Entanglement Entropy from Spacetime Correlations DIAS STP Seminar Yasaman K. Yazdi Imperial College London October 07, 2020
Outline • General remarks on entanglement entropy (EE). • Entropy of a chain of harmonic oscillators. • The need for a spacetime definition of EE and the cutoff. • Spacetime EE of a Gaussian scalar field and example applications. • Extensions to non-Gaussian and interacting theories. • Future directions and summary.
Entanglement Entropy Entanglement entropy is a measure of our limited access to a quantum system. For density matrix ρ on a spatial hypersurface Σ S = − Tr ρ ln ρ (1) If Σ is divided into complementary subregions A and B , then the reduced density matrix for subregion A is ρ A = Tr B ρ (2) and its entanglement entropy with region B is S A = − Tr ρ A ln ρ A (3)
Complementarity and Area Laws Complementarity of EE: we get the same answer whether we trace out the degrees of freedom in region A or region B. S A = − Tr ρ A ln ρ A = − Tr ρ B ln ρ B = S B (4) EE often obeys spatial area laws S A ∝ | ∂ A | /ℓ d (5) UV
Is Black Hole Entropy Entanglement Entropy? • R D Sorkin, On the Entropy of the Vacuum outside a Horizon , (1983), arXiv:1402.3589. EE is a promising candidate for the origin of black hole and cosmological horizon entropy, which is still a major open question.
Applications of Entanglement Entropy (EE) • Quantum Gravity: key insight to connect QM & GR • Fundamental origin of (Bekenstein-Hawking) black hole and cosmological horizon entropy. (Next slide) • T. Jacobson, EE and the Einstein Equation , (2015). • Lashkari, McDermott, Van Raamsdonk, Comments on QG and Entanglement , (2013). • Theorems in QFT (e.g. c-theorems) & AdS/CFT • Ryu, Takayanagi, Holographic Derivation of EE from AdS/CFT , (2006). • Myers, Sinha, Seeing a c-theorem with Holography , (2010). • Casini, Huerta, A c-theorem for the EE, (2007). • Condensed Matter Physics: topological order & properties of Fermi surfaces. • Kitaev, Preskill, Topological EE , (2006). • B. Swingle, EE and the Fermi Surface , (2010). • Quantum Information: Teleportation & Firewalls • Vidal, Werner, A Computable Measure of Entanglement , (2001). • AMPS, Black Holes: Complementarity or Firewalls? , (2012).
Numerous Papers on Entanglement Entropy As mentioned, EE has been extensively studied in many different fields (QG, QFT, CMT, QI, ...) and it has had many important uses. # of Papers 350 300 250 200 150 100 50 Year 2005 2010 2015 2020 Figure: The number of papers per year with “entanglement entropy” in the abstract, from arXiv.org.
Calculating Entanglement Entropy S A = − Tr ρ A ln ρ A • Numerically using mode expansions or lattice discretizations (eg. in condensed-matter systems) • Analytically in CFTs, e.g. via the Replica Trick: ∂ ∂ n Tr ( ρ n S A = − lim A ) n → 1 • Ryu-Takayanagi formula in holography, using areas of minimal surfaces γ : S A = Area of γ A 4 G • Using the Euclidean path integral (e.g. perturbatively in interacting QFTs) • Using spatial correlation functions � φ ( � x ) φ ( � x ′ ) � (will review next) • Using spacetime correlation functions � φ ( � x ′ , t ′ ) � (focus of x , t ) φ ( � this talk)
Entropy of a 1d Chain of Harmonic Oscillators Consider a chain of oscillators, with nearest-neighbour couplings. To find the EE associated to a subchain, we can do following 1 : The Lagrangian is N max N max � � = 1 q 2 ˆ L ˙ N − V MN ˆ q N ˆ q M (6) 2 N =1 N , M =1 N max � = 1 [ˆ q 2 N − m 2 ˆ q 2 q N ) 2 ] , ˙ N − k (ˆ q N +1 − ˆ (7) 2 N =1 where k is the coupling strength between the oscillators, and in terms of the spatial UV cutoff a , k = 1 / a 2 . √ C ≡ V (8) 1 Bombelli, Koul, Lee, Sorkin, Quantum Source of Entropy for Black Holes, PRD 34, 373, 1986.
Entropy of a Chain of Harmonic Oscillators Now consider the division of the chain into a subchain and its complement (Greek and Latin indices resp.). It is convenient to rewrite C in terms of blocks referring to these two subsets: � C ab � � C ab � C a β C a β and its inverse C AB = C AB = , C α b C αβ C α b C αβ and the inverse of each block will be expressed with tildes (for C ab is the inverse of C ab ). example � � det ( � C ab ) e − 1 1 4 � 2 C ab ( q a q b + q ′ a q ′ b ) e C αβ C α a C β b ( q + q ′ ) a ( q + q ′ ) b ρ red ( q a , q ′ b ) = π Finally, S = − Tr ρ red ln ρ red can be expressed in terms of the b ≡ C ac C c α � eigenvalues λ n of the operator Λ a C αβ C β b , as � � � � � { ln(1 S = λ n ) + 1 + λ n ln( 1 + 1 /λ n + 1 / λ n ) } . (9) 2 n
Spatial Area Laws in 1 + 1d We can compare to CFT results for the EE between a shorter line-segment and a longer one containing it. S for m = 0 and a sub-interval with two boundaries takes the asymptotic form for a → 0 of 2 S ∼ 1 3 ln[ L π a sin( πℓ L )] + c 1 , (10) where a is the UV cutoff, ℓ and L are the lengths of the shorter and longer intervals, and c is a non-universal constant. In the limit ℓ L → 0: � ℓ � S ∼ 1 3 ln + c 1 . (11) a 2 P. Calabrese and J. Cardy, Entanglement Entropy and Conformal Field Theory, JPA: Mathematical and Theoretical 42 (2009), no. 50.
Periodic Boundary Conditions: q 1 = q N +1 With finite mass (for IR regularity), 1 / m 2 = k = 10 6 , the entropy obeys the expected logarithmic scaling with the UV cutoff. 3 S 5.4 5.2 5.0 4.8 4.6 � � � 5 10 15 20 25 30 Figure: S vs. ℓ/ a , along with a fit to S = b 1 ln( ℓ/ a ) + c 1 . The best fit parameters are b 1 = 0 . 3337 and c 1 = 5 . 9316. 3 YKY, Zero Modes and EE, JHEP 04 (2017) 140.
Seeking a Spacetime Definition of EE • EE requires a UV cutoff to render it finite. Need a covariant cutoff in gravitational systems, esp. those with horizons. • Quantum fields are too singular to always admit meaningful restrictions to hypersurfaces. (Next slide) • Causal sets, which provide a fundamental and covariant UV cutoff, require a spacetime formulation of EE.
Quantum Fields Live more Happily in Spacetime Consider the normal-ordered φ 2 operator, and smear it with a test function with compact support on a time t ′ = const slice, � � � � d 3 pd 3 k k e i ( p + k ) · x + . . . : φ 2 ( t ′ , f ) := d 4 xf ( � x ) δ ( x 0 − t ′ ) 2(2 π ) 6 � E � a � p a � p E � k � p | 2 + m 2 . We then square the result and compute where E � p = | � its expectation value in the Minkowski vacuum, � d 3 p d 3 k | ˜ f ( � p ) | 2 k + � � 0 | : φ 2 ( t ′ , f ) :: φ 2 ( t ′ , f ) : | 0 � = (12) 2(2 π ) 12 E � p E � k � � d 3 k d 3 p ′ | ˜ p ′ ) | 2 ∝ f ( � � . k ) 2 + m 2 � � p ′ − � k 2 + m 2 ( � f is the Fourier inverse of f and p ′ = p + k . The k -integral where ˜ diverges linearly in the large | � k | limit.
Causal Set Theory: Spacetime is Fundamentally Discrete 4 A causal set is a locally finite partially ordered set. It is a set C along with an ordering relation � that satisfy: • Reflexivity: for all X ∈ C , X � X . • Antisymmetry: for all X , Y ∈ C , X � Y � X implies X = Y . • Transitivity: for all X , Y , Z ∈ C , X � Y � Z implies X � Z . • Local finiteness: for all X , Y ∈ C , | I ( X , Y ) | < ∞ , where | · | denotes cardinality and I ( X , Y ) is the causal interval defined by I ( X , Y ) := { Z ∈ C| X � Z � Y } . Order + Number = Geometry 4 Bombelli, L., Lee, J. H., Meyer, D. and Sorkin, R. D., 1987, Space-Time as a Causal Set, Phys. Rev. Lett. 59, 521.
Hasse Diagrams for a Sample of 6 Element Causal Sets Figure: Relations not implied by transitivity are drawn in as lines. Time goes upwards.
Causal Set Sprinklings Sprinkling : generates a causal set from a given Lorentzian manifold M , by placing points at random in M via a Poisson process with “density” σ , such that P ( N ) = ( σ V ) N N ! e − σ V . N ∼ V Lorentz invariant and nonlocal.
Spacetime Definition of S for Gaussian Theory 5 Express S directly in terms of the spacetime correlation functions. The entropy can be expressed as a sum over the solutions λ of the generalized eigenvalue problem W v = i λ ∆ v , (∆ v � = 0) (13) as � S = λ ln | λ | . (14) λ W and i ∆ are the Wightman W ( x , x ′ ) = � 0 | φ ( x ) φ ( x ′ ) | 0 � and Pauli-Jordan i ∆( x , x ′ ) = [ φ ( x ) , φ ( x ′ )] functions. Also i ∆ = i ( G R − G A ) = 2 Im ( W ). 5 Sorkin, Expressing Entropy Globally inTerms of (4D) Field Correlations , 2012, arXiv:1205.2953.
Spacetime Definition of S for Gaussian Theory First we calculate the entropy with the replica trick. The field theory entropy can be broken up into a sum of entropies of single degrees of freedom { q , p } . ρ ( q , q ′ ) ≡ � q | ρ | q ′ � = N e − A / 2( q 2 + q ′ 2 )+ iB / 2( q 2 − q ′ 2 ) − C / 2( q − q ′ ) 2 , (15) Replica trick: ∂ ∂ n Tr ( ρ n ) , S = − Tr ( ρ log ρ ) = − lim (16) n → 1 � Tr ( ρ n ) = N n dq 1 ... dq n ρ ( q 1 , q 2 ) ρ ( q 2 , q 3 ) ...ρ ( q n , q 1 ) � � � n n � � = | 1 − µ | n N n d n q exp q 2 = − ( A + C ) i + C q i q i +1 | 1 − µ n | i =1 i =1 √ 1+2 C / A − 1 √ where µ = 1+2 C / A +1 . We insert this into (16) and take the limit to obtain the entropy.
Recommend
More recommend
