---
title: Displaced-Basis Exact Diagonalization
url: https://www.emergentmind.com/topics/displaced-basis-exact-diagonalization
type: topic
---

# Displaced-Basis Exact Diagonalization

to here.
Searching arXiv for recent and foundational papers on displaced-basis exact diagonalization and related basis-adapted ED methods.
Displaced-basis exact diagonalization denotes a class of exact diagonalization workflows in which the Hamiltonian is first represented in a basis adapted to the dominant correlations, conserved quantities, or impurity dressing, and only then diagonalized in a truncated but physically targeted subspace. In the materials considered here, the most explicit realization is the construction of many-body eigenstates by successive displacement transformations that nearly diagonalize an interacting fermionic Hamiltonian in terms of local integrals of motion, followed by exact diagonalization in a projected excitation subspace [1901.10368]. Related formulations replace the generic Fock basis by Bethe Ansatz eigenstates for mobile-impurity problems [2112.06627], while extended dynamical mean-field theory solvers expose the same general issue from the bosonic side, where displaced oscillator bases are discussed as a possible alternative to direct occupation-number truncation, even though the reported implementation remains in the Fock basis [1709.09176].

## 1. Concept and scope

Within many-body numerical methods, exact diagonalization is limited primarily by the exponential growth of Hilbert-space dimension. The basis-adapted strategy summarized here seeks to reduce that burden by selecting a representation in which the target states are closer to product states, or in which the off-diagonal structure of the Hamiltonian is substantially simplified. In the displacement-transformation formulation, the target is a Hamiltonian of the form
\[
\tilde{H} = U^\dagger H U = \sum_i \epsilon_i \tau_i + \epsilon_{ij}\tau_i \tau_j + \cdots
\]
with \(\tau_i = U^\dagger n_i U\), so that eigenstates become occupation configurations of transformed number operators [1901.10368].

A closely related idea appears in the mobile-impurity problem, where exact diagonalization is performed not in a lattice or oscillator basis but in a truncated basis of Bethe Ansatz states of the integrable equal-mass limit [2112.06627]. In the EDMFT impurity problem, the large bosonic Hilbert space motivates discussion of basis optimization; the reported solver uses a direct-product Fock basis with boson-number cutoffs, but notes that displaced oscillator bases are known in the literature as a way to reduce the required cutoff in strongly coupled electron-boson problems [1709.09176].

This suggests that “displaced-basis exact diagonalization” is best understood not as a single algorithmic identity, but as a methodological family in which diagonalization is deferred until the basis has been transformed or selected to reflect the problem’s emergent structure.

## 2. Displacement transformations and operator-diagonalization

The central operator construction is given for a general interacting fermionic Hamiltonian in second quantization,
\[
H = \sum_X V_X (X^\dagger + X),
\]
where each \(X\) is a string of number operators and creation/annihilation operators,
\[
X = n_{\alpha_1} \ldots n_{\alpha_k} c^\dagger_{\beta_1} c_{\gamma_1} \ldots c^\dagger_{\beta_l} c_{\gamma_l},
\]
with distinct indices and conserved particle number [1901.10368].

A displacement transformation associated with an operator \(X\) is defined by
\[
\mathcal{D}_X(\lambda) = \exp\left[\lambda (X^\dagger - X)\right].
\]
The compact representation reported in the source is
\[
\mathcal{D}_X(\lambda) = \mathbf{1} + \sin\lambda (X^\dagger - X) + (\cos\lambda-1)(X^\dagger X + X X^\dagger),
\]
and these transformations are canonical, preserving fermionic anti-commutation relations [1901.10368].

The coefficient of the selected quantum term \(X^\dagger + X\) can be canceled by choosing
\[
\tan(2\lambda) = \frac{2V_X}{\Delta\epsilon_X},
\]
where \(V_X\) is the coefficient of \(X^\dagger + X\) and \(\Delta\epsilon_X\) is the energy difference between the two configurations connected by \(X\), as constructed from the diagonal entries of the Hamiltonian. Under this transformation, the remaining contribution is the density-like combination
\[
V_X \tan(\lambda) \left(X^\dagger X - X X^\dagger \right)
\]
[1901.10368].

Successive transformations therefore suppress quantum terms and move the Hamiltonian toward a diagonal, “classical” form in a transformed basis. The source emphasizes that this constructs local integrals of motion and provides a systematic way to extend Hartree-Fock and configuration interaction theories to higher order [1901.10368].

## 3. Reference states, variational targeting, and the exact diagonalization step

The practical implementation is organized around a reference Slater determinant,
\[
|\Phi_0\rangle = c^\dagger_{\beta_1} \cdots c^\dagger_{\beta_N} |0\rangle,
\]
and only displacement transformations that affect expectation values for \(|\Phi_0\rangle\) are retained. This restriction is the main efficiency mechanism: it avoids generating the full operator algebra and instead targets the neighborhood of a chosen state [1901.10368].

The method alternates between two criteria. The first is energy minimization or maximization relative to the reference state. For admissible transitions \(X\), the parameter is chosen according to
\[
\tan 2\lambda_X = \frac{2V_{X,1}}{\Delta E_X},
\]
where \(V_{X,1} = \langle\Phi_0 | X^\dagger H|\Phi_0\rangle\), and \(\Delta E_X\) is the energy difference between \(|\Phi_0\rangle\) and \(X|\Phi_0\rangle\) [1901.10368]. The second is minimization of the energy variance,
\[
\sigma^2 = \langle H^2 \rangle - \langle H \rangle^2,
\]
which is used when pure energy optimization stalls, especially for excited states or in delocalized regimes [1901.10368].

After substantial suppression of off-diagonal terms, the residual Hamiltonian is projected onto a subspace of low-lying electron-hole excitations around the reference state, and that projected Hamiltonian is exactly diagonalized. The paper explicitly characterizes this as a step akin to a limited configuration interaction calculation [1901.10368]. In this workflow, exact diagonalization is not the primary approximation engine; rather, it is the final refinement after basis adaptation has already compressed the relevant state space.

## 4. Relation to Hartree-Fock, configuration interaction, and conventional exact diagonalization

The displacement-transformation scheme is presented as a systematic extension of both Hartree-Fock and configuration interaction. Hartree-Fock is identified with including only second-order displacement transformations, producing gaussian, single-Slater-determinant ground states. Higher-order transformations incorporate correlation effects and non-gaussian correlations [1901.10368].

Compared with configuration interaction, the distinction is that the basis is transformed before the CI-like diagonalization is applied. Standard CI augments Hartree-Fock by adding excitations, whereas the displacement method first makes the Hamiltonian nearly diagonal by controlled unitary transformations and only then performs the projected diagonalization [1901.10368]. This suggests a different allocation of computational effort: rather than enlarging the excitation manifold in a fixed basis, the method reshapes the basis to reduce the importance of high-order mixing.

Relative to standard exact diagonalization, the source states that exact diagonalization is exact but scales exponentially with system size, restricting practical calculations to \(L \lesssim 20\) sites for fermionic models, whereas the displacement method can scale to considerably larger systems, up to \(L=30\), particularly when higher-order terms can be neglected or systematically controlled [1901.10368]. The reported computational cost depends on the highest order of transformations included; up to fourth order, the CPU time per sample scales as \(L^{10}\) [1901.10368].

A related but distinct comparison appears in the mobile-impurity literature. There, standard exact diagonalization in a generic basis is contrasted with exact diagonalization in a basis of Bethe Ansatz eigenstates of the integrable model. The latter is physically adapted, truncated by overlap with a chosen initial state, and allows controlled study of non-integrable perturbations such as mass imbalance [2112.06627]. Although this is not a displacement-transformation method, it exemplifies the same general principle that basis adaptation can be more important than raw diagonalization power.

## 5. Excited states, integrals of motion, and truncation logic

A defining feature of the displacement-transformation construction is that it keeps track of the local integrals of motion forming the reference state. In the transformed basis, eigenstates correspond to occupation patterns of the \(\tau_i\), so targeting an excited state amounts to choosing a different occupancy configuration of transformed creation operators and then applying the same energy- and variance-based optimization [1901.10368].

The source emphasizes that energy and variance minimization are equally valid for ground and excited states; only the reference configuration changes [1901.10368]. This is significant because many approximate diagonalization methods are optimized almost exclusively for the ground state, whereas here excited-state construction is built into the operator formalism.

The benchmark claims are specific. For one-dimensional spinless fermion models with disorder and interactions, ground-state energies for system sizes up to \(L=16\) agree extremely well with exact diagonalization, and for \(L=30\) observables such as number variance are reported to be very good [1901.10368]. For excited states, the paper reports that a large proportion of eigenstates can be constructed explicitly, with benchmarks stating up to \(96\%\) of excited states found [1901.10368].

Truncation logic in the Bethe-basis approach follows a different criterion but serves a similar purpose. The truncated set \(\mathcal{S}\) is selected so that the sum rule
\[
\sum_{\psi} |\langle \Psi_0 | \psi \rangle|^2 = 1
\]
is saturated to accuracy \(\epsilon\), and the preferred implementation uses a Metropolis-type Markov-chain algorithm in partition space to sample states with probability proportional to the squared overlap [2112.06627]. The source notes that ordering states by descending overlap, rather than by energy, yields much faster convergence of the truncated sum rule [2112.06627]. A plausible implication is that displaced-basis exact diagonalization, in its broadest sense, is often defined less by a particular transformation than by a physically informed truncation criterion.

## 6. Bosonic Hilbert spaces and the displaced-oscillator question

In impurity models with bosonic baths, the basis problem reappears in a different form. The EDMFT exact diagonalization solver is built for a Holstein-Anderson impurity Hamiltonian containing both fermionic bath orbitals and bosonic modes,
\[
\begin{aligned}
H_{\text{imp}} ={}\ &\sum_{\sigma} \varepsilon_d d^\dagger_\sigma d_\sigma  + U n_{d\uparrow} n_{d\downarrow} + \sum_{k\sigma} \left[ \mathcal{V}_k d^\dagger_\sigma f_{k\sigma} + \text{h.c.} \right] + \sum_{k \sigma} \varepsilon_k f^\dagger_{k\sigma} f_{k\sigma} \\
&+ \sum_{p} \Omega_p b_p^\dagger b_p + \sum_{p} \mathcal{W}_p (b^\dagger_p + b_p)\, \overline{n}_d .
\end{aligned}
\]
The bosonic Hilbert space is infinite, so the implementation imposes an occupation cutoff \(N_p\) for each mode and uses a direct-product Fock basis [1709.09176].

The source is explicit that the paper does not report the use of displaced oscillator basis or similar transformations in the actual implementation. Instead, it states that the standard basis truncation is sufficient for the presented parameter regions and low temperatures [1709.09176]. At the same time, it notes that displaced oscillator bases can be used in the literature to reduce the required boson cutoff for strongly coupled electron-boson problems [1709.09176].

That distinction is methodologically important. Displaced-basis exact diagonalization is not automatically synonymous with any bosonic ED solver involving phonons or retarded interactions. In the EDMFT case, the displaced basis appears as a possible optimization strategy rather than a realized component of the reported algorithm. The implemented solver relies on sparse Hermitian matrices, iterative Arnoldi/Krylov diagonalization, block diagonalization from particle-number and spin conservation, and bosonic truncation justified by the exponential decay of boson-number probabilities at low temperature [1709.09176].

## 7. Applications, advantages, and limitations

The three strands of work collectively identify several domains in which basis-adapted exact diagonalization is effective. In disordered interacting fermion systems, displacement transformations are used to construct local integrals of motion and to compute both ground and excited states for systems reaching \(L=30\) sites [1901.10368]. In one-dimensional mobile-impurity physics, exact diagonalization in a truncated Bethe basis allows nonperturbative treatment of unequal masses and reveals spectral flow, avoided crossings, and level-statistical signatures of the transition from integrable to non-integrable behavior in a \(5+1\) particle system [2112.06627]. In EDMFT, exact diagonalization is used as an impurity solver with real-axis access and favorable convergence in the deep Mott insulator, while the basis question is handled through occupation cutoffs rather than displaced oscillators [1709.09176].

The principal advantages stated across these works are controlled truncation, physical relevance of the selected basis, and improved access to targeted states or observables. The displacement-transformation method is described as particularly efficient for excited states because it keeps track of the local integrals of motion forming the reference state [1901.10368]. The Bethe-basis approach provides error control through the missing weight in the overlap sum rule and studies non-integrable perturbations in a basis inherited from the integrable limit [2112.06627]. The EDMFT solver provides noise-free real-frequency quantities and better convergence behavior than a strong-coupling CTQMC solver in the deep insulator [1709.09176].

The limitations are equally specific. In the displacement-transformation method, the number of generated terms grows rapidly, and the computational cost depends strongly on the highest order retained [1901.10368]. In the Bethe-basis formulation, the method is less suited for the thermodynamic limit, depends on the chosen initial state, and still becomes demanding at higher particle number or accuracy [2112.06627]. In the EDMFT solver, bath discretization artifacts and bosonic Hilbert-space growth remain the major constraints, and the displaced oscillator basis is discussed only as a possible future optimization [1709.09176].

Taken together, these results place displaced-basis exact diagonalization within a broader research program: exact diagonalization becomes substantially more powerful when preceded by a representation change that encodes local integrals of motion, integrable eigenstates, or oscillator displacements. The strongest explicit realization in the cited literature is the displacement-transformation construction of many-body eigenstates [1901.10368]; the other cases clarify how the same structural idea extends, with varying terminology and implementation, across continuum impurity physics and impurity solvers with bosonic baths [2112.06627; 1709.09176].

Source: https://www.emergentmind.com/topics/displaced-basis-exact-diagonalization