Valley Physics in High Magnetic Fields

week ending 5 AUGUST 2005

PHYSICAL REVIEW LETTERS

PRL 95, 066809 (2005)

ε[010]

hωc

E[

E [100]

(a)

piezo University of California, Berkeley

∆Ev ∆EZ

010]

(b)

ε[010]

-1

1.5

∆Ev (meV) 0 1

2 3

0.15

0.5

0 (d)

E

[e2/4πεlB]

1.0

1∆

1∆

[e2/4πεlB]

B

B 0

0.02

0.04

∆EZ [e2/4πεlB]

hωc 2

0.10 ∆EZ

1

0.05 0 (c)

-0.05

0

0.05

0.10

0 0.15

∆Ev [e2/4πεlB]

FIG. 1 (color online). (a) In-plane strain breaks the valley energy degeneracy; e.g., tension along [100] results in a transfer of electrons from the [100] valley to the [010] valley. (b) Single-particle fan diagram for the AlAs 2DES using band parameters. (c) Single-particle (dashed lines) and measured (circles) # ! 1 QHS gap, 1 !, as a function of the single-particle valley splitting, !Ev , for n ! 2:55 ' 1011 cm%2 . 1 ! and !Ev are given both in meV (top and right axes) and also in units of the Coulomb energy e2 =4$"lB , where " ! 10"0 is the dielectric constant of AlAs and lB ! $@=eB&1=2 is the magnetic length; in our experiments, B ! 10:5 T and e2 =4$"lB ! 18:2 meV. (d) 1 ! vs !EZ for spin Skyrmions in a single-valley 2DES. The dashed line is the single-particle Zeeman splitting. The theoretical prediction for the Skyrmion excitation energy is shown by the solid curve and the solid symbols show the experimentally determined gaps at # ! 1 (see text).

j

from the oscillations of the sample’s resistance at high integer filling factors as a function of strain [12] and corroborate it with the Fourier analysis of Shubnikov– de Haas oscillations, the zero-field piezoresistance data [14], and our calibration of the piezoinduced strain [12]. Before presenting the experimental data, it is instructive to describe the fan diagram of the AlAs 2DES as a function of strain-induced valley splitting in a noninteracting, single-particle picture [Fig. 1(b)]. The magnetic field perpendicular to the plane of the 2DES quantizes the orbital motion of the electrons and forces them to occupy a discrete set of energy levels separated by the cyclotron energy, @!c ! @eB=m" , where m" ! 0:46 is the cyclotron effective mass in AlAs [15]. There are four sets of these Landau levels, one for each spin and valley combination. The energy splitting between oppositely polarized spins is con-

trolled through !EZ ! gb !B B (band g factor gb ! 2), while the levels corresponding to different valleys are separated by !Ev . When "xy ! 0, in the single-particle picture of Fig. 1(b), the energy degeneracy of the Landau levels associated with different valleys implies that there is no # ! 1 QHS. As the applied strain increases and breaks the valley degeneracy, the # ! 1 QHS gap (1 !) develops. For !Ev < !EZ , 1 ! should increase linearly with strain and be equal to !Ev . For sufficiently large strains such that !Ev exceeds !EZ , 1 ! should be equal to !EZ , independent of strain. The results of our measurements are in sharp contrast to the simple, noninteracting picture described above. Experimentally, we determine 1 ! from the activated T dependence of the longitudinal resistance, Rxx , according to R # exp$%1 !=2k T&. This gap is plotted in Fig. 1(c)

Impurities and Textures in Unconventional Magnets, MPI-PKS Dresden, 2012

Monday, December 3,

(meV)

ε[100]

010]

1∆

Sid Parameswaran AlAs

ε[100]

E[

E [100]

Collaborators

Dmitry Abanin

Akshay Kumar

Princeton→Harvard→Perimeter

IIT Delhi→Princeton

Shivaji Sondhi

Steve Kivelson

Princeton

Stanford

Phys. Rev. B., 82, 035428 (2010) Phys. Rev. Lett., 103, 076802 (2009) Monday, December 3,

Outline 0. Motivation: quantum Hall ferromagnets in GaAs I. Ising-nematic order in multivalley quantum wells symmetry analysis random-field Ising phases domains & transport II. Charge 2e valley skyrmions in bilayer graphene ‘orbital’ degeneracy level splittings near ν=0 skyrmion binding; skyrme crystals

Monday, December 3,

What’s ‘unconventional’?

Monday, December 3,

What’s ‘unconventional’?

c. 6 BCE

Monday, December 3,

What’s ‘unconventional’?

c. 6 BCE

Monday, December 3,

c. 1930

What’s ‘unconventional’?

c. 6 BCE

Monday, December 3,

c. 1930

c.1990-present

What’s ‘unconventional’?

c. 6 BCE

c. 1930

c.1990-present

This talk will focus on relatively ‘conventional’ magnetism, but in an unconventional setting

Monday, December 3,

What’s ‘unconventional’?

c. 6 BCE

c. 1930

c.1990-present

This talk will focus on relatively ‘conventional’ magnetism, but in an unconventional setting valley ferromagnets in 2DEGs in the quantum Hall regime Monday, December 3,

Spin in the Quantum Hall Effect B

E

j

Electrons in magnetic field: Landau levels

Monday, December 3,

Spin in the Quantum Hall Effect B

E

j

Electrons in magnetic field: Landau levels N states per LL

~!c

Monday, December 3,

Spin in the Quantum Hall Effect B

E

j

Electrons in magnetic field: Landau levels N states per LL N filling factor: ⌫ = N

Monday, December 3,

~!c

Spin in the Quantum Hall Effect B

E

j

Electrons in magnetic field: Landau levels N states per LL N filling factor: ⌫ = N

add spin

~!c ~!c Z

Monday, December 3,

⇠ g ⇤ µB B

Spin in the Quantum Hall Effect B

E

j

Electrons in magnetic field: Landau levels N states per LL N filling factor: ⌫ = N

add spin

~!c ~!c Z

GaAs:

Monday, December 3,

Z

⌧ ~!c

⇠ g ⇤ µB B

Spin in the Quantum Hall Effect B

E

j

Electrons in magnetic field: Landau levels N states per LL N filling factor: ⌫ = N

add spin

GaAs:

Monday, December 3,

Z

⌧ ~!c

~!c

~!c

Z

⇠ g ⇤ µB B

Spin in the Quantum Hall Effect B

E

j

Electrons in magnetic field: Landau levels N states per LL N filling factor: ⌫ = N

add spin

GaAs:

Monday, December 3,

Z

⌧ ~!c

~!c

~!c

Z

⇠ g ⇤ µB B

Spin in the Quantum Hall Effect B

E

j

Electrons in magnetic field: Landau levels N states per LL N filling factor: ⌫ = N

~!c

add spin

GaAs: ~!c Monday, December 3,

Texpt

Z

⌧ ~!c Z

~!c

Z

⇠ g ⇤ µB B

=) need interactions to split spin states

lowest Landau level limit: ~!c ! 1. degenerate spin states (g ⇡ 0)

Monday, December 3,

⌫=1

lowest Landau level limit: ~!c ! 1. degenerate spin states

⌫=1

(g ⇡ 0)

Add repulsive e-e interactions

Monday, December 3,

lowest Landau level limit: ~!c ! 1. degenerate spin states

⌫=1

(g ⇡ 0)

Add repulsive e-e interactions Spin polarization + Pauli exclusion minimizes interaction energy

Monday, December 3,

e2 ⇠ "`B

lowest Landau level limit: ~!c ! 1. degenerate spin states

⌫=1

(g ⇡ 0)

Add repulsive e-e interactions Spin polarization + Pauli exclusion minimizes interaction energy Hund’s Rule

Monday, December 3,

e2 ⇠ "`B

lowest Landau level limit: ~!c ! 1. degenerate spin states

⌫=1 e2 ⇠ "`B

(g ⇡ 0)

Add repulsive e-e interactions Spin polarization + Pauli exclusion minimizes interaction energy Hund’s Rule very effective since kinetic energy quenched in B field Monday, December 3,

Low-energy theory E[n] ⇡

Z

Monday, December 3,

2

d r

h⇢

s

2

2

(rn) +

Zn

z

1 2

Z

d2 r0 V (r

r0 )q(r)q(r0 )

]

e2 ⇢s / "`B

[Sondhi et. al., PRB 47, 16419 (1993)]

Low-energy theory E[n] ⇡

Z

2

d r

h⇢

s

2

2

(rn) +

Zn

z

1 2

Z

d2 r0 V (r

Pontryagin density 1 abc a b c q(r) = " n (rn ⇥ rn ) 8⇡

Monday, December 3,

r0 )q(r)q(r0 )

]

e2 ⇢s / "`B

[Sondhi et. al., PRB 47, 16419 (1993)]

Low-energy theory E[n] ⇡

Z

2

d r

h⇢

s

2

2

(rn) +

Zn

z

1 2

Z

d2 r0 V (r

Pontryagin density 1 abc a b c q(r) = " n (rn ⇥ rn ) 8⇡ skyrmion number Z 2 Qtopo = q(r)d r

Monday, December 3,

r0 )q(r)q(r0 )

]

e2 ⇢s / "`B

[Sondhi et. al., PRB 47, 16419 (1993)]

Low-energy theory E[n] ⇡

Z

2

d r

h⇢

s

2

2

(rn) +

Zn

z

1 2

Z

d2 r0 V (r

r0 )q(r)q(r0 )

]

[Sondhi et. al., PRB 47, 16419 (1993)]

Pontryagin density 1 abc a b c q(r) = " n (rn ⇥ rn ) 8⇡ skyrmion number Z 2 Qtopo = q(r)d r

spin textures carry electrical charge

Monday, December 3,

e2 ⇢s / "`B

Low-energy theory E[n] ⇡

Z

2

d r

h⇢

s

2

2

(rn) +

Zn

z

1 2

Z

d2 r0 V (r

r0 )q(r)q(r0 )

]

e2 ⇢s / "`B

[Sondhi et. al., PRB 47, 16419 (1993)]

Pontryagin density 1 abc a b c q(r) = " n (rn ⇥ rn ) 8⇡ skyrmion number Z 2 Qtopo = q(r)d r

spin textures carry electrical charge ⇢, Sˆµ ] 6= 0 in LLL b/c restricted Hilbert space: roughly, [ˆ Monday, December 3,

Doping with Skyrmions ⌫=1

Monday, December 3,

adding charge must involve unfilled up spin-LL single flip or skyrmion? need to compare energies

Doping with Skyrmions ⌫=1

adding charge must involve unfilled up spin-LL single flip or skyrmion? need to compare energies

g=0 gap

Monday, December 3,

Zeeman both couples to and changes total spin

Doping with Skyrmions adding charge must involve unfilled up spin-LL single flip or skyrmion? need to compare energies

⌫=1

g=0 gap

Zeeman both couples to and changes total spin

Skyrmion gap at g=0 : 1/2 single-flip gap so for sufficiently small g: smallest gap is for skyrmions

Monday, December 3,

Doping with Skyrmions adding charge must involve unfilled up spin-LL single flip or skyrmion? need to compare energies

⌫=1

g=0 gap

Zeeman both couples to and changes total spin

Skyrmion gap at g=0 : 1/2 single-flip gap so for sufficiently small g: smallest gap is for skyrmions Moving away from ν=1 Monday, December 3,

adding skyrmions/antiskyrmions

Experimental evidence for spin skyrmions in GaAs

Monday, December 3,

Experimental evidence for spin skyrmions in GaAs 71Ga

NMR: Knight Shift

[Barrett et. al., PRL 74, 5112 (1994)] Ks / hS z i

rapid drop in polarization ~ skyrmion proliferation

Monday, December 3,

Experimental evidence for spin skyrmions in GaAs 71Ga

NMR: Knight Shift

Tilted-field Activation Gap

[Barrett et. al., PRL 74, 5112 (1994)]

[Schmeller et. al., PRL 75, 4290 (1995)]

Ks / hS z i

rapid drop in polarization ~ skyrmion proliferation

Monday, December 3,

Fixed interaction energy / B?

Varying Zeeman splitting / Btot

Experimental evidence for spin skyrmions in GaAs 71Ga

NMR: Knight Shift

Tilted-field Activation Gap

[Barrett et. al., PRL 74, 5112 (1994)]

[Schmeller et. al., PRL 75, 4290 (1995)]

Ks / hS z i

rapid drop in polarization ~ skyrmion proliferation

Fixed interaction energy / B?

Varying Zeeman splitting / Btot

can work for valleys, if ‘valley Zeeman’ can be applied Monday, December 3,

Quantum Hall Ferromagnets topological order quantized xy

spontaneously broken symmetry elec. charged topo. defects

Symmetries determine low-energy NLσM + defects

Monday, December 3,

Quantum Hall Ferromagnets topological order quantized xy

spontaneously broken symmetry elec. charged topo. defects

Symmetries determine low-energy NLσM + defects SU(2) (Heisenberg)

skyrmions

e.g. spin [Sondhi et. al., PRB 47, 16419 (1993)]

Monday, December 3,

Quantum Hall Ferromagnets topological order quantized xy

spontaneously broken symmetry elec. charged topo. defects

Symmetries determine low-energy NLσM + defects SU(2) (Heisenberg)

skyrmions

e.g. spin [Sondhi et. al., PRB 47, 16419 (1993)]

easy-plane (XY)

merons

e.g. layer [Moon et. al., PRB 51, 5138 (1995)]

Monday, December 3,

Quantum Hall Ferromagnets topological order quantized xy

spontaneously broken symmetry elec. charged topo. defects

Symmetries determine low-energy NLσM + defects SU(2) (Heisenberg)

skyrmions

e.g. spin [Sondhi et. al., PRB 47, 16419 (1993)]

easy-plane (XY)

merons

e.g. layer [Moon et. al., PRB 51, 5138 (1995)]

easy-axis (Ising)

domain walls e.g subbands/LL crossing [Muraki et. al., PRL 87, 196801 (2001)] [Toyama et. al., PRL 101, 016805 (2008)]

Monday, December 3,

Quantum Hall Ferromagnets topological order quantized xy

spontaneously broken symmetry elec. charged topo. defects

Symmetries determine low-energy NLσM + defects SU(2) (Heisenberg)

skyrmions

e.g. spin [Sondhi et. al., PRB 47, 16419 (1993)]

easy-plane (XY)

merons

e.g. layer [Moon et. al., PRB 51, 5138 (1995)]

easy-axis (Ising)

domain walls e.g subbands/LL crossing [Muraki et. al., PRL 87, 196801 (2001)] [Toyama et. al., PRL 101, 016805 (2008)]

What about valley degrees of freedom? Monday, December 3,

Valley degeneracy

Monday, December 3,

crystalline point-group symmetry

Valley degeneracy

crystalline point-group symmetry

In general, spontaneously breaking valley symmetry must also break the point group symmetry (except in special cases...)

Monday, December 3,

Valley degeneracy

crystalline point-group symmetry

In general, spontaneously breaking valley symmetry must also break the point group symmetry (except in special cases...)

nematic order (usually Ising)

Monday, December 3,

Valley degeneracy

crystalline point-group symmetry

In general, spontaneously breaking valley symmetry must also break the point group symmetry (except in special cases...)

nematic order (usually Ising)

Characteristic features:

Monday, December 3,

transport anisotropies, disorder effects

Valley QHFM: Experimental Motivation

• • •

AlAs: indirect-gap multivalley semiconductor Shayegan group: transport in AlAs quantum wells Uniaxial strain≈valley Zeeman

[Shayegan et. al., phys. stat. sol. (b), 243, 3629 (2006)] Monday, December 3,

Monday, December 3,

•

Monday, December 3,

Clear ν=1 plateau

Monday, December 3,

•

Clear ν=1 plateau

•

Activated transport is ‘skyrmion-like’

spins in GaAs

[Schmeller et. al., PRL 75, 4290 (1995)] Monday, December 3,

•

Clear ν=1 plateau

•

Activated transport is ‘skyrmion-like’

•

Clear ν=1 plateau week ending 5 AUGUST 2005

PHYSICAL REVIEW LETTERS

•

AlAs

ε[100]

ε[100]

Activated transport is ‘skyrmion-like’ E[

E [100]

hωc

E[

E [100]

(a)

piezo

(b)

spins in GaAs ε[010]

-1

1.5

010]

∆Ev ∆EZ

010]

valleys in AlAs ∆Ev (meV) 0

1

2

3

[e2/4πεlB]

1.0

0.5

0 (d)

1∆

1∆

[e2/4πεlB]

0.15

0

0.02 ∆EZ

[e2/4πεl

0.04 B]

hωc

measured 2

0.10

∆EZ 0.05

single qp 1 (theory)

0

0 0.15

(c)

-0.05

0

0.05

0.10

(meV)

ε[010]

1∆

PRL 95, 066809 (2005)

∆Ev [e2/4πεlB]

[Schmeller et. al., PRL 75, 4290 (1995)] [Shkolnikov et. al., PRL 95, 066809 (2005)] FIG. 1 (color online). (a) In-plane strain breaks the valley energy degeneracy; e.g., tension along [100] results in a transfer of

Monday, December 3,

But let’s look a bit more closely at AlAs...

Monday, December 3,

But let’s look a bit more closely at AlAs... [100], [010] axes selected by growth direction

Monday, December 3,

But let’s look a bit more closely at AlAs... [100], [010] axes selected by growth direction very anisotropic dispersion

Monday, December 3,

But let’s look a bit more closely at AlAs... [100], [010] axes selected by growth direction very anisotropic dispersion

symmetry: picking a valley breaks C4 down to C2

Monday, December 3,

But let’s look a bit more closely at AlAs... [100], [010] axes selected by growth direction very anisotropic dispersion

symmetry: picking a valley breaks C4 down to C2 Incompatible with SU(2)-symmetric theory - no skyrmions

Monday, December 3,

But let’s look a bit more closely at AlAs... [100], [010] axes selected by growth direction very anisotropic dispersion

symmetry: picking a valley breaks C4 down to C2 Incompatible with SU(2)-symmetric theory - no skyrmions

Need an alternative scenario. Monday, December 3,

A Simple Model ky

B

mx 1 = 2 my

mx = my

• Two valleys at X-points of 2D BZ • Effective mass tensor rotated in two valleys

2

kx

• Symmetry: 2

λ ~5 for AlAs

x

y

[Abanin, SAP, Kivelson, Sondhi, PRB 82, 035428 (2010)]

with modifications, also other systems w/ anisotropic valleys : graphene/bilayer + trigonal warping (2 valleys) 3D Bi? (3 valleys) Si (111)? (4/6 valleys) Monday, December 3,

ν=1: The QH Ising Nematic •

LLL eigenstates in the two valleys have different shapes

Monday, December 3,

ν=1: The QH Ising Nematic •

LLL eigenstates in the two valleys have different shapes

•

Isotropic interaction projected to LLL acquires anisotropy

Monday, December 3,

ν=1: The QH Ising Nematic •

LLL eigenstates in the two valleys have different shapes

•

Isotropic interaction projected to LLL acquires anisotropy

V =V ≠V

Monday, December 3,

(reflects underlying C4)

ν=1: The QH Ising Nematic •

LLL eigenstates in the two valleys have different shapes

•

Isotropic interaction projected to LLL acquires anisotropy

V =V ≠V

•

Hartree-Fock: q=0 valley order

Monday, December 3,

(reflects underlying C4)

|✓, 'i = cos ✓

+ sin ✓ei'

ν=1: The QH Ising Nematic •

LLL eigenstates in the two valleys have different shapes

•

Isotropic interaction projected to LLL acquires anisotropy

V =V ≠V

(reflects underlying C4)

•

Hartree-Fock: q=0 valley order

•

Energy minimized for ✓ = 0, ⇡ ; breaks C4 to C2

Monday, December 3,

|✓, 'i = cos ✓

+ sin ✓ei'

ν=1: The QH Ising Nematic •

LLL eigenstates in the two valleys have different shapes

•

Isotropic interaction projected to LLL acquires anisotropy

V =V ≠V

(reflects underlying C4)

•

Hartree-Fock: q=0 valley order

•

Energy minimized for ✓ = 0, ⇡ ; breaks C4 to C2

|✓, 'i = cos ✓

+ sin ✓ei'

y x Monday, December 3,

ν=1: The QH Ising Nematic •

LLL eigenstates in the two valleys have different shapes

•

Isotropic interaction projected to LLL acquires anisotropy

V =V ≠V

(reflects underlying C4)

•

Hartree-Fock: q=0 valley order

•

Energy minimized for ✓ = 0, ⇡ ; breaks C4 to C2 Ising-nematic ordering.

|✓, 'i = cos ✓

+ sin ✓ei'

y x

Monday, December 3,

Ising-nematic order occurs via finite-temperature transition Z Z ⇢s ↵ 2 2 2 2 E[n(r)] = d r(rn(r)) + d rnz 2 2

Monday, December 3,

Ising-nematic order occurs via finite-temperature transition Z Z ⇢s ↵ 2 2 2 2 E[n(r)] = d r(rn(r)) + d rnz 2 2 Rough estimate: B = 10 T

Monday, December 3,

2

⇡5

" ⇡ 10

Ising-nematic order occurs via finite-temperature transition Z Z ⇢s ↵ 2 2 2 2 E[n(r)] = d r(rn(r)) + d rnz 2 2 Rough estimate: B = 10 T ⇢s ⇠ 5 K

Monday, December 3,

2

⇡5

↵ ⇠ 3K

" ⇡ 10

Ising-nematic order occurs via finite-temperature transition Z Z ⇢s ↵ 2 2 2 2 E[n(r)] = d r(rn(r)) + d rnz 2 2 Rough estimate: B = 10 T ⇢s ⇠ 5 K

2

⇡5

" ⇡ 10

↵ ⇠ 3K

Relatively large anisotropy; estimate Tc ⇠ ⇢s ⇠ 5 K (experiments at 10-100 mK)

Monday, December 3,

Ising-nematic order occurs via finite-temperature transition Z Z ⇢s ↵ 2 2 2 2 E[n(r)] = d r(rn(r)) + d rnz 2 2 Rough estimate: B = 10 T ⇢s ⇠ 5 K

2

⇡5

" ⇡ 10

↵ ⇠ 3K

Relatively large anisotropy; estimate Tc ⇠ ⇢s ⇠ 5 K (experiments at 10-100 mK)

Nematic order parameter couples strongly to disorder Monday, December 3,

Adding Disorder

•

Two sources: random potential + random strain

Monday, December 3,

Adding Disorder

•

Two sources: random potential + random strain

•

Local strains prefer one of the valleys

Monday, December 3,

Adding Disorder

•

Two sources: random potential + random strain

•

Local strains prefer one of the valleys

•

Couples as a random Zeeman field @u(r) hst (r) / @x

Monday, December 3,

@u(r) @y

Est

1 = 2

Z

d2 r hst (r)nz (r)

u ~ strain field

Adding Disorder

•

Two sources: random potential + random strain

•

Local strains prefer one of the valleys

•

Couples as a random Zeeman field @u(r) hst (r) / @x

•

@u(r) @y

What about random potential?

Monday, December 3,

Est

1 = 2

Z

d2 r hst (r)nz (r)

u ~ strain field

From Random Potentials to Random Fields

•

Smooth disorder potential from donor impurities

y

donor plane

2DEG

Monday, December 3,

⇠d

x

⇠d

From Random Potentials to Random Fields

•

Smooth disorder potential from donor impurities

y

donor plane

2DEG

•

⇠d

⇠d

x

Isotropic distribution, but local anisotropies are valley-selective

Monday, December 3,

From Random Potentials to Random Fields

•

Smooth disorder potential from donor impurities

y

donor plane

2DEG

•

⇠d

⇠d

x

Isotropic distribution, but local anisotropies are valley-selective

potential also couples as random field

Monday, December 3,

From Random Potentials to Random Fields

•

Smooth disorder potential from donor impurities

y

donor plane

2DEG

•

⇠d

⇠d

x

Isotropic distribution, but local anisotropies are valley-selective

potential also couples as random field "✓ ◆2 2 (mx my )`B @Vpot hpot (r) = 2⇡~2 @x Monday, December 3,

✓

@Vpot @y

◆2 #

curvature anisotropy

Ising Model in a Random Field Effect of h on ordered phase at T=0 hr = 0

hr hr0 = cr,r0

[Y. Imry and S. Ma, PRL 35, 1399 (1975)] Monday, December 3,

Ising Model in a Random Field Effect of h on ordered phase at T=0 hr = 0

hr hr0 = cr,r0 droplet of linear dimension L

L

he↵ ⇠ Ld/2

[Y. Imry and S. Ma, PRL 35, 1399 (1975)] Monday, December 3,

Ising Model in a Random Field Effect of h on ordered phase at T=0 hr = 0

hr hr0 = cr,r0 droplet of linear dimension L

L

he↵ ⇠ Ld/2 energy gain:

Eh ⇠

c1 Ld/2

[Y. Imry and S. Ma, PRL 35, 1399 (1975)] Monday, December 3,

Ising Model in a Random Field Effect of h on ordered phase at T=0 hr = 0

hr hr0 = cr,r0 droplet of linear dimension L

L

he↵ ⇠ Ld/2 energy gain:

Eh ⇠

c1 Ld/2

d E ⇠ c L domain-wall cost: W 2

1

[Y. Imry and S. Ma, PRL 35, 1399 (1975)] Monday, December 3,

Ising Model in a Random Field Effect of h on ordered phase at T=0 hr = 0

hr hr0 = cr,r0 droplet of linear dimension L

L

he↵ ⇠ Ld/2 energy gain:

Eh ⇠

c1 Ld/2

d E ⇠ c L domain-wall cost: W 2

1

ordered phase unstable to weak random field if Eh + EW < 0

[Y. Imry and S. Ma, PRL 35, 1399 (1975)] Monday, December 3,

Ising Model in a Random Field Effect of h on ordered phase at T=0 hr = 0

hr hr0 = cr,r0 droplet of linear dimension L

L

he↵ ⇠ Ld/2 energy gain:

Eh ⇠

c1 Ld/2

d E ⇠ c L domain-wall cost: W 2

1

ordered phase unstable to weak random field if Eh + EW < 0 always the case for d < dc = 2 [Y. Imry and S. Ma, PRL 35, 1399 (1975)] Monday, December 3,

“Clean” vs. “Dirty”

•

d=2 RFIM breaks up into domains, size set by variance of h and DW energy per unit length DW surface tension disorder strength

•

Compare domain size ξIM to sample size Ls , get two regimes:

Monday, December 3,

“Clean” vs. “Dirty”

•

d=2 RFIM breaks up into domains, size set by variance of h and DW energy per unit length DW surface tension disorder strength

•

Compare domain size ξIM to sample size Ls , get two regimes: ξIM

•

“Dirty” (ξIM ≪ Ls, hr “strong” ) transport via domain walls Ls

Monday, December 3,

“Clean” vs. “Dirty”

•

d=2 RFIM breaks up into domains, size set by variance of h and DW energy per unit length DW surface tension disorder strength

•

Compare domain size ξIM to sample size Ls , get two regimes: ξIM

•

“Dirty” (ξIM ≪ Ls, hr “strong” ) transport via domain walls Ls

•

“Clean” (ξIM ≫ Ls, hr “weak” ) transport by bulk quasiparticles ξIM ~ Ls

Monday, December 3,

“Clean” vs. “Dirty”

•

d=2 RFIM breaks up into domains, size set by variance of h and DW energy per unit length DW surface tension disorder strength

•

Compare domain size ξIM to sample size Ls , get two regimes: ξIM

•

“Dirty” (ξIM ≪ Ls, hr “strong” ) transport via domain walls Ls

•

“Clean” (ξIM ≫ Ls, hr “weak” ) transport by bulk quasiparticles ξIM ~ Ls

Shayegan’s samples: Monday, December 3,

;

dirty limit

Domain Diagnostics Need experimental probe of single vs. multiple domains - in single domain, transport is by bulk qp hopping - look at finite-T longitudinal conductivity near ν=1: xy

2e2 /h e2 /h y x

Monday, December 3,

=1

n

Domain Diagnostics Need experimental probe of single vs. multiple domains - in single domain, transport is by bulk qp hopping - look at finite-T longitudinal conductivity near ν=1: xx

>

xy

yy

2e2 /h e2 /h y x

Monday, December 3,

=1

n

Domain Diagnostics Need experimental probe of single vs. multiple domains - in single domain, transport is by bulk qp hopping - look at finite-T longitudinal conductivity near ν=1: xy

2e2 /h e2 /h y x

Monday, December 3,

=1

n

Domain Diagnostics Need experimental probe of single vs. multiple domains - in single domain, transport is by bulk qp hopping - look at finite-T longitudinal conductivity near ν=1: xx

<

xy

yy

2e2 /h e2 /h y x

Monday, December 3,

=1

n

Domain Diagnostics Need experimental probe of single vs. multiple domains - in single domain, transport is by bulk qp hopping - look at finite-T longitudinal conductivity near ν=1: xx

<

xy

yy

2e2 /h e2 /h y

=1

x

xx xx Monday, December 3,

yy

+

yy

changes sign across ⌫ = 1

n

Domain Walls: Weak Anisotropy residual U(1) broken by DW (restored by fluctuations)

[Fal’ko and Iordanskii, PRL 82, 402 (1999)] Monday, December 3,

Domain Walls: Weak Anisotropy residual U(1) broken by DW (restored by fluctuations)

T =0:

[Fal’ko and Iordanskii, PRL 82, 402 (1999)] Monday, December 3,

Domain Walls: Weak Anisotropy residual U(1) broken by DW (restored by fluctuations)

T =0: gapless ‘almost-Goldstone mode’

[Fal’ko and Iordanskii, PRL 82, 402 (1999)] Monday, December 3,

Domain Walls: Weak Anisotropy residual U(1) broken by DW (restored by fluctuations)

T =0: gapless ‘almost-Goldstone mode’ power-law correlations along DW

[Fal’ko and Iordanskii, PRL 82, 402 (1999)] Monday, December 3,

Domain Walls: Weak Anisotropy residual U(1) broken by DW (restored by fluctuations)

T =0: gapless ‘almost-Goldstone mode’ power-law correlations along DW 2π kink excitations

[Fal’ko and Iordanskii, PRL 82, 402 (1999)] Monday, December 3,

Domain Walls: Weak Anisotropy residual U(1) broken by DW (restored by fluctuations)

T =0: gapless ‘almost-Goldstone mode’ power-law correlations along DW 2π kink excitations Weak intervalley scattering gaps out DW, exponential correlations characteristic scale ξiv [Fal’ko and Iordanskii, PRL 82, 402 (1999)] Monday, December 3,

Domain Walls: Strong Anisotropy DW ≃ 2 copies of QH edge counterpropagating modes nonchiral Luttinger liquid

[Fal’ko and Iordanskii, PRL 82, 402 (1999)] [Mitra & Girvin Phys. Rev. B., 67, 245311 (2003)] Monday, December 3,

Domain Walls: Strong Anisotropy DW ≃ 2 copies of QH edge counterpropagating modes nonchiral Luttinger liquid

Intervalley scattering

backscattering between chiral modes

[Fal’ko and Iordanskii, PRL 82, 402 (1999)] [Mitra & Girvin Phys. Rev. B., 67, 245311 (2003)] Monday, December 3,

Domain Walls: Strong Anisotropy DW ≃ 2 copies of QH edge counterpropagating modes nonchiral Luttinger liquid

Intervalley scattering

backscattering between chiral modes

Localizes the 1D DW, σ→0

[Fal’ko and Iordanskii, PRL 82, 402 (1999)] [Mitra & Girvin Phys. Rev. B., 67, 245311 (2003)] Monday, December 3,

Domain Walls: Strong Anisotropy DW ≃ 2 copies of QH edge counterpropagating modes nonchiral Luttinger liquid

Intervalley scattering

backscattering between chiral modes

Localizes the 1D DW, σ→0 (Luttinger liquid picture smoothly connects to weak-anisotropy limit) [Fal’ko and Iordanskii, PRL 82, 402 (1999)] [Mitra & Girvin Phys. Rev. B., 67, 245311 (2003)] Monday, December 3,

Domain Walls: Microscopics Can also examine DW microscopically (Hartree-Fock) |DW i =

Y⇣ X

uX c†X, + vX c†X,

⌘

|0i

n

z

h

pin wall by h gradient; determine texturing, surface tension

[Kumar, SAP, Sondhi, Kivelson, work in progress] Monday, December 3,

Domain Walls: Microscopics Can also examine DW microscopically (Hartree-Fock) |DW i =

Y⇣

uX c†X, + vX c†X,

X

⌘

|0i

n

z

h

Lx = 10⇡`B Ly = 20`B

hS x i/Ly

in-plane pseudospin per unit length

pin wall by h gradient; determine texturing, surface tension

field gradient Monday, December 3,

for typical parameters wall is textured

[Kumar, SAP, Sondhi, Kivelson, work in progress]

Domain Walls: Dipole Moment Due to nematic order parameter, DWs have dipole moments Microsopic origin: anisotropy of wavefunctions n

z

background charge

Monday, December 3,

Domain Walls: Dipole Moment Due to nematic order parameter, DWs have dipole moments Microsopic origin: anisotropy of wavefunctions n

z

background charge

Monday, December 3,

Domain Walls: Dipole Moment Due to nematic order parameter, DWs have dipole moments Microsopic origin: anisotropy of wavefunctions n

z

background charge

Monday, December 3,

Domain Walls: Dipole Moment Due to nematic order parameter, DWs have dipole moments Microsopic origin: anisotropy of wavefunctions n

z

background charge

Monday, December 3,

Domain Walls: Dipole Moment Due to nematic order parameter, DWs have dipole moments Microsopic origin: anisotropy of wavefunctions n

z

background charge

Monday, December 3,

dipole moment

Domain Walls: Dipole Moment Due to nematic order parameter, DWs have dipole moments Microsopic origin: anisotropy of wavefunctions n

z

dipole moment

background charge

Dipole-dipole interactions affect DW geometry (complicated) [eg in ferrofluids: Oliveira et. al., PRE 77, 016304 (2008)] [Kumar, SAP, Sondhi, Kivelson, work in progress] Monday, December 3,

QHE in the Dirty Limit

•

For samples with Ls ≫ ξIM, domain wall network percolates at Δv=0

Monday, December 3,

QHE in the Dirty Limit

•

For samples with Ls ≫ ξIM, domain wall network percolates at Δv=0

•

Low-energy charged excitations at DW dominate transport

Monday, December 3,

QHE in the Dirty Limit

•

For samples with Ls ≫ ξIM, domain wall network percolates at Δv=0

•

Low-energy charged excitations at DW dominate transport

•

No intervalley scattering: 2 copies of Chalker-Coddington model

Monday, December 3,

QHE in the Dirty Limit

•

For samples with Ls ≫ ξIM, domain wall network percolates at Δv=0

•

Low-energy charged excitations at DW dominate transport

•

No intervalley scattering: 2 copies of Chalker-Coddington model metallic behavior

Monday, December 3,

xx

⇠ e2 /h

QHE in the Dirty Limit

•

For samples with Ls ≫ ξIM, domain wall network percolates at Δv=0

•

Low-energy charged excitations at DW dominate transport

•

No intervalley scattering: 2 copies of Chalker-Coddington model metallic behavior

•

xx

⇠ e2 /h

Intervalley scattering: DW transport is suppressed, QHE restored

Monday, December 3,

QHE in the Dirty Limit

•

For samples with Ls ≫ ξIM, domain wall network percolates at Δv=0

•

Low-energy charged excitations at DW dominate transport

•

No intervalley scattering: 2 copies of Chalker-Coddington model metallic behavior

•

xx

⇠ e2 /h

Intervalley scattering: DW transport is suppressed, QHE restored

quantum Hall random-field paramagnet

Monday, December 3,

QHE in the Dirty Limit

•

For samples with Ls ≫ ξIM, domain wall network percolates at Δv=0

•

Low-energy charged excitations at DW dominate transport

•

No intervalley scattering: 2 copies of Chalker-Coddington model metallic behavior

•

xx

⇠ e2 /h

Intervalley scattering: DW transport is suppressed, QHE restored

quantum Hall random-field paramagnet [somewhat analogous to ‘QH spin glass’ S. Rapsch et. al. PRL, 88, 036801 (2002)]

Monday, December 3,

Transport in Multi-Domain Samples

•

Absence of valley Zeeman: percolating DW network, transport minigap due to intervalley scattering Δv = 0

B

Monday, December 3,

Transport in Multi-Domain Samples

•

Absence of valley Zeeman: percolating DW network, transport minigap due to intervalley scattering Δv = 0

B

Monday, December 3,

Transport in Multi-Domain Samples

•

Absence of valley Zeeman: percolating DW network, transport minigap due to intervalley scattering Δv > 0

B

Monday, December 3,

Transport in Multi-Domain Samples

•

Absence of valley Zeeman: percolating DW network, transport minigap due to intervalley scattering Δv > 0

B

Monday, December 3,

Transport in Multi-Domain Samples

•

Absence of valley Zeeman: percolating DW network, transport minigap due to intervalley scattering Δv > 0

B

•

Monday, December 3,

With valley Zeeman: rapid shift away from percolation; transport dominated by DW-DW tunneling; σxx rises sharply (as does measured Δact.trans. )

Monday, December 3,

(b) -1

∆Ev (meV) 0 1

2 3

measured 2

0.10

∆EZ

single qp (theory)

0.05 0 (c)

-0.05

0

0.05

0.10

1

(meV)

hωc

1∆

1∆

[e2/4πεlB]

0.15

0 0.15

∆Ev [e2/4πεlB]

energy degeneracy; e.g., tension along [100] results in a transfer of le-particle fan diagram et. for al., the PRL AlAs 95, 2DES using band parameters. [Shkolnikov 066809 (2005)] 1 QHS gap, !, as a function of the single-particle valley splitting, !Ev , (top and right axes) and also in units of the Coulomb energy e2 =4$"lB , @=eB&1=2 is the magnetic length; in our experiments, B ! 10:5 T and a single-valley 2DES. The dashed line is the single-particle Zeeman n energy is shown by the solid curve and the solid symbols show the

h d – a

trolled through !EZ ! gb !B B (band g factor gb ! 2), while the levels corresponding to different valleys are separated by !Ev . When "xy ! 0, in the single-particle picture of Fig. 1(b), the energy degeneracy of the Landau levels associated with different valleys implies that there is

Monday, December 3,

(b) -1

∆Ev (meV) 0 1

2 3

measured 2

0.10

∆EZ

single qp (theory)

0.05 0 (c)

-0.05

0

0.05

0.10

1

(meV)

hωc

1∆

1∆

[e2/4πεlB]

0.15

0 0.15

∆Ev [e2/4πεlB]

energy degeneracy; e.g., tension along [100] results in a transfer of le-particle fan diagram et. for al., the PRL AlAs 95, 2DES using band parameters. [Shkolnikov 066809 (2005)] 1 QHS gap, !, as a function of the single-particle valley splitting, !Ev , (top and right axes) and also in units of the Coulomb energy e2 =4$"lB , @=eB&1=2 is the magnetic length; in our experiments, B ! 10:5 T and a single-valley 2DES. The dashed line is the single-particle Zeeman n energy is shown by the solid curve and the solid symbols show the

h d – a

trolled through !EZ ! gb !B B (band g factor gb ! 2), while the levels corresponding to different valleys are separated by !Ev . When "xy ! 0, in the single-particle picture of Fig. 1(b), the energy degeneracy of the Landau levels associated with different valleys implies that there is

Monday, December 3,

Our picture: domain wall transport, not skyrmions

(b) -1

∆Ev (meV) 0 1

2 3

measured 2

0.10

∆EZ

single qp (theory)

0.05 0 (c)

-0.05

0

0.05

0.10

1

(meV)

hωc

1∆

1∆

[e2/4πεlB]

0.15

0 0.15

∆Ev [e2/4πεlB]

energy degeneracy; e.g., tension along [100] results in a transfer of le-particle fan diagram et. for al., the PRL AlAs 95, 2DES using band parameters. [Shkolnikov 066809 (2005)] 1 QHS gap, !, as a function of the single-particle valley splitting, !Ev , (top and right axes) and also in units of the Coulomb energy e2 =4$"lB , @=eB&1=2 is the magnetic length; in our experiments, B ! 10:5 T and a single-valley 2DES. The dashed line is the single-particle Zeeman n energy is shown by the solid curve and the solid symbols show the

h d – a

trolled through !EZ ! gb !B B (band g factor gb ! 2), while the levels corresponding to different valleys are separated by !Ev . When "xy ! 0, in the single-particle picture of Fig. 1(b), the energy degeneracy of the Landau levels associated with different valleys implies that there is

Monday, December 3,

Our picture: domain wall transport, not skyrmions

More work (expt./theory) needed to decide which one

Valley skyrmions in Bilayer Graphene

Monday, December 3,

Valley skyrmions in Bilayer Graphene Quadratic touching + 2π Berry’s phase

Monday, December 3,

Valley skyrmions in Bilayer Graphene Quadratic touching + 2π Berry’s phase Noninteracting problem : zero-energy LLL ‘octet’

Monday, December 3,

Valley skyrmions in Bilayer Graphene Quadratic touching + 2π Berry’s phase Noninteracting problem : zero-energy LLL ‘octet’ 8=

Monday, December 3,

Valley skyrmions in Bilayer Graphene Quadratic touching + 2π Berry’s phase Noninteracting problem : zero-energy LLL ‘octet’ 8 = (2 spins) ↑,↓

Monday, December 3,

Valley skyrmions in Bilayer Graphene Quadratic touching + 2π Berry’s phase Noninteracting problem : zero-energy LLL ‘octet’ 8 = (2 spins) x (2 valleys) ↑,↓

Monday, December 3,

K,K’

Valley skyrmions in Bilayer Graphene Quadratic touching + 2π Berry’s phase Noninteracting problem : zero-energy LLL ‘octet’ 8 = (2 spins) x (2 valleys) x (2 ‘orbitals’) ↑,↓

Monday, December 3,

K,K’

0,1

Valley skyrmions in Bilayer Graphene Quadratic touching + 2π Berry’s phase Noninteracting problem : zero-energy LLL ‘octet’ 8 = (2 spins) x (2 valleys) x (2 ‘orbitals’) ↑,↓

K,K’

0,1

‘orbital’ degeneracy is between n=0, 1 magnetic oscillator eigenstates n=0

Monday, December 3,

n=1

Valley skyrmions in Bilayer Graphene Quadratic touching + 2π Berry’s phase Noninteracting problem : zero-energy LLL ‘octet’ 8 = (2 spins) x (2 valleys) x (2 ‘orbitals’) ↑,↓

K,K’

0,1

‘orbital’ degeneracy is between n=0, 1 magnetic oscillator eigenstates n=0

within restricted subspace, valley≡layer

Monday, December 3,

n=1

⟂ E field acts as valley Zeeman

Valley skyrmions in Bilayer Graphene Quadratic touching + 2π Berry’s phase Noninteracting problem : zero-energy LLL ‘octet’ 8 = (2 spins) x (2 valleys) x (2 ‘orbitals’) ↑,↓

K,K’

0,1

‘orbital’ degeneracy is between n=0, 1 magnetic oscillator eigenstates n=0

within restricted subspace, valley≡layer valleys are isotropic (ignoring warping)

Monday, December 3,

n=1

⟂ E field acts as valley Zeeman SU(2) symmetry

Interactions lift degeneracy: sequence of splitting from inter-orbital exchange + Zeeman

[Abanin, SAP, Sondhi, PRL103, 076802 (2009)] Monday, December 3,

Interactions lift degeneracy: sequence of splitting from inter-orbital exchange + Zeeman 1K’↓ 0K’↓ 1K↓ 0K↓ 1K’↑ 0K’↑ 1K↑ 0K↑

ν=+4 ν=+3 ν=+2 ν=+1 ν=0 ν=-1 ν=-2 ν=-3 ν=-4

[Splittings partially confirmed by expts. B. Feldman et. al. Nat. Phys. 5, 889 (2009); Y. Zhao et. al. PRL 104, 066801 (2010)]

[Abanin, SAP, Sondhi, PRL103, 076802 (2009)] Monday, December 3,

Effective Action for a Spin Texture

Interactions lift degeneracy: sequence of splitting from ! +K Zeeman action for a spin texture inter-orbital in the valleyexchange indices (K, ). Note that throughout thi

LLs: ν=-2 ley-polarizedTwo N =filled 2 state. Thisstate is the state ! † |ψ0 ! = c0,K (X)c†1,K (X)|0! X

1K’↓ 0K’↓ 1K↓

ν=+4 ν=+3

ν=+2(51

ν=+1 0K↓ the following points: (i tate being both polarized and fully occupied, we have ν=0 µ Sˆab (q)|ψ0 ! = 0 for µ "= z (we define the z axis to be the axis of quantization of the 1K’↑ ν=-1 ψ0 ! ∝ δq,0 for a "= b. 0K’↑ by rotating away from the z-axis. An arbitrary such rotation is generated by the ν=-2 1K↑ ν=-3 # µ µ 0K↑ ˆ Ωab (q)Sab (−q) (52 O= ν=-4 q,a,b

ng a spin-texture is given by " " iO† −iO |ψ0 ! − ψ0 |H|ψ0 ! E = ψ0 |e He

[Splittings partially confirmed by expts. B. Feldman et. al. Nat. Phys. 5, 889 (2009); Y. Zhao et. al. PRL 104, 066801 (2010)]

(53

plies that the Hermitian conjugate of O is # µ∗ # µ∗ µ µ [Abanin, SAP, Sondhi, PRL103, 076802 (2009)] Ωba (q)Sˆab (q) = Ωba (−q)Sˆab (−q) (54 O† = Monday, December 3,

Effective Action for a Spin Texture

Interactions lift degeneracy: sequence of splitting from ! +K Zeeman action for a spin texture inter-orbital in the valleyexchange indices (K, ). Note that throughout thi

LLs: ν=-2 ley-polarizedTwo N =filled 2 state. Thisstate is the state ! † |ψ0 ! = c0,K (X)c†1,K (X)|0! X

1K’↓ 0K’↓ 1K↓

ν=+4 ν=+3

ν=+2(51

ν=+1 0K↓ the following points: (i tate being both polarized and fully occupied, we have ν=0 µ valley symmetry + z axis to be the Sˆab (q)|ψ0 ! = 0SU(2) for µ "= z (we define the axis of quantization of the 1K’↑ ν=-1 ψ0 ! ∝ δq,0 inter-orbital for a "= b. exchange stabilizes 0K’↑ charge-2e skyrmion by rotating away from the z-axis. An arbitrary such rotation is generated by the ν=-2 1K↑ ν=-3 # µ µ 0K↑ ˆ Ωab (q)Sab (−q) (52 O= ν=-4 q,a,b

ng a spin-texture is given by " " iO† −iO |ψ0 ! − ψ0 |H|ψ0 ! E = ψ0 |e He

[Splittings partially confirmed by expts. B. Feldman et. al. Nat. Phys. 5, 889 (2009); Y. Zhao et. al. PRL 104, 066801 (2010)]

(53

plies that the Hermitian conjugate of O is # µ∗ # µ∗ µ µ [Abanin, SAP, Sondhi, PRL103, 076802 (2009)] Ωba (q)Sˆab (q) = Ωba (−q)Sˆab (−q) (54 O† = Monday, December 3,

Valley Texture Crystals in Bilayer Graphene e/2-meron crystal

2e skyrmion crystal ν = 2.2

skyrme crystal slightly higher in energy; but picture of coupled excitations in two orbitals verified

(cf also Dima Kovrizhin’s talk earlier) Monday, December 3,

[ R. Côté et.al. PRB 82, 245307 (2010)]

Valley Texture Crystals in Bilayer Graphene e/2-meron crystal

2e skyrmion crystal ν = 2.2

skyrme crystal slightly higher in energy; but picture of coupled excitations in two orbitals verified

valley-layer equivalence: imaging by STM? (cf also Dima Kovrizhin’s talk earlier) Monday, December 3,

[ R. Côté et.al. PRB 82, 245307 (2010)]

A d=3 example?

Resistance anisotropy with ‘onset’ at low T/large B

Monday, December 3,

Conclusions •

Multivalley QH Ferromagnets can entangle internal and rotational degrees of freedom

• •

Resulting QH state generically has nematic order

• • • •

Disorder leads to domain formation

• •

Single domains: transport anisotropies Multiple domains: domain-wall transport, sensitive to valley field

Longitudinal transport near ν=1 probes valley order/domain structure DW transport can explain ‘skyrmion-like’ gap scaling Bilayer graphene: SU(2)-symmetric case, charge-2e skyrmions at ν=±2 Valley ordering in d=3? Intriguing magnetotransport in 3D Bi...

Monday, December 3,