We study the deformation of the thermofield double (TFD) under evolution by a doubly traced operator by calculating its entanglement entropy. From the AdS/CFT correspondence, in which TFD is dual for a two-sided black hole, such distortions correspond to the coherent shrinkage or expansion of the black hole. We then calculate the entanglement entropy after perturbatively coupling a system to the bath as a function of 𝜅, the system-bath coupling.
Furthermore, the black hole fulfills the laws of thermodynamics, in that the energy of the system is equal to the black hole's mass and entropy is proportional to the area of the horizon. The AdS/CFT duality implies that the partition function of the black hole is equal to that of the limit theory, and consequently the entanglement entropy associated with one side is equal to the entropy of the black hole in the bulk. When calculating the two-point function for leading order in the coupling, the theory's four-point function is necessary.
GEOMETRY OF THE CHARGED BLACK HOLES IN FOUR DIMENSIONS
The charged black holes
On the other hand, for 𝑟 𝑟𝐸 the spacetime becomes flat. Therefore, as we move from the outer horizon to infinity, one can imagine that the geometry of 𝐴𝑑𝑆 will change. 2.2 The spherical reduction and the JT gravity. To study the near horizon boundary of the black hole1, we will be the metric axis. Therefore, the conserved charge is the sum of the above charges for the left and right sides.
In the rest of the section, we calculate the leading contribution of the Schwarzian action to the four-point function. On the other hand, in the case where 𝜃. The contribution of the Schwarzian modes to the four-point function takes the following form Now, the important feature of F𝑂𝑇 𝑂 is that in the Lorentzian time where𝜃 = 2𝜋𝑖𝑡. Also note that from the form of the exponentially growing term in F𝑂𝑇 𝑂, we will take the following ansatz for the general case early on [23]:.
THE SACHDEV-YE-KITAEV MODEL
The green solid circle is the self-energyΣ and the black solid circle is the two-point function𝐺. We are interested in studying the theory's response to turning on the perturbation𝜎. Therefore, we expand the fields around the conformal value, i.e. 𝐺 =𝐺𝑐+𝛿𝐺 , Σ = Σ𝑐+𝛿Σand extends the action to second order:. 3.24). A set of eigenfunctions for this equation is𝑠(𝜑 .. 3.30) In particular, the source|𝑠corresponding to the responses with eigenvalue𝑘𝑐(ℎ) =1 produces resonances that reach IR physics.
So we can think of the Majorana fermions in 3.1 as the UV degrees of freedom that describe the theory of gravity, the Schwarzian action.
DISENTANGLING THE THERMOFIELD-DOUBLE STATE
Now consider two copies of the thermofield dual with the insertions of the unit operator 4.3. The second order correction comes from the connected part of the four-point functions in the OPE limit. The expression associated with the second configuration is equal to:. where the second line corresponds to T. Both expressions are with respect to the inverse temperature 𝑠 𝛽. Note that in the limit 𝑠 →1 both 4.17 and 4.18 vanish, which means that we can calculate their contribution to the entanglement entropy separately. 4.19) and the anti-time-ordered four-point function is simply the complex conjugate of the above expression.
In this section, we will calculate the temperature of the deformed thermofield double mode directly by studying the two-point function of two probing fields inserted on the right side. We expect the 𝑡1+2𝑡2 dependence of the two-point function to come from the second type, from the exponentially growing term in the OTOC configuration [39]. 2Δ due to exponential decrease in the strength of the interaction Hamiltonian, and the system will equilibrate with the new temperature much earlier than the scrambling time.
Then the value of the entropy can be derived from 𝑆(𝑢) = 2𝜋. Note that the relationship between entanglement entropy and the coarse-grained quantity 4.43 is more than the equality of final answers. For g<0, the operator will decrease the entropy of the black hole, the size of the black hole will decrease, and so part of the region beyond the horizon will now be visible to the outside observer. You could argue that the effect of the double traced operator for 𝑔 < 0 is that the black hole, the Einstein-Rosen Bridge, shrinks, so that part of the area beyond the horizon is now visible to the outside observer.
As a result, the wormhole will shrink between the two sides and part of the interior can be examined from the outside; see Figure 4.4. To quantify this, we define the length of the wormhole at time 𝑡 as the length of the geodesic connecting the two points at 𝑟 = 1. To send a message from one side to the other, we also need the feedback from the message to the black hole leading to the expansion of the wormhole.
We can also interpret 4.37 as the conservation of energy for a particle whose trajectory is the boundary.
PERTURBATIVE CALCULATIONS OF THE ENTANGLEMENT ENTROPY
Then each term in the expansion (5.4) will look like this (where T,eT and subscripts are omitted):. 5.6) The diagram element in the middle is the Keldyshev contour for the heat bath. B∗B(𝑡))𝑠 can be represented by gluing such 𝑠 diagrams (describing different replicas of the density matrix) in cyclic order — see Figure 5.1, where the replicas are depicted in different colors. The main contributions are due to the purple loops; the shorter and longer loops correspond to equations (5.11) and (5.12), respectively. b) Simplified diagram obtained by omitting bath replicas.
As already mentioned, there are two types of loops that contribute to ln Tr(𝜌. Loops of length 𝑑 =2 themselves can be of two forms, one of which is shown in Figure 5.1. Letting the system interact with the bath, we must we see a behavior similar to the peak in the Page curve. In fact, a non-trivial effect will be seen in the second order and only when combined with a subsequent interaction with the environment.
In the next section we study partial derivatives of 𝑆(𝜌), assuming that 𝜌 depends in some way on parameters. To cover this case, we will use a trick called “locking two operators in the same replica”, see section 5.1. But this last identity is false because in the expression for hB, 𝑌 , 𝑍i the operators 𝑌 and 𝑍 can appear in different replicates, but inhB, 𝑌 𝑍i cannot.
Although the transformation just described changes the operator𝑊(𝛽, 𝑥 , 𝜖) in a non-trivial way, we will find an agreement in the Ω → ∞ limit. The last equation can be interpreted as the operators𝑌(𝑡) and𝑌†(𝑡) in the expansion of Tr𝑊𝑠 are locked in the same replica. We now take the 𝑠derivative of both sides at 𝑠 = 1 and consider the 𝑥2𝑦2 term in the Taylor expansion.
In the 𝑠 → 1 limit, the geometry is given by a smooth metric on the disc and the position of the branch point. The only property we need is that if there is large time separation, 𝑝+ and 𝑝− only depend on the fields inserted in the past and the future respectively. Thus, the change in the entropy must factor in two quantities, dependent on the corresponding fields.
THE ADS 2 SPACE IN DIFFERENT COORDINATES
A BRIEF REVIEW OF THE THERMOFIELD-DOUBLE STATE AND ITS PROPERTIES
It is clear that such states are in fact orthogonal to all other states of pre-Hilbert space. Therefore, we can take the quotient of our pre-Hilbert space relative to. J, or equivalently, define the state|𝐴ias follows:. The final part of our construction is considering the completion of the space above.
Before we study the representation of the operators on this condition, we provide some definitions: we call a condition to be cyclic for an operator algebra A, if the operation of the operators on this condition is a dense subset of the Hilbert will give space. It is clear that the operator algebra [A𝑅,A𝐿] =0 and they are in fact each other's commutator. Therefore, under the evolution with 𝑈, the time directions in the left and right sides are opposite.
In AdS/CFT correspondence, it was suggested by Maldacena that given a holographic CFT, the thermofield doublet state represents a two-sided black hole in the bulk. To construct the Tomita operator, it should be noted that such black holes have a time-like Killing vector which is in the Schwarzschild coordinate with the generator H . Near the horizon, the geometry of the black hole is similar to the geometry of the Rindler space to first order and H is the boost generator.
To construct the Tomita operator, we consider𝑈(𝑖 𝜋) generated by the modular Hamiltonian, which is the Euclidean rotation in the (𝑇 , 𝑋) plane, and use the CRT4 operator to bring it back to the same point, B.1.
THE MAXIMAL EXTENSION OF THE BLACK HOLE FINAL STATES
Then, in a small neighborhood of N, denoted by O, we can transport the vectors parallel along 𝑟𝜇 and so it will be coordinated by (𝑣, 𝑟 , 𝑥𝐴), called the Eddington-Finkelstein (EF) coordinate, and N is described by 𝑟 =0. From our construction of the coordinate system, the metric components are not a function of 𝑣, since on the horizon by going along 𝑣 geometry does not change, and the u component of the coordinate system does not change as we go along the r direction. To summarize, we have so far proved that there is a neighborhood |𝑈𝑉| < 𝜖(𝑥𝐴)of the Killing horizon N where the metric will take the form C.16 with the constraint that𝑉 >0.
We can define our manifold in a straightforward way: since we assume that the cross-sections of N are compact, we know that every cover with open cross-section sets is reducible to a finite one. We will define the new time coordinate𝑡(𝑣, 𝑟, 𝑥𝐴) that the Killing vector is orthogonal to the constant time surfaces. In fact, the role of Einstein's equation and the state of matter is to confirm that surface gravity is constant at the horizon.
BIBLIOGRAPHY
