# On the characteristic polynomial of an effective Hamiltonian

Yong Zheng<EMAIL_ADDRESS>School of Physics and Electronics, Qiannan
Normal University for Nationalities, Duyun 558000, China

###### Abstract

The characteristic polynomial of the effective Hamiltonian for a general model
has been discussed. It is found that, compared with the associated energy
eigenvalues, this characteristic polynomial generally has better analytical
properties and larger convergence radius when being expanded in powers of the
interaction parameter, and hence is more suitable for a perturbation
calculation. A form of effective Hamiltonian which has the same singularities
(branch points) as such characteristic polynomial has also been constructed.

###### keywords:

characteristic polynomial , effective Hamiltonian , branch points ,
perturbation calculation

##  1 Introduction

Many quantum interaction models, such as the Hubbard model and nuclear-shell
model, involve states with different energy scales. If one is only interested
in some special states, say, the lower-energy-scale states, the full Hilbert
space of the model can be partitioned into two disjoint subspaces: the so-
called $P$\- and $Q$-spaces, which are spanned by the states we are interested
and uninterested in respectively. For a full Hamiltonian $H(\lambda)$ with an
interaction parameter $\lambda$, an effective elimination of the states we are
uninterested in, via rigorous perturbation treatment or canonical
transformation, etc., can yield a dimension-reduced equivalent Hamiltonian
which only acts on the $P$-space or the remaining states, the so-called
effective Hamiltonian $H_{\mathrm{eff}}(\lambda)$ (For recent and historical
references of this, see Refs. [1, 2, 3, 4, 5, 6, 7]).

The obtained $H_{\mathrm{eff}}(\lambda)$ generally is complicated and always
in an infinite series form of the interaction parameter $\lambda$, from which
the further solving process for interesting physical quantities such as the
energy eigenvalues is still nontrivial. On the other hand, the convergence
problem of such series-form of $H_{\mathrm{eff}}(\lambda)$ itself is difficult
to discuss [1, 2]. Especially, the effective Hamiltonian can be derived by
several ways, e.g., by block-diagonalization of $H(\lambda)$ via similarity
transformation, Rayleigh-Schrödinger or Brillouin-Wigner Perturbation.
Different derivation procedures usually lead to different series-forms of
$H_{\mathrm{eff}}(\lambda)$, which can be energy-dependent or energy-
independent, Hermitian or Non-Hermitian [4, 5]. Mathematically, even for the
same problem, there are infinitely many $H_{\mathrm{eff}}(\lambda)$ which are
similar matrices to each other [8, 9]. This makes the theory of effective
Hamiltonian lack unity in some sense.

It is interesting whether the effective Hamiltonian can be discussed in a more
unified and efficient way. Here we want to study $H_{\mathrm{eff}}(\lambda)$
from another point of view, i.e., via discussing its characteristic polynomial
$\det[E-H_{\mathrm{eff}}(\lambda)]$. One consideration of our discussion is
that though the effective Hamiltonian can be in different Matrix forms similar
to each other, the characteristic polynomial is unique. This may enable us to
study some essential properties of the effective Hamiltonian itself, rather
than those only associated with its some particular form. Additionally, we
will find that the characteristic polynomial itself has some very important
characteristics, such as that it always has less singularities in the complex
plane of $\lambda$ than the $P$-space energy eigenvalues, which would allow us
to perform a more effective perturbation treatment even when the detailed form
of $H_{\mathrm{eff}}(\lambda)$ is unknown.

##  2 Formulation

We consider a system of dimension $\tilde{N}=N+M$, of which the model space
can be decoupled into two subspaces: a $P$-space spanned by $N$ states of
interesting and a $Q$-space spanned by the other $M$ states. The Hamiltonian
can be written as

$H(\lambda)=H_{0}+\lambda H_{I},$ (1)

where $H_{0}$ and $H_{I}$ both are Hermitian:

$\displaystyle
H_{0}=\sum_{n=1}^{N}\epsilon_{n}^{{}_{P}}|\psi^{{}_{P}}_{n}\rangle\langle\psi^{{}_{P}}_{n}|+\sum_{n=1}^{M}\epsilon_{n}^{{}_{Q}}|\psi^{{}_{Q}}_{n}\rangle\langle\psi^{{}_{Q}}_{n}|,$
$\displaystyle
H_{I}=\sum_{n,m}h^{\\!{}_{P}}_{mn}|\psi^{{}_{P}}_{n}\rangle\langle\psi^{{}_{P}}_{m}|+\sum_{n,m}h^{\\!{}_{Q}}_{mn}|\psi^{{}_{Q}}_{n}\rangle\langle\psi^{{}_{Q}}_{m}|+\sum_{n,m}\left[h^{\\!{}_{P\\!Q}}_{mn}|\psi^{{}_{P}}_{n}\rangle\langle\psi^{{}_{Q}}_{m}|+\text{h.c}\right],$

Without losing generality, we assume that the unperturbed energies of $P$\-
and $Q$-space states all are non-degenerate, i.e.,
$\epsilon_{1}^{{}_{P}}\neq\epsilon_{2}^{{}_{P}}$,
$\epsilon_{1}^{{}_{P}}\neq\epsilon_{1}^{{}_{Q}}$, etc. For small $\lambda$,
the eigenvalues of $P$-space states,
$E_{1}^{{}_{P}}(\lambda),E_{2}^{{}_{P}}(\lambda),\cdots,E_{N}^{{}_{P}}(\lambda)$
can be expanded as

$\displaystyle E_{1}^{{}_{P}}=\epsilon_{1}^{{}_{P}}+\lambda
h^{\\!{}_{P}}_{11}+\lambda^{2}\left(\sum_{n\neq
1}\frac{|h^{\\!{}_{P}}_{1n}|^{2}}{\epsilon_{1}^{{}_{P}}-\epsilon_{n}^{{}_{P}}}+\sum_{n}\frac{|h^{\\!{}_{P\\!Q}}_{1n}|^{2}}{\epsilon_{1}^{{}_{P}}-\epsilon_{n}^{{}_{Q}}}\right)+\cdots$
(2.1) $\displaystyle E_{2}^{{}_{P}}=\epsilon_{2}^{{}_{P}}+\lambda
h^{\\!{}_{P}}_{22}+\lambda^{2}\left(\sum_{n\neq
2}\frac{|h^{\\!{}_{P}}_{2n}|^{2}}{\epsilon_{2}^{{}_{P}}-\epsilon_{n}^{{}_{P}}}+\sum_{n}\frac{|h^{\\!{}_{P\\!Q}}_{2n}|^{2}}{\epsilon_{2}^{{}_{P}}-\epsilon_{n}^{{}_{Q}}}\right)+\cdots$
(2.2) $\displaystyle\quad\cdots\cdots$ $\displaystyle
E_{N}^{{}_{P}}=\epsilon_{N}^{{}_{P}}+\lambda
h^{\\!{}_{P}}_{N\\!N}+\lambda^{2}\left(\sum_{n\neq
N}\frac{|h^{\\!{}_{P}}_{Nn}|^{2}}{\epsilon_{N}^{{}_{P}}-\epsilon_{n}^{{}_{P}}}+\sum_{n}\frac{|h^{\\!{}_{P\\!Q}}_{Nn}|^{2}}{\epsilon_{N}^{{}_{P}}-\epsilon_{n}^{{}_{Q}}}\right)+\cdots.$
(2.N)

Similar expressions can also be written out for the eigenvalues of $Q$-space
states
$E_{1}^{{}_{Q}}(\lambda),E_{2}^{{}_{Q}}(\lambda),\cdots,E_{M}^{{}_{Q}}(\lambda)$.
These expansion series always break down when $\lambda$ is adequately large.
This is due to the fact that the $E_{n}^{{}_{P}}(\lambda)$ and
$E_{m}^{{}_{Q}}(\lambda)$, as functions of $\lambda$, generally have
singularities in the complex plane of $\lambda$. The convergence radius of the
expansion series is determined by the module of the singularity closest to the
origin, for each $E_{n}^{{}_{P}}(\lambda)$ or $E_{m}^{{}_{Q}}(\lambda)$.
Mathematically, since $E_{n}^{{}_{P}}$ and $E_{m}^{{}_{Q}}$ are solutions of
the algebraic equation $\det[E-H(\lambda)]=0$, the only singularities are
branch points [10, 11, 12]. Here,

$\det[E-H(\lambda)]=\prod_{n=1}^{N}[E-E_{n}^{{}_{P}}(\lambda)]\prod_{m=1}^{M}[E-E_{m}^{{}_{Q}}(\lambda)]\\\
=E^{\tilde{N}}+p_{1}(\lambda)E^{\tilde{N}-1}+\cdots+p_{\tilde{N}-1}(\lambda)E+p_{\tilde{N}}(\lambda),$
(3)

with the coefficients $p_{n}(\lambda)$ being polynomials of $\lambda$.

As is well-known, these branch points are characterized by “level-crossing”,
where two (or more) eigenvalues, accompanying their eigenfunctions, coincide.
Level-crossings can occur between two $P$-space states ($PP$-crossing), two
$Q$-space states ($QQ$-crossing), or one $P$-space state and one $Q$-space
state ($PQ$-crossing). Hence, if some $\lambda$ in the complex plane is a
branch point of eigenvalue $E_{n}^{{}_{P(Q)}}(\lambda)$, it must also be a
branch point of at least one other eigenvalue. For each
$E_{n}^{{}_{P}}(\lambda)$ or $E_{m}^{{}_{Q}}(\lambda)$, the branch-point set
can be denoted by
$\\{\lambda^{{}_{P}}_{n,l_{n}},\lambda^{{}_{PQ}}_{n,l^{\prime}_{n}}\\}$ or
$\\{\lambda^{{}_{Q}}_{m,l_{m}},\lambda^{{}_{QP}}_{m,l^{\prime}_{m}}\\}$
respectively, with the $\lambda^{{}_{P(Q)}}_{i,l_{i}}$ or
$\lambda^{{}_{PQ(QP)}}_{i,l^{\prime}_{i}}$ representing the branch points due
to the level-crossing of $E_{i}^{{}_{P(Q)}}(\lambda)$ with the other
$E_{j}^{{}_{P(Q)}}(\lambda)$ or $E_{j}^{{}_{Q(P)}}(\lambda)$ respectively,
which are numbered by $l_{i}(l^{\prime}_{i})=1,2,\cdots$. For example, if
$E_{1}^{{}_{P}}(\lambda)$ only has five branch points, three due to
$PP$-crossing and two due to $PQ$-crossing, we have
$\\{\lambda^{{}_{P}}_{1,l_{1}},\lambda^{{}_{PQ}}_{1,l^{\prime}_{1}}\\}=\\{\lambda^{{}_{P}}_{1,1},\lambda^{{}_{P}}_{1,2},\lambda^{{}_{P}}_{1,3},\lambda^{{}_{PQ}}_{1,1},\lambda^{{}_{PQ}}_{1,2}\\}$.

Then, the convergence radius of the $E_{n}^{{}_{P}}$-series is determined by
$r_{n}^{{}_{P}}=\min_{l_{n},l^{\prime}_{n}}\\{|\lambda^{{}_{P}}_{n,l_{n}}|,|\lambda^{{}_{PQ}}_{n,l^{\prime}_{n}}|\\}$.
We can further introduce a common convergence radius for these
$E_{n}^{{}_{P}}$-series,

$\bar{r}^{{}_{P}}=\min_{n}\\{r_{n}^{{}_{P}}\\}=\min_{l_{1},l^{\prime}_{1},\cdots,l_{N},l^{\prime}_{N}}\\{\lambda^{{}_{P}}_{1,l_{1}},\lambda^{{}_{PQ}}_{1,l^{\prime}_{1}},\cdots,\lambda^{{}_{P}}_{N,l_{N}},\lambda^{{}_{PQ}}_{N,l^{\prime}_{N}}\\}.$
(4)

Obviously, $\bar{r}^{{}_{P}}$ specifies the maximum $|\lambda|$ for which all
the $E_{n}^{{}_{P}}$-series converge.

Mathematically, if some branch points, say these due to $PP$-crossing, can be
removed, enlarged convergence radii may be obtained for the perturbation
calculation of $E_{n}^{{}_{P}}$. Our strategy is to use the characteristic
polynomial of $H_{\mathrm{eff}}(\lambda)$,

$\det[E-H_{\mathrm{eff}}(\lambda)]=\prod_{n=1}^{N}[E-E_{n}^{{}_{P}}(\lambda)]=E^{N}-P_{1}(\lambda)E^{N-1}\\\
+\cdots+(-1)^{N-1}P_{N-1}(\lambda)E+(-1)^{N}P_{N}(\lambda),$ (5)

where the coefficients are symmetric polynomials of $P$-space eigenvalues,

$\displaystyle
P_{1}(\lambda)=E_{1}^{{}_{P}}(\lambda)+E_{2}^{{}_{P}}(\lambda)+\cdots+E_{N}^{{}_{P}}(\lambda),$
(6.1) $\displaystyle P_{2}(\lambda)=\sum_{1\leq j_{1}<j_{2}\leq
N}E_{j_{1}}^{{}_{P}}(\lambda)E_{j_{2}}^{{}_{P}}(\lambda),$ (6.2)
$\displaystyle\quad\cdots\cdots$ $\displaystyle
P\\!_{N}(\lambda)=E_{1}^{{}_{P}}(\lambda)E_{2}^{{}_{P}}(\lambda)\cdots
E_{N}^{{}_{P}}(\lambda).$ (6.N)

Unlike what in Eq. (3), since the detailed form of $H_{\mathrm{eff}}(\lambda)$
has not been specified, the coefficients $P_{n}(\lambda)$ in Eq. (5) is
unknown; however, we can calculate them using Eqs. (6.1)–(6.N), i.e., by their
relationship with $E_{n}^{{}_{P}}(\lambda)$, since $E_{n}^{{}_{P}}(\lambda)$
can be calculated via the perturbation expansion as shown above. We first
discuss the availability of such calculation for $P_{n}(\lambda)$.

Noting that both $\det[E-H_{\mathrm{eff}}(\lambda)]$ and $P_{n}(\lambda)$ are
unchanged under the exchange of any two $P$-space eigenvalues, say,
$E_{i}^{{}_{P}}(\lambda)$ and $E_{j}^{{}_{P}}(\lambda)$, one can expect that
the $PP$-crossing of $P$-space eigenvalues would not cause branch points for
them. This can be demonstrated in more detail as follows. For any finite
$\lambda$-region we want to discuss, Eq. (3) can be further written as

$\prod_{n=1}^{N}[E-E_{n}^{{}_{P}}(\lambda)]=\frac{\det[E-H(\lambda)]}{\prod_{m=1}^{M}[E-E_{m}^{{}_{Q}}(\lambda)]},$
(7)

where $E$ is viewed as a constant and takes its value large enough, say,
outside the Gerschgorin disks of the $E_{m}^{{}_{Q}}(\lambda)$, to ensure a
nonzero denominator of the right-hand side of the equation. One should note
that the left-hand side of this equation just has the form of
$\det[E-H_{\mathrm{eff}}(\lambda)]$.

Although $E_{n}^{{}_{P}}(\lambda)$ and $E_{m}^{{}_{Q}}(\lambda)$ each may have
several branch points in the complex plane of $\lambda$, the true branch
points $\lambda_{c}$ of the right or left side of Eq. (7) can only be the
those which are the common branch points of the both sides. Noting that here
$\det[E-H(\lambda)]$, as shown in Eq. (3), has no branch point in the complex
plane of $\lambda$, the only common branch points certainly are these due to
the $PQ$-crossing of $E_{n}^{{}_{P}}(\lambda)$ and $E_{m}^{{}_{Q}}(\lambda)$,
which constitute a set
$S_{PQ}\equiv\\{\lambda^{{}_{PQ}}_{1,l_{1}},\lambda^{{}_{PQ}}_{2,l_{2}},\cdots,\lambda^{{}_{PQ}}_{N,l_{N}}\\}$,
with the serial numbers $l_{n}=1,2,\cdots$ as above. Namely, unlike in the
case of $E_{n}^{{}_{P}}(\lambda)$ and $E_{m}^{{}_{Q}}(\lambda)$, the $PP$\- or
$QQ$-crossing does not cause any branch points for
$\prod_{n=1}^{N}[E-E_{n}^{{}_{P}}(\lambda)]$ (i.e.,
$\det[E-H_{\mathrm{eff}}(\lambda)]$) and
$\prod_{m=1}^{M}[E-E_{m}^{{}_{Q}}(\lambda)]$.

Hence, the characteristic polynomial $\det[E-H_{\mathrm{eff}}(\lambda)]$, as a
function of $\lambda$, is “more analytical” than $E_{n}^{{}_{P}}(\lambda)$,
and more suitable for a perturbation calculation.

To illustrate this more simply, we can first discuss the case of $N=2$:

$\det[E-H_{\mathrm{eff}}(\lambda)]=E^{2}-P_{1}(\lambda)E+P_{2}(\lambda),$ (8)

for which, the set of the only branch points becomes
$S_{PQ}=\\{\lambda^{{}_{PQ}}_{1,l_{1}},\lambda^{{}_{PQ}}_{2,l_{2}}\\}$.

The branch-point set of $P_{1}(\lambda)$ and $P_{2}(\lambda)$ must be the same
as $S_{PQ}$111We can let $E=a$ and $b$ ($a,b$ are two unequal constants)
respectively to construct two functions being analytic at the branch points of
$E_{1}^{{}_{P}}(\lambda)$ and $E_{2}^{{}_{P}}(\lambda)$ due to $PP$-crossing:
$f_{1}(\lambda)=a^{2}-[E_{1}^{{}_{P}}(\lambda)+E_{2}^{{}_{P}}(\lambda)]a+E_{1}^{{}_{P}}(\lambda)E_{2}^{{}_{P}}(\lambda)$
and
$f_{2}(\lambda)=b^{2}-[E_{1}^{{}_{P}}(\lambda)+E_{2}^{{}_{P}}(\lambda)]b+E_{1}^{{}_{P}}(\lambda)E_{2}^{{}_{P}}(\lambda)$.
Then
$P_{1}(\lambda)=E_{1}^{{}_{P}}(\lambda)+E_{2}^{{}_{P}}(\lambda)=\frac{1}{b-a}[f_{1}(\lambda)-f_{2}(\lambda)]+b+a$
and
$P_{2}(\lambda)=E_{1}^{{}_{P}}(\lambda)E_{2}^{{}_{P}}(\lambda)=\frac{1}{b-a}[(b-a+1)f_{1}(\lambda)-f_{2}(\lambda)]+ab$
certainly are also analytic at these branch points. Such procedure can
obviously be extended to the case of $N>2$.. Namely, unlike
$E^{{}_{P}}_{1}(\lambda)$ and $E^{{}_{P}}_{2}(\lambda)$, $P_{1}(\lambda)$ and
$P_{2}(\lambda)$ also do not have branch points due to the $PP$-crossing.

Therefore, unlike $E^{{}_{P}}_{1}(\lambda)$\- or
$E^{{}_{P}}_{2}(\lambda)$-series, the expansions

$\displaystyle P_{1}(\lambda)$
$\displaystyle=E_{1}^{{}_{P}}(\lambda)+E_{2}^{{}_{P}}(\lambda)$
$\displaystyle=\epsilon_{1}^{{}_{P}}+\epsilon_{2}^{{}_{P}}+\lambda(h^{\\!{}_{P}}_{11}+h^{\\!{}_{P}}_{22})+\lambda^{2}\sum_{i=1,2}\sum_{n}\frac{|h^{\\!{}_{P\\!Q}}_{in}|^{2}}{\epsilon_{i}^{{}_{P}}-\epsilon_{n}^{{}_{Q}}}+\cdots,$
$\displaystyle P_{2}(\lambda)$
$\displaystyle=E_{1}^{{}_{P}}(\lambda)E_{2}^{{}_{P}}(\lambda)=\epsilon_{1}^{{}_{P}}\epsilon_{2}^{{}_{P}}+\lambda\left(\epsilon^{{}_{P}}_{1}h^{{}_{P}}_{22}+\epsilon^{{}_{P}}_{2}h^{{}_{P}}_{11}\right)$
$\displaystyle\quad+\lambda^{2}\Big{[}h^{{}_{P}}_{11}h^{{}_{P}}_{22}-|h^{\\!{}_{P}}_{12}|^{2}+\sum_{n}\Big{(}\frac{\epsilon^{{}_{P}}_{1}|h^{\\!{}_{P\\!Q}}_{2n}|^{2}}{\epsilon_{2}^{{}_{P}}-\epsilon_{n}^{{}_{Q}}}+\frac{\epsilon^{{}_{P}}_{2}|h^{\\!{}_{P\\!Q}}_{1n}|^{2}}{\epsilon_{1}^{{}_{P}}-\epsilon_{n}^{{}_{Q}}}\Big{)}\Big{]}+\cdots,$

only become divergent for $|\lambda|\geq r^{{}_{P}}$, where
$r^{{}_{P}}=\min_{l_{1},l_{2}}\\{|\lambda^{{}_{PQ}}_{1,l_{1}}|,|\lambda^{{}_{PQ}}_{2,l_{2}}|\\}$.
The characteristic polynomial in Eq. (8) is further obtained and then the
eigenvalues can be solved from $\det[E-H_{\mathrm{eff}}(\lambda)]=0$ as

$E_{1,2}^{{}_{P}}(\lambda)=\frac{P_{1}(\lambda)\pm\sqrt{P_{1}(\lambda)^{2}-4P_{2}(\lambda)}}{2}.$
(9)

Such procedure can be directly extended to a general case of $N$. Similar to
the case of $N=2$, we can use the $\lambda$-expansion of
$E^{{}_{P}}_{n}(\lambda)$ in Eqs. (2.1)–(2.N) to calculate $P_{n}(\lambda)$
via Eqs. (6.1)– (6.N). Also, since the branch-point set of these
$P_{n}(\lambda)$ is the same as that of $\det[E-H_{\mathrm{eff}}(\lambda)]$,
the convergence radius of the obtained expansions can be generally determined
as

$r^{{}_{P}}=\min_{l_{1},l_{2},\cdots,l_{N}}\\{|\lambda^{{}_{PQ}}_{1,l_{1}}|,|\lambda^{{}_{PQ}}_{2,l_{2}}|,\cdots,|\lambda^{{}_{PQ}}_{N,l_{N}}|\\}.$
(10)

Once the expansion of $P_{n}(\lambda)$ has been obtained, the detailed form of
$\det[E-H_{\mathrm{eff}}(\lambda)]$ can be calculated with Eq. (5). Then the
$P$-space eigenvalues $E^{{}_{P}}_{n}(\lambda)$ can be solved just from

$E^{N}-P_{1}(\lambda)E^{N-1}+\cdots+(-1)^{N-1}P_{N-1}(\lambda)E+(-1)^{N}P_{N}(\lambda)=0.$
(11)

Obviously, for $N<5$, Eq. (11) can be solved analytically; while for $N\geq
5$, numerical methods can be employed.

Although such procedure is still a perturbative one, we can calculate
$E_{n}^{{}_{P}}(\lambda)$ for $|\lambda|<r^{{}_{P}}$ in principle. Comparing
Eqs. (4) and (10), we have $r^{{}_{P}}\geq\bar{r}^{{}_{P}}$. Actually, for
cases that an effective Hamiltonian can be well-defined [1], the $P$\- and
$Q$-space eigenvalues are always “well-separated”: the $PQ$-crossing of
eigenvalues only occurs when $|\lambda|$ becomes relatively large, compared
with the $PP$-crossing, i.e., $r^{{}_{P}}>\bar{r}^{{}_{P}}$ or even
$r^{{}_{P}}\gg\bar{r}^{{}_{P}}$; then, one can expect that with our procedure,
$E_{n}^{{}_{P}}(\lambda)$ can be calculated for a larger $\lambda$-region,
even far beyond the applicable scope of a direct perturbation calculation.

Additionally, $\det[E-H_{\mathrm{eff}}(\lambda)]$ also can be calculated
directly with $H_{\mathrm{eff}}(\lambda)$. Certainly, this requires the the
specific form of $H_{\mathrm{eff}}(\lambda)$. However, due to the uniqueness
of characteristic polynomial, any expanded form of
$H_{\mathrm{eff}}(\lambda)$, regardless of the convergence radius, can be used
to calculate the coefficients $P_{n}(\lambda)$ and hence
$\det[E-H_{\mathrm{eff}}(\lambda)]$ uniquely in powers of $\lambda$.

##  3 Simple example

As a simple example, we consider a case of $N=2$ and $M=1$. Due to the small
dimension, it is convenient to write $H(\lambda)$ in a matrix form, which we
specify as

$H(\lambda)=\begin{bmatrix}\epsilon_{1}^{{}_{Q}}&0&0\\\
0&\epsilon_{2}^{{}_{P}}&0\\\
0&0&\epsilon_{1}^{{}_{P}}\end{bmatrix}+\lambda\begin{bmatrix}0&h^{\\!{}_{Q\\!P}}_{12}&0\\\
h^{\\!{}_{P\\!Q}}_{21}&0&h^{\\!{}_{P}}_{21}\\\
0&h^{\\!{}_{P}}_{12}&0\end{bmatrix},$ (12)

where we can take $\epsilon_{1}^{{}_{Q}}=2$, $\epsilon_{1}^{{}_{Q}}=1$,
$\epsilon_{2}^{{}_{Q}}=1.1$ and
$h^{\\!{}_{P\\!Q}}_{12}=h^{\\!{}_{P\\!Q}}_{21}=h^{\\!{}_{P}}_{21}=h^{\\!{}_{P}}_{12}=1$
(in arbitrary energy unit).

One purpose for us to value these quantities in this way is that in a
perturbation expansion, the case of nearly-degenerate $\epsilon_{2}^{{}_{P}}$
and $\epsilon_{1}^{{}_{P}}$ is very important, since terms such as those
containing “$\frac{1}{\epsilon_{2}^{{}_{P}}-\epsilon_{1}^{{}_{P}}}$” would
become very large and cause divergence. Such kind of divergence problem due to
the quasi-degenerate unperturbed energies is well-known, especially for the
multireference perturbation theory (MRPT) in studying multiconfigurational
quantum-chemical systems, and is often called the “intruder state problem”
(ISP) [13, 14, 15, 16, 17]. As for our study of $H_{\mathrm{eff}}(\lambda)$,
unperturbed energies $\epsilon_{n}^{{}_{P}}$ are always apart from
$\epsilon_{m}^{{}_{Q}}$ far enough, and hence the ISP can occur mainly due to
the quasi-degenerate $\epsilon_{n}^{{}_{P}}$, or terms containing factors such
as “$\frac{1}{\epsilon_{2}^{{}_{P}}-\epsilon_{1}^{{}_{P}}}$”, which we can
call the ISP terms.

This simple model can be analytically solved, and actually, a similar model
has been extensively discussed in previous studies [18, 19]. To the order of
$10^{-6}$, the branch points due to the $PP$-crossing of
$E_{1}^{{}_{P}}(\lambda)$ and $E_{2}^{{}_{P}}(\lambda)$ are $\lambda=\pm
0.051392i$, and that due to the $PQ$-crossing of $E_{1}^{{}_{P}}(\lambda)$ and
$E_{2}^{{}_{P}}(\lambda)$ with $E_{1}^{{}_{Q}}(\lambda)$ are
$\lambda=-0.238116\pm 0.502871i$ and $0.238116\pm 0.502871i$ respectively.
Then, the convergence radius of the $E_{1}^{{}_{P}}(\lambda)$\- and
$E_{2}^{{}_{P}}(\lambda)$-series is $r_{1}^{{}_{P}}=r_{2}^{{}_{P}}\approx
0.051392$, while that for the valid calculation using the characteristic
polynomial is $r^{{}_{P}}\approx 0.556397$.

We have calculated $E_{1}^{{}_{P}}(\lambda)$ and $E_{2}^{{}_{P}}(\lambda)$ via
our characteristic polynomial respectively for cases in which only terms
through 2nd, 4th or 6th order are retained in the $\lambda$-expansion of
coefficients $P_{1}(\lambda)$ and $P_{2}(\lambda)$. The results are shown in
Fig. 1, and with increasing order, the obvious deviation from the exact value
of $E_{1}^{{}_{P}}(\lambda)$ and $E_{2}^{{}_{P}}(\lambda)$ does tend to occur
at $\lambda=r^{{}_{P}}$. As a comparison, we have also shown the result of
$E_{1}^{{}_{P}}(\lambda)$ and $E_{2}^{{}_{P}}(\lambda)$ by the usual
perturbation theory, i.e., via retaining terms in Eqs. (2.1) and (2.2) through
2nd, 4th and 6th respectively; with increasing order, the obvious deviation
from the exact value tends to occur at
$\lambda=r_{1}^{{}_{P}}=r_{2}^{{}_{P}}=\bar{r}^{{}_{P}}$, as expected.

Figure 1: Results of $E_{1}^{{}_{P}}(\lambda)$ and $E_{2}^{{}_{P}}(\lambda)$
calculated respectively by characteristic polynomial (CP) for different-order
(2nd, 4th and 6th) $\lambda$-expansion of coefficients, and by different-order
(2nd, 4th and 6th) perturbation theory (PT), for the example model. Also shown
is the exact value of $E_{1}^{{}_{P}}(\lambda)$ and $E_{2}^{{}_{P}}(\lambda)$.
The position of $\lambda=\bar{r}^{{}_{P}}$ and $r^{{}_{P}}$ has been marked by
vertical dash lines.

An interesting thing should be noted is that in our calculation of the
characteristic polynomial, ISP terms always cancel in the coefficients
$P_{1}(\lambda)$ and $P_{2}(\lambda)$, no matter for 2nd, 4th or 6th order
$\lambda$-expansion, leading to a final result being free from the ISP.

Furthermore, as far as our simple example ($N=2$) is concerned, the branch
points of $E_{1}^{{}_{P}}(\lambda)$ and $E_{2}^{{}_{P}}(\lambda)$ lying
nearest to the origin can be alternatively determined by
$P_{1}(\lambda)^{2}-4P_{2}(\lambda)=0$, i.e., by the condition for
$E_{1}^{{}_{P}}(\lambda)=E_{2}^{{}_{P}}(\lambda)$ from Eq. (9). When we retain
terms in $P_{1}(\lambda)$ and $P_{2}(\lambda)$ to the order of $\lambda^{2}$,
$\lambda^{4}$, $\lambda^{6}$, such branch points are obtained as $\pm
0.0513922i$, $\pm 0.0513924i$, $\pm 0.0513922i$, respectively, of which the
difference is vanish small. These values are in good agreement with the exact
one $\pm 0.051392i$ obtained above.

##  4 Discussion

One advantage of using characteristic polynomial is its uniqueness. There are
infinitely many similar matrices that can be chosen as
$H_{\mathrm{eff}}(\lambda)$; of which, the convergence radii for
$\lambda$-expansion obviously can be different, but all should not exceed that
of the characteristic-polynomial coefficients $P_{n}(\lambda)$, i.e.,
$r^{{}_{P}}$. Mathematically, the most convergent form of
$H_{\mathrm{eff}}(\lambda)$ can be obtained via the least action of the
unitary transformation to block diagonalize $H(\lambda)$, i.e., via changing
$H(\lambda)$ as little as possible to bring it into a block diagonal form [9].
Such “least action” condition generally is hard to be satisfied. Hence, one
may expect that it is difficult to construct an $H_{\mathrm{eff}}(\lambda)$
which has the same convergence radius for $\lambda$-expansion as $r^{{}_{P}}$.
Actually, it has been shown that the convergence radii of $\lambda$-series
expansion for those $H_{\mathrm{eff}}(\lambda)$ constructed by methods such as
Rayleigh-Schrödinger or Brillouin-Wigner perturbation are mainly determined by
the smallest module of the so-called “exceptional points” in the complex plane
of $\lambda$ [1, 20], where some $P$\- and $Q$-state eigenvalues, say
$E_{1}^{{}_{P}}(\lambda)$ and $E_{1}^{{}_{Q}}(\lambda)$, coincide. These
points should be distinguished from the branch points, since they become the
same only if the associated eigenfunctions also coincide [10, 20]. Obviously,
each branch point must also be an exceptional point, but the reverse is not
necessarily true. This means that for those $H_{\mathrm{eff}}(\lambda)$, the
convergence radius of $\lambda$-expansion indeed cannot exceed $r^{{}_{P}}$.

It is interesting whether there exists some unified way to construct
$H_{\mathrm{eff}}(\lambda)$ so that its convergence radius for
$\lambda$-expansion is always equal to $r^{{}_{P}}$. The answer is yes. In
fact, we can construct an effective Hamiltonian directly using the
characteristic-polynomial coefficients $P_{n}(\lambda)$ as follows

$\bar{H}_{\mathrm{eff}}(\lambda)=\begin{bmatrix}\bar{P}_{1}(\lambda)&\bar{P}_{2}(\lambda)&\cdots&\bar{P}_{N-1}(\lambda)&\bar{P}_{N}(\lambda)\\\
1&0&\cdots&0&0\\\ 0&1&\cdots&0&0\\\ \vdots&\vdots&\ddots&\vdots&\vdots\\\
0&0&\cdots&1&0\end{bmatrix},$ (13)

where $\bar{P}_{n}(\lambda)\equiv(-1)^{n+1}P_{n}(\lambda)$. One can easily
verify that the characteristic polynomial of $\bar{H}_{\mathrm{eff}}(\lambda)$
is just in the same form shown in Eq. (5) [21, 22], and hence it indeed can be
viewed an effective Hamiltonian. Obviously, due to its relationship with
$P_{n}(\lambda)$, $\bar{H}_{\mathrm{eff}}(\lambda)$ must have a convergence
radius same as $r^{{}_{P}}$ for $\lambda$-expansion.

Another point we want to discuss is whether our calculation of
$P_{n}(\lambda)$ and $\det[E-H_{\mathrm{eff}}(\lambda)]$ can apply to the case
with degenerate or nearly-degenerate $\epsilon_{n}^{{}_{P}}$. In the simple
example above, we have already mentioned that in finite-order calculation of
$P_{n}(\lambda)$, ISP terms tend to cancel. It is interesting whether this
cancellation still occurs when the order is very large or even tends to
infinity, since as shown in Eqs (2.1)–(2.N), the
$E^{{}_{P}}_{n}(\lambda)$-series used to calculate $P_{n}(\lambda)$ always
have such terms .

We note that, due to the uniqueness of characteristic polynomial, if one can
construct any effective Hamiltonian $H_{\mathrm{eff}}(\lambda)$ which can be
expanded in powers of $\lambda$ without the appearance of ISP terms,
regardless of the convergence radius, the complete cancellation of these terms
in the $\lambda$-expansion of $\det[E-H_{\mathrm{eff}}(\lambda)]$ or
$P_{n}(\lambda)$ can be proved.

Actually, the expanded form of $H_{\mathrm{eff}}(\lambda)$ constructed in
usual ways always does not have ISP terms, at least when the first few orders
of $\lambda$-expansion are retained [1, 2, 3, 4, 5, 6, 7]. A form of
$H_{\mathrm{eff}}(\lambda)$ which does not have any ISP terms for all orders
of $\lambda$-expansion is constructed in the Appendix.

Then, we find that ISP terms indeed are canceled completely in the
$\lambda$-expansion of $\det[E-H_{\mathrm{eff}}(\lambda)]$ or
$P_{n}(\lambda)$. This means that although non-degenerate unperturbed energies
have been assumed for $P$-space states, our discussion is free from the ISP,
and can directly apply to the cases with these energies being nearly-
degenerate or degenerate. Hence, our discussion of the characteristic
polynomial of $H_{\mathrm{eff}}(\lambda)$ may find its potential use in MRPT
in quantum chemistry or other effective-Hamiltonian problems.

##  5 Conclusion

In conclusion, we have discussed the properties of the characteristic
polynomial of $H_{\mathrm{eff}}(\lambda)$ for a general model Hamiltonian.
Unlike $P$-space energy eigenvalues, $\det[E-H_{\mathrm{eff}}(\lambda)]$ and
the coefficients $P_{n}(\lambda)$ do not have the branch points due to the
$PP$-crossing, and hence are more suitable for a perturbation calculation. We
can calculate $P_{n}(\lambda)$, and hence $\det[E-H_{\mathrm{eff}}(\lambda)]$
in a perturbative way, which can be further used to solve for the eigenvalues
$E^{{}_{P}}_{n}(\lambda)$. Such procedure generally possesses a larger
convergence radius for $\lambda$ expansion, comparing with the direct
perturbation calculation of $E^{{}_{P}}_{n}(\lambda)$. An effective
Hamiltonian $\bar{H}_{\mathrm{eff}}(\lambda)$, which has the same branch
points and convergence radius for $\lambda$ expansion as our characteristic
polynomial, has also been constructed.

The uniqueness of $\det[E-H_{\mathrm{eff}}(\lambda)]$ also brings convenience
to our discussion. We find that our procedure is free from the ISP which
arises when the unperturbed energies of $P$-space states become nearly-
degenerate or degenerate, as frequently encountered in cases such as the MRPT
study of multiconfigurational quantum-chemical systems. Due to this advantage,
one can expect that our treatment of $\det[E-H_{\mathrm{eff}}(\lambda)]$ may
find more potential use in the future study of effective-Hamiltonian problems.

##  Appendix A Constructing an effective Hamiltonian without the appearance of
ISP terms

To construct an effective Hamiltonian without any ISP terms for our
$H(\lambda)$, we follow the procedure in Refs. [2, 23, 24]. We can denote the
eigenstate associated with $E_{n}^{{}_{P}}(\lambda)$ by
$\left|\Psi_{n}\right\rangle$, i.e.,

$H(\lambda)\left|\Psi_{n}\right\rangle=E_{n}^{{}_{P}}(\lambda)\left|\Psi_{n}\right\rangle,\quad
n=1,\ldots,N.$

We let
$\hat{P}\equiv\sum_{n}|\psi^{{}_{P}}_{n}\rangle\langle\psi^{{}_{P}}_{n}|$ and
introduce

$|\Psi^{{}_{P}}_{n}\rangle\equiv\hat{P}\left|\Psi_{n}\right\rangle,$

where $|\Psi^{{}_{P}}_{n}\rangle$ is required to be normalized, but
$\left|\Psi_{n}\right\rangle$ is not. One can further introduce the so-called
wave operator $\hat{\Omega}$ by requiring that,

$|\Psi_{n}\rangle=\hat{\Omega}\left|\Psi^{{}_{P}}_{n}\right\rangle.$

It has been shown that $\hat{\Omega}\hat{P}=\hat{\Omega}$ and
$\hat{P}\hat{\Omega}=\hat{P}$, from which we have

$\displaystyle\langle\psi^{{}_{P}}_{m}|\hat{\Omega}|\psi^{{}_{P}}_{n}\rangle=\langle\psi^{{}_{P}}_{m}|\hat{P}\hat{\Omega}|\psi^{{}_{P}}_{n}\rangle=\langle\psi^{{}_{P}}_{m}|\hat{P}|\psi^{{}_{P}}_{n}\rangle=\delta_{n,m},$
(14)
$\displaystyle\langle\psi^{{}_{P(Q)}}_{m}|\hat{\Omega}|\psi^{{}_{Q}}_{n}\rangle=\langle\psi^{{}_{P(Q)}}_{m}|\hat{\Omega}\hat{P}|\psi^{{}_{Q}}_{n}\rangle=0.$
(15)

An effective Hamiltonian can then be constructed as

$H_{\mathrm{eff}}(\lambda)=\hat{P}H_{0}\hat{P}+\lambda\hat{P}H_{I}\hat{\Omega}\\\
=\hat{P}H_{0}\hat{P}+\lambda\hat{P}H_{I}(\hat{\Omega}^{(0)}+\hat{\Omega}^{(1)}+\hat{\Omega}^{(2)}+\cdots)$
(16)

where $\hat{\Omega}$ has be expanded in powers of $\lambda$ as
$\hat{\Omega}=\hat{\Omega}^{(0)}+\hat{\Omega}^{(1)}+\hat{\Omega}^{(2)}+\cdots$,
with $\Omega^{(k)}$ standing for the $k$th order term. Obviously,
$\hat{\Omega}^{(0)}=\hat{P}$.

One can further derive an equation for $\hat{\Omega}$,

$\left[\hat{\Omega},H_{0}\right]=\lambda
H_{I}\hat{\Omega}-\lambda\hat{\Omega}\hat{P}H_{I}\hat{\Omega},$

with which an iterative formula for $\hat{\Omega}^{(k)}$ can be derived:

$\left[\hat{\Omega}^{(k)},H_{0}\right]=\hat{A}_{k}\equiv\lambda
H_{I}\hat{\Omega}^{(k-1)}-\lambda\sum_{j=0}^{k-1}\hat{\Omega}^{(j)}\hat{P}H_{I}\hat{\Omega}^{(k-j-1)},$
(17)

where $k=1,2,\cdots$.

We can calculate $\hat{\Omega}^{(k)}$ in the bases of
$\\{|\psi^{{}_{P}}_{n}\rangle,|\psi^{{}_{Q}}_{m}\rangle\\}$. Due to Eqs. (14)
and (15), the nonzero matrix elements of $\hat{\Omega}^{(k)}$ ($n\geq 1$) can
only be those such as
$\langle\psi^{{}_{Q}}_{m}|\hat{\Omega}^{(k)}|\psi^{{}_{P}}_{n}\rangle$.

Now, we show that ISP terms, i.e., these containing
“$\frac{1}{\epsilon_{n_{1}}^{{}_{P}}-\epsilon_{n_{2}}^{{}_{P}}}$”, would not
appear in the $H_{\mathrm{eff}}(\lambda)$ constructed above. Actually, all our
procedure till now can be found in Refs. [2, 23, 24]. Noting that
$\langle\psi^{{}_{Q}}_{m}|[\hat{\Omega}^{(k)},H_{0}]|\psi^{{}_{P}}_{n}\rangle=(\epsilon_{n}^{{}_{P}}-\epsilon_{m}^{{}_{Q}})\langle\psi^{{}_{Q}}_{m}|\hat{\Omega}^{(k)}|\psi^{{}_{P}}_{n}\rangle$,
we have

$\langle\psi^{{}_{Q}}_{m}|\hat{\Omega}^{(k)}|\psi^{{}_{P}}_{n}\rangle=\frac{\langle\psi^{{}_{Q}}_{m}|\hat{A}_{k}|\psi^{{}_{P}}_{n}\rangle}{\epsilon_{n}^{{}_{P}}-\epsilon_{m}^{{}_{Q}}},$

from which, it follows that if
$\langle\psi^{{}_{Q}}_{m}|\hat{A}_{k}|\psi^{{}_{P}}_{n}\rangle$, or
equivalently, the matrix elements of $\hat{\Omega}^{(j)}$,
($j=0,1,\cdots,k-1$), do not contain any ISP terms, neither does
$\langle\psi^{{}_{Q}}_{n}|\hat{\Omega}^{(k)}|\psi^{{}_{P}}_{m}\rangle$.

Noting that ISP terms do not appear in $\hat{\Omega}^{(0)}=\hat{P}$, we
conclude that $\hat{\Omega}^{(1)}$, and hence
$\hat{\Omega}^{(2)},\hat{\Omega}^{(3)},\cdots$, all would not have any ISP
terms appear in their matrix elements. Hence, when being expanded in powers of
$\lambda$, the $H_{\mathrm{eff}}(\lambda)$ given by Eq. (16) indeed is an
effective Hamiltonian without any ISP terms.

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