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
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.
Keywo word rds: Ferromagnetism, Hubbard model, The Nagaoka-Thouless theroem
Photo：Gakken Kids’ net
Plan of the talk
✓ Defin inition ition of the Hubbar ard d model ✓ Pro roblem m to be be so solved ✓ Th The Nagaoka
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
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
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
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
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
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
✓Lieb’s theorem（Lieb’s ferrimagnetism）
✓ The flat-band ferromagnetism
Rigorous results
Key results
✓The Nagaoka-Thouless theorem
✓Lieb’s theorem（Lieb’s ferrimagnetism）
✓ The flat-band ferromagnetism
Today’s talk
The Nagaoka-Thouless theorem
The Nagaoka-Thouless theorem
The Nagaoka-Thouless theorem
D.J. Thouless, Proc. Phys. Soc. Lond. 86, 893–904 (1965)
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
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
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
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．