Pekar s Ansatz and the Ground-State Symmetry of a Bound Polaron

Transcription

1 Pekar s Ansatz and the Ground-State Symmetry of a Bound Polaron Rohan Ghanta Georgia Institute of Technology March 9, 018

2 The Polaron Model In 1937 H. Fröhlich proposed a model known today as the Fröhlich polaron of an electron interacting with the quantized optical modes (phonons) of an ionic crystal (Proc. R. Soc. Lond. A, 160 (1937)). We consider a Fröhlich polaron bound in an external electric potential (impurity in a crystal). The Hilbert space associated with the system is H = L (R 3 ) F, where F := n 0 n s L (R 3 ) is a symmetric (phonon) Fock space over L (R 3 ).

3 Fröhlich Hamiltonian It is described by the Hamiltonian H V a = p a V (ax)+ a R 3 k a kdk p a (p) 3/ R3 h a k e ik x + a k e ik xi dk k p = i x is the electron momentum V L 3/ (R 3 )+L (R 3 ) is nonnegative and vanishes at infinity a k and a k are creation and annihlation operators on F satisfying [a k,a k 0 ]=d(k k 0 ) a > 0 is the electron-phonon coupling parameter

4 The Ground State The Fröhlich Hamiltonian is H V a = p a V (ax)+ a R 3 k a kdk p a (p) 3/ R3 h a k e ik x + a k e ik xi dk k The ground-state energy is E F a (V )= inf k k H =1, H V a A normalized function in H that achieves the ground-state energy is called a ground-state wave function.

5 Existence and Uniqueness of a Ground State The Fröhlich Hamiltonian is H V a = p a V (ax)+ a R 3 k a kdk p a (p) 3/ R3 h a k e ik x + a k e ik xi dk k Fix a > 0. Suppose the external potential satisfies the following two conditions: 1 p a V (ax) has a negative energy bound state in L (R 3 ),and a V (ax) is relatively form-bounded with respect to p with form bound strictly less than one. Then there exists a unique ground-state wave function V a H.

6 Pekar s Produkt-Ansatz The Fröhlich Hamiltonian is H V a = p a V (ax)+ a R 3 k a kdk E F a (V )= inf k k H =1 p a (p) 3/, H V a R3 h a k e ik x + a k e ik xi dk k The electron-phonon interaction term in the Hamiltonian makes it di to calculate the ground-state energy of the polaron. S.I. Pekar s Ansatz (Issledovaniya po Elektronnoi Teorii Kirstallov. Gostekhizdat, Moskow, 1951) When a is large, the ground-state wave function is a product of a normalized electronic wave function in L (R 3 ) and a normalized coherent state in Fock space: (x, ):=f(x) ( ). cult

7 Pekar s Produkt-Ansatz S.I. Pekar s Ansatz (Issledovaniya po Elektronnoi Teorii Kirstallov. Gostekhizdat, Moskow, 1951) When a is large, the ground-state wave function is a product of a normalized electronic wave function in L (R 3 ) and a normalized coherent state in Fock space: (x, ):=f(x) ( ). = exp k! z(k) + z(k)a k 0i 0i is the vacuum in F, z(k) L (R 3 ). a k =z(k) and h, i F = 1.

8 Pekar s Produkt-Ansatz Note: = exp k Writing rˆ f (k)= 1 z(k) (p) 3/ R + z(k)a k 0i and a k =z(k). R eik x 3 f dx, wecalculatewithpekar sansatz: D E f, Ha V f = R R 3 f dx a R R 3 f V (ax)dx " + dk z(k) p ap rˆ f (k) z(k) R 3 k p ap rˆ # f (k) z(k) k = R R 3 f dx a R R 3 f V (ax)dx 0 z(k) p ap rˆ f (k) k R 3 4pa 1 rˆ f (k) Adk k

9 Pekar s Produkt-Ansatz Note: = exp k Writing rˆ f (k)= 1 z(k) (p) 3/ R + z(k)a k 0i and a k =z(k). R eik x 3 f dx, wecalculatewithpekar sansatz: D E f, Ha V f = R R 3 f dx a R R 3 f V (ax)dx " + dk z(k) p ap rˆ f (k) z(k) R 3 k p ap rˆ # f (k) z(k) k = R R 3 f dx a R R 3 f V (ax)dx 0 z(k) p ap rˆ f (k) k R 3 4pa 1 rˆ f (k) Adk k

10 Pekar s Produkt-Ansatz It follows that E F a (V ) apple inf kfk =k k F =1 D f = a e P (V ),, H V a f E where e P (V ) given by the Pekar minimization problem: inf kfk =1 8 < : e P (V ):= R 3 f f V (x) dx R 3 R 3 9 f(x) f(y) = dx dy x y ;. P.L. Lions showed that the Pekar minimization problem admits a minimizer (Ann. Inst. H. Poin. Non-Linear. 1, 1984).

11 Pekar s Ground State Pekar s Product State Let j V (x) be a minimizer for the Pekar energy e P (V ). Fora large, the ground-state wave function is a 3 jv (ax) V a, where and V a = exp k za V (k)= p ap k (p) 3/ za V (k)! + za V (k)a k 0i R 3 e ik x a 3 jv (ax) dx.

12 Exact Ground-State Energy in the Strong-Coupling Limit Recall that Ea F (V ) apple a e P (V ),where 8 < e P (V )= inf kfk =1: R 3 f f V (x) dx 9 f(x) f(y) = dx dy x y ;. R 3 R 3 However, it was shown by M.D. Donsker and S.R.S. Varadhan that Pekar s upper bound is in fact exact in the strong-coupling limit (Comm. Pure. Appl. Math. 36, 1983): Ground-State Energy Ea F (V ) lim a! a = e P (V ). A simple proof was later given by E.H. Lieb and L.E. Thomas (Comm. Math. Phys. 183, 1995).

13 What about the Ground-State Wave Function? Pekar s Product State Let j V (x) be a minimizer for the Pekar energy e P (V ). Fora large, the ground-state wave function is a 3 jv (ax) V a, where and V a = exp k za V (k)= p ap k (p) 3/ za V (k)! + za V (k)a k 0i R 3 e ik x a 3 jv (ax) dx.

14 If Pekar Minimizer is Unique... Adapting an argument of E.H. Lieb and B. Simon in Thomas-Fermi theory (Adv. Math. 3(1), 1977), we know: Convergence of Electron Density Suppose the Pekar minimization problem for the energy e P (V ) admits a unique minimizer j V.ThenforallW L 3/ (R 3 )+L (R 3 ), But... lim 1 a! a 3 R 3 k V x a k F W (x)dx = j V (x) W (x)dx. a R 3 What if the minimizers for the Pekar energy e P (V ) are not unique?

15 A Discrepancy in Spherical Symmetry Mexican Hat-type Potential Let R >. Consider the potential V R C c (R 3 ),where0applev R apple 1and 8 < 0 when x apple 1 V R (x)= 1 when apple x apple R. : 0 when x R + 1 For all a > 0andR >, the ground-state is unique and its electron density k (x) is therefore a radial function! V R a k F Nonradiality of Pekar Minimizers For R large, the Pekar problem for the energy e P (V R ) admits only nonradial minimizers. V R a

16 A Discrepancy in Spherical Symmetry Nonradiality of Pekar Minimizers (RG) For R large, the Pekar problem for the energy e P (V R ) admits only nonradial minimizers. If j VR (x) is a minimizer, then j VR (Rx) is also a minimizer for all R SO(3). The radial ground-state electron density k converge weakly to a nonradial minmimizer j VR (x). V R a k F (x) cannot

17 Assuming the Minimizers are Unique up to Rotation... Convergence to Rotational Average Let R be large enough so that the minimization problem for e P (V R ) admits only nonradial minimizers, and assume that the nonradial minimizer j VR (x) is unique up to rotation. Let g be the Haar measure on SO(3). Then,forallW L 3/ (R 3 )+L (R 3 ) 1 V lim a! a 3 k R x apple R 3 a k F W (x)dx = a R 3 SO(3) j VR (Rx) dg(r) W (x)dx