Density matrix renormalization group method                                                                

Swapan K Pati

Theoretical Sciences UnitJNCASR, Bangalore 

DMRG

Introduction (DMRG and Quantum many body)

Numerical renormalization group  (K G Wilson)

DMRG method ( S. R. White)

Extension and a few applications

Outline

What is it?

Density matrix renormalization group (DMRG) is a numericaltechnique for finding accurate approximations of the ground state and the low-energy excited states of strongly interactingquantum lattice 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 thetruncation of the Hilbert space used to represent the Hamiltonianin a controlled way, keeping the most probable eigenstates!

The physical understanding of quantum many­body 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 resultsVariational and non-PerturbativeNo problems with frustration or fermions

It can calculate: 

All ground state properties (energies, correlation functions, gaps, moments) Finite temperature propertiesClassical systems at finite temperatureDynamical quantitiesTime evolution of a quantum system

Convergence depends on details of the system (dimensionality, boundary conditions, range of interactions) and efficient programming is very complicated.

Limitations:

Low­dimensional materials are very different from their 3D counterpart

Quantum Fluctuations

Lack of true long-range order

Quantum Confinement

New Physics that can not be derived from Old

Particularly important for today’s materials world

Even a 3­dimensional object with at least one of its dimension very small oreffective interactions only along one or two dimensions

                                                                                                  Low­dimensional

As an example…..

A  macroscopic magnetic material (antiferromagnetic):

Neel TemperatureLong-range Order (Neel Configuration)

Curie Paramagnetism

Critical exponents, M-H plot

A one­dimensional antiferromagnetic system:

Neel Configuration:

Not a ground state (or for that matter not even an eigenstate!) => Quantum entanglement

Spin­flip (AF superexchange ) interaction destroys any ordering

Quantum Many­Body 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

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.

|↑> |↓> |↑↓>

( )i j j i i iij i

H t a a a a U n n+ +↑ ↓

< >

= − + +∑ ∑

Localized QM spin degrees of freedom: (2S+1)N for N spin­S objects.

i jij

H J S S< >

= − ⋅∑

Heisenberg model:

Strong coupling limit of the Hubbard model at n=1Antiferromagnetic exchange J=4t2/U

A model to describe quantum magnetism in most of the oxide materials orany 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 many­body interactions?

Analytic:

Mean-field theoriesStrong and weak coupling expansions (perturbative methods)Field theoretical methods

Mostly uncontrolledNumerical:

Exact diagonalizationQuantum Monte CarloDynamical 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

Numerical Renormalization Group (K. G. Wilson, 1974).

Integrate out the degrees of freedom numerically for obtaininglow­energy properties.

Isolate a finite system (N) Diagonalize numerically Keep m lowest energy eigenstates Add another finite system (N) Solve (2N) system and iterate the process.

REAL Space algorithm

Idea behind all lattice renormalization group methods is to enlarge the systemiteratively but keeping only a constant number of basis states.

Low­energy states are most important for low­energy behaviorof larger system

Kondo problem and Anderson Impurity problem

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.

Extremely successful method for solving

The message:

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; Boundaryconditions seem not important.

Just have a look for a tight-binding model: 2tii – ti, i+1 – t i, i-1

−−−

−−−

=

2100

1210

0121

0012

H

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.

Diagonalize superblock composed of n blocks (each block size L)Project wavefunctions onto size 2L block, orthogonalizeExact results as n becomes large

HL HL HL HL

HL

Fixed-Fixed Fixed-Free Free-Free Free-Fixed

2L

Relative error vs no of states kept

The total number of states is 228

This amazing accuracy is achieved by just keeping only  ~100 states!

Use of density matrix

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 todescribe the subsystem block best?

Reduced density matrix for the subsystem block:

Trace over states of the environment block, all many­body states.

Steven White, 1992

Density matrix:

Eigenstates of density matrix form complete basis for subsystem blockEigenvalues give the weight of a stateKeep the m eigenstates corresponding to m highest eigenvaluesEigenstate of the whole system thus given by:

Schmidt decomposition

The optimal approximation

Entanglement states (mutual quantum information):

0| | |wα α αα

ψ φ χ>≈ > >∑

( ) ( log ) logS Tr w wα αα

ρ ρ ρ= − = −∑DM can be defined for pure, coherent superposition or statistical averaged states

Density matrix renormalization group: Formulation 

Diagonalization of a small finite latticeDivision of systemReduction of the subsystem block via density matrixRenormalize the matrix formulation of all the operatorsAdd one or two sites (few possible degrees of freedom)Construct the bigger latticeRepeat all the steps

Environment block:

a) Exact sites onlyb) Reflection of subsystem blockc) Stored block from a previous step

Technical Details:

For nn-Heisenberg Hamiltonian: .i jij

H S S= ∑

Given that the ground state of n-sites are known

Let |µ> with µ=1,2,3,……..l specify the complete set of statesfor the subsystem L and |ν>, with ν=1,2,3,……k for the rest of thesystem (n-L). Generally they are considered to be same (L=R=n-L)

L R

The wavefunction then can be written as ,

| |µνµ ν

ψ ψ µ ν= > >∑We like to find optimal states to represent the subsystem with less numberof states than µ α < l.

°| | |a Lααν

αν

ψ ψ ν>≈ = > >∑

° 2|| | |d ψ ψ= > − >

We wish to minimize

°| | |a Lααν

αν

ψ ψ ν>≈ = > >∑

by varying aαν and Lα subjected to <Lα| Lα’>= δ α α’

Without loss of generality, we can write °| | |a L Rα αα

α

ψ >= > >∑In terms of matrix notation, we thus have

2

1,

( )m

d a L Rα αµν α µ ν

µν α

ψ=

= −∑ ∑And we minimize d over all Lα, Rα and aα . Use Linear algebra: The solution is produced by singular value decomposition

TUDVψ =

Minimization of d: the m largest-magnitude of diagonal elements of D are aαand thecorresponding columns of U and V are Lα and Rα

*' 'µµ µν µ ν

ν

ρ ψ ψ= ∑

Each eigenvalue (ωα) of the density matrix describes the probability of the subsystem being in Lα , with Σα ωα=1

The error then is the deviation of Σα=1,,m ωα from unitybecause of truncation..

Thus, the description of the optimal states, Lα, is best represented by keeping the eigenvectors corresponding to highest eigenvalues of the reduced density matrix.

These optimal states Lα are also eigenvectors of the reduced density matrix

of the subsystem as part of the whole system. SVD is now 2 TUD Uψ = so that U diagonalizes ρ

Transformation of operators:

L R

Density matrix for subsystem(L): *' 'µµ µν µ ν

ν

ρ ψ ψ= ∑Diagonalize the density matrix and obtain a transformation matrixO µµ’;α composed of m eigenvectors of density matrix

Φαµµ’ with α=1,……….m.

Any operator A ij;i’j’ is transformed as

' ; ; ' ' ' '; ', , ', '

ij ij i j i ji j i j

X O A Oαα α α= ∑Dimension of X m times m

L l p p+1 r R

' ' ' ' ' ' ' 1 ';

' ' ' 1 '

[ ] [ ] [ ] [ ] [ ]

[ ] [ ]

tot L R p p jjii jj ii jj ii jj ii

l p ii jj ii p r jj

H H H S S

S S S S

δ δ

δ δ+

+

= + +

+ +

i and i’ are the basis states of left block with new site pj and j’ are the basis of right block with new site p+1

Iteration:

Add two spin-sites in the middle. Left block now consists of L+1 sites andRight block R+1 sites (generally L=R).

The basis for left block is now |i> = |Lα>*degrees of freedom for the new site (σ )Similarly for right block, the basis is |j> In this basis, the total Hamiltonian for the L+R+2 sites is:

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: Ψ (µ,σ,τ,ν) Expectation values of various quantities can be calculated at this point

Form the reduced density matrix for two-block subsystem (2 sites) using

ρ( µ ,σ; µ ‘ σ ‘)=Στν Ψ( µ, σ, τ, ν) Ψ ( µ ‘, σ ‘, τ, ν)

Diagonalize ρ to find a set of eigenvalues ωα and eigenvectors, u (µ,σ; α). Discard

all but the largest m eigenvalues and the corresponding eigenvectors: O=u (µ,σ; α)

Form matrix representations of various operators of the two-block subsystem.

Form a new block-1 by changing the basis to the u α 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.

DMRG Algorithm

System size

2 ways of building up superblock; choice of environment block

Infinite system algorithm:

Environment block: reflection of the subsystem blockSuperblock grows by 2 lattice sites per iteration

Finite system algorithm:

Starting point:      infinite system methodSize of whole system same:     environment block size shrinksZipping back and forth:     iterative convergenceEnvironment block:    use from previous iteration

Infinite system DMRG

Systemsizegrowsby2sitesevery iteration

Finite systemDMRG

Left block growsRight block shrinks

Reflect the whole system

Left block growsRight block shrinks

First iteration ends

Start: Infinite DMRG generated system

Most accurate method for interacting systems in low­dimensions.

Higher dimension: if X states are required for obtaining accurate results inone­dimensional systems, X d states are necessary to obtain comparableaccuracy in d­dimensional 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. 

Few Remarks

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 definitequantum numbers.

Eg. Sz total and N total (fermionic case)

Operators can be stored in sparse matrix form. Do all arithmetic'susing sparse matrices.

Non-abelian quantum numbers, like S2, also possible (though complicated)

Disk usage: Information not used in current loop may be writtento disk (eg, infinite DMRG generated blocks for finite DMRG)

Applications:

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 GroupDesign Methodology Research GroupDirect Marketing Resource GroupData Management Resource Group

Groupe de Renormalisation de la Matrice Densite (GRMD) !!!

All are DMRG indeed !

Strongly correlated electronic systemsNuclear PhysicsQuantum information theoryQuantum ChemistryClassical Statistical PhysicsSoft condensed matter Physics

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 ComputationalPhysics 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).

TMRG: Transfer matrix renormalization group

Transfer Matrix: (Bursill et al, 1996, Xiang et al, 1998).

Trotter-Suzuki decomposition:

“Checkerboard” decomposition of magnetic lattice.

DMRG is used to compute quantum transfer matrix in space direction.

Z= Lt L infinity Tr(T1T2)L/2

Infinite system and finite temperature propertiesLimited to 1D systemsTransfer matrix not hermitianConstruction of density matrix not uniqueLocal dynamics possible, however, analytic continuation necessary

So far:Frustrated spin chain (1998, Maisinger, Schollwock)Kondo lattice model (1998, Shibata)Spin-orbital model (Sirker, 2003).

/ 2limH NM

MZ Tre TrTβ−

→∞= =

2

1 2 1 23 3 1 1

2 1 2 2 1 2 2 2 1 2 2 13 3 2 2 2 2 1 1

1{ }

....... | | .........

( | )* ( | )k

M MM

Mk k k k k k k k

k

T

t tσ

σ σ σ σ

σ σ σ σ σ σ σ σ− − + +

=

< >

= ∑∏1 1 1 1

1 1 1 1( | ) , | | ,ihk k k k k k k ki i i i i i i it e τσ σ σ σ σ σ σ σ−+ + + ++ + + +=< − − >

/ Mτ β=| |z k k ki i i iS σ σ σ>= >

TMRG method starts by mapping a 1D quantum system onto a 2D classical one with Trotter-Suzuki decomposition.

The partition function is represented as a trace of power function ofvirtual transfer matrix-----

M is the Trotter numbers.

TM = inner product of 2M local transfer matrices

where

Superscript= Trotter space and subscript = real space

max

1 1lim ln lim ln

2N MF Z

Nλ

β β→∞ →∞= − = −

Where λ max is the maximum eigenvalue of T M

Apply DMRG to obtain QTM in the real space direction

The QTM is enlarged in the imaginary time direction and in eachiteration the temperature is lowered at fixed Trotter time steps

The free energy and other thermodynamic properties can be directlyobtained from the largest eigenvalue of QTM and the correspondingeigenvectors.

Accurate and unbiased. Errors are mainly due to finite size of the Trotter time steps.

Can be used for Fermions. Better method than QMC.

DDMRG: Dynamical DMRG method

Dynamics of quantum systems:

Correlation function:1

0 0( , ) | ( ) |q qG q A i H Aω ψ ω η ψ+ −=< + − >

Require to keep more density eigenstates

Lanczos vector method (Hallberg, 1995):

Target vectors: Krylov subspace, then do continued fraction

20 0 0 0| , | , | , | ,......q q qA HA H Aψ ψ ψ ψ> > > >

Quantities required for Lanczos iteration

0 021

0 22

12

| |1( ) Im

......

A AI

bz a

bz a

z a

ψ ψωπ

+

< > = − − − − − − −

Spin dynamics of a S=1/2 Heisenberg model (K. Hallberg, PRB, 1995)

DMRG coupled with Lanczos

Correction Vector Method (Pati, Ramasesha, 1996, 1997):

Target vectors: all the excitations in terms of one vector

10 0 0| , | , ( ) |qA i Hψ ψ ω η ψ−> > + − >

More accurate than Lanczos; But different run for every ω.

Quantities required:

°

°

(1)0 1 1 0

(2) (1)0 2 2 1

( ) | ( ) |

( ) | ( ) | ( )

i i

ij i j

H E i

H E i

ω ε φ ω µ ψ

ω ε φ ω µ φ ω

− + + >= >

− + + >= >

°0

1

( 1)N

i k kk

e n r iψµ µ=

= − − −

∑

where

Use of symmetries are crucial: all spatial symmetries

SDW­CDW crossover

Explains the reason behind large nonlinear absorption cross section observed in Cuprate systems.

Nonlinear optics in correlated models (Pati and Ramasesha, PRB 1998):

DMRG for Quantum Chemistry: (White, 1999)

, , , , ' , ' ,, , , , , '

1

2ij i j ijkl i j k l

i j i j k l

H t c c V c c c cσ σ σ σ σ σσ σσ

+ + += +∑ ∑ ∑

Doing full CI using DMRGVery similar to 2D methods (momentum space)                                                                                                      

21| |

2ij extt i v j=< − ∇ + >

12

1| |ijklV ij jir

=< >where

&

Given a molecular geometry, choose a basis set.Usually, real­space 1­electron functions  atom­centered GaussianCompute the integralsSolve for the many­body wavefunction   DMRG 

So far…….

Ordering of orbitals and use of symmetries crucial

White and Martin, 1999H2O

Chan and M. Head­Gordon            JCP (2002)

“ Better ( in terms of both time & accuracy) than any other Quantum Chemical methods”

Time evolution of a quantum system:

/| ( ) | (0)iHtt eψ ψ−>= >

Approaches using DMRG:

Runga-Kutta integration of |ψ0>DMRG

Only good for small times, t.

Division of e-iHt into 2-site terms (White and Feiguin, 2005)

Trotter­Suzuki decomposition, 2­sites parts exactly applied (quantum gates)

Applications:

Transport current in quantum dots, molecules…..mesoscopic physics

Bose-Hubbard model, Bose condensation, atom traps.

Difficulty is in dealing with exponential operator

ET

Charge transfer isfacilitated by formationof exciton, volume of whichis of the order of thesystem size

Long-range electron transfer across long-chain conjugated polymer

Lakshmi, Datta, Pati (2005)

Best method for dealing with low­dimensional interacting systems

Atoms, molecules, clusters, condensed phase materials

The idea can be adapted to many other methods

Immense scope for various applications.

DMRG