An exact real-space renormalization method and applications

TL;DR

This paper introduces an exact real-space renormalization method for frustration-free Hamiltonians, enabling efficient computation of ground spaces, with applications to spin systems showing novel ground state properties and scalability advantages.

Contribution

The paper presents a novel polynomial-time method for exactly determining ground spaces of frustration-free Hamiltonians, outperforming existing techniques especially for highly degenerate cases.

Findings

Successfully computed ground spaces for large spin systems up to 160 spins.

Discovered a triplet ground state in a spin-1/2 Heisenberg model with cyclic exchange.

Demonstrated quadratic scaling of ground-space degeneracy in certain spin chains.

Abstract

We present a numerical method based on real-space renormalization that outputs the exact ground space of "frustration-free" Hamiltonians. The complexity of our method is polynomial in the degeneracy of the ground spaces of the Hamiltonians involved in the renormalization steps. We apply the method to obtain the full ground spaces of two spin systems. The first system is a spin-1/2 Heisenberg model with four-spin cyclic-exchange interactions defined on a square lattice. In this case, we study finite lattices of up to 160 spins and find a triplet ground state that differs from the singlet ground states obtained in C.D. Batista and S. Trugman, Phys. Rev. Lett. 93, 217202 (2004). We characterize such a triplet state as consisting of a triplon that propagates in a background of fluctuating singlet dimers. The second system is a family of spin-1/2 Heisenberg chains with uniaxial exchange anisotropy and next-nearest neighbor interactions. In this case, the method finds a ground-space degeneracy that scales quadratically with the system size and outputs the full ground space efficiently. Our method can substantially outperform methods based on exact diagonalization and is more efficient than other renormalization methods when the ground-space degeneracy is large.