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arXiv:2101.12140 · Physical review. B./Physical review. B

Exactly solvable spin- 12 XYZ models with highly degenerate partially ordered ground states

Grgur Palle, Owen Benton

Published Jun 16, 2021• 7 citations
arXivHTMLPDF

Abstract

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 $S=\frac{1}{2}$ 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.

DOI
10.1103/physrevb.103.214428

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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.
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 $\frac{{J}_{z}}{{J}_{\ensuremath{\perp}}}>\ensuremath{-}\frac{1}{2}$.
It is noted that the pair correlation matrix χ of the nearest neighbor Ising model on periodic three-dimensional (d = 3) kagome-like lattices of corner-sharing triangles can be calculated partially exactly.Specifically, a macroscopic number 1/3 N + 1 out of N eigenvalues of χ are degenerate at all temperatures T , and correspond to an eigenspace L -of χ, independent of T .Degeneracy of the eigenvalues, and L -are an exact result for a complex d = 3 statistical physical model.It is further noted that the eigenvalue degeneracy describing the same Lis exact at all T in an infinite spin dimensionality m limit of the isotropic m-vector approximation to the Ising models.A peculiar match of the opposite m = 1 and m → ∞ limits can be interpreted that the m → ∞ considerations are exact for m = 1.It is not clear whether the match is coincidental.It is then speculated that the exact eigenvalues degeneracy in L -in the opposite limits of m can imply their quasi-degeneracy for intermediate 1 m < ∞.For an anti-ferromagnetic nearest neighbor coupling, that renders kagome-like models highly geometrically frustrated, these are spin states largely from L -that for m 2 contribute to χ at low T .The m → ∞ formulae can be thus quantitatively correct in description of χ and clarifying the role of perturbations in kagome-like systems deep in the collective paramagnetic regime.An exception may be an interval of T , where the order-by-disorder mechanisms select sub-manifolds of L -.
Frustrated quantum magnets are a central theme in condensed matter physics due to the richness of their phase diagrams. They support a panoply of phases including various ordered states and topological phases. Yet, this problem has defied a solution for a long time due to the lack of controlled approximations which make it difficult to distinguish between competing phases. Here we report the discovery of a special quantum macroscopically degenerate point in the XXZ model on the spin-1/2 kagome quantum antiferromagnet for the ratio of Ising to antiferromagnetic transverse coupling J_{z}/J=-1/2. This point is proximate to many competing phases explaining the source of the complexity of the phase diagram. We identify five phases near this point including both spin-liquid and broken-symmetry phases and give evidence that the kagome Heisenberg antiferromagnet is close to a transition between two phases.
We analyze the antiferromagnetic spin-1/2 XXZ model on the kagome lattice at finite external magnetic field with the help of a non-perturbative zero-temperature renormalization group (RG) technique. The exact nature of the ground and excited state properties (e.g. gapped or gapless spectrum etc) of this system are still debated. Approximate methods have typically been adopted towards understanding the low-energy spectrum. Following the work of Kumar et al (2014 Phys. Rev. B 90 174409), we use a Jordan–Wigner transformation to map the spin problem into one of spinless fermions (spinons) in the presence of a statistical gauge field, and with nearest-neighbor interactions. While the work of Kumar et al was confined mostly to the plateau at 1/3-filling (magnetization per site) in the XY regime, we analyze the role of inter-spinon interactions in shaping the phases around this plateau in the entire XXZ model. The RG phase diagram obtained contains three spin liquid phases whose position is determined as a function of the exchange anisotropy and the energy scale for fluctuations arising from spinon scattering. Two of these spins liquids are topologically ordered states of matter with gapped, degenerate states on the torus. The gap for one of these phases corresponds to the one-spinon band gap of the Azbel–Hofstadter spectrum for the XY part of the Hamiltonian, while the other arises from two-spinon interactions. The Heisenberg point of this problem is found to lie within the interaction gapped spin liquid phase, in broad agreement with a recent experimental finding. The third phase is an algebraic spin liquid with a gapless Dirac spectrum for spinon excitations, and possess properties that show departures from the Fermi liquid paradigm. The three phase boundaries correspond to critical theories, and meet at a SU (2)-symmetric multicritical point. This special critical point agrees well with the gap-closing transition point predicted by Kumar et al . We discuss the relevance of our findings to various recent experiments, as well as results obtained from other theoretical analyses.
We introduce quantum dimer models on lattices made of corner-sharing triangles. These lattices include the kagome lattice and can be defined in arbitrary geometry. They realize fully disordered and gapped dimer-liquid phase with topological degeneracy and deconfined fractional excitations, as well as solid phases. Using geometrical properties of the lattice, several results are obtained exactly, including the full spectrum of a dimer liquid. These models offer a very natural-and maybe the simplest possible-framework to illustrate general concepts such as fractionalization, topological order, and relation to Z2 gauge theories.