Background Stability of Lieb’s ferrimagnetism Stability of Nagaoka’s ferromagnetism Stability of magnetism in the Hubbard model Quantissima II in Venice Aug. 21–25, 2017 Tadahiro Miyao Dept. Math. Hokkaido Univ. 1 / 32
Background Stability of Lieb’s ferrimagnetism Stability of Nagaoka’s ferromagnetism Abstract I will report the stabilities of the Nagaoka theorem and Lieb theorem in the Hubbard model, even if the influence of phonons and photons is taken into account. Picture: Gakken kidsnet 2 / 32
Background Stability of Lieb’s ferrimagnetism Stability of Nagaoka’s ferromagnetism Table of Contents I 1. Background I 2. Stability of Lieb’s ferrimagnetism I 3. Stability of Nagaoka’s ferromagnetism 3 / 32
Background Stability of Lieb’s ferrimagnetism Stability of Nagaoka’s ferromagnetism Background 4 / 32
Background Stability of Lieb’s ferrimagnetism Stability of Nagaoka’s ferromagnetism A brief history I Magnets have a long history, e.g., chinese writing dating back to 4000 B.C. mention magnetite, ancient greeks knew magnetite, etc. I The origin of ferromagnetism in material has been a mystery. I Modern approach was initiated by Kanamori, Gutzwiller and Hubbard. They studied a simple tight-binding model, called the Hubbard model . I Nagaoka’s ferromagnetism (1965): A first rigorous result about ferromagnetism in the Hubbard model. (Cf. D. J. Thouless, 1965) I Lieb’s ferrimagnetism (1989): A rigorous example of ferrimagnetism in the Hubbard model. I Mielke, Tasaki’s ferromagnetism (1991–): Construction of flat-band ferromagnetism 5 / 32
Background Stability of Lieb’s ferrimagnetism Stability of Nagaoka’s ferromagnetism Motivation I Electrons always interact with phonons (or photons) in actual metals. I On the other hand, ferromagnetism is experimentally observed in various metals and has a wide range of uses in daily life. ✓ ✏ Motivation If Nagaoka’s and Lieb’s theorems contain an essence of real ferromagnetism, their theorems should be stable under the in- fluence of the electron-phonon(or electron-photon) interaction. ✒ ✑ 6 / 32
Background Stability of Lieb’s ferrimagnetism Stability of Nagaoka’s ferromagnetism The Hubbard model ✓ ✏ The Hubbard model on Λ : U xy � � t xy c ∗ � H H = xσ c yσ + 2 ( n x − 1)( n y − 1) x,y ∈ Λ σ = ↑ , ↓ x,y ∈ Λ ✒ ✑ I Λ : finite lattice I c xσ : the electron annihilation operator at site x ; { c xσ , c ∗ yσ ′ } = δ xy δ σσ ′ . I n x : the electron number operator at site x ∈ Λ given by σ = ↑ , ↓ n xσ , n xσ = c ∗ n x = � xσ c xσ . I t xy : the hopping matrix. I U xy : the energy of the Coulomb interaction. 7 / 32
Background Stability of Lieb’s ferrimagnetism Stability of Nagaoka’s ferromagnetism I { t xy } and { U xy } are real symmetric | Λ | × | Λ | matrices. I N -electron Hilbert space: N � ( ℓ 2 (Λ) ⊕ ℓ 2 (Λ)) . E N = � n � ℓ 2 (Λ) ⊕ ℓ 2 (Λ) � indicates the n -fold antisymmetric tensor product of ℓ 2 (Λ) ⊕ ℓ 2 (Λ) . 8 / 32
Background Stability of Lieb’s ferrimagnetism Stability of Nagaoka’s ferromagnetism The Holstein-Hubbard model ✓ ✏ The Holstein-Hubbard model on Λ � g xy n x ( b ∗ � ωb ∗ H HH = H H + y + b y ) + x b x x,y ∈ Λ x ∈ Λ ✒ ✑ I H H is the Hubbard Hamiltonian. I b ∗ x and b x are phonon creation- and annihilation operators at site x ∈ Λ , respectively: [ b x , b ∗ y ] = δ xy , [ b x , b y ] = 0 . I g xy is the strength of the electron-phonon interaction. We assume that { g xy } is a real symmetric matrix. I The phonons are assumed to be dispersionless with energy ω > 0 . 9 / 32
Background Stability of Lieb’s ferrimagnetism Stability of Nagaoka’s ferromagnetism I Hilbert space E N ⊗ P , P = � ∞ n =0 ⊗ s ℓ 2 (Λ) , the bosonic Fock space over ℓ 2 (Λ) ; ⊗ n s indicates the n -fold symmetric tensor product. I H HH is self-adjoint on dom( N b ) and bounded from below, x ∈ Λ b ∗ where N b = � x b x . 10 / 32
Background Stability of Lieb’s ferrimagnetism Stability of Nagaoka’s ferromagnetism A many-electron system coupled to the quantized radiation field I We suppose that Λ is embedded into the region V = [ − L/ 2 , L/ 2] 3 ⊂ R 3 with L > 0 . ✓ ✏ Hamiltonian � � � � � c ∗ H rad = t xy exp i dr · A ( r ) xσ c yσ C xy x,y ∈ Λ σ = ↑ , ↓ U xy � + 2 ( n x − 1)( n y − 1) x,y ∈ Λ � � ω ( k ) a ( k, λ ) ∗ a ( k, λ ) . + k ∈ V ∗ λ =1 , 2 ✒ ✑ 11 / 32
Background Stability of Lieb’s ferrimagnetism Stability of Nagaoka’s ferromagnetism I Hilbert space E N ⊗ R , where R is the bosonic Fock space over ℓ 2 ( V ∗ × { 1 , 2 } ) with V ∗ = ( 2 π L Z ) 3 . I a ( k, λ ) ∗ and a ( k, λ ) are photon creation- and annihilation operators, respectively: [ a ( k, λ ) , a ( k ′ , λ ′ ) ∗ ] = δ λλ ′ δ kk ′ , [ a ( k, λ ) , a ( k ′ , λ ′ )] = 0 . I A ( r ) ( r ∈ V ) is the quantized vector potential given by A ( r ) χ κ ( k ) � e ik · r a ( k, λ ) + e − ik · r a ( k, λ ) ∗ � = | V | − 1 / 2 � � ε ( k, λ ) . � 2 ω ( k ) k ∈ V ∗ λ =1 , 2 12 / 32
Background Stability of Lieb’s ferrimagnetism Stability of Nagaoka’s ferromagnetism I χ κ is the indicator function of the ball of radius 0 < κ < ∞ , where κ is the ultraviolet cutoff. I The dispersion relation: ω ( k ) = | k | for k ∈ V ∗ \{ 0 } , ω (0) = m 0 with 0 < m 0 < ∞ . I C xy is a piecewise smooth curve from x to y . I For concreteness, the polarization vectors are chosen as ε ( k, 1) = ( k 2 , − k 1 , 0) ε ( k, 2) = k , | k | ∧ ε ( k, 1) . � k 2 1 + k 2 2 (To avoid ambiguity, we set ε ( k, λ ) = 0 if k 1 = k 2 = 0 . ) I H rad is essentially self-adjoint and bounded from below. We denote its closure by the same symbol. I This model was introduced by Giuliani et al. in [GMP]. 13 / 32
Background Stability of Lieb’s ferrimagnetism Stability of Nagaoka’s ferromagnetism Stability of Lieb’s ferrimagnetism 14 / 32
Background Stability of Lieb’s ferrimagnetism Stability of Nagaoka’s ferromagnetism Basic definitions Definition 3.1 Let Λ be a finite lattice. Let { M xy } be a real symmetric | Λ | × | Λ | matrix. (i) We say that Λ is connected by { M xy } , if, for every x, y ∈ Λ, there are x 1 , . . . , x n ∈ Λ such that M xx 1 M x 1 x 2 · · · M x n y � = 0 . (ii) We say that Λ is bipartite in terms of { M xy } , if Λ can be divided into two disjoint sets A and B such that M xy = 0 whenver x, y ∈ A or x, y ∈ B . ♦ 15 / 32
Background Stability of Lieb’s ferrimagnetism Stability of Nagaoka’s ferromagnetism Lieb’s ferrimagnetism I Since we are interested in the half-filled system, we will study the Hamiltonian ˜ H H = H H ↾ E N = | Λ | . I Let S (+) x ↑ c x ↓ and let S ( − ) S (+) � ∗ . The spin = c ∗ � = x x x operators are defined by S (3) = 1 S (+) = S ( − ) = � � � S (+) S ( − ) ( n x ↑ − n x ↓ ) , , . x x 2 x ∈ Λ x ∈ Λ x ∈ Λ I The total spin operator is defined by tot = ( S (3) ) 2 + 1 2 S (+) S ( − ) + 1 S 2 2 S ( − ) S (+) with eigenvalues S ( S + 1) . 16 / 32
Background Stability of Lieb’s ferrimagnetism Stability of Nagaoka’s ferromagnetism Definition 3.2 If ϕ is an eigenvector of S 2 tot with S 2 tot ϕ = S ( S + 1) ϕ , then we say that ϕ has total spin S . Assumptions: (B. 1) Λ is connected by { t xy } ; (B. 2) Λ is bipartite in terms of { t xy } ; (B. 3) { U xy } is positive definite. Theorem 3.3 (Lieb’s ferrimagnetism) Assume that | Λ | is even. Assume (B. 1) , (B. 2) and (B. 3) . The ground state of ˜ H H has total spin S = 1 � and is unique � � � | A | − | B | 2 apart from the trivial (2 S + 1) -degeneracy. 17 / 32
Background Stability of Lieb’s ferrimagnetism Stability of Nagaoka’s ferromagnetism Corollary 3.4 � = c | Λ | , then the ground state of ˜ � � � | A | − | B | H H exhibits If ferrimagnetism. Example: copper oxide lattice Picture: W.Tsai et.al., New Jour. Phys. 17, 2015. 18 / 32
Background Stability of Lieb’s ferrimagnetism Stability of Nagaoka’s ferromagnetism Stability of Lieb’s theorem I I We will study the half-filled case: ˜ H HH = H HH ↾ E N = | Λ | ⊗ P . I We continue to assume (B. 1) and (B. 2) . I As to the electron-phonon interaction, we assume the following: (B. 4) � x ∈ Λ g xy is a constant independent of y ∈ Λ . 19 / 32
Background Stability of Lieb’s ferrimagnetism Stability of Nagaoka’s ferromagnetism I The effective Coulomb interaction is defined by U eff ,xy = U xy − 2 � g xz g yz . ω z ∈ Λ (B. 5) { U eff ,xy } is positive definite. Theorem 3.5 (T.M., 2017) Assume that | Λ | is even. Assume (B. 1) , (B. 2) , (B. 4) and (B. 5) . Then the ground state of ˜ H HH has total spin S = 1 � � � | A | − | B | � 2 and is unique apart from the trivial (2 S + 1) -degeneracy. 20 / 32
Recommend
More recommend
