Computing the eigenstate localisation length at very low energies from Localisation Landscape Theory
Sophie S. Shamailov, Dylan J. Brown, Thomas A. Haase, Maarten D. Hoogerland
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Computing the eigenstate localisation length fromLocalisation Landscape Theory
S.S. Shamailov , D.J. Brown , T.A. Haase , M.D. Hoogerland Dodd-Walls Centre for Photonic and Quantum Technologies, Department of Physics,University of Auckland, Private Bag 92019, Auckland 1142, New Zealand.* [email protected] 13, 2020
Abstract
While Anderson localisation is largely well-understood, its description has tra-ditionally been rather cumbersome. A recently-developed theory – LocalisationLandscape Theory (LLT) – has unparalleled strengths and advantages, both com-putational and conceptual, over alternative methods. To begin with, we demon-strate that the localisation length cannot be conveniently computed starting di-rectly from the exact eigenstates, thus motivating the need for the LLT approach.Then, we reveal the physical significance of the effective potential of LLT, jus-tifying the crucial role it plays in our new method. We proceed to use LLTto calculate the localisation length, as defined by the length-scale of exponentialdecay of the eigenstates, (manually) testing our findings against exact diagonalisa-tion. We place our computational scheme in context by explaining the connectionto the more general problem of multidimensional tunnelling and discussing theapproximations involved. The conceptual approach behind our method is notrestricted to a specific dimension or noise type and can be readily extended toother systems.
Contents a r X i v : . [ c ond - m a t . d i s - nn ] A ug ciPost Physics Submission7 Conclusions and future work 23References 24 Anderson localisation [1] is a universal wave interference phenomenon, whereby transport (i.e.wave propagation) is suppressed in a disordered medium due to dephasing upon many scat-tering events from randomly-positioned obstacles. This can be understood from Feynman’sinterpretation of quantum mechanics, where one must sum over all possible paths from theinitial to the final points of interest to obtain the total transmission probability. The randompositions of the scatterers guarantee dephasing between the different paths, leading to an at-tenuation of the amplitude of the wavefunction. First discovered in the context of quantisedelectron conduction and spin diffusion [2], Anderson localisation of particles thus providesdirect evidence for the quantum-mechanical nature of the universe at a small scale.In particular, Anderson localisation is characterised by an exponential decay in the tailsof the wavefunction with a length scale known as the localisation length [2]. The computationof this key variable is not straight-forward. For continuous systems, a rough estimate canbe obtained by setting the renormalised diffusion coefficient, derived in the limit of weakscattering where it is only slightly reduced from its classical value, to zero [3, 4]. While theresulting analytical formula is not expected to be accurate, it is of course convenient, and isthus used by many researchers [5–8]. The diffusive picture is in general often employed todescribe Anderson localisation, even though it is strictly inapplicable in this limit [6, 8]. Arigorous calculation can be performed using Green’s functions [3, 4, 9], but it requires manyassumptions regarding the nature of the disorder and is quite involved. On the other hand,Green’s functions can be used to extend the classical diffusive picture into the weakly-localisedregime by computing the correction to the diffusion coefficient [3, 4, 6], and even push thispicture into the strongly localised limit by making the renormalised diffusion integral equationself-consistent [3, 4, 6, 10].Another approach to obtain the localisation length is the Born approximation, commonlyutilised for weak scattering [5, 7, 11]: here, one takes the total wave in the extended scat-tering body as the incident wave only, assuming that the scattered wave is negligibly smallin comparison. Understandably, this method is inaccurate for strong disorder. Exact time-dependent simulations with the Schr¨odinger [5, 7, 12, 13] or Gross-Pitaevskii [14, 15] equationscan be used instead, but this approach is very time-consuming and yields little insight intothe physics. Finally, access to the localisation length directly through the eigenstates of theHamiltonian is hampered by practical considerations (as we shall show below).Other, more model-specific methods have also been employed in the literature: [16] solvedthe Schr¨odinger equation via a random walk on a hyperboloid, [17] derived a non-linearwave equation to extract the Lyapunov exponents corresponding to the linear problem ofinterest, [18] solved the kicked-rotor model analytically, and [19] derived analytical expressionsrelevant for the weak disorder limit.For discrete models, a plethora of methods to calculate the localisation length likewise ex-2 ciPost Physics Submission ists. The most renowned is of course the transfer matrix method, allowing for the calculationof Lyapunov exponents and thus the localisation length [20–28]. Such calculations have com-monly been used to confirm the predictions of finite scaling theory [23, 26]. While often usedtogether, transfer matrices and Lyapunov exponents have been combined with other elementsto obtain the localisation length: the former with analytical continuation [29] to computemoments of resistance and the density of states, and the latter in a perturbative expansion,with numerical simulations of a quantum walker [30]. The Kubo-Greenwood formalism hasalso proved highly successful [23, 31, 32].Green’s functions have been as invaluable for discrete systems as for continuous [9, 11, 12,23, 33, 34], allowing for renormalisation techniques to be applied [34, 35], or alternatively scat-tering matrices, treated with the Dyson equation [9]. Out of these references, [33] examinedthe off-diagonal elements of the Green’s matrix as a localisation order parameter, [12] thedistribution of eigenstates which was related to the spatial extent of the eigenstates, [9] thecharacteristic determinant related to the poles of the Green’s function, and Ref. [34] developeda renormalised perturbation expansion for the self energy. Recursion formulae encoding theexact solution [36, 37] can also sometimes allow one to calculate the localisation length (andthe density of states [37]).Out of the studies above, one-dimensional (1D) [5, 12, 16–18, 20–22, 29, 30, 33, 34, 36, 37]and two-dimensional (2D) [5–7, 9, 11, 13, 19–27, 35] models have been numerically exploredfar more thoroughly than three-dimensional (3D) [5, 26, 33], simply because of the increasedcomputational requirements of higher-dimensional spaces. Possibly the most heavily studiedmodel of localisation is the Anderson model, also known as the tight-binding Hamiltonian[3, 9, 11, 12, 21–26, 28, 29, 31, 33–36, 38–43], but other examples include the kicked rotor [18](formally equivalent to the Anderson model), the Lloyd model [12,20], the Peierls chain [37], aquantum walker [30], and the continuous Schr¨odinger equation [12,13,16], with either a specklepotential [6, 15], delta-function point scatterers [5, 9], or more realistic Gaussian scatterers [7].Meantime, a break-through new theory – coined Localisation Landscape Theory (LLT)[44–50] – was developed recently, completely revolutionising the field. It allows for intuitiveand transparent new insights into the physics, as well as a practical, efficient way of performingcalculations. To give a brief overview, this theory relies on the construction of a function,the localisation landscape, which governs all the low-energy, localised physics. One can treatfinite problems so that boundary effects are accounted for, and yet push the algorithms tovery large system sizes, where alternative methods are completely impractical. The validity ofthis theory is not restricted to a specific noise type, making it widely applicable to a range ofproblems. An effective potential can be constructed, such that quantum interference effectscan be captured instead by quantum tunnelling through this effective potential. One canpredict the main regions of existence (referred to as “domains”) of the low-energy localisedeigenstates, reconstruct the eigenstates on these domains, as well as compute the associatedenergy eigenvalues. Thus, Anderson localisation can be fully reinterpreted in this picture,including the energy dependence of the localisation length. Very recently, LLT has been usedto support an experimental study of Anderson localisation [51].In this paper, we search for a way to calculate the localisation length for an arbitrarydisordered, continuous potential. We begin by showing that the localisation length cannot beefficiently extracted from the eigenstates of the Hamiltonian. From there, we turn to LLT,and directly reveal the physical significance of the effective potential, which justifies its use inthe work that follows. Indeed, we demonstrate how the localisation length can be obtainedfrom LLT, a method that can be applied to continuous systems with any potential (as long3 ciPost Physics Submission as it satisfies the basic applicability requirements of LLT), for any strength of the disorder,and which will provide accurate results for a range of (reasonably low-lying) energies. Ourdescription is in 2D, a 1D version is much simpler and can be implemented with no additionaleffort, while a 3D version can be eventually developed by a direct analogy.Thus, we extend LLT by developing a method for the computation of the localisationlength in 2D systems. The main achievement lies in finding an efficient way of evaluating theso-called “Agmon distance”, an exponential decay cost for crossing domain walls – barriers inthe effective potential of LLT separating neighbouring domains. We discuss how our methodfits in to the extensive literature on multidimensional tunnelling, and then test it against theresults of exact diagonalisation.For the rest of the article, familiarity with LLT is certainly helpful. This backgroundknowledge can be obtained by reading the original papers [44–50], or a succinct review of thekey facts found in [52] (we use the same notation here as in [52]). Note that the present articleis one out of five related publications, available as a single, coherent report [52].The paper is structured as follows. We begin by introducing the system of interest insection 2, and proceed to demonstrate what can and cannot be learned from an exact diago-nalisation of the Hamiltonian in section 3. In section 4, we expose new aspects of the physicalsignificance of the effective potential of LLT. Then, in section 5, we extend known LLT tocalculate the localisation length, as defined by the length scale of exponential decay in thetails of the eigenstates of the Hamiltonian, and directly test the method by comparison toexact eigenstates. In the process, we develop a simple and practical approximation to multi-dimensional tunnelling, discussed in section 6, which has many potential applications in othercontexts.Conclusions are presented in section 7 and several ideas are discussed as directions fora possible forthcoming investigation. Details about our computational techniques and thenumerical methods employed can be found in the appendices of [52].
We consider a (non-interacting) particle of mass m confined to a 2D plane, whose motionis restricted to a rectangular region defined by x ∈ [0 , L ] and y ∈ [0 , W ] with Dirichletboundary conditions. The particle moves in an external potential V ( x, y ), taken as a sum of N s randomly-placed Gaussian peaks of amplitude V and standard deviation σ , as suggestedin [7]. This system could be experimentally realised with cold atoms as in [14]. A moredetailed description of the system can be found in [52].Next, we must introduce a set of dimensionless units, to be used throughout the paper.Let (cid:96) be a typical physical length scale relevant for the problem (for example, (cid:96) ∼ σ ). Lengthswill be measured in units of (cid:96) , energy is units of E = (cid:126) / (2 m(cid:96) ), and time in t = (cid:126) /E .Typically, for a cold-atom experiment such as [14], (cid:96) ∼ µ m, E ∼ × k B , and t ∼ f : f = N s (cid:96) LW . (1)4 ciPost Physics Submission
Our overall aim is to predict the localisation length for the system in section 2. Since thesystem size is finite, the potential is continuous, and does not have the required statistics forthe Green’s functions method to be helpful, we must search for an alternative approach. Now,the problem of interest is linear, so all the information is contained in the Hamiltonian andits spectrum. We therefore begin our investigation by directly diagonalising the Hamiltonianand inspecting the eigenstates and energies, with the goals of (a) gaining intuition for oursystem and (b) checking whether useful quantitative predictions may be readily obtained inthis framework. Details on the numerical implementation are given in appendix A of [52].In general, our results support the well-known fact that the localisation length increaseswith energy. We find that the localised eigenstates lie at low energies, and the degree of local-isation decreases as the energy increases. This can be easily seen by eye when inspecting theeigenstates, plotting | ψ | . An example is shown in Fig. 1, depicting nine low-energy eigenstatesfor a particular noise realisation. Overall, as energy increases, the weight of the eigenstatesspreads out over a larger area (see Fig. 3 of [44] for another example). This process, however,is not monotonic: occasionally we encounter very localised states with a fairly high energy,where most of the energy comes from the rapidly changing wavefunction rather than the spa-tial extent and the associated potential energy. Also quite intuitively, if f or V are increased,the strength of localisation increases and the area within which the weight of the eigenstatesis contained shrinks. Figure 2 demonstrates this by visually comparing the lowest energyeigenvector for different combinations of f and V . We see that both the fill factor and thescatterer height are equally important parameters, influencing localisation properties just asstrongly.Increasing the width of the scatterers σ also leads to stronger localisation (not illustrated),because the area occupied by the Gaussian peaks increases, but the dependence on the scat-terer width is not explored here. The shape of the scatterers also plays a role, of course, but aslong as the (“volume”) integral over a single scatterer is kept constant, the specific functionalform is expected to have a much weaker effect on the physics than f and V . The shape of thescatterers influences the spectral properties of the disordered potential, the relation of whichto a (possible) mobility edge could be investigated in the future.Note that one may wonder whether the low-energy, localised states seen in Figs. 1 and 2 aresimply trapped in local minima of the potential V , formed by surrounding Gaussian scatterers.This can be easily ruled out by visual inspection of the potential, as was in fact done in [44],confirming that the localised nature of these states arises from quantum interference and notclassical trapping.Next, let us consider how the localisation length may be extracted from the exact eigen-states of the Hamiltonian. By definition, the localisation length is the length scale on whichthe localised states decay exponentially, far away from the region where their main weight isconcentrated. This decay can be seen in Fig. 2 as a change of colour from dark red to red toorange to yellow to green to blue, as the wavefunction gradually drops by orders of magnitude.The localisation length increases with energy, depends on the strength of the disorder, andshould only be discussed in a configuration-averaged context.If we inspect any one given eigenstate, assuming the energy is sufficiently low or locali-sation is strong enough, there is usually only one peak – one local maximum – in | ψ | . If wetemporarily place our origin there and vary the azimuthal angle θ , then the curve | ψ ( r ) | along5 ciPost Physics Submission Figure 1: Nine low-energy eigenstates of the Hamiltonian for a given noise realisation with L = W = 25 (cid:96) , f = 0 . V = 20 E , σ = (cid:96)/
2, showing the absolute value of the eigenstatesas a colour-map. Note that all eigenstates are normalised such that the maximum is one sothat the values can be read on the same colour bar. We see that overall, the spatial extentof the eigenstates increases with energy, quoted above each panel. However, occasionally,very localised states are encountered at higher energies, on account of the considerable kineticenergy such eigenstates carry. 6 ciPost Physics Submission
Figure 2: The lowest eigenstate of the Hamiltonian for some noise realisations with L = W =25 (cid:96) and σ = (cid:96)/
2, showing the logarithm of the absolute value of the eigenstates as a colour-map. Top left: f = 0 . V = 10 E , top right: f = 0 . V = 10 E , bottom left: f = 0 . V = 20 E , bottom right: f = 0 . V = 20 E . We observe that the degree of localisationis controlled both by the density of the scatterers and their height. The energy eigenvalue isquoted above each panel: it increases as the area of the (node-free) localised mode decreases.different directions will certainly be different depending on θ . Still, we could average thesecurves over θ , and attempt fitting an exponential function to the tail of the resultant. If thepeak is located in a corner of our rectangular system, for example, the average should onlybe taken over those angles along which one has reasonable extent along r .However, as energy increases (or localisation decreases due to changes in parameters), theeigenstates develop a multi-peak structure: there are several “bumps” (see Fig. 1), and it is notclear where to place our origin. Furthermore, the energy eigenvalues are of course quantised, soany extracted localisation lengths from single-peak eigenstates need to be averaged over noiserealisations, only using eigenstates of roughly the same energy (binning within a reasonablerange). This makes such an approach very limited.Now, a very common solution to this problem – heavily used in the literature (e.g. [13,15, 53–57]) – is to compute the spatial variance of the localised states instead. Since we areworking in 2D, we could tentatively examine the quantity (cid:2) ∆ x ∆ y (cid:3) / , (2)where the variance along x is∆ x = (cid:10) x (cid:11) − (cid:104) x (cid:105) = L (cid:90) dx W (cid:90) dy x | ψ | − L (cid:90) dx W (cid:90) dy x | ψ | , (3)assuming the wavefunction is normalised to one, and ∆ y is defined similarly.7 ciPost Physics Submission Figure 3 shows a typical low energy eigenstate, plotting | ψ | on a linear scale. The small-amplitude yet large-scale structure seen on the logarithmic plots of Fig. 2, capturing theexponential decay of the eigenstates away from their main region of existence, is completelyinvisible on such a plot. The variance-based length scale of (3) reports only on the width ofthe main peak seen in Fig. 3 – analogous to the full-width-at-half-maximum or the standarddeviation of a Gaussian peak. It measures the size of the main bump, but carries no infor-mation on the exponential decay in the tails, and thus does not report on the localisationlength, as such. We therefore advise caution when using the variance to quantify localisationproperties, a common practice in the literature.Figure 3: The lowest eigenstate of the Hamiltonian for a given noise realisation with L = W = 25 (cid:96) , f = 0 . V = 20 E , σ = (cid:96)/
2, plotting (cid:96) | ψ | as a colour-map. The exponential decayaway from the main region of existence of the eigenstate is unresolvable on a linear scale. Since exact diagonalisation cannot help us to efficiently extract the localisation length, weturn to LLT for a solution. The method we develop heavily relies on the effective potentialintroduced in this theory, so before describing our approach, we explicitly validate the use ofthe effective potential in place of the real one in quantum-mechanical calculations. This isdone in the present section.The key object of LLT is the localisation landscape u , defined by the partial differentialequation Hu = 1, where H is the Hamiltonian [44]. The associated effective potential W E is simply given by W E = 1 /u . So far, LLT has produced several extremely useful resultsinvolving W E which allow to make physical predictions for a system with real potential V –in our case, a disordered one. In particular, W E controls the regions of localisation of theeigenstates at different energies, the density of states according to Weyl’s law, and the decay ofthe eigenstates through the valley lines of u according to the Agmon distance [47]. While theauthors of [47, 48] motivate this remarkable success of the effective potential by an auxiliarywave equation, it appears that W E may, to a good approximation, be able to replace V in thereal Schr¨odinger equation, directly in the Hamiltonian. In this section we test to what degreethis statement is valid.First, we check whether the eigen-states and -energies of H with W E are similar to thoseof H with V . To some extent, this is indeed the case, as demonstrated in Fig. 4. The energyspectrum seems very similar up to a global energy shift, while the eigenstates themselves areclosely correlated for sufficiently low energies. Now, according to LLT, the valley lines of u –8 ciPost Physics Submission collectively referred to as the “valley network” – divide the system into “domains” [44] (seethe top panel of Fig. 7 for an illustrative example). The valley lines of u are of course thepeak ranges of the effective potential, simply due to the inverse relationship between u and W E . Therefore, the domains are surrounded by potential barriers and constitute the regionsof localisation of low-energy eigenstates. Increasing the energy of the eigenstates enables thewavefunction to cross some of the potential barriers separating the domains and spread outfurther [47]. We find that for eigenstates that are localised to a handful of domains, involvingfundamental local modes (i.e. there is only one density peak per domain), the similaritybetween eigenstates obtained using V and W E is immediately obvious. Once localisation isweakened (e.g. by increasing the energy) to allow the occupation of many domains (possiblyin excited local states), the correlation is lost. If Anderson localisation is strengthened (byincreasing either or all of V , f , σ ), more low-energy eigenstates match between the spectraof H with V and H with W E , and the agreement between the eigenstates is improved.Returning to the energy shift between the eigenvalues in Fig. 4, the energies arising fromdiagonalising H with W E always lie higher than their counterparts using H with V . Preciselythe same trend is seen in Fig. 8 of [52], where the approximate eigenstates and eigenvaluesare reconstructed from the localisation landscape u , avoiding numerical diagonalisation. Thisis very likely linked to the fact that in both cases, the approximate LLT eigenstates area little more spread out than the exact. Since both V and W E are positive functions, ifan eigenstate has additional non-zero weight in some region of the system, its contributionwould be to increase the potential energy. On the other hand, the more tightly-localisedexact eigenstates would have more rapidly changing wavefunctions (as they decay to zerowithin a smaller area), and consequently, higher kinetic energy. It would thus appear thatthe difference in potential energy between exact and approximate eigenstates is larger thanin the kinetic energy. Curiously, we observe that the energy shift seen in the top panel ofFig. 4 seems roughly equal to the value of W E in its local basins, which was tested for manysets of parameters and several noise realisations. As a final note, we will see shortly thattransmission in the effective potential always happens more readily than in the real. Thismay be explained by the observation that the eigenstates of H with W E are somewhat moreextended than the exact and have higher overlaps.Next, let us consider time evolution (see appendix C of [52] for details on implementa-tion). In light of the apparent physical significance of W E , one would hope that a low energywavefunction would evolve similarly in W E and in V . We begin by placing a 2D Gaussianwavepacket at the centre of the system. The initial condition (up to normalisation) reads ψ = exp (cid:18) − r σ (cid:19) , (4)where r is the radial coordinate centred on ( L/ , W/ σ = (cid:96) (for this example), and thestate has energy E ≈ . E . Snap shots of the density are shown in Fig. 5 and reveal thatindeed there is a visible similarity between the expansion of the wavefunction in the twopotentials, although the state overlap drops quite rapidly. The effective potential generallyallows for a better transmission than the real one, but a strong correlation is undeniable. Ofcourse, as time goes on, the two evolving states become less similar. More faithful agreementcan be obtained if localisation is strengthened by changing parameters, or if a lower energywavefunction is used. An important aspect is the energy distribution of the wavefunction: in The use of similar probing waves was independently suggested by [5] and used in the experiment [58]. ciPost Physics Submission VW E Figure 4: Low-energy eigenspectrum (top) and six of the lowest eigenstates with L = W =25 (cid:96) , f = 0 . V = 10 E , σ = (cid:96)/
2, showing the logarithm of the absolute value of theeigenstates as a colour-map (bottom). A direct comparison is drawn between the spectrum ofthe Hamiltonian with potential V and with W E for the same noise realisation. The eigenvaluesseem very similar, up to a global energy shift. In the bottom panel, going across the rows, weplot consecutively the n th eigenstate using V and the n th eigenstate using W E , alternatingbetween the potentials before increasing n . Thus the first and second panels can be directlycompared, the third and fourth, etc. Up to the fifth eigenstate, the correlation between themode shapes is clear. From the sixth eigenstate onward, there is no visible relation betweenthe eigenmodes of the Hamiltonian with the two potentials.10 ciPost Physics Submission our case here, the Gaussian is well-localised in position space, and therefore covers quite abroad momentum range. The behaviour of the high energy components will not be capturedwell by evolution in the effective potential, as we will see shortly.In the case just considered, a wavefunction with stationary centre of mass (CoM) dynamicswas initiated inside the disordered potential and allowed to expand into it. Now we introducea transmissive scenario, studied in more detail in [52, 59]. First we have to slightly modify thegeometry of the system we are examining. The region occupied by the potential scatterersremains precisely the same, x ∈ [0 , L ] , y ∈ [0 , W ], but we add empty “reservoirs” on eitherside of the disorder where the potential is zero. These occupy x ∈ [ − R, , y ∈ [0 , W ](first reservoir, R ) and x ∈ [ L, L + R ] , y ∈ [0 , W ] (second reservoir, R ). Usually, wechoose R = 30 (cid:96) , just large enough to contain the initial condition that will be used. In thetransmissive scenario, a wavefunction with CoM translation starts out in R and goes throughthe disorder, finally arriving in R .The initial condition we will use in this set up is a 1D Gaussian wavepacket (Gaussianalong x and uniform along y ), which is fairly wide in position space and therefore has a ratherlocalised energy distribution. The functional form is simply ψ = exp( − ik x ) exp (cid:20) − ( x + R/ σ (cid:21) , (5)where we leave out the normalisation constant. Figure 6 demonstrates the transmission of sucha wavepacket with ¯ σ = 5 (cid:96) , k = 1 /(cid:96) , so that the momentum distribution is quite localised andthe mean energy is E ≈ . E . Only at fairly late times significant differences arise betweensimulations using V and W E for the potential, but the state overlap of the two wavefunctionsdecreases rather quickly. Whenever there is a strong difference between the two potentials, W E always allows the wavefunction to spread / transmit farther and more freely. By varying k we can easily change the energy of the probing wavepacket to address the question underwhat conditions can W E approximate V well? The most accurate, although perhaps not souseful, answer we have been able to find is that this substitution works well as long as thedynamics are fairly localised. In other words, as energy increases, the validity of replacing V by W E becomes questionable. Of course for weaker or sparser disorder, the range of energieswhere the replacement works well is much smaller.To conclude, we have shown that W E can to some degree replace V directly in theSchr¨odinger equation, both in terms of the eigen-values and -vectors, and in terms of timeevolution in expansion and transmission. This understanding explains why general quantum-mechanical results based on the external potential serve to give useful physical predictions fora particle moving in V if W E is used in these formulae instead of V . In this section we extend LLT to compute the localisation length, defined as the length scale ofexponential decay in the tails of the eigenstates of the Hamiltonian. A combination of severalLLT concepts allows for the development of a general methodology that can be applied toother systems, with other kinds of disorder, or in other dimensions. Technical details regardingthe implementation can be found in appendix D of [52]. We explicitly test our ideas by directcomparison to exact eigenstates. 11 ciPost Physics Submission t/t | V | W E | Figure 5: Top panel: Density profiles (all normalised such that the maximum is one so thatthe values can be read on the same colour bar) during time evolution of the initial condition(4) centred on ( L/ , W/
2) with ¯ σ = (cid:96) , for the same parameters and noise realisation as usedfor Fig. 4. Columns 1,3,5 show evolution in V and 2,4,6 in W E . Time starts at t = 0 andadvances by t / V and W E as a function of time for the simulation in the top panel.12 ciPost Physics Submission t/t | V | W E | Figure 6: Top panel: Density profiles (all normalised such that the maximum is one so thatthe values can be read on the same colour bar) during time evolution of the initial condition(5) with ¯ σ = 5 (cid:96) , k = 1 /(cid:96) , R = 30 (cid:96) for the same noise realisation and parameters as inFig. 4. Columns 1,3,5 show evolution in V and 2,4,6 in W E . Time starts at t = 4 (after theatoms enter the region with the scatterers) and advances by t / V and W E as a function of timefor the simulation in the top panel. 13 ciPost Physics Submission Recall that LLT has taught us that the low-energy eigenstates are localised inside domains ofthe valley network [44], and must tunnel through the peaks of the effective potential in orderto spread to neighbouring domains. Within any given domain, there is nothing to induceexponential decay – the decay does not happen continuously (as commonly believed), but indiscrete steps, every time the wavefunction crosses a valley line [47]. Furthermore, valley lineswhich are not part of a closed domain (referred to as “open” valley lines below) are irrelevant,as the wavefunction simply goes around them without losing amplitude.Because of its prime importance to this section, we repeat here the definition of the energy-dependent quantity known as the Agmon distance [47, 48], which controls the decay of theeigenstates outside of their main domain of existence: ρ E ( x , x ) = min γ (cid:90) γ (cid:60) (cid:112) m [ W E ( x ) − E ] / (cid:126) ds . (6)Because only the real part of the square root is used, the integrand is zero if E exceeds W E at position x . The integral should be minimised over all possible paths γ going from x to x , and ds is the arc length. If we have an eigenstate peaked at position x inside some givendomain, then it will have amplitude at position x outside of this main domain bounded by | ψ ( x ) | (cid:46) | ψ ( x ) | exp [ − ρ E ( x , x )] . (7)As the authors of [47] point out, the formula (6) is commonly encountered in the context ofthe Wentzel–Kramers–Brillouin (WKB) approximation in 1D (and higher dimensions).If we approximate the domains on average as circular in shape and denote the diameter D ,then every distance D , the wavefunction undergoes a decay. The cost of crossing a valley linewill be determined by the Agmon distance ρ E , such that the amplitude of the wavefunctiondrops by a factor of exp( − ρ E ) on average every time. Combining these two quantities, we seethat the localisation length is simply given by ξ E = D/ρ E , (8)where the subscript E on ξ stands for “eigenstate”. Remarkably, the difference between D and ξ E was already realised in [34].Now, evaluating ρ E between any two arbitrary points in the x − y plane is extremelydifficult, as discussed in section 6. However, this is not strictly necessary for our purposes.With the understanding that the system is divided into network domains, with every closeddomain containing a unique maximum of u , we can estimate the Agmon distance between theminima of W E (equivalently, the maxima of u ), considering only nearest neighbour domains.In other words, if we have two neighbouring domains (which share some common segmentof domain walls), we aim to find the least-cost path, according to the Agmon measure, thatconnects the two unique maxima of u which reside in these domains. Evaluating ρ E alongthis path would then be straight-forward.Again, formally, finding the true least-cost path is a difficult task. We have found anapproximate solution to this problem that seems much simpler to implement compared toall currently known alternatives, while not sacrificing much in terms of accuracy at all (seesection 6 to gain perspective). As explained in appendix B of [52], the valley lines are the14 ciPost Physics Submission paths of steepest descent, starting from each saddle point and ending at minima of u (valleylines may also terminate by exiting the system). Consider now curves that start from thesaddle points and follow paths of steepest ascent , ending at maxima of u . Each saddle pointthus links two maxima of u , and the curve formed in this way is the lowest-lying path on theinverse landscape W E that connects the two minima of W E in question. Figure 7 first showsan example of the valley network as originally defined [44], and then with open valley linesremoved (as they do not matter for eigenstate confinement and decay) and the minimal pathsconnecting maxima of u through the saddle points overlaid.We will use these paths to compute ρ E between any two neighbouring maxima of u . Firstof all, we highlight that the Agmon distance is an energy-dependent quantity. Thus, alongeach path, the integral must be done separately at each energy of interest, E . Now, generallyspeaking, any two neighbouring domains have several common saddles on the shared sectionof their domain walls (see Fig. 7 for an example). At each energy, we must choose the minimalpath which has the smallest Agmon integral out of the finite, discrete number of availableoptions (which is computationally trivial). The path integral along that curve then becomesthe Agmon distance ρ E between the domain maxima in question at the energy considered.This must be done for all neighbouring domains and at all energies in any given landscape u .As pointed out, ρ E between neighbouring domains is an intrinsically energy-dependentquantity. Once the energy is so high that the saddle point of the minimal path on theeffective potential W E is below E , the cost of crossing from one domain to the other vanishes: ρ E becomes zero as a break develops in the domain wall separating the two maxima of u (valleylines only effectively constrain eigenstates if u < /E , evaluated on the valley lines [44]). Forour computation of ξ E , we need the average of all non-zero ρ E across the 2D system as afunction of energy, but we also need to compute the domain area to extract the diameter, D . This requires integrating over the individual domain areas (at E = 0), averaging overall domains, assuming the area is that of a circle, and computing the diameter. However,as energy goes up and domain walls break down, domains effectively merge , so that thearea increases with energy as well. This domain merging is fully taken into account in ourcalculations.To summarise, the main steps of the calculation are as follows. Take a precomputedvalley network, remove any open valley lines and calculate all the “minimal paths” connectingsaddles to maxima of u . Next, identify the valley lines (and potentially segments of thesystem boundary) that form the domain walls for each domain and perform local, on-domainintegrals (e.g. to find the domain area, in which case the integrand is one). From here, identifyall saddles linking any two neighbouring domains, calculate the path integral of the Agmondistance over all linking paths between them, and finally obtain ρ E by choosing the smallest ofthe integrals at every energy. Then for each noise configuration, the mean of ρ E is computedover all neighbouring domain pairs, and the mean domain area yields the diameter D . Bothof these quantities are energy dependent: zero-cost links are excluded from the average of ρ E and domain areas are merged as the walls between them break down. Finally, many noiseconfigurations need to be averaged over to get a reasonable estimate of the localisation length.We remark that this calculation can be performed for any given localisation landscape aslong as it has extrema. This includes, in particular, cases when the potential V is regularand Anderson localisation is impossible. The resulting “localisation length” is then of coursemeaningless. It is up to the researcher performing the calculation to identify cases when one isdealing with localisation before attaching any significance to the result. This can be done by15 ciPost Physics Submission Figure 7: The original valley network (top) for some given noise realisation with L = W =25 (cid:96) , f = 0 . V = 5 E , σ = (cid:96)/
2, and the same network after all “open” valley lines havebeen removed (bottom). Both panels plot the valley lines in red and blue. The extrema of u are also shown as symbols (maxima in blue, minima in red, saddles in green). The bottompanel displays in addition all candidate approximate paths of least cost with respect to theAgmon metric as green and black lines, connecting neighbouring maxima of u through thelinking saddle points. 16 ciPost Physics Submission examining the fundamental on-domain eigen-energies, and ensuring that they are randomised,as explained in detail in [44, 52, 60]. We have just outlined a proposed method for computing the localisation length. While therecan be no question that the areas of the domains and the derived mean distance betweenthe valley lines really give us the desired physical quantities (as long as they are calculatedcorrectly, which has been tested), the decay constant from one domain to another, ρ E , is adifferent matter entirely. As will be discussed in section 6, the level of approximation involvedis very high, and there is no a priori assurance that our method yields numbers which faithfullycapture the decay of the eigenstates. Therefore, a direct test is in order. This can be done asfollows: for the same noise realisation, we perform the full LLT calculation, as well as find thelow energy eigenstates by exact diagonalisation. Now, we know that within each domain, thewavefunction remains roughly constant (same order of magnitude). Therefore, we integrate | ψ | over the domains, and divide by the domain areas to get the average of the wavefunctionamplitude on each domain.Then, by visual inspection of the eigenstates, we find examples of eigenstates and domainpairs where it is clear that the wavefunction tunnels from one domain to the other, as opposedto an independent occupation of the two domains. We also avoid higher local modes thanthe fundamental (excited local states involve nodes of the wavefunction within a domain).Having identified suitable candidates, we take the ratio of the mean amplitudes on the twodomains and compute the logarithm. The resulting number is equivalent to ρ E from LLT, theexponential cost of going specifically between these two domains (in this noise realisation), atan energy equal to the eigenvalue corresponding to the eigenstate examined.We have performed this test, and the results are shown in Fig. 8. A clear correlation is seen,whether the predictions of LLT are compared to the eigenstates of H with potential V or W E .The performance of the LLT method is equally good for arbitrary strengths of localisation(compare sparse and dense scatterer results), simply because the only numbers included inthe test are those for which the eigenstates and domains chosen are sensible (sufficiently lowenergy, correct local modes, decay as opposed to independent occupation, etc.). Of coursethere is scatter about the identity function, but since much averaging is performed duringthe calculation of ξ E , this scatter will disappear in the mean. This gives us confidence in thevalidity of our novel computational method. Let us examine the localisation length obtained via the prescription given in this section.Figure 9 shows ξ E computed from LLT for different densities of the scatterers (the samedensities are examined in both panels), comparing low and high scatterers between the panels.The higher f , the smaller ξ E , as expected. The system length in the bottom panel is twicethat in top, which has the effect of increasing the localisation length due to finite size effects,as shown in Fig. 10. Finite size effects are studied methodically in [52, 61], where we find thatthese are visible when at least one dimension of the system is smaller than the mean distancebetween the valley lines. Furthermore, localisation weakens with increasing system size, butthis trend is not strong and can easily be obscured by fluctuations arising from either workingin a regime where finite size effects are very small, or where localisation is weak and much17 ciPost Physics Submission Figure 8: Exponential decay cost linking two neighbouring domains, plotting the valuesmeasured from exact eigenstates and LLT against each other. There is a very clear correlationbetween them: the data points fall nicely around the identity map, shown as a black solidline. All data points presented were obtained for a system with L = W = 25 (cid:96) , V = 21 . E , σ = 0 . (cid:96) . Blue and red circles have f = 0 .
02, with blue coming from diagonalising H with W E and red with V , while green squares used the real potential V and f = 0 . ξ E as L is increased at constant scatterer density,but there is always an initial increase for L changing from 25 (cid:96) to 50 (cid:96) . This initial increasepersists at higher V and higher fill factors. Despite this, it is absolutely obvious that at low V the localisation length is much larger than at high V (see Fig. 9). Increasing the widthof the scatterers also decreases the localisation length, but we do not simulate this directly inthis paper.Each of the curves in Figs. 9 and 10 is only shown over the range of low energies whereit can be trusted, i.e. where the curve is fairly smooth and monotonically increasing. Wehave verified that the structure seen at higher energies (in particular, the local maximum, thediscontinuous jumps, etc. – see the inset of the bottom panel of Fig. 9) is all simply due tothe fact the system has a finite size, combined with insufficient averaging (we use 20 noiserealisations) because the network thins out so much by that point (ineffective valley linesare removed as the energy goes up). To explain, as energy increases, domains merge andtheir area grows in discontinuous jumps every time a domain wall breaks down. Once theaverage merged domain area becomes limited by system size (i.e. if the system was larger,more domains would have joined each cluster, but because there aren’t any more domains,the cluster area stops growing), the calculation cannot be trusted anymore. At this point, thecalculated ξ E ( E ) deviates from the expected monotonically increasing trend.Furthermore, as energy increases, more and more of the domains merge and the Agmondistances linking neighbouring domains vanish. Thus the number of measurements beingaveraged necessarily decreases, which deteriorates the quality of the final curve. Note also18 ciPost Physics Submission that once E exceeds all saddle points, ξ E diverges to infinity and ceases to exist, at whichpoint our curves must terminate. This is a predicted mobility edge, and it is studied in [52,61],where we find evidence suggesting that this prediction is unphysical. Thus, even if one couldhandle infinite systems numerically and remove the noise in ξ E , we conclude in Refs. [52, 61]that LLT cannot be trusted at high energies, and with it, the extracted localisation length.As already pointed out (and demonstrated in Fig. 9), ξ E ( E ) depends strongly on both f and V , so that one might wonder as to the precise functional form of this dependence. Thisis a highly non-trivial question. There is no guarantee in general that an analytical expressioncan be written down at all, let alone a simple one. Perhaps an expansion in an asymptoticlimit could yield a simple, analytical formula for the localisation length as a function of theparameters of the noise, but obtaining accurate numerical data in these regimes is envisionedto be rather difficult. For the purpose of the present article, we mostly leave this investigationfor future work, only conducting a single, simple test of the analytical formula in 2D ξ ∼ (cid:96) e exp (cid:16) π k e (cid:96) e (cid:17) , (9)where (cid:96) e is the mean free path and k e the wavenumber associated with the energy at whichthe localisation length is evaluated. We recall that this formula is not expected to be entirelycorrect as it is derived by first assuming weak localisation and then forcing the diffusioncoefficient to zero [3, 4] (in addition, we do not have white noise or an infinite system).One may relate the mean free path to the fill factor rather trivially by simple geometricalarguments, yielding (cid:96) e ∝ / √ f , and then fit the numerically-obtained ξ E as a function of fillfactor to ξ ∼ a √ f exp (cid:18) b √ f (cid:19) (10)with energy held fixed. By examining the dependence of the fitted parameters a and b on E ∝ k e , we can judge whether the formula (9) is supported by the numerical data. We havecarried out this test for a large system ( L = 75 (cid:96), W = 25 (cid:96) , well beyond the regime of visiblefinite-size effects [52, 61]) with high scatterers ( V = 21 . E ), varying fill factor over a widerange ( f ∈ [0 . , . ξ E may be evaluated by computing the standarderror in the domain area A and the decay coefficient ρ E , and then propagating them to findthe standard error in ξ E [see equation(8)]. Upon performing standard nonlinear fitting , wefound that the coefficients a and b indeed varied smoothly with E , which was encouraging.Moreover, the b coefficient was fairly consistent with a b ∝ √ E dependence, as expectedfrom equation (9). On the other hand, a was not independent of E , as (9) predicts, butappeared to vary linearly with 1 /E . This is not only contradictory to the formula (9), butalso dimensionally inconsistent, which suggests that this functional form is incorrect in ourcase. This is not alarming, however, because one cannot expect this formula to be applicabledue to the way and the conditions under which it was derived. Thus, the true functionaldependence of ξ E on f and V in our system remains an open question. In order to ensure the quality of each individual fit was of sufficiently high quality, we had to remove (avariable number of) the lowest fill-factor data points. ciPost Physics Submission Figure 9: The eigenstate localisation length ξ E computed for different scatterer densities (thesame colour is used for the same density in both panels; see legend) and different scattererheights: V = 5 E in the top and V = 20 E in the bottom panel. Other parameters are W = 25 (cid:96) , σ = (cid:96)/ L = 25 (cid:96) in the top and L = 50 (cid:96) inthe bottom panel. The inset in the bottom panel shows the f = 0 .
04 curve over a largerenergy range to demonstrate the numerical noise obtained from the calculation, and the axeslabels are the same as for the main figure. The localisation length increases with energy: thebehaviour at high E is artificial (see inset and the text for details) and therefore is not shownfor the majority of the data. In addition, ξ E strongly decreases with increasing scattererdensity and height. 20 ciPost Physics Submission Figure 10: The eigenstate localisation length ξ E computed for different system lengths(see legend). Other parameters are W = 25 (cid:96) , σ = (cid:96)/ V = 5 E , f = 0 .
06. Initially thelocalisation length certainly increases as L is increased (this has been confirmed in many othercases), but then there is no consistent pattern: the differences that are seen at higher L aresimply fluctuations (see discussion in the main text). The Agmon distance of LLT, including minimisation over all paths connecting the two pointsin space, gives a prescription to predict the decay of eigenstates through the barriers of W E as they tunnel out of each domain – a local potential well – and spread across the system.In the previous section we have heuristically outlined and tested a method to quantitativelyestimate ρ E between neighbouring domain minima of W E , avoiding the path minimisationstage, but using the usual expression for the integrand along the path.Multidimensional tunnelling is in fact an old and thoroughly-investigated problem. Ofcourse, brute force quantum mechanical calculations are possible, but physicists have beenstriving to obtain insight into the process by generalising the WKB approximation to di-mensions higher than one to describe it. In 1D, WKB is a straight-forward and methodicalapproach (see, e.g., [62]) – a controlled approximation that is fully understood. The general-isation to several dimensions is a different matter entirely: there is a large body of literaturedeveloping and discussing different methods, their limitations, suggesting improvements, andutilising these techniques to solve practical problems. In this section, we will provide anoverview of this topic, to place our method of section 5 in perspective.Let us see where the Agmon distance equation (6) comes from. The starting point of thederivation is usually the Feynman propagator, none other than the Green’s function of thesystem. One has to go through a series of approximations, listed below, in order to arrive atthis semi-classical formalism:1. The propagator is expanded in powers of (cid:126) , and only the zeroth order term is retained An equivalent approach is to write the wavefunction in polar form and expand the phase similarly. ciPost Physics Submission [63, 64].2. Next, one usually assumes that Hamilton’s principle function is pure imaginary [63, 65,66].3. In principle, if we want to use the Feynman propagator to describe tunnelling from oneregion of space where the wavefunction is initially contained to another, we must considerall source points, all target points, and all possible paths to arrive from each source toeach target point. In the simplest approximation, one uses the fact that the contributionof the classical path is the largest, and as we move away from it in configuration space,the contribution of the other paths is exponentially suppressed. Therefore, one usuallyonly examines the classical path, or at most a “tube” of paths around the classical one.Moreover, it is common to only consider one source point (at which the wavefunction ismaximal) and one target point (say the minimum in the potential on the other side ofthe barrier). The classical trajectory method was developed and used in many papers,e.g. [64, 67–69], and relies on minimising the action via the Euler-Lagrange equations.Assumption 1 is already a strong limitation, and to the best of our knowledge, first ordersolutions were only ever obtained in the classically allowed region [63]. However, taking (cid:126) → k -vector is arbitrarily predetermined. It has also been argued thatthis approximation can even fail for tunnelling out of a potential well [66]. The geometricalconstruction proposed in these papers is extremely involved, and completely impractical forour purposes.While in principle, accuracy could be improved by including more than one source andtarget point, as well as considering multiple paths as in [64], all three simplifications of thethird assumption are essential for our case: we cannot afford (computationally) to calculatemany paths or to describe each domain by anything more than the point at which W E attainsits minimum. This is simply because the calculation needs to be done so many times that itis simply impractical.The usual final form of the semi-classical approximation in the forbidden region involvessolving the classical equations of motion with negative the potential and the energy, or equiv-alently, in imaginary time. The differential equations are based on Newton’s laws, imposingenergy conservation as a constraint, and seek out the path of minimal action. In the contextof tunnelling out of a potential well, the trajectory is usually required to pass through theturning surface (where the kinetic energy vanishes) normally, so that it can connect smoothlyto a classical trajectory in the allowed region. On the turning surface, the velocity is alignedalong the gradient of the potential [64, 69]. An alternative constraint was used in [67]: theauthors required their escape paths to pass through the saddles of the potential and be alignedalong the correct axis of the saddle at those points (which is closer in spirit to our approach,but is less rigorous). Essentially, if the direction of the incoming wave is predetermined and itimpinges on the turning surface at any angle other than normally, the action must be taken22 ciPost Physics Submission as complex and the classical equations are insufficient. This is the chief difference betweentunnelling out of a local well and the transmission of an incoming wave through a barrier.We highlight that in the final form of the semi-classical approximation, the minimal pathis energy-dependent: one must solve the set of ordinary differential equations defining theminimal path for each energy separately. If we wish to find the classical path that connectstwo specific points, knowledge of the energy gives us the magnitude of the velocity vector,but its direction is unknown. Trial and error is called for to discover the latter: one needs totry different initial directions of motion until a path that arrives at the desired end point isfound. This makes the traditional (and formally correct) solution of the semi-classical problemimpractical for our purposes.Our method of section 5 overcomes these problems: no differential equations need to besolved at all (one only needs to know the localisation landscape u ), one path is computed forall energies, and there is no need to guess the initial condition. As we have seen in Fig. 8, itperforms well, which justifies its use despite the many approximations in deriving the semi-classical formulation, as well as our heuristic way of computing the escape paths. In eithercase, no other level of approximation is practical for our purposes, as we need to computethe Agmon distance between every two neighbouring domains at all energies for many noiserealisations (twenty are used in practice), at each set of parameters investigated.A few final notes are in order, without which any review of multidimensional tunnellingwould be incomplete. References [70, 71] have developed the path decomposition expansionmethod, which allows one to divide space into separate regions, minimise the action in eachregion using whatever method happens to be optimal in that region (chosen based on physicalconsiderations), and then collate the solutions using global consistency equations. Whilenot used in our work, it is clear that our problem would fit nicely into such a formalism:our system is naturally divided into domains (which are local basins in W E ). It shouldbe possible to use the path decomposition expansion formalism to predict tunnelling acrosslarge distances, spanning several domains, by combining local information through globalcollocation equations.Reference [64] deserves special attention, as an exceptional effort was made to considermany classical paths from many source points, deriving the tunnelling current and transmis-sion coefficient through the potential barrier.For a more comprehensive review of the topic, the reader is referred to [72], as well as theoriginal literature cited above. In this paper we used LLT to calculate the eigenstate localisation length, quantifying thedecay length scale of the eigenstates, as a function of energy. This required us to develop apractical approximation to multidimensional tunnelling and a formidable extension of LLTtechniques and machinery. It also involved considerable conceptual progress, linking togetherdomain size and the decay exponent (the “cost”) of tunnelling through the peak ranges of W E separating domains through the saddle points. We accounted for the effect of increasingenergy by merging domains as the domain walls separating them broke down. Crucially, weexplicitly tested the decay coefficients computed from LLT against exact eigenstates, vali-dating our computational method and the many approximations involved. We also reviewed23 ciPost Physics Submission multidimensional tunnelling to set our method in context.In addition, we highlighted the difficulty in extracting the localisation length out of ex-act diagonalisation calculations. We further demonstrated that the effective potential W E can replace the real potential V in the Hamiltonian in terms of reproducing the low-energyeigenspectrum as well as for time-evolution of expanding or transmitting wavepackets.Some ideas for future work that naturally came up during this investigation are:1. First of all, it would be excellent to generalise LLT to 3D, where the logic and conceptualpicture are largely unchanged, but the practical framework and the technology arenot yet in place (everything beyond obtaining u and performing simple mathematicaloperations on it). This would open the door to a large number of possible studies in3D.2. One should also investigate the functional dependence of ξ E on the fill factor and V . Atthe moment, this can only be done by running large numbers of simulations at differentparameters and examining the dependence explicitly, hoping to discover the functionalform by inspection.3. What effect does the shape of the scatterers have? We have limited ourselves to 2DGaussian peaks (of more or less constant width) for this paper. What would happen ifwe changed the width, or even made the scatterers, say, square? Acknowledgements
S.S.S. warmly thanks the following researchers for extremely helpful discussions on the topicsindicated in parentheses after each name: Daniel V. Shamailov (the entire project), AntonioMateo-Mun˜oz (spectral methods in exact diagonalisation), Xiaoquan Yu (importance of thedensity of states for Anderson localisation), Marcel Filoche and Svitlana Mayboroda (theAgmon distance). Jan Major is further gratefully acknowledged for reading the manuscriptand providing useful comments.
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