andrew lucas - ima.umn.edu · solid-state physics at the ballistic-to-hydrodynamic crossover 1....

Solid-state physics at the ballistic-to-hydrodynamic crossover

1. Introduction

Andrew Lucas

Stanford Physics

Theory and Computation for Transport Properties in 2D Materials; IMA, UMN

March 26-27, 2018

Collaborators 2

Kin Chung FongRaytheon BBN

Subir SachdevHarvard Physics & Perimeter Institute

Jesse CrossnoHarvard Physics/SEAS

Philip KimHarvard Physics/SEAS

Hydrodynamics 3

Classical Many-Body Dynamics

I what is the emergent dynamics of N � 1 particles?

I compute by solving N equations? (if N . 104)

F1 = m1a1, F2 = m2a2, . . . FN = mNaN .

Hydrodynamics 3

I what is the emergent dynamics of N � 1 particles?

I compute by solving N equations? (if N . 104)

F1 = m1a1, F2 = m2a2, . . . FN = mNaN .

Hydrodynamics 4

I what if N ∼ 1023?

I check for collective, hydrodynamic description

Hydrodynamics 4

I what if N ∼ 1023?

I check for collective, hydrodynamic description

Hydrodynamics 5

Quantum Fluids

I what about quantum systems?

d|Ψ〉dt

= −iH|Ψ〉

I since dimH ∼ dN , exact solution for N . 20I quantum systems can also be fluids

quark-gluon plasma:

cold atoms: condensed matter:

Hydrodynamics 5

Quantum Fluids

d|Ψ〉dt

= −iH|Ψ〉

I since dimH ∼ dN , exact solution for N . 20

I quantum systems can also be fluids

quark-gluon plasma:

Hydrodynamics 5

Quantum Fluids

d|Ψ〉dt

= −iH|Ψ〉

quark-gluon plasma:

Hydrodynamics 5

Quantum Fluids

d|Ψ〉dt

= −iH|Ψ〉

quark-gluon plasma:

Hydrodynamics 5

Quantum Fluids

d|Ψ〉dt

= −iH|Ψ〉

quark-gluon plasma: cold atoms:

condensed matter:

Hydrodynamics 5

Quantum Fluids

d|Ψ〉dt

= −iH|Ψ〉

quark-gluon plasma: cold atoms: condensed matter:

Hydrodynamics 6

A Billiard Ball Thought Experiment

I so what is hydrodynamics? consider the following twoinitial conditions for classical chaotic billiards:

I can we distinguish these after a time ∆t?I not if ∆t� tmfp – computational rounding errors...

Hydrodynamics 6

I can we distinguish these after a time ∆t?

I not if ∆t� tmfp – computational rounding errors...

Hydrodynamics 6

I can we distinguish these after a time ∆t?I not if ∆t� tmfp – computational rounding errors...

Hydrodynamics 7

I what about these?

I can we distinguish these after a time ∆t?

I yes, for any t – count whether there are 13 or 14 balls!

Hydrodynamics 7

I what about these?

I can we distinguish these after a time ∆t?I yes, for any t – count whether there are 13 or 14 balls!

Hydrodynamics 8

Hydrodynamics as an Effective Theory

slow variables of a chaotic, thermalizing many-body system aredensities of conserved quantities

I if only ball number (charge) is conserved, EOM is

∂tρ+ ∂iJi = 0.

I on time/length scales t� tmfp, x� `mfp,

Ji = Ji (∂tρ, ∂jρ) ≈ −D∂iρ︸︷︷︸∼`mfp/x

+ D′∂2∂iρ︸︷︷︸∼(`mfp/x)3

+D′′∂t∂iρ+ · · ·

I universal late time dynamics: Fick’s law!

∂tρ ≈ D∂i∂iρ.

I stability, or second law of thermodynamics:

D ≥ 0.

Hydrodynamics 8

∂tρ+ ∂iJi = 0.

+ D′∂2∂iρ︸︷︷︸∼(`mfp/x)3

+D′′∂t∂iρ+ · · ·

D ≥ 0.

Hydrodynamics 8

∂tρ+ ∂iJi = 0.

+ D′∂2∂iρ︸︷︷︸∼(`mfp/x)3

+D′′∂t∂iρ+ · · ·

D ≥ 0.

Hydrodynamics 8

∂tρ+ ∂iJi = 0.

+ D′∂2∂iρ︸︷︷︸∼(`mfp/x)3

+D′′∂t∂iρ+ · · ·

D ≥ 0.

Hydrodynamics 8

∂tρ+ ∂iJi = 0.

+ D′∂2∂iρ︸︷︷︸∼(`mfp/x)3

+D′′∂t∂iρ+ · · ·

D ≥ 0.

Quantum Fluids and Correlation Functions 9

What is a Quantum Fluid?

I this story doesn’t depend on classical vs. quantummicroscopic dynamics

I but, in quantum systems, density ρ(x, t) is an operatoracting on many-body Hilbert space!

I a ‘careful’ way to define a quantum fluid is by expectationvalues in thermal equilibrium:

GRρρ(x, t) ≡

~Θ(t)〈[ρ(x, t), ρ(0, 0)]〉T .

Fourier transform as k, ω → 0:

GRρρ(k, ω) ≈ χρρ︸︷︷︸

compressibility

Dk2 − iω

[Kadanoff, Martin (1963)]

GRρρ(x, t) ≡

~Θ(t)〈[ρ(x, t), ρ(0, 0)]〉T .

compressibility

Dk2 − iω

GRρρ(x, t) ≡

~Θ(t)〈[ρ(x, t), ρ(0, 0)]〉T .

compressibility

Dk2 − iω

Deriving the Green’s Function

I why does GRρρ take this form? consider perturbing quantum

system by small point-source of charge:

Hpert = H0 −Θ(−t)eεt∫

ddx µ(x)︸︷︷︸classical source

· ρ(x)︸︷︷︸operator

I linear response theory (first order time-dependentperturbation theory) gives us

〈ρ(x, t)〉pert = 〈ρ〉0 −i

−∞

∫ddx′ µ(x′)eεs〈[ρ(x′, s), ρ(x, t)]〉

= 〈ρ〉0 +0∫

−∞

∫ddx′ µ(x′)eεsGR

ρρ(x− x′, t− s)

I why does GRρρ take this form? consider perturbing quantum

system by small point-source of charge:

Hpert = H0 −Θ(−t)eεt∫

ddx µ(x)︸︷︷︸classical source

· ρ(x)︸︷︷︸operator

I linear response theory (first order time-dependentperturbation theory) gives us

〈ρ(x, t)〉pert = 〈ρ〉0 −i

−∞

∫ddx′ µ(x′)eεs〈[ρ(x′, s), ρ(x, t)]〉

= 〈ρ〉0 +0∫

−∞

∫ddx′ µ(x′)eεsGR

ρρ(x− x′, t− s)

I take the Fourier transform in space:

〈ρ(k, t)〉pert =

∫dω

0∫−∞

ds µ(k)e(ε+iω)sGRρρ(k, ω)e−iωt

and Laplace transform in time (only want t > 0):

〈ρ(k, z)〉pert = µ(k)GRρρ(k, z)−GR

ρρ(k, 0)

I we can sensibly write

∂t〈ρ(k, t)〉pert ≈ −Dk2〈ρ(k, t)〉pert

and thus find

(iz −Dk2)〈ρ(k, z)〉pert = −〈ρ(k, t = 0)〉pert

= −µ(k)GRρρ(k, ω = 0)

I take the Fourier transform in space:

〈ρ(k, t)〉pert =

∫dω

0∫−∞

ds µ(k)e(ε+iω)sGRρρ(k, ω)e−iωt

and Laplace transform in time (only want t > 0):

〈ρ(k, z)〉pert = µ(k)GRρρ(k, z)−GR

ρρ(k, 0)

I we can sensibly write

∂t〈ρ(k, t)〉pert ≈ −Dk2〈ρ(k, t)〉pert

and thus find

(iz −Dk2)〈ρ(k, z)〉pert = −〈ρ(k, t = 0)〉pert

= −µ(k)GRρρ(k, ω = 0)

Higher Derivative Corrections?

I using that susceptibility χρρ = ∂〈ρ〉/∂µ = GRρρ(k, ω = 0):

GRρρ ≈ χρρ

Dk2 − iω.

I higher order terms in effective theory:

GRρρ = χρρ

Dk2 +D′k4 − iω + · · · .

Im(!) Re(!) = 0

di↵usion poles

non-hydrodynamic poles/branch cuts

`�1mfp

⌧�1mfp

GRρρ ≈ χρρ

Dk2 − iω.

GRρρ = χρρ

Dk2 +D′k4 − iω + · · · .

Im(!) Re(!) = 0

di↵usion poles

`�1mfp

⌧�1mfp

GRρρ ≈ χρρ

Dk2 − iω.

GRρρ = χρρ

Dk2 +D′k4 − iω + · · · .

Im(!) Re(!) = 0

di↵usion poles

`�1mfp

⌧�1mfp

Fluids from (Quantum) Kinetic Theory 13

The Distribution Function

I kinetic theory: a historical perspective on fluid dynamics

I f(x,k, t) = “the number of particles at point x atmomentum k” can be defined through correlators 〈φφ〉

I f obeys the (quantum) Boltzmann equation

∂tf + veff · ∂xf + Feff · ∂pf︸︷︷︸free particle streaming

+ O (~)︸︷︷︸∼λF∂x

= −C[f ]︸︷︷︸(weak) collisions

where veff is effective quasiparticle velocity w/ self-energycorrections, etc.

+ O (~)︸︷︷︸∼λF∂x

Collisions

I example collision integral: electron-electron scattering:

C[f ] ∼ fpIm (Σp)

∼∑

p1,q1,q2

q2q1 2

fpfp1(1− fq1)(1− fq2)

“outgoing states empty?”

“incoming states occupied?”

T -matrix (perturbative QFT)

I kinetic theory: classical equations, coefficients from QFT

Collisions

I example collision integral: electron-electron scattering:

C[f ] ∼ fpIm (Σp)

∼∑

p1,q1,q2

q2q1 2

fpfp1(1− fq1)(1− fq2)

“outgoing states empty?”

“incoming states occupied?”

T -matrix (perturbative QFT)

I kinetic theory: classical equations, coefficients from QFT

Example: a Two-Dimensional Fermi Liquid

I Fermi liquid: a toy model of ametal [at low temperatures]

I distribution function:

f ≈ Θ(kF − k) + δ(kF − k)Φ(θ)

Φ(θ) =∑n∈Z

einθΦn

I collision integral w/ charge,momentum conservation:

C[Φ] = − 1

∑|n|≥2

einθΦn

viscous corrections

f ≈ Θ(kF − k) + δ(kF − k)Φ(θ)

Φ(θ) =∑n∈Z

einθΦn

C[Φ] = − 1

∑|n|≥2

einθΦn

viscous corrections

f ≈ Θ(kF − k) + δ(kF − k)Φ(θ)

Φ(θ) =∑n∈Z

einθΦn

C[Φ] = − 1

∑|n|≥2

einθΦn

viscous corrections

The Hydrodynamic Limit

I the kinetic equations read

∂tΦn +vF

2∂x (Φn−1 + Φn+1) = −Φn

τeeδ|n|≥2

I the hydrodynamic limit is τee∂t � 1:

∂tΦ0︸︷︷︸∂tρ

+ vF∂xΦ1︸︷︷︸ρ∂xv

= 0, ∂tΦ1︸︷︷︸∂tv

2∂xΦ0︸︷︷︸∂xP

− v2Fτee

4∂2xΦ1︸︷︷︸

−η∂2xv

viscous corrections

I analogous effects happen in more complicated models

∂tΦn +vF

2∂x (Φn−1 + Φn+1) = −Φn

τeeδ|n|≥2

∂tΦ0︸︷︷︸∂tρ

= 0, ∂tΦ1︸︷︷︸∂tv

2∂xΦ0︸︷︷︸∂xP

− v2Fτee

4∂2xΦ1︸︷︷︸

−η∂2xv

viscous corrections

∂tΦn +vF

2∂x (Φn−1 + Φn+1) = −Φn

τeeδ|n|≥2

∂tΦ0︸︷︷︸∂tρ

= 0, ∂tΦ1︸︷︷︸∂tv

2∂xΦ0︸︷︷︸∂xP

− v2Fτee

4∂2xΦ1︸︷︷︸

−η∂2xv

viscous corrections

Is the Viscosity Large or Small?

large lee: fast momentum di↵usion small lee: slow momentum di↵usion

I for general quantum systems:

η ∼ (ε+ P )τee

{ ∞ no interactions

& (ε+ P ) ~kBT

strong interactions.

Green’s Functions

I a diffusive Green’s function: (D = 14v

2Fτee)

GRJyJy (kx, ω) =

χρρDk2x

Dk2x − i2ω(1− iωτee +

√(1− iωτee)2 + (vFτeekx)2

I diffusive form is corrected by terms ∼ ωτee and ∼ Dk2xτee!(though in this toy model, pole persists at ω = −iDk2x...)

I Green’s function that couples to sound wave:

GRρρ(kx, ω) =

χρρk2x

k2x − ω(ω − iτ−1

ee + i√k2 − (ω − iτ−1

ee )2) .

I sound pole as ω → 0; zero sound w/ branch cuts as ω →∞

Green’s Functions

2Fτee)

GRJyJy (kx, ω) =

χρρDk2x

GRρρ(kx, ω) =

χρρk2x

k2x − ω(ω − iτ−1

ee + i√k2 − (ω − iτ−1

ee )2) .

Green’s Functions

2Fτee)

GRJyJy (kx, ω) =

χρρDk2x

GRρρ(kx, ω) =

χρρk2x

k2x − ω(ω − iτ−1

ee + i√k2 − (ω − iτ−1

ee )2) .

Green’s Functions

2Fτee)

GRJyJy (kx, ω) =

χρρDk2x

GRρρ(kx, ω) =

χρρk2x

k2x − ω(ω − iτ−1

ee + i√k2 − (ω − iτ−1

ee )2) .

Hydrodynamics of Electrons in Metals 19

Quantum Dynamics of Electrons in Metals

I the rest of my talks will focus on a specific problem:electrons in metals

Ne∑i=1

Ne∑i<j=1

4πε0|ri − rj |−

Ni∑I=1

Ne∑i=1

4πε0|ri −RI |

I for simplicity – ions (RI) frozen

I this is a canonical (quantum) many-body problem – surelythe hydrodynamic limit is rather trivial?

I lack of translation invariance, due to RI (especiallydefects/impurities)

I quasiparticle interactions might be negligibleI in small samples with few impurities, all quasiparticle

scattering might be negligible!

Ne∑i=1

Ne∑i<j=1

Ni∑I=1

Ne∑i=1

4πε0|ri −RI |

Ne∑i=1

Ne∑i<j=1

Ni∑I=1

Ne∑i=1

4πε0|ri −RI |

Ne∑i=1

Ne∑i<j=1

Ni∑I=1

Ne∑i=1

4πε0|ri −RI |

I quasiparticle interactions might be negligible

I in small samples with few impurities, all quasiparticlescattering might be negligible!

Ne∑i=1

Ne∑i<j=1

Ni∑I=1

Ne∑i=1

4πε0|ri −RI |

Electronic Hydrodynamics

I collision time scales in most metals disfavor electron fluid:

tee ⌧ timp tee � timp

ordinary metal (iron etc.)ultraclean metal (GaAs, graphene?)

tee ⇠ 10�11 s timp ⇠ 10�14 s

@t+ r · J = 0

@t= mess

I “electron hydrodynamics” limit is tee � timp

Electronic Hydrodynamics

I collision time scales in most metals disfavor electron fluid:

tee ⌧ timp tee � timp

ordinary metal (iron etc.)ultraclean metal (GaAs, graphene?)

tee ⇠ 10�11 s timp ⇠ 10�14 s

@t+ r · J = 0

@t= mess

I “electron hydrodynamics” limit is tee � timp

Navier-Stokes with “Friction”

I conservation of charge:

∇ · (nv) ≈ n∇ · v = 0.

I “conservation” of momentum: Navier-Stokes equation...

∇P − η∇2v = n∇µ− η∇2v ≈ −Γv

...with “friction” term Γ ∼ 1/`imp, from momentumrelaxation due to electron-impurity scattering

I conservation of energy is “not needed”:

∇ · (εv + · · · ) ≈ EF∇ · v = 0.

Ohmic diffusion:

`�√η

viscous hydrodynamic:

`�√η

∇ · (nv) ≈ n∇ · v = 0.

∇P − η∇2v = n∇µ− η∇2v ≈ −Γv

∇ · (εv + · · · ) ≈ EF∇ · v = 0.

Ohmic diffusion:

`�√η

∇ · (nv) ≈ n∇ · v = 0.

∇P − η∇2v = n∇µ− η∇2v ≈ −Γv

∇ · (εv + · · · ) ≈ EF∇ · v = 0.

Ohmic diffusion:

`�√η

∇ · (nv) ≈ n∇ · v = 0.

∇P − η∇2v = n∇µ− η∇2v ≈ −Γv

∇ · (εv + · · · ) ≈ EF∇ · v = 0.

Ohmic diffusion:

`�√η

Flows through a Channel: Analytic Solution

vx(y) wx

ρE − Γvx = −η∂2yvx

I assuming that vx = 0 at boundaries (no-slip), the electricalresistance per unit length:

σdc(w − 2λ tanh w2λ)

, σdc =ρ2

(viscous “skin depth” of size λ enhances resistance)I if λ� w, Poiseuille flow:

R =12η

ρ2w3.

vx(y) wx

, σdc =ρ2

(viscous “skin depth” of size λ enhances resistance)

I if λ� w, Poiseuille flow:

R =12η

ρ2w3.

vx(y) wx

, σdc =ρ2

(viscous “skin depth” of size λ enhances resistance)I if λ� w, Poiseuille flow:

R =12η

ρ2w3.

Experimental Channel Flow

I some evidence for R ∼ w−3 has been seen in PdCoO2 (e.g.)[Moll et al, 1509.05691]

(they are defining “ρ = Rw”)

Viscous Flows in Constrictions

I we also predict, in an ‘ordinary’ metal

R ∼ η ∼ `ee ∼ T−2

I ∂R/∂T < 0 in graphene constrictions:[Kumar et al; 1703.06672]

Viscous Flows in Constrictions

I we also predict, in an ‘ordinary’ metal

R ∼ η ∼ `ee ∼ T−2

I ∂R/∂T < 0 in graphene constrictions:[Kumar et al; 1703.06672]

Viscous Backflow

I vortices (“whirlpools”) – a more dramatic hydro effect:NATURE PHYSICS DOI: 10.1038/NPHYS3667 LETTERS

10Viscous flow

Ohmic flow

Source

Potential (a.u.)Potential (a.u.)

−0.8

−0.4

Figure 1 | Current streamlines and potential map for viscous and ohmicflows. White lines show current streamlines, colours show electricalpotential, arrows show the direction of current. a, Mechanism of a negativeelectrical response: viscous shear flow generates vorticity and a backflowon the side of the main current path, which leads to charge buildup of thesign opposing the flow and results in a negative nonlocal voltage.Streamlines and electrical potential are obtained from equation (5) andequation (6). The resulting potential profile exhibits multiple sign changesand ±45� nodal lines, see equation (7). This provides directly measurablesignatures of shear flows and vorticity. b, In contrast, ohmic currents flowdown the potential gradient, producing a nonlocal voltage in theflow direction.

Corbino disc9. These proposals, however, rely on fairly complexa.c. phenomena originating from high-frequency dynamics in theelectron system. In each of these cases, as well as in those ofrefs 8,20, a model-dependent analysis was required to delineate thee�ects of viscosity from ‘extraneous’ contributions. In contrast, thenonlocal d.c. response considered here is a direct manifestationof the collective momentum transport mode which underpinsviscous flow, therefore providing an unambiguous, almost textbook,diagnostic of the viscous regime.

A nonlocal electrical response mediated by chargeless modeswas found recently to be uniquely sensitive to quantities whichare not directly accessible in electrical transport measurements, inparticular spin currents and valley currents25–27. In a similarmanner,the nonlocal response discussed here gives a diagnostic of viscoustransport, which is more direct and powerful than any approachesbased on local transport.

There are several aspects of the electron system in graphene thatare particularly well suited for studying electronic viscosity. First,the momentum-nonconserving Umklapp processes are forbiddenin two-body collisions because of graphene’s crystal structure andsymmetry. This ensures the prominence of momentum conserva-tion and associated collective transport. Second, although carrierscattering is weak away from charge neutrality, it can be enhancedby several orders of magnitude by tuning the carrier density to theneutrality point. This allows one to cover the regimes of high and lowviscosity, respectively, in a single sample. Last, the two-dimensionalstructure and atomic thickness makes the electronic states ingraphene fully exposed and amenable to sensitive electric probes.

To show that the timescales are favourable for thehydrodynamical regime, we will use parameter values estimatedfor pristine graphene samples which are almost defect free, suchas freestanding graphene28. Kinematic viscosity can be estimatedas the momentum di�usion coe�cient ⌫ ⇡ (1/2)v2

F��1ee , where

�ee is the carrier–carrier scattering rate, and vF = 106 m s�1 forgraphene. According to Fermi-liquid theory, this rate behaves as�ee ⇠ (kBT )2/EF in the degenerate limit (that is, away from chargeneutrality), which leads to large ⌫ values. Near charge neutrality,

−0.4 −0.2−0.2

Distance x/w

0.0 0.2 0.4

(enw)2/ = 200ρ η (enw)2/ = 150ρ η (enw)2/ = 100ρ η (enw)2/ = 50 = 0

Figure 2 | Nonlocal response for di�erent resistivity-to-viscosity ratios⇢/⌘. Voltage V(x) is plotted at a distance x from current leads obtainedfrom equation (12) for the set-up shown in the inset. The voltage is positivein the ohmic-dominated region at large |x| and negative in theviscosity-dominated region closer to the leads (positive values at evensmaller |x| reflect the finite contact size a⇡0.05w used in simulation).Viscous flow dominates up to fairly large resistivity values, resulting in thenegative response persisting up to values as large as ⇢(enw)2/⌘⇡ 120.Positive and negative voltage regions are marked by blue and pink,respectively. Nodal points, marked by arrows, are sensitive to the ⇢/⌘ value,which provides a way to directly measure viscosity (see text).

however, the rate �ee grows and ⌫ approaches the AdS/CFT limit—namely s~/4⇡kB, where s is entropy density. Refs 12,23 estimatethis rate as �ee ⇡A↵2kBT/~, where ↵ is the interaction strength.For T = 100K, assuming EF = 0 and approximating the prefactoras A⇡ 1 (refs 12,23), this predicts characteristic times as short as� �1ee ⇡80fs. Disorder scattering can be estimated from themeasured

mean free path values, which reach a few microns at large doping29.Using the momentum relaxation rate square-root dependenceon doping, �p / n�1/2, and estimating it near charge neutrality,n. 1010 cm�2, gives times � �1

p ⇠ 0.5 ps which are longer than thevalues � �1

ee estimated above. The inequality �p ⌧ �ee justifies ourhydrodynamical description of transport.

Momentum transport in the hydrodynamic regime is describedby the continuity equation for momentum density,

@t pi +@jTij =��ppi, Tij =P�ij +µvivj +T (v)ij (1)

where Tij is the momentum flux tensor, P and µ are pressure andmass density, and v is the carrier drift velocity. The quantity �p,introduced above, describes electron-lattice momentum relaxationdue to disorder or phonons, which we will assume to be smallcompared to the carrier scattering rate. We can relate pressure tothe electrochemical potential � through P=e

�(n0)dn0. Herewe work at degeneracy, EF � kBT , ignoring the entropic/thermalcontributions, and approximating P ⇡ e(n � n0)� , with n theparticle number density. While carrier scattering is suppressed atdegeneracy as compared to its value at EF = 0, here we assumethat the carrier-carrier scattering remains faster than the disorderscattering, as required for the validity of hydrodynamics. Viscositycontributes to the momentum flux tensor through

T (v)ij =⌘(@ivj +@jvi)+(⇣ �⌘)@kvk�ij (2)

where ⌘ and ⇣ are the first and second viscosity coe�cients.For drift velocities smaller than plasmonic velocities, transport incharged systems is described by an incompressible flow with a

NATURE PHYSICS | VOL 12 | JULY 2016 | www.nature.com/naturephysics

[Torre et al, 1508.00363]; [Levitov, Falkovich, 1508.00836]

Viscous Backflow

I experiment in single layer (SLG) and bilayer (BLG)graphene (RV < 0 =⇒ sign change in relative potential)

[Bandurin et al, 1509.04165]

Thermal and Electrical Conductivity at Charge Neutrality

E�rT

I in a clean charge neutral metal, such as neutral graphene:

κ =Qx−∂xT

=∞, σ =JxEx

= finite.

Thermal and Electrical Conductivity at Charge Neutrality

E�rT

I in a clean charge neutral metal, such as neutral graphene:

κ =Qx−∂xT

=∞, σ =JxEx

= finite.

Wiedemann-Franz Law Violations in Experiment

−10−15 15−5 0 5 10

n (109 cm-2)

disorder-limited

phonon-limited

[Crossno et al; 1509.04713]

Summary 29

I hydrodynamics is the effective theory of thermalizingsystems – degrees of freedom are densities of conservedquantities

I in quantum systems, hydrodynamics is formally establishedvia sound and diffusion poles in correlation functions

I in solid-state systems, electronic hydrodynamics isimperfect because of impurities. evidence forhydrodynamics (so far) is pretty indirect...

I tomorrow: two important future directions which I expectto be computationally interesting

I thermoelectric transport phenomena for complicatedmaterials

I nonlinear “hydrodynamics” at the ballistic crossover

Summary 29

andrew lucas - ima.umn.edu · solid-state physics at the ballistic-to-hydrodynamic crossover 1....

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