t–J Model Preview

The t–J model describes spin-$1/2$ fermions moving through a lattice under the strict constraint that no site may be doubly occupied. Its standard nearest-neighbor Hamiltonian is

$$H_{tJ} ={} -t \sum_{\langle i,j\rangle,\sigma} \left( \widetilde c_{i\sigma}^\dagger \widetilde c_{j\sigma} + \widetilde c_{j\sigma}^\dagger \widetilde c_{i\sigma} \right) + J \sum_{\langle i,j\rangle} \left( \mathbf s_i\cdot\mathbf s_j - \frac14\widetilde n_i\widetilde n_j \right).$$

The projected hopping bilinear moves an electron only when its destination is empty. The exchange term rewards a spin singlet on an occupied bond. These processes compete: hole motion tends to rearrange the spin background, while antiferromagnetic exchange favors local singlet correlations.

For the single-band repulsive Hubbard model with $U\gg\lvert t\rvert$, the leading exchange is

$$J = \frac{4\lvert t\rvert^2}{U} >0.$$

The relation is controlled only within a specified strong-coupling expansion. A phenomenological t–J model may treat $t$, $J$, longer-range hopping, and additional interactions as independent effective parameters.

Helpful background. Hubbard model supplies the parent interaction; Heisenberg exchange explains singlet and triplet bond energies.

Empty sites and singly occupied spins

An unconstrained spinful fermion site has four states,

$$\lvert0\rangle_i, \qquad \lvert\uparrow\rangle_i, \qquad \lvert\downarrow\rangle_i, \qquad \lvert\uparrow\downarrow\rangle_i.$$

The t–J site retains only

$$\mathcal H_i^{(tJ)} = \operatorname{span} \left\{ \lvert0\rangle_i, \lvert\uparrow\rangle_i, \lvert\downarrow\rangle_i \right\}.$$

The doublon $\lvert\uparrow\downarrow\rangle_i$ is not a high-energy state still present in the model. It has been removed from the low-energy Hilbert space.

Define the global no-double-occupancy projector

$$P = \prod_i \left( 1-n_{i\uparrow}n_{i\downarrow} \right).$$

Physical states satisfy

$$P\lvert\Psi\rangle = \lvert\Psi\rangle.$$

The constraint can also be written locally as

$$n_{i\uparrow}n_{i\downarrow} =0$$

on the physical subspace.

Hilbert-space dimension

For $L$ sites, $N_\uparrow$ up-spin electrons, and $N_\downarrow$ down-spin electrons with $N_\uparrow+N_\downarrow\le L$, the constrained basis has dimension

$$\dim\mathcal H_{L,N_\uparrow,N_\downarrow}^{(tJ)} = \binom{L}{N_\uparrow} \binom{L-N_\uparrow}{N_\downarrow}.$$

If only the total electron number $N_e$ is fixed, summing over spin assignments gives

$$\dim\mathcal H_{L,N_e}^{(tJ)} = \binom{L}{N_e}2^{N_e}.$$

The full constrained space has dimension $3^L$, smaller than the Hubbard space $4^L$ but still exponentially large.

The t–J model retains the empty and singly occupied states while removing the doublon. Projected hopping moves charge at scale $t$ only into an empty site. Exchange acts on two occupied sites at scale $J$ and lowers the spin singlet relative to the triplet.

Projected Fermion Operators

The globally projected annihilation and creation operators are

$$\begin{aligned} \widetilde c_{i\sigma} &= Pc_{i\sigma}P = c_{i\sigma} \left( 1-n_{i\bar\sigma} \right)P, \\ \widetilde c_{i\sigma}^\dagger &= Pc_{i\sigma}^\dagger P = \left( 1-n_{i\bar\sigma} \right) c_{i\sigma}^\dagger P. \end{aligned}$$

Here $\bar\sigma$ denotes the opposite spin. The final global $P$ can be omitted when acting only on physical states; on the full Hubbard space, a local factor alone would fail to remove doublons at other sites. Acting on the local states,

$$\begin{aligned} \widetilde c_{i\uparrow}^\dagger \lvert0\rangle_i &= \lvert\uparrow\rangle_i, \\ \widetilde c_{i\uparrow}^\dagger \lvert\downarrow\rangle_i &=0. \end{aligned}$$

The second line is the constraint in action: creating an up electron on a down-occupied site would make a forbidden doublon.

The algebra is not canonical

Projection changes the local operator algebra. For one spin component,

$$\left\{ \widetilde c_{i\sigma}, \widetilde c_{i\sigma}^\dagger \right\} = 1-n_{i\bar\sigma},$$

not the identity. This identity is understood on the constrained space. For distinct spin labels there is also a nonzero local anticommutator:

$$\{\widetilde c_{i\sigma},\widetilde c_{i\tau}^\dagger\} =X_i^{\tau\sigma},\qquad \sigma\ne\tau.$$

It flips a local spin. The Hubbard operators below make both relations transparent.

It is unsafe to manipulate $\widetilde c$ as though it were an ordinary canonical fermion and impose the constraint only at the end. The constraint and algebra are coupled.

Hubbard Operators

An exact local notation uses Hubbard operators

$$X_i^{ab} = \lvert a\rangle_i {}_i\!\langle b\rvert, \qquad a,b\in\{0,\uparrow,\downarrow\}.$$

They satisfy

$$X_i^{ab}X_i^{cd} = \delta_{bc}X_i^{ad}$$

and the completeness relation

$$X_i^{00} +X_i^{\uparrow\uparrow} +X_i^{\downarrow\downarrow} =I_i.$$

Projected fermions become

$$\widetilde c_{i\sigma} = X_i^{0\sigma}, \qquad \widetilde c_{i\sigma}^\dagger = X_i^{\sigma0}.$$

The local density and spin operators are

$$\begin{aligned} \widetilde n_i &= X_i^{\uparrow\uparrow} +X_i^{\downarrow\downarrow}, \\ s_i^z &= \frac12 \left( X_i^{\uparrow\uparrow} -X_i^{\downarrow\downarrow} \right), \\ s_i^+ &= X_i^{\uparrow\downarrow}, \qquad s_i^- = X_i^{\downarrow\uparrow}. \end{aligned}$$

The odd operators $X_i^{0\sigma}$ and $X_i^{\sigma0}$ change fermion parity. Across different sites they must carry fermionic grading: in an ordinary tensor-product implementation, multiply each such local matrix by the parity string over preceding sites. Bare three-by-three matrices on distinct tensor factors commute and would give the wrong hopping signs. Even density and spin operators need no such string. Locally $\mathbf s_i^2=3\widetilde n_i/4$, so an empty site carries no spin.

This representation enforces the local space exactly. Auxiliary-particle representations can also encode the constraint, but they enlarge the Hilbert space and introduce a gauge redundancy that must be handled rather than ignored.

Exchange is an occupied-bond singlet projector

The combination in the Hamiltonian has a simple spectrum:

$$\mathbf s_i\cdot\mathbf s_j-\frac14\widetilde n_i\widetilde n_j =-P_{ij}^{\mathrm{singlet}}.$$

Here the projector selects the normalized singlet when both sites are occupied and vanishes if either site is empty. The triplets have eigenvalue zero; the singlet has eigenvalue minus one. Thus a positive $J$ lowers precisely the occupied-bond singlet by $J$.

At one electron per site the density product is one, so its contribution is a constant per bond. With mobile holes it varies from configuration to configuration and cannot be discarded. For ordered distinct sites, the normalized pair creator below also gives $P_{ij}^{\mathrm{singlet}}=\Delta_{ij}^\dagger\Delta_{ij}$.

Large-U Hubbard Origin

Begin with the repulsive Hubbard Hamiltonian

$$H_{\mathrm{Hub}} = T+UD,$$

where

$$D = \sum_i n_{i\uparrow}n_{i\downarrow}.$$

Let $P$ project onto $D=0$ and $Q=I-P$. For the uniform repulsive Hubbard parent, small hopping relative to the doublon cost permits perturbative elimination of $Q$. At second order, one hop from $D=0$ reaches only $D=1$, so its energy denominator is $U$ even though $Q$ also contains sectors with more doublons. To second order in the nearest-neighbor hopping,

$$H_{\mathrm{eff}} = PTP - \frac1U PTQTP + O\!\left( \frac{t^3}{U^2} \right).$$

The first term is projected hopping. The second contains bond exchange and correlated processes involving three sites.

The complete derivation, including the sign of the energy denominator and the separation of half-filled and doped sectors, is in Effective Hamiltonians in Many-Body Systems. Here the result fixes the interpretation of the model parameters.

Two distinct low-energy scales

The projected hopping survives at order $t$, while exchange arises at order $t^2/U$:

$$\lvert J\rvert \sim \frac{t^2}{U} \ll \lvert t\rvert \ll U.$$

For the simple one-band repulsive model,

$$\frac{J}{\lvert t\rvert} = \frac{4\lvert t\rvert}{U} \ll1.$$

Treating $J/t$ as order unity can define a useful phenomenological t–J model, but it is not the asymptotic large-$U$ limit of this simplest Hubbard parent.

Why Doping Restores First-Order Motion

At half filling within the constrained space, every site contains one electron. Any single hop would place an electron on an occupied site, so

$$PTP=0.$$

Hopping contributes only through virtual excursions to the doublon sector, producing exchange.

With a hole present, an electron can move into the empty site without creating a doublon. Both the initial and final states remain in $P$, and the matrix element survives at order $t$.

Define the hole concentration for $N_e\le L$ by

$$\delta = \frac{N_h}{L} = 1-\frac{N_e}{L}.$$

Then

$$\delta=0$$

is half filling, while $\delta>0$ introduces mobile holes. The standard no-doublon t–J representation is naturally adapted to hole doping. Electron-doped applications require an explicit particle–hole transformation or a correspondingly chosen low-energy representation; they should not be inferred by silently allowing doublons back into the same Hilbert space.

Half-Filled Limit

At half filling,

$$\widetilde n_i=1$$

on every site and the projected hopping vanishes. If $N_b$ is the number of nearest-neighbor bonds,

$$H_{tJ} \longrightarrow J \sum_{\langle i,j\rangle} \mathbf s_i\cdot\mathbf s_j - \frac{J}{4}N_b.$$

The last term is a constant. The dynamical Hamiltonian is therefore the antiferromagnetic Heisenberg model for $J>0$.

This limit is an exact reduction inside the minimal t–J model. The statement that a particular microscopic material reduces to that model still depends on the validity of the preceding scale separation and orbital reduction.

Three-Site Terms

A systematic Hubbard expansion away from half filling produces correlated three-site hopping at the same order $t^2/U$ as $H_J$:

$$H_{\mathrm{eff}} = H_t+H_J+H_{3\mathrm s} + O\!\left( \frac{t^3}{U^2} \right).$$

Schematically, an electron can hop through an occupied intermediate site by visiting a virtual doublon state and then return to the constrained sector on a different bond. The amplitude is of order

$$\frac{t_{ij}t_{jk}}{U}.$$

The commonly named “minimal t–J model” sets $H_{3\mathrm s}=0$. That omission is an additional approximation, not a consequence of no double occupancy. It may be reasonable for a chosen observable and parameter range, but it must be stated.

At half filling, three-site charge motion is blocked and the spin-only reduction is cleaner. At finite doping, omitting $H_{3\mathrm s}$ while retaining every exchange term is not a uniform truncation solely by powers of $t/U$.

Symmetries

For uniform real hopping and isotropic exchange, the standard model has:

• global $U(1)$ charge conservation;
• global $SU(2)$ spin rotation;
• lattice translations and point-group symmetries when the geometry permits;
• time-reversal symmetry, with complex conjugation of orbital amplitudes and reversal of spin;
• the no-double-occupancy restriction on its Hilbert space, which is a constraint rather than an additional physical symmetry.

The total electron number

$$N_e = \sum_i\widetilde n_i$$

commutes with $H_{tJ}$, as does total spin for the isotropic model:

$$\left[ H_{tJ}, \mathbf S_{\mathrm{tot}}^2 \right] =0, \qquad \mathbf S_{\mathrm{tot}} = \sum_i\mathbf s_i.$$

Longer-range hopping, magnetic flux, spin–orbit coupling, anisotropic exchange, disorder, or boundaries reduce this symmetry set in model-specific ways.

Sign and Gauge Conventions

For nearest-neighbor hopping on a bipartite lattice, the sign of $t$ can be reversed by a staggered phase transformation. Relative signs and loop fluxes are physical when they cannot be removed by site phases. A staggered phase still reverses all intersublattice hoppings in a bipartite graph with flux, preserving its even-loop fluxes; same-sublattice links and odd loops obstruct that simple global sign change.

The sign of $J$ is independently meaningful. The repulsive single-band Hubbard limit gives

$$J>0,$$

favoring antiferromagnetic singlets. A ferromagnetic $J<0$ defines a different effective model and is not generated by the simple $4t^2/U$ mechanism.

Physical Competition

The t–J model is not a free hole moving through an inert magnet. Every projected hop changes which sites carry spins and can disrupt or rearrange short-range antiferromagnetic correlations. Conversely, the spin background changes the coherent paths available to a hole.

Three tendencies are useful for orientation:

Tendency	Favored by	Diagnostic examples
coherent carrier motion	larger $\lvert t\rvert$ and available holes	kinetic energy, charge stiffness, spectral weight
antiferromagnetic singlets	$J>0$ near half filling	spin structure factor, bond energy
charge clustering or phase separation	sufficiently strong effective attraction in some parameter regimes	density curvature, real-space inhomogeneity

This table is not a universal phase diagram. Dimension, lattice, $J/t$, longer-range hopping, boundary conditions, and omitted terms all matter.

Relation to Strongly Correlated Electrons

The model is a canonical idealization of a doped Mott insulator. It keeps low-energy spin and charge motion but removes explicit doublon excitations whose cost is assumed to be large.

For cuprate materials, a one-band t–J description is itself the endpoint of a material-dependent reduction from copper and oxygen orbitals. The Zhang–Rice construction supplies one influential route by identifying a local low-energy singlet, but its quantitative range depends on orbital energies, hybridization, interactions, and the observable under study.

Realistic effective descriptions may require

$$t', \qquad t'', \qquad H_{3\mathrm s}, \qquad \text{ring exchange},$$

as well as electron–phonon, disorder, interlayer, or multiorbital effects. The minimal model is valuable because it isolates constrained motion and exchange, not because every strongly correlated material is exactly described by two parameters.

The model has played a central role in theories of unconventional superconductivity, pseudogap behavior, magnetic polarons, and spin–charge separation. Those connections are active research programs. The t–J Hamiltonian alone does not constitute a settled derivation of any one material phase diagram or pairing mechanism.

Controlled and Special Limits

Half filling

At $\delta=0$, the model reduces to Heisenberg exchange plus a constant.

Empty lattice

At $N_e=0$, both hopping and exchange annihilate the vacuum.

One electron

With one electron, the exchange term vanishes and projected hopping reduces to an ordinary one-particle tight-binding problem.

Infinite-U one-hole limit

The $U\to\infty$ Hubbard limit gives $J\to0$. Under restrictive lattice-connectivity and hopping-sign assumptions, Nagaoka’s theorem establishes saturated ferromagnetism for a single hole. The theorem is an exact corner case, not a generic finite-doping phase statement.

One-dimensional special points

The one-dimensional model has Bethe-ansatz-solvable supersymmetric points. With the conventional choice $t>0$, the commonly discussed point $J/t=2$ is not the asymptotic large-$U$ Hubbard regime, where $J/t\ll1$. Exact solvability at a special phenomenological ratio should not be confused with the controlled parent-model expansion.

Observables

Hole density

The local hole operator is

$$h_i = 1-\widetilde n_i.$$

Density profiles and correlations

$$C_h(i,j) = \langle h_i h_j\rangle - \langle h_i\rangle \langle h_j\rangle$$

diagnose clustering, repulsion, or spatial inhomogeneity.

Spin correlations

The static spin structure factor can be defined as

$$S(\mathbf q) = \frac1L \sum_{i,j} e^{-i\mathbf q\cdot(\mathbf R_i-\mathbf R_j)} \langle \mathbf s_i\cdot\mathbf s_j \rangle.$$

Near half filling on a square bipartite lattice, weight near $(\pi/a,\pi/a)$ diagnoses antiferromagnetic correlations. A peak on a finite cluster is not by itself proof of spontaneous order.

Projected momentum distribution

The electron momentum distribution uses projected correlators:

$$n_\sigma(\mathbf k) = \frac1L \sum_{i,j} e^{i\mathbf k\cdot(\mathbf R_i-\mathbf R_j)} \left\langle \widetilde c_{i\sigma}^\dagger \widetilde c_{j\sigma} \right\rangle.$$

The projected spectral weight is not the unit weight of a canonical electron; the exercise below gives its exact local sum. Moreover, when comparing with a finite-$U$ Hubbard observable $O$, use the same unitary elimination as for the Hamiltonian:

$$O_{\mathrm{eff}}=P e^S O e^{-S}P.$$

Bare projection $POP$ is only its leading term. In particular, the displayed distribution is exact for the chosen projected-electron observable of the minimal model, but it need not equal the physical Hubbard momentum distribution through the order retained in the effective Hamiltonian. The effective-Hamiltonian construction explains these observable corrections.

Singlet pairing correlations

On a bond $ij$, define a projected singlet creator

$$\Delta_{ij}^\dagger = \frac1{\sqrt2} \left( \widetilde c_{i\uparrow}^\dagger \widetilde c_{j\downarrow}^\dagger - \widetilde c_{i\downarrow}^\dagger \widetilde c_{j\uparrow}^\dagger \right).$$

Pair correlations such as

$$\left\langle \Delta_{ij}^\dagger \Delta_{kl} \right\rangle$$

can compare spatial symmetries and decay laws. A negative pair-binding energy or enhanced short-range correlator on one finite cluster is not sufficient to establish bulk superconducting order.

Charge response

Finite-size addition energies, compressibility, Drude weight, and twist sensitivity probe charge motion. They must be extrapolated at fixed density and with declared boundary conditions.

Finite-Size Formulation

At fixed $N_\uparrow$ and $N_\downarrow$, enumerate only configurations with no shared up- and down-spin site. This enforces the constraint before matrix construction and avoids carrying forbidden states with an artificial penalty.

The hopping matrix element between allowed occupation states retains the fermionic sign fixed by the global mode ordering. Projection removes some destinations but does not remove anticommutation signs.

The exchange term can be written as diagonal $s_i^zs_j^z-\widetilde n_i\widetilde n_j/4$ contributions plus spin flips,

$$\frac{J}{2} \left( s_i^+s_j^- + s_i^-s_j^+ \right).$$

This decomposition is useful for sparse exact diagonalization and tensor-network implementations.

Worked Example: One Hole on Two Sites

In the one-electron sector, use basis

$$\left\{ \lvert\sigma,0\rangle, \lvert0,\sigma\rangle \right\}.$$

The exchange term vanishes because no bond has two occupied endpoints. The projected hopping matrix is

$$[H_{tJ}]_\sigma = \begin{pmatrix} 0 & -t\\ -t & 0 \end{pmatrix},$$

with eigenvalues

$$E_\pm = \pm t.$$

This elementary sector isolates first-order charge motion without any virtual doublon.

Exercises

Why three-site terms are not higher order than exchange

An electron hops from site $i$ to $j$, creates a virtual doublon of cost $U$, and then a fermion hops from $j$ to a distinct neighbor $k$. Estimate the amplitude and compare it with $J$.

Solution

Each hop contributes one hopping matrix element, and the virtual state contributes an energy denominator $1/U$. The process therefore has scale

$$\frac{t_{ij}t_{jk}}{U}.$$

For uniform hopping this is $t^2/U$, the same order as

$$J = \frac{4t^2}{U}.$$

The three-site process may have a smaller numerical coefficient or reduced phase space, but it is not suppressed by an additional power of $t/U$ relative to exchange.

Where the projected electron weight goes

In a homogeneous spin-balanced state with hole fraction $\delta$, find the local integrated removal weight $\langle\widetilde c_\sigma^\dagger\widetilde c_\sigma\rangle$, addition weight $\langle\widetilde c_\sigma\widetilde c_\sigma^\dagger\rangle$, and their sum.

Solution

The local matrix products are $X^{\sigma\sigma}$ and $X^{00}$, respectively. Thus removal has weight $(1-\delta)/2$, addition has weight $\delta$, and the sum is $(1+\delta)/2$. At half filling there is no allowed low-energy electron addition, even though removal remains possible. The full canonical Hubbard electron has total weight one; the excluded doublon addition lies outside this projected local space. At finite $U$, quantitative low-energy weights also receive the transformed-operator corrections discussed above.

Spin-flip anticommutator

Using only $X^{ab}X^{cd}=\delta_{bc}X^{ad}$, compute $\{X^{0\uparrow},X^{\downarrow0}\}$. Explain why a local canonical two-fermion algebra cannot describe the three-state site.

Solution

The first product vanishes because $\uparrow\ne\downarrow$. The reversed product is $X^{\downarrow\uparrow}$, which turns an up spin into a down spin. Canonical annihilation and creation operators of distinct modes would instead have zero anticommutator. Removing the doublon state changes the algebra as well as the allowed states.

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