L 2 localization landscape for highly-excited states
LL localization landscape for highly-excited states Lo¨ıc Herviou and Jens H. Bardarson Department of Physics, KTH Royal Institute of Technology, Stockholm, 106 91 Sweden
The localization landscape gives direct access to the localization of bottom-of-band eigenstates innon-interacting disordered systems. We generalize this approach to eigenstates at arbitrary energiesin systems with or without internal degrees of freedom by introducing a modified L -landscape, andwe demonstrate its accuracy in a variety of archetypal models of Anderson localization in one andtwo dimensions. This L -landscape function can be efficiently computed using hierarchical methodsthat allow evaluating the diagonal of a well-chosen Green function. We compare our approach toother landscape methods, bringing new insights on their strengths and limitations. Our approachis general and can in principle be applied to both studies of topological Anderson transitions andmany-body localization. Introduction. —The theoretical discussion of Andersonlocalization, the strict confinement of matter waves toa finite subspace due to destructive quantum interfer-ence, dates back to 1958 . Progress since then hascome in bursts, often separated by long intervals. Onlyafter over twenty years did mathematical proofs startto appear and the scaling theory of localization wasintroduced , suggesting that all eigenstates in low dimen-sions are localized. Two major recent modern develop-ments involve the interplay of localization with topologyand interactions: Surfaces of topological insulators re-sist localization and extended bulk states are obtainedat the transition between two topologically distinct insu-lating phases . Interactions give rise to many-bodylocalization, in which an eigenstate phase transition isobtained at energies high above the ground state .These phenomena only started to be understood in thelast couple of decades.One reason for this slow progress may be that local-ization is due to nontrivial interference patterns that arenot easily guessed from the random potential the parti-cles move in. There is generally no obvious correlationbetween the localization centers of wave functions and thepotential extrema. In a sense, this means that there isno obvious classical starting point from which one can dosimple perturbation theory. Now, in a series of fascinat-ing work, such a starting point may have been identifiedin the so-called localization landscape . The localiza-tion landscape is an effective potential obtained from theinitial random potential, and it has the property that itspeaks and valleys predict the location of the few lowestenergy localized wave functions. It furthermore gives thecorrect integrated density of states at low energy from asimple Weyl law, which otherwise badly fails when usingthe original potential.The original formulation of the localization land-scape is for scalar field theories with a real and posi-tive Green function, and applies strictly only to low en-ergy states close to the bottom of the energy spectrum.These constraints prevent direct applications to many ofthe modern approaches mentioned above, where the in-teresting physics often takes place in states at or nearthe middle of the spectrum. Here, we introduce an ex- tension of the localization landscape, which we coin the L -landscape, that faithfully captures the localization ofeigenstates at all energies and in the presence of internaldegrees of freedom. The L -landscape can be efficiently numerically obtained in generic physical models, in theabsence of long-range hopping. We exemplify its validityand reliability through several archetypal models of local-ization in one and two dimensions. This new landscapeis applicable to both topological models and many-bodyHamiltonians and can therefore be used to analyze mostlocalization problems.An alternative extension of the localization landscapeto Dirac fermions was recently introduced by Lemut etal. in Ref. 30. This method, based on the comparison-matrix , has the advantage that is retains the simplic-ity of the original landscape, and can be applied to DiracHamiltonians with inner degrees of freedom. Neither theoriginal landscape nor the one based on the comparisonmatrix can, however, describe a generic high-energy stateas does our L -landscape, albeit at the cost of a slightlyreduced efficiency. We conclude our work by briefly com-paring our method to these alternatives, bringing insightsinto the strengths and weaknesses of conventional local-ization landscape approaches. L localization landscape. —In their original paper ,Filoche and Mayboroda considered the localization of ascalar field, or equivalently of spinless fermions. Let H bethe corresponding single-particle Hamiltonian and (cid:12)(cid:12) φ β (cid:11) an eigenstate of H with eigenvalue E β . We denote by φ βj = (cid:10) j | φ β (cid:11) its amplitude at site j . By application of theinverse of the Hamiltonian, one straightforwardly obtains | φ βj | = | E β (cid:88) m ( H − ) j,m φ βm | (1) ≤ | E β | (cid:13)(cid:13) φ β (cid:13)(cid:13) ∞ (cid:88) m | ( H − ) j,m | ≡ | E β | (cid:13)(cid:13) φ β (cid:13)(cid:13) ∞ u j . (2) u is called the localization landscape. The key insightof Ref. 20 was to realize that in a wide class of models, H − can have all components positive, implying that u is a solution to the differential equation Hu = 1 . (3) a r X i v : . [ c ond - m a t . d i s - nn ] A p r The requirement of element-wise positivity of H − en-forces strong restriction on H : it must be a monotonematrix , a class of matrices that is generally hard tocharacterize. In the case of a real symmetric matrix withall off-diagonal (hopping) terms negative, such as in thestandard Anderson model, a necessary and sufficient con-dition is that H is positive definite. The localizationlandscape proves to tightly bound bottom-of-band eigen-states, almost saturating Eq. (2), in a wide variety ofmodels . This saturation implies that the lowest-energyeigenstates are localized at the peaks of the landscapeand different eigenstates are separated by landscape min-ima. Indeed, we can rewrite the localization landscapeas u j = (cid:88) β φ βj E β (cid:88) m φ βm . (4)By construction, due to the inverse energy factor, eigen-states with the lowest energy will contribute more to thelocalization landscape than ones at higher energies. Onthe other hand, high-energy states are not accurately lo-calized by the landscape. This landscape can thereforenot be used to study center-of-band properties.We propose to overcome this limitation by slightlymodifying the definition of the localization landscape.Starting from Eq. (1), we apply the Cauchy-Schwartzinequality to obtain | φ βj | ≤ | E β | (cid:13)(cid:13) φ β (cid:13)(cid:13) (cid:115)(cid:88) n ( H − ) j,n ( H − ) ∗ j,n (5)= | E β | (cid:113) ( M − ) j,j , (6)where M = H † H is a Hermitian positive definite matrixand we assume normalized eigenfunctions with || φ β || =1. The L -landscape u (2) is then defined by u (2) j = (cid:113) ( M − ) j,j . (7) M is invertible as long as H is invertible, and the inequal-ities are valid whether H is Hermitian or non-Hermitian.The largest contributions to the landscape u (2) are fromthe eigenstates with the smallest absolute energy. Withthis definition, there is no requirement that H be pos-itive definite, and we can therefore explore localizationat all energies by simply shifting the Hamiltonian by aconstant real factor E . Note also that the normaliza-tion by the largest element of φ β has vanished, replacedby its 2-norm (equal to 1 by convention). The change innormalization can conveniently help to differentiate local-ized and delocalized regimes. In the original formulation,several tightly localized but close-in-energy eigenstateswould have exactly the same landscape signatures as astate delocalized on a subpart of the system (with wellseparated peaks) as the difference in amplitude of thewave functions is not taken into account. Eq. (7) is valid in the continuum limit, and can be straightforwardly ap-plied to systems with internal degrees of freedom.To ensure that M can be inverted, it is convenient tointroduce a complex energy shift ε and work with thematrix ˜ H = H + iε Id. The energy in the bound is thenrenormalized to E βε = | E β + iε | = (cid:112) ( E β ) + ε . ε can betaken as small as required (though too small a value mayaffect the coordination number of M and therefore thenumerical precision of certain computations), and needsto be smaller than the level spacing at the probed energyrange in order to resolve different eigenstates. We cangain an intuition for this by writing, for an HermitianHamiltonian, the square of the landscape as( u (2) j ) = (cid:88) β | φ βj | ( E β − E ) + ε , (8)where we have now also explicitly included the real en-ergy shift E . We therefore have that ε ( u (2) j ) −−−→ ε → ρ j ( E ) , (9)where ρ j ( E ) = (cid:80) β | φ βj | δ ( E − E β ) is the local density ofstates at site j and energy E . This explains why the L -landscape provides an efficient description of states closeto E , while the presence of the factor of ε on the lefthand side of relation (9) means that states further awayfrom E also contribute to the landscape.The L localization landscape can be computedefficiently, even if it does not satisfy a simple (discrete)differential equation. Indeed, for short ranged Hamil-tonians, numerous methods have been developed tocompute the diagonal of the Green functions efficiently,such as hierarchical algorithms (that can also takeadvantage of the positive definiteness of M ). Morerefined algorithms in two dimensions can computethe diagonal of the inverse in O ( L ) operations, where L is the linear dimension of the two-dimensional system.Moreover, several methods exist to numericallyderive upper bounds on the components of the inverseof Hermitian definite positive matrices, that can readilybe applied here. Anderson model. —We first illustrate our method in theprototypical one-dimensional Anderson model for local-ization, with Hamiltonian H = − t (cid:88) j ( c † j c j +1 + c † j +1 c j ) + (cid:88) j V j c † j c j . (10) c j ( c † j ) is the fermionic annihilation (creation) operatoron site j , t is the hopping amplitude (set to 1 in thefollowing) and V j is a random on-site potential uniformlydistributed in [ − W, W ]. An arbitrarily weak disorderis enough to localize all eigenstates at all energies inthe thermodynamic limit, including in the middle ofthe spectrum. In Fig. 1 we show the L localization FIG. 1. L -landscape and the four eigenstates closest to zeroenergy ( E = 0) in the Anderson model for disorder strengths W = 25 (a) and W = 2 (b). The eigenstates are normalizedby their energy and ε = 10 − is fixed to be smaller thanthe typical mean level spacing. The different peaks in thelocalization landscape coincide with the different eigenstatesand their location. The low minima form domain boundsthat separate different eigenstates at low-energy. u (2) predictsaccurately the localization and ordering of the states in allcases, and tightly bounds the localization of these states. landscape at zero energy in a chain of L = 100 sites, andcompare it with the few eigenstates nearest in energy.Taking the cut-off ε to be smaller than the typical levelspacing, u (2) accurately describes the localization of thestates close to E = 0 at both strong and weak disorder.As with the conventional landscape, many eigenstatesare captured by a single computation of the landscape,whether at strong or weak disorder. The ordering ofpeak amplitudes matches the eigenstate ordering. Chiral Anderson model. —The ability to access arbi-trary energies allows us to access more refined propertiesof localization, such as due to the presence of symme-tries. In one dimension, the presence of chiral symmetryleads to an even-odd effect in terms of the number of channels ; indeed, due to the symmetry, states eithercome in pairs ( E, − E ) or have zero energy. For an oddnumber of channels (and an odd number of sites at finitesizes) there therefore must exist a symmetry protectedzero energy eigenstate. This zero energy state is delocal-ized even in the presence of strong disorder. In Fig. 2a,we compare the L localization landscape and eigenstatesof the minimal single-channel chiral Hamiltonian H = − (cid:88) j t j c † j c j +1 + h.c. (11)with t j taken uniformly in [ − V , V ].Though the landscape may appear similar to the oneobtained in Fig. 1, we can identify that most peaks arecontributions of a zero mode by varying the cutoff ε . In-deed, the energy in Eq. (6) is given by | E β + iε | . When ε →
0, the contributions to the landscape of states withnonzero energy are suppressed compared to the divergentcontribution of the zero energy eigenstates and we canidentify the zero mode contributions by computing thelandscape for two different cut-offs: bounds of the zeromodes will scale as the inverse of ε . In Fig. 2, we showan example with such a delocalized state. Additionally,around j = 60, one can see a few peaks where the land-scape does not scale linearly with ε ; this is where the firstexcited states are localized. In the absence of degenera-cies, it is then immediate to identify that the zero modespans large part of the system. One can verify by shiftingthe energy reference E that bulk states are localized.Conversely, in the case of an even number of channels,the symmetry no longer guarantees the presence of a zeromode, and the eigenstates close to zero energy are alllocalized. The Hamiltonian H = − t (cid:88) j (cid:126)c j † σ z (cid:126)c j +1 + h.c. − (cid:88) j V j (cid:126)c j † σ y (cid:126)c j +1 + h.c. (12)with (cid:126)c = ( c ↑ , c ↓ ) two fermionic species, σ α with α = x, y, z the Pauli matrices and V j ∈ [ − V , V ], is an ex-ample of two-channel chiral Anderson model. The chiralsymmetry is realized by σ x Hσ x = − H (13)As shown in Fig. 2b, there are no zero modes andthe eigenstate closest to zero energy is now localized.The localization landscape bounds the eigenstates lesstightly than in the previous examples due to the chiralsymmetry: states comes in pairs of opposite energieswhich have exactly the same renormalized energiesand similar local polarization. These two contributionstherefore sum up constructively and strongly relax theusual tightness of the bound. This can be remediated bya small breaking of the chiral symmetry with a nonzero E smaller than the mean level spacing. Dirac fermions in two dimensions. —Finally, wedemonstrate that the L -landscape also captures the (ab- FIG. 2. The L localization landscape at E = 0 for twovalues of ε and the lowest lying eigenstate in the one-channel(a) and two-channel (b) chiral Anderson model, for V = 4and L = 101 sites. We only plot the spin up component of thelandscapes and wave functions in the two-channel case for sim-plicity; the other component can be obtained by symmetry.For one channel, there exists an extended zero mode whichgives a clear contribution to the landscape, with an amplitudethat scales as ε − . Conversely, the part of the landscape thatdoes not scale with ε (e.g., around j = 60) corresponds tohigher energy states. In the inset we show, for reference, thewave functions of the four states in the bulk of the band with E = V . For two channels, there is no zero mode, and thelowest energy states are localized. The rescaled landscapedoes not match its initial counterpart. The landscape is aless tight bound than usual due to the chiral symmetry whichdoubles the number of states. To get a tighter bound onecan split the pairs of states at ± E by a weak breaking of thesymmetry. sence of) localization of Dirac fermions in two dimen-sions. Single Dirac cones with time reversal are not lo-calized at any energy , and belong to different uni-versality classes depending on the form of the disorder. Aconvenient lattice model to simulate a single Dirac coneis a critical two-dimensional Chern insulator on a square lattice H = − t (cid:88) (cid:104) (cid:126)r,(cid:126)r (cid:48) (cid:105) (cid:0) (cid:126)c (cid:126)r † σ z (cid:126)c (cid:126)r (cid:48) + h.c. (cid:1) − µ (cid:88) (cid:126)r (cid:126)c (cid:126)r † σ z (cid:126)c (cid:126)r (14)+ ∆ x (cid:88) (cid:126)r (cid:0) i(cid:126)c (cid:126)r † σ x (cid:126)c (cid:126)r + (cid:126)e x + h.c. (cid:1) (15)+ ∆ y (cid:88) (cid:126)r (cid:0) i(cid:126)c (cid:126)r † σ y (cid:126)c (cid:126)r + (cid:126)e y + h.c. (cid:1) (16) t , ∆ x and ∆ y act as different flavors of spin-orbit cou-pling, and µ is a chemical potential. The system fallsinto class D with the particle-hole symmetry σ x H ∗ σ x = − H. (17)For µ = ± t and ∆ x and ∆ y nonzero, the Hamiltonian isat a critical point between a topological phase with Chernnumber ± (cid:126)k = (0 ,
0) for µ = − t and (cid:126)k = ( π, π )for µ = 4 t . We place ourselves at this phase transitionand introduce all possible random local perturbations V a = (cid:88) (cid:126)r,a V a(cid:126)r (cid:126)c (cid:126)r † σ a (cid:126)c (cid:126)r , (18)with a ∈ { , x, y, z } and V a(cid:126)r taken uniformly in[ − V a , V a ]. V is a random scalar potential, V z arandom mass and V x/y random chiral hoppings. Therandom mass V z preserves the particle-hole symmetry,while the other potential terms break the symmetry suchthat the system falls directly into class A . In class D ,at weak disorder, the system would fall into the thermalquantum hall transition fixed point, before transitioningat higher disorder to a metallic phase, as long as thedisorder averages to zero. In class A on the otherhand, the model flows towards the integer quantum halltransition fixed point, though with strong finite-sizeeffects that will lead to apparent localization at strongdisorder and higher-energies. In Fig. 3, wecompare the prediction of the localization landscape forthe critical Chern insulator and the actual low-energyeigenstates in the presence of all types of disorder, forthe two spin components. Similar results are obtained inthe D class. Peaks and valleys in the landscape matchthe ones in the eigenstates, both exactly at zero energywhere the gap closes, but also deep in the band. We doobserve the absence of localization close to zero energy,as is evident by looking at the spin-down component. Discussion. —We have introduced the L -landscape, anextension of the localization landscape that can be usedto characterize eigenstates of a Hamiltonian in the bulkspectrum of arbitrary models. This requires the com-putation of the diagonal of the inverse of the positive-definite matrix M = H † H . It provides an accurate andtight bound, in the absence of degeneracies, on the local-ization or delocalization of eigenstates at an arbitrary en-ergy. We have demonstrated the power of this new land-scape in a variety of models in one and two dimensions, FIG. 3. L localization landscape for the critical two-dimensional Chern insulator in the presence of all types ofdisorder for both spin components (left: spin up, right: spindown). We fix V α to 2 t . In both graphs, the vertical compo-nent depicts the low-level eigenstates, while the colorscale isthe corresponding normalized value of the landscape. Peaksand valleys in the landscape match the lowest energy eigen-states. with and without internal degrees of freedom, and pre-senting mobility edges and other nontrivial localizationproperties. In all these examples, our method success-fully and accurately pinpointed the eigenstates closest toany target energy.It is pertinent to compare our results to otherlandscape-based approaches. In particular, thecomparison-matrix landscape introduced in Ref. 30 canin principle be used to study states in the middle ofthe spectrum. In practice, the comparison matrix needsto be positive-definite, which, in the models we consid-ered, requires the introduction of the same shift ε weintroduced. Instead of being a small control parameter,however, this shift is much larger than the mean levelspacing and sometimes even of the order of the band-width. The energy denominator in Eq. (4) (replacing theHamiltonian by the comparison matrix) is then stronglyflattened, with all eigenstates contributing with similaramplitudes. The obtained landscape is then no longer agood predictor of the localization of the eigenstates clos-est to the target energies (see the Appendix for moredetails). These are generic limitations in conventionallandscape methods, as long as the energy gap to thelowest eigenstate is much larger than the level spacing.This problem can be alleviated by certain types of disor-der that make the Hamiltonian diagonal dominant, andtherefore allow for small ε , such as discrete disorder dis-tributions or disorder of the form V (cid:126)n.(cid:126)σ , with V a largeconstant amplitude and (cid:126)n a random unit vector, repre-senting strongly disordered magnetic impurities.The generality of our approach—including both inter-acting systems (in configuration space for example), non-Hermitian models and continuous models—is straightfor-ward as it only requires the invertibility of the Hamilto-nian, that can be shifted by a small ε ∈ C . In particu- lar, the possibility of targeting accurately highly excitedstates may prove useful for applications to many-bodylocalization , though the high coordination number ofthe equivalent Anderson lattice may limit a purely nu-merical computation. For possible future directions, wenote that wave functions at the Anderson transition pointare known to exhibit multifractal behavior . Theproperties of the critical point can be identified by com-puting the fractal dimension of the wave function. Howto generalize these ideas to the localization landscape,as the latter does not describe a single eigenstate, but asuperposition of several with weight depending on theirenergies, is an interesting open question. ACKNOWLEDGMENTS
We thank Carlo Beenakker, Vardan Kaladzhyan, Ed-win Langmann and Bj¨orn Sbierski for useful discussions.This work was supported by the ERC Starting GrantNo. 679722, the Roland Gustafsson’s Foundation for The-oretical Physics and the Karl Engvers foundation.
Appendix A: Comparison-matrix landscape methodfor highly excited states
In this appendix we discuss the limitation of traditionallocalization landscape methods to study excited states.For concreteness we focus on the method of Ref. 30,which introduced a variation on the localization land-scape based on the comparison matrix in order to studysystems with inner degrees of freedom. While it can inprinciple also be applied to middle of spectrum states,it generically fails at characterizing the localization ofthese states. Here we discuss the reasons for this failureas it reveals some limitations of conventional localiza-tion landscape methods. There exist two natural waysto study highly excited states using the comparison ma-trix, by introducing the Hamiltonians H ( ε ) and H ( ε )defined by H ( ε ) = H + iε Id , (A1) H ( ε ) = H † H + ε Id . (A2)These two Hamiltonians admit the same eigenstates as H (for H Hermitian) and are both invertible for ε (cid:54) = 0.They satisfy Eq. (1) with renormalized energies given by E β = (cid:113) ( E β ) + ε and E β = ( E β ) + ε . (A3)Both Green functions H − α are generally not real posi-tive, despite H being definite positive. In particular, H − is generally complex-valued. Ostrowski’s compari-son matrix can be introduced to solve this issue andavoid the need to compute the full inverse . The com-parison matrix H of an Hamiltonian H is defined by H m,n = 2 | H m,m | δ m,n − | H m,n | . (A4)If it is positive definite, then it verifies | H − m,n | ≤ H − m,n . (A5)We then have | φ βm | ≤ E βα max n | φ βn | u CM α,m , where the local-ization landscape can be efficiently obtained by solvingthe equation H α u CM α = 1.The key limitation of the method, like the originallandscape method, is the need for H α to be positive def-inite, and the consequences of such a requirement on u α .A naive but informative sufficient condition for definitepositiveness for a real symmetric matrix A is (cid:88) m A m,n > n. (A6)For the comparison matrix, this translates into having H be diagonally dominant. This condition can alwaysbe satisfied by choosing ε large enough. On the otherhand, if ε is too large, i.e., much larger than the typicalmean level spacing or of the order of the bandwidth, the renormalized energies E α become comparable for all low-energy states. The localization landscapes u α are then nolonger a good predictor of the localization of low-energyeigenstates as too many eigenstates contribute with sim-ilar amplitudes. An alternative interpretation is that theeigenvectors of the comparison matrix are no longer closeto those of the original Hamiltonian, and the landscapeobtained from H α , which describes the localization of itseigenvectors, no longer describes the eigenstates of H α .Conversely, when H is already diagonally dominant be-fore introducing ε —for example, for well-chosen disorderdistributions in the strong disorder limit—it proves tobe a very efficient way to study the localization of thelow-energy eigenstates.Let us illustrate these statements in the Andersonmodel introduced in Eq. (10). Fig. 4 summarizes ourresults studying eigenstates at zero energy in a chain of L = 100 sites, looking at the same disorder realizationsas in Fig. 1. When the disorder is strong enough, thecomparison matrix can typically be definite positive for ε smaller than the typical level spacing. The localizationlandscapes are then good predictors of the localization ofthe eigenstates. Note that this is a finite-size effect: asthe system size increases, one requires larger and largerdisorder to reach that limit. On the other hand, at lowdisorder, ε needs to be much larger than the level spac-ing in order for H to be positive definite and the land-scapes completely fail to predict the localization of thelow-energy eigenstates. P. W. Anderson, “Absence of diffusion in certain randomlattices,” Phys. Rev. , 1492–1505 (1958). I. Goldshtein, Stanislav Molchanov, and Leonid Pas-tur, “Pure point spectrum of stochastic one dimensionalschrdinger operators,” Functional Analysis and Its Appli-cations , 1–8 (1977). H. Kunz and B. Souillard, “Sur le spectre des oprateursaux diffrences finies alatoires,” Comm. Math. Phys. ,201–246 (1980). E. Abrahams, P.W. Anderson, D. C. Licciardello, andT. V. Ramakrishnan, “Scaling theory of localization: Ab-sence of quantum diffusion in two dimensions,” Phys. Rev.Lett. , 673–676 (1979). M. Z. Hasan and C. L. Kane, “Colloquium: Topologicalinsulators,” Rev. Mod. Phys. , 3045–3067 (2010). A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W.Ludwig, “Classification of topological insulators and su-perconductors in three spatial dimensions,” Phys. Rev. B , 195125 (2008). F. Evers and A. D. Mirlin, “Anderson transitions,” Rev.Mod. Phys. , 1355–1417 (2008). A. W. W. Ludwig, “Topological phases: classificationof topological insulators and superconductors of non-interacting fermions, and beyond,” Physica Scripta
T168 ,014001 (2015). P. W. Brouwer, C. Mudry, B. D. Simons, and A. Altland,“Delocalization in coupled one-dimensional chains,” Phys. Rev. Lett. , 862–865 (1998). T. Senthil, M. P. A. Fisher, L. Balents, and C. Nayak,“Quasiparticle transport and localization in high- T c su-perconductors,” Phys. Rev. Lett. , 4704–4707 (1998). I. A. Gruzberg, A. W. W. Ludwig, and N. Read, “Ex-act exponents for the spin quantum hall transition,” Phys.Rev. Lett. , 4524–4527 (1999). T. Senthil, J. B. Marston, and M. P. A. Fisher, “Spinquantum hall effect in unconventional superconductors,”Phys. Rev. B , 4245–4254 (1999). N. Read and D. Green, “Paired states of fermions in twodimensions with breaking of parity and time-reversal sym-metries and the fractional quantum hall effect,” Phys. Rev.B , 10267–10297 (2000). M. Titov, P. W. Brouwer, A. Furusaki, and C. Mudry,“Fokker-planck equations and density of states in disor-dered quantum wires,” Phys. Rev. B , 235318 (2001). B. L. Altshuler, Y. Gefen, A. Kamenev, and L. S. Levitov,“Quasiparticle lifetime in a finite system: A nonperturba-tive approach,” Phys. Rev. Lett. , 2803–2806 (1997). I. V. Gornyi, A. D. Mirlin, and D. G. Polyakov, “Inter-acting electrons in disordered wires: Anderson localizationand low- t transport,” Phys. Rev. Lett. , 206603 (2005). D. M. Basko, I. L. Aleiner, and B. L. Altshuler, “Metalinsulator transition in a weakly interacting many electronsystem with localized single particle states,” Ann. Phys. , 1126 – 1205 (2006).
FIG. 4. Localization landscapes ((a-b) u CM1 , (c-d): u CM2 )and the four eigenstates closest to zero energy in the Ander-son model for different disorder strengths ((a, c): W = 25,(b, d): W = 2). We consider the same disorder realizationsas in Fig. 1. At large disorder, due to the small size of thesystem, the comparison matrices can be (close to) positive def-inite and we can take ε smaller than the typical level spacing.Then peaks in u CM and u CM do correspond to the low-energyeigenstates, albeit the bound is not tight and the ordering ofthe height of the peaks might not correspond to the orderingof eigenstates. At lower disorder, typical realizations requiremuch larger shift for the comparison matrix to be positivedefinite and ε becomes of the order of the bandwidth. Thepeaks of the localization landscape are then no longer well-correlated with the localization of the low-lying states. Notealso that we had to normalize the landscape in order to rep-resent it at the same scale as the normalized eigenstates. D. A. Abanin and Z. Papi, “Recent progress in many-bodylocalization,” Ann. Phys. (Berlin) , 1700169 (2017). F. Alet and N. Laflorencie, “Many-body localization: Anintroduction and selected topics,” C. R. Phys. (2018). M. Filoche and S. Mayboroda, “Universal mechanism foranderson and weak localization,” Proc. Natl. Acad. Sci.USA , 14761 (2012). M. Filoche and S. Mayboroda, “The landscape of ander-son localization in a disordered medium,” Contemp. Math. , 103 (2013). M. L. Lyra, S. Mayboroda, and M. Filoche, “Dual hiddenlandscapes in anderson localization on discrete lattices,”Euro. Phys. Lett. , 47001 (2014). D. N. Arnold, G. David, D. Jerison, S. Mayboroda, andM. Filoche, “Effective confining potential of quantumstates in disordered media,” Phys. Rev. Lett. , 056602(2016). S. Steinerberger, “Localization of Quantum States andLandscape Functions,” Proc. Amer. Math. Soc. , 2895(2017). M. Filoche, M. Piccardo, Y.-R. Wu, C.-K. Li, C. Weisbuch,and S. Mayboroda, “Localization landscape theory of dis-order in semiconductors. i. theory and modeling,” Phys.Rev. B , 144204 (2017). M. Piccardo, C.-K. Li, Y.-R. Wu, JamesJ. S. Speck,B. Bonef, R. M. Farrell, M. Filoche, L. Martinelli,J. Peretti, and C. Weisbuch, “Localization landscape the- ory of disorder in semiconductors. ii. urbach tails of dis-ordered quantum well layers,” Phys. Rev. B , 144205(2017). C.-K. Li, M. Piccardo, L.-S. Lu, S. Mayboroda, L. Mar-tinelli, J. Peretti, J. S. Speck, C. Weisbuch, M. Filoche,and Y.-R. Wu, “Localization landscape theory of disorderin semiconductors. iii. application to carrier transport andrecombination in light emitting diodes,” Phys. Rev. B ,144206 (2017). D. N. Arnold, G. David, M. Filoche, D. Jerison, andS. Mayboroda, “Computing spectra without solving eigen-value problems,” SIAM Journal on Scientific Computing , B69–B92 (2019). G. David, M. Filoche, and S. Mayboroda, “The landscapelaw for the integrated density of states,” arXiv:1909.10558. G. Lemut, M. J. Pacholski, O. Ovdat, A. Grabsch,J. Tworzyd(cid:32)lo, and C. W. J. Beenakker, “Localizationlandscape for dirac fermions,” Phys. Rev. B , 081405(2020). A. Ostrowski, “ ¨Uber die determinanten mit ¨uberwiegenderhauptdiagonale,” Comm. Math. Helvetici , 69–96(1937). A. Ostrowski, “Determinanten mit ¨uberwiegender haupt-diagonale und die absolute konvergenz von linearen it-erationsprozessen,” Comm. Math. Helvetici , 175–210(1956). L. Collatz,
Functional Analysis and Numerical Mathemat-ics (Academic Press, New York, 1966). B. L. Buzbee, G. H. Golub, and C. W. Nielson, “On directmethods for solving poissons equations,” SIAM Journal onNumerical Analysis , 627–656 (1970). A. George, “Nested dissection of a regular finite elementmesh,” SIAM Journal on Numerical Analysis , 345–363(1973). M. P. L. Sancho, J. M Lopez Sancho, J. M. L. Sancho, andJ. Rubio, “Highly convergent schemes for the calculationof bulk and surface green functions,” Journal of Physics F:Metal Physics , 851–858 (1985). A. Svizhenko, M. P. Anantram, T. R. Govindan, B. Biegel,and R. Venugopal, “Two-dimensional quantum mechanicalmodeling of nanotransistors,” Journal of Applied Physics , 2343–2354 (2002). C. H. Lewenkopf and Eduardo R. Mucciolo, “The recursivegreen’s function method for graphene,” Journal of Compu-tational Electronics , 203–231 (2013). S. Li, S. Ahmed, G. Klimeck, and E. Darve, “Computingentries of the inverse of a sparse matrix using the find al-gorithm,” Journal of Computational Physics , 9408 –9427 (2008). L. Lin, J. Lu, L. Ying, R. Car, and E. Weinan, “Fast algo-rithm for extracting the diagonal of the inverse matrix withapplication to the electronic structure analysis of metallicsystems,” Commun. Math. Sci. , 755–777 (2009). S. Li and E. Darve, “Extension and optimization of thefind algorithm: Computing greens and less-than greensfunctions,” Journal of Computational Physics , 1121– 1139 (2012). P. D. Robinson and A. J. Wathen, “Variational boundson the entries of the inverse of a matrix,” IMA Journal ofNumerical Analysis , 463–486 (1992). G. Golub and G. Meurant, “Matrices, moments andquadrature,” Numerical Analysis 1993 (1994). M. Benzi and G. H. Golub, “Bounds for the entries of ma- trix functions with applications to preconditioning,” BITNumerical Mathematics , 417–438 (1999). T. Morimoto, A. Furusaki, and C. Mudry, “Anderson lo-calization and the topology of classifying spaces,” Phys.Rev. B , 235111 (2015). J. H. Bardarson, J. Tworzyd(cid:32)lo, P. W. Brouwer, andC. W. J. Beenakker, “One-parameter scaling at the diracpoint in graphene,” Phys. Rev. Lett. , 106801 (2007). K. Nomura, M. Koshino, and S. Ryu, “Topological de-localization of two-dimensional massless dirac fermions,”Phys. Rev. Lett. , 146806 (2007). P. M. Ostrovsky, I. V. Gornyi, and A. D. Mirlin, “Quan-tum criticality and minimal conductivity in graphene withlong-range disorder,” Phys. Rev. Lett. , 256801 (2007). T. Senthil and M. P. A. Fisher, “Quasiparticle localizationin superconductors with spin-orbit scattering,” Phys. Rev.B , 9690–9698 (2000). J. H. Bardarson, M. V. Medvedyeva, J. Tworzyd(cid:32)lo, A. R.Akhmerov, and C. W. J. Beenakker, “Absence of a metal-lic phase in charge-neutral graphene with a random gap,”Phys. Rev. B , 121414 (2010). M. V. Medvedyeva, J. Tworzyd(cid:32)lo, and C. W. J. Beenakker,“Effective mass and tricritical point for lattice fermionslocalized by a random mass,” Phys. Rev. B , 214203(2010). M. Wimmer, A. R. Akhmerov, M. V. Medvedyeva,J. Tworzyd(cid:32)lo, and C. W. J. Beenakker, “Majorana boundstates without vortices in topological superconductors withelectrostatic defects,” Phys. Rev. Lett. , 046803 (2010). A. W. W. Ludwig, M. P. A. Fisher, R. Shankar, andG. Grinstein, “Integer quantum hall transition: An al- ternative approach and exact results,” Phys. Rev. B ,7526–7552 (1994). A. Altland, “Spectral and transport properties of d-wavesuperconductors with strong impurities,” Phys. Rev. B ,104525 (2002). P. M. Ostrovsky, I. V. Gornyi, and A. D. Mirlin, “Conduc-tivity of disordered graphene at half filling,” The EuropeanPhysical Journal Special Topics , 63–72 (2007). S. Balasubramanian, Y. Liao, and V. Galitski, “Many-body localization landscape,” Phys. Rev. B , 014201(2020). M. Janssen, “Multifractal analysis of broadly-distributedobservables at criticality,” International Journal of ModernPhysics B , 943–984 (1994). B. Huckestein, “Scaling theory of the integer quantum halleffect,” Rev. Mod. Phys. , 357–396 (1995). “Two-dimensional conformal field theory for disorderedsystems at criticality,” Nucl. Phys. B , 383 – 443(1996). C. Chamon, C. Mudry, and X.-G. Wen, “Localization intwo dimensions, gaussian field theories, and multifractal-ity,” Phys. Rev. Lett. , 4194–4197 (1996). F. Evers and A. D. Mirlin, “Fluctuations of the inverseparticipation ratio at the anderson transition,” Phys. Rev.Lett. , 3690–3693 (2000). F. Evers, A. Mildenberger, and A. D. Mirlin, “Multifrac-tality of wave functions at the quantum hall transition re-visited,” Phys. Rev. B , 241303 (2001). T. Nakayama and K. Yakubo, “Multifractals in the ander-son transition,” in