Luận án Chuyển pha kim loại - điện môi từ trong một số hệ điện tử tương quan trao đổi kép

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[182] Jotzu G., Messer M., Desbuquois R. et al (2014), "Experimental realiza- tion of the topological Haldane model with ultracold fermions", Nature, 515, 237. 119 PHỤ LỤC A. Phân tích Hamiltonian trong mô hình Kane - Mele Đầu tiên, Chúng ta xét Hamiltonian dựa trên mô hình Kane - Mele với sự hiện diện của liên kết spin - quỹ đạo H =− t ∑ ⟨i,j⟩,σ c†iσcjσ + iλ ∑ ⟨⟨i,j⟩⟩,s,s′ νijc † isσ z ss′cjs′ (3.43) =H1 +H2. Chúng ta phân tích số hạng đầu tiên H1 = −t ∑ ⟨i,j⟩,σ c†iσcjσ (3.44) = −t ∑ i ∑ j⟨i⟩ ( C†i↑ C † i↓ )Cj↑ Cj↓  = −t ∑ i ∑ j⟨i⟩ (C†i↑.Cj↑ + C † i↓.Cj↓) = −t( ∑ i∈A + ∑ i∈B ) ∑ j⟨i⟩ (C†i↑.Cj↑ + C † i↓.Cj↓) = −t[ ∑ i∈A ∑ j⟨i⟩ (C†i↑.Cj↑ + C † i↓.Cj↓) + ∑ i∈B ∑ j⟨i⟩ (C†i↑.Cj↑ + C † i↓.Cj↓)] = −t[ ∑ i∈A ∑ j⟨i⟩ (a†i↑.bj↑ + a † i↓.bj↓) + ∑ i∈B ∑ j⟨i⟩ (b†i↑.aj↑ + b † i↓.aj↓)]. Thực hiện phép biến đổi Fourier ta có aiσ = 1√ N ∑ k e−ik.Riakσ, (3.45) 120 a†iσ = 1√ N ∑ k eik.Ria†kσ. (3.46) Chúng ta viết lại số hạng thứ nhất H1 = − t N [( ∑ i∈A ∑ j⟨i⟩ ( ∑ k eik.Ria†k↑ ∑ k′ e−ik ′.Rj bk′↑ + ∑ k eik.Ria†k↓ ∑ k′ e−ik ′.Rj bk′↓) (3.47) + ∑ i∈B ∑ j⟨i⟩ ( ∑ k′ eik ′.Rj b†k′↑ ∑ k e−ik.Riak↑ + ∑ k′ eik ′.Rj b†k′↓ ∑ k e−ik ′.Riak↓))], trong đó i, j là hai nút lân cận gần nhất nên ta có Rj = Ri + δi. Trong đó δi là các vector lân cận gần nhất của mạng tổ ong. H1 = − t N ∑ Ri ∑ kk′ ∑ δi [ei(k−k ′).Ri .e−ik ′.δi(a†k↑.bk′↑ + a † k↓.bk′↓) (3.48) + ei(k−k ′).Ri .eik ′.δi(b†k↑.ak′↑ + b † k↓.ak′↓)] = −t ∑ kk′ ∑ δi δ(k − k′).e−ik′.δi(a†k↑.bk′↑ + a†k↓.bk′↓) + δ(k − k′).eik′.δi(b†k↑.ak′↑ + b†k↓.ak′↓)] = −t ∑ k γk(a † k↑.bk↑ + a † k↓.bk↓)− t ∑ k γ∗k(b † k↑.ak↑ + b † k↓.ak↓), trong đó, γk = ∑ δi eik ′.δi là tổng theo tất cả các vector lân cận gần nhất. γ∗k là liên hợp phức của γk. Chúng ta phân tích số hạng thứ hai: H2 = iλ ∑ ⟨⟨i,j⟩⟩,s,s′ νijc † isσ z ss′cjs′ (3.49) = iλ ∑ i ∑ j⟨⟨i⟩⟩ νij ( C†i↑ C † i↓ )1 0 0 −1 Cj↑ Cj↓  = iλ( ∑ i∈A + ∑ i∈B ) ∑ j⟨⟨i⟩⟩ νij ( C†i↑ C † i↓ ) Cj↑ −Cj↓  = iλ( ∑ i∈A + ∑ i∈B ) ∑ j⟨⟨i⟩⟩ νij(C † i↑.Cj↑ − C†i↓.Cj↓) = iλ[ ∑ i∈A ∑ j⟨⟨i⟩⟩ νij(C † i↑.Cj↑ − C†i↓.Cj↓) + ∑ i∈B ∑ j⟨⟨i⟩⟩ νij(C † i↑.Cj↑ − C†i↓.Cj↓)] 121 = iλ[ ∑ i∈A ∑ j⟨⟨i⟩⟩ νij(a † i↑.bj↑ − a†i↓.bj↓) + ∑ i∈B ∑ j⟨⟨i⟩⟩ νij(b † i↑.aj↑ − b†i↓.aj↓)]. Thực hiện phép biến đổi Fourier ta được: H2 = iλ 1 N [ ∑ i∈A ∑ j⟨⟨i⟩⟩ νij( ∑ k .eik.Ria†k↑. ∑ k′ e−ik ′Rjak′↑ − ∑ k .eik.Ria†k↓. ∑ k′ e−ik ′Rjak′↓) (3.50) + ∑ i∈B ∑ j⟨⟨i⟩⟩ νij( ∑ k′ .eik ′.Rj b†k′↑. ∑ k .e−ik.Ri .bk↑ − ∑ k′ .e−ik ′.Rj .b†k′↓. ∑ k .e−ik.Ribk↓)]. Vì i, j là hai nút lân cận gần nhất thứ hai nên ta có: Rj = Ri + ηi. Trong đó γi là các vector lân cận gần nhất thứ hai của mạng tổ ong. H2 = iλ. 1 N [ ∑ Ri ∑ kk′ ∑ ηi [νije i(k−k′).Ri .e−ik ′.ηi .(a†k↑.ak′↑ − a†k↓.ak′↓ + b†k↑.bk′↑ − b†k↓.bk′↓) (3.51) = iλ.[ ∑ kk′ ∑ ηi νijδ(k − k′).e−ik′.ηi .(a†k↑.ak′↑ − a†k↓.ak′↓ + b†k↑.bk′↑ − b†k↓.bk′↓)] = iλ.[ ∑ k ∑ ηi νij .e −ik.ηi .(a†k↑.ak↑ − a†k↓.ak↓ + b†k↑.bk↑ − b†k↓.bk↓)] = λ[ ∑ k ξk.(a † k↑.ak↑ − a†k↓.ak↓ + b†k↑.bk↑ − b†k↓.bk↓)], trong đó ξk = i ∑ ηi νij .e −ik.ηi được lấy tổng theo các vector lân cận gần nhất thứ hai. H = ∑ k ψ†kH0(k)ψk (3.52) = ( a†k↑ b † k↑ a † k↓ b † k↓ )  H11 H12 H13 H14 H21 H22 H23 H24 H31 H32 H33 H34 H41 H42 H43 H44   ak↑ bk↑ ak↓ bk↓  = H11a † k↑ak↑ +H12a † k↑bk↑ +H13a † k↑ak↓ +H14a † k↑bk↓ +H21ak↑b † k↑ +H22b † k↑bk↑ +H23ak↓b † k↑ +H24b † k↑bk↓ 122 +H31a † k↓ak↑ +H32a † k↓bk↑ +H33a † k↓ak↓ +H34a † k↓bk↓ +H41ak↑b † k↓ +H42b † k↓bk↑ +H43ak↓b † k↓ +H44b † k↓bk↓. Ta tìm được H0(k) =  λξk −tγk 0 0 −tγ∗k λξk 0 0 0 0 −λξk −tγk 0 0 −tγ∗k λξk  (3.53) Chúng ta xét thêm số hạng mô tả thế ion có dạng H4 = −∆ 2 ∑ iσ ϵiC † iσCiσ (3.54) = −∆ 2 ∑ i ϵi ( C†i↑ Ci↓ )Ci↑ Ci↓  = −∆ 2 ( ∑ i∈A + ∑ i∈B )ϵk(C † i↑Ci↑ + C † i↓Ci↓) = −∆ 2 ∑ i∈A ϵk(C † i↑Ci↑ + C † i↓Ci↓) + ∑ i∈B ϵk(C † i↑Ci↑ + C † i↓Ci↓) = −∆ 2 ∑ i (a†i↑ai↑ + a † i↓ai↓ − b†i↑bi↑ − b†i↓bi↓). Thực hiện phép biến đổi Fourier ta được H4 = −∆ 2 . 1 N ∑ i ( ∑ k eik.Ri .a†k↑. ∑ k′ e−ik ′.Riak′↑ + ∑ k eik.Ri .a†k↓. ∑ k′ e−ik ′.Riak′↓ (3.55) − ∑ k eik.Ri .b†k↑. ∑ k′ e−ik ′.Ribk′↑ − ∑ k eik.Ri .b†k↓. ∑ k′ e−ik ′.Ribk′↓ = −∆ 2 . 1 N ∑ i ∑ k,k′ ei(k−k ′).Ri .(a†k↑.ak′↑ + a † k↓.ak′↓ − b†k↑.bk′↑ − b†k↓.bk′↓) = −∆ 2 . ∑ k,k′ δ(k − k′).(a†k↑.ak′↑ + a†k↓.ak′↓ − b†k↑.bk′↑ − b†k↓.bk′↓) = −∆ 2 . ∑ k (a†k↑.ak↑ + a † k↓.ak↓ − b†k↑.bk↑ − b†k↓.bk↓). 123 Do đó: H0(k) =  λξk −∆/2 −tγk 0 0 −tγ∗k λξk −∆/2 0 0 0 0 −λξk +∆/2 −tγk 0 0 −tγ∗k −λξk +∆/2  . (3.56) B. Phân tích Hamiltonian trao đổi trong mô hình trao đổi kép Hamiltonian của mô hình có dạng Hex = J ∑ i,ss′ Sic † isσss′cis′ . (3.57) Spin tạp cổ điển có thể được biểu thị qua góc phương vị φi và góc cực θi Sxi = S sinφi cos θi, S x i = S sinφi sin θi, S z i = S cos θi. Tích vô hướng Si.σ có thể được viết lại dưới dạng ma trận vuông cấp hai được xác định trong không gian spin của điện tử Si.σ = S x i σ x + Syi σ y + Szi σ z, (3.58) trong đó các ma trận Pauli được xác định như sau σx = 0 1 1 0  , σy = 0 −i i 0  , σz = 1 0 0 −1  . Chúng ta thu được Si.σ = S x i 0 1 1 0 + Syi 0 −i i 0 + Szi 1 0 0 −1  . Si.σ = S  cos θi cosφi sin θi − i sinφi sin θi cosφi sin θi + i sinφi sin θi −cosθi  =  Szi Sxi − iSyi Sxi + iS y i −Szi  . 124 Trị riêng của ma trận trên thỏa mãn hệ thức det  S cos θi − λ S cosφi sin θi − iS sinφi sin θi S cosφi sin θi + iS sinφi sin θi −S cos θi − λ  = 0. Giải phương trình trị riêng chúng ta thu được hai trị riêng: λ1,2 = ±S. Từ đó, chúng ta thu được hai vector riêng ứng với hai trị riêng λ1,2 có dạng e1 =  cos θi2 sin θi2 .e iφi  và e2 = − sin θi2 .e−iφi cos θi2  . Từ đó chúng ta có thể xây dựng ma trận Ui =  cos θi2 − sin θi2 .e−iφi sin θi2 .e iφi cos θi2  . (3.59) Chúng ta xác định ma trận U†i là ma trận liên hợp Hermit của ma trận Ui(U † i Ui = I), trong đó I là ma trận đơn vị. U†i =  cos θi2 sin θi2 .e−iφi − sin θi2 .eiφi cos θi2  . (3.60) Hamiltonian trao đổi kép có thể chéo hóa bằng cách cho các ma trận U†i và Ui tác động bên trái và bên phải nó tương ứng.fi↑ fi↓  = U†i ci↑ ci↓  =  cos θi2 sin θi2 .e−iφi − sin θi2 .eiφi cos θi2 ci↑ ci↓  (3.61) =  cos θi2 ci↑ + sin θi2 .e−iφici↓ − sin θi2 .eiφici↑ + cos θi2 ci↓  . ( f†i↑ f † i↓ ) = ( c†i↑ c † i↓ ) Ui = ( c†i↑ c † i↓ ) cos θi2 − sin θi2 .e−iφi sin θi2 .e iφi cos θi2  (3.62) = ( c†i↑ cos θi 2 + c † i↓ sin θi 2 .e iφi −c†i↑ sin θi2 .e−iφi + c†i↓ cos θi2 ) . Cuối cùng, chúng ta thu được số hạng Hamiltonian trao đổi Hex có dạng chéo H0 = −JS ∑ i,σ σf†iσfiσ, (3.63) trong đó σ = ±1. Số hạng trao đổi spin là nguyên nhân gây ra từ hóa tự phát. 125