Entanglement entropy at quantum critical points: Can you ... - INFN

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Entanglement Entropy in Quantum Mechanics • Consider a quantum mechanical system with two sets of degrees of freedom, {ϕ A } and {ϕ B }. Given the pure state Ψ[ϕ A , ϕ B ] and the trivial density matrix for the combined system A ∪ B, 〈ϕ A , ϕ B |ρ A∪B |ϕ ′ A,ϕ ′ B〉 = Ψ[ϕ A , ϕ B ]Ψ ∗ [ϕ ′ A, ϕ ′ B] • The reduced density matrix for A is constructed by tracing over the degrees of freedom in B: 〈ϕ A |ρ A |ϕ ′ A〉 = tr B ρ A∪B • The von Neumann entanglement entropy is S A = −tr A (ρ A ln ρ A ) = −tr B (ρ B ln ρ B ) = S B • The von Neumann entropy gives a measurement of how entangled (or correlated) the degrees of freedom of A are with those of B. 4
Quantum Phase Transitions and Quantum Criticality M order g c disorder g • Phase transitions at zero temperature tuned by a parameter g of a quantum Hamiltonian • Scale Invariance at Quantum Critical Points: Correlation functions of local operators exhibit scaling. • As the transition there is a diverging correlation length ξ ∼ |g − g c | −ν • The energy gap ∆ closes with a power law ∆ ∼ |g − g c | νz , where z is the dynamic critical exponent • For systems which exhibit an effective Lorentz invariance at criticality z = 1; in general z ≠ 1. 5
Page 1 and 2: Entanglement entropy at quantum cri
Page 3: Motivation • Entanglement entropy
Page 7 and 8: Entanglement and Conformal Invarian
Page 9 and 10: Spin Liquids and Topological States
Page 11 and 12: Challenges • To develop a consist
Page 13 and 14: The Quantum Dimer Model H RK = ∑
Page 15 and 16: Scale Invariant Ground State Wave F
Page 17 and 18: Phase diagram for a quantum eight v
Page 19 and 20: Entanglement Entropy of 2D Quantum
Page 21 and 22: • 〈{φ A 1 }|ρ A |{φ A 2 }〉
Page 23 and 24: • The boundary terms impose conti
Page 25 and 26: • For the QCPs we are discussing
Page 27 and 28: If the A and B share a common bound
Page 29: Conclusions • We examined the beh

Entanglement entropy at quantum critical points: Can you ... - INFN

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