Di usion hypothesis and the Green-Kubo-Streda formula

Transcription

1 Di usion hypothesis and the Green-Kubo-Streda formula In this section we present the Kubo formula for electrical conductivity in an independent electron gas model of condensed matter. We shall not discuss its full justification, and/or limitations, but will show how the formula is arrived at within the linear response approximation. Not surprisingly, the formula shows vanishing of the direct conductance at energies within the localization regime. Also presented is a proof of the asymptotic vanishing of the o -diagonal elements of the one-particle density function for the corresponding system of noninteracting Fermions if the Fermi-energy falls within the localization regime. We start however with the general di usion conjecture for disordered systems, where the di usion constant is linked via an Einstein relation to the dc conductivity.. The di usion hypothesis Some of the relevant information on the dynamics generated by a self-adjoint operator H in the Hilbert space over a graph with metric d may be encoded in the rate of growth, in time, of the position moments ( >0) in generic states 2 `2(G), i.e. M (, t) := d(x, 0) (x; t) 2. (.) x2g The corresponding Abel average over times of order t / is given by: bm (, ) := 2 Z 0 e 2 t M (, t) dt = Z (H d(x, 0) E i ) (x) 2 de. (.2) x2g 26

2 . The di usion hypothesis 27 The RAGE theorem, or more precisely (2.37), implies that bm (, )! as # 0 for any 2H c. The asymptotic growth rates can indicate qualitative di erences between systems. Particularly worthy of mentioning are the two cases: 8 >< M (, ) / C ballistic growth (.3) >: D /2 di usive growth More generally, one refers to the growth rate M (, ) / C r as subdi usive if r < /2, and superdi usive if r > /2. Following are some general observations. The free motion generated by the Laplacian on Z d is ballistic for generic initial states 2 `2(Z d ), and all values of >0. 2. For Hamiltonians with tempered hopping amplitudes, such as Schrödinger operators (2.2), the growth rate is at most ballistic (i.e. r apple ), cf. Exercise Lower bounds on the moments M (, t), for 2H c, can be obtained by bounding above the probability of lingering: Prob (t)( x < bt r ). At r = 0 these tend to to zero for any b > 0. More explicit bounds can be derived based on the finer distinction among spectral types along the lines of Definition 2.2. Such bounds were due originally to I. Guarneri [06] with generalization found in [52, 73, 42]. They were presented here in Exercise For disordered systems it is expected that the motion is generally di usive, provided the generator s continuous spectrum is not empty. Thus, starting in the localized state = 0 the growth rate of the second moment is expected to be M 0 (2, t) td 0 with a finite di usion constant D 0. To expand on the last point, if indeed for = 0 (x; t) 2 h x, e td 0 x 0 i, bm (, ) (.4) then under the time average one is led to the following di usion hypothesis for the Green function h x, (H E i ) 0i 2 2 h x, ( D 0 (E) +2 ) 0i. (.5) For the total di usion constant this yields D 0 = R (E)D 0 (E)dE, with (E) the density of states (defined in Chapter 5). The above considerations lead to the Kubo-Greenwood formula for the conjectured relation between the di usion constant and the Green function: 2 (E) D 0 (E) = lim #0 d(x, 0) 2 h x, (H E i ) 0i 2. (.6) x Random Operators c M. Aizenman & S. Warzel DRAFT

3 28 Di usion hypothesis and the Green-Kubo-Streda formula We will see later in this Chapter that the right side here coincides with the dc conductance in linear response theory. Establishing the di usion hypothesis for a random Schrödinger operator on any finite-dimensional configuration space is an open challenge. Existing partial result in this direction include: i) proof of di usive/ballistic behavior on tree graphs, where di usion is ballistic (discussed here in Chapter 4), ii) proofs of di usive behavior up to a bounded times scale which depends on the disorder [87, 88, 89], iii) results in this vane for certain supersymmetric models [73]. The latter go beyond the framework of operators discussed here, yet an analog of (.5) formulated in terms of the relevant correlation function (substituting the Green function) provides there the defining feature of di usive behavior..2 Heuristic linear response theory The Kubo formula expresses the electrical conductivity within the framework of linear response theory. Here one considers the e ects of an electric field E 2 R d which is adiabatically switched on. For non-interacting particles with unit charge q 2 R the time evolution is thus governed by a time-dependent operator on some Hilbert space of the form H(t) = H qe xe t, ( < t apple 0, # 0). The adiabatic time evolution of the system which is described by a density matrix W, is given in terms of the following initial value problem of the Liouville equation: d W(t) = i [H(t), W(t)], lim dt t! eith W(t) e ith = W. (.7) The state at t = is thereby chosen as the one which would generate W at t = 0 under the time evolution generated by H if startet at t =. Simplifications arise if the initial state W is invariant under the time evolution of H. A physically relevant example is the thermal equlilibrium state of a noninteracting system of Fermions at inverse temperature 2 [0, ], i.e., W = EF (H), EF ( ) := 8 >< >: + e ( E F ), 2 (0, ) (,EF ]( ), =, (.8)

4 .2 Heuristic linear response theory 29 where E F 2 R is the Fermi energy. Then, to first oder in E, the approximate solution of (.7) is: Z 0 W(0) W = iq e ith [E x, W] e ith e t dt. (.9) This equation constitutes the linear response ansatz. The current density caused by the application of the electric field in a system with elementary charges q 2 R which is translation invariant on average, is given by j := q Tr (v (W(0) where v := i [H, x] is the velocity operator and Tr ( ) := lim L! L tr L ( ) denotes the trace per unit volume. In the ergodic set-up, the trace-per volume can be replaced by an average over the disorder, W)) Tr ( ) = E [h 0, ( ) 0 i]. If H is a magnetic Schrödinger operator (A) + V on `2(Z d ), the components of the velocity operator are given by (v ) (x) = i e iqa(x,y) (x y) (y). (.0) y x = In case A = 0, they coincide up to a factor of i with the components of the discrete derivative. Within the linear response ansatz (.9), the relation between the components of the current density and the electric field is hence linear, j = P µ,µ E µ, with, µ = lim iq 2 Tr Z 0 v e ith [x µ, W] e ith e t dt (.) #0 defining the Kubo-Greenwood conductivity tensor. The above arguments constitute the customary heuristic motivation for the definition of the conductivity tensor. From a mathematical point of view, several questions arise:. In which sense can one give a meaning to the solutions of the Liouville equation in the ergodic setup? 2. Can one justify the linear response ansatz (.9) as a first order approximation to the exact solution of the Liouville equation? 3. Is there a mathematical meaning to the quantities in the right side of (.) and does the limit exist? In the following, we will sketch an answer to the first question and provide Random Operators c M. Aizenman & S. Warzel DRAFT

5 30 Di usion hypothesis and the Green-Kubo-Streda formula the answer to the third question. Both answers go back to the work of Bellisard, van Elst and Schulz-Baldes [32]. We skip the justification of the linear response ansatz and refer the reader to the remarks in the Notes..3 The Green-Kubo-Streda formulae As discussed in Chapter 4, disordered systems are modeled in terms of random, translation-covariant operators. Let us restrict our discussion to the lattice case Z d and, in order to incorporate e ects of a constant magnetic field B, implement the group of translations on `2(Z d ) in terms of magnetic translations given by U x (A) (y) = e ibq(y x 2 x y 2 ) (y x). In such a setup, all operators entering Liouville s equation (.7) may be taken to be bounded and linear operators M(!) on `2(Z d ). In their dependence on!, they are required to be weakly measurable and satisfy: U (A) x M(!) U (A) x = M(S x!). (.2) Such operators constitute the linear space K mc of measurable covariant operators. Of particular interest is the subspace K 2 := n M 2K mc E [hm 0, M 0 i] < o (.3) which together with the scalar product hh M, M 2 ii := E [hm 0, M 2 0 i] (.4) forms a Hilbert space. (The reader is encouraged to prove this assertion). Solutions of the Liouville equation can now be discussed on this space. Namely, given a bounded self-adjoint H 2K mc with the linear operator ess sup!2 kh(!)k <, L H : K 2!K 2, M 7! [H, M], (.5) called the Liouville operator, is self-adjoint on K 2. It therefore generates a unitary group e itl H on K 2 and its resolvent at >0is a bounded operator given by Z 0 (L H i ) = i e itl H e t dt. (.6)

6 .3 The Green-Kubo-Streda formulae 3 This sketches the answer to the first question. For future reference, let us mention that, as expected for a trace, the trace per unit volume as given by E [h 0, ( ) 0 i] is cyclic, i.e., for any M, M 2 2K 2 : E [h 0, M M 2 0 i] = E [h 0, M x ih x, M 2 0 i] x2z d = E [h x, M 0 ih 0, M 2 x i] x2z d = E [h 0, M 2 M 0 i]. (.7) Here the first and third equality use Fubini s theorem together with the completeness of ( x ) x2z d. The second equality is based on the covariance relation (.2). As a preparation of an answer to the third question in the above list, we note that within the above framework the conductivity tensor (.) can formally be reexpressed as,µ Z 0 = lim iq 2 Tr v e itl H [xµ, W] e t dt #0 = q 2 lim Tr v (L H i ) [x µ, W] #0 = q 2 lim #0 hh v, (L H i ) [x µ, W] ii. (.8) Clearly, in the standard situation (.0) the components of the velocity operator v are elements of K 2. However, to provide meaning to the expression on the right side we also require [x µ, W] 2K 2. (.9) Even if this applies, the existence of the limit cannot be taken for granted. To further discuss these issues we restrict ourselves to the following situation: The Hamiltonian is given by the Laplacian corresponding to a constant magnetic field plus a bounded random potential, H(!) = (A) + V(!), on `2(Z d ), with ess sup!2 kv(!)k <. The initial state is given by the thermal equilibrium, W = EF (H), cf. (.8). We now distinguish the cases of positive temperature, temperature, =. 2 (0, ), and zero Random Operators c M. Aizenman & S. Warzel DRAFT

7 32 Di usion hypothesis and the Green-Kubo-Streda formula Zero temperature limit In the zero temperature limit, the thermal equilibrium state is a projection, E F (H) = P (,EF ](H) =: P on `2(Z d ). The requirement (.9) then translates to E h k[x, P] 0 k 2i h apple x 2 E h 0, P x i 2i < (.20) x2z d for all 2{,...,d}. This is a localisation condition on the two-point function in the ground-state of non-interacting Fermions whose Fermi level is at E F. In Subsection.4 we will show that the exponential decay of a fractional moment of the Green function at E F is su cient for the validity of (.20). The condition (.20) ensures that the prelimit in (.8) is well-defined. This makes the representation (.8) of the conductivity available. The remaining question about the existence of the limit # 0 is resolved in the following theorem. Under somewhat more restrictive conditions it was first presented in [32]. The present version and its proof are taken from [8]. Theorem. (Streda-Kubo formula) Under assumption (.20) the conductivity tensor for a system of Fermions with unit charge q 2 R at zero temperature as given by the right side in (.8) is well defined and can be expressed as follows:,µ = iq 2 E h h 0, P h [x, P], [x µ, P] i 0i i. (.2) As a consequence, the direct conductivity vanishes,, = 0 for any 2 {,...,d}. The evaluation of the limit # 0 in (.8) will be based on the formula lim (L H + i ) L H (M) = M (.22) #0 valid for any M 2 (ker L H )?, the orthogonal complement of the kernel of the Liouville operator. Since the latter is self-adjoint, this identity follows from spectral calculus. We will apply it to M = [[x, P], P] and hence need the following Lemma.2 Under assumption (.20), one has [[x, P], P] 2 (ker L H )?. Proof For any M 2 ker L H we have hh M, [[x, P], P] ii = E h h 0, M [x, P] P 0 i i E h h 0, M P [x, P] 0 i i = E h h 0, [P, M ][x, P] 0 i i = hh[m, P], [x, P] ii = 0. (.23)

8 .3 The Green-Kubo-Streda formulae 33 Here the second equality uses the cyclicity of the trace per unit volume (.7). The vanishing of the right side follows from M 2 ker L H, which implies [M, P] = 0. In order to establish the existence of the limit in (.8), it is thus useful to rewrite its right side. This will be done in the subsequent Proof of Theorem.: Denoting by P? := P, we have [x µ, P] = P[x µ, P]P? + P? [x µ, P]P il H ([[x, P], P]) = ip? [H, x ]P + ip[h, x ]P? = P? v P + Pv P?. The cyclic invariance of the trace per volume hence yields hh v, (L H i ) [x µ, P] ii = hh P? v P + Pv P?, (L H i ) [x µ, P] ii = i hh L H ([x, P], P]), (L H i ) [x µ, P] ii = i hh (L H + i ) L H ([x, P], P]), [x µ, P] ii. (.24) Since [[x, P], P] 2 (ker L H )? by Lemma.2, the spectral calculus of L H implies the validity of (.22) in case M = [[x, P], P]. Inserting the above results into (.8), we hence obtain,µ = iq 2 hh [[x, P], P], [x µ, P] ii = iq 2 E h h 0, P h [x, P], [x µ, P] i 0i i. (.25) This finishes the proof of the Streda-Kubo formula (.2). The fact that the direct conductivity vanishes is an immediate consequence thereof. In the special case d = 2 with a constant perpendicular magnetic field, the transversal, i.e. Hall, conductivity as given by the o -diagonals in (.2) turn out to be an integer divided by 2. There is deeper topological reason, which applies more generally than to the mentioned quantum Hall situation, that the number given by the o -diagonals in (.2) (modulo 2 ) turns out to be integer. As has been noticed in [24, 32, 23] in certain situations, these o -diagonals can be identified with an index or Chern number; see also [27]. A notable property of indices is that they appear to vanish under careless application of standard principles, such as commutativity of the trace. Yet they come to life if one is careful to pay attention to convergence issues. Related observation applies to the above argument. Let that be a lesson for the uninitiated. Another lesson is: nature tends not to waste beautiful mathematical principles. We will come back to this in the next section. Random Operators c M. Aizenman & S. Warzel DRAFT

9 34 Di usion hypothesis and the Green-Kubo-Streda formula Having established conditions for the existence of the Streda-Kubo formula and the vanishing of the direct conductivity in the zero temperature limit, we now turn to the Kubo formula at positive temperatures. Positive temperatures At positive temperature, it is reasonable to rewrite (.8) using the spectral representation for H:, µ = q 2 lim #0 = q 2 lim #0 Z Z Z Z E E 0 i E h h 0, v P de (H)[x µ, EF (H)] P de 0(H) 0 i i i EF (E) EF (E 0 ) E E 0 i E E 0 m, µ (de de 0 ). (.26) The measure appearing in the right side is the velocity correlation measure associated with H. Definition.3 The velocity correlation measure of H(!) = (A) + V(!) on `2(Z d ) is the finite, complex-valued Borel measure on R 2 which is defined through: Z Z f (E) g(e) m,µ (de de 0 ) := E h h 0, v f H v µ g H 0 i i, for all f, g 2 C 0 (R) and, µ 2{,...,d}. The measure is well-defined through the Riesz-Markov representation theorem. There is a wealth of additional information on this measure available in the literature. For example, using the techniques of Section 3.3, one may show that the velocity-correlation measure is self-averaging in the sense that it coincides almost surely with the infinite-volume limit of finite-volume counterparts. We refer the interested reader to the Notes at the end of this section. It is elementary to check that m,µ (I J) = m µ, (I J) = m,µ (J I) for any Borel set I, J R. In the special case H(!) = +V(!), the velocity correlation measure is real, i.e., m,µ = m,µ. Consequently, the Kubo conductivity tensor is symmetric,,µ = µ,, and the left side in (.26) can be simplified: sym,µ :=,µ + 2 Z Z = q 2 lim #0 µ, (E E 0 ) E F (E) EF (E 0 ) E E 0 m, µ (de de 0 ), (.27) where (u) := (u ) is an approximate identity. The above expression for corresponds to the familiar form of the Kubo formula. It is sym,µ

10 .3 The Green-Kubo-Streda formulae 35 the starting point for another customary formula for the symmetric part of the conductivity involving the pointwise limit 2 e,µ (E) := lim #0 For energies E in the localization regime in which sup >0 h x µ x E i G(0, x; E + i ) 2. (.28) x2z d x2z d x 2 E G(0, x; E + i ) s < (.29) for some s 2 (0, ), this limit is in fact zero, s e,µ (E) apple lim x µ x E G(0, x; E + i ) s = 0 #0 x2z d where the inequality results from G(0, x; E + i ) apple. Since e,µ is related to the di usion tensor of the system (cf. (.6)), the limit should exists in the presence of disorder regardless of localization. Theorem.4 (Existence of the Kubo conductivity) Suppose that h sup x 2 (Im z) 2 E i G(0, x; z) 2 <. (.30) z2c + x2z d and that the limit (.28) exists for all E 2 R. Then for any 2 (0, ) and E F 2 R: Z sym,µ = q 2 e,µ (E) d E F (E) de. (.3) de Proof We start by rewriting the prelimit in the right side of (.28): 2 = h x µ x E i G(0, x; E + i ) 2 x2z d 2 E h h 0, h x µ, (H E i ) ih x, (H E + i ) i 0i i = 2 E h h 0, (H E i ) v µ H E i 2 v (H E + i ) 0i i Z Z = (E u) (E u 0 ) m µ, (du du 0 ), (.32) where (u) := (u ) is an approximate identity. The second equality is based on the equality h x, (H z) i = (H z) [H, x ](H z) Random Operators c M. Aizenman & S. Warzel DRAFT

11 36 Di usion hypothesis and the Green-Kubo-Streda formula for all z 2 C +, and the last inequality relies on the cyclic invariance of the trace per unit volume and the spectral theorem. The dominated convergence theorem, which is applicable thanks to (.30), guarantees that the limit # 0 may be interchanged with the E-integration in (.3). We hence conclude from (.27) and (.32) that with sym,µ + q 2 Z e µ,µ (E) d E F (E) de Z Z de = q 2 lim #0 g (u, u 0 ) m µ (du du 0 ), (.33) g (u, u 0 ) = Z 0 d EF (u + x) (x) B@ du (u u 0 + x) EF (u) EF (u 0 ) u u 0 (u u 0 ) CA dx. Using the fact that EF has bounded derivatives (to arbitrary order) for 2 (0, ), it is a tedious but elementary exercise to show that i) g (u, u 0 ) is bounded uniformly in (u, u 0 ), and ii) lim!0 g (u, u 0 ) = 0 for all (u, u 0 ) 2 R 2. Since the velocity correlation measure m µ,µ is finite, the dominated convergence theorem yields Z Z lim g (u, u 0 ) m µ (du du 0 ) = 0, (.34)!0 which is equivalent to the assertion. It is not hard to see that the low-temperature limit of (.3) exists and is given by! of right side lim! sym,µ = q 2 e,µ (E F ). (.35) In fact, an inspection of the previous proof shows that this identity for the double limit of first taking # 0 and then!in (.27) may be established under the weaker assumption that the limit (.28) exists for a neighborhood of the Fermi energy E F..4 Localization and decay of the two-point function We will now come back to the question whether localization implies condition (.20) on the decay of the kernel of the Fermi projection. This kernel represents the two-point function in the ground-state (corresponding to = )

12 .4 Localization and decay of the two-point function 37 of non-interacting Fermions. More generally, the two-point function of a system of non-interacting Fermions in the thermal equilibrium state corresponding to inverse temperature 2 [0, ] and Fermi level at E F is h (y) (x)i = h x, EF (H) y i (.36) using the notation (x) for the Fock-space fermionic annihilation operator at x. At positive temperature <, this kernel always exhibits exponential decay. This follows from the fact that the function EF is meromorphic with poles at E F + i(2n + ) /, n 2 Z. It is hence analytic and bounded within the strip Im z < / about the real axis. For any such function the decay of the kernel is exponential. Proposition.5 For a function F which is analytic and bounded in the strip Im z < and a bounded self-adjoint operator H on `2(Z d ) with the property sup x Py H(x, y) e µ x y apple /2 for some µ>0, there is some C > 0 such that for all x, y 2 Z d : h x, F(H) y i apple Ce µ x y. (.37) Proof By analyticty of F in the strip, we may represent F(E) for any E 2 (H) a contour integral with a contour C which encircles the spectrum counterclockwise at a distance /2 from (H). By the spectral theorem this yields h x, F(H) y i = I h x, (z H) yi F(z) dz. (.38) 2 i C The asserted bound is then a simple consequence of the Combes-Thomas estimate (Theorem B.), the fact that F is bounded on C, and the boundedness of the length of the contour. The above proposition may be generalised to unbounded self-adjoint operators. This requires to a more careful estimate on the contour integral which then consists of two infinite path above and below the real line. The key here is a variant of the Combes-Thomas estimate, which also leads an improved upper bound in the bounded case [8]. In case F = EF and H is a bounded random Schrödinger operator H(!) = (A) + V(!) on `2(Z d ), we hence conclude that the two point function has an exponentially decay with exponent µ proportional to. This bound hence does not survive the zero temperature limit =. However, if the spectrum of H below the Fermi Random Operators c M. Aizenman & S. Warzel DRAFT

13 38 Di usion hypothesis and the Green-Kubo-Streda formula level consists only of localized states and the eigenfunction correlator decays exponentially, the bound h x, P (,EF ](H) y i apple Q(x, y;(, E F ]) implies the decay of the two-point function. At zero temperature in a Fermi system only those states can participate in any dynamics which are at the Fermi energy. Physical intuition therefore suggests that the exponential localization of states with energies in a neighborhood of E F (instead of the full half line (, E F ) should be enough to guarantee the exponential decay of the two-point function. Theorem.6 For a random Schrödinger operator H(!) (A) + V(!) on `2(Z d ) with ess sup!2 kv(!)k <, if for some s 2 (0, ) and C, < and all x, y 2 Z d : sup,0 E G(x, y; E F + i ) s apple Ce x y / then there is some A < such that for all x, y 2 Z d : E h h x, P (,EF )(H) y i i apple Ae x y /. (.39) Proof We represent the Fermi projection in terms of a contour integral with a contour C EF which consists of the segments connecting E F i, E F + i, E F +i and E F +i where >0 is chosen such that dist (E F, (H(!))) with >0chosen later, cf. Figure??. Splitting the integral and noting that Figure. The choice of the contour for the representation of the Fermi projection the Hamiltonian is bounded from below. the assumption implies that with probability one E F is not an eigenvalue, we obtain P (,EF )(H) = Q (H E F ) + Q 2 (H E F ) + Q 3 (H E F ) (.40)

14 Exercises 39 where Q (x) = Z 2 Q 2 (x) = 2 i Z 0 d i x u i x u + i x! du Q 3 (x) = Q (x + ). (.4) The contribution of the second and third term is estimated using the Combes- Thomas estimate (Theorem B.) which for dist(z, (H(!))) reads: h x, (H(!) z) yi apple Ce c x y with some constants c > 0, C < which are independent of!. Since the length of the contours on the segments one and three are bounded and we may choose = (c ), this establishes the claimed exponential decay for those terms. In order to bound the first term, we use the localization assumption E h h x, Q (H E F ) y i i apple 2 apple 2 Z Z E h h x, (H E F i ) yi i d E h h x, (H E F i ) yi si d s apple C s s e x y /. (.42) Here the second inequality results from the bound h x, (H z) yi apple Im z. In the setting of Theorem.6 one may also show that E h h x, EF (H) y i i apple Ae x y /. (.43) where the constants in the right side are independent of the inverse temperature 2 [0, ]. Moreover, similarly as Proposition.5 the theorem can be extended to cover unbounded random operators [8]. Exercises. Let H be a self-adjoint operator on `2(Z d ) with the property that S := sup H(x, y) e x y < x for some >0. y Random Operators c M. Aizenman & S. Warzel DRAFT