Entanglement Entropy in Quantum Field Theories

= ln 2 (6.8)

which equals, also as expected, the natural logarithm of the dimension of the Hilbert space H_A. One can also straightforwardly check that indeed SA = SB for this system.

(Strong) Subadditivity: Another important property that entanglement entropy satisfies is called (strong) subadditivity. A bipartite system with partsAand B is called subadditive if it satisfies the following inequality

SA+SB≥SA∪B+SA∩B. (6.9)

A tripartite system with partsA,B andCis called strong subadditive if it satisfies SA∪B∪C +S_B ≤SA∪B+SB∪C, (6.10) or equivalently [103]

S_A+S_C ≤SA∪B+SB∪C. (6.11)

These strong subadditivity inequalities for quantum systems have been proven in [104]. I will also review later how these inequalities can be derived in a simple way using a holographic approach.

Entanglement Entropy in Quantum Field Theories

Consider a relativistic quantum field theory on a lattice with lattice spacingain d+1 dimensions with d≥2. At a given fixed timet=t0 one can again divide the total system into a subsystemAand its complementB, where∂Ais the boundary of the subsystem. In [105,106] it was then shown that the entanglement entropy of such a system diverges in the continuum limita→0as

SA∼ Area(∂A)

a^d+1 +. . . (6.12)

This behavior is calledarea lawand is very similar to the behavior of the Bekenstein-Hawking entropy of a black hole, which also scales with the area of the black hole horizon. This similarity will play an important role for motivating a holographic de-scription of entanglement entropy but first I want to elaborate on one key technique which is used for calculating entanglement entropy in quantum field theories, the replica trick.

Since the replica trick is by now a well established technique for calculating

entan-glement entropy there exists also excellent literature explaining the trick and its application to entanglement entropy in detail e.g. [107] (for arbitrary dimensions and finite temperature) or [108] (for 2d conformal field theories) whose structure I will also follow in this section. For the sake of simplicity and also relevance for this thesis I will stick to two dimensions and zero temperature in what follows.

Assume that the 2d system is defined on a (euclidean) plane with coordinates (x, tE) and can be described in terms of an euclidean actionS_E[φ(x, t_E)]. The entangling regionAwill be an interval inxdirection att_E = 0.

Calculating entanglement entropy for such a 2d quantum field theory using (6.2) is often not very efficient since calculating the logarithm of the density matrix can be potentially very involved. The replica trick allows one to circumvent this problem by effectively reducing the problem of calculating entanglement entropy to determining the partition function of ann-sheeted Riemann surface. The starting point of this trick are the so-called Tsallis¹entropies which are given by

S_n^Tsallis= TrAρⁿ_A−1

1−n . (6.13)

The entanglement entropy can be obtained from the Tsallis entropies in the limit² n→1which can also be written as³

S_A= lim

n→1S_n^Tsallis =−∂_nTr_Aρⁿ_A

n=1 =−∂_nln (Tr_Aρⁿ_A)

n=1, (6.14)

which at this point shifted the problem of calculating entanglement entropy from calculating a logarithm to calculating powers of the density matrix, which is com-putationally easier to perform. Now in order to make contact with my statement aboutn-sheeted Riemann surfaces in the beginning one has to calculate Tr_Aρⁿ_Aand perform some path integral gymnastics.

As in usual quantum mechanics the density matrix of the whole system is given by the ground state waveΨfunction and its hermitian conjugateΨ^†asρ = ΨΨ^†. In a path integral representationΨ[φ(x,0)]is given by path integrating fromt_E =∞ up tot_E = 0and the hermitian conjugateΨ^†[φ⁰(x,0)]by integrating fromt_E = 0to tE =∞i.e.

Ψ[φ(x,0)] =

φ(x,0)

Z

tE=−∞

Dφ e^−SE[φ], Ψ^†[φ⁰(x,0)] =

tE=∞

Z

φ⁰(x,0)

Dφ e^−SE[φ]. (6.15)

See Fig. 6.1 for a pictorial interpretation of this path integration. The reduced density matrixρ_Acan now be obtained by combiningΨ[φ(x,0)]andΨ^†[φ⁰(x,0)]into

1The Tsallis entropies are related to the maybe better known Rényi entropies S^Rényin as Sn^Tsallis= _1−n¹

h

e^{(1−n)SRényin} −1 i

[109].

2TrAρⁿ_Ais analytic for Re(n)>1. Thus the limitn→1⁺is well defined.

3Assuming TrAρA= 1.

t_E

x t_E =−∞

φ(x,0) tE = 0 Ψ[φ(x,0)] =

tE =∞ tE =∞

φ⁰(x,0)

tE = 0 = Ψ^†[φ⁰(x,0)]

t_E =−∞

Fig. 6.1.: Pictorial interpretation of the ground state written as a path integral.1

[ρ_A]_φ+φ⁻, such that∀x ∈B →φ(x,0) =φ⁰(x,0)and ∀x ∈A







φ(x,0)≡φ⁻ φ⁰(x,0)≡φ⁺

. In therms of a path integral this is equivalent to gluing together the two patches in Fig.6.1along the regionB as shown in Fig.6.2leaving an open cut along regionA with boundariesφ^±.

tE

0

∞

−∞

x B A B

φ⁺ φ⁻

= [ρA]_φ⁺_φ−

A φ⁺ φ⁻

Fig. 6.2.: Path integrating out the degrees of freedom of the system1 BfromAamounts to gluing together the two sheets as depicted, leaving a plane with a cut alongA.

Then in order to determine TrAρⁿ_Aone makesnreplicas[ρ]_φ⁺

1φ⁻₁[ρ]_φ⁺

2φ⁻₂ . . .[ρ]_φ⁺

nφ⁻n of such a reduced density matrix and takes the trace successively alongside those copies.

Again, this can be done in a very elegant way in the path integral formalism where this operation amounts to gluing together theφ^±_j’s asφ⁻_j =φ⁺_j+1, whereφ⁻_n =φ⁺₁ and then integrating overφ⁺_j, see Fig.6.3for another pictorial interpretation. This procedure effectively then corresponds to a partition functionZncalculated on an n-sheeted Riemann surfaceR_n.

TrAρⁿ_A= (Z1)⁻ⁿ Z

(x,tE)∈Rn

Dφe^−SE[φ]≡ Zn

(Z₁)ⁿ. (6.16)

φ

⁺₃

φ

⁻₃

φ

⁺₂

φ

⁻₂

t

E

x φ

⁺₁

φ

⁻₁

Fig. 6.3.: Determining TrAρⁿ_A using a path integral, where I set1 n = 3 for illustrative purposes.

With this knowledge one can rewrite (6.14) as SA=−(∂_n−1) lnZn

n=1. (6.17)

Hence all one needs to know in order to determine entanglement entropy is the partition functionZ_non the givenn-sheeted Riemann surface.

Example: 2d Conformal Field Theory

In a 2d conformal field theory with central chargeccalculating entanglement entropy is particularly easy to perform because of the infinite number of symmetries in a conformally invariant system. I will review this for three different examples of conformal field theories.

Infinitely Long System at Zero Temperature: Here I will consider a conformal field theory which corresponds to a quantum system which extends from−∞< x <∞ at zero temperature. First I will change to complex coordinatesw =x+itE and map the n-sheeted Riemann surface R_n to the complex plane Cvia a conformal transformation as

z=

w−u w−v

¹

n, (6.18)

where|u−v|=lare the boundary points of the entangling intervalA.

Using the transformation properties of the energy-momentum tensor one can relate the expressions of the energy momentum tensor onR_nwith the ones on the plane as

T(w) = dz

dw 2

T(z) + c

12{z, w}, (6.19)

where {·,·} denotes the Schwarzian derivative {z, w} = ^z_z⁰⁰⁰0 − ³₂^z_z⁰⁰0

2

. Since hT(z)i_C = 0 by translational and rotational invariance once finds for the expec-tation value of the energy-momentum tensor onR_n

hT(w)i_R In [107] it has then been shown that the conformal ward identities imply

hT(w)i_R

n = hT(w)Φn(u)Φ−n(v)i_C

hΦ_n(u)Φ−n(v)i_C , (6.21) where Φ_n(u) and Φ_−n(v) are two primary operators with scaling dimensions⁴

∆_n= ¯∆_n= ₂₄^c 1−_n¹₂. The key observation made in [107] then was that Tr_Aρⁿ_A behaves like the n^th power of the two-point function of a primary operator Φ_n with scaling dimensions ∆_n = ¯∆_n as specified before. Thus one has Tr_Aρⁿ_A ∝ hΦ_n(u)Φ−n(v)iⁿ_C. NormalizinghΦ_n(u)Φ−n(v)iⁿ_C=|v−u|^{−2n(∆n+ ¯∆n)}, and introduc-ing an infinitesimal parameter a(e.g. the lattice spacing) in order to render the resulting expression dimensionless one obtains

TrAρⁿ_A∝

|v−u|

a

−₆^c(n−_n¹)

. (6.22)

Using (6.14) and|u−v|=lone immediately obtains the famous expression of the entanglement entropy of an infinitely long 2d conformally invariant system at zero temperature [107,110]

S_A= c 3ln l

a. (6.23)

Other Geometries: The fact that TrAρⁿ_A transforms under a general coordinate transformation as a product ofntwo-point function of primary operators with specific scaling dimensions also means that it is particularly easy to compute that quantity in other geometries, simply by using conformal transformationsz→z˜=w(z)and [95]

hΦ(z₁,z¯₁)Φ(z₂,z¯₂). . .i=^Y

j

|w⁰(z_j)|^2∆nhΦ(w₁,w¯₁)Φ(w₂,w¯₂). . .i. (6.24) Assuming that the x-direction is compactified to a circle of circumference L one can use the conformal mapw = tan^πw_L⁰ to map fromw ∈ R_n to ann-sheeted cylinder with complex coordinates w⁰. This yields the following entanglement entropy [107,110]

4These kind of primary operators are also often calledtwist operatorsin the literature.

In the same way one can immediately obtain the entanglement entropy for a sys-tem on the circle in a thermal mixed state at finite inverse sys-temperature β. This corresponds to the conformal mapw=e^2πβw0 which maps each sheet ofR_nonto an infinitely long cylinder with circumferenceβ. This yields then the following entropy [107]

SA= c 3ln

β πasinh

πl β

. (6.26)

Im Dokument How General Is Holography? (Seite 63-68)