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Matrix Product States for frustrated spin chains, lattices with an extended Hilbert space and constrained models in 1D Natalia Chepiga Swiss National Science Foundation University of Amsterdam, The Netherlands 23 July 2019 Natalia Chepiga

  1. MPO construction For a given site j write all possible terms in the Hamiltonian: Five non-trivial entries in the MPO Look at the left and right basis in which the MPO is going to be written I ...Full... I I ... IJS x j − 1 I ... I S x I ... I j +1 I ... I I ...Full... I Natalia Chepiga (SNF, UvA) 23 July 2019 16 / 68

  2. MPO construction For a given site j write all possible terms in the Hamiltonian: Five non-trivial entries in the MPO Look at the left and right basis in which the MPO is going to be written Fill-in the matrix: I ...Full... I I 0 0 I ... IJS x S x 0 0 j − 1 j hS z JS x I ... I I j j S x I ... I j +1 I ... I I ...Full... I Natalia Chepiga (SNF, UvA) 23 July 2019 16 / 68

  3. MPO exercise: Heisenberg nearest-neighbor: � H = J S j · S j +1 j 1 � � � S + j S − j +1 + S − j S + + S z j S z = J (1) j +1 j +1 2 j J 1 − J 2 model: � � H J 1 − J 2 = J 1 S j · S j +1 + J 2 S j · S j +2 (2) j j Three-site interaction: � H = H J 1 − J 2 + J 3 [( S j · S j +1 )( S j +1 · S j +2 ) + h . c . ] (3) j Natalia Chepiga (SNF, UvA) 23 July 2019 17 / 68

  4. MPO answers: Heisenberg nearest-neighbor: 1 � � � S + j S + j S − j +1 + S − + S z j S z H = J (4) j +1 j +1 2 j   I . . . S − . . .   j   S + . . . H j =   j   S z . . .   j   J 2 S + 2 S − J JS z . I j j j Natalia Chepiga (SNF, UvA) 23 July 2019 18 / 68

  5. MPO answers: J 1 − J 2 model: � � H J 1 − J 2 = J 1 S j · S j +1 + J 2 S j · S j +2 (5) j j  I . . . . . .  S − . . . . . . j   S +  . . . . . .  j     S z . . . . . .   j H j =   . I . . . . .     . . I . . . .     . . . I . . .   J 1 2 S + J 1 J 1 S z J 2 2 S + J 2 J 2 S z . 2 S − 2 S − I j j j j j j Natalia Chepiga (SNF, UvA) 23 July 2019 19 / 68

  6. MPO answers: Three-site interaction: � H = H J 1 − J 2 + J 3 [( S j · S j +1 )( S j +1 · S j +2 ) + h . c . ] (6) j   I . . . . . . S − . . . . . .  i  S +  . . . . . .  i   S z . . . . . .   i H i = ,   J 2 2 I + J 3 Q + − J 3 Q − z . J 3 Q −− . . .     J 2 J 3 Q + z . J 3 Q ++ 2 I + J 3 Q + − . . .     J 3 Q + z J 3 Q − z J 2 I + J 3 Q zz . . . .   J 1 2 S + J 1 S + J 1 S z S z . 2 S − S − I i i i i i i i with S α = { S + where Q αβ i S β i + S β = S α i S α 2 , S − 2 , S z } √ √ i Natalia Chepiga (SNF, UvA) 23 July 2019 20 / 68

  7. DMRG / variational MPS Natalia Chepiga (SNF, UvA) 23 July 2019 21 / 68

  8. DMRG sweep Left-to-right: diag SVD Matrix Product Operators diag SVD Right-to-left: diag SVD diag SVD ... Natalia Chepiga (SNF, UvA) 23 July 2019 21 / 68

  9. DMRG sweep Left-to-right: Group the legs and treat this rank-8 tensor as a matrix Right-to-left: Natalia Chepiga (SNF, UvA) 23 July 2019 21 / 68

  10. DMRG sweep Left-to-right: diag SVD diag SVD Right-to-left: diag SVD diag SVD ... Natalia Chepiga (SNF, UvA) 23 July 2019 21 / 68

  11. DMRG sweep Left-to-right: diag SVD diag SVD Right-to-left: diag SVD diag SVD ... Natalia Chepiga (SNF, UvA) 23 July 2019 21 / 68

  12. DMRG sweep Left-to-right: diag SVD diag SVD Right-to-left: diag SVD diag SVD ... Natalia Chepiga (SNF, UvA) 23 July 2019 21 / 68

  13. DMRG sweep Left-to-right: diag SVD diag SVD diag SVD Right-to-left: diag SVD ... Natalia Chepiga (SNF, UvA) 23 July 2019 21 / 68

  14. DMRG sweep Left-to-right: diag SVD diag SVD diag SVD Right-to-left: diag SVD ... Natalia Chepiga (SNF, UvA) 23 July 2019 21 / 68

  15. DMRG sweep Left-to-right: diag SVD diag SVD diag SVD Right-to-left: diag SVD ... Natalia Chepiga (SNF, UvA) 23 July 2019 21 / 68

  16. DMRG sweep Left-to-right: diag SVD diag SVD diag SVD Right-to-left: diag SVD ... Natalia Chepiga (SNF, UvA) 23 July 2019 21 / 68

  17. DMRG sweep Left-to-right: diag SVD diag SVD diag SVD Right-to-left: diag SVD ... Natalia Chepiga (SNF, UvA) 23 July 2019 21 / 68

  18. Initial guess Product state Random state Infinite-size DMRG Natalia Chepiga (SNF, UvA) 23 July 2019 22 / 68

  19. Infinite-size DMRG diag SVD diag SVD ... Natalia Chepiga (SNF, UvA) 23 July 2019 22 / 68

  20. Infinite-size DMRG diag SVD diag SVD ... Natalia Chepiga (SNF, UvA) 23 July 2019 22 / 68

  21. Infinite-size DMRG diag SVD diag SVD ... Natalia Chepiga (SNF, UvA) 23 July 2019 22 / 68

  22. Infinite-size DMRG diag SVD diag SVD ... Natalia Chepiga (SNF, UvA) 23 July 2019 22 / 68

  23. Abelian symmetry A S B A Assign quantum numbers - labels to physical bonds of MPS Using fusion rules of the symmetry, find quantum numbers on auxiliary legs When local basis is sorted according to the quantum number of states, the MPS takes a block-diagonal form Natalia Chepiga (SNF, UvA) 23 July 2019 23 / 68

  24. Abelian symmetry. Examples A A A Natalia Chepiga (SNF, UvA) 23 July 2019 24 / 68

  25. Abelian symmetry. Examples A A A Natalia Chepiga (SNF, UvA) 23 July 2019 24 / 68

  26. Abelian symmetry. Examples A A A Natalia Chepiga (SNF, UvA) 23 July 2019 24 / 68

  27. Observables Natalia Chepiga (SNF, UvA) 23 July 2019 25 / 68

  28. Observables Nearest-neighbor correlations � Ψ | S i · S i +1 | Ψ � Natalia Chepiga (SNF, UvA) 23 July 2019 25 / 68

  29. Observables Nearest-neighbor correlations � Ψ | S i · S i +1 | Ψ � Natalia Chepiga (SNF, UvA) 23 July 2019 25 / 68

  30. Observables Nearest-neighbor correlations � Ψ | S i · S i +1 | Ψ � Natalia Chepiga (SNF, UvA) 23 July 2019 25 / 68

  31. Observables Nearest-neighbor correlations � Ψ | S i · S i +1 | Ψ � Natalia Chepiga (SNF, UvA) 23 July 2019 25 / 68

  32. Observables Nearest-neighbor correlations � Ψ | S i · S i +1 | Ψ � Natalia Chepiga (SNF, UvA) 23 July 2019 25 / 68

  33. Observables Nearest-neighbor correlations � Ψ | S i · S i +1 | Ψ � Natalia Chepiga (SNF, UvA) 23 July 2019 25 / 68

  34. Observables Nearest-neighbor correlations � Ψ | S i · S i +1 | Ψ � Natalia Chepiga (SNF, UvA) 23 July 2019 25 / 68

  35. Observables Nearest-neighbor correlations � Ψ | S i · S i +1 | Ψ � Natalia Chepiga (SNF, UvA) 23 July 2019 25 / 68

  36. Observables Nearest-neighbor correlations � Ψ | S i · S i +1 | Ψ � Natalia Chepiga (SNF, UvA) 23 July 2019 25 / 68

  37. Observables Nearest-neighbor correlations � Ψ | S i · S i +1 | Ψ � On-site measures � Ψ | O i | Ψ � or Natalia Chepiga (SNF, UvA) 23 July 2019 26 / 68

  38. Observables Nearest-neighbor correlations � Ψ | S i · S i +1 | Ψ � On-site measures � Ψ | O i | Ψ � or Long range correlations � Ψ | O i · O j | Ψ � Natalia Chepiga (SNF, UvA) 23 July 2019 26 / 68

  39. Dimerization transition in frustrated spin-1 chain in collaboration with Frederic Mila (EPFL) and Ian Affleck (UBC) Natalia Chepiga (SNF, UvA) 23 July 2019 26 / 68

  40. Spin chains Heisenberg Hamiltonian: � H = J 1 S i · S i +1 i Spin-1/2 chain is critical Spin-1 chain: finite bulk gap topologically non-trivial ground state spin-1/2 edge states Haldane, Phys. Lett. A 93 , 464 ’83 Affleck, Kennedy, Lieb, Tasaki, PRL 59 , 799 ’87 Kennedy J. Phys: Cond. Mat 2 , 5737 ’90 Natalia Chepiga (SNF, UvA) 23 July 2019 27 / 68

  41. Introduction Add biquadratic interaction: � J 1 S i · S i +1 + J b ( S i · S i +1 ) 2 H = i Critical spin-1 chains: WZW SU(2) 2 SU(3) Affleck, Nucl.Phys.B 265 , 409 ’86 F´ ath, S´ olyom, PRB 44 ,11836 ’91 Takhtajan-Babudjian Schollw¨ ock, Jolicœur, Garel, PRB 53 , 3304 ’96 L¨ auchli, Schmid, Trebst, PRB 74 , 144426 ’06 Natalia Chepiga (SNF, UvA) 23 July 2019 28 / 68

  42. The model Hamiltonian: � H = ( J 1 S i · S i +1 + J 2 S i − 1 · S i +1 ) i � + J 3 [( S i − 1 · S i )( S i · S i +1 ) + H . c . ] i Three-site term: Appears in next-to-leading order in the strong coupling expansion of the two-band Hubbard model Induces spontaneous dimerization Reduces to next-nearest-neighbor interaction for spin-1/2 Natalia Chepiga (SNF, UvA) 23 July 2019 29 / 68

  43. Motivation � H = ( J 1 S i · S i +1 + J 2 S i − 1 · S i +1 ) i � + J 3 [( S i − 1 · S i )( S i · S i +1 ) + H . c . ] i 1 0.8 1 st order 1 st order transition 0.6 between two topologically different phases 0.4 Kolezhuk, Roth, Scholw¨ ock, 0.2 PRL 77 , 5142 ’96 Kolezhuk, Scholw¨ ock, PRB 65 , 100401 ’01 0 Natalia Chepiga (SNF, UvA) 23 July 2019 30 / 68

  44. Motivation � H = ( J 1 S i · S i +1 + J 2 S i − 1 · S i +1 ) Generalization of the i Majumdar-Ghosh model: � + J 3 [( S i − 1 · S i )( S i · S i +1 ) + H . c . ] fully dimerized state is an i exact ground state at J 3 /J 1 = 1 / 6 1 0.8 1 st order Continuous WZW 0.6 SU(2) k =2 transition at J 3 /J 1 = 0 . 111 0.4 or 0.2 Michaud, Vernay, Manmana, Mila, PRL 108 , 127202 ’12 1/6 + 0 0.1 0.15 0 0.05 Natalia Chepiga (SNF, UvA) 23 July 2019 31 / 68

  45. Motivation � H = ( J 1 S i · S i +1 + J 2 S i − 1 · S i +1 ) There is a line in J 2 − J 3 i parameter space, at � + J 3 [( S i − 1 · S i )( S i · S i +1 ) + H . c . ] which the fully dimerized state is exact eigenstate i 1 First order phase 0.8 transition has to appear 1 st order between the Haldane and 0.6 dimerized phases E 0.4 x a c Michaud, Vernay, t or l y d i Manmana, Mila, PRL 108 , m e r i z 127202 ’12 e d l i n e 0.2 Wang, Furuya, Nakamura, Komakura, PRB 88 , 224419’13 1/6 + 0 0.1 0.15 0 0.05 Natalia Chepiga (SNF, UvA) 23 July 2019 32 / 68

  46. Phase diagram 1 c=1/2 0.8 0.6 or 0.4 0.2 c=3/2 0 0 0.05 0.1 0.15 The transition between the Haldane and dimerized phases is continuous WZW SU(2) 2 below and including at the end point The transition between the NNN-Haldane phase and the dimerized phase is in the Ising universality class Topological transition between the Haldane and NNN-Haldane phases is always first order Natalia Chepiga (SNF, UvA) 23 July 2019 33 / 68

  47. Phase diagram. Field theory 1 Effective Hamiltonian: c=1/2 0.8 H = H WZW + λ 1 (tr g ) 2 + λ 2 � J R · � J L 0.6 λ 2 < 0 Continuous SU(2) 2 or λ 2 = 0 End point 0.4 λ 2 > 0 First order Free boson and Ising fields: 0.2 tr g ∝ σ sin √ πθ c=3/2 √ 0 (tr g ) 2 ∝ ǫ − C 1 cos 4 πθ 0 0.05 0.1 0.15 √ J L · � � J R ∝ ǫ cos 4 πθ + C 2 ∂ x φ L ∂ x φ R Second order transition between the Haldane and dimerized phases occurs simultaneously in Ising and boson sectors . Far from the WZW critical end point the Ising and boson critical lines could split Natalia Chepiga (SNF, UvA) 23 July 2019 34 / 68

  48. Dimerization Local dimerization: D ( j, N ) = |� S j · S j +1 � − � S j − 1 · S j �| 2 2 2 1 1 1 0 0 0 0 0.1 0.1 0.2 0 0.1 Natalia Chepiga (SNF, UvA) 23 July 2019 35 / 68

  49. Ising transition. Dimerization Local dimerization: D ( j, N ) = |� S j · S j +1 � − � S j − 1 · S j �| Finite-size scaling of the middle-chain dimerization in log-log scale , The separatrix is associated 0.057 with the phase transition 0.0575 0.058 The slope corresponds to the 0.0585 critical exponent 0.059 Critical exponent in Ising chain: d = 1 / 8 Natalia Chepiga (SNF, UvA) 23 July 2019 36 / 68

  50. Ising transition. Dimerization Local dimerization: D ( j, N ) = |� S j · S j +1 � − � S j − 1 · S j �| Open boundary favors dimerization DMRG 2 Fit In the transverse-field Ising chain it is , 800 , equivalent to the applied boundary 1.5 magnetic field At the critical point the magnetization decays away from the edges as 1 200 400 600 800 σ ( x ) ∝ [ N sin( πj/N )] − 1 / 8 In spin-1 chain the dimerization decays away from the boundaries in the same way and with the same critical exponent Natalia Chepiga (SNF, UvA) 23 July 2019 37 / 68

  51. Conformal towers = fingerprints of a critical theory Ising critical theory is minimal model of CFT It is described by a finite number of primary fields Combining boundary CFT with DMRG we can probe all of them numerically! Natalia Chepiga (SNF, UvA) 23 July 2019 38 / 68

  52. Conformal towers from entanglement spectrum Transverse field Ising model: Ising CFT c=1/2 pbc: 0 + 1/2 obc: 0 + 1/2 3 3 1/2+6 6.5 4 3 3 0+6 6 2 2 1/2+5 5.5 3 2 2 0+5 5 2 2 1/2+4 4.5 2 2 degeneracy 2 0+4 4 1 1 1/2+3 3.5 2 1 1 0+3 3 1 1 1/2+2 2.5 1 1 1 0+2 2 1 1 1/2+1 1.5 1 1 1 1 1/2 0.5 1 1 1 0 0 0.1 0.1 0 0.2 0.3 0 1/2 1/16 0 0.2 0.3 primary f e lds 1/Ln(L) 1/Ln(L) +de sce ndants A.L¨ auchli, arxiv:1303.0441 Natalia Chepiga (SNF, UvA) 23 July 2019 39 / 68

  53. Energy excitation spectrum with DMRG Natalia Chepiga (SNF, UvA) 23 July 2019 40 / 68

  54. Excitation spectrum with DMRG/MPS 1 The excited state is the ’ground-state’ of the different symmetry sector Natalia Chepiga (SNF, UvA) 23 July 2019 41 / 68

  55. Excitation spectrum with DMRG/MPS 1 The excited state is the ’ground-state’ of the different symmetry sector 2 Conventional DMRG: Mixed states The ground-state is spoilt Heavy memory usage Natalia Chepiga (SNF, UvA) 23 July 2019 41 / 68

  56. Excitation spectrum with DMRG/MPS 1 The excited state is the ’ground-state’ of the different symmetry sector 2 Conventional DMRG: Mixed states No longer variational Heavy memory usage 3 MPS: Construct the lowest-energy state orthogonal to the previously constructed ones Time consuming Accumulation of the error Natalia Chepiga (SNF, UvA) 23 July 2019 41 / 68

  57. Excitation spectrum with DMRG/MPS 1 The excited state is the ’ground-state’ of the different symmetry sector 2 Conventional DMRG: Mixed states No longer variational Heavy memory usage 3 MPS: Construct the lowest-energy state orthogonal to the previously constructed ones Time consuming Accumulation of the error 4 MPS: Domain wall or special tensor (see Laurens’ talk) Translation invariant MPS Natalia Chepiga (SNF, UvA) 23 July 2019 41 / 68

  58. Excitation spectrum with DMRG/MPS 1 The excited state is the ’GS’ of the different symmetry sector 2 Conventional DMRG: Mixed states No longer variational & Heavy memory usage 3 MPS: Construct the lowest-energy state orthogonal to the previously constructed ones Time consuming & Accumulation of the error 4 MPS: Domain wall or special tensor (see Laurens’ talk) Translation invariant MPS There is a cheaper option: Sometimes it is sufficient to target multiple eigenstates of the effective Hamiltonian and keep track of the energies as a function of iterations [NC, Mila, Phys.Rev.B 96 , 054425 (2017)] Natalia Chepiga (SNF, UvA) 23 July 2019 41 / 68

  59. Effective Hamiltonian D D Approximate basis Hamiltonian Truncation and rotation of the basis The Hamiltonian is written in a truncated and rotated basis. This basis is selected for the ground state. Could this basis be suitable for other low-energy states? Natalia Chepiga (SNF, UvA) 23 July 2019 42 / 68

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