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Introduction to Dynamic DMRG Methods S. Ramasesha Solid State and Structural Chemistry Unit

Introduction to Dynamic DMRG Methods S. Ramasesha Solid State and Structural Chemistry Unit Indian Institute of Science Bangalore 560 012 Collaborators Zoltan G. Soos Swapan Pati Zhigang Shuai Tirthankar Dutta H.R. Krishnamurthy Institute for Mathematical Sciences

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Introduction to Dynamic DMRG Methods S. Ramasesha Solid State and Structural Chemistry Unit

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  1. Introduction to Dynamic DMRG Methods S. Ramasesha Solid State and Structural Chemistry Unit Indian Institute of Science Bangalore 560 012 Collaborators Zoltan G. Soos Swapan Pati Zhigang Shuai Tirthankar Dutta H.R. Krishnamurthy Institute for Mathematical Sciences Chennai, March 19-21 2012.

  2. Dynamic response to external perturbations Response can be viewed as - a function of frequency or - a function of time. The two are related but, more accurate to compute them separately Unperturbed Hamiltonian is an Interacting Hamiltonian In Physics – Hubbard Hamiltonian, Heisenberg Spin Hamiltonians and their many variants. In Chemistry – Long range interacting models like Pariser-Parr-Pople (PPP) Model or restricted Configuration Interaction (CI) matrices like single CI, singles and doubles CI etc.

  3. To test the technique, we compare the rotationally averaged linear polarizability and THG coefficient Computed at  = 0.1t model exact valuesfor a Hubbard chain of 12 sites at U/t=4 compared with DMRG computation with m=200 The dominant xx) is 14.83 (exact) and 14.81 (DMRG) and xxxx) 2873 (exact) and 2872 (DMRG).  in 10-24 esu and  in 10-36 esu in all cases

  4. (a) (b) THG coefficient in Hubbard models as a function of chain length, L and dimerization : Superlinear behavior diminishes both with increase in U/t and increase in .

  5. gav.vs Chain Length and d in U-V Model For U > 2V, (SDW regime)av. shows similar dependence on L as the Hubbard model, independent of d. U=2V (SDW/CDW crossover point) Hubbard chains have larger av. than the U-V chains PRB, 59, 14827 (1999).

  6. Time evolution operator: U(0,t) = exp[-iHt/ħ] • Discretized unitary form of time evolution is U (t, t+t)  t 2ħ [1 - iH ] [1 + iH ] t 2ħ t 2ħ t 2ħ iH iH • Time evolution of (t) by t is given by [1 + ] (t + t) = [1 - ] (t) • Expressing (t) in an appropriate basis (eg.Slater Determinants), r.h.s. can be converted to a vector b, with (t + t) being expressed as an unknown x, the above equation can be converted to a set of linear inhomogeneous algebraic equations Ax = b

  7. Multistep Differencing (MSD)Techniques MSD4: Fast - involves only one sparse matrix multiplication for time propagation. Time dependent quantities evaluated as <O(t)> = <(t)|O|(t)>. 19

  8. Full Hilbert-space 𝝍(0) 𝝍(0) DMRG-space for 𝝍(0) 𝝍(tp) 𝝍(tp) DMRG-space for 𝝍(tp) 𝝍(T) 𝝍(T) DMRG-space for 𝝍(T) DMRG space of 𝝍(0) (initial wave packet) adapted to follow the time evolving wave packet |𝝍(t)> td-DMRG method: Fundamental quantity in td-DMRG: weighted average reduced density matrix

  9. “Sliding window” pace-Keeping (LXW) td-DMRG algorithm Instead of retaining ALL time-dependent wave packets, retain ONLY ‘p’ of them (sliding time window) (each “sliding time window” has length 𝜟t = p𝜟τ) Computational time reduces compared to parent LXW scheme T. Dutta and SR ,Computing Letters, 3, 457 (2007).

  10. Time Step Targeting (TST) td-DMRG algorithm (Phys. Rev. B, 72, 020404, 2005) • Combination of infinite and finite-system DMRG algorithms; accuracy < LXW; computational time ≈ parent LXW scheme • One or several finite-system ½-sweeps are required to update Hilbert space for time step 𝜟t ; evolution time step = 𝜟τ = 𝜟t/p

  11. Double Time Window Targeting (DTWT) td-DMRG algorithm (our development;Phys. Rev. B, 82, 035115, 2010 ) • A hybrid of LXW and TST schemes, but at least twice as fast and more accurate than either • A completely generalized td-DMRG algorithm for any interacting one-dimensional system

  12. a) Pace-Keeping or LXW algorithm(Liu, Xiang, Wang) (PRL, 91, 049701, 2003) b) “Sliding window” LXW algorithm (Dutta, SR) (Comput. Lett., 3, 457, 2007 ) • Time-step targeting (TST) algorithm (Feiguin, White) (PRB, 72, 020404, 2005) d) Double time window targeting (DTWT) technique (Dutta, SR) (PRB, 82, 035115, 2010)

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