Pith. sign in

REVIEW 3 major objections 5 minor 46 references

Non-Abelian gauge potential driven localization transition in quasiperiodic optical lattices

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The central claim is that adding a non-Abelian SU(2) gauge potential to a quasiperiodic AAH optical lattice makes the model self-dual and yields four localization phases — pure delocalization, two coexistence phases, and pure localization.

desk verdict A genuinely new self-dual non-Abelian AAH model, but the four-phase diagram rests on an unvalidated 1e-7 band-width cutoff and needs stronger numerics before I'd trust it. read the letter →

arxiv 1908.06839 v1 pith:KXTJQ4VY submitted 2019-08-19 cond-mat.quant-gas

En Guo Guan , Hang Yu , Gang Wang This is my paper · ORCID
classification cond-mat.quant-gas PACS 03.75.-b72.15.Rn
keywords non-AbeliangaugeAubry-André-HarpermodelquasiperiodicopticallatticeAndersonlocalizationmobilityedgeself-dualityphasediagramultracoldatoms
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to establish that adding a non-Abelian SU(2) gauge potential to a quasiperiodic Aubry-André-Harper (AAH) optical lattice changes the localization transition from the familiar two-phase picture into a four-phase sequence. For a tunable gauge strength $q$, increasing the modulation strength $\lambda$ moves the system through a pure delocalized phase, a coexistence phase I, a coexistence phase II, and a pure localized phase, with the two coexistence phases separated by the self-dual point $\lambda_{c0}=1$. The coexistence phases, where extended and localized states appear at the same parameters, are absent at $q=0$, $\pi/2$, and $\pi$, where the model reduces to Abelian cases. A sympathetic reader would care because the non-Abelian gauge becomes a controllable handle for creating and moving mobility edges in a one-dimensional quasiperiodic system, something the Abelian AAH model cannot do.

What carries the argument

The central object is the non-Abelian AAH Hamiltonian obtained from the Hofstadter Hamiltonian with a constant SU(2) gauge potential $\mathbf{A} = (q\sigma_y,\, 2\pi\Omega m\,\sigma_0 + q\sigma_x,\, 0)$ via Peierls substitution. The argument is carried by the exact Fourier duality between the real-space Hamiltonian Eq. (6) and its dual Eq. (7): the two are the same form with $\lambda$ replaced by $1/\lambda$, so a state extended at $(\lambda,q)$ maps to a state localized at $(1/\lambda,q)$. The phase diagram is extracted from the total band widths $W_r$ and $W^*$ of the real- and dual-space spectra, computed on Fibonacci superlattice approximants; a coexistence phase is identified when both widths remain above a numerical threshold of $1.0\times10^{-7}$.

What would settle it

Perform a finite-size scaling study at a point inside the claimed coexistence region (for example, $\lambda = 1.2$, $q = 0.3\pi$): compute $W_r$ and $W^*$ for Fibonacci approximants with increasing period $F_l$; if either width decays to zero as the period grows, the coexistence phase is a finite-size artifact and the central claim fails. Alternatively, directly search for an energy-resolved mobility edge in the density of participation ratios at that point; if no energy separates localized from extended states in the thermodynamic limit, the claim fails.

Watch

Extended reading notes

Core claim

The central claim is that the non-Abelian AAH Hamiltonian of Eq. (6) is self-dual under Fourier transformation: its dual, Eq. (7), has the same matrix structure with the hopping and modulation strengths interchanged, so $\lambda \leftrightarrow 1/\lambda$ is a symmetry. This fixes $\lambda_{c0}=1$ as a transition line, but unlike the Abelian AAH model, the phase diagram is not simply metal versus insulator. Numerical IPR and spectral band-width calculations show four regions in the $(\lambda,q)$ plane — pure delocalization, coexistence I, coexistence II, and pure localization — where the coexistence regions are diagnosed by simultaneously nonzero real-space and dual-space band widths. The paper concludes that the non-Abelian gauge drives a metal-to-coexistence-to-insulator transition and that the $q = 0$, $\pi/2$, and $\pi$ limits reproduce the Abelian behavior.

Load-bearing premise

The existence and location of the coexistence phases rest on the numerical criterion that a phase is coexisting only when both the real-space and dual-space band widths exceed $1.0\times10^{-7}$ at the largest Fibonacci approximant studied, together with the assumption that these approximants have converged to the infinite quasiperiodic limit.

Editorial extensions

If this is right

  • For $0<q<\pi/2$ and $\pi/2<q<\pi$, tuning $\lambda$ across the two critical lines switches the system between pure metal, coexistence with mobility edges, and pure insulator, so the non-Abelian gauge acts as a tunable source of mobility edges.
  • The self-dual line $\lambda_{c0}=1$ separates two distinct coexistence phases, meaning the same physical parameters support two different mixed phases that can be probed separately.
  • At $q=0$, $\pi/2$, and $\pi$, the model reduces to decoupled Abelian replicas or an Abelian flux model, so the coexistence phases disappear and the standard AAH transition is recovered.
  • The simultaneous band-width criterion ($W_r>0$ and $W^*>0$) provides a practical, general diagnostic for identifying coexistence phases in other quasiperiodic systems.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the $\lambda\leftrightarrow 1/\lambda$ symmetry of the phase diagram is exact, then the two coexistence boundaries should satisfy $\lambda_{c2}(q) = 1/\lambda_{c1}(q)$; measuring one boundary would determine the other.
  • In a cold-atom implementation, the coexistence phases should appear in time-of-flight images as a mixture of ballistic and localized components, with the ratio controlled by $\lambda$ and $q$.
  • The same construction — adding a constant SU(2) gauge to a self-dual quasiperiodic model — may generate coexistence phases in other self-dual families (e.g., power-law hopping), offering a general route to mobility-edge engineering.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The manuscript studies a non-Abelian generalization of the Aubry-André-Harper (AAH) model, obtained from a two-component Hofstadter Hamiltonian with an SU(2) gauge potential. The authors derive a Fourier self-duality (Sec. III A), and then use inverse participation ratios and the total band widths of the real-space and dual-space spectra, Wr and W*, to construct a λ–q phase diagram (Fig. 4). They claim that increasing λ at fixed q drives the system through pure delocalization, coexistence phase I, coexistence phase II, and pure localization, with the two coexistence phases separated by the self-dual line λc0=1 and absent at q=0, π/2, and π.

Significance. The self-duality derivation is clean, and the physical motivation in terms of non-Abelian gauge potentials for ultracold atoms is sound. If the phase diagram is correct, the paper establishes a qualitatively new localization scenario relative to the Abelian AAH model and makes falsifiable predictions that could be probed in cold-atom experiments. The numerical calculations are deterministic and contain no fitted parameters, and the use of spectral band widths in both real and dual space is a sensible diagnostic idea. However, the central phase diagram is only as reliable as the convergence and threshold assumptions behind the Wr/W* classification, and those assumptions are not established in the manuscript.

major comments (3)
  1. [Sec. III B, Fig. 4] The phase diagram is obtained by declaring Wr > 1.0×10^{-7} and W* > 1.0×10^{-7} to define the coexistence region, with the threshold described only as a matter of computational accuracy. This threshold is not tested for sensitivity. Because the band widths in Fig. 3 fall by many orders of magnitude over the Fibonacci sequence, a tail that is merely small at the largest computed Fl would be classified as zero, moving the boundaries of the yellow coexistence region; the isolated reentrant islands in Fig. 4 are exactly the kind of fine structure that a hard cutoff can create or remove. The authors should provide threshold-sweep results (for example, thresholds of 10^{-6} and 10^{-8}) and finite-size scaling or extrapolation of Wr and W* to establish the coexistence regions.
  2. [Sec. III B, Fig. 3] Convergence of the Fibonacci superlattice approximants to the quasiperiodic limit is asserted but not quantified. The text states that Wr for λ = 0.6, 0.8, and 1.2 converges to a finite value, that λ = 1 shows algebraic decay, and that λ = 1.6 shows exponential decay, but no largest Fibonacci index, no fitting form, and no extrapolation rule are given. The same convergence analysis is not shown for W*, although the phase diagram depends on both quantities. Without this information, the classification of a band width as finite or vanishing is not reproducible.
  3. [Sec. III B, p. 6] The criterion that simultaneously finite Wr and W* serves as an order parameter for coexistence is an assumption rather than a proven equivalence. A finite total real-space band width together with a vanishing dual-space width is claimed to indicate a pure metal, but the text does not directly verify this diagnostic against the actual presence of mobility edges. The authors should validate the criterion on a model with a known mobility edge, or against their own IPR data at fixed λ, before using it to locate the phase boundaries.
minor comments (5)
  1. [Eq. (3)] The IPR definition does not specify how the two-component spinor wave function is normalized; please state explicitly the norm and the summation over spin components and lattice sites.
  2. [Sec. III B] The symbol W* is introduced verbally but never defined precisely; please state whether it is the total band width of Eq. (7) computed with the same Fibonacci approximants and boundary conditions used for Wr.
  3. [Fig. 1] Figure 1 labels λc1 and λc2, but the text does not give their numerical values for q = 0.3π; please quote the values obtained from the spectral analysis.
  4. [Sec. III B, q = π/2] The statement that q = π/2 reduces Eq. (5) to the Abelian flux model is asserted without derivation; a short demonstration or a precise reference to [39] would make the special-case argument self-contained.
  5. [General] The manuscript contains several grammatical and typographical issues (for example, 'separatively', 'firstly', and 'the non-Abelian gauge involved drives'); a careful language edit is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the self-duality is derived from the Hamiltonian, and the phase diagram is obtained by direct spectral computation.

full rationale

The paper's central derivations are self-contained. The NA-AAH Hamiltonian (Eq. 6) is transformed by Fourier transformation into Eq. 7, and the paper explicitly notes that Eq. 7 takes the same form as Eq. 6 with coefficient interchange, establishing self-duality without importing any target result. The phase diagram (Fig. 4) is extracted from computed total band widths Wr and W* of the real- and dual-space Hamiltonians, using a stated numerical threshold of 1.0e-7. This threshold is a diagnostic convention ('Due to the computational accuracy we take 1.0×10−7'), not a parameter fitted to reproduce the claimed phases; the existence of extended and localized states is independently supported by the IPR data (Fig. 1) and typical-state plots (Fig. 2). The only overlapping self-citation is Ref. [30] (Sun, Wang, Li, Nakayama), which appears in a list of prior works on coexistence phases in one-dimensional quasiperiodic models; it is not used as the basis for any conclusion here. No uniqueness theorem or ansatz is imported from the authors' prior work, and no known empirical result is renamed as a new prediction. The arbitrary threshold and the lack of explicit Fibonacci-convergence analysis are numerical-robustness concerns, not circularity.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The only hand-chosen numerical input is the 1e-7 threshold used to define phase boundaries; λ and q are physical control parameters, not fitted. The main unproved modeling assumptions are the Peierls-substitution description and the Wr/W* coexistence criterion. No new particles or forces are introduced.

free parameters (1)
  • band-width threshold epsilon = 1.0e-7
    Chosen by hand as the numerical zero for Wr and W*; every phase boundary in Fig. 4 is defined by crossing this threshold, so the claimed phase diagram depends on it.
assumptions (4)
  • domain assumption Peierls substitution with the non-Abelian gauge A = (qσ_y, 2πΩm σ0 + qσ_x, 0) yields the tight-binding Hamiltonian in Eq. (5), and this Hamiltonian describes ultracold atoms in a laser-assisted optical superlattice.
    Invoked when deriving Eq. (5) from the gauge potential; it assumes the single-band tight-binding description and the experimental realizability of the gauge.
  • domain assumption Spectral decomposition: absolutely continuous spectrum corresponds to extended states, pure point spectrum to localized states, and singular continuous spectrum to the transition, as in Refs [43,44].
    Used in Sec. III B to map band widths in real and dual space onto localization phases.
  • ad hoc to paper Simultaneously finite Wr and W* implies coexistence of extended and localized states in the same spectrum.
    This is the paper's central diagnostic, stated in Sec. III B; it is not proven and is validated only indirectly through IPR plots.
  • standard math Fibonacci rational approximants Omega' = F_{l-1}/F_l converge to the golden ratio so that the band widths of periodic superlattices asymptotically reproduce the quasiperiodic spectrum.
    Standard approximation used in Sec. III B to compute Wr; relies on known convergence of continued fraction approximants.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Non-Abelian gauge potential driven localization transition in quasiperiodic optical lattices." pith.science (2026). https://pith.science/paper/KXTJQ4VY

@misc{pith2026190806839,
  author       = {Pith},
  title        = {Pith review of: Non-Abelian gauge potential driven localization transition in quasiperiodic optical lattices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KXTJQ4VY}},
  note         = {Machine review of arXiv:1908.06839}
}
read the original abstract

Gauge potential is an emergent concept in systems of ultracold atomic gases. Derived from quantum waves immersed in an \emph{Abelian} gauge, the quasiperiodic Aubry-Andre-Harper (AAH) model is a simple yet powerful Hamiltonian to study the Anderson localization of ultracold atoms. In this work, we investigate the localization properties of ultracold atoms trapped in quasiperiodic optical lattices subject to a non-Abelian gauge, which can be depicted by a family of non-Abelian AAH models. We identify that the non-Abelian AAH models can bear the self-duality under the Fourier transformation. We thus analyze the localization transition of this self-dual non-Abelian quasiperiodic optical lattices, revealing that the non-Abelian gauge involved drives a transition from a pure delocalization phase, then to coexistence phases, and then finally to a pure localization phase. This is in stark contrast to the Abelian AAH model that does not support the coexistence phases. Our results thus comprise a new insight on the fundamental aspects of Anderson localization in quasiperiodic systems, from the perspective of non-Abelian gauge.

Figures

Figures reproduced from arXiv: 1908.06839 by the authors.

Figure 1
Figure 1. FIG. 1. (Color online) The IPR distribution of states as a function of [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (Color online) For a fixed non-Abelian gauge component [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (Color online) (a) Total width [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. (Color online) Localization phase diagram of the quasiperiodic NA-AAH lattice obtained from the band width data [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

46 extracted references · 46 canonical work pages

  1. [45]

    Phase diagram of a non-Abelian Aubry-Andr´ e-Harper model with p-wave superfluidity,

    Jun Wang, Xia-Ji Liu, Gao Xianlong, and Hui Hu, “Phase diagram of a non-Abelian Aubry-Andr´ e-Harper model with p-wave superfluidity,” Phys. Rev. B 93, 104504 (2016)

    work page 2016

  2. [46]

    Metal-Insulator Transition Revisited for Cold Atoms in Non-Abelian Gauge Potentials,

    I. Satija, D. Dakin, and C. Clark, “Metal-Insulator Transition Revisited for Cold Atoms in Non-Abelian Gauge Potentials,” Phys Rev. Lett. 97, 216401 (2006)

    work page 2006

  3. [1]

    Significance of electromagnetic potentials in the quantum theory,

    Y. Aharonov and D. Bohm, “Significance of electromagnetic potentials in the quantum theory,” Phys. Rev. 115, 485 (1959)

    work page 1959

  4. [2]

    Artifical Gauge Fields with Ultracold Atoms,

    V. Galitski, Ian Spielman, and G. Juzeli¯ unas, “Artifical Gauge Fields with Ultracold Atoms,” Physics Today72, 39 (2019)

    work page 2019

  5. [3]

    Creation of effective magnetic fields in optical lattices: the Hofstadter butterfly for cold neutral atoms,

    D. Jaksch and P. Zoller, “Creation of effective magnetic fields in optical lattices: the Hofstadter butterfly for cold neutral atoms,” New J. Phys. 5, 56 (2003)

    work page 2003

  6. [4]

    Cold Atoms in Non-Abelian Gauge Potentials: From the Hofstadter Moth to Lattice Gauge,

    K. Osterloh, M. Baig, L. Santos, P. Zoller, and M. Lewenstein, “Cold Atoms in Non-Abelian Gauge Potentials: From the Hofstadter Moth to Lattice Gauge,” Phys. Rev. Lett. 95, 010403 (2005)

    work page 2005

  7. [5]

    Non-Abelian Gauge Potentials for Ultracold Atoms with Degenerate Dark States,

    J. Ruseckas, G. Juzeli¯ unas, P. ¨Ohberg, and M. Fleischhauer, “Non-Abelian Gauge Potentials for Ultracold Atoms with Degenerate Dark States,” Phys. Rev. Lett. 95, 010404 (2005)

    work page 2005

  8. [6]

    Artificial gauge potentials for neutral atoms,

    Jean Dalibard, Fabrice Gerbier, Gediminas Juzeli¯ unas, and Patrik¨Ohberg, “Artificial gauge potentials for neutral atoms,” Rev. Mod. Phys. 83, 1523 (2011)

    work page 2011

Show all 46 references
  1. [7]

    Light-induced gauge fields for ultracold atoms,

    N. Goldman, G. Juzeli¯ unas, P.¨Ohberg, I. B. Spielman, “Light-induced gauge fields for ultracold atoms,” Rep. Prog. Phys. 77, 126401 (2014)

    2014

  2. [8]

    Artificial gauge fields in materials and engineered systems,

    M. Aidelsburger, S. Nascimbene, and N. Goldman, “Artificial gauge fields in materials and engineered systems,” C. R. Phys. 19, 394 (2018)

    2018

  3. [9]

    Appearance of Gauge Structure in Simple Dynamical Systems,

    F. Wilczek and A. Zee, “Appearance of Gauge Structure in Simple Dynamical Systems,” Phys. Rev. Lett. 52, 2111 (1984)

    1984

  4. [10]

    Absence of Diffusion in Certain Random Lattices,

    P. W. Anderson, “Absence of Diffusion in Certain Random Lattices,” Phys. Rev. 109, 1492 (1958)

    1958

  5. [11]

    Scaling Theory of Localization: Absence of Quantum Diffusion in Two Dimensions,

    E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, “Scaling Theory of Localization: Absence of Quantum Diffusion in Two Dimensions,” Phys. Rev. Lett. 42, 673 (1979)

    1979

  6. [12]

    Disordered electronic systems,

    P. A. Lee and T. V. Ramakrishnan, “Disordered electronic systems,” Rev. Mod. Phys. 57, 287 (1985)

    1985

  7. [13]

    Localization theory and experiment,

    B. Kramer and A. MacKinnon, “Localization theory and experiment,” Rep. Prog. Phys. 56, 1469 (1993)

    1993

  8. [14]

    Anderson transitions,

    F. Evers and A. D. Mirlin, “Anderson transitions,” Rev. Mod. Phys. 80, 1355 (2008)

    2008

  9. [15]

    Many-body physics with ultracold gases,

    I. Bloch, J. Dalibard, and W. Zwerger, “Many-body physics with ultracold gases,” Rev. Mod. Phy. 80, 885 (2008). 9

    2008

  10. [16]

    Coherent Backscattering of Ultracold Atoms,

    F. Jendrzejewski et al ., “Coherent Backscattering of Ultracold Atoms,” Phys. Rev. Lett. 109, 195302 (2012)

    2012

  11. [17]

    Suppression and Revival of Weak Localization through Control of Time-Reversal Symmetry,

    K. M¨ ulleret al., “Suppression and Revival of Weak Localization through Control of Time-Reversal Symmetry,” Phys. Rev. Lett. 114, 205301 (2015)

    2015

  12. [18]

    Direct observation of Anderson localization of matter waves in a controlled disorder ,

    J. Billy et al ., “Direct observation of Anderson localization of matter waves in a controlled disorder ,” Nature 453, 891 (2008)

    2008

  13. [19]

    Anderson localization of a non-interacting Bose Einstein condensate,

    G. Roati et al ., “Anderson localization of a non-interacting Bose Einstein condensate,” Nature 453, 895 (2008)

    2008

  14. [20]

    Three-Dimensional Anderson Localization of Ultracold Matter,

    S. S. Kondov, W. R. McGehee, J. J. Zirbel, and B. DeMarco, “Three-Dimensional Anderson Localization of Ultracold Matter,” Science 334, 66 (2011)

    2011

  15. [21]

    Three-dimensional localization of ultracold atoms in an optical disordered potential,

    F. Jendrzejewski et al ., “Three-dimensional localization of ultracold atoms in an optical disordered potential,” Nature Phys. 8, 398 (2012)

    2012

  16. [22]

    Measurement of the mobility edge for 3D Anderson localization,

    G. Semeghini et al ., “Measurement of the mobility edge for 3D Anderson localization,” Nat. Phys. 11, 554 (2015)

    2015

  17. [23]

    Analyticity breaking and Anderson localization in incommensurate lattices,

    S. Aubry and G. Andr´ e, “Analyticity breaking and Anderson localization in incommensurate lattices,” Ann. Isr. Phys. Soc. 3, 133 (1980)

    1980

  18. [24]

    Single Band Motion of Conduction Electrons in a Uniform Magnetic Field,

    P. G. Harper, “Single Band Motion of Conduction Electrons in a Uniform Magnetic Field,” Proc. Phys. Soc. A 68, 874 (1955)

    1955

  19. [25]

    Mobility Edge in a Model One-Dimensional Potential,

    S. Das Sarma, S. He, and X. C. Xie, “Mobility Edge in a Model One-Dimensional Potential,” Phys.Rev. Lett. 61, 2144 (1988)

    1988

  20. [26]

    Localization by a Potential with Slowly Varying Period,

    D. J. Thouless, “Localization by a Potential with Slowly Varying Period,” Phys. Rev. Lett. 61, 2141 (1988)

    1988

  21. [27]

    Mobility edges in bichromatic optical lattices,

    Dave J. Boers, Benjamin Goedeke, Dennis Hinrichs, and Martin Holthaus, “Mobility edges in bichromatic optical lattices,” Phys. Rev. A. 75, 063404 (2007)

    2007

  22. [28]

    Predicted Mobility Edges in One-Dimensional Incommensurate Optical Lattices: An Exactly Solvable Model of Anderson Localization,

    J. Biddle and S. Das Sarma, “Predicted Mobility Edges in One-Dimensional Incommensurate Optical Lattices: An Exactly Solvable Model of Anderson Localization,” Phys. Rev. Lett. 104, 070601 (2010)

    2010

  23. [29]

    Nearest Neighbor Tight Binding Models with an Exact Mobility Edge in One Dimension,

    S. Ganeshan, J. H. Pixley, and S. Das Sarma, “Nearest Neighbor Tight Binding Models with an Exact Mobility Edge in One Dimension,” Phys. Rev. Lett. 114, 146601 (2015)

    2015

  24. [30]

    Localization-delocalization transition in self-dual quasiperiodic lattices,

    M. L. Sun, G. Wang, N. B. Li, and T. Nakayama, “Localization-delocalization transition in self-dual quasiperiodic lattices,” Europhys. Lett. 110, 57003 (2015)

    2015

  25. [31]

    Self-dual model for one dimensional incommensurate crystals including next-nearest- neighbor hopping, and its relation to the Hofstadter model,

    M. Johansson and R. Riklund, “Self-dual model for one dimensional incommensurate crystals including next-nearest- neighbor hopping, and its relation to the Hofstadter model,” Phys. Rev. B 43, 13468 (1991)

    1991

  26. [32]

    Self-dual quasiperiodic systems with power-law hopping,

    S. Gopalakrishnan, “Self-dual quasiperiodic systems with power-law hopping,” Phys. Rev. B 96, 054202 (2017)

    2017

  27. [33]

    Quantum nonergodicity and fermion localization in a system with a single-particle mobility edge,

    X. Li, J. H. Pixley, D. L. Deng, S. Ganeshan, and S. Das Sarma, “Quantum nonergodicity and fermion localization in a system with a single-particle mobility edge,” Phys. Rev. B 93, 184204 (2016)

    2016

  28. [34]

    Mobility edges in one-dimensional bichromatic incommensurate potentials,

    X. Li, X. Li and S. Das Sarma, “Mobility edges in one-dimensional bichromatic incommensurate potentials,” Phys. Rev. B 96, 085119 (2017)

    2017

  29. [35]

    Anderson localization in one-dimensional quasiperiodic lattice models with nearest- and next-nearest-neighbor hopping,

    L. Gong, Y. Feng, and Y. Ding, “Anderson localization in one-dimensional quasiperiodic lattice models with nearest- and next-nearest-neighbor hopping,” Phys. Lett. A 381, 588 (2017)

    2017

  30. [36]

    Single-Particle Mobility Edge in a One-Dimensional Quasiperiodic Optical Lattice,

    H. P. L¨ uschenet al., “Single-Particle Mobility Edge in a One-Dimensional Quasiperiodic Optical Lattice,” Phys. Rev. Lett. 120, 160404 (2018)

    2018

  31. [37]

    Engineering a Flux-Dependent Mobility Edge in Disordered Zigzag Chains,

    F. A. An, E. J. Meier, and B. Gadway, “Engineering a Flux-Dependent Mobility Edge in Disordered Zigzag Chains,” Phys. Rev. X 8, 031045 (2018)

    2018

  32. [38]

    Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields,

    D. R. Hofstadter, “Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields,” Phys. Rev. B 14, 2239 (1976)

    1976

  33. [39]

    Non-Abelian Optical Lattices: Anomalous Quantum Hall Effect and Dirac Fermions,

    N. Goldman, A. Kubasiak, A. Bermudez, P. Gaspard, M. Lewenstein, and M. A. Martin-Delgado, “Non-Abelian Optical Lattices: Anomalous Quantum Hall Effect and Dirac Fermions,” Phys. Rev. Lett. 103, 035301 (2009)

    2009

  34. [40]

    Ultracold atomic gases in non-Abelian gauge potentials: The case of constant Wilson loop,

    N. Goldman, A. Kubasiak, P. Gaspard, and M. Lewenstein, “Ultracold atomic gases in non-Abelian gauge potentials: The case of constant Wilson loop,” Phys. Rev. A 79, 023624 (2009)

    2009

  35. [41]

    Massless Dirac fermions in a square optical lattice,

    J. M. Hou, W. X. Yang, and X. J. Liu, “Massless Dirac fermions in a square optical lattice,” Phys. Rev. A 79, 043621 (2009)

    2009

  36. [42]

    Wilson Fermions and Axion Electrodynamics in Optical Lattices,

    A. Bermudez et al. , “Wilson Fermions and Axion Electrodynamics in Optical Lattices,” Phys. Rev. Lett. 105, 190404 (2010)

    2010

  37. [43]

    Metal-Insulator Transition and Scaling for Incommensurate Systems,

    M. Kohmoto, “Metal-Insulator Transition and Scaling for Incommensurate Systems,” Phys. Rev. Lett. 51, 1198 (1983)

    1983

  38. [44]

    New Localization in a Quasiperiodic System,

    H. Hiramoto and M. Kohmoto, “New Localization in a Quasiperiodic System,” Phys. Rev. Lett. 62, 2714 (1989)

    1989

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.