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Density matrix renormalization group method
DMRG
Swapan K Pati
Theoretical Sciences Unit
JNCASR, Bangalore
Outline
Introduction (DMRG and Quantum many body)
Numerical renormalization group (K G Wilson)
DMRG method ( S. R. White)
Extension and a few applications
What is it?
Density matrix renormalization group (DMRG) is a numerical
technique for finding accurate approximations of the ground
state and the low-energy excited states of strongly interacting
quantum systems. Its accuracy is remarkable for one-dimensional
systems with very little amount of computational effort. It is however
limited by the dimensionality or range of interactions.
The method is kind of “iterative method” and is based on the
truncation of the Hilbert space used to represent the Hamiltonian
in a controlled way, keeping the most probable eigenstates!
The physical understanding of quantum many-body systems is hindered
by the fact that the number of parameters describing the physical states
grows exponentially with the number of particles, or size of the system.
More formally in a nutshell: DMRG method
For large systems
Accuracy comparable to exact results
Variational and non-Perturbative
No problems with frustration or fermions
It can calculate:
All ground state properties (energies, correlation functions, gaps, moments)
Finite temperature properties
Classical systems at finite temperature
Dynamical quantities (frequency dependent)
Time evolution
Limitations:
Convergence depends on details of the system (dimensionality, boundary
conditions, range of interactions) and efficient programming is very
complicated.
Quantum Many-Body Problem:
System of N quantum mechanical subsystems
Examples:
Hubbard Model: an effective model for electrons in narrow band (eg, -d or –f
electron metal ions). It is applicable for atoms, clusters, molecules, solids,…..
One-tight binding band, local Coulomb interactions
4N degrees of freedom – states: |0>, | , | and |
H  t  (ai a j  a j ai )  U  ni ni
ij 
i
Interesting ground state properties: (AFM at half-band filling, n=1),
1D: Luttinger liquid;
2D: d-wave superconductivity (n <1?)
Dynamical properties like conductivity and temperature dependence are also
quite interesting.
Heisenberg model:
H   J  Si  S j
 ij 
Strong coupling limit of the Hubbard model at n=1
Antiferromagnetic exchange J=4t2/U
Localized QM spin degrees of freedom: (2S+1)N for N spin-S objects.
A model to describe quantum magnetism in most of the oxide materials or
any system with localized spin orbitals
Bethe-ansatz (closed form exact) solution exist only in 1D
A good model for describing the parent phase of high-Tc cuprates
Ground state, dynamics and low-temperature properties quite interesting.
How does one study many-body interactions?
Analytic:
Mean-field theories
Strong and weak coupling expansions (perturbative methods)
Field theoretical methods
Mostly uncontrolled
Numerical:
Exact diagonalization
Configuration Interactions and Coupled Cluster
Quantum Monte Carlo
Dynamical mean-field theory (DMFT)
DMRG
Extremely involved, each method has its own difficulties
Microscopic understanding of systems for applications in magnetic, optical,
electrical, mechanical, transport…..phenomena
What is RG Method?
The basic idea behind a renormalization group method is to apply
a transformation to the Hamiltonian which eliminates unimportant
degrees of freedom for the description of the system within a given
energy range.
For example, if we are interested in the low-energy states of a system with a energy cut-off
E, one integrates out energy modes with energy
E d E    E
dE is a small energy interval
Then we rescale the parameters of the new system so that it reproduces
the previous one.
Given a H of a system with N variables, a RG transformation Ra is a
mapping in the Hamiltonian space which maps H to H’ : H’=Ra(H).
H’ now has N’ variables where N’=N/a which is less than N.
RG transformation must be unitary; i.e., it has to preserve
Z=Tr exp(-H/kT) so that ZN’[H’] = ZN [H].
However, an exact transformation is not possible.
Wilson and others:
Work in Fourier space and use a perturbative scheme in order
to analytically solve this problem.
Extremely successful method for solving
Kondo problem and Anderson Impurity problem
K. G. Wilson and F. Kogut, J. Phys. Rep C 12, 75 (1974).
K. G. Wilson, Rev. Mod. Phys. 43, 773 (1975).
H. R. Krishnamurthy, J. W. Wilkins and K. G. Wilson, Phys. Rev. B
21, 1044 (1980).
Numerical Renormalization Group (K. G. Wilson, 1974).
Can it be applied for Correlated Lattice problem?
Idea behind all lattice renormalization group methods is to enlarge the system
iteratively but keeping only a constant number of basis states.
Integrate out the degrees of freedom numerically for obtaining
low-energy properties.
Let H be a Hamiltonian describing an interacting electronic systems
on a lattice with L sites. Each site has four states: |0>, |down>, |up> & |2>.
The dimension of the Hilbert space for L=100 with Nup=Ndown=50 is
1058, which is not intractable numerically. The idea is to obtain the
Low-energy eigen-states of this system keeping only a small number of
states, say 100.
REAL Space algorithm
Let Bl be a block describing the first l sites for
which we only keep m states to describe the H.
The same goes for Bl’ block also with m’ states.
When we put these two blocks together, the
H of the new block Bl+l’ has dimensions mm’.
Hl l '  Hl  Hl '  C s As  Bs
Solve H l+l’ and keep only lowest m energy eigenstates





Isolate a finite system (N)
Diagonalize numerically
Keep m lowest energy eigenstates
Add another finite system (N)
Solve (2N) system and iterate the process.
By iterating this procedure, one obtains recursion relations
on the set of coupling constants which define the Hamiltonian
and the properties in the thermodynamic limit.
The message:
Low-energy states are most important for low-energy behavior
of larger system
However, only for Kondo lattice or Anderson impurity models
Very bad for other quantum lattice models:
Hubbard (Bray, Chui, 1979)
Heisenberg (White, 1992, Xiang and Gehring, 1992)
Anderson localization (Lee, 1979)
Kondo impurity problem: Hierarchy in the matrix elements; Boundary
conditions seem not important.
Just have a look for a tight-binding model: 2tii – ti, i+1 – t i, i-1
 2 1 0 0 


 1 2 1 0 
H 
0  1 2  1


 0 0 1 2 


If one puts two blocks together: does not represent the full system
Another way….
Two same size boxes (of length L, 1D, 1-electron problem)
Will putting together the ground states of L-length box
give rise to the ground state of box of size 2L? NO
Treatment of boundary
becomes critical:
Gr states of a chain of 16 atoms (open) and two 8 atoms chains (filled)
When boundary becomes critical: (White and Noack, 1992)
Use combinations of boundary conditions (BCs):
Diagonalize a block, HL with different combinations of BCs.
Use orthogonalized set of states as new basis.
Fluctuations in additional blocks allow general behavior at boundaries.
HL
Fixed-Fixed
HL
Fixed-Free
HL
HL
Free-Free
Free-Fixed
HL
2L
Diagonalize superblock composed of n blocks (each block size L)
Project wavefunctions onto size 2L block, orthogonalize
Exact results as n becomes large
The total number of states is 228
This amazing accuracy is
achieved by just keeping
only ~100 states!
Relative error vs no of states kept
Use of density matrix
Steven White, 1992
Divide the many-body system: how?
Density matrix projection
Divide the whole system into a subsystem and an environment
We know the eigenfunction for the whole system: how to
describe the subsystem block best?
Reduced density matrix for the subsystem block:
Trace over states of the environment block, all many-body states.
Density Matrix
“When we solve a quantum-mechanical problem, what we really do is
divide the universe into two parts – the system in which we are
interested and the rest of the universe.”
- Richard P. Feynman
(Statistical Mechanics : A set of lectures; Westview press, 1972)
When we include the part of the universe outside the system, the
motivation of using the density matrices become clear.
So what does this mean ?
Let |  i  be a complete set of vectors in the vector space describing the
system, and let |  i  be a complete set for the rest of the universe.
The most general way to write the wavefunction for the total system is
|     Cij | i  |  j 
ij
Now let A be an operator that acts only on the system, ie A does not act
on |  i 
A | i |  j   Aii ' | i  |  j  j | i ' |
ii ' j
Now we have,
 | A |     C Ci ' j i | A | i ' 
*
ij
iji'
  i | A | i '  i 'i
ii '
 i 'i   Cij* Ci ' j
j
Density Matrix
We define the operator ρ to be such that,
i 'i  i ' |  | i 
Again,
Note that ρ is Hermitian.
 | A |     i | A | i '  i 'i
ii '
  i | A | i ' i ' |  | i 
i
i'
  i | A | i 
i
= Tr Aρ
Due to the Hermitian nature of ρ it can be diagonalized with a complete
orthonormal set of eigenvectors | i with real eigenvalues wi
   wi | i i |
i
So, we have <|A|> = Tr A and
   wi | i i |
i
If we let A be 1, we obtain
w
i
 Tr   A   |    1
i
If we let A be | i ' i ' | , we have
wi  TrA   A   | A |    ( | i ' |  j  )( j | i ' |  )
j
  | (i ' |  j |) |  |2
j
Therefore, we have
wi  0
and
w
i
1
i
Orthonormal set of eigenvectors and real eigenvalues.
Any system is described by a density matrix ρ, where ρ is of the form
 wi | ii | and
i
(a) The set | i is a complete orthonormal set of vectors.
(b) wi  0
(c)
w
i
1
i
(d) Given an operator A, the expectation of A is given by
 A  TrA
Notice that,
 A  Tr   i' | A | i'    wi i' | ii |A | i'    wi i | A | i
i'
i 'i
i
Thus, wi is the probability that the system is in state | i . If all but one
wi are zero, we say that the system is in a pure state; otherwise it is
in a mixed state .
1

Consider a pure state: |    1 | |
2


 |   | 



1 

2  1
1  2

2
1
1  2

2   1
2
1

2
1

2

Notice that ρ2 = ρ
and
Tr ρ = 1
2 Consider a mixed state: 50% | and 50% |




1
1
  ||  ||
2
2
1 1
1 0
   1 0      0 1
2 0
2 1
1 1 0
 
2  0 1 
Notice that ρ2  ρ
and
Tr ρ = 1
3 50% and 50% mixture of |   
1
2
| | 
& |  
1
2
| | 
1
1
  |   |  |   |
2
2
1
1 1
 1
Notice that ρ2  ρ





1
1
2
and
 2 2  2
2  1 1  2  1 1 
Tr ρ = 1




2 2
 2 2 
1 1 0
Diagonal elements = Populations

 
2 0 1
Off-diagonal elements = Coherence
It is important to note that in 2 and 3, both cases we describe
a system about which we know nothing, ie, A state of total ignorance.
Density matrix:
Eigenstates of density matrix form complete basis for subsystem block
Eigenvalues give the weight of a state
Keep the m eigenstates corresponding to m highest eigenvalues
Eigenstate of the whole system thus given by:
| 0   w |  |  
Schmidt decomposition

The optimal approximation
Entanglement states (mutual quantum information):
S (  )  Tr (  log  )   w log w

DM can be defined for pure, coherent superposition or statistical averaged states
Density matrix renormalization group: Formulation







Diagonalization of a small finite lattice
Division of system
Reduction of the subsystem block via density matrix
Renormalize the matrix formulation of all the operators
Add one or two sites (few possible degrees of freedom)
Construct the bigger lattice
Repeat all the steps
Environment block:
a) Exact sites only
b) Reflection of subsystem block
c) Stored block from a previous step
Technical Details:
For nn-Heisenberg Hamiltonian: H   Si .S j
ij
Given that the ground state of n-sites are known
Let |m> with m=1,2,3,……..l specify the complete set of states
for the subsystem L and |n>, with n=1,2,3,……k for the rest of the
system (n-L). Generally they are considered to be same (L=R=n-L)
L
R
The wavefunction then can be written as
   mn | m | n 
m ,n
We like to find optimal states to represent the subsystem with less number
of states than l   < l.
|      an | L | n 
n
|      an | L | n 
n
We wish to minimize
d ||    |  |
2
by varying an and L subjected to <L| L’>= d ’


a
|
L

|
R


Without loss of generality, we can write |  

In terms of matrix notation, we thus have
d   ( mn 
mn
a


1, m

 m
 2
n
L R )
And we minimize d over all L, R and a. Use Linear algebra: The solution is
produced by singular value decomposition
  UDV T
Minimization of d: the m largest-magnitude of diagonal elements of D are aand the
corresponding columns of U and V are L and R
These optimal states L are also eigenvectors of the reduced density matrix
*
 mm '   mn
 m 'n
n
2
T
of the subsystem as part of the whole system. SVD is now   UD U
so that U diagonalizes 
Thus, the description of the optimal states, L, is best represented by
keeping the eigenvectors corresponding to highest eigenvalues of the
reduced density matrix.
Each eigenvalue () of the density matrix describes
the probability of the subsystem being in L , with S 1
The error then is the deviation of S1,,m  from unity
because of truncation..
4 Site Problem
Basis states for S Ztot  0
1) 
2) 
| i
3) 
4) 
5) 
6) 
6
Ground state |  G    Ci | i
i 1
Density matrix for 2 sites from 4 sites ψG
*
 LL '   C LR
C L 'R
R
Basis states for two sites
i ) 
ii ) 
| L or | R
iii ) 
iv ) 
|  G    C LR | L | R
LR
and since
*
 LL '   C LR
C L 'R
R
4 sites ψG can be written in a general way as
1)()() | L(i ) | R(iv )
2)()() | L(ii ) | R(ii )
3)()() | L(ii ) | R(iii )
4)()() | L(iii ) | R(ii )
5)()() | L(iii ) | R(iii )
6)()() | L(iv ) | R(i )
Reduced density matrix: How does it look?
In Matrix representation, the  for 2 sites within 4 sites is
 X 11
 0

 0

 0
0
X 22
X 32
0
0
X 23
X 33
0
0 

0 
0 

X 44 
Xab are the nonzero elements
Transformation of operators:
L
R
Density matrix for subsystem(L):  mm ' 
*

 mn m 'n
n
Diagonalize the density matrix and obtain a transformation matrix
O mm’; composed of m eigenvectors of density matrix
Fmm’ with =1,……….m.
Any operator A ij;i’j’ is transformed as
X  ' 

Oij ; Aij ;i ' j 'Oi ' j '; '
i , j ,i ', j '
Dimension of X 
m times m
Iteration:
Add two spin-sites in the middle. Left block now consists of L+1 sites and
Right block R+1 sites (generally L=R).
L
l
p
p+1
r
R
The basis for left block is now |i> = |L>*degrees of freedom for the new site (s)
Similarly for right block, the basis is |j>
In this basis, the total Hamiltonian for the L+R+2 sites is:
[ H tot ]ii' ; jj '  [ H L ]ii' d jj '  d ii ' [ H R ] jj '  [ S p ]ii ' [ S p 1 ] jj '
[ Sl S p ]ii ' d jj '  d ii ' [ S p 1Sr ] jj '
i and i’ are the basis states of left block with new site p
j and j’ are the basis of right block with new site p+1
DMRG Algorithm

Make four initial blocks, each consisting of a single site. Set up relevant matrices
representing each block. Form the Hamiltonian matrix for the whole system

Diagonalize the Hamiltonian to get target state: Y m,s,t,n
Expectation values of various quantities can be calculated at this point

Form the reduced density matrix for two-block subsystem (2 sites) using
 m ,s; m ‘ s ‘Stn Y m, s, t, n Y  m ‘, s ‘, t, n

Diagonalize  to find a set of eigenvalues  and eigenvectors, u m,s; ). Discard
all but the largest m eigenvalues and the corresponding eigenvectors: O=u (m,s; )

Form matrix representations of various operators of the two-block subsystem.

Form a new block-1 by changing the basis to the u  and truncating to m states using
P’ = O P O+. Its dimension is m X m.

Replace old block-1 with this new block 1.

Replace the old block-4 with the reflection of this new block 1.

Go to step -2.
System size
2 ways of building up superblock; choice of environment block
Infinite system algorithm:
Environment block: reflection of the subsystem block
Superblock grows by 2 lattice sites per iteration
Finite system algorithm:
Starting point: infinite system method
Size of whole system same: environment block size shrinks
Zipping back and forth: iterative convergence
Environment block: use from previous iteration
Infinite system DMRG
System
size
grows
by
2
sites
every
iteration
Start: Infinite DMRG generated system
Finite system
DMRG
Left block grows
Right block shrinks
Reflect the whole system
Left block grows
Right block shrinks
First iteration ends
Few Remarks
Most accurate method for interacting systems in low-dimensions.
Higher dimension: if X states are required for obtaining accurate
results in one-dimensional systems, X d states are necessary to
obtain comparable accuracy in d-dimensional systems.
All kinds of Static, Dynamical, Thermodynamic properties can
be calculated.
It is not a RG technique in strict sense. No fixed point or RG flow.
The Hamiltonian matrix that one encounters from iteration to
iteration, remains roughly of the same order, but the matrix
elements keep changing. In this sense it can be called
a renormalization procedure
 Coupling constants keep changing while the system size
increases, as in the RG procedure in a blocking technique.
While writing the code:
Use quantum numbers:
Any basis states can be partitioned by various quantum numbers.
All operators, in general, consist of blocks connecting definite
quantum numbers.
Eg. Sz total and N total (fermionic case)
Operators can be stored in sparse matrix form. Do all arithmetic's
using sparse matrices.
Non-abelian quantum numbers, like S2, also possible (though complicated)
Disk usage: Information not used in current loop may be written
to disk (eg, infinite DMRG generated blocks for finite DMRG)
Applications:
Strongly correlated electronic systems
Nuclear Physics
Quantum information theory
Quantum Chemistry
Classical Statistical Physics
Soft condensed matter Physics
Total number of papers published with the string “density matrix renormalization”
in their title or abstract from 1993 to 2005 is more than 5,000 (obtained from
ISI database)
Distributed Multimedia Research Group
Design Methodology Research Group All are DMRG indeed !
Direct Marketing Resource Group
Data Management Resource Group
Groupe de Renormalisation de la Matrice Densite (GRMD) !!!
Original Reference and some review articles
S. R. White, “Density matrix formulation for quantum renormalization group”
Phys. Rev. Lett. 69, 2863 (1992).
S. R. White, “Density matrix algorithm for quantum renormalization group”
Phys. Rev. B48, 10345 (1993).
Steven White was awarded the 2003 Aneesur Rahman Prize for Computational
Physics by the American Physical Society for the development of DMRG method.
S. Ramasesha, S K Pati, Z. Shuai, J.L. Bredas, “The DMRG method: application
to the low-lying electronic states of conjugated polymers”, Adv. in Quant 38,
120 (2000).
S K Pati, S. Ramasesha and D. Sen, “Exact and approximate techniques for
quantum magnetism in low-dimensions”, as a chapter in “Magnetism: from
molecules to materials”, edited by M. Drillon and J. Miller, 2002.
U. Schollwock, “The density matrix renormalization group”, Rev. Mod. Phys.
77, 259 (2005).
Symmetrized DMRG Methods
Try to exploit different symmetries of Model Hamiltonians
Some possible symmetries: [H, J] = [H, C2] = [H, P] =0
Electron – Hole Symmetry (J): only for ½-filled fermionic system
End – to – End interchange or 180o rotational symmetry ( C2 )
Spin-Parity symmetry ( P ): Only for Sztot=0
How do they operate?
Electron – Hole symmetry operator interchanges the creation and
annihilation operators at a site with a phase.
Example : ai† = ( -1 )i bi
Transformation of the Fock space of a single site under J i
Ji | 0 > = | ↑↓ > ;
Ji | ↑ > = ( -1 )i | ↑ >
Ji | ↓ > = (-1 )i | ↓ > ; Ji | ↑↓ > = (-1 ) | 0 >
J   Ji
i
C2 symmetry operation :
 Rotation through an angle 1800
 Interchanges the states of the left and right halves of the system
with a phase factor.
Consider a state for system+Env: | µ, σ, σ’, µ’ >
C2 | µ, σ, σ’, µ’ > = (-1 )γ | µ’ , σ’ , σ , µ >
γ = Phase= ( nµ + nσ ) ( nµ’ + nσ’ )
Where m (m’) refers to the eigenvectors of the left (right) block DM and
s (s’) refers to the Fock space states of the new left and right sites added.
nµ , nµ’ , nσ and nσ’ are the occupancies in the states | µ > , | µ’ >, | σ > , | σ’ >
Spin-Parity operation :
 Flips the electron spin at a particular site.
Transformation under spin-parity operation :
Pi | 0 > = | 0 > ,
Pi | ↓ > = | ↑ > ,
Pi | ↑ > = | ↓ >
Pi | ↑↓ > = | ↓↑ >
The full spin-parity operator ( P ) is the Direct Product of single site
parity operators (Pi ) :
P = ∏ i Pi
This operator bifurcates the Sztot=0 spin states into Even and Odd total Spin
states.
If we use all the three symmetry operators, together with Identity, they
Form an Abelian group with 8 irreducible representations.
eA+, eA-
, oA+ , oA- , eB+ , eB-,
oB+ , oB-
Projection operator for a given irrudecible representation Г
PГ = 1/h ∑ χГ ( R ) R
Where R ‘s are the symmetry operations, χГ ( R ) is the character of R
in Г and h is the order of the group.
Since the symmetries do not exist for “only system” or “only environment”
blocks, the implementaion of the symmetries require “symmetrization” and
then “unsymmetrization” steps in every DMRG iteration.
1. The Matrix represenatation for symmetry operators of the full system with (2n +2 ) sites in
direct product space are obtained as the direct product of corresponding matrices.
< µ , σ , σ’ µ’ | R2n+2 | ν , ‫ ح‬, ‫ ’ ح‬, ν’ >
= < µ | Rn | ν > < σ | R1 | ‫ < > ح‬σ’ | R1 | ‫ < > ’ ح‬µ ’ | Rn | ν ’ >
2. Matrix representation in the basis of eigen vectors of new density matrix : Matrix ( R )
as a direct product of Rn and R1 given as
< µ, σ | Rn+1 | ν, ‫ < = > ح‬µ | Rn | ν > < σ | R1 | ‫> ح‬
3. This matrix is Renormalized by the Transformation
Rn+1 = O† Rn+1 O
O is the Matrix whose columns are the chosen eigenvectors of the density matrix
4. The Hamiltonian matrix in the direct product basis can be transformed:
H2n+2 = S† H2n+2 S
where S is the symmetry elements matrix in the
direct product basis.
5. The wave function then is back transformed to unsymmetrized basis:
|symm> = S-1 |unsymm>
This symmetrized DMRG scheme has been implemented in Hubbard Model
for the calculation of excitation gap.
Ref : Phys. Rev. B 54, 7598 (1996)
Energy gaps ( measured from the
ground state ) of the lowest state in
each subspace for chain length
varying 40 to 50 , for two different
values of U/t.
Splitting of the Hubbard bands