Field Tensor Network States

We define a class of tensor network states for spin systems where the individual tensors are functionals of fields. The construction is based on the path integral representation of correlators of operators in quantum field theory. These tensor network states are infinite dimensional versions of matrix product states and projected entangled pair states. We find the field-tensor that generates the Haldane-Shastry wave function and extend it to two dimensions. We give evidence that the latter underlies the topological chiral state described by the Kalmeyer-Laughlin wave function.

Tensor networks (TN) are becoming a key tool to describe many-body quantum systems [1]. On the one hand, they can efficiently approximate quantum states of local Hamiltonians in thermal equilibrium, which has led to powerful numerical algorithms with applications in condensed matter and, to some extent, in high-energy physics [2]. On the other hand, they provide us with paradigmatic examples of strongly correlated states and thus allow us to investigate intriguing many-body quantum phenomena. For instance, they offer us a guide to classify symmetry protected topological phases [3,4], or to understand a large variety of topologically ordered behavior. In fact, states (or models) like the AKLT [5], string-net states [6], or resonating valence-bond states have a very simple description in terms of TN. By simple we mean with a small bond dimension, D, which limits the number of coefficients describing the tensors generating the many-body states. The description of such states in terms of TN automatically opens up the possibility of using powerful tools in order to describe their physical properties by just inspecting a simple tensor. In 1D, one can easily describe symmetries and string order parameters [7], or even gauge symmetries [8]. In 2D, apart from obtaining the physical symmetries, one can directly identify the topological properties or type of anyon excitations of the parent Hamiltonian [9].
There exist, however, some classes of states for which no exact expressions in terms of tensor network states of finite bond dimensions exist. Two prominent examples are critical states [10], and chiral topological states of gapped Hamiltonians in one and two dimensional spin lattices, respectively [11,12]. The reason behind the lack of description as TN for the first stems from the fact that critical states violate the area law [13,14]. Specifically, the entanglement entropy of a connected region containing L spins scales as ∝ ln(L) [15,16], whereas for a matrix product state (MPS), the one-dimensional version of tensor network states, it is bounded by 2 ln(D); therefore, in the thermodynamic limit for any finite D, there always exists some L for which an MPS cannot cope with the amount of entanglement and thus it is impossible that it describes a critical state. The reason for the second class is more subtle and yet not fully understood; however, there are good reasons to believe that there exist obstructions due to the non-existence of local Wannier states [17] (see, however [18]). In fact, for Gaussian fermionic states, it is not possible to describe gapped chiral topological insulators [19] 1 . We emphasize that here we mean an exact description; in fact, both classes of states may well be approximated efficiently with an error that decreases as D increases [20][21][22].
The arguments above do not prevent the existence of exact descriptions of critical or chiral topological states with TN of infinite bond dimensions. In [23], it was noted that the conformal field theory (CFT) formulation [24] of the Haldane-Shastry state has similarities with MPS, and in [20][21][22], the CFT formulation was used to obtain MPS with a discrete, infinite bond dimension describing chiral topological states in 2D. From the tensor network perspective it is, however, desirable to use projected entangled pair states (PEPS) to deal with 2D systems. Furthermore, although the approach of [20][21][22] can, in principle, be used to describe critical states in 1D with open boundary conditions, it is more appropriate to use periodic boundary conditions for translationally invariant systems.
In this letter we define Field Tensor Networks (FTN) for spin lattices in any dimension, where the bonds in the tensors are functions, the corresponding contractions are accomplished by a path integration, and the tensors themselves are functionals. The virtual space is hence continuous. We show how this approach can be used to describe translationally invariant critical systems, as well as the analogs of PEPS for two dimensional systems. Our construction is reminiscent of recent proposals for constructing tensor networks for quantum fields, where path integration is also employed [25,26]. In our case, however, we deal with discrete spin lattices and the con-struction is quite different. We give a procedure to compute the FTN for states whose coefficients in the spin basis can be written as vacuum correlators of a quantum field theory with a local action. In particular, we give an explicit construction for free boson CFTs and vertex operators. We also take advantage of the fact that the Haldane-Shastry state [27,28], a prominent critical state, can be expressed in that form [23,29] to compute a FTN generating that state. The description allows both periodic and open boundary conditions. We also propose a FTN in 2D and give strong evidence that it represents a Kalmeyer-Laughlin state [11], a prototypical representative of chiral topological order.
FTN in 1D: We consider a spin chain of N spins of dimension d, and a translationally invariant state Let us start recalling MPS, where D is the bond dimension, and for each value of s = 1, . . . , d, A s n,m is a D×D matrix. In an analogous way we define (translationally invariant) Field Tensor Network States (FTNS) as Here, α n : R → R belong to the set of square-integrable functions and also include a constant function, and for each value of s = 1, . . . , d, A s α,β are functionals of α, β. Note that Ψ has the same structure as a MPS, where the indices of the matrices are replaced by the functions α n , and the sum over repeated indices is replaced by a path integral.
We also define the functionals A s1,...,sn which we will call "sewing" and "closing" conditions. Example: Particularly interesting examples of FTNS are those for which the coefficient can be written in terms of correlators of a simple CFT in 1+1 dimensions and a local action. Specifically, we shall here study a family of critical states in 1D with c s1,...,s N ∝ δ n sn,0 n Here, x n = π(n− 1 2 ), q is a real number, χ sn is a phase factor that may depend on s n , and s n = ±1. The member of the family with q = 1/2 and χ sn = e inπ(sn−1)/2 is the ground state of the Haldane-Shastry model, which has been extensively studied in the literature as a paradigm of criticality. The wave function (6) has also been used as an ansatz for the ground state of the Hamiltonian of the XXZ spin 1/2 chain in the critical regime with anisotropy parameter ∆ = − cos(4πq 2 ) [23].
The states (6) violate the area law, so that they cannot be written as MPS with finite bond dimension. Nevertheless, we will show here how they can be expressed as FTNS.
It is not difficult to show that [30] c s1,...,s N ∝ χ s1 : e iqs1ϕ(r1) : . . . χ s N : e iqs N ϕ(r N ) : 0 , (7) where ϕ is a real scalar field defined on a cylinder of circumference πN , :: denotes normal ordering, the r n = (x n , 0) are points in cylindrical coordinates (see Fig. 1), and the expectation value is taken in the vacuum. In this case, using the path integral representation we have where is the Euclidean action of the boson field. Notice that (9) vanishes if ϕ is a constant ϕ 0 which upon integration generates the constraint n s n = 0 appearing in (6). In order to find the FTNS representation of (7), we rewrite (10) Here, α n (t) is a function of t only, and ϕ n (x, t) is defined in the interval (x, t) ∈ [x n − δ, x n + δ] × R with δ = π/2, and the indicates that it fulfills the boundary condition (see Fig. 1) Thus, we simply identify where the indicates that ϕ has α = α 1 and α = α L+1 as boundary conditions, and S is defined as in (9) but with the integral in x restricted to the interval [x 1 − δ, x L + δ]. Note that (3) trivially follows. The sewing and closing conditions (5) are represented for this case in Fig. 1(c,d).
In the supplemental material [31], using Green's function techniques, we explicitly compute where S (0) S (1) and ϕ 0 is the zero mode of the boson field that has been subtracted from the functions α, α to guarantee their normalizability. Here α = (α, α ), U is a 2 × 2 matrix with elements and v L, where is the derivative of the principal value distribution P 1 t . Chiral version: We aim at also being able to describe chiral states, and as a test case, we next consider a chiral formulation of the critical states (6). In this formulation the states are defined in terms of a chiral free boson field ϕ(z), which depends on z, but not on its conjugatez. The states are again given by (7), except that the vertex operators now take the form : e iqsnϕ(zn) :, where q ∈ R and z n = t + ix n (the wave function obtained with these chiral vertex operators coincides with (6) except that √ 2q is replaced by q). This correlator can be written as in (8) with a chiral action [32] employed to study the edge excitations in the Quantum Hall effect [33]. However, the slicing of the path integral into the intervals (x, t) ∈ [x n − δ, x n + δ] × R introduces boundaries that mix the left and right moving modes of the bosonic field, which in turn complicates the approach.
We notice, however, that in (16) there are two parts related by complex conjugation. Moreover, (14) comes from a Green function with four terms where only one of them is analytic in the location of the vertex operators. It is therefore natural to expect that one obtains the chiral state by selecting only one of those parts. We shall use this property to define the new tensorŝ The factor L − 1 2 Lq 2 guarantees that (19) satisfies the sewing condition (5a).
The wave function that one obtains using (19) coincides with (6) (although with q replaced by √ 2q, so that the Haldane-Shastry state now corresponds to q = 1/ √ 2). This is not evident when comparing Eqs. (19)(20)(21) with (13)(14)(15)(16) as they look very different, and it is not obvious that they are related by a gauge transformation [1]. However, one can prove this statement by showing that (19) fulfills the sewing condition (5a) and that by closing (5b) one indeed obtains the Haldane-Shastry wavefunction. We show that in the supplementary material [31], where we use the translational invariance of the action restricted to the strip along the t-coordinate, which allows us to diagonalize in k-space. Thus, the procedure leading to (19) provides a field theory version of the chiral vertex operator in CFT.
FTN in 2D: The constructions presented above can be straightforwardly extended to represent states in two dimensions, corresponding to, e.g., a square spin lattice. We construct it on a cylinder, although one can similarly use a torus. The strip [x n −δ, x n +δ]×R considered above is replaced by the rectangle [x n −δ, δ ]. We define the functional A s αn,m,βn,m,γn,m,δn,m , which depends on the functions α n,m , β n,m : [t m − δ , t m + δ ] → R and δ n,m , γ n,m : [x n − δ, x n + δ] → R (see Fig. 2a). By arranging the functionals along the cylinder (Fig. 2b), identifying functions like in Fig. 2a, and integrating over them, one can construct states very much in the same way as one builds PEPS in two dimensions.
Example: Again, an illustrative example is provided by states that are expressed in terms of a simple CFT with a local action. For instance, we can define a spin state as (see Fig. 2) We then carry out the path integral in (22). The main technical tool to do this is to apply a conformal map that transforms the rectangle [x n − δ, x n + δ] × [t m − δ , t m + δ ] into the complex upper-half plane in terms of Jacobi elliptic functions. In the case (n, m) = (1, 0) one finds (see the supplemental material [31]) where with α = (α, β, γ, δ), ξ 1 = ξ 2 = t and ξ 3 = ξ 4 = x. The integration domain I is adapted to the type of variables involved. U is a 4 × 4 matrix some of whose elements are (the complete matrix is given in the supplemental material [31]) Here, sn(t), cn(t), and dn(t) are Jacobi elliptic functions of modulus k and sn(t), cn(t), and dn(t) of modulus k = √ 1 − k 2 , where k is determined from the aspect ratio δ/δ of the considered rectangle [31]. In the limit k → 1, the rectangle degenerates into a strip and we recover the matrix elements (17).
As in the previous example we can truncate this functional to a chiral onê where the phase factor χ s can be chosen at will and We here consider the system on a cylinder, and to remove the virtual degrees of freedom on the boundaries, we take the rectangular regions for the boundary spins to go all the way to infinity. These boundary tensors can be obtained following the same approach as for the tensors in the bulk. We conjecture that sewing these amplitudes we get the wave function (30) with z n = t n + ix n . When q = 1/ √ 2, Eq. (30) is a 2D topological state in the same universality class as the bosonic Laughlin state at filling fraction 1/2, and the Kalmeyer-Laughlin wave function is obtained for N → ∞ [34].
Conclusions: We introduced a new class of TN constructed using functionals of fields that are contracted by means of the path integral of the functions defined on the links of the network. These tensors satisfy sewing and closing conditions that are similar to those employed in the construction of the scattering amplitudes in string theory [35,36].
We illustrate our approach using a massless boson in 2D that allows us to derive the Haldane-Shastry wave function that describes a critical state in the universality class given by the WZW model SU (2) 1 . We also conjecture the field-tensor that generates the Kalmeyer-Laughlin state, which suggests that the chiral PEPS underlying topological chiral states in 2D require infinite bond dimension. The latter suggestion could be further studied by truncating the field variables to a finite number of modes in which case the field-tensor provides a PEPS with finite bond dimension. We have here focused on lattice states, but utilizing the techniques in [37] to approach the continuum limit of the states, one could similarly describe continuum states. The definition of field tensor network states applies equally well to other types of lattices than those considered here. Our approach also allows a way to study topological chiral states based on the symmetry properties of the field-tensors. Let φ(x 1 , x 2 ) be a real massless scalar field in a simply connected region M of the two dimensional spacetime R 2 . The Euclidean free action of this field is given by (S1) Let f (x) be the values that φ(x) takes at the boundary ∂M of the region M , and {q j } N j=1 a set of N charges located at the positions x j ∈ M that corresponds to the charge density We shall associate to these data the path integral To compute (S3) we write the scalar field as φ(x) = φ 0 +φ(x), where φ 0 is a constant andφ is an orthogonal integrable function. Similarly, we write f (x) = f 0 +f (x) so that (S3) becomes where δ(φ 0 − f 0 ) implements the constraint φ(x)| ∂M = f (x) between the zero modes. The second factor in (S5) can be computed explicitly obtaining where φ cl satisfies the Poisson equation, that is solved by where dx i is the line element along the curve ∂M oriented anti-clockwise, 12 = − 21 = 1, and G M is the Green's function with Dirichlet BCs From (S4) we get where we have performed a partial integration using the Gauss theorem, Inserting eq.(S8) into (S10) gives and then This expression can also be applied to the case when M is the sphere S 2 . Since S 2 has no boundary the last two terms of (S12) are absent. The Green's function on S 2 is where z = x 1 + ix 2 and z = x 1 + ix 2 . In the rest of the SM we shall use the variable z = x + iy, that corresponds to t + ix in the main text. We also have to integrate over f 0 . The final result is that does not vanish under the neutrality condition N j=1 q j = 0 . (S16) In the case discussed in the main text the charges are given by and hence eq.(S16) becomes N j=1 s j = 0.

THE MPS FUNCTIONAL
We shall use below the results obtained above to construct the MPS functional. First of all, we shall find the Green's function that solves eq.(S9). This can be done using the Riemann's mapping theorem that asserts the existence of a conformal map g from M to the upper-half plane H , when M is a simply connected region of the complex plane C such that the boundary of M , is mapped into the real axis, that is g : ∂M → R. This map allows us to construct G M from the Green's function G H in H , that is given by (S19) Notice that G H vanishes if ζ or ζ are real satisfying the Dirichlet BCs (S9). The Green's function G M can be obtained replacing ζ and ζ by g(z) and g(z ) respectively, To construct the MPS functional we take the strip M = R × [0, π]. The corresponding conformal map (S18) is given by that replaced into (S20) gives In order to prove the sewing and closing conditions, given in eqs.(5a) and (5b) of the main text, we shall consider a generic strip M = R × π[a, b] (a < b), that can be mapped into H by the conformal map The associated Green's function is Choosing z = x + iy, z = x + iy we get .

The non chiral MPS functional
The boundary ∂M consists of the straight lines in the plane with y = πa and y = πb. We define the real functions (see fig. S1) The constant mode f 0 is common to both lines and will be treated separately. Its contribution to the functional is simply the phase factor in eq.(S13). The functions f ± (x) correspond to α(t) and β(t) in the main text. The definitions (S26) lead us to write eq.(S12) as where y + = π(a + ε), y + = π(a + ε ), are a regularization of y = πa, πb. Eqs.(S25) lead to In equations (S31) and (S32) we have taken the limit ε, ε → 0 that gives ordinary functions, while in (S30) we have replaced ε ± ε by that in the limit → 0 becomes a generalized function, namely a distribution. To discuss this issue in more detail we define the function that is obtained from (S30) after rescaling the variables. Let us also define the function whose first term is regular at x = 0. We aim at showing that f (x) → g (x) in the limit → 0. Expanding the difference between (S33) and (S34) around = 0 yields The term proportional to 2 , has the series expansion − 2 15 + O(x 2 ) around x = 0, so that one can take safely the limit → 0 obtaining The terms depending of in (S34) can be expressed using the principal value distribution, whose derivative respect to x is that together with (S34) and (S36) yields Rescaling the variables leads finally to that coincides with the Eq. (14) given in the main text with ∆ = L and x − x → t.

The chiral MPS functional
Eq.(S29) is the basis of our proposal for a chiral version of the MPS functional. It is obtained by keeping the terms that depend exclusively on z or z in the Green's function and in the piece proportional to ρ(x ). The terms quadratic in f ± do not possess a chiral/antichiral factorization and stay the same. The chiral functional is defined by truncating S M to .
We have introduced a constant µ whose value will be fixed later on. Replacing the charge density (S2) in (S41) gives where we have eliminated the divergent terms arising when j = k in the sum over the charges. The charges are located in the strip M = R × π[a, b], that is πa < Im z j < πb, j = 1, . . . , N . We shall define the functional that is represented in fig.S1. Observe that for N = 1 the first term in eq.(S42) does not appear. A M [f + , g, {q, z}] is the basic building block to construct the functionals with N > 1. This is a consequence of the sewing condition illustrated in fig.S1, .
If L = 1 the log term does not appear. Similarly, the functional for M 1 is given by where from (S30) one has In the last term of eq.(S47) we used that The chiral MPS functional in momentum space To perform the path integral (S45), we exploit the translation invariance of u ±,L (x − x ) working in momentum space. First of all, we define the Fourier transformf ± (k) of the integrable functions f ± (x), The reality of f ± (x) implies thatf ± (−k) =f * ± (k). The Fourier transform of the functions (S48) is The functionals (S46) and (S47) become in momentum space and where ω +,L (k) = k coth(πkL), ω −,L (k) = −k/ sinh(πkL) .
Upon integration, the LHS of (S55) can be written as e −R M 1 ∪M L with Our goal is to show that e −R M 1 ∪M L coincides with e −R M 1 ∪M L . The last integral in (S66) splits as The integrand of 11 behaves as 1/k 2 for k ∼ 0. To obtain a finite value we use the following regularization method.
The integrands of these function is even, so one can replace the integration interval from (0, ∞) to R. Next, we replace R by a contour R in the complex plane that runs along the negative real axis until the point (−ε, 0), encircles the origin clockwise until the point (0, ) and continues along the positive real axis, The regularized integrals (S71) are given by sinh(πk) cos(kz) sinh(πkL) sinh(πk(L + 1)) , 0 < |Im(z)| < πL (S73) cosh(ikz + πk(L + 1)) sinh(πk(L + 1)) , 0 < Im(z) < π(L + 1) (S74) sinh(πk) sinh(πkL) sinh(πk(L + 1)) , (S75) where we included the ranges of the variable z that come from their relation to z j and z 0 . The simplest integral is that can be computed using residue calculus. Similarly To compute I 2,L (z) by residue calculus we close the contour R on a half circle of radius R in the upper half plane, and take the limit R → ∞. Next, we find the conditions under which the integration along the half circle vanishes. The modulus square of the integrand of (S74) is given by (not including 1/k that does not contribute to the integral along the half-circle) cosh(ikz + πk(L + 1)) sinh(πk(L + 1)) where R has been replaced by R/(π(L + 1)) and x = x π(L + 1) , x = Re z .

Collecting termsR
q j e ikzj sinh(πk(L + 1)) (e πkLf + (k) −f − (k)e −πk ) , that using (S52) can be written as where we have included the zero mode f 0 (recall eq.(S13)). The sewing equation (S45) can finally be written as that is equivalent to the equation (5a) in the main text.
where .. cyl is the vacuum expectation value of the product of chiral vertex operators, at positions z j = iy j in a cylinder of length πL. This result provides a proof of equation (5b) in the main text.

PEPS FUNCTIONAL
The conformal map The PEPS functional is obtained when the region M of the path integral is a rectangle in the complex plane. The conformal map g : M → H can be constructed using the Schwarz-Christoffel formula, where F (φ, k) is the incomplete elliptic integral of the first kind defined as and k is the elliptic modulus that satisfies 0 < k 2 < 1. The points z = ±1, ±1/k, on the real axis are mapped into the vertices of the rectangle with coordinates where K(k) is the complete elliptic integral of the first kind, and K (k) = K(k ), with k = √ 1 − k 2 is the complementary modulus. Moving along the real axis in the z-plane one goes through the points − 1 k , −1, 1, 1 k , that correspond in the u-plane to the points −K + iK , −K, K, K + iK that form the vertices of a rectangle of width 2K and height K . Figure S2 shows these constants that intersect at k = 0.171573 . . . . In the limit k → 1 one has The inverse function of F is the Jacobi amplitude φ = F −1 (u, k) = am(u, k) , (S109) in terms of which the Jacobi elliptic functions are defined sin φ = sin(am(u, k)) = sn(u, k) , (S110) cos φ = cos(am(u, k)) = cn(u, k) , 1 − k 2 sin 2 φ = 1 − k 2 sin 2 (am(u, k)) = dn(u, k) .
Since k is a real parameter, we used that sn(x + iy) = sn(x − iy). Taking the derivatives respect to x and y and using d sn(z) dz = cn(z)dn(z) (S119) where we have used the charge density (S2). Equations (S108) shows that in the limit k → 1, the rectangle M = [−K, K] × [0, K ] becomes the strip R × [0, π 2 ]. Therefore, the PEPS functional (S125) must be closely related to the MPS functional (S42) with a = 0 and ∆ = 1/2. The reason for this fact is the following. In the limit k → 1, the conformal map (S112) becomes g 1 (z) = tanh(z) = e 2z − 1 e 2z + 1 (S126) where we used (S111). On the other hand, the conformal map (S23), with a = 0, ∆ = 1/2, is g(z) = e 2z . Notice that g 1 (z) is a Möbius transformation of g(z), so we expect the wave functions constructed with both functionals to be the same. This issue will be considered elsewhere in more detail.