Thermal Transport in =5/2 Fractional Quantum Hall Edges S. H. - - PowerPoint PPT Presentation

Thermal Transport in ν=5/2 Fractional Quantum Hall Edges

• S. H. Simon, PRB 97, 121406, 2018
• S. H. Simon and Bernd Rosenow PRL 124, 126801, 2020
• S. H. Simon, M. Ippoliti, M. Zaletel, E. H. Rezayi PRB 2020

• E

B ⊗ Edge of sample

Quantum Hall Edge States

Halperin 1982; Wen Early 1990s; Kane+Fisher mid 90’s; + many many more

v

• Edge States Carry Electrical Current
• Edge States Carry Heat

See …

Case of Noninteracting Fermions

Thermal Conductance Tells Something More “Central Charge” Electric Hall Conductance Always Matches Filling Fraction

Thermal Conductance proposed as a probe: 1997 (Kane-Fisher) No one really thought this could be measured! 1011 e’s ... .then came twenty years of innovation...

Develops technique of noise thermometry Measurement of thermal conductance. Applied to mesoscopic n-channel ballistic wire K = 1 unit thermal conductance per channel G =1 unit electrical conductance per channel

Forward – backwards (bosonic) edge modes

Abelian vs NonAbelian Statistics

time ≈ Abelian U(1) vs nonabelian Lie Group Chern-Simons theory Abelian: Wavefunction is a scalar Exchange incurs a phase (recent experimental proof!) NonAbelian: Wavefunction is a vector in a space of degenerate states Exchange applies a unitary matrix to this space. Applications to Topological Quantum Computing!

ν = 5/2 FQHE

Data: Pan et al, ‘99

• Unusual even denominator
• Probably has Majorana Quasiparticles

(Nonabelian statistics!)

• Topological Quantum Computation?!

ν =5/2: Expect Majorana Edge (?)

Majorana = Half-Fermion ⇒ Half-Unit of Thermal Conductance

ν = 5/2 FQHE

Proposed States of Matter Majorana Mode has ½ unit of thermal conductance Morf 1998 2007 Pfaffian vs AntiPfaffian

ν = 5/2 FQHE

Proposed States of Matter Majorana Mode has ½ unit of thermal conductance

Rezayi, Simon, 2009… Rezayi 2017

... And the experiment says:

Banerjee, Heiblum et al, 2018

K ≈ 2.5 quanta

Half-Victory: Half-integer = Majorana mode But also Half-Puzzling: …

• D. T. Son 2015

(Also Fidkowski et al, 13 Bonderson, et al 13; Barkeshli et al, 15, Wan and Yang, 16)

How could the (very clear) numerics get the answer so wrong? Numerics have no disorder!

Mixed Domains of Pfaffian and AntiPfaffian ... network of neutral internal edge modes.

Mross, Oreg, Stern, Margalit, Heiblum ’17 Wang, Halperin, Vishwanath ’17 Lian, Jiang ‘18

• One possibility: neutral internal edge modes percolate

Quantized electrical conductivity Unquantized thermal conductivity Thermal Metal

4 majoranas per domain wall

... And the experiment says:

Banerjee, Heiblum et al, 2018

K ≈ 2.5 quanta

Mixed Domains of Pfaffian and AntiPfaffian ... network of neutral internal edge modes.

Mross, Oreg, Stern, Margalit, Heiblum ’17 Wang, Halperin, Vishwanath ’17 Lian, Jiang ‘18

Can domain walls gap to create PH-Pfaffian? Yes… but… Wang, Halperin,Vishwanath: “Generically our network model does not favor this behavior” Requires fine tuning of scattering matrix at saddles

Mixed Domains of Pfaffian and AntiPfaffian ... network of neutral internal edge modes.

Mross, Oreg, Stern, Margalit, Heiblum ’17 Wang, Halperin, Vishwanath ’17 Lian, Jiang ‘18

Either Thermal Metal or (unlikely) PH-Pfaffian But why should we ever expect domains?

Mixed Domains of Pfaffian and AntiPfaffian ... network of neutral internal edge modes. But why should we ever expect domains?

• In a clean system at ν=5/2, APf is lower energy
• But quasielectrons of APf are higher energy than those of Pf
• With enough quasielectrons (ν slightly away from 5/2)

Pf + quasielectrons is lower energy than Apf + quasielectrons APf Pf

• In a clean system APf is lower energy
• But quasielectrons of APf are higher energy than those of Pf
• With enough quasielectrons (ν slightly away from 5/2)

Pf + quasielectrons is lower energy than Apf + quasielectrons Mixed Domains of Pfaffian and AntiPfaffian ... network of neutral internal edge modes.

• Phys. Rev. B 101, 041302(R) 2020

See however… (full disclosure)

Domain walls have dipole

• Moment. If the moment

aligns with local impurity electric field, the wall could be stabilized

Is there another solution?

Yes

( +Extension: Steven H. Simon and Bernd Rosenow PRL 124, 126801, 2020)

ν = 5/2 FQHE

Proposed States of Matter Is there a way we can get K=2.5 quanta from Anti-Pfaffian?

Detailed Structure of APf Edge

LL0 ↑ LL1↑ LL0↓ Boson Majorana

K = 3 – 1.5 = 1.5

Why subtract?

Forward – backwards (bosonic) edge modes

Edges that thermal equilibrate : conductances add with their signs Edges that don’t talk to each other: conductances add in absolute value

How Thermal Conductances Add

Forward – backwards edge modes

Non equilibration

Detailed Structure of APf Edge

LL0 ↑ LL1↑ LL0↓ Boson Majorana Suppose these thermally equilibrate

K=3-1

But not this

K=0.5

Ktotal = 2.0 + 0.5 = 2.5

... is it that easy?

Detailed Structure of APf Edge

LL0 ↑ LL1↑ LL0↓ Boson Majorana Suppose these thermally equilibrate

K=3-1

But not this

K=0.5

Scattering must occur to equilibrate charge...(cf Kane, Fisher, Polchinski) Necessary for quantized Electrical conductivity! .... But the heat must go only into the boson mode only.

Mechanisms to get equilibration of charge without equilibration of heat (therefore K=2.5)

• 1. Large ratio of bose velocity to majorana velocity

(plus long enough wavelength disorder) Simon, PRB 2018

Proposal 1: (Simon PRB 2018) Long length scale disorder + Large ratio between Bose and Majorana velocities

Integer mode Bose mode Majorana mode

Distribution of wavevectors of disorder creates scattering Can scatter to this region → Energy in Majorana is very small

Mechanisms to get equilibration of charge without equilibration of heat (therefore K=2.5)

• 1. Large ratio of bose velocity to majorana velocity

(plus long enough wavelength disorder) Simon, PRB 2018 (+comment from Feldman)

• 2. Ma and Feldman, 2019 similar bose/majorana velocity

contrast mechanism

• 3. Asasi and Mulligan 2004.04161: A novel stable fixed point
• 4. Localized Majorana assisted tunneling (Simon + Rosenow)

LL1↑ Boson Majorana Simon+Rosenow Ignore all disorder

?

LL1↑ Boson Majorana Simon+Rosenow Ignore all disorder Position Displacement = Momentum Displacement Forbids Scattering Unless x-Translation Invariance is Broken

Forbidden

LL1↑ Boson Majorana Simon+Rosenow Majorana impurity assisted tunneling

Remote localized Majorana in bulk

Tunnel strength λ

Add one impurity

At high energy impurity is irrelevant At low energy impurity disorders edge

Majorana Simon+Rosenow Majorana impurity assisted tunneling

Remote localized Majorana in bulk

Tunnel strength λ

Add one impurity

At high energy impurity is irrelevant At low energy impurity disorders edge

LL1↑ Boson Majorana Simon+Rosenow Majorana impurity assisted tunneling

Remote localized Majorana in bulk

Tunnel strength λ

Add one impurity

Detailed Result: Scattering only occurs if majorana gets energy

Gives Natural Scenario To Get

LL0 ↑ LL1↑ LL0↓ Boson Majorana These thermally equilibrate

K=3-1

But not this

K=0.5

Ktotal ⇒ 2.5

Scattering Rate ~ T

At very low T

LL0 ↑ LL1↑ LL0↓ Boson Majorana these equilibrate and not this

Ktotal ⇒ 4.5

But not this (and eventually G becomes nonquantized too.. But much slower )

K=0.5 K=3 K=1

Theory vs Experiment

v = vi = vb= 106 cm/sec ; vm =105 cm/sec

λ = 4mK (coupling of majorana to edge)

Number of scatters = nimp = 10/micron Edge momentum mismatch = p = 0.1 /lB Edge Length = L = 150 micron lB = 16nm

How to test non-equilibration theories

Experiment suggested by Halperin Add (thermally and electrically) floating contacts along edge to force equilibration of all edge modes Fully thermal equilibrated AntiPfaffian edge gives K = 1.5

How The Actual Calculation is done:

LL1↑ Boson Majorana

1. Exact solution of Majorana+Impurity 2. Fermi’s Golden Rule for Electron Tunneling 3. Green’s function of fractionalized edge is made of Boson and Majorana Greens functions.

X = e or E for charge or heat current

Summary

• Measurement of thermal edge conductance is tour-de-force
• ν=5/2 does not match expectations
• Probably not Pf/APf domains ⇒ Not thermal metal or PH-Pfaff
• Could be AntiPfaffian (predicted state) with Majorana

mode remaining out of thermal equilibrium

• Majorana impurity assisted scattering naturally gives a

mechanism for this to happen

Thermal Transport in ν=5/2 Fractional Quantum Hall Edges

• S. H. Simon, PRB 97, 121406, 2018
• S. H. Simon and Bernd Rosenow PRL 124, 126801, 2020
• S. H. Simon, M. Ippoliti, M. Zaletel, E. H. Rezayi PRB 2020

Case of Noninteracting Fermions Electric current

Case of Noninteracting Fermions Thermal current

Edges in simple quantum Hall states: Laughlin State ν =1/m with P a homogeneous symmetric polynomial ⇒ Laughlin state has a chiral boson edge

Fermions=Bosons Ground state E – EGS = 0 First excited state E – EGS = 1 Fermions Bosons

Fermions=Bosons E – EGS = 2 E – EGS = 2 Fermions Bosons

Fermions=Bosons E – EGS = 3 E – EGS = 3 Fermions Bosons

Fermions=Bosons E – EGS = 3 E – EGS = 3 Fermions Bosons

2 Majorana Modes = 1 Boson Mode = 1 Fermion Mode Majoranas have positive half integer momentum Parity constraint: Majoranas come in pairs

• E

B ⊗ Edge of sample

Quantum Hall Edge States

Halperin 1982; Wen Early 1990s; Kane+Fisher mid 90’s; + many many more

v

• Edge States Carry Electrical Current
• Edge States Carry Heat