slides - iwcse 2013

Report
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)
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 = ℋ ⊗ ℋ
• Product state
–  =  | 〉
–  = 0 |0〉
• Entangled state
–  =
1
√2
0 0 + 1 |1〉 ≠  | 〉
Reduced Density Matrix
• Partition of the Hilbert space
– ℋ = ℋ ⊗ ℋ
• Start from a pure state
– 
• Trace out ℋ to get the reduce density matrix
–  = Tr  〈|
• Product state  is pure
–  = 0 0 →  = 0 〈0|
• Entangled state  is mixed
–
1
√2
0 0 + 1 |1〉 →  =
1
2
0 〈0| + 1 〈1|
Entropy as a Measure of Entanglement
• Entanglement entropy=von Neumann entropy
– 1 = −Tr ( log  )
• Renyi entropy
– ≥2 =
1
ln
1−
– 1 = lim 
→1
Tr 
Entanglement Area Law
• Local Hamiltonian + Gapped ground state
–  ∝ 
• Violation of area law
– Logarithmic correction
– Fermi surface
– Conformal field theory
– Permutation symmetry
Entanglement Entropy
 = −Tr  log 
B
 = Tr |  |
B
A

B
A

A

B
Entanglement Entropy Scaling
With Conformal Invariance
• Periodic boundary condition (PBC)



c
′
1 ,  = log
sin
+ 1 → logL
3


3
• Open boundary condition (OBC)



c
′
1 (, ) = log
sin
+ 1 +  → logL
6


6
• Off-critical spin chain with correlation length ξ

1 ()~ log 
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
• =
 1
=1 2
+
−

+ +1
+ − +1
+ ∆ +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
• , = 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)
–
–
–
–
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 , = 0
• Vladislav Popkov, Mario Salerno, PRA 71, 012301 (2005)
• Olalla A. Castro-Alvaredo, Benjamin Doyon,
JSTAT/2011/P02001d
–
–
1
DMRG: 1 = log 
2
1
iDMRG: 1 = log 
2
• Fit 1 ,  =
→
→

= log
3

= log
6



log sin
3


3
2
 →= >1
 →=3>1
+ 1′ to get c(m,L)
Finite-Size DMRG
iDMRG
 Δ
Identify CFT without
Using Entanglement Scaling
Finite-Size Scaling of
Ground and Excited States Energies
• Finite-size correction of ground state energy
•
 ()

= ∞ −


2
6
• Finite-size correction of excited state energy
•   −   =
2


• Spin-wave velocity  =
 sin 
, ∆=
2
cos 
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  =
lim
→∞,→∞
 (, )
• 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)

1


 (, ) =
1+
log
sin
+ 1′
6



• Open boundary condition (OBC)

1


 (, ) =
1+
log
sin
+ 1′ + 
12



• Off-critical spin chain with correlation length ξ

1
 ()~
1+
log 
12

Renyi Entropy Scaling
of Permutation-Invariant States
• Olalla A. Castro-Alvaredo, Benjamin Doyon, JSTAT/2011/P02001
–
–
–

1


CFT:  ,  = 1 + log sin
6



1
FM:  ,  ∝ log 
2
1

1

=
1 + ⇒  = 3
2
6

+1
• Renyi entropy scaling
• Calculate  (, )
• Fit CFT scaling to obtain  ()
• Expect that   →  as  → ∞
, = 1
Spin ½ XXZ Model, Δ = −0.5
Observations
•
•
•
•
1 is monotonically decreasing
≥2 are monotonically increasing
 → 1 as  → ∞
1 > 2 > 3 > ⋯
Spin ½ XXZ Model, Δ = −0.9
Spin ½ XXZ Model, Δ = −0.99
,
,
Observations
• 1 is monotonically decreasing
• ≥2
– first increase to some maximal value , at ,
– then decrease monotonically
•  → 1 as  → ∞
•  > ⋯ > 3 > 2 > 1 for  > ,
, v.s. Δ
1 
1

=
1+
⇒  = 3
2
6

+1
 , ≥2, 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
–
–
–
–
–
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:  > 
– 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?
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

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