2D and time dependent DMRG

1. Implementation of Real space DMRG in 2D

2. Time dependent DMRG

Tao Xiang

Institute of Theoretical PhysicsChinese Academy of Sciences

Extension of the DMRG in 2D

• Direct extension of the real space DMRG in 2D

• Momentum space DMRG: (T. Xiang, PRB 53, 10445 (1996))

momentum is a good quantum number, more states can be retained,but cannot treat a pure spin system, e.g. Heisenberg model

• Trial wavefunction: Tensor product state(T Nishina, Verstraete and Cirac)

extension of the matrix product wavefunction in 1D still not clear how to combine it with the DMRG

Real Space DMRG in 2D

Remark 1:

• should be a single site, not a row of sites, to reduce the truncation error

• To perform DMRG in 2D, one needs to map a 2D lattice onto a 1D one, this is equivalent to taking a 2D system as a 1D system with long rang interactions 2D Real space DMRG does not have a good starting point

superblock

How to map a 2D lattice to 1D?

B

H

G

I

D

F

Multi-chain mapping:

The width of the lattice is fixed

A 2D mapping:

Lattice grows in both directions

T Xiang, J Z Lou, Z B Su, PRB 64, 104414 (2001)

From a 2x2 to a 3x3 lattice

1

2 4

3 1

2

33 34 41

25

6 7

8

9

5

6 7

8

9

2 x 2 3 x 3 3 x 3

(a) (b) (c)

From 3x3 to 4x4 lattice

1 3 4 10 1 3 4 10

2 5 9 11 2 5 9 11

6 8 12 15 6 8 12 15

7 13 14 16 7 13 14 16

(a) (b)

From 4x4 to 5x5

X2

X1

X1

X2

X1

X2

1 3 4 10 11

2 5 9 12 19

6 8 13 18 20

7 14 17 21 24

15 16 22 23 25

(a)

(b)

(c)

A triangular lattice can be treated as a square lattice with next nearest neighbor interactions

(a) (b)

Comparison of the ground state energy: multichain versus 2d mapping

Symmetry of the total spin S2 is considered

Two limits: m and N

How to take these two limits?

1. taking the limit m first and then the limit N

2. taking the limit N first and then the limit m

How to extrapolate the result to the limit m ?

The limit m is equivalent to the limit the truncation error 0

-0.3622

-0.362

-0.3618

-0.3616

-0.3614

-0.3612

10-7 10-6 10-5 0.0001 0.001

E2d

Truncation Error

6x6 square lattice

Heisenberg model

Converging Speed of DMRG

1/ 4

~ mDMRG ExactE E e

decreases with increasing L

10-7

10-6

10-5

0.0001

0.001

0.01

1.5 2 2.5 3 3.5 4 4.5

Heisenberg model with free boundary conditions

E(m) - E(300) per bond

Trun Error

6

8

10

12

E(m

) -

E(3

00)

per

bond

m1/4

Error vs truncation error

True error is approximately proportional to the truncation error

0.0000 0.0002 0.0004 0.0006 0.0008 0.00100.00

0.01

0.02

0.03

Err

or

Truncation Error

6x6 Heisenberg model with periodic boundary conditions

Remark 2

• The truncation error is not a good quantity for measuring the error of the result

an extreme example is the following superblock system

its truncation error is exactly zero at every step of DMRG iteration

• A right quantity for directly measuring the error is unknown but required

superblock m m

m

iiii esw

10

Ground state energy of the 2D Heisenberg model

Extrapolation with respect to 1/L

-0.4

-0.38

-0.36

-0.34

0 0.1 0.2 0.3 0.4

Gro

und

Sta

te E

nerg

y

1/L

Square Lattice

-0.26

-0.24

-0.22

-0.2

-0.18

0 0.1 0.2 0.3

Gro

und

Sta

te E

nerg

y

1/L

Y = M0 + M1*x + ... M8*x8 + M9*x9

-0.18144M0

-0.12382M1

-0.39072M2

0.34025M3

1R

Triangle Lattice

Square Triangle

DMRG -0.3346 -0.1814

MC -0.334719 -0.1819

SW -0.33475 -0.1822

Free boundary conditions

E(L) ~1/L

Periodic boundary conditions:

E(L) ~ (1/L)3

Staggered magnetization

0.74

0.745

0.75

0.755

0.76

0.765

0.77

10-7 10-6 10-5 0.0001 0.001 0.01

6 by 6 free boundary conditions

Sta

gger

ed M

agne

tizat

ion

Truncation Error

line: Mst=M

0-

22 2 2 2 2

2

2

2

2 2

4

8

4 1

8

A B tot A B s c

c

S S S S S m M

N NN even

MN N

N odd

22 A B

s

S Sm

N

In the thermodynamic limit

In an ideal Neel state, ms=1 independent on N

Staggered magnetization vs 1/N

0.6

0.7

0.8

0.9

0 0.02 0.04 0.06

Sta

gger

ed M

agne

tiza

tion

ms

1/N

N = L2 square lattice

ms ~ 0.617 DMRG

0.615 QMC and series expansion

0.607 spin-wave theory

For triangular lattice, the DMRG result of the staggered magnetization is poor

Summary

• A LxL lattice can be built up from two partially overlapped (L-1)x(L-1) lattices

• The 2D1D mapping introduced here preserves more of the symmetries of 2D lattices than the multichain approach

• The ground state energy obtained with this approach is generally better than that obtained with the multichain approach in large systems

2. Time dependent DMRG

How to solve time dependent problems in highly correlated systems?

1. pace-keeping DMRG

2. Adaptive DMRG

(S R White, U Schollwock)

Physical background

)()()( ttHti t

Vtt0

lead

Quantum Dot

leadV

many body effects + non-equilibrium

)()(exp)( 0

0

ttdtHitt

t

formal solution

Possible methods for solving this problem

1. closed time path Green’s function method

2. solve Lippmann-Schwinger equation (t)

3. solve directly the Schrodinger equation using the density-matrix renormalization group

Example: tunneling current in a quantum dot system

Quantum Dott ttL tR

( ) ( )L R d T vH t H H H H H t

, 1,

0 0

1 0 0 1

. .

' ' .

( ) ( )( )

L R i ii L R

d d

T

v L R

H t c c h c

H c c

H t c c t c c h c

H t t N N

External bias term

Interaction representation

)()()( ttHti t

, 1,

0 0

1 0 0 1

. .

' .

( ) ( ) )

( ) ( )

(

L R i ii L R

d d

T

v L R

L R d T v

H t c c h c

H c c

H t c c

H t H H H H H t

c c h c

H t t N N

( )( )( ) ( )

( ) ( )

L Ri t N N

t

t e t

t d

( ) '( ) ( )ti t H t t ( )

1 0 0 1

'( ) ( )

' .

L R d T

i tT

H t H H H H t

H t e c c c c h c

Solution of the Schrodinger equation

( ) '( ) ( )ti t H t t

( ) exp '( ) ( )t t

tt t i d H t

2 3 41 1( ) ( ) 1 ... ( )

2 6 24iA i

t t e t iA A A A t

Straightforward extension of the DMRG

Cazalilla and Marston, PRL 88, 256403 (2002)

1. Run DMRG to determine the ground state wavefunction ψ0, the truncated Hamiltonian Htrun and truncated Hilbert space before applying a bias voltage:

2. Evaluate the time dependent wavefunction by solving directly the Schordinger equation within the truncated Hilbert space, starting from time t0

0

0( ) exp ( ) ( )t

trun

t

t i dtH t t

( ) 1 ( ) ... ( )trunt t iH t t t

Comparison with exact result

0 10 20 30 40 50 60 700.000

0.005

0.010

0.015

0.020

DMRG Exact Solution

cu

rren

t J(t

)

time t

L = 64, M = 256

The reduced density matrix contains only the information of the ground state. But after the bias is applied, high energy excitation states are present, these excitation states are not considered in the truncation of Hilbert space

The problem of the above approach

0 0sys envTr

Pace-keeping DMRG

0

0

| ( ) ( ) |

1

t

t

N

env l l ll

N

ll

Tr t t

t0: start time of the bias

Nt: number of sampled points

Luo, Xiang and Wang, PRL 91, 049701 (2003)

1. Calculate the ground state wavefunction 0 and (t) in the whole time range

2. Construct the reduced density matrix

3. Truncate Hilbert space according to the eigenvalues of the above extended density matrix

Pace-keeping DMRG

sys env

L/2 L/2

sys env

L/2 L/2

superblock

0

| ( ) ( ) |tN

sys env l l ll

Tr t t

Add two sites

Variation of the results with Nt

Reflection currentCurrent

Free boundary

Finite Size Effects

Echo time ~ 70

0 10 20 30 40 50 60 700.000

0.005

0.010

0.015

0.020

Nt = 0

Nt = 5

Nt = 30

Nt = 60

Exact evolution

curr

ent J

(t)

time t

L = 64, M = 256

Length and time dependence of the tunneling current

Exact result

How does the result depend on the weight α0 of the ground state in the density matrix?

0

0

| ( ) ( ) |

0

10

tN

env l l ll

l

t

Tr t t

l

lN

Variation with the number of states retained

Real and complex density matrix

0

| ( ) ( ) |tN

env l l ll

Tr t t

0

Re | ( ) ( ) |tN

env l l ll

Tr t t

Complex reduced density matrix

real reduced density matrix

Example 2: Tunneling junction between two Luttinger liquids (LL)

t tt’Junction

Luttinger liquid Luttinger liquid

( )L R LL T vH H H H H H t

, 1, ,

1 112 2

,

1 1

. .

' . .

( ) ( )( )

L R i ii L R

LL i ii L R

T

v L R

H t c c h c

H V n n

H t c c h c

H t t N N

V: interaction in the LL

Metallic regimes:V = 0.5w, 0, -0.5w

• The current I(t) is enhanced by attractive interactions, but suppressed by repulsive interactions, consistent with the analytic result. (Kane and Fisher, PRB 46, 15233 (1992))

• The Fermi velocity is enhanced by repulsive interactions and suppressed by attractive interactions

Vbias = 6.25 x 10-2 w

Echo time from the boundary

The current grows faster in the attractive interaction case

Vbias = 6.25 x 10-2 w

Nonlinear response

V = -0.5w, L = 160, m = 1024

Summary

• The long-time behavior of a non-equilibrium system can be accurately determined by extending the density matrix to include the information of time evolution of the ground state wavefunction

• With increasing m, this method converges slower than the adaptive DMRG method. But unlike the latter approach, this method can be used for any system.