\documentstyle[twocolumn,aps,epsbox]{revtex}
\makeatletter
\makeatother
\begin{document}
\title{Basis Optimization Renormalization Group 
for Quantum Hamiltonian}
\author{Takanori Sugihara}
\address{Department of Physics, 
Nagoya University, Chikusa, Nagoya 464-8602, Japan}
%\date{\today}
\maketitle
\begin{abstract}
We propose a renormalization group algorithm 
for quantum Hamiltonian to calculate energy and wavefunctions 
of a small arbitrary number of low-lying states 
in finite dimensional Hilbert space with high accuracy. 
This algorithm is free from limitations associated with 
dimensionality and interaction types of models. 
We also find analytic forms of renormalization group equations for 
the ground state that provide convergence to the exact solution. 
\end{abstract}
\pacs{PACS numbers: 11.10.Hi, 11.10.Kk}

It is a common thing to use lattice models based on the 
Monte Carlo method when we consider nonperturbative aspects 
of relativistic field theories such as QCD. 
In such models,  calculations of ground state energy 
are relatively straight forward, 
whereas its wavefunction and excited states run against 
a stump because the models are based on path integral and 
its time variable is Euclidean. 
Physical quantities associated with wavefunctions, such as 
structure functions and form factors, are inevitable for 
providing an intermediary between models and the real world. 
We cannot get by with avoiding this difficulty 
if we hope to make clear unreached areas 
of the existing standard model and find the key to new physics. 
Hamiltonian formulation is best suitable to 
this purpose because it provides pairs of 
energy and wavefunction of its eigenstates 
by way of diagonalization of a Hamiltonian matrix. 
However, direct diagonalization of Hamiltonian is despairing 
since realistic models --- not limited to relativistic field 
theories --- have generally huge or infinite degrees of freedom. 
We need an algorithm for renormalization group (RG) 
that provides an effective Hamiltonian. 

Hamiltonian-based RG has been discussed by Wilson \cite{wilson} 
and White \cite{white,nw}. 
Wilson applied his RG scheme to the Kondo problem 
and succeeded in explaining critical phenomena. 
However, it is not applicable to other lattice models 
such as Heisenberg and Hubbard models \cite{flaw}. 
In this scheme, a spin is added in each RG step to enlarge 
Hilbert space and only the low-lying states are left to form 
the next optimized basis set approximately. This truncation may 
cause decoupling between low and high energy scales \cite{white2}. 
To eliminate this defect, 
White found an RG scheme called DMRG 
(density matrix renormalization group). 
It uses a density matrix to control the calculation accuracy 
and fix the decoupling problem of the former scheme. 
However, it is the class of one-dimensional quantum 
spin models to which his scheme is fairly applicable, 
and its extensions to other dimensions or interaction types 
have not been completed \cite{ext}. 
Both the schemes are not applicable, just as they are, 
to realistic relativistic field theories that have 
spatial dimension three and 
particle-number-changing interactions. 
In this letter, we propose an RG algorithm to create 
an effective Hamiltonian using optimized basis states. 
In each RG step, a basis state is added to the optimized basis set 
to obtain the low-lying states of a Hamiltonian, 
and then a part of them is chosen to form 
the next optimized basis set. 
By iterating this RG step, we obtain convergence 
to the exact solutions of the low-lying states. 
By virtue of the variational principle, 
the RG transformation restores missing interactions 
between low and high energy scales. 
We can control the calculation accuracy 
by changing the order of the initial Hamiltonian matrix 
and the number of RG steps. 
Since our algorithm is independent of dimensionality 
and interaction types, it is not limited to 
relativistic field theories and also applicable to 
other systems with many degrees of freedom, 
such as quantum lattice, quantum chemistry, 
and nuclear physics, 
if a canonically quantized Hamiltonian is available. 

An exact eigenstate of Hamiltonian is a superposition 
of the infinite number of basis states. 
Since we cannot treat such infinite dimensional problem 
numerically, we consider Hilbert space with finite dimension $N$. 
\begin{equation}
    \{|b_i\rangle|i=1,2,...,N\}, 
\end{equation}
where $|b_i\rangle$ are orthonormal basis states 
in an arbitrary representation. 
We can choose a representation 
considering the convenience of calculating 
matrix elements of the Hamiltonian. 
We call $|b_i\rangle$ a {\it fundamental basis}. 
In addition, it is assumed that $N$ is finite 
but sufficiently large to reproduce the true values 
in the infinite dimension. 
The problem is further reduced into smaller one 
since the Hamiltonian with the large $N$ cannot 
be directly diagonalized. 
We attempt to obtain convergence of the low-lying states 
to the exact solutions in the dimension $N$ 
by diagonalizing small order Hamiltonian matrix iteratively 
and optimizing the basis set. 
Our purpose is to obtain energy $E_i$ and wavefunctions 
$|\Psi_i\rangle$ of the low-lying states ($i=1,2,...,M\ll N$) 
with high accuracy. 
\begin{equation}
  \langle \Psi_i|H|\Psi_j \rangle = E_i \delta_{ij}, 
\end{equation}
where $\langle \Psi_i|\Psi_j \rangle = \delta_{ij}$. 
Our RG process is realized by the following 
basis-optimization algorithm. 
\begin{enumerate}
\item Choose a basis set composed of $M$-piece 
fundamental basis states $\{|b_i\rangle|i=1,2,...,M\}$ 
and diagonalize the $M$-th order Hamiltonian 
in this $M$-dimensional space to obtain energy and wavefunctions. 
The obtained $M$ states are used to compose the initial set of 
{\it optimized basis} $\{|\Psi_i\rangle|i=1,2,...,M\}$. 
\item Calculate the initial expansion coefficients $T_{ij}$, 
where $|\Psi_i\rangle = \sum_{j=1}^M T_{ij}|b_j\rangle$. 
Set $I=1$. 
\item Set $l=M+1$ if $I=1$ or $l=1$ if $I\ge2$. 
\item Add a basis $|b_l\rangle$ to the 
optimized basis set $\{|\Psi_i\rangle|i=1,2,...,M\}$ 
to form an enlarged basis set 
$\{|\Psi_1\rangle,|\Psi_2\rangle,...,|\Psi_M\rangle,|b_l\rangle\}$, 
and then diagonalize the ($M$+1)-th order Hamiltonian 
to obtain updated energy $E_i'$ and 
wavefunctions $|\Psi_i'\rangle$, $i=1,2,...,M+1$. 
\item Choose the $M$ low-lying states 
$\{|\Psi_i'\rangle|i=1,2,...,M\}$ 
for the next optimized basis set and calculate new 
expansion coefficients $T_{ij}'$. 
\item If $l<N$, increase $l$ by one and 
go to step 4 regarding 
$E_i'$, $|\Psi_i'\rangle$, and $T_{ij}'$ 
as $E_i$, $|\Psi_i\rangle$, and $T_{ij}$, respectively. 
Otherwise, increase $I$ by one and go to step 3. 
\end{enumerate}
The algorithm has two loops for $l$ and $I$. 
The outer loop for $I$ between the steps 3 and 6 
is called {\it sweep}. 
We obtain convergence to the exact values by iterating sweeps. 
The details of calculation differ slightly 
between the first $I=1$ and later $I\ge 2$ sweeps. 
The optimized basis states are expanded as 
\begin{equation}
  |\Psi_i\rangle = \sum_{j=1}^{N_I} T_{ij} |b_j\rangle, 
  \label{t}
\end{equation}
where $N_I=l-1$ for $I=1$ and $N_I=N$ for $I\ge 2$. 
In the first sweep $I=1$, contributions of all the fundamental 
basis states to the low-lying states are only contained 
after the last loop process $l=N$ is completed. 
It is a sweep to create approximately optimized 
basis states. On the other hand, in the later sweep $I\ge 2$, 
all the fundamental basis states already exist as component of 
the expansion (\ref{t}). It is a sweep to bring 
the $M$ optimized basis states convergent to the exact solutions. 

The followings are the details of diagonalization process 
in the $I$-th sweep. 
$|\Psi_i'\rangle$ are ($M$+1)-dimensional vector states
and expressed as superposition of 
the $M$ optimized states and an added fundamental basis. 
\begin{eqnarray}
  |\Psi_i'\rangle
  &=& \sum_{n=1}^M c_n^{(i)}|\Psi_n\rangle 
  + c_{M+1}^{(i)}|b_l\rangle 
  \nonumber
  \\
  &=& \sum_{j=1}^N T_{ij}'|b_j\rangle, 
\end{eqnarray}
Then, the optimized expansion coefficients are 
\begin{equation}
    T_{ij}' = \sum_{n=1}^M c_n^{(i)} T_{nj} 
       + c_{M+1}^{(i)}\delta_{jl}, 
\end{equation}
which satisfies the following orthonormal relation. 
\begin{equation}
    \sum_{n=1}^N T_{in}'T_{jn}' = \delta_{ij}. 
\end{equation}
The component calculation in this RG algorithm is an 
eigenvalue problem of a small order Hamiltonian matrix. 
\begin{equation}
    \sum_{j=1}^{M+1} H_{ij} c_j
    = E \sum_{j=1}^{M+1} A_{ij} c_j, 
\end{equation}
where $H_{ij}$ and $A_{ij}$  are Hamiltonian and norm matrices, 
respectively. We have the following Hamiltonian matrix 
for arbitrary $I$ using $E_i$ and $|\Psi_i\rangle$ 
obtained in the previous RG step. 
\begin{equation}
  H_{ij} = 
  \pmatrix{
    E_i\delta_{ij}\hfill & H_{i,M+1}\hfill \cr 
    H_{M+1,j} & \langle b_l|H|b_l\rangle \cr
  },
  \label{h}
\end{equation}
where 
\begin{equation}
    H_{M+1, i} = H_{i, M+1}^* =
    \langle b_l|H|\Psi_i\rangle. 
    \label{offd}
\end{equation}
Note that the $M$ states $|\Psi_i\rangle$ diagonalize 
the Hamiltonian 
$\langle \Psi_i|H|\Psi_j \rangle = E_i \delta_{ij}$
but do not satisfy $H|\Psi_i\rangle = E_i|\Psi_i\rangle$ 
in the finite $N$-dimensional Hilbert space. 
The norm matrix for $I=1$ is $A_{ij}=\delta_{ij}$. 
When $I\ge 2$, we have 
\begin{equation}
  A_{ij} = 
  \pmatrix{
    \delta_{ij} & T_{il}^* \cr 
    T_{jl} & 1 \cr
  }. 
  \label{a}
\end{equation}
In this case, we need to diagonalize both the matrices 
$A_{ij}$ and $H_{ij}$ to obtain updated energy $E_i'$ 
and optimized states $|\Psi_i'\rangle$. 
\begin{eqnarray}
    &\langle \Psi_i'|H|\Psi_j' \rangle = E_i' \delta_{ij}, &
    \\
    &\langle \Psi_i'|\Psi_j' \rangle = \delta_{ij}. &
\end{eqnarray}
In this manner, we create an effective Hamiltonian 
and optimized basis set for the next RG step 
in terms of the previously obtained energy and wavefunctions. 
This RG step is iterated till convergence is 
obtained with the desired accuracy. 
In numerical calculations based on this RG algorithm, 
diagonalization requires little memory. 
It is storing of the expansion coefficients $T_{ij}$ 
that consumes memory. 
The necessary volume of memory is proportional to 
$N$ and $N^2$ for this RG algorithm and direct diagonalization, 
respectively. If we have one G bytes of memory, 
more than $10^8$ dimensional problem is soluble 
in our RG scheme. 

The most important point of this RG algorithm is 
the coupling between the optimized basis states and 
an added fundamental basis (\ref{offd}). 
\begin{equation}
    H_{M+1, i} = H_{i, M+1}^* 
    = \sum_{n=1}^{N_I} T_{in}\langle b_l|H|b_n\rangle. 
\end{equation}
Various couplings among the fundamental basis states are 
effectively renormalized into the low-lying states 
through $H_{M+1, i} = H_{i, M+1}^*$ 
of the Hamiltonian matrix (\ref{h}). 
The first sweep $I=1$ cannot retain interactions 
between the $M$ fundamental basis states provided 
in the initial basis set and basis states newly added 
in the late of the $l$-loop processes 
since the number of fundamental basis states 
associated with the loop calculations are less than $N$. 
Therefore, the first sweep $I=1$ gives values 
with a certain accuracy relatively close to the exact ones 
when the initial basis states have significant contribution to 
the low-lying states. 
On the other hand, we just obtain values far from 
the exact ones when basis states added in the 
late loops are dominant. 
The first sweep cannot describe physical situations 
such as spontaneous symmetry breaking, 
which various energy scales are concerned with. 
The situation is same as the Wilson's scheme. 
The sweeps $I\ge 2$ can restore such missing couplings. 
Since the sweeps $I\ge 2$ perform calculations 
containing all the $N$ fundamental basis states, 
iterated RG transformations make the optimized basis states 
converge to the exact solutions. 
In $I\ge 2$, optimizations are performed by reusing 
the fundamental basis states and all the couplings 
among the various scales are effectively renormalized 
to the optimized basis states. 
The convergence of the low-lying states 
to the exact solutions is ensured 
by the variational principle. 

\begin{figure}[h]
  \begin{center}
    \epsfile{file=fig1.eps,scale=0.7}
  \end{center}
  \caption{
  The energy difference $\Delta E_n=E_n-E_n^{\rm exact}$ 
  between the RG and exact values are plotted as a 
  function of the sweep number $I$ for $a=10$, $N=107$, and $M=8$. 
  \label{fig1}}
\end{figure}

In order to see whether this RG scheme can really 
reproduce circumstances which many particle states 
are involved with, 
we apply it to the following one-dimensional 
anharmonic quantum mechanical system. 
\begin{equation}
    H = \frac{1}{2}p^2 + \frac{1}{8a^2}(q^2-a^2)^2, 
    \label{anh}
\end{equation}
where $[q,p]=i$. In a representation where 
the particle number is diagonalized, 
the fundamental basis states are 
\begin{equation}
  |b_n\rangle = \frac{1}{\sqrt{n!}}
  (a^\dagger)^{n-1} |0\rangle, \quad n=1,2,..,N
\end{equation}
where $[a,a^\dagger]=1$ and $a|0\rangle=0$. 
The initial basis set is composed of 
$M$ pieces of fundamental basis states 
$\{|b_1\rangle,|b_2\rangle,...|b_M\rangle\}$. 
Since this model is very simple, 
the exact solutions are available by direct diagonalization 
with a numerical accuracy proper to a computer and 
numerical routines. 

Table \ref{table1} shows energy spectra of the six low-lying 
states obtained by diagonalizing the $N$-th order 
Hamiltonian matrix directly. 
Fock space is composed of both even- and odd-number sectors of 
composite particles. These two sectors decouple each other 
since the Hamiltonian (\ref{anh}) has $Z_2$ symmetry. 
$E_0$, $E_2$, and $E_4$ belong to the even sector, 
and $E_1$, $E_3$, and $E_5$ to the odd sector. 
The parameter $N$ is the minimum dimension of Fock space that 
gives convergence of all the six low-lying states in eleven digits. 
Fock vacuum $|0\rangle$ is dominant for $a=1$, 
whereas many-particle components for $a=10$. 
When $a$ is large, we see degeneracy of adjacent states 
each of which separately belong to even and odd sectors. 
The double-well shaped potential induces degeneracies 
for large $a$ because of the two bottoms of 
the classical potential being isolated, 
resulting in dominant contribution of many-particle 
components to the low-lying states. 

\begin{figure}[h]
  \begin{center}
    \epsfile{file=fig2.eps,scale=0.6}
  \end{center}
  \caption{
  Squared wavefunction $T_{1,n+1}^2$ of the even-sector 
  ground state is plotted as a function of 
  the particle number $n$ 
  for the initial eight sweeps $I=1,2,...,8$ 
  with parameters $a=10$, $N=107$, and $M=8$. 
  The exact solution is plotted with the solid curve. 
  \label{fig2}}
\end{figure}

Figure \ref{fig1} plots the energy difference 
$\Delta E_n=E_n-E_n^{\rm exact}$ between the RG and exact 
values of the six low-lying states for $a=10$, 
$N=107$, and $M=8$ as a function of the sweep number $I$. 
When $I$ passes over 90, 
all the six RG values for $E_n$ coincide with 
the exact results in eleven digits. 
Figure \ref{fig2} plots the squared wavefunction 
$T_{1,n+1}^2$ of the even-sector ground state 
as a function of the particle number $n$ 
for the initial eight sweeps $I=1,2,...,8$ 
with parameters $a=10$, $N=107$, and $M=8$. 
The exact solution is also shown 
with the solid curve for reference. 
The even-sector ground state degenerates with 
the lowest state of the odd-sector. 
Both the two wavefunctions of the ground state 
have almost the same shape because of their degeneracy. 
The probability distribution of the exact solution 
is centered around the 50-body state and 
has a relatively broad width. 
The RG optimized wavefunction quickly approaches to 
the exact one in the initial about ten sweeps. 
A remarkable feature is that this RG algorithm can 
provide convergence of the low-lying states 
to the exact solutions even if the $M$ fundamental basis states 
chosen for the initial basis set are not dominant 
in the resultant low-lying states. 

Table \ref{table2} shows the $M$ dependence of 
the sweep number $I$ required to reproduce 
the eleven digits of the exact energy 
of the six (or $M$ for $M<6$) low-lying states 
shown in Table \ref{table1} 
for three values of $a=1$, $7$, and $10$. 
For any $a$, it is sufficient to choose $M$ 
equal to or slightly larger than 
the number of the states (here six) to be calculated. 
An excessively large $M$ is futile since it does not 
decrease the required sweep number significantly. 

Interestingly, the $M=1$ case also gives convergence to 
the exact solution of the ground state in eleven digits. 
In other words, if we would like to know only 
the ground state, diagonalization of the second order matrices 
is sufficient to reproduce the exact energy and wavefunction. 
Therefore, we can write analytic expressions of 
the RG equations for the ground state by doing 
trivial diagonalization of 2-by-2 matrices. 
The two-dimensional basis set expands a state as 
\begin{equation}
  |\Psi'\rangle = c_1|\Psi\rangle + c_2|b_l\rangle, 
\end{equation}
which gives the following Hamiltonian and norm matrices 
\begin{equation}
  H_{ij} = 
  \left(
    \begin{array}{@{\,}ll@{\,}}
      E & h \\
      h & e
    \end{array}
  \right), 
  \quad
  A_{ij} = 
  \left(
    \begin{array}{@{\,}ll@{\,}}
      1 & T \\
      T & 1
    \end{array}
  \right), 
\end{equation}
where $h$, $e$, $T$ are matrix elements 
dependent on the loop index $l$ of the added basis $|b_l\rangle$. 
\begin{equation}
  h\equiv \langle b_l|H|\Psi\rangle, \quad
  e\equiv \langle b_l|H|b_l\rangle, \quad
  T\equiv T_l, 
\end{equation}
where $|\Psi\rangle=\sum_{i=1}^{N_I}T_i|b_i\rangle$. 
Using the energy and wavefunctions obtained in the previous 
step, the updated energy is expressed as 
\begin{equation}
  E' = \frac{E_+-2hT -
  \sqrt{(2h-E_+ T)^2+(1-T^2)E_-^2}}{2(1-T^2)}
\end{equation}
where $E_\pm\equiv E\pm e$. 
The optimized wave vector is 
\begin{equation}
  c_1 = c\left(E'+\frac{h-e}{1-T}\right), \quad
  c_2 = c\left(E'+\frac{h-E}{1-T}\right), 
\end{equation}
where 
\begin{equation}
 c \equiv 
   \left[
      \frac{E_-^2}{2(1-T)}
      +2(1+T)\left(E'+\frac{2h-E_+}{2(1-T)}\right)^2 
   \right]^{-1/2}.
\end{equation}
Evidently, this analytic expression for RG transformation 
reproduces the exact results of direct diagonalization 
as shown in Table \ref{table2}. 
These RG formulas for the ground state are applicable 
to other models independent of dimension and interaction types. 

This work was supported by the Grant-in-Aid for Scientific 
Research Fellowship, No. 11000480. 

\begin{thebibliography}{30}
\bibitem{wilson}
K. G. Wilson, 
Rev. Mod. Phys. {\bf 47}, 773 (1975).
\bibitem{white}
S. R. White, 
Phys. Rev. Lett. {\bf 69}, 2863 (1992), 
Phys. Rev. B {\bf 48}, 10345 (1993), 
Phys. Rep. {\bf 301}, 187 (1998). 
\bibitem{nw}
S. Noack and S. R. White, 
in {\it Density Matrix Renormalization, Lecture Notes in Physics}, 
edited by I. Peschel et al. 
(Springer-Verlag, New York, 1999). 
\bibitem{flaw}
J. W. Bray and S. T. Chui, 
Phys. Rev. B {\bf 19}, 4876 (1979); 
T. Xiang and G. A. Gehring, 
Phys. Rev. B {\bf 48}, 303 (1993). 
\bibitem{white2}
S. R. White and R. M. Noack, 
Phys. Rev. Lett. {\bf 68}, 3487 (1992). 
\bibitem{ext}
T. Xiang, 
Phys. Rev. B {\bf 53}, 445 (1996); 
M. A. Mart\'{\i}n-Delgado and G. Sierra, 
Phys. Rev. Lett. {\bf 83}, 1514 (1999). 
\end{thebibliography}

\begin{table}
\caption{
Energy spectra of the six low-lying states 
for $a=1$, $7$, and $10$ 
obtained by direct diagonalization of the Hamiltonian 
in a representation that diagonalizes the particle number. 
The parameter $N$ is the minimum order of matrices 
to reproduce the convergent values in eleven digits. 
For convenience, we call these eleven-digit values {\it exact}. 
For the upper eight digits, 
we have seen their agreement with the results from 
another diagonalization calculation, 
where the Schr\"odinger equation is written 
in the $x$ representation and discretized with respect to the $x$. 
The latter calculation does not give convergence for 
numbers in parentheses. 
\label{table1}}
\begin{tabular}{cccc}
 & $a=1$ & $a=7$ & $a=10$\\
\hline
$E_0$ & 0.2939806(208) & 0.4947744(431) & 0.4974711(541)\\
$E_1$ & 0.9313683(815) & 0.4947744(431) & 0.4974711(541)\\
$E_2$ & 1.9559364(194) & 1.4624211(089) & 1.4820766(467)\\
$E_3$ & 3.1503922(635) & 1.4624211(089) & 1.4820766(467)\\
$E_4$ & 4.4923065(136) & 2.3944961(484) & 2.4506263(590)\\
$E_5$ & 5.9553778(901) & 2.3944961(537) & 2.4506263(590)\\
$N$ & 43 & 65 & 107 \\
\end{tabular}
\end{table}

\begin{table}
\caption{
The minimum sweep number $I$ that reproduces eleven 
digits of the exact values for the six (or $M$ for $M< 6$) 
low-lying states shown in Table I. 
\label{table2}}
\begin{tabular}{rrrr}
$M$ & $a=1$ & $a=7$ & $a=10$\\
& ($N=43$) & ($N=65$) & ($N=107$)\\
\tableline
1 &  9 & 83 & 173 \\
2 & 10 & 84 & 173 \\
4 & 15 & 82 & 179 \\
6 & 25 & 82 & 177 \\
8 & 22 & 36 & 90 \\
10 & 21 & 28 & 54 \\
12 & 19 & 23 & 47 \\
14 & 15 & 21 & 39 
\end{tabular}
\end{table}

\end{document}

