Dualities for Ising networks

In this note, we study the equivalence between planar Ising networks and cells in the positive orthogonal Grassmannian. We present a microscopic construction based on amalgamation, which establishes the correspondence for any planar Ising network. The equivalence allows us to introduce two recursive methods for computing correlators of Ising networks. The first based on duality moves, which generate networks belonging to the same cell in the Grassmannian. This leads to fractal lattices where the recursion formulas become the exact RG equations of the effective couplings. For the second, we use amalgamation where each iteration doubles the size of the seed lattice. This leads to an efficient way of computing the correlator where the complexity scales logarithmically with respect to the number of spin sites.


INTRODUCTION
Recent years there has been a fascinating interplay between the physics of observables in quantum field theories and geometries in mathematics. Consistency conditions of the observables, arising from fundamental principles of unitarity, locality and symmetries, are often connected to the defining properties of certain mathematical objects. For instance, scattering amplitudes of gauge theories are connected to the positive Grassmannian [1,2] and further into the Amplituhedron [3], couplings of higherdimensional operators in effective field theories, or fourpoint functions of a conformal field theory, are bounded by cyclic polytopes [4]. In these examples, the mathematical object of interest has an intrinsic definition that does not make any direct reference to physics. Thus the principles that determine these physical observables are in effect equivalent to the mathematical properties that define the object. Put in another way, the physical principle becomes emergent.
Recently a fascinating new connection was revealed by Galashin and Pylyavskyy [5]. There, the observable in question is the correlation functions of 2D planar Ising network, which were shown to be equivalent to cells in the positive Orthogonal Grassmannian (OG). The latter also describes scattering amplitudes of supersymmetric Chern-Simon matter theories [2,6].
In the examples mentioned in the beginning, it is the factorization properties of the physical observables that ensure their identification with the mathematical objects. In this note, we identify the corresponding physical property for Ising correlators: under the amalgamation, where two of the external spin sites are identified, the correlation function of the new network can be written as a function of the former. The map is non-linear. However, when cast into the OG, the map is linearised and manifestly preserves positivity. Since any planar network can be built through amalgamation, this ensures that all Ising networks are dual to cells in the positive OG.
Importantly, for fixed boundary sites the cells in OG are finite, whereas there are infinite possible Ising networks. This implies that there are duality transformations that translate between the networks of the same equivalence class. We identify the duality transforms which act locally on the Ising network, and express the effective couplings of the new network in terms of the original one. More precisely, the correlation functions for the original network, is equivalent to that computed in the dual network using the effective coupling. We use the Ising network on the Sierpinski triangle as an example. Note that since this is a self-replicating network, the duality transform becomes the exact Renormalization group (RG) equation for the couplings, and fixed points correspond to phase transitions. We use this correspondence to demonstrate the lack of finite temperature phase transition for general anisotropic couplings.
MAP ISING NETWORK TO OG ≥0 (n, 2n) We begin by considering two-point correlation functions of a planar Ising network with n boundary spin sites. Higher-point functions can be written as products of two-point functions, and thus the correspondence generalizes easily [5]. The two point function is defined as where σ i represents spin sites taking value ±1, E is the set of edges in the Ising network and J ab is the coupling of the edge connecting sites a and b, taking any positive value. Intuitively, since we are considering ferromagnetic couplings we expect that the correlator to be nonnegative. However, as it is given by a summation over arXiv:1809.01231v1 [hep-th] 4 Sep 2018 many terms with alternating signs, its positivity is far from obvious. Remarkably, it was recently proven in [7] that not only is eq.(1) positive, all minors of the n × n unit symmetric matrix (with 1 in the diagonal) σ i σ j are positive (with definite sign)! This is referred to as total positivity. Thus it appears that Ising networks enjoy more positivity than implied by the underlying physics. As suggested in [5], this n × n unit symmetric matrix can be naturally embedded in a n × 2n matrix m ij with the following map: The resulting n × 2n matrix contains rows that are 2ndimensional mutually null vectors, an Orthogonal Grassmannian OG(n, 2n): 1 the moduli space of a null n-plane in a 2n-dimensional space. The correlation functions can then be recovered by the inverse map, written as where ∆ I denotes the n × n minors of OG(n, 2n) with columns I = {i 1 , i 2 , · · · , i n }. Note that for OG(n, 2n), ∆ I = ∆Ĩ whereĨ is the complement of I. Note that the correlation function is given by ratios of the minors and hence GL(n) invariant. As the simplest example where two boundary spin sites are connected by a single edge, the corresponding OG(2, 4) is where With ∆ 12 = c(J), ∆ 13 = 1, ∆ 14 = s(J), using eq.(3) we indeed recover the correct two-point correlation function.
From the above discussion one can see that any unit symmetric matrix can be embedded in a OG(n, 2n). Surprisingly, Galashin and Pylyavskyy showed [5] that 1 The metric in 2n dimensions has an alternating signature. when applied the correlation function of a Ising network, the corresponding OG(n, 2n) is totally non-negative, OG ≥0 (n, 2n)!
The totally non-negative orthogonal Grassmannian has the property that all ordered minors ∆ I s are positive. Note that through eq.(3), total positivity of σ i σ j can be inferred from the positivity of OG ≥0 (n, 2n).

ESTABLISH THE CORRESPONDENCE THROUGH AMALGAMATION
In this section we will establish the correspondence by utilizing the fact that any planar Ising network can be built from the following two basic moves: The first move is somewhat trivial, as one simply declares one of the external spin sites to be internal. Let us then focus on the second operation which corresponds to identifying two boundary spin sites, n and n−1. In terms of correlation functions, there is a simple relation between correlators before and after the "amalgamation": where the · · · amal represents the correlation function of the amalgamated network. To understand this relation, note that as we identify the two external spins, we are essentially subtracting σ n = −σ n−1 from σa∈{±1} . This is easily achieved by writing which leads to eq.(8). The four-point function in the formula can be further recast as sum of products of twopoint functions [8]: Given that the correlation functions can be embedded in OG(n, 2n), let us now see what is the image of the amalgamation operation in the Grassmannian. Since we are identifying two external sites, this operation correspond to reducing OG(n, 2n) to OG(n−1, 2n−2). We would like to find an expression for the minors of OG(n−1, 2n−2), which computes σ i σ j amal , in terms of that of OG(n, 2n), which computes the correlator of the pre-amalgamated network. Let us take an explicit example, from OG (3,6) to OG (2,4). Begin with the matrix where we label the columns that will be removed by the amalgamation as a, b. 2 From eq.(3) we see that the twopoint function for the amalgamated network is simply where the minors above are that of OG (2,4). On the other hand the RHS of eq.(8) tells us that the same twopoint function can also be written as the two-point function of the pre-amalgamated Ising network which resides in OG (3,6). Equating the two leads us to the following identification: One can straight forwardly check that this generalizes to higher points, where the image of eq.(8) in the OG is This is precisely the amalgamation operation of Grassmannian [1]. Note the same is also true for the first case of eq.(7) with the position of columns a, b shifted. 3 Importantly, since the minors of the amalgamated Ising network is simply a positive sum of the pre-amalgamated network, the positivity of the Grassmannian is preserved! Thus as we iteratively build up more and more complicated Ising network, when mapped in to the Grassmannian via eq.(3), it will always reside in OG ≥0 (n, 2n). For higher-point correlators, the translation between the amalgamated and pre-amalgamated is simply modified to: where {A} labels the set of spin sites defining the correlator. Using the map given in [5]: we see that it again translates to the amalgamation of the minors.
2 At this moment the choice of the positions of columns a and b is simply that it is required by eq.(8), but it becomes evident if we map Ising networks fig.(7) to on-shell diagrams using the rules presented in the next section. 3 We thank Pavel Galashin for pointing this out. For example the path shown in the graph indicates that column 5 is spanned by (2,3,4). Since each column is three-dimensional that implies that all ∆I = 0 and this is a top cell. (b): here we see that 1 is spanned by (2,3) indicating that ∆1,2,3 = 0, and thus a co-dimension one cell.

THE STRUCTURE OF OG ≥0 (n, 2n)
The space of OG ≥0 (n, 2n) consists of cells defined by the set of vanishing minors ∆ I s. This forms a stratification: starting from the top cell with all ∆ I s non-zero, one has the co-dimension one boundaries where one of the consecutive minors vanishes, and etc. As shown in [5], the combinatorics of the cell structure for OG ≥0 (n, 2n) is topologically a ball. Each cell can be represented by an on-shell diagram constructed by quartic vertices, which encodes the information of which ∆ I vanishes. For each diagram with 2n-boundary sites, one can associate it with permutation paths that are determined by the rule that one never turns when passing through a vertex. If a path connects boundary sites i, j, then the column vector j in the n × 2n matrix is spanned by the columns of i and those between i and j. For example, cells of OG ≥0 (3, 6) are shown in fig.(1). The total number of vertices represent a dimension of the cell. For more detailed discussion, please see [9].
Since each Ising network should correspond to a cell in OG ≥0 (n, 2n), it must correspond to an on-shell diagram. Indeed starting with an Ising network, one can identify the on-shell diagram by adding a vertex to each Ising edge and two vertices on the two sides of the boundary spin site i, labeled by (2i−1, 2i). If the vertices sit on edges that intersect, they are connected by a line. If an edge extends to the boundary, then the vertex on the edge must connect to the neighboring vertex on the boundary. See examples in fig.(2). It is straightforward to see that the resulting graph will be quartic in nature, as required for OG ≥0 (n, 2n).
Armed with the on-shell diagrams, we can interpret the operation of amalgamation as adding an extra edge and taking the coupling to infinity to identify the spin sites. Take a simple example where one has trivial Ising . (17) Now adding a vertex to two legs ( a and b) in the on-shell diagram correspond to taking a linear combination of the corresponding two columns, witĥ where α 2 − β 2 = 1 to maintain the orthogonal condition of the new cell. Going to the boundary then correspond to α, β → ∞, which sets (â,b) collinear. This produces the on-shell diagram for the amalgamated network.

LOCAL DUALITY TRANSFORMATIONS
Now since a given OG(n, 2n) has finite number of cells, while there are infinite possibilities for planar Ising net-works, this implies that the different networks are secretly dual to each other and fall into equivalence classes. In other words there exist duality transformations that relate different Ising networks belonging to the same cell! Indeed two different looking on-shell diagrams can actually be equivalent if they are related by equivalence moves [1]. For OG ≥0 (n, 2n), the equivalence moves consist of tadpole reductions, bubble reductions and triangle move [2,9]. The equivalence moves reflect the fact that the diagrams are just different, in some cases redundant, parameterizations of the same cell, and thus there exists map that translate between the two charts. When translated into Ising networks means that different Ising networks related by equivalence moves yield the same boundary correlator, with duality transformations applied on the coupling constants.
We begin with the simplest tadpole reductions. It is the case where on-shell diagrams contain a tadpole, whereas the Ising networks contain an external spin coupled to itself or an internal one coupled to the rest via only one edge: It is easy to see that such an Ising spin is decoupled from the rest, therefore can be removed leaving correlation functions invariant.
Let us now consider the Bubble reductions. It is the case where on-shell diagrams contain a bubble. There are two kinds of Ising networks whose corresponding onshell diagrams contain a bubble. Begin with the first case: As shown in the above picture, the Bubble reductions allow us to remove the bubble of the on-shell diagram, and the effect on the Ising network is to remove the isolated spin a and to define an effective coupling via [9] c(J 12 ) = c(J 1a )c(J 2a ) 1 + s(J 1a )s(J 2a ) , Another kind of bubble reduction is given by, As evident from the graph, for this type of bubble we simply have where we denote two couplings of Ising graph on LHS of fig.(21) as J (1) 12 and J (2) 12 , and J 12 the coupling of Ising graph on RHS. On-shell diagrams are reducible if they contain bubbles (or equivalently Ising networks contain sub-diagrams as LHS of fig.(19) and fig.(21), and they can be reduced by applications of bubble reductions. Reducible sub-diagrams may actually be hidden, and can be made visible using another equivalence move as we will discuss below.
Indeed Ising networks exist the triangle move, which is to relate two triangle on-shell diagrams: a (23) The duality transformation which preserves all minors, and therefore correlators, is given by [9] for i = 1, 2, 3 with i + 3 := i is understood. As we have emphasized, Ising networks related by equivalence moves yield the same boundary correlation functions after applying the above duality transformations on the couplings. In particular, there is a unique top cell for a given OG ≥0 (n, 2n), and one can generate all lower cells by taking boundaries which corresponds to the limit J → 0 or J → ∞.

Exact RG and the Sierpinski triangle
When the equivalence move is applied to a network with self-similar structure, the resulting map for the effective coupling becomes an exact RG equation. In this (25) Using the duality transformations discussed previously, it is straightforward to obtain that the effective couplings of the final network in terms of the couplings of the original Ising network. Begin with the simplest case where all the couplings are identical, we find the transformation is given by This recursion relation has appeared previously in [10,11] derived using different methods. As mentioned previously the Sierpinski triangle can be brought into a single triangle by applying the transformation eq.(26) iteratively. In such case, eq.(26) actually contains the information of RG, controlling how the coupling behaves as we go through each iteration. The fixed point to the RG equation then represents the point of the phase transition. Indeed the fixed point solutions to eq.(26) are simply c = 0 and 1, trivial solutions, consistent with known results [11]. Instead of homogenous coupling, we also consider more general anisotropic couplings with (J 1 , J 2 , J 3 ) on the three different sides of each triangle. This is shown as where edges with the same colors have the same coupling. Requiring that the duality map comes back to itself yields three 2-dimensional manifold in the 3-dimensional space of coupling, and the fixed points are given as the intersection points. We show the result of this plotted in fig.(4). One sees that the manifolds only intersect at c(J i ) = 1, 0, thus ruling out finite temperature phase transitions.

CONCLUSIONS AND OUTLOOK
In this note, we explore the dualities between positive orthogonal Grassmannian and 2D planar Ising networks. The dualities are established via the amalgamation, which is a basis move to build any Ising networks and corresponding on-shell diagrams. The classification of cells in positive orthogonal Grassmannian is applied to classify Ising networks, which leads to duality transformations that relate Ising networks of the same equivalence class. As an example, the equivalence moves and duality transformations are utilized to study Ising model on the Sierpinski triangle. Clearly, the idea can be applied to Ising model on other interesting lattice shapes, and one can ask whether or not the classification in terms of cells can be the determination factor of whether an Ising network exhibit a finite temperature phase transition, which may ultimately lead to a completely geometric understanding of planar Ising networks. We will leave this as well as other aspects of the dualities as the future projects.