Lattice study of quantum entanglement in SU(3) Yang-Mills theory at zero and finite temperatures

Transcription

1 Lattice study of quantum entanglement in SU(3) Yang-Mills theory at zero and finite temperatures Yoshiyuki Nakagawa Graduate School of Science and Technology, Niigata University, Igarashi-2, Nishi-ku, Niigata , Japan in collaboration with A. Nakamura (RIISE, Hiroshima University) S. Motoki (KEK) V.I. Zakharov (ITEP, MPI) ExtreMexico QCD(XQCD11) San Carlos, Mexico, July 18-20, 2011

2 Outline Motivation Entanglement entropy in quantum mechanics Entanglement entropy in quantum field theory How to measure the entanglement entropy Replica method Holographic approach Lattice QCD simulations Summary and outlook

3 Motivation T deconfinement phase effective d.o.f. = gluons (colorful) SU(N c ) pure Yang-Mills theory T c ~ 300[MeV] ( in SU(3)) confinement phase effective d.o.f. = glueballs (colorless) color confinement effective d.o.f. = gluons (colorful) asymptotic freedom l -1

4 Motivation T deconfinement phase effective d.o.f. = gluons (colorful) SU(N c ) pure Yang-Mills theory T c ~ 300[MeV] ( in SU(3)) confinement phase effective d.o.f. = glueballs (colorless) color confinement lc? effective d.o.f. = gluons (colorful) asymptotic freedom l -1

5 Entanglement entropy Measures how much a given quantum state is entangled quantum mechanically. A non-local quantity like the Wilson loop as opposed to correlation functions We can probe the quantum properties of the ground state for a quantum system (quantum spin system, quantum Hall liquid,... ). Proportional to the degrees of freedom The entanglement entropy can be used as an order parameter Applications: quantum information and computing, condensed matter physics,...

6 Entanglement entropy : definition Divide a system into two subsystems A and B The density matrix on A is defined by ρ A = Tr B (ρ AB ), i.e., by tracing over the states of the subsystem B. ρ A can be regarded as the density matrix for an observer who can only access to the subsystem A. Entanglement entropy as the von Neumann entropy S A = Tr A (ρ A ln ρ A ) B A

7 Entanglement entropy : a simple example Consider two spin 1/2 particles ( a 2 + b 2 + c 2 + d 2 = 1) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ψ = a + b + c + d A B A The density matrix ρ AB = ψ ψ is given by ( ) ( ) ( ) ( ) ρ AB = aa + ab A B A B The reduced density matrix ρ A = is ( ) aa + bb ac + bd ρ A = Tr B (ρ AB ) = ca + db cc + dd B A B A B

8 Entanglement entropy : a simple example The eigenvalues of the reduced density matrix ρ A are λ ± = 1 (1 ± ) 1 4 ad bc 2 2 The entanglement entropy is S A = Tr A (ρ A ln ρ A ) = i λ i ln λ i For a separable (product) state, e.g., ψ = 1 2 ( A + A ) B = S A = 0 For an entangled state, e.g., ψ = 1 ( ) A B + A B 2 = S A = ln 2

9 Entanglement entropy in a quantum field theory Divide spacetime into two regions A and B Entanglement between two subregions = correlations between separeted systems ρ A of the ground state t B c A ρ A = Tr B ρ = Tr B 0 0 x l The entanglement entropy S A (l) = Tr A (ρ A ln ρ A ) How to calculate the entanglement entropy? 1. replica method [Carabrese and Cardy (2004), Callan and Wilczek (1994)] 2. holography [Ryu and Takayanagi (2006)]

10 Replica method The entanglement entropy (Tr A ρ A = 1) S A (l) = Tr A (ρ A ln ρ A ) = lim α 1 α ln Tr A ρ α A Entanglement entropy can be expressed as S A (l) = ( ) Z(l, α) lim α 1 α ln Z α = lim F [l, α] + F α 1 α where Z(l, α) is the partition function on the α-sheeted Riemann surface. Z(l,α) = Z = B B A A αβ β

11 Some examples For (1 + 1)-dimensional model at the critical point (CFT), [Holzhey, Larsen and Wilczek, 1994, Calabrese and Cardy, 2004] S A (l) = c 3 log l a + c 1, where c is the central charge of CFT, a the UV cutoff.

12 Some examples For (1 + 1)-dimensional model at the critical point (CFT), [Holzhey, Larsen and Wilczek, 1994, Calabrese and Cardy, 2004] S A (l) = c 3 log l a + c 1, where c is the central charge of CFT, a the UV cutoff. Not in the critical regime, S A (l) l ξ c 3 log ξ a, B ξ A where ξ is the correlation length of the system. l

13 Some examples For (1 + 1)-dimensional model at the critical point (CFT), [Holzhey, Larsen and Wilczek, 1994, Calabrese and Cardy, 2004] S A (l) = c 3 log l a + c 1, where c is the central charge of CFT, a the UV cutoff. Not in the critical regime, S A (l) l ξ c 3 log ξ a, B ξ A where ξ is the correlation length of the system. l S A is the amount of quantum correlations between subregions. S A can serve as an order parameter for a quantum phase transition.

14 Entanglement entropy : holographic approach AdS/CFT correspondence argues that the supergravity on (d + 2)-dimensional anti-de Sitter space AdS d+2 is equivalent to a (d + 1)-dimensional conformal field theory living on the boundary of AdS d+2 [Maldacena, 1998].

15 Entanglement entropy : holographic approach AdS/CFT correspondence argues that the supergravity on (d + 2)-dimensional anti-de Sitter space AdS d+2 is equivalent to a (d + 1)-dimensional conformal field theory living on the boundary of AdS d+2 [Maldacena, 1998]. It has been proposed that the entanglement entropy S A in (d + 1)-dimensional CFT can be computed from the area law [Ryu and Takayanagi, 2006] S A (l) = area of γ A 4G (d+2), N where γ A is the d-dimensional static minimal surface in AdS d+2 with γ A = A, and G (d+2) N is the (d + 2)-dimensional Newton constant (cf. Bekenstein-Hawking formula for black hole entropy)

16 Holographic approach : example The gravitational theories on AdS 3 space of radius R are dual to (1 + 1)-dimensional CFTs with the central charge c = 3R/2G 3 N t 2$l/L Metric of AdS 3 ds 2 = R 2 ( cosh ρ 2 dt 2 + dρ 2 + sinh ρ 2 dθ 2 ) The subsystem A is the region 0 θ 2πl/L. γ A is the static geodesic which connects the boundary of A traveling inside AdS 3. B #A A "! The geodesic distance L γa is given by ( ) LγA cosh = sinh 2 ρ 2 0 sin 2 πl R L

17 Holographic approach : example Assuming exp(ρ 0 ) 1, the entanglement entropy is S A (l) c ( )] πl [exp(ρ 3 log 0 ) sin, exp(ρ 0 ) L/a, L which coincides with the result obtained by using the replica trick.

18 Holographic approach : example Assuming exp(ρ 0 ) 1, the entanglement entropy is S A (l) c ( )] πl [exp(ρ 3 log 0 ) sin, exp(ρ 0 ) L/a, L which coincides with the result obtained by using the replica trick. In (3 + 1)-dimensional N = 4 SYM considered to be dual to AdS 5 S 5 background in type IIB string theory [Ryu and Takayanagi, 2006] c (2 + 6) }{{} gauge + real scalar 1 A S A(l) = c N c 2 a 2 c N 2 c l 2, AdS result + } {{} = free field Majorana

19 holographic approach : confining background Example: AdS soliton solution [Witten, 1998] [Nishioka and Takayanagi, 2007] ds 2 = R 2 dr 2 r 2 f(r) + r2 ( dt 2 R 2 + f(r)dχ 2 + dx dx 2 ) r 4 2, f(r) = 1 0 r 4 dual to N = 4 SYM on R 1,2 S 1 the SUSY broken due to the APBC for fermions along the χ direction

20 holographic approach : confining background Example: AdS soliton solution [Witten, 1998] [Nishioka and Takayanagi, 2007] ds 2 = R 2 dr 2 r 2 f(r) + r2 ( dt 2 R 2 + f(r)dχ 2 + dx dx 2 ) r 4 2, f(r) = 1 0 r 4 dual to N = 4 SYM on R 1,2 S 1 the SUSY broken due to the APBC for fermions along the χ direction scalar fields acquire non-zero masses from radiative corrections

21 holographic approach : confining background Example: AdS soliton solution [Witten, 1998] [Nishioka and Takayanagi, 2007] ds 2 = R 2 dr 2 r 2 f(r) + r2 ( dt 2 R 2 + f(r)dχ 2 + dx dx 2 ) r 4 2, f(r) = 1 0 r 4 dual to N = 4 SYM on R 1,2 S 1 the SUSY broken due to the APBC for fermions along the χ direction scalar fields acquire non-zero masses from radiative corrections almost the same as the (2+1)-dimensional pure YM theory

22 holographic approach : confining background Example: AdS soliton solution [Witten, 1998] [Nishioka and Takayanagi, 2007] B γa disconnected surface A l IR cutoff connected surface S A (l) = area of γ A 4G (d+2), N S(l) lc connected 2 solution ~ O(Nc ) disconnected solution ~ O(1) l

23 holographic approach : confining background Example: AdS soliton solution [Witten, 1998] [Nishioka and Takayanagi, 2007] B γa disconnected surface A l IR cutoff connected surface S A (l) = area of γ A 4G (d+2), N S(l) lc connected 2 solution ~ O(Nc ) disconnected solution ~ O(1) l Rapid transition at l c from O(N 2 c ) solution (gluons) to O(1) solution (glueballs)

24 holographic approach : confining background Generalization of the area law formula to non-conformal theories [Klebanov, Kutasov,and Murugan, 2008] S A = 1 4G (10) N d 8 σe 2φ G (8) ind.

25 holographic approach : confining background Generalization of the area law formula to non-conformal theories [Klebanov, Kutasov,and Murugan, 2008] S A = 1 4G (10) N d 8 σe 2φ G (8) ind. nonanalytic behavior for backgrounds: Klebanov-Strassler D4-branes on a circle D3-branes on a circle ((2 + 1)-dim. theory) Soft wall model (e z2 dilaton) Other behavior for backgrounds: Maldacena-Nunez Hard wall model Soft wall model (e zn, n < 2) 0 S / l ~ N c 2 / l 3 l l c ~ O(1)

26 Aim of this study Entanglement entropy proportional to effective degrees of freedom Holographic approach predicts a sharp transition at zero temperature from O(Nc 2 ) solution (gluons) at small l to O(1) solution (glueballs) at large l gluons 0 S / l ~ N c 2 / l 3 l l c glueballs ~ O(1)

27 Aim of this study Entanglement entropy proportional to gluons effective degrees of freedom Holographic approach predicts a sharp 2 3 ~ N c / l transition at zero temperature from O(Nc 2 ) solution (gluons) at small l ~ O(1) to O(1) solution (glueballs) at large l l c l Investigate the properties of the QCD vacuum via a gauge invariant 0 S / l glueballs quantity Can we find such a sharp transition from gluon phase to glueball phase by measuring the entanglement entropy?

28 Entanglement entropy in the lattice gauge theory (Exactly solvable) (1 + 1)-dimensional SU(N c ) lattice gauge theory analytic behavior (independent of l) [Velytsky, 2008] (3 + 1)-dimensional SU(2) lattice gauge theory treated within Migdal-Kadanoff approximation nonanalytic change at l c T c (1.56, 1.66) [Velytsky, 2008]

29 Entanglement entropy in the lattice gauge theory (Exactly solvable) (1 + 1)-dimensional SU(N c ) lattice gauge theory analytic behavior (independent of l) [Velytsky, 2008] (3 + 1)-dimensional SU(2) lattice gauge theory treated within Migdal-Kadanoff approximation nonanalytic change at l c T c (1.56, 1.66) [Velytsky, 2008] SU(2) quenched simulation [Buividovich and Polikarpov, 2008] 1/ da ds f (l)/dl, fm a = fm a = 0.10 fm a = 0.11 fm a = 0.12 fm a = 0.14 fm C l -3 fit C(l) a = fm a = 0.10 fm a = 0.11 fm a = 0.12 fm a = 0.14 fm l, fm l, fm

30 Lattice QCD simulations SU(3) quenched lattice simulations 5000 sweeps for thermalization, measurement every 100 sweeps, confs. Pseudo heat-bath MC update: Wilson plaquette action S W = β ( 1 1 ) Tr(U p + U 2N p) p c Mesure the derivative of S A (l) with respect to l S A (l) = [ ( )] Z(l, α, T ) lim dl l α 1 α ln Z α (T ) = lim F [l, α, T ] α 1 l α

31 Estimate the derivative by Lattice QCD simulations S A (l) dl = lim α 1 l F [A, α] α l lim (F [l, α + 1] F [l, α]) α 1 F [l + a, 2] F [l, 2] a

32 Lattice QCD simulations Differneces of free energies [Endrodi et al., PoS LAT2007] Z(λ) = Dφ exp ( (1 λ)s 1 [φ] λs 2 [φ]) F 2 F 1 = 1 0 dλ ln Z(λ) = λ 1 0 dλ S 2 [φ] S 1 [φ] λ S2 S1

33 Action difference , β=5.7, l = a ~ 0.17[fm] 2000 < S 2 - S 1 > λ Contribution from λ > 0.5 and λ < 0.5 almost cancel.

34 Results: derivative of S A (l) wrt l 70 ( 1 / A ) S / l [fm -3 ] , β= , β= , β= , β= , β= , β=6.00 c / l d, c=0.161(49), d=3.03(18) 0 S / l ~ N c 2 / l 3 l l c ~ O(1) l [fm] S A / l A N 2 c /l 3 for N = 4 SYM in(3+1)-dim.

35 Results: derivative of S A (l) wrt l ( 1 / A ) S / l [fm -3 ] , β= , β= , β= , β= , β= l [fm] No clear discontinuity has been observed.

36 Results: entropic C-function C(l) , β= , β= , β= , β= , β= l [fm] C(l) = l3 S A (l) A l

37 Entanglement entropy at finite temperature Entanglement entropy A thermal state is a mixed state, S A = Tr A (ρ A ln ρ A ), ρ A = Tr B ρ, ρ AB = n exp ( E ) n n n T

38 Entanglement entropy at finite temperature Entanglement entropy S A = Tr A (ρ A ln ρ A ), ρ A = Tr B ρ, A thermal state is a mixed state, ρ AB = ( exp E ) n n n T n For (1 + 1)-dimensional CFT at finite temperature T = 1/β [Calabrese and Cardy, 2004] S A (l) = c 3 log ( β πa ) πl sinh + c 1 = β c 3 log l a + c 1 l β, πc 3β l + c 1 l β.

39 Entanglement entropy at finite temperature Entanglement entropy S A = Tr A (ρ A ln ρ A ), ρ A = Tr B ρ, A thermal state is a mixed state, ρ AB = ( exp E ) n n n T n For (1 + 1)-dimensional CFT at finite temperature T = 1/β [Calabrese and Cardy, 2004] c S A (l) = c ( ) β 3 log πl sinh + c 3 log l a + c 1 l β, 1 = πa β πc 3β l + c 1 l β. At sufficiently large l (or in the high temperature limit), S A reduces to the thermal entropy.

40 Results: ds A /dl at finite temperatures (below T c ) 50 ( 1 / A ) ds / dl [fm -3 ] , β=5.70, T/T c ~ , β=5.82, T/T c ~ , β=5.86, T/T c ~ l [fm]

41 Results: ds A /dl at finite temperatures (above T c ) 150 ( 1 / A ) ds / dl [fm -3 ] , β=6.03, T/T c ~ , β=5.84, T/T c ~1.44 a / l 3 + b, a=0.187(14), b=61.8(15) a / l 3 + b, a=0.180(16), b=14.1(8) l [fm]

42 Results: ds A /dl at finite temperatures Rough estimates : Figures taken from Boyd et al., NPB469, 419 (1996) ( ) (8) s ( ) (15)

43 Summary We discussed the entanglement entropy in SU(3) pure YM theory. Entanglement entropy measures amount of quantum correlations between subregions S/ l behaves as 1/l 3 at small l, and vanishes at large l. Clear discontinuity has not been observed at zero temperature. In the deconfinement phase, entanglement entropy approaches to a finite value at large l, comparable to the thermal entropy.

44 The entanglement entropy can be written as (Tr A ρ A = 1) Replica method S A (l) = Tr A (ρ A ln ρ A ) S A (l) = lim α 1 α ln Tr A ρ α A The total density matrix ρ is (Z(T ) = Tr exp( βh)) ρ[φ ( x), φ ( x)] = Z 1 (T ) φ ( x) exp( βh) φ ( x), or, in the path integral expression, ρ[φ ( x), φ ( x)] = Z 1 (T ) φ( x,t=β)=φ ( x) φ( x,t=0)=φ ( x) Dφ exp ( S E )

45 Replica method contd. The total density matrix ρ[φ ( x), φ ( x)] = Z 1 (T ) φ ( x) exp( βh) φ ( x) = Z 1 (T ) φ( x,t=β)=φ ( x) φ( x,t=0)=φ ( x) Dφ exp ( S E ) Z(T ) = Tr exp( βh) is found by setting φ ( x) = φ ( x) and integrating over φ ( x). The reduced density matrix, ρ A [φ ( x), φ ( x)] = Tr B ρ, can be obtained by imposing the boundary conditions φ( x, t = 0) = φ ( x) and φ( x, t = β) = φ ( x) if x A φ( x, t = 0) = φ( x, t = β) if x B

46 c Replica method contd. α-th power of the reduced density matrix ρ α A [φ ( x), φ ( x)] = x A Dφ 1 φ α 1 ρ A [φ ( x), φ 1 ( x)]ρ A [φ 1 ( x), φ 2 ( x)] ρ A [φ α 1 ( x), φ ( x)] B A The trace of ρ α A is found to be Tr ρ α A = x A Dφ 1 φ α β ρ A [φ 1 ( x), φ 2 ( x)]ρ A [φ 2 ( x), φ 3 ( x)] ρ A [φ α ( x), φ 1 ( x)], Tr ρ α A is obtained by imposing the periodic boundary condition in time with period αβ if x A and with period β if x B.

47 Replica method contd. The entanglement entropy (Tr A ρ A = 1) S A (l) = Tr A (ρ A ln ρ A ) = lim α 1 α ln Tr A ρ α A Entanglement entropy can be expressed as S A (l) = ( ) Z(l, α) lim α 1 α ln Z α = lim F [l, α] + F α 1 α where Z(l, α) is the partition function on the α-sheeted Riemann surface. Z(l,α) = Z = B B A A αβ β

48 Analyticity of Tr ρ A See [Calabrese and Cardy, quant-ph/ ] Tr ρ α A = Z(l, α, T ) Z α (T ), Tr ρ α A = i λ i, 0 λ i < 1 Tr ρ α A is absolutely convergent and analytic for all Re α > 1. The derivative with respect to α exists and analytic in the region. If ρ A = i λ i ln λ i is finite, then the limit as α 1 + of the first derivative coverges to ρ A. Z(l, α, T )/Z α (T ) has a unique analytic continuation to Re α > 1.