# Hankel method for energy level extraction In hadron: Analysis Framework for Monte Carlo Simulation Data in Physics

library(hadron)


# Hankel Matrix for a Correlation Function

A $n\times n$ Hankel matrix $H$ corresponding to a vector $x=(a_0, a_1, a_2, \ldots, a_{2n-2})$ is given by $$H[x, n] = \begin{pmatrix} a_0 & a_1 & a_2 & \ldots & a_{n-1} \ a_1 & a_2 & a_3 & \ldots & a_{n} \ a_2 & & & & \vdots \ \vdots & & & & \vdots \ a_{n-1} & \ldots & & & a_{2n-2} \ \end{pmatrix}$$ Lets for simplicity consider now a single correlation function $C(t)$ for $t = 0, \ldots, T/2$ with $T$ the temporal extent of the lattice. Define a time shift $\delta t > 0$, an initial time $t_0\geq 0$ and chose $n < (T/2 - t_0 -\delta t)/2$. Now define two vectors $$\begin{split} x_1 &= (C(t_0), C(t_0+1), \ldots, C(T/2))\ x_2 &= (C(t_0+\delta t), C(t_0+\delta t+1), \ldots, C(T/2))\ \end{split}$$ and the following two $n\times n$ Hankel matrices $$H_1 = H[x_1, n]\,,\qquad H_2 = H[x_2, n]\,.$$ With these we can define the following generalised eigenvalue problem (GEVP) $$H_2\, v(t_0, \delta t)\ =\ H_1\, \lambda(t_0, \delta t)\, v(t_0, \delta t)$$ with eigenvectors $v(t_0, \delta t)$ and eigenvalues $\lambda(t_0, \delta t)$.

If the correlator $C(t)$ is given by a sum of exponentials, i.e. [ C(t)\ =\ \sum_{i=1}^n a_i \exp(-E_i t)\,, ] one can see that the eigenvalues $\lambda(t_0, \delta t)$ correspond to the exponentials as follows $$\lambda_i(t_0, \delta t)\ =\ \exp(-E_i \delta t)\,.$$ This method is known as the method of Prony \cite{prony:1795} in the literature, see also \cite{Lin:2007iq}. For an improved method see \cite{gardner:1959}.

This method is implemented in hadron as follows: we first load the sample correlator matrix, solve the $4\times 4$ GEVP and determine the first principal correlator:

data(correlatormatrix)
correlatormatrix <- bootstrap.cf(correlatormatrix, boot.R=99, boot.l=1, seed=132435)
correlatormatrix.gevp <- bootstrap.gevp(cf=correlatormatrix, t0=4, element.order=c(1,2,3,4))
pc1 <- gevp2cf(gevp=correlatormatrix.gevp, id=1)


Note that the data is for the pion. Next, we use the first principal correlator as input to the hankel method. For this, we call

pc1.hankel <- bootstrap.hankel(cf=pc1, t0=2, n=2)


Thus, $n=2$ and $t_0=2$ in this case. Next, we extract the lowest eigenvalue by converting into a \texttt{cf} object

hpc1 <- hankel2cf(hankel=pc1.hankel, id=1)


which looks as follows

plot(hpc1, log="y", ylab="lambda(delta t)", xlab="delta t + t0")


which could, for instance be analysed using the \texttt{matrixfit} hadron function. However, we can also cast the eigenvalues directly into effective masses, since the eigenvalues are generalisations of those.

heffectivemass1 <- hadron:::hankel2effectivemass(hankel=pc1.hankel, id=1)


For comparison, we also compute the effective masses of the original principal correlator

pc1.effectivemass <- bootstrap.effectivemass(cf=pc1)


and compare in a plot

plot(pc1.effectivemass, pch=21, col="red", ylim=c(0,1.1), xlim=c(0,18),
xlab="t", ylab="M(t)")
plot(heffectivemass1, rep=TRUE, pch=22, col="blue")
legend("topright", legend=c("pc1", "hankel1"), bty="n", pch=c(21,22), col=c("red", "blue"))


The result depends strongly on the choice of $t_0$, of course.

plot(pc1.effectivemass, pch=21, col="red", ylim=c(0,1.1), xlim=c(0,18),
xlab="t", ylab="M(t)")
plot(heffectivemass1, pch=22, col="blue", rep=TRUE)
pc1.hankel <- bootstrap.hankel(cf=pc1, t0=1, n=2)
plot(heffectivemass1, pch=23, col="darkgreen", rep="TRUE")
pc1.hankel <- bootstrap.hankel(cf=pc1, t0=3, n=2)
plot(heffectivemass1, pch=24, col="black", rep="TRUE")
legend("topright", legend=c("pc1", "t0=2", "t0=1", "t0=3"), bty="n", pch=c(21:24),
col=c("red", "blue", "darkgreen", "black"))


One observes increasing statistical errors, but also earlier plateaus with increasing $t_0$-values. We can aso apply this method to the original correlation functions directly without using the GEVP before

ppcor <- extractSingleCor.cf(cf=correlatormatrix, id=1)
ppcor.effectivemass <- bootstrap.effectivemass(cf=ppcor)
ppcor.hankel <- bootstrap.hankel(cf=ppcor, t0=3, n=2)
plot(ppcor.effectivemass, pch=21, col="red", ylim=c(0,1.1), xlim=c(0,18),
xlab="t", ylab="M(t)")
plot(heffectivemass1, pch=22, col="blue", ylim=c(0,1.1), rep=TRUE)
legend("topright", legend=c("ppcor", "hankel1"), bty="n", pch=c(21,22), col=c("red", "blue"))


which works not as well as the method applied to the principal correlator.

# Proof

In order to proof the GEVP relation from above, we first introduce a more general Hankel matrix [ H = \begin{pmatrix} s_1 & s_2 & \ldots & s_n\ s_2 & s_3 & \ldots & \ \vdots & & & \ s_k & \ldots & & s_m\ \end{pmatrix} ] with the signal vector $$\label{eq:signal} s_k\ =\ \sum_{i=1}^r c_i z_i^{k-1}\,;\qquad z_j\ =\ e^{\mathrm{i} \omega_j}\,,$$ (complex frequencies $\omega_j$ and coefficients $c_i$) and [ k=m-n+1 > n \geq 1\,. ] In case the sum does not start with $z_i^0$ for $k=1$ but with some $z_i^{t_0}$, we can simply re-define the coefficients $c_i\to c_i' = c_i z_i^{t_0}$ to bring $s_k$ into the form Eq. (\ref{eq:signal}). Define further [ e\ =\ \begin{pmatrix} 1\ 1\ \vdots\ 1\ \end{pmatrix} ] and [ D_c = \mathrm{diag}(c_1, \ldots, c_r)\,,\quad D_z\ =\ \mathrm{diag}(z_1, \ldots, z_r)\,. ] Then, by multiplying out, one can see that [ H = \begin{pmatrix} e^T\ e^T D_z\ \vdots\ e^T D_z^{k-1}\ \end{pmatrix}\cdot D_c\cdot (e\ D_z e\ \ldots\ D_z^{n-1} e)\,. ] With this one has shown implicitly that the rank of $H$ is $r$. Now write [ H\ =\ \begin{pmatrix} g_1 \ H_1\ \end{pmatrix}\ =\ \begin{pmatrix} H_2\ g_2\ \end{pmatrix} ] with [ g_1\ =\ \begin{pmatrix} s_1 & s_2 & \ldots & s_n\ \vdots & & & \vdots \ s_{\delta t+1} & & & s_{\delta t+n}\ \end{pmatrix} \,,\qquad g_2\ =\ \begin{pmatrix} s_{k-\delta t} & & \ldots & s_{m-\delta t}\ \vdots & & & \vdots \ s_{k} & & & s_{m}\ \end{pmatrix}\,. ] Define two Vandermonde matrices, which have full rank [ V_1\ =\ \begin{pmatrix} e^T\ e^T D_z\ \vdots\ e^T D_z^{k-1-\delta t} \end{pmatrix}\,,\qquad V_2\ =\ \begin{pmatrix} e^T\ e^T D_z\ \vdots\ e^T D_z^{n-1} \end{pmatrix}\,. ] With these we can re-write $H$ as [ H\ =\ \begin{pmatrix} V_1 \ e^T D_z^{k-\delta t}\ \vdots\ e^T D_z^{k-1}\ \end{pmatrix} \cdot D_c\cdot V_2^T ] From this follows $$H_2\ =\ V_1 D_c V_2^T\,,\quad H_1\ =\ V_1 D_z^{\delta t} D_c V_2^T\,.$$ Now, perform a $QR$ decomposition, with $Q$ unitary and $R$ upper triangular, of the Vandermonde matrices $V_i =Q_i R_i\,,\ i=1,2$, which is always possible due to the full rank property. This means [ H_2\ =\ Q_1 R_1 D_c R_2^T Q_2^T\,,\quad H_1\ =\ Q_1 R_1 D_z^{\delta t} D_c R_2^T Q_2^T ] and, since $D_z$ is diagonal, by multiplying both equations with $(R_2^T Q_2^T)^{-1}$ from the right one obtains $$Q_1 R_1 D_c\ =\ H_2 Q_2 R_2^{-T}\ =\ D_z^{-\delta t} H_1 Q_2 R_2^{-T} \,.$$ The last equation is the desired generalised eigenvalue relation with eigenvalues the diagonal elements of $D_z^{\delta t}$ and the eigenvectors the columns of the matrix $Q_2 R_2^{-T}$.

## Alternative

We assume $n$ states contributing, with $E_k\neq 0$ for $k=0, \ldots, n-1$ and all the $E_k$ distinct. Let $H(t)$ be a $n\times n$ Hankel matrix for $i,j=0, 1, 2, \ldots, n-1$ defined as $$H_{ij}(t)\ =\ \sum_{k=0}^{n-1} e^{-E_k (t + i + j)} c_k\ =\ \sum_{k=0}^{n-1} e^{-E_k t} e^{-E_k i} e^{-E_k j} b_k^2\,,$$ with $b_k$ the (complex) root of $c_k$ with positive real part. Now define $$\chi_{ki}\ =\ b_k e^{-E_k i}\,.$$ Now introduce the dual vectors $u_k$ with [ (u_k, \chi_l)\ =\ \sum_{i=0}^{n-1} (u_{k}^)i \chi{li}\ =\ \delta_{kl}\,. ] This means H(t)\, u_l\ =\ \sum_{k=0}^{n-1} e^{-E_k t}\chi_k \chi_k^ u_l\ =\ e^{-E_l t} \chi_l = e^{-E_l(t-t_0)}\ e^{-E_l t_0} \chi_l\ =\ e^{-E_l(t-t_0)} H(t_0)\, u_l Thus, $$H(t)\, u_l\ =\ e^{-E_l(t-t_0)} H(t_0)\, u_l\,.$$ Moreover, we get the orthogonality [ (u_l,\, H(t) u_k)\ =\ e^{-E_l t}\delta_{lk}\,, ] because $H(t) u_k\propto \chi_k$.