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arXiv:2208.10754 · SciPost Physics

Nonsymmorphic spin-space cubic groups and SU(2)$_1$ conformal invariance in one-dimensional spin-1/2 models

Wang Yang, Alberto Nocera, Chao Xu, Ian K. Affleck

Published Oct 1, 2024• 3 citations
arXivHTMLPDF

Abstract

Recently, extended gapless phases with emergent SU(2) _1 1 conformal invariance occupying finite regions in the phase diagrams have been found in one-dimensional spin-1/2 models with nonsymmorphic O_h O h symmetry groups. In this work, we investigate the question of whether the conditions for emergent SU(2) _1 1 invariance can be loosened. We find that besides the nonsymmorphic O_h O h group, the other four smaller nonsymmorphic cubic groups including O O , T_h T h , T_d T d and T T can also give rise to emergent SU(2) _1 1 invariance. Minimal spin-1/2 models having these nonsymmorphic cubic groups as symmetry groups are constructed, and numerical evidences for the emergent SU(2) _1 1 invariance are provided. Our work is useful for understanding gapless phases in one-dimensional spin systems with nonsymmorphic symmetries.

DOI
10.21468/scipostphys.17.4.097

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Nonsymmorphic spin-space cubic groups and SU(2)1_11​ conformal invariance in one-dimensional spin-1/2 models
Recently, extended gapless phases with emergent SU(2) _1 1 conformal invariance occupying finite regions in the phase diagrams have been found in one-dimensional spin-1/2 models with nonsymmorphic O_h O h symmetry groups. In this work, we investigate the question of whether the conditions for emergent SU(2) _1 1 invariance can be loosened. We find that besides the nonsymmorphic O_h O h group, the other four smaller nonsymmorphic cubic groups including O O , T_h T h , T_d T d and T T can also give rise to emergent SU(2) _1 1 invariance. Minimal spin-1/2 models having these nonsymmorphic cubic groups as symmetry groups are constructed, and numerical evidences for the emergent SU(2) _1 1 invariance are provided. Our work is useful for understanding gapless phases in one-dimensional spin systems with nonsymmorphic symmetries.
We study the one-dimensional spin-1/2 Kitaev-Gamma model with a bond-dependent Dzyaloshinskii-Moriya (DM) interaction, which can be induced by an electric field applied in the third direction where the first and second directions refer to the two bond directions in the model. By a combination of field theory and symmetry analysis, extended gapless phases with an emergent SU(2)1\mathrm{SU}{(2)}_{1}SU(2)1​ conformal symmetry are found in the phase diagram of the spin-1/2 Kitaev-Gamma-DM chain. The analytic predictions are in good agreements with numerical results obtained from density matrix renormalization group simulations.
We perform a detailed study on the consequences of the nonsymmorphic symmetries in the Luttinger phase of the one-dimensional (1D) spin-12\frac{1}{2}21​ Kitaev-Heisenberg-Gamma model with an antiferromagnetic Kitaev interaction. Although the low-energy Hamiltonian has an emergent U(1) symmetry, the bosonization formulas for the spin operators contain ten nonuniversal coefficients which only respect the discrete nonsymmorphic symmetries and can be determined by our density matrix renormalization group simulations to a high degree of accuracy. Using the nonsymmorphic bosonization formulas, the response to weak magnetic field is analyzed, and the zigzag phase in the 2D Kitaev-Heisenberg-Gamma model is recovered by weakly coupling an infinite number of 1D chains. Furthermore, we find a line of critical points with an emergent SU(2)1\mathrm{SU}{(2)}_{1}SU(2)1​ conformal symmetry located on the boundary of the Luttinger liquid phase. Our work reveals the rich physics related to nonsymmorphic symmetries in strongly correlated quantum magnetic models, and demonstrates the usefulness of 1D analysis for understanding the 2D Kitaev physics.
We study a one-dimensional spin-orbital model using both analytical and numerical methods. Renormalization group calculations are performed in the vicinity of a special integrable point in the phase diagram with SU(4)\mathrm{SU}(4)SU(4) symmetry. These indicate the existence of a gapless phase in an extended region of the phase diagram, missed in previous studies. This phase is SU(4)\mathrm{SU}(4)SU(4) invariant at low energies apart from the presence of different velocities for spin and orbital degrees of freedom. The phase transition into a gapped dimerized phase is in a generalized Kosterlitz-Thouless universality class. The phase diagram of this model is sketched using the density matrix renormalization group technique.
We study the phase diagram of a one-dimensional version of the Kitaev spin-1/2 model with an extra "Γ term," using analytical, density matrix renormalization group and exact diagonalization methods. Two intriguing phases are found. In the gapless phase, although the exact symmetry group of the system is discrete, the low energy theory is described by an emergent SU(2)_{1} Wess-Zumino-Witten (WZW) model. On the other hand, the spin-spin correlation functions exhibit SU(2) breaking prefactors, even though the exponents and the logarithmic corrections are consistent with the SU(2)_{1} predictions. A modified non-Abelian bosonization formula is proposed to capture such exotic emergent "partial" SU(2) symmetry. In the ordered phase, there is numerical evidence for an O_{h}→D_{4} spontaneous symmetry breaking.
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