Anderson Localization from Classical Trajectories Piet Brouwer Laboratory of Atomic and Solid State Physics Cornell University Support: NSF, Packard Foundation With: Alexander Altland (Cologne)
Quantum Transport Manifestations of the wave nature of electrons in electrical transport shot noise weak localization conductance fluctuations Anderson localization Originally discovered for disordered conductors. This talk: no scattering off point-like impurities cavity I antidot lattice sample 1 μm
Weak localization disordered metals Nonzero (negative) ensemble average δ G at zero magnetic field G = μ A μ 2 + μ ν A μ A ν G [e 2 /h] δg Hikami box B [10-4 T] Mailly and Sanquer (1991) in Hik out Hik = + + Cooperon
Weak localization disordered metals Nonzero (negative) ensemble average δ G at zero magnetic field G = μ μ A μ 2 + μ ν A μ A ν ν Hik δg G [e 2 /h] Hik = + B [10-4 T] + permutations Hikami box Cooperon
Weak localization Theory based on diffractive scattering off point-like impurities not possible; Hik = + disordered ballistic
Weak localization Theory based on diffractive scattering off point-like impurities not possible; Instead: Semiclassics g = A A e i(s S )/,, and have equal angles upon entrance/exit S, : classical action A, : stability amplitudes Needed: Careful summation over classical trajectories,. Jalabert, Baranger, Stone (1990) Argaman (1995) Aleiner, Larkin (1996) Richter, Sieber (2001,2002) Heusler, Müller, Braun, Haake (2006)
Weak localization g, A A e i(s S )/, in out Weak localization: Trajectory pairs with small-angle self encounter Sieber, Richter (2001) also: Aleiner, Larkin (1996) t enc = 1 λ ln S cl ΔS Encounter duration t enc = τ E = 1 λ ln S cl If τ E << dwell time: Recover weak localization correction of disordered metal Aleiner, Larkin (1996) Richter, Sieber (2002) Heusler et al. (2006) Brouwer (2007) ballistic Hikami box
Beyond weak localization g, A A e i(s S )/, One or more small-angle self encounters shot noise conductance fluctuations quantum pump full counting statistics time delay Braun et al. (2006) Whitney and Jacquod (2006) 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 Braun et al. (2006)
Beyond weak localization g, A A e i(s S )/, One or more small-angle self encounter shot noise conductance fluctuations quantum pump full counting statistics time delay Braun et al. (2006) Whitney and Jacquod (2006) Brouwer and Rahav (2006) Rahav and Brouwer (2006) Berkolaiko et al. (2007) Kuipers and Sieber (2007) Braun et al. (2006) If τ E << dwell time: Recover quantum corrections of disordered metals But all of these are 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 Today: Anderson localization Heusler et al. (2007) inspired by theory of Anderson localization in disordered metals one-dimensional nonlinear sigma model scaling approach Efetov and Larkin (1983) 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)
S(n) = Anderson localization ( ) S11 (n) S 12 (n) S 21 (n) S 22 (n) disordered metals T (n) =S 12 (n)s 12 (n) interdot conductance: g c T m (n) =trt (n) m 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) = 1 g c T 1 (n 1) 2 + O(g 2 c ) Replace difference equation by differential equation: L T 1 = 2 ξ T 2 1 L/ξ = n/2g c : conductance of array of n dots ξ: localization length (no time-reversal symmetry, =2)
S(n) = Anderson localization disordered metals ( ) S11 (n) S 12 (n) S 21 (n) S 22 (n) T (n) =S 12 (n)s 12 (n) T T m (n) =trt (n) m interdot conductance: g c general recursion relation: n δ T im ( n = 1 g c k=1 + 1 n g c k=1 i k ) i k 1 j=1 T 1 n T im i k (T j (T ik j T ik j+1)) + 2 n k 1 i k i l (T ik +i g l T ik +i l +1) c k=1 l=1 n T im m k n m k,l T im + O(gc 2 )
S(n) = ( ) S11 (n) S 12 (n) S 21 (n) S 22 (n) Anderson localization disordered metals T (n) =S 12 (n)s 12 (n) T T m (n) =trt (n) m n ( n ) δ T im = 1 n i k T 1 T im g c k=1 + 1 n ik 1 n i k (T j (T ik j T ik j+1)) T im g c Transform into differential equation for generating function: F 2 (θ 1,θ 3 ) = L F 2 k=1 j=1 = 2 ξ J(θ 1,θ 3 )= + 2 n k 1 i k i l (T g T ik+il ik+il+1) c k=1 l=1 det j=1,3 m k n T im + O(g 2 c ) ( 2+(cos(θ3 ) 1)T 2+(cosh(θ 1 ) 1)T ) 1 J(θ 1,θ 3 ) F 2, J(θ 1,θ 3 ) θ j θ j sin(θ 3 )sinh(θ 1 ) (cosh(θ 1 ) cos(θ 3 )) 2. interdot conductance: g c Description equivalent to existing theory of localization in quantum wires Efetov and Larkin (1983) Dorokhov (1982) Mello, Pereyra, Kumar (1988)
S(n) = ( ) S11 (n) S 12 (n) S 21 (n) S 22 (n) Anderson localization disordered metals T (n) =S 12 (n)s 12 (n) T T m (n) =trt (n) m n ( n ) δ T im = 1 n i k T 1 T im g c k=1 + 1 n ik 1 n i k (T j (T ik j T ik j+1)) T im g c k=1 j=1 + 2 n k 1 i k i l (T g T ik+il ik+il+1) c k=1 l=1 m k n T im + O(g 2 c ) interdot conductance: g c Can one derive the same set of recursion relations from semiclassics?
S(n) = Anderson localization ( S11 (n) S 12 (n) S 21 (n) S 22 (n) ) T (n) =S 12 (n)s 12 (n) interdot conductance: g c T m (n) =trt (n) m Can we show that δ T 1 = T 1 (n) T 1 (n 1) = 1 g c T 1 (n 1) 2 + O(g 2 c ) from semiclassical expression for T 1? T 1 =, A A e i(s S )/
Anderson localization T 1 =, A A e i(s S )/ and each have m segments in n th dot, 1,, m ; 1,, m. first n-1 dots n first n-1 dots n = = m=2 = m=3
Anderson localization T 1 =, A A e i(s S )/ and each have m segments in n th dot, 1,, m ; 1,, m. To leading order in g c : diagonal approximation in n th dot pair i with i, i=1,,m No restriction on number of small-angle self encounters in first n-1 dots first n-1 dots n first n-1 dots n = = = 1 = 1 1 = 1 2 = 2 2 = 2 1 = 1 2 = 2 2 = 2 1 = 1 3 = 3 1 = 3 = 3 1 1 = 1 m=2 m=3
Anderson localization 1 2 T 1(n 1) first n-1 dots n first n-1 dots n = 1 = 1 1 = 1 1 4g c T 1 (n 1)R 1 (n 1) = 1 = 1 1 = 1 m=2 reflection: R 1 = g c T 1 1 8g 2 c T 1 (n 1)R 1 (n 1) 2 = 2 = 2 2 = 2 2 = 2 2 = 2 1 = 1 3 = 3 3 = 3 m=3 1 = 1
Anderson localization 1 2 T 1(n 1) = reflection: R 1 = g c T 1 1 4g c T 1 (n 1)R 1 (n 1) 1 T 8g 2 1 (n 1)R 1 (n 1) 2 c 1 T 1 (n) = T1 (n 1)R 2 m g m 1 1 (n 1) m 1 c gc T 1 (n 1) = 2g c R 1 (n 1) + = = m=2 m=3 = T 1 (n 1) 1 g c T 1 (n 1) 2 + O(g 2 c )
Anderson localization Beyond diagonal approximation in n th dot: contribution of order g c -2 first n-1 dots n m=2 1 tr S 8gc 2 12 S 22 S 22 S 12 = 1 T 8gc 2 2 (n 1) T 1 (n 1) Pairing i with j, i = j: contribution of order g c -2 1 8g 2 c tr S 12 (S 22S 22 ) 2 S 12 = = 1 T 8gc 2 1 (n 1) 2T 2 (n 1) + T 3 (n 1) 2 = 1 3 = 3 1 = 2 m=3
Summarizing Anderson localization δ T 1 = T 1 (n) T 1 (n 1) = 1 g c T 1 (n 1) 2 + O(g 2 c ) interdot conductance: g c Extension to higher moments or =1 (time-reversal symmetry): Need to consider up to one encounter in n th dot; Need to go (slightly) beyond pairing i with i, i=1,,m.
Anderson localization Summarizing δ T 1 = T 1 (n) T 1 (n 1) = 1 g c T 1 (n 1) 2 + O(g 2 c ) interdot conductance: g c Extension to higher moments or =1 (time-reversal symmetry): Need to consider up to one encounter in n th dot; Need to go (slightly) beyond pairing i with i, i=1,,m.
Summarizing Anderson localization δ T 1 = T 1 (n) T 1 (n 1) = 1 g c T 1 (n 1) 2 + O(g 2 c ) interdot conductance: g c Extension to higher moments and =1 (time-reversal symmetry): Need to consider up to one encounter in n th dot; Need to go (slightly) beyond pairing i with i, i=1,,m. full set of recursion relations n ( = 1 n n n ) δ T δ,1 i k T ik +1 T im 1 im i k g c g k=1 c k=1 m k + 1 n i k 1 n i k (T j (T ik j T ik j+1)) T im g c + 4 g c k=1 j=1 n k 1 i k i l (T ik +i l T ik +i l +1) k=1 l=1 m k n m k,l T 1 n T im T im + O(gc 2 ),
Summarizing Anderson localization δ T 1 = T 1 (n) T 1 (n 1) = 1 g c T 1 (n 1) 2 + O(g 2 c ) interdot conductance: g c Extension to higher moments or =1 (time-reversal symmetry): Theory of Anderson localization in array of Need to consider up to one encounter in n th dot; ballistic Need to go chaotic (slightly) beyond cavities, pairing formulated i with i, i=1,,m. in terms of classical trajectories only. full set of recursion relations from Brouwer classical and trajectories Altland, only arxiv:0802.0976 n ( = 1 n n n ) δ T δ,1 i k T ik +1 T im 1 im i k g c g k=1 c k=1 m k + 1 n i k 1 n i k (T j (T ik j T ik j+1)) T im g c + 4 g c k=1 j=1 n k 1 i k i l (T ik +i l T ik +i l +1) k=1 l=1 m k n m k,l T 1 n T im T im + O(gc 2 ),