Entanglement entropy through conformal interfaces in the 2D Ising model
We consider the entanglement entropy for the 2D Ising model at the conformal fixed point in the presence of interfaces. More precisely, we investigate the situation where the two subsystems are separated by a defect line that preserves conformal invariance. Using the replica trick, we compute the entanglement entropy between the two subsystems. We observe that the entropy, just like in the case without defects, shows a logarithmic scaling behavior with respect to the size of the system. Here, the prefactor of the logarithm depends on the strength of the defect encoded in the transmission coefficient. We also comment on the supersymmetric case.
1 Introduction and summary
Entanglement entropy in quantum systems has been investigated in recent years in many different fields, ranging from quantum information theory to black hole physics. It encodes the information of the entanglement of a subsystem with the rest of the system. In 1+1 dimensional systems at the critical point, the vacuum entanglement entropy for a subsystem which is obtained by geometrically singling out an interval is given by calabrese_entanglement_2009 ; holzhey_geometric_1994 ; vidal_entanglement_2003
where is the reduced density matrix and the length of the interval specifying .
In this paper, we consider situations where the separation of the full system is not merely geometric. Rather, we investigate the case where there is a physical separation given by a conformal interface (to which we also refer as “defect”). This is a one-dimensional domain wall, localized in the time direction, separating the two-dimensional space-time into two parts. An interface CFT naturally consists of two sub-systems – the two theories that are joined by the defect line. The domain wall can be fully or partially transmissive, such that the quantum field theories living on the two sides are related nontrivially across the defect line. A useful and interesting probe to the whole system is the quantum correlation, i.e. entanglement, between the two sub-systems. The subsystem in (1) thus consists of the CFT on one side of the interface.
In the present paper, our discussion focusses on the two-dimensional Ising model, where defects have been analysed by integrability henkel_ising_1989 ; abraham_transfer_1989 and conformal field theory oshikawa_defect_1996 ; oshikawa_boundary_1997 techniques. There are altogether three classes of defects preserving conformal invariance. Two of them have a simple description in terms of a square lattice model. At the position of the interface, the couplings between the spins are different than in the bulk of the lattice. In formulas, the energy-to-temperature ratio is given by
where are the spin variables, and so that the bulk theory is critical. Along the (vertical) interface, couplings are rescaled by the factor that parametrizes deformations of the interface. In the special case the situation reduces to the case without any defect, on the other hand, for or , one obtains two isolated subsystems, separated by a totally reflective defect. One furthermore distinguishes between ferromagnetic interfaces for which the parameter takes values and anti-ferromagnetic interfaces which are parametrized by .
In a spin chain interpretation, the defect sits on a particular link of the spin chain, and we consider the entanglement entropy of the subsystems located left and right of the defect link. When the system propagates in time, the defect link sweeps out a one-dimensional line in two-dimensional space-time, which is the defect line of the conformal field theory.
One now expects the entanglement entropy between two subsystems to reduce to (1) in the totally transmissive case at and to vanish in the totally reflective case. For generic , the entanglement entropy will depend on , as this parameter determines the “strength” of the defect.
Indeed, we will show that the entanglement entropy is given by
where is a constant independent of and parametrizes the transmission of the defect. For the defects (2) can be expressed in terms of and the constant vanishes. The formula for the case of the remaining class of defects, which is not described by (2), is similar. This class can be obtained by appyling order-disorder duality to one side of the interface, which does not change the transmissivity and also not the prefactor . On the other hand, it does shift the constant .
Entropy formulas of the form (3) have appeared in several different contexts in 2D CFT. First of all, the entropy through a conformal interface was considered in the example of the free boson compactified on a circle in sakai_entanglement_2008 with a result precisely of the form (3). Indeed, our analysis in sections 3, 4 ,5 is similar to their computations.
The entanglement entropy for subsystems separated by a defect in the Ising model as well as other fermionic chains was studied before with different methods. Numerical results were presented in 2009PhRvB..80b4405I , subsequently an analytical analysis appeared in 2010arXiv1005.2144E . In particular, the form of the entanglement entropy (3) was derived using the spectrum of the reduced density matrix in the lattice model. The paper 2010arXiv1005.2144E initiated a series of following papers addressing related topics, see e.g. 2012JPhA…45j5206C ; 2012JPhA…45o5301P ; 2012EL…..9920001E ; 2013JPhA…46q5001C ; 2015arXiv150309116E .
In the present paper, we investigate the defects of the Ising model from the point of view of conformal field theory. Here, one associates one class of defect lines to each primary of the Ising model. Within each class, the elements differ by marginal perturbations and as a result also by their transmission and we verify (3) by a conformal field theory computation. The different classes differ physically by properties such as their -factor and RR-charge. From the point of view of the entanglement entropy, this changes the constant contribution in (3). The constant shifts are particularly interesting in the supersymmetric case, which we analyze by combining the Ising-model results with those of sakai_entanglement_2008 .
Constant shifts in the entanglement entropy were also observed in Nozaki:2014hna ; Nozaki:2014uaa ; He:2014mwa . In those papers, the shift in the entanglement entropy of excited states, rather than the vacuum, was determined. The excited states were obtained by acting with local operators on a CFT vacuum. The physical difference to our situation is that the defects we consider extend in one dimension, hence are not local operators. In the case of the papers Nozaki:2014hna ; Nozaki:2014uaa ; He:2014mwa the logarithmic term remains the same (compared to the situation of the vacuum), whereas the constant term gets shifted. In the case of rational conformal field theories, the constant shifts have an interpretation in terms of quantum dimensions. In PandoZayas:2014wsa ; Das:2015oha the entanglement entropy was considered for conformal field theories with boundary. These systems are related to ours by the folding trick, where one folds along the defect line to obtain a tensor product theory with a boundary, see figure 1. However, the division in subsystems is different in their case, as they consider the division of the system into left and rightmovers and compute the left-right entropy.
This paper is organized as follows: In section 2 we review the construction of conformal interfaces for the Ising model. Using the folding trick, conformal interfaces can be mapped to boundary conditions for a free boson on a circle orbifold. Hence, they are given in terms of D0 branes (ferromagnetic and anti-ferromagnetic) and D1 branes (order-disorder) parametrized by their position and Wilson lines, respectively. While this description offers an intuitive interpretation of the possible interfaces, it is less useful for calculations in our context. We hence go back to a formulation in terms of a GSO projected free fermion theory. The ferromagnetic and anti-ferromagnetic interfaces are then given by interfaces charged under RR-charge whereas the order-disorder interface is a neutral interface.
In section 3 we explain the basics of how to compute the entanglement entropy as a derivative of a partition function involving defects. Sections 4 and 5 contain the concrete calculations for the Ising model, in particular and in equation (3) is derived for all classes of interfaces of the Ising model. In section 6 we comment on the supersymmetric model, combining our results with those of sakai_entanglement_2008 . We show that simplifies due to cancellations between oscillator modes. We also express the result in a way that is suggestive for generalizations to higher dimensional tori and possibly other supersymmetric models. Finally, in section 7 we draw some conclusions and point out open problems.
2 Conformal interfaces of the Ising model
2.1 Interfaces and boundary conditions
A convenient description of defects in the Ising model arises, when we employ the folding trick. Here, as illustrated in figure 1, an Ising model interface is mapped to a boundary condition of the tensor product of two Ising models. The latter is well known to be equivalent to a orbifold of a free boson compactified on a circle of radius oshikawa_defect_1996 ; oshikawa_boundary_1997 .
Dirichlet conditions with ,
Neumann conditions with .
In string theory language, is the position of a D0-brane on the circle with a identification, whereas is the Wilson line on a D1-brane which belongs to the position of the dual D0-brane on the dual circle (of radius ). Unfolding converts the boundary states of (Ising) to interfaces of the Ising model.
For Dirichlet interfaces one can relate and the parameter of the interface model (2) as in oshikawa_defect_1996 ; oshikawa_boundary_1997 by comparing the CFT spectrum with the exact diagonalization of the transfer matrix delfino_scattering_1994 :
A special case is corresponding to which means there is no interface. Hence the interface operator is given by the identity operator. Another special case is which belongs to . This operator belongs to the -symmetry of the Ising model.
At the special values and , corresponding to and , respectively, the interfaces reduce to separate boundary conditions for the two Ising models given by
where dennote the three conformal boundary conditions of the Ising model, namely spin-up, spin-down and free cardy_boundary_1989 .
For Neumann interfaces, or order-disorder interfaces, the relation between and is similar but with and thus (see e.g. in bachas_fusion_2013 ). Again, for the special value we have . This means that this Neumann interface is topological. On the other hand, at the values the interfaces reduces to separate boundary conditions
Other two interesting quantities that characterise all conformal interfaces are the reflection coefficient and the transmission coefficient which are given by 2-point functions of the energy momentum tensor as follows quella_reflection_2007
where are the components of the energy momentum tensor at the point and are evaluated at the corresponding point reflected at the interface. For the interfaces we considered previously the reflection and transmission coefficients are given by
It is easy to see that . Note that for topological Dirichlet interfaces, where or , there is no reflection, namely . On the other hand, for the reflection coefficient is , and thus the interface reduces to a totally reflecting boundary conditions. For Neumann interfaces the statements are alike.
2.2 Free fermion description
While the description in terms of the free boson provides an overview over the possible interfaces, to construct the explicit interface operator one needs to undo the folding. This is best done in the language of free fermions. Recall that the Ising model can be regarded as a system of a free real Majorana fermion, where modular invariance is achieved by a projection on even fermion number (where the fermion number is the sum of left and right fermion number). In a free fermion theory one distinguishes between the NS-sector and the R-sector. In the NS sector, the fermions (denoting left and rightmovers) are half integer moded and there is a non-degenerate ground state. In the R-sector the fermions are integer moded and the ground state degenerates. The Ising model has three primary fields with respect to the Virasoro algebra, of left-right conformal dimensions . In terms of the free fermion is the NS-vacuum, the first excited state of the NS-sector and a R-ground state (after the degeneracy of the ground states has been lifted by the GSO-projection).
where we have split the operator into two factors; is a map of the ground states of the free fermion theory, whereas contains the higher oscillator modes. The latter can be factorized further; in only the th modes of the fermion field appear pairwise, such that all commute. It is given by
where the are the modes of CFT1/CFT2 which are acting from the left/right on – the ground state operators. The matrix specifies the interface and can be given in terms of a boost matrix which guarantees that the interface preserves conformal invariance, see bachas_worldsheet_2012 ; bachas_fusion_2013 for more details. Their exact relation is given by
The matrices with det correspond to Dirichlet boundary conditions in the orbifold theory whereas det corresponds to the Neumann boundary conditions. For the relation gives
Indeed, and precisely correspond to the parameters describing the D0 and D1 brane moduli space. From now on we omit the tildes. Then we can write in both cases. To obtain the interface operators of the Ising model from those of the free fermion theory, one still has to GSO-project on total even fermion number. This requires taking linear combinations of the free fermion interfaces. In the Ising model the type-0-GSO projection allows us to distinguish three cases: The interface operators for that carry either positive or negative RR charge and can be written as 111 See bachas_fusion_2013 for more details on the construction.
and these for which are the neutral operators
The operators and act on the Neveu-Schwarz and Ramond sector of the free fermion theory, respectively. They are given by
Here, denote R-ground states and is the spinor representation of .
3 How to derive the entanglement entropy
In this section we briefly want to show the derivation of formulas we use in the following sections. It is an adaptation of the procedure used to derive the entanglement entropy through interfaces in the free boson theory. Thus, for a more detailed derivation we recommend sakai_entanglement_2008 , but also holzhey_geometric_1994 .
Our goal is to derive the (ground state) entanglement entropy between two 2D GSO-projected free fermionic CFTs connected by the interfaces (14) or (15). We formally define CFT1 to live on a half complex plane and CFT2 on , respectively. The interface then lies on the imaginary axis . The EE is defined by the von Neumann entropy of the reduced density matrix for the ground state as (see e.g. calabrese_entanglement_2004 ; calabrese_entanglement_2009 )
The trace of the -th power of the reduced density matrix is also given by the partition function on a -sheeted Riemann surface with a branch cut along the positive real axis holzhey_geometric_1994 ; sakai_entanglement_2008
To evaluate the partition function we change coordinates to and introduce the cutoffs and . The -sheet then looks as on the left part in figure 2. As in sakai_entanglement_2008 we impose periodic boundary conditions in Re and choose for simplicity. We then end up with the torus partition function with interfaces inserted which is given by
with and .
4 Derivation of the partition function
In the following we explicitly derive the partition function (21) for the interface operators introduced in section 2.2. We start with a single NS operator which is the simplest case. Step by step we show how to derive for the more complicated R, neutral, and charged operator.
4.1 The partition function for a single NS operator
In the NS sector of the free fermion theory we can formally write the Hilbert space of the theory as a tensor product where . Then each as in (10) has a matrix representation in given by
and we can write . In this notation the propagator is given by with
Using the above notation with CFT1 = CFT2 the partition function (21) of the -sheet can be written as
where , , are the eigenvalues of .
4.1.1 Explicit calculation
We have two distinguishable interfaces: the Dirichlet-interface with and the Neumann-interface with . However, in both cases the matrices are similar and their eigenvalues are given by
so that the partition function of the -sheet is given by
At this stage one could proceed further by directly using formula (20) on the latter result for . Through the logarithm the infinite product simplifies to a sum. Taking the derivative w.r.t. in every summand it is then easy to write down a result for the entanglement entropy by means of an infinite sum. One could then evaluate the sum – and thus the entanglement entropy – numerically for every and up to arbitrary accuracy. However, we are mainly interested in small – which means large – behaviour of the entanglement entropy, since is introduced as a UV cutoff. In this limit we can derive the EE analytically by proceeding as in the following.
For odd the partition function (26) can be written as
Thus we can conclude that the -sheet partition function for odd can be expressed as
Using the behaviour of and under -transformations we can write
where , and is constant in .
4.2 The partition function for a single R-operator
In the Ramond sector the modes are integer and the zero-mode map is slightly more difficult, . The latter allows us to write
where vanishes. Proceeding similar to the case of NS-operators one gets
where again , , are the eigenvalues of .
4.2.1 Explicit calculation
The eigenvalues for the R-interface are similar to the eigenvalues for the NS interface but with . Thus, for odd we can write
where again . This time the latter is given in terms of because of being integer. One important difference between the -function we use here and the -function used in the case of the NS-interface is that there appears an additional factor of . One can show that
for odd so that in this case the partition function reduces to
The same steps as for the NS interface now lead us to
in the limit .
In the Ramond sector only the Dirichlet interfaces have non-trivial components, so we do not consider Neumann boundary conditions, although they would not make any difference for .
4.3 The partition function for the neutral interface operator
The neutral interface operator is given by with Neumann boundary conditions, i.e. . Some simple algebra leads to
with given by the tensor product of the matrices
where we here used the reflection coefficient and the transmission coefficient as introduced in (7). We can see that both and can be diagonalized simultaneously. With straight forward linear algebra we can now calculate the partition function for the -sheet with a neutral interface insertion. It is given by
The important factor of can be understood with the following simpler example: Consider the matrices . It is now easy to convince oneself that
which allows us to directly conclude that . The generalization to and is straight forward.
The first summand in (38) is the same as for the single NS-interface. In a very similar way as in section 4.1, the second summand can also be written in terms of . We here only state the result in the limit :
where with .
4.4 The partition function for the charged interface operator
The charged interface operator is given by with Dirichlet boundary conditions, i.e. . With similar considerations as for the neutral interface we can write
where corresponds to the vacuum and corresponds to in a similar way to the single Ramond interface. There is no difference between the positively and the negatively charged interface. In matrix representation all the ’s are tensor products of matrices similar to (37) but with integers for the Ramond operators. Thus the partition function of the -sheet with a charged interface can be written as
Using the same logic as in the previous sections, the partition function for odd reduces to
in the limit . The functions and are given as before and is given by
5 Derivation of the entanglement entropy
Before we explicitly derive the entanglement entropy we want to show that – for – it is the same in all previous cases up to an additional term for the neutral interface. Our formula of choice (20) is
for which it is easy to check that any overall factor in with constant in does not contribute to the entanglement entropy.
At first we want to write down the preliminary result for the single NS-interface where is given by (30) so that the entanglement entropy can be written as
Next we consider the single R-interface where the partition function is given by (35). Its EE is simply the same as for the NS-interface
where we can simplify to the second line because we are in the limit and because .
At last we want to derive the EE for the charged interface operator where is given by (42). As in the case of the neutral interface operator every term with a factor can be neglected. Consequently also in (42) has no contribution to the entanglement entropy when we are in the limit . It again simply reduces to the EE for the single NS-interface:
5.1 Explicit derivation of
To derive the entanglement entropy explicitly we have to calculate . Therefore we proceed similar as in sakai_entanglement_2008 and write , which can be written as a Taylor series around and further massaged as
where are the Bernoulli polynomials and Bernoulli numbers, respectively, as given in Appendix A.3. Its derivative in the limit is then given by
At this stage we use the formula (76) to obtain