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Effective eigenstates in a dynamical disordered tight-binding model
Physica D 168–169 (2002) 258–265
Effective eigenstates in a dynamical disordered tight-binding model A. Karina Chattah a,∗ , Manuel O. Cáceres b a
b
Facultad de Matemática Astronom´ıa y F´ısica, Universidad Nacional de Córdoba, 5000 Córdoba, Argentina Centro Atómico Bariloche and Instituto Balseiro, CNEA and Universidad Nacional de Cuyo, Av. Bustillo Km 9.5, 8400 Bariloche, Argentina
Abstract We study effective eigenstates—in mean value—for a 1D tight-binding (TB) Hamiltonian in the presence of diagonal dynamical disorder (DD). The averaged density of states (DOS) and localization length are shown as a function of the intensity of disorder and the correlation time of its temporal fluctuations. A delocalization electronic phenomenon is interpreted in terms of the localization length of the wave function, as a function of the time correlation of the DD. © 2002 Elsevier Science B.V. All rights reserved. PACS: 71.55; 02.50.Ey; 02.50.Wp; 05.30.−d Keywords: Tight-binding; Dynamical disorder; Kubo–Anderson process; Delocalization
1. Introduction In classical statistical mechanics the effects of dynamical disorder (DD) on the transport coefficient is well understood. A strong quenched disorder leads to sub-diffusion, but the presence of DD breaks down this classical localization, restoring a finite static conductivity [1–3]. The mathematical reason for this phenomenon is that DD introduces a shift in the generalized diffusion coefficient, which is ultimately the one responsible for reestablishing the analyticity at zero frequency. This shift is proportional to the inverse of the correlation time of the stochastic process which mimics the DD. This problem can be solved in a perturbative way, e.g. using the effective medium approximation (EMA) [4] for dichotomic disorder. This approximate solution interpolates smoothly between static disorder and the limit of rapid temporal fluctuations, thus giving support to what physical intuition expects of the model. In quantum mechanics the conditions for localization are not as simple as in classical mechanics. The simplest model goes back to Anderson’s pioneer work, where a spinless non-interacting particle moving in a quenched random potential was studied. Since then, a lot of work has been done concerning transport in random media [5]. But the interesting quantum problem of DD was not studied until 1975 by Ovchinnikov and Érikhman [6], later on Madhukar and Post [7] reported an exact solution for the motion of a particle in a system with site diagonal and nearest-neighbor off-diagonal DD, using Gaussian white noises. Even when this result was of particular importance some doubts concerning the mobility were reported later [8,9]. This continuous quantum model was “exactly” solved ∗
Corresponding author. E-mail addresses: [email protected] (A.K. Chattah), [email protected] (M.O. C´aceres). 0167-2789/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 2 7 8 9 ( 0 2 ) 0 0 5 1 4 - 6
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thanks to the white-Gaussian character of the stochastic potential. Later on, both models [7,8] gave insight into the problem of quantum dissipation. As a matter of fact a stochastic Hamiltonian can help to imitate irreversibility, so non-equilibrium theories were also inspired in those pioneer works [10]. The success of a stochastic Hamiltonian lies on the “approximation” of taking a large number of fast degrees of freedom (unknown in detail) into account, and the correct emulation of those degrees of freedom by suitable noises [11]. A related model of DD was also used to study the evolution of electron states in terms of quasi-stationary solutions of a time-dependent Hamiltonian [12]. A breakthrough in the analysis of quantum diffusion in the presence of a temporally fluctuating potential, to emulate the random excitation of phonons, was achieved introducing non-white stochastics modulations. In particular many dichotomous processes (noises) were introduced in a tight-binding (TB) Hamiltonian to represent the DD; this model was solved by approximate methods to calculate the transport coefficient [13–15]. Nevertheless, the character of the associated effective eigenstates has not been explored in detail. By effective eigenstates we mean averaged eigenstates over the temporal realizations of the DD. Note that in a DD model the time dependence of the disorder is taken to be parametric, in the sense that the electron–phonon interaction, responsible of the DD, does not introduce additional degrees of freedom into the Hamiltonian. Thus the effective eigenstates can be analyzed by studying the noise average of the density of states (DOS) and of the localization length. These quantities can be calculated from the mean value of the associated Green function of the system. As we mentioned earlier if the DD is not white-Gaussian the resulting problem cannot be solved in an exact way. Thus in the present paper we use a reported theorem concerning the Kubo–Anderson process (KAP) [16] and the coherent potential approximation (CPA) [17] to tackle a site disorder dynamical model. In the case of having only one stochastic impurity the effective properties were helpful to understand the phenomenon of the dynamical localization for the excitations of the system [18]. In that reference we were able to calculate the exact mean propagator of the system. In this way we could define localized states with finite life-times: Im[xk ]−1 (i.e. the dynamical localization), where xk are the poles of the corresponding Green function, which depend on the impurity stochastic correlation time ν −1 . Therefore an infinite life-time is only obtained in the limit of quenched disorder. For the one impurity case the effective eigenstates—characterized by the averaged DOS and by the inverse of the localization length—gave a complementary point of view to analyze the delocalization phenomena. This type of model mimics the stochastic transitions of a single impurity ion between different equivalent positions (i.e. a temperature-dependent trap in a polymer sample [19]). Also our model of one stochastic impurity can be related to the problem of delocalization due to measurement (i.e. interaction with a detector apparatus). In [20] the decoherence—generated by the interaction with the detector—was analyzed by studying the stationary state of the reduced density matrix. There it was shown that the environment destroys the localization, therefore affecting the time dependence of the observed system.
2. Dynamical disorder Here we are interested in the averaged properties coming from the stochastic Schrödinger equation iψ˙ = H (t)ψ ≡ (H0 + εH1 (t))ψ. The total Hamiltonian is composed of an ordered part H0 given by the hopping term of a TB model (written in the Wannier basis {|n }), and a stochastic contribution H1 (t) which introduces an energy-site DD model, i.e. for n ∈ [−∞, ∞]: H (t) = υ0 |n n ± 1| − ε mn (t)|n n|. (1) n
n
Here ε is the amplitude of the disorder and {mn (t)} represents a KAP set (leading to a “renewal” model for the disordered energies). The KAP stochastic set is constituted by independent dichotomous processes mn (t) all having
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the same jumping times ti which are uniformly and independently distributed with a density ν (Poisson distribution). Then each mn (ti ) = m is a random variable that takes one of the values ±1 with probability p, q, respectively. In the stationary regime the mean value is mn (t) = m , ∀n and the correlations are given by
mn (t)mk (t ) = δn,k
m2 exp[−ν|t − t |]. In general we assume that ν is an increasing function of the temperature T . Now we look for an expression for the mean-value Green function G(z) , this means taking the average over the DD. For this we take into account that the set {mn (t)} is a KAP one, thus the Hamiltonian H (t) is a KAP matrix as defined in [16]. In terms of the propagator defined by ψ(t) = G(t)ψ(0), and using the initial condition G(0) = I the exact average is t
G(t) = GS (t) exp[−νt] + ν dt exp[−ν(t − t )] GS (t − t ) G(t ) . (2) 0
Here GS (t) = exp[tHS ] is the t-dependent Green function associated to the Hamiltonian (1) in the static limit (quenched disorder). This means that when ν = 0, H ≡ HS = H0 − ε n mn |n n|, where mn are statistically independent random variables taking values mn = ±1 with probability p, q. We now ∞ define the averaged Green function introducing the Laplace transform of the propagator G(z) = −(i/) 0 dt G(t) exp[izt/] (denoted by its argument z). Thus from (2) the averaged Green function is given by
G(z) = [I − iν GS (z + iν) ]−1 GS (z + iν) ,
(3)
where GS (z) = [z − HS ]−1 . It means that G(z) is given in terms of the averaged Green function for the corresponding quenched disorder problem. From G(z) we are now able to calculate single-particle properties, for example, the averaged DOS and localization length; both quantities give insight into the localization character of the eigenstates in presence of DD.
3. CPA for dynamical disorder Eq. (3) is important because it gives the exact expression for G(z) in terms of GS (z) . So the difficulty in averaging over the DD has been moved to the problem of calculating GS (z) ; of course this is not a trivial task. Here the static case (ν = 0) corresponds to a binary alloy with concentrations p, q of atoms A, B, respectively; even in this case GS (z) cannot be calculated exactly [17,21]. In what follows we use the CPA to calculate the averaged static Green function. Then we can approximate
GS (z) ∼ = [z − HC (z)]−1 ,
(4)
where the coherent Hamiltonian is given by HC (z) = −ωC (z) n |n n| + H0 . Here we have chosen the minus sign in front of the coherent medium ωC (z) for future convenience. Inserting (4) in (3) we finally arrive at a closed expression for G(z) CPA for DD
G(z) CPA = [z − HC (z + iν)]−1 ,
(5)
where HC (z + iν) is defined through a dynamical coherent medium ωD (z) = ωC (z + iν). This is one of the goals of our work, allowing us to calculate several effective single-particle properties for a given system under the influence of DD. The shift in the dynamical coherent medium is equivalent to the result predicted by Harrison and Zwanzig [1] when studying, with EMA, DD in a classical master equation (leading to the suppression of the percolation threshold). In our case the shift comes from the analytically solved KAP model (3) and coincides, in the CPA approximation, with the Inaba results for individual fluctuating impurities [15].
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Thus the coherent G(z) CPA is a homogeneous Green function and its matrix elements can be read from ( G(z) CPA )n,m = F (z)ρ(z)|n−m| , where F −1 (z) = 2v0 x(z)2 − 1 and ρ(z) = x(z) − x(z)2 − 1 with x(z) = (z + ωD (z))/2v0 .
(6)
4. The static coherent medium ωC (z) In order to give a final expression for the dynamical coherent medium ωD (z), we have to perform the CPA for the Hamiltonian HS . The basic assumption of static CPA tells us that the diagonal part of the Green function of a real atom (impurity) embedded in a coherent medium, averaged over the possible values of the random variables, should equal the corresponding Green function of the coherent medium itself [17]; it leads to ω − ωC (z) = 0, (7) 1 + (ω − ωC (z)) ( GS (z) CPA )0,0 P (ω)
where P (ω) = pδ(ω − ε) + qδ(ω + ε). When averaged over P (ω), the last equation leads to a cubic equation for the coherent medium ωC (z) ωC3 + a1 ωC2 + a2 ωC + a3 = 0,
(8)
with coefficients a1 =
z2 − 4v02 + 2ε 2 + ∆2 − 4∆z , 2(z − ∆)
a2 =
2∆(z∆ − (z2 − 4v02 )) , 2(z − ∆)
a3 =
∆2 (z2 − 4v02 ) − ε 4 , 2(z − ∆)
where we have denoted ∆ = ε(p−q). Introducing the quantities R = a1 a2 /6−a3 /2−a13 /27, Q = a2 /3−a12 /9 and s = R + Q3 + R 2 we can write the three roots in the form Xk = Sk +Tk −a1 /3, where Sk = |s|1/3 ei(arg[s]+2kπ)/3 , Tk = −Q/Sk for k = 0, 1, 2. It is possible to see that the real solutions do not contribute to the DOS. So we look for a range in energies z = E ∈ Re when there could be complex roots; in that range we choose the root having positive imaginary part, Im[ωC (z)] ≥ 0. In this paper, we will consider two concentrations from the interval 1/2 ≤ p ≤ 1 related to different physical situations. (a) The same concentration for the two kind of atoms, p = 1/2. Using the symmetry of this problem we are going to analyze only z = E ≥ 0, thus the coherent medium can be written in the form ωC (z) = X1 (z),
z = E ≥ 0.
(9)
(b) Concentration p = 0.8. In this case atoms B play the role of few random impurities in a homogenous basis represented by atoms A. Here the coherent medium cannot be defined from a single root for the whole E range.
5. Results for dynamical disorder Now we use ωD (z) = ωC (z + iν) to extend the static coherent medium to the dynamical case. First of all we should remark that for the dynamical case the three roots are complex, so in order to do the correct selection we have to consider the static (ν = 0) and the ν 1 limit. In the ν 1 limit we obtain perturbatively from (3) the corresponding dynamical coherent medium ωD (z)|ν1 ≈ ε m + iε 2
m2 /ν (which happens to be independent of z).
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Having found the dynamical coherent medium ωD (z), we can calculate the corresponding effective DOS at an arbitrary site, which is given by the imaginary part of the lateral averaged Green function for energy z = E π D(E) = −Im( G+ (E) CPA )0,0 .
(10)
We have calculated the DOS using G+ (E) = limη→0 G(E + iη) , because the dynamical coherent medium ωD (z) always has a positive imaginary part added to z, see x(z) in (6). The effective inverse localization length λ−1 (E) (or Liapunov coefficient) is approximated (upper bound) in terms of the off-diagonal elements of G+ (E) CPA : λ−1 (E) − lim
1
n→∞ n
ln |( G+ (E) CPA )n,0 |.
Thus using (6) we arrive at the expression λ−1 (E) ln |x(E) + x(E)2 − 1|. In the following we analyze separately the two concentrations defined above: (a) Concentration p = 1/2: In the dynamical case it is possible to see that the selection ωD (z) = X1 (z + iν) for z = E ≥ 0 is the correct one for all frequencies, and X1 (z + iν)|ν1 → iε 2 /ν. In Fig. 1 we show the effective DOS and λ−1 for several values of the frequency ν (the correlation time of the DD is ν −1 ). We only show positive values of energy E because of the symmetry D(E) = D(−E) and λ−1 (E) = λ−1 (−E). Note that from a white-Gaussian DD of intensity Γ , as in the diagonal case of [7], it is possible to obtain the exact average G(z) W = [z − HW (z)]−1 with HW (z) = −(iΓ /2) n |n n| + H0 . In Fig. 1c and d we show
Fig. 1. (a) and (b) Effective DOS and inverse localization length vs. energy for several frequencies ν of the KAP model of DD, with p = 1/2 and ε = 1. In (c) and (d) we also show these results for a white-Gaussian DD with intensity Γ = 2ε 2 /ν. We use = 1 and the half band width 2υ0 = 1.
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Fig. 2. Effective DOS vs. energy for several frequencies ν of the KAP model of DD, with p = 1/2 (ε = 1, 2υ0 = = 1). For ν → 0 we reobtain the static disordered case, and for ν 1 the effective DOS approaches the corresponding ordered one.
2 the corresponding DW (E) and λ−1 W (E). In order to compare both models of DD we have used Γ = 2ε /ν. We can see that for Γ / 4v0 both types of DD show a tendency of restoring—in mean value—a DOS with the usual characteristics of an “ordered medium”. When Γ / > 4v0 the white-Gaussian and the dichotomic disorder show different characteristics, as can be seen in Fig. 1c. For dichotomic DD when ν → 0, Fig. 1a shows the expected quenched disorder results, which cannot be achieved by white-Gaussian DD. Concerning the effective λ−1 , for small frequencies ν (slow fluctuating potential) we expect to find localized states surviving during a long life-time, then in this case λ−1 can be interpreted as a measure of the inverse of localization length for these states. For high frequencies ν the fluctuating potential “erases” the disorder and the effective states are mostly extended; then λ−1 has the sense of the inverse coherence length. In Fig. 2 we present for dichotomic DD, a 3D plot showing with realism the tendency of D(E) for several values of ν, we see the expected behavior for low and high ν (the stochastic correlation time is ν −1 ). Similar conclusions come from changing the intensity of the impurity ε. We remark that the noise-averaged DOS and λ−1 both characterize the effective eigenstates, in the sense that we expect to find dynamical localized excitations with finite life-times if ν ≡ 0. (b) Concentration p = 0.8: In this case it is not possible to define the dynamical coherent medium from a single root for all frequencies ν and energies z = E, so ωD (z) is obtained for each particular frequency and energy from the cubic equation (8). For the high frequency limit ν 1, the correct selection is ωD (z) = X1 (z + iν) for all z = E. In Fig. 3 we show the effective DOS and λ−1 for intensity ε = 1 and several values of frequency ν. When ν ∼ 0 we reobtain the result given by CPA for this particular concentration [21]. In the figure we can see two separated areas for the effective DOS: one is centered at −ε corresponding to the homogeneous part of the alloy (atoms A) and the other part centered near the energy 1 + (ε/2v0 )2 showing the presence of few random impurities B. Fig. 3a indicates that for very small frequency, also the effective states corresponding to the homogeneous like part of the DOS have a certain degree of localization due to the presence of the disorder. In the high frequency limit ν 1, the system behaves as having, at each site, an averaged energy. Then the characteristics of the static disordered problem are erased, and an homogeneous TB is restored showing a zero λ−1 inside the band of states. Changing ε we see a similar behavior.
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Fig. 3. Effective DOS and inverse localization length vs. energy for several frequencies ν of the KAP model of DD, with p = 0.8 and ε = 1 (we use = 1 and the half band width 2υ0 = 1).
6. Discussion The purpose of this communication has been to present a systematic method that enables us to obtain analytical solutions, for single-particle quantities, of a TB in presence of DD represented by a KAP matrix. We emphasize that in our paper we use a dynamical CPA to calculate the averaged Green function for the stochastic problem in terms of the static disorder solution. More complex objects like two-particle correlation can in principle be analyzed in a similar way [16], so the information concerning the density matrix in the presence of DD could also be studied in the context of our dynamical CPA. Therefore the Laplace analysis of the generalized diffusion coefficient (second moment of the displacement) could be obtained. As matter of fact, a perturbative analysis of this second moment was analyzed by Inaba [14] (in the case of individual random impurities) predicting, therefore, a delocalization of the particle when ν ≡ 0 and a completely “ordered” TB behavior for ν → ∞. Our results are complementary in the sense that we predict an effective λ−1 which qualitatively decreases inside the band (where D(E) ≡ 0) when ν increases (increasing temperature T ), and finally we find extended states (λ−1 ∼ 0) in the rapid fluctuations limit −1 ν → ∞. Then using Borland’s conjecture to relate λ−1 to the conductivity σ ∝ e−const. λ , we qualitatively see that diffusion constant decreases when ν increases (via Einstein’s relation). Note that a similar relation between σ and λ−1 can also be obtained working with a white-Gaussian DD, in this case the calculation of the diffusion coefficient is exact [7]. We expect that the effective localized states are dynamical in the sense that a localized excitation in the system disappears after a life-time which is related to ν. Our conjecture concerning a dynamical localization is supported by the fact that solving a TB with one dynamical impurity leads to a representation in terms of quasi-particles
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(related to the effective localized states) having a characteristic energy and life-time which is longer when ν is small [18]. In the one stochastic impurity case the dynamical localization phenomenon was a clear consequence of the occurrence of complex poles in the averaged Green function, thus indicating irreversible processes in the system. Unfortunately the dynamical CPA solution, for the N -stochastic impurities case, cannot lead to the interesting picture of quasi-particles nor to the exact calculation of the delocalization life-times. We also show that the presence of DD broadens the sharp-functions that appear in the DOS when using quenched disorder, consequently we believe that a dynamical CPA improves the calculation of the DOS in the case of DD. This fact can also be seen from our previous analysis concerning a TB with one dynamical impurity; in that case the problem has an exact solution [18]. On the contrary a model with N dynamical impurities or a KAP matrix can only be solved perturbatively [13,14], or using a dynamical CPA as we present here. The effective DOS, predicted by our dynamical CPA is reliable due to the spreading phenomena occurring in presence of DD. The case when the DD is of the off-diagonal form can be worked out in a similar way. Also the interesting general kangaroo process (with a power-law temporal correlation function) can be used as DD model. We also note that if we think that the stochastic character of the Hamiltonian represents internal fluctuations produced by a thermal bath at temperature T (that can be related to the frequency ν), our averaged DOS helps to calculate effective equilibrium thermodynamics quantities; for example, the specific heat, etc. Several quantum many-body problems, to which the dynamical mean field theory was successfully applied [22], motivate us to believe that our one-particle dynamical CPA can also be useful to study dynamical spin fluctuating systems.
Acknowledgements MOC thanks CONICET for Grant 4948/96. We also thank to Prof. V. Grünfeld for the English revision. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
A.K. Harrison, R. Zwanzig, Phys. Rev. A 32 (1985) 1072. S.D. Druger, M.A. Ratner, A. Nitzan, Phys. Rev. B 31 (1985) 3938. C.E. Budde, M.O. Cáceres, Phys. Rev. Lett. 60 (1988) 2712. T. Odagaki, M. Lax, Phys. Rev. B 24 (1981) 5284. B. Kramer, A. MacKinnon, Rep. Prog. Phys. 56 (1993) 1469. A.A. Ovchinnikov, N.S. Érikhman, Sov. Phys. JETP 40 (1975) 733. A. Madhukar, W. Post, Phys. Rev. Lett. 39 (1977) 1424. A.M. Jayannavar, M. Kumar, Phys. Rev. Lett. 48 (1982) 553. A.M. Jayannavar, Phys. Rev. E 48 (1993) 837. W. Fischer, H. Leschke, P. Müller, Phys. Rev. Lett. 73 (1994) 1578. A.A. Budini, A.K. Chattah, M.O. Cáceres, J. Phys. A: Math. Gen. 32 (1999) 631. D. Berchier, K. Maschke, J. Phys.: Cond. Matter 3 (1991) 2881. K. Kitahara, J.W. Haus, Z. Phys. B 32 (1979) 419. Y. Inaba, J. Phys. Soc. Jpn. 50 (1981) 2473. Y. Inaba, J. Phys. Soc. Jpn. 52 (1983) 3144. A. Brissaud, U. Frisch, J. Math. Phys. 15 (1974) 524. A. Gonis, Green Functions for Ordered and Disordered Systems, North-Holland, New York, 1992. M.O. Cáceres, A.K. Chattah, Phys. Lett. A 276 (2000) 272; A.K. Chattah, M.O. Cáceres, J. Phys. A: Math. Gen. 34 (2001) 5795. J.G. Hunt, D. Bloor, B. Movaghar, J. Phys. C: Solid State Phys. 16 (1983) L623–L628. S.A. Gurvitz, Phys. Rev. Lett. 85 (2000) 812. B. Velický, S. Kirkpatrick, H. Ehrenreich, Phys. Rev. 175 (1968) 747. A. Georges, G. Kotliar, W. Krauth, M.J. Rozenberg, Rev. Mod. Phys. 68 (1996) 13.
Effective eigenstates in a dynamical disordered tight-binding model A. Karina Chattah a,∗ , Manuel O. Cáceres b a
b
Facultad de Matemática Astronom´ıa y F´ısica, Universidad Nacional de Córdoba, 5000 Córdoba, Argentina Centro Atómico Bariloche and Instituto Balseiro, CNEA and Universidad Nacional de Cuyo, Av. Bustillo Km 9.5, 8400 Bariloche, Argentina
Abstract We study effective eigenstates—in mean value—for a 1D tight-binding (TB) Hamiltonian in the presence of diagonal dynamical disorder (DD). The averaged density of states (DOS) and localization length are shown as a function of the intensity of disorder and the correlation time of its temporal fluctuations. A delocalization electronic phenomenon is interpreted in terms of the localization length of the wave function, as a function of the time correlation of the DD. © 2002 Elsevier Science B.V. All rights reserved. PACS: 71.55; 02.50.Ey; 02.50.Wp; 05.30.−d Keywords: Tight-binding; Dynamical disorder; Kubo–Anderson process; Delocalization
1. Introduction In classical statistical mechanics the effects of dynamical disorder (DD) on the transport coefficient is well understood. A strong quenched disorder leads to sub-diffusion, but the presence of DD breaks down this classical localization, restoring a finite static conductivity [1–3]. The mathematical reason for this phenomenon is that DD introduces a shift in the generalized diffusion coefficient, which is ultimately the one responsible for reestablishing the analyticity at zero frequency. This shift is proportional to the inverse of the correlation time of the stochastic process which mimics the DD. This problem can be solved in a perturbative way, e.g. using the effective medium approximation (EMA) [4] for dichotomic disorder. This approximate solution interpolates smoothly between static disorder and the limit of rapid temporal fluctuations, thus giving support to what physical intuition expects of the model. In quantum mechanics the conditions for localization are not as simple as in classical mechanics. The simplest model goes back to Anderson’s pioneer work, where a spinless non-interacting particle moving in a quenched random potential was studied. Since then, a lot of work has been done concerning transport in random media [5]. But the interesting quantum problem of DD was not studied until 1975 by Ovchinnikov and Érikhman [6], later on Madhukar and Post [7] reported an exact solution for the motion of a particle in a system with site diagonal and nearest-neighbor off-diagonal DD, using Gaussian white noises. Even when this result was of particular importance some doubts concerning the mobility were reported later [8,9]. This continuous quantum model was “exactly” solved ∗
Corresponding author. E-mail addresses: [email protected] (A.K. Chattah), [email protected] (M.O. C´aceres). 0167-2789/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 2 7 8 9 ( 0 2 ) 0 0 5 1 4 - 6
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thanks to the white-Gaussian character of the stochastic potential. Later on, both models [7,8] gave insight into the problem of quantum dissipation. As a matter of fact a stochastic Hamiltonian can help to imitate irreversibility, so non-equilibrium theories were also inspired in those pioneer works [10]. The success of a stochastic Hamiltonian lies on the “approximation” of taking a large number of fast degrees of freedom (unknown in detail) into account, and the correct emulation of those degrees of freedom by suitable noises [11]. A related model of DD was also used to study the evolution of electron states in terms of quasi-stationary solutions of a time-dependent Hamiltonian [12]. A breakthrough in the analysis of quantum diffusion in the presence of a temporally fluctuating potential, to emulate the random excitation of phonons, was achieved introducing non-white stochastics modulations. In particular many dichotomous processes (noises) were introduced in a tight-binding (TB) Hamiltonian to represent the DD; this model was solved by approximate methods to calculate the transport coefficient [13–15]. Nevertheless, the character of the associated effective eigenstates has not been explored in detail. By effective eigenstates we mean averaged eigenstates over the temporal realizations of the DD. Note that in a DD model the time dependence of the disorder is taken to be parametric, in the sense that the electron–phonon interaction, responsible of the DD, does not introduce additional degrees of freedom into the Hamiltonian. Thus the effective eigenstates can be analyzed by studying the noise average of the density of states (DOS) and of the localization length. These quantities can be calculated from the mean value of the associated Green function of the system. As we mentioned earlier if the DD is not white-Gaussian the resulting problem cannot be solved in an exact way. Thus in the present paper we use a reported theorem concerning the Kubo–Anderson process (KAP) [16] and the coherent potential approximation (CPA) [17] to tackle a site disorder dynamical model. In the case of having only one stochastic impurity the effective properties were helpful to understand the phenomenon of the dynamical localization for the excitations of the system [18]. In that reference we were able to calculate the exact mean propagator of the system. In this way we could define localized states with finite life-times: Im[xk ]−1 (i.e. the dynamical localization), where xk are the poles of the corresponding Green function, which depend on the impurity stochastic correlation time ν −1 . Therefore an infinite life-time is only obtained in the limit of quenched disorder. For the one impurity case the effective eigenstates—characterized by the averaged DOS and by the inverse of the localization length—gave a complementary point of view to analyze the delocalization phenomena. This type of model mimics the stochastic transitions of a single impurity ion between different equivalent positions (i.e. a temperature-dependent trap in a polymer sample [19]). Also our model of one stochastic impurity can be related to the problem of delocalization due to measurement (i.e. interaction with a detector apparatus). In [20] the decoherence—generated by the interaction with the detector—was analyzed by studying the stationary state of the reduced density matrix. There it was shown that the environment destroys the localization, therefore affecting the time dependence of the observed system.
2. Dynamical disorder Here we are interested in the averaged properties coming from the stochastic Schrödinger equation iψ˙ = H (t)ψ ≡ (H0 + εH1 (t))ψ. The total Hamiltonian is composed of an ordered part H0 given by the hopping term of a TB model (written in the Wannier basis {|n }), and a stochastic contribution H1 (t) which introduces an energy-site DD model, i.e. for n ∈ [−∞, ∞]: H (t) = υ0 |n n ± 1| − ε mn (t)|n n|. (1) n
n
Here ε is the amplitude of the disorder and {mn (t)} represents a KAP set (leading to a “renewal” model for the disordered energies). The KAP stochastic set is constituted by independent dichotomous processes mn (t) all having
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the same jumping times ti which are uniformly and independently distributed with a density ν (Poisson distribution). Then each mn (ti ) = m is a random variable that takes one of the values ±1 with probability p, q, respectively. In the stationary regime the mean value is mn (t) = m , ∀n and the correlations are given by
mn (t)mk (t ) = δn,k
m2 exp[−ν|t − t |]. In general we assume that ν is an increasing function of the temperature T . Now we look for an expression for the mean-value Green function G(z) , this means taking the average over the DD. For this we take into account that the set {mn (t)} is a KAP one, thus the Hamiltonian H (t) is a KAP matrix as defined in [16]. In terms of the propagator defined by ψ(t) = G(t)ψ(0), and using the initial condition G(0) = I the exact average is t
G(t) = GS (t) exp[−νt] + ν dt exp[−ν(t − t )] GS (t − t ) G(t ) . (2) 0
Here GS (t) = exp[tHS ] is the t-dependent Green function associated to the Hamiltonian (1) in the static limit (quenched disorder). This means that when ν = 0, H ≡ HS = H0 − ε n mn |n n|, where mn are statistically independent random variables taking values mn = ±1 with probability p, q. We now ∞ define the averaged Green function introducing the Laplace transform of the propagator G(z) = −(i/) 0 dt G(t) exp[izt/] (denoted by its argument z). Thus from (2) the averaged Green function is given by
G(z) = [I − iν GS (z + iν) ]−1 GS (z + iν) ,
(3)
where GS (z) = [z − HS ]−1 . It means that G(z) is given in terms of the averaged Green function for the corresponding quenched disorder problem. From G(z) we are now able to calculate single-particle properties, for example, the averaged DOS and localization length; both quantities give insight into the localization character of the eigenstates in presence of DD.
3. CPA for dynamical disorder Eq. (3) is important because it gives the exact expression for G(z) in terms of GS (z) . So the difficulty in averaging over the DD has been moved to the problem of calculating GS (z) ; of course this is not a trivial task. Here the static case (ν = 0) corresponds to a binary alloy with concentrations p, q of atoms A, B, respectively; even in this case GS (z) cannot be calculated exactly [17,21]. In what follows we use the CPA to calculate the averaged static Green function. Then we can approximate
GS (z) ∼ = [z − HC (z)]−1 ,
(4)
where the coherent Hamiltonian is given by HC (z) = −ωC (z) n |n n| + H0 . Here we have chosen the minus sign in front of the coherent medium ωC (z) for future convenience. Inserting (4) in (3) we finally arrive at a closed expression for G(z) CPA for DD
G(z) CPA = [z − HC (z + iν)]−1 ,
(5)
where HC (z + iν) is defined through a dynamical coherent medium ωD (z) = ωC (z + iν). This is one of the goals of our work, allowing us to calculate several effective single-particle properties for a given system under the influence of DD. The shift in the dynamical coherent medium is equivalent to the result predicted by Harrison and Zwanzig [1] when studying, with EMA, DD in a classical master equation (leading to the suppression of the percolation threshold). In our case the shift comes from the analytically solved KAP model (3) and coincides, in the CPA approximation, with the Inaba results for individual fluctuating impurities [15].
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Thus the coherent G(z) CPA is a homogeneous Green function and its matrix elements can be read from ( G(z) CPA )n,m = F (z)ρ(z)|n−m| , where F −1 (z) = 2v0 x(z)2 − 1 and ρ(z) = x(z) − x(z)2 − 1 with x(z) = (z + ωD (z))/2v0 .
(6)
4. The static coherent medium ωC (z) In order to give a final expression for the dynamical coherent medium ωD (z), we have to perform the CPA for the Hamiltonian HS . The basic assumption of static CPA tells us that the diagonal part of the Green function of a real atom (impurity) embedded in a coherent medium, averaged over the possible values of the random variables, should equal the corresponding Green function of the coherent medium itself [17]; it leads to ω − ωC (z) = 0, (7) 1 + (ω − ωC (z)) ( GS (z) CPA )0,0 P (ω)
where P (ω) = pδ(ω − ε) + qδ(ω + ε). When averaged over P (ω), the last equation leads to a cubic equation for the coherent medium ωC (z) ωC3 + a1 ωC2 + a2 ωC + a3 = 0,
(8)
with coefficients a1 =
z2 − 4v02 + 2ε 2 + ∆2 − 4∆z , 2(z − ∆)
a2 =
2∆(z∆ − (z2 − 4v02 )) , 2(z − ∆)
a3 =
∆2 (z2 − 4v02 ) − ε 4 , 2(z − ∆)
where we have denoted ∆ = ε(p−q). Introducing the quantities R = a1 a2 /6−a3 /2−a13 /27, Q = a2 /3−a12 /9 and s = R + Q3 + R 2 we can write the three roots in the form Xk = Sk +Tk −a1 /3, where Sk = |s|1/3 ei(arg[s]+2kπ)/3 , Tk = −Q/Sk for k = 0, 1, 2. It is possible to see that the real solutions do not contribute to the DOS. So we look for a range in energies z = E ∈ Re when there could be complex roots; in that range we choose the root having positive imaginary part, Im[ωC (z)] ≥ 0. In this paper, we will consider two concentrations from the interval 1/2 ≤ p ≤ 1 related to different physical situations. (a) The same concentration for the two kind of atoms, p = 1/2. Using the symmetry of this problem we are going to analyze only z = E ≥ 0, thus the coherent medium can be written in the form ωC (z) = X1 (z),
z = E ≥ 0.
(9)
(b) Concentration p = 0.8. In this case atoms B play the role of few random impurities in a homogenous basis represented by atoms A. Here the coherent medium cannot be defined from a single root for the whole E range.
5. Results for dynamical disorder Now we use ωD (z) = ωC (z + iν) to extend the static coherent medium to the dynamical case. First of all we should remark that for the dynamical case the three roots are complex, so in order to do the correct selection we have to consider the static (ν = 0) and the ν 1 limit. In the ν 1 limit we obtain perturbatively from (3) the corresponding dynamical coherent medium ωD (z)|ν1 ≈ ε m + iε 2
m2 /ν (which happens to be independent of z).
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Having found the dynamical coherent medium ωD (z), we can calculate the corresponding effective DOS at an arbitrary site, which is given by the imaginary part of the lateral averaged Green function for energy z = E π D(E) = −Im( G+ (E) CPA )0,0 .
(10)
We have calculated the DOS using G+ (E) = limη→0 G(E + iη) , because the dynamical coherent medium ωD (z) always has a positive imaginary part added to z, see x(z) in (6). The effective inverse localization length λ−1 (E) (or Liapunov coefficient) is approximated (upper bound) in terms of the off-diagonal elements of G+ (E) CPA : λ−1 (E) − lim
1
n→∞ n
ln |( G+ (E) CPA )n,0 |.
Thus using (6) we arrive at the expression λ−1 (E) ln |x(E) + x(E)2 − 1|. In the following we analyze separately the two concentrations defined above: (a) Concentration p = 1/2: In the dynamical case it is possible to see that the selection ωD (z) = X1 (z + iν) for z = E ≥ 0 is the correct one for all frequencies, and X1 (z + iν)|ν1 → iε 2 /ν. In Fig. 1 we show the effective DOS and λ−1 for several values of the frequency ν (the correlation time of the DD is ν −1 ). We only show positive values of energy E because of the symmetry D(E) = D(−E) and λ−1 (E) = λ−1 (−E). Note that from a white-Gaussian DD of intensity Γ , as in the diagonal case of [7], it is possible to obtain the exact average G(z) W = [z − HW (z)]−1 with HW (z) = −(iΓ /2) n |n n| + H0 . In Fig. 1c and d we show
Fig. 1. (a) and (b) Effective DOS and inverse localization length vs. energy for several frequencies ν of the KAP model of DD, with p = 1/2 and ε = 1. In (c) and (d) we also show these results for a white-Gaussian DD with intensity Γ = 2ε 2 /ν. We use = 1 and the half band width 2υ0 = 1.
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Fig. 2. Effective DOS vs. energy for several frequencies ν of the KAP model of DD, with p = 1/2 (ε = 1, 2υ0 = = 1). For ν → 0 we reobtain the static disordered case, and for ν 1 the effective DOS approaches the corresponding ordered one.
2 the corresponding DW (E) and λ−1 W (E). In order to compare both models of DD we have used Γ = 2ε /ν. We can see that for Γ / 4v0 both types of DD show a tendency of restoring—in mean value—a DOS with the usual characteristics of an “ordered medium”. When Γ / > 4v0 the white-Gaussian and the dichotomic disorder show different characteristics, as can be seen in Fig. 1c. For dichotomic DD when ν → 0, Fig. 1a shows the expected quenched disorder results, which cannot be achieved by white-Gaussian DD. Concerning the effective λ−1 , for small frequencies ν (slow fluctuating potential) we expect to find localized states surviving during a long life-time, then in this case λ−1 can be interpreted as a measure of the inverse of localization length for these states. For high frequencies ν the fluctuating potential “erases” the disorder and the effective states are mostly extended; then λ−1 has the sense of the inverse coherence length. In Fig. 2 we present for dichotomic DD, a 3D plot showing with realism the tendency of D(E) for several values of ν, we see the expected behavior for low and high ν (the stochastic correlation time is ν −1 ). Similar conclusions come from changing the intensity of the impurity ε. We remark that the noise-averaged DOS and λ−1 both characterize the effective eigenstates, in the sense that we expect to find dynamical localized excitations with finite life-times if ν ≡ 0. (b) Concentration p = 0.8: In this case it is not possible to define the dynamical coherent medium from a single root for all frequencies ν and energies z = E, so ωD (z) is obtained for each particular frequency and energy from the cubic equation (8). For the high frequency limit ν 1, the correct selection is ωD (z) = X1 (z + iν) for all z = E. In Fig. 3 we show the effective DOS and λ−1 for intensity ε = 1 and several values of frequency ν. When ν ∼ 0 we reobtain the result given by CPA for this particular concentration [21]. In the figure we can see two separated areas for the effective DOS: one is centered at −ε corresponding to the homogeneous part of the alloy (atoms A) and the other part centered near the energy 1 + (ε/2v0 )2 showing the presence of few random impurities B. Fig. 3a indicates that for very small frequency, also the effective states corresponding to the homogeneous like part of the DOS have a certain degree of localization due to the presence of the disorder. In the high frequency limit ν 1, the system behaves as having, at each site, an averaged energy. Then the characteristics of the static disordered problem are erased, and an homogeneous TB is restored showing a zero λ−1 inside the band of states. Changing ε we see a similar behavior.
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Fig. 3. Effective DOS and inverse localization length vs. energy for several frequencies ν of the KAP model of DD, with p = 0.8 and ε = 1 (we use = 1 and the half band width 2υ0 = 1).
6. Discussion The purpose of this communication has been to present a systematic method that enables us to obtain analytical solutions, for single-particle quantities, of a TB in presence of DD represented by a KAP matrix. We emphasize that in our paper we use a dynamical CPA to calculate the averaged Green function for the stochastic problem in terms of the static disorder solution. More complex objects like two-particle correlation can in principle be analyzed in a similar way [16], so the information concerning the density matrix in the presence of DD could also be studied in the context of our dynamical CPA. Therefore the Laplace analysis of the generalized diffusion coefficient (second moment of the displacement) could be obtained. As matter of fact, a perturbative analysis of this second moment was analyzed by Inaba [14] (in the case of individual random impurities) predicting, therefore, a delocalization of the particle when ν ≡ 0 and a completely “ordered” TB behavior for ν → ∞. Our results are complementary in the sense that we predict an effective λ−1 which qualitatively decreases inside the band (where D(E) ≡ 0) when ν increases (increasing temperature T ), and finally we find extended states (λ−1 ∼ 0) in the rapid fluctuations limit −1 ν → ∞. Then using Borland’s conjecture to relate λ−1 to the conductivity σ ∝ e−const. λ , we qualitatively see that diffusion constant decreases when ν increases (via Einstein’s relation). Note that a similar relation between σ and λ−1 can also be obtained working with a white-Gaussian DD, in this case the calculation of the diffusion coefficient is exact [7]. We expect that the effective localized states are dynamical in the sense that a localized excitation in the system disappears after a life-time which is related to ν. Our conjecture concerning a dynamical localization is supported by the fact that solving a TB with one dynamical impurity leads to a representation in terms of quasi-particles
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(related to the effective localized states) having a characteristic energy and life-time which is longer when ν is small [18]. In the one stochastic impurity case the dynamical localization phenomenon was a clear consequence of the occurrence of complex poles in the averaged Green function, thus indicating irreversible processes in the system. Unfortunately the dynamical CPA solution, for the N -stochastic impurities case, cannot lead to the interesting picture of quasi-particles nor to the exact calculation of the delocalization life-times. We also show that the presence of DD broadens the sharp-functions that appear in the DOS when using quenched disorder, consequently we believe that a dynamical CPA improves the calculation of the DOS in the case of DD. This fact can also be seen from our previous analysis concerning a TB with one dynamical impurity; in that case the problem has an exact solution [18]. On the contrary a model with N dynamical impurities or a KAP matrix can only be solved perturbatively [13,14], or using a dynamical CPA as we present here. The effective DOS, predicted by our dynamical CPA is reliable due to the spreading phenomena occurring in presence of DD. The case when the DD is of the off-diagonal form can be worked out in a similar way. Also the interesting general kangaroo process (with a power-law temporal correlation function) can be used as DD model. We also note that if we think that the stochastic character of the Hamiltonian represents internal fluctuations produced by a thermal bath at temperature T (that can be related to the frequency ν), our averaged DOS helps to calculate effective equilibrium thermodynamics quantities; for example, the specific heat, etc. Several quantum many-body problems, to which the dynamical mean field theory was successfully applied [22], motivate us to believe that our one-particle dynamical CPA can also be useful to study dynamical spin fluctuating systems.
Acknowledgements MOC thanks CONICET for Grant 4948/96. We also thank to Prof. V. Grünfeld for the English revision. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
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