Gap Estimation by means of Hyperbolic Deformation

TL;DR

This paper introduces a numerical method using hyperbolic deformation and DMRG to accurately estimate the excitation gap in one-dimensional quantum systems, demonstrated on the S=1 Heisenberg chain.

Contribution

The paper proposes a novel hyperbolic deformation approach combined with DMRG for precise bulk gap estimation in quantum chains, improving boundary insensitivity.

Findings

Estimated Haldane gap in [0.41047905, 0.41047931]

Method effectively isolates bulk properties from boundary effects

Demonstrated efficiency on the S=1 antiferromagnetic Heisenberg chain

Abstract

We present a way of numerical gap estimation applicable for one-dimensional infinite uniform quantum systems. Using the density matrix renormalization group method for a non-uniform Hamiltonian, which has deformed interaction strength of $j$-th bond proportional to $\cosh \lambda j$, the uniform Hamiltonian is analyzed as a limit of $\lambda \to 0$. As a consequence of the deformation, an excited quasi-particle is weakly bounded around the center of the system, and kept away from the system boundary. Therefore, insensitivity of an estimated excitation gap of the deformed system to the boundary allows us to have the bulk excitation gap $\Delta(\lambda)$, and shift in $\Delta(\lambda)$ from $\Delta(0)$ is nearly linear in $\lambda$ when $\lambda \ll 1$. Efficiency of this estimation is demonstrated through application to the S=1 antiferromagnetic Heisenberg chain. Combining the above estimation and another one obtained from the technique of convergence acceleration for finite-size gaps estimated by numerical diagonalizations, we conclude that the Haldane gap is in $[0.41047905,~0.41047931]$.