Entanglement entropy: From Field Theory to Condensed Matter Pasquale Calabrese SISSA-Trieste ERG 2016, Trieste, 23/9/2016 Based on collaboration with: J. Cardy, V. Alba, M. Fagotti, E. Tonni....
Why entanglement entropy? hep-th arXiv preprints with “entanglement” in the title 250" 220" 200" 170" 150" 119" 100" 56" 62" 76" 40" 39" 37" 35" 50" 0" 2" 0" 0" 4" 0" 0" 6" 3" 3" 3" 0" 11" 13" 22" 0" 91" 92" 93" 94" 95" 96" 97" 98" 99" 00" 01" 02" 03" 04" 05" 06" 07" 08" 09" 10" 11" 12" 13" 14" 15" Year%
Why entanglement entropy? hep-th arXiv preprints with “entanglement” in the title 250" 220" 200" 170" 150" 119" 100" 56" 62" 76" 40" 39" 37" 35" 50" 0" 2" 0" 0" 4" 0" 0" 6" 3" 3" 3" 0" 11" 13" 22" 0" 91" 92" 93" 94" 95" 96" 97" 98" 99" 00" 01" 02" 03" 04" 05" 06" 07" 08" 09" 10" 11" 12" 13" 14" 15" Year%
Many-body quantum systems When many particles do not interact, their properties follow straightforwardly from those of few Free Fermions Free Bosons = metals = superfluids Interactions dramatically change this paradigm especially in low dimensions
“More is different” PW Anderson 1972 Interactions give rise to new phases of matter Spin-Charge separation Non-abelian Statistics Unconventional Superconductors Interacting particles Confined phases (QCD) Mott insulators Topological states The properties of many do not follow simply from those of few: “more is truly different!”
“The complexity frontier” How to describe these many-body systems? Numerically? Too difficult, e.g. for a spin-chain ⇤ | Ψ ⇤ = A s 1 s 2 ...s N | s 1 , s 2 , . . . s N ⇤ s i = ± 2 N coefficients: too many for a classical PC
“The complexity frontier” How to describe these many-body systems? Numerically? Too difficult, e.g. for a spin-chain ⇤ | Ψ ⇤ = A s 1 s 2 ...s N | s 1 , s 2 , . . . s N ⇤ s i = ± 2 N coefficients: too many for a classical PC We need a criterion that sets physical states apart from the others Entanglement is this criterion
Entanglement entropy Consider a system in a quantum state | ψ 〉 ( ρ =| ψ 〉〈 ψ |) H = H A ⇥ H B A Alice can measure only in A, while Bob in the remainder B B Alice measures are entangled with Bob’s ones: Schmidt deco � � c 2 | Ψ � = c n | Ψ n � A | Ψ n � B c n ⌅ 0 , n = 1 n n ● If c 1 =1 ⇒ | ψ 〉 unentagled ● If c i all equal ⇒ | ψ 〉 maximally entangled
Entanglement entropy Consider a system in a quantum state | ψ 〉 ( ρ =| ψ 〉〈 ψ |) H = H A ⇥ H B A Alice can measure only in A, while Bob in the remainder B B Alice measures are entangled with Bob’s ones: Schmidt deco � � c 2 | Ψ � = c n | Ψ n � A | Ψ n � B c n ⌅ 0 , n = 1 n n ● If c 1 =1 ⇒ | ψ 〉 unentagled ● If c i all equal ⇒ | ψ 〉 maximally entangled A natural measure is the entanglement entropy ( ρ A =Tr B ρ ) S A ≡ - Tr ρ A ln ρ A = S B basis independent = - ∑ c n ln c n 2 2
Entanglement in extended systems If | ψ 〉 is the ground state of a local Hamiltonian B Area Law A [Srednicki ’93 S A ∝ Area separating A and B +many more] If the Hamiltonian has a gap
Entanglement in extended systems If | ψ 〉 is the ground state of a local Hamiltonian B Area Law A [Srednicki ’93 S A ∝ Area separating A and B +many more] If the Hamiltonian has a gap In a 1+1 D CFT Holzhey, Larsen, Wilczek ’94 B A B l S A = c ln l _ 3 This is the most effective way to determine the central charge
Importance Only a tiny fraction of states satisfy the area Area law states law (or small violations) If we can limit the search for the ground state to this small subset, the Full Hilbert space complexity of the problem is exponentially reduced One meaning of S A : S A gives the amount of classical information required to specify | Ψ ⟩
Tensor network states A new and powerful set of numerical methods based on entanglement content of quantum states MPS MERA PEPS “Alphabet soup of proposals” Subir Sachdev
Matrix Product States (MPS) h i X A [1] s 1 . . . A [ N ] | ⇤ � = | s 1 . . . s N � . Tr s N s 1 ,...,s N = ± • For each site there are two matrices A [i] of finite dimension χ×χ . ± More entanglement can be stored as χ increases. • The famous DMRG is a practical way to find a variational MPS • At fixed χ , the maximum entanglement entropy of an MPS is ln χ • 1D area is a number ⇒ entanglement entropy constant ⇒ an MPS with finite χ can describe it • In d dimensions, area law N d − 1 ⇒ χ needs to be χ ∼ exp (N d − 1 )
Entanglement entropy and path integral PC, J Cardy 2004 The density matrix at temperature β -1 Z [ d φ ( x , τ )] ⟨ Φ 1 | ρ | Φ 2 ⟩ = Y Y δ ( φ ( x , β ) � φ 1 ( x )) e − S E = δ ( φ ( x , 0) � φ 2 ( x )) Z x x R The trace sews together the edges along τ = 0 and τ = β to form a cylinder of circumference β . A = (u, v) : ρ A sews together only those points x which are not in A, leaving an open cut along the τ = 0. ⇤ Φ 1 ( x ) | ρ A | Φ 2 ( x ) ⌅ =
Replicas and Riemann surfaces PC, J Cardy 2004 ∂ ∂ n Tr ρ n S A = − Tr ρ A log ρ A = − lim A n → 1 n For n integer, Tr ρ A is obtained by sewing cyclically n cylinders above. This is the partition function on a n- sheeted Riemann surface Tr ρ n A = n Renyi EE: S A ≡ 1/(1- n ) ln Tr ρ A
Riemann surfaces and CFT This Riemann surface is mapped to the plane by PC, J Cardy 2004 ⇥ 1 / n � w → � = w − u w − v ; � → z = � 1 / n ⇒ w → z = w − u w − v = c n | u − v | − c 6 ( n − 1 / n ) n Tr ρ A = | u-v | = l ⇤ A = c ⇤ n Tr ⇥ n S A = − lim 3 log ⌅ n → 1 n Tr ρ A is equivalent to the 2-point function of twist fields ⇤ T T ⌅ ⇧ n � 1 ⌃ ∆ T n = c A = ⌅ T n ( u ) ¯ Tr � n with scaling dimension T n ( v ) ⇧ 12 n
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