pith. sign in

arxiv: 2605.13259 · v1 · pith:N3R552ZPnew · submitted 2026-05-13 · ❄️ cond-mat.supr-con · cond-mat.str-el

Multiband Superconductivity in the Exactly Solvable Hatsugai-Kohmoto Model

Nico Hahn , R. Matthias Geilhufe This is my paper

Pith reviewed 2026-05-14 18:42 UTC · model grok-4.3

classification ❄️ cond-mat.supr-con cond-mat.str-el
keywords multiband superconductivityHatsugai-Kohmoto modelpairing symmetryD4h symmetrymean-field theorycorrelated electronsexact solvabilityorbital degrees of freedom
0
0 comments X

The pith

The orbital Hatsugai-Kohmoto model supplies an exactly solvable platform for classifying multiband superconducting gap structures and computing their mean-field critical temperatures.

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

The paper studies superconductivity in the multiband extension of the Hatsugai-Kohmoto model, whose momentum-local interactions keep the normal state exactly solvable while permitting orbital degrees of freedom. For a two-orbital system with D4h point-group symmetry the authors enumerate all symmetry-allowed gap structures that respect the combined spin, orbital and momentum selection rules. They then solve the mean-field equations to obtain the critical temperature and the magnitude of the order parameter for representative pairing channels as explicit functions of the interaction and pairing strengths. A sympathetic reader cares because the construction isolates the interplay of strong correlations and multiband pairing in a setting where the underlying Hamiltonian remains tractable, offering a controlled laboratory for ideas that usually require heavy numerics.

Core claim

Focusing on a two-orbital system with point-group symmetry D4h, we classify the symmetry-allowed superconducting gap structures, taking into account spin, orbital and momentum degrees of freedom. We further compute the critical temperature and the superconducting order parameter for selected pairing channels as functions of interaction and pairing strength within a mean-field treatment.

What carries the argument

The momentum-local interaction of the orbital Hatsugai-Kohmoto model, which renders the normal-state Hamiltonian exactly diagonalizable in momentum space while still supporting orbital structure for symmetry classification of pairing.

If this is right

  • All symmetry-allowed gap structures for the two-orbital D4h case are enumerated by group theory.
  • Critical temperature and order-parameter magnitude become explicit functions of the interaction parameters for each channel.
  • The normal-state correlations enter the gap equations exactly because the Hatsugai-Kohmoto Hamiltonian is diagonal in momentum.
  • The same symmetry analysis immediately extends to any other point group once the orbital basis is fixed.

Where Pith is reading between the lines

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

  • The classification supplies a template that can be reused for other exactly solvable correlated models once orbital degrees of freedom are added.
  • Because the normal state is solvable, one can in principle compute fluctuation corrections around the mean-field solution without additional approximations.
  • Materials whose low-energy bands map onto two orbitals with D4h symmetry become direct testing grounds for the predicted channel dependence of Tc.

Load-bearing premise

The mean-field decoupling of the pairing interaction remains quantitatively reliable for the superconducting transition even when the normal state is strongly correlated.

What would settle it

Observation of a critical temperature or gap anisotropy in a two-orbital D4h material that lies outside the set of mean-field solutions obtained for every symmetry-allowed channel would falsify the framework's predictive reach.

Figures

Figures reproduced from arXiv: 2605.13259 by Nico Hahn, R. Matthias Geilhufe.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Mott gap [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Positions of the extrema of the free energy as functions of [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Critical temperature [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Illustration of nearest- and second-nearest-neighbour [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
read the original abstract

Multiband superconductivity gives rise to a rich landscape of possible pairing states. Here we study superconductivity in the multiband extension of the Hatsugai-Kohmoto model, an exactly solvable model of correlated electrons with momentum-local interactions, which provides a minimal framework to explore the interplay of strong correlations, orbital structure and pairing symmetry. Focusing on a two-orbital system with point-group symmetry $\rm D_{4h}$, we classify the symmetry-allowed superconducting gap structures, taking into account spin, orbital and momentum degrees of freedom. We further compute the critical temperature and the superconducting order parameter for selected pairing channels as functions of interaction and pairing strength within a mean-field treatment. Our results provide a systematic framework for analyzing superconductivity in the orbital Hatsugai-Kohmoto model and extend symmetry-based approaches to correlated multiband settings.

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 paper studies multiband superconductivity in the orbital extension of the Hatsugai-Kohmoto model, which is exactly solvable in the normal state due to momentum-local interactions. For a two-orbital system with D_{4h} point-group symmetry, the authors classify symmetry-allowed superconducting gap structures that incorporate spin, orbital, and momentum degrees of freedom. They then compute the critical temperature T_c and the superconducting order parameter for selected pairing channels as functions of interaction and pairing strength within a mean-field treatment.

Significance. The exact solvability of the normal-state HK model offers a controlled setting in which to examine the interplay between strong correlations, orbital degrees of freedom, and pairing symmetry. If the mean-field results are reliable, the symmetry classification and the explicit T_c curves provide a useful systematic framework that extends conventional symmetry-based approaches to correlated multiband superconductors. The work is strongest in its group-theoretic classification; its quantitative predictions rest on the mean-field approximation whose validity in the strongly correlated regime is not benchmarked.

major comments (2)
  1. [§3] §3 (Mean-field decoupling): The superconducting instability is obtained exclusively from a mean-field treatment of the added pairing term. Although the normal state is exactly diagonalizable per momentum, the manuscript provides no fluctuation correction, exact benchmark, or self-consistency check demonstrating that the mean-field gap equation remains accurate when the on-site interaction strength becomes comparable to or larger than the bandwidth. This assumption is load-bearing for all reported T_c values and order-parameter magnitudes.
  2. [§4.2] §4.2, Fig. 3: The plotted T_c versus interaction strength shows a monotonic increase, but the text does not discuss the expected suppression of T_c at very large U due to the projected Hilbert space or the competition with the normal-state Mott-like physics already present in the HK model. Without this analysis the physical regime of the mean-field results remains unclear.
minor comments (2)
  1. [Table 1] The explicit matrix form of the orbital-dependent gap functions for the D_{4h} irreps (Table 1) should be written out; the current listing by irrep label alone makes it difficult to verify the stated momentum and orbital structure.
  2. [Eq. (12)] The definition of the effective pairing strength in Eq. (12) mixes the bare pairing amplitude with a renormalization factor from the normal-state self-energy; a short appendix deriving this renormalization would improve reproducibility.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading of our manuscript and for the positive assessment of the symmetry classification. We address the major comments point by point below. Where appropriate, we have revised the manuscript to clarify the regime of validity of the mean-field results.

read point-by-point responses

  1. Referee: [§3] §3 (Mean-field decoupling): The superconducting instability is obtained exclusively from a mean-field treatment of the added pairing term. Although the normal state is exactly diagonalizable per momentum, the manuscript provides no fluctuation correction, exact benchmark, or self-consistency check demonstrating that the mean-field gap equation remains accurate when the on-site interaction strength becomes comparable to or larger than the bandwidth. This assumption is load-bearing for all reported T_c values and order-parameter magnitudes.

    Authors: We agree that the mean-field treatment is an approximation whose accuracy in the strong-coupling regime requires justification. In the revised manuscript we have expanded the discussion in Section 3 to explicitly state the limitations of the mean-field gap equation when U becomes comparable to or larger than the bandwidth. We emphasize that the exact solvability of the normal state still provides a controlled platform for identifying pairing instabilities, but we acknowledge that fluctuation corrections or exact benchmarks lie beyond the present scope and would require separate methodological development. revision: partial

  2. Referee: [§4.2] §4.2, Fig. 3: The plotted T_c versus interaction strength shows a monotonic increase, but the text does not discuss the expected suppression of T_c at very large U due to the projected Hilbert space or the competition with the normal-state Mott-like physics already present in the HK model. Without this analysis the physical regime of the mean-field results remains unclear.

    Authors: We thank the referee for this observation. In the revised version we have added a paragraph in Section 4.2 and updated the caption of Fig. 3 to discuss the expected suppression of T_c at large U. We now explicitly state that our calculations are performed in the regime where the normal state remains metallic and that, at sufficiently large U, competition with Mott-like physics in the projected Hilbert space is expected to suppress superconductivity. The range of U values displayed in the figure has been clarified accordingly. revision: yes

standing simulated objections not resolved
  • Exact benchmarks or fluctuation corrections that would rigorously validate the mean-field gap equation for large U in the superconducting phase of the Hatsugai-Kohmoto model.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained from model definitions and symmetry analysis

full rationale

The paper starts from the exactly solvable Hatsugai-Kohmoto Hamiltonian with momentum-local interactions, performs a standard mean-field decoupling for the added pairing term, and classifies allowed gap functions under D4h symmetry using group theory. Critical temperatures and order parameters are then obtained by direct solution of the resulting self-consistent equations as explicit functions of the interaction and pairing strengths. No equation reduces to a prior fitted parameter renamed as a prediction, no uniqueness theorem is imported from the authors' own prior work to force the result, and the central claims rest on explicit computation rather than self-referential definitions or load-bearing self-citations.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The work rests on the standard Hatsugai-Kohmoto interaction form, point-group symmetry D4h, and the validity of mean-field theory for the superconducting instability.

free parameters (2)
  • interaction strength
    Varied parametrically to compute Tc and order parameter as functions of interaction and pairing strength.
  • pairing strength
    Varied parametrically to compute Tc and order parameter as functions of interaction and pairing strength.
axioms (2)
  • domain assumption Mean-field approximation suffices to capture the superconducting transition
    Invoked for computing critical temperature and order parameter.
  • standard math D4h point-group symmetry constrains the allowed gap structures
    Used to classify symmetry-allowed superconducting gap structures.

pith-pipeline@v0.9.0 · 5441 in / 1188 out tokens · 57544 ms · 2026-05-14T18:42:36.052364+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

50 extracted references · 50 canonical work pages

  1. [1]

    The free energy is F = −T ln Z = N g |∆|2 − T ∑ k∈HBZ ln Zk

    The partition function is Z = tr e−β ˆH = e− β N g |∆|2 ∏ k∈HBZ Zk, Zk = trke−β ˆHk , (17) where we denote the trace over a single k-point as trk and we call Zk the local partition function. The free energy is F = −T ln Z = N g |∆|2 − T ∑ k∈HBZ ln Zk. (18) In the following, we consider the intensive free energy density f = F/N. The critical temperature TC...

    work page arXiv 2023

  2. [2]

    Souma, Y

    S. Souma, Y . Machida, T. Sato, T. Takahashi, H. Matsui, S.-C. Wang, H. Ding, A. Kaminski, J. C. Campuzano, S. Sasaki, and K. Kadowaki, The Origin of Multiple Superconducting Gaps in MgB2, Nature 423, 65 (2003)

    work page 2003

  3. [3]

    H. Suhl, B. T. Matthias, and L. R. Walker, Bardeen-Cooper- Schrieffer Theory of Superconductivity in the Case of Overlap- ping Bands, Phys. Rev. Lett. 3, 552 (1959)

    work page 1959

  4. [4]

    H. J. Choi, D. Roundy, H. Sun, M. L. Cohen, and S. G. Louie, The Origin of the Anomalous Superconducting Properties of MgB2, Nature 418, 758 (2002)

    work page 2002

  5. [5]

    Linder and A

    J. Linder and A. V . Balatsky, Odd-Frequency Superconductiv- ity, Rev. Mod. Phys.91, 045005 (2019)

    work page 2019

  6. [6]

    A. M. Black-Schaffer and A. V . Balatsky, Odd-Frequency Su- perconducting Pairing in Multiband Superconductors, Phys. Rev. B 88, 104514 (2013)

    work page 2013

  7. [7]

    Triola, J

    C. Triola, J. Cayao, and A. M. Black-Schaffer, The Role of Odd-Frequency Pairing in Multiband Superconductors, Ann. Phys. 532, 1900298 (2020)

    work page 2020

  8. [8]

    Raghu, X.-L

    S. Raghu, X.-L. Qi, C.-X. Liu, D. J. Scalapino, and S.-C. Zhang, Minimal Two-Band Model of the Superconducting Iron Oxyp- nictides, Phys. Rev. B 77, 220503 (2008)

    work page 2008

  9. [9]

    P. J. Hirschfeld, M. M. Korshunov, and I. I. Mazin, Gap Sym- metry and Structure of Fe-Based Superconductors, Rep. Prog. Phys. 74, 124508 (2011)

    work page 2011

  10. [10]

    Chubukov, Pairing Mechanism in Fe-Based Superconduc- tors, Annu

    A. Chubukov, Pairing Mechanism in Fe-Based Superconduc- tors, Annu. Rev. Condens. Matter Phys. 3, 57 (2012)

    work page 2012

  11. [11]

    Zehetmayer, A Review of Two-Band Superconductivity: Materials and Effects on the Thermodynamic and Reversible Mixed-State Properties, Supercond

    M. Zehetmayer, A Review of Two-Band Superconductivity: Materials and Effects on the Thermodynamic and Reversible Mixed-State Properties, Supercond. Sci. Technol. 26, 043001 (2013)

    work page 2013

  12. [12]

    P. M. C. Rourke, M. A. Tanatar, C. S. Turel, J. Berdeklis, C. Petrovic, and J. Y . T. Wei, Spectroscopic Evidence for Mul- tiple Order Parameter Components in the Heavy Fermion Su- perconductor CeCoIn5, Phys. Rev. Lett. 94, 107005 (2005)

    work page 2005

  13. [13]

    Seyfarth, J

    G. Seyfarth, J. P. Brison, M.-A. Méasson, J. Flouquet, K. Izawa, Y . Matsuda, H. Sugawara, and H. Sato, Multiband Supercon- ductivity in the Heavy Fermion Compound PrOs 4Sb12, Phys. Rev. Lett. 95, 107004 (2005)

    work page 2005

  14. [14]

    R. W. Hill, S. Li, M. B. Maple, and L. Taillefer, Multiband Or- der Parameters for the PrOs 4Sb12 and PrRu4Sb12 Skutterudite Superconductors from Thermal Conductivity Measurements, Phys. Rev. Lett. 101, 237005 (2008)

    work page 2008

  15. [15]

    Majumdar, D

    A. Majumdar, D. VanGennep, J. Brisbois, D. Chareev, A. V . Sadakov, A. S. Usoltsev, M. Mito, A. V . Silhanek, T. Sarkar, A. Hassan, O. Karis, R. Ahuja, and M. Abdel-Hafiez, Inter- play of Charge Density Wave and Multiband Superconductiv- ity in Layered Quasi-Two-Dimensional Materials: The Case of 2H −NbS2 and 2H −NbSe2, Phys. Rev. Mater. 4, 084005 (2020)

    work page 2020

  16. [16]

    Y . Noat, J. A. Silva-Guillén, T. Cren, V . Cherkez, C. Brun, S. Pons, F. Debontridder, D. Roditchev, W. Sacks, L. Cario, P. Ordejón, A. García, and E. Canadell, Quasiparticle Spectra of 2H − NbSe2: Two-Band Superconductivity and the Role of Tunneling Selectivity, Phys. Rev. B92, 134510 (2015)

    work page 2015

  17. [17]

    Zhang, Y

    X. Zhang, Y . Zhou, B. Cui, M. Zhao, and F. Liu, Theoreti- cal Discovery of a Superconducting Two-Dimensional Metal– Organic Framework, Nano Letters 17, 6166 (2017)

    work page 2017

  18. [18]

    Huang, S

    X. Huang, S. Zhang, L. Liu, L. Yu, G. Chen, W. Xu, and D. Zhu, Superconductivity in a Copper(II)-Based Coordination Polymer with Perfect Kagome Structure, Angew. Chem. Int. Ed.57, 146 (2018)

    work page 2018

  19. [19]

    Takenaka, K

    T. Takenaka, K. Ishihara, M. Roppongi, Y . Miao, Y . Mizukami, T. Makita, J. Tsurumi, S. Watanabe, J. Takeya, M. Yamashita, K. Torizuka, Y . Uwatoko, T. Sasaki, X. Huang, W. Xu, D. Zhu, N. Su, J.-G. Cheng, T. Shibauchi, and K. Hashimoto, Strongly Correlated Superconductivity in a Copper-Based Metal-Organic Framework with a Perfect Kagome Lattice, Sci. Adv...

    work page 2021

  20. [20]

    M. F. Ohlrich, E. M. Makaresz, H. L. Nourse, and B. J. Pow- ell, Flat Bands and Unconventional Superconductivity in a Sim- ple Model of Metal-Organic Frameworks, Phys. Rev. B 111, L100503 (2025)

    work page 2025

  21. [21]

    Hatsugai and M

    Y . Hatsugai and M. Kohmoto, Exactly Solvable Model of Cor- related Lattice Electrons in Any Dimensions, J. Phys. Soc. Jpn. 61, 2056 (1992)

    work page 2056

  22. [22]

    Lidsky, J

    D. Lidsky, J. Shiraishi, Y . Hatsugai, and M. Kohmoto, Simple Exactly Solvable Models of non-Fermi Liquids, Phys. Rev. B 57, 1340 (1998)

    work page 1998

  23. [23]

    E. W. Huang, G. L. Nave, and P. W. Phillips, Discrete Symmetry Breaking Defines the Mott Quartic Fixed Point, Nat. Phys. 18, 511 (2022)

    work page 2022

  24. [24]

    P. W. Phillips, L. Yeo, and E. W. Huang, Exact Theory for Su- perconductivity in a Doped Mott Insulator, Nat. Phys. 16, 1175 (2020)

    work page 2020

  25. [25]

    Y . Li, V . Mishra, Y . Zhou, and F.-C. Zhang, Two-stage Super- conductivity in the Hatsugai–Kohomoto-BCS Model, New J. Phys. 24, 103019 (2022)

    work page 2022

  26. [26]

    J. Zhao, L. Yeo, E. W. Huang, and P. W. Phillips, Thermody- namics of an Exactly Solvable Model for Superconductivity in a Doped Mott Insulator, Phys. Rev. B 105, 184509 (2022). 8

    work page 2022

  27. [27]

    Bácsi and B

    A. Bácsi and B. Dóra, Nonequilibrium Dynamics of Supercon- ductivity in the Hatsugai-Kohmoto Model, Phys. Rev. B 111, 075115 (2025)

    work page 2025

  28. [28]

    C. E. S. Corsino and H. Freire, Exact Analysis of the Inter- play of Charge Order and Unconventional Pairings in the 2D Hatsugai-Kohmoto Model, Phys. Lett. A 563, 131070 (2025)

    work page 2025

  29. [29]

    P. Mai, J. Zhao, B. E. Feldman, and P. W. Phillips, 1/4 is the New 1/2 when Topology is Intertwined with Mottness, Nat. Commun. 14, 5999 (2023)

    work page 2023

  30. [30]

    P. Mai, B. E. Feldman, and P. W. Phillips, Topological Mott Insulator at Quarter Filling in the Interacting Haldane Model, Phys. Rev. Res. 5, 013162 (2023)

    work page 2023

  31. [31]

    P. Mai, J. Zhao, T. A. Maier, B. Bradlyn, and P. W. Phillips, Topological Phase Transition without Single Particle Gap Clos- ing in Strongly Correlated Systems, Phys. Rev. B 110, 075105 (2024)

    work page 2024

  32. [32]

    Leeb and J

    V . Leeb and J. Knolle, Quantum Oscillations in a Doped Mott Insulator Beyond Onsager’s Relation, Phys. Rev. B108, 085106 (2023)

    work page 2023

  33. [33]

    Zhong, Notes on Quantum Oscillation for Hat- sugai–Kohmoto Model, Mod

    Y . Zhong, Notes on Quantum Oscillation for Hat- sugai–Kohmoto Model, Mod. Phys. Lett. B 38, 2450027 (2024)

    work page 2024

  34. [34]

    Y . Ma, J. Zhao, E. W. Huang, D. Kush, B. Bradlyn, and P. W. Phillips, Charge Susceptibility and Kubo Response in Hatsugai- Kohmoto-Related Models, Phys. Rev. B 112, 045109 (2025)

    work page 2025

  35. [35]

    Guerci, G

    D. Guerci, G. Sangiovanni, A. J. Millis, and M. Fabrizio, Elec- trical Transport in the Hatsugai-Kohmoto Model, Phys. Rev. B 111, 075124 (2025)

    work page 2025

  36. [36]

    Zhao, W.-W

    M. Zhao, W.-W. Yang, and Y . Zhong, Hatsugai–Kohmoto Mod- els: Exactly Solvable Playground for Mottness and non-Fermi Liquid, J. Phys. Condens. Matter 37, 183005 (2025)

    work page 2025

  37. [37]

    Manning-Coe and B

    D. Manning-Coe and B. Bradlyn, Ground State Stability, Sym- metry, and Degeneracy in Mott Insulators with Long-Range In- teractions, Phys. Rev. B 108, 165136 (2023)

    work page 2023

  38. [38]

    P. Mai, J. Zhao, G. Tenkila, N. A. Hackner, D. Kush, D. Pan, and P. W. Phillips, Twisting the Hubbard Model into the Momentum-Mixing Hatsugai–Kohmoto Model, Nat. Phys. 22, 81 (2026)

    work page 2026

  39. [39]

    Tenkila, J

    G. Tenkila, J. Zhao, and P. W. Phillips, Dynamical Spectral Weight Transfer in the Orbital Hatsugai-Kohmoto Model, Phys. Rev. B 111, 045126 (2025)

    work page 2025

  40. [40]

    Sigrist and K

    M. Sigrist and K. Ueda, Phenomenological Theory of Uncon- ventional Superconductivity, Rev. Mod. Phys.63, 239 (1991)

    work page 1991

  41. [41]

    Skolimowski, Real-Space Analysis of Hatsugai-Kohmoto In- teraction, Phys

    J. Skolimowski, Real-Space Analysis of Hatsugai-Kohmoto In- teraction, Phys. Rev. B 109, 165129 (2024)

    work page 2024

  42. [42]

    Rosenhaus, An Introduction to the SYK Model, J

    V . Rosenhaus, An Introduction to the SYK Model, J. Phys. A: Math. Theor. 52, 323001 (2019)

    work page 2019

  43. [43]

    Panchenko, The Sherrington-Kirkpatrick Model: An Overview, J

    D. Panchenko, The Sherrington-Kirkpatrick Model: An Overview, J. Stat. Phys. 149, 362 (2012)

    work page 2012

  44. [44]

    Yang, Exactly Solvable Model of Fermi Arcs and Pseudo- gap, Phys

    K. Yang, Exactly Solvable Model of Fermi Arcs and Pseudo- gap, Phys. Rev. B 103, 024529 (2021)

    work page 2021

  45. [45]

    Wang and K

    R. Wang and K. Yang, Non-Fermi Liquid Behavior in a Simple Model of Fermi Arcs and Pseudogap, Mod. Phys. Lett. B 37, 2350119 (2023)

    work page 2023

  46. [46]

    J. C. Slater and G. F. Koster, Simplified LCAO Method for the Periodic Potential Problem, Phys. Rev. 94, 1498 (1954)

    work page 1954

  47. [47]

    Hergert and R

    W. Hergert and R. M. Geilhufe, Group Theory in Solid State Physics and Photonics: Problem Solving with Mathematica (Wiley-VCH, 2018) ISBN : 978-3-527-41133-7

    work page 2018

  48. [48]

    R. M. Geilhufe and W. Hergert, GTPack: A Mathematica Group Theory Package for Application in Solid-State Physics and Photonics, Front. Phys. 6, 86 (2018)

    work page 2018

  49. [49]

    W.A.Harrison, Electronic Structure and the Properties of Solids (Dover Publications Inc., 1989)

    work page 1989

  50. [50]

    H. J. Monkhorst and J. D. Pack, Special Points for Brillouin- Zone Integrations, Phys. Rev. B 13, 5188 (1976). Appendix A: Two-Band Model FIG. 4. (a) Illustration of nearest- and second-nearest-neighbour hopping between p-orbitals on the square lattice. (b) Band struc- ture of the two-band model along an irreducible path in the square Brillouin zone. The ...

    work page 1976