2d and time dependent dmrg 1.implementation of real space dmrg in 2d 2.time dependent dmrg tao xiang...
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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 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
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
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
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.