Skip to main content
SpringerLink
Log in
Book cover

Physics and Mathematics of Quantum Many-Body Systems pp 371–455Cite as

The Origin of Ferromagnetism

  • Chapter
  • First Online:

  • 3826 Accesses

Part of the book series: Graduate Texts in Physics ((GTP))

Abstract

In the present chapter we focus on ferromagnetism (or, more precisely, saturated ferromagnetism) in the ground states of the Hubbard model. Recalling that both the non-interacting models and the non-hopping models exhibit paramagnetism, we see that ferromagnetism can be generated only through nontrivial “competition” between wave-like nature and particle-like nature of electrons. After discussing basic properties of saturated ferromagnetism in the Hubbard model in Sect. 11.1, we present some rigorous results which establish that the ground states of certain versions of the Hubbard models exhibit ferromagnetism. They include Nagaoka’s ferromagnetism for systems with infinitely large U (Sect. 11.2), flat-band ferromagnetism by Mielke and by Tasaki (Sect. 11.3), and ferromagnetism in nearly-flat-band Hubbard model due to Tasaki (Sect. 11.4). The final result is of particular importance since it deals with a situation where ferromagnetism is intrinsically a non-perturbative phenomenon, and above mentioned competition plays an essential role. We end the chapter by briefly discussing the fascinating but extremely difficult problem of metallic ferromagnetism in Sect. 11.5.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   89.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   119.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    One uses the same complex rotation (4.4.26), regarding the spin operator as written in terms of the fermion operators.

  2. 2.

    Lieb’s example in Sect. 10.2.3 satisfies this criterion, but it should better be interpreted as ferrimagnetism rather than partial ferromagnetism.

  3. 3.

    Proof Since \(\hat{H}\) and \(\hat{S}_\mathrm{tot}^{(3)}\) are simultaneously diagonalizable, there is a ground state in \({\mathscr {H}}_{S_{\mathrm {max}},M_0}\) for some \(M_0\). Then Theorem A.16 (p. 473) implies that there is a ground state in \({\mathscr {H}}_{S_{\mathrm {max}},M}\) for each M.

  4. 4.

    The constants \(n_{0}\) and d represent the degeneracy of the single-electron ground states and the dimension of the system, respectively.

  5. 5.

    See the Proof of Theorem 11.5 in Sect. 11.2 for the treatment of the \(U\uparrow \infty \) limit.

  6. 6.

    If the transition amplitude between two states is negative (resp., positive), one superposes the two states with the same (resp., opposite) signs.

  7. 7.

    In Japanese institutions it often happens that the presence of a single foreign participant in a seminar room makes everybody shift from Japanese to English. I used to mention this phenomenon in the introduction to talks about Nagaoka’s ferromagnetism. I remember one seminar at RIMS in Kyoto, where I had a perfect situation for this joke; the topic was about my refinement of Nagaoka’s theorem, there was exactly one foreign participant at the seminar, and among many Japanese participants was Yosuke Nagaoka!

  8. 8.

    To be precise, \({\mathscr {H}}_{N}^\mathrm{hc}\) is spanned by the basis states (9.2.35) with the constraint that \(x_j\ne x_k\) when \(j\ne k\).

  9. 9.

    If this is not the case, we redefine \(ia\{\varphi (x,\varvec{\sigma })-\varphi (x,\varvec{\sigma })^{*}\}\) as \(\varphi (x,\varvec{\sigma })\), where \(a\in \mathbb {R}\) is a normalization factor. The corresponding \(|\varPhi _\mathrm {GS}\rangle \) is also a ground state.

  10. 10.

    A lattice (or a graph) is biconnected (or non-separable) if and only if one cannot make it disconnected by removing a single site.

  11. 11.

    Those who read Japanese might enjoy a short article by Nagaoka entitled “The 15 Puzzle” [43].

  12. 12.

    The sufficient condition is, in a sense, physical since it only makes use of basic exchange processes caused by local motions of the hole. But, after all, we should note that Nagaoka’s ferromagnetism itself is not quite physical.

  13. 13.

    For a proof of this property, see “Proof of the property (iii)” in p. 41.

  14. 14.

    This point is relevant to the extension of Nagaoka’s ferromagnetism to the SU(n) Hubbard model [4, 20].

  15. 15.

    It was very early days of arXiv, and neither Mielke nor myself were posting papers. I learned about Mielke’s work from my colleague some time after it was published in the journal. I was at that time working on a draft of my paper [67].

  16. 16.

    We believe it fair to say that rigorous results preceded numerical works in the study of ferromagnetism in the Hubbard model. There also appeared numerical works in various versions of the Hubbard model, but we shall not try to list them.

  17. 17.

    This is in fact not entirely obvious because \(\varvec{\beta }_u\) with \(u\in {\mathscr {I}}\) are not mutually orthogonal. Let us give a careful proof. Define the Gramm matrix \(\mathsf {G}\) by \((\mathsf {G})_{u,v}=\langle \varvec{\beta }_u,\varvec{\beta }_v\rangle =\{\hat{b}_{u,\sigma },\hat{b}^\dagger _{v,\sigma }\}\) for \(u,v\in {\mathscr {I}}\). The linear independence of \(\{\varvec{\beta }_u\}_{u\in {\mathscr {I}}}\) implies that \(\mathsf {G}\) is invertible. We define the dual operators by \(\hat{b}'_{u,\sigma }=\sum _{v\in {\mathscr {I}}}(\mathsf {G}^{-1})_{u,v}\hat{b}_{v,\sigma }\), which clearly satisfy \(\{\hat{b}'_{u,\sigma },\hat{b}^\dagger _{v,\tau }\}=\delta _{u,v}\delta _{\sigma ,\tau }\). We also have \(\{\hat{b}'_{u,\sigma },\hat{a}^\dagger _{p,\tau }\}=0\) for any \(u\in {\mathscr {I}}\) and \(p\in {\mathscr {E}}\). Since (11.3.11) implies \(\hat{b}'_{u,\sigma }|\varPhi _\mathrm {GS}\rangle =0\) for any \(u\in {\mathscr {I}}\) and \(\sigma =\uparrow ,\downarrow \), we get the desired property.

  18. 18.

    This should be obvious, but see “Proof of the property (iii)” in p. 41 for a rigorous proof.

  19. 19.

    By recalling (2.4.11), one can directly show that the superposition of basis states (with a common \(S_\mathrm{tot}^{(3)}\)) with equal weights leads to a ferromagnetic state.

  20. 20.

    In [68] we conjectured that a version of the Brandt-Giesekus model may exhibit superconductivity. We now believe that this is (unfortunately) not the case.

  21. 21.

    See footnote 28 in p. 33 for the definition of connectedness.

  22. 22.

    “Kagomé” is a Japanese word that means the mesh of woven bamboo. Thus “me” is pronounced like “mesh” (without “sh”).

  23. 23.

    See p. 37 and Fig. 2.1 for the definition of bipartiteness.

  24. 24.

    A lattice (or a graph) is biconnected (or two-fold connected) if and only if one cannot make it disconnected by removing a single site.

  25. 25.

    A lattice in which all the sites are connected is called a complete graph.

  26. 26.

    We note that this is by no means the unique general method to construct tight-binding models with flat-bands.

  27. 27.

    A large part of the present proof is due to Akinori Tanaka (private communication).

  28. 28.

    Unfortunately the lemma, as far as we know, has not been applied much to concrete problems. It is possible that the lemma will be useful when one studies totally different aspects of flat-band systems.

  29. 29.

    This characterization of r is known as the determinantal rank. Its equivalence to the standard characterization in terms of the dimension of \(\ker \mathsf {A}\) is a well-known theorem.

  30. 30.

    This may be obvious, but let us see a proof. If \(\mu _z(x)=0\) for some \(x\in \varLambda \) and all \(z\in I\), then it is obvious that \(\psi _j(x)=0\) for all \(j=1,\ldots ,D_0\), where \(\{\varvec{\psi }_j\}_{j=1,\ldots ,D_0}\) is an orthonormal basis of \(\mathfrak {h}_0\). (See (11.3.44).) Thus \((\mathsf {P}_0)_{x,x}=\sum _{j=1}^{D_0}(\psi _j(x))^*\psi _j(x)=0\). Next, suppose that \(\mu _z(x)\ne 0\) for some \(x\in \varLambda \) and some \(z\in I\). Let us denote by \(\mathsf {P}_z\) the projection matrix onto the unit vector proportional to \(\varvec{\mu }_z\). Then since \(\mathsf {P}_0\ge \mathsf {P}_z\), we have \((\mathsf {P}_0)_{x,x}\ge (\mathsf {P}_z)_{x,x}\ne 0\).

  31. 31.

    In fact we only need to assume that \(\gamma \) is sufficiently away from \(-1\).

  32. 32.

    In fact we obtained (11.4.5) by demanding that \(\langle \varvec{\omega }_p,\varvec{\omega }_{p+1}\rangle =0\) and that \(\mathsf {T}\varvec{\omega }_p=\tau _0\,\varvec{\omega }_p+\tau _1\{\varvec{\omega }_{p-1}+\varvec{\omega }_{p+1}\}+O(\nu ^3)\) holds for some \(\tau _0\) and \(\tau _1\).

  33. 33.

    This is not really the second order perturbation which corresponds to the first order that we saw above. Here we are treating \(\tau \) as a perturbation.

  34. 34.

    In [71] general models obtained by the cell construction (see the end of Sect. 11.3.1) are treated.

  35. 35.

    \(\hat{A}\) is translation invariant if \((\hat{\tau }_z)^{-1}\hat{A}\hat{\tau }_z=\hat{A}\), which means \([\hat{\tau }_z,\hat{A}]=0\).

  36. 36.

    We expect (or hope) that the model represents a specially tractable class of tight-binding systems, and may play the role analogous to that played by matrix product states (or the VBS state) in quantum spin chains.

  37. 37.

    When \(d=1\) and \(\nu =1/\sqrt{2}\), one finds from (11.4.40) that \(t_{x,x}=t-s\) for all \(x\in \varLambda \). Thus the model is equivalent to that with \(t_{x,x}=0\) for all \(x\in \varLambda \).

  38. 38.

    One needs to set \(\kappa >0\) in Lemma 11.22. Numerical results for \(d=1\) indicate that one should set \(\kappa =0\) to get the largest range in the parameter space \((\nu ,t/s,U/s)\) in which the existence of ferromagnetism is provable. (Kensuke Tamura and Hosho Katsura, private communication. See also [57].)

  39. 39.

    In [74], general models obtained by the cell construction (see the end of Sect. 11.3.1) are also treated. The Proof of Lemma 6.1 in [74] (which corresponds to our Lemma 11.23) is highly technical. We still do not know any simplifications of the proof for models other than the simplest models that we are studying here.

  40. 40.

    The converse is, of course, not true. There are states of the form (11.4.61) that have \(S_\mathrm{tot}=n/2\).

  41. 41.

    One may check these relations by applying the left-hand and right-hand sides to relevant basis states and see that they yield the same results.

  42. 42.

    In fact it is also easy to write down the ground states explicitly. See [78].

  43. 43.

    Recall that our aim here is not to build realistic models of existing materials, but to understand fundamental and universal mechanism of various physical phenomena including metallic ferromagnetism.

  44. 44.

    In the ground state of Mielke’s flat-band model (Sect. 11.3.2), the empty (dispersive) second lowest band touches the lowest flat-band, which is filled by ferromagnetically coupled electrons. It is likely that the model exhibits electric conduction, but the situation is slightly different from standard metals. In what follows we only focus on metals which have a partially filled band.

  45. 45.

    Recall that we are not even able to prove the existence of ferromagnetic order in the three dimensional ferromagnetic Heisenberg model at low enough temperatures. See Sect. 4.4.4.

  46. 46.

    The flat-band models of Sect. 11.3 with electron numbers less than the half-filling were also studied in [35, 40, 67], and it was proved that the models still exhibit ferromagnetism in two or higher dimensions when the electron fillings are sufficiently large. We however believe that the models (as they are) are not relevant to metallic ferromagnetism since the electrons in the lowest flat bands do not contribute to conduction.

  47. 47.

    We are here relying on the standard criterion (which can be found in any textbook in condensed matter physics) that a non-interacting fermion system with a partially filled band describes a metallic state. Although we are dealing with a system of strongly interacting electrons, it behaves as a non-interacting system within the space of states with \(S_\mathrm{tot}=N/2\). There must be electric conduction, at least within the ferromagnetic sector.

  48. 48.

    When there are vacant sites there appears new contribution from the “second order” perturbation which involves three sites. See, e.g., Sect. 5.1 of [10], Appendix 2.A of [9], and Sect. 3.2 of [2]. We can neglect this term since we assumed that \(\tau ^2/U\) is negligible.

  49. 49.

    We have subtracted a trivial term \(2\tau \sum _{p=1}^L\hat{\tilde{n}}_p\) from the effective Hamiltonian.

  50. 50.

    For convenience, we here use a different sign convention for the hopping term.

  51. 51.

    A hop between sites 1 and L is exceptional, but it does not produce any sign if N is odd. When N is even, such a hop generates the “wrong” sign for the Perron–Frobenius theorem.

  52. 52.

    Instead of the Gutzwiller projection, we use the projection operator on states in which an up-spin electron and a down-spin electron never come to neighboring sites.

  53. 53.

    Recall that the ground states can be expressed as Slater determinant states as in (11.5.5).

  54. 54.

    This explanation should not be taken too literally since all electrons are exactly identical, and a ground state should be determined once and for all to minimize the total energy.

  55. 55.

    Of course, by continuity, one can show that the ground states are ferromagnetic for sufficiently large \(u_2\) and U for each L. But we do not have any estimates that are independent of L.

  56. 56.

    The biggest difference is that, in insulating systems, ferromagnetic ground states may be expressed in the k-space basis as in (11.5.5) or in the real space basis as in, e.g., (11.3.9). In a metallic system, one only has the former expression.

References

  1. R. Arita, Y. Suwa, K. Kuroki, H. Aoki, Gate-induced band ferromagnetism in an organic polymer. Phys. Rev. Lett. 88, 127202 (2002), arxiv:cond-mat/0108413

  2. A. Auerbach, Interacting Electrons and Quantum Magnetism (Springer, New York, 1994)

    Book  Google Scholar 

  3. A. Barbieri, J.A. Riera, A.P. Young, Stability of the saturated ferromagnetic state in the one-band Hubbard model. Phys. Rev. B 41, 11697 (1990)

    ADS  Google Scholar 

  4. E. Bobrow, K. Stubis, Y. Li, Exact results on itinerant ferromagnetism and the 15-puzzle problem. Phys. Rev. B 98, 180101 (2018), arxiv:1804.02347

  5. U. Brandt, A. Giesekus, Hubbard and anderson models on perovskitelike lattices: exactly solvable cases. Phys. Rev. Lett. 68, 2648 (1992)

    ADS  Google Scholar 

  6. O. Derzhko, J. Richter, Dispersion-driven ferromagnetism in a flat-band Hubbard system. Phys. Rev. B 90, 045152 (2014), arxiv:1404.2230v1

  7. B. Douçot, X.-G. Wen, Instability of the Nagaoka state with more than one hole. Phys. Rev. B 40, 2719 (1989)

    ADS  Google Scholar 

  8. F.J. Dyson, E.H. Lieb, B. Simon, Phase transitions in quantum spin systems with isotropic and nonisotropic interactions. J. Stat. Phys. 18, 335–382 (1978)

    Article  ADS  MathSciNet  Google Scholar 

  9. F.H.L. Essler, H. Frahm, F. Göhmann, A. Klümper, V.E. Korepin, The One-Dimensional Hubbard Model (Cambridge University Press, Cambridge, 2010)

    Google Scholar 

  10. P. Fazekas, Lecture Notes on Electron Correlation and Magnetism. Series in Modern Condensed Matter Physics, vol. 5 (World Scientific, Singapore, 1999)

    Google Scholar 

  11. M. Garnica, D. Stradi, S. Barja, F. Calleja, C. Diaz, M. Alcami, N. Martin, A.L.V. de Parga, F. Martin, R. Miranda, Long-range magnetic order in a purely organic 2D layer adsorbed on epitaxial graphene. Nat. Phys. 9, 368 (2013)

    Google Scholar 

  12. D. Ghosh, Nonexistence of magnetic ordering in the one- and two-dimensional Hubbard model. Phys. Rev. Lett. 27, 1584 (1971)

    ADS  Google Scholar 

  13. Z. Gulácsi, A. Kampf, D. Vollhardt, Exact many-electron ground states on the diamond Hubbard chain. Phys. Rev. Lett. 99, 026404 (2007), arxiv:cond-mat/0703818

  14. T. Hanisch, E. Müller-Hartmann, Ferromagnetism in the Hubbard model: instability of the Nagaoka state on the square lattice. Ann. Phys. 2, 381 (1993)

    Google Scholar 

  15. T. Hanisch, G.S. Uhrig, E. Müller-Hartmann, Lattice dependence of saturated ferromagnetism in the Hubbard model. Phys. Rev. B 56, 13960 (1997), arxiv:cond-mat/9707286

  16. W.J. Heisenberg, Zur theorie des ferromagnetismus. Z. Phys. 49, 619–636 (1928)

    ADS  MATH  Google Scholar 

  17. M. Ichimura, K. Kusakabe, S. Watanabe, T. Onogi, Flat-band ferromagnetism in extended \(\Delta \)-chain Hubbard models. Phys. Rev. B 58, 9595 (1998)

    ADS  Google Scholar 

  18. H. Katsura, I. Maruyama, How to construct tight-binding models with flat bands (in Japanese). Kotai Butsuri (Solid State Phys.) 50, 257–270 (2015)

    Google Scholar 

  19. H. Katsura, I. Maruyama, A. Tanaka, H. Tasaki, Ferromagnetism in the Hubbard model with topological/non-topological flat bands. Europhys Lett. 91, 57007 (2010), arxiv:0907.4564

  20. H. Katsura, A. Tanaka, Nagaoka states in the SU(\(n\)) Hubbard model. Phys. Rev. A 87, 013617 (2013), arxiv:1210.4774

  21. T. Koma, H. Tasaki, Decay of superconducting and magnetic correlations in one- and two-dimensional Hubbard models. Phys. Rev. Lett. 68, 3248–3251 (1992), arxiv:cond-mat/9709068

  22. K. Kubo, Note on the ground states of systems with the strong hund-coupling. J. Phys. Soc. Jpn. 51, 782–786 (1982)

    ADS  Google Scholar 

  23. K. Kusakabe, H. Aoki, Robustness of the ferromagnetism in flat bands. Phys. B 194–196, 215–216 (1994)

    ADS  Google Scholar 

  24. K. Kusakabe, H. Aoki, Ferromagnetic spin-wave theory in the multiband Hubbard model having a flat band. Phys. Rev. Lett. 72, 144 (1994)

    ADS  Google Scholar 

  25. K. Kusakabe, H. Aoki, Scaling properties of the ferromagnetic state in the Hubbard model. Phys. Rev. B 50, 12991 (1994), arxiv:cond-mat/9407036

  26. K. Kusakabe, M. Maruyama, Magnetic nanographite. Phys. Rev. B 67, 092406 (2003), arxiv:cond-mat/0212391

  27. S. Liang, H. Pang, Ferromagnetism in the infinite-\(U\) Hubbard model. Europhys. Lett. 32, 173 (1995)

    ADS  Google Scholar 

  28. H.-H. Lin, T. Hikihara, H.-T. Jeng, B.-L. Huang, C.-Y. Mou, X. Hu, Ferromagnetism in armchair graphene nanoribbons. Phys. Rev. B 79, 035405 (2009)

    ADS  Google Scholar 

  29. R.-J. Liu, W. Nie, W. Zhang, Flat-band ferromagnetism of the SU(N) Hubbard model on the Tasaki lattice (2019), arxiv:1901.07004

  30. L. Lu, The stability of ferromagnetism in the Hubbard model on two-dimensional line graphs. J. Phys. A 42, 265002 (2009)

    ADS  MathSciNet  MATH  Google Scholar 

  31. M. Maksymenko, O. Derzhko, J. Richter, Localized states on triangular traps and low-temperature properties of the antiferromagnetic Heisenberg and repulsive Hubbard models. Eur. Phys. J. B 84, 397–408 (2011), arxiv:1108.5453

  32. M. Maksymenko, A. Honecker, R. Moessner, J. Richter, O. Derzhko, Flat-band ferromagnetism as a Pauli-correlated percolation problem. Phys. Rev. Lett. 109, 096404 (2012), arxiv:1204.2110

  33. A. Mielke, Ferromagnetic ground states for the Hubbard model on line graphs. J. Phys. A 24, L73 (1991)

    ADS  MathSciNet  Google Scholar 

  34. A. Mielke, Ferromagnetism in the Hubbard model on line graphs and further considerations. J. Phys. A 24, 3311 (1991)

    ADS  MathSciNet  Google Scholar 

  35. A. Mielke, Exact ground states for the Hubbard model on the Kagome lattice. J. Phys. A 25, 4335 (1992)

    ADS  MathSciNet  Google Scholar 

  36. A. Mielke, Ferromagnetism in the Hubbard model and Hund’s rule. Phys. Lett. A 174, 443 (1993)

    ADS  MathSciNet  Google Scholar 

  37. A. Mielke, Ferromagnetism in single-band Hubbard models with a partially flat band. Phys. Rev. Lett. 82, 4312 (1999), arxiv:cond-mat/9904258

  38. A. Mielke, Stability of Ferromagnetism in Hubbard models with degenerate single-particle ground states. J. Phys. A: Mat. Gen. 32, 8411 (1999), arxiv:cond-mat/9910385

  39. A. Mielke, Properties of Hubbard models with degenerate localised single-particle eigenstates. Eur. Phys. J. B 85, 184 (2012), arxiv:abs/1202.3158

  40. A. Mielke, H. Tasaki, Ferromagnetism in the Hubbard model. Examples from models with degenerate single-electron ground states. Commun. Math. Phys. 158, 341–371 (1993), https://projecteuclid.org/euclid.cmp/1104254245

  41. T. Miyao, Stability of ferromagnetism in many-electron systems. J. Stat. Phys. 176, 1211–1271 (2019), arxiv:1712.05529

  42. Y. Nagaoka, Ferromagnetism in a narrow, almost half-filled s-band. Phys. Rev. 147, 392 (1966)

    ADS  Google Scholar 

  43. Y. Nagaoka, The15 puzzle (in Japanese). Kotai Butsuri (Solid State Phys.) 22, 756 (1987)

    Google Scholar 

  44. S. Nishino, M. Goda, Three-dimensional flat-band models. J. Phys. Soc. Jpn. 74, 393 (2005)

    ADS  Google Scholar 

  45. K. Penc, H. Shiba, F. Mila, T. Tsukagoshi, Ferromagnetism in multi-band Hubbard models: from weak to strong Coulomb repulsion. Phys. Rev. B 54, 4056 (1996), arxiv:cond-mat/9603042

  46. P. Pieri, S. Daul, D. Baeriswyl, M. Dzierzawa, P. Fazekas, Low density ferromagnetism in the Hubbard model. Phys. Rev. B 45, 9250 (1996), arxiv:cond-mat/9603163

  47. W.O. Putikka, M.U. Luchini, M. Ogata, Ferromagnetism in the two-dimensional \(t\)-\(J\) model. Phys. Rev. Lett. 69, 2288 (1992)

    ADS  Google Scholar 

  48. J.A. Riera, A.P. Young, Ferromagnetism in the one-band Hubbard model. Phys. Rev. B 40, 5285 (1989)

    ADS  Google Scholar 

  49. L.M. Roth, Spin wave stability of the ferromagnetic state for a narrow s-band. J. Phys. Chem. Solids 28, 1549 (1967)

    ADS  Google Scholar 

  50. H. Sakamoto, K. Kubo, Metallic ferromagnetism in a one-dimensional hubbard model; study using the density-matrix renormalization-group method. J. Phys. Soc. Jpn. 65, 3732–3735 (1996)

    ADS  Google Scholar 

  51. T. Sekizawa, An extension of Tasaki’s flat-band Hubbard model. J. Phys. A: Math. Gen. 42, 10451 (2003), arxiv:cond-mat/0304295

  52. B.S. Shastry, H.R. Krishnamurthy, P.W. Anderson, Instability of the Nagaoka ferromagnetic state of the \(U=\infty \) Hubbard model. Phys. Rev. B 41, 2375 (1990)

    ADS  Google Scholar 

  53. S.-Q. Shen, Ferromagnetic ground state in the Kondo lattice model. Eur. Phys. J. B 2, 11–16 (1998)

    ADS  Google Scholar 

  54. A. Sütő, Absence of highest-spin ground states in the Hubbard model. Commun. Math Phys. 140, 43–62 (1991), https://projecteuclid.org/euclid.cmp/1104247906

  55. Y. Suwa, R. Arita, K. Kuroki, H. Aoki, Flat-band ferromagnetism in organic polymers designed by a computer simulation. Phys. Rev. B 68, 174419 (2003), arxiv:cond-mat/0308154

  56. M. Takahashi, \(I=\infty \) Hubbard model on finite lattices. J. Phys. Soc. Jpn. 51, 3475 (1982)

    ADS  Google Scholar 

  57. K. Tamura, H. Katsura, Ferromagnetism in the SU(\(n\)) Hubbard model with a nearly flat band. Phys. Rev. B 100, 214423 (2019), arxiv:1908.06286

  58. A. Tanaka, Ferromagnetism in the Hubbard model with a gapless nearly-flat band. J. Stat. Phys. 170, 399–420 (2018), arxiv:1707.03944

  59. A. Tanaka, An Extension of the Cell-Construction Method for the Flat-Band Ferromagnetism, preprint (2020), arxiv:2004.07516

  60. A. Tanaka, T. Idogaki, Nonperturbative ferromagnetism in the Hubbard model for quarter-filling with one extra electron. J. Phys. Soc. Jpn. 67, 401–404 (1998)

    ADS  Google Scholar 

  61. A. Tanaka, T. Idogaki, Metallic ferromagnetism in the Hubbard model on three coupled chains. J. Phys. A: Math. Gen. 32, 4883–4895 (1999)

    ADS  MathSciNet  MATH  Google Scholar 

  62. A. Tanaka, T. Idogaki, Ferromagnetism in multi-band Hubbard models. Phys. A 297, 441–484 (2001)

    MATH  Google Scholar 

  63. A. Tanaka, H. Tasaki, Metallic ferromagnetism in the Hubbard model: a rigorous example. Phys. Rev. Lett. 98, 116402 (2007), arxiv:cond-mat/0611318

  64. A. Tanaka, H. Tasaki, Metallic ferromagnetism supported by a single band in a multi-band Hubbard model. J. Stat. Phys. 163, 1049–1068 (2016), arxiv:1601.04265

  65. A. Tanaka, H. Ueda, Stability of ferromagnetism in the Hubbard model on the Kagome lattice. Phys. Rev. Lett. 90, 067204 (2003), arxiv:cond-mat/0209423

  66. H. Tasaki, Extension of Nagaoka’s theorem on the large-\(U\) Hubbard model. Phys. Rev. B 40, 9192 (1989)

    ADS  Google Scholar 

  67. H. Tasaki, Ferromagnetism in the Hubbard models with degenerate single-electron ground states. Phys. Rev. Lett. 69, 1608 (1992)

    ADS  MathSciNet  MATH  Google Scholar 

  68. H. Tasaki, Exact resonating-valence-bond ground state and possibility of superconductivity in repulsive Hubbard models. Phys. Rev. Lett. 70, 3303 (1993)

    ADS  Google Scholar 

  69. H. Tasaki, Uniqueness of the ground state in exactly solvable Hubbard, periodic Anderson, and Emery models. Phys. Rev. B 49, 7763 (1994)

    ADS  Google Scholar 

  70. H. Tasaki, Stability of ferromagnetism in the Hubbard model. Phys. Rev. Lett. 73, 1158 (1994)

    ADS  Google Scholar 

  71. H. Tasaki, Stability of ferromagnetism in Hubbard models with nearly flat bands. J. Stat. Phys. 84, 535 (1996), arxiv:1901.02617

  72. H. Tasaki, Ferromagnetism in Hubbard models. Phys. Rev. Lett. 75, 4678 (1995), arxiv:cond-mat/9509063

  73. H. Tasaki, From Nagaoka’s ferromagnetism to flat-band ferromagnetism and beyond — An introduction to ferromagnetism in the Hubbard model. Prog. Theor. Phys. 99, 489–548 (1998), arxiv:cond-mat/9712219

  74. H. Tasaki, Ferromagnetism in the Hubbard model: a constructive approach. Commun. Math. Phys. 242, 445–472 (2003), arxiv:cond-mat/9712219

  75. D.J. Thouless, Exchange in solid \({}^3\)He and the Heisenberg Hamiltonian. Proc. Phys. Soc. Lond. 86, 893 (1965)

    ADS  Google Scholar 

  76. B. Tóth, Failure of saturated ferromagnetism for the Hubbard model with two holes. Lett. Math. Phys. 22, 321 (1991)

    ADS  MathSciNet  MATH  Google Scholar 

  77. H. Ueda, A. Tanaka, T. Idogaki, Ferromagnetism in the Hubbard model on a lattice of three coupled chains. J. Magnetism Mag Mater. 272–276, 950–951 (2004)

    ADS  Google Scholar 

  78. J.A. Vergés, F. Guinea, J. Galán, P.G.J. van Dongen, G. Chiappe, E. Louis, Ground state of the \(U=\infty \) Hubbard model with infinite-range hopping. Phys. Rev. B 49, 15400(R) (1994)

    ADS  Google Scholar 

  79. Y. Watanabe, S. Miyashita, New saturated ferromagnetism in the flat-band Hubbard model in high electron filling. J. Phys. Soc. Jpn. 66, 2123 (1997)

    ADS  Google Scholar 

  80. Y. Watanabe, S. Miyashita, Properties of new saturated ferromagnetism in the 1D flat-band Hubbard model as the spiral state. J. Phys. Soc. Jpn. 66, 3981 (1997)

    ADS  Google Scholar 

  81. R.M. Wilson, Graph puzzles, homotopy, and the alternating group. J. Combin. Theory, (B) 16, 86–96 (1974), https://www.sciencedirect.com/science/article/pii/0095895674900987

Download references

Author information

Authors and Affiliations

  1. Gakushuin University, Tokyo, Japan

    Hal Tasaki

Authors
  1. Hal Tasaki
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hal Tasaki .

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Tasaki, H. (2020). The Origin of Ferromagnetism. In: Physics and Mathematics of Quantum Many-Body Systems. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-41265-4_11

Download citation

Publish with us

Policies and ethics

Search

Navigation