Hubbard Model

Represent many-body Fock states using integer bit-strings for computational implementation.

Construct the Hubbard Hamiltonian matrix for a finite cluster using the bit-string basis.

Block-diagonalize the Hamiltonian by exploiting conservation of total particle number and total spin projection (Sz).

Explain the core iterative procedure of the Lanczos algorithm for finding extremal eigenvalues of a large sparse matrix.

Implement matrix-free Hamiltonian-vector multiplication for use with a numerical Lanczos solver.

Compute static observables, such as ground-state energy and double occupancy, from the Lanczos-computed ground state.

Compute static spin-spin correlation functions ⟨Si · Sj⟩ to probe magnetic tendencies.

Analyze finite-size effects by comparing ground-state observables calculated for several small cluster sizes.

Formulate the mean-field decoupling of the Hubbard on-site interaction term.

Derive the self-consistent Hartree-Fock equations for the Hubbard model.

Solve the Hartree-Fock equations for the paramagnetic state and compute the renormalized band structure.

Derive the Stoner criterion for ferromagnetic instability from the paramagnetic Hartree-Fock solution.

Apply the Hartree-Fock approximation assuming a staggered magnetization on a bipartite lattice to derive the antiferromagnetic mean-field Hamiltonian.

Diagonalize the antiferromagnetic mean-field Hamiltonian and derive the self-consistency equation for the sublattice magnetization at half-filling.

Solve the self-consistency equation to compute the sublattice magnetization and the mean-field energy gap as a function of U/t.

Explain why the Hartree-Fock approximation qualitatively fails to describe the Mott insulating state for U > 0.

Describe the atomic limit (t = 0) of the Hubbard model and enumerate the ground-state degeneracy at half-filling.

Partition the Hubbard Hamiltonian into an unperturbed part (U) and a perturbation (t) for a strong-coupling expansion.

Apply second-order degenerate perturbation theory in t/U to derive the antiferromagnetic Heisenberg model as the effective low-energy Hamiltonian.

Explain the superexchange mechanism as the physical origin of the effective antiferromagnetic coupling.

Derive the t-J model as the effective low-energy Hamiltonian for the strongly-correlated Hubbard model away from half-filling.

State the no-double-occupancy constraint in the t-J model and explain its physical origin.

Define the retarded single-particle Green's function for a many-body system at zero temperature.

Derive the Lehmann spectral representation of the Green's function and interpret it in terms of particle addition/removal energies.

Define the spectral function A(k, ω) and relate it to the local density of states and photoemission experiments.

Define the self-energy and write Dyson's equation, relating the interacting and non-interacting Green's functions.

Compute the Green's function and spectral function for the non-interacting tight-binding model.

Apply the equation-of-motion technique to compute the exact Green's function of the single-site (atomic) Hubbard model.

From the atomic limit Green's function, identify the upper and lower Hubbard bands and determine the Mott gap.

Define a Mott insulator and distinguish it from a band insulator using the concept of the charge gap.

Describe the evolution of the spectral function with increasing U/t, showing the transfer of spectral weight from a quasiparticle peak to the Hubbard bands.

Construct the Gutzwiller variational wave function and explain its physical motivation for controlling double occupancy.

Apply the Gutzwiller approximation to compute the ground-state energy and double occupancy as a function of U/t.

Derive the Brinkman-Rice transition from the Gutzwiller approximation, showing the vanishing of the quasiparticle weight Z at a critical U/t.

Explain the divergence of the effective mass at the Brinkman-Rice transition and its physical significance.

Identify key experimental signatures of the Mott transition, such as the resistivity jump and opening of an optical gap.

Define the Néel state and the staggered magnetization as an order parameter for antiferromagnetism.

Interpret spin-spin correlation functions ⟨Si · Sj⟩ and the spin structure factor S(q) in terms of magnetic ordering patterns.

State the Mermin-Wagner theorem and explain its implication that the 2D Hubbard model cannot exhibit long-range magnetic order at finite temperature.

Describe Nagaoka's theorem and the physical mechanism for ferromagnetism at U = ∞ with a single hole.

Sketch the qualitative phase diagram of the half-filled Hubbard model as a function of U/t and temperature, incorporating both the Mott transition and magnetic ordering.

Describe the competition between antiferromagnetism and paramagnetism as a function of doping away from half-filling.

Explain the central idea of DMFT: mapping a lattice model to a self-consistent quantum impurity problem in the limit of infinite dimensions.

Write the Anderson impurity model Hamiltonian and identify its parameters (impurity level, hybridization function, bath).

Define the Weiss effective medium (dynamical mean field) that determines the non-interacting impurity Green's function.

Express the local lattice Green's function by summing the momentum-space Green's function over the Brillouin zone.

State the core DMFT self-consistency condition: the impurity self-energy equals the local lattice self-energy.

Describe the iterative DMFT self-consistency loop and the role of the impurity solver.

Qualitatively describe the DMFT solution of the Mott transition, including the coexistence region and hysteresis.

Identify the primary limitation of single-site DMFT as the neglect of non-local spatial correlations.

Write the extended Hubbard model with nearest-neighbor interaction V and describe the charge-density-wave instability it can introduce.

Describe multi-orbital Hubbard-Kanamori models and identify physical situations where they are necessary (e.g., transition metal oxides with degenerate d-orbitals).

Explain the widely-held belief that the doped 2D Hubbard model contains the essential physics of high-temperature superconductivity in cuprates.

Identify d-wave pairing symmetry as the leading candidate for the superconducting order parameter in the 2D Hubbard model.

Describe the concept of a quantum spin liquid as an exotic ground state without magnetic order in frustrated Hubbard or Heisenberg models.

Outline current open problems and active research directions involving the Hubbard model and its variants.