Universality of topological physics inquantum and classical systems
Tomoki Ozawa
AIMR, Tohoku University
Interdisciplinary Science Conference in Okinawa (ISCO 2023)— Physics and Mathematics meet Medical Science —February 28, 2023

In this course, our focus will be on the IQ
in terms of single-particle physics, but we shal
more complicated and richer FQHE later on. N
primarily from a solid-state perspective, as th

Quantum Hall effectVy

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(a)

(b)

In 1980, it was found that the Hall resistivity undera higheld shows
unexpectedof
plateauFigure2: magnetic(a) Typicalmeasurement
the inte- Integer Quantum
Hall Effectfrom Schurr
et al. IEEE Trans. Instrum. Mefi

the following, we use the symbol ⌫ to label theMain goal of my talk today are twofold:surement of the fractional quantum Hall e↵ect,• To discuss that this effect is related to “topology”
of the Stormer,“wave function”
and H.L.
Science, 1510, (1990). Here,• To show that this effect is more general thanclassicaljust in semiconductor
electronphysics
Hall resistanceandthe solid diagonal
resistance.

Landau level, topology, and bulk-edge correspondence
Possible energy of an electron in a magnetic eld takes only discrete values called Landau levels
Energy is an eigenvalue of an operator called Hamiltonian
We can also consider its eigenvectors, which represent thecorresponding quantum states

Eigenvectors represented as arrows

Chern number (calculated in momentum space) turns out tobe equal to the number of edge-localized state in real space- bulk-edge correspondence -Hall resistivity is inversely proportional to
the number of edge-localized modesy<latexit sha1_base64="tmEvmksfb9cWDAa5xB1ckWUhUbQ=">AAAB6HicbVDLSgNBEOz1GeMr6tHLYBA8hV3xdQx68ZiAeUCyhNlJbzJmdnaZmRXCki/w4kERr36SN//GSbIHTSxoKKq66e4KEsG1cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dZwqhg0Wi1i1A6pRcIkNw43AdqKQRoHAVjC6m/qtJ1Sax/LBjBP0IzqQPOSMGivVx71S2a24M5Bl4uWkDDlqvdJXtx+zNEJpmKBadzw3MX5GleFM4KTYTTUmlI3oADuWShqh9rPZoRNyapU+CWNlSxoyU39PZDTSehwFtjOiZqgXvan4n9dJTXjjZ1wmqUHJ5ovCVBATk+nXpM8VMiPGllCmuL2VsCFVlBmbTdGG4C2+vEya5xXvqnJZvyhXb/M4CnAMJ3AGHlxDFe6hBg1ggPAMr/DmPDovzrvzMW9dcfKZI/gD5/MH6v2NBw==</latexit>x<latexit sha1_base64="ZwKkaJcdU3hHlnhMyVD4tdz55QM=">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHaNryPRi0dI5JHAhswODYzMzm5mZo1kwxd48aAxXv0kb/6NA+xBwUo6qVR1p7sriAXXxnW/ndzK6tr6Rn6zsLW9s7tX3D9o6ChRDOssEpFqBVSj4BLrhhuBrVghDQOBzWB0O/Wbj6g0j+S9Gcfoh3QgeZ8zaqxUe+oWS27ZnYEsEy8jJchQ7Ra/Or2IJSFKwwTVuu25sfFTqgxnAieFTqIxpmxEB9i2VNIQtZ/ODp2QE6v0SD9StqQhM/X3REpDrcdhYDtDaoZ60ZuK/3ntxPSv/ZTLODEo2XxRPxHERGT6NelxhcyIsSWUKW5vJWxIFWXGZlOwIXiLLy+TxlnZuyxf1M5LlZssjjwcwTGcggdXUIE7qEIdGCA8wyu8OQ/Oi/PufMxbc042cwh/4Hz+AOl5jQY=</latexit>fiTopological property

Eigenvectors can form a nontrivial shape in momentum space
How many times the eigenvectors rotate is an integer called
the Chern number, and this is 1 for Landau levels

Topological photonics
Eigenvectors in momentum space rotate also in other systems
PHYSICAL REVIEW A 78, 033834 !2008"

Analogs of quantum-Hall-effect edge states in photonic crystals
S. Raghu*
Department of Physics, Stanford University, Stanford, California 94305-4045, USA

F. D. M. Haldane
Department of Physics, Princeton University, Princeton, New Jersey 08544-0708, USA!Received 14 April 2008; published 23 September 2008"

NATURE461 | 82009
October 2009
NATURE| Vol 461 || 8Vol
October

LETTERSLETTERS

Photonic crystals built with time-reversal-symmetry-breaking Faraday-effect media can exhibit chiral edgemodes
that propagate unidirectionally along boundaries across which the Faraday
axis reverses.
These modes
A
a
A
a
a
a
ScattererofAntennaB
Scatterer ofof the electronic
are precise
analogsedge states of quantum-Hall-effect !QHE" systems, and are also immune
Antenna B
variable lengthlvariablelength l
to backscatteringand localization by disorder. The Berry curvature of the photonic bands plays a role analoCES waveguide
CES waveguidegous Antenna
to thatof the magnetic field in the QHE. Explicit calculations demonstrating the existence of suchA
AntennaA
unidirectionallypropagating
photonic edge states are presented.
Metal wall
Metal wall

DOI: 10.1103/PhysRevA.78.033834

PACS number!s": 42.70.Qs, 03.65.VfAbTopological Photonics:
Aharonov phase of an electronic wave packet
is related to theenergy eigenstates in photonics
Topologicalmagnetic flux density in real space.
While the uniform propagation of wave
packets in persystems
fectly translationally invariant systems does not involve theb

I. INTRODUCTIONzzcontrolb of the yflow xof light using photonic-band-gap
y considerablexmaterials has received
attention over thebade #1$. Moreover, the potential for using artificially
ed metamaterials, such as the recently discovered
ded media #2$, has shown considerable technological. In the past, significant progress has been achieved
eld of photonics by making use4 cmof analogies withigure
1 | Microwave
waveguide
supporting
Schematicof thematerial,ic
systems.
For
instance,
theCESs.
ideaa, ofa PBG
waveguide composed of an interface between a gyromagnetic photonic4 cm
mrystalwitha spatiallyvaryingandslab (bluerods) and a metal
wall (yellow).
The periodic
structure is dielectric

A

B

c

Berry curvature, the semiclassical description of the accelB
c l packets in media with spatial inhomogeneity
eration of waveof lengtha scales large comparedto the underlying latticelWang,spacing is incomplete if it is not taken into
account.Chong,
Recently, Joannopoulos, & Soljačić (MIT),a
Onoda et al. #7$ have pointed out the role of Berry curvature
Nature 461, 772 (2009)in photonic
crystals without inversion symmetry; while theseE
0 this Positive
authors Negative
characterize
as a “Hall effect of light,” the Hall
FigureCESs
andeffectsofalarge
scatterer.
a, CES field
Figure1 | Microwave
supporting
CESs.
a, Schematicof the 2 | Photonic
andwichedbetweentwo parallelwaveguidecopper plates(yellow) for
confinementin
effect in electronic
E systems is associated with broken timez

Topological photonics in synthetic dimensions
recently the first experimental realization of a synthetic frequency dimension was demonstrated using a macroscale
optical fiber loop [20] and later a synthetic Hall ladder was realized by harnessing the direction of propagation
within a ring as an additional dimension [21]. However, to enable robust platforms for researching complex
multidimensional effects and creating practical optical devices, it is vital to miniaturize and transfer photonic
realizations of synthetic dimensions into a monolithically integrated format. Early efforts towards this goal
involved modelling thin-film lithium niobate electro-optic frequency combs as a multi-dimensional tight-binding
Theory:Price,
Goldman,Zilberberg,& Carusotto, PRA 93, 043827 (2016)
lattice [22]. However, utilizing a more scalable
silicon TO,
photonic
chip platform
would provide
a considerable
advancement, as it on one hand would allow photonics with synthetic dimensions to benefit from the mature andYuan,
Shi, create& Fan,
Opt.forLett.
41, 741 (2016)sophisticated CMOS commercial fabrication toolbox, andon the other,
the meansmulti-dimensional
topological phenomena to be introduced into novel device applications.
2experiment:
Dutt,onMinkov,
Yuan,
Miller,a &custom
Fan, Nature Comm. 10, 3122 (2019)
We report the realization of Firsta synthetic
frequency dimensiona Si CMOSLin,
platformby employinga the resonator at its free-spectral rangebdynamically modulated integrated ring resonator cavity design. By driving(FSR) rate of ΩR = 20.4 GHz a frequency lattice spanning a 280 GHz bandwidth was established and sideband
Here H(t) is the Hamiltonian with the matrix elements,(a)intensity enhancement was observed. The equidistant ring modes and periodic modulation of the cavityi(n m)⌦thm|H(t)|ni.
respectively represent lattice sites and their coupling mechanism, hence could beH
mappeda 1Dtight-binding = Jn m (t) emn (t)to=model. By detecting the time resolved transmittance during modulation at the FSR
we were able toThisrateHamiltonian
H(t) has two symmetries. The firstexperimentally measure the band structure involving a synthetic frequency dimension in an integrated photonic
is the modal translational symmetry along the frequency
device for the first time. Furthermore, we demonstrate an ability to engineer inter-site couplings and induce nonaxis between
equally spaced modes, since the matrix
reciprocity through photonic gauge potentials. By respectively employing FSR-detuned
modulation thefrequencies
or multiple modulation signals we were able to attain time-resolved demonstrationsof band structuresindicative only on m n. This symmetry perelement
Hmn dependsof behaviors analogous to effective electric and magnetic field effects for photons.
We suggest that such CMOS
Resonatorsmits
the withdefinition
process fabricated ring resonator devices – supporting an extensive frequency dimension
versatile ofanda conserved Bloch quasimomentum
thebuilding
associatedreciprocal space. Since the reciproreconfigurable
coupling mechanisms,
robustblocks for(b)(c)as well as two synthetic forces – can act kas inscalable integrated circuits with nontrivial topology, which are promising forcal
bothspaceansweringfundamental
here Realis conjugate to the frequency dimension,questions in physics and realizing novel device prototypes.

One can use non-spatial degrees of freedom as dimensions: Synthetic dimensions

Synthetic≡

Frequency

We can use resonant modes of a resonator as a dimension and realize topological lattice models

we expect it to be identified with time. We will formallyFirst experiment in silicon ring resonator:| Prospects of synthetic dimensions. a | Schematic illustration of a 1D chain of opticalFig.
4
Results
show that this is indeed the case below. The second symtors, simulatinga 2D topologicallattice. Theedge mode travelling along the frequency axis a
Balčytis, TO, Ota, Iwamoto, Maeda, & Baba, Science
Advances8, eabk0468(2022)
is dimension
thetime-translation
symmetry= H(t+T
Device design. Our approach to produce a lattice of coupled states along frequency
a syntheticmetry
frequency
involves
converter,
whereasthe edge mode
travelling H(t)along thereal axis).
acts as an opticalthe use of a ring resonator [20, 22]. Provided the waveguiding medium comprising
the cavity
exhibits
near-zero bands with quasienergies ✏ that
This
leads
to Floquetb | Schematic image of a 4D lattice made of three real dimensions and kone synthetic dimension
aretodefinedin theintervalHall[ ⌦/2,
can be usedrealize the4D quantum
effect.⌦/2]. The relationship
FIG. 1. Dynamically modulated ring cavity and its band
between the quasienergy ✏k and the quasimomemtum k
structure in the synthetic frequency dimension. (a) A sinuis the band structure.
soidal signal VM modulates the index of a part of the ringP
periodic tableof topologicalphasesfor
TABLE 1, the idea ofAssummarized
in|mi.
im⌦k
Define
the
Bloch
modes
|ki
=
e
(left) at the mode spacing ⌦R , creating coupling J betweenm3,121,122, |bi
it is known
non-interacting
fermions
synthetic dimensions
The
state
vectorcan be written
as |bi =has already led t
the di↵erent frequency modes (right). EOM: electro-optic
that there shouldRbe
a varietywealth of proposals and experiments,⇡/⌦
phase modulator. (b) Band structure of the nearest-neighbor
˜ of topologicalTopological mechanics
Situations similar to quantum Hall e ect can also be realizedwith Newtonian mechanics
ETH Zurich: Süsstrunk & Huber, Science 349, 47 (2015)ff

https://www.youtube.com/watch?v=__TGJEtFD-E

Two-pendulum experimentTwo-pendulum experiment in collaboration with Gadway group @ University of Illinois
Natural frequency of a pendulum is 3Hz
Pendulums are coupled to electromagnets, through which
forces can be applied

Measurement of non-Hermitian
Berry phase(a)delta J = 6

Experiment(b)delta J = 5
delta J = 4

(a)

Time
TheoryPT

PT Symmetric

J

J• Martello, Singhal, Gadway, TO, Price, arXiv:2302.03572• Singhal, Martello, Agrawal, TO, Price, Gadway, arXiv:2205.02700• Anandwade, Singhal, Paladugu, Martello, Castle, Agrawal, Carlson,
Battle-McDonald, TO, Price, Gadway, arXiv:2107.09649

PT Brokendelta J

There are ocean and atmospheric waves called Kelvin wave and Yanai-Maruyama wave,
which ow along the equator to the east

Earth

霜
Ocean

Kelvin wave
Yanai-Maruyama wave

flfi

Equatorial wave

Land

Moving objects on the earth feel the Coriolis
force due to the rotation of the earth~v<latexit sha1_base64="J4lTN6uubHUaU6cWKqiSKo+7Eqs=">AAAB73icbVBNS8NAEJ3Ur1q/qh69LBbBU0mkoMeiF48VrCm0oWy2k3bpZhN2N4US+iO8eFAQr/4db/4bt20O2vpg4PHeDDPzwlRwbVz32yltbG5t75R3K3v7B4dH1eOTJ51kimGbJSJRnZBqFFxi23AjsJMqpHEo0A/Hd3Pfn6DSPJGPZppiENOh5BFn1FjJ702Q5ZNZv1pz6+4CZJ14BalBgVa/+tUbJCyLURomqNZdz01NkFNlOBM4q/QyjSllYzrErqWSxqiDfHHujFxYZUCiRNmShizU3xM5jbWexqHtjKkZ6VVvLv7ndTMT3QQ5l2lmULLloigTxCRk/jsZcIXMiKkllClubyVsRBVlxiZUsSF4qy+vE/+q7jXqnvfQqDVvizzKcAbncAkeXEMT7qEFbWAwhmd4hTcndV6cd+dj2VpyiplT+APn8wdHj5AB</latexit>

ObjectCoriolis force

NorthernhemisphereF = mvf

Chargedparticle

Magneticeld

Lorentz force<latexit sha1_base64="kEodtEVB+8ATAFzUMNc6z1XtKQo=">AAAB73icdVBNS8NAEN3Ur1q/qh69LBbBU8jW0NaDUBTEYwVrCm0om+2mXbqbhN1NoYT+CC8eFMSrf8eb/8ZNW0FFHww83pthZl6QcKa043xYhZXVtfWN4mZpa3tnd6+8f3Cv4lQS2iYxj2UnwIpyFtG2ZprTTiIpFgGnXjC+yn1vQqVicXSnpwn1BR5GLGQEayN51/ACiknYL1cc+7xRq7o16NiOU0dVlJNq3T1zITJKjgpYotUvv/cGMUkFjTThWKkuchLtZ1hqRjidlXqpogkmYzykXUMjLKjys/m5M3hilAEMY2kq0nCufp/IsFBqKgLTKbAeqd9eLv7ldVMdNvyMRUmqaUQWi8KUQx3D/Hc4YJISzaeGYCKZuRWSEZaYaJNQyYTw9Sn8n3hVG7k2QrdupXm5zKMIjsAxOAUI1EET3IAWaAMCxuABPIFnK7EerRfrddFasJYzh+AHrLdPJS6PRw==</latexit>

F = evB<latexit sha1_base64="+PrJL56KTU6Q6tbojN2qzAL6qnU=">AAAB73icdVBNS8NAEN34WetX1aOXxSJ4Ctka2noQSgXxWMGaQhvKZrtpl242YXdTKKE/wosHBfHq3/Hmv3HTVlDRBwOP92aYmRcknCntOB/Wyura+sZmYau4vbO7t186OLxXcSoJbZOYx7ITYEU5E7Stmea0k0iKo4BTLxhf5b43oVKxWNzpaUL9CA8FCxnB2kjeNbyEdNLsl8qOfVGvVtwqdGzHqaEKykml5p67EBklRxks0eqX3nuDmKQRFZpwrFQXOYn2Myw1I5zOir1U0QSTMR7SrqECR1T52fzcGTw1ygCGsTQlNJyr3ycyHCk1jQLTGWE9Ur+9XPzL66Y6rPsZE0mqqSCLRWHKoY5h/jscMEmJ5lNDMJHM3ArJCEtMtEmoaEL4+hT+T7yKjVwboVu33Ggu8yiAY3ACzgACNdAAN6AF2oCAMXgAT+DZSqxH68V6XbSuWMuZI/AD1tsn4jOPGw==</latexit>

Coriolis force and Lorentz force are very similar
Kelvin wave and Yanai-Maruyama waves turned
out to be the topological edge modes
Delplace, Marston, & Venaille, Science 358, 1075 (2017)

Neural stem cell
Stem cells of mice can move aroundflfi• They seem to bend in one direction (perhaps due to left-rightasymmetry of cells)• This motion looks similar to particles under Lorentz or Coriolis force• When the cells are con ned to move in a region with edges, they ow
along the edges

Topological band structure of cell motion
One can model the system using the language of active matter
Linearizing the uid equation, the resulting equation looks like the Schrödinger equation

i@t | i = H| i + |si<latexit sha1_base64="4QJckdYcXDoPi1JSUECmWB674G8=">AAACHnicbZDLSsNAFIYnXmu9VV26GSyCIJRELLoRim66rGAv0IQwmZ62QyeTMDMRStsnceOruHGhiOBK38ZJm4Vt/WHg4z/ncOb8QcyZ0rb9Y62srq1vbOa28ts7u3v7hYPDhooSSaFOIx7JVkAUcCagrpnm0IolkDDg0AwGd2m9+QhSsUg86GEMXkh6gnUZJdpYfqHM3JhIzQj3NR67NcVcSUSPA77B1XnjHI9Vxn6haJfsqfAyOBkUUaaaX/hyOxFNQhCacqJU27Fj7Y3SzZTDJO8mCmJCB6QHbYOChKC80fS8CT41Tgd3I2me0Hjq/p0YkVCpYRiYzpDovlqspeZ/tXaiu9feiIk40SDobFE34VhHOM0Kd5gEqvnQAKGSmb9i2ieSUG0SzZsQnMWTl6FxUXLKJfv+sli5zeLIoWN0gs6Qg65QBVVRDdURRU/oBb2hd+vZerU+rM9Z64qVzRyhOVnfv6dHoiw=</latexit>

Order parameter

Source term

5-by-5 matrix

The “Hamiltonian” H resembles the one in the case of equatorial waves

Left

5.0

c

15.0

15.0
Edge

10.0

LocalizationLocalization
Left edge 5.0Right edge
5.0

5.0

0.0
0.0
-5.0

Density component
Density component
0
-5.0
0 1
1
0

Right -10.0 -10.0-10.0

-10.0

0.0

-5.0

-5.0

-15.0
-15.0 -15.0
-15.0
-10.0
-5.0
15.0
-15.0 -10.0-5.0
0.0 -5.0
5.0 0.0
10.0 5.0
15.0
-15.0
-10.0
0.0
5.0 10.010.0
15.0(kyv0W
Wavenumber(kyv0W
Wavenumber
Wavenumber (kyv0W

Momentum

b

b

10.0

d

d

city v0

0.5

velocity v0

elocity v0

Bulk
Edge1.0
Edgeb Bulk Bulk
ddrug1.01.0No drugNo
Yamauchi, Hayata, Uwamichi, TO,& Kawaguchi,
arXiv:2008.10852
Edge

30 nM
Jaspl 0.530 nM Jaspl No
drug

1 -5.0-10.0

No drug
15.0

Experiment(edge & bulk)
No drug No drug

Edge

Edge

10.0
10.0
Localization

5.0

edge
Left edge Left
Rightedge Right edge
0.0
0.0
Density component

Momentumflc10.015.0

Energy (Re(E)W)
Energy(real part)

10.0
Energy (Re(E)W)

Energy

15.0

Left

Right

a

15.0

5.0
0.0
-5.0

15.0

15.0
Bulk

10.0

10.0

5.0

5.0

-10.0

0.0
-5.0

-10.0

0.0

Bulk

10.0

Bulk

Frequency (ZWacMore realistic situation

Frequency (ZW

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Frequency (ZWaBand structure of the above Hamiltonian

v0：free velocity of cells
f ：strength of chirality⌫ ：odd viscosityv 0 pyi(f ⌫p2 )A0Frequency (ZWv 0 px0i(f ⌫p2 )Frequency (ZW0
H3x3 = @v0 px
v 0 py1Frequency (ZW

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-5.0

5.0
0.0
-5.0

-10.0

-10.0

-15.0
-15.0
-15.0
-15.0
-15.0
-15.0
-5.05.0 0.0
5.0
10.0 15.0
-15.0
10.0
-15.0 -10.0-15.0-5.0-10.0
0.0
10.0
15.0
-15.0
-10.0
-5.0-10.0
0.0 -5.05.0 0.0
10.05.015.0
-15.0
-10.0
-5.0
0.0
5.0
10.0
15.0
-15.0
-10.0
(kyv0W(kyv0W
Wavenumber
Wavenumber(kyv0W(kyv0W
Wavenumber
Wavenumber
Wavenumber (kyv0W
W

f

f

15.0

15.0
Edge

g

f

Jasplakinolide
Jasplakinolide
3.75 nM 3.75 nM
Edge

15.0

g

15.0
15.0 3.75
Jasplakinolide
nM

Edge

g

Jasplakinolide
30 nM
Jasplakinolide
30 nM
Edge

15.0

Ja

Topological phases• Electrons• Electromagnetic waves• Newtonian mechanics• Geophysics• Active matter
︙

Mathematics
Index theorem, K theory
Kähler geometry

Fundamental Science
Bulk-edge correspondence

Topological
Eigenstates• Topological laser• Frequency converter• Optical delay line
• Loss-free waveguide︙

Applied Science

Some reviews I wrote:- Topological photonics: TO et al., Rev. Mod. Phys. 91, 015006 (2019).
- Synthetic dimensions: TO and Price, Nature Reviews Physics 1, 349–357 (2019).
- Active topological photonics: Ota et al., Nanophotonics 9, 547 (2020).
- Roadmap on topological photonics: Price et al., J. Phys. Photonics 4 032501 (2022).