# Multi-path fading and interference mitigation with Reconfigurable
Intelligent Surfaces

Jean Baptiste Gros, Geoffroy Lerosey
Greenerwave, 75005 Paris, France
&Fabrice Lemoult
ESPCI Paris, PSL University, CNRS
The Institut Langevin, 75005 Paris, France
&Mir Lodro, Steve Greedy, Gabriele Gradoni
Radio Frequency Laboratory
George Green Institute for Electromagnetic Research-GGIEMR
University of Nottingham, NG72RD, United Kingdom

###### Abstract

We exploit multi-path fading propagation to improve both the signal-to-
interference-plus-noise-ratio and the stability of wireless communications
within electromagnetic environments that support rich multipath propagation.
Quasi-passive propagation control with multiple binary reconfigurable
intelligent surfaces is adopted to control the stationary waves supported by a
metallic cavity hosting a software-defined radio link. Results are
demonstrated in terms of the error vector magnitude minimization of a
quadrature phase-shift modulation scheme under no-line-of-sight conditions. It
is found that the magnitude of fluctuation of received symbols is reduced to a
stable constellation by increasing the number of individual surfaces, or
elements, thus demonstrating channel hardening. By using a second software-
defined radio device as a jammer, we demonstrate the ability of the RIS to
mitigate the co-channel interference by channel hardening. Results are of
particular interest in smart radio environments for mobile network
architectures beyond 5G.

_K_ eywords SDR, IoT, RIS, Overmoded Enclosure, Interference, Channel
Hardening

Reconfigurable intelligent surface (RIS) technology has opened up a new
approach to achieving efficient, i.e., high data-rate and high capacity,
wireless communications [1, 2]. Quasi-passive planar structures equipped with
tunable reflecting metallic patches were first designed and engineered in [3,
4], to achieve electromagnetic energy focusing within wave chaotic
environments [5]. The concept of smart environments was first introduced in
[6, 7], where the authors proposed to optimize the usage of various access
points inside a building, by controlling the electromagnetic reflectivity of
its walls in order to distribute the signal in an optimal way within the
building. Later on, the idea of using electronically reconfigurable surfaces
for improved wireless communications was proposed in [3, 8], where the fields
incident on the surface were locally controlled in order to focus them on a
given antenna. Follow up works were then published over the years, for example
[9, 10, 11], but the topic remained of little interest to the wireless
community, until a series of papers were published in late 2018 and 2019 [11,
12, 13, 14, 15]. From there, the idea of smart environments using
electronically reconfigurable surfaces, coined Reconfigurable Intelligent
Surfaces (RIS) emerged as a credible major technological advancement for 6G
and has attracted an enormous and growing interest in the wireless
communications community. Associated to RIS there are numerous research topics
ranging from energy efficient wireless communication[16] to channel modeling
[17, 18], RIS based signal modulation and encoding [19], MIMO channel
estimation and beamforming [20, 21, 22], telecommunication performance
evaluation [23], mathematical model and optimization method for wavefront
shaping with RIS [24, 25, 26] and even stochastic analysis approaches [27]. Of
the numerous works proposed, a very large proportion are concerned with
theoretical and mathematical approaches, and very few deal with experimental
demonstrations of RIS [28]. However, the technology has been patented as an
innovative antenna for use in space communication links [29, 30]. Other, but
limited experimental realisations of meta-surfaces exist in the scientific
literature and we mention here the spatially modulated meta-surfaces adopted
in beam-forming [31] and large printed dipoles antenna arrays for long range
focusing [32]. Previous works looked to develop a beam-steering approach based
on wave emission, while in this work we propose to control the propagation of
waves emitted by a simple antenna. Furthermore, while those realisations
remain a valid tool for beam-forming in free-space, the RIS adopted in the
present study possess dual polarisation unit cells controlled by an in-house
sequential optimisation algorithm [33, 29, 34] that accounts for the back-
reflection from the multi-path propagation environment [35]. Of significant
interest is the fact that the meta-surface can be optimised globally through
embedded electronics, e.g., FPGA platforms, integrated within the unit cell
forming the planar meta-surface [36]. Further RIS applications include energy
harvesting [37], motion detection [38], as well as environment based digital
processing and computing [39]. More recently, the ability of the binary RIS to
engineer the eigen-space of the environment has been exploited to create wave
chaos in integrable environments [35] and electronic mode-stirring in static
reverberation chambers [40]. The electromagnetic (EM) environment that we
consider in this study is inherently an overmoded metallic cavity (MC),
equipped with two wall-mounted binary RIS elements. The MC offers a flexible
environment for the emulation of a real-world multi-path propagation
environment [41, 42, 43]. This is depicted in Fig. 1.

Figure 1: Metallic cavity equipped with wall mounted binary RISs. The objects
and absorbers inside the cavity create non-uniform wave field distribution by
reflection and scattering.

For instance, the power delay profile (PDP) of any environment that one could
encounter in practice can be reproduced statistically inside an MC by
tailoring the losses within the metallic cavity sustaining the wireless energy
[44] where a two-antenna universal software radio peripheral (USRP) is
operated inside the MC to create a digital transmission link and an additional
two-antenna USRP is used to create intentional co-channel interference
disturbing the legitimate MC link. Of additional interest is the experimental
work on RIS empowered wireless communications that reports path loss (PL) as
well as scattering parameter studies in anechoic and indoor scenarios. It is
worth mentioning, for examples, the PL model validation in [45] as well as the
transmission performance of static metallic reflectors in [46]. Here, we focus
on the study of performance of the digital transmission based on parameters
monitored by the USRP devices. The study fills the gap between proofs of
principle achieved in the applied physics community and the system predictions
obtained by the wireless communications community. This is a fundamental step
towards transferring the technology towards practical application in mobile
network architectures. Notable work that show related studies is present in
the literature. A recent RIS-based transceiver has been developed, building on
conventional embedded electronics systems in [45]. Furthermore, modulation of
commodity Wi-Fi signals has been used to deliver a wireless digital
transmission based on SDR [47]. Authors in [48] have demonstrated a
programmable metasurface-based transmitter to show comparable BER performance
and transmission rates as more conventional approaches.

The scope of this article is to show how RIS can be used to improve the
signal-to-interference-plus-noise ratio (SINR) by reducing the effect of
inference in, and due to, the environment while increasing the local energy
density focusing. We then widen this scope by considering intentional,
alongsides unintentional and environment driven interference mechanisms.
Achieved results include an experimental proof of channel hardening in RIS-
assisted single-input single-output (SISO) wireless links [49] and that
robustness against arbitrary dynamic intense interference can be achieved by
deploying multiple RISs.

The article is organised as follows. In Section I we revisit the RIS
technology, the software defined radio system, the controlled propagation
environment, as well as the practical realisation of the wireless link in
Section II. In Section III, we provide a detailed account of the results
obtained with and without intentional interference. Finally, in Section IV, we
draw some conclusions and future perspective.

##  1 RECONFIGURABLE TECHNOLOGY AND METHODS

The wireless link is a single-input single output wireless channel operating
within a metallic cavity with variable losses. Following tradition, and as
adopted in [50, 51], the transmitter is hereby referred to as Alice, the
receiver is referred to as Bob, while the intentional interferer in the
external environment is referred to as Eve.

###  1.1 RIS assisted multipath environment

The EM environment within the MC is presented in Figure 2.

Figure 2: Metallic cavity equipped with wall mounted binary RISs. The URSP
X310 supporting the Alice-Bob link operates in presence of a second USRP
supporting the co-channel interferer Eve.

The MC that we have used, is a large metallic enclosure, i.e., of
characteristic size that is much larger than the wavelength of the excitation
(carrier) frequency. In electromagnetic compatibility (EMC) applications, an
MC is often operated under mechanical mode-stirring by moving an irregular
metallic structure though a number of statistically independent positions
[44]. In this work, we leverage the multipath fading generated by a high
density of cavity eigenmodes excited at the carrier frequency within the MC.
The MC is loaded with EM absorbing material, depicted in Fig. 2 as blue cones.
A variable amount of absorbing material is typically used to emulate the PDP
of real-life outdoor/indoor environments.

An USRP X310 maintains the wireless digital link between two antennas in the
same device. Two antennas are chosen to achieve a cross-polarized
configuration, thus minimising the direct coupling within the device, i.e., a
no line of sight situation is created. Also, while a second USRP is present,
its operation is initially switched off in order to study the unperturbed
propagation channel. This means that Alice and Bob can communicate without
disturbance from Eve. Once switched on, Eve creates a real attack to the
wireless link by sending random noise within the same frequency band
supporting the Alice-Bob link.

###  1.2 RIS technology and optimisation

Figure 3: Picture of the $5.2$ GHz RIS used in this experiment. The surface
consists of 76 phase-binary pixels. And each pixel can be configured
electronically to control independently, via the states of two PIN diodes,
both components of the reflected electric field tangential to the pixel plane
thus leading to a total number of 152 effective pixels.

The RIS is a 6G enabling technology that can configure the wireless
propagation environment by using low-cost passive elements. Those elements
perform independent fine tuning of the phase shifts, as well as precise
control of amplitude levels of the reflected waves. The collective response of
an array of passive elements can be engineered for example to beamform, focus,
or cancel the wireless signal propagating in complex EM environments. The RIS
structure that we consider here is based on a binary unit cell that has been
demonstrated in free-space [29] and closed environments [8]. Most of the
theoretical work considers higher phase shift values and fine amplitude level
control, however, in practical terms this may introduce a lot of overhead. In
this work we use a maximum of 2 RISs operating at sub-6GHz frequencies, in the
layout represented in Figure 3. The RIS are connected to a PC via USB that is
used to simultaneously provide power to the integrated control board of the
RIS and send commands to configure the RIS by controlling the bias voltage of
a set of PIN diodes through a set of shift registers. Each surface consists of
76 phase-binary pixels, which are based on hybridizing two resonances [3, 4].
Each pixel can be configured electronically to control independently, via the
states of two PIN diodes, both components of the reflected electric field
tangential to the pixel plane by imposing a 0 or $\pi$ phase shift, thus
leading to a total number of 152 effective pixels per RIS. Since the design of
the RIS is based on resonant effects, the frequency range over which they are
efficient is limited to 1 GHz around 5.2 GHz. The RIS design shown in Figure 3
has already featured in several experimental investigative works ranging from
wave chaos physics to EMC and wireless communications [35, 29, 52] and its
working principles are described in the following papers [3, 4]. The RISs are
used to improve the communication between Alice and Bob and to that end, we
use an optimisation procedure to find the RIS configurations that will improve
the quality of the data link. This procedure, described later in the paper,
requires the measurement of quantities related to the performance of the data
links. Different approaches exist, for instance a pre-optimisation of the RIS
configurations, through channel sounding, via the evaluation of the scattering
matrices between Alice and Bob using a vectorial network analyser [34, 53],
that is subsequently replaced by an actual communication set-up. However as we
are using a SDR based techniques and a USRP to establish the link, we will
have direct access to key performance indicators (KPI) of the communication
link such as bit-error-rate (BER) or error vector magnitude (EVM). Therefore,
we can use these KPIs to simultaneously probe the quality of the communication
channel and optimize it by adjusting the RIS configuration.

###  1.3 Software Defined Radio Setup

We used two USRPs, hereafter referred to as USRP I and USRP II, for creating a
wireless communication data link and intentional interference respectively. A
X310 USRP (USRP I) with UBX-160 RF daughterboard is a high performance full-
duplex SDR that has been used to sustain a high data-rate 2x2 MIMO
communication link. Another SDR, again from Ettus Research, a B210 USRP (USRP
II) with integrated RF front-end is used to generate the interference signals.
The USRP II is also a full-duplex SDR that can also maintain 2x2 MIMO link,
however at low sampling rates in contrast to the USRP I. We connected the Tx
antenna (Alice) with channel 1 of USRP I and Rx antenna (Bob) with channel 2
of USRP I, in order to avoid intra-board leakage. More precisely, a log
periodic antenna was used for Alice and a dipole antenna was used for Bob.
Both the antennas are placed in cross-polarized orientation to avoid direct
coupling between them.Therefore, there is no direct link between Alice and
Bob, neither via internal SDR cross-talk, nor through direct LOS.

The experiment begins with a degraded channel and performs RIS-assisted
optimization of the link. USRP I was set to operate at sampling rate of 400
kS/s and the USRP configuration parameters are summarized in Table 1. The QPSK
modulated signals transmitted in the main communication link were generated
using Simulink according to previous work which details the Simulink baseband
models [54][55]. The base-band interference signals were generated using GNU
Radio and transmitted using USRP II at 5.2 GHz. Typical waveform sent by Alice
is shown in Fig.4a and the typical noise waveform sent by Eve is shown in Fig.
4b. Furthermore, typical received and optimized waveforms at Bob are shown in
Fig. 5 with relative time frame. The waveforms shown at Bob are at different
levels of interferences.

(a)

(b)

Figure 4: Alice and EVE waveforms (a) typical QPSK modulated waveform as sent
by Alice (b) snapshot of EVE noise waveform.

(a)

(b)

(c)

Figure 5: Received waveforms as shown in blue and optimized waveforms as shown
in orange for various interference levels (a) -10 dB (b) -5 dB (c) 0 dB.

We avoided any packet loss risk in the communication between the SDR and host
PC over Ethernet cable by performing IQ data acquisition at lower sampling
rates. Packet loss can be observed by monitoring the presence and absence of
long string of ones and zeros at the terminal. Furthermore, we monitored the
transmitter and receiver system object, as the presence of overflows may lead
to inconsistent performance. USRP I and II were set to stream data in real-
time, hence reliable EVM measurements were recorded that could only be
influenced by the propagation environment in the cavity. The EVM was regularly
measured in a time window of 10 s from an stable QPSK constellation diagram
after all the hardware impairments from the received waveform were removed.

Parameters | Values
---|---
Sampling Rate | 400 kS/s
Master Clock Rate | 200 MHz
Interpolation/Decimation | 500
Tx Channel | Channel 1
Rx Channel | Channel 2
Tx Gain | 1 dB
Rx Gain | 0 dB
Table 1: Main communication link USRP parameters

##  2 RESULTS

###  2.1 RIS-assisted channel hardening: NLOS without intentional interference

Figure 6: Constellation diagrams between Alice and Bob for three different
scenario. In red, both RIS are switched off. In light green, only one RIS is
used to optimize the mean EVM. In dark green, both RIS are used to optimize
the mean EVM. The gold circle correspond to ideal symbols for QPSK modulation.
In all scenario EVE is switched off.

In presence of environment related interference generated from multi-path
propagation, the wireless link between Alice and Bob operates unperturbed as
Eve does not interfere with QPSK symbols. Starting from a random configuration
for both the two RISs, we measure the constellation diagram, red cloud of
points in Fig. 6, and in each quadrant of the complex plane, we evaluate

$\sigma_{\mathrm{k}}=\frac{\left|\sigma_{\mathrm{k}}-\sigma_{\mathrm{I}}\right|}{\sigma_{\mathrm{I}}}$
(1)

where $\sigma_{\mathrm{k}}$ is the error vector magnitude between the
$\mathrm{k}$-th actual symbol sent associated to the complex value
$s_{\mathrm{k}}$ and the ideal QPSK symbol $s_{\mathrm{I}}$, gold circles in
Figure 6. Then, we compute the mean EVM, hereafter denoted by
$\sigma_{\mathrm{QPSK}}$, and defined as the root mean square of the ensemble
of $\left\\{\sigma_{\mathrm{k}}\right\\}$

$\mathrm{EVM}=\sqrt{\frac{1}{N}\sum_{\mathrm{k}}^{N}\sigma_{\mathrm{k}}^{2}}.$
(2)

We are interested in the mean EVM as it is a KPI that probes the quality of
the digital communication link, and it is directly obtained using EVM block in
the baseband Simulink model. In other words, we do not need to post-process
data externally and the EVM is measured in real-time in the baseband model.
The red cloud shown in Fig. 6 corresponds to the constellation diagram
measured with the initial random configuration of the RIS, and its mean EVM is
$\mathrm{EVM}=90\,\%$. Starting from this initial situation, we then use the
RIS to improve the communication link between Alice and Bob by minimizing the
mean EVM. To do so, we use the iterative optimisation algorithm described in
[56]. The state of the first pixel is flipped to its opposite state and kept
in this new state, if the goal function (here the average EVM) decreased.
Otherwise, the pixel is flipped again back to its previous state. This
procedure is repeated with all pixels one by one and typically requires
multiple loops across the RIS reflection matrix. The optimization is
terminated when the goal function no longer decrease during one loop of the
optimization. Because it based on root mean square, the minimization of the
mean EVM simultaneously allows us to reduced the distance between the
barycentres of the measured constellation and the ideal symbols (gold circle
in Figure 6) and the spreading of the clouds of points around those
barycentres. It is worth noticing that we are optimizing measured quantities
that are related to the performance of the _digital_ wireless data link,
instead of the SNR. In particular, we obtain the simultaneous increase of both
the average and the variance of symbol energy by minimizing the average EVM.
For this experiment we first optimize the configuration of only one RIS,
keeping the state of the second one unchanged. The next step, starting from
the final optimized configuration, is to include both RISs in the optimization
process. The final constellation resulting from the optimization involving one
or two RIS are shown in Fig. 6. The light green clouds of points correspond to
the configuration where only one RIS is used. The mean EVM in this case moves
from $\mathrm{EVM}=90\,\%$ to $\mathrm{EVM}=36.4\,\%$. The dark green clouds
is the constellation diagram resulting from the optimization with both RIS. In
this case, the mean EVM is $\mathrm{EVM}=8\,\%$ and the measured symbols are
really close to the ideal symbols. In both scenarios, when we use the RIS we
are able to improve the communication link between Alice and Bob by optimizing
in real-time the mean EVM given by the SDR set-up. Of course, the larger the
controllable surfaces the higher the degrees of freedom to control the EM
field inside the chamber, and the closer we can get to ideal constellation
symbols. Indeed, it is known from [8], that the amount of power we are able to
focus with RIS is directly linked to the number of controllable degrees of
freedom. Therefore, in comparison with the scenario using two RIS elements,
against only one RIS in the present experiment, we achieve enough wave field
control to separate the symbols. However, the power that we are able to focus
on Bob is limited, and the distance between cloud barycenters and ideal
symbols is bounded. This is mainly due to the limited number of RIS pixels, as
well as to the presence of short orbits [57] that do not intercept the RIS,
and thus couple transmitter and receiver through the multi-path propagation
environment. Moreover, the width of the cloud associated to the constellation
symbols after RIS iterative optimization is also bounded. Those fundamental
limitations to the RIS operation within a reverberating environment are mainly
ascribed to: i) the finite loss factor of the cavity, established by
imperfectly conducting walls; ii) the finite RIS area covering the cavity
walls, which limits spatial self-averaging underpinning the channel hardening
effect. Both the effects contribute to form the residual fluctuation of the
constellation symbols, in addition to thermal noise fluctuations at receiver.
In particular, based on [8] and the fluctuation of the quality factor of
overmoded enclosures [58], we expect that the radius of the fluctuation cloud
is proportional to the standard deviation of the cavity quality factor. Hence,
$\sigma_{Q}\propto\sqrt{1/M}$ where $M$ is the maximum number of ergodic
cavity eigenmodes controlled by the RIS in the optimal configuration. The
maximum data-rate is subject to the same limitations of the average SNR
achievable by a prescribed number of RIS pixels. However, the CH effect
accompanying optimal RISs makes the wireless communication link robust against
multipath fading and co-channel interference, as explained in the next
subsection, even with a moderate number of controlled ergodic eigenmodes. This
has been recently ascribed to the law of large numbers for RISs with an
increasing number of independent pixels [59].

###  2.2 RIS-assisted channel hardening: NLOS with intentional interference

Figure 7:  Constellation diagrams between Alice and Bob when EVE is switched
on and it is acting as a source of co-channel interference. From top panel to
bottom panel, the intensity of interferences is sequentially increased by step
of $5$ dB, starting from interferences with an intensity of $10$ dB lower than
the signal between Alice and Bob. The red constellations correspond with the
constellation at the time of activation of EVE. The dark green constellations
are the constellation obtained after re-optimization of the mean EVM using
both RIS. The gold circle correspond to ideal symbols for QPSK modulation.

In this section, we report the ability of RIS to cancel unwanted co-channel
interference caused, for instance, by a jammer or simply other emitters in the
environment perturbing the channel. To do this, we use as the sources of
interference the signal provided by USRP II (Eve)in Figure 2. When Eve is
switched on, the transmission between Alice and Bob is attacked by the co-
channel interference presented by Eve who emitted in the environment Gaussian
noise with tunable intensity. The waveform transmitted by Eve can be seen in
Fig. 4b which was transmitted for three levels of unintended interference. The
ratio between the signal between Alice and Bob and Eve interference is defined
as Int. Starting from a random RIS configuration and $\textrm{Int}=10$ dB, we
first optimize the mean EVM using both RIS, then when the optimisation reaches
a plateau, we increase the interference intensity by $5$ dB and re-optimized
the mean EVM until a new plateau is reached. This procedure is repeated until
the signal and interference strength are equal. The corresponding
constellation diagrams are shown in Fig.7. From top to bottom panel, the
interference intensity is increased at steps of $5$ dB. The red constellations
on each correspond to the constellation measured at the time when Eve power is
increased. From top to the bottom , the mean EVM associated to these red
constellations are

  * •

$\mathrm{EVM}=90\,\%$ for interference strength $10$ dB lower than the signal
strength and random configuration of RIS

  * •

$\mathrm{EVM}=45\,\%$ after first optimization and an interference to signal
power ratio $\textrm{Int}=-5$ dB

  * •

$\mathrm{EVM}=37\,\%$ after the second optimization and an interference to
signal power ratio $\textrm{Int}=0$ dB.

Let us note that the more the SINR increase the more the constellation symbols
are spread through the complex plane. In each panel of Fig.7, dark green
constellations are the ones obtained after the optimizations that follow the
increase of interferences’ power. Whatever the interference to signal power
ratio, the optimized constellation barycenter returns to be close to ideal
symbols, with a mean EVM $\mathrm{EVM}=8\,\%$. We thus show the ability of RIS
to mitigate and even cancel the co-channel interference operated by Eve.
Indeed, we would like to draw the attention of the reader to the fact that all
optimized constellation diagrams shown in Figure 7 are similar to the one
shown in Figure 6, which are associated to the case with no interference and
both RISs used for the optimization.

##  3 Conclusion

We have shown that robust multi-path fading mitigation can be achieved by
using reconfigurable intelligent surfaces within an overmoded metallic
enclosure supporting rich multipath fading. The adopted environment mimics
complex wave propagation in wireless indoor channels, while controlling
distributed losses within its interior volume. We have found that RIS
optimization results in channel hardening, besides energy maximization widely
reported in the scientific literature. A software defined radio system makes
it possible to observe channel hardening directly in the constellation of a
QPSK digital modulation. Achievements show the RIS capability of optimizing
wireless links under combined multi-path fading and intentional/unintentional
co-channel interference, making it an appealing technology for operation in
contested electromagnetic environments such as multi-operator mobile wireless
networks.

##  4 Acknowledgment

The work of GG was supported in part by the Royal Society under Grant
INF\R2\192066; in part by the EPSRC under Grant EP/V048937/1, and in part by
the European Commission through the H2020 RISE-6G Project under Grant
101017011. Greenerwave acknowledges funding from the French “Ministere des
Armées, Direction G␣́ne␣́rale de l’Armement” and “Agence de l’Innovation de
de␣́fense” through the RAPID “3SFA” project and the RAPID “M3SFA” project.

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##  Appendix A USRP Architecture and performance

Figure 8: X310 USRP architecture

Figure 8 shows an architecture of USRP I. The USRP I has: i) modular structure
where RF cards can be separately interfaced with the motherboard; ii) high
performance FPGA which is interfaced with RF frontends, host interfaces, and
DDR3 memory. When used with UBX-160 cards, the USRP I can provide a maximum
bandwidth of up to 160 MHz, and it is tunable from 10 MHz to 6 GHz. It has a
high-speed 14-bit ADC and 16-bit DACs per RF chains with maximum clock rates
of 200 MS/s and 800 MS/s respectively. Additionally, USRP I has a clock and
timing distribution circuitry to distribute external 10 MHz and 1 PPS
synchronization signals, and has out-of-the-box support for 1 GigE and 10GigE
that can be harnessed using SFP+ adapters. However, the exploitation of
maximum sampling rates depends upon host capabilities, including waveform
complexity, host PC speed, and writing speed of memory.


# Two-dimensional topological order and operator algebras

Yasuyuki Kawahigashi
Graduate School of Mathematical Sciences
The University of Tokyo, Komaba, Tokyo, 153-8914, Japan
e-mail<EMAIL_ADDRESS>
Kavli IPMU (WPI), the University of Tokyo
5–1–5 Kashiwanoha, Kashiwa, 277-8583, Japan
Trans-scale Quantum Science Institute
The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan
and
iTHEMS Research Group, RIKEN
2-1 Hirosawa, Wako, Saitama 351-0198,Japan

###### Abstract

We review recent interactions between mathematical theory of two-dimensional
topological order and operator algebras, particularly the Jones theory of
subfactors. The role of representation theory in terms of tensor categories is
emphasized. Connections to 2-dimensional conformal field theory are also
presented. In particular, we discuss anyon condensation, gapped domain walls
and matrix product operators in terms of operator algebras.

##  1 Introduction to operator algebras

In quantum mechanics, an observable is represented by a self-adjoint operator
on a complex Hilbert space of states. It would be convenient if we are allowed
to make algebraic operations such as addition and multiplication on them, so
we consider a set of operators on a complex Hilbert space which is closed
under addition and multiplication. In order to add and multiply them freely,
we restrict our attention to bounded linear operators, though many observables
appearing quantum mechanics are unbounded. One way to get a bounded linear
operator from an unbounded one is exponentiation. We exponentiate an unbounded
self-adjoint operator $A$ to get a (bounded) unitary operator $\exp(iA)$.
Since self-adjointness of an observable is important, we also require the set
of operators is closed in the $*$-operation. (In mathematics, we write $A^{*}$
for the adjoint of $A$ rather than $A^{\dagger}$ which is more common in
physics literature.) We further require that it is closed in a certain
topology, and then we call it an operator algebra.

We have two natural choices for this topology. One is the norm topology which
arises from uniform convergence on the unit ball of the Hilbert space and the
other is the strong operator topology which arises from pointwise convergence
on the Hilbert space. (The strong operator topology is weaker than the norm
topology.) When an operator algebra is closed in the norm topology, it is
called a $C^{*}$-algebra. When an operator algebra is closed in the strong
operator topology, it is called a von Neumann algebra. The latter was
introduced by von Neumann in connection to quantum mechanics and
representation theory as the name shows. Both naturally appear in mathematical
physics. Being closed in a weaker topology is a stronger condition, so a von
Neumann algebra is automatically a $C^{*}$-algebra by definition, but it is
usually not very convenient to regard a von Neumann algebra as a
$C^{*}$-algebra since a typical von Neumann algebra is often quite different
from a typical $C^{*}$-algebra. For example, a typical $C^{*}$-algebra
appearing in applications is often separable (as a Banach space), but a von
Neumann algebra is never separable unless it is finite dimensional.

Finite dimensional $C^{*}$-algebras and finite dimensional von Neumann
algebras are the same, and they are finite direct sums of
$M_{n}({\mathbb{C}})$, $n\times n$-matrix algebras with complex entries. From
this viewpoint, one can say that an infinite dimensional operator algebra is
an infinite dimensional analogue of a matrix algebra.

A commutative $C^{*}$-algebra is isomorphic to $C(X)$ for some compact
Hausdorff space $X$. A commutative von Neumann algebra is isomorphic to
$L^{\infty}(Y,\mu)$ for some measure space $(Y,\mu)$. From this viewpoint, one
can say that a general operator algebra is a noncommutative version of a
function algebra. We sometimes say for this that a general operator algebra is
a function algebra on a noncommutative space. Noncommutative geometry of
Connes[15] is a far advancement of this idea.

A simple von Neumann algebra, in the sense that it has only trivial closed
two-sided ideals in the strong operator topology, is called a factor. A
general von Neumann algebra is decomposed into a direct integral of factors, a
generalized version of a direct sum, just like a representation decomposes
into a direct integral irreducible representations. A von Neumann algebra is a
factor if and only if it does not decompose into a direct sum of two nonzero
von Neumann algebras. A von Neumann algebra is a factor if and only if its
center is the complex number field ${\mathbb{C}}$.

The von Neumann algebra $B(H)$, the set of all bounded linear operators on a
Hilbert space $H$, is a factor, but this is not very exciting. A much more
interesting factor is constructed as follows. A matrix $x\in
M_{n}({\mathbb{C}})$ is mapped to $\left(\begin{array}[]{cc}x&0\\\
0&x\end{array}\right)\in M_{2n}({\mathbb{C}})$. With this embedding, we have
an increasing sequence of matrix algebras,

$M_{2}({\mathbb{C}})\subset M_{4}({\mathbb{C}})\subset
M_{8}({\mathbb{C}})\subset\cdots.$

We take an increasing union $\bigcup_{n=1}^{\infty}M_{2^{n}}({\mathbb{C}})$.
This has a natural representation on a Hilbert space and we take the closure
of the image in the strong operator topology. This gives a factor and is
called the hyperfinite II1 factor.

When one factor is $N$ contained in another factor $M$, we say it is a
subfactor. Its study is somehow analogous to an algebraic study of a subgroup
$H\subset G$ and a subfield $K\subset L$. We have a notion of the index
$[G:H]$ for a subgroup $H\subset G$ and the degree of extension $[L:K]$ for a
field extension $L\supset K$. We have a similar notion of the Jones index
$[M:N]$ for a subfactor $N\subset M$. One important new feature is that it is
a positive real number larger than or equal to 1, or infinity, and can easily
take a non-integer value. We are interested in algebraic studies of subfactors
with finite Jones index. Subfactors of the hyperfinite II1 factor have very
rich structures.

Theory of subfactors produced a new topological invariant, the Jones
polynomial[33] for knots, which is expected to be useful for topological
quantum computations[61].

##  2 Topological phases of matter and tensor categories

We consider a certain 2-dimensional status of matters and a typical example is
thin liquid on a large plane. A point on the plane can have a special status
by excitation. An excited point behaves like a particle (quasi-particle) and
is called an anyon.

Suppose we have finitely many anyons and study exchanges among them. The
natural group for exchanges is the braid group and we have braid group
statistics. (If the original dimension where quasi-particles live is 3, then
the natural group representing such an exchange is the permutation group, and
we have a boson or a fermion.) An anyon is a more general version of a boson
and a fermion and we have this name because a phase arising from an exchange
of quasi-particle can take any value. A modular tensor category gives a
mathematical description of such a system of finitely many anyons. Each
irreducible object of a modular tensor category corresponds to an anyon. Two
anyons are fused and produce new anyons. This is a fusion of anyons. (See
Bakalov-Kirillov[4] for details of modular tensor categories.) A so-called
non-abelian anyon is expected to produce a topological quantum computer, but
such an anyon has not been observed experimentally yet.

Noncommutative geometry[15] is a certain generalization of Riemannian geometry
using operator algebras. An operator algebra with a certain extra structure is
regarded as a noncommutative manifold. It has been used to study the
fractional quantum Hall effect. This is also related to anyons.

A topological phase is also understood in terms of homotopic classification of
gapped Hamiltonians and we have a version called symmetry protected
topological phases. It has been recently studied extensively in the context of
$C^{*}$-algebras and index theorems by Ogata[54].

We next explain the notion of a fusion category, which is a more general
version than a modular tensor category. Recall that a finite group $G$
consists of the following ingredients.

  1. 1.

An associative multiplication

  2. 2.

The identity element

  3. 3.

The inverse elements

For a finite group $G$, the set of its finite dimensional unitary
representations have the following structures.

  1. 1.

Irreducible decomposition into finitely many ones

  2. 2.

An associative tensor product

  3. 3.

The identity representation

  4. 4.

The contragredient representation

Abstract axiomatization of such a set gives a notion of a fusion category.
That is, each object behaves like a representation, and we have only finitely
many irreducible objects. We have direct sums and irreducible decompositions,
tensor products of objects, and the trivial object and the dual object of each
object, which behaves like a contragredient representation. We have morphisms
between objects, like intertwiners between representations. For two
representations $\pi$ and $\sigma$ of a group $G$, the two tensor products
$\pi\otimes\sigma$ and $\sigma\otimes\pi$ are trivially unitarily equivalent,
but such equivalence is not assumed for a general fusion category. In this
sense, the tensor product operation in a fusion category is noncommutative in
general. Fusion categories give a special subclass of tensor categories. (See
Etingof-Nikshych-Ostrik[20] for a general theory of fusion categories. See
Bischoff-Kawahigashi-Longo-Rehren[7] for tensor categories and operator
algebras.) Each object of a fusion category has a dimension, which is a
positive real number larger than or equal to 1. A dimension is additive with
respect to a direct sum and multiplicative with respect to a tensor product.

We have an important class of fusion categories for which the above
commutativity of tensor products holds in some mathematically nice way. Such
commutativity is called braiding because it is similar to reversing a crossing
of two wires. (A wire labels an object in a fusion category.) A braiding
naturally comes in a pair — overcrossing and undercrossing. It is more
interesting if these two are really different. If this is the case in some
appropriate sense, this gives a notion of a modular tensor category, as
mentioned above for anyons. So objects in a modular tensor category behave
more like representations of a group, and they also behave like particles.

The Kitaev toric code[34] gives one example of a modular tensor category. In
this example, we have four irreducible objects representing four anyons. Each
irreducible object has dimension 1 in this example, and the tensor product
rules (also called fusion rules) are given by the group multiplication of
${\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/2{\mathbb{Z}}$. (This group
structure can give a trivial braiding, but here we have a different,
nontrivial braiding structure.)

Another example is the Fibonacci category and it has two anyons, the trivial
one labeled as $1$ and another one labeled as $\tau$. The fusion rules are
given by $\tau^{2}=1\oplus\tau$. This is expected to be related to fractional
quantum Hall liquids. This is also related to the Jones polynomial at the
deformation parameter $q=\exp(2\pi i/5)$. The name Fibonacci comes from the
fact that certain intertwiner spaces have dimensions given by the Fibonacci
numbers.

Suppose we have a modular tensor category having $n$ irreducible objects. The
braiding produces an $n$-dimensional unitary representation $\pi$ of
$SL(2,{\mathbb{Z}})$, the modular group. (See Bakalov-Kirillov[4].) We set
$S=\displaystyle\pi\left(\begin{array}[]{cc}0&1\\\ -1&0\end{array}\right)$ and
$T=\displaystyle\pi\left(\begin{array}[]{cc}1&1\\\ 0&1\end{array}\right)$.

A matrix $Z$ is called a modular invariant if it satisfies the following,
where the index $0$ means the trivial object, often called the vacuum, which
behaves like a trivial representation.

  1. 1.

$Z_{\lambda\mu}\in\\{0,1,2,\dots\\}$.

  2. 2.

$Z_{00}=1$.

  3. 3.

$ZS=SZ$, $ZT=TZ$.

When a modular tensor category is given, the number of modular invariants is
always finite, and they are sometimes explicitly classified. See Cappelli-
Itzykson-Zuber[12] for such a classification. A typical appearance of a
modular invariant is 2-dimensional conformal field theory. Use of modular
invariants in classification in operator algebraic conformal field theory has
been given in Kawahigashi-Longo[42][43].

##  3 Algebraic quantum field theory

We now recall a mathematical (axiomatic) framework of quantum field theory.
Our basic ingredients are as follows (as in the Wightman axioms).

  1. 1.

The spacetime

  2. 2.

The spacetime symmetry group

  3. 3.

Quantum fields (operator-valued distributions) on the spacetime

This setting is well-studied and has a long history, but is difficult to
handle from a mathematical/technical viewpoint, and we consider a different
approach based on operator algebras instead here. We now consider a family of
von Neumann algebras parameterized by spacetime regions. Each von Neumann
algebra is generated by observables on a spacetime region. Though observables
can be easily unbounded operators, we consider von Neumann algebras of bounded
linear operators as before. Our idea is that relative positions of this family
of von Neumann algebras encode physical information of a quantum field theory.
This approach is called algebraic quantum field theory as in Haag[28].

A 2-dimensional conformal field theory is a quantum field theory with
conformal symmetry on the 2-dimensional Minkowski space. It decomposes into an
extension of a tensor product of two quantum field theories living on the
compactification of the light rays $\\{x=\pm t\\}$ and each is called a chiral
conformal field theory (a chiral half). It is a quantum field theory on the
spacetime $S^{1}$ with ${\mathrm{Diff}}(S^{1})$-symmetry. (Now the space and
the time are mixed into one dimension).

An operator algebraic formulation of a chiral conformal field theory is called
a (local) conformal net in Kawahigashi-Longo[42]. This is based on operator
algebras of observables on $S^{1}$. We have another mathematical formulation
of conformal field theory, based on Fourier expansions of operator-valued
distributions on the circle $S^{1}$, which is called a vertex operator
algebra. (This name means an algebra of vertex operators and is not directly
related to operator algebras.) See Kawahigashi[36][39] as general references
on conformal nets and vertex operator algebras. These two are supposed to be
different mathematical formulations of the same physical theory, so they
should be equivalent, at least under some nice assumptions. We have shown that
we can pass from a vertex operator algebra to a local conformal net and come
back under some mild assumptions as in Carpi-Kawahigashi-Longo-Weiner[13].
This relation of the two has been much studied recently.

Under the standard set of axioms of a conformal net, each von Neumann algebra
appearing in one is always isomorphic to the unique Araki-Woods factor of type
III1. So each von Neumann algebra contains no information about a conformal
field theory, but as a family of von Neumann algebras, it does contain
information about conformal field theory.

Representation theory of a conformal net is called a theory of superselection
sectors. Each representation (of the family of von Neumann algebras on another
Hilbert space) is called a superselection sector as in Doplicher-Haag-
Roberts[18][19]. They gave a tensor product structure there as a composition
of DHR endomorphisms and it has a braiding in the setting of conformal field
theory as in Fredenhagen-Rehren-Schroer[24][25]. Under some finiteness
assumption, called complete rationality, we get a modular tensor category of
representations as in Kawahigashi-Longo-Müger[44].

Representation theory of a vertex operator algebra is theory of modules. We
have a tensor product structure and a braiding for modules. Under some
finiteness assumption, we also get a modular tensor category of modules as in
Huang[29][30].

Under nice identification of a local conformal net and a vertex operator
algebra as in Carpi-Kawahigashi-Longo-Weiner[13], Gui[27] has identification
of the corresponding representation categories for many examples.

##  4 Topological phases of matter and gapped domain walls

Recall a 2-dimensional topological order is described with a modular tensor
category. Suppose we have two topological orders described with two modular
tensor categories ${\mathcal{C}}_{1}$ and ${\mathcal{C}}_{2}$. The exterior
tensor product ${\mathcal{C}}_{1}\boxtimes{\mathcal{C}}_{2}^{\mathrm{opp}}$,
where “opp” means reversing the braiding, gives a new modular tensor category.
The number of new anyons is the product of the two numbers of anyons.

We have a physical notion of a gapped domain wall between the two topological
orders and it is mathematically defined to be an irreducible local Lagrangian
Frobenius algebra with the base space $\bigoplus
Z_{\lambda\mu}\lambda\boxtimes\bar{\mu}$ in
${\mathcal{C}}_{1}\boxtimes{\mathcal{C}}_{2}^{\mathrm{opp}}$, where
$Z_{\lambda\mu}=0,1,2,\dots$ gives a generalized modular invariant. (See
Kawahigashi[37] for a mathematically precise formulation.)

Lan-Wang-Wen[46] conjectured that if we have a generalized modular invariant
matrix $Z$ with $Z_{00}=1$ for modular tensor categories ${\mathcal{C}}_{1}$
and ${\mathcal{C}}_{2}$, and the entries of matrix $Z$ satisfy some
inequalities about multiplicities, then there would exist a corresponding
gapped domain wall.

However, subfactor theory easily disproves this conjecture. Actually, the
charge conjugation matrix $(\delta_{\lambda\bar{\mu}})$ gives a counterexample
for some modular tensor category ${\mathcal{C}}_{1}={\mathcal{C}}_{2}$ arising
from a finite group by a recent work of Davydov[16]. We have also given a
correct form of the conjecture using Witt equivalence in Kawahigashi[37]

In physics literature, we have a notion of composition of two gapped domain
walls and its irreducible decomposition. We would like to formulate this
notion mathematically.

Suppose we have three topological orders described with three modular tensor
categories ${\mathcal{C}}_{1}$, ${\mathcal{C}}_{2}$ and ${\mathcal{C}}_{3}$,
respectively. We further assume to have two irreducible local Lagrangian
Frobenius algebras with the base space $\bigoplus
Z^{1}_{\lambda\mu}\lambda\boxtimes\bar{\mu}$ in
${\mathcal{C}}_{1}\boxtimes{\mathcal{C}}_{2}^{\mathrm{opp}}$ and $\bigoplus
Z^{2}_{\mu\nu}\mu\boxtimes\bar{\nu}$ in
${\mathcal{C}}_{2}\boxtimes{\mathcal{C}}_{3}^{\mathrm{opp}}$.

We would like to have a new Frobenius algebra with the base space
$\bigoplus(\sum_{\mu}Z^{1}_{\lambda\mu}Z^{2}_{\mu\nu})\lambda\boxtimes\bar{\nu}$.
That is, the matrix part is just given by a matrix multiplication.

We can construct a Frobenius algebra with the base space
$\bigoplus(\sum_{\mu}Z^{1}_{\lambda\mu}Z^{2}_{\mu\nu})\lambda\boxtimes\bar{\nu}$
by considering a tensor product functor and taking an intermediate Frobenius
algebra but this is reducible in general. We have a notion of irreducible
decomposition of a Frobenius algebra corresponding to irreducible
decomposition of an operator algebra.

We show that after irreducible decomposition, each Frobenius algebra is local
and Lagrangian in Kawahigashi[38]. Being Lagrangian is shown to be equivalent
to modular invariance property in Müger[51].

On the matrix level, we thus have a decomposition
$\sum_{\mu}Z^{1}_{\lambda\mu}Z^{2}_{\mu\nu}=\sum_{i}Z^{3,i}_{\lambda\nu}$.

##  5 The $\alpha$-induction and anyon condensation

For a modular tensor category ${\mathcal{C}}$ and a Frobenius algebra, we have
a machinery of $\alpha$-induction, similar to the induction procedure in the
classical representation theory. If ${\mathcal{C}}$ corresponds to a rational
conformal field theory and it has an extension (like a conformal embedding),
then the vacuum representation of the larger conformal field theory gives a
commutative Frobenius algebra in ${\mathcal{C}}$. Our setting is a
generalization of this, and then from a representation of the smaller
conformal field theory, we get something like a representation of the larger
conformal field theory. This process of $\alpha$-induction depends on a choice
of a positive or negative braiding, and we use symbols
$\alpha^{\pm}_{\lambda}$ for the induced “representations” arising from an
object $\lambda$ in the modular tensor category $\mathcal{C}$. See
Böckenhauer-Evans-Kawahigashi[8][9] for an operator algebraic formulation. The
Frobenius algebra corresponds to the dual canonical endomorphism in
Böckenhauer-Evans-Kawahigashi[8][9]. The intersection of the images of
positive and negative $\alpha$-inductions give a new modular tensor category.
In the case of an extension of a conformal field theory like a conformal
embedding, this new modular tensor category is the representation category of
the larger conformal field theory. That is, positive and negative
$\alpha$-inductions produce only something like a representation, but the
intersection of their images really give a genuine representation. This
procedure corresponds to condensation of anyons, as in the branching rule of
Bais-Slingerland[3], and objects in the Frobenius algebra corresponds to
condensed anyons in Bais-Slingerland[3]. (If we have a commutative Frobenius
algebra, then the condensed anyons have to be bosons.) They make a new vacuum
in the new modular tensor category.

Many researchers are interested in commutative Frobenius algebras, but the
results and methods in Böckenhauer-Evans-Kawahigashi[8][9] apply also to
noncommutative Frobenius algebras. Commutativity of a Frobenius algebra is
called locality in Böckenhauer-Evans-Kawahigashi[8][9], since this corresponds
to locality of an extended chiral conformal field theory. The results in
Böckenhauer-Evans-Kawahigashi[8][9] are stated in terms of operator algebras,
but they can be translated into a language of modular tensor categories alone.

The $\alpha$-induction machinery produces a modular invariant
$Z_{\lambda\mu}=\langle\alpha^{+}_{\lambda},\alpha^{-}_{\mu}\rangle$, where
$\lambda,\mu$ are irreducible objects in $\mathcal{C}$ as in Böckenhauer-
Evans-Kawahigashi[8] and an irreducible local Lagrangian Frobenius algebra in
${\mathcal{C}}\boxtimes{\mathcal{C}}^{\mathrm{opp}}$ as in Rehren[58]. The
latter coincides with the one constructed in Fröhlich-Fuchs-Runkel-
Schweigert[26] in a more categorical language, as shown in Bischoff-
Kawahigashi-Longo[6].

A general fusion category has no braiding, but we have a general procedure
called the Drinfel′d center as in Drinfel′d[17] which produces a new modular
tensor category from a given fusion category.

The $\alpha$-induction produces a new fusion category for one choice of $+$ or
$-$ braidings. Its Drinfel′d center Drinfel′d[17] is given by the tensor
product of the original modular tensor category and the extended one as in
Böckenhauer-Evans-Kawahigashi[10]. This mathematical result coincides with a
result called boundary-bulk duality in physical literature like Kong[45]. See
Table 1 of Kong[45] for more interpretation of $\alpha$-induction in terms of
anyon condensation.

See Bischoff-Jones-Lu-Penneys[5] for another aspect of anyon condensation and
fusion categories.

##  6 Tensor networks and matrix product operators

A vector $(v_{j})$ has one index $j$. This is pictorially represented with one
circle with one leg labeled with $j$ as in Fig. 1. A matrix $(a_{jk})$ has two
indices $j,k$. This is pictorially represented with one circle with two legs
labeled with $j,k$ as in Fig. 1. Similarly, we can consider circles with 3
legs, 4 legs, and so on. Such an object is called a tensor.

$v$$j$$a$$j$$k$ Figure 1: A vector $v_{j}$ and a matrix $a_{jk}$

The $(j,l)$ entry of the matrix product of $(a_{jk})$ and $(b_{kl})$ is given
by $\sum_{k}a_{jk}b_{kl}$. This is pictorially represented with two circles
corresponding to $(a_{jk})$ and $(b_{kl})$ where one leg of one and one leg of
the other, both labeled with $k$, are concatenated as in Fig. 2. Such
concatenation is called a contraction. We are interested in tensors with 3, 4
and 5 legs here.

$a$$b$$j$$l$ Figure 2: A matrix product $(ab)_{jl}$

Consider a tensor with 3 legs and denote it by $a^{l}_{jk}$. We consider a
state represented as

$\sum_{l_{1},l_{2},\cdots,l_{n}}\mathrm{Tr}(a^{l_{1}}a^{l_{2}}\cdots
a^{l_{n}})\left|l_{1}l_{2}\cdots l_{n}\right\rangle$

where $a^{l_{m}}$ is a matrix and a 3-tensor as in Fig. 3. Such a state was
first considered by Fannes-Nachtergaele-Werner[23] and is called a matrix
product state (MPS) today.

$a$$a$$a$$l_{1}$$l_{2}$$l_{n}$$\displaystyle\sum_{l_{1},l_{2},\cdots,l_{n}}$$\left|l_{1}l_{2}\cdots
l_{n}\right\rangle$ Figure 3: A matrix product state

Next consider a tensor with 4 legs and denote it by $b^{lm}_{jk}$. We consider
a matrix represented as

$\sum_{l_{1},l_{2},\dots,l_{n},m_{1},m_{2},\dots,m_{n}}\mathrm{Tr}(b^{l_{1}m_{1}}\cdots
b^{l_{n}m_{n}})\left|l_{1}l_{2}\cdots l_{n}\right\rangle\left\langle
m_{1}m_{2}\cdots m_{n}\right|$

as in Fig. 4 and we call it a matrix product operator (MPO).

$\displaystyle\sum_{l_{1},l_{2},\dots,l_{n},m_{1},m_{2},\dots,m_{n}}$$a$$a$$a$$m_{1}$$m_{2}$$m_{n}$$l_{1}$$l_{2}$$l_{n}$$\left|m_{1}m_{2}\cdots
m_{n}\right\rangle\left\langle l_{1}l_{2}\cdots l_{n}\right|$ Figure 4: A
matrix product operator

Bultinck-Mariëna-Williamson-Şahinoğlu-Haegeman-Verstraete[11] considered an
algebra of matrix product operators arising from a system of physically nice
tensor networks of 4-tensors. Such an algebra is called a matrix product
operator algebra (MPOA). They considered a category of matrix product
operators and presented a graphical method to construct an interesting system
of anyons. They have shown that we get a modular tensor category and discussed
its physical consequence.

This paper has caught much attention of physicists. They are aware of its
similarity to an old work of Ocneanu[21]. We now show that their construction
is really the same as Ocneanu’s mathematically as in Kawahigashi[40].

Let $N\subset M$ be a subfactor with finite Jones index and finite depth. We
set $N=M_{-1}$, $M=M_{0}$ and apply the Jones tower/tunnel construction as in
Definitions 9.24, 9.43 of Evans-Kawahigashi[22] to get

$\cdots\subset M_{-2}\subset M_{-1}\subset M_{0}\subset M_{1}\subset
M_{2}\cdots,$

where the prime denotes the commutants. We then consider a double sequence
$A_{jk}=M^{\prime}_{-k}\cap M_{j}$ of finite dimensional $C^{*}$-algebras.
They form commuting squares as in Section 9.6 of Evans-Kawahigashi[22].

Recall that a sequence

$\begin{array}[]{ccc}A&\subset&B\\\ \cap&&\cap\\\ C&\subset&D\end{array}$

is called a commuting square if the restriction of $E_{B}$ to $C$ is equal to
$E_{A}$ where $E_{A}$ and $E_{B}$ are conditional expectations as in Theorem
5.25 of Evans-Kawahigashi[22].

Look at the Bratteli diagrams of a commuting square. (An inclusion of finite
dimensional $C^{*}$-algebras gives a Bratteli diagram as in page 67 of Evans-
Kawahigashi[22].) Choose one edge from each diagram corresponding to each of
the four inclusions. We then get a complex number from the 4 edges. This is a
notion of a bi-unitary connection as in Definition 11.3 of Evans-
Kawahigashi[22] considered by Ocneanu and Haagerup. (Also see Asaeda-
Haagerup[2].) In the case as above, we have an especially nice bi-unitary
connection called a flat connection as in Ocneanu[52], Kawahigashi[35],
Section 11.4 of Evans-Kawahigashi[22]. (We also get this by fixing two labels
of quantum $6j$-symbols. See Chapter 12 of Evans-Kawahigashi[22] for relations
between quantum $6j$-symbols and flat connections.) If the original subfactor
is hyperfinite and has finite Jones index and finite depth, then this flat
conenction recovers the original subfactor entirely by Popa’s theorem[56].

We further have a finite system of flat connections corresponding to the
direct summands of $N^{\prime}\cap M_{2n+1}$. They produce a tensor of 4 legs.
(We have the number 4 since a commuting square has 4 inclusions.) Though the
normalization conventions for a bi-unitary connection and a 4-tensor are
slightly different, the difference is only up to the Perron-Frobenius
eigenvector entries as in Fig. 11 of Kawahigashi[40]. Such a tensor exactly
fits in the setting of Bultinck-Mariëna-Williamson-Şahinoğlu-Haegeman-
Verstraete[11].

Let all the four inclusion diagrams be the Dynkin diagram $A_{n}$. Number the
vertices as $1,2,\dots,n$ and set
$\varepsilon=\sqrt{-1}\exp\left(\pi\sqrt{-1}/{2(n+1)}\right)$. We define a bi-
unitary connection as in Fig. 5, where $\mu$ denotes the Perron-Frobenius
weight, that is, $\mu(j)=\sin(j\pi/(n+1))/\sin(\pi/(n+1))$. (This also works
for the Dynkin diagrams $D_{n}$, $E_{6}$, $E_{7}$ and $E_{8}$. See Fig. 11.32
of Evans-Kawahigashi[22].) This is actually a “square root” version of the one
considered in Bultinck-Mariëna-Williamson-Şahinoğlu-Haegeman-Verstraete[11].

$W$$l$$j$$m$$k$$\displaystyle=\delta_{kl}{\varepsilon}+\sqrt{\frac{\mu(k)\mu(l)}{\mu(j)\mu(m)}}\delta_{jm}\bar{\varepsilon}$
Figure 5: A flat connection on the Dynkin diagram $A_{n}$

The above situations are very similar to interaction round the face (IRF)
solvable lattice models, like the one due to Andrews-Baxter-Forrester[1]. For
an IRF model, we fix a diagram, choose four edges from it and make a square
with them. Then a complex value called the Boltzmann weight, depending on a
spectral parameter, is assigned to each such square. These values satisfy
certain compatibility conditions such as the Yang-Baxter equation. The
crossing symmetry of an IRF model also corresponds to a flat connection. When
the diagram is the Dynkin diagram of type $A_{n}$, our formula for the flat
connection is essentially the same as the restricted solid-on-solid (RSOS)
model in Pasquier[55].

We have a construction of the Drinfel′d center as in Drinfel′d[17] which gives
a modular tensor category from a fusion category. In subfactor theory, the
first such construction was Ocneanu’s asymptotic inclusion as in Evans-
Kawahigashi[21], Section 12.6 of Evans-Kawahigashi[22] which means
$N\vee(N^{\prime}\cap M_{\infty})\subset M_{\infty}$ constructed from a
hyperfinite II1 subfactor $N\subset M$ with finite Jones index and finite
depth. We later have the Longo-Rehren subfactor construction[49] and Popa’s
symmetric enveloping algebra construction[57].

In connection to 3-dimensional topological quantum field theory, Ocneanu
further introduced a tube algebra as in Evans-Kawahigashi[21], Section 12.6 of
Evans-Kawahigashi[22], Izumi[31] which is a finite dimensional $C^{*}$-algebra
arising from a fusion category. Its each direct summand describes an “anyon”
of the modular tensor category given by the Drinfel′d center construction as
in Drinfel′d[17].

Bultinck-Mariëna-Williamson-Şahinoğlu-Haegeman-Verstraete[11] have a similar
construction of the anyon algebra and it describes a system of anyons.

Suppose we start with a subfactor $N\subset M$ with finite Jones index and
finite depth. We then have a family of flat connections as above. (The number
of flat connections is equal to the dimension of the center of $N^{\prime}\cap
M_{2n+1}$ for sufficiently large $n$.) We can next show that this family
produces a 4-tensor satisfying all the settings of Bultinck-Mariëna-
Williamson-Şahinoğlu-Haegeman-Verstraete[11] simply by changing the
normalization constants arising from the fourth root of the Perron-Frobenius
eigenvector entries as in Fig. 11 of Kawahigashi[40].

###### Theorem 6.1

Ocneanu’s tube algebra for the fusion category arising from a subfactor and
the anyon algebra of Bultinck et al. arising from its flat connections are
isomorphic. In particular, the two fusion rules are identical and the Verlinde
formula also holds for the setting of anyon algebra.

We have seen that we can construct a tensor network satisfying the
requirements of Bultinck-Mariëna-Williamson-Şahinoğlu-Haegeman-Verstraete[11]
from a subfactor. It is known any fusion category is realized from a
subfactor. Usually a “representation” of such a category is a bimodule over
II1 factors or an endomorphism of a type III factor as in Longo[47][48], but
we can also realize such a “representation” as a flat connection due to the
open string bimodule construction of Asaeda-Haagerup[2].

###### Theorem 6.2

Suppose a tensor satisfies a setting of Bultinck et al. If it realizes a
fusion category, then this fusion category is realized by a tensor of flat
connections arising from a subfactor as above. Then the resulting tube
algebras and the modular tensor categories are the same for the original and
new systems.

In studies of topological phases, a gapped Hamiltonian has been important. It
means a family of self-adjoint matrices where the lowest eigenvalues are
always separated (“gapped”) from the next eigenvalues. Bultinck-Mariëna-
Williamson-Şahinoğlu-Haegeman-Verstraete[11] studied such a gapped Hamiltonian
where the ground state is realized with a projected entangled-pair state
(PEPS), which is a two-dimensional version of a matrix product state. They
have a certain expression of a matrix product operator, which is called a
projector matrix product operator (PMPO) since it is also a projection, and
the PEPS naturally lives in the range of several PMPOs. We are thus interested
in identification of these PMPOs from an operator algebraic viewpoint. (See
Cirac-Perez-Garcia-Schuch-Verstraete[14] for a general theory of PEPS.)

Our setting is as follows. We have a 4-tensor corresponding to a bi-unitary
connection. With this bi-unitary connection, we construct a double sequence of
finite-dimensional string algebras $A_{kl}$ with $A_{00}={\mathbb{C}}$,
$A_{kl}\subset A_{k+1,l}$ and $A_{kl}\subset A_{k,l+1}$ as in Section 11.3 of
Evans-Kawahigashi[22] (with any choice of the starting vertex $*$) and this
produces hyperfinite II1 factors $A_{\infty,l}$ and $A_{k,\infty}$ as the
closures of $\bigcup_{k}A_{kl}$ and $\bigcup_{l}A_{kl}$ in the strong operator
topology on appropriate Hilbert spaces. In particular, we have two subfactors
$A_{0,\infty}\subset A_{1,\infty}$ and $A_{\infty,0}\subset A_{1,\infty}$,
both with finite Jones index as in Theorem 11.9 of Evans-Kawahigashi[22].
Suppose one of the two subfactors has finite depth. Then so does the other by
Sato[59]. In this case, we get a finite family of bi-unitary connections
$\\{W_{a}\\}$ as in Section 3 of Asaeda-Haagerup[2], Section 2 of
Kawahigashi[41], corresponding to irreducible $A_{0,\infty}$-$A_{0,\infty}$
bimodules arising from the subfactor $A_{0,\infty}\subset A_{1,\infty}$. Each
$W_{a}$ gives a corresponding 4-tensor and they give a PMPO of length $k$ as
in Section 3.1 of Bultinck-Mariëna-Williamson-Şahinoğlu-Haegeman-
Verstraete[11], Section 3 of Kawahigashi[41]. We are interested in identifying
the range of this projection. (Note that the range of this projection is much
smaller than the entire Hilbert space.) Then with subfactor technique based on
flat fields of strings of Ocneanu[53], Theorem 11.15 of Evans-Kawahigashi[22],
and Definition 3 of Asaeda-Haagerup[2] gives the following as in Theorem 3.3
of Kawahigashi[41].

###### Theorem 6.3

The range of the $k$th PMPO in this setting is naturally identified with the
$k$th higher relative commutant $A^{\prime}_{\infty,0}\cap A_{\infty,k}$ for
the subfactor $A_{\infty,0}\subset A_{\infty,1}$ arising from the original bi-
unitary connection.

The ranges of the $k$th PMPOs give an increasing sequence of finite
dimensional Hilbert spaces indexed by $k$ and they are Hilbert spaces we would
like to study in connection to 2-dimensional topological order. The $k$th
higher relative commutants give an increasing sequence of finite dimensional
$C^{*}$-algebras with a positive definite inner product arising from a trace
and they have been extensively studied for almost 40 years in subfactor
theory. The above identification is expected to be useful to study further
relations between the two theories.

Take the Dynkin diagram $A_{5}$ as an example and consider the bi-unitary
connection as in Fig. 5. Since the graph $A_{5}$ has four edges, the dimension
of the Hilbert space on which the $k$th PMPO acts is $4^{2k}$. Since this bi-
unitary connection is flat, the Bratteli diagram of the higher relative
commutants is given as in Fig. 6. For example, the row for the 6th higher
relative commutant says $5,9,4$, which means the algebra is
$M_{5}({\mathbb{C}})\oplus M_{9}({\mathbb{C}})\oplus M_{4}({\mathbb{C}})$.
(The top row is counted as the 0th.) The dimension of this algebra is
$5^{2}+9^{2}+4^{2}=122$. An elementary computation shows that the dimension of
the $k$th higher relative commutants is $(3^{k-1}+1)/2$ and this number is
much smaller than $16^{k}$. If we take the Dynkin diagram $A_{n}$ for a large
$n$, the difference is even more drastic.

$1$$1$$2$$5$$1$$2$$5$$1$$3$$9$$1$$4$$1$$4$ Figure 6: The Bratteli diagram of
the higher relative commutants

Note that the subfactor for the higher relative commutants here is
$A_{\infty,0}\subset A_{\infty,1}$ while we had the other subfactor
$A_{0,\infty}\subset A_{1,\infty}$ for the initial finite depth assumption
above. The relation between the two subfactor was studied in Sato[60], and
they are characterized as mutually anti-Morita equivalent subfactors. (In the
above example of the Dynkin diagram $A_{5}$, the two subfactors are the same.)

A recent general treatment of such tensors and tensor categories is given in
Lootens-Fuchs-Haegeman-Schweigert-Verstraete[50]. Though the language is a
little bit different, this is also closely related to subfactor theory.

## Acknowledgements

This work was partially supported by JST CREST program JPMJCR18T6 and Grants-
in-Aid for Scientific Research 19H00640 and 19K21832.

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