Density Matrix Renormalisation Group

David Weir

Student Seminars 2007-08

Not my own work

I Andy Parry suggested I might want to investigate this technique inmy spare time.

I General narrative borrowed from cond-mat/0510321.I I haven’t had the time (yet) to fully code my own DMRG simulation.I I focus (mostly) on the algorithms, not the models that inspired them.I The aim of this student seminar is to point out applications of DMRG

that I think are interesting . . .I . . . but I won’t mention any links to quantum information theory.

I Algorithmic optimisation by studying Hilbert space.I von Neumann entropy for density matrix with eigenvalues wα:

S = −X

α

wα logwa

is a useful measure of the importance of the discarded states.I S has interesting critical behaviour

Not my own work

I Andy Parry suggested I might want to investigate this technique inmy spare time.

I General narrative borrowed from cond-mat/0510321.I I haven’t had the time (yet) to fully code my own DMRG simulation.I I focus (mostly) on the algorithms, not the models that inspired them.I The aim of this student seminar is to point out applications of DMRG

that I think are interesting . . .I . . . but I won’t mention any links to quantum information theory.

I Algorithmic optimisation by studying Hilbert space.I von Neumann entropy for density matrix with eigenvalues wα:

S = −X

α

wα logwa

is a useful measure of the importance of the discarded states.I S has interesting critical behaviour

Introduction

I Exact diagonalisationI Numerical renormalisation group (NRG)

I Notions of RG for discrete quantum systemsI NRG algorithmI Limitations and extensions

I Density matrix renormalisation group (DMRG)I AlgorithmI Extensions (scary bit)I Applications: (λφ4)1 and massive Schwinger model

Exact Diagonalisation

I We want to study the ‘flow’ of eigenvalues and energies for a 1Dsystem as we vary the system size L.

I Obviously, the simplest way of finding the ground state for a 1Ddiscretised quantum system of finite size is through exactdiagonalisation of the Hamiltonian.

I Advantages:I Intuitive, it’s the first thing you learned in quantum mechanics.I Retains the full Hilbert space of the system.I Can be fast (using algorithms such as the Lanczos algorithm)

I And big disadvantages:I Doesn’t truncate the Hilbert space, matrices get very big: (dimH)2L

entries for a spin chain . . .I . . . when we can do without some of the states if we are happy

approximating.

“In practice, the spin- 12 Heisenberg model can be diagonalized on up to

about 36 sites.” – Noack and White – takes weeks.

Exact Diagonalisation

I We want to study the ‘flow’ of eigenvalues and energies for a 1Dsystem as we vary the system size L.

I Obviously, the simplest way of finding the ground state for a 1Ddiscretised quantum system of finite size is through exactdiagonalisation of the Hamiltonian.

I Advantages:I Intuitive, it’s the first thing you learned in quantum mechanics.I Retains the full Hilbert space of the system.I Can be fast (using algorithms such as the Lanczos algorithm)

I And big disadvantages:I Doesn’t truncate the Hilbert space, matrices get very big: (dimH)2L

entries for a spin chain . . .I . . . when we can do without some of the states if we are happy

approximating.

“In practice, the spin- 12 Heisenberg model can be diagonalized on up to

about 36 sites.” – Noack and White – takes weeks.

Enter the renormalisation group

I Key idea is always to ‘weed out unimportant degrees of freedom’iteratively.

I Wilson wanted to study the Kondo problem by iteratively addingextra sites to the system and looking at the changes to theeigenvalues (‘RG flow’) of H .

I Degrees of freedom are added at every iteration, so we won’t beable to study many RG steps.

I Solution (heuristically):I Retain lowest m eigenvectors at each step, and discard the rest.I Make a similarity transformation to the new, undercomplete basis.I Incorporate the ‘downsampled’ system into a new, larger system by

forming new Hamiltonian.

I The first big difference between NRG and DMRG concerns how wechoose the “lowest m eigenvectors”.

Connection with Wilsonian, continuous momentum-spaceRG

I In the analytic RG for a scalar field, we rescale the distances andmomenta by b:

k′ = k/b, x′ = x/b

and integrate over the high-momentum states to create approximatevertices in the effective Lagrangian. Instead, we resize the lattice:

L′ = L+M

then truncate the Hilbert space and perform a similaritytransformation to create an approximate Hamiltonian H̃ .

I In fact, if we use a momentum-space formulation of our discretesystem, we will find that the truncation removes thehighest-momentum states!

I In both cases, we study the flow of operators under therescaling/truncation, but must be careful to flow to the right fixedpoint:

I Analytic RG: trivialityI DMRG/NRG: finite-volume fixed point

The Numerical RG algorithm: NRG

L M

H̃L+ Hint + HM

HL

{un} eigenbasis

OL matrix

diagonalise choose m

1. Diagonalise the L-site Hamiltonian HL exactly, retaining m eigenvalues andeigenvectors.

2. Make a similarity transformation using these m states to a truncated Hilbertspace of the L-site system.

3. Add M sites to the chain and construct HL+M , the next Hamiltonian in theiterative sequence.

4. Repeat the above steps until (say) fixed-point behaviour is seen in theeigenvalue spectrum.

In most early applications M = 1 was used, i.e. a single site was added to thechain at each step.

Simple example: Single particle on a chain

This is what Noack and Manmana call it, but it’s basically the particle in abox. Unfortunately, we will find that the NRG algorithm doesn’t work,unless we are even more clever. The Hamiltonian is:

H = −L−1∑l=1

(|l〉 〈l + 1|+ |l + 1〉 〈l|) + 2L∑

l=1

|l〉 〈l|

I This is a single particle on a chain, “equivalent to ... −∂2/∂x2 ”.I There are fixed boundary conditions, the wavefunction vanishes at

the ends.I But when we make the chain longer, the wavefunction needs to

vanish at the ends of the longer chain!I Otherwise we have an expensive dislocation in our chain that the

RG transformation can’t remove.

The failure of NRG and the post-mortem

I Since we keep only the lowest few eigenvalues in each block, wecan’t construct the ground state for the whole system when weshove two blocks together.

I We have totally ignored the short-range interactions at theboundaries!

I We need to treat the boundaries differently, recognising that thesystem is destined for greater things in the long run.

I When we are building our new Hamiltonian H2L we should choosethe most important states, not necessarily those of lowest energy.

I One method of doing this is artificially expanding the Hilbert spaceto include eigenfunctions for the system with free boundaryconditions, but this isn’t very general.

We want a method of constructing H2L that lets us take account ofshort-range interactions at the edge of any system.

The superblock and fluctuations

I We instead use an algorithm where we diagonalise an over-enlargedHamiltonian Hp=4

2L :

Hp=42L =

H̃L H̃int 0 H̃†

int

H̃†int H̃L H̃int 00 H̃†

int H̃L H̃int

H̃int 0 H̃†int H̃L

I The lowest-lying m eigenvalues we retain are the ones we would

find for a system where there are 4L sites.I Trace out 2L of the sites, to leave a set of states for H̃2L.I This gives a better set of states to approximate interactions at the

boundary.

The superblock approach is promising for interacting systems. Weare now ready to present DMRG in its full horror.

The infinite-system DMRG algorithm...

L/2 L/2

HsuperL = H̃L/2

+ H̃int+ H̃R

L/2

1. We begin with a L-site superblock that we can diagonalise, anddenote the corresponding Hamiltonian Hsuper

L .

2. Obtain the ground state of HsuperL , ρgs = |ψ〉 〈ψ|.

3. For the first L/2 sites in the superblock, we deduce HL/2 and formthe reduced density matrix ρ′gs.

4. Diagonalise ρ′gs, retaining m eigenvectors with the largesteigenvalues.

5. Transform HL/2 to the new truncated eigenbasis, forming H̃L/2.

6. The rescaled H̃L/2 is reflected onto the remaining L/2 sites.

7. Add two sites to the superblock L→ L+ 2, form HsuperL+2 and repeat

from step 2.

Some notes

I We need to use a iterative diagonalisation algorithm that canterminate after the first step, to ‘bootstrap’ the DMRG.

I For finite systems:I Keep a fixed number L of sites.I The number of sites in the sub-block varies from 1 to L− 3, then a

new sub-block of 1 site is formed.I Like ‘zipping and unzipping a coat’.

l L− l − 2

I Why we use density matrices:I It can be shown variationally that the states that are most important in

approximating the ground state are the eigenvectors of ρgs with thelargest eigenvalues.

I The generalisation to finite temperatures or mixed states is trivial, andagain it is the eigenvectors with the largest eigenvalues that are best atreconstructing any of the underlying pure states.

Extension to 2D via matrix-product states (scary)

I If we have a system that is long and thin, we can map the 2Dproblem onto a 1D problem where the short-range interactionsremain (fairly) short-range.

I For a convincing treatment of 2D systems, and for succesfulgeneralisation to higher dimensions, we need something else.

I In 1D we can swap our description on L lattice sites, for a dualdescription in terms of a reduced basis of entangled states ofnearest-neighbour sites. The original system is recovered by applyinga stack of L operators Ak – think of these as 2-index tensors thatproject out part of the state from bonds pointing in two directions.

I The variational problem is now to determine the Ak that return the bestapproximation to the ground state |ψ〉.

I In 2D the generalisation is clear: we still have a dual lattice, but thestack of operators becomes a stack of tensors

I Neighbouring sites are pairwise entangled, so these are called‘pairwise entangled product states’ (PEPS).

I Details are much more complicated than this, see cond-mat/0407066

I We are now a very long way away from DRMG.

Application of DMRG to a (λφ4)1 theory in 1+1dimensions

This is the work of Sugihara, which uses the finite-system algorithmapplied to a discretised (λφ4)1 system to obtain the critical coupling βC .

I We start with the well-known lattice Hamiltonian:

HS =L∑

n=1

( 12aπ2

n +µ2

0a

2φ2

n +λa

4!φ4

n

)︸ ︷︷ ︸

hn

+L−1∑n=1

12a

(φn − φn+1

)2︸ ︷︷ ︸hn,n+1

for λφ4 in 1+1 dimensions.I Only the spatial coordinate is discretised.I We split the superblock into left and right blocks, with a single block

between them:

HS = HL + hn−1,n + hn + hn,n+1 +HR

I It is with this arrangement that the finite system DMRG ‘zipping’procedure can be performed.

(λφ4)1 DMRG: technical details

Sugihara uses the finite-system algorithm with a single site between thetwo blocks. Therefore the overall size of the system remains the sameand we sweep the ‘zipper’ up and down.

I The operators φn and πn are rewritten with the creation a†n andannihilation an operators for the nth site

I These are not sharp in momentum-space, so they do not create orannihilate particles of given momentum.

I Since this is a bosonic system, we are going to throw away lots ofdegrees of freedom each time we make an undercomplete changeof basis.

I The basis for the superblock is a tensor product of the basis statesfor the left block {|u(L)

i 〉}, the central site {|j〉}, and the right block{|v(R)

k 〉}.

(λφ4)1 DMRG: results

I As in lattice Monte Carlo, the critical coupling is determined byextrapolation to the continuum limit a→ 0.

I L = 250, L = 500 and L = 1000 used (largest MC study usedL = 512 and took far longer!).

I Only retain 10 basis states for each subsystem.I Sugihara does not discuss the effect of criticality on DMRG’s

performance.I Claimed to have smaller errors than – and is consistent with – the

best MC result:

Method ResultDMRG (Sugihara) 59.89± 0.01Lattice MC (Loinaz and Willey) 61.56+0.48

−0.24

Connected Green’s function (Hauser et al.) 58.704

Sugihara expresses hope that the method can be applied, one day, toQCD. We will finish by looking at a very simple lattice gauge field theorywhere DMRG has met with success.

The massive Schwinger model

I This is a ‘gauge field theory’ in 1D which Coleman said wouldharbour ‘half-asymptotic particles’ – deconfined fermions

I The Lagrangian is basically QED in 1+1 dimensions, ψ is a2-component spinor:

L = −14FµνF

µν + ψ̄(i∂ − g /A−m)ψ

I The electric field E is given by F 10, for which we can find anexpression by integrating the equations of motion:

E = g

∫dx ψ̄γ0ψ + F

where F is our chosen constant of integration, which is interpretedas an applied electric field.

I F = g/2 corresponds to a first-order phase transition, a discontinuityin the energy. This is where the half-asymptotic behaviour takesplace – see Coleman’s original paper.

DMRG treatment of the massive Schwinger model

I The lattice formulation of the model puts the gauge field on the links,and staggers the fermions across two adjacent sites:

H =N∑

n=1

[− i

2a

(φ†(n)eiθ(n)φ(n+ 1)− h.c.

)+m(−1)nφ†(n)φ(n) +

g2a

2L2(n)

]I L(n) is our rescaled external applied field, and with θ corresponds

to the continuum limit gauge field Aµ.I Since the fermions are staggered across two adjacent sites, we

must add two sites to the sub-block at each DMRG iteration.I Even with a tiny number of basis states retained the ground state is

determined to very high accuracy:I Byrnes at al. go on to find the location of the critical point of the first

order phase transition by standard methods, and then to probe theproperties of the half-asymptotic fermions.

Conclusions

Main idea: to find the undercomplete basis that can bestapproximate the ground state, and to study the properties of theground state as the system size increases.

I NRG convergence is very slow unless ‘dirty tricks’ are applied.I Sometimes NRG doesn’t work at all (unless we artificially overinflate

the system size).I DMRG is excellent for 1D quantum systems giving rapid, very

accurate results.I 2D systems require very sophisticated extensions, which are not

intuitive.

Q: DMRG, is it useful for QFTs?A: Not really∗.

∗ but variational state-approximation methods might be.

Conclusions

Main idea: to find the undercomplete basis that can bestapproximate the ground state, and to study the properties of theground state as the system size increases.

I NRG convergence is very slow unless ‘dirty tricks’ are applied.I Sometimes NRG doesn’t work at all (unless we artificially overinflate

the system size).I DMRG is excellent for 1D quantum systems giving rapid, very

accurate results.I 2D systems require very sophisticated extensions, which are not

intuitive.

Q: DMRG, is it useful for QFTs?A: Not really∗.

∗ but variational state-approximation methods might be.

Sources and further reading

I On NRG: Costi in Density-Matrix Renormalization, A New NumericalMethod in Physics, Springer.

I On DMRG:I Noack and White, ibid.I Noack and Manmana: cond-mat/0510321

I Entangled product states: Verstraete and Cirac: cond-mat/0407066I On (λφ4)1: Sugihara, JHEP 05 (2004) 007I On massive Schwinger model: Byrnes et al., Phys. Rev. D 66,

013002