Site Basis Excitation Ansatz for Matrix Product States

Abstract

We introduce a simple and efficient variation of the tangent-space excitation ansatz used to compute elementary excitation spectra of one-dimensional quantum lattice systems using matrix product states (MPS). A small basis for the excitation tensors is formed based on a single diagonalization analogous to a single site DMRG step but for multiple states. Once overlap and Hamiltonian matrix elements are found, obtaining the excitation for any momentum only requires diagonalization of a tiny matrix, akin to a non-orthogonal band-theory diagonalization. The approach is based on an infinite MPS description of the ground state, and we introduce an extremely simple alternative to variational uniform matrix product states (VUMPS) based on finite system DMRG. For the $S=1$ Heisenberg chain, our method -- site basis excitation ansatz (SBEA) -- efficiently produces the one-magnon dispersion with high accuracy. We also examine the role of MPS gauge choices, finding that not imposing a gauge condition -- leaving the basis nonorthogonal -- is crucial for the approach, whereas imposing a left-orthonormal gauge (as in prior work) severely hampers convergence. We also show how one can construct Wannier excitations, analogous to the Wannier functions of band theory, where one Wannier excitation, translated to all sites, can reconstruct the single magnon modes exactly for all momenta.

Figures (4)

• Figure 1: Lowest eigenvalues of the effective single-site excitation Hamiltonian $\mathcal{H}$ for the spin-1 chain, obtained via Lanczos. The horizontal axis is the index of the eigenvector (with eigenvalues sorted by increasing energy). Shown are results without pre-truncation (bond dimension $\chi=126$, red circles) and with truncation to $\chi_{\text{tr}}=30$ (blue, with $\lambda^2=10^{-7}$ as a cutoff) and 8 (green, $10^{-4}$). The inset shows the untruncated case, where the upper curve has the left gauge condition imposed, resulting in all the local states being much higher in energy and much less useful.
• Figure 2: Spin profile of single site excitations $\langle S_z(x) \rangle$ versus site $x$. (a) Excitation profiles for the lowest energy single-site state, where a single tensor is different from the ground state. The numbers beside each curve indicate the cutoff $\lambda^2$ for the pretruncation step; these cutoffs result in $\chi_{\text{tr}}=8, 24, 46$, and $90$ for the curves top to bottom. (b) Comparison of excitation profile for states with and without restriction to the left null space, both with $\lambda^2=10^{-10}$.
• Figure 3: One‑magnon dispersion of the spin‑1 Heisenberg chain, as a function of the maximum single-site energy eigenvalue included in the site-basis. In all cases, $\chi_{\rm tr}=8$; for each curve, $N_\alpha$ of the $B_\alpha$ were used in the basis, with $E_{\rm max}$ indicating the maximum energy. We see excellent convergence in the spectra for energies below $E_{\rm max}$, which agree with the tDMRG results of White and Affleck.WhiteAffleck08 The one magnon line is inside the two-magnon continuum for $k$ smaller than about 0.72; in that region, the EA is less appropriate and convergence in $N_\alpha$ is slower.
• Figure 4: Spin profile of a Wannier excitation for the Heisenberg chain, using pretruncation $\chi_{\text{tr}}=8$, and $N_\alpha=7$. The profile shows that the excitation is well-localized. Its lack of reflection symmetry is tied to the choice of local states $|L_j\rangle$.