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) E. Ercolessi, S. Evangelisti, F. Ravanini; - Quant. Inf. Proc. 10: 325 (2011) - JPA 41: 2530 (2008) V. E. Korepin, A. R. Its, L. A. Takhtajan … - JPA 40: 8467 (2007) Entanglement Entropy in 1-D exactly solvable models
Fabio Franchini(M.I.T./SISSA)
Collaborators: - arXiv:1205:6426 - PRB 85: 115428 (2012) A. De Luca; - PRB 83: 12402 (2011) E. Ercolessi, S. Evangelisti, F. Ravanini; - Quant. Inf. Proc. 10: 325 (2011) - JPA 41: 2530 (2008) V. E. Korepin, A. R. Its, L. A. Takhtajan … - JPA 40: 8467 (2007) Motivation
• 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
Entanglement Entropy in 1D Exactly Solvable Models n. 3 Fabio Franchini Aims
• 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
Entanglement Entropy in 1D Exactly Solvable Models n. 4 Fabio Franchini Outline
• 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
Entanglement Entropy in 1D Exactly Solvable Models n. 5 Fabio Franchini Introduction
• 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. …
Entanglement Entropy in 1D Exactly Solvable Models n. 6 Fabio Franchini Understanding Entanglement: A simple Example
• Two spins 1/2 in triplet state → Sz = 1 :
No entanglement
• Middle component with Sz = 0:
Maximally entangled
Entanglement Entropy in 1D Exactly Solvable Models n. 7 Fabio Franchini Entanglement Entropy
• 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 in 1D Exactly Solvable Models n. 8 Fabio Franchini Entropy as a measure of entanglement
• 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)
Entanglement Entropy in 1D Exactly Solvable Models n. 9 Fabio Franchini More Entanglement Estimators
• Von Neumann Entropy:
• Renyi Entropy: (equal to Von Neumann for α → 1)
• Concurrence (Two-Tangle)
• ...
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 General Behavior (Area Law)
• Asymptotic behavior (block size l → ∞)
(Double scaling limit: 0 << l << N )
• 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 Subleading corrections
• Integers Powers of r accessible in CFT (replica) (Cardy, Calabrese 2010)
−1 From cut-off • Close to criticality: x ~ Δ , n → ∞ regularization (Calabrese, Cardy, Peschel 2010) Conjecture
Entanglement Entropy in 1D Exactly Solvable Models n. 13 Fabio Franchini Corner Transfer Matrices
• 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 y, t D C
x
Entanglement of one half-line A B with the other
Entanglement Entropy in 1D Exactly Solvable Models n. 14 Fabio Franchini Entanglement & Integrability
• Baxter diagonalized CTM’s of integrable models
⇒ regular structure of the entanglment spectrum
y, t D C
x A B a real (or even complex)!
Entanglement Entropy in 1D Exactly Solvable Models n. 15 Fabio Franchini CTM & Integrability
• CTM spectrum in integrable models same as certain Virasoro representations (unknown reason!)
y, t D C
x Only formal: q measures “mass gap”, not same as CFT! A B
Entanglement Entropy in 1D Exactly Solvable Models n. 16 Fabio Franchini Integrable Models
• Restricted Solid-On-Solid (RSOS) Models
→ Minimal & Parafermionic CFTs with Andrea De Luca • Two integrable chains (8-vertex model)
1) XY in transverse field (Jz = 0) with Korepin, Its, Takhtajan 2) XYZ in zero field (h = 0) with Stefano Evangelisti, Ercolessi, Ravanini
Entanglement Entropy in 1D Exactly Solvable Models n. 17 Fabio Franchini Restricted Solid-On-Solid Models
• Specified by 3 parameters: r, p, v W(l1 ,l2 ,l3 ,l4) • 2-D square lattice l1 l2
• Heights at vertices:
l4 l3
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)
Entanglement Entropy in 1D Exactly Solvable Models n. 18 Fabio Franchini RSOS Phase Diagram
• At fixed r l1 l2
l4 l3
•4 Phases:
Entanglement Entropy in 1D Exactly Solvable Models n. 19 Fabio Franchini RSOS: Phases I & III Phase I
• 1 ground state → Disordered
•For p → 0: parafermion CFT (Virasoro + )
Phase III:
• r -2ground states → Ordered
•For p → 0: minimal CFT
Entanglement Entropy in 1D Exactly Solvable Models n. 20 Fabio Franchini Sketch of the calculation y, t • Diagonal reduced r depends on
b.c. at origin ( a ) & infinity ( b )
x
: at criticality: Poisson Summation formula (S-Duality)
Entanglement Entropy in 1D Exactly Solvable Models n. 21 Fabio Franchini Regime III: Minimal models
• Fixing a & b: single minimal model character:
• After S-Duality (Poisson) duality and logarithm
where h dimension of most relevant operator here
(generally )
Entanglement Entropy in 1D Exactly Solvable Models n. 22 Fabio Franchini Regime III: Minimal models
• Fixing a equivalent to projecting Hilbert space
• True ground state by summing over a:
• dicates most relevant operator vanishes (odd):
Entanglement Entropy in 1D Exactly Solvable Models n. 23 Fabio Franchini Regime III: conclusion
• RSOS as integrable deformation of minimal models
• Integrability fixes coefficients:
• Corrections from relevant operators
•Same scaling function in x & l ?
• role at criticality?
Entanglement Entropy in 1D Exactly Solvable Models n. 24 Fabio Franchini Regime I: Parafermions
• 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?)
Entanglement Entropy in 1D Exactly Solvable Models n. 25 Fabio Franchini RSOS Round-up
• 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
Entanglement Entropy in 1D Exactly Solvable Models n. 26 Fabio Franchini Subtle Puzzle •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)
Entanglement Entropy in 1D Exactly Solvable Models n. 27 Fabio Franchini XYZ Spin Chain
• Commutes with transfer matrices of 8-vertex model
• Use of Baxter’s Corner Transfer Matrices (CTM) y, t D C
x A B
Entanglement Entropy in 1D Exactly Solvable Models n. 28 Fabio Franchini Phase Diagram of XYZ model
• Gapped in bulk of plane
• Critical on dark lines (rotated XXZ paramagnetic phases)
• 4 “tri-critical” points:
C1,2 conformal
E1,2 quadratic spectrum
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 The Reduced Density Matrix
• All spin 1/2 integrable chain systems have the same
diagonal structure for r (Baxter’s Book, Peschel et al 2009, …):
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 Entanglement Spectrum
• 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 Entropy in 1D Exactly Solvable Models n. 36 Fabio Franchini Characters
• For integrable models: entropy reads characters
• CTM spectrum = Virasoro representation (Tokyo Group, Cardy...)
• For XYZ:
• Close to QPT: expansion in the S-dual variable:
Entanglement Entropy in 1D Exactly Solvable Models n. 37 Fabio Franchini Entropy & Characters
Divisor function
• Need to express as universal paramter
•In scaling limit:
Entanglement Entropy in 1D Exactly Solvable Models n. 38 Fabio Franchini Entropy & Characters
• Partition function of a Bulk Ising Model !
Entanglement Entropy in 1D Exactly Solvable Models n. 39 Fabio Franchini Lattice effects
• For a finite ultra-violet cut-off a0:
• New subleading corrections to entanglement entropy
• Low energy states: free excitations
Entanglement Entropy in 1D Exactly Solvable Models n. 40 Fabio Franchini Lattice effects
• For a finite ultra-violet cut-off a0:
• New subleading corrections to entanglement entropy
• Low energy states: bound states
→ direction dependent
Entanglement Entropy in 1D Exactly Solvable Models n. 41 Fabio Franchini Lattice effects
• For a finite ultra-violet cut-off a0:
• New subleading corrections to entanglement entropy
• Low energy states: bound states
→ direction dependent
Entanglement Entropy in 1D Exactly Solvable Models n. 42 Fabio Franchini Lattice effects
• For a finite ultra-violet cut-off a0:
• New subleading corrections to entanglement entropy
• Low energy states: bound states
→ direction dependent
Entanglement Entropy in 1D Exactly Solvable Models n. 43 Fabio Franchini Conclusions & Outlook
•Analyticalstudy 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? Thank you!
Entanglement Entropy in 1D Exactly Solvable Models n. 44 Fabio Franchini