1 / 40

Entanglement entropy scaling of the XXZ chain

Entanglement entropy scaling of the XXZ chain. Pochung Chen 陳柏中 National Tsing Hua University, Taiwan 10/14/2013, IWCSE, NTU. Acknowledgement. Collaborators Zhi -Long Xue (NTHU) Ian P. McCulloch (UQ, Australia) Ming-Chiang Chung (NCHU) Miguel Cazalilla (NTHU)

watson
Download Presentation

Entanglement entropy scaling of the XXZ chain

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Entanglement entropy scaling of the XXZ chain Pochung Chen 陳柏中 National TsingHua University, Taiwan 10/14/2013, IWCSE, NTU

  2. Acknowledgement • Collaborators • Zhi-Long Xue (NTHU) • Ian P. McCulloch (UQ, Australia) • Ming-Chiang Chung (NCHU) • Miguel Cazalilla (NTHU) • Chao-Chun Huang (IoP, Sinica) • Sung-Kit Yip (IoP, Sinica) • Reference • J. Stat. Mech. (2013) P10007. (arXiv:1306.5828) • Funding • NSC, NCTS

  3. Outline • Introduction • Entanglement, entropy, area law • Entropy scaling • Conformal field theory • Ferromagnetic point • Spin-1/2 XXZ model • Entanglement entropy scaling • Renyi entropy scaling • Summary

  4. Introduction

  5. Quantum Entanglement • Partition of the Hilbert space • Product state • Entangled state

  6. Reduced Density Matrix • Partition of the Hilbert space • Start from a pure state • Trace out to get the reduce density matrix • Product state  is pure • Entangled state  is mixed

  7. Entropy as a Measure of Entanglement • Entanglement entropy=von Neumann entropy • Renyi entropy

  8. Entanglement Area Law • Local Hamiltonian + Gapped ground state • Violation of area law • Logarithmic correction • Fermi surface • Conformal field theory • Permutation symmetry

  9. Entanglement Entropy B B A B A B A

  10. Entanglement Entropy Scaling With Conformal Invariance • Periodic boundary condition (PBC) • Open boundary condition (OBC) • Off-critical spin chain with correlation length ξ P. Calabrese and J. Cardy, JSTAT/2004/P06002

  11. DMRG for Entanglement Entropy Scaling SU(3) Heisenberg model M. Führinger, S. Rachel, R. Thomale, M. Greiter, P. Schmitteckert, Ann. Phys. 17, 922 (2008)

  12. Spin-1/2 XXZ Model Entanglement Entropy Scaling

  13. Case 1: Spin-1/2 XXZ Model • : Neel phase • : Ferromagnetic Ising phase • : Gapless critical XY phase with c=1 • U(1) symmetry • Unique ground state • : Ferromagnetic point • Hamiltonian has enlarged SU(2) symmetry • Infinite degenerate ground state • Particular ground state that is smoothly connected to the ground date in the critical XY phase

  14. Entanglement Entropy Scaling of Spin ½ XXZ Model -0.75 L=200 G. De Chiara, S. Montangero, P. Calabrese, R. Fazio, JSTAT/2006/P03001

  15. Entanglement Entropy Scaling Without Conformal Invariance • Spin chain with random interaction • G. Refael and J. E. Moore, J. Phys. A: Math. Theor. 42 (2009) 504010. • Lipkin-Meshkov-Glick model • José I. Latorre, Román Orús, Enrique Rico, Julien Vidal, Phys. Rev. A 71, 064101 (2005) • Permutation-invariant states (Ferromagnetic point) • Vladislav Popkov, Mario Salerno, PRA 71, 012301 (2005) • Olalla A. Castro-Alvaredo, Benjamin Doyon, JSTAT/2011/P02001 • Olalla A. Castro-Alvaredo, Benjamin Doyon, PRL 108,120401 (2012) • Vincenzo Alba, MasudulHaque, Andreas M Lauchli, JSTAT/2012/P08011 • Olalla A. Castro-Alvaredo, Benjamin Doyon, JSTAT/2013/P02016

  16. Entanglement Scaling of Permutation-Invariant States • Ground state at ferromagnetic point with • Vladislav Popkov, Mario Salerno, PRA 71, 012301 (2005) • Olalla A. Castro-Alvaredo, Benjamin Doyon, JSTAT/2011/P02001d • DMRG: • iDMRG: • Fit to get c(m,L)

  17. Finite-Size DMRG

  18. iDMRG

  19. Identify CFT without Using Entanglement Scaling

  20. Finite-Size Scaling ofGround and Excited States Energies • Finite-size correction of ground state energy • Finite-size correction of excited state energy • Spin-wave velocity

  21. Finite-Size Scaling of Ground State Energy

  22. Spin-Wave Velocity & Scaling Dimension

  23. Some Remarks • c(m,L) is a decreasing function of L • c(m,L) is an increasing function of m • True • Be careful about the error cancelation • Crossover behavior is observed in iDMRG • How to measure the ferromagnetic length scale?

  24. Spin-1/2 XXZ Model Renyi Entropy Scaling

  25. How to Measure the Entropy of a Finite System? • Not easy to measure entanglement entropy • Possible to measure Renyi entropy • Possible reconstruct entanglement entropy from Renyi entropy

  26. Renyi Entropy Scaling With Conformal Invariance • Periodic boundary condition (PBC) • Open boundary condition (OBC) • Off-critical spin chain with correlation length ξ

  27. Renyi Entropy Scaling of Permutation-Invariant States • Olalla A. Castro-Alvaredo, Benjamin Doyon, JSTAT/2011/P02001 • CFT: • FM: • Renyi entropy scaling • Calculate • Fit CFT scaling to obtain • Expect that as

  28. Spin ½ XXZ Model,

  29. Observations • is monotonically decreasing • are monotonically increasing • as

  30. Spin ½ XXZ Model,

  31. Spin ½ XXZ Model, 9

  32. Observations • is monotonically decreasing • first increase to some maximal value at • then decrease monotonically • as • for

  33. v.s.

  34. v.s.

  35. Renyi Entropy Scaling from IDRMG

  36. Rényi Entropy Scaling (Spin-1/2 XXZ)

  37. Rényi Entropy Scaling (Spin-1/2 XXZ)

  38. How to Determine the CFT? • Use all possible methods to extract c and make sure they are consistent with each other • Entanglement entropy scaling of finite system • Entanglement entropy scaling of infinite system • Finite-size scaling of ground state energy • Finite-size scaling of excited state energy • Energy spectrum from exact diagonalization • May have strong finite-size; finite-truncation effects, especially near ferromagnetic phase • May observe cross-over effects due to ferromagnetic phase

  39. Conformal Invariance v.s.Permutation Symmetry • Case-1: • When  ceff from permutation symmetry • When  c from CFT • Case-2: • When  ceff from permutation symmetry • When  c from CFT • When c' from some approximated CFT?

  40. Measuring theFerromagnetic Entanglement • When the critical system is close to the ferromagnetic boundary, the groundstatewavefunction looks "ferromagnetic" at small length scale • It is possible to detect this ferromagnetic length scale and the ferromagnetic scaling via measuring the Renyi entropy of a finite system • Clear signature in iDMRG calculation

More Related