2D and time dependent DMRG

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, 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

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, 104414 (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. Heisenberg model 6x6 square lattice How to extrapolate the result to the limit m ? 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. superblock m m 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

15. Square Lattice Triangle Lattice Ground state energy of the 2D Heisenberg model Extrapolation with respect to 1/L 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

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 ~ 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

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

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. Quantum Dot tL tR t t Example: tunneling current in a quantum dot system External bias term

23. Interaction representation

24. Solution of the Schrodinger equation

25. Straightforward extension of the DMRG Cazalilla and Marston, PRL 88, 256403 (2002) • Run DMRG to determine the ground state wavefunction ψ0, the truncated Hamiltonian Htrunand 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 Luo, Xiang and Wang, PRL 91, 049701 (2003) t0: start time of the bias Nt: number of sampled points

29. sys env L/2 L/2 Pace-keeping DMRG • 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. Reflection current Current Variation of the results with Nt Free boundary Finite Size Effects Echo time ~ 70

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

32. How does the result depend on the weight α0of 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) Luttinger liquid Luttinger liquid t t’ t Junction 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, 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

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.