Site Basis Excitation Ansatz (SBEA)

• SBEA is a method that defines a local excitation basis from a ground-state MPS to compute elementary excitations in one-dimensional quantum systems.
• It first obtains a small basis via a Lanczos diagonalization similar to a single-site DMRG step, and then recovers momentum dependence through Fourier transformations of overlap and Hamiltonian kernels.
• The approach efficiently compresses the excitation space by retaining non-orthogonality in the local basis, enabling accurate representation of the single-magnon band with low computational cost.

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 (White, 7 Sep 2025).

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 $A$0, 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 $A$1 to a single “multi-state single-site” diagonalization, reusing a small basis of local excitation tensors $A$2 for all $A$3, and avoiding the left-gauge or null-space condition on $A$4, which is reported to be detrimental to basis truncation (White, 7 Sep 2025).

2. Construction of the local excitation basis

After obtaining a uniform ground-state MPS $A$5, SBEA defines

$A$6

so that inserting an arbitrary tensor at the orthogonality center preserves correct normalization. The variational object is introduced through the state

$A$7

An initial choice of $A$8 can be, for example, $A$9 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 $d$0 and $d$1 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 $d$2 acting on $d$3,

$d$4

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 $d$5 to $d$6 by inserting isometries $d$7 on the bonds adjacent to $d$8. One then works in the reduced basis

$d$9

obtains the lowest $\chi$0 eigenpairs $\chi$1, and transforms back via

$\chi$2

The integer $\chi$3 is chosen so that the resulting local energies $\chi$4 span the single-magnon band up to its maximum (White, 7 Sep 2025).

3. Overlap kernels, Hamiltonian kernels, and the non-orthogonal band formulation

For basis tensors $\chi$5 and $\chi$6 located at sites separated by $\chi$7, SBEA defines the overlap and Hamiltonian kernels

$\chi$8

Because the underlying ground state is gapped, with correlation length $\chi$9, both kernels decay as $A$0. In practice, the formulation truncates to $A$1, so that neglected terms are $A$2. Once these arrays of size $A$3 have been computed by blocking $A$4 and $A$5 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,

$A$6

followed by the generalized eigenvalue problem

$A$7

This is explicitly identified with a band-theory diagonalization in a non-orthogonal orbital basis $A$8. If $A$9 is near-singular, directions with very small eigenvalues of $j$0 are projected out. The outputs are the lowest few solutions $j$1 and amplitudes $j$2; for a single-magnon band, only the lowest branch $j$3 is physical (White, 7 Sep 2025).

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 $j$4-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 $j$5, with the center of the chain effectively translationally invariant but carrying random gauges on its bonds. At the central bond one performs an SVD,

$j$6

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 $j$7 sites. This yields a new three-index tensor $j$8 with identical left and right bond spaces. The corresponding infinite-MPS building blocks are then

$j$9

so that $B$0 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 (White, 7 Sep 2025).

The most distinctive algorithmic point in SBEA concerns gauge. In the original excitation ansatz one often imposes the left gauge condition

$B$1

which implies $B$2 for $B$3. This null-space projection produces an orthogonal basis of plane-wave MPS. In SBEA, however, imposing that gauge is reported to push all local single-site energies $B$4 in the Lanczos diagonalization very high, so that one would require $B$5 states to span the low-energy band. The method therefore works by not imposing any gauge on $B$6, accepting that $B$7 at different sites are non-orthogonal and allowing the generalized eigenvalue problem to treat the overlap explicitly (White, 7 Sep 2025).

A recurrent misconception in MPS excitation methods is that orthogonality of the local variational basis is automatically numerically advantageous. In the SBEA formulation, the opposite conclusion is reported for basis truncation: a non-orthogonal basis is not an incidental by-product but a crucial ingredient of efficient convergence.

5. Wannier excitations

SBEA also provides a Wannier-space representation of the single-magnon sector. The excitation manifold

$B$8

is a true vector space, and the analogy with electronic band theory motivates the construction of localized, orthonormal Wannier excitations $B$9 spanning the same space as the plane-wave states $$\bigl|B\bigr\rangle_j = \cdots A\,A\,\underline{B}\,A\,A\cdots,$$0.

The projection operator onto the single-magnon subspace is written as

$$\bigl|B\bigr\rangle_j = \cdots A\,A\,\underline{B}\,A\,A\cdots,$$1

Applying $$\bigl|B\bigr\rangle_j = \cdots A\,A\,\underline{B}\,A\,A\cdots,$$2 to localized trial states $$\bigl|B\bigr\rangle_j = \cdots A\,A\,\underline{B}\,A\,A\cdots,$$3, for example

$$\bigl|B\bigr\rangle_j = \cdots A\,A\,\underline{B}\,A\,A\cdots,$$4

produces projected localized states. In practice, a small admixture of other $$\bigl|B\bigr\rangle_j = \cdots A\,A\,\underline{B}\,A\,A\cdots,$$5 is used to break parity so that the overlap matrix is nonsingular. Symmetric orthonormalization is then performed: $$\bigl|B\bigr\rangle_j = \cdots A\,A\,\underline{B}\,A\,A\cdots,$$6 which gives $$\bigl|B\bigr\rangle_j = \cdots A\,A\,\underline{B}\,A\,A\cdots,$$7.

Under lattice translation,

$$\bigl|B\bigr\rangle_j = \cdots A\,A\,\underline{B}\,A\,A\cdots,$$8

the dispersion is recovered from

$$\bigl|B\bigr\rangle_j = \cdots A\,A\,\underline{B}\,A\,A\cdots,$$9

Equivalently, one constructs the finite Hamiltonian matrix $$\bigl|\Psi_k(B)\bigr\rangle =\sum_j e^{\,i k j}\;\bigl|B\bigr\rangle_j, \qquad \min_B\frac{\langle\Psi_k(B)|H|\Psi_k(B)\rangle} {\langle\Psi_k(B)|\Psi_k(B)\rangle}=E(k).$$0, diagonalizes it by an ordinary band-theory step, and obtains exactly the same single-magnon band as in SBEA. White further states that one Wannier excitation, translated to all sites, can reconstruct the single magnon modes exactly for all momenta (White, 7 Sep 2025).

6. Benchmark on the $$\bigl|\Psi_k(B)\bigr\rangle =\sum_j e^{\,i k j}\;\bigl|B\bigr\rangle_j, \qquad \min_B\frac{\langle\Psi_k(B)|H|\Psi_k(B)\rangle} {\langle\Psi_k(B)|\Psi_k(B)\rangle}=E(k).$$1 Heisenberg chain

The main application in the original presentation is the spin-1 Heisenberg chain,

$$\bigl|\Psi_k(B)\bigr\rangle =\sum_j e^{\,i k j}\;\bigl|B\bigr\rangle_j, \qquad \min_B\frac{\langle\Psi_k(B)|H|\Psi_k(B)\rangle} {\langle\Psi_k(B)|\Psi_k(B)\rangle}=E(k).$$2

which is gapped with correlation length $$\bigl|\Psi_k(B)\bigr\rangle =\sum_j e^{\,i k j}\;\bigl|B\bigr\rangle_j, \qquad \min_B\frac{\langle\Psi_k(B)|H|\Psi_k(B)\rangle} {\langle\Psi_k(B)|\Psi_k(B)\rangle}=E(k).$$3 and one-magnon gap $$\bigl|\Psi_k(B)\bigr\rangle =\sum_j e^{\,i k j}\;\bigl|B\bigr\rangle_j, \qquad \min_B\frac{\langle\Psi_k(B)|H|\Psi_k(B)\rangle} {\langle\Psi_k(B)|\Psi_k(B)\rangle}=E(k).$$4. The ground state is built using the finite-DMRG insertion procedure to obtain an iMPS of bond dimension $$\bigl|\Psi_k(B)\bigr\rangle =\sum_j e^{\,i k j}\;\bigl|B\bigr\rangle_j, \qquad \min_B\frac{\langle\Psi_k(B)|H|\Psi_k(B)\rangle} {\langle\Psi_k(B)|\Psi_k(B)\rangle}=E(k).$$5, converged to double-precision accuracy. For the local excitation basis, the left and right environments are pre-truncated to $$\bigl|\Psi_k(B)\bigr\rangle =\sum_j e^{\,i k j}\;\bigl|B\bigr\rangle_j, \qquad \min_B\frac{\langle\Psi_k(B)|H|\Psi_k(B)\rangle} {\langle\Psi_k(B)|\Psi_k(B)\rangle}=E(k).$$6, and Lanczos yields $$\bigl|\Psi_k(B)\bigr\rangle =\sum_j e^{\,i k j}\;\bigl|B\bigr\rangle_j, \qquad \min_B\frac{\langle\Psi_k(B)|H|\Psi_k(B)\rangle} {\langle\Psi_k(B)|\Psi_k(B)\rangle}=E(k).$$7 local eigenmodes up to $$\bigl|\Psi_k(B)\bigr\rangle =\sum_j e^{\,i k j}\;\bigl|B\bigr\rangle_j, \qquad \min_B\frac{\langle\Psi_k(B)|H|\Psi_k(B)\rangle} {\langle\Psi_k(B)|\Psi_k(B)\rangle}=E(k).$$8, identified as the top of the magnon band (White, 7 Sep 2025).

The overlap and Hamiltonian kernels are computed for $$\bigl|\Psi_k(B)\bigr\rangle =\sum_j e^{\,i k j}\;\bigl|B\bigr\rangle_j, \qquad \min_B\frac{\langle\Psi_k(B)|H|\Psi_k(B)\rangle} {\langle\Psi_k(B)|\Psi_k(B)\rangle}=E(k).$$9. The stated cost is

$k$0

with MPO bond dimension $k$1; on a laptop this takes $k$2. The generalized eigenvalue problem is then solved at 500 values of $k$3, each solve being a dense generalized eigensolve of a $k$4 matrix and taking a few $k$5.

The resulting one-magnon dispersion $k$6 agrees to within $k$7 of time-dependent DMRG results of White–Affleck 2008 for all $k$8. Below that threshold, the magnon enters the two-magnon continuum and the single-particle excitation ansatz is no longer strictly valid. At $k$9, SBEA gives

$A$00

The computational profile reported for this benchmark consists of one Lanczos diagonalization for the local basis, with cost $A$01; a single small $A$02 generalized eigensolve per momentum; no per-$A$03 MPS update; no VUMPS sweeps; and controllable error in the single-magnon regime through $A$04 and the pre-truncation cutoff (White, 7 Sep 2025).

Within the scope explicitly demonstrated, SBEA is therefore a momentum-independent local-basis construction combined with a non-orthogonal band-theory solve, specialized to elementary excitations above an infinite-MPS ground state. Its main conceptual departure from earlier excitation-ansatz practice is the deliberate retention of non-orthogonality in the local excitation basis, and its main numerical result is that this choice permits the full single-magnon band of the $A$05 Heisenberg chain to be represented with a very small basis.