Germán Sierra,

Instituto de Física Teórica UAM-CSIC, Madrid

9th Bolonia Workshop in CFT and Integrable Systems

Bolonia, 15-18 Sept 2014

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S = A∪B →H = HA ⊗HB

:a pure state in H choosen at random

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ψ

EE is almost maximal (Page)

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SA ≈ logdimHA − cte ≈ nA − cte

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nA :number of sites of A

Volumen law like for the thermodynamic entropy

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ρA = trB ψ ψ

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SA = −Tr ρ A log ρ A

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SA ∝ ∂A = ∂B

If is the ground state of a local Hamiltonian

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ψ

Physics happens at a corner of the Hilbert space

Experiments occur in the Lab not in a Hilbert space (A. Peres)

Basis of Tensor Networks (MPS, PEPS, MERA,..)

(c) MERA

Hastings theorem (2007):

Conditions:

-Finite range interactions

-Finite interaction strengths

-Existence of a gap in the spectrum

In these cases the GS can be well approximated by a MPS

In 1D

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SA ∝ cte

Violations of the area law in 1D require one of the following

-non local interactions

-divergent interactions

-gapless systems

Best well known examples are CFT and quenched disordered systems

-> Log violations of entropy

Here we shall investigate a stronger violation

Entanglement entropy satisfies a volumen law

Part I:

The rainbow model: arXiv:1402.5015 G. Ramírez, J. Rodríguez-Laguna, GS

Part II:

Infinite Matrix Product States: arXiv:1103.2205

A.E.B. Nielsen, GS, J.I.Cirac

PART I : The Rainbow Model

Inhomogenous free fermion model in an open chain with 2L sites

Introduced by Vitigliano, Riera and Latorre (2010)

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0 < α ≤1

Other inhomogenous Hamiltonians

-Smooth boundary conditions (Vekic and White 93)

-Quenched disordered: J’s random (Fisher, Refael-Moore 04)

- Scale free Hamiltonian and Kondo (Okunishi, Nishino 10)

-Hyperbolic deformations (Nishino, Ueda, Nakano, Kusabe 09)

Dasgupta-Ma method (1980)

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Ji >>Ji±1

At the i-th bond there is a bonding state

In second order perturbation

This method is exact for systems with quenched disorder (Fisher, …)

Choosing the J’s at random -> infinite randomness fixed point

Average entanglement entropy and Renyi entropies

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SL ≈c log2

3log L

Refael, Moore 04Laflorencie 05Fagotti,Calabrese,Moore 11Ramirez,Laguna,GS 14

CFT Renyi

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SL(n ) ≈

c (n +1/n)

6logL

If the strongest bond is between sites i=1,-1

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0 < α <1

RG gives the effective coupling:

This new bond is again the strongest one because

Repeating the process one finds the GS: valence bond state

It is exact in the limit

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α → 0+ (fixed point of the RG)

Density matrix of the rainbow state

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ρB = TrB c RL RL

B: a block

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nB number of bonds joining B with the rest of the chain

has an eigenvalue with multiplicity

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ρB

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λk = 2−nB

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2nB

von Neumann entropy

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SB = − λ k logλ k = nB log2k

∑

Moreover all Renyi entropies are equal to von Neumann

Take B to be the half-chain then

Maximal entanglement entropy for a system of L qubits€

nB = L

The energy gap is proportional to the effective coupling of the last effective bond

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gap ∝ α 2 L →0, L →∞

Hasting’s theorem is satisfied

Define

Uniform case

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α =1→z = 0

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z = −L logα ≥ 0

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SB = L log2

Hopping matrix

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Ti, j = −J0 δ ij,−1 −Ji δ | i− j |,1 , i, j = ±1,K ,±L

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Ti, j φ jk = Ek φi

k

Particle-hole symmetry

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φi →(−1)i sign(i)φi : E →−E

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Ek = −E−k −1 k = 0,±1,K ,±(L −1),−L

Ground state at half-filling

Non uniform model

scaling behaviour

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Ek (L,α ) ≈ ez(k /L) ≅ vF (z)k

L,

k

L<<1

Uniform model

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Ek = 2sinπ (2k +1)

2(2L +1)→

πk

L,

k

L<<1

The Fermi velocity only depends on

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z = −L logα

Correlation method (Peschel,…)

Two point correlator in the block B of size

Diagonalize finding its eigenvalues

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l

Reduced density matrix of the block

von Neumann entropy

For small and L large there is a violation of the arealaw that becomes a volumen law.

This agrees with the analysis based on the Dasgupta-Ma RG

What about the limit ? €

α

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α → 1−

The proximity of the CFT:

Half-chain entropy

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z = 0

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z = 0.2

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z = 0.4

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z =K€

z = 2.0€

α ≈1, L >>1, z = cte

CFT formula for open chain

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SLCFT ≈

c

6log

2L

π+ c1'+ 2g + f cos(πL) L−K

Boundary entropy Luttinger parameter

Fitting curve

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SL ≈c(z)

6log L + d(z) + f (z)cos(πL) L−K

The fits have

c(z) decreases with z: similar to the c-theorem

d(z) increases with z: the g-theorem does not apply because the bulk is not critical

Origin of the volumen law

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d(z) ≈ 0.318 z →SL ≈ −0.318 L logα

(z similar to mR)

Entanglement Hamiltonian

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ρB = e− HE

For free fermions

In the rainbow state ( )

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ν p =1

2→ε p = 0, ∀p

Entanglement energies

L=40 L=41

L: even

L: odd

Make the approximation one can estimate the EE

- Critical model :

Peschel, Truong (87), Cardy, Peschel (88), …Corner Transfer Matrix

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ρA = e− HE , HE = HCTM = n hn,n +1n

∑

ES: energy spectrum of a boundary CFT (Lauchli, 14)

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SL ≈c

6logξ +K →ΔL ∝ 1/ξ

- Rainbow model for for L sufficiently large

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α <1

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Δ L ∝ 1/L →SL ∝ 1/ΔL ∝ L

- Massive models in the scaling limit

Cardy, Calabrese (04) using CTMErcolessi, Evangelisti, Francini, Ravanini 09,…14Castro-Alvaredo, Doyon, Levi, Cardy, 07,…14

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α =1

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α =1.1

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α =0.9

Entanglement spacing for constant

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α

Based on equations

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SL ≈π 2

3ΔL

one is lead to the ansatz for the entanglement spacing

depend on the parity of L

And

Entanglement spacing for z constant

evenodd

The fit has

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χ2 ≈10−12 z∈[0,1]

Fitting functions

Entropy/gap relation

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π 2

3

Generalization to other models

Local hamiltonian

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hi,i+1

AF Heisenberg

Continuum limit of the rainbow model (work in progress)

Uniform model

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α =e−h , h ≥ 0

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α =1, h = 0

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H ≈ HR + HL = iψ R ∂x ψ R −iψ L ∂x ψ L

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cn ≈ e iπ n / 2 ψ L (x) + e−iπ n / 2 ψ R (x), x = n a

Fast-low factorization

CFT with c=1

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Hε ,ε ' = iψ ε ,ε ' fε ,ε ' (x) +1

2f 'ε ,ε ' (x)

⎛

⎝ ⎜

⎞

⎠ ⎟ψε ,ε '

fε ,ε ' (x) ≈ ε '(1+ h + h2 − 2h(1+ h)εx + 2h2x 2)

Non uniform model

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α <1 h > 0

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cn ≈ e iπ n / 2 ψ A ,L (x) + e−iπ n / 2 ψ A ,R (x), x < 0

≈ e iπ n / 2 ψ B ,L (x) + e−iπ n / 2 ψ B ,R (x), x > 0

wave functionsnear E=0

numerical

theory

It is expected to predict some of the scaling functions c(z)

PART II : Infinite MPS

MPS

Infinite MPS

Vertex operators in CFT (Cirac, GS 10)

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Vs(z) =:exp i s α ϕ(z)( ), s = ±1

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ψ(s1,K ,sN ) = (zn − zm )α sn sm ,n<m

∏ zn = e2π n i / N , n =1,2,K ,N

Renyi 2 entropy

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0 < α ≤1

2→SL

(2) ≈1

4log

N

πsin

π L

N

⎛

⎝ ⎜

⎞

⎠ ⎟+ fluctuations

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1

2< α →SL

(2) →cte L >>1

Good variational ansatz for the XXZ model

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Δ =−cos(2π α ) −1 < Δ ≤1

Truncate the vertex operator to the first M modes (Nielsen,Cirac,GS)

The wave function is

Renyi entropy

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SL(2) = a sin

π L

N

⎛

⎝ ⎜

⎞

⎠ ⎟b

+ c

b

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N = 4000

α = 0.15

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b ≈ 0.967e−0.246 M

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SL ∝ Lb

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M →∞ b →0 SL →logL

M →0 b →1 SL →L

Experimental implementation

We have shown that rather simple local Hamiltonianscan give rise to ground states that violate the area law.

They can be thought of as conformal transformation on a criticalmodel that preserves some of the entanglement properties.

In the strong coupling limit they become valence bond states:provide a way to interpolate continuously between the CFT and the VBS.

The infinite MPS based on CFT lie in the boundary of the statesthat satisfy the area law.

Thank you

Grazie mille