Localisation in quasiperiodic chains: a theory based on convergence of local propagators
LLocalisation in quasiperiodic chains:a theory based on convergence of local propagators
Alexander Duthie, ∗ Sthitadhi Roy,
1, 2, † and David E. Logan
1, 3, ‡ Physical and Theoretical Chemistry, Oxford University,South Parks Road, Oxford OX1 3QZ, United Kingdom Rudolf Peierls Centre for Theoretical Physics, Clarendon Laboratory,Oxford University, Parks Road, Oxford OX1 3PU, United Kingdom Department of Physics, Indian Institute of Science, Bangalore 560012, India
Quasiperiodic systems serve as fertile ground for studying localisation, due to their propensityalready in one dimension to exhibit rich phase diagrams with mobility edges. The deterministicand strongly-correlated nature of the quasiperiodic potential nevertheless offers challenges distinctfrom disordered systems. Motivated by this, we present a theory of localisation in quasiperiodicchains with nearest-neighbour hoppings, based on the convergence of local propagators; exploitingthe fact that the imaginary part of the associated self-energy acts as a probabilistic order parameterfor localisation transitions and, importantly, admits a continued-fraction representation. Analysingthe convergence of these continued fractions, localisation or its absence can be determined, yieldingin turn the critical points and mobility edges. Interestingly, we find anomalous scalings of theorder parameter with system size at the critical points, consistent with the fractal character ofcritical eigenstates. The very nature of the theory implies that it goes far beyond the leading-orderself-consistent framework introduced by us recently [Phys. Rev. B 103, L060201 (2021)]. Self-consistent theories at high orders are in fact shown to be conceptually connected to the theorybased on continued fractions, and in practice converge to the same result. Results are exemplifiedby analysing the theory for three families of quasiperiodic models covering a range of behaviour.
I. INTRODUCTION
The physics of localisation in disordered quantum sys-tems [1] has been a cornerstone of condensed matter the-ory and statistical mechanics for well over half a century.It is however well understood by now that the paradigmof localisation goes beyond systems with quenched ran-dom disorder: systems with quasiperiodicity comprise afamily of non-random and deterministic systems whichhost localisation and related phenomena such as mobilityedges, robust multifractality, ‘mixed phases’ with bothextended and localised eigenstates, and anomalous trans-port [2–24].In phenomenological terms, localisation in quasiperi-odic systems is quite different from, and arguably richerthan, that occurring in their disordered cousins. Forinstance, the simplest quasiperiodic model, the Aubry-Andr´e-Harper (AAH) model [2, 3], hosts a localisationtransition already in one-dimension, and variants of themodel have genuine mobility edges in their spectra [4, 6–11, 16, 22–24]. In fact, mobility edges appear quite typi-cally in systems where the quasiperiodicity arises from acontinuous periodic potential incommensurate with theunderlying periodic lattice. In this regard the AAHmodel itself is a special case in which, due to an ex- ∗ [email protected] † [email protected] ‡ [email protected] There are of course other quasiperiodic models which are criticalthroughout their spectra and parameter space, such as Fibonaccichains [25, 26]. act energy-independent duality, eigenstates at all ener-gies undergo a localisation transition at the same point,such that there is no genuine mobility edge.From a theoretical point of view, quasiperiodic sys-tems are also qualitatively different to disordered ones,because the potential in the former is deterministic andhence infinite-range correlated. Much of the remarkabletheoretical progress in disordered systems over the years,stems from the ability to average over uncorrelated dis-order in an independent and unbiased fashion. In thisregard, quasiperiodic systems pose a unique challenge:the deterministic nature of the potential implies the needto account for the potential at all points in space si-multaneously. One thus expects the analysis involvedto be bespoke to the specific model considered. Thisis indeed typically the case; examples include model-specific energy-dependent generalised duality transfor-mations [9–11], or duality transformations relating mod-els with known phase diagrams [24], and Lyapunov expo-nent calculations based on global theories of Schr¨odingeroperators [23].It is therefore of importance to develop a general theo-retical framework to predict and analyse the localisationphase diagrams for essentially arbitrary quasiperiodicmodels. A step towards that was taken by us in a recentwork [27], where a self-consistent theory of mobility edgeswas developed. The theory, of a self-consistent mean-fieldnature, was rooted in analysis of the local propagatorsand in particular the imaginary part of their self-energies,using a renormalised perturbation series (RPS) [28–30] atleading order. In spirit, the theory was inspired heavilyfrom self-consistent approaches to localisation in disor-dered systems [30–32]. a r X i v : . [ c ond - m a t . d i s - nn ] F e b In this work, we undertake a complementary approachto the analysis of the local self-energies. Exploiting thefact that the RPS for the self-energies can be recast asa continued fraction (CF), we analyse the latter’s con-vergence. The very nature of the analysis means that itgoes far beyond the leading-order treatment of Ref. [27].This treatment is closer in spirit to that of Anderson’soriginal work [1] on localisation. Additionally, we alsotruncate the continued fraction at arbitrary order, andperform a self-consistent analysis at that order; whichyields quantitatively the same results for the localisationphase diagram as the convergence of the continued frac-tion, as well as very good agreement with the leading-order theory [27]. Going to such high orders not onlyuncovers finer structure within the phases, such as hier-archical spectral gaps, but also makes a case for the ro-bustness of the leading-order theory. We note here thatthe results from all three approaches – the leading-orderself-consistent theory, convergence of continued fractions,and higher-order self consistent theory – are found to bein excellent agreement with numerical results obtainedfrom exact diagonalisation for a broad class of quasiperi-odic chains.We turn now to an overview of the paper.
Overview
In order to test the theory, we employ three modelswith exactly known mobility edges. Defined and de-scribed briefly in Sec. II, these are chosen to span a widerange of behaviour, from no mobility edge (transition atsame critical point for all energies) to multiple mobilityedges in the spectra.Section III is devoted to setting up the basic formalismfor the local propagator and the self-energies which un-derpin the work. In particular, we lay special emphasison the imaginary part of the self-energy (∆( ω )), and inSec. III A discuss its importance as a probabilistic orderparameter for a localisation transition. With unit prob-ability, ∆( ω ) is respectively finite and vanishing in anextended and localised phase [30]. Crucially, in the lat-ter, it vanishes ∝ η → + , the regulator in the theory (orimaginary part of the energy); this allows one to definea further probabilistic order parameter y ( ω ) = ∆( ω ) /η ,which is finite in the localised phase and divergent inthe extended. Before delving into a detailed analysis,Sec. III B gives numerical results for ∆( ω ) and y ( ω ) ob-tained via exact diagonalisation, as a demonstration oftheir validity as order parameters. In Sec. III C, we dis-cuss how the RPS for ∆( ω ) and y ( ω ) can be recast as acontinued fraction, which forms the basis of the analysisin subsequent sections.Analysis of the convergence of the continued fractionsconstitutes Sec. IV. We show that in the localised phase,the continued fraction for y ( ω ) converges and has a finitetypical value, y typ ( ω ) (the geometric mean of its distri-bution); whereas in the extended phase it does not, and L´evy-tailed distributionfor y ( ω )CF for y ( ω ) convergent y typ ( ω ) finite∆ typ ( ω ) finite y typ ( ω ) divergentCF for y ( ω ) divergent ω ME Vω FIG. 1. Schematic localisation phase diagram in the spaceof energy ( ω ) and quasiperiodic potential strength ( V ). Amobility edge (red dashed line) separates the extended phase(green) from the localised (blue). The salient features of theimaginary part of the self-energy in the two phases (∆( ω ) and y ( ω ) = ∆( ω ) /η ) are indicated. the typical y ( ω ) is divergent in the thermodynamic limit.Using this diagnostic for the quasiperiodic chains consid-ered, one can determine the presence of localised or ex-tended states at any point in the parameter and energyspace. This is the first main result of the work.In Sec. V we present a self-consistent theory at arbi-trary orders by truncating the continued fractions; whichis the second main result of the work. Consistent withthe results obtained in Sec. IV, the self-consistent typical y ( ω ) is respectively finite and divergent in the localisedand extended phases. Equivalently, the typical ∆( ω ) ob-tained self-consistenly is finite and vanishing in the ex-tended and localised phases respectively. An importantpoint to note is that the convergence of the continuedfraction for y ( ω ) is tied to the convergence of the self-consistent typical y typ ( ω ) to a finite value. The analysispresented places the recently investigated leading-ordertheory [27] within a broader framework, encompassingboth higher-order self-consistent theories as well as theconvergence of the underlying RPS for y ( ω ).As mentioned above, ∆( ω ) and y ( ω ) are probabilisticorder parameters. Their distributions are thus of funda-mental interest, and this forms the subject of Sec. VI.We obtain the distributions using both the continuedfractions as well as self-consistently; both of which showexcellent mutual agreement, as well as with results ob-tained from exact diagonalisation. One main result hereis that the distributions of y ( ω ) in the localised phasehave L´evy tails ∝ y − / ; which likewise arise in localisedphases of disordered systems, for both uncorrelated [30]and correlated disorder [33], suggesting they are ratheruniversal in localised systems. It is also of course be-cause of these fat-tails that the geometric mean ( y typ ) isa suitable measure of typicality for the distribution.We close in Sec. VII with some discussion and direc-tions for future work. Summary of results:
The essential outcome of thiswork is a theory of localisation in quasiperiodic chainswith nearest-neighbour hoppings, based on the conver-gence properties of the local propagators and associatedself-energies. The extended and localised regimes can bediagnosed in terms of ∆( ω ) and y ( ω ) as follows. A fi-nite ∆ typ in the thermodynamic limit at any point in the( V, ω )-plane where eigenstates exist, implies the presenceof extended states; y typ is naturally divergent at sucha point. Conversely, a finite y typ (and vanishing ∆ typ )indicates the presence of localised states. The criticalpoints/mobility edges in the ( V, ω )-plane can thus be un-ambiguously identified as points where, simultaneosuly,∆ typ vanishes and y typ diverges. How these characteris-tics are reflected in the convergence properties of theircontinued fractions is the central point of this work. Aschematic summary is given in in Fig. 1, showing a lo-calisation phase-diagram with a mobility edge, with thesalient features of the two phases indicated. II. MODELS
We consider quasiperiodic chains of length L withnearest-neighbour hoppings, described generally by aHamiltonian of form H = V L − (cid:88) j =0 (cid:15) j c † j c j + J L − (cid:88) j =0 [ c † j c j +1 + H . c . ] , (1)with c † j ( c j ) the creation (annihilation) operator on site j . The model-specific quasiperiodic potential is encodedin (cid:15) j , with V its strength, and J the hopping amplitude;without loss of generality we consider V, J ≥
0. We focuson three models which between them display a wide rangeof behaviour and for which the mobility edges ω ME ( V ), orequivalently the energy-dependent critical points V c ( ω ),are exactly known. The Aubry-Andr´e-Harper model:
The first is the fa-miliar and much-studied AAH model [2, 3] defined via (cid:15) j = cos(2 πκj + φ ) , (2)with κ an irrational number, reflecting the incommensu-rability of the potential relative to the underlying lattice(here we take κ to be the golden mean); and φ ∈ [0 , π )is a random but uniform phase shift employed to performthe analogue of disorder averaging. The model does nothost a genuine mobility edge, as all eigenstates undergoa localisation transition at the critical V c = 2 J (localisedfor V > V c ) [2, 3]. Equivalently, the mobility edge canbe viewed as a line parallel to the energy axis at V = V c . The β -model: The second model was introduced inRef. [11], and we refer to it as the β -model. It is a variantof the AAH model, and is described by (cid:15) j = cos(2 πκj + φ )1 − β cos(2 πκj + φ ) , (3) with 0 ≤ | β | <
1. The potential in Eq. (3) breaks theexact duality of the AAH model leading to a genuinemobility edge given by [11] ω ME = (2 J − V ) /β . (4)In the limit β →
0, the potential in Eq. (3) reduces tothe AAH potential Eq. (2) and, consistently, the mobilityedge in Eq. (4) becomes a straight line parallel to the ω -axis at V = 2 J . The mosaic models:
The third model is from the fam-ily of so-called mosaic models [23], the name arising fromthe fact that regularly spaced blocks of sites have (cid:15) j = 0while the complementary sites experience a quasiperiodicpotential. Explicitly, (cid:15) j = (cid:40) cos(2 πκj + φ ) : j = lk k ∈ Z . The system thus consists of blocks of size l within which only the first site has a non-zero (AAH)potential. Throughout the rest of the paper we will fo-cus on l = 2, for which the system has two symmetricmobility edges [23] ω ME = ± J /V . (6)Note from Eq. (6) that states in the centre of the spec-trum ( ω = 0) are extended for all finite values of V . III. LOCAL PROPAGATORS ANDSELF-ENERGIES
In this section we set up the basic formalism for thelocal propagator and associated self-energies, which arethe central quantities of interest in our theory.
A. Imaginary part of local self-energy
In the time domain, the local propagator at site j isdefined as G j ( t ) = − i Θ( t ) (cid:104) j | e − iHt | j (cid:105) , which is simplythe return probability amplitude. Since a localisationtransition is an eigenstate phase transition – and the verynotion of a mobility edge is intrinsically energy dependent– it is more natural to consider the local propagator inthe energy domain, G j ( ω ) = [ ω + − V (cid:15) j − S j ( ω )] − , (7)where ω + = ω + iη with η = 0 + the regulator, and S j ( ω )is the local self-energy of site j . The self-energy has realand imaginary parts S j ( ω ) = X j ( ω ) − i ∆ j ( ω ) , (8)where the real part can be interpreted physically as theperturbative shift in the eigenvalues from the bare (cid:15) j ’s. Inphysical terms the imaginary part, ∆ j ( ω ), gives the rateof loss of probability amplitude from site j into eigen-states of energy ω which overlap that site. It thus actsas a probabilistic diagnostic for the localisation transi-tion, and our analysis centres on it. As mentioned pre-viously, an extended (localised) phase is signified by afinite (vanishing) ∆ j ( ω ) with unit probability. In partic-ular, in the localised phase ∆ j ( ω ) ∝ η , and as such onecan define an equivalent probabilistic order parameter y j ( ω ) = ∆ j ( ω ) /η , which is finite with unit probabilityin the localised phase and diverges as the transition isapproached from the localised side.The RPS for S j ( ω ) [28–30] is given for a chain withnearest neighbour hoppings by S j ( ω ) = (cid:80) k J jk G ( j ) k ( ω ),with J jk the hopping amplitude between sites j and k ,and G ( j ) k ( ω ) the local propagator for site k with site j removed. Since any site j only has two neighbours, j ± J , S j ( ω ) thus reads S j ( ω ) = J [ G ( j ) j − ( ω ) + G ( j ) j +1 ( ω )] . (9)Quite crucially, G ( j ) j ± are the local propagators of sites j ± j removed, and hence depend onlyon the sites to the right/left of j . They are the end-site propagators of two semi-infinite half-chains . The decom-position in Eq. 9 thus allows S j ( ω ) to be constructed en-tirely from the knowledge of the half-chain propagatorsalone, which will be the focus of our analysis henceforth.To simplify notation, in the following we will denotethe end-site propagator as G ( ω ) and the associated self-energy as S ( ω ). Since the chain end-site, j = 0, is con-nected to only one site, j = 1, the RPS for S ( ω ) has onlyone term, S ( ω ) = J G (0)1 ( ω ) where G (0)1 ( ω ) is again theend-site propagator of a chain, now starting at j = 1.This recursive structure allows us to write G ( ω ) as acontinued fraction G ( ω ) = 1 ω + − V (cid:15) − J ω + − V (cid:15) − J . . . , (10)from which the continued fraction for S ( ω ) follows as S ( ω ) = J ω + − V (cid:15) − J ω + − V (cid:15) − J . . . . (11)Equations (10) and (11) are formally exact and, impor-tantly, take into account the site energies at all sites j ,crucial for the deterministic and infinite-range correlatedquasiperiodic potentials. We will turn to a detailed anal-ysis of the continued fractions in Secs. III C and IV.It is important to emphasise that ∆ ( ω ) and y ( ω ) areprobabilistic order parameters, as they are characterised by distributions over values of φ and kinds of end sites; and that it is their typical (and not average) values thatare appropriate order parameters. These can be obtainedfrom their distributions P ∆ (∆ ) as∆ typ ( ω ) = exp (cid:20)(cid:90) d ∆ P ∆ (∆ ) ln ∆ (cid:21) , (12)and similarly for y typ ( ω ). Note further that ∆ ( ω ) is pro-portional to the local density of states (LDoS), which isfurther suggestive that it is the typical value of ∆ ( ω )which is an appropriate order parameter. Indeed, thetypical LDoS has served as an order parameter for lo-calisation transitions in disordered [32, 34, 35] as well asquasiperiodic systems [11]. The average value of ∆ ( ω )by contrast is finite in both phases – it merely givesthe average density of states/eigenvalues – and does nottherefore discriminate between them. B. Results from exact diagonalisation
To demonstrate the validity of ∆ typ ( ω ) and y typ ( ω )as suitable diagnostics of localised and extended phases,we present numerical results for them obtained via exactdiagonalisation. In terms of the half-chain eigenstates | ψ n (cid:105) and eigenvalues E n , the local propagator G ( ω ) is G ( ω ) = L (cid:88) n =1 | (cid:104) | ψ n (cid:105) | ω + − E n , (13)from which S ( ω ) can be trivially extracted using G ( ω ) = [ ω + − V (cid:15) − S ( ω )] − . Since the regulator η should be on the order of the mean level spacing, wetake η = c/L with c ∼ O (1).Fig. 2 shows the resultant ∆ typ ( ω ) and y typ ( ω ) as acolour-map in the ( V - ω )-plane for all three models de-scribed in Sec. II. The top row shows ∆ typ ( ω ), a finitevalue of which indicates an extended phase. On the‘other’ side of exactly known mobility edges (shown byblack dashed lines), ∆ typ drops to a vanishingly smallvalue, indicating a localised phase. This is consistentwith the behaviour of y typ ( ω ), shown in the bottom row.In the localised phase, y typ ( ω ) is finite and diverges atthe mobility edge, indicating a transition to an extendedphase. While the results in Fig. 2 were for a single sys-tem size, in Fig. 3 we show the system-size dependence of∆ typ ( ω ) at exemplary ( V, ω ) points in the extended andlocalised phases, for each model. In the extended phase∆ typ ( ω ) saturates with increasing L to a finite value,whereas in the localised phase it decays to zero as L − . We refer to a model as possessing multiple kinds of end sitesif, depending on the configuration of the chain, the end sitesexperience different potentials. For instance, the l = 2 mosaicmodel has two kinds of end sites, one with zero potential, andone with an AAH potential. − − ω AAH model β − model l = 2 mosaic model0 2 V − − ω V V . . . . ∆ t y p − . − . . l n y − t y p FIG. 2. Spectra obtained from exact diagonalisation for thethree models described in Sec. II, colour-coded with ∆ typ ( ω )(top row) and y typ ( ω ) (bottom row), reflecting the localisationphase diagram and the mobility edges (black dashed lines).The columns correspond respectively to the AAH, the β (=0 . l = 2 mosaic model. A finite and vanishinglysmall value of ∆ typ in the upper row indicates respectivelyextended and localised states. Consistently, y typ is divergentand finite in the two phases as can be seen in the lower row.For clarity, note that we show ln y − , which vanishes in theextended phase. All data shown for L = 2500 with η = 1 /L ,and statistics accumulated over 5000 values of φ ∈ [0 , π ).Here, and in all subsequent figures, we set the hopping J ≡ Consistently, y − ( ω ) decays to zero in the former and is L -independent in the latter as y typ = ∆ typ /η ∝ L ∆ typ ∼ O (1). The numerical results thus confirm the validityand applicability of ∆ typ ( ω ) and y typ ( ω ) as diagnosticsfor the localisation transitions and mobility edges in thequasiperiodic models considered. C. Continued fraction
We now discuss in more detail the continued fractionin Eq. (11). Using ∆ = − Im[ S ], this yields a series for∆ as∆ = ∞ (cid:88) n =1 ∆ ( n ) with ∆ ( n ) ≡ η n (cid:89) k =1 J Ω k (14)and Ω k = ( ω − V (cid:15) k − X ( k − k ) + ( η + ∆ ( k − k ) . The ap-parent simplicity of this series is deceptive, as Ω k explic-itly depends on X ( k − k and ∆ ( k − k which have their owncontinued-fraction representations. Physically, ∆ ( n ) isthe contribution to ∆ of a process where the particlegoes out to site n from the root site 0 and retraces itspath back to the root [33]. Such path contributions arealso present in the forward-scattering approximation [36],albeit in an un-renormalised fashion. Here by contrast,the presence of X ( k − k ’s and ∆ ( k − k ’s in the Ω k denomi-nators reflects that the theory is fully (and exactly) renor-malised. L − − − ∆ t y p AAH model V = 1 . , ω = 0 V = 1 . , ω = 0 V = 2 . , ω = 0 V = 2 . , ω = 0 V = 3 . , ω = 0 10 − β − model V = 1 . , ω = 0 V = 2 . , ω = 02 L − − − l = 2 mosaic model V = 1 . , ω = 0 . V = 2 . , ω = 0 . V = 2 . , ω = 1 . V = 3 . , ω = 2 . FIG. 3. System-size ( L ) dependence of ∆ typ ( ω ) at represen-tative ( V, ω )-points in the extended and localised regimes forall three models, obtained via exact diagonalisation. Data inblue/cyan corresponds to the extended regime where ∆ typ isfinite as L → ∞ ; whereas the localised regime data, shown inred/orange, shows that ∆ typ decays to zero as L − (indicatedby the red dashed lines). For the AAH model, we also showdata for the critical point, where ∆ typ again decays to zerobut with an anomalous power, L − . , shown by the blackdashed line. In analysing these continued fractions, it proves con-venient to introduce a notation S [ (cid:96) ]0 = J ω + − V (cid:15) − J ω + − V (cid:15) − J . . . J ω + − V (cid:15) (cid:96) − S (cid:96) , (15)where the superscript [ (cid:96) ] denotes that we have merely it-erated the continued fraction to level (cid:96) , but not truncatedit. The S (cid:96) on the right-hand side has its own continuedfraction representation, such that Eq. (15) is formally ex-act and continues ad infinitum for a thermodynamicallylarge system. By a truncation at order (cid:96) , we shall meansetting S (cid:96) = 0, which is physically equivalent to truncat-ing the half-chain at site (cid:96) .The localised phase offers significant simplifications inthe structure of the continued fractions. Since ∆ ∝ η inthis phase and η → + , the series for y reads y = ∞ (cid:88) n =1 y ( n ) with y ( n ) ≡ n (cid:89) k =1 J Ω k (16)where Ω k = ( ω − V (cid:15) k − X ( k − k ) now. Crucially, thecoresponding continued fraction for X ( k − k is X ( k − k = J ω − V (cid:15) k +1 − J ω − V (cid:15) k +2 − J . . . , (17) L − − − − ∆ t y p ∼ L − . ∼ L − AAH model V = 1 . , ω = 0 V = 1 . , ω = 0 V = 2 . , ω = 0 V = 2 . , ω = 0 V = 2 . , ω = 0 V = 3 . , ω = 0 10 L ∼ L − . ∼ L − l = 2 mosaic model V = 1 . , ω = 0 . V = 2 . , ω = 0 . V = 1 . , ω = 1 . V = 2 . , ω = 1 . V = 3 . , ω = 2 . FIG. 4. Results for ∆ typ obtained from the exact continuedfraction for S [ (cid:96) ]0 in Eq. (15) with (cid:96) = L − L sites, for the AAH (left) and l = 2 mosaic (right) models.Data in blue/cyan and pink/red/orange correspond respec-tively to extended and localised regimes, while data in greycorresponds to a critical point/mobility edge. In the extendedregime, ∆ typ is finite as L → ∞ whereas in the localisedregime it decays to zero as L − . At the critical point/mobilityedge, the scaling of ∆ typ with L is anomalous, reflecting thefractal nature of the eigenstates therein. Statistics for all datawere obtained over 5000 values of φ ∈ [0 , π ). and no longer depends on the ∆ k ’s. Hence, the Ω k ’s havea closed set of recursion relationsΩ k = ω − V (cid:15) k − J / Ω k +1 . (18)Truncation of the series Eq. (16) at level (cid:96) by setting S l = 0 corresponds to a boundary condition of the recur-sion relation Eq. (18) as Ω (cid:96) = ω − V (cid:15) (cid:96) . Note from Eq. (16)and Eq. (18) that the framework governing y ( ω ) in thelocalised phase does not depend on η , and hence the con-vergence properties of the series can be analysed directlyin the thermodynamic limit, not just formally but alsoin practice. IV. CONVERGENCE OF CONTINUEDFRACTION
Results for ∆ typ and y typ are now presented, obtainedvia the continued fractions discussed in Sec. III C. We be-gin with ∆ typ obtained from Eq. (15). For a finite systemof size L (with sites i = 0 , , · · · , L − S [ L − is exact,and our approach is to study ∆ typ thereby obtained asa function of L for different points in the ( V, ω )-plane.Representative results are shown in Fig. 4 for the AAHand l = 2 mosaic model (results for the β -model are omit-ted for brevity, as they are qualitatively similar to thosefor AAH). In the extended regime, ∆ typ as expected isfinite in the limit of L → ∞ . In the localised regimeon the other hand, it decays to zero ∝ L − . It is infact important that the L − decay is universal through-out the localised regime, since this in turn implies that ‘ y [ ‘ ] t y p / y [ ] t y p AAH model V = 1 . , ω = 0 V = 1 . , ω = 0 V = 2 . , ω = 0 V = 2 . , ω = 0 V = 3 . , ω = 0 10 ‘ l = 2 mosaic model V = 1 . , ω = 0 . V = 2 . , ω = 0 . V = 2 . , ω = 1 . V = 3 . , ω = 2 . β − model V = 1 , ω = 0 V = 3 , ω = 0 FIG. 5. Results for y [ (cid:96) ]typ = (cid:80) (cid:96)n =1 y ( n ) vs (cid:96) , obtained for theseries in Eq. (16). Different panels correspond to the threemodels, as indicated. In the localised regime y [ (cid:96) ]typ saturatesto a finite value as (cid:96) → ∞ . In the extended regime, y [ (cid:96) ]typ thus computed diverges, indicating both the breakdown oflocalisation and the validity of Eq. (16). y typ = ∆ typ /η ∝ L ∆ typ is finite.At the critical point/mobility edges shown in grey inFig. 4, ∆ typ again vanishes as L → ∞ but with a powerthat is anomalous, ∆ typ ∼ L − α with α <
1. Thiswe attribute to the (multi)fractal nature of the criticaleigenstates. Fractal eigenstates exhibit behaviour inter-mediate between perfectly extended and exponentiallylocalised states, as indeed is reflected in the anomalousscaling of ∆ typ with L at the critical point. As a consis-tency check, we mention that the anomalous exponentsshown in Fig. 4 are identical (within fitting errors) tothose obtained from the scaling of ∆ typ obtained via ex-act diagonalisation (see e.g. Fig. 3 for the AAH model).We turn next to the continued fraction for y , Eq. (16),which is valid specifically in the localised phase. As notedin Sec. III C in formulating Eqs. (16) and (18), the com-putation of y can be interpreted as one directly in thethermodynamic limit. One therefore studies the conver-gence of the series with the truncation level (cid:96) , writing(see Eq. (16)) y [ (cid:96) ]0 = (cid:80) (cid:96)n =1 y ( n ) and studying the con-vergence of the series with increasing (cid:96) . Saturation ofthe series to a finite value as (cid:96) → ∞ indicates a localisedregime, whereas in an extended regime the framework it-self breaks down, leading to a divergent y [ (cid:96) ]0 with increas-ing (cid:96) . The results of Fig. 5, which show the (cid:96) -dependenceof the resultant typical value y [ (cid:96) ]typ , indeed confirm this be-haviour.In the localised regime, while y typ saturates to a finitevalue as (cid:96) → ∞ , Fig. 5 shows that the saturation value,as well as the length-scale at which the saturation oc-curs, grows with proximity to the critical point/mobilityedge. This in turn reflects physically the fact that thetypical localisation length in the ( V, ω )-plane increasesas one moves closer to a transition. In order to under-stand this in the simplest fashion, consider an eigenstate | ψ n (cid:105) of some energy ω , which is localised on the rootsite with a localisation length ξ . The (normalised) wave-function densities | ψ ( r ) | = |(cid:104) r | ψ n (cid:105)| for such a statecan then be written as | ψ ( r ) | = (1 − e − /ξ ) e − r/ξ . Atthe same time, it is straightforward to show [37] that | ψ (0) | = (1 + y ( ω )) − , which leads to a relation be-tween y ( ω ) and ξ , viz. y ( ω ) = (1 − e − /ξ ) − − ξ (cid:29) ∼ ξ ;showing explicitly that an increasing typical ξ implies anincreasing y typ . Moreover near a critical point, where ξ (cid:29) y ( ω ) ∼ ξ shows that the typical localisationlength and y typ each diverge with the same critical ex-ponent ( ν = 1, see e.g. [27]) on approaching a transitionfrom the localised side.The same physical picture can be used to understandthat y [ (cid:96) ]typ saturates when (cid:96) exceeds a length-scale set bythe typical localisation length. As discussed previously, y ( n ) in Eq. (16) is the contribution to y of processeswhere the particle goes out to site n and retraces itspath on the chain back to the root. However, since thewavefunction density decays exponentially with r with alength-scale ξ , then naturally for all n (cid:29) ξ the contribu-tions are negligibly small. The sum in Eq. (16) thereforesaturates for n (cid:38) ξ which, crucially, is independent of L .Finally here, as explained in Sec. III C, the seriesEq. (16) is not applicable in the extended regime. Never-theless, studying the series in an extended regime indeedsignals the breakdown of localisation [33], via the factthat y [ (cid:96) ]typ diverges as (cid:96) → ∞ . V. SELF-CONSISTENT THEORY
In the previous section, truncation of the continuedfraction was enacted by setting the terminal S (cid:96) to zero,and analysing its convergence as a function of trunca-tion order (cid:96) . An alternative approach [27] is to assign atypical value to the terminal self-energy S (cid:96) → − i ∆ typ ,and analyse the series self-consistently. This amounts tohaving a continued fraction for ∆ which depends on allthe quasiperiodic site energies up to site (cid:96) , together with∆ typ on site (cid:96) . Compiling statistics over φ , one thus ob-tains a distribution of ∆ , P ∆ (∆ , ∆ typ ), which dependsparametrically on ∆ typ . Self-consistency is then imposedby requiring that ∆ typ obtained from this distribution isequal to the parametric ∆ typ ,log ∆ typ = (cid:90) d ∆ P ∆ (∆ , ∆ typ ) log ∆ . (19)This constitutes a self-consistent analysis at order (cid:96) . Onecan analogously construct a similar self-consistent frame-work for y typ . In Ref. [27] we analysed the self-consistenttheory at leading order ( (cid:96) = 1); resulting, for the mod-els specified in Sec. II, in analytical results for mobilityedges and phase diagrams that are in very good agree-ment with previous results. Here by contrast, our focuswill be on the higher-order theories. First, however, we point out briefly that self-consistency imposed separately in the extended andlocalised phases breaks down at the same point inthe ( V, ω )-plane, demonstrating a consistent criticalpoint/mobility edge within the theory. In terms of theseries in Eqs. (14) and (16), one obtains respectively∆ = ∆ [ (cid:96) − + ∆ ( (cid:96) )(1 + ∆ typ /η ) ,y = y [ (cid:96) − + y ( (cid:96) )(1 + y typ ) . (20)Near a localisation critical point, where a typical ∆ → k → ( ω − V (cid:15) k − X ( k − k ),which reduces the two equations in Eq. (20) to the sameequation. The same critical point thus arises as the tran-sition is approached from either phase. This is also re-flected in the fact that the distribution P ∆ (∆) in the ex-tended regime smoothly evolves to P y ( y ) in the localisedregime (after a trivial rescaling by η ). At leading order( (cid:96) = 1) it can in fact be shown explicitly [27] that theself-consistent equations in the extended and localisedregimes are respectively (cid:104)(cid:104) ln[( ω − V (cid:15) ) + ∆ ] (cid:105)(cid:105) − ln J = 0 , ln(1 + y − ) = (cid:104)(cid:104) ln( ω − V (cid:15) ) (cid:105)(cid:105) − ln J , (21)where (cid:104)(cid:104)·(cid:105)(cid:105) denotes an average over φ and kinds of end-sites. Requiring ∆ typ , y typ > is physically arate), one obtains an expression for the mobility edge as (cid:104)(cid:104) ln[( ω ME − V (cid:15) ) ] (cid:105)(cid:105) − ln J = 0 . (22)While such a simple and tractable expression for the mo-bility edge is not yielded by higher-order self-consistenttheories, they can be readily implemented numerically;and we will be particularly interested in showing thatthey converge to the same results as the analysis usingthe continued fractions.Let us now establish the conceptual connection be-tween the two approaches. In the extended phase, theseries for ∆ typically converges to a finite value. Inorder for this to happen, successive terms ∆ ( n ) in theseries must decay with increasing n sufficiently rapidly,such that ∆ ( n ) → n → ∞ . The effect on ∆ of mul-tiplying (1 + ∆ typ /η ) by ∆ ( (cid:96) ) in Eq. (20), is thereforenegligible provided (cid:96) is large enough that ∆ ( (cid:96) ) /η → typ is then governed en-tirely by the series, and one naturally expects the twoapproaches to yield identical results. In the localisedregime, ∆ typ →
0. Since the localisation length, ξ , is fi-nite, it is obvious that ∆ is insensitive to either cuttingoff the chain at a site (cid:96) or endowing S (cid:96) with an imaginarypart (provided (cid:96) (cid:29) ξ ), again leading to an equivalence ofthe two approaches.The connection between the two appoaches can like-wise be understood in terms of y typ , which is the quan-tity of interest in the localised regime. There, the seriesfor y in Eq. (16) saturates to a finite value, implyingthat y ( n ) decays sufficiently rapidly with increasing n . ‘ − − − − − δ y [ ‘ ] t y p AAH model β − model l = 2 mosaic model 0 25 ‘ y [ ‘ ] t y p AAH modelSCCF0 25 ‘ y [ ‘ ] t y p β − model SCCF0 25 ‘ y [ ‘ ] t y p l = 2 mosaic modelSCCF FIG. 6. Convergence of y typ , obtained via the self-consistenttheory (SC) and continued fraction (CF), to the same resultwith increasing order of truncation ( (cid:96) ) in the localised regime.Left panel shows that the difference, δy [ (cid:96) ]typ = y [ (cid:96) ] , SCtyp − y [ (cid:96) ] , CFtyp ,between the SC and CF decays exponentially with (cid:96) , whilethe right panels show the raw data for y [ (cid:96) ]typ for each model.Results are shown for V = 3 and ω = 0 for the AAH and β = 0 . V = 1 . ω = 1 . Analogous to the argument given above, capping y ( (cid:96) )with (1 + y typ ) is thus immaterial for y in Eq. (20) pro-vided (cid:96) (cid:29) ξ . The two approaches are therefore bound togive the same result for y typ . We confirm this as shown inFig. 6, by comparing the (cid:96) -dependence of the numericallyevaluated y [ (cid:96) ]typ , to the y typ arising from the self-consistenttheory at order (cid:96) . Defining δy [ (cid:96) ]typ = y [ (cid:96) ] , SCtyp − y [ (cid:96) ] , CFtyp as thedifference between the results from the self-consistent(SC) and continued fraction (CF) approaches, we findthat δy [ (cid:96) ]typ decays to zero exponentially with (cid:96) (Fig. 6, leftpanel), showing that the two approaches converge to thesame solution. This is also directly evident in the rightpanels of Fig. 6, where the (cid:96) -dependence of y [ (cid:96) ]typ is shownfor each approach, and all three models considered. VI. DISTRIBUTIONS OF IMAGINARY PARTOF SELF-ENERGY
Since ∆ typ and y typ are probabilistic order parame-ters, their distributions are naturally important objectsto study. We will focus mainly on the distributions P y ( y ),owing to their apparent universality for disordered sys-tems (with both uncorrelated [30] and correlated [33] dis-order) as well as quasiperiodicity [27]. Within the self-consistent theory these distributions are byproducts ofthe framework, because once the self-consistent y typ isobtained the distribution P y ( y, y typ ) is fixed. δy [ (cid:96) ]typ = y ( (cid:96) ) y typ is necessarily positive, since y typ , y ( (cid:96) ) > y − − − − − − P y ( y ) AAH model 10 − − − β − model ‘ = 1 ‘ = 2 ‘ = 20ED10 y − − − l = 2 mosaic model5 10 15 ‘ − − − D K L FIG. 7. Distributions P y ( y ) vs y obtained from the self-consistent (SC) theory truncated at order (cid:96) (shown for (cid:96) =1 , , L = 5000 (shaded grey with black edging). Note that the (cid:96) = 20 SC results and those from ED are barely distinguish-able. With increasing (cid:96) the additional structures on top of the ∝ y − / L´evy tail (shown by the red dashed line) are recov-ered. The inset in the left panel shows the Kullback–Leiblerdivergence between the distribution obtained from the SCmethod and that obtained from ED. It decays rapidly with (cid:96) ,indicating rapid convergence of the distributions to the exactresult. Results are shown for V = 3 for all three models, with ω = 0 for the AAH and β = 0 . ω = 2 for themosaic model. In Ref. [27], the distributions P y ( y ) were obtained ana-lytically at leading-order level, revealing a characteristicL´evy-tailed distribution ( ∝ y − / for y (cid:29) φ -averaging.Hence we expect that the higher-order theories, whichexplicitly take into account the potentials on all sites (upto the truncation order), will reveal them.Fig. 7 shows the distributions P y ( y ) obtained from theself-consistent theory in the localised phase, for differ-ent values of truncation order (cid:96) . While the y − / tail isuniversally present at all orders and for all three mod-els, with increasing (cid:96) finer structures are indeed seento emerge in the distributions, which match very wellwith the exact result obtained from ED. This confirmsthat these structures are manifestations of the highly-correlated quasiperiodic potential, and as such they arecaptured better at higher orders of the theory.To quantify the convergence of the distributions ob-tained to those obtained exactly from ED, we compute . . . P ∆ ( ∆ ) st orderEDSCCF 500 2500 5000 ‘ − − − − D K L FIG. 8. Distributions P ∆ (∆) for the AAH model in the ex-tended phase (shown for V = 1 and ω = 0), obtained fromthe continued fraction (CF) and self-consistent (SC) treat-ment at order (cid:96) = 5000, together with results obtained fromED for L = 5000. The three are barely distinguishable. In-set panel shows the Kullback-Leibler divergence D KL betweenSC distributions at various orders (cid:96) and the ED distribu-tion. D KL decays with increasing (cid:96) , indicating convergenceof the self-consistent distributions to the ED result. The ana-lytic leading-order ( (cid:96) = 1) SC result [27] is also shown (blackdashed line), and bar an overall shift captures the distributionrather well. the Kullback–Leibler divergence [38] D KL ( (cid:96) ) = (cid:90) dy P ED y ( y ) log (cid:32) P ED y ( y ) P [ (cid:96) ] y ( y ) (cid:33) , (23)where P [ (cid:96) ] y is the distribution obtained from the self-consistent theory at order (cid:96) . As shown in the inset inFig. 7, D KL ( (cid:96) ) decays rapidly to zero with (cid:96) showingthe rapid convergence of the distributions. We add thatthe distributions obtained from the CF approach likewiseconverge to the ED results at large (cid:96) .Finally, to illustrate results in the extended regime,Fig. 8 shows representative distributions of P ∆ (∆) forthe AAH model. Obtained as specified in the captionfrom the truncated continued fraction method, the self-consistent approach, and ED calculations, the three areseen to be barely distinguishable from each other. Thefigure also shows the analytical result arising [27] fromthe leading order ( (cid:96) =1) self-consistent theory (with char-acteristic square-root singularities evident at the hardedges of the distribution). Excepting an overall shift,this is seen to capture the full distribution rather well. VII. DISCUSSION
In summary, we have presented a theory for localisa-tion in quasiperiodic chains with nearest-neighbour hop-pings, based upon the continued-fraction representationof the local propagators, and in particular the end-sitepropagators for a semi-infinite half-chain. Our focus was on the imaginary part, ∆( ω ), of the associated self-energy, which likewise admits a continued-fraction rep-resentation and which acts as a probabilistic order pa-rameter. In a regime of extended states, we showed thatthe typical value of the self-energy distribution, ∆ typ ( ω ),converges to a finite value; while in a localised regimeit decays to zero as the level of the continued fractionis increased. Analogously, the continued fraction for the‘complementary’ order parameter y typ ( ω ) = ∆ typ ( ω ) /η was shown to converge to a finite value in the localisedregime but diverges in the extended regime. Together,they can be used to map out the localisation phase dia-gram in the space of energy and Hamiltonian parameters,thus giving access to the critical points/mobility edges.Interestingly, it was also found that at the critical points,∆ typ ( ω ) has an anomalous scaling with the system sizeand decays to zero as L − α with α <
1, which we at-tribute to the fractal nature of the critical eigenstates.The continued-fraction method was moreover shown tobe intimately connected to higher-order self-consistenttheories; such that the two approaches are asymptoti-cally equivalent, and give quantitative agreement withresults arising from exact diagonalisation. Finally, weshowed how going to higher orders in the theory revealsfiner structures in the distributions of ∆ and y , reflect-ing the deterministic and highly correlated nature of thequasiperiodic potential. These substructures appear ontop of the universal features such as a L´evy-tailed distri-bution of y in the localised regime.Quasiperiodic systems of the kind considered here alsohave fine structures in their eigenvalue spectra (or to-tal DoS), in the form of hierarchical gaps; seen e.g. inthe ED results shown in Fig. 2. The theory presentedhere can also be used to probe these and related struc-tures. While an in-depth exposition of such is not ouraim here, we give one illustrative example in Fig. 9. Forthe AAH model, this shows the φ -averaged local DoS forthe root site (proportional to the imaginary part of thelocal self-energy); obtained from the continued-fractionapproach, and compared to ED results. At the lowest-order level ( (cid:96) = 1) the mean LDoS is smooth, and devoidof the peaks obtained by ED reflecting the actual energybands. On increasing (cid:96) however, the peaks begin to ap-pear, and by a modest value of (cid:96) = 32 the mean LDoSobtained from the continued fraction is well converged tothe numerically exact result.In the continued-fraction approach, the series wastruncated by setting the terminal self-energy S (cid:96) → S (cid:96) →− i ∆ typ was set to a typical value. In both approaches,the regulator played a central role. It is worth notingan approach similar in spirit that has been employed fordisordered systems [39–41], in which one avoids making ω complex via the regulator, and self-consistency is notimposed. The imaginary part of the self-energy is simplyseeded by assigning a finite and constant imaginary ter-minal self-energy, S (cid:96) → − i . Physically, this correspondsto connecting the terminal site to a conducting lead. The0 − − − − ω − − − L D o S ‘ = 1 ‘ = 2 ‘ = 32ED FIG. 9. The average LDoS for the root site (shown for theAAH model, with V = 3), showing hierarchical structure inthe mean local spectrum. While these features are not cap-tured by the continued fraction at low orders ( (cid:96) ), the higher-order theories capture them well, and are well converged tothe ED result. The latter is shown here for L = 1024. quantity of interest then becomes the imaginary part ∆ that is induced at the root site. In a localised regime, afinite localisation length implies that the imaginary partin S (cid:96) does not in effect propagate to S , such that ∆ → (cid:96) (cid:29) ξ ; while in an extended regime the effect of cou- pling to the conducting lead induces a non-vanishing ∆ .One qualitatively new result found from the higher-order theories is the anomalous scaling of ∆ typ with L at critical points/mobility edges. Looking to the future,establishing a quantitative connection between this andthe anomalous scaling of generalised inverse participa-tion ratios of critical eigenstates, is naturally a questionof substantial interest, and well within the capabilitiesof the framework we have presented. Such an advancewould shed light not only on the critical points of themodels considered here, but also on other models whichare critical throughout, such as Fibonacci chains [25, 26].Finally, a further interesting direction for future workis the case of models with longer-ranged hopping [9, 17,24, 42], or quasiperiodic models in higher-dimensions [43–45]; where the general RPS framework for the self-energies continues to hold, albeit with a substantiallydifferent underlying structure. ACKNOWLEDGMENTS
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