We discuss ground-state projector simulations of a modified two-dimensional $S=1/2$ Heisenberg model in the valence bonds basis. Tuning matrix elements corresponding to the diagonal and off-diagonal terms in the quantum dimer model, we show that there is a quantum phase transition from the antiferromagnet into a columnar valence-bond solid (VBS). There are no signs of discontinuities, suggesting a continuous or very weak first-order transition. The ${\text{Z}}_{4}$ -symmetric VBS order parameter exhibits an emergent U(1) symmetry as the phase transition is approached. We extract the associated length-scale governing the $\text{U}(1)\ensuremath{-}{\text{Z}}_{4}$ cross-over inside the VBS phase.We discuss ground-state projector simulations of a modified two-dimensional $S=1/2$ Heisenberg model in the valence bonds basis. Tuning matrix elements corresponding to the diagonal and off-diagonal terms in the quantum dimer model, we show that there is a quantum phase transition from the antiferromagnet into a columnar valence-bond solid (VBS). There are no signs of discontinuities, suggesting a continuous or very weak first-order transition. The ${\text{Z}}_{4}$-symmetric VBS order parameter exhibits an emergent U(1) symmetry as the phase transition is approached. We extract the associated length-scale governing the $\text{U}(1)\ensuremath{-}{\text{Z}}_{4}$ cross-over inside the VBS phase.

arXiv:0811.0837Physical Review BDec 31, 20096 citations

Criticality enabled long-range order in a U(1)-symmetric spin-1 Heisenberg chain with biquadratic interactions

Natalia Chepiga, Fabian H. L. Essler

In a recent paper Nahum argued on the basis of a renormalization-group analysis that, contrary to standard lore, ground states of 1D spin chains with short-range interactions can spontaneously break U(1) ``easy-plane'' spin rotation symmetry. True long-range order of $(S^x,S^y)$ arises at the phase transition between two quasi-long-range-ordered phases. Here we present detailed numerical results for an anisotropic spin-1 chain model. We find a magnetization exponent $β\approx0.29$ , consistent with the $ε$ -expansion prediction $β\approx0.28$ , a divergence of the Luttinger parameter on approaching the transition from both sides with an exponent that we relate to Nahum's transition, and finite-size spectra consistent with a dynamical critical exponent $z\approx2$ . Our results support the critical behavior predicted by Nahum's field theory.In a recent paper Nahum argued on the basis of a renormalization-group analysis that, contrary to standard lore, ground states of 1D spin chains with short-range interactions can spontaneously break U(1) ``easy-plane'' spin rotation symmetry. True long-range order of $(S^x,S^y)$ arises at the phase transition between two quasi-long-range-ordered phases. Here we present detailed numerical results for an anisotropic spin-1 chain model. We find a magnetization exponent $β\approx0.29$, consistent with the $ε$-expansion prediction $β\approx0.28$, a divergence of the Luttinger parameter on approaching the transition from both sides with an exponent that we relate to Nahum's transition, and finite-size spectra consistent with a dynamical critical exponent $z\approx2$. Our results support the critical behavior predicted by Nahum's field theory.

cond-mat.str-elCross-listedarXiv:2609.31319Sep 28, 20260 citations

Ordering of Energy Levels in Heisenberg Models and Applications

Bruno Nachtergaele, Shannon L. Starr

arXiv:math-ph/0503056Lecture notes in physicsJan 1, 20066 citations

Phase transitions for the anisotropic Heisenberg model

Harvey S. Leff, Michael Flicker, Kenneth Millard

Physics Letters AMay 1, 19713 citations

We discuss ground-state projector simulations of a modified two-dimensional

S=1/2

Heisenberg model in the valence bonds basis. Tuning matrix elements corresponding to the diagonal and off-diagonal terms in the quantum dimer model, we show that there is a quantum phase transition from the antiferromagnet into a columnar valence-bond solid (VBS). There are no signs of discontinuities, suggesting a continuous or very weak first-order transition. The

{\text{Z}}_{4}

-symmetric VBS order parameter exhibits an emergent U(1) symmetry as the phase transition is approached. We extract the associated length-scale governing the $\text{U}(1)\ensuremath{-}{\text{Z}}_{4}$ cross-over inside the VBS phase.

(S^x,S^y)

arises at the phase transition between two quasi-long-range-ordered phases. Here we present detailed numerical results for an anisotropic spin-1 chain model. We find a magnetization exponent

β\approx0.29

, consistent with the

ε

-expansion prediction

β\approx0.28

, a divergence of the Luttinger parameter on approaching the transition from both sides with an exponent that we relate to Nahum's transition, and finite-size spectra consistent with a dynamical critical exponent

z\approx2

. Our results support the critical behavior predicted by Nahum's field theory.