14d. Spin models

[math] \newcommand{\mathds}{\mathbb}[/math]

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In order to explain the connection between the Hadamard matrices and the subfactors, we will need some more subfactor theory, regarding the commuting squares. Consider a commuting square in the sense of subfactor theory, denoted as follows:

[[math]] \xymatrix@R=35pt@C35pt{ C_{01}\ar[r]&C_{11}\\ C_{00}\ar[u]\ar[r]&C_{10}\ar[u]} [[/math]]

The idea is that any such square [math]C[/math] produces a subfactor of the hyperfinite [math]{\rm II}_1[/math] factor [math]R[/math]. And, we will see in what follows that, when applying this construction to the commuting square [math]C[/math] associated to a complex Hadamard matrix [math]H[/math], the planar algebra of the corresponding subfactor will appear as the planar algebra [math]P[/math] of the associated quantum permutation group [math]G\subset S_N^+[/math], according to the following scheme:

[[math]] \xymatrix@R=50pt@C=100pt{ G\ar@{-- \gt }[r]&P\\ H\ar@{-- \gt }[u]\ar@{-- \gt }[r]&C\ar@{-- \gt }[u]} [[/math]]

Let us begin with some basics. Given a commuting square [math]C[/math] as above, under suitable assumptions on the inclusions [math]C_{00}\subset C_{10},C_{01}\subset C_{11}[/math], we can perform the basic construction for them, in finite dimensions, and we obtain a whole array of commuting squares:

[[math]] \xymatrix@R=35pt@C35pt{ A_0&A_1&A_2&\\ C_{02}\ar[r]\ar@.[u]&C_{12}\ar[r]\ar@.[u]&C_{22}\ar@.[r]\ar@.[u]&B_2\\ C_{01}\ar[r]\ar[u]&C_{11}\ar[r]\ar[u]&C_{21}\ar@.[r]\ar[u]&B_1\\ C_{00}\ar[u]\ar[r]&C_{10}\ar[u]\ar[r]&C_{20}\ar[u]\ar@.[r]&B_0} [[/math]]

Here the various [math]A,B[/math] letters stand for the von Neumann algebras obtained in the limit, which are all isomorphic to the hyperfinite [math]{\rm II}_1[/math] factor [math]R[/math]. The point now is that the planar algebra of the associated subfactor can be computed explicitely, as follows:

Theorem

In the context of the above diagram, the following happen:

• [math]A_0\subset A_1[/math] is a subfactor, and [math]\{A_i\}[/math] is the Jones tower for it.
• The corresponding planar algebra is given by the following formula:
  [[math]] A_0'\cap A_k=C_{01}'\cap C_{k0} [[/math]]
• A similar result holds for the “horizontal” subfactor [math]B_0\subset B_1[/math].

Show Proof

This is something very standard in subfactor theory, with the result itself being the starting point for various explicit constructions of subfactors, out of concrete combinatorial data, such as the construction of the ADE subfactors mentioned in the above, in the context of the Jones index theorem, the idea being as follows:

(1) This is something quite routine, obtained by working out first the axiomatics of the Jones basic construction, and then using this result.

(2) This is a subtle result, called Ocneanu compactness theorem , which follows by working out the linear algebra of the basic construction.

(3) This simply follows from (1,2), by flipping the diagram.

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Getting back now to the Hadamard matrices, we can extend our lineup of results on the associated von Neumann algebraic aspects, namely Theorem 14.17 and Theorem 14.18, with an advanced statement, regarding subfactors, as follows:

Theorem

Given a complex Hadamard matrix [math]H\in M_N(\mathbb C)[/math], the diagram formed by the associated pair of orthogonal MASA, namely

[[math]] \xymatrix@R=35pt@C35pt{ \Delta\ar[r]&M_N(\mathbb C)\\ \mathbb C\ar[u]\ar[r]&H\Delta H^*\ar[u] } [[/math]]

is a commuting square in the sense of subfactor theory, and the associated planar algebra [math]P=(P_k)[/math] is given by the following formula, in terms of [math]H[/math] itself,

[[math]] T\in P_k\iff T^\circ G^2=G^{k+2}T^\circ [[/math]]

where the objects on the right are constructed as follows:

• [math]T^\circ=id\otimes T\otimes id[/math].
• [math]G_{ia}^{jb}=\sum_kH_{ik}\bar{H}_{jk}\bar{H}_{ak}H_{bk}[/math].
• [math]G^k_{i_1\ldots i_k,j_1\ldots j_k}=G_{i_ki_{k-1}}^{j_kj_{k-1}}\ldots G_{i_2i_1}^{j_2j_1}[/math].

Show Proof

We have two assertions here, the idea being as follows:

(1) The fact that we have indeed a commuting square is something that we already know, coming from the orthogonal MASA result, explained in Theorem 14.18.

(2) The computation of the associated planar algebra is possible thanks to the Ocneanu compactness theorem, corresponding to the formula in Theorem 14.22 (2). To be more precise, by doing some direct computations, which are quite similar to those in the proof of Theorem 14.9, we obtain the formula in the statement. See Jones .

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The point now is that all the above is very similar to Theorem 14.9. To be more precise, by comparing the above result with the formula obtained in Theorem 14.9, which is identical, we are led to the following result, clarifying the situation:

Theorem

Let [math]H\in M_N(\mathbb C)[/math] be a complex Hadamard matrix.

• The planar algebra associated to [math]H[/math] is given by
  [[math]] P_k=Fix(u^{\otimes k}) [[/math]]
  where [math]G\subset S_N^+[/math] is the associated quantum permutation group.
• The corresponding Poincaré series [math]f(z)=\sum_k\dim(P_k)z^k[/math] is
  [[math]] f(z)=\int_G\frac{1}{1-z\chi} [[/math]]
  which is the Stieltjes transform of the law of the main character [math]\chi=\sum_iu_{ii}[/math].

Show Proof

This follows by comparing the quantum group and subfactor results:

(1) As already mentioned above, this simply follows by comparing Theorem 14.9 with the subfactor computation in Theorem 14.23. For full details here, we refer to .

(2) This is a consequence of (1), and of the Peter-Weyl type results from , which tell us that fixed points can be counted by integrating characters.

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Summarizing, we have now a clarification of the various quantum algebraic objects associated to a complex Hadamard matrix [math]H\in M_N(\mathbb C)[/math], the idea being that the central object, which best encodes the “symmetries” of the matrix, and which allows the computation of the other quantum algebraic objects as well, such as the associated planar algebra, is the associated quantum permutation group [math]G\subset S_N^+[/math].

The above results, which are of purely algebraic nature, do not close the discussion, because we still have to understand how the subfactor itself appears from the quantum group. The result here, which is something a bit more technical, is as follows:

Theorem

The subfactor associated to [math]H\in M_N(\mathbb C)[/math] is of the form

[[math]] A^G\subset(\mathbb C^N\otimes A)^G [[/math]]

with [math]A=R\rtimes\widehat{G}[/math], where [math]G\subset S_N^+[/math] is the associated quantum permutation group.

Show Proof

This is something more technical, the idea being that the basic construction procedure for the commuting squares, explained before Theorem 14.22, can be performed in an “equivariant setting”, for commuting squares having components as follows:

[[math]] D\otimes_GE=(D\otimes(E\rtimes\widehat{G}))^G [[/math]]

To be more precise, starting with a commuting square formed by such algebras, we obtain by basic construction a whole array of commuting squares as follows, with [math]\{D_i\},\{E_i\}[/math] being by definition Jones towers, and with [math]D_\infty,E_\infty[/math] being their inductive limits:

[[math]] \xymatrix@R=35pt@C35pt{ D_0\otimes_GE_\infty&D_1\otimes_GE_\infty&D_2\otimes_GE_\infty\\ D_0\otimes_GE_2\ar@.[u]\ar[r]&D_1\otimes_GE_2\ar@.[u]\ar[r]&D_2\otimes_GE_2\ar@.[u]\ar@.[r]&D_\infty\otimes_GE_2\\ D_0\otimes_GE_1\ar[u]\ar[r]&D_1\otimes_GE_1\ar[u]\ar[r]&D_2\otimes_GE_1\ar[u]\ar@.[r]&D_\infty\otimes_GE_1\\ D_0\otimes_GE_0\ar[u]\ar[r]&D_1\otimes_GE_0\ar[u]\ar[r]&D_2\otimes_GE_0\ar[u]\ar@.[r]&D_\infty\otimes_GE_0} [[/math]]

The point now is that this quantum group picture works in fact for any commuting square having [math]\mathbb C[/math] in the lower left corner. In the Hadamard matrix case, that we are interested in here, the corresponding commuting square is as follows:

[[math]] \xymatrix@R=35pt@C35pt{ \mathbb C\otimes_G\mathbb C^N\ar[r]&\mathbb C^N\otimes_G\mathbb C^N\\ \mathbb C\otimes_G\mathbb C\ar[u]\ar[r]&\mathbb C^N\otimes_G\mathbb C\ar[u] } [[/math]]

Thus, the subfactor obtained by vertical basic construction appears as follows:

[[math]] \mathbb C\otimes_GE_\infty\subset\mathbb C^N\otimes_GE_\infty [[/math]]

But this gives the conclusion in the statement, with the [math]{\rm II}_1[/math] factor appearing there being by definition [math]A=E_\infty\rtimes\widehat{G}[/math], and with the remark that we have [math]E_\infty\simeq R[/math].

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All this is of course quite heavy, with the above results being subject to several extensions, and with all this involving several general correspondences between quantum groups, planar algebras, commuting squares and subfactors, that we will not get into.

As a technical comment here, it is possible to deduce Theorem 14.24 directly from Theorem 14.25, via some routine quantum group computations. However, Theorem 14.25 and its proof involve some heavy algebra and functional analysis, coming on top of the heavy algebra and functional analysis required for the general theory of the commuting squares, and this makes the whole thing quite unusable, in practice.

Thus, while being technically weaker than Theorem 14.25, and dealing with pure algebra only, Theorem 14.24 above remains the main result on the subject.

As already mentioned in the beginning of this book, all this is conjecturally related to statistical mechanics. Indeed, the Tannakian category/planar algebra formula from Theorem 14.23 has many similarities with the transfer matrix computations for the spin models, and this is explained in Jones' paper , and known for long before that, from his 1989 paper . However, the precise significance of the Hadamard matrices in statistical mechanics, or in related areas such as link invariants, remains a bit unclear.

From a quantum group perspective, the same questions make sense. The idea here, which is old folklore, going back to the 1998 discovery by Wang of the quantum permutation group [math]S_N^+[/math], is that associated to any 2D spin model should be a quantum permutation group [math]G\subset S_N^+[/math], which appears by factorizing the flat representation [math]C(S_N^+)\to M_N(\mathbb C)[/math] associated to the [math]N\times N[/math] matrix of the Boltzmann weights of the model, and whose representation theory computes the partition function of the model.

This is supported on one hand by Jones' theory in , , via the connecting results presented above, and on the other hand by a number of more recent results, such as those in , having similarities with the computations for the Ising and Potts models. However, the whole thing remains not axiomatized, at least for the moment, and in what regards the Hadamard matrices, their precise physical significance remains unclear.

General references

Banica, Teo (2024). "Invitation to Hadamard matrices". arXiv:1910.06911 [math.CO].