Matrix product state methods for excitations

Chen Liu, Anders W. Sandvik

Jesse J. Osborne, Ian P. McCulloch

Numerical methods based on matrix product states (MPSs) are currently the standard for calculating the ground-state properties of (quasi-)one-dimensional quantum many-body systems. While the properties of the low-lying excitations in such systems are often studied in this MPS framework through by means of time-evolution simulations, we can also look at their by directly calculating eigenstates corresponding to these excitations. The so-called MPS excitation ansatz is a powerful method for finding such eigenstates with a single-particle character in the thermodynamic limit. Although this excitation ansatz has been used quite extensively, a general method for calculating expectation values for these states is lacking in the literature: We aim to fill this gap by presenting a recursive algorithm to calculate arbitrary observables expressed as matrix product operators. This method concisely encapsulates existing methods for—as well as extensions to—the excitation ansatz, such as excitations with a larger spatial support and multiparticle excitations, and is robust enough to handle further innovations. We demonstrate the versatility of our method by studying the low-lying excitations in the spin-1 Heisenberg chain and the one-dimensional Hubbard model, looking at how the excitations converge in the former, while in the latter, we present a refined method of targeting single-particle excitations inside a continuum by minimizing the energy rather than the energy itself. We hope that this technique will foster further advancements with the excitation ansatz.

Assa Auerbach

Steven R. White

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 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.

Miodrag V. Mihailović, Mitja Rosina

Xing-Yu Zhang, Runze Chi, Yang Liu +1 more

Understanding the excitation spectrum in two-dimensional quantum many-body systems has long been a formidable challenge. In this study, we propose an innovative approach by introducing an excitation Ansatz based on an infinite matrix product state (MPS) with a helix structure. The use of infinite uniform MPS states allows us to accurately extract key properties such as energy, degeneracy, spectrum weight, and scaling behavior of low-energy excited states simultaneously. To validate the effectiveness of our method, we apply it to the critical point of the transverse-field Ising model. The scaling exponent of the energy gap extracted aligns closely with conformal bootstrap results. Subsequently, we extend our method to the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on a square lattice. Our findings reveal that the degeneracy of lowest-energy excitations serves as a reliable metric for distinguishing different phases. The phase boundaries identified by our method are consistent with previous findings. The present method provides a promising avenue for studying the excitation spectrum of two-dimensional quantum many-body systems.