In this work we investigate some aspects of density matrix renormalization group (DMRG) method. We
intuitively show why DMRG works better for open boundary conditions and why the number of sweeps in a
periodic system is greater than an open one. We also describe reduction of the Hilbert space dimension using
symmetries. Finally, we show that eliminating the repetitious states may help as much as symmetries
to reduce the Hilbert space and thus increase the DMRG speed

Solaimani, M.

English

SocioEconomic Challenges Journal (Sumy State University)

JOURNAL OF NANO- AND ELECTRONIC PHYSICS ЖУРНАЛ НАНО- ТА ЕЛЕКТРОННОЇ ФІЗИКИ 
Vol. 7 No 4, 04101(4pp) (2015) Том 7 № 4, 04101(4cc) (2015) 
 
 
2077-6772/2015/7(4)04101(4) 04101-1  2015 Sumy State University 
Number of Repetitious States in One Dimensional Hubbard Model: a Density Matrix  
Renormalization Group Perspective 
 
M. Solaimani* 
 
Department of Physics, Qom University of technology, Qom, Iran 
 
(Received 20 March 2015; revised manuscript received 10 June 2015; published online 24 December 2015) 
 
In this work we investigate some aspects of density matrix renormalization group (DMRG) method. We 
intuitively show why DMRG works better for open boundary conditions and why the number of sweeps in a 
periodic system is greater than an open one. We also describe reduction of the Hilbert space dimension us-
ing symmetries. Finally, we show that eliminating the repetitious states may help as much as symmetries 
to reduce the Hilbert space and thus increase the DMRG speed. 
 
Keywords: Density matrix renormalization group, Repetitious states, Reduction of the Hilbert space  
dimension. 
 
 PACS numbers: 78.20.Bh, 71.10.Fd, 
71.10. – w, 71.10.Pm 
 
 
                                                 
* solaimani.mehdi@gmail.com 
1. INTRODUCTION 
 
Density Matrix Renormalization Group Method 
(DMRG) is one of powerful methods to investigate the 
one dimensional system (nanowires). By increase the 
dimension to two and three, DMRG accuracy decreases 
considerably but still is the most powerful existing 
method. There are different type of DMRG and among 
its different types, the Lanczos-DMRG which was in-
troduced by Hallberg, uses the Lanczos method in 
DMRG to calculate the dynamical properties of lattice 
quantum many body systems [1]. Then it was used in 
other branches of physics such as calculation of the 
dynamical correlation functions [2]. Different types of 
DMRG are generally time-consuming but a fast version 
(has been called Dynamical-DMRG (DDMRG)) present-
ed by jeckelmann et al [3-4]. It is very similar to correc-
tion vector-DMRG which was proposed by Ramasesha 
et al. to calculate some types of dynamical correlation 
functions [5]. Despite the fact that researchers are 
working on DMRG to extend it to reach more precision 
in their calculations [6], some authors complain that 
DMRG does not satisfy them. When Kampf et al. ap-
plied the DMRG to one-dimensional ionic Hubbard 
model, They showd that the accuracy of the currently 
available DMRG data is not enough to provide a strin-
gent argument in their studies and with the available 
chain length and DMRG accuracy limitations it was 
not possible to precisely identify and locate the second 
transition point in their system [7]. In two dimensions, 
the accuracy in k-DMRG and r-DMRG decreases with 
the system size. The accuracy becomes rapidly worse 
with increasing interaction and is not significantly 
better at half filling [8]. In the meantime, boundary 
condition had a very important role in the works uses 
from the DMRG method and people consistently try to 
modify the boundary scheme to extract more precise 
results [9]. Sometimes, the boundary conditions influ-
ence the spectrum of the reduced density matrix (RDM) 
and generally DMRG performs substantially worse for 
systems with periodic rather than open boundary con-
ditions [3-4]. In addition, periodic boundary conditions 
are normally highly preferable to the open ones, as 
surface effects are eliminated and finite size extrapola-
tion gives better results for smaller system sizes. Lege-
za et al. found that because of the increase in the en-
tanglement of their periodic states the number of states 
kept tends to be the square of that required for open 
boundary conditions [10, 15]. To keep the accuracy of 
the results, people need much more basis states in each 
block of the DMRG method. The reduction of the accu-
racy comes from the boundary condition and when the 
interactions across the boundaries are weak, the accu-
racy of the DMRG calculation will be improved [11]. 
In this work, it is shown why DMRG with doubling 
the block size is less satisfactory than DMRG with 
adding a single site at a time. It is quantitatively 
shown that when we add a single site to the chain in 
each step, some repetitious states appear that lead to 
some decrease in the speed of DMRG method. Getting 
rid of these repetitious states may help us to find more 
flexible DMRG variants. We have also investigated the 
reduction of the Hilbert space dimension by using of 
symmetries. In our previous work we have used some 
these aspects [12]. 
 
2. THEORY 
 
There are different types of DMRG computer im-
plementation. Doubling the block sizes in each itera-
tion or adding single site to the left and right block is 
two approaches among them. Doubling the block size in 
DMRG leads to rough results. An obvious test is that, 
when we are generating the desired states of a 8-sites 
chain from tensor product of the possible states of a  
4-sites chain in quarter-filling in DMRG method, for 
simplicity all spins up, 
 
 
00
0 0

 
 (II-1) 
 
the state 0 0 00    can not be obtained from the all 
 M. SOLAIMANI J. NANO- ELECTRON. PHYS. 7, 04101 (2015) 
 
 
04101-2 
possible tensor product of the (II-1) configurations un-
less we use the tensor product of 0 0   and 0 0   
and a translation. Possible states of a 4-sites chain in 
quarter-filling in DMRG method, is written as (II-1) 
equations because the following states have the same 
probability as above ones. 
 
 
00
0 0
0 0


 
 (II-2) 
 
When we use the periodic boundary condition 
0 0 00    may be identical with 0 0 0 0    that 
is generated from (II-2), the translation may is not 
necessary but these tasks in DMRG lead to lose of 
speed (because we have to retain the repetitious states 
II-2 that we discuss below) and when we use the open 
boundary condition this state can not be generated 
(using doubling the block size method and II-1) and we 
are not able to consider its probability and finally 
DMRG loses its accuracy. 
When we use adding a single site in each iteration, 
we should also note that the repetitious configurations 
may lead to some decrease in the speed of DMRG meth-
od. For a periodic chain an obvious example of repeti-
tious configurations appears when we reach 8 sites 
chain by adding sites to an initial 4 site chain in quarter 
filling. ( 0 0) ( 00)     and ( 00) ( 0 0)     lead 
to the same configurations. These two configurations 
even for an open boundary condition have the same 
probabilities since in the first case atoms are aligned 
leftward and in the next case they are aligned right-
ward. Note that rotating left ward has not any ad-
vantage to the rightward rotation. Another intuitive 
example is presented in the Table I and a quantitative 
diagram is presented in the Fig. 1.  
 
Table 1 – Schematic presentation of the configurations ob-
tained from adding a single site in a RG procedure in a boson-
ic Hubbard chain in half filling. ‘1’ is a symbol of site with an 
electron and ‘0’ represent an empty site. L is the chain length, 
N is number of repeated configurations at half filling for a 
Hubbard ring, M is the number of all possible configurations 
at quarter filling, O is the number of configurations at half 
filling and P is number of configurations after reduction of the 
Hilbert space dimension using symmetries 
 
 L  4 
N  0,  
M  2, 
O  2 
P  2 
L  6 
N  0,  
M  8,  
O  4 
P  3 
L  8 
N  2,  
M  32,  
O  12 
P  8 
L  10 
N  15,  
M  128,  
O  41 
P  16 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
1100 
1010 
 
1 1 0 0 1 0 
1 1 0 0 0 1 
1 0 1 0 1 0 
1 0 1 0 0 1 
 
1 1 0 0 1 1 0 0 
1 1 0 0 1 0 1 0 
1 1 0 0 1 0 0 1 
1 1 0 0 0 1 1 0 
1 1 0 0 0 1 0 1 
1 1 0 0 0 0 1 1 
1 0 1 0 1 1 0 0 
1 0 1 0 1 0 1 0 
1 0 1 0 1 0 0 1 
1 0 1 0 0 1 1 0 
1 0 1 0 0 1 0 1 
1 0 1 0 0 0 1 1 
 
1 1 0 0 1 1 1 0 0 0 
1 1 0 0 1 1 0 1 0 0 
1 1 0 0 1 0 1 0 1 0 
1 1 0 0 1 0 1 0 0 1 
1 1 0 0 1 0 1 1 0 0 
1 1 0 0 0 1 0 1 1 0 
1 1 0 0 0 1 0 1 0 1 
1 1 0 0 1 0 0 0 1 1 
1 1 0 0 0 1 1 1 0 0 
1 1 0 0 0 1 1 0 0 1 
1 1 0 0 1 0 0 1 1 0 
1 1 0 0 1 0 0 1 0 1 
1 1 0 0 0 1 0 0 1 1 
1 1 0 0 0 0 0 1 1 1 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
1 1 0 0 0 0 1 0 1 1 
1 1 0 0 0 0 1 1 1 0 
1 1 0 0 0 0 1 1 0 1 
1 0 1 0 1 1 1 0 0 0 
1 0 1 0 1 1 0 1 0 0 
1 0 1 0 1 0 0 1 1 0 
1 0 1 0 1 0 1 1 0 0 
1 0 1 0 1 0 1 0 1 0 
1 0 1 0 1 0 1 0 0 1 
1 0 1 0 1 1 0 0 1 0 
1 0 1 0 1 1 0 0 0 1 
1 0 1 0 1 0 1 1 0 0 
1 0 1 0 1 0 0 1 1 0 
1 0 1 0 1 0 1 0 1 0 
1 0 1 0 1 0 1 0 0 1 
1 0 1 0 1 0 0 1 0 1 
1 0 1 0 1 0 0 0 1 1 
1 0 1 0 0 1 1 1 0 0 
1 0 1 0 0 1 1 0 1 0 
1 0 1 0 0 1 1 0 0 1 
1 0 1 0 0 1 0 1 1 0 
1 0 1 0 0 1 0 1 0 1 
1 0 1 0 0 1 0 0 1 1 
1 0 1 0 0 0 0 1 1 1 
1 0 1 0 0 0 1 0 1 1 
1 0 1 0 0 0 1 1 1 0 
1 0 1 0 0 0 1 1 0 1 
 
 
 
Fig. 1 – Hilbert space dimension of a Heisenberg spin chain 
and the amount of reduction due to Symmetries and repeti-
tious states 
 
States mentioned in the table is ones remained after 
reduction by translational symmetry. In this case we 
have not the spin inversion symmetry. A rapid increase 
in the number of repetitious configurations is clear from 
the Table 1. As we have mentioned in the previous sec-
tion, one has to do greater number of sweeps for fermi-
onic systems to reach satisfactory accuracy than spin 
system. Here we can construct fermionic system using 
two bosonic one. 
Let the total number of electrons in the fermionic 
system be even. At first to generate the desired states of 
a one-dimensional fermionic system in a given filling, 
we can construct two set of configurations. The first set 
contains the probable Configurations with one half of 
the entire electrons with spin up and the other set is 
include the same configurations but with spin down. By 
selecting a configuration from each mentioned set and 
defining the following summations,  
 NUMBER OF REPETITIOUS STATES IN ONE DIMENSIONAL… J. NANO- ELECTRON. PHYS. 7, 04101 (2015) 
 
 
04101-3 
 
  0
  0
0
0
  
  
  
  
  
 (II-3) 
 
we construct the states of the fermionic system. For a 
4-site chain the two mentioned sets are, 
 
 
00
0 0

 
 (II-4) 
 
And 
 
 
00
0 0

 
 (II-5) 
 
All possible states of the 4-sites chain at half filling 
can be calculated using the Eqs. (II-4 and II-5). For 
example 00 0 0 0       is a possible state for a 
4-sites chain at half-filling. 
If we do not discard the repetitious configurations in 
DMRG method and generate the states of the fermionic 
system, it is easy to see that this number of repetitious 
configurations grows more rapidly. This extremely more 
repetitious configuration enforces one to do more 
sweeps for fermionic systems to get a satisfactory accu-
racy. Note the in this picture we break a fermionic  
4-stite chain into two chains that all electrons has the 
same spin. 
When we are to generate the most appropriate state 
of an eight sites Hubbard chain another difficulty may 
be seen. To generate the most probable states of a linear 
chain in routine DMRG method, we have to store the 
most probable state of a seven sites Hubbard chain with 
6, 7 and 8 electrons since adding ,   or   and an 
empty site to them, respectively, may lead to a probable 
state for the 8 site Hubbard chain and this number of 
state increases logarithmically by growing the number 
of atoms in the chain. Usually people do not do this 
work and retain only Sz  0 in each iteration. 
The repetitious states in DMRG generate more diffi-
culties because lead to a greater superblock dimension. 
Number of repetitious states of the superblock is pre-
sented in the Fig. 2. 
 
 
 
Fig. 2 – Number of repetitious states of a fermionic Hubbard 
Chain 
Calculating the amount of reduction of the Hilbert 
space is not so straightforward since even two symmet-
ric states of the left and right block may lead to an 
asymmetric state of the superblock. 
When we generate the desired states of a fermionic 
system by using equations (II-4) and (II-5), each repeti-
tious state from each set leads to a repetitious state of 
the fermionic system. Thus to evaluate the above infor-
mation for fermionic NRG system when you use these 
equations to generate the desired states of a fermionic 
chain from spin chain, we note that eliminating the 
repetitious states of spin chain before generating the 
desired states of the fermionic chain may lead to loss 
some non-repetitious states of the fermionic chain. For 
example 0   can not be generated from equations (II-
4) and (II-5), because the translation has eliminated
0 0 . Using rotation of 00  on 00  we are able 
to generate this state, but it leads to more repetitious 
states. According to the mentioned facts, counting the 
number of repetitious state of a fermionc chain is a little 
more difficult than the spin system. Here we evaluate a 
minimum number of repetitious states may appear in 
the calculations. If we do not rotate the states of the 
group with spin up on the group with spin down, we loss 
some non-repetitious states, if we ignore them (an ap-
proximation) the number of repetitious states of the 
fermionic chain is equal to the number of repetitious 
states of the superblock in DMRG method. Thus we can 
use the Fig.1 for the fermionic system. For a linear chain 
with L sites like Hubbard that each site can have four 
different states (empty, one spin up electron and a dou-
bly occupies site), the number of repetitious states grows 
more rapid than a Heisenberg like system that each site 
of it can have two different states(spin up and down). 
Entanglement of the states is also an important fac-
tor that should be considered. When the entanglement 
increases and therefore the wave function related to 
each atom overlap with that of the neighbor ones, 
roughly speaking, they are able to see each other. In 
this situation, to get the accurate eigen-states of the 
system, they should be anti-symmetrized [13]. This 
means we have to include the anti-symmetrization in 
DMRG. Ref [10, 15] is an obvious example of this situa-
tion where by increase in entanglement of their periodic 
states, Legeza et al. had to do more sweeps to reach the 
desired accuracy. Paying attention to the necessity of 
the anti-symmetrization of the obtained eigen-states 
and ignoring it when its usage is not so important lead 
to some increase in the speed of the DMRG method.  
In the last step we try to reduce the Hilbert space 
dimension by means of symmetries and appropriate 
quantum numbers. Symmetries we propose to reduce 
the dimension of the Hilbert space for a Hubbard ring is 
the spin inversion symmetry, L time’s rotation which L 
is the ring (or chain) length and mirror reflection with 
respect to a diameter of the ring(center of the chain). 
Reduction of the Hilbert space dimension using symme-
tries like has also done previously. For an example that 
has used constrains such as Sz  0, spin inversion, 40 
translations, 4 rotations on a S  1/2 Heisenberg model 
on a tilted square lattice with 40 40  sites you may 
see the Ref [14]. 
 
 M. SOLAIMANI J. NANO- ELECTRON. PHYS. 7, 04101 (2015) 
 
 
04101-4 
3. CONCLUSION 
 
In this work some aspects of DMRG is investigated. 
Influence of the boundary condition on this method is 
described. Within a sample ring, reduction of the Hil-
bert space dimensions using symmetries is specified. 
We present some repetitious states appear in the all 
RG methods that lead to reduction of the speed and 
accuracy of the method. In this paper we do not present 
a new DMRG type that resolves the difficulty of the 
repetitious states but we have shown that importance 
of the repetitious states is as much as using the sym-
metries. The reason why DMRG works better for fer-
mionic system that bosonic one is somewhat investi-
gated. The main goal of the paper is to find the difficul-
ties of the DMRG thus at first we try to find the au-
thors complain that DMRG do not satisfy them. Then 
we took a glance on the problem under investigation to 
find the difficulties. We find that when the system size 
increases the number of repetitious states also increase 
rapidly and in the periodic boundary condition this is 
worse. More information is presented in the text. 
 
ACKNOWLEDGMENT 
 
We are grateful for the financial supports of the 
Iranian Nanotechnology Initiative Council (INIC) and 
Qom University of Technology. 
 
 
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Number of Repetitious States in One Dimensional Hubbard Model: a Density Matrix Renormalization Group Perspective

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Number of Repetitious States in One Dimensional Hubbard Model: a Density Matrix Renormalization Group Perspective

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SocioEconomic Challenges Journal (Sumy State University)