\nu}{2\pi}\,  {\Gamma}_{\omega\nu}\alpha_{\nu} \,.
\end{equation}
We distinguish the Minkowski modes by their arguments, $\nu,\kappa$.  Then
\begin{equation}
b_{\omega k} =\frac{1}{\sqrt{N}}  \int\!\!\!\!\int_{-\infty}^\infty \frac{d\nu_+ d\nu_-}{(2\pi)^2}  {\Gamma}_{\omega_+\nu_+}  {\Gamma}_{\omega_-\nu_-}\alpha_{\nu_+} {\cdot}\tilde\alpha_{\nu_-} \,, \label{eq:pre1}
\end{equation}
where $\nu_{\pm} = \frac12(\nu \pm \kappa)$.
Our goal now is to show that this is equivalent to the precursor found from the Poincar\'e smearing function.

The precursor for a Poincar\'e mode is
\begin{equation} \label{eq:bpoin}
b_{\nu\kappa} = \frac{1}{\sqrt{N}}\alpha_{\nu_+} {\cdot} \tilde\alpha_{ \nu_-} \,.
\end{equation}
Unlike the Rindler case, where there are evanescent modes with $|\omega|<|k|$ \cite{RR}, the Minkowski modes are restricted to $|\nu|>|\kappa|$, or $\nu_+ \nu_- >0$.
We want to express our AdS Rindler mode in terms of Poincar\'e modes, and then use the precursor (\ref{eq:bpoin}) for each of the Poincar\'e modes. To do this we need to do a bulk Bogoliubov transformation,
\begin{eqnarray}
b_{\omega k} &=&\int\!\!\!\!\int_{-\infty}^\infty \frac{d\nu_+ d\nu_-}{(2\pi)^2}\theta(\nu_+ \nu_-)  \Gamma_{\omega k, \nu\kappa} b_{\nu\kappa}
\nonumber\\
&=&  \frac{1}{\sqrt{N}}    \int\!\!\!\!\int_{-\infty}^\infty \frac{d\nu_+ d\nu_-}{(2\pi)^2}\theta(\nu_+ \nu_-)  \Gamma_{\omega k, \nu\kappa} \alpha_{\nu_+} {\cdot} \tilde\alpha_{ \nu_-} . \quad \label{eq:pre2}
\end{eqnarray}
In the second line we have inserted the Poincar\'e precursor (\ref{eq:bpoin}).

The Rindler and Poincar\'e precursors~(\ref{eq:pre1}) and (\ref{eq:pre2}) differ in their ranges of integration, the latter restricted to $\nu_+\nu_- > 0$.  The smaller momentum range translates into a larger coordinate range: the Rindler precursor by construction includes only bilinears with both fields in one Rindler patch of AdS, while for the Poincar\'e precursor each field can be in either patch.  The equivalence of Eqs.~(\ref{eq:pre1}) and (\ref{eq:pre2}) requires that
\begin{equation}
\Gamma_{\omega k, \nu\kappa} =  {\Gamma}_{\omega_+\nu_+}  {\Gamma}_{\omega_-\nu_-}  \label{bogobogo}
\end{equation}
 for $\nu_+ \nu_- > 0$.   The  remaining difference, from modes with $\nu_+ \nu_- < 0$, is then pure gauge to leading order in $1/N$.  This was shown in Eq.~(\ref{normord}) for the global case; the same argument applies for Poincar\'e.

For completeness we verify Eq.~(\ref{bogobogo}). First, we note that near the boundary AdS-Rindler asymptotes to Rindler. To see this, let the AdS-Rindler metric be
\begin{equation}
ds^2 = - (r^2 - 1) d\tau^2 + \frac{dr^2}{r^2 - 1} + r^2 dx^2
\end{equation}
and consider the coordinate transformation
\begin{equation}
u = -\frac{\sqrt{r^2-1}}{r} e^{x- \tau}\,,  \ \ v =  \frac{\sqrt{r^2-1}}{r} e^{x+ \tau}\,, \ \
z = e^{x}/r \,,
\end{equation}
which yields the Poincar\'e metric,
\begin{equation}
ds^2 = \frac{- du dv + dz^2}{z^2} \,.
\end{equation}
Notice that as $r\rightarrow \infty$, we have that
\begin{equation}
u\rightarrow - e^{x - \tau}, \ \ \ \ v\rightarrow e^{x +\tau} \,,
\end{equation}
which is  the standard transformation between Minkowski and Rindler coordinates.

The bulk Bogoliubov coefficients are given by the Klein-Gordon inner product between an AdS-Rindler mode and a Poincar\'e mode. This  is usually computed on a spacelike codimension-one surface. We will instead compute the Klein-Gordon inner product  on a timelike codimension-one surface: the boundary of AdS. The boundary limit of the Poincar\'e/AdS-Rindler modes is, respectively,
\begin{equation}
e^{-i u\nu_+ - i v\nu_-}~,  \ \ \ (-u)^{i \omega_+} v^{i \omega_-}
\end{equation}
where we have rescaled by the conformal factor $z^{\Delta}$ and have chosen a nonstandard normalization for the modes. The Bogoliubov coefficients are thus
\begin{eqnarray}
\Gamma_{\omega k, \nu\kappa} &=& \int du dv \, e^{-i u\nu_+ - i v\nu_-} (-u)^{i \omega_+} v^{i \omega_-} \nonumber\\
&=& \Gamma_{\omega_+, \nu_+}\Gamma_{\omega_-, \nu_-} \,,
\end{eqnarray}
 confirming the needed relation (\ref{bogobogo}) between the AdS-Rindler/Poincar\'e Bogoliubov coefficients and the boundary Rindler/Minkowski Bogoliubov coefficients.


\section{\label{sec:intro}Introduction}

Since its synthesis a decade ago, graphene has been the subject of intense research\cite{graphenereview,transportrev}.  In the long-wavelength limit, graphene satisfies the massless Dirac equation, where its electrons follow a linear band dispersion given by $E = \pm \hbar v_f q$, where the $\pm$ refers to the conduction/valence bands, $v_f \sim 10^8 \mathrm{cm/s} \approx c/300$ is the Fermi velocity, and $q$ is the electron momentum (in 2D)\cite{wallace,dirac}.  As a result of this dispersion, the bare electron-electron interaction strength -- as defined by the fine-structure constant, $\alpha = U/K \approx 2.2/\epsilon$, where $\epsilon$ is the dielectric constant of the substrate on which the graphene is mounted ($\alpha \approx 2.2$ for suspended graphene) -- is of order unity, in contrast to the interaction strength in QED, where $\alpha = 1/137$; thus, many-body effects should, in principle, play a significant role in graphene\cite{kotov}.  In particular, the screened value of $\alpha$, which one could call a renormalized or effective fine structure constant, $\alpha^\ast$, is a subject of great theoretical interest\cite{ulybyshev,fogler,kotov,non-fermi_renorm,mishchenko}, and is relevant to all Dirac systems including topological insulator surface states\cite{kane,qi}, some classes of transition metal dichalcogenides\cite{rossnagel}, three-dimensional Dirac and Weyl semimetals\cite{ashvin,young}, etc.

Experiment and theory, however, give conflicting views on the value of $\alpha^\ast$.  Using inelastic X-ray scattering (IXS) experiments performed on graphite\cite{reed}, we previously found that, for freestanding graphene, $\alpha^\ast \approx 0.15$ as $q \rightarrow 0$ and $\omega \rightarrow 0$, and cited excitonic shifts in the $\pi \rightarrow \pi^*$, particle-hole continuum as the origin of the reduced value of $\alpha^\ast$. Interpretation of this experiment was based on the assumption that the interaction between graphene sheets in graphite is primarily Coulombic, with interlayer hopping playing a secondary role\cite{reed}. The experiment, however, had an energy resolution of only 0.3 eV, when the features of primary interest range from 1 - 15 eV.  Additionally, the lowest momenta measured were at 0.238 \AA$^{-1}$ and 0.476 \AA$^{-1}$ so the extrapolation of $q \rightarrow 0$ relied on only a few points.  Subsequent calculations in the random-phase-approximation\cite{katsnelson} (RPA) using the full $\pi$-bands and including the interlayer hopping parameters $\gamma_3$ and $\gamma_1$ in graphite (see Fig. \ref{fig:structure}) suggested that this discrepancy was due to graphitic effects, i.e., the presence of interlayer hopping effects ignored in our analysis.

In this paper, we present a refined measurement of $\alpha^\ast$ in graphene as measured in graphite, using an improved version of the previous method\cite{reed}, with higher energy and momentum resolution, as well as reduced elastic background.  We evaluate the importance of interlayer hopping, excitonic corrections, and the higher energy $\sigma$-bands to the effective fine-structure constant.  We compare to theoretical results obtained within the RPA, and show that for $q \sim 0.212$ \AA$^{-1}$ (length scale $\sim$ 30 \AA), $\alpha^\ast$ falls in the range 0.25 to 0.35, which is well below the RPA value for graphene. We show the origin of this anomalously low value is not graphitic (i.e., interlayer hopping) effects, but a combination of screening from the $\sigma$-bonded electrons, which contribute significantly at finite momentum, and excitonic effects in the $\pi$ bands, which comprise a beyond-PRA correction to the screening. While the effect of $\sigma$ band screening decreases with decreasing $q$, we argue such effects remain relevant to length scales up to $\sim 30 \AA$.

\section{\label{sec:experiment}Experiment}

IXS measurements were carried out in Sector 9 at the Advanced Photon Source. Energy analysis was done with a diced, Si-444, spherical backscattering analyzer operating at 7.81 keV, which imaged scattered photons onto a MYTHEN microstrip detector. The total energy resolution of the instrument was 175 meV, which is a factor of two better than our previous study\cite{reed}.

For these measurements, we constructed a new sample chamber equipped with a moving, in-vacuum beam stop, providing access to scattering angles as low as $1^{\circ}$. The momentum resolution was set to 0.15-0.3 \AA$^{-1}$, depending upon the scan region. The chamber was designed for windowless operation with the synchrotron beam pipes, reducing scattering from upstream vacuum windows. The final setup exhibited greatly reduced background in the 1-5 eV range, particularly from the quasielastic line, a key source of error as described in Appendix \ref{app:sumfit}.

Experiments were done on both ZYA-grade, highly-ordered pyrolitic graphite (HOPG) and high quality, commercially obtained graphite single-crystals. The spectra of the two were found to be indistinguishable in the momentum range in this study. The experimental data were placed in absolute units of $(eV \AA^3)^{-1}$ by normalizing using the $F$ sum rule, as described in Ref. \cite{reed} and Appendix \ref{app:sumfit}.

\section{\label{sec:conversion}Conversion of Experimental Data}

In order to obtain information about $\alpha^\ast$, we need to determine the two-dimensional, density-density response function of graphene, $\chi_{2D}(\ve{q},\omega)$, which is the most general description of the collective charge dynamics of the system\cite{reed}. IXS measurements cannot be carried out on single-layer graphene, however, so a means of determining properties of graphene from measurements on graphite is required.

The measured intensity in an IXS experiment at a given momentum and energy is proportional to the dynamic structure factor, $S(\ve{q},\omega)$,
which is related to the density-density response function of three-dimensional graphite, $\chi_{3D}(q,\omega)$, by the quantum mechanical version of the fluctuation-dissipation theorem,

\begin{equation}
\label{eq:f-d}
S(\ve{q},\omega)=-\frac{1}{\pi}\frac{1}{1-e^{-\beta\hbar\omega}}\operatorname{Im}\chi_{3D}(\ve{q},\omega).
\end{equation}
We note that $\chi_{3D}$ is causal and satisfies the Kramers-Kronig relation
\begin{equation}
	\operatorname{Re}\chi_{3D}(\ve{q},\omega)=
		\frac{2}{\pi}\mathcal{P}\int_0^{\infty}{d\omega'
		\frac{\omega'\operatorname{Im}\chi_{3D}(\ve{q},\omega')}{(\omega')^2-\omega^2}}.
	\label{eq:kramers}
\end{equation}

\noindent
So, by extrapolating to $\omega \rightarrow \infty$, interpolating the discrete points in $\omega$, and integrating, it is possible to determine the full, complex $\chi_{3D}(\ve{q},\omega)$ for graphite in absolute units of $(eV \AA^3)^{-1}$, as shown in Fig. \ref{fig:imchi3d}.  Note that the onset of $\pi \rightarrow \pi^*$ particle-hole excitations is visible in the spectra near $\omega= v_f q$, indicating that we are probing the relevant, low-energy valence excitations. The peak at $\omega \approx 8$ eV is the well-known $\pi$-plasmon, which is not a free carrier plasmon but an effect of the Van Hove singularity at the top of the $\pi$ band\cite{reed}.

\begin{figure*}
	\subfloat[][]{
		\includegraphics[width=0.45\textwidth]{chi3dplot_redux.pdf}
		\label{fig:imchi3d}
	}
	\subfloat[][]{
		\includegraphics[width=0.45\textwidth]{chi2dplot_redux.pdf}
		\label{fig:imchi2d}
	}
	\caption[]{The imaginary part of the experimental density response of graphite and graphene compared to results from RPA calculations. (a) $-\operatorname{Im}\chi_{3D}(\ve{q},\omega)$ from IXS measurements (blue points), RPA calculations including interlayer hopping (green line) and omitting interlayer hopping (red line), plotted against energy for selected in-plane momenta, $q$. The RPA curves had to be multiplied by an arbitrary scale factor to allow visual comparison to the experiment. The black, vertical lines indicate the energy, $\hbar v_F q$, at which the onset of the particle-hole continuum is expected in RPA. (b) $-\operatorname{Im}\chi_{2D}(\ve{q},\omega)$ determined from IXS measurements from Eq. 3 (blue points), RPA calculations including interlayer hopping (red line) and omitting interlayer hopping (orange line), plotted against energy for selected in-plane momenta, $q$. The black line, again, indicates the onset energy of the continuum expected from RPA. Experimental error bars are derived from both Poisson statistics and our fitting parameters (see Appendix \ref{app:sumfit}).}
	\label{fig:chiplots}
\end{figure*}

Learning about graphene requires finding a relationship between its density response, $\chi_{2D}(\ve{q},\omega)$, and the measured, three-dimensional response of graphite, $\chi_{3D}(\ve{q},\omega)$. This can be accomplished by making three assumptions: (1) that the primary interaction between layers in graphite is electrostatic, with interlayer hopping playing a negligible role, (2) that the layers are arbitrarily thin, and (3) that the electrons are distributed homogeneously within each layer. In this case, the two response functions are related by \cite{reed}

\begin{equation}
	\chi_{2D}(\ve{q},\omega) = \frac{\chi_{3D}(\ve{q},\omega) d}{1 - V_{2D}(\ve{q})[1-F(\ve{q})]\chi_{3D}(\ve{q},\omega) d}.
	\label{eq:2d3d}
\end{equation}

\noindent
Here, $V_{2D}(\ve{q}) = 2\pi e^2 /q$ is the two-dimensional Coulomb interaction, and $F(\ve{q}) = \sinh(q d)/(\cosh(q d)-\cos(q_z d))$ is a structure factor that describes the Coulomb interaction in a layered, three-dimensional system, i.e., $V_{3D}(\ve{q}) = V_{2D}(\ve{q}) F(\ve{q}) d$, where $d=3.35\AA$ is the interlayer spacing\cite{shung,reed,kotov}.

Eq. \ref{eq:2d3d} makes use of the fact that, if the above three assumptions are valid, the polarization function of graphene, $\Pi_{2D}(\ve{q},\omega)$, is (apart from units) identical to that for graphite,\cite{reed} i.e.,

\begin{equation}
	\Pi_{2D}(\ve{q},\omega) = \Pi_{3D}(\ve{q},\omega) d.
	\label{eq:pi2d3d}
\end{equation}

\noindent
These quantities are related to the respective susceptibilities by $\chi_{3D}=\Pi_{3D}/\epsilon_{3D}$ and $\chi_{2D}=\Pi_{2D}/\epsilon_{2D}$, where $\epsilon_{3D}=1-V_{3D}\Pi_{3D}$ and $\epsilon_{2D}=1-V_{2D}\Pi_{2D}$ are the dielectric functions.  The spectra for $\operatorname{Im}\chi_{2D}$ obtained in this manner are shown in Fig. \ref{fig:imchi2d}, and the associated polarization functions in Fig. \ref{fig:piplots}. Crucially, the $\chi_{2D}(\ve{q},\omega)$ obtained from Eq. \ref{eq:2d3d} exhibits the correct asymptotic properties expected for two-dimensional graphene (see Fig. \ref{fig:chi0} and Section VI).

Written in terms of the other quantities, the screened, effective fine structure constant of graphene $\alpha^\ast$ is

\begin{equation}
	\alpha^\ast(\ve{q},\omega) = \frac{\alpha}{\epsilon_{2D}} = \alpha \cdot \left[1 + V_{2D}(\ve{q})\chi_{2D}(\ve{q},\omega)\right].
	\label{eq:alphastar}
\end{equation}

\noindent
This quantity can be thought of as the value of the fine structure constant incorporating screening corrections to all orders in perturbation theory. Note that, although Eq. \ref{eq:alphastar} appears similar to an RPA expression, we have at no point assumed that only RPA polarization bubbles contribute to the susceptibility; indeed, the $\chi$ we recover from experiment includes all screening processes including excitonic effects and other corrections beyond the RPA.

\section{\label{sec:rpasect}RPA Calculations}

\begin{figure}
	\subfloat[][]{
		\includegraphics[width=0.35\columnwidth]{hopgpic.pdf}
		\label{fig:structure}
	}
	\subfloat[][]{
		\includegraphics[width=0.55\columnwidth]{graphpic.pdf}
		\label{fig:brillouin}
	}
	\caption[]{\subref{fig:structure} Crystal structure of ABA-stacked graphite defining the hopping parameters $t$, $\gamma_1$, and
			$\gamma_3$. \subref{fig:brillouin} Brillouin zone of graphene defining the in-plane azimuthal angle of the basal plane. The spectra were found to be independent of this angle in the range of momenta measured in this study.}
\end{figure}

To aid interpretation of the experimental data, we performed a $\pi$-electron tight binding calculation of the susceptibilities of graphite and graphene in the random phase approximation (RPA), following the approach of Yuan et al. \cite{katsnelson}. In this approach, one calculates the polarization functions for single-layer graphene (SLG), $\Pi_{2D}$, and multilayer graphite (MLG), $\Pi_{3D}$, using the Lindhard formula, as described in Appendix B. The polarization functions are then used to determine the dielectric functions and susceptibilities via the relationships discussed in Section III.

The RPA approach neglects excitonic effects, which have been argued to be crucial for interpreting the particle-hole excitation spectrum of graphene\cite{reed,makHeinzExc}. However, RPA has the advantage of allowing one to switch on and off the interlayer hopping parameters, $\gamma_1$ and $\gamma_3$ defined in Fig. \ref{fig:structure}, allowing one to evaluate the consequences of both interlayer hopping and excitonic effects by comparing to the experimental IXS data.

Like Yuan \cite{katsnelson}, our calculations are based on a tight binding theory that includes only the electrons in the $\pi$ bands (see Appendix B). The effects of the deeper $\sigma$ bands are incorporated phenomenologically as a frequency-independent, background dielectric constant, \citep{kappakats,kappaorig}

\begin{equation}
	\kappa_N(\ve{q}) = \frac{\kappa_0 + 1 - (\kappa_0 -1)e^{-q L_N}}{\kappa_0 + 1 + (\kappa_0-1)e^{-q L_N}}\kappa_0
	\label{eq:kappa}
\end{equation}

\noindent
where $\kappa_0 = 2.4$ is the background dielectric constant of graphite\cite{dresselhaus}, $L_N=d_m + (N - 1) d$ is the height of a multilayer graphite system with $N$ layers, and $d_m = 2.8 \AA$ is the thickness of a single graphene sheet \cite{katsnelson}. Eq. \ref{eq:kappa} exhibits the proper scaling with the number of layers at low momenta, and was argued to correctly describe screening by excitations involving the sigma bands\cite{katsnelson}. Note that for infinite, MLG, this expression reduces to $\kappa_\infty(\ve{q})=\kappa_0$ for any nonzero momentum.

We differ from Yuan\cite{katsnelson}, however, in our use of $\kappa_N(\ve{q})$. In Ref. \cite{katsnelson}, $\kappa_N(\ve{q})$ was used only to reduce the strength of the Coulomb interaction, by using a screened $W(\ve{q})=V(\ve{q})/\kappa(\ve{q})$. As shown in Appendix C, this usage neglects interference between screening by $\pi$ and $\sigma$ electrons. The correct use of $\kappa_N(\ve{q})$ is in the relationship between the polarization and dielectric functions,

\begin{equation}
	\epsilon(\ve{q},\omega) = \kappa_N(\ve{q}) - V(\ve{q})\Pi(\ve{q},\omega),
	\label{eq:epsilon}
\end{equation}

\noindent
where $N$ is chosen to match the dimensionality of the system.

Having determined the polarization functions for MLG, $\Pi_{3D}(\ve{q},\omega)$, and SLG, $\Pi_{2D}(\ve{q},\omega)$, we calculated four different susceptibilities for comparison to experiment. The first,

\begin{equation}
	\chi_{3D}^{RPA}(\ve{q},\omega) = \frac{\Pi_{3D}(\ve{q},\omega)}{\kappa_\infty(\ve{q}) - V_{2D}(\ve{q})F(\ve{q}) d \, \Pi_{3D}(\ve{q},\omega)},
\end{equation}

\noindent
is the three-dimensional susceptibility for MLG in the RPA, including the effects of both interlayer hopping and electrostatic coupling between the layers. The imaginary part of this quantity could be compared directly to the experimental data to assess the importance of excitonic effects. The second,

\begin{equation}
	\chi_{2D}^{RPA}(\ve{q},\omega) = \frac{\Pi_{2D}(\ve{q},\omega)}{\kappa_1(\ve{q}) - V_{2D}(\ve{q})\Pi_{2D}(\ve{q},\omega)}
\end{equation}

\noindent
is the two-dimensional susceptibility for SLG. This quantity could be compared to the converted experimental data resulting from the application of Eq. 3 to assess the combined importance of excitonic and interlayer hopping effects. The third quantity

\begin{equation}
	\tilde{\chi}_{2D}^{RPA}(\ve{q},\omega) = \frac{\Pi_{3D}(\ve{q},\omega) d}{\kappa_1(\ve{q}) - V_{2D}(\ve{q})\Pi_{3D}(\ve{q},\omega) d}
\end{equation}

\noindent
is a two-dimensional propagator describing, in RPA, how the physics of graphene would be modified if $\gamma_{1,3} \neq 0$, i.e., if it were subjected to the same interlayer hopping effects as graphite. Finally,

\begin{equation}
	\tilde{\chi}_{3D}^{RPA}(\ve{q},\omega) = \frac{\Pi_{2D}(\ve{q},\omega)/d}{\kappa_\infty(\ve{q}) - V_{2D}(\ve{q})F(\ve{q})\Pi_{2D}(\ve{q},\omega)},
\end{equation}

\noindent
is a three-dimensional propagator for graphite in which the interlayer hopping is switched off, i.e., $\gamma_{1,3} = 0$, but the layers are still coupled by electrostatic effects. By comparing these quantites to one another, to the experimental data, and to the converted data determined from Eq. 3, it should be possible to disentangle the contribution of excitonic effects, interlayer hopping, and $\sigma$-band screening on the effective fine structure constant of graphene.

\section{Results}
\label{sec:results}

We are now in a position to evaluate the source of the anomalous screening observed in our original study\cite{reed}. In particular, we wish to know whether this screening is due to the graphitic effects discussed recently\cite{katsnelson}, i.e. the interlayer hopping parameters $\gamma_1$ and $\gamma_3$, background screening from the $\sigma$-bands, or excitonic effects as we claimed previously\cite{reed}.

In Fig. \ref{fig:chiplots}a we compare the experimental data to the imaginary part of $\chi_{3D}$ computed from RPA both with and without interlayer hopping (Eqs. 8 and 11, respectively). The first thing to note is that RPA does a poor job at reproducing the energy of the so-called ``$\pi$-plasmon" at $\sim 8$ eV. Note that this excitation is not a free carrier plasmon, but a result of the Van Hove singularity at the top of the $\pi$ band\cite{reed}. We attempted to improve the agreement by adjusting the parameters $t$, $\gamma_{1,3}$, and $\kappa_0$, but any parameters that produced the correct plasmon dispersion were dramatically different from the commonly accepted values\cite{kotov,taft} of $t=3$ eV, $\gamma_1 = 0.4$ eV, $\gamma_3 =0.3$ eV, and $\kappa_0 = 2.4$.  Changes to these parameters also drastically distorted the qualitative shape of the $\pi \rightarrow \pi^\ast$ continuum. The discrepancy is most likely due to excitonic effects not captured by RPA\cite{reed,MakHeinzExc}. We therefore show only the spectra calculated using the commonly accepted values for the hopping parameters and $\kappa_0$. Note that, in agreement with Ref. \cite{katsnelson}, the effect of a nonzero $\gamma_{1,3}$ is not to qualitatively change the spectra, but to smear some of the features near threshold. Such effects are subtle and not easily distinguished from broadening due to experimental resolution.

In addition, a significant discrepancy is observed between the {\it size} of the experimental response and that computed with RPA (Fig. 1a). The latter is significantly larger in magnitude and had to be multiplied by an arbitrary scale factor, ranging from 0.15 to unity, to allow visual comparison with the experiment. This magnitude discrepancy is fundamental, and arises from the significant spectral weight in the $\sigma$-bands in the experimental data that is absent from the RPA result. Despite residing at $\sim$ 25 eV, the $\sigma$-bands are quite polarizable and have significant influence on the value of $\chi$ at low energy, because $\chi$ is a nonlinear function of $\Pi$\cite{taft,mari}. In other words, our RPA approach treats the system as if all the screening is accomplished by the $\pi$ electrons, while in reality some of the work is done instead by the $\sigma$-bonded electrons. RPA therefore predicts that the response from the $\pi$ electrons is anomalously large, since the added $\sigma$ screening channel is absent. The influence of the $\sigma$-bands increases with increasing $\ve{q}$, reaching a maximum at around 2 $\AA^{-1}$. Note that this problem is not corrected by including a background $\kappa_N(q)$, which does not replace spectral weight that is missing from the $F$ sum rule (note that using $\kappa_N(q)$ inside a screened $W(q)$, as was done in Ref. \cite{katsnelson}, does not improve the situation).

\begin{figure}
	\includegraphics[width=0.7\columnwidth]{pi3dplot_redux}
	\caption[]{Imaginary part of the polarization function of graphite, $-\operatorname{Im}\Pi_{3D}(\ve{q},\omega)$, from experiment (blue points), compared to the results of RPA calculations both with (green line) and without (red line) interlayer hopping, $\gamma_{1,3}$. Within the assumptions described in Section III, $\operatorname{Im}\Pi_{3D}(\ve{q},\omega)$ should be the same as $\operatorname{Im}\Pi_{2D}(\ve{q},\omega)$ apart from an overall factor of the interlayer spacing, $d$. Note that the magnitude discrepancy observed in Fig. 1 is not observed here, because polarization functions from different excitations combine additively (Eq. 7)}
	\label{fig:piplots}
\end{figure}

Still looking at Fig. 1a, another key observation is that the onset of the $\pi \rightarrow \pi^*$ continuum in the experiment is consistently lower in energy than in the RPA calculation. These shifts are even better visible in the imaginary part of the polarization function (Fig. 3), which is better reflective of single particle excitations than the susceptibility, which emphasizes the collective modes\cite{caigraphite}. For momenta $q > 0.35 \AA^{-1}$, these shifts are as much as 1 eV compared to RPA, which is too large to be explained by graphitic effects, and must be attributed to excitonic interaction between the valence hole and photoexcited electron, as discussed previously\cite{reed,MakHeinzExc}. At lower momenta, the shifts are smaller and similar in magnituide to what is expected from interlayer hopping effects, though excitonic effects surely still play some role.

We turn now to the two-dimensional response, $\operatorname{Im}\chi_{2D}(\ve{q},\omega)$, shown in Fig. 1b, which compares the experimental data converted using Eq. 3 to the two-dimensional RPA response calculated with and without interlayer hopping (Eqs. 9 and 10). The discrepancy in the $\pi$ plasmon energy observed in Fig. 1a, as well as the excitonic shifts in the $\pi \rightarrow \pi^*$ continuum, are visible also in the two-dimensional response. However, the magnitude discrepancy that is so pronounced in three dimensions is much less significant in two dimensions and fades as $q$ decreases. At the lowest momentum point, $q=0.212 \AA^{-1}$, the magnitude discrepancy is absent entirely. This means that in two dimensions, in contrast to the three dimensional case, the polarizability of the high energy $\sigma$ bands ceases to be important in the limit of low momentum. This confirms one's intuition that screening from high energy excitations should be unimportant in two dimensions as $q \rightarrow 0$.

\section{discussion}

We now turn to the fundamental question of how the effects of interlayer hopping, excitonic shifts in the spectra, and the polarizability of the $\sigma$ bands influence the value of the screened, effective fine structure constant, $\alpha^*(\ve{q},\omega) = \alpha_0 [1 + V_{2D}(\ve{q}) \chi_{2D}(\ve{q},\omega)]$. The quantity $\alpha^*(\ve{q},\omega)$ is a strong function of both energy and momentum, and its value differs greatly depending upon the relative size of the energy, $\omega$, and the momentum, $q v_F$. For transport experiments in a micron-sized device at low temperature, $\omega/v_F q \sim k_B T L / 2 \pi \hbar v_F \sim 10^{-3}$. So the quantity of primary interest is the zero-frequency quantity $\alpha^*(\ve{q},0)$ in the limit of small momentum. We note that this quantity is purely real, and could be either larger or smaller than the bare value, $\alpha_0 = e^2/\epsilon_0 \hbar v_F$.

\begin{figure*}
	\subfloat[][]{
		\includegraphics[width=0.53\textwidth]{asymptotic_chi_ehopg2_big_redux.pdf}
		\label{fig:chi0}
	}
	\subfloat[][]{
		\includegraphics[width=0.47\textwidth]{alphaplot_ehopg2_big_redux.pdf}
		\label{fig:alpha0}
	}
	\caption[]{\subref{fig:chi0} Asymptotic screening and the effective fine structure constant of graphene. (a) Combined plot of the static ratios $-\operatorname{Im}\chi_{3D}(q,0)/q$ and $-\operatorname{Im}\chi_{2D}(q,0)/q d$ from experiment and from RPA calculations both with ($\gamma_{1,3} \neq 0$) and without ($\gamma_{1,3} = 0$) interlayer hopping. The results from our previous study\cite{reed} are also plotted, for comparison. This plot shows that all quantities exhibit the proper asymptotic properties in the regime of small $q$. (b) Momentum dependence of the magnitude of the static value of the fine structure constant of graphene. This plot suggests a limiting value in the range 0.25-0.35.
	}
	\label{fig:statics}
\end{figure*}

In Fig. 4a, we plot the momentum dependence of the ratios $\chi_{2D}(\ve{q},0)/q d$ and $\chi_{3D}(\ve{q},0)/q$ for both the experimental data as well as the RPA calculations both with ($\gamma_{1,3} \neq 0$) and without ($\gamma_{1,3} = 0$) interlayer hopping. Note that both of these quantities are real and have units $(eV \AA^2)^{-1}$, allowing them to be compared on a single scatter plot. As expected, the quantity $\chi_{3D}(\ve{q},0)/q$ tends toward 0 as $q \rightarrow 0$, since the susceptibility in three dimensions is expected to scale like $q^2$ at small $q$. All three $\chi_{2D}(\ve{q},0)/q d$ curves, on the other hand, converge to a constant as $q \rightarrow 0$, indicating that the quantity $\chi_{2D}(\ve{q},0)$ scales linearly in $q$ at low momenta, which is expected in two dimensions. This scaling is crucially important for two reasons. First, it demonstrates that our conversion expression, Eq. 3, gives the proper asymptotic behavior of the susceptibility in two dimensions, which is a powerful validation of our method. The second is that the product $V_{2D}(\ve{q}) \cdot \chi_{2D}(\ve{q},0)$ converges to a constant at low $q$ for all momenta, indicating that graphene exhibits a finite screening strength, $\epsilon(\ve{q},0) = [1+V_{2D}(\ve{q})\chi_{2D}(\ve{q},0)]^{-1}$, in the long-wavelength limit.

An important conclusion about interlayer hopping can immediately be reached from Fig. 4a by comparing the RPA calculations of $\chi_{2D}(\ve{q},0)/q d$ with and without interlayer hopping. Notice that the curves are only distinguishable for momenta less than about 0.2 $\AA^{-1}$, which is below the lowest momentum measured in our experiment. What this implies is that, while graphitic effects can in principle influence the value of $\alpha^*$ at low momenta, such effects are likely irrelevant in the regime of our experiment, and that excitonic corrections are in part responsible for deviations compared to RPA visible in Figs. 1a,b at momenta $q < 0.35 \AA^{-1}$.

Fig. 4a also reveals a major discrepancy between the experiment and theoretical curves at large momenta. For $q > 0.6 \AA^{-1}$, the RPA curves drop off gradually, while the experimental points increase dramatically. This increase is a consequence of screening from the $\sigma$ bands, which exhibits a broad maximum at $q \sim 2 \AA^{-1}$, which is of the order the carbon-carbon bond length\cite{reed}. This discrepancy is, again, a consequence of the absence of $\sigma$ band electrons in the RPA calculation. Hence, the quantity $\kappa(\ve{q})$ taken from Ref. \cite{katsnelson}, while argued to be a good description at small $q$, fails to describe any of the major features of $\sigma$ band screening at larger momenta. Fortunately, the influence of these high energy excitations on $\chi_{2D}$ fades at small momentum, suggesting that we should still get useful information about the low-$q$ value of $\alpha^*$ from our study.

Finally, in Fig. 4b we plot the magnitude of the static, effective fine structure constant, $\alpha^*(\ve{q},0)$, as a function of momentum for the experiment (i.e., deduced from Eq. 3), and for RPA both with and without interlayer hopping. For reference, the points from our previous study \cite{reed} are also shown. The difference between the two experiments can be attributed to better resolution, better statistics, and lower experimental background in the current study. For momenta $q > 0.5 \AA^{-1}$ the value of $\alpha^*$ is dominated by screening from the $\sigma$ bands. Unfortunately, the error bars on the lowest momentum point, at which $\sigma$-band effects are negligible, are extremely large due to extrapolation of the $F$ sum rule (see Appendix A). The remaining points are converging toward a value in the range of 0.25 to 0.35. This quantity is larger than given in our previous study, but still smaller than the value expected from RPA, and implies an asymptotic dielectric constant in the range 6.2 to 8.8. Because interlayer hopping effects are irrelevant in this regime, we conclude that the anomalously low value of $\alpha^*$ is a combined consequence of $\sigma$ band screening and excitonic shifts in the $\pi \rightarrow \pi^*$ continuum.

\section{\label{sec:con}Conclusions}

We presented significantly improved measurements of the momentum- and energy- dependent effective fine structure constant of graphene, $\alpha^\ast(\ve{q},\omega)$, using inelastic x-ray scattering measurements of graphite. We deduce an asymptotic value of $\alpha^*$ in the range 0.25--0.35, which is larger than stated in our previous study \cite{reed} but smaller than the RPA value of 0.49, and implies an asymptotic value of $\epsilon$ in the range 6.6--8.8. We also performed $\pi$-band RPA calculations with and without interlayer hopping, and found that graphitic effects from interlayer hopping have no significant effect on screening in the range of momenta we studied. Screening by $\sigma$ band electrons, on the other hand, contribute significantly at finite wave vector. We therefore conclude that the anomalously low value of $\alpha^*$ may be attributed to a combination of excitonic effects in the $\pi \rightarrow \pi^*$ continuum and $\sigma$-band effects. The latter are very large at finite $q$, and should strongly screen short-ranged, Hubbard-like interactions in graphene and carbon allotropes more generally.

We close by commenting on the discrepancies between our conclusions and the RPA calculation described in Ref. \cite{katsnelson}. That study reported a calculated static dielectric constant at $q = 0.238 \AA^{-1}$ that was not far from our earlier result derived from IXS experiments on graphite\cite{reed}. They attributed the anomalously low value of $\alpha^*$ to graphitic effects from the interlayer hopping. Here we have shown that the interlayer hopping parameters, $\gamma_1$ and $\gamma_3$, have no effect on the dielectric response at the lowest momenta measured in our experiments. The discrepancy between these two conclusions arises from the way the Coulomb interaction was handled in the two studies, which differ in two important ways. First, Ref. \cite{katsnelson} ignores the interaction between screening in the $\sigma$ and $\pi$ channels, which must be accounted for as described in our Appendix C. Second, in comparing to our result, they failed to eliminate the Coulomb interaction between the layers, which we have done through the use of Eq. 3. These differences result significantly larger values of $\epsilon(q,0)$ in Ref. \cite{katsnelson}, which they atribute to interlayer hopping effects. What we have shown is that, if the Coulomb effects are handled properly, the apparent effects of interlayer hopping are in fact small, and the primary causes of increased screening are excitonic effects and $\sigma$ band screening.

\begin{acknowledgments}
The authors acknowledge helpful discussions with L. K. Wagner. This work was supported by the U.S. Department of Energy grant DE-FG02-06ER46285, with use of the Advanced Photon Source supported by DEAC02-06CH11357. B.U. acknowledges NSF CAREER grant No. DMR-1352604 for support. E. F. acknowledges support from DOE award No. DE-SC001236. P.A. acknowledges support from the EPiQS Initiative of the Gordon and Betty Moore Foundation, through Grant GBMF4542.
\end{acknowledgments}


\section{Introduction}

Nine different  formulations of non-relativistic quantum mechanics exist  \cite{nine}. These are  the wavefunction,
matrix, path integral, phase space, density matrix, second quantization, variational, pilot wave, and
Quantum Hamilton-Jacobi (QHJ) formulations.
It is interesting to note the close resemblance between random matrix theory and the QHJ formalism. The $n$ stationary points of the random matrix ensemble action \cite{met} $H=V(x)-\frac{1}{2}\sum_{k\neq l}log(\vert x_k-x_l\vert)$ where $V(x)$ is the potential. In QHJ formalism, the problem is solved using the quantum momentum function (QMF) and it plays an important role .  The quantum momentum function  is defined  as \cite{bhal,bhal1}  $p=\sum_{k=1}^n\frac{-i}{x-x_k}+Q(x)$ here the moving poles are simple poles with residue $-i\hbar$ (we take here $\hbar=m=1$) and $Q(x)$ is the singular part of the quantum momentum function.  The moving poles are described as  in the   $n$ first-order poles in the QMF of quantum mechanical origin. The positions of the  poles are and same as the zeros of the wave function. As the zeros of the
wave function change their positions with energy, so do the
location of the corresponding poles in the QMF  \cite{bhal}.  The moving poles behave like random numbers as they are energy dependent (as the energy changes the position of these moving poles will change). The above statement clearly captures the connection between the  $n$ stationary random points and the $n$ moving poles of QMF.

There are other models in the literature whose equations resemble the random matrix ensemble action. They are the Stieltjes electrostatic problem \cite{st,st1} with $n$ moving unit charges,
interacting  through a logarithmic potential, are placed
between  two fixed charges $p$ and $q$ at $a$ and $b$ respectively on a real line
is given by $V(x_k)+\sum_{k\neq l}\frac{1}{x_k-x_l}=0$ where $V(x_k)$ is
the position of the two fixed charges. The motion of point vortices in hydrodynamics is governed by the Helmholtz’s equations\cite{aref} $\frac{dz_k}{dt}=\frac{1}{2\pi i}\sum_{j=1}^{n}\frac{\Gamma_j}{z_k-z_j}+W(z_k)$ here the $\Gamma_j$ is circulations or strengths of point vortices at $z_j$ respectively and $W(z_k)$ is the back ground flow. The equation for low-energy spectrum of the Heisenberg ferromagnetic chain \cite{sas} $\frac{1}{u_l}=\pi d+\frac{2d}{l}\sum_{m\neq l}\frac{1}{u_l-u_m}$.  The analogy between the Stieltjes electrostatic problem  and the quantum momentum function is established by the author him self \cite{kvs} and the connection between the Heisenberg ferromagnetic chain and
the Stieltjes electrostatic problem was established in ref \cite{sas}.

In this paper, we show  that the solving quantum mechanics using QHJ formalism is analogous to solving the quantum mechanics problem using the random matrix theory. The paper is organised as follows, in the following section of the introduction we briefly describe the RMT (part A) followed by QHJ (part B) and Supersymmetric quantum mechanics (part C). In the section II, we give the equivalence between the QHJ and RMT followed by conclusion in section III.

\subsection{Random Matrix Theory}

In 1951, Wigner proposed the use of random matrix theory (RMT) to describe the properties of excited states of atomic nuclei \cite{wig}. This
was the first time RMT was used to model physical reality. In random matrix theory, the dynamics of the ensemble of a random infinite dimensional Hermitian matrix is described by the probability distribution function
\begin{equation}\label{pdfo}
{\cal P}(\lambda_1,\ldots,\lambda_N)\,d\Lambda
= c_ne^{-\beta H}
\,\,d\Lambda
\end{equation}
where  $d\Lambda=d\lambda_1 \ldots d\lambda_N$, $c_n$ is constant of proportionality and the index $\beta=1, 2, 4$  characterizes the real parameters of the symmetry class of orthogonal, unitary and symplectic respectively for the random  matrix
\begin{equation}
H=-\sum_{i=1}^NV(\lambda_i)-\sum_{i<1}^Nln\vert\lambda_i-\lambda_j\vert.
\end{equation}
 As the random matrix is invariant under the symmetry group these ensembles are called the invariant ensembles. In the random matrix $H$,  $V(\lambda_i)$ is the potential, $\lambda_i$ are the eigenvalues with $i$ being a free index running from $1,2,\cdots N$ and $\vert\lambda_i-\lambda_j\vert$ is the Vandermonde determinant. For details please refer to \cite{met}. The probability distribution function can be rewritten as
\begin{equation}\label{pdf}
{\cal P}(\lambda_1,\ldots,\lambda_N)\,d\Lambda
=c_n e^{-\sum_{i=1}^N\beta V(\lambda_i)}
\prod_{i<j}\vert\lambda_i-\lambda_j\vert^\beta
\,\,d\Lambda
\end{equation}
By manipulating the Vandermonde determinant, that is, by adding and deleting columns or rows, the equation (\ref{pdf}) is written as
\begin{equation}\label{pdf1}
{\cal P}(\lambda_1,\ldots,\lambda_N)\,d\Lambda
=c_n \prod_{i=1}^N w^{\frac{1}{2}} (\lambda_i)
\prod_{i<j}\vert\lambda_i-\lambda_j\vert
\,\,d\Lambda
\end{equation}
where $w_\beta (\lambda_i)$ is the weight function of classical orthogonal polynomials.
The classical orthogonal polynomials are classified into three different
categories depending upon the range of the polynomials. The polynomials in the intervals $(-\infty;\infty)$ with weight function $w=e^{-\lambda^2}$ are the Hermite polynomials.
In the intervals $[0;\infty)$ with weight function $w=\lambda^be^{-\lambda}$ are the Laguerre polynomials. In the intervals $[-1;1]$ with weight function $w=(1+\lambda)^{a}(1-\lambda)^b$ are the Jacobi polynomials.

The invariant ensembles in random matrix theory are classified into three the Gaussian ensembles, the Wishart ensembles and the two Wishart ensembles.  In the case of Gaussian, the matrices are known as the Wigner matrices. The Gaussian ensembles are further classified the  Gaussian orthogonal ensemble (GOE), the Gaussian
unitary ensemble (GUE), and the Gaussian symplectic ensemble (GSE).  In stationary case, for the Gaussian ensembles, Wigner law states that the empirical distribution of eigenvalues $\lambda_1,\ldots,\lambda_N$ as $N$ tends to infinity converges to a semicircle in the interval [a,b]. In non-stationary case the equation (\ref{pdf}) is the stationary solution of a Fokker-Planck equation \cite{met}.

In this paper we only deal with the Gaussian ensembles and show that the bound state problems in quantum mechanics are analogous to solving Gaussian unitary ensemble of random matrix theory. This study helps in identifying the potential that appear in the joint probability distribution function in the random matrix theory as a super potential. This approach allows to extend the random matrix theory to the newly discovered exceptional polynomials.

\subsection{Quantum Hamilton Jacobi}

In this section, a brief review of  Quantum Hamilton Jacobi formalism is presented below.
For details see the references \cite{sree}.
The Schr\"odinger equation is given by,
\begin{equation}
- \frac{\hbar^2}{2m}\nabla^2\psi(x,y,z)+ V(x,y,z) \psi(x,y,z) = E
  \psi(x,y,z).    \label{sc}
\end{equation}
One defines a function $S$ analogous to the classical characteristic function by the relation
\begin{equation}
\psi(x,y,z) = \exp\left(\frac{iS}{\hbar}\right)       \label{ac}
\end{equation}
which, when substituted in (\ref{sc}), gives
\begin{equation}
(\vec{\nabla}S)^2 -i \hbar \vec{\nabla}.(\vec{\nabla}S) = 2m (E
  - V(x,y,z)).   \label{qhj0}
\end{equation}
the quantum momentum function $p$ is defined in terms of the function $S$ as
\begin{equation}
\vec{p} = \vec{\nabla} S. \label{mp}
\end{equation}
Substituting (\ref{mp}) in (\ref{qhj0}) gives the QHJ equation
for $\vec{p} $ as
\begin{equation}
(\vec{p})^2 - i \hbar \vec{\nabla}.\vec{p} = 2m (E - V(x,y,z))
  \label{bhy}
\end{equation}
and from (\ref{sc}) and (\ref{mp}), one can see that $\vec{p}$
is the the logarithmic derivative of $\psi(x,y,z)$ {\it i. e}.
\begin{equation}
\vec{p} = -i \hbar \vec{\nabla} ln \psi(x,y,z) \label{lg}
\end{equation}
The above discussion of the QHJ formalism is done in three
dimensions the same equation in one dimension takes the following form
\begin{equation}
p^2 - i \hbar \frac{dp}{dx} = 2m (E - V(x)),       \label{qhj1}
\end{equation}
which is also known as the Riccati equation. In one dimension the eq (\ref{lg}) take the form
\begin{equation}
p = -i \hbar \frac{d}{dx}ln \psi(x). \label{lg1}
\end{equation}
It is shown by Leacock and Padgett \cite{qhj, qhj1} that the action angle variable
gives rise to exact quantization condition
\begin{equation}
J(E) \equiv  \frac{1}{2\pi} \oint_C{pdx} = n\hbar.       \label{act}
\end{equation}
\subsection{SUSY Quantum mechanics}
In supersymmetry, the superpotential $\mathcal{W}(x)$ is defined in terms of the intertwining operators $ \hat{A}$ and $\hat{A}^{\dagger}$  as
\begin{equation}
  \hat{A} = \frac{d}{dx} + \mathcal{W}(x), \qquad \hat{A}^{\dagger} = - \frac{d}{dx} + \mathcal{W}(x),
\label{eq:A}
\end{equation}
This allows one to define a pair of factorized Hamiltonians $H^{\pm}$ as
\begin{eqnarray}
   H^{+} &=& 	\hat{A}^{\dagger} \hat{A} 	= - \frac{d^2}{dx^2} + \mathcal{V}^{+}(x) - E, \label{vp}\\
  H^{-} &=& 	\hat{A}  {\hat A}^{\dagger} 	= - \frac{d^2}{dx^2} + \mathcal{V}^{-}(x) - E, \label{vm}
\end{eqnarray}
where $E$ is the factorization energy.
The partner potentials $\mathcal{V}^{\pm}(x)$ are related to $\mathcal{W}(x)$ by
\begin{equation}\label{gh}
 \mathcal{V}^{\pm}(x) = \mathcal{W}^2(x) \mp \mathcal{W}'(x) + E,
\end{equation}
where  prime denotes differentiation with respect to $x$,refer the reader to \cite{kharebook} for more details.

\section{Equivalence between RMT and QHJ}
To establish the relation between the RMT and QHJ we start with the quantum momentum function given in equation (\ref{qhj1}). In equation (\ref{qhj1}) $p$ is the QMF and is given by equation (\ref{lg1}). It is evident  form the equation  (\ref{lg1}) the QMF is related to wave function. Further more the relation between the QMF and the wave function in the equation (\ref{lg1}) captures the randomness arising from the moving poles in QMF.
Then the quantum momentum function in terms of $n$ moving poles and fixed poles \cite{bhal, bhal1, sree, geo1} is given by
\begin{equation}
p(x)=\sum_{k=1}^n\frac{-i\hbar}{x-x_k}+Q(x).\label{uf}
\end{equation}
By introducing the polynomial
\begin{eqnarray}
f(x)=(x-x_1)(x-x_2)\cdots (x-x_n),\label{poly}
\end{eqnarray}
then the quantum momentum function ( $\hbar=1$) in terms of the polynomial reads as
\begin{equation}
p=\sum_{k=1}^n i\frac{f'(x)}{f(x)}+Q(x)\label{uf4}
\end{equation}
and substituting in (\ref{qhj1}) then one gets
\begin{eqnarray}
f''(x) + 2iQ(x)f'(x)+[Q^2(x)-iQ'(x)-E+V(x)]f(x)=0.\label{dif11}
\end{eqnarray}
The search for the polynomial solutions to the equation (\ref{dif11}) leads to quantization. This is equivalent to demanding $ [Q^2(x)-iQ'(x)-E + V(x)]$ to be constant. This will only be possible  if
$Q(x)=i\mathcal{W}(x)$.  In QHJ it turns out that the $Q(x)$ is the super potential.  It should be pointed out here that the energy in the equation (\ref{dif11}) id different from the energy in equation (\ref{gh}). Then the equation (\ref{dif11}) reduces to
\begin{eqnarray}
-f''(x) +2\mathcal{W}(x)f'(x)=-E_nf(x).\label{dif111}
\end{eqnarray}
It is evident from the above equation that the left hand side ( \textit{lhs}) balances the right hand side ( \textit{rhs}).
By dividing equation (\ref{dif111}) with $2f'(x_k)$ and defining the \textit{lhs} of equation (\ref{dif111}) as T(x) one gets
\begin{eqnarray}
T(x)=-\frac{f''(x_k)}{2_n'(x_k)} +\mathcal{W}(x).\label{di1}
\end{eqnarray}

Now, we will show that QHJ is analogous to  the Steiljes electrostatic model  and then show that QHJ is  equivalent to RMT.
By following the argument in the Metha's book \cite{met}
\begin{eqnarray}
\sum_{k=1}\frac{1}{x-x_k}=\frac{f'(x)}{f(x)}-\frac{1}{x-x_j}=\frac{(x-x_j)f'(x)-f(x)}{(x-x_j)f(x)},\label{dig}
\end{eqnarray}
by taking the following limit $x\rightarrow x_j$ and using l'Hospital rule one gets
\begin{eqnarray}
\sum_{1\leq j\leq n,j\neq k} \frac{1}{x_j-x_k}&=&\lim_{x\rightarrow x_j}
\left[\frac{f'(x)}{f(x)}-\frac{1}{x-x_j}\right]\nonumber\\&=&\lim_{x\rightarrow x_j}\frac{(x-x_j)f'(x)-f(x)}{(x-x_j)f(x)}
\nonumber \\&=& \frac{f''(x_j)}{2f'(x_j)}.\label{lhp}
\end{eqnarray}
Substituting (\ref{lhp}) in (\ref{di1}) one gets
\begin{equation}
T(x)=\sum_{1\leq j\leq n,j\neq k} \frac{1}{x_k-x_j} + \mathcal{W}(x),\label{ufo}
\end{equation}
which is minimum of the Lagrangian considered for the the Steiljes electrostatic model \cite{st,st1}.

By re-substituting $Q(x)=i\mathcal{W}(x)$
in equation (\ref{ufo})  the quantum momentum function reduces to
\begin{equation}
p(x_j)=\sum_{1\leq j\leq n,j\neq k} \frac{-i}{x_k-x_j}+Q(x_j).\label{uf1}\;\; k=1,2\cdots n.
\end{equation}

In order to compare the RMT and QHJ, we define the probability distribution function in RMT as the wave function
\begin{equation}
\psi(\lambda_k)={\cal P}(\lambda_1,\ldots,\lambda_N)=C_n exp^{-\beta H}\label{rwf}
\end{equation}
where
\begin{equation}
H=V(\lambda_k)-\frac{1}{2}\sum_{1\leq j\leq n,j\neq k}ln(\vert \lambda_k-\lambda_j\vert)
\end{equation}
Substituting (\ref{rwf}) in (\ref{lg1}) the quantum momentum function is given by
\begin{eqnarray}\label{uf11}
p(\lambda_k)=\sum_{1\leq j\leq n,j\neq k}\frac{\beta}{\lambda_k-\lambda_j}-S(\lambda_k)\;\; k=1,2\cdots n
\end{eqnarray}
where $S(x_k)=\beta\frac{dV(\lambda_l)}{d\lambda_k}$. By comparing equation (\ref{uf11}) and (\ref{uf}) one gets $\beta=-i \hbar$ and
$\mathcal{W}(x)=\frac{1}{\beta}S(x)=Q(x)$. It is therefore clear that the potential used in the random matrix theory is the super potential $\mathcal{W}(x)$.
Substituting the equation (\ref{poly}) in  equation (\ref{uf11}) and the substuting this in equation (\ref{qhj1}) one gets the following differential equation  as
\begin{eqnarray}
f''(x) + 2iQ(x)f'(x)+[Q^2(x)-iQ'(x)-E+V(x)]f(x)=0.\label{fgy}
\end{eqnarray}

By exploting the supersymmetric quantum mechanics one gets the solutions for the above differential equation as the classical orthogonal polynomials for the given range fixed by the potential.
 For the Harmonic oscillator \cite{kharebook} the super potential is given by  $Q(x)=\frac{1}{2}\omega x$ then the differential equation (\ref{fgy}) becomes
\begin{eqnarray}
f''(\zeta) - 2\zeta f'(x)+2nf(\zeta)=0 \label{herm} ~~~~1\leq k\leq n.
\end{eqnarray}
where $\zeta=\alpha x$ and $\alpha^2= \frac{\omega}{2}$. The differential equation (\ref{herm}) is the Hermite differential equation and the solutions are the Hermite polynomials given by $\psi(x)= exp[-\frac{1}{2}x^2] H_n(x)$.  One can show by supersymmetric arguments the potential is given by $V(x)=\frac{1}{2}\omega x^2$. This model in RMT is known as the Dyson gas model.

One can extend the RMT models to the recently discovered exceptional polynomials, as quantum Hamiltonian Jacobi formalism for the deformed oscillator is studied in ref \cite{pkp} and the momentum function is given by
\begin{eqnarray}\label{hj}
p=\sum_{k=1}^{n}\frac{1}{x-x_k}-x-\frac{1}{x-x_1}+\frac{g+l}{x}.
\end{eqnarray}
and thus the solutions are exceptional Laguerre polynomials
we obtain the solution as exceptional polynomials
\begin{eqnarray}
\psi=\frac{x^{g+l}e^{-\frac{1}{2}x^2}}{L^{g+l - \frac{3}{2}}_l(−x^2)}\hat{L}^n_l(x^2,g).
\end{eqnarray}

It is clear from quantum mechanic the wave function is given by
$\psi = w^{\frac{1}{2}}(x)P_l(x)$
where $w(x)$ is the weight function for the corresponding orthogonal polynomial $P_l$. The probability of finding the particle in an interval is given by
\begin{eqnarray}
\rho=\int_a^b \psi(x)^*\psi(x)dx=\int_a^b w(x)P_l(x)P_m(x)dx.
\end{eqnarray}
In random matrix theory this turns out to be the Joint probability distribution function
Gaussian unitary ensemble in terms of eigenvalues
\begin{eqnarray}\label{pdf23}
{\cal P}(\lambda_1,\ldots,\lambda_N)\,d\Lambda
=C_n w\vert\Delta(x)\vert^2\,\,d\Lambda.
\end{eqnarray}
where $ \vert\Delta(x)\vert =\prod_{i<j}\vert\lambda_i-\lambda_j\vert$ is the Vandermonde determinant.  This the Joint probability distribution function for exceptional Laguerre polynomials in terms of eigenvalues is given by
\begin{eqnarray}\label{pdf11}
{\cal P}(\lambda_1,\ldots,\lambda_N)\,d\Lambda
=C_n \frac{x^{2(g+l)}e^{- x^2}}{L^{2(g+l)-3}_l(−x^2)}
\prod_{i<j}\vert\lambda_i-\lambda_j\vert^2
\,\,d\Lambda.
\end{eqnarray}
where $\hat{L}^n_l(x^2,g)$ are exceptional Laguerre polynomials.

\section{Conclusion}

In this paper, we given random matrix theory approach to the quantum mechanics using the quantum
Hamilton-Jacobi formalism. We shown that the bound state problems in quantum mechanics are analogous to solving Gaussian unitary ensemble of random matrix theory. We have also shown that the  potential appear in the joint probability distribution function in the random matrix theory as a super potential. Using this approach we have extend the random matrix theory to the newly discovered exceptional polynomials.

\section{Acknowledgements}
KVSSC acknowledges the Department of Science and technology, Govt of India, (fast track
scheme (D. O. No: SR/FTP/PS-139/2012)) for financial support. Author thank  A. K. Kapoor, V. Srinivasan, Kannan Ramaswamy and P K Thirivikraman for stimulating conversations.


\section{Introduction}
Higher-order flow coefficients $v_n$ are useful observables to constrain the properties of the quark-gluon plasma, such as a shear viscosity over entropy density ratio, in heavy ion collisions~\cite{chvn_phnx,CGCvsVn}.
The $v_n$ are found to largely fluctuate in each event even if the collision centrality is fixed~\cite{ebeVn_atlas},
which is due to the initial spatial fluctuation of participating nucleons.
To control such event-by-event initial fluctuations, the event shape engineering was suggested~\cite{ESE}.
The event shape engineering can be performed by selecting the magnitude of the event flow vector $Q_{2}$:
\begin{eqnarray}
Q_2 &=& \sqrt{q_{2,x}^{2}+q_{2,y}^{2}}/\sqrt{M}, \\ \label{eq_qv}
q_{2,x} &=& \sum_i w_{i} \cos(2\phi_i), \,\,\,\,\, q_{2,y} = \sum_i w_{i} \sin(2\phi_i), \nonumber
\end{eqnarray}
where $M$ is the particle multiplicity, $\phi_i$ is the azimuthal angle of particles, and $w_i$ is a weight for particle $i$.
The magnitude of $Q_2$ is proportional to the strength of event-by-event $v_2$ and also reflects the resolution of the second order event plane.
This technique could allow us to test the effect of the initial geometry on the final state distribution of emitted particles,
which help to understand the medium response through the system evolution.

The event shape engineering focuses on the fluctuations in the transverse plane.
It is naturally considered that there would be the fluctuations not only in the transverse plane but also in the longitudinal direction.
The presence of such fluctuations could cause a twisted source along the longitudinal direction~\cite{Bozek}.
The number of participants going to the forward and backward directions would be different,
and therefore participant planes might also be different.
As a result, the event plane angle in the final state could be different between the forward and backward angles.
Thus the initial spatial twist may survive in the final state as well as a twisted flow~\cite{ETW1,ETW2}.

In this proceedings, we present results on HBT measurements using charged pions and applying the event shape engineering technique for Au+Au collisions at $\sqrt{s_{_{NN}}}$=200 GeV recorded with the PHENIX experiment.
Also, we examine the possibility of a spatially twisted source in the final state using HBT interferometry in AMPT model.

\section{The event shape engineering at the PHENIX}
The flow vector was determined by the Reaction Plane Detector (RXN, $1<|\eta|<2.8$).
In our analysis, the $w_i$ in~\refeq{eq_qv} reflects the multiplicity or energy measured in a segment $i$ of the RXN and M is $\sum w_i$.
The tracking for charged particles was performed by the Drift Chamber and the Pad Chamber, and the particle identification was done
by the electromagnetic calorimeter, where they have the acceptance of $|\eta|<0.35$ and $|\phi|<\pi/2$ in the west and east central arms.

The measured $Q_2$ distributions were fitted with the Bessel-Gaussian function~\cite{ESE}, and then the events were classified with the magnitude of $Q_2$. Figure~\ref{fig1} shows charged hadron $v_2$ for the higher 20\% and the lower 30\% $Q_2$ events. Results without $Q_2$ selection are also plotted for the comparison.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.90\textwidth, clip, trim=0 0 0 0]{v2_QrxnSN}
\caption{\label{fig1}Charged hadron $v_2$ as a function of $p_{\rm T}$ for four centrally bins with and without the $Q_2$ selection in Au+Au 200 GeV collisions. The lower and higher $Q_{2}$ events were selected.}
\end{center}
\end{figure}
The effect of $Q_2$ selection on the $v_2$ is clearly observed, that is, the higher (lower) $Q_2$ enhances (decreases) the strength of $v_2$ compared to the $v_2$ without the $Q_2$ selection.

Then the $Q_2$ selection was applied to the HBT measurements using charged pion pairs.
In the HBT analysis, pion pairs were analyzed with the out-side-long parameterization~\cite{OSL1,OSL2} in the longitudinally co-moving system. The effect of the event plane resolution was also corrected for both cases with and without $Q_2$ selection.
Figure~\ref{fig2}(left) shows the extracted pion HBT radii, $R_{\rm s}^2$ and $R_{\rm o}^2$, as a function of azimuthal pair angle $\phi$ relative to the second-order event plane $\Psi_2$. Results show that the higher $Q_2$ selection increases the oscillation strength compared to the case without $Q_2$ selection.

These oscillations of HBT radii are supposed to be sensitive to the final source eccentricity at freeze-out, $\varepsilon_{\rm final}$. Blast-wave studies suggest that the quantity of $2R_{\rm s,2}^{2}/R_{\rm s,0}$ would be a good probe of $\varepsilon_{\rm final}$ in the limit of $k_{\rm T}=0$, where $k_{\rm T}$ denotes a mean transverse momentum of pairs. The oscillation amplitudes of $R_{\rm s}^2$ and $R_{\rm o}^2$ in a form of the final eccentricity are plotted as a function of the number of participants calculated by Glauber model in Fig.~\ref{fig2}(right). The higher $Q_2$ selection enhances the measured $\varepsilon_{\rm final}$ as well as $v_2$.
It could be originating from a larger initial eccentricity, although there should be a bias from selecting the larger $v_2$ events
because the radii modulations also depend on the anisotropy in the momentum space~\cite{bw,psi3}.
\begin{figure}[bht]
\begin{minipage}{0.6\textwidth}
\includegraphics[width=\textwidth,trim=0 10 0 0]{RsRo_wESE}
\end{minipage}\hspace{0.5pc}%
\begin{minipage}{0.4\textwidth}
\includegraphics[width=0.85\textwidth,clip,trim=0 10 5 0]{e2_wESE}
\end{minipage}
\caption{\label{fig2} (Left and center) Azimuthal angle dependence of $R_{\rm s}^{2}$ and $R_{\rm o}^{2}$ relative to $\Psi_2$. (Right) 2$^{\rm nd}$-order azimuthal oscillation on $R_{s}^{2}$ and $R_{o}^{2}$. In both panels, results with and without higher $Q_2$ selection are shown.}
\end{figure}

\section{Study on the event twist effect}
The data of Pb+Pb 2.76 TeV collisions simulated using a Multi Phase Transport Model (AMPT, v2.25 with string melting)
were used for the study on the twist effect of the particle-emitting source in the final state. The impact parameter was fixed to $8$ fm.
For the HBT study, the interference effect between two identical particles,
$1+\cos(\Delta {\bf r} \cdot \Delta {\bf p})$, was calculated and weighted to the relative pair momentum distributions. Then the correlation functions were reconstructed by taking a ratio of the distributions with and without the weight. Also, all charged pions were allowed to make a pair with each other including $\pi^+\pi^-$ to increase the statistics (the consistency between results for positive and negative pairs was checked).
The event plane was determined using particles in $4<|\eta|<6$, where particles were divided into two sub-groups.
A set of forward and backward event planes ($\Psi_2^{\rm F}, \Psi_2^{\rm B}$) were used for the event cut which requires finite difference between $\Psi_2^{\rm F}$ and $\Psi_2^{\rm B}$ as a event twist selection, whereas the other set of event planes was used for a reference angle of azimuthal HBT measurements. The effect of the event plane resolution was not taken into account in this study, assuming the resolution hardly affects the phase of the radii oscillations but the magnitude of the oscillations.

Figure~\ref{fig3} shows $R_{\rm s}^2$, $R_{\rm o}^2$, and $R_{\rm os}^2$ as a function of azimuthal pair angle relative to the $\Psi_2^{\rm B}$ ($\Delta\phi$) for four $\eta$ regions, where $(\Psi_2^{\rm B}-\Psi_2^{\rm F})>0.6$ was required. The oscillations of the HBT radii measured in $\eta>0$ are shifted to the direction of $\Delta\phi<0$, which is the direction of $\Psi_2^{\rm F}$ in the current event cut. This phase shift can be understood to be a possible twist effect in the final source distribution.
The radii oscillations measured in $\eta<0$ have no phase shift because the negative $\eta$ is closer to the backward angle used for a reference of the HBT measurement and therefore the twist effect is supposed to be small.
\begin{figure}[b]
\begin{center}
\includegraphics[width=0.92\textwidth,trim=0 20 0 20]{R_tw_zoom.pdf}
\caption{\label{fig3} Azimuthal angle dependence of $R_{\rm s}^{2}$, $R_{\rm o}^{2}$, and $R_{os}^2$ relative to the backward $\Psi_2$ with the event cut of $(\Psi_2^{\rm B}-\Psi_2^{\rm F})>0.6$, where the dashed lines show $R_{\rm os}^2=0$ and the dotted lines show $\Delta\phi=\pi/2$. }
\end{center}
\end{figure}
Figure~\ref{fig4} shows all the extracted radii parameters measured with respect to $\Psi_{2}^{B}$ and $\Psi_{2}^{F}$
with the event cut of $(\Psi_{2}^{B}-\Psi_{2}^{F})>0.6$. The phase shift can be seen in the transverse radii (not in the longitudinal radii including the cross terms) when comparing the results for $\Psi_{2}^{B}$ and $\Psi_{2}^{F}$ in the same $\eta$ region. It indicates the twist of the event plane angles in the backward and forward angles.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.9\textwidth,trim=0 0 0 10]{Rall_PbPb_kt02-1_tw06_bEPvsfEP_nEta_v2.pdf}
\caption{\label{fig4} Azimuthal angle dependence of the radii parameters relative to the backward and forward $\Psi_2$ with the event cut of $(\Psi_2^{\rm B}-\Psi_2^{\rm F})>0.6$, where the dashed lines show $R_{\rm os}^2=0$.}
\end{center}
\end{figure}
These oscillations were fitted with the following functions:
\begin{eqnarray}
R_{\mu}^2 (\Delta\phi) &=& R_{\mu,0}^2 + 2R_{\mu,2}^2 \cos(2 \Delta\phi + \alpha) \,\; ({\rm for} \; \mu=o, s),\\
R_{\mu}^2 (\Delta\phi) &=& R_{\mu,0}^2 + 2R_{\mu,2}^2 \sin( 2\Delta\phi + \alpha) \;\; ({\rm for} \; \mu=os),
\end{eqnarray}
to extract the magnitude of the phase shift ($\alpha$). The phase shift parameter $\alpha$ obtained from the results with respect to $\Psi_2^{\rm F}$ and $\Psi_2^{\rm B}$ is plotted as a function of $\eta$ in Fig.~\ref{fig4}. The $\alpha$ increases going from backward to forward angle in all cases. The variation of $\alpha$ in the $\eta$ dependence is comparable to the difference between results relative to $\Psi_2^{\rm F}$ and $\Psi_2^{\rm B}$ at the same $\eta$. The finite slope in the $\eta$ dependence of $\alpha$ indicates
the twisted source in the final state due to the initial longitudinal fluctuations, as well as the twisted event plane and flow as discussed in Ref.~\cite{ETW1,ETW2}.
In this study, the effect of the event plane resolution is assumed to be negligible,
but there should be an uncertainty derived from the event cut of $(\Psi_2^{\rm B}-\Psi_2^{\rm F})$ and their finite event plane resolutions.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.98\textwidth,clip,trim=0 0 0 0]{phase_PbPb_kt02-1_twp06_bEPvsfEP_v2.pdf}
\caption{\label{fig5} Phase shift parameters $\alpha$ obtained from the $R_{\rm s}^2$, $R_{\rm o}^2$, and $R_{\rm os}$ as a function of $\eta$. Results measured with respect to forward and backward event planes ($\Psi_2^{\rm F}$, $\Psi_2^{\rm B}$) are shown, with the event cut of $(\Psi_2^{\rm B}-\Psi_2^{\rm F})>0.6$.}
\end{center}
\end{figure}
\section{Summary}
We presented the results of HBT measurements using the event shape engineering for Au+Au collisions at $\sqrt{s_{_{\rm NN}}}=200$ GeV at the PHENIX experiment. We found that the higher $Q_2$ selection enhances the measured final source eccentricity as well as $v_2$. Although the model comparison is needed to disentangle both spatial and dynamical effects on the HBT radii, this study would clarify the relation between the initial and final source eccentricities and constrain better the system dynamics.

We also studied the event twist effect using AMPT model. When selecting events with finite difference between forward and backward event plane angles, the oscillations of HBT radii are shifted in the phase and the phase shift increases with $\eta$. The results indicate a possible twisted source in the final state preserving the initial twist due to the longitudinal fluctuations. This effect might be measured in experiments at RHIC and the LHC.

Both techniques could be useful to probe and control initial fluctuations in transverse plane and longitudinal directions,
as well as to study the response of the system to the space-time evolution.