Thermal entanglement close to a Quantum phase transition

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Thermal entanglement close to a Quantum phase
transition
Luigi AmicoMATIS INFM DMFCI Università di
Catania
Superconductivity Mesoscopics Theory
group
Materials and Technologies for Information and
communication Sciences
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• In collaboration with
• G. Falci (MATIS-DMFCI,Catania)
• R. Fazio (NEST-SNS, Pisa)
• Osterloh (ITP, Hannover)
• D. Patane (DMFCI, Catania)

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Outline

• Quantum phase transitions.
• Ising model in a trasverse field.
• Entanglement QPT
• Summary at T0
• In collaboration with G. Falci (Catania), R.
  Fazio (Pisa),
• A. Osterloh
  (Hannover).
• Thermal entanglement close to QPT In
  collaboration with D. Patanè (Catania).
• Acknoledgement. A Fubini.

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Phase Transitions
Thermal fluctuations
Quantum fluctuations
Es controlled by a parameter in the Hamiltonian
Universality class
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1d-Anisotropic XY models

• Exactly solved
• Correlation functions accessible

Lieb, Schulz, Mattis Ann. Phys.NY 16, 407 (1961)
Barouch, McCoy, Dresden PRA 2, 1075 (1970)
Barouch, McCoy PRA 3, 786
(1971) Pfeuty Ann.
Phys.NY 57, 79 (1970)
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Cross-over phase diagram for the quantum Ising
models
T
T0
ac1
a
Ordered phase
Chakravarty, Halperin, Nelson 1989 Chubukov,
Sachdev, Ye (1994) Sachdev 1996 Kopp, Chakravarty
(2005)
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Aim We analize the Entanglement close to QPT
Possible questions Correlation Vs
Entanglement ?
Critical properties ?
Universality ?
Quantum computation ?
Preskill 2000 Arnesen, Bose, Vedral
2001 Gunlicke, Bose, Kendon, Vedral 2001
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Used Entanglement measures

• One-tangle
• von Neumann Entropy
• E-Tr r1 log2 r1 ? 4 det r1

Bennet et al. 1996. Coffman, Kundu,Wootters
2000. Osterloh, Siewert 2004.

• Concurrence

C(R) maxl1- l2 - l3 - l4 ,0
are square roots of the eigenvalues of
Wootters 1998.
Pairwise entanglement
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Quantum Phase Transitions

• T0 phase transition driven by a coupling
  constant ?
• Correlation length diverges as x g - gc-n
  ?1
• log-divergence of the bipartite entanglement
  ?finite size scaling

Osterloh, Amico, Falci, Fazio Nature 416, 608
(2002)
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Summary at T0

• Critical change of Concurrence at the quantum
  critical point
• Finite size scaling
• In general NO long range Concurrence.
• Range NOT UNIVERSAL

QPT
Osterloh, Amico, Falci e Fazio, Nature (2002)

• Exsistence of a non trivial factorizing field
• Ground state factorized in the real space for

R1, g1
short ranged!
The One tangle is zero
Kurman et al., Physica A 112 (1982) Roscilde,Verru
cchi,Fubini,Haas, Tognetti 2004 Fubini,
Roscilde, Tognetti, Tusa, Verrucchi, 2005
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Value of the minimum
Must be lt -1
Position of the minimum
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Coffman-Kundu-Wootters
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Non-universal Entanglement range
asymptotics ng ? const.

1/n
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(No Transcript)
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Factorized ground state
The One tangle is zero
Kurman et al., Physica A 112 (1982) Roscilde,Verru
cchi,Fubini,Haas, Tognetti 2004 Fubini,
Roscilde, Tognetti, Tusa, Verrucchi, 2005
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T
TTcross
T0
Vidal, et al., Phys. Rev. Lett.90
(2003) Verstraete, et al. Phys. Rev. Lett
2004 Calabrese, Cardy JSTAT 2004 Its, Jin,
Korepin 2004 Roscilde et al., Phys. Rev. Lett. 93
(2004) Somma, Ortiz, Barnum, Knill, Viola 2004
a
Osterloh, Amico, Falci e Fazio, Nature 416
(2002) Osborne, Nielsen PRA 2002
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Main results at T?0
Scaling of entanglement close to the quantum
critical point Phase diagram of the
(pairwise) entanglement Entanglement
Quantum Criticality
Amico, Patanè 2005
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Critical change of the concurrence.
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Scaling of the Entanglement at finite T
R1, g1
T
Energy scale
QPT, Finite T
Sx
Dx
Data Collapse
Does the entanglement show a scaling behaviour
at finite temperature ?
a
Scaling ansatz
Raw data funct(T)
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Universality
g0.5
Data Collapse with the same critical
indices 0lt?1
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Phase diagram of the concurrence
R1
C0
g0.5
g0.3
g0.1
The parallel concurrence is suppressed by
decreasing the anisotropy
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g0.5
R1
R3
R4
Non monotonous switch from parallel-entanglement
to antiparallel-entanglement . This is due to
certain oscillation of correlation functions
(Barouch, McCoy 1971)
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Non monotonous switch from parallel-entanglement
to antiparallel-entanglement
antiparallel
parallel
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Entanglement Vs Quantum Critical
Range Entanglement ------------------------------
--------------------------------------------------
-- Correlation lenght
Quantum Critical
g1
Sachdev, Quantum Phase Transition, (Cambridge
2000)
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How is the entanglement affected by the
combination of thermal and quantum fluctuations ?
Anomalies at the gait to the quantum critical
region
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Conclusions
? Finite temperature Scaling of
Entanglement Derivative of the concurrence
scales keeping the universal features of the
quantum Ising model
? Phase diagram of pairwise Entanglement
Reentrant behaviour of the Parallel/Antiparalle
l Entanglement

• ? Entanglement Quantum Critical matter
• Entangled droplets
• Anomalies of the pairwise entanglement at the
• cross-over temperature

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Procedimento
R
simmetrie di H
C(R)
parte puramente quantistica delle f. di
correlazione
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Concurrence
misura di entanglement per coppie di spin-1/2

• stati puri

• stati misti

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Entanglement Vs Quantum Critical
Range Entanglement ------------------------------
--------------------------------------------------
-- Lunghezza di Correlazione
g0.7
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Scaling dell Entanglement
R2, g1
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Correlazione Classica
Stato Misto
Misura di B
Stati separabili
A e B correlati classicamente