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Approaching critical points through entanglement: why take one, when you can take them all? Fabio Franchini (M.I.T./SISSA) Collaborators: - arXiv:1205:6426 - PRB 85: 115428 (2012) A. De Luca; - PRB 83: 12402 (2011) - Quant. Inf. Proc. 10: 325

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SLIDE 1

Approaching critical points through entanglement: why take

  • ne, when you can take them all?

Collaborators:

  • A. De Luca;
  • E. Ercolessi, S. Evangelisti, F. Ravanini;
  • V. E. Korepin, A. R. Its, L. A. Takhtajan …

Fabio Franchini (M.I.T./SISSA)

  • arXiv:1205:6426
  • PRB 85: 115428 (2012)
  • PRB 83: 12402 (2011)
  • Quant. Inf. Proc. 10: 325 (2011)
  • JPA 41: 2530 (2008)
  • JPA 40: 8467 (2007)
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SLIDE 2

Entanglement Entropy in 1-D exactly solvable models

Collaborators:

  • A. De Luca;
  • E. Ercolessi, S. Evangelisti, F. Ravanini;
  • V. E. Korepin, A. R. Its, L. A. Takhtajan …
  • arXiv:1205:6426
  • PRB 85: 115428 (2012)
  • PRB 83: 12402 (2011)
  • Quant. Inf. Proc. 10: 325 (2011)
  • JPA 41: 2530 (2008)
  • JPA 40: 8467 (2007)

Fabio Franchini (M.I.T./SISSA)

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SLIDE 3
  • Entanglement Entropy: non-local correlator → area law
  • 1+1-D CFT prediction (universal behavior):

where c central charge,

h dimension of (relevant) operator

  • Exactly solvable, lattice models efficient testing tools

Motivation

Entanglement Entropy in 1D Exactly Solvable Models

  • n. 3

Fabio Franchini

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SLIDE 4
  • Gapped systems: entropy saturates
  • We’ll test:

1. Expected simple scaling law: with the same dimension h ?

  • 2. Close to non-conformal points: competition between

different length scales → essential singularity

Aims

Entanglement Entropy in 1D Exactly Solvable Models

  • n. 4

Fabio Franchini

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SLIDE 5
  • Introduction: Von Neumann and Renyi Entropy

as a measure of Entanglement

  • Entanglement Entropy in 1-D systems
  • Integrability & Corner Transfer Matrices
  • Restriced Solid-On-Solid Models: integrable

deformation of minimal & parafermionic CFT

  • Essential Critical Point for the entropy: XYZ chain
  • Conclusions

Outline

Entanglement Entropy in 1D Exactly Solvable Models

  • n. 5

Fabio Franchini

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SLIDE 6
  • Entanglement: fundamental quantum property
  • Different reasons for interest:

1. Quantum information → quantum computers 2. Quantum Phase Transitions → universality 3. Condensed matter → non-local correlator 4. Integrable Models → new playground 5. Cosmology → Black Holes 6. …

Introduction

Entanglement Entropy in 1D Exactly Solvable Models

  • n. 6

Fabio Franchini

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SLIDE 7
  • Two spins 1/2 in triplet state → Sz = 1 :
  • Middle component with Sz = 0:

Understanding Entanglement: A simple Example

No entanglement Maximally entangled

Entanglement Entropy in 1D Exactly Solvable Models

  • n. 7

Fabio Franchini

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SLIDE 8
  • Whole system in a pure quantum state
  • Compute Density Matrix of subsystem:
  • Entanglement for pure state as Quantum Entropy

(Bennett, Bernstein, Popescu, Schumacher 1996):

Von Neumann Entropy

Entanglement Entropy

Entanglement Entropy in 1D Exactly Solvable Models

  • n. 8

Fabio Franchini

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SLIDE 9
  • Quantum analog of Shannon Entropy: Measures the

amount of “quantum information” in the given state

  • Assume Bell State as unity of Entanglement:
  • Von Neumann Entropy measures how many

Bell-Pairs are contained in a given state (i.e. closeness of state to maximally entangled one)

Entropy as a measure of entanglement

Entanglement Entropy in 1D Exactly Solvable Models

  • n. 9

Fabio Franchini

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SLIDE 10
  • Von Neumann Entropy:
  • Renyi Entropy:

(equal to Von Neumann for α → 1)

  • Tsallis Entropy
  • Concurrence (Two-Tangle)
  • ...

More Entanglement Estimators

Entanglement Entropy in 1D Exactly Solvable Models

  • n. 10

Fabio Franchini

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SLIDE 11

Bi-Partite Entanglement

  • Consider the Ground state of a Hamiltonian H
  • Space interval [1, l ] is subsystem A
  • The rest of the ground state is subsystem B.

→ Entanglement of a block of spins in the space interval [1, l ] with the rest of the ground state as a function of l

Entanglement Entropy in 1D Exactly Solvable Models

  • n. 11

Fabio Franchini

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SLIDE 12
  • Asymptotic behavior (block size l → ∞)

(Double scaling limit: 0 << l << N )

General Behavior (Area Law)

  • For gapped phases: (Vidal, Latorre, Rico, Kitaev 2003)
  • For critical conformal phases: (Calabrese, Cardy 2004)

Entanglement Entropy in 1D Exactly Solvable Models

  • n. 12

Fabio Franchini

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SLIDE 13
  • Integers Powers of r accessible in CFT (replica)

(Cardy, Calabrese 2010)

  • Close to criticality: x ~ Δ−1, n → ∞

(Calabrese, Cardy, Peschel 2010)

Subleading corrections

Conjecture

Entanglement Entropy in 1D Exactly Solvable Models

  • n. 13

Fabio Franchini

From cut-off regularization

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SLIDE 14
  • Consider 2-D classical system whose transfer matrices

commutes with Hamiltonian of 1-D quantum model

  • Use of Corner Transfer Matrices (CTM) to compute

reduced density matrix

Corner Transfer Matrices

A B D C

x y, t

Entanglement Entropy in 1D Exactly Solvable Models

  • n. 14

Fabio Franchini

Entanglement of one half-line with the other

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SLIDE 15
  • Baxter diagonalized CTM’s of integrable models

⇒ regular structure of the entanglment spectrum

Entanglement & Integrability

A B D C

x

Entanglement Entropy in 1D Exactly Solvable Models

  • n. 15

Fabio Franchini

a real (or even complex)!

y, t

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SLIDE 16
  • CTM spectrum in integrable models same as

certain Virasoro representations (unknown reason!)

CTM & Integrability

A B D C

x

Entanglement Entropy in 1D Exactly Solvable Models

  • n. 16

Fabio Franchini

Only formal: q measures “mass gap”, not same as CFT! y, t

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SLIDE 17

Integrable Models

  • Restricted Solid-On-Solid (RSOS) Models

→ Minimal & Parafermionic CFTs

  • Two integrable chains (8-vertex model)

1) XY in transverse field (Jz = 0) 2)XYZ in zero field (h = 0)

Entanglement Entropy in 1D Exactly Solvable Models

  • n. 17

Fabio Franchini

with Andrea De Luca with Korepin, Its, Takhtajan with Stefano Evangelisti, Ercolessi, Ravanini

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SLIDE 18
  • Specified by 3 parameters: r, p, v
  • 2-D square lattice
  • Heights at vertices:

with local constraint

  • Interaction Round-a-Face: weight for each plaquette
  • Choice of weights makes model integrable

(satisfy Yang-Baxter of 8-vertex model: p, v parametrize weights)

Restricted Solid-On-Solid Models

Entanglement Entropy in 1D Exactly Solvable Models

  • n. 18

Fabio Franchini

l1 l2 l4 l3

W(l1 ,l2 ,l3 ,l4)

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SLIDE 19
  • At fixed r
  • 4 Phases:

RSOS Phase Diagram

Entanglement Entropy in 1D Exactly Solvable Models

  • n. 19

Fabio Franchini

l1 l2 l4 l3

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SLIDE 20

RSOS: Phases I & III

Entanglement Entropy in 1D Exactly Solvable Models

  • n. 20

Fabio Franchini

Phase I

  • 1 ground state → Disordered
  • For p → 0: parafermion CFT (Virasoro + )

Phase III:

  • r - 2 ground states → Ordered
  • For p → 0: minimal CFT
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SLIDE 21
  • Diagonal reduced r depends on

b.c. at origin ( a ) & infinity ( b )

Sketch of the calculation

x

Entanglement Entropy in 1D Exactly Solvable Models

  • n. 21

Fabio Franchini

y, t : at criticality: Poisson Summation formula (S-Duality)

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SLIDE 22
  • Fixing a & b: single minimal model character:
  • After S-Duality (Poisson) duality and logarithm

where h dimension of most relevant operator here (generally )

Regime III: Minimal models

Entanglement Entropy in 1D Exactly Solvable Models

  • n. 22

Fabio Franchini

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SLIDE 23
  • Fixing a equivalent to projecting Hilbert space
  • True ground state by summing over a:
  • dicates most relevant operator vanishes (odd):

Regime III: Minimal models

Entanglement Entropy in 1D Exactly Solvable Models

  • n. 23

Fabio Franchini

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SLIDE 24
  • RSOS as integrable deformation of minimal models
  • Integrability fixes coefficients:
  • Corrections from relevant operators
  • Same scaling function in x & l ?
  • role at criticality?

Regime III: conclusion

Entanglement Entropy in 1D Exactly Solvable Models

  • n. 24

Fabio Franchini

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SLIDE 25
  • b.c. at infinity factorize out
  • a selects a combination of operators neutral for
  • In general: h = 4 / r (most relevant neutral op)
  • b can give logarithmic corrections (marginal fields?)

Regime I: Parafermions

Entanglement Entropy in 1D Exactly Solvable Models

  • n. 25

Fabio Franchini

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SLIDE 26
  • RSOS as integrable deformations of CFT
  • CTM spectrum mimics critical theory (accident?)

⇒ same scaling function for entanglement in x & l ?

  • Logarithmic corrections for parafermions?

Let’s look directly at some 1-D quantum models

RSOS Round-up

Entanglement Entropy in 1D Exactly Solvable Models

  • n. 26

Fabio Franchini

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SLIDE 27
  • For c=1 CFT, it is by now established: h = K
  • Off criticality, expected?
  • Close to Heisenberg AFM point, observed

(Calabrese, Cardy, Peschel 2010)

→ h=2 ? (K=1/2)

Subtle Puzzle

Entanglement Entropy in 1D Exactly Solvable Models

  • n. 27

Fabio Franchini

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SLIDE 28
  • Commutes with transfer matrices of 8-vertex model
  • Use of Baxter’s Corner Transfer Matrices (CTM)

XYZ Spin Chain

A B D C

x y, t

Entanglement Entropy in 1D Exactly Solvable Models

  • n. 28

Fabio Franchini

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SLIDE 29
  • Gapped in bulk of plane
  • Critical on dark lines

(rotated XXZ paramagnetic phases)

  • 4 “tri-critical” points:

C1,2 conformal E1,2 quadratic spectrum

Phase Diagram of XYZ model

Entanglement Entropy in 1D Exactly Solvable Models

  • n. 29

Fabio Franchini

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SLIDE 30

3-D plot of entropy

Entanglement Entropy in 1D Exactly Solvable Models

  • n. 30

Fabio Franchini

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SLIDE 31

Iso-Entropy lines

  • Conformal point:

entropy diverges close to it

  • Non-conformal

point (ECP): entropy goes from 0 to ∞ arbitrarily close to it (depending on direction)

Entanglement Entropy in 1D Exactly Solvable Models

  • n. 31

Fabio Franchini

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SLIDE 32

Close-up to non-conformal point

  • Isotropic

Ferromagnetic Heisenberg: quadratic spectrum

  • Curves of constant

entropy pass through it

  • Similar physics as XY

model

Entanglement Entropy in 1D Exactly Solvable Models

  • n. 32

Fabio Franchini

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SLIDE 33

Conformal check

  • Expansion close to

conformal points agree with expectations:

  • Plus the corrections...

Entanglement Entropy in 1D Exactly Solvable Models

  • n. 33

Fabio Franchini

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SLIDE 34

1st round-up

  • Gapped phases saturate
  • Close to conformal points: logarithmic divergence
  • Close to non-conformal points: essential singularity

(entropy depends on direction of approach) → Entanglement to discriminate non-conformal QPTs (finite size scaling?)

Entanglement Entropy in 1D Exactly Solvable Models

  • n. 34

Fabio Franchini

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SLIDE 35
  • All spin 1/2 integrable chain systems have the same

diagonal structure for r (Baxter’s Book, Peschel et al 2009, …):

The Reduced Density Matrix

where e is characteristic of the model

  • Origin in CTM of 8-vertex model

Entanglement Entropy in 1D Exactly Solvable Models

  • n. 35

Fabio Franchini

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SLIDE 36
  • Eigenvalues form a geometric series
  • Degeneracies from partitions of integers

(Okunishi et al. 1999; Franchini et al. 2010; ... )

  • All these models have the same entanglement spectrum

→ HEntanglement: free fermions with spectrum e

  • Microscopic of the model only in e

Entanglement Spectrum

Entanglement Entropy in 1D Exactly Solvable Models

  • n. 36

Fabio Franchini

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SLIDE 37
  • For integrable models: entropy reads characters
  • CTM spectrum = Virasoro representation (Tokyo Group, Cardy...)
  • For XYZ:
  • Close to QPT: expansion in the S-dual variable:

Characters

Entanglement Entropy in 1D Exactly Solvable Models

  • n. 37

Fabio Franchini

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SLIDE 38
  • Need to express as universal paramter
  • In scaling limit:

Entropy & Characters

Entanglement Entropy in 1D Exactly Solvable Models

  • n. 38

Fabio Franchini

Divisor function

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SLIDE 39

Entropy & Characters

Entanglement Entropy in 1D Exactly Solvable Models

  • n. 39

Fabio Franchini

  • Partition function of a Bulk Ising Model !
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SLIDE 40
  • For a finite ultra-violet cut-off a0:
  • New subleading corrections to entanglement entropy
  • Low energy states: free excitations

Lattice effects

Entanglement Entropy in 1D Exactly Solvable Models

  • n. 40

Fabio Franchini

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SLIDE 41
  • For a finite ultra-violet cut-off a0:
  • New subleading corrections to entanglement entropy
  • Low energy states: bound states

→ direction dependent

Lattice effects

Entanglement Entropy in 1D Exactly Solvable Models

  • n. 41

Fabio Franchini

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SLIDE 42
  • For a finite ultra-violet cut-off a0:
  • New subleading corrections to entanglement entropy
  • Low energy states: bound states

→ direction dependent

Lattice effects

Entanglement Entropy in 1D Exactly Solvable Models

  • n. 42

Fabio Franchini

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SLIDE 43
  • For a finite ultra-violet cut-off a0:
  • New subleading corrections to entanglement entropy
  • Low energy states: bound states

→ direction dependent

Lattice effects

Entanglement Entropy in 1D Exactly Solvable Models

  • n. 43

Fabio Franchini

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SLIDE 44

Conclusions & Outlook

Thank you!

  • Analytical study of bipartite entanglement of 1-D integrable models:

RSOS, the XY and XYZ models

  • CTM/ reduced r spectrum & CFT: unusual corrections
  • Logarithmic corrections in parafermions?
  • Near non-conformal points, entropy has an essential singularity

→ Universality close to ECPs? Finite size scaling?

  • Lattice corrections (logarithmic)
  • Subleading corrections from (bulk) Ising model
  • Relation between CTM & critical theory?

Entanglement Entropy in 1D Exactly Solvable Models

  • n. 44

Fabio Franchini