11.1 Hubbard model

Hubbard Model Fundamentals

The Hubbard model describes interacting electrons on a lattice and is one of the most important models in condensed matter physics. It captures the competition between two tendencies: electrons wanting to hop between sites (lowering kinetic energy) and electrons repelling each other when they share a site (raising potential energy). This tension gives rise to rich physics including metal-insulator transitions, magnetism, and potentially high-temperature superconductivity.

Band theory treats electrons as independent particles and works well for many metals and semiconductors. But for materials where electron-electron interactions are strong, band theory fails badly. The Hubbard model is the simplest framework that includes these interactions, making it the starting point for understanding strongly correlated electron systems.

Origins and development

John Hubbard introduced the model in 1963 to study electron correlations in narrow energy bands, building on earlier work by Nevill Mott and Conyers Herring on transition metals. It was first applied to ferromagnetism in metals like nickel and iron. The model gained major renewed attention in the 1980s after the discovery of high-temperature superconductors, since the parent compounds of cuprate superconductors are Mott insulators that the Hubbard model naturally describes.

Key assumptions and limitations

• Electrons interact only when they occupy the same lattice site. All long-range Coulomb interactions between electrons on different sites are neglected.
• In its simplest form, only a single orbital per site is included.
• Electron-phonon coupling is not accounted for, so the model struggles with systems where lattice vibrations play a major role.
• Complex crystal structures and multi-orbital effects require extensions beyond the basic model.

These are significant simplifications, but they make the model tractable while still capturing the essential competition between kinetic energy and interaction energy.

Hamiltonian Formulation

The full Hubbard Hamiltonian combines three terms representing kinetic energy, on-site interaction, and chemical potential:

$H = -t \sum_{\langle i,j \rangle,\sigma} (c^\dagger_{i\sigma}c_{j\sigma} + c^\dagger_{j\sigma}c_{i\sigma}) + U \sum_i n_{i\uparrow}n_{i\downarrow} - \mu \sum_{i,\sigma} n_{i\sigma}$

Each term controls a different aspect of the physics, and the interplay between them determines the ground state and excitations of the system.

Kinetic energy term

$H_t = -t \sum_{\langle i,j \rangle,\sigma} (c^\dagger_{i\sigma}c_{j\sigma} + c^\dagger_{j\sigma}c_{i\sigma})$

This term describes electrons hopping between neighboring lattice sites. The hopping integral $t$ sets the energy scale for delocalization and determines the bandwidth of the resulting energy band. The operators $c^\dagger_{i\sigma}$ and $c_{i\sigma}$ create and annihilate an electron with spin $\sigma$ at site $i$, and the sum runs over nearest-neighbor pairs $\langle i,j \rangle$. When $t$ dominates, electrons spread out across the lattice and the system behaves like a metal.

On-site interaction term

$H_U = U \sum_i n_{i\uparrow}n_{i\downarrow}$

This term penalizes double occupancy: it costs energy $U$ whenever two electrons (one spin-up, one spin-down) sit on the same site. Here $n_{i\sigma} = c^\dagger_{i\sigma}c_{i\sigma}$ is the number operator counting electrons of spin $\sigma$ at site $i$. When $U$ is large, electrons avoid sharing sites, which tends to localize them and can drive the system into an insulating state.

Chemical potential term

$H_\mu = -\mu \sum_{i,\sigma} n_{i\sigma}$

The chemical potential $\mu$ controls the total electron number (the filling). At half-filling (one electron per site on average), particle-hole symmetry holds when $\mu = U/2$. Tuning $\mu$ away from this value corresponds to doping the system with extra electrons or holes, which is crucial for exploring superconducting and metallic phases.

Electron Correlation Effects

Electron correlations refer to the ways that electron-electron interactions modify material behavior beyond what independent-particle theories predict. These effects drive emergent phenomena like Mott insulating states, magnetic ordering, and unconventional superconductivity.

Strong vs weak coupling

The ratio $U/t$ is the key dimensionless parameter:

• Weak coupling ($U \ll t$): Kinetic energy wins. Electrons are itinerant and the system is metallic. Band theory gives a reasonable starting point, with correlations treated as perturbative corrections. Spin fluctuations in this regime can mediate unconventional superconductivity.
• Strong coupling ($U \gg t$): Interaction energy wins. Electrons localize to avoid the energy cost of double occupancy, producing a Mott insulator. At half-filling, the system develops antiferromagnetic order because virtual hopping processes (second-order in $t$) favor antiparallel spin alignment on neighboring sites.
• Intermediate coupling ($U \sim t$): Neither limit applies cleanly. This is where the most complex behavior emerges, including pseudogap phases and the crossover between metallic and insulating states. This regime is also the hardest to treat theoretically.

Mott-Hubbard transition

The Mott-Hubbard transition is a metal-to-insulator transition driven purely by electron correlations, not by band structure. As $U$ increases past a critical value $U_c$, a gap opens in the density of states at the Fermi level, splitting the original band into a lower Hubbard band (singly occupied states) and an upper Hubbard band (doubly occupied states).

This transition is accompanied by dramatic changes in transport (resistivity jumps by orders of magnitude) and optical properties (spectral weight shifts). In real materials like vanadium oxides ($\text{V}_2\text{O}_3$), the transition can be triggered by pressure, temperature, or chemical doping.

Single-Band Hubbard Model

The single-band model considers one orbital per lattice site. Despite its simplicity, it captures the essential physics of strong correlations and serves as the foundation for understanding more complex multi-band systems.

Half-filled case

At half-filling, there is exactly one electron per site on average. This case has special significance:

• Particle-hole symmetry holds at $\mu = U/2$, simplifying the analysis.
• For large $U/t$, the ground state is an antiferromagnetic Mott insulator. Electrons are localized, and neighboring spins align antiparallel due to superexchange.
• As $U/t$ decreases, the system undergoes a Mott transition to a paramagnetic metal.
• The half-filled case directly describes the undoped parent compounds of cuprate superconductors (e.g., $\text{La}_2\text{CuO}_4$).

Doped systems

Doping means moving away from half-filling by adding or removing electrons. This introduces mobile charge carriers (holes or electrons) into the Mott insulator. Doped Hubbard models exhibit:

• Metallic behavior as carriers become mobile
• Superconducting instabilities, particularly d-wave pairing in 2D
• Charge ordering and stripe formation at certain doping levels
• An asymmetry between electron-doped and hole-doped behavior, mirroring what's observed in cuprate phase diagrams

Understanding the doped Mott insulator is one of the central challenges in condensed matter theory, directly relevant to high-$T_c$ superconductivity.

Multi-Band Extensions

Real materials often have multiple relevant orbitals per atom. Multi-band Hubbard models extend the framework to include these, adding new interactions and new physics.

Orbital degeneracy

When multiple orbitals are nearly degenerate (as in transition metals with partially filled d-shells), the model must track which orbital each electron occupies. This introduces:

• Inter-orbital Coulomb repulsion ($U'$), the cost of putting two electrons in different orbitals on the same site
• Orbital ordering, where the system spontaneously breaks the symmetry between orbitals, analogous to how magnetic ordering breaks spin symmetry
• A much larger Hilbert space, making calculations significantly harder

Orbital degeneracy is essential for describing materials like manganites and iron pnictide superconductors.

Hund's coupling

Hund's coupling ($J_H$) is the intra-atomic exchange interaction that favors parallel spin alignment for electrons in different orbitals on the same atom. This is the same Hund's rule you encounter in atomic physics.

In a multi-orbital Hubbard model, $J_H$ competes with crystal field splitting (which may favor certain orbital occupancies) and with the on-site Coulomb repulsion $U$. Depending on the relative magnitudes:

• Large $J_H$ produces high-spin configurations
• Small $J_H$ relative to crystal field splitting produces low-spin configurations

Hund's coupling has been recognized as a key ingredient in "Hund's metals," a class of correlated metals (including iron-based superconductors) where $J_H$ rather than $U$ drives the strong correlation effects.

Numerical Methods

The Hubbard model is exactly solvable only in one dimension (via the Bethe ansatz) and in certain limiting cases. For realistic 2D and 3D systems, numerical methods are essential.

Exact diagonalization

This approach directly constructs and diagonalizes the Hamiltonian matrix.

• Gives exact results: all eigenstates, energies, and dynamical properties are accessible.
• The Hilbert space grows exponentially with system size, limiting calculations to roughly 16–20 sites.
• Primarily useful for benchmarking other methods and studying small clusters where finite-size effects can be analyzed.

Quantum Monte Carlo

Quantum Monte Carlo (QMC) methods use stochastic sampling to evaluate the partition function and expectation values.

• Can handle much larger systems than exact diagonalization (hundreds of sites).
• Provides accurate thermodynamic properties at finite temperature.
• Suffers from the fermionic sign problem: for many parameter regimes (especially away from half-filling or at low temperature), statistical errors grow exponentially, making results unreliable. This is one of the major unsolved computational challenges in the field.

Dynamical mean-field theory

Dynamical mean-field theory (DMFT) maps the full lattice problem onto a single impurity site coupled to a self-consistent bath. The impurity problem is then solved numerically (using QMC, exact diagonalization, or other impurity solvers).

• Becomes exact in the limit of infinite dimensions or infinite coordination number.
• Captures all local quantum fluctuations, making it excellent for describing the Mott transition.
• Treats spatial correlations at a mean-field level, which is its main limitation.
• Extensions like cluster DMFT and the dynamical cluster approximation (DCA) reintroduce short-range spatial correlations by solving a cluster of sites instead of a single site.

Applications in Condensed Matter

High-temperature superconductivity

The Hubbard model (and its strong-coupling descendant, the t-J model) is the leading theoretical framework for cuprate superconductors. The basic picture:

1. The undoped parent compound is a half-filled antiferromagnetic Mott insulator.
2. Doping introduces holes that destroy long-range antiferromagnetic order.
3. Residual antiferromagnetic spin fluctuations mediate an attractive interaction between holes.
4. This attraction leads to Cooper pairing with d-wave symmetry ($d_{x^2-y^2}$), consistent with experiments.

The model also captures competing orders like charge stripes and the mysterious pseudogap phase, though a complete quantitative theory of cuprate superconductivity remains an open problem.

Magnetism in transition metals

The Hubbard model explains how local magnetic moments form in partially filled d-orbitals and how they order:

• At large $U/t$ and half-filling, superexchange gives antiferromagnetism with exchange coupling $J \sim 4t^2/U$.
• Away from half-filling, ferromagnetism can be favored (Nagaoka ferromagnetism in certain limits, or Stoner-type instabilities at weak coupling).
• The model provides a unified framework for understanding the competition between itinerant and localized magnetism, which is central to the physics of iron, nickel, cobalt, and their compounds.

Hubbard Model vs Other Models

Heisenberg model comparison

The Heisenberg model describes localized spins interacting via exchange:

$H = J \sum_{\langle i,j \rangle} \mathbf{S}_i \cdot \mathbf{S}_j$

It can be derived from the half-filled Hubbard model in the strong coupling limit ($U \gg t$) via second-order perturbation theory, with $J = 4t^2/U$. The Heisenberg model retains only spin degrees of freedom and is appropriate for insulators with well-defined local moments. The Hubbard model is more general because it also describes charge fluctuations and the transition to metallic behavior.

t-J model relationship

The t-J model is obtained from the large-$U$ Hubbard model by projecting out doubly occupied states:

$H_{t\text{-}J} = -t \sum_{\langle i,j \rangle,\sigma} \tilde{c}^\dagger_{i\sigma}\tilde{c}_{j\sigma} + J \sum_{\langle i,j \rangle} \left(\mathbf{S}_i \cdot \mathbf{S}_j - \frac{n_i n_j}{4}\right)$

The tilde on the operators indicates projection to the no-double-occupancy subspace. This model captures the low-energy physics of the doped Mott insulator and is widely used to study cuprate superconductivity. It's computationally more tractable than the full Hubbard model because the restricted Hilbert space is smaller.

Experimental Realizations

Ultracold atoms in optical lattices

Ultracold atomic gases trapped in optical lattices (periodic potentials created by interfering laser beams) provide a remarkably clean realization of the Hubbard model.

• Lattice geometry, depth, and dimensionality are tunable by adjusting the laser configuration.
• Interaction strength can be controlled via Feshbach resonances.
• Both bosonic and fermionic versions of the Hubbard model can be studied.
• The superfluid-to-Mott-insulator transition has been directly observed for bosons (Greiner et al., 2002), and antiferromagnetic correlations have been measured for fermions.
• These systems also enable the study of non-equilibrium dynamics and quantum quenches with single-site resolution.

Transition metal oxides

Transition metal oxides are the primary solid-state systems where Hubbard model physics is realized:

• Cuprates (e.g., $\text{YBa}_2\text{Cu}_3\text{O}_{7-\delta}$): doped Mott insulators exhibiting high-$T_c$ superconductivity
• Vanadates (e.g., $\text{V}_2\text{O}_3$): classic Mott-Hubbard systems showing pressure- and temperature-driven metal-insulator transitions
• Manganites (e.g., $\text{La}_{1-x}\text{Sr}_x\text{MnO}_3$): exhibit colossal magnetoresistance tied to the interplay of charge, spin, and orbital degrees of freedom

Parameters in these materials can be tuned through chemical doping, applied pressure, or electric fields, allowing systematic comparison with theoretical predictions.

Recent Developments

Topological Hubbard models

A major current research direction combines topological band theory with strong correlations. By starting from a band structure with nontrivial topology (e.g., a Chern insulator or topological insulator) and adding Hubbard-type interactions, researchers investigate:

• Whether topological phases survive in the presence of strong correlations
• The possibility of interaction-induced topological phases that have no non-interacting analog
• Fractional topological insulators, which would host fractionalized excitations similar to the fractional quantum Hall effect
• Topological Mott insulators, where the Mott gap coexists with topologically protected edge states

Non-equilibrium dynamics

Studying Hubbard models out of equilibrium is increasingly important, driven by advances in ultrafast spectroscopy and cold-atom experiments:

• Quantum quenches: suddenly changing $U$ or $t$ and watching the system evolve reveals information about thermalization and many-body localization.
• Floquet engineering: periodic driving (e.g., shaking an optical lattice) can create effective Hamiltonians with novel properties, including topological bands.
• Prethermalization: before reaching thermal equilibrium, strongly interacting systems can get stuck in long-lived prethermal states with distinct properties.
• These studies connect directly to pump-probe experiments on correlated materials, where ultrafast laser pulses can transiently modify the effective $U/t$ ratio and induce non-equilibrium phase transitions.