Transcript Shot noise spectrum of SN tunneling junction

2D and time dependent DMRG
1. Implementation of Real
space DMRG in 2D
2. Time dependent DMRG
Tao Xiang
Institute of Theoretical Physics
Chinese 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
superblock
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
How to map a 2D lattice to 1D?
T Xiang, J Z Lou, Z B Su, PRB 64, 104414 (2001)
Multi-chain mapping:
A 2D mapping:
The width of the lattice is fixed
Lattice grows in both directions
From a 2x2 to a 3x3 lattice
6
7
9
6
7
9
2
4
2
5
8
2
5
8
1
3
1
3
4
1
3
4
2x2
3x3
3x3
(a)
(b)
(c)
From 3x3 to 4x4 lattice
7
13
6
8
12
2
5
9
1
3
14
4
(a)
16
7
13
15
6
8
12
15
11
2
5
9
11
10
1
3
4
(b)
14
16
10
X
2
15
16
22
23
25
7
14
17
21
24
6
8
13
18
20
2
5
9
12
19
1
3
4
10
11
(a)
X
From 4x4 to 5x5
X
2
(b)
X
X
(c)
2
X
1
1
1
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 ?
Heisenberg model
-0 .3 61 2
6x6 square lattice
The limit m is
equivalent to the
limit the truncation
error   0
-0 .3 61 4
E
2d
-0 .3 61 6
-0 .3 61 8
-0.36 2
-0 .3 62 2
-7
10
10
-6
10
-5
0.0 0 0 1
T runcatio n E rro r
0.00 1
Converging Speed of DMRG
H eisen b erg m od el w ith free b o u n d ary con d ition s
E (m ) - E (30 0) p er b on d
0.01
EDMRG  EExact ~ e
0.001
 m
1/ 4
0.0 001
10
-5
E (m ) - E (3 0 0 ) p e r b o n d
T ru n E rro r
 decreases with increasing L
6
10
-6
8
10
12
10
-7
1.5
2
2.5
3
m
1 /4
3.5
4
4.5
Error vs truncation error
6x6 Heisenberg model with
periodic boundary conditions
0.03
True error is
approximately
proportional to the
truncation error
Error
0.02
0.01
0.00
0.0000
0.0002
0.0004
0.0006
0.0008
Truncation Error
0.0010
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
superblock
m
m
m
 0   wi si ei
i 1
its truncation error is exactly zero at every step of DMRG
iteration
• A right quantity for directly measuring the error is unknown but
required
Ground state energy of the 2D Heisenberg model
Extrapolation with respect to 1/L
Ground State Energy
Ground State Energy
Square Lattice
-0.34
-0.36
-0.38
Triangle Lattice
-0.18
-0.2
-0.22
-0.24
-0.26
-0.4
0
0.1
0.2
0.3
0.4
1/L
Square
Triangle
DMRG
-0.3346
-0.1814
MC
-0.334719
-0.1819
SW
-0.33475
-0.1822
0
0.1
0.2
0.3
1/L
Free boundary conditions
E(L) ~1/L
Periodic boundary conditions:
E(L) ~ (1/L)3
Staggered magnetization
S
A
 SB

2


 Stot  2 S A  S B  2ms M c
2
2
2
2
6 b y 6 fr ee b oun d ary co n d ition s
S taggered M agnetization
0.77
 N 2  4N


8
2
Mc  
2
 N  4N 1

8

0.7 65
line: M = M -  
0.76
st
 
0
0.7 55
N  even
N  odd
0.75
In an ideal Neel state, ms=1
independent on N
0.7 45
0.74
10
-7
10
-6
10
-5
0.0 001
T ru nc ation E rror
0.001
0.01
In the thermodynamic limit
ms 
2

S A  SB
N

2
2
S tagge red M agnetization m
s
Staggered magnetization vs 1/N
0.9
N = L2 square lattice
0.8
ms ~ 0.617 DMRG
0.7
0.6
0.615 QMC and
series expansion
0
0.0 2
0.0 4
0.06
0.607 spin-wave theory
1 /N
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 2D1D 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
formal solution
 t

 (t )  exp  i  dtH (t )  (t0 )
 t0

i t (t )  H (t ) (t )
many body effects + non-equilibrium
V
lead
lead
Quantum Dot
V
t0
t
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 Dot
tL
t
tR
t
H (t )  H L  H R  H d  HT  H v (t )
H L ,R   t

 c i c i 1  h .c . 

i L , R

H d   d c0 c0


H T   t ' c  1 c 0  t ' c 0 c1  h .c
H v ( t )   ( t )( N L  N R )
External bias term
Interaction representation
H (t )  H L  H R  H d  H T  H v (t )
i t (t )  H (t ) (t )
 c

i
H L , R  t
ci 1  h.c.
iL , R

H d   d c0 c0
H T  t '  c1c0  c0 c1   h.c

 (t )  e
i ( t )( N L  N R )
 (t )
t

H v (t )   (t )( N L  N R )
 (t )    d ( )

it  (t )  H '(t )  (t )
H '( t )  H L  H R  H d  H T ( t )
H T  t ' e
i ( t )
 c  1c 0  c 0 c1   h .c


Solution of the Schrodinger equation
it  (t )  H '(t )  (t )
t t

 (t  t )  exp i  d H '( )   (t )
 t

1 2 i 3
1 4


 (t  t )  e  (t )  1  iA  A  A 
A  ...   (t )
2
6
24


iA
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
 t

 (t )  exp  i  dtH trun (t )   (t0 )
 t0

(t  t )  1  iHtrun (t )t  ... (t )
Comparison with exact result
0.020
current J(t)
0.015
0.010
DMRG
Exact Solution
0.005
0.000
0
10
20
30
L = 64, M = 256
40
time t
50
60
70
The problem of the above approach
sys  Trenv  0  0
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
Pace-keeping DMRG
Luo, Xiang and Wang, PRL 91, 049701 (2003)
Nt
  Trenv   l 
| (tl ) (tl ) |
l 0
Nt

l
1
l 0
t0: start time of the bias
Nt: number of sampled points
Pace-keeping DMRG
sys
env
L/2
1.
Calculate the ground state wavefunction
0 and (t) in the whole time range
2.
Construct the reduced density matrix
L/2
Add two sites
Nt
 sys  Trenv  l 
| (tl ) (tl ) |
l 0
superblock
sys
env
L/2
L/2
3.
Truncate Hilbert space according to the
eigenvalues of the above extended density
matrix
Variation of the results with Nt
0.020
current J(t)
0.015
0.010
0.005
0.000
Free boundary
Nt = 0
Nt = 5
Nt = 30
Nt = 60
Exact evolution
0
10
20
30
L = 64, M = 256
40
50
60
Finite Size Effects
70
Echo time ~ 70
time t
Reflection current
Current
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?
Nt
  Trenv   l 
| (tl ) (tl ) |
l 0
0

l   1
N
 t
l 0
l0
Variation with the number of states retained
Real and complex density matrix
Complex reduced density matrix
Nt
  T renv   l 
| ( t l )   ( t l ) |
l0
real reduced density matrix
Nt
  R e T renv   l |  ( t l )   ( t l ) |
l0
Example 2: Tunneling junction between two Luttinger liquids (LL)
t
Luttinger liquid
t’
Junction
Luttinger liquid
t
H  H L  H R  H LL  HT  H v (t )
H L , R  t
 c

i
ci 1  h.c.
iL , R ,
H LL  V
 n
i
 12  ni 1  12 
iL , R
H T    t ' c1 c1  h.c.


H v (t )   (t )( N L  N R )
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.