Schrieffer–Wolff Transformation

The Schrieffer–Wolff transformation is a perturbative unitary change of basis that removes couplings between a chosen low-energy subspace and its complement. Its output is an effective Hamiltonian that acts only inside the low-energy subspace while retaining the leading virtual effects of the eliminated states.

It is the right tool when the states outside the model space are not occupied in the low-energy description, but still shift energies and generate indirect couplings.

Projection Methods gives the exact energy-dependent reduction of the same P/Q block problem. Folded Effective Hamiltonians removes that energy dependence by Q-box derivative feedback. The Schrieffer–Wolff construction instead seeks a perturbative unitary change of basis, which keeps the effective Hamiltonian energy independent and supplies a consistent transformation for observables.

Its original impurity application starts from the charge-fluctuating Anderson Impurity Model Preview, removes empty and doubly occupied local-orbital states, and produces the antiferromagnetic exchange used in the Kondo Model Preview.

Effective Hamiltonians in Many-Body Systems applies this formalism to extensive projected sectors and compares the Hubbard-to-Heisenberg and Anderson-to-Kondo reductions, including generated same-order operators.

Setup

Let the Hilbert space be split by orthogonal projectors

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

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

$$H=H_0+V,$$

where $H_0$ is block diagonal with respect to $P$ and $Q$:

$$PH_0Q=0, \qquad QH_0P=0.$$

The perturbation can be separated into block-diagonal and block-off-diagonal parts:

$$V_{\mathrm d}=PVP+QVQ, \qquad V_{\mathrm{od}}=PVQ+QVP.$$

The off-diagonal part is what mixes the retained and eliminated subspaces.

Goal

The goal is to find a unitary operator

$$U=e^S, \qquad S^\dagger=-S,$$

such that the transformed Hamiltonian

$$H'=e^SHe^{-S}$$

is block diagonal to a chosen order:

$$PH'Q=0$$

up to controlled corrections. The effective Hamiltonian in the retained space is then

$$H_{\mathrm{eff}}=PH'P.$$

Because $U$ is unitary, this is not merely a projection after throwing away information. It is a perturbative change of basis that first folds virtual high-energy processes into the low-energy block.

Generator at Leading Order

Using the Baker-Campbell-Hausdorff expansion,

$$H' = H+[S,H] +\frac12[S,[S,H]] +\cdots.$$

At first order in $V$, choose $S$ to cancel the off-diagonal coupling:

$$V_{\mathrm{od}}+[S,H_0]=0.$$

In an eigenbasis of $H_0$, let $\lvert a\rangle$ and $\lvert b\rangle$ lie in the retained space, and let $\lvert r\rangle$ lie in the eliminated space. Then

$$\langle a|S|r\rangle = \frac{\langle a|V|r\rangle} {E_a^{(0)}-E_r^{(0)}}.$$

The denominator is the energy cost of the virtual excursion into the eliminated subspace. If that denominator is small, the chosen model space is not well separated.

Effective Hamiltonian to Second Order

With the leading generator, the retained-space Hamiltonian through second order is

$$H_{\mathrm{eff}} = PH_0P +PVP +\frac12P[S,V_{\mathrm{od}}]P +O\!\left(\frac{V^3}{\Delta^2}\right),$$

where $\Delta$ denotes the relevant energy separation between $P$ and $Q$ sectors.

For low-energy basis states $\lvert a\rangle,\lvert b\rangle$, this gives

$$\begin{aligned} \langle a|H_{\mathrm{eff}}|b\rangle =& E_a^{(0)}\delta_{ab} +V_{ab} \\ &+ \frac12 \sum_{r\in Q} V_{ar}V_{rb} \left( \frac{1}{E_a^{(0)}-E_r^{(0)}} + \frac{1}{E_b^{(0)}-E_r^{(0)}} \right) +\cdots. \end{aligned}$$

This form makes Hermiticity transparent when $V$ is Hermitian. It also shows why the method is closely related to degenerate perturbation theory: both keep a model space and account for coupling to outside states perturbatively.

Two-Level Example

Consider

$$H = \begin{pmatrix} 0&g\\ g^*&\Delta \end{pmatrix}, \qquad |\Delta|\gg |g|.$$

Take $P$ to project onto the lower bare state and $Q$ onto the upper bare state. The second-order effective Hamiltonian in $P$ is the number

$$H_{\mathrm{eff}} = - \frac{|g|^2}{\Delta} +O\!\left(\frac{|g|^4}{\Delta^3}\right).$$

The eliminated state is not populated in the effective description, but virtual visits to it lower or raise the retained energy depending on the sign of $\Delta$.

This is the simplest version of dispersive shifts, level repulsion, and perturbative elimination.

Three-Level Example

Suppose two nearly degenerate states $\lvert1\rangle,\lvert2\rangle$ are coupled only through a far-detuned state $\lvert e\rangle$:

$$H_0 = \Delta\lvert e\rangle\langle e\rvert, \qquad V = \Omega_1\lvert e\rangle\langle1\rvert +\Omega_2\lvert e\rangle\langle2\rvert +\mathrm{h.c.}$$

For $|\Omega_i|\ll|\Delta|$, eliminating $\lvert e\rangle$ gives the retained-space Hamiltonian

$$H_{\mathrm{eff}} \approx - \frac{1}{\Delta} \begin{pmatrix} |\Omega_1|^2&\Omega_1^*\Omega_2\\ \Omega_2^*\Omega_1&|\Omega_2|^2 \end{pmatrix}$$

in the basis $\{\lvert1\rangle,\lvert2\rangle\}$. The diagonal terms are light shifts or level shifts. The off-diagonal terms are effective couplings mediated by a virtual excursion through $\lvert e\rangle$.

The same algebra appears in Raman transitions, superexchange, dispersive qubit couplings, and many perturbative gadget constructions.

Validity

The control parameter is not simply the size of $V$. It is the ratio of off-diagonal matrix elements to energy denominators:

$$\epsilon \sim \frac{\|V_{\mathrm{od}}\|}{\Delta}.$$

The Schrieffer–Wolff expansion is controlled when $\epsilon\ll1$ and the retained subspace includes all states that are close enough to mix strongly. If a supposedly eliminated state is nearly resonant, it should be moved into $P$.

The method also assumes that the desired observable or process is insensitive to the discarded high-energy amplitudes beyond the order retained. When observables are transformed, they must be transformed consistently:

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

Using $H_{\mathrm{eff}}$ while leaving other operators untransformed can produce inconsistent matrix elements.

Physical Interpretation

The transformation replaces direct access to high-energy states by virtual processes. A state in $P$ can briefly enter $Q$ and return, shifting its energy. Two different states in $P$ can communicate through the same eliminated state, producing an effective coupling.

This is why the second-order term contains products such as

$$V_{ar}V_{rb}.$$

The intermediate state $\lvert r\rangle$ is not a final state. It is a virtual state whose contribution is weighted by an energy denominator.

Relation to Other Methods

Degenerate perturbation theory diagonalizes the perturbation inside a near-degenerate subspace and then treats outside states perturbatively. Schrieffer–Wolff does the same job in a unitary block-diagonal language.

Feshbach projection methods produce energy-dependent effective Hamiltonians using resolvents. Schrieffer–Wolff instead produces an energy-independent perturbative Hamiltonian after a change of basis.

Adiabatic elimination starts from equations of motion and slaves a fast amplitude to slow variables. It reproduces the same leading virtual coupling when its timescale assumptions hold, while Adiabatic Elimination owns the initial transient, memory, derivative corrections, and decaying-sector extension.

The Foldy–Wouthuysen transformation in relativistic quantum mechanics is a close cousin: it block diagonalizes positive- and negative-energy sectors perturbatively. In effective field theory, analogous transformations remove redundant high-energy effects and organize low-energy operators by powers of a scale separation.

Common Mistakes

• Eliminating a state that is nearly degenerate with the retained subspace.
• Forgetting the factor of $1/2$ in the commutator form of the second-order Hamiltonian.
• Treating the projected Hamiltonian $PHP$ as the effective Hamiltonian when virtual corrections are important.
• Comparing bare and effective states without applying the same unitary transformation to observables.
• Using the method without naming the energy gap or small dimensionless ratio.

Cross-Links

• Effective Hamiltonians and Scale Separation
• Folded Effective Hamiltonians
• Adiabatic Elimination
• Projectors
• Unitary Operators
• Degenerate Perturbation Theory
• Quasi-Degenerate Perturbation Theory
• Small Parameters and Error Estimates
• Rotating-Wave Approximation
• Effective Hamiltonians in Quantum Information
• Effective Hamiltonians in Quantum Matter
• Exchange Interactions in Quantum Matter
• Kondo Model Preview
• Two-Level System

References

• J. R. Schrieffer and P. A. Wolff, “Relation between the Anderson and Kondo Hamiltonians,” Physical Review 149, 491-492, 1966.
• S. Bravyi, D. P. DiVincenzo, and D. Loss, “Schrieffer–Wolff transformation for quantum many-body systems,” Annals of Physics 326, 2793–2826, 2011.
• A. Messiah, Quantum Mechanics, Dover, 1999.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics, Wiley, 1977.

Exercises

1. For the two-level Hamiltonian

$$H = \begin{pmatrix} 0&g\\ g&\Delta \end{pmatrix}, \qquad \Delta\gt0,$$

show that the lower exact eigenvalue agrees with the Schrieffer–Wolff result to order $g^2/\Delta$.

Solution

The exact eigenvalues are

$$E_\pm = \frac{\Delta}{2} \pm \frac12\sqrt{\Delta^2+4g^2}.$$

For $|g|\ll\Delta$,

$$\sqrt{\Delta^2+4g^2} = \Delta + \frac{2g^2}{\Delta} +O\!\left(\frac{g^4}{\Delta^3}\right).$$

Thus

$$E_- = - \frac{g^2}{\Delta} +O\!\left(\frac{g^4}{\Delta^3}\right),$$

which matches the effective Hamiltonian in the retained lower subspace.

2. Why is the second-order matrix element symmetric in the two retained energies $E_a^{(0)}$ and $E_b^{(0)}$?

Solution

Hermiticity requires

$$\langle a|H_{\mathrm{eff}}|b\rangle = \langle b|H_{\mathrm{eff}}|a\rangle^*$$

when $H$ is Hermitian. The symmetrized denominator

$$\frac12 \left( \frac{1}{E_a^{(0)}-E_r^{(0)}} + \frac{1}{E_b^{(0)}-E_r^{(0)}} \right)$$

is the form produced by the unitary block diagonalization and preserves this Hermitian structure. A formula using only one of the two denominators generally corresponds to a different, energy-dependent perturbative convention and must be interpreted carefully.