t–J Model

The t–J model is a low-energy theory of mobile, spinful carriers in a Hilbert space from which local double occupancy has been removed. In quantum materials it is used most often as a deliberately compressed description of a doped Mott or charge-transfer insulator: charge moves at a scale set by hopping, while occupied neighboring sites exchange spin at a lower scale.

Required background. Mott Insulators supplies the charge-localization and doping context, while Hubbard Physics in Materials supplies the retained-orbital and parameter choices for a material reduction. t–J Model Preview supplies projected fermion operators and the no-double-occupancy Hilbert space. Effective Hamiltonians in Many-Body Systems supplies the elimination of virtual charge states and the generated operators needed to assess the truncation.

Projected Carriers in Quantum Materials

A material may require more than nearest-neighbor hopping and exchange. Write the extended projected Hamiltonian as

$$H_{\mathrm{mat}} ={} -\sum_{i<j,\sigma} \mathcal P_G \left( t_{ij}c_{i\sigma}^{\dagger}c_{j\sigma} +\mathrm{h.c.} \right) \mathcal P_G + \sum_{i<j} J_{ij} \left( \mathbf S_i\cdot\mathbf S_j -\frac14 n_i n_j \right) +H_{\mathrm{3s}} +H_{\mathrm{ring}} +H_{\mathrm{other}} .$$

Here $\mathcal P_G=\prod_i(1-n_{i\uparrow}n_{i\downarrow})$ projects onto empty or singly occupied effective sites. The terms $H_{\mathrm{3s}}$, $H_{\mathrm{ring}}$, and $H_{\mathrm{other}}$ stand for correlated three-site motion, cyclic exchange, longer-range interactions, phonon couplings, disorder, or other operators generated by the reduction. The Hermitian conjugate includes the complex conjugate of $t_{ij}$, so the kinetic term also applies to hopping with phases. We use dimensionless spins $\mathbf S_i=\tfrac12\sum_{\alpha\beta}\widetilde c_{i\alpha}^{\dagger}\boldsymbol\sigma_{\alpha\beta}\widetilde c_{i\beta}$ and $n_i=\sum_\sigma\widetilde c_{i\sigma}^{\dagger}\widetilde c_{i\sigma}$, with $\widetilde c_{i\sigma}=\mathcal P_Gc_{i\sigma}\mathcal P_G$. Here $\boldsymbol\sigma$ denotes the Pauli matrices. Which additional terms matter depends on the material and the observable.

Strong-Coupling Route from the Hubbard Model

Split the one-band Hubbard Hamiltonian into an interaction $H_U=U D$, where $D=\sum_i n_{i\uparrow}n_{i\downarrow}$ counts doublons, and hopping pieces $T_m$ that change $D$ by $m=0,\pm1$. Let $P$ project onto the no-doublon sector and $Q=1-P$. For $U$ larger than every retained hopping scale and every target excitation energy, a Schrieffer–Wolff transformation gives (Chao, Spałek, and Oleś, 1978; MacDonald, Girvin, and Yoshioka, 1988)

$$H_{\mathrm{eff}} ={} P T_0 P - \frac{1}{U} P T_{-1}T_{+1}P + O\!\left( \frac{\lVert T\rVert^3}{U^2} \right).$$

The operator-norm remainder shown here is a finite-cluster estimate. Since the total hopping norm grows with system size, it is not a uniform thermodynamic bound. At fixed coordination, bulk strong-coupling reasoning instead organizes local or linked-cluster contributions in hopping-to-charge-gap ratios, with the low-energy window separated from discarded charge excitations.

The two displayed terms have distinct physical roles.

• $P T_0P$ describes an allowed, rather than virtual, hop within the low-energy sector. It survives at first order when an empty site is available.
• $-PT_{-1}T_{+1}P/U$ describes a virtual excursion through a doublon. On one bond it produces antiferromagnetic exchange with $J_{ij}=4\lvert t_{ij}\rvert^2/U$ in the simplest one-band convention.

At exactly one electron per site, projected hopping is blocked and the leading nonconstant dynamics is spin exchange. Upon hole doping, projected hopping immediately reappears at order $t$, while exchange remains of order $t^2/U$. This separation explains why a single doped carrier can strongly disturb an antiferromagnetic background even when $J/t$ is numerically modest.

Controlled does not mean complete

On a fixed finite cluster, $\lVert T\rVert/U\ll1$ controls the displayed norm expansion; the local $t/U$ hierarchy is the appropriate starting point for an extensive system. In either case, truncating the result to projected hopping plus pairwise exchange makes an additional approximation. A two-step path $i\to j\to k$ through a virtual doublon generates correlated three-site hopping at order $t_{ij}t_{jk}/U$, the same order as $J$. At fourth order, one obtains corrections to pairwise exchange and cyclic exchange around plaquettes. Longer-range parent hoppings produce $t'$, $t''$, and mixed exchange processes.

Two examples of second-order hopping. Both pass through a doubly occupied site at energy cost $U$ and return to the projected space. Returning to the same bond can exchange spins; continuing to a third site moves the hole. Each pictured path contributes at scale $t^2/U$. The full effective operator includes the other allowed paths and their fermionic signs; the drawing does not specify its numerical coefficient.

Consequently, these statements are different:

1. the no-doublon effective theory is organized in powers of $t/U$;
2. the nearest-neighbor two-parameter t–J Hamiltonian is accurate for a chosen observable;
3. a particular material is represented by a one-band Hubbard parent.

The first can be controlled while the second or third fails.

No Double Occupancy Is a Statement About the Effective Orbital

Each projected site has the three states

$$\lvert 0\rangle, \qquad \lvert\uparrow\rangle, \qquad \lvert\downarrow\rangle.$$

The excluded fourth state is a doublon of the chosen effective orbital. It need not correspond to two electrons on one atom, nor does the projection say that all microscopic charge fluctuations vanish. A Wannier orbital may contain metal and ligand weight, and a local low-energy state may be a composite of several atomic orbitals.

This distinction is essential in cuprates. In a hole representation, the insulating parent has approximately one $d_{x^2-y^2}$ hole on each Cu site, whereas an added hole has substantial O $2p$ character. If a ligand hole and the neighboring Cu spin bind into a low-energy singlet, that composite can be represented as an empty site of an effective one-band model. Projecting the resulting band does not project oxygen out of the real charge density.

Filling and the projected sum rule

For $N_s$ effective sites, define the hole concentration

$$x = 1-\frac{1}{N_s}\sum_i\langle n_i\rangle .$$

The projected space then contains $(1-x)N_s$ electrons on average and $xN_s$ empty sites. Summing over spin, the zero-temperature spectral weights of a strictly projected electron operator obey

$$\begin{array}{rcl} W_{\mathrm{rem}}&=&1-x,\\[3pt] W_{\mathrm{add}}&=&2x,\\[3pt] W_{\mathrm{low}}&=&1+x. \end{array}$$

Removal acts on occupied sites; addition may place either spin on an empty site. The missing weight belongs to high-energy sectors absent from the t–J Hilbert space. A strict t–J model therefore cannot by itself predict the upper Hubbard band or the transfer of weight between retained and discarded sectors. Those questions require the parent Hubbard or charge-transfer model together with a consistently transformed observable.

Antiferromagnetic Exchange in a Material

In the one-band nearest-neighbor limit, $J=4t^2/U$ is a clean benchmark. In a material, however, the measured exchange can contain several virtual paths:

• metal–ligand–metal superexchange;
• direct overlap and intersite Coulomb exchange;
• Hund coupling on metal or ligand orbitals;
• multiple active metal orbitals;
• spin–orbit-generated anisotropic exchange;
• cyclic processes around a plaquette.

For a three-band Cu–O model written in hole language, let $\Delta_{pd}=\epsilon_p-\epsilon_d>0$ be the cost of moving a hole from Cu to O, $t_{pd}$ the Cu–O hopping, and $U_d,U_p$ the local repulsions. In a simple Cu–O–Cu cluster, take the positive excitation energies $\Delta_{pd}$, $U_d$ and $2\Delta_{pd}+U_p$ much larger than $\lvert t_{pd}\rvert$. Omitting intersite repulsion and oxygen–oxygen hopping, fourth-order perturbation theory gives the scale

$$J_{\mathrm{CuO}} \sim \frac{4t_{pd}^{\,4}}{\Delta_{pd}^{\,2}} \left( \frac{1}{U_d} + \frac{2}{2\Delta_{pd}+U_p} \right),$$

The two denominators in parentheses represent intermediate states with two holes on Cu or on O. This expression illustrates the virtual paths; it is not a generally accurate numerical formula for a cuprate. Oxygen–oxygen hopping, geometry and higher orders can change the answer substantially (Eskes and Jefferson, 1993).

Inelastic neutron scattering and resonant inelastic x-ray scattering constrain the spin dispersion. For La$_2$CuO$_4$, the zone-boundary dispersion is evidence that nearest-neighbor Heisenberg exchange alone is insufficient; longer-range or cyclic exchange can change this dispersion. Coldea and collaborators (2001) found a consistent description with ring exchange; the dispersion alone does not identify a unique microscopic Hamiltonian. A fit value called $J$ is therefore meaningful only together with the fitted Hamiltonian and momentum range. The broader taxonomy belongs in Exchange Interactions.

From a Cu–O Model to One Effective Band

The minimal orbital starting point for a CuO$_2$ plane retains one Cu $d_{x^2-y^2}$ orbital and the neighboring O $p_x,p_y$ orbitals. Schematically,

$$H_{pd} ={} \epsilon_d\sum_i n_i^d + \epsilon_p\sum_{\ell}n_\ell^p + H_{t_{pd}} + H_{t_{pp}} + U_d\sum_i n_{i\uparrow}^d n_{i\downarrow}^d + U_p\sum_\ell n_{\ell\uparrow}^p n_{\ell\downarrow}^p + H_{V_{pd}} .$$

This is a charge-transfer problem, not merely a one-band Hubbard model with a large onsite penalty. A one-band reduction requires an additional local-state argument.

Zhang–Rice singlet

Let $L_{i\sigma}^{\dagger}$ create a normalized, symmetry-adapted ligand-hole combination surrounding Cu site $i$ (Zhang and Rice, 1988). The local spin singlet has the form

$$\lvert\mathrm{ZR};i\rangle = \frac{1}{\sqrt2} \left( d_{i\uparrow}^{\dagger}L_{i\downarrow}^{\dagger} - d_{i\downarrow}^{\dagger}L_{i\uparrow}^{\dagger} \right) \lvert\mathrm{vac}\rangle .$$

If this state is separated from local triplets, nonbonding oxygen states, and other orbital configurations, it can serve as the spinless “empty” state of a projected one-band description. Motion and overlap of these composite states then generate effective hoppings and interactions.

To use this singlet as an effective site, check the following physical conditions:

1. Local spectral isolation: the singlet must remain the relevant low-energy charged state over the doping, pressure, and temperature range of interest.
2. Orthogonalization: ligand combinations on neighboring Cu sites overlap, so constructing orthogonal effective orbitals changes hopping amplitudes and produces longer-range terms.
3. Orbital stability: apical oxygen, Cu $4s$, other Cu $3d$ orbitals, and lattice distortions must not enter the target window strongly.
4. Observable transformation: oxygen-resolved spectroscopy cannot be compared with a bare one-band operator without transforming that operator through the same reduction.
5. Energy-window discipline: a model that reproduces sub-eV spin and carrier dynamics need not reproduce several-eV charge-transfer excitations.

Extended hopping is not cosmetic

With square-lattice spacing set to one, the Fourier transform of the bare hopping is often written

$$\varepsilon(\mathbf k) ={} -2t(\cos k_x+\cos k_y) -4t'\cos k_x\cos k_y - 2t''(\cos2k_x+\cos2k_y) +\cdots .$$

This is a kinetic-energy parametrization, not the interacting quasiparticle dispersion; the projection and correlations still have to be treated. The ratios $t'/t$ and $t''/t$ change the location and curvature of low-energy states, magnetic frustration, stripe energetics, and pairing correlations in finite-cylinder calculations. Material trends in hopping range are tied to orbitals outside the nominal Cu $d_{x^2-y^2}$–O $p_\sigma$ manifold, including axial-orbital chemistry. Thus “same $J/t$” does not imply equivalent cuprates.

Doped Mott Insulators

At half filling, a projected carrier cannot hop to an occupied neighbor. A hole opens allowed first-order paths, but its motion permutes spins and accumulates fermionic signs. The resulting problem is not a dilute gas moving on a rigid antiferromagnet: charge motion reconstructs the spin correlations that in turn control charge coherence.

Possible effects depend on the parameters and phase:

• the coherent bandwidth can be much smaller than the bare hopping scale;
• spectral weight is redistributed over energies of order both $t$ and $J$;
• antiferromagnetic correlations may become incommensurate or short-ranged;
• holes may form polarons, stripes, bound pairs, or more homogeneous fluids depending on parameters and geometry;
• the signs and ranges of hopping matter, not only $J/t$;
• a finite cluster can favor an order that does not survive the two-dimensional and thermodynamic limits.

The t–J model is therefore a controlled descendant of a strong-coupling parent but generally a nonperturbative many-body problem in its own right.

Hole and electron doping are not mirror images

For real hopping with no Peierls flux on a bipartite lattice, a particle–hole transformation can preserve the sign of nearest-neighbor hopping after a sublattice gauge choice. Same-sublattice hopping behaves differently:

$$t_{\mathrm{nn}}^{(h)}=t_{\mathrm{nn}}, \qquad t_{\mathrm{nnn}}^{(h)}=-t_{\mathrm{nnn}}.$$

For complex oriented hopping, reversing the bond also complex-conjugates its amplitude; the simple sign rule above assumes real amplitudes. This is one reason $t'$ is central to electron–hole asymmetry in one-band studies. Real cuprates add a deeper asymmetry: a doped electron tends toward a spinless Cu $d^{10}$ configuration, whereas a doped hole has strong ligand character and may form a Zhang–Rice-like state. A single projected band can encode some of this difference through parameters and operators, but it should not erase the distinct microscopic charge distributions.

What doping changes

Doping is not just a chemical-potential shift. It can change screening, effective interactions, orbital occupancies, lattice constants, disorder, and the energy separation to discarded states. A fixed-parameter t–J calculation answers a conditional question:

Given this projected Hamiltonian, how does its state evolve with carrier number?

It does not by itself prove that one Hamiltonian with fixed $t_{ij}$ and $J_{ij}$ describes an entire experimental doping series.

Singlet Correlations and Superconductivity

Projected models provide a useful setting for studying the competition among spin, charge and pair correlations. That competition need not select the same phase for every hopping range, filling or geometry. The microscopic charge-transfer excitations removed by projection cannot be recovered by solving the projected model more accurately.

Why exchange suggests singlet pairing

Define the projected bond-singlet creator

$$B_{ij}^{\dagger} = \widetilde c_{i\uparrow}^{\dagger} \widetilde c_{j\downarrow}^{\dagger} - \widetilde c_{i\downarrow}^{\dagger} \widetilde c_{j\uparrow}^{\dagger}.$$

Within the projected space,

$$\mathbf S_i\cdot\mathbf S_j -\frac14n_i n_j = -\frac12 B_{ij}^{\dagger}B_{ij}.$$

The exchange interaction is therefore attractive in the occupied-bond singlet channel. A mean-field decoupling can produce a $d_{x^2-y^2}$ pattern because horizontal and vertical bond amplitudes may take opposite signs. But a local attractive channel is not yet superconductivity. A reliable bulk claim requires long-distance pair correlations or phase stiffness, careful comparison with charge order, control of truncation and boundary errors, and an extrapolation in both system length and width.

For example, Jiang, Weng, and Kivelson (2018) and Lu and collaborators (2024) found regimes with intertwined charge and pairing correlations on finite cylinders and, for selected model parameters, algebraically decaying $d$-wave pair correlations. Their continuation to the isotropic two-dimensional thermodynamic limit and their quantitative material interpretation remain open questions. Read such results together with the cylinder width, boundary conditions, couplings and extrapolation method.

Comparing with Measurements

Different probes constrain different parts of the effective description. Agreement across several probes is more informative than refitting a different Hamiltonian to each one.

Measurement	What the comparison tests
Neutron or resonant x-ray spin spectra	Exchange range, anisotropy and cyclic terms; compare the whole dispersion, not only its slope.
Angle-resolved photoemission	Carrier dispersion and spectral weight, including interaction effects and photoemission matrix elements.
Optical and x-ray spectral weights	The measured energy window and transformed operators; discarded charge states may be essential.
Charge or spin diffraction	Ordering wave vector and correlation length, with temperature, disorder and finite-size effects.
Phase stiffness or Josephson response	Coherent superconductivity, beyond a short-range bond-singlet preference.

A practical example is a model fitted to the long-wavelength spin-wave velocity. It can still fail at the zone boundary or in oxygen-resolved spectra. The first discrepancy may require additional exchange terms; the second may require the oxygen content of the measured operator or an explicit Cu–O description. Exercise 5 develops this comparison.

Exercises

1. Order counting in the strong-coupling reduction

For a doped one-band Hubbard model, identify the leading orders in $t/U$ of projected nearest-neighbor hopping, exchange, three-site hopping, and plaquette ring exchange.

Solution

Projected hopping connects two states in the no-doublon sector directly, so its amplitude is order $t$.

Exchange and three-site hopping each require two hops with one virtual doublon denominator. Their scales are

$$J\sim\frac{t^2}{U}, \qquad t_{\mathrm{3s}}\sim\frac{t^2}{U}.$$

A four-site cyclic permutation requires four hops and three high-energy denominators in the simplest one-band process, so

$$K_{\mathrm{ring}} \sim \frac{t^4}{U^3}.$$

Numerical coefficients and additional denominators depend on the lattice and microscopic model. The central conclusion is that omitting three-site hopping is not justified by one extra power of $t/U$ relative to exchange.

2. Projected spectral-weight sum rule

A projected lattice has hole concentration $x$. Derive the total low-energy electron-removal and electron-addition weights per site, summed over spin.

Solution

An electron can be removed only from an occupied site. The average occupied fraction is $1-x$, hence

$$W_{\mathrm{rem}}=1-x.$$

An electron can be added only to an empty site. Each empty site accepts either spin, so

$$W_{\mathrm{add}}=2x.$$

Thus the projected low-energy total is

$$W_{\mathrm{low}} = W_{\mathrm{rem}}+W_{\mathrm{add}} = 1+x.$$

The result differs from the total two-spin-orbital weight $2$ of an unprojected one-band model. The missing $1-x$ resides in discarded high-energy addition states. At finite $U$, virtual mixing also transfers spectral weight dynamically, which is beyond the strict projected counting.

3. Particle–hole transformation of t′

On a square bipartite lattice with real hopping and no Peierls flux, take $c_{i\sigma}\mapsto\eta_i h_{i\sigma}^{\dagger}$ with $\eta_i=+1$ on sublattice A and $-1$ on sublattice B. Show why nearest-neighbor hopping preserves its sign while next-nearest-neighbor hopping changes sign in the hole Hamiltonian.

Solution

For $i\ne j$,

$$c_{i\sigma}^{\dagger}c_{j\sigma} \mapsto \eta_i\eta_j h_{i\sigma}h_{j\sigma}^{\dagger} = -\eta_i\eta_j h_{j\sigma}^{\dagger}h_{i\sigma}.$$

Comparing with the standard hopping form shows that the transformed amplitude is

$$t_{ij}^{(h)} = -\eta_i\eta_j t_{ij}.$$

Nearest neighbors lie on opposite sublattices, so $\eta_i\eta_j=-1$ and $t^{(h)}=t$. Next-nearest neighbors lie on the same sublattice, so $\eta_i\eta_j=+1$ and $t'^{(h)}=-t'$. For a complex coefficient in $-t_{ij}c_i^\dagger c_j+\mathrm{h.c.}$, the corresponding oriented hole coefficient is instead $t_{ij}^{(h)}=-\eta_i\eta_j t_{ij}^{*}$. Longer-range hopping therefore carries physically important electron–hole asymmetry.

4. Bond exchange and the singlet channel

Verify the identity

$$\mathbf S_i\cdot\mathbf S_j -\frac14 n_i n_j = -\frac12 B_{ij}^{\dagger}B_{ij}$$

on the empty, singly occupied, triplet, and singlet bond sectors.

Solution

If fewer than two electrons occupy the bond, both sides vanish: there is no pair for $B_{ij}$ to remove, and either $n_in_j=0$ or one spin operator is zero.

For two occupied sites, use

$$\mathbf S_i\cdot\mathbf S_j = \frac12 \left[ S_{\mathrm{tot}}(S_{\mathrm{tot}}+1) -\frac32 \right].$$

On a triplet, $S_{\mathrm{tot}}=1$, so the left side is $1/4-1/4=0$. The singlet annihilator also kills a triplet.

On a singlet, $S_{\mathrm{tot}}=0$, so the left side is $-3/4-1/4=-1$. With the unnormalized definition used above, $B_{ij}$ maps the normalized singlet to $\sqrt2\lvert0\rangle$, hence $B_{ij}^{\dagger}B_{ij}$ has eigenvalue $2$ and the right side is also $-1$.

5. Which parts of a material does the model explain?

Suppose a one-band t–J calculation reproduces the low-energy spin-wave velocity of a cuprate after fitting $J$, but misses the zone-boundary dispersion, oxygen-resolved x-ray weight, and doping dependence of the optical sum. Explain what each agreement or discrepancy tells us, and propose a useful model revision.

Solution

Matching the spin-wave velocity calibrates one low-energy combination of exchange parameters. It does not validate the full Hamiltonian.

The zone-boundary residual is structured in momentum and points toward longer-range or cyclic exchange. Missing oxygen-resolved weight is expected if the observable embedding or the oxygen orbitals themselves were discarded. Failure of the optical doping trend indicates that fixed projected-space sum rules or fixed parameters are inadequate over the stated window.

A useful revision would first add the exchange operators demanded by the spin dispersion and transform the x-ray and optical operators consistently. If the high-energy and doping-dependent residuals remain, the active space should be enlarged to a Cu–O charge-transfer model, with doping-dependent screening and orbital content tested rather than absorbed into an unconstrained fit.

Related Topics

• Charge and Spin Density Waves connects stripes and other finite-wavevector orders to material measurements.
• Equal-Time Correlations explains finite-size tests for spin, charge and pair order.
• Variational Many-Body States develops projected trial states and their limitations.

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