6.4.3 Zero temperature limit
The second analytically tractable limit is the one where one of the temperatures is zero, say TR = 0. In this case the geodesic length (6.30) is extremised by
x= πtTL(`−t) coth (πtTL) + (`+t)
2 + 2πTL(`−t) coth (πtTL) , zc2 = `(`−t
1 +πTL(`−t) coth (πtTR). (6.39) For a high temperature of the left heat bath TL`1 the entanglement entropy of the interval A with length` increases linearly in time
SA= L
4GNπtTL. (6.40)
It is tempting to discuss this result in the context of entanglement tsunamis, see e.g. [127, 131–133], which refers to the linear growth of the entanglement entropy after a global quench of a sufficiently large region and an initial quadratic growth [127–129,131–133,158]. Note however that our setup does not admit a quasiparticle picture.
6.5. Enganglement entropy: numerical approach 121
point s0 = 0. However, at the start of the procedure their numerical values are unknown. Thus the solving procedure is the search for the correct set of numerical values for xµ(s0) and (xµ)0(s0), which result in the desired endpoints (6.24). On a technical level this is implemented with a ‘find root’ method which requires an acceptable initial guess. For similar temperatures TL,R, a reasonable choice is to take the analytically known values of xµ(s0) and (xµ)0(s0) for a geodesic in the simple AdS-Schwarzschild spacetime given in equations (3.25)-(3.27) in [147].
Those initial conditions used in the dynamical spacetime (6.19) do not result in a geodesic which satisfies the boundary conditions, but the endpoints will not be far off either. This works very reliably for small temperature differences ∆T ≡ (|TL − TR|)/(TL +TR) ∼ 0.05 and for a not too sharp transition between the three regions, i.e. α ≤ 30. Those requirements restrict the applicability of the the shooting method in this setup. For the shooting method we used the minimal substraction renormalisation scheme (6.7).
The second method, which we refer to as matching method is the numerical com-pletion of the analytic approach in the previous section 6.4. Recall that the idea is to construct the geodesic ranging from one point at the boundary xµL, located in one region to the other xµR, located at a different but neighbouring region, as a piecewise defined trajectory. The expression for the resulting length of the trajec-tory as a function of the boundary conditions xµL,R and the pointxµc at which the two pieces meet is given in (6.30). This is the furthest we get analytically. From here it remains to minimise dL,R with respect to the matching point to obtain the trajectory of minimal length, the actual geodesic. This requires to find a numer-ical solution of (6.32). Note that to derive this expression we used the minimal subtraction scheme as well. Briefly, this undertaking can be split into two steps.
The first step is to find approximate solutions and check which of them result in a positive length. The approximate solutions passing this test are then refined us-ing Newton’s method. The final answer is given by the minimal positive geodesic length between the two endpoints xµL,R. Note that solutions which correspond to a geodesic going beyond the horizon were shown to be unphysical, i.e. they do not result in a positive geodesic length. Furthermore the minimal distance dL,R becomes complex valued for solutions which correspond to a null or timelike sep-aration. The matching method is not constrained by restrictions similar to the ones of the shooting method. It can be employed for large temperature differences in relative and absolute values as well. It does however, rely on the assumption that the coordinates xµ smoothly parametrise the border between the different
0.2 0.4 0.6 0.8 1.0ρ 0.2
0.4 0.6 0.8 1.0fA
3ρ2-2ρ3 TL=0.4 TL=10.
0.2 0.4 0.6 0.8 1.0ρ -0.05
0.05 0.10 fA-(3ρ2-2ρ3)
TL=0.2 TL=0.4 TL=10.
Figure 6.3: Normalised entanglement entropy fA for the interval A in temper-atures TL = 0.4 and TL = 10 compared to the universal formula (6.38) (left) and deviations from it for TL = 0.2, 0.4, 10. (right). The interval is chosen to
∆xA∈[0.175,1.35] and TR = 1.95. Figure taken from [1].
regions and that the metric components take care of possible discontinuities. The Mathematica code used for this method in a supplementary file together with [1].
In the parameter regimes where both methods are applicable, the results agree, providing a consistency check for each other. Note that the interval length is understood to be measured in units of the AdS radiusL and the temperatures in units of its inverse L−1. For the numerical analysis we set L=GN= 1.
6.5.2 Corrections to the universal behaviour
Let us first address the universal behaviour for small temperatures and temperature differences. In section 6.4 we presented an analytic formula for the universal time evolution of the normalised entanglement entropyfA(ρ) of a region A (6.38). The normalised time 0≤ρ≤1 (6.37) parametrises the time interval during which the shock wave passes through the interval of interest. This formula is restricted to the limit of small temperatures`TL,R <1 and temperature differences`|TL−TR|<1.
The universality refers to the fact that this expression is independent of the interval length ` and the location of the interval. The numerical analysis confirms this universal behaviour and in addition to it allows to go beyond the small temperature and temperature difference limit. Note that is was actually those numerical results which inspired the analytical treatment.
Figure 6.3 shows the deviation from the universal time evolution upon increasing the temperature of the left heat bath and thereby also increasing the difference between the two temperatures, as we set TR = 1.95. Moreover at TL = 10 the data hints at the linear time dependence of the normalised entanglement entropy (6.40). Recall that the latter was derived at TR = 0. This situation is mimicked
6.5. Enganglement entropy: numerical approach 123
0.2 0.4 0.6 0.8 1.0 1.2 1.4 t 1.005
1.010 1.015 1.020 1.025
(SA+SB)/(2SA(t=∞))
TL=0.2 TL=0.5 TL=0.7 TL=1.
TL=10.
0.2 0.4 0.6 0.8 1.0 1.2 1.4 t 1.005
1.010 1.015 1.020 1.025
(SA+SB)/(2SA(t=∞))
TL=10.
TL=11.
TL=12 TL=15.
Figure 6.4: The normalised sum SA+SB (6.42) as a function of boundary time.
The deviation of the sum’s initial and final value seems to be bounded. Figure taken from [1].
here by TL TR.
Another way to quantify the deviations from the universal behaviour (6.38) is to look at the sum of the entanglement entropies of two intervals A and B. The two intervals are assumed to be of the same length ` and are arranged symmetrically around x= 0. If the time evolution were truly independent of also the location of the interval, the sum of SA and SB would have to be constant. This conclusion becomes clear by inverting the definition of the normalised entanglement entropy f (6.37)
SA(ρ) =SA(0) + [SA(t→ ∞)−SA(0)]f(ρ), (6.41) where SA(t → ∞) = SA(ρ = 1) for any interval A, since S is constant as soon as the shockwave passed through the interval. Now we know from the analytical perspective that the sums of the entanglement entropies of two symmetric intervals AandBatt= 0 andt → ∞agree (6.29) andSA(t)+SB(t) = const iffA=fB =f. The fact that this is not the case is illustrated in figure 6.4. Interestingly, the normalised sum
SA(t) +SB(t)
2SA(∞) (6.42)
appears to be bounded from above by approximately 1.025.
6.5.3 Entanglement inequalities
Let us now turn to a very interesting usage of the matching method, explained in the beginning of this section. When it comes to the discussion of entanglement entropy, inequalities often play a role. They arise in the context of more than just
one regionAand its complementAc. In 6.2 we already mentioned two examples in the case of two different regions: subadditivity and triangle or Araki-Lieb inequal-ity. Three regions,A, B andC, already give rise to more complicated inequalities.
Among those the strong subadditivity is one of the most prominent examples S(AB) +S(BC)−S(ABC)−S(B)≥0, (6.43) where combinations of A, B and C indicate the union of the respective regions.
Note that there is a second inequality which sometimes is also referred to strong subbadditivity. It is given in equation (6.10) in [1]. Another common example is the monogamy of mutual information, also referred to as negativity of tripartite information
I3(A:B :C)≡S(A) +S(B) +S(C)−S(AB)−S(BC)
−S(AC) +S(ABC)≤0. (6.44)
Inequalities for the entanglement entropies of several regions are of particular in-terest from the perspective of holography. Given that they have to be obeyed from a quantum physics point of view, it is interesting to see whether they are obeyed in a holographic setup as well. It turned out that those inequalities are intricately related to the energy conditions in the bulk. Thus they promise to give further inside into both the details and the bigger picture of the holographic duality.
The matching method allows us to study inequalities in the context of the steady state setup whose peculiarity it is to provide a simple yet nontrivial time dependent framework. In this work we are interested in checking the strong subadditivity (6.43) and monogamy of mutual information (6.44) forn= 3 regions, the positivity of the four-partite information given in equation (6.18) in [1] for n= 4 regions as well as the negativity of the five partite function (6.19) and further inequalities given in equations (6.12)-(6.16) for n = 5 regions. This is the maximum number of regions studied in [1].
The first challenge for that purpose is to construct the holographic duals of the different unions of regions or intervals, as our setup only has one spatial dimension at the boundary. Figure 6.2 illustrates that even in the case of two intervals one has to think carefully about which of theoretically possible ways to connect the intervals’ boundaries through bulk geodesics. The authors of [1] developed an algorithm which allows to enumerate all of the possible configurations taking