--- title: Site Basis Excitation Ansatz (SBEA) url: https://www.emergentmind.com/topics/site-basis-excitation-ansatz-sbea type: topic --- # Site Basis Excitation Ansatz (SBEA) Site Basis Excitation Ansatz (SBEA) is a variation of the tangent-space excitation ansatz for computing elementary excitation spectra of one-dimensional quantum lattice systems using matrix product states (MPS). It is formulated on top of an infinite MPS description of the ground state and replaces the conventional momentum-by-momentum large generalized eigenvalue workflow with a two-stage procedure: first, a small basis of local excitation tensors is obtained from a single diagonalization analogous to a single-site DMRG step but for multiple states; second, momentum dependence is recovered from overlap and Hamiltonian matrix elements in that basis through diagonalization of a tiny generalized eigenvalue problem, akin to a non-orthogonal band-theory diagonalization. In the formulation introduced by S. R. White, SBEA is also accompanied by an extremely simple alternative to variational uniform matrix product states (VUMPS) based on finite-system DMRG, and by a construction of Wannier excitations that can reconstruct the single-magnon modes exactly for all momenta [2509.06241]. ## 1. Relation to the tangent-space excitation ansatz SBEA is introduced as a modification of the tangent-space or excitation ansatz associated with Haegeman et al. In the standard formulation, the ground state is represented by a uniform infinite MPS in canonical form, \[ \cdots \lambda\,\Gamma\,\lambda\,\Gamma\,\lambda\,\Gamma\cdots \quad\longleftrightarrow\quad \cdots A\,A\,A\cdots, \quad A = \lambda^{1/2}\Gamma\,\lambda^{1/2}. \] The tensor \(A\) carries one physical index of dimension \(d\) and two bond indices of dimension \(\chi\). A single local excitation is generated by replacing one \(A\) tensor at site \(j\) by a variational tensor \(B\), \[ \bigl|B\bigr\rangle_j = \cdots A\,A\,\underline{B}\,A\,A\cdots, \] and a Bloch superposition is formed as \[ \bigl|\Psi_k(B)\bigr\rangle =\sum_j e^{\,i k j}\;\bigl|B\bigr\rangle_j, \qquad \min_B\frac{\langle\Psi_k(B)|H|\Psi_k(B)\rangle} {\langle\Psi_k(B)|\Psi_k(B)\rangle}=E(k). \] In that tangent-space scheme, one must solve a separate generalized eigenvalue problem for each momentum \(k\). The implementation also requires a converged uniform MPS, for example via VUMPS, together with a gauge choice for \(B\), often a left-orthonormal one. White’s formulation of SBEA is motivated by three specific objectives: reducing the number of large diagonalizations from one per \(k\) to a single “multi-state single-site” diagonalization, reusing a small basis of local excitation tensors \(\{B_\alpha\}\) for all \(k\), and avoiding the left-gauge or null-space condition on \(B\), which is reported to be detrimental to basis truncation [2509.06241]. ## 2. Construction of the local excitation basis After obtaining a uniform ground-state MPS \(\cdots A A A\cdots\), SBEA defines \[ A_L = A\,\lambda^{-1/2},\qquad A_R = \lambda^{-1/2}\,A, \] so that inserting an arbitrary tensor at the orthogonality center preserves correct normalization. The variational object is introduced through the state \[ |\tilde B\rangle = \cdots A\,A_L\,\underline{\tilde B}\,A_R\,A\cdots. \] An initial choice of \(\tilde B\) can be, for example, \(\tilde B=S^+(\lambda^{1/2}A\lambda^{1/2})\) in order to target a triplet magnon. The Hamiltonian is written as a matrix product operator in infinite uniform form, or long-distance terms are truncated beyond some cut-off. Left and right environment tensors \(E_L\) and \(E_R\) are then computed by contracting all MPO tensors to the left and right of the center site into single objects. This yields an effective single-site operator \(\mathcal H\) acting on \(\tilde B\), \[ \bigl(\mathcal H\,\tilde B\bigr) = \mathrm{Contract}\bigl(E_L,\;H,\;\tilde B,\;E_R\bigr). \] A Lanczos diagonalization of this effective operator produces a small basis of local excitation tensors. Optionally, the left and right bond environments can first be pre-truncated from dimension \(\chi\) to \(\chi_{\rm tr}\) by inserting isometries \(U\,U^\dagger\) on the bonds adjacent to \(B\). One then works in the reduced basis \[ \tilde B' = U^\dagger\,\tilde B\,U, \qquad \mathcal H' = U^\dagger\,\mathcal H\,(U\,\cdot\,U^\dagger)\,U, \] obtains the lowest \(N_\alpha\) eigenpairs \((B'_\alpha,E_\alpha)\), and transforms back via \[ B_\alpha = U\,B'_\alpha\,U^\dagger. \] The integer \(N_\alpha\) is chosen so that the resulting local energies \(E_\alpha\) span the single-magnon band up to its maximum [2509.06241]. ## 3. Overlap kernels, Hamiltonian kernels, and the non-orthogonal band formulation For basis tensors \(B_\alpha\) and \(B_{\alpha'}\) located at sites separated by \(d=j'-j\), SBEA defines the overlap and Hamiltonian kernels \[ O_{\alpha\alpha'}(d) = {}_{j}\!\langle B_\alpha \mid B_{\alpha'}\rangle_{j'}, \qquad H_{\alpha\alpha'}(d) = {}_{j}\!\langle B_\alpha \bigl|\,H\,\bigr| B_{\alpha'}\rangle_{j'}. \] Because the underlying ground state is gapped, with correlation length \(\xi\), both kernels decay as \(e^{-|d|/\xi}\). In practice, the formulation truncates to \(|d|\le L_c\sim150\), so that neglected terms are \(\lesssim10^{-12}\). Once these arrays of size \(N_\alpha^2\times(2L_c+1)\) have been computed by blocking \(A\) and \(B_\alpha\) into left and right environments, they encode all pairwise overlaps and Hamiltonian matrix elements between excitations localized on different sites. Momentum dependence is then obtained through Fourier transformation, \[ \widetilde O_{\alpha\alpha'}(k) = \sum_{d=-L_c}^{+L_c} e^{\,i k d}\;O_{\alpha\alpha'}(d), \qquad \widetilde H_{\alpha\alpha'}(k) = \sum_{d=-L_c}^{+L_c} e^{\,i k d}\;H_{\alpha\alpha'}(d), \] followed by the generalized eigenvalue problem \[ \widetilde H(k)\,c = E(k)\;\widetilde O(k)\,c. \] This is explicitly identified with a band-theory diagonalization in a non-orthogonal orbital basis \(\{B_\alpha\}\). If \(\widetilde O(k)\) is near-singular, directions with very small eigenvalues of \(\widetilde O\) are projected out. The outputs are the lowest few solutions \(E_n(k)\) and amplitudes \(c^k_{n,\alpha}\); for a single-magnon band, only the lowest branch \(n=0\) is physical [2509.06241]. The formal significance of this construction is that the expensive optimization is shifted entirely into the momentum-independent local basis generation. This suggests an interpretation of SBEA as a compressed excitation-space representation in which the full \(k\)-resolved problem is reduced to band formation within a small non-orthogonal local basis. ## 4. Infinite-MPS ground states from finite-system DMRG and the role of gauge choice White introduces an alternative to VUMPS for constructing the infinite MPS ground state needed by SBEA. The procedure begins with an ordinary two-site or single-site DMRG calculation on an open chain of length \(N\gg\xi\), with the center of the chain effectively translationally invariant but carrying random gauges on its bonds. At the central bond one performs an SVD, \[ \tilde D_{\;\alpha\beta}^{m} \quad\longleftrightarrow\quad \tilde \lambda_{\alpha}, \qquad \tilde\Gamma_{\alpha\beta}^m = (\tilde\lambda^{-1})_{\alpha}\,\tilde D_{\alpha\beta}^m\,(\tilde\lambda^{-1})_{\beta}. \] A new site is inserted at that bond, its tensor is initialized randomly, and a single high-accuracy Lanczos/DMRG update is performed to minimize the energy, analogous to a single-site DMRG step on \(N+1\) sites. This yields a new three-index tensor \(\tilde D\) with identical left and right bond spaces. The corresponding infinite-MPS building blocks are then \[ \lambda = \tilde\lambda,\qquad \Gamma=\tilde\lambda^{-1}\,\tilde D\,\tilde\lambda^{-1}, \qquad A = \lambda^{1/2}\,\Gamma\,\lambda^{1/2}, \] so that \(\cdots A\,A\,A\cdots\) represents the desired uniform state up to DMRG accuracy. Optionally, one Orús–Vidal canonicalization sweep can be applied to ensure perfect left/right orthonormality of the half-chains [2509.06241]. The most distinctive algorithmic point in SBEA concerns gauge. In the original excitation ansatz one often imposes the left gauge condition \[ \sum_s A_L^{s,\alpha\gamma}\,B^{s}_{\gamma\beta}=0 \quad\forall\,\alpha,\beta, \] which implies \(\langle B\rvert_j B\rangle_{j'}=0\) for \(j'