| t-J model | |
|---|---|
| Name | t–J model |
| Caption | Schematic of electrons on a lattice with hopping t and exchange J |
| Field | Condensed matter physics |
| Introduced | 1970s |
| Derived from | Hubbard model |
| Notable person | Phil Anderson |
| Keywords | strongly correlated electrons, high-temperature superconductivity |
t-J model
The t–J model is an effective lattice model in condensed matter physics describing strongly correlated electrons with kinetic hopping (t) and antiferromagnetic exchange (J) under a no-double-occupancy constraint. It is widely used to study the low-energy physics of doped Mott insulators and is central to theoretical approaches to high-temperature superconductivity, quantum magnetism, and non-Fermi liquid behavior.
The t–J model captures the competition between electron itinerancy and local spin interactions on a crystalline lattice such as the square lattice relevant for cuprate superconductors. It emerges as a minimal effective model for carriers introduced into an antiferromagnet and is applied to study phenomena including d-wave superconductivity, spin-charge separation, and stripe order. The model is closely associated with research by P. W. Anderson on resonating valence bond theory and the physics of Mott insulators.
The canonical Hamiltonian of the t–J model on a lattice is H = -t \sum_{\langle i,j\rangle,\sigma} (\tilde c^\dagger_{i\sigma}\tilde c_{j\sigma} + h.c.) + J \sum_{\langle i,j\rangle} \left(\mathbf{S}_i\cdot\mathbf{S}_j - \tfrac{1}{4} n_i n_j\right), where \tilde c_{i\sigma} are projected fermion operators forbidding double occupancy, \mathbf{S}_i are spin-1/2 operators, and n_i is the number operator. The sums run over nearest-neighbor bonds \langle i,j\rangle on lattices such as the square lattice or triangular lattice. Parameters t and J are effective amplitudes: t governs single-particle hopping and J (typically \sim 4t^2/U) sets the superexchange energy scale. The projection is often implemented by the Gutzwiller operator or by slave-particle representations such as slave boson or slave fermion techniques.
The t–J model is derived as the low-energy effective Hamiltonian of the single-band Hubbard model in the limit of large on-site repulsion U relative to t. Using a canonical Schrieffer–Wolff transformation or degenerate perturbation theory, virtual hoppings that would create doubly occupied sites are integrated out, yielding an effective antiferromagnetic exchange J = 4t^2/U between neighboring spins plus projected hopping terms. This derivation links the t–J model to microscopic parameters of transition-metal oxides and provides justification for its use in modeling doped cuprate planes studied at institutions such as Bell Labs and research groups including those of P. W. Anderson and Patrick A. Lee.
Analytical and semi-analytical approaches to the t–J model include variational wavefunctions (e.g., Gutzwiller projection applied to BCS states), mean-field theory and renormalized mean-field approximations, slave-boson and slave-fermion gauge theories, and large-N expansions based on SU(N) generalizations. Field-theoretic techniques invoke bosonization in one dimension and spin-charge separation concepts related to Luttinger liquid theory. Perturbative expansions around ordered states and linear spin-wave theory are used in the low-doping, magnetically ordered regime. Variational Monte Carlo and analytic continuation methods bridge between variational and dynamical properties.
The t–J model exhibits rich phase behavior as a function of doping and J/t, including antiferromagnetism at half-filling, superconducting tendencies (notably d-wave pairing on the square lattice), stripe and charge-density-wave order, and possible spin-liquid phases. Elementary excitations include spinons and holons in fractionalized descriptions, magnons in ordered phases, and quasiparticle-like Bogoliubov excitations in superconducting states. Competing orders and quantum critical points arise, connecting to experimental probes such as angle-resolved photoemission spectroscopy (ARPES), neutron scattering, and nuclear magnetic resonance (NMR) measurements on cuprate materials like La2-xSrxCuO4 and YBa2Cu3O7.
Numerical investigations employ exact diagonalization on small clusters, quantum Monte Carlo methods where sign problems allow, density matrix renormalization group (DMRG) on quasi-one-dimensional geometries, and tensor network states including projected entangled pair states (PEPS). Dynamical mean-field theory (DMFT) and cluster extensions (CDMFT, DCA) are used to capture local correlations. Computational studies conducted at centers such as Oak Ridge National Laboratory and in collaborations across universities have mapped phase diagrams, spectral functions, and pairing correlations, confronting experimental data and testing analytical predictions.
The t–J model remains a paradigmatic framework for understanding strongly correlated electron systems, particularly the mechanism of high-temperature superconductivity in cuprate superconductors. It informs theories of quantum magnetism, correlated insulators in transition metal oxides, and emerging platforms such as cold atoms in optical lattices where Hubbard-type physics and effective t–J regimes can be engineered experimentally by groups working with quantum simulation platforms. Insights from t–J studies contribute to broader themes in quantum many-body physics, including entanglement structure, non-Fermi-liquid behavior, and mechanisms for unconventional superconductivity.
Category:Condensed matter physics Category:Strongly correlated electron systems