# Entanglement of low-energy excitations in Conformal Field Theory

###### Abstract

In a quantum critical chain, the scaling regime of the energy and momentum of the ground state and low lying excitations are described by conformal field theory (CFT). The same holds true for the von Neumann and Rényi entropies of the ground state, which display a universal logarithmic behaviour depending on the central charge. In this letter we generalize this result to those excited states of the chain that correspond to primary fields in CFT. It is shown that the -th Rényi entropy is related to a 2-point correlator of primary fields. We verify this statement for the critical and chains. This result uncovers a new link between quantum information theory and CFT.

Entanglement is one of the central concepts
in quantum physics since Schroedinger used the term in an answer to
the Einstein-Podolsky-Rosen article in 1935. A particularly active
line of research is concerned with the role played by entanglement
in the physics of many-body systemsAmico . One is
typically interested in the amount of entanglement between two spatial
partitions, say and , of a many-body system in its ground
state. For a pure ground state the amount of entanglement is usually
quantified with the entanglement entropy, or the von Neumann
entropy of the reduced density matrix :
. Alternatively, the Rényi
entropies are also used: . One
of the most important results in this topic is the celebrated
area lawSrenidcki93 ; Eisert ; Cirac09 , which, roughly
speaking, states that ground states of gapped many-body systems with
short-range interactions have an entanglement entropy proportional to the area of the hypersurface separating both
partitions. The area law restricts the fraction of
the Hilbert space accessible to ground states of local Hamiltonians in
an essential way, allowing for their efficient numerical simulation Cirac09 .

Violations of the area law occur in gapless (critical) systems. In
one dimension most of critical systems, as well as being gapless, are
also conformal invariant. The attention to the entanglement properties
on these systems came after the seminal result of Holzhey, Larsen and
Wilczek HLW , who showed that the leading behavior of the ground
state entropies is proportional to the central charge of the
underlying conformal field theory (CFT) governing the long-distance
physics of the discrete quantum chain. If and are the
lengths of the partition and of the total system, both measured in
lattice spacing units, then the Rényi entropy of the ground state,
with periodic boundary conditions, is HLW ; Calabrese04 ; Vidal2003 , the entanglement entropy being

(1) |

where is the central charge of the CFT and is a non-universal constant.

In a critical model, the finite-size scaling of the energy of excitations is
given by the scaling dimension of the corresponding conformal operators
Cardy . This fact suggests that also the entanglement entropy could be related
to properties of these operators.
Entanglement of excited states has been considered previously. In Alcaraz08 it was
shown that the negativity of the excited states in the XXZ critical model
shows a universal scaling. In Masanes it was shown that a violation of area
law should be expected for the low lying excited states of critical quantum
chains, and in Alba09 , it was considered the entanglement of very large energy
excitations in XY and XXZ spin chains.

In this letter we show that the entropy of excited
states associated to primary fields exhibits a universal behaviour
that generalizes (1). The energy of these low-lying states degenerate as
in the bulk limit . We prove that the
excess of entanglement, , is a finite-size
scaling function related to the -point correlator of the primary field. These results are verified in two models: the and spin chains.

Entanglement of generic primary states. Let us consider a system of length with
periodic boundary conditions. To describe it, we introduce the complex
variable , where is the spatial
coordinate and is the time coordinate. is split into
two subsystems , with and , and where is a short-distance cutoff HLW . The world sheet of the past (), is a cylinder with
two semidisks of radius cut out (denoted and in figure 1). The boundary of the world sheet
of figure 1 is given by the union . After the conformal transformations :

(2) |

the cylinder gets mapped into a strip of height and width

(3) |

The wave function of this state is given by the path integral

(4) |

where denotes the local field whose Euclidean action is
. The field is a functional of , that is evaluated
at the infinite past in equation (4) (recall equation
(3)). and denote the values of the
field in the subsystems and respectively. Periodic
boundary conditions are imposed on the and edges
HLW . If were not primary, then equation (4) would include additional terms generated by the conformal transformations (2).

The density matrix of subsystem is obtained by tracing over the variables in :

(6) |

where represents the point at the infinite future. The functional integral is over a strip of height and width , with boundary conditions on the lower edge and on the upper edge. The normalization factor is determined by the condition , which implies that is the functional integral with no operator insertion and the top and bottom edges of the strip being identified (i.e. a torus partition function), and is the two point correlator on this torus. To compute the entanglement entropy one first computes the trace of , which is given by

where denotes the partition function on a torus of lengths and , so that the moduli parameter is given by , and where denotes the expectation value in the -torus. Notice that the -point correlator of fields depends on the ratio and on the moduli parameter.

To further proceed one uses the expression of the partition function of a general CFT with central charge for chiral and antichiral sectors of the theory,

(8) |

as anticipated in (1). In the general case, equation (Entanglement of low-energy excitations in Conformal Field Theory) depends on a -point correlator of the fields and on a cylinder of length along the time direction. It is now convenient to rescale this length to . Afterwards, we shift the coordinates where . Finally, we exchange and coordinates in such a way that for and for . The ratio between the excited and the ground state traces, , becomes, from (Entanglement of low-energy excitations in Conformal Field Theory):

where denotes the expectation value in a cylinder of length . Note that . The dependence of the entropies of the excited states on the -point correlation functions was also observed in the ground state entropies of two disjoint segments of the quantum critical chains cala-et-al . The entanglement entropy for the excited state can then be computed using the replica trick:

(10) |

In the limit , the terms appearing in (Entanglement of low-energy excitations in Conformal Field Theory) can be approximated by the operator product expansion (OPE) , finding:

(11) |

where is the operator with the smallest scaling dimension,
. The term of order depends on the OPE
constants and on the expectation values
. If , this
term is as the first one in equation (11), and
eventually they may cancel one another, as we shall see in an example
below. If one could use (11) to infer the quantities
, and from the numerical computation of the entanglement.

Using equation (10) one
finds, for the low- behaviour of the entanglement entropy
():

(12) |

Equations (Entanglement of low-energy excitations in Conformal Field Theory-12) are the main results of
this letter. They relate the von Neumann and -Rényi entropy of the excitation
represented by the primary operator to the -point
correlators of and in the cylinder. Notice that the ratio
does not depend on the non-universal constant , which is therefore common to and .

As an example of the laws (Entanglement of low-energy excitations in Conformal Field Theory,10) we shall
consider a CFT given by a massless boson compactified on a
circle. The primary fields are given by the vertex operators
(being
, chiral and antichiral boson fields) where
, is the compactification ratio, and . The scaling dimensions of these operators are . Using the chiral correlator of vertex operators on the cylinder, ,
it turns out that

(13) |

Hence, all the excitations represented by vertex operators
have the same entropy as the ground state. This result is not in contradiction with (11) because, in this case, and both terms in (11) cancel out due to the properties of the OPE constants. In fact, the cancellation happens in all order of .

Next, we study the operator
. Using its correlator on the cylinder
and the Wick theorem, we get (in
terms of ):

(14) |

and a more lengthy expression for . In the low- limit, one finds that , which leads to an excess of entanglement entropy given by (12) with .

Realizations of both types of excitations in particular models will be now shown, and their amount of entanglement compared with the CFT predictions (13,14).

Excitations in the and models. The Hamiltonian of the spin-1/2 model is given by

(15) |

where is even and periodic boundary conditions are assumed (for
we get the model). This model is integrable
Lieb1961 and gapless for . The corresponding
CFT is given by the aforementioned bosonic CFT with
. The model in the sector
with magnetization can be mapped, through a
Jordan-Wigner transformation, into a system with free
fermions in a lattice of sites. We computed the entanglement and
Rényi entropies of several types of excitations in these models. This
task was achieved using the methods of references
Peschel2003 ; Vidal2003 in the free fermion problem and through
numerical exact diagonalization in the case.

Let us first consider the vertex operator
. In the free fermion model, the result (13) is
exact and can be proved analytically. Indeed, corresponds
to the umklapp excitation
, where is the Fermi momentum, and where is the Fermi state and
the fermionic creation operator with momentum
. This state can be obtained from the Fermi state shifting all the momenta as . Such a shift produces a global phase factor in
the wavefunction in real space and, consequently, the entropy remains
unchanged. In the model, the state corresponds to the
ground state in the sector with spins up and total
momentum . We observe that the prediction
holds, up to the oscillations expected for Calabrese2010 , which in this case are of the order of for systems with spins.

We will now consider the excitation . In a
system of free fermions the resulting state corresponds to
the addition of two fermions at the right of the Fermi point, i.e., to
the state . Figure 2 shows that ground and excited states
entropies coincide, up to oscillations. In the model, is the lowest eigenstate with total momentum . Again in this case, oscillations of around one are observed (see figure 2).

Finally, figure 3 shows some numerical results for the entanglement of the excitation . In the free fermion problem, corresponds to a particle-hole excitation: , while in the model it corresponds to the lowest eigenstate with . We observe an excellent agreement with the theoretical prediction (14) for . Similar results hold for . Moreover, we have checked, for up to 6, that the low- formula (12) is very well satisfied for fermions.

In summary, we have obtained an expression for the Rényi entropies of excitations associated to any primary field. We verified the results with finite-size realizations of the and models up to 30 sites in the latter case, finding very good agreement with the theory.

As explained earlier, equation (Entanglement of low-energy excitations in Conformal Field Theory) can be used as a numerical method to extract information
about correlators, conformal dimensions and OPE coefficients of primary fields. An interesting problem is to
generalize these results to the descendent states in CFT. We expect
that the Rényi entropies, at a given level of a conformal tower will
depend on the particular state targeted. This can provide a method to
establish the correspondence between degenerated excited
states of a critical lattice model, and the descendent fields in the
underlying CFT.

Equation (Entanglement of low-energy excitations in Conformal Field Theory) further suggests a generalization of the Rényi entropies in terms of traces of different density matrices
. This
object would be related to the correlator:
in the very same fashion as in (Entanglement of low-energy excitations in Conformal Field Theory). The numerical computation of
the associated generalized entropies would then provide
information on more general correlators in CFT, and vice-versa. Applications of the present work to other models and to non-primary fields are in progress.

This work represents a further step along the direction of deriving CFT data using quantum information methods.

Acknowledgements.
We thank P. Calabrese, R. Pereira and V. Rittenberg for discussions. This work was supported by the Spanish project FIS2009-11654 and by FAPESP and CNPq (Brazilian agencies).

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