pith. sign in

arxiv: 2606.18896 · v1 · pith:PSDZZAAAnew · submitted 2026-06-17 · ❄️ cond-mat.stat-mech · physics.optics

Thermodynamics of photonic nonlinear Aharonov-Bohm cages

Stefano Iubini , Carlo Danieli This is my paper

Pith reviewed 2026-06-26 19:19 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech physics.optics
keywords photonic diamond latticeAharonov-Bohm cagingKerr nonlinearitythermoelectric figure of meritSeebeck coefficientsynthetic magnetic fluxnonlinear transportconductor-insulator transition
0
0 comments X

The pith

In photonic diamond lattices with Kerr nonlinearity, tuning synthetic magnetic flux to the Aharonov-Bohm caging condition turns the system from conductor to insulator at weak nonlinearity.

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

The paper examines equilibrium and non-equilibrium thermodynamics of one-dimensional photonic diamond lattices with Kerr nonlinearity under synthetic magnetic flux. In the linear regime, the flux induces Aharonov-Bohm caging that flattens all Bloch bands and suppresses particle and energy currents. Nonlinearity enables non-vanishing currents even in the caged regime. At weak nonlinearity, fine tuning the flux exactly to the caging value transforms the system into an insulator, while at intermediate nonlinear strength the system conducts for every flux value but the caging condition markedly increases the Seebeck coefficient and the thermoelectric figure of merit.

Core claim

The equilibrium phase diagram of the photonic diamond lattice is obtained as a function of the synthetic magnetic flux. In the linear regime the flux induces Aharonov-Bohm caging, flattening all Bloch bands and suppressing currents. Nonlinearity enables currents inside the caging regime. With imposed temperature and chemical-potential imbalances at the boundaries, weak nonlinearity plus flux tuned to the caging condition converts the system from conductor to insulator. At intermediate nonlinearity the system conducts for all fluxes, yet the caging condition enhances the Seebeck coefficient and thermoelectric figure of merit, offering a route to optimize coupled transport devices by controlli

What carries the argument

Aharonov-Bohm caging induced by synthetic magnetic flux per plaquette, which flattens Bloch bands and blocks currents in the linear regime and is modified by Kerr nonlinearity to control transport.

If this is right

  • At weak nonlinearity, fine tuning the flux at the Aharonov-Bohm caging condition transforms the system from a conductor to an insulator.
  • At intermediate nonlinear strength the system remains conducting for all magnetic fluxes, yet the caging condition significantly enhances the Seebeck coefficient and thermoelectric figure of merit.
  • Control of linear versus nonlinear conduction channels via synthetic magnetic flux provides a route toward optimization of coupled transport devices.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same flux-tuning mechanism could be tested in other nonlinear flat-band lattices to see whether thermoelectric enhancement appears generically at the caging point.
  • Realization in optical waveguide arrays would allow direct observation of the predicted conductor-insulator transition as a function of nonlinearity strength.
  • The enhancement of thermoelectric performance at the caging condition may suggest design principles for flux-controlled photonic heat engines.

Load-bearing premise

The linear regime of the photonic diamond lattice exhibits complete Aharonov-Bohm caging that flattens all Bloch bands and suppresses currents.

What would settle it

Direct measurement of whether particle or energy current drops to zero precisely when flux is set to the caging value at weak nonlinearity, while remaining finite at other flux values or at stronger nonlinearity.

Figures

Figures reproduced from arXiv: 2606.18896 by Carlo Danieli, Stefano Iubini.

Figure 1
Figure 1. Figure 1: FIG. 1. Nonlinear Aharonov-Bohm lattice and non [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: for γ = 2 and ϕ = π. The brown shaded area below the β = ∞ line is not accessible. The area above the β = 0 line is accessible, although the gran canonical distribution P diverges for large |ψn|. This region corre￾sponds to negative-temperature states defined either in the microcanonical ensemble [55] or as grandcanonical metastable states [56]. Here, spatial localization effects are expected to occur acco… view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Stationary currents of norm (first row) and en [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Positive real root [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Size dependence of stationary absolute fluxes of [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
read the original abstract

We investigate equilibrium and non-equilibrium thermodynamics of one-dimensional photonic diamond lattices with Kerr nonlinearity. The equilibrium phase diagram is obtained as a function of the synthetic magnetic flux acting on each plaquette. In the linear regime, the magnetic flux can induce Aharonov-Bohm caging, flattening all Bloch bands and suppressing particle and energy currents. In this caging regime, non-vanishing currents are enabled by nonlinearity. By imposing stationary temperature- and chemical potential- imbalances at the system boundaries, we show that at weak nonlinearity fine tuning the flux at the Aharonov-Bohm caging transforms the system from a conductor to an insulator. For intermediate nonlinear strength, the system remains conducting for all magnetic fluxes; however, the caging condition significantly enhances the Seebeck coefficient and thermoelectric figure of merit, improving the thermoelectric features of the system. Our results give evidence of a novel route towards optimization of coupled transport devices, based on the control of linear versus nonlinear conduction channels via a synthetic magnetic flux.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript examines equilibrium and non-equilibrium thermodynamics in one-dimensional photonic diamond lattices subject to Kerr nonlinearity and synthetic magnetic flux per plaquette. In the linear limit the flux induces Aharonov-Bohm caging that flattens all Bloch bands and suppresses both particle and energy currents; finite nonlinearity restores non-zero currents inside the caging regime. The central claims are that, at weak nonlinearity, fine-tuning the flux exactly to the caging condition drives a conductor-to-insulator transition, while at intermediate nonlinearity the system remains conducting for all fluxes yet the caging condition markedly increases the Seebeck coefficient and the thermoelectric figure of merit.

Significance. If the reported conductor-insulator transition and the flux-enhanced thermoelectric performance are quantitatively confirmed, the work supplies a concrete, flux-tunable mechanism for separating linear and nonlinear transport channels in photonic lattices. This route to thermoelectric optimization is distinct from conventional band-engineering approaches and could be relevant for synthetic-gauge-field platforms.

major comments (2)
  1. [§3] §3 (or the section presenting the non-equilibrium steady-state calculation): the conductor-to-insulator transition at weak nonlinearity is asserted to occur precisely at the linear caging flux, yet no explicit expression for the nonlinear current or the critical nonlinearity strength is given; the transition appears to rest on a numerical observation whose convergence with system size and boundary conditions is not quantified.
  2. The definition of the Seebeck coefficient and figure of merit (presumably in the transport-coefficient section) is not shown; it is unclear whether these quantities are extracted from the full nonlinear steady-state solution or from a linearized response around the caging point, which would affect the claimed enhancement.
minor comments (2)
  1. The abstract states that 'all Bloch bands' are flattened, but the diamond lattice dispersion is only sketched; an explicit band-structure plot or analytic expression for the linear spectrum would clarify the caging condition.
  2. Notation for the Kerr coefficient and the flux per plaquette is introduced without a dedicated table of symbols; consistency with standard photonic-lattice conventions should be checked.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. Below we respond point-by-point to the major remarks and indicate the revisions that will be incorporated.

read point-by-point responses

  1. Referee: [§3] §3 (or the section presenting the non-equilibrium steady-state calculation): the conductor-to-insulator transition at weak nonlinearity is asserted to occur precisely at the linear caging flux, yet no explicit expression for the nonlinear current or the critical nonlinearity strength is given; the transition appears to rest on a numerical observation whose convergence with system size and boundary conditions is not quantified.

    Authors: We agree that the presentation of the conductor-to-insulator transition can be strengthened. The nonlinear current is obtained by solving the steady-state Lindblad master equation for the open system (Sec. III); an analytic closed-form expression for the critical nonlinearity strength is not available because of the many-body character of the Kerr interaction. In the revised manuscript we will (i) write the explicit expression for the steady-state particle current in terms of the density-matrix elements and (ii) add a paragraph plus supplementary data quantifying finite-size convergence (N = 4, 6, 8) and the effect of different boundary reservoirs. revision: yes

  2. Referee: The definition of the Seebeck coefficient and figure of merit (presumably in the transport-coefficient section) is not shown; it is unclear whether these quantities are extracted from the full nonlinear steady-state solution or from a linearized response around the caging point, which would affect the claimed enhancement.

    Authors: The Seebeck coefficient is defined as S = Δμ/ΔT evaluated at vanishing particle current J = 0 and is obtained directly from the full nonlinear steady-state solution by imposing a temperature bias and adjusting the chemical-potential bias until J vanishes. The figure of merit ZT is then constructed from this S together with the nonlinear electrical and thermal conductances computed in the same steady state. We will add the explicit definitions and a sentence clarifying that all quantities are extracted from the nonlinear solution, not from linear-response theory around the caging point. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The abstract and available summary describe a standard linear Aharonov-Bohm caging reference (flat bands, suppressed currents) perturbed by Kerr nonlinearity, with transport coefficients extracted from boundary imbalances. No equations, self-citations, fitted parameters renamed as predictions, or ansatzes are presented that reduce the central claims (conductor-insulator transition or Seebeck enhancement) to inputs by construction. The linear reference is invoked as an external benchmark rather than defined circularly from the nonlinear results, and the derivation chain appears self-contained against external physical assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities can be identified.

pith-pipeline@v0.9.1-grok · 5697 in / 1062 out tokens · 20320 ms · 2026-06-26T19:19:42.135368+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

77 extracted references · 3 canonical work pages · 3 internal anchors

  1. [1]

    Leykam, A

    D. Leykam, A. Andreanov, and S. Flach, Artificial flat band systems: from lattice models to experiments, Adv. Phys.: X3, 1473052 (2018)

    2018

  2. [2]

    Rhim and B.-J

    J.-W. Rhim and B.-J. Yang, Singular flat bands, Ad- vances in Physics: X6, 1901606 (2021)

    2021

  3. [3]

    Danieli and S

    C. Danieli and S. Flach, Progress on artificial flat band systems: classifying, perturbing, applying, Advances in Physics0, 1 (2026)

    2026

  4. [4]

    Leykam and S

    D. Leykam and S. Flach, Perspective: Photonic flat- bands, APL Photonics3, 070901 (2018)

    2018

  5. [5]

    Vicencio, Photonic flat band dynamics, Advances in Physics: X6, 1878057 (2021)

    R. Vicencio, Photonic flat band dynamics, Advances in Physics: X6, 1878057 (2021)

    2021

  6. [6]

    Danieli, A

    C. Danieli, A. Andreanov, D. Leykam, and S. Flach, Flat band fine-tuning and its photonic applications, Nanopho- tonics13, 3925 (2024)

    2024

  7. [7]

    R. A. Vicencio, C. Cantillano, L. Morales-Inostroza, B. Real, C. Mej´ ıa-Cort´ es, S. Weimann, A. Szameit, and M. I. Molina, Observation of localized states in Lieb pho- tonic lattices, Phys. Rev. Lett.114, 245503 (2015)

    2015

  8. [8]

    Mukherjee, A

    S. Mukherjee, A. Spracklen, D. Choudhury, N. Goldman, P. ¨Ohberg, E. Andersson, and R. R. Thomson, Obser- vation of a localized flat-band state in a photonic Lieb lattice, Phys. Rev. Lett.114, 245504 (2015)

    2015

  9. [9]

    S. Xia, Y. Hu, D. Song, Y. Zong, L. Tang, and Z. Chen, Demonstration of flat-band image transmission in opti- cally induced lieb photonic lattices, Opt. Lett.41, 1435 (2016)

    2016

  10. [10]

    Brosco, L

    V. Brosco, L. Pilozzi, R. Fazio, and C. Conti, Non-abelian thouless pumping in a photonic lattice, Phys. Rev. A103, 063518 (2021)

    2021

  11. [11]

    Danieli, L

    C. Danieli, L. Pilozzi, C. Conti, and V. Brosco, Thouless quantum walks in topological flat bands, Light: Science & Applications15, 244 (2026)

    2026

  12. [12]

    Mukherjee and R

    S. Mukherjee and R. R. Thomson, Observation of robust flat-band localization in driven photonic rhombic lattices, Opt. Lett.42, 2243 (2017)

    2017

  13. [13]

    Sun, X.-L

    Y.-K. Sun, X.-L. Zhang, F. Yu, Z.-N. Tian, Q.-D. Chen, and H.-B. Sun, Non-abelian thouless pumping in pho- tonic waveguides, Nature Physics18, 1080 (2022)

    2022

  14. [14]

    R. A. Vicencio and M. Johansson, Discrete flat-band soli- tons in the kagome lattice, Phys. Rev. A87, 061803 (2013)

    2013

  15. [15]

    Johansson, U

    M. Johansson, U. Naether, and R. A. Vicencio, Compact- ification tuning for nonlinear localized modes in sawtooth lattices, Phys. Rev. E92, 032912 (2015)

    2015

  16. [16]

    Danieli, A

    C. Danieli, A. Maluckov, and S. Flach, Compact discrete breathers on flat-band networks, Low Temp. Phys.44, 678 (2018)

    2018

  17. [17]

    Vidal, R

    J. Vidal, R. Mosseri, and B. Doucot, Aharonov-Bohm cages in two-dimensional structures, Phys. Rev. Lett.81, 5888 (1998)

    1998

  18. [18]

    Creutz, End states, ladder compounds, and domain- wall fermions, Phys

    M. Creutz, End states, ladder compounds, and domain- wall fermions, Phys. Rev. Lett.83, 2636 (1999)

    1999

  19. [19]

    Vidal, B

    J. Vidal, B. Dou¸ cot, R. Mosseri, and P. Butaud, Interac- tion induced delocalization for two particles in a periodic 6 potential, Phys. Rev. Lett.85, 3906 (2000)

    2000

  20. [20]

    Brosco, L

    V. Brosco, L. Pilozzi, and C. Conti, Two-flux tunable aharonov-bohm effect in a photonic lattice, Phys. Rev. B 104, 024306 (2021)

    2021

  21. [21]

    Mukherjee, M

    S. Mukherjee, M. Di Liberto, P. ¨Ohberg, R. R. Thomson, and N. Goldman, Experimental observation of Aharonov- Bohm cages in photonic lattices, Phys. Rev. Lett.121, 075502 (2018)

    2018

  22. [22]

    C´ aceres-Aravena, D

    G. C´ aceres-Aravena, D. Guzm´ an-Silva, I. Salinas, and R. A. Vicencio, Controlled transport based on multior- bital aharonov-bohm photonic caging, Phys. Rev. Lett. 128, 256602 (2022)

    2022

  23. [23]

    R. A. Vicencio, Multi-orbital photonic lattices, APL Pho- tonics10, 071101 (2025)

    2025

  24. [24]

    N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Discrete solitons in photorefrac- tive optically induced photonic lattices, Phys. Rev. E66, 046602 (2002)

    2002

  25. [25]

    Heinrich, R

    M. Heinrich, R. Keil, F. Dreisow, A. Tunnermann, A. Szameit, and S. Nolte, Nonlinear discrete optics in femtosecond laser-written photonic lattices, Applied Physics B104, 469 (2011)

    2011

  26. [26]

    Gligoric, P

    G. Gligoric, P. P. Beliˇ cev, D. Leykam, and A. Maluckov, Nonlinear symmetry breaking of Aharonov-Bohm cages, Phys. Rev. A99, 013826 (2019)

    2019

  27. [27]

    Di Liberto, S

    M. Di Liberto, S. Mukherjee, and N. Goldman, Nonlinear dynamics of Aharonov-Bohm cages, Phys. Rev. A100, 043829 (2019)

    2019

  28. [28]

    Danieli, A

    C. Danieli, A. Andreanov, T. Mithun, and S. Flach, Non- linear caging in all-bands-flat lattices, Phys. Rev. B104, 085131 (2021)

    2021

  29. [29]

    Chang, S

    N. Chang, S. Gundogdu, D. Leykam, D. G. Angelakis, S. Kou, S. Flach, and A. Maluckov, Nonlinear Bloch wave dynamics in photonic Aharonov–Bohm cages, APL Pho- tonics6, 030801 (2021)

    2021

  30. [30]

    Y. Kim, O. I. Utesov, A. Andreanov, M. V. Fistul, and S. Flach, Nematic phase transitions and density mod- ulations in 1d flat band condensates, arXiv:2604.05258 (2026)

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  31. [31]

    L. G. Wright, F. O. Wu, D. N. Christodoulides, and F. W. Wise, Physics of highly multimode nonlinear optical sys- tems, Nature Physics18, 1018 (2022)

    2022

  32. [32]

    Y. Sun, P. Parra-Rivas, G. P. Agrawal, T. Hansson, C. Antonelli, A. Mecozzi, F. Mangini, and S. Wabnitz, Multimode solitons in optical fibers: a review, Photon. Res.12, 2581 (2024)

    2024

  33. [33]

    A. E. Kraych, D. Agafontsev, S. Randoux, and P. Suret, Statistical properties of the nonlinear stage of modulation instability in fiber optics, Phys. Rev. Lett.123, 093902 (2019)

    2019

  34. [34]

    Krupa, A

    K. Krupa, A. Tonello, A. Barth´ el´ emy, V. Couderc, B. M. Shalaby, A. Bendahmane, G. Millot, and S. Wabnitz, Observation of geometric parametric instability induced by the periodic spatial self-imaging of multimode waves, Phys. Rev. Lett.116, 183901 (2016)

    2016

  35. [35]

    Mangini, M

    F. Mangini, M. Gervaziev, M. Ferraro, D. S. Kharenko, M. Zitelli, Y. Sun, V. Couderc, E. V. Podivilov, S. A. Babin, and S. Wabnitz, Statistical mechanics of beam self-cleaning in grin multimode optical fibers, Opt. Ex- press30, 10850 (2022)

    2022

  36. [36]

    Ferraro, F

    M. Ferraro, F. Mangini, M. Zitelli, and S. Wabnitz, On spatial beam self-cleaning from the perspective of optical wave thermalization in multimode graded-index fibers, Advances in Physics: X8, 2228018 (2023)

    2023

  37. [37]

    L. G. Wright, D. N. Christodoulides, and F. W. Wise, Spatiotemporal mode-locking in multimode fiber lasers, Science358, 94 (2017)

    2017

  38. [38]

    Ferraro, K

    M. Ferraro, K. Baudin, M. Gervaziev, A. Fusaro, A. Pi- cozzi, J. Garnier, G. Millot, D. Kharenko, E. Podivilov, S. Babin, F. Mangini, and S. Wabnitz, Wave turbulence, thermalization and multimode locking in optical fibers, Physica D: Nonlinear Phenomena481, 134758 (2025)

    2025

  39. [39]

    F. O. Wu, A. U. Hassan, and D. N. Christodoulides, Thermodynamic theory of highly multimoded nonlinear optical systems, Nature Photonics13, 776 (2019)

    2019

  40. [40]

    K. G. Makris, F. O. Wu, P. S. Jung, and D. N. Christodoulides, Statistical mechanics of weakly nonlin- ear optical multimode gases, Opt. Lett.45, 1651 (2020)

    2020

  41. [41]

    M. A. Selim, F. O. Wu, H. Ren, M. Khajavikhan, and D. Christodoulides, Thermodynamic description of the near- and far-field intensity patterns emerging from mul- timode nonlinear waveguide arrays, Phys. Rev. A105, 013514 (2022)

    2022

  42. [42]

    Pourbeyram, P

    H. Pourbeyram, P. Sidorenko, F. O. Wu, N. Bender, L. Wright, D. N. Christodoulides, and F. Wise, Direct observations of thermalization to a rayleigh–jeans distri- bution in multimode optical fibres, Nature Physics18, 685 (2022)

    2022

  43. [43]

    H. M. Dinani, G. G. Pyrialakos, A. M. Berman Bradley, M. Monika, H. Ren, M. A. Selim, U. Peschel, D. N. Christodoulides, and M. Khajavikhan, Universal routing of light via optical thermodynamics, Nature Photonics 19, 1116 (2025)

    2025

  44. [44]

    Kurnosov, L

    A. Kurnosov, L. J. Fern´ andez-Alc´ azar, A. Ramos, B. Shapiro, and T. Kottos, Optical kinetic theory of non- linear multimode photonic networks, Phys. Rev. Lett. 132, 193802 (2024)

    2024

  45. [45]

    Optical Thermodynamics Beyond the Weak Nonlinearity Limit

    E. Kabat, S. Mohapatra, P. Kevrekidis, and T. Kottos, Optical thermodynamics beyond the weak nonlinearity limit, arXiv:2602.13161 (2026)

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  46. [46]

    Wen, H.-Y

    K. Wen, H.-Y. Xie, A. Auerbach, and B. Uchoa, Thermal and thermoelectric transport in flat bands with nontrivial quantum geometry, Phys. Rev. B111, 205140 (2025)

    2025

  47. [47]

    T. F. O. Lara, E. B. Barros, M. Godoy, R. N. Costa Filho, T. A. S. Pereira, and D. R. da Costa, Interconvertibility in the thermodynamic properties of lieb-kagome lattices, Phys. Rev. B113, 045413 (2026)

    2026

  48. [48]

    F. J. Pena, R. Garcia-Zamora, G. De Chiara, J. Flores, S. Henriquez, F. Barra, and P. Vargas, Bosonic work- ing media in a frustrated rhombi chain: Otto and stir- ling cycles from flat bands, caging, and flux control, arXiv:2604.13185 (2026)

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  49. [49]

    Mishra, R

    S. Mishra, R. Das, and C. Benjamin, Majorana ther- moelectrics and refrigeration, Journal of Applied Physics 136, 234401 (2024)

    2024

  50. [50]

    Garcia, S

    D. Garcia, S. Arias, and R. Lopez, High-efficiency thermoelectric transport in aharonov-bohm-casher rings, Phys. Rev. Res.8, 023020 (2026)

    2026

  51. [51]

    Ramachandran, A

    A. Ramachandran, A. Andreanov, and S. Flach, Chiral flat bands: Existence, engineering, and stability, Phys. Rev. B96, 161104(R) (2017)

    2017

  52. [52]

    See Supplemental Material at [URL will be inserted by publisher] for additional information

  53. [53]

    (6) reduce toi˙p n = 0,i ˙fn = 0, andi˙c n = 0 – hence, the 7 solution of each of the three complex fields is constant at zero

    If trimer is initially empty,p n(0) = 0,c n(0) = 0, and fn−1(0) = 0, the nonlinear equations of motion Eq. (6) reduce toi˙p n = 0,i ˙fn = 0, andi˙c n = 0 – hence, the 7 solution of each of the three complex fields is constant at zero

  54. [54]

    K. O. Rasmussen, T. Cretegny, P. G. Kevrekidis, and N. Grønbech-Jensen, Statistical mechanics of a discrete nonlinear system, Phys. Rev. Lett.84, 3740 (2000)

    2000

  55. [55]

    Gradenigo, S

    G. Gradenigo, S. Iubini, R. Livi, and S. N. Majum- dar, Localization transition in the discrete nonlinear schr¨ odinger equation: ensembles inequivalence and nega- tive temperatures, Journal of Statistical Mechanics: The- ory and Experiment2021, 023201 (2021)

    2021

  56. [56]

    Iubini and A

    S. Iubini and A. Politi, Effective grand canonical descrip- tion of condensation in negative-temperature regimes, Phys. Rev. Lett.134, 097102 (2025)

    2025

  57. [57]

    Szavits-Nossan, M

    J. Szavits-Nossan, M. R. Evans, and S. N. Majumdar, Constraint-driven condensation in large fluctuations of linear statistics, Phys. Rev. Lett.112, 020602 (2014)

    2014

  58. [58]

    Sheng, Introduction to wave scattering, localization, and mesoscopic phenomena, Vol

    P. Sheng, Introduction to wave scattering, localization, and mesoscopic phenomena, Vol. 88 (Springer-Verlag, Berlin, 1996)

    1996

  59. [59]

    R. S. MacKay and S. Aubry, Proof of existence of breathers for time-reversible or hamiltonian networks of weakly coupled oscillators, Nonlinearity7, 1623 (1994)

    1994

  60. [60]

    Lepri, R

    S. Lepri, R. Livi, and A. Politi, Thermal conduction in classical low-dimensional lattices, Physics Reports377, 1 (2003)

    2003

  61. [61]

    Iubini, S

    S. Iubini, S. Lepri, and A. Politi, Nonequilibrium discrete nonlinear schr¨ odinger equation, Phys. Rev. E86, 011108 (2012)

    2012

  62. [62]

    Iubini, A

    S. Iubini, A. Politi, and P. Politi, Onsager coefficients in a coupled-transport model displaying a condensation transition, New Journal of Physics25, 063020 (2023)

    2023

  63. [63]

    Lepri, Thermal transport in low dimensions: from sta- tistical physics to nanoscale heat transfer, Springer New York921(2016)

    S. Lepri, Thermal transport in low dimensions: from sta- tistical physics to nanoscale heat transfer, Springer New York921(2016)

    2016

  64. [64]

    A. F. Ioffe, L. S. Stil’bans, E. K. Iordanishvili, T. S. Stavitskaya, A. Gelbtuch, and G. H. Vineyard, Semi- conductor thermoelements and thermoelectric cooling (1957)

    1957

  65. [65]

    (9,10) in vector form read j=Lf, where the 2×2 matrixL= −Laa Lah −Lha Lhh en- capsulates the Onsager coefficients, whilej= (j a, jh)t andf= (m y, βy)t

    Onsager relations in Eqs. (9,10) in vector form read j=Lf, where the 2×2 matrixL= −Laa Lah −Lha Lhh en- capsulates the Onsager coefficients, whilej= (j a, jh)t andf= (m y, βy)t

  66. [66]

    G. D. Mahan and J. O. Sofo, The best thermoelectric., Proceedings of the National Academy of Sciences93, 7436 (1996)

    1996

  67. [67]

    Shakouri, Thermoelectric, thermionic and thermopho- tovoltaic energy conversion, ICT 2005

    A. Shakouri, Thermoelectric, thermionic and thermopho- tovoltaic energy conversion, ICT 2005. 24th International Conference on Thermoelectrics , 507 (2005)

    2005

  68. [68]

    Benenti and G

    G. Benenti and G. Casati, Increasing thermoelectric ef- ficiency: dynamical models unveil microscopic mecha- nisms, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences369, 466 (2011)

    2011

  69. [69]

    Gayner and Y

    C. Gayner and Y. Amouyal, Energy filtering of charge carriers: Current trends, challenges, and prospects for thermoelectric materials, Advanced Functional Materials 30, 1901789 (2020)

    2020

  70. [70]

    Masci, E

    A. Masci, E. Dimaggio, N. Neophytou, D. Narducci, and G. Pennelli, Large increase of the thermoelectric power factor in multi-barrier nanodevices, Nano Energy132, 110391 (2024)

    2024

  71. [71]

    Ishibe, T

    T. Ishibe, T. Hinakawa, S. Sakane, S. Ishigaki, K. Suzuki, K. Sato, T. Fujita, Y. Yamashita, E. Kobayashi, and Y. Nakamura, Robust thermoelectric power factor en- hanced by energy filtering effect in b20-type cosi films, Phys. Rev. Mater.9, 075402 (2025)

    2025

  72. [72]

    Garmroudi, S

    F. Garmroudi, S. D. Cataldo, M. Parzer, J. Coul- ter, Y. Iwasaki, M. Grasser, S. Stockinger, S. P´ azm´ an, S. Witzmann, A. Riss, H. Michor, R. Podloucky, S. Khmelevskyi, A. Georges, K. Held, T. Mori, E. Bauer, and A. Pustogow, Energy filtering–induced ultrahigh thermoelectric power factors in Ni3Ge, Science Advances 11, eadv7113 (2025)

    2025

  73. [73]

    J. Zhou, R. Yang, G. Chen, and M. S. Dresselhaus, Opti- mal bandwidth for high efficiency thermoelectrics, Phys. Rev. Lett.107, 226601 (2011)

    2011

  74. [74]

    R. Luo, G. Benenti, G. Casati, and J. Wang, Onsager reciprocal relations with broken time-reversal symmetry, Phys. Rev. Res.2, 022009(R) (2020)

    2020

  75. [75]

    Iubini, L

    S. Iubini, L. Chirondojan, G.-L. Oppo, A. Politi, and P. Politi, Dynamical freezing of relaxation to equilibrium, Phys. Rev. Lett.122, 084102 (2019)

    2019

  76. [76]

    Campbell, I

    S. Campbell, I. D’Amico, M. A. Ciampini, J. Anders, N. Ares, S. Artini, A. Auff` eves, L. Bassman Oftelie, L. P. Bettmann, M. V. S. Bonan¸ ca, T. Busch, M. Campisi, M. F. Cavalcante, L. A. Correa, E. Cuestas, C. B. Dag, S. Dago, S. Deffner, A. Del Campo, A. Deutschmann- Olek, S. Donadi, E. Doucet, C. Elouard, K. Ensslin, P. Erker, N. Fabbri, F. Fedele, G....

    2026

  77. [77]

    W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical recipes 3rd edition (Cambridge uni- versity press, New York, 2007). 8 SUPPLEMENT AL MA TERIAL ROT A TING THE HAMIL TONIAN Let us recall the unitary transformation recapped    pn = an+bn√ 2 fn = an−bn√ 2 cn =c n ⇔    an = pn+fn√ 2 bn = pn−fn√ 2 cn =c n (11) In the new c...

    2007