The Schrieffer Wolff Transformation
Blog Post 1-19-25
Introduction
Systems in many-body physics frequently involve a weak perturbation \(\epsilon V\) to a decoupled system \(H_0\) which mixes these systems. It is common to call the uncoupled system the “bare" system and the inclusion of the weak perturbation the “dressed" system, with bare/dressed eigenstates and energy levels respectively. Examples of systems where this situation arises include Rydberg atoms, with the Hamiltonian [1] \[\begin{aligned} H = -\Delta\sum_{i}n_i + \frac{C}{2}\sum_{i,j}\frac{n_in_j}{r_{ij}^6} + \frac{\Omega}{2} \sum_i \sigma^x_i \end{aligned}\] This describes an array of neutral atoms at positions \(r_i\) which interact via the \(\sim \frac{1}{r^6}\) Van der Waals force, where \(\Omega\) is the laser power and \(\Delta\) is the laser detuning from resonance. If \(\Omega\) is small compared to \(\frac{C}{r^6}\), then this leads to the Rydberg blockade effect, where the subspace of states with no adjacent excitations is separated from the rest of the spectrum by a gap and weakly coupled by \(\Omega\). Another relevant situation is the interaction between a multilevel atom and a resonant cavity [2]: \[\begin{aligned} H = \sum_{j}\omega_j\ketbra{j}{j} + \omega_r a^\dagger a + \sum_{ij}g_{ij}\ketbra{i}{j}a^\dagger + \text{h.c.} \end{aligned}\] Although formally the coupling is unbounded, for small enough excitation numbers these couplings can be treated as small with respect to the bare system (see Appendix B of [2]). Lastly, the original use of Schrieffer and Wolff was to relate the Kondo and Anderson model Hamiltonians [3]. The Anderson model Hamiltonian for a single localized \(d\) orbital is \[\begin{aligned} H = \sum_{ks}\epsilon_k n_{ks} + \sum_s \epsilon_d n_{ds} + U n_{d\uparrow} n_{d\downarrow} + \sum_{ks}V_{kd} c_{ks}^\dagger c_{ds} + \text{h.c.} \end{aligned}\] where \(\epsilon_k\) and \(\epsilon_d\) are the single-particle energies of the localized and conduction orbitals respectively. The potential \(V\) mixes these states, and \(U\) is the Coulomb repulsion between two \(d\)-orbital electrons. The idea of the Schrieffer-Wolff transformation is to construct a Hamiltonian \(H_{\text{eff}}\) which acts on the low-energy subspace of \(H_0\) but reproduces the eigenvalues of \(H = H_0 + \epsilon V\), obtaining an effective interaction between the bare states.
Direct rotations
The essence is to define a canonical rotation between two subspaces, which will be the low-energy subspaces of the unperturbed and perturbed Hamiltonian respectively. This canonical rotation is called a “direct rotation," which we will define in the following section.
Def 1. Given two subspaces \(P_0, P_1\) of the same dimension, the unitary \(U\) closest to the identity that maps \(P_0 \to P_1\) is called the direct rotation relating the two subspaces.
Def 2. Given a subspace \(P_0\), we define a reflection \(R_0\) about that subspace via \(R_0 = 2\pi_0-1\), where \(\pi_0\) is a projector onto \(P_0\).
Lemma 1. \(\Vert \pi_1 - \pi_0 \Vert < 1\) iff no vector in \(P_0\) is orthogonal to \(P_1\) and vice versa.
Proof. For the forward direction, suppose the contrary. Then WLOG \(v \in P_1 \perp P_0\), so \(\Vert(\pi_1 - \pi_0)v\Vert = 1\), which is a contradiction. For the reverse direction, again suppose the contrary. Then the maximum eigenvalue of \(\pi_1 - \pi_0\) is \(\pm 1\), so there exists \(v\) such that \((\pi_1 - \pi_0)v = \pm v\). This is possible only if \(\pi_0v = 0\) or \(\pi_1v = 0\), which completes the proof. ◻
Lemma 2. We may block-diagonalize \(R_1\) and \(R_0\) with blocks of size less than or equal to two on the diagonal.
Proof. TBD. ◻
Lemma 3. Suppose that \(P_0\) and \(P_1\) are subspaces and \(\Vert \pi_1 - \pi_0 \Vert < 1\). Then \(T = \pi_1 - \pi_0\) satisfies \(\Vert T \Vert = \Vert \pi_0 \pi_1^\perp \Vert = \Vert \pi_0^\perp \pi_1\Vert\).
Proof. Considering again the block-diagonal form, we need only prove the claim for each block. We have \(T^\dagger T = \pi_0 + \pi_1 - \pi_0\pi_1 - \pi_1\pi_0\), and thus \(\pi_0^\perp T^\dagger T \pi_0 = 0\), so \(T^\dagger T\) is block-diagonal with respect to \(\pi_0\). Thus the largest eigenvalue of \(T^\dagger T\) is the largest eigenvalue of either \(\pi_0 T^\dagger T \pi_0 = \pi_0 \pi_1^\perp \pi_0\) or \(\pi_0^\perp T^\dagger T \pi_0^\perp = \pi_0^\perp \pi_1 \pi_0^\perp\), which is either \(\Vert \pi_0 \pi_1^\perp\Vert\) or \(\Vert \pi_0^\perp\pi_1 \Vert\). Since \(P_0, P_0^\perp, P_1,P_1^\perp\) are all one-dimensional subspaces, it is clear that \(\Vert \pi_0^\perp \pi_1 \Vert = \Vert \pi_0 \pi_1^\perp \Vert\). ◻
Prop 1. If \(\Vert \pi_1 - \pi_0 \Vert < 1\), then the unitary \(U_{01}\) given by \(\sqrt{R_1R_0}\) maps \(P_0\) to \(P_1\), where this is well-defined when the branch cut of the square root is taken along \(\mathbb R^{<0}\).
Proof. Since \(P_0\) and \(P_1\) may not contain any subspaces orthogonal to the other, we focus on the \(2\times 2\) blocks. Then \(P_0\) is spanned by \(\ket{0}\) and \(P_1\) is spanned by \(\cos(\theta)\ket{0} + \sin(\theta)\ket{1}\), where \(\ket{0}, \ket{1}\) are normalized kets spanning the block in question. Explicitly, we have \(R_0 = \sigma^z\) and \[\begin{aligned} R_1 &= 2(\cos(\theta)\ket{0} + \sin(\theta)\ket{1})(\cos(\theta)\bra{0} + \sin(\theta)\bra{1}) - 1 \\ &= 2\cos^{2}(\theta)\ketbra{0}{0} + 2\sin^{2}(\theta)\ketbra{1}{1} + 2\sin(\theta)\cos(\theta)\sigma^x-1 \\ &= 2\ketbra{1}{1} + 2\cos^{2}(\theta)\sigma^z + \sin(2\theta)\sigma^x-1 \\ &= 2\ketbra{1}{1} +\cos(2\theta)\sigma^z + \sigma^z + \sin(2\theta)\sigma^x-1 \\ &= \cos(2\theta)\sigma^z + \sin(2\theta)\sigma^x \end{aligned}\] Therefore \[\begin{aligned} R_1R_0 &= \exp(-2i\theta \sigma^y) \end{aligned}\] This shows that \(U_{01} = \exp(-i\theta \sigma^y)\), and therefore performs the desired rotation. Furthermore, since \(\theta < \pi/2\), the square root is well-defined. ◻
Cor 1. \(U_{01}\) satisfies \(U_{01}P_1 U_{01}^\dagger = P_0\).
It is more useful later on to work directly with the generator, whose properties we address in the following lemma:
Lemma 4. The generator \(S\) of \(U_{01}\) is block-off-diagonal with respect to \(P_0\), satisfies \(\exp(S)P_1\exp(-S) = P_0\) and \(\Vert S \Vert < \frac{\pi}{2}\). Furthermore, \(S\) satisfying these properties is unique.
Proof. We will not prove uniqueness. The properties follow from the previous proof. ◻
With uniqueness, we can formally identify the construction above with the initial definition of a direct rotation.
The Schrieffer-Wolff transformation
The Schrieffer-Wolff transformation is a second-order perturbation theory technique to block-diagonalize a Hamiltonian into low-energy and high-energy subspaces. Given a bare Hamiltonian \(H_0\) and eigenspaces \(V_{<}, V_{>}\), let \(H = H_0 + \epsilon V\) be the system Hamiltonian, where \(V\) does not preserve the decomposition \(V_{<}\oplus V_{>}\). Then we wish to find a unitary transformation \(U\), called the Schrieffer-Wolff transformation, such that \(UHU^\dagger\) reproduces the spectrum of \(H\) when restricted to \(V_{<}\). This will be the direct rotation relating the low-energy subspaces of \(H\) and \(H_0\). We say that \(H_0\) has a spectral gap \(\Delta\) if there exists an interval \(I_0\) such that any eigenvalue of \(H_0\) in \(I_0\) is separated from the rest of the spectrum by at least \(\Delta\). Define \(P_0\) to be the space spanned by eiegenvalues of \(H_0\) within \(I_0\). Then let \(I\) be the interval obtained by thickening \(I_0\) by \(\Delta/2\). Define \(P_1\) to be the subspace spanned by eigenvectors of \(H\) within \(I\). Intuitively, the perturbation \(\epsilon V\) can shift the spectrum by at most \(\epsilon\), so for \(\epsilon < \Delta/2\), all of the eigenvalues of \(H_0\) within \(I_0\) will correspond to eigenvalues of \(H\) within \(I\). This is expressed in the following lemma:
Lemma 5. Suppose \(\Vert V \Vert = 1\) and \(\epsilon < \Delta /2\). Then \(\Vert \pi_1 - \pi_0 \Vert \leq 2\epsilon/\Delta < 1\)
Proof. Put \(T = \pi_1 - \pi_0\). By a proposition in the previous section, we have \(\Vert T \Vert = \Vert \pi_0 \pi_1^\perp \Vert\). Then we observe that \[\begin{aligned} (\pi_0 H_0 \pi_0)(\pi_0 \pi_1^\perp) - (\pi_0 \pi_1^\perp)(\pi_1^\perp H \pi_1^\perp) = \pi_0(H_0 - H)\pi_1^\perp = -\epsilon \pi_0 V \pi_1^\perp \end{aligned}\] We freely subtract a constant from \(H_0\) so that \(I_0\) is centered at zero. Since the spectrum of \((\pi_0 H_0 \pi_0)\) is separated from the spectrum of \((\pi_1^\perp H \pi_1^\perp)\) by at least \(\Delta/2\), by the lemma that follows we have \(\Vert \pi_0 \pi_1^\perp \Vert \leq \frac{2\epsilon}{\Delta}\). ◻
Lemma 6. Let \(A, B\) be normal operators such that \(\operatorname{spec}(A) \subseteq D_\rho\) and \(\operatorname{spec}(B) \subseteq D_{\rho+\delta}^c\). Then the solution of the equation \(BX - XA = Y\) (called a Sylvester equation) satisfies \(\Vert X \Vert \leq \frac{1}{\delta}\Vert Y \Vert\).
Proof. First, observe that \[\begin{aligned} X = \sum_{n=0}^\infty B^{-n-1}YA^n \end{aligned}\] Satisfies the given equation. Furthermore, the operator norm is submultiplicative, so \[\begin{aligned} \Vert X \Vert &\leq \sum_{n=0}^\infty \Vert B^{-1} \Vert^{n+1} \Vert Y \Vert \Vert B \Vert^n \\ &\leq \frac{\Vert Y \Vert}{\rho + \delta} \sum_{n=0}^\infty \qty(\frac{\rho}{\rho + \delta})^n \\ &= \frac{1}{\delta}\Vert Y \Vert \end{aligned}\] ◻
Def 3. The direct rotation relating \(P_0\) and \(P_1\) (as defined at the beginning of the section) is called the Schrieffer-Wolff transformation, and the operator \(H_{\text{eff}} = \pi_0U_{01}H U_{01}^\dagger \pi_0\) is called an effective Hamiltonian.
Perturbation Theory
Without diagonalizing both Hamiltonians, it is difficult to construct the transformation directly. Instead, we can use the fact (proved in the last section) that \(S\) is off-diagonal in \(\pi_0\) and \(H\) is block-diagonal in \(\pi_0\).
Def 4. Define the super-operators \(\mathcal O\) and \(\mathcal D\) as projecting onto the off-diagonal and diagonal blocks respectively with respect to \(\pi_0\). Let \(\mathcal L\) be a superoperator defined by \(\mathcal L(\ketbra{i}{j}) = \mathcal O(\ketbra{i}{j})/(E_i - E_j)\), where \(H_0\ket{i} = E_i\ket{i}\).
Prop 2. If \(X\) is an operator, then \[\begin{aligned} \mathcal L([H_0, X]) = [H_0, \mathcal L(X)] = \mathcal O(X) \end{aligned}\]
Proof. \[\begin{aligned} \mathcal L([H_0, X]) &= \sum_{i,j}\frac{\bra{i}\mathcal O([H_0, X])\ket{j}}{E_i - E_j}\ketbra{i}{j} \\ &= \sum_{i,j}\frac{\bra{i}H_0\mathcal O(X) - \mathcal O(X) H_0)\ket{j}}{E_i - E_j}\ketbra{i}{j} \\ &=\sum_{i,j}\frac{(E_i-E_j)\bra{i}\mathcal O(X)\ket{j}}{E_i - E_j}\ketbra{i}{j} = \sum_{i,j}\bra{i}\mathcal O(X)\ket{j}\ketbra{i}{j} = \mathcal O(X) \end{aligned}\] ◻
Lemma 7. \(S\) satisfies the equation \(S = \epsilon \qty[(\mathcal D(V)) + \hat S \coth(\hat S)\mathcal O(V)]\)
Proof. Write \(\hat S\) for the superoperator action of \(S\) by the commutator. Then we may write \(H_{\text{eff}} = UHU^\dagger = \exp(S)H\exp(-S) = \exp(\hat S)H\), which is block-diagonal with respect to \(\pi_0\) as shown previously. Then \[\begin{aligned} \mathcal O(\exp(\hat S)H) = \sinh(\hat S)\mathcal D(H) + \cosh(\hat S)\mathcal O(H) = \sinh(\hat S)\mathcal D(H) + \epsilon\cosh(\hat S)\mathcal O(V) = 0 \end{aligned}\] Since \(S\) is block-off-diagonal, \[\begin{aligned} S = \mathcal O(S) = \mathcal L([H_0, S]) = -\mathcal L(\hat S H_0) \end{aligned}\] Assuming that \(\epsilon\) is sufficiently small such that \(\sinh(\hat S)\) is invertible, we have \(\mathcal D(H) = H_0 + \epsilon \mathcal D(V) = -\epsilon \coth(\hat S)\mathcal O(V)\) from before, so \[\begin{aligned} \mathcal L(\hat S H_0) = -\epsilon\mathcal L(\hat S\mathcal D(V) - \hat S\coth(\hat S)\mathcal O(V)) \end{aligned}\] ◻
Lemma 8. If \(S = \sum_{n=0}^\infty\epsilon^n S_n\) is the Taylor series expansion of \(S\), then \(S_n\) satisfies \(S_{n+1} = \mathcal L \hat S_n \mathcal D(V) + \sum_{j=1}^\infty a_{2j}\mathcal L(\hat S^{2j}_{n-1}\mathcal O(V))\), where we define the symbol \[\begin{aligned} \hat S^k_m = \sum_{\substack{n_1, \dots, n_k=1 \\ n_1 + \dots + n_k = m}}\hat S_{n_1}\dots \hat S_{n_k} \end{aligned}\] and \(a_m\) are the coefficients in the Taylor series expansion of \(x\coth(x)\).
Proof. We solve the equation from the previous lemma with \(S = \sum_{n=0}^\infty \epsilon^n S_n\): \[\begin{aligned} \sum_{n=0}^\infty\epsilon^n S_n &= \sum_{n=0}^\infty \epsilon^{n+1}\mathcal L\hat S_n \mathcal D(V) + \epsilon\mathcal L \sum_{m=0}^\infty a_mS^m\mathcal O(V) \\ &= \sum_{n=0}^\infty \epsilon^{n+1}\mathcal L\hat S_n \mathcal D(V) + \epsilon\mathcal L \sum_{m=0}^\infty a_m\qty(\sum_{n}\epsilon^n \hat S_n)^m\mathcal O(V) \\ &= \sum_{n=0}^\infty \epsilon^{n+1}\mathcal L\qty[\hat S_n \mathcal D(V) + \sum_{m=0}^\infty a_m\hat S_k^n\mathcal O(V)] \end{aligned}\] Equating powers of \(\epsilon\) on each side gives the desired result. ◻
Lemma 9. The effective Hamiltonian is given by \[\begin{aligned} H_{eff} = P_0HP_0 + P_0\sum_{n=2}^\infty\epsilon_n\qty[\sum_{j = 1}^\infty b_{2j-1}\hat S^{2j-1}_{n-1}\mathcal O(V)]P_0 \end{aligned}\] Where \(b_m\) are the coefficients in the Taylor series of \(\tanh(x/2)\).
Proof. Assuming \(\cosh(\hat S)\) is invertible, which holds for sufficiently small \(S\), we have \(\tanh(\hat S)\mathcal D(H) = -\epsilon \mathcal O(V)\). Then the transformed Hamiltonian can be rewritten as \[\begin{aligned} \exp(\hat S)H = \mathcal D(\exp(\hat S)H) &= \cosh(\hat S)\mathcal D(H) + \sinh(\hat S)\mathcal O(H) \\ &= \mathcal D(H) +\frac{\cosh(\hat S)-1}{\tanh(\hat S)}\tanh(\hat S)\mathcal D(H) + \epsilon\sinh(\hat S)\mathcal O(V) \\ &= \mathcal D(H) + \epsilon\qty[\sinh(\hat S) - \frac{\cosh(\hat S)-1}{\tanh(\hat S)}]\mathcal O(V) \\ &= \mathcal D(H) + \epsilon\tanh(\hat S/2)\mathcal O(V) \end{aligned}\] Combining this with the series expression for \(S\) from the previous lemma gives the desired result. ◻
Locality bounds
For the remainder of this section, consider a bipartition of the system into subsystems \(A\) and \(B\), and let \(O^A\) denote an operator that acts nontrivially only on subsystem \(A\).
Lemma 10. If \(\Vert \pi^A_1 - \pi^A_0\Vert < 1\) and \(\Vert \pi^B_1 - \pi^B_0 \Vert < 1\), then this implies \(\Vert \pi^A_1 \otimes \pi^B_1 -\pi^A_0 \otimes \pi^B_0\Vert < 1\).
Proof. By assumption, we have \(\pi^A_1 - \pi^A_0 \leq (1-\alpha)I\) and \(\pi^B_1 - \pi^B_0 \leq (1-\beta)I\) for \(\alpha, \beta > 0\) (We interpret \(M \geq N\) as saying that \(M-N\) is positive semi-definite). Rearranging, we find \(\pi^A_1\pi^A_0 \pi^A_1 \geq \alpha \pi^A_1\) and \(\pi^B_1 \pi^B_0 \pi^B_1 \geq \beta \pi^B_1\), which implies that \[\begin{aligned} (\pi^A_1 \otimes \pi^B_1)(\pi^A_0 \otimes \pi^B_0)(\pi^A_1 \otimes \pi^B_1) \geq \alpha \beta \pi^A_1 \otimes \pi^B_1 \end{aligned}\] where we use the fact that the tensor product preserve positive semi-definiteness. However proceeding by contradiction, if \(\Vert \pi^A_1 \otimes \pi^B_1 -\pi^A_0 \otimes \pi^B_0\Vert = 1\), then there is a vector \(v\) such that \(\pi^A_1\otimes \pi^B_1 v = v\) (or likewise for \(0\)) and \(\pi^A_0 \otimes \pi^B_0v = 0\). However applying the equation above to \(v\) gives \(0 \geq \alpha \beta\), which is a contradiction. ◻
Lemma 11. \[\begin{aligned} (U^{AB})^\dagger (\pi^A_1 \otimes \pi^B_1) = (U^A \otimes U^B)^\dagger(\pi^A_1 \otimes \pi^B_1) \end{aligned}\]
Proof. The above lemma implies that we can block-diagonalize the projectors \(\pi^A_1, \pi^A_0\), and \(\pi^B_1, \pi^B_1\), we only need to prove it for a \(2\times 2\) block. However, this follows quickly from the fact that each block is a projector onto a one-dimensional subspace, i.e. \((U^A)^\dagger\ket{\psi^A_1} = \ket{\psi^A_0}\) and \((U^B)^\dagger\ket{\psi^B_1} = \ket{\psi^B_0}\) implies that \((U^A \otimes U^B)^\dagger\ket{\psi^A_1}\otimes \ket{\psi^B_1} = \ket{\psi^A_0} \otimes \ket{\psi^B_0}\). ◻
Lemma 12. Let \(H_0 = H_0^A + H_0^B\) and \(V = V^A + V^B\). Then \[\begin{aligned} H_{eff} = H_{eff}^A + H_{eff}^B \end{aligned}\] where \(H_{eff}^A\) is the SW transformation obtained for \(H_0^A\) to \(H_0^A + \epsilon V^A\) and similarly for \(B\).
Proof. Now we have \[\begin{aligned} H^{AB}_{eff} &= (\pi^A_0 \otimes \pi^B_0) U^{AB}H(U^{AB})^\dagger(\pi^A_0 \otimes \pi^B_0) \\ &= (\pi^A_0 \otimes \pi^B_0) (U^A \otimes U^B)H(U^{A}\otimes U^B)^\dagger(\pi^A_0 \otimes \pi^B_0) \\ &= H_{eff}^A + H_{eff}^B \end{aligned}\] ◻
Def 5. Consider an interaction graph \((\Lambda, E)\). Expand \(V = \sum_{e \in E}\epsilon_e V^e\), where \(e \in E\) indexes the terms \(\epsilon_e\) and \(V^e\). Then we can expand \[\begin{aligned} H_{eff} = \sum_{n = 0}^\infty\sum_{C \subset E}^\infty K_n^C \end{aligned}\] where \(K^C_n \equiv \sum_{m_1 + m_2 + \dots + m_n = n}\prod_{e_i \in C}\epsilon_{e_i}^{m_i}K^C_{m_1, \dots, m_n}\) for some operators \(K^C_{m_1, \dots, m_n}\)
Prop 3. \(K^C_n\) acts only on \(C\) and \(K_n^C = 0\) unless \(C\) is a connected set and \(|C| \leq n\).
Proof. Since \(K_n^C\) does not depend on \(\epsilon_e\) for \(e \notin C\), we can freely take these to be zero. Then the system factorizes into \(\Lambda(C)\) and its compliment, so the additivity property shows that \(K_n^C\) only acts on \(\Lambda(e) \in C\). For the second claim, suppose \(C = C_1 \sqcap C_2\). Then in a similar method, we set \(\epsilon_e = 0\) for any \(e \notin C\), and the system factorizes into a product of \(C_1\) and \(C_2\). Thus \(H_{eff} = H_{eff}^A + H_{eff}^B\), which an affords an expansion in which no term contains variables \(\epsilon^e\) from both \(C_1\) and \(C_2\). The claim that \(K_n^C = 0\) for \(|C| > n\) is just by definition. ◻
Note that the above proof is very general, and only relies on the additivity property of the transformation.