1. 2D and time dependent DMRG
Implementation of Real space DMRG in 2D
Time dependent DMRG
Tao Xiang
Institute of Theoretical Physics
Chinese Academy of Sciences

2. Extension of the DMRG in 2D
Direct extension of the real space DMRG in 2D
Momentum space DMRG: (T. Xiang, PRB 53, (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

3. 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

4. How to map a 2D lattice to 1D?
T Xiang, J Z Lou, Z B Su, PRB 64, (2001)
Multi-chain mapping:
The width of the lattice is fixed
A 2D mapping:
Lattice grows in both directions

5. From a 2x2 to a 3x3 lattice

6. From 3x3 to 4x4 lattice

7. From 4x4 to 5x5

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

9. Comparison of the ground state energy: multichain versus 2d mapping
Symmetry of the total spin S2 is considered

10. 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 

11. How to extrapolate the result to the limit m ?
6x6 square lattice
Heisenberg model
The limit m is equivalent to the limit the truncation error   0

12. Converging Speed of DMRG
 decreases with increasing L

13. Error vs truncation error
6x6 Heisenberg model with periodic boundary conditions
True error is approximately proportional to the truncation error

14. 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

15. Ground state energy of the 2D Heisenberg model
Extrapolation with respect to 1/L
Square Lattice
Triangle Lattice
Square Triangle
DMRG MC SW
Free boundary conditions
E(L) ~1/L
Periodic boundary conditions:
E(L) ~ (1/L)3

16. Staggered magnetization
In an ideal Neel state, ms=1 independent on N
In the thermodynamic limit

17. Staggered magnetization vs 1/N
N = L2 square lattice
ms ~ DMRG
QMC and series expansion
spin-wave theory
For triangular lattice, the DMRG result of the staggered magnetization is poor

18. 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

19. How to solve time dependent problems in highly correlated systems?
2. Time dependent DMRG
How to solve time dependent problems in highly correlated systems?
pace-keeping DMRG
Adaptive DMRG
(S R White, U Schollwock)

20. many body effects + non-equilibrium
Physical background
formal solution
many body effects + non-equilibrium
lead
Quantum Dot V V t t0

21. Possible methods for solving this problem
closed time path Green’s function method
solve Lippmann-Schwinger equation (t)
solve directly the Schrodinger equation using the density-matrix renormalization group

22. Example: tunneling current in a quantum dot systemtL tR
External bias term

23. Interaction representation

24. Solution of the Schrodinger equation

25. Straightforward extension of the DMRG
Cazalilla and Marston, PRL 88, (2002)
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

26. Comparison with exact result

27. The problem of the above approach
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

28. Pace-keeping DMRG t0: start time of the bias
Luo, Xiang and Wang, PRL 91, (2003)
t0: start time of the bias
Nt: number of sampled points

29. Pace-keeping DMRG
sys
env
L/2
Calculate the ground state wavefunction 0 and (t) in the whole time range
Construct the reduced density matrix
Truncate Hilbert space according to the eigenvalues of the above extended density matrix
Add two sites
superblock
sys
env
L/2
L/2

30. Variation of the results with Nt
Free boundary
Finite Size Effects
Echo time ~ 70
Reflection current
Current

31. Length and time dependence of the tunneling current
Exact result

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

33. Variation with the number of states retained

34. Real and complex density matrix
Complex reduced density matrix
real reduced density matrix

35. Example 2: Tunneling junction between two Luttinger liquids (LL)
V: interaction in the LL

36. 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, (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

37. The current grows faster in the attractive interaction case
Vbias = 6.25 x 10-2 w

38. Nonlinear response
V = -0.5w, L = 160, m = 1024

39. 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.