Transcript slides - iwcse 2013

Entanglement entropy scaling of
the XXZ chain
Pochung Chen 陳柏中
National Tsing Hua University, Taiwan
10/14/2013, IWCSE, NTU
Acknowledgement
• Collaborators
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–
–
–
–
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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
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
Introduction
Quantum Entanglement
• Partition of the Hilbert space
– ℋ = ℋ𝐴 ⊗ ℋ𝐵
• Product state
– 𝜓 = 𝜓𝐴 |𝜓𝐵 〉
– 𝜓 = 0 |0〉
• Entangled state
– 𝜓 =
1
√2
0 0 + 1 |1〉 ≠ 𝜓𝐴 |𝜓𝐵 〉
Reduced Density Matrix
• Partition of the Hilbert space
– ℋ = ℋ𝐴 ⊗ ℋ𝐵
• Start from a pure state
– 𝜓
• Trace out ℋ𝐵 to get the reduce density matrix
– 𝜌𝐴 = Tr𝐵 𝜓 〈𝜓|
• Product state  is pure
– 𝜓 = 0 0 → 𝜌𝐴 = 0 〈0|
• Entangled state  is mixed
–
1
√2
0 0 + 1 |1〉 → 𝜌𝐴 =
1
2
0 〈0| + 1 〈1|
Entropy as a Measure of Entanglement
• Entanglement entropy=von Neumann entropy
– 𝑆1 = −Tr𝐴 (𝜌𝐴 log 𝜌𝐴 )
• Renyi entropy
– 𝑆𝑛≥2 =
1
ln
1−𝑛
– 𝑆1 = lim 𝑆𝑛
𝑛→1
Tr𝐴 𝜌𝐴𝑛
Entanglement Area Law
• Local Hamiltonian + Gapped ground state
– 𝑆𝐴 ∝ 𝜕𝐴
• Violation of area law
– Logarithmic correction
– Fermi surface
– Conformal field theory
– Permutation symmetry
Entanglement Entropy
𝑆𝐴 = −Tr 𝜌𝐴 log 𝜌𝐴
B
𝜌𝐴 = Tr𝐵 |𝜙𝑔𝑠 𝜙𝑔𝑠 |
B
A
𝑙
B
A
𝐿
A
𝜉
B
Entanglement Entropy Scaling
With Conformal Invariance
• Periodic boundary condition (PBC)
𝑐
𝐿
𝜋𝑙
c
′
𝑆1 𝑙, 𝐿 = log
sin
+ 𝑐1 → logL
3
𝜋
𝐿
3
• Open boundary condition (OBC)
𝑐
𝐿
𝜋𝑙
c
′
𝑆1 (𝑙, 𝐿) = log
sin
+ 𝑐1 + 𝑔 → logL
6
𝜋
𝐿
6
• Off-critical spin chain with correlation length ξ
𝑐
𝑆1 (𝜉)~ log 𝜉
6
P. Calabrese and J. Cardy, JSTAT/2004/P06002
DMRG for
Entanglement Entropy Scaling
SU(3) Heisenberg model
M. Führinger, S. Rachel, R. Thomale, M. Greiter, P. Schmitteckert, Ann. Phys. 17, 922 (2008)
Spin-1/2 XXZ Model
Entanglement Entropy Scaling
Case 1: Spin-1/2 XXZ Model
• 𝐻=
𝐿 1
𝑖=1 2
+
−
𝑧
𝑆𝑖+ 𝑆𝑖+1
+ 𝑆𝑖− 𝑆𝑖+1
+ ∆𝑆𝑖𝑧 𝑆𝑖+1
– 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
• 𝑆𝑧,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
Entanglement Entropy Scaling
of Spin ½ XXZ Model
-0.75
L=200
G. De Chiara, S. Montangero, P. Calabrese, R. Fazio, JSTAT/2006/P03001
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)
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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, Masudul Haque, Andreas M Lauchli,
JSTAT/2012/P08011
– Olalla A. Castro-Alvaredo, Benjamin Doyon, JSTAT/2013/P02016
Entanglement Scaling
of Permutation-Invariant States
• Ground state at ferromagnetic point with 𝑆𝑧,𝑡𝑜𝑡 = 0
• Vladislav Popkov, Mario Salerno, PRA 71, 012301 (2005)
• Olalla A. Castro-Alvaredo, Benjamin Doyon,
JSTAT/2011/P02001d
–
–
1
DMRG: 𝑆1 = log 𝐿
2
1
iDMRG: 𝑆1 = log 𝜉
2
• Fit 𝑆1 𝑙, 𝐿 =
→
→
𝑐
= log
3
𝑐
= log
6
𝑐
𝐿
𝜋𝑙
log sin
3
𝜋
𝐿
3
2
𝐿 →𝑐= >1
𝜉 →𝑐=3>1
+ 𝑐1′ to get c(m,L)
Finite-Size DMRG
iDMRG
𝜉𝑐𝐹 Δ
Identify CFT without
Using Entanglement Scaling
Finite-Size Scaling of
Ground and Excited States Energies
• Finite-size correction of ground state energy
•
𝐸𝑔 (𝐿)
𝐿
= 𝜀∞ −
𝜋𝑣
𝑐
2
6𝐿
• Finite-size correction of excited state energy
• 𝐸𝑛 𝐿 − 𝐸𝑔 𝐿 =
2𝜋𝑣
𝑥𝑛
𝐿
• Spin-wave velocity 𝑣 =
𝜋 sin 𝜇
, ∆=
2𝜇
cos 𝜇
Finite-Size Scaling
of Ground State Energy
Spin-Wave Velocity & Scaling Dimension
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?
Spin-1/2 XXZ Model
Renyi Entropy Scaling
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
Renyi Entropy Scaling
With Conformal Invariance
• Periodic boundary condition (PBC)
𝑐
1
𝐿
𝜋𝑙
𝑆𝑛 (𝑙, 𝐿) =
1+
log
sin
+ 𝑐1′
6
𝑛
𝜋
𝐿
• Open boundary condition (OBC)
𝑐
1
𝐿
𝜋𝑙
𝑆𝑛 (𝑙, 𝐿) =
1+
log
sin
+ 𝑐1′ + 𝑔
12
𝑛
𝜋
𝐿
• Off-critical spin chain with correlation length ξ
𝑐
1
𝑆𝑛 (𝜉)~
1+
log 𝜉
12
𝑛
Renyi Entropy Scaling
of Permutation-Invariant States
• Olalla A. Castro-Alvaredo, Benjamin Doyon, JSTAT/2011/P02001
–
–
–
𝑐
1
𝐿
𝜋𝑙
CFT: 𝑆𝑛 𝑙, 𝐿 = 1 + log sin
6
𝑛
𝜋
𝐿
1
FM: 𝑆𝑛 𝑙, 𝐿 ∝ log 𝐿
2
1
𝑐𝑛
1
𝑛
=
1 + ⇒ 𝑐𝑛 = 3
2
6
𝑛
𝑛+1
• Renyi entropy scaling
• Calculate 𝑆𝑛 (𝑙, 𝐿)
• Fit CFT scaling to obtain 𝑐𝑛 (𝐿)
• Expect that 𝑐𝑛 𝐿 → 𝑐 as 𝐿 → ∞
,𝑐 = 1
Spin ½ XXZ Model, Δ = −0.5
Observations
•
•
•
•
𝑐1 is monotonically decreasing
𝑐𝑛≥2 are monotonically increasing
𝑐𝑛 → 1 as 𝐿 → ∞
𝑐1 > 𝑐2 > 𝑐3 > ⋯
Spin ½ XXZ Model, Δ = −0.9
Spin ½ XXZ Model, Δ = −0.99
𝑐𝑛,𝑚𝑎𝑥
𝐿𝑛,𝑐
Observations
• 𝑐1 is monotonically decreasing
• 𝑐𝑛≥2
– first increase to some maximal value 𝑐𝑛,𝑚𝑎𝑥 at 𝐿𝑛,𝑐
– then decrease monotonically
• 𝑐𝑛 → 1 as 𝐿 → ∞
• 𝑐𝑖𝑛𝑓 > ⋯ > 𝑐3 > 𝑐2 > 𝑐1 for 𝐿 > 𝐿𝑛,𝑐
𝑐𝑛,𝑚𝑎𝑥 v.s. Δ
1 𝑐𝑛
1
𝑛
=
1+
⇒ 𝑐𝑛 = 3
2
6
𝑛
𝑛+1
𝜉𝑐 , 𝐿𝑛≥2,𝑐 v.s. Δ
Renyi Entropy Scaling
from IDRMG
Rényi Entropy Scaling (Spin-1/2
XXZ)
Rényi Entropy Scaling (Spin-1/2 XXZ)
How to Determine the CFT?
• Use all possible methods to extract c and make
sure they are consistent with each other
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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
Conformal Invariance v.s.
Permutation Symmetry
• Case-1: 𝜉𝑐𝐹 > 𝜉𝑐
– When 𝜉 < 𝜉𝑐𝐹  ceff from permutation symmetry
– When 𝜉 > 𝜉𝑐𝐹  c from CFT
• Case-2: 𝜉𝑐𝐹 < 𝜉𝑐
– When 𝜉 < 𝜉𝑐𝐹  ceff from permutation symmetry
– When 𝜉 > 𝜉𝑐  c from CFT
– When 𝜉𝑐 > 𝜉 > 𝜉𝑐𝐹  c' from some approximated CFT?
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