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, for a periodic chain of [MATH] sites, as [EQUATION] The phases [MATH] have the same properties as before, being uniformly distributed in [MATH] . We can reduce this definition to our formulation by writing the Bloch wave-numbers as |
[EQUATION] so that Eq. 18 becomes [EQUATION] Since this sum is carried only over the positive half of the first Brillouin zone (i.e., [MATH] [MATH] ), it can be rewritten as |
[EQUATION] with [MATH] defined as [EQUATION] and the independent random phases obeying the constraint [MATH] The [MATH] term is excluded as before, and we have introduced a normalization factor [MATH] that will define a finite variance for the local disorder. |
To study the thermodynamic limit ( [MATH] ) in the previous case (gaussian), we replaced all the sums over [MATH] by integrals. In this case, since [MATH] [MATH] |
we could try to do the same, but this turns out to be quite tricky due to the possibility of generating low- [MATH] singularities. Consider, as an example, the calculation of the disorder’s local variance, |
[EQUATION] for [MATH] the corresponding integral does not have a low- [MATH] singularity and the situation is be very similar to a system with uncorrelated disorder. A more interesting case happens for [MATH] where the integrals will have low- [MATH] singularities with a natural cut-off of [MATH] . At the same time, in... |
[EQUATION] is found to converge as [MATH] . These two facts mean that, no matter how large [MATH] is, the number of terms contributing to the sum is always of [MATH] . Hence, we can never approximate it by an integral. Luckily, the infinite sum in Eq. 24 |
is known to define the Riemann Zeta function [EQUATION] Finally, in the same limit, the local variance of the disorder can be written as |
[EQUATION] allowing us to express [MATH] in terms of [MATH] as follows, [EQUATION] The correlation function of this potential can also be calculated using, |
[EQUATION] Writing [MATH] and taking the thermodynamic limit in the last sum, we can express the result in terms of a polylogarithm function |
[MATH] as follows, [EQUATION] A plot of this space correlation function is shown in the Figure for several values of the exponent [MATH] |
As a last remark, we note that to ensure a finite local variance, [MATH] , we had to choose [MATH] (see Eq. 27 ). This weird fact implies that [MATH] as [MATH] (for [MATH] ), which will have important consequences in what follows. |
2.2 The Kernel Polynomial Method The spectral function of a large disordered quantum system can be efficiently computed by a polynomial expansion-based technique — the Kernel Polynomial Method (KPM) |
In this approach, a function of an operator with spectrum normalized to the interval [MATH] is approximated by a truncated Chebyshev series. The expansion coefficients can be computed either by the stochastic evaluation of a trace |
or by the expectation values of Chebyshev polynomials in a given basis. Furthermore, the accuracy and numerical convergence of the KPM estimates are controlled by employing an optimized Gibbs damping factor and using sufficient number of Chebyshev polynomials |
The Chebyshev polynomial of the first kind, [MATH] , is an [MATH] -degree polynomial in [MATH] , defined as [EQUATION] where [MATH] takes values in the interval [MATH] . Moreover, the [MATH] ’s are |
[EQUATION] and also satisfy the orthogonality relation [EQUATION] In our case, we consider a free electron gas hopping on a finite cyclic chain of size [MATH] , under the influence of on-site correlated disorder. Suppose that the [MATH] Hamiltonian matrix [MATH] (Eq. ), has eigenvalues [MATH] with corresponding eigenst... |
[EQUATION] where [MATH] is a Bloch state of one electron as defined in last section. Notice also that, in the absence of disorder [MATH] and by summing [MATH] over [MATH] one obtains the density of states. |
To calculate [MATH] we must normalize the Hamiltonian, so that its spectrum fits inside the interval [MATH] . The KPM approximation to the spectral function is written as |
[EQUATION] where the expansion coefficients [MATH] are determined as [EQUATION] The recursion relations obeyed by the Chebyshev polynomials carry over to these moments, and greatly simplify their calculation. The expression Eq. 33 , represents the truncated sum of the Chebyshev series. It is known that the abrupt trunc... |
Jackson Kernel [MATH] defined as follows [EQUATION] The use of this kernel does not alter the series’ convergence to the intended function, as [MATH] goes to infinity. Furthermore, this makes the KPM approximations always non-negative, which is particularly relevant when approximating a non-negative function, like [MAT... |
Numerical Results and Discussion We have performed numerical computations of the spectral functions for the 1D non-interacting system in the presence of on-site gaussian and power-law correlated disorder with periodic boundary conditions, at zero temperature. The computations were carried out by using the KPM. For comp... |
3.1 Gaussian Correlated Disorder We start by presenting results for the spectral function in the uncorrelated Anderson model. For a rectangular distribution of site energies, |
[EQUATION] and [MATH] . The strength of disorder is commonly characterized by [MATH] , but as we are interested in other types of distributions for the site energies, in this paper we use [MATH] instead. |
In Figure we show the approximated spectral function for various values of the local variance [MATH] at the band center, i.e. [MATH] |
[MATH] . The data is well fitted by a lorentzian, as expected from perturbation theory. In the inset, we show a comparison between the half-width of the lorentzian, obtained from the fits, and the value calculated from the Born approximation. |
[EQUATION] This perturbative result seems to give a good account of the data until values [MATH] The spectral function, at the band center ( [MATH] for a gaussian correlated disorder with different values of the parameter [MATH] is shown in Figure for [MATH] The magenta dashed curves are the corresponding fits. For [MA... |
[Figure (a)], the best fit of the numerical data can be found with a lorentzian of width [MATH] When [MATH] , the scattering becomes local in momentum space, and the spectral function is seen to be a gaussian [Figure (b)]. Its width is just the variance of the site energies, [MATH] |
as can be seen in Figure , where the spectral functions for different values of [MATH] are scaled to show that [EQUATION] In Eq. 38 [MATH] |
is the normal distribution of mean [MATH] and variance [MATH] This result calls to mind the classical limit of the spectral function discussed by Trappe et. al. |
. In that limit, the disordered potential dominates, and the spectral function merely reflects the probability distribution of local potential values. This is, in fact, what is observed here. Since |
[EQUATION] in the thermodynamic limit (i.e. [MATH] ), the energy at each site is a sum of a large number of random independent variables, and by the central limit theorem, it is normally distributed. But what is significant here is that this limit can be obtained even when the disorder strength is small enough to be co... |
3.2 Power-Law Correlated Disorder A power-law correlated disorder is characterized by the exponent [MATH] that determines how fast the Fourier transform of [MATH] |
decays with the wavenumber [MATH] [EQUATION] As [MATH] increases, scattering becomes increasingly dominated by small values of [MATH] [MATH] . In Figure we see that a transition for a lorentzian to a gaussian shape (with unit variance) of the spectral function at the band center and for |
[MATH] , occurs at [MATH] . This transition seems to hold for other values of [MATH] as well, as can be seen from the left panels of Figure On closer scrutiny, however, a perfect gaussian fit is only possible for [MATH] , in the large [MATH] limit, and deviations become increasingly obvious as [MATH] increases; the spe... |
Even though the form of the spectral function is not a gaussian, one still observes (Figure ) a universal behavior, for different disorder strengths, similar to the one found for gaussian disorder, namely |
[EQUATION] with the [MATH] ) depending on [MATH] , but not on the disorder variance [MATH] As for the gaussian disorder case, we will show that the results of Figs. b and reveal the emergence of the classical limit, as a consequence of the local character of scattering in momentum space. |
3.3 Statistical properties of the spectral function in the thermodynamic limit Thus far we have discussed the disorder-averaged spectral function. It is not however clear if this quantity represents a typical value for measurable quantity of macroscopic systems. This becomes specially concerning in the case of the powe... |
To investigate this issue, we calculated the standard deviation of [MATH] for increasing number of sites and different values of the exponent [MATH] . These results are shown for two examples in Figure |
From the numerical data, we conclude that for [MATH] the standard deviation scales as [MATH] , which clearly indicates a self-averaging behavior. On the other hand, for [MATH] there seems to be a finite standard deviation for [MATH] , even in the thermodynamic limit, i.e. [MATH] still fluctuates from sample to sample i... |
In Figure , we can also see an example of the same calculation done for [MATH] , where no qualitative changes in the scaling behavior of [MATH] can be seen. Obviously, for very large values of [MATH] , these persistent fluctuations start to decrease, since the system is approaching an ordered limit ( [MATH] ). To sum u... |
Analytical Results and Discussion If the state at [MATH] is [MATH] , the probability amplitude that the state at time [MATH] is still the same is [MATH] Using a complete set of energy eigenstates [MATH] we can see that this amplitude is the Fourier transform of the spectral function defined in Eq . 32 |
[EQUATION] Expanding both sides in powers of [MATH] and averaging over disorder, we get the following expression for the [MATH] -moment of the disorder-averaged spectral function [MATH] |
[EQUATION] The Hamiltonian is the one defined in Eq. and can be written as [MATH] where [EQUATION] with the band Hamiltonian [MATH] being diagonal in the Bloch basis, and the disordered potential, [MATH] |
in the local Wannier basis. In the calculation of [MATH] we will assume that [MATH] . This is strictly true for the states in the center of the band (i.e [MATH] ), for which we calculated numerically the spectral function. However, this assumption implies no loss of generality, since for an arbitrary value [MATH] , we ... |
[EQUATION] such that [MATH] , remains true. The calculation will show that changing [MATH] only shifts the spectral function in energy, by the value of [MATH] |
4.1 Gaussian case As a justification for our numerical results, we managed to calculate the average spectral function for the infinite chain, with a gaussian model of correlated disorder. Generally, our analytical results will be valid in the limits when [MATH] and [MATH] |
4.1.1 Lowest Order Terms To illustrate the gist of the argument, we begin by looking at the lowest order moments, using the Eq. 43 |
It is obvious that for [MATH] the result is zero, because [MATH] and [MATH] . For [MATH] [EQUATION] and resolving the identity in the Bloch basis, |
[EQUATION] Recalling Eq. [EQUATION] By the same arguments, in the third moment only one term survives: [EQUATION] In the thermodynamic limit, the sum over [MATH] turns into an integral and if [MATH] , we can extend the integration range to [MATH] |
and expand [MATH] . In this case, the integrand is odd in [MATH] and the right-hand side of Eq. 49 vanishes upon integration. Finally, we tackle the [MATH] -moment (the last, before presenting the general argument), whose the only non-zero terms are |
[EQUATION] Using the same technique as above, the first term is [EQUATION] which is a complete gaussian integral (in the limit [MATH] ), whose value is |
[EQUATION] On the other hand, the term containing the [MATH] power of [MATH] is [EQUATION] The averages of these random phase factors are discussed in the In particular, we show that, in the thermodynamic limit ( [MATH] ), the expression above reduces to |
[EQUATION] Finally, by looking at the Eqs. 52 and 53 we see that, as long as [MATH] we can ignore terms that have insertions of [MATH] . Then, we simply write [MATH] |
as: [EQUATION] 4.1.2 General Expression for the Moments of [MATH] Inspired on the results above, we argue that the general form of the terms in Eq. 43 is: |
[EQUATION] [EQUATION] Furthermore, in the we show that the averages [MATH] have the following general form [EQUATION] Using the Eqs. 55 57 , in the thermodynamic limit ( [MATH] ), we can rebuild the entire Taylor series for the averaged diagonal propagator, and re-sum it as follows: |
[EQUATION] The spectral function is the time-domain Fourier transform of this last expression, yielding [EQUATION] which agrees with the results found in our numerical calculations, using the KPM. |
For the sake of completeness, we also state the result for a general value of [MATH] , which can be obtained from Eq. 58 simply by shifting the energy variable by the corresponding band energy |
[MATH] of that state, i.e. [EQUATION] In conclusion, we found that, if [MATH] and [MATH] then the disorder-averaged spectral function, in the thermodynamic limit, will have a gaussian shape. This is true, even if the disorder strength (measured by [MATH] ) is small, as long as this is matched by a decrease of [MATH] an... |
, where [MATH] is the disorder correlation length. 4.1.3 Emergence of the Classical Limit for the Spectral Function We were able to establish precise conditions in which the classical limit of the spectral function, found by Trappe et. |
al appears. The statement of this limit is equivalent to Eq. 55 and reads ( [MATH] [EQUATION] so that [EQUATION] Using the Wannier basis (eigenbasis of [MATH] ) and its transformation law to the Bloch basis [MATH] we can rewrite the above equation (with [MATH] ) as |
[EQUATION] where [MATH] is the probability distribution of a site energy. Comparing the above with Eq. 42 , we have [EQUATION] Thus, the averaged spectral function is just the probability distribution of a single site energy. As it is clear for the definition of the disordered potential (Eq. ), the distribution |
[MATH] must be a gaussian according of the Central Limit Theorem. 4.2 Power-Law Correlated Disorder 4.2.1 Validity of the Classical Limit |
In the case of Power-law correlated disorder, the argument leading to the Eq. 55 still holds, as long as [MATH] but requires a slightly different formulation. To see how this comes about, let us consider Eq. 51 as an example. In this case, we have |
[EQUATION] As before, if we expand [MATH] in powers of [MATH] , we get terms of the form [EQUATION] If [MATH] the sum above is convergent and the result vanishes, in the large- [MATH] limit, as [MATH] On the other hand, if [MATH] |
the sum diverges, but instead it can be written as an integral over the First Brillouin Zone, as follows [EQUATION] Both terms in the equation above go to zero in the thermodynamic limit, since [MATH] and [MATH] . This argument is obviously true for every term in [MATH] , containing insertions of [MATH] . Hence, in the... |
In this limit the spectral function can only depend on the parameters of the disordered potential, namely [MATH] and [MATH] Since [MATH] is dimensionless, there is a single energy scale, |
[MATH] , in [MATH] . The scaling of Eq. 41 , illustrated in Figure follows at once. It should be noted, however, that as [MATH] gets closer to 1, this scaling is not observed numerically. This is due to finite size effects that we have not accounted for. An example is the very slow convergence of [MATH] to |
[MATH] . For [MATH] for instance, the truncation error is still of order 10% for [MATH] 4.2.2 The Limiting Cases ( [MATH] and [MATH] And The Double-Peaked Shape |
Despite the validity of the classical limit for the averaged spectral function, we have shown in the that it is not clear how to obtain a closed form for the [MATH] -moment of [MATH] |
even in this limit. Nevertheless, the limit [MATH] revealed itself as very special case, where the exact averaged spectral function is found to be a gaussian, |
[EQUATION] This result is consistent with the numerical results obtained in the last section (see Figure ). For [MATH] , however, the higher cumulants of the spectral function cease to be zero, and [MATH] drifts away from a gaussian shape. For illustration, we have calculated the [MATH] -cumulant of the averaged spectr... |
. This has the following definition: [EQUATION] and can be directly computed using the expressions obtained in the i.e [EQUATION] |
Other than explaining the deviations from the gaussian shape that we found in the numerical plots of [MATH] , these effects have another striking consequence. According to our earlier remarks, in the classical limit, the averaged spectral function is the same as the probability distribution of the site energies. Since ... |
[EQUATION] When [MATH] , this sum is convergent in the [MATH] limit, which means that only a number of [MATH] of terms actually contribute to the variance of the local disorder [MATH] Furthermore, as [MATH] increases, this sum is dominated by less and less terms, meaning that we are never in the conditions of the centr... |
This becomes particularly clear in the extreme case [MATH] In this limit, the local value of the disordered potential is dominated by a single term, [MATH] , and the disorder is a static cosine potential with a wavelength [MATH] and a random phase, |
[EQUATION] The corresponding probability density function can be calculated, yielding the expression: [EQUATION] As an illustration, we depict in Figure |
the KPM calculated the spectral function for [MATH] and the normalized histogram of site energies for a single realization of disorder. As [MATH] increases above [MATH] , the spectral function smoothly approaches the limiting form of Eq. 72 , by first displaying a two peaked shape as illustrated in Figure 10 a. |
The expression of Eq. 72 also corresponds to the one we obtain numerically for gaussian disorder case when [MATH] (see Figure 10 b). In either case, of course, a single value [MATH] dominates the sum |
[EQUATION] and the two models of disorder cannot be distinguished. Conclusions We have studied the spectral function of Bloch states in a tight-binding chain, with two models of correlated disorder: the gaussian model (with a correlation length given by [MATH] ) and the power-law model (with an algebraic decay of corre... |
The analytical calculations of [MATH] were done in the thermodynamic limit, by resuming the short-time expansion of the diagonal propagator in momentum space. For the gaussian case, we found out that, in the regimes when [MATH] and the correlation length of the disorder is much larger than the lattice spacing ( [MATH] ... |
applied to our lattice system. In the power-law model, where there is no energy scale associated with the space-correlations, we still found that the averaged spectral function is given by its classical limit, but only if the exponent |
[MATH] , characterizing the algebraic decay of the power-spectrum, exceeds unity (while the delocalization of the eigenstates occurs only at [MATH] ). The mean spectral function is a gaussian in the limit [MATH] , but develops non-zero higher cumulants for larger values of [MATH] , reflecting the actual distribution of... |
[MATH] is a dimensionless parameter); hence, [MATH] must be a function of [MATH] . All these results are confirmed by our numerical calculations of [MATH] |
For the later model, we discovered that the standard deviation of the spectral function, for [MATH] , does not go to zero in the thermodynamic limit. This means that in the non-perturbative regime, the spectral function is not a self-averaging quantity and remains sample dependent in the infinite size system. While thi... |
[MATH] is a crossover point for these potentials. More surprisingly, the results on the single-particle spectral function do not seem to give any indication that [MATH] is a special point for these models, as was argued by Petersen et al |
in relation to the predicted delocalization transition. Granted that there is no obvious relation between the spectral function and the localization/delocalization of the eigenstates, one could still expect that a qualitative change in the disordered potential might show up at the transition point. Yet, we found no suc... |
In conclusion, we studied the spectral function in a 1D band model with correlated disorder. Through a combination of numerical and analytical work we were able to obtain results in a non-perturbative regime, and show explicitly how the classical limit of the spectral function emerges |
. In the case of power-law disorder, this happens when the local distribution of site energies is not gaussian, due to inapplicability of the central limit theorem. The localization transition in these models occurs deep in the region where the spectral function is classical, and that raises the question of whether som... |
Acknowledgments For this work, N. A. Khan was supported by the grants ERASMUS MUNDUS Action 2 Strand 1 Lot 11, EACEA/42/11 Grant Agreement 2013-2538 / 001-001 EM Action 2 Partnership Asia-Europe and research scholarship UID/FIS/04650/2013 of Fundação da Ciência e Tecnologia; J. P. Santos Pires was supported by the MAP-... |
The work at Centro de Física do Porto, as a whole, is supported by the grant UID/FIS/04650/2013 of Fundação da Ciência e Tecnologia. |
We would also like to thank the referee for its suggestions about the pathological properties of the power-law correlated model, which drove us to extend the paper with the data shown in the Subsection |
3.3 Appendix A Random Phase Averages In section , we needed to calculate terms of the form [EQUATION] where [MATH] are independent random phases with an uniform distribution in the circle and obeying the constraint [MATH] . These expressions appear inside sums over momenta, of the form |
[EQUATION] where [MATH] Clearly, since these phases are uniformly distributed independent variables (except in the case [MATH] ), we have |
[EQUATION] Therefore, we can only obtain a non zero result if all the phase factors are paired. This means that [MATH] is zero unless [MATH] |
General Procedure To actually calculate the phase averages, we may start with the following illustrative case: [EQUATION] To prevent lengthy notation, we define |
[EQUATION] such that [MATH] . Note also, that since [MATH] , the contraction of two momenta is equivalent to a Kronecker delta in the momentum sums. |
Hence, we can write [EQUATION] and repeat the process until we exhaust all possibilities. In this case, we just need to do it once, |
[EQUATION] so [EQUATION] Finally, if we express everything in terms of Kronecker deltas (using [MATH] ), we get [EQUATION] The left-hand side of the above equation can be divided in three groups of terms: |
1. The first three terms correspond to all the pairwise contractions of momenta, which gives a contribution of the form: [EQUATION] |
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