In the early studies of the planar limit of the N=4 super-Yang-Mills theory, it turned out that the anomalous dimensions of single-trace operators can be obtained from the spectrum of an integrable Hamiltonian with long-range interaction. At one loop, the dilatation operator corresponds to an integrable nearest-neighbor interacting model [1]. For higher loops, the interaction range increases; more precisely, the interaction range is ℓ+1 at ℓ loops.

In the region where the spin chain length J is bigger than the loop order (the asymptotic region), the Hamiltonian can be written as a sum of local densities. For these local operators, the integrability condition can be generalized, and it was shown that the Hamiltonian of the SU(2) sector preserves integrability for higher loops [2]. These local Hamiltonians can be diagonalized with the asymptotic Bethe ansatz [3], and the result can be generalized to the full psu(2,2|4) spectrum [4]. However, this result is correct only in the asymptotic region. In the region where the spin chain length J is smaller than the loop order (the wrapping region), wrapping corrections appear [5]. So far, it was not clear whether good spin chain toy models, which mimic the wrapping corrections, could be found—i.e., even if an asymptotic Hamiltonian were given, we could not define the corresponding finite-size Hamiltonian.

The solution for the wrapping corrections came from holographic duality. In the string theory side, the scaling dimensions correspond to the energy spectrum of strings which can be described as a 1+1-dimensional integrable field theory [6]. In field theory, if we know the dispersion relation and the scattering matrix at infinite volume, then we can calculate the finite-volume spectrum as well (at least in principle). The finite-size corrections can be obtained from the thermodynamic Bethe ansatz [7–10], and it has been shown that they agree with the wrapping corrections [11–13].

Since the asymptotic data of the string theory (dispersion relation, scattering matrix) completely defines the finite-size corrections, a natural conclusion is that the asymptotic data on the spin chain side should also define the wrapping corrections. In other words, there must be a procedure that gives the finite-size Hamiltonians from the asymptotic ones. The aim of this Letter is to present such a method.

Recently, an algebraic framework was developed for integrable medium-range spin chains (with an interaction range bigger than 2, but finite) [14]. This method gives a recipe for how to define transfer matrices which are the generating functions of the conserved quantities, including the Hamiltonians. An interesting observation is that this transfer matrix is well defined even when the length of the spin chain is smaller than the interaction range; therefore, generalizing this method to long-range spin chains, we obtain transfer matrices which define the finite-length Hamiltonians even for the lengths where the wrapping corrections appear.

In this section, we summarize the definition of the long-range spin chain following Refs. [3,15] and specify our goals.

An integrable long-range spin chain has a tower of coupling constant λ-dependent commuting charges Qr(λ)≡Qr [16] which have the following series expansions: Qr=Qr(0)+∑j=1∞λjQr(j),(1)where r≥2, and the λ-independent operators Qr(ℓ) are sums of local operators with range r+ℓ: Qr(ℓ)=∑j=-∞∞qjr,ℓ=∑j=-∞∞qj,j+1,…,j+r+ℓ-1r,ℓ,(2)where the local densities qjr,ℓ≡qj,j+1,…,j+r+ℓ-1r,ℓ act on the sites j,j+1,…,j+r+ℓ-1. The Hamiltonian is the charge Q2.

It turns out that, for a fixed nearest-neighbor model Qk(λ=0), a large class of integrable deformations exists. The moduli space is given by four sets of parameters: αr(λ), βr,s(λ), γr,s(λ), and εk(λ). The last two sets are unphysical parameters, and they correspond to the linear combinations of the charges Qr→∑γr,s(λ)Qs and the similarity transformations Qr→eXQre-X,X=∑j=-∞∞∑kεk(λ)Xjk,(3)where Xjk≡Xj,…,j+ℓk-1k’s are local operators with range ℓk. The remaining parameters are the physical ones. The αr and βr,s parameters appear in the rapidity map and the scattering phase [15].

It is clear that the operators Qk(ℓ) can also be defined on a finite length J for J≥ℓ+k. More concretely, the Hamiltonian H on size J is defined up to order λJ-2 (asymptotic region). Our goal is to find an integrability-preserving method which defines the finite-volume version of the asymptotic Hamiltonians even for higher orders than λJ-2 (wrapping region).

In this section, we generalize the construction of Ref. [14] (the basics appeared first in Ref. [17]) to obtain transfer matrices for perturbative long-range spin chains [3]. In Ref. [14], an algebraic framework was introduced for integrable spin chains with the interaction range ℓ+2, which is defined by the Hamiltonian H(ℓ)=∑j=1Jhj,j+1,…,j+ℓ+1=∑jhj(ℓ),(4)where hj(ℓ) is the Hamiltonian density which acts on the sites j,j+1,…,j+ℓ+1. We use a periodic boundary condition. The construction of Ref. [14] is based on the existence of the Lax and the R operators: Lˇj(ℓ)(u)=Lˇj,j+1,…,j+ℓ+1(ℓ)(u)=1+uhj(ℓ)+O(u2),(5)Rˇj(ℓ)(u,v)=Rˇj,j+1,…,j+2ℓ+1(ℓ)(u,v),(6)which satisfy the RLL relation Rˇ2(ℓ)(u,v)Lˇ1(ℓ)(u)Lˇℓ+2(ℓ)(v)=Lˇ1(ℓ)(v)Lˇℓ+2(ℓ)(u)Rˇ1(ℓ)(u,v).(7)In this Letter, we choose to write these operators in the “checked” form (the R matrix is multiplied by a permutation), which might be less familiar to some readers [18], although it has the advantage that the Lax operator has a simpler expansion in the spectral parameter [Eq. (5)]. In the alternative “unchecked” convention, the quantum and auxiliary spaces are separated. Figure 1 shows graphical presentations of Lax operators and RLL relations, and the colored legs denote the auxiliary spaces of the “unchecked” convention.

The consequence the RLL relation is that the transfer matrix Tˇ(ℓ)(u)=Tr^J,ℓ+1(LˇJ(ℓ)(u)…Lˇ1(ℓ)(u))(8)defines commuting quantities: [Tˇ(ℓ)(u),Tˇ(ℓ)(v)]=0 [18]. In Eq. (8), we define the twisted trace operator Tr^J,ℓ, which acts on an operator X as Tr^J,ℓ(X)=TrJ+1,…,J+ℓ(XPℓ,J+ℓPℓ-1,J+ℓ-1…P1,J+1),(9)where Pj,k is the permutation operator and TrJ+1,…,J+ℓ is the usual trace on the sites J+1,…,J+ℓ. The transfer matrix generates the local conserved charges Qk+1(ℓ)=∂k∂uklogTˇ(ℓ)(u)|u=0.(10)The interaction range of Qk(ℓ) is (k-1)ℓ+k, and H(ℓ)=Q2(ℓ).

Let us turn to the long-range spin chains. At first, we have to introduce the coupling constant λ-dependent truncated operators Lˇ1(ℓ)(u,λ)≡Lˇ1(ℓ)(u) and Rˇ1(ℓ)(u,v,λ)≡Rˇ1(ℓ)(u,v) with range ℓ+2 and 2ℓ+2 as Lˇ1(ℓ)(u)=Lˇ1(0)(u)+∑j=1ℓλjLˇ1(j)(u),(11)Rˇ1(ℓ)(u,v)=Rˇ1(ℓ,0)(u,v)+∑j=1ℓλjRˇ1(ℓ,j)(u,v),(12)which satisfy the RLL relation up to order O(λℓ+1): Rˇ2(ℓ)Lˇ1(ℓ)(u)Lˇℓ+2(ℓ)(v)=Lˇ1(ℓ)(v)Lˇℓ+2(ℓ)(u)Rˇ1(ℓ)+O(λℓ+1),(13)where we use the shorthand notation Rˇj(ℓ)≔Rˇj(ℓ)(u,v). We also require that Lˇ1(ℓ)(u)=1+uh1(ℓ)+O(u2),h1(ℓ)=∑j=0ℓλjh1(j),(14)where h1(j) are λ- and u-independent operators with interaction range j+2.

At first sight, we might think that the truncated RLL relation [Eq. (13)] and the matrices Rˇ1(ℓ,j) are completely independent for every order ℓ, but this is not true. It turns out that Eq. (13) up to order O(λℓ) is equivalent with the truncated RLL relation for ℓ-1. We can show that Rˇ1(ℓ)+O(λℓ)=Lˇℓ+1(ℓ-1)(u)Rˇ1(ℓ-1)Jˇℓ+1(ℓ-1)(v),(15)where we define the perturbative inverse Jˇ1(ℓ-1)(u) as Jˇ1(ℓ-1)(u)Lˇ1(ℓ-1)(u)=1+O(λℓ) [18].

The consequence of Eq. (15) is that the matrices Rˇ1(ℓ,j) are determined by Rˇ1(ℓ-1,j) for j=1,…ℓ-1; therefore, the full truncated R matrix Rˇ1(ℓ) is completely determined by the matrices Rˇ1(j,j) for j=1,…,ℓ. By fixing the leading-order Lˇ1,2(0),Rˇ1,2(0,0) to already known L and R matrices of a nearest-neighbor interacting model, we can calculate the matrices Lˇ1(ℓ)(u), Rˇ1(ℓ,ℓ)(u,v) order by order from the highest order λℓ of the truncated RLL relation [Eq. (13)].

As in the medium-range case, the transfer matrix Tˇ(ℓ)(u)=Tr^J,ℓ+1(LˇJ(ℓ)(u)…Lˇ1(ℓ)(u))+O(λℓ+1)(16)defines commuting quantities up to order λℓ+1: [Tˇ(ℓ)(u),Tˇ(ℓ)(v)]=O(λℓ+1). The transfer matrix generates the conserved charges up to order O(λℓ+1) Qk+1(ℓ)=∂k∂uklogTˇ(ℓ)(u)|u=0+O(λℓ+1),(17)where Qk(ℓ)=Qk(0)+∑j=1ℓλjQk(j)(u)+O(λℓ+1).(18)It turns out that the charge Qk(ℓ) has the interaction range ℓ+k.

Since the Lax operators have the property in Eq. (14), the Hamiltonian reads as Q2(ℓ)=∑j=1JTr^J,ℓ+1(hj(ℓ))+O(λℓ+1).(19)Since hj(ℓ)=∑k=0ℓλkhj(k), we obtain that Q2(ℓ)=∑j=1JTr^J,ℓ+1(hj(ℓ)).(20)For the asymptotic region (i.e., J>ℓ+1), we have the identity Tr^J,ℓ+1(h1(ℓ))=h1(ℓ); therefore, this charge has the usual form Q2(ℓ)=∑j=1Jhj(ℓ).

Above, we have shown that the solutions of the RLL relations [Eq. (13)] define long-range charges [Eq. (1)] in the asymptotic limit. An important question is whether the reverse statement is also true: i.e., do there exist Lax operators for every integrable long-range charge Q2? At this point, we do not know the answer. However, I investigated the long-range gl(N) spin chains of Ref. [3] up to order O(λ3). After fixing the unphysical parameters γ(λ), ε(λ), I found the matrices Lˇ1(2)(u) which give the Q2(2) values for every set of physical parameters α(λ), β(λ) [18].

The main advantage of the algebraic construction of the previous section is that the transfer matrix is well defined and satisfies [Tˇ(ℓ)(u),Tˇ(ℓ)(v)]=O(λℓ+1) [20] even for J<ℓ+2—i.e., the wrapping region. So far, it has not been clear how to define the Hamiltonian in the wrapping region in an integrability-preserving way, but our transfer matrix gives a recipe. We emphasis that the Lax operator [Eq. (11)] is an asymptotic, density-like quantity (since it is defined on an infinite chain and contains the asymptotic Hamiltonian density); therefore, it describes the elementary physical interaction. The transfer matrix shows a consistent possibility how to define this interaction for finite sizes in a translation-invariant and integrability-preserving way.

To obtain the integrable Hamiltonian for the wrapping region (J≤ℓ+1), we only have to use the definition in Eq. (17). We can repeat the previous calculation up to Eq. (20); i.e., the Hamiltonian reads as Q2(ℓ)=∑j=1Jh˜j(J,ℓ), where we introduce a “wrapped” Hamiltonian density (see Fig. 2) h˜1(J,ℓ)≡h˜12…J(J,ℓ)≔Tr^J,ℓ+2-J(h1(ℓ)),(21)and the periodic boundary condition is prescribed—i.e., h˜j(J,ℓ)=h˜j,j+1,…,J,1,2,…j-1(J,ℓ). We see that the twisted trace acts as an identity for the asymptotic region, but for the wrapping region it defines a new operator which “’fits” with the length of the chain.

Let us summarize what we have learned from this analysis. Let us take an asymptotic integrable long-range Hamiltonian H∞=∑j=-∞∞hj∞=∑j=-∞∞∑ℓ=0∞Xj(ℓ),(22)where Xj(ℓ)≡Xj,…,j+ℓ+1(ℓ) is a coupling-constant-dependent operator with interaction range ℓ+2. We see that our method defines a unique integrability-preserving Hamiltonian for every finite length J as HJ=∑j=1J∑ℓ=0∞X˜j(J,ℓ),X˜1(J,ℓ)={X1(ℓ),ℓ+2≤J,Tr^J,ℓ+2-J(X1(ℓ)),ℓ+2>J.(23)

In this section, we demonstrate that our finite-volume Hamiltonian is consistent with a naive physical argument. Let us take a long-range interaction and assume that we already know its manifestation for every length J; i.e., for every length J we have the Hamiltonians HJ which correspond to the same physical interaction. Since we know the Hamiltonians for every length, we can obtain the asymptotic model by the limit H∞=limJ→∞HJ.(24)A natural requirement is that our procedure in Eq. (23) should return to the original models HJ.

In the following, we validate this requirement on Inozemtsev’s spin chain [21], for which the finite-volume Hamiltonian reads as HJ=∑1≤j,k≤J(℘(k-j)+2ωζ(ω2))Pj,k,(25)where ℘(z), ζ(z) are the Weierstrass functions defined on the torus C/ZJ+Zω, ω=i(π/κ), and the local Hilbert spaces are CN. The J is length of the spin chain, and κ∈R is the coupling. In the asymptotic limit, we obtain the hyperbolic Inozemtsev spin chain [21]: H∞=∑-∞<j<k<∞V(k-j)Pj,k,V(j)=(κsinh(jκ))2.(26)After a renormalization of the coupling constant κ(λ), this Hamiltonian is compatible with the perturbative long-range description [22]. Let us rewrite the Hamiltonian as H∞=∑-∞<j<∞hj∞=∑ℓ=0∞∑-∞<j<∞Xj(ℓ),X1(ℓ)=V(ℓ+1)P1,ℓ+2.(27)Now, let us apply Eq. (23) on X1(ℓ). At first, let us wrap the permutations P˜1,k={N,k≡1  mod  JP1,kJ,k≡2,…,J  mod  J,(28)where 1<kJ≤J and kJ≡k  mod  J. Now, we can wrap the full h1∞=∑ℓ=0∞X1(ℓ) as h1J≔∑ℓ=0∞X˜1(J,ℓ)=∑1<k≤JP1,k∑0≤l<∞V(k+lJ-1)+c,(29)where c=N∑1≤l<∞V(lJ). The full finite-volume Hamiltonian is HJ=∑1≤j≤JhjJ=∑1≤j<k≤JPj,k∑-∞<l<∞V(k-j+lJ)+Jc.(30)The infinite sum can be written in the following closed form [Eq. (23.8.3) in Ref. [23] ]: ∑-∞<l<∞V(k+lJ)=℘(k)+2ωζ(ω2).(31)Substituting back and dropping the irrelevant identity operator, we simply obtain the original Inozemtsev Hamiltonian [Eq. (25)]. We can see that our wrapping method gives the finite-volume Inozemtsev spin chain from the infinite-volume hyperbolic Inozemtsev spin chain, which is an expectation for a consistent wrapping procedure.

In this section, we summarize some properties of the wrapping corrections in the planar N=4 super Yang-Mills theory (SYM). We show that our finite-volume Hamiltonians are compatible with these requirements.

Argument 1: On the string theory side (1+1-dimensional field theory description), we know that the asymptotic data (dispersion relation and scattering matrix) uniquely define the wrapping corrections. This fact is in agreement with our method, which uniquely defines finite-size Hamiltonians [Eq. (23)] for a given asymptotic Hamiltonian [Eq. (22)].

Argument 2: In the dilatation operator of the N=4 SYM, there are unfixed parameters coming from the free choice of the renormalization scheme [24,25]. These are unphysical parameters which disappear from the spectrum. On the asymptotic level, these parameters correspond to εk(λ)—i.e., the global rotations in Eq. (3); therefore, it is clear that they have no effect on the spectrum. The disappearance on finite volume is a nontrivial condition for the physical finite-size Hamiltonians. It turns out that the spectrum of our finite-volume Hamiltonians is free from εk(λ) as well [18].

Argument 3: In the asymptotic limit, the spectrum of the closed sectors is completely independent from the full theory. To be more concrete, let us consider three asymptotic Hamiltonians HN=4∞, HSU(N)∞, and HSU(2)∞, which correspond to the N=4 SYM, one of the SU(N) and the SU(2) long-range models for which the restriction to the SU(2) sector is the same—i.e., HN=4∞|SU(2)=HSU(N)∞|SU(2)=HSU(2)∞.Clearly, the spectrum of a closed sector does not know about the full model in which it is embedded. However, we know that for proper wrapping corrections, we have to consider contributions from the full spectrum (for Lüsher corrections, we have to sum for all virtual particles of the mirror model [11]); therefore, HN=4J|SU(2)≠HSU(2)J.This is an important requirement for the definition of the finite-size long-range Hamiltonians. Let us take our definition in Eq. (23). We can see that the wrapped operator X˜1(J,ℓ) contains a sum for a tensor product of the full local Hilbert spaces. Therefore, these wrapped operators, even in the closed subsectors, explicitly depend on the full asymptotic Hamiltonian; therefore, our definition satisfies HSU(N)J|SU(2)≠HSU(2)J.

Argument 4: We also know that, in the wrapping corrections, extra transcendental numbers appear. For example, let us consider the Konishi operator [a length-4 operator in the SU(2) sector]. At four loops, the asymptotic dilatation operator of the SU(2) sector contains only one transcendental number, ζ(3) [24,25]. However, the length-4 Hamiltonian at four loops [24,25] contains an extra ζ(5) compared to the asymptotic Hamiltonian. We already mentioned that our finite-volume Hamiltonian includes a sum for the full one-site Hilbert space in the wrapping region; therefore, extra transcendental numbers can appear in the finite-size Hamiltonians if the one-site Hilbert space is infinite-dimensional, which is the case for N=4 SYM. We note that transcendental numbers appear in the spectrum of higher chargers of nearest-neighbor spin chains with infinite-dimensional local Hilbert spaces [26].

In this Letter, we generalized the algebraic framework of medium-range spin chains [14] to perturbative long-range spin chains [3]. Using this method, we were able to define finite-volume Hamiltonians [Eq. (23)] for every asymptotic long-range model. We demonstrated that this definition is physically relevant by showing that our definition is in agreement with several physical requirements coming from Inozemtsev’s spin chain and AdS/CFT.

We saw that our wrapping procedure [Eq. (23)] leads to wrapping corrections with similar properties to what we expect from the N=4 SYM. This is an important result, because previously, the finite-size corrections under simpler conditions could be studied using integrable field theories. From now on, the wrapping corrections can be also tested on spin chains, which can be simpler in many ways.

I believe that this result could open up a number of new research directions. One possible direction is to generalize the integrable boundary states [27,28] for long-range spin chains as well. Combining this with the method of this Letter, we could investigate the wrapping corrections of the overlaps between boundary and Bethe states which describe certain one- and three-point functions in N=4 SYM [29–35] and ABJM theories [36,37].

It would be interesting to apply the algebraic Bethe ansatz, although it is not clear how this should be done due to the increasing number of auxiliary spaces. However, there are other ways to diagonalize the transfer matrices—e.g., functional techniques [38] (quantum spectral curve [39] for simpler long-range models?) and the separation of variables [40–42].

Another interesting direction would be to give some nonperturbative definitions of the quantities appearing in this Letter (Lax operators, transfer matrix), the derivation of the Yangian symmetry [43] from our framework, and connection for the TT¯ deformations of spin chains [44].

Finally, we need to address a major shortcoming of our method. The spin chain which appears in the perturbation theory of N=4 SYM is dynamical, which means that the Hamiltonian can change the length of the spin chain. Our method in its present form is not suitable for describing such models. In the future, we plan to extend the process to dynamic spin chains, but in the meantime, the nondynamical Hamiltonians like Eq. (23) can serve as good toy models of wrapping effects.

It is worth mentioning that parallel research has also been started on the topic of long-range spin chains [45].