# Excitation spectrum and Density Matrix Renormalization Group iterations

## TL;DR

This paper reveals that flat excitation spectrum modes during DMRG sweeps can accurately identify excited states, enabling efficient analysis of critical, topological, and impurity systems with minimal additional computational cost.

## Contribution

It introduces a novel method to extract excited states from ground state DMRG calculations by analyzing flat modes in the effective Hamiltonian spectrum.

## Key findings

- Accurate low-energy spectra for critical systems like Ising and Potts models.
- Effective detection of topological quasi-degenerate excitations.
- Assessment of excited state accuracy using Hamiltonian variance.

## Abstract

We show that, in certain circumstances, exact excitation energies appear as locally site-independent (or flat) modes if one records the excitation spectrum of the effective Hamiltonian while sweeping through the lattice in the variational Matrix Product State formulation of the Density Matrix Renormalization Group (DMRG), a remarkable property since the effective Hamiltonian is only constructed to target the ground state. Conversely, modes that are very flat over several consecutive iterations are systematically found to correspond to faithful excitations. We suggest to use this property to extract accurate information about excited states using the standard ground state algorithm. The results are spectacular for critical systems, for which the low-energy conformal tower of states can be obtained very accurately at essentially no additional cost, as demonstrated by confirming the predictions of boundary conformal field theory for two simple minimal models - the transverse-field Ising model and the critical three-state Potts model. This approach is also very efficient to detect the quasi-degenerate low-energy excitations in topological phases, and to identify localized excitations in systems with impurities. Finally, using the variance of the Hamiltonian as a criterion, we assess the accuracy of the resulting Matrix Product State representations of the excited states.