6-site Hubbard model
Large Fock space:  dim 212
Use conservation of Sz : (s1, s2) sectors of dim
For example a basis function from (1,2) sector:
in binary code (10000|101000)
binom_6_s_1_bino.pdf Image

6-site Hubbard model
Sector (3,3)
Image
0
U
2U
3U
Matrix elements of the interaction part (diagonal in present basis):
implementation of
(101000|100000) -> (101000).(100000) = 1
Image
implementation of
(101000|100000) -> (101000)-(100000) = (001000)
langle_S_iz_rang.pdf
i=2

6-site Hubbard model
Matrix elements of the hopping part:
in binary code (10100|100000)
->                   -(01100|100000)
✓                     (11000|000000)
Image
c^phantom_dagger.pdf
Signs:

6-site Hubbard model
Sector (3,3)
Image
Spectrum of eigenenergies:
t=1
U
U=10
t
1
0

6-site Hubbard model
Sector (3,3)
Image
Spectrum of eigenenergies:
t=1
U
U=10
t
1
0
Non-interacting limit
(weak-coupling expansion)
(strong-coupling expansion)
atomic limit

H=_sum_a,b_h_ab_.pdf c^phantom_dagger.pdf H=_sum_i_epsilon.pdf c^phantom_dagger.pdf
|_phi_rangle=c^d.pdf H|_phi_rangle=_l.pdf
Canonical commutation relations!
6-site Hubbard model
Non-interacting (canonical) bosons or fermions
=> We can find all eigenstates by diagonalizing the 1-p Hamiltonian (= hopping matrix)

6-site Hubbard model
Sector (3,3)
Image
0
U
2U
3U
Large U >> t limit:
Degeneracy:
binom_6_3_=20.pdf 6_cdot_5_cdot_bi.pdf binom_6_2_cdot_b.pdf binom_6_3_=20.pdf
180
180

6-site Hubbard model
Image
Expectation values/correlation functions:
The average value one gets when many measurement on site i are performed. Possible result of each
individual
measurement is 0, 1 and -1.
langle_S_iz_rang.pdf
At half filling (N=6)
Fluctuations of Sz
langle_S_iz^2_ra.pdf langle_n_i_uparr.pdf
Total moment (occupation number):
S_z_equiv_sum_i_.pdf
Conserved quantities (corresponding operators commute with
Hamiltonian)
langle_S_iz_rang.pdf |_psi_g_rangle=_.pdf
Simple form for operator diagonal in a given basis

6-site Hubbard model
Image
Expectation values/correlation functions:
Double occupancy: (probability to find two electrons in a given site)
langle_n_i_uparr.pdf
?
U

6-site Hubbard model
Image
Expectation values/correlation functions:
(Fluctuating) local moment
?
U
Screenshot 2020-11-05 at 12.24.48.png langle_S_iz^2_ra.pdf langle_n_i_uparr.pdf

6-site Hubbard model
Image
Expectation values/correlation functions:
Non-local spin-spin correlation function
1
-1
0
langle_S_iz_S_jz.pdf
Weighted sum over configurations like
Which one has the largest weight?

6-site Hubbard model
Image
Expectation values/correlation functions:
1
-1
0
langle_S_iz_S_jz.pdf Screenshot 2020-11-05 at 12.46.07.png
12
13
14
U

6-site Hubbard model
Image
Why correlation functions?
•Contributions to interaction energy of the system
•Response to small perturbations
langle_n_i_uparr.pdf
external local field
delta_langle_S_i.pdf delta_langle_S_i.pdf
U
Screenshot 2020-11-16 at 10.37.55.png chi_text_loc_=2_.pdf
Correction: Factor 2 missing in the recorded
presentation

6-site Hubbard model
Image
What about symmetry?
•We have used conservation of N and Sz when constructing the basis
•We did not use translation symmetry
This would require a bit more 'brain' input

6-site Hubbard model
Image
Translation symmetry is reflected in the correlation functions:
1
-1
0
12
13
14
U
S_z(_mathbf_k_)=.pdf langle_S_z(_math.pdf

6-site Hubbard model
Image
Translation symmetry is reflected in the correlation functions:
S_z(_mathbf_k_)=.pdf langle_S_z(_math.pdf
invariant: (k''=0)
invariant
k+k'+(k"-k")=0 (mod 2𝝅)
langle_S_z(_math.pdf
=>
k''=𝝅
Correction:
The ground state is dominated by the states
of the type
The ground state phase does not matter as pointed out correctly.

6-site Hubbard model
Image
Translation symmetry is reflected in the correlation functions:
S_z(_mathbf_k_)=.pdf
U
1
-1
0
U
mathbf_k_=0.pdf mathbf_k_=_pi.pdf mathbf_k_=_pi∕3.pdf mathbf_k_=2_pi∕3.pdf