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arXiv:2204.05441 · Physical review. B./Physical review. B

Origin of nonsymmorphic bosonization formulas in generalized antiferromagnetic Kitaev spin- 12 chains from a renormalization-group perspective

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

Published Aug 22, 2022• 8 citations
arXivHTMLPDF

Abstract

Recently, in the Luttinger liquid phase of the one-dimensional generalized antiferromagnetic Kitaev spin-1/2 model, it has been found that the Abelian bosonization formulas of the local spin operators only respect the exact discrete nonsymmorphic symmetry group of the model, not the emergent U(1) symmetry. In this work, we perform a renormalization group (RG) study to provide explanations for the origin of the U(1) breaking terms in the bosonization formulas. We find that the lack of U(1) symmetry originates from the wave-function renormalization effects in the spin operators along the RG flow induced by the U(1) breaking interactions in the microscopic Hamiltonian. In addition, the RG analysis can give predictions to the signs and order of magnitudes of the coefficients in the bosonization formulas. Our work is helping to understand the rich nonsymmorphic physics in one-dimensional Kitaev spin models.

DOI
10.1103/physrevb.106.064425

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Recently, in the Luttinger liquid phase of the one-dimensional generalized antiferromagnetic Kitaev spin-1/2 model, it has been found that the Abelian bosonization formulas of the local spin operators only respect the exact discrete nonsymmorphic symmetry group of the model, not the emergent U(1) symmetry. In this work, we perform a renormalization group (RG) study to provide explanations for the origin of the U(1) breaking terms in the bosonization formulas. We find that the lack of U(1) symmetry originates from the wave-function renormalization effects in the spin operators along the RG flow induced by the U(1) breaking interactions in the microscopic Hamiltonian. In addition, the RG analysis can give predictions to the signs and order of magnitudes of the coefficients in the bosonization formulas. Our work is helping to understand the rich nonsymmorphic physics in one-dimensional Kitaev spin models.
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.
Based on a renormalization-group (RG) analysis, we study the bosonization formulas for spin-SSS Kitaev-Gamma and Kitaev-Heisenberg-Gamma chains in the parameter regime K<0K<0K<0, Γ>0Γ>0Γ>0, and J>0J>0J>0, with SSS being a half-odd integer. We find that the effects associated with the breaking of emergent continuous symmetries in the bosonization formulas scale as 1/S1/S1/S in the large-SSS limit, in qualitative agreement with density-matrix renormalization-group (DMRG) results for Kitaev-Gamma chains. For Kitaev-Heisenberg-Gamma chains, the low-energy excitations are characterized by incommensurate gapless wave vectors, and symmetry analysis yields ten independent bosonization coefficients, five of which are predicted by the RG analysis to be independent of the Heisenberg coupling up to linear order. Our results provide useful input for determining magnetic ordering tendencies in two-dimensional Kitaev spin models within a quasi-one-dimensional approach.
The properties of stable Luttinger liquid phases in models with a nonconserved number of particles are investigated. We study the Luttinger liquid phases in one-dimensional models of hard-core boson and spinless fermion chains where particles can be created and annihilated three by three on adjacent sites. We provide an intuitive and systematic method based on the flow equation approach, which accounts for additional terms in the correlations generated by the Z3{\mathbb{Z}}_{3}Z3​-symmetric interactions. We find that despite the emergence of U(1) symmetry under renormalization, the observables are still affected by its breaking in the bare Hamiltonian. In particular, the standard bosonization mapping becomes insufficient to capture the full behavior of correlation functions.
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.
A minimal Kitaev-Gamma model has been recently investigated to understand various Kitaev systems. In the one-dimensional Kitaev-Gamma chain, an emergent SU(2)1\mathrm{SU}{(2)}_{1}SU(2)1​ phase and a rank-1 spin ordered phase with ${O}_{h}\ensuremath{\rightarrow}{D}_{4}$ symmetry breaking were identified using non-Abelian bosonization and numerical techniques. However, puzzles near the antiferromagnetic Kitaev region with finite Gamma interaction remained unresolved. Here we focus on this parameter region and find that there are two new phases, namely, a rank-1 ordered phase with an ${O}_{h}\ensuremath{\rightarrow}{D}_{3}$ symmetry breaking, and a peculiar Kitaev phase. Remarkably, the ${O}_{h}\ensuremath{\rightarrow}{D}_{3}$ symmetry breaking corresponds to the classical magnetic order, but appears in a region very close to the antiferromagnetic Kitaev point where the quantum fluctuations are presumably very strong. In addition, a two-step symmetry breaking ${O}_{h}\ensuremath{\rightarrow}{D}_{3d}\ensuremath{\rightarrow}{D}_{3}$ is numerically observed as the length scale is increased: At short and intermediate length scales, the system behaves as having a rank-2 spin nematic order with ${O}_{h}\ensuremath{\rightarrow}{D}_{3d}$ symmetry breaking; and at long distances, time reversal symmetry is further broken leading to the ${O}_{h}\ensuremath{\rightarrow}{D}_{3}$ symmetry breaking. Finally, there is no numerical signature of spin orderings nor Luttinger liquid behaviors in the Kitaev phase whose nature is worth further studies.
In this work, we analyze the nonsymmorphic symmetry-group structures for a variety of generalized Kitaev spin chains and ladders with bond alternations, including Kitaev-Gamma chain, Kitaev-Heisenberg-Gamma chain, beyond-nearest-neighbor interactions, and two-leg spin ladders. The symmetry analysis is applied to determine the symmetry-breaking patterns of several magnetically ordered phases in the bond-alternating Kitaev-Gamma spin chains, as well as the dimerization order parameters for spontaneous dimerizations. Our work is useful in understanding the magnetic phases in related models and may provide guidance for the symmetry classifications of mean field solutions in further investigations.