Second quantization and lattice QFT
Quantum mechanics:
Quantum field theory:
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vector graphics (.ps)
•we follow each particle (r is dynamical variable)
•impractical for many electrons
•Pauli statistics causes complications (Slater det.)
•cannot capture states with fractional occupation
•Fock space is artificial construct
      ‘product’ of Hilbert spaces of each particle
bitmap (.bmp)
•we follow the state of space points (lattice sites)
•r (=site index) is a parameter
•general approach
•Pauli statistics is simple (commutation rules)
•no problem with fractional occupation
•Fock space is very natural
      ‘product’ of Hilbert spaces of lattice sites
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Hubbard model
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t
Fock space in lattice QFT
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lattice Fock space
local Fock space for fermions
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Fock space in lattice QFT
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lattice Fock space
local Fock space
for bosons
Hubbard model
U
t
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Hubbard model
U
t
Fock space in lattice QFT
2-flavors per site    and
local Fock space:
Pauli statistics:
•Fock space can be constructed by acting with creation operators on vacuum
•One can use binary code to index the states
•Order of operators is crucial
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Definition:
Flavor = (orbital, spin)
Hilbert space of each flavor is
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Hubbard model
U
t
Fock space in lattice QFT
2-flavors per site    and
local Fock space:
Pauli statistics:
•Fock space can be constructed by acting with creation operators on vacuum
•One can use binary code to index the states
•Order of operators is crucial
Image
Definition:
Flavor = (orbital, spin)
Hilbert space of each flavor is
Image Image Image n=a^dagger_a.pdf &n|_emptyset_ran.pdf
Density (number) operator:

H=_sum_a,b_h_ab_.pdf c^phantom_dagger.pdf c^phantom_dagger.pdf H=_sum_i_epsilon.pdf
|_phi_rangle=c^d.pdf H|_phi_rangle=_l.pdf
Canonical commutation relations!
Non-interacting problem
Aufbau principle

Example of wave functions
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•Total size of fermionic Fock space is 4N. (bosonic is infinite)
•Any state can be written as a linear combination of the states in occupation number basis

Action of an operator
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(_pm).pdf (_pm).pdf (_pm).pdf (_pm).pdf (_pm).pdf +_ldots.pdf
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H|_phi_rangle=.pdf t_times_(.pdf
•Move annihilation operators to the right ->calculate number of (anti)commutators
•Put creations operators to standard order (for fermions only)-> determine signs
Action of an operator
Meaning of the Hamiltonian:
Hamiltonian generates the time evolution of the system!

Remarks:
•number of 1-p states N=4
•dimension of the Fock space
•dimension of an M-particle sector
•density/particle number operator
Hubbard molecule
A
B
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Construction of the Hamiltonian (in occupation number basis):
•sign convention, e.g.
•order the 1-p states:
Two options: Construct the matrices of the elementary creation/anihilation
         operators. (computer - sparse matrices)

         Construct the basis states and compute the matrix elements of
         H using commutation relations. (pen&paper)

Hubbard molecule
A
B
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Construction of the Hamiltonian (in occupation number basis):
•sign convention, e.g.
•order the 1-p states:
Let us focus on the 2 electron sector (the rest is trivial)
The basis:        index  i2i1      state                  Hamiltonian:
Hubbard molecule
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Hubbard molecule
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The basis:        index  i2i1      state                  Hamiltonian:
Spectrum:                   Energy            Eigenfunctions       Total spin
Ground state:

Various operators:
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Hubbard molecule
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Various operators:
•
Hubbard molecule
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Some expectation values
Ground state:
Lowest excitation energy:
Total spin (conserved):
Spin per atom (non-conserved):
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large u=U/t
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Some physics
Ground state:
Non-interacting limit (μ=1):
Bonding—anti-bonding picture:
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2t
atom A
atom B

Some physics
Ground state:
Non-interacting limit (μ=1):
Bonding—anti-bonding picture:
Image Image
2t
atom A
atom B