Competing States in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>S</mi></mrow><mo>=</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></math> Triangular-Lattice <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>J</mi></mrow><mrow><mn>1</mn></mrow></msub><mtext>−</mtext><msub><mrow><mi>J</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math> Heisenberg Model: A Dynamical Density-Matrix Renormalization Group Study

Shengtao Jiang (蒋晟韬; Steven R. White; Steven A. Kivelson; Hong-Chen Jiang

doi:10.1103/zmnz-tkq2

Competing States in the $S = 1 / 2$ Triangular-Lattice $J_{1} − J_{2}$ Heisenberg Model: A Dynamical Density-Matrix Renormalization Group Study

Shengtao Jiang (蒋晟韬)^1,*, Steven R. White ², Steven A. Kivelson ³, and Hong-Chen Jiang ^1,†

¹Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory and Stanford University, Menlo Park, California 94025, USA
²Department of Physics and Astronomy, University of California, Irvine, California 92697, USA
³Department of Physics, Stanford University, Stanford, California 94305, USA

^*Contact author: stjiang@stanford.edu
^†Contact author: hcjiang@stanford.edu

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Phys. Rev. Lett. 137, 056703 – Published 28 July, 2026

DOI: https://doi.org/10.1103/zmnz-tkq2

Abstract

Previous studies of the $S = 1 / 2$ triangular-lattice $J_{1} − J_{2}$ Heisenberg antiferromagnet have inferred the existence of a nonmagnetic ground-state phase for an intermediate range of $J_{2}$ , but disagree concerning whether it is a gapped $Z_{2}$ quantum spin liquid (QSL), a gapless (Dirac) QSL, or a weakly symmetry-broken phase. Using an improved dynamical density-matrix renormalization group method, we investigate the relevant intermediate $J_{2}$ regime for cylinders with circumferences from 6 to 9. Depending on the initial state and boundary conditions, we find two distinct variational states. The higher energy state is consistent with a Dirac QSL. In the lower-energy state, both the static and dynamical properties are qualitatively similar to the magnetically ordered state at $J_{2} = 0$ , suggestive of either a weakly magnetically ordered non-QSL or a gapped QSL proximate to a continuous transition to such an ordered state.

Physical Review Letters

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Competing States in the $S = 1 / 2$ Triangular-Lattice $J_{1} − J_{2}$ Heisenberg Model: A Dynamical Density-Matrix Renormalization Group Study

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Competing States in the S=1/2 Triangular-Lattice J1−J2 Heisenberg Model: A Dynamical Density-Matrix Renormalization Group Study

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Competing States in the $S = 1 / 2$ Triangular-Lattice $J_{1} − J_{2}$ Heisenberg Model: A Dynamical Density-Matrix Renormalization Group Study