Entanglement Entropy in 2+1 Chern-Simons Theory Shiying Dong UIUC With: Eduardo Fradkin, Rob Leigh, Sean Nowling arxiv: hep-th/0802.3231 4/27/2008 Great Lakes String Conference @ University of Wisconsin-Madison
Motivation Candidate of black hole entropy 2+1 Gravity, BTZ Order parameter of topological states Fractional quantum Hall effect p+ip superconductors Quantum computing with anyons
Definition For a system consisting of two subsystems A, B, from any pure state φ, the density matrix ρ = φ φ, define the reduced density matrix on A by ρ A = tr B ρ, and the entanglement entropy iss A = tr(ρ A ln ρ A ). For a pure state, S A = S B. The entanglement entropy should depend on the common features of A and B.
Scale Dependence It depends on the interface length scale L, the correlation length ξ, and the ultraviolet cutoff ε. Area law: When the interface is rotational symmetric, the leading term is proportional to the area of the interface. In general, for spatial dimension d, S A = g d 1 ( L ɛ )d 1 + g d 2 ( L ɛ )d 2 + + g 0 ln L ɛ + S 0. H.Casini and M. Huerta 06
Universal Terms In odd d dimensions, or even d dimensions with non-smooth interfaces, the entanglement entropy has a logarithmic divergent term, which is universal. Otherwise, there is a universal constant term. In particular, d=1, S A = β ln L ɛ δ, d=2, S A = αl γ. H.Casini and M. Huerta 06
Calculation Define Z n = tr(ρ n A ), for integer n. There is an unambiguous analytic continuation to real n 1. S A = lim n 1 n Z n. In practice we usually have to normalize it, S A = lim n 1 n Z n Z 1 n. P. Calabrese and J. Cardy 04
2D Free Boson CFT ρ = lim β e βh ρ A = tr B ρ
Z n = tr(ρ A n ) e.g., n=3 u v n-sheeted w
T (w) = ( dz dw )2 T (z) + c 12 Where z = ( w u w v ) 1 n = c(1 1/n2 ) 24 = T (w)φ n(u)φ n (v) Φ n (u)φ n (v) ±n = c 24 (1 1 n 2 ). S A = c 3 {z, w} (v u) 2 (w u) 2 (w v) 2 And so ln( u v ɛ ). P. Calabrese and J. Cardy 04
2D Massive Free Boson A R m 2 ln ZB n = 1 2 B R ξ=1/m R. The Green function is defined on the n-sheeted complex plane. m 2 ln ZB n = 1 2m 2 ( 1 12n + n 2 (mr 1 2 )2 ), S A = 1 6 ln 1 mɛ. d 2 rg n ( r, r) P. Calabrese and J. Cardy 04
2D Massive Free Fermion m ln ZF n = d 2 r trs n ( r, r). m 2 ln ZF n = 1 2m 2 ( 1 24n n 2 (mr 1 2 )2 ), S A = 1 12 ln 1 mɛ. The linear divergence is cancelled between the bosons and fermions. m 2 ln(zb n Zn F ) = 1 24m 2 n (1 + 1 2 ),
Summary for 2D The logarithmic term in the entanglement entropy of 2D free QFT is universal. It is proportional to the conformal anomaly of the system. It is also proportional to the number of interfaces between A and B subsystems.
2+1 Chern-Simons The Hilbert space on a 2d closed surface is spanned by the conformal blocks of the WZW CFT living on that surface. The wavefunctions can be written as the partition function of the gauged WZW model, ψ J(A z )=exp[ ik Tr( 2π JA)] [Dg]exp[ikS + (g, A z, J)] E. Witten 89, 92, Elitzur, Moore, Schwimmer and Serberg, 89
We define a state by doing path integral on a 3D manifold enclosed by the surface. And the density matrix has two manifolds with opposite orientations. Trace over B means to identify the boundary value of the Chern-Simons fields on the two B surfaces, and sum over them properly. This means ρ A = tr B ρ is generated by gluing the two manifolds along their B surfaces. To calculate Z n = tr(ρ n A ), we need to glue n pieces of the ρ A manifolds, and study the CS partition function of the final manifold.
A Simplest Example: S 2 ρ = φ φ ρ A
Tr(ρ A n ) = Z n Z 1 n = Z(S3 ) Z(S 3 ) n S A = lim n 1 = ln S 00 = ln D n Z(S3 ) 1 n = ln Z(S 3 ) E. Witten 89 D = i d 2 i = i ( S 0i S 00 ) 2 = 1 S 00 quantum dimension modular S matrix
Remarks In 2+1 theory, we have S A = αl γ in general. Since Chern-Simons theory is topological, there is no scale dependence, only the topological piece survives. If we move away the topological phase, we can still calculate the topological entropy by computing D γ 0 = S A + S B + S C S AB S AC S BC + S ABC. A C B A. Kitaev and J. Preskill 06
S 2 With Two Interfaces ρ = B2 A B1 φ B1* A* B2* φ b2 b1 b1 b2 A b1 A* ρ A A A* b1 b2 = A b2 A*
Now the final manifold is the connected sum of two S 3 s along n S 2 s, Tr(ρ A n ) = Z n Z 1 n = Z(S3, S 3, n) Z(S 3 ) n = Z(S3 ) 2 Z(S 3 ) 2n. The entanglement entropy is doubled, S A = lim n 1 n Z(S3 ) 2(1 n) = 2 ln D. In general, S 2 with I interfaces gives us S A = I ln D.
Useful Facts The key to generalize the operation is that, if any three manifold is a connected sum of two submanifolds, with their interface supporting only one state, we can cut it into two pieces. On S 2 with two punctures, the Hilbert space is one dimensional if they are a conjugate pair, zero otherwise. A single link inside S 3 has expectation value S 0 j.
General Manifolds and States Finding all the conformal blocks Squeeze all the interfaces Cut and glue around each interface The ears will cancel after the normalization
An Example: 2-Tori A b B Ψ = {i,j,k} φ {i,j,k} {i, j, k} j i k A b B
D 2 A b B k1 S 2 i1 j k2 i2 2n j Z n = {{i,k},j} S 0 j n φ {it,j,k t }φ ψ A ψ A {it,j} ψ B ψ B {kt,j} {i t,j,k t+1 } (S j 0 ) 2 t=1 S 3
General Result Normalize the basis states Z n = n (S j 0 ) 1 n Ψ Ψ = ψ A ψ A ψ B ψ B S 0 j =1 (S 0 j ) 1 n tr(ρ j ) n {{i,k},j} t=1 φ {it,j,k t }φ {i t,j,k t+1 } = j S A = I ln S 0 0 {j i }( # of interfaces I d ji )tr i=1 quantum dimensions all possible configurations around interfaces projected density matrix ( ρ{ji } I i=1 d ln ρ ) {j i } I j i i=1 d. j i
Summary for Chern-Simons The entanglement entropy has a vacuum contribution, which is proportional to the number of interfaces. The nontrivial part comes from the sewing law of CFT. The total entropy is a sum of the traditional entanglement entropy from all the sewing channels. There is a microscopic degeneracy for all the states, associated with the quantum dimensions of the states defined on loops.
Thank you.