Phys 711 Topics in Particles & Fields | Spring 2013 | Lecture 8 | v0.1.2 Lecture notes: Fermi-Hubbard model Jeffrey Yepez Department of Physics and Astronomy University of Hawai`i at Manoa Watanabe Hall, 2505 Correa Road Honolulu, Hawai`i 96822 E-mail: [email protected] www.phys.hawaii.edu/∼yepez (Dated: March 7, 2013) Contents I. CRYSTALLOGRAPHIC POTENTIAL I. Crystallographic potential 1 Consider a nonrelativistic particle in a crystallographic A. Bloch waves and quasimomentum 1 periodic potential with period a. For simplicity, we will B. Sine squared potential 2 C. Localized Wannier functions 2 treat this problem in 1+1 dimensions. Let us begin by writing down the single-particle Hamiltonian II. One-band Hubbard Hamiltonian 3 2 A. Method of solution 3 H = − ~ @ + V (x); (1) 1. Block diagonalization 3 2m xx crys. 2. Site interchange operator 5 2 3. Implementation of the S operator 6 where Vcrys.(x) = Vcrys.(x + a). B. Exact solutions 6 1. Double-empty symmetry above and below half filling 7 A. Bloch waves and quasimomentum 2. Check of Nagaoka's theorem 7 3. Quantum informational derivation of the ground state 9 The eigenstates of (1) may be written in the form 4. Triangular cluster above half filling 9 5. Predicted energy eigenstates for the triangular (n) ikx (n) φk (x) = e uk (x); (2) cluster 9 (n) (n) III. Acknowledgements 11 where uk (x) = uk (x + a) is a periodic function known as the Bloch wave for the crystal and where k is a real- A. Triangular cluster solutions 11 valued wave number. Since 1 1. half filling: Ne = 3 and Sz = 2 11 a. A State with S = 1 13 (n) (2) ikx 2 (n) 1 2 − @xxφ (x) = e −@xx − 2ik@x + k u (x) (3) 1 3 k k b. A2 States with S = 2 ; 2 13 1 c. E States with S = 2 13 and since 2. Above half filling: Ne = 4 and Sz = 0 13 2 2 a. A1 States with S = 0 13 (−i@x + k) = −@xx − 2ik@x + k (4) b. A2 State with S = 1 13 c. E States with S = 0; 1 14 the energy eigenequation 3. Above half filling: Ne = 4 and Sz = 1 14 (n) (n) (n) a. A2 State with S = 1 14 Hφ (x) = E φ (x) (5) b. E state with S = 1 14 k k k 4. Below half filling: Ne = 2 and Sz = 0 14 reduces to an equation the Bloch waves a. A1 states with S = 0 14 b. A2 State with S = 1 15 2 ! c. E States with S = 0; 1 15 (−i~@x + ~k) (n) (n) (n) + Vcrys.(x) uk (x) = Ek uk (x); 5. Below half filling: Ne = 2 and Sz = 1 15 2m a. A2 State with S = 1 15 (6) b. E State with S = 1 15 where ~k is called the quasimomentum of the crystallo- B. Square cluster solutions 15 graphic potential. The wave number range k 2 (π=2; π=a] 1. Half filling: Ne = 4 and Sz = 0 16 defines the primitive cell in reciprocal space1, also known a. A1 States with S = 0; 1 16 b. A2 States with S = 0; 1 16 c. B1 States with S = 0; 1 17 d. B2 States with S = 0; 1; 2 17 e. E States with S = 0; 1 17 1 The wave number k-space (the Fourier transform of position x- space) is called reciprocal space in the condensed matter litera- References 18 ture. A Bloch waves and quasimomentum I CRYSTALLOGRAPHIC POTENTIAL as the first Brillouin zone. The eigenvalue problem (6) has a family of solutions (a discrete energy spectrum) la- 2.0 belled by the quantum number n for each wave number (n) 1.5 k. The Bloch wave solutions uk (x) represents the band (n) 2 f structure of the crystal with energy bands Ek . È 1.0 0.5 B. Sine squared potential 0.0 Let us consider a scalar quantum particle in a sine 0 2 4 6 8 squared potential x 2 Vcrys.(x) = V◦ sin (kx): (7) FIG. 1 A Mathieu function for a = 0:4 and q = −0:563078 plotted over the range x 2 [0; 3π]. Both the sine squared Then (1) may be written as potential (red) and the periodic Block wave (blue) are shown. 2 1 − cos 2kx − ~ @ φ(n) +V φ(n) = E(n)φ(n); (8a) 2m xx k ◦ 2 k k k 8 8 which after rescaling x 7! kx ≡ w becomes a differential 6 6 R R E E (n) L L n 4 n 4 H H equation in w for the function y(w) ≡ φk (kx) k k E E 2 2 2k2 1 − cos 2w − ~ @ y + V y = E(n)y: (8b) 0 0 2m ww ◦ 2 k -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 k ap k ap 2 2 (a)No gaps for q = 0 (b)1st gap appears at q = − 1 With the atomic recoil energy ER ≡ ~ k =(2m), the 8 eigenequation may be written as Mathieu's differential (V◦=ER = 0) (V◦=ER = 1=2) equation 8 8 6 @ y + [a − 2q cos(2w)] y = 0; (8c) 6 R ww R E E L L 4 n n H k H 4 k E with constant coefficients E 2 2 0 E(n) − V =2 0 a ≡ k ◦ (9a) -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 ER k ap k ap (c)2nd gap appears at q = − 1 (d)Gaps widen at q = −2 V◦ 2 q ≡ − : (9b) (V◦=ER = 2) (V◦=ER = 8) 4ER According to Floquet's theorem, any Mathieu function FIG. 2 Band structure for period potential Vcrys.(x) = 2 (i.e. solution to (8c)) can be written in the form V◦ sin (kx) obtained using Mathieu exponent solutions for q = 0; 1=8; 1=2; 2. First (blue), second (red), and third (green) y(w) = eirwf(w); (10) energy bands for the sine squared potential. The energy bands are shown using the reduced zone scheme. where f(w) has period 2π and the integer-valued param- eter r is known as the Mathieu exponent. This is the mathematician's version of the physicist's Block wave so- lution with quasimomentum rk discussed above. are delocalized states. If one is interested in localized An example Bloch wave solution is shown in Fig. 1. orthogonal states2, then these can be constructed as a The band structure of the quantum particle is shown in superposition of Bloch waves Fig. 2 for the lowest three energy eigenstates. Band gaps appear and then widen as the V◦ increases. For very large 1 N−1 V◦ the bands become flat (i.e. the band width of each X −ikν xi (n) wn(x − xi) = e φ (x); (11) energy band approaches zero and the forbidden energy 2N kν regions become maximally large). ν=−N C. Localized Wannier functions 2 A localized state is one where the quantum particle's wave func- tion is narrow and primarily positioned near the minimum of one The advantage of the Bloch wave solutions is that they well in the periodic potential. The quantum particle's average represent orthogonal energy eigenstates. However, they position is centered at the well's minimum. 2 II ONE-BAND HUBBARD HAMILTONIAN ^ for kν = πν=(Na) with a crystal cellsize a. Such localized where (r) is the field operator at the space point r. states are Wannier functions (Wannier, 1937) when we The last two terms of (15) are obtained from (16) if one make the assumption that the Bloch wave is independent substitutes for the field operator a linear combination of wave number. Hence, writing the energy eigenstate of Wannier states, denoted φiσ(r), as follows, (r) = (n) ik x (0) P as φ (x) = e ν u (x), then we express the Wannier iσ aiσφiσ(r). U denotes the intra-site matrix elements kν functions as of the Wannier states so that the significant short-ranged N−1 part of the Coulomb interaction is modeled. Then (16) u(0)(x) X w (x − x ) = eikν (x−xi) (12a) reduces to n i 2N ν=−N int X X X 2 H = U niσniσ0 = U ni"ni# + U n : (17) N−1 iσ (0) 0 u (x) X πiν (x−x ) iσσ i iσ = e Na i : (12b) 2N ν=−N The last term of (17) is a constant self-energy, UNe, 2 We can make use of finite geometric series identity where Ne is the total number of electrons, because niσ = niσ for fermions. Since this term only causes a constant N−1 1 X 1 zN − z−N energy shift of the spectrum it is typically ignored. zν = (13) 2N 2N z − 1 Obtaining analytic solutions of the Hubbard model is ν=−N difficult. Exact solutions are possible for certain cases, to rewrite (12b) as for example when U=t = 0 or when U=t = 1, the latter being the strong-coupling limit where doubly occupied (0) πi (x−x ) − πi (x−x ) u (x) e a i − e a i w (x − x ) = sites are prohibited. A well known approximation ap- n i πi (x−x ) 2N e Na i − 1 proach involves altering the Hubbard Hamiltonian by a (14a) unitary transformation to express it in a t=U expansion (Gros et al., 1987). The lowest order terms of this ex- iu(0)(x) sin(π(x − x )=a) = i (14b) pansion are kept in the large-U limit, both large on-site πi (x−x ) N e Na i − 1 mutual interaction and self-interaction.