Andrew Lucas - qpt.physics.harvard.eduqpt.physics.harvard.edu/physicsnext/Andrew_Lucas.pdf · Kinetic theory of transport for inhomogeneous electron uids Andrew Lucas Stanford Physics

Kinetic theory of transport for inhomogeneous electron fluids

Andrew Lucas

Stanford Physics

Physics Next: From Quantum Fields to Condensed Matter; Hyatt Place Long Island

August 25, 2017

Collaborators 2

Sean HartnollStanford Physics

I A. Lucas and S. A. Hartnoll. “Kinetic theory of transport forinhomogeneous electron fluids”, 1706.04621

Introduction to Transport 3

Transport in Metals

I Ohm’s law – the “simplest” experiment...

E = ρJ (V = IR)

I ...yet ρ hard to compute in interesting systems:

transportgases

fluids

QFT

chaosstat.mech. black holes

I textbooks on canonical Boltzmann/kinetic transport:

Ziman, Electrons and Phonons: Theory of Transport in Solids (1960)

57 years later, we have ‘completed’ this theory

Introduction to Transport 3

Transport in Metals

I Ohm’s law – the “simplest” experiment...

E = ρJ (V = IR)

I ...yet ρ hard to compute in interesting systems:

transportgases

fluids

QFT

chaosstat.mech. black holes

I textbooks on canonical Boltzmann/kinetic transport:

Ziman, Electrons and Phonons: Theory of Transport in Solids (1960)

57 years later, we have ‘completed’ this theory

Introduction to Transport 3

Transport in Metals

I Ohm’s law – the “simplest” experiment...

E = ρJ (V = IR)

I ...yet ρ hard to compute in interesting systems:

transportgases

fluids

QFT

chaosstat.mech. black holes

I textbooks on canonical Boltzmann/kinetic transport:

Ziman, Electrons and Phonons: Theory of Transport in Solids (1960)

57 years later, we have ‘completed’ this theory

Introduction to Transport 3

Transport in Metals

I Ohm’s law – the “simplest” experiment...

E = ρJ (V = IR)

I ...yet ρ hard to compute in interesting systems:

transportgases

fluids

QFT

chaosstat.mech. black holes

I textbooks on canonical Boltzmann/kinetic transport:

Ziman, Electrons and Phonons: Theory of Transport in Solids (1960)

57 years later, we have ‘completed’ this theory

Old Boltzmann Transport 4

The Boltzmann Equation

I assume quasiparticles:

〈ψ†(k, ω)ψ(k, ω)〉 ∼ 1

ω − ε(k) + icω2 + · · · .

I non-equilibrium distribution function:

f(x,p) ∼ “particles of momentum p at position x”

I weak interactions + ∆x∆p� ~ =⇒ kinetic theory:

∂tf + v · ∂xf + F · ∂pf︸ ︷︷ ︸free-particle streaming

= C[f ]︸︷︷︸collisions

.

Old Boltzmann Transport 4

The Boltzmann Equation

I assume quasiparticles:

〈ψ†(k, ω)ψ(k, ω)〉 ∼ 1

ω − ε(k) + icω2 + · · · .

I non-equilibrium distribution function:

f(x,p) ∼ “particles of momentum p at position x”

I weak interactions + ∆x∆p� ~ =⇒ kinetic theory:

∂tf + v · ∂xf + F · ∂pf︸ ︷︷ ︸free-particle streaming

= C[f ]︸︷︷︸collisions

.

Old Boltzmann Transport 4

The Boltzmann Equation

I assume quasiparticles:

〈ψ†(k, ω)ψ(k, ω)〉 ∼ 1

ω − ε(k) + icω2 + · · · .

I non-equilibrium distribution function:

f(x,p) ∼ “particles of momentum p at position x”

I weak interactions + ∆x∆p� ~ =⇒ kinetic theory:

∂tf + v · ∂xf + F · ∂pf︸ ︷︷ ︸free-particle streaming

= C[f ]︸︷︷︸collisions

.

Old Boltzmann Transport 5

Computing Transport

I assume:I inversion and time reversal symmetryI thermodynamic equilibrium feq is not unstable

I static linear response: f = feq + δf ,

v · ∂xδf + F · ∂pδf︸ ︷︷ ︸streaming terms

+ eE · v∂feq

∂ε︸ ︷︷ ︸source term

=δCδf

∣∣∣∣eq

· δf︸ ︷︷ ︸collision operator

I BIG linear algebra problem...schematically:

L|Φ〉+W|Φ〉 = Ei|Ji〉, δf = −∂feq

∂εΦ︸ ︷︷ ︸

Φ not singular

I choose inner product 〈Jx|Φ〉 = Javgx :

σxx = 〈Jx|(W + L)−1|Jx〉.

Old Boltzmann Transport 5

Computing Transport

I assume:I inversion and time reversal symmetryI thermodynamic equilibrium feq is not unstable

I static linear response: f = feq + δf ,

v · ∂xδf + F · ∂pδf︸ ︷︷ ︸streaming terms

+ eE · v∂feq

∂ε︸ ︷︷ ︸source term

=δCδf

∣∣∣∣eq

· δf︸ ︷︷ ︸collision operator

I BIG linear algebra problem...schematically:

L|Φ〉+W|Φ〉 = Ei|Ji〉, δf = −∂feq

∂εΦ︸ ︷︷ ︸

Φ not singular

I choose inner product 〈Jx|Φ〉 = Javgx :

σxx = 〈Jx|(W + L)−1|Jx〉.

Old Boltzmann Transport 5

Computing Transport

I assume:I inversion and time reversal symmetryI thermodynamic equilibrium feq is not unstable

I static linear response: f = feq + δf ,

v · ∂xδf + F · ∂pδf︸ ︷︷ ︸streaming terms

+ eE · v∂feq

∂ε︸ ︷︷ ︸source term

=δCδf

∣∣∣∣eq

· δf︸ ︷︷ ︸collision operator

I BIG linear algebra problem...schematically:

L|Φ〉+W|Φ〉 = Ei|Ji〉, δf = −∂feq

∂εΦ︸ ︷︷ ︸

Φ not singular

I choose inner product 〈Jx|Φ〉 = Javgx :

σxx = 〈Jx|(W + L)−1|Jx〉.

Old Boltzmann Transport 5

Computing Transport

I assume:I inversion and time reversal symmetryI thermodynamic equilibrium feq is not unstable

I static linear response: f = feq + δf ,

v · ∂xδf + F · ∂pδf︸ ︷︷ ︸streaming terms

+ eE · v∂feq

∂ε︸ ︷︷ ︸source term

=δCδf

∣∣∣∣eq

· δf︸ ︷︷ ︸collision operator

I BIG linear algebra problem...schematically:

L|Φ〉+W|Φ〉 = Ei|Ji〉, δf = −∂feq

∂εΦ︸ ︷︷ ︸

Φ not singular

I choose inner product 〈Jx|Φ〉 = Javgx :

σxx = 〈Jx|(W + L)−1|Jx〉.

Old Boltzmann Transport 6

Conservation Laws

I exact solution hard. common approximations:

W =1

τ.

relaxation time approximation

L→ 0.

homogeneous fluid

I however there are conservation laws:

C[feq

(ε(p)− µ− δµ

T

)]= 0

I conservation of charge: W|Φ = 1〉 = 0

I and conservation of momentum: if density ρ 6= 0:

W|Φ = px〉 = 0 =⇒ σxx = 〈Jx|W−1|Jx〉 =∞

Old Boltzmann Transport 6

Conservation Laws

I exact solution hard. common approximations:

W =1

τ.

relaxation time approximation

L→ 0.

homogeneous fluid

I however there are conservation laws:

C[feq

(ε(p)− µ− δµ

T

)]= 0

I conservation of charge: W|Φ = 1〉 = 0

I and conservation of momentum: if density ρ 6= 0:

W|Φ = px〉 = 0 =⇒ σxx = 〈Jx|W−1|Jx〉 =∞

Old Boltzmann Transport 6

Conservation Laws

I exact solution hard. common approximations:

W =1

τ.

relaxation time approximation

L→ 0.

homogeneous fluid

I however there are conservation laws:

C[feq

(ε(p)− µ− δµ

T

)]= 0

I conservation of charge: W|Φ = 1〉 = 0

I and conservation of momentum: if density ρ 6= 0:

W|Φ = px〉 = 0 =⇒ σxx = 〈Jx|W−1|Jx〉 =∞

Old Boltzmann Transport 6

Conservation Laws

I exact solution hard. common approximations:

W =1

τ.

relaxation time approximation

L→ 0.

homogeneous fluid

I however there are conservation laws:

C[feq

(ε(p)− µ− δµ

T

)]= 0

I conservation of charge: W|Φ = 1〉 = 0

I and conservation of momentum: if density ρ 6= 0:

W|Φ = px〉 = 0 =⇒ σxx = 〈Jx|W−1|Jx〉 =∞

Strange Metals in Experiment 7

Viscous Flows in Constrictions

I at finite T , more scattering? ∂∂TW > 0 =⇒

∂∂T ρ = ∂

∂T (〈Jx|W−1|Jx〉)−1 > 0?

I however, ∂∂T ρ < 0 in graphene constrictions!

[Kumar et al; 1703.06672]

Strange Metals in Experiment 7

Viscous Flows in Constrictions

I at finite T , more scattering? ∂∂TW > 0 =⇒

∂∂T ρ = ∂

∂T (〈Jx|W−1|Jx〉)−1 > 0?I however, ∂

∂T ρ < 0 in graphene constrictions!

[Kumar et al; 1703.06672]

Strange Metals in Experiment 8

Electron-Electron Interaction Limited Resistivity in Fermi Liquids

I in a Fermi liquid:

τee ∼~µ

(kBT )2, ρ = AT 2 ∼ 1

τee. . .

I A depends on thermodynamics (not disorder?):[Jacko, Fjaerestad, Powell; 0805.4275]

Strange Metals in Experiment 8

Electron-Electron Interaction Limited Resistivity in Fermi Liquids

I in a Fermi liquid:

τee ∼~µ

(kBT )2, ρ = AT 2 ∼ 1

τee. . .

I A depends on thermodynamics (not disorder?):[Jacko, Fjaerestad, Powell; 0805.4275]

Strange Metals in Experiment 9

Linear Resistivity: A Challenge

I in a theory without quasiparticles:

τee &~kBT

.

I “Drude” ρ =m

ne2

1

τee∼ m

ne2

kBT

~:

[Bruin, Sakai, Perry, Mackenzie; (2013)]

Strange Metals in Experiment 9

Linear Resistivity: A Challenge

I in a theory without quasiparticles:

τee &~kBT

.

I “Drude” ρ =m

ne2

1

τee∼ m

ne2

kBT

~:

[Bruin, Sakai, Perry, Mackenzie; (2013)]

New Boltzmann Transport 10

Classical Disorder

I charge puddles: H = Hclean +

∫ddx n(x)Vimp(x):

⇠

�F

µ(x) = µ0 � Vimp(x)

I for simplicity: neglect umklapp, phononsI W|P 〉 = 0 – ee collisions conserve momentumI must include streaming terms: L 6= 0I Vimp perturbatively small – efficient analytical/numerical

algorithm to exactly solve the transport problem:

ρ = V 2imp × · · ·

New Boltzmann Transport 10

Classical Disorder

I charge puddles: H = Hclean +

∫ddx n(x)Vimp(x):

⇠

�F

µ(x) = µ0 � Vimp(x)

I for simplicity: neglect umklapp, phonons

I W|P 〉 = 0 – ee collisions conserve momentumI must include streaming terms: L 6= 0I Vimp perturbatively small – efficient analytical/numerical

algorithm to exactly solve the transport problem:

ρ = V 2imp × · · ·

New Boltzmann Transport 10

Classical Disorder

I charge puddles: H = Hclean +

∫ddx n(x)Vimp(x):

⇠

�F

µ(x) = µ0 � Vimp(x)

I for simplicity: neglect umklapp, phononsI W|P 〉 = 0 – ee collisions conserve momentum

I must include streaming terms: L 6= 0I Vimp perturbatively small – efficient analytical/numerical

algorithm to exactly solve the transport problem:

ρ = V 2imp × · · ·

New Boltzmann Transport 10

Classical Disorder

I charge puddles: H = Hclean +

∫ddx n(x)Vimp(x):

⇠

�F

µ(x) = µ0 � Vimp(x)

I for simplicity: neglect umklapp, phononsI W|P 〉 = 0 – ee collisions conserve momentumI must include streaming terms: L 6= 0

I Vimp perturbatively small – efficient analytical/numericalalgorithm to exactly solve the transport problem:

ρ = V 2imp × · · ·

New Boltzmann Transport 10

Classical Disorder

I charge puddles: H = Hclean +

∫ddx n(x)Vimp(x):

⇠

�F

µ(x) = µ0 � Vimp(x)

I for simplicity: neglect umklapp, phononsI W|P 〉 = 0 – ee collisions conserve momentumI must include streaming terms: L 6= 0I Vimp perturbatively small – efficient analytical/numerical

algorithm to exactly solve the transport problem:

ρ = V 2imp × · · ·

New Boltzmann Transport 11

Free Fermions (L 6= 0, W = 0)

Vimp ⇠

Vimp Vimp

x

y

x

y

x

y

I simple argument:

1

τDrude=vF

ξ×∆θ2

I formally:

ρ ∼ν(µ)V 2

imp

n2e2vFξ

I arrow of time =⇒ entropyproduction when W = 0!

New Boltzmann Transport 11

Free Fermions (L 6= 0, W = 0)

Vimp ⇠

Vimp Vimp

x

y

x

y

x

y

I simple argument:

1

τDrude=vF

ξ×∆θ2

I formally:

ρ ∼ν(µ)V 2

imp

n2e2vFξ

I arrow of time =⇒ entropyproduction when W = 0!

New Boltzmann Transport 11

Free Fermions (L 6= 0, W = 0)

Vimp ⇠

Vimp Vimp

x

y

x

y

x

y

I simple argument:

1

τDrude=vF

ξ×∆θ2

I formally:

ρ ∼ν(µ)V 2

imp

n2e2vFξ

I arrow of time =⇒ entropyproduction when W = 0!

New Boltzmann Transport 12

Toy Model: Single Fermi Surface

empty

filled

empty

filled

filled

kx

ky

kx

ky

I low T limit:

Φ ≈∑j∈Z

Φj(x)eijθ

I conserve charge (j = 0),momentum (j = ±1):

W ∼ vF

`ee

∑|j|≥2

|j〉〈j|

I analytically compute ρ!interactions alwaysdecrease ρ:

0 0.2 0.4 0.6 0.8 10.2

0.4

0.6

0.8

1

ξ/`ee

ρ/ρ

res

kTFξ � 1kTFξ � 1

New Boltzmann Transport 12

Toy Model: Single Fermi Surface

empty

filled

empty

filled

filled

kx

ky

kx

ky

I low T limit:

Φ ≈∑j∈Z

Φj(x)eijθ

I conserve charge (j = 0),momentum (j = ±1):

W ∼ vF

`ee

∑|j|≥2

|j〉〈j|

I analytically compute ρ!interactions alwaysdecrease ρ:

0 0.2 0.4 0.6 0.8 10.2

0.4

0.6

0.8

1

ξ/`ee

ρ/ρ

res

kTFξ � 1kTFξ � 1

New Boltzmann Transport 12

Toy Model: Single Fermi Surface

empty

filled

empty

filled

filled

kx

ky

kx

ky

I low T limit:

Φ ≈∑j∈Z

Φj(x)eijθ

I conserve charge (j = 0),momentum (j = ±1):

W ∼ vF

`ee

∑|j|≥2

|j〉〈j|

I analytically compute ρ!interactions alwaysdecrease ρ:

0 0.2 0.4 0.6 0.8 10.2

0.4

0.6

0.8

1

ξ/`ee

ρ/ρ

res

kTFξ � 1kTFξ � 1

New Boltzmann Transport 12

Toy Model: Single Fermi Surface

empty

filled

empty

filled

filled

kx

ky

kx

ky

I low T limit:

Φ ≈∑j∈Z

Φj(x)eijθ

I conserve charge (j = 0),momentum (j = ±1):

W ∼ vF

`ee

∑|j|≥2

|j〉〈j|

I analytically compute ρ!interactions alwaysdecrease ρ:

0 0.2 0.4 0.6 0.8 10.2

0.4

0.6

0.8

1

ξ/`ee

ρ/ρres

kTFξ � 1kTFξ � 1

New Boltzmann Transport 13

Comparison with Viscous Hydrodynamics

I `ee � ξ =⇒ viscoushydrodynamic transport

ρ ∼ η∫

d2x

V

(∇ 1

ne

)2

with η ∼ `ee

I QP random walks – seesVimp slower:

ρ ∼ 1

τ∼ vF`ee

ξ2

Vimp ⇠

Vimp Vimp

x

y

x

y

x

y

New Boltzmann Transport 13

Comparison with Viscous Hydrodynamics

I `ee � ξ =⇒ viscoushydrodynamic transport

ρ ∼ η∫

d2x

V

(∇ 1

ne

)2

with η ∼ `ee

I QP random walks – seesVimp slower:

ρ ∼ 1

τ∼ vF`ee

ξ2

Vimp ⇠

Vimp Vimp

x

y

x

y

x

y

New Boltzmann Transport 14

Toy Model: Two Fermi Surfaces

empty

filled

empty

filled

filled

kx

ky

kx

ky

I pockets have different vF

I conservation laws:I charge in pocket 1I charge in pocket 2I total momentum

I numerical computationgives:

0 0.5 1 1.5 20

2

4

6

⇠/`ee

⇢/⇢

res

kTF⇠ ⌧ 1

vF,1/vF,2 = 0.3

vF,1/vF,2 = 0.5

vF,1/vF,2 = 0.7

vF,1/vF,2 = 1

0 0.5 1 1.5 2

0.5

1

1.5

⇠/`ee

⇢/⇢

res

kTF⇠ � 1

I similar to Baber scattering,but novel mechanism

New Boltzmann Transport 14

Toy Model: Two Fermi Surfaces

empty

filled

empty

filled

filled

kx

ky

kx

ky

I pockets have different vF

I conservation laws:I charge in pocket 1I charge in pocket 2I total momentum

I numerical computationgives:

0 0.5 1 1.5 20

2

4

6

⇠/`ee

⇢/⇢

res

kTF⇠ ⌧ 1

vF,1/vF,2 = 0.3

vF,1/vF,2 = 0.5

vF,1/vF,2 = 0.7

vF,1/vF,2 = 1

0 0.5 1 1.5 2

0.5

1

1.5

⇠/`ee

⇢/⇢

res

kTF⇠ � 1

I similar to Baber scattering,but novel mechanism

New Boltzmann Transport 14

Toy Model: Two Fermi Surfaces

empty

filled

empty

filled

filled

kx

ky

kx

ky

I pockets have different vF

I conservation laws:I charge in pocket 1I charge in pocket 2I total momentum

I numerical computationgives:

0 0.5 1 1.5 20

2

4

6

⇠/`ee

⇢/⇢

res

kTF⇠ ⌧ 1

vF,1/vF,2 = 0.3

vF,1/vF,2 = 0.5

vF,1/vF,2 = 0.7

vF,1/vF,2 = 1

0 0.5 1 1.5 2

0.5

1

1.5

⇠/`ee

⇢/⇢

res

kTF⇠ � 1

I similar to Baber scattering,but novel mechanism

New Boltzmann Transport 14

Toy Model: Two Fermi Surfaces

empty

filled

empty

filled

filled

kx

ky

kx

ky

I pockets have different vF

I conservation laws:I charge in pocket 1I charge in pocket 2I total momentum

I numerical computationgives:

0 0.5 1 1.5 20

2

4

6

⇠/`ee⇢/⇢

res

kTF⇠ ⌧ 1

vF,1/vF,2 = 0.3

vF,1/vF,2 = 0.5

vF,1/vF,2 = 0.7

vF,1/vF,2 = 1

0 0.5 1 1.5 2

0.5

1

1.5

⇠/`ee

⇢/⇢

res

kTF⇠ � 1

I similar to Baber scattering,but novel mechanism

New Boltzmann Transport 14

Toy Model: Two Fermi Surfaces

empty

filled

empty

filled

filled

kx

ky

kx

ky

I pockets have different vF

I conservation laws:I charge in pocket 1I charge in pocket 2I total momentum

I numerical computationgives:

0 0.5 1 1.5 20

2

4

6

⇠/`ee⇢/⇢

res

kTF⇠ ⌧ 1

vF,1/vF,2 = 0.3

vF,1/vF,2 = 0.5

vF,1/vF,2 = 0.7

vF,1/vF,2 = 1

0 0.5 1 1.5 2

0.5

1

1.5

⇠/`ee

⇢/⇢

res

kTF⇠ � 1

I similar to Baber scattering,but novel mechanism

New Boltzmann Transport 15

Why Do Interactions Increase the Resistivity?

I ∇ · J1 = ∇ · J2 = 0; QPspushed out of equilibrium:

J

Jimb

`ee

x

µimbalance

I all excited QPs relaxmomentum:

1

τ∼ `eevF

ξ2︸ ︷︷ ︸diffusion

× ξ2

`2ee︸︷︷︸imbalance

∼ vF

`ee

Vimp ⇠

Vimp Vimp

x

y

x

y

x

y

New Boltzmann Transport 15

Why Do Interactions Increase the Resistivity?

I ∇ · J1 = ∇ · J2 = 0; QPspushed out of equilibrium:

J

Jimb

`ee

x

µimbalance

I all excited QPs relaxmomentum:

1

τ∼ `eevF

ξ2︸ ︷︷ ︸diffusion

× ξ2

`2ee︸︷︷︸imbalance

∼ vF

`ee

Vimp ⇠

Vimp Vimp

x

y

x

y

x

y

New Boltzmann Transport 16

Bounds

I we proved a unified transport bound:

ρ ≤ T s

J2=〈Φodd|W|Φodd〉〈Φodd|E〉2

, subject to ∇ ·J [Φodd] = 0︸ ︷︷ ︸conservation of

charge, energy, imbalance...

upon integrating out non-conserved even modes:

W = Wodd + LTW−1evenL

I hydrodynamic limit:

T S ∼∫

ddx

(J − nv) ·Σ−1 · (J − nv)︸ ︷︷ ︸imbalance diffusion

+η(∇v)2

.I perturbative results from above become non-perturbative

bounds on ρ

New Boltzmann Transport 16

Bounds

I we proved a unified transport bound:

ρ ≤ T s

J2=〈Φodd|W|Φodd〉〈Φodd|E〉2

, subject to ∇ ·J [Φodd] = 0︸ ︷︷ ︸conservation of

charge, energy, imbalance...

upon integrating out non-conserved even modes:

W = Wodd + LTW−1evenL

I hydrodynamic limit:

T S ∼∫

ddx

(J − nv) ·Σ−1 · (J − nv)︸ ︷︷ ︸imbalance diffusion

+η(∇v)2

.

I perturbative results from above become non-perturbativebounds on ρ

New Boltzmann Transport 16

Bounds

I we proved a unified transport bound:

ρ ≤ T s

J2=〈Φodd|W|Φodd〉〈Φodd|E〉2

, subject to ∇ ·J [Φodd] = 0︸ ︷︷ ︸conservation of

charge, energy, imbalance...

upon integrating out non-conserved even modes:

W = Wodd + LTW−1evenL

I hydrodynamic limit:

T S ∼∫

ddx

(J − nv) ·Σ−1 · (J − nv)︸ ︷︷ ︸imbalance diffusion

+η(∇v)2

.I perturbative results from above become non-perturbative

bounds on ρ

Return to Experiments 17

Phenomenology: Enhanced Resistivity near Criticality

I sample phase diagram:

doping

Tρ ∼ T

ρ ∼ T 2

I imbalance diffusion + smooth disorder =⇒ entire phasediagram ?

I layered materials with chemical doping =⇒ smootherpotentials...

I many ways to get this effect besides two Fermi surfaces...

Return to Experiments 17

Phenomenology: Enhanced Resistivity near Criticality

I sample phase diagram:

doping

Tρ ∼ T

ρ ∼ T 2

I imbalance diffusion + smooth disorder =⇒ entire phasediagram ?

I layered materials with chemical doping =⇒ smootherpotentials...

I many ways to get this effect besides two Fermi surfaces...

Return to Experiments 17

Phenomenology: Enhanced Resistivity near Criticality

I sample phase diagram:

doping

Tρ ∼ T

ρ ∼ T 2

I imbalance diffusion + smooth disorder =⇒ entire phasediagram ?

I layered materials with chemical doping =⇒ smootherpotentials...

I many ways to get this effect besides two Fermi surfaces...

Return to Experiments 17

Phenomenology: Enhanced Resistivity near Criticality

I sample phase diagram:

doping

Tρ ∼ T

ρ ∼ T 2

I imbalance diffusion + smooth disorder =⇒ entire phasediagram ?

I layered materials with chemical doping =⇒ smootherpotentials...

I many ways to get this effect besides two Fermi surfaces...