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.

The transformed low-energy states lie in the retained block, but their physical counterparts contain small admixtures of eliminated bare states. This distinction is essential when calculating populations or transition matrix elements.

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.

Required background. Unitary operators supplies the change of representation; projectors defines the blocks; degenerate perturbation theory explains why strongly mixed levels must be retained together.

Separated Hamiltonian blocks

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.

Unitary block diagonalization

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,

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

Choose a block-off-diagonal generator, $PSP=QSQ=0$, fixing the freedom to rotate separately within the blocks. At first order in $V$, write $S=S_1+O(V^2)$ and choose $S_1$ to cancel the off-diagonal coupling (denoted $S$ in the leading formulas below):

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

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 gap and the remainder notation assumes bounded perturbations with $\|V\|/\Delta\ll1$. A second-order generator cancels the remaining off-diagonal terms when $V_{\mathrm d}\ne0$, but does not alter the displayed retained block through second order.

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

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

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 retain the first bare state and $Q$ the second. For $\Delta>0$ the retained state is the lower one; for $\Delta<0$ the same algebra describes the separated upper 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 $\lvert e\rangle$:

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

Here $\Omega_i$ denote energy matrix elements, not angular frequencies; a frequency convention would include the appropriate factors of $\hbar$. 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:

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

This ratio is necessary but does not by itself control arbitrarily large block-diagonal perturbations. Such terms can close the gap even when $V_{\mathrm{od}}$ is small; either bound the full perturbation relative to the original gap or include the large diagonal terms in the unperturbed blocks and reassess their separation. The retained subspace must contain all strongly mixed levels. Extensive many-body systems require local or linked-cluster estimates rather than an unqualified volume-independent bound on the total perturbation norm. 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:

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

The physical state corresponding to an effective state $|\phi\rangle\in P\mathcal H$ is $e^{-S}|\phi\rangle$, to the calculated order. Consequently a bare $Q$-population measurement can be nonzero even though the effective state lies entirely in $P$. 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. The perturbative products encode virtual admixture through $Q$, shifting the energy; they do not assign a measurable duration to an intermediate stationary state. 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 explains initial transients, memory, derivative corrections, and decaying sectors.

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.

Exercises

1. In the two-level example with $\Delta>0$, calculate the physical upper-bare-state population of the dressed lower eigenstate to leading order. Compare it with the maximum upper-state population when the initial state is the bare retained vector.

Solution

Reconstruction gives $|\Psi_-\rangle\simeq|p\rangle-(g^*/\Delta)|q\rangle$, so the dressed eigenstate has upper-state probability $|g|^2/\Delta^2+O(|g/\Delta|^4)$. Direct two-level evolution of $|p\rangle$ instead gives

$$P_q(t)=\frac{4|g|^2}{\Delta^2+4|g|^2} \sin^2\!\left(\frac{\sqrt{\Delta^2+4|g|^2}\,t}{2\hbar}\right).$$

Its maximum is approximately $4|g|^2/\Delta^2$. The bare preparation includes both dressed eigenstates and has a fast transient that a stationary one-block description alone omits.

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.

References

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