The Polaron at Strong Coupling Robert Seiringer IST Austria - - PowerPoint PPT Presentation

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The Polaron at Strong Coupling Robert Seiringer IST Austria Quantissima in the Serenissima III Venice, August 1923, 2019 R. Seiringer The Polaron at Strong Coupling August 21, 2019 # 1 The Polaron Model of a charged particle

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SLIDE 1

The Polaron at Strong Coupling

Robert Seiringer IST Austria

Quantissima in the Serenissima III

Venice, August 19–23, 2019

  • R. Seiringer — The Polaron at Strong Coupling — August 21, 2019

# 1

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SLIDE 2

The Polaron

Model of a charged particle (electron) interacting with the (quantized) phonons of a polar crystal. Polarization proportional to the electric field created by the charged particle.

  • R. Seiringer — The Polaron at Strong Coupling — August 21, 2019

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SLIDE 3

The Fr¨

  • hlich Model

On L2(R3) ⌦ F (with F the bosonic Fock space over L2(R3)), Hα = ∆ pα Z

R3

1 |k| ⇣ akeikx + a†

keikx⌘

dk + Z

R3 a† kak dk

with α > 0 the coupling strength. The creation and annihilation operators satisfy the usual CCR [ak, al] = 0 , [ak, a†

l ] = δ(k l)

This models a large polaron, where the electron is spread over distances much larger than the lattice spacing. Note: Since k 7! |k|1 is not in L2(R3), Hα is not defined on the domain of H0. It can be defined as a quadratic form, however. Similar models of this kind appear in many places in physics, e.g., the Nelson model, spin-boson models, etc., and are used as toy models of quantum field theory.

  • R. Seiringer — The Polaron at Strong Coupling — August 21, 2019

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SLIDE 4

Strong Coupling Units

The Fr¨

  • hlich model allows for an “exact solution” in the strong coupling limit α ! 1.

Changing variables x ! α1x , ak ! α1/2aα−1k we obtain α2Hα ⇠ = hα := ∆ Z

R3

1 |k| ⇣ akeikx + a†

keikx⌘

dk + Z

R3 a† kak dk

where the CCR are now [ak, a†

l ] = α2δ(k l).

Hence α2 is an effective Planck constant and α ! 1 corresponds to a classical limit. The classical approximation amounts to replacing ak by a complex-valued function

  • zk. We write it as a Fourier transform

zk = Z

R3 (ϕ(x) + iπ(x)) eikxdk

  • R. Seiringer — The Polaron at Strong Coupling — August 21, 2019

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SLIDE 5

The Pekar Functional(s)

The classical approximation leads to the Pekar functional E(ψ, ϕ, π) = Z

R3 |rψ(x)|2dx 2

Z

R6

|ψ(x)|2ϕ(y) |x y|2 dx dy + Z

R3

  • ϕ(x)2 + π(x)2

dx Minimizing with respect to ϕ and π gives EP(ψ) = min

ϕ,π E(ψ, ϕ, π) =

Z

R3 |rψ(x)|2dx

Z

R6

|ψ(x)|2|ψ(y)|2 |x y| dx dy Lieb (1977) proved that there exists a minimizer of EP(ψ) (with kψk2 = 1) and it is unique up to translations and multiplication by a phase. In particular, the classical approximation leads to self-trapping of the electron due to its interaction with the polarization field. Let eP < 0 denote the Pekar energy eP = min

kψk2=1 EP(ψ)

  • R. Seiringer — The Polaron at Strong Coupling — August 21, 2019

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SLIDE 6

Asymptotics of the Ground State Energy

Donsker and Varadhan (1983) proved the validity of the Pekar approximation for the ground state energy: lim

α!1 inf spec hα = eP

They used the (Feynman 1955) path integral formulation of the problem, leading to a study of the path measure exp α Z

R

dse|s| 2 Z T dt |ω(t) ω(t + s)| ! dWT (ω) as T ! 1, where WT denotes the Wiener measure of closed paths of length T. Lieb and Thomas (1997) used operator techniques to obtain the quantitative bound eP inf spec hα eP O(α1/5) for large α. Note that the upper bound follows from a simple product ansatz.

  • R. Seiringer — The Polaron at Strong Coupling — August 21, 2019

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SLIDE 7

Quantum Fluctuations

What is the leading order correction of inf spec hα compared to eP? With FP(ϕ) = min

ψ,π E(ψ, ϕ, π) = inf spec

  • ∆ 2ϕ ⇤ |x|2

+ Z

R3 ϕ(x)2dx

we expand around a minimizer ϕP FP(ϕ) ⇡ eP + hϕ ϕP|HP|ϕ ϕPi + O(kϕ ϕPk3

2)

with HP the Hessian at ϕP. We have 0  HP  1, and HP has exactly 3 zero-modes due to translation invariance (Lenzmann 2009). Reintroducing the field momentum and studying the resulting system of harmonic oscilla- tors leads to the conjecture inf spec hα = eP + 1 2α2 Tr ⇣p HP 1 ⌘ + o(α2) predicted in the physics literature (Allcock 1963).

  • R. Seiringer — The Polaron at Strong Coupling — August 21, 2019

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SLIDE 8

A Theorem for a Confined Polaron

Allcock’s conjecture was recently proved for a confined polaron with Hamiltonian hα,Ω = ∆Ω Z

Ω

(∆Ω)1/2(x, y)

  • ay + a†

y

  • dy +

Z

Ω

a†

yay dy

for (nice) bounded sets Ω ⇢ R3. Assuming coercivity of the corresponding Pekar functional EP

Ω(ψ) =

Z

Ω

|rψ(x)|2dx Z

Ω2 |ψ(x)|2(∆Ω)1(x, y)|ψ(y)|2dx dy

i.e., EP

Ω(ψ) EP Ω(ψP Ω) + KΩ min θ

Z

Ω

|r(ψ(x) eiθψP

Ω(x))|2dx

for some KΩ > 0 (which can be proved for Ω a ball [FeliciangeliS19]), one has Theorem [FrankS19]: As α ! 1 inf spec hα,Ω = eP

Ω +

1 2α2 Tr ✓q HP

Ω 1

◆ + o(α2)

  • R. Seiringer — The Polaron at Strong Coupling — August 21, 2019

# 8

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SLIDE 9

Effective Mass

The Fr¨

  • hlich Hamiltonian Hα is translation invariant and commutes with the total

momentum P = irx + Z

R3 k a† kak dk

Hence there is a fiber-integral decomposition H = R

R3 HP α dP. In fact,

HP

α ⇠

= ✓ P Z

R3 k a† kak dk

◆2 pα Z

R3

1 |k| ⇣ ak + a†

k

⌘ dk + Z

R3 a† kak dk

(acting on F only). With Eα(P) = inf spec HP

α , the effective mass m 1/2 is defined

as 1 m := 2 lim

P !0

Eα(P) Eα(0) |P|2 A simple argument based on the Pekar approximation suggests m ⇠ α4 as α ! 1. The best rigorous result so far is Theorem [LiebS19]: lim

α!1 m = 1

  • R. Seiringer — The Polaron at Strong Coupling — August 21, 2019

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SLIDE 10

Summary and Open Problems

We derived the quantum corrections to the (classical) Pekar asymptotics of the ground state energy of a confined polaron, and showed that the polaron’s effective mass diverges in the strong coupling limit. Many open problems remain:

  • Quantum corrections to the Pekar approximation in the unconfined case Ω = R3
  • Divergence rate of the effective mass, conjectured to satisfy

lim

α!1

m α4 = 8π 3 Z

R3 |ψPek(x)|4dx

  • The Pekar approximation can also be applied in a dynamic setting. It should be

possible to derive the corresponding time-dependent Pekar equations from the Schr¨

  • dinger equation with the Fr¨
  • hlich Hamiltonian.

Recent partial results by Frank & Schlein, Frank & Gang and Griesemer, as well as [Leopold,Rademacher,Schlein,S,2019].

  • R. Seiringer — The Polaron at Strong Coupling — August 21, 2019

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SLIDE 11

Ideas in the Proof

Theorem [FrankS19]: As α ! 1 inf spec hα,Ω = eP

Ω +

1 2α2 Tr ✓q HP

Ω 1

◆ + o(α2)

  • electron in instantaneous ground state of potential generated by (fluctuating) field
  • ϕ 62 L2, hence not close to ϕP; need ultraviolet cutoff Λ
  • quantify effect of cutoff using commutator method of [Lieb, Yamazaki, 1958]:

Z

|k|>Λ

eikx |k| akdk = " irx, Z

|k|>Λ

keikx |k|3 akdk #

  • we apply, in fact, three commutators, and a Gross transformation, to conclude

that the ground state energy is affected at most by Λ5/2

  • IMS localization in Fock space, use Hessian close to ϕP + global coercivity
  • R. Seiringer — The Polaron at Strong Coupling — August 21, 2019

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SLIDE 12

Ideas in the Proof

Theorem [LiebS19]: lim

α!1 m = 1

  • decomposing the Pekar product ansatz into fibers suggests for the fiber ground

states ΦP ΦP ⇡ ˆ ψPek

α

(P Pf)ea†(ϕPek

α

)|Ωi ⇡ Φ0 + P · r ˆ

ψPek

α

(Pf) ˆ ψPek

α

(Pf) Φ0

  • use this as a trial state for HP , with Φ0 the actual ground state of H0, yielding

1 2m  1 + hΦ0|Oα|Φ0i for some explicit operator Oα built from Pf and H0.

  • Prove that limα!1hΦ0|Oα|Φ0i = 1 by suitably perturbing H and redoing the

Lieb-Thomas proof with perturbation terms.

  • R. Seiringer — The Polaron at Strong Coupling — August 21, 2019

# 12