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The understanding and predicting of novel phenomena in magnetic materials is an important theme in condensed matter physics. The most interesting phenomena among them are the ones that exhibit intrinsic quantum behavior, where high degree of entanglement gives rise to macroscopic quantum effect not described in a traditional theory of symmetry breaking. This thesis is a collection of our efforts to combine the theoretical toolkit in analyzing exotic quantum states with recent experimental progress in realizing and finding quantum magnetic materials. We present our study in three parts:The first part presents a study of the frustrated triangular lattice antiferromagnet NaYbO2. Both spin liquid signatures in zero field and quantum-induced ordering in intermediate fields are observed, suggesting the existence of an intrinsically quantum disordered ground state. Through symmetry analysis and spin wave calculations, we determine the microscopic model relevant to NaYbO2 and map out the phase diagram of magnetic orders in presence of a magnetic field. Our result indicates that NaYbO2 is a promising platform for exploring spin liquid physics with full tunability of field and temperature.The second part presents an investigation of a chemically related compound LiYbO2, which has instead a stretched diamond lattice structure. Experiments reveal a rich magnetic phase diagram of LiYbO2 that includes a low field incommensurate spiral order and a high field commensurate order. We first show that the former is largely captured by a J1–J2 Heisenberg model, and then employ a phenomenological model to understand the incommensurate-to-commensurate transition at intermediate magnetic fields. Finally, several effects are addressed in order to understand a small variance between the observed and predicted phasing of Yb moments.The last part is devoted to the classification of symmetric Z2 and U(1) spin liquids on the three-dimensional pyrochlore lattice. We first analyze the magnetic orders linked to specific Z2 quantum spin liquids. We find that under certain conditions, seemingly unrelated orders are intertwined and the conventional orders detected in experiments are accompanied by hidden orders. We then turn to the study of U(1) spin liquid classes and observe that, surprisingly, a large family of them is described by a U(1) gauge field coupled to symmetry protected gapless multi-nodal line spinons, hence uncovering a new prototype of quantum spin liquid beyond the standard example of pyrochlore quantum spin ice. The low temperature specific heat receives a T^{3/2} contribution with logarithmic corrections from the gauge–spinon coupling and the spinon bands, which serves as a simple criterion for the existence of these U(1) nodal line spin liquids.

The adsorbed atoms exhibit tendency to occupy a triangular lattice formed by periodic potential of the underlying crystal surface. Such a lattice is formed by, e.g., a single layer of graphane or the graphite surfaces as well as (111) surface of face-cubic center crystals. In the present work, an extension of the lattice gas model to S=1/2 fermionic particles on the two-dimensional triangular (hexagonal) lattice is analyzed. In such a model, each lattice site can be occupied not by only one particle, but by two particles, which interact with each other by onsite U and intersite W1 and W2 (nearest and next-nearest-neighbor, respectively) density-density interaction. The investigated hamiltonian has a form of the extended Hubbard model in the atomic limit (i.e., the zero-bandwidth limit). In the analysis of the phase diagrams and thermodynamic properties of this model with repulsive W1&gt, 0, the variational approach is used, which treats the onsite interaction term exactly and the intersite interactions within the mean-field approximation. The ground state (T=0) diagram for W2≤0 as well as finite temperature (T&gt, 0) phase diagrams for W2=0 are presented. Two different types of charge order within 3×3 unit cell can occur. At T=0, for W2=0 phase separated states are degenerated with homogeneous phases (but T&gt, 0 removes this degeneration), whereas attractive W2&lt, 0 stabilizes phase separation at incommensurate fillings. For U/W1&lt, 0 and U/W1&gt, 1/2 only the phase with two different concentrations occurs (together with two different phase separated states occurring), whereas for small repulsive 0&lt, U/W1&lt, 1/2 the other ordered phase also appears (with tree different concentrations in sublattices). The qualitative differences with the model considered on hypercubic lattices are also discussed.

The existence of a gapped chiral spin liquid has been recently suggested in the vicinity of the metal-insulator transition of the Hubbard model on the triangular lattice, by intensive density-matrix renormalization group (DMRG) simulations [A. Szasz, J. Motruk, M.P. Zaletel, and J.E. Moore, Phys. Rev. X {\bf 10}, 021042 (2020)]. Here, we report the results obtained within the variational Monte Carlo technique based upon Jastrow-Slater wave functions, implemented with backflow correlations. As in DMRG calculations, we consider $N$-leg cylinders. For $N=4$ and in the presence of a next-nearest neighbor hopping, a chiral spin liquid emerges between the metal and the insulator with magnetic quasi-long-range order. Within our approach, the chiral state is gapped and breaks the reflection symmetry. By contrast, for both $N=5$ and $N=6$, the chiral spin liquid is not the state with the lowest variational energy: in the former case a nematic spin liquid is found in the entire insulating regime, while for the less frustrated case with $N=6$ the results are very similar to the one obtained on two-dimensional clusters [L.F. Tocchio, A. Montorsi, and F. Becca, Phys. Rev. B {\bf 102}, 115150 (2020)], with an antiferromagnetic phase close to the metal-insulator transition and a nematic spin liquid in the strong-coupling regime., Comment: 9 pages, 7 figures

The extended Hubbard model in the atomic limit, which is equivalent to lattice $S=1/2$ fermionic gas, is considered on the triangular lattice. The model includes onsite Hubbard $U$ interaction and both nearest-neighbor ($W_{1}$) and next-nearest-neighbor ($W_{2}$) density-density intersite interactions. The variational approach treating the $U$ term exactly and the $W_l$ terms in the mean-field approximation is used to investigate thermodynamics of the model and to find its finite temperature ($T>0$) phase diagrams (as a function of particle concentration) for $W_{1}>0$ and $W_{2}0$. The evolution of the diagrams with increasing of $|W_{2}|/W_{1}$ is investigated. It is found that some of the PS states are stable only at $T>0$. Two different critical values of $|W_{2}|/W_{1}$ are determined for the PS states, in which two ordered phases of the same type (i.e., two domains of the DCO or TCO phase) coexist., 7 pages, 5 figures, 57 references; pdf-RevTeX class used. This is the author created version of an article accepted for publication in J. Magn. Magn. Matter (Elsevier) journal. The article has been published on a gold open access basis under a CC BY 4.0 licence

This work investigates dense packings of congruent hard infinitesimally--thin circular arcs in the two-dimensional Euclidean space. It focuses on those denotable as major whose subtended angle $\theta \in \left ( \pi, 2\pi \right ]$. Differently than those denotable as minor whose subtended angle $\theta \in \left [0, \pi \right]$, it is impossible for two hard infinitesimally-thin circular arcs with $\theta \in \left ( \pi, 2\pi \right ]$ to arbitrarily closely approach once they are arranged in a configuration, e.g. on top of one another, replicable ad infinitum without introducing any overlap. This makes these hard concave particles, in spite of being infinitesimally thin, most densely pack with a finite number density. This raises the question as to what are these densest packings and what is the number density that they achieve. Supported by Monte Carlo numerical simulations, this work shows that one can analytically construct compact closed circular groups of hard major circular arcs in which a specific, $\theta$-dependent, number of them (anti-)clockwise intertwine. These compact closed circular groups then arrange on a triangular lattice. These analytically constructed densest-known packings are compared to corresponding results of Monte Carlo numerical simulations to assess whether they can spontaneously turn up., Comment: 9 pages, 9 figures; to be published in Physical Review E

Contains fulltext : 195552pub.pdf (Publisher’s version ) (Open Access) Contains fulltext : 195552pre.pdf (Author’s version preprint ) (Open Access) {In the theory of antiferromagnetism, the staggered field-an external magnetic field that alternates in sign on atomic length scales-is used to select the classical Neel state from a quantum magnet but justification is missing. This work examines, within the decoherence framework, whether repeated local measurement can replace a staggered field. Accordingly, the conditions under which local decoherence can be considered a continuous measurement are studied. The dynamics of a small magnetic system is analyzed to illustrate that local decoherence can lead to (symmetry-broken) order similar to order resulting from a staggered field.} 10 p.

We study the effects of quantum fluctuations on the dynamical generation of a gap and on the evolution of the spin-wave spectra of a frustrated magnet on a triangular lattice with bond-dependent Ising couplings, analog of the Kitaev honeycomb model. The quantum fluctuations lift the subextensive degeneracy of the classical ground-state manifold by a quantum order-by-disorder mechanism. Nearest-neighbor chains remain decoupled and the surviving discrete degeneracy of the ground state is protected by a hidden model symmetry. We show how the four-spin interaction, emergent from the fluctuations, generates a spin gap shifting the nodal lines of the linear spin-wave spectrum to finite energies., 9 pages, 3 figures, 4 panels

We investigate non-equilibrium properties of the frustrated Heisenberg antiferromagnets on the triangular lattice. Nonequilibrium critical relaxation of frustrated Heisenberg antiferromagnets shows a dynamic transition (or, at least, sharp crossover) at the same temperature $T_{\rm u}=0.282 J$ as for static properties due to unbinding of $\mathbb{Z}_2$-vortices. We show that starting from the high-temperature initial state, due to presence of $\mathbb{Z}_2$-vortices in the considering system, in a broad temperature range $T, Comment: 9 pages, 9 figures

We use the Wang-Landau algorithm to calculate a density of states for an asymmetric Ising model on a triangular lattice with two- and three-body interactions in an external field. An accurate density of states allows us to determine the phase diagram and to study the critical behavior of this model at and near the critical endpoint. We observe a divergence of the curvature of the spectator phase boundary at the critical endpoint in accordance with theoretical predictions.

The role of topological excitations in frustrated Heisenberg antiferromagnets between two and three spatial dimensions is considered. In particular, the antiferromagnetic Heisenberg model on a stacked triangular geometry with a finite number of layers is studied using Monte Carlo methods. A phase transition that is purely topological in nature occurs at a finite temperature for all film thicknesses. The results indicate that topological excitations are important for a complete understanding of the critical properties of the model between two and three dimensions. PACS Nos.: 75.10.Hk, 75.40.Cx, 75.40.Mg

We present the real-space block renormalization group equations for fermion systems described by a Hubbard Hamiltonian on a triangular lattice with hexagonal blocks. The conditions that keep the equations from proliferation of the couplings are derived. Computational results are presented including the occurrence of a first-order metal-insulator transition at the critical value of U/t ≈ 12.5.

We describe a simulation method for a quantum spin model of a generic, general purpose quantum computer. The use of this quantum computer simulator is illustrated through several implementations of Grover’s database search algorithm. Some preliminary results on the stability of quantum algorithms are presented.

The breakdown of magnons, the quasiparticles of magnetic systems, has rarely been seen. By using an inelastic neutron scattering technique, we report the observation of spontaneous magnon decay in multiferroic ${\mathrm{LuMnO}}_{3}$, a simple two dimensional Heisenberg triangular lattice antiferromagnet, with large spin $S=2$. The origin of this rare phenomenon lies in the nonvanishing cubic interaction between magnons in the spin Hamiltonian arising from the noncollinear 120\ifmmode^\circ\else\textdegree\fi{} spin structure. We observed all three key features of the nonlinear effects as theoretically predicted: a rotonlike minimum, a flat mode, and a linewidth broadening, in our inelastic neutron scattering measurements of single crystal ${\mathrm{LuMnO}}_{3}$. Our results show that quasiparticles in a system hitherto thought of as ``classical'' can indeed break down.

We study the longitudinal excitations of quantum antiferromagnets on a triangular lattice by a recently proposed microscopic many-body approach based on magnon-density waves. We calculate the full longitudinal excitation spectra of the antiferromagnetic Heisenberg model for a general spin quantum number in the isotropic limit. Similar to the square lattice model, we find that, at the center of the first hexagonal Brillouin zone $\Gamma(\mathbf q=0)$ and at the magnetic ordering wavevectors $\pm[\mathbf Q= (4\pi/3,0)]$, the excitation spectra become gapless in the thermodynamic limit, due to the slow, logarithmic divergence of the structure factor. However, these longitudinal modes on two-dimensional models may be considered as quasi-gapped, as any finite-size effect or small anisotropy will induce a large energy gap, when compared with the counterpart of the transverse spin-wave excitations. We also discuss a possible second longitudinal mode in the triangular lattice model due to the noncollinear nature of its magnetic order.

International audience; We consider some classical and frustrated lattice spin models with global O(3) spin symmetry. No general analytical method to find a ground state exists when the spin dependence of the Hamiltonian is more than quadratic (i.e., beyond the Heisenberg model) and/or when the lattice has more than one site per unit cell. To deal with these cases, we introduce a family of variational spin configurations, dubbed "regular magnetic orders" (RMO's), which respect all the lattice symmetries modulo global O(3) spin transformations (rotations and/or spin flips). The construction of these states is made explicit through a group-theoretical approach, and all the RMO's on the square, triangular, honeycomb, and kagome lattices are listed. Their equal-time structure factors and powder averages are shown for comparison with experiments. Well known Néel states with one, two, or three sublattices on various lattices are RMO's, but the RMO's also encompass exotic nonplanar states with cubic, tetrahedral, or cuboctahedral geometry of the T=0 order parameter. Regardless of the details of the Hamiltonian (with the same symmetry group), a large fraction of these RMO's are energetically stationary with respect to small deviations of the spins. In fact, these RMO's appear as exact ground states in large domains of parameter space of simple models that we have considered. As examples, we display the variational phase diagrams of the J1-J2-J3 Heisenberg model on all the previous lattices as well as that of the J1-J2-K ring-exchange model on square and triangular lattices.

We show that the liquid-to-crystal quantum phase transition in the Rokhsar--Kivelson dimer model on the two-dimensional triangular lattice occurs as a condensation of vortex-like excitations called ``visons''. This conclusion is drawn from the numerical studies of the vison spectrum in the liquid phase by using the Green's function Monte Carlo method. We find that visons remain the lowest excitation throughout the liquid phase and that their gap decreases continuously to zero at the phase transition. The nature of the crystal phase and the second order of the phase transition are in agreement with the earlier prediction of Moessner and Sondhi [Phys. Rev. B 63, 224401 (2001)]., 4 pages, 4 figures

We study and solve the ground-state problem of a microscopic model for a family of orbitally degenerate quantum magnets. The orbital degrees of freedom are assumed to have directional character and are represented by static Potts-like variables. In the limit of vanishing Hund's coupling, the ground-state manifold of such a model is spanned by the hard-core dimer (spin singlet) coverings of the lattice. The extensive degeneracy of dimer coverings is lifted at a finite Hund's coupling through an order-out-of-disorder mechanism by virtual triplet excitations. The relevance of our results to several experimentally studied systems is discussed., Final version, minor changes

We study the phase diagram of two-dimensional Bose-Fermi mixtures of ultracold atoms on a triangular optical lattice, in the limit when the velocity of bosonic condensate fluctuations is much larger than the Fermi velocity. We contrast this work with our previous results for a square lattice system in Phys. Rev. Lett. {\bf 97}, 030601 (2006). Using functional renormalization group techniques we show that the phase diagrams for a triangular lattice contain exotic superconducting phases. For spin-1/2 fermions on an isotropic lattice we find a competition of $s$-, $p$-, extended $d$-, and $f$-wave symmetry, as well as antiferromagnetic order. For an anisotropic lattice, we further find an extended p-wave phase. A Bose-Fermi mixture with spinless fermions on an isotropic lattice shows a competition between $p$- and $f$-wave symmetry. These phases can be traced back to the geometric shapes of the Fermi surfaces in various regimes, as well as the intrinsic frustration of a triangular lattice., Comment: 6 pages, 4 figures, extended version, slight modifications

We study the zero-temperature partition function of the Potts antiferromagnet (i.e., the chromatic polynomial) on a torus using a transfer-matrix approach. We consider square- and triangular-lattice strips with fixed width L, arbitrary length N, and fully periodic boundary conditions. On the mathematical side, we obtain exact expressions for the chromatic polynomial of widths L=5,6,7 for the square and triangular lattices. On the physical side, we obtain the exact ``phase diagrams'' for these strips of width L and infinite length, and from these results we extract useful information about the infinite-volume phase diagram of this model: in particular, the number and position of the different phases., Comment: 72 pages (LaTeX2e). Includes tex file, three sty files, and 26 Postscript figures. Also included are Mathematica files transfer6_sq.m and transfer6_tri.m. Final version to appear in Nucl. Phys. B

We present a detailed study of the low-energy excitations of two existing finite-size realizations of the planar kagome Heisenberg antiferromagnet on the sphere, the cuboctahedron and the icosidodecahedron. After highlighting a number of special spectral features (such as the presence of low-lying singlets below the first triplet and the existence of localized magnons) we focus on two major issues. The first concerns the nature of the excitations above the plateau phase at 1/3 of the saturation magnetization Ms. Our exact diagonalizations for the s=1/2 icosidodecahedron reveal that the low-lying plateau states are adiabatically connected to the degenerate collinear ``up-up-down'' ground states of the Ising point, at the same time being well isolated from higher excitations. A complementary physical picture emerges from the derivation of an effective quantum dimer model which reveals the central role of the topology and the intrinsic spin s. We also give a prediction for the low energy excitations and thermodynamic properties of the spin s=5/2 icosidodecahedron Mo72Fe30. In the second part we focus on the low-energy spectra of the s>1/2 Heisenberg model in view of interpreting the broad inelastic neutron scattering response reported for Mo72Fe30. To this end we demonstrate the simultaneous presence of several broadened low-energy ``towers of states'' or ``rotational bands'' which arise from the large discrete spatial degeneracy of the classical ground states, a generic feature of highly frustrated clusters. This semiclassical interpretation is further corroborated by their striking symmetry pattern which is shown, by an independent group theoretical analysis, to be a characteristic fingerprint of the classical coplanar ground states., 22 pages Added references Corrected typos

We study the chromatic polynomial P_G(q) for m \times n square- and triangular-lattice strips of widths 2\leq m \leq 8 with cyclic boundary conditions. This polynomial gives the zero-temperature limit of the partition function for the antiferromagnetic q-state Potts model defined on the lattice G. We show how to construct the transfer matrix in the Fortuin--Kasteleyn representation for such lattices and obtain the accumulation sets of chromatic zeros in the complex q-plane in the limit n\to\infty. We find that the different phases that appear in this model can be characterized by a topological parameter. We also compute the bulk and surface free energies and the central charge., Comment: 55 pages (LaTeX2e). Includes tex file, three sty files, and 22 Postscript figures. Also included are Mathematica files transfer4_sq.m and transfer4_tri.m. Journal version

We establish the most general form of the discrete elasticity of a two-dimensional triangular lattice embedded in three dimensions, taking into account up to next-nearest-neighbour interactions. Besides crystalline system, this is relevant to biological physics (e.g. red blood cell cytoskeleton) and soft matter (e.g. percolating gels, etc.). In order to correctly impose the rotational invariance of the bulk terms, it turns out to be necessary to take into account explicitly the elasticity associated with the vertices located at the edges of the lattice. We find that some terms that were suspected in the literature to violate rotational symmetry are, in fact, admissible.

We construct a quantum extension of the (classical) three-coloring model introduced by Baxter [J.Math.Phys.11, 784 (1970)] for which the ground state can be computed exactly along a continuous line of Rokhsar-Kivelson solvable points. The quantum model, which admits a local spin representation, displays at least three different phases; an antiferromagnetic (AF) phase, a line of quantum critical points, and a ferromagnetic (F) phase. We argue that, in the ferromagnetic phase, the system cannot reach dynamically the quantum ground state when coupled to a bath through local interactions, and thus lingers in a state of quantum glassiness., Comment: (6 pages, 6 figures) - revised and expanded with additional explanatory paragraphs; published version

We study the q-state Potts model with nearest-neighbor coupling v=e^{\beta J}-1 in the limit q,v \to 0 with the ratio w = v/q held fixed. Combinatorially, this limit gives rise to the generating polynomial of spanning forests; physically, it provides information about the Potts-model phase diagram in the neighborhood of (q,v) = (0,0). We have studied this model on the square and triangular lattices, using a transfer-matrix approach at both real and complex values of w. For both lattices, we have computed the symbolic transfer matrices for cylindrical strips of widths 2 \le L \le 10, as well as the limiting curves of partition-function zeros in the complex w-plane. For real w, we find two distinct phases separated by a transition point w=w_0, where w_0 = -1/4 (resp. w_0 = -0.1753 \pm 0.0002) for the square (resp. triangular) lattice. For w > w_0 we find a non-critical disordered phase, while for w < w_0 our results are compatible with a massless Berker-Kadanoff phase with conformal charge c = -2 and leading thermal scaling dimension x_{T,1} = 2 (marginal operator). At w = w_0 we find a "first-order critical point": the first derivative of the free energy is discontinuous at w_0, while the correlation length diverges as w \downarrow w_0 (and is infinite at w = w_0). The critical behavior at w = w_0 seems to be the same for both lattices and it differs from that of the Berker-Kadanoff phase: our results suggest that the conformal charge is c = -1, the leading thermal scaling dimension is x_{T,1} = 0, and the critical exponents are \nu = 1/d = 1/2 and \alpha = 1., Comment: 131 pages (LaTeX2e). Includes tex file, three sty files, and 65 Postscript figures. Also included are Mathematica files forests_sq_2-9P.m and forests_tri_2-9P.m. Final journal version

We study the dynamics of topological defects of a frustrated spin system displaying spiral order. As a starting point we consider the SO(3) nonlinear sigma model to describe long-wavelength fluctuations around the noncollinear spiral state. Besides the usual spin-wave magnetic excitations, the model allows for topologically non-trivial static solutions of the equations of motion, associated with the change of chirality (clockwise or counterclockwise) of the spiral. We consider two types of these topological defects, single vortices and vortex-antivortex pairs, and quantize the corresponding solutions by generalizing the semiclassical approach to a non-Abelian field theory. The use of the collective coordinates allows us to represent the defect as a particle coupled to a bath of harmonic oscillators, which can be integrated out employing the Feynman-Vernon path-integral formalism. The resulting effective action for the defect indicates that its motion is damped due to the scattering by the magnons. We derive a general expression for the damping coefficient of the defect, and evaluate its temperature dependence in both cases, for a single vortex and for a vortex-antivortex pair. Finally, we consider an application of the model for cuprates, where a spiral state has been argued to be realized in the spin-glass regime. By assuming that the defect motion contributes to the dissipative dynamics of the charges, we can compare our results with the measured inverse mobility in a wide range of temperature. The relatively good agreement between our calculations and the experiments confirms the possible relevance of an incommensurate spiral order for lightly doped cuprates., Comment: 22 pages, 7 figures, final published version

In the Li{sub 1-x}Na{sub x}NiO{sub 2} solid solutions, three different single-phase regions exist: for x{>=}0.9, for x{approx_equal}0.7, and for x{

Hard constraints imposed in statistical mechanics models can lead to interesting thermodynamical behaviors, but may at the same time raise obstructions in the thoroughfare to thermal equilibration. Here we study a variant of Baxter's three-color model in which local interactions and defects are included, and discuss its connection to triangular arrays of Josephson junctions of superconductors with broken time-reversal symmetry and kagom\'e networks of superconducting wires. The model is equivalent to an Ising model in a hexagonal lattice with the additional constraint that the magnetization of each hexagon is $\ifmmode\pm\else\textpm\fi{}6$ or $0.$ Defects in the superconducting models correspond to violations of this constraint, and include fractional and integer vortices, as well as open strings within two-color loops. In the absence of defects, and for ferromagnetic interactions, we find that the system is critical for a range of temperatures (critical line) that terminates when it undergoes an exotic first-order phase transition with a jump from a zero magnetization state into the fully magnetized state at finite temperature. Dynamically, however, we find that the system becomes frozen into domains. The domain walls are made of perfectly straight segments, and domain growth appears frozen within the time scales studied with Monte Carlo simulations, with the system trapped into a ``polycrystalline'' phase. This dynamical obstruction has its origin in the topology of the allowed reconfigurations in phase space, which consist of updates of closed loops of spins. Only an extreme rare-event dominated proliferation of confined defects may overcome this obstruction, at much longer time scales. Also as a consequence of the dynamical obstruction, there exists a dynamical temperature, lower than the (avoided) static critical temperature, at which the system is seen to jump from a ``supercooled liquid'' to the polycrystalline phase within our Monte Carlo time scale. In contrast, for antiferromagnetic interactions, we argue that the system orders for infinitesimal coupling because of the constraint, and we observe no interesting dynamical effects.

We show that one-dimensional topological objects (kinks) are natural degrees of freedom for an antiferromagnetic Ising model on a triangular lattice. Its ground states and the coexistence of spin ordering with an extensive zero-temperature entropy can be easily understood in terms of kinks forming a hard-sphere liquid. Using this picture we explain effects of quantum spin dynamics on that frustrated model, which we also study numerically., Comment: 5 pages, 3 figures

We present exact calculations of the Potts model partition function Z(G,q,v) for arbitrary q and temperature-like variable v on n-vertex strip graphs G of the triangular lattice for a variety of transverse widths equal to L vertices and for arbitrarily great length equal to m vertices, with free longitudinal boundary conditions and free and periodic transverse boundary conditions. These have the form Z(G,q,v)=\sum_{j=1}^{N_{Z,G,\lambda}} c_{Z,G,j}(\lambda_{Z,G,j})^{m-1}. We give general formulas for N_{Z,G,j} and its specialization to v=-1 for arbitrary L. The free energy is calculated exactly for the infinite-length limit of the graphs, and the thermodynamics is discussed. It is shown how the internal energy calculated for the case of cylindrical boundary conditions is connected with critical quantities for the Potts model on the infinite triangular lattice. Considering the full generalization to arbitrary complex q and v, we determine the singular locus {\cal B}, arising as the accumulation set of partition function zeros as m\to\infty, in the q plane for fixed v and in the v plane for fixed q., Comment: 59 pages, LaTeX2e. Self-unpacking archive containing the tex file, three sty files, 51 ps files, and a gziped Mathematica file (transfer_Tutte_tri.m.gz). Version published in journal

Motivated by the lack of an obvious spectroscopic probe to investigate non-conventional order such as quadrupolar orders in spin S>1/2 systems, we present a theoretical approach to inelastic light scattering for spin-1 quantum magnets in the context of a two-band Hubbard model. In contrast to the S=1/2 case, where the only type of local excited state is a doubly occupied state of energy $U$, several local excited states with occupation up to 4 electrons are present. As a consequence, we show that two distinct resonating scattering regimes can be accessed depending on the incident photon energy. For $\hbar\omega_{in}\lesssim U$, the standard Loudon-Fleury operator remains the leading term of the expansion as in the spin-1/2 case. For $\hbar\omega_{in}\lesssim4U$, a second resonant regime is found with a leading term that takes the form of a biquadratic coupling $\sim({\bf S}_{i}\cdot{\bf S}_{j)^{2}$. Consequences for the Raman spectra of S=1 magnets with magnetic or quadrupolar order are discussed. Raman scattering appears to be a powerful probe of quadrupolar order., Comment: 10 Pages, 6 Figures, Submitted to PRB