Quantum Entanglement of Fermionic Local Operators

In this paper we study the time evolution of (Renyi) entanglement entropies for locally excited states in four dimensional free massless fermionic field theory. Locally excited states are defined by being acted by various local operators on the ground state. Their excesses are defined by subtracting (Renyi) entanglement entropy for the ground state from those for locally excited states. They finally approach some constant if the subsystem is given by half of the total space. They have spin dependence. They can be interpreted in terms of quasi-particles.


Introduction
Quantum entanglement is an essential idea that distinguishes quantum physics from classical physics.There has been a lot of work done to investigate various subfields of physics from a perspective of quantum entanglement.For instance, the universal properties of quantum entanglement are used to characterize conformal field theories [1,2,3,4], as well as topologically ordered phases [5,6].The quantum entanglement also plays an essential role to understand the Hilbert space of gravity [7,8].While the most commonly used quantities for characterizing states are correlation functions of operators, quantum entanglement focuses more directly on the Hilbert space.
Entanglement entropy is defined as the von Neumann entropy of a reduced density matrix ρ A for a subsystem A. It has been observed that the dominant contribution of entanglement entropy in ground states of gapped systems comes from near boundary region of A. This leads to the area law of entanglement entropy [9].For ground states of gapless systems, the contribution of the entanglement from a long distance does not necessarily become subdominant.Indeed, in 2d conformal field theories (CFTs), entanglement entropy behaves universally as c 3 log l where c is the central charge and l is the size of the subsystem A, suggesting that entanglement of any distance equally contributes to the entropy.Recently some entanglement measures are introduced to describe the more detail structure of the entanglement entropy [10,11].
The above story is only for static (ground) states and much less is understood for the dynamical aspects of the entanglement entropy as well as the entanglement entropy for excited states.There are interesting questions about the dynamics of quantum entanglement, such as how the quantum entanglement is created and propagates, how they distributes in space time.Those are essential for understanding far-equilibrium states and their thermalization [12], how efficiently quantum dynamics can be simulated in classical computers, and where and how the black hole information goes.
In this paper, we study the time evolution of (Rényi) entanglement entropies by being acted by a fermionic local operator on the ground state.First we explain quantum quenches which are a useful protocol to investigate the dynamical aspects of the entanglement.Suppose a system is prepared for a Hamiltonian H(λ 0 ) which has an experimentally controllable parameter λ 0 .The state prepared for this system is given by a ground state.Then at a certain time the parameter is shifted from λ 0 to λ 1 .The prepared state is no longer a ground state of H(λ 1 ) and the state undergoes a time evolution.This kind of control is indeed possible in experiments such as cold atom systems [13].When parameters are changed globally, these protocols are called as global quenches [14].On the other hand, it is called as local quenches if parameters are changed locally [15].
Our protocol is similar to local quenches.In our setup, parameters of Hamiltonian are not changed.Instead of being changed them, a local operator O(t 0 , x 0 ) acts on the ground state at certain time and creates entangled quasi-particle.In CFTs they propagate spherically at the speed of light.If this local operator is inserted outside of A, a part of quasi-particles eventually enters into the subsystem A and the rest part stays outside.Since they are entangled, it may create an entanglement between the subsystem A and the rest of the system.Indeed this kind of behavior, the increase of the entanglement and the saturation, has been observed in various quantum field theories [16,17,18,19,20,21,22].On the other hand, in holographic field theories the excesses of (Rényi) entanglement entropies do not saturate and keep to increase logarithmically even if time passes efficiently [19,24,23].
The standard method for computing the entanglement entropy for ground states or thermal states in path integral is the replica method: we compute the (Rényi) entanglement entropy S (n) A first and then take n → 1 limit, which gives entanglement entropy S A = −tr A ρ A log ρ A .Here ρ A = tr B ρ and we trace out the degrees of freedom outside A (the region B).
We consider free massless fermionic field theory in 4 dimensional spacetime and take a half space as the subsystem A. We generate an excitation by being acted by local operator O at a distance l away from the entangling surface and time −t.The state prepared is given by a locally excited state.
where x = (x 2 , x 3 ).We define the excesses of (Rényi) entanglement entropies ∆S A by subtracting (Rényi) entanglement entropies for the ground state from those for locally excited states.
They do not change for t < l, as expected from the causality.The main difference is the final values of ∆S (n) A : In the case of free massless fermionic field case, ∆S A have spin dependence.Their density matrices can depend on the direction of spin because the probability with which (anti-)particles are included in A can depend on the direction of spin.In the free massless scalar field theory, of course ∆S (n) A do not have such a dependence.
This paper is organized as follows.
In Sec.2, we explain our setup.In Sec.3, we explain the replica method and the analytic continuation which we perform.In Sec.4 we derive the Green function in 4d free massless fermionic field theory on the replica space.In Sec.5 we compute the time evolution of ∆S (n) A .In Sec.6 we explain the same thing in terms of quasi-particles.In Sec.7, we conclude and discuss our results.

Setup
We study the time evolution of excesses of (Rényi) entanglement entropies for locally excited states which are defined by being acted by various local operators on the ground states in following setup.We consider the 4 dimensional free massless fermionic theory, where γ µ = {γ t , γ 1 , γ 2 , γ 3 } and ψ = iψ † γ t .
where x = (x 2 , x 3 ) and N is a normalization constant.The subsystem A is given by a half of the total space, x 1 ≥ 0 as in Figure .1.We trace out the degrees of freedom in the complement space B outside A and define a reduced density matrix, where ρ = |Ψ Ψ|.By using this reduced density matrix, (Rényi) entanglement entropy for locally excited state is defined by By using the reduced density matrix for the ground state ρ G A = tr B |0 0|, (Rényi) entanglement entropy for the ground state is defined by We define the excess of (Rényi) entanglement entropy by subtracting We will study the time evolution of ∆S (n) A in following sections.

Locally Excited States
In this section, we explain the replica method for locally excited states.The states we are considering are given as follows: Here O(−l, x) is a local operator in Shrödinger picture and we introduce the regularization factor e −ǫH because the norm of the state |Ψ is divergent without e −ǫH and |Ψ does not belong to the Hilbert space.This corresponds to smearing the point like excitation.
The density matrix ρ is given by Here O(τ, −l, x) is a local operator in Heisenberg picture and in the second line we introduce complex times τ e = −ǫ−it and τ l = ǫ−it.In the calculation of (Rényi) entanglement entropy, we first treat complex times τ e and τ l as if they are real parameters and finally we analytically continue to complex values.

Replica method
Now we explain the replica method for locally excited states.In the path integral formalism, we can express the wave functional for locally excited states is given by In the same manner, the we can express the bra vector as follows: Then, the density matrix for total system is given by the path integral on the space which has boundary at τ = +0 and τ = −0: where Z EX 1 appears in order to keep trρ = 1 and it is given by Partial trace corresponds to sawing the region which was traced out, so the reduced density matrix is given by From this we can see that the only difference between the reduced density matrix for ground states and that for locally excited states is the insertion of local operators O(τ e ) and O(τ l ).We need to insert two local operators in each sheet, so finally we need to insert 2n local operators in the n-sheeted manifold Σ n which is constructed of n flat spaces and it has a conical singularity on the entangling surface as in Figure .2.Then, the tr(ρ EX A ) n is given by the partition function with the insertion of 2n local operators: where we introduce the polar coordinate (r, θ) on (τ, x 1 ) plane , the region of θ is given by 0 < θ < 2πn and θ k e,l = θ 1 e,l + 2π(k − 1), see Figure .2. ( 15) is almost the correlation function and the only difference is that right hand side is divided by (Z EX 1 ) n , not by Z n where Z n is the partition function on n-sheeted manifold Σ n without any operator insertion.If we consider the difference between Rényi entanglement entropy for excited states and that for the ground state, we can find that it is expressed by the correlation function on n-sheeted manifold: In this way, we can express the difference of (Rényi) entanglement entropy using the correlation function on n-sheeted manifold.

Propagator
In conformal field theories, (Rényi) entanglement entropy for a ground state is invariant under the conformal transformation.In order to preserve its conformal symmetry in the replica trick, the action on Σ n is given by where Γ i (i = 0, 1, 2, 3) obeys the clifford algebra ({Γ i , Γ j } = 2δ i,j 1) and 1 is the identity.In this case, Σ n is given by the flat space except for the origin.We introduce a polar coordinate and this geometry is described by The two point function of ψ and ψ is defined by S ab obeys the equation of motion as follows, S ab can be rewritten by g is defined by where H(x, x ′ ) is given by G(r, r 2 ) obeys the following equation of motion, Next we consider the boundary condition for ψ(x) 1 .It is given by, Then the boundary condition for two point function of ψ is given by The boundary condition for the green function G is given by

Computation of Proapgator
Let's compute the propagator G(r, r ′ , θ, θ ′ ).It can be expanded by the eigenfunctions V (r, θ, x) which are defined by where where J ν (x) is the Bessel function of the first kind.We can rewrite this green function as in [17,18,19,27,28,29].It is given by where t 0 is defined by If n is odd, the green function is given by In this case, the boundary condition ( 27) is given by The green function given by (32) obeys this boundary condition.
If n is even, the green function is In this case, boundary condition given by ( 27) is given by

Analytic Continuation to Real Time
Up to here, we consider propagators in the Euclidean space.Two local operators are located as in Figure .3.We would like to study the time evolution of (Rényi) entanglement entropy.Therefore we perform an analytic continuation to real time as follows, where in lorentzian spacetime ǫ is a cutoff parameter which regulates the divergence when a local operator contacts with another.After performing it, parameters in Euclidean space are related to those in lorentzian spacetime as follows, (37)

Dominant Propagators
After performing the analytic continuation in (36), we take the limit ǫ → 0. A few propagators dominantly contribute to n point functions.We call them dominant propagators.They are O(ǫ −3 ).If t ≤ l, dominant propagators on Σ 1 are given by On the other hand, If t > l, they are given by If t ≤ l, dominant propagators on Σ n>1 are red arrows in Figure .4.They are given by If t > l, they are red arrows and blue arrows in Figure .4.They are given by

Propagators in the Cartesian Coordinate
Given locally excited states are defined by being acted by local operators on the ground state in Cartesian coordinate.Therefore we explain the relation between propagators in (17) and them in the Cartesian coordinate.In the Cartesian coordinate, the action in free massless fermionic theory is given by where γ µ = {γ 0 , γ 1 , γ 2 , γ 3 }.
When we introduce a polar coordinate x 1 = r cos θ, x 0 = r sin θ and redefine the fermionic field by ψ ′ (r, θ) = e − γ 1 γ 0 2 θ ψ(r, θ), it is rewritten by where After perfprming this map, the boundary condition for ψ ′2 is mapped to We define propagators by After performing this map, we also perform the analytic continuation in (36) and take the limit ǫ → 0. After that, only a few propagators can contribute to n point function dominantly.They are O(ǫ −3 ).In any time, dominant propagators on Σ 1 are given by where 1 is identity.γ 0 is changed to iγ t ((γ t )2 = −1).
On the other hand, if t ≤ l those on Σ n>1 are the same as in (47).In the region t > l, they are given by and Let's study the time evolution of ∆S (n) A in the next section.

A for Various Local Operators
In this section, we study the time evolution of ∆S (n) A for various operators by the replica trick.Especially, we focus on their behavior in the late time region (t ≫ l).

∆S (n)
A for ψ ′ a The locally exited state is given by The time evolution of ∆S (n) A is as follows.

∆S (n)
A for it is given by ∆S When we take the limit ǫ → 0, ∆S A vanishes in the early time region (t ≤ l).When t is greater than l, two diagram in Figure.5

dominantly contribute to ∆S (n)
A .In this region the denominator is given by Therefore ∆S (n) A is given by where (γ t γ 1 ) aa is real because γ t γ 1 is a hermitian matrix 3 .If we take the late time limit (t ≫ l), ∆S A is given by where as in Figure .5,A 1 and A 2 are respectively given by If we take the Von Neumann limit n → 1, ∆S A is given by ((γ t γ 1 ) aa + 2) log((γ t γ 1 ) aa + 2) + log( 4).( 56) If (γ t γ 1 ) aa vanishes, ∆S A is given by log 2 which is entanglement entropy for the EPR state.The lower value of ∆S A in (56) is given by log 4/3 3 4 ((γ t γ 1 ) aa = 1 or −1).

Reduced Density Matrix
If we identify A 1 and A 2 as the diagonal components of ρ n A respectively, it is expected that the density matrix for this state is given by where tr A ρ A and each diagonal components of ρ A are positive.
It is expected that the density matrix for |Ψ = N ψ′ a |0 is given by Figure .5: The schematic description of A 1 and A 2 .

∆S (n)
A for ψ′ ψ ′ A locally excited state is given by which is invariant for SL(2, C) transformation.Let's study the time evolution of ∆S (n) A .

∆S (2)
A for (59) is given by ∆S (2) In the early time region (t ≤ l), ∆S A vanishes when we take the limit ǫ → 0. On the other hand, in the region t > l, ∆S A is given by ∆S (2) where c = 4.
If we take the limit t → ∞, ∆S A is given by ∆S (2) If t ≤ l, for arbitrary n, ∆S A vanishes.In the region t > l, ∆S A is given by where A 1 , A 2 , A 3 are given by If we take the late time limit t → ∞, ∆S A is given by

Reduced Density Matrix
By using results up to here, we are able to guess the reduce density matrix for |Ψ = N ψ′ ψ ′ as follows.In the limit ǫ → 0, only four diagrams in (66) In the replica trick, (A 1 ) n , (A 2 ) n and 8 • tr(A n ) respectively correspond to the red-lined diagram, blue lined diagram and green lined diagrams in Figure .6.Therefore it is expected that the reduced density matrix for this state is given by the 10 × 10 matrix, 12 0 0 0 0 0 0 Ã 0 0 0 0 0 0 Ã 0 0 0 0 0 0 Ã 0 0 0 0 0 0 Ã 0 0 0 0 0 0 12 where Ã is given by the 2 × 2 matrix, If we diagonalize the matrix Ã, it is given by For any n, (Rényi) entanglement entropy is given by If we take the von Neumann entropy limit n → 1, entanglement entropy is given by Min entropy is given by (Rényi) entanglement entropy for this state monotonically decreases when the replica number n increases.Therefore we can consider ∆S (∞) A as the lower bound of (Rényi) entanglement entropies.
If we take the large N limit in the U(N) or SU(N) free massless fermionic field theory, the number of diagrams which can dominantly contribute to ∆S (n) A decreases.Because the number of trace in the green-lined diagram is less than that in the others, the blue-lined and red-lined diagram dominantly contribute to ∆S (n) A in the large N limit.Therefore it is given by After that, if we take n → ∞ limit, Min entropy is given by This result describes that Min entropy is same as (72) even if we take the large N limit.
Figure .6: The schematic explanation of the correspondence between diagrams and components of a reduce density matrix.

∆S (n)
A for ψ ′ † ψ ′ A locally excited state is given by which is variant under the SL(2, C) transformation.

∆S
A for the state in (75) is given by ∆S

Reduced Density Matrix
We define B 1 , B 2 , B by In the replica trick, (B 1 ) n , (B 2 ) n and 8 • tr(B n ) respectively correspond to the red dashed lined diagram, blue dashed lined diagram and green dashed lined diagrams in Figure .7.Therefore it is expected that the reduced density matrix for this state is given by the 10 × 10 matrix, where B is given by 2 × 2 matrix, If we take Von Neumann entropy limit n → 1 in (81) entanglement entropy is given by Although the local operator A is invariant under this transformation.
If we take the limit n → ∞, it is given by If we take the large N limit in the U(N) or SU(N) free massless fermionic field theory, the number of diagrams which can contribute to ∆S (n) A decreases.Because the number of trace in green dashed lined diagram is less than that in the others, blue dashed lined and red dashed lined diagram dominantly contribute to ∆S (n) A in the large N limit.Therefore it is given by ∆S If we take n → ∞ limit, ∆S Even if we take the large N limit, the lower bound of ∆S (n) A agrees with that in (86) similarly to the result in the previous subsection.

Spin dependence
Here we discuss the spin dependence of the reduced density matrices in (57) and (58).Their components have (γ t γ 1 ) aa .Therefore they depend on the direction of the spin.These spin dependences can be understood as follows.To explain why matrix components depend on the spin, we diagonalize γ t γ 1 and it is given by We can derive (p 0 + γ t γ i p i ) Φ(p) = 0 from the equation of motion for Φ(p).The modes propagating along x 1 direction (p 1 = 0, p 2 = 0, p 3 = 0) obey The equation in (90) describes that that (anti-)particles can propagate along only the left or right direction as follow.An anti-particle Φ + (p) (γ t γ 1 Φ + (p) = Φ + (p)) can not propagate in the right direction parallel to x 1 axis (x 1 > 0).On the other hand, the anti-particle Φ − (p) (γ t γ 1 Φ − (p) = −Φ − (p)) is not able to propagate in the left direction (x 1 < 0).Therefore if we define a locally excited state by acting a component of ψ or ψ on the ground state, their reduced density matrices depend on the representation of gamma matrices as in (57) and (58).For example, we choose the basis which diagonalizes γ t γ 1 as in (89) and ψ 1 acts on the ground state.This local operator creates an anti-particle and it propagates with time.At the late time (t ≫ l), it is necessarily included in the region A or B. As explained above, it is not able to propagate in the right direction parallel to x 1 axis.Therefore it is included in A (B) with the probability which is bigger than 1 2 (less than 1  2 ).The reduced density matrix in (57) is given by The result in (91) describes that the antiparticle which created by ψ 1 are included in A (B) with probability 3 4 ( 1 4 ) 45 .If we choose a basis in which (γ t γ 1 ) 11 is given by 0, ∆S is given by log 2.
6 Quasi-Particle Interpretation In [17,18,19], we have interpreted the late time value of ∆S (n) A in free massless scalar field theories in terms of entanglement between quasi-particles.
In free fermionic field theory, it is expected that the late time values of ∆S A can be interpreted in terms of entanglement between quasi-particles.Therefore we decompose local operators into left moving modes and right moving modes, where ψK † is defined by The left moving mode and right moving mode of the anti-particle and particle are defined as follows.We decompose ψ, ψ † into the momentum modes.Their right moving modes (left moving modes) are defined by the sum of the momentum modes whose p 0 p 1 is positive ( p 0 p 1 is negative).As we explained in the previous section, the number of momentum modes which the right moving modes (left moving modes) have depends on the choice of spin's direction.Therefore instead of ordinary anti-commutation relationships ({Φ K a , Φ K ′ † b } = δ KK ′ δ ab ), we impose the following exotic anti-commutation relationship on the particle and anti-particle, 4 If we act ψ 1 + ψ 2 on the ground state, the late time value of ∆S A is given by log 2 as we expected 5 If you choose the bases which diagonalize γ t γ 1 in 2d massless free fermionic field theory, a component of ψ ( ψ) is able to propagate in the only left or right direction parallel to x 1 .Therefore ∆S The vacuum where K = L, R. The number of momentum modes which the left and right moving modes have depends on the choice of γ t γ 1 (the choice of the spin's direction).Therefore, it is reasonable that the anti-commutation for them is given by that in (94) 6 .We claim that the exotic anti-commutation relationship in (94) can be applied to ∆S A in 4 dimensional free massless fermionic field theory.However in d( = 4) dimensional spacetime, that in (94) should be deformed.For example, in 2 dimensional spacetime the reduced density matrix for the state excited by a local operator constructed of chiral and anti-chiral operators should become Therefore the anti-commutation relation in (94) should change to another one.This way, we can construct reduced density matrices which agree with those which are obtained by the replica trick as we will see later.In following subsections we will compute them under this decomposition in various examples in 4 dimensional free massless fermionic field theory.

Various Examples
Here we compute reduced density matrices for various locally excited states under the quasi-particle interpretation.

ρ A for ψ a
A locally excited states is given by which is variant under the SL(2, C) transformation.Under the decomposition in (92), this state is given by after the normalization constant N is determined.Normalized orthogonal left and right moving states are given by where (N L ) 2 and (N R ) 2 are given by A reduced density matrix ρ A is defined by tr L ρ.It is given by which agrees with the one obtained by the replica trick.When (γ t γ 1 ) aa vanishes, ∆S (n) A are given by log 2. In the n → 1 limit, it is given by entanglement entropy for maximally entangled state (EPR state).If we choose gamma matrices as in (89), the anti-particles created by ψ 1 is not able to propagate in the right direction parallel to x 1 axis.Therefore the probability with which it is finally included in B is larger than the one with which it is included in A at the late time.The reduced density matrix in (101) agrees with our intuitive expectation.

ρ A for ψψ
A locally excited states is given by which is invariant under the SL(2, C) transformation.By using an unitary transformation, γ t γ 1 can be diagonalized.After diagonalize it, the reduced density matrix for this state can be computed easily.Here it is given by that in (89) Under the decomposition in (92), the state in (102) is given by after the normalization constant N is determined.Normalized orthogonal states are given by where normalization factors are given by (105) A reduce density matrix is given by where M i and Mi are given by This reduced density matrix agrees with the one obtained by the replica trick.
-It is important to study how we should deform the relationship in (94) in d( = 4) dimensional free massless fermionic field theory.
-It is interesting to study the time evolution of ∆S (n) A in non-relativistic field theory.
-It would be expected that gamma matrices included in reduced density matrices is related to the modular Hamiltonian.It is interesting to clarify the relationship between the spin dependence of the reduced matrices and the entanglement Hamiltonian.

Figure. 1 :
Figure.1: The location of an local operator and the subsystem A in the Minkowski spacetime.

Figure. 3 :
Figure.3: The location of operators in Euclidean space.

Figure. 7 :
Figure.7: The schematic explanation of the correspondence between diagrams and components of a reduce density matrix.
(n)A vanish for the locally excited states generated by acting a component of ψ ( ψ) on the ground state. {φ