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Germán Sierra, Instituto de Física Teórica UAM-CSIC, Madrid 9th Bolonia Workshop in CFT and...
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Transcript of Germán Sierra, Instituto de Física Teórica UAM-CSIC, Madrid 9th Bolonia Workshop in CFT and...
Germán Sierra,
Instituto de Física Teórica UAM-CSIC, Madrid
9th Bolonia Workshop in CFT and Integrable Systems
Bolonia, 15-18 Sept 2014
€
S = A∪B →H = HA ⊗HB
:a pure state in H choosen at random
€
ψ
EE is almost maximal (Page)
€
SA ≈ logdimHA − cte ≈ nA − cte
€
nA :number of sites of A
Volumen law like for the thermodynamic entropy
€
ρA = trB ψ ψ
€
SA = −Tr ρ A log ρ A
€
SA ∝ ∂A = ∂B
If is the ground state of a local Hamiltonian
€
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
€
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)
€
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)
€
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
€
SL ≈c log2
3log L
Refael, Moore 04Laflorencie 05Fagotti,Calabrese,Moore 11Ramirez,Laguna,GS 14
CFT Renyi
€
SL(n ) ≈
c (n +1/n)
6logL
If the strongest bond is between sites i=1,-1
€
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
€
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
€
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
€
2nB
von Neumann entropy
€
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
€
gap ∝ α 2 L →0, L →∞
Hasting’s theorem is satisfied
Define
Uniform case
€
α =1→z = 0
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z = −L logα ≥ 0
€
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
€
Ti, j φ jk = Ek φi
k
Particle-hole symmetry
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φi →(−1)i sign(i)φi : E →−E
€
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
€
Ek = 2sinπ (2k +1)
2(2L +1)→
πk
L,
k
L<<1
The Fermi velocity only depends on
€
z = −L logα
Correlation method (Peschel,…)
Two point correlator in the block B of size
Diagonalize finding its eigenvalues
€
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
€
α → 1−
The proximity of the CFT:
Half-chain entropy
€
z = 0
€
z = 0.2
€
z = 0.4
€
z =K€
z = 2.0€
α ≈1, L >>1, z = cte
CFT formula for open chain
€
SLCFT ≈
c
6log
2L
π+ c1'+ 2g + f cos(πL) L−K
Boundary entropy Luttinger parameter
Fitting curve
€
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
€
d(z) ≈ 0.318 z →SL ≈ −0.318 L logα
(z similar to mR)
Entanglement Hamiltonian
€
ρB = e− HE
For free fermions
In the rainbow state ( )
€
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
€
ρA = e− HE , HE = HCTM = n hn,n +1n
∑
ES: energy spectrum of a boundary CFT (Lauchli, 14)
€
SL ≈c
6logξ +K →ΔL ∝ 1/ξ
- Rainbow model for for L sufficiently large
€
α <1
€
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
€
α =1
€
α =1.1
€
α =0.9
Entanglement spacing for constant
€
α
Based on equations
€
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
€
χ2 ≈10−12 z∈[0,1]
Fitting functions
Entropy/gap relation
€
π 2
3
Generalization to other models
Local hamiltonian
€
hi,i+1
AF Heisenberg
Continuum limit of the rainbow model (work in progress)
Uniform model
€
α =e−h , h ≥ 0
€
α =1, h = 0
€
H ≈ HR + HL = iψ R ∂x ψ R −iψ L ∂x ψ L
€
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
€
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
€
α <1 h > 0
€
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)
€
Vs(z) =:exp i s α ϕ(z)( ), s = ±1
€
ψ(s1,K ,sN ) = (zn − zm )α sn sm ,n<m
∏ zn = e2π n i / N , n =1,2,K ,N
Renyi 2 entropy
€
0 < α ≤1
2→SL
(2) ≈1
4log
N
πsin
π L
N
⎛
⎝ ⎜
⎞
⎠ ⎟+ fluctuations
€
1
2< α →SL
(2) →cte L >>1
Good variational ansatz for the XXZ model
€
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
€
SL(2) = a sin
π L
N
⎛
⎝ ⎜
⎞
⎠ ⎟b
+ c
b
€
N = 4000
α = 0.15
€
b ≈ 0.967e−0.246 M
€
SL ∝ Lb
€
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