entanglement entropy of disconnected regions - Staff.city.ac.uk - City ...
entanglement entropy of disconnected regions - Staff.city.ac.uk - City ... entanglement entropy of disconnected regions - Staff.city.ac.uk - City ...
– The maximally entangled state |gs〉 = 1 √ 2 (| ↑ ↓ 〉 + | ↓ ↑ 〉): ρ 1 st spin = 1 ( 2 (| ↑ 〉〈 ↑ |+| ↓ 〉〈 ↓ |) ⇒ S 1 st spin = −2× − log( √ ) 2) = log(2) • In general the entanglement entropy is not “directional”, that is S A = S A . • We can think of the entropy as counting the number of “connections” between A and Ā.
Scaling limit • Say |gs〉 is a ground state of some local spin-chain Hamiltonian, and that the chain is infinitely long. • An important property of |gs〉 is the correlation length ξ: 〈gs|ˆσ iˆσ j |gs〉 ∼ e −|i−j|/ξ as |i − j| → ∞ • If there are parameters in the Hamiltonian (e.g. temperature, magnetic field etc) such that for certain values ξ → ∞. This is a quantum critical point. • We may adjust these parameters in such a way that L, ξ → ∞ in such as way that the length L of A is proportional to ξ: L/ξ = mr. • This is the scaling limit, and what we obtain is a quantum field theory. m here is a mass scale – we may have many masses m α associated to many correlation lengths – and r is the dimensionful length of region A in the scaling limit. • The resulting entanglement entropy has a universal part: a part that does not depend very much on the details of the Hamiltonian.
- Page 1: Entanglement entropy in 1+1 dimensi
- Page 4 and 5: How and why to measure (or quantify
- Page 8 and 9: Short- and large-distance entanglem
- Page 10 and 11: Partition functions on multi-sheete
- Page 12 and 13: • Another twist field ˜T is asso
- Page 14 and 15: Hence we have Short- and large-dist
- Page 16 and 17: The entropy of disconnected regions
- Page 18 and 19: The entropy of disconnected regions
- Page 20 and 21: Some simple limit cases • When r
- Page 22 and 23: Non-extensivity • In recent years
Scaling limit<br />
• Say |gs〉 is a ground state <strong>of</strong> some local spin-chain Hamiltonian, and that the chain is<br />
infinitely long.<br />
• An important property <strong>of</strong> |gs〉 is the correlation length ξ:<br />
〈gs|ˆσ iˆσ j |gs〉 ∼ e −|i−j|/ξ as |i − j| → ∞<br />
• If there are parameters in the Hamiltonian (e.g. temperature, magnetic field etc) such that<br />
for certain values ξ → ∞. This is a quantum critical point.<br />
• We may adjust these parameters in such a way that L, ξ → ∞ in such as way that the<br />
length L <strong>of</strong> A is proportional to ξ: L/ξ = mr.<br />
• This is the scaling limit, and what we obtain is a quantum field theory. m here is a<br />
mass scale – we may have many masses m α associated to many correlation lengths –<br />
and r is the dimensionful length <strong>of</strong> region A in the scaling limit.<br />
• The resulting <strong>entanglement</strong> <strong>entropy</strong> has a universal part: a part that does not depend<br />
very much on the details <strong>of</strong> the Hamiltonian.
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