Approaching critical points
through entanglement: why take
one, 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)
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)
• 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
• 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
• Introduction: Von Neumann and Renyi Entropyas a measure of Entanglement
• Entanglement Entropy in 1-D systems
• Integrability & Corner Transfer Matrices
• Restriced Solid-On-Solid Models: integrabledeformation 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
• 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
• 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
• 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
• 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
• 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
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
• 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
• 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
• 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 Cx
y, t
Entanglement Entropy in 1D Exactly Solvable Models n. 14 Fabio Franchini
Entanglement of one half-linewith the other
• Baxter diagonalized CTM’s of integrable models
⇒ regular structure of the entanglment spectrum
Entanglement & Integrability
A B
D Cx
Entanglement Entropy in 1D Exactly Solvable Models n. 15 Fabio Franchini
a real (or even complex)!
y, t
• CTM spectrum in integrable models same as
certain Virasoro representations (unknown reason!)
CTM & Integrability
A B
D Cx
Entanglement Entropy in 1D Exactly Solvable Models n. 16 Fabio Franchini
Only formal: q measures “mass gap”, not same as CFT!
y, t
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
• 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)
• At fixed r
• 4 Phases:
RSOS Phase Diagram
Entanglement Entropy in 1D Exactly Solvable Models n. 19 Fabio Franchini
l1 l2
l4 l3
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
• 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)
• 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
• 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
• 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
• 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
• 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
• 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
• Commutes with transfer matrices of 8-vertex model
• Use of Baxter’s Corner Transfer Matrices (CTM)
XYZ Spin Chain
A B
D Cx
y, t
Entanglement Entropy in 1D Exactly Solvable Models n. 28 Fabio Franchini
• 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
3-D plot of entropy
Entanglement Entropy in 1D Exactly Solvable Models n. 30 Fabio Franchini
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
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
Conformal check
• Expansion close to conformal points agree with expectations:
• Plus the corrections...
Entanglement Entropy in 1D Exactly Solvable Models n. 33 Fabio Franchini
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
• 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
• 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
• 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
• Need to express as universal paramter
• In scaling limit:
Entropy & Characters
Entanglement Entropy in 1D Exactly Solvable Models n. 38 Fabio Franchini
Divisor function
Entropy & Characters
Entanglement Entropy in 1D Exactly Solvable Models n. 39 Fabio Franchini
• Partition function of a Bulk Ising Model !
• 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
• 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
• 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
• 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
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