Bell-paired states inducing volume law for Entanglement Entropy in - - PowerPoint PPT Presentation

Bell-paired states inducing volume law for Entanglement Entropy in fermionic lattices

Giacomo Gori (SISSA) Auditya Sharma (Tel Aviv University) Pasquale Sodano (IIP , Natal) Andrea Trombettoni (SISSA)

in collaboration with:

• G. Gori et al. PRB 91, 245138 (2015)

Simone Paganelli

Outlook

• Entanglement entropy in many-body systems: area law
• Free fermion’s entanglement entropy
• Bell-paired states, nontrivial Fermi surface and violation of the area-law
• Examples
• Conclusions

Black holes, holographic principle Analogy with Bekenstein-Hawking area law: a black hole carries an entropy that is proportional to its horizon area A.

Relevance of the EE and its scaling

Efficiency of simulations Efficiency of 1D algorithms such as DMRG is based on the Area law behaviour

(U. Schollwöck, Rev. Mod. Phys. 77 259 (2005))

Characterization of quantum phases Universal behavior near critical points. Indicator of phases that cannot be described by local

• rder parameters (informations about central charge, topological charge, Kondo length …)

(X. G. Wen, Phys. Rev. B 40 7387 (1989)), (E. Witten, Adv. Theory Math. 2 253 (1998)), (E. Nussinov et al. , Ann. Phys. 322 977 (2009))

Nonequilibrium dynamics, quenches Propagation of excitation after a quench

SA(t) ∝ ∂(A)

• Measure of the entanglement of a pure state
• Bipartition into two subsystems: and

A ¯ A SA = −TrA(ρA ln ρA) ρA = Tr ¯

Aρ

SA = 0

In QM positive entropies may arise in a subsystem even at T=0. Entanglement produces a loss of information if one observes only a subsystem

SA 6= 0

Entanglement Entropy (EE)

ρ = ρAρ ¯

A

ρ 6= ρAρ ¯

A

ρ

• Scaling of the entropy. Thermal/classical states: extensive entropy (volume law).
• Typical ground states: area law. The EE grows as the boundary of the subsystem

How does the EE scale with the size L of the subsystem?

SA ∼ Ld−1

Free fermions

Logarithmic deviation to the area law

SA ∼ Ld−1 log L

• 1D gapped systems , short range, finite interaction strengths: area law proved by Hastings

(M. B. Hastings, Phys. Rev. B 69 104431 (2007))

• Gapless systems: CFT, logarithmic divergence. EE proportional to the central charge

(C. F. Holzhey et al., Nucl. Phys. B 424 443 (1994)), (G. J. Vidal et al., Phys. Rev. Lett. 424 227902 (2003))

Entanglement entropy for free fermions

H = − X

I,J

c†

ItIJcJ

SA = −

L

X

γ=1

[(1 − Cγ) ln (1 − Cγ) + Cγ ln Cγ] Cij = hΨ|c†

icj|Ψi.

Cγ

eigenvalues of the correlation matrix EE in terms of correlation matrix

i,j: index of the subsystem’s sites

CIJ = Z

E<EF

dk 2π eik(I−J)

Independent of the values of the energy translational invariance: plane wave solutions

ψk(J) = 1 √NS eikJ

Non-local power law hopping

• 1D free fermions with PBC, half filling
• Long range hopping: can it determine a violation of the area law beyond

the logarithmic correction?

tI,J = ( I = J,

t |I−J|α

p

I 6= J, |I − J|p = min(|I − J|, NS − |I − J|)

Non-local power law hopping

• 1D free fermions with PBC, half filling
• Long range hopping: can it determine a violation of the area law beyond

the logarithmic correction?

tI,J = ( I = J,

t |I−J|α

p

I 6= J, |I − J|p = min(|I − J|, NS − |I − J|)

EE does not depend on

α Still Area law!

Non-local power law hopping

• 1D free fermions with PBC, half filling
• Long range hopping: can it determine a violation of the area law beyond

the logarithmic correction?

tI,J = ( I = J,

t |I−J|α

p

I 6= J, |I − J|p = min(|I − J|, NS − |I − J|)

EE does not depend on

α Still Area law!

A A

• Solid line: nearest neighbors
• dashed line:
• Inset: nearest and next-nearest neighbors model

with different hopping’s signs

α = 2

Non-local power law hopping

• 1D free fermions with PBC, half filling
• Long range hopping: can it determine a violation of the area law beyond

the logarithmic correction?

tI,J = ( I = J,

t |I−J|α

p

I 6= J, |I − J|p = min(|I − J|, NS − |I − J|)

EE does not depend on

α Still Area law!

A A

• Solid line: nearest neighbors
• dashed line:
• Inset: nearest and next-nearest neighbors model

with different hopping’s signs

α = 2

EE depends on the topology of the Fermi surface

CIJ = Z kd

ka

dk 2π eik(I−J) − Z kc

kb

dk 2π eik(I−J)

Let’s try all the possible filling configurations

2 4 6 8 10 12

L

1 2 3 4 5 6 7

S/ ln 2

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NS = 12

• Maximal EE for alternating filling of the wave vectors
• Periodicity in k-space: piecewise linear behavior

Half filling

States with maximal EE

• Single particle vector space
• Basis in the complementary subsystems

V = A ⊕ ¯ A

• single particle occupied states (not necessarily wave vectors) constructed in

this way maximize the EE for half filling

• Also all the Rényi entropies
• Bell-paired states

|β1i = 1 p 2(|α1i + |¯ α1i), |β2i = 1 p 2(|α2i + |¯ α2i), . . . , |β|A|i = 1 p 2(|α|A|i + |¯ α|A|i)

|αji 2 A, | ¯ αji 2 ¯ A

hJ|αki = (

1

p

NS/2e2πinkJ/(NS/2)

for J  NS/2 for J > NS/2 hJ|¯ αki = ( for J  NS/2 ±

1

p

NS/2e2πinkJ/(NS/2)

for J > NS/2,

alternate occupation of the momentum space

• A. contiguous occupation of k
• B. “zigzag” state
• C. two filled and two empty momenta alternates
• D. Bell-paired state with random connection

between the complementary subsystems

|CIJ|

tI,J = ⇢ t for |I − J|p = NS

2

• therwise,

A model can be constructed for the GS to have the structure of the Bell- paired states.

Examples of models violating the area law

H = − X

I,J

c†

ItIJcJ

Long-Range with a Magnetic flux

tI,J = t · eiφdI,J |I − J|α

p

φ = 2π NS Φ

N=100

Φ = 0.1 α = 0.1 α = 0.4

Φ = 0.001 Φ = 0.1 Φ = 0.3

fitting function: S=a L

εk = −t · sin ✓ 1 kα ◆ , dbox = α α + 1

Fermi surface: Set with box counting dimension:

S = a + bLβ

Fit function:

{± 1 π , ± 1 π2α ± 1 π3α . . .}

L=1…128. Short range: S=a+b ln L

Fermi surface with an accumulation point

tI,J = t · ηI,J |I − J|α

p

ηI,J = ±1

Random Long-Range hopping

• Breaking of translational symmetry
• Logarithm behavior for
• Linear behavior for

α 1 α ⌧ 1 S = a + bLβ

Fit function:

Conclusions

• Violation of the area law has been investigated for free-fermions
• Long range hopping is not a sufficient condition. A more complex structure of the

spectrum and of the Fermi surface is needed and/or a breaking of the translational invariance.

• Explicit construction of the states maximizing the EE in terms of Bell pairs.
• Explicit examples: magnetic phase; FS with an accumulation point
• Preliminary results for a disordered lattice