Chiral entanglement in massive quantum field theories in 1+1 dimensions
We determine both analytically and numerically the entanglement between chiral degrees of freedom in the ground state of massive perturbations of 1+1 dimensional conformal field theories quantised on a cylinder. Analytic predictions are obtained from a variational Ansatz for the ground state in terms of smeared conformal boundary states recently proposed by J. Cardy, which is validated by numerical results from the Truncated Conformal Space Approach. We also extend the scope of the Ansatz by resolving ground state degeneracies exploiting the operator product expansion. The chiral entanglement entropy is computed both analytically and numerically as a function of the volume. The excellent agreement between the analytic and numerical results provides further validation for Cardy’s Ansatz. The chiral entanglement entropy contains a universal term for which an exact analytic result is obtained, and which can distinguish energetically degenerate ground states of gapped systems in 1+1 dimensions.
Entanglement entropy plays a central role both in Quantum Field Theory (QFT) and condensed matter theory AFO (); Eis_rev (); Nis_rev (); Laflor_rev (). Universal behaviour of entanglement can be used to deduce information about the conformal field theory underlying quantum phase transitions HLW (); VLRK (); CC (); CCcft (), while studying the entanglement spectrum leads to deeper insights of topological properties KP (); LevinWen () of quantum Hall states LH (). In non-equilibrium situations entanglement and entropy production are deeply connected CC2 (); KTLRSPG (); AC (), and are also diagnostic of non-perturbative effects such as confinement KCTP () and thresholds in the quasi-particle spectrum DynGibbs1 (); DynGibbs2 (). Black hole entropy was identified with the entanglement of modes across the horizon BKLS (); Srednicki (); DasBH (), and entanglement entropy was also connected to gravitational space-time geometry in a holographic context RT (); NRT ().
Most of the above developments concern entanglement between spatially separated subsystems. In contrast, the present work proposes a characterisation of the Renormalisation Group (RG) flow in 1+1 dimensional quantum field theories through a quantification of the entanglement between left and right moving excitations, which are decoupled in the ultraviolet limit and become entangled during the flow to the infrared.
Non-trivial fixed points of RG RG () trajectories in QFT are characterised by scale invariance and consequently a gapless spectrum in infinite volume. In relativistic quantum field theories, scale invariance is generally promoted to conformal symmetry Polyakov (); Polchinski (), which is extended to an infinite dimensional symmetry in the 1+1 dimensional case BPZ (). 1+1 dimensional Conformal Field Theories (CFTs) obey holomorphic (chiral) factorisation: the excitation spectrum consists of left and right movers which transform under a separate chiral symmetry algebra and do not interact. When quantised on a cylinder of circumference , the eigenstates of the CFT Hamiltonian span irreducible representations of the tensor product of two identical Virasoro algebras, one for each chirality BPZ (). The Hilbert space of the CFT is further restricted by physical requirements such as modular invariance on the torus Cardy_mod (). In this paper we only consider the simplest case when only left and right movers coming from the same representation are paired in the tensor product (the so-called diagonal or more precisely -type modular invariant theories CIZ ()).
Coupling the CFT to a relevant scalar field with coupling constant according to the formal Euclidean action
results in breaking conformal invariance. Here we consider the case when this leads to a finite mass gap , related to the coupling as
where is the scaling dimension of the field at the RG fixed point associated to the CFT. The action (1) describes an RG flow from an ultraviolet (UV) massless fixed point for to a (trivial) infrared (IR) fixed point when . In general, the theory is regularised using a high energy (equivalently short distance) cut-off which is sent to infinity to obtain physical predictions.
Properties of RG flows in 1+1 dimensions have been extensively studied with sophisticated analytical and numerical techniques, inspired by the seminal contributions Zam_RG (); ZamPotts (); YurZam (), with most of the analysis focused on the evolution of the QFT spectrum. Real space entanglement was also computed for integrable flows using exact form factors of branch point twist fields FFentanglement (); FFentanglement_review (); CastroA_E8 (); FFentanglement_new () and was studied numerically in Palmai ().
However, universal information in the IR limit can also be obtained determining overlaps between eigenstates of the QFT Hamiltonian associated to (1) and states of the UV conformal basis GaiottoDW (); Kon (). More explicitly, assume that is the ground state of the theory (1) with a finite cut-off, and define the left/right chiral Hilbert spaces
as a direct sum of inequivalent irreducible representations of the symmetry chiral algebra. Then the Hilbert space of the CFT corresponding to a diagonal modular invariant is the diagonal subspace
of . The finite volume ground state of the perturbed conformal field theory can be fully characterised by the overlaps
where and enumerate two orthonormal basis of and . Note that the overlaps (5) vanish when the left and right chiral vectors transform under different irreducible representations of the chiral algebra.
Based on the observation that can be considered as a density matrix on the chiral space of states, the right chiral reduced density matrix can be constructed tracing out the left chiral sector with matrix elements
The spectrum of the right reduced density matrix (or equivalently the left ) quantifies the entanglement of the UV chiral degrees of freedom along the RG flow. We refer to the corresponding von Neumann entropy
as the chiral entanglement entropy.
The chiral entanglement entropy vanishes at the UV fixed point and remains finite along the RG flow; i.e. the limit in Eq. (5) always exists. We will demonstrate that for the chiral entanglement entropy grows linearly with with a non-universal (i.e. mass dependent) slope
Moreover, we conjecture that the sub-leading term , which is in the system size, is universal and provides an independent characterisation of the ground state of the (infinite volume) massive theory, which is especially useful when it is degenerate. Such a term first appeared in the analysis of chiral entanglement of regularised conformal boundary states in KL (); DubailReadRezayi () (see also LREEfreebos (); DD (); LREEDp ()), and was suggested as a possible benchmark for topological states in two spatial dimensions following KP (); LevinWen (). These claims were tested numerically using matrix product states in MPS1 (); MPS2 ().
The idea of KP (); LevinWen () regarding the universality of the term can be extended to the ground state of massive perturbations of conformal field theories based on a variational Ansatz in finite volume proposed recently by J. Cardy Cardy17 (), which describes the exact ground state
in terms of smeared conformal boundary states.
Here we use this Ansatz to determine analytically both the reduced chiral density matrix and its chiral entanglement entropy. In cases where the variational Ansatz predicts degenerate ground states, we resolved the degeneracies utilising the conformal OPEs. As argued in Kon () and proved there for free Ising Field Theory, a representation of the ground state of the massive QFT as a conformal boundary state should be exact in the IR limit . Based on this assumption, of which we provide numerical tests in the rest of the paper, Cardy’s variational Ansatz should be exact (apart from a divergent normalisation) in infinite volume. This implies that the term of the chiral entanglement entropy obtained from such an Ansatz in the limit is exact and indeed a universal feature of the RG flow. In addition, the Ansatz and the results derived for the chiral entanglement entropy do not depend on any special properties of the flow such as integrability at all. To provide numerical support and verification of our claims, we use the Truncated Conformal Space Approach (TCSA) introduced in YurZam () (cf. JKLRT () for a recent review). First we validate the variational Ansatz, using the scaling Ising Field Theory (IFT) and Tricritical Ising Field Theory (TIFT) as examples, and then analyse the overlaps with the conformal states and the large volume behaviour of the chiral entanglement entropy.
There are some important differences between the properties of the chiral and real space entanglements. Strictly speaking, the real space ground state entanglement entropy of 1+1 dimensional QFTs does not contain any universal term. Indeed, the leading term is logarithmically divergent HLW () in the limit and requires a regularisation, e.g. through introduction of a relative entropy CH04 (); Cas08 (); L14 (); Wrev (). For a spatial interval of length embedded into an infinite system, the coefficient of the leading cut-off dependence of real space entanglement entropy is universal for both and , and related to the central charge of the UV theory CC (); FFentanglement (). This feature has found important applications in statistical mechanics models such as spin systems VLRK (); JK () where the lattice spacing provides a natural regularisation. However, since the leading cut-off dependence is logarithmic, there cannot be any universal term in contrast to defined from the chiral entanglement entropy in Eq. (8).
The real space entanglement spectrum was also analysed recently in lr (). For a bipartition in two semi-infinite halves and in the limit of a large mass gap, the entanglement Hamiltonian BW () is the generator of translation around an annulus with certain conformal boundary conditions Cardy89 (). Similarly to the entanglement entropy, the entanglement spectrum also exhibits universal logarithmic short-distance cut-off dependence, but it is not finite in the limit . In contrast, the chiral entanglement spectrum is finite for ; we return to comment on this issue in Section 3.
We finally note that the chiral entanglement entropy can be calculated from TCSA in a rather simple and straightforward way, in marked contrast to the real space entanglement entropy Palmai ().
The outline of the paper is as follows. In Section 2 we review and slightly extend Cardy’s variational Ansatz for the QFT ground state in finite volume. In Section 3 the variational ground state is used to compute the chiral entanglement entropy and in particular the universal term characterising the RG flow. In Section 4 we discuss applications to Ising field theory and the Tricritical Ising field theory. In Section 5 all our claims contained in the previous sections are subjected to numerical tests based on TCSA. Our conclusions are presented in Section 6. In order to keep the main line of the argument focused, technical details regarding the implementation of numerical computations were relegated to two Appendices.
2 Variational Ansatz for the ground state of the perturbed CFT
The QFT Hamiltonian on the cylinder associated to the Euclidean action in Eq. (1) is
where the relevant scalar field has conformal dimension at the UV fixed point (see also Eq. (64)). The IR regime is reached for , which is equivalent to either increasing the volume or the coupling constant . The UV fixed point is a CFT with central charge and a torus partition function which is assumed to correspond to a diagonal modular invariant. In finite volume any eigenstate of the QFT Hamiltonian can be expanded in the eigenstates of which form the so-called conformal basis (cf. Appendix A).
2.1 The Ansatz for the non-degenerate case
If we now imagine to turn on the relevant perturbation only for negative imaginary times , in the limit , the perturbation flows in the far infrared to a conformal invariant boundary condition for the theory in the upper half CCQuench (); Kon (); DubailReadRezayi (). In the crossed channel, the conformal boundary condition is represented by a boundary state, while evolution in imaginary time asymptotically projects any state of the QFT onto the ground state of the Hamiltonian. Therefore for a large but finite the ground state of the massive QFT is expected to correspond to a deformation of the asymptotic conformal boundary state by boundary irrelevant operators DubailReadRezayi (). The simplest such operator is given by the stress tensor , and Cardy suggested in Cardy17 () that it should be possible to approximate the exact ground state of (10) by a suitable linear combination of smeared conformal boundary states
where the are conformal boundary states (defined in detail later) and is a smearing parameter proportional to the inverse of the mass gap. The normalisation of the smeared boundary states is given by
which is the conformal partition function on the annulus of Figure 1 with boundary conditions on both sides. In general we will denote by the same conformal partition function with boundary condition and .
The coefficients of the linear combination in Eq. (11) and the smearing parameter are determined by minimising the energy density in infinite volume
For , the matrix elements of are diagonal on the smeared boundary states up to exponentially small corrections Cardy17 (). In particular it turns out that
where the functions are given by
Eq. (14) is minimized keeping in Eq. (11) only the boundary state such that has a global minimum at which is smaller than all the other minima (if any) of the functions in Eq. (15). As discussed in Cardy17 (), in the case of a single relevant perturbation with the sign of positive (negative), the variational Ansatz selects such that is minimum (maximum). If there is a unique solution of the variational equations, the best approximation of the QFT ground state is then obtained normalising the unique smeared boundary state selected by the variational principle, i.e.
We return to the case of degenerate solutions at the end of the this section.
For perturbations with several relevant fields with couplings and one must find the minimum of
however we will not consider this case here.
For an explicit evaluation of the variational functions (15), we recall some basic properties of the conformal boundary states in CFTs with diagonal modular invariant partition functions. They can be expressed as a linear combination of Ishibashi states
where the sum runs over all allowed irreducible representations of the Virasoro algebra. The Ishibashi states themselves are in one-to-one correspondence with irreducible representations and have the form
where the vectors form an orthonormal basis of the irreducible Virasoro representation of highest weight , with denoting their descendant level and a further index enumerating linearly independent vectors at a fixed level. The character of such a representation is given by
For a diagonal CFT, the physical conformal boundary states (also called Cardy states) satisfy further constraints Cardy89 () and are in one-to-one correspondence with the scalar primary fields generating the operator content of the theory. Introducing the chiral weight of the primary field as , the corresponding conformal boundary state is denoted as . For a Cardy state the coefficients of the linear combination in Eq. (19) are
where is the orthogonal symmetric modular matrix Verlinde (), which describes the linear transformation of Virasoro characters
under the modular transformation
which can be inserted into the variational functions (15) to obtain them explicitly in terms of the modular matrix.
2.2 The degenerate case
So far it was assumed that one of the variational functions (15) had a minimum lower than all the others. Suppose that, on the contrary, the minimisation of Eq. (15) leads to two different Cardy states and with such that and have the the same minimum at , which can happen e.g. when the two functions and are identical.
To resolve this problem, note that while the functional (14) is obtained for , the degeneracy between candidate smeared conformal boundary states is generally lifted in finite volume. The way the degeneracy is lifted is constrained by the block diagonal structure of the QFT Hamiltonian dictated by the OPE of the perturbing field with the primaries belonging to the CFT. The Hilbert space spanned by the eigenstates of in Eq. (10) (see also Eq. (68)) then splits into a direct sum
The vector spaces are direct sums of tensor products of left/right irreducible Virasoro representations as in Eq. (4). They are composed of modules corresponding to sets of scalar primaries which form a closed set under operator product with the the perturbing field in Eq. (1). The decomposition (26) follows from the explicit expression of the interaction matrix in Eq. (69) and reflects residual discrete symmetries of the perturbed CFT.
In all the cases considered in this paper the index can have at most two values. For the degenerate case we denote them as and the Hilbert space (resp. ) is identified to be the sector which contains (resp. does not contain) the highest weight representation associated to the identity field. Provided there is no level crossing involving the ground state in finite volume (which is generally the case) the exact QFT ground state belongs to the subspace , whereas in general the degenerate variational states selected by minimizing Eq. (15) do not. Consistency requires that the best approximation of the ground state must be the smeared boundary state constructed from a linear combination of the degenerate Cardy states belonging to . It is also possible to construct a linear combination of the degenerate Cardy states belonging to , which gives the best approximation of a state with an energy gap that decays exponentially with the volume of the system.
Finally we note that the linear combinations of smeared Cardy states obtained from the argument above should be reproduced directly analysing finite size effects in the QFT Hamiltonian restricted to the degenerate subspace spanned by the variational states. If and are the degenerate Cardy states, one must examine the matrix
with given in (10) and . The true ground state corresponds to the eigenvector of (27) with the lower eigenvalue in the limit and belongs to , while the other eigenstate of (27) is in . Constructing the above matrix is quite involved due to the matrix elements of the perturbing field which also require the bulk-boundary and boundary OPE structure constants calculated in Lew (); Run (). However, one can consider a simplified version of this computation by neglecting the perturbing field and replacing by , resulting in the matrix
which has the leading large volume behaviour
where is the minimum (chiral) conformal weight of the fields that occur in the OPE . Note that while for , i.e. the off-diagonal elements are exponentially suppressed with the volume Cardy17 (), and therefore the same is true for the level splitting generated by them.
We checked by explicit computation that in all cases considered in this work (29) gives the same eigenvectors lying in the subspaces as the previous consideration, with the lower level always lying in the subspace . In fact, this result is easy to understand: replacing by corresponds to setting , but once the appropriate eigenstates are determined to lie in the subspaces , switching on the perturbation cannot mix them. In addition, it cannot change their ordering in energy either due to the absence of level crossings along the RG flow generated by the perturbation.
3 Chiral entanglement spectrum and entropy
The Ansatz (11) immediately implies an analytic expression for the chiral reduced density matrix and the chiral entanglement entropy introduced in Section 1. To include the possibility of degenerate solutions and linear combinations of physical boundary states, we consider a variational state
which is a linear combination of Ishibashi states as in Eq. (19), but with a priori arbitrary coefficients (i.e. not necessarily a conformal boundary state). The normalization is fixed by and is a suitable linear combination of Virasoro characters, cf. Eq. (21). The overlaps of the variational state with the conformal basis are then
and we used Eq. (64) for the conformal Hamiltonian. For a scalar perturbation the actual ground state of the QFT Hamiltonian (10) has zero conformal spin and therefore can be expanded in the conformal basis as
It is then clear from Eq. (31) that the variational state can only provide a valid approximation of the QFT ground state if off-diagonal elements of the matrices are small and diagonal ones satisfy Eq. (31). This is verified explicitly in Section 5 using the TCSA.
The chiral reduced density matrix for the variational state defined in Eq. (6) is then diagonal on the orthonormal basis :
The modular property (23) of the conformal characters allows an analytic evaluation of physical quantities in the limit as we now demonstrate for the chiral entanglement entropy. It is convenient to start with the -th chiral Rényi entropy for the variational state defined as
The limit can be determined applying the modular transformation (23), and retaining only the leading order contribution which consists of only that of the conformal vacuum state:
The chiral entanglement entropy of the variational ground state (17) is then obtained in the limit :
which shows an extensive dependence on the volume as anticipated in Eq. (8). The term in the large volume expansion is independent on the smearing parameter and reads
Since the variational Ansatz (17) is supposed to be exact in the limit , we conjecture that in Eq. (38) is a universal number characterising the ground state of a perturbed CFT in infinite volume. In Section 5 TCSA is used to verify numerically the validity of Eqs. (37, 38) which were derived assuming the variational Ansatz.
We close this section with a few comments and remarks. First, analogous expressions for the density matrix (33) and the universal term in Eq. (38) contained in this section already appeared in KL (); DubailReadRezayi () (see also DD ()), in the context of topological states of matter KP (); LevinWen (). In KL (); DubailReadRezayi (), Eq. (33) was in particular conjectured to explain the conformal nature of the entanglement spectrum LH () in a d Quantum Hall fluid. Here we derived these results in the context of d massive QFT RG flows, as direct consequences of the variational Ansatz (17). Secondly, it is important that in our context the validity of these results is numerically testable with TCSA.
Finally, notice that is obtained in the opposite limit compared to the better-known Affleck–Ludwig boundary entropy AL (). This observation allows establishing a partial analogy with recent studies lr () concerning the real space entanglement spectrum in the ground state of a perturbed CFT. In lr () the entanglement Hamiltonian is the generator of translation around the annulus (i.e. in the periodic direction) in Figure 1 with aspect ratio
where is the inverse of the mass gap and is an UV cut-off. The boundary conditions on the annulus are left free on one side and are fixed by a conformal boundary state on the other, although the boundary states are not explicitly determined in lr (). The entanglement spectrum in real space displays universal features in the limit which corresponds to , and the term in the corresponding entanglement entropy is the Affleck–Ludwig boundary entropy, however due to the logarithmic dependence on the cut-off this result is not generally universal CC ().
In contrast, the chiral entanglement Hamiltonian following from (33) is the chiral part of the generator of translation along the annulus (i.e. in the open direction), with the same (linear combination of) conformal boundary states on both sides. As a result, the chiral entanglement spectrum displays universal features in the opposite limit , while the term in the entropy is given by Eq. (38) and is universal. Note that albeit the two settings are related by a modular transformation corresponding to crossing between the open and closed channels, the relevant boundary conditions are generally different.
We now present applications of the formalism we discussed in the previous two sections. In the examples below the UV fixed point of the RG flow (1) is a Virasoro minimal model BPZ () with diagonal partition function on the torus. The central charge of the minimal model is
while the primary fields have scaling dimensions with
for and . Taking into account the symmetry , there are exactly primary fields. The CFT describes the th Ising multicritical point, with discrete symmetry Zam_LG (), and explicit expressions for the modular matrices can be found for instance in Difra ():
4.1 Ising Field Theory
The simplest example is the perturbed Ising CFT with the fixed point . The non-trivial relevant fields that could be considered as perturbations are
odd sector: spin field (),
even sector: energy field ()
The coupling to the spin field is denoted by and the one to the energy field by , where one can always choose since can be obtained by symmetry . As noticed in Cardy17 (), for the Ansatz does not reproduce the logarithmic singularity of the ground state energy correctly. However this contribution is a universal shift of all the energy eigenvalues and does not affect the state vectors themselves (cf. also Section 5). The Cardy boundary states are given in terms of Ishibashi states as Cardy89 ()
and have a simple interpretation in terms of spins in the lattice model Cardy89 (): the first two correspond to the boundary spins fixed in up/down position, while the third one describes the free boundary condition. The variational analysis leads to the following results Cardy17 ():
For and , the ground state is best approximated by , for which . In the infinite volume limit the variational state coincides with the ground state in the Neveu–Schwarz sector of a free Majorana fermion Kon ().
The case and corresponds to the spontaneously broken phase and the variational Ansatz is degenerate: the two solutions are given by and . Their degeneracy is lifted in a finite volume and the corresponding eigenvectors can be identified following the method discussed in Subsection 2.2. The OPEs of the three conformal modules with the perturbing field are BPZ ()
Therefore the Hilbert space splits into the -even (Neveu–Schwarz) sector built upon the primaries and , and the -odd (Ramond) sector built upon the primary .
The finite volume ground state is the unique linear combination of the Cardy states and which belongs to :
for which the term of the chiral entanglement entropy vanishes:
The antisymmetric linear combination
belongs to and corresponds to the first excited state which has a level spacing from the ground state which decays exponentially with the volume . The smeared boundary state obtained form Eq. (50) has a different universal constant in the chiral entanglement entropy:
The same conclusions could be reached using the Kramers-Wannier duality WuPotts () which maps into and transforms the state into Eq. (48), thereby mapping the variational solution for into the result for .
Note that the two degenerate ground states in the ferromagnetic phase of the free Ising field theory are distinguished by a different term in the chiral entanglement entropy, in contrast to their real space entanglement.
4.2 Tricritical Ising Field Theory
odd sector: (), ()
even sector: (), (), ()
The couplings to the odd fields and are denoted by and , while couplings to the even fields and are denoted by and , respectively (note that the field is irrelevant). The microscopic interpretation of the different physical boundary states was given in Affleck_tri () in the context of the dilute Ising model.
For the sake of brevity we only consider a perturbations by a single field; the generalization to more than one perturbing fields is straightforward. The variational analysis leads to the following results (by convention all couplings not explicitly specified are assumed to be zero):
: the variational solution is obtained by selecting in Eq. (17) which corresponds to a boundary condition with all spins fixed.
: the unique solution is , corresponding to a symmetric ground state.
: analogously to the Ising case there are two degenerate solutions of (14) corresponding to the Cardy states and . The two sectors correspond again to even/odd primaries. In a finite volume the true ground state is in the even sector and is given by the smeared boundary state (17) obtained from
where is the golden ratio and a numerical coefficient. The lowest energy state in the odd sector is given by
where is another numerical coefficient. The two states are degenerate in the thermodynamic limit and their splitting decays exponentially with the volume.
Note that the even resp. odd combinations only contain fields in the Neveu–Schwarz resp. Ramond sectors of the super-Virasoro algebra of the minimal model DFQ (), respectively (indeed this explains our choice of labelling for them). The difference of the chiral entanglement entropies is given by
which coincides with the analogous expression in the Ising case.
: again there are two degenerate solutions to the variational equations corresponding to the Cardy states and with associated primaries and . The way the degeneracy is lifted in a finite volume can be obtained by examining the relevant OPEs Difra ():
from which we can infer that sector contains representations built from the primaries , while consists of . The unique linear combination of degenerate Cardy states belonging to is
while the one giving the lowest energy state in is
or in terms of Ishibashi states as
Moreover, using Eq. (38), the difference between their chiral entanglement entropy is given by
Comparing to Eq. (43) one can note that and are effectively two Ising-type boundary states; in particular, the fields realize an OPE isomorphic to that of the Ising model. This observation confirms that the tricritical Ising model perturbed by describes a phase separation between two magnetized Ising pure phases that are not distinguished by symmetry alone Cardy_book ().
From the QFT point of view this conclusion is quite remarkable since it implies a double degeneracy of the ground state of different nature respect to the most familiar one observed in Ising field theory, which is potentially interesting to the study of topological phases of matter FTLTKWF () (cf. also Section 6).
So far we only gave the difference of the chiral entanglement entropy between degenerate ground states where applicable. To complete this information, the values of the universal contributions for the true ground states themselves are summarised in Table 1.
|Perturbation||Coupling||for the true ground state|
5 Numerical Results from TCSA
Now we turn to a numerical computation of the chiral entanglement using the Truncated Conformal Space Approach (TCSA), which is a variational method originally introduced to construct the approximate low energy spectrum of field theories in finite volume in YurZam (). Since then it was applied for various models and its scope extended to extracting a range of quantities as well as determining phase diagrams and simulating non-equilibrium time evolution; for a recent review we refer the interested reader to JKLRT (). The ingredients necessary for the present computations are briefly reviewed in Appendices A and B.
The TCSA consists of considering the Hamiltonian (10) in finite volume with appropriate boundary conditions, which are chosen to be periodic for the subsequent calculations. The finite volume spectrum is discrete and introducing an upper energy cut-off reduces the Hilbert space to a finite-dimensional subspace, in which all states and operators are represented as finite-dimensional vectors and matrices, respectively. Units are chosen by expressing the coupling in terms of the mass gap (i.e. the mass of the lightest excitation), and the volume is parameterized by the dimensionless combination , while energies are measured in units of . The determination of the spectrum is reduced to the diagonalization of the finite truncated Hamiltonian matrix, with the components of eigenstates numerically known in the conformal basis. As a result, the reduced density matrix and the chiral entanglement entropy can be computed in a straightforward way. The results obtained from TCSA depend on the cut-off: the more relevant the perturbing operator the faster the convergence with the cut-off. The convergence can be improved further by eliminating the leading cut-off dependence using a renormalization group approach, explained in Appendix B for the case of the chiral entanglement entropy.
5.1 Verifying the Ansatz: energy densities and overlaps
Before considering the chiral entanglement entropy, we first test Cardy’s variational Ansatz. The simplest test is to compare the predicted ground state energy density obtained by substituting the variational solution into (14) to exact results coming from integrability Fat94 (), or from the one determined numerically from TCSA when the perturbation is non-integrable. The results of this comparison are summarized in Table 2. Note that for the thermal perturbation of the Ising model the energy density predicted by the Ansatz is not meaningful since there is a logarithmic divergence with the UV cut-off. The results demonstrate that the Ansatz predicts the energy density for strongly relevant perturbations quite precisely, however its accuracy gets progressively worse when the weight of the perturbation increases.
One can also compare the overlaps calculated from TCSA to the prediction from (31) using the coefficients obtained by the variational method. The results are demonstrated in Figures 2, 3, 4 and 5. The exponential fall-off of the coefficients with the conformal level is clearly visible and agrees with the predictions of the Ansatz. However, it is also clear that there are some discrepancies since the boundary states only have diagonal components in an orthonormal basis according to the structure of Ishibashi states (19), which should be exactly the same for components with a given highest weight and level . The TCSA shows that this is violated: there are non-zero non-diagonal components, and the diagonal ones have slightly different values as well, in some cases they clearly deviate from the predicted exponential curve. These deviations indicate that the Ansatz is not perfect and is only strictly valid in the limit , and for a more precise description in finite volume it is necessary to include more irrelevant operators around the boundary fixed point as pointed out in Cardy17 (). However as we show in Appendix B (cf. Figure 11) both the (relative) difference between the diagonal coefficients and the non-diagonal coefficients converge to zero with increasing volume.
The variational Ansatz (13) and its extension in presence of degeneracy are quite poor for Ising field theory perturbed by the energy operator; see Figure 3. This is not surprising since it was already pointed out in Cardy17 () that the Ansatz cannot account for the logarithmic contribution to the ground state energy either. Exact calcuation performed in Kon () shows that the ratio of overlaps between the true QFT ground state and two states in the CFT basis can converge to the IR value (obtained for ) with a power law behaviour, rather than exponentially as predicted instead by (31). It is worth noticing, however, that the agreement obtained for the chiral entanglement entropy remains remarkable; see Figure 7.
5.2 Chiral entanglement entropy
Now we turn to the predictions for the chiral entanglement entropy, which for large enough volume can be written in the form (8):
The prediction therefore consists of two quantities: the linear slope , which according to (37) can be obtained as
5.2.1 Ising Field Theory
In the case of the Ising field theory we studied numerically three different perturbations of the fixed point CFT.
(magnetic perturbation): as already demonstrated in Figure 2 the variational state resulting from Cardy’s Ansatz provides an excellent approximation to the ground state. The chiral entanglement entropy as a function of the volume is showed in Figure 6, and it is clear that the predictions are in excellent agreement with the extrapolated TCSA value.
(thermal perturbation): the positive/negative signs of the coupling correspond to the paramagnetic/ferromagnetic phases, for which the chiral entanglement entropy as a function of the volume is shown in Figure 7. In the paramagnetic phase () the unique ground state is in the Neveu–Schwarz sector, while in the ferromagnetic case there are two degenerate ground states in large but finite volume, namely in the Neveu–Schwarz and in the Ramond sector. In all cases the predictions from Cardy’s Ansatz combined with eq. (37) are in agreement with the extrapolated TCSA data. In Table 3 we present the difference between the chiral entanglement in the two ground states in different volumes, which shows that it converges to the predicted value with increasing volume.
5.2.2 Tricritical Ising Field Theory
perturbation: Cardy’s Ansatz predicts with the (cf. Table 2). The predicted chiral entanglement is compared to the TCSA reuslt in Figure 9 which shows an excellent agreement. The universal constants and the slope obtained from a linear fit of the TCSA data agrees with the theoretical prediction up to four and three digits respectively as shown in Table 4.
perturbation: for there is a unique ground state corresponding to , while for the two degenerate ground states are and given in (52) and (53). The theoretical prediction and the TCSA results for the chiral entanglement entropy are shown in Figure 8, and the extrapolated data are in good agreement with the predictions for each case.
perturbation: for there are two ground states (58) and (59) which are degenerate in infinite volume. The theoretical prediction and the TCSA results for the chiral entanglement entropy are compared in Figure10. For there are again two ground states, but their expressions differ from the result (58,59) by flipping sign of the / terms, respectively, as discussed in Section 4. This sign change has no effect neither on the slope nor the intercept , the prediction is the same for both signs, while the overlaps obtained from TCSA are also the same up to a sign flip that has no effect on the chiral entanglement entropy.
5.2.3 Detailed numerical comparison
Beyond the graphical comparison presented so far, we also extracted the values of and by fitting the volume dependence of the chiral entanglement entropy obtained from TCSA with a linear function. For any quantity determined from TCSA there is a so-called scaling regime where finite size effects (which decrease with the volume) are of the same order of magnitude as truncation effects (which increase with the volume). For the entanglement entropy, since the predicted behaviour in large volume is linear, the scaling regime can be found by examining the numerically computed second derivative and choosing an interval in where it is sufficiently small. This cannot be done completely automatically, as sometimes a blind search would find the minimum of the second derivative in a range that is clearly at too small volume still dominated by finite size effects, and can be aided by the graphical comparison between prediction and TCSA data discussed above. The detailed numerical matching of the predicted to the results obtained from TCSA is presented in Table 4.
|Perturbed CFT with||theory||TCSA||theory||TCSA||TCSA fit|
|ground state||range in|
In this paper we analysed entanglement production among UV chiral degrees of freedom along a massive renormalisation group flow in dimensions for theories with invariant fixed points, belonging to the -series of Minimal Models and perturbed by a relevant scalar field. We studied the reduced density matrix, obtained from the finite volume ground state projector after tracing out one chiral sector of the Hilbert space, and showed that contrary to real space entanglement, chiral entanglement is finite when the short-distance cut-off is sent to zero. Moreover, in the large volume limit, the chiral entanglement entropy grows linearly with the system size and contains a subleading constant term , which was argued to be a universal property of the RG flow and can characterise uniquely the QFT ground state also in presence of degeneracies. We considered examples in Ising Field Theory and Tricritical Ising Field Theory. In the latter theory, we also pointed out the existence of a two-fold ground state degeneracy in presence of a odd perturbation of weight . Interestingly, in such a case, the QFT Hilbert space splits into the same two sectors discussed in FTLTKWF () when describing lattice realisations of Fibonacci anyons.
For an analytic determination of chiral entanglement we employed a recently proposed variational Ansatz Cardy17 () for the QFT ground state in terms of smeared conformal boundary states. We tested the variational Ansatz extensively with TCSA, and our detailed numerical analysis of the energy and the overlaps suggests that the variational Ansatz for the QFT ground state is exact in the infinite volume limit. We then computed both the volume density and the constant term of the chiral entanglement from the Ansatz and demonstrated that it matches excellently the TCSA data, which also provides a further validation of Cardy’s Ansatz.
There are several interesting directions for future investigation. Among them, we could mention: the extension of the present investigation to non-diagonal theories, such as e.g. the invariant Potts Field Theory ZamPotts () or to massive deformations of free bosonic theories, such as e.g. the Sine-Gordon model. The behaviour of the chiral entanglement and its universal features along a massless RG flow Zam_massless_flow () is an important challenge as well; in this case a variational Ansatz for the ground state wave function is currently missing.
It would also be interesting to extract the coefficient in Eq. (8) (cf. also (38)) directly from lattice calculations, either analytically or numerically. This requires the construction of the conformal basis KS () on the lattice, and also the determination of the ground state overlaps. Both are challenging tasks; nevertheless, some progress has been made recently in MV (). We hope our results could trigger more activity in this direction.
There is also a number of potential applications to quantum quenches in field theory since Cardy’s Ansatz determines the ground state of a massive quantum field theory, and so it can be used as a starting point to study the time evolution. There has been considerable progress in both numerical 2016NuPhB.911..805R (); 2017PhLB..771..539H (); 2018ScPP….5…27H (); 2018PhRvL.121k0402K (); 2018arXiv180906789H () and analytical approaches 2012JSMTE..04..017S (); 2014JSMTE..10..035B (); 2017JSMTE..10.3106C (); 2014JPhA…47N2001D (); 2017JPhA…50h4004D (); 2016PhRvE..93f2101K (); 2016NuPhB.902..508H (); 2018JHEP…08..170H (). However, finding the overlaps of the initial state with the post-quench energy levels and characterising the time evolution especially for long times is still largely an open problem. Chiral entanglement also looks interesting as a potential tool to analyse and understand the temporal dynamics.
We are grateful to F. Essler for a useful discussion, and also to J. Dubail and H. Katsura for discussions and for drawing our attention to the papers DubailReadRezayi () and KL (). M.L. was supported by the Brazilian Ministry of Education. G.T. was supported by the National Research Development and Innovation Office (NKFIH) under a K-2016 grant no. 119204 by the Quantum Technology National Excellence Program (Project No. 2017-1.2.1-NKP-2017- 00001), and also by the BME-Nanotechnology FIKP grant of EMMI (BME FIKP-NAT).
Appendix A CFT and TCSA on a cylinder
We briefly recall some relevant notions regarding diagonal CFTs on a cylinder to set up our conventions. The Euclidean space-time cylinder of circumference is parameterised by the Euclidean time and the spatial coordinate with the identification , which can be mapped onto the conformal plane by the exponential function
The conformal Hamiltonian on the cylinder can be writen in terms of the left/right moving (/) Virasoro generators on the plane as
and the eigenvalues of are called the scaling dimensions. For a diagonal CFT the states in the conformal basis are of the form