Non-Hermitian disorder in two-dimensional optical lattices

Tzortzakakis, A. F.; Makris, Konstantinos G.; Economou, E. N.

doi:10.1103/physrevb.101.014202

Cited by 114 publications

(64 citation statements)

References 103 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Similar to Ref. 16 we found that eigenvalues near the rectangular border of the complex spectrum correspond to more localized states. Therefore, to deal with eigenvalues with similar localization properties we calculated r of 25% smallest-modulus eigenvalues selected by a rectangular window whose sides are roughly twice smaller than the whole spectrum.…”

supporting

confidence: 89%

“…A simple and elegant extension of the 2D Anderson localization problem was proposed in a recent paper by Tzortzakakis, Makris and Economou (TME) 16 . They studied 50 × 50 tight-binding square lattices with real overlap energy I ij = I, and random complex onsite energies E i whose real and imaginary parts are independent random variables distributed uniformly between −W/2 and W/2.…”

mentioning

confidence: 99%

“…For the TME model, where eigenvalues are points in the complex plane (see, for example, Ref. 16) we have chosen the parameter r(W ) = s 1 /s 2 , where s 1 and s 2 are distances from a given eigenvalue to its first and second nearest eigenvalues. 20 Our parameter r(W ) is the modulus of the more informative complex parameter introduced in Ref.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Anderson transition in three-dimensional systems with non-Hermitian disorder

2020

View full text Add to dashboard Cite

We study the Anderson transition for three-dimensional (3D) N × N × N tightly bound cubic lattices where both real and imaginary parts of onsite energies are independent random variables distributed uniformly between −W/2 and W/2. Such a non-Hermitian analog of the Anderson model is used to describe random-laser medium with local loss and amplification. We employ eigenvalue statistics to search for the Anderson transition. For 25% smallest-modulus complex eigenvalues we find the average ratio r of distances to the first and the second nearest neighbor as a function of W . For a given N the function r(W ) crosses from 0.72 to 2/3 with a growing W demonstrating a transition from delocalized to localized states. When plotted at different N all r(W ) cross at Wc = 6.0±0.1 (in units of nearest neighbor overlap integral) clearly demonstrating the 3D Anderson transition. We find that in the non-Hermitian 2D Anderson model, the transition is replaced by a crossover.Anderson localization is the central concept of solid state physics for more than 60 years 1-3 . It determines electron conductivity of doped crystalline and amorphous semiconductors and many other disordered systems and is observed in experiments 4,5 .In recent years the problem of localization attracted renewed interest as research moved to formerly unexplored area of non-Hermitian systems. Random lasers 6-9 with random dissipation and amplification regions are such prototypical non-Hermitian systems. The other parts of non-Hermitian disorder physics are related to Hatano-Nelson matrices 10 , their biological applications 11,12 or to spin chains [13][14][15] . All these works focus on onedimensional systems.A simple and elegant extension of the 2D Anderson localization problem was proposed in a recent paper by Tzortzakakis, Makris and Economou (TME) 16 . They studied 50 × 50 tight-binding square lattices with real overlap energy I ij = I, and random complex onsite energies E i whose real and imaginary parts are independent random variables distributed uniformly between −W/2 and W/2. The Hamiltonian readswhere i, j in the second term are nearest neighbors, and the hard-wall boundary is employed (no bonds extended out the boundary). Below we call this non-Hermitian Hamiltonian the TME model. By calculating the participation ratio of eigenfunctions of such non-Hermitian Hamiltonian, TME noticed that they become progressively more localized when W (in units of I) grows from 1 to 5. Simultaneously the distribution function P (s) of nearest neighbor distances s between eigenvalues in the complex plane widens, which shows that the repulsion of eigenvalues weakens due to the progressive localization of eigenfunctions. This behavior is similar to what happens in the Anderson model 17 . TME, however, did not raise a question whether there is an Anderson transition or a crossover in the limit of large system.In this paper, we focus on the question of the existence of the Anderson transition in the TME model for 3D cubic and 2D square lattices. We show that in the TME m...

show abstract

supporting

confidence: 89%

mentioning

confidence: 99%

See 1 more Smart Citation

Anderson transition in three-dimensional systems with non-Hermitian disorder

2020

View full text Add to dashboard Cite

show abstract

“…The non-Hermitian Anderson Hamiltonian [26,27] H = −∇ 2 + V 1 (r) + iV 2 (r) (16) has been studied in the context of a random laser [28]: a disordered optical lattice with randomly varying absorption and amplication rates, described by a complex dielectric function V 1 + iV 2 . On a d-dimensional square lattice (lattice constant a), the discretization of −∇ 2 → a −2 d i=1 (2−2 cos k i a) produces a spectral band width of W 0 = 4d/a 2 .…”

Section: Introduction -mentioning

confidence: 99%

Localization landscape for Dirac fermions

et al. 2020

View full text Add to dashboard Cite

In the theory of Anderson localization, a landscape function predicts where wave functions localize in a disordered medium, without requiring the solution of an eigenvalue problem. It is known how to construct the localization landscape for the scalar wave equation in a random potential, or equivalently for the Schrödinger equation of spinless electrons. Here we generalize the concept to the Dirac equation, which includes the effects of spin-orbit coupling and allows to study quantum localization in graphene or in topological insulators and superconductors. The landscape function u(r) is defined on a lattice as a solution of the differential equation H u(r) = 1, where H is the Ostrowsky comparison matrix of the Dirac Hamiltonian. Random Hamiltonians with the same (positive definite) comparison matrix have localized states at the same positions, defining an equivalence class for Anderson localization. This provides for a mapping between the Hermitian and non-Hermitian Anderson model.

show abstract

“…The spectral statistics of random Hermitian Hamiltonians usually exhibits universal behavior, depending only on symmetries of the system [68]. An interesting question then naturally arises whether the spectral statistics of disordered non-Hermitian Hamiltonians also exhibits universal behavior [69][70][71][72][73][74][75][76][77][78][79][80][81][82][83].…”

Section: Introductionmentioning

confidence: 99%

Statistical properties of eigenvalues of the non-Hermitian Su-Schrieffer-Heeger model with random hopping terms

et al. 2020

View full text Add to dashboard Cite

We explore the eigenvalue statistics of a non-Hermitian version of the Su-Schrieffer-Heeger model, with imaginary on-site potentials and randomly distributed hopping terms. We find that owing to the structure of the Hamiltonian, eigenvalues can be purely real in a certain range of parameters, even in the absence of parity and time-reversal symmetry. As it turns out, in this case of purely real spectrum, the level statistics is that of the Gaussian orthogonal ensemble. This demonstrates a general feature which we clarify that a non-Hermitian Hamiltonian whose eigenvalues are purely real can be mapped to a Hermitian Hamiltonian which inherits the symmetries of the original Hamiltonian. When the spectrum contains imaginary eigenvalues, we show that the density of states (DOS) vanishes at the origin and diverges at the spectral edges on the imaginary axis. We show that the divergence of the DOS originates from the Dyson singularity in chiral-symmetric one-dimensional Hermitian systems and derive analytically the asymptotes of the DOS which is different from that in Hermitian systems.

show abstract

Non-Hermitian disorder in two-dimensional optical lattices

Cited by 114 publications

References 103 publications

Anderson transition in three-dimensional systems with non-Hermitian disorder

Anderson transition in three-dimensional systems with non-Hermitian disorder

Localization landscape for Dirac fermions

Statistical properties of eigenvalues of the non-Hermitian Su-Schrieffer-Heeger model with random hopping terms

Contact Info

Product

Resources

About