Quasi-Degenerate Perturbation Theory

Quasi-degenerate perturbation theory treats several nearby states as one model space. Their small internal splittings and mutual couplings are diagonalized together, while coupling to well-separated states is expanded perturbatively.

The method is needed when levels are not exactly degenerate, yet their detuning is comparable to the perturbation that mixes them. A one-state formula then contains small energy denominators and predicts large state corrections. Exact diagonalization of the nearby cluster removes those artificial singularities.

The central rule is:

Retain every strongly mixed nearby state in $P$, diagonalize within $P$, and perturb only the coupling between $P$ and $Q$.

Here $P$ projects onto the retained model space and $Q=I-P$ onto its complement. The method interpolates between Nondegenerate Perturbation Theory when $P$ is one-dimensional and Degenerate Perturbation Theory when the unperturbed levels inside $P$ coincide exactly.

Required background. Use degenerate perturbation theory for matrix-valued level shifts and projectors to split the Hilbert space into $P$ and $Q$.

Mixing near an avoided crossing

Suppose two unperturbed states satisfy

$$H_0\lvert a\rangle =E_a^{(0)}\lvert a\rangle, \qquad H_0\lvert b\rangle =E_b^{(0)}\lvert b\rangle,$$

and are coupled by $V_{ab}=\langle a\rvert V\lvert b\rangle$. Nondegenerate perturbation theory tries to assign the mixing amplitude

$$c_{b\leftarrow a}^{(1)} = \lambda \frac{V_{ba}} {E_a^{(0)}-E_b^{(0)}}.$$

This is small only if

$$\lvert\lambda V_{ba}\rvert \ll \left|E_a^{(0)}-E_b^{(0)}\right|.$$

When the two scales are comparable, the exact eigenvectors are order-$1$ mixtures of $\lvert a\rangle$ and $\lvert b\rangle$. No higher-order correction to a state that remains “mostly $a$” can repair a zeroth-order labeling that has already failed.

Near-degeneracy is therefore a power-counting statement, not merely visual closeness on an energy plot. A small spacing matters when an allowed matrix element is large enough to compete with it. Symmetry can make $V_{ab}=0$, in which case nearby levels may cross without mixing.

The Two-State Cure

Retain both states and diagonalize their projected Hamiltonian. Write

$$H_P(\lambda) = \begin{pmatrix} \varepsilon_1(\lambda) & w(\lambda) \\ w^*(\lambda) & \varepsilon_2(\lambda) \end{pmatrix},$$

where the diagonal entries may already include perturbative shifts from remote states. Define

$$\begin{aligned} \bar\varepsilon &= \frac{\varepsilon_1+\varepsilon_2}{2}, \\ d &= \frac{\varepsilon_1-\varepsilon_2}{2}. \end{aligned}$$

The two eigenvalues are

$$E_\pm = \bar\varepsilon \pm \sqrt{d^2+\lvert w\rvert^2}.$$

At zero detuning, $d=0$, the gap is

$$E_+-E_- = 2\lvert w\rvert.$$

The eigenstates are equal-weight mixtures there. Away from the crossing, write $w=\lvert w\rvert e^{i\phi}$ and choose a mixing angle satisfying

$$\tan(2\theta) = \frac{2\lvert w\rvert} {\varepsilon_1-\varepsilon_2}.$$

The phase $\phi$ fixes the relative phase of the two basis components. For eigenvector conventions and the Bloch-sphere interpretation, see Two-State Hamiltonians.

If $\varepsilon_1\gt\varepsilon_2$ and $\lvert w\rvert$ is small compared with their difference, define $\Delta_\varepsilon=\varepsilon_1-\varepsilon_2$. Expansion of the exact roots gives

$$\begin{aligned} E_+ &= \varepsilon_1 + \frac{\lvert w\rvert^2} {\Delta_\varepsilon} + O\!\left( \frac{\lvert w\rvert^4}{\Delta_\varepsilon^3} \right),\\ E_- &= \varepsilon_2 - \frac{\lvert w\rvert^2} {\Delta_\varepsilon} + O\!\left( \frac{\lvert w\rvert^4}{\Delta_\varepsilon^3} \right). \end{aligned}$$

Thus ordinary second-order level repulsion is the far-detuned expansion of an avoided crossing. The exact square root, rather than a divergent inverse detuning, is the uniform local description.

Model Space and Complement

Let the model space be spanned by $d$ orthonormal unperturbed states,

$$P = \sum_{a=1}^{d} \lvert a\rangle\langle a\rvert, \qquad Q=I-P.$$

Block the Hamiltonian as

$$H = \begin{pmatrix} PHP & PHQ \\ QHP & QHQ \end{pmatrix}.$$

The internal block $PHP$ is not expanded into separate one-state answers. It is diagonalized as a matrix. Remote states influence it through induced diagonal shifts and induced off-diagonal couplings.

The model space $P$ contains the close levels and all strong internal mixing. Remote $Q$ states are eliminated perturbatively, producing an effective Hamiltonian on $P$. The final diagonalization is exact within that retained space.

For a cluster with unperturbed width

$$W_P = \max_{a\in P}E_a^{(0)} - \min_{a\in P}E_a^{(0)},$$

and distance to the complementary spectrum

$$G_{PQ} = \operatorname{dist} \left( \operatorname{spec}(PH_0P), \operatorname{spec}(QH_0Q) \right),$$

the useful regime is schematically

$$\begin{aligned} W_P &\lesssim \lvert\lambda\rvert\lVert PVP\rVert, \\ \eta_{PQ} &\equiv \frac{ \lvert\lambda\rvert\lVert PVQ\rVert }{G_{PQ}} \ll1. \end{aligned}$$

The first relation is only a rough motivation for retaining a cluster: a common diagonal shift makes $\|PVP\|$ large without mixing anything. Compare actual off-diagonal couplings with the corresponding internal detunings. The second relation is a leading diagnostic, not a guarantee: diagonal perturbations can close the $P$–$Q$ gap even when $PVQ$ is small. Monitor the separation of the perturbed diagonal blocks as well. For unbounded operators, these norm expressions are only a finite-model diagnostic; domain and relative-boundedness hypotheses require separate analysis.

Exact Energy-Dependent Effective Equation

Project an exact eigenstate as

$$\lvert\Psi\rangle = \lvert\Psi_P\rangle + \lvert\Psi_Q\rangle.$$

If $E-QHQ$ is invertible, the $Q$ equation gives

$$\lvert\Psi_Q\rangle = \frac{1}{E-QHQ} QHP\lvert\Psi_P\rangle.$$

Substitution into the $P$ equation yields

$$H_{\mathrm{eff}}(E) \lvert\Psi_P\rangle = E\lvert\Psi_P\rangle,$$

with

$$\begin{aligned} H_{\mathrm{eff}}(E) &= PHP+\Sigma_P(E), \\ \Sigma_P(E) &= PHQ \frac{1}{E-QHQ} QHP. \end{aligned}$$

This identity is exact where the resolvent exists. Its energy dependence makes the eigenvalue problem nonlinear, and poles signal that a supposedly eliminated state is dynamically relevant. Projection methods derive this Schur complement and the reconstruction of the full state; Feshbach projection extends it to continua and resonances.

Quasi-degenerate perturbation theory expands this exact structure in the coupling to $Q$ while preserving matrix diagonalization inside $P$.

Hermitian Effective Hamiltonian Through Second Order

Let

$$H=H_0+\lambda V,$$

and choose $P$ from eigenstates $\lvert a\rangle$ of $H_0$ with nearby, not necessarily equal, energies $E_a^{(0)}$. Let Greek indices $\mu$ label $Q$ states, and define

$$D_{a\mu} \equiv \frac{1}{E_a^{(0)}-E_\mu^{(0)}}.$$

A standard Hermitian energy-independent effective Hamiltonian through second order has matrix elements

$$\bigl(H_{\mathrm{eff}}\bigr)_{ab} ={} E_a^{(0)}\delta_{ab} +\lambda V_{ab} + \lambda^2 \bigl(H_{\mathrm{eff}}^{(2)}\bigr)_{ab} +O(\lambda^3).$$

Here

$$\bigl(H_{\mathrm{eff}}^{(2)}\bigr)_{ab} = \frac12 \sum_{\mu\in Q} V_{a\mu}V_{\mu b} \bigl(D_{a\mu}+D_{b\mu}\bigr).$$

This is often called a canonical Van Vleck or Löwdin effective Hamiltonian. It is Hermitian because the two denominators are symmetrized between the bra and ket states.

For $a=b$, the second-order term reduces to

$$\bigl(H_{\mathrm{eff}}^{(2)}\bigr)_{aa} = \sum_{\mu\in Q} \frac{\lvert V_{a\mu}\rvert^2} {E_a^{(0)}-E_\mu^{(0)}}.$$

This is the usual second-order energy correction, but the sum excludes the other model-space states. Their effects are generated by diagonalizing $H_{\mathrm{eff}}$ and must not be added again with small denominators.

For $a\ne b$, this same matrix element can be nonzero even when $V_{ab}=0$. Virtual excursions into the eliminated space can therefore shift levels, open or modify avoided-crossing gaps, and generate interactions absent in the directly projected block.

Different effective-Hamiltonian conventions can differ in their off-shell matrix elements. Bloch, canonical Van Vleck, Löwdin, and Schrieffer–Wolff constructions impose different normalization or block-diagonalization choices. When carried out consistently, they reproduce the target exact eigenvalues through the claimed order; some forms are manifestly Hermitian and others are not. Do not combine matrix elements from different conventions term by term.

Exactly Degenerate Limit

If all model-space states have the same unperturbed energy $E_0$, the second-order matrix simplifies to

$$\bigl(H_{\mathrm{eff}}\bigr)_{ab} ={} E_0\delta_{ab} +\lambda V_{ab} + \lambda^2 \sum_{\mu\in Q} \frac{V_{a\mu}V_{\mu b}} {E_0-E_\mu^{(0)}} +O(\lambda^3).$$

At first order, this is precisely the instruction to diagonalize $PVP$. At second order, virtual $Q$ states add another Hermitian matrix that may split any degeneracy left by $PVP$.

Quasi-degenerate theory therefore does not compete with degenerate perturbation theory. It extends the same model-space logic to clusters whose internal splittings are small but nonzero.

Choosing the Model Space

A useful $P$ space is large enough to remove every dangerous denominator and small enough that eliminating $Q$ is worthwhile.

Use the following tests:

1. Energy-window test. Include levels whose unperturbed detuning from the target cluster is comparable to retained direct or induced couplings.
2. Matrix-element test. A nearby state with a symmetry-forbidden coupling need not be included for that perturbation, though another allowed interaction may change the conclusion.
3. Stability test. Enlarge $P$ and verify that target eigenvalues and observables change only beyond the claimed order.
4. Symmetry test. Build $P$ from complete symmetry multiplets when the perturbation preserves that symmetry. Cutting a multiplet can create basis-dependent results.
5. Intruder-state test. Monitor the smallest $P$–$Q$ denominator. If a $Q$ level approaches the cluster, promote it into $P$.
6. State-character test. Track eigenvectors by overlaps or symmetry labels rather than sorting eigenvalues independently at each parameter value.

There is no universal energy cutoff. The correct model space depends on both detunings and couplings.

Reconstructing States and Observables

An eigenvector $|u\rangle=\sum_a c_a|a\rangle$ of the Hermitian effective Hamiltonian represents the full state in a transformed model space. It is not exactly the bare projection $P|\Psi\rangle$: the two already differ in normalization at second order. Undoing the block-diagonalizing transformation gives the leading complementary component,

$$\lvert\Psi_Q\rangle \approx \lambda\sum_{\mu\in Q}\sum_{a\in P} \frac{V_{\mu a}c_a}{E_a^{(0)}-E_\mu^{(0)}}|\mu\rangle.$$

The leakage probability is therefore schematically

$$p_Q = \langle\Psi_Q\vert\Psi_Q\rangle = O(\eta_{PQ}^{2}).$$

Normalization, transition matrix elements, and expectation values require this dressing. An effective observable generally differs from the bare projection $PAP$ because the same transformation that removes $P$–$Q$ coupling also dresses operators. Using a corrected Hamiltonian with undressed observables can lose terms of the very order one intended to retain.

For an energy-dependent Feshbach Hamiltonian, the derivative of the self-energy also enters normalization. For an energy-independent unitary construction, transform both states and operators with the same unitary map.

Applications

Biased double wells

Localized states $\lvert L\rangle$ and $\lvert R\rangle$ form a two-dimensional model space. Bias supplies the detuning and tunneling supplies the off-diagonal coupling. Diagonalizing the pair gives the avoided crossing. For the spatial construction, see Coupled Wells and Avoided Crossings.

The tunneling amplitude itself may be exponentially small in a semiclassical parameter. Finite-order power-series perturbation theory need not generate that nonperturbative scale; quasi-degenerate theory organizes the mixing once the amplitude has been calculated or matched.

Atomic and molecular level crossings

External electric or magnetic fields can tune two levels close together. Selection rules determine whether the crossing is exact or avoided. Remote levels contribute Stark, Zeeman, and spin–orbit shifts to the effective matrix, while the nearby pair or multiplet is diagonalized together. This is the setting in which smooth adiabatic potential curves and reliable state tracking matter more than assigning a fixed unperturbed label.

Two-level atoms and dressed-state truncations

For a static two-level approximation, the retained pair must be separated from all other atomic levels by gaps large compared with the relevant couplings. Near resonant driving, the quasi-degenerate pair often lives in a rotating or Floquet representation rather than the bare atomic spectrum. The rotating-wave approximation explains that change of frame and its limits.

Band-crossing preview

Near a crossing momentum $\mathbf k_0$, retain the few bands whose energies coincide or nearly coincide and eliminate remote bands. A two-band effective Hamiltonian often has the form

$$\begin{aligned} H_{\mathrm{eff}}(\mathbf q) &= d_0(\mathbf q)I + \mathbf d(\mathbf q)\cdot\boldsymbol\sigma, \\ \mathbf q &= \mathbf k-\mathbf k_0. \end{aligned}$$

Symmetry may force components of $\mathbf d$ to vanish and protect a crossing. An allowed constant component can open a gap. The two-band form alone does not establish topological protection; that requires the momentum dependence and symmetry constraints on $\mathbf d$.

Exercises

1. Uniform two-level result

Let

$$H = \begin{pmatrix} \delta/2 & g \\ g^* & -\delta/2 \end{pmatrix}.$$

Find the eigenvalues, the minimum gap as $\delta$ is varied, and the far-detuned expansion.

Solution

The characteristic equation is

$$E^2 = \frac{\delta^2}{4} +\lvert g\rvert^2,$$

so

$$E_\pm = \pm \sqrt{ \frac{\delta^2}{4} +\lvert g\rvert^2 }.$$

The gap is

$$E_+-E_- = 2\sqrt{ \frac{\delta^2}{4} +\lvert g\rvert^2 },$$

whose minimum at $\delta=0$ is $2\lvert g\rvert$. For $\delta\gt0$ and $\lvert g\rvert\ll\delta$,

$$\begin{aligned} E_+ &= \frac{\delta}{2} + \frac{\lvert g\rvert^2}{\delta} + O\!\left( \frac{\lvert g\rvert^4}{\delta^3} \right), \\ E_- &= -\frac{\delta}{2} - \frac{\lvert g\rvert^2}{\delta} + O\!\left( \frac{\lvert g\rvert^4}{\delta^3} \right). \end{aligned}$$

The inverse detuning is valid only in the far-detuned regime; the square root remains finite through the crossing.

2. Remote-state-mediated coupling

Let $P$ contain $\lvert1\rangle$ and $\lvert2\rangle$, and let $Q$ contain one state $\lvert3\rangle$. Assume $V_{12}=0$, but $V_{13}=g_1$ and $V_{32}=g_2$. Find the induced off-diagonal matrix element through second order.

Solution

The Hermitian second-order formula gives

$$\bigl(H_{\mathrm{eff}}^{(2)}\bigr)_{12} = \frac{g_1g_2}{2} \Biggl[ \frac{1}{E_1^{(0)}-E_3^{(0)}} + \frac{1}{E_2^{(0)}-E_3^{(0)}} \Biggr].$$

The physical contribution to the Hamiltonian is $\lambda^2(H_{\mathrm{eff}}^{(2)})_{12}$. Even though there is no direct coupling, the path $1\to3\to2$ mediates mixing. Hermiticity gives the reverse element as its complex conjugate.

3. Degenerate limit

Take $E_1^{(0)}=E_2^{(0)}=E_0$ in Exercise 2. Show how the symmetrized denominator simplifies.

Solution

Both denominators become $E_0-E_3^{(0)}$, so

$$\bigl(H_{\mathrm{eff}}^{(2)}\bigr)_{12} = \frac{g_1g_2} {E_0-E_3^{(0)}}.$$

The factor $1/2$ is canceled by the two equal terms. This is the usual second-order effective matrix within an exactly degenerate subspace.

4. Choosing a model space

Three unperturbed levels have energies $0$, $0.03\Delta$, and $\Delta$. At the physical coupling, the first two are connected by a matrix element of magnitude $0.05\Delta$, while their couplings to the third have magnitude at most $0.02\Delta$. Which states should be in the first model-space trial?

Solution

The detuning between the first two states is $0.03\Delta$, smaller than their coupling scale $0.05\Delta$. They must be retained and diagonalized together. The third state is separated by a gap of order $\Delta$, while its coupling is at most $0.02\Delta$, giving a mixing ratio of order $0.02$. It is a reasonable first $Q$-space state.

This is a starting choice, not a proof. One should add the third state to $P$ as a stability check and verify that target quantities change only beyond the intended order.

5. Leading leakage probability

Let $|u\rangle$ be the normalized eigenvector of the Hermitian effective Hamiltonian. Write the leading $Q$-space probability after reconstructing the full state.

Solution

Let $c_a=\langle a\vert u\rangle$ and let $\lvert\mu\rangle$ label $Q$ states. To first order,

$$\langle\mu\vert\Psi_Q\rangle \approx \lambda \sum_{a\in P}\frac{V_{\mu a}c_a}{E_a^{(0)}-E_\mu^{(0)}}.$$

Therefore

$$p_Q \approx \lambda^2 \sum_{\mu\in Q} \left| \sum_{a\in P}\frac{V_{\mu a}c_a}{E_a^{(0)}-E_\mu^{(0)}} \right|^2.$$

Interference between amplitudes from different model-space components is physical and can enhance or suppress leakage.

6. Biased double well

For

$$H_{LR} = \begin{pmatrix} \epsilon & -K \\ -K & -\epsilon \end{pmatrix},$$

find the energies and state character in the limits $\lvert\epsilon\rvert\gg K$ and $\epsilon=0$.

Solution

The energies are

$$E_\pm = \pm\sqrt{\epsilon^2+K^2}.$$

For $\lvert\epsilon\rvert\gg K$, each eigenstate is predominantly localized in one well and receives a small admixture of the other. At $\epsilon=0$, the eigenstates are the symmetric and antisymmetric combinations

$$\frac{ \lvert L\rangle\pm\lvert R\rangle }{\sqrt2},$$

with a splitting $2|K|$. The exact two-state result is uniform across the bias value where a one-state expansion would fail.

References

• P.-O. Löwdin, “A Note on the Quantum-Mechanical Perturbation Theory,” Journal of Chemical Physics 19, 1396–1401 (1951), doi:10.1063/1.1748067.
• C. Bloch, “Sur la théorie des perturbations des états liés,” Nuclear Physics 6, 329–347 (1958), doi:10.1016/0029-5582(58)90116-0.
• S. Bravyi, D. P. DiVincenzo, and D. Loss, “Schrieffer–Wolff Transformation for Quantum Many-Body Systems,” Annals of Physics 326, 2793–2826 (2011), doi:10.1016/j.aop.2011.06.004.
• J. des Cloizeaux, “Extension d’une formule de Lagrange à des problèmes de valeurs propres,” Nuclear Physics 20, 321–346 (1960), doi:10.1016/0029-5582(60)90177-2.
• T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Springer, 1976; reprint 1995, doi:10.1007/978-3-642-66282-9.
• I. Lindgren and J. Morrison, Atomic Many-Body Theory, 2nd ed., Springer, 1986.
• A. Messiah, Quantum Mechanics, Vol. II, North-Holland, 1962, Chapter XVII.
• J. R. Schrieffer and P. A. Wolff, “Relation between the Anderson and Kondo Hamiltonians,” Physical Review 149, 491–492 (1966), doi:10.1103/PhysRev.149.491.