This chapter discusses several models of second quantization including the Jordan-Wigner transformation, Hubbard model, and non-interacting particles in thermal equilibrium. The Jordan-Wigner transformation describes how to map systems of interacting spins onto systems of fermions by introducing a phase string. The Hubbard model is introduced as a simplified model of interacting electrons in solids. The section on non-interacting particles discusses the properties of fermion and boson gases at thermal equilibrium, including the concept of a degenerate Fermi liquid and Fermi surface.

1. z Chapter 4:
Simple examples of second quantization
Felipe Brevis

2. z
4.1 Jordan-Wigner transformation
“Down” and “up” states of a single spin Empty and singly occupied fermion states
Spin-raising and spin-lowering operators

3. z
The 𝑧 component of the spin operator
4.1 Jordan-Wigner transformation
𝑓†
𝑓 −
1
2
=
0 1
0 0
0 0
1 0
−
1 2 0
0 1 2
=
1 0
0 0
−
1 2 0
0 1 2
=
1
2
1 0
0 −1
The transverse spin operators
They satisfy commutation and anticommutation relations.

4. z
For more than one spin
4.1 Jordan-Wigner transformation
Independent spin operators commute
Modification of the representation.
The representation of the spin operator at site 𝑗
Solving the problem in 1D (chain of spins):
Independent fermions anticommute
The problem:
Attach a phase factor called a “string” to the fermions
The phase operator 𝜙𝑗 contains the sum over all
fermion occupancies at sites to the left of 𝑗
string
operator

5. z
For more than one spin
4.1 Jordan-Wigner transformation
Modification of the representation.
Attach a phase factor called a “string” to the fermions

6. z
Important properties of the string:
4.1 Jordan-Wigner transformation
• Anticommutes with any fermion operator to the left of its free end.
• Commutes with any fermion operator to the left of its free end.
• The transverse spin operators satisfy the correct commutation algebra.
By multiplying a fermion by the string operator, it is transformed into a boson.

7. z
Example of the application of this method: One-dimensional Heisenberg model
4.1 Jordan-Wigner transformation
Interaction of local magnetic moments
Ferromagnetic Antiferromagnetic
Arise from direct exchange
Coulomb repulsion energy is lowered if…
Electrons: Triplet state
(wavefunction is spatially antisymmetric)
Arise from super exchange
Electrons on neighboring sites form singlets
Lower their energy through virtual quantum
fluctuations into high-energy states
(same orbital)

8. z
Example of the application of this method: One-dimensional Heisenberg model
4.1 Jordan-Wigner transformation
“Fermionizing” the first term:
The transverse component of the interaction induces a “hopping” term in the fermionized Hamiltonian.
The z component of the Hamiltonian

9. z
Example of the application of this method: One-dimensional Heisenberg model
4.1 Jordan-Wigner transformation
Transforming it to momentum space (for a most compact form)
𝑠𝑘
†
: creates a spin excitation in momentum space with momentum 𝑘
The anisotropic Heisenberg Hamiltonian
Magnon excitation energy.

10. z
Example of the application of this method: One-dimensional Heisenberg model
4.1 Jordan-Wigner transformation
Casting the second term in momentum space:
Interaction is function of 𝑖 − 𝑗 (short range): −𝐽𝑧/2 for 𝑖 − 𝑗 = ±1 (0 otherwise)
The Fourier transform of this short-range interaction is
This transformation holds for both the ferromagnet and the antiferromagnet
Ferromagnet: fermionic spin excitations Magnons
Antiferromagnet: fermionic spin excitations Spinons

11. z
Example of the application of this method: One-dimensional Heisenberg model
4.1 Jordan-Wigner transformation
Let’s see what this Hamiltonian means neglecting the interactions (limiting cases):
a) Heisenberg ferromagnet, 𝐽𝑧 = 𝐽
The spectrum goes like: ≥ 0
¡There are no magnons in the ground state!
Ground state:
Magnetization

12. z
Example of the application of this method: One-dimensional Heisenberg model
4.1 Jordan-Wigner transformation
b) 𝑥-𝑦 ferromagnet
𝐽𝑧 is reduced from 𝐽 Spectrum:
Negative part
Magnon states (𝐸 < 0)
Occupied
Pure x-y ferromagnet: 𝐽𝑧 = 0
Ground state 𝑛𝑓 = 1/2
There is no
ground-state
magnetization.

13. z
4.2 The Hubbard model
Real electronic system Apparent complexity Model electrons
Complex systems
BUT
Low energies A few electronic degrees of freedom
are excited
One such model is the Hubbard model, introduced by Philip W. Anderson, John Hubbard,
Martin Gutziller and Junjiro Kanamori.
Closely tied up to with the idea of renormalization

14. z
4.2 The Hubbard model
Consider a lattice of atoms where electrons are almost localized in atomic orbitals at each site.
Using a basis of atomic orbitals
Operator which creates
a particle at site 𝑗
Φ 𝑥 : wavefunction of a particle in the localized atomic orbital.
The Hamiltonian: the motion and interactions between the particles
(in general)
One-particle matrix element
Interaction matrix element between two-particle states
𝑖, 𝑗 : states

15. z
4.2 The Hubbard model
For highly localized orbitals
Just nearest-neighbor hopping
Interaction between electrons at different sites
For well localized states
Dominates the onsite interaction between
two electrons in a single orbital
Approximating
(Exclusion principle)

16. z
4.2 The Hubbard model
Number of electrons of
spin 𝜎 at site 𝑗.
Rewriting this in momentum space
Kinetic energy of the electron excitations Applications
Theory of magnetism, metal–insulator transitions and
also the description of electron motion in high-
temperature superconductors.
An important prediction: Mott insulator (large interactions
produce localization of electrons).
The complete understanding of doped Mott insulator
could guide to understand high-temp. superconductivity.

17. z
4.3 Non-interacting particles in thermal
equilibrium
Ground-state (GS) properties of free particles
For non-interacting but identical particles
GS is a highly correlated many-body state.
Effect of turning on the interactions adiabatically.

18. z
4.3 Non-interacting particles in thermal
equilibrium
Ground-state (GS) properties of non-interacting gases of identical particles.
Quantum effects Influence a fluid of identical particles
Characteristic wavelength
∼ separation inter-particles
Characteristic de Broglie wavelength Characteristic temperature
If 𝑇 < 𝑇∗
Identical particles start to interfere with one another.
Fermi fluid: Exclusion statistics ⊳ particles apart ⊳ Pressure grows.
Bose fluid: Exclusion statistics ⊳ the motion is correlated ⊳ Pressure decreases.
Electron fluids: T∗
∼ 102
𝑇𝑅 ⊳ Electricity ⊳ Example of quantum physics in everyday phenomena.

19. z
4.3 Non-interacting particles in thermal
equilibrium
4.3.1 Fluid of non-interacting fermions
Thermal equilibrium: Number of fermions in a state with momentum 𝒑 = ℏ𝒌
Degenerate Fermi liquid
• GS and excitations of metals
• Low-energy physics of liquid 𝐻𝑒3
• Degenerate Fermi gas of neutrons, electrons and protons in neutron stars.
General Hamiltonian
Single free energy
functional

20. z
4.3 Non-interacting particles in thermal
equilibrium
4.3.1 Fluid of non-interacting fermions Degenerate Fermi liquid
𝑇 → 0 𝐾
States with are completely occupied and states above this energy are empty.
Ground state Where all fermion states with momentum 𝑘 < 𝑘𝐹 (Fermi momentum).
To excite this ground state: a) Add particles at energies above the Fermi wavevector or
b) create holes beneath the Fermi wavevector.

21. z
4.3 Non-interacting particles in thermal
equilibrium
4.3.1 Fluid of non-interacting fermions
Describing the excitations: Particle–hole transformation:
Beneath the Fermi surface:
In terms of particle and hole excitations

22. z
4.3 Non-interacting particles in thermal
equilibrium
4.3.1 Fluid of non-interacting fermions
• To create a hole with momentum 𝒌 and spin 𝜎, we must destroy a fermion with
momentum −𝒌 and spin −𝜎.
• The excitation energy of a particle or hole is given by 𝜖𝒌
∗
= |𝑬𝑘 − 𝜇| ⇒ “reflecting”
the excitation spectrum of the negative energy fermions about the Fermi energy.

23. z Chapter 4:
Simple examples of second quantization
Felipe Brevis