Effective Hamiltonians in Many-Body Systems

An effective Hamiltonian acts in a deliberately retained subspace and reproduces specified low-energy predictions of a more detailed parent Hamiltonian to a declared accuracy. High-energy states are not simply forgotten. Virtual excursions through them generate energy shifts, exchange interactions, constrained hopping, potential scattering, multispin terms, and corrections to observables.

The central structure is

$$\text{low-energy state} \longrightarrow \text{virtual high-energy state} \longrightarrow \text{low-energy state}.$$

If the coupling for each step is $v$ and the intermediate-state cost is $\Delta$, the induced scale is typically

$$\frac{v^2}{\Delta}.$$

This estimate is only a beginning. A trustworthy reduction must specify:

• the parent Hamiltonian and all sign conventions;
• the retained projector $P$ and eliminated projector $Q$;
• the gap or energy denominators separating the sectors;
• the order and local parameter of the expansion;
• every generated operator retained or discarded;
• the energy, temperature, frequency, and filling window of use;
• the transformation of states and observables;
• the benchmark used to test the truncation.

Merely replacing $H$ by $PHP$ is generally not enough. That replacement keeps processes that never leave the model space but misses the virtual processes responsible for superexchange and Kondo exchange.

Helpful background. Projection methods explain the block elimination, and the Schrieffer–Wolff transformation derives its unitary form.

Retained and Eliminated Sectors

Let

$$P^\dagger=P,\quad Q^\dagger=Q,\quad P+Q=I, \qquad P^2=P, \qquad Q^2=Q, \qquad PQ=0.$$

The model space is $P\mathcal H$. The eliminated space is $Q\mathcal H$. Write

$$H=H_0+V,$$

where $H_0$ preserves both sectors:

$$[H_0,P]=0.$$

It is useful to split the perturbation into block-diagonal and block-off-diagonal pieces:

$$\begin{aligned} V_{\mathrm d} &= PVP+QVQ, \\ V_{\mathrm{od}} &= PVQ+QVP. \end{aligned}$$

$V_{\mathrm d}$ acts without changing sector. $V_{\mathrm{od}}$ creates or removes the high-energy configurations that will appear virtually.

The gap is between sectors

Suppose the retained manifold has reference energy $E_0$ and

$$Q(H_0-E_0)Q \ge \Delta Q, \qquad \Delta>0.$$

This does not require $P\mathcal H$ to be internally gapped. A half-filled Mott system can have a charge gap of order $U$ between $P$ and $Q$ while its retained spin sector is gapless in the thermodynamic limit.

The projection controls off-resonant excursions, while the retained sector still has its own low-energy dynamics. In an extended system, the entire many-body spectra of $P$ and $Q$ can overlap at high total energy. A local strong-coupling expansion then concerns specified low-energy states and local charge costs, not a uniform spectral separation of every state in the two full sectors.

Projectors encode physical assumptions

Examples include:

Parent problem	Retained sector $P$	Eliminated sector $Q$
half-filled large-$U$ Hubbard model	one fermion per site	states with doublon–hole pairs
doped large-$U$ Hubbard model	no double occupancy, holes allowed	states with at least one doublon
Anderson impurity in local-moment regime	singly occupied impurity doublet	empty and doubly occupied impurity
crystal-field multiplet	selected low multiplet	higher local multiplets
multiband lattice model	active bands or orbitals	remote bands or orbitals

A projector is not justified merely because its states are intuitively interesting. The omitted sector must remain sufficiently costly over the parameter and kinematic range of interest.

Second-Order Effective Hamiltonian

For a degenerate retained manifold at $E_0$, the leading energy-independent result is

$$H_{\mathrm{eff}} ={} E_0P + PVP - PVQ \left[ Q(H_0-E_0)Q \right]^{-1} QVP + O\!\left( \frac{V^3}{\Delta^2} \right).$$

The inverse acts only in $Q\mathcal H$. The second-order term has the path structure

$$P \xrightarrow{\ V\ } Q \xrightarrow{\ V\ } P.$$

For retained states $\lvert a\rangle$ and $\lvert b\rangle$ with a common energy $E_0$,

$$\langle a| H_{\mathrm{eff}}^{(2)} |b\rangle = \sum_{m\in Q} \frac{ \langle a|V|m\rangle \langle m|V|b\rangle }{ E_0-E_m }.$$

Every allowed intermediate state contributes. Relative phases, fermionic signs, spin selection rules, and unequal denominators can make virtual paths add or cancel.

Quasi-degenerate retained states

If $\lvert a\rangle$ and $\lvert b\rangle$ have nearby but unequal unperturbed energies, a standard Hermitian second-order form is

$$\langle a| H_{\mathrm{eff}}^{(2)} |b\rangle ={} \frac12 \sum_{m\in Q} \langle a|V|m\rangle \langle m|V|b\rangle \times \left[ \frac1{E_a-E_m} + \frac1{E_b-E_m} \right].$$

This is one consistent convention. Bloch, Löwdin, Van Vleck, Brillouin–Wigner, and Schrieffer–Wolff effective Hamiltonians can differ off shell or by a unitary transformation inside $P$. Non-Hermitian Bloch forms can additionally require a similarity transformation and an induced model-space metric; they are not all related by an ordinary unitary matrix in the original basis. Their coefficients should not be mixed term by term.

Quasi-Degenerate Perturbation Theory derives the general model-space treatment.

Schrieffer–Wolff View

Choose an anti-Hermitian generator

$$S^\dagger=-S$$

and transform

$$\widetilde H = e^SHe^{-S}.$$

The generator is chosen so that

$$P\widetilde H Q = 0$$

through the desired order. With $S=S_1+O(V^2)$, a convenient first-order convention is

$$[H_0,S_1] = V_{\mathrm{od}}.$$

Then the retained block through second order is

$$\begin{aligned} H_{\mathrm{eff}} ={}& P\widetilde H P \\ ={}& P(H_0+V_{\mathrm d})P + \frac12 P[S_1,V_{\mathrm{od}}]P + O(V^3). \end{aligned}$$

The projected-resolvent and Schrieffer–Wolff forms organize the same virtual physics differently. The unitary form has an important advantage: it also tells us how to transform states and observables.

The detailed construction, including sign conventions for $S$, belongs to Schrieffer–Wolff Transformation.

Why Many-Body Control Is Local

For an $L$-site lattice, the norm of the total perturbation often scales with $L$. A condition such as

$$\frac{\lVert V\rVert}{\Delta} \ll1$$

can therefore fail even when every local virtual process is weak. The useful control parameter is instead built from local matrix elements, gaps, coordination, and connected clusters.

For local coupling scale $v$ and coordination $z$, a rough diagnostic is

$$\epsilon_{\mathrm{loc}} \sim \frac{zv}{\Delta}.$$

This is not a universal theorem. It reminds us that one site can access several virtual channels. The actual coefficient depends on geometry, operator norms, statistics, and cancellations.

Linked support

At order $n$, a local perturbation can generate operators supported on connected clusters reached by $n$ elementary couplings. Thus:

• second order commonly generates bond exchange or correlated hopping;
• third order can detect triangular loops or complex hopping phases;
• fourth order can generate longer-range exchange and four-site ring exchange;
• operator range and body order generally grow with perturbative order.

Disconnected virtual processes contribute to extensive constants or factorized pieces. A linked-cluster organization is what keeps local effective interactions meaningful in the thermodynamic limit.

Energy density versus whole-spectrum accuracy

A truncated local Hamiltonian can approximate low-energy densities and local dynamics even though it does not approximate every eigenvalue of the exponentially large spectrum. Claims of control must name the quantity:

• ground-state energy density;
• a low-energy band of states;
• local observables for finite times;
• thermal observables below a cutoff;
• matrix elements within a symmetry sector.

“The effective Hamiltonian is accurate” is incomplete without this target.

Hubbard to Heisenberg

Consider the repulsive single-band Hubbard Hamiltonian

$$H = H_U+T,$$

with

$$H_U = U \sum_i n_{i\uparrow}n_{i\downarrow}, \qquad U>0,$$

and

$$T = - \sum_{\langle i,j\rangle,\sigma} \left( t_{ij} c_{i\sigma}^\dagger c_{j\sigma} + t_{ij}^* c_{j\sigma}^\dagger c_{i\sigma} \right).$$

Hubbard Model defines the model and its symmetries.

Half filling and the no-doublon sector

At one fermion per site, let $P_0$ project onto states with no double occupancy. With the total particle number fixed to the number of sites, every retained site is singly occupied.

A single hop necessarily creates one doublon and one hole, so

$$P_0TP_0 = 0.$$

The first nonzero dynamics is second order:

$$H_{\mathrm{eff}}^{(2)} = - P_0TQ \frac1{QH_UQ} QTP_0.$$

For the simplest uniform model, the relevant intermediate states cost $U$, giving

$$H_{\mathrm{eff}}^{(2)} = - \frac1U P_0TQT P_0.$$

One bond fixes the exchange

On a bond $\langle i,j\rangle$, define the dimensionless spin operator

$$\mathbf s_i = \frac12 \sum_{\alpha,\beta} c_{i\alpha}^\dagger \boldsymbol\sigma_{\alpha\beta} c_{i\beta}.$$

The singlet projector for two singly occupied sites is

$$P_{S,ij} = \frac14 - \mathbf s_i\cdot\mathbf s_j.$$

The spin singlet couples to the two virtual doublon–hole configurations. Fermionic antisymmetry makes the corresponding triplet amplitudes vanish or cancel. The shifts are

$$\Delta E_S = - \frac{4|t_{ij}|^2}{U}, \qquad \Delta E_T = 0.$$

Therefore the bond operator is

$$\begin{aligned} H_{ij}^{(2)} &= -J_{ij}P_{S,ij} \\ &= J_{ij} \left( \mathbf s_i\cdot\mathbf s_j - \frac14 \right), \end{aligned}$$

with

$$J_{ij} = \frac{4|t_{ij}|^2}{U} >0.$$

Positive $J_{ij}$ is antiferromagnetic in this convention. The exact dimer matrix, singlet–triplet spectrum, and finite-$t/U$ benchmark belong to Effective Hamiltonians in Quantum Matter.

Lattice Hamiltonian

Summing bond contributions gives

$$H_{\mathrm{spin}} = \sum_{\langle i,j\rangle} J_{ij} \left( \mathbf s_i\cdot\mathbf s_j - \frac14 \right) + \delta H.$$

At fixed half filling, the constant $-J_{ij}/4$ per bond can be dropped when only state differences and dynamics are needed. The remaining operator is the antiferromagnetic Heisenberg model. Retain the constant for absolute free energies and parameter derivatives, including the double occupancy obtained by differentiating with respect to $U$.

The hierarchy of scales is

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

Charge excitations become costly at scale $U$, while spin dynamics survives at the parametrically smaller scale $J$. A system can therefore display well-formed local moments at temperatures below $U$ but still remain thermally disordered relative to $J$.

Virtual does not mean absent

The block-diagonal low-energy state has no doublon by construction. The physical state is

$$|\Psi\rangle = e^{-S} |\Psi_{\mathrm{eff}}\rangle.$$

Its doublon–hole amplitude is generally

$$O\!\left( \frac{t}{U} \right),$$

so its double-occupancy probability is

$$O\!\left( \frac{t^2}{U^2} \right).$$

The spin Hamiltonian freezes real low-energy charge motion at half filling; it does not claim that the parent Hubbard eigenstate has exactly zero virtual charge fluctuation.

Away from Half Filling

Below half filling, a fermion can hop into an empty site without creating a doublon. Consequently,

$$P_0TP_0 \ne 0.$$

Define projected fermion operators

$$\widetilde c_{i\sigma} = P_0c_{i\sigma}P_0 = c_{i\sigma} \left( 1-n_{i\bar\sigma} \right)$$

within the no-doublon Hilbert space. Electron doping above one particle per site needs the corresponding particle–hole-transformed retained space; $P_0$ would annihilate every state at $N>L$. The leading hole-doped strong-coupling Hamiltonian has the structure

$$H_{t\text{–}J} ={} - \sum_{\langle i,j\rangle,\sigma} \left( t_{ij} \widetilde c_{i\sigma}^\dagger \widetilde c_{j\sigma} + \mathrm{h.c.} \right) + \sum_{\langle i,j\rangle} J_{ij} \left( \mathbf s_i\cdot\mathbf s_j - \frac14 n_in_j \right) + H_{\mathrm{3site}} + \cdots.$$

The projected hopping is first order in $t$. Exchange and three-site correlated hopping have energy scale $t^2/U$, second order in the hopping. Omitting $H_{\mathrm{3site}}$ defines an additional approximation; it is not part of the projection identity.

t–J Model Preview develops the constrained Hilbert space and projected-operator algebra.

Why the density term matters

At half filling, $n_in_j=1$ on every retained bond, so

$$-\frac14J_{ij}n_in_j$$

is a constant. With holes, it depends on configuration and cannot be dropped without changing the Hamiltonian.

This is a common example of a term that looks unimportant in one sector and becomes dynamical when the retained Hilbert space changes.

Corrections to the Simplest Hubbard Reduction

The formula

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

has a precise domain: repulsive one-band Hubbard dynamics, a well-separated no-doublon sector, and spin-independent hopping in the stated convention.

Beyond that limit:

• site-energy differences produce unequal doublon denominators;
• bond-dependent hopping produces bond-dependent $J_{ij}$;
• multiple orbitals and Hund coupling can alter the sign and tensor structure;
• spin–orbit coupling can generate anisotropic and Dzyaloshinskii–Moriya exchange;
• complex hopping around loops can generate flux-sensitive cyclic or scalar-chirality terms;
• triangular loops can contribute at third order;
• on bipartite lattices with real nearest-neighbor hopping, leading half-filled corrections commonly appear at fourth order as longer-range and ring exchange;
• near a charge-transfer resonance, the relevant ligand or orbital states must be retained rather than hidden in one denominator.

Writing only the Heisenberg term is a truncation decision. Its omitted operator content should be stated together with its order.

Anderson Impurity to Kondo

The single-impurity Anderson Hamiltonian is

$$H_{\mathrm A} ={} \sum_{k,\sigma} \epsilon_k c_{k\sigma}^\dagger c_{k\sigma} + \epsilon_d \sum_\sigma n_{d\sigma} + U n_{d\uparrow}n_{d\downarrow} + \sum_{k,\sigma} \left( V_k c_{k\sigma}^\dagger d_\sigma + \mathrm{h.c.} \right).$$

In the local-moment regime,

$$\epsilon_d<0, \qquad \epsilon_d+U>0,$$

the singly occupied impurity doublet is retained. The empty and doubly occupied impurity states have charge costs

$$\Delta_0 = -\epsilon_d, \qquad \Delta_2 = \epsilon_d+U.$$

Hybridization changes impurity charge, so

$$PH_{\mathrm{hyb}}P = 0.$$

Second-order virtual paths visit both charge sectors.

The Anderson reduction retains the singly occupied doublet. Both virtual charge paths contribute to $J_K=2|V|^2(\Delta_0^{-1}+\Delta_2^{-1})$, while $W=|V|^2(\Delta_0^{-1}-\Delta_2^{-1})/2$ measures their imbalance. The charge costs include the exchanged bath particle at the Fermi energy; low external energies justify neglecting its small energy in these denominators.

Normalized local-orbital convention

Take

$$V_k = \frac{V}{\sqrt{\mathcal N}}, \qquad f_\sigma = \frac1{\sqrt{\mathcal N}} \sum_k c_{k\sigma}.$$

Define

$$\mathbf s_0 = \frac12 \sum_{\alpha,\beta} f_\alpha^\dagger \boldsymbol\sigma_{\alpha\beta} f_\beta.$$

Within the singly occupied impurity sector, define the dimensionless impurity spin

$$\mathbf S_d = \frac12 \sum_{\alpha,\beta} d_\alpha^\dagger \boldsymbol\sigma_{\alpha\beta} d_\beta.$$

At energies small compared with both charge costs, the effective interaction is

$$H_{\mathrm{int}}^{\mathrm{eff}} = J_K \mathbf S_d\cdot\mathbf s_0 + W \sum_\sigma f_\sigma^\dagger f_\sigma.$$

The leading coefficients are

$$J_K = 2|V|^2 \left( \frac1{\Delta_0} + \frac1{\Delta_2} \right) >0$$

and

$$W = \frac{|V|^2}{2} \left( \frac1{\Delta_0} - \frac1{\Delta_2} \right).$$

The empty and doubly occupied paths add in the spin-exchange channel and subtract in the potential-scattering channel. With the displayed spin density $\mathbf s_0=f^\dagger\boldsymbol\sigma f/2$, the factor $1/2$ in $W$ is required. There is also an additive shift $C=-|V|^2/\Delta_0$ in the simple zero-energy local-bath benchmark.

One can check the factors without a spin-flip calculation. With the bath orbital empty, only the impurity electron can leave virtually, giving $C$. With bath and impurity spins parallel, Pauli blocking forbids hybridization, so the total shift is

$$C+W+\frac{J_K}{4}=0.$$

The displayed coefficients satisfy this identity. Omitting the half in $W$ fails it away from particle–hole symmetry.

At particle–hole symmetry,

$$\epsilon_d = -\frac U2,$$

so

$$\Delta_0=\Delta_2=\frac U2,$$

and therefore

$$J_K = \frac{8|V|^2}{U}, \qquad W=0.$$

The numerical factors depend on the normalization of the local conduction field and on whether spin density includes the factor $1/2$. Quoting $J_K$ without those conventions is incomplete.

What the mapping keeps and loses

The Kondo Hamiltonian keeps:

• the impurity spin;
• conduction-electron dynamics;
• spin exchange;
• potential scattering when symmetry allows it.

It removes explicit impurity charge states. It therefore cannot reproduce the Anderson model’s charge-transfer peaks, mixed-valence crossover, or impurity occupancy over a wide gate range without additional matching.

The Anderson impurity treatment develops the charge spectrum and thermodynamics; Kondo physics develops the subsequent low-energy screening.

Control condition

A useful local-moment condition is

$$\Gamma \ll \min(\Delta_0,\Delta_2),$$

where $\Gamma$ is the hybridization width in a declared bath convention. The effective model is used at energies

$$E, \quad k_{\mathrm B}T, \quad \hbar\omega \ll \min(\Delta_0,\Delta_2).$$

Near $\epsilon_d=0$ or $\epsilon_d=-U$, one denominator becomes small. The impurity charge then participates in low-energy dynamics, and the Kondo-only Hilbert space is too small.

Generated Operators Are Part of the Answer

Projection can generate operators compatible with the retained symmetries and the available virtual paths. Symmetry allowance alone does not require a nonzero coefficient: path interference or microscopic restrictions can make it vanish. Depending on the problem, this can include:

• additive constants and chemical-potential shifts;
• density interactions;
• spin exchange;
• anisotropic exchange;
• Dzyaloshinskii–Moriya interactions;
• potential scattering;
• correlated and pair hopping;
• three-site terms;
• multispin and ring exchange;
• longer-range couplings;
• boundary operators.

A derivation is incomplete if it keeps the attractive headline term and silently drops other terms of the same order.

Power counting each operator

For every generated term, record:

1. the number of sector-changing vertices;
2. the virtual denominators;
3. the spatial support;
4. the symmetry channel;
5. the filling or state dependence;
6. whether another retained term is parametrically larger.

An operator of order $t^2/U$ can be negligible for one observable and essential for another.

Symmetry and Gauge Checks

If $H$, $P$, and the transformation preserve a symmetry, $H_{\mathrm{eff}}$ must represent that symmetry within the retained space.

Useful checks include:

• Hermiticity;
• particle-number conservation;
• spin rotation or the declared spin anisotropy;
• lattice translations and point-group symmetries;
• time reversal;
• gauge covariance under rephasing of microscopic orbitals;
• particle–hole symmetry when present.

For example, $J_{ij}$ depends on

$$|t_{ij}|^2$$

at second order and is invariant under local orbital rephasing. At higher order, gauge-invariant loop phases can survive and generate flux-sensitive interactions.

An unexpected symmetry-breaking term may indicate:

• an algebraic sign error;
• omission of a symmetry-related virtual path;
• a projector that does not preserve the symmetry;
• a basis choice whose symmetry action was not transformed.

Locality and Internal Unitary Freedom

Eliminating local high-energy states generally produces quasi-local low-energy interactions whose range grows with order. This has two consequences.

First, a finite-order effective Hamiltonian is not usually restricted to the same operator range as its parent. Truncating generated long-range or many-body terms requires a separate estimate.

Second, two valid effective Hamiltonians can differ by a unitary transformation $U_P$ acting only inside $P$:

$$H_{\mathrm{eff}}' = U_P H_{\mathrm{eff}} U_P^\dagger.$$

Their individual coefficients can look different while spectra and consistently transformed matrix elements agree through the target order.

Before declaring two derivations inconsistent:

1. align the retained basis;
2. align the energy-zero convention;
3. align normal ordering and operator definitions;
4. transform observables consistently;
5. compare invariant predictions rather than isolated coefficients.

Effective States and Observables

If

$$\widetilde H=e^SHe^{-S},$$

then an operator must be transformed as

$$O_{\mathrm{eff}} = Pe^SOe^{-S}P.$$

Expanding,

$$O_{\mathrm{eff}} ={} POP + P[S,O]P + \frac12 P[S,[S,O]]P + \cdots.$$

Using $POP$ alone can miss the same virtual admixtures that generated $H_{\mathrm{eff}}$.

Hubbard example

For the bare doublon operator

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

one has

$$P_0DP_0=0.$$

Yet $D_{\mathrm{eff}}$ acquires nonzero terms at order $t^2/U^2$. Physical low-energy Hubbard states therefore have small but finite double occupancy.

Anderson example

After projecting onto the impurity doublet,

$$Pn_dP=P,$$

but the physical Anderson state still contains virtual empty and doubly occupied components. A bare Kondo spin model does not by itself reconstruct gate-dependent impurity charge or high-energy spectral weight.

Currents and response

Current, polarization, and probe operators can acquire additional terms. Matching only the energy spectrum does not guarantee correct optical weights, susceptibilities, or transition amplitudes.

Static Projection Is Not Every Form of Elimination

Several related procedures answer different questions.

Energy-dependent projection

Exact $Q$ elimination gives

$$H_{\mathrm{eff}}(E) = PHP + PHQ \frac1{E-QHQ} QHP.$$

Near a $Q$-sector threshold, this energy dependence may be essential. Replacing it by a constant denominator can fail.

Effective action

Eliminating dynamical fields can produce frequency-dependent or retarded interactions. A time-local Hermitian Hamiltonian may not capture that memory without adding auxiliary degrees of freedom.

Open channels

When eliminated states form an on-shell continuum, outgoing retarded boundary conditions can produce an imaginary self-energy. A finite closed eliminated sector instead has discrete poles and recurrences; it does not by itself create irreversible decay. A closed Hermitian low-energy Hamiltonian is then not the whole answer.

Renormalization group

A Schrieffer–Wolff transformation removes a specified off-resonant sector. Renormalization group methods repeatedly change a cutoff and track scale-dependent couplings. The Anderson-to-Kondo transformation derives the initial Kondo exchange; the subsequent flow to the Kondo scale is a separate problem.

Exercises

1. One virtual level and two retained states. Let $\lvert a\rangle$ and $\lvert b\rangle$ be degenerate at energy zero. One eliminated state $\lvert m\rangle$ has energy $\Delta>0$, with

   $$\langle m|V|a\rangle=g_a, \qquad \langle m|V|b\rangle=g_b.$$

   Derive the second-order $2\times2$ effective Hamiltonian. Identify its bright and dark combinations.

Solution

The degenerate second-order formula gives

$$H_{\mathrm{eff}}^{(2)} = - \frac1\Delta \begin{pmatrix} |g_a|^2 & g_a^*g_b \\ g_b^*g_a & |g_b|^2 \end{pmatrix}.$$

Define

$$g = \sqrt{|g_a|^2+|g_b|^2}.$$

A normalized bright state is

$$|B\rangle = \frac{ g_a^*|a\rangle + g_b^*|b\rangle }{g}.$$

It has shift

$$\Delta E_B = - \frac{g^2}{\Delta}.$$

An orthogonal dark state is

$$|D\rangle = \frac{ g_b|a\rangle - g_a|b\rangle }{g},$$

up to an overall phase. It has zero second-order shift because its amplitudes to $\lvert m\rangle$ cancel.

2. Ionic offset on a Hubbard bond. Give site $1$ energy $-\delta/2$ and site $2$ energy $+\delta/2$. At half filling, the two virtual doublon configurations cost $U-\delta$ and $U+\delta$. Derive the antiferromagnetic exchange, assuming $|\delta|<U$.

Solution

The two virtual paths contribute separate denominators. The singlet–triplet splitting is

$$J = 2|t|^2 \left( \frac1{U-\delta} + \frac1{U+\delta} \right).$$

Combining fractions,

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

For $|\delta|<U$, both charge costs are positive and $J>0$. As $|\delta|\to U$, one denominator becomes small and the projection fails before the formal expression can be trusted.

3. Effective double occupancy. A Hubbard-dimer low-energy singlet has leading energy

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

   Use the Hellmann–Feynman theorem to estimate its total double occupancy to leading order.

Solution

Because

$$\frac{\partial H}{\partial U} = \sum_i n_{i\uparrow}n_{i\downarrow},$$

the total double occupancy is

$$\langle D\rangle = \frac{\partial E_S}{\partial U}.$$

Thus

$$\langle D\rangle = \frac{4t^2}{U^2} + O\!\left( \frac{t^4}{U^4} \right).$$

Although $P_0DP_0=0$, the physical state has nonzero virtual double occupancy of the expected order $(t/U)^2$.

4. Potential scattering away from symmetry. In the zero-energy local-bath benchmark take $\Delta_0=2$, $\Delta_2=6$ and $V=1$ in a common energy unit. Find $J_K$, $W$ and $C$. Compare the empty bath, two parallel spins and a filled bath.

Solution

$J_K=4/3$, $W=1/6$ and $C=-1/2$. The empty bath has shift $C=-1/2$. Parallel spins have $C+W+J_K/4=0$, as Pauli blocking requires. A filled bath has zero spin and shift $C+2W=-1/6=-|V|^2/\Delta_2$, the cost of the allowed virtual doublon path.

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