Germán Sierra(IFT, Madrid)

Anyons in Quantum Many-Body SystemsMPI-PKS Dresden 21-25 January 2019

Topological rainbow chains

Collaborators

- Giovanni Ramírez (U. Guatemala)

- Nadir Samos (IFT, Madrid)

- Silvia Santalla (U. Carlos III, Madrid)

- Javier Rodríguez-Laguna (UNED, Madrid)

- Jerôme Dubail (CNRS, Nancy)

- Vincenzo Alba (SISSA, Trieste)

- Paola Ruggiero (SISSA, Trieste)

- Erik Tonni (SISSA, Trieste)

- Pasquale Calabrese (SISSA, Trieste)

Critical spin chain described by CFT

Entanglement entropy

Critical spin chain described by CFT in curved spacetime

Ground State is a Valence Bond

- Concentric singlet phase (Vitigliano et al. 2010)

- Rainbow state (Ramirez et al. 2015)

- Matrioska state (Di Franco et al 2008)

- Nested Bell state (Alkurtass, et al 2014)

- The parity of the number of sites the chain (even/odd)

- Symmetry of the couplings (bond center/site center)

Symmetry Protected Topological phases

Su-Schrieffer-Heeger and Haldane phases

Topological properties of the rainbow model

Outline

- Review of the rainbow XX chain

- Topological properties of XX and Heisenberg rainbows

- Relations to other works

Vitigliano, Riera and Latorre (2010) Ramirez, Rodriguez-Laguna, GS (2015)

Rainbow XX model

𝐻 = ∑ 𝐽&(𝑆&)𝑆&*+)& + 𝑆&-𝑆&*+- )

JordanWigner

0 1Uniform model

Exact diagonalization

Dasgupta-Ma RG Field Theory

Dasgupta-Ma RG

XX Hamiltonian

Effective coupling:

This new bond is again the strongest one because

In the limit

Iterate the RG GS: valence bond state

Exact in the limit

Half-block entanglement entropy

0

2

4

6

8

10

12

14

16

0 5 10 15 20 25 30

Blo

cken

trop

ySℓ(α)

Size of the block ℓ

(a)α

0.10.20.30.40.50.60.70.80.91.0

All inhomogeneities : Exact Diagonalization

Fast-slow separation of degrees of freedom

Weak inhomogeneity : Field theory

Massless Dirac fermion in curved spacetime

Curvature

Half-block entropy

𝑆. ≈ 16 ln 𝐿

5 = 16 ln

𝑒7. − 1ℎ

𝑆. ≈ 16 ln 𝐿Uniform chain

ℎ𝐿 ≫ 1 → 𝑆. ≈ ℎ𝐿6 Volumen law

More results

- General expressions of entanglement entropies

- Entanglement spectrum and Entanglement Hamiltonian

- Finite temperature

- Random + Rainbow

Thermofield double

ℎ𝐿 ≫ 1 → 𝑆. ≈ ℎ𝐿6

Entropy at finite temperature T

𝐻<=>|𝑛⟩ = 𝐸&|𝑛⟩

Finite temperature

0

5

10

15

20

25

30

35

40

0 10 20 30 40 50 60 70 80 90 100

Entr

opy

Block size

𝑇 = 𝐽(𝑥E)

rainbow + random = ranbow

V. Alba, S. Santalla, P. Ruggiero, J. Rodriguez-Laguna, P. Calabrese, G.S 2018

𝑆G ∝ 𝐿XX model:

Open odd chain

(N. Samos, S. Santallana, J.Rodriguez-Laguna, GS, 2018)

2L sites 2L+1 sites

block entanglement entropies

𝑆G = ℓ ln 2 𝑆G = (ℓ +12) (2ln2 − 1)

XX chain

Strong inhomogeneity

AF Heisenberg model

ground stateS= 1/2

+Resonatingvalence bond

Renormalization

Folding the chain

Folding the chain

Folding the chain

Folding the chain

Folding the chain

AKLT state spin 1 end spin

Folding the even chain

Folding the even chain

Even rainbow = product state in the folded basis

Even rainbow = trivial

Odd rainbow = non trivial

delocalized spinon localized edge spin

uniform strong inhomogeneity

Folding

long-range entanglement short-range entanglement

𝑆. ≈ ℎ𝐿6

𝑥-𝑥

𝑆 𝑥 < 13 ln

4ℎ

even/odd chains bond/site centered symmetry

Middle Block Entropy (XX model)

00.20.40.60.81

1.21.41.61.82

0 10 20 30 40 50

S(x)

x

scs: h = 0bcs: h = 0

0.10.1

0.50.5

11

77

−30

−20

−10

0

10

20

30

0 1 2 3 4 5 6 7

ϵp

h

Entanglement spectrum (XX model)

site-centered symmetry

zero entanglement energy degeneracy of EE

Time reversal + Particle hole preserve the degeneracy

SSH phase ( BDI class)

Middle Block Entropy and Entanglement Spectrum (Heisenberg)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

3 6 9 12 15 18 21 24 27

0.1

0.2

0.3

0.4

0.5

S(x)

x

h = 0h = 0.25h = 0.5h = 1

h = 1.5h = 2log(2)

λk

h = 0h = 0.5h = 1

String order parameter (Heisenberg)

−0.45

−0.3

−0.15

0

0 0.5 1 1.5 2 2.5 3 3.5 4

g(L

)

h

scs, L = 7scs, L = 8scs, L = 9

bcs, L = 7bcs, L = 8bcs, L = 9

−49

AKLT

Higher spin chains

Heisenberg chainsS = 1 /2, 3/2, 5/2 …

WZW SU(2)@k=1

AKLT S =1, 3, 5,..

Non trivial SPT

Heisenberg chainsS = 1, 2, 3 …

gapped

AKLT S =2, 4, 6,..

Trivial SPT

Relation between gapped and “gapless” SPT phases

Global anomalies Furuya and Oshikawa (2017)

SU 2 @𝑘 k: even (trivial), odd (non trivial)

Cormack, Liu, Nozaki, Ryu (2018)

Holography and the rainbow

Rainbow: CFT spacetime with curvature 𝑅 𝑥 = −ℎR

𝐴𝑑𝑆Ris a foliation of 𝐴𝑑𝑆U

bulk

Ryu-Takayanagi

(2018)

(2004)

Works

From conformal to volume-law for the entanglement entropy in exponentially deformed critical spin 1/2 chainsG. Ramírez, J. Rodríguez-Laguna, G.S. (2015).

Entanglement over the rainbowG. Ramírez, J. Rodríguez-Laguna, G.S. (2015).

More on the rainbow chain: entanglement, space-time geometryand thermal statesJ. Rodríguez-Laguna, J. Dubail, G. Ramírez, P. Calabrese, G.S. (2016).

Entanglement hamiltonian and entanglement contour in inhomogeneous 1D critical systemsE. Tonni, J. Rodríguez-Laguna, G.S. (2018).

Unusual area-law violation in random inhomogeneous systemsV. Alba, S. Santalla, P. Ruggiero, J. Rodriguez-Laguna, P. Calabrese, G.S. (2018)

Symmetry protected phases in inhomogeneous spin chainsN. Samos Sáenz de Buruaga, S. Santalla, J. Rodriguez-Laguna, G.S. (2018)

Thanks for your attention