Superexchange and the t–J Projection

At large positive $U$, a hop can create a virtual doublon–holon pair and a second hop can remove it. This process lowers a bond singlet but not the Pauli-blocked triplet, producing the antiferromagnetic scale $J=4t^2/U$. The derivation also reveals what the familiar t–J shorthand can hide: projected hopping, three-site motion, transformed observables, and a precise energy range of validity.

Required background. Use the Hubbard model and its strong-coupling sector. Helpful background. Integrating out heavy fields supplies the general matching logic.

Doublon sectors and the Schrieffer–Wolff step

Work at fixed particle number, so the chemical-potential term is an irrelevant constant. Use the ordered-pair hopping convention of the prerequisite page, with $t_{ji}=t_{ij}^*$, and let $\bar\sigma$ denote the spin opposite to $\sigma$. Write the remaining Hubbard Hamiltonian as $H=H_U+T$, where

$$H_U=U D,\qquad D=\sum_i n_{i\uparrow}n_{i\downarrow}.$$

Decompose the hopping according to how it changes the doublon number:

$$T=T_{-1}+T_0+T_{+1}, \qquad [D,T_m]=mT_m.$$

$T_{+1}$ creates one extra doublon, $T_{-1}=T_{+1}^\dagger$ removes one, and $T_0$ preserves $D$ MacDonald, Girvin, and Yoshioka 1988, p. 9753, Eqs. (3)–(4).

For example, with $h_{i\sigma}=1-n_{i\sigma}$,

$$\begin{aligned} T_{+1} &=-\sum_{ij\sigma}t_{ij}\, n_{i\bar\sigma}c_{i\sigma}^\dagger c_{j\sigma}h_{j\bar\sigma},\\ T_0 &=-\sum_{ij\sigma}t_{ij}\left( h_{i\bar\sigma}c_{i\sigma}^\dagger c_{j\sigma}h_{j\bar\sigma} +n_{i\bar\sigma}c_{i\sigma}^\dagger c_{j\sigma}n_{j\bar\sigma} \right). \end{aligned}$$

The occupation factors show which process changes the doublon count. Let $P$ project onto states with no doublons and $Q=1-P$. The control parameter is set by the separation to the eliminated sector, not by the thermodynamic charge gap. More precisely, choose a low-energy window inside $P$, let $\Delta_{PQ}$ be its smallest positive energy detuning to a $Q$ state connected by hopping, and require the relevant local hopping matrix elements to obey

$$\lambda\sim\frac{t_{\mathrm{mix}}}{\Delta_{PQ}}\ll1, \qquad E,\,k_{\mathrm B}T_{\mathrm{phys}}\ll\Delta_{PQ}.$$

Here $T_{\mathrm{phys}}$ is the physical temperature; the symbol $T$ without a subscript continues to denote the hopping operator. For the one-band Hubbard model at leading order, $\Delta_{PQ}$ is replaced by $U$; a conservative lattice-scale shorthand is that the bandwidth is small compared with $U$. A hole-doped projected sector can be compressible while this additional-doublon detuning remains large.

Choose an anti-Hermitian generator, $S^\dagger=-S$, for the Schrieffer–Wolff transformation

$$\overline H=e^SHe^{-S}, \qquad S^{(1)}=\frac{T_{+1}-T_{-1}}{U},$$

so that $[S^{(1)},H_U]=-(T_{+1}+T_{-1})$ cancels the hopping that mixes $P$ and $Q$ to first order. In the projected Baker–Campbell–Hausdorff expansion, $P[S^{(1)},T_0]P=0$ because that commutator changes the doublon number. The surviving second-order step is therefore

$$\begin{aligned} H_{\mathrm{eff}} &=P\overline H P\\ &=PT_0P +\frac12P[S^{(1)},T_{+1}+T_{-1}]P +O(t^3/U^2)\\ &=PT_0P-\frac1U\,PT_{-1}T_{+1}P +O(t^3/U^2). \end{aligned}$$

The first line exposes where the factor $1/2$ enters; the second uses $T_{-1}P=0$. This fixes the sign and coefficient rather than importing them from the final t–J Hamiltonian MacDonald, Girvin, and Yoshioka 1988, pp. 9753–9754, Eqs. (5)–(8), (12), and (17).

This is degenerate perturbation theory written as a unitary change of variables. With $H_U=UD$, the unperturbed no-doublon sector has $E_0=0$. If all intermediate states cost $U$,

$$H_{\mathrm{eff}}^{(2)} =-PTQ\frac1{H_U-E_0}QTP.$$

The negative sign is the usual lowering from virtual states above the low-energy sector. This resolvent form also shows what changes in an extended or multiorbital model: when intermediate energies are inequivalent, keep the denominator as an operator rather than replacing it by $U$. Canonical, resolvent, and properly orthonormalized degenerate-perturbation constructions agree through a low-energy unitary transformation Chernyshev et al. 2004, § I, p. 235111-2; § IV.D, p. 235111-6; § VIII, p. 235111-10.

Bond singlet and the factor of four

The two-site Hubbard model supplies an exact check. Choose a local gauge in which the bond hopping is real and positive, and use the normalized states

$$|s\rangle=\frac{|\uparrow,\downarrow\rangle-|\downarrow,\uparrow\rangle}{\sqrt2}, \qquad |d\rangle=\frac{|\uparrow\downarrow,0\rangle+|0,\uparrow\downarrow\rangle}{\sqrt2}.$$

The relative Fock-state phases are chosen so that the off-diagonal matrix element below is negative; changing the phase of $|d\rangle$ flips that sign without changing any energy. At two electrons, all three triplets have energy zero relative to the singly occupied sector. The orthogonal ionic singlet

$$|d_-\rangle =\frac{|\uparrow\downarrow,0\rangle-|0,\uparrow\downarrow\rangle}{\sqrt2}$$

decouples and has energy $U$. The coupled ground-state singlet subblock is

$$\left.H\right|_{\{s,d\}}= \begin{pmatrix} 0&-2t\\ -2t&U \end{pmatrix}.$$

Its lower eigenvalue is

$$E_s=\frac{U-\sqrt{U^2+16t^2}}{2} =-\frac{4t^2}{U} +\frac{16t^4}{U^3} +O(t^6/U^5),$$

whereas $E_t=0$. After fixing the triplet correction to zero and requiring the correction to vanish on an empty bond, the spin-rotation-invariant bond operator with this leading $O(t^2/U)$ splitting is

$$H_J=\sum_{\langle ij\rangle}J_{ij} \left(\mathbf S_i\cdot\mathbf S_j-\frac14n_in_j\right), \qquad J_{ij}=\frac{4\lvert t_{ij}\rvert^2}{U}.$$

Here $\mathbf S_i=\tfrac12\sum_{\alpha\beta}c_{i\alpha}^\dagger\boldsymbol\sigma_{\alpha\beta}c_{i\beta}$, and all operators are understood inside $P$. For two occupied sites, the bracket is $-1$ in the singlet and $0$ in a triplet. This checks the sign, the factor of four, and the density term independently of the canonical-transformation algebra; the same second-order operator follows directly from the strong-coupling expansion Eskes and Eder 1996, p. R14226, Eq. (2). At exactly one electron per site the density term is a constant; with holes it is essential.

The exact dimer splitting also calibrates the first omitted correction:

$$J_{\mathrm{dimer}} =E_t-E_s =\frac{4t^2}{U}\left[1-4\left(\frac tU\right)^2 +O\!\left(\frac{t^4}{U^4}\right)\right].$$

Thus the leading Heisenberg result overestimates the exact dimer splitting by a relative $4(t/U)^2+O((t/U)^4)$.

Projected hopping and three-site motion

At half filling, $PT_0P=0$ because every neighboring site is occupied. With holes, hopping inside the no-doublon sector is allowed. Define

$$\widetilde c_{i\sigma}=c_{i\sigma}(1-n_{i\bar\sigma}).$$

For uniform real nearest-neighbor hopping, the effective Hamiltonian through order $t^2/U$ is

$$\begin{aligned} H_{tJ}={}&-t\sum_{\langle ij\rangle\sigma} \left(\widetilde c_{i\sigma}^\dagger\widetilde c_{j\sigma} +\mathrm{h.c.}\right)+H_J+H_{3\mathrm s},\\ H_{3\mathrm s}={}&-\frac{t^2}{U} \sum_j\sum_{\substack{i,k\in N(j)\\i\ne k}}\sum_\sigma \left( \widetilde c_{i\sigma}^\dagger n_{j\bar\sigma}\widetilde c_{k\sigma} -\widetilde c_{i\sigma}^\dagger \widetilde c_{j\bar\sigma}^\dagger \widetilde c_{j\sigma}\widetilde c_{k\bar\sigma} \right). \end{aligned}$$

The sum runs over ordered length-two paths $k\to j\to i$, so it already includes the reverse path. Under $i\leftrightarrow k$, the first operator maps to its adjoint; the second does so under $(i,k,\sigma)\leftrightarrow(k,i,\bar\sigma)$. No extra h.c. or factor $1/2$ is therefore needed. The first term moves a carrier through an oppositely occupied intermediate site, while the second includes a spin exchange during that motion. Both have coefficient magnitude $t^2/U=J/4$, with the displayed relative sign. They vanish in the fully occupied projected sector because there is no destination hole, but at finite hole density they are of the same formal order as $H_J$. Dropping them can be a useful model choice only after checking the observable and doping dependence; it is not justified by the $t/U$ power count alone.

There is a direct doped-sector calibration. On an ordered path $k\to j\to i$ with a hole at $i$, spin $\bar\sigma$ at $j$, and spin $\sigma$ at $k$, the density-assisted term moves the $\sigma$ carrier from $k$ to $i$ with matrix-element magnitude

$$\frac{t^2}{U}=\frac J4.$$

The extensive contribution still depends on the available holes and spin correlations, but the local amplitude is not of higher order than exchange. This full second-order form and its possible importance away from half filling are displayed in Eskes and Eder 1996, pp. R14226–R14227, Eq. (2) and the following discussion; a one-dimensional numerical stress test is given by Ammon, Troyer, and Tsunetsugu 1995, pp. 629–631.

For nonuniform hopping, derive the coefficient from the ordered product of the two hop amplitudes and the actual intermediate-state denominator; do not replace it by $t^2/U$ by inspection. Open-path phases are gauge covariant together with the carrier operator, whereas physical flux dependence is encoded by interference around closed paths. The chapter reduction map and correlated-electron validity table place these projected terms beside, rather than in place of, the parent Hubbard charge sector.

Matching observables as well as energies

The same unitary transformation acts on every observable. Matching the Hamiltonian without matching the operators preserves low-energy eigenvalues but generally gives the wrong matrix elements Chernyshev et al. 2004, § IV.C, p. 235111-6:

$$\mathcal O_{\mathrm{eff}} =Pe^S\mathcal Oe^{-S}P =P\left(\mathcal O+[S^{(1)},\mathcal O]+\cdots\right)P.$$

For example, the $P$–$P$ block of the electron annihilation operator describes removal that stays within the low-energy no-doublon sector—the lower-Hubbard-band contribution at half filling:

$$\left(c_{i\sigma}\right)_{\mathrm{eff}} =\widetilde c_{i\sigma} +\frac1U P[T_{+1}-T_{-1},c_{i\sigma}]P +O(t^2/U^2).$$

The commutator is a nearest-neighbor composite correction of relative order $t/U$: it describes the virtual rearrangement of charge and spin while an electron is removed. A neighbor sum can make its effect on spectral weight much larger than the prefactor alone suggests Eskes and Eder 1996, p. R14227, Eq. (3) and the following discussion.

The displayed $P$–$P$ block does not contain an addition transition into the extra-doublon sector. Algebraically, that response lives in the separate block

$$Qe^S c_{i\sigma}^\dagger e^{-S}P$$

and its adjoint, which connect the low-energy sector to states containing a doublon. A Hamiltonian projected entirely into $P$ can predict low-energy dynamics, but it cannot reproduce the full spectrum across energy $U$ merely by using a bare projected electron operator Eskes and Eder 1996, p. R14227, text following Eq. (3). The Hubbard-band and spectral-weight page owns the full pole, sum-rule, and doping-transfer analysis.

Control, higher orders, and breakdown

The actual small parameter is the matrix element that mixes $P$ and $Q$ divided by their separation. The familiar $t/U\ll1$ criterion is the uniform one-band shorthand, and the effective theory is used only for energies and temperatures well below the eliminated scale.

• Hole doping activates projected motion but does not by itself close the gap to states with an additional doublon. The low-energy thermodynamic charge gap may be zero while the $P$–$Q$ separation remains of order $U$.
• The no-doublon space contains at most one electron per site. Electron doping beyond that filling therefore requires a particle–hole reformulation where available, or a different retained sector; it cannot be inserted into the same $P$ space unchanged.
• Large double occupancy or a small $P$–$Q$ separation means that $Q$ is not safely eliminable. If another local orbital lies near the retained band, return to the effective-Hamiltonian construction and enlarge the low-energy space.
• At generic filling the omitted Hamiltonian begins at $O(t^3/U^2)$. At exactly one electron per site with real nearest-neighbor hopping on a bipartite lattice, odd orders vanish and the leading $O(t^4/U^3)$ terms correct exchange, generate longer-range exchange, and—where four-site loops exist—generate ring exchange. Triangular loops can contribute at third order; Peierls flux modifies the first closed-loop term permitted by the lattice geometry.
• A projected theory assumes the strong-coupling separation; it cannot describe the interaction-driven closing of that separation or determine the full chemical-potential jump. Its observables must also possess an order-by-order expansion Chernyshev et al. 2004, § VIII, p. 235111-10.
• Solving the t–J Hamiltonian remains a nonperturbative many-body problem. Control over its derivation is not control over an approximate solver used afterward.

Common pitfalls

Treating the sign of the generator as physical. A source using $e^{-S}He^S$ instead of $e^SHe^{-S}$ reverses the displayed sign of $S$. Compare the cancelled off-diagonal block and the resulting $H_{\mathrm{eff}}$, not the generator in isolation.

Calling the bare t–J model the complete second-order Hubbard theory. Away from half filling, $H_{3\mathrm s}$ is present at the same order as $H_J$. Omitting it defines a simpler model whose consequences must be checked rather than a more accurate power counting.

Using bare projected observables. The transformation changes states and operators together. A bare $\widetilde c_{i\sigma}$ misses virtual-charge contributions even within the low-energy spectral weight and cannot create the upper-Hubbard-band response.

Adding exchange twice. The exchange contribution $H_J$ is generated, together with $H_{3\mathrm s}$ away from half filling, by $-PT_{-1}T_{+1}P/U$. At half filling $H_{3\mathrm s}$ vanishes and this second-order operator reduces to $H_J$. Adding the same $J$ to an unprojected Hubbard Hamiltonian counts that virtual process twice unless the extra term represents a separately matched direct exchange and the overlap is subtracted.

Exercises

Diagonalize the displayed two-state Hubbard-dimer matrix and recover the leading singlet–triplet splitting.

Solution

The characteristic equation is $E(E-U)-4t^2=0$, so

$$E_\pm=\frac{U\pm\sqrt{U^2+16t^2}}2.$$

Expanding the lower root at $t/U\ll1$ gives $E_-=-4t^2/U+16t^4/U^3+\cdots$. Since the triplet remains at zero, the leading splitting is $J=4t^2/U$ with the singlet lower.

Why is the density term $-Jn_in_j/4$ necessary in the projected bond Hamiltonian?

Solution

For two occupied sites, $\mathbf S_i\cdot\mathbf S_j$ is $1/4$ in a triplet and $-3/4$ in a singlet. Subtracting $n_in_j/4$ makes the triplet shift zero and the singlet shift $-J$, matching the dimer. If either site is empty, both $\mathbf S_i\cdot\mathbf S_j$ and $n_in_j$ vanish, so the virtual bond correction also vanishes.

At what order do projected hopping, exchange, and three-site motion enter, and which survive in the no-hole half-filled sector?

Solution

Projected hopping is order $t$, while exchange and three-site motion are order $t^2/U$. At exactly one electron per site, projected hopping has no empty destination and the three-site term has no final hole, so both vanish. Exchange survives because two hops can create and then remove a virtual doublon. With holes, all three terms act; the three-site contribution cannot be discarded relative to exchange by power counting alone.

Starting from $S^{(1)}=(T_{+1}-T_{-1})/U$, derive the projected second-order term in the Baker–Campbell–Hausdorff expansion. Why is its sign negative?

Solution

Because $T_0$ preserves the doublon number while $S^{(1)}$ changes it, $P[S^{(1)},T_0]P=0$. The remaining commutator is

$$\frac12P[S^{(1)},T_{+1}+T_{-1}]P =\frac1U P[T_{+1},T_{-1}]P.$$

Now $T_{-1}P=0$: a state with no doublons cannot lose one. Therefore $PT_{+1}T_{-1}P=0$, leaving

$$-\frac1U PT_{-1}T_{+1}P.$$

The rightmost hop creates a virtual doublon and the leftmost hop removes it. The negative sign is the energy lowering produced by mixing with states an energy $U$ above the retained sector.

Consider a hole-doped large-$U$ state that is compressible, so its thermodynamic charge gap is zero, while creating an additional doublon still costs an energy of order $U$. A student rejects the projection and then uses the bare operator $\widetilde c_{i\sigma}$ to calculate the entire one-particle spectrum. Diagnose both mistakes.

Solution

Compressibility concerns particle addition and removal within the retained low-energy sector. It does not by itself close the $P$–$Q$ separation to states with an additional doublon, so the transformation can remain controlled when the mixing matrix element is small compared with that separation.

The bare projected operator gives only the leading low-energy contribution. It misses the commutator correction in $Pe^Sc_{i\sigma}e^{-S}P$ and cannot create the separate $Q$–$P$ transitions that carry upper-Hubbard-band weight. A low-energy calculation must use the matched projected operator; a calculation of the full spectrum must also retain or reconstruct the high-energy sector and its matched transition blocks.

Continue the strong-coupling route

The Hubbard-band and spectral-weight analysis follows how the eliminated charge sector still controls high-energy poles and sum rules. The exchange and spin-Hamiltonian chapter develops the magnetic models generated at half filling, while the insulator comparison separates this Mott mechanism from band, Slater, charge-transfer, and Anderson alternatives.

References

• Beat Ammon, Matthias Troyer, and Hirokazu Tsunetsugu, “Effect of the Three-Site Hopping Term on the t–J Model,” Physical Review B 52 (1995) 629–636, doi:10.1103/PhysRevB.52.629; Open PDF.
• A. L. Chernyshev, D. Galanakis, P. Phillips, A. V. Rozhkov, and A.-M. S. Tremblay, “Higher Order Corrections to Effective Low-Energy Theories for Strongly Correlated Electron Systems,” Physical Review B 70 (2004) 235111, doi:10.1103/PhysRevB.70.235111; Open PDF.
• Henk Eskes and Robert Eder, “Hubbard Model versus t–J Model: The One-Particle Spectrum,” Physical Review B 54 (1996) R14226–R14229, doi:10.1103/PhysRevB.54.R14226; Open PDF.
• Allan H. MacDonald, Steven M. Girvin, and Daijiro Yoshioka, “$t/U$ Expansion for the Hubbard Model,” Physical Review B 37 (1988) 9753–9756, doi:10.1103/PhysRevB.37.9753.