Real-space renormalization-group treatment of quadratic chains

We have recently proposed a one-dimensional nonperiodic chain with lattice positions at 02 d, 12 d, 22 d, ... with length d a constant. The spectrum is singular-continuous, and for weak potential, the states are all extended apart from a trivial set of localized states. In this study, we obtain the exact extended-state spectrum of the quadratic chain in a nearest-neighbor tight-binding model where the quadratic modulation is in the onsite matrix elements. Then, a real-space renormalization-group method (RSRG) is used by decimation to reduce the transfer matrix for the chain into self-similar matrix products. The RSRG decimation scheme is used here to organize the calculation and facilitate numerical computation. The extended-state spectrum appears as minibands broken by numerous gaps. Previous work on quadratic chains shows that the structure factor is singular-continuous and given by a dense set of states with wavevectors with scaling exponent γ(k) = 2 as in periodic and quasi-periodic chains. The origin of extended states in this nonperiodic lattice appears to arise from a type of mechanism not yet identified in deterministic nonperiodic lattices, and is based on a hidden symmetry giving rise to an energy-dependent translational invariance of the transfer matrix.


Introduction
Interest in nonperiodic lattices has undergone explosive growth since Shechtman first identified an Al-Mn alloy as a quasicrystal based on x-ray diffraction experiments [1].Shortly thereafter, intensive work began on the Fibonacci chain, which for the proper parameter choice is a one-dimensional quasicrystal [2][3][4][5].Many studies of other nonperiodic lattices followed suit, with much attention focused on semiconductor superlattices, which provide a platform to realize nonreriodic chains [6,7].A quantity associated with lattices-periodic and nonperiodic-is the structure factor (SF) S 0 (k) with wavevector k which gives the x-ray and neutron diffraction scattering strength in the first Born approximation.It also provides insight into other physical properties.One of the key pieces of information obtained from the SF is the scaling exponent γ(k) given by L γ( k) = S 0 (k) as the lattice length L increases (assuming power-law behavior).For example, for periodic and quasi-periodic (such as Fibonacci) lattices, γ(k) = 2, while for Thue-Morse chains, γ(k) is between 1 and 2.
The existence of extended states in nonperiodic lattices of various sorts has attracted considerable attention in view of the localization of all states in random one-dimensional chains [8].As noted above, the Fibonacci chain has undergone intensive investigation, but many other types of nonperiodic lattices that have been studied.Because a number of these lattice can be considered as intermediate between periodic (in which case the Bloch theorem holds) and random (in which all states are localized in one dimension), the extended or localized nature of states has attracted interest, also with the possibility of critical states [9].In certain random lattices, the existence extended states relies on correlations between the impurity atoms on the host lattice [10][11][12].This type of origin for extended states also applies to certain deterministic nonperioidc lattices, such as the copper mean chain [10].[Some authors refer to states that are not exponentially localized as extended states where half the absolute value of the trace of the transfer matrix is less than or equal to unity [13] (this is equivalent to the eigenvalues of the transfer matrix lying on the unit circle of the complex plane), while others distinguish extended and critical states from localized states in terms of the scaling properties of the Landauer resistance [14,15].In the present work, we use the former definition and do not differentiate between extended and critical states.]Types of random chains with correlated diagonal and off-diagonal disorder also exhibit extended states [10][11][12][13][16][17][18][19][20][21][22][23][24].Certain aspects of these issues are reviewed in Refs.[2][3][4][5][6][7].
The quadratic chain (QC) is a one-dimensional nonperiodic lattice with B-atom sites at z j = 0 2 d, 1 2 d, 2 2 d, Lwith d the lattice constant and A atoms at other integer multiples of d, as shown in figure 1.Only B atoms have nonzero scattering amplitude/onsite potential.The QC was recently introduced and shown to have a singularcontinuous SF indicating a dense set of extended states with γ(k) = 2.In [25] the QC was treated in a nearestneighbor tight-binding model (NNTBM) with QC modulation on the diagonal, while in [26] a Kronig-Penney model was adopted.Many of the features found are generic and not tied to a specific model.A key result is the hidden symmetry of the transfer matrix, which for certain energies reduces to a periodic product of matrices [27].The SF is found to be singular-continuous and composed of a set of peaks at values of the wavevector k in the host lattice given by = p k r s r s d , with r and s mutually prime integers of opposite parity.As for periodic and quasi-periodic lattices, γ(k) = 2 for all k = k r,s for which the SF exists.The states were found to break up into minibands separated by minigaps [28].As the chain length increased, those minibands become a dense set of discrete states.In this study, we implement a real-space renormalization-group (RSRG) treatment of the QC.By means of a decimation process, the transfer matrix for the chain is step-by-step reduced to a self-similar product of transfer matrices, but with renormalized parameters.Ultimately, by iterating the decimation process, the transfer matrix is reduced to a simple expression for the product of matrices for a pair of sites in the QC.Our focus here is on using an RSRG approach to organize the numerical calculation of extended states in the QC.In addition, we rigorously show that an extended-state spectrum is exhibited that is singular-continuous.
The remainder of this paper is organized as follows.Section 2 discusses the SF to highlight the properties of QCs.The material presented in section 2 largely review results from [25].In section 3, we lay the groundwork for the tight-binding model, including the construction of the transfer matrix and the exact solution for the extended-state spectrum.Section 4 implements a RSRG approach to reduce the transfer matrix, while in section 5 we conclude.

Structure factor
To introduce important basic features of the QC, we first review the QC SF as discussed in [25].The QC can be characterized by the distribution with z j the positions of B-atom sites on the chain along the z axis, as mentioned above with z j = j 2 d; δ 0j is the Kroenecker delta, while δ(z − z j ) is the Dirac delta.The atomic form factor of the site j = 0 being 1  2 is due to the definition of p(z); we can dispense with the factor of 1  2 by equivalently writing for z j = j 2 d.A atoms present zero atomic form factor, and thus do not appear in equation (1).It proves convenient to impose a window w(z) on p(z) and consider the SF associated with the windowed QC f 0 (z) = w(z)p(z); w(z) will be chosen to be (i , and 0 otherwise is the unit rectangle function.The factor e − α| z| describes a spatially decreasing scattering amplitude with increasing z.We shall use the window (i) with a sharp cutoff at site N 2 when we move to a NNTBM; however, to obtain a closed-form expression for the SF, window (ii) is more convenient.In [25] we have shown that in the long-chain limit, these two choices for windows lead to the same results.
The SF S 0 (k) is the squared modulus of the Fourier transform F 0 (k) of f 0 (z) with k the wavevector.In general, the Fourier transform G(k) of function g(z) and the associated inverse Fourier transform (if they exist) are . Schematic diagram of the QC.z j = j 2 d labels positions of lattice sites (or atoms) [pink for j = 0, red for Î j  (B atoms)] comprising the QC.The atomic form factor of the sites in black (A atoms) is zero.The Fourier transform of this chain is the Jacobi theta function ϑ 3 (q) with a = + [ ( ) ] q i k i d exp the nome and α goes to zero a convergence factor (see below) leading to a highly complex structure factor despite the QC's simplicity.
i.e. the power spectrum of f 0 (z).
Having dispensed with preliminaries, we now obtain the SF for the QC using window function (ii).Substituting the expression for f 0 (t), we have with ϑ 3 (q) the theta function In [25], the zeros and peaks of S 0 (k) were found.Zeros are at wavevector = p k r s r s d , with r and s mutually prime odd integers and s > 0. Peaks occur at wavevectors k r,s where r and s are mutually prime integers of opposite parity and s > 0. Closed-form expressions in terms of Gauss sums are also found there for the peak heights.S 0 (k) is periodic and even in k with period 2πdue to the fact that QC B sites lie on sites of the underlying periodic lattice with lattice constant d (here taken to be 1).The scaling exponent is γ(k) = 2 as in periodic and quasi-periodic chains [25].This suggests (when we consider the electronic structure; see below) the existence of extended states.A scaling exponent γ(k) = 2 for all k for which the SF is defined.This is also the case for periodic and Fibonacci chains [6,7].Figure 2 shows the SF for a QC with (a), (b) j 2 = 25 and (c), (d) 100 sites; for (b) and (d), the vertical scale is logarithmic to emphasize the smaller peaks.S 0 (k) is composed of numerous sharp peaks, the number of visually evident peaks increasing with chain length.As discussed in [25], the peaks become dense in the limit j → ∞ .

Nearest-neighbor tight-binding model
The SF is closely related to x-ray and neutron diffraction experiments on crystals, and gives the scattering strength of the lattice in the first Born approximation.But, the SF provides insight into other physical properties, such as the electronic, vibrational, and transport, though a rigorous connection between the SF and other physical properties may not always be clear.The presence of interaction with a potential, however, means that the solution to the single-particle Schrödinger equation is more complicated than obtaining the SF.For numerical calculations, we therefore consider finite QCs within a NNTBM.The QC B atoms lie on sites z j as above.These sites are assigned an onsite matrix element.In between the QC sites are equally space A-atom sites with zero onsite matrix element.The hopping matrix element is taken to be −1 between all neighboring sites.
The Hamiltonian is ,perfect square ,0 2 The subscripts n and ¢ n here to refer to tight-binding sites n, while subscript j is reserved for values of ¢ = n n j , 2 , i.e. when n is a perfect square.λ n , which will be chosen below to be negative (attractive) for the sake of illustrating the method, is the onsite matrix element; λ n = λ if n = j 2 is a perfect square (other than 0), i.e. a B-atom site on the QC, l 2 if n = 0, and 0 otherwise.(The sign of λ is not important; if we change the sign of λ and E, all properties are preserved though the bandedge from which an effect is observed is flipped.)As the nearest-neighbor hopping matrix element is taken to be −1, setting λ = 0 results in the energy eigenvalues to be = =- We employ a transfer-matrix approach as discussed in [18].We can use the Hamiltonian to propagate a state from one site to the next as where M n is the transfer matrix for site n, the negative sign in the upper left entry of M n is due to the fact that the hopping matrix element is −1.We take M 0 to be the matrix with l l = = l n 0 2 , M 1 to be the matrix with λ n = λ 1 = λ, and M 2 to be the matrix with λ n = λ 2 = 0 for n greater than zero; these are the three distinct building blocks of the QC transfer matrix.
By iterating, we obtain for the QC (leaving off the site j = 0) The extended states thus satisfy which is the condition that the eigenstates neither grown nor decay in space.(The last M 0 can be replaced by M 1 without affecting the extended-state spectrum, and will be changed as convenient.)In figure 3 are plotted the extended-state energy eigenvalues as a function of λ < 0 for (a) j = 5, (b) 10, (c) 15, and (d) 20.We comment on the observed features below when we repeat the calculation using the RSRG procedure.We just note here that some numerical noise is evident, particularly in figure 3(a).To understand the energies at which extended states may occur, it is useful to explore a hidden symmetry of the transfer matrix, as discussed in [27] for a continuum Kronig-Penney model for the QC.Inasmuch as M 2 is unimodular, we have via a special case of the Cayley-Hamilton theorem [18,30,31] = --- where U ν (x) is a Chebyshev polynomial of the second kind and = =- 2 .Suppose j is very large, and write the transfer matrix in equation (10) for the QC as with Î s .If for a given s we can choose x so that =  M I s 2 2 , then the transfer matrix is periodic giving the possibility of extended states.Extended states correspond to when half the absolute value of the trace of the transfer matrix is less than or equal to unity.
As above, = --- . The zeros of U s−1 (x) and local extrema (with values±1) of U s−2 (x) occur when [32] Now the above can be carried out for all Î s , and therefore, extended-state energies are given by  with increasing m max .For larger m max , despite the visual disappearance of the many of the minigaps, the extendedstate spectrum is singular-continuous and only has values of k that are certain classes of rational multiples of p d as discussed above.Note that these plots are similar to the energy eigenvalues including localized states found numerically in [25]; = m 10 max corresponds roughly to the QC-chain length 81 considered in figure 4 of [25], which shows a similar minigap structure as in figure 4(a).To reiterate: figure 4 does not show the localized states found numerically in [25].In that work, all strongly localized states were found for sufficiently large |λ| in finite-length QCs; here we prove that in infinite-length QCs, there are extended states (as well as localized states that are not shown).

Real-space renormalization group
While the transfer matrix above has been simplified using the Cayley-Hamilton theorem to obtain the values of K for which extended states may exist, additional insight can be gained by adopting an alternative approach to evaluating the transfer matrix using a RSRG-inspired method.The key use to which we here put the RSRG is to organize the numerical computation.(15).There are a total of m max presented in figure 3 despite the fact that the calculations upon which that figure is based did not include ( ) M 0 0 .Without making any precise quantitative claims, the computations for figure 3 took about thrice the time needed for those upon which figure 6 is based.For a given j, the extended-state miniband widths decrease as |λ| increases since increasing |λ| tends to localize states.As |λ| increases, more and more minigaps appear breaking up the minibands.It appears that for a given stack structure, there is a maximum |λ| beyond which extended states largely disappear.This maximum |λ| depends sensitively on the stack.For example, using 4 ,all of which persist up to values of |λ| greater than for other stacks.There is also a tendency of stacks to fade and re-emerge at higher |λ|, an effect we have called reëntrance [26].In addition, the energies of the minibands tend to decrease as |λ| increases (λ decreases).For small j and small |λ|, the finite chain length plays a significant rôle in determining the energies.As |λ| increases with λ < 0, the QC potential pulls down the energies.This effect is less pronounced for larger j due to the fact that the confinement effect due to the finite chain length is of relatively little importance.

Conclusion
In this study, we have implemented a RSRG treatment of the transfer matrix for the QC.The SF is known to be simply related to the Jacobi theta function 2 with q = e ikd .The SF S 0 (k) has is singularcontinuous with peaks at wavevectors = p k r s r s d , where r and s are mutually prime integers of opposite parity.In addition, S 0 (k) has scaling exponent γ(k) = 2, as for periodic and quasi-periodic lattices, suggesting the existence of extended states.An onsite NNTBM is implemented from which we obtain the extended-state spectrum that depends on wavevectors given by Chebyshev nodes.These wavevectors are also products of rational fractions and p d .The appearance of the Jacobi theta function ϑ 3 (q) and of Chebyshev modes brings out the number theoretic aspect to the problem.
We consider the extended-state spectrum for various parameters.As the magnitude |λ| of the QC onsite energy increases for a given finite chain length, minibands become increasingly broken up by minigaps.Increasing the chain length also increases the number of minigaps.We also show rigorously that the singularcontinuous spectrum is indeed dominated by extended states at low onsite potential.The exact extended eigenstates and associated energies are identified for the infinite-length QC.Dinstinct from previously studied nonperiodic lattices, the extended states emerge from a specific k-dependent hidden translational invariance of the QC.Finally, the RSRG treatment reveals the self-similar nature of the QC and moreover speeds up computation of the electronic structure.

Figure 2 .
Figure 2. S 0 (k) for a QC with (a) and (b) j 2 = 25 and (c) and (d) 100 sites.Note that for (b) and (d) the vertical scale is logarithmic.

Figure 3 .
Figure 3. Extended-state energies as a function of λ < 0 for (a) j = 5, (b) 10, (c) 15, and (d) 20 obtained by finding the eigenvalues of the transfer matrix lying on the unit circle of the complex plane.

Figure 4 . 10 max,
Figure 4. Extended-state dispersion (the 90°rotation of what is often called the cumulative density of states) for (a) = m 10 max , (b) 20, and (c) 30 obtained from equation (15).There are a total of m max