Translation-Invariant Parent Hamiltonians of Valence Bond Crystals.

Daniel Huerga,Alejandro Muramatsu,Andrés Greco,Claudio Gazza

doi:10.1103/physrevlett.118.167202

Abstract

We present a general method to construct translation-invariant and SU(2) symmetric antiferromagnetic parent Hamiltonians of valence bond crystals (VBCs). The method is based on a canonical mapping transforming S=1/2 spin operators into a bilinear form of a new set of dimer fermion operators. We construct parent Hamiltonians of the columnar and the staggered VBCs on the square lattice, for which the VBC is an eigenstate in all regimes and the exact ground state in some region of the phase diagram. We study the departure from the exact VBC regime upon tuning the anisotropy by means of the hierarchical mean field theory and exact diagonalization on finite clusters. In both Hamiltonians, the VBC phase extends over the exact regime and transits to a columnar antiferromagnet (CAFM) through a window of intermediate phases, revealing an intriguing competition of correlation lengths at the VBC-CAFM transition. The method can be readily applied to construct other VBC parent Hamiltonians in different lattices and dimensions.

Highlights

We study the depart from the exact valence bond crystals (VBCs) regime upon tuning the anisotropy by means of the hierarchical mean field theory and exact diagonalization on finite clusters
In this Letter, we present a general method to construct SU(2) symmetric and translation-invariant VBC parent Hamiltonians based on a canonical mapping that exactly identifies a VBC state with the vacuum of a new set of dimer fermions (DF)
Phase diagram.— As the four-spin AF interactions present in (12) and (13) pose sign-problems to stateof-the-art quantum Monte Carlo (QMC) simulations [28], we study the depart from the exact VBC GS by combining exact diagonalization (ED) on finite clusters with periodic boundary conditions and hierarchical mean field theory (HMFT)

Summary

Bij j ij

Phase diagram.— As the four-spin AF interactions present in (12) and (13) pose sign-problems to stateof-the-art QMC simulations [28], we study the depart from the exact VBC GS by combining exact diagonalization (ED) on finite clusters with periodic boundary conditions and hierarchical mean field theory (HMFT). Both methods involve the diagonalization of finite clusters of N sites, and their combined use provide complementary information about the thermodynamic limit. The HMFT-Gutzwiller consists in using an homogeneus product of cluster states as an Ansatz for the GS in the thermodynamic limit. Its variational determination reduces to perform ED on a cluster with open boundary conditions and a set of self-consistently defined mean-fields acting on its boundaries that allow for the breakdown of symmetries and stabilization of different long-range orders. For appropriate choices of the cluster shape —romboid and diamond for the SVBC, and square for the CVBC— the HMFT wave function contains the exact VBC state

Dy Dx

THE DIMER FERMION MAPPING

Applying the third component to the vacuum and doubly occupied states

LOCAL HAMILTONIANS OF THE SVBC AND CVBC PARENT HAMILTONIANS

HIERARCHICAL MEAN FIELD THEORY AND EXACT DIAGONALIZATION