Topological order,tensor networks and subfactors

Dedicated to the memory of Vaughan Jones

Yasu Kawahigashi

Harvard (online), December 2020Yasu Kawahigashi (Univ. Tokyo) Topological order and subfactors Harvard (online), December 2020 0 / 20

Topological order and operator algebras

Physicists study fusion categories in condensed matterphysics in connection to topological order and gappedHamiltonians. We clarify relations between theirapproaches and subfactor theory of Jones and identifythe role of higher relative commutants.

Outline of the talk:

1 Representation theory, particles and subfactors2 Gapped Hamiltonian, MPS and PEPS3 Work of Bultinck et al.4 Subfactors and bi-unitary connections5 MPO, a tube algebra and a modular tensor category6 PMPO, PEPS and higher relative commutants

Yasu Kawahigashi (Univ. Tokyo) Topological order and subfactors Harvard (online), December 2020 1 / 20

Group representations and particles

A particle behaves like a group representation. For twoirreducible representations π, σ of a group G, we have atensor product representation π ⊗ σ of G and itsirreducible decomposition. This is a fusion of particles.The trivial representation corresponds to the vacuum.The contragredient representation corresponds to anantiparticle. We have pair creation and annihilation of aparticle and an antiparticle.

If the group G is finite, we have only finitely manyirreducible representations of G, and they are closed intaking tensor products and contragredientrepresentations. The trivial/contragredientrepresentation is like the identity/inverse element.Yasu Kawahigashi (Univ. Tokyo) Topological order and subfactors Harvard (online), December 2020 2 / 20

Group representations and bimodules

Let M be a factor, which is a simple von Neumannalgebra on a Hilbert space. A representation theory ofM is not exciting. The correct setting is a theory ofbimodules over M . An M -M bimodule MHM is aHilbert space H with left and right actions of M . For

MHM and MKM , we have a natural notion of arelative tensor product MH ⊗M KM . We also haveirreducible decomposition, the trivial representation

MMM and the contragredient representation MHM .

For a finite group G, its representation theory iscompletely imitated by that of bimodules over a certainfactor M . However, the latter theory is much moregeneral and diverse.Yasu Kawahigashi (Univ. Tokyo) Topological order and subfactors Harvard (online), December 2020 3 / 20

Subfactors and bimodules

Study of a subfactor N ⊂ M (with finite Jones index)was initiated by Jones and many rich structures inconnection to low-dimensional topology and quantumphysics have been discovered in the last 40 years. It givesa “quantum” version of Galois theory throughrepresentation theory. The correct representation theoryof a subfactor N ⊂ M is given by theory of bimodules.A subfactor N ⊂ M gives a bimodule NMN . (Weshould actually take a Hilbert space completion of M .)

We take a relative tensor power of NMN and look atirreducible N -N bimodules arising in this way. If wehave only finitely many such bimodules, we say thesubfactor is of finite depth.Yasu Kawahigashi (Univ. Tokyo) Topological order and subfactors Harvard (online), December 2020 4 / 20

Fusion categories and subfactors

Irreducible N -N bimodules arising from a subfactorN ⊂ M with finite Jones index and finite depth give afusion category which is similar to that of representationsof a finite group.

That is, we have relative tensor products, the trivialrepresentation and the contragredient representation.However, we do not have commutativity of relativetensor products. That is, NH ⊗N KN and

NH ⊗N KN are not isomorphic in general, whileπ ⊗ σ and σ ⊗ π are always unitarily equivalent. Wehave an abstract axiomatization of a (unitary) fusioncategory and any such a category is realized withbimodules over a certain factor.Yasu Kawahigashi (Univ. Tokyo) Topological order and subfactors Harvard (online), December 2020 5 / 20

Gapped Hamiltonian, MPS and PEPS

From a viewpoint of operator algebras, gappedHamiltonians are studied in the context of an infinitetensor product of matrix algebras Md(C). A gappedHamiltonian is a sequence of self-adjoint matrices infinite tensor products and they have a fixed gap betweenthe lowest eigenvalues and the next eigenvalues. Variousfunctional analytic studies have been made.

A matrix product state (MPS) is a certain expression ofa state using a trace of a matrix product. It is useful toexpress a ground state of a gapped Hamiltonian explicitlyfor a one-dimensional system. A projected entangled pairstate (PEPS) is its higher dimensional analogue. We arenow interested in the 2-dimensional case here.Yasu Kawahigashi (Univ. Tokyo) Topological order and subfactors Harvard (online), December 2020 6 / 20

Topological order and anyons

A topological phase is a homotopy equivalence class ofgapped Hamiltonians. One topological phase hastopological order and we have finitely manyquasi-particles called anyons there.

Such a system of anyons is described with a fusioncategory with a non-degenerate braiding (a modulartensor category). That is, the tensor product operation iscommutative in a subtle way. Such a system is expectedto be useful for constructing a topological quantumcomputer. One anyon corresponds to one irreducibleobject of a modular tensor category. Such a modulartensor category also arises from an operator algebraicconformal field theory (K-Longo-Muger).Yasu Kawahigashi (Univ. Tokyo) Topological order and subfactors Harvard (online), December 2020 7 / 20

Tensor networks

A vector (vj) has one index j and a matrix (ajk) hastwo indices j, k. They are graphically represented asfollows.

Similarly, we can consider circles with 3 legs, 4 legs, andso on. Such an object is called a tensor.

The (j, l) entry of the matrix product of (ajk) and(bkl) is given by

∑k ajkbkl. This is pictorially

represented as follows and called a contraction.

We are now interested in tensors with 3 and 4 legs.Yasu Kawahigashi (Univ. Tokyo) Topological order and subfactors Harvard (online), December 2020 8 / 20

Work of Bultinck et al.

Bultinck-Mariena-Williamson-Sahinoglu-Haegemana-Verstraete (2017) considered an algebra of matrixproduct operators (MPOs) arising from a system ofphysically nice tensor networks. Such an algebra is calleda matrix product operator algebra (MPOA). Theyconsidered a fusion category of matrix product operatorsand presented a graphical method to construct aninteresting system of anyons. They get a modular tensorcategory and discuss its physical significance.

They are aware of its similarity to an old work ofOcneanu in subfactor theory. We clarify the role ofsubfactor theory in their work and its extensions.

Yasu Kawahigashi (Univ. Tokyo) Topological order and subfactors Harvard (online), December 2020 9 / 20

Commuting squares and subfactors

Fix a subfactor N ⊂ M with finite Jones index andfinite depth. The diagramM ′ ∩ Mk ⊂ M ′ ∩ Mk+1

∩ ∩N ′ ∩ Mk ⊂ N ′ ∩ Mk+1

gives a square of finite

dimensional C∗-algebras. Such a square satisfies aspecial property called a commuting square. If N or Mhas a nice approximation property with finite dimensionalsubalgebras, then the above commuting square for asingle, sufficiently large k has complete information torecover the original N ⊂ M (Popa). So classification ofsubfactors is reduced to the classification of certaincommuting squares of finite dimensional C∗-algebras.Yasu Kawahigashi (Univ. Tokyo) Topological order and subfactors Harvard (online), December 2020 10 / 20

From a commuting square to a subfactor

Consider the following commuting square of finite

dimensional C∗-algebras.A ⊂ B∩ ∩C ⊂ D.

We can apply the Jones basic construction horizontallyand vertically to get a double sequence {Ak,l}k,l=0,1,2,...

of finite dimensional C∗-algebras with Ak,l ⊂ Ak+1,l

and Ak,l ⊂ Ak,l+1. Then we get two subfactorsA0,∞ ⊂ A1,∞ and A∞,0 ⊂ A∞,1 as certain limitalgebras. If the original commuting square arises fromN ⊂ M as in the previous slide, then the former isanti-isomorphic to N ⊂ M and the latter is isomorphicto N ⊂ M .Yasu Kawahigashi (Univ. Tokyo) Topological order and subfactors Harvard (online), December 2020 11 / 20

Bi-unitary connections

For an inclusion A ⊂ B of finite dimensionalC∗-algebras, we draw the Bratteli diagram as follows,where a dot represents a direct summand.

Look at the Bratteli diagrams of a commuting square.Choose one edge from each diagram corresponding toeach of the four inclusions. We then get a complexnumber from the 4 edges. This is a notion of a bi-unitaryconnection considered by Ocneanu and Haagerup.Yasu Kawahigashi (Univ. Tokyo) Topological order and subfactors Harvard (online), December 2020 12 / 20

Bi-unitary connections and 4-tensors

If one of the two subfactors A0,∞ ⊂ A1,∞ andA∞,0 ⊂ A∞,1 arising from a bi-unitary connection is offinite depth, so is the other (Sato). From now on, weassume this holds.

We would like to define a 4-tensor so that the labels fora wire are given by the edges of a Bratteli diagram andthe value of a 4-tensor with four wires labeled is given bythat of the square for a bi-unitary connection.

Yasu Kawahigashi (Univ. Tokyo) Topological order and subfactors Harvard (online), December 2020 13 / 20

Bi-unitary connections, a fusion category and a 4-tensor

We have natural notions of a tensor product andirreducible decomposition for such bi-unitary connectionscorresponding to those for bimodules. In this way, wehave a fusion category of bi-unitary connectionsequivalent to that of bimodules (Asaeda-Haagerup).

Yasu Kawahigashi (Univ. Tokyo) Topological order and subfactors Harvard (online), December 2020 14 / 20

MPO and PMPO

Using such 4-tensors arising from a bi-unitaryconnection, we define a matrix product operator (MPO)and a projector matrix product operator (PMPO) asfollows, where the length of a circle happens to be 4.

The product structure of these MPOs is isomorphic tothat of relative tensor products of bimodules. We have aso-called zipper condition for these MPOs arising fromthis product structure.Yasu Kawahigashi (Univ. Tokyo) Topological order and subfactors Harvard (online), December 2020 15 / 20

An anyon algebra

Bultinck et al. introduced a new algebra called an anyonalgebra for a family of 4-tensors and obtained a modulartensor category describing a topological order.

They have physical arguments to show braiding and theVerlinde formula in this setting and work out examplesarising from a finite group and a 3-cycle. They are awareof its similarity to Ocneanu’s tube algebra in subfactortheory.Yasu Kawahigashi (Univ. Tokyo) Topological order and subfactors Harvard (online), December 2020 16 / 20

Identification of the two machineries

Suppose we start with a subfactor N ⊂ M with finiteJones index and finite depth. We then have a family ofbi-unitary connections as above. We can next show thatthis family produces 4-tensors satisfying all the settingsof Bultinck et al. simply by changing the normalizationconstants arising from the Perron-Frobenius eigenvectorentries.

Theorem Ocneanu’s tube algebra for the fusioncategory arising from a subfactor and the anyon algebraof Bultinck et al. arising from its bi-unitary connectionsare isomorphic. In particular, the two fusion rules areidentical and the Verlinde formula also holds for thesetting of anyon algebra of Bultinck et al.Yasu Kawahigashi (Univ. Tokyo) Topological order and subfactors Harvard (online), December 2020 17 / 20

PEPS arising from a bi-unitary connection

For a PMPO of length 4, we have 4 exterior wires and 4internal wires. By combining the internal 4 wires intoone, we get a 5-tensor from this diagram.

Bultinck et al. studied a PEPS on a square lattice as inthe above picture, where the square lattice is of size2 × 2 now. This gives a ground state of a localHamiltonian.Yasu Kawahigashi (Univ. Tokyo) Topological order and subfactors Harvard (online), December 2020 18 / 20

PEPS and PMPO

For the PEPS in the right figure in the previous slide, aPMPO of length 8 acts from the exterior wires andPMPOs of length 4 acts from the internal wires. So thisPEPS lives in the intersections of the ranges of thesePMPOs.

The range of a PMPO of length k has physicalsignificance and we would like to identify this range withsomething important and well-studied in subfactortheory: higher relative commutants of a subfactor.Yasu Kawahigashi (Univ. Tokyo) Topological order and subfactors Harvard (online), December 2020 19 / 20

PMPO and the higher relative commutants

Theorem The range of a PMPO of length k isnaturally identified with the higher relative commutantsA′

∞,0 ∩ A∞,k of the subfactor A∞,0 ⊂ A∞,1 since anelement in this range is identified with a flat field ofstrings in subfactor theory.

Note that we have two subfactors A0,∞ ⊂ A1,∞A∞,0 ⊂ A∞,1 in this study. The former appears in theanyon algebra and the latter appears here. They areknown to be complex conjugate Morita equivalent.Yasu Kawahigashi (Univ. Tokyo) Topological order and subfactors Harvard (online), December 2020 20 / 20