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Genevois recently classified which graph braid groups are word hyperbolic. In the 3-strand case, he asked whether all such word hyperbolic groups are actually free; this reduced to checking two infinite classes of graphs: sun and pulsar graphs. We prove that 3-strand braid groups of sun graphs are free. On the other hand, it was known to experts that 3-strand braid groups of most pulsar graphs contain surface subgroups. We provide a simple proof of this and prove an additional structure theorem for these groups.
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Word-Representable Graphs: Orientations, Posets, and Bounds
Word-representable graphs were originally introduced by Kitaev and Pyatkin, motivated by work of Kitaev and Seif in algebra. Since their introduction, however, there has been a great deal of work in understanding their graph theoretical properties. In this paper, we introduce tools from partially ordered sets, Ramsey theory as well as probabilistic methods to study them. Through these, we settle a number of open problems in the field, regarding both the existence and length of word-representations for various classes of graphs.
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- Award ID(s):
- 2015425
- PAR ID:
- 10601294
- Publisher / Repository:
- Electronic Journal of Combinatorics
- Date Published:
- Journal Name:
- The Electronic Journal of Combinatorics
- Volume:
- 31
- Issue:
- 4
- ISSN:
- 1077-8926
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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null (Ed.)Here we revisit the thermodynamics of the Kitaev quantum spin liquid realized on the honeycomb lattice. We address two main questions: First, we investigate whether there are observable thermodynamic signatures of the topological Majorana boundary modes of the Kitaev honeycomb model. We argue that for the time-reversal invariant case the residual low-temperature entropy is the primary thermodynamic signature of these Majorana edge modes. Using large-scale Monte Carlo simulations, we verify that this residual entropy is present in the full Kitaev model. When time-reversal symmetry is broken, the Majorana edge modes are potentially observable in more direct thermodynamic measurements such as the specific heat, though only at temperatures well below the bulk gap. Second, we study the energetics, and the corresponding thermodynamic signatures, of the flux excitations in the Kitaev model. Specifically, we study the flux interactions on both cylinder and torus geometries numerically and quantify their impact on the thermodynamics of the Kitaev spin liquid by using a polynomial fit for the average flux energy as a function of flux density and extrapolating it to the thermodynamic limit. By comparing this model to Monte Carlo simulations, we find that flux interactions have a significant quantitative impact on the shape and the position of the low-temperature peak in the specific heat.more » « less
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α-RuCl3 is considered to be the top candidate material for the experimental realization of the celebrated Kitaev model, where ground states are quantum spin liquids with interesting fractionalized excitations. It is, however, known that additional interactions beyond the Kitaev model trigger in α-RuCl3 a long-range zigzag antiferromagnetic ground state. In this work, we investigate a nanoflake of α-RuCl3 through guarded high impedance measurements aimed at reaching the regime where the system turns into a zigzag antiferromagnet. We investigated a variety of temperatures (1.45–175 K) and out-of-plane magnetic fields (up to 11 T), finding a clear signature of a structural phase transition at ≈160 K as reported for thin crystals of α-RuCl3, as well as a thermally activated behavior at temperatures above ≈30 K, with a characteristic activation energy significantly smaller than the energy gap that we observe for α-RuCl3 bulk crystals through our angle resolved photoemission spectroscopy (ARPES) experiments. Additionally, we found that below ≈30 K, transport is ruled by Efros–Shklovskii variable range hopping (VRH). Most importantly, our data show that below the magnetic ordering transition known for bulk α-RuCl3 in the frame of the Kitaev–Heisenberg model (≈7 K), there is a clear deviation from VRH or thermal activation transport mechanisms. Our work demonstrates the possibility of reaching, through specialized high impedance measurements, the thrilling ground states predicted for α-RuCl3 at low temperatures in the frame of the Kitaev–Heisenberg model and informs about the transport mechanisms in this material in a wide temperature range.more » « less
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null (Ed.)The Kitaev honeycomb model has attracted significant attention due to its exactly solvable spin-liquid ground state with fractionalized Majorana excitations and its possible materialization in magnetic Mott insulators with strong spin-orbit couplings. Recently, the 5d-electron compound H3LiIr2O6 has shown to be a strong candidate for Kitaev physics considering the absence of any signs of a long-range ordered magnetic state. In this work, we demonstrate that a finite density of random vacancies in the Kitaev model gives rise to a striking pileup of low-energy Majorana eigenmodes and reproduces the apparent power-law upturn in the specific heat measurements of H3LiIr2O6. Physically, the vacancies can originate from various sources such as missing magnetic moments or the presence of nonmagnetic impurities (true vacancies), or from local weak couplings of magnetic moments due to strong but rare bond randomness (quasivacancies). We show numerically that the vacancy effect is readily detectable even at low vacancy concentrations and that it is not very sensitive either to the nature of vacancies or to different flux backgrounds. We also study the response of the site-diluted Kitaev spin liquid to the three-spin interaction term, which breaks time-reversal symmetry and imitates an external magnetic field. We propose a field-induced flux-sector transition where the ground state becomes flux-free for larger fields, resulting in a clear suppression of the low-temperature specific heat. Finally, we discuss the effect of dangling Majorana fermions in the case of true vacancies and show that their coupling to an applied magnetic field via the Zeeman interaction can also account for the scaling behavior in the high-field limit observed in H3LiIr2O6.more » « less
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Abstract Spin-orbit coupled honeycomb magnets with the Kitaev interaction have received a lot of attention due to their potential of hosting exotic quantum states including quantum spin liquids. Thus far, the most studied Kitaev systems are 4 d /5 d -based honeycomb magnets. Recent theoretical studies predicted that 3 d -based honeycomb magnets, including Na 2 Co 2 TeO 6 (NCTO), could also be a potential Kitaev system. Here, we have used a combination of heat capacity, magnetization, electron spin resonance measurements alongside inelastic neutron scattering (INS) to study NCTO’s quantum magnetism, and we have found a field-induced spin disordered state in an applied magnetic field range of 7.5 T < B (⊥ b -axis) < 10.5 T. The INS spectra were also simulated to tentatively extract the exchange interactions. As a 3 d -magnet with a field-induced disordered state on an effective spin-1/2 honeycomb lattice, NCTO expands the Kitaev model to 3 d compounds, promoting further interests on the spin-orbital effect in quantum magnets.more » « less
- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.37236/12806
