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

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

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 # 2

The Fr¨ ohlich Model On L 2 ( R 3 ) ⌦ F (with F the bosonic Fock space over L 2 ( R 3 ) ), H α = � ∆ � p α Z 1 Z a k e ikx + a † ⇣ k e � ikx ⌘ R 3 a † dk + k a k dk | k | R 3 with α > 0 the coupling strength. The creation and annihilation operators satisfy the usual CCR [ a k , a † [ a k , a l ] = 0 , 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 L 2 ( R 3 ) , H α is not defined on the domain of H 0 . 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 # 3

Strong Coupling Units The Fr¨ ohlich model allows for an “exact solution” in the strong coupling limit α ! 1 . Changing variables x ! α � 1 x , a k ! α � 1 / 2 a α − 1 k we obtain Z 1 Z a k e ikx + a † ⇣ k e � ikx ⌘ α � 2 H α ⇠ R 3 a † = h α := � ∆ � dk + k a k dk | k | R 3 where the CCR are now [ a k , a † l ] = α � 2 δ ( k � l ) . Hence α � 2 is an e ff ective Planck constant and α ! 1 corresponds to a classical limit . The classical approximation amounts to replacing a k by a complex-valued function z k . We write it as a Fourier transform Z R 3 ( ϕ ( x ) + i π ( x )) e � ikx dk z k = R. Seiringer — The Polaron at Strong Coupling — August 21, 2019 # 4

The Pekar Functional(s) The classical approximation leads to the Pekar functional | ψ ( x ) | 2 ϕ ( y ) Z Z Z ϕ ( x ) 2 + π ( x ) 2 � R 3 | r ψ ( x ) | 2 dx � 2 � E ( ψ , ϕ , π ) = dx dy + dx | x � y | 2 R 6 R 3 Minimizing with respect to ϕ and π gives | ψ ( x ) | 2 | ψ ( y ) | 2 Z Z E P ( ψ ) = min R 3 | r ψ ( x ) | 2 dx � ϕ , π E ( ψ , ϕ , π ) = dx dy | x � y | R 6 Lieb (1977) proved that there exists a minimizer of E P ( ψ ) (with k ψ k 2 = 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 e P < 0 denote the Pekar energy e P = k ψ k 2 =1 E P ( ψ ) min R. Seiringer — The Polaron at Strong Coupling — August 21, 2019 # 5

Asymptotics of the Ground State Energy Donsker and Varadhan (1983) proved the validity of the Pekar approximation for the ground state energy: α !1 inf spec h α = e P lim They used the (Feynman 1955) path integral formulation of the problem, leading to a study of the path measure Z T ! dse � | s | Z dt d W T ( ω ) exp α 2 | ω ( t ) � ω ( t + s ) | 0 R as T ! 1 , where W T denotes the Wiener measure of closed paths of length T . Lieb and Thomas (1997) used operator techniques to obtain the quantitative bound e P � inf spec h α � e P � O ( α � 1 / 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 # 6

Quantum Fluctuations What is the leading order correction of inf spec h α compared to e P ? With Z F P ( ϕ ) = min � ∆ � 2 ϕ ⇤ | x | � 2 � R 3 ϕ ( x ) 2 dx � ψ , π E ( ψ , ϕ , π ) = inf spec + we expand around a minimizer ϕ P F P ( ϕ ) ⇡ e P + h ϕ � ϕ P | H P | ϕ � ϕ P i + O ( k ϕ � ϕ P k 3 2 ) with H P the Hessian at ϕ P . We have 0  H P  1 , and H P 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 ⇣ p 1 inf spec h α = e P + ⌘ H P � 1 + o ( α � 2 ) 2 α 2 Tr predicted in the physics literature (Allcock 1963). R. Seiringer — The Polaron at Strong Coupling — August 21, 2019 # 7

A Theorem for a Confined Polaron Allcock’s conjecture was recently proved for a confined polaron with Hamiltonian Z Z ( � ∆ Ω ) � 1 / 2 ( x, y ) a y + a † a † � � h α , Ω = � ∆ Ω � dy + y a y dy y Ω Ω for (nice) bounded sets Ω ⇢ R 3 . Assuming coercivity of the corresponding Pekar functional Z Z E P | r ψ ( x ) | 2 dx � Ω 2 | ψ ( x ) | 2 ( � ∆ Ω ) � 1 ( x, y ) | ψ ( y ) | 2 dx dy Ω ( ψ ) = Ω i.e., Z E P Ω ( ψ ) � E P Ω ( ψ P | r ( ψ ( x ) � e i θ ψ P Ω ( x )) | 2 dx Ω ) + K Ω min θ Ω for some K Ω > 0 (which can be proved for Ω a ball [FeliciangeliS19] ), one has Theorem [FrankS19] : As α ! 1 ✓q ◆ 1 inf spec h α , Ω = e P + o ( α � 2 ) H P Ω + 2 α 2 Tr Ω � 1 R. Seiringer — The Polaron at Strong Coupling — August 21, 2019 # 8

Effective Mass The Fr¨ ohlich Hamiltonian H α is translation invariant and commutes with the total momentum Z R 3 k a † P = � i r x + k a k dk R � R 3 H P Hence there is a fiber-integral decomposition H = α dP . In fact, ◆ 2 � p α ✓ 1 Z Z Z ⇣ ⌘ α ⇠ R 3 k a † a k + a † R 3 a † H P P � k a k dk dk + k a k dk = k | k | R 3 (acting on F only). With E α ( P ) = inf spec H P α , the e ff ective mass m � 1 / 2 is defined as 1 E α ( P ) � E α (0) m := 2 lim | P | 2 P ! 0 A simple argument based on the Pekar approximation suggests m ⇠ α 4 as α ! 1 . The best rigorous result so far is Theorem [LiebS19] : α !1 m = 1 lim R. Seiringer — The Polaron at Strong Coupling — August 21, 2019 # 9

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 e ff ective mass diverges in the strong coupling limit. Many open problems remain: • Quantum corrections to the Pekar approximation in the unconfined case Ω = R 3 • Divergence rate of the e ff ective mass, conjectured to satisfy α 4 = 8 π m Z R 3 | ψ Pek ( x ) | 4 dx lim 3 α !1 • 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¨ odinger equation with the Fr¨ ohlich 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 # 10

Ideas in the Proof Theorem [FrankS19] : As α ! 1 1 ✓q ◆ inf spec h α , Ω = e P + o ( α � 2 ) H P Ω + 2 α 2 Tr Ω � 1 • electron in instantaneous ground state of potential generated by (fluctuating) field • ϕ 62 L 2 , hence not close to ϕ P ; need ultraviolet cuto ff Λ • quantify e ff ect of cuto ff using commutator method of [Lieb, Yamazaki, 1958]: " # e ikx ke ikx Z Z | k | a k dk = � i r x , | k | 3 a k dk | k | > Λ | k | > Λ • we apply, in fact, three commutators, and a Gross transformation , to conclude that the ground state energy is a ff ected at most by Λ � 5 / 2 • IMS localization in Fock space, use Hessian close to ϕ P + global coercivity R. Seiringer — The Polaron at Strong Coupling — August 21, 2019 # 11

Ideas in the Proof Theorem [LiebS19] : α !1 m = 1 lim • decomposing the Pekar product ansatz into fibers suggests for the fiber ground states Φ P ) | Ω i ⇡ Φ 0 + P · r ˆ ψ Pek ( � P f ) ( P � P f ) e a † ( ϕ Pek Φ P ⇡ ˆ ψ Pek α Φ 0 α α ˆ ψ Pek ( � P f ) α • use this as a trial state for H P , with Φ 0 the actual ground state of H 0 , yielding 1 2 m  1 + h Φ 0 |O α | Φ 0 i for some explicit operator O α built from P f and H 0 . • Prove that lim α !1 h Φ 0 |O α | Φ 0 i = � 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