Thouless conductance, Landauer-Büttiker currents and spectrum

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1 , Landauer-Büttiker currents and spectrum (Collaboration with V. Jakšić, Y. Last and C.-A. Pillet) L. Bruneau Univ. Cergy-Pontoise Marseille - June 13th, 2014 L. Bruneau, Landauer-Büttiker currents and spectrum

2 Plan L. Bruneau, Landauer-Büttiker currents and spectrum

3 Introduced by Edwards and Thouless in 1972 as a criterion for localization in disordered (finite) systems (scaling theory of localization). Idea: if a state is localized (not too close to the boundary) it should be insensitive to boundary conditions (B.C.). L. Bruneau, Landauer-Büttiker currents and spectrum

4 Introduced by Edwards and Thouless in 1972 as a criterion for localization in disordered (finite) systems (scaling theory of localization). Idea: if a state is localized (not too close to the boundary) it should be insensitive to boundary conditions (B.C.). (or parameter) is defined as : g T := δe E where δe is a measure of sensitivity to B.C. and E the level spacing of the system (at the Fermi energy where conduction takes place). L. Bruneau, Landauer-Büttiker currents and spectrum

5 g T as a (dimensionless) conductance If the material as conductivity and if the size of the system is L >> l atom, the material exhibits diffusive behavior. L. Bruneau, Landauer-Büttiker currents and spectrum

6 g T as a (dimensionless) conductance If the material as conductivity and if the size of the system is L >> l atom, the material exhibits diffusive behavior. The uncertainty in energy is related to the time t = L2 D from one end to the other : δe t. needed to diffuse L. Bruneau, Landauer-Büttiker currents and spectrum

7 g T as a (dimensionless) conductance If the material as conductivity and if the size of the system is L >> l atom, the material exhibits diffusive behavior. The uncertainty in energy is related to the time t = L2 D from one end to the other : δe t. needed to diffuse Einstein relation gives σ = e 2 D dn, with σ the conductivity and de dn de = 1 V E the density of states (per unit volume and unit energy). L. Bruneau, Landauer-Büttiker currents and spectrum

8 g T as a (dimensionless) conductance If the material as conductivity and if the size of the system is L >> l atom, the material exhibits diffusive behavior. The uncertainty in energy is related to the time t = L2 D from one end to the other : δe t. needed to diffuse Einstein relation gives σ = e 2 D dn, with σ the conductivity and de dn de = 1 V E the density of states (per unit volume and unit energy).the conductance is thus (A = cross-sectional arera, i.e. V = A L) G = σa L e2 g T, with equality iff the uncertainty relation is an equality. Even if it is heuristic, g T is widely accepted as a measure of conductance. L. Bruneau, Landauer-Büttiker currents and spectrum

9 A mathematical definition for g T Main issue: define δe L. Bruneau, Landauer-Büttiker currents and spectrum

10 A mathematical definition for g T Main issue: define δe Consider a finite system described by a Jacobi matrix (h(k)ψ)(x) = J x ψ(x + 1) + λ x ψ(x) + J x 1 ψ(x 1) from x = 1 to x = L with Bloch type B.C. ψ(l + 1) = e ikl ψ(1), ψ(0) = e ikl ψ(l). L. Bruneau, Landauer-Büttiker currents and spectrum

11 A mathematical definition for g T Main issue: define δe Consider a finite system described by a Jacobi matrix (h(k)ψ)(x) = J x ψ(x + 1) + λ x ψ(x) + J x 1 ψ(x 1) from x = 1 to x = L with Bloch type B.C. ψ(l + 1) = e ikl ψ(1), ψ(0) = e ikl ψ(l). Various definitions have been proposed for δe, the variation of energy level from periodic to antiperiodic B.C., i.e. from E(0) to E( π L ) where E(k) is an eigenvalue of h(k), the curvature 1 d 2 E L 2 dk 2 of an eigenvalue at the edge of a band. k=0 L. Bruneau, Landauer-Büttiker currents and spectrum

12 A mathematical definition for g T How to define δe? We shall consider the first proposal. L. Bruneau, Landauer-Büttiker currents and spectrum

13 A mathematical definition for g T How to define δe? We shall consider the first proposal. Let h per be the periodic extension of the sample to l 2 (Z). Its spectrum consists in L bands, each corresponding to an eigenvalue E(k) of h(k) as k varies from 0 (periodic) to π/l (antiperiodic). L. Bruneau, Landauer-Büttiker currents and spectrum

14 A mathematical definition for g T How to define δe? We shall consider the first proposal. Let h per be the periodic extension of the sample to l 2 (Z). Its spectrum consists in L bands, each corresponding to an eigenvalue E(k) of h(k) as k varies from 0 (periodic) to π/l (antiperiodic). We fix an interval I (e.g. small and around the Fermi energy) and consider algebraic averages within I E = I number of levels in I, δe = sp(h per ) I number of levels in I. L. Bruneau, Landauer-Büttiker currents and spectrum

15 A mathematical definition for g T How to define δe? We shall consider the first proposal. Let h per be the periodic extension of the sample to l 2 (Z). Its spectrum consists in L bands, each corresponding to an eigenvalue E(k) of h(k) as k varies from 0 (periodic) to π/l (antiperiodic). We fix an interval I (e.g. small and around the Fermi energy) and consider algebraic averages within I E = I number of levels in I, δe = sp(h per ) I number of levels in I. This leads to the following Definition The for the energy interval I is g T (I ) = sp(h per) I. I L. Bruneau, Landauer-Büttiker currents and spectrum

16 g T as a (dimensionless) conductance II Let I = (µ l, µ r ) and assume the sample is connected to a left and right reservoir, at zero temperature and chemical potentials µ l and µ r. e T (I ) := sp(h per ) I, called Thouless energy, is identified with the current through the sample driven by the voltage difference between these reservoirs. Hence g T (I ) = e T (I ) µ r µ l is the conductance. L. Bruneau, Landauer-Büttiker currents and spectrum

17 g T as a (dimensionless) conductance II Let I = (µ l, µ r ) and assume the sample is connected to a left and right reservoir, at zero temperature and chemical potentials µ l and µ r. e T (I ) := sp(h per ) I, called Thouless energy, is identified with the current through the sample driven by the voltage difference between these reservoirs. Hence g T (I ) = e T (I ) µ r µ l is the conductance. This identification holds provided some optimal feeding assumption holds: the time it takes for an electron to leave the right reservoir and occupy a vacant state is smaller than the time it takes to go through the sample. Otherwise the current should be less than e T (I ). L. Bruneau, Landauer-Büttiker currents and spectrum

18 g T as a (dimensionless) conductance II Let I = (µ l, µ r ) and assume the sample is connected to a left and right reservoir, at zero temperature and chemical potentials µ l and µ r. e T (I ) := sp(h per ) I, called Thouless energy, is identified with the current through the sample driven by the voltage difference between these reservoirs. Hence g T (I ) = e T (I ) µ r µ l is the conductance. This identification holds provided some optimal feeding assumption holds: the time it takes for an electron to leave the right reservoir and occupy a vacant state is smaller than the time it takes to go through the sample. Otherwise the current should be less than e T (I ). Our goal: make these statements precise, i.e. derive Thouless conductance formula from first principles of quantum statistical mechanics. L. Bruneau, Landauer-Büttiker currents and spectrum

19 Quasi-free systems Fermi gas in the independent electrons approximation: h = one particle space. h = hamiltonian of one fermion. The total Hilbert space is then H = Γ (h) = n=0 n h. The hamiltonian is H = dγ(h), i.e. H f 1 f n = n f 1 hf j f n. j=1 The algebra of observables O = CAR(h): the C -algebra generated by annihilation/creation operators a(f )/a (f ). L. Bruneau, Landauer-Büttiker currents and spectrum

20 Electronic black box model A sample S coupled to 2 reservoirs of electrons R l/r described by a free fermi gas at thermal equilibrium (at inverse temperature β and chemical potential µ l/r ) and in the independent electron approximation. L. Bruneau, Landauer-Büttiker currents and spectrum

21 Electronic black box model A sample S coupled to 2 reservoirs of electrons R l/r described by a free fermi gas at thermal equilibrium (at inverse temperature β and chemical potential µ l/r ) and in the independent electron approximation. We thus consider a quasi-free system with One particle space: h = h l h S h r. L. Bruneau, Landauer-Büttiker currents and spectrum

22 Electronic black box model A sample S coupled to 2 reservoirs of electrons R l/r described by a free fermi gas at thermal equilibrium (at inverse temperature β and chemical potential µ l/r ) and in the independent electron approximation. We thus consider a quasi-free system with One particle space: h = h l h S h r. One particle hamiltonian: h = h 0 + h T where h 0 = h l h S h r, h T = χ l ψ l + ψ l χ l + χ r ψ r + ψ r χ r, (h S, h S ): finite-dimensional system (the sample), ψ l/r h S. (h l/r, h l/r ): free reservoirs. Without loss of generality we may assume that δ l/r are cyclic vectors for h l/r. If ν l/r is the spectral measure for h l/r and δ l/r we may now take h l/r = L 2 (R, dν l/r (E)), h l/r = mult par E, δ l/r (E) 1. L. Bruneau, Landauer-Büttiker currents and spectrum

23 Electronic black box model The hamiltonian of the system is H = dγ(h) = dγ(h 0 )+a (χ l )a(ψ l )+a (ψ l )a(χ l )+a (χ r )a(ψ r )+a (ψ r )a(χ r ), and for any A O, τ t (A) := e ith Ae ith, e.g. for f h one has τ t (a # (f )) = a # (e ith f ). L. Bruneau, Landauer-Büttiker currents and spectrum

24 Electronic black box model The hamiltonian of the system is H = dγ(h) = dγ(h 0 )+a (χ l )a(ψ l )+a (ψ l )a(χ l )+a (χ r )a(ψ r )+a (ψ r )a(χ r ), and for any A O, τ t (A) := e ith Ae ith, e.g. for f h one has τ t (a # (f )) = a # (e ith f ). Initial state of the system: quasi-free state ω 0 associated to the density matrix ϱ = (1 + e β(h l µ l ) ) 1 ρ S (1 + e β(hr µr ) ) 1, i.e. ω 0 is such that ω 0 (a (g n ) a (g 1 )a(f 1 ) a(f m )) = δ nm det( f i, ϱg j ) i,j. L. Bruneau, Landauer-Büttiker currents and spectrum

25 The current observable The observable which describes the flux of particles out of R r is J r := i[h, N r ] = a (iχ r )a(ψ r ) + a (ψ r )a(iχ r ), where N r = dγ(1l r ) is the number of fermions in reservoir R r (1l r is the projection onto h r 0 0 h r ). L. Bruneau, Landauer-Büttiker currents and spectrum

26 The current observable The observable which describes the flux of particles out of R r is J r := i[h, N r ] = a (iχ r )a(ψ r ) + a (ψ r )a(iχ r ), where N r = dγ(1l r ) is the number of fermions in reservoir R r (1l r is the projection onto h r 0 0 h r ). We are interested in ω + (J r ) := 1 T lim ω 0 τ t (J r )dt = ω + (J r ), T + T 0 where ω + = w lim 1 T exists). T 0 ω 0 τ t is the NESS of the system (if it L. Bruneau, Landauer-Büttiker currents and spectrum

27 The wave operators w ± := s lim t ± eith e ith0 1l ac (h 0 ) exist and are complete. The scattering matrix s = w+w acts as multiplication by ( ) sll (E) s s(e) = lr (E). s rl (E) s rr (E) T (E) := s lr (E) 2 = s rl (E) 2 is the transmission probability between the reservoirs at energy E. L. Bruneau, Landauer-Büttiker currents and spectrum

28 The wave operators w ± := s lim t ± eith e ith0 1l ac (h 0 ) exist and are complete. The scattering matrix s = w+w acts as multiplication by ( ) sll (E) s s(e) = lr (E). s rl (E) s rr (E) T (E) := s lr (E) 2 = s rl (E) 2 is the transmission probability between the reservoirs at energy E. Theorem (AJPP 07) If sp sc (h) =, ω + (A) := lim 1 T ω + (J r ) = 1 ( T (E) 2π R T 0 ω 0 τ t (A)dt exists A O and ) de e β(e µr ) e β(e µ l ) Remark : T (E) = 4 ψ l, (h E i0) 1 ψ r 2 ImF l (E) ImF r (E) where F l/r (E) := χ l/r, (h l/r E i0) 1 χ l/r. Hence T (E) = 0 if E / sp ac (h l ) sp ac (h r ). L. Bruneau, Landauer-Büttiker currents and spectrum

29 Landauer conductance Assume we are at zero temperature and µ l < µ r, the Landauer conductance is then g L (I ) = ω +(J r ) 1 µr = T (E) de. µ r µ l 2π(µ r µ l ) µ l L. Bruneau, Landauer-Büttiker currents and spectrum

30 Landauer conductance Assume we are at zero temperature and µ l < µ r, the Landauer conductance is then g L (I ) = ω +(J r ) 1 µr = T (E) de. µ r µ l 2π(µ r µ l ) µ l Take now as a sample our finite system, i.e. h S = l 2 ([1, L]), and (h S ψ)(x) = J x ψ(x + 1) + λ x ψ(x) + J x 1 ψ(x 1) with Dirichlet boundary condition and ψ l = δ 1, ψ r = δ L. L. Bruneau, Landauer-Büttiker currents and spectrum

31 Crystal EBB Assume that the left/right reservoir are such that h = h per, i.e. the left, resp. right, reservoir is the restriction of h per to l 2 ((, 0]), resp. l 2 ([L + 1, )), with Dirichlet B.C. and χ l/r = J 0/L δ 0/L+1, so-called matching wire (Economou-Soukoulis 91) or optimal feeding. L. Bruneau, Landauer-Büttiker currents and spectrum

32 Crystal EBB Assume that the left/right reservoir are such that h = h per, i.e. the left, resp. right, reservoir is the restriction of h per to l 2 ((, 0]), resp. l 2 ([L + 1, )), with Dirichlet B.C. and χ l/r = J 0/L δ 0/L+1, so-called matching wire (Economou-Soukoulis 91) or optimal feeding. One then have T (E) = 1 for E sp(h per ) while T (E) = 0 for E / sp(h per ) so that g L (I ) = where I = (µ l, µ r ). 1 µr T (E) de = 1 sp(h per ) I = 1 2π(µ r µ l ) µ l 2π I 2π g T (I ), Remark: if optimal feeding fails then the current should be less, i.e. g T (I ) should give an upper bound. L. Bruneau, Landauer-Büttiker currents and spectrum

33 Crystallined We come back to arbitrary reservoirs but now consider the following sample. Given an integer N: is the restriction of h per to l 2 ([1, NL]), i.e. a finite Jacobi matrix whose coefficients are L-periodic, with Dirichlet B.C. and with ψ l = δ 1 and ψ r = δ NL. h (N) S We denote by ω (N) + (J r (N) ) the corresponding current expectation value in the NESS. We are interested in its large N limit. Remark: Similar idea of crystallizing or periodizing appears in a band random matrix approach by Cassati-Guarneri-Maspero ( 97). L. Bruneau, Landauer-Büttiker currents and spectrum

34 Crystallined Let h per, (l) resp. h per, (r) denote the restriction of h per to l 2 ((, 0]), resp. l 2 ([1, )), with Dirichlet B.C. We denote by m l/r its Weyl m-function: m l (E) := δ 0, (h (l) per E i0) 1 δ 0, m r (E) := δ 1, (h (r) per E i0) 1 δ 1. L. Bruneau, Landauer-Büttiker currents and spectrum

35 Crystallined Let h per, (l) resp. h per, (r) denote the restriction of h per to l 2 ((, 0]), resp. l 2 ([1, )), with Dirichlet B.C. We denote by m l/r its Weyl m-function: m l (E) := δ 0, (h (l) per E i0) 1 δ 0, m r (E) := δ 1, (h (r) per E i0) 1 δ 1. The crystalline transmission coefficient is defined, for E sp(h per ), as T crys (E) = 4J0 2 1 J2 0 m l(e)m r (E) 2 ImF l (E) ImF r (E) J0 4 1 m r (E)F l (E) 2 1 m l (E)F r (E) 2 J0 2m l(e) F l (E) 2 J0 2m r (E) F r (E). 2 L. Bruneau, Landauer-Büttiker currents and spectrum

36 Crystallined Let h per, (l) resp. h per, (r) denote the restriction of h per to l 2 ((, 0]), resp. l 2 ([1, )), with Dirichlet B.C. We denote by m l/r its Weyl m-function: m l (E) := δ 0, (h (l) per E i0) 1 δ 0, m r (E) := δ 1, (h (r) per E i0) 1 δ 1. The crystalline transmission coefficient is defined, for E sp(h per ), as T crys (E) = 4J0 2 1 J2 0 m l(e)m r (E) 2 ImF l (E) ImF r (E) J0 4 1 m r (E)F l (E) 2 1 m l (E)F r (E) 2 J0 2m l(e) F l (E) 2 J0 2m r (E) F r (E). 2 Property: 0 T crys (E) 1. For a crystal EBB F l/r (E) = J 2 0 m l/r (E) and T crys (E) 1. L. Bruneau, Landauer-Büttiker currents and spectrum

37 Crystallined Theorem J r crys := lim 1 = 2π N ω(n) + (J r (N) ) R ( T crys (E) In particular, at zero temperature, J r crys = 1 2π µr e 1 β(e µr ) 1 + e β(e µ l ) µ l T crys (E)dE 1 2π e T ((µ l, µ r )). ) de. Thouless energy gives an upper bound for Landauer current, with equality iff T crys (E) 1 on (µ l, µ r ). Optimal feeding is identified to reflectionless transport between the reservoirs. It holds for example for crystal EBB (matching wires). L. Bruneau, Landauer-Büttiker currents and spectrum

38 vs ac spectrum Thouless s idea is that a system exhibits localization if g T (I ) decays to 0 with L. Let h be a Jacobi matrix on l 2 (Z + ) and hper L its periodic approximant of size L, i.e. hper L coincide with h on [1, L] and then is extended periodically. How does gt L (I ) := sp(hl per) I behaves as L? I L. Bruneau, Landauer-Büttiker currents and spectrum

39 vs ac spectrum Thouless s idea is that a system exhibits localization if g T (I ) decays to 0 with L. Let h be a Jacobi matrix on l 2 (Z + ) and hper L its periodic approximant of size L, i.e. hper L coincide with h on [1, L] and then is extended periodically. How does gt L (I ) := sp(hl per) I behaves as L? I Theorem If I is an open interval such that sp ac (h ) I = then gt L (I ) 0. If moreover h is ergodic then it is an equivalence. First part follows from Gesztesy-Simon ( 96) results, the second part was proven by Last in is thesis ( 94). L. Bruneau, Landauer-Büttiker currents and spectrum

40 Landauer conductance vs ac spectrum In the Landauer approach, instead of first crystallizing and then taking the size of the sample to infinity, one may simply take the size of the sample to infinity. That is, consider the EBB model where the sample is the restriction of h to [1, L] and let ω +,L (J r,l ) denote the corresponding expectation value of the charge current in the steady state. L. Bruneau, Landauer-Büttiker currents and spectrum

41 Landauer conductance vs ac spectrum In the Landauer approach, instead of first crystallizing and then taking the size of the sample to infinity, one may simply take the size of the sample to infinity. That is, consider the EBB model where the sample is the restriction of h to [1, L] and let ω +,L (J r,l ) denote the corresponding expectation value of the charge current in the steady state. Theorem If I is an open interval such that lim inf L ω +,L(J r,l ) = 0 then sp ac (h ) I =. Remark: The reservoirs have to contain I in their ac spectra but otherwise arbitrary. The proof uses the connection between ω +,L (J r,l ) and the transfer matrices of h combined with results by Last-Simon ( 99). L. Bruneau, Landauer-Büttiker currents and spectrum

Landauer-Büttiker formula and Schrödinger conjecture

Landauer-Büttiker formula and Schrödinger conjecture L. Bruneau 1, V. Jakšić 2, C.-A. Pillet 3 1 Département de Mathématiques and UMR 888 CNRS and Université de Cergy-Pontoise 95 Cergy-Pontoise, France 2 Department of Mathematics and Statistics McGill University