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arXiv:2012.05930 · Physical review. B./Physical review. B

Colorful points in the XY regime of XXZ quantum magnets

Santanu Pal, P. A. Sharma, Hitesh J. Changlani, Sumiran Pujari

Published Apr 9, 2021• 13 citations
arXivHTMLPDF

Abstract

In the XY regime of the XXZ Heisenberg model phase diagram, we demonstrate that the origin of magnetically ordered phases is influenced by the presence of solvable points with exact quantum coloring ground states featuring a quantum-classical correspondence. Using exact diagonalization and density matrix renormalization group calculations, for both the square and the triangular lattice magnets, we show that the ordered physics of the solvable points in the extreme XY regime, at $\frac{{J}_{z}}{{J}_{\ensuremath{\perp}}}=\ensuremath{-}1$ and $\frac{{J}_{z}}{{J}_{\ensuremath{\perp}}}=\ensuremath{-}\frac{1}{2}$, respectively, with ${J}_{\ensuremath{\perp}}>0$, adiabatically extends to the more isotropic regime $\frac{{J}_{z}}{{J}_{\ensuremath{\perp}}}\ensuremath{\sim}1$. We highlight the projective structure of the coloring ground states to compute the correlators in fixed magnetization sectors which enables an understanding of the features in the static spin structure factors and correlation ratios. These findings are contrasted with an anisotropic generalization of the celebrated one-dimensional Majumdar-Ghosh model, which is also found to be (ground-state) solvable. For this model, both exact dimer and three-coloring ground states exist at $\frac{{J}_{z}}{{J}_{\ensuremath{\perp}}}=\ensuremath{-}\frac{1}{2}$ but only the two dimer ground states survive for any $\frac{{J}_{z}}{{J}_{\ensuremath{\perp}}}>\ensuremath{-}\frac{1}{2}$.

DOI
10.1103/physrevb.103.144414

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In the XY regime of the XXZ Heisenberg model phase diagram, we demonstrate that the origin of magnetically ordered phases is influenced by the presence of solvable points with exact quantum coloring ground states featuring a quantum-classical correspondence. Using exact diagonalization and density matrix renormalization group calculations, for both the square and the triangular lattice magnets, we show that the ordered physics of the solvable points in the extreme XY regime, at JzJ\ensuremath⊥=\ensuremath−1\frac{{J}_{z}}{{J}_{\ensuremath{\perp}}}=\ensuremath{-}1J\ensuremath⊥​Jz​​=\ensuremath−1 and JzJ\ensuremath⊥=\ensuremath−12\frac{{J}_{z}}{{J}_{\ensuremath{\perp}}}=\ensuremath{-}\frac{1}{2}J\ensuremath⊥​Jz​​=\ensuremath−21​, respectively, with J\ensuremath⊥>0{J}_{\ensuremath{\perp}}>0J\ensuremath⊥​>0, adiabatically extends to the more isotropic regime JzJ\ensuremath⊥\ensuremath∼1\frac{{J}_{z}}{{J}_{\ensuremath{\perp}}}\ensuremath{\sim}1J\ensuremath⊥​Jz​​\ensuremath∼1. We highlight the projective structure of the coloring ground states to compute the correlators in fixed magnetization sectors which enables an understanding of the features in the static spin structure factors and correlation ratios. These findings are contrasted with an anisotropic generalization of the celebrated one-dimensional Majumdar-Ghosh model, which is also found to be (ground-state) solvable. For this model, both exact dimer and three-coloring ground states exist at JzJ\ensuremath⊥=\ensuremath−12\frac{{J}_{z}}{{J}_{\ensuremath{\perp}}}=\ensuremath{-}\frac{1}{2}J\ensuremath⊥​Jz​​=\ensuremath−21​ but only the two dimer ground states survive for any JzJ\ensuremath⊥>\ensuremath−12\frac{{J}_{z}}{{J}_{\ensuremath{\perp}}}>\ensuremath{-}\frac{1}{2}J\ensuremath⊥​Jz​​>\ensuremath−21​.
Exactly solvable models play a special role in condensed matter physics, serving as secure theoretical starting points for investigation of new phenomena. Changlani et al. [Phys. Rev. Lett. 120, 117202 (2018)] have discovered a limit of the XXZ model for S=12S=\frac{1}{2}S=21​ spins on the kagome lattice, which is not only exactly solvable, but features a huge degeneracy of exact ground states corresponding to solutions of a three-coloring problem. This special point of the model was proposed as a parent for multiple phases in the wider phase diagram, including quantum spin liquids. Here, we show that the construction of Changlani et al. can be extended to more general forms of anisotropic exchange interaction, finding a line of parameter space in an XYZ model which maintains both the macroscopic degeneracy and the three-coloring structure of solutions. We show that the ground states along this line are partially ordered, in the sense that infinite-range correlations of some spin components coexist with a macroscopic number of undetermined degrees of freedom. We therefore propose the exactly solvable limit of the XYZ model on corner-sharing triangle-based lattices as a tractable starting point for discovery of quantum spin systems which mix ordered and spin-liquid-like properties.
Motivated by the recent discovery of a macroscopically degenerate exactly solvable point of the spin-1/2XXZ1/2\phantom{\rule{4pt}{0ex}}XXZ1/2XXZ model for ${J}_{z}/J=\ensuremath{-}1/2$ on the kagome lattice [H. J. Changlani et al. Phys. Rev. Lett. 120, 117202 (2018)]---a result that holds for arbitrary magnetization---we develop an exact mapping between its exact ''quantum three-coloring'' wave functions and the characteristic localized and topological magnons. This map, involving ''resonating two-color loops,'' is developed to represent exact many-body ground state wave functions for special high magnetizations. Using this map we show that these exact ground state solutions are valid for any ${J}_{z}/J\ensuremath{\ge}\ensuremath{-}1/2$. This demonstrates the equivalence of the ground-state wave function of the Ising, Heisenberg, and XYXYXY regimes all the way to the ${J}_{z}/J=\ensuremath{-}1/2$ point for these high magnetization sectors. In the hardcore bosonic language, this means that a certain class of exact many-body solutions, previously argued to hold for purely repulsive interactions $({J}_{z}\ensuremath{\ge}0)$, actually hold for attractive interactions as well, up to a critical interaction strength. For the case of zero magnetization, where the ground state is not exactly known, we perform density matrix renormalization group calculations. Based on the calculation of the ground state energy and measurement of order parameters, we provide evidence for a lack of any qualitative change in the ground state on finite clusters in the Ising $({J}_{z}\ensuremath{\gg}J)$, Heisenberg (Jz=J)({J}_{z}=J)(Jz​=J), and XYXYXY (Jz=0)({J}_{z}=0)(Jz​=0) regimes, continuing adiabatically to the vicinity of the macroscopically degenerate ${J}_{z}/J=\ensuremath{-}1/2$ point. These findings offer a framework for recent results in the literature and also suggest that the ${J}_{z}/J=\ensuremath{-}1/2$ point is an unconventional quantum critical point whose vicinity may contain the key to resolving the spin-1/21/21/2 kagome problem.
Based on exact diagonalization and density matrix renormalization group method, we show that an anisotropic triangular lattice Heisenberg spin model has three distinct quantum phases. In particular, a spin-liquid phase is present in the weak interchain coupling regime, which is characterized by an anisotropic spin structure factor with an exponential-decay spin correlator along the weaker coupling direction, consistent with the Cs2CuCl4{\mathrm{Cs}}_{2}\mathrm{Cu}{\mathrm{Cl}}_{4}Cs2​CuCl4​ compounds. In the obtained phase diagram, the spin-liquid phase is found to persist up to a relatively large critical anisotropic coupling ratio ${J}^{\ensuremath{'}}∕J=0.78$, which is stabilized by strong quantum fluctuations, with a parity symmetry distinct from two magnetic ordered states in the stronger coupling regime.
We study the zero-temperature phase diagram of the J 1 XXZ – J 2 XXZ Heisenberg model for spin-1 particles on an infinite square lattice interacting via nearest-neighbour ( ) and next-nearest-neighbour ( J 2 >0) bonds. The two bonds have the same XXZ -type anisotropy in spin space. The effects on the quasiclassical Néel-ordered and collinear stripe-ordered states of varying the anisotropy parameter Δ are investigated using the coupled cluster method carried out up to high orders. By contrast with the case for spin- particles studied previously, no intermediate disordered phase between the Néel and collinear stripe phases, for any value of the frustration J 2 / J 1 , for either the z -aligned (Δ>1) or xy -planar-aligned (0≤Δ<1) states, is predicted here. The quantum phase transition is determined as first order for all values of J 2 / J 1 and Δ. The position of the phase boundary J 2 c (Δ) is determined accurately. It is observed to deviate most from its classical position (for all values of Δ>0) at the Heisenberg isotropic point (Δ = 1), where J 2 c (1) = 0.55 ± 0.01. By contrast, at the XY isotropic point (Δ = 0), we find J 2 c (0) = 0.50 ± 0.01. In the Ising limit ( ), as expected.
We report the exact dimer phase, in which the ground states are described by product of singlet dimer, in the extended XYZ model by direct generalizing the isotropic Majumdar-Ghosh model to the fully anisotropic regime. This model also supports three different ferromagnetic (FM) phases, denoted as xxx-FM, yyy-FM and zzz-FM. The boundaries between the dimer phase and FM phases are infinite-fold degenerate. The breaking of this infinite-fold degeneracy by either translational symmetry or Z2\mathbb{Z}_2Z2​ symmetry lead to dimer phase and FM phases, respectively. Moreover, the boundaries between the three FM phases are critical with central charge c=1c=1c=1. We characterize the properties of these boundaries using entanglement entropy, excitation gap and long-range spin-spin correlations. These results are relevant to a large number of quasi-one-dimensional magnets, in which anisotropy is necessary to isolate a single chain out from the bulk material. We discuss the possible experimental signatures in true materials with magnetic field along different directions, and show that the anisotropy may resolve the long-standing disagreements between theory and experiments.
We use a combination of numerical density matrix renormalization group calculations and several analytical approaches to comprehensively study a simplified model for a spatially anisotropic spin-12\frac{1}{2}21​ triangular lattice Heisenberg antiferromagnet: the three-leg triangular spin tube (TST). The model is described by three Heisenberg chains, with exchange constant JJJ, coupled antiferromagnetically with exchange constant ${J}^{\ensuremath{'}}$ along the diagonals of the ladder system, with periodic boundary conditions in the shorter direction. Here, we determine the full phase diagram of this model as a function of both spatial anisotropy (between the isotropic and decoupled chain limits) and magnetic field. We find a rich phase diagram, which is remarkably dominated by quantum states: the phase corresponding to the classical ground state appears only in an exceedingly small region. Among the dominant phases generated by quantum effects are commensurate and incommensurate coplanar quasiordered states, which appear in the vicinity of the isotropic region for most fields, and in the high-field region for most anisotropies. The coplanar states, while not classical ground states, can at least be understood semiclassically. Even more strikingly, the largest region of phase space is occupied by a spin density wave phase, which has incommensurate collinear correlations along the field. This phase has no semiclassical analog, and may be ascribed to enhanced one-dimensional fluctuations due to frustration. Cutting across the phase diagram is a magnetization plateau, with a gap to all excitations and ``up-up-down'' spin order, with a quantized magnetization equal to 13\frac{1}{3}31​ of the saturation value. In the TST, this plateau extends almost but not quite to the decoupled chains limit. Most of the above features are expected to carry over to the two-dimensional system, which we also discuss. At low field, a dimerized phase appears, which is particular to the one-dimensional nature of the TST, and which can be understood from quantum Berry phase arguments.