exact diagonalization: introduction · symmetries in ed ii consider the simplest parent...

Exact Diagonalization: Introduction

C o n d e n s e d M a t t e r S u m m e r S c h o o l 9 / 5 - 9 / 9

H o n g - Y i C h e n National Taiwan Normal University, Taipei, Taiwan1

Matrix

2

U

Ut

Ut

Utt

tU

tt

t

tU

tt

t

ttU

t

t

H

2000000000000000

00000000000000

00000000000000

0000000000000

00000000000000

0000000000000000

00000000000000

000000000000000

00000000000000

00000000000000

0000000000000000

000000000000000

0000000000000

000000000000000

000000000000000

0000000000000000

Ingredients

Hilbert space

Basis representation, Lookup techniques

Symmetries

Hamiltonian Matrix

Sparse Matrix representation (memory/disk)

Matrix recalculation on the fly (matrixfree)

Linear Algebra : Eigensolver / Time propagation

LAPACK full diagonalization

Lanczos type diagonalization (needs only operations)

More exotic eigensolver techniques, imaginary-time propagation,

Observables

Static quantities (multipoint correlation functions, correlation density

matrices,...)

Dynamic observables (spectral functions, density of states,...)

Real-time evolution

3

Lattice model

Direct diagonalization of Hamiltonian matrix on finite clusters

Hilbert space, Symmetries

Hamiltonian matrix

Linear Algebra

Observables

4

Representation of Many-Body States

Mapping to (binary) integers:

Spin1/2 Heisenberg:

For example

Hubbard:

For example

mj,

5101

61001

5

Symmetries in ED I

The inclusion of symmetries in an ED code has two major

advantages:

Quantum number resolved energies and states.

Reduction of the Hilbert space to be diagonalized.

Given group G with generators {gp}

[H, gp] 0 H block diagonal (Hilbert space can be divided)

6

Symmetries in ED II

Consider the simplest parent Hamiltonian, the Heisenberg

model on a chain.

The dimension of Hilber space is 2N

For exmaple N 20 220 1048576 @ 106

To store a real number (double precision) in computer we need 8 Bytes.

To store the matrix element of the above Hamiltonian we need:

Number of bytes 8 (106)2 @ 8000GB

Considering the Sz symmetry

For example, the most amount of basis is total Sz 0

10 upspin, 10 downspin

Number of bytes 8 (105)2 @ 80GB

i

iiJH 1SS

520

10 10184756!10!10

!20@C

7

Symmetries in ED III

Continuous:

U(1) related symmetries

Conservation of particle number(s)

Conservation of total Sz

SU(2) symmetry

Difficult to implement together with spatial symmetries

For Sz 0 an operation called “spin inversion (Z2)” splits the Hilbert

space in even and odd spin sectors.

Spatial symmetry groups:

Translation symmetry (abelian symmetry, therefore 1D irreps)

Pointgroup symmetries (in general nonabelian)

8

Sparse Matrix

U

Ut

tU

Ut

tU

Utt

tt

tt

ttU

t

t

t

t

H

2

0

0

0

0

0

0

0

0

0

9

Hamiltonian Matrix Storage

Different possibilities exist:

Store Hamiltonian matrix elements in memory in a sparse matrix

format

Fast matrix vector multiplies, but obviously limited by available memory.

Store Hamiltonian matrix elements on disk in a sparse matrix

format.

In principle possible due to the vast disk space available, but I/O speed

is much slower than main memory access times. Difficult to parallelize.

Recalculate the Hamiltonian matrix elements in each iterations

“on the fly”.

Needed for the cutting edge simulations, where the whole memory is

used by the Lanczos vectors. Can be parallelized on most architectures.

10

Diagonalization routine

If H is dense or system small enough

Use

Jacobi

Householder

LAPACK

...

(all these apply orthogonal transformations to H until tridiagonal

form, then quickly diagonalize)

If H is sparse

Use

ARPACK

IETL/ALPS

DiagHam

(these are iterative solvers based on variants of Lanczos algorithm

which preserves the sparseness of H)

11

LAPACK

Complete diagonalization routine written in F77 (Fortran)

Support all kind of data structure

Real, Real (double precision), complex, complex (double precision)

Support most of matrix formation

Symmetric, Hermitian, Tridigonal, ...

Useful for:

Simple problems, testing

Matrix H dense

Many eigenstates required

But

H must be stored

Entire matrix must be diagonalized

12

Linear Algebra I

Lanczos algorithm:

C. Lanczos, J. Res. Natl. Bur. Stand. 45, 255 (1950).

Iterative “Krylov” space method which brings matrices into

tridiagonal form.

Method of choice in many largescale ED programs.

Works well for sparse, shortrange H

Very rapid convergence but numerically unstable (can be controlled by a

suitable techniques).

Easy to implement and memory efficient only 3 vectors needed at once

If the ground state (energy) or some low energy spectrum is

interesting, the Lanczos method may help us. Using Lanczos and

considering the Sz 0 symmetry of the model above, the required

memory to get the ground state (energy) and few low lying states is:

Number of bytes @ 30×184756 5542680Bytes @ 55MB

The required memory is available as RAM in any personal computer.

13

Linear Algebra II

“modified Lanczos” algorithm:

E. Gagliano, et al. Phys. Rev. B 34, 1677 (1986).

actually more like a Powermethod, therefore rather slow

convergence.

needs only two vectors in memory

At each step the approximate groundstate wavefunction is

available

Difficult to get excited states

Jacobi-Davidson Algorithm:

E. R. Davidson, Comput. Phys. 7, 519 (1993).

subspace expanded by diagonal approximation to inverse iteration

varying number of vectors in memory

higher-order convergence than Lanczos (usually)

Rapid convergence, especially for diagonally dominant matrices

(Hubbard model)

Often used in DMRG programs as well.

14

Lanczos

11

1

1

/

nn

n

nn

nn

nnn

b

b

a

a

bH

NN

NNN

ab

bab

bab

bab

ba

N

N

T

11

433

322

21

1

3

2

1

00 b

space Krylov , , 0

NN

N

NN

N NaH

At each step n, find the lowest

eigenvalue of that matrix

Stop when the lowest eigenvalue

E0 has converged (ΔE0/E0 < 1012)

15

Roundoff errors and ghosts

In exact arithmetic the vectors {i} are orthogonal and the

Lanczos iterations stop after at most N 1 steps. The

eigenvalues of T are then the exact eigenvalues of H.

Roundoff errors in finite precision cause a loss of

orthogonality so that the matrix T contains extra spurious

eigenvalues, called “ghosts”.

16


There are two ways to deal with that:

Reorthogonalization of the vectors after every step. This requires

storing all of the vectors {i} and is memory intensive. (you don’t

want to do it)

Control of the effects of roundoff.

We will discuss the second solution as it is faster and needs

less memory.

The main effect of roundoff errors is that the matrix T

contains extra spurious eigenvalues, called “ghosts”. These

ghosts are not real eigenvalues of H.

However they converge towards real eigenvalues of H over

time and increase their multiplicities.

17


A simple criterion distinguishes ghosts from real eigenvalues.

Ghosts are caused by roundoff errors. Thus they do not

depend on the starting vector i. As a consequence these

ghosts are also eigenvalues of the matrix T’, which can be

obtained from T’ by deleting the first row and column:

Criterion to distinguish ghosts from real eigenvalues:

All multiple eigenvalues are real, but their multiplicities might be

too large.

All single eigenvalues of T which are not eigenvalues of T’ are also

real.

NN

NNN

ab

bab

bab

bab

ba

T

11

433

322

21

'

18

Degenracy

19

Green’s functionContinued fraction

Zero temperature Green’s function

Consider the diagonal element

Use the expansion

jiij

jiij

cEH

cG

ctcitG

0

1)(

)0()()(

iiiiiiEH

Gc

0

1

2

32

1111H

zH

zzHz

20


Truncated expansion evaluated exactly in Krylov subspace

generated by i

We perform a Lanczos procedure on i, then Gii is given by

Jacobi continued fraction

2

2

21

2

10

a

ba

ba

Gii

ii

21


Non-diagonal element

Define k (cicj) , we obtain

Since Gij Gji, then

)(1

)()(0

jijiij ccEH

ccG

2

2

21

2

10

a

ba

ba

Gkk

ij

][2

1jjiiij GGGG

ij

22

Green’s functionLehmann representation

Lehmann representation

Extended Krylov space form a band Hamiltonian (2L1

diagonals)

n n

i

m m

i

i

nn

jj

mm

iij

EE

cn

EE

mc

cnEE

nccmEE

mcG

0

2

0

2

00

11)(

...L ic ii 1 ,

LiaH N

i

N

ii

N ...1 , )(

23

Green’s functionRecursion

Green’s function

HQ QT

where ji is calcuated by band Lanczos method

...L ic ii 1 ,

[ ]321 φφφQ

1

0

1

0

11)(

EHc

EHcG jiij

NN

NNN

ab

bab

b

babb

bbab

bba

1,

1,11,1

42

4133132

4231221

32211

T

24

Green’s functionRecursion

Recursion

where

0

0

0

1

1

3

2

1

1 φφ

φ

φ

φQ

110

1

0

1

1

0

1

1

0

1

1

1

1

1)(

E

E

E

EG ij

T

φQT

Qφ

φQQH

QQφ

φH

φ

25

Finite Temperature Lanczos Methods

These methods (FTLM, LTLM) combine the Lanczos method

and random sampling to go to larger systems.

For high to moderate temperatures these methods can obtain

results basically in the thermodynamic limit.

The Low Temperature Lanczos Method can also go to low

temperatures to get correct results on a given sample. Finite

size effects however persist.

Like the T 0 ED method it is most useful, where QMC or T

DMRG etc fail, such as frustrated and fermionic models.

26

FTLM

In the canonical ensemble, the thermal average of a physical

quantity A at finiteT in orthonormal basis n is defined as

High temperature expansion gives

From two successive Lanczos procedures for two operators

N

n

H

N

n

H

nenZ

nAenZ

A

1

N

n

k

k

k

N

n

k

k

k

nHnk

Z

nAHnkZ

A

0

0

!

)(

!

)(1

l

j

k

ij

M

i

M

j

jii

lk nABnnABHHn )~()(~~

0 0

27

FTLM

We obtain (k < M)

The superscript (n) means that i(n) and i

(n) are obtained in Lanczos

algorithm with the initial state i(n)

Extend the results to k > M, the approximate expressions are

It is very expensive to calculate all n

M

i

kn

i

n

i

n

i

k nAnnAHn0

)()()( )(

N

n

M

i

n

i

n

i

N

n

M

i

kn

i

n

i

n

i

k

k

N

n

M

i

n

i

n

i

N

n

M

i

kn

i

n

i

n

i

k

k

nneZ

nnk

Z

nAneZ

nAnkZ

A

ni

ni

0

)()(

0

)()()(

0

0

)()(

0

)()()(

0

)(

)(

1)(

!

)(

1)(

!

)(1

28

FTLM

Stochastic sampling of Krylov space: Replace the summation

of the full Hilbert space by a sum over random number of

states

Ss over symmetry sector of dimension Ns

Sr over R random starting vectors r

s

R

r

M

i

r

i

s

R

r

M

i

r

i

r

i

reZ

rAreZ

A

ri

ri

2)(

)()(

)(

)(1

29

LTLM

Since FTLM is related to highT expansion, i.e., T limit

correct, the limit of T 0 does not lead to the ground state

expectation value.

One more Lanczos procedure for each sampling

In the limit of T 0

s

R

r

M

i

r

i

r

i

s

R

r

M

i

r

i

r

j

r

j

M

j

reZ

rAreeZ

A

ri

rj

ri

2)(

)()()()(2/2/

)(

)()(1

00 AA

30

Strongly Correlated Systems

Hubbard model

Heisenberg model

High-Tc superconductor

Quantum Phase Transitions

31

Applications

Quantum Magnets: nature of novel phases, critical points in

1D, dynamical correlation functions in 1D & 2D

Fermionic models (Hubbard/tJ): gaps, pairing properties,

correlation exponents, etc

Fractional Quantum Hall states: energy gaps, overlap with

model states, entanglement spectra

Quantum dimer models or other constrained models (anyon

chain..)

Full Configuration Interaction in Quantum Chemistry

32

Largest system reached

S 1/2 spin models

square lattice: N 40

triangular lattice: N 39

maximum dimension of basis: 1.5 billion

tJ models

checkerboard lattice with 2 holes: N 32

square lattice with 2 holes: N 32

maximum dimension of basis: 2.8 billion

Hubbard models

square lattice at half filling: N 20

quantum dot structure: N 20

maximum dimension of basis: 3 billion

Holstein models

chain with N 14 phonon pseudo-sites

maximum dimension of basis: 30 billion

33

Exact diagonalization is expensive!

Why use ED then?

Robust, unbiased and completely versatilealmost anything can be

calculated!

There are models which are not easy to access via other models

(e.g. frustrated magnets)

Error is at least as low as 1014 numerical precision

Exploiting symmetries reduces computational effort and gives

physical information about eigenstates (good quantum numbers)

34

The best we can do: Low-lying states

Low-lying states sufficient for lowtemperature physics;

Low-lying states important for quantum phase transitions, e.g.

magnetization jumps.

Several methods available to obtain extreme eigenvalues:

projection, Lanczos, DMRG.

35

Exact Diagonalization: Application

Spin1/2


Heisenberg model

i

ij

ij

i hJ BSSS

H

37

Spin1/2 system

Define ladder operators:

The basis of Hilbert space

The Hamiltonian becomes

y

i

x

ii iSSS

y

i

x

ii iSSS

mj,

ij

jiji

z

j

z

i

ij

y

j

y

i

x

j

x

i

z

j

z

i SSSSSSJSSSSSSJH ][2

1][

38

Three spin1/2 Heisenberg model

Non-periodic boundary condition

The size of the system is 23 8

The trial basis states are

][2

1][

2

1323232212121

SSSSJSJSSSSSJSJSH zzzz

7111

6110

5101

4100

3011

2010

1001

0000

39

The Hamiltonian matrix

The Hamiltonian in the given basis is

2

02

220

22

0000

2000

200

22

20

2

J

J

JJJJ

J

JJJ

J

J

H

40

Magnetization

The magnetization is

The size of the largest block is

Mz 1/2 or 1/2

The new basis states become

74

62

51

30

23

21

21

21

21

21

21

23

zz

zz

zz

zz

MM

MM

MM

MM

i

z

iz SM

33

1 C

41

Magnetization

The Hamiltonian becomes

2

02

222

20

02

222

20

2

2321212121212123

J

J

JJJ

J

J

JJJ

J

J

H

42

How does the computer work

Declare a vector A[m] to store basis states

For example:

Next, act H on the basis states

For example:

Then, we need to find out the corresponding index with

respect to the basis

7]8[6]7[5]6[4]5[

3]4[2]3[1]2[0]1[

AAAA

AAAA

42

1222

2]3[JJJJ

HHAH

5 , ]5[4

2 , ]2[1

2

1

jA

jA

43

How does the computer work

The most efficient way: declare a vector B[m], m 0 to 7

Therefore we have

and

The matrix elements are

8]7[7]6[6]5[5]4[

4]3[3]2[2]1[1]0[

BBBB

BBBB

2

1

4]]4[[

2]]2[[

jAB

jAB

2)4,3(),(

2)2,3(),(

2

1

JHjiH

JHjiH

44

Two-Table method

Divide lattice to two sublattice

For example: site1,2 and site3

83351

72251

61151

50051

43310

32210

21110

10010

2,13

2,13

22

jjj

iii

45

Two spin3/2 Heisenberg model

The size of the system is 42 16

][2

1212121

SSSSJSJSH zz

23

23

23

21

21

23

21

21

21

23

21

21

23

23

23

21

23

21

23

23

21

21

21

23

21

21

21

23

23

21

23

23

46

Two-bit per-site

23

21

23

23

11

10

01

00

11 1111 01

10 1110 01

01 1101 01

00 1100 01

11 1011 00

10 1010 00

01 1001 00

00 1000 00

23

23

23

21

21

23

21

21

21

23

21

21

23

23

23

21

23

21

23

23

21

21

21

23

21

21

21

23

23

21

23

23

47

Magnetization Mz 3, 2, 1, 0, 1, 2, 3

The Basis and lookup tables

11 11

10 1111 10

01 1110 1011 01

00 1101 1010 0111 00

00 1001 0110 00

00 0101 00

00 00

23

23

21

23

23

21

21

23

21

21

23

21

23

23

21

21

21

21

23

23

23

21

21

21

21

23

23

21

21

23

23

23

15]16[14]15[11]14[13]13[10]12[7]11[12]10[9]9[

6]8[3]7[8]6[5]5[2]4[4]3[1]2[0]1[

AAAAAAAA

AAAAAAAA

16]15[15]14[13]13[10]12[14]11[12]10[9]9[6]8[

11]7[8]6[5]5[3]4[7]3[4]2[2]1[1]0[

BBBBBBBB

BBBBBBBB

48

Hamiltonian matrix elements

J

JJ

JJ

JJ

JJJ

JJ

JJ

JJJ

JJJ

JJ

JJ

JJJ

JJ

JJ

JJ

J

H

49

43

23

23

43

43

41

43

49

23

23

41

41

23

23

49

43

43

43

43

23

23

43

49

30

33

03

00

20

02

00

30

33

03

15

14

11

13

10

7

12

9

6

3

8

5

2

4

1

0

49

Exact Diagonalization: Application

Hubbard Model


Hubbard Model

i

iii

ij

i nnUcct ˆˆ

H

1 2 3 4

UHO

LHO

U

51

Atomic Ground State (t 0)

Four possible occupation of a single site

U

H

000

0000

0000

0000

2

0

52

A. Non-interacting electrons

Assume j to be Wannier wave functions (Fourier transform of

Bloch orbitals)

Assume tij is nonzero only when i,j are nearest neighbors

Diagonalize the Hamiltonian by a Fourier transformation

ij

jiij cctH0

jiij vm

t ˆ2

22

( ) ij

ijji cccctH0

j

j

ikj

k ceL

c1

53


Then the Hamiltonian in j basis becomes

where εk 2t cos(k)

k

kkk ccH 0

<

Fkkkk

ccGS 0

54


Particle-hole excitation

where εk(q) εk+q εk

As the energy of the particle-hole excitation εk(q) can be made

arbitrarily small,︱GS〉characterizes a metal.

nkqkqkn GSccHH

00

55

B. Electron-electron interaction

Two-site case

)()(110001100110

nnnnUcccccccctHHH Ut

U

Ut

Ut

Utt

tU

tt

t

tU

tt

t

ttU

t

t

H

222

2

2

20

2

0

2

0

02

0

0

00

56

B. Electron-electron interaction

57

N 2 and Sz 0

Declare a matrix A(1:4)

0101

1001

0110

1010

→ Decimal

569

10→ Binary

States Basis

10)4(

9)3(

6)2(

5)1(

A

A

A

A

Site 0Site 1

58

N 2 and Sz 0 : Diagonal elements

Basis states

Acting on-site Coulomb interaction HU on the basis states

01015)1(1 Aj

U

U

HU0

0

1010

0110

1001

0101

59

N 2 and Sz 0 : Off-Diagonal elements

Acting hopping terms Ht on the basis states

How does the computer work to come out the same results?

001015)1(001

ccAj

32

00010010

00010001

00010001

00011001101

00

00

00

0)(

jj

j

tt

ccctcccctc

ccctcccctc

ccctcccctc

cccccccccctH t

60


Acting hopping terms Ht on the basis states

Count the number of electrons between site i and site j

0][0][

11 NN rrrrcccc

ji

NjiN

NNji

jiji

rr

rrrr

NN

rr

rrrrrr

rrrr

cc

cccccc

cccccc

cccc

0][)1()1(0][

0][0][

11

11

3

0

0001101

2

0

0001101

)1(0

)1(0

cccccc

cccccc

61


Similar to other basis states

Therefore in these basis states, the Hamiltonian has the

following matrix representation.

324

413

412

321

ttH

ttH

ttH

ttH

t

t

t

t

tt

tt

tt

tt

H t

1010

0110

1001

0101

62

N 2 and Sz 1 , 1

For Sz 1 , the basis state is

For Sz 1 , the basis state is

0 and 0 Ut HH

00

00

0 and 0 Ut HH

1100

0011

63

N 0 Block Diagonal Structure

0

0

0011

1010

0110

1001

0101

1100

Utt

tt

tt

ttU

H t

64

The Block Diagonal Structure

U

Ut

tU

Ut

tU

Utt

tt

tt

ttU

t

t

t

t

H

2

0

0

0

0

0

0

0

0

0

22

2

2

2

2

20

02

0

0

0

0

00

65

Two-Table method

For example: Up configuration and Down configuration

610005311

522103210

411013210

322101101

211011101

113110000

21

21

22

jjj

iii

66

How to implement

Use subroutine (in Fortran 77 or c), module (in Fortran

90/95/2003), or object (in c or python) to structure your code

Write an object (here it means subroutine, module, or object

in different language) to store the symmetrized basis states

and lookup table

Write an object to calculate the matrix elements and assign it

to a matrix

Call the diagonalization routine using LAPACK/ARPACK/...

Write an object to calculate the observable quantities

67

Main piece of wisdom

DO NOT start writing your own code from the scratch (unless

really forced to)

68

exact diagonalization: introduction · symmetries in ed ii consider the simplest parent...

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