Generalized DMRG with Tree Tensor Network Sujay Ray (Int. PhD) Department of Physics, IISc Quantum Condensed Matter Journal Club 11/12/2018 Date: 11/12/2018

Outline Brief introduction of DMRG. Matrix Product State. Tensor network and Tree network. Generalized DMRG with TTNS. Application to problems. 11/12/2018 Generalized DMRG with Tree Tensor Network Date: 11/12/2018

Outline Brief introduction of DMRG. Matrix Product State. Tensor network and Tree network. Generalized DMRG with TTNS. Application to problems. 11/12/2018 Generalized DMRG with Tree Tensor Network Date: 11/12/2018

Brief introduction of DMRG Density Matrix Renormalization Group (DMRG) was introduced by Steve White in 1992. One of the dominating numerical techniques to simulate strongly correlated systems. It relies on exact diagonalization and numerical renormalization group ideas. It is widely used to study Fermionic spin chain problems in 1D Higher dimensional problems can be mapped to 1D chain and DMRG can be used. Generalized DMRG with Tree Tensor Network Date: 11/12/2018

Brief introduction of DMRG Block growth 2 sites basis with dim 4: For 3 sites: For i sites: Generalized DMRG with Tree Tensor Network Date: 11/12/2018

Brief introduction of DMRG Some basic concepts Reduced Density Matrix: U S V Singular Value Decomposition: Generalized DMRG with Tree Tensor Network Date: 11/12/2018

Brief introduction of DMRG DMRG process Hilbert space for single site has dimension d Block A Block B Keep creating Hamiltonian until blocks become larger than d x m. Obtain full spectrum and eigen- vectors of reduced density matrices. Truncate basis keeping m eigen- Vectors with largest eigenvalues. Rotate Hamiltonian and operators to new basis and reiterate the process. Generalized DMRG with Tree Tensor Network Date: 11/12/2018

Brief introduction of DMRG DMRG process Schmidt decomposition: Generalized DMRG with Tree Tensor Network Date: 11/12/2018

Outline Brief introduction of DMRG. Matrix Product State. Tensor network and Tree network. Generalized DMRG with TTNS. Application to problems. 11/12/2018 Generalized DMRG with Tree Tensor Network Date: 11/12/2018

Matrix product state Introduction The quantum state is described by: ith site L th site Local basis of dimension d The quantum state is described by: where Through a sequence of singular value decomposition we get: Generalized DMRG with Tree Tensor Network Date: 11/12/2018

Matrix product state Introduction Then we have the diagrammatic representation: Physical index Dummy index Generalized DMRG with Tree Tensor Network Date: 11/12/2018

Matrix product state Operations Some other forms: Dot product as sum over physical indices: Generalized DMRG with Tree Tensor Network Date: 11/12/2018

Matrix product state Matrix product Operators Most general MPO Generalized DMRG with Tree Tensor Network Date: 11/12/2018

Matrix product state DMRG revisited DMRG as variational problem: Minimize using Lagrange multiplier DMRG as variational problem: Considering in MPO language and after minimization: dimension: Generalized DMRG with Tree Tensor Network Date: 11/12/2018

Matrix product state DMRG revisited Finite size DMRG: Sweeping is important during sweeping we add one extra site to one block and shrink one site from other block. Right to left sweep MPS site by site optimization of M matrices are same. Both lead to the same algorithm. Left to right sweep Generalized DMRG with Tree Tensor Network Date: 11/12/2018

Outline Brief introduction of DMRG. Matrix Product State. Tensor network and Tree network. Generalized DMRG with TTNS. Application to problems. 11/12/2018 Generalized DMRG with Tree Tensor Network Date: 11/12/2018

Tensor network and Tree network Introduction What is a tree network? A network of tensors that has a tree structure. Bipartition Weight of the index Edges represent qudits. Canonical form of bipartition: When weight is set of Schmidt coefficient and bases are Schmidt bases. Generalized DMRG with Tree Tensor Network Date: 11/12/2018

Tensor network and Tree network Introduction : Generalized DMRG with Tree Tensor Network Date: 11/12/2018

Tensor network and Tree network Introduction Two random qudits are connected through O(log(n)) tensors in binary tree and through O(n) in MPS. Time required is a factor log(n)/n lower when Tree TN is used. Red dots represent atoms and red dots are interaction pattern. Generalized DMRG with Tree Tensor Network Date: 11/12/2018

Outline Brief introduction of DMRG. Matrix Product State. Tensor network and Tree network. Generalized DMRG with TTNS. Application to problems. 11/12/2018 Generalized DMRG with Tree Tensor Network Date: 11/12/2018

Generalized DMRG with TTNS Introduction How tree network can be related to generalized DMRG with z blocks? In the approach we are considering all the sites represent physical sites and entanglement is transferred through virtual bonds . Physical bond Virtual bond Generalized DMRG with Tree Tensor Network Date: 11/12/2018

Generalized DMRG with TTNS Long range correlation where Generalized DMRG with Tree Tensor Network Date: 11/12/2018

Generalized DMRG with TTNS Long range correlation Logarithmic scaling counteracts this exponential decay. Correlation scales polynomially with number of sites M. Generalized DMRG with Tree Tensor Network Date: 11/12/2018

Generalized DMRG with TTNS Separation of blocks At each site we can associate with z +1 indices. D: virtual dimension. d: physical dimension. Separation of states into z blocks: Basis of environment block i. State of site m. Generalized DMRG with Tree Tensor Network Date: 11/12/2018

Generalized DMRG with TTNS Network optimization Orthonormality: Network optimization: Minimise - Generalized DMRG with Tree Tensor Network Date: 11/12/2018

Generalized DMRG with TTNS Interaction Suppose two body interaction acting on site 7 and 15 : Network is contracted from leaves and working in the inward direction. Numerical effort: number of sites x Generalized DMRG with Tree Tensor Network Date: 11/12/2018

Generalized DMRG with TTNS Fermions In case of fermions: Interactions with local support turned into interactions with non local supports via Jordan –Wigner transformation. Suppose for two particle interaction: Generalized DMRG with Tree Tensor Network Date: 11/12/2018

Generalized DMRG with TTNS Fermions In case of fermions: Introducing an index pair tensor is made block diagonal. Generalized DMRG with Tree Tensor Network Date: 11/12/2018

Generalized DMRG with TTNS Fermions Fermionic number parity: Z can be moved to virtual bond: Generalized DMRG with Tree Tensor Network Date: 11/12/2018

Generalized DMRG with TTNS Fermions The calculation of effective Hamiltonian with supports on a few sites simplifies. Z is moved to virtual bond and only one Z remains. So it is sufficient for two site interaction to take only the path connecting the sites. Generalized DMRG with Tree Tensor Network Date: 11/12/2018

Outline Brief introduction of DMRG. Matrix Product State. Tensor network and Tree network. Generalized DMRG with TTNS. Application to problems. 11/12/2018 Generalized DMRG with Tree Tensor Network Date: 11/12/2018

Application to problems 2D Heisenberg model 4 x 4 4 x 4 6 x 6 Z =3 Z =4 Z =3 6 x 6 Z =4 Virtual bond dimension was kept fixed D=4 Generalized DMRG with Tree Tensor Network Date: 11/12/2018

Application to problems 2D interacting spinless fermions 4 x 4 6 x 6 J = 1 U = 0.5 N = 3 Periodic boundary condition assumed. Virtual bond dimension was kept fixed D=4 Generalized DMRG with Tree Tensor Network Date: 11/12/2018

Conclusion and outlook Tree tensor network state method is used to treat strongly correlated systems with long range interactions with an arbitrary coordination number z. This tree tensor network method has low computational cost compared to DMRG-based method. The long range correlation deviates from mean field value polynomially with distance in contrast to MPS states which deviates exponentially. Calculations done in numerical problems with fixed coordination number z but it can be varied from site to site and may give better results. Generalized DMRG with Tree Tensor Network Date: 11/12/2018

Thanks to Tanmoy Das, Kausik Ghosh, Adhip Agarwala, Abhas Mallick for valuable discussions. References: V. Murg and F. Verstraete, Ö. Legeza and R. M. Noack, PHYSICAL REVIEW B 82, 205105 (2010) Y.-Y. Shi, L.-M. Duan, and G. Vidal, PHYSICAL REVIEW A 74, 022320 (2006) Ö. Legeza and J. Solyom, Phys. Rev. B 68, 195116 (2003). F. Verstraete, J. I. Cirac, and V. Murg, Adv. Phys. 57, 143 (2008). F. Verstraete and J. Cirac, Phys. Rev. B 73, 094423 (2006). N. Schuch, M. M. Wolf, F. Verstraete, and J. I. Cirac, Phys. Rev. Lett. 100, 030504 (2008). A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States, Roman Orus. Generalized DMRG with Tree Tensor Network Date: 11/12/2018

Thank you Generalized DMRG with Tree Tensor Network Date: 11/12/2018

MPS MPS operations: Order of contraction is important: MPS Date: 11/12/2018 MPS Date: 11/12/2018