Hubbard Model Analysis and Its Extensions in Solid-State Physics
Explore the intricacies of the Hubbard model through an analysis of solutions for a 2-site system and its Hartree-Fock extension to a linear chain. Delve into the Hubbard Hamiltonian, possible configurations on a single site, and the implications of electron hopping and repulsion. Discover the matrix elements of the Hamiltonian for all 2-particle states with spin 0 and the corresponding eigenvalues. Uncover the eigenvalues of the Hubbard model and gain insights into the complexities of solid-state physics.
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PHY 752 Solid State Physics 11-11:50 AM MWF Olin 107 Plan for Lecture 33: The Hubbard model Analysis of solution for a 2 site system Hartree-Fock approximation Extension to linear chain 4/17/2015 PHY 752 Spring 2015 -- Lecture 33 1
4/17/2015 PHY 752 Spring 2015 -- Lecture 33 2
The Hubbard Hamiltonian: two particle contribution single particle contribution = { , } 0 = c c ' ' l l c { , } 0 c ' ' l l = { , } c c ' ' l l ll 4/17/2015 PHY 752 Spring 2015 -- Lecture 33 3
l = 1 2 3 4 Possible configurations on a single site 0 0 c 0 c c 4/17/2015 PHY 752 Spring 2015 -- Lecture 33 4
Hubbard model -- continued t represents electron hopping between sites, preserving spin U represents electron repulsion on a single site 4/17/2015 PHY 752 Spring 2015 -- Lecture 33 5
Two-site Hubbard model ( 1 2 H t c c = ) ( ) + + + + + c c c c c c U n n n n 2 1 1 2 2 1 1 1 2 2 where n c c l l l Consider all possible 2 particle states with zero spin: 0 A c c 1 1 0 B c c 2 2 1 ( ) 0 C c c c c 1 2 1 2 2 4/17/2015 PHY 752 Spring 2015 -- Lecture 33 6
Two-site Hubbard model ( 1 H t c = ) ( ) + + + + + c c c c c c c U n n n n 2 2 1 1 2 2 1 1 1 2 2 Matrix elements of Hamiltonian for all 2 particle states with spin 0: = 0 2 U t 0 2 H U t 2 2 0 t t Eigenvectors of the Hamiltonia n: Eigenvalues of Hamiltonian: + = + 2 1 1 2 U U 2 ( ) U U = + + + 1 C A B = 2 1 E t 1 4 4 t t 2 1 4 4 t t 1 ( ) E U = A B 2 2 2 2 U U = + + 2 1 E t 2 1 1 2 U U ( ) 3 4 4 t t = + + + 1 C A B 3 4 4 t t 2 4/17/2015 PHY 752 Spring 2015 -- Lecture 33 7
Eigenvalues of the Hubbard model Eigenvalues of Hamiltonian: + = + 2 U U = 2 1 E t 1 4 4 t t E3 E U 2 2 U U = + + 2 1 E t 3 4 4 t t E2 E1 4/17/2015 PHY 752 Spring 2015 -- Lecture 33 8
Two-site Hubbard model ( 1 2 H t c c = ) ( ) + + + + + c c c c c c U n n n n 2 1 1 2 2 1 1 1 2 2 Ground state of the two-site Hubbard model 2 2 1 1 2 U U U U ( ) = + + + = + 1 C A B 2 1 E t 1 1 4 4 4 4 t t t t 2 Single particle limit (U 0) 2 E t = 1 1 2 ( ) = + + C A B 1 1 2 1 ( ) 0 0 0 A c c B c c C c c c c 1 1 2 2 1 2 1 2 2 1 2 ( )( ) = + + 0 c c c c 1 1 2 1 2 4/17/2015 PHY 752 Spring 2015 -- Lecture 33 9
Single particle limit (U0) Two-site Hubbard model Full spectrum for spin 0 eigenstates 2 E t = 1 2 ( )( ) = + + 0 c c c c 1 1 1 2 1 2 1 4 ( )( ) = = + 0 0 E c c c c 2 2 1 2 1 2 1 4 ( )( ) + + 0 c c c c 1 2 1 2 1 2 ( )( ) = + = 2 t 0 E c c c c 3 1 1 2 1 2 Single particle picture: +t E2 E1 E3 -t 4/17/2015 PHY 752 Spring 2015 -- Lecture 33 10
Two-site Hubbard model -- Hartree-Fock approximation ( 1 2 2 1 H t c c c c c = + + ) ( ) + + + c c c U n n n n 1 2 2 1 1 1 2 2 Wave function assumed to be pr oduct of single particle states Zero order approximati on: 1 ( ) + Define: a c c 1 2 2 = HF Let 0 a a 1 1 2 = = + HF HF HF 2 E H t U 1 1 1 4/17/2015 PHY 752 Spring 2015 -- Lecture 33 11
Two-site Hubbard model -- Hartree-Fock approximation ( 1 2 2 1 H t c c c c c = + + ) ( ) + + + c c c U n n n n 1 2 2 1 1 1 2 2 Variational search for lower energy solutions High spin solution = Spin 1 2 L et 0 c c H 1 = = S in p Spin Sp in 0 E 1 1 1 4/17/2015 PHY 752 Spring 2015 -- Lecture 33 12
Ground state energy for 2-site Hubbard model Hartree Fock high spin exact 4/17/2015 PHY 752 Spring 2015 -- Lecture 33 13
Two-site Hubbard model -- continued ( 1 2 2 H t c c c = + ) ( ) + + + + c c c c c U n n n n 1 1 2 2 1 1 1 2 2 Ground state of the two-site Hubbard model 2 2 1 1 2 U U U U ( ) = + + + = + 1 C A B 2 1 E t 1 1 4 4 4 4 t t t t 2 Isolated particle limit (t 0) 2 4 U t E 1 1 2 t ( ) ( ) + + + 0 0 c c c c c c c c 1 1 2 2 1 1 1 2 2 U 4/17/2015 PHY 752 Spring 2015 -- Lecture 33 14
One-dimensional Hubbard chain 4/17/2015 PHY 752 Spring 2015 -- Lecture 33 15
Approximate solutions in terms of single particle states; broken symmetry Hartree-Fock type solutions 4/17/2015 PHY 752 Spring 2015 -- Lecture 33 16
4/17/2015 PHY 752 Spring 2015 -- Lecture 33 17
In the following slides, u represents U/t: 4/17/2015 PHY 752 Spring 2015 -- Lecture 33 18
k = -2 cos(ka) k k a kF a -kF 4/17/2015 PHY 752 Spring 2015 -- Lecture 33 19
4/17/2015 PHY 752 Spring 2015 -- Lecture 33 20
1.5 SHF=FHF (m=0) 1.0 0.5 0.0 E -0.5 -1.0 -1.5 0 2 4 6 8 10 u 4/17/2015 PHY 752 Spring 2015 -- Lecture 33 21
1.5 SHF=FHF (m=0) 1.0 FHF (m=1/2) 0.5 FHF (m=1) 0.0 E -0.5 -1.0 -1.5 0 2 4 6 8 10 u 4/17/2015 PHY 752 Spring 2015 -- Lecture 33 22
4/17/2015 PHY 752 Spring 2015 -- Lecture 33 23
Antiferromagnetic form: a (x cos k) For spin (x sin k) (x cos k) For spin (x sin k) 4/17/2015 PHY 752 Spring 2015 -- Lecture 33 24
4/17/2015 PHY 752 Spring 2015 -- Lecture 33 25
Solutions to AHF equation for spin magnetization m as a function of u m u 4/17/2015 PHY 752 Spring 2015 -- Lecture 33 26
4/17/2015 PHY 752 Spring 2015 -- Lecture 33 27
SDW form: a For spin (x cos k) For spin (x sin k) 4/17/2015 PHY 752 Spring 2015 -- Lecture 33 28
4/17/2015 PHY 752 Spring 2015 -- Lecture 33 29
1.5 SHF=FHF (m=0) 1.0 0.5 0.0 E -0.5 -1.0 -1.5 0 2 4 6 8 10 u 4/17/2015 PHY 752 Spring 2015 -- Lecture 33 30
1.5 SHF=FHF (m=0) 1.0 FHF (m=1/2) 0.5 FHF (m=1) 0.0 E -0.5 -1.0 -1.5 0 2 4 6 8 10 u 4/17/2015 PHY 752 Spring 2015 -- Lecture 33 31
1.5 SHF=FHF (m=0) 1.0 FHF (m=1/2) 0.5 FHF (m=1) 0.0 AHF E Exact -0.5 -1.0 -1.5 0 2 4 6 8 10 4/17/2015 PHY 752 Spring 2015 -- Lecture 33 32 u
Single partical spectra 4/17/2015 PHY 752 Spring 2015 -- Lecture 33 33
