Entanglement entropy scaling of the XXZ chain

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