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

2. Acknowledgement Collaborators Reference Funding 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) P (arXiv: )
Funding
NSC, NCTS

3. Outline Introduction Entropy scaling Spin-1/2 XXZ model Summary
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
ℋ= ℋ 𝐴 ⊗ ℋ 𝐵
Product state
𝜓 = 𝜓 𝐴 | 𝜓 𝐵 〉
𝜓 = 0 |0〉
Entangled state
𝜓 = 1 √ |1〉 ≠ 𝜓 𝐴 | 𝜓 𝐵 〉

6. Reduced Density Matrix
Partition of the Hilbert space
ℋ= ℋ 𝐴 ⊗ ℋ 𝐵
Start from a pure state 𝜓
Trace out ℋ 𝐵 to get the reduce density matrix
𝜌 𝐴 = Tr 𝐵 𝜓 〈𝜓|
Product state  is pure
𝜓 = 0 0 → 𝜌 𝐴 = 0 〈0|
Entangled state  is mixed
1 √ |1〉 → 𝜌 𝐴 = 〈0|+ 1 〈1|

7. Entropy as a Measure of Entanglement
Entanglement entropy=von Neumann entropy
𝑆 1 =− Tr 𝐴 ( 𝜌 𝐴 log 𝜌 𝐴 )
Renyi entropy
𝑆 𝑛≥2 = 1 1−𝑛 ln Tr 𝐴 𝜌 𝐴 𝑛
𝑆 1 = lim 𝑛→1 𝑆 𝑛

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

9. Entanglement Entropy 𝜌 𝐴 = Tr 𝐵 | 𝜙 𝑔𝑠 𝜙 𝑔𝑠 | 𝑆 𝐴 =−Tr 𝜌 𝐴 log 𝜌 𝐴 𝑙 𝐿B A B 𝑙 A B 𝐿 𝜉 A B

10. Entanglement Entropy Scaling With Conformal Invariance
Periodic boundary condition (PBC)
Open boundary condition (OBC)
Off-critical spin chain with correlation length ξ
𝑆 1 𝑙,𝐿 = 𝑐 3 log 𝐿 𝜋 sin 𝜋𝑙 𝐿 + 𝑐 1 ′ → c 3 logL
𝑆 1 (𝑙,𝐿)= 𝑐 6 log 𝐿 𝜋 sin 𝜋𝑙 𝐿 + 𝑐 1 ′ +𝑔 → c 6 logL
𝑆 1 (𝜉)~ 𝑐 6 log 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. Entanglement Entropy Scaling
Spin-1/2 XXZ Model
Entanglement Entropy Scaling

13. Case 1: Spin-1/2 XXZ Model
𝐻= 𝑖=1 𝐿 𝑆 𝑖 + 𝑆 𝑖+1 − + 𝑆 𝑖 − 𝑆 𝑖 ∆ 𝑆 𝑖 𝑧 𝑆 𝑖+1 𝑧
Δ>+1: Neel phase
Δ<−1: Ferromagnetic Ising phase
−1<Δ≤+1: Gapless critical XY phase with c=1
U(1) symmetry
Unique ground state
𝑆 𝑧,𝑡𝑜𝑡 =0
Δ=−1: 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)
Lipkin-Meshkov-Glick model
José I. Latorre, Román Orús, Enrique Rico, Julien Vidal, Phys. Rev. A 71, (2005)
Permutation-invariant states (Ferromagnetic point)
Vladislav Popkov, Mario Salerno, PRA 71, (2005)
Olalla A. Castro-Alvaredo, Benjamin Doyon, JSTAT/2011/P02001
Olalla A. Castro-Alvaredo, Benjamin Doyon, PRL 108, (2012)
Vincenzo Alba, Masudul Haque, 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 𝑆 𝑧,𝑡𝑜𝑡 =0
Vladislav Popkov, Mario Salerno, PRA 71, (2005)
Olalla A. Castro-Alvaredo, Benjamin Doyon, JSTAT/2011/P02001d
DMRG: 𝑆 1 = 1 2 log 𝐿 → = 𝑐 3 log 𝐿 →𝑐= 3 2 >1
iDMRG: 𝑆 1 = 1 2 log 𝜉 → = 𝑐 6 log 𝜉 →𝑐=3>1
Fit 𝑆 1 𝑙,𝐿 = 𝑐 3 log 𝐿 𝜋 sin 𝜋𝑙 𝐿 + 𝑐 1 ′ to get c(m,L)

17. Finite-Size DMRG

18. iDMRG
𝜉 𝑐 𝐹 Δ

19. Identify CFT without Using Entanglement Scaling

20. Finite-Size Scaling of Ground and Excited States Energies
Finite-size correction of ground state energy
𝐸 𝑔 (𝐿) 𝐿 = 𝜀 ∞ − 𝜋𝑣 6 𝐿 2 𝑐
Finite-size correction of excited state energy
𝐸 𝑛 𝐿 − 𝐸 𝑔 𝐿 = 2𝜋𝑣 𝐿 𝑥 𝑛
Spin-wave velocity 𝑣= 𝜋 sin 𝜇 2𝜇 , ∆= cos 𝜇

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 𝑐= lim 𝐿→∞,𝑚→∞ 𝑐 (𝑚,𝐿)
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
𝑆 𝑛 (𝑙,𝐿)= 𝑐 𝑛 log 𝐿 𝜋 sin 𝜋𝑙 𝐿 + 𝑐 1 ′
𝑆 𝑛 (𝑙,𝐿)= 𝑐 𝑛 log 𝐿 𝜋 sin 𝜋𝑙 𝐿 + 𝑐 1 ′ +𝑔
𝑆 𝑛 (𝜉)~ 𝑐 𝑛 log 𝜉

27. Renyi Entropy Scaling of Permutation-Invariant States
Olalla A. Castro-Alvaredo, Benjamin Doyon, JSTAT/2011/P02001
CFT: 𝑆 𝑛 𝑙,𝐿 = 𝑐 𝑛 log 𝐿 𝜋 sin 𝜋𝑙 𝐿 ,𝑐=1
FM: 𝑆 𝑛 𝑙,𝐿 ∝ 1 2 log 𝐿
1 2 = 𝑐 𝑛 𝑛 ⇒ 𝑐 𝑛 =3 𝑛 𝑛+1
Renyi entropy scaling
Calculate 𝑆 𝑛 (𝑙,𝐿)
Fit CFT scaling to obtain 𝑐 𝑛 (𝐿)
Expect that 𝑐 𝑛 𝐿 →𝑐 as 𝐿→∞

28. Spin ½ XXZ Model, Δ=−0.5

29. Observations 𝑐 1 is monotonically decreasing
𝑐 𝑛≥2 are monotonically increasing
𝑐 𝑛 →1 as 𝐿→∞
𝑐 1 > 𝑐 2 > 𝑐 3 >⋯

30. Spin ½ XXZ Model, Δ=−0.9

31. Spin ½ XXZ Model, Δ=−0.99
𝑐 𝑛,𝑚𝑎𝑥
𝐿 𝑛,𝑐

32. Observations 𝑐 1 is monotonically decreasing 𝑐 𝑛≥2 𝑐 𝑛 →1 as 𝐿→∞
first increase to some maximal value 𝑐 𝑛,𝑚𝑎𝑥 at 𝐿 𝑛,𝑐
then decrease monotonically
𝑐 𝑛 →1 as 𝐿→∞
𝑐 𝑖𝑛𝑓 >⋯ >𝑐 3 > 𝑐 2 > 𝑐 1 for 𝐿> 𝐿 𝑛,𝑐

33. 𝑐 𝑛,𝑚𝑎𝑥 v.s. Δ
1 2 = 𝑐 𝑛 𝑛 ⇒ 𝑐 𝑛 =3 𝑛 𝑛+1

34. 𝜉 𝑐 , 𝐿 𝑛≥2,𝑐 v.s. Δ

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: 𝜉 𝑐 𝐹 > 𝜉 𝑐
When 𝜉< 𝜉 𝑐 𝐹  ceff from permutation symmetry
When 𝜉> 𝜉 𝑐 𝐹  c from CFT
Case-2: 𝜉 𝑐 𝐹 < 𝜉 𝑐
When 𝜉> 𝜉 𝑐  c from CFT
When 𝜉 𝑐 >𝜉> 𝜉 𝑐 𝐹  c' from some approximated CFT?

40. Measuring the Ferromagnetic Entanglement
When the critical system is close to the ferromagnetic boundary, the groundstate wavefunction 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