Entanglement Entropy with Current & Chemical Potential Bom Soo Kim Department of Physics and Astronomy, University of Kentucky, Lexington 40506

Great Lakes Strings conference University of Cincinnati May 5-7, 2017 Based on 1705.01859

Partition Function & EE with J & µ (3 slides) - Twisted boundary condition & background gauge fields - Partition Function with J & µ - General formula for EE & R´enyi entropy with J & µ

Applications (6 slides)

˜µ Twisted B.C. & A • Dirac fermion in the presence of background gauge fields : S=

1 2π

Z

¯ µ (∂µ + iAµ ) ψ , d 2 x i ψγ

– on torus with τ = τ1 + iτ2 1 – for coordinates ζ = 2π (s + it) is identified as ζ ≡ ζ + 1 ≡ ζ + τ .

• twisted boundary conditions ψ(t, s) = e −2πia ψ(t, s + 2π) = e −2πib ψ(t + 2πτ2 , s + 2πτ1 ) . ˜ µ = (˜ ˜ ≡ periodic Dirac fermion with flat gauge connection A µ, J) n a=J ˜ = πi [(b−τ a)d ζ¯−(b − τ¯a)dζ] = ads + b−aτ1 dt → A . τ2 τ2 b = τ1 J˜ +iτ2 µ ˜ Di Francesco, Mathieu & Senechal, “Conformal Field Theory,” 1997; Hori, et. al. “Mirror symmetry,” 2003

˜µ Twisted B.C. & A • Dirac fermion in the presence of background gauge fields : S=

1 2π

Z

¯ µ (∂µ + iAµ ) ψ , d 2 x i ψγ

– on torus with τ = τ1 + iτ2 1 – for coordinates ζ = 2π (s + it) is identified as ζ ≡ ζ + 1 ≡ ζ + τ .

• twisted boundary conditions ψ(t, s) = e −2πia ψ(t, s + 2π) = e −2πib ψ(t + 2πτ2 , s + 2πτ1 ) . ˜ µ = (˜ ˜ ≡ periodic Dirac fermion with flat gauge connection A µ, J) ˜ = πi [(b−τ a)d ζ¯−(b − τ¯a)dζ] = ads + b−aτ1 dt A τ2 τ2

n

a = J˜ . b = τ1 J˜ +iτ2 µ ˜

Di Francesco, Mathieu & Senechal, “Conformal Field Theory,” 1997; Hori, et. al. “Mirror symmetry,” 2003

Partition function 2   −2  1/2−a  2H Z[a,b] = |{z} Tr e −2πi(b−1/2)FA e −2πiτ1 P e| −2πτ ϑ b−1/2 (0, τ ) , {z } = |η(τ )|

trace over H (twisted B.C. a for s ∼ s + 2π)

euclidean time evolution t → t +2πτ2 with H – s → s −2πτ1 with e −2πiτ1 P , P momentum – phase rotation e −2πi(b−1/2)FA , fermion # FA ,

Q∞ n 2πiτ n=1 (1 − q ) with q = e h i P (n+α)2 /2 2πi(z+β)(n+α) Jacobi theta function ϑ α (z|τ ) = q e n∈Z β h i h i NS(R)-NS sector: ϑ3 (z|τ ) = ϑ 00 (z|τ ), ϑ2 (z|τ ) = ϑ 1/2 (z|τ ) 0

– Dedekind function η(τ ) = q 1/24 – –

Di Francesco, Mathieu & Senechal, “Conformal Field Theory,” 1997; Hori, et. al. “Mirror symmetry,” 2003

• including current J and chemical potential µ: 2     (τ1 J +iτ2 µ|τ ) . Tr e 2πi(τ1 J+iτ2 µ+b−1/2)FA e −2πiτ1 P e −2πτ2 H = |η(τ )|−2 ϑ 1/2−a−J b−1/2

Partition function 2   −2  1/2−a  2H Z[a,b] = |{z} Tr e −2πi(b−1/2)FA e −2πiτ1 P e| −2πτ ϑ b−1/2 (0, τ ) , {z } = |η(τ )|

trace over H (twisted B.C. a for s ∼ s + 2π)

euclidean time evolution t → t +2πτ2 with H – s → s −2πτ1 with e −2πiτ1 P , P momentum – phase rotation e −2πi(b−1/2)FA , fermion # FA ,

Q∞ n 2πiτ n=1 (1 − q ) with q = e h i P (n+α)2 /2 2πi(z+β)(n+α) Jacobi theta function ϑ α (z|τ ) = q e n∈Z β h i h i NS(R)-NS sector: ϑ3 (z|τ ) = ϑ 00 (z|τ ), ϑ2 (z|τ ) = ϑ 1/2 (z|τ ) 0

– Dedekind function η(τ ) = q 1/24 – –

Di Francesco, Mathieu & Senechal, “Conformal Field Theory,” 1997; Hori, et. al. “Mirror symmetry,” 2003

• including current J and chemical potential µ: 2     (τ1 J +iτ2 µ|τ ) . Tr e 2πi(τ1 J+iτ2 µ+b−1/2)FA e −2πiτ1 P e −2πτ2 H = |η(τ )|−2 ϑ 1/2−a−J b−1/2

Partition function 2   −2  1/2−a  2H Z[a,b] = |{z} Tr e −2πi(b−1/2)FA e −2πiτ1 P e| −2πτ ϑ b−1/2 (0, τ ) , {z } = |η(τ )|

trace over H (twisted B.C. a for s ∼ s + 2π)

euclidean time evolution t → t +2πτ2 with H – s → s −2πτ1 with e −2πiτ1 P , P momentum – phase rotation e −2πi(b−1/2)FA , fermion # FA ,

Q∞ n 2πiτ n=1 (1 − q ) with q = e h i P (n+α)2 /2 2πi(z+β)(n+α) α Jacobi theta function ϑ β (z|τ ) = e n∈Z q h i h i NS(R)-NS sector: ϑ3 (z|τ ) = ϑ 00 (z|τ ), ϑ2 (z|τ ) = ϑ 1/2 (z|τ ) 0

– Dedekind function η(τ ) = q 1/24 – –

Di Francesco, Mathieu & Senechal, “Conformal Field Theory,” 1997; Hori, et. al. “Mirror symmetry,” 2003

• including current J and chemical potential µ: 2     1/2−a−J Tr e 2πi(τ1 J+iτ2 µ+b−1/2)FA e −2πiτ1 P e −2πτ2 H = |η(τ )|−2 ϑ b−1/2+τ (0|τ ) 1 J+iτ2 µ 2   (τ1 J +iτ2 µ|τ ) . = |η(τ )|−2 ϑ 1/2−a−J b−1/2

EE & R´ enyi entropy Calabrese and Cardy 2004, 2009; Casini, Fosco and Huerta 2005; Nishioka, Ryu and Takayanagi 2009

– For a subsystem A with `t in total system 2πL – replica of n-copies in Path integral: Zn orbifold theory on torus – signs of Tr and connections of copies: σ±k operators with ∆k = k 2 /2n2 Sn =

1 1 h [log Tr(ρA )n ] = log 1−n 1−n

hσk (`t )σ−k (0)i =

n−1/2

Y

i hσk (`t )σ−k (0)i = Sn0 + Snµ,J ,

k=−(n−1)/2

2k 2 /n2 ϑ[1/2−a−J ]( k `t + τ1 J + iτ2 µ|τ ) 2 b−1/2 n 2πL × . 1/2−a−J `t ϑ[1/2 ]( ϑ[ ](τ J + iτ µ|τ ) |τ ) 1 2 1/2 2πL b−1/2 2πη(τ )3

– n → 1 to obtain EE – factorize into two parts: one independent of (J, µ): well studied – EE(µ) previously studied in Ogawa, Takayanagi and Ugajin 2011; Herzog and Nishioka 2013 – we study the R´ eny entropy and EE depending both on (J, µ)! – straightforward to generalize for multiple intervals. Herzog and Nishioka 2013 – NS-NS sector: a + J = 1/2, b = 1/2; R-NS sector: a + J = 0, b = 1/2.

EE & R´ enyi entropy Calabrese and Cardy 2004, 2009; Casini, Fosco and Huerta 2005; Nishioka, Ryu and Takayanagi 2009

– For a subsystem A with `t in total system 2πL – replica of n-copies in Path integral: Zn orbifold theory on torus – signs of Tr and connections of copies: σ±k operators with ∆k = k 2 /2n2 Sn =

1 1 h [log Tr(ρA )n ] = log 1−n 1−n

hσk (`t )σ−k (0)i =

n−1/2

Y

i hσk (`t )σ−k (0)i = Sn0 + Snµ,J ,

k=−(n−1)/2

2k 2 /n2 ϑ[1/2−a−J ]( k `t + τ1 J + iτ2 µ|τ ) 2 b−1/2 n 2πL × . 1/2−a−J `t ϑ[1/2 ]( ϑ[ ](τ J + iτ µ|τ ) |τ ) 1 2 1/2 2πL b−1/2 2πη(τ )3

– n → 1 to obtain EE – factorize into two parts: one independent of (J, µ): well studied – EE(µ) previously studied in Ogawa, Takayanagi and Ugajin 2011; Herzog and Nishioka 2013 – we study the R´ eny entropy and EE depending both on (J, µ)! – straightforward to generalize for multiple intervals. Herzog and Nishioka 2013 – NS-NS sector: a+J = 1/2, b = 1/2; R-NS sector: a+J = 0, b = 1/2.

Partition Function & EE with J & µ (3 slides)

Applications – Entropies & Chemical potential in T → 0 – Entropies & Current in T → 0

(2 slides)

– Entropies & large radius limit

(1 slides)

– Mutual information with J & µ

(1 slide)

(2 slides)

EE (µ) for NS-NS sector – NS-NS (a = b = 1/2) for J = τ1 = 0: Snµ =

1 h 1−n

iβµ ϑ[1/2−a−J b−1/2 ](τ1 J + iτ2 µ|τ ) → ϑ3 ( 2π |iβ)

n−1/2

`t ϑ3 ( kn 2πL + iβµ |iβ) 2 i 2π , log iβµ ϑ3 ( 2π |iβ) k=−(n−1)/2

X

m m−1/2 )(1+y −1 q m−1/2 ), m=1 (1−q )(1+yq ` −βµ+i k t − βµ − β n L , y2 = e y1 = e , q =e

– ϑ3 (z|τ ) =

Q∞

– – taking β = 2πτ2 → ∞ limit

Snµ =

=

1 h 1−n

n−1/2

∞ Y (1−q m )(1+y1 q m−1/2 )(1+y1−1 q m−1/2 ) 2 i log m−1/2 )(1+y −1 q m−1/2 ) m 2 m=1 (1−q )(1+y2 q k=−(n−1)2

X

∞ i 2 h X (−1)l−1 e −lβµ +e lβµ  l`t  l`t  sin csc −n , lβ(m−1/2) 1−n l,m=1 l 2L 2Ln e

µ Sn=1 =2

∞ X l`t i (−1)l−1 cosh (lβµ) h l`t 1− cot . l sinh (lβ/2) 2L 2L l=1

– valid for y2 q 1/2 , y2−1 q 1/2 < 1 ⇒ −1/2 < µ < 1/2 & S → 0 for β → ∞. – same for other constant µ due to the periodicity S(µ) = S(µ + 1) = S(µ − 1).

EE (µ) for NS-NS sector – NS-NS (a = b = 1/2) for J = τ1 = 0: Snµ =

1 h 1−n

iβµ ϑ[1/2−a−J b−1/2 ](τ1 J + iτ2 µ|τ ) → ϑ3 ( 2π |iβ)

n−1/2

`t ϑ3 ( kn 2πL + iβµ |iβ) 2 i 2π , log iβµ ϑ3 ( 2π |iβ) k=−(n−1)/2

X

m m−1/2 )(1+y −1 q m−1/2 ), m=1 (1−q )(1+yq ` −βµ+i k t − βµ − β n L , y2 = e y1 = e , q =e

– ϑ3 (z|τ ) =

Q∞

– – taking β = 2πτ2 → ∞ limit

Snµ =

=

1 h 1−n

n−1/2

∞ Y (1−q m )(1+y1 q m−1/2 )(1+y1−1 q m−1/2 ) 2 i log m−1/2 )(1+y −1 q m−1/2 ) m 2 m=1 (1−q )(1+y2 q k=−(n−1)2

X

∞ i 2 h X (−1)l−1 e −lβµ +e lβµ  l`t  l`t  sin csc −n , lβ(m−1/2) 1−n l,m=1 l 2L 2Ln e

µ Sn=1 =2

∞ X l`t i (−1)l−1 cosh (lβµ) h l`t 1− cot . l sinh (lβ/2) 2L 2L l=1

– valid for y2 q 1/2 , y2−1 q 1/2 < 1 ⇒ −1/2 < µ < 1/2 & S → 0 for β → ∞. – same for other constant µ due to the periodicity S(µ) = S(µ + 1) = S(µ − 1).

EE (µ) for NS-NS sector – NS-NS (a = b = 1/2) for J = τ1 = 0: Snµ =

1 h 1−n

iβµ ϑ[1/2−a−J b−1/2 ](τ1 J + iτ2 µ|τ ) → ϑ3 ( 2π |iβ)

n−1/2

`t ϑ3 ( kn 2πL + iβµ |iβ) 2 i 2π , log iβµ ϑ3 ( 2π |iβ) k=−(n−1)/2

X

m m−1/2 )(1+y −1 q m−1/2 ), m=1 (1−q )(1+yq ` −βµ+i k t − βµ − β n L , y2 = e y1 = e , q =e

– ϑ3 (z|τ ) =

Q∞

– – taking β = 2πτ2 → ∞ limit

Snµ =

=

1 h 1−n

n−1/2

∞ Y (1−q m )(1+y1 q m−1/2 )(1+y1−1 q m−1/2 ) 2 i log m−1/2 )(1+y −1 q m−1/2 ) m 2 m=1 (1−q )(1+y2 q k=−(n−1)2

X

∞ i 2 h X (−1)l−1 e −lβµ +e lβµ  l`t  l`t  sin csc −n , lβ(m−1/2) 1−n l,m=1 l 2L 2Ln e

µ Sn=1 =2

∞ X l`t i (−1)l−1 cosh (lβµ) h l`t 1− cot . l sinh (lβ/2) 2L 2L l=1

– valid for y2 q 1/2 , y2−1 q 1/2 < 1 ⇒ −1/2 < µ < 1/2 & S → 0 for β → ∞. – same for other constant µ due to the periodicity S(µ) = S(µ + 1) = S(µ − 1).

EE (µ) for NS-NS sector – NS-NS (a = b = 1/2) for J = τ1 = 0: Snµ =

1 h 1−n

iβµ ϑ[1/2−a−J b−1/2 ](τ1 J + iτ2 µ|τ ) → ϑ3 ( 2π |iβ)

n−1/2

`t ϑ3 ( kn 2πL + iβµ |iβ) 2 i 2π , log iβµ ϑ3 ( 2π |iβ) k=−(n−1)/2

X

m m−1/2 )(1+y −1 q m−1/2 ), m=1 (1−q )(1+yq ` −βµ+i k t − βµ − β n L , y2 = e y1 = e , q =e

– ϑ3 (z|τ ) =

Q∞

– – taking β = 2πτ2 → ∞ limit

Snµ =

1 h 1−n

=

µ Sn=1 =2

n−1/2

∞ Y (1−q m )(1+y1 q m−1/2 )(1+y1−1 q m−1/2 ) 2 i log m−1/2 )(1+y −1 q m−1/2 ) m 2 m=1 (1−q )(1+y2 q k=−(n−1)2

X

∞ i 2 h X (−1)l−1 e −lβµ +e lβµ  l`t  l`t  sin csc −n , lβ(m−1/2) 1−n l,m=1 l 2L 2Ln e ∞ X l`t i (−1)l−1 cosh (lβµ) h l`t 1− cot . l sinh (lβ/2) 2L 2L l=1

– valid for y2 q 1/2 , y2−1 q 1/2 < 1 ⇒ −1/2 < µ < 1/2 & S → 0 for β → ∞. – same for other constant µ due to the periodicity S(µ) = S(µ + 1) = S(µ − 1).

EE (µ) for NS-NS sector – NS-NS (a = b = 1/2) for J = τ1 = 0: Snµ =

1 h 1−n

iβµ ϑ[1/2−a−J b−1/2 ](τ1 J + iτ2 µ|τ ) → ϑ3 ( 2π |iβ)

n−1/2

`t ϑ3 ( kn 2πL + iβµ |iβ) 2 i 2π , log iβµ ϑ3 ( 2π |iβ) k=−(n−1)/2

X

Q∞ m m−1/2 )(1+y −1 q m−1/2 ), m=1 (1−q )(1+yq ` −βµ+i k t − βµ − β n L , y2 = e y1 = e , q =e

– ϑ3 (z|τ ) =

– – taking β = 2πτ2 → ∞ limit

Snµ =

=

1 h 1−n

n−1/2

∞ Y (1−q m )(1+y1 q m−1/2 )(1+y1−1 q m−1/2 ) 2 i log m−1/2 )(1+y −1 q m−1/2 ) m 2 m=1 (1−q )(1+y2 q k=−(n−1)2

X

∞ i 2 h X (−1)l−1 e −lβµ +e lβµ  l`t  l`t  sin csc −n , lβ(m−1/2) 1−n l,m=1 l 2L 2Ln e

µ Sn=1 =2

∞ X l`t i (−1)l−1 cosh (lβµ) h l`t . 1− cot l sinh (lβ/2) 2L 2L l=1

– valid for y2 q 1/2 , y2−1 q 1/2 < 1 ⇒ −1/2 < µ < 1/2 & S → 0 for β → ∞. – same for other constant µ due to the periodicity S(µ) = S(µ + 1) = S(µ − 1). Ogawa, Takayanagi and Ugajin 2011; Herzog and Nishioka 2013

• Consider β(µ − 1/2) = M = const. for β → ∞ & µ → 1/2 Q (1 − q m )(1 + yq m−1/2 ) (1 + y −1 q m+1/2 ) – ϑ3 (z|τ ) = (1 + y −1 q 1/2 ) ∞ | | {z } m=1 {z } separate contribution for EE validity extended to −1/2 < µ < 3/2 lβ

µ Sn=1 =2

    ∞ X e− 2 l`t (−1)l−1 −Ml l`t e + cot . 1 − l 2L 2L sinh( lβ ) l=1 2

– µ = 1/2: one of the energy levels of the anti-periodic fermion! – red contribution exist for general value M (here M < 1.)

– in general, non-zero contribution exists for    N N = ±1, ±3, ±5, · · · β µ− = const. N = 0, ±2, ±4, · · · 2

anti-periodic fermion, NS periodic fermion, R

– EE (R´ enyi entropy) is useful for probing the energy level!

• Consider β(µ − 1/2) = M = const. for β → ∞ & µ → 1/2 Q (1 − q m )(1 + yq m−1/2 ) (1 + y −1 q m+1/2 ) – ϑ3 (z|τ ) = (1 + y −1 q 1/2 ) ∞ | | {z } m=1 {z } separate contribution for EE validity extended to −1/2 < µ < 3/2 lβ

µ Sn=1 =2

    ∞ X e− 2 l`t (−1)l−1 −Ml l`t e + cot . 1 − l 2L 2L sinh( lβ ) l=1 2

– µ = 1/2: one of the energy levels of the anti-periodic fermion! – red contribution exist for general value M (here M < 1.)

– in general, non-zero contribution exists for    N N = ±1, ±3, ±5, · · · β µ− = const. N = 0, ±2, ±4, · · · 2

anti-periodic fermion, NS periodic fermion, R

– EE (R´ enyi entropy) is useful for probing the energy level!

• Consider β(µ − 1/2) = M = const. for β → ∞ & µ → 1/2 Q (1 − q m )(1 + yq m−1/2 ) (1 + y −1 q m+1/2 ) – ϑ3 (z|τ ) = (1 + y −1 q 1/2 ) ∞ | | {z } m=1 {z } separate contribution for EE validity extended to −1/2 < µ < 3/2 lβ

µ Sn=1 =2

    ∞ X e− 2 l`t (−1)l−1 −Ml l`t e + cot . 1 − l 2L 2L sinh( lβ ) l=1 2

– µ = 1/2: one of the energy levels of the anti-periodic fermion! – red contribution exist for general value M (here M < 1 .)

– in general, non-zero contribution exists for    N N = ±1, ±3, ±5, · · · β µ− = const. N = 0, ±2, ±4, · · · 2

anti-periodic fermion, NS periodic fermion, R

– EE (R´ enyi entropy) is useful for probing the energy level!

• Consider β(µ − 1/2) = M = const. for β → ∞ & µ → 1/2 Q (1 − q m )(1 + yq m−1/2 ) (1 + y −1 q m+1/2 ) – ϑ3 (z|τ ) = (1 + y −1 q 1/2 ) ∞ | | {z } m=1 {z } separate contribution for EE validity extended to −1/2 < µ < 3/2 lβ

µ Sn=1 =2

    ∞ X e− 2 l`t (−1)l−1 −Ml l`t e + cot . 1 − l 2L 2L sinh( lβ ) l=1 2

– µ = 1/2: one of the energy levels of the anti-periodic fermion! – red contribution exist for general value M (here M < 1 .)

– in general, non-zero contribution exists for    N N = ±1, ±3, ±5, · · · β µ− = const. N = 0, ±2, ±4, · · · 2

anti-periodic fermion, NS periodic fermion, R

– EE (R´ enyi entropy) is useful for probing the energy level!

EE (J) • R´enyi entropy derived in the introduction SnJ,µ =

1 h log 1−n

ϑ[1/2−a−J ]( k `t + τ1 J + iτ2 µ|τ ) 2 i b−1/2 n 2πL . 1/2−a−J ϑ[ ](τ J + iτ µ|τ ) 1 2 k=−(n−1)/2 b−1/2 n−1/2

Y

• simple case: a = b = 1/2 & µ = α = 2πτ1 = 0 at β → ∞ limit  P ∞    2 l=1 S=    2 P∞ l=1

(−1)l−1 l sinh(lβ/2)

h

(−1)l−1 l tanh(lβ/2)

h

1− 1−

l`t 2L

cot



l`t 2L

cot



l`t 2L l`t 2L

i

,

J =0

(NS) ,

i

,

J = 12

(R) .

– J = 0 → J = 1/2: visible effects in EE (R´ enyi entropy)! • mode expansion in the presence of the current J X ψ= ψr (t)e i˜r s . ˜ r ∈Z+a+J

– changing J is equivalent to changing a.

EE (J) • R´enyi entropy derived in the introduction SnJ,µ =

1 h log 1−n

ϑ[1/2−a−J ]( k `t + τ1 J + iτ2 µ|τ ) 2 i b−1/2 n 2πL . 1/2−a−J ϑ[ ](τ J + iτ µ|τ ) 1 2 k=−(n−1)/2 b−1/2 n−1/2

Y

• simple case: a = b = 1/2 & µ = α = 2πτ1 = 0 at β → ∞ limit  P ∞    2 l=1 S=    2 P∞ l=1

(−1)l−1 l sinh(lβ/2)

h

(−1)l−1 l tanh(lβ/2)

h

1− 1−

l`t 2L

cot



l`t 2L

cot



l`t 2L l`t 2L

i

,

J =0

(NS) ,

i

,

J = 12

(R) .

– J = 0 → J = 1/2: visible effects in EE (R´ enyi entropy)! • mode expansion in the presence of the current J X ψ= ψr (t)e i˜r s . ˜ r ∈Z+a+J

– changing J is equivalent to changing a.

EE (J) • R´enyi entropy derived in the introduction SnJ,µ =

1 h log 1−n

ϑ[1/2−a−J ]( k `t + τ1 J + iτ2 µ|τ ) 2 i b−1/2 n 2πL . 1/2−a−J ϑ[ ](τ J + iτ µ|τ ) 1 2 k=−(n−1)/2 b−1/2 n−1/2

Y

• simple case: a = b = 1/2 & µ = α = 2πτ1 = 0 at β → ∞ limit  P ∞    2 l=1 S=    2 P∞ l=1

(−1)l−1 l sinh(lβ/2)

h

(−1)l−1 l tanh(lβ/2)

h

1− 1−

l`t 2L

cot



l`t 2L

cot



l`t 2L l`t 2L

i

,

J =0

(NS) ,

i

,

J = 12

(R) .

– J = 0 → J = 1/2: visible effects in EE (R´ enyi entropy)! • mode expansion in the presence of the current J X ψ= ψr (t)e i˜r s . ˜ r ∈Z+a+J

– changing J is equivalent to changing a.

• anti-periodic fermion at small temperature limit: a + J = 1/2 ∞ X

   (−1)l−1 l`t l`t cos([m − 1/2]αl) cos(αJl) × 1 − cot (m−1/2)βl 2L 2L le l,m=1    `t `t = 4e −β/2 cos (α/2) cos(αJ) 1 − cot + ··· 2L 2L

SJ = 4

– EE is a periodic function of J and α. – EE depends on α in two different ways, producing ‘interference pattern’ – Dominant contribution (l = m = 1) has a ‘beat frequency’ J/π for J < 1/2

• periodic fermion at small temperature limit: a + J = 0 ∞ X (−1)l−1 S =2 l l=1 J

"

#    ∞ X l`t l`t cos(mαl) 1+ 2 cos(αJl) × 1 − cot . mβl e 2L 2L m=1

– similar to anti-periodic fermions for temperature dependent parts – at β = ∞, EE has non-zero contributions. – for αJ = π, negative! (there is infinite contribution independent of J)

• anti-periodic fermion at small temperature limit: a + J = 1/2 ∞ X

   (−1)l−1 l`t l`t cos([m − 1/2]αl) cos(αJl) × 1 − cot (m−1/2)βl 2L 2L le l,m=1    `t `t = 4e −β/2 cos (α/2) cos(αJ) 1 − cot + ··· 2L 2L

SJ = 4

– EE is a periodic function of J and α. – EE depends on α in two different ways, producing ‘interference pattern’ – Dominant contribution (l = m = 1) has a ‘beat frequency’ J/π for J < 1/2

• periodic fermion at small temperature limit: a + J = 0 ∞ X (−1)l−1 S =2 l l=1 J

"

#    ∞ X l`t l`t cos(mαl) 1+ 2 cos(αJl) × 1 − cot . mβl e 2L 2L m=1

– similar to anti-periodic fermions for temperature dependent parts – at β = ∞, EE has non-zero contributions. – for αJ = π, negative! (there is infinite contribution independent of J)

Large radius limit, `t /L  1 • NS-NS sector with only chemical potential for simplicity. – ϑ3 (z|τ ) =

Q∞

m=1 (1−q

m ) (1+yq m−1/2 )(1+y −1 q m−1/2 ),

| {z } (1+q 2m−1 +2 cos(2πz2 )q m−1/2 ) – for z1 = i βµ , z2 = i βµ + 2π 2π 1+q

2m−1

+2 cos(2πz2 )q

k `t n L

m−1/2

and `t /L  1

= 1+q − 2q

Snµ =

=

1 h 1−n

2m−1

+2 cos(2πz1 )q

m−1/2 

i

m−1/2

 1 k `t 2 k `t sinh(βµ)+ ( ) cosh(βµ) +· · · . n L 2 n L

n−1/2

∞ Y (1−q m )(1+y1 q m−1/2 )(1+y1−1 q m−1/2 ) 2 i log m−1/2 )(1+y −1 q m−1/2 ) m 2 m=1 (1−q )(1+y2 q k=−(n−1)2

X

 4 ∞ (n + 1)π 2 `2t X 4 + 4 cosh(βµ) cosh(βm) `t +O , 2 2 12n L m=1 (cosh(βm) + cosh(βµ)) L

– linear terms identically 0 upon summing k – similar for all J & µ dependent parts of EE & R´ enyi entropy

Large radius limit, `t /L  1 • NS-NS sector with only chemical potential for simplicity. – ϑ3 (z|τ ) =

Q∞

m=1 (1−q

m ) (1+yq m−1/2 )(1+y −1 q m−1/2 ),

| {z } (1+q 2m−1 +2 cos(2πz2 )q m−1/2 ) – for z1 = i βµ , z2 = i βµ + 2π 2π 1+q

2m−1

+2 cos(2πz2 )q

k `t n L

m−1/2

and `t /L  1

= 1+q

2m−1

+2 cos(2πz1 )q

m−1/2

 1 k `t 2 m−1/2  k `t sinh(βµ)+ ( ) cosh(βµ) +· · · . − 2q i n L 2 n L

Snµ =

=

1 h 1−n

n−1/2

∞ Y (1−q m )(1+y1 q m−1/2 )(1+y1−1 q m−1/2 ) 2 i log m−1/2 )(1+y −1 q m−1/2 ) m 2 m=1 (1−q )(1+y2 q k=−(n−1)2

X

 4 ∞ `t (n + 1)π 2 `2t X 4 + 4 cosh(βµ) cosh(βm) + O , 12n L2 m=1 (cosh(βm) + cosh(βµ))2 L

– linear terms identically 0 upon summing k – similar for all J & µ dependent parts of EE & R´ enyi entropy

Large radius limit, `t /L  1 • NS-NS sector with only chemical potential for simplicity. – ϑ3 (z|τ ) =

Q∞

m=1 (1−q

m ) (1+yq m−1/2 )(1+y −1 q m−1/2 ),

| {z } (1+q 2m−1 +2 cos(2πz2 )q m−1/2 ) – for z1 = i βµ , z2 = i βµ + 2π 2π 1+q

2m−1

+2 cos(2πz2 )q

k `t n L

m−1/2

and `t /L  1

= 1+q

2m−1

+2 cos(2πz1 )q

m−1/2

 1 k `t 2 m−1/2  k `t sinh(βµ)+ ( ) cosh(βµ) +· · · . − 2q i n L 2 n L

Snµ =

=

1 h 1−n

n−1/2

∞ Y (1−q m )(1+y1 q m−1/2 )(1+y1−1 q m−1/2 ) 2 i log m−1/2 )(1+y −1 q m−1/2 ) m 2 m=1 (1−q )(1+y2 q k=−(n−1)2

X

 4 ∞ (n + 1)π 2 `2t X 4 + 4 cosh(βµ) cosh(βm) `t , + O 12n L2 m=1 (cosh(βm) + cosh(βµ))2 L

– linear terms identically 0 upon summing k – Previously EE is independent of µ for an interval in infinite space based on symmetry Cardy, “Entanglement in CFTs at Finite Chemical Potential,” presentation at the Yukawa International Seminar 2016 (YKIS2016) “Quantum Matter, Spacetime and Information.”

– similar for all J & µ dependent parts of EE & R´ enyi entropy

Mutual (R´ enyi) information In (A, B) = Sn (A) + Sn (B) − Sn (A ∪ B) – measuring entanglement between two intervals, `A and `B , separated by `C – free of UV divergence and finite.

• NS-NS sector at zero temperature limit: In = In0 + InJ,µ Inµ (A, B) =

=

2 n−1

(n−1)/2

X k=−(n−1)/2

log

`A +`B + iβµ |iβ) ϑ3 ( iβµ |iβ) 2 2πL 2π 2π ` ` iβµ iβµ A B ϑ3 ( kn 2πL + 2π |iβ) ϑ3 ( kn 2πL + 2π |iβ)

ϑ3 ( kn

 ∞ sin 2 X (−1)l−1 cosh(lβµ) n− n−1 l=1 l sinh(lβ/2) sin

 l(`A +`B )  l`A  sin l`2LB sin 2L 2L − + .   +`B  l `A sin nl `2LB sin nl `A2L n 2L

– for simplicity, α = 2πτ1 = 0, J = 0 and a+J = 1/2, b = 1/2 – dependence on µ (also on J) are the same as R´ nyi entropy & EE – independent of separation `C between two intervals `A and `B .

Mutual (R´ enyi) information In (A, B) = Sn (A) + Sn (B) − Sn (A ∪ B) – measuring entanglement between two intervals, `A and `B , separated by `C – free of UV divergence and finite.

• NS-NS sector at zero temperature limit: In = In0 + InJ,µ Inµ (A, B) =

=

2 n−1

(n−1)/2

X k=−(n−1)/2

∞ X

log

`A +`B + iβµ |iβ) ϑ3 ( iβµ |iβ) 2 2πL 2π 2π ` ` iβµ iβµ A B ϑ3 ( kn 2πL + 2π |iβ) ϑ3 ( kn 2πL + 2π |iβ)

ϑ3 ( kn

 sin 2 (−1)l−1 cosh(lβµ) n− n−1 l=1 l sinh(lβ/2) sin

 l(`A +`B )  l`A  sin l`2LB sin 2L 2L − + .   +`B  l `A sin nl `2LB sin nl `A2L n 2L

– for simplicity, α = 2πτ1 = 0, J = 0 and a+J = 1/2, b = 1/2 – dependence on µ (also on J) are the same as R´ enyi entropy & EE – independent of separation `C between two intervals `A and `B .

Mutual (R´ enyi) information In (A, B) = Sn (A) + Sn (B) − Sn (A ∪ B) – measuring entanglement between two intervals, `A and `B , separated by `C – free of UV divergence and finite.

• NS-NS sector at zero temperature limit: In = In0 + InJ,µ Inµ (A, B) =

=

2 n−1

(n−1)/2

X k=−(n−1)/2

log

`A +`B + iβµ |iβ) ϑ3 ( iβµ |iβ) 2 2πL 2π 2π ` ` iβµ iβµ A B ϑ3 ( kn 2πL + 2π |iβ) ϑ3 ( kn 2πL + 2π |iβ)

ϑ3 ( kn

 ∞ sin 2 X (−1)l−1 cosh(lβµ) n− n−1 l=1 l sinh(lβ/2) sin

 l(`A +`B )  l`A  sin l`2LB sin 2L 2L − + .   +`B  l `A sin nl `2LB sin nl `A2L n 2L

– for simplicity, α = 2πτ1 = 0, J = 0 and a+J = 1/2, b = 1/2 – dependence on µ (also on J) are the same as R´ enyi entropy & EE – independent of separation `C between two intervals `A and `B .

Summary & future directions • Generalized R´enyi & EE in the presence of J and µ • Compute the entropies with J and µ in zero T & large radius limits. – The entropies are periodic in J, which has the role of ‘beat frequency’ – The entropies vanish as fast as O(`t /L)2 – The entropies is non-zero for β(µ − N/2) = const. as β → 0, µ → N/2 • Mutual (R´enyi) information has same dependences on J & µ, which is further independent of the separation between the intervals.

• Future directions : – seeking possible experimental signatures of ‘beat frequency’ – applying this to three dimensional static system .. .

## Entanglement Entropy with Current & Chemical Potential

May 7, 2017 - trace over H euclidean time evolution t ât+2ÏÏ2 with H. (twisted B.C. a for s â¼ s + 2Ï). â s âsâ2ÏÏ1 with eâ2ÏiÏ1P , P momentum.

#### Recommend Documents

The elusive chemical potential
22 with respect to T, interchange the order of differentia- tion on the right-hand side, and note that .... the electronic charge. The electron's intrinsic chemical po-.

pdf entropy
Page 1 of 1. File: Pdf entropy. Download now. Click here if your download doesn't start automatically. Page 1 of 1. pdf entropy. pdf entropy. Open. Extract.

Entanglement in a quantum annealing processor
Jan 15, 2014 - 3Center for Quantum Information Science and Technology, University of Southern California. 4Department of ... systems reach equilibrium with a thermal environment. Our results ..... annealing processor during its operation.

Moment-entropy inequalities
MOMENT-ENTROPY INEQUALITIES. Erwin Lutwak, Deane Yang and Gaoyong Zhang. Department of Mathematics. Polytechnic University. Brooklyn, NY 11201.

IBPSGuide - Important Current Affairs MCQs March_2017 with ...
Immigration Crime Engagement. Trump gave. examples of Americans in the audience whose loved. ones were killed by illegal immigrants. He. condemned the shooting of an Indian engineer while. adding that the US would be protected against. "radical Islam

Quizz: Targeted Crowdsourcing with a Billion (Potential) - CiteSeerX
As our analysis shows, a key component for the success .... face the problem of sparse data, especially during the early ...... In Data Mining Workshops, 2006.

How to Convert Potential Audience into Customers with Banners in ...
places must attract immediate attention, draw curiosity, entice Internet users or web. surfers and make them click on it. The banners must have a right set of mix visuals,. words as well as colors that would stand apart from millions of others. As a

Maximum-entropy model
Test set: 264 sentences. Noisy-channel. 63.3. 50.247.1. 75.3. 64.162.1. 80.9. 72.069.5. Maximum EntropyMaximum Entropy with Bottom-up. F-measure. Bigram F-measure. BLEU score. 10. 20. 30. 40. 50. 60. 70. 80. 90. 100. S. NP. VP. NP. PP. The apple on t