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Quantum entanglement in de Sitter space from Stringy Axion: An analysis using α vacua Sayantan Choudhury, 1a,b Sudhakar Panda c,d,e a Quantum Gravity and Unified Theory and Theoretical Cosmology Group, Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am M¨ uhlenberg 1, 14476 Potsdam-Golm, Germany. b Inter-University Centre for Astronomy and Astrophysics, Post Bag 4, Ganeshkhind, Pune 411007, India. c Institute of Physics, Sachivalaya Marg, Bhubaneswar, Odisha - 751005, India. d National Institute of Science Education and Research, Jatni, Bhubaneswar, Odisha - 752050, India. e Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai-400085, India. E-mail: [email protected],[email protected], [email protected] Abstract: In this work, we study the phenomena of quantum entanglement by computing de Sitter entanglement entropy from von Neumann measure. For this purpose we consider a bipartite quantum field theoretic set up for axion field, previously derived from Type II B string theory compactified to four dimensions. We consider the initial vacuum to be CPT invariant non-adiabatic α vacua state under SO(1, 4) isometry, which is characterized by a real one-parameter family. To implement this technique we use a S 2 which divide the de Sitter into two exterior and interior sub-regions. First, we derive the wave function of axion in an open chart for α vacua by applying Bogoliubov transformation on the solution for Bunch-Davies vacuum state. Further, we quantify the density matrix by tracing over the contribution from the exterior region. Using this result we derive entanglement entropy, R´ enyi entropy and explain the long-range quantum effects in primordial cosmological correlations. Our results for α vacua provides the necessary condition for generating non zero entanglement entropy in primordial cosmology. Keywords: De-Sitter space, α vacua, Quantum Entanglement, Cosmology of Theories beyond the SM. 1 Alternative E-mail: [email protected]. arXiv:1712.08299v3 [hep-th] 24 Mar 2019

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Page 1: Quantum entanglement in de Sitter space from Stringy Axion ... · E-mail: sayantan@aei.mpg.de,sayantan.choudhury@aei.mpg.de, panda@iopb.res.in Abstract: In this work, we study the

Quantum entanglement in de Sitterspace from Stringy Axion: An analysisusing α vacua

Sayantan Choudhury, 1a,b Sudhakar Panda c,d,e

aQuantum Gravity and Unified Theory and Theoretical Cosmology Group, Max Planck Institute for GravitationalPhysics (Albert Einstein Institute), Am Muhlenberg 1, 14476 Potsdam-Golm, Germany.bInter-University Centre for Astronomy and Astrophysics, Post Bag 4, Ganeshkhind, Pune 411007, India.cInstitute of Physics, Sachivalaya Marg, Bhubaneswar, Odisha - 751005, India.dNational Institute of Science Education and Research, Jatni, Bhubaneswar, Odisha - 752050, India.eHomi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai-400085, India.

E-mail: [email protected],[email protected],

[email protected]

Abstract: In this work, we study the phenomena of quantum entanglement by computing de Sitterentanglement entropy from von Neumann measure. For this purpose we consider a bipartite quantumfield theoretic set up for axion field, previously derived from Type II B string theory compactified tofour dimensions. We consider the initial vacuum to be CPT invariant non-adiabatic α vacua state underSO(1,4) isometry, which is characterized by a real one-parameter family. To implement this techniquewe use a S2 which divide the de Sitter into two exterior and interior sub-regions. First, we derive thewave function of axion in an open chart for α vacua by applying Bogoliubov transformation on the solutionfor Bunch-Davies vacuum state. Further, we quantify the density matrix by tracing over the contributionfrom the exterior region. Using this result we derive entanglement entropy, Renyi entropy and explain thelong-range quantum effects in primordial cosmological correlations. Our results for α vacua provides thenecessary condition for generating non zero entanglement entropy in primordial cosmology.

Keywords: De-Sitter space, α vacua, Quantum Entanglement, Cosmology of Theories beyond the SM.

1Alternative E-mail: [email protected].

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Contents

1 Introduction 1

2 Basic setup: Brief review 3

3 Quantum entanglement for axionic pair using α vacua 43.1 Wave function of axion in open chart 4

3.1.1 Model for axion effective potential 43.1.2 Wave function for Axion using α vacua 4

3.2 Construction of density matrix using α vacua 103.3 Computation of entanglement entropy using α vacua 123.4 Computation of Renyi entropy using α vacua 18

4 Summary 26

A Wave function for Axion using Bunch Davies vacuum 28

1 Introduction

It is well accepted fact that von Neumann entropy is a measure of quantum entanglement to quantify longrange correlation in condensed matter physics [1–3] and cosmology [4–21]. In condensed matter physicsentanglement entropy exactly mimics the role of an order parameter and the corresponding phase transitionphenomena can be characterized by correlations at quantum level. Also, it is expected that, from thisunderstanding of long range effects in quantum correlations, we can understand the underlying physics ofthe theory of multiverse, bubble nucleation etc. in de Sitter space [22]. As a consequence, we can observea prompt response due to the local measurement in quantum physics, by violating causal structure ofthe space-time. In quantum theory such causality violation is known as Einstein-Podolsky-Rosen (EPR)paradox [23]. But in such type of local measurement causality remains unaffected as the required quantuminformation is not propagating. In this context Schwinger effect in de Sitter space [24, 25] is one of theprominent examples of quantum entanglement. In Schwinger effect, particle pair creation takes place witha finite separation in de Sitter space-time in presence of a constant electric field [15] and the quantum statesexhibit long range correlation.

To quantify entanglement entropy in the context of quantum field theory, one requires to have a bipartitesystem. In strong coupling regime of a such a theory we can compute entanglement entropy by using theprinciples of gauge gravity duality in the bulk [26–35]. Using this technique many issues have been addressedin the context of holographic entanglement entropy. Further, in ref. [10] the authors have constructed acompletely different computational algorithm to quantify entanglement entropy using Bunch Davies initialstate1 in de Sitter space. Later, in ref. [15] this has been extended to the computation of entanglemententropy using α vacua initial state in de Sitter space by following the techniques presented in ref. [10].Moreover, using Bunch Davies initial state in de Sitter space, we have computed entanglement entropy ina field theory where the effective action contains a linear source term [14]. We have shown that this resultcomplements the necessary condition for the violation of Bell’s inequality in primordial cosmology.

In quantum field theory in de Sitter background, one can construct a one parameter family of initialvacuum states which are CPT invariant under SO(1,4) isometry group [36, 37]. These class of states arecharacterized by a real parameter α, known as α vacua. For α = 0 these states reduce to the usual BunchDavies vacuum state. In a more technical terms, the α vacua can be treated as squeezed quantum states,

1We note that, Bunch Davies initial state is exact equivalent to the Euclidean or adiabatic vacuum state in quantum fieldtheory.

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which are created by an unitary operator acting on the Bunch Davies vacuum state. This leads to thegeneralisation of Wick’s theorem in interacting quantum field theory, which allows us to describe any freequantum field theory Green’s function computed using α vacua in terms of the products of the Green’sfunctions computed using Bunch Davies vacuum [38]. See refs. [37, 39, 40] for more details on the quantumfield theory of α vacua. For a specific α vacuum state as a quantum initial condition, it is possible todescribe the long range correlations within the framework of quantum field theory. As a consequence, thenon-local quantum phenomena can be associated with the long range effects, which is described by quantumentanglement of vacuum state as an initial condition. We note that till date no such experiment is availableusing which one can able to test the local behavior of quantum field theory in cosmological scale (Hubblescale). However, it is expected that in future it may be possible to test such prescriptions. Additionally, itis important to note that, propagators in free quantum field theory of de Sitter space computed in presenceof adiabatic Bunch Davies vacuum state manifest Hadamard singularity which is consistent with the resultobtained in the context of Minkowski flat space-time limit [41, 42].

However, for interacting quantum field theory in the de Sitter space-time background, such singularpropagators applicable for adiabatic Bunch Davies vacuum are dubious. Hence α vacuum state plays signif-icant role, using which one can express the propagators in interaction picture. In the quantum field theorydescribed by the α vacua state the real parameter α plays the role of super-selection number associatedwith a quantum state of a different bipartite Hilbert space. But it is still a debatable issue that whetherthe interaction picture of the quantum field theory with any arbitrary α vacua with any super-selection ruleare consistent with the physical requirements of quantum mechanics or not [39]. In general, one can treatthe α vacua as a family of quantum initial state, where we have quantum fluctuation around an excitedstate. Here it might be possible that the Hilbert space corresponding to excited state (for α vacua) andthe adiabatic Bunch Davies vacuum coincides with each other. In such a situation it is perfectly consistentto describe quantum field theory of excited state in de Sitter space in terms of the adiabatic Bunch Daviesvacuum in the ultraviolet regime2 As a result this specific identification allows us to write an effective fieldtheory description in the ultraviolet regime . This implies that identifying the correct and more appropriatequantum α vacuum state is fine tuned. However, this fine tuning only allows us to describe the quantumfield theory with any excited states compared to ground state described by the adiabatic Bunch Daviesvacuum. Using this prescription apart from inflationary paradigm, one may be able to explain a lot ofunexplored late time physical phenomena of nonstandard vacuum state. In this work, we further generalisethe computational strategy of entanglement entropy for axion field using α vacua initial state in de Sitterspace. This result will establish the generation of quantum entanglement entropy in early universe in amore generalised fashion. In this setup, while the possibility of EPR pair creation increases, it appears thatthe quantum long range correlation will increase naturally. As a consequence, the amount of entanglemententropy increases as the parameter α increases. In this report, we have investigated this possibility with aspecific model of axion field theory previously derived from Type IIB string theory [43–46] setup. Here,we will demonstrate the Bell’s inequality violation from nonzero entanglement entropy of axion field. Thisconnection also will be helpful in future to provide a theoretical tool to compare various models of inflation[47–50]. We have also commented on Renyi entropy using the same setup which will finally give rise to acomplete new interpretation to long range quantum correlation for the case of α vacua.

We note that for cosmology, it is crucial to know the observational constraints on the α (non- BunchDavies) vacua from CMB maps [19, 51, 52]. It is expected that the (auto and cross) correlation functionsof primordial fluctuations get modified significantly, which is an important information to understand theunderlying new physics of α vacua. Also this will help us to discriminate between the physical outcomes ofα vacua and the adiabatic Bunch Davies vacuum state.

Similarly, for the case of gravity, it is also important to understand the physical implications of the newphysics originated from α vacua described in a specific curved gravitational background. Note that EinsteinGeneral Theory of relativity is a classical field theoretic description, which describes the interactions inastrophysical scales and constrained by galaxy rotation curves, dynamics of clusters etc. [39, 53]. However,in the infrared regime of the gravity sector, we do not have observational probes to test the infrared correctionto the classical field theory of gravity. From the theoretical perspective, if we describe the fluctuation in

2In the infrared regime, due to the nonremoval of divergences appearing from various interaction in quantum field theory,explaining the physics of excited states with the adiabatic vacuum is not a good approximation.

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the metric in terms of spin 2 transverse, traceless degrees of freedom in de Sitter space-time, then using αvacua and the non local field redefinition in metric ,one can express the scalar degrees of freedom also. Thisscheme needs to be developed in future.

This paper is organised as follows. In section 2, we briefly review the basic set up using which we willcompute the entanglement entropy and Renyi entropy using α vacua. In section 3.1.1, we introduce theaxion model from string theory. Then using this model we compute the expression for the wave functionin a de Sitter hyperbolic open chart in presence of Bunch Davies vacuum in section A. Further usingBogoliubov transformation we express the solution in terms of new basis, called α vacua in section 3.1.2.After that in section 3.2, we construct the density matrix in presence of α vacua. Using this result furtherin section 3.3, we derive the expression for the von Neumann entropy which is the measure of entanglemententropy in presence of α vacua. Next in section 3.3, we compute Renyi entropy using the result of densitymatrix as derived in section 3.2. Finally we conclude in section 4 with some future prospects of thepresent work.

2 Basic setup: Brief review

In this section we briefly review the computational method to derive entanglement entropy in de Sitterspace following the work performed in ref. [10] and ref. [14]. We consider a time preserving space-likehypersurface S2 for this purpose. As a result S2 is divided into two sub regions, interior and exterior, whichare characterized by RI (≡L) and RII (≡R). In terms of the Lorentzian signature an open chart in deSitter space is described by three different subregions Further[10, 14]:

R(= RII)/L(= RI) :

τE = π/2− itR/τE = −π/2 + itL for tR/L ≥ 0

ρE = −irR/L for rR/L ≥ 0.(2.1)

C :

τE = tC for −π/2 ≤ tC ≤ π/2

ρE = π/2− irC for −∞ < rc <∞.(2.2)

Also in open chart the metric with Lorentzian signature can be written as [10, 14]:

R(= RII)/L = (RI) :ds2R/L = H−2

[−dt2R/L + sinh2 tR/L

(dr2

R/L + sinh2 rR/L dΩ22

)], (2.3)

C :

ds2

C = H−2[dt2C + cos2 tC

(−dr2

C + cosh2 rC dΩ22

)], (2.4)

where H = a/a is the Hubble parameter and dΩ22 represents angular part of the metric on S2.

Now let us assume that the total Hilbert space of the local quantum mechanical system is described byH, which can be written using bipartite decomposition in a direct product space [54] as, H = HINT⊗HEXT.Here HINT and HEXT are the Hilbert space associated with interior and exterior region and describe thelocalised modes in RI and RII respectively. Consequently, one can construct the reduced density matrixfor the internal RI region by tracing over the external RII region and the Von Neumann entropy measure,the entanglement entropy in de Sitter space can be expressed as:

ρ(α) = TrR|α〉〈α| =⇒ S(α) = −Tr [ρ(α) ln ρ(α)] . (2.5)

Here the vacuum state |α〉 is the α vacuum. The reduced density matrix, which is a key ingredient forcomputing entanglement entropy, is obtained by tracing over the exterior (R) region. Also it is important tonote that the total entanglement entropy can be expressed as a sum of UV divergent and finite contribution.In 3 + 1 D, the UV-divergent part of the entropy can be written as [10, 14, 15]:

SUV−divergent = c1ε−2UVAENT +

[c2 +

(c3m

2 + c4H2)AENT

]ln (εUVH) , (2.6)

SUV−finite = c5AENTH2 − c6 ln

(√AENTH

)+ finite terms. (2.7)

– 3 –

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where εUV is the short distance lattice UV cut-off, AENT is the proper area of the entangling region ofS2 and ci∀i = 1, 2, 3, 4 are the coefficients. Here we restrict ourself only within the UV-finite part whichcontains the information of long range effects of quantum state. Here c6 quantify the long range effect. Ingeneral, c6 can be expressed as [10, 14, 15], c6 ≡ Sintr, where Sintr is the UV-finite relevant part which wequantify in later sections.

3 Quantum entanglement for axionic pair using α vacua

3.1 Wave function of axion in open chart

3.1.1 Model for axion effective potential

In this section our prime objective is to compute de Sitter entanglement entropy for axion field. Such axionfield appears from RR sector of Type IIB string theory compactified on CY3 in presence of NS 5 brane.For details, see refs. [43–46, 55]. Let us start with the following effective action for axion field:

Saxion =

∫d4x√−g[−1

2(∂φ)2 + µ3φ+ Λ4

G cos

fa

)]=

∫d4x√−g[−1

2(∂φ)2 + µ3

[φ+ bfa cos

fa

)]],(3.1)

where µ3 is the mass scale, fa is the decay constant of axion and we introduce a parameter b, is defined

as, b =Λ4G

µ3fa. Here ΛG can be expressed as, ΛG =

√mSUSY L3√

α′gse−cSinst , where Sinst is the instanton action,

factor c ∼ O(1), mSUSY is SUSY breaking scale, α′

represents Regge slope parameter, gs characterisesthe string coupling constant and L6 is the world volume factor. Here we restrict up to the linear termof the effective potential as given by V (φ) ≈ µ3φ, which can be interpreted as a massless source in theequation of motion. In the limit φ << fa, the total effective potential for axion can be approximated as,

V (φ) ≈ µ3 (bfa + φ)− m2axion

2 φ2, where we introduce the effective mass of the axion as, m2axion = µ3b

fa=

Λ4G

f2a.

3.1.2 Wave function for Axion using α vacua

Here our prime objective is to derive results for α-vacua, which can be interpreted as a quantum state filledwith particles defined by some hypothetical observer who initially belongs to the Bunch Davies vacuum state(α = 0). Here the α vacua are invariant under SO(1,4) isometry group of de Sitter space. Consequentlywe use the equivalent prescription followed in case of Bunch Davies vacuum by defining two subspaces inde Sitter space, RI and RII respectively. In general α-vacua is CPT invariant, which is parametrised bya single real positive parameter α which plays the role of super-selection quantum number. We use theresults obtained for the solution of the EOM where we expand the field in terms of creation and annihilationoperators in Bunch Davies vacuum, and further using Bogoliubov transformation the mode functions forthe α-vacua can be written as:

Φ(r, t, θ, φ) =

∫ ∞0

dp∑σ=±1

∞∑l=0

+l∑m=−l

[dσplmE(α)

σplm(r, t, θ, φ) + d†σplm(E(α)σplm)∗(r, t, θ, φ)

], (3.2)

where the α-vacua state are defined as, dσplm|α〉 = 0∀σ = (+1,−1); 0 < p < ∞; l = 0, · · · ,∞,m =

−l, · · · ,+l. In this context, the α-vacua mode function E(α)σplm can be expressed in terms of Bunch Davies

mode function Uσplm(r, t, θ, φ) using Bogoliubov transformation as:

E(α)σplm =

[coshα Uσplm(r, t, θ, φ) + sinhα U∗σplm(r, t, θ, φ)

]. (3.3)

After substituting Eq (3.3) in Eq (3.2) we get the following expression for the wave function:

Φ(r, t, θ, φ) =H

sinh t

∫ ∞0

dp

∞∑l=0

+l∑m=−l

∑σ=±1

[dσplm coshα χp,σ(t) + d†σplm sinhα χ∗p,σ(t)

]Yplm(r, θ, φ), (3.4)

Finally, the solution of the time dependent part of the wave function can be recast as:

χp,σ(t) =∑q=R,L

1

Np[ασq Pq + βσq Pq∗

]+

∞∑n=0

1

Npn (p2n − p2)

[ασq,n Pq,n + βσq,n Pq∗,n

], (3.5)

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where we use the following shorthand notation, Pq,n = µ3 sinh2 t∫dt′χ

(c)pn,σ,q(t

′) Pq,n. Additionally,

here we use the shorthand notations Pq, P∗q, Pq,n, P∗q,n for the Legendre polynomial, which is definedin ref. [14]. Also the coefficient functions (ασq , β

σq ) and (ασq,n, β

σq,n), normalisation constants Np, Npn are

explicitly mentioned in ref. [14] .For further computation α-vacua are defined in terms of Bunch Davies vacuum state as:

|α〉 = exp

(1

2tanhα

∑σ=±1

a†σaσ

)exp

1

2

∑i,j=R,L

mij b†i b†j +

1

2

∑i,j=R,L

∞∑n=0

mij,n b†i,n b

†j,n,

(|R〉 ⊗ |L〉),(3.6)

Further one can also write the the R and L vacua as [14]:

|R〉 = |R〉(c) +

∞∑n=0

|R〉(p),n, |L〉 = |L〉(c) +

∞∑n=0

|L〉(p),n, (3.7)

with (c) and (p) representing the complementary and particular part respectively. Here the matrices mij

and mij,n corresponding to complementary and particular part of the solution are explicitly computed inref. [14] for Bunch Davies vacuum. Also the creation and annihilation operators a†σ and aσ for the R andL vacuum are defined as [14]:

aσ =∑q=R,L

[γqσbq + δ∗qσb

†q

]+

∞∑n=0

[γqσ,nbq,n + δ∗qσ,nb

†q,n

], (3.8)

a†σ =∑q=R,L

[γ∗qσb

†q + δqσbq

]+

∞∑n=0

[γ∗qσ,nb

†q,n + δqσ,nbq,n

], (3.9)

with σ = ±1. Here it is important to note that, the coefficient matrices for the Bogoliubov transformationγqσ, δqσ, γqσ,n and δqσ,n helps us to write the a type of oscillators in terms of a new b type of oscillators.For more details see ref. [14] where all the symbols are explicitly defined. Here it is important to notethat, the newly introduced b type of oscillators exactly satisfy the harmonic oscillator algebra, providedthe oscillators corresponding to the solution of complementary and particular part of the time dependentsolution of the wave function are not interacting with each other. This surely helps us to set up the rulesfor the operation of creation and annihilation operators of these oscillators in this context [14].

Below, we use the definition of α-vacuum state as given in Eq (3.6), which is very useful to computeentanglement entropy in de Sitter space. However, note that the technical steps for the computation of theentanglement entropy in de Sitter space from α-vacua are exactly similar to the steps followed for BunchDavies vacuum. The difference will only appear when we use the creation and annihilation operators in thecontext of α-vacua, which can be written in terms of the creation and annihilation operators defined for Ror L vacuum state as:

dσ =∑q=R,L

[(coshα γqσ − sinhα δqσ) bq +

(coshα δ∗qσ − sinhα γ∗qσ

)b†q]

+

[(coshα

∞∑n=0

γqσ,nbq,n − sinhα

∞∑n=0

δqσ,nbq,n

)+

(coshα

∞∑n=0

δ∗qσ,nb†q,n − sinhα

∞∑n=0

γ∗qσ,nb†q,n

)], (3.10)

d†σ =∑q=R,L

[(coshα γ∗qσ − sinhα δ∗qσ

)b†q + (coshα δqσ − sinhα γqσ) bq

]+

[(coshα

∞∑n=0

γ∗qσ,nb†q,n − sinhα

∞∑n=0

δ∗qσ,nb†q,n

)+

(coshα

∞∑n=0

δqσ,nbq,n − sinhα

∞∑n=0

γqσ,nbq,n

)], (3.11)

where we use the definition of creation and annihilation operators in Bunch Davies vacuum as mentioned inEq (3.9) and Eq (3.8). In this computation it is important to note that, under Bogoliubov transformationthe original matrix γqσ, δqσ, γqσ,n and δqσ,n used for Bunch Davies vacuum are transformed in the context

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of α-vacua as:

γqσ −→ (coshα γqσ − sinhα δqσ) , δqσ −→ (coshα δqσ − sinhα γqσ) , (3.12)

γqσ,n −→(coshα γqσ,n − sinhα δqσ,n

), δqσ,n −→

(coshα δqσ,n − sinhα γqσ,n

).

Considering this fact, after Bogoliubov transformation α-vacua state can be written in terms of R and Lvacua as:

|α〉 = exp

1

2

∑i,j=R,L

mij b†i b†j +

1

2

∑i,j=R,L

∞∑n=0

¯mij,n b†i,n b

†j,n

(|R〉 ⊗ |L〉), (3.13)

Here mij and ¯mij,n represents the entries of the matrices corresponding to the complementary and particularsolution in presence of α vacuum which we will compute in this paper.

Further one can write the annihilation of α vacuum in terms of the annihilations of the direct productstate of R and L vacuum as:

dσ|α〉 =∑q=R,L

4∑s=1

J (q)s = 0, (3.14)

where neglecting contribution from the powers of creation operators, J (q)s ∀s = 1, 2, 3, 4, q = R,L are defined

as:∑q=R,L

J (q)1 =

∑q=R,L

(coshα γqσ − sinhα δqσ) bq eˆO (|R〉 ⊗ |L〉) ≈

∑i,j=R,L

mij (coshα γjσ − sinhα δjσ) b†i (|R〉 ⊗ |L〉) ,(3.15)

∑q=R,L

J (q)2 =

∑q=R,L

(coshα δ∗qσ − sinhα γ∗qσ

)b†q e

ˆO (|R〉 ⊗ |L〉) ≈∑q=R,L

(coshα δ∗qσ − sinhα γ∗qσ

)b†q (|R〉 ⊗ |L〉) , (3.16)

∑q=R,L

J (q)3 =

∑q=R,L

(coshα

∞∑n=0

γqσ,nbq,n − sinhα

∞∑n=0

δqσ,nbq,n

)e

ˆO (|R〉 ⊗ |L〉)

≈∑

i,j=R,L

(coshα

∞∑n=0

¯mij,nγjσ,nb†i,n − sinhα

∞∑n=0

¯mij,nδjσ,nb†i,n

)(|R〉 ⊗ |L〉) , (3.17)

∑q=R,L

J (q)4 =

∑q=R,L

(coshα

∞∑n=0

δ∗qσ,nb†q,n − sinhα

∞∑n=0

γ∗qσ,nb†q,n

)e

ˆO (|R〉 ⊗ |L〉)

≈∑q=R,L

∞∑n=0

(coshα

∞∑n=0

δ∗qσ,nb†q,n − sinhα

∞∑n=0

γ∗qσ,nb†q,n

)(|R〉 ⊗ |L〉) . (3.18)

This directly implies that:

[mij (coshα γjσ − sinhα δjσ) + (coshα δ∗iσ − sinhα γ∗iσ)] b†i

+

[(coshα

∞∑n=0

¯mij,nγjσ,nb†i,n − sinhα

∞∑n=0

mij,nδjσ,nb†i,n

)+

(coshα

∞∑n=0

δ∗iσ,nb†i,n − sinhα

∞∑n=0

γ∗iσ,nb†i,n

)]= 0. (3.19)

As we have already mentioned that the complementary and particular part of the solutions are completelyindependent of each other and hence vanish individually. Consequently, we get the following constraints incase of α vacuum:

[mij (coshα γjσ − sinhα δjσ) + (coshα δ∗iσ − sinhα γ∗iσ)] = 0, (3.20)[(coshα ¯mij,nγjσ,n − sinhα mij,nδjσ,n

)+(coshα δ∗iσ,n − sinhα γ∗iσ,n

)]= 0∀ n. (3.21)

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Further using Eq (3.20) and Eq (3.21), the matrices corresponding to the complementary and particularpart of the solution can be expressed as:

mij = − (coshα δ∗iσ − sinhα γ∗iσ) (coshα γ − sinhα δ)−1σj

= −Γ(ν + 1

2 − ip)

Γ(ν + 1

2 + ip) 2 D(ν)

ij

e2πp (coshα− sinhα e−2πp)2

+ e2iπν (coshα+ sinhα e−2iπν)2 , (3.22)

¯mij,n = −(coshα δ∗iσ,n − sinhα γ∗iσ,n

) (coshα γ − sinhα δ

)−1

σj,n

= −Γ(ν + 1

2 − ipn)

Γ(ν + 1

2 + ipn) 2 D(ν,n)

ij

e2πpn (coshα− sinhα e−2πpn)2

+ e2iπν (coshα+ sinhα e−2iπν)2 ∀ (i, j) = R,L. (3.23)

Here we define the D matrices as:

D(ν)ij =

D(ν)RR D(ν)

RL

D(ν)LR D(ν)

LL

, D(ν,n)ij =

D(ν,n)RR D(ν,n)

RL

D(ν,n)LR D(ν,n)

LL

. (3.24)

and the corresponding entries of the D matrices are given by:

D(ν)RR = D(ν)

LL =(cosh2 α eiπν + sinh2 α e−iπν

)cosπν − sinh 2α sinh2 πp, (3.25)

D(ν)RL = D(ν)

LR = i(cosh2 α eiπν + sinh2 α e−iπν + sinh 2α cosπν

)sinhπp, (3.26)

D(ν,n)RR = D(ν,n)

LL =(cosh2 α eiπν + sinh2 α e−iπν

)cosπν − sinh 2α sinh2 πpn, (3.27)

D(ν,n)RL = D(ν,n)

LR = i(cosh2 α eiπν + sinh2 α e−iπν + sinh 2α cosπν

)sinhπpn. (3.28)

Before further discussion here we point out few important features from the obtained results:

• We see that for the complementary and particular part of the solution

mRR = mLL = −Γ(ν + 1

2 − ip)

Γ(ν + 1

2 + ip) 2

[(cosh2 α eiπν + sinh2 α e−iπν

)cosπν − sinh 2α sinh2 πp

]e2πp (coshα− sinhα e−2πp)

2+ e2iπν (coshα+ sinhα e−2iπν)

2 , (3.29)

¯mRR,n = ¯mLL,n = −Γ(ν + 1

2 − ipn)

Γ(ν + 1

2 + ipn) 2

[(cosh2 α eiπν + sinh2 α e−iπν

)cosπν − sinh 2α sinh2 πpn

]e2πpn (coshα− sinhα e−2πpn)

2+ e2iπν (coshα+ sinhα e−2iπν)

2 .(3.30)

which is non vanishing for 0 < ν ≤ 3/2 and ν > 3/2. For ν = 3/2 we get the non vanishing resultusing α-vacuum and this result is significantly different from the result obtained for Bunch Daviesvacuum state.

Finally to implement numerical analysis we use the following approximated expressions for the entriesof the coefficient matrices as given by 3:

mRR = eiθ√

2 e−pπ cosπν√cosh 2πp+ cos 2πν

[(cosh2 α+ sinh2 α e−2iπν

)− sinh 2α sinh2 πp e−iπν secπν

](cosh2 α+ sinh2 α e−2π(p+iν)

) , (3.31)

¯mRR,n = eiθ√

2 e−pnπ cosπν√cosh 2πpn + cos 2πν

[(cosh2 α+ sinh2 α e−2iπν

)− sinh 2α sinh2 πpn e

−iπν secπν](

cosh2 α+ sinh2 α e−2π(pn+iν)) . (3.32)

• We see that for the complementary and particular part of the solution:

mRL = mLR = −Γ(ν + 1

2 − ip)

Γ(ν + 1

2 + ip) 2 i

[(cosh2 α eiπν + sinh2 α e−iπν + sinh 2α cosπν

)sinhπp

]e2πp (coshα− sinhα e−2πp)

2+ e2iπν (coshα+ sinhα e−2iπν)

2 , (3.33)

¯mRL,n = ¯mLR,n = −Γ(ν + 1

2 − ip)

Γ(ν + 1

2 + ip) 2 i

[(cosh2 α eiπν + sinh2 α e−iπν + sinh 2α cosπν

)sinhπpn

]e2πpn (coshα− sinhα e−2πpn)

2+ e2iπν (coshα+ sinhα e−2iπν)

2 . (3.34)

3For rest of the analysis we absorb this overall phase factor eiθ.

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Additionally, the non vanishing entries of the off diagonal components of the coefficient matrix for bothof the cases in presence of α-vacuum indicates the existence of quantum entanglement in the presentcomputation, which we will explicitly show that finally give rise to a non vanishing entanglemententropy.

Finally to interpret the result numerically we use the following approximated expressions for theentries of the coefficient matrices as given by:

mRL = ei(θ+π2 )

√2 e−pπ sinhπp√

cosh 2πp+ cos 2πν

[cosh2 α+ sinh2 α e−2iπν + sinh 2α cosπν e−iπν

](cosh2 α+ sinh2 α e−2π(p+iν)

) , (3.35)

¯mRL,n = ei(θ+π2 )

√2 e−pnπ sinhπpn√

cosh 2πpn + cos 2πν

[cosh2 α+ sinh2 α e−2iπν + sinh 2α cosπν e−iπν

](cosh2 α+ sinh2 α e−2π(pn+iν)

) . (3.36)

To find a suitable basis first of all we trace over all possible contributions from R and L region. Toimplement this we need to perform another Bogoliubov transformation introducing new sets of operatorsas given by:

cR = u bR + v b†R, cL = ¯u bL + ¯v b†L, CR,n = Un bR,n + Vn b†R,n, CL,n = ¯Un bL,n + ¯Vn b

†L,n,(3.37)

where following conditions are satisfied:

|u|2 − |v|2 = 1, |¯u|2 − |¯v|2 = 1, |Un|2 − |Vn|2 = 1, | ¯Un|2 − | ¯Vn|2 = 1. (3.38)

Using these new sets of operators one can write the α-vacuum state in terms of new basis represented bythe direct product of R

′and L

′vacuum state as:

|α〉 = eˆO (|R〉 ⊗ |L〉) =

(N (α)p

)−1

eˆQ(|R′〉 ⊗ |L

′〉)(α)

, (3.39)

where we introduce a new composite operator ˆQ which is defined in the new transformed basis as:

ˆQ = γ(α)p c†R c†L +

∞∑n=0

Γ(α)p,n C

†R,n C

†L,n, (3.40)

where γ(α)p and Γ

(α)p,n are defined corresponding to the complementary and particular solution, which we will

explicitly compute further for α vacuum. Additionally, it is important to note that the overall normalization

factor N (α)p is defined as:

N (α)p =

∣∣∣∣e ˆQ(|R′〉 ⊗ |L

′〉)(α)

∣∣∣∣ ≈[

1−

(|γ(α)p |2 +

∞∑n=0

|Γ(α)p,n|2

)]−1/2

, (3.41)

which reduces to the result obtained for Bunch Davies vacuum in ref. [14] for α = 0. In this calculation dueto the second Bogoliubov transformation the direct product of the R and L vacuum state is connected tothe direct product of the new R

′and L

′vacuum state as:

(|R〉 ⊗ |L〉)→(|R′〉 ⊗ |L

′〉)(α)

= N (α)p e−

ˆQ eˆO (|R〉 ⊗ |L〉) . (3.42)

Let us now mention the commutation relations of the creation and annihilation operators corresponding tothe new sets of oscillators describing the R

′and L

′vacuum state as:[

ci, c†j

]= δij , [ci, cj ] = 0 =

[c†i , c

†j

],[Ci,n, C

†j,m

]= δijδnm,

[Ci,n, Cj,m

]= 0 =

[C†i,mC

†j,m

]. (3.43)

Here, for α vacuum, the oscillator algebra is exactly same as that obtained for Bunch Davies vacuum.However for α vacuum the structure of these operators are completely different and also they are acting in

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a different Hilbert space Hα, which is characterised by one parameter α. Here it is important to note thatfor α = 0 these oscillators will act on Bunch Davies vacuum state where the corresponding Hilbert space,HBD is the subclass of Hα.

The action of creation and annihilation operators defined on the α vacuum state are appended bellow:

cR|α〉 = γ(α)p c†L|α〉, cL|α〉 = γ(α)

p c†R|α〉, CR,n|α〉 = Γ(α)p,n C

†L,n|α〉, CL,n|α〉 = Γ(α)

p,n C†R,n|α〉. (3.44)

Further, one can express the new c type annihilation operators in terms of the old b type annihilationoperators as:

cJ = bI GIJ = bI

Uq V ∗q

Vq U∗q

, CJ(n) = bJ(n)

(G(n)

)IJ

= bJ(n)

¯Uq,n¯V ∗σq,n

¯Vq,n¯U∗q,n

. (3.45)

Here the entries of the matrices for α vacuum are given by, Uq ≡ diag (u, ¯u),Vq ≡ diag (v, ¯v), ¯Uq,n ≡diag

(Un,

¯Un

), ¯Vq,n ≡ diag

(Vn,

¯Vn

). Further using Eq (3.37), in Eq (3.44), we get the following sets of

homogeneous equations:

For complementary solution :

mRRu+ v − γ(α)p mRL

¯v∗ = 0, mRR¯u+ ¯v − γ(α)

p mRLv∗ = 0, (3.46)

mRLu− γ(α)p

¯u∗ − γ(α)p mRR

¯v∗ = 0, mRL¯u− γ(α)

p u∗ − γ(α)p mRRv

∗ = 0, (3.47)

For particular solution :

mRR,nUn + Vn − Γ(α)p,nmRL,n

¯V ∗n = 0, mRR,n¯Un + ¯Vn − Γ(α)

p,nmRL,nV∗n = 0, (3.48)

mRL,nUn − Γ(α)p,n

¯U∗n − Γ(α)p,nmRR,n

¯V ∗n = 0, mRL,n¯Un − Γ(α)

p,nU∗n − Γ(α)

p,nmRR,nV∗n = 0,(3.49)

Now with α vacuum, it is not sufficient to use v∗ = ¯v, u∗ = ¯u for particular part and also V ∗n = ¯Vn, U∗n = ¯Un

for the complementary part. In this case, the system of four equations, each for complementary andparticular part will not be reduced to two sets of simplified equations. This is an outcome of the fact thatin case of α vacuum, the entries of the coefficient matrices mij and ˜mij,n are complex in nature. On theother hand, they are either real or imaginary for Bunch Davies vacuum state. To solve these equations for

γ(α)p and Γ

(α)p,n, we also need to use the normalisation conditions, |u|2 − |v|2 = 1 and |Un|2 − |Vn|2 = 1.

Finally, the non trivial solutions obtained from these system of equations can be expressed as:

γ(α)p =

1√2|mRL|

[(1 + |mRL|4 + |mRR|4 − 2|mRR|2 − m2

RR(m∗RL)2 − m2RL(m∗RR)2

)±(−1− |mRL|4 − |mRR|4

+2|mRR|2 + m2RR(m∗RL)2 + m2

RL(m∗RR)2)2 − 4|mRL|4

12

] 12

, (3.50)

Γ(α)p,n =

1√2|mRL,n|

[(1 + |mRL,n|4 + |mRR,n|4 − 2|mRR,n|2 − m2

RR,n(m∗RL,n)2 − m2RL,n(m∗RR,n)2

)±(−1− |mRL,n|4 − |mRR,n|4 + 2|mRR,n|2 + m2

RR,n(m∗RL,n)2 + m2RL,n(m∗RR,n)2

)2 − 4|mRL,n|4 1

2

] 12

, (3.51)

where the components mRR = mLL, mRL = mLR and mRR,n = mLL,n, mRL,n = mLR,n are defined in

Eqn (3.29), Eqn (3.30) and Eqn (3.33), Eqn (3.34) respectively. Note that, in both the solutions for γ(α)p

and Γ(α)p,n we absorb the overall phase factor.

After further simplification we get the following expressions for the non trivial solutions for arbitrary νcan be written as:

γ(α)p ≈ i

√2√

cosh 2πp+ cos 2πν ±√

cosh 2πp+ cos 2πν + 2

[cosh2 α+ sinh2 α e2iπν + sinh 2α cosπν eiπν

](cosh2 α+ sinh2 α e−2π(p−iν)

) ,(3.52)

Γ(α)p,n ≈ i

√2√

cosh 2πpn + cos 2πν ±√

cosh 2πpn + cos 2πν + 2

[cosh2 α+ sinh2 α e2iπν + sinh 2α cosπν eiπν

](cosh2 α+ sinh2 α e−2π(pn−iν)

) .(3.53)

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3.2 Construction of density matrix using α vacua

In this subsection our prime objective is construct the density matrix using the α vacuum state which isexpressed in terms of another set of annihilation and creation operators in the Bogoliubov transformedframe. Here the Bunch Davies vacuum state can be expressed as a product of the quantum state for eachoscillator in the new frame after Bogoliubov transformation. Each oscillator is labeled by the quantumnumbers p, l and m in this calculation. After tracing over the right part of the Hilbert space we getthe following expression for the density matrix and the left part of the Hilbert space can be written as,(ρL(α))p,l,m = TrR|α〉〈α|, where the α vacuum state can be written in terms of c type of oscillators as:

|α〉 ≈

[1−

(|γ(α)p |2 +

∞∑n=0

|Γ(α)p,n|2

)]1/2

exp

[γ(α)p c†R c†L +

∞∑n=0

Γ(α)p,n C

†R,n C

†L,n

](|R′〉 ⊗ |L

′〉)(α)

, (3.54)

which is already derived in the earlier section. Further using Eq (3.54), we find the following simplifiedexpression for the density matrix for the left part of the Hilbert space for α vacuum as:

(ρL(α))p,l,m =(

1− |γ(α)p |2

) ∞∑k=0

|γ(α)p |2k ˜|k; p, l,m〉 ˜〈k; p, l,m|+(f (α)

p )2∞∑n=0

∞∑r=0

|Γ(α)p,n|2r ˜|n, r; p, l,m〉 ˜〈n, r; p, l,m|,

(3.55)

where γ(α)p and Γ

(α)p,n are derived in the earlier section. Also we define the α parameter dependent source

normalisation factor f (α)p as, f

(α)p =

(∑∞n=0

(1− |Γ(α)

p,n|2)−1

)−1

. In this computation the states ˜|k; p, l,m〉

and ˜|n, r; p, l,m〉 are defined in terms of the quantum state |L′〉 as:

˜|k; p, l,m〉 =1√n!

(c†L)k|L′〉, ˜|n, r; p, l,m〉 =

1√r!

(C†L,n)r|L′〉. (3.56)

Here we note that:

1. For α vacuum density matrix is diagonal for a given set of the SO(1,3) quantum numbers p, l,m andadditionally depends on the parameter α explicitly. This leads to the total density matrix to take thefollowing simplified form as:

ρL(α) =(

1− |γ(α)p |2

)diag

(1, |γ(α)

p |2, |γ(α)p |4, |γ(α)

p |6 · · ·)

+ (f (α)p )2

∞∑n=0

diag(

1, |Γ(α)p,n|2, |Γ(α)

p,n|4, |Γ(α)p,n|6 · · ·

), (3.57)

2. To find out an acceptable normalization of the total density matrix in presence of α vacuum state, weuse the following limiting results:

∞∑k=0

|γ(α)p |2k = lim

k→∞

1− |γ(α)p |2k

1− |γ(α)p |2

|γ(α)p | < 1∀α−−−−−−−−→

1

1− |γ(α)p |2

, (3.58)

∞∑n=0

∞∑r=0

|Γ(α)p,n|2r =

∞∑n=0

limr→∞

1− |Γ(α)p,n|2r

1− |Γ(α)p,n|2

|Γ(α)p,n| < 1∀n, α−−−−−−−−−−→

∞∑n=0

1

1− |Γ(α)p,n|2

=(f (α)p

)−1

. (3.59)

Consequently using these results for α vacuum we get:

Tr[(

1− |γ(α)p |2

)diag

(1, |γ(α)

p |2, |γ(α)p |4, |γ(α)

p |6 · · ·)]

=(

1− |γ(α)p |2

) ∞∑k=0

|γ(α)p |2k = 1, (3.60)

Tr

[(f (α)p )2

∞∑n=0

diag(

1, |Γ(α)p,n|2, |Γ(α)

p,n|4, |Γ(α)p,n|6 · · ·

)]= (f (α)

p )2∞∑n=0

∞∑r=0

|Γ(α)p,n|2r = f (α)

p , (3.61)

Consequently the normalisation condition of this total density matrix in presence of α vacuum state

is given by, TrρL(α) = 1 + f(α)p . This result is consistent with the ref. [10] where f

(α)p = 0∀α and also

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ref. [14] where α = 0 and f(0)p = fp. But for simplicity it is better to maintain always TrρL(α) = 1 and

to get this result for α vacuum the total density matrix can be redefined by changing the normalisationconstant as:

(ρL(α))p,l,m =

(1− |γ(α)

p |2)

1 + f(α)p

∞∑k=0

|γ(α)p |2k ˜|k; p, l,m〉 ˜〈k; p, l,m|+ (f

(α)p )2

1 + f(α)p

∞∑n,r=0

|Γ(α)p,n|2r ˜|n, r; p, l,m〉 ˜〈n, r; p, l,m|, (3.62)

In this context equivalent convention for normalisation factors can also be chosen such that it alwayssatisfies TrρL(α) = 1, even in the presence of source contribution 4.

3. For each set of values of the SO(1,3) quantum numbers p, l,m, the density matrix yields (ρL)p,l,mand so that the total density matrix can be expressed as a product of all such possible contributions:

ρL(α) =

∞∏p=0

p−1∏l=0

+l∏m=−l

(ρL(α))p,l,m. (3.64)

This also indicates that in such a situation entanglement is absent among all states which carries nonidentical SO(1,3) quantum numbers p, l,m.

4. Finally, the total density matrix can be written in terms of entanglement modular Hamiltonian of theaxionic pair as, ρL(α) = e−βHENT , where at finite temperature TdS of de Sitter space β = 2π/TdS.If we further assume that the dynamical Hamiltonian in de Sitter space is represented by entangledHamiltonian then for a given principal quantum number p the Hamiltonian for axion can be expressedas:

Hp(α) =

[E(α)p c†pcp +

∞∑n=0

E(α)p,n C

†p,nCp,n

]. (3.65)

Acting this Hamiltonian on the α vacuum state we find:

Hp(α)|α〉 ≈

[1−

(|γ(α)p |2 +

∞∑n=0

|Γ(α)p,n|2

)]1/2 [E(α)p c†pcp +

∞∑n=0

E(α)p,n C

†p,nCp,n

exp

[γ(α)p c†R c†L +

∞∑m=0

Γ(α)p,m C†R,m C†L,m

](|R′〉 ⊗ |L

′〉)(α)

= E(α)T,p|α〉, (3.66)

where the total energy spectrum of this system for α vacuum can be written as:

E(α)T,p = E(α)

p +

∞∑n=0

E(α)p,n ∀α ,with E(α)

p = − 1

2πln(|γ(α)p |2

), E(α)

p,n = − 1

2πln

(1

1− |Γ(α)p,n|2

.

)(3.67)

where spectrum for the complementary and particular part for α vacuum state is defined by E(α)p and

E(α)p,n .

4 Here one can choose the following equivalent ansatz for total density matrix in presence of α vacuum as:

(ρL(α))p,l,m =

[1

1− |γ(α)p |2

+ f(α)p

]−1 [ ∞∑k=0

|γ(α)p |2k ˜|k; p, l,m〉 ˜〈k; p, l,m|+ (f

(α)p )2

∞∑n=0

∞∑r=0

|Γ(α)p,n|2r ˜|n, r; p, l,m〉 ˜〈n, r; p, l,m|

]. (3.63)

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Now if we consider any arbitrary mass parameter ν and any arbitrary value of the parameter α, theSO(1,3) principal quantum number p dependent spectrum can be expressed as:

E(α)p = − 1

2πln

1

2|mRL|2[(

1 + |mRL|4 + |mRR|4

−2|mRR|2 − m2RR(m∗RL)2 − m2

RL(m∗RR)2)±(−1− |mRL|4 − |mRR|4

+2|mRR|2 + m2RR(m∗RL)2 + m2

RL(m∗RR)2)2 − 4|mRL|4

12

], (3.68)

E(α)p,n =

1

2πln

1− 1

2|mRL,n|2[(

1 + |mRL,n|4 + |mRR,n|4

−2|mRR,n|2 − m2RR,n(m∗RL,n)2 − m2

RL,n(m∗RR,n)2)±(−1− |mRL,n|4 − |mRR,n|4

+2|mRR,n|2 + m2RR,n(m∗RL,n)2 + m2

RL,n(m∗RR,n)2)2 − 4|mRL,n|4

12

]. (3.69)

Here the components mRR = mLL, mRL = mLR and mRR,n = mLL,n, mRL,n = mLR,n are definedin Eqn (3.29), Eqn (3.30) and Eqn (3.33), Eqn (3.34) respectively.

Further using Eq (3.50) and Eq (3.51), we get the following simplified expressions:

E(α)p ≈ − 1

2π ln

[2

(√

cosh 2πp+cos 2πν±√

cosh 2πp+cos 2πν+2)2

∣∣∣∣ [cosh2 α+sinh2 α e2iπν+sinh 2α cosπν eiπν ](cosh2 α+sinh2 α e−2π(p−iν))

∣∣∣∣2],

E(α)p,n ≈ 1

2π ln

[1− 2

(√

cosh 2πpn+cos 2πν±√

cosh 2πpn+cos 2πν+2)2

∣∣∣∣ [cosh2 α+sinh2 α e2iπν+sinh 2α cosπν eiπν ](cosh2 α+sinh2 α e−2π(pn−iν))

∣∣∣∣2].

(3.70)

These results imply that for arbitrary parameter ν and α the entangled Hamiltonian (HENT) andthe Hamiltonian for axion (Hp)R×H3 are significantly different, compared to the result obtained inabsence of linear source term in case of Bunch Davies vacuum.

3.3 Computation of entanglement entropy using α vacua

In this subsection our prime objective is to derive the expression for entanglement entropy in de Sitter spacein presence of α vacuum state. In general the entanglement entropy with arbitrary α can be expressed as:

S(p, ν, α) = −Tr [ρL(p, α) ln ρL(p, α)] , (3.71)

where the parameter ν and the corresponding α vacuum state are defined in the earlier section. In thiscontext the expression for entanglement entropy for a given SO(1,3) principal quantum number p can beexpressed as 5:

S(p, ν, α) = −

(1 +

f(α)p

1 + f(α)p

)ln(

1− |γ(α)p |2

)+

|γ(α)p |2(

1− |γ(α)p |2

) ln(|γ(α)p |2

)− (1− f (α)p

)ln(

1 + f (α)p

). (3.73)

Then the final expression for the entanglement entropy in de Sitter space can be expressed as a sum overall possible quantum states which carries SO(1,3) principal quantum number p. Consequently, the final

5If we follow the equivalent ansatz of density matrix as mentioned in Eq (3.63), the expression for entanglement entropyfor a given SO(1,3) principal quantum number p can be expressed as:

S(p, ν, α) = − ln a(α)p − a(α)

p|γ(α)p |2(

1− |γ(α)p |2

)2ln(|γ(α)p |2

)(1 + f

(α)p

(1− |γ(α)

p |2))

−a(α)p f

(α)p ln

(1 + f

(α)p

(1− |γ(α)

p |2))− a(α)

p (f(α)p )2

(1− |γ(α)

p |2)

ln(

1 + f(α)p

). (3.72)

For our computation we will not use this ansatz any further.

– 12 –

Page 14: Quantum entanglement in de Sitter space from Stringy Axion ... · E-mail: sayantan@aei.mpg.de,sayantan.choudhury@aei.mpg.de, panda@iopb.res.in Abstract: In this work, we study the

expression for the entanglement entropy in de Sitter space is given by the following expression:

S(ν, α) =∑

States

∞∑p=0

S(p, ν, α)→ VH3

∫ ∞p=0

dp D3(p) S(p, ν, α) = c6(α, ν)VH3/V REGH3 , (3.74)

where D3(p) = p2/2π2 characterize the density of states for radial functions on the Hyperboloid H3.Additionally, it is important to note that the volume of the hyperboloid H3 is denoted by the overall factorVH3 . Here the regularized volume of the hyperboloid H3 for r ≤ Lc can be written as:

VH3 = VS2

∫ Lc

r=0

dr sinh2 r large Lc−−−−−−→π

2

[e2Lc − 4Lc

]=[AENT − π ln AENT + π ln

(π2

)]= V REG

H3

[1

4η2+ ln η

]. (3.75)

where AENT is the entangling area and we use VS2 = 4π. Here the cutoff Lc can be written as, Lc ∼ − ln η.In this context we define regularized volume of the hyperboloid H3 as, V REG

H3 = VS3/2 = 2π.In 3 + 1 D , for the case of α vacuum, long range quantum correlation is measured by c6(α, ν), which

is defined as:

c6(α, ν) ≡ Sintr(α, ν) =

[(1 +

f(α)p

1 + f(α)p

)I(α) +

(1− f (α)

p

)ln(

1 + f (α)p

)V

], (3.76)

where the integrals I(α) and V can be written in 3 + 1 dimensional space-time as:

I(α) = − 1

π

∫ ∞p=0

dp p2

ln(

1− |γ(α)p |2

)+

|γ(α)p |2(

1− |γ(α)p |2

) ln(|γ(α)p |2

) , V = − 1

π

∫ ∞p=0

dp p2.(3.77)

α=0

α=0.03

α=0.1

α=0.3

-4 -2 0 2 4 6

0.0

0.5

1.0

1.5

2.0

ν2

Sintr(α)/S

ν=1

/2(α)

Entanglement entropy in dS (D=4) with fp(α)=0

(a) Normalized entanglement entropy vs ν2 in 3 + 1 Dde Sitter space in presence of α vacuum and in absence

axionic source (f(α)p = 0).

α=0

α=0.03

α=0.1

α=0.3

-4 -2 0 2 4 6

0.9998

0.9999

1.0000

1.0001

1.0002

ν2

Sintr(α)/S

ν=1

/2(α)

Entanglement entropy in dS (D=4) with fp(α)=10-7

(b) Normalized entanglement entropy vs ν2 in 3 + 1 D deSitter space in presence of α vacuum and axionic source

(f(α)p = 10−7).

Figure 1. Normalized entanglement entropy Sintr/Sν=1/2 vs mass parameter ν2 in 3 + 1 D de Sitter space in

absence of axionic source (f(α)p = 0) and in presence of axionic source (f

(α)p = 10−7) for ‘+′ branch of solution of α

vacuum i.e |γ(α)p | and |Γ(α)

p,n|. In both the situations we have normalized with conformal ν = 1/2 result in presenceof α vacuum.

Here it is important to mention that:

• The integral V is divergent. To make it finite, we need to regularise it by introducing a change invariable by using x = 2πp and by introducing a UV cut off ΛC leading to:

V = − 1

8π4

∫ ΛC

x=0

dx x2 = − Λ3C

24π4= −L

3

3π= − 1

4π2VS2 where VS2 =

4

3πL3 =

1

6π2Λ3

C. (3.78)

The magnitude of this integral represents the finite volume of the configuration space in which weare computing the entanglement entropy from Von Neumann measure. In principle, this volume can

– 13 –

Page 15: Quantum entanglement in de Sitter space from Stringy Axion ... · E-mail: sayantan@aei.mpg.de,sayantan.choudhury@aei.mpg.de, panda@iopb.res.in Abstract: In this work, we study the

ν2=0

ν2=1/16,9/16,25/16

ν2=1/4,9/4,25/4

0.0 0.2 0.4 0.6 0.8 1.00.00

0.01

0.02

0.03

0.04

0.05

0.06

α

Sintr(α)

Entanglement entropy in dS (D=4) with fp(α)=0

(a) Entanglement entropy vs α in 3 + 1 D de Sitter space

without axionic source (f(α)p = 0).

ν2=0

ν2=1/16,9/16,25/16

ν2=1/4,9/4,25/4

0.0 0.2 0.4 0.6 0.8 1.0

42.78

42.79

42.80

42.81

42.82

42.83

α

Sintr(α)

Entanglement entropy in dS (D=4) with fp(α)=10-7

(b) Entanglement entropy vs α in 3 + 1 D de Sitter space

with axionic source (f(α)p = 10−7).

ν2=-1/16,-9/16,-25/16

ν2=-1/4,-9/4,-25/4

ν2=-1/2

0.0 0.2 0.4 0.6 0.8 1.0

0.005

0.010

0.015

0.020

0.025

α

Sintr(α)

Entanglement entropy in dS (D=4) with fp(α)=0

(c) Entanglement entropy vs α in 3 + 1 D de Sitter space

with axionic source (f(α)p = 10−7).

ν2=-1/16,-9/16,-25/16

ν2=-1/4,-9/4,-25/4

ν2=-1/2

0.0 0.2 0.4 0.6 0.8 1.0

42.780

42.785

42.790

42.795

42.800

α

Sintr(α)

Entanglement entropy in dS (D=4) with fp(α)=10-7

(d) Entanglement entropy vs α in 3 + 1 D de Sitter space

with axionic source (f(α)p = 10−7).

Figure 2. Entanglement entropy Sintr(α) vs parameter α in 3 + 1 D de Sitter space in absence of axionic source

(f(α)p = 0) and in presence of axionic source (f

(α)p = 10−7) for ‘+′ branch of solution of |γ(α)

p | and |Γ(α)p,n|. Here we

fix the value of the parameter ν2 at different positive and negative values including zero.

be infinite, but after fixing the cut-off, this integral actually proportional to a finite volume of asphere of radius ΛC. Actually, the cut-off of the parameter x fix the highest accessible value of thecharacteristic length scale p , which mimics the role of some sort of principal quantum number asappearing in hydrogen atom problem. Only the difference is, here p is restricted within the window,0 < p < ΛC/2π and it takes all possible values within this window (i.e. continuous). After fixing the

cut-off, the length of the system under consideration can be expressed as, L =∫ ΛC/2π

0dp = ΛC/2π.

Now it is obvious that if the cut-off ΛC →∞, the entanglement entropy is diverging, which physicallyimplies that the effect of long range quantum correlation is very large at very large length scale,L =

∫∞0dp→∞ i.e. at late time scales in de Sitter space.

• Further we analyse the integral I(α) using both of the solutions obtained for arbitrary ν and ν = 3/2.Following the previous argument, we also put a cut-off ΛC to perform the integral on the rescaledSO(1,3) quantum number x = 2πp and after performing the integral we can study the behaviour ofboth of the results. First of all we start with the following integral with “± ” signature, as given by:

I(α) = − 1

8π4

∫ ΛC

x=0

dx x2

[ln (1− 2G±(x, ν, α)) +

2G±(x, ν, α)

(1− 2G±(x, ν, α))ln (2G±(x, ν, α))

], (3.79)

where G±(x, ν, α) for any arbitrary value of the parameter α is defined as:

G±(x, ν, α) =1

4|mRL|2[(

1 + |mRL|4 + |mRR|4

−2|mRR|2 − m2RR(m∗RL)2 − m2

RL(m∗RR)2)±(−1− |mRL|4 − |mRR|4

+2|mRR|2 + m2RR(m∗RL)2 + m2

RL(m∗RR)2)2 − 4|mRL|4

12

], (3.80)

– 14 –

Page 16: Quantum entanglement in de Sitter space from Stringy Axion ... · E-mail: sayantan@aei.mpg.de,sayantan.choudhury@aei.mpg.de, panda@iopb.res.in Abstract: In this work, we study the

where the components mRR = mLL and mRL = mLR are redefined in terms of the new variablex = 2πp.

Here small axion mass (ν2 > 0) limiting situations are considered in ν = 1/2 conformal mass as wellin ν = 3/2 case in presence of an additional arbitrary parameter α. Additionally, we consider largeaxion mass (ν2 < 0 where ν → −i|ν|) limiting situation. In this large axion mass limiting situation weconsider the window of SO(1,3) principal quantum number is 0 < p < |ν|. Consequently, the entriesof the coefficient matrix m can be approximated as:

mRR = −

√cosh(|ν| − p)cosh(|ν|+ p)

2[cosh 2α cosh2 π|ν| − sinh 2α sinh2 πp+ 1

2 sinh 2π|ν|]

(e2πp + e2π|ν|) cosh2 α+ (e2πp + e2π|ν|) sinh2 α, (3.81)

mRL = −

√cosh(|ν| − p)cosh(|ν|+ p)

2 i [(cosh 2α+ sinh 2α) coshπ|ν|+ sinhπ|ν|](e2πp + e2π|ν|) cosh2 α+ (e2πp + e2π|ν|) sinh2 α

. (3.82)

This implies that for α vacuum if we consider the large axion mass (ν2 < 0 where ν → −i|ν|) limitingsituation we get always real value for mRR and imaginary value for mRL. Consequently one can esailyreduce the four sets of Eqn. (3.46) and Eqn. (3.47) into two sets of equations as exactly we have donein ref. [14] for Bunch Davies vacuum. In this large axion mass (ν2 < 0 where ν → −i|ν|) limiting

situation the two solutions for the γ(α)p for α vacuum are given by:

γ(α)p ≈ 1

2|mRL|

[(1 + |mRL|2 − m2

RR

)±√

(1 + |mRL|2 − m2RR)

2 − 4|mRL|2]. (3.83)

Small mass limiting situations are explicitly appearing in ν = 1/2 and ν = 3/2 case. For our studyhere we consider large mass limiting situation which is important to study the physical outcomes.In this situation we divide the total window of p into two regions, as given by 0 < p < |ν| and

|ν| < p < ΛC. Here in these region of interests the two solutions for γ(α)p in presence of α vacuum can

be approximately written as:

|γ(α)p | ≈

e−π|ν| (1 + tanα) for 0 < p < |ν|

e−πp (1 + tanα)(1 + tanα e2π|ν|)(

1 + tan2 α e−2πp) for |ν| < p < ΛC/2π.

(3.84)

and

|γ(α)p | =

eπ|ν| (1 + tanα) for 0 < p < |ν|

eπp (1 + tanα)(1 + tanα e2π|ν|)(

1 + tan2 α e−2πp) for |ν| < p < ΛC/2π.

(3.85)

As a result, for large mass limiting range the α parameter dependent regularised integral I(α)1 for the

first solution for |γ(α)p | can be written as:

I(α)1 =

−A(ν)

8π4

ln(

1− e−2πν (1 + tanα)2)

+(2 ln (1 + tanα)− 2πν) e−2πν (1 + tanα)2(

1− e−2πν (1 + tanα)2)

for 0 < x < 2π|ν|

−B(ν, α,ΛC)

8π4for 2π|ν| < x < ΛC.

(3.86)

and for the second solution for |γ(α)p | we get:

I(α)1 =

−A(ν)

8π4

ln(

1− e2πν (1 + tanα)2)

+(2 ln (1 + tanα) + 2πν) e2πν (1 + tanα)2(

1− e2πν (1 + tanα)2)

for 0 < x < 2π|ν|

−C(ν, α,ΛC)

8π4for 2π|ν| < x < ΛC.

(3.87)

– 15 –

Page 17: Quantum entanglement in de Sitter space from Stringy Axion ... · E-mail: sayantan@aei.mpg.de,sayantan.choudhury@aei.mpg.de, panda@iopb.res.in Abstract: In this work, we study the

In Eq. (3.86) and Eq (3.87) coefficients A(ν), B(ν, α,ΛC) and C(ν, α,ΛC) are defined by the followingexpressions:

A(ν) =

∫ 2πν

x=0

dx x2 =8π3

3ν3, (3.88)

B(ν, α,ΛC) ≈∫ ΛC

x=2πν

dx x2

[ln

(1−

e−x (1 + tanα)2 (

1 + tanα e2π|ν|)2(1− tan2 α e−x

)2)

+

e−x(1+tanα)2(1+tanα e2π|ν|)2

(1−tan2 α e−x)2(1− e−x(1+tanα)2(1+tanα e2π|ν|)

2

(1−tan2 α e−x)2

) 2 ln(

(1 + tanα)(

1 + tanα e2π|ν|))

− 2 ln(1− tan2 α e−x

)− x], (3.89)

C(ν, α,ΛC) =

∫ ΛC

x=2πν

dx x2

[ln

(1−

ex (1 + tanα)2 (

1 + tanα e2π|ν|)2(1− tan2 α e−x

)2)

+

ex(1+tanα)2(1+tanα e2π|ν|)2

(1−tan2 α e−x)2(1− ex(1+tanα)2(1+tanα e2π|ν|)

2

(1−tan2 α e−x)2

) 2 ln(

(1 + tanα)(

1 + tanα e2π|ν|))

− 2 ln(1− tan2 α e−x

)+ x]. (3.90)

Further within the window 0 < x < 2π|ν| we take the large mass limit |ν| >> 1 in the first solution

for |γ(α)p | in presence of α vacuum:

lim|ν|>>1

I(α)1 ≈ 2ν4

3e−2πν (1 + tanα)

2

1− 1

πνln (1 + tanα)

[1 + (1 + tanα)

2O(ν−1

)]. (3.91)

Similarly the integral V can be written as:

V =

− 1

8π4

∫ 2π|ν|

x=0

dx x2 = −|ν|3

3πfor 0 < x < 2π|ν|

− 1

8π4

∫ ΛC

x=2π|ν|dx x2 = − 1

24π4

(Λ3

C − 8π3|ν|3). for 2π|ν| < x < ΛC.

(3.92)

Consequently in the large mass limiting situation (0 < x < 2π|ν|) we get the following expression forthe entanglement entropy:

lim|ν|>>1

c6(α, ν) ≈

(1 +

f(α)p

1 + f(α)p

)2ν4

3e−2πν (1 + tanα)

2

×

1− 1

πνln (1 + tanα)

[1 + (1 + tanα)

2O(ν−1

)]−(

1− f (α)p

)ln(

1 + f (α)p

) ν3

3π, (3.93)

Further in absence of the source contribution in the large mass limit the long range quantum correlationcan be expressed as:

lim|ν|>>1,fp→0

c6(α, ν) ≈ 2ν4

3e−2πν (1 + tanα)

2

1− 1

πνln (1 + tanα)

[1 + (1 + tanα)

2O(ν−1

)].(3.94)

For the second solution of |γ(α)p | in presence of α vacuum, we get:

lim|ν|>>1

I(α)1 = − ν

3

ln(

1− e2πν (1 + tanα)2)

+(2 ln (1 + tanα) + 2πν) e2πν (1 + tanα)

2(1− e2πν (1 + tanα)

2)

. (3.95)

– 16 –

Page 18: Quantum entanglement in de Sitter space from Stringy Axion ... · E-mail: sayantan@aei.mpg.de,sayantan.choudhury@aei.mpg.de, panda@iopb.res.in Abstract: In this work, we study the

Consequently in the large mass limiting situation (0 < x < 2π|ν|) we get the following expression forthe entanglement entropy:

lim|ν|>>1

c6(α, ν) ≈ −

(1 +

f(α)p

1 + f(α)p

)ν3

[ln(

1− e2πν (1 + tanα)2)

+(2 ln (1 + tanα) + 2πν) e2πν (1 + tanα)

2(1− e2πν (1 + tanα)

2)

− (1− f (α)p

)ln(

1 + f (α)p

) ν3

3π.(3.96)

Further in absence of the source contribution in the large mass limit the long range quantum correlationcan be expressed as:

lim|ν|>>1,fp→0

c6(α, ν) ≈ − ν3

ln(

1− e2πν (1 + tanα)2)

+(2 ln (1 + tanα) + 2πν) e2πν (1 + tanα)

2(1− e2πν (1 + tanα)

2)

. (3.97)

In fig. (1(a)) and fig. (1(b)), we have demonstrated the behaviour of entanglement entropy in D = 4

de Sitter space in absence (f(α)p = 0) and in presence (f

(α)p = 10−7) of axionic source with respect

to the mass parameter ν2. In both the cases we have normalized the entanglement entropy with theresult obtained from conformal mass parameter ν = 1/2 in presence of α vacuum. In fig. (1(a)), itis clearly observed that in absence of axionic source in the large mass regime (where ν2 < 0) thenormalised entanglement entropy Sintr(α)/Sν=1/2(α) asymptotically approaches towards zero. In thelarge mass regime the measure of long range correlation (or more precisely the entanglement entropy)

in presence of α vacuum for axion can be expressed for γ(α)p = e−π|maxion/H| (1 + tanα) as:

c6

(α, |ν| ≈ maxion

H

)= Sintr

(α, |ν| ≈ maxion

H

)≈ 2

3

(maxion

H

)4

e−2πmaxion

H (1 + tanα)2

1− H

πmaxionln (1 + tanα)

[1 + (1 + tanα)

2O(

H

maxion

)], (3.98)

If we further use Eq (3.98) then it is clearly observed that in presence of α vacuum one is able to getconsiderably large entanglement compared to the result obtained for Bunch Davies vacuum (α = 0)for large mass regime (ν2 < 0). To demonstrate this clearly we have depicted the numerical val-ues of the entanglement entropy for α = 0 (red), α = 0.03 (blue), α = 0.1 (green) and α = 0.3(violet). Now from the fig. (1(a)) it is observed that in ν2 > 0 region Sintr(α)/Sν=1/2(α) reachesits maximum value at α = 0.1 (green) and α = 0.3 (violet) with ν = 0 (or maxion = 3H/2),as given by,

(Sintr(0.1)/Sν=1/2(0.1)

)max,ν=0

∼ 1.2 and(Sintr(0.3)/Sν=1/2(0.3)

)max,ν=0

∼ 2.1. On

the other hand, at α = 0.03 (blue) and α = 0 and α = 0.3 (red) with ν = 1/2 (or maxion =√2H) the maximum value of Sintr(α)/Sν=1/2(α) is given by,

(Sintr(0.03)/Sν=1/2(0.03)

)max,ν=1/2

∼(Sintr(0)/Sν=1/2(0)

)max,ν=1/2

∼ 1. Further if we consider the interval 3/2 < ν < 5/2 then Sintr(α)/Sν=1/2(α)

show one oscillation with different amplitude for all values of the parameter α. After that it reachesits maximum value for α = 0 and α = 0.03, as given by,

(Sintr(0.03)/Sν=1/2(0.03)

)max,3/2<ν<5/2

∼(Sintr(0)/Sν=1/2(0)

)max,3/2<ν<5/2

∼ 1. On the other hand, Sintr(α)/Sν=1/2(α) reaches its mini-

mum value for α = 0.1 and α = 0.3, as given by,(Sintr(0.1)/Sν=1/2(0.1)

)min,3/2<ν<5/2

∼ 1 and(Sintr(0.3)/Sν=1/2(0.3)

)min,3/2<ν<5/2

∼ 1. Similarly in the interval 5/2 < ν < 7/2 we can observe the

same feature for the same values of α with larger period of oscillation. In fig. (1(b)), the significantrole of axionic source term is explicitly shown. In both ν2 < 0 and ν2 > 0 regime the behaviour ofSintr(α)/Sν=1/2(α) is exactly same as depicted in fig. (1(a)). But in presence of axionic source termthe amount of Sintr(α)/Sν=1/2(α) increase for α = 0, α = 0.03 and decrease for α = 0.1, α = 0.3 com-pared to fig. (1(a)). Also it is important to note that, the amplitude of the maximum and minimumof the oscillations change in presence of axionic source term.

– 17 –

Page 19: Quantum entanglement in de Sitter space from Stringy Axion ... · E-mail: sayantan@aei.mpg.de,sayantan.choudhury@aei.mpg.de, panda@iopb.res.in Abstract: In this work, we study the

On the other hand, for γp = eπ|maxion/H| (1 + tanα) the entanglement entropy for axion in the largemass limiting range is given by the following expression:

c6

(α, |ν| ≈ maxion

H

)= Sintr

(α, |ν| ≈ maxion

H

)≈ − 1

(maxion

H

)3

ln(

1− e2πmaxion

H (1 + tanα)2)

+

(2 ln (1 + tanα) + 2πmaxion

H

)e

2πmaxionH (1 + tanα)

2(1− e

2πmaxionH (1 + tanα)

2)

, (3.99)

Next, in fig. (2), we have depicted the behaviour of entanglement entropy Sintr(α) with respect to

the parameter α in absence (f(α)p = 0) and presence (f

(α)p = 10−7) of axionic source for the mass

parameter ν2 < 0 and ν2 > 0 respectively. In fig. (2(a)) and fig. (2(b)) it is observed that a crossovertakes place for ν2 = 1/4, 9/4, 25/4 (green), ν2 = 1/16, 9/16, 25/16 (blue) and ν2 = 0 (red) with smallvalues of the parameter α. We also observe that for ν2 = 1/4, 9/4, 25/4 (green) entanglement entropydecreases with increasing value of the parameter α. On the other hand, for ν2 = 1/16, 9/16, 25/16(blue) and ν2 = 0 (red) entanglement entropy increases with increasing value of the parameter α.Additionally, we observe that, in presence of axionic source the entanglement entropy is significantlylarger compared to the result obtained in absence of source contribution. In fig. (2(c)) and fig. (2(d))it is observed that no crossover takes place for ν2 = −1/2 (green), ν2 = −1/4,−9/4,−25/4 (blue)and ν2 = −1/16,−9/16,−25/16 (red) with all values of the parameter α. Also it is important to notethat, for all values of ν2 < 0 entanglement entropy increases with increasing value of the parameterα.

3.4 Computation of Renyi entropy using α vacua

In this context one can further use the density matrix to compute the Renyi entropy for α vacuum, whichis defined as:

Sq(p, ν, α) =1

1− qln Tr [ρL(p, α)]

q. with q > 0. (3.100)

The obtained solution for α vacuum with a given SO(1,3) principal quantum number p can be written as:

Sq(p, ν, α) =1

1− q

[q ln

(1− |γ(α)

p |2

1 + f(α)p

)− ln

(1− |γ(α)

p |2q)]

+ ln

1 +

q∑k=1

qCk(f (α)p )k

(1− |γ(α)

p |2)−k

(1− |γ(α)

p |−2k) ,(3.101)

using which the interesting part of the Renyi entropy in de Sitter space for α vacuum can be written as:

Sq,intr(α, ν) =1

π

∫ ∞p=0

dp p2 Sq(p, ν, α). (3.102)

Now to study the properties of the derived result we check the following physical limiting situations asgiven by:

• If we take the limit q → 1 limit then from the Renyi entropy in α vacuum we get, limq→1 Sq(p, ν, α) 6=S(p, ν, α). which shows that in presence of axionic source, the entanglement entropy and Renyi entropyare not same in the limit q → 1. Now if we take further fp → 0 then entanglement entropy and Renyientropy both are same.

• Further if we take the limit q →∞ limit then from the Renyi entropy in α vacuum we get:

limq→∞

Sq(p, ν, α) = − ln[ρL]max ≈ ln

(1 + f

(α)p

1− |γ(α)p |2

). (3.103)

which directly implies the largest eigenvalue of density matrix. Now if we take further fp → 0 inEqn (3.103) then entanglement entropy and Renyi entropy both are same.

– 18 –

Page 20: Quantum entanglement in de Sitter space from Stringy Axion ... · E-mail: sayantan@aei.mpg.de,sayantan.choudhury@aei.mpg.de, panda@iopb.res.in Abstract: In this work, we study the

Further substituting the expression for entanglement entropy S(p, ν, α) computed in presence of axion forα vacuum and integrating over all possible SO(1,3) principal quantum number, lying within the window0 < p <∞, we get:

Sq,intr(α, ν) =[M1,q + ln

(1 + f (α)

p

)M(α)

2,q +M(α)3,q

], (3.104)

where the integrals M1,q, M(α)2,q and M(α)

3,q can be written as:

M1,q =1

π

∫ ∞p=0

dp p2

[q

1− qln(

1− |γ(α)p |2

)− 1

1− qln(

1− |γ(α)p |2q

)], (3.105)

M(α)2,q = − 1

π

q

1− q

∫ ∞p=0

dp p2, (3.106)

M(α)3,q =

1

π

1

1− q

∫ ∞p=0

dp p2 ln

1 +

q∑k=1

qCk(f (α)p )k

(1− |γ(α)

p |2)−k

(1− |γ(α)

p |−2k) . (3.107)

Here it is important to note that:

• Here the integral M(α)2,q diverges. Further introducing a change in variable to x = 2πp along with a

cut-off ΛC the regularised version of this integral can be written as:

M(α)2,q = − 1

8π4

q

1− q

∫ ΛC

x=0

dx x2 = − Λ3C

24π4

q

1− q. (3.108)

• On the other hand for arbitrary ν and α we get:

M(α)1,q =

1

8π4

∫ ΛC

x=0

dx x2

[q

1− qln (1− 2G±(x, ν, α))− 1

1− qln (1− (2G±(x, ν, α))

q)

], (3.109)

M(α)3,q =

1

8π4

1

1− q

∫ ΛC

x=0

dx x2 ln

[1 +

q∑k=1

qCk(f (α)p )k

(1− 2G±(x, ν, α))−k

(1− (2G±(x, ν, α))−k)

], (3.110)

where G±(x, ν, α) is defined in Eqn (3.80). We consider large axion mass (ν2 < 0 where ν → −i|ν|)limiting situation which is important to study the physics from this case. In this large axion masslimiting situation we consider the window of SO(1,3) principal quantum number is 0 < p < |ν|.

As a result, the regularized integral M(α)1,q and M(α)

3,q for the first solution for |γ(α)p | in presence of α

vacuum can be expressed as:

M(α)1,q =

A(ν)

8π4

[q

1− qln(

1− e−2πν (1 + tanα)2) 1

1− qln(

1− e−2πνq (1 + tanα)2q)]

for 0 < x < 2π|ν|D(ν, α,ΛC, q)

8π4for 2π|ν| < x < ΛC.

(3.111)

M(α)3,q =

A(ν)

8π4

1

1− qln

1 +

q∑k=1

qCk(f(α)p )k

(1− e−2πν (1 + tanα)2

)−k(1− e2πνk (1 + tanα)−2k

) for 0 < x < 2π|ν|

1

8π4

1

1− q

∫ ΛC

x=2πνdx x2 ln

1 +

q∑k=1

qCk(f(α)p )k

(1− e−x(1+tanα)2(1+tanα e2πν)2

(1+tan2 α e−x)2

)−k(

1− exk(1+tanα)−2k(1+tanα e2πν)−2k

(1+tan2 α e−x)−2k

) for 2π|ν| < x < ΛC.

(3.112)

– 19 –

Page 21: Quantum entanglement in de Sitter space from Stringy Axion ... · E-mail: sayantan@aei.mpg.de,sayantan.choudhury@aei.mpg.de, panda@iopb.res.in Abstract: In this work, we study the

and for the second solution for |γ(α)p | in presence of α vacuum we get:

M(α)1,q =

A(ν)

8π4

[q

1− qln(

1− e2πν (1 + tanα)2)− 1

1− qln(

1− e2πνq (1 + tanα)2q)]

for 0 < x < 2π|ν|

W(ν, α,ΛC, q)

8π4for 2π|ν| < x < ΛC.

(3.113)

M(α)3,q =

A(ν)

8π4

1

1− qln

1 +

q∑k=1

qCk(f(α)p )k

(1− e2πν (1 + tanα)2

)−k(1− e−2πνk (1 + tanα)−2k

) for 0 < x < 2π|ν|

1

8π4

1

1− q

∫ ΛC

x=2πνdx x2 ln

1 +

q∑k=1

qCk(f(α)p )k

(1− ex(1+tanα)2(1+tanα e2πν)2

(1+tan2 α e−x)2

)−k(

1− e−xk(1+tanα)−2k(1+tanα e2πν)−2k

(1+tan2 α e−x)−2k

) for 2π|ν| < x < ΛC.

(3.114)

Here the coefficient function A(ν) is defined in Eq (3.89) and other α parameter dependent functionsD(ν, α,ΛC) and W(ν, α,ΛC) are defined as:

D(ν, α,ΛC, q) =

∫ ΛC

x=2πν

dx x2

[q

1− qln

(1−

e−x (1 + tanα)2 (

1 + tanα e2πν)2(

1 + tan2 α e−x)2

)

− 1

1− qln

(1−

e−xq (1 + tanα)2q (

1 + tanα e2πν)2q(

1 + tan2 α e−x)2q

)], (3.115)

W(ν, α,ΛC, q) =

∫ ΛC

x=2πν

dx x2

[q

1− qln

(1−

ex (1 + tanα)2 (

1 + tanα e2πν)2(

1 + tan2 α e−x)2

)

− 1

1− qln

(1−

exq (1 + tanα)2q (

1 + tanα e2πν)2q(

1 + tan2 α e−x)2q

)]. (3.116)

α=0

α=0.03

α=0.1

α=0.3

-4 -2 0 2 4 6

0.0

0.5

1.0

1.5

2.0

ν2

Sq,intr(α)/Sq,ν=1/2(α)

Renyi entropy in dS (D=4) with q=0.9 and fp=0

(a) Normalized Renyi entropy vs ν2 in 3 + 1 D de Sitter

space without axionic source (f(α)p = 0).

α=0

α=0.03

α=0.1

α=0.3

-2 0 2 4 60

2

4

6

8

10

12

ν2

Sq,intr(α)/Sq,ν=1/2(α)

Renyi entropy in dS (D=4) with q=0.9 and fp=10-7

(b) Normalized Renyi entropy vs ν2 in 3 + 1 D de Sitter

space without axionic source (f(α)p = 10−7).

Figure 3. Normalized Renyi entropy Sq,intr(α)/Sq,ν=1/2(α) vs mass parameter ν2 in 3 + 1 D de Sitter space

in absence of axionic source (f(α)p = 0) and in presence of axionic source (f

(α)p = 10−7) for q = 0.9 and α = 0

(red),α = 0.03 (blue),α = 0.1 (green),α = 0.3 (violet) with ‘+′ branch of solution of |γ(α)p | and |Γ(α)

p,n|. Here weset the cut-off ΛC = 300 for numerical computation.

Further, using the results obtained from the first solution for |γ(α)p |, within the range 0 < x < 2π|ν|

– 20 –

Page 22: Quantum entanglement in de Sitter space from Stringy Axion ... · E-mail: sayantan@aei.mpg.de,sayantan.choudhury@aei.mpg.de, panda@iopb.res.in Abstract: In this work, we study the

α=0

α=0.03

α=0.1

α=0.3

-4 -2 0 2 4 6

0.0

0.5

1.0

1.5

2.0

ν2

Sq,intr(α)/Sq,ν=1/2(α)

Renyi entropy in dS (D=4) with q=0.7 and fp=0

(a) Normalized Renyi entropy vs ν2 in 3 + 1 D de Sitter

space without axionic source (f(α)p = 0).

α=0

α=0.03

α=0.1

α=0.3

-4 -2 0 2 4 6

0.0

0.5

1.0

1.5

2.0

ν2

Sq,intr(α)/Sq,ν=1/2(α)

Renyi entropy in dS (D=4) with q=0.7 and fp=10-7

(b) Normalized Renyi entropy vs ν2 in 3 + 1 D de Sitter

space without axionic source (f(α)p = 10−7).

Figure 4. Normalised Renyi entropy Sq,intr(α)/Sq,ν=1/2(α) vs mass parameter ν2 in 3 + 1 D de Sitter space in

absence of axionic source (f(α)p = 0) and in presence of axionic source (f

(α)p = 10−7) for q = 0.7 and α = 0

(red),α = 0.03 (blue),α = 0.1 (green),α = 0.3 (violet) with ‘+′ branch of solution of |γ(α)p | and |Γ(α)

p,n|. Here weset the cut-off ΛC = 300 for numerical computation.

α=0

α=0.03

α=0.1

α=0.3

-4 -2 0 2 4 6

0.5

1.0

1.5

ν2

Sq,intr(α)/Sq,ν=1/2(α)

Renyi entropy in dS (D=4) with q=0.5 and fp=0

(a) Normalized Renyi entropy vs ν2 in 3 + 1 D de Sitter

space without axionic source (f(α)p = 0).

α=0

α=0.03

α=0.1

α=0.3

-4 -2 0 2 4 60.0

0.5

1.0

1.5

ν2

Sq,intr(α)/Sq,ν=1/2(α)

Renyi entropy in dS (D=4) with q=0.5 and fp=10-7

(b) Normalized Renyi entropy vs ν2 in 3 + 1 D de Sitter

space without axionic source (f(α)p = 10−7).

Figure 5. Normalised Renyi entropy Sq,intr(α)/Sq,ν=1/2(α) vs mass parameter ν2 in 3 + 1 D de Sitter space in

absence of axionic source (f(α)p = 0) and in presence of axionic source (f

(α)p = 10−7) for q = 0.5 and α = 0

(red),α = 0.03 (blue),α = 0.1 (green),α = 0.3 (violet) with ‘+′ branch of solution of |γ(α)p | and |Γ(α)

p,n|. Here weset the cut-off ΛC = 300 for numerical computation.

with ν2 < 0, we take q → 1 limit. This gives the following simplified expression for the integralM(α)1,q :

limq→1M(α)

1,q =ν3

3

2(1 + tanα)2ν − 1

π ln(1 + tanα)

(e2πν − (1 + tanα)2)−

ln(

1− e−2πν (1 + tanα)2)

π

. (3.117)

Now further using |ν| >> 1 approximation in Eq (3.117) we get:

lim|ν|>>1,q→1

M(α)1,q =

2ν4

3e−2πν (1 + tanα)

2

1− 1

πνln (1 + tanα)

[1 + (1 + tanα)

2O(ν−1

)], (3.118)

– 21 –

Page 23: Quantum entanglement in de Sitter space from Stringy Axion ... · E-mail: sayantan@aei.mpg.de,sayantan.choudhury@aei.mpg.de, panda@iopb.res.in Abstract: In this work, we study the

α=0

α=0.03

α=0.1

α=0.3

-4 -2 0 2 4 6

0.4

0.6

0.8

1.0

1.2

1.4

ν2

Sq,intr(α)/Sq,ν=1/2(α)

Renyi entropy in dS (D=4) with q=0.3 and fp=0

(a) Normalized Renyi entropy vs ν2 in 3 + 1 D de Sitter

space without axionic source (f(α)p = 0).

α=0

α=0.03

α=0.1

α=0.3

-4 -2 0 2 4 6

0.4

0.6

0.8

1.0

1.2

1.4

ν2

Sq,intr(α)/Sq,ν=1/2(α)

Renyi entropy in dS (D=4) with q=0.3 and fp=10-7

(b) Normalized Renyi entropy vs ν2 in 3 + 1 D de Sitter

space without axionic source (f(α)p = 10−7).

Figure 6. Normalised Renyi entropy Sq,intr(α)/Sq,ν=1/2(α) vs mass parameter ν2 in 3 + 1 D de Sitter space in

absence of axionic source (f(α)p = 0) and in presence of axionic source (f

(α)p = 10−7) for q = 0.3 and α = 0

(red),α = 0.03 (blue),α = 0.1 (green),α = 0.3 (violet) with ‘+′ branch of solution of |γ(α)p | and |Γ(α)

p,n|. Here weset the cut-off ΛC = 300 for numerical computation.

α=0

α=0.03

α=0.1

α=0.3

-4 -2 0 2 4 6

0.95

1.00

1.05

1.10

1.15

ν2

Sq,intr(α)/Sq,ν=1/2(α)

Renyi entropy in dS (D=4) with q=0.1 and fp=0

(a) Normalized Renyi entropy vs ν2 in 3 + 1 D de Sitter

space without axionic source (f(α)p = 0).

α=0

α=0.03

α=0.1

α=0.3

-4 -2 0 2 4 6

0.95

1.00

1.05

1.10

1.15

ν2

Sq,intr(α)/Sq,ν=1/2(α)

Renyi entropy in dS (D=4) with q=0.1 and fp=10-7

(b) Normalized Renyi entropy vs ν2 in 3 + 1 D de Sitter

space without axionic source (f(α)p = 10−7).

Figure 7. Normalised Renyi entropy Sq,intr(α)/Sq,ν=1/2(α) vs mass parameter ν2 in 3 + 1 D de Sitter space in

absence of axionic source (f(α)p = 0) and in presence of axionic source (f

(α)p = 10−7) for q = 0.1 and α = 0

(red),α = 0.03 (blue),α = 0.1 (green),α = 0.3 (violet) with ‘+′ branch of solution of |γ(α)p | and |Γ(α)

p,n|.Here we setthe cut-off ΛC = 300 for numerical computation.

In this context further if we take the source less limit f(α)p → 0 then the integral M(α)

3,q vanishes:

limq→1,|ν|>>1,fp→0

M(α)3,q = 0. (3.119)

As a result in the large mass limiting situation with q → 1 the long range correlation can be expressedin terms of Renyi entropy as:

limq→1,|ν|>>1,f

(α)p →0

Sq,intr(α) ≈ 2ν4

3e−2πν (1 + tanα)

2

1− 1

πνln (1 + tanα)

[1 + (1 + tanα)

2O(ν−1

)]= Sintr(α) = lim

|ν|>>1,f(α)p →0

c6(α, ν). (3.120)

– 22 –

Page 24: Quantum entanglement in de Sitter space from Stringy Axion ... · E-mail: sayantan@aei.mpg.de,sayantan.choudhury@aei.mpg.de, panda@iopb.res.in Abstract: In this work, we study the

α=0

α=0.03

α=0.1

α=0.3

-4 -2 0 2 4 6

0.0

0.5

1.0

1.5

2.0

2.5

ν2

Sq→∞,intr(α)/Sq→∞,ν=1/2(α)

Renyi entropy in dS (D=4) with q→∞ and fp(α)=0

(a) Normalized Renyi entropy vs ν2 in 3 + 1 D de Sitterspace without axionic source (fp = 0).

α=0

α=0.03

α=0.1

α=0.3

-4 -2 0 2 4 6

0.5

1.0

1.5

2.0

ν2

Sq→∞,intr(α)/Sq→∞,ν=1/2(α)

Renyi entropy in dS (D=4) with q→∞ and fp(α)=10-7

(b) Normalised Renyi entropy vs ν2 in de Sitter spacewith axionic source (fp = 10−7).

Figure 8. Normalized Renyi entropy Sq→∞,intr(α)/Sq→∞,ν=1/2(α) vs mass parameter ν2 in 3 + 1 D de Sitter space

in absence of axionic source (f(α)p = 0) and in presence of axionic source (f

(α)p = 10−7) for q → ∞ and α = 0

(red),α = 0.03 (blue),α = 0.1 (green),α = 0.3 (violet) with ‘+′ branch of solution of |γ(α)p | and |Γ(α)

p,n|, whichquantifies largest eigenvalue of density matrix. Here we set the cut-off ΛC = 300 for numerical computation.

ν2=0

ν2=1/16,9/16,25/16

ν2=1/4,9/4,25/4

0.0 0.2 0.4 0.6 0.8 1.0

0.01

0.02

0.03

0.04

0.05

0.06

α

Sq=0.9,intr(α)

Renyi entropy in dS (D=4) with fp(α)=0

(a) For q = 0.9 and ν2 > 0.

ν2=0

ν2=1/16,9/16,25/16

ν2=1/4,9/4,25/4

0.0 0.2 0.4 0.6 0.8 1.0

0.02

0.04

0.06

0.08

α

Sq=0.7,intr(α)

Renyi entropy in dS (D=4) with fp(α)=0

(b) For q = 0.7 and ν2 > 0.

ν2=0

ν2=1/16,9/16,25/16

ν2=1/4,9/4,25/4

0.0 0.2 0.4 0.6 0.8 1.0

0.02

0.04

0.06

0.08

0.10

0.12

α

Sq=0.5,intr(α)

Renyi entropy in dS (D=4) with fp(α)=0

(c) For q = 0.5 and ν2 > 0.

ν2=0

ν2=1/16,9/16,25/16

ν2=1/4,9/4,25/4

0.0 0.2 0.4 0.6 0.8 1.0

0.10

0.15

0.20

0.25

0.30

α

Sq=0.3,intr(α)

Renyi entropy in dS (D=4) with fp(α)=0

(d) For q = 0.3 and ν2 > 0.

Figure 9. Renyi entropy Sq,intr(α) vs parameter α plot in 3 + 1 D de Sitter space in absence of axionic source

(f(α)p = 0) for q = 0.1, q = 0.3, q = 0.5, q = 0.7, q = 0.9 with ‘+′ branch of solution of |γ(α)

p | and |Γ(α)p,n|.

Similarly using the results obtained from the second solution for |γ(α)p |, within the range 0 < x < 2π|ν|

– 23 –

Page 25: Quantum entanglement in de Sitter space from Stringy Axion ... · E-mail: sayantan@aei.mpg.de,sayantan.choudhury@aei.mpg.de, panda@iopb.res.in Abstract: In this work, we study the

ν2=0

ν2=1/16,9/16,25/16

ν2=1/4,9/4,25/4

0.0 0.2 0.4 0.6 0.8 1.0

0.01

0.02

0.03

0.04

0.05

0.06

α

Sq=0.9,intr(α)

Renyi entropy in dS (D=4) with fp(α)=10-7

(a) For q = 0.9 and ν2 > 0.

ν2=0

ν2=1/16,9/16,25/16

ν2=1/4,9/4,25/4

0.0 0.2 0.4 0.6 0.8 1.0

0.02

0.04

0.06

0.08

α

Sq=0.7,intr(α)

Renyi entropy in dS (D=4) with fp(α)=10-7

(b) For q = 0.7 and ν2 > 0.

ν2=0

ν2=1/16,9/16,25/16

ν2=1/4,9/4,25/4

0.0 0.2 0.4 0.6 0.8 1.0

0.02

0.04

0.06

0.08

0.10

0.12

α

Sq=0.5,intr(α)

Renyi entropy in dS (D=4) with fp(α)=10-7

(c) For q = 0.5 and ν2 > 0.

ν2=0

ν2=1/16,9/16,25/16

ν2=1/4,9/4,25/4

0.0 0.2 0.4 0.6 0.8 1.0

0.10

0.15

0.20

0.25

0.30

α

Sq=0.3,intr(α)

Renyi entropy in dS (D=4) with fp(α)=10-7

(d) For q = 0.3 and ν2 > 0.

Figure 10. Renyi entropy Sq,intr(α) vs parameter α plot in 3 + 1 D de Sitter space in presence of axionic source

(f(α)p = 10−7) for q = 0.1, q = 0.3, q = 0.5, q = 0.7, q = 0.9 with ‘+′ branch of solution of |γ(α)

p | and |Γ(α)p,n|.

with ν2 < 0, we take q → 1 limit. This gives the following simplified expression for the integralM(α)1,q :

limq→1M(α)

1,q =ν3

3

2e2πν (1 + tanα)2 ν + ln (1 + tanα)

e2πν tan2 α+ 2e2πν tanα+ e2πν − 1−

ln(

1− e2πν (1 + tanα)2)

π

, (3.121)

limq→1,|ν|>>1,fp→0

M(α)3,q = 0. (3.122)

As a result in the large mass limiting situation with q → 1 the long range correlation can be expressedin terms of Renyi entropy as:

limq→1,|ν|>>1,f

(α)p →0

Sq,intr(α) ≈ ν3

3

2e2πν (1 + tanα)2 ν + ln (1 + tanα)

e2πν tan2 α+ 2e2πν tanα+ e2πν − 1−

ln(

1− e2πν (1 + tanα)2)

π

= Sintr(α) = lim

|ν|>>1,f(α)p →0

c6(α, ν). (3.123)

In fig. (3(a)), fig. (4(a)), fig. (5(a)), fig. (6(a)), fig. (7(a)), we have demonstrated the behaviour of Renyientropy for q = 0.9, q = 0.7, q = 0.5, q = 0.3 and q = 0.1 with respect to the mass parameter ν2. Here

we did the computation in D = 4 de Sitter space in absence (f(α)p = 0) of axionic source. Similarly in

fig. (3(b)), fig. (4(b)), fig. (5(b)), fig. (6(b)), fig. (7(b)), we have demonstrated the behaviour of Renyientropy for q = 0.9, q = 0.7, q = 0.5, q = 0.3 and q = 0.1 with respect to the mass parameter ν2.Additionally, the largest eigenvalue of the density matrix (q →∞) in absence and presence of axionicsource are plotted in fig. (8(a)) and fig. (8(b)). Here we did the computation in D = 4 de Sitter space

in presence (f(α)p = 10−7) of axionic source. In this both the cases we also have normalised the Renyi

entropy with the result obtained from conformal mass parameter ν = 1/2 in presence of α vacuum.For a given value of the parameter q we have shown the plots for α = 0 (red), α = 0.03 (blue),α = 0.1 (green) and α = 0.3 (violet) in both the cases. Here we observe the following features:

– 24 –

Page 26: Quantum entanglement in de Sitter space from Stringy Axion ... · E-mail: sayantan@aei.mpg.de,sayantan.choudhury@aei.mpg.de, panda@iopb.res.in Abstract: In this work, we study the

ν2=-1/16,-9/16,-25/16

ν2=-1/4,-9/4,-25/4

ν2=-1/2

0.0 0.2 0.4 0.6 0.8 1.0

0.005

0.010

0.015

0.020

0.025

0.030

α

Sq=0.9,intr(α)

Renyi entropy in dS (D=4) with fp(α)=0

(a) For q = 0.9 and ν2 < 0.

ν2=-1/16,-9/16,-25/16

ν2=-1/4,-9/4,-25/4

ν2=-1/2

0.0 0.2 0.4 0.6 0.8 1.0

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0.045

α

Sq=0.7,intr(α)

Renyi entropy in dS (D=4) with fp(α)=0

(b) For q = 0.7 and ν2 < 0.

ν2=-1/16,-9/16,-25/16

ν2=-1/4,-9/4,-25/4

ν2=-1/2

0.0 0.2 0.4 0.6 0.8 1.0

0.03

0.04

0.05

0.06

0.07

0.08

α

Sq=0.5,intr(α)

Renyi entropy in dS (D=4) with fp(α)=0

(c) For q = 0.5 and ν2 < 0.

ν2=-1/16,-9/16,-25/16

ν2=-1/4,-9/4,-25/4

ν2=-1/2

0.0 0.2 0.4 0.6 0.8 1.0

0.14

0.16

0.18

0.20

0.22

α

Sq=0.3,intr(α)

Renyi entropy in dS (D=4) with fp(α)=0

(d) For q = 0.3 and ν2 < 0.

Figure 11. Renyi entropy Sq,intr(α) vs parameter α plot in 3 + 1 D de Sitter space in absence of axionic source

(f(α)p = 0) for q = 0.1, q = 0.3, q = 0.5, q = 0.7, q = 0.9 with ‘+′ branch of solution of |γ(α)

p | and |Γ(α)p,n|.

– For q = 0.9 case in absence of the axionic source (see fig. (3(a))) in the large mass parameterrange (ν2 < 0) normalised Renyi entropy asymptotically approaches towards zero value. On theother hand in the small mass parameter range (ν2 > 0) it show oscillations in a periodic fashion.Here the amplitude of the oscillation is larger for α = 0.3 compared to the other values of α.Also it is important to note that, at ν = 1/2, ν = 3/2 and ν = 5/2 we get extrema for theoscillation. Further in presence of the axionic source (see fig. (3(b))) in the large mass parameterrange (ν2 < 0) normalised Renyi entropy rapidly approaches to zero value for all values of theparameter α considered in this paper. Also in the small mass parameter range (ν2 > 0) theamplitude of the oscillation is significantly large for α = 0.3. Also it is observed that for ν2 > 0the long range correlation is larger in presence of the axionic source. But for ν2 < 0 the long

range correlation is rapidly decaying with f(α)p = 10−7 and asymptotically decaying with f

(α)p = 0

for all values of α.

– For other values of the parameter q i.e. q = 0.7, q = 0.5, q = 0.3 and q = 0.1 cases in absenceof the axionic source (see fig. (4(a)), fig. (5(a)), fig. (6(a)) and fig. (7(a))) in the large massparameter range (ν2 < 0) normalized Renyi entropy asymptotically approaches towards zerovalue. On the other hand in the small mass parameter range (ν2 > 0) it show oscillations ina periodic fashion. Here the amplitude of the oscillation is larger for α = 0.3 compared to theother values of α. Also it is important to note that, at ν = 1/2, ν = 3/2 and ν = 5/2 we getextrema for the oscillation. Further in presence of the axionic source (see fig. (4(b)), fig. (5(b)),fig. (6(b)) and fig. (7(b))) one can observe the exact behaviour as observed without any sourcecontribution. It also implies that for all q < 0.9 the normalised Renyi entropy is insensitive tothe source contribution.

– For q → ∞ case in absence (see fig. (8(a))) and presence of the axionic source (see fig. (8(b)))variation of normalised Renyi entropy with ν2 for all values of the parameter α is similar. It isimportant to note that the amplitudes of the oscillations in ν2 > 0 region and the saturationvalue in ν2 < 0 region is larger in presence of axionic source.

– 25 –

Page 27: Quantum entanglement in de Sitter space from Stringy Axion ... · E-mail: sayantan@aei.mpg.de,sayantan.choudhury@aei.mpg.de, panda@iopb.res.in Abstract: In this work, we study the

ν2=-1/16,-9/16,-25/16

ν2=-1/4,-9/4,-25/4

ν2=-1/2

0.0 0.2 0.4 0.6 0.8 1.0

0.010

0.015

0.020

0.025

0.030

α

Sq=0.9,intr(α)

Renyi entropy in dS (D=4) with fp(α)=10-7

(a) For q = 0.9 and ν2 < 0.

ν2=-1/16,-9/16,-25/16

ν2=-1/4,-9/4,-25/4

ν2=-1/2

0.0 0.2 0.4 0.6 0.8 1.0

0.015

0.020

0.025

0.030

0.035

0.040

0.045

α

Sq=0.7,intr(α)

Renyi entropy in dS (D=4) with fp(α)=10-7

(b) For q = 0.7 and ν2 < 0.

ν2=-1/16,-9/16,-25/16

ν2=-1/4,-9/4,-25/4

ν2=-1/2

0.0 0.2 0.4 0.6 0.8 1.0

0.03

0.04

0.05

0.06

0.07

0.08

α

Sq=0.5,intr(α)

Renyi entropy in dS (D=4) with fp(α)=10-7

(c) For q = 0.5 and ν2 < 0.

ν2=-1/16,-9/16,-25/16

ν2=-1/4,-9/4,-25/4

ν2=-1/2

0.0 0.2 0.4 0.6 0.8 1.0

0.14

0.16

0.18

0.20

0.22

α

Sq=0.3,intr(α)

Renyi entropy in dS (D=4) with fp(α)=10-7

(d) For q = 0.3 and ν2 < 0.

Figure 12. Renyi entropy Sq,intr(α) vs parameter α plot in 3 + 1 D de Sitter space in presence of axionic source

(f(α)p = 10−7) for q = 0.1, q = 0.3, q = 0.5, q = 0.7, q = 0.9 with ‘+′ branch of solution of |γ(α)

p | and |Γ(α)p,n|.

Next, in fig. (9(a)), fig. (9(b)), fig. (9(c)), fig. (9(d)) and fig. (10(a)), fig. (10(b)), fig. (10(c)),fig. (10(d)), we have depicted the behaviour of Renyi entropy with respect to the parameter α in

absence (f(α)p = 0) and presence (f

(α)p = 10−7) of axionic source for the mass parameter ν2 > 0.

In all figures it is observed that a crossover takes place for ν2 = 1/4, 9/4, 25/4 (green), ν2 =1/16, 9/16, 25/16 (blue) and ν2 = 0 (red) with small values of the parameter α. We also observethat for ν2 = 1/4, 9/4, 25/4 (green) Renyi entropy decreases with increasing value of the parameterα. On the other hand, for ν2 = 1/16, 9/16, 25/16 (blue) and ν2 = 0 (red) Renyi entropy increaseswith increasing value of the parameter α. Additionally, in presence of axionic source the Renyi en-tropy is slightly larger compared to the result obtained in absence of source contribution. Also it isobserved that no crossover takes place for ν2 = −1/2 (green), ν2 = −1/4,−9/4,−25/4 (blue) andν2 = −1/16,−9/16,−25/16 (red) with all values of the parameter α. Also it is important to notethat, for all values of ν2 < 0 Renyi entropy increases with increasing value of the parameter α. Furtherin fig. (13(a)), fig. (13(b)), fig. (13(c)), fig. (13(d)), we have shown the variation of Renyi entropy withrespect to the parameter q in absence and presence of axionic source for α = 0 (purple), α = 0.03(orange), α = 0.1 (cyan) and α = 0.3 (brown) respectively. It is observed that for small values ofthe parameter q the value of the Renyi entropy for a given value of α always increase. On the otherhand for small values of the parameter q Renyi entropy saturates to a finite small value.

4 Summary

To summarize, in this paper, we have addressed the following issues:

• First we have presented the computation of entanglement entropy in de Sitter space in presence ofaxion with a linear source contribution in the effective potential as originating from Type IIB stringtheory. To demonstrate this we have derived the axion wave function in an open chart.

– 26 –

Page 28: Quantum entanglement in de Sitter space from Stringy Axion ... · E-mail: sayantan@aei.mpg.de,sayantan.choudhury@aei.mpg.de, panda@iopb.res.in Abstract: In this work, we study the

α=0

α=0.03

α=0.1

α=0.3

0 5 10 15 200.000

0.002

0.004

0.006

0.008

0.010

q

Sq,intr(α)

Renyi entropy in dS (D=4) with ν2=1 and fp

(α)=0

(a) For ν2 = 1 and f(α)p = 0.

α=0

α=0.03

α=0.1

α=0.3

0 5 10 15 200.000

0.002

0.004

0.006

0.008

0.010

q

Sq,intr(α)

Renyi entropy in dS (D=4) with ν2=1 and fp

(α)=10-7

(b) For ν2 = 1 and f(α)p = 10−7.

α=0

α=0.03

α=0.1

α=0.3

0 5 10 15 200.0000

0.0002

0.0004

0.0006

0.0008

0.0010

q

Sq,intr(α)

Renyi entropy in dS (D=4) with ν2=-1 and fp

(α)=0

(c) For ν2 = −1 and f(α)p = 0.

α=0

α=0.03

α=0.1

α=0.3

0 5 10 15 200.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

q

Sq,intr(α)

Renyi entropy in dS (D=4) with ν2=-1 and fp

(α)=10-7

(d) For ν2 = −1 and f(α)p = 10−7.

Figure 13. Renyi entropy Sq,intr(α) vs q plot in 3 + 1 D de Sitter space in absence and presence of axionic source

for α = 0, α = 0.03 , α = 0.1 and α = 0.3 with ‘+′ branch of solution of |γ(α)p | and |Γ(α)

p,n|.

• Next using the α vacuum state we have expressed the wave function in terms of creation and annihila-tion operators. Further applying Bogoliubov transformation on α vacuum state we have constructedthe expression for reduced density matrix.

• Further, using reduced density matrix we have derived the entanglement entropy, which is consistentwith ref. [10] if we set α = 0. In the ν2 < 0 range we have derived analytical result for the entanglemententropy. Finally, we have used numerical approximations to estimate entanglement entropy with anyvalue of ν2.

• We have also computed the Renyi entropy in presence of axion source. In absence of the source thisresult is consistent with ref. [10] in q → 1 limit. Here in ν2 < 0 region we have provided the analyticalexpression for the Renyi entropy. We have also used numerical techniques to study the behaviour ofRenyi entropy and largest eigenvalue of the density matrix with any value of ν2.

• Our result provides the necessary condition to generate non vanishing entanglement in primordialcosmology due to axion.

The future directions of this paper are appended below:

• Using the derived expression for density matrix for generalised α vacua, one can further computeany n point long range quantum correlation to study the implications in the context of primordialcosmology. It is expected that this result will surely help to understand the connection between theBell’s inequality violation, quantum entanglement and primordial non-Gaussianity.

• Till now we have studied the necessary condition for generating non zero entanglement entropy inprimordial cosmology. In this connection one can further compute quantum discord, entanglementnegativity etc., which play a significant role to quantify long range quantum correlations withoutnecessarily involving quantum entanglement.

– 27 –

Page 29: Quantum entanglement in de Sitter space from Stringy Axion ... · E-mail: sayantan@aei.mpg.de,sayantan.choudhury@aei.mpg.de, panda@iopb.res.in Abstract: In this work, we study the

Acknowledgments

SC would like to thank Quantum Gravity and Unified Theory and Theoretical Cosmology Group, MaxPlanck Institute for Gravitational Physics, Albert Einstein Institute and Inter University Center for As-tronomy and Astrophysics, Pune for providing the Post-Doctoral Research Fellowship. SP acknowledgesthe J. C. Bose National Fellowship for support of his research. Last but not the least, We would all like toacknowledge our debt to the people of India for their generous and steady support for research in naturalsciences.

A Wave function for Axion using Bunch Davies vacuum

Further using Eqn (3.1) the field equation of motion for the axion can be written as [14]:[(a(t))−3∂t

(a3(t)∂t

)− (Ha(t))−2L2

H3 +m2axion

]φ = µ3, (A.1)

where the scale factor a(t) in de Sitter open chart is given by, a(t) = sinh t/H. Here the Laplacian operator

L2H3 in H3 satisfies the following eigenvalue equation [56]:

L2H3Yplm(r, θ, φ) =

1

sinh2 r

[∂r(sinh2 r ∂r

)+

1

sin θ∂θ (sin θ ∂θ) +

1

sin2 θ∂2φ

]Yplm(r, θ, φ) = −(1 + p2)Yplm(r, θ, φ), (A.2)

where Yplm(r, θ, φ) represents orthonormal eigenfunctions which can be written as:

Yplm(r, θ, φ) =Γ (ip+ l + 1)

Γ (ip+ 1)

p√sinh r

P−(l+ 12 )

(ip− 12 )

(cosh r)Ylm(θ, φ). (A.3)

The total solution of the equations of motion can be written as:

Φ(t, r, θ, φ) =

∫ ∞0

dp∑σ=±1

∞∑l=0

+l∑m=−l

[aσplmUσplm(t, r, θ, φ) + a†σplmU

∗σplm(t, r, θ, φ)

], (A.4)

where Uσplm(t, r, θ, φ) forms complete basis of mode function, Uσplm(t, r, θ, φ) = Hsinh tχp,σ(t)Yplm(r, θ, φ).

Here χp,σ(t) forms a complete set of positive frequency function. Also this can be written as a sum of

complementary (χ(c)p,σ(t)) and particular integral (χ

(p)p,σ(t)) part, as given by χp,σ(t) = χ

(c)p,σ(t) + χ

(p)p,σ(t).

Explicitly the complementary and particular integral part can be expressed as [14]:

χ(c)p,σ(t) = χ

(c)−p,σ(t) =

1

2 sinhπp

[(eπp − iσ e−iπν

)Γ(ν + 1

2 + ip) Pip

(ν− 12 )

(cosh tR/L)−(e−πp − iσ e−iπν

)Γ(ν + 1

2 − ip) P−ip

(ν− 12 )

(cosh tR/L)

], (A.5)

χ(p)p,σ(t) = µ3 sinh2 t

∞∑n=0

1

(p2 − p2n)χ(c)pn,σ(t)

∫dt′χ(c)pn,σ(t

′), (A.6)

where the parameter ν is defined as, ν =

√94 −

m2axion

H2 =√

94 −

µ3bfaH2 =

√94 −

Λ4G

f2aH

2 .

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