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

• Product state

• Entangled state

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–

Entropy as a Measure of Entanglement

• Entanglement entropy=von Neumann entropy

• Renyi entropy

Entanglement Area Law

• Local Hamiltonian + Gapped ground state

• Violation of area law– Logarithmic correction– Fermi surface– Conformal field theory– Permutation symmetry

Entanglement Entropy

A

𝑆 𝐴=−Tr (𝜌𝐴 log 𝜌 𝐴 ) 𝜌𝐴=Tr𝐵¿

B

A BB

𝑙

𝐿

A B

𝜉

Entanglement Entropy Scaling With Conformal Invariance

• Periodic boundary condition (PBC)

• Open boundary condition (OBC)

• Off-critical spin chain with correlation length ξ

𝑆1 (𝑙 ,𝐿 )=𝑐3log ( 𝐿𝜋 sin 𝜋 𝑙𝐿 )+𝑐1′→ c

3logL

𝑆1(𝑙 ,𝐿)=𝑐6log( 𝐿𝜋 sin 𝜋 𝑙𝐿 )+𝑐1′ +𝑔→c

6logL

𝑆1(𝜉 )𝑐6log (𝜉 ) P. Calabrese and J. Cardy, JSTAT/2004/P06002

DMRG for Entanglement Entropy Scaling

M. Fuhringer, S. Rachel, R. Thomale, M. Greiter, P. Schmitteckert, Ann. Phys. 17, 922 (2008)

SU(3) Heisenberg model

Spin-1/2 XXZ Model

Entanglement Entropy Scaling

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

Entanglement Entropy Scaling of Spin ½ XXZ Model

G. De Chiara, S. Montangero, P. Calabrese, R. Fazio, JSTAT/2006/P03001

L=200-0.75

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 • 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)

Finite-Size DMRG

iDMRG

𝜉𝑐𝐹 (Δ )

Identify CFT without Using Entanglement Scaling

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

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 • 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)

• Open boundary condition (OBC)

• Off-critical spin chain with correlation length ξ

𝑆𝑛(𝑙 ,𝐿)=𝑐6 (1+ 1𝑛 ) log( 𝐿𝜋 sin 𝜋 𝑙𝐿 )+𝑐1′

𝑆𝑛(𝑙 ,𝐿)= 𝑐12 (1+ 1𝑛 ) log( 𝐿𝜋 sin 𝜋 𝑙𝐿 )+𝑐1′ +𝑔

𝑆𝑛(𝜉 ) 𝑐12 (1+ 1𝑛 ) log (𝜉 )

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

Spin ½ XXZ Model,

Observations

• is monotonically decreasing• are monotonically increasing• as •

Spin ½ XXZ Model,

Spin ½ XXZ Model, 9

𝐿𝑛 , 𝑐

𝑐𝑛 ,𝑚𝑎𝑥

Observations

• is monotonically decreasing• – first increase to some maximal value at – then decrease monotonically

• as • for

v.s.

12=𝑐𝑛6 (1+ 1𝑛 )⇒𝑐𝑛=3

𝑛𝑛+1

v.s.

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 theFerromagnetic 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