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Scattering effect on entanglement propagation in RCFTs
Authors
Tokiro Numasawa,Stephen Campbell
S. Kalyana Rama,Marco CANTAMESSA,Sandra Signorella,Arman Shafieloo,Timothy Paulitz,Vladimir Makarenko,Marina Avdonina,Sándor Vajna,Jose Vega,Celso Martinez Rivero,Tonia Moore,Mariya Repenkova,Eduard Dzhagityan,Alex Singleton,Dalimil Mazáč,Martin Goodchild,Trevor Glasbey,Valentin Hirschi,Wieland Staessens,Jonathan Gaunt,Hartmut Gimpel,Srinivasan Jagannathan,Kunal Vyas,Kristen Smith,Timothy Cohen,Aurelien Bigot,Pramod Shukla,Maria Sazanova,Aqeel Ahmed,Zoran Jovanović,Mahendra Ramachandran,Alexandru Ciubotaru,Veroljub Dugali,Milan Rapajić,Michele Portolan,Garrett Poe,Ljiljana Dukic,Aleksandar Antić,Stefanos Katmadas,Alain Jaccard,Carolyn Dzierba,Steven Shannon,Andrew Degnan,Andy Van Brocklin,Anatoly Dymarsky,Joseph Fotsing,Vasilis Niarchos,Digamber Porob,Nikolay Bobev,Danuta Kolozyn-Krajewska,Seok Ki Choi,Isabel Maria Ratola Duarte,Oleg Lunin,Jovana Brašić Stojanović,Felipe Lumbreras,Tetsuji Kimura,Jonathan Wall,Marie Connett,Aaron Balog,Ezio Maina,Ka Ming Tsui,Thorsten Alexander Kern,Ryutarou Ohbuchi,Christopher Herzog,Martin Köhne,Claudia de Rham,David Clarke,David Kotecki,António Pinho,Bo Feng,Dan Andersen,Kari Anne Bråthen,María Jesús Souto Blanco,Kingsley Anukam,Наталия Егорова,Asger Reinstrup Bihlet,Francois Delduc,Juan Mateos Guilarte,Miquel Sànchez-Marrè,Krzysztof Dmytrów,Eva López Terrada,Eduardo Garcia-Valdecasas Tenreiro,Bruno Dufour,Md. 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arXiv:1610.06181v2 [hep-th] 25 Oct 2016
YITP-16-116
Scattering effect on entanglement propagation in RCFTs
Tokiro Numasawa 1,2
1 Yukawa Institute for Theoretical Physics,
Kyoto University, Kyoto 606-8502, Japan
2 Kavli Institute for Theoretical Physics,
University of California, Santa Barbara, CA 93117, USA
Abstract
In this paper we discuss the scattering effect on entanglement propagation in
RCFTs. In our setup, we consider the time evolution of excited states created by
the insertion of many local operators. Our results show that because of the finiteness
of quantum dimension, entanglement is not changed after the scattering in RCFTs.
In this mean, entanglement is conserved after the scattering event in RCFTs, which
reflects the integrability of the system. Our results are also consistent with the free
quasiparticle picture after the global quenches.
YITP-16-116
Scattering effect on entanglement propagation in RCFTs
Tokiro Numasawa 1,2
1 Yukawa Institute for Theoretical Physics,
Kyoto University, Kyoto 606-8502, Japan
2 Kavli Institute for Theoretical Physics,
University of California, Santa Barbara, CA 93117, USA
Abstract
In this paper we discuss the scattering effect on entanglement propagation in
RCFTs. In our setup, we consider the time evolution of excited states created by
the insertion of many local operators. Our results show that because of the finiteness
of quantum dimension, entanglement is not changed after the scattering in RCFTs.
In this mean, entanglement is conserved after the scattering event in RCFTs, which
reflects the integrability of the system. Our results are also consistent with the free
quasiparticle picture after the global quenches.
Contents
1 Introduction 1
2 Review of single operator case 4
2.1 Construction of states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Replica method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Example: RCFT case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 Excitations by multiple operators 11
4 Many operators excitations in RCFT 12
4.1 Example: 2nd R´enyi entropy in Ising CFT . . . . . . . . . . . . . . . . . . . 12
4.2 n-th R´enyi entropies of general RCFT . . . . . . . . . . . . . . . . . . . . . 15
4.2.1 2nd R´enyi entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.2.2 n-th R´enyi entropies . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.3 scattering effect on entanglement entropy . . . . . . . . . . . . . . . . . . . . 21
5 Discussion 22
1 Introduction
AdS/CFT correspondence[1], which is one of the realizations of holographic principle[2, 3],
relates string theory on AdSd+1 spacetime to d dimensional conformal field theory (CFTd).
In the Einstein gravity regime, the number of fields in CFTd should be very large (large N)
and the coupling between them should be strong. Related to these properties, recently the
chaotic nature of holographic CFTs is the focus of attention[4, 5, 6, 7, 8]. Out-of-time order
correlation function (OTOC), or equally the square of the commutator of operators, is one of
the useful quantities to diagnose the chaotic behavior of many body systems[9, 10]. In chaotic
system, we can see the chaotic behavior such as Lyapnov behavior, scrambling and Ruelle
resonance[11, 12]. On the other hand, in integrable CFTs such as RCFT, the behavior
is different and we cannot see such chaotic behavior[13, 14, 15]. The non integrability of
boundary theory seems to be important to create black holes in the bulk1[13, 16, 17].
These differences between integrable CFTs and chaotic CFTs can be seen by the time
evolution of entanglement in excited states. For example, let us consider the time evolution
of entanglement entropy after the global quench in 1 + 1d CFTs[18]. First we consider the
entanglement entropy of the single interval. In this case, the results are universal and depend
only on the central charge c of the CFT when time t and the length of interval L is sufficiently
large compared to the initial correlation length ξ. At early time, entanglement entropy grows
1N =4 SYM is believed to be integrable at large N , but this integrability is broken by the finite N
correction or the introduction of thermal background. We thank to P. Caputa for pointing out this.
1
1 Introduction 1
2 Review of single operator case 4
2.1 Construction of states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Replica method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Example: RCFT case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 Excitations by multiple operators 11
4 Many operators excitations in RCFT 12
4.1 Example: 2nd R´enyi entropy in Ising CFT . . . . . . . . . . . . . . . . . . . 12
4.2 n-th R´enyi entropies of general RCFT . . . . . . . . . . . . . . . . . . . . . 15
4.2.1 2nd R´enyi entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.2.2 n-th R´enyi entropies . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.3 scattering effect on entanglement entropy . . . . . . . . . . . . . . . . . . . . 21
5 Discussion 22
1 Introduction
AdS/CFT correspondence[1], which is one of the realizations of holographic principle[2, 3],
relates string theory on AdSd+1 spacetime to d dimensional conformal field theory (CFTd).
In the Einstein gravity regime, the number of fields in CFTd should be very large (large N)
and the coupling between them should be strong. Related to these properties, recently the
chaotic nature of holographic CFTs is the focus of attention[4, 5, 6, 7, 8]. Out-of-time order
correlation function (OTOC), or equally the square of the commutator of operators, is one of
the useful quantities to diagnose the chaotic behavior of many body systems[9, 10]. In chaotic
system, we can see the chaotic behavior such as Lyapnov behavior, scrambling and Ruelle
resonance[11, 12]. On the other hand, in integrable CFTs such as RCFT, the behavior
is different and we cannot see such chaotic behavior[13, 14, 15]. The non integrability of
boundary theory seems to be important to create black holes in the bulk1[13, 16, 17].
These differences between integrable CFTs and chaotic CFTs can be seen by the time
evolution of entanglement in excited states. For example, let us consider the time evolution
of entanglement entropy after the global quench in 1 + 1d CFTs[18]. First we consider the
entanglement entropy of the single interval. In this case, the results are universal and depend
only on the central charge c of the CFT when time t and the length of interval L is sufficiently
large compared to the initial correlation length ξ. At early time, entanglement entropy grows
1N =4 SYM is believed to be integrable at large N , but this integrability is broken by the finite N
correction or the introduction of thermal background. We thank to P. Caputa for pointing out this.
1
A
A
1
2
23
3
4
Figure 1: The schematic picture of scattering of EPR pairs. The scattering is represented
the red star, which corresponds to the unitary matrix U on the Hilbelt space of particle 1
and 3.
linearly and saturates at some time determined by the length L. This can be explained by
the freely propagating quasiparticle model. On the other hand, the entanglement entropy
of disjoint region is not universal[19]. Let us consider the case of two intervals. When the
theory is integrable, we find that there is a regime that entanglement entropy decreases. In
other words, we can see a dip in the time evolution of entanglement entropy. This phenomena
can be explained by the model of freely propagating quasiparticles. On the other hand, in
non-integrable theories, such quasiparticle dip becomes smaller. We can think of the size of
the quasiparticle dip as a degree of scrambling, which is a quantum information theoretic
signature of quantum chaos. In holographic CFTs, which are the maximally chaotic CFTs
[6], the quasiparticle dip vanishes.
We can also see such difference in the time evolution of entanglement entropy in local
excited states. Consider the excited states that are created by the insertion of local operators
on the ground states. If the theory is integrable, we can see the propagation of quasiparticles.
At the late time, the change of entanglement entropy saturates and the value is given by
the entanglement between the propagating quasiparticles[20, 21, 22, 23]. On the other hand,
in the case of holographic CFTs, the excess of entanglement entropy does not saturate and
grows logarithmically in time[24, 25, 26]. This growth of entanglement is caused by the
chaotic interaction of holographic CFTs and can be seen as a kind of scrambling.
This difference of entanglement growth after the excitations depends on the property of
interaction (i.e. integrable or chaotic) of the interaction of the systems. Then, how can we
see the scattering effect on the propagation of entanglement? This is the motivation of this
paper. For example, in the paper [27] they consider the effect of scattering between two
EPR pairs on the propagation of entanglement (Figure 1). The initial state is given by the
tensor product of two EPR pairs:
|ψ〉 = 1
1 + |α|2 (|00〉12 + α |11〉12) ⊗ (|00〉34 + α |11〉34), (1.1)
where the index of vectors means the label of particles. The initial entanglement entropy
2
A
1
2
23
3
4
Figure 1: The schematic picture of scattering of EPR pairs. The scattering is represented
the red star, which corresponds to the unitary matrix U on the Hilbelt space of particle 1
and 3.
linearly and saturates at some time determined by the length L. This can be explained by
the freely propagating quasiparticle model. On the other hand, the entanglement entropy
of disjoint region is not universal[19]. Let us consider the case of two intervals. When the
theory is integrable, we find that there is a regime that entanglement entropy decreases. In
other words, we can see a dip in the time evolution of entanglement entropy. This phenomena
can be explained by the model of freely propagating quasiparticles. On the other hand, in
non-integrable theories, such quasiparticle dip becomes smaller. We can think of the size of
the quasiparticle dip as a degree of scrambling, which is a quantum information theoretic
signature of quantum chaos. In holographic CFTs, which are the maximally chaotic CFTs
[6], the quasiparticle dip vanishes.
We can also see such difference in the time evolution of entanglement entropy in local
excited states. Consider the excited states that are created by the insertion of local operators
on the ground states. If the theory is integrable, we can see the propagation of quasiparticles.
At the late time, the change of entanglement entropy saturates and the value is given by
the entanglement between the propagating quasiparticles[20, 21, 22, 23]. On the other hand,
in the case of holographic CFTs, the excess of entanglement entropy does not saturate and
grows logarithmically in time[24, 25, 26]. This growth of entanglement is caused by the
chaotic interaction of holographic CFTs and can be seen as a kind of scrambling.
This difference of entanglement growth after the excitations depends on the property of
interaction (i.e. integrable or chaotic) of the interaction of the systems. Then, how can we
see the scattering effect on the propagation of entanglement? This is the motivation of this
paper. For example, in the paper [27] they consider the effect of scattering between two
EPR pairs on the propagation of entanglement (Figure 1). The initial state is given by the
tensor product of two EPR pairs:
|ψ〉 = 1
1 + |α|2 (|00〉12 + α |11〉12) ⊗ (|00〉34 + α |11〉34), (1.1)
where the index of vectors means the label of particles. The initial entanglement entropy
2
xA
t
Oa Ob
lb
la
1
2
3 2 4
Figure 2: The figure of the setup we consider in this paper. Oa and Ob are primary operators.
The index of operators means the sector of each primary operator. At t = 0, these operators
are inserted apart from the entangling surface (in this case actually a point) and entangling
quasiparticles are emitted. la and lb represent the length from entangling surface. We
consider the case that A is given by the half of space {x ∈ R|x > 0}.
between particles 1, 3 and 2, 4 is given by the twice of entanglmenet entropy of EPR pairs
1/√1 + |α|2(|00〉12 +α |11〉12). The scattering effect is given by the action of a unitary matrix
U ∈ U(4) on the Hilbert space of particle 2 and 3. Then the state after the scattering is
given by
|ψf 〉 = (1 ⊗ U ⊗ 1) |ψi〉 . (1.2)
For general U, entanglement entropy between particles 1, 3 and 2, 4 changes after the scat-
tering event. In quantum field theory, the scattering effect U should be determined by the
Hamiltonian of the system. Then, we expect that the scattering effect on the entanglement
reflects the property of system, especially the integrability or chaotic nature. In this pa-
per, we consider the scattering of local excitations in 2d conformal field theory, especially
in RCFT that describes integrable systems. In the case of RCFT, we can create the pair
of quasi-particles by the action of local operator Oa on the ground state Oa |0〉, where the
index a is label of the conformal family that the primary operator belongs to. In RCFT,
as shown in the paper [21, 28, 29], entanglement between quasi-particles are given by log da
where da is so called quantum dimension. To see the interaction effect on entanglement, first
we need to prepare two entangling quasi-particles. This is done by the insertion of two local
operators:
|ψi〉 = OaOb |0〉 (1.3)
Then, if we can calculate the entanglement entropy after the scattering, we can see the
scattering effect on entanglement (Figure.2). This can be done if we follow the time evolution
of entanglement entropy of the state e−iHt |ψi〉 = e−iHtOaOb |0〉. Therefore the problem
reduces to the calculation of time evolution of entanglement entropy after the insertion of
two local operators. We study these problems in this paper.
This paper is organized as follows. In section 2 we briefly review the replica method with
3
t
Oa Ob
lb
la
1
2
3 2 4
Figure 2: The figure of the setup we consider in this paper. Oa and Ob are primary operators.
The index of operators means the sector of each primary operator. At t = 0, these operators
are inserted apart from the entangling surface (in this case actually a point) and entangling
quasiparticles are emitted. la and lb represent the length from entangling surface. We
consider the case that A is given by the half of space {x ∈ R|x > 0}.
between particles 1, 3 and 2, 4 is given by the twice of entanglmenet entropy of EPR pairs
1/√1 + |α|2(|00〉12 +α |11〉12). The scattering effect is given by the action of a unitary matrix
U ∈ U(4) on the Hilbert space of particle 2 and 3. Then the state after the scattering is
given by
|ψf 〉 = (1 ⊗ U ⊗ 1) |ψi〉 . (1.2)
For general U, entanglement entropy between particles 1, 3 and 2, 4 changes after the scat-
tering event. In quantum field theory, the scattering effect U should be determined by the
Hamiltonian of the system. Then, we expect that the scattering effect on the entanglement
reflects the property of system, especially the integrability or chaotic nature. In this pa-
per, we consider the scattering of local excitations in 2d conformal field theory, especially
in RCFT that describes integrable systems. In the case of RCFT, we can create the pair
of quasi-particles by the action of local operator Oa on the ground state Oa |0〉, where the
index a is label of the conformal family that the primary operator belongs to. In RCFT,
as shown in the paper [21, 28, 29], entanglement between quasi-particles are given by log da
where da is so called quantum dimension. To see the interaction effect on entanglement, first
we need to prepare two entangling quasi-particles. This is done by the insertion of two local
operators:
|ψi〉 = OaOb |0〉 (1.3)
Then, if we can calculate the entanglement entropy after the scattering, we can see the
scattering effect on entanglement (Figure.2). This can be done if we follow the time evolution
of entanglement entropy of the state e−iHt |ψi〉 = e−iHtOaOb |0〉. Therefore the problem
reduces to the calculation of time evolution of entanglement entropy after the insertion of
two local operators. We study these problems in this paper.
This paper is organized as follows. In section 2 we briefly review the replica method with
3
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