Main

The entanglement structure of ground states of many-body systems plays a crucial role in our understanding of quantum phases of matter. It can be used to detect quantum phase transitions1,2, and much work has been invested in obtaining a deep understanding of the entanglement structure of gapped ground states dominated by the area law3. An important outcome of this work has been the introduction of tensor networks as both numerical and theoretical tools to understand such systems4,5,6. Gapped phases of matter exhibit rich entanglement properties, for example, encoded in their topological orders with the associated anyonic excitations and quantum error-correcting codes7,8,9,10.

The entanglement of ground states at quantum phase transitions shows a very distinct behaviour from the gapped phases. Critical many-body systems are expected to have a scaling limit described by a conformal field theory (CFT)11 (see refs. 12,13 for recent rigorous results in free-fermion systems). On large scales, one can hence expect similar behaviour as in conformal field theories, for which various local quantifiers of entanglement, such as the geometric entropy of single or multiple intervals, have been computed14,15,16,17,18,19,20. This has been confirmed in specific models (see, for example, refs. 2,21). In one spatial dimension, a hallmark feature of such critical ground states is that the entanglement entropy of a connected region grows logarithmically with the size of the region instead of the constant upper bound implied by the area law. Therefore, if we partition an infinite chain into two half-chains, they are necessarily infinitely entangled.

In this work, we explain how the infinite entanglement in critical free-fermionic many-body systems can be characterized using the information-theoretic task of embezzlement of entanglement introduced in ref. 22. The recent result that embezzlement provides an operational interpretation for different ways of being infinitely entangled23,24 prompts an application to many-body physics. In the task of embezzlement, two agents (say Alice and Bob) can each access one half of a shared entangled resource as well as a local subsystem. They are asked to produce, via local operations and without communication, an entangled state on their two local subsystems while perturbing the resource state as little as possible. The resource is called an embezzling state if this is possible for every entangled state (of arbitrary dimension) to arbitrary precision while perturbing the resource state arbitrarily little23,24. More precisely, a pure state \({| \varOmega \left.\right\rangle }_{AB}\) on a Hilbert space \({\mathcal{H}}\) modelling a bipartite system is embezzling if for every ε > 0 and any finite-dimensional entangled state \({| \varPsi \left.\right\rangle }_{AB}\in {{\mathbb{C}}}^{d}\otimes {{\mathbb{C}}}^{d}\) (for any d) there exist unitaries \({u}_{A{A}^{{\prime} }}\) and \({u}_{B{B}^{{\prime} }}\) of Alice and Bob, respectively, such that

$${u}_{A{A}^{{\prime} }}{u}_{B{B}^{{\prime} }}\left({| \varOmega \left.\right\rangle }_{AB}\otimes {| 0\left.\right\rangle }_{{A}^{{\prime} }}{| 0\left.\right\rangle }_{{B}^{{\prime} }}\right){\approx }_{\varepsilon }{| \varOmega \left.\right\rangle }_{AB}\otimes {| \varPsi \left.\right\rangle }_{{A}^{{\prime} }{B}^{{\prime} }},$$
(1)

where \({| 0\left.\right\rangle }_{{A}^{{\prime} }}{| 0\left.\right\rangle }_{{B}^{{\prime} }}\in {{\mathbb{C}}}^{d}\otimes {{\mathbb{C}}}^{d}\) is a product state.

A bipartite physical system on which every pure state is an embezzling state is called a universal embezzler. It is far from obvious that embezzling states, much less universal embezzlers, exist. Indeed, such systems clearly require infinite amounts of entanglement. Ref. 24 showed that universal embezzlers can be constructed mathematically and are deeply linked to the classification of so-called von Neumann algebras. The results imply that, unsurprisingly, a universal embezzler must have infinitely many degrees of freedom. A quantum system with infinitely many degrees of freedom is best described on an operator algebraic level, associating commuting von Neumann algebras to distinct subsystems and interpreting quantum states as functionals associating expectation values to observables from these algebras. Von Neumann algebras are classified into different types, which are in one-to-one correspondence with operational entanglement properties25. A bipartite quantum system is a universal embezzler if and only if the local observable algebras of the two parts of the system are so-called type III1 von Neumann algebras. Thus, type III1 von Neumann algebras can be operationally characterized via embezzlement of entanglement.

One may wonder whether universal embezzlers (respectively type III1 algebras) are purely mathematical objects or actually appear in physics. It has long been known that algebras of local observable in quantum field theory are generically of type III1 (refs. 26,27,28). Together with the general results from ref. 24, this led to the conclusion that relativistic quantum field theories provide examples of universal embezzlers23. While this transparently explains the maximum Bell inequality violations of the vacuum29, it currently appears to be necessary that both agents must control exceedingly large portions of their respective halves of spacetime to achieve acceptable errors for non-trivial target states, which renders this result rather impractical.

In this work, we show that critical, (non-interacting) fermionic many-particle systems provide an abundance of examples of universal embezzlers: on a one-dimensional lattice, any free-fermionic, translation-invariant, local model with a non-trivial Fermi surface is a universal embezzler in the ground-state sector.

Moreover, we show that the large-scale structure of entanglement required for universal embezzlers has strong implications for the finite-size ground states: For any specified finite error ε > 0 and any fixed dimension d, there is a sufficiently large system size N(ε, d) such that all entangled states of local dimension d may be embezzled up to error ε from the ground state of the system. The latter result is true beyond free-fermionic systems and, in fact, holds for all universally embezzling many-body systems with a unique ground state.

Thus, the relationship between embezzling states and embezzling families is roughly similar to that between the sharp (and idealized) notion of phase transitions in infinite systems and their approximate manifestation in large but finite systems. This shows that the study of the entanglement in the thermodynamic limit is also relevant for finite systems.

To obtain our results, we have to overcome several obstacles. First, we have to argue that the parity super-selection rule for fermions does not prohibit the use of the results of ref. 24 on embezzlement. This requires new operator algebraic results on the structure of fermionic quantum theory. In particular, we need to establish a property called Haag duality for the physically relevant even operators. Second, we need to show that the local von Neumann algebras in the ground state sectors are of type III1 for the full class of models that we consider. This result generalizes a prior result of Matsui, who showed the type III1 property for the XX spin chain30,31. To do so, we combine general operator algebraic results on quasi-free fermionic states by Powers and Størmer32 with the theory of Toeplitz operators. Toeplitz theory has previously been used to obtain precise calculations of the asymptotic behaviour of local entanglement quantifiers in spin chains and free-fermionic models33,34,35,36,37 and is closely connected to the classification of one-dimensional topological phases of quantum walks38,39. Third, we have to relate the entanglement structure in the thermodynamic limit to that in finite systems. In the following, we discuss these points in more detail with a focus on providing intuition and a high-level understanding. The precise mathematical arguments are given in Supplementary Sections A to D.

Results

We consider fermionic many-body Hamiltonians on the infinite one-dimensional lattice with b orbitals per lattice site in second quantization. The algebra of operators is generated by the creation and annihilation operators a(ξ), a†(η) with \(| \xi \left.\right\rangle ,| \eta \left.\right\rangle \in {\mathfrak{h}}:={\ell }^{2}({\mathbb{Z}})\otimes {{\mathbb{C}}}^{b}\) obeying the canonical anti-commutation relations

$$\{a(\xi\; ),{a}^{\dagger }(\eta )\}=\langle \xi | \eta \rangle ,\quad \{a(\xi\; ),a(\eta )\}=0.$$
(2)

We use the standard bases \(| x\left.\right\rangle \in {\ell }^{2}({\mathbb{Z}})\) and \(|\; j\left.\right\rangle \in {{\mathbb{C}}}^{b}\) to define the short-hand \({a}_{j}(x):=a(| x\left.\right\rangle \otimes |\; j\left.\right\rangle )\), that is, aj(x) = a(ξ) with \(| \xi \left.\right\rangle =| x\left.\right\rangle \otimes |\; j\left.\right\rangle \in {\mathfrak{h}}\). The class of Hamiltonians that we consider are second-quantized one-particle Hamiltonians h on \({\mathfrak{h}}\) of the form

$$\begin{array}{l}H=\sum _{x,y\in {\mathbb{Z}}}\mathop{\sum }\limits_{l,m=1}^{b}{h}_{lm}(x-y){a}_{l}^{\dagger }(x){a}_{m}(\;y),\\\qquad {h}_{lm}(x-y)=\left\langle \right.x| \otimes \left\langle \right.l| h|\; y\left.\right\rangle \otimes | m\left.\right\rangle ,\end{array}$$
(3)

where we already made use of translation invariance. We assume that the Fourier transforms \({\widehat{h}}_{lm}(k)\) for k ∈ S1 ≅ [0, 2π) exist. Then, the hermitian matrix \(\widehat{h(k)}\) can be diagonalized for every k ∈ S1. We denote the projection onto the strictly positive eigenvalues of \(\widehat{h(k)}\) by \({\widehat{p}}_{+}(k)\) and require that h is sufficiently local in space so that \({\widehat{p}}_{+}\) is a piecewise continuous function with at most finitely many discontinuities. This is guaranteed if the Hamiltonian is short-ranged but also allows long-range couplings. We say that the Hamiltonian H is critical if \({\widehat{p}}_{+}\) has at least one discontinuity. In the case of a short-ranged Hamiltonian, we can think of energy bands described by eigenvalue functions ϵj that are analytic on S1 up to (at most) a single point of discontinuity and the same is true for the associated spectral projections40. It is then sufficient that at least one of the energy bands ϵj is negative on a non-trivial interval I ⊂ S1 and positive on a non-trivial interval in S1\I (see Fig. 1 for an illustration). We emphasize that being critical in our sense is a stronger condition than merely being gapless: if an energy band has isolated points at which it vanishes, it is not critical, but the Hamiltonian does not have a gap above the ground state. An example is provided in the Supplementary Information. The simple hopping Hamiltonian gives a paradigmatic example for the given class of critical Hamiltonians

$${H}_{{\rm{XX}}}=\sum _{x}\left({a}^{\dagger }(x)a(x+1)+{a}^{\dagger }(x+1)a(x)\right).$$
(4)

A different example is the Su–Schrieffer–Heeger model41 at the topological phase-transition point, given by

$$\begin{array}{l}{H}_{{\rm{SSH}}}=\sum _{x}\left({a}_{1}^{\dagger }(x){a}_{2}(x)+{a}_{2}{(x)}^{\dagger }{a}_{1}(x)\right.\\\qquad\;\,\left.+{a}_{1}^{\dagger }(x){a}_{2}(x+1)+{a}_{2}{(x+1)}^{\dagger }{a}_{1}(x)\right),\end{array}$$
(5)

which features a topologically non-trivial band structure (Supplementary Section C). The topology of the band structure is not important for our results.

Fig. 1: Energy bands.
figure 1

We consider translation-invariant free-fermionic models on a one-dimensional lattice, where the energy bands εj(k) are piecewise continuous functions of the wave number k and at least one of the energy bands is critical, that is, exhibits a non-trivial Fermi surface, here illustrated in solid green. The energy bands depicted in dashed magenta do not count as critical in our sense because the subsets of wave numbers k for which they are not strictly positive or not strictly negative, respectively, consist of only finitely many points: the bands do not cross the Fermi energy at 0.

Full size image

In systems with infinitely many degrees of freedom, states are most usefully defined as expectation value functionals on the algebra of local observables describing the system. If \({{\mathfrak{h}}}_{0}\) is the kernel of h, and p+ is the spectral projection onto the strictly positive part of h, the pure ground states of H are parametrized by projections p0 onto subspaces of \({{\mathfrak{h}}}_{0}\). They are given by the quasi-free (fermionic Gaussian) states ωp on the algebra of canonical anti-commutation relations \(\,\text{CAR}\,({\mathfrak{h}})\) on \({\mathfrak{h}}\) determined by

$${\omega }_{p}\left({a}_{l}(x){a}_{m}{(\;y)}^{\dagger }\right)=\left\langle \right.x| \otimes \left\langle \right.l| p|\; y\left.\right\rangle \otimes | m\left.\right\rangle ,\quad p={p}_{+}+{p}_{0},$$
(6)

and Wick’s theorem for higher-order correlators. In the following, we set p0 = 0, but our results also hold for a wide range of choices of p0 (if \({{\mathfrak{h}}}_{0}\) is non-trivial) (Supplementary Section C).

A bipartition of the system is defined by a projection q onto a subspace \(q{\mathfrak{h}}\): the observables generated by a(ξ), a†(η) with \(| \xi \left.\right\rangle ,| \eta \left.\right\rangle \in q{\mathfrak{h}}\) belong to Alice, and those generated by \(| \xi \left.\right\rangle ,| \eta \left.\right\rangle \in {q}^{\perp }{\mathfrak{h}}=(1-q){\mathfrak{h}}\) belong to Bob. We consider a bipartition of the spatial lattice into two half-infinite chains so that q is the projection onto \({\ell }^{2}({{\mathbb{Z}}}_{+})\otimes {{\mathbb{C}}}^{b}\). The quantum state associated with Alice is simply the restriction of ωp to the corresponding operators, which we can identify with ωqpq.

Physical observables of a fermionic system are necessarily even operators, that is, invariant under parity transformations, as dictated by the parity super-selection rule42. We write \({\text{CAR}}_{e}({\mathfrak{h}})\) for the algebra of even operators generated by the creation and annihilation operators with vectors from \({\mathfrak{h}}\). The even operators \({\text{CAR}}_{e}({\mathfrak{h}})\) operate on the even part \({{\mathcal{H}}}_{e}\) of the full Fock-space \({\mathcal{H}}\) associated with ωp, which is represented by a vector \(| \Omega \left.\right\rangle \in {{\mathcal{H}}}_{e}\). The local observable algebras \({\text{CAR}}_{e}(q{\mathfrak{h}})\) and \({\text{CAR}}_{e}({q}^{\perp }{\mathfrak{h}})\) act on \({{\mathcal{H}}}_{e}\). Their closures in the weak topology are the von Neumann algebras \({{\mathcal{M}}}_{A}\) and \({{\mathcal{M}}}_{B}\), belonging to Alice and Bob, respectively. The weak topology makes the expectation value functionals with respect to vectors on H continuous, as physically expected. The triple \(\left({{\mathcal{M}}}_{A},{{\mathcal{M}}}_{B},{{\mathcal{H}}}_{e}\right)\) defines a bipartite system in the sense of ref. 24.

Recall from the introduction that ref. 24 showed that the bipartite system \(({{\mathcal{M}}}_{A},{{\mathcal{M}}}_{B},{{\mathcal{H}}}_{e})\) is a universal embezzler if \({{\mathcal{M}}}_{A}\) and \({{\mathcal{M}}}_{B}\) are type III1 von Neumann algebras. In fact the results of ref. 24 require two further technical conditions: (1) that \({{\mathcal{M}}}_{A}\) and \({{\mathcal{M}}}_{B}\) together generate all bounded operators on \({{\mathcal{H}}}_{e}\) and (2) that the two operator algebras exhaust all quantum degrees of freedom, which is captured by a condition called Haag duality: it says that the commutant \({{\mathcal{M}}}_{A}^{{\prime} }:=\{x\in {\mathcal{B}}({{\mathcal{H}}}_{e}):[x,y]=0\,\forall y\in {{\mathcal{M}}}_{A}\}\) is precisely given by \({{\mathcal{M}}}_{B}\). If \({\mathfrak{h}}\) is infinite-dimensional, the condition 1 does a priori not ensure condition 2. Settling this issue is our first result.

Theorem 1

If ωp is a quasi-free pure state on \({\text{CAR}}_{e}({\mathfrak{h}})\), and q a projection on \({\mathfrak{h}}\), then \({{\mathcal{M}}}_{A}^{{\prime} }={{\mathcal{M}}}_{B}\).

The relation \({{\mathcal{M}}}_{A}^{{\prime} }={{\mathcal{M}}}_{B}\) also ensures that, if \({{\mathcal{M}}}_{A}\) is of type III1, then the same holds true for \({{\mathcal{M}}}_{B}\). The type of \({{\mathcal{M}}}_{A}\) is determined by the properties of Alice’s reduced state ωqpq, which in turn is completely determined by the operator qpq on \({\mathfrak{h}}\). In fact, it was shown by Powers and Størmer in ref. 32 (section 5) in conjunction with the seminal results of Araki and Woods26 that the type of \({{\mathcal{M}}}_{A}\) is determined entirely by the spectrum of the operator qpq. To get an intuition for this, assume that \({\mathfrak{h}}\) is finite-dimensional. If qpq has eigenvectors \(| {\xi }_{j}\left.\right\rangle \in q{\mathfrak{h}}\) with eigenvalues λj, we find from Wick’s theorem that

$$\begin{array}{l}{\omega }_{qpq}\left({a}^{\dagger }({\xi }_{{j}_{1}})\cdots {a}^{\dagger }({\xi }_{{j}_{m}})a({\xi }_{{j}_{m}})\cdots a({\xi }_{{j}_{1}})\right)\\\quad=\mathop{\prod }\limits_{k=1}^{m}{\omega }_{qpq}\left({a}^{\dagger }({\xi }_{{j}_{k}})a({\xi }_{{j}_{k}})\right)=\mathop{\prod }\limits_{k=1}^{m}(1-{\lambda }_{{j}_{k}}).\end{array}$$
(7)

Each normal mode \(| {\xi }_{j}\left.\right\rangle\) corresponds to a single fermionic mode, and we just saw that all of them are uncorrelated. We can think of Alice’s state ωqpq as describing a tensor product of a finite number of uncorrelated two-level systems, each represented by a density matrix ρj = diag(λj, 1 − λj). The λj measure how mixed the local state of Alice is. In the bipartite pure state, each normal mode of Alice is entangled with precisely one normal mode of Bob and carries at most one ebit of entanglement, which happens if λj = 1/2. This entanglement structure is the characteristic property of Gaussian fermionic states.

In our case, \({\mathfrak{h}}\) is infinite-dimensional. The type of the local observable algebra of an infinite spin chain described by an infinite tensor product ⊗ jρj has been studied by Araki and Woods43. The resulting algebra has type III1 if and only if the ratios of products of eigenvalues of the ρj are dense in \({{\mathbb{R}}}_{+}\). Intuitively, this means that all amounts of entanglement occur with arbitrary multiplicity. The results have been transferred to quasi-free fermionic states by Powers and Størmer, where the spectrum of qpq can be continuous. These results allow us to show the following.

Theorem 2

If H is critical, there exists a gauge-invariant quasi-free, pure and translation-invariant ground state such that \({{\mathcal{M}}}_{A}\) is a type III1 factor.

In the case of a linear, gapless dispersion relation, the scaling limit of H is expected to be well described by a CFT11,44,45. It is known that the local observable algebras in CFTs are of type III1 (refs. 46,47). Hence, their vacua yield embezzling states. We, therefore, expect that any local Hamiltonian with a CFT as its scaling limit has a universally embezzling ground-state sector.

Theorem 2 also applies to spin systems that are dual to the corresponding fermionic Hamiltonians by the Jordan–Wigner transformation48,49,50. This needs to be clarified because the relevant local observable algebras \({{\mathcal{M}}}_{A}\) are different due to the missing parity super-selection rule (see remark 12 in the Supplementary Information). We also show that Haag duality holds. Thus, these spin systems are also universal embezzlers.

Our final result shows that the embezzling property of ground states in the thermodynamic limit also descends to finite systems. We call a family of pure states \(| {\varOmega }_{n}\left.\right\rangle\) an embezzling family if for every dimension d and every error ε > 0 there exists a number n(ε, d) such that every state \(| \varPsi \left.\right\rangle \in {{\mathbb{C}}}^{d}\otimes {{\mathbb{C}}}^{d}\) can be embezzled from \(| {\varOmega }_{m}\left.\right\rangle\) with m ≥ n(ε, d) up to error ε in the sense of equation (1)22,51.

Theorem 3

(informal) Consider a family of local Hamiltonians Hn on increasingly long chains with a unique ground state \(| \varOmega \left.\right\rangle\) in the thermodynamic limit. If \(| \varOmega \left.\right\rangle\) is an embezzling state, then there exists a sequence of ground states \(| {\varOmega }_{n}\left.\right\rangle\) of Hn that is an embezzling family. Informal indicates that Theorem 3 is not stated in a formal mathematical fashion. The formal statement can be found in the Supplementary Information (see Cor. 40).

More generally, this result follows from a theorem showing that any embezzling state on an approximately finite-dimensional (also called hyperfinite) von Neumann algebra gives rise to an embezzling family. A von Neumann algebra is approximately finite-dimensional if there is a dense increasing sequence of finite-dimensional subalgebras, that is, every operator may be approximated by a finite-dimensional one (in the weak topology). The von Neumann algebras appearing in physics are typically approximately finite-dimensional, with the increasing sequence of finite-dimensional subalgebras related to a natural physical property, such as a localization length or an energy scale. In our context, the relevant parameter is the system size, and the result implies that, if we only want to embezzle states of a fixed dimension and up to a fixed error, we do not actually need the full thermodynamic limit. The thermodynamic limit provides a sharp distinction but is not physically required. We note that assuming the uniqueness of the ground state \(| \varOmega \left.\right\rangle\) is not strictly necessary (see remark 41 in the Supplementary Information).

Discussion

We have shown that a very large class of effective models for critical, fermionic many-body systems have a remarkable large-scale structure of entanglement by acting as universal embezzlers when interpreted as bipartite systems.

We have formulated our results for Hamiltonians in one spatial dimension that are particle-number conserving, that is, do not contain pairing terms, which, for example, appear in effective models describing superconductors. We show in the Supplementary Information that our results also transfer to the case of non-particle-number-conserving models, which we illustrate for the fermionic dual of the critical transverse-field Ising spin chain52,53,54. It also yields a universal embezzler (see examples 23 and 32 in the Supplementary Information). In higher dimensions, it is known that particle-number-conserving, translation-invariant and critical free-fermionic systems with a finite Fermi surface lead to logarithmic corrections to the area law for the entanglement entropy55,56. We expect that one obtains embezzling ground states of type III1 in all these cases case when considering a bipartition into two infinite subsystems with sufficiently smooth boundary. Indeed, the embezzlement property crucially relies on the choice of bipartition. For example, suppose, instead of considering two half-chains, Alice had access to the odd sublattice, and Bob had access to the even sublattice. In that case, it follows from the results of ref. 57 and ref. 32 that the algebras \({{\mathcal{M}}}_{A}\) have type II1 instead of III1 for any sublattice symmetric Hamiltonian, such as in equation (4). According to ref. 24, no type II von Neumann algebra can host embezzling states.

The embezzlement property extends to large but finite systems in an approximate sense by theorem 3. A natural question to ask is how the achievable error scales with the system size. In Supplementary Section D, we show that the half-chain entanglement entropy of the ground state is lower bounded as

$$H({\omega }_{A})\ge \log (1/\varepsilon )+\log (1-1/d\;),$$
(8)

if the system can embezzle entangled states with Schmidt rank d up to error ε. For critical systems, the universal relation \(H({\omega }_{A}) \approx \frac{c}{3}\log L/a\) as L → ∞ with central charge c and lattice spacing a for critical systems resulting from CFT2,16,17 implies that

$$L\ge \,\text{const.}\,\times {\left(1/\varepsilon \right)}^{\frac{3}{c}}.$$
(9)

On the other hand, if the half-chain entanglement entropy scales as Lα for some α > 0, it is in principle sufficient to have L only scale as \({\rm{polylog}}(1/\varepsilon )\) (ref. 22). This shows that, while critical systems are the only ones for which we know that ground states are embezzling, they are not particularly efficient for this task compared with arbitrary quantum states.

Making use of the conjectured universal distribution of eigenvalues in critical systems by Calabrese and Lefevre58, it may be possible to compute the precise scaling of errors in the scaling limit even for interacting critical systems. We leave this interesting open problem for future work.

We have not specified the unitary operators that can be used for embezzlement. In finite systems, it is easy to describe the optimal choice of unitary explicitly (Supplementary Section D). However, they require performing a Schmidt decomposition of the embezzling resource Ω, which is infeasible in general. For quasi-free systems it can be achieved via the normal mode decomposition of the system. We caution, however, that the quantum circuit complexity of such unitaries must diverge as ε → 0 (ref. 59). Thus, the problem is to find embezzling unitaries with minimal circuit complexity given an achievable error ε. We leave it as a further open problem to carry out such an analysis for quasi-free systems.

Gapped, one-dimensional many-body systems cannot be universal embezzlers as a consequence of the area law, which implies that \({{\mathcal{M}}}_{A}\) has type I (ref. 60). This raises the natural question of whether every gapless, translation-invariant, local many-body Hamiltonian has a ground state that leads to a universal embezzler in the thermodynamic limit. We provide a counter-example in the Supplementary Information. Nevertheless, it is tempting to conjecture that this is always the case if the system is critical in the sense of a CFT scaling limit, meaning that the large-scale observables of a half-space form a III1 factor. Interestingly, the Motzkin spin chain61 provides an example of a system where the gap closes too quickly for a CFT scaling limit to exist, and where the half-chain entanglement entropy grows as \(O(\sqrt{L})\). It would be of great interest to determine the type of the associated half-chain von Neumann algebra. Is it also of type III1? Does Haag duality hold?

More generally, in conjunction with ref. 24, our results motivate the study of the large-scale structure of entanglement in interacting critical many-body systems. For example, it would be interesting to know whether the type of the half-chain algebras can be deduced from scaling laws of entanglement entropies.

Finally, in ref. 62 the existence of multipartite universal embezzlers was established. It would be fascinating to know whether, for example, a critical system on a ring provides a tripartite universal embezzler in the scaling limit if it is subdivided into three intervals.