an electron-phonon

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Quantum chaos in an electron-phonon bad metal

Yochai Werman

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Quantum chaos in an electron-phonon bad metal

Yochai Werman

With: Erez Berg, Steve Kivelson.

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Metallic transport, resistivity saturation, and bad metals.

1 Ioffe-Regel limit

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Quasiparticle description

◉ Boltzmann theory:

○ Electrons as coherent quasiparticles

○ Distribution function 𝑓_𝑝(𝑟)

○ wavepackets in both 𝑟 and 𝑝.

○ 𝑙_𝑚𝑓𝑝 ≫ 𝜆_𝐹 or 𝐸_𝐹𝜏 ≫ 1.

Ashcroft and Mermin

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◉

Drude formula

○ Electrons as coherent quasiparticles.

○ 𝜎_2𝐷 = ^𝑒²

ℎ 𝐸_𝐹𝜏.

◉

Electron – phonon coupling about 𝜔_𝐷

○ 𝜏⁻¹ ∝ 𝑇

○ 𝜌 ∝ 𝑇

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𝜌 ∝ 𝑇

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◉ Key concept:

Ioffe-Regel Limit

○ Limit on the resistivity in the qp picture:

• 𝐸_𝐹𝜏 > 1 and 𝜎 = ^𝑒²

ℎ 𝐸_𝐹𝜏.

• 𝜎_2𝐷 ≳ ^𝑒²

ℎ

• 𝜌_3𝐷 ≲ 100𝜇Ω𝑐𝑚

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Resistivity saturation

◉ Resistivity saturation

Gunnarsson et al., RMP 75 (2003)

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Bad Metals

◉ Breakdown of qp picture:

o

Bad metals violate the Ioffe-Regel limit

o

Strong el-el interactions.

o

Linear-T resistivity.

Hussey, (2001)

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◉ Saturating metals

○ Obey the MIR limit.

○ Electron – phonon systems.

○ Good BCS

superconductors.

Saturating vs. bad metals

◉ Bad metals

○ Violate the MIR limit.

○ Electron – electron systems.

○ High Tc

superconductors.

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Outline

◉ Large-N electron – phonon model

◉ Transport Coefficients

○ Resistivity – saturation / bad metal.

○ Thermal conductivity.

◉ Chaos

○ Chaos is linked with thermal transport.

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Large N model which displays resistivity saturation or bad metallic behavior.

2 Electron-phonon

model

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Electron-phonon model

◉ 𝑁 ≫ 1 bands of electrons 𝑐

_𝑎

𝑘 , 𝑎 ∈ [1, 𝑁] .

○ Fermi energy 𝐸_𝐹.

○ Bandwidth Λ.

◉ 𝑁

²

≫ 𝑁 phonon modes 𝑋

_𝑎𝑏

𝑘 , 𝑎, 𝑏 ∈ [1, 𝑁] .

○ Dispersionless optical phonons.

○ Einstein frequency 𝜔₀.

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Single particle properties

◉ 𝑁 electron bands, 𝑁

²

≫ 𝑁 phonon modes:

○ Strongly renormalized electrons

• Σ^′′ ∝ 𝑂(1).

○ Weakly renormalized phonons

• Π^′′ ∝ ¹

𝑁.

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◉ Motivation for large N:

o

Realistic (?)

•

E.g. 𝐴₃𝐶₆₀ has 𝑁_𝑝ℎ = 189, 𝑁_𝑒𝑙 = 3.

o

Strong coupling in controlled manner.

o

Suppresses lattice instabilities.

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◉

Electrons degenerate:

○ ^𝐸𝐹 ≫ 𝑇.

◉

Phonons in the classical limit:

○ 𝜔₀ ≪ 𝑇.

◉

Strong coupling:

○ 𝜆 = ^𝛼²^𝜈

𝐾 ≫ 1 – strong coupling.

○ 𝜆𝑇 may be larger than 𝐸_𝐹, Λ.

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Single particle properties

◉ Strongly renormalized electrons

• Σ^′′ = 𝜆𝑇 × 𝐸_𝐹 > 𝐸_𝐹 for 𝑇 > 𝐸_𝐹/𝜆

◉ Weakly renormalized phonons

• Π^′′ = ¹

𝑁 𝜔₀²

𝑇 for 𝑇 > 𝐸_𝐹/𝜆

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On-site coupling (“Holstein”)

𝐻_𝑖𝑛𝑡 = 𝛼

𝑁 𝑖,𝑎𝑏

𝑋_{𝑖,𝑎𝑏}𝑐_𝑖,𝑎^† 𝑐_𝑖,𝑏 Electron-phonon model

Bond-density coupling (“SSH”)

𝐻_𝑖𝑛𝑡 = 𝛼

𝑁𝑖,𝑎𝑏

𝑋_{𝑖,𝑎𝑏}𝑐_𝑖,𝑎^† 𝑐_𝑖+1,𝑏

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Transport: resistivity

◉

Resistivity depends on coupling.

◉

Saturation of resistivity in SSH model due to phonon- assisted channel.

On site

Bond density

𝜆𝑇/𝐸_𝐹

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On-site coupling (“Holstein”)

𝐽 = 𝑑^𝑑𝑘 𝑣_𝒌𝑐_𝒌^†𝑐_𝒌 ≡ 𝐽₀ Current operator

Bond-density coupling (“SSH”)

𝐽 = 𝐽₀ + 𝑖 𝛼 𝑁 𝑖

𝑋_𝑖𝑐_𝑖+1^† 𝑐_𝑖

≡ 𝐽₀ + 𝐽₁

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Conductivity

◉ Kubo formula: 𝜎 = lim

𝜔→0

ℑΠ^𝐽𝐽 𝜔 𝜔

𝐽₀ 𝐽₀ 𝐽₁ 𝐽₀ 𝐽₁ 𝐽₁

𝐽₀ = 𝑖

𝑖

𝑐_𝑖+1^† 𝑐_𝑖−𝑐_𝑖^† 𝑐_𝑖+1

𝐽₁ = 𝑖 𝜆

𝑁 _𝑖 𝑋_𝑖 𝑐_𝑖+1^† 𝑐_𝑖−𝑐_𝑖^† 𝑐_𝑖+1

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Conductivity

◉ 00 channel decays as ¹

𝑇.

◉ 11 channel saturates to 𝜎₀ ∼ 𝜎_𝑀𝐼𝑅.

◉ 01 channel is negligible.

𝜆𝑇/Λ

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Transport: resistivity

◉

Resistivity depends on coupling.

◉

Saturation of resistivity in SSH model due to phonon- assisted channel.

On site

Bond density

𝜆𝑇/𝐸_𝐹

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Transport: optical conductivity

◉

Optical conductivity for bond density coupling.

◉

At high 𝑇, the relevant energy scale is Δ𝐸

= 𝜆𝑇 × 𝐸_𝐹.

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◉

Interaction term dominant at high T:

H =

^𝛼

𝑁 𝑎𝑏=1𝑁

𝑋

_𝑎𝑏

𝑐

_𝑎^†

𝑐

_𝑏

◉

^𝑋𝛼𝛽2 ≈ 𝑇/𝐾.

◉

Phonons act as a random matrix in 𝑎𝑏 space.

E E

dos dos

Λ

−Λ

ΔE

−ΔE

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◉

Interaction term dominant at high T:

H =

^𝛼

𝑁 𝛼𝛽=1𝑁

𝑋

_𝛼𝛽

𝑐

_𝛼^†

𝑐

_𝛽

◉

Random matrix theory:

E E

dos dos

Λ

−Λ

ΔE

Δ𝐸

_𝑒𝑓𝑓

= 𝜆𝑇 × 𝐸

_𝐹

−ΔE

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◉

Effective bandwith

○ W ∼ 𝜆𝑇/𝜈

◉

Electron decay rate

○ ^{Σ′′ ∼} ^{𝜆𝑇/𝜈}

◉

Compressibility:

○ 𝜒 ∼ 1/ 𝜆𝑇/𝜈

E E

dos dos

Λ

−Λ

ΔE

−ΔE

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◉

Incoherent transport

◉

Diffusion constant given by Fermi’s Golden rule:

𝐷 ∼ 𝑡_ℎ𝑜𝑝² 𝜈_𝑒𝑓𝑓 ∼ 𝑡_ℎ𝑜𝑝² / 𝜆𝑇 × 𝐸_𝐹

site i+1 site i

1 𝜈_𝑒𝑓𝑓 t

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𝐷 ∼ 𝑡_ℎ𝑜𝑝² 𝜈_𝑒𝑓𝑓 ∼ 𝑡_ℎ𝑜𝑝² / 𝜆𝑇 × 𝐸_𝐹

◉

On site model: 𝑡_ℎ𝑜𝑝 = 𝑡 → 𝐷 ∝ 1/ 𝑇

◉

Bond density model: 𝑡_ℎ𝑜𝑝 ∝ 𝑇 → 𝐷 ∝ 𝑇

site i+1 site i

1 𝜈_𝑒𝑓𝑓 t

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◉

Einstein relation:

○ 𝜎 = 𝜒𝐷

◉

Resistivity:

○ Site density: 𝜌 ∼ 𝑇

○ Bond density: 𝜌 ∼ 𝑐𝑜𝑛𝑠𝑡.

◉

Temperature dependence originates from both 𝜒 and 𝐷.

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Thermal conductivity

◉

Thermal conductivity is dominated by the phonons:

o

There are 𝑁² phonons and only 𝑁 electrons.

o

The phonons are long lived, Π^′′ ∼ ¹

𝑁.

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◉

If the phonons are dispersive, Boltzmann theory gives

𝜅 = 𝑁² v_ph²

Π′′ = N³v_ph² T 𝜔₀²

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◉

Non-dispersive phonons:

○ The el-ph interaction gives a correction to phonon velocity:

𝑣_𝑝ℎ = 1 𝑁

𝜔₀ 𝑇

𝐸_𝐹 Σ′′𝑣_𝑒𝑙

◉

we use the Kubo formula to find:

𝜅 = 𝑁² 𝑣_ph² Π′′

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Electron-phonon model - summary

◉

^𝑁 electrons, 𝑁² phonons.

◉

Decay rate:

○ ^Σ^′′ ⁼ ^{𝜆𝑇 × 𝐸}𝐹

○ ^Π^′′ ⁼ _𝑁¹^𝜔_𝑇⁰²

◉

Resistivity:

○ Site density: 𝜌 ∼ 𝑇 above the MIR limit

○ Bond density: 𝜌 ∼ 𝑐𝑜𝑛𝑠𝑡.

◉

Phonons responsible for 𝜅.

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3 Chaos

Scrambling rate and butterfly velocity in the el-ph model.

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Quantum chaos

𝑉 𝑥, 𝑡 , 𝑊 0

²

∝ 𝑒

^𝜆^𝐿 ^𝑡−

𝑥 𝑣𝐵

◉

𝑉, 𝑊 – generic local observables.

◉

𝜆_𝐿 - rate at which local information is scrambled.

◉

𝑣_𝐵 - velocity of the effects of a local perturbation

.

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Quantum chaos

𝑉 𝑥, 𝑡 , 𝑊 0

²

∝ 𝑒

^𝜆^𝐿 ^𝑡−

𝑥 𝑣_𝐵

◉ Bound on 𝜆

_𝐿

: (Maldacena et al., JHEP, 2016)

𝜆

_𝐿

≤ 2𝜋𝑇/ℏ

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Quantum chaos

◉

Non quasiparticle systems are expected to saturate the bound.

○ SYK model.

◉

Systems with quasiparticles have parametrically smaller 𝜆_𝐿:

○ Fermi liquids: 𝜆_𝐿 ∝ 𝑇²

○ Weakly coupled large-𝑁 systems: 𝜆_𝐿 ∝ 1/𝑁. (Chowdhury and Swingle, 2017; Julia’s talk yesterday)

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Quantum chaos and transport

◉

The quantities of chaos define a diffusion constant

𝐷_𝐿 = 𝑣_𝐵² 𝜆_𝐿

◉

Question: in a generic system, can we bound 𝐷 ≥ 𝐷_𝐿 > ^𝑣^𝐵²

𝑇 ?

(Hartnoll, Nature 2015; Blake, PRL, 2016)

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Quantum chaos and bad metals

This conjecture is pertinent to bad metals:

◉

Non-quasiparticle transport

◉

Universality in bad metals:

o

Many display 𝜌 𝑇 ∝ 𝑇 over large range of temperatures.

o

Many display 𝜏⁻¹ = 𝑇 – dissipative timescale. (Bruin et al., Science)

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Bruin et al., Science

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We ask:

◉

Is quantum chaos related to the transport coefficients in bad metals?

◉

does the bound on the scrambling rate lead to a universal bound on transport in these materials?

◉

Is a strongly incoherent metal necessarily also strongly chaotic?

(43)

◉

Solvable model for a high-𝑇 bad metal.

On site

Bond density

𝜆𝑇/𝐸_𝐹

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Chaos in el-ph model

◉

We calculate

○ The scrambling rate 𝜆_𝐿

○ The butterfly velocity 𝑣_𝐵

◉ For

○ Dispersive phonons

○ Non-dispersive phonons.

◉ In the bad metallic regime of the on-site model.

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◉

We find that

○ The scrambling rate is the same for phonons and electrons.

○ Chaos is carried by the phonons.

○ Chaos is weak (𝜆_𝐿 ≪ 𝑇).

○ Chaos is related to thermal conductivity.

(46)

◉

We calculate the Fourier transform of

(47)

Quantum chaos

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Quantum chaos

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We find that in the bad metallic regime

◉

𝜆_𝐿 = Π^′′ = ¹

𝑁 𝜔₀²

𝑇

- the phonon decay rate

◉

𝑣_𝐵 =

○ 𝑣_𝑝ℎ for dispersive phonons

○ 𝑣_𝑝ℎ for non-dispersive phonons

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We find that in the bad metallic regime

◉

𝜆_𝐿 = Π^′′ = ¹

𝑁 𝜔₀²

𝑇

- the phonon decay rate

◉ Phonons are the longest lived excitations

○ Bottleneck for scrambling.

(51)

We find that in the bad metallic regime

◉

𝐷_𝐿 = 𝑣_𝐵²/𝜆_𝐿 = 𝐷_𝐸

◉

𝐷_𝐿

○ > 𝐷_𝐶 for dispersive phonons

○ < 𝐷_𝐶 for non-dispersive phonons

Thermal diffusion constant

Charge diffusion constant

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We find that 𝜆_𝐿, 𝑣_𝐵 are the same for phonons and electrons.

electrons

phonons

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We find that 𝜆_𝐿, 𝑣_𝐵 are the same for phonons and electrons.

electrons

phonons

(54)

𝜆

_𝐿

= min(Σ

^′′

, Π

^′′

)

(55)

Conclusions

◉

Charge transport and chaos are not connected.

○ In the limit 𝜔0 → 0, phonons act as disorder.

○ ^𝜆𝐿 ∝ 𝜔₀² → 0.

○ 𝜌 is finite as 𝜔₀ → 0.

◉

There is no bound on charge transport.

◉

Bad metals are not necessarily strongly chaotic.

(56)

Conclusions

◉ Thermal transport and chaos are connected.

◉ Both are carried by the phonons.

(57)

Conclusions

◉ Chaos is not related to the decay of any physical quantity

○ Single particle Green’s function.

○ Current operator.

◉ Chaos is related to thermalization

○ Slowest decay rate in the system.

(58)

Any questions ?

Thanks!