6-site Hubbard model
Large Fock space:  dim 212
Use conservation of Sz : (s1, s2) sectors of dim
binom_6_s_1_bino.pdf Image
We study all sectors with N=6 simultaneously (6,0), (5, 1), (4,2), (3,3), (2,4), (1,5), (0,6)

Thermal equilibrium
Image
Thermal averages (finite temperature):
system
reservoir
energy and/or
particles
Coupling is very weak !

Image
Thermal averages (finite temperature):
system
reservoir
energy and/or
particles
Coupling is very weak !
Thermal equilibrium

Image
ensemble averaging:
p4
p3
p2
p1
...
t1
t3
t2
t4
p1
p3
p2
p4
Measurement involves different (weakly coupled)
parts of the system,
e.g., total moment is a sum of local moments
Measurement involves averaging over (long) time (=duration
of measurement)
Ergodic hypothesis
Averaging over various realizations of the system
Thermal equilibrium

Image
ensemble averaging:
t1
t3
t2
t4
p1
p3
p2
p4
Measurement involves averaging over (long) time (=duration
of measurement)
Ergodic hypothesis
Averaging over various realizations of the system
system
reservoir
Coupling is very weak !
The slow dynamics due to system-reservoir
interaction is replaced  by ensemble averaging
(the fast dynamics of the system itself is retained)
Thermal equilibrium

Image
ensemble averaging:
p1
p3
p2
p4
Averaging over various realizations of the system
How do we get the probabilities?
Statistical considerations => in thermal equilibrium:
Z(T)=_sum_l_exp(.pdf langle_langle_O_.pdf
avg. at temperature T
sum over eigenstates l
Boltzmann weight/factor
Partition function
Thermal equilibrium

Image
ensemble averaging:
p1
p3
p2
p4
Averaging over various realizations of the system
How do we get the probabilities?
Statistical considerations => in thermal equilibrium:
Thermal equilibrium
langle_langle_O_.pdf rho=_frac_1_Z_ex.pdf Z=_operatorname_.pdf beta=_frac_1_k_B.pdf
density matrix
(statistical operator)
Hamiltonian (operator)
(number)
inverse temperature

Image
ensemble averaging:
p1
p3
p2
p4
Averaging over various realizations of the system
Thermal equilibrium
rho=_frac_1_Z_ex.pdf langle_langle_O_.pdf langle_psi_g|e^i.pdf G_AB_(t)=.pdf =&_frac_1_Z_oper.pdf
Ground state:
Finite temperature:

Thermodynamic observables
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Thermodynamic observables:
U=_langle_langle.pdf F=-T_ln_Z(T).pdf F=U-TS.pdf c=_frac_partial_.pdf
internal energy
free energy
entropy S
specific heat (heat capacity)
Screenshot 2020-11-06 at 10.35.16.png
T
U
F
E
Energy spectrum (N=6)
U=1.23

Image
Thermodynamic observables:
U=_langle_langle.pdf F=-T_ln_Z(T).pdf F=U-TS.pdf c=_frac_partial_.pdf
internal energy
free energy
entropy S
specific heat (heat capacity)
Screenshot 2020-11-06 at 10.37.19.png
T
c
Screenshot 2020-11-06 at 10.38.39.png
T
S
E
Energy spectrum (N=6)
U=1.23
Thermodynamic observables

Correlation functions
Image
Expectation values/correlation functions:
langle_S_iz_S_jz.pdf Screenshot 2020-11-05 at 12.46.07.png
12
13
14
U
Screenshot 2020-11-05 at 16.31.50.png
U=1.23
T=0
T

Linear response
Image
Why correlation functions?
•Contributions to interaction energy of the system
•Response to small perturbations
langle_n_i_uparr.pdf
external uniform field
delta_langle_lan.pdf langle_langle_S_.pdf S_z(l)=_langle_l.pdf frac_partial_Z(h.pdf
frac_partial_lan.pdf

Image
Why correlation functions?
•Contributions to interaction energy of the system
•Response to small perturbations
langle_n_i_uparr.pdf
external uniform field
delta_langle_lan.pdf
T
frac_partial_lan.pdf frac_partial_lan.pdf
Linear response

Image
Does it work?
Let's calculate a response to finite h.
delta_langle_lan.pdf
T
h
Sz
h
Sz
Linear response

Linear response
Image
Why correlation functions?
•Contributions to interaction energy of the system
•Response to small perturbations
external uniform field
delta_langle_lan.pdf langle_langle_S_.pdf S_z(l)=_langle_l.pdf frac_partial_Z(h.pdf
frac_partial_lan.pdf chi_text_loc_(_o.pdf
General case (e.g. local susceptibility):
Uniform susceptibility is a special case because
S_z_,H_=0.pdf

Kubo formula
Linear response (perturbative) regime
H=H_0+V(t).pdf U(t)=e^-i(H_0+V_.pdf
External time-dependent field (el.-mag. field, photon)
Evolution operator
Initial state (ground state)
H0 and V do not commute and thus is it not possible to split the exponential even if V did not
depend on time!
color_red_e^(H_0.pdf U(t)=e^-iH_0_tau.pdf
Standard trick: discretise the time into small steps and use the fact
that                                            ,
i.e., we can split the exponential on each time (the error can be made arbitrarily small):
e^i(H_0+V)_tau_=.pdf
Next we expand the exponentials containing the external field:

Kubo formula
Linear response (perturbative) regime
H=H_0+V(t).pdf U(t)=e^-i(H_0+V_.pdf
External time-dependent field (el.-mag. field, photon)
Evolution operator
Initial state (ground state)
H0 and V do not commute and thus is it not possible to split the exponential even if V did not
depend on time!
color_red_e^(H_0.pdf U(t)=e^-iH_0_tau.pdf
Standard trick: discretise the time into small steps and use the fact
that                                            ,
i.e., we can split the exponential on each time (the error can be made arbitrarily small):
e^i(H_0+V)_tau_=.pdf
Next we expand the exponentials containing the external field:
+
+
+
+
+
+
+
+
….
….
….
….
Now we arrange the terms in powers of V:

Kubo formula
Linear response (perturbative) regime
H=H_0+V(t).pdf U(t)=e^-i(H_0+V_.pdf
External time-dependent field (el.-mag. field, photon)
Evolution operator
Initial state (ground state)
H0 and V do not commute and thus is it not possible to split the exponential even if V did not
depend on time!
color_red_e^(H_0.pdf U(t)=e^-iH_0_tau.pdf
Standard trick: discretise the time into small steps and use the fact
that                                            ,
i.e., we can split the exponential on each time (the error can be made arbitrarily small):
e^i(H_0+V)_tau_=.pdf
Next we expand the exponentials containing the external field:
tilde_V_(t)_equi.pdf U(t)=&e^-iH_0N_t.pdf

U(t)=&e^-iH_0N_t.pdf
Kubo formula
Linear response (perturbative) regime
H=H_0+V(t).pdf U(t)=e^-i(H_0+V_.pdf
External time-dependent field (el.-mag. field, photon)
Evolution operator
Initial state (ground state)
H0 and V do not commute and thus is it not possible to split the exponential even if V did not
depend on time!
color_red_e^(H_0.pdf U(t)=e^-iH_0_tau.pdf
Standard trick: discretise the time into small steps and use the fact
that                                            ,
i.e., we can split the exponential on each time (the error can be made arbitrarily small):
e^i(H_0+V)_tau_=.pdf
Next we expand the exponentials containing the external field:
tilde_V_(t)_equi.pdf
T is so called time ordering symbol (sometimes called an time-ordering operator). Note that it is
not an operator! The meaning of T is the expansion on the line above.
You cannot interpret this expression as “evaluate the integral in the bracket -> exponentiate the
result -> apply T on the result”!
tilde_V_(t)_equi.pdf

U(t)=&e^-iH_0N_t.pdf
Kubo formula
Linear response (perturbative) regime
H=H_0+V(t).pdf U(t)=e^-i(H_0+V_.pdf
External time-dependent field (el.-mag. field, photon)
Evolution operator
Initial state (ground state)
H0 and V do not commute and thus is it not possible to split the exponential even if V did not
depend on time!
color_red_e^(H_0.pdf U(t)=e^-iH_0_tau.pdf
Standard trick: discretise the time into small steps and use the fact
that                                            ,
i.e., we can split the exponential on each time (the error can be made arbitrarily small):
e^i(H_0+V)_tau_=.pdf
Next we expand the exponentials containing the external field:
tilde_V_(t)_equi.pdf tilde_V_(t)_equi.pdf
int_dt'_dt''.pdf =_frac_1_2_int_d.pdf
neq_frac_1_2_int.pdf

Kubo formula
Linear response (perturbative) regime
Now we can evaluate the expectation value of operator A in the system evolving with time
We keep only terms linear in V.
In compact form. Note that operators are in so called interaction representation with respect to
H0 (This is indicated by <>0)
The formula can be used in thermal equilibrium, in which case <>0 refers to thermal average -
trace with                  .
langle_A(t)_rang.pdf
We have shifted origin of time integral to infinity, assuming that the perturbation is
adiabatically (slowly) turned on (this excludes the transient states that takes place after sudden
turning of finite external field)
langle_n(t)|A|n(.pdf

Kubo formula
langle_A(t)_rang.pdf V(t)=B_phi(t).pdf
A typical external field has the form of a (classical) function (electric field, Zeeman field,
vector
potential, …) coupled to a typically (semilocal) operator (charge-, spin-, current-density):
In the linear response regime the amplitude of the response is linearly proportional to the
amplitude of the external field. We want to investigate the trivial temporal (or frequency)
relationship.
Note, that the expectation value of the commutator depends only on the difference t-t’,
because H0 does not depend on time.
Linear response (perturbative) regime