Feb. 13 th, 2024

Electron dynamics at boundaries
of quantum many-body systems

Masayuki Hashisaka
ISSP, The University of TokyoMasayuki Hashisaka @ ISSP, The Univ. of Tokyo

https://hashisaka.issp.u-tokyo.ac.jp/Research Interest
 Experiments
 Condensed matter physics
 Mesoscopic physics
Keywords:
Quantum Hall, Topological materials,
Transport measurements, Electronics,etc.

Master, Ph.D. @京⼤化研

Prof. Ono

Prof. Kobayashi(Supervisor)

20232017-2023
2009-2017

Associate Prof. @ISSPNTT Basic Research Labs.
Assistant Prof. @Tokyo Tech.

Mar. 2009 Ph.D from Kyoto University

Assistant prof. @東⼯⼤

Prof. Fujisawa

Reseacher @NTT BRL

Dr. Muraki

Dr. Kumada

Boundary of a material

 Measurement for bulk properties
Reaction from a boundary Mismatch between systems forming the boundary
A boundary between a sample and a well-known system Bulk property of the sample

An example:

Fresnel law

1Measurement for a quantum many-body system
An example:

 Andreev reflection at a Normal metal/Superconductor junction
Mismatch between carrier charges
 Normal metal: single particles (well-known)
 Superconductor: Cooper pairCharge conservation
Injection

Transmission

Reflectione (electron) = 2e (Cooper pair) - e (hole)

ActionBulk property

Reaction
2

Quantum many-body systems
 Superconductor

Some analogies appear in electron dynamics at boundaries

 Fractional quantum Hall state Tomonga-Luttinger liquid Kondo effect ・・・
Superconductor

Wikipedia

Quantum Hall system
Two-dimensional electron system(2DES) at a high magnetic field

FQH

Feldman et al., Science 2012.

1D electron system

Wikipedia

3

Contents
Background Quantum many-body effects in quantum Hall systems
 Fractional quantum Hall (FQH) effect Tomonaga-Luttinger liquid nature of edge channels
Main Electron dynamics
at boundaries of quantum many-body systems
 Fractional-Integer quantum Hall junction
Hashisaka et al., Nat. Commun. 12, 2794 (2021).

 A boundary of a Tomonaga-Luttinger liquid
Hashisaka et al., Nat. Phys. 13, 559 (2017).

4

Two-dimensional topological systems
S. Oh, Science 2013.

 Integer quantum Hall (IQH) effectKlaus von Klitzing 1980.

Nobel prize 1985.

 Fractional quantum Hall (FQH) effectTsui, Stomer, Gossard 1982.
Laughlin 1983.

Nobel prize 1998.
5

Quantum Hall effect Integer quantum Hall (IQH) effectvon Klitzing 1980.

 Landau-level Filling factor︓v = (integer)
・Charge carrier: e* = e
・Quantum statistics: θ = π（Fermion）
Quantum many-body state Fractional QH (FQH) effectTsui, Stomer, Gossard 1982.
 Laughlin state: v = 1/3, etc.
・Charge carrier: e* = e/3
・Quantum statistics: θ = π/3（Abelian）
 Moore-Read Pfaffian state: v = 5/2
・Charge carrier: e* = e/4
・Quantum statistics: Non-abelian (Ising)

R. Willett et al.,Phys. Rev. Lett. 59, 1776 (1987).

6

Fractional quasiparticles
FQH state

Ground state（v = 1/3）

Bulk: incompressible (insulating)Edge: compressible (metallic)
B

Edge state

FQH state

ElectronFlux
Repulsive Coulomb Interaction

Uniform electron density:
Three flux quanta per an electron.

Elementary excitation (quasiparticle)
 Laughlin state: v = 1/3, etc.

Minimal electrondensity modulation

 Charge carrier: e* = e/3
 Quantum statistics: θ = π/3

Fractional charge
Local excitation

e* = e/3
7

Edge channels
Integer quantum Hall edge:
Chiral 1D channels

Edge state

B

QH state

Fractional quantum Hall edge:
Chiral or helical 1D channels

 Tomonaga-Luttinger liquidInteracting 1D electron systemc.f. Fermi liquid
Real space

Strong screening

Real space

Correlation function:Long range (power law)

Correlation function:
Local (exp. decay)

8Luttinger-liquid nature of edge channels
Quantum many-body state Fractional QH edge channel
Chiral Luttinger liquid

v = 1/3(Short-range interaction)

 Power-law behaviors.

 Integer QH edge channel
Non-interacting 1D system(Short-range interaction)

Artificial Luttinger liquid(Long-range interaction)

 Charge fractionalization
 Spin-charge separation(Discussions later)

Power-law behavior
Phys.Rev. Lett. 77, 2538 (1996).

9

ContentsBackground Quantum many-body effects in quantum Hall systems
 Fractional quantum Hall (FQH) effect Tomonaga-Luttinger liquid nature of edge channels
Main Electron dynamics
at boundaries of quantum many-body systems
 Fractional-Integer quantum Hall junction
Hashisaka et al., Nat. Commun. 12, 2794 (2021).

 A boundary of a Tomonaga-Luttinger liquid
Hashisaka et al., Nat. Phys. 13, 559 (2017).

10

Andreev-reflection-like scatterings
Electron dynamics at boundaries ofquantum many-body systems
Electron scattering reflects the mismatch
between systems forming the boundary.

1. Fractional-Integer quantum Hall junction(Q Many-body)(Normal)Hashisaka et al., Nat. Commun. 12, 2794 (2021).

2. A boundary of a Tomonaga-Luttinger liquid
Hashisaka et al., Nat. Phys. 13, 559 (2017).

11

Fractional-Integer QH junction
Andreev reflection of fractional quantum Hall quasiparticles
M. Hashisaka, T. Jonckheere, T. Akiho, S. Sasaki, J. Rech, T. Martin, K. Muraki
Nature Communications, 12, 2794 (2021).

NTT members
Experiments
CPT, CNRS
Calculations

T. Akiho

S. Sasaki

K. Muraki

T. Jonckheere

J. RechT. MartinAcknowledgements
N. Kumada, T. Ito,
N. Shibata T. Fujisawa,
H. Murofushi, & M. Imai

12

Fractional-Integer QH junction
Andreev reflection

My first thought:

Fractional AR?

Charge conservation at a
FQH／normal metal interface.

???

Fractionally-charged
carrier dynamics
Conventional Andreev reflection at a Metal/Superconductor interface
Injection

Transmission

Reflectione (electron) = 2e (Cooper pair) - e (hole)
Similar scattering process at an FQH/Metal interface (?)
Injection

Transmission

Reflection

e/3 (quasiparticle) = e (electron) - 2e/3 (quasihole)

13

Theoretical proposals
Conductance through the 1/3-1 junction:
tunneling between Luttinger liquids.
Analytical result of thesingle-impurity case:

1/3

1

1/2

1/2

G =1.5 x e2/3h （Strong coupling）
Safi & Schulz, PRB 52, R17040 (1995).
Chamon & Fradkin, PRB 56, 20120(1997).
Chklovskii & Halperin, PRB 57, 3781 (1998).
Sandler, Chamon & Fradlkin, PRB 59, 12521 (1999).

Conductance exceeding e2/3h:Interpretation:
Andreev reflection of fractional quasiparticles.
Sandler, Chamon, Fradkin PRB 57, 12324 (1998).

No experimental demonstration since thetheoretical prediction in the 1990s.

14SampleSplit-gate aperture:

2DES in a GaAs quantum well（with a back gate: VBG ）Electron density ～ 1.8 x 1011 cm-2

300 nm
600 nm
900 nmMagnetic field B = 9.0T
(Bulk filling: v＝ 1 @ VBG = +1.29 V)
Electron temperature ~10 mK

15

Boundary of the FQH and IQH systems
Hashisaka et al., Phys. Rev. X 13, 031024 (2023).

Electron-density profile(Wrong)?

?

We need to consider
electron correlation in the v = 1/3 region.
16

Boundary of the FQH and IQH systems
Hashisaka et al., Phys. Rev. X 13, 031024 (2023).

Electron-density profile
δv = 1δv = 1/3?

Density-Matrix Renormalization Group(DMRG) calculation
Ito & Shibata (Tohoku Univ.)c.f. Ito, T. & Shibata, PRB 103, 115107 (2021).

17

Tunneling between edge channels
Hashisaka et al., Phys. Rev. X 13, 031024 (2023).

Schematic of the 1/3-1 junctionv = 0 strip between
the v = 1/3 and 1 channels.

18

Experimental result 1:

Conductance oscillations with G > e2/3h
Maximum conductance: ~ 1.2 × e2/3h (1.5e2/3h: strong coupling limit)
Oscillation amplitude increases with decreasing the junction width.

測定

Reflection

Evidence of the Andreev process

Transmission

G > e2/3h

Conductance G (e2/3h)

Injection

G = e2/3h

Nanoscale

Wide

Split-gate voltage VS (V)19Experimental result 2:

Negative reflection signal
Negative output voltage of the reflected channel.
Similar oscillations are observed as a function of VS.

Vref
Reflection

Evidence of the Andreev process

Transmission

Vref < 0Normalized output voltage

Injection

Nanoscale

Vref < 0

Wide

Vref = 0

Split-gate voltage VS (V)20

Conductance oscillations
Why does the conductance oscillate?

Multiple impurities

Multiple impurities
G oscillates between 0 and 1.5 x e2/3h.
Theory:Chamon, Fradlkin PRB 56, 20120(1997).
Ponomarenko et al., PRB 67, 035314 (2003).
Zülicke and Shimshoni, PRB 69, 085307 (2004).
Our Experiment:(GaAs 2DES) Hashisaka et al., Nat. Commun. 2021.

Single impurity
G =1.5 x e2/3h （Strong coupling）
Recent Experiment:(Graphene) Cohen et al., Science 2023.

21

Summary of the first topic
Andreev-reflection-like scatterings at an FQH-IQH interface
Nature Communications, 12, 2794 (2021).
Theory: Safi & Schulz, PRB 52, R17040 (1995).

Chamon & Fradkin, PRB 56, 20120(1997).
Sandler, Chamon, Fradkin PRB 57, 12324 (1998).
Chklovskii & Halperin, PRB 57, 3781 (1998).
Sandler, Chamon & Fradlkin, PRB 59, 12521 (1999).

c.f.

Hashisaka et al., Phys. Rev. X 13, 031024 (2023).
Cohen et al., Science 382, 542 (2023).

22

ContentsBackground Quantum many-body effects in quantum Hall systems
Fractional quantum Hall (FQH) effect
Edge channel (Tomonaga-Luttinger liquid)Main Electron dynamics
at boundaries of quantum many-body systems
 Fractional-Integer quantum Hall junction
Hashisaka et al., Nat. Commun. 12, 2794 (2021).

 A boundary of a Tomonaga-Luttinger liquid
Hashisaka et al., Nat. Phys. 13, 559 (2017).

23

Boundary of a Tomonaga-Luttinger liquid
Waveform measurement of charge- and spin-density wavepackets
in a chiral Tomonaga-Luttinger liquid
M. Hashisaka, N. Hiyama, T. Akiho, K. Muraki, and T. Fujisawa
Nature Physics 13, 559 (2017).

Tokyo Tech.
Measurements
NTT members
Crystal growth

N. Hiyama

T. FujisawaT. Akiho

K. Muraki

Acknowledgements
H. Kamata, N. Kumada,
Y. Tokura24

Cooper-pair splitting
Andreev reflection at a
Y-shaped Metal-Superconductor junction
A promising device for generating
pairwise entangled electrons in solids.
Experiments:
Das et al., Nat. Commun. 2012.
Deacon et al., Nat. Commun. 2015.
Rani et al., Nat. Commun. 2021.
Bordoloi et al., Nature 2022.
etc.

Hofstetter et al., Nature 2009.

Mode transformation at a boundary of a Tomonaga-Luttinger liquid
Electron fractionalization (Spin-Charge separation)
Hashisaka et al., Nat. Phys. 13, 559 (2017).

25

Artificial Luttinger liquid Integer QH edge channel(Short-range interaction)

Non-interacting 1D system(Long-range interaction)

Artificial Luttinger liquid Charge fractionalization
 Spin-charge separation

RF transport

S. Oh, Science 2013.

26

Low-energy charge dynamics
Quantum Hall system Bulk: (Energy gap) > ~100 GHz
 Edge: gapless Edge magneto-plasmon (EMP)••

Change in the boundary shape
Almost linear dispersion relation

2-dimensional electron system(2DES)

n=1

n = 0 Landau level

27

Inter-channel interaction
 Circuit model of edge magnetoplasmons
Hashisaka et al., PRB 2012; PRB 2013.

・1D continuity equation
・Electron correlation
・Equation of motion

Inter-channel Coulomb interaction
 1D wave equation:v = …, UX = …
Hashisaka & Fujisawa, Rev. Phys. 2018.

28

Artificial Luttinger liquidInteracting integer edge channels show Luttinger-liquid-like behaviors
Theory: Berg, Oreg, Kim, von Oppen, Phys. Rev. Lett. 102, 236409 (2009).

Charge conservation:e = er + e(1  r)electronCharge fractionalization

Mode A + Mode B

Spin-Charge separation
Spin
ChargeKamata, Hashisaka, et al., Nat. Nanotechnol. 2014.
Hashisaka et al., PRB 2012.

29

Waveforms of charge & spin excitations
Direct time-domain measurement of the spin-charge separation
in a Tomonaga-Luttinger liquid.

Hashisaka et al., Nat. Phys. 13, 559 (2017).

Optical micrograph of the sample (v = 2 IQH system in a GaAs quantum well).

30

Spin-charge separation
Hashisaka et al., Nat. Phys. 13, 559 (2017).

 Spin-up charge injection

 Spin-down charge injection

31

Spin-charge separation
Hashisaka et al., Nat. Phys. 13, 559 (2017).

 Spin-up charge injection

 Spin-down charge injection

32Summary of the second topic
Waveform measurement of charge- and spin-density wavepackets
in a chiral Tomonaga-Luttinger liquid
Nature Physics 13, 559 (2017).

Theory: Berg, Oreg, Kim, von Oppen, Phys. Rev. Lett. 102, 236409 (2009).

Mode transformations at boundaries
Singleelectron

Multiple modes(Fractionalization)

SingleCooper pairMultiple electrons
Hashisaka et al., Nat. Phys. 13, 559 (2017).

Hofstetter et al., Nature 2009.

33

SummaryBackground Quantum many-body effects in quantum Hall systems
Fractional quantum Hall (FQH) effect
Edge channel (Tomonaga-Luttinger liquid)Main Electron dynamics
at boundaries of quantum many-body systems
 Fractional-Integer quantum Hall junction
Hashisaka et al., Nat. Commun. 12, 2794 (2021).

 A boundary of a Tomonaga-Luttinger liquid
Hashisaka et al., Nat. Phys. 13, 559 (2017).

Appendix Quantum many-body phase of interacting 1D channels
Hashisaka et al., Phys. Rev. X 13, 031024 (2023).

34

Edge-mode transformation
Coherent-Incoherent Crossover of Charge and Neutral Mode Transport
as Evidence for the Disorder-Dominated Fractional Edge Phase
Hashisaka et al., Phys. Rev. X 13, 031024 (2023).

NTT members
ExperimentsT. Akiho

S. Sasaki

Tohoku Univ.

N. KumadaK. Muraki

Acknowledgements

Calculations
T. ItoN. Shibata

T. Jonckheere, J. Rech,
T. Martin, T. Fujisawa,
H. Murofushi, & M. Imai35

Appendix: Coulomb interaction & Tunnel couplings
Example: Helical edge channels

 2D topological insulator+Coulomb interaction(HgTe, etc.)

Tunnel couplings

Quantized
electrical conductance (?)

 Equilibration in the coupled helical channels.

Tunnel couplings

Strong coupling

Weak coupling

StrongWeak

Phase diagram of thecoupled helical channels(from renormalization group theory)
The 1D phase can be identified by theElectrical and thermal transport meas.
MacDonald, PRL 1990.
Kane, Fisher, Polchinski, PRL 1994.
Kane, Fisher, PRB 1995.
Protopopov, Gefen, Mirlin, Ann. Phys. 2017.

36

Appendix: Coulomb interaction
Mechanism: How do the charge and charge-neutral modes emerge?
Wave equation:(Coulomb interaction only)
Hashisaka & Fujisawa, Rev. Phys. 3, 32 (2018).

Coulomb interaction only
Right- and left-moving modes
carrying finite charges.

qL

qR

Renormalization toward
the strong coupling limit

Coulomb + Tunnel couplings
Downstream charge andupstream charge-neutral modes

q=0q2/3

Kane, Fisher, Polchinski 1994.
Kane, Fisher 1995.
Protopopov, Gefen, Mirlin 2017.

37

Appendix: Renormalization
Disorder-dominated FQH edge phase
q=0q2/3

Charge and neutral modes.
Weak coupling phase (Δ > 3/2)

Right- and left-moving
modes both carrying charges.
Kane, Fisher, Polchinski 1994.
Kane, Fisher 1995.
Protopopov, Gefen, Mirlin 2017.

38

Appendix: Upstream heat transport
Experimental evidence (?) of the strong coupling phase Electrical conductance: 2e2/3h Upstream heat transport

ChargeCurrentHeatv = 2/3(Charge-neutral mode)“Standard model” forfractional QH edge transport••

Downstream charge mode
Upstream neutral mode

Banerjee et al., Nature 545, 75 (2017).
Banerjee et al., Nature 559, 205 (2018).
Nakamura et al., Nat. Phys. 15, 563 (2019).
Dutta et al., Science 375, 193 (2022).
Dutta et al., Science 377, 1198 (2022). etc.

Bid et al.,Nature 2010.

Bid et al., Nature 466, 585 (2010).
Venkatachalam et al., Nat. Phys. 8, 676 (2012).
Altimiras et al., PRL 109, 026803 (2012).
Inoue et al., Nat. Commun. 5, 4067 (2014).
Sabo et al., Nat. Phys. 13, 491 (2017).
Kumar et al., Nat. Commun. 13, 213 (2022).
Melcer et al., Nat. Commun. 13, 376 (2022).
Breton et al., PRL 129, 116803 (2022).
Srivastav et al., Nat. Commun. 13, 5185 (2022).
Nakamura et al., PRL 130, 076205 (2023). etc.

39Appendix: Problems in experiments and theories
 Non-interacting “leads” near ohmic contacts
Protopopov, Gefen, Mirlin, Ann. Phys. 385, 287 (2017).
“Strong” and “Weak” coupling phases havevery similar electrical & thermal transport properties
except for the mesoscopic regime.

Non-interacting “leads”

40Appendix: Coherence in a disordered system
Previous experiments:

Protopopov, Gefen, Mirlin, Ann. Phys. 385, 287 (2017).

Long counter-propagating channels with inelastic scatterings.
Bid et al., Nature 2010, etc.

This study: crossover between long and short channel regimes.

Short counter-propagating channels with only elastic scatterings.
Mesoscopic regime

Coherent Mesoscopic Incoherent

L < Lin
L: 1/3-1 channel length
Lin: Inelastic scattering length

Cocurrence of Electrical conductance fluctuations
 Quantized thermal conductancec.f.
Cohen et al., Nat. Commun. 2019.
Kumar et al., Nat. Commun. 2022.
Melcer et al., Nat. Commun. 2022.

41

Appendix: Half-quantized thermal conductanceExperimental hallmark

 Strong coupling phase Electrical conductance: 2e2/3h
 Upstream heat transport

Half-quantized thermal conductancein the dissipationless transport regime.

Bid et al., Nature 2010.
Inoue et al., Nat. Commun. 2014.
・・・

Protopopov, Gefen, Mirlin 2017.

e2/h
e2/3h

Electrical transport 2e2/3h(1 – 1/3)

Tunnel couplings

Upstream heat transport

StrongWeak42

Appendix: Thermal conductance measurement
Joule heat

2DES in a GaAs quantum well

HeatCharge

Thermal Noise(Thermal conductance from top to bottom)

Half-quantized thermal conductance GQ/2:

Evidence of the strong coupling phase43

Appendix: Particle-hole symmetry
Coulomb interaction + Tunneling

M. Hashisaka et al., PRX 13, 031024 (2023).

Generalized quantum Hall (QH) edge system
Conventional v = 2/3 device

Fractional-Integer QH junction(FQH-IQH)
v(x)  1  v(x)Particle-holesymmetry

Spontaneously
appeared
1/3-1 channels

44Appendix: Bulk-edge correspondence Counter-propagating v = 1/3 and v = 1 edge channels
BulkSpin-polarized v = 2/3 state:
vh = 1/3 fractional quantum Hall state(v = 2/3 = 1  1/3)
Bulk-Edge correspondence

Edge
Counter-propagating 1/3 and 1 channels(MacDonaldʼs model: 1-1/3 channels)
MacDonald, PRL 64, 220 (1990).
Johnson, MacDonald, PRL 67, 2060 (1991).
Meir, PRL 72, 2624 (1994).
Chamon, Wen, PRB 49, 8227 (1994).
…

45

Summary of the Appendix partExperiment:
Coherent-Incoherent Crossover of Charge and Neutral Mode Transport
as Evidence for the Disorder-Dominated Fractional Edge Phase
M. Hashisaka, T. Ito, T. Akiho, S. Sasaki, K. Kumada, N. Shibata, K. Muraki
Physical Review X 13, 031024 (2023).

Theory:Transport in a disordered v = 2/3 fractional quantum Hall junction
I. V. Protopopov, Y. Gefen, A. D. Mirlin
Annals of Physics 385, 287 (2017).

Key background theory:
Neutral modes in fractional QH systems.
Kane, Fisher, Polchinski, PRL 72, 4129 (1994).
Kane, Fisher, PRB 51, 13449 (1995).

46