ISSP workshop on 27/July/2006

Is DMRG a renormalizationgroup?
---unconventional introduction to DMRG--Department of Physics, Niigata University
Kouichi Okunishi

outline transfer matrix in 2D classical system
 variational principle& matrix product connection to 1D quantum systems
 White’s DMRG
 Is DMRG a renormalization group?
 Wilson’s like NRG

DMRG prehistory
1941 Kramers-Wannier Approx.

1968 Wilson’s RG

1968 Baxter’s variational Approx
matrix product eigenvector
corner transfer matrix

1971 Wilson’s NRG
Kondo problem
1987 VBS state/AKLT modelmatrix product state
1992 White’s DMRG

Transfer matrix
W ( s1′s2′ s1s2 ) = e− β2 ( s1s 2 + s1′ s ′2 + s1s1′ + s2 s′2 )WBolztmann weightZ =∑all •T=row-to-row
transfer matrix

Thermodynamic behaviors => the largest eigenvalue of T

Z = lim N →∞ Tr (T N ) ≈ ΛN

1D quantum and 2D classicalTrotter formula — path integral(summation) representation---e− βH−βHe−β= lim N →∞ [e M e M

τWoweWowe]M[H e , H o ] = 0We ,o = eWoweHo− Mβ H e ,o i ,i+1

we
WoweweM.Suzuki,Prog.Theor.Phys.56.1454(1976)

xKramers-Wannier Approx.
Eigenvector of the transfer matrix can be approximatedby the Ising model itself in an effective magnetic field.

Ψ ( s1 , s2 L) = const × eL

L~~− K ( s1s2 + s2 s3 + ) + h ( s1 + s2 + )= φ ( s1 , s2 )φ ( s2 , s3 ) L

φ ( s, s ' ) = ce~~
Kss ' + h ( s + s ') / 2~ ~K , h : effective couplings maximizingΛ = Ψ T ΨΨ Ψ

The result is much better than the well-known mean field approx,
Bethe approx, etc…
eigenvector can be represented as a product of scalar function

Baxter’s idea---matrix product form of the eigenvector--Ψ = lim N →∞ TN

Ψ0.
.
si + 2
si +1
sisi −1.
.

Max eigenvalue-eigenvector
block spin variables µ

si +1

µi +1

siµi

= Fµi , µi+1 ( si , si +1 )

Ψ = ∑ ∏ Fµi , µi+1 ( si , si +1 )N{ µ } i =1TN

Ψ0He write the eigenvectoras a matrix product form.

Corner transfer matrix ---variational approx. again--ΨT Ψ =∏[ Fµ µ ( s' , s ' )W ( s' , s' | s , s ) Fµ µ ( s , s )]∑µ µN'i , 'i +1{ , ', s , s '} i =1ii +1ii +1ii +1i , i +1ii +1~
We can regard FWF as a renormalized transfer matrix T~
and use the variational approximation for T again.

~TF

Wµ’~
ΨF
µs’ sAAGν’~
T ≡ FWF~ ~~ ~T Ψ =ΛΨ~Ψ = ∑ Aµ ',ν ' ( s' )Gν ',ν ( s' , s) Aν , µ ( s )ν ,ν 'νA : corner transfer matrix(CTM)~~ ~ ~~ ~Λ= ΨT Ψ / Ψ ΨReduced density matrix
The previous “renormalized transfer matrix” is very similar
to the block Hamiltonian in DMRG.
The connection in the eigenvector levelAΨ≈Aquadrant
2

half infinite planeρ DMRG ≈ Ψ ≈ A
24

Z = Trρ DMRG = TrA4

whole 2D plane witha half infinite cutIterative method
prepare initial matrices
extend size of CTM
diagonalize CTMkeep the larger eigenvalue eigenstates of CTM
result is very good and convergence is rapid

Keeping larger λ gives a good approximation ofthe partition function.

~ ~4444Z ≈ Ψ Ψ ≈ Tr A = λ1 + λ2 + λ3 L

λ： eigenvalues of CTM in decreasing orderkeeping larger λ,variational principlerole of the renormalized transfer matrix is rather auxiliary.

2D classical v.s.1D quantum
2D classicalWe can treat the two spatial dimension equivalently.
This is of great benefit in thinking in the 2D world
1D quantum
This is a kind of anisotropic limit of 2D classical caseZero temperature & infinite trotter number limit
in Suzuki-Trotter transformation
if the zero temp & Trotter limits is properly taken.
the eigenvector of the transfer matrix in the mapped system
becomes the groundstate wavefunction of the original system.

How to approach 1D quantum systems?
In principle, it would be possible to formulate a numericalalgorithm to solve the 1D quantum system, based on thetransfer matrix/corner transfer matrix approach.
But, the history did not go so. Why?
1. Boltzmann weight becomes singular as T ->0 !
2. We are not able to know the MP form of the groundstate
wavefunction without the transfer matrix.
3. Timing! computer resources those days is insufficient for
the transfer matrix computation.

We need some break through!

White’s DMRG
Hamiltonian formulation

HLHR

1.The wavefunction is obtained by the direct diagonalization of super block
Hamiltonian(full Hamiltonain including both left and right block)lanczos diagonalization
2. maximize the norm of the wavefunction within the kept number of basisΨ = ΨLωΨR

singular value decomposition
reduced density matrix

3. renormalization is performed for left/right block Hamiltonian

How did he get such an idea?
The answer is partly described by himself in “Density-matrix
renormalization group”, eds. by Peschel etal, springer(1999)

This question is just out of curiosity(^_^)
NRG: particle in a box(Wilson’s approach)
QMC: for fermion(path integral)
Exact diagonalization
QMC: negative sign(block variables)
QMC: zero temp.
NRG: in momentum space
NRG: Wilson’s perturbative approach
NRG: boundary condition

He got the idea of “projection of super block into a block”
after a lot of trials and failures.

Can I get such an idea?
This question is just for fun(^_^), because DMRG had already beenpresented when I started my research in the graduate school.

I started my research from the variational principle ofthe MP state and CTM.
In 2D classical lattice statistics, we can construct
the eigenvector directly by using CTMs
I would try to a method which can directly manipulate
the wavefunction in the 1D quantum case

I would have never gotten the idea that I diagonalize thesuperblock to obtain the groundstate wavefunction.

Matrix product & DMRG
In contrast to 2D classical case, we can not decompose
the wavefunction into a MP state directly.

Ostlund and Rommer reconstructed the wavefunction ofDMRG, by analyzing the iteration process in DMRG.

Ψ ( s1 , s2 , L) = ∑∏ A[ si ]µi µi+1

A[s] : projection(transformation) matrix in DMRGIn the bulk limit, the matrix in MP state should be uniform.
For a finite size system, MP is position dependent.

finite system size method
finite system size method of DMRG is more useful.
•site resolved information: spin profile•excitation, time evolution, etc..
Various improvements and applications of DMRG
are based on the finite system size method, whichwere included in Prof. Jackelmann’s talk
But, here…
In the context of statistical mechanics,
infinite system size DMRG is more interesting.
we are now interested in critical phenomena in DMRG……

Is DMRG a renormalization group?
questions•How does DMRG behave in the critical limit?
What is a relevant/irrelevant operator in DMRG?
Can we extract a critical index through DMRG?
•What is the energy/length scale in DMRG?
•What is the meaning of eigenvalue spectrum of the
reduced density matrix at critical point?
Comparison to Wilson’s real space renormalization group
Wilson NRG for Kondo problem is only the method which
can deal with critical properties of the 1D quantum system.

block spin transformation RG

trace out

rescaleThe reason for the failure of this procedure wasanalyzed by WhiteThe “boundary” of the block spin is important.

Wilson’s NRG
Kondo impurity problem

1D quantum systemwith the boundary

Add free electrons and project out the higher energy states

H ΛN +1 = H ΛN + Λt (c N+ cN +1 + c N+ +1cN )Λ(>1) is a cut off parameter, which controls the energy
scale of the system.
Λ itself comes from log-discretizationof fermi sea

Testing Wilson-like NRG
HN +1
Λr r
= H + S N ⋅ S N +1N
ΛFree electrons
Cutoff Λr r
Interacting spins S ⋅ S

Λ =1
We do not touch the energy
scale of the systemboundary

free boundary condition

cf. T. Xiang,(1991) PRB (almost the same timing as DMRG!)

Results[excitation gap]
S=1/2 Heisenberg/XY

S=1 Heisenbergn=5Haldane gap

n=1gapless; 1/N dependence

m=16004 fold degenerating ground state
due to edge spins

m=800

The ground state energy is consistent with the known value

some implications
The results are qualitatively reasonable, but the accuracyof DMRG is much better.
To improve the accuracy of NRG may be difficult.

The meaning of the density-matrix-based transformationin DMRG should be discussed again.
The spectrum of Wilson-like NRG with Λ=1 looks like thefinite size scaling (S=1/2 critical case)

The role of cutoff is significant in Wilson’s NRG!

Wilson and WhiteWh

reduced density matrix/larger eigenvalues
angular quantization in 2D space-time
add the center sites

WiHamiltonian/lower energy
add the boundary site(the other side of the impurity)
cutoff parameter controlling the energy scaleconclusion at this stage
Is DMRG a renormalization group? => No?
Renormalization group see a response of the system
when changing cutoff paramer(energy scale/length scale)(not only real space approaches
but also field theoretical renormalization group
In terms of “renormalization group”, the theory shouldhave an actively controllable cutoff parameter.
Although the block spin transformation is used in DMRG,
the theory has no explicit scale transformation.
In order to see critical phenomena, a cutoff parameter
controlling the energy scale should be implemented in DMRG

corner Hamiltonian
Is there any possible approach to transform DMRG
into a Wilson-like NRG?

partially YES!
corner Hamiltonian approachρ ∝eplease see symposium− aH CTM

Quantum information

Entanglement entropy
S = − ρ ln ρ∂ nS = − kB
ρ
= − k B ρ ln ρ∂nn =1