Anderson Localization from Classical Trajectories Piet Brouwer - PowerPoint PPT Presentation

Anderson Localization from Classical Trajectories Piet Brouwer Laboratory of Atomic and Solid State Physics Cornell University With: Alexander Altland (Cologne) Support: NSF, Packard Foundation

Quantum Transport Manifestations of the wave nature of electrons in electrical transport “cavity” “antidot lattice” • shot noise • weak localization • conductance fluctuations … 1 μ m • Anderson localization Originally discovered for sample sample disordered conductors. This talk: ballistic conductors no scattering off point-like impurities I

Weak localization disordered metals Nonzero (negative) ensemble average δ G at zero magnetic field G [ e 2 / h ] | A µ | 2 + � � A µ A ∗ G = ν µ µ � = ν δG B [10 -4 T] Mailly and Sanquer (1991) “Hikami box” in Hik out Hik = + + “Cooperon”

Weak localization disordered metals Nonzero (negative) ensemble average δ G at zero magnetic field G [ e 2 / h ] | A µ | 2 + � � A µ A ∗ G = = ν µ µ � = ν δG B [10 -4 T] ν Hik = + Hik μ + permutations ‘Hikami box’ “Cooperon”

Weak localization ballistic conductors • Theory based on diffractive scattering off point-like impurities not possible; Hik = + … “ballistic” “disordered”

Weak localization ballistic conductors • Theory based on diffractive α scattering off point-like β impurities not possible; Instead: Semiclassics g = � A α A β e i ( S α −S β ) / � , α,β • α and β have equal angles upon entrance/exit • S α,β : classical action • A α,β : stability amplitudes Jalabert, Baranger, Stone (1990) Needed: Careful summation over Argaman (1995) classical trajectories α , β . Aleiner, Larkin (1996) Richter, Sieber (2001,2002) Heusler, Müller, Braun, Haake (2006)

Weak localization ballistic conductors g � A α A β e i ( S α −S β ) / � , ∼ in out α,β t enc = 1 λ ln S cl Weak localization: Trajectory pairs | ∆ S| with small-angle self encounter Sieber, Richter (2001) α also: Aleiner, Larkin (1996) β Encounter duration t enc = τ E = 1 λ ln S cl � If τ E << dwell time: Recover weak localization correction of disordered metal “ballistic Hikami box” Aleiner, Larkin (1996) Richter, Sieber (2002) Heusler et al . (2006) Brouwer (2007)

Beyond weak localization ballistic conductors g � A α A β e i ( S α −S β ) / � , ∼ α,β One or more small-angle self encounters • shot noise • conductance fluctuations Braun et al. (2006) • quantum pump • full counting statistics Braun et al. (2006) Whitney and Jacquod (2006) • time delay Brouwer and Rahav (2006) • … Rahav and Brouwer (2006) Berkolaiko et al. (2007) Kuipers and Sieber (2007) … If τ E << dwell time: Recover quantum corrections of disordered metals

Beyond weak localization ballistic conductors g � A α A β e i ( S α −S β ) / � , ∼ α,β One or more small-angle self encounter • shot noise • conductance fluctuations Braun et al. (2006) • quantum pump • full counting statistics Braun et al. (2006) Whitney and Jacquod (2006) • time delay Brouwer and Rahav (2006) • … Rahav and Brouwer (2006) Berkolaiko et al. (2007) Kuipers and Sieber (2007) If τ E << dwell time: Recover quantum But all of these are corrections of disordered metals perturbative effects!

Non-perturbative effects Level correlations: Form factor K ( t ) for | t | > τ H Heusler, Müller, Altland, Braun, Haake (2007) “inspired by field theoretical formulation of RMT correlation functions” Heusler et al. (2007)

Non-perturbative effects Level correlations: Form factor K ( t ) for | t | > τ H Heusler, Müller, Altland, Braun, Haake (2007) “inspired by field theoretical formulation of RMT correlation functions” Heusler et al. (2007) Today: Anderson localization … inspired by theory of Anderson localization in disordered metals • one-dimensional nonlinear sigma model Efetov and Larkin (1983) • scaling approach Dorokhov (1982) Mello, Pereyra, Kumar (1988)

Anderson localization disordered metals Model system: array of “quantum dots” Dots are connected via ballistic contacts with conductance g c >> 1. → ∞ Take limit g c while keeping ratio g c / n fixed. Disordered quantum dots: random matrix theory Localization in quantum dot array: Mirlin, Müller-Groeling, Zirnbauer (1994) Brouwer, Frahm (1996)

Anderson localization disordered metals � � S 11 ( n ) S 12 ( n ) S ( n ) = S 21 ( n ) S 22 ( n ) interdot conductance: g c T ( n ) = S 12 ( n ) S † 12 ( n ) T m ( n ) = tr T ( n ) m : conductance of array of n dots g ( n ) = T 1 ( n ) random matrix theory: recursion relation for moments of the T i : δ � T 1 � = � T 1 ( n ) � − � T 1 ( n − 1) � (no time-reversal symmetry, = − 1 β =2) � T 1 ( n − 1) 2 � + O ( g − 2 c ) g c Replace difference equation by differential equation: ∂L � T 1 � = − 2 ∂ ξ : “localization length” ξ � T 2 1 � L/ξ = n/ 2 g c

Anderson localization disordered metals � S 11 ( n ) S 12 ( n ) � S ( n ) = S 21 ( n ) S 22 ( n ) T ( n ) = S 12 ( n ) S † 12 ( n ) T interdot conductance: g c T m ( n ) = tr T ( n ) m general recursion relation: � n � � n � � n � = − 1 � � � i k T 1 T i m δ T i m g c k =1 m =1 m =1 i k − 1 n � n � + 1 � � � i k ( T j ( T i k − j − T i k − j +1 )) T i m g c k =1 j =1 m =1 m � = k n k − 1 � n � + 2 � � � + O ( g − 2 i k i l ( T i k + i l − T i k + i l +1 ) T i m c ) g c k =1 l =1 m =1 m � = k,l

Anderson localization disordered metals � S 11 ( n ) S 12 ( n ) � S ( n ) = S 21 ( n ) S 22 ( n ) T ( n ) = S 12 ( n ) S † 12 ( n ) T T m ( n ) = tr T ( n ) m � n � � n � interdot conductance: g c � n � = − 1 � � � i k T 1 T i m δ T i m g c k =1 m =1 m =1 n i k − 1 � n � + 1 � � � i k ( T j ( T i k − j − T i k − j +1 )) T i m g c k =1 j =1 m =1 m � = k n k − 1 � n � + 2 � � � + O ( g − 2 i k i l ( T i k + i l − T i k + i l +1 ) T i m c ) g c k =1 l =1 m =1 Transform into differential equation for generating function: � 2 + (cos( θ 3 ) − 1) T � �� F 2 ( θ 1 , θ 3 ) = det Description equivalent to 2 + (cosh( θ 1 ) − 1) T existing theory of localization ∂LF 2 = 2 ∂ 1 ∂ J ( θ 1 , θ 3 ) ∂ � F 2 , in quantum wires ξ J ( θ 1 , θ 3 ) ∂θ j ∂θ j j =1 , 3 Efetov and Larkin (1983) sin( θ 3 ) sinh( θ 1 ) J ( θ 1 , θ 3 ) = (cosh( θ 1 ) − cos( θ 3 )) 2 . Dorokhov (1982) Mello, Pereyra, Kumar (1988)

Anderson localization disordered metals � S 11 ( n ) S 12 ( n ) � S ( n ) = S 21 ( n ) S 22 ( n ) T ( n ) = S 12 ( n ) S † 12 ( n ) T T m ( n ) = tr T ( n ) m � n � � n � interdot conductance: g c � n � = − 1 � � � i k T 1 T i m δ T i m g c k =1 m =1 m =1 n i k − 1 � n � + 1 � � � i k ( T j ( T i k − j − T i k − j +1 )) T i m g c k =1 j =1 m =1 m � = k n k − 1 � n � + 2 � � � + O ( g − 2 i k i l ( T i k + i l − T i k + i l +1 ) T i m c ) g c k =1 l =1 m =1 Can one derive the same set of recursion relations from semiclassics?

Anderson localization ballistic conductors � � S 11 ( n ) S 12 ( n ) S ( n ) = S 21 ( n ) S 22 ( n ) T ( n ) = S 12 ( n ) S † 12 ( n ) interdot conductance: g c T m ( n ) = tr T ( n ) m Can we show that δ � T 1 � = � T 1 ( n ) � − � T 1 ( n − 1) � = − 1 � T 1 ( n − 1) 2 � + O ( g − 2 c ) g c from semiclassical expression for T 1 ? � A α A β e i ( S α −S β ) / � T 1 = α,β

Anderson localization ballistic conductors first n -1 dots first n -1 dots n n � A α A β e i ( S α −S β ) / � T 1 = α α,β α=β m =1 β • α and β each have m segments in n th dot, α 1 ,…, α m ; β 1 ,…, β m . α α=β m =2 β α α=β β m =3

Anderson localization ballistic conductors first n -1 dots first n -1 dots n n � A α A β e i ( S α −S β ) / � T 1 = α α,β α=β α 1 = β 1 α 1 = β 1 m =1 β • α and β each have m segments in n th dot, α 1 = β 1 α 1 ,…, α m ; β 1 ,…, β m . α 1 = β 1 α α=β m =2 β To leading order in g c : α 2 = β 2 • diagonal approximation in α 2 = β 2 n th dot α 2 = β 2 α 2 = β 2 • pair α i with β i , i =1,…, m α α=β α 1 = β 1 β m =3 α 3 = β 3 No restriction on number of α 3 = β 3 α 1 = β 1 small-angle self encounters in first n -1 dots

Anderson localization ballistic conductors first n -1 dots first n -1 dots n n α α=β 1 α 1 = β 1 α 1 = β 1 m =1 β 2 � T 1 ( n − 1) � α 1 = β 1 α 1 = β 1 α α=β 1 m =2 β � T 1 ( n − 1) R 1 ( n − 1) � 4 g c α 2 = β 2 α 2 = β 2 reflection: R 1 = g c − T 1 α 2 = β 2 α 2 = β 2 α α=β α 1 = β 1 1 β � T 1 ( n − 1) R 1 ( n − 1) 2 � m =3 8 g 2 c α 3 = β 3 α 3 = β 3 α 1 = β 1

Anderson localization ballistic conductors α α=β 1 β m =1 2 � T 1 ( n − 1) � α α=β 1 β m =2 � T 1 ( n − 1) R 1 ( n − 1) � 4 g c reflection: R 1 = g c − T 1 α α=β 1 β � T 1 ( n − 1) R 1 ( n − 1) 2 � m =3 8 g 2 c … + ∞ 1 � T 1 ( n − 1) R 1 ( n − 1) m − 1 � � � T 1 ( n ) � = 2 m g m − 1 c m =1 � g c T 1 ( n − 1) � = 2 g c − R 1 ( n − 1) − 1 � T 1 ( n − 1) 2 � + O ( g − 2 = � T 1 ( n − 1) � c ) g c