Fermi Hubbard ModelEdit

The Fermi-Hubbard model, often discussed under the umbrella of the Hubbard model, is a foundational lattice model in condensed matter physics. It captures the essential tension between the itinerant motion of electrons and their mutual repulsion on a crystal lattice. In its simplest form, electrons hop between neighboring lattice sites with amplitude t and experience an on-site repulsion U whenever two electrons occupy the same site. The model is a compact, yet richly predictive framework that has guided thinking about metal-insulator transitions, magnetism, and possible routes to unconventional superconductivity. For a broad treatment within the field, see condensed matter physics and related discussions of high-temperature superconductivity and Mott insulator phenomena.

The Fermi-Hubbard model is usually formulated on a lattice with two spin states (often denoted up and down). At half-filling, where the average number of electrons per site is one, the model exhibits a competition between kinetic energy, which favors delocalized states, and repulsive interaction energy, which favors localization. This competition gives rise to distinct regimes: a metallic or weakly correlated regime at small U/t, and a Mott insulating regime at large U/t, in which electron motion is suppressed and local magnetic moments emerge. The interplay of these tendencies has made the model a central reference point for understanding strongly correlated electron systems, including connections to the physics of cuprate superconductors and related materials.

History and development - The model was introduced in the early 1960s as a minimal description of electron correlations in narrow energy bands. Early formulations credit J. Hubbard, with independent and complementary work by other researchers around the same period. The basic ideas were then extended and refined through decades of theoretical work, numerical studies, and experimental probes of correlated matter. See the historical discussions surrounding the development of the Hubbard model and its various extensions. - Over the years, the model has been studied on a range of lattice geometries, from one dimension to two and three dimensions, and with increasingly sophisticated methods. A pivotal development was the use of dynamical mean-field theory (DMFT) and its cluster extensions to treat local correlations in a controlled way, bridging exactly solvable limits and more realistic systems. See dynamical mean field theory and its variants for more on these approaches.

Formal definition and basic properties - The standard Fermi-Hubbard Hamiltonian on a lattice is written as H = -t ∑{⟨i,j⟩,σ} (c†{iσ} c_{jσ} + h.c.) + U ∑i n{i↑} n_{i↓}, where i and j label lattice sites, ⟨i,j⟩ denotes nearest-neighbor pairs, σ ∈ {↑, ↓} is the spin, c†{iσ} creates a fermion of spin σ at site i, and n{iσ} = c†{iσ} c{iσ} is the number operator. - The model preserves particle number and spin, and its physics emerges from the competition between the hopping term (controlled by t) and the on-site repulsion (controlled by U). In many discussions, the lattice structure (square, triangular, honeycomb, etc.) and filling (number of electrons per site) are crucial for the resulting phases.

Phases, phenomena, and typical behaviors - At half-filling and large U/t, the system tends toward a Mott insulating state: electrons localize to avoid paying the on-site repulsion, and charge degrees of freedom acquire a gap while spins remain dynamic. In two dimensions on common lattices (e.g., the square lattice), this regime is closely associated with antiferromagnetic correlations between localized moments, and, in the low-temperature limit, long-range antiferromagnetic order can emerge in appropriate conditions. The low-energy physics of the half-filled, large-U regime maps to the Heisenberg model with an exchange coupling J ∝ t^2/U. - At weak coupling (small U/t), the electrons behave more like a conventional metal, with itinerant quasiparticles and a Fermi surface that resembles that of noninteracting electrons, albeit with renormalized parameters. - Doping away from half-filling introduces charge carriers into the localized spin background, and the resulting phase diagram is rich and subtle. In two dimensions, these doped regimes have been studied intensively due to their potential relevance to unconventional superconductivity. The possibility of pairing instabilities with non-s conventional symmetry (for example, d-wave) arises in certain parameter ranges, though a universal, fully established mechanism within the pure two-dimensional Hubbard model remains a matter of active research. See discussions of high-temperature superconductivity and d-wave superconductivity for related threads. - In one dimension, the Hubbard model is exactly solvable via the Bethe ansatz, and its ground state and excitation spectrum display characteristic features such as spin-charge separation and a Mott gap at half-filling for any U>0. In higher dimensions, numerical and approximate methods are essential, and the precise nature of phases often depends on lattice geometry, dimensionality, and additional interactions.

Methods and computational approaches - Exact diagonalization provides precise results for small systems but scales poorly with size due to exponential growth of the Hilbert space. - Quantum Monte Carlo (QMC) methods offer powerful statistical approaches, but they encounter a notorious sign problem away from special cases (such as half-filling on bipartite lattices), which restricts accessible temperatures and sizes in many important settings. - Dynamical mean-field theory (DMFT) treats local quantum fluctuations exactly while approximating nonlocal correlations, becoming exact in the limit of infinite coordination. Extensions such as cluster DMFT and the dynamical cluster approximation (DCA) incorporate short-range correlations and are routinely used to study two-dimensional systems in the Hubbard model family. - Density matrix renormalization group (DMRG) and related tensor-network techniques excel in one dimension and have been extended to quasi-two-dimensional geometries, providing highly accurate insight into ground-state properties and low-energy excitations. - Experimental quantum simulators, especially ultracold fermionic atoms loaded into optical lattices, realize realized realizations of the Fermi-Hubbard model in a clean and highly controllable setting, enabling direct tests of theoretical predictions about Mott physics and spin correlations. See ultracold atoms in optical lattices for more.

Extensions, variations, and related models - The t-J model arises as an effective description in the strong-coupling limit (large U/t) when double occupancy is projected out, emphasizing exchange interactions and constrained hopping. - Multi-orbital and Kanamori-type generalizations incorporate more electron flavors and interaction channels, broadening the range of materials and phenomena that can be modeled. See Kanamori-type models and multi-band Hubbard models for related frameworks. - The extended Hubbard model adds interactions beyond the on-site term, such as nearest-neighbor Coulomb repulsion V, which can influence charge ordering, phase separation, and superconducting tendencies. Lattice geometry and longer-range hopping terms (such as t′, t″) are also important for capturing realistic Fermi surfaces in particular materials. - Spin-fermion and related hybrid models attempt to couple itinerant electrons to collective spin fluctuations, providing alternative perspectives on how magnetism and superconductivity may emerge in correlated systems. See spin-fermion model for a broader family of approaches.

Controversies and debates - Relevance to real materials: While the Fermi-Hubbard model captures essential ingredients of strong electron correlations, its applicability to complex materials like the cuprates remains debated. Some researchers argue that a minimal one-band Hubbard description captures universal aspects of the physics, while others emphasize the importance of additional orbitals, lattice distortions, electron-phonon coupling, or longer-range interactions. See discussions on high-temperature superconductivity and Mott insulator phenomena for context. - Origin of superconductivity in the doped regime: In two dimensions, the possibility of superconductivity arising from purely electronic mechanisms within the Hubbard framework has generated extensive debate. Proponents point to numerical and analytical evidence of pairing tendencies in certain parameter regimes, while skeptics highlight sensitivity to model details, finite-size effects, and the need to include other interactions to obtain robust, experimentally verifiable superconducting states. The broader question remains, in part, about what minimal, realistic ingredients are required to reproduce observed superconducting behavior. - Computational limitations and interpretation: The sign problem in QMC and finite-size effects in exact diagonalization or tensor-network studies complicate definitive conclusions about phase boundaries and pairing mechanisms. This has driven ongoing efforts to develop better algorithms, scalable simulations, and reliable extrapolation techniques, as well as to explore alternative models that retain essential physics while offering tractable analyses. - Role of extensions and realism: Some researchers advocate starting from more elaborate models (multi-orbital, longer-range interactions, or explicit electron-phonon coupling) to explain specific materials, while others defend the value of the minimal Hubbard description as a probe of universal strong-correlation physics. Both perspectives contribute to a productive debate about where simplicity serves understanding and where realism is indispensable.

See also - Hubbard model - Fermi-Hubbard model - Mott insulator - antiferromagnetism - high-temperature superconductivity - d-wave superconductivity - ultracold atoms in optical lattices - dynamical mean field theory - Bethe ansatz - t-J model