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

Quantum Hall Edge States Halperin 1982; Wen Early 1990s; Kane+Fisher mid 90’s; + many many more -E B ⊗ Edge of sample v • Edge States Carry Electrical Current • Edge States Carry Heat See …

Case of Noninteracting Fermions

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

Thermal Conductance proposed as a probe: 1997 (Kane-Fisher) No one really thought this could be measured! 10 11 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 ≈ Abelian U(1) vs nonabelian Lie Group Chern-Simons theory time 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 • Unusual even denominator • Probably has Majorana Quasiparticles (Nonabelian statistics!) • Topological Quantum Computation?! Data: Pan et al, ‘99

ν =5/2: Expect Majorana Edge (?) Majorana = Half-Fermion ⇒ Half-Unit of Thermal Conductance

ν = 5/2 FQHE Proposed States of Matter Pfaffian vs AntiPfaffian Morf 1998 2007 Majorana Mode has ½ unit of thermal conductance

ν = 5/2 FQHE Proposed States of Matter Rezayi, Simon, 2009… Rezayi 2017 Majorana Mode has ½ unit of thermal conductance

... And the experiment says: K ≈ 2.5 quanta Banerjee, Heiblum et al, 2018 Half- Victory: Half- integer = Majorana mode But also Half -Puzzling: …

How could the (very clear) numerics get the answer so wrong? Numerics have no disorder! D. T. Son 2015 (Also Fidkowski et al, 13 Bonderson, et al 13; Barkeshli et al, 15, Wan and Yang, 16)

Mixed Domains of Pfaffian and AntiPfaffian ... network of neutral internal edge modes. 4 majoranas per domain wall Mross, Oreg, Stern, Margalit, Heiblum ’17 Wang, Halperin, Vishwanath ’17 Lian, Jiang ‘18 • One possibility: neutral internal edge modes percolate Quantized electrical conductivity Thermal Metal Unquantized thermal conductivity

... And the experiment says: K ≈ 2.5 quanta Banerjee, Heiblum et al, 2018

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 Yes… but… Can domain walls gap to create PH-Pfaffian? 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. APf Pf 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

Mixed Domains of Pfaffian and AntiPfaffian ... network of neutral internal edge modes. See however… (full disclosure) •In a clean system APf is lower energy Domain walls have dipole Phys. Rev. B 101 , 041302(R) 2020 Moment. If the moment •But quasielectrons of APf are higher energy than those of Pf aligns with local impurity electric field, the wall could •With enough quasielectrons ( ν slightly away from 5/2) be stabilized Pf + quasielectrons is lower energy than Apf + quasielectrons

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 ↑ LL0 ↓ LL1 ↑ Boson Majorana K = 3 – 1.5 = 1.5 Why subtract?

Forward – backwards (bosonic) edge modes

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

Non equilibration Forward – backwards edge modes

Detailed Structure of APf Edge LL0 ↑ LL0 ↓ Suppose these K=3-1 thermally LL1 ↑ equilibrate Boson But not Majorana K=0.5 this K total = 2.0 + 0.5 = 2.5 ... is it that easy?

Detailed Structure of APf Edge LL0 ↑ LL0 ↓ Suppose these K=3-1 thermally LL1 ↑ equilibrate Boson But not Majorana K=0.5 this 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 Bose mode mode Majorana mode Distribution of Can scatter to this region → wavevectors of disorder Energy in Majorana creates scattering 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)

Simon+Rosenow LL1 ↑ ? Ignore all disorder Boson Majorana

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

Simon+Rosenow Majorana impurity assisted tunneling LL1 ↑ Add one impurity Boson Majorana Tunnel strength λ Remote localized Majorana in bulk

Simon+Rosenow Majorana impurity assisted tunneling Add one impurity Majorana Tunnel strength λ Remote localized Majorana in bulk At high energy impurity is irrelevant At low energy impurity disorders edge

Simon+Rosenow Majorana impurity assisted tunneling LL1 ↑ Add one impurity Boson Majorana Tunnel strength λ Remote localized Majorana in bulk At high energy impurity is irrelevant At low energy impurity disorders edge Scattering only occurs if majorana gets energy Detailed Result:

Gives Natural Scenario To Get LL0 ↑ LL0 ↓ These thermally K=3-1 equilibrate LL1 ↑ Scattering Rate ~ T Boson But not Majorana K=0.5 this K total ⇒ 2.5

At very low T LL0 ↑ these LL0 ↓ K=3 equilibrate LL1 ↑ But not K=1 Boson this and not Majorana K=0.5 this K total ⇒ 4.5 (and eventually G becomes nonquantized too.. But much slower )

Theory vs Experiment v = v i = v b = 10 6 cm/sec ; v m =10 5 cm/sec λ = 4mK (coupling of majorana to edge) Number of scatters = n imp = 10/micron Edge momentum mismatch = p = 0.1 / l B Edge Length = L = 150 micron l B = 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 ↑ 1. Exact solution of Majorana+Impurity Boson 2. Fermi’s Golden Rule for Electron Tunneling Majorana X = e or E for charge or heat current 3. Green’s function of fractionalized edge is made of Boson and Majorana Greens functions.

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 – E GS = 0 First excited state E – E GS = 1 Fermions Bosons

Fermions=Bosons E – E GS = 2 E – E GS = 2 Fermions Bosons