Entanglement in Fock space of
random QFT states
Javier M. Magán and Stefan Vandoren
[8mm] Institute for Theoretical Physics and Center for Extreme Matter and Emergent Phenomena,
Utrecht University, 3508 TD Utrecht, The Netherlands
j.martinezmagan,
[3mm]
Entanglement in random states has turned into a useful approach to quantum thermalization and black hole physics. In this article, we refine and extend the ‘random unitaries framework’ to quantum field theories (QFT), and to include conserved charges. We show that in QFT, the connection between typical states, reduced subsystems and thermal dynamics is more transparent within the Fock basis. We provide generic formulae for the typical reduced density matrices and entanglement entropies of any given subset of particles. To illustrate our methods, we apply the generic framework to the simplest but non trivial cases, a massless scalar field in two dimensions and its generalization to the case of scalar fields, including the large limit. We find the effective temperature, by matching the reduced dynamics to a Gibbs ensemble, and derive the equation of state of the QFT. The deviations from perfect thermality are shown to be of order instead of , a result which might be relevant for black hole physics. Finally we describe the analogue of the socalled ‘Page curve’ in the QFT scenario as a function of the energy scale which divides high from low energy degrees of freedom.
Contents
1 Introduction and summary of results
1.1 Quantum thermalization and random matrices
In its simplest formulation, the black hole information paradox [1] follows from the statement that black hole radiation [2] is precisely thermal, with the temperature depending just on the conserved charges defining the black hole. Given that there is an exponentially big number of initial states that could collapse to the given black hole, where is the BekensteinHawking entropy [3, 2], it is concluded that after the black hole has evaporated the information about the initial state is lost. Mathematically, the paradox arises since under unitary evolution of the initial quantum state, we can never produce a thermal ensemble,
(1.1) 
where is an appropriate Gibbs distribution for the problem at hand. Interestingly, this problem is strikingly similar to one old problem in quantum mechanics, the problem of ‘quantum thermalization’, see [4, 5, 6] and references therein. The problem of quantum thermalization can be easily stated as:

In the context of microscopic unitarity, and given (1.1), how can Gibbs ensembles emerge?
In what follows we will assume the Hilbert space to be factorizable^{1}^{1}1The ideas and results concerning the problem quantum thermalzation are most probably not sensitive to this condition. But in this framework the concepts can be explained more transparently, and a plethora of results can be derived. Besides, the models we will explicitly consider in this article fall in this factorizable class. :
(1.2) 
The number of factors in (1.2) is unconstrained, it can be finite or infinite. Besides, the Hilbert space factors do not need to be equal to each other. If the algebra of operators acting on is given by , with a discrete parameter, the algebra of operators acting on is
(1.3) 
Reduced subalgebras/subsystems are defined by specifying a set of Hilbert factors, and considering nonunity operators only in those subfactors:
(1.4) 
As is well known, given a unitarily evolving pure state , the expectation value of any operator belonging to such subalgebra A is completely characterized by its associated reduced density matrix:
(1.5) 
where
(1.6) 
and is the complement subalgebra/subsystem of A.
Although perfect thermality cannot be attained within unitary evolution, we might still expect certain approximate thermality for the correlation functions of the theory. The condition stating that all measurements (1.5) are approximately thermal
(1.7) 
can be more rigorously stated as
(1.8) 
with small errors, and where
(1.9) 
is the reduced thermal density matrix associated to A.
Given these preliminaries, some immediate questions concerning quantum thermalization are:

What is the corresponding for the specific theory and state considered? Or equivalently, how the parameters defining are related to the parameters defining ?

How does the error scale in the thermodynamic limit?
Considering (1.8) and the previous questions, one modern approach to tackle this problem is to compute the entanglement entropy of the reduced subsystem A, which is defined by
(1.10) 
If our state is such that relation (1.8) holds, the next insightful relation follows as well:
(1.11) 
which states that the origin of thermal entropy in quantum mechanics lies in quantum entanglement.
The discussion has been until now entirely generic. To proceed we should take a specific Hamiltonian and find the exact evolved quantum state:
(1.12) 
This is certainly a difficult task, since we are usually interested in high energy states, with many particles excited^{2}^{2}2Some important progress has been done in the context of integrable systems, see [7] and references therein, conformal field theories, see [8], and in the context of holographic thermalization, see [9, 10]. In this article we will be interested in other approach, which is more general in certain aspects and it is able to provide the scaling of the deviations from thermality with the size of the system.. But due to these complicated dynamics, we expect that some approximations can be done. In particular, one common procedure is to assume that after a certain time we can approximate the evolved state as a random vector^{3}^{3}3See [11] and [12] for two nice expanded reviews on the subject.:
(1.13) 
which can be equivalently thought as being created by the application of a random unitary matrix to a given initial state:
(1.14) 
The question now is obvious:

How do we define or ?
If we could answer this question generically for any theory and initial state, it would allows us to compute typical values of physical quantities in a unitary framework. The physical quantities in which we are interested in this article are the typical^{4}^{4}4We use the words ”typical”and ”average” interchangeably. For example, the typical reduced density is by definition the same as the average density matrix. reduced density matrix of a given subsystem A, given by
(1.15) 
and the typical entanglement entropies
(1.16) 
This approach was pioneered in [13, 5]. In those articles a bipartite Hilbert space was considered, with dimension , where and are the dimensions of the corresponding factors of the total Hilbert space. For this Hilbert space, was defined by averaging over the continuum manifold of quantum states [13], with a constant measure, i.e the Haar ensemble of quantum states in the full Hilbert space. Again, this is equivalent to applying a random unitary picked from the Haar ensemble of random unitaries. With this specific definition of randomness in the given Hilbert space, the typical reduced density matrix (1.15) was found in [13] and, using those results, the typical entanglement entropy (1.16) was found in [5]^{5}^{5}5See [12], and references therein, for an analogous but somewhat different approach coming from quantum information theory.. In the light of the previous questions about quantum thermalization, their results for this specific case are the following:

The typical reduced density matrix of a subsystem A is proportional to the identity matrix. Therefore, this reduced density matrix might be associated to the thermal density matrix of any Hamiltonian (defined for the finite dimensional Hilbert space) in the infinite temperature limit.

The error depends on the size of both subsystems A and B. For , the error in formula (1.11) is of , where is the total entropy of the system. For the error is of .
The first bullet point expresses the fact that unitary evolution typically leads to thermal behavior, with an infinite temperature density matrix for this specific case. The second point expresses the deviations from thermality, showing that the error size when is exponentially suppressed. This result has been used repeatedly in the past to argue against the Hawking information paradox [2], since the available computations of correlation functions during black hole evaporation are insensitive to the expected corrections, of if we trust the previous random approximations.
We remark here, see [11, 12] for recent reviews and references therein, that the intuition coming from the random unitaries/states framework has had a strong impact in the context of black physics and quantum thermalization, given a plethora of new concepts, problems and solutions. But we believe that the framework deserves further development, since its present status has also deficiencies. Firstly, the random unitary framework has not been formulated generically in the context of QFT. Secondly, the random unitary evolution defined in [13] does not conserve any charge, such as energy, momentum, electric charge, etc. Besides, the framework just provides a Gibbs distribution with infinite temperature. It is then interesting to ask the following questions:

Can the framework be generalized systematically so as to include conservation of any given charge?

Can it be formalized in a QFT context?

Can noninfinite effective temperatures emerge in the framework?

Do the deviations from thermality differ from the previous case?
1.2 Summary of results
The natural answer for the first question is not to choose the Haar ensemble of random unitaries/states, but to choose the appropriate ensemble of unitary matrices conserving any given set of charges. The problem with this approach is to construct such an ensemble, and we do not see a systematic way of doing so. In this article we will use the Haar ensemble in an appropriate subspace of the Hilbert space, defined by the given initial charges. This procedure can also be seen as implictly defining an esemble of random unitaries different from the Haar ensemble in the full Hilbert space. Within this framework we will provide generic formulae for the typical reduced density matrix and entanglement entropy of any given subsystem and for any given set of conserved charges. These objects turn out to be written generically in terms of ‘constrained multiplicities’, number of different configurations with a given set of constraints. The results connect in a direct and transparent way modern approaches of quantum information theory, concerning for example the computation of entanglement entropies, with more traditional frameworks of microstate counting.
For the second question we argue that the natural way to generalize the framework to a QFT context is to use the Fock space formalism. Fock space provides an ideal setup to define the appropriate subspaces corresponding to the given set of conserved charges. Besides, it is also an ideal setup to define partitions/subsystems by tracing out different set of particles. One possibility is to integrate out all momentum modes but a given one, as done in [18] for the case of vacuum states. We can also integrate out positively charged particles, left movers of a CFT, etc. Interestingly, the formulas we find in the case of random states are generic for all types of subsystems.
For the third and fourth questions we need to sacrifice generality of the analysis and choose concrete models in which to apply the developed formulas. We choose the simplest but nontrivial cases, in which many things can be computed analytically: a massless scalar field theory in 1+1 dimensions and its generalization to the case of scalar fields. For a given initial total energy, we derive the typical density matrix and prove that in the thermodynamic limit it is equal to the thermal density matrix. It turns out that the effective temperature is finite, and depends on the initial energy in the way expected from usual thermal considerations. In this way we are able to derive the equation of state of the QFT directly from random dynamics. The equation of state is seen as the typical macroscopic configuration of an underlying unitary quantum model.
We also compute the deviations from thermality, obtaining an important insight. As commented before, for the socalled Page case, developed in [13, 5], the typical density matrix is exactly the thermal distribution withouth the need of taking the thermodynamic limit, and the deviations of the typical entropy from the entropy of the typical reduced density matrix are exponentially suppressed. This means that deviations from thermality are exponentially supressed, see Appendix A for a complete discussion of all cases. On the other hand, as we will show, in the QFT context the typical density matrix and thererefore the entanglement entropy of the typical density matrix are not directly equal to their thermal counterparts, the difference being of . Therefore, to compute the leading deviations from thermality we just need to compute the entanglement entropy of the typical density matrix, a feature which will simplify matters in future developments. From other perspective, our results pinpoint the microcanonical ensemble as the right ensemble to use when considering pure state unitary evolution.
At any rate, the important upshot is that the errors expected in a scenario in which some charges are conserved are much bigger than the ones commented before, being of instead of . The applications of these results to conceptual problems involving black holes is still unclear and deserves further development. At any rate it is interesting that:

The corrections affect directly the diagonal entries of the density matrix, and therefore should be seen in observables such as mode occupation numbers. They are not nonperturbative, and should be visible in the first corrections.

For any QFT, deviations from thermality can be computed exactly within our framework, and it would be interesting if this can guide the computations of these corrections directly from black hole geometric considerations.
At the end of the article we speculate on potential applications of this framework to the relation between geometry and entanglement [19, 20]. In particular, our framework can compute entanglement entropies for any desired subsystem, and it is imaginable that we could get important insights into the emergence of near horizon regions through entanglement in the field theory. In previous literature there has been problems with the ‘integration out’ procedure of high momentum modes, or the socalled ‘holographic renormalization’, when trying to get to the near horizon region of black hole backgrounds, and therefore only the asymptotic region is well understood. It seems that our framework may overcome such difficulties in the specific context of random evolution, and might teach us important lessons about event horizons, stretched horizons and near horizon regions of black holes.
2 Reduced density matrices in Fock space and multiplicities counting
Our initial quantum state will be a pure quantum state within a Fock space, which is the natural basis of QFT and certain spin systems:
(2.1) 
The notation works as follows. Labels denote the number of particles in a given representation. Labels specify the representation. For example, in a certain theory we could have , where is the momentum, the spin and some charges labelled by . As usual we assume that the Fock states in (2.1) are attached with a total conserved energy and total conserved charges of the free theory, where label different possible charges. Finally, when there is no need to write all the previous labels we will use greek letters to simplify notation. They are just natural numbers running over the states^{7}^{7}7We suppose we have a discrete theory, probably with an infinite, but countable number of states such as in QFT. The examples we work out in the article fall in this class, since they are QFT defined on bounded domains.. For example, normalization of the basis vector is expressed as .
We now follow the logic described in the introduction. Consider that the full interacting Hamiltonian of the system commutes with the total charges, so that and are conserved. If we begin with an initial Fock space state , with definite values of and , the various charge conservation laws force the unitary evolution of the initial state to be of the following form:
(2.2) 
where the sum does not run over all Fock states (2.1), but only for those with definite values of and . By definition there are of those, i.e the microcanonical number of states for a given total energy and total charges.
As explained in the previous section, a useful approximation when considering chaotic and highly entropic sectors of the theory is to consider . In this article, with the objective of extending the random unitary framework to systems with conserved charges, we will use Haar randomness in the previous subspace, which is indeed defined by the given set of charges. Up to second order corrections (doubly exponentially suppressed in the entropy of the given sector) this approximation can be defined by the following averaging procedure^{8}^{8}8In Appendix A we show how to derive Page’s formula for the average entanglement entropy [5] with these simple Gaussian relations.
(2.3) 
Imposing average normalization of the state we obtain
(2.4) 
We want to remark that the ensemble of random states will always be the same, the previous gaussian ensemble of random states, which can be obtained by an analogue ensemble of random unitaries with random gaussian matrix entries. What will change is the sector of states in which gaussian randomness is considered. The sector of states will be the mathematical object which needs to be analyzed in several ways. From the perspective of the full Fock space, the relations (2.3) implicitly define the action of an ensemble of random unitaries, preserving any given number of conserved charges, on some state of the Hilbert space.
With (2.3) we can now ask questions about typical properties of the field theoretic state . For example, the average of the global density matrix is given by
(2.5) 
which is just the microcanonical density matrix for a theory with a generic number of conserved charges, obtained here in a straightforward manner by applying random dynamics in the appropriate subspace. Random dynamics seem to pinpoint the maximally mixed microcanonical state, instead of the canonical ensemble.
The next natural step, given the structure provided by Fock space, is to choose one particle type, say , and integrate out all the others. Notice that the case of momentum space entanglement, analyzed in [18] for vacuum states, is going to be seen here as a special case, arising when we consider all types of particles within a specified momentum shell. But otherwise the Fock space formalism provides naturally more possibilities, such as the entanglement between positively charged particles and the rest, particles with a given spin and the rest, left and right movers of a CFT, etc. All possible types of sets and reduced dynamics within the Fock basis of the Hilbert space are constructed by joining the appropriate set of conforming the type of subsystem we want to study. We thus begin with the simplest one, given by one particle type . We first write the state in the following form
(2.6) 
so that now represents all other particle types different from , but the sum still runs over the previously specified subspace. The reduced density matrix is
(2.7) 
This is directly a diagonal density matrix, without the need of averaging. Indeed, for a subsystem with one particle type , it just happens that for two states and with the same energy and charge, they must obey . It will no longer be true for subsystems with more than one particle type. Using (2.3) we can obtain the typical reduced density matrix
(2.8)  
where is the number of states with energies , charges , and a fixed number of particles of type . The maximum value of the number of particles of type is denoted by and needs to be determined case by case. We discuss this in the example presented in the next section.
With the previous density matrix, the average entanglement entropy of a particle of a given type , in the fixed and sector is given by:
(2.9) 
Being rigorous this is the entanglement entropy of the average density matrix, usually denoted by . The full average entropy is computed in Appendix A, where we show that the difference between the two is exponentially suppressed in the microcanonical entropy. The entropy of the typical density matrix will be enough for us, since we are not interested in the deviations between the typical entropy and the entropy of the typical density matrix, but instead, on the deviations from thermality. In the usual case of [5], typicality and thermality turn out to be exactly the same, and therefore deviations between thermal entropy and average entanglement entropy are equal to deviations between average entanglement entropy and entanglement entropy of the average (see Appendix A for the complete discussion). We will show that when conserved charges are taken into account this is not longer true. The leading deviations from thermal entropy are those already accounted for by the entropy of the average density matrix, a feature which simplifies present and further developments. At any rate, if needed, the full computations are described in Appendix A.
Relations (2.8) and (2.9) are generic formulas which turn out to be written just in terms of ‘constrained multiplicities’, the number of different states with a given set of constraints. Notice that the result for the entanglement entropy is finite, and no divergences occur, even taking into account that we are dealing with quantum field theories. This is obviously because we are carefully applying conservation of charges. Although the multiplicities might be difficult to compute in general, there might be a class of theories in which they can be computed, and explicit connections with thermal density matrices (in particular generalized Gibbs ensembles) might be established. We do this in the following section for the case of certain scalar field theories. At the same time the generality of (2.8) and (2.9) suggests there might be a generic way to prove that the probabilities are well approximated by the Generalized Gibbs Ensemble. Also it would be interesting to compute this multiplicities for integrable theories. In those theories Generalized Gibbs Ensembles are expected to govern the dynamics but deviations from them might be bigger. This might aliviate several problems encountered in previous literature, see [7]. We leave these interesting paths for future work.
Generalizing the procedure to include any desired subset of particles is straightforward. We first form the set of in which we are interested, and write as:
(2.10) 
where now labels all particles types different from . The reduced density matrix then reads:
where labels the set of particles which are traced out. As opposed to the previous oneparticle case, this is not a diagonal density matrix^{9}^{9}9Although it is not diagonal, it has a nice block diagonal structure, as used and described in Appendix A.. However, taking the average we find the following generic formula:
(2.12)  
which is diagonal. The average probabilities are ready to be compared with thermal expectations, i.e with probabilities coming from Gibbs distributions. We will discuss examples in the next section.
The average entanglement entropy  i.e. the entanglement entropy of the average density matrix (see Appendix A) is finally
(2.13) 
We concldue that the generic equations (2.12) and (2.13) apply to any theory with a Fock basis structure, such as a QFT or certain spin systems. In particular it is applicable to Holographic Field theories. It seems a rigorous framework to study entanglement between infrared and ultraviolet domains at finite temperature. An exciting possibility is that we could potentially extract physics from the near horizon regions of quantum black holes from the structure of entanglement in Fock space of the field theory. At any rate it is expected to give more insights into the connections between entanglement and quantum gravity [19, 20], an exciting direction which we leave for future work.
3 Examples: massless scalar fields in two dimensions
In this section we analyze in detail two specific examples. The first example is a massless scalar field theory in 1+1 dimensions defined on a finite line segment. The second one is its generalization to the case of scalar fields. For both cases we will show the emergence of Gibbs distributions as effective states for reduced subsystems. The procedure also allows the computation of deviations from precise thermality. Finally, entanglement entropy for single modes as a function of the momentum, and entanglement between high and low energy momentum modes for a given energy cutoff will be discussed. We will end with the analogue of Page curve [5] for the case at hand. This enlarges the program spelled out in [18] to the case of random states.
3.1 Massless real scalar field on a finite segment
In this case the energy/momentum dispersion relation for excitations over the vacuum, together with the quantization condition reads
(3.1) 
for a segment of length and . The Fock space is spanned by vectors of the type:
(3.2) 
where is the number of particles with momentum , and is just a natural number running over all the infinite but countable eigenstates, used here to simplify notation. These are eigenstates of the free Hamiltonian with Dirichlet boundary conditions. If the true Hamiltonian conserves the total energy, which for a Fock state reads
(3.3) 
where is defined by , then any initial state with definite total energy will evolve towards states of the form
(3.4) 
where the are all the states belonging to (3.2) with total energy . By definition these are , where is the microcanonical entropy at energy . Noticing that (3.3) can be written as
(3.5) 
we conclude that the number of states with a given is equal to the number of different partitions of the natural number , which is given asymptotically for large by
(3.6) 
The global entropy can now easily be extracted and for large we find
(3.7) 
So, in the limit , this agrees with the Cardy formula for a CFT with central charge and . If the interacting theory would be a conformal theory on a line segment of length , the conformal dimension of the pure state would then be .
Now we can directly apply the generic formulas derived in the previous section. The approximation is operationally defined by (2.3), which we repeat here to emphasize that it does not change from one case to another:
(3.8) 
Imposing average normalization, we obtain . The average of the global density matrix is given by
(3.9) 
which is just the microcanonical density matrix, obtained here in a straightforward manner by applying random dynamics in the appropriate subspace.
3.1.1 Entanglement of a single momentum cell
We now study the entanglement entropy in a pure state of a single particle specie. In our example, choosing one particle type just amounts to choosing a definite momentum for a fixed . The momentum cell can still be multiply occupied, so the subsystem has a finite dimension determined by the size of the cell, i.e. , where the maximum occupation number is given by , the integer closest to from below. Writing the state in the form
(3.10) 
where now represents all other particles types different from , we arrive at
(3.11) 
This is directly a diagonal density matrix, due to energy conservation. The average density matrix is
(3.12) 
where is the number of states with total energy and particles with momentum . Therefore, the average entanglement entropy of momentum cell is given by
(3.13) 
Formulas (3.12) and (3.13) are the analogues of (2.8) and(2.9) for the case at hand. To compute these quantitities we need to find . In this example this number can be analytically computed. The procedure is explained in detail in Appendix B. Here we just quote the result:
(3.14) 
where is the number of partitions of the number .
Now we are ready to make the first crosscheck of our computations, which constitutes one of the main results of the article. The typical probability of finding particles with momentum is given by
(3.15) 
It is easy to check that all probabilities add up to unity. The leading term when , or equivalently , is given by
(3.16) 
This is a thermodynamic limit. Indeed, means that the energy in momentum cell is small compared to the total energy, or equivalently, momentum cell is not well occupied. This implies that most of the energy is sitting in the other momentum cells, and that these cells form a heat bath for momentum cell . Another way of saying this, is that this limit is a good approximation for low values of . For higher values of , the energy in this momentum cell will typically be too large, or will be too small for the thermodynamic limit to be a good approximation. If we would have used the usual Gibbs ensemble this probability would be given by
(3.17) 
So in the limit , of for CFT’s, large conformal dimensions , the pure state is typically seen by the momentum cell as a thermal bath at a temperature given by
(3.18) 
We thus see that it is possible to derive generic Gibbs ensembles, with any desired effective temperature, using random dynamics in the approppriate subspace. Notice that (3.18) implies on the one hand, and on the other hand there is the general relation . Combining the two, we can determine the pressure density:
(3.19) 
This is the same equation of state as for a twodimensional CFT. In this way we expect to recover the known relation , valid for dimensional CFT’s, using just random dynamics. More generically it should be possible to derive any equation of state by choosing the appropriate field theory and using the same procedure. The equation of state of a fluid system, at least for this case, is the typical macroscopic configuration of the true evolving pure quantum state.
The next step is to compute the deviations form thermality. The next to leading term in the previous expansion is given by
(3.20) 
To proof this, we only needed the HardyRamanujan asymptotic formula for the number of partitions, see (B.12) in Appendix B.
The error we find here is thus much bigger than the one is usually expected. Within the usual random unitary formalism, the errors are typically of , so exponentially suppressed in the entropy. In our case, if we take a momentum cell with energy , we are finding errors of . This seems just to be due to energy conservation, which is explicitly ensured in our formalism. That we find such large errors may have implications for black hole physics and bulk locality in AdS/CFT, see [11, 12] and references therein, and we address this issue further in the next subsection when we discuss the large limit.
Finally, the average entanglement entropy of the momentum mode is given by^{10}^{10}10In the sum it might happen that some probabilities are zero. These terms simply do not contribute to the sum, since . :
(3.21) 
and we remind that . The dominant contributions to this sum come from the low occupation numbers, where the probabilities are approximately thermal. For high occupation numbers, , the probabilities are exponentially suppressed in the entropy, as one can easily compute from (3.15).
At present we have not found a way to evaluate the sum algebraically in closed form as a function of , though one can evaluate the sums explicitly for any given and , e.g. on Mathematica. However, for high values of this is possible. For example, in the case of , the highest momentum possible, there are only two terms in the sum in the entanglement entropy, and , and the result is
(3.22)  
where in the second line, we took the leading term in the limit . The relation (3.22) shows that there are extremely small entanglement entropies in random QFT states^{11}^{11}11Notice that this is not the case when considering the common Page case [5]. Indeed, in that case the minimum entanglement entropies are of in the thermodynamic limit.. For conformal field theories, and in the context of the AdS/CFT correspondence, we do not expect these entanglement entropies to be captured by some geometric quantities in the bulk. The proposal for deriving entanglement entropy of CFT’s holographically, developed in [19], is expected to capture entanglement entropies with a minimum size of , since this would correspond to surfaces of Planckian size. Entanglement entropies of are clearly of nonperturbative nature from the point of view of AdS/CFT dualities [21].
On the other hand, the thermal entropy of a single mode is based on the Gibbs probability distribution (3.17). The result is wellknown and can be directly computed using and Shannon’s expression for the entropy. It reads
(3.23) 
with inverse temperature given by (3.18), and we remind that , such that , which is independent of the size . The function (3.23) is a monotonically decaying function.
The two functions, the entanglement entropy and the thermal entropy, are plotted against the momentum in Figure 1. The dependence against provides some short of entanglement ‘running’ on the energy scale of a thermallike state.
It would be interesting to find an asymptotic formula for the entanglement entropy for large and generic momentum . This would make easier the comparison with the thermal approximation. Numerical analysis, as can be seen from Figure 2, shows that for large values of , the entanglement entropy approaches the thermal entropy, but the corrections to the thermal result are not exponentially suppressed in the entropy, and go like . This was obviously expected from the corresponding errors in the probabilities themselves. At any rate, notice that since corrections die in the thermodynamic limit, the expectations coming from the simple analytical thermal expression (3.23) might be ready to compare with geometric dual formulations of the QFT, a very interestring direction to explore.
3.1.2 Entanglement entropy of lowenergy degrees of freedom and Page curve
As described before we can also consider any subset of particles we wish. In this case a natural property to analyze is the entanglement between highmomentum and lowmomentum modes for a given threshold . More explicitly we can integrate out all particles with momentum higher than . The reduced density matrix is then given by
(3.24) 
where labels the set of particles with momentum greater than . As in the previous section, this is not a diagonal density matrix. Taking the average we find
(3.25)  
The average probabilities can be computed using the same previous method of constrained partitions. This is explained in detail in Appendix B. The result is given by (B.17) divided by the total number of partitions :
(3.26)  
This is a quite complicated expression for generic , though one can evaluate these sums using Mathematica rather easily. The situation becomes more cumbersome when evaluating the entanglement entropy, since we now have to perform additional sums over multiple occupation numbers. In practice this turns out to be rather hard using Mathematica, and we only succeeded to evaluate the entanglement entropy of a system of up to the first eight modes. However, a simplification occurs whenever , namely when the subsystem has much smaller energy then the total energy. Notice that if , then we can approximate the partitions
(3.27) 
which we can use in the probabilities (3.26). Indeed, if holds, we can apply (3.27) to all terms in (3.26), so that we obtain
where is the probability given by the Gibbs ensemble. One can again compute the subleading corrections, using (B.11) for each term in (3.1.2) and find powerlaw suppressed terms in the energy of the subsystem divided by the total energy. So the error analysis is similar to the case of a single momentum cell, and the leading corrections scale like . Hence for any subsystem, up to computable corrections, the entanglement entropy associated to is just the sum of thermal entropies of each momentum mode. In this way we can derive the analogous curve in a QFT setup to the socalled Page’s curve [5]. Obviously, if we express the entanglement entropy as a function of the corresponding thermal entropy we obtain directly Page’s curve, with somewhat different deviations from thermality. But in this context it is more interesting to paint the entanglement entropy as a function of the energy scale used to divide the highenergy modes from the lowenergy ones, given by . Hence we define two complementary subsystems and in momentum space, as depicted in Figure 3.
Up to subleading corrections in the thermodynamic limit, we can use the thermal entropy for the lowenergy modes, which becomes simply the sum of the thermal entropy of the individual modes
(3.29) 
The plot of the thermal entropy is given in Figure 4.
It is hard to perform the sum in (3.29) analytically, except in the continuum limit where the modes become infinitesimally spaced. Since , the continuum limit is , or keeping temperature fixed. In this limit we can approximate the sum by the integral
where we used the indefinite integral
(3.31) 
and is the total entropy (3.7). As a crosscheck one can verify that in the limit , one obtains the total entropy. This can either be seen from the asymptotic expansion of the dilogarithm, or by directly doing the integral
(3.32) 
We can also determine the thermal entropy of the complementary system, which we denote by , namely the system of momentum modes that lie between a given and . This is given by
(3.33) 
Applying the previous continuum limit we can obtain also analytical formulas for the entropy of . The plot of the thermal entropy of subsystem is given in Figure (5).
The thermal entropies associated to and have different functional structures as we vary the subsystem sizes. Given that the entanglement entropy of should be equal to that of it might naively seem there is a contradiction here. But indeed there is no contradiction, since the thermal entropy is a good approximation for the entanglement entropy of for values up to (half the entropy), and for the entanglement entropy of for larger values of .
Therefore, what is meaningful here is the critical momentum mode for which the thermal entropy of the reduced subsystem is precisely half of the total entropy. This happens when the corrections to the first term in (3.1.2) cancel between each other. Remarkably, in the large limit, there is an exact solution of this, due to the identity of dilogarithms
(3.34) 
which we will use for and in (3.1.2). For these choices, all other terms cancel except for the first one. Hence the critical momentum dividing the QFT in two halfs maximally entangled with each other is given by
(3.35) 
giving a different interpretation of temperature in the QFT. From this perspective, the temperature provides the energy scale wich divides the QFT into two equal parts maximally entangled with each other, with a entanglement entropy equal to .
3.2 The large N limit
As commented in the introduction, part of the reasons to generalize the random unitary framework to the context of quantum field theory is to get closer to the physics of black holes^{12}^{12}12This formalism might be also interesting in the context of integrable theories and Generalized Gibbs Ensembles, see [7].. In particular our formalism can be useful to describe black holes in anti de Sitter spacetimes via AdS/CFT [21]. In these theories we have a large gauge group, with a corresponding large number of field species. In AdS/CFT this number of field species is counted by the central charge of the CFT, which is taken to be large to have a smooth gravitational dual background. With this in mind we would like to repeat the previous exercise for the case of scalar fields, in the large limit.
For each scalar field, labelled by , we have the same energy and momentum dispersion relation for excitations over the vacuum
(3.37) 
for a segment of length and . The Fock space is spanned by states of the type
(3.38) 
where is the number of particles of the scalar field with momentum , and is again a natural number running over the infinite but countable set of eigenstates, used here to simplify notation. Conservation of total energy