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Register a new userXXZ and Ising Spins on the Triangular Kagome Lattice
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The recently fabricated two-dimensional magnetic materials Cu9X2(cpa)6.xH2O (cpa=2-carboxypentonic acid; X=F,Cl,Br) have copper sites which form a triangular kagome lattice (TKL), formed by introducing small triangles (``a-trimers) inside of each kagome triangle (``b-trimer). We show that in the limit where spins residing on b-trimers have Ising character, quantum fluctuations of XXZ spins residing on the a-trimers can be exactly accounted for in the absence of applied field. This is accomplished through a mapping to the kagome Ising model, for which exact analytic solutions exist. We derive the complete finite temperature phase diagram for this XXZ-Ising model, including the residual zero temperature entropies of the seven ground state phases. Whereas the disordered (spin liquid) ground state of the pure Ising TKL model has macroscopic residual entropy ln72=4.2767... per unit cell, the introduction of transverse(quantum) couplings between neighboring $a$-spins reduces this entropy to 2.5258... per unit cell. In the presence of applied magnetic field, we map the TKL XXZ-Ising model to the kagome Ising model with three-spin interactions, and derive the ground state phase diagram. A small (or even infinitesimal) field leads to a new phase that corresponds to a non-intersecting loop gas on the kagome lattice, with entropy 1.4053... per unit cell and a mean magnetization for the b-spins of 0.12(1) per site. In addition, we find that for moderate applied field, there is a critical spin liquid phase which maps to close-packed dimers on the honeycomb lattice, which survives even when the a-spins are in the Heisenberg limit.
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We derive exact results for close-packed dimers on the triangular kagome lattice (TKL), formed by inserting triangles into the triangles of the kagome lattice. Because the TKL is a non-bipartite lattice, dimer-dimer correlations are short-ranged, so that the ground state at the Rokhsar-Kivelson (RK) point of the corresponding quantum dimer model on the same lattice is a short-ranged spin liquid. Using the Pfaffian method, we derive an exact form for the free energy, and we find that the entropy is 1/3 ln2 per site, regardless of the weights of the bonds. The occupation probability of every bond is 1/4 in the case of equal weights on every bond. Similar to the case of lattices formed by corner-sharing triangles (such as the kagome and squagome lattices), we find that the dimer-dimer correlation function is identically zero beyond a certain (short) distance. We find in addition that monomers are deconfined on the TKL, indicating that there is a short-ranged spin liquid phase at the RK point. We also find exact results for the ground state energy of the classical Heisenberg model. The ground state can be ferromagnetic, ferrimagnetic, locally coplanar, or locally canted, depending on the couplings. From the dimer model and the classical spin model, we derive upper bounds on the ground state energy of the quantum Heisenberg model on the TKL.
We study the thermodynamics of Ising spins on the triangular kagome lattice (TKL) using exact analytic methods as well as Monte Carlo simulations. We present the free energy, internal energy, specific heat, entropy, sublattice magnetizations, and susceptibility. We describe the rich phase diagram of the model as a function of coupling constants, temperature, and applied magnetic field. For frustrated interactions in the absence of applied field, the ground state is a spin liquid phase with integer residual entropy per spin $s_0/k_B={1/9} ln 72approx 0.4752...$. In weak applied field, the system maps to the dimer model on a honeycomb lattice, with irrational residual entropy 0.0359 per spin and quasi-long-range order with power-law spin-spin correlations that should be detectable by neutron scattering. The power-law correlations become exponential at finite temperatures, but the correlation length may still be long.
In this paper and its sequel, we construct topologically invariant defects in two-dimensional classical lattice models and quantum spin chains. We show how defect lines commute with the transfer matrix/Hamiltonian when they obey the defect commutation relations, cousins of the Yang-Baxter equation. These relations and their solutions can be extended to allow defect lines to branch and fuse, again with properties depending only on topology. In this part I, we focus on the simplest example, the Ising model. We define lattice spin-flip and duality defects and their branching, and prove they are topological. One useful consequence is a simple implementation of Kramers-Wannier duality on the torus and higher genus surfaces by using the fusion of duality defects. We use these topological defects to do simple calculations that yield exact properties of the conformal field theory describing the continuum limit. For example, the shift in momentum quantization with duality-twisted boundary conditions yields the conformal spin 1/16 of the chiral spin field. Even more strikingly, we derive the modular transformation matrices explicitly and exactly.
We introduce and analyze a quantum spin/Majorana chain with a tricritical Ising point separating a critical phase from a gapped phase with order-disorder coexistence. We show that supersymmetry is not only an emergent property of the scaling limit, but manifests itself on the lattice. Namely, we find explicit lattice expressions for the supersymmetry generators and currents. Writing the Hamiltonian in terms of these generators allows us to find the ground states exactly at a frustration-free coupling. These confirm the coexistence between two (topologically) ordered ground states and a disordered one in the gapped phase. Deforming the model by including explicit chiral symmetry breaking, we find the phases persist up to an unusual chiral phase transition where the supersymmetry becomes exact even on the lattice.
On Archimedean lattices, the Ising model exhibits spontaneous ordering. Three examples of these lattices of the majority-vote model with noise are considered and studied through extensive Monte Carlo simulations. The order/disorder phase transition is observed in this system. The calculated values of the critical noise parameter are q_c=0.089(5), q_c=0.078(3), and q_c=0.114(2) for honeycomb, Kagome and triangular lattices, respectively. The critical exponents beta/nu, gamma/nu and 1/nu for this model are 0.15(5), 1.64(5), and 0.87(5); 0.14(3), 1.64(3), and 0.86(6); 0.12(4), 1.59(5), and 1.08(6) for honeycomb, Kagome and triangular lattices, respectively. These results differs from the usual Ising model results and the majority-vote model on so-far studied regular lattices or complex networks. The effective dimensionalities of the system D_{eff}= 1.96(5) (honeycomb), D_{eff} =1.92(4) (Kagome), and D_{eff}= 1.83(5) (triangular) for these networks are just compatible to the embedding dimension two.
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