Extensions of the Nagaoka- Thouless Theorem Venice ce 2019 9 - - PowerPoint PPT Presentation

Extensions of the Nagaoka- Thouless Theorem Venice ce 2019 9 - Quantis tissima sima in the Sereniss ssim ima a II III Palazzo azzo Pesaro ro-Pa Papafav afava, a, Venice ce, 19 August-23 August 2019 Tada dahir hiro Miyao Hokkai aido do Un Univer versi sity ty

Abstract I will report several extensions of the Nagaoka- Thouless theorem. Photo ： Gakken Kids’ net Keywo word rds: Ferromagnetism, Hubbard model, The Nagaoka-Thouless theroem

Plan of the talk ✓ Defin inition ition of the Hubbar ard d model ✓ Pro roblem m to be be so solved ✓ Th The Nagaoka oka-Thoule Thouless theore rem m ✓ Stabil ilitie ities of the N-T T theore rem ✓ Th Therm rmal al stabil iliti ities es Keneth, Neodymium magnet

Plan of the talk ✓ Defin inition ition of the Hubbar ard d model ✓ Pro roblem m to be be so solved ✓ Th The Nagaoka oka-Thoule Thouless theore rem m ✓ Stabil ilitie ities of the N-T T theore rem ✓ Th Therm rmal al stabil iliti ities es Keneth, Neodymium magnet

Rigorous study of ferromagnetism ✓ Metallic magnets （ E.g. ： Fe ） ✓ The magnet has been a well-known material since the age of ancient Greek ✓ The origin gin of ferrom omag agne netis tism remains ns a mystery. ery.

Defintion of the Hubbard model

Defintion of the Hubbard model

Defintion of the Hubbard model ✓ Parameters:

Defintion of the Hubbard model ✓ The simplest model describing the strongly correlated electron system. Hubbard said “Oversimplified model” ✓ Nevertheless the model can explain various phenomena: ferromagnetism, anti-ferromagnetism, charge density wave, Mott transition, Bond order, Tomonaga-Luttinger liquid,……

Properties of the Hubbard model

Properties of the Hubbard model We expect that an interplay of four basic properties induces ferromagnetism ． Very difficult to clarify the interplay ・・・ Mathematically, the difficulty comes from the fact that we cannot use the naive perturbation theory.

Plan of the talk ✓ Defin inition ition of the Hubbar ard d model ✓ Pro roblem m to be be so solved ✓ Th The Nagaoka oka-Thoule Thouless theore rem m ✓ Stabil ilitie ities of the N-T T theore rem ✓ Th Therm rmal al stabil iliti ities es Keneth, Neodymium magnet

Spin operators

Spin operators

Problem to be solved ✓ By controlling the parameters, we will try to show the above goal. ✓ We need several estimates of the parameters which are uniform in |Λ|．

Plan of the talk ✓ Defin inition ition of the Hubbar ard d model ✓ Pro roblem m to be be so solved ✓ Th The Nagaoka oka-Thoule Thouless theore rem m ✓ Stabil ilitie ities of the N-T T theore rem ✓ Th Therm rmal al stabil iliti ities es Keneth, Neodymium magnet

Rigorous results Key results ✓ The Nagaoka-Thouless theorem Y. Nagaoka, Solid State Commun. 3 (1965), 409-412. D. J. Thouless, Proc. Phys. Soc. London 86 (1965), 893-904. H. Katsura, A. Tanaka, Phys. Rev. A 87 (2013), 013617. ✓ Lieb ’s theorem （Lieb’s ferrimagnetism ） E. H. Lieb, Phys. Rev. Lett. 62 (1989), 1201-1204. ✓ The flat-band ferromagnetism A. Mielke, J. Phys. A: Math. Gen. 24 (1991), 3311-3321. H. Tasaki, Comm. math. phys. 242 (2003), 445-472. A. Tanaka, Jour. Stat.Phys. 170 (2018). 399-420.

Rigorous results Today’s talk Key results ✓ The Nagaoka-Thouless theorem Y. Nagaoka, Solid State Commun. 3 (1965), 409-412. D. J. Thouless, Proc. Phys. Soc. London 86 (1965), 893-904. H. Katsura, A. Tanaka, Phys. Rev. A 87 (2013), 013617. ✓ Lieb ’s theorem （Lieb’s ferrimagnetism ） E. H. Lieb, Phys. Rev. Lett. 62 (1989), 1201-1204. ✓ The flat-band ferromagnetism A. Mielke, J. Phys. A: Math. Gen. 24 (1991), 3311-3321. H. Tasaki, Comm. math. phys. 242 (2003), 445-472. A. Tanaka, Jour. Stat.Phys. 170 (2018). 399-420.

The Nagaoka-Thouless theorem

The Nagaoka-Thouless theorem

The Nagaoka-Thouless theorem Y. Nagaoka, Solid State Commun. 3, 409 – 412 (1965) D.J. Thouless, Proc. Phys. Soc. Lond. 86, 893 – 904 (1965) H. Tasaki, Phys. Rev. B 40, 9192 – 9193 (1989)

The Nagaoka-Thouless theorem

Plan of the talk ✓ Defin inition ition of the Hubbar ard d model ✓ Pro roblem m to be be so solved ✓ Th The Nagaoka oka-Thoule Thouless theore rem m ✓ Stabil ilitie ities of the N-T T theore rem ✓ Th Therm rmal al stabil iliti ities es Keneth, Neodymium magnet

Stability Motivation: ✓ The electrons in actual metals are always perturbed by environment, e.g., phonons, electromagnetic fields and so on. ✓ Ferromagnetism is experimentally observed in various metals and has a wide range of uses in daily life. ✓ It appears that ferromagnetism is stable against the perturbations originating from the environment. ✓ Can we prove the stability?

Stability of the Nagaoka-Thouless theorem I

Stability of the Nagaoka-Thouless theorem I

Stability of the Nagaoka-Thouless theorem I

Stability of the Nagaoka-Thouless theorem II A. Giuliani, V. Mastropietro, M. Porta, Ann. Phys. 327, 461 – 511 (2012)

Stability of the Nagaoka-Thouless theorem II

Stability of the Nagaoka-Thouless theorem II

Plan of the talk ✓ Defin inition ition of the Hubbar ard d model ✓ Pro roblem m to be be so solved ✓ Th The Nagaoka oka-Thoule Thouless theore rem m ✓ Stabil ilitie ities of the N-T T theore rem ✓ Th Therm rmal al stabil iliti ities es Keneth, Neodymium magnet

Thermal stabilities

Thermal stabilities

Thermal stabilities

Thermal stabilities

Thermal stabilities

Summary ✓ The Nagaoka-Thouless ferromagnets are stable against the electron-phonon and the electron- photon interactions. ✓ The Nagaoka-Thouless theorem and its generalizations can be extended to the positive temperature. ✓ The origin of ferromagnetism has been still mystery ．

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